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--- author: - Dafei Jin - Yang Xia - Thomas Christensen - Siqi Wang - King Yan Fong - Matthew Freeman - 'Geoffrey C. Gardner' - Saeed Fallahi - Qing Hu - Yuan Wang - Lloyd Engel - 'Michael J. Manfra' - 'Nicolas X. Fang' - Xiang Zhang bibliography: - 'References.bib' title: 'Magnetically-defined topological edge plasmons in edgeless electron gas' --- =8000 [**Topological materials bear gapped excitations in bulk yet protected gapless excitations at boundaries [@Qi2011RMP; @Lu2014NatPhoton]. Magnetoplasmons (MPs), as high-frequency density excitations of two-dimensional electron gas (2DEG) in a perpendicular magnetic field [@Ando1982RMP; @Kushwaha2001SSR], embody a prototype of band topology for bosons [@Jin:2016; @Jin2017PRL]. The time-reversal-breaking magnetic field opens a topological gap for bulk MPs up to the cyclotron frequency [@Zudov2003PRL; @Gao2016NatCommun]; topologically-protected edge magnetoplasmons (EMPs) bridge the bulk gap and propagate unidirectionally along system’s boundaries [@Mast1985PRL; @Glattli1985PRL; @Fetter:1986; @Volkov1988JETP]. However, all the EMPs known to date adhere to physical edges where the electron density terminates abruptly [@Ashoori1992PRB; @Balev1997PRB; @Kumada2014PRL]. This restriction has made device application extremely difficult. Here we demonstrate a new class of topological edge plasmons – domain-boundary magnetoplasmons (DBMPs), within a uniform edgeless 2DEG. Such DBMPs arise at the domain boundaries of an engineered sign-changing magnetic field and are protected by the difference of gap Chern numbers ($\pm1$) across the magnetic domains. They propagate unidirectionally along the domain boundaries and are immune to domain defects [@Jin:2016]. Moreover, they exhibit wide tunability in the microwave frequency range under an applied magnetic field or gate voltage. Our study opens a new direction to realize high-speed reconfigurable topological devices [@Mahoney2017PRX; @Fang2012NatPhoton; @Bahari2017Science].** ]{} In this work, we present the first experimental observation of a new class of topological edge plasmons, domain-boundary magnetoplasmons (DBMPs), at microwave frequencies in a high-mobility GaAs/AlGaAs heterojunction. In contrast to the traditional wisdom, where edge magentoplasmons (EMPs) must rely on a space-varying electron density $n({\mathbf{r}})$, in our scenario, the DBMPs are defined by a space-varying magnetic field $B({\mathbf{r}})=B({\mathbf{r}}){\hat{\mathbf{e}}_{z}}$ embedded into a uniform 2DEG [@Ye1995PRL; @Nogaret2000PRL; @Reijniers2000JPCM; @Yasuda2017Science]. A custom-shaped NdFeB strong permanent magnet, placed immediately above the heterojunction, produces a sign-changing magnetic field around $0.15$ T in magnitude, sufficient to gap bulk MPs in each magnetic domain. The $10^7$ cm$^2$ V$^{-1}$ s$^{-1}$ high electron mobility in this system affords an ultra-long relaxation time of hundreds of picoseconds and ultra-low damping rate of only a few gigahertz, superior to any other existing 2DEG systems [@Mast1985PRL; @Bolotin2008PRL; @Ohtomo2004Nature]. By measuring microwave resonant spectra, we clearly verify the existence and nonreciprocal nature of DBMPs. Their excitation frequencies display a unique dependence on both an applied magnetic field and gate voltage, differing substantially from the conventional EMPs in several intriguing aspects. Our theoretical prediction and experimental observation show excellent mutual agreement. [**System design**]{} Figure \[fig:device\] illustrates the layout of our magnetoplasmonic device. Conceptually (Fig. \[fig:device\]a), a 2DEG in a GaAs/AlGaAs heterojunction (see Methods) is cladded above and below by a fused silica (glass) spacer and a GaAs substrate, respectively, of thicknesses ${d_{{\scriptscriptstyle}\text{A}}}= 100~\mu$m and ${d_{{\scriptscriptstyle}\text{B}}}= 150~\mu$m, and permittivities ${\epsilon_{{{\scriptscriptstyle}\text{A}}}}=3.8$ and ${\epsilon_{{{\scriptscriptstyle}\text{B}}}}=12.8$. This dielectric-2DEG-dielectric structure is enclosed in a metallic cavity along $z$, terminating at the spacer’s top and substrate’s bottom. A holed NdFeB permanent magnet, atop the upper cavity wall, projects a circular magnetic field $\mathbf{B}_{\text{m}}({\mathbf{r}}) = B_{\text{m}}(r){\hat{\mathbf{e}}_{z}}$ onto the 2DEG. The sign of $B_{\text{m}}(r)$ changes abruptly across the projection of the hole’s radius, $a = 0.75~$mm, producing adjacent oppositely-signed magnetic domains (see Methods). The entire 2DEG is additionally exposed to a tunable homogeneous magnetic field $\mathbf{B}_0({\mathbf{r}}) = B_0{\hat{\mathbf{e}}_{z}}$ from a superconducting coil, allowing an overall shift of the magnetic field profile.  In practice (Fig. \[fig:device\]b), the heterojunction sample has a 12 mm $\times$ 6 mm rectangular footprint. A 9 mm $\times$ 3 mm Hall bar is fabricated atop of it, allowing *in situ* measurements and control of the 2DEG electron concentration $n_0$. The fused silica spacer is topped by a 100 nm thick e-beam evaporated Cr-coating, serving simultaneously as upper cavity wall and gate electrode [@Hatke2015NatCommun; @Mi2017arXiv]. A gate voltage of $V_\text{g} \sim \pm 100~$V can be applied across the Cr-coating–Hall bar junction to tune the electron concentration. The sample-spacer-magnet assembly is glued by Poly(methyl methacrylate) (PMMA) onto a customized Cu printed circuit board (PCB) with a 5 $\mu$m Ni and 200 nm Au surface finish. The PCB hosts a coplanar waveguide (CPW) connecting RF Ports 1 and 2 with mini-SMP connectors [@Hatke2015NatCommun]. By design, the CPW has a 50 $\Omega$ impedance with the sample-magnet assembly loaded. The CPW signal line is aligned tangentially to the projected circle from the hole of magnet so as to maximize the microwave-DBMP coupling. [**Theoretical prediction**]{} The main physics of MP system can be captured by the continuity equation and a constitutive equation containing the longitudinal Coulomb force and transverse Lorentz force: \[eqs:governing\] $$\begin{aligned} &\omega\rho({\mathbf{r}},\omega) = -{{\rm i}}\nabla\cdot {\mathbf{j}}({\mathbf{r}},\omega),\\ &\omega{\mathbf{j}}({\mathbf{r}},\omega) = -{{\rm i}}\frac{e^2}{m_*}n({\mathbf{r}})\nabla\Phi({\mathbf{r}},\omega) - {\omega_{\text{c}}}({\mathbf{r}}){\mathbf{j}}({\mathbf{r}},\omega)\times\hat{\mathbf{e}}_{z}. \end{aligned}$$ Here, ${\mathbf{j}}$ and $\rho$ are the surface current and charge densities, evaluated at frequencies $\omega$ and in-plane positions ${\mathbf{r}}$. $\Phi({\mathbf{r}},\omega) = \int V({\mathbf{r}}-{\mathbf{r}}')\rho({\mathbf{r}}',\omega)\,{{\rm d}}^2{{\mathbf{r}}'}$ is the self-consistent potential due to the (screened) Coulomb interaction $V$. ${\omega_{\text{c}}}({\mathbf{r}})=eB({\mathbf{r}})/cm_*$ is a space-varying cyclotron frequency, with $m_*$ the electron effective mass. As elaborated below, even with a constant electron density $n({\mathbf{r}})=n_0$, topologically-protected DBMPs can reside at boundaries of sign-changing magnetic domains solely defined by the spatial profile $B({\mathbf{r}})$ and ${\omega_{\text{c}}}({\mathbf{r}})$ [@Jin:2016]. The total magnetic field, $B(r) = B_0 + B_{\text{m}}(r)$, is the sum of a tunable, uniform field $B_0$ from the superconducting coil, and a fixed, $r$-dependent field $B_{\text{m}}(r)$ from the holed NdFeB permanent magnet. The latter is well-approximated by a step function, $$\label{eq:Bm} B_{\text{m}}(r) \simeq {\bar{B}_{\text{m}}}+ \operatorname{sgn}(r-a){\Delta B_{\text{m}}}.$$ Here, ${\Delta B_{\text{m}}}$ contributes an equal-magnitude sign-changing jump at $r=a\approx 0.75$ mm, while ${\bar{B}_{\text{m}}}$ accounts for a small, overall shift due to the small distance between magnet and 2DEG. By a combination of finite-element simulations and room-temperature Hall-probe measurements on the surface of magnet, we infer the low-temperature values of each as ${\Delta B_{\text{m}}}\approx0.14$ T and ${\bar{B}_{\text{m}}}\approx0.01$ T (see Methods). The presence of cladding dielectrics and encapsulating metals in this system significantly influences the Coulomb interaction and frequency scale of the problem. In momentum space, the Coulomb interaction takes a screened form, $$V(q) = \frac{2\pi}{q} \beta(q) = \frac{2\pi}{q} \frac{2}{{\epsilon_{{{\scriptscriptstyle}\text{A}}}}\coth(q {d_{{\scriptscriptstyle}\text{A}}})+{\epsilon_{{{\scriptscriptstyle}\text{B}}}}\coth(q {d_{{\scriptscriptstyle}\text{B}}})} , \label{eq:coulomb}$$ with $\beta(q)$ being the $q$-dependent screening function [@Fetter:1986; @Volkov1988JETP]. The scalar potential and surface charge density are related by $\Phi(q)=V(q)\rho(q)$. The eigenmodes consistent with Eqs.  are eigenstates of a $3\times3$ Hamiltonian $\boldsymbol{\mathcal{\hat{H}}}$ with operator elements [@Jin:2016; @Jin2017PRL]. In the circularly symmetric “potential” of Eq. , the eigenmodes decompose according to $\mathbf{R}_{m}(r){{\rm e}}^{{{\rm i}}m\varphi}$ with azimuthal angle $\varphi$ and angular wavenumber $m\in\mathbb{Z}$. The radial function $\mathbf{R}_{m}(r)$ can be expanded by the Bessel functions with radial wavenumbers $q_{mn}$, $n\in\mathbb{Z}^+$, which enter the Coulomb interaction Eq. (\[eq:coulomb\]) (see Methods).  The resulting plasmonic properties are explored in Fig. \[fig:theory\]. Figure \[fig:theory\]a illustrates the magnetoplasmonic dispersion of bulk MP and DBMP modes for $n_0=1\times 10^{11}$ cm$^{-2}$, $B_0={\bar{B}_{\text{m}}}=0$ T, and ${\Delta B_{\text{m}}}=0.15$ T. The spectrum exhibits particle-hole symmetry, [i.e.]{}$\omega_{nm} = -\omega_{n,-m}$, with a zero-frequency band describing static modes [@Jin:2016]. The bulk MPs in each magnetic domain exhibits a gap from zero frequency to approximately $|{\omega_{\text{c}}}(r)|=e|B(r)|/m_*c$. The band topology of each domain, considered as an extended bulk, is characterized by a topological invariant, the Chern number, equaling $C = -\operatorname{sgn}B(r) = \operatorname{sgn}(a-r)=\pm 1$ [@Jin:2016]. The associated gap Chern number $\bar{C}$, also equaling $\pm 1$ in this case, can be identified, whose difference across the domains, $\Delta\bar{C} = 2$, dictates the existence of two unidirectional edge states localized at $r=a$. These conclusions are clearly manifest in Fig. \[fig:theory\]a and \[fig:theory\]b from the existence of quasi-even and quasi-odd DBMP branches (so named due to their asymptotic association with the even and odd DBMPs of a linear domain boundary). Both are unidirectional and exhibit increasing localization with incrementing angular wavenumbers $m$. We emphasize that these DBMPs drastically differ from the conventional EMPs by being solely magnetically-engineered. Interestingly, they are topologically equivalent to the equatorial Kelvin and Yanai waves of the ocean and atmosphere (with a Coriolis parameter in place of $B(r)$) [@Delplace:2017]. Figure \[fig:theory\]c investigates the dispersion for increased ${\Delta B_{\text{m}}}$ (from 0.15 T to 0.18 T) and $B_0$ (from 0 to 0.1 T). Comparing to Fig. \[fig:theory\]a, increasing ${\Delta B_{\text{m}}}$ widens the bandgap and decreases the frequencies of the quasi-even DBMPs. Conversely, increasing $B_0$ (but maintaining $B_0<{\Delta B_{\text{m}}}$) reduces the overall gap—since the cyclotron frequency is lowered in the inner domain—and increases the excitation frequencies of the quasi-even DBMP. This latter behavior further distinguishes our new DBMPs from the traditional EMPs which shift in the opposite direction with increasing $B_0$ [@Fetter:1986; @Mast1985PRL]. The quasi-odd DBMP branch in Fig. \[fig:theory\]c appears non-gapless and hence non-topological in the considered range of $m$: this, however, is remedied at larger $|m|$ where the dispersion bends downwards (not shown), instating an asymptotically gapless behavior in deference to the topological requirements.  [**Experimental observation**]{} We next seek experimental evidence for the theoretically predicted DBMPs. The device is inserted into a He-3 cryostat running at 0.5 K. An Agilent E5071C Network Analyzer (NA) is used to acquire power transmission $S_{21}$ (Port 1 to Port 2) and $S_{12}$ (Port 2 to Port 1) in the frequency range 300 kHz to 20 GHz [@Hatke2015NatCommun; @Mi2017arXiv; @Mahoney2017PRX]. In practice, however, our focused frequency range is limited to 1 to 10 GHz, beyond which the cables and NA bear too high loss and noise, prohibiting acquisition of clear signals. Referring to Figs. \[fig:theory\]a and \[fig:theory\]c, we consequently expect to observe characteristic absorption associated with only the $m=1$ and $2$ quasi-even DBMPs. In the first series of measurements, we keep the gate grounded, ${V_{\text{g}}}= 0$ V, and investigate the influence of the applied magnetic field $B_0$ on the resonant absorption of quasi-even DBMPs in $S_{21}$ (Fig. \[fig:measurements\]a). All signals are divided by a reference (denoted baseline). Here, we choose $B_0=0.2$ T as baseline, which provides a high suppression of unwanted low-frequency bulk modes, without exerting too great a torque on the magnet–sample assembly. For every $S_{21}$-spectrum in Fig. \[fig:measurements\]a, each reflecting a single applied field in the range $B_0=0$ to $0.1$ T, we observe two well-defined absorptive resonances, corresponding to the $m=1\text{ and }2$ right-circulating quasi-even DBMPs. Spanning frequencies from 3 to 4 GHz and 6 to 8 GHz, they exhibit linewidths of approximately 1 to 2 GHz, roughly consistent with the Hall-probe inferred DC damping rate $\gamma\sim2.6$ GHz. Figure \[fig:measurements\]c compares the measured and theoretically predicted resonance frequencies. First, we observe the excellent mutual agreement in the absence of fitting parameters. Second, we emphasize the monotonously increasing excitation frequencies with increasing $B_0$, which unambiguously differentiates our magnetically-defined DBMPs from the conventional EMPs. In the second series of measurements, we fix the applied magnetic field $B_0=0$ T, and explore the DBMPs’ dependence on the gate voltage ${V_{\text{g}}}$ (Fig. \[fig:measurements\]b). The baseline is chosen at ${V_{\text{g}}}=-80$ V, which corresponds to an essentially electron-depleted 2DEG supporting no plasmonic modes. Once more, every spectrum in Fig. \[fig:measurements\]b, each now corresponding to distinct gate voltages in the range ${V_{\text{g}}}=-20$ to $+20$ V, exhibits two clear absorptive resonances associated with the $m=1\text{ and }2$ quasi-even DBMPs. Increasing the gate voltage (or, equivalently, the electron concentration $n_0$) increases the DBMP frequency, as expected. Moreover, the extinction depth of each resonance also increases with the ${V_{\text{g}}}$. This is consistent with the $f$-sum rule [@YangKall:2015] which dictates a linear increase of integrated extinction with increased $n_0$ (disregarding the negligible spectral dispersion in the microwave-DBMP coupling). Comparing theoretical and experimental observations, in Fig. \[fig:measurements\]d, we once again find excellent agreement. ![ **Nonreciprocal transmission near DBMPs.** Measured power transmission along $S_{21}$ (“easy-coupling”) and $S_{12}$ (“hard-coupling”) directions, respectively, at $B_0 = 0$ T and gate voltage ${V_{\text{g}}}= 40$ V. The distinct absorption depths manifests the nonreciprocal nature of the $m=\text{1}$ and 2 quasi-even DBMPs. \[fig:isolation\] ](OneWayShrunkRevised.pdf) Finally, in Fig. \[fig:isolation\], we examine the nonreciprocal properties of the DBMPs in order to explicitly demonstrate the underlying unidirectional character of the DBMPs. Since the DBMPs are right-circulating in the bandgap (Fig. \[fig:theory\]), $S_{21}$ and $S_{12}$ correspond to the “easy-coupling” and “hard-coupling" directions, respectively, of our device (Fig. \[fig:device\]b). Each coupling direction is normalized separately, with baselines taken at $B_0=0.2$ T. The 2DEG is gated by ${V_{\text{g}}}=40$ V, ensuring a pronounced extinction depth, and the applied magnetic field is turned off $B_0=0$ T. In this configuration, the $m=1$ and $2$ quasi-even DBMPs exist at 4.2 GHz and 8.0 GHz, respectively. Comparing $S_{21}$ and $S_{12}$ we observe distinct asymmetry of extinction depth at each resonance, with $S_{12}$ exhibiting shallower extinction. This asymmetry is indicative of the unidirectional character of the DBMPs. The observed isolation ratio $S_{21}/S_{12} = (S_{21}-S_{12})|_\text{dB}$ is small because of the wavelength mismatch between the microwaves in CPW and the DBMPs along the circle. This is mainly a limitation from the CPW evanescent-coupling technique. A fuller assessment of the isolation capabilities could more naturally be enabled by point-source excitation [@Ashoori1992PRB; @Wang2009Nature; @Fei:2011; @Kumada2014PRL]. [**Conclusion**]{} We have for the first time realized a new class of topologically-protected edge plasmons, domain-boundary magnetoplasmons, embedded in an edgeless 2DEG. They situate at magnetically-defined domain boundaries, and are topologically distinct from the conventional edge magnetoplasmons. We have experimentally observed and characterized these new DBMPs at microwave frequencies in a high-mobility GaAs/AlGaAs heterojunction under a custom-shaped NdFeB permanent magnet. Our experimental results show remarkable agreement with theoretical calculations. The demonstrated DBMP architecture, if packed with denser magnetic patterns, can be extended to higher frequencies and finer scales, and shall pave the way towards high-speed reconfigurable topological devices [@Mahoney2017PRX; @Fang2012NatPhoton; @Bahari2017Science]. [**Acknowledgements**]{} D.J., Y.X., S.W., K.Y.F., Y.W. and X.Z. acknowledge support from AFOSR MURI (Grant No. FA9550-12-1-0488) and Office of Sponsored Research (OSR) (Award No. OSR-2016-CRG5-2950-03). T.C. acknowledges support from the Danish Council for Independent Research (Grant No. DFF–6108-00667). The National High Magnetic Field Laboratory (NHMFL) is supported by NSF Cooperative Agreement (No. DMR-0654118), the State of Florida, and the DOE. M.F. and L.E., and Microwave Spectroscopy Facility are supported by the DOE (Grant No. DE-FG02-05-ER46212). G.C.G. and M.J.M. acknowledge support from the DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (Award No. [DE[-]{}SC0006671]{}), the W. M. Keck Foundation, and Microsoft Station Q. Q.H. and N.X.F. are supported by AFOSR MURI (Grant No. FA9550-12-1-0488). [**Methods**]{} [**2DEG sample growth and characterization**]{} Our sample is a single-interface GaAs/Al$_{x}$Ga$_{1-x}$As ($x=0.22$) heterojunction grown by molecular beam epitaxy (MBE) on a 500 $\mu$m thick GaAs wafer. After the growth, the sample is back-polished down to 100 $\mu$m thick in order to enhance the evanescent microwave coupling. The MBE growth consists of a 500 nm thick GaAs layer followed by a 170 nm thick Al$_{x}$Ga$_{1-x}$As ($x=0.22$) spacer and a 20 nm GaAs cap layer to prevent oxidization of the AlGaAs barrier. It is delta-doped with Si doping concentration $1.6\times 10^{12}$ cm$^{-2}$ at a setback of 120 nm above the GaAs/AlGaAs interface containing 2DEG. The 2DEG lies 190 nm below top surface. The electron concentration $n_{0}=0.95\times10^{11}$ cm$^{-2}$ and mobility $\mu=8.6\times10^6$ cm$^2$V$^{-1}$s$^{-1}$ are extracted from our Hall measurement at $T=0.3$ K in dark. In our actual microwave experiment at 0.5 K, the typical zero-gate electron concentration is measured to be about $1\times10^{11}$ cm$^{-2}$. This number is used in our calculation. The uniform magnetic field $B_0$ is supplied by a superconductor coil. In the absence of the holed NdFeB magnet, it can safely reach above 7 T, enabling a quantum-Hall measurement to characterize the sample (see Fig. \[efig:qhe\]). When the NdFeB magnet is present, the applied field is limited by practical concerns to at most 0.5 T, beyond which a huge magnetic torque is exerted onto the magnet, risking damage to the sample underneath. ![Quantum Hall measurement of the GaAs/AlGaAs 2DEG sample at 0.3 K in dark with zero gate voltage. The inset shows the layer structure of the sample. \[efig:qhe\]](HallOnly.pdf) [**NdFeB magnet design and characterization**]{} The NdFeB magnet is 10 mm long, 4 mm wide, and 1 mm thick, and the hole radius is 0.75 mm. It is produced by sintering NdFeB powders in a custom mold and subsequently magnetizing it along the thickness direction. At room temperature, Hall-probe measurements indicate that the holed magnet provides approximately $\pm0.18$ T remanent magnetic field in the surface area inside and outside the hole. With this value, and taking into account the known anisotropic reduction of the magnetism of NdFeB at cryogenic temperatures [@Garcia2000PRL; @Strnat1985Proc; @TECHNotes], we are able to simulate out the magnetic field profile over the entire magnet at low temperature (see Fig. \[efig:magnet\]) using a finite-element software (Comsol Multiphysics). From the results, we infer that the two key parameters of Eq. (2) in the main text, namely, a sign-changing field strength ${\Delta B_{\text{m}}}\approx\pm0.14$ T and a overall shift ${\bar{B}_{\text{m}}}\approx0.01$ T. These are the values used in our theoretical calculations in Fig. 2c and 2d of the main text, demonstrating excellent agreement between theory and experiment with no fitting parameters. ![Calculated magnetic field profile of the holed NdFeB magnet at low temperature. The key parameter is imported from the room-temperature Hall-probe measurement and the well-known temperature-dependence in literature. \[efig:magnet\]](MagnetOnlyRevised.pdf) [**Theoretical development and calculation scheme**]{} The domain-boundary magnetoplasmon (DBMP) modes in our device can be accurately calculated. The evanescent nature of DBMPs and the presence of encapsulating metals, which screen away the long-range part of Coulomb interaction, allow us to focus on the region around and inside the circle $r\lesssim a=0.75$ mm. We can legitimately take a circularly-symmetric model system cut off at a radius $R=10$ mm $\gg a$, where the scalar potential $\Phi$ is grounded $\Phi(r=R,\varphi)=0$. This rigid boundary condition does not affect the DBMPs far inside. The associated eigenproblem is most conveniently expressed in the chiral representation [@Jin:2016], $$\begin{aligned} \omega {j_{{\scriptscriptstyle}\text{R}}}(r,\varphi) &= + {\omega_{\text{c}}}(r) {j_{{\scriptscriptstyle}\text{R}}}(r,\varphi) \\ & + \frac{e^2n_0}{\omega_0m_*}\frac{{{\rm e}}^{-{{\rm i}}\varphi}}{{{\rm i}}\sqrt{2}} \left[\partial_r - \frac{{{\rm i}}}{r}\partial_\varphi \right] {j_{{\scriptscriptstyle}\text{D}}}(r,\varphi) , \nonumber\\ \omega {j_{{\scriptscriptstyle}\text{D}}}(r,\varphi) & = \omega_0\hat{V} \frac{{{\rm e}}^{+{{\rm i}}\varphi}}{{{\rm i}}\sqrt{2}} \left[\partial_r + \frac{{{\rm i}}}{r}\partial_\varphi \right] {j_{{\scriptscriptstyle}\text{R}}}(r,\varphi) \\ & + \omega_0\hat{V} \frac{{{\rm e}}^{-{{\rm i}}\varphi}}{{{\rm i}}\sqrt{2}} \left[\partial_r - \frac{{{\rm i}}}{r}\partial_\varphi \right] {j_{{\scriptscriptstyle}\text{L}}}(r,\varphi), \nonumber\\ \omega {j_{{\scriptscriptstyle}\text{L}}}(r,\varphi) &= - {\omega_{\text{c}}}(r) {j_{{\scriptscriptstyle}\text{L}}}(r,\varphi) \\ & + \frac{e^2n_0}{\omega_0m_*}\frac{{{\rm e}}^{+{{\rm i}}\varphi}}{{{\rm i}}\sqrt{2}} \left[\partial_r + \frac{{{\rm i}}}{r}\partial_\varphi \right] {j_{{\scriptscriptstyle}\text{D}}}(r,\varphi) . \nonumber\end{aligned}$$ The basic field components are the right-circulating current ${j_{{\scriptscriptstyle}\text{R}}}\equiv\frac{1}{\sqrt{2}}(j_{r}-{{\rm i}}j_{\varphi}){{\rm e}}^{-{{\rm i}}\varphi}$, the left-circulating current ${j_{{\scriptscriptstyle}\text{L}}}\equiv\frac{1}{\sqrt{2}}(j_{r}+{{\rm i}}j_{\varphi}){{\rm e}}^{+{{\rm i}}\varphi}$, and the “scalar-potential (density-fluctuation)" current ${j_{{\scriptscriptstyle}\text{D}}}\equiv\omega_0\Phi$. Here, $\omega_0\equiv\sqrt{ e^2n_0 / m_*R}$ is a characteristic plasmon frequency, ${\omega_{\text{c}}}(r) = eB(r)/m_*c$ is the $r$-dependent cyclotron frequency, and $\hat{V}$ is the Coulomb integration operator, $$\hat{V} \rho(r,\varphi) = \int_0^R r'{{\rm d}}r' \int_0^{2\pi} {{\rm d}}\varphi'\ V(|\bm{r}-\bm{r}'|) \rho(r',\varphi'),$$ with $V(|\bm{r}-\bm{r}'|)$ being the screened Coulomb interaction in real space, which does not have a simple form. The eigensolutions with a given angular wavenumber $m$ and obeying the hard-wall boundary condition are linear expansion of Bessel functions, $$j_{s}(r,\varphi) =\Bigg[ \sum_{n=1}^{N\rightarrow\infty} A_{n,s} \mathrm{J}_{m+s} (q_{m n} r)\Bigg] {{\rm e}}^{+{{\rm i}}(m+s)\varphi}.$$ Here $s=-1,0,+1$ resembles a spin index referring to the ${j_{{\scriptscriptstyle}\text{R}}}$, ${j_{{\scriptscriptstyle}\text{D}}}$, ${j_{{\scriptscriptstyle}\text{L}}}$ components, respectively. $q_{m n}=\zeta_{m n}/R$ are discretized radial wavenumbers in which $\zeta_{m n}$ is the $n$th zero of the $m$th order Bessel function $\mathrm{J}_m(\zeta)$. The expansion is practically cutoff at a large finite $N$ up to a desired spectral resolution. (In our calculations, we use $N=2000$.) With such discretized cylindrical-wave bases, we can rigorously prove that the screened Coulomb interaction relates $\rho$ and $\Phi$ by $\Phi(q_{m n}) = V(q_{m n}) \rho(q_{m n})$, where $V(q_{m n})$ follows Eq. (3) in the main text. The matrix-form eigenequation in the cylindrical-wave bases reads $$\frac{\omega}{\omega_0} \begin{pmatrix} \mathbf{A}_{+1} \\ \mathbf{A}_{0} \\ \mathbf{A}_{-1} \end{pmatrix} = \begin{pmatrix} + \mathbf{W} & +\frac{q_{m n}R}{{{\rm i}}\sqrt{2}} \mathbf{I} & 0 \\ - \frac{2\pi\beta(q_{m n})}{{{\rm i}}\sqrt{2}} \mathbf{I} & 0 & + \frac{2\pi\beta(q_{m n})}{{{\rm i}}\sqrt{2}} \mathbf{I} \\ 0 & -\frac{q_{m n}R}{{{\rm i}}\sqrt{2}} \mathbf{I} & - \mathbf{W} \end{pmatrix} \begin{pmatrix} \mathbf{A}_{+1} \\ \mathbf{A}_{0} \\ \mathbf{A}_{-1} \end{pmatrix} .$$ Here $\mathbf{A}_s=(\mathcal{A}_{1,s},\mathcal{A}_{2,s},\dots,\mathcal{A}_{N,s})^{\text{T}}$, $\mathbf{I}$ is an $N\times N$ identity matrix, $\mathbf{W}$ is an $N\times N$ full matrix determined by the magnetic-field profile. If $B(r)=B_0$, then $\mathbf{W}=({\omega_{\text{c}}}/\omega_0)\mathbf{I}$ is diagonal with the constant cyclotron frequency ${\omega_{\text{c}}}= eB_0 / m_*c$, and the usual bulk MP modes can be recovered [@Jin:2016]. The generally radially-varying magnetic field in our problem results in scattering between different radial index $n$ (within the $s=-1$ and $+1$ chiral subspace though) and hence localization of new edge modes at the magnetic-domain boundary. We can find the matrix $\mathbf{W}$ via the expansion $$\begin{aligned} {\omega_{\text{c}}}(r) \mathrm{J}_{m-1}(q_{m n}r) \equiv \omega_0 \sum_{n'} W_{nn'}\mathrm{J}_{m-1}(q_{m n'}r),\\ {\omega_{\text{c}}}(r) \mathrm{J}_{m+1}(q_{m n}r) \equiv \omega_0 \sum_{n'} W_{nn'}\mathrm{J}_{m+1}(q_{m n'}r),\end{aligned}$$ in which $W_{nn'}$ are the elements of $\mathbf{W}$. Upon laborious calculations involving Bessel integrals [@Abramowitz2013Book], we can get $$\begin{aligned} \mathbf{W} = \frac{e}{\omega_0m_*c} \left[ 2{\Delta B_{\text{m}}}\mathbf{Y}^{-1} \mathbf{X} + (B_0+{\bar{B}_{\text{m}}}-{\Delta B_{\text{m}}}) \mathbf{I} \right] ,\end{aligned}$$ where $\mathbf{X}$ and $\mathbf{Y}$ are $N\times N$ matrices too, whose elements are $$\begin{aligned} X_{nn'} &= \int_0^{\tilde{a}} \tilde{r}{{\rm d}}\tilde{r}\ \mathrm{J}_{m-1}(\zeta_{m n}\tilde{r}) \mathrm{J}_{m-1}(\zeta_{m n'}\tilde{r}) \\ &= \begin{cases} \!\begin{aligned} {\displaystyle}\frac{\tilde{a}}{\zeta_{m n}^2-\zeta_{m n'}^2} \Big\{& \zeta_{m n} \mathrm{J}_{m}(\zeta_{m n}\tilde{a}) \mathrm{J}_{m-1}(\zeta_{m n'}\tilde{a}) \\ & - \zeta_{m n'} \mathrm{J}_{m}(\zeta_{m n'}\tilde{a}) \mathrm{J}_{m-1}(\zeta_{m n}\tilde{a}) \Big\} \end{aligned} \quad &\text{for }n\neq n',\\ \!\begin{aligned} {\displaystyle}\frac{\tilde{a}}{2\zeta_{m n}} \Big\{& \tilde{a}\zeta_{m n} \mathrm{J}^2_{m}(\zeta_{m n}\tilde{a}) + \tilde{a}\zeta_{m n} \mathrm{J}^2_{m-1}(\zeta_{m n}\tilde{a}) \\ &- 2(m-1) \mathrm{J}_{m}(\zeta_{m n}\tilde{a}) \mathrm{J}_{m-1}(\zeta_{m n}\tilde{a}) \Big\} \end{aligned} \quad &\text{for }n=n', \end{cases}\nonumber\\ Y_{nn'} &= \delta_{nn'}\frac{1}{2} \mathrm{J}^2_{m-1}(\zeta_{m n}),\end{aligned}$$ where $\tilde{a}\equiv a/R$ and $\tilde{r}\equiv r/R$.   [**Author Contributions**]{} D.J. and Y.X. designed the experiment and fabricated the device. D.J. and T.C. performed the calculation and drafted the manuscript. Y.X., S.W. and K.Y.F. tested the device. D.J. and M.F. carried out the measurement. G.C.G., S.F. and Q.H. grew and processed the MBE samples. Y.W., L.E., M.J.M., N.X.F. and X.Z. provided the guidance and joined the discussion. X.Z. led the project. All authors contributed to the manuscript.

--- abstract: 'The studies of hyperon production performed at COSY-11 are summarized. The results of the experiments in the reaction channels $pp \rightarrow pK^+\Lambda$, $pp \rightarrow pK^+\Sigma^0$, and $pp \rightarrow nK^+\Sigma^+$ are shown. Excitation functions from threshold up to about 90 MeV excess energies have been evaluated with high precision for the $\Lambda$ and $\Sigma^0$ production. The $\Lambda p$ and $\Sigma^0 p$ final state interactions were extracted. The $\Sigma^+$ production was measured at 13 and 60 MeV excess energies.' author: - | D. Grzonka\ for the COSY-11 collaboration title: | Summary of the COSY-11 Measurements\ of Hyperon Production --- [ address=[Institut f[ü]{}r Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany]{}]{} Introduction ============ The hyperon-nucleon interaction is less known than the one for the nucleon-nucleon system due to the difficulties in performing scattering experiments with the unstable hyperons. The existing YN-scattering data are rather limited [@eng66; @sec68; @ale68; @eis71] and for a better understanding of the strong interaction in the nonperturbative region of the QCD an extension of the data base in the strangeness sector is very important. Besides hyperon-nucleon scattering, reactions into 3-body exit channels like: $NN \rightarrow NKY$ can be used to extract detailed information on the NY-subsystem. The YN interaction is only one aspect covered by these kinds of experiments which can be separated into three stages, the initial state interaction of the incoming nucleons, the associated strangeness production process and the final state interaction. Final state interaction happens between all exit particles and by separating a suitable kinematic region also the KN and KY interaction can be studied. Furthermore information about the contributing reaction mechanisms are obtained including the excitation of nucleon resonances which is also directed to the structure of these resonances. Most favourable for these studies are experiments close to the reaction threshold due to the low relative momenta and therefore long interaction times between the ejectiles. In order to get a detailed understanding of these elementary interactions involving strangeness differential cross sections as a function of spin and isospin degrees of freedom are required. A significant contribution to this kind of physics has been done by the hyperon production experiments at COSY-11. The experimental setup for strangeness production at COSY-11 ============================================================ The internal COSY-11 installation [@bra96] at COSY [@mai97] was designed for near threshold meson production studies. It used a COSY machine dipole as magnetic spectrometer and included scintillation detectors and drift chambers to reconstruct particle tracks of positively charged particles and measure their velocities in order to determine their four-momentum components with high precision. A sketch of the setup is given in fig. \[c11setup\] and for more details see [@bra96]. ![ \[c11setup\] The COSY-11 detection system installed at a COSY machine dipole with the detector components relevant for the hyperon production studies. The left side shows a 3-d view of the arrangement and on the right side is a sketch of the detector components to illustrate the principle of operation. The S8 scintillator was only used for the $pp \rightarrow n K^+ \Sigma^+$ reaction.](c11detsw.ps "fig:"){width=".6\textwidth"} ![ \[c11setup\] The COSY-11 detection system installed at a COSY machine dipole with the detector components relevant for the hyperon production studies. The left side shows a 3-d view of the arrangement and on the right side is a sketch of the detector components to illustrate the principle of operation. The S8 scintillator was only used for the $pp \rightarrow n K^+ \Sigma^+$ reaction.](c11det_sketch.eps "fig:"){width=".4\textwidth"} In the case of hyperon production via the reaction channels $pp \rightarrow pK^+ \Lambda / \Sigma^0$ the proton velocities are measured with the scintillator hodoscopes S1 and S3 but for the kaon the $\sim$ 9 m flight path to S3 is too long. Most of the kaons would decay before reaching S3. Here the flight path from the target to S1 is used where the start time is calculated from the measured proton momentum. The particle identification is worse than in the proton case due to the much shorter flight path but its still sufficient to separate most of the pions and protons from the kaons. The hyperon four-momentum $P_{hyperon}$ is determined by the missing mass technique: $P_{\Lambda} = P_{beam} - P_{p} - P_{K^+}$ with the known beam $P_{beam}$ and the measured proton $P_{p}$ and kaon $P_{K^+}$ four-momenta. This method results in a rather clean separation of the hyperon production events as can be seen from fig. \[hypexp\] left for an event sample of $\Lambda$ production at 7 MeV excess energy. ![\[hypexp\]Invariant mass of the second track as a function of the missing mass for $\Lambda$ production at 7 MeV excess energy (left). Missing mass squared distribution for $\Sigma^0$ production at 7 MeV excess energy (up right). Missing mass squared distributions for $\Sigma^+$ production at 13 and 60 MeV excess energy (down right) with the applied background subtraction and the expected distributions of $\Sigma^+$ production from Monte Carlo.](pkl_reac_idn2.eps){width="\textwidth"} ![\[hypexp\]Invariant mass of the second track as a function of the missing mass for $\Lambda$ production at 7 MeV excess energy (left). Missing mass squared distribution for $\Sigma^0$ production at 7 MeV excess energy (up right). Missing mass squared distributions for $\Sigma^+$ production at 13 and 60 MeV excess energy (down right) with the applied background subtraction and the expected distributions of $\Sigma^+$ production from Monte Carlo.](mmsigma.eps "fig:"){width="\textwidth"} ![\[hypexp\]Invariant mass of the second track as a function of the missing mass for $\Lambda$ production at 7 MeV excess energy (left). Missing mass squared distribution for $\Sigma^0$ production at 7 MeV excess energy (up right). Missing mass squared distributions for $\Sigma^+$ production at 13 and 60 MeV excess energy (down right) with the applied background subtraction and the expected distributions of $\Sigma^+$ production from Monte Carlo.](mm_nks.eps "fig:"){width="\textwidth"} In the case of the $\Sigma^0$ production its similar but the background level is higher as can seen from fig. \[hypexp\] up right which shows the missing mass squared distribution in the kaon band at an excess energy of 7 MeV for $pp \rightarrow pK^+ \Sigma^0$. In parallel to the $\Sigma^0$ production also $\Lambda$ production at about 80 MeV higher excess energies is measured. With the addition of a neutron detector, installed for studies at a deuteron target, another hyperon channel was accessible at COSY-11, namely the $pp \rightarrow nK^+ \Sigma^+$ reaction. Here the peak to background ratio was less favourable, see fig. \[hypexp\] down right, because no proton is in the exit channel to produce a precise timing signal. The neutron detector provided the time and position of the point of the first neutron interaction producing a charged ejectile from which the neutron momentum was calculated. The absolute time calibration was performed with $\gamma$’s by selecting $pp \rightarrow pp \pi^0$ events with the $\pi^0$ decaying within the target into two $\gamma$’s from which the event start time was calculated. For the $K^+$ time of flight measurement the additional S8 scintillator was used with a distance of only 1.9 m to S1. The $\Sigma^+$ with a $c\tau$ of 2.4 cm couldn’t be measured directly but its four-momentum was determined by a missing mass analysis. Further hyperon channels are not feasible at COSY-11. In principle also the ($n \Lambda$) and ($n \Sigma^0$) system could be studied by using a deuteron target but the additional detection of the spectator proton would reduce the efficiency drastically. Also hyperon decay products could in principle be measured but the efficiency was extremely low. In all measurements the luminosity was determined by elastic $pp$-scattering detected in parallel to the hyperon production. For the detection of the second proton a Si-pad detector combined with a scintillator ($Si_{mon} / S4$ in fig. \[c11setup\] ) was installed. For studies with a polarized beam the polarization has to be determined. Two detection systems served for this aim: In addition to the COSY polarimeter a pair of wire chambers and scintillators were installed above and below the beam close to the target to measure the elastic $pp$-scattering at $\phi = 90 ^{\circ}$ which is independent of the polarisation. Experimental results ==================== When COSY-11 went into operation in 1996 no data were available for $\Lambda$ and $\Sigma$ hyperon production close to the reaction threshold. For the reaction channel $pp \rightarrow pK^+ \Lambda$ above 300 MeV/c excess energy data were existing mostly from bubble chamber measurements at CERN [@bal88]. On the theoretical side parametrizations of the excitation function were on the market which differ close to threshold by several orders of magnitude [@ran80; @sch88]. For the $\Sigma$ production the situation was similar, there were no data available below a few hundred MeV excess energy. The first hyperon production studies at COSY-11 were performed for the $\Lambda$ channel. The excitation function for the $pp \rightarrow pK^+ \Lambda$ reaction was measured in several beam times for excess energies between 0.7 MeV and 90 MeV [@bal96; @bal98; @sew99; @kow04]. Compared to the parametrization of [@ran80; @sch88] the data differ by more than an order of magnitude but with the COSY activities in the hyperon channel several theory groups were triggered to develop improved models which describe the data much better [@sib95; @lik95; @lik98; @fal97; @tsu97; @tsu99]. In fig. \[c11data\] the threshold data are shown in a linear excess energy scale including the expected phase space behavior with and without $NY$-FSI adjusted to the data. It is clearly seen that close to threshold a pure 3-body phase space description is insufficient for the case of the $\Lambda$ hyperon production. The final state interaction between proton and $\Lambda$ has to be taken into account. To include FSI the Fäldt-Wilkin parametrization has been used [@fal97; @wil07]. ![ \[c11data\] The $pp \rightarrow pK^+ \Lambda$, $pp \rightarrow pK^+ \Sigma^0$ and $pp \rightarrow nK^+ \Sigma^+$ cross sections as a function of the excess energy $Q$  [@bal98; @sew99; @kow04; @bil98]. The lines show the calculations corresponding to 3-body phase space with (solid line) and without (dashed line) final state interaction. ](fw_ps_all.eps){width=".7\textwidth"} Similar data were taken for the $\Sigma^0$ production channel $pp \rightarrow p K^+ \Sigma^0$ [@sew99; @kow04]. In a first study for excess energies around 15 MeV the obtained results (fig. \[lsratio\]) show the remarkable feature that the cross section ratio $\sigma _{\Lambda} / \sigma_ {\Sigma^0}$ gives a factor of 28 instead of $\sim$ 2.5 known from high energy data above 300 MeV excess energy expected from isospin relations. The first step to understand this behavior was the extension to higher excess energies to see the transition from this unexpected high ratio to the value of 2.5. In order to reduce systematical errors further data were taken in the supercycle mode of COSY operation which allows to switch the beam momentum betweeen two successive COSY cycles. One cycle for $\Lambda$ production was followed by a few cycles for $\Sigma^0$ production at the same excess energy. The data are also given in fig.\[c11data\]. In the $\Sigma^0$ case there is no need for any FSI, an inclusion of (p$\Sigma$) FSI doesn’t give any improvement of the $\chi ^2$ in the fit. The difference between the (p$\Sigma$) and (p$\Lambda$) system is best seen in the cross section ratio which is shown in fig. \[lsratio\]. The cross section ratio $\sigma _{\Lambda} / \sigma_ {\Sigma^0}$ goes smoothly down to the low energy value. The question arises whether the whole enhancement in the $\Lambda$ channel is due to the strong (p$\Lambda$) FSI as proposed by [@sib06] or the production process itself gives some enhancement for the $\Lambda$ channel. If its a pure FSI effect why is the (p$\Lambda$) so much larger than the (p$\Sigma$) FSI? When the first data on the cross section ratio were published several theory groups tried to describe the data in different models: coherent or incoherent $\pi$ and $K$ exchange or in addition an intermediate resonance excitation. Most of them indeed show a trend of increasing ratio towards the threshold, see fig. \[lsratio\], but there is no clear preference of any description suggested. More data are needed to understand the hyperon production process as for instance data in other isospin channels. To illustrate this lets consider a specific model. In the Jülich model [@gas01] the $pp \rightarrow p K^+ \Lambda / \Sigma^0$ reactions are described by only pion and kaon exchange where a reduction of the $\Sigma^0$ cross section results from a destructive interference of the pion and kaon amplitudes. Calculations for the $pp \rightarrow n K^+ \Sigma^+$ channel within this model show a big difference between a destructive ($\sigma_{pp\rightarrow nK^+ \Sigma^+} / \sigma_{pp\rightarrow pK^+ \Sigma^0} \ = 3.1$) and a constructive ($\sigma_{pp\rightarrow nK^+ \Sigma^+} / \sigma_{pp\rightarrow pK^+ \Sigma^0} \ = 0.34$) interference. A similar high sensitivity is expected also in other models which include nucleon resonances. Therefore the study was extended to the $\Sigma^+$ production in order to disentangle the different production mechanisms [@roz06]. From the experimental point of view a clean separation of the $pp \rightarrow n K^+ \Sigma^+$ channel was difficult due to the high background level which resulted in large error bars for the cross section determination. However, in spite of the large errors the measured cross sections exceed the highest predictions from models [@gas01; @sib99; @tsu99; @sib07] by at least an order of magnitude. The data points are shown in fig. \[c11data\]. This interesting but not yet understood result confirms the need for more data in the hyperon sector. As mentioned in the introduction the three body final state allows to study within some approximation the individual two body subsystems. This was done by a Dalitz plot analysis of the COSY-11 $\Lambda$ production data in order to extract the $\Lambda$-p scattering length [@bal98b]. Within such an analysis with unpolarized beam and target it is not possible to separate the spin singlet and spin triplet components of the scattering length. Only spin averaged values for scattering length $\bar{a}$ and effective range $\bar{r_0}$ could be determined with values of $\bar{a}\, = \, -2.0 fm$ , $\bar{r_0} = 1.0 fm$. The analysis was done in an effective range expansion which, however, is only applicable for systems where the scattering length is significantly larger than the effective range. Furthermore the procedure exhibits strong correlations between the effective range parameters $a$ and $r$ that can only be disentangled by including $\Lambda N$ elastic cross sections data. A new method to determine the scattering length was developed by the Jülich group which is based on a dispersion relation technique [@gas04; @gas05]. The advantage of the technique is that the error in the relation between scattering length and experimental observable due to the derivation in the model is well under control and is below 0.2 fm. Furthermore observables have been derived where spin singlet (($1+A_{xx} + A_{yy} + A_{zz}) \cdot \sigma _{\Theta (K) = 90 ^{\circ}}$) and spin triplet ($A_{0y} \cdot \sigma _{\Theta (K) = 90 ^{\circ}}$) contributions are separated. The measurement of a separate spin singlet contribution requires all spin correlation coefficients ($A_{ii}$), i.e. polarized beam and polarized target, but for the spin triplet component the asymmetry $A_{0y}$ is sufficient if a special kinematic condition, kaon emission around 90 deg. is selected. In order to extract the spin triplet scattering length of the $\Lambda p$ system a measurement with polarized beam has been performed at COSY-11. The experiment aimed for a precision of the scattering length in the same order as the theoretical uncertainty of about 0.2 fm. The error is given by the statistical uncertainty and by the size of the asymmetry which is not known in this excess energy region. From first results of the analysis the asymmetry seems to be rather small which would result in large errors on the scattering length but the data evaluation is still going on and final results have to be awaiten. Besides COSY-11 also other COSY experiments studied hyperon production. The TOF collaboration uses a large acceptance non magnetic detection system with a decay spectrometer for the delayed strange particle decays resulting in a high selectivity for the hyperon reaction channels. Various reactions were studied like $pp \rightarrow p K^+ \Lambda$ [@bil98; @abd06], $pp \rightarrow p K^+ \Sigma^0$, $pp \rightarrow p K^0 \Sigma^+$ [@abd07], and $pp \rightarrow n K^+ \Sigma^+$ but at somewhat higher excess energies than COSY-11. For the $pp \rightarrow p K^+ \Lambda$ reaction channel at excess energies of 85, 115 and 171 MeV Dalitz plot analyses have been performed which show clearly the excitation of nucleon resonances contributing to the production process [@abd06]. But to what extent the resonance excitation contributs to the COSY-11 data close to threshold is not clear. At the BIG KARL magnetic spectrometer an inclusive measurement of reaction was done by detecting the kaon with high momentum resolution in order to determine the $p\Lambda$ scattering parameters. The analysis gives constrains on the singlet and triplet scattering length where $|a_s| > |a_t|$ [@hin05] but to separate singlet and triplet contribution measurements with polarized beam and target are needed. Furthermore at ANKE, an internal magnetic spectrometer, hyperon production studies have started. The kaon detection is done by measuring the delayed decay of stopped kaons which gives a very high selectivity for kaons. Data of the $pp \rightarrow n K^+ \Sigma^+$ reaction have been taken at 93 and 128 MeV excess energy which don’t show such a high cross section as the COSY-11 data at lower excess energies but are consistent with model predictions [@val072]. These studies will be continued at lower excess energies [@val07]. Conclusion ========== The hyperon production studies of the COSY-11 collaboration resulted in precise cross section data of the reaction channels $pp \rightarrow p K^+ \Lambda$ and $pp \rightarrow p K^+ \Sigma^0$ from the production threshold up to about 90 MeV excess energy where no data were available at all. These cross section data are important ingrediants for calculations in different fields like heavy ion collision studies, hypernuclei production or neutron star formation. The excitation functions allowed to extract the $NY$ final state interaction which is related to the $NY$ interaction strength parameterized by the scattering length. Models of the hyperon-nucleon scattering mostly rely on SU(3) symmetry relations for the coupling constants and fit the model to the data. There is need for an improved data base in this strangeness sector for a better understanding of the strong interaction in this domain of non perturbative QCD. For the $p\Lambda$ system a strong $p\Lambda$ FSI is clearly seen whereas the $p\Sigma^0$ seems to show no FSI although the quark content of both systems is the same. But for a clear picture also the other $\Sigma$ isospin channels have to be studied. A first attempt has been done with the measurement of the $pp \rightarrow n K^+ \Sigma^+$ reaction cross sections which exceed largely the model predictions. The large error bars indicate somehow the limit of the COSY-11 facility for these studies. An improved detection technique has to be applied for further investigations. Studies of the hyperon channels have to be continued. There are still many open questions and more differential observables including the spin and isospin degrees of freedom are needed to disentangle the contributing production mechnisms and to improve the knowledge on the $YN$-interaction. At ANKE and at the TOF experiment [@tof07] further studies are proposed and in the near future certainly also WASA at COSY will be used for such investigations. With its large solid angle coverage for charge and neutral particles in principle all hyperon channels can be studied in detail. Acknowledgments {#acknow} =============== The hyperon production studies at COSY-11 were supported by the following grants: European Community - Access to Research Infrastructure action of the Improving Potential Programme (FP5 and FP6 within HadronPhysics RII3-CT-2004-506078, FFE grants (41266606 and 41266654, 41324880) from the Research Center Jülich, DAAD Exchange Programme (PPP-Polen), Polish State Committee for Scientific Research (grant No. PB1060/P03/2004/26, 2P03B-047-13), the German Research Foundation (DFG) grant No. GZ: 436 POL 113/117/0-1, the International B[üro]{} IB-DLR , and the Verbundforschung of the BMBF (06MS881I). [9]{} R. Engelmann et al., *Phys. Lett.* **21**, 587 (1966). B. Sechi-Zorn et al., *Phys. Rev*. **175**, 1735 (1968). G. Alexander et al., *Phys. Rev.* **173**,1452 (1968). F. Eisele et al., *Phys. Lett.* **37B**,204 (1971). S. Brauksiepe et al., *Nucl. Instr. & Meth.* [**A376**]{},397 (1996). R. Maier, *Nucl. Instr. & Meth.* **A390**,1 (1997). A. 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--- abstract: 'The upgrade of the DA$\Phi$NE machine layout requires a modification of the size and position of the inner focusing quadrupoles of KLOE$^2$ thus asking for the realization of two new calorimeters covering the quadrupoles area. To improve the reconstruction of $K_L\to 2\pi^0$ events with photons hitting the quadrupoles, a tile calorimeter, QCALT, with high efficiency to low energy photons (20-300 MeV), time resolution of less than 1 ns and space resolution of few cm, is needed. We propose a tile calorimeter with a high granularity readout corresponding to about 2500 silicon photomultipliers (SiPM) of $1\times 1$ mm$^2$ area. Moreover, the low polar angle regions need the realization of a dense crystal calorimeter with very high time resolution performances to extend the acceptance for multiphotons events. Best candidates for this calorimeter are LYSO crystals with APD readout or PbWO$_4$ crystals with large area SIPM readout.' address: - 'Laboratori Nazionali di Frascati dell’INFN' - 'Dipartimento di Energetica Univ. Roma La Sapienza' author: - 'F.Happacher, M.Martini$^{a,}$[^1], S.Miscetti$^{a}$, I.Sarra$^{a}$' title: Tile and crystal calorimeters for the KLOE$^2$ experiment --- The KLOE$^2$ proposal ===================== In the last decade a wide experimental program has been carried out at Da$\Phi$ne[@dafne], the $e^+e^-$ collider of the Frascati National Laboratories, running at a center of mass energy of 1020 MeV, the $\Phi$ resonance mass. During KLOE run, Da$\Phi$ne delivered a peak luminosity of 1.5$\times$10$^{32}$ cm$^{-2}$s$^{-1}$ which granted about 1 fb$^{-1}$ per year in the last data taking period. A new machine scheme has been recently proposed by the Frascati accelerator group aiming at increasing the luminosity of the machine up to a factor 5. This scheme has been succesfully tested at Da$\Phi$ne, and these encouraging results push for a new data taking compaign for the KLOE experiment to complete its physics program and to perform a new interesting set of measurements. The new experiment, named KLOE$^2$, expects to collect 5 fb$^{-1}$/year. We are now working to improve the performances of our detector[@kloe_all] adding: an inner tracker, a tagger system to study $\gamma\gamma$ physics, a new small angle calorimeter and a new quadrupole calorimeter. In this paper we explain the project and the R&D for these last two items. Quadrupole tile calorimeter, QCALT ================================== In Fig.\[figkloe\], we show a section of the KLOE detector in which is visible the old position of the focalizing quadrupoles and the surrounding calorimeters QCAL [@oldqcal] which have a polar angle coverage of 0.94$\,<|\cos\theta|<\break$0.99. Each calorimeter consists of 16 azimuthal sectors composed by alternating layers of 2 mm lead and 1 mm BC408 scintillator tiles, for a total thickness of $\sim$5X$_0$. The readout is done by wavelength shifter fibers (Kuraray Y11-200) and photomultipliers. The fiber arrangement allows the measurement of the longitudinal coordinate by time differences. These calorimeters are characterized by a low light response ($\sim$3 pe/mip/tile at zero distance from the photomultiplier) due to the coupling in air, to the fiber lenght ($\sim$2 m for each tile) and to the quantum efficiency of the used photomultipliers (standard bialkali with $\sim$20% QE). The project of the new QCAL consists in a dodecagonal structure, 1 m long, covering the region of the quadrupoles. The structure consists in a sampling of 5 layers of 5 mm thick scintillator plates alternated with 3.5 mm thick tungsten plates, for a total depth of 4.75 cm (5.5 X$_0$). The active part of each plane is divided into twenty tiles of 5$\times$5 cm$^2$ area with 1 mm diameter WLS fibers embedded in circular grooves. Each fiber is then optically connected to a silicon photomultiplier of 1 mm$^2$ area, SiPM, for a total of 2400 channels. We have done some R&D studies on SiPM, fibers and tiles to choose the combination which optimizes the response of our system. Test on MPPC ------------ We have compared the characteristics of two different SiPM produced by Hamamatsu (multi pixel photon counter, MPPC): 100 (S10362-11-100U) and 400 pixels (S10362-11-050U), both with 1$\times 1$ mm$^2$ active area. To manage the signals, the electronic service of the Frascati Laboratory (SELF) has developed a custom electronics composed by a 1$\times$2 cm$^2$ card, containing the pre-amplifier, and a multifunction NIM board. For these tests, we have set the pre-amplifier gain to 50. The NIM board supplies the voltage to the photodetector (Vbias) with a precision of 2 mV and a stability at the level of 0.03 permill. A low threshold discriminator and a fanout are also present in the board. To determine the gain, we have prepared a setup based on a blue light LED and a polaroid filter to change light intensity. We have measured the gain and the dark rate variation as a function of the applied Vbias and the temperature of the photodetector. The readout electronics was based on CAMAC, with a charge sensitivity of 0.25 pC/count and a time of 125 ps/count. Our tests confirm the performances declared by Hamamatsu and show a significative variation of the detector gain as a function of the temperature. The 400 pixels shows a temperature dependence of the gain which is a factor four smaller than the 100 pixels (3% versus 12%), with a gain reduction of a factor five. Tests on fibers --------------- We have studied the light response of three different, 1 mm$^2$, fibers optically connected to MPPC: - Kuraray SCSF81 (blue scintillating) - Saint Gobain BCF92 single cladding (WLS from blue to green) - Saint Gobain BCF92 multi cladding (WLS from blue to green) The test is done firing the fiber with a Sr$^{90}$ source. The trigger is provided by a NE110 scintillator finger (1$\times$5$\times$0.5 cm$^3$) connected to a bialkali photomultiplier positioned below the fiber. As expected, a large light yield is shown for SCSF81 while the WLS fibers have a reduced response. However, the BCF92 multi cladding has a reasonable light yield as shown in Fig.\[figsgmc\]. For this fiber we have: maximum light yield, fast response (5 ns/pe) and high attenuation length (3.5 m). Tests on tiles -------------- Light response and time resolution of tiles have been measured using cosmic rays. The system was prepared connecting the fiber to the MPPC and using two NE110 fingers to trigger the signal. We have prepared two different tiles: A : 3 mm thick tile with 400 pixels MPPC, B : 5 mm thick tile with 100 pixels MPPC. Using ADC distribution we find: 14 pe/mip for tile “A” and 26 pe/mip for tile “B” (See Fig.\[figtile\]). These results are comparable taking into account the thickness ratio between tiles and the photon detection efficiency of the two detectors (40% for 400 pixels and 45% for 100 pixels). Correcting for the time dependence on pulse height, we find a preliminary time resolution of 1000 ps (750 ps) for tile “A” (“B”). Next plans ---------- We are now assembling two small dimension multi-tiles prototypes of the QCAL, to study signal transportation and to measure the effective radiation length. In 2009, we plan also to construct a “module 0” consisting of a complete slice of the dodecagon (1/12 of one calorimeter) with final material and electronics. Crystal calorimeter with timing, CCALT ====================================== In the new design of Da$\Phi$ne interaction region, the position of the quadrupoles increases the acceptance of the central calorimeter from 21$^{\circ}$ to 18$^{\circ}$. Below this limit we can safely insert few crystals to improve the acceptance for photons coming from $\eta$ and $K_S$ decays. This detector could work as veto detector for photons down to 8$^{\circ}$. The particular region is visible in Fig.\[figkloe\] and is delimited between the focalizing quadrupoles and the spherical interaction region of the KLOE detector. The proposed solution is to insert an homogeneus calorimeter based on LYSO ($Lu_{18}Y_{.2}SiO_5:Ce$) crystals. The most important characteristics of these crystals are a very high light yield, a time emission, $\tau$, of 40 ns, high density and X$_0$ without beeing hygroscopic (See Tab.\[tablfs\]). These crystals well match the request of high efficiency to low energy photons and excellent time resolution for the CCALT, which will help in fighting the high level of machine background events present in the low energy region. The preliminary project consists in a dodecagonal barrel for each side of the interaction region, composed by LYSO crystals (2$\times$2$\times$13 cm$^3$) readout using avalanche photodiodes (APD). Preliminary tests on LYSO ------------------------- We have tested one LYSO crystals from Saint Gobain (2$\times$2$\times$15 cm$^3$) using both cosmic rays and electrons from Frascati beam test facility (BTF). For cosmic rays test, the crystal was readout using standard bialkali photomultiplier, while in the test beam we used a 5$\times$5 mm$^2$ avalanche photodiode provided by Hamamatsu (S8664-55). In both cases, the crystal was simply wrapped with mylar and the photodevice coupled with optical grease and simple mechanical arrangement. After correcting the time dependence on pulse height, we obtain, from cosmic rays, a preliminary time resolution $\sigma_T=360$ ps, which corresponds to 12000 pe/mip ($N_{pe}=(\tau/\sigma_T)^2$). Assuming a MIP to deposit 10 MeV/cm this correponds to 600 pe/MeV. At the BTF, we have used electrons from 50 to 400 MeV to measure the time resolution of the crystals which is well parametrized by: $$\sigma_T=\frac{82\,ps}{\sqrt{E(GeV)}}\oplus 293\,ps$$ The constant term is probably related to the different arrival times of the showering photons along the crystal axis. From the statistical term we derive a light yield of 240 pe/MeV which is in rough agreement with the results from cosmic rays taking into account quantum and collection efficiency. Next plans ---------- A dedicated test beam with electrons will be held in Frascati beam test facility in the first months of 2009. The aim of the test is to characterize energy and time resolution of a 3$\times$3 matrix of high quality crystals surrounded by an outer leakage detector done with PbWO. This test will also allow to compare the response of different kind of LYSO crystals (Saint Gobain and Scionix) and a similar crystal (LFS by Zecotek) which could be an alternative candidate (See Tab.\[tablfs\]). LYSO LFS ---------------------------- ------ --------- Density 7.1 7.2-7.3 Attenuation lenght (cm) 1.2 1.12 Decay constant (ns) 41 35-36 Max emission (nm) 420 435-438 Light yield (relative NaI) 75 80-85 Energy resolution 8 9-12 Hygroscopic no no Refractive index 1.81 1.78 : Comparison between LYSO and LFS characteristics[]{data-label="tablfs"} Conclusions =========== The new scheme proposed for the Da$\Phi$ne machine allows a factor 5 increase in the delivered luminosity. Some R&D are in progress to add new components to the KLOE apparatus. We have presented the proposal for two low angle calorimeters. QCALT is a tile calorimeter surronding the focalizing quadrupoles to increase the coverage of the electromagnetic calorimeter. CCALT is a LYSO calorimeter aimed to be a good veto for low angle photons. We have presented the preliminary measurement on tiles and crystals and the characterization of the photodevices used with tiles. Acknowledgement =============== We want to thank G.Corradi, D.Tagnani and C.Paglia to have developed the readout electronics of the SiPM; M.Cordelli for his help during the preparation of test setup; M.Arpaia, G.Bisogni, A.Cassarà, A.Di Virgilio, U.Martini and A.Olivieri for their help in the mechanical preparation of the setup and in the preparation of the tiles. [90]{} S. Guiducci, in: P. Lucas, S. Weber (Eds.), Proceedings of the 2001 Particle Accelerator Conference, Chicago, Il., USA, 2001. KLOE collaboration, LNF-92/019(IR) (1992) and LNF-93/002(IR) (1993). M. Adinolfi [*et al.*]{} (KLOE collaboration), [*NIM*]{} [**A483**]{} (2002) 649. The KLOE collaboration, NIM [**A 482**]{} (2002) 364-386. [^1]: Corresponding author. Mail: matteo.martini@lnf.infn.it

--- abstract: 'We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group $G$ equipped with an arbitrary compatible left-invariant metric $d$, the Lipschitz-free space over $G$, ${\mathcal{F}}(G,d)$, satisfies the metric approximation property. We show also that, given a finitely generated group $G$, with its word metric $d$, from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, ${\mathcal{F}}(G,d)$ has a Schauder basis. Examples and applications are discussed. In particular, for any net $N$ in a real hyperbolic $n$-space $\mathbb{H}^n$, ${\mathcal{F}}(N)$ has a Schauder basis.' address: - | Institute of Mathematics\ Czech Academy of Sciences\ Žitná 25\ 115 67 Praha 1\ Czech Republic - 'Instituto de Ciência e Tecnologia da Universidade federal de São Paulo, Av. Cesare Giulio Lattes, 1201, ZIP 12247-014 São José dos Campos/SP, Brasil' author: - Michal Doucha - 'Pedro L. Kaufmann' bibliography: - 'references.bib' title: 'Approximation properties in Lipschitz-free spaces over groups' --- [^1] Introduction ============ Lipschitz-free spaces form by now a fundamental class of Banach spaces, whose study has been revitalized since the appearance of the seminal paper of Godefroy and Kalton ([@godefroy2003lipschitz]). There are two main important properties that both characterize these spaces. Namely, they are free objects in the category of Banach spaces over the metric spaces. Second, they are canonical isometric preduals to the Banach spaces of pointed Lipschitz real-valued functions. Another appealing feature is that their study connects Banach space theory to several other areas of mathematics, including optimal transport and geometry, and, as we demonstrate here, also harmonic analysis. We recall some basic facts about Lipschitz-free spaces in Section \[section:preliminaries\]. Approximation properties in Lipschitz-free spaces have been one of the main research directions since the publication of [@godefroy2003lipschitz]. It has become clear since then that there are metric spaces such that the corresponding Lipschitz-free space does not have the approximation property, since by [@godefroy2003lipschitz Theorem 5.3], a Banach space $X$ has the bounded approximation property if and only if the Lipschitz-free space ${\mathcal{F}}(X)$ does. The attention was therefore shifted to certain amenable classes of metric spaces, in particular compact metric spaces and, to some extent, also to uniformly discrete metric spaces. The compact metric case is particularly important since it has been shown by Godefroy in [@God2015] that the bounded approximation property of Lipschitz-free space over a compact metric space $M$ is equivalent to the existence of linear almost extension operators of Lipschitz functions over subsets of $M$, a subject currently receiving high attention in both geometry and computer science (see e.g. [@BruBru] and [@LN05]). The question whether Lipschitz-free space over any uniformly discrete metric space has the bounded approximation property is perhaps the most serious and still open, we refer to [@godefroy2014free Question 1] for a motivation and to [@Kalton] for the proof that such a space has the approximation property. Regarding compact metric spaces, the first compact metric space such that the corresponding Lipschitz-free space fails the approximation property has been found in [@godefroy2014free], and later, even a compatible metric on the Cantor space has been found so that the Lipschitz-free space lacks the approximation property (see [@HaLaPe]). Positive results have been however obtained in [@Dalet-compact] and [@dalet2015free] when one restricts to countable compact, resp. proper metric spaces. The goal of this paper is to consider certain fundamental classes of compact metric spaces, resp. uniformly discrete metric spaces, which are amenable to methods of harmonic analysis, resp. geometry. Namely, compact groups with left-invariant (or right-invariant) metrics, resp. finitely generated groups with word metrics. It turns out that harmonic analytic methods, resp. certain combinatorial and geometric methods, go hand in hand with our goal of showing approximation properties in Lipschitz-free spaces over compact metric groups, resp. finitely generated groups. In the compact group case we obtain a satisfactory definitive solution. \[thm:intro1\] Let $G$ be a compact metrizable group with an arbitrary compatible left-invariant (or right-invariant) metric $d$. Then ${\mathcal{F}}(G,d)$ has the metric approximation property. Just to show a meager application, we recall that there has been interest in for which compatible metrics of the Cantor space the corresponding Lipschitz-free space has some approximation property. Godefroy and Ozawa show in [@godefroy2014free] that for certain ‘small Cantor spaces’, the free space has the metric approximation property. On the other hand, H' ajek, Lancien, and Perneck' a show in [@HaLaPe] that there are ‘fat Cantor spaces’ for which the free space does not have the approximation property. We recall that there is a very large and thoroughly studied class of compact (metrizable) groups, the *profinite (metrizable) groups*, which are inverse limits of finite groups. So in the infinite metrizable case, they are topologically totally disconnected uncountable metrizable spaces without isolated points - therefore homeomorphic to the Cantor space (see the monograph [@profinite] for more information on profinite groups). It turns out that for any compatible and left-invariant metric on any such group structure on the Cantor space we get a free space with the metric approximation property. In case of finitely generated groups, the metric approximation property follows from known results (see Section \[section:fingengrps\]), so we aim for much stronger property, having the Schauder basis, at the cost of having less general result that applies just to a certain subclass of finitely generated groups. We state the result and postpone the definition of the new notions to the corresponding section. \[thm:intro2\] Let $G$ be a shortlex combable group with its word metric $d$. Then ${\mathcal{F}}(G,d)$ has the Schauder basis [*(see Theorem \[thm:shortlex\])*]{}. We mention that the theorem applies in particular to hyperbolic groups and Artin groups of large type. One of the applications (see Corollary \[cor:hyperbolicnet\]) is that the Lipschitz-free space over any net in the real hyperbolic $n$-space ${\mathbb{H}}^n$ has the Schauder basis.\ The paper is organized as follows. In Section \[section:preliminaries\] we present a characterization of the $\lambda$-bounded approximation property tailored for Lipschitz-free spaces (Proposition \[tool\]). In Section \[section:cpt\] we prove Theorem \[thm:intro1\]; first we tackle Lie groups using harmonic analysis tools in Subsection, then in Subsection \[subsection:generalCpt\] we prove the general case by approximating compact groups by compact Lie ones. In this last subsection we also generalize of Theorem \[thm:intro1\] to compact homogeneous spaces equipped with quotient metrics (Theorem \[thm:homogeneousspace\]). Section \[section:fingengrps\] is dedicated to finitely generated groups; we prove Theorem \[thm:intro2\] and provide some examples and applications. We conclude with some remarks and questions in Section \[section:problems\], and presenting in Appendix \[appendSphere\] a generalization of Theorem \[thm:homogeneousspace\] for the specific case of the Euclidean sphere. Preliminaries {#section:preliminaries} ============= Lipschitz-free spaces --------------------- Let $M$ be a metric space and $0$ be some distinguished point in $M$. Let $\mathrm{Lip}_0(M)$ denote the Banach space of real-valued Lipschitz functions defined on $M$ which vanish at $0$, equipped with the norm $\|\cdot\|_{\mathrm{Lip}}$ which assigns to each function its minimal Lipschitz constant. The Lipschitz-free space over $M$, denoted by ${\mathcal{F}}(M)$, is the canonical isometric predual of $\mathrm{Lip}_0(M)$ given by the closed linear span of $\{\delta(x):x\in M\}$ in $\mathrm{Lip}_0(M)^*$, where each $\delta(x)$ is the evaluation functional defined by $\delta(x)(f):=f(x)$. This gives $\mathrm{Lip}_0(M)$ a $w^*$ topology which coincides, on bounded sets of $\mathrm{Lip}_0(M)$, with the topology of pointwise convergence. ${\mathcal{F}}(M)$ satisfies a powerful universal property: given a Banach space $X$ and a Lipschitz function $F:M\to X$ vanishing at $0$, there exists a unique bounded linear operator $T:{\mathcal{F}}(M)\to X$ such that $T\circ \delta = F$. Its operator norm coincides with the Lipschitz constant of $F$. We refer to Weaver’s book [@weaver1999lipschitz] for a thorough introduction to the subject. There, Lipschitz-free spaces are denominated Arens-Eells spaces. Verifying if a Lipschitz-free space has the BAP ----------------------------------------------- In this subsection we present a characterization of $\lambda$-bounded approximation property suited for Lipschitz-free spaces (Proposition \[tool\] below). Let us briefly recall and comment the definition of this property: Let $X$ be a Banach space, and $\lambda\geq 1$. We say that $X$ has the $\lambda$-approximation property ($\lambda$-BAP for short) if one of the following equivalent assertions holds: 1. For each compact subset $K$ of $X$ and each $\epsilon>0$, there is a $\lambda$-bounded finite rank operator $T$ on $X$ such that $\|Tx-x\|<\epsilon$, for each $x\in K$. 2. For each finite subset $F$ of $X$ and each $\epsilon>0$, there is a $\lambda$-bounded finite rank operator $T$ on $X$ such that $\|Tx-x\|<\epsilon$, for each $x\in F$. 3. There is a $\lambda$-bounded net of finite rank operators $(T_\alpha)$ on $X$ such that $\langle \varphi, T_\alpha x\rangle \rightarrow \langle \varphi,x\rangle$ for each $\varphi\in X^*$ and each $x\in X$ (that is, $T_\alpha$ converges to the identity operator in the *weak operator topology*). If a Banach space $X$ has the 1-BAP, we say that $X$ has the metric approximation property (MAP for short). \[defBAP\] Formulations (1) and (2) are classic; their equivalence with (3) is shown for instance in [@kim2008characterizations]. Recall that a Banach space $X$ has the $\lambda$-BAP if and only if, for each $\delta>0$, $X$ has the $(\lambda+\delta)-BAP$. To see this, fix a compact set $K\subset X$ and $\varepsilon>0$, and take $\delta>0$ small enough so that $M\delta(\lambda+\delta)/\lambda < \epsilon/2$, where $M=\sup_{x\in K}\|x\|$. Let $T$ be a finite rank, $(\lambda+\delta)$-bounded operator on $X$ such that $\|Tx - x\|<\epsilon/2$, for all $x\in K$. Then it is immediately verified that the $\lambda$-bounded operator $S=\lambda T/\|T\|$ satisfies, for each $x\in K$, $\|Sx - x\| <\varepsilon.$\ In what follows $\mathrm{Lip}(M)$ denotes the space of real-valued functions defined on the metric space $M$. We will still use $\|f\|_{\mathrm{Lip}}$ to denote the Lipschitz constant of $f\in \mathrm{Lip}(M)$, keeping in mind that $\|\cdot\|_{\mathrm{Lip}}$ defines only a seminorm in $\mathrm{Lip}(M)$. Let $K$ be a compact metric space and $\lambda\geq 1$. The following assertions are equivalent: 1. ${\mathcal{F}}(K)$ has the $\lambda$-BAP; 2. for each $\varepsilon>0$ there is a net $T_\alpha$ of bounded operators on $C(K)$ such that 1. $T_\alpha$ are of finite rank, 2. each $T_\alpha$ maps Lipschitz functions to Lipschitz functions, 3. $\|T_\alpha f\|_{\mathrm{Lip}}\leq (\lambda+\varepsilon)\|f\|_{\mathrm{Lip}}$, for each $\alpha$ and each $f\in \mathrm{Lip}(K)$, and 4. $T_\alpha f$ converges pointwise to $f$, for each $f\in \mathrm{Lip}(K)$. \[tool\] For the proof we will need the following: Let $K$ be a compact metric space, fix $0\in M$, and let $T$ be a bounded operator on $C(K)$ such that $T(B_{\mathrm{Lip}_0(K)})\subset A.B_{ \mathrm{Lip}_0(K)}$ for some $A>0$. Then $T$ restricted to $\mathrm{Lip}_0(K)$, when seen as an operator on $\mathrm{Lip}_0(K)$, is $w^*$-continuous. \[Trest\] Consider $S=T|_{\mathrm{Lip}_0(K)}$ as an operator on $\mathrm{Lip}_0(K)$. Let $f_\nu$ be a bounded net $w^*$-converging to $f$ in $B_{\mathrm{Lip}_0(K)} $. Since on bounded sets the $w^*$ topology coincides with the topology of pointwise convergence and $f_\nu$ are equicontinuous, it follows that $f_\nu$ converges to $f$ uniformly. By the norm continuity of $T$, $Sf_\nu$ converges to $Sf$ uniformly and, since $\{Sf_\nu\}$ is $A$-bounded in the Lipschitz norm, $Sf_\nu$ $w^*$-converges to $Sf$. That is, $S$ is $w^*$-continuous when restricted to $B_{\mathrm{Lip}_0(K)}$. It follows by Banach-Dieudonné’s theorem that $S$ is $w^*$-continuous in $\mathrm{Lip}_0(K)$. ((1)$\Rightarrow$(2)) Let $T_\alpha$ be a net of $\lambda$-bounded finite rank operators on ${\mathcal{F}}(K)$ converging to the identity in the weak operator topology. Their adjoints $T_\alpha^*:\mathrm{Lip}_0(K)\rightarrow \mathrm{Lip}_0(K)$ can be extended to operators on $C(K)$ that clearly satisfy properties (a)–(d). ((2)$\Rightarrow$(1)) Choose some $0\in K$, and $T_\alpha$ satisfying (a)–(d). It is easily checked that the operators $S_\alpha$ defined on $C(K)$ by $$S_\alpha (f)(x) = T_\alpha (f)(x) - T_\alpha (f)(0)$$ are bounded, satisfy the same assumptions (a)–(d) and $S_\alpha$ are moreover ($\lambda+\varepsilon$)-bounded operators on $\mathrm{Lip}_0(K)$. Each $S_\alpha$ thus satisfies the assumptions of Lemma \[Trest\], and it follows that there are finite-rank, ($\lambda+\varepsilon$)-bounded operators $R_\alpha$ on ${\mathcal{F}}(K)$ with $R_\alpha^*=S_\alpha$, that is: $$\langle S_\alpha f,\gamma\rangle = \langle f, R_\alpha \gamma\rangle,\,\forall f\in \mathrm{Lip}_0(K),\,\forall \gamma \in {\mathcal{F}}(K).$$ Now since $S_\alpha f$ is a bounded net in $\mathrm{Lip}_0(K)$ converging pointwise to $f$, it must converge $w^*$, thus $R_\alpha$ converges to $Id_{{\mathcal{F}}(M)}$ in the weak operator topology. The conclusion follows from the remark after Definition \[defBAP\]. We point out that the above characterization can be generalized to an arbitrary metric space $M$ if we forget the part of $T_\alpha$ being bounded operators on some $C(K)$, and assume that $T_\alpha$ are ($\lambda+\varepsilon)$-bounded operators on $\mathrm{Lip}(M)$ which are moreover pointwise continuous. We omit the proof, which follows the same lines. Approximation properties of Lipschitz-free spaces over compact groups {#section:cpt} ===================================================================== The classical Birkhoff-Kakutani theorem says that a topological group is metrizable if and only if it is metrizable by a left-invariant (equivalently right-invariant) metric if and only if it is Hausdorff and first countable. Compatible left-invariant metrics on a fixed metrizable group are easily seen to be unique up to uniform equivalence. However, a group can admit compatible and left-invariant metrics which are not bi-Lipschitz equivalent. It is even possible that the corresponding Lipschitz-free spaces are not isomorphic. For example, if $d$ is the arc length metric on the torus $\mathbb{T}$, the Lipschitz-free space ${\mathcal{F}}(\mathbb{T},d)$ is isomorphic to $L^1$, but if we substitute $d$ by $d^\alpha$ with $0<\alpha<1$ (in the process often referred to as *snowflaking*), ${\mathcal{F}}(\mathbb{T},d^\alpha)$ is isomorphic to $\ell_1$ (see [@Kalton Theorem 6.5]). In this section we focus on compact metrizable groups (or equivalently, compact first countable groups) and the issue of many non-Lipschitz equivalent left-invariant metrics does not bother us. We use methods of harmonic analysis that are robust enough to make our proofs work for arbitrary compactible left or right-invariant metrics on any compact metrizable group. The main goal of the section is to prove Theorem \[thm:intro1\], however we will have something to say also about free spaces over certain homogeneous spaces of compact groups (see Theorem \[thm:homogeneousspace\]). The section is divided into two subsections. One is dealing with compact Lie groups where the machinery of harmonic analysis is used. The other is dealing with general compact metrizable groups using the fact each such a group can be approximated by compact Lie groups. This approximation then lifts to the corresponding Lipschitz-free space; indeed, we shall show that Lipschitz-free space over a compact metric group can be approximated by Lipschitz-free spaces over compact Lie groups. Lipschitz-free spaces over compact Lie groups {#subsection:cptLie} --------------------------------------------- \[cpt\] From this point on, unless stated otherwise, we always assume that each locally compact group $G$ is equipped with a left-invariant Haar measure, denoted by $\mu$. When $G$ is compact, $\mu$ is assumed to be bi-invariant and of course normalized. Let us start some basic definitions and facts from representation theory of compact groups, which can be found in any standard textbook on harmonic analysis or representation theory of compact groups. Let $G$ be a compact group. A *unitary representation* $U$ is a strongly continuous group homomorphism $x\in G\mapsto U_x \in B(H_U)$ (i.e. on $B(H_U)$ we consider the strong operator topology), where $H_U$ is a complex Hilbert space, such that each operator $U_x$ is unitary on $H_U$. Such unitary representation is said to be *irreducible* if the only invariant subspaces of $H_U$, i.e. subspaces preserved by all $U_x$, are $\{0\}$ and $H_U$. In our specific case where $G$ is compact, all continuous irreducible unitary representations $x\in G\mapsto U_x \in B(H_U)$ satisfy $\dim H_U <\infty$ (see [@hewitt2012abstract Theorem 22.13]). Given an irreducible unitary representation $U:G\rightarrow B(H_U)$ and non-zero vectors $\xi,\eta\in H_U$, the function of the form $$x\in G\mapsto \langle U_x \xi,\eta\rangle \in \mathbb{C}$$ is called a *matrix coefficient* (associated to $U$). If $\xi=\eta$, we call such matrix coefficient a *positive-definite function*. If $H_U$ is equipped with a fixed orthonormal basis $\{\zeta^U_1,...,\zeta^U_{d_U}\}\subseteq H_U$, then the matrix coeeficients $$\varphi^U_{jk}(x) = \langle U_x \zeta^U_j,\zeta^U_k\rangle$$ are called *coordinate functionals* (associated to $U$ and $\{\zeta^U_1,...,\zeta^U_{d_U}\}$). They satisfy the following property, as a consequence of [@hewitt2013abstract Theorem 27.20]: $$\begin{aligned} \forall f\in C(G)\,\forall x\in G,\, f\ast \varphi^U_{jk}(x) = \sum_{r=1}^{d_U} \int f(y)\,\overline{\varphi^U_{rj}(y)}\,d\lambda(y)\, \varphi^U_{rk}(x). \label{coordfunct}\end{aligned}$$ A *trigonometric polynomial* on $G$ is a linear combination of matrix coefficients. It is straightforward to see that all trigonometric polynomials are in fact linear combinations of coordinate functionals, independently of the choice of bases $\{\zeta^U_1,...,\zeta^U_{d_U}\}$, which are assumed to be fixed. Thus, for any given trigonometric polynomial $P$, there is a finite set $F$ of continuous irreducible unitary representations such that $P$ is in the linear span of $\{\varphi^U_{jk}| U \in F,\, j,k = 1,...,d_U\}$. It follows from (\[coordfunct\]) that the operator $f\in C(G) \mapsto f\ast P\in C(G)$ has its range contained in the linear span of $\{\varphi^U_{jk}| U \in F,\, j,k = 1,...,d_U\}$, thus in particular it is of finite rank. The following proposition is our crucial tool from harmonic analysis. \[crucialharmonic\] Suppose that $G$ is a compact Lie group. Then there exists a sequence $F_n$ of positive real functions on $G$ satisfying 1. each $F_n$ is a positive definite *central* (commutes under convolution with any function in $L_1(G)$) trigonometric polynomial, 2. $F_n(g^{-1}) = F_n(g)$, $g\in G$, for each $n$, 3. for each $n$, $\int F_n \,d\lambda =1$, and 4. $f\ast F_n(x) \rightarrow f(x)$ $\lambda$-almost everywhere for every $f\in L_p(G),\,1\leq p <\infty$. By [@Knapp Corollary IV.4.22], every compact Lie group is isomorphic to a matrix group, so we may suppose that $G\subseteq \mathrm{GL}(n,\mathbb{C})$ for some $n\in {\mathbb{N}}$. By the following standard ‘unitarization trick’, we may assume that $G$ is a (necessarily closed) subgroup of the unitary group $U(n)$: let $\langle \cdot,\cdot\rangle'$ be an arbitrary inner product on $\mathbb{C}^n$. Define a new inner product $\langle \cdot,\cdot\rangle$ by setting $$\langle \xi,\eta\rangle:=\int_{g\in G} \langle g\xi,g\eta\rangle'd\mu(g),$$ where $\mu$ is an invariant probability Haar measure on $G$. It is standard and straightforward to check that $\langle \cdot,\cdot\rangle$ is still an inner product, which is moreover invariant by the action of $G$. That implies that $G$ is a subgroup of a finite-dimensional unitary group. Now by [@hewitt2013abstract Theorem 44.29], $G$ satisfies all conditions needed to apply directly [@hewitt2013abstract Theorem 44.25], which gives us positive real functions $F_n$ on $G$ satisfying the conditions (1)–(4). Another ingredient we will need is the following version of Young’s convolution inequality, suitable for Lipschitz functions defined on a locally compact group: Let $G$ be a locally compact group equipped with a compatible left-invariant metric $d$. Suppose that $f\in L^1(G)$ and $g\in \mathrm{Lip}(G)$. Then $f\ast g\in \mathrm{Lip}(G)$ and $$\|f\ast g \|_{\mathrm{Lip}} \leq \|f\|_{L^1}\|g\|_{\mathrm{Lip}}.$$ \[lipconv\] Given arbitrary $x,y\in G$, $$\begin{aligned} |f*g(x) - f*g(y)| & = \left| \int f(z)[g(z^{-1}x) - g(z^{-1}y)]\,d\mu(z) \right| \\ & \leq \int |f(z)| |g(z^{-1}x) - g(z^{-1}y)|\,\mu(z)\\ & \leq \|g\|_{\mathrm{Lip}}\int |f(z)| d(z^{-1}x,z^{-1}y)\, \mu(z)\\ & \leq \|g\|_{\mathrm{Lip}}\int |f(z)| d(x,y) \,\mu(z)\\ &\leq \|f\|_{L^1}\|g\|_{\mathrm{Lip}} d(x,y).\end{aligned}$$ We are now ready to prove the main result from this subsection. We emphasize that in the following we are equipping a Lie group with an *arbitrary* left-invariant metric inducing its topology, not Riemannian metric as it is common in Lie theory. Suppose that $G$ is a compact Lie group equipped with a compatible left-invariant metric. Then ${\mathcal{F}}(G)$ has the MAP. \[proptorus\] Define $T_n: C(G) \rightarrow C(G)$ by $T_n(f) = f\ast F_n$, where $F_n$ are the functions established in Theorem (\[crucialharmonic\]). It suffices to see that these operators satisfy conditions (a)–(d) from Proposition (\[tool\]) with $\lambda=1$, from which follows that ${\mathcal{F}}(G)$ satisfies the MAP. Indeed, each $T_n$ has finite rank, since $F_n$ is a trigonometric polynomial. Thus condition (a) is satisfied. Young’s inequality gives us that $\|T_n f \|_{\mathrm{Lip}} \leq \|F_n\|_{L^1}\|f\|_{\mathrm{Lip}}\leq \|f\|_{\mathrm{Lip}}$, for each $f\in Lip(G)$. Thus condition (b) and (c) are satisfied with constant $\lambda = 1$. (d) follows from condition (4) and the fact that, for each fixed $f\in Lip(G)$, $\{T_n f\}_n$ is equicontinuous, which yields uniform convergence since $G$ is compact. Lipschitz-free spaces over general compact metric groups and homogeneous spaces {#subsection:generalCpt} ------------------------------------------------------------------------------- We start with some remarks on quotient metrics on homogeneous spaces. Let $G$ be a topological group equipped with a compatible left-invariant metric $d$ and let $H$ be a closed subgroup. We want to define a quotient metric on the homogeneous space $G/H$ of left-cosets of $H$. There are two cases. Either $H$ is normal, so $G/H$ is a group. Then the formula $$\begin{aligned} \label{quotmetric} D(gH,fH):=\inf_{h_1,h_2\in H} d(gh_1,fh_2)\end{aligned}$$ defines a compatible left-invariant metric on the quotient group $G/H$, which we refer to as *quotient metric*. We leave the easy verification to the reader. Or $H$ is not normal, so $G/H$ is just a $G$-homogeneous space, i.e. a homogeneous space equipped with a continuous transitive action of $G$. Then the same formula $D(gH,fH):=\inf_{h_1,h_2\in H} d(gh_1,fh_2)$ does not in general define a metric; it does define a compatible $G$-invariant metric provided that $d$ is additionally right $H$-invariant, i.e. $d(g,f)=d(gh,fh)$ for $g,f\in G$ and $h\in H$. We show this. We only verify the triangle inequality, the compatibility and $G$-invariance are easier and left to the reader. Pick $g_1,g_2,g_3\in G$ and let us check that $D(g_1H,g_3H)\leq D(g_1H,g_2H)+D(g_2H,g_3H)$. Choose an arbitrary $\varepsilon>0$ and and some $h_1,h_2, h'_2,h_3\in H$ such that $D(g_1H,g_2H)\geq d(g_1h_1,g_2h_2)-\varepsilon$ and $D(g_2H,g_3H)\geq d(g_2h'_2,g_3h_3)-\varepsilon$. Then $$D(g_1H,g_3H)\leq d(g_1h_1,g_3h_3(h'_2)^{-1}h_2)\leq d(g_1h_1,g_2h_2)+d(g_2h_2,g_3h_3(h'_2)^{-1}h_2)=$$ $$d(g_1h_1,g_2h_2)+d(g_2h'_2,g_3h_3)\leq D(g_1H,g_2H)+D(g_2H,g_3H)-2\varepsilon.$$ Since $\varepsilon$ was arbitrary, we are done. We note that when $G$ is a metrizable group and $H$ is a compact subgroup, then a compatible left-invariant and right $H$-invariant metric on $G$ always exists. Indeed, let $d$ be an arbitrary compatible left-invariant metric on $G$. We define, for $g,f\in G$, $D(g,f):=\max_{h\in H} d(gh,fh)$. Alternatively, using a normalized invariant Haar measure $\mu$ on $H$, we can define $D$ by averaging as follows: $D(g,f):=\int_H d(gh,fh)d\mu(h)$. We leave to the reader to check that both formulas define a compatible left-invariant and right $H$-invariant metrics. Our main tool in this subsection will be the following proposition. We note that simultaneously while writing this paper, the content of the proposition is being developed into a more general form in [@AACD]. \[prop:projection\] Let $G$ be a topological group equipped with a compatible metric $d$ and a compact subgroup $H$. Suppose, additionally, that at least one of the following conditions holds: (i) $d$ is left-invariant and $H$ is normal, (ii) $d$ is right-invariant and $H$ is normal, or (iii) $d$ is left-invariant and right $H$-invariant. Then there exists a norm one projection $P:{\mathcal{F}}(G,d)\rightarrow {\mathcal{F}}(G,d)$ ranging onto a linearly isometric copy of ${\mathcal{F}}(G/H,D)$, where $D$ is the quotient metric as defined in . By the discussion preceding the statement of the proposition, it is verified that $D$ is a well-defined metric. Let $\mu$ be the normalized invariant Haar measure on $H$. If (ii) or (iii) holds, we define a map $P':G\rightarrow {\mathcal{F}}(G)$ by setting for any $g\in G$ $$\begin{aligned} P'(g):=\int_H \delta(g\cdot h)-\delta(h)d\mu(h). \label{P'}\end{aligned}$$In case only (i) holds, we define for any $g\in G$ $$P'(g):=\int_H \delta(h\cdot g)-\delta(h)d\mu(h).$$ We will treat only the case where (ii) or (iii) holds and $P'$ is defined as in , the remainder case is completely analogous. We claim that $P'$ is a $1$-Lipschitz map preserving the distinguished point. The latter is clear, we show that it is $1$-Lipschitz. Pick $g,f\in G$, we have $$\|P'(g)-P'(f)\|=\|\int_H \delta(g\cdot h)-\delta(f\cdot h)d\mu(h)\|\leq \int_H \|\delta(g\cdot h)-\delta(f\cdot h)\|d\mu(h)=d(g,f),$$ where in the last equality we used that $\mu$ is probability and $d$ is invariant. It follows that $P'$ extends to a norm one linear operator $P:{\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G)$. We claim it is the desired projection. First we show that it is a projection. For every $h\in H$ let $P'_h:G\rightarrow {\mathcal{F}}(G)$ be the map defined for every $g\in G$ by $P'_h(g):=\delta(g\cdot h)-\delta(h)$. The following are straightforward to verify: - $P'_h$ is an isometry with $P'_h(1)=0$, thus it extends to a norm one linear map $P_h:{\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G)$. - For every $g\in G$ we have $P'(g)=\int_H P'_h(g)d\mu(h)$ and so also for every $x\in{\mathcal{F}}(G)$ we have $P(x)=\int_H P_h(x)d\mu(h)$. It follows that in order to show that $P^2=P$ it suffices to check that for every $g\in G$ and $h\in H$ we have $P_h\circ P(\delta(g))=P(\delta(g))$. Indeed, the previous equality implies $$P^2(\delta(g))=\int_H P_h\circ P(\delta(g))d\mu(h)=\int_H P(\delta(g))d\mu(h)=P(\delta(g)).$$ Since the set $\{x\in{\mathcal{F}}(G)\colon P^2(x)=P(x)\}$ is a closed linear subspace, we get that $P^2(x)=P(x)$ for all $x\in{\mathcal{F}}(G)$ since ${\mathcal{F}}(G)$ is the closed linear span of $\{\delta(g)\colon g\in G\}$. Let us thus fix $g\in G$ and $h\in H$. We have $$\begin{aligned} P_h\circ P(\delta(g))&=P_h\circ\int_H \delta(g\cdot f)-\delta(f)d\mu(f)\\ &=\int_H \Big(\delta(g\cdot f\cdot h)-\delta(h)\Big)-\Big(\delta(f\cdot h)-\delta(h)\Big)d\mu(f)\\ &=\int_H \delta(g\cdot f\cdot h)-\delta(f\cdot h)d\mu(f\cdot h)=P(\delta(g)),\end{aligned}$$ which finishes the claim. Let $X$ denote the range of $P$ and let us show that it is linearly isometric with ${\mathcal{F}}(G/H,D)$. We define a map $T':G/H\rightarrow {\mathcal{F}}(G)$ by setting for any left coset $gH$ $$T'(gH):=P'(g).$$ We check that it is correctly defined and $1$-Lipschitz. For the former, we need to check that for any $g\in G$ and $h\in H$ we have $P'(g)=P'(gh)$, i.e. $\int_H \delta(g\cdot f)d\mu(f)=\int_H \delta(gh\cdot f)d\mu(f)$. But the equality follows from the invariance of $\mu$. To check that $T'$ is $1$-Lipschitz, pick two cosets $g_1H$ and $g_2H$ and suppose that $f\in H$ is such that $D(g_1H,g_2H)=d(g_1,g_2f)$. Then we have $$\begin{aligned} \|T'(g_1H)-T'(g_2H)\|&=\|\int_H \delta(g_1\cdot h)-\delta(g_2 f\cdot h)d\mu(h)\| \\ &=\int_H D(g_1H,g_2H)d\mu(f)=D(g_1H,g_2H),\end{aligned}$$ showing that $T'$ is actually isometric. It follows that $T'$ extends to a norm one linear surjection $T:{\mathcal{F}}(G/H)\rightarrow X$. In order to show that $T$ is isometric, it suffices to prove that for any finite linear combination $x=\sum_i \alpha_i \delta(g_i H)$ we have $\|x\|_{{\mathcal{F}}(G/H)}=\|T(x)\|_{{\mathcal{F}}(G)}$. One inequality already follows from the fact that $\|T\|=1$, so we only need to prove $\|x\|_{{\mathcal{F}}(G/H)}\leq\|T(x)\|_{{\mathcal{F}}(G)}$. Let $f\in\rm{Lip}_0(G/H)$ be a $1$-Lipschitz function satisfying $\|x\|_{{\mathcal{F}}(G/H)}=|\sum_i \alpha_i f(g_iH)|$. Let $\tilde f$ denote its lift to $G$. That is, for any $g\in G$ and $h\in H$, $\tilde f(gh)=f(gH)$. It is clear that $\tilde f$ is $1$-Lipschitz. In the following, we shall not notationally distinguish between $\rm{Lip}_0(G/H)$-functions and their unique extension to linear functionals. Since we have $\|T(x)\|\geq \tilde f(T(x))$, it suffices to check that $\tilde f(T(x))=f(x)$. For that, in turn, it suffices to check that for every $gH\in G/H$ we have $f(gH)=\tilde f(T'(g))$. We have $$\tilde f(T'(g))=\langle \int_H \delta(g\cdot h)-\delta(h)d\mu(h),\tilde f\rangle=\int_H \tilde f(g\cdot h)-\tilde f(h)d\mu(h)=f(gH)-f(H)=f(gH),$$ which finishes the proof. \[thm:MAPcompactgrp\] Let $G$ be a compact group with a compatible left-invariant metric $d$. Then ${\mathcal{F}}(G,d)$ has the MAP. Before embarking on the proof, we state the following folklore fact, which we prove for the convenience of the reader. \[fact2\] For every compatible left-invariant metric $d$ on a compact metrizable group $G$ there exists a compatible bi-invariant metric $D$ such that $d(g,h)\leq D(g,h)$ for all $g,h\in G$. Let $d$ be an arbitrary compatible left-invariant metric on $G$. We define a compatible bi-invariant metric $D$ by setting, for any $g,f\in G$, $$D(g,f):=\max_{h\in G} d(gh,fh).$$ Clearly, $d\leq D$ and $D$ is bi-invariant. Let us check that $D$ satisfies the triangle inequality and it is compatible with the topology of $G$. For the triangle inequality, pick $g,h,f\in G$ and let us show that $D(g,f)\leq D(g,h)+D(h,f)$. Let $x\in G$ be such that $D(g,f)=d(gx,fx).$ Then we have $$D(g,f)=d(gx,fx)\leq d(gx,hx)+d(hx,fx)\leq D(g,h)+D(h,f),$$ showing the traingle inequality. In order to show that it is compatible with the topology, using left-invariance, it suffices to show that for every sequence $(g_n)_n\subseteq G$ we have $d(g_n,1)\to 0$ if and only if $D(g_n,1)\to 0$. One direction follows immediately from the fact that $d\leq D$. Thus we only need to show that if $d(g_n,1)\to 0$, then $D(g_n,1)\to 0$. Suppose it is not the case and assume without loss of generality that $\lim_n D(g_n,1)=r>0$. For each $n$, let $h_n\in G$ be such that $D(g_n,1)=d(g_nh_n,h_n)$. Again, without loss of generality, we may assume that $(h_n)_n$ converges to some $h\in G$. Then, using that $(g_n)_n$ converges to $1$ since $d$ is compatible, we have $$r=\lim_n D(g_n,1)=\lim_n d(g_nh_n,h_n)=d(h,h)=0,$$ a contradiction finishing the proof. Let $G$ be a compact metrizable group with compatible left-invariant metric $d$. We shall also fix some compatible bi-invariant metric $D$ on $G$ which majorizes $d$, i.e. $d\leq D$, which exists by Fact \[fact2\]. It is a standard consequence of Peter-Weyl’s theorem ([@Knapp Theorem 4.20]) that $G$ can be topologically embedded into an infinite direct product $\prod_{i\in\mathbb{N}} U_i$, where each $U_i$ is a finite-dimensional unitary group. Indeed, by [@Knapp Theorem 4.20] finite-dimensional unitary representations of $G$ separate points. Since $G$ is separable, one can find a sequence $\{\pi_n:G\rightarrow B(H_n)\}_{n\in{\mathbb{N}}}$ of finite dimensional unitary representations separating the points, and their product $\prod_{n\in{\mathbb{N}}} \pi_n$ is then, using also compactness of $G$, a topological embedding of $G$ into a countable product of unitary groups. In particular, $G$ is an inverse limit of a sequence of compact Lie groups $(G_n)_n$. Indeed, let $\Psi:=\prod_{n\in{\mathbb{N}}} \pi_n: G\rightarrow \prod_{i\in\mathbb{N}} U_i$ be the embedding and let, for each $n\in\mathbb{N}$, $P_n:\prod_{i\in\mathbb{N}} U_i\rightarrow \prod_{i\leq n} U_i$ be the projection on the first $n$-coordinates. Then for each $n\in\mathbb{N}$, $G_n:=P_n\circ \Psi[G]$ is a compact Lie group (a closed subgroup of $\prod_{i\leq n} U_i$), and $(G_n)_n$ form an inverse system whose limit is $G$. It follows that there exists a decreasing sequence of compact normal subgroups $(H_n)_n$ of $G$ such that $\bigcap_n H_n=\{1\}$, and for every $n\in\mathbb{N}$, $G/H_n=G_n$. For each $n$, we denote by $d_n$ the quotient metric on $G_n$, which is then compatible and left-invariant on $G_n$. In the following, ${\mathcal{F}}(G)$ is meant to be ${\mathcal{F}}(G,d)$ and ${\mathcal{F}}(G_n)$ is meant to be ${\mathcal{F}}(G_n,d_n)$, for all $n$. Notice that in the following claim we rather work with the bi-invariant metric $D$ majorizing $d$, established in Fact \[fact2\].\ [**Claim 1.**]{} $\rm{diam}_D(H_n)\to 0$, where $\rm{diam}_D(H_n):=\sup_{g,h\in H_n} D(g,h)=\sup_{g\in H_n} D(g,1)$. Suppose on the contrary that there exist $\varepsilon>0$ and a sequence $(h_n)_n\subseteq G$ such that $h_n\in H_n$ and $D(h_n,1)\geq \varepsilon$. Since $G$ is compact, the sequence without loss of generality converges to some $h\in G$. By the continuity of the metric, we have $D(h,1)\geq \varepsilon$, thus in particular $h\neq 1$. On the other hand, since the sequence $(H_n)_n$ is decreasing and each of the subgroups is closed, $h\in \bigcap_n H_n$. This contradicts that $\bigcap_n H_n=\{1\}$. For each $n$, let $\mu_n$ be the normalized invariant Haar measure on the compact subgroup $H_n$, and let $P'_n$, resp. $P_n$ be the map, resp. projection from Proposition \[prop:projection\] applied to the groups $G$ and $H_n$ with the metric $d$ and the quotient metric $d_n$ on $G/H_n$. [**Claim 2.**]{} For every $x\in {\mathcal{F}}(G)$, we have $x=\lim_{n\to\infty} P_n(x)$. Suppose first that $x\in {\mathcal{F}}(G)$ is a finite linear combination of Dirac elements. That is, there are $m\in {\mathbb{N}}$, $g_1,\dots,g_m \in G$ and $\alpha_1,\dots,\alpha_m\in{\mathbb{R}}$ with $x=\sum_{i=1}^m \alpha_i \delta(g_i)$. For a fixed $n$, let us compute $\|x-P_n(x)\|$. We have $$\begin{aligned} \|x-P_n(x)\|& = \|x-\int_{H_n} h\cdot x - (\sum_{i=1}^m \alpha_i)\delta(h)\,d\mu_n(h)\|\\ &\leq \int_{H_n}\|x- h\cdot x \|\,d\mu_n(h) + (\sum_{i=1}^m |\alpha_i|)\|\int_{H_n}\delta(h)\,d\mu_n(h)\|.\end{aligned}$$ Suppose that $\rm{diam}_D(H_n)= \varepsilon_n$. Then for every $h\in H_n$ we have $$\begin{aligned} \|x-h\cdot x\|& =\|\sum_{i=1}^m \alpha_i \delta(g_i)-\sum_{i=1}^m \alpha_i \delta(h\cdot g_i)\|\leq \sum_{i=1}^m |\alpha_i| d(h\cdot g_i,g_i)\\ & \leq \sum_{i=1}^m |\alpha_i| D(h\cdot g_i,g_i)\leq \sum_{i=1}^m |\alpha_i| D(h,1)\leq \sum_{i=1}^m |\alpha_i| \varepsilon_n,\end{aligned}$$ and it follows that $\int_{H_n}\|x-h\cdot x\|d\mu_n(h)\leq \sum_{i=1}^m |\alpha_i| \varepsilon_n$. On the other hand, for each $f\in B_{\rm{Lip}_0(G)}$, $$|\langle \int_{H_n}\delta(h)\,d\mu_n(h),f \rangle| = |\int_{H_n}f(h)\,d\mu_n(h)|\leq \int_{H_n}\rm{diam}_d(H_n)\,d\mu_n(h)\leq \int_{H_n} \rm{diam}_D(H_n)\leq \varepsilon_n,$$ so $\|\int_{H_n}\delta(h)\,d\mu_n(h)\|\leq \varepsilon_n$. It follows that $$\|x-P_n(x)\|\leq 2\sum_{i=1}^m |\alpha_i|\varepsilon_n,$$ so by **Claim 1**, $\lim_{n\rightarrow\infty} P_n(x)=x$.\ Now let $x\in {\mathcal{F}}(G)$ be an arbitrary element and note that, for each finitely supported $y$, $$\begin{aligned} \|x-P_n(x)\|&\leq \|x-y\|+\|y-P_n(y)\| + \|P_n(y) - P_n(x)\|\\ & = \|x-y\| +\|y-P_n(y)\| + \|P_n\|\|y - x\|=2\|x-y\| +\|y-P_n(y)\|.\end{aligned}$$ Hence, $$\limsup_{n\rightarrow\infty}\|x-P_n(x)\|\leq 2\|x-y\| +\lim_{n\rightarrow\infty}\|y-P_n(y)\| = 2\|x-y\|.$$ Since $y$ can be chosen arbitrarily close to $x$, the result follows. We are ready to show that ${\mathcal{F}}(G)$ has MAP. Pick finitely many $x_1,\ldots,x_m\in {\mathcal{F}}(G)$ and some $\varepsilon$. By the previous claim we can find $n$ so that for all $i\leq m$ we have $\|x_i-P_n(x_i)\|<\varepsilon/2$. By Theorem \[proptorus\], $F(G_n)$ has the MAP. Thus there exists a norm one finite rank operator $T':F(G_n)\rightarrow F(G_n)$ such that for all $i\leq m$ we have $\|P_n(x_i)-T'\circ P_n(x_i)\|<\varepsilon/2$. Set $T:=T'\circ P_n$. It is a norm one finite rank operator from $F(G)$ into $F(G_n)\subseteq F(G)$ such that for all $i\leq n$ we have $$\|T(x_i)-x_i\|\leq \|x_i-P_n(x_i)\|+\|P_n(x_i)-T(x_i)\|<\varepsilon/2+\varepsilon/2=\varepsilon.$$ This finishes the proof. Finally, we show the metric approximation property also for Lipschitz-free spaces over homogeneous spaces of compact metrizable groups. We recall that a *homogeneous space for a group* $G$ is a topological space on which $G$ acts transitively. It can be identified with the left coset space $G/H$, where $H$ is a subgroup, with the quotient topology. To ensure some regularity, we must restrict to compatible metrics which are $G$-invariant, i.e. those so that the action of $G$ is by isometries. \[thm:homogeneousspace\] Let $M$ be a $G$-homogeneous space, where $G$ is a compact group. Suppose that $D$ is a $G$-invariant metric on $M$ that is a quotient of some bi-invariant metric $d$ on $G$. Then ${\mathcal{F}}(M,D)$ has the MAP. We have that $M$ is isomorphic to $G/H$, where is a stabilizer of some point $0\in M$. $H$ is then a closed subgroup. By Theorem \[thm:MAPcompactgrp\], ${\mathcal{F}}(G,d)$ has the MAP. By Proposition \[prop:projection\], there exists a norm one projection $P:{\mathcal{F}}(G)\rightarrow X$, where $X$ is linearly isometric to ${\mathcal{F}}(G/H,D)$. Since MAP is inherited by $1$-complemented subspaces, we are done. Lipschitz-free spaces over finitely generated groups {#section:fingengrps} ==================================================== The goal of this section is to prove Theorem \[thm:intro2\], show examples of situations to which the theorem applies, and provide applications. We mention that the class of Lipschitz-free spaces for which it is known they have the Schauder basis is still rather limited. It is proved in [@HaPe] that ${\mathcal{F}}({\mathbb{R}}^n)$ and ${\mathcal{F}}(\ell_1)$ have a Schauder basis, in [@CuDo] that free spaces over any separable ultrametric space have a monotone Schauder basis, and in [@hajek2017some] that ${\mathcal{F}}(N)$ has a Schauder basis if $N$ is a net in a separable $C(K)$-space. Obviously, it also follows that free spaces isomorphic to these Banach spaces have basis as well, so e.g. by [@kaufmann2015products], ${\mathcal{F}}(B_{\ell_1})$ and ${\mathcal{F}}(B_{{\mathbb{R}}^n})$ have bases. We start by recalling the fundamental idea of geometric group theory - how to view finitely generated groups as metric spaces. Our standard reference for geometric group theory is [@DruKap]. In contrast to the case of compact metrizable groups, we will not consider arbitrary compatible left-invariant metrics on such groups, just certain canonical and maximal, in a sense, ones, called word metrics. Let $G$ be a finitely generated group. Let $S\subseteq G$ be a finite symmetric generating subset (recall that ‘symmetric’ means that for each $s\in S$, also $s^{-1}\in S$). Recall that we can then define a (left-invariant) metric $d_S$, called *word metric*, on $G$ by defining, for $g\neq h\in G$, $$d_S(g,h):=\min\{n\in{\mathbb{N}}\colon \exists s_1,\ldots,s_n\in S\; (g=hs_1\ldots s_n)\}.$$ It is well known and easy to check that by chosing another finite symmetric generating set $T\subseteq G$, the identity map between $(G,d_S)$ and $(G,d_T)$ is bilipschitz. In particular, the isomorphism class of the Banach space ${\mathcal{F}}(G)$ is well-defined. Since every finitely generated group with its word metric is a countable proper metric space, it immediately follows from [@dalet2015free] that ${\mathcal{F}}(G)$ has the MAP. Therefore we will aim for stronger properties and indeed we shall present a class of finitely generated group such that free spaces over any of them has the Schauder basis. Fix now some finitely generated group $G$ and a finite symmetric generating set $S$. Next choose arbitrarily a linear order on the set $S$. Consider now $S$ as an alphabet, i.e. its elements are considered to be letters, and denote by $W$ the set of all *reduced words* over the alphabet $S$. That is, each element $w$ of $W$ is a string of symbols $s_1 s_2\ldots s_n$ from $S$ such that for no $i<n$, the letters $s_i$ and $s_{i+1}$ are inverses of each other when viewed as group elements. For each $w\in W$, - by $|w|$ we denote the length of the word, i.e. the number of its letters; - by $w_G$ we denote the corresponding group element of $G$, i.e. we evaluate the letters of $w$ as elements of $G$; - for every $i<|w|$, by $w(i)$, we denote the $i$-th letter of $w$, and by $w(\leq i)$, we denote the word obtained from $w$ by taking the first $i$ letters. The fixed linear order on the set $S$ defines a lexicographical order on $W$ which we shall denote by $\preceq'$. We define another linear order $\preceq$ on $W$, called the *shortlex* order, by setting, for $w,v\in W$, $$w\preceq v \text{ if either }|w|<|v|, \text{ or }|w|=|v|\text{ and }w\preceq' v.$$ For an element $g\in G$, by $W_g$ we denote the set $\{w\in W\colon w_G=g\}$ and by $w_g$ the minimal element of the set $W_g$, which is easily verified to exist, in $\preceq$. If there is no danger of confusion, for an element $g\in G$ we shall denote by $|g|$ the number $|w_g|$ which is equal to $d_S(g,1_G)$. Finally, we use the linear order $\preceq$ on $W$ to define a linear order $\leq$ on $G$. For $g,h\in G$ we set $$g\leq h\text{ if }w_g\preceq w_h.$$ We call $G$ *shortlex combable*, with respect to a fixed symmetric generating set $S$ and a linear order on $S$, if there exists a constant $K\geq 1$ such that for every $g,h\in G$ with $d_S(g,h)=1$ and for every $i\leq \min\{d_S(g,1_G),d_S(h,1_G)\}$ we have $$d_S((w_g(\leq i))_G, (w_h(\leq i))_G)\leq K.$$ The constant $K$ will be called the *combability constant* of $G$. First we show how such groups are useful for our purposes. Then we provide examples and show some applications. The following is the main result of this section. \[thm:shortlex\] Let $G$ be a finitely generated shortlex combable group (with respect to $S$ equipped with some linear order). Then ${\mathcal{F}}(G,d_S)$ has a Schauder basis. Since the linear order $\leq$ on $G$ is clearly isomorphic to the standard order on ${\mathbb{N}}$, we can use it to enumerate $G$ as $(g_n)_{n\in{\mathbb{N}}}$. For each $n\in{\mathbb{N}}$, set $G_n:=\{g_i\colon i\leq n\}$. For each $n$ we now define maps $P_n: G\rightarrow G_n$ as follows. First, set $m=\max\{|h|\colon h\in G_n\}$, then for $g\in G$, set $$P_n(g):=\begin{cases} g & \text{if }g\in G_n\\ (w_g(\leq m))_G & \text{if }g\notin G_n, (w_g(\leq m))_G\in G_n\\ (w_g(\leq m-1))_G & \text{otherwise}. \end{cases}$$ We leave to the reader the straightforward verification that $P_n$ is well defined.\ [**Claim.**]{} The maps $(P_n)_{n\in{\mathbb{N}}}$ are uniformly bounded Lipschitz commutting retractions.\ First we check that for every $n,m\in{\mathbb{N}}$, $P_n\circ P_m=P_m\circ P_n$, i.e. the maps commute. For every $g\in G$ and $0\leq i\leq |g|$, denote by $g_i$ the element $(w_g(\leq i))_G$. That is, $g_0=1_G$, $g_{|g|}=g$, and $g_0,g_1,\ldots,g_{|g|}$ is a geodesic path in $G$ from the identity element $1_G$ to $g$. It is easy to see that for each $n\in{\mathbb{N}}$, $P_n(g)=g_i$, where $i$ is the largest integer so that $g_i\in G_n$. With this observation it is now clear that the maps $(P_n)_n$ commute. Let $K$ be the combability constant of $G$. We show that each $P_n$ is a $K+1$-Lipschitz retraction on its image. It is obviously a retraction, so it suffices to show that for every $n\in{\mathbb{N}}$ and $g,h\in G$ with $d_S(g,h)=1$ we have $d_S(P_n(g),P_n(h))\leq K+1$. Let such $n$ and $g,h\in G$ be fixed. We distinguish two cases.\ [*Case 1.*]{} At least one of $g,h$ lies in $G_n$: Say that $g\in G_n$. If also $h\in G_n$, then there is nothing to prove since $P_n(g)=g$ and $P_n(h)=h$. So suppose that $h\notin G_n$. Necessarily we have $h_{|h|-1}\leq g$ since otherwise the path $1_G, g_1,\ldots, g, h$ would be a geodesic path from $1_G$ to $h$ of length $|h|$ smaller in the lexicographical ordering than the path $1_G, h_1,\ldots,h_{|h|-1},h$. It follows that $h_{|h|-1}\in G_n$, so $P_n(h)=h_{|h|-1}$. Since $P_n(g)=g$ we get $$d_S(P_n(g),P_n(h))\leq 2.$$\ [*Case 2.*]{} We have $g,h\notin G_n$. Note that then $P_n(g)=g_i$ and $P_n(h)=h_j$, for some $i,j< \max\{|g|,|h|\}$, where $|i-j|\leq 1$. Since by the definition of shortlex combability, $d_S(g_i,h_i)\leq K$ and $d_S(g_j,h_j)\leq K$, we get that $$d_S(P_n(g),P_n(h))=d_S(g_i,h_j)\leq K+1.$$ This finishes the proof of the claim.\ Finally, for each $n\in{\mathbb{N}}$ we denote by $L_n: {\mathcal{F}}(G)\rightarrow {\mathcal{F}}(G_n)$ the lift of $P_n$, the unique linear operator extending $P_n$. The properties of $(P_n)_n$ imply that - for each $n\in{\mathbb{N}}$, the map $L_n$ is a linear projection onto a finite-dimensional subspace of norm bounded by $K+1$; - the projections $(L_n)_n$ commute; - the dimension of the range $L_n[{\mathcal{F}}(G)]$ is the dimension of ${\mathcal{F}}(G_n)$, which is equal to $n$. Since now obviously for every $x\in {\mathcal{F}}(G)$ (notice that it suffices to verify it for the dense set $\mathrm{span}\{\delta_g\colon g\in G\}$) we have $$\lim_{n\to\infty} L_n(x)=x,$$ then by [@AlbiacKalton Proposition 1.1.7] ${\mathcal{F}}(G)$ possesses a Schauder basis. Examples {#subsec:examples} -------- [*Finitely generated abelian groups*]{}. Recall that every infinite finitely generated abelian group $A$ is of the form ${\mathbb{Z}}^n\oplus F$, where $n\geq 1$ and $F=\{0,f_1,\ldots,f_m\}$ is a finite abelian group. Let $e_1,\ldots,e_n$ be the canonical generators of ${\mathbb{Z}}^n$. Then $e_1\leq -e_1\leq e_2\leq -e_2\leq\ldots\leq e_n\leq -e_n\leq f_1\leq \ldots\leq f_m$ is a linearly ordered finite symmetric generating set. We leave to the reader to verify that with this ordered generating set $A$ is shortlex combable.\ [*Free groups*]{}. Let $n\geq 1$ and let $F_n$ be a free group on $n$ generators (which is ${\mathbb{Z}}$ for $n=1$). Let $a_1,\ldots,a_n$ be the free generators. It is again an exercise that with the order $a_1\leq a^{-1}_1\leq\ldots\leq a_n\leq a_n^{-1}$, the group $F_n$ is shortlex combable. We note however, that ${\mathcal{F}}(F_n)$ is linearly isometric to $\ell_1$, thus admits monotone Schauder basis. Indeed, this can be verified directly by noticing that the set $\{\delta(g)-\delta(h)\in{\mathcal{F}}(F_n)\colon d(g,h)=1, d(g,1)>d(h,1)\}$ is equivalent to the standard basis of $\ell_1$.\ [*Hyperbolic groups*]{}. Recall that a geodesic metric space $(M,d)$ is (Rips)-hyperbolic if there exists a constant $K\geq 0$, *hyperbolicity constant*, such that for any triple of points $x,y,z\in M$ and geodesic segments $S_1,S_2,S_3$ connecting each pair of the triple we have $d_{H}(S_i,S_j\cup S_k)\leq K$, where $d_H$ is the Hausdorff distance and $i,j,k$ are pairwise different from $\{1,2,3\}$. In other words, for each $i\leq 3$ and each point $x\in S_i$, $d(x,S_j\cup S_k)\leq K$, where $S_j$ and $S_k$ are the other geodesics besides $S_i$. The notion of hyperbolicity makes sense even for metric spaces which are not literally geodesic, but when a reasonable notion of geodesic segment can be defined. This is the case e.g. for finitely generated groups with word metrics, where a geodesic segment between elements $x,y\in G$ is a sequence $g_0=x,\ldots,g_n=y$, where $n=d(x,y)$, and $d(g_i,g_{i+1})=1$, for $i<n$. Let $G$ be a finitely generated hyperbolic group (with hyperbolicity constant $K$), generated by elements $a_1,\ldots,a_n$. We claim that $G$ with the ordered generating set $a_1\leq a^{-1}_1\leq\ldots\leq a_n\leq a_n^{-1}$ is shortlex combable. Indeed, pick $g,h\in G$ with $d(g,h)=1$, and $i<\max\{|g|,|h|\}$. We show that $d(g_i,h_i)\leq 2K+2$. We have two cases. 1. Either $g_i=g$ or $h_i=h$ (or both). Then it is clear that $d(g_i,h_i)\leq 2$. 2. We have $g_i\neq g$ and $h_i\neq h$. There are geodesic segments $g_0=1_G,\ldots,g_i,\ldots,g$, $h_0=1_G,\ldots,h_i,\ldots,h$, and $g,h$ (of length $1$). We consider thr triple of points $1_G,x,y$ and the geodesic segments between them as above. By definition of hyperbolicity with constant $K$, there exists point $z\in\{1_G=h_0,\ldots, h,g\}$ such that $d(g_i,z)\leq K$. Assume first that $z=h_j$, for some $j\leq |h|$. - $j\geq i$: Since $d(g_i,h_j)\leq K$, by triangle inequality we get that $j\leq i+K$, therefore $d(g_i,h_i)\leq 2K$. - $j<i$: Again by triangle inequality we get that $j\geq i-K$, so $d(g_i,h_i)\leq 2K$. Finally, if $z=g$, then $d(g_i,h)\leq K+1$, so again by triangle inequality we get $|h|\leq i+K+1$, so $d(g_i,h_i)\leq 2(K+1)$.\ [*Large-type Artin groups.*]{} Holt and Rees in [@HoRe] prove that large-type Artin groups are shortlex automatic which immediately from the definition implies being shortlex combable. Artin groups in general belong to one of the most studied classes of groups in geometric group theory. We refer the reader to [@HoRe] for the notion of shortlex automaticity and for the definition of large-type Artin groups. We do not know whether there are groups that are shortlex combable but not shortlex automatic. We refer to [@HoRe] for details. Applications ------------ Our main goal is to prove that for any net $N$ in a real hyperbolic $n$-space ${\mathbb{H}}^n$ (whose definition we recall later), we have that ${\mathcal{F}}(N)$ has the Schauder basis. This will be an immediate consequence of Theorem \[thm:shortlex\] and several standard more general results that we present below. We recall that an action of a group $G$ on a metric space $X$ by isometries is *free* if for every $g\in G$ and $x\in X$, $g\cdot x\neq x$, and *cocompact* if there exists a compact set $K\subseteq X$ such that $\bigcup_{g\in G} g\cdot K=X$. These actions, or rather more generally properly discontinuous cocompact actions (see [@DruKap Chapter 5]), are one of the most studied in geometric group theory. \[prop:actinggroup\] Let $G$ be a finitely generated group acting freely and cocompactly on a proper geodesic metric space $X$ by isometries. Then for every net $N\subseteq X$, we have ${\mathcal{F}}(N)\simeq {\mathcal{F}}(G)$. Let $G$, $X$ and an action of $G$ on $X$ as in the statement of the proposition be fixed. First we invoke [@hajek2017some Proposition 5] which says that for any two nets $N_1,N_2\subseteq X$ we have ${\mathcal{F}}(N_1)\simeq{\mathcal{F}}(N_2)$. It follows that it suffices to find one net $N\subseteq X$ such that ${\mathcal{F}}(N)\simeq{\mathcal{F}}(G)$. Let $0\in X$ be a distinguished point and let $N$ be the $G$-orbit of $0$. Since $X$ is proper and the action is cocompact, it follows that $N$ is a net in $X$. To show that ${\mathcal{F}}(N)\simeq{\mathcal{F}}(G)$ it suffices to prove that $N$ with the restriction of the metric on $X$ is bi-Lipschitz to $G$ (with its word metric). By the Milnor-Schwarz lemma (see [@DruKap Theorem 8.37]), the map $\phi:G\rightarrow N$ defined by $\phi(g):=g\cdot 0$ is a quasi-isometry (we refer reader not familiar with quasi-isometries again to [@DruKap]). Since the action is free, it is also a bijection. It is easy to check that a bijective quasi-isometry between two uniformly discrete metric spaces is in fact a bi-Lipschitz equivalence. This finishes the proof. We now recall the definition of the real hyperbolic $n$-space. There are many definitions and we refer the reader to [@DruKap Chapter 4] for a more thorough discussion. We define ${\mathbb{H}}^n$, for $n\geq 2$, as follows. First we define the following quadratic form; for $x,y\in{\mathbb{R}}^{n+1}$: $$\langle x,y\rangle:=\sum_{i=1}^n x_iy_i-x_{n+1}y_{n+1},$$ and we set $${\mathbb{H}}^n:=\{x\in{\mathbb{R}}^{n+1}\colon \langle x,x\rangle=-1,x_{n+1}>0\}.$$ A metric $d$ on ${\mathbb{H}}^n$ can be defined using the formula, for $x,y\in{\mathbb{H}}^n$, $$\cosh d(x,y)=-\langle x,y\rangle.$$ We now state the main result of this subsection. \[cor:hyperbolicnet\] Let $n\geq 2$ and let $N\subseteq {\mathbb{H}}^n$ be a net. Then ${\mathcal{F}}(N)$ has a Schauder basis. It suffices to find a group $G$ acting freely and cocompactly on ${\mathbb{H}}^n$. Indeed, again by the Milnor-Schwarz lemma ([@DruKap Theorem 8.37]), such $G$ is then finitely generated and quasi-isometric to ${\mathbb{H}}^n$. It follows that $G$ is hyperbolic ([@DruKap Observation 11.125]). Therefore, as we demonstrated in Subsection \[subsec:examples\], $G$ is shortlex combable. Applying Theorem \[thm:shortlex\], we get that ${\mathcal{F}}(G)$ has a Schauder basis. Finally, an application of Proposition \[prop:actinggroup\] finishes the argument. In order to find a group acting freely and cocompactly on ${\mathbb{H}}^n$, one can use several standard results from Riemannian geometry. Let $M$ be an arbitrary $n$-dimensional compact Riemannian manifold without boundary of constant sectional curvature $-1$ equipped with some Riemannian metric. By [@BrHa Theorem 3.32], its universal cover (refer to any standard textbook on algebraic topology, e.g. [@Hatch]) is isometric to ${\mathbb{H}}^n$. Another standard argument from algebraic topology (see e.g. [@Hatch Proposition 1.39]) shows that the fundamental group $\pi_1(M)$ acts on the universal cover ${\mathbb{H}}^n$ by deck transformations, which is a free cocompact action by isometries. This finishes the proof. We remark that in contrast to the Euclidean space ${\mathbb{R}}^n$, in which there exist two non Lipschitz equivalent nets ([@BuKl]), it has been proved in [@Bog] that all nets in ${\mathbb{H}}^n$ are bi-Lipschitz equivalent. Let $\mathcal{G}$ denote the set of all finitely generated hyperbolic groups. A fair question is how many isomorphism types of Banach spaces the set $\{{\mathcal{F}}(G)\colon G\in\mathcal{G}\}$ contains. As we mentioned, since the free group $F_n$ is hyperbolic, it contains ${\mathcal{F}}(F_n)\simeq \ell_1$. It is unknown to us whether or not there exists $G\in\mathcal{G}$ such that ${\mathcal{F}}(G)\not\simeq \ell_1$. Thus it is relevant at this point to reiterate [@candido2019isomorphisms Question 1], which asks precisely about an example of $G\in\mathcal{G}$ with ${\mathcal{F}}(G)\not\simeq\ell_1$, and the discussion following the question. Either answer would bring interesting consequences. If there are such $G$, then we have, potentially many, new examples of Lipschitz-free spaces with Schauder basis. If on the other hand for every $G\in\mathcal{G}$, ${\mathcal{F}}(G)\simeq \ell_1$, then we have an example of a group with Kazhdan’s property (T) that has a metrically proper action by isometries on a renorming of $\ell_1$. We refer to [@candido2019isomorphisms Question 1] where this is properly discussed and the importance of such a result is explained. Problems and Notes {#section:problems} ================== Let us finish by posing some natural questions that follow up this work. The first is whether or not we can generalize Theorem \[thm:intro1\] to locally compact groups: Let $G$ be a locally compact group equipped with a compatible and left-invariant metric. Does ${\mathcal{F}}(G)$ have the MAP? In the noncompact case, we have positive answer for finite dimensional Banach spaces with their norm induced metrics. Although this would be a consequence of the mentioned result from [@godefroy2003lipschitz] which states that a Banach space has $\lambda$-BAP if and only if ${\mathcal{F}}(X)$ does, actually the order of the proofs is reversed. First, Godefroy and Kalton prove that, for finite dimensional Banach spaces $X$, ${\mathcal{F}}(X)$ has the MAP, and then use this to prove the result for general Banach spaces. The proof of the finite dimensional part also involves harmonic averaging. Since Lipschitz-free spaces over finite dimensional Banach spaces even possess a Schauder basis ([@HaPe]), we suggest to turn the attention to locally compact metric groups that are very closely related to such Banach spaces; that is, Carnot groups. We note that a Carnot group is both analytically and algebraically a mild generalization of a finite dimensional Banach space and many results from geometric analysis on Euclidean spaces have been generalized to the setting of Carnot groups (see [@LeD]). The duals of Lipschitz-free spaces over Carnot groups have been already investigated in [@candido2019isomorphisms] and their Lipschitz-free spaces in [@albiac2020lipschitz]. Let $G$ be a Carnot group with a Carnot-Carath' eodory metric. Does ${\mathcal{F}}(G)$ has the Schauder basis? In the case of connected Lie groups, one has a canonical compatible left-invariant metric: the left-invariant Riemannian metric. Since such a metric is locally bi-Lipschitz equivalent to the Euclidean metric, and isomorphic properties of Lipschitz-free spaces often depend only on the local behaviour of the metric, the following question is very natural. We note that one could even replace Lie group there with a general connected Riemannian manifold (without boundary). \[quest:liegrp\] Let $G$ be a connected (real) Lie group equipped with a left-invariant Riemannian metric. Do we have ${\mathcal{F}}(G)\simeq {\mathcal{F}}({\mathbb{R}}^n)$, where $n$ is the dimension of the (real) Lie algebra $\mathfrak{g}$ of $G$? Still in the compact setting, we note that we only required in Theorem \[thm:homogeneousspace\] that the metric in $G$-homogeneous space is a quotient of a metric in $G$ so that we could apply directly Proposition \[prop:projection\]. So one could ask if we can drop this condition. Let $G$ be a (locally) compact group, $M$ be a compact $G$-homogeneous space, and $d$ be a compatible $G$-invariant metric on $M$. Does ${\mathcal{F}}(G)$ have the MAP? In Appendix A, the reader will find a positive answer in the specific case of the sphere ${\mathbb{S}}^{n-1}=O(n)/O(n-1)$. It is natural also to ask about generalizations of Theorem \[thm:intro2\]. Let $G$ be a finitely generated group equipped with a word metric. Does ${\mathcal{F}}(G)$ admit a Schauder basis, or at least a finite dimensional decomposition? In Corollary \[cor:hyperbolicnet\], we have shown that for any net $N\subseteq {\mathbb{H}}^n$, ${\mathcal{F}}(N)$ has a Schauder basis. Given the prominence of the space ${\mathbb{H}}^n$ in geometry and beyond, it is important to understand the Lipschitz-free space of ${\mathbb{H}}^n$ itself. We also note that the hyperbolic spaces ${\mathbb{H}}^n$ together with the Euclidean spaces ${\mathbb{R}}^n$ and Euclidean spheres ${\mathbb{S}}^n$ are important as the model spaces of spaces of constant curvature (see e.g. [@BrHa] for a thorough treatment). Since for ${\mathbb{R}}^n$ and ${\mathbb{S}}^n$ with their canonical Euclidean metrics we have by [@albiac2020lipschitz Theorem 4.21], ${\mathcal{F}}({\mathbb{R}}^n)\simeq{\mathcal{F}}({\mathbb{S}}^n)$, the answer to the following would be desirable. Does ${\mathcal{F}}({\mathbb{H}}^n)$ have the Schauder basis? Do we have ${\mathcal{F}}({\mathbb{H}}^n)\simeq {\mathcal{F}}({\mathbb{R}}^n)$? We note that the previous question is related to the stronger version of Question \[quest:liegrp\] that considers a general Riemannian manifold since ${\mathbb{H}}^n$ is a Riemannian manifold with Riemannian metric (see [@BrHa Proposition 6.17]). On the MAP for ${\mathcal{F}}({\mathbb{S}}^n)$ {#appendSphere} ============================================== Let $d\geq 2$ and let ${\mathbb{S}}^{d-1}=O(d)/O(d-1)$ be the $(d-1)$-dimensional sphere equipped with a rotation-invariant metric $D$ which is compatible with the usual topology. Here, we are not assuming that $D$ is a quotient metric, as the ones described in Subsection \[subsection:generalCpt\]. Let us show that ${\mathcal{F}}({\mathbb{S}}^{d-1},D)$ has the MAP. The proof will also follow from Proposition \[tool\] and summability results. We recall the definitions and results from harmonic analysis that we will use, and establish again a Lipschitz version of Young’s convolution inequality for our setting. Equip ${\mathbb{S}}^{d-1}$ with its surface area measure $\sigma$, and denote $\omega_d=\sigma({\mathbb{S}}^{d-1})$. Convolution on ${\mathbb{S}}^{d-1}$ can be defined as follows. First let $\Lambda = (d-2)/2$, and consider the weighted $L^1$ space $L^1(w_\Lambda,[-1,1])$, where $w_\Lambda (x) = (1-x^2)^{\Lambda - 1/2}$ for each $x\in ]-1,1[$. Recall that the norm in $L^1(w_\Lambda,[-1,1])$ is defined by $$\|f\|_{\Lambda,1} = c_\Lambda \int_{-1}^1 |f(x)|w_\Lambda (x)dx,$$ where $c_\Lambda$ is the normalization constant such that $c_\Lambda \int_{-1}^1w_\Lambda (x)dx =1$. For each $f\in L^1({\mathbb{S}}^{d-1})$ and $g\in L^1(w_\Lambda,[-1,1])$, the convolution $f\ast g: {\mathbb{S}}^{d-1}\rightarrow {\mathbb{R}}$ is defined by $$(f\ast g)(x)= \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} f(y)g(x\cdot y)d\sigma(y).$$ We now prove the validity of Young’s inequality. Let $f\in \mathrm{Lip}({\mathbb{S}}^{d-1})$ and $g \in L^1(w_\Lambda,[-1,1])$, and let $x,y\in {\mathbb{S}}^{d-1}$. There is a rotation $R$ with $y=Rx$, thus $$\begin{aligned} |f*g(x) - f*g(y)| & = \left| \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \frac{1}{\omega_n}\int_{{\mathbb{S}}^{d-1}} f(z)g(Rx\cdot z)d\sigma(z) \right|\\ & = \frac{1}{\omega_d}\left| \int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \int_{{\mathbb{S}}^{d-1}} f(Rz)g(Rx\cdot Rz)d\sigma(z) \right|\\ & = \frac{1}{\omega_d}\left| \int_{{\mathbb{S}}^{d-1}} f(z)g(x\cdot z)d\sigma(z) - \int_{{\mathbb{S}}^{d-1}} f(Rz)g(x\cdot z)d\sigma(z) \right|\\ & \leq \frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} |f(z)-f(Rz)||g(x\cdot z)|d\sigma(z) \\ & \leq \|f\|_{\mathrm{Lip}}\frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} D(z,Rz)|g(x\cdot z)|d\sigma(z)\\ & = \|f\|_{\mathrm{Lip}}D(x,y)\frac{1}{\omega_d}\int_{{\mathbb{S}}^{d-1}} |g(x\cdot z)|d\sigma(z)\\ & = \|f\|_{\mathrm{Lip}}D(x,y)\frac{1}{\omega_n}\int_{-1}^1 |g(t)|w_\Lambda (t) dt \leq \|f\|_{\mathrm{Lip}}\|g\|_{\Lambda,1}D(x,y).\end{aligned}$$ It follows that $f\ast g\in\mathrm{Lip}({\mathbb{S}}^{d-1})$, and $\|f\ast g\|_{\mathrm{Lip}}\leq \|f\|_{\mathrm{Lip}}\|g\|_{\Lambda,1}.$ The finite dimensional space of real homogeneous harmonic polynomials of degree $n$ on ${\mathbb{R}}^d$ restricted to ${\mathbb{S}}^{d-1}$ is denoted by $\mathcal{H}^d_n$. These spaces are mutually orthogonal with respect to the inner product $$\langle f,g\rangle_{{\mathbb{S}}^{d-1}} = \frac{1}{\omega_n}\int_{{\mathbb{S}}^{d-1}} f(x)g(x)d\sigma(x)$$ and they densely span $L^2({\mathbb{S}}^{d-1})$ (see e.g. [@dai2013approximation], Theorem 2.2.2). Denoting by proj$_n$ the corresponding projection operators, we can associate the partial sum operators $$S_n f=\sum_{k=0}^n \mbox{proj}_nf.$$ These are finite-rank and satisfy $S_nf = f\ast K_n$, where $$K_n(t) = \sum_{k=0}^n \frac{k+\Lambda}{\Lambda} C_k^\Lambda(t)$$ and $C_k^\Lambda$ are the Gegenbauer polynomials ([@dai2013approximation], Proposition 2.2.1). Fix $\delta\geq d-1$ and consider the averages $$K^\delta_n(t) = \frac{1}{A_n^\delta}\sum_{k=0}^n A_{n-k}^\delta \frac{k+\Lambda}{\Lambda} C_k^\Lambda(t),$$ where $A_k^\delta = \binom{k+\delta}{k}=\frac{(\delta+k)(\delta+k-1)\dots(\delta+1)}{k!}$. These give rise to a sequence of finite-rank operators on $L^2({\mathbb{S}}^{d-1})$ defined by $S_n^\delta: f\mapsto f\ast K_n^\delta$. Write $\Lambda_n^\delta:=\|S_n^\delta\|_1= \sup \{\|S_n^\delta h\|_1: h\in B_{L^1({\mathbb{S}}^{d-1})}\}$. By [@dai2013approximation Theorem 2.4.3], for each $n\in {\mathbb{N}}$, $S_n^\delta$ is a nonnegative operator, which implies that $K_n^\delta(t)\geq 0$, $t\in [-1,1]$. It follows that $$\Lambda_n^\delta=\sup_{\|f\|_1 \leq 1} \|f\ast K_n^\delta\|_1 \geq \|1\ast K_n^\delta\|_1 = \|K_n^\delta\|_{\Lambda,1}.$$ It is clear that also $S_n^\delta(C({\mathbb{S}}^{d-1}))\subset C({\mathbb{S}}^{d-1})$ and that its restriction to $C({\mathbb{S}}^{d-1})$ is continuous in the uniform norm. Moreover, by [@dai2013approximation Corollary 2.4.5], when $f$ is continuous (and in particular when it is Lipschitz) on ${\mathbb{S}}^{d-1}$, we have that $S_n^\delta f$ converges to $f$ uniformly. On the other hand, by Young’s inequality, for each $f\in \mathrm{Lip}({\mathbb{S}}^{d-1})$ we have that $$\|S_n^\delta f\|_{\mathrm{Lip}} = \|f\ast K_n^\delta \|_{\mathrm{Lip}} \leq \|f\|_{\mathrm{Lip}} \| K_n^\delta \|_{\Lambda,1}\leq \Lambda_n^\delta \|f\|_{\mathrm{Lip}}.$$ To conclude, let $\varepsilon>0$. By [@dai2013approximation Theorem2.4.4], there is some $n_\varepsilon$ such that $\Lambda_n^\delta<1+\varepsilon$, for each $n\geq n_\varepsilon$. Thus $S_n^\delta$, $n\geq n_\varepsilon$ satisfy conditions (a)–(d) in Proposition \[tool\] and we are done. [**Acknowledgement.**]{} We would like to thank to Gilles Godefroy who suggested to us to study these problems. [^1]: P. L. Kaufmann was supported by grants 2017/18623-5 and 2016/25574-8, São Paulo Research Foundation (FAPESP). M. Doucha was supported by the GAČR project 19-05271Y and RVO: 67985840.

--- abstract: 'We consider a radiation from a uniformly accelerating harmonic oscillator whose minimal coupling to the scalar field changes suddenly. The exact time evolutions of the quantum operators are given in terms of a classical solution of a forced harmonic oscillator. After the jumping of the coupling constant there occurs a fast absorption of energy into the oscillator, and then a slow emission follows. Here the absorbed energy is independent of the acceleration and proportional to the log of a high momentum cutoff of the field. The emitted energy depends on the acceleration and also proportional to the log of the cutoff. Especially, if the coupling is comparable to the natural frequency of the detector ($e^2/(4m) \sim \omega_0$) enormous energies are radiated away from the oscillator.' address: - ' Dept. of Physics, Sungkyunkwan Univ., SUWON 440-746, KOREA ' - ' Dept. of Physics, KAIST , Taejon 305-701, KOREA ' author: - 'Hyeong-Chan Kim[^1]' - 'Jae Kwan Kim[^2]' date: 'April 8, 1997' title: Radiation from a uniformly accelerating harmonic oscillator --- [      pacs number 04.60.+n, 03.70.+k ]{} Introduction ============ It is well known that the concept of a particle depends on the motion of an observer [@birrell]. Especially, the Minkowski vacuum is a canonical ensemble with the temperature $a/2\pi$ from the point of view of a uniformly accelerated observer with the acceleration $a$ (the thermalization theorem) [@unruh]. This observer dependence is most easily shown if one use a particle detector model invented by Unruh [@unruh] and DeWitt [@dewitt79]. It consists of an idealized point particle with internal energy levels labeled by $E$, coupled via a monopole interaction with a scalar field $\phi$ (Unruh-DeWitt model). Following these, many works emerged in the literature. Letaw [@Letaw] exhibited the stationary world lines, on which the detectors in a vacuum have a time-independent excitation spectra. Grove and Ottewill [@Grove:Otte] studied the problem of a non-extended detector, and clarified the radiation effect arising both from the walls of the detector and from the interaction with the field. Several authors [@hinton; @takagi] discussed the anisotropic nature of the thermal radiation of the accelerated detector. A full review for this thermal character was given by Takagi [@takagi86]. The vacuum noise seen by a uniformly accelerated observer in flat space-times of arbitrary dimensions was investigated and was shown to exhibit the phenomenon of the apparent inversion of statistics in odd dimensions, which was discussed precisely by Unruh [@unruh86] and Fukazawa [@fukazawa]. A few years ago, the excitation rate associated with a uniformly accelerated finite-time detector interacting with the Minkowski vacuum has been analyzed in an inertial frame by Svaiter and Svaiter [@svaiter]. They found a logarithmic ultraviolet divergences on the transition amplitude, which was due to the instantaneous switching of the detector [@higuchi]. This UV divergence does not occur in lower dimensions. Grove argue that a macroscopic constantly accelerating object will emit negative energy radiation until equilibrium with the Minkowski vacuum is achieved [@grove]. Several years ago a new particle detector model–a harmonic oscillator coupled to a scalar field in $1+1$ dimensions–was introduced by Raine, Sciama, and Grove(RSG) [@raine]. Several aspects of this model was discussed in connection with the ‘open quantum system’ [@unruhZurek; @unruhWald; @anglin]. Hinterleitner [@hin] and Massar, Parentani, and Brout [@massar] shown that there is a polarization cloud which surrounds the detector at all times and energy is exchanged with it locally. Audretsch and Müller [@aud] explored nonlocal pair correlations in accelerated detector. Recently stochastic aspects of this detector were discussed by Raval, Hu, and Anglin [@raval]. These works mainly interested on the asymptotic states with time independent coupling. In this paper we consider the intermediate region during the equilibrium achieved between the detector and the field. We show that this is not a simple energy absorption process but there are two main stages after the two systems in contact with. First stage is a fast absorption of energy of the oscillator from the field. This occurs shortly after the change of the coupling in a time which is much smaller than the inverse of the characteristic frequency of the oscillator. The total energy absorbed during this period is independent of the acceleration and depends on the log of a high momentum cutoff. Second stage is slow emission of radiation which exponentially decrease in a time scale of the coupling constant. The total radiated energy during this period depends on the acceleration of the oscillator. If the coupling constant is small then the total radiation is smaller than the inertial one. But if the coupling is comparable to the characteristic frequency, enormous energies are radiated away from the oscillator. In the case of a weakly coupled system, the absorbed energy during the first stage is larger than the emitted one during the second stage. In Sec. II-A, we describe the model in Minkowski space and give the general form of the solution for the field and the oscillator. These evolutions of the operators are given by use of the inhomogeneous solution $G(\omega,t)$ of a forced harmonic oscillator. Similarly, all physical quantities like the correlation function or the stress tensor can be expressed with this single function. In Sec. II-B, the model is generalized to incorporate the uniformly accelerating oscillators. Sec. III is devoted to present two solvable models. $G(\omega,t)$ is obtained in the asymptotic region. We obtain the stress tensor in Sec. IV when the detector is turned on suddenly. Sec. V is summary and discussions. There are two appendices where we describe the details of the calculation of the stress tensor. Models for the particle detector ================================ Let us consider a minimally coupled system of a massless real scalar field $\phi(t,x)$ in two dimensions and a detector of a harmonic oscillator $q(t)$ with mass $m$. The action for this system is $$\begin{aligned} \label{ac} S &=& \int \mbox{d}x \mbox{d}t \frac{1}{2}\left\{ \left(\frac{\partial}{\partial t} \phi(t,x) \right)^2 - \left( \frac{\partial}{\partial x} \phi(t,x)\right)^2 \right\} \\ &+& \int d\tau \left\{ \frac{1}{2} m \left(\frac{d q(\tau)}{d \tau} \right)^2 -\frac{1}{2} m \omega_0^2 q^2(\tau) -e(\tau)q(\tau) \frac{ d\phi}{d \tau} \left(t(\tau),x(\tau)\right) \right\} . \nonumber\end{aligned}$$ The oscillator follows the explicitly given path $(t(\tau),x(\tau))$ where $\tau$ is the proper time of the oscillator along the path. In this paper, we select two paths through which the oscillator moves: the inertial and the uniformly accelerated. Varying Eq. (\[ac\]) with respect to $\phi(t,x)$ and $q(\tau)$ we get the Heisenberg equation of motion for the field and the oscillator $$\begin{aligned} \Box \phi(t,x) &=& \frac{de(\tau)q(\tau)}{d \tau} \delta (\rho), \label{2}\\ m \left( \frac{d}{d \tau}\right)^2 q(\tau) &+& m \omega_0^2 q(\tau) = - e(\tau)\frac{d \phi}{d \tau}(t(\tau),x(\tau)), \label{3}\end{aligned}$$ where $\rho$ is an appropriate space coordinate which is orthogonal to $\tau$ and the path of the oscillator can be represented as $\rho=0$. Eq. (\[2\]) can be integrated to give $$\begin{aligned} \label{phi:uv} \phi(t,x) = \phi^0(t,x) + \frac{e(\tau_{ret})}{2} q(\tau_{ret}),\end{aligned}$$ where $\tau_{ret}$ is the value of $\tau$ at the intersection of the past lightcone of $(t,x)$ and the detector trajectory. where we have used the explicit retarded propagator of a massless field $$\begin{aligned} G_{\mbox{ret}}(t,x;0,0) = \frac{1}{2} \theta(t+x) \theta(t-x).\end{aligned}$$ Substituting the solution (\[phi:uv\]) into Eq. (\[3\]), one get $$\begin{aligned} \label{qeq} m \ddot{q}(\tau) + \frac{1}{2} e^2(\tau) \dot{q}(\tau) + m \left( \omega_0^2 + \frac{\dot{e}^2(\tau)}{4 m}\right) q(\tau) = -e(\tau) \dot{\phi}^0(t(\tau),x(\tau)).\end{aligned}$$ The redefinitions $$\begin{aligned} M(\tau) &=& m \exp\left(\int_{\tau_0}^{\tau} \mbox{d}\tau \frac{e^2(\tau)}{2m } \right), \label{M:t} \\ \omega^2(\tau) &=& \omega_0^2 + \frac{\dot{e}^2(\tau)}{4m}, \\ F(\tau) &=& - \frac{e(\tau)}{m} \frac{d\phi^0}{d\tau} (t(\tau),x(\tau)).\end{aligned}$$ make Eq. (\[qeq\]) into the equation of motion of the forced harmonic oscillator with the effective mass $M(t)$, and the frequency $\omega^2(t)$ $$\begin{aligned} \label{q'':F} \ddot{q}(\tau) + \frac{d\ln M(\tau)}{d \tau}\dot{q}(\tau) + \omega^2(\tau) q(\tau) = F(\tau).\end{aligned}$$ Here $F(\tau)$ is the force density per unit effective mass. We take the normalization of the effective mass as $M(\tau_0)= m$ at some initial time $\tau_0$. As one see from Eq. (\[q”:F\]), we can arbitrarily choose the normalization of the effective mass. Note that we can rewrite this equation into quadratic form: $$\begin{aligned} \label{quad} \left[ \left(\frac{d}{d\tau}\right)^2 + \Omega^2(\tau) \right] \sqrt{M(\tau)} q(\tau) = \sqrt{M(\tau)} F(\tau),\end{aligned}$$ where $$\begin{aligned} \label{Omega} \Omega^2(\tau) = \omega_0^2- \left( \frac{e^2(\tau)}{4m}\right)^2.\end{aligned}$$ The behavior of a homogeneous solution of Eq. (\[quad\]) changes from oscillatory to exponential decay according to the value of $\Omega^2(t)$. We restrict our discussion into $\Omega^2(t)$ greater than zero. If $\epsilon(\tau) \ll \omega_0$ then $\Omega$ is natural positive frequency mode of the oscillator. The behavior of the homogeneous solution, in this case, is $$\begin{aligned} f(t) = \frac{1}{\sqrt{M(\tau)}} exp\left[{\pm i \int^\tau \Omega(\tau') d\tau'}\right].\end{aligned}$$ Let the initial Heisenberg operators for the oscillator to be $q(\tau_0)$ and $p(\tau_0)=m \dot{q}(\tau_0)$. Then the exact quantum motion of $q(\tau)$ which is subjected to the external force $F(\tau)$ in the Heisenberg picture is given by [@kim] $$\begin{aligned} \label{q:0A} q(\tau) &=& q_O(\tau)+ q_F(\tau) \nonumber \\ &=& q(\tau_0) \frac{\sqrt{g_-(\tau) g_+(\tau_0) }}{ \omega_I} \cos \left[\Theta(\tau) - \chi(\tau_0)\right] + p(\tau_0) \frac{ \sqrt{g_-(\tau)g_-(\tau_0)}}{\omega_I} \sin\Theta(\tau) \\ &+& A_F(\tau) + A_F^{\dagger}(\tau). \nonumber\end{aligned}$$ In this equations we use the following definitions: $$\begin{aligned} g_-(\tau) &=& f(\tau) f^*(\tau), \label{g_-(t)} \\ g_0(\tau) &=& - \frac{M(\tau)}{2} \dot{g}_-(\tau) , \nonumber \\ g_+(\tau) &=& M^2(\tau)\left|\dot{f}(\tau)\right|^2, \nonumber \\ \label{phase} \Theta(\tau) &=& \int^\tau_{\tau_0}\mbox{d}\tau \frac{\omega_I}{M(\tau) g_{-}(\tau)}, \end{aligned}$$ where $f(\tau)$ is a homogeneous solution of Eq. (\[q”:F\]) and $\omega_I = \sqrt{g_{+}(\tau) g_{-}(\tau) - g_{0}^2(\tau)}$ is invariant under the time evolution. For the definition of $g_i(\tau)$ $(i= \pm,0)$ see [@kim] and references therein. If $\tau$ is Killing time, we can expand the free field solution into its positive solutions and negative solutions. Let us classify its solution by $\omega$ and set the positive solution as $u_\omega$. Therefore $$\begin{aligned} \frac{\partial}{\partial \tau} u_\omega = -i |\omega| u_\omega\end{aligned}$$ and The free field solution in two dimension can be written as $$\begin{aligned} \phi^0(t,x) &=& \int_{-\infty}^{\infty} \mbox{d}k [ a_k u_k(\tau, \rho) + a_k^{\dagger} u_k^*(\tau,\rho) ] \end{aligned}$$ where $a_k$ and $a_k^\dagger$ is the corresponding creation and annihilation operators and $u_k$ is proportional to $ e^{-i\omega \tau}$ . With this choice, we can write the annihilation part of the inhomogeneous solution $q_F(\tau)= A_F(\tau)+ A_F^{\dagger}(\tau)$ as $$\begin{aligned} A_F(\tau) = \int_0^{\infty} \mbox{d}\omega \omega G(\omega,\tau)( a_{\omega} + a_{-\omega}).\end{aligned}$$ Where $G(\omega,\tau)$ is the classical inhomogeneous solution of the forced harmonic oscillator equation $$\begin{aligned} \ddot{G}(\omega,\tau) + \frac{d \ln M(\tau)}{d\tau} \dot{G}(\omega,\tau) + \omega^2(\tau) G(\omega,\tau) = -i \frac{e(\tau)}{m} u_\omega\end{aligned}$$ with the initial condition $$\begin{aligned} \label{condition1} G(\omega,0)=0 \hspace{1cm} \dot{G}(\omega,0) = 0.\end{aligned}$$ If we analyze $G(\omega,\tau)$, we can know all time evolutions of the operators in principle. The general solution for $G(\omega,\tau)$ can be written as $$\begin{aligned} \label{G:g} G(\omega,\tau) = g(\omega, \tau) - g^*(-\omega,\tau),\end{aligned}$$ where $$\begin{aligned} \label{g:tau} g(\omega,\tau) = e^{i \Theta(\tau)}\frac{\sqrt{g_{-}(\tau)}}{2 m \omega_I} \int_{\tau_0}^\tau \mbox{d}\tau' \sqrt{g_-(\tau')} M(\tau') e(\tau') e^{-i \Theta(\tau')} u_\omega (x(\tau'),t(\tau')).\end{aligned}$$ One can show that Eq. (\[q:0A\]) satisfies (\[q”:F\]) by direct substitution. The high momentum behavior of $G(\omega,t)$ is $O(1/\omega^{5/2})$ except some special case like the sudden jumping of the coupling constant, which we consider in Sec. IV. In the case of large $\omega$ the integral of (\[g:tau\]) is approximately given by the contributions around $\tau_0$, which makes the arguments of the exponential of $u_\omega(\tau)$ vanishes. Therefore the first approximation of $g(\omega,\tau)$ is of the form $\int d\tau' f(\tau_0) e^{\pm i \omega \tau}$. But this term is canceled in $G(\omega,\tau)= g(\omega,\tau) -g^*(-\omega,\tau)$, and leaves only the $O(1/\omega^{5/2})$ and higher terms. The inertial oscillator {#sec:II-1} ----------------------- At first let us consider the simplest inertial path: $x= 0$ and $t = \tau$. Moreover the mode solution is $u_k = 1/\sqrt{4 \pi |k|} e^{-i (|k| t -k x)}$. Let the initial state of the combined system to be $$\begin{aligned} \label{instate} \left|i\right> =\left|n \right> \left|0\right>_M,\end{aligned}$$ the $n$th excited state for the oscillator and the Minkowski vacuum state for the field. The correlation functions of $q(t)$ for state (\[instate\]) is $$\begin{aligned} \left<q_O(t) q_O(t')\right> &=& (2n+1) \frac{\sqrt{g_{-}(t)g_{-}(t')}}{2 \omega_I} \exp\left\{-i[ \Theta(t)- \Theta(t')] \right\},\\ \left<q_F(t) q_F(t')\right> &=& 2 \int \mbox{d}\omega \omega^2 G(\omega,t) G^*(\omega, t'),\\ \left<q_O(t) q_F(t')\right> &=& 0. \end{aligned}$$ The correlation of the homogeneous part decrease because $M(t)$ increase monotonically. Therefore for a large enough time the correlation is governed by the inhomogeneous term. If there is absent of $1/(\omega)^{3/2}$ term in $G(\omega,t)$ there is no UV contribution to the correlation function. As we see in the previous section, this is normally true. Therefore in the case of a slowly varying coupling, the main contribution to the correlation comes from the frequency region around the resonance frequence $\Omega(t)$ (See Eq. (\[quad\]) ). The system is symmetric about $x=0$. Therefore, it is enough to obtain the correlations of the field in the area $x,x'<0$. In this area Eq. (\[phi:uv\]) becomes $$\begin{aligned} \phi(t,x) = \phi^0_R(u) + \phi^0_L(v) + \frac{1}{2} e(v) [q_O(v)+q_F(v)].\end{aligned}$$ Therefore the correlations of the field and the oscillator is for the state (\[instate\]) are $$\begin{aligned} \left<\phi^0_R(u) q_F(v')\right> &=& \left<q_F(v') \phi^0_R(u)\right>^* = \int \mbox{d}\omega \omega G^*(\omega, v') u_{\omega}(u), \\ \left<\phi^0_L(v) q_F(v')\right> &=& \left<q_F(v') \phi^0_L(v)\right>^* = \int \mbox{d}\omega \omega G^*(\omega, v') u_{\omega}(v).\end{aligned}$$ From these, one get the renormalized correlation function $$\begin{aligned} \label{correlation} &&\left<\phi(t,x) \phi(t',x')\right> - \left<\phi^0(t,x)\phi^0(t',x')\right> = \frac{e(v)}{2} \left\{ \left<\phi^0_R(u) q_F(v')\right> + \left<\phi^0_L(v) q_F(v')\right> \right\} \nonumber \\ &&~~~+ \frac{e(v')}{2}\left\{\left<q_F(v)\phi^0_R(u')\right> + \left<q_F(v)\phi^0_L(v')\right> \right\} + \frac{e(v)e(v')}{4} \left\{\left<q_O(v) q_O(v')\right> + \left<q_F(v)q_F(v')\right> \right\} \nonumber \\ &&~=(2n+1) \frac{e(v)e(v')}{4} \frac{ \sqrt{g_{-}(v)g_{-}(v')}}{2 \omega_I} \exp\left\{-i[ \Theta(v)- \Theta(v')] \right\} \\ &&~~+ \frac{e(v)e(v')}{2} \int \mbox{d}\omega \omega^2 G(\omega,v) G^*(\omega, v') \nonumber \\ &&~~+ \frac{e(v)}{2} \int \mbox{d}\omega\omega G^*(\omega,v') \left[ u_\omega(v) +u_\omega(u)\right] + \frac{e(v')}{2} \int \mbox{d}\omega\omega G(\omega,v) \left[ u_\omega^*(v') +u_\omega^*(u') \right]. \nonumber \end{aligned}$$ The uniformly accelerated oscillator ------------------------------------- Now let us consider a uniformly accelerating trajectory $x = \frac{1}{a} \cosh a\tau, t= \frac{1}{a}\sinh a \tau $. Rindler space $(\tau,\rho)$ is given by $$\begin{aligned} x = \frac{1}{a} ~e^{a\rho} \cosh a\tau, ~~ t = \frac{1}{a} ~e^{a\rho} \sinh a \tau.\end{aligned}$$ In this system the retarded time is $$\begin{aligned} \tau_{ret} &=&\tau-\rho ~~~\mbox{for} ~~ \rho > 0,\\ &=&\tau+\rho ~~~\mbox{for} ~~ \rho < 0 .\nonumber \end{aligned}$$ At the right Rindler wedge, the free field $\phi^0(t,x)$ can be expanded with the normal modes of Rindler space-time as $$\begin{aligned} \phi^0(t,x) &=& \sum_{k=-\infty}^{\infty} [b_k \xi_k + H.C.] \\ &=& \sum_{\lambda =0}^{\infty}[b_\lambda \xi_\lambda(U) + b_{-\lambda} \xi_{\lambda}(V) + H.C.], \nonumber\end{aligned}$$ where $U = \tau - \rho=-\mbox{ln}(-a u)/a $, $V = \tau + \rho= \mbox{ln}(av)/a$, and $\xi_\lambda = 1/\sqrt{4 \pi |\lambda|} e^{-i \lambda U} = 1/\sqrt{4 \pi |\lambda|} (-a u)^{i \lambda/a}$, and $b_\lambda, b_\lambda^\dagger$ is the creation and annihilation operator in the Rindler spacetime. Therefore we can set $u_\omega \rightarrow \xi_\lambda$ and $\omega \rightarrow \lambda$ in Sec. II. Let us consider the initial state to be (\[instate\]). The expectation value of $q(\tau)$ for $|i>$ is zero. The correlation functions for $|i>$ are $$\begin{aligned} \left<q_O(\tau)\right.&&\left. q_O(\tau')\right> = (2n+1) \frac{\sqrt{g_{-}(\tau)g_{-}(\tau')}}{2 \omega_I} \exp\left\{-i[ \Theta(\tau)- \Theta(\tau')] \right\},\\ \left<q_F(\tau)\right.&&\left. q_F(\tau')\right> = 2 \int \mbox{d}\lambda \lambda^2 \left[ \left\{1 + N(\lambda/a)\right\} G(\lambda, \tau) G^*( \lambda, \tau') \right. \\ +&& \left. N(\lambda/a) G^*(\lambda, \tau)G(\lambda, \tau') \right], \nonumber \\ \left<\phi^0_R(U)\right.&&\left. q_F(\tau'_{ret})\right> = \left<q_F(\tau'_{ret}) \phi^0_R(U) \right>^* \\ =&& \int \mbox{d}\lambda \lambda \left[ \xi_{\lambda}(U) G^*(\lambda, \tau') \left\{ 1+N(\lambda/a) \right\} + \xi^*_{\lambda} (U) G(\lambda, \tau'_{ret}) N(\lambda/a) \right], \nonumber \\ \left<\phi^0_L(V)\right.&& \left.q_F(\tau'_{ret})\right> = \left<q_F(\tau'_{ret}) \phi^0_L(V)\right>^* \\ =&& \int \mbox{d}\lambda \lambda \left[\xi_\lambda(V) G^*(\lambda, \tau_{ret}') \left\{ 1+N(\lambda/a) \right\} + \xi_\lambda^*(V) G(\lambda, \tau'_{ret}) N(\lambda/a) \right], \nonumber\end{aligned}$$ where $N(\Omega) = 1/(e^{2\pi \Omega}-1)$. We use the fact that the Minkowski vacuum is FDU thermal bath with temperature $a/(2\pi)$ to the accelerating observer. The renormalized correlation function can be obtained with the same method of inertial case. These correlation functions can be devided into two classes: First is those of zero temperature and second is thermal contributions. The high momentum behavior of the second is cut off by the presence of the exponential in the denominator. Therefore it is evident that the first term dominate the correlation if there is UV divergences due to the existence of the $1/\omega$ term in $G(\lambda,\tau)$. The sudden jump case of the coupling is exactly that case. Exactly solvable models {#sec:solvable} ======================= Constant coupling ----------------- The most easiest problem is, of course, the case of constant coupling ($e(\tau)=e$). In this case $$\begin{aligned} M(\tau) &=& \frac{e^2}{2} (\tau-\tau_0) \\ f(\tau) &=& \frac{1}{\sqrt{m}} e^{\pm (i \Omega +e^2/(2m))(\tau-\tau_0)} \end{aligned}$$ where $\Omega ^2 = \omega_0^2 - \epsilon^2$. $G(\omega, \tau)$ satisfies the following equation: $$\begin{aligned} \ddot{G}(\omega,\tau) + \frac{e^2}{2m} \dot{G}(\omega,\tau) + \omega_0^2 G(\omega,\tau) = -i \frac{e}{m} u_\omega\end{aligned}$$ The general solution to this equation is given by $$\begin{aligned} \label{sol1} G(\omega,\tau) &=& a \exp(-\epsilon \tau) \exp(i\sqrt{\omega_0^2-\epsilon^2} \tau+ \alpha) + i e \chi(\omega) u_\omega (\tau,0) \end{aligned}$$ where $$\begin{aligned} \chi(\omega) = \frac{1}{m \left[\omega_0^2 -\omega^2 -2i \epsilon \omega \right]} \end{aligned}$$ and $\epsilon = e^2/4m$. The coefficients $a$ and $\alpha$ must be chosen $G(\omega,\tau)$ to satisfy the initial condition (\[condition1\]). The first term exponentially decay therefore there remain only the second term asymptotically. This is exactly the same result with Massar, Parentani, and Brout [@massar]. Turn on of the coupling {#sec:turnon} ----------------------- The next example is given by the coupling $$\begin{aligned} \label{cha} 2\epsilon(\tau) =\frac{e^2(\tau)}{2 m} &=& \frac{e_-^2}{4m} \left(1- \tanh \frac{\tau}{d} \right)+ \frac{e_+^2}{4m} \left(1+ \tanh \frac{\tau}{d}\right) \\ & =& \epsilon_{-}\left(1- \tanh \frac{\tau}{d}\right)+\epsilon_{+} \left(1+ \tanh \frac{\tau}{d} \right). \nonumber\end{aligned}$$ The limit $d \rightarrow 0$ corresponds to the sudden jump and $d \rightarrow \infty$ to the adiabatic one. Eq. (\[Omega\]) $$\begin{aligned} \Omega^2(\tau) = \omega_0^2 - \epsilon^2(\tau) = \frac{\omega_-^2}{2}\left(1- \tanh \frac{\tau}{d} \right) + \frac{\omega_+^2}{2}\left(1+ \tanh \frac{\tau}{d} \right) + \frac{(\epsilon_+- \epsilon_-)^2/4}{\cosh^2(\tau/d)}\end{aligned}$$ has two limiting values $\omega_\pm^2 = \omega_0^2 -\epsilon_\pm^2$ at the positive and negative infinity. These two values define a natural positive frequency modes of the oscillator in the past and the future asymptotic region. The effective mass (\[M:t\]) becomes $$\begin{aligned} \label{mass} M(\tau) = m \exp \int 2 \epsilon(\tau) \mbox{d}\tau = m \left( \frac{\cosh \tau/d}{ \cosh \tau_0/d}\right)^{(\epsilon_+-\epsilon_-)d} \exp \left[ (\epsilon_+ +\epsilon_-)(\tau-\tau_0) \right].\end{aligned}$$ From these we get the homogeneous solution for the classical equation of motion (\[q”:F\]) $$\begin{aligned} \label{f:t} f(\tau) = \frac{1}{\sqrt{M(\tau)}} e^{-i(\omega_+ + \omega_-)\tau/2} \left(\cosh \frac{\tau}{d} \right)^{-i(\omega_+-\omega_-)d/2} ~_2F_1(\alpha_-, \alpha_+;1-i \omega_- d; y),\end{aligned}$$ where $$\begin{aligned} y &=& \frac{1+ \tanh \tau/d}{2}, \\ \alpha_\pm &=& \frac{1 \pm \sqrt{1+ (\epsilon_+-\epsilon_-)^2d^2}}{2} + i \frac{(\omega_+ - \omega_-)d}{2},\end{aligned}$$ and $_2F_1$ is the hypergeometric function [@morse]. We choose Eq. (\[f:t\]) to be pure positive frequency mode at the past infinity. On the other hand, it becomes generally mixture of the positive and negative modes at the future: $$\begin{aligned} \lim_{\tau \rightarrow -\infty} f(\tau) &=& \frac{2^{i(\omega_+-\omega_-)d/2}}{ \sqrt{M(\tau)}} e^{-i \omega_-\tau} , \label{f:-} \\ \lim_{\tau \rightarrow \infty} f(\tau) &=& \frac{2^{i(\omega_+-\omega_-)d/2}}{ \sqrt{M(\tau)}} \left( \alpha e^{-i \omega_+ \tau} + \beta e^{i \omega_+ \tau} \right). \label{f:+}\end{aligned}$$ where $$\begin{aligned} \alpha &=&\frac{\Gamma(1-i\omega_- d) \Gamma(1-i\omega_-d -\alpha_- -\alpha_+) }{\Gamma(1-i\omega_-d -\alpha_-) \Gamma(1-i\omega_-d -\alpha_+)} , \\ \beta &=&\frac{\Gamma(1- i\omega_-d) \Gamma(\alpha_- +\alpha_+ -1+i\omega_-d) }{\Gamma(\alpha_-)\Gamma(\alpha_+)}. \end{aligned}$$ The absolute squares of $\alpha$ and $\beta$ $$\begin{aligned} |\alpha|^2 &=& \frac{1}{2}\frac{\omega_-}{\omega_+} \frac{ \cosh\pi(\omega_+ + \omega_-)d + \cos 2\pi x}{\sinh \pi \omega_- d \sinh \pi \omega_+ d}, \\ |\beta|^2 &=& \frac{1}{2}\frac{\omega_-}{\omega_+} \frac{ \cosh\pi(\omega_+ - \omega_-)d + \cos 2\pi x}{\sinh \pi \omega_- d \sinh \pi \omega_+ d}\end{aligned}$$ satisfy a Bogolubov type relation $$\begin{aligned} |\alpha|^2 -|\beta|^2 =\frac{\omega_-}{\omega_+},\end{aligned}$$ where $x = \sqrt{1 + (\epsilon_+ -\epsilon_-)^2 d^2}/2$. The factor $\omega_-/\omega_+$ comes from the change of the natural frequency of the oscillator [@jyji2]. At the present problem the initial homogeneous solution for the oscillator decays by $1/\sqrt{M(\tau)}$ factor so the asymptotic form for large $\tau$ is given by $q_F$. Therefore the particle creation or other related topics must be discussed with the inhomogeneous solution $G(\omega,\tau)$ with respect to the positive frequency mode at the future asymptotic region. Since our primary purpose is not the oscillator state but the radiation from the oscillator, we do not discuss it further. In the adiabatic limit $\beta$ vanishes, on the other hand, in the sudden jump limit it becomes $(1-\omega_-/\omega_+)/2$. From (\[f:t\]) one get $$\begin{aligned} \label{g_-:on} g_-(\tau) = f(\tau) f^*(\tau) = \frac{|_2F_1(\alpha_-, \alpha_+;1-i \omega_-d;y)|^2}{M(\tau)},\end{aligned}$$ and the invariant frequency $\omega_I= \omega_-$. The integral of generalized frequency (\[phase\]) is $$\begin{aligned} \label{phase1} \Theta(\tau) &=& \omega_I \int^\tau \mbox{d}\tau' \frac{1}{|_2F_1(\alpha_-, \alpha_+; 1-i \omega_-d;y)|^2} \nonumber \\ &=& \omega_I \int^\tau \mbox{d}\tau'\frac{1}{R^2(\tau)} - \omega_I \int^\tau \mbox{d}\tau'\frac{1}{R^2(\tau)} \left(1- \frac{R^2(\tau)}{|_2F_1(\alpha_-, \alpha_+;1-i \omega_-d;y)|^2} \right) \\ &=& \theta(\tau) -\theta(\tau_0) - \theta_f(\tau), \nonumber\end{aligned}$$ where $R(\tau)$ and $\theta(\tau)$ are the absolute value and the real phase of $$\begin{aligned} R e^{-i \theta(\tau)}=\alpha e^{-i \omega_+ \tau} +\beta e^{+i \omega_+\tau},\end{aligned}$$ and $\theta_f(\tau)$ approaches to some finite value as $t \rightarrow \infty$. Eqs. (\[q:0A\] and (\[g\_-:on\]) shows $q_O(t)$ decreases exponentially for $\tau \gg d$. Using these and Eq. (\[G:g\]) we get the asymptotic form $$\begin{aligned} \label{Ginfty:chi} G(\omega,\tau) &=& i e_+ \chi(\omega) u_\omega(\tau,0) \\ &-& \frac{1}{2 m \omega_I \sqrt{M(\tau)}} \left[\left\{\alpha \chi_f(-\omega) - \beta^* \chi_f(\omega)\right\} e^{-i \omega_+ \tau} + \left\{\beta \chi_f(-\omega) -\alpha^* \chi_f(\omega) \right\} e^{i \omega_+ \tau} \right]\nonumber\end{aligned}$$ where $$\begin{aligned} \chi(\omega) &=& \frac{1}{m [\omega_0^2 - \omega^2 -2i \epsilon_+\omega]},\end{aligned}$$ and $\chi_f(\omega) = \lim_{\tau\rightarrow \infty} \chi_f(\omega,\tau)$. $$\begin{aligned} \label{chif} \chi_f(\omega,\tau) &=& -e_+ \int_{\tau_0}^\tau d\tau'\sqrt{M(\tau')} R(\tau') e^{-i\omega_+\tau'} u_\omega(\tau' ,0) \\ & & \cdot \left[1- \frac{\sqrt{M(\tau')g_-(\tau')}}{R(\tau')} \frac{e(\tau')}{e_+} e^{i[\omega_+\tau' - \theta(\tau')]} \right]. \nonumber\end{aligned}$$ This result is similar with that of the constant coupling except $\chi_f(\omega)$ is determined by the integral. Stress Energy tensor in the Sudden Jump Limit {#sec:jump} ============================================= Now let us study the stress tensor of the scalar field in the presence of the oscillator. We consider instant switching process ($d \rightarrow 0$ limit of Sec. \[sec:turnon\]). We solve this problem up to zeroth order on $d$ or $e^{-2|\tau|/d}$, where $\tau$ is the proper time seen by the oscillator. We calculate $G(\omega,\tau)$ without explicit choice of coordinates system because it is common both the inertial and the uniformly accelerating oscillator. Let us set $\tau_0=-\infty$ and rescale the mass to be $M(0) =m$. Similarly we also set $\Theta(0) =0$. After carrying out the integral (\[chif\]) explicitly in the limit $|t| \gg d$ we get $$\begin{aligned} \label{G:-+} G(\omega, \tau) &=& i e_- \chi_-(\omega) u_\omega(\tau,0), \hspace{5.9cm} ~~~~~ \mbox{for} ~~ \tau \ll -d, \\ &=& G_\infty(\omega,\tau) + \frac{i e_+}{2\sqrt{4 \pi \omega}} e^{-\epsilon_+ \tau} \left\{ \chi_+^*(-\omega) e^{i\omega_+\tau}+\chi_+(\omega)e^{-i\omega_+\tau} \right\}, \mbox{ for}~~ \tau \gg d, \nonumber \end{aligned}$$ where $$\begin{aligned} \label{chip:chida} \chi_+(\omega) = \chi_d(\omega)+\frac{e_-}{e_+} \chi_-(\omega),\end{aligned}$$ and $$\begin{aligned} \chi_d(\omega) &=& \frac{1}{m \omega_+} \left[\frac{1}{\omega- \omega_+ +i \epsilon_+} - \frac{1- e_-/e_+ }{ + \omega-\omega_++i(\epsilon_+ - 2/d)} \right. \\ &-&\left. \frac{e_-}{2e_+} \left(\frac{1}{\omega-\omega_-+i\epsilon_- }+ \frac{1}{\omega +\omega_- +i \epsilon_-} \right)\right], \label{chi_d} \nonumber \\ \chi_-(\omega) &=& \chi_-^*(-\omega)= \frac{1}{m}\frac{1}{\omega_0^2-\omega^2- 2i \omega \epsilon_-}. \label{chi_a}\end{aligned}$$ Here needs some remarks. All physical quantities like the coupling, classical solution, and effective mass must be continuous at $\tau=0$. This constraints demands the second term in $\chi_d$, which makes $G(\omega,\tau)$ to be quadratically decrease for large $\omega$. $G_\infty(\omega,\tau) = i e_+ \chi(\omega,\tau) u_\omega(\tau,0) $ dominates the asymptotic form of $G(\omega,\tau)$ and the second term, which is the effect of the change of the coupling, decrease exponentially on the time scale of $1/\epsilon_+$. At $\tau <0$ the inhomogeneous solution $G(\omega,\tau)$ is that of the equilibrium. Therefore our system represent a system which is in equilibrium at $\tau<0$ become dynamic due to the change of the coupling at $\tau=0$. The solution $G(\omega,\tau) $describe this dynamic approaching process to equilibrium. If we restrict the region of $\omega$ as $0 < \omega < \Gamma \ll 1/d$, we can ignore the $d$ dependent term in $\chi_d(\omega)$. In this limit $\chi_d(\omega)$ becomes $O(1/\omega)$ and gives cutoff dependent UV behaviors. On the other hands in the case of $\Gamma \gg 1/d$, $\chi_d(\omega)$ is $O(1/\omega^2)$ which makes the theory UV finite. But we cannot get a sensible theory because the asymptotic form of the stress tensor crucially depends on $1/d$, which is unphysical. Therefore we restrict the cutoff $\Gamma \ll 1/d$ and also restrict our attention to $|\tau| > 1/\Gamma$. The Stress tensor in the presence of a inertial oscillator {#sec:stressiner} ---------------------------------------------------------- In Sec. (\[sec:II-1\]) we obtain the renormalized correlation function (Eq. (27)) in the region $x , x'<0$. In this region there is no $u, u'$ dependent terms. Therefore $T_{uu}$ component of the stress tensor vanishes. Moreover, $\left<T_{uv}\right> = tr T/4 = 0$ since we are dealing with a massless field in two dimension and there is no trace anomaly because the curvature is zero. The stress tensor vanishes in the region $v<0$ since $G(\omega,t)$ has the same form with the asymptotic case (\[Ginfty:chi\]) and there is no radiation asymptotically. In the region $t > 0$ we must calculate the stress tensor explicitly. We restrict our attention to $v > 1/\Gamma \gg d$ since we are interested in the radiation after turn on the coupling. After taking differentiation of the correlation function with respect to $v$ and $v'$ followed by the limit $v' \rightarrow v$ we get $$\begin{aligned} \label{Tvv:G} T_{vv} &=& T_1 + \frac{e_+}{2} T_2 + \frac{e_+^2}{2} T_3,\end{aligned}$$ where $$\begin{aligned} T_1 &=& (2n+1) \frac{e_+^2}{8 \omega_I} \lim_{v'\rightarrow v} \left(\partial_v \partial_v'\sqrt{g_-(v)g_-(v')} \exp \left[-i\left\{ \Theta(v) - \Theta(v')\right\}\right] \right) \\ T_2 &=& \int d \omega \omega \left[ \partial_v G^*(\omega,v) \partial_v u_\omega(v) + \partial_v G(\omega,v)\partial_v u_\omega^*(v) \right], \\ T_3 &=& \int d \omega \omega^2 \partial_v G(\omega,v) \partial_v G^*(\omega,v).\end{aligned}$$ Where we ignore terms related with $\dot{e}(v)$ which is important for $t \sim d$. Since we consider only the region $t > 1/\Gamma \gg d$, it is safe to ignore such terms. Now let us write down only the dominant terms of the stress tensor. (For details see appendix.) For small $v$, it is dominant the interference ($T_2$) between the oscillator and the field. $$\begin{aligned} T_{vv}= -\frac{e^2_+ \omega_0}{8 \pi m \omega_+} e^{-\epsilon_+ v} \left(1- \frac{e_-}{e_+} \right) \frac{1}{v} \cos (\omega_+v + \theta) , \hspace{0.5cm} \mbox{for} \hspace{0.3cm} \frac{1}{\omega_0} \gg v > \frac{1}{\Gamma}.\end{aligned}$$ In this region the energy is absorbed into the oscillator from the field with the amount $$\begin{aligned} \label{Eabs} E_{absorbed} = \frac{e^2_+}{8 \pi m } \left(1- \frac{e_-}{e_+}\right) \ln\frac{\Gamma}{\omega_0} + \mbox{smaller terms}.\end{aligned}$$ For $v \gg 1/\Gamma$, energy is radiated away from the oscillator and $T_3$ is dominant. $$\begin{aligned} \label{Tvvasym} T_{vv} &=& \frac{e_+^4}{8 \pi} \left(\frac{\omega_0}{m \omega_+} \right)^2 e^{-2\epsilon_+ v} \cos^2(\omega_+ v + \theta) \left(1-\frac{e_-}{e_+}\right)^2 \mbox{ln}\left(\Gamma/\omega_0\right) \hspace{0.5cm} \mbox{for} \hspace{.3cm} v \gg \frac{1}{\Gamma}.\end{aligned}$$ where $\tan \theta = \epsilon_+/\omega_+$. The total radiated energy in this region is $$\begin{aligned} \label{Erad} E_{radiated} = \frac{e^2_+}{8 \pi m } \left(1- \frac{e_-}{e_+}\right)^2 \ln\frac{\Gamma}{\omega_0}\end{aligned}$$ Therefore we can conclude that in general the absorbed energy into the oscillator is greater than the radiated one. The Stress tensor in the presence of a uniformly accelerating oscillator {#sec:stressacc} ------------------------------------------------------------------------ With the same reason given at the previous subsection, $T^A_{uu}$ and $T^A_{uv}$ are zero. Similarly, the stress tensor vanishes for $ V < 0$. In the region $V>0$, we also restrict to $V > 1/\Gamma \gg 1/d$, and we ignored $\dot{e}(\tau)$ related terms. In this region, $T^A_{vv}$ components can be written as follows: $$\begin{aligned} T^A_{vv} = T^A_{1}+ \frac{e_+}{2} T^A_{2} + \frac{e_+^2}{2} T^A_{3},\end{aligned}$$ where $$\begin{aligned} T^A_{1} &=& (2n+1)\frac{e_+^2}{8 \omega_I} \lim_{v' \rightarrow v} \left( e^{-a (V + V')} \partial_V \partial_V' \sqrt{g_-(V)g_-(V')} \exp \left[-i\left\{ \Theta(V) - \Theta(V')\right\}\right] \right) \\ T^A_{2} &=& \int d \lambda \lambda \left\{1 + 2 N\left( \frac{\lambda}{a} \right) \right\} \left[\partial_V \xi_{\lambda}(V) \partial_V G^*(\lambda,V) + \partial_V \xi_{\lambda}^*(V) \partial_V G(\lambda,v) \right] e^{-2a V}, \\ T^A_{3} &=& \int d \lambda \lambda^2 \left\{1 + 2 N\left( \frac{\lambda}{a} \right) \right\} \partial_V G(\lambda,V) \partial_V G^*(\lambda, V) e^{-2 a V}.\end{aligned}$$ For small $V$ the stress tensor is dominated by $T_2^A$ which is given by $$\begin{aligned} T_{vv} ^A &=& \frac{e^2_+ }{8 \pi m \omega_+} \left(1- \frac{e_-}{e_+} \right) \frac{a}{\ln av } \\ & &\cdot \left\{ \beta_+ (av)^{i \frac{ \omega_+}{a}} + \beta_+^* ( av)^{-i \frac{\omega_+}{a}} \right\} (av)^{-2- \epsilon_+/a} \hspace{.5cm} \mbox{for} \hspace{.2cm} \frac{1}{\omega_0} \gg V > \frac{1}{\Gamma}, \nonumber\end{aligned}$$ This is exactly the same form with the inertial oscillator except the retardation factor $(a v)^{-2}$ due to the acceleration and the mere coordinate change $v \rightarrow \ln av/a$. The total absorbed energy is given by $$\begin{aligned} E^A_{absorbed} = \frac{e^2_+}{8 \pi m } \left(1- \frac{e_-}{e_+}\right) \ln\frac{\Gamma}{\omega_0} + \mbox{smaller terms}.\end{aligned}$$ This is exactly the same with Eq. (\[Eabs\]). Physically, it is natural because there is no enough time the acceleration to act on the short time interference. For $V \gg 1/\Gamma$, $T_3^A$ is dominant. $$\begin{aligned} T_{vv}^A = \frac{e_+^2}{8 \pi m \omega_+} \left( 1- \frac{e_-}{e_+}\right)^2 \ln \frac{\Gamma}{ \omega_0} \left\{ \beta_+ (av)^{i \frac{ \omega_+}{a}} + \beta_+^* ( av)^{-i \frac{\omega_+}{a}} \right\}^2 (av)^{-2- 2\epsilon_+/a} \hspace{.2cm} \mbox{for} \hspace{.2cm} V \gg \frac{1}{\Gamma},\end{aligned}$$ This equation is quite similar to Eq.(\[Tvvasym\]) except the retardation effect and the coordinate change ($v \rightarrow \ln{a v}/a $) due to acceleration. This is because the main effect to the radiation comes from the high momentum region. The total energy radiated away from the oscillator is $$\begin{aligned} E^A_{radiated} &=& \frac{e_+^2}{8 \pi m} \left(1-\frac{e_-}{e_+} \right)^2 \ln \left(\frac{\Gamma}{\omega_0} \right) \tan \theta \left[\frac{1}{(x+ \sin \theta) \cos \theta} + \frac{\cos 2 \theta x - \tan \theta}{x^2 + 2 \sin \theta x + 1} \right] ,\end{aligned}$$ where $x = a/(2 \omega_0)$. Summary and Discussion ====================== We have discussed the influence of a harmonic oscillator on a scalar quantum field in $1+ 1$ dimensions. These are illustrated by calculating the radiation of the scalar field from the oscillator. The first step to do this is to express the time evolutions with the classical inhomogeneous solution $G(\omega,t)$ of a damping forced harmonic oscillator. Then we applied this result to the sudden jumping limit of the coupling and obtain the change of the stress tensor in the presence of the oscillator. There are two main effects on the radiation. The first is due to sudden change of the coupling which is described by the correlation between the oscillator and the field. This effect rapidly die out but the oscillator absorbs large energy from the field through this correlation. Moreover this absorption is independent of the acceleration. Subsequently, slow radiation from the oscillator take place. In case of an inertial oscillator this radiation is smaller than the absorbed energy through the first stage. The behavior of the total radiated energy become nontrivial if the detector is accelerated. In the case of a small coupling constant ( $\epsilon_+ \ll \omega_0$), the radiated energy is maximized by $a=0$. But there is a peak of the radiated energy at a non-zero acceleration if $\theta $ greater than some value $\theta_0 \sim 1.07702$ or $$\begin{aligned} \left(\frac{\epsilon_+}{\omega_0}\right)^2 > \frac{- \omega_+/\omega_0 + \sqrt{(\omega_+/\omega_0)^2 - 8 \omega_+/\omega_0 +8}}{4(1-\omega_+/\omega_0)}\end{aligned}$$ In this case the radiated energy can greater than the absorbed one during the first stage. Especially, if $\epsilon \rightarrow \omega_0$ then the radiated energy becomes extremely large for non-zero acceleration. If the acceleration is large enough, the radiation decrease according to the inverse of the acceleration. The following two points can help to understand this phenomena. First, the radiation is not due to the acceleration but due to the change of the coupling. Second, as acceleration grows the unit proper time of an accelerating oscillator correspond to a larger coordinate time to the Minkowski observer. Therefore the coordinate time which takes to vary the coupling becomes larger for the larger acceleration. [Fig. 1. Total radiated energy\ Total radiated energy is plotted according to the acceleration. The acceleration of each time is given by $ a/(2\omega_0)= \{\pi/6, \pi/3, \pi/2-0.3, \pi/2-0.2, \pi/2-0.1\}$ from the below. The unit for energy is $\frac{e_+^2}{8 \pi m} \left(1-\frac{e_-}{e_+} \right)^2 \ln \left(\frac{\Gamma}{\omega_0} \right)$. ]{}  \  \   acknowledgements ================ This work was supported by the Korea Science and Engineering Foundation (KOSEF). One of the authors thanks to Min-Ho Lee for his helpful discussions. Appendix {#appendix .unnumbered} ======== The Stress Energy Tensor of the field in the sudden jump of the coupling – Inertial Case ---------------------------------------------------------------------------------------- In this appendix we obtain the stress tensor for the model of Sec. \[sec:turnon\] in the sudden jump limit. It is easy to know that the stress tensor simply vanishes for $v<0$, from (\[G:-+\]). Therefore we calculate it only for $v \geq 0$ in the left hand side of the oscillator. The stress tensor (\[Tvv:G\]) is composed of three terms. The first term can be evaluated easily to become $$\begin{aligned} \label{T1} T_1 &=&\frac{\epsilon_+}{4 \omega_{-}\omega_{+}^2} e^{-2 \epsilon_+ v} \left[(\omega_+^2+ \omega_-^2)\omega_0^2 +(\epsilon_+^2-\omega_+^2 )(\omega_+^2-\omega_-^2) \cos 2\omega_+ v \right. \\ &&+ \left. 2\epsilon_+ \omega_+ ( \omega_+^2- \omega_-^2) \sin2 \omega_+ v \right]. \nonumber\end{aligned}$$ $T_2$ is sum of two terms which are mutually complex conjugate. One of these is $$\begin{aligned} \label{Gu} \int d \omega \omega \partial_v G(\omega,v) \partial_{v} u_\omega^*(v) = \int d \omega \omega \partial_v G_\infty(\omega,v) \partial_{v} u_\omega^*(v) +\frac{i e_+}{8 \pi}e^{-\epsilon_+ v} T_{2a},\end{aligned}$$ where $$\begin{aligned} \label{T2a} T_{2a} = -\beta_+ e^{i\omega_+v} \int d \omega \omega \chi_+^*(-\omega) e^{i\omega v} + \beta_+^* e^{-i \omega_+v} \int d\omega \omega \chi_+(\omega) e^{i \omega v} ,\end{aligned}$$ and we define the constant $$\begin{aligned} \beta_\pm = \omega_\pm + i \epsilon_\pm.\end{aligned}$$ Therefore $$\begin{aligned} T_{2a}-T_{2a}^* &=& -\frac{1}{m \omega_+} \beta_+ e^{i \omega_+ v} \left[ \left(1-\frac{e_-}{e_+} \right) \frac{2i}{v} + 2 \beta_+ e^{-i\beta_+ v} Ei(i \beta_+ v) \right. \\ &-& \left. \frac{e_-}{e_+}\left\{ \left(1+ \frac{\omega_+}{\omega_-} \right) \beta_- e^{-i \beta_- v} Ei(i \beta_- v) - \left(1+ \frac{\omega_+}{\omega_-} \right) \beta_-^* e^{i \beta_-^*v} Ei(-i \beta_-^*v) \right\} \right] \nonumber \\ &-& C.C. \nonumber\end{aligned}$$ Rather than use this complex form, lets us extract only its limiting form for small and large $v$ . In the region $ d \ll 1/\Gamma < v \ll 1/\omega_0 $ $$\begin{aligned} T_2 &=& \frac{e_+}{4 \pi m \omega_+} e^{-\epsilon_+ v}\left(1- \frac{e_-}{e_+} \right) \frac{ \omega_0}{v} \cos (\omega_+v+ \theta)\end{aligned}$$ and for large $v \gg 1/\omega_0$, $T_2 = O(e^{-2 \epsilon_+ v})$. Where $ \tan \theta = \epsilon_+/\omega_+$. Finally, let us evaluate $T_3$. If we define the following integrals $$\begin{aligned} I_1(v) &=& \int_0^\Gamma d \omega \omega^2 \chi(\omega) \chi_d(-\omega) e^{-i \omega v}, \\ I_2(v) &=& \int_0^\Gamma d \omega \omega^2 \chi(\omega) \chi^*_d (\omega) e^{-i \omega v}, \\ J(v) &=& \int_0^\Gamma d \omega \omega^2 \chi(\omega) \chi_-(-\omega) e^{-i \omega v} ,\end{aligned}$$ then $T_3$ becomes $$\begin{aligned} T_3 = && \lim_{v' \rightarrow v} \int d \omega \omega^2 \partial_v G_\infty(\omega,v) \partial_{v'} G_\infty^*(\omega,v') \\ &+& \frac{e_+^2}{8 \pi} \left( e^{-i\beta_+^* v} \beta_+^* T_{3a} + e^{i \beta_+ v} \beta_+ T_{3a}^* \right)+ \frac{e_+^2}{16 \pi} e^{-2\epsilon_+ v} T_{3b}. \label{T3:T3ab} \nonumber\end{aligned}$$ where $$\begin{aligned} T_{3a} &=& -I_1(v) + I_2^*(v)+ \frac{e_-}{e_+}[-J(v)+ J^*(v)] \label{T3a:int} \\ T_{3b} &=& \omega_0^2 \int _0^\Gamma d \omega \omega \left( |\chi_+(\omega)|^2+|\chi_+(-\omega)|^2 \right) \label{T3b:int} \\ &&-\beta_+^2 e^{2i\omega_+ v} \int _0 ^\Gamma d \omega \omega\chi_+^*(-\omega)\chi_+^*( \omega)-\beta_+^{*2} e^{-2i\omega_+ v} \int_0^\Gamma d \omega \omega \chi_+(-\omega)\chi_+(\omega). \nonumber \end{aligned}$$ where we have introduced explicit high momentum cut-off $\Gamma$ to regularize the UV behaviors. The first term of $T_3$ is canceled by the $\lim_{v\rightarrow v'}\int d \omega \omega (\partial_vG_\infty \partial_{v'}u^* +\partial_v u^* \partial_{v'}G_\infty)$ term of $T_2$ (The detail of the calculation can be consulted in Ref. [@massar].) As one can see in $T_{3b} $ major contribution to the stress tensor comes from the ultra-violet region. As one can easily see $T_{3b}$ don’t have UV contribution. Therefore the major contribution comes from $T_{3b}$. Let us examine $T_{3b}$ in detail. $\chi_-$ is of order $O(1/\omega^2)$ for large $\omega$, therefore only the first term of the integral $$\begin{aligned} \label{intchip} \int d&& \omega \omega |\chi_+(\omega)|^2 \\ &&= \int d \omega \omega |\chi_d|^2 +\frac{e_-}{e_+} \int d \omega \omega \left( \chi_d \chi_-^* + \chi_-\chi_d^* \right) +\frac{e_-^2}{e_+^2} \int d \omega \omega |\chi_-|^2 \nonumber\end{aligned}$$ can have important ultra-violet contribution. If one try to extract only the high momentum part it is ease to show that $$\begin{aligned} \int d \omega \omega |\chi_d|^2 &\cong & \frac{1}{m^2 \omega_+^2} \left(1-\frac{e_-}{e_+} \right)^2 \int^\Gamma d\omega \frac{1}{\omega} \\ &=& \frac{1}{m^2 \omega_+^2} \left(1-\frac{e_-}{e_+} \right)^2 \mbox{ln}\left(\Gamma/\omega_0\right). \nonumber\end{aligned}$$ Therefore $$\begin{aligned} T_3 = \frac{e_+^2}{4 \pi} \left(\frac{\omega_0}{m \omega_+} \right)^2 e^{-2\epsilon_+ v} \cos^2(\omega_+ v + \theta) \left(1-\frac{e_-}{e_+}\right)^2 \mbox{ln}\left(\Gamma/\omega_0\right).\end{aligned}$$ where $\tan \theta = \epsilon_+/\omega_+$. The stress tensor of the field in the sudden jump limit of the coupling – Accelerating Case ------------------------------------------------------------------------------------------- The stress tensor for $V<0$ vanishes. In the region $V>0$, the term $T^A_{1}$ is $$\begin{aligned} T^A_1 (v) &=& \frac{\epsilon_+}{4 \omega_- \omega_+^2} \left[ (\omega_+^2+ \omega_-^2)\omega_0^2 \right.\\ &-& \left. \frac{1}{2} (\omega_+^2-\omega_-^2) \left(\beta_+^{*2} (av)^{-2i\omega_+/a} + \beta_+^2 (av)^{2i \omega_+/a} \right) \right] (av)^{-2(1+\epsilon_+/a)}. \nonumber\end{aligned}$$ The integral for $T^A_{2,3}$ are of the form $\int d \lambda \lambda (1 + 2 N(\lambda/a) )f(\lambda,V)$. One can separate the ultra-violet (UV) $2\int d \lambda \lambda f(\lambda, V)$ from its thermal contributions $ \int d \lambda \lambda N(\lambda/a) f(\lambda,V)$. Let us look at each terms more closely. The UV term of $T^A_3$ is $$\begin{aligned} T^A_{3UV} = && \int^\Gamma d \lambda \lambda^2 \partial_v G_\infty(\lambda,V) \partial_{v} G_\infty^*(\lambda,V) \\ &+& \frac{e_+^2}{8 \pi} e^{-2aV} \left( e^{-i\beta_+^* V} \beta_+^* T^A_{3a} + e^{i \beta_+ V} \beta_+ T^{A*}_{3a} \right)+ \frac{e_+^2}{16 \pi} e^{-2(\epsilon_++a) V} T^A_{3b}. \label{T^A_3},\nonumber\end{aligned}$$ where $T^A_{3a}$ and $T^A_{3b}$ are given by Eqs. (\[T3a:int\]) and (\[T3b:int\]) if one replace $\omega \rightarrow \lambda $, $v \rightarrow V$. The first term of $T^A_3$ is canceled by the $\int d \lambda \lambda \left\{\partial_vG_\infty(\lambda,V) \partial_{v} \xi^*_\lambda(V) +\partial_v \xi _\lambda(V) \partial_{v}G_\infty^*(\lambda,V)\right\}$ term of $T^A_2$ [@massar]. The dominant term for this UV contribution is $$\begin{aligned} \label{T_3^AUV} \frac{e_+^2}{4 \pi}&& \left(\frac{1-e_-/e_+}{m \omega_+} \right)^2 \ln \frac{\Gamma}{\omega_0} \omega_0^2 \cos^2(\omega_+ V + \theta) e^{-2(\epsilon_++a) V} \\ &&=\frac{e_+^2}{16 \pi}\left(\frac{1-e_-/e_+}{m \omega_+} \right)^2 \ln \frac{ \Gamma}{\omega_0} \left[ 2 \omega_0^2 + \beta_+^2 (av)^{2i \frac{\omega_+}{ a}} + \beta_+^{*2} (av)^{-2i \frac{\omega_+}{a}} \right] (av)^{-2(1+ \epsilon/a)} \nonumber\end{aligned}$$ There are thermal contributions in $T^A_3$ but we can argue that it does not give comparable contribution to the UV term. The general form of the integral of the thermal part is $$\begin{aligned} \int d \lambda \frac{2\lambda^2}{e^{\lambda/a}-1} \partial _V G(\lambda, V) \partial_V G^*(\lambda, V).\end{aligned}$$ As one can easily see, there is no UV divergence because of the thermal factor in the denominator. Moreover there is no IR contributions which comes from $\lambda \sim 0$. Therefore it do not give terms depends on the cutoff $\Gamma$ which is the main contribution of the $T^A_{3UV}$. In case of $T^A_2$ the situation is much different to $T^A_3$ because there are no UV contributions and the main contribution of it is only for small $V$. So we must calculate it exactly in the region $\frac{1}{\Gamma} < V \ll 1/\omega_0, 2/a$. $T_2^A$ is sum of two terms which are mutually complex conjugate. One of these is $$\begin{aligned} &&\int d \lambda \lambda \coth (2\pi \lambda/a) \partial_V G(\lambda, V) \partial_V \xi_{\lambda}^*(V) e^{-2a V} \\ &&= \int d \lambda \lambda \coth (2\pi \lambda/a) \partial_V G_\infty(\lambda, V) \partial_V \xi_{\lambda}^*(V) e^{-2aV} + \frac{i e_+}{8 \pi} e^{-(\epsilon_+ + 2 a)v} T^A_{2a}, \nonumber\end{aligned}$$ where $$\begin{aligned} \label{T^A_{2a}} T^A_{2a} &=& -\beta_+ e^{i\omega_+V} \int d \lambda \lambda \coth{2 \pi \lambda/a }\chi_+^*(-\lambda) e^{i\lambda V} \\ &+& \beta_+^* e^{-i \omega_+V} \int d\lambda \lambda \coth{2 \pi \lambda /a} \chi_+(\lambda) e^{i \lambda V} . \nonumber\end{aligned}$$ Therefore we must calculate $T^A_{2a}- T^{A*}_{2a}$. After change of variable and using the fact $\lambda \coth{2 \pi \lambda/a} $ is even function on $\lambda$, we get $$\begin{aligned} T^A_{2a} &-& T^{A*}_{2a} = -\beta_+ e^{-i \omega_+ V} \int_{-\infty}^{\infty} d \lambda \left[ \lambda \coth{2 \pi \lambda/a} \right] \xi_+^*(-\lambda)e^{i \lambda V} \\ &-& C.C. \nonumber\end{aligned}$$ Now we use $$\begin{aligned} \coth \pi x = \frac{1}{\pi x} + \frac{2 x}{\pi}\sum_{k=1}^{\infty} \frac{1}{x^2+ k^2}\end{aligned}$$ and do the residue integral along the upper half plane of the $\lambda$ plane, then we get $$\begin{aligned} T^A_{2a} &-& T^{A*}_{2a} = ia \beta_+ e^{-i \omega_+ V} \left[ S(-2i \beta_+) - \frac{e_-}{2 e_+} \left\{ \left(1+ \frac{\omega_+}{\omega_-} \right) S(-2i \beta_-) + \left(1- \frac{\omega_+}{\omega_-}\right) S(2i \beta_-^*) \right\} \right] \nonumber \\ &-& C.C\end{aligned}$$ where $$\begin{aligned} S(\beta)= \sum_{k=1}^{\infty} \frac{ak e^{-akV/2}}{ak + \beta}.\end{aligned}$$ If we restrict $V$ to $V \ll 2/a, 1/\omega_0$, we get $$\begin{aligned} S(\beta) \cong \frac{2}{a V} + \frac{\beta}{a} \ln\left(\frac{V}{\omega_0}\right).\end{aligned}$$ The second term is much smaller than the first. Therefore we can write $$\begin{aligned} T^A_2 &=& - \frac{e_+}{4 \pi} \left(1- \frac{e_-}{e_+} \right) e^{-(\epsilon_+ + 2a)V } \frac{\omega_0}{V} \cos(\omega_+ V+ \theta) \\ &=& - \frac{e_+}{4 \pi m \omega_+} \left( 1- \frac{e_-}{e_+} \right) \frac{ a}{ \ln av} \left[ \beta_+ (av)^{i \frac{ \omega_+}{a}} + \beta_+^* ( av)^{-i \frac{\omega_+}{a}} \right] (av)^{-(2+ \epsilon_+/a)}. \nonumber\end{aligned}$$ [8]{} N. D. Birrell and P. C. W. Davis, [*Quantum Fields in Curved Space*]{} (Cambridge Univ. Press, Cambridge, 1984). W. G. Unruh, Phys. Rev. D [**14**]{}, 870, (1976). B. S. DeWitt, in General Relativity, eds. S. W. Hawking and W. Israel (Cambridge: Cambridge University Press) (1979). J. R. Letaw, Phys. Rev. D [**23**]{} 1709 (1981). J. Phys. A: Math. Gen. [**16**]{} 3905 (1983). K. Hinton, P. C. W. Davis, and J. Pfautsch, Phys. Lett. B [**120**]{} 88 (1983). S. Takagi, Phys. Lett. B [**148**]{} 116 (1984). S. Takagi, Prog. Theor. Phys. 88 (1986). W. G. Unruh, Phys. Rev. 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Morse and H. Feshbach, Method of Theoretical Physics Part I 1659 (1953). N. N. Lebedev and R. A. Silverman, Special functions and their applications, Dover Publications, INC. 32 (1972). Hyeong-Chan Kim, Min-Ho Lee, J. Y. Ji, and Jae Kwan Kim, Phys. Rev. A [**53**]{}, 3767 (1996). ; J. Y. Ji, Jae Kwan Kim, and Sang Pyo Kim, Phys. Rev. A [**51**]{}, 4268, (1995); H. R. Lewis. Jr., and W. B. Riesenfeld, J. of Math. Phys. [**10**]{}, 1458, (1969). [^1]: me@taegeug.skku.ac.kr [^2]: jkkim@chep5.kaist.ac.kr

--- abstract: 'The time evolution of the local field in [*symmetric*]{} $Q$-Ising neural networks is studied for arbitrary $Q$. In particular, the structure of the noise and the appearance of gaps in the probability distribution are discussed. Results are presented for several values of $Q$ and compared with numerical simulations.' author: - | D. Bollé [^1] [^2]\ Instituut voor Theoretische Fysica, K.U. Leuven,\ B-3001 Leuven, Belgium\ \ and G. M. Shim [^3]\ Department of Physics, Chungnam National University\ Yuseong, Taejon 305-764, R.O. Korea title: 'Local field dynamics in symmetric $Q$-Ising neural networks' --- Symmetric networks; $Q$-Ising neurons; parallel dynamics; local field; probabilistic approach Introduction ============ In a number of papers in the nineties (cfr. [@PZ]-[@BJS99] and references therein) the parallel dynamics of $Q$-Ising type neural networks has been discussed for several architectures –extremely diluted, layered feedforward, recurrent– using a probabilistic approach. For the asymmetric extremely diluted and layered architectures the dynamics can be solved exactly and it is known that the local field only contains Gaussian noise. For networks with symmetric connections, however, things are quite different. Even for extremely diluted versions of these systems feedback correlations become essential from the second time step onwards, complicating the dynamics in a nontrivial way. A complete solution for the parallel dynamics of symmetric $Q$-Ising networks at zero-temperature taking into account all feedback correlations, has been obtained only recently using a probabilistic signal-to-noise ratio analysis [@BJSF]-[@BJS99]. Thereby it is seen that both for the fully connected and the extremely diluted symmetric architectures, the local field contains a discrete and a normally distributed noise part. The difference between the two architectures is that for the diluted model the discrete part at a certain time $t$ does not involve the spins at all previous times $t-1, t-2, \ldots$ up to $0$ but only the spins at time step $t-1$. Even so, this discrete part prevents a closed-form solution of the dynamics but a recursive scheme can be developed in order to calculate the complete time evolution of the order parameters, i.e., the retrieval overlap and the activity. In the work above the focus has been on the non-equilibrium behavior of the order parameters of the network. But, since the local field itself is a basic ingredient in the development of the relevant recursive scheme it is interesting to study also the non-equilibrium behavior of the local field distribution. The more so since this distribution does not convergence to a simple sum of Gaussians as is frequently thought, but it develops a gap structure. This is precisely one of the points studied in detail in the present communication. Moreover, the analogies and differences between the fully connected architecture and the symmetrically diluted one are highlighted. Finally, numerical simulations are presented confirming the analytic study and giving additional insight in the structure of these local field distributions. The model ========= Consider a neural network $\Lambda$ consisting of $N$ neurons which can take values $\sigma_i$ from a discrete set $ {\cal S} = \lbrace -1 = s_1 < s_2 < \ldots < s_Q = +1 \rbrace $. The $p$ patterns to be stored in this network are supposed to be a collection of independent and identically distributed random variables (i.i.d.r.v.), $\{{\xi}_i^\mu \in {\cal S}\}$, $\mu \in {\cal P}=\{1,\ldots,p\}$ and $i \in \Lambda$, with zero mean, $E[\xi_i^\mu]=0$, and variance $A=\Var[\xi_i^\mu]$. The latter is a measure for the activity of the patterns. Given the configuration ${\bsigma}_\Lambda(t)\equiv\{\sigma_j(t)\}, j\in\Lambda=\{1,\ldots,N\}$, the local field in neuron $i$ equals $$\label{eq:h} h_i({\bsigma}_{\Lambda}(t))= \sum_{j\in\Lambda} J_{ij}(t)\sigma_j(t)$$ with $J_{ij}$ the synaptic coupling from neuron $j$ to neuron $i$. In the sequel we write the shorthand notation $h_{\Lambda,i}(t) \equiv h_i({\bsigma}_{\Lambda}(t))$. For the extremely diluted symmetric (SED) and the fully connected (FC) architectures the couplings are given by the Hebb rule $$\begin{aligned} J_{ij}^{SED}&=&\frac{c_{ij}}{CA} \sum_{\mu \in {\cal P}} \xi_i^\mu \xi_j^\mu \quad \mbox{for} \quad i \not=j \,, \quad J_{ii}^{SED}=0 \, , \label{eq:JED} \\ J_{ij}^{FC}&=&\frac{1}{NA} \sum_{\mu \in {\cal P}} \xi_i^\mu \xi_j^\mu \quad \mbox{for} \quad i \not=j \,, \quad J_{ii}^{FC}=0 \, , \label{eq:JFC} \end{aligned}$$ with the $\{c_{ij}=0,1\}, i,j \in \Lambda$ chosen to be i.i.d.r.v. with distribution $\mbox{Pr}\{c_{ij}=x\}=(1-C/N)\delta_{x,0} + (C/N) \delta_{x,1}$ and satisfying $c_{ij}=c_{ji} $. For the diluted symmetric model the architecture is a local Cayley-tree but, in contrast with the diluted asymmetric model, it is no longer directed such that it causes a feedback from $t \geq 2$ onwards. In the limit $N \rightarrow \infty$ the probability that the number of connections $T_i=\{j\in \Lambda |c_{ij}=1\}$ giving information to the site $i \in \Lambda$, is still a Poisson distribution with mean $C=E[|T_i|]$. Thereby it is assumed that $ C \ll \log N$ and in order to get an infinite average connectivity allowing to store infinitely many patterns one also takes the limit $C \rightarrow \infty$ [@BJS99]. At zero temperature all neurons are updated in parallel according to the rule $$\begin{aligned} \label{eq:gain} \sigma_i(t+1) & = & \mbox{g}_b(h_{\Lambda,i}(t)) \nonumber \\ \mbox{g}_b(x) &\equiv& \sum_{k=1}^Qs_k \left[\theta\left[b(s_{k+1}+s_k)-x\right]- \theta\left[b(s_k+s_{k-1})-x\right] \right]\end{aligned}$$ with $s_0\equiv -\infty$ and $s_{Q+1}\equiv +\infty$. Here $\mbox{g}_b(\cdot)$ is the gain function and $b>0$ is the gain parameter of the system. For finite $Q$, this gain function is a step function. The gain parameter $b$ controls the average slope of $\mbox{g}_b(\cdot)$. Local field dynamics ==================== In order to measure the retrieval quality of the system one can use the Hamming distance between a stored pattern and the microscopic state of the network $$d({\bxi}^\mu,{\bsigma}_\Lambda(t))\equiv \frac{1}{N} \sum_{i\in \Lambda}[\xi_i^\mu-\sigma_i(t)]^2 \,.$$ This introduces the main overlap and the arithmetic mean of the neuron activities $$\label{eq:mdef} m_\Lambda^\mu(t)=\frac{1}{NA} \sum_{i\in\Lambda}\xi_i^\mu\sigma_i(t), \quad \mu \in {\cal P}\, ; \quad a_\Lambda(t)=\frac{1}{N}\sum_{i\in\Lambda}[\sigma_i(t)]^2 \,.$$ The key question is then how these quantities evolve in time under the parallel dynamics specified before. For a general time step we find from eq. (\[eq:gain\]) using the law of large numbers (LLN) that in the thermodynamic limit $$\begin{aligned} m^1(t+1) \ustr{Pr}{=} \frac{1}{A} \langle\!\langle \xi_i^1\mbox{g}_b(h_i(t)) \rangle\!\rangle , \quad a(t+1) \ustr{Pr}{=} \langle\!\langle \mbox{g}_b^2(h_i(t)) \rangle\!\rangle \, , \label{eq:a}\end{aligned}$$ where the convergence is in probability [@SH]. In the above $\langle\!\langle \cdot \rangle\!\rangle$ denotes the average both over the distribution of the embedded patterns $\{\xi_i^\mu\}$ and the initial configurations $\{\sigma_i(0)\}$. The average over the latter is hidden in an average over the local field through the updating rule (\[eq:gain\]). Some remarks are in order. For the symmetric diluted model the sum over the sites $i$ is restricted to $T_j$, the part of the tree connected to neuron $j$. Moreover, for that model the thermodynamic limit contains the limit $C \rightarrow \infty$ besides the $N \rightarrow \infty$ limit. In this thermodynamic limit $C, N \rightarrow \infty$ all averages have to be taken over the treelike structure, viz. $\frac{1}{N}\sum_{i \in \Lambda} \rightarrow \frac{1}{C} \sum_{i \in T_j}$, and the capacity defined by $\alpha =p/N$ has to be replaced by $\alpha =p/C$. In (\[eq:a\]) the local field is the main ingredient. Suppose that the initial configuration of the network $\{\sigma_i(0)\},{i\in\Lambda}$, is a collection of i.i.d.r.v. with mean $\E[\sigma_i(0)]=0$, variance $\Var[\sigma_i(0)]=a_0$, and correlated with only one stored pattern, say the first one $\{\xi^1_i\}$: $$\label{eq:init1} \E[\xi_i^\mu\sigma_j(0)]=\delta_{i,j}\delta_{\mu,1}m^1_0 A$$ with $m^1_0>0$. By the LLN one gets for the main overlap and the activity at $t=0$ $$\begin{aligned} m^1(0)&\equiv&\lim_{(C),N \rightarrow \infty} m^1_\Lambda(0) \ustr{Pr}{=}\frac1A \E[\xi^1_i \sigma_i(0)] = m^1_0 \label{eq:mo} \\ a(0)&\equiv&\lim_{(C),N \rightarrow \infty} a_\Lambda (0) \ustr{Pr}{=} \E[\sigma_i^2(0)]=a_0 \label{eq:a0}\end{aligned}$$ where the notation should be clear. In order to obtain the configuration at $t=1$ we have to calculate the local field (\[eq:h\]) at $t=0$. To do this we employ the probabilistic signal-to-noise ratio analysis ([@PZ]-[@BJS99]). Recalling the learning rule (\[eq:JFC\]) we separate the part containing the signal from the part containing the noise. In the limit $N \rightarrow \infty$ we then arrive at $$h_i(0) \equiv \lim_{N \rightarrow \infty} h_{{\Lambda},i}(0) \stackrel{{\cal D}}{=} \xi_i^1 m^1(0) + {\cal N}(0,\alpha a(0)) \label{eq:F16}$$ where the convergence is in distribution [@SH] and with ${\cal N}(0,V)$ representing a Gaussian random variable with mean $0$ and variance $V$. We note that this structure of the distribution of the local field at time zero – signal plus Gaussian noise – is typical for all architectures treated in the literature. For a general time step $t+1$, a tedious study reveals that the distribution of the local field is given by [@BJSF], [@BJS99] $$h_i(t+1)=\xi_i^1m^1(t+1) + {\cal N}(0,\alpha a(t+1)) + \chi(t) [F(h_i(t)-\xi_i^1m^1(t))+\alpha\sigma_i(t)] \label{eq:hrec}$$ where $F=1$ for the fully connected architecture and $F=0$ for the symmetrically diluted one. So, the local field at time $t$ consists out of a discrete part and a normally distributed part, viz. $$h_i(t)=M_i(t) + {\cal N}(0, V(t))$$ where $M_i(t)$ and $V(t)$ satisfy the recursion relations $$\begin{aligned} && M_i(t+1)=\chi(t) [F(M_i(t)-\xi_i^1m^1(t))+\alpha\sigma_i(t)] + \xi_i^1m^1(t+1) \label{eq:Mrec} \\ \label{eq:Drec} && V(t+1)= \alpha a(t+1)A+F\chi^2(t)V(t)+ 2 F \alpha A \chi(t) {Cov}[\tilde r^\mu(t),r^\mu(t)] \,. \end{aligned}$$ The quantity $\chi (t)$ reads $$\chi(t) = \sum_{k=1}^{Q-1} f_{ h_i^\mu (t)}(b(s_{k+1}+s_k)) (s_{k+1}-s_k) \label{eq:chi}$$ where $f_{ h_i^\mu (t)}$ is the probability density of $ h_i^\mu (t) $ in the thermodynamic limit. Furthermore, $r^\mu(t)$ is defined as $$r^\mu(t) \equiv \lim_{N \rightarrow \infty} \frac1{A\sqrt{N}}\sum_{i\in \Lambda} \xi_i^\mu \sigma_i(t), \quad \mu \in {\cal P}\setminus\{1\} \, , \label{eq:w}$$ and $\tilde r^\mu(t)$ is given by a similar expression with $\sigma_i(t)$ replaced by $\mbox{g}_b(h_{\Lambda,i}(t) - \frac{1}{\sqrt{N}}\xi_i^\mu r_\Lambda^\mu(t) )$. Finally, as can be read off from eq. (\[eq:Mrec\]) the quantity $M_i(t)$ consists out of a signal term and a discrete noise term, viz. $$M_i(t)=\xi _i^1 m^1(t) + \alpha \chi(t-1)\sigma _i(t-1) + F\sum_{t'=0}^{t-2} \alpha \left[\prod_{s=t'}^{t-1} \chi(s)\right] \, \sigma _i(t') \,. \label{eq:MM}$$ Since different architectures contain different correlations not all terms in these final equations are present, as is apparent through $F$. We remark that for the asymmetric diluted and the layered feedforward architecture $M_i(t)=\xi _i^1 m^1(t)$ so that in these cases the local field consists out of a signal term plus Gaussian noise for [*all*]{} time steps [@BSVZ],[@BSV]. For the architectures treated here we still have to determine the probability density $f_{h_i(t)}$ in eq. (\[eq:chi\]). This can be done by looking at the form of $M_i(t)$ given by eq. (\[eq:MM\]). The evolution equation tells us that $\sigma _i(t')$ can be replaced by $g_b(h_i(t'-1))$ such that the second and third terms of $M_i(t)$ are the sums of stepfunctions of correlated variables. These are also correlated through the dynamics with the normally distributed part of $h_i(t)$. Therefore, the local field can be considered as a transformation of a set of correlated normally distributed variables $x_s$, which we choose to normalize. Defining the correlation matrix $W = \left(\rho(s,s')\equiv \E[x_s x_{s'}] \right)$ we arrive at the following expression for $f_{h_i(t)}$ for the fully connected model $$\begin{aligned} f_{h_i(t)}(y)&=&\int dx_t \,\prod_{s=0}^{t-2} dx_s ~ \delta \left(y - M_i(t)-\sqrt{V(t)}\,x_t\right) \nonumber\\ &\times& \frac{1}{\sqrt{\mbox{det}(2\pi W)}} ~\mbox{exp}\left(-\frac{1}{2}{\bf x} W^{-1} {\bf x}^T\right) \label{eq:fhdisfc}\end{aligned}$$ with ${\bf x}=\{x_s\}=(x_0,\ldots x_{t-2},x_t)$. For the symmetric diluted case this expression simplifies to $$\begin{aligned} f_{h_i(t)}(y)&=&\int\prod_{s=0}^{[t/2]} dx_{t-2s} ~ \delta \left(y -\xi^1_i m^1(t)- \alpha \chi(t-1)\sigma_i(t-1) -\sqrt{\alpha a(t)}\,x_t\right) \nonumber\\ &\times& \frac{1}{\sqrt{\mbox{det}(2\pi W)}} ~\mbox{exp}\left(-\frac{1}{2}{\bf x} W^{-1} {\bf x}^T \right) \label{eq:fhdisd}\end{aligned}$$ with ${\bf x}=(\{x_s\})=(x_{t-2[t/2]},\ldots x_{t-2},x_t)$. The brackets $[t/2]$ denote the integer part of $t/2$. Gap structure ============= The equilibrium distribution of the local field can be obtained by eliminating the time dependence in the evolution equations (\[eq:hrec\]) $$\label{eq:hfix} h_i=\xi_i^1m^1 + \eta{\cal N}(0,\alpha a) +\alpha \chi \eta \sigma_i$$ with $\eta= 1/(1-\chi)$ for the fully connected architecture and $\eta=1$ for the extremely diluted one. The corresponding updating rule (\[eq:gain\]) $$\sigma_i = g_b(\tilde {h_i} + \alpha \chi \eta \sigma_i) \, , \quad \tilde{h_i} = \xi_i^1 m_i^1 + \eta{\cal N}(0,\alpha a) \label{eq: res1}$$ in general admits more than one solution. A Maxwell construction (see, e.g., refs. [@BJSF],[@BJS99],[@SF]) can be made leading to a unique solution $$\sigma_i = g_{\tilde{b}}(\tilde{h_i})\, , \quad \tilde{b}= (b - \frac{\alpha \eta \chi}{2}) \label{eq: res2}$$ such that we have $$\sigma_i = s_k \quad \mbox{if} \quad \tilde{b}(s_k+s_{k-1})+\alpha \chi\eta s_k < h_i < \tilde{b}(s_k+s_{k+1}) +\alpha \chi \eta s_k \, . \label{eq: res3}$$ for ${\tilde b} > 0 $. This unique solution can be used to obtain fixed-point equations for the main overlap and activity (\[eq:a\]). Those equations which we choose not to write down explicitly here (see refs. [@BJSF],[@BJS99]) are equal to the equations derived from a thermodynamic replica-symmetric mean-field theory approach [@BRS],[@BCS]. We remark that for analog networks ($Q \to \infty$) such a Maxwell construction is not necessary because eq. (\[eq: res1\]) has only one solution. Next, we calculate the probability density of the local field by plugging this result (\[eq: res1\])-(\[eq: res3\]) into (\[eq:hfix\]) to obtain, forgetting about the site index $i$ and the pattern index $1$ $$\begin{aligned} f(h) &=& \sum_{k=1}^Q \frac{1}{\eta\sqrt{2\pi \alpha a}} \exp\biggl( -\frac{(h-\xi m -\alpha\chi\eta s_k)^2}{2\alpha a \eta^2} \biggr) \nonumber \\ &\times& \biggl( \theta[ \tilde{b}(s_k+s_{k+1})+\alpha \chi\eta s_k -h] -\theta[ \tilde{b}(s_k+s_{k-1})+\alpha \chi\eta s_k -h] \biggr) \label{eq:distri}\end{aligned}$$ meaning that (Q-1) gaps occur respectively at $ \tilde{b}(s_k+s_{k-1})+ \alpha\chi\eta s_{k-1} < h < \tilde{b}(s_k+s_{k+1}) +\alpha\chi\eta s_k $ with width $\Delta h= 2\alpha\chi\eta/(Q-1)$. For analog networks no gaps occur. When ${\tilde b} \leq 0$ the effective gain function (\[eq: res2\]) becomes two-state Ising-like as in the Hopfield model such that case only one gap occurs. For $Q=2$ this expression simplifies to $$\begin{aligned} f(h) &=& \frac{1}{\eta \sqrt{2\pi \alpha a}} \exp\biggl( -\frac{(h-\xi m -\alpha\chi\eta )^2}{2\alpha a \eta^2} \biggr) \theta(h-\alpha\chi\eta) \nonumber \\ &+& \frac{1}{\eta\sqrt{2\pi \alpha a}} \exp\biggl( -\frac{(h-\xi m +\alpha\chi\eta )^2}{2\alpha a \eta^2} \biggr) \theta(-h-\alpha\chi\eta) \end{aligned}$$ and for Q=3 we have $$\begin{aligned} f(h) &=& \frac{1}{\eta\sqrt{2\pi \alpha a}} \exp\biggl( -\frac{(h-\xi m -\alpha\chi\eta )^2}{2\alpha a \eta^2} \biggr) \theta(h-\tilde{b}-\alpha\chi\eta) \nonumber \\ &+& \frac{1}{\eta\sqrt{2\pi \alpha a}} \exp\biggl( -\frac{(h-\xi m )^2}{2\alpha a \eta^2} \biggr) \theta(\tilde{b}^2-h^2) \nonumber \\ &+& \frac{1}{\eta\sqrt{2\pi \alpha a}} \exp\biggl( -\frac{(h-\xi m +\alpha\chi\eta )^2}{2\alpha a \eta^2} \biggr) \theta(-\tilde{b}-\alpha\chi\eta-h) \, .\end{aligned}$$ Similar formula can be written down for bigger values of $Q$. For $Q=2$ this result seems to be consistent with the gap in the internal-field distribution for an infinite range spin glass found by a Bethe-Peierls-Weiss approach [@SK] (see also [@ZC]-[@CS]). We have investigated this probability distribution numerically using the corresponding fixed-point equations mentioned before, for several values of $Q$ and compared them with those obtained from numerical simulations of the dynamics for networks of $N=6000$ neurons. Some typical results are shown in figs. 1-6. In figs. 1-2 the local field distribution for the fully connected $Q=2$ network is shown for a retrieval state ($\alpha=0.13,m_0=0.5$) just below the critical capacity and a non-retrieval spin-glass state ($\alpha=0.14, m_0=0.2$) just above it. Both the first few time steps and the equilibrium result derived above are compared with numerical simulations. They are in agreement. For the retrieval state there is, typically, a small gap in the equilibrium distribution around h=0. For small $\alpha$ the gap is very narrow. Furthermore, in the simulations one sees that this gap shows up very quickly. For the non-retrieval state the gap is typically much bigger. Again in the simulations one quickly sees the gap but it is extremely difficult numerically to find points touching the zero axis because of finite size effects. Figure 3 shows the gap width at equilibrium, $\Delta h$, for the non-retrieval state as a function of $Q$ with $b=0.5$. It scales as $\Delta h \sim 1/(Q-1)$ and, hence, decreases to zero for $Q \to \infty$. This constant behaviour of $(Q-1) \Delta h$ attains already for values of $Q \geq 20$ and is also seen for the retrieval state. These results are insensitive to the structure of the symmetric architecture. In figure 4 the gap boundaries in $h$ as a function of $\alpha$ are compared for retrieval and non-retrieval states in the symmetric diluted $Q=3, b=0.2$ model. We remark that in this case the spin-glass states do not exist for $\alpha \leq 0.04$ [@BCS] so that there is no gap for these $\alpha$-values. For $\alpha$ large enough ($\alpha > 0.465$ for retrieval states and $\alpha > 0.252$ for spin-glass states) there exists one gap only since the effective gain function becomes Ising-like [@BCS]. More gaps with smaller widths are formed when increasing $Q$ for both the fully connected and diluted models. For $Q \to \infty$ the gaps disappear. Figure 5 compares the gaps for the spin-glass states in the fully connected and symmetric diluted $Q=3$ models with $b=0.5$. For $\alpha \leq 0.25 $ there exist no spin-glass states in the diluted model [@BCS] and for $\alpha \leq 0.004 $ there are none in the fully connected model [@BRS]. When both do exist the gap widths are almost equal. So the dilution has some influence on the existence of the gap but, again, not on its width. Finally, fig. 6 presents the local field distribution for the symmetric diluted $Q=3, b=0.5$ model for a retrieval state ($\alpha =0.6, m_0=0.7$) just below the critical capacity. Only the distribution with pattern values $+1$ is shown. It is asymmetric and two gaps are found at equilibrium. For pattern values $0$ the distribution is symmetric and the gap locations and widths are the same (see eq. (\[eq:distri\])) but their height is different. In conclusion, we have studied the time evolution of the local field in symmetric $Q$-Ising neural networks both in the retrieval and spin-glass regime. We have found a gap structure in the local field distribution depending on the specific architecture and on the value of $Q$. The results agree with the numerical simulations we have performed. Acknowledgments {#acknowledgments .unnumbered} =============== This work has been supported in part by the Fund of Scientific Research, Flanders-Belgium and the Korea Science and Engineering Foundation through the SRC program. The authors are indebted to A. Coolen, G. Jongen and V. Zagrebnov for constructive discussions. [99]{} A.E. Patrick and V.A. Zagrebnov, Parallel dynamics for an extremely diluted neural network, : L1323 (1990); : 1009 (1992). A.E. Patrick and V.A. Zagrebnov, On the parallel dynamics for the Little-Hopfield model, : 59 (1991). T.L.H. Watkin and D. Sherrington, The parallel dynamics of a dilute symmetric neural network, : 5427 (1991). A.E. Patrick and V.A. Zagrebnov, A probabilistic approach to parallel dynamics for the Little-Hopfield model, : 3413 (1991). D. Bollé, B. Vinck, and V.A. Zagrebnov, On the parallel dynamics of the $Q$-state Potts and $Q$-Ising neural networks, : 1099 (1993). 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D. Bollé, H. Rieger and G.M. Shim, Thermodynamic properties of fully connected $Q$-Ising neural networks, : 3411 (1994). D. Bollé, D. Carlucci and G.M. Shim, Thermodynamic properties of extremely diluted $Q$-Ising neural networks, : 6481 (2000). L.J. Schowalter and M.W. Klein, Analytic treatment of the hole in the internal field distribution for an infinite-range spin glass, [*J.Phys.C: Solid State Physics*]{} [**12**]{}: L935 (1979). V.A. Zagrebnov and A.S. Chvyrov, The Little-Hopfield model: recurrence relations for retrieval-pattern errors, [*Sov.Phys.JETP*]{} [**68**]{}: 153 (1989) A.C.C. Coolen and D. Sherrington, Order parameter flow in the fully connected Hopfield model near saturation, : 1921 (1994). [^1]: e-mail: desire.bolle@fys.kuleuven.ac.be. [^2]: Also at Interdisciplinair Centrum voor Neurale Netwerken, K.U.Leuven, Belgium. [^3]: e-mail: gmshim@cnu.ac.kr.

--- abstract: 'This is a brief survey of the research performed by Grandata Labs in collaboration with numerous academic groups around the world on the topic of human mobility. A driving theme in these projects is to use and improve Data Science techniques to understand mobility, as it can be observed through the lens of mobile phone datasets. We describe applications of mobility analyses for urban planning, prediction of data traffic usage, building delay tolerant networks, generating epidemiologic risk maps and measuring the predictability of human mobility.' author: - | Carlos Sarraute[^1,\*^]{} and Martin Minnoni[^1^]{}\ [[^1^]{}Grandata Labs, San Francisco, CA, USA]{}\ [[^^]{}Corresponding author: charles@grandata.com]{}\ bibliography: - '../GD\_works.bib' title: Brief survey of Mobility Analyses based on Mobile Phone Datasets --- Introduction ============ The mission of Grandata’s research lab is to improve our understanding of Human Dynamics through the analysis of massive datasets coming from mobile phone companies and other industries. This research has been performed in collaboration with numerous academic groups at MIT, INRIA and ENS Lyon, LNCC, UBA and many others. We provide here a brief review of our research, intended to serve as an introduction and guideline to the research papers on mobility aspects. This brief survey focuses on the analysis of mobility in space, investigating the important locations of the users’ trajectories, and how they can be used to infer their participation in large social events. The study of mobility has numerous applications, such as urban planning (Section \[urban-planning\]), data traffic usage prediction (Section \[data-traffic-usage\]), building delay tolerant networks (Section \[delay-tolerant-networks\]), epidemiology (Section \[epidemiology\]), and predictability of human mobility (Section \[mobility-predictability\]).

--- abstract: 'We investigate the suppression of the baryon density fluctuations compared to the dark matter in the linear regime. Previous calculations predict that the suppression occurs up to a characteristic mass scale of $\sim 10^6$ M$_\odot$, which suggests that pressure has a central role in determining the properties of the first luminous objects at early times. We show that the expected characteristic mass scale is in fact substantially lower (by a factor of $\sim 3$–10, depending on redshift), and thus the effect of baryonic pressure on the formation of galaxies up to reionization is only moderate. This result is due to the influence on perturbation growth of the high pressure that prevailed in the period from cosmic recombination to $z\sim 200$, when the gas began to cool adiabatically and the pressure then dropped. At $z\sim10$ the suppression of the baryon fluctuations is still sensitive to the history of pressure in this high-redshift era. We calculate the fraction of the cosmic gas that is in minihalos and find that it is substantially higher than would be expected with the previously-estimated characteristic mass. Expanding our investigation to the non-linear regime, we calculate in detail the spherical collapse of high-redshift objects in a $\Lambda$CDM universe. We include the gravitational contributions of the baryons and radiation and the memory of their kinematic coupling before recombination. We use our results to predict a more accurate halo mass function as a function of redshift.' author: - | S. Naoz and R. Barkana$^{1}$ [^1]\ $^{1}$School of Physics and Astronomy, The Raymond and Beverly Sackler Faculty of Exact Sciences,\ Tel Aviv University, Tel Aviv 69978, ISRAEL title: The formation and gas content of high redshift galaxies and minihalos --- \[firstpage\] galaxies:high-redshift – cosmology:theory – galaxies:formation Introduction {#intro} ============ The detection of the cosmic microwave background (CMB) temperature anisotropies [@bennett] confirmed the notion that the present-day galaxies and large-scale structure (LSS) evolved from the primordial inhomogeneities in the density distribution at very early times. After cosmic recombination, the gas decoupled from its mechanical drag on the CMB, and the baryons subsequently began to fall into the pre-existing gravitational potential wells of the dark matter. Regions that were denser than average collapsed and formed bound halos. First the smallest, least massive objects collapsed, and later, larger objects formed through a mixture of mergers and accretion. The formation and properties particularly of early galaxies at high redshift are being actively studied in anticipation of many expected observational probes [e.g., @BL01; @R06]. A well-known solution for the collapse of a halo that consists of dark matter *only* in an Einstein de Sitter (EdS) universe was presented by @gg. This solution considers a spherical region initially with a small uniform overdensity compared to the background universe. As the universe expands, the overdensity expands slower than the background until it reaches a maximum radius, turns around, and collapses. The critical overdensity, in the corresponding linearly-extrapolated calculation, marks the collapse time of a dark matter halo in this case as $\delta_c=1.686$, a value that does not depend on the halo mass or collapse redshift. The mathematical solution gives a singularity as the final state, but physically we know that even a small initial asymmetry will make the object stabilize with a finite size after reaching a virial equilibrium between motion and gravity. Extensive work has been done on spherical collapse models, especially models that include a cosmological constant or a dark energy background [e.g., @lahav; @desh; @hoffman; @H; @lahav2; @wang]; in particular, the cosmological constant $\Lambda$ changes the above value of overdensity ($\delta_c$) by about $0.6\%$. In addition, many numerical simulations of the formation of primordial objects at $z\sim 20$–30 have been performed. However, the earliest stars formed at $z \sim 65$ [@NNB], and even for a halo that collapsed at $z\sim30$, $\delta$ must have started significantly non-linear ($\sim 9\%$) even for a simulation that begins as early as $z \sim 600$. With the $\Lambda$CDM cosmological parameters [@Spergel06], the contribution of the photons to the expansion of the universe cannot be neglected when considering the formation of the first objects [@NNB]. Moreover, the baryons have a non-negligible contribution compared to the dark matter, and their different evolution must be included in the collapse process. When considering the formation and properties of the first luminous objects, we must investigate the relation between the baryon and the dark matter fluctuations. @cs defined a fiducial “filtering mass” that describes the highest mass at which the baryonic pressure still manages to suppress the linear baryonic fluctuations significantly. @gnedin00 extended the usefulness of the filtering mass to the fully non-linear regime by showing that it is also related to another characteristic mass scale – the largest halo mass for which the gas content is substantially suppressed compared to the cosmic fraction. As we show below, if we follow previous calculations [@cs; @gnedin00; @gnedin03], we find a characteristic mass at high redshift of $\sim 10^6 M_\odot$, approximately constant at $z\ga 60$ and decreasing only slowly with time afterwards. This is somewhat larger than the mass scale of the first objects and suggests a potent effect on the formation of the first objects. Here we present an improved calculation of the characteristic mass that is mainly based on the improved calculation of the baryon density and temperature fluctuations that we presented in @NB. We first review the basic equations of linear perturbation growth (Section \[sec:scales\]). We then divide the power spectrum into several different ranges of scales that are associated with large-scale structure (Section \[LS\]), the filtering scale (Section \[sec:TmTran\]), and small scales (Section \[small\]). Note that we define the filtering mass with a different normalization than in previous works, as explained in Section \[sec:TmTran\]. For completeness we compare our calculation to the older, inaccurate approximation of a spatially-uniform sound speed along with other approximations (Sections \[sec:CS\] and \[sec:CSMF\]). We use our results for the filtering mass to estimate the gas fraction in minihalos (Section \[sec:gas\]). In Section \[non-linear\] we calculate in detail the critical overdensity for collapse of halos that form at very high redshifts, following the evolution of perturbations outside the horizon (Section \[sec:outH\]) and inside it (Section \[sec:delc\]). We also predict the halo abundance at different redshifts (Section \[mass\_a\]). Finally, we summarize and discuss our results in Section 4. Our calculations are made in a $\Lambda$CDM universe, including dark matter, baryons, radiation, and a cosmological constant. We assume cosmological parameters matching the three year WMAP data together with weak lensing observations [@Spergel06], i.e., $\Omega_m=0.299$, $\Omega_\Lambda=0.74$, $\Omega_b=0.0478$, $h=0.687$, $n=0.953$ and $\sigma_8=0.826$. We also consider the effect of current uncertainties in the values of cosmological parameters on some of our results, by comparing to the results with a different cosmological parameter set specified by @Viel: $\Omega_m=0.253$, $\Omega_\Lambda=0.747$, $\Omega_b=0.0425$, $h=0.723$, $n=0.957$ and $\sigma_8=0.785$. These parameters represent typical 1-$\sigma$ errors, in terms of the parameter uncertainties given by @Spergel06. Linear Growth of Perturbations {#sec:linear} ============================== The Basic Equations {#sec:scales} ------------------- @NB showed that the baryonic sound speed varies spatially, so that the baryon temperature and density fluctuations must be tracked separately. Thus, the evolution of the linear density fluctuations of the dark matter ($\delta_{\mathrm {dm}}$) and the baryons ($\delta_{\mathrm {b}}$) is described by two coupled second-order differential equations: $$\begin{aligned} \label{g_T} \ddot{\delta}_{{\mathrm {dm}}} + 2H \dot {\delta}_{{\mathrm {dm}}} & = & \frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3} \left(f_{{\mathrm {b}}} \delta_{{\mathrm {b}}} + f_{{\mathrm {dm}}} \delta_{{\mathrm {dm}}}\right)\ , \\\ddot{\delta}_{{\mathrm {b}}}+ 2H \dot {\delta}_{{\mathrm {b}}} & = & \frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3} \left(f_{{\mathrm {b}}} \delta_{{\mathrm {b}}} + f_{{\mathrm {dm}}} \delta_{{\mathrm {dm}}}\right)-\frac{k^2}{a^2}\frac{k_B\bar{T}}{\mu} \left(\delta_{{\mathrm {b}}}+\delta_{T}\right)\ ,\nonumber\end{aligned}$$ where $\Omega_m$ is the present matter density as a fraction of the critical density, $k$ is the comoving wavenumber, $a$ is the scale factor, $\mu$ is the mean molecular weight, $H_0$ marks the present value of the Hubble constant $H$, and $\bar{T}$ and $\delta_T$ are the mean baryon temperature and its dimensionless fluctuation, respectively. These equations can be derived by linearizing the continuity, Euler, and Poisson equations. The baryon equation includes a pressure term whose form comes from the equation of state of an ideal gas. The linear evolution of the temperature fluctuations is given by [@BL05; @NB] $$\label{gamma} \frac{d \delta_T} {d t} = \frac{2}{3} \frac{d \delta_{\mathrm {b}}} {dt} + \frac{x_e(t)} {t_\gamma}a^{-4} \left\{ \delta_\gamma\left( \frac{\bar{T}_\gamma}{\bar{T}} -1\right) +\frac{\bar{T}_\gamma} {\bar{T}} \left(\delta_{T_\gamma} -\delta_T \right) \right\}\ ,$$ where $x_e(t)$ is the electron fraction out of the total number density of gas particles at time $t$, $\delta_\gamma$ is the photon density fluctuation, $t_\gamma=8.55 \times 10^{-13} {\mathrm{yr}}^{-1}$, and $T_\gamma$ and $\delta_{T_\gamma}$ are the mean photon temperature and its dimensionless fluctuation, respectively. Equation (\[gamma\]) results from the first law of thermodynamics, where in the post-recombination era before the formation of galaxies, the only external heating arises from Compton scattering of the remaining free electrons with CMB photons. The first term of equation (\[gamma\]) comes from the adiabatic cooling or heating of the gas, while the second term is the result of the Compton interaction. Note that prior analyses [e.g., @Peebles; @Ma] had assumed a spatially uniform speed of sound for the gas, but this assumption is inaccurate. It is often useful to express the evolution in terms of two linear combinations of the dark matter and baryon fluctuations. Defining $\delta_{\mathrm{tot}}=f_{\mathrm {b}}\delta_{\mathrm {b}}+f_{\mathrm {dm}}\delta_{\mathrm {dm}}$ (in terms of the cosmic baryon and dark matter mass fractions $f_{\mathrm {b}}$ and $f_{\mathrm {dm}}$) and $\Delta=\delta_{\mathrm {b}}-\delta_\mathrm{tot}$ [following the derivation in @BL05], and using equations (\[g\_T\]) we can write two differential equations that describe the evolution of $\delta_\mathrm{tot}$ and $\Delta$: $$\begin{aligned} \label{tot_ev_T} \ddot{\delta}_{\mathrm{tot}} + 2H \dot {\delta}_{\mathrm{tot}}& =& \frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3} \delta_\mathrm{tot}-f_{\mathrm {b}}\frac{k^2}{a^2}\frac{k_B\bar{T}}{\mu} \left(\delta_{\mathrm{tot}}+\Delta+\delta_T\right)\ ,\nonumber \\ \ddot{\Delta}+2H\dot{\Delta}&=&-f_{\mathrm {dm}}\frac{k^2}{a^2} \frac{k_B\bar{T}}{\mu}\left(\delta_{\mathrm{tot}}+\Delta+\delta_T\right)\ .\end{aligned}$$ Note that the gravitational force depends directly on $\delta_\mathrm{tot}$, which is the fluctuation in the total matter density, while $\Delta$ describes the difference between the baryon fluctuations and $\delta_\mathrm{tot}$. Before recombination, the baryons are dynamically strongly coupled to the photons while the dark matter fluctuations continue to grow independently. At lower redshifts the dominant contribution to the baryon fluctuation growth is the gravitational attraction to the dark matter gravitational wells (e.g., see Figure 1 in @NB). Large-Scale Structure: Small-$k$ Limit {#LS} --------------------------------------- In the small-$k$ limit, the pressure terms can be neglected and equations (\[tot\_ev\_T\]) can be approximated as [following the derivation in @BL05] $$\begin{aligned} \label{tot_ev2} \ddot{\delta}_{\mathrm{tot}} + 2H \dot {\delta}_{\mathrm{tot}}& =& \frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3}\delta_\mathrm{tot}\ ,\\ \nonumber \ddot{\Delta}+2H\dot{\Delta}&=&0\ .\end{aligned}$$ Each of these equations has two independent solutions. Assuming that the universe is accurately described as EdS before reionization, these solutions are simple. The solutions for the $\delta_\mathrm{tot}$ equation are growing and decaying modes $D_{\mathrm{tot},1}\propto a$ and $D_{\mathrm{tot},2}\propto a^{-3/2}$, while for the $\Delta$ equation we obtain $D_{\Delta,1}\propto 1$ and $D_{\Delta,2}\propto a^{-1/2}$. Moreover, with standard inflationary initial conditions, equations (\[tot\_ev2\]) satisfy $\Delta(k) \propto \delta_{\mathrm{tot}}(k)$ at a given redshift. In other words, the relation between $\Delta$ and $\delta_{\mathrm{tot}}$ is independent of wavenumber, for the range of redshifts and $k$ values that we are considering here. We therefore find it useful to define $$r_{\rm LSS} \equiv \frac{\Delta}{\delta_{\mathrm{tot}}}\ , \label{r_LSS}$$ in terms of the solutions of equations (\[tot\_ev2\]) in the large-scale structure regime. The ratio $r_{\rm LSS}$ (which is negative) is independent of $k$ in this regime, and its magnitude decreases in time approximately $\propto 1/a$, since $\Delta$ is roughly constant and $\delta_{\mathrm{tot}}$ is dominated by the growing mode $\propto a$. Figure \[fig:db\_tot\] (top panel) shows $|r_{\rm LSS}|$ as a function of redshift in the regime of large-scale structure. Although $|r_{\rm LSS}|$ decreases as $z\to 0$, the initial difference between the dark matter and the baryon fluctuations has a large effect on the filtering mass even at $z<10$, as we show below. The small-$k$ regime ($k \ll k_F$) can be seen in Figure \[fig:db\_tot\] (bottom panel), in terms of the filtering wavenumber $k_F$ which is defined precisely in the following subsection. Note that remnants of the acoustic oscillations in the photon-baryon plasma can be seen on the largest scales ($k/k_F \sim 10^{-4}$, which corresponds to comoving $k \sim 0.01$–0.1 Mpc$^{-1}$), but these are much larger scales than those of high-redshift halos and the oscillations do not affect our results. ![The top panel shows $|r_{\rm LSS}|$ (equation (\[r\_LSS\])) versus redshift. The bottom panel shows the baryon to total fluctuation ratio versus $k/k_F$ for various redshifts. We consider, from bottom to top, $z=100$, 65, 20, and 7.[]{data-label="fig:db_tot"}](db_tot_rLSS_final.eps){width="84mm"} A good numerical fit for $r_{\rm LSS}$ in the redshift range of $z=7-150$ is given by $$r_{\rm LSS}=\frac{\alpha_1(\Omega_m)}{a}+\frac{\alpha_2(\Omega_m)}{a^{3/2}}+ \alpha_3(\Omega_m)\ , \label{rLSS_fit}$$ where the form is motivated by the definition of $r_{\rm LSS}$ (equation (\[r\_LSS\])) and the EdS solutions of equations (\[tot\_ev2\]). The fitted coefficients are $$\begin{aligned} \label{a_b_r_LSSfit} \alpha_1(\Omega_m)&=&10^{-4}\times(-1.99\,\Omega_m^2+2.41\, \Omega_m+0.21)\ ,\\ \nonumber \alpha_2(\Omega_m)&=&10^{-3}\times(6.37\,\Omega_m^2-6.99\, \Omega_m-1.76)\ ,\\ \nonumber \alpha_3(\Omega_m)&=&10^{-2}\times(-1.83\,\Omega_m^2+2.4\, \Omega_m-0.54)\ ,\end{aligned}$$ where the maximum residual error of the fit is $0.2 \%$, over the range $\Omega_m=$0.25–0.4 in $\Lambda$CDM. The Filtering Scale {#sec:TmTran} ------------------- In a $\Lambda$CDM universe, virialized CDM halos form on extremely small scales at extremely early times. The minimum mass scale on which galaxies form within these halos is determined through a combination of two physical properties of the infalling gas. These are cooling and pressure, and each produces a characteristic minimum scale, so that the larger of the two scales becomes the dominant factor that determines the minimum mass of star-forming halos. During the gravitational collapse, the gas potential energy is transformed into kinetic and thermal energy through virialization and adiabatic compression. Unless the gas is able to dissipate its thermal energy with an effective cooling mechanism, the temperature will continue to rise, allowing the increasing pressure to halt the collapse. At very high redshifts, before metals were produced in supernovae and efficiently distributed in the intergalactic medium, the only cooling mechanism at temperatures below 10,000 K was cooling by molecular hydrogen, which itself is efficient only above a few hundred K. Thus, the first objects are expected to be fairly massive. Numerical calculations and simulations have shown that the minimum cooling mass is $\sim 10^5 M_\odot$ at $z\sim 100$ and increases with time [@th2; @Abel; @fuller; @Yoshida]. An object more massive than this minimum mass, after it goes through the virialization process, will cool and collapse further, producing high-density clumps in which stars can form. @Haiman showed that the UV flux needed to dissociate [H$_2$ ]{}in a collapsing environment is lower by more than two orders of magnitude than the flux that is necessary to reionize the universe. Therefore, once stars reach a certain abundance, the formation of stars through [H$_2$ ]{}cooling is suppressed, and atomic cooling becomes the only available mechanism. This mechanism requires a much higher virial temperature ($T_{vir}\geq 10^4K$) which is associated with a halo mass of $\sim10^8M_\odot$. Therefore, in order to study the first generations of objects we must consider halo masses of $\sim 10^5-10^8M_\odot$ (see also the review by @R06). On large scales (small wavenumbers) gravity dominates halo formation and pressure can be neglected. On small scales, on the other hand, the pressure dominates, and the baryon density fluctuations are suppressed compared to the dark matter fluctuations. The relative force balance at a given time can be characterized by the Jeans scale, which is the minimum scale on which a small perturbation will grow due to gravity overcoming the pressure gradient. If the gas has a uniform sound speed $c_s$, then the comoving Jeans wavenumber is $$\label{eq:jeans} k_J=\frac{a}{c_s}\sqrt{4\pi G\bar{\rho}}\ .$$ Once the universe is matter-dominated, the Jeans scale is constant in time as long as $T\sim1/a$, i.e., as long as the Compton scattering of the CMB with the residual free electrons after cosmic recombination keeps the gas temperature coupled to that of the CMB. At redshift $z\sim 200$ the gas temperature decouples from the CMB temperature and the Jeans scale decreases with time as the gas cools adiabatically. Any halo more massive than the Jeans mass can begin to collapse despite the pressure gradients. Figure \[fig:kf\] is a reminder that the Jeans mass (dotted curve) is in the range $10^4$– $10^5 M_\odot$ during the formation of the earliest generations of galaxies. The Jeans mass is related only to the evolution of perturbations at a given time. When the Jeans mass itself varies with time, the overall suppression of the growth of perturbations depends on a time-averaged Jeans mass. Following @cs, we define a “filtering” scale and use it to identify the largest scale on which the baryon fluctuations are substantially suppressed compared to those of the dark matter. While @cs assumed a spatially-uniform sound speed and made a number of other approximations (see Section \[sec:CS\]), we calculate here the fitering scale obtained from the exact numerical solution of equations (\[g\_T\]). In particular, our initial conditions at high redshift account for the coupling of the baryons and photons, i.e., $\delta_{\mathrm {b}}\ll \delta_{\mathrm {dm}}$ up to cosmic recombination. We define the filtering wavenumber $k_F$ based on the scale at which the baryon-to-total fluctuation ratio drops substantially below its value on large scales. Thus, we expand this ratio up to linear order in $k^2$, and write the expansion in the following form: $$\frac{\delta_{\mathrm {b}}}{\delta_{\mathrm{tot}}}=1-\frac{k^2}{k_F^2}+r_{\rm LSS}\ , \label{kf_btot}$$ where $r_{\rm LSS}$ was defined in equation (\[r\_LSS\]). Our definition of $k_F$ is a generalization of that by @cs, who did not include the $r_{\rm LSS}$ term. To find $k_F$ in our more general case we first write it in the form: $$k^2_F(t)=\frac{\delta_{\mathrm{tot}}(t)}{u(t)}\ , \label{kf}$$ where $u(t)$ is to be determined. Substituting the expansion into equations (\[g\_T\]) and (\[tot\_ev2\]), we obtain an equation for $u$: $$\ddot{u}+2H\dot{u}=f_{\mathrm {dm}}\frac{1}{a^2}\frac{k_B\bar{T}}{\mu} \left(\delta_{\mathrm{tot}}+r_{\rm LSS}\delta_{\mathrm{tot}}+\delta_T\right)\ , \label{ratio1}$$ where we have neglected terms of higher order in $k^2$. We can solve this equation to find the parameter $u$: $$\begin{aligned} \label{u_eq} \lefteqn{ u(t)=\int^{t}_{t_{\rm rec}}\frac{dt^{\prime\prime}}{a^2(t^{\prime\prime})} \int_{t_{\rm rec}}^{t^{\prime\prime}} dt^{\prime}f_{\mathrm {dm}}\frac{k_B\bar{T}(t^\prime)}{\mu}} \\ \nonumber && \ \ \ \times\left(\delta_{\mathrm{tot}}(t^\prime)+r_{\rm LSS}(t^\prime)\delta_{\mathrm{tot}}(t^\prime)+\delta_T(t^\prime)\right)\ . $$We have started the integral from the time of cosmic recombination ($t_{\rm rec}$), since before recombination the contribution of the baryon density and temperature fluctuations is negligible, i.e., the integrand essentially vanishes (Note that $\delta_{\mathrm{tot}}+r_{\rm LSS}\delta_{\mathrm{tot}}+\delta_T=\delta_{\mathrm {b}}+\delta_T$). As we discuss in Section \[sec:CSMF\], this integral at high redshift (before cosmic reionization) gives a significantly different result from the approximate formula of @cs. Figure \[fig:db\_tot\] (bottom panel) shows the baryon-to-total ratio versus $k/k_F$ at several different redshifts. The different values in the large-scale structure regime for different redshifts are due to the different values of $r_{\rm LSS}$ (see top panel, Figure \[fig:db\_tot\]). The ratio drops on small scales. We have found a functional form that can be used to produce a good fit for the drop with wavenumber: $$\frac{\delta_{\mathrm {b}}}{\delta_{\mathrm{tot}}}=(1+r_{\rm LSS})\left(1+\frac{1}{n} \frac{x}{1+r_{\rm LSS}}\right)^{-n}\ ,$$ where $x=k^2/k^2_F$, and $n$ must be adjusted at each redshift. Defining $$\label{def_y_eta} \eta=\frac{1}{1+r_{\rm LSS}}\, \frac{k^2}{k_F^2}\ ; \quad y=\frac{1}{1+r_{\rm LSS}}\, \frac{\delta_{\mathrm {b}}} {\delta_{\mathrm{tot}}}\ ,$$ we can write the same fit as $$\label{y_eta} y=\left(1+\frac{1}{n}\eta\right)^{-n}\ .$$ To second order, this gives: $y\approx 1-\eta+(n+1)\eta^2/(2n)$. We show in Figure \[fig:y\_eta\] both the first and second-order approximations, as well as the full formula of equation (\[y\_eta\]), compared to the exact numerical results. We choose two redshifts that bracket the interesting range, $z=100$ (bottom panel) for the dark ages and $z=7$ (top panel) for the latest possible beginning of reionization. From equation (\[y\_eta\]) we can see that the first-order approximation is independent of redshift. The figure also shows the exponential approximation, generalized from the suggestion made by @gnedin03 for the post-reionization regime. In the universe prior to reionization, the exponential approximation $y=\exp(-\eta)$ (which is also independent of redshift) becomes highly inaccurate at low redshifts. The figure makes clear that the drop of the fluctuations with $k$ (or with $\eta$) has a functional form that varies with redshift. With equation (\[y\_eta\]) we find that the power index at $z=100$ is $n=23$ (with behavior similar to an exponential which would correspond to $n\to \infty$), while at $z=7$ it is $n=0.5$. The values of $n$ were found by matching the numerical result up to second order in $\eta$. ![$y$ versus $\eta$ for two redshifts, $z=100$ (top panel) and $z=7$ (bottom panel). We compare the exact numerical results (dotted curves) to several approximations: the formula of equation (\[y\_eta\]) (solid curves), the redshift-independent first-order approximation (dot-dashed curves), the second-order approximation (short-dashed curves), and the redshift-independent exponential approximation (long-dashed curves) suggested by @gnedin03 for the post-reionization regime.[]{data-label="fig:y_eta"}](y_eta2.eps){width="84mm"} ![The filtering mass versus redshift. The exact calculation (solid curve), which can be fitted by equations (\[Mf\_fit\]) and (\[a\_b\_fit\]), is compared to a couple versions of the mean sound speed approximation (see Sections \[sec:CS\] and \[sec:CSMF\]): where the integrals in equation (\[u\_cs\]) are begun at time zero as written (long-dashed curve), or at cosmic recombination (short-dashed curve). Also shown is the Jeans mass (dotted curve), which is constant at $z\ga 150$.[]{data-label="fig:kf"}](Mass_filter.eps){width="84mm"} We define the filtering mass in terms of the filtering wavenumber using the traditional convention used to define the Jeans mass: $$M_F=\frac{4\pi}{3}\bar{\rho_0}\left(\frac{1}{2} \frac{2\pi}{k_F}\right)^3\ . \label{Mf}$$ Note that this relation, which we use consistently in this paper, is one eighth of the definition in @gnedin00. The filtering mass essentially describes the largest mass scale on which pressure must be taken into account. Thus, in comparing different scenarios, those with higher gas temperatures will tend to have higher pressures, leading to higher values of the filtering mass. Figure \[fig:kf\] shows the evolution of the filtering mass with redshift. Since the baryon fluctuations are very small before cosmic recombination, the gas pressure (which depends on $\delta_b$) is small compared to gravity (which depends on $\delta_{\mathrm{tot}}$; see equations (\[g\_T\])). Thus, the filtering mass starts from low values and rises with time at $z \sim 100$. At lower redshifts the gas cools and the pressure drops. Therefore, even at $z \sim 10$ the integral in equation (\[u\_eq\]) receives a large contribution from much higher redshifts ($z > 100$). We have found a simple, accurate fit for the evolution of the filtering mass in the redshift range of $z=7-150$. Using the notation $L_M \equiv \log(M_F/M_\odot)$ and $L_z \equiv \log(1+z)$, the fit is of the form $$L_M=\beta_1(\Omega_m)L_z^3+\beta_2(\Omega_m)L_z^2+ \beta_3(\Omega_m)L_z+\beta_4(\Omega_m). \label{Mf_fit}$$ The fitted coefficients are $$\begin{aligned} \label{a_b_fit} \beta_1(\Omega_m)&=&-0.38\, \Omega_m^2+0.41\, \Omega_m-0.16 \ ,\\ \nonumber \beta_2(\Omega_m)&=&3.3\, \Omega_m^2-3.38\, \Omega_m+1.15 \ ,\\ \nonumber \beta_3(\Omega_m)&=&-9.64\, \Omega_m^2+9.75\, \Omega_m-2.37 \ ,\\ \nonumber \beta_4(\Omega_m)&=&9.8\, \Omega_m^2-10.68\, \Omega_m+11.6\ ,\end{aligned}$$ where the maximum residual error of the fit is $0.2\%$, over the range $\Omega_m=$0.25–0.4 in $\Lambda$CDM. Small Scale, Large-$k$ Limit {#small} ---------------------------- In this limit the pressure dominates and makes $\delta_b\ll [\delta_{\mathrm{tot}},\delta_{\mathrm {dm}}]$, and therefore the first equation (\[g\_T\]) becomes: $$\label{largek_x} \ddot{\delta}_{{\mathrm {dm}}} + 2H \dot {\delta}_{{\mathrm {dm}}} \cong f_{\mathrm {dm}}\frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3}\delta_{\mathrm {dm}}\ .$$ Equation (\[largek\_x\]) has a simple analytical solution in the EdS regime: $\delta_{\mathrm {dm}}\propto t^\alpha$, where $\alpha=\left(-1\pm\sqrt{1+24f_{\mathrm {dm}}}\right)/6$. In particular, the growing mode of the dark matter is reduced compared to the usual $\delta_{\mathrm {dm}}\propto t^{2/3}$ solution, since while the baryons contribute to the cosmic expansion rate (in the Friedmann equation) they do not, on small scales, contribute to perturbation growth. Thus, since the dark matter fluctuations on large scales grow by a factor of $\sim 300$ between recombination and $z=10$, then for $f_{\mathrm {dm}}= 0.84$ the total growth of the dark matter fluctuations is reduced on small scales by a factor of $\sim 1.8$ relative to large scales. This linear calculation is limited to redshifts $\ga 10$, since even without reionization, by $z=10$ the filtering scale becomes non-linear. Mean Sound Speed Approximation {#sec:CS} ------------------------------ @NB showed that the presence of spatial fluctuations in the sound speed modifies the calculation of perturbation growth significantly. Nevertheless, for completeness and ease of comparison with previous results we compare the above analysis to earlier, approximate calculations. Thus, we proceed by applying a similar derivation as in the previous sections. In this approximation of a uniform sound speed, however, the evolution of the density fluctuations is described by a different set of coupled second order differential equations: $$\begin{aligned} \label{g_cs} \ddot{\delta}_{{\mathrm {dm}}} + 2H \dot {\delta}_{{\mathrm {dm}}} & = & \frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3} \left(f_{{\mathrm {b}}} \delta_{{\mathrm {b}}} + f_{{\mathrm {dm}}} \delta_{{\mathrm {dm}}}\right)\ , \\ \ddot{\delta}_{{\mathrm {b}}}+ 2H \dot {\delta}_{{\mathrm {b}}} & = & \frac{3}{2}H_0^2\frac{\Omega_{m}}{a^3} \left(f_{{\mathrm {b}}} \delta_{{\mathrm {b}}} + f_{{\mathrm {dm}}} \delta_{{\mathrm {dm}}}\right)-\frac{k^2}{a^2}c^2_s\delta_{{\mathrm {b}}}\ ,\nonumber\end{aligned}$$ where $c_s^2=dp/d\rho$ is assumed to be spatially uniform (i.e., independent of $k$). With this assumption, the temperature fluctuations (as a function of $k$) are simply proportional at any given time to the gas density fluctuations: $$\frac{\delta_T}{\delta_{{\mathrm {b}}}}=\frac{c_s^2}{k_B \bar{T}/\mu}-1\ .$$ This leads to different expressions for the filtering mass. Mean Sound Speed Approximation: The Filtering Scale {#sec:CSMF} --------------------------------------------------- We again expand the ratio of the baryon fluctuations to dark matter fluctuations in powers of $k^2$. In this case we follow the derivation in @cs and thus make several additional assumptions: that the baryon fraction is small $(f_{\mathrm {b}}\ll f_{\mathrm {dm}})$, that the dark matter perturbation growth is dominated by the growing mode ($\delta_{\mathrm {dm}}\propto D_+$), and that there is no initial difference between the baryon and the dark matter fluctuations. Although this derivation includes several different approximations, for simplicity we refer to it as the “mean sound speed approximation”. With these assumptions, $$\frac{\delta_{\mathrm {b}}} {\delta_{\mathrm {dm}}}=1-\frac{k^2}{k_{F,c_s}^2}\ . \label{kf_cs_bdm}$$ Writing this as $$\label{kf_cs} k^2_{F,c_s}(t)=\frac{\delta_{\mathrm {dm}}(t)}{u_{c_s}(t)}\ ,$$ and substituting into equations (\[g\_cs\]), we obtain an equation for the evolution of the parameter $u_{c_s}$ (analogous to equation (\[ratio1\])): $$\ddot{u}_{c_s}+2H\dot{u}_{c_s}=\frac{c_s^2}{a^2}D_+(t)\ . \label{ratio1_cs}$$ The solution (in analogy with equation (\[u\_eq\])) is $$\label{u_cs} u_{c_s}(t)=\int^{t}_0\frac{dt^{\prime\prime}}{a^2(t^{\prime\prime})}\int _0^{t^{\prime\prime}} dt^{\prime}c^2_s(t^\prime)D_+(t^\prime)\ .$$ We note that the lower limit of the integral here is $z\to\infty$, and not recombination as in equation (\[u\_eq\]). Before recombination, the coupling of the baryons with the radiation suppresses the baryon density fluctuations, but this is unaccounted for in this approximate calculation. Indeed, this formula implicitly assumes that the baryon perturbations grow like those of the dark matter, except for the effect of pressure. In particular, the filtering scale in this approximation does not depend on the relative contributions of the baryons and the dark matter to the total matter density. In Figure \[fig:kf\], we show the filtering mass with this approximation from @cs. The correct filtering mass at $z \sim 10$ is substantially lower than predicted by the mean sound speed approximation. As noted before, this is mainly due to the fact that pressure forces depend on gradients in the gas density, and the baryon perturbations are reduced on all scales until they catch up with the dark matter fluctuations (at $z \ll 100$). In the approximate model, on the other hand, the initial difference between the baryon and the dark matter fluctuations is incorrectly neglected. In the Figure, we consider also the mean sound speed approximation where we start the integral in equation (\[u\_cs\]) at recombination ($z=1200$). This should be more realistic than starting at time zero, since the baryon density fluctuation before recombination were negligible. This causes a rise in the filtering mass at $z \sim 100$, as in the correct calculation (i.e., the solid curve). However, the rise at high redshift is still much too fast, since this approximation still assumes that the baryons catch up with the dark matter fluctuations immediately after recombination. To summarize, the Figure shows that the difference between the the correct calculation and the approximate one persists with time. This difference is remembered through the integrated effect of pressure, since at low redshift the Jeans mass is lower and thus the pressure is lower as well. Therefore, the high redshifts contribute most to the integrated pressure, and thus even though the difference between the baryon and the dark matter fluctuations declines with time (e.g., see Figure \[fig:db\_tot\] top panel), the system still remembers the initial difference. Gas Fraction {#sec:gas} ------------ One useful application of the filtering mass is to the estimation of the fraction of gas inside halos. @gnedin00 [his equation (8)] estimated the mean baryonic mass $M_g$ in halos of total mass $M_{\mathrm{tot}}$ using a formula fitted to his post-reionization simulation: $$\bar{M}_g(M_{\mathrm{tot}},t)\approx \frac{f_{\mathrm {b}}M_{\mathrm{tot}}} {\left[1+(2^{1/3}-1)M_c(t)/M_{\mathrm{tot}}\right]^3}\ ,$$ where $M_c(t)$ is a characteristic halo mass that corresponds to a gas fraction of $50\%$ of the cosmic baryon fraction. It is natural to expect a close relation between the characteristic halo mass and the filtering mass, since the gas fraction in a collapsing halo reflects the amount of gas that was able to accumulate in the central, collapsing region, during the entire extended collapse process. In particular, if the Jeans mass changes suddenly, this does not immediately affect then-collapsing halos. A change of pressure immediately begins to affect gas motions (through the pressure-gradient force), but has only a gradual, time-integrated effect on the overall amount of gas in a given region. As mentioned before, our definition of the filtering mass (equation (\[Mf\])) is one eighth of the previous definition of @gnedin00. Thus the characteristic mass that matches the @gnedin00 simulations corresponds to about 8 times our filtering mass. Assuming that this is true also prior to reionization, we can evaluate the total gas fraction in halos as a function of redshift as $$F_g(z)=\int{f_{ST}(M_{\mathrm{tot}},z)\,\frac{\bar{M}_g(M_{\mathrm{tot}},t)}{M_{\mathrm{tot}}}\,dS} \ ,$$ where $S=\sigma^2(M,z)$ is the variance and $f_{ST}$ is the @Sheth function for the fraction of mass associated with halos of mass $M$ (explicitly given in equation (\[sheth\]), below). The total fraction of gas that is in halos is shown in Figure \[fig:gas\_frac\], for our correct calculation of the filtering mass as well as for the previous calculation (which we have referred to as the mean sound speed approximation). We also compare to the fraction of gas in halos above the minimum $H_2$ cooling mass, and to the fraction above the minimum atomic cooling mass. In the correct calculation, a significant part (10–$50\%$) of the total gas in halos arises from halos that are below the characteristic mass. The prediction of the gas fraction in halos in our correct calculation is higher than that based on the previous approximation, by a factor $>2$ at high redshifts and still by $10\%$ at redshift 7. In this redshift range, around half the gas in halos is in potentially star-forming halos (i.e., those with efficient $H_2$ cooling), and the rest is in gas minihalos. In particular, this means that the smallest star-forming halos (i.e., those with a mass equal to the minimum $H_2$ cooling mass) are moderately affected by pressure, and have their gas content reduced to around half the cosmic baryon ratio. The importance of the pressure in halos of mass equal to the minimum H$_2$ cooling mass is illustrated in Figure \[fig:fg\_fb\]. We consider the improved calculation (solid curve) and the mean sound speed approximation (short-dashed curve). This Figure shows that for the very first stars the previous calculation underestimates the gas fraction in these halos by more than an order of magnitude. E.g., at $1+z=66$ (the formation of the first star) the improved calculation predicts that the gas is about $35\%$ of the cosmic baryon fraction in halos with mass equal to the H$_2$ cooling mass, while the mean sound speed approximation predicts only $1.3\%$. Thus, we conclude that the effect of pressure on the very first stars is only moderate, unlike the result suggested by the mean sound speed approximation. The discrepancy decreases with the redshift. However, at $1+z=20$ the previous calculation still underestimates the gas fraction by a factor of 2, and even at $1+z=8$ the previous calculation underestimates the gas fraction by about $10\%$ compared with the improved calculation. ![Top panel: Gas fraction in halos versus redshift. We show our correct calculation (solid curve) and the previous calculation using the filtering mass in the mean sound speed approximation (dotted curve). We also compare to the fraction of gas in halos above the $H_2$ cooling mass (long-dashed curve), and the fraction above the atomic cooling mass (short-dashed curve). Bottom panel: Ratio of gas fractions in different cases. We consider the gas fraction in halos above the characteristic mass (solid curve), the gas fraction in halos above the $H_2$ cooling mass (long-dashed curve), and the gas fraction in the mean sound speed approximation (dotted curve); each is divided by the total gas fraction in halos in our correct calculation.[]{data-label="fig:gas_frac"}](gas_frac_ratio8.eps){width="84mm"} ![The gas fraction in halos of mass equal to the minimum H$_2$ cooling mass compared to the cosmic fraction. We plot the improved calculation (solid curve) and the mean sound speed approximation (short-dashed curve). In the latter case we have used the previous estimate, starting the integrals in eq. (\[u\_cs\]) at $t=0$.[]{data-label="fig:fg_fb"}](fg_fb.eps){width="84mm"} Formation of Non-Linear Objects {#non-linear} =============================== The small amplitude density fluctuations probed by the CMB grow over time as described by equations (\[g\_T\]) and (\[gamma\]), as long as the perturbations are linear. However, when $\delta$ in some region becomes of order unity, the full non-linear problem must be considered. The standard calculation that describes the formation of spherical non-linear objects was done for a dark matter halo *only*, as was explained in Section \[intro\]. Here, we generalize the calculation to the high-redshift regime, including the gravitational effects of the baryons and the radiation. The initial conditions in N-body simulations of the first galaxies are set long after the recombination epoch (usually at $z\sim200$ or later). Halos that collapse early ($z\sim 20$ and earlier) thus cannot arise from fluctuations that are linear at the beginning of the simulation, regardless of the size of the corresponding perturbed region. Now, perturbations that correspond to galaxies or clusters start from a comoving length below 100 Mpc and are thus expected to have entered the horizon at some time $t_{enter}$ in the radiation dominated era. The perturbations then do not grow significantly until equality ($z\sim3000$). If we follow the evolution of a collapsing halo, then the linear $\delta$ grows approximately $\propto a$ after equality. Thus a region that collapses to a halo at $z\sim100$ must have entered equality already in the weakly non-linear regime ($\delta\sim 1.686/30 \sim 5\%$); since growth was slow before then, even when the halo entered the horizon in the radiation dominated universe, the perturbation was not extremely small. In practical applications, we find that the largest $\delta$ gets at horizon crossing is $\sim 10^{-3}$, for halos hosting the very first stars [@NNB], so that non-linear corrections are always small. For completeness, however, we first consider the behavior of spherical fluctuations outside the horizon. Fluctuation Growth Outside the Horizon {#sec:outH} -------------------------------------- We consider a spherical top-hat fluctuation bigger than the Hubble radius at an early cosmic time. Since the fluctuation is outside the horizon, we cannot use Newtonian perturbation theory, and must apply General Relativity. We consider a spherical overdensity with a uniform density $\rho=\bar{\rho}\left(1+\delta\right)$. Birkhoff’s theorem implies that the Friedmann equation is the correct solution within the spherical region (which has a curvature $\tilde{k}$): $$\label{gr} H_{sph}^2 = H_0^2\left[ \frac{\Omega_m}{a^3}\left(1+\delta\right)+ \frac{\Omega_r}{a^4}\left(1+\delta\right)^\frac{4}{3} + \frac{\Omega_{\tilde{k}}}{a_{sph}^2}+\Omega_\Lambda\right]\ ,$$ where $H_{sph}$ and $a_{sph}$ are the Hubble constant and expansion parameter, respectively, associated with the evolution of the perturbed region, while $a$ describes the cosmic expansion of the mean universe. The $4/3$ power is a result of the difference between the non-relativistic (dark matter and baryon) density fluctuation $\delta$ and the radiation density fluctuation. Just as in the regular top-hat collapse, we wish to compare the exact non-linear evolution (given by equation (\[gr\])) to the linearly-extrapolated evolution. Comparing the above non-linear equation to the evolution of the background universe and linearizing, we obtain: $$2H\alpha\dot{\delta}=I^2\delta-\frac{\tilde{k}}{a^2}, \label{gr_lin}$$ where $I^2=H_0^2[\Omega_m/a^3+(4/3)\Omega_r/a^4]$, and $\alpha=0.5\,I^2/\dot{H}$; at high redshift, in the radiation-dominated regime, $\alpha\approx-0.25$ and then $\delta \propto a^2$ (corresponding to the synchronous gauge; @Ma; @pad). We start early enough so that $\delta=1\times 10^{-4}$ initially, and assume the growing mode for the initial perturbation. Solving numerically the non-linear and linear equations (equations (\[gr\]) and (\[gr\_lin\])), until the fluctuation enters the horizon, yields the initial value of the fluctuation entering the horizon in the radiation dominated universe. As noted above, in practice, linear initial conditions at horizon crossing would suffice for the parameter space we consider in this paper. Fluctuation Growth Inside the Horizon {#sec:delc} ------------------------------------- In the Newtonian regime the non-linear growth is described by the Newtonian equation (or, more precisely, the equation for the acceleration that results from the Einstein equations): $$\label{E2} \ddot{r}=-\frac{GM}{r^2}-\frac{4\pi G}{3}\left(\rho+3P\right)_{rest}r\ ,$$ where the [*rest*]{} stands for all matter that does not participate in the collapse, and thus only contributes to the expansion of the universe. We define $r_{\mathrm {dm}}$ and $r_{\mathrm {b}}$ to be physical radii that enclose a fixed mass of dark matter and of baryons, respectively, assuming a tophat perturbation in each. Then we obtain two coupled non-linear equations of motion: $$\begin{aligned} \label{non_r} \ddot{r}_{\mathrm {dm}}&=&\frac{-1}{r^2_{\mathrm {dm}}}\frac{4\pi G }{3}r_{\mathrm {dm}}^3\left(\rho_{\mathrm {dm}}+\rho_{\mathrm {b}}\right)+H_0^2\Omega_\Lambda r_{\mathrm {dm}}- \frac{8\pi G}{3}\rho_{r}r_{\mathrm {dm}}\ ,\nonumber \\ \ddot{r}_{\mathrm {b}}&=&\frac{-1}{r^2_{\mathrm {b}}}\frac{4\pi G }{3}r_{\mathrm {b}}^3\left(\rho_{\mathrm {dm}}+\rho_{\mathrm {b}}\right)+H_0^2\Omega_\Lambda r_{\mathrm {b}}-\frac{8\pi G}{3}\rho_{r} r_{\mathrm {b}}\ .\end{aligned}$$ We have assumed that the radiation is kept smooth by its own pressure and does not participate in the collapse; the factor $8\pi/3$ is the result of inserting $P_r=\rho_r/3$ in equation (\[E2\]). Since the time when the fluctuation enters the horizon is substantially early in the radiation-dominated universe, the baryon-photon coupling yields $\delta_{\mathrm {b}},\delta_\gamma\ll\delta_{\mathrm {dm}},\delta_\mathrm{tot}$ initially. We calculate separately the linear and non-linear growth of the fluctuations. The resulting critical (linear) overdensity at the time of collapse is shown in Figure \[fig:delta\]. For the cosmological parameters that we use in this paper, we find that $\delta_c$ is essentially independent of $M$, and is lower than the EdS value by $\sim 1\%\times(1+z)/20$ in the range of $z=9-100$. When dealing with very rare halos, even a change of a few percent in $\delta_c$ can change the halo abundance at a given redshift by over an order of magnitude (see Figure 2 of @NNB). The results are insensitive to the cosmological parameters: also shown in Figure \[fig:delta\] are the results for the set of other parameters (Section 1) specified by @Viel, which differ by 1-$\sigma$ from our standard WMAP parameters; the results are almost identical in the EdS regime ($1+z=2-10$), with only a $\la 0.2\%$ difference at high redshift (and even less at $z<1$). ![Critical overdensity versus collapse redshift. We compare $\delta_c$ for the full calculation (solid curves) to the results with other assumptions. The top panel compares to several cases that include only some of the physical ingredients that affect the spherical collapse calculation. We show the results of previous calculations that did not properly include the baryons and photons at high redshift (dotted curve), the results if the baryons are treated correctly but the contribution of the uniform radiation background is neglected (long-dashed curve), and the results if the radiation is included but the baryon-photon coupling is neglected (i.e., the baryons are treated like dark matter: short-dashed curve). The bottom panel shows results also for the @Viel set of parameters (dashed curve).[]{data-label="fig:delta"}](fig_sphericnWMAP_Viel.eps){width="84mm"} The effect of various physical ingredients on $\delta_c$ can be illustrated using the different cases shown in the upper panel of Figure \[fig:delta\]. In general, collapse is most efficient in the EdS case where all the matter participates in the collapse (resulting in $\delta_c=1.686$); any smooth component that does not collapse (a cosmological constant at low redshift, radiation and baryons at high redshift) reduces the collapse efficiency since only the dark matter component takes part in the collapse throughout. Now, any reduction in the matter fraction that collapses depresses the linear evolution of the density perturbation more strongly, while the non-linear perturbation is larger and is thus less affected by the components that do not help in the collapse. Therefore the linear perturbation reaches a lower value of $\delta_c$ when the non-linear perturbation collapses. As shown in the upper panel of Figure \[fig:delta\], the extended period at high redshift when the baryon perturbations remain suppressed is the main cause of the reduction of the value of $\delta_c$, but the contribution of the photons to the expansion of the universe also makes a significant contribution. The Mass Function and Halo Abundance {#mass_a} ------------------------------------ In addition to characterizing the properties of individual halos, it is important for any model of structure formation to predict also the abundance of halos. We calculate in this section the number density of halos as a function of mass, at any redshift. We start with the simple analytical model of @Press, which is based on Gaussian random fields, linear growth and spherical collapse. It predicts that the abundance of halos depends on mass and redshift through the two functions $\sigma(M,z)$ and $\delta_c(z)$, where $\sigma^2(M,z)$ is the variance (calculated from the power spectrum) as a function of halo mass at $z$, and $\delta_c(z)$ is the critical collapse overdensity from Section \[sec:delc\]. In Figure \[fig:sigma\_dc\] we show the number of $\sigma$ that a fluctuation must be in order for it to collapse, at various redshifts from the first star until the present. For example, a $10^5 M_\odot$ halo that collapses and forms the first star is a 9.5-$\sigma$ fluctuation according to this criterion (but see below). Halos of mass $10^5 M_\odot$ become 1-$\sigma$ fluctuations only at $z\sim5$, while today the 1-$\sigma$ collapsing fluctuations are $10^{13}M_\odot$ group halos. ![Rarity of fluctuations that produce halos at various redshifts. We show the size of a fluctuation that produces a collapsed halo on the scale $M$ at redshift $z$, in terms of the typical fluctuation level (i.e., measured as a number of $\sigma$). This includes $\sigma(M,z)$ from the correct calculation of the power spectrum [@NB], and $\delta_c(z)$ from the correct spherical collapse calculation (Section \[sec:delc\]). We consider, from bottom to top, $z=0$, 1.2, 6.5, 11, 47 and 66. Redshift 66 corresponds to the formation of the first star, while $z=47$ corresponds to the redshift of the second generation of stars, i.e., the first collapse via atomic cooling. Redshift 11 corresponds to the first halo as massive as that of the Milky Way and $z=1.2$ is associated with the formation of the first cluster as massive as Coma (see @NNB). We also show the results at $z=6.5$ associated with observations of the most distant quasars (and perhaps with the end of reionization).[]{data-label="fig:sigma_dc"}](prob_delta_S_first.eps){width="84mm"} The growth of the density fluctuations in our correct calculation can be described by an effective growth factor which depends on the mass scale: $D_{\rm eff} \equiv \sigma(M,z)/\sigma(M,z=0)$. We compare this effective growth factor to the traditional growing mode $D(z)$ [@Peebles] in Figure \[fig:S\_D\], where we show $D_{\rm eff}/D-1$ as a function of mass, illustrated at the same redshifts as in Figure \[fig:sigma\_dc\]. The source of the difference is in the smoothness of the radiation and baryons at high redshifts; in addition, pressure suppresses growth on small mass scales. These effects make $D_{\rm eff}$ larger at high redshift compared to its value today (since there has been less growth). ![The fractional difference between the effective growth factor of our correct calculation, $\sigma(M,z)/\sigma(M,z=0)$, and the traditional growth factor $D(z)$, as a function of mass at various redshifts. We consider, from bottom to top, $z=0$, 1.2, 6.5, 11, 47 and 66, as in Figure \[fig:sigma\_dc\].[]{data-label="fig:S_D"}](S_ratio_first.eps){width="84mm"} The comoving number density of halos of mass $M$ at redshift $z$ is $$\label{dndM} \frac{dn}{dM}=\frac{\rho_0}{M}f_{ST}\left|\frac{dS}{dM}\right|\ ,$$ were we have used the @Sheth mass function that fits simulations, and includes non-spherical effects on the collapse. The function $f_{ST}$ is the fraction of mass associated with halos of mass $M$: $$\label{sheth} f_{ST}(\delta_c,S)=A'\frac{\nu}{S}\sqrt{\frac{a'}{2\pi}} \left[1+\frac{1}{\left(a'\nu^2\right)^{q'}} \right] \exp \left[\frac{-a'\nu^2}{2} \right]\ ,$$ where $\nu = \delta_c/\sqrt{S}$. We use best-fit parameters $a'= 0.75$ and $q'= 0.3$ [@Sheth02], and ensure normalization to unity by taking $A'= 0.322$. We apply this formula with $\delta_c(z)$ and $\sigma^2(M,z)$ as the arguments. The @Sheth mass function makes it easier for rare fluctuations to collapse compared to @Press. This, for instance, makes the first star-forming halo effectively only an 8.3-$\sigma$ density fluctuation on the mass scale of $10^5 M_{\odot}$, in terms of the total cosmic mass fraction contained in halos above this mass. The cumulative comoving number density of halos $n(>M_{\rm min})$ is given by $$\label{nofM} n(>M_{\rm min})=\int_{M_{\rm min}}^\infty \frac{dn}{dM}\,dM\ .$$ In Figure \[fig:NofM\] we plot the cumulative number density of halos. In addition to our standard WMAP cosmological parameter set, we have also compared to the @Viel set of parameters (Section 1). We obtained for these parameters that the first star formed at $z=60.0$ (instead of 65.8) and the first Coma size cluster formed at $z=1.06$ (instead of 1.24). The difference between the two sets of cosmological parameter arises mainly from the difference in $\sigma_8$. The difference in the values of $\delta_c$ is negligible (see Figure \[fig:delta\]). ![The halo abundance as a function of the halo mass at different redshifts. From top to bottom (at the high-mass end): $z=0,1.2,6.5,11,47$ and finally $z=66$ which corresponds to the formation of the first star [@NNB]. We consider our standard WMAP cosmological parameters (solid curves), and compare to the @Viel set of parameters (dotted curves).[]{data-label="fig:NofM"}](NofM_first.eps){width="84mm"} Conclusions =========== We have calculated the filtering mass correctly at high redshift and compared it to previous estimates. We have found that at high redshift the filtering mass is lower by about an order of magnitude compared to previous calculations. The difference declines with time but remains a factor of $\sim 3$ even at $z=7$ (Figure \[fig:kf\]). Our calculation predicts a lower filtering mass because it includes the initial difference between the dark matter and baryon fluctuations, due to the baryon-photon coupling, and this lowers the pressure. This means that in contrast with the previous prediction, pressure has only a moderate effect on the formation of the earliest luminous objects. Before recombination the baryon fluctuations were suppressed compared to the dark matter fluctuations due to tight coupling with the radiation. The filtering mass rises with time at high redshift because after recombination the baryon pressure gradients start to increase. However, after the baryon temperature decouples from the CMB temperature the gas cools adiabatically, the Jeans mass drops, and eventually (at lower redshifts) the filtering mass drops as well (see Figure \[fig:kf\]). The high pressure at very high redshift still contributes significantly to the filtering mass at redshift below 10. The delay in the drop of the filtering mass compared to the Jeans mass is a signature of the continuing contribution of the memory of the pressure from very high redshifts. We have found numerical fits to the difference between the baryon and the total density fluctuations on large scales (equations (\[rLSS\_fit\])); we have also fit the filtering mass as a function of $\Omega_m$ and $z$ (equations (\[Mf\_fit\]) and (\[a\_b\_fit\])). Using the new prediction of the filtering mass we have shown that at high redshift there is more gas in halos than in previous estimates. This difference is as large as a factor of 2, but remains $\sim 10\%$ even at redshift 7. Around half of the gas is in halos with efficient $H_2$ cooling, and the rest is in gas minihalos. The previous calculation also suggests a greatly reduced gas fraction in the halo that hosts the first star, while we find only a moderate effect (Figure \[fig:fg\_fb\]). In addition, we have computed the evolution of linear as well as non-linear spherical overdensities, outside and inside the horizon, in a $\Lambda$CDM universe. Previous analyses showed that the cosmological constant contribution to the expansion of the universe results in a drop of the value of $\delta_c$ by $\sim 0.6 \%$ today [e.g., @lahav]. This makes a significant difference in the abundance of clusters. Considering structure formation at high redshift, we investigated the effect on $\delta_c$ and on the halo abundance of the contribution of radiation to the expansion of the universe, and of the contribution of the baryons to the collapsing halo, given their different initial conditions compared to the dark matter. We have found that there is a $3\%$ change in the value of the overdensity $\delta_c$ at $z \sim 60$ (Figure \[fig:delta\]). This changed value translates to a much larger difference in the halo abundance at high redshift (Figure \[fig:NofM\]); these differences decline at low redshift. The large difference in the halo abundance is the result of the difference in the exponent in the mass function (equation (\[sheth\])), and shows that even small effects on halo formation can in some cases be very important. Acknowledgments {#acknowledgments .unnumbered} =============== The authors acknowledge support by Israel Science Foundation grant 629/05 and U.S. - Israel Binational Science Foundation grant 2004386. 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--- abstract: | Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol. **Keywords:** Ornstein-Uhlenbeck processes; coupling property; Liouville theorem; gradient estimates. **MSC 2010:** 60J25; 60J75. author: - 'René L. Schilling Jian Wang' title: '**On the Coupling Property and the Liouville Theorem for Ornstein-Uhlenbeck Processes**' --- [^1] [^2] Main Results {#section1} ============ Let $(X^x_t)_{t{\geqslant}0}$ be an $n$-dimensional Ornstein-Uhlenbeck process, which is defined as the unique strong solution of the following stochastic differential equation $$\label{ou1} dX_t = AX_t\,dt + B\,dZ_t,\qquad X_0=x\in{\mathds R}^n.$$ Here $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ matrix and $Z_t$ is a Lévy process in ${\mathds R}^d$; note that we allow $Z_t$ to take values in a proper subspace of ${\mathds R}^d$. It is well known that $$X_t^x =e^{tA}x + \int_0^t e^{(t-s)A}B\,dZ_s.$$ The characteristic exponent or symbol $\Phi$ of $Z_t$, defined by $${\mathds E}\bigl(e^{i{\langle\xi,Z_t\rangle}}\bigr) =e^{-t\Phi(\xi)},\quad \xi\in{\mathds R}^d,$$ enjoys the following Lévy-Khintchine representation: $$\label{ou2} \Phi(\xi) =\frac{1}{2}{\langleQ\xi,\xi\rangle} +i{\langleb,\xi\rangle} +\int_{z\neq 0} \Bigl(1-e^{i{\langle\xi,z\rangle}}+i{\langle\xi,z\rangle}{\mathds 1}_{B(0,1)}(z)\Bigr)\nu(dz),$$ where $Q=(q_{j,k})_{j,k=1}^d$ is a positive semi-definite matrix, $b\in{{\mathds R^d}}$ is the drift vector and $\nu$ is the Lévy measure, i.e.a $\sigma$-finite measure on ${\mathds R}^d\setminus\{0\}$ such that $\int_{z\neq 0}(1\wedge |z|^2)\,\nu(dz)<\infty$. For every $\varepsilon>0$, define ${\nu}_\varepsilon$ on ${\mathds R}^d$ as follows: $${\nu}_\varepsilon(C) = \begin{cases} \nu(C), & \text{if\ \ } \nu({\mathds R}^d)<\infty;\\ \nu(C\setminus \{z: |z|<\varepsilon\}), & \text{if\ \ } \nu({\mathds R}^d)=\infty. \end{cases}$$ Let $(Y_t)_{t{\geqslant}0}$ be a Markov process on ${\mathds R}^n$ with transition function $P_t(x,\cdot)$. Then, according to [@Li; @T; @SW], we say that $(Y_t)_{t{\geqslant}0}$ admits a *successful coupling* (also: enjoys the *coupling property*) if for any $x,y\in{\mathds R}^n$, $$\label{prex1} \lim_{t\rightarrow\infty}\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}=0,$$ where $\|\cdot\|_{{\mathrm{Var}}}$ stands for the total variation norm. If a Markov process admits a successful coupling, then it also has the Liouville property, i.e. every bounded harmonic function is constant; in this context a function $f$ is harmonic, if $Lf=0$ where $L$ is the generator of the Markov process. See [@CG; @CW] and the references therein for this result and more details on the coupling property. Let $A$ be an $n\times n$ matrix. We say that an eigenvalue $\lambda$ of $A$ is *semisimple* if the dimension of the corresponding eigenspace is equal to the algebraic multiplicity of $\lambda$ as a root of characteristic polynomial of $A$. Note that for symmetric matrices $A$ all eigenvalues are real and semisimple. Recall that for any two bounded measures $\mu$ and $\nu$ on $({\mathds R}^d,{\mathscr{B}}({\mathds R}^d))$, $\mu\wedge\nu:=\mu-(\mu-\nu)^+$, where $(\mu-\nu)^{\pm}$ refers to the Jordan-Hahn decomposition of the signed measure $\mu-\nu$. In particular, $\mu\wedge\nu=\nu\wedge\mu$, and $\mu\wedge \nu\,({\mathds R}^d)=\frac{1}{2}\big[\mu({\mathds R}^d)+\nu({\mathds R}^d)-\|\mu-\nu\|_{{\mathrm{Var}}}\big].$ One of our main results is the following \[th1\] Let $P_t(x,\cdot)$ be the transition probability of the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ given by . Assume that ${\operatorname{Rank}}(B)=n$ (which implies $n{\leqslant}d$), and that there exist $\varepsilon,\delta>0$ such that $$\label{th2233} \inf_{z\in{\mathds R}^d,|z|{\leqslant}\delta}\nu _\varepsilon\wedge (\delta_z*\nu_\varepsilon)({\mathds R}^d)>0.$$ If the real parts of all eigenvalues of $A$ are non-positive and if all purely imaginary eigenvalues are semisimple, then there exists a constant $C=C(\varepsilon,\delta,\nu,A,B)>0 $ such that for all $x,y\in{\mathds R}^n$ and $t>0$, $$\label{th21} \|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}} {\leqslant}\frac{C(1+|x-y|)}{\sqrt{t}}\wedge2.$$ As a consequence of Theorem \[th1\], we immediately obtain the following result which partly answers the following question about Liouville theorems for non-local operators from [@PZ page 458]: *A challenging task would be to apply other probabilistic techniques, based on ... coupling to non-local operators*. Under the conditions of Theorem \[th1\], the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ admits a successful coupling and has the Liouville property. \[remarkth1\] (1) If $A=0$, $d=n$ and $B={\operatorname{id}}_{{\mathds R}^n}$, then $X_t$ is just a Lévy process on ${\mathds R}^n$. The condition is one possibility to guarantee sufficient jump activity such that the Lévy process $X_t$ admits a successful coupling. To see that is sharp, we can use the example in [@SW Remark 1.2]. \(2) Let $Z_t$ be a (rotationally symmetric) $\alpha$-stable Lévy process $Z_t$, $0<\alpha<2$, and denote by $X_t$ the $n$-dimensional Ornstein-Uhlenbeck process driven by $Z_t$, i.e.$$dX_t = AX_t\,dt + dZ_t.$$ If at least one eigenvalue of $A$ has positive real part, then $X_t$ does not have the coupling property. Indeed, according to [@PZ Example 3.4 and Theorem 3.5], we know that $X_t$ does not have the Liouville property, i.e. there exists a bounded harmonic function which is not constant. According to [@Li Theorem 21.12] or [@CG Theorem 1 and its second remark], $X_t$ does not have the coupling property. This example indicates that the non-positivity of the real parts of the eigenvalues of $A$ is also necessary. In [@SW Theorem 4.1 and Corollary 4.2] we show that Lévy processes which have the strong Feller property admit the coupling property. A similar conclusion, however, does not hold for general Ornstein-Uhlenbeck processes. Consider, for instance, the one-dimensional Ornstein-Uhlenbeck process given by $$dX_t=X_t\,dt+dZ_t,\qquad X_0=x\in{\mathds R},$$ where $Z_t$ is an $\alpha$-stable Lévy process $Z_t$ on ${\mathds R}$. According to [@PZ2 Theorem 1.1] (or [@NS Theorem A]) and [@PZ2 Proposition 2.1], we know that $X_t$ has the strong Feller property. However, the argument used in Remark \[remarkth1\] shows that this process fails to have the coupling property. Recently, F.-Y. Wang [@wang1] has studied the coupling property of an Ornstein-Uhlenbeck process $X_t$ defined by . Assume that ${\operatorname{Rank}}(B)=n$ and ${\langleAx,x\rangle}{\leqslant}0$ holds for $x\in{\mathds R}^n$. In [@wang1 Theorem 3.1] it is proved that is satisfied for some constant $C>0$, whenever the Lévy measure of $Z_t$ satisfies $\nu(dz){\geqslant}\rho_0(z)dz$ such that $$\label{wang22} \int_{\{|z-z_0|{\leqslant}\varepsilon\}} \frac{dz}{\rho_0(z)}<\infty$$ holds for some $z_0\in{\mathds R}^d$ and some $\varepsilon>0$. Let us compare F.-Y. Wang’s result with our Theorem \[th1\]. \[improvement\] Assume that holds for some $\rho_0\in L^1_{\mathrm{loc}}({\mathds R}^d\setminus\{0\})$, some $z_0\in{\mathds R}^d$ and some $\varepsilon>0$. Then, there exist a closed subset $F\subset \overline{B(z_0,\varepsilon)}=\{z\in{\mathds R}^d: |z-z_0|{\leqslant}\varepsilon\}$ and a constant $\delta>0$ such that $$\inf_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F\big(\rho_0(z)\wedge\rho_0(z-x)\big)\,dz>0.$$ We postpone the technical proof of Proposition \[improvement\] to Section \[subsec-appendix2\] in the appendix. Proposition \[improvement\] shows that Theorem \[th1\] improves [@wang1 Theorem 3.1], even if the Lévy measure $\nu$ of $Z_t$ has an absolutely continuous component as we will see in the following example. Let $C_{3/4}$ be a Smith-Volterra-Cantor set in $[0,1]$ with Lebesgue measure ${\operatorname{Leb}}(C_{3/4})=3/4$, i.e. $C_{3/4}$ is a perfect set with empty interior, see e.g. [@AB Chapter 3, Section 18]. Consider the following one-dimensional Ornstein-Uhlenbeck process $$dX_t=-X_t\,dt + dZ_t,\qquad X_0=x\in{\mathds R},$$ where $Z_t$ is a real-valued Lévy process with Lévy measure $\nu(dz)={\mathds 1}_{C_{3/4}}(z)\,dz$. We will see that we can use Theorem \[th1\] to show the coupling property of the process $X_t$ while the criterion from [@wang1 Theorem 3.1] fails. Let $\delta \in (0, 1/8)$ and $z\in [-\delta, \delta]$. Then $$\begin{aligned} \nu _\varepsilon\wedge (\delta_z*\nu_\varepsilon)({\mathds R}) &=\int\Bigl({\mathds 1}_{C_{3/4}}(x)\wedge{\mathds 1}_{C_{3/4}}(x+z)\Bigr)dx\\ &={\operatorname{Leb}}\bigl( C_{3/4}\cap (C_{3/4}-z)\bigr)\\ &={\operatorname{Leb}}(C_{3/4}) + {\operatorname{Leb}}(C_{3/4}-z) - {\operatorname{Leb}}\bigl( C_{3/4}\cup (C_{3/4}-z)\bigr)\\ &{\geqslant}\frac 64 - {\operatorname{Leb}}[-|z|,1+|z|]{\geqslant}\frac 14. \end{aligned}$$ This shows that the conditions of Theorem \[th1\] are satisfied. On the other hand, since $C_{3/4}$ contains no intervals, we see that for all $z_0\in{\mathds R}$ and $\varepsilon>0$, $$\int_{\{|z-z_0|{\leqslant}\varepsilon\}}\frac{dz}{{\mathds 1}_{C_{3/4}}(z)}=\infty$$ (here we use the convention $\frac 10 = +\infty$). This means that does not hold. Now we are going to estimate $\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}$ for large values of $t$ with the help of the characteristic exponent $\Phi(\xi)$ of the Lévy process $Z_t$. We restrict ourselves to the case where $Q=0$ in , i.e. to Lévy process $(Z_t)_{t{\geqslant}0}$ without a Gaussian part. For $t, \rho>0$, define $$\varphi_t(\rho) :=\sup_{|\xi|{\leqslant}\rho}\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds,$$ where $M^\top$ denotes the transpose of the matrix $M$. \[coup\] Let $P_t(x,\cdot)$ be the transition function of the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ on ${\mathds R}^n$ given by . Assume that there exists some $t_0 > 0$ such that $$\label{coup1} \liminf\limits_{|\xi|\rightarrow\infty}\frac{\int_0^{t_0}{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds}{\log (1+|\xi|)} >2n+2.$$ If $$\label{coup2} \int \exp\left(-\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)ds\right) |\xi|^{n+2} \,d\xi = \mathsf{O}\left(\varphi_t^{-1}(1)^{2n+2}\right) \quad\text{as\ \ } t\to\infty,$$ then there exist $t_1,C>0$ such that for any $x,y\in{\mathds R}^n$ and $t{\geqslant}t_1$, $$\label{coup3} \|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}} {\leqslant}C|e^{tA}(x-y)|\,\varphi^{-1}_t(1).$$ In particular, when $$\label{coup4}\xi\mapsto\int_0^\infty{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds \quad\textrm{ is locally bounded},$$ we only need the condition to get . Note that is, e.g. satisfied, if the real parts of all eigenvalues of $A$ are negative and $$\limsup_{|\xi|\rightarrow0}\frac{{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top \xi\big)}{|\xi|^\kappa}<\infty$$ for some constant $\kappa>0$. The remaining part of this paper is organized as follows. In Section \[section2\] we first present the proof of Theorem \[th1\], where a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by a compound Poisson process are used. Then, we follow the approach of our recent paper [@SSW] to prove Theorem \[coup\]. As a byproduct, we also derive explicit gradient estimates for Ornstein-Uhlenbeck processes, cf. the Appendix \[subsec-appendix1\]. Proofs of Theorems {#section2} ================== We begin with the proof of Theorem \[th1\]. The proof is split into six steps. *Step 1*. For any $\varepsilon>0$, let $(Z_t^\varepsilon)_{t{\geqslant}0}$ be a compound Poisson process on ${\mathds R}^d$ whose Lévy measure is $\nu_\varepsilon$. Then, $(Z_t^\varepsilon)_{t{\geqslant}0}$ and $(Z_t-Z_t^\varepsilon)_{t{\geqslant}0}$ are independent Lévy processes. It follows, in particular, that the random variables $$X_t^{\varepsilon,x}:=e^{tA}x+\int_0^t e^{(t-s)A}B\,dZ_s^\varepsilon$$ and $$X_t^x-X_t^{\varepsilon,x}:=\int_0^t e^{(t-s)A}B\,d(Z_s-Z_s^\varepsilon)$$ are independent for any $\varepsilon>0$ and $t{\geqslant}0$. *Step 2*. Denote by $\mu_{\varepsilon,t}$ the law of random variable $$X_t^{\varepsilon,0} := X_t^{\varepsilon,x}-e^{tA}x = \int_0^te^{(t-s)A}B\,dZ_s^\varepsilon.$$ We will compute $\mu_{\varepsilon,t}$, which coincides with the law of $\int_0^te^{sA}B\,dZ_s^\varepsilon$, cf. Lemma \[lemmaproofth11\] below. Our argument follows the proof of [@PZ2 Theorem 1.1], which is motivated by [@SA Theorem 27.7]. The law of the compound poisson process $Z_t^\varepsilon$ is given by $$e^{-C_\varepsilon t}\bigg[\delta_0+\sum_{k=1}^\infty \frac{(C_\varepsilon t)^k}{k!}\,\bar{\nu}_\varepsilon^{*k}\bigg],$$ where $C_\varepsilon=\nu_\varepsilon({\mathds R}^d)$, $\bar{\nu}_\varepsilon=\nu_\varepsilon/C_\varepsilon$ and $\bar{\nu}_\varepsilon^{*k}$ is the $k$-fold convolution of $\bar{\nu}_\varepsilon$. Construct a sequence $(\xi_i)_{i{\geqslant}1}$ of iid random variables which are exponentially distributed with intensity $C_\varepsilon$, and introduce a further sequence $(U_i)_{i{\geqslant}1}$ of iid random variables on ${\mathds R}^d$ with law $\bar{\nu}_\varepsilon$. We will assume that the random variables $(U_i)_{i{\geqslant}1}$ are independent of the sequence $(\xi_i)_{i{\geqslant}1}$. It is not difficult to check that the random variable $$\label{proofs0} 0\cdot{\mathds 1}_{\{\xi_1>t\}} +\sum_{k=1}^\infty{\mathds 1}_{\{\xi_1+\cdots+\xi_k{\leqslant}t<\xi_1+\cdots+\xi_{k+1}\}} \Big(e^{\xi_1A}BU_1+\cdots+e^{(\xi_1+\cdots+\xi_k)A}BU_k\Big)$$ also has the probability distribution $\mu_{\varepsilon,t}$. Using we find for any $f\in B_b({\mathds R}^n)$, $$\label{proofs1} {\mathds E}f\bigl(X_t^{\varepsilon,x}\bigr) =\int f\bigl(e^{tA }x+z\bigr)\,\mu_{\varepsilon,t}(dz) =f\bigl(e^{tA}x\bigr)\,e^{-C_\varepsilon t}+Hf(x), $$ where $$\begin{aligned} &Hf(x)\\ &:={\mathds E}f\left(\sum_{k=1}^\infty{\mathds 1}_{\{\xi_1+\cdots+\xi_k{\leqslant}t<\xi_1+\cdots+\xi_{k+1}\}} \Bigl(e^{tA}x+e^{\xi_1A}BU_1+\cdots+e^{(\xi_1+\cdots+\xi_k)A}BU_k\Bigr)\right)\\ &=\sum_{k=1}^\infty{\mathds E}f\left({\mathds 1}_{\{\xi_1+\cdots+\xi_k{\leqslant}t<\xi_1+\cdots+\xi_{k+1}\}} \Bigl(e^{tA}x+e^{\xi_1A}BU_1+\cdots+e^{(\xi_1+\cdots+\xi_k)A}BU_k\Bigr)\right)\\ &=\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}{\;\;\times}\\ &\qquad\quad{\times}\;\int\limits_{{\mathds R}^d}\cdots\int\limits_{{\mathds R}^d} f\bigl(e^{tA}x+e^{t_1A}By_1+\cdots+e^{(t_1+\cdots+t_k)A}By_k\bigr) \,\bar{\nu}_\varepsilon(dy_1)\cdots \bar{\nu}_\varepsilon(dy_k)\\ &=\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\ &\qquad\quad\times\;\int\limits_{{\mathds R}^n} f\bigl(e^{tA}x+z\bigr)\,\mu_{t_1,\cdots,t_k}(dz).\end{aligned}$$ Here $\mu_{t_1,\cdots,t_k}$ is the probability measure on ${\mathds R}^n$ which is the image of the $k$-fold product measure $\bar{\nu}_\varepsilon\times \cdots\times \bar{\nu}_\varepsilon$ under the linear transformation $J_{t_1,\ldots,t_k}$ (independent of $\varepsilon$) acting from $({\mathds R}^{d})^k$ into ${\mathds R}^n$: $$J_{t_1,\ldots,t_k}(y_1,\ldots,y_k) =e^{t_1A}B y_1 + \cdots +e^{(t_1+\cdots+t_k)A}B y_k,$$ for $y_i\in{\mathds R}^d$ and $i=1,\cdots,k$. *Step 3*. Let $P_t(x,\cdot)$ and $P_t$ be the transition function and the transition semigroup of the Ornstein-Uhlenbeck process $(X^x_t)_{t{\geqslant}0}$. Similarly, we denote by $P^\varepsilon_t(x,\cdot)$ and $P^\varepsilon_t$ the transition function and the transition semigroup of $(X_t^{\varepsilon,x})_{t{\geqslant}0}$, and by $Q^\varepsilon_t(x,\cdot)$ and $Q^\varepsilon_t$ the transition function and the transition semigroup of $(X_t^x-X_t^{\varepsilon,x})_{t{\geqslant}0}$. By the independence of the processes $(X_t^{\varepsilon,x})_{t{\geqslant}0}$ and $(X_t^x-X_t^{\varepsilon,x})_{t{\geqslant}0}$, we get $$\label{proofs2}\begin{aligned} \|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}} &= \sup_{\|f\|_\infty{\leqslant}1}\big|P_tf(x)-P_tf(y)\big|\\ &=\sup_{\|f\|_\infty{\leqslant}1} \big|P_t^\varepsilon Q_t^\varepsilon f(x)-P_t^\varepsilon Q_t^\varepsilon f(y)\big|\\ &{\leqslant}\sup_{\|h\|_\infty{\leqslant}1} \big|P^\varepsilon_th(x)-P^\varepsilon_th(y)\big|. \end{aligned}$$ Furthermore, it follows from that $$\label{proofs3}\begin{aligned} &\sup_{\|h\|_\infty{\leqslant}1} \big|P^\varepsilon_th(x)-P^\varepsilon_th(y)\big|\\ &{\leqslant}2e^{-C_\varepsilon t}+\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\ &\qquad\times\sup_{\|h\|_\infty{\leqslant}1} \bigg|\int\limits_{{\mathds R}^n} h\big(e^{tA}x+z\big)\,\mu_{t_1,\cdots,t_k}(dz) - \int\limits_{{\mathds R}^n} h\big(e^{tA}y+z\big)\,\mu_{t_1,\cdots,t_k}(dz)\biggr|\\ &= 2e^{-C_\varepsilon t}+\sum_{k=1}^\infty \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\ &\qquad\times\sup_{\|h\|_\infty{\leqslant}1} \bigg|\int\limits_{{\mathds R}^n} h\big(e^{tA}(x-y)+z\big)\,\mu_{t_1,\cdots,t_k}(dz) - \int\limits_{{\mathds R}^n} h(z)\,\mu_{t_1,\cdots,t_k}(dz)\bigg|\\ &{\leqslant}2e^{-C_\varepsilon t}+\sum_{k=1}^\infty\mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\;\;\times\\ &\qquad\times\|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k}-\mu_{t_1,\cdots,t_k}\|_{{\mathrm{Var}}}. \end{aligned}$$ *Step 4*. For any $a\in{\mathds R}^n$, $a\neq 0$, let $R_a$ be the non-degenerate rotation such that $R_a a=|a|e_1$. Then, by [@SW Lemma 3.2], $$\begin{aligned} \big\|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k} &-\mu_{t_1,\cdots,t_k}\big\|_{{\mathrm{Var}}}\\ &=\big\|\delta_{|e^{tA}(x-y)|e_1}*\big(\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big)-\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big\|_{{\mathrm{Var}}}.\end{aligned}$$ Since $\mu_{t_1,\cdots,t_k}$ is the law of the random variable $$\sum_{i=1}^ke^{(t_1+\cdots+t_i)A}BU_i,$$ $\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}$ is the law of the random variable $$\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big).$$ To estimate $\big\|\delta_{|e^{tA}(x-y)|e_1}*\big(\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big)-\mu_{t_1,\cdots,t_k}\circ R_{e^{tA}(x-y)}^{-1}\big\|_{{\mathrm{Var}}}$, we will use the Mineka and Lindvall-Rogers couplings for random walks. The remainder of this part is based on the proof of [@SW Proposition 3.3]. In order to ease notations, we set ${\mathsf{n}}:=\bar{\nu}_\varepsilon$ and ${\mathsf{n}}^{a}:=\delta_a*\bar\nu_\varepsilon$ for any $a\in{\mathds R}^d$. Since ${\operatorname{Rank}}(B)=n$, there exists a real $d\times n$ matrix $\bar{B}$ such that $B\bar{B}={\operatorname{id}}_{{\mathds R}^n}$, see e.g. [@Bern Theorem 2.6.1, Page 35]. For any $i{\geqslant}1$, let $(U_i,\Delta U_i)\in {\mathds R}^d \times {\mathds R}^d$ be a pair of random variables with the following distribution $${\mathds P}\big((U_i,\Delta U_i)\in C\times D\big) = \begin{cases} \qquad \frac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i})(C), & \text{if\ \ } D=\{a_i\};\\ \qquad \frac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{a_i})(C), & \text{if\ \ } D= \{-a_i\};\\ \big({\mathsf{n}}- \frac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}+{\mathsf{n}}\wedge{\mathsf{n}}^{a_i})\big)(C), & \text{if\ \ } D=\{0\}; \end{cases}$$ where $C\in{\mathscr{B}}({\mathds R}^d)$, $a_i=\bar B\, e^{(t-(t_1+\cdots+t_i))A}\,(x-y)$ and $D$ is any of the following three sets: $\{-a_i\}$, $\{0\}$ or $\{a_i\}$. Again by [@SW Lemma 3.2], $$\begin{aligned} {\mathds P}\big(\Delta U_i=-a_i\big) &=\frac{1}{2}\big({\mathsf{n}}\wedge\big(\delta_{a_i}*{\mathsf{n}})\big)({\mathds R}^d)\\ &=\frac{1}{2}\big({\mathsf{n}}\wedge\big(\delta_{-a_i}*{\mathsf{n}})\big)({\mathds R}^d)\\ &={\mathds P}(\Delta U_i=a_i).\end{aligned}$$ It is clear that the distribution of $U_i$ is ${\mathsf{n}}$. Let $U_i'=U_i+\Delta U_i$. We claim that the distribution of $U_i'$ is also ${\mathsf{n}}$. Indeed, for any $C\in\mathscr{B}({\mathds R}^d)$, $$\begin{aligned} &{\mathds P}(U_i'\in C)\\ &={\mathds P}(U_i-a_i\in C, \Delta U_i=-a_i) + {\mathds P}(U_i+a_i\in C, \Delta U_i=a_i) +{\mathds P}(U_i \in A, \Delta U_i=0)\\ &= \frac 12\left(\delta_{-a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{a_i})\right)(C) +\! \frac12\left(\delta_{a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i})\right)(C) \! +\! \left(\!\!{\mathsf{n}}-\!\! \frac 12\,\big({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}+{\mathsf{n}}\wedge{\mathsf{n}}^{a_i}\big)\!\!\right)(C)\\ &={\mathsf{n}}(C),\end{aligned}$$ where we have used that $$\delta_{a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i})={\mathsf{n}}\wedge{\mathsf{n}}^{a_i}\quad\textrm{ and }\quad\delta_{-a_i}*({\mathsf{n}}\wedge{\mathsf{n}}^{a_i})={\mathsf{n}}\wedge{\mathsf{n}}^{- a_i}.$$ Without loss of generality, we can assume that the pairs $(U_i,U_i')$ are independent for all $i{\geqslant}1$. Now we construct the coupling $$(S_k,S_k')_{k{\geqslant}1} =\left(\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big), \sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU'_i\big)\right)_{k{\geqslant}1}$$ of $$S_k:=\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big).$$ Since $U'_i-U_i=\Delta U_i$ is either $\pm a_i$ or $0$, we know that $$\begin{aligned} (S_k-& S_k')_{k{\geqslant}1}\\ &=\left(\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU'_i\big) -\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big)\right)_{k{\geqslant}1}\\ &=\left(\sum_{i=1}^kR_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}B(U'_i-U_i)\big)\right)_{k{\geqslant}1}\end{aligned}$$ is a random walk on ${\mathds R}^n$ whose steps are symmetrically (but not necessarily identically) distributed and take only the values $\pm |e^{tA}(x-y)| e_1$ and $0$. Set $S^j_{k}=\sum_{i=1}^k\eta^j_{i}$ and $S^{j\,\prime}_{k}=\sum_{i=1}^k\eta^{j\,\prime}_{i}$ for $1{\leqslant}j{\leqslant}n$, where $$(\eta^{1}_{i},\ldots,\eta^{n}_{i})=R_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i\big)$$ and $$(\eta_{i}^{1\,\prime},\ldots,\eta^{n\,\prime}_{i})=R_{e^{tA}(x-y)}\big(e^{(t_1+\cdots+t_i)A}BU_i^{\prime}\big).$$ Then $(S^1_k-S^{1\,\prime}_k)_{k{\geqslant}1}$ is a random walk on ${\mathds R}$ whose steps are independent and attain the values $-|e^{tA}(x-y)|$, $0$ and $|e^{tA}(x-y)|$ with probabilities $\frac 12(1-p_i)$, $p_i$ and $\frac 12(1-p_i)$, respectively; the values of the $p_i$ are given by $$\begin{aligned} p_i :&={\mathds P}(\eta^{1\,\prime}_i-\eta^1_{i}=0)\\ &= \left({\mathsf{n}}- \tfrac 12 ({\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}+{\mathsf{n}}\wedge{\mathsf{n}}^{a_i})\right)({\mathds R}^d)\\ &= 1-{\mathsf{n}}\wedge{\mathsf{n}}^{-a_i}({\mathds R}^d).\end{aligned}$$ Since $S^j_{k}=S^{j\,\prime}_{k}$ for $2{\leqslant}j{\leqslant}n$, we get $$\label{proofs6} \|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k}-\mu_{t_1,\cdots,t_k}\|_{{\mathrm{Var}}} {\leqslant}2\,{\mathds P}(T^S>k),$$ where $$T^S=\inf\{i{\geqslant}1\::\: S^1_{i}=S^{1\,\prime}_{i}+|e^{tA}(x-y)|\}.$$ *Step 5*. Since the real parts of all eigenvalues of $A$ are non-positive and since all purely imaginary eigenvalues are semisimple, we know from [@Bern Proposition 11.7.2, Page 438] that $C_A:=\sup_{t{\geqslant}0}\|e^{tA}\|<\infty$. In particular, when $t{\geqslant}t_1+\cdots+t_i$, $$\big|e^{(t-(t_1+\cdots+t_i))A}(x-y)\big|{\leqslant}C_A|x-y|.$$ From we get that for all $i{\geqslant}1$ and $x, y\in{\mathds R}^n$ with $|x-y|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $$\label{proofcon}\begin{aligned} \frac 12(1-p_i) &= \frac{1}{2}\big({\mathsf{n}}\wedge\big(\delta_{-a_i}*{\mathsf{n}})\big)({\mathds R}^d)\\ &{\geqslant}\frac{1}{2}\inf_{|a|{\leqslant}C_A\|\bar{B}\||x-y|}{\mathsf{n}}\wedge (\delta_a*{\mathsf{n}})({\mathds R}^d)\\ &{\geqslant}\frac{1}{2}\inf_{|a|{\leqslant}\delta}{\mathsf{n}}\wedge (\delta_a*{\mathsf{n}})({\mathds R}^d)\\ &=:\frac{1}{2}\,\gamma(\delta)>0. \end{aligned}$$ We will now estimate ${\mathds P}(T^S>k)$. Let $V_i$, $i{\geqslant}1$, be independent symmetric random variables on ${\mathds R}$, whose distributions are given by $${\mathds P}(V_i=z) = \begin{cases} \frac 12(1-p_i), &\text{if\ \ } z=-|e^{tA}(x-y)|;\\ \frac 12(1-p_i), &\text{if\ \ } z=|e^{tA}(x-y)|;\\ \qquad p_i, &\text{if\ \ } z=0. \end{cases}$$ Set $Z_k:=\sum_{i=1}^k V_i$. We have seen earlier that $$T^S=\inf\{k{\geqslant}1\::\: Z_k=|e^{tA}(x-y)|\}.$$ For any $k{\geqslant}1$, let $$\eta=\eta(k):=\#\big\{i\::\: i{\leqslant}k\textrm{ and }V_i\neq 0\big\}$$ and set $\tilde{Z}_k :=\sum_{i=1}^k\tilde{V}_i$, where $\tilde{V}_i$ denotes the $i$th $V_j$ such that $V_j\neq 0$. Then, $\tilde{Z}_k$ is a symmetric random walk with iid steps which are either $-|e^{tA}(x-y)|$ or $|e^{tA}(x-y)|$ with probability $1/2$. Define $$T^{\tilde{Z}}:=\inf\{k{\geqslant}1\::\: \tilde{Z}_k=|e^{tA}(x-y)|\}.$$ By , $$\label{lll1}\begin{aligned} {\mathds P}(T^S>k) &={\mathds P}\left(T^S>k,\; \eta{\geqslant}\frac12\,\gamma(\delta)k\right) +{\mathds P}\left(T^S>k,\, \eta{\leqslant}\frac12\, \gamma(\delta)k\right)\\ &{\leqslant}{\mathds P}\left(T^{\tilde{Z}}> \frac12\,\gamma(\delta)k\right) +{\mathds P}\bigg(\eta{\leqslant}\frac{1}{2}\sum_{i=1}^k(1-p_i)\bigg)\\ &{\leqslant}{\mathds P}\left(T^{\tilde{Z}}>\frac12\,\gamma(\delta)k\right) +{\mathds P}\bigg(\Big|\eta-\sum_{i=1}^k(1-p_i)\Big|{\geqslant}\frac{1}{2}\sum_{i=1}^k(1-p_i)\bigg). \end{aligned}$$ Note that $$\eta=\eta(k)=\sum_{i=1}^k\zeta_i,$$ where $\zeta_i = {\mathds 1}_{\{V_i\neq 0\}}$, ${1{\leqslant}i{\leqslant}k}$, are independent random variables with ${\mathds P}(\zeta_i = 0) = p_i$ and ${\mathds P}(\zeta_i=1)=1-p_i$. Chebyshev’s inequality shows that $$\label{lll2}\begin{aligned} {\mathds P}\bigg(\Big|\eta-\sum_{i=1}^k(1-p_i)\Big|{\geqslant}\frac{1}{2}\sum_{i=1}^k(1-p_i)\bigg) &{\leqslant}\frac{4{\mathrm{Var}}(\eta)}{\Big(\sum_{i=1}^k(1-p_i)\Big)^2}\\ &=\frac{4\sum_{i=1}^kp_i(1-p_i)}{\Big(\sum_{i=1}^k(1-p_i)\Big)^2}\\ &{\leqslant}\frac{4(1-\gamma(\delta))\sum_{i=1}^k(1-p_i)}{\Big(\sum_{i=1}^k(1-p_i)\Big)^2}\\ &{\leqslant}\frac{4(1-\gamma(\delta))}{\gamma(\delta)k}. \end{aligned}$$ For the second and the last inequality we have used . On the other hand, by Lemma \[lemmaproofth13\] below, $$\begin{aligned} {\mathds P}\bigg(T^{\tilde{Z}}>\frac{\gamma(\delta)k}{2}\bigg) &={\mathds P}\bigg(\max_{i{\leqslant}\big[\frac{\gamma(\delta)k}{2}\big]}\tilde{Z}_i< |e^{tA}(x-y)|\bigg)\\ &{\leqslant}2\,{\mathds P}\left(0{\leqslant}\tilde{Z}_{\big[\frac{\gamma(\delta)k}{2}\big]} {\leqslant}|e^{tA}(x-y)|\right).\end{aligned}$$ From the construction above, we know that $(\tilde{Z}_k)_{k{\geqslant}1}$ is a symmetric random walk with iid steps with values $\pm |e^{tA}(x-y)|$. Using the central limit theorem we find for sufficiently large values of $k{\geqslant}k_0$ and some constant $C=C(k_0)$ $$\label{lll3}\begin{aligned} {\mathds P}\left(T^S>\frac12\,\gamma(\delta)k\right) &= 2\,{\mathds P}\left(0{\leqslant}\frac{Z_k}{|e^{tA}(x-y)|\sqrt{\big[\frac{\gamma(\delta)k}{2}\big]}} {\leqslant}{\left[\frac{\gamma(\delta)k}{2}\right]}^{-1/2}\right)\\ &{\leqslant}\frac{C}{\sqrt{2\pi}} \int_{0}^{ {\left[\frac{\gamma(\delta)k}{2}\right]}^{-1/2}} e^{-u^2/2}\,du\\ &{\leqslant}\frac{C_{\gamma(\delta)}}{\sqrt k}. \end{aligned}$$ Combining , and gives for all $x,y\in{\mathds R}^n$ with $|x-y| {\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $t{\geqslant}t_1+\cdots+t_k$ and $k{\geqslant}k_0$ that $${\mathds P}\big(T^S>k\big) {\leqslant}\frac{C_{\gamma(\delta)}}{\sqrt k}+\frac{4(1-\gamma(\delta))}{\gamma(\delta)k}.$$ Finally, yields for all $x,y\in{\mathds R}^n$ with $|x-y|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $t{\geqslant}t_1+\cdots+t_k$ and $k{\geqslant}1$, that $$\label{proofs7} \|\delta_{e^{tA}(x-y)}*\mu_{t_1,\cdots,t_k}-\mu_{t_1,\cdots,t_k}\|_{{\mathrm{Var}}} {\leqslant}\frac{C_{1,\delta,{\mathsf{n}}}}{\sqrt k}.$$ *Step 6*. If we combine , and , we obtain that for all $x, y\in{\mathds R}^n$ with $|x-y|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$, $$\begin{aligned}\label{proofs8} \|&P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}\\ &{\leqslant}2e^{-C_\varepsilon t} +C_{1,\delta,{\mathsf{n}}}\sum_{k=1}^\infty\frac{1}{\sqrt{k}} \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t<t_1+\cdots+t_{k+1}} \!\!\!\!\!\!\!\!C_\varepsilon^{k+1}e^{-C_\varepsilon(t_1+\cdots+t_{k+1})}\,dt_1\cdots dt_{k+1}\\ &{\leqslant}2e^{-C_\varepsilon t}+C_{1,\delta,{\mathsf{n}}}e^{-C_\varepsilon t}\sum_{k=1}^\infty \frac{C_\varepsilon^{k+1}}{\sqrt{k}} \mathop{\int\cdots\int}\limits_{t_1+\cdots+t_k{\leqslant}t}\!\!dt_1\cdots dt_{k}\\ &{\leqslant}2e^{-C_\varepsilon t}+C_{1,\delta,{\mathsf{n}}}C_\varepsilon \sum_{k=1}^\infty\frac{C_\varepsilon^k t^k}{\sqrt{k}\,k!}e^{-C_\varepsilon t}\\ &{\leqslant}2e^{-C_\varepsilon t}+\frac{\sqrt{2}C_{1,\delta,{\mathsf{n}}} C_\varepsilon (1-e^{-C_\varepsilon t})}{\sqrt{C_\varepsilon t}}\\ &{\leqslant}\frac{C_{2,\epsilon,\delta,{\mathsf{n}}}}{\sqrt{t}}, \end{aligned}$$ where the penultimate inequality follows as in [@SW Proposition 2.2]. For any $x,$ $y\in{\mathds R}^n$, set $k=\left[\frac{C_A\|\bar{B}\||x-y|}{\delta}\right]+1$. Pick $x_0, x_1, \ldots, x_k\in{\mathds R}^n$ such that $x_0=x$, $x_k=y$ and $|x_i-x_{i-1}|{\leqslant}\delta(C_A\|\bar{B}\|)^{-1}$ for $1{\leqslant}i{\leqslant}k$. By , $$\begin{aligned} \|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}} &{\leqslant}\sum_{i=1}^k\|P_t(x_i,\cdot)-P_t(x_{i-1},\cdot)\|_{{\mathrm{Var}}}\\ &{\leqslant}\frac{C_{\epsilon,\delta,{\mathsf{n}},A,B}(1+|x-y|)}{\sqrt{t}},\end{aligned}$$ which finishes the proof of . The following two lemmas have been used in the proof of Theorem \[th1\] above. For the sake of completeness we include their proofs. \[lemmaproofth12\] Let $B\in{\mathds R}^{n\times d}$ and $(Z_t)_{t{\geqslant}0}$ be a $d$-dimensional Lévy process with characteristic exponent $\Phi$ as in . Then, $(Z^B_t)_{t{\geqslant}0}:=(BZ_t)_{t{\geqslant}0}$ is a Lévy process on (a subspace of) ${\mathds R}^n$, and the corresponding characteristic exponent is $${\mathds R}^n\ni\xi \mapsto \Phi_B(\xi):=\Phi(B^\top\xi).$$ The Lévy triplet $(Q_B,b_B,\nu_B)$ of $(Z^B_t)_{t{\geqslant}0}$ is given by $ Q_B=BQB^\top$, $\nu_B(C) = \nu\{y : By\in C\}$ and $$b_B=Bb+\int_{x\neq 0} Bx\,\big({\mathds 1}_{\{z\in{\mathds R}^d:|z|{\leqslant}1\}}(Bx) - {\mathds 1}_{\{z\in{\mathds R}^d:|z|{\leqslant}1\}}(x)\big)\, \nu(dx).$$ For all $\xi\in{\mathds R}^n$ and $t{\geqslant}0$, we have $${\mathds E}(e^{i{\langle\xi,Z^B_t\rangle}}) ={\mathds E}(e^{i{\langle\xi,BZ_t\rangle}}) ={\mathds E}(e^{i{\langleB^\top\xi,Z_t\rangle}}) =e^{-t\Phi(B^\top\xi)}.$$ The assertion follows from and some straightforward calculations. \[lemmaproofth11\] Let $A\in{\mathds R}^{n\times n}$, $B\in{\mathds R}^{n\times d}$ and $(Z_t)_{t{\geqslant}0}$ be a $d$-dimensional Lévy process with the characteristic exponent $\Phi$ as in . For all $t>0$ the random variables $\int_0^t e^{(t-s)A}B\,dZ_s$ and $\int_0^t e^{sA}B\,dZ_s$ have the same probability distribution. Furthermore, both random variables are infinitely divisible, and the characteristic exponent (log-characteristic function) is given by $${\mathds R}^n\ni\xi \mapsto \Phi_t(\xi):=\int_0^t\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds.$$ We first assume that $n=d$ and $B={\operatorname{id}}_{{\mathds R}^d}$. For any $t>0$, we can use Lemma \[lemmaproofth12\] and follow the proof of [@SA (17.3)] to deduce $${\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{(t-s)A}\,dZ_s\right\rangle}\right)\right] =\exp\left[-\int_0^t\Phi(e^{(t-s)A^\top}\xi)\,ds\right]$$ for all $\xi\in{\mathds R}^d$. Similarly, for every $\xi\in{\mathds R}^d$, $${\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{sA}\,dZ_s\right\rangle}\right)\right] =\exp\left[-\int_0^t\Phi(e^{sA^\top}\xi)\,ds\right].$$ Since $$\exp\left[-\int_0^t\Phi(e^{(t-s)A^\top}\xi)\,ds\right] =\exp\left[-\int_0^t\Phi(e^{sA^\top}\xi)\,ds\right],$$ it follows that $\int_0^t e^{(t-s)A}B\,dZ_s$ and $\int_0^t e^{sA}B\,dZ_s$ have the same law. Now replace in the preceding calculations $A$ with $\frac 1k\,A$, $k{\geqslant}1$, and set $Y_k := \int_0^t e^{s\,\frac 1k\, A}\,dZ_s$. Denote by $Y_k^{(j)}$, $1{\leqslant}j{\leqslant}k$, independent copies of $Y_k$. It is straightforward to see that $\sum_{j=1}^k Y_k^{(j)}$ and $\int_0^t e^{sA}\,dZ_s$ have the same law. This proves the infinite divisibility. If $n\neq d$, we consider, as in Lemma \[lemmaproofth12\], the Lévy process $(Z^B_t)_{t{\geqslant}0}:=(BZ_t)_{t{\geqslant}0}$ on (a subspace of) ${\mathds R}^n$. Then, for any $\xi\in{\mathds R}^n$, $${\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{(t-s)A}B\,dZ_s\right\rangle}\right)\right] ={\mathds E}\left[\exp\left(i{\left\langle\xi,\int_0^t e^{(t-s)A}\,dZ^B_s\right\rangle}\right)\right],$$ and the claim follows from the first part of our proof. The following result presents the upper estimate for the distribution of the maximum of a symmetric random walk, by using the reflection principle. Since we could not find a precise reference in the literature, we include the complete proof for the readers’ convenience. \[lemmaproofth13\] Consider a random walk $(S_i)_{i{\geqslant}1}$ on ${\mathds Z}$ with iid steps, which attain the values $-1$, $1$ and $0$ with probabilities $(1-r)/2$, $(1-r)/2$ and $r$ $(0{\leqslant}r<1)$, respectively. Then for any positive integers $a$ and $k$, we have $$\label{lemmaproofth1311} 2{\mathds P}(S_k> a) {\leqslant}{\mathds P}\Big(\max_{i{\leqslant}k}S_i{\geqslant}a\Big) {\leqslant}2{\mathds P}(S_k{\geqslant}a)$$ and $$2{\mathds P}\big(0<S_k<a\big) {\leqslant}{\mathds P}\Big(\max_{1{\leqslant}i{\leqslant}k}S_i < a\Big){\leqslant}2{\mathds P}\big(0{\leqslant}S_k{\leqslant}a\big).$$ Fix any positive integer $a$ and define $\tau :=\tau_a := \inf\{i{\geqslant}1\::\: S_i=a\}$. Since the random walk has iid steps, it is obvious that $(S_{i+\tau}-S_\tau)_{i{\geqslant}0}$ and $(S_i)_{i{\geqslant}0}$ are independent random walks having the same law. Observing that $S_\tau = a$ and $\left\{\max_{i{\leqslant}k}S_i{\geqslant}a\right\} = \{\tau {\leqslant}k\}$ we find, therefore, $$\begin{aligned} {\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a\Bigr) &={\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a,\; S_k{\geqslant}a\Bigr) + {\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a,\; S_k < a\Bigr)\\ &={\mathds P}(S_k{\geqslant}a)+{\mathds P}(\tau{\leqslant}k,S_k<S_\tau)\\ &={\mathds P}(S_k{\geqslant}a)+{\mathds P}(\tau{\leqslant}k,S_k>S_\tau)\\ &={\mathds P}(S_k{\geqslant}a)+{\mathds P}(S_k>a). \end{aligned}$$ From this we conclude that $$2{\mathds P}(S_k{\geqslant}a) {\geqslant}{\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a\Bigr) {\geqslant}2{\mathds P}(S_k>a).$$ Since ${\mathds P}(S_k{\geqslant}0)={\mathds P}(S_k{\leqslant}0){\geqslant}1/2$, we see $$\begin{aligned} {\mathds P}\Bigl(\max_{i{\leqslant}k}S_i< a\Bigr) &=1-{\mathds P}\Bigl(\max_{i{\leqslant}k}S_i{\geqslant}a\Bigr)\\ &{\leqslant}1-2{\mathds P}(S_k>a)\\ &{\leqslant}2\big({\mathds P}(S_k{\geqslant}0)-{\mathds P}(S_k>a)\big)\\ &=2{\mathds P}(0{\leqslant}S_k{\leqslant}a); \end{aligned}$$ the other inequality follows similarly if we use ${\mathds P}(S_k>0)={\mathds P}(S_k<0){\leqslant}1/2$. Next, we turn to the proof of Theorem \[coup\]. *Step 1*. As in the proof of Lemma \[lemmaproofth11\] we may, without loss of generality, assume that $n=d$ and $B={\operatorname{id}}_{{\mathds R}^d}$. For $t>0$, denote by $\mu_t$ the law of $X_t^0:=\int_0^t e^{(t-s)A}\,dZ_s$. According to Lemma \[lemmaproofth11\], the law $\mu_t$ is an infinitely divisible probability distribution, and the characteristic exponent of $\mu_t$ is given by $$\Phi_t(\xi):=\int_0^t\Phi\big(e^{sA^\top}\xi\big)\,ds.$$ Since the driving Lévy process $(Z_t)_{t{\geqslant}0}$ has no Gaussian part, the Lévy [triplet]{} $(0,b_t,\nu_t)$ of $\Phi_t$ is given by, cf. [@SAT Theorem 3.1], $$\begin{gathered} \nu_t(C)= \int_0^t\nu(e^{-sA}C)\,ds,\qquad C\in\mathscr{B}({\mathds R}^d\setminus\{0\}),\\ b_t= \int_0^t e^{sA}b\,ds + \int_{z\neq 0} \int_0^t e^{sA}z\Big({\mathds 1}_{\{|z|{\leqslant}1\}} \big(e^{sA}z\big) - {\mathds 1}_{\{|z|{\leqslant}1\}}(z) \Big)\,ds\,\nu(dz). \end{gathered}$$ For every $r>0$, let $\{\mu_t^r, t{\geqslant}0\}$ be the family of infinitely divisible probability measures on ${\mathds R}^d$ whose Fourier transform is of the form $\widehat{\mu}^r_t(\xi)=\exp(-\Phi_{t,r}(\xi))$, where $$\Phi_{t,r}(\xi) = \int_{|z|{\leqslant}r} \left(1-e^{i{\langle\xi,z\rangle}}+i{\langle\xi,z\rangle}\right)\,\nu_t(dz)$$ with $\nu_t$ as above. Set $h(t):=1\big/\varphi_t^{-1}(1)$. Following the proof of [@SSW Propostion 2.2], the conditions and ensure that there exists $t_1>0$ such that for all $t{\geqslant}t_1$, the measure $\mu^{h(t)}_t$ has a density $p^{h(t)}_t\in C^{n+2}_b({\mathds R}^d)$; moreover, $$\label{proofcoup1} \big|\nabla p^{h(t)}_t(y)\big| {\leqslant}c(n,\Phi)\,h(t)^{-(n+1)}\big(1+h(t)^{-1}|y|\big)^{-(n+1)}$$ holds for all $y\in{\mathds R}^d$. *Step 2*. For $r>0$ and $\xi\in{\mathds R}^d$, define $$\Psi_{t,r}(\xi) := \Phi_t(\xi)-\Phi_{t,r}(\xi) = \int_{|z|>r}\left(1-e^{i{\langle\xi,z\rangle}}\right)\,\nu_t(dz) - i{\left\langle\xi,\int_{1<|z|{\leqslant}r}z\,\nu_t(dz)-b_t\right\rangle}.$$ Since $\Psi_{t,r}$ is given by a Lévy-Khintchine formula, it is the characteristic exponent of some $d$-dimensional infinitely divisible random variable. Let $\{{\mathsf{\pi}}_t^{r}, t{\geqslant}0\}$ be the family of infinitely divisible measures whose Fourier transforms are of the form $\widehat{{\mathsf{\pi}}}^{r}_t(\xi)=\exp(-\Psi_{t,r}(\xi))$. Clearly, $\mu_t=\mu_t^r*{\mathsf{\pi}}_t^{r}$ for all $t,r>0$. Let $P_t(x,\cdot)$ and $P_t$ be the transition function and the transition semigroup of the Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ given by . For all $f\in B_b({\mathds R}^d)$ we have $$\begin{aligned} P_tf(x) &=\int f\bigl(e^{tA}x+z\bigr)\,\mu_t(dz)\\ &=\int f\bigl(e^{tA}x+z\bigr)\,\mu^r_t*{\mathsf{\pi}}_t^{r}(dz)\\ &=\iint f\bigl(e^{tA}x+z_1+z_2\bigr)\,{\mathsf{\pi}}_t^{r}(dz_1)\,\mu^r_t(dz_2). \end{aligned}$$ Taking $r=h(t)$ we get, using the conclusions of step 1, that for all $t{\geqslant}t_1$ and $x\in{\mathds R}^d$, $$\begin{aligned} P_tf(x) &= \int p^{h(t)}_t(z_2)\,dz_2 \int f\bigl(e^{tA}x+z_1+z_2\bigr)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\\ &= \int p^{h(t)}_t\bigl(z_2-e^{tA}x\bigr)\,dz_2 \int f(z_1+z_2)\,{\mathsf{\pi}}_t^{h(t)}(dz_1). \end{aligned}$$ If $\|f\|_\infty{\leqslant}1$, then $$\bigg\|\int f(z_1+\cdot)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\bigg\|_\infty {\leqslant}\|f\|_\infty\, {\mathsf{\pi}}_t^{h(t)}({\mathds R}^d) {\leqslant}1.$$ *Step 3*. For all $x,y\in{\mathds R}^d$, $$\label{proofcoup2}\begin{aligned} \|P_t(x,&\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}\\ &= \sup_{\|f\|_\infty{\leqslant}1}\big|P_tf(x)-P_tf(y)\big|\\ &= \sup_{\|f\|_\infty{\leqslant}1} \bigg|\int p^{h(t)}_t\bigl(z_2-e^{tA}x\bigr)\,dz_2 \int f(z_1+z_2)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\\ &\qquad\qquad\quad\mbox{}-\int p^{h(t)}_t\bigl(z_2-e^{tA}y\bigr)\,dz_2 \int f(z_1+z_2)\,{\mathsf{\pi}}_t^{h(t)}(dz_1)\bigg|\\ &{\leqslant}\sup_{\|g\|_\infty{\leqslant}1} \bigg|\int g(z)p^{h(t)}_t\bigl(z-e^{tA}x\bigr)\,dz - \int g(z)p^{h(t)}_t\bigl(z-e^{tA}y\bigr)\,dz\bigg|\\ &= \sup_{\|g\|_\infty{\leqslant}1} \bigg|\int g(z)\Big(p^{h(t)}_t\bigl(z-e^{tA}x\bigr)-p^{h(t)}_t\bigl(z-e^{tA}y\bigr)\Big)\, dz\bigg|\\ &= \int \Big|p^{h(t)}_t\bigl(z-e^{tA}x\bigr)-p^{h(t)}_t\bigl(z-e^{tA}y\bigr)\Big|\,dz. \end{aligned}$$ With the argument used in the proof of [@SSW Theorem 3.1], follows from and . *Step 4*. By assumption , $$\varphi_\infty(\rho):=\sup_{|\xi|{\leqslant}\rho}\int_0^\infty{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds$$ is finite on $(0,\infty)$; in particular, $\varphi_\infty^{-1}(1)\in(0,\infty]$. On the other hand, for any $t{\geqslant}t_0$, according to , $$\begin{aligned} \int \exp\left(-\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds\right)& |\xi|^{n+2} \,d\xi\\ &{\leqslant}\int \exp\left(-\int_0^{t_0}{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds\right) |\xi|^{n+2} \,d\xi\\ &=:C(t_0)<\infty. \end{aligned}$$ Since the function $t\mapsto \varphi^{-1}_t(1)$ is decreasing on $(0,\infty]$, holds. This finishes the proof. Appendix {#sec-appendix} ======== Gradient Estimates for Ornstein-Uhlenbeck Processes {#subsec-appendix1} --------------------------------------------------- Motivated by [@SSW Theorem 1.3], we have the following results for gradient estimates of an Ornstein-Uhlenbeck process. This is the counterpart of Theorem \[coup\]. For $t, \rho>0$, define $$\varphi(\rho) :=\sup_{|\xi|{\leqslant}\rho}{\ensuremath{\operatorname{Re}}}\Phi\bigl(B^\top \xi\bigr)\quad\textrm{ and }\quad \varphi_t(\rho) :=\sup_{|\xi|{\leqslant}\rho}\int_0^t{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds,$$ where $\Phi$ is the characteristic exponent of the driving Lévy process $(Z_t)_{t{\geqslant}0}$ from . \[strong\] Let $P_t(x,\cdot)$ be the transition function of the $n$-dimensional Ornstein-Uhlenbeck process $\{X_t^x\}_{t{\geqslant}0}$ given by . Assume that $$\label{strong1} \liminf_{|\xi|\rightarrow\infty} \frac{{\ensuremath{\operatorname{Re}}}\Phi\bigl(B^\top \xi\bigr)}{\log (1+|\xi|)} =\infty.$$ If for any $C>0$, $$\label{strong2} \int \exp\left[-C t{\ensuremath{\operatorname{Re}}}\Phi\bigl(B^\top\xi\bigr)\right] |\xi|^{n+2} \,d\xi = \mathsf{O}\left(\varphi^{-1}\Big(\frac{1}{t}\Big)^{2n+2}\right) \qquad\text{as\ \ } t\to 0,$$ then there exists $c>0$ such that for all $t>0$ and $f\in{B}_b({\mathds R}^n)$, $$\label{strong3} \|\nabla P_t f\|_\infty{\leqslant}c\|f\|_\infty \,\varphi^{-1}\Big(\frac{1}{t\wedge1}\Big).$$ If, in addition, $$\xi\mapsto\int_0^\infty{\ensuremath{\operatorname{Re}}}\Phi\big(B^\top e^{sA^\top}\xi\big)\,ds\quad \textrm{ is locally bounded},$$ then there exist $t_1,c>0$ such that for $t{\geqslant}t_1$ and $f\in{B}_b({\mathds R}^n)$, $$\label{strong4} \|\nabla P_t f\|_\infty {\leqslant}c\,\|f\|_\infty \bigg[\|e^{tA}\|\,\varphi_t^{-1}(1)\bigg],$$ where $\|M\|=\sup_{|x|{\leqslant}1}|Mx|$ denotes the norm of the matrix of $M$. To illustrate the power of Theorem \[strong\], we consider \[examplegradient\] Let $\mu$ be a finite nonnegative measure on the unit sphere ${\mathds{S}}\subset {\mathds R}^n$ and assume that $\mu$ is nondegenerate in the sense that its support is not contained in any proper linear subspace of ${\mathds R}^n$. Let $\alpha\in(0,2)$, $\beta\in (0,\infty]$ and assume that the Lévy measure $\nu$ satisfies $$\nu(C) {\geqslant}\int_0^{r_0}\int_{{\mathds{S}}}{\mathds 1}_C(s\theta)s^{-1-\alpha} \,ds\, \mu(d\theta) + \int_{r_0}^\infty\int_{{\mathds{S}}}{\mathds 1}_C(s\theta)s^{-1-\beta}\,ds\, \mu(d\theta)$$ for some constant $r_0>0$ and all $C\in {\mathscr{B}}({\mathds R}^n\setminus\{0\})$. Consider the following Ornstein-Uhlenbeck process $X_t$ on ${\mathds R}^n$ given by $$dX_t=AX_t\,dt+dZ_t,$$ where $(Z_t)_{t{\geqslant}0}$ is a Lévy process on ${\mathds R}^n$ with the Lévy measure $\nu$. By Theorem \[strong\] there exists a constant $c>0$ such that for all $t>0$ and $f\in B_b({\mathds R}^n)$, $$\|\nabla P_tf\|_\infty {\leqslant}c\,\|f\|_\infty \, (t\wedge 1)^{-1/\alpha}.$$ Furthermore, if the real parts of all eigenvalues of $A$ are negative, then there exists a constant $c>0$ such that for all $t>0$ and $f\in B_b({\mathds R}^n)$, $$\|\nabla P_tf\|_\infty {\leqslant}c\,\|f\|_\infty \,\frac{\|e^{tA}\|\quad}{(t\wedge 1)^{1/\alpha}}.$$ Recently, F.-Y. Wang [@W2 Theorem 1.1] has presented explicit gradient estimates for Ornstein-Uhlenbeck processes, by assuming that the corresponding Lévy measure has absolutely continuous (*with respect to Lebesgue measure*) lower bounds. Since lower bounds of Lévy measure in Example \[examplegradient\] could be much irregular, Theorem \[strong\] is more applicable than [@W2 Theorem 1.1]. Assuming the conditions and , we can mimic the proof of [@SSW Theorem 3.2] to show that there exist $t_1, C>0$ such that for all $x,y\in{\mathds R}^n$ and $t{\leqslant}t_1$, $$\label{proofstrong} \|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}} {\leqslant}C\,|e^{tA}(x-y)|\,\varphi^{-1}\Big(\frac{1}{t}\Big).$$ Thus we can apply to find for all $f\in B_b({\mathds R}^n)$ with $\|f\|_\infty= 1$, $$\label{proofstrong}\begin{aligned} |\nabla P_tf(x)| &{\leqslant}\limsup_{y\to x}\frac{|P_tf(x)-P_tf(y)|}{|y-x|}\\ &{\leqslant}\limsup_{y\to x}\frac{\sup_{\|w\|_\infty{\leqslant}1}|P_tw(x)-P_tw(y)|}{|y-x|}\\ &{\leqslant}\limsup_{y\to x}\frac{\|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}}{|y-x|}\\ &{\leqslant}C\,\|e^{tA}\|\,\varphi^{-1}\Big(\frac{1}{t}\Big)\\ &{\leqslant}\Big[C\,\sup_{s{\leqslant}t_1}\|e^{sA}\|\Big]\,\varphi^{-1}\Big(\frac{1}{t}\Big). \end{aligned}$$ Because of the Markov property of the semigroup $P_t$, the function $$t\mapsto \sup_{f\in B_b({\mathds R}^n),\, \|f\|_\infty =1}{ \|\nabla P_t f\|_\infty}$$ is deceasing. Combining this and yields . The assertion follows if we combine the above argument with : there exist $t_2,C>0$ such that for all $x,y\in{\mathds R}^n$ and $t{\geqslant}t_2$, $$\begin{gathered} \|P_t(x,\cdot)-P_t(y,\cdot)\|_{{\mathrm{Var}}}{\leqslant}C\,|e^{tA}(x-y)|\,\varphi_t^{-1}(1). \qedhere\end{gathered}$$ Proof of Proposition \[improvement\] {#subsec-appendix2} ------------------------------------ Because of , we can choose a closed subset $F\subset \overline{B(z_0, \varepsilon)}$ such that $0\notin F$ and $$\int_F \frac{dz}{\rho_0(z)} < \infty.$$ By the Cauchy-Schwarz inequality, we have $$\left(\int_F \rho_0(z)\,dz\right)^{-1} {\leqslant}\frac{1}{{\operatorname{Leb}}(F)^2} \int_F \frac{dz}{\rho_0(z)} < \infty.$$ Hence, $$K:=\int_F\rho_0(z)\,dz>0.$$ Since $F$ is a compact set and $0\notin F$, there exists some $\delta_0>0$ such that $ 0\notin F+\overline{B(0,\delta_0)}, $ where $F+\overline{B(0,\delta_0)}:=\{a+b:a\in F, |b|{\leqslant}\delta_0\}.$ Since $\rho_0$ is locally integrable, we know that $$K{\leqslant}\int_{F+\overline{B(0,\delta_0)}}\rho_0(z)\,dz<\infty.$$ The remainder of the proof is now similar to the argument which shows that the shift $x\mapsto \|f(\cdot - x)-f\|_{L^1}$, $f\in L^1({\mathds R}^d,{\operatorname{Leb}})$, is continuous, see e.g. [@STR Lemma 6.3.5] or [@RSCC Theorem 14.8]: choose $\chi\in C_c^\infty ({\mathds R}^d)$ such that ${\operatorname{supp}}\chi\subset F+\overline{B(0,\delta_0)}$ and $$\int_{F+\overline{B(0,\delta_0)}}|\rho_0(z)-\chi(z)|\, dz{\leqslant}\frac{K}{4}.$$ Therefore, for any $x\in{\mathds R}^d$ with $|x|{\leqslant}\delta_0$, we obtain $$\begin{aligned} \int_F & |\rho_0(z)-\rho_0(z-x)|\,dz\\ &{\leqslant}\int_F|\rho_0(z)-\chi(z)|\,dz +\int_F|\chi(z)-\chi(z-x)|\,dz +\int_F|\rho_0(z-x)-\chi(z-x)|\,dz\\ &= \int_F|\rho_0(z)-\chi(z)|\,dz +\int_F|\chi(z)-\chi(z-x)|\,dz +\int_{F+x}|\rho_0(z)-\chi(z)|\,dz\\ &{\leqslant}2\int_{F+\overline{B(0,\delta_0)}}|\rho_0(z)-\chi(z)|\,dz +\int_F|\chi(z)-\chi(z-x)|\,dz\\ &{\leqslant}\frac{K}{2}+\int_F|\chi(z)-\chi(z-x)|\,dz. \end{aligned}$$ By the dominated convergence theorem we see that $$x\mapsto\int_F|\chi(z)-\chi(z-x)|\,dz$$ is continuous on ${\mathds R}^d$. Therefore, there exists $0<\delta{\leqslant}\delta_0$ such that $$\sup_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F|\chi(z)-\chi(z-x)|\,dz{\leqslant}\frac{K}{4}$$ and, in particular, $$\sup_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F|\rho_0(z)-\rho_0(z-x)|\,dz{\leqslant}\frac{3K}{4}.$$ Using $2(a\wedge b)=a+b-|a-b|$ for all $a,b{\geqslant}0$, we get $$\begin{aligned} \inf_{x\in{\mathds R}^d, |x|{\leqslant}\delta} & \int_F\big(\rho_0(z)\wedge\rho_0(z-x)\big)\,dz\\ &=\frac{1}{2}\inf_{x\in{\mathds R}^d, |x|{\leqslant}\delta} \bigg[\int_F\big(\rho_0(z)+\rho_0(z-x)\big)\,dz - \int_F\big|\rho_0(z)-\rho_0(z-x)\big|\,dz\bigg]\\ &{\geqslant}\frac{1}{2}\int_F\rho_0(z)\,dz - \frac{1}{2}\sup_{x\in{\mathds R}^d, |x|{\leqslant}\delta}\int_F\big|\rho_0(z)-\rho_0(z-x)\big|\,dz\\ &{\geqslant}\frac{K}{8}>0. \end{aligned}$$ This finishes the proof. Financial support through DFG (grant Schi 419/5-1) and DAAD (PPP Kroatien) (for René L. Schilling) and the Alexander-von-Humboldt Foundation and the Natural Science Foundation of Fujian $($No. 2010J05002$)$ (for Jian Wang) is gratefully acknowledged. [99]{} Aliprantis, C.D. and Burkinshaw, O.: *Principles of Real Analysis (3rd ed.)*, Academic Press, San Diego, California 1998, Bernstein, D.S.: *Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory*, Princeton University Press, Princeton 2005. Cranston, M. and Greven, A.: Coupling and harmonic functions in the case of continuous time Markov processes, *Stoch.Proc. Appl.* **60** (1995), 261–286. Cranston, M. and Wang, F.-Y.: A condition for the equivalence of coupling and shift-coupling, *Ann. Probab.* **28** (2000), 1666–1679. Lindvall, T.: *Lectures on the Coupling Method*, Wiley, New York 1992. 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See also arXiv 1011.1067 Schilling, R.L. and Wang, J.: On the coupling property of Lévy processes, to appear in *Ann. Inst. Henri Poincaré: Probab. Stat.*, 2010. See also arXiv 1006.5288 Stroock, D.W.: *A Concise Introducation to the Theory of Integration (2nd ed.)*, Birkhäuser, Boston 1994. Thorisson, H.: *Coupling, Stationarity and Regeneration*, Springer, New York 2000. Wang, F.-Y.: Coupling for Ornstein-Uhlenbeck jump processes, to appear in *Bernoulli*, 2010. See also arXiv:1002.2890v5 Wang, F.-Y.: Gradient estimate for Ornstein-Uhlenbeck jump processes, *Stoch. Proc. Appl.* **121** (2011), 466–478. [^1]: *R. Schilling:* TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany. `rene.schilling@tu-dresden.de` [^2]: *J. Wang:* School of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, P.R. China *and* TU Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany. `jianwang@fjnu.edu.cn`

--- abstract: 'The design objectives for an automatic transcription system are to produce text readable by humans and to minimize the impact on manual post-editing. This study reports on a recognition system used for transcribing speeches in the Icelandic parliament - Althingi. It evaluates the system performance and its effect on manual post-editing. The results are compared against the original manual transcription process. 239 total speeches, consisting of 11 hours and 33 minutes, were processed, both manually and automatically, and the editing process was analysed. The dependence of word edit distance on edit time and the editing real-time factor has been estimated and compared to user evaluations of the transcription system. The main findings show that the word edit distance is positively correlated with edit time and a system achieving a 12.6% edit distance would match the performance of manual transcribers. Producing perfect transcriptions would result in a real-time factor of 2.56. The study also shows that 99% of low error rate speeches received a medium or good grade in subjective evaluations. On the contrary, 21% of high error rate speeches received a bad grade.' address: | Reykjavik University\ Language and Voice Lab\ Menntavegur 1, 101 Reykjavik, Iceland bibliography: - 'refs.bib' title: 'Manual Post-editing of Automatically Transcribed Speeches from the Icelandic Parliament - Althingi' --- speech recognition, parliamentary transcription, manual editing, human-computer interaction, Icelandic Introduction ============ In the last 5 years, automatic speech recognition (ASR) technology has advanced enough to be used in real-life applications. Recognition technology has been used extensively to transcribe speeches for large languages such as English, German or Spanish [@miro2015efficiency; @munteanu2008collaborative; @kolkhorst2012evaluation]. These systems are often composed of an ASR module to produce audio-to-text transcription and several natural language processing modules to improve text formatting. The main issue, however, is that neither module performs with perfect accuracy so manual post-processing is needed to produce final transcriptions. The system for automatically transcribing university lectures in Spanish [@miro2015efficiency] compared three post-processing approaches: one involving automatic corrections, another using lecturer corrections, and a third using a mixture of both. The system was tested on twenty lectures and the was compared to a real-time factor of the post-editing time versus the total duration of the lecture. The authors conclude that the edit time is directly correlated with the transcription accuracy, but the relationship between the real-time factor and was weak, perhaps due to the low range produced by the ASR. In the English transcription system [@munteanu2008collaborative], the challenge of achieving a low error rate in transcribing university lectures was handled using collaborative editing. The authors’ findings conclude that correcting transcripts with $\mbox{\it WER}$ lower than 25% increases the editing effort. The transcription errors for lectures in German [@munteanu2008collaborative] were corrected using student edits and this error correction was studied. During the transcription process they noted that their ASR made errors caused by uncommon and non-German terms in the lectures. Their analysis showed that the corrections of inexperienced editors tend to bring a high WER down to about 25%, corroborating the findings of [@munteanu2008collaborative]. Evaluation of post-editing transcribed speech was studied in [@sperber2017transcribing], where authors observe a strong variation in editing accuracy and speed among editors. Authors also note that low $\mbox{\it WER}$ transcripts require advanced editing strategies to achieve error rate improvements comparable to improvements for high $\mbox{\it WER}$ transcripts. Different transcription strategies were compared in [@Sperber2016]; namely a fully manual post-editing of ASR transcripts and confidence-enhanced post-editing of ASR transcripts. The authors conclude that post-editing automatic transcripts results in more accurate and faster transcripts, when compared to manually transcribing from scratch. This conclusion was further corroborated in [@Miro2018], which dealt with automatic subtitling of videos. This paper evaluates an ASR system in the context of transcribing speeches for the Icelandic parliament - Althingi. The system has only recently been developed for Icelandic [@gudhnason2012almannaromur; @Helgadottir2017corpus; @gudhnason2017building; @NIKULASDOTTIR2018], and is now being incorporated into the transcription process of Althingi. The current manual procedure is done in two stages: an initial manuscript is obtained from a contracted transcription service, which is then post-edited by in-house specialists. The main objective of the current project is to replace the initial manual transcription process with an automatic speech recognizer. This is the first time an ASR system is used as a core component in transcription for the Icelandic language, and the purpose of this paper is threefold; to introduce the evaluation procedure, to present the first measurements of the manual post-editing and to report on the performance of the system. Transcription System for Althingi ================================= The current transcription procedure for the Icelandic parliament is done in two stages, illustrated in Figure \[fig:althingi\_workflow\]. The speeches are first created in the Althingi document management system, Documentum, as XML documents, with only the speech meta-data and a link to the speech in the MP3 format. Then, in the manual transcription stage the transcribers listen to the audio and transcribe the speech into the XML document (Text A). The initial transcript is meant to reflect the spoken record as accurately as possible but the transcribers might also enrich the text with minor changes. For example, they might add in different formatting for poems and remove repetitions. Next, in the manual editing stage the XML speech document is sent to the editors who modify the speech to be fit for publication and record their editing time. It is common that an editor corrects transcription errors, fixes grammar or enriches the text with context to make the parliamentary record clearer. Finally, the speeches (Text C) are published to their website. ![Diagram of the transcription and editing process for Althingi. t(d) indicates editing time.[]{data-label="fig:althingi_workflow"}](Althingicurrentprocess.pdf){width="\linewidth"} The main objective of the current project is to replace the first stage, manual transcription, with an automatic speech recognizer. Before the experiment, the in-house specialists gave suggestions regarding relevant data to gather and discussed the important differences between the ASR and manual transcriptions. For the experiment, the manual transcription and ASR transcriptions were done in parallel. With the intent of using Text A as reference material, only the ASR transcriptions received manual post-editing. The experiment was performed for a week; on the first day, only the first stage was tested, to ensure the integration worked as intended. For that week the Icelandic parliament was in session for four days. It is from the last three days that this data was gathered. The ASR system -------------- The details about the preparation of the ASR training data and the development of the ASR can be found in [@Helgadottir2017corpus]. The acoustic model is a deep neural network, based on a recipe developed for the Switchboard corpus[^1], using the Kaldi ASR toolkit [@povey2011kaldi]. It is a sequence trained neural network based on lattice-free MMI [@povey2016purely]. It consists of seven time delay deep neural network layers [@waibel1989phoneme] and three long-short term memory layers [@sak2014long]. The network takes 40 dimensional LDA feature vectors and a 100 dimensional i-vector as input. Two n-gram language models were trained using the KenLM toolkit [@heafield2011kenlm]. The first one is a pruned trigram model, used in the decoding. The other one is a 5-gram language model, trained on the total parliamentary text set, 55M tokens, and is used for re-scoring decoding results. The lexicon is based on the pronunciation dictionary from the Hjal project [@rognvaldsson2003icelandic], available at M[á]{}lf[ö]{}ng[^2]. We added words from the language model training data, which appeared three or more times, with some constraints, resulting in roughly a dictionary containing 200k words. Inconsistencies in the pronunciation dictionary were also fixed. The WER of the ASR, before any post-processing is done, is $9.63\%$ on the test set, using 1500 hours of parliamentary speeches and corresponding text, for training. In real life, this number is going to be higher, partly because of imperfect punctuation reconstruction and disparate casing of many words in our texts, and partly because the ASR test set had been manually cleaned to better match the audio. Automatic post-processing ------------------------- The ASR returns a stream of words with no punctuation or formatting. Since the purpose of the system is to publish parliamentary speeches, human readability needs to be factored into the final transcription. The OpenGrm Thrax Grammar Development tool [@tai2011thrax; @roark2012opengrm] was used to compile grammars into weighted finite-state transducers, in order to denormalize numbers and abbreviations, according to parliamentary conventions. The Punctuator toolkit [@tilk2016bidirectional] is used to restore punctuations in the text, specifically periods, question marks and colons. There are no clear rules for the use of commas in Icelandic, making learning their position difficult. Therefore, no commas are added to the ASR transcripts. Punctuator is a bidirectional recurrent neural network model with an attention mechanism. It can both be trained on punctuation annotated text only, or additionally, take in pause annotated text. Both versions were tested, with the text-only training giving better results, an overall $F1$-score of $86.7$ versus a score of $83.7$ for the two stage training. These errors are obtained on well structured text and are likely higher in the automatically transcribed speeches. The training text set contains roughly 50M words. The development and test set contain 114k and 111k words, respectively. The pause annotated text set contains 1.3M words. The pause annotated development and test sets are each around 81k words. The pause information is obtained from existing alignment lattices, from earlier data preparations before the ASR training. Apart from punctuation, formatting also plays a large factor in human readability. Therefore, Thrax grammar rules are used to capitalize the start of sentences and to collapse expanded acronyms. They are also used for other small formatting, such as timestamps, regulations, time intervals, and websites. Another important task for long texts is adding paragraph insertions. Currently, a new paragraph is only started whenever the speaker of the house is addressed. Integration with the Althingi system ------------------------------------ The ASR needs to connect with the Althingi servers in four different instances. This is enabled for the first three occurrences via a representational state transfer application programming interface (RESTful API). The API first takes in the timestamp of when the speech ends through a GET request. Then, using the ending timestamp, the ASR server queries Althingi’s metadata server to obtain the timestamp of when the speech started. With the two timestamps, the ASR server queries Althingi’s audio server for the audio segment, which the ASR server then downloads. The rest of the experiment is semi-automatic. With the ASR, the audio is then transcribed. Next, the ASR TXT document (Text B of Figure \[fig:user\_test\_1b\]) is batched and wrapped in the speech metadata as well as XML tags. Finally, they are copied from the ASR server and manually entered into the Documentum editor queue. After the speeches are post-edited (Text D), they are posted onto the Althingi website[^3]. Currently, the ASR is housed on its own server and interacts with the rest of the Althingi servers through the RESTful API. The ASR is built on a Ubuntu 16.04 server with 4 CPUs and 16 GB of RAM. The number of parallel transcriptions are limited by the number of CPUs within the server. During the test, 3-4 speeches were processed in parallel because members of parliament tended to deliver speeches faster than the ASR could transcribe. ASR integration concerns ------------------------ Since this integration was only for the experiment, not permanent, the ASR speeches needed to be delivered to the editors while still keeping the existing transcription procedure intact. In order to manage this, automatically transcribed speeches were manually entered into Documentum. To accomplish the goal of keeping the test procedure separate but integrated, Althingi put the ASR speeches in their own separate folder which was then integrated with the normal procedure at the post-editing stage through the Documentum lifecycles. The in-house specialists’ queue only showed the ASR transcriptions. Despite familiarity with the technical details, a deeper understanding of the Althingi speech publishing procedure was needed. Thus, several of their in-house specialists were requested to give valuable insight on important details in the post-editing procedure which could not be gleaned from data. For example, the idea of inserting a new paragraph when parliamentarians address the speaker of the house as it will usually signal a change in topic originated from these specialists. Methodology =========== ![Diagram of the transcription and editing process for the experiment. WED is the word edit distance of Text B from Text A. t(d) is the edit time \[s\] the editors take to post-process the speech. []{data-label="fig:user_test_1b"}](ASRTest2Diagram-manualTrueToAudio.pdf){width="\linewidth"} The primary objective for this experiment is to discover the impact of switching from manual transcriptions to automatic transcriptions on the Althingi publication department’s work. Figure \[fig:user\_test\_1b\] illustrates the experimental setup. First, the speech segment is sent to both the manual transcription stage and the ASR stage. Wherein, Text A and Text B are created. Then, only Text B is sent to the editor queue for the in-house specialists to post-edit and produce the final transcription, Text D. Over a three day period, the Icelandic parliament delivered 279 speeches. However, at the conclusion of the experiment, 35 speeches still hadn’t been processed by the publications department, and 5 speeches were duplicates. Therefore, only 239 speeches could be analysed. The data collected includes the following: speech length, word count, edit time, editor feedback and calculations of the subsequent measures. The system was evaluated using the following measures: 1) word edit distance () \[%\], 2) edit time per word () \[s/w\], and 3) real-time factor () \[-\]. All three measures reflect on an editor’s effort in processing a transcribed speech. The was calculated using the following formula: $$WED = \frac{S+I+D}{N}*100$$ where , , , is the number of substitutions, insertions, deletions and total words respectively, obtained by aligning the texts. This formula is identical to , but since it also reports on the edit distance between transcription and final text, the term word edit distance is preferred. The was computed as the edit time in seconds divided by the speech length in seconds. For Text A, the transcribers were asked to transcribe the audio as true as possible, leaving in speaker errors, in order to get good reference texts. Since this is contrary to the work they normally do, some of the texts were not true-to-audio. The manual transcriptions tended to have small corrections since repetitions are removed, badly structured sentences are corrected, and three words common to parliamentary speeches are abbreviated. In addition, they also contained spelling mistakes and word substitutions due to malformed speech. Despite these flaws, Text A is still the better reference than Text D when estimating errors as both automatic and manual transcription aim to produce audio-to-text transcription. However, it is true that the DB alignment better reflects the work the editors do to make an automatically transcribed text publishable. Hence, one would expect the between Text B and Text D to better explain ET/W than the edit distance between Text A and Text B. The DB results are obtained under the verification and guidance of the AB results. Editors gave feedback in the form of comments and grades for the whole system and on individual speeches. While recording the edit time for a speech the editors were also asked to grade and comment on the speech based on their own perceptions. There were no guidelines other than giving the speeches a grade (Good, Medium, or Bad). Not giving them guidelines better simulates their day-to-day feelings. After the experiment they filled out a short survey with their evaluations of the current procedure versus the inclusion of the automatic transcription system. In the succeeding week speeches were edited with the procedure illustrated in Figure \[fig:althingi\_workflow\] to produce the results for the fully manual procedure, referenced as ’Fully Manual’ later in the the text. They lend insight into the speed of the fully manual transcription process. Results ======= The ultimate goal of the automatic transcription system is to replace human transcribers, so the obvious benchmark to compare against are the results for the fully manual transcription process. This would include matching the and , but not necessarily the word edit distance. **ET/W** \[s/w\] **RTF** \[-\] -------------- ------------------ ----------------- Fully Manual 1.32 $\pm$ 0.51 2.66 $\pm$ 1.05 Automatic 1.52 $\pm$ 0.53 3.26 $\pm$ 1.24 : Influence of the fully manual and automatic transcription on editing effort.[]{data-label="tab:summary"} The DB alignment results are summarized in Table \[tab:summary\]. The analysis shows that the automatic process under-performs when compared to the fully manual process. The is higher by 0.20 s/w, and the by 0.60. The initial hypothesis was that the edit distance would be the main factor affecting the edit time and that the higher the distance, the higher the edit time. Also, that both RTF and ET/W would positively correlate with . In order to confirm this hypothesis, the linear correlation analysis was performed to model the dependence of and on . Also, the Pearson’s correlation coefficient (PCC) between the two variables was computed. The results are as follows: - ET/W = [*0.019*]{} \* WED + [*1.08*]{} - RTF = [*0.03*]{} \* WED + [*2.56*]{} - PCC(WED,ET/W) = 0.33 - PCC(WED,RTF) = 0.22 These are the observations from this analysis: **1)** reducing the ET/W to the manual level would require lowering the to 12.6%, **2)** producing perfect automatic transcriptions can only outperform manual transcription RTF by $\approx4\%$, since, the cost of reading through the transcription, independent of any errors, far outweighs the impact of errors, **3)** the correlation between the variables is low, indicating that WED might not be the best predictor of editing efforts. The following assessment focused on system performance in terms of several error types: ASR, punctuation, capitalization and abbreviation mismatches. The analysis showed that the majority of transcription errors occur due to the ASR and wrong punctuation by far, followed by capitalization, and abbreviation respectively. As a consequence, an improvement to the ASR appears to be of the highest priority. However, in the post-experiment survey the editors frequently commented on inaccurate punctuation which prompted a singling out of punctuation from other errors and to study its affect on editing time. This approach also helped answer the question of whether a certain type of error takes more time to correct than others. The following results distinguish between with punctuation (WED\_wp) and without punctuation (WED\_wop). The data was categorized into two groups, highs (H) and lows (L), representing speeches with high or low WED, with and without punctuation. Table \[tab:avgWED\_DB\] summarizes average edit distance values for DB alignment, and Table \[tab:avgWED\_AB\] for AB alignment. High was chosen as $\mu+0.25*\sigma$ and low as $\mu-0.25*\sigma$. The assumption is that speeches with both high WED\_wp and low WED\_wop will give insight into the effect of punctuation errors on editing time. Likewise, assuming speeches with low WED\_wp and high WED\_wop gives insight into the effect of all other errors on the editing time. The H-H group provides an opportunity to study deficiencies of our system that need to be addressed. The L-L group, on the other, gives an impression of the current performance ceiling. The table shows that excluding punctuation from transcripts improved the WED metric by $\approx6\%$ for both AB and DB alignment in absolute terms, likely due to the removal of the start of sentence capitalization errors introduced by the punctuation module. However, the editors always get transcripts with punctuation, so the values of WED\_wp are more relevant to editing effort. The same assumption is true with regards to DB alignment. **WED\_wp** **WED\_wop** --------- ------------------ ------------------ Average 24.20 $\pm$ 9.56 19.58 $\pm$ 9.59 High $>$ 26.59 $>$ 21.97 Low $<$ 21.81 $<$ 17.18 : The average WED \[%\] for DB alignment and the cut-off points for the High and Low groups.[]{data-label="tab:avgWED_DB"} **WED\_wp** **WED\_wop** --------- ------------------ ------------------ Average 20.12 $\pm$ 5.80 14.35 $\pm$ 4.78 High $>$ 21.57 $>$ 15.45 Low $<$ 18.67 $<$ 13.15 : The average WED \[%\] for AB alignment and the cut-off points for the High and Low groups.[]{data-label="tab:avgWED_AB"} The average values of , and speech count for DB alignment are summarized in Table \[tab:resultsDB\]. Singling out the L-L group also showed that when the transcription system is doing as well as it can, the corresponding (1.36) is comparable to the effort for manual transcripts (1.32). This corresponds to a 3% relative difference. The relative difference for RTF reached about 10%. On the other hand, the relative differences between the H-H group and the Fully Manual process reached 22.8% and 31.9% for ET/W and RTF respectively. The direct comparison between H-H and L-L shows a similar dramatic increase in both measures, clearly proving that the higher the edit distance, the higher the editing effort. The immediate concern for in-between groups is a lack of data to draw statistically significant conclusions. **WED\_wp** **WED\_wop** **\# speeches** **ET/W** **RTF** ------------- -------------- ----------------- ---------- --------- High High 84 1.71 3.51 High Low 1 1.14 1.6 Low High 0 - - Low Low 101 1.36 2.96 : ET/W and RTF results for selected groups for DB alignment.[]{data-label="tab:resultsDB"} The average values of , and speech count for AB alignment is summarized in Table \[tab:resultsAB\]. The general trends for the H-H and L-L groups are identical to the DB alignment, confirming our initial hypothesis. This time, however, there were some points for in-between categories. The data shows that high WED\_wop leads to higher edit times. Therefore, from the numbers themselves one might conclude that punctuation errors take less time to correct than other errors, most being ASR-based errors. This observation is further supported by fixing WED\_wop as H and changing WED\_wp. But the H-L cluster shows lower and than even the manual process, which indicates that the punctuation errors are less severe than the other errors. The main issue, however, with these conclusions are too few data point so further experiments are needed to confirm or deny the validity of this finding. **WED\_wp** **WED\_wop** **\# speeches** **ET/W** **RTF** ------------- -------------- ----------------- ---------- --------- High High 73 1.80 3.86 High Low 4 1.25 2.14 Low High 5 1.43 3.02 Low Low 91 1.37 2.90 : ET/W and RTF results for selected groups for AB alignment.[]{data-label="tab:resultsAB"} Part of the experiment was also to obtain subjective evaluations of the system from the editors. The editors’ independent evaluation of most speeches showed that for the 234 graded speeches 26 were graded as Bad, 105 as Medium, and 103 as Good. Comparing the grades to the H-H/L-L groups, shows that 68%/70% of the L-L speeches were graded as Good and only 2%/1% as Bad, for Texts AB and Texts DB, respectively. In the H-H group 27%/21% of the speeches were graded as Bad and 18%/18% as Good, for Texts AB and Texts DB, respectively. These numbers are highly subjective and vary between editors but give an otherwise hard to obtain insight on the how in-house editors’ opinions line up with the other data. Reading the comments the editors wrote about the speeches in these two groups show some differences. More of the speeches in the H-H groups have comments and the comments are longer. Most prominent are complaints about word substitutions and deletions, as well as incorrect punctuation. For these speeches the editors often mention that the speaker is hard to understand. Comments in the L-L group are fewer. However, complaints about incorrect capitalization are more prominent. Multiple factors also contributed to a higher edit time and WED: dealing with Roman numerals, differences in repetitions or incorrect capitalization. But the edit time alone also had many factors contributing to it other than WED. Editors often formatted the text, such as splitting the speech into paragraphs. Sometimes, editors researched references to bills mentioned in the speeches. Other times, they had to pay close attention to the audio. Members of parliament would sometimes mention named-entities from different languages, which is outside of the scope of a monolingual ASR. Conclusion ========== This paper evaluated the transcription system for the Icelandic Parliament, Althingi. The purpose of the system is to automatically transcribe parliamentary speeches, which are then manually edited by in-house editors and published on the Althingi website. The objective of the analysis was to gain insight on the system performance with respect to editing effort. The analysis focused on determining the relationship of the word edit distance with respect to edit time per word and the real-time factor of edits. The secondary focus was to evaluate the contribution of punctuation-related errors and the quality of automatically produced transcriptions as perceived by the editors. The study shows that editors currently take more time to edit automatic transcripts than manual transcripts, as both observed measures, ET/W and RTF, were higher. Further analysis shows that high WED negatively affects edit time. Improving the automatic transcription to the level exhibited by manual transcription process would require lowering DB WED\_wp to 12.6%. This conclusion is further supported by selectively looking at results for low error rate speeches, as the corresponding ET/W and RTF are similar to the resultant values for the fully manual transcription process. Despite the comments from editors, analyses do not show punctuation having a significant contribution to edit time. Further analysis is warranted before a decisive conclusion on this matter is reached. Based on the fact that only 11% of transcriptions received a bad grade, the Althingi in-house editors were satisfied with the experimental transcription system and its integration. Further teasing out and grouping of the factors would provide useful insights into what else an ASR integration to an existing transcription process requires. [^1]: https://github.com/kaldi-asr/kaldi/blob/master/egs/swbd/s5c/\ local/chain/tuning/run\_tdnn\_lstm\_1e.sh [^2]: http://www.malfong.is [^3]: http://www.althingi.is

--- author: - 'G. Ruffini - Jan 2017 — Starlab Technical Note, TN00344 (v1.0)' bibliography: - 'kolmogorov.bib' title: 'Lempel-Ziv complexity reference' --- Abstract {#abstract .unnumbered} ======== The aim of this note is to provide some reference facts for LZW—mostly from Thomas and Cover [@Cover:2006aa]—adapted to the needs of the Luminous project. LZW is an algorithm to compute a Kolmogorov Complexity estimate derived from a limited programming language that only allows copy and insertion in strings (not Turing complete set). Despite its delightful simplicity, it is rather powerful and fast. We then focus on definitions of LZW derived complexity metrics consistent with the notion of descriptive length, and discuss different normalizations, which result in a set of metrics we call $\rho_0$, $\rho_1$ and $\rho_2$, in addition to the Description Length $l_{LZW}$ and the Entropy Rate. LZW compression: the main concept ================================= The main idea in LZW is to look for repeating patterns in the data, and instead of rewriting repeating sequences, refer to the last one seen [@Lempel:1976aa]. As Kaspar clearly states, LZW is the Kolmogorov Complexity computed with a limited set of programs that only allow copy and insertion in strings [@Kaspar:1987aa; @Ruffini:2016ad].\ “We do not profess to offer a new absolute measure for complexity which, as mentioned already, we believe to be nonexistent. Rather, we propose to evaluate the complexity of a finite sequence from the point of view of a simple self- delimiting learning machine which, as it scans a given n- digit sequence $S=s_{1}\cdot s_{1} \cdot ... s_{n}$ , from left to right, adds a new word to its memory every time it discovers a substring of consecutive digits not previously encountered. The size of the vocabulary, and the rate at which new words are encountered along $S$, serve as the basic ingredients in the proposed evaluation of the complexity of $S$.”\ We consider a string of characters in and alphabet with $A$ symbols (typically binary) of length $n$. From wikipedia: A high level view of the encoding algorithm is shown here: %[frame=single] % Start your code-block %P = [set of parameters] %B= [set of backgrounds] 1. Initialize the dictionary to contain all strings of length one. 2. Find the longest string W in the dictionary that matches the current input. 3. Emit the dictionary index for W to output and remove W from the input. 4. Add W followed by the next symbol in the input to the dictionary. 5. Go to Step 2. After applying LZW, we will end up with a set of words (or phrases, as they are sometimes called) $c(n)$ that go into a dictionary. The length of the compressed string will be $l_{LZW} \leq n$ (the analog of Kolmogorov or algorithmic complexity).\ The [**description length of the sequence encoded by LZW**]{} would have length less or equal to the number of phrases times the number of bits needed to identify a seen phrase plus the bits to specify a new symbol (to form a new phrase), hence[^1] $$l_{LZW} \le c(n) \log_{2} \left[ c(n)+ \log_{2} A \right] \approx c(n) \log_{2} \left[ c(n)\right]$$ The process of digitization =========================== When we digitize (e.g., binarize) a signal prior LZW, we are creating a new string from the data, and we make an explicit choice on what aspects of the data we wish to compress. In this process we destroy information—we are going to do lossy compression. Thus, the choice of digitization results in us having access to a subset of the features of the original string.\ A reasonable strategy is to preserve as much information as possible in the resulting transformed string. In this sense, using methods that maximize the entropy of the resulting series are recommended, such as using the median for thresholding (this is guaranteed to result in $H_{0}=1$)[^2].\ On the other hand, other methods that destroy more information may tap and highlight other, also relevant features of the data. At this stage, then, how to binarize or preprocess (e.g., filter) the original string is an empirical question. The same applies to the choice of compression method, of course, as LZW is just one framework for compression. LZW and entropy rate for stochastic processes ============================================= The main fact from Thomas and Cover [@Cover:2006aa] refers to stochastic random processes $\{X_{i}\}$. A key concept is the [**entropy rate**]{} of the stochastic process, given by $$\mathcal H(X)= \lim_{n\rightarrow \infty} {1\over n} H(X_1, ..., X_n),$$ when this limit exists, with $H$ denoting the usual multivariate entropy of $X$, $ H(X)=-E_{X}[\log(P(X)] $. It is an important theorem that for stationary processes, $$\mathcal H(X) = \lim_{n\rightarrow \infty} H(X_n|X_{n-1}, X_{n-2} ..., X_1).$$ Let also $$H_{0}(p) = -p\log p -(1-p)\log (1-p)$$ denote the [**univariate entropy**]{}, with $p$ the probability of a Bernoulli (binary) process (Markov chain[^3] of order zero).\ We note that entropy rate of a stochastic processes is non-increasing as a function of order, that is, $0\leq \mathcal H \leq .. \leq H_{q} \leq ... \leq H_{0} \leq 1$.\ The fundamental relation is that description length is closely related to entropy rate, $$l_{LZW}= c(n) \log_{2} \left[ c(n)+ \log_{2} A \right] \approx c(n) \log_{2} \left[ c(n)\right] \longrightarrow {n}{\mathcal H}$$ Another important result in what follows is that with probability 1 (Thomas and Cover Theorem 13.5.3) $$\lim_{n\rightarrow \infty} \sup l_{LZW} \leq n \mathcal H$$ which can rewrite as $$\lim_{n\rightarrow \infty} \sup c(n) \log_{2} c(n) \leq n \mathcal H \leq n H_{0}$$ and use to rewrite (in the limit above) $$c(n) \leq \frac{n \mathcal H}{\log_{2} c(n) } \leq \frac{n \mathcal H}{\log_{2} \frac{n \mathcal H}{\log_{2} c(n) }} \sim \frac{n \mathcal H}{\log_{2} n } \leq \frac{n H_{0}}{\log_{2} n }$$ which we use below for normalization purposes. Metrics ======= Two metrics are used in the field, one is $c(n)$ and the other $l_{LZW}$. Of the two the latter is more closely related to Kolmogorov complexity or description length. Both contain similar information (in fact one is a monotonic function of the other). Fundamental Normalization of LZW ================================ The purest way to normalize this metric is to normalize by the original string length $n$ $$\rho_{0} = l_{LZW} / n = \frac{c(n) \log_{2} [ c(n) +A] } {n} \rightarrow \mathcal H$$ with units of bits per character. This is the [**LZW compression ratio**]{}. Other normalizations or measures ================================ A typical normalization adopted by the literature is to “divide by entropy”. By this we mean $\rho_{1}= l_{LZW} / H_{0}$. In the literature this is usually defined through $c(n)$, $$\rho_{1} = \frac{c(n)}{\frac{n H_{0}}{\log_{2} n } } \sim \frac{\mathcal H}{H_{0}} \sim \frac{l_{LZW} }{nH_{0}} \rightarrow \frac{\mathcal H}{H_{0}}$$ (with units of bits per character). Essentially the same can be computed from the randomly reshuffled data series, which with high probability forces $l_{LZW} \sim n H_{0}$ by destroying 2nd order interactions. Hence, $$\rho_{1} \approx \frac{l_{LZW}} { H_{0}} \approx \frac{l_{LZW} }{l^{shuf}_{LZW}}$$ This ratio tells us how much information density is hidden in 2nd and higher order entropy rate as compared to first order one.\ We can think of this a being the comparison of “first order apparent complexity” (entropy) and an estimate of the entropy rate (which provides and upper bound to algorithmic complexity). This is an important comparison, as the proposal in [@Ruffini:2016ac] is that conscious level may be associated to systems that exhibit high apparent entropy with low algorithmic complexity. Alternatively, we could define $$\rho_{2}= H_{0} - \rho_{0} >0$$ which can be interpreted as the extra apparent extra entropy (bits/char) incurred by using first order methods instead estimating the true entropy rate.\ At any rate, from a machine learning point of view it is probably best to compute $\rho_{0}$, $H_{0}$, ..., $H_{q}$ as separate measures.\ Also, it is known that LZW or entropy estimates are sensitive to string length. When comparing metrics across datasets make sure you keep string length constant and as long as possible. LZW and Kolmogorov complexity ============================= As mentioned above, LZW description length is only an estimate of algorithmic complexity. We can easily provide examples of sequences that have low algorithmic complexity, but which are rather hard to compress using LZW. For example, consider the firs $n$ digits of $\pi$. Or consider “the digits of the smallest prime with $10^9$ digits”. The algorithmic complexity of these numbers is very low, but LZW won’t compress them much if at all. Does this mean that LZW is useless? No, but we should keep in mind that it provides only [**an upper bound on algorithmic complexity.**]{} In order to get better bounds, we may consider a family of compressors and use the lowest complexity that any of them can find as a better upper bound. Annex: StarLZW.py {#annex-starlzw.py .unnumbered} ================= [^1]: Actually, we can do a bit better than this. In practice, not all dictionary entries are used. We can use the max dictionary key ID and state that “n bits are needed to describe any key entry, and there are m of them (and here they are)", leading to $ l_{LZW} \le \log_{2}(\log_{2} \max(output))+ \mbox{length}(output) * \log_{2} \left[ \max(output) \right] $, since we need $ \log_{2}(\log_{2} \max(output))$ bits to describe $n$. This is how it is implemented in the appended code. [^2]: Can we generalize this idea? Can we, e.g., binarize the data so that it has maximal $H_{0}$ and $H_{1}$? [^3]: We denote a [**Markov chain of order $m$**]{} to be one where the future state depends on the past $m$ states (time-translation invariantly), $P(X_{n}|X_{n-1}, ..., X_{1})= P(X_{n}|X_{n-1}, ..., X_{n-m})$ for $n>m$.

--- abstract: 'The magnetic structures and the magnetic phase transitions in the Mn-doped orthoferrite TbFeO$_3$ studied using neutron powder diffraction are reported. Magnetic phase transitions are identified at $T^\mathrm{Fe/Mn}_N \approx$ 295 K where a paramagnetic-to-antiferromagnetic transition occurs in the Fe/Mn sublattice, $T^\mathrm{Fe/Mn}_{SR} \approx$ 26 K where a spin-reorientation transition occurs in the Fe/Mn sublattice and $T^\mathrm{R}_N \approx$ 2 K where Tb-ordering starts to manifest. At 295 K, the magnetic structure of the Fe/Mn sublattice in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ belongs to the irreducible representation $\Gamma_4$ ($G_xA_yF_z$ or $Pb''n''m$). A mixed-domain structure of ($\Gamma_1 + \Gamma_4$) is found at 250 K which remains stable down to the spin re-orientation transition at $T^\mathrm{Fe/Mn}_{SR}\approx$ 26 K. Below 26 K and above 250 K, the majority phase ($> 80\%$) is that of $\Gamma_4$. Below 10 K the high-temperature phase $\Gamma_4$ remains stable till 2 K. At 2 K, Tb develops a magnetic moment value of 0.6(2) $\mu_\mathrm{B}/$f.u. and orders long-range in $F_z$ compatible with the $\Gamma_4$ representation. Our study confirms the magnetic phase transitions reported already in a single crystal of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ and, in addition, reveals the presence of mixed magnetic domains. The ratio of these magnetic domains as a function of temperature is estimated from Rietveld refinement of neutron diffraction data. Indications of short-range magnetic correlations are present in the low-$Q$ region of the neutron diffraction patterns at $T < T^\mathrm{Fe/Mn}_{SR}$. These results should motivate further experimental work devoted to measure electric polarization and magnetocapacitance of TbFe$_{0.5}$Mn$_{0.5}$O$_3$.' author: - 'Harikrishnan S. Nair' - Tapan Chatterji - 'C. M. N. Kumar' - Thomas Hansen - Hariharan Nhalil - Suja Elizabeth - 'André M. Strydom' title: 'Magnetic structures and magnetic phase transitions in the Mn-doped orthoferrite TbFeO$_3$ studied by neutron powder diffraction' --- \[INTRO\]Introduction ===================== The orthoferrite \[$R$FeO$_3$; $R$ = rare earth\] oxides have been recently re-investigated experimentally and theoretically from the fascinating perspective of multiferroicity. [@mandal2011spin; @shang2013multiferroic; @deng2015magnetic; @pavlov2012optical; @zhao2014creating] Pursuing the recent line of multiferroics research, theoretical work on $R$FeO$_3$ thinfilms has identified that strain can convert paraelectric phase of orthoferrites in to ferroelectrics thus rendering them multiferroic[@zhao2014creating]. It has been found theoretically that for large values of strain on $R$FeO$_3$ with large rare earth ion, giant polarization is realized. In fact, with increasing strain, a new ferroelectric phase, not observed in any perovskite before, is realized. Multifunctional properties like large magnetoelectric coupling and ultrafast optical control of spins have been observed in the orthoferrites [@tokunaga2008magnetic; @yamaguchi2013terahertz; @mikhaylovskiy2014terahertz]. The $R$FeO$_3$ realize high Néel temperature, $T_N \approx$ 623 -740 K[@marezio1970crystal; @eibschutz1967mossbauer] however, in bulk form they are paraelectric rather than ferroelectric suggesting weak multiferroic effects. Weak ferroelectric polarization has been recently reported in Gd and Sm orthoferrites [@tokunaga2008magnetic; @lee2011spin] which are thought to have “improper” origin induced by magnetic order. In TbFeO$_3$, an unusual incommensurate magnetic phase was discovered through neutron diffraction [@artyukhin2012solitonic] – it was shown that the exchange of spin waves between extended topological defects could result in novel magnetic phases drawing parallels with the Yukawa forces that mediate between protons and neutrons in a nucleus. The Fe$^{3+}$ moments in TbFeO$_3$ exhibit $G_xA_yF_z$ ($Pb'n'm$) spin configuration at room temperature [@bouree1975mise; @bertaut1967structures; @tejada1995quantum] which is accompanied by a spin-reorientation to $F_xC_yG_z$ ($Pbn'm'$). At 3 K, another spin re-orientation occurs to revert to the $G_xA_yF_z$ ($Pb'n'm$) structure. It is considered that the Tb$^{3+}$ spins order in $F_xC_y$ structure in 10 - 3 K interval and in the $A_xG_y$ structure below 3 K. Doping the $R$-site in $R$FeO$_3$ with another rare earth is found to be profitable to realize electric field induced generation and reversal of ferromagnetic moments [@tokunaga2012electric; @tokunaga2014magnetic]. Chemical substitution at the Fe-site in $R$FeO$_3$ also brings about interesting multiferroic effects. For example, in the case of Mn-substituted YFeO$_3$, magnetoelectric and magnetodielectric effects at different temperatures were reported[@mandal2011spin]. First-order spin-reorientation effects were observed as a result of Mn-substitution however, the magnetodielectric effects were observed at lower temperatures than $T_N$ or $T_{SR}$. Giant magnetodielectric coupling is also observed in another doped-orthoferrite, DyMn$_{0.33}$Fe$_{0.67}$O$_3$ [@hong2012temperature]. Spin-reorientation effects and magnetic sublattice effects were also observed in doped-orthoferrites with large $R$[@nagata2001magnetic; @mihalik2013magnetic]. $G$-type magnetic ordering of Mn$^{3+}$ and Cr$^{3+}$ spins were observed below $T_N \approx$ 84 K in the case of TbMn$_{0.5}$Cr$_{0.5}$O$_3$[@staruch2014magnetic], in addition to signatures of short-range magnetic correlations observed below 40 K which was attributed to the ferromagnetic component from canting of magnetic moments along the $c$-axis. In the case of Mn-substituted compound TbFe$_{0.75}$Mn$_{0.25}$O$_3$, the $T_N$ was determined to be 550 K and the $T_{SR}$ as 180 K through magnetic studies and Mößbauer spectroscopy[@kim2011spin].\ In our previous investigation using magnetometry it was inferred that TbFe$_{0.5}$Mn$_{0.5}$O$_3$ orders in $A_xG_yC_z$ ($\Gamma_1$) structure at $T^\mathrm{Fe/Mn}_N \approx$ 286 K followed by a spin re-orientation at $T^\mathrm{Fe/Mn}_{SR} \approx$ 28 K to the structure $G_xA_yF_z$ ($\Gamma_4$)[@hariharan2015reorientation]. No signature of Tb ordering was obtained in the previous study. In the present manuscript, we make a detailed investigation of the magnetic structures and spin re-orientation transitions in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ using neutron powder diffraction in order to confirm the magnetic structure arrived at through macroscopic magnetization earlier. We update the magnetic structures as a function of temperature and observe that they evolve between $\Gamma_1$ and $\Gamma_4$ through mixed-domains of ($\Gamma_1$ + $\Gamma_4$). \[EXP\]Experimental details =========================== Polycrystalline samples of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ were prepared by conventional solid state reaction methods employing the oxides Tb$_2$O$_3$, FeO, MnO$_2$ (all from Sigma Aldrich, 99.9$\%$) as precursors. The thoroughly-mixed powder was heated at 1300$^{\circ}$ C for 4 days with intermediate grinding. The phase-purity of the black powder that resulted was checked using x ray diffraction employing a Philips X’pert diffractometer with Cu-$K\alpha$ radiation. The chemical composition of the prepared sample was determined using the Inductively Coupled Plasma emission Spectroscopy (ICPAES) method. Magnetization measurements were performed on a sintered pellet of the sample in a magnetic property measurement system (MPMS, Quantum Design, San Diego). Neutron powder diffraction experiments were performed on 8 g of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ powder at the instrument D1B in ILL, Grenoble. A wavelength of 2.52 Å was used for the experiment. Diffractograms were recorded at 2 K, 5 K, 10 K, 26 K, 50 K to 100 K in 10 K interval and 100 K to 300 K in 50 K interval. The diffraction data was analyzed using FullProf suite of programs[@fullprof] employing the Rietveld method[@rietveld]. Magnetic structure was determined using the software SARA$h$[@sarah_wills] and was refined using FullProf. \[RESULTS\] Results =================== \[mag\] Magnetization --------------------- The experimentally measured magnetization curves, $M(T)$, in zero field-cooled and field-cooled protocols at 0.02 T are plotted in Fig \[fig\_mag\] (a) and the isothermal magnetization curves of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 2 K and 40 K are presented in (b). Three magnetic phase transitions are identified in (a) [*viz.,*]{} $T^\mathrm{Fe/Mn}_{N} \approx$ 295 K, $T^\mathrm{Fe/Mn}_{SR} \approx$ 26 K and $T^\mathrm{R}_{N} \approx$ 2 K. The magnetic phase transition at $T^\mathrm{Fe/Mn}_{N}$ marks the paramagnetic (PM) to antiferromagnetic (AFM) phase transition. TbFe$_{0.5}$Mn$_{0.5}$O$_3$ adopts $G_xA_yF_z$ magnetic structure below this temperature [@hariharan2015reorientation]. The apparent difference seen in the transition temperatures of the single crystal and the polycrystalline sample might stem from differences in sample quality. As determined by the ICPAES method, the chemical composition of the sample is Tb$_{1.97}$Fe$_{0.51}$Mn$_{0.49}$O$_3$ which is very close to the nominal value. The second transition occurring at $T^\mathrm{Fe/Mn}_{SR} \approx$ 26 K corresponds to the spin-reorientation transition where the structure transforms from $G_xA_yF_z$ to $A_xG_yC_z$. At temperatures close to 2 K, signatures of Tb-order are observed which is reflected in the magnetization measurements at 0.02 T as a irreversibility between the ZFC and FC plots at $\approx$ 5 K. The panel (c) in Fig \[fig\_mag\] magnifies the $M(T)$ curves around $T^\mathrm{Fe/Mn}_{SR}$ where a “loop”-like feature is observed. It can be observed that the “loop”-like feature begins at $\approx$ 36 K and extends till about 18 K. The $M(T)$ at a higher applied field of 1 T was measured where the signs of magnetic phase transitions or bifurcation between ZFC and FC were absent (data not shown). The panel (d) shows the presence of irreversibility extends over a wide temperature range from 36 K to $T^\mathrm{Fe/Mn}_N$. In the panel (e), the derivative $dM/dT$ versus $T$ is plotted to show the $T^\mathrm{Fe/Mn}_{N}$ transition more clearly. It is seen that the $T^\mathrm{Fe/Mn}_{N}$ is a very broad transition with a spread in temperature from about 280 K extending to 303 K. The transition temperature of 295 K is estimated approximately at the point of steepest slope of $dM/dT$.\ 300 K 150 K 26 K 2 K ----------- ------------ ------------ ------------ ------------ -- $a (\AA)$ 5.3055(5) 5.3046(5) 5.3124(4) 5.3138(1) $b (\AA)$ 5.6877(8) 5.6772(6) 5.6676(8) 5.6692(6) $c (\AA)$ 7.5391(9) 7.5270(8) 7.5221(2) 7.5242(4) Tb: $x$ -0.0173(7) -0.0193(7) -0.0223(6) -0.0202(6) $y$ 0.0715(6) 0.0714(6) 0.0732(5) 0.0712(5) $z$ 0.25 0.25 0.25 0.25 O1: $x$ 0.1089(8) 0.1084(9) 0.1091(9) 0.1094(8) $y$ 0.4669(7) 0.4659(8) 0.4665(7) 0.4668(7) $z$ 0.25 0.25 0.25 0.25 O2: $x$ -0.2999(6) -0.3007(6) -0.3007(5) -0.3000(5) $y$ 0.3152(6) 0.3149(5) 0.3145(5) 0.31420(5) $z$ 0.0509(6) 0.0513(6) 0.0549(6) 0.0520(6) : \[tab1\] The refined lattice parameters and fractional coordinates of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 300 K, 150 K, 26 K and at 2 K. These parameters are obtained through Rietveld refinement of the neutron powder diffraction data obtained from the instrument D1B, ILL, Grenoble. The nuclear structure model used was $Pbnm$ with Fe/Mn at $4b$ ($\frac{1}{2}$, 0, 0) and Tb at $4c$ ($x$,$y$,$z$). At 2 K, the maximum magnetization attained with application of 5 T is about $4~\mu_\mathrm{B}/$f.u. This value is lower than the value obtained for ferromagnetic alignment of Fe$^{3+}$, Mn$^{3+}$ and Tb$^{3+}$ moments. The observed maximum moment at 2 K, 5 T is comparable to the moment value obtained on the single crystal of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ in the case of $H \parallel b$ at 25 K[@hariharan2015reorientation]. The first-order spin-flip-like transition observed at $H_c \pm$ 26 kOe is not clear in the present measurement on polycrystalline sample. Though a weak hysteresis is observed at 2 K in Fig \[fig\_mag\] (b), prominent hysteresis loops as observed for $H \parallel c$ in single crystals are absent. These features underline the magnetic anisotropy in the crystal sample of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ [@hariharan2015reorientation]. \[npd\] Neutron diffraction --------------------------- The neutron diffraction patterns at different temperature points were refined using FullProf. The values of refined lattice parameters and fractional atomic coordinates of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 300 K, 150 K, 26 K and 2 K are collected in Table \[tab1\]. The temperature-dependent variation of the lattice parameters $a$, $b$ and $c$ are shown in Fig \[fig\_latt\]. No significant anomalies are observed for the unit cell parameters as a function of temperature except for a change-of-slope at 150 K and 250 K. However, lack of enough data points makes any inference unreliable here. The trend of thermal evolution of $b$ and $c$ are comparable while that of $a$ is opposite to the other two. Jahn-Teller (JT) or pseudo-JT distortion has been correlated with the observation of multiferroicity in perovskites[@bersuker2012pseudo] including TbMnO$_3$ which are known to show significant JT effect[@zhou2007evidence]. In order to investigate the presence of JT effect in TbFe$_{0.5}$Mn$_{0.5}$O$_3$, estimates of the distortion parameters and bond angles and bond distances were obtained from the refined structural data, Table \[tab2\]. However, with the substitution of 50$\%$ Fe at the Mn-site, the effects of JT-distortion are found to have diminished in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ however, effects of distortion of the perovskite structure are clearly seen. To facilitate comparison, in Table \[tab2\], the values of the JT-parameters of TbMnO$_3$ at 300 K taken from Ref.\[28\] [@zhou2007evidence] are given in parenthesis.\ 300 K 150 K 26 K 2 K ----------------------------------- ------------------------- ------------------------- ------------------------ ------------------------ -- Mn–O(2) $l$=2.123(4) $\times$ 2 $l$=2.112(5) $\times$2 $l$=2.114(6) $\times$2 $l$=2.111(5) $\times$2 $s$=1.947(6) $\times$ 2 $s$=1.949(6) $\times$2 $s$=1.956(7) $\times$2 $s$=1.951(7) $\times$2 Mn–O(1) $m$=1.977(8) $\times$ 2 $m$=1.977(7) $\times$ 2 $m$=1.977(8) $\times$2 $m$=1.978(6) $\times$2 Mn–O(2)–Mn 145.81(8) 146.06(9) 145.14(7) 145.81(11) Mn–O(1)–Mn 144.9(12) 144.2(10) 144.04(8) 144.9(9) O(1)–Mn–O(1) 180 180 180 180 O(2)–Mn–O(2) 90.57(12) $\times$ 2 90.60(8) $\times$ 2 90.16(11) $\times$ 2 90.41(8) $\times$ 2 89.53(9) $\times$ 2 89.40(6) $\times$ 2 89.83(9) $\times$ 2 89.59(7) $\times$ 2 $Q_2 = 2(l -s)/\sqrt(2)$ 0.2489 (0.45) 0.2305 0.2234 0.2263 $Q_3 = 2(2m-l -s)/\sqrt(6)$ -0.0947 (-0.2) -0.0874 -0.0947 -0.0865 $\phi$ = tan($Q_3/Q_2$) -20.84$^{\circ}$ (-24) -20.77$^{\circ}$ -22.98$^{\circ}$ -20.92$^{\circ}$ $\rho_0$ = $\sqrt{Q_3^2 + Q_2^2}$ 0.266 (0.5) 0.2465 0.2426 0.2423 A symmetry analysis of $R$FeO$_3$ in $Pbnm$ space group with Fe$^{3+}$ in $4b$ and $R^{3+}$ in $4c$ Wyckoff positions leads to eight irreducible representations, $\Gamma_1$ through $\Gamma_8$, for magnetic structure. For the $4b$ position, the configurations $\Gamma_5$ to $\Gamma_8$ are not allowed and hence $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ and $\Gamma_4$ are selected as the final possibilities. Table \[tab3\] lists these irreducible representations, the Shubnikov space groups and the magnetic structure notations used for $R$FeO$_3$ in general. The neutron diffraction pattern of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 300 K is presented in Fig \[fig\_npd\_300\] as black plus signs. The nuclear structure at 300 K is refined in $Pbnm$ space group. In Fig \[fig\_npd\_300\], the calculated pattern is shown as red solid line, difference curve as gray dotted line and the allowed Bragg peaks for $Pbnm$ space group as vertical bars. From the magnetization measurements presented in Fig \[fig\_mag\] (a, e), it is clear that TbFe$_{0.5}$Mn$_{0.5}$O$_3$ undergoes a magnetic phase transition very close to 300 K (notice from Fig \[fig\_mag\] (e) that the transition extends over a wide range from 280 K to 303 K). Hence, the diffraction data at 300 K is refined with additional magnetic phase. Irreps Space group $4b$ $4c$ ------------ ------------- ------------- ---------- -- -- $\Gamma_1$ $Pbnm$ $A_xG_yC_z$ $C_z$ $\Gamma_2$ $Pbn'm'$ $F_xC_yG_z$ $F_xC_y$ $\Gamma_3$ $Pb'nm'$ $C_xF_yA_z$ $C_xF_y$ $\Gamma_4$ $Pb'n'm$ $G_xA_yF_z$ $F_z$ : \[tab3\] The possible magnetic structures of $R$FeO$_3$ allowed by symmetry. The space group is chosen as $Pbnm$ and $x$, $y$ and $z$ denote orientations parallel to the crystallographic directions $a$, $b$ and $c$. $R$ occupies $4c$ and Fe, $4b$ Wyckoff positions in this structure. In order to solve the magnetic structure at 300 K, the peaks below 2$\Theta \approx$ 40$^{\circ}$ were used to perform a $\bf k$-search to find the propagation vector. The utility called $\bf k$-search within FullProf suite of programs was used for this purpose. Thus, $\bf k$=(000) was identified as the propagation vector. Representation analysis using $\bf k$(000) and $Pbnm$ nuclear cell lead to the listing of four possibilities – $\Gamma_1$ ($Pbnm$), $\Gamma_2$ ($Pbn'm'$), $\Gamma_3$ ($Pb'nm'$) and $\Gamma_4$ ($Pb'n'm$) matching with the selection in Table \[tab3\]. They also match with the magnetic structures of orthoferrites already reported in the literature [@deng2015magnetic]. From the refinement trials it was noted that the representation $\Gamma_3$ contributes zero intensity to the Bragg peaks at (101) and (011) and hence can be excluded. A better visual fit to the experimental data and reasonable agreement factors were obtained for $\Gamma_4$ (the $R_\mathrm{mag}$ factors were, $\Gamma_1\approx$ 70; $\Gamma_2\approx$ 21; $\Gamma_3\approx$ 20 and $\Gamma_4\approx$ 5) and hence was accepted as the solution to the magnetic structure at 300 K. In Fig \[fig\_npd\_300\], the lower set of vertical tick marks correspond to the magnetic Bragg positions.\ After confirming the room-temperature crystal structure to be $Pbnm$ and the magnetic structure as $\Gamma_4$ ($Pb'n'm$), we now discuss the low temperature diffraction data. In Fig \[fig\_npd\], an expanded view of the reflections in the 2$\Theta$-range 31–35$^{\circ}$ is given for 2 K, 90 K and 300 K. At 2 K, the intensity of the (011) reflection is observed to increase, compared to the value at 300 K. The relative intensity, $I_{(011)}/I_{(101)} > 1$ at 2 K and 300 K where as the opposite is true for 90 K. Macroscopic magnetic characterization of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ [@hariharan2015reorientation] clearly suggested spin-reorientations and magnetic phase transformations as a function of temperature and magnetic fields. At 250 K, the nuclear structure was refined in $Pbnm$ and the magnetic structure in $\Gamma_4$ ($Pb'n'm$) similar to the case of 300 K-data. However, at 200 K mixed magnetic domains consisting of ($\Gamma_1$ + $\Gamma_4$) is found to reproduce the experimental data faithfully. Neither of the representations $\Gamma_1$ or $\Gamma_4$ alone could give a satisfactory fit, while the mixed-domain model significantly returned lower values of reliability factors ($R\sim$ 5.3 for mixed-domains whereas $\sim$ 6 for pure phases). A better fit using the mixed-domain model could result from utilizing a bigger parameter space for least-squares however, we also notice the presence of clear irreversibility in the magnetization profiles that are in support of the claim of presence of mixed-domains. The mixed-domains of ($\Gamma_1$ + $\Gamma_4$) were found to exist down till $T^\mathrm{Fe/Mn}_{SR}$ at 26 K. Fig \[fig\_mag\] shows that “loop-like” anomalies are present in the $M(T)$ which correspond to the spin-reorientation transition at $T^\mathrm{Fe/Mn}_{SR}$. In fact, from the panel (c) of Fig \[fig\_mag\] it can be seen that the irreversibility in ZFC and FC curves commences at 36 K itself. Finally at 10 K, the $\Gamma_4$ representation observed at 300 K re-emerges. It was observed that the inclusion of Tb moments in the refinement at 10 K and 5 K did not lead to any appreciable values of refined magnetic moment on the Tb site. At 2 K, the magnetic structure remains in the $\Gamma_4$ representation however, Tb develops a magnetic moment value of 0.6(2) $\mu_\mathrm{B}/$f.u. indicating that Tb is magnetically ordered at this temperature. The irreversibility in $M (T)$ suggesting magnetic ordering in the rare earth sublattice occurs $\approx$ 5 K. However, attempts to refine the 5 K-data assuming magnetic contribution from Tb did not yield better agreement factors. Hence magnetic contributions from Mn/Fe were only considered. The refinement of the magnetic structure of Tb was performed in $F_z$ representation following the symmetry analysis[@deng2015magnetic]. In Fig \[fig\_npd\_refine\] (a)–(e), the refined diffraction patterns of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ at 200, 50, 26, 10 K and 2 K are shown. Though no structural distortions are observed in the entire temperature range of the study, the magnetic structure is seen to evolve between the $\Gamma_4$ and $\Gamma_1$ representation through mixed-domain regions. A schematic of the magnetic structures $\Gamma_1$ and $\Gamma_4$ are shown in Fig \[fig\_magstr\]. In the case of $\Gamma_4$, the $F_z$ type ordering of Tb ions are also depicted (Tb moments are shown in blue). Next, we make an attempt to quantify the volume fraction of the two magnetic domains $\Gamma_1$ and $\Gamma_4$. For this calculation we assume that the total Fe/Mn ordered moment at a particular temperature is equal for both $\Gamma_1$ and $\Gamma_4$ domains. With this assumption and by setting the scale factors of both $\Gamma_1$ and $\Gamma_4$ domains equal to that of the nuclear phase, the ratio of refined ordered moments of $\Gamma_1$ and $\Gamma_4$ should give the percentage of each domains. The result of this calculation is presented as Fig \[fig\_domains\]. As clear from the figure, mixed domain structure is present in the wide temperature range of 250 K to 26 K. Outside this temperature window, the major magnetic phase is that of $\Gamma_4$. \[DISCUSSION\]Discussion ======================== Our investigation of the magnetic structure of TbFe$_{0.5}$Mn$_{0.5}$O$_3$ shows that the predominant structures are $\Gamma_4$ and $\Gamma_1$ however, a competition between these two magnetic phases is evident from the presence of a mixed-domains in certain temperature range. At 295 K, the high temperature paramagnetic phase transforms to the magnetic structure $\Gamma_4$ ($G_xA_yF_z$ or $Pb'n'm$). It is seen that down till $\approx$ 250 K, the $\Gamma_4$ structure remains stable. In the intermediate temperature range close to 200 K a mixed-domains of ($\Gamma_1$ + $\Gamma_4$) is observed which remains down till 26 K. A precise phase space study to determine the boundaries of the mixed phase would demand many more temperature points, but such a detailed study was outside the design and scope of our present experiments. As evidenced by the $M(T)$ curve (Fig \[fig\_mag\] (a)), the $T^\mathrm{Fe/Mn}_{SR}$ is associated with a clear bifurcation of ZFC and FC forming a “loop-like” region between 18 K and 36 K. We deduce that the mixed-domains extend over this temperature window. Further at 10 K, the $\Gamma_4$ structure reemerges and remains till 2 K. In the temperature range 2 - 10 K, the magnetic representation $\Gamma_1$ leads to a high $R_\mathrm{mag}\approx$ 60. It did not adequately account for the magnetic peaks and hence, $\Gamma_4$ ($R_\mathrm{mag}\approx$ 3.5) was chosen as the solution. Note that in both $\Gamma_1$ and $\Gamma_4$, Fe/Mn are constrained to have magnetic moments in $x$, $y$ and $z$ whereas the $R$-moment is constrained to the $z$-direction. We found that in whole temperature range in $\Gamma_1$ setting the Fe/Mn magnetic moments had negligible $x$ and $z$ components (practically zero) and aligned along y-axis (Fig. 6). Also the whole temperature range in $\Gamma_4$ setting Fe/Mn moments have negligible y-component and aligned along $x$-axis with small canting along $y$-direction.\ Different rare earths in $R$FeO$_3$ are observed to have different magnetic structures at low temperatures. For example, DyFeO$_3$ has a $Pb'n'm'$ space group for Dy and $Pnma$ for Fe whereas TbFeO$_3$ has $Pbnm'$ for Tb. In the present case of TbFe$_{0.5}$Mn$_{0.5}$O$_3$, it is found that in the low temperature region significant diffuse scattered intensity is present arising from the short-range magnetic order of Tb. Supporting this feature is the fact that no enhancement on quality of the fit was obtained in refining the 5 K or 10 K diffraction data by introducing a magnetic moment for Tb. Only at 2 K does Tb begin to order as $F_z$ as evident by the development of 0.6(2) $\mu_\mathrm{B}$ for Tb moment. A notable difference of the magnetic structure of Fe/Mn in TbFe$_{0.5}$Mn$_{0.5}$O$_3$ compared to that of the parent compound TbFeO$_3$ is that the latter compound transforms to $\Gamma_2$ ($F_xC_yG_z$) structure at the spin-reorientation transition. In both the cases, the high temperature structure pertaining to Fe/Mn is $\Gamma_4$. In TbFeO$_3$, the Tb moment undergoes two types of ordering below 10 K in to $F_xC_y$ and to $A_xG_y$ whereas in TbFe$_{0.5}$Mn$_{0.5}$O$_3$, Tb develops no significant magnetic moment until 2 K where it orders $F_z$. \[CONCLUSION\]Conclusions ========================= In conclusion, the Mn-doped orthoferrite compound TbFe$_{0.5}$Mn$_{0.5}$O$_3$ orders antiferromagnetically at $T^\mathrm{Fe/Mn}_{N}\approx$ 300 K and undergoes spin-reorientation transition at $T^\mathrm{Fe/Mn}_{SR}$ 26 K. Further, at 2 K the rare earth Tb is found to be magnetically ordered, however, the exact ordering temperature is no confirmed through our study. It is found that in the intermediate temperatures 250 K to 26 K, a mixed-domain model best describe the neutron diffraction data. At 2 K and at 300 K $\Gamma_4$ representation is stable while at the spin-reorientation transition and around 200 K mixed-domains of ($\Gamma_1$ + $\Gamma_4$) exist. An estimate of the domain concentration as a function of temperature is made. Clear indication of diffuse magnetic scattering from Tb is present especially below the $T^\mathrm{Fe/Mn}_{SR}$ while long-range order emerges at 2 K. Acknowledgements {#acknowledgements .unnumbered} ================ H. S. N. acknowledges FRC/URC of UJ for a postdoctoral fellowship. A. M. S. thanks the SA-NRF (93549) and the FRC/URC of UJ for financial assistance. 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--- abstract: | This paper elaborates on relationalism about space and time as motivated by a minimalist ontology of the physical world: there are only matter points that are individuated by the distance relations among them, with these relations changing. We assess two strategies to combine this ontology with physics, using classical mechanics as example: the Humean strategy adopts the standard, non-relationalist physical theories as they stand and interprets their formal apparatus as the means of bookkeeping of the change of the distance relations instead of committing us to additional elements of the ontology. The alternative theory strategy seeks to combine the relationalist ontology with a relationalist physical theory that reproduces the predictions of the standard theory in the domain where these are empirically tested. We show that, as things stand, this strategy cannot be accomplished without compromising a minimalist relationalist ontology. *Keywords*: relationalism, parsimony, atomism, matter points, ontic structural realism, Humeanism, classical mechanics author: - 'Antonio Vassallo[^1], Dirk-André Deckert[^2], Michael Esfeld[^3]' bibliography: - 'references\_fundont.bib' title: Relationalism about mechanics based on a minimalist ontology of matter --- *Accepted for publication in European Journal for Philosophy of Science* From atomism to relationalism about space and time {#sec:motivation} ================================================== Atomism, going back to the pre-Socratic philosophers Leucippus and Democritus and turned into a precise physical theory by Newton, is the most successful paradigm in both classical physics and traditional natural philosophy. On the one hand, it is a proposal for a fundamental ontology that is most parsimonious and most general, applying to everything in the universe. On the other hand, it offers a clear and simple explanation of the realm of our experience. Macroscopic objects are composed of fundamental, indivisible particles. All the differences between the macroscopic objects – at a time as well as in time – are accounted for in terms of the spatial configuration of these particles and its change. However, there is no straightforward answer to the question of what are the atoms. Both Democritus and Newton adopt the view of the atoms being inserted into an absolute space. Consequently, they are committed to a dualism between on the one hand space and on the other hand matter in the guise of atoms filling space. But what is it that fills space? In other words, what makes up the difference between a location in space being occupied by an atom and its being empty? Physical properties taken to characterize matter – such as mass, charge, or spin, etc. – are introduced in physical theories in terms of their causal role for the evolution of the configuration of matter. Consequently, invoking these properties cannot answer the question of what it is that evolves in space as described or prescribed by these properties (see [@Blackburn:1990aa]). To put it differently, the parameters that figure in the equations of a physical theory *presuppose* a spatial configuration of matter to which they are applied. This is particularly evident in the case of the quantum state as represented by the wave function, which is defined on configuration space – that is, the mathematical space each of whose points represents a possible configuration of matter in physical space. Hence, the quantum state presupposes a configuration of objects in physical space to which it is applied. But also in classical mechanics, parameters such as mass and charge presuppose objects given in terms of their spatial location to which they are applied. If one reacts to this situation by taking the objects in space to be bare substrata (cf. [@Locke:1690aa], book II, chapter XXIII, § 2), one runs into the problem that a bare substratum or primitive stuff-essence of matter is mysterious. The same goes for a primitive thisness (haecceity) of the material objects. However, this impasse of not being able to come up with a characterization of matter that stands up to scrutiny arises only if one accepts a dualism of an absolute space and matter as that what fills space. If one abandons this dualism and conceives atomism in terms of relationalism about space, then the spatial relations are available to answer the question of what the atoms are. This is the idea that we shall pursue in this paper, making use of the Cartesian conception of matter in terms of (spatial) extension and the stance of ontic structural realism according to which objects are individuated by the relations in which they stand – i.e. distance relations in our case. In other words, our claim is that atomism, if set out in terms of point particles being inserted into an absolute space, fails to achieve the aim of being a parsimonious ontology. The consequence of this failure is that atomism, thus conceived, is unable to formulate a cogent answer to the question of what the atoms are. To meet the requirement of parsimony, one has to abandon points of space and retain only point particles (matter points), with these point particles standing in distance relations that individuate them. Atomism hence motivates relationalism about space and time instead of being tied to the commitment of an absolute space and time. Furthermore, relationalism is motivated by the fact that the commitment to absolute space and time introduces a surplus structure that is not needed to account for the empirical evidence. Thus, Leibniz points out in his famous objections to Newton’s substantivalism that there are many different possibilities to place or to transform the whole configuration of matter in an absolute space that leave the spatial relations among the material objects unchanged so that there is no physical difference between them (see notably Leibniz’ third letter, §§ 5-6, and fourth letter, § 15, in [@Leibniz:1890aa], pp. 363-364, 373-374, English translation [@Leibniz:2000aa]). However, Leibniz’ objection does not apply to all forms of substantivalism in classical physics, not to mention relativity physics. For instance, it can be circumvented in neo-Newtonian space-time (see e.g. [@Maudlin:1993aa], p. 192; see furthermore [@Pooley:2013aa], for a recent and comprehensive overview of the substantivalism/relationalism debate). Nonetheless, the general objection of introducing a commitment to surplus structure hits any form of space-time substantivalism: assume, as is well supported by all the available physical evidence, that the configuration of matter consists in finitely many discrete objects, such as point particles. If that configuration is embedded in an absolute space, then that space will stretch out to infinity, unless an arbitrary boundary is imposed (at least in a classical setting, since in general relativity the global matter distribution might determine a compact geometry); in any case, it will stretch out far beyond the actual particle configuration. However, all the experimental evidence is one of relative particle positions and change of particle positions, that is, motion. Thus, space is needed in physics only to describe the configuration of matter and, notably, the change in that configuration. Consequently, subscribing to the existence of an absolute space in which that configuration is embedded amounts to inflating the ontology. Against this background, our claim is that in order to accomplish the task of elaborating on a parsimonious ontology of the physical world – at least as far as the setting of classical, pre-relativistic physics is concerned –, only the following two axioms are required: \[a1\] There are distance relations that individuate objects, namely matter points. \[a2\] The matter points are permanent, with the distances between them changing. We submit that these two axioms are necessary and minimally sufficient to formulate an ontology of the physical world in the context of classical, pre-relativistic physics that is empirically adequate, given that all the empirical evidence comes down to relative particle positions and change of these positions. Why should one single out the distance relations? If there is a plurality of objects, there has to be a certain type of relations in virtue of which these objects make up a configuration that then is the world. Generally speaking, one can conceive different types of relations making up different sorts of worlds. For instance, one may imagine thinking relations that individuate mental substances making up a world of minds, etc. Lewis’s hypothetical basic relations of like-chargedness and opposite-chargedness, by contrast, would not pass the test, since, as Lewis notes himself, these relations fail to individuate the objects that stand in them as soon as there are at least three objects ([@Lewis:1986ab], p. 77). When it comes to the natural world, the issue are relations that qualify as providing for extension. That is the reason to single out distance relations. In a future theory of quantum gravity, these relations may be conceived in a different manner than in our current and past physical theories. Nonetheless, we submit that relations providing for extension – namely distances – are indispensable for an ontology of the natural world that is to be empirically adequate. Change in these relations then is sufficient to obtain empirical adequacy. That is the reason to pose the two above mentioned axioms, and only these two ones. Accordingly, distances individuating point-objects that then are matter points and change of these distances are the primitives of a minimalist ontology of the physical world, again at least as far as classical, pre-relativistic physics is concerned. To convey what axiom \[a1\] means, we have to choose a representation. Let us consider a universe consisting of a finite number of $N\in\mathbb{N}$ matter points. Taking the number of matter points to be finite is sufficient for empirical adequacy and will make the following discussion much easier. In order to obtain an ontology that is necessary and minimally sufficient for empirical adequacy, we assume that the set $\Omega$ of all possible configurations of distance relations between $N\in\mathbb{N}$ matter points can be represented as follows: \[def:dist\] Let $\mathcal M=\{1,2,\dots,N\}$ and $\mathcal E=\{(i,j)\,|\,i,j\in\mathcal M, i\neq j\}$. The set $\Omega$ comprises elements $\Delta=(\Delta_{ij})_{(i,j)\in\mathcal E}$ that can be represented by numerical assignments fulfilling the following requirements: (i) \[one\] $\Delta=(\Delta_{ij})_{(i,j)\in\mathcal E}$ is a $\frac{N}{2}(N-1)$-tuple of positive values $\Delta_{ij}\in\mathbb R^{+}$ for each $(i,j)\in\mathcal E$. (ii) \[two\] For all $(i,j)\in\mathcal E$ one has $\Delta_{ij}=\Delta_{ji}$. (iii) \[three\] For all $i,j,k\in \mathcal M$, it is the case that $\Delta_{ij}\leq\Delta_{ik}+\Delta_{kj}$. (iv) \[four\] For all $i \neq j$ there is a $k \neq i,j$ such that $\Delta_{ik} \neq \Delta_{jk}$. Due to requirements (\[one\]) and (\[two\]), the distance relation is irreflexive, symmetric, and connex. Requirement (\[three\]) is the triangle inequality in virtue of which the relation is a distance relation. Requirement (\[four\]) states that the distance relations individuate the matter points: if matter point $i$ is distinct from matter point $j$, there exists at least one other matter point $k$ such that matter points $i$ and $j$ are distinguished by their relation to $k$. Formulating these requirements in terms of numerical assignments is a means to express the features in virtue of which a relation is a distance relation. However, the numerical assignments do not belong to the ontology, let alone the notion of absolute scale that comes with them; they are just introduced for representational purposes. Consequently, the fact that the values assignable to $\Delta_{ij}$ are real numbers does not smuggle in any infinity in this ontology: there is a finite number of $N$ matter points and, hence, finitely many distance relations. Nonetheless, by introducing a labelling $\mathcal M$ of the matter points, this definition can be taken to suggest that their numerical plurality is a primitive fact. But this is just an artifact of the set-theoretical language. The elements of a set $\mathcal{M}$ *qua* set-theoretical objects have to be numerically distinct for $\mathcal{M}$ to be a well-defined set of $N$ objects. However, the referents of this formalism – the matter points *qua* physical objects – are individuated by the distance relations given by $\Delta$, so that these relations account for their numerical plurality. To emphasize the indistinguishability of the matter points in the formalism, it is possible to make the above definition of $\Omega$ independent of the labelling by introducing the following equivalence relation: Take $\Delta,\Delta'\in\Omega$, and consider $\mathbb S_N$ as the set of all possible permutations of elements of $\mathcal M$. We define $\Delta\simeq\Delta'$ if and only if there is a permutation $\sigma\in \mathbb S_N$ such that for all $(i,j)\in\mathcal E$ it is the case that $\Delta'_{ij}=\Delta_{\sigma(i)\sigma(j)}$. The set $$\begin{aligned} \widetilde \Omega = \Omega / {\simeq} := \left\{ [\Delta]_{\simeq} \, \big| \, \Delta \in \Omega \right\}, \qquad [\Delta]_{\simeq} = \{\Delta'\in\Omega\,|\,\Delta'\simeq\Delta\}\end{aligned}$$ then comprises all possible configurations of distance relations independently of a labelling of the matter points. One way to envision an element of $[\Delta]_{\simeq}\in \widetilde \Omega$ is by a representative $\Delta\in\Omega$ that can be viewed as a coloured graph $G(\Delta)=(\mathcal M,\mathcal E,\Delta)$ in which $\mathcal M$ are the nodes, $\mathcal E$ are the edges, and to each edge $(i,j)\in\mathcal E$ the colour $\Delta_{ij}$ is attached. Also the graphs $G(\Delta)$ can be made label-independent by considering the equivalence classes $[G(\Delta)]_{\simeq}=\{G(\Delta')\,|\, \Delta'\simeq\Delta\}$ and treating $G(\Delta)$ only as the corresponding representative of the class. This ontology follows Leibniz’ relationalism about space. According to Leibniz, distances make up the order of what coexists (third letter, § 4, in [@Leibniz:1890aa], p. 363). Distances are able to distinguish objects, thus respecting Leibniz’ principle of the identity of indiscernibles. Hence, in virtue of these relations, there is a configuration of matter points that is constituted through variation in the distance relations that connect the matter points and that make it that these are matter points (in contrast to, say, mind points that are individuated by thinking relations). This ontology furthermore accounts for the impenetrability of matter without having to invoke a notion of mass: for any two matter points to overlap it would have to be the case that there is no distance between them. Consequently, matter is structurally individuated, namely by the distances among the material objects. As the literature on ontic structural realism has made clear, structures in the sense of concrete physical relations – such as distances – can individuate physical objects (see e.g. [@Ladyman:2007aa]). Indeed, structures in this sense can do exactly the same what properties are supposed to do: if one holds that objects are bundles of properties, then the corresponding view is what is known as *radical* ontic structural realism, namely the view that objects are constituted by relations, being the nodes in a network of relations (see [@Ladyman:2007b], chapters 2 and 3, and [@French:2014aa], chapters 5-7). If one thinks that there are underlying substances that instantiate properties, then the corresponding view is what is known as *moderate* ontic structural realism, namely the view that objects and relations are on a par, being mutually ontologically dependent: relations require relata in which they stand, but all there is to the relata is given by the relations that obtain among them (see [@Esfeld:2011aa]). In any case, the fundamental objects do not have an intrinsic nature, but a relational one. In order to obtain the result that the distance relations individuate the matter points and thus distinguish them, one has to require that these relations establish what is known as absolute discernibility in today’s literature: each of the matter points distinguishes itself from all the other ones by at least one distance relation that it bears to another matter point. What is known as weak discernibility in today’s literature would not be enough, since weak discernibility does not avoid having to endorse a given numerical plurality of objects (*contra* our parsimony requirement): for weak discernibility to be satisfied, it is sufficient that objects stand in an irreflexive relation, without there being anything that distinguishes one object from the other ones. Hence, weak discernibility indicates that there is a numerical plurality of objects, but is too weak to individuate the objects (the debate about weak discernibility goes back to [@Saunders:2006aa]; as regards the distance relations, see the exchange between [@Wuthrich:2009aa] and [@Muller:2011aa]). By way of consequence, any model of a theory that qualifies as genuinely relationalist by these standards has to include at least three matter points and has to comply with requirement (\[four\]), thus ruling out notably symmetrical configurations. However, this is no objectionable restriction: having empirical adequacy in mind, there is no need to admit, e.g., worlds with only one or two objects or entirely symmetrical worlds as physically possible worlds (and see [@Hacking:1975aa], and [@Belot:2001aa], for an argument not to admit these as metaphysically possible worlds either). Furthermore, there is no need to abandon absolute discernibility in quantum physics either (since Bohmian mechanics solves the measurement problem by, among other things, respecting the individuality of the quantum particles; we will briefly mention Bohmian mechanics at the end of this paper). In a nutshell, its position in the network of distance relations distinguishes each matter point from all the other ones (absolute discernibility), but there is no fact as to which matter point has the position in question in that network (permutation invariance in the sense that labelling the matter points has no significance). Since all there is to the matter points are the distance relations in which they stand, this is, like Cartesianism, a geometrical conception of matter. However, it is not to be confused with super-substantivalism, that is, the view that space (or space-time) is the only substance and matter a property of space. The main problem for this view is to account for motion, if there are only points of space (or space-time) and their topological and metrical properties, since these cannot move. Indeed, [@Wheeler:1962aa] tried super-substantivalism out in his programme of geometrodynamics, but failed in the attempt to reduce dynamical parameters to geometrical properties of points of space-time (see [@Misner:1973aa], § 44.3-4, in particular p. 1205). By contrast, if there are no points of space or space-time, but only distance relations between sparse points that hence are matter points, all the dynamical parameters that figure in physical theories can then be construed in terms of the role that they play in accounting for the change in these relations, that is, the motion of these points. In short, in a geometrical conception of matter by distance relations between sparse points, there is a clear sense in which there is motion and dynamical parameters capturing motion. However, the substantivalist who accepts a dualism of matter and space can retort that by endorsing an absolute space that underlies the spatial configuration of matter, the substantivalist ontology, although being less simple than the relationalist one, gains in explanatory value. Thus, [@Maudlin:2007aa], pp. 87-89, takes length of a path in space as the primitive notion and derives the notion of distance of point particles from that notion, claiming that he is thus able to explain the constraints on the distance relation (such as the triangle inequality). But there is no gain in explanation here: the relationalist endorses certain relations as primitive. These relations exhibit constraints such as satisfying the triangle inequality. Consequently, in virtue of these relations being subject to these constraints, the world is one of matter points connected by distance relations (by contrast to e.g. a Cartesian world of mind points connected by thinking relations). The substantivalist, to the contrary, traces these distances back to an underlying space. However, this space is construed as the space such that these constraints are satisfied: it comes with a metric in terms of, for example, paths of geodesic motion. Any metric defining a physical space is such that it fulfills all the constraints of three dimensional geometry. Hence, there is no additional explanatory value here in comparison to the relationalist who just presupposes that the relations admitted as primitive fulfill certain constraints; there only is the disadvantage that substantival space contains more structure than is needed to account for the experimental evidence, which consists in relative particle positions and change of these positions. In a nutshell, the substantivalist creates the illusion of giving a deeper explanation of something that, in fact, comes in a package with the postulation of a substantival space. By the same token, the relationalist ontology stated in terms of the two axioms above accepts the whole change in the configuration of matter as primitive – that is, the entire evolution of the distances among the matter points throughout the history of the universe. Again, this is no loss in explanation. Retracing this change to properties of the particles such as mass or charge, to forces or fields or wave functions, etc. does not provide a deeper explanation, since these are dynamical parameters that are *defined* through the causal role that they play for the evolution of the particle configuration. Thus, one does not give a deeper explanation of attractive particle motion in terms of mass or the gravitational force, because these are defined through the effect that they have (or can have or are the power to have) on the motion of the particles. Taking explanations to end in the distance relations among matter points and their change endows this relationalist ontology with all the explanatory value that one can reasonably demand, namely to explain all the other phenomena in terms of the fundamental physical entities. Hence, the argument against admitting dynamical parameters over and above change in distance relations to the ontology is the same as the argument against absolute space: doing so amounts to a commitment to a surplus structure that does not yield an additional explanatory value. Quite to the contrary, it leads to new drawbacks: in the case of absolute space, the commitment to a dualism of matter and space results in the impasse of not being able to come up with a cogent answer to the question of what it is that fills space. In the case of the dynamical parameters, it results in having to answer questions such as how a particle can reach out to other particles and change their motion in virtue of properties that are intrinsic to it (e.g. mass) or how a wave function, being defined on configuration space instead of being an entity in physical space, can influence the motion of matter in physical space. The argument for axiom \[a1\] as well as axiom \[a2\] boils therefore down to this one: bringing in more than what is admitted as primitive in these axioms not only amounts to a commitment to surplus structure, but this surplus structure also introduces new drawbacks, like for example the already mentioned Leibnizian arguments for ontological differences that make no physically observable difference. Let us now consider the second axiom. If there are only matter points connected by distances, all change is change in the distance relations among the permanent matter points. The change in the distance relations (and, hence, the motion of the matter points) can be represented as a parametrized list of states of the configuration of matter, that is, a map $$\begin{aligned} \label{chng} \Delta_{(\cdot)}:\mathbb R \to \widetilde\Omega, \qquad \lambda\mapsto[\Delta_\lambda]_{\simeq}, \end{aligned}$$ which, by means of $\lambda\mapsto[\Delta_\lambda]_{\simeq}$, denotes the change of the distance relations in the configuration of matter independently of a labelling. When formulating a dynamics on this basis, we have to assume that, following axiom \[a1\], the change takes place in such a way that the configuration cannot evolve into a state that violates requirements (\[one\]) to (\[four\]) in definition \[def:dist\]: the distance relations individuate the matter points in any given state of the configuration, and the identity of the matter points across different states of the configuration is provided by their continuous trajectories. Note that, in order for the dynamics to make sense of the notion of particle trajectory, particles *have to be* impenetrable and distinguishable (see [@Bach:1997aa section 1.2.3.], for a rigorous proof of this statement). As this minimalist ontology does not imply absolutism about space, in the same vein it does not imply absolutism about time: time derives from change. Again, one can follow Leibniz for whom time is the order of succession (see notably Leibniz’ third letter, § 4, and fourth letter, § 41 in [@Leibniz:1890aa], pp. 363, 376). Hence, there is no time without change; but change exhibits an order, and what makes this order temporal is that it is unique and has a direction. Although Leibnizian relationalism thus implies that the topology of time induced by the unique ordering of the elements in $\widetilde \Omega$ is absolute, there is no external measure of time: the idea that the global dynamics unfolds according to the ticking of a universal clock is meaningless. If the entire universe could evolve at different external time rates, then two such evolutions would be physically indistinguishable, given that they would consist in the very same sequence of states of the universal configuration of spatial relations. Hence, they would exhibit the very same change in the distances among the matter points. In short, a commitment to an universal external clock would introduce ontological differences that would not make any physical difference (this point is made also in e.g. [@Barbour:1982aa], see especially pp. 296-297). Consequently, there is no absolute metric of time. What we call “time” in this context is just an arbitrary parametrization of the curve $\lambda\mapsto[\Delta_\lambda]_{\simeq}$ on $\widetilde \Omega$ and not, as in the Newtonian case, an additional external variable. For this reason, the only meaningful way to define a clock is to choose a reference subsystem within the universe relative to which time is measured. An example of a simple reference subsystem is the circular motion of a pointer on a dial of a watch, the arc length drawn by the pointer being directly related to the parameter $\lambda$ in the definition of the map $\lambda\mapsto[\Delta_\lambda]_{\simeq}$. Taking the topology of time to be absolute by no means contradicts relationalism, since time depends on the change in the configuration of matter points. This is in line with Mach’s idea that “time is an abstraction, at which we arrive by means of the change of things” [@Mach:1919aa p. 224]. Moreover, endorsing such a unique and directed order allows this relationalism to be compatible with both the $A$-series and the $B$-series view of time. On the latter, there is an objective, directed sequence of states of the configuration of matter so that the change in the configuration is ordered according to “earlier” and “later”; but there is no past, present and future. The $A$-series view designates one state of the configuration as the present one. By the same token, this relationalism is compatible with both eternalism and presentism: one may take the whole stack of states of the configuration of matter to exist, or, endorsing the $A$-series view of time, maintain that only the present state of the configuration exists. The commitment to a unique order of the change in the universal configuration of matter is a necessary condition also for the weaker, minimalist $C$-series view of time that only imposes an order on the change in the configuration of matter, but no direction (see [@McTaggart:1908aa], in particular pp. 458, 461-462). Hence, one may relax the Leibnizian relationalism about time by abandoning the requirement that the order in the change of the universal configuration of matter has a direction. The direction of time may not originate in that order as such, but in a particular initial configuration of matter (such as e.g. a low entropy initial configuration). However, if one also abandons the criterion of that order being unique, one arguably ends up in rejecting any notion of time, since not even the $C$-series can be recovered in this case. Such a radical stance is sometimes taken to be required for a proper understanding of time and change in general relativity (especially in its Hamiltonian formulation, as argued by [@Earman:2002]; but see [@Maudlin:2002aa] against [@Earman:2002]). It is notably defended by [@Rovelli:2004aa] (in particular sections 1.3.1, 2.4, 3.2.4) in the context of quantum gravity (but see [@Gryb:2015aa], in particular pp. 23-24, 27-29, 35-38, for an argument that the relationalist stance to which we subscribe also covers the domain of – quantum – gravity). From ontology to physics: two strategies {#sec:physics} ======================================== Although our concern here is not with necessary connections, we can formulate the minmalist ontology advocated in the preceding section in terms of Humeanism: the distance relations among points, making that these are matter points, and their change are the Humean mosaic. Everything else supervenes on that mosaic in the sense that, as far as physics is concerned, everything else comes in as the means to achieve the description of the distance relations among the matter points and their change throughout the entire history of the universe that strikes the best balance between simplicity and informational content. The argument for this Humean stance is its parsimony together with its empirical adequacy: less than distance relations individuating sparse points that then are matter points and the change of these relations would not do for an ontology of the physical world. Bringing in more is not only not necessary, but would also create new drawbacks instead of providing additional explanatory value. Putting this ontology in terms of Humeanism paves the way for introducing the simplest strategy to link it up with physics. This strategy consists in adopting physical theories as they stand and interpreting them as being committed to no more than this parsimonious ontology. When it comes to describing the distance relations and their change, it is appropriate to represent them as being embedded in a geometrical space (such as Euclidean space) and to introduce dynamical parameters that are constant (such as e.g. mass, charge, total energy, constants of nature) or that have an initial value (such as e.g. momenta, forces, fields, a wave function). This is appropriate because a physical theory seeks for dynamical laws that are such that by specifying an initial configuration of matter and putting that configuration into the law, all the – past and future – change in the configuration of matter is fixed. However, there is nothing about the mere distance relations in a given configuration of matter that yields such a law. In other words, there is nothing in a given configuration that fixes the – past and future – development of that configuration. That is why when seeking for a dynamical law, one usually attributes further parameters – both geometrical and dynamical ones, over and above relative distances that change – to the configuration of matter. However, the minimalist maintains that these parameters are nothing that pertains to the configuration of matter in addition to the distance relations; considering the evolution that these relations take throughout the history of the universe, these parameters are introduced when seeking for a law that describes that history in a simple and informative manner. In brief, whatever geometry and dynamics a physical theory employs, all this apparatus is there only as the means to achieve the description of the change in the distance relations that strikes the best balance between being simple and being informative. Consequently, this apparatus does not introduce ontological commitments that go beyond distance relations individuating matter points and their change. In general, simplicity in ontology and simplicity in representation pull in opposite directions. Using only the concepts that describe what there is on the simplest ontology (matter points individuated by distance relations), the description of the evolution of the configuration of matter would not be simple at all, since one could only dress an extremely long list that enumerates all the change. Reading one’s ontological commitments off from the simplest description – such as e.g. Newtonian mechanics –, the ontology would not be simple at all: it would in this case be committed to absolute space and time, to momenta, gravitational masses, forces, etc. Humeanism allows us to have the best of these two worlds: simplicity in ontology achieved through parsimony and simplicity in description achieved through buying into the simplest physical theory that is empirically adequate. [@Huggett:2006aa] shows how one can understand Euclidean geometry and Newtonian mechanics in a package as the Humean best system for a world of classical mechanics that consists only in distances among point particles and their change (see [@Belot:2011aa], pp. 60-77, for a criticism that spells out how the general objections against Humeanism apply in this case). The idea is that if one considers the entire history of the change in the distance relations in the configuration of matter points of the universe, the spatio-temporal geometry best suited to describe the universe is fixed together with the dynamical laws by the history of these relations as a whole. Given the fact that the change in the distance relations manifests certain salient patterns, Huggett singles out the notion of inertial motion as the idea of a particularly regular and simple motion. He then ties the notion of reference frame to that of a material body at rest at the origin of the frame. This makes it possible to relate different frames by means of continuous spatially rigid transformations. An inertial frame can then be defined as the frame in which the dynamical laws that supervene on the history of relations for the entire universe hold. In this way, the notion of inertial motion neither requires a substantival affine structure that singles out straight trajectories nor an absolute external time to which the uniformity of motion should be referred. Rather, these two structures supervene on purely relational facts. By the same token, absolute acceleration is reduced to the history of change of the spatial relations holding between an inertial and a non-inertial frame. Similarly, the regularities in the history of relations make it that Euclidean geometry is the simplest and most informative geometry representing that history. Such a framework clearly vindicates Leibnizian relationalism about space (spatial relations and their change are the ontological bedrock) and time (temporal facts supervene on the history, i.e. an ordered sequence, of instantaneous distance relations). In the same vein, @Hall:2009aa [§ 5.2] sketches out how dynamical parameters such as both inertial and gravitational mass as well as charge can be introduced as variables that figure in the laws of classical mechanics achieving the simplest and most informative description of the change in the relative particle positions throughout the history of the universe. Furthermore, this strategy has recently been applied to the quantum theories that admit a primitive ontology of matter in physical space in order to solve the measurement problem, such as Bohmian mechanics: in a nutshell, the universal wave function only has a nomological role, being a variable in the law that achieves the simplest and most informative description of the change in the primitive ontology (e.g. relative particle positions) throughout the history of the universe (see [@Miller:2014aa], [@Esfeld:2014aa], [@Callender:2014aa], [@Bhogal:2015aa]). The availability of this Humean strategy shows that there is no point in reading off one’s ontological commitments from the formalism of physical theories: one can be a scientific realist and yet be committed only to distance relations individuating matter points and the change of these relations, whatever other variables may figure in a physical theory that captures that change. To mention a particularly striking example of this fact, consider a model of Newtonian mechanics with an angular momentum of the universe $\mathbf{J}$ that is greater than zero in the centre-of-mass rest frame. Obviously, a rotating universe is not conceivable in a relationalist ontology that admits only distance relations among matter points, but no space in which the configuration of matter is embedded. However, also in a Newtonian ontology of a universe rotating in absolute space, the rotation of the universe would manifest itself in certain changes in the distance relations among the point particles (e.g. in inhomogeneities in the cosmic microwave background with respect to our point of view). Consequently, the relationalist is free to interpret a value of angular momentum of the universe that is greater than zero as a convenient means to capture those changes in a simple and informative manner, without being committed to a space in which the universe rotates: describing those changes by using only variables for the change of relative distances would lead to a law of motion that is extremely complicated. Introducing further variables – such as an angular momentum of the universe that can have a value greater than zero –, by contrast, would enable the formulation of a law of motion that is simple and elegant, but that is there only to capture the change in the relative distances among the point particles. In this manner, the relationalist can handle all kinds of bucket-like challenges. Nonetheless, this strategy cannot recognize all the possible mathematical solutions of the dynamical equations of a physical theory as describing physically possible situations. As already mentioned at the end of the preceding section when discussing time, axiom \[a1\] and requirement (\[four\]) of definition \[def:dist\] pose also a constraint on the dynamics: for instance, an evolution of the distance relations among the matter points that ends up in an entirely symmetrical configuration of the matter points is excluded in the same way as is a symmetrical initial configuration. Such solutions are a mathematical surplus of the formalism; the corresponding points in configuration space do not represent physically possible configurations of matter points. As argued in the preceding section, this is no objectionable restriction: having empirical adequacy in mind, there is no need to admit e.g. entirely symmetrical worlds as physically possible worlds. This Humean strategy is a purely philosophical one: it is always available to vindicate the minimalist ontology set out in the previous section, physics be as it may, as long as all the evidence in fundamental physics comes down to relative particle positions and their change. In a nutshell, on this strategy, *there is a relationalist ontology, but a non-relationalist physical theory*. This is no problem, since the non-relationalist theory can be interpreted in a cogent manner that is consistent with scientific realism as being committed to no more than the parsimonious relationalist ontology. However, one may wonder whether buying into all the formal apparatus of, say, Newtonian mechanics is necessary to achieve a description of the change in the distance relations that meets the standards of simplicity and informational content. This reflection opens up the way for another strategy that can be dubbed *alternative theory strategy*: instead of endorsing physical theories as they stand – such as Newtonian mechanics – and refusing to read ontological commitments off from their formalisms, one constructs alternative physical theories whose formal apparatus stays as close as possible to the ontology of there being only distance relations among point particles and their change and that matches the standard theories in their testable empirical predictions. In a nutshell, this strategy consists in *building a relationalist physical theory on a relationalist ontology*. It is important to be clear about what the alternative theory strategy can achieve and what it cannot achieve: even if the ontology is exhausted by distance relations individuating matter points and the change of these relations, when it comes to formulating a dynamical law capturing that change, further dynamical parameters have to be introduced for the reason given above. Consequently, the alternative theory strategy has to admit dynamical parameters such as mass, constants of nature, initial momenta, etc. and cannot but resort to the Humean strategy in order to ban these parameters from the ontology. However, when it comes to space and time, the aim of this strategy is to avoid quantities that are tied to absolute space and time, such as empty space-time points, absolute velocities, absolute accelerations, or absolute rotations. As regards classical mechanics, the alternative theory strategy goes back at least to [@Mach:1919aa]. In the last three decades, it has been quite exhaustively worked out by Julian Barbour and collaborators (see [@Barbour:1982aa], for the seminal paper that laid down the “best-matching” framework). Furthermore, @Belot:1999aa and @Saunders:2013aa have also proposed each a relationalist theory of classical mechanics. Let us discuss Belot’s proposal first (see also [@Pooley:2002aa], section 5, for an appraisal of the framework). Belot starts by considering the Hamiltonian formulation of classical mechanics given in the language of symplectic geometry. Simply speaking, if we take $\mathcal{Q}$ to be the configuration space of a given system, then the cotangent bundle $T^{*}\mathcal{Q}$ is the phase space of the system (see [@Frankel:1997aa section 2.3c], for technical details). This space comes equipped with a smooth function $H$, called the Hamiltonian, which – roughly – represents the total energy of the system, and a 2-form $\omega$, called the symplectic form. Glossing over the technical aspects, $\omega$ renders it possible to define a map $H\mapsto X_{H}$ that associates to the Hamiltonian a smooth vector field $X_{H}$ over $T^{*}\mathcal{Q}$: such a map is nothing but an intrinsic representation of the usual Hamilton’s equations. Hence, by integrating $X_{H}$ given some initial conditions $(\mathbf{q}_{0},\mathbf{\dot{q}}_{0})$, we get the unique curve in $T^{*}\mathcal{Q}$ that represents the dynamical evolution of the system under scrutiny. In the case of $N$ gravitating particles, $T^{*}\mathcal{Q}$ will be nothing but $\mathbb{R}^{6N}$. The justification of this fact is straightforward, if we consider what it takes to determine an initial condition $(\mathbf{q}_{0},\mathbf{\dot{q}}_{0})$: for each particle, we have to specify three numbers that give its position and further three numbers that specify the velocity vector “attached” to it. We immediately see in what sense this framework naturally fits a substantivalist understanding of space: two $N$-particle states that agree on all relational facts about the configuration (not only the relative positions, but also the relative orientations of the velocities), but disagree on how such a configuration is embedded in Euclidean space, would count as physically distinct possibilities. Then, the natural relationalist move would be to construct a relational configuration space $\mathcal{Q}_{0}$ by quotienting out from $\mathcal{Q}$ all the degrees of freedom associated with an embedding in Euclidean space, such as rigid translations and rotations. If we call $E(3)$ the set of isometries of Euclidean $3$-space, then $\mathcal{Q}_{0}=\mathcal{Q}/E(3)$: this construction assures us that distinct points in $\mathcal{Q}$ that represent the same relational configuration “collapse” to the same point in $\mathcal{Q}_{0}$. Note that (i) $\mathcal{Q}_{0}$ admits a well-defined cotangent bundle $T^{*}\mathcal{Q}_{0}$, which is equipped with a well-behaved symplectic structure, and (ii) that the starting Hamiltonian defined on $\mathcal{Q}$ admits a smooth projection $H_{0}$ to $T^{*}\mathcal{Q}_{0}$ because it is invariant under the action of $E(3)$. Belot’s theory qualifies as relational, since the ontological facts making up a set of initial data do not encode any notion of position in absolute space or absolute velocity, and the laws of motion specify how these initial data evolve; furthermore, the dynamical laws of the theory are fully defined on relational phase space. However, there are at least three concerns that one can raise about Belot’s proposal. In the first place, this theory still has a notion of absolute time inherent in the dynamical laws. In fact, there is nothing in the quotienting out procedure that leads from a dynamics over $\mathcal{Q}$ to one over $\mathcal{Q}_0$ that eliminates the absolute temporal metric of Newtonian mechanics, which means that the very same succession of purely relational configurations can unfold at different rates depending on the ticking of an universal external clock. Secondly, there is a clear sense in which spatial relations are Euclidean from the beginning: they are just equivalence classes of embedding degrees of freedom as encoded in $E(3)$. Thirdly, Belot’s theory, despite being very close to Newtonian mechanics, is not as empirically predictive as its absolute counterpart. This is obvious because, if we think about all the initial data that are needed in the Newtonian theory, we realize that they must include the rate of change in the orientation of the configuration of $N$ particles with respect to absolute space: in the passage from $\mathcal{Q}$ to $\mathcal{Q}_{0}$, this information is simply washed away. In particular, the Newtonian theory admits models with non vanishing total angular momentum $\mathbf{J}$ of the universe. We repeat that this is not just a metaphysical aspect, but a physical one in the sense that the condition $\mathbf{J}\neq 0$ carries with it empirically testable consequences. Belot’s proposal to overcome the problem is just to bite the bullet: his relational reduction indeed recovers only a part of the Newtonian one, but – given that up to now we have reliable experimental evidence that the universe is not rotating – it recovers exactly the empirically adequate part. Let us turn to Barbour’s proposal (see [@Barbour:1982aa; @Barbour:2003aa; @Barbour:2012aa] for the original resources; [@Pooley:2002aa], sections 6-7, give an excellent overview of the framework, together with some cogent philosophical considerations). Barbour’s relationalist motivations are the same as Belot’s, that is, to eliminate all the spatial degrees of freedom that produce no observable difference. However, Barbour extends this requirement to temporal degrees of freedom as well. By way of consequence, the construction of his framework involves two steps, namely the implementation of (i) spatial and (ii) temporal relationalism. As regards the first step, Barbour adopts the same strategy as Belot: he takes standard configuration space $\mathcal{Q}$ and quotients out all Euclidean isometries, comprised of scale transformations, which means that he quotients out also the degrees of freedom related to “stretchings” or “shrinkings” of configurations that preserve the ratio of distances. This means that he considers a wider group than $E(3)$, namely the similarity group $Sim(3)$. Hence, his relational configuration space $\mathcal{Q}_{0}=\mathcal{Q}/Sim(3)$ is aptly called *shape* space, because each configuration in there is individuated by its form and not by its size. The second step is technically more complicated: firstly he defines an “intrinsic” difference that measures how similar two shapes are. This difference is expressed in terms of “best-matching” coordinates. Intuitively, we imagine the two shapes laid down over two distinct Cartesian coordinate grids $O$ and $O'$; then we hold fixed the first shape and grid and “move” the second by applying transformations in $Sim(3)$ until the two shapes are juxtaposed as close as possible. The best-matching coordinates are then defined as the overlap deficit $O-O'$ between the two coordinate grids. Secondly, he uses this intrinsic metric to define a Jacobi action, thus setting a Jacobi variational principle on $\mathcal{Q}_{0}$ (see [@Lanczos:1970], pp. 132-140, for a technical introduction to the Jacobi principle in classical mechanics). The Jacobi action is reparametrization invariant, that is, it does not change whatever “time” parameter we choose. With this machinery in place, carrying out the variation of the action with respect to the best-matching coordinates, we obtain a set of generalized Euler-Lagrange equations whose integral curves are nothing but the geodesics of $\mathcal{Q}_{0}$. Given some initial conditions, one of these curves is singled out, which represents the dynamical evolution of the system. This evolution is given in fully relationalist terms: the curve singled out by the equations plus the initial conditions represents a list of relational configurations, which is parametrized by an arbitrary monotonically increasing parameter: hence, there is no external clock that measures dynamical change; on the contrary, it is the change in the list of configurations that enables an (arbitrary) parametrization. The important point is that there exists a particular parametrization of the curve for which the generalized Euler-Lagrange equations take the usual Newtonian form. Thus, if we adopt this (again, arbitrary) parametrization, we obtain a dynamical description that matches the Newtonian one. In this sense, Newtonian mechanics comes out of Barbour’s framework by means of something closely resembling a gauge fixing. The descriptive simplicity of the Newtonian formulation then explains why, historically, classical physics was framed in these terms. There are at least three critical points about this framework worth being highlighted. Firstly, no usual Newtonian potential is compatible with the condition of scale-invariance. Even if it is always possible to reproduce the form of the most usual classical potentials in the appropriate gauge by a clever mathematical manoeuvre, still this mimicking strategy might lead to unwanted physical restrictions, such as no angular momentum exchange between subsystems (see [@Anderson:2013aa section 5.1.2], for a technical discussion of this point). Secondly, the implementation of a geodesic principle on a general shape space might not always be that straightforward. In general cases, in fact, the quotienting out procedure sketched above leads to a shape space whose global geometry is that of a stratified manifold, where each “stratum” is a sub-manifold that can differ from the others in many respects, including the dimensionality. It is then quite intuitive to understand that, if $\mathcal{Q}_{0}$ is a stratified manifold, it is problematic to account for a dynamical evolution given in terms of a geodesic trajectory that hits different strata of $\mathcal{Q}_{0}$ (see [@Anderson:2015aa section 9.4 and references mentioned therein], for discussion). The moral is that Barbour’s framework works well in a suitably small region of $\mathcal{Q}_{0}$, but might break down on a larger scale, depending on the particular geometrical structure of $\mathcal{Q}_{0}$. Thirdly, as a result of quotienting out the group of rotations from $\mathcal{Q}$, one gets the condition $\mathbf{J}=0$ as a constraint on shape space dynamics. However, even if the actual universe satisfies the condition of $\mathbf{J}=0$, it is desirable that a relationalist theory should be able, as far as possible, to accommodate observable consequences ascribable in absolute terms to a non-vanishing total angular momentum of the universal configuration of matter. Finally, let us consider the proposal spelled out in @Saunders:2013aa. Like Belot and Barbour, Saunders’ aim is to dispense with absolute quantities of motion. However, unlike the former two, he also seeks to save the core conceptual structure of Newton’s *Principia*. In order to do so, he shows that the Newtonian laws can be cast in terms of directed distances representing inter-particle separations. This is possible because the absolute notion of “straight trajectory” needed to make sense of inertial motion – which, in turn, is required for rendering Newton’s first and second law meaningful – involves too much structure, namely, a privileged affine connection (that of neo-Newtonian space-time). Instead, Newton’s laws can make perfect sense even if we replace the talk of straight trajectories with that of relative velocities not changing over time, and this can be accounted for not just by a single preferred connection, but by a whole class of affine connections whose time-like geodesics are mutually non-rotating. This is all that Saunders needs in order to account for accelerations and rotations in relationalist terms: a space-time manifold equipped with enough structure to allow for the comparison of spatial directions (and related angles) at different times (what Saunders calls “Newton-Huygens” space-time, see also [@Earman:1989aa], pp. 31-32, for a formal characterization of this space-time). In the Newton-Huygens space-time, unlike the neo-Newtonian one, it is meaningless to talk about the absolute acceleration of a particle, or even its inertial motion (two notions that are tied to a privileged affine connection), while it is perfectly meaningful to ask questions about the change of orientation of a configuration in time. The huge virtue of this framework is that it is able to recover the full spectrum of Newtonian models, thus accounting also for $\mathbf{J}\neq0$ cases, without invoking absolute notions. Given, in fact, that differences in direction can be defined relationally by admitting a primitive notion of parallelism, and then defining change in direction by comparison of spatial relations at different times, Saunders’ theory can account for global rotations in terms of relational quantities. However, the fact that this theory makes it meaningful to compare spatial directions at different times represents a substantial weakening of the relationalist programme. This framework is much less relationalist than Belot’s and Barbour’s, for which the excision of any physical meaning attached to global rotations is a constitutive feature. In this sense, Saunders’ theory is a “halfway house” form of a relationalism, as he himself notes ([@Saunders:2013aa], p. 44). In sum, there are well-grounded reservations whether these relationalist theories fully implement a relationalist ontology. That ontology is relationalist both with respect to space and with respect to time, whereas Belot’s and Saunders’ theories are relationalist only with respect to space. Furthermore, that ontology is not tied to a particular geometry such as Euclidean geometry: a configuration of matter points satisfying axiom \[a1\] and definition \[def:dist\] does not carry with it any primitive geometrical fact that singles out a distinguished space in which it has to be embedded. By contrast, there is a clear sense in which Belot’s and Barbour’s spatial relations are Euclidean from the outset: each point in $\mathcal{Q}_0$ can be seen as an equivalence class of Euclidean configurations; there is no way, by fixing a certain gauge, to end up with a configuration embedded in a non-Euclidean space. Also Saunders’ relations are inherently Euclidean, since Newton-Huygens space-time encodes the structure of a series of instantaneous $3$-dimensional affine spaces equipped with an Euclidean metric. Even worse, all these theories rely on more primitive structure than just distances. The very concept of shape requires primitive facts about angles to be meaningful, so that Barbour’s ontology has to include a conformal structure. Saunders’ ontology requires not only distances, but *directed* distances, which means that some primitive geometrical facts have to be postulated (especially those making up a standard of space-like parallel transport), which are encoded in an affine structure. At this point, one may legitimately ask whether it is possible to resort to the Humean strategy to argue that the additional structure of these theories is just part of the package we get when seeking for the best description of motion. The answer is that such a move, in this case, raises substantial worries. Put simply, a Humean justification of the surplus structure would imply that what these theories do is basically to “embellish” a set of relational initial data $\Delta_0$ by embedding them in a more structured set $Q_0$ and then using the equations of motion cast in terms of this surplus structure to evolve these data until reaching the result $Q_t$, from which the relational solution $\Delta_\lambda$ would be read off. But that would have unwanted implications. The first of these is that, in this way, neither of the above theories could be considered even mildly relationalist anymore, being parasitic on a dynamical description that involves an irreducible surplus structure. In such a case, it would be awkward to prefer these mathematically elaborated theories to Newtonian gravitation, given that Huggett’s Humean strategy works perfectly for the purposes of a relationalist ontology in that context. The second implication is that the strategy of evolving relational data by “stealing a ride” to a non-relationalist dynamics and then discarding the surplus structure as a mere representational means would suspiciously look like a trivial instrumentalist move, as discussed, e.g., by @Earman:1989aa [p. 128] and @Belot2000:aa [p. 10]. In short, combining a minimalist relationalist ontology with the alternative theory strategy faces a dilemma: either one insists that angles or directions are just part of the Humean package, thus ending up with – using Earman’s words – a “cheap instrumentalist rip-off” of a theory that in any case does not qualify as a genuine relationalist competitor to Newtonian mechanics; or one bites the bullet and introduces more primitive structure in the ontology (be it a conformal or an affine one), thus compromising the original motivations for relationalism from ontological parsimony. Let us now briefly consider how the Humean and the alternative theory strategy fare when it comes to quantum mechanics. Adopting relationalism about space and time is a reasonable option if one pursues a solution to the quantum measurement problem in terms of being committed to what is known as a primitive ontology of matter being distributed in ordinary space-time (and not just the quantum state defined on configuration space). Bohmian mechanics is the most prominent primitive ontology theory of quantum mechanics, setting out a primitive ontology of point particles moving in a three-dimensional, Euclidean space (see [@Durr:2013aa]). Given that the laws of Bohmian dynamics, although being different from the Newtonian ones, are nonetheless formulated over an absolute Newtonian spatio-temporal background, all the options for relationalism discussed in this section can be applied to Bohmian mechanics as well. Concerning the strategy to interpret a non-relationalist theory as being committed only to a relationalist ontology, we already mentioned the Humean treatment of the wave function at the beginning of this section. Space-time in Bohmian mechanics poses no problem for Humeanism: the manner in which [@Huggett:2006aa] deals with space-time in the Newtonian case can simply be applied to the Bohmian case. As regards the alternative theory strategy, the proposal of [@Belot:1999aa] would require forcing the Bohmian formalism in a Hamiltonian context; this is possible, but results in a quite unreasonably complicated formalism with a huge amount of surplus descriptive structure, which is not needed by Bohmian dynamics (see [@Holland:2001aa; @Holland:2001ab] for a decently worked out Hamiltonian version of Bohmian mechanics). Barbour’s framework, by contrast, can in a natural way be applied to Bohmian mechanics (see [@Vassallo:2015aa], [@Vassallo:2016aa]). The same goes for the milder relationalist approach of [@Saunders:2013aa], since this approach would basically amount to rewrite the Bohmian theory in a way that makes it meaningless to refer distances and directions to any point taken as “origin”. We cannot go into relativity physics in this paper for lack of space. We would just like to make two brief remarks concerning quantum field theory and general relativity theory, respectively. (i) The relationalist ontology set out in section 1 can be carried on to quantum field theory: this theory is hit by the measurement problem in the same way as quantum mechanics (see [@Barrett:2014aa]). The Bohmian solution to the measurement problem can be applied to quantum field theory in the same way as to quantum mechanics, even in the form of an ontology of permanent point particles moving according to a deterministic law that explains the statistics of the appearances of particle creation and annihilation phenomena in the experiments (this approach is known as Dirac sea Bohmian quantum field theory; see [@Colin:2007aa], [@Deckert:2010aa], chs. 6-7). However, it is an open issue whether and how the relationalist approaches discussed in this section can be carried on from Bohmian mechanics to such a Bohmian quantum field theory. \(ii) When it comes to general relativity theory, also in this domain, as in any other field theory, fields are tested by the motion of particles. There is no direct evidence of fields. All the evidence is one of relative particle motion (cf. [@Einstein:1949aa], p. 209). Against this background, a Humean strategy of treating the formal apparatus of general relativity theory as being the means to achieve a description of the overall relative particle motion that strikes the best balance between being simple and being informative about that motion seems to be an option that would be worth trying out (see [@Vassallo:2016ab], for a concrete proposal in this sense). As regards the alternative theory strategy, Barbour and collaborators have developed an alternative theory of gravitation in the relativistic regime (see notably [@Barbour:2002aa], [@Barbour:2012aa]), which has recently attracted attention in the philosophical literature (see e.g. [@Gryb:2015aa]). In conclusion, we have seen that, when vindicating a relationalist ontology for classical mechanics consisting of matter points individuated by distance relations and the change of these relations only, the Humean strategy of combining a relationalist ontology with a non-relationalist physical theory and interpreting that latter theory as being committed to no more than the minimalist relationalist ontology is coherent and always available as a fall back option for the relationalist. Pursuing the more ambitious strategy of developing a relationalist physical theory that is an alternative to the standard non-relationalist physics can, as things stand, not be carried out without compromising the relationalist ontology – at least by including a primitive conformal structure, or even by including a primitive affine structure (when seeking to recover all models of Newtonian mechanics that may have observable consequences). #### Acknowledgements. We are grateful to Vincent Lam, Dustin Lazarovici, Andrea Oldofredi and Christian Wüthrich for helpful discussions. A. Vassallo’s work on this paper was supported by the Swiss National Science Foundation, grant no. 105212\_149650, while D.-A. Deckert’s work was funded by the junior research group grant *Interaction between Light and Matter* of the Elite Network of Bavaria. [^1]: Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: <Antonio.Vassallo@unil.ch> [^2]: Ludwig-Maximilians-Universität München, Mathematisches Institut, Theresienstrasse 39, 80333 München, Germany. E-mail: <deckert@math.lmu.de> [^3]: Université de Lausanne, Faculté des lettres, Section de philosophie, 1015 Lausanne, Switzerland. E-mail: <Michael-Andreas.Esfeld@unil.ch>

--- abstract: 'We present measurements of the fluctuation superconductivity in an underdoped thin film of La$_{1.905}$Sr$_{0.095}$CuO$_4$ using time-domain THz spectroscopy. We compare our results with measurements of diamagnetism in a similarly doped crystal of La$_{2-x}$Sr$_x$CuO$_4$. We show through a vortex-plasma model that if the fluctuation diamagnetism solely originates in vortices, then they must necessarily exhibit an anomalously large vortex diffusion constant, which is more than two orders of magnitude larger than the Bardeen-Stephen estimate. This points to either the extremely unusual properties of vortices in the under-doped $d$-wave cuprates or a contribution to the diamagnetic response that is not superconducting in origin.' author: - 'L.S. Bilbro' - 'R.Valdés Aguilar' - 'G. Logvenov' - 'I. Bozovic' - 'N.P. Armitage' title: 'On the possibility of fast vortices in the Cuprates: A vortex plasma model analysis of the THz conductivity and diamagnetism in La$_{2-x}$Sr$_{x}$CuO$_4$' --- Nearly 25 years after the demonstration of high-temperature superconductivity in the cuprate superconductors and more than 15 years since the discovery of the anomalous pseudogap in underdoped compounds, the microscopic physics of the superconducting phase and its relationship to the pseudogap remain hotly debated. Due to their low superfluid densities, it is generally agreed that superconducting fluctuations will be large and prominent in these materials [@Emery95a]. What is less agreed upon is the temperature range above $T_c$ in which superconducting correlations are truly significant and their contributions to the physics of the pseudogap. Experimental probes such as photoemission, tunneling, NMR spin relaxation, heat capacity, the Nernst effect, and diamagnetic susceptibility have shown evidence for a gaplike structure reminiscent of $d$-wave superconductivity in the density of states implying a strong connection of the pseudogap to superconductivity and/or superconducting correlations at temperatures well above $T_c$ [@Timusk99a; @Norman05a; @Xu00a; @Wang05a; @Li10a]. However, other mechanisms exist that can create such structures in the density of states [@Hlubina95a; @Chakravarty01a]. {width="2.1\columnwidth"} Interestingly, perhaps the most essential probe of the electronic properties – charge transport – does not show an extended range of superconducting fluctuations in temperature or field. [@Corson99a; @Miura02a; @Ando95a]. In La$_{2-x}$Sr$_x$CuO$_4$ the region of enhanced diamagnetism extends almost 100 K above $T_c$ [@Li10a] while the THz fluctuation conductivity has an extent limited to 10 - 20 K above $T_c$ [@Bilbro11a]. This is surprising as one might expect a close correspondence between these quantities [@Halperin79a]. Similarly, it has been argued from Nernst and diamagnetism measurements that $H_{c2}$ may be as high as 150 T [@Li10a], while the resistive transition is essentially complete in optimally and underdoped LSCO by 45 T [@Miura02a; @Ando95a]. In this Rapid Communication we present results of our detailed THz time-domain spectroscopy (TTDS) study of the fluctuation superconductivity in LSCO. The THz fluctuation conductivity shows an onset approximately only 10 K above $T_c$, which contrasts strongly with measurements like diamagnetism in which the onset is approximately 100K above $T_c$. We analyze our data in the context of a vortex plasma model and show, however that it is not the functional dependences of these data that are in strongest contrast, but their overall scales. Conventional vortex dynamics would predict a much larger fluctuation conductivity given the size of diamagnetism. We demonstrate that if the regime of enhanced diamagnetism originates in vortices, then the vortex diffusion constant $D$ must be anomalously large and in the range of 10-30 cm$^2/$sec above $T_c$. This is more than two orders of magnitude larger than conventional benchmarks based on the Bardeen-Stephen model [@Stephen65a]. It is then a well-posed theoretical challenge to explain a $D$ this large. This points to either extremely unusual vortex properties in the underdoped $d$-wave cuprates or a contribution to the diamagnetic response that is not superconducting in origin. We begin with the observation that the ratio $\chi_{2D}/\mu_0 G$ of the two-dimensional (2D) susceptibility over the conductance has units of length squared over time, i.e., diffusion [@Torron94a]. One can show that in a diffusive vortex plasma this ratio gives a unique measure of the vortex diffusion constant[@Orenstein06a]. Using the notation of Halperin and Nelson [@Halperin79a], but in SI units, the 2D susceptibility and conductance of a conventional thin superconducting film at temperatures above a vortex unbinding transition are $$\begin{aligned} \chi_{2D} = - \frac{ c_2 \pi^2\mu_0 k_B T}{\phi_0^2} \xi^2 \label{Suscp}\\ G_S = \frac{1}{\phi_0^2 n_f \mu. } \label{Cond}\end{aligned}$$ Here $\xi$ is a correlation length, $\phi_0$ is the flux quantum, and $\mu$ is the vortex mobility. $n_f$ is the areal density of thermally excited free vortices, which is related to the correlation length by the relation $n_f =1 / 2\pi c_1 \xi^{2}$. $c_1$ and $c_2$ are small dimensionless constants. It is reasonable to expect that very close to $T_c$ vortices are the principal degrees of freedom in even quasi-2D materials. Note that these are essentially model-free forms constrained only by dimensional analysis, Maxwell equations, and immutable properties of superfluid vortices like the Josephson relation. Using accepted values for $c_1$ and $c_2$ [@Halperin79a], and the Einstein relation $D = \mu k_B T$, the expression $$D(T) = - \frac{6}{ \mu_0 } \frac{\chi_{2D}}{G_S}. \label{ratio}$$ follows [@Orenstein06a] and in principle may be used to give a determination of the vortex diffusion constant $D$ using only experimentally determined quantities. Interestingly, this treatment using the analogous equations within the Gaussian approximation and in the dirty limit gives the diffusion constant of the normal state $electrons$. This is potentially useful as a diagnostic considering that electronic diffusion is proportional to the normal-state conductance while vortex diffusion is conventionally proportional to the normal-state resistance. One may also heuristically motivate Eq. \[ratio\] through the fact that correlations in length (diamagnetism in 2D $\propto \xi^2$) probed by a thermodynamic measurement like susceptibility and the correlations in time ($1/\Omega$) probed by a dynamic measurement like conductivity are related within diffusive dynamics as $\xi^2 \propto D/ \Omega$, where $\Omega$ is the characteristic fluctuation rate. A problem with applying Eq. \[ratio\] to real type-II superconductors is that, in general, the motion of vortices is limited by both dissipative (viscous) flux-flow and pinning forces. In 2D, the classical equation of motion for a single vortex is $ \dot{x} / \mu + k_p x = K_y \phi_0$ where $K_y$ is a driving sheet current, $x$ is the vortex displacement and $k_p$ is a pinning constant [@Gittleman66a]. Here the complex physics of pinning and flux-flow are represented by phenomenological parameters. This leads to an expression for the 2D resistance from moving vortices as $R_v = \phi_0^2 n_f \mu [ 1 / (1+ i \omega_{d}/\omega)]$ where $\omega_{d} = k_p \mu $ is the “depinning frequency". This expression shows that at frequencies well above $\omega_{d}$, viscous forces dominate and the motion of vortices becomes predominately dissipative. This is a considerable simplification. In this limit the expression for $R_v$ reduces to the inverse of Eq. 2 for the vortex conductance. In cuprate superconductors, $\omega_{d}$ is generally of the order of a few GHz [@Golosovsky94a]. This puts the appropriate frequency regime to probe purely dissipative vortex transport in the range of our TTDS measurements. We have measured the THz range optical conductivity of molecular beam epitaxy (MBE) grown LSCO films using a homebuilt transmission-based time-domain THz spectrometer. With this technique the complex transmission function can be directly inverted to get the complex conductivity [@Comment2]. In Fig. 1(a) and (b) we present the real ($\sigma_1$) and imaginary ($\sigma_2$) THz conductivity of one particular LSCO film (x=0.095, $T_c$=23.5K) out of a large series we have recently studied [@Bilbro11a]. At high temperature $\sigma_1$ is fairly constant in frequency. As the temperature is lowered, $\sigma_1$ increases, develops a frequency dependence near $T_c$, and then decreases as spectral weight is shifted into a delta function at zero frequency. The $\sigma_2$ vs. frequency data in Fig. 1(b) show a small imaginary part of the conductivity at high temperatures, which is enhanced dramatically as temperature is reduced near $T_c$. At the lowest displayed temperatures $\sigma_2$ shows the 1/$\omega$ dependence expected for the superfluid response of a superconductor. While the low and high-temperature limits are easily understood, we are most interested in the fluctuation regime near $T_c$. The enhancement of the conductivity in this fluctuation regime is more clear in Fig. 1(c) and (d), where we plot $\sigma_1$ and $\sigma_2$ vs. temperature. One can see clearly the slow increase and subsequent decrease in $\sigma_1$ as temperature is lowered below $T_c$. At low frequency there is a well-defined peak around $T_c$. The location of this peak shifts to lower temperature as frequency is reduced corresponding to the slowing down of fluctuations as the temperature decreases. Above $T_c$, we see a sudden onset in $\sigma_2$ at a temperature $T$ $\approx$ 30K. In earlier work, we found that the second derivative with respect to temperature of the quantity $\omega\sigma_2$ (which is related to the phase stiffness) showed a clear and dramatic onset from a near-zero high-temperature signal [@Bilbro11a]. We denoted this temperature as $T_o$, and defined it as the onset of superconducting fluctuations in the charge conductivity (for this film $T_o\approx$ 31K). Note that there is no sign of conductivity enhancement at the high temperatures of the Nernst or diamagnetism onset [@Xu00a; @Wang05a; @Li10a]. ![(a) Magnitude of the conductivity ($|\sigma|$) as function of temperature. The filled region is a fit of the normal state background conductivity at 300 GHz [@Comment2]. The fluctuation conductivity $\sigma_S$ is obtained by subtracting this background from $|\sigma|$. (b) A comparison of fluctuation conductivity with the diamagnetism in similarly doped La$_{2-x}$Sr$_x$CuO$_4$ crystals [@Ong10a].](PRBratioFig2.pdf){width="1\columnwidth"} As mentioned above, in conventional models where vortices are the principal degree of freedom in the region above $T_c$, one expects that correlations in length and time scale together as a diffusionlike relation with vortex diffusion constant $D$. Therefore, the large difference in the temperature of the inferred onset of superconducting correlations $T_o$ above $T_c$ between our experiments (10 - 20K) and for instance, diamagnetism measurements ($\approx$100K) [@Wang05a; @Li10a] begs an explanation. Here we evaluate the relative size of the signals in terms of the diffusion constant derived in the above analysis and show that conventional vortex dynamics would predict a much larger fluctuation conductivity given the size of diamagnetism. {width="1\columnwidth"} In Fig. 2(a), we plot the magnitude of the conductivity $|\sigma|$. To isolate the superconducting fluctuation contribution $\sigma_S$, we define a normal-state contribution that fits the conductivity well at temperatures above the onset of the diamagnetism (T$_D \approx$ 75 - 110 K in this doping range [@Li10a]), extrapolate to low temperatures, and take the difference. Although we fit the background through a temperature-dependent Drude model [@Comment2], our final conclusions are not sensitive to the precise background choice as we are only concerned with the temperature region up to about 10 K above $T_c$, where the fluctuations are obvious. In previous work [@Bilbro11a] we have performed a scaling analysis that allowed us to extract the characteristic frequency scale $\Omega$ of the fluctuation superconductivity in the region above T$_c$ [@Comment2]. In the analysis that follows we evaluate $|\sigma_S (\omega,T)|$ at a frequency $\omega=\Omega(T)/2$ for each temperature. This conductivity differs formally from the conductivity in Eq. \[Cond\] by a constant of order unity, which we set to one below. The use of THz frequencies eliminates the effects of pinning and the scaling analysis essentially connects the response of the system at finite frequency to the dc response that the system $would$ have had in the absence of vortex pinning. In Fig. 2(b), on the left axis, we plot the magnitude of the fluctuation conductivity contribution, evaluated at $\omega=\Omega(T) / 2$, vs. temperature. On the right axis, we include diamagnetic susceptibility $\chi/\mu_0$ at 1 and 10 Tesla of a single-crystal LSCO sample [@Ong10a] with a similar doping and $T_c$ ($x=0.9$ and 23 K, respectively). In this data, one can see how the larger field suppresses the susceptibility near $T_c$. Although there is some correspondence between the form of the lower-field susceptibility with the conductivity, we now show that in fact it is the relative scale of these quantities which is particularly remarkable. We now apply Eq. 3 with the data in Fig. 2(b) to extract $D$ for a small range of temperatures above $T_c$. As shown in Fig. 3 we find that $D$ is of the order of 10’s of cm$^2$/sec throughout the range above $T_c$. This is at least 2 orders of magnitude larger than a simple Bardeen-Stephen (BS) estimate $D = (2 k_B T e^2 \xi^2_{c}) / (\pi \hbar^2 \sigma_n t)$ [@Stephen65a] (here $\sigma_n$ is the extrapolated normal state background conductivity, $t$ is the spacing between CuO$_2$ layers, and $\xi_{c} $ is the vortex core size [@Kato04a; @Pan00a]). The BS approximation appears to work well to model flux-flow dissipation in conventional $s$-wave materials [@Peroz05a; @Fiory68a; @Berghuis93a], where the majority of dissipation occurs through quasiparticle motion in the vicinity of the vortex cores. Note that the magnetic susceptibility appears to become singular as $B \rightarrow$ 0 near T$_c$ (“fragile London rigidity") [@Li10a], so that evaluating $D$ at lower fields (corresponding to our $B=0$ TTDS experiment) will only increase the ratio of $\chi/G$ and the discrepancy with the BS estimate. Although there is an expectation that due to their $d$-wave nature, short coherence lengths, gapped vortex core, and proximity to the Mott insulator, the cuprate vortices may be “fast" as compared to the BS estimate [@Volovik93a; @Maggio95a; @Ioffe02a; @Melikyan05a; @Lee06a; @Nikolic06a; @Fanfarillo2011a], the discrepancy we find is extreme. It is an open question whether a diffusion constant as large as we have found can be reconciled. We have currently performed this analysis for one underdoped sample due to difficulty in obtaining compatible diamagnetism data. However, we anticipate similar behavior for the entire underdoped part of the phase diagram, since signals of conductivity and diamagnetism vary smoothly as a function of doping [@Li10a; @Bilbro11a]. There are two obvious possible conclusions from our data and analysis. If in fact the large diamagnetic response in the cuprates comes entirely from superconducting correlations, then we have shown that their vortex motion must be anomalously fast and their dissipation anomalously small to reconcile the behavior with charge transport. One expects that above $T_c$ an effective two fluid model may apply where the total conductivity has contributions from both normal electron and superconducting degrees of freedom in the form of $\sigma_T = \sigma_N + G_S / t$ where $G_S$ is given by Eq. 2 in the vortex regime. Our results show the manner in which superconducting correlations may persist far above $T_c$ but be invisible to the charge response; the fast vortices are shorted out by the normal electrons. It is a separate, but well posed theoretical challenge to explain vortex motion this fast. Although detailed calculations must be performed, it is possible that such anomalously fast diffusion may arise as a consequence of the cuprates’ $d$-wave nature and small gapped cores [@Melikyan05a; @Geshkenbein98a], inhomogeneities [@Martin10a], the existence of a competing state nucleated in the vicinity of a vortex [@Lee06a], or the proximity to a Mott insulator [@Ioffe02a; @Fanfarillo2011a]. Alternatively, if calculations show that vortex dissipation must always be at least parametrically related to the BS estimate by numbers of order unity, then our analysis shows that there must be another large contribution to the diamagnetic response that is not superconducting in origin (See Ref. [@Sau2010a] for one such possibility). We thank L. Li, I. Martin, A. Millis, P. Nikolic, V. Oganesyan, N.P. Ong, O. Pelleg, Z. Tesanovic, and S. Tewari for helpful discussions and/or correspondences. We also would like to thank L. Li and N.P. Ong for access to their unpublished data. 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A broadband THz range pulse is emitted by one antenna, transmitted through the LSCO film, and measured at the other antenna. By varying the length-difference of the two paths, the electric field of the transmitted pulse is measured as a function of time. Ratioing the Fourier transform of the transmission through the LSCO film on a substrate to that of a bare reference substrate, we resolve the frequency dependent complex transmission. The transmission is inverted to obtain the complex conductivity by the standard formula for thin films on a substrate: $\tilde{T}(\omega)=[(1+n)/(1+n+Z_0\tilde{\sigma}(\omega)d)] e^{i\Phi_s}$ where $\Phi_s$ is the phase accumulated from the small difference in thickness between the sample and reference substrates and $n$ is the substrate index of refraction. #### Scaling analysis of the fluctuation conductivity - In previous work [@Bilbro11aSI] we have also performed a scaling analysis that allowed us to extract out a characteristic fluctuation rate of the superconductivity. This analysis follows from the fact that for a fluctuating superconductor one expects that the relation $$\sigma_S(\omega)= \frac{G_Q}{t} \frac{k_B T_\theta^0}{\hbar \Omega} \mathcal S (\frac{\omega}{\Omega}) \label{scaling}$$ holds for the portion of the conductivity, $\sigma_S$, due to superconducting fluctuations. Here $G_Q = e^2/ \hbar$ is the quantum of conductance, $t$ is the inter-CuO$_2$ plane spacing, $T_\phi^0$ is a temperature dependent prefactor and $\Omega$ is the characteristic fluctuation rate. This scaling function is similar to the one proposed by Fisher, Fisher, and Huse [@Fisher91aSI] and is identical to the one used in previous THz measurements on underdoped BSCCO [@Corson99aSI]. In Fig. 4a we show the collapsed phase $\varphi =$ tan$^{-1}\sigma_2/\sigma_1$ from the data in Fig. 1 as a function of reduced frequency $\omega/\Omega$ at temperatures from 22 K to 30 K. The phase is an increasing function of $\omega/\Omega$, with the metallic limit $\varphi = 0$ reached at $\omega/\Omega \rightarrow 0$ and $\varphi$ becoming large (but bounded by $\pi/2$) as $\omega/\Omega \rightarrow \infty$. We plot the extracted fluctuation rate $\Omega$ as a function of temperature in Fig. 4b. As noted previously [@Bilbro11aSI], we continue to obtain good scaling and data collapse if we push the analysis 3-4 K above the temperature of the obvious onset in $\sigma_2$. This region shows a linear dependence of $\Omega$ on temperature. {width="1\columnwidth"} In the main text, we use $\Omega(T)$ to evaluate the magnitude of the fluctuation contribution to the conductivity. We evaluate the magnitude at a frequency of fixed proportionality of 1/2 the characteristic fluctuation frequency at each temperature, i.e. $\sigma_S(\omega=\Omega(T)/2 ,T)$. The $S$ function in Eq. 4 then becomes the same constant of order unity for all temperatures; in the analysis in this paper we set it equal to one. #### Modeling of normal state conductivity - We estimate the normal state contribution to the conductivity by using the Drude model $\sigma = \frac{\omega_p^2}{4 \pi} \frac{\tau}{1 - i \omega \tau}$, which should be valid in the cuprate normal state at low enough frequencies. In this model note that $\sigma_2(\omega)/\sigma_1(\omega) = \omega\tau$. Assuming $\tau$ is relatively frequency independent in this frequency range, we extract $\tau$ at high temperatures by finding the slope of $\frac{\sigma_2}{\sigma_1}(\omega)$. Using the high temperature $\tau$ and $|\sigma|$, we find the plasma frequency $\omega_p$. We then fit $1/\tau$ to a power law form a$_1$+b$_1$T$^n$ and $\omega_p$ to a linear form a$_2$+b$_2$T at high temperature and extrapolate these fits to low temperature. Since $\omega\tau << 1$ in this frequency regime, the normal state background contribution is well approximated as $|\sigma_{bg}| = \omega_p^2\tau/4\pi$. The error bars on the vortex diffusion and fluctuation conductivity come from our uncertainty in fitting $1/\tau$ and $\omega_p$ in the high temperature range. The upper limit of $|\sigma_{bg}|$ was set with the values of the $1/\tau$ fitting parameters (in THz units) of a$_1$=4.08, b$_1$=1.33$\times 10^{-6}$, and n=2.82; the lower limit of $|\sigma_{bg}|$ used the values a$_1$=3.72, b$_1$=2$\times 10^{-4}$, and n=1.85. In fitting $\omega_p$ we allowed a$_2$ to have a very small frequency dependence (varying by $\approx 2\%$ of the average value of 4.5 GHz), while b$_2$ was kept constant at b$_2$=-1.09 MHz/K. #### Molecular beam epitaxy of La$_{1.905}$Sr$_{0.095}$CuO$_4$ films - The LSCO films were deposited on 1-mm-thick single-crystal LaSrAlO$_4$ substrates, epitaxially polished perpendicular to the (001) direction, by atomic-layer-by-layer molecular-beam-epitaxy (ALL-MBE) [@Bozovic01aSI]. The samples were characterized by reflection high-energy electron diffraction, atomic force microscopy, X-ray diffraction, and resistivity and magnetization measurements, all of which indicate excellent film quality. The thickness is known accurately by counting atomic layers and RHEED oscillations, as well as from so-called Kiessig fringes in small-angle X-ray reflectance and from finite thickness oscillations observed in XRD pattern. For further details see Refs. [@Bilbro11aSI; @Bozovic01aSI]. [10]{} L. S. Bilbro [*et al.*]{}, Nature Physics (2011) doi:10.1038/nphys1912. D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B, [**[43]{}**]{}, 130 (1991). J. Corson [*[et al.]{}*]{}, Nature, [**[398]{}**]{}, 221 (1999). I. Bozovic, IEEE T Appl. Supercon., [**[11]{}**]{}, 2686 (2001).

--- author: - title: '**Comparison of potential ASKAP HI survey source finders** ' --- Introduction ============ Radio astronomy is facing a new era, acquiring extremely large data volumes with the coming of the Square Kilometre Array (SKA) [@dewdney2009] and precursors such as MeerKAT [@jonas2009] in South Africa, APERTIF [@verheijen2008] in the Netherlands and the Australian SKA Pathfinder (ASKAP) [@deboer2009] in Australia. Various continuum (2D) and spectral line (3D) surveys, which cover large fractions of the sky, will be conducted with these telescopes. The surveys are expected to detect millions of objects, accelerating the need for reliable automated source finders. A good source finder should have high [*completeness*]{} and high [ *reliability*]{}, ie. a low rate of false detections. Choosing a suitable trade-off between both parameters is necessary and depends on both the algorithm and the rms uniformity of the data. Detecting objects is relatively easy in the case of (strong) point sources, but becomes more complicated in the case of irregular shapes and diffuse or extended emission in one or more dimensions and at low signal to noise ratios. The work presented in this paper aims to highlight the strengths and weaknesses of potential 3D source finders for the Deep Investigations of Neutral Gas Origins (DINGO) survey [@meyer2009] and the Widefield ASKAP L-band Legacy All-sky Blind Survey (WALLABY) [@koribalski2009]. These are two of the large survey science projects for ASKAP [@johnston2008]. To achieve the respective science goals, we aim to develop source finding algorithms which reliably and efficiently recover 3D sources. We have identified five different source finders that will be subjected to testing and comparison; 1) the [Duchamp]{} source finder [@whiting2011a], 2) the [Gamma-finder]{} [@boyce2003] 3) the CNHI source finder [@jurek2011], 4) the 2D-1D Wavelet Reconstruction source finder [@floer2011] and 5) the [S+C finder]{} [@serra2011a]. Testing of each algorithm was done on the same set of data cubes. The first containing 961 point sources with varying peak flux and a Gaussian velocity profile. The second cube contains 1024 modelled galaxies with more realistic properties such as extended disks, inclinations and rotation profiles. Here we compare their performance in terms of completeness and reliability. In section 2 we briefly summarise the main properties of the source finding algorithms and in section 3 we describe the testing method and the two model cubes that have been used for the testing. The test results are presented in Section 4, followed by a discussion in Section 5. We compare in detail the performance and reliability of the source finders, to understand where the strong and weak points of the different source finders are and to highlight possible improvements. We finish with a short conclusion in the final section. Source Finders ============== Here we provide a short description of the five source finders compared throughout the paper. For a more extended review of the individual algorithm we refer to the reference papers describing each method in detail. Duchamp source finder --------------------- [Duchamp]{} [@whiting2011a] is a source finder designed for 3D data, although it can be used for 2D and even 1D datasets. The source finder has been developed by Matthew Whiting at CSIRO.[^1] [Duchamp]{} identifies sources by simply applying a specified flux or signal-to-noise threshold and searching for signals above that threshold. In a second step, detections are merged or rejected based on several criteria specified by the user. To improve its performance, [Duchamp]{} offers several methods of preconditioning and filtering of the input data, including spatial and spectral smoothing as well as wavelet reconstruction of the entire image or cube. In a final step, [Duchamp]{} measures several basic parameters for each detected source, including position, radial velocity, size, line width, and integrated flux. The performance of the [Duchamp]{} source finder is tested in @westmeier2011. CNHI source finder ------------------ The Characterised Noise (CNHI) source finder [@jurek2011] is being developed as part of the WALLABY design study. The CNHI source finder treats spectral datacubes as a collection of spectra, using the Kuiper test, which is a variant of the Kolmogorov-Smirnov test, to identify regions in each spectrum that do not look like noise. The Kuiper test is used to calculate the probability that the test region and the rest of the spectrum come from the same distribution of voxel flux values. If the probability is sufficiently low, then the test region is flagged as an object section. The probability threshold is specified by the user. Once all of the spectra have been processed, the object sections are combined into objects. Object sections are combined using a variant of Lutz’s one pass algorithm. There are two caveats to using the CNHI source finder. Firstly, the CNHI source finder assumes that each spectrum is dominated by noise. This is a safe assumption as spectral datacubes are generally sparsely populated by sources. The presence of ripples, artifacts and continuum signal will potentially invalidate this assumption though. The second caveat is that the test region needs to be at least four channels wide for the Kuiper test to be reliable. This matches the smallest channel extent expected of WALLABY sources. Spectral datacubes with a poorer velocity resolution than WALLABY will be affected by this. For a more detailed description of the CNHI source finder see @jurek2011. Gamma-finder ------------ [Gamma-finder]{} is a [Java]{} application developed by [@boyce2003] which automatically searches for objects in 3-dimensional data cubes. The searching algorithm of [Gamma-finder]{} is based on the [Gamma-test]{} [@Stefansson1997], and a full description can be found in [@jones2002]. The [Gamma-test]{} is a near-neighbour data analysis routine which estimates the noise variance in a continuous dataset. This estimate is known as the [*Gamma Statistic*]{}, denoted by $\Gamma$. When using the [Gamma-finder]{} a Gamma signal-to-noise ratio can be defined which is used as a clipping for objects to be qualified as a detection. The output of the [Gamma-finder]{} is limited compared to other source finders (eg. [Duchamp]{} and CNHI), because it does not do any parametrisation, but only gives the three dimensional position of a detection and the sigma level. 2D-1D Wavelet Reconstruction source finder ------------------------------------------ The 2D-1D Wavelet Reconstruction source finder is described in detail in @floer2011, they have adapted a multi-dimensional wavelet denoising scheme first used by @starck2009. It takes into account that 3D data from spectroscopic surveys have two angular dimensions and one spectral dimension, in which the shape of the sources is vastly different than in the angular dimensions. The algorithm therefore performs a two-dimensional wavelet transform in all planes of the cubes and a subsequent one-dimensional wavelet transform along each line of sight, i.e. each pixel. Once the image has been de-noised by thresholding of the wavelet coefficients, reconstructing the data from only the significant coefficients yields a noise-free cube. The latter can be used to create a mask for the sources in the original data. Smooth plus clip (S+C) finder ----------------------------- @serra2011a developed a source finder which uses a limited number of filters in order to optimise the signal-to-noise ratio of objects present in a data cube. For each dataset, the finder looks for sources in the original cube and in the cubes obtained by smoothing the original cube either on the sky, or in velocity, or along all three axes. In this study we use a Gaussian filter of FWHM=60 arcsec for smoothing on the sky, and a box filter of width 2, 4, 8, 16, and 32 channels for smoothing in velocity. For each smoothed cube a mask is built including all voxels brighter (in absolute value) than a chosen threshold. The final mask is the union of all masks (i.e., a voxel is included in the total mask if it is included in at least one of the individual masks), a value of 1 is allocated to all masked voxels and 0 to all unmasked voxels. A size filter is applied to the final binary mask by convolving it with a 30 arcsec Gaussian kernel, equal to the original angular resolution of the cube and to 3 channels in velocity. Subsequently the mask is shrunk again by taking only voxels in the convolved mask brighter than 0.5. This procedure removes a large number of noise peaks included in the mask whose size is of the order of the cube resolution. Testing method ============== When comparing the five 3D source finders, we concentrate on two main parameters, the completeness and the reliability of a source finder. Completeness is defined as the number of detected sources divided by the total number of sources. While this number is known for simulated cubes, in reality we usually have a much harder problem: we neither know the number of detectable sources in a cube, nor their shape, size or velocity extent. There are a few examples of real datacubes where there is a much deeper datacube of the same region of sky, for example the HIPASS region that is covered by the HIDEEP survey [@minchin2003]. The completeness can be given as a single number, but can also be measured as a function of a certain parameter such as integrated flux or velocity-width. Raw reliability is defined as the number of true detections divided by the total number of detections. In a good scenario the number of false detections is very low, so the reliability is close to 100%. We have to stress that although completeness is a general parameter for a simulation, reliability is highly dependent on the size of a cube. When making a cube twice as large but keeping the number of sources constant, the completeness will not change. However as the noise voxels approximately double, so do the number of false detections. In practise this is complicated by the non-linear steps used by some source finders, and the number of false detections does not necessarily scale linearly with the size of a data cube. The reliability of different source finders can only be compared if the finders are applied on exactly the same data sets. In many cases the reliability of a source finder can be improved upon by applying a threshold for one or more measured parameters like integrated flux. We only concentrate on the capability of source finders to determine detections. Not all source finders have the capability of parametrizing detections, this however is a different problem that can be addressed in the post-processing of detections once they have been identified. Input Models ------------ For the testing and comparison of the different source finders we have used 2 data cubes containing: 1) 961 artificial point sources with Gaussian spectra and 2) 1024 artifical model galaxies with a range of orientation parameters. [ASKAP]{}-specific noise has been added to the cubes, which was generated by the [Uvgen]{} task within [Miriad]{} and is based on the [ASKAP]{} telescope configuration, a system temperature of $T_{sys}$=50K and an integration time of 8 hours. The [rms]{} in the cubes is 1.95 mJy/beam ($30''$) per channel (3.9 km/s). The cubes are similar to the cubes that have been used for the testing of the [ Duchamp]{} source finder by [@westmeier2011]. In the first cube with point sources each source was randomly assigned a peak flux in the range of 1 to 20 $\sigma$, spectral line widths (FWHM) range from approximately 0.4 to 40 km s$^{-1}$. While in reality sources with line widths as small as 0.4 km s$^{-1}$ do not occur, they are included to test the performance of source finders on objects that are spectrally unresolved. In the second cube with model galaxies all sources have an infinitely thin discs with varying inclination (0$^{\circ}$ to 89$^{\circ}$), position angle (0$^{\circ}$ to 180$^{\circ}$), and rotation velocity (20 to 300 km s$^{−1}$). For a more detailed description of the cubes and the input parameters we refer to the paper describing the [Duchamp]{} testing [@westmeier2011]. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Cross-Matching -------------- To properly compare the five source finders, they have to be analysed in exactly the same manner to exclude any discrepancies based on different methods or interpretations. Apart from the [Gamma-Finder]{}, all source finders produce a 3-dimensional mask containing all the voxels that belong to a detection. Although some source finders such as [Duchamp]{} have the capability to determine source parameters, we have chosen to extract the source parameters from the produced masks, using a separate script. In this way the results of all source finders are treated in exactly the same manner and we are able to make an objective comparison of the results. Using the mask, we have merged all detections that were separated by one pixel in the two spatial dimensions and seven channels in the spectral dimension. Furthermore we required detections to be apparent in at least three channels of the cube to reject spurious detections. The way in which detections are merged can effect the results significantly. For example double-horned, unresolved sources are often split up into two separate sources. They can be recovered as one source, however this depends on the scale that is used for the merging, and it is inevitable that in the merging process not all split sources are recovered properly. Some basic object parameters that have been extracted are the position of the source, the velocity width, the peak flux and the integrated flux. Crucial but not trivial is how the cross-matching is done between the implemented input catalogues and the results of the different source finders. Measuring the central position of a source can be difficult, however in the case of the model cube with point sources the position of the objects is very well determined, both in spatial and velocity direction. The list of input objects is compared with the detections of the source finders, and pairs are sought within $\pm$1 pixel in the spatial direction and $\pm$2 pixels in the spectral direction. As the synthesised beam at FWHM of the used models is described by only three pixels, this is a very robust method. For the cube with disk galaxies the measured centre of a certain object is not always trivial to determine as the sources can be very extended. Due to the rotation, for many objects several components are detected, without detecting emission in the actual centre of the object. As the objects can have line widths of up to several hundred km/s, the central velocity is difficult to estimate and might differ significantly amongst the different source finders. To do the cross-matching we have used a Python script that is used and described in the paper on testing of the [Duchamp]{} source finder [@westmeier2011]. We created a three-dimensional mask containing all voxels containing emission from the model galaxies. For each detection we assess whether the central position $\pm$1 pixel overlaps with one of the voxels in the mask and then determine to which object from the input model catalogue it belongs. Results ======= The range of source properties is large and well documented in many published galaxy catalogs (e.g. @koribalski2004, @meyer2004, @springob2005, @haynes2011) as well as catalogs of high velocity clouds (HVCs; e.g. @putman2002) and peculiar features (@hibbard2001, Rogues Gallery). The shape of spectra ranges from simple Gaussian profiles to steep double-horn profiles and almost everything in between. The distribution of in disk galaxies is often symmetric and regular, but many irregular sources exist, from peculiar dwarf galaxies and rings to plumes/filaments and clouds. As typically only the highest column density gas is detected, it is likely that the low column density gas is more pervasive and irregular. In the following we present a comparison of source finding algorithms applied to the two cubes described in Section 3.1. We start with the simple point sources with Gaussian profiles, then progress to extended disks with more complex profiles. Point sources ------------- Point sources with a Gaussian velocity profile are ideal sources in the sense that they do not have any complicated structures and are relatively easy to detect. Fig. \[point\_comp\_flux\] shows the completeness as a function of integrated flux ($F_{int}$), integrated signal-to-noise ratio, peak flux ($F_{peak}$) and 50% velocity width ($W_{50}$). The integrated flux and integrated signal-to-noise ratio are plotted on a logarithmic scale, to highlight the differences between the source finders. All parameters are the true parameters determined from the input models. For $F_{int}$ we use the same definition as [@westmeier2011] (their equation 4). \[duchamp\_setup\] Parameter Value Comment -------------------- -------- --------------------- threshold (test 1) 0.0039 $2\times $[rms]{} threshold (test 2) 0.0029 $1.5\times $[rms]{} minPix 5 minChannels 3 flagAdjacent true flagATrous true Wavelet reconstr. reconDim 3 in 3 dimensions snrRecon 3 scaleMin (test 1) 1 scaleMin (test 2) 2 : [Duchamp]{} input parameters for the data cube with point sources. We have plotted two results for each of the individual source finders on this particular cube, apart from the 2D-1D wavelet reconstruction method which only produced one output. For [Duchamp]{} the input parameters are given in Table. \[duchamp\_setup\]. For the [ Gamma-finder]{} we use a 3$\sigma$ and a 4$\sigma$ clipping threshold and for the CNHI source finder we use a probability of $10^{-3}$ and $3\cdot10^{-4}$. The [S+C finder]{} has been tested using clipping levels of 3$\sigma$ and 4$\sigma$. For each test, the raw reliability is given as a percentage in the legend of the figure. Here the completeness is the principal value to compare the source finders as the single value for raw reliability can be a misleading number. The number of possible settings or input parameters for each source finder is very large and we experimented with each source finder until we found a set of parameters that was representative for its performance. We emphasise that the scope of this paper is to compare the results of the different source finders, rather than to test them individually which has been done in other papers in this special issue. [Duchamp]{} performs very well on point sources, and the completeness is superior to the other source finders for all plotted parameters. The completeness starts at very low values, but rapidly increases to a completeness of about $\sim$50% at an integrated flux of $\sim$0.08 Jy km s$^{-1}$. There is a turnover in the plot reaching full completeness around $\sim$0.2 Jy km s$^{-1}$. The completeness does not stay at 100% as some of the bright sources become merged due to the wavelet reconstruction and multiple objects are counted as one. The other source finders show a very similar behaviour however the completeness levels are lower. There is a large variation in the reliability numbers, but apart from CNHI the reliabilities for all source finders have values above 70%. We have to stress here again that the raw reliability is an initial estimate of the quality of a source finder, but is likely to be improved upon in post-processing of the data. We will explain this in more detail in the discussion. The reliability will go down however with more realistic noise containing unpredictable features such as e.g. continuum sources and solar interference. In the top-right panel of Fig. \[point\_comp\_flux\] the completeness is plotted as a function of integrated signal-to-noise ratio $(F_{int}/{\sigma}_{int})$. The integrated noise is calculated as: $$\sigma_{int} = rms \cdot dV \cdot \sqrt{2.35 \cdot W_{50}/dV}.$$ where $dV$ is the spectral resolution of the cube and the coefficient is used to convert the $W_{50}$ value to the line width of a Gaussian. The general trend is very similar, here for the better performing source finders in terms of completeness, about 50% completeness can be achieved at an integrated signal-to-noise ratio of $\sim4-5$. The completeness increases very rapidly and for several source finders 100% completeness is achieved at an integrated signal-to-noise ratio around 10 while for the best [Duchamp]{} run this result is achieved at an integrated signal-to-noise ratio close to 6. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Model galaxies -------------- For the testing of the source finders on the cube with model galaxies, we analyse again two different runs for each of the source finders apart from the 2D-1D wavelet source finder. The tested parameters for [ Duchamp]{} are almost identical to two results as presented in [@westmeier2011], in table \[model\_duchamp\] we summarise the parameters that were used. The only difference between the two runs is that in the second run the objects are [*“grown”*]{} to a lower threshold once detected. When doing this, objects that are broken up into multiple detections can get merged. The [Gamma-finder]{} has been used with a 3$\sigma$ and a $5\sigma$ clipping level, while for the CNHI source finder we have used probability thresholds of $5\cdot10^{-4}$ and $5\cdot10^{-5}$ respectively. In the case of the [S+C finder]{} clipping levels of 3.5$\sigma$ and 4$\sigma$ have been used. Parameter Run 1 Run 2 Comment ----------------------- --------- --------- ------------------------ threshold 0.00186 0.00186 $1.0 \times$[rms]{} minPix 10 10 minChannels 3 3 flagAdjacent true true flagGrowth false true growthThreshold – 0.00093 $0.5 \times$[rms]{} flagRejectBeforeMerge false true flagATrous true true Wavelet reconstruction reconDim 3 3 in 3 dimensions snrRecon 2 2 scaleMin 3 3 In Fig. \[model\_completeness\] we plot again the completeness of the source finders as function of integrated flux, integrated signal-to-noise ratio, peak flux and velocity width ($W_{50}$). The integrated flux of the model galaxies is defined as: $$F_{int} [\textrm{Jy km s$^{-1}$}] = F_{peak} \cdot (2\pi)^{1.5} \cdot disp \cdot B_{maj} \cdot B_{min}$$ where $F_{peak}$ is the peak flux, $disp$ is the velocity dispersion and $B_{maj}$ and $B_{min}$ are the FWHM major and minor axis respectively of the 2-dimensional Gaussian describing the galaxy. The integrated noise is given by: $$\sigma_{int} = \sqrt{\frac{2.35 \cdot W_{50}}{dV}} \cdot \sqrt{\frac{1.13 \cdot B_{maj} \cdot B_{min}}{b_{min} \cdot b_{maj}}} \cdot rms \cdot dV \cdot 2.35$$ where $W_{50}$ is the velocity width FWHM given by the model catalogue, $dV$ is the channel separation, $b_{maj}$ and $b_{min}$ are the major and minor axis of the synthesised beam and [rms]{} is the noise in the cube.  \ The general results are slightly different to the results as obtained from the cube with point sources. The performance of the different source finders is quite comparable, however in general both completeness and reliability levels are slightly lower than for the point sources. Sources that are extended in space or velocity can be almost hidden in the noise and hard to detect. For the better performing source finders, we reach 50% completeness around an integrated signal-to-noise ratio between 4 and 6 and 100% completeness for a signal-to-noise ratio between 10 and 15. These are very promising results given that the achieved completeness values are very close to the completeness of the point sources which should be much easier to detect. Compared to the point sources the S+C finder is performing much better and seems the best algorithm here in terms of completeness. This is due to the fact that with smoothing to different spatial or spectral scales the real shape of an object is matched as close as possible. In the case of point sources smoothing to a larger scale does not increase the signal to noise and hence the S+C finder does not benefit as much. The [Gamma-finder]{} performs much worse for model galaxies as this source finder is most sensitive to sudden changes in the spectrum, which are not as apparent in the case of extended sources. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"}\ {width="48.00000%"}\ {width="48.00000%"}\ {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"}\ {width="48.00000%"}\ {width="48.00000%"}\ {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Discussion ========== A different way of demonstrating the performance of the source finders is by plotting the completeness of the source finders on a two dimensional plot as a function of integrated flux and velocity width. As a reference the total number of objects in both cubes is shown on this grid in Fig. \[total\_detections\_2D\]. Point sources ------------- In Fig. \[point\_2D\] we plot the completeness and the reliability results of the different source finders when applied to the point sources on a two-dimensional grid. For each result, completeness is plotted as a function of integrated flux and velocity width (represented by FWHM ($W_{50}$)) of the modelled point sources in the top panels. In the middle panels the ratio is shown between number of objects detected by the tested source finder and the number of sources detected by any source finder. Instead of showing the overall completeness this plot shows how a particular source finder performs compared to the other source finding results. Regions in the parameter space that appear blue in this plot are regions that can be improved upon, as other source finders do detect objects within this parameter space. Apart from showing how one source finder performs compared to the others, this plot also shows the parameter space that is covered by all the source finders combined. In the bottom panel of Fig. \[point\_2D\] reliability is plotted as function of measured integrated flux and velocity width $(W_{50})$. These panels are not included for the [Gamma-finder]{} as this source finder does not parameterise sources. The completeness plots in the top two panels all have the same scale as the parameters are based on the intrinsic parameters of the input catalogue. The scaling of the reliability panels is different in each plot as this is determined by the measured parameters of the different source finders. We have to emphasise here that the measured parameters are not by definition correct values as this depends on the capability to parameterise sources properly. Different parameterisation algorithms are used by the different source finders. We have not compared the parameters obtained from the source finders, but a possible difference has to be taken into account when comparing the plots. [Duchamp]{} is incomplete for small integrated fluxes, but is basically 100% complete for fluxes above 0.3 Jy km s$^{-1}$. It is expected that very low flux values are difficult to detect, however in quite a large area of the parameter space sources are detected which are not recovered by [Duchamp]{}. This indicates that although in Fig. \[point\_comp\_flux\] [Duchamp]{} appears to be the best performing source finder, another source finder is needed to detect the very low fluxes, or [Duchamp]{} has to be improved here. For both [Duchamp]{} tests the reliability is reasonable as most detections are true detections and the false detections are especially concentrated at very small fluxes. The [CNHI]{} source finder does not perform very well on the tested point sources, it misses almost all sources with a FWHM velocity width below 12 km s$^{-1}$. Apart from that this source finder also misses a very significant fraction of the bright sources. The number of false detections is relatively large and spread over the whole parameter range. Many of the false detections have low fluxes and very broad line widths, much broader than any of the real line widths. The [S+C finder]{} detects sources down to very low integrated fluxes, lower than most of the other source finders. As can be seen in the middle panel of the first [S+C finder]{} results, some of the sources with a low integrated flux are only recovered by this source finder. On the opposite side, the [S+C finder]{} is not 100% complete at either large fluxes or large line widths. False detections are quite difficult to distinguish when using this source finder, as the false detections are not clustered in a narrow range of the parameter space. For a large region in the plot the reliability fluctuates around 50%, indicating that the determined parameters of false detections are very similar to that of true detections. For the 2D-1D wavelet finder there seems to be a clear trend from 0% completeness at low fluxes to almost full completeness at high integrated flux values, very similar to the [Duchamp]{} results. In the parameter space covering the largest fluxes and line widths, the finder is not 100% complete. This could be caused by the fact that our model cube is very dense with many sources, and for the largest wavelet scales these sources start to merge. The wavelet finder can be improved here, as [Duchamp]{} also uses wavelet reconstructions, but appears to be less sensitive to this problem. The reliability of the 2D-1D wavelet finder is very good and 100% in most of the parameter space, although there are some false detections with a high integrated flux, we have no good explanation for why the reliability decreases here. The [Gamma-finder]{} seems to perform well on sources with a strong integrated flux and narrow line width. In fact it is the best finder for objects with a narrow line width below 5 km/s, although we have to question how realistic such sources are when observing real galaxies. As the [Gamma-finder]{} does not give a mask or parameters of the detected sources, we cannot make reliability plots for this source finder. Model galaxies -------------- In Fig. \[model\_2D\] we show very similar plots as in the previous figure, but now for the model galaxies. In the top panels the completeness of the different source finders is plotted, while the middle panels compare the completeness of the source finders with respect to each other. In the bottom panels the reliability of the source finders is plotted. These modelled galaxies have more complex structures compared to the point sources, and the completeness and reliability results are very different. Note the different scales in both integrated flux and velocity compared to the point sources in Fig. \[point\_2D\]. [Duchamp]{} is complete for objects with high flux in the first run, but in the second run misses a few sources that should be easy to detect due to their high flux. The only difference between the two [Duchamp]{} runs is the [*growth*]{} parameter, which has merged some of the extended sources. As can be seen in the plot, the missed sources have a large integrated flux but relatively narrow line width, which indicates that they are spatially extended. As the objects were all placed at a similar radial velocity in the cube, there is a high risk of merging. There is a clear transition phase between non-detected and detected objects and [Duchamp]{} misses objects with low integrated fluxes that are detected by at least one other source finder. The reliability of [Duchamp]{} looks very good, as almost all false detections are clustered in a limited area of the parameter space at small fluxes and narrow line widths. The CNHI finder also shows a transition phase from non-detected objects with a low flux to detected objects with high fluxes, however the transition is much broader than for [Duchamp]{}. The CNHI finder is less likely here to miss sources with a narrow line width as the velocity profiles of the model galaxies are much broader and more realistic than for the point sources. When compared to the other source finders, this finder detects a significant fraction of the objects at low integrated flux. The reliability is worse than for the other source finders, but a large fraction of the detected objects have very low fluxes, covering a large range in line width. For the [S+C finder]{} the results are very impressive as it even detects many of the sources with small flux and narrow line width. This finder also has a very small number of false detections that appear to be concentrated in a rather limited range of the parameter space. Although currently this appears to be the best source finder on the tested cube with model galaxies, it is not the best source finder on the full parameter range. In particular, objects with a small integrated flux and broad line width are missing, which in some cases are detected by the Gamma-finder. The 2D-1D wavelet source finder has a very narrow transition between detected and non-detected sources where almost all objects with a flux below 0.5 Jy km/s are missed, while almost all objects with an integrated flux above 1.5 Jy km/s are detected. The reliability of this source finder is very good and the completeness can probably be improved upon by decreasing the clipping threshold used on the reconstructed wavelet scales. The parameter space covered in the reliability plots is very different to the other source finders, the 2D-1D wavelet method seems to detect higher fluxes and smaller line-widths. The [Gamma-finder]{} has a relatively good performance in completeness as it detects the objects with high fluxes, but also a significant number of objects with low flux values. Interesting to see is that the first [Gamma-finder]{} results gives the best result for objects with a low flux and broad line width. Although not plotted in this figure, this good performance in completeness probably comes at the cost of reliability as the reliability of the first run is very low at 12%. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Figure \[point\_scatter\]: Continued {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"}\ Figure \[model\_scatter\]: Continued Reliability of Source Finders ----------------------------- In Fig. \[point\_comp\_flux\] and \[model\_completeness\] the reliability of the different source finders is given by a single number, which can be misleading. This number could be completely dominated by a large number of false detections at a very low threshold, while the source finder is very reliable for high flux values. To better understand where the bulk of the false detections are, all detections are plotted in a scatter plot in Fig. \[point\_scatter\] for the point sources and Fig. \[model\_scatter\] for the model galaxies. The detections are again plotted as a function of velocity width and integrated flux, where true detections are plotted in green and false detections are plotted in red. In the [Duchamp]{} results for the point sources shown in the top panels of Fig. \[point\_scatter\] there are barely any false detections in the first run. In the second run, all false detections have low fluxes. A possibility that can improve the reliability is to apply a cut in integrated flux after the parametrisation of detections. In this example a cut a 0.05 Jy km/s would increase the reliability to $\sim$100% while the number of missed real detections is still limited. For the [CNHI]{} finder the difference between true and false detections is not so obvious. False detections are not clustered in a clearly defined parameter space, but rather mixed with real detections, making it more difficult to eliminate them after post-processing. There is however a very large bulge of detections with a low flux and broad line width. The [S+C finder]{} has a large number of false detections in the first run, however a very large fraction can be eliminated by applying a cut in integrated flux. In the second run the number of false detections is much lower, however they are very well mixed with true detections and difficult to eliminate. Although not shown in the plot, particular for this source finder is that it also reports negative fluxes. These are by default all considered false detections. Assuming that the noise is symmetric, the reliability of positive detections can be determined based on the properties of the negative detections. This method is further explored and explained in @serra2011b. The 2D-1D wavelet source finder is very reliable for point sources as shown before, with barely any false detections. The false detections are however difficult to eliminate as they are concentrated toward high fluxes and line widths. As mentioned this could be a consequence of the used test cube which has a very high source density. Especially in the case of strong sources the largest wavelet scales will merge sources, decreasing the number of detected objects and hence the completeness.  \ A very similar set of plots is given in Fig. \[model\_scatter\], where true and false detections of the model galaxies are plotted for all the source finders apart from the [Gamma-finder]{}. The behaviour of the different algorithms is very similar to before, where the false detections of [Duchamp]{} tend to have a low integrated flux, although it is difficult to completely isolate them. The CNHI finder has a very large number of false detection with low flux and broad line-width, many of which can be rejected to refine the reliability. The performance of the [S+C finder]{} is very good when it comes to reliability as the number of false detections is relatively low. Also the reliability of the 2D-1D wavelet finder is very good, however the false detections are mixed with true detections.  \ The reliability of the source finders can be dramatically improved upon through simple cuts in parameter space. To be able to do this, it is crucial to properly parameterise the detections which has not been done sufficiently at this stage. Nevertheless, to illustrate the concept, we applied a cut on the detections at different integrated flux levels. In Fig. \[cut\_results\] the results are shown, where completeness is plotted as function of reliability for the different source finders after applying cuts at different flux levels. For the point sources cuts have been applied at $F_{int}$ = 0.0, 0.01, 0.02, 0.03, 0.04 and 0.05 Jy km s$^{-1}$ while for the model galaxies at $F_{int}$ = 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5 Jy km s$^{-1}$. The results move from high completeness and low reliabiliy when not applying a cut to low completeness and high realiability when applying the most extreme cut. Although the improvements in reliability vary amongst the different source finders, for each of them the raw reliability can be improved by tens of percent, while only losing a few percent in completeness. In the case of the second [Duchamp]{} test on the point sources the reliability increases from 72% to 96%, while the completeness drops by only 0.6% from 83% to 82% at the fourth data point. On the model galaxies the most impressive result is achieved with the [S+C finder]{} where in the second run the reliability increases to above 95%, while still maintaining a completeness of almost 70%. {width="48.00000%"} {width="48.00000%"} Conclusions =========== In this paper we have compared the performance of five potential ASKAP source finders. The tested source finders are 1) the [ Duchamp]{} source finder, 2) the [Gamma-finder]{}, 3) the [CNHI]{} source finder, 4) 2D-1D Wavelet reconstruction source finder and 5) the [S+C finder]{}, a source finder based on sigma clipping of smoothed versions of the original data cube. The source finders have been applied to two data cubes with model sources, the first containing point sources with a relatively narrow Gaussian line profile and the second containing extended galaxies with inclinations and rotation curves. We have to stress that apart from the [Gamma-finder]{} the tested source finders are not final products but are still under active development. In this paper we want to present the current status of the different source finders, however there is significant room for improvement as is also discussed in other papers in this issue describing some of the tested source finders individually. The testing of different source finding algorithms on different data cubes has proven that it is very difficult to find a good source finder which is reliable for many types of objects. Source finders perform very differently depending on the type of object that is detected. An important feature of a source finder is its reliability, which has not yet been fully explored. Although a number for the raw reliability can be given, in many cases the false detections are clustered within a certain range of flux and line width. We are confident that a large fraction of the false detections can be rejected through simple cuts in parameter space as has been demonstrated in the discussion, however to be able to do this properly all detections have to be parameterised accurately which has not been done yet. For the current source finders and datasets, we find that for point sources 50% completeness can be achieved at an integrated signal-to-noise ratio of $\sim$4-5 sigma, and 100% completeness can be achieved around an integrated signal-to-noise ratio of $\sim$10. For the extended sources the completeness estimates are very similar: for the best results 50% completeness is achieved at an integrated signal-to-noise ratio of $\sim$4-6 and 100% completeness is achieved at an integrated signal-to-noise ratio of $\sim$10. It is interesting to see that the different source finders achieve a different performance, depending on the type of object. Currently none of the source finders excels at being able to achieve the best result in the full parameter space when looking at integrated flux and line width. Nevertheless we have pointed out the strong and weak points of the different source finders, which provides input for future development and testing. For the tested parameters, currently [Duchamp]{} gives the best results on point sources, while the [S+C finder]{} gives the best result for extended objects when looking at the completeness. Due to the different smoothing levels that have been applied in the [S+C finder]{}, this algorithm is best capable of matching the true shape of an object. As the [S+C finder]{} concept is simple yet powerful, we recommend that the other source finders improve their performance by incorporating smoothing on multiple scales. Currently all the tested source finders perform reasonably well, however there is significant room for improvement to meet our goals. All of the source finders have a certain area in parameter space where they perform best and we will combine the algorithms of different source finders to optimise the result. @duffy2011 give predictions of the number of objects that will be detected with WALLABY and DINGO. They predict that at an angular resolution of $30''$, 14% of the WALLABY sources will be unresolved and the bulk of the remainder will be marginally resolved, while for DINGO 93.3% of the sources will be unresolved. This means that many of the unresolved sources in DINGO will have very different profiles to the ones tested in this paper. At an angular resolution of $10''$ these numbers change dramatically, as for WALLABY none of the sources will be point sources as all sources will be larger than one beam, and for DINGO 7.4% of the objects will be smaller than one beam. Although the two cubes that have been used for testing cover a large area in parameter space, they do not sample the full signal-to-noise ratio range properly. We have started efforts to test the source finders on models covering a large range of parameters, keeping integrated signal-to-noise values constant. These tests should give accurate estimates of how many sources can be detected by WALLABY and DINGO. We have a fairly good understanding of the different source finders on simulated objects as presented in this paper. The cubes that have been tested are ideal cubes in the sense that the noise is Gaussian and does not have any systematic artefacts caused by continuum sources, solar ripples, phase errors, radio frequency interference, etc. These contributions have not been taken into account but will have a very significant effect on the performance of source finders, especially in terms of reliability. The simulated model sources are perfectly symmetric sources without any weird or unexpected shapes or extended tails. To have a better understanding of the performance of the source finders, the next step will be to test the source finders on a cube containing data from real galaxies as they occur in the Universe. Boyce, P. 2003, GammaFinder: a Java application to find galaxies in astronomical spectral line datacubes, M.Sc. Dissertation, Cardiff University Deboer, D. R., et al. 2009, IEEE Proceedings, 97, 1507 Dewdney, P. E., Hall, P. J., Schilizzi, R. T., & Lazio, T. J. L. 2009, IEEE Proceedings, 97, 1482 Driver, S. P.et al. 1999, A&G, 50, 12 Duffy, A., R. et al. 2011, MNRAS in prep. Haynes, M. P. at al. 2011, AJ, 142, 170 Hibbard, J. E., van Gorkom, J. H., Rupen, M. P.& Schiminovich, D. 2001, ASPC, 240, 657 Johnston, S., et al. 2008, ExA 22, 151 Jonas, J. L. 2009, IEEE Proceedings, 97, 1522 Jones, A. J., Evans, D., Margetts, S. & Durrant, P. J., 2002,. The Gamma test. In: Sarker, R et al. eds. Heuristic and Optimization for Knowledge Discovery. Hershey: Idea Group Publishing. Jurek, R. 2011, PASA, special issue, arXiv1112.1561J Flöer, L., & Winkel, B. 2011, PASA, special issue, arXiv1112.3807F Koribalski, B. S. et al. 2004, AJ, 128, 16 Koribalski, B. S. & Staveley-Smith, L. 2009, ASKAP Survey Science Proposal Meyer, M. J.et al. 2004, MNRAS, 350, 1195 Meyer, M. 2009, Panoramic Radio Astronomy Proceedings, 15 Minchin, R. F. et al. 2003, MNRAS, 346, 787 Putman, M. E. et al. 2002, AJ, 123, 873 Starck, J. L., Fadili, J. M., Digel, S., Zhang, B., & Chiang, J., 2009, A&A, 504, 641 Serra, P. et al. 2011, MNRAS, submitted Serra, P. , Jurek, R. & Flöer, L. 2011, PASA, special issue, arXiv1112.3162S Springob, C. M. et al. 2005, ApJS, 160, 149 Stefansson, A., Koncar, N. & Jones, A. J. 1997, A note on the Gamma test. Neural Computing and Applications, 5, 131. 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--- abstract: | In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: \[abstract-thm\] Let ${\mathbf{K}}$ be a tame AEC with a monster model. Assume that ${\mathbf{K}}$ is stable in a proper class of cardinals. The following are equivalent: 1. \[abstract-1\] For all high-enough $\lambda$, ${\mathbf{K}}$ has no long splitting chains. 2. \[abstract-2\] For all high-enough $\lambda$, there exists a good $\lambda$-frame on a skeleton of ${\mathbf{K}}_\lambda$. 3. \[abstract-3\] For all high-enough $\lambda$, ${\mathbf{K}}$ has a unique limit model of cardinality $\lambda$. 4. \[abstract-4\] For all high-enough $\lambda$, ${\mathbf{K}}$ has a superlimit model of cardinality $\lambda$. 5. \[abstract-5\] For all high-enough $\lambda$, the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 6. \[abstract-6\] There exists $\mu$ such that for all high-enough $\lambda$, ${\mathbf{K}}$ is $(\lambda, \mu)$-solvable. This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model. address: - 'Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA' - 'Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA' author: - Rami Grossberg - Sebastien Vasey bibliography: - 'superstab-defs.bib' date: | \ AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75, 03E55. title: Equivalent definitions of superstability in tame abstract elementary classes --- Introduction ============ In the context of classification theory for abstract elementary classes (AECs), a notion analogous to the first-order notion of *stability* exists: let us say that an AEC ${\mathbf{K}}$ is *stable in $\lambda$* if ${\mathbf{K}}$ has at most $\lambda$-many Galois types over every model of cardinality $\lambda$ (a justification for this definition is Fact \[stab-spectrum\], showing that it is equivalent, under tameness, to failure of the order property). However it has been unclear what a parallel notion to superstability might be. Recall that for first-order theories we have: \[fo-superstab\] Let $T$ be a first-order complete theory. The following are equivalent: 1. $T$ is stable in every cardinal $\lambda \ge 2^{|T|}$. 2. For all infinite cardinals $\lambda$, the union of an increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 3. $\kappa(T)=\aleph_0$ and $T$ is stable. 4. $T$ has a saturated model of cardinality $\lambda$ for every $\lambda\geq 2^{|T|}$. 5. $T$ is stable and ${\operatorname{D}}^n[{\bar{x}}={\bar{x}},L (T),{\infty}]<{\infty}$. 6. There does not exists a set of formulas $\Phi=\{{\varphi}_n({\bar{x}};{\bar{y}}_n)\mid n<\omega\}$ such that $\Phi$ can be used to code the structure $(\omega^{\leq\omega},<,<_{lex})$ $(1) \implies (2)$ and $(1) \iff (\ell)$ for $\ell \in \{3, 4, 5, 6\}$ all appear in Shelah’s book [@shelahfobook]. Albert and Grossberg [@agchains 13.2] established $(2)\implies (6)$. In the last 30 years, in the context of classification theory for non elementary classes, several notions that generalize that of first-order superstablity have been considered. See papers by Grossberg, Shelah, VanDieren, Vasey and Villaveces: [@grsh238; @gr88], [@sh394], [@shvi635], [@vandierennomax; @nomaxerrata], [@gvv-mlq], [@ss-tame-jsl; @indep-aec-apal]. Reasons for developping a superstability theory in the non-elementary setup include the aesthetic appeal (guided by motivation from the first-order case) and recent applications such as Shelah’s eventual categoricity conjecture in universal classes [@ap-universal-v10; @categ-universal-2-v3-toappear] or the fact that (in an AEC with a monster model) the model in a categoricity cardinal is saturated [@categ-saturated-v2]. In [@sh394 p. 267] Shelah states that part of the program of classification theory for AECs is to show that all the various notions of first-order saturation (limit, superlimit, or model-homogeneous, see Section \[sat-def-subsec\]) are equivalent under the assumption of superstablity. A possible definition of superstability is *solvability* (see Definition \[solvability-def\]), which appears in the introduction to [@shelahaecbook] and is hailed as a true counterpart to first-order superstability. Full justification is delayed to [@sh842] but [@shelahaecbook Chapter IV] already uses it. Other definitions of superstability analogous to the ones in Fact \[fo-superstab\] can also be formulated. The main result of this paper is to show that, at least in tame AECs with a monster model, several definitions of superstability that previously appeared in the literature are equivalent (see the preliminaries for precise definitions of some of the concepts appearing below). Many of the implications have already been proven in earlier papers, but here we complete the loop by proving two theorems. Before stating them, some notation will be helpful: \[hanf-notation\] Given a fixed AEC ${\mathbf{K}}$, set $H_1 := {\beth_{\left(2^{{\operatorname{LS}}({\mathbf{K}})}\right)^+}}$. **Theorem \[ss-from-chainsat\].** Let ${\mathbf{K}}$ be an ${\operatorname{LS}}({\mathbf{K}})$-tame AEC with a monster model. There exists $\chi < H_1$ such that for any $\mu \ge \chi$, if ${\mathbf{K}}$ is stable in $\mu$, there is a saturated model of cardinality $\mu$, and every limit model of cardinality $\mu$ is $\chi$-saturated, then ${\mathbf{K}}$ has no long splitting chains in $\mu$. **Theorem \[strong-solvable-thm\].** Let ${\mathbf{K}}$ be an ${\operatorname{LS}}({\mathbf{K}})$-tame AEC with a monster model. There exists $\chi < H_1$ such that for any $\mu \ge \chi$, if ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$ then ${\mathbf{K}}$ is uniformly $(\mu', \mu')$-solvable, where $\mu' := \left(\beth_{\omega + 2} (\mu)\right)^+$. These two theorems prove (\[sc0-3\]) implies (\[sc0-1\]) and (\[sc0-1\]) implies (\[sc0-6\]) of our main corollary, proven in detail after the proof of Corollary \[main-cor-unbounded\]. \[main-cor\] Let ${\mathbf{K}}$ be a ${\operatorname{LS}}({\mathbf{K}})$-tame AEC with a monster model. Assume that ${\mathbf{K}}$ is stable in some cardinal greater than or equal to ${\operatorname{LS}}({\mathbf{K}})$. The following are equivalent: 1. \[sc0-1\] There exists $\mu_1 < H_1$ such that for every $\lambda\geq\mu_1$, ${\mathbf{K}}$ has no long splitting chains in $\lambda$. 2. \[sc0-2\] There exists $\mu_2 < H_1$ such that for every $\lambda\geq\mu_2$, there is a good $\lambda$-frame on a skeleton of ${\mathbf{K}}_\lambda$ (see Section \[skeleton-sec\]). 3. \[sc0-3\] There exists $\mu_3 < H_1$ such that for every $\lambda\geq\mu_3$, ${\mathbf{K}}$ has a unique limit model of cardinality $\lambda$. 4. \[sc0-4\] There exists $\mu_4 < H_1$ such that for every $\lambda\geq\mu_4$, ${\mathbf{K}}$ has a superlimit model of cardinality $\lambda$. 5. \[sc0-5\] There exists $\mu_5 < H_1$ such that for every $\lambda\geq\mu_5$, the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 6. \[sc0-6\] There exists $\mu_6 < H_1$ such that for every $\lambda \geq \mu_6$, ${\mathbf{K}}$ is $(\lambda, \mu_6)$-solvable. Moreover any of the above conditions also imply: 7. \[sc0-7\] There exists $\mu_7 < H_1$ such that for every $\lambda \geq \mu_7$, ${\mathbf{K}}$ is stable in $\lambda$. The main corollary has a global assumption of stability (in some cardinal). While stability is implied by some of the equivalent conditions (e.g. by (\[sc0-2\]) or (\[sc0-6\])) other conditions may be vacuously true if stability fails (e.g. (\[sc0-1\])). Thus in order to simplify the exposition we just require stability outright. In the context of the main corollary, if $\mu_1 \ge {\operatorname{LS}}({\mathbf{K}})$ is such that ${\mathbf{K}}$ is stable in $\mu_1$ and has no long splitting chains in $\mu_1$, then for any $\lambda \ge \mu_1$, ${\mathbf{K}}$ is stable in $\lambda$ and has no long splitting chains in $\lambda$ (see Fact \[ss-stable\]). In other words, superstability defined in terms of no long splitting chains transfers up. In (\[sc0-3\]), one can also require the following strong version of uniqueness of limit models: if $M_0, M_1, M_2 \in {\mathbf{K}}_{\lambda}$ and both $M_1$ and $M_2$ are limit over $M_0$, then $M_1 \cong_{M_0} M_2$ (i.e. the isomorphism fixes the base). This is implied by (\[sc0-2\]): see Fact \[good-frame-uq-limit\]. \[stab-ss-rmk\] At the time this paper was first circulated (July 2015), we did not know whether (\[sc0-7\]) implied the other conditions. This has recently been proven by the second author [@stab-spec-v4]. Note that in Corollary \[main-cor\], we can let $\mu$ be the maximum of the $\mu_\ell$’s and then each property will hold above $\mu$. Interestingly however, the proof of Corollary \[main-cor\] does *not* tell us that the *least* cardinals $\mu_\ell$ where the corresponding properties holds are all equal. In fact, it uses tameness heavily to move from one cardinal to the next and uses e.g. that one equivalent definition holds below $\lambda$ to prove that another definition holds at $\lambda$. Showing equivalence of these definitions cardinal by cardinals, or even just showing that the least cardinals where the properties hold are all equal seems much harder. We also show that we can ask only for each property to hold in *a single high-enough* cardinals below $H_1$ (but again the least such cardinal may not be the same for each property, see Corollary \[main-cor-unbounded\]). In general, we suspect that the problem of computing the minimal value of the cardinals $\mu_\ell$ could play a role similar to the computation of the first stability cardinal for a first-order theory (which led to the development of forking, see e.g. the introduction of [@primer]). We discuss earlier work. Shelah [@shelahaecbook Chapter II] introduced good $\lambda$-frames (a local axiomatization of first-order forking in a superstable theory, see more in Section \[good-frame-subsec\]) and attempts to develop a theory of superstability in this context. He proves for example the uniqueness of limit models (see Fact \[good-frame-uq-limit\], so (\[sc0-2\]) implies (\[sc0-3\]) in the main theorem is due to Shelah) and (with strong assumptions, see below) the fact that the union of a chain (of length strictly less than $\lambda^{++}$) of saturated models of cardinality $\lambda^+$ is saturated [@shelahaecbook II.8]. From this he deduces the existence of a good $\lambda^+$-frame on the class of $\lambda^+$-saturated models of ${\mathbf{K}}$ and goes on to develop a theory of prime models, regular types, independent sequences, etc. in [@shelahaecbook Chapter III]. The main issue with Shelah’s work is that it does not make any global model-theoretic hypotheses (such as tameness or even just amalgamation) and hence often relies on set-theoretic assumptions as well as strong local model-theoretic hypotheses (few models in several cardinals). For example, Shelah’s construction of a good frame in the local setup [@shelahaecbook II.3.7] uses categoricity in two successive cardinals, few models in the next, as well as several diamond-like principles. By making more global hypotheses, building a good frame becomes easier and can be done in ZFC (see [@ss-tame-jsl] or [@shelahaecbook Chapter IV]). Recently, assuming a monster model and tameness (a locality property of types introduced by VanDieren and the first author, see Definition \[shortness-def\]), progress have been made in the study of superstability defined in terms of no long splitting chains. Specifically, [@ss-tame-jsl 5.6] proved (\[sc0-1\]) implies (\[sc0-7\]). Partial progress in showing (\[sc0-1\]) implies (\[sc0-2\]) is made in [@ss-tame-jsl] and [@indep-aec-apal] but the missing piece of the puzzle, that (\[sc0-1\]) implies (\[sc0-5\]), is proven in [@bv-sat-v3]. From these results, it can be deduced that *(\[sc0-1\]) implies (\[sc0-2\])-(\[sc0-5\])* (see [@bv-sat-v3 7.1]). Shelah has shown that (\[sc0-2\]) implies (\[sc0-3\]), see Fact \[good-frame-uq-limit\]. Some implications between variants of (\[sc0-3\]), (\[sc0-4\]) and (\[sc0-5\]) are also straightforward (see Fact \[local-implications\]), though one has to be careful about where the class is stable (the existence of a limit model of cardinality $\lambda$ implies stability in $\lambda$, but not the fact that the union of a chain of $\lambda$-saturated models is $\lambda$-saturated). In the proof of Corollary \[main-cor-unbounded\], we end up using a single technical condition, ($\ref{sc0-3}^\ast$), asserting that limit models have a certain degree of saturation. It is quite easy to see that (\[sc0-3\]), (\[sc0-4\]), and (\[sc0-5\]) all imply ($\ref{sc0-3}^\ast$). Finally, (\[sc0-6\]) directly implies (\[sc0-4\]) from its definition (see Section \[solvability-subsec\]). Thus as noted before the main contributions of this paper are (\[sc0-3\]) (or really ($\ref{sc0-3}^\ast$)) implies (\[sc0-1\]) and (\[sc0-1\]) implies (\[sc0-6\]). In Theorem \[solv-transfer\] it is shown that, assuming a monster model and tameness, solvability in *some* high-enough cardinal implies solvability in *all* high-enough cardinals. Note that Shelah asks (inspired by the analogous question for categoricity) in [@shelahaecbook Question N.4.4] what the solvability spectrum can be (in an arbitrary AEC). Theorem \[solv-transfer\] provides a partial answer under the additional assumptions of a monster model and tameness. The proof notices that a powerful results of Shelah and Villaveces [@shvi635] (deriving no long splitting chains from categoricity) can be adapted to our setup (see Fact \[ns-lc-fact\] and Corollary \[ns-lc-cor\]). Shelah also asks [@shelahaecbook Question N.4.5] about the superlimit spectrum. In our context, we can show that if there is a high-enough *stability* cardinal $\lambda$ with a superlimit model, then ${\mathbf{K}}$ has a superlimit on a tail of cardinals (see Corollary \[main-cor-unbounded\]). We do not know if the hypothesis that $\lambda$ is a stability cardinal is needed (see Question \[superlimit-question\]). Since this paper was first circulated (July 2015), several related results have been proven. VanDieren [@vandieren-symmetry-apal; @vandieren-chainsat-apal] gives some relationships between versions of (\[sc0-3\]) and (\[sc0-5\]) in a single cardinal (with (\[sc0-1\]) as a background assumption). This is done without assuming tameness, using very different technologies than in this paper. This work is applied to the tame context in [@vv-symmetry-transfer-v3], showing for example that (\[sc0-1\]) implies (\[sc0-3\]) holds cardinal by cardinal. A recent preprint of the second author [@stab-spec-v4] studies the model theory of strictly stable tame AECs, establishing in particular that stability on a tail implies no long splitting chains (see Remark \[stab-ss-rmk\]). We do not know how to prove analogs to the last two properties of Fact \[fo-superstab\]. Note also that, while the analogous result is known for stability (see Fact \[stab-spectrum\]), we do not know whether no long splitting chains should hold below the Hanf number: Let ${\mathbf{K}}$ be a ${\operatorname{LS}}({\mathbf{K}})$-tame AEC with a monster model. Assume that there exists $\mu \ge {\operatorname{LS}}({\mathbf{K}})$ such that ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$. Is the least such $\mu$ below $H_1$? The background required to read this paper is a solid knowledge of AECs (for example Chapters 4-12 of Baldwin’s book [@baldwinbook09] or the upcoming [@grossbergbook]). We rely on the first ten sections of [@indep-aec-apal], as well as on the material in [@sv-infinitary-stability-afml; @bv-sat-v3]. At the beginning of Sections \[forking-sec\] and \[solvability-sec\], we make *global* hypotheses that hold until the end of the section (unless said otherwise). This is to make the statements of several technical lemmas more readable. We will repeat the global hypotheses in the statement of major theorems. This paper was written while the second author was working on a Ph.D. thesis under the direction of the first author at Carnegie Mellon University. He would like to thank Professor Grossberg for his guidance and assistance in his research in general and in this work specifically. We also thank Will Boney and a referee for feedback that helped improve the presentation of the paper. Preliminaries ============= We assume familiarity with a basic text on AECs such as [@baldwinbook09] or [@grossbergbook] and refer the reader to the preliminaries of [@sv-infinitary-stability-afml] for more details and motivations on the concepts used in this paper. We use ${\mathbf{K}}$ (boldface) to denote a class of models together with an ordering (written ${{\le_{{\mathbf{K}}}}}$). We will often abuse notation and write for example $M \in {\mathbf{K}}$. When it becomes necessary to consider only a class of models without an ordering, we will write $K$ (no boldface). Throughout all this paper, ${\mathbf{K}}$ is a fixed AEC. Most of the time, ${\mathbf{K}}$ will have amalgamation, joint embedding, and arbitrarily large models. In this case we say that *${\mathbf{K}}$ has a monster model*. The notion of tameness was introduced by Grossberg and VanDieren [@tamenessone] as a useful assumption to prove upward stability results. In [@tamenesstwo; @tamenessthree], several cases of Shelah’s eventual categoricity conjecture were established in tame AECs. Boney [@tamelc-jsl] derived from the existence of a class of strongly compact cardinals that all AECs are tame. In a forthcoming paper Boney and Unger [@lc-tame-v3-toappear] establish that if every AEC is tame then a proper class of large cardinals exists. Thus tameness (for all AECs) is a large cardinal axioms. We believe that this is evidence for the assertion that tameness is a new interesting model-theoretic property, a new dichotomy[^1], that should follow (see [@tamenessthree Conjecture 1.5]) from categoricity in a “high-enough” cardinal. \[shortness-def\] Let $\chi \ge {\operatorname{LS}}({\mathbf{K}})$ be a cardinal. ${\mathbf{K}}$ is *$\chi$-tame* if for any $M \in {\mathbf{K}}_{\ge \chi}$ and any $p \neq q$ in ${\text{gS}}(M)$, there exists $M_0 \in {\mathbf{K}}_{\chi}$ such that $p {\upharpoonright}M_0 \neq q {\upharpoonright}M_0$. We similarly define $(<\chi)$-tame (when $\chi > {\operatorname{LS}}({\mathbf{K}})$). We say that *${\mathbf{K}}$ is tame* provided there exists a cardinal $\chi$ such that[^2] ${\mathbf{K}}$ is $\chi$-tame. If ${\mathbf{K}}$ is $\chi$-tame for $\chi > {\operatorname{LS}}({\mathbf{K}})$, the class ${\mathbf{K}}' := {\mathbf{K}}_{\ge \chi}$ will be an ${\operatorname{LS}}({\mathbf{K}}')$-tame AEC. Hence we will usually directly assume that ${\mathbf{K}}$ is ${\operatorname{LS}}({\mathbf{K}})$-tame. We will use the equivalence between stability and the order property under tameness [@sv-infinitary-stability-afml 4.13]: \[stab-spectrum\] Assume that ${\mathbf{K}}$ is ${\operatorname{LS}}({\mathbf{K}})$-tame and has a monster model. The following are equivalent: 1. ${\mathbf{K}}$ is stable in some cardinal greater than or equal to ${\operatorname{LS}}({\mathbf{K}})$. 2. There exists $\mu < H_1$ such that ${\mathbf{K}}$ is stable in all $\lambda \ge {\operatorname{LS}}({\mathbf{K}})$ such that $\lambda = \lambda^{\mu}$. 3. ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property. Superstability and no long splitting chains ------------------------------------------- A definition of superstability analogous to $\kappa (T) = \aleph_0$ in first-order model theory has been studied in AECs (see [@shvi635; @gvv-mlq; @vandierennomax; @nomaxerrata; @ss-tame-jsl; @indep-aec-apal]). Since it is not immediately obvious what forking should be in that framework, the more rudimentary independence relation of $\lambda$-splitting is used in the definition. Since in AECs, types over models are much better behaved than types over sets, it does not make sense in general to ask for every type to not split over a finite set[^3]. Thus we require that every type over the union of a chain does not split over a model in the chain. For technical reasons (it is possible to prove that the condition follows from categoricity), we require the chain to be increasing with respect to universal extension. Definition \[loc-card-def\] rephrases (\[sc0-1\]) in Corollary \[main-cor\]: \[loc-card-def\] Let $\lambda \ge {\operatorname{LS}}(K)$. We say ${\mathbf{K}}$ has *no long splitting chains in $\lambda$* if for any limit $\delta < \lambda^+$, any increasing ${\langle M_i : i < \delta \rangle}$ in ${\mathbf{K}}_\lambda$ with $M_{i + 1}$ universal over $M_i$ for all $i < \delta$, any $p \in {\text{gS}}(\bigcup_{i < \delta} M_i)$, there exists $i < \delta$ such that $p$ does not $\lambda$-split over $M_i$. The condition in Definition \[loc-card-def\] first appears in [@sh394 Question 6.1]. In [@baldwinbook09 15.1], it is written as[^4] $\kappa ({\mathbf{K}}, \lambda) = \aleph_0$. We do not adopt this notation, since it blurs out the distinction between forking and splitting, and does not mention that only a certain type of chains are considered. A similar notation is in [@indep-aec-apal 3.16]: ${\mathbf{K}}$ has no long splitting chains in $\lambda$ if and only if ${\kappa_{1}} ({{\mathfrak{i}}_{\lambda\text{-ns}}} ({\mathbf{K}}_\lambda), <_{\text{univ}}) = \aleph_0$. In tame AECs with a monster model, no long splitting chains transfers upward: \[ss-stable\] Let ${\mathbf{K}}$ be an AEC with a monster model and let ${\operatorname{LS}}({\mathbf{K}}) \le \lambda \le \mu$. If ${\mathbf{K}}$ is stable in $\lambda$ and has no long splitting chains in $\lambda$, then ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$. Definitions of saturated {#sat-def-subsec} ------------------------ The search for a good definition of “saturated” in AECs is central. We quickly review various possible notions and cite some basic facts about them, including basic implications. Implicit in the definition of no long splitting chains is the notion of a *limit model*. It plays a central role in the study of AECs that do not necessarily have amalgamation [@shvi635] (their study in this context was continued in [@vandierennomax; @nomaxerrata]). We use the notation and basic definitions from [@gvv-mlq]. The two basic facts about limit models (in an AEC with a monster model) are: 1. Existence: If ${\mathbf{K}}$ is stable in $\lambda$, then for every $M$ and every limit $\delta < \lambda^+$ there is a $(\lambda, \delta)$-limit over $M$. 2. Uniqueness: Any two limit models of the same length are isomorphic. Uniqueness of limit models that are *not* of the same cofinality is a key concept which is equivalent to superstability in first-order model theory. A second notion of saturation can be defined using Galois types (when ${\mathbf{K}}$ has a monster model): for $\lambda > {\operatorname{LS}}({\mathbf{K}})$, say $M$ is *$\lambda$-saturated* if every type over a ${{\le_{{\mathbf{K}}}}}$-substructure of $M$ of size less than $\lambda$ is realized inside $M$. We will write ${{\mathbf{K}}^{\lambda\text{-sat}}}$ for the class of $\lambda$-saturated models in ${\mathbf{K}}$. A third notion of saturation appears in [@sh88 3.1(1)][^5]. The idea is to encode a generalization of the fact that a union of saturated models should be saturated. \[sl-def\] Let $M \in {\mathbf{K}}$ and let $\lambda \ge {\operatorname{LS}}({\mathbf{K}})$. $M$ is called *superlimit in $\lambda$ if:* 1. $M \in {\mathbf{K}}_\lambda$. 2. $M$ is “properly universal”: For any $N \in {\mathbf{K}}_\lambda$, there exists $f: N \rightarrow M$ such that $f[N] {{<_{{\mathbf{K}}}}}M$. 3. Whenever ${\langle M_i : i < \delta \rangle}$ is an increasing chain in ${\mathbf{K}}_\lambda$, $\delta < \lambda^+$ and $M_i \cong M$ for all $i < \delta$, then $\bigcup_{i < \delta} M_i \cong M$. The following local implications between the three definitions are known: \[local-implications\] Assume that ${\mathbf{K}}$ has a monster model. Let $\lambda \ge {\operatorname{LS}}({\mathbf{K}})$ be such that ${\mathbf{K}}$ is stable in $\lambda$. 1. \[local-1\] If $\chi \in [{\operatorname{LS}}({\mathbf{K}})^+, \lambda]$ is regular, then any $(\lambda, \chi)$-limit model is $\chi$-saturated. 2. \[local-2\] If $\lambda > {\operatorname{LS}}({\mathbf{K}})$ and $\lambda$ is regular, then $M \in {\mathbf{K}}_{\lambda}$ is saturated if and only if $M$ is $(\lambda, \lambda)$-limit. 3. \[local-3\] If $\lambda > {\operatorname{LS}}({\mathbf{K}})$, then any two limit models of size $\lambda$ are isomorphic if and only if every limit model of size $\lambda$ is saturated. 4. \[local-4\] If $M \in {\mathbf{K}}_{\lambda}$ is superlimit, then for any limit $\delta < \lambda^+$, $M$ is $(\lambda, \delta)$-limit and (if $\lambda > {\operatorname{LS}}({\mathbf{K}})$) saturated. 5. \[local-5\] Assume that $\lambda > {\operatorname{LS}}({\mathbf{K}})$ and there exists a saturated model $M$ of size $\lambda$. Then $M$ is superlimit if and only if in ${\mathbf{K}}_\lambda$, the union of any increasing chain (of length strictly less than $\lambda^+$) of saturated models is saturated. (\[local-1\]), (\[local-2\]), and (\[local-3\]) are straightforward from the basic facts about limit models and the uniqueness of saturated models. (\[local-4\]) is by [@drueckthesis 2.3.10] and the previous parts. (\[local-5\]) then follows. (\[local-3\]) is stated for $\lambda$ regular in [@drueckthesis 2.3.12] but the argument above shows that it holds for any $\lambda$. Skeletons {#skeleton-sec} --------- The notion of a skeleton was introduced in [@indep-aec-apal Section 5] and is meant to be an axiomatization of a subclass of saturated models of an AEC. It is mentioned in (\[sc0-2\]) of the main corollary. Recall the definition of an abstract class, due to the first author [@grossbergbook] (or see [@sv-infinitary-stability-afml 2.7]): it is a pair ${\mathbf{K}}' = (K', {\le_{{\mathbf{K}}'}})$ so that $K'$ is a class of $\tau$-structures in a fixed vocabulary $\tau = \tau ({\mathbf{K}}')$, closed under isomorphisms, and ${\le_{{\mathbf{K}}'}}$ is a partial order on $K'$ which respects isomorphisms and extends the $\tau$-substructure relation. \[skel-def\] A *skeleton* of an abstract class ${\mathbf{K}}^\ast$ is an abstract class ${\mathbf{K}}'$ such that: 1. $K' \subseteq K^\ast$ and for $M, N \in {\mathbf{K}}'$, $M {\le_{{\mathbf{K}}'}} N$ implies $M {\le_{{\mathbf{K}}^\ast}} N$. 2. ${\mathbf{K}}'$ is dense in ${\mathbf{K}}^\ast$: For any $M \in {\mathbf{K}}^\ast$, there exists $M' \in {\mathbf{K}}'$ such that $M {\le_{{\mathbf{K}}^\ast}} M'$. 3. If $\alpha$ is a (not necessarily limit) ordinal and ${\langle M_i : i < \alpha \rangle}$ is a strictly ${\le_{{\mathbf{K}}^\ast}}$-increasing chain in ${\mathbf{K}}'$, then there exists $N \in {\mathbf{K}}'$ such that $M_i {\le_{{\mathbf{K}}'}} N$ and[^6] $M_i \neq N$ for all $i < \alpha$. Let $\lambda \ge {\operatorname{LS}}({\mathbf{K}})$. Assume that ${\mathbf{K}}$ is stable in $\lambda$, has amalgamation and no maximal models in $\lambda$. Let $K'$ be the class of limit models of size $\lambda$ in ${\mathbf{K}}$. Then $(K', {{\le_{{\mathbf{K}}}}})$ (or even $K'$ ordered with “being equal or universal over”) is a skeleton of ${\mathbf{K}}_\lambda$. If ${\mathbf{K}}'$ is a skeleton of ${\mathbf{K}}_\lambda$ and ${\mathbf{K}}'$ itself generates an AEC, then $M {\le_{{\mathbf{K}}'}} N$ if and only if $M, N \in {\mathbf{K}}'$ and $M {{\le_{{\mathbf{K}}}}}N$. This is because of the third clause in the definition of a skeleton (used with $\alpha = 2$) and the coherence axiom. We can define notions such as amalgamation and Galois types for any abstract class (see the preliminaries of [@sv-infinitary-stability-afml]). The properties of a skeleton often correspond to properties of the original AEC: \[skeleton-facts\] Let $\lambda \ge {\operatorname{LS}}({\mathbf{K}})$ and assume that ${\mathbf{K}}$ has amalgamation in $\lambda$. Let ${\mathbf{K}}'$ be a skeleton of ${\mathbf{K}}_\lambda$. 1. \[skel-1\] For $P$ standing for having no maximal models in $\lambda$, being stable in $\lambda$, or having joint embedding in $\lambda$, ${\mathbf{K}}$ has $P$ if and only if ${\mathbf{K}}'$ has $P$. 2. \[skel-2\] Assume that ${\mathbf{K}}$ has joint embedding in $\lambda$ and for every limit $\delta < \lambda^+$ and every $N \in {\mathbf{K}}'$ there exists $N' \in {\mathbf{K}}'$ which is $(\lambda, \delta)$-limit over $N$ (in the sense of ${\mathbf{K}}'$). 1. \[skel-2a\] Let $M, M_0 \in {\mathbf{K}}'$ and let $\delta < \lambda^+$ be a limit ordinal. Then $M$ is $(\lambda, \delta)$-limit over $M_0$ in the sense of ${\mathbf{K}}'$ if and only $M$ is $(\lambda, \delta)$-limit over $M_0$ in the sense of ${\mathbf{K}}$. 2. \[skel-2b\] ${\mathbf{K}}'$ has no long splitting chains in $\lambda$ if and only if ${\mathbf{K}}$ has no long splitting chains in $\lambda$. (\[skel-1\]) is by [@indep-aec-apal 5.8]. As for (\[skel-2a\]), (\[skel-2b\]), note first that the hypotheses of (\[skel-2\]) imply (by (\[skel-1\])) that ${\mathbf{K}}$ is stable in $\lambda$ and has no maximal models in $\lambda$. In particular, limit models of size $\lambda$ exist in ${\mathbf{K}}$. Let us prove (\[skel-2a\]). If $M$ is $(\lambda, \delta)$-limit over $M_0$ in the sense of ${\mathbf{K}}'$, then it is straightforward to check that the chain witnessing it will also witness that $M$ is $(\lambda, \delta)$-limit over $M_0$ in the sense of ${\mathbf{K}}$. For the converse, observe that by assumption there exists a $(\lambda, \delta)$-limit $M'$ over $M_0$ in the sense of ${\mathbf{K}}'$. Furthermore, by what has just been observed $M'$ is also limit in the sense of ${\mathbf{K}}$, hence by uniqueness of limit models of the same length, $M' \cong_{M_0} M$. Therefore $M$ is also $(\lambda, \delta)$-limit over $M_0$ in the sense of ${\mathbf{K}}'$. The proof of (\[skel-2b\]) is similar, see [@indep-aec-apal 6.7]. Good frames {#good-frame-subsec} ----------- Good frames are a local axiomatization of forking in a first-order superstable theories. They are introduced in [@shelahaecbook Chapter II]. We will use the definition from [@indep-aec-apal 8.1] which is weaker and more general than Shelah’s, as it does not require the existence of a superlimit (as in [@jrsh875]). As opposed to [@indep-aec-apal], we allow good frames that are *not* type-full: we only require the existence of a set of well-behaved basic types satisfying some density property (see [@shelahaecbook Chapter II] for more). Note however that Remark \[type-full-rmk\] says that in the context of the main theorem the existence of a good frame implies the existence of a *type-full* good frame (possibly over a different class). In [@indep-aec-apal 8.1], the underlying class of the good frame consists only of models of size $\lambda$. Thus when we say that there is a good $\lambda$-frame on a class ${\mathbf{K}}'$, we mean the underlying class of the good frame is ${\mathbf{K}}'$, and the axioms of good frames will require that ${\mathbf{K}}'$ generates a non-empty AEC with amalgamation in $\lambda$, joint embedding in $\lambda$, no maximal models in $\lambda$, and stability in $\lambda$. The only facts that we will use about good frames are: \[good-frame-uq-limit\] Let $\lambda \ge {\operatorname{LS}}({\mathbf{K}})$. If there is a good $\lambda$-frame on a skeleton of ${\mathbf{K}}_{\lambda}$, then ${\mathbf{K}}$ has a unique limit model of size $\lambda$. Moreover, for any $M_0, M_1, M_2 \in {\mathbf{K}}_{\lambda}$, if both $M_1$ and $M_2$ are limit over $M_0$, then $M_1 \cong_{M_0} M_2$ (i.e. the isomorphism fixes $M_0$). Let ${\mathbf{K}}'$ be the skeleton of ${\mathbf{K}}_\lambda$ which is the underlying class of the good $\lambda$-frame. By [@shelahaecbook II.4.8] (see [@ext-frame-jml 9.2] for a detailed proof), ${\mathbf{K}}'$ has a unique limit model of size $\lambda$ (and the moreover part holds for ${\mathbf{K}}'$). By Fact \[skeleton-facts\](\[skel-2a\]), this must also be the unique limit model of size $\lambda$ in ${\mathbf{K}}$ (and the moreover part holds in ${\mathbf{K}}$ too). \[good-frame-existence\] Assume that ${\mathbf{K}}$ has a monster model and is ${\operatorname{LS}}({\mathbf{K}})$-tame. If $\mu < H_1$ is such that ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$, then there exists $\lambda_0 < H_1$ such that for all $\lambda \ge \lambda_0$, there is a good $\lambda$-frame on ${{\mathbf{K}}^{\lambda\text{-sat}}}_\lambda$. Moreover, ${{\mathbf{K}}^{\lambda\text{-sat}}}_\lambda$ is a skeleton of ${\mathbf{K}}_\lambda$, ${\mathbf{K}}$ is stable in $\lambda$, any $M \in {{\mathbf{K}}^{\lambda\text{-sat}}}_\lambda$ is superlimit, and the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. First assume that ${\mathbf{K}}$ has no long splitting chains in ${\operatorname{LS}}({\mathbf{K}})$ and is stable in ${\operatorname{LS}}({\mathbf{K}})$. By [@bv-sat-v3 7.1], there exists $\lambda_0 < \beth_{\left(2^{\mu^+}\right)^+}$ such that for any $\lambda \ge \lambda_0$, any increasing chain of $\lambda$-saturated models is $\lambda$-saturated and there is a good $\lambda$-frame on ${{\mathbf{K}}^{\lambda\text{-sat}}}_\lambda$. That any $M \in {{\mathbf{K}}^{\lambda\text{-sat}}}_\lambda$ is a superlimit (Fact \[local-implications\](\[local-5\])) and ${{\mathbf{K}}^{\lambda\text{-sat}}}_\lambda$ is a skeleton of ${\mathbf{K}}_\lambda$ easily follows, and stability in $\lambda$ is given (for example) by Fact \[skeleton-facts\](\[skel-1\]). Now by [@bv-sat-v3 6.12], we more precisely have that if ${\mathbf{K}}$ has no long splitting chains in $\mu$ and is stable in $\mu$ (with $\mu \ge {\operatorname{LS}}({\mathbf{K}})$) and $(<{\operatorname{LS}}({\mathbf{K}}))$-tame (tameness being defined using types over sets), then the same conclusion holds with $\beth_{\left(2^{\mu^+}\right)^+}$ replaced by $H_1$. Now the use of $(<{\operatorname{LS}}({\mathbf{K}}))$-tameness is to derive that there exists $\chi < H_1$ so that ${\mathbf{K}}$ does not have a certain order property of length $\chi$, but [@bv-sat-v3] relies on an older version of [@sv-infinitary-stability-afml] which proves Fact \[stab-spectrum\] assuming $(<{\operatorname{LS}}({\mathbf{K}}))$-tameness instead of ${\operatorname{LS}}({\mathbf{K}})$-tameness. In the current version of [@sv-infinitary-stability-afml], it is shown that ${\operatorname{LS}}({\mathbf{K}})$-tameness suffices, thus the arguments of [@bv-sat-v3] go through assuming ${\operatorname{LS}}({\mathbf{K}})$-tameness instead of $(<{\operatorname{LS}}({\mathbf{K}}))$-tameness. Solvability {#solvability-subsec} ----------- Solvability appears as a possible definition of superstability for AECs in [@shelahaecbook Chapter IV]. The definition uses Ehrenfeucht-Mostowski models and we assume the reader has some familiarity with them, see for example [@baldwinbook09 Section 6.2] or [@shelahaecbook IV.0.8]. \[em-def\] 1. A countable set $\Phi = \{p_n : n < \omega\}$ is *proper for linear orders* if the $p_n$’s are an increasing sequence of $n$-variable quantifier-free types in a fixed vocabulary $\tau (\Phi)$ which are satisfied by a sequence of indiscernibles. As usual, such a set $\Phi$ determines an EM-functor from linear orders into $\tau (\Phi)$-structures, mapping a linear order $I$ to ${\operatorname{EM}}(I, \Phi)$ and taking suborders to substructures. 2. [@shelahaecbook IV.0.8] For $\mu \ge {\operatorname{LS}}({\mathbf{K}})$, let $\Upsilon_{\mu}[{\mathbf{K}}]$ be the set of $\Phi$ proper for linear orders with $\tau ({\mathbf{K}}) \subseteq \tau (\Phi)$, $|\tau (\Phi)| \le \mu$, and such that the $\tau ({\mathbf{K}})$-reduct ${\operatorname{EM}}_{\tau ({\mathbf{K}})} (I, \Phi)$ is a functor from linear orders into members of ${\mathbf{K}}$ of cardinality at most $|I| + \mu$. Such a $\Phi$ is called an *EM blueprint for ${\mathbf{K}}$*. \[solvability-def\] Let ${\operatorname{LS}}({\mathbf{K}}) \le \mu \le \lambda$. 1. [@shelahaecbook IV.1.4(1)] We say that *$\Phi$ witnesses $(\lambda, \mu)$-solvability* if: 1. $\Phi \in \Upsilon_{\mu}[{\mathbf{K}}]$. 2. If $I$ is a linear order of size $\lambda$, then ${\operatorname{EM}}_{\tau ({\mathbf{K}})} (I, \Phi)$ is superlimit in $\lambda$ for ${\mathbf{K}}$, see Definition \[sl-def\]. ${\mathbf{K}}$ is *$(\lambda, \mu)$-solvable* if there exists $\Phi$ witnessing $(\lambda, \mu)$-solvability. 2. ${\mathbf{K}}$ is *uniformly $(\lambda, \mu)$-solvable* if there exists $\Phi$ such that for all $\lambda' \ge \lambda$, $\Phi$ witnesses $(\lambda', \mu)$-solvability. \[EM-existence\] Let ${\mathbf{K}}$ be an AEC and let $\mu \ge {\operatorname{LS}}({\mathbf{K}})$. Then ${\mathbf{K}}$ has arbitrarily large models if and only if $\Upsilon_\mu[{\mathbf{K}}] \neq \emptyset$. We give some more manageable definitions of solvability ((\[equiv-cond-3\]) is the one we will use). Shelah already mentions one of them on [@shelahaecbook p. 61] (but does not prove it is equivalent). \[solvability-equiv\] Let ${\operatorname{LS}}({\mathbf{K}}) \le \mu \le \lambda$. The following are equivalent. 1. \[equiv-cond-1\] ${\mathbf{K}}$ is \[uniformly\] $(\lambda, \mu)$-solvable. 2. \[equiv-cond-2\] There exists $\tau' \supseteq \tau ({\mathbf{K}})$ with $|\tau'| \le \mu$ and $\psi \in {\mathbb{L}}_{\mu^+, \omega} (\tau')$ such that: 1. $\psi$ has arbitrarily large models. 2. [\[]{}For all $\lambda' \ge \lambda$\], if $M \models \psi$ and $\|M\| = \lambda$ \[$\|M\| = \lambda'$\], then $M {\upharpoonright}\tau ({\mathbf{K}})$ is in ${\mathbf{K}}$ and is superlimit. 3. \[equiv-cond-3\] There exists $\tau' \supseteq \tau ({\mathbf{K}})$ and an AEC ${\mathbf{K}}'$ with $\tau({\mathbf{K}}') = \tau'$, ${\operatorname{LS}}({\mathbf{K}}') \le \mu$ such that: 1. ${\mathbf{K}}'$ has arbitrarily large models. 2. [\[]{}For all $\lambda' \ge \lambda$\], if $M \in {\mathbf{K}}'$ and $\|M\| = \lambda$ \[$\|M\| = \lambda'$\], then $M {\upharpoonright}\tau ({\mathbf{K}})$ is in ${\mathbf{K}}$ and is superlimit. - : Let $\Phi$ witness $(\lambda, \mu)$-solvability and write $\Phi = \{p_n \mid n < \omega\}$. Let $\tau' := \tau (\Phi) \cup \{P, <\}$, where $P$, $<$ are symbols for a unary predicate and a binary relation respectively. Let $\psi \in {\mathbb{L}}_{\mu^+, \omega} (\tau')$ say: 1. $(P, <)$ is a linear order. 2. For all $n < \omega$ and all $x_0 < \cdots < x_{n - 1}$ in $P$, $x_0 \ldots x_{n - 1}$ realizes $p_n$. 3. For all $y$, there exists $n < \omega$, $x_0 < \cdots < x_{n - 1}$ in $P$, and $\rho$ an $n$-ary term of $\tau (\Phi)$ such that $y = \rho (x_0, \ldots, x_{n - 1})$. Then if $M \models \psi$, $M {\upharpoonright}\tau = {\operatorname{EM}}_{\tau ({\mathbf{K}})} (P^M, \Phi)$ (and by solvability if $\|M\| = \lambda$ then $M$ is superlimit in ${\mathbf{K}}$). Conversely, if $M = {\operatorname{EM}}_{\tau ({\mathbf{K}})} (I, \Phi)$, we can expand $M$ to a $\tau'$-structure $M'$ by letting $(P^{M'}, <^{M'}) := (I, <)$. Thus $\psi$ is as desired. - : Given $\tau'$ and $\psi$ as given by (\[equiv-cond-2\]), Let $\Psi$ be a fragment of ${\mathbb{L}}_{\mu^+, \omega} (\tau')$ containing $\psi$ of size $\mu$ and let ${\mathbf{K}}'$ be ${\operatorname{Mod}}(\psi)$ ordered by $\lee_{\Psi}$. Then ${\mathbf{K}}'$ is as desired for (\[equiv-cond-3\]). - : Directly from Fact \[EM-existence\]. Forking and averages in stable AECs {#forking-sec} =================================== In the introduction to his book [@shelahaecbook p. 61], Shelah asserts (without proof) that in the first-order context solvability (see Section \[solvability-subsec\]) is equivalent to superstability. We aim to give a proof (see Corollary \[solvability-fo\]) and actually show (assuming amalgamation, stability, and tameness) that solvability is equivalent to any of the definitions in the main theorem. First of all, if there exists $\mu$ such that ${\mathbf{K}}$ is $(\lambda, \mu)$-solvable for all high-enough $\lambda$, then in particular ${\mathbf{K}}$ has a superlimit in all high-enough $\lambda$, so we obtain (\[sc0-4\]) in the main corollary. We work toward a converse. The proof is similar to that in [@fcp-saturation]: we aim to code saturated models using their characterization with average of sequences (the crucial result for this is Lemma \[saturation-charact\]). In this section, we use the theory of averages in AECs (as developed by Shelah in [@shelahaecbook2 Chapter V.A] and by Boney and the second author in [@bv-sat-v3]) to give a new characterization of forking (Lemma \[forking-charact\]). We also prove the key result for (\[sc0-5\]) implies (\[sc0-1\]) in the main corollary (Theorem \[ss-from-chainsat\]). All throughout, we assume: \[global-nf-hyp\] 1. ${\mathbf{K}}$ has a monster model ${\mathfrak{C}}$ (we work inside it). 2. ${\mathbf{K}}$ is ${\operatorname{LS}}({\mathbf{K}})$-tame. 3. ${\mathbf{K}}$ is stable in some cardinal greater than or equal to ${\operatorname{LS}}({\mathbf{K}})$. We set $\kappa := {\operatorname{LS}}({\mathbf{K}})^+$ and work in the setup of [@bv-sat-v3 Section 5]. In particular we think of Galois types of size ${\operatorname{LS}}({\mathbf{K}})$ as formulas and think of bigger Galois types as set of such formulas. That is, we work inside the Galois Morleyization of ${\mathbf{K}}$ (see [@sv-infinitary-stability-afml 3.3, 3.16]). We encourage the reader to have a copy of both [@sv-infinitary-stability-afml] and [@bv-sat-v3] open, since we will cite from there freely and use basic notation and terminology ($\chi$-convergent, $\chi$-based, $(\chi_0, \chi_1, \chi_2)$-Morley, ${\operatorname{Av}}_\chi ({\mathbf{I}}/ A)$ etc.) often without even an explicit citation. We will say that $p \in {\text{gS}}^{<\kappa} (M)$ *does not syntactically split* over $M_0 {{\le_{{\mathbf{K}}}}}M$ if it does not split in the syntactic sense of [@bv-sat-v3 5.7] (that is, it does not split in the usual first-order sense when we think of Galois types of size ${\operatorname{LS}}({\mathbf{K}})$ as formulas). Note that several results from [@bv-sat-v3] that we quote assume $(<{\operatorname{LS}}({\mathbf{K}}))$-tameness (defined in terms of Galois types over sets). However, as argued in the proof of Fact \[good-frame-existence\], ${\operatorname{LS}}({\mathbf{K}})$-tameness suffices. We will define several other cardinals $\chi_0 < \chi_0' < \chi_1 < \chi_1' < \chi_2$ (see Notation \[notation-1\], \[notation-2\], and \[notation-3\]). The reader can simply see them as “high-enough” cardinals with reasonable closure properties. If $\chi_0$ is chosen reasonably, we will have $\chi_2 < H_1$. The letters ${\mathbf{I}}$, ${\mathbf{J}}$ will denote sequences of tuples of length strictly less than $\kappa$. We will use the same conventions as in [@bv-sat-v3 Section 5]. Note that the sequences *may be indexed by arbitrary linear orders*. By Facts \[stab-spectrum\] and [@sh394 I.4.5(3)] (recalling that there is a global assumption of stability in this section), we have: \[stab-op\] There exists $\chi_0 < H_1$ such that ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property of length $\chi_0$. Another property of $\chi_0$ is the following more precise version of Fact \[stab-spectrum\] (see [@sv-infinitary-stability-afml] on how to translate Shelah’s syntactic version to AECs): \[stab-spectrum-2\] If $\lambda = \lambda^{\chi_0}$, then ${\mathbf{K}}$ is stable in $\lambda$. In particular, ${\mathbf{K}}$ is stable in $\chi_0'$. The following notation will be convenient: \[notation-1\] Let $\chi_0$ be any regular cardinal such that $\chi_0 \ge 2^{{\operatorname{LS}}({\mathbf{K}})}$ and ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property of length $\chi_0^+$. For a cardinal $\lambda$, let $\gamma (\lambda) := (2^{2^{\lambda}})^+$. We write $\chi_0' := \gamma (\chi_0)$. \[rmk-hanf\] By Fact \[stab-op\], one can take $\chi_0 < H_1$. In that case also $\chi_0' < H_1$. For the sake of generality, we do *not* require that $\chi_0$ be least with the property above. Recall [@bv-sat-v3 5.21] that if ${\mathbf{I}}$ is a $(\chi_0^+, \chi_0^+, \gamma (\chi_0))$-Morley sequence, then ${\mathbf{I}}$ is $\chi$-convergent. We want to use this to relate average and forking: Let $M_0, M \in {{\mathbf{K}}^{(\chi_0')^+\text{-sat}}}$ be such that $M_0 {{\le_{{\mathbf{K}}}}}M$. Let $p \in {\text{gS}}(M)$. We say that *$p$ does not fork over $M_0$* if there exists $M_0' \in {\mathbf{K}}_{\chi_0'}$ such that $M_0' {{\le_{{\mathbf{K}}}}}M_0$ and $p$ does not $\chi_0'$-split over $M_0'$. We will use without comments: \[forking-props\] Forking has the following properties: 1. Invariance under isomorphisms and monotonicity: if $M_0 {{\le_{{\mathbf{K}}}}}M_1 {{\le_{{\mathbf{K}}}}}M_2$ are all $(\chi_0')^+$-saturated and $p \in {\text{gS}}(M_2)$ does not fork over $M_0$, then $p {\upharpoonright}M_1$ does not fork over $M_0$ and $p$ does not fork over $M_1$. 2. \[forking-props-2\] Set local character: if $M \in {{\mathbf{K}}^{(\chi_0')^+\text{-sat}}}$ and $p \in {\text{gS}}(M)$, there exists $M_0 \in {{\mathbf{K}}^{(\chi_0')^+\text{-sat}}}$ of size $(\chi_0')^+$ such that $M_0 {{\le_{{\mathbf{K}}}}}M$ and $p$ does not fork over $M_0$. 3. Transitivity: Assume $M_0 {{\le_{{\mathbf{K}}}}}M_1 {{\le_{{\mathbf{K}}}}}M_2$ are all $(\chi_0')^+$-saturated and $p \in {\text{gS}}(M_2)$. If $p$ does not fork over $M_1$ and $p {\upharpoonright}M_1$ does not fork over $M_0$, then $p$ does not fork over $M_0$. 4. \[forking-props-4\] Uniqueness: If $M_0 {{\le_{{\mathbf{K}}}}}M$ are all $(\chi_0')^+$-saturated and $p, q \in {\text{gS}}(M)$ do not fork over $M_0$, then $p {\upharpoonright}M_0 = q {\upharpoonright}M_0$ implies $p = q$. Moreover $p$ does not $\lambda$-split over $M_0$ for any $\lambda \ge \left(\chi_0'\right)^+$. 5. \[forking-props-5\] Local extension over saturated models: If $M_0 {{\le_{{\mathbf{K}}}}}M$ are both saturated, $\|M_0\| = \|M\| \ge (\chi_0')^+$, $p \in {\text{gS}}(M_0)$, there exists $q \in {\text{gS}}(M)$ such that $q$ extends $p$ and does not fork over $M_0$. Use [@indep-aec-apal 7.5]. The generator used is the one given by Proposition 7.4(2) there. For the moreover part of uniqueness, use [@bgkv-apal 4.2] (and [@bgkv-apal 3.12]). Note that the extension property need not hold in general. However if the class has no long splitting chains we have: \[union-sat\] If ${\mathbf{K}}$ has no long splitting chains in $\chi_0'$, then: 1. ([@indep-aec-apal 8.9] or [@ss-tame-jsl 7.1]) Forking has: 1. The extension property: If $M_0 {{\le_{{\mathbf{K}}}}}M$ are $(\chi_0')^+$-saturated and $p \in {\text{gS}}(M_0)$, then there exists $q \in {\text{gS}}(M)$ extending $p$ and not forking over $M_0$. 2. The chain local character property: If ${\langle M_i : i < \delta \rangle}$ is an increasing chain of $(\chi_0')^+$-saturated models and $p \in {\text{gS}}(\bigcup_{i < \delta} M_i)$, then there exists $i < \delta$ such that $p$ does not fork over $M_i$. 2. [@bv-sat-v3 6.9] For any $\lambda > (\chi_0')^+$, ${{\mathbf{K}}^{\lambda\text{-sat}}}$ is an AEC with ${\operatorname{LS}}({{\mathbf{K}}^{\lambda\text{-sat}}}) = \lambda$. For notational convenience, we “increase” $\chi_0$: \[notation-2\] Let $\chi_1 := (\chi_0')^{++}$. Let $\chi_1' := \gamma (\chi_1)$. We obtain a characterization of forking that adds to those proven in [@indep-aec-apal Section 9]. A form of it already appears in [@shelahaecbook IV.4.6]. Again, we define more cardinal parameters: \[notation-3\] Let $\chi_2 := \beth_{\omega} (\chi_0)$. \[rmk-hanf-2\] We have that $\chi_0 < \chi_0' < \chi_1 < \chi_1' < \chi_2$, and $\chi_2 < H_1$ if $\chi_0 < H_1$. \[forking-charact\] Let $M_0, M$ be $\chi_2$-saturated with $M_0 {{\le_{{\mathbf{K}}}}}M$. Let $p \in {\text{gS}}(M)$. The following are equivalent: 1. \[forking-char-1\] $p$ does not fork over $M_0$. 2. \[forking-char-2\] $p {\upharpoonright}M_0$ has a nonforking extension to ${\text{gS}}(M)$ and there exists $M_0' {{\le_{{\mathbf{K}}}}}M_0$ with $\|M_0'\| < \chi_2$ such that $p$ does not syntactically split over $M_0'$. 3. \[forking-char-3\] $p {\upharpoonright}M_0$ has a nonforking extension to ${\text{gS}}(M)$ and there exists $\mu \in [\chi_0^+, \chi_2)$ and ${\mathbf{I}}$ a $(\mu, \mu, \gamma (\mu)^+)$-Morley sequence for $p$, with all the witnesses inside $M_0$, such that ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M) = p$. When ${\mathbf{K}}$ has no long splitting chains in $\chi_0'$, forking has the extension property (Fact \[union-sat\]) so the first part of (\[forking-char-2\]) and (\[forking-char-3\]) always hold. However in Theorem \[ss-from-chainsat\] we apply Lemma \[forking-charact\] in the strictly stable case (i.e. ${\mathbf{K}}$ may only be *stable* in $\chi_0'$ and not have no long splitting chains there). We recall more definitions and facts before giving the proof of Lemma \[forking-charact\]: \[ns-lc\] If $p \in {\text{gS}}(M)$ and $M$ is $\chi_0^+$-saturated, there exists $M_0 \in {\mathbf{K}}_{\le \chi_0}$ with $M_0 {{\le_{{\mathbf{K}}}}}M$ such that $p$ does not syntactically split over $M_0$. \[average-splitting\] Let $M_0 {{\le_{{\mathbf{K}}}}}M$ be both $(\chi_1')^+$-saturated. Let $\mu := \|M_0\|$. Let $p \in {\text{gS}}(M)$ and let ${\mathbf{I}}$ be a $(\mu^+, \mu^+, \gamma (\mu))$-Morley sequence for $p$ over $M_0$ with all the witnesses inside $M$. Then if $p$ does not syntactically split or does not fork over $M_0$, then ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M) = p$. For syntactic splitting, this is [@bv-sat-v3 5.25]. The Lemma is actually more general and the proof of [@bv-sat-v3 6.9] shows that this also works for forking. Before starting, note that if $\mu < \chi_2$, then ${\mathbf{K}}$ is stable in $2^{{\mu + \chi_0}} < \chi_2$ by Fact \[stab-spectrum-2\]. Thus there are unboundedly many stability cardinals below $\chi_2$, so we have “enough space” to build Morley sequences. - : By Fact \[ns-lc\], we can find $M_0' {{\le_{{\mathbf{K}}}}}M_0$ such that $p {\upharpoonright}M_0$ does not syntactically split over $M_0'$ and $\|M_0'\| \le \chi_1$. Taking $M_0'$ bigger, we can assume $M_0'$ is $\chi_1$-saturated and $p {\upharpoonright}M_0$ does not fork over $M_0'$. Thus by transitivity $p$ does not fork over $M_0'$. Let ${\mathbf{I}}$ be a $(\chi_1^+, (\chi_1')^+, (\chi_1')^+)$-Morley sequence for $p {\upharpoonright}M_0$ over $M_0'$ inside $M_0$. By [@bv-sat-v3 5.21], ${\mathbf{I}}$ is $\chi_1'$-convergent. By [@bv-sat-v3 5.20], ${\mathbf{I}}$ is $\chi_1'$-based on $M_0'$. Note also that ${\mathbf{I}}$ is a $(\chi_1^+, (\chi_1')^+, (\chi_1')^+)$-Morley sequence for $p$ over $M_0'$ and by Fact \[average-splitting\], ${\operatorname{Av}}_{\chi_1'} ({\mathbf{I}}/ M_0) = p$ so as ${\mathbf{I}}$ is $\chi_1'$-based on $M_0'$, $p$ does not syntactically split over $M_0'$. - : As in the proof of (\[forking-char-1\]) implies (\[forking-char-2\]) (except $\chi_1$ could be bigger). - : By Fact [@bv-sat-v3 5.21], ${\mathbf{I}}$ is $\gamma (\mu)$-convergent. Pick any ${\mathbf{J}}\subseteq {\mathbf{I}}$ of length $\gamma (\mu)$ and use [@bv-sat-v3 5.10] to find $M_0' {{\le_{{\mathbf{K}}}}}M_0$ of size $\gamma (\mu)$ such that ${\mathbf{J}}$ is $\gamma (\mu)$-based on $M_0'$. Since also ${\mathbf{J}}$ is $\gamma (\mu)$-convergent, we have that ${\mathbf{I}}$ is $\gamma (\mu)$-based on $M_0'$. Thus ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M) = p$ does not syntactically split over $M_0'$. - : Without loss of generality, we can choose $M_0'$ to be such that $p {\upharpoonright}M_0$ also does not fork over $M_0'$. Let $\mu := \|M_0'\| + \chi_0$. Build a $(\mu^+, \mu^+, \gamma (\mu))$-Morley sequence ${\mathbf{I}}$ for $p$ over $M_0'$ inside $M_0$. If $q$ is the nonforking extension of $p {\upharpoonright}M_0$ to $M$, then ${\mathbf{I}}$ is also a Morley sequence for $q$ over $M_0'$ so by the proof of (\[forking-char-1\]) implies (\[forking-char-2\]) we must have ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M) = q$, but also ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M) = p$, since $p$ does not syntactically split over $M_0'$ (Fact \[average-splitting\]). Thus $p = q$. The next result is a version of [@shelahfobook III.3.10] in our context. It is implicit in the proof of [@bv-sat-v3 5.27]. \[saturation-charact\] Let $M \in {{\mathbf{K}}^{\chi_2\text{-sat}}}$. Let $\lambda \ge \chi_2$ be such that ${\mathbf{K}}$ is stable in unboundedly many $\mu < \lambda$. The following are equivalent. 1. \[sat-cond-1\] $M$ is $\lambda$-saturated. 2. \[sat-cond-2\] If $q \in {\text{gS}}(M)$ is not algebraic and does not syntactically split over $M_0 {{\le_{{\mathbf{K}}}}}M$ with $\|M_0\| < \chi_2$, there exists a $((\|M_0\| + \chi_0)^+, (\|M_0\| + \chi_0)^+, \lambda)$-Morley sequence for $p$ over $M_0$ inside $M$. (\[sat-cond-1\]) implies (\[sat-cond-2\]) is trivial using saturation. Now assume (\[sat-cond-2\]). Let $p \in {\text{gS}}(N)$, $\|N\| < \lambda$, $N {{\le_{{\mathbf{K}}}}}M$. We show that $p$ is realized in $M$. Let $q \in {\text{gS}}(M)$ extend $p$. If $q$ is algebraic, we are done so assume it is not. Let $M_0 {{\le_{{\mathbf{K}}}}}M$ have size $(\chi_1')^+$ such that $q$ does not fork over $M_0$. By Lemma \[forking-charact\], we can increase $M_0$ if necessary so that $q$ does not syntactically split over $M_0$ and $\mu := \|M_0\| \ge \chi_0$. Now by (\[sat-cond-2\]), there exists a $(\mu^+, \mu^+, \lambda)$-Morley sequence ${\mathbf{I}}$ for $q$ over $M_0$ inside $M$. Now by Fact \[average-splitting\], ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M) = q$. Thus ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ N) = p$. By [@bv-sat-v3 5.6] and the hypothesis of stability in unboundedly many cardinals below $\lambda$, $p$ is realized by an element of ${\mathbf{I}}$ and hence by an element of $M$. We end by showing that if high-enough limit models are sufficiently saturated, then no long splitting chains holds. A similar argument already appears in the proof of [@shelahaecbook IV.4.10]. We start with a more local version, \[forking-chain-lc\] Let $\lambda \ge \chi_2$. Let $\delta < \lambda^+$ be a limit ordinal and let ${\langle M_i : i < \delta \rangle}$ be an increasing chain of saturated models in ${\mathbf{K}}_\lambda$. Let $M_\delta := \bigcup_{i < \delta} M_i$. If $M_\delta$ is $\chi_2$-saturated, then for any $p \in {\text{gS}}(M_\delta)$, there exists $i < \delta$ such that $p$ does not fork over $M_i$. Without loss of generality, $\delta$ is regular. If $\delta \ge \chi_2$, by set local character (Fact \[forking-props\](\[forking-props-2\])), there exists $M_0'$ of size $\chi_1$ such that $p$ does not fork over $M_0'$ and $M_0' {{\le_{{\mathbf{K}}}}}M_\delta$, so pick $i < \delta$ such that $M_0' {{\le_{{\mathbf{K}}}}}M_i$ and use monotonicity. Now assume $\delta < \chi_2$. By assumption, we have that $M_\delta$ is $\chi_2$-saturated. We also have that $p$ does not fork over $M_\delta$ (by set local character) so by Lemma \[forking-charact\], there exists $\mu \in [\chi_0^+, \chi_2)$ and ${\mathbf{I}}$ a $(\mu, \mu, \gamma (\mu)^+)$-Morley sequence for $p$ with all the witnesses inside $M_\delta$ such that ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}/ M_\delta) = p$. Since $M_\delta$ is $\chi_2$-saturated (and there are unboundedly many stability cardinals below $\chi_2$), we can increase ${\mathbf{I}}$ if necessary to assume that $|{\mathbf{I}}| \ge \chi_2$. Write ${\mathbf{I}}_i := |M_i| \cap {\mathbf{I}}$. Since $\delta < \chi_2$, there must exists $i < \delta$ such that $|{\mathbf{I}}_i| \ge \chi_2$. Note that ${\mathbf{I}}_i$ is a $(\mu, \mu, \chi_2)$-Morley sequence for $p$. Because ${\mathbf{I}}$ is $\gamma (\mu)$-convergent and $|{\mathbf{I}}_i| \ge \chi_2 > \gamma (\mu)$, ${\operatorname{Av}}_{\gamma (\mu)} ({\mathbf{I}}_i / M_\delta) = p$. Letting $M' {{\ge_{{\mathbf{K}}}}}M_\delta$ be a saturated model of size $\lambda$ and using local extension over saturated models (Fact \[forking-props\](\[forking-props-5\])), $p {\upharpoonright}M_i$ has a nonforking extension to ${\text{gS}}(M')$ and hence to ${\text{gS}}(M_\delta)$. By Lemma \[forking-charact\], $p$ does not fork over $M_i$, as desired. \[ss-from-chainsat\] Assume that ${\mathbf{K}}$ has a monster model, is ${\operatorname{LS}}({\mathbf{K}})$-tame, and stable in some cardinal greater than or equal to ${\operatorname{LS}}({\mathbf{K}})$. Let $\chi_0 \ge {\operatorname{LS}}({\mathbf{K}})$ be such that ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property of length $\chi_0$, and let $\chi_2 := \beth_\omega (\chi_0)$. Let $\lambda \ge \chi_2$ be such that ${\mathbf{K}}$ is stable in $\lambda$ and there exists a saturated model of cardinality $\lambda$. If every limit model of cardinality $\lambda$ is $\chi_2$-saturated, then ${\mathbf{K}}$ has no long splitting chains in $\lambda$. Let ${\mathbf{K}}'$ be ${K^{\chi_2\text{-sat}}}_\lambda$ ordered by being equal or universal over. Note that, by stability in $\lambda$, ${\mathbf{K}}'$ is a skeleton of ${\mathbf{K}}_\lambda$ (see Definition \[skel-def\]). Moreover since every limit model of cardinality $\lambda$ is $\chi_2$-saturated, for any limit $\delta < \lambda^+$, one can build an increasing continuous chain ${\langle M_i : i \le \delta \rangle}$ in ${\mathbf{K}}_\lambda$ such that for all $i \le \delta$, $M_i$ is $\chi_2$-saturated and (when $i < \delta$) $M_{i + 1}$ is universal over $M_i$. Therefore limit models exist in ${\mathbf{K}}'$, so the assumptions of Fact \[skeleton-facts\](\[skel-2b\]) are satisfied. So it is enough to see that ${\mathbf{K}}'$ (not ${\mathbf{K}}$) has no long splitting chains in $\lambda$. Let $\delta < \lambda^+$ be limit and let ${\langle M_i : i < \delta \rangle}$ be an increasing chain of models in ${\mathbf{K}}'$, with $M_{i + 1}$ universal over $M_i$ for all $i < \delta$. Let $M_\delta := \bigcup_{i < \delta} M_i$. By assumption, $M_\delta$ is $\chi_2$-saturated. By uniqueness of limit models of the same length, we can assume without loss of generality that $M_{i + 1}$ is saturated for all $i < \delta$. Let $p \in {\text{gS}}(M_\delta)$. By Lemma \[forking-chain-lc\] (applied to ${\langle M_{i + 1} : i < \delta \rangle}$), there exists $i < \delta$ such that $p$ does not fork over $M_i$. By the moreover part of Fact \[forking-props\](\[forking-props-4\]), $p$ does not $\lambda$-split over $M_i$, as desired. No long splitting chains implies solvability {#solvability-sec} ============================================ From now on we assume no long splitting chains: \[ns-hyp\] 1. Hypothesis \[global-nf-hyp\], and we fix cardinals $\chi_0 < \chi_0' < \chi_1 < \chi_1' < \chi_2$ as defined in Notation \[notation-1\], \[notation-2\], and \[notation-3\]. Note that by Fact \[stab-spectrum-2\] ${\mathbf{K}}$ is stable in $\chi_0'$. 2. ${\mathbf{K}}$ has no long splitting chains in $\chi_0'$. In Notation \[notation-4\], we will define another cardinal $\chi$ with $\chi_2 < \chi$. If $\chi_0 < H_1$, we will also have that $\chi < H_1$. Note that no long splitting chains in $\chi_0'$ and stability in $\chi_0'$ implies (Fact \[ss-stable\]) that ${\mathbf{K}}$ is stable in all $\lambda \ge \chi_0'$. Further, forking is well-behaved in the sense of Fact \[union-sat\]. This implies that Morley sequences are closed under unions (here we use that they are indexed by arbitrary linear orders, as opposed to just well-orderings). Recall that we say ${\mathbf{I}}\smallfrown {\langle N_i : i \le \delta \rangle}$ is a Morley sequence when ${\mathbf{I}}$ is a sequence of elements and the $N_i$’s are an increasing chain of sufficiently saturated models witnessing that ${\mathbf{I}}$ is Morley, see [@bv-sat-v3 5.14] for the precise definition. \[morley-union\] Let ${\langle I_\alpha : \alpha < \delta \rangle}$ be an increasing (with respect to substructure) sequence of linear orders and let $I_\delta := \bigcup_{\alpha < \delta} I_\alpha$. Let $M_0, M$ be $\chi_2$-saturated such that $M_0 {{\le_{{\mathbf{K}}}}}M$. Let $\mu_0, \mu_1, \mu_2$ be such that $\chi_2 < \mu_0 \le \mu_1 \le \mu_2$, $p \in {\text{gS}}(M)$ and for $\alpha < \delta$, let ${\mathbf{I}}_\alpha := {\langle a_i : i \in I_\alpha \rangle}$ together with ${\langle N_i^\alpha : i \in I_\alpha \rangle}$ be $(\mu_0, \mu_1, \mu_2)$-Morley for $p$ over $M_0$, with $N_i^\alpha {{\le_{{\mathbf{K}}}}}N_i^{\beta} {{\le_{{\mathbf{K}}}}}M$ for all $\alpha \le \beta < \delta$ and $i \in I_\alpha$. For $i \in I_\alpha$, let $N_i^\delta := \bigcup_{\beta \in [\alpha, \delta)} N_i^{\beta}$. Let ${\mathbf{I}}_\delta := {\langle a_i : i \in I_\delta \rangle}$. If $p$ does not fork over $M_0$, then ${\mathbf{I}}_\delta \smallfrown {\langle N_i^\delta : i \in I_\delta \rangle}$ is $(\mu_0, \mu_1, \mu_2)$-Morley for $p$ over $M_0$. By Lemma \[forking-charact\], $p$ does not syntactically split over $M_0$. Therefore the only problematic clauses in [@bv-sat-v3 Definition 5.14] are (4) and (7). Let’s check (4): let $i \in I_\delta$. By hypothesis, ${\bar{a}}_i$ realizes $p {\upharpoonright}N_i^\alpha$ for all sufficiently high $\alpha < \delta$. By local character of forking, there exists $\alpha < \delta$ such that ${\text{gtp}}({\bar{a}}_i / N_i^\delta)$ does not fork over $N_i^\alpha$. Since ${\text{gtp}}({\bar{a}}_i / N_i^\delta) {\upharpoonright}N_i^\alpha = p {\upharpoonright}N_i^\alpha$ and $p$ does not fork over $M_0 {{\le_{{\mathbf{K}}}}}N_i^\alpha$, we must have by uniqueness that $p {\upharpoonright}N_i^\delta = {\text{gtp}}({\bar{a}}_i / N_i^\delta)$. The proof of (7) is similar. For convenience, we make $\chi_2$ even bigger: \[notation-4\] Let $\chi := \gamma (\chi_2)$ (recall from Notation \[notation-1\] that $\gamma (\chi_2) = \left(2^{2^{\chi_2}}\right)^+$). A Morley sequence means a $(\chi_2^+, \chi_2^+, \chi)$-Morley sequence. \[rmk-hanf-3\] By Remark \[rmk-hanf-2\], we still have $\chi < H_1$ if $\chi_0 < H_1$. We are finally in a position to prove solvability (in fact even uniform solvability). We will use condition (\[equiv-cond-3\]) in Lemma \[solvability-equiv\]. We define a class of models $K'$ and a binary relation ${\le_{{\mathbf{K}}'}}$ on $K'$ (and write ${\mathbf{K}}' := (K', {\le_{{\mathbf{K}}'}})$) as follows. - $K'$ is a class of $\tau' := \tau ({\mathbf{K}}')$-structures, with: $$\tau' := \tau ({\mathbf{K}}) \cup \{N_0, N, F, R\}$$ where: - $N_0$ and $R$ are binary relations symbols. - $N$ is a ternary relation symbol. - $F$ is a binary function symbol. - A $\tau'$-structure $M$ is in $K'$ if and only if: 1. $M {\upharpoonright}\tau ({\mathbf{K}}) \in {{\mathbf{K}}^{\chi\text{-sat}}}$. 2. $R^M$ is a linear ordering of $|M|$. We write $I$ for this linear ordering. 3. \[cond-3-k’\] For[^7] all $a \in |M|$ and all $i \in I$, $N^M (a, i) {{\le_{{\mathbf{K}}}}}M {\upharpoonright}\tau ({\mathbf{K}})$ (where we see $N^M (a, i)$ as an $\tau ({\mathbf{K}})$-structure; in particular, $N^M (a, i) \in {\mathbf{K}}$; it will follow from (\[morley-seq-cond\]) that the $N^M (a, i)$’s are increasing with $i$, $N_0^M (a) {{\le_{{\mathbf{K}}}}}N^M (a, i)$, and $N_0^M (a)$ is saturated of size $\chi_2$. 4. \[cond-4-k’\] There exists a map $a \mapsto p_a$ from $|M|$ onto the non-algebraic Galois types (of length one) over $M {\upharpoonright}\tau ({\mathbf{K}})$ such that for all $a \in |M|$: 1. $p_a$ does not fork[^8] over $N_0^M (a)$. 2. \[morley-seq-cond\] ${\langle F^M (a, i) : i \in I \rangle} \smallfrown {\langle N^M (a, i) : i \in I \rangle}$ is a Morley sequence for $p_a$ over $N_0^M (a)$. - $M {\le_{K'}} M'$ if and only if: 1. $M \subseteq M'$. 2. $M {\upharpoonright}\tau ({\mathbf{K}}) {{\le_{{\mathbf{K}}}}}M' {\upharpoonright}\tau ({\mathbf{K}})$. 3. For all $a \in |M|$, $N_0^M (a) = N_0^{M'} (a)$. We show in Lemma \[k’-aec\] that ${\mathbf{K}}'$ is an AEC, but first let us see that this suffices: \[k’-expansion\] Let $\lambda \ge \chi$. 1. If $M \in {\mathbf{K}}_\lambda$ is saturated, then there exists an expansion $M'$ of $M$ to $\tau'$ such that $M' \in {\mathbf{K}}'$. 2. If $M' \in {\mathbf{K}}'$ has size $\lambda$, then $M' {\upharpoonright}\tau ({\mathbf{K}})$ is saturated. <!-- --> 1. Let $R^{M'}$ be a well-ordering of $|M|$ of type $\lambda$. Identify $|M|$ with $\lambda$. By stability, we can fix a bijection $p \mapsto a_p$ from ${\text{gS}}(M)$ onto $|M|$. For each $p \in {\text{gS}}(M)$ which is not algebraic, fix $N_p {{\le_{{\mathbf{K}}}}}M$ saturated such that $p$ does not fork over $N_p$ and $\|N_p\| = \chi_2$. Then use saturation to build ${\langle a_p^i : i < \lambda \rangle} \smallfrown {\langle N_p^i : i < \lambda \rangle}$ Morley for $p$ over $N_p$ (inside $M$). Let $N_0^{M'} (a_p) := N_p$, $N^{M'} (a_p, i) := N_p^i$, $F^{M'} (a, i) := a_p^i$. For $p$ algebraic, pick $p_0 \in {\text{gS}}(M)$ nonalgebraic and let $N_0^{M'} (a_p) := N_0^{M'} (a_{p_0})$, $N^{M'} (a_{p_0}) := N^{M'} (a_{p_0})$, $F^{M'} (a_{p}) := F^{M'} (a_{p_0})$. 2. By Lemma \[saturation-charact\]. \[k’-aec\] ${\mathbf{K}}'$ is an AEC with ${\operatorname{LS}}({\mathbf{K}}') = \chi$. It is straightforward to check that ${\mathbf{K}}'$ is an abstract class with coherence. Moreover: - : Let ${\langle M_i : i < \delta \rangle}$ be increasing in ${\mathbf{K}}'$. Let $M_\delta := \bigcup_{i < \delta} M_i$. - $M_0 {\le_{{\mathbf{K}}'}} M_\delta$, and if $N {\ge_{{\mathbf{K}}'}} M_i$ for all $i < \delta$, then $N {\ge_{{\mathbf{K}}'}} M_\delta$: Straightforward. - $M_\delta \in {\mathbf{K}}'$: $M_\delta {\upharpoonright}\tau ({\mathbf{K}})$ is $\chi$-saturated by Fact \[union-sat\]. Moreover, $R^{M_\delta}$ is clearly a linear ordering of $M_\delta$. Write $I_i$ for the linear ordering $(M_i, R_i)$. Condition \[cond-3-k’\] in the definition of ${\mathbf{K}}'$ is also easily checked. We now check Condition \[cond-4-k’\]. Let $a \in |M_\delta|$. Fix $i < \delta$ such that $a \in |M_i|$. Without loss of generality, $i = 0$. By hypothesis, for each $i < \delta$, there exists $p_a^i \in {\text{gS}}(M_i {\upharpoonright}\tau ({\mathbf{K}}))$ not algebraic such that ${\langle F^{M_i} (a, j) \mid j \in I_i \rangle} \smallfrown {\langle N^{M_i} (a, j) \mid j \in I_i \rangle}$ is a Morley sequence for $p_a^i$ over $N_0^{M_i} (a) = N_0^{M_0} (a)$. Clearly, $p_a^i {\upharpoonright}N_0^{M_0} (a) = p_a^0 {\upharpoonright}N_0^{M_0} (a)$ for all $i < \delta$. Moreover by assumption $p_a^i$ does not fork over $N_0^{M_0}$. Thus for all $i < j < \delta$, $p_a^j {\upharpoonright}M_i = p_a^i {\upharpoonright}M_i$. By extension and uniqueness, there exists $p_a \in {\text{gS}}(M_\delta {\upharpoonright}\tau ({\mathbf{K}}))$ that does not fork over $N_0^{M_0} (a)$ and we have $p_a {\upharpoonright}M_i = p_a^i$ for all $i < \delta$. Now by Lemma \[morley-union\], ${\langle F^{M_\delta} (a, j) \mid j \in I_\delta \rangle} \smallfrown {\langle N^{M_\delta} (a, j) \mid j \in I_\delta \rangle}$ is a Morley sequence for $p_a$ over $N_0^{M_0} (a)$. Moreover, the map $a \mapsto p_a$ is onto the nonalgebraic Galois types over $M_\delta {\upharpoonright}\tau ({\mathbf{K}})$: let $p \in {\text{gS}}(M_\delta {\upharpoonright}\tau ({\mathbf{K}}))$ be nonalgebraic. Then there exists $i < \delta$ such that $p$ does not fork over $M_i$. Let $a \in |M_i|$ be such that ${\langle F^{M_i} (a, j) \mid j \in I_i \rangle} \smallfrown {\langle N^{M_i} (a, j) \mid j \in I_i \rangle}$ is a Morley sequence for $p {\upharpoonright}M_i$ over $N_0^{M_i} (a)$. It is easy to check it is also a Morley sequence for $p$ over $N_0^{M_i} (a)$. By uniqueness of the nonforking extension, we get that the extended Morley sequence is also Morley for $p$, as desired. - : An easy closure argument. \[strong-solvable\] ${\mathbf{K}}$ is uniformly $(\chi, \chi)$-solvable. By Lemma \[k’-aec\], ${\mathbf{K}}'$ is an AEC with ${\operatorname{LS}}({\mathbf{K}}') = \chi$. Now combine Lemma \[k’-expansion\] and Lemma \[solvability-equiv\]. Note that by Fact \[union-sat\], for each $\lambda \ge \chi$ there is a saturated model of size $\lambda$, and it is also a superlimit. For the convenience of the reader, we give a more quotable version of Theorem \[strong-solvable\]. For the next results, we drop Hypothesis \[ns-hyp\]. \[strong-solvable-thm\] Assume that ${\mathbf{K}}$ has a monster model, is ${\operatorname{LS}}({\mathbf{K}})$-tame, and is stable in some cardinal greater than or equal to ${\operatorname{LS}}({\mathbf{K}})$. There exists $\chi < H_1$ such that for any $\mu \ge \chi$, if ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$ then ${\mathbf{K}}$ is uniformly $(\mu', \mu')$-solvable, where $\mu' := \left(\beth_{\omega + 2} (\mu)\right)^+$. Hypothesis \[global-nf-hyp\] holds. Let $\chi < H_1$ be such that ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property of length $\chi$ (see Fact \[stab-op\]). Let $\mu \ge \chi$ be such that ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$. We apply Theorem \[strong-solvable\] by letting $\chi_0$ in Notation \[notation-1\] stand for $\mu$ here. By Fact \[ss-stable\], ${\mathbf{K}}$ is stable in $\mu_1$ and has no long splitting chains in $\mu_1$ for every $\mu_1 \ge \mu$, thus Hypothesis \[ns-hyp\] holds. Moreover $\chi_2$ in Notation \[notation-3\] corresponds to $\beth_\omega (\mu)$ here, and $\chi$ in Notation \[notation-4\] corresponds to $\mu'$ here. Thus the result follows from Theorem \[strong-solvable\]. Assume that ${\mathbf{K}}$ has a monster model and is ${\operatorname{LS}}({\mathbf{K}})$-tame. If there exists $\mu < H_1$ such that ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$, then there exists $\mu' < H_1$ such that ${\mathbf{K}}$ is uniformly $(\mu', \mu')$-solvable. Let $\mu < H_1$ be such that ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$. Fix $\chi < H_1$ as given by Theorem \[strong-solvable-thm\]. Without loss of generality, $\mu \le \chi$. By Fact \[ss-stable\], ${\mathbf{K}}$ is stable in $\chi$ and has no long splitting chains in $\chi$, so apply the conclusion of Theorem \[strong-solvable-thm\]. Superstability below the Hanf number {#hanf-section} ==================================== In this section, we prove the main corollary. In fact, we prove a stronger version that instead of asking for the properties to hold on a tail asks for them to hold only in a single high-enough cardinal. Toward this end, we start by explaining why no long splitting chains follows from categoricity in a high-enough cardinal. In fact, categoricity can be replaced by solvability. All the ingredients for this result are contained in [@shvi635] and this specific form has only appeared recently [@shvi-notes-v3-toappear Theorem 3]. Note also that Shelah states a similar result in [@sh394 5.5] but his definition of superstability is different. \[ns-lc-fact\] Let ${\mathbf{K}}$ be an AEC with arbitrarily large models and amalgamation[^9] in ${\operatorname{LS}}({\mathbf{K}})$. Let $\lambda > {\operatorname{LS}}({\mathbf{K}})$ be such that ${\mathbf{K}}_{<\lambda}$ has no maximal models. If ${\mathbf{K}}$ is $(\lambda, {\operatorname{LS}}({\mathbf{K}}))$-solvable, then ${\mathbf{K}}$ is stable in ${\operatorname{LS}}({\mathbf{K}})$ and has no long splitting chains in ${\operatorname{LS}}({\mathbf{K}})$. \[ns-lc-cor\] Let ${\mathbf{K}}$ be an AEC with a monster model. Let $\lambda > {\operatorname{LS}}({\mathbf{K}})$. If ${\mathbf{K}}$ is categorical in $\lambda$, then ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$ for all $\mu \in [{\operatorname{LS}}({\mathbf{K}}), \lambda)$. By Fact \[ns-lc-fact\] applied to ${\mathbf{K}}_{\ge \mu}$ for each $\mu \in [{\operatorname{LS}}({\mathbf{K}}), \lambda)$. Note that, since ${\mathbf{K}}$ has arbitrarily large models, categoricity in $\lambda$ implies $(\lambda, {\operatorname{LS}}({\mathbf{K}}))$-solvability. We conclude that solvability is equivalent to superstability in the first-order case: \[solvability-fo\] Let $T$ be a first-order theory and let ${\mathbf{K}}$ be the AEC of models of $T$ ordered by elementary substructure. Let $\mu \ge |T|$. The following are equivalent: 1. \[ss-cond-1\] $T$ is stable in all $\lambda \ge \mu$. 2. \[ss-cond-2\] ${\mathbf{K}}$ is $(\lambda, \mu)$-solvable, for some $\lambda > \mu$. 3. \[ss-cond-3\] ${\mathbf{K}}$ is uniformly $(\mu, \mu)$-solvable. (\[ss-cond-3\]) implies (\[ss-cond-2\]) is trivial. (\[ss-cond-2\]) implies (\[ss-cond-1\]) is by Corollary \[ns-lc-cor\] together with Fact \[ss-stable\]). Finally, (\[ss-cond-1\]) implies (\[ss-cond-3\]) is as in the proof of Theorem \[strong-solvable-thm\]. We can also use the ZFC Shelah-Villaveces theorem to prove the following interesting result, showing that the solvability spectrum satisfies an analog of Shelah’s categoricity conjecture in tame AECs (Shelah asks what the behavior of the solvability spectrum should be in [@shelahaecbook Question N.4.4]). \[solv-transfer\] Assume that ${\mathbf{K}}$ has a monster model and is ${\operatorname{LS}}({\mathbf{K}})$-tame. There exists $\chi < H_1$ such that for any $\mu \ge \chi$, if ${\mathbf{K}}$ is $(\lambda, \mu)$-solvable for *some* $\lambda > \mu$, then ${\mathbf{K}}$ is uniformly $(\mu', \mu')$-solvable, where $\mu' := \left(\beth_{\omega + 2} (\mu)\right)^+$. Let $\chi < H_1$ be as given by Theorem \[strong-solvable-thm\]. Let $\mu \ge \chi$ and fix $\lambda > \mu$ such that ${\mathbf{K}}$ is solvable in $\lambda$. By Fact \[ns-lc-fact\], ${\mathbf{K}}$ is stable in $\mu$ and has no long splitting chains in $\mu$. Now apply Theorem \[strong-solvable-thm\]. We are now ready to prove the stronger version of the main corollary where the properties hold only in a single high-enough cardinal below $H_1$ (but the cardinal may be different for each property). \[main-cor-unbounded\] Assume that ${\mathbf{K}}$ has a monster model, is ${\operatorname{LS}}({\mathbf{K}})$-tame, and is stable in some cardinal greater than or equal to ${\operatorname{LS}}({\mathbf{K}})$. Then there exists $\chi \in ({\operatorname{LS}}({\mathbf{K}}), H_1)$ such that the following are equivalent: - \[ssm-1\] For some $\lambda_1 \in [\chi, H_1)$, ${\mathbf{K}}$ is stable in $\lambda_1$ and has no long splitting chains in $\lambda_1$. - \[ssm-2\] For some $\lambda_2 \in [\chi, H_1)$, there is a good $\lambda_2$-frame on a skeleton of ${\mathbf{K}}_{\lambda_2}$. - \[ssm-3\] For some $\lambda_3 \in [\chi, H_1)$, ${\mathbf{K}}$ has a unique limit model of cardinality $\lambda_3$. - \[ssm-4\] For some $\lambda_4 \in [\chi, H_1)$, ${\mathbf{K}}$ is stable in $\lambda_4$ and has a superlimit model of cardinality $\lambda_4$. - \[ssm-5\] For some $\lambda_5 \in [\chi, H_1)$, the union of any increasing chain of $\lambda_5$-saturated models is $\lambda_5$-saturated. - \[ssm-6\] For some $\lambda_6 \in [\chi, H_1)$, for some $\mu < \lambda_6$, ${\mathbf{K}}$ is $(\lambda_6, \mu)$-solvable. \[type-full-rmk\] In $(\ref{sc0-2})^-$, we do *not* assume that the good frame is type-full (i.e. it may be that there exists some nonalgebraic types which are not basic, so fork over their domain). However if $(\ref{sc0-1})^-$ holds, then the proof of $(\ref{sc0-1})^-$ implies $(\ref{sc0-2})^-$ (Fact \[good-frame-existence\]) actually builds a *type-full* frame. Therefore, in the presence of tameness, the existence of a good frame implies the existence of a *type-full* good frame (in a potentially much higher cardinal, and over a different class). By Fact \[stab-spectrum\], ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property. By Fact \[stab-op\], there exists $\chi_0 < H_1$ such that ${\mathbf{K}}$ does not have the ${\operatorname{LS}}({\mathbf{K}})$-order property of length $\chi_0$. Let $\chi := \beth_{\omega}\left(\chi_0 + {\operatorname{LS}}({\mathbf{K}})\right)$. We will use the following auxiliary condition, which is a weakening of $(\ref{sc0-3})^-$ (the problem is that we do not quite know that $(\ref{sc0-5})^{-}$ implies $(\ref{sc0-3})^-$ as ${\mathbf{K}}$ might not be stable in $\lambda_5$): - For some $\lambda_3^\ast \in [\chi, H_1)$, ${\mathbf{K}}$ is stable in $\lambda_3^\ast$, has a saturated model of cardinality $\lambda_3^\ast$, and every limit model of cardinality $\lambda_3^\ast$ is $\chi$-saturated. We will prove the following claims, which put together give us what we want: : $(\ref{sc0-1})^- \Leftrightarrow (\ref{sc0-6})^-$. : $(\ref{sc0-3})^\ast \Rightarrow (\ref{sc0-1})^-$. : For $\ell \in \{1,2, 3, 4, 5\}$, $(\ell)^- \Rightarrow (\ref{sc0-3})^\ast$. : By Theorem \[strong-solvable-thm\] and Fact \[ns-lc-fact\]. $\dagger_{\text{Claim 1}}$ : This is Theorem \[ss-from-chainsat\], where $\chi_2$ there stands for $\chi$ here. $\dagger_{\text{Claim 2}}$ : It is enough to prove the following subclaims: - : $(\ref{sc0-1})^- \Rightarrow (\ref{sc0-2})^- \Rightarrow (\ref{sc0-3})^-$. - : $(\ref{sc0-4})^- \Rightarrow (\ref{sc0-3})^-$. - : $(\ref{sc0-3})^- \Rightarrow (\ref{sc0-3})^\ast$. - : $(\ref{sc0-5})^- \Rightarrow (\ref{sc0-3})^\ast$. <!-- --> - : By Fact \[good-frame-existence\]. $\dagger_{\text{Subclaim 1}}$ - : By Fact \[local-implications\](\[local-4\]). $\dagger_{\text{Subclaim 2}}$ - : By Fact \[local-implications\](\[local-3\]). $\dagger_{\text{Subclaim 3}}$ - : Let $\lambda_3^\ast \in [\lambda_5, H_1)$ be a regular stability cardinal. Then ${\mathbf{K}}$ has a saturated model of cardinality $\lambda_3^\ast$, and from $(\ref{sc0-5})^-$ it is easy to see that any limit model of cardinality $\lambda_3^\ast$ is $\lambda_5$-saturated, hence $\chi$-saturated. $\dagger_{\text{Subclaim 4}}$ We can now prove the main result of this paper (Corollary \[main-cor\]): Let $\chi$ be as given by Corollary \[main-cor-unbounded\]. By Fact \[stab-spectrum\], there exists unboundedly-many regular stability cardinals in $(\chi, H_1)$. This implies that for $\ell \in \{1, 2, 3, 4, 5, 6\}$, $(\ell)$ (from Corollary \[main-cor\]) implies $(\ell)^-$ (from Corollary \[main-cor-unbounded\]). Moreover $(\ref{sc0-1})^-$ implies both (\[sc0-1\]) and (\[sc0-7\]) by Fact \[ss-stable\]. Since Corollary \[main-cor-unbounded\] tells us that $(\ell_1)^-$ is equivalent to $(\ell_2)^-$ for $\ell_1, \ell_2 \in \{1, 2, 3, 4, 5 ,6\}$, it follows that $(\ell_1)$ is equivalent to $(\ell_2)$ as well, and (\[sc0-7\]) is implied by any of these conditions. \[superlimit-question\] Is stability in $\lambda_4$ needed in condition $(\ref{sc0-4})^-$ of Corollary \[main-cor-unbounded\]? That is, can one replace the condition with: - For some $\lambda_4 \in [\chi, \theta)$, ${\mathbf{K}}$ has a superlimit model of cardinality $\lambda_4$. The answer is positive when ${\mathbf{K}}$ is an elementary class [@sh868 3.1]. Future work {#end-section} =========== While we managed to prove that some analogs of the conditions in Fact \[fo-superstab\] are equivalent, much remains to be done. For example, one may want to make precise what the analog to (5) and (6) in \[fo-superstab\] should be in tame AECs. One possible definition for (6) could be: Let $\lambda, \mu > {\operatorname{LS}}({\mathbf{K}})$. We say that ${\mathbf{K}}$ has the *$(\lambda,\mu)$-tree property* provided there exists $\{p_n(\x;\y_n) \mid n<\omega\}$ Galois-types over models of size less than $\mu$ and $\{M_\eta \mid \eta\in {\sideset{^{\leq\omega}}{}{\operatorname{\lambda}}}\}$ such that for all $n<\omega, \nu\in{\sideset{^{n}}{}{\operatorname{\lambda}}}$ and every $\eta\in{\sideset{^{\omega}}{}{\operatorname{\lambda}}}$: $$\langle M_\eta,M_\nu\rangle\models p_n \iff \nu \text{ is an initial segment of }\eta.$$ We say that ${\mathbf{K}}$ has the *tree property* if it has it for all high-enough $\mu$ and all high-enough $\lambda$ (where the “high-enough” quantifier on $\lambda$ can depend on $\mu$). We can ask whether no long splitting chains (or any other reasonable definition of superstability) implies that ${\mathbf{K}}$ does not have the tree property, or at least obtain many models from the tree property as in [@grsh238]. This is conjectured in [@sh394] (see the remark after Claim 5.5 there). As for the D-rank in \[fo-superstab\](5), perhaps a simpler analog would be the $U$-rank defined in terms of $(<\kappa)$-satisfiability in [@bg-v11-toappear 7.2] (another candidate for a rank is Lieberman’s $R$-rank, see [@liebermanrank]). By [@bg-v11-toappear 7.9], no long splitting chains implies that the $U$-rank is bounded but we do not know how to prove the converse. Perhaps it is possible to show that $U[p] = \infty$ implies the tree property. [^1]: Consider, for example, the statement that in a monster model for a first-order theory $T$, for every sufficiently long sequence ${\mathbf{I}}$ there exists a subsequence ${\mathbf{J}}\subseteq {\mathbf{I}}$ such that ${\mathbf{J}}$ is indiscernible. In general, this is a large cardinal axiom, but it is known to be true when $T$ is on the good side of a dividing line (in this case stability). We believe that the situation for tameness is similar. [^2]: As opposed to [@tamenessone 3.3], we do *not* require that $\chi < H_1$. [^3]: But see [@ap-universal-v10 C.14] where a notion of forking over set is constructed from categoricity in a universal class. [^4]: Of course, the $\kappa$ notation has a long history, appearing first in [@sh3]. [^5]: We use the definition in [@shelahaecbook N.2.4(4)] which requires in addition that the model be universal. [^6]: Note that if $\alpha$ is limit this follows. [^7]: For a binary relation $Q$ we write $Q (a)$ for $\{b \mid Q (a, b)\}$, similarly for a ternary relation. [^8]: Note that by Lemma \[forking-charact\] this also implies that it does not syntactically split over some $M_0' {{\le_{{\mathbf{K}}}}}N_0^M (a)$ with $\|M_0'\| < \chi_2$. [^9]: In [@shvi635], this is replaced by the generalized continuum hypothesis (GCH).

--- abstract: 'We report experimental studies of the influence of symmetric dual-loop optical feedback on the RF linewidth and timing jitter of self-mode-locked two-section quantum dash lasers emitting at 1550 nm. Various feedback schemes were investigated and optimum levels determined for narrowest RF linewidth and low timing jitter, for single-loop and symmetric dual-loop feedback. Two symmetric dual-loop configurations, with balanced and unbalanced feedback ratios, were studied. We demonstrate that unbalanced symmetric dual loop feedback, with the inner cavity resonant and fine delay tuning of the outer loop, gives narrowest RF linewidth and reduced timing jitter over a wide range of delay, unlike single and balanced symmetric dual-loop configurations. This configuration with feedback lengths 80 and 140 m narrows the RF linewidth by $\sim$ 4-67x and $\sim$ 10-100x, respectively, across the widest delay range, compared to free-running. For symmetric dual-loop feedback, the influence of different power split ratios through the feedback loops was determined. Our results show that symmetric dual-loop feedback is markedly more effective than single-loop feedback in reducing RF linewidth and timing jitter, and is much less sensitive to delay phase, making this technique ideal for applications where robustness and alignment tolerance are essential.' address: 'Department of Physics and Tyndall National Institute, University College Cork, Ireland T12 YN60' author: - 'Haroon Asghar, Wei Wei, Pramod Kumar, Ehsan Sooudi, and John. G. McInerney' title: 'Stabilization of self-mode-locked quantum dash lasers by symmetric dual-loop optical feedback' --- [99]{} V. Vujicic, C. Calò, R. Watts, F. Lelarge, C. Browning, K. Merghem, A. Martinez, A. Ramdane, and L. P. Barry, “ Quantum dash mode-locked lasers for data centre applications ,”  IEEE J. Sel. Top. Quantum Electron. [**21**]{}(6), 53–60 (2015). J. P-Cetina, S. Latkowski, R. 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Top. Quantum Electron. [**45**]{}(11), 1456–1464 (2009). H. Asghar, and John. G. McInerney, “ Asymmetric dual-loop feedback to suppress spurious tones and reduce timing jitter in self-mode-locked quantum dash lasers emitting at 1.55 $\mu$m,” Opt. Letter. [**42**]{}(18), 3714–3717 (2017). J.-H. Cho, H. Kim, and H.-K. Sung, “Performance optimization of an optically combined dual-loop optoelectronic oscillator based on optical interference analysis,” Optical Engineering. [**56**]{}(6), 066111 (2017). Introductions ============= [Quantum nanostructure based mode-locked semiconductor lasers are of increasing interest for many applications as frequency comb sources in data centers \[1\], optical clock recovery \[2\] and high capacity coherent terabit communication systems \[3\]. While picosecond pulses from these lasers have been demonstrated routinely, these pulses have significant chirp and poor timing jitter. The latter is usually determined by measuring the linewidth of the repetition-rate peak in the RF intensity fluctuation spectrum. Several techniques such as single-loop external optical feedback \[4-8\], coupled optoelectronic oscillators (OEOs) \[9-11\], hybrid mode-locking \[12\], injection-locking \[13-15\] and dual-loop feedback \[16-20\] have been proposed and demonstrated to improve this key parameter of mode-locked lasers (MLLs). Among feedback-based techniques, optoelectronic feedback requires optical-to-electrical conversion, while hybrid mode-locking requires high speed electrical modulation of the gain and/or absorber sections. On the other hand, optical injection based techniques require a stable external laser, making these techniques less attractive in practice where low cost, simplicity and reliability are paramount. Of all stabilization techniques, external optical feedback is the simplest and most cost-effective demonstrated to date both experimentally \[4-8\] and numerically \[21-24\]. Most recently, 99% reduction in RF linewidth and 23 fs pulse-to-pulse jitter was reported using single cavity feedback for a 40 GHz quantum-dot MLL \[8\]. Five different feedback regimes were identified along with the regime of resonant optical feedback most favorable and desirable for practical applications. In this demonstration \[8\], linewidth and timing jitter were shown to be very sensitive to small delay adjustments, with optimum performance being limited to one narrow regime (&lt;15 ps delay). In practice, MLLs require reduced sensitivity of RF linewidth and timing jitter to detuning and drift in the delay phase. In particular, changing delay length should not cause switching into unstable or unwanted dynamical regimes.]{} [Recently we have achieved lowest RF linewidth and reduced timing jitter over the widest delay range, introducing balanced asymmetric dual-loop (unequal arms of external loops) feedback such that the delay time of the exterior (shorter) loop was set to half that of the interior (longer) loop \[20\]. However, for ease of production and reliability of stable robust MLLs, using dual-loops with no or minimal delay difference is more advantageous and desirable. In this paper we propose symmetric dual-loop (SDL) feedback (equal arms of external loops), and demonstrate its efficacy over a much wider feedback range, for two-section self-mode-locked quantum dash (QDash) lasers emitting at $\sim$ 1550 nm and operating at $\sim$ 21 GHz pulse repetition rate. The feedback ratio for narrowest RF linewidth and lowest timing jitter was determined for single and SDL feedback schemes. We demonstrate that unbalanced SDL with fine tuning of the delay of the weaker feedback cavity (lowest feedback strength) produces narrowest RF spectra and reduced timing jitter across the widest delay range, unlike single-loop and balanced SDL feedback. Under stable resonant conditions and feedback level (-22 dB), RF linewidth is reduced from 100 kHz free-running to 3 kHz for single-loop feedback, 1.5 kHz for unbalanced SDL with 80 m loop and 1 kHz (instrument limited) for 140 m loop length. Moreover, RMS timing jitter is also reduced from 3.9 ps free-running to 0.6 ps for single-loop, 0.45 ps for unbalanced SDL with 80 m loop, 0.4 ps for 140 m loop (integrated 10 kHz-100 MHz). Our proposed unbalanced SDL scheme provides an effective regime of resonant feedback parameters much wider than single-loop and balanced SDL feedback, making it ideal for practical applications.]{} Experimental Setup ================== [Devices under investigation were two-section InAs/InP QDash MLLs with active regions consisting of nine InAs quantum dash monolayers grown by gas source molecular beam epitaxy (GSMBE) embedded within two barrier layers (dash-in-barrier device), and separate confinement heterostructure layers of InGaAsP, emitting at $\sim$ 1.55 $\mu$m \[25\]. Cavity length was 2030 $\mu$m, 11.8% (240 $\mu$m) of which formed the absorber section, giving pulsed repetition frequency $\sim$ 20.7 GHz ($I_{Gain}$ = 300 mA, $V_{Abs}$: Floating). Gain and absorber sections were electrically isolated by 9 k$\Omega$. The lasers were mounted p-side up (substrate down) on an AlN submount and a copper block with active temperature control. Electrical contacts were formed by wire bonding, and heat sink temperature was fixed at $19^0$C. Mode-locking was obtained without reverse bias applied to the absorber section. This was a two-section device but was packaged similarly to a single-section self-mode-locked laser since the absorber was unbiased: its minimal absorption does not affect the self-mode-locking mechanism \[13\]. Recently, 1.55 $\mu$m InAs/InP based QDash single-section self-mode-locked lasers have demonstrated promising high speed, narrow pulse generation, specifically GHz pulse repetition rate and very low RF linewidth \[26, 27\] with a low timing jitter. The absence of any obvious active/passive mode-locking scheme in these devices looks surprising at first glance. However, strong four-wave mixing \[25\] in the cavity has been proposed as the reason for this coherent self-pulsing behavior.]{} {width="11cm"} [Our experiment is depicted in Fig. 1. For single and dual-loop feedback, a calibrated fraction of light was fed back through port 1 of an optical circulator, then injected into the laser cavity via port 2. Optical coupling loss between adjacent ports was -0.64 dB. The output of the circulator was sent to a semiconductor optical amplifier (SOA, gain 9.8 dB) then split into two arms by a 50/50 coupler. Half the amplified signal went to an RF spectrum analyzer (Keysight E-series, E4407B) via a 21 GHz photodiode, and to optical spectrum analyzers (Ando AQ6317B and Advantest Q8384). The other half of the power was directed to the feedback arrangement. For a single feedback loop, all power passed through loop-I in Fig. 1. For SDL configurations, power was split equally into two loops via a 3-dB splitter, each containing an optical delay line, a variable optical attenuator and polarization controller. For SDL configurations, two combinations of feedback ratios were studied. For SDL with balanced feedback ratios, equal power was coupled to both external cavities. To implement unbalanced SDL feedback, more power (-20 dB) was coupled to loop-I than to loop-II (-26 dB). The lengths of the fiber loops were fine-tuned by optical delay lines based on stepper-controlled stages with delay resolution 1.67 ps. Polarization controllers in each loop and one polarization controller before port 1 of the circulator ensured the light from both loops matched the emitted light polarizations to maximize feedback effectiveness.]{} [RMS timing jitter is calculated from the single sideband (SSB) phase noise spectra measured for the fundamental RF frequency ($\approx$ 20.7 GHz) using: $$\sigma_{RMS}=\frac{1}{2 \pi f_{ML}}\sqrt{2 \int_{f_{d}}^{f_{u}} L(f)\,df}$$ where $f_{ML}$ is the pulse repetition rate, $f_{u}$ and $f_{d}$ are the upper and lower integration limits. *L(f)* is the single sideband (SSB) phase noise spectrum, normalized to the carrier power per Hz. To measure RMS timing jitter of the laser in more detail, single-sideband (SSB) noise spectra for the fundamental harmonic repetition frequency were measured. To assess this, RF spectra at several spans around the repetition frequency were measured from small (finest) to large (coarse) resolution bandwidths. The corresponding ranges for frequency offsets were then extracted from each spectrum and superimposed to obtain SSB spectra normalized for power and per unit of frequency bandwidth. The higher frequency bound was set to 100 MHz (instrument limited).]{} [We investigated the effects of three key parameters: external feedback level, optical loop length (80 and 140 m) and optical delay phase tuning, on the timing stability of our QDash MLL. The laser was subjected to single-loop and SDL (for both balanced and unbalanced power ratios) feedback into the gain section.]{} Results and discussions ======================= Effects of feedback strength on RF linewidth and integrated timing jitter ------------------------------------------------------------------------- To investigate the effects of feedback on the RF linewidth and timing jitter, attenuation in the feedback loops was varied from -46 dB to -22 dB, after which the laser became unstable. At   -46 dB the RF linewidth was 73 kHz for single-loop, 69 kHz for unbalanced SDL and 75 kHz for balanced SDL feedback, with respective timing jitter 3, 2.9, 3.1 ps (10 kHz-100 MHz). At this weak feedback level, upon tuning of the optical delay, no deviation in the fundamental frequency occurs, and no major reduction in RF linewidth and timing jitter were seen relative to free-running. However, at higher feedback -29 dB, RF linewidth and timing jitter began to decrease: the former was 28.7 kHz for single-loop, 21.5 kHz for unbalanced SDL, and 29 kHz for balanced SDL; RMS timing jitter was 1.75 ps for single-loop, 1.6 ps for unbalanced SDL and 1.8 ps for balanced SDL feedback. Further increase in feedback ratio to -22 dB resulted in significant reduction in RF linewidth and timing jitter for single and SDL configuration subject to both balanced and unbalanced feedback ratios. Lowest achieved RF linewidth and timing jitter for single and SDL configurations as functions of feedback ratio at integer resonances are depicted in Figs. 2(a) and 2(b) respectively. From these data, we have identified the feedback ratio to be -22 dB for single-loop and SDL feedback, limited by self-pulsation above this level. For practical applications, the relatively flat characteristics of RF linewidth versus feedback ratio (-26 dB, -24 dB, -23 dB and -22 dB) are favorable. Furthermore, variation in RF linewidth and timing jitter in all feedback schemes studied follows similar trends when feedback approaches the optimal value, which agrees well with reported analytical results (square root dependence of the RF linewidth on integrated timing jitter) \[28\]. {width="13cm"} Recently, for a quantum dot MLL operating at 5.1 GHz, minimum RF linewidth was obtained at relatively low feedback -36 dB \[6\]. On the other hand, a passively mode-locked QDash laser emitting at 1580 nm and operating at 17 GHz repetition rate saw marked reduction in RF linewidth at significantly stronger feedback -22 dB \[7\], in agreement with our studies. These differences are explicable by the likelihood that the anti-guiding (phase-amplitude coupling) factor, the primary measure of nonlinearity in these lasers, is lower in quantum dashes. RF linewidth and integrated timing jitter versus delay for single-loop feedback ------------------------------------------------------------------------------- [To study the effects of single-loop feedback on RF linewidth and timing jitter over a wide delay range (0-84 ps), loop-II was disconnected and maximum feedback to the gain section was set to -22 dB via a single 60 m fiber span, stable resonance being achieved by optimizing optical delay line ODL-I adjustable from 0-84 ps in steps of 1.67 ps. The resulting RF linewidth (black squares) and timing jitter (blue triangles) are shown in Fig. 3 versus delay. Clearly stabilization effectiveness depends strongly on feedback delay, most likely because detuning of optical delay from exact resonance changes synchronization conditions between pulses in the laser cavity and feedback loops \[29\]. The periodicity in RF linewidth versus delay tuning is 48 ps, in agreement with the fundamental mode-locked frequency (20.7 GHz) of our laser. Furthermore, this optimization of the single-loop delay reduced the RF linewidth and corresponding timing jitter considerably, as for other reported experiments \[5,6\] and theoretical predictions \[30\]. Effective stable mode-locking occurs when the external cavity optical length is close to an integer multiple of that of the laser cavity. When fully resonant, the RF linewidth decreased from 100 kHz free-running to 3 kHz, and integrated timing jitter from 3.9 ps to 0.6 ps (10 kHz-100 MHz). Measured RF spectra and phase noise traces at this feedback delay with single-loop feedback (blue line) and free-running (gray line) are shown in Figs. 4(a) and 4(b) respectively. Upon tuning of the loop delay by 6-54 ps, synchronization of the optical pulses between the laser cavity and external cavity did not occur, the RF spectra become highly deformed and non-resonant feedback was observed.]{} {width="10cm"} [Experimental results for single-loop feedback show that for practical use of QDash MLLs the most stable, delay-insensitive ranges are near 5 and 53 ps. Optimum stabilization using conventional single-loop feedback is very sensitive to phase adjustment, and limits the region of optimum performance to a narrow parameter space. For practical applications of MLLs, it is desirable to extend the range of resonant feedback condition over a much wider range of delay times, such that environment changes maintain stable pulse trains with narrow linewidth and low timing jitter.]{} {width="13cm"} RF linewidth and integrated timing jitter versus delay for balanced and unbalanced symmetric dual-loop (SDL) feedback arrangements ---------------------------------------------------------------------------------------------------------------------------------- [For dual-loop experiments, the optical feedback was split into two fiber cavities whose lengths were calibrated by measurement of RF spectra with each loop unblocked separately. For the first set, cavity spacing was 2.53 MHz consistent with 80 m nominal length of both equal loops. RF spectral measurements with this arrangement (using frequency span 10 MHz with resolution bandwidth 10 kHz and video bandwidth 1 kHz) are shown in Fig. 5(a). ]{} {width="13cm"} [To study the effects of SDL feedback on laser stability, fine adjustment of optical attenuator (Att-I) and polarization controller (PC-I) was made and equal feedback power (-22 dB) coupled to both loops. Optical delay line ODL-I was set to full resonance (integer number of times the laser cavity delay) and ODL-II tuned over its full range. RF linewidth (black squares) and timing jitter (black squares) are presented as functions of delay in Figs. 6(a) and 6(b). We see that SDL feedback yields results comparable to those using a single-loop, and is similarly sensitive to delay. Only with both loops resonant does stability improve, giving RF linewidth 12 kHz and timing jitter 0.85 ps, versus 3 kHz and 0.6 ps for optimized single-loop feedback, refer to Fig. 5(b). Recently, in a separate series of experiments, we achieved 0.97 kHz linewidth (instrument limited) and timing jitter 0.45 ps with both cavities resonant \[17\], confirming that balanced SDL produces effective stabilization, but only at a specific delay with tolerance $\sim$ 1 ps, a very stringent requirement in practice.]{} {width="13cm"} [Next, we explored unbalanced SDL feedback, in which the power split between the two loops was varied. In these experiments the inner feedback cavity (loop-I) was fully resonant and the outer feedback cavity fine-tuned around the resonance. Measured RF spectra at different loop power splits are shown in Fig. 7, with corresponding RF linewidths in Table 1. Minimum RF linewidth occurred when both external cavities were fully resonant, as expected. Values of 1.6 and 1.5 kHz were achieved when resonant loop-I had feedback -20.6 and -20 dB, and fine-tuned loop-II had -24.3 and -26 dB, respectively. This combination of feedback ratios was particularly effective and was investigated further.]{} {width="13cm"} **Loop-I** **Loop-II (c)** **Feedback into Gain** **RF-Linewidth** ------------ ----------------- ------------------------ ------------------ -26 dB -20 dB -22 dB 4.1 kHz -24.3 dB -20.6 dB -22 dB 3.4 kHz -23 dB -21.3 dB -22 dB 2.1 kHz -22 dB -22 dB -22 dB 12 kHz -21.3 dB -23 dB -22 dB 30 kHz -20.6 dB -24.3 dB -22 dB 1.6 kHz -20 dB -26 dB -22 dB 1.5 kHz : Calculated RF linewidth as a function of power split ratio (in dB) through two external feedback loops using SDL feedback configuration [The next experiments concerned the effects of unbalanced SDL feedback on timing stability of the laser: feedback strength in loop-I and loop-II were -20 and -26 dB, a ratio of 4:1 resulting in overall feedback -22 dB to the gain section. Delay in loop-I was then fine-tuned to full resonance, and loop-II tuned over its entire available delay range 0-84 ps. This yielded much more stable dynamics: narrow RF spectra and reduced timing jitter were maintained over the full delay range, unlike single-loop and balanced SDL feedback. This dual-loop scheme with 4:1 power ratio between loops was most successful, reducing RF linewidth by up to two orders of magnitude (70x) compared to free-running, 2-5x over single-loop and 5-8x relative to balanced SDL feedback. Measured RF linewidths (blue triangles) and timing jitter (blue triangles) for this unbalanced SDL scheme are given in Figs. 6(a) and 6(b). Furthermore, with this feedback configuration, measured RF linewidth and integrated timing jitter ranged from as high as 28 kHz and 1.5 ps, to as low as 1.5 kHz and 0.45 ps (free running values are 100 kHz and 3.9 ps). Again, most effective and robust linewidth narrowing and lowest timing jitter occurred when both external cavities were fully resonant. The RF spectrum under double resonance is shown in Fig. 5(b) (blue line). RF spectra (with 10 MHz frequency span, 10 kHz resolution bandwidth and 1 kHz video bandwidth) and phase noise for unbalanced SDL feedback versus frequency offset are given in Figs. 8(a) and 8(b). Side-modes were spaced by 2.53 MHz, corresponding to the feedback loop length used (80 m). Most recently, we have used asymmetric dual-loop feedbacks with optimized inner loop delay to suppress spurious tones and timing jitter in self-mode-locked lasers \[31\]. Furthermore, when free-running the peak power of RF noise spectra is -20 dB. For single-loop and unbalanced SDL feedback the noise peak is 30 dB higher (see Fig. 4(a) and Fig. 5(b)) due to the reduced RF linewidth and also lower threshold current with feedback, increasing the optical power emitted at fixed bias. ]{} {width="13cm"} [Measured RF linewidths versus delay for a single-loop, with -20 dB feedback through loop-I (blue triangles), are shown in Fig. 9(a) for comparison. At stable resonance, single-loop feedback at -20 and -26 dB narrows the linewidth to 8 kHz and 68 kHz, respectively. When dual-loops were unbalanced, measured RF linewidth as a function of delay was as in Fig. 9(b), showing that unbalanced dual-loops are more effective in stabilizing the linewidth. Here optimization of ODL-II yields better linewidth stabilization (blue triangles) than optimization of ODL-I (black squares), see Fig. 9(b). For SDL feedback, fine tuning of ODL-I yields narrow RF linewidth at an integer resonance, but linewidth broadens significantly when delay is tuned away from this point. Our results show that the most effective algorithm for stable linewidth reduction over a broad range of phase delay is to set the stronger cavity to an integer resonance then fine-tune the weaker cavity. Optimizing loop-II in SDL feedback (blue triangles in Fig. 9(b)), changes the linewidth similarly to single-loop feedback (black squares in Fig. 9(a)) but almost 1-2 orders of magnitude (6-64x) narrower.]{} {width="13cm"} [Effects of longer delay times on RF linewidth were also investigated subject to both balanced and unbalanced SDL feedback. For this purpose, the 60 m fiber loop was replaced with 120 m fiber. Measured RF linewidth versus phase tuning is shown in Fig. 10 for SDL with balanced (black squares) and unbalanced (blue triangles) feedback ratios. Unbalanced SDL feedback again reduced the RF linewidth by 10-100x across a broader range of delay than the free-running laser. However, balanced SDL was less sensitive to delay, though the linewidth was 8-16x broader than unbalanced SDL (Fig. 10). Narrowest linewidth obtained was 1 kHz under full resonance (25 ps delay in Fig. 10) which is the limit of resolution of our spectrum analyzer. Timing jitter was also minimized at 0.4 ps. The RF spectrum under these conditions is shown in Fig. 8(a) (gray line). We observed 1.45 MHz external cavity mode spacing as expected for a 140 m loop. Measured phase noise versus frequency offset from the fundamental mode-locked frequency is shown in Fig. 8(b) (gray line). RF linewidth and timing jitter were lower over a wider delay range with the longer cavity, due to its higher quality factor (Q).]{} {width="10cm"} [In summary, our experiments demonstrate that unbalanced SDL is more effective than balanced SDL and single-loop feedback in stabilizing the laser over a wider delay range. Recent experiments have shown asymmetric dual-loop feedback improves timing jitter of a QDash MLL and suppresses unwanted spurious side-bands. With this resonant arrangement, sub-kHz RF linewidth, sub-picosecond integrated timing jitter and 30 dB side-mode suppression were simultaneously achieved. Recently, it was theoretically predicted that dual-loop optoelectronic oscillators could be optimized by controlling the phase delay and power split ratio \[32\]. Our unbalanced (asymmetric power split) SDL configuration produces narrower linewidth, lower timing jitter over a delay range limited to 0-84 ps by the variable delay lines available, with only one loop requiring fine-tuning. This promises to be a robust and effective means to stabilize mode-locked QDash lasers emitting at $\sim$ 1550 nm.]{} Conclusion ========== [We investigated the effectiveness of single-loop and SDL optical feedback as means of robust stabilization of self-mode-locked QDash lasers operating at 21 GHz pulse repetition rate and emitting at 1550 nm wavelength. Mode-locking occurred without reverse bias applied to the absorber section, and robust stabilization was achieved with predictable delay difference between the two external feedback cavities, which simplifies product design and packaging. We demonstrated that unbalanced SDL feedback provides best stability, maintaining stable RF spectra with narrow linewidth and low timing jitter over a range of delay detuning  80 ps, which means it would be insensitive to temperature, vibration and other common environmental variations. Unbalanced SDL is significantly better than conventional single-loop feedback and balanced SDL feedback, producing up to two orders of magnitude reduction in RF linewidth to 1 kHz (instrument limited) and RMS timing jitter 0.4 ps, compared to free-running. Longer 140 m fiber loops are more effective than shorter 80 m loops. For SDL feedback, we have studied the effects of varying the power split between the loops. The proposed scheme is effective in overcoming the primary drawback of mode-locked diode lasers, their lack of dynamical stability and robustness, in practical applications such as frequency comb generation, optical sampling, signal timing and regeneration, metrology, lidar and many others.]{} Acknowledgments {#acknowledgments .unnumbered} =============== The authors acknowledge financial support from Science Foundation Ireland (grant 12/IP/1658) and the European Office of Aerospace Research and Development (grant FA9550-14-1-0204).

--- abstract: 'We present a numerical study of dephasing of electron spin ensembles in a diffusive quasi-one-dimensional GaAs wire due to the D’yakonov-Perel’ spin-dephasing mechanism. For widths of the wire below the spin precession length and for equal strength of Rashba and linear Dresselhaus spin-orbit fields a strong suppression of spin-dephasing is found. This suppression of spin-dephasing shows a strong dependence on the wire orientation with respect to the crystal lattice. The relevance for realistic cases is evaluated by studying how this effect degrades for deviating strength of Rashba and linear Dresselhaus fields, and with the inclusion of the cubic Dresselhaus term.' author: - 'J. Liu' - 'T. Last' - 'E. J. Koop' - 'S. Denega' - 'B. J. van Wees' - 'C. H. van der Wal' title: 'Spin-dephasing anisotropy for electrons in a diffusive quasi-1D GaAs wire' --- \[sec:Introduction\]INTRODUCTION ================================ The ability to maximize the spin-dephasing time $T_{2}^*$ of an electron spin ensemble is one of the key issues for developing semiconductor-based spintronic devices [@review; @Kikkawa1998]. However, in all III-V semiconductor materials spin ensembles rapidly dephase due to the D’yakonov-Perel’ (DP) spin-dephasing mechanism [@D'yakonov1986; @Miller2003]. For the case of electron ensembles in a heterojunction two-dimensional electron gas (2DEG), two distinct contributions to DP spin-dephasing have to be considered: the inversion asymmetry of the confining potential (structural inversion asymmetry) and the bulk inversion asymmetry of the crystal lattice. The former results in an effective Rashba field and the latter in an effective Dresselhaus field, which includes linear and cubic contributions [@D'yakonov1986; @Bychkov1984; @Lommer1985; @Miller2003]: $$\begin{aligned} \vec{B}_{R} \ &=& \ C_{R}~(\hat{x}k_{y}-\hat{y}k_{x}), \\ \vec{B}_{D1} \ &=& \ C_{D1}(-\hat{x}k_{x}+\hat{y}k_{y}), \\ \vec{B}_{D3} \ &=& \ C_{D3}(\hat{x}k_xk_{y}^2-\hat{y}k_{x}^2k_y), %\vec{B}_{D3} \ &=& \ \alpha_{D3}(\hat{x}k_x[k_{y}^2-k_{z}^2]+\hat{y}k_y[k_{z}^2-k_{x}^2]+\hat{z}k_z[k_{x}^2-k_{y}^2]),\end{aligned}$$ where $\hat{x}$, $\hat{y}$ are the unit vectors along the \[100\] and \[010\] crystal directions, $k_{x}$, $k_{y}$ are the components of the in-plane wave vector, and $C_{R,D1,D3}$ are the spin-orbit coupling parameters. The total effective spin-orbit field $\vec{B}_{eff}$ is the vector sum of all three contributions. For 2D and quasi-1D electron systems, the direction and magnitude of these effective spin-orbit fields can be illustrated as arrows on the Fermi circle. Figure \[1\] presents this for selected points in the 2D momentum space, for the Rashba (a) and linear Dresselhaus (b) field alone, and their sum (c) for the case of equal strength of Rashba and linear Dresselhaus field. In contrast to the individual cases (Figure \[1\] (a), (b)), the magnitude of the vector sum shows a strong anisotropy in momentum space (Figure \[1\] (c)). ![A schematic representation of the direction and magnitude of the effective magnetic field for selected points in a two-dimensional $k$-space, sketched for (a) the Rashba field, (b) the linear Dresselhaus field and for (c) the symmetric case of the sum of equal Rashba and linear Dresselhaus field. Both the magnitude and direction are depicted as arrows on the Fermi circle with radius $k_F$ in the ($k_x$, $k_y$)-plane.[]{data-label="1"}](fig1.png){width="8cm"} This already suggests that spin dephasing in very narrow wires in which electron motion is restricted to the \[110\] direction can be strongly reduced as compared free 2DEG or such wires oriented along other crystal directions. However, it is harder to analyze whether such a dephasing anisotropy also occurs for wider quasi-1D wires, where the motion in the 2DEG plane is still completely random and diffusive, but where the width of the wire is less than length scales as the spin precession length or the mean free path. For the latter case the transport regime could be named quasi-ballistic, but we consider the case of a large ensemble where transport along the wire is still diffusive, and where the width of the wire in the 2DEG plane is much wider than the regime of quantum confinement. Initial studies of such spin-dephasing anisotropy include a recent experiment [@Folk2008] on wires, and theoretical work on drifting ensembles in free 2DEG [@Loss2007]. However, until now most emphasis was on work related to the spin field-effect transistor [@Datta1990], using InAs-based systems or highly asymmetrical heterojunctions, where structural inversion asymmetry dominates the spin-orbit interaction [@Kiselev2000; @Bruno2002; @Mireles2001; @Pramanik2003; @Holleitner2006].\ We report here how the D’yakonov-Perel’ spin dephasing mechanism can be strongly suppressed in diffusive quasi-1D electron systems based on GaAs heterojunction material, for which Rashba and linear Dresselhaus spin-orbit contributions can be indeed of comparable magnitude [@Miller2003; @Folk2008]. The dephasing is studied for spin ensembles initially aligned perpendicular to the plane of the wire (\[001\] direction). This situation reflects the method of preparing and interrogating a spin population via optical pump-probe techniques [@Kikkawa1998]. Our numerical calculation is first performed for conditions with equal Rashba and linear Dresselhaus contributions and the cubic Dresselhaus term set to zero. For widths of the diffusive quasi-1D wires smaller than the spin precession length the DP spin-dephasing mechanism can be strongly suppressed and the spin-dephasing time $T_2^*$ is considerably enhanced if the wire is aligned along the direction of zero effective spin-orbit field. Moreover, we want to point out that the value of our numerical tool lies in the opportunity to study such phenomena also for more realistic conditions. Thus, we can study how breaking the equality of the Rashba and linear Dresselhaus spin-orbit fields, or adding the cubic Dresselhaus term leads to a degradation of the spin-dephasing anisotropy. \[Method\]Method ================ We apply a Monte Carlo method [@Koop2008] to study the temporal evolution of the normalized spin orientation (average spin expectation value) in an elongated quasi-1D wire. Our numerical tool is based on a semiclassical approach. We use a classical description for the electron motion, and a quantum mechanical description of the dynamics of the electron spin. The wire is treated as a rectangular box of aspects 1 $\mu$m and 200 $\mu$m. The electron density and mobility are set to 4$\cdot10^{15}$ m$^{-2}$ and 100 m$^{2}$/Vs, which are typical values for a GaAs/AlGaAs heterojunction material. All electrons are assumed to have the same Fermi velocity $\upsilon_{F}$ of 2.7$\cdot10^{5}$ m/s. This is a valid approximation for $k_BT$, $\Delta E_{Z,SO}$ $\ll$ $E_F$ (with respect to the bottom of the conduction band), where $\Delta E_{Z,SO}$ is the Zeeman splitting due to the spin-orbit fields alone. Electron-electron interaction and inelastic scattering mechanisms are neglected.\ The electron is regarded as a point particle which moves on a classical trajectory between scatter events on impurities (randomly determined at a rate to obtain an average scatter time of 38 ps) yielding diffusive behavior in the ensemble (electron mean free path $L_{p}$ = $\upsilon_{F}\tau_{p}$ = 10 $\mu$m), and specular scattering on the edges of the wire. For each electron moving on such a ballistic trajectory we calculate the spin evolution in the effective spin-orbit fields quantum mechanically, and we then take the ensemble average on a set of electrons with random initial position and momentum direction.\ Within a straight ballistic segment of an electron trajectory the spin rotates around $\vec{B}_{eff}$ over a precession angle given by $$\begin{aligned} \phi_{prec} \ = \ \frac{g \mu_B |\vec{B}_{eff}|}{\hbar} \ t,\end{aligned}$$ where $\hbar$ is the reduced Planck’s constant and $t$ the time of traveling through the segment. The spin rotation operator $\widehat{U}$ for rotation over the precession angle $\phi_{prec}$ about the direction $\vec{u}$ (unit vector) of the effective magnetic field $\vec{B}_{eff}$ is obtained by (see e.g. [@Claude]) $$\begin{aligned} \widehat{U} \ = \ \text{exp}(-i \ \frac{\phi_{prec}}{2} \ \vec{\sigma} \ \vec{u}) \ = \ \left( \begin{array}{*{2}{c}} \text{cos} \frac{\phi_{prec}}{2}-iu_z\text{sin} \frac{\phi_{prec}}{2} & -(iu_x+u_y)\text{sin}\frac{\phi_{prec}}{2} \\ -(iu_x-u_y)\text{sin}\frac{\phi_{prec}}{2} & \text{cos} \frac{\phi_{prec}}{2}+iu_z\text{sin} \frac{\phi_{prec}}{2} \\ \end{array} \right),\end{aligned}$$ where $\vec{\sigma}$ represents the vector of Pauli spin matrices (x, y, z components of the spin). $\widehat{U}$ acting on a spin state $| \Psi_{initial} \rangle$ at the beginning of a ballistic trajectory yields the spin state $ | \Psi_{final} \rangle \ = \widehat{U} \ | \Psi_{initial} \rangle$ at the end of the trajectory. Thus, we can follow the spin state of each electron (labeled $i$), and we use this to define a semiclassical spin vector to present its orientation, $ \vec{S}_i = ( \langle S_x \rangle_i, \langle S_y \rangle_i, \langle S_z \rangle_i ) $ from its spin expectation values in $x$, $y$, $z$ directions. For an electron experiencing multiple scattering events, the orientation of the effective magnetic field is changed at each scatter event. For each trajectory a rotation is applied. Once a scattering event takes place, the wave vector state is updated, $\vec{B}_{eff}$ is recalculated based on the new wave vector, resulting in a new rotation operator, and the evolution of the spin state will carry on. For spin ensembles, the randomization will bring a reduction of the normalized spin orientation (average spin expectation value) $\langle S_r \rangle \ = \ \left| ( \sum_{i=1}^N \vec{S}_i )/ N \right|$ for the ensemble. $\langle S_r \rangle$ is obtained by averaging over an ensemble of $N$ = 1000 spins, independent of their positions within the system, and calculated as a function of time. We choose to study the spin coherence in the ensemble here as $\langle S_r \rangle$. The advantage is that $\langle S_r \rangle$ gives the magnitude of the residual spin orientation in the direction that is maximum (automatically evaluating the envelope in case the ensemble average shows precession). However, in the present study without externally applied fields, there was no development of average spin orientation in the $x$ and $y$ directions, and the decay of $\langle S_r \rangle$ always equaled the decay of ensemble average $\langle S_z \rangle \ = \ \left| ( \sum_{i=1}^N \langle S_z \rangle_i )/ N \right|$. The spin-dephasing time $T_2^*$ of the ensemble is defined as the decay time over which $\langle S_r \rangle$ reduces to 1/e of its initial value. Note, however, that decay traces of $\langle S_r \rangle$ were not always mono exponential in our simulations. Results and Discussion ====================== The temporal evolution of the normalized spin orientation $\langle S_r \rangle$ is studied for out-plane initial spin states oriented along the \[001\] direction. The effective spin-orbit field resulting from equal magnitudes of the Rashba and linear Dresselhaus fields is always parallel to the \[110\] axis ($C_R$ = $C_{D1}$ = $-1.57\cdot$10$^{-8}$ Tm, [@Miller2003]). These values of the spin-orbit parameters give rise to an average spin precession length of about 3 $\mu$m. The width of our wire of 1 $\mu$m is chosen to be smaller than this length scale. ![The spin-dephasing time $T_2^*$ of a spin ensemble is plotted as a function of the wire orientation with respect to the \[100\] lattice direction. The ensemble is initially oriented along the \[001\] direction. $T_2^*$ is strongly enhanced for a quasi-1D wire oriented in the \[110\] direction (black) as compared to an ensemble in a 2D system (gray). The arrows (in the top insets, horizontal axes span $\pm 5^{\circ}$) for data at 45$^{\circ}$ and 225$^{\circ}$ indicate that here $T_2^*$ is larger than could be calculated ($C_R$ = $C_{D1}$ = $-1.57\cdot$10$^{-8}$ Tm, $C_{D3}$ = 0). Inset: Ensemble spin expectation value $\langle S_r \rangle$ as a function of time for different wire orientations.[]{data-label="5"}](fig5.png){width="8cm"} The inset of Figure \[5\] shows the temporal evolution for five different orientations of the quasi-1D wire with respect to the \[100\] direction. A distinct anisotropy of the spin dephasing times is observed. The peak value of $T_2^*$ is reached when the wire is oriented exactly along the \[110\] direction. It is found to be in excess of 10$^{4}$ ps, but the exact value could not be calculated in a reasonable computation time. Yet, a deviation of only 5$^\circ$ in the wire orientation with respect to the \[110\] direction leads to a reduction of $T_2^*$ of more than an order of magnitude towards a value of about 200 ps. For angles close to \[110\] $\pm$ 15$^\circ$ the spin-dephasing time drops already to 43 ps and it reaches a minimum of 9 ps for wires oriented along the \[$\overline{1}$10\] and \[1$\overline{1}$0\] direction. A detailed summary of these findings is presented in Figure \[5\] where the extracted spin-dephasing times are plotted as a function of the wire orientation. This plot reveals the strong anisotropy of $T_2^*$ with respect to the crystal axes.\ The anisotropy of $T_2^*$ is directly connected to the motion of single electrons within the ensemble. Specular edge scattering implies that electrons with a solely transverse momentum component to the wire orientation (traveling less than the spin precession length between scatter events) almost do not contribute to the spin-dephasing because of motional narrowing. Only electrons with a strong momentum component longitudinal to the wire orientation are contributing to the dephasing of the spin ensemble. This results in the strong enhancement of $T_2^*$ for wires in the \[110\] direction.\ Spin-dephasing times for non-confined spin ensembles (wire width taken much wider than the mean free path and spin precession length) are also calculated (gray, Figure \[5\]). In contrast to the quasi-1D wire case, no spin-dephasing anisotropy is found. We investigated the crossover from 2D to quasi 1D behavior for a wide range of values for the spin precession length and mean free path (with respect to the wire width), and found that the spin precession length is the crucial length scale which is governing this crossover.\ Next, we discuss whether this distinct spin-dephasing anisotropy in quasi-1D wires (with a strong enhancement of $T_2^*$ for wires in the \[110\] direction as a finger print) can be maintained under more realistic circumstances. First we study the influence of adding the cubic Dresselhaus term on the enhancement of $T_2^*$. Secondly, the influence of deviating strength of Rashba and linear Dresselhaus fields will be discussed. To avoid the difficulty that $T_2^{*}$ cannot be calculated for 45$^\circ$ with respect to the \[100\] direction within reasonable calculation time, we take 43$^\circ$ as a test case, as this already shows a very strong $T_2^{*}$ enhancement, while $T_2^{*}$ is limited to nanoseconds. Hence, the calculated spin-dephasing times in the following part can be seen as lower bounds. $T_2^*$ for the exact \[110\] direction is expected to be distinctively higher.\ The influence of the additional cubic Dresselhaus term on the spin dephasing time in the quasi-1D wire is presented in Figure \[6\] (a). Again, the 2D case is plotted as a reference (gray). Without the cubic Dresselhaus term the calculation results in a $T_2^*$ of 1.3 ns. However, the spin-dephasing time is decreasing rapidly with increasing Dresselhaus parameter. For $C_{D3}$ = -7$\cdot$10$^{-25}$ Tm$^{3}$, $T_2^{*}$ is already reduced to 78 ps. To estimate whether the spin-dephasing anisotropy is still present in a more realistic situation, experimentally deduced parameters are applied for comparison with our calculations. In [@Miller2003] a value of $C_{D3}$ = -1.18$\cdot$10$^{-24}$ Tm$^{3}$ is evaluated which results in a $T_2^*$ of only 40 ps. A value which is less than an order of magnitude higher than the value calculated for the 2D case. This points to the conclusion that the cubic Dresselhaus term nearly annihilates the spin-dephasing anisotropy. However, $C_{D3}$ depends strongly on the electron density of the system. For samples with lower densities $C_{D3}$ is orders of magnitude smaller [@Miller2003; @Jusserand1995; @Jusserand1993]. Considering those values, it turns out, that there is a much weaker decay of the peak value of the spin-dephasing time, even when the cubic Dresselhaus term is included. ![Data shows how the $T_2^*$ enhancement of Figure \[5\] reduces when the cubic Dresselhaus term is added and when the symmetry between $C_R$ and $C_{D1}$ is lifted. The initial spin state is chosen to be along \[001\] direction and the quasi-1D wire is set at 43$^{\circ}$ with respect to the \[100\] direction. The case of 43$^{\circ}$ wire orientation (rather than 45$^{\circ}$) avoids the need to deal in calculations with extremely long $T_2^*$ for the symmetric case of $C_R$ = $C_{D1}$ and $C_{D3}$ $\approx$ 0, while still clearly showing the 1D $T_2^*$ enhancement. (a) The effect of the cubic Dresselhaus term on $T_2^*$ is plotted for the 2D case (gray) and for the quasi-1D case (black) ($C_R$ = $C_{D1}$ = -1.57$\cdot$10$^{-8}$ Tm). (b) $T_2^*$ is plotted here against the difference $C_R$ - $C_{D1}$, at the fixed value of $C_{D1}$ = -1.57$\cdot$10${^{-8}}$ Tm for the 2D case (gray) and for the quasi-1D case (black) ($C_{D3}$ = 0).[]{data-label="6"}](fig67.png){width="8cm"} Finally, $T_2^*$ is investigated for deviating Rashba and linear Dresselhaus contributions. This dependence is summarized in Figure \[6\] (b) where $T_2^*$ is plotted as a function of the difference in strength of the Rashba and linear Dresselhaus parameter, $(C_{R} - C_{D1})$, in the interval $\pm$1$\cdot$10$^{-8}$ Tm for $C_{D1}$ fixed at $-1.57 \cdot$10$^{-8}$ Tm. At $C_{R} - C_{D1} = 0$ this results in the previously calculated $T_2^*$ of around 1.3 ns. With either increasing $|C_{R}|$ or increasing $|C_{D1}|$, $T_2^*$ is decaying equally fast. $C_R$ and $C_{D1}$ taken from [@Miller2003] result in a difference $C_{R} - C_{D1}$ of 0.4$\cdot$10$^{-8}$ Tm. For this value our calculated $T_2^*$ is already considerably reduced to about 110 ps. However, this spin dephasing time is still an order of magnitude higher than the one resulting from the 2D case and in addition, $C_R$ can be tuned with a gate or heterostructure design to equalize it to the linear Dresselhaus field. Therefore, in summary, it can be stated that the spin-dephasing anisotropy which is very distinct for $C_{R}$ and $C_{D1}$ exactly equal can still prevail under less ideal conditions. \[Conclusion\]Conclusion ======================== A useful numerical tool is developed for studying spin-dephasing in device structures due to the D’yakonov-Perel’ spin-dephasing mechanism. The Rashba, linear and cubic Dresselhaus contributions can be taken into account. With this tool is was demonstrated that quasi-1D wires (narrower than the spin precession length, but with diffusive 2D motion for the electron ensemble) can show very clear signatures of spin-dephasing anisotropy, with a strong suppression of spin dephasing for wires in the \[110\] crystal direction. 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--- abstract: | The muon anomalous magnetic moment is one of the most precisely measured quantities in particle physics. In a recent experiment at Brookhaven it has been measured with a remarkable 14-fold improvement of the previous CERN experiment reaching a precision of 0.54ppm. Since the first results were published, a persisting “discrepancy” between theory and experiment of about 3 standard deviations is observed. It is the largest “established” deviation from the Standard Model seen in a “clean” electroweak observable and thus could be a hint for New Physics to be around the corner. This deviation triggered numerous speculations about the possible origin of the “missing piece” and the increased experimental precision animated a multitude of new theoretical efforts which lead to a substantial improvement of the prediction of the muon anomaly $a_\mu=(g_\mu-2)/2$. The dominating uncertainty of the prediction, caused by strong interaction effects, could be reduced substantially, due to new hadronic cross section measurements in electron-positron annihilation at low energies. Also the recent electron $g-2$ measurement at Harvard contributes substantially to the progress in this field, as it allows for a much more precise determination of the fine structure constant $\alpha$ as well as a cross check of the status of our theoretical understanding. In this report we review the theory of the anomalous magnetic moments of the electron and the muon. After an introduction and a brief description of the principle of the muon $g-2$ experiment, we present a review of the status of the theoretical prediction and in particular discuss the role of the hadronic vacuum polarization effects and the hadronic light–by–light scattering correction, including a new evaluation of the dominant pion-exchange contribution. In the end, we find a 3.2 standard deviation discrepancy between experiment and Standard Model prediction. We also present a number of examples of how extensions of the electroweak Standard Model would change the theoretical prediction of the muon anomaly $a_\mu$. Perspectives for future developments in experiment and theory are briefly discussed and critically assessed. The muon $g-2$ will remain one of the hot topics for further investigations. address: - 'Humboldt-Universität zu Berlin, Institut für Physik, Newtonstrasse 15, D-12489 Berlin, Germany' - 'Institute of Physics, University of Silesia, ul. Uniwersytecka 4, PL-40007 Katowice, Poland' - | Regional Centre for Accelerator-based Particle Physics, Harish-Chandra Research Institute,\ Chhatnag Road, Jhusi, Allahabad - 211 019, India author: - Fred Jegerlehner - Andreas Nyffeler title: 'The Muon g-2' ---   --------------------------------------------------- HU-EP-09/07, HRI-P-09-02-001, RECAPP-HRI-2009-003 --------------------------------------------------- , muon, anomalous magnetic moment, precision tests 14.60.Ef,13.40.Em

--- author: - | Jin Guo[^1] and [Tongsuo Wu[^2]]{}\ [Department of Mathematics, Shanghai Jiaotong University]{} title: ' **Monomial ideals under ideal operations[^3]** ' --- [3mm]{}[**Abstract.**]{} [In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure ${\overline}{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I, P)$ is studied, which also gives a clear description of the primary decomposition of $I^{(k)}$. Then a new simplicial complex $_J\bigtriangleup$ of a monomial ideal $J$ is defined, and it is shown that $I_{_J\bigtriangleup^{\vee}} = \sqrt{J}$. Finally, we show under an additional weak assumption that a monomial ideal is universal lexsegment if and only if its polarization is a squarefree strongly stable ideal. ]{} [3mm]{}[Key Words:]{} [Borel type monomial ideal; $k^{th}$ symbolic power; integral closure; polarization; universal lexsegment monomial ideal]{} [4mm]{} Introduction ============= [3mm]{}Throughout the paper, $K$ is an infinite field and let $S=K[x_1,\,x_2,\,\ldots,\,x_n]$ be the polynomial ring with $n$ indeterminants over $K$. If an ideal $I$ is generated by $u_1,\ldots,u_s$, then we denote it by $I=\langle u_1,\ldots,u_s\rangle$. For a monomial ideal $I$ of $S$, recall that $I$ is called [*strongly stable*]{} if for any monomial $u$ in $I$ and any $i<j\le n$, $x_j\mid u$ implies $x_i(u/x_j)\in I$. Recall that $I$ is called [*Borel-fixed*]{}, if ${\alpha}(u)\in I$ holds for any invertible upper $n\times n$ matrix ${\alpha}$ over $K$. Recall that $I$ is called [*of Borel type*]{} if $$I:x_i^{\infty}=I: \langle x_1,\,x_2,\,\ldots,\,x_i \rangle^\infty\quad \quad (*)$$ holds for every $i=1,\ldots,n$. It is known that each strongly stable monomial ideal is Borel-fixed, and the converse holds under the additional assumption $char(K)=0$. Bayer and Stillman in [@BS] noted that Borel-fixed ideals satisfy condition $(*)$. Herzog et al. in [@HPV] gave the definition of a Borel type monomial ideal, and they proved among other things that a Borel type monomial ideal is sequentially Cohen-Macaulay, see also [@Popescu]. Furthermore, there are other two classes of strongly stable monomial ideals, namely, monomial ideals which are lexsegment or universal lexsegment, see [@AHH] or [@HH]. We have the following relations for conditions on a monomial ideal: [3mm]{} universal lexsegment${\Longrightarrow}$lexsegment${\Longrightarrow}$strongly stable${\Longrightarrow}$ Borel-fixed${\Longrightarrow}$ of Borel type. [2mm]{} The following is the fundamental characterization of Borel type monomial ideals: \[BT\] ([@HH Proposition 4.2.9]) For a monomial ideal $I$ of $S$, the following conditions are equivalent: $(1)$ $I$ is of Borel type. $(2)$ For each monomial $u\in I$ and all positive integers $i,j,s$ with $i<j\le n$ such that $x_j^s\mid u$, there exists an integer $t\ge 0$ such that $x_i^t(u/x_j^s)\in I$. $(3)$ Each associated prime ideal $P$ of $I$ has the form $\langle x_1,x_2,\ldots, x_r \rangle$ for some $r\le n$. In [@MC Proposition 1], Mircea Cimpoeas observed that the afore mentioned property is preserved under several operations, such as sum, intersection, product, colon. For a monomial ideal $I$ of Borel type, note that $I:\mathfrak{m}^\infty=I:\mathfrak{m}^r$ holds for $r>>0$, thus the saturation $I:\mathfrak{m}^\infty$ is a monomial ideal of Borel type. The root ideal $\sqrt{I}$ is a prime ideal of the form $\langle x_1,x_2,\ldots,x_r \rangle$, and is thus universal lexsegment. Some parts of the following proposition are well known, the others are direct to check, so we omit the verification. \[operation\] Let $I,J, L$ be monomial ideals of $S$. \(1) If further $I,J$ are of Borel type (strongly stable, respectively), then each of the following is a monomial ideal of Borel type ( strongly stable, respectively): $$I\cap J ,\, I+J,\,\,I:L,\,\, IJ.$$ In particular, the saturation $I:\mathfrak{m}^\infty $ of $I$ is of Borel type (strongly stable, respectively) if $I$ has the same property. \(2) If further $I,J$ are Borel-fixed ideals, then each of $ I\cap J ,\, I+J,\,\, I:J,\,\, IJ$ is again Borel-fixed. In particular, the saturation $I:\mathfrak{m}^\infty $ of $I$ is Borel-fixed. \(3) If further $I,J$ are lexsegment (universal lexsegment, respectively) ideals, then each of $ I\cap J ,\, I+J,\,\, I:L$ is again lexsegment (universal lexsegment, respectively). [3mm]{} Let $I$ be a Borel-fixed monomial ideal, and $L$ a monomial ideal which need not to be Borel-fixed. The following example shows that the colon $I : L$ may be not Borel-fixed. \[colon not Borel-fixed\] Let $K$ be a field with $char(K)=2$, and let $S=K[x_1, \ldots, x_n]$. If $I= \langle x_1^3, x_1x_2^2 \rangle$. It is direct to check that $I$ is Borel-fixed. Set $L = \langle x_2 \rangle$. It is easy to see that $I : L = \langle x_1^3, x_1x_2 \rangle$, which is not Borel-fixed. [3mm]{} The following example shows that $IJ$ may be not lexsegment, even though $I, J$ are lexsegment. \[product not lexsegment\] Let $S=K[x_1, x_2, x_3]$, and let $I= \langle x_1^3, x_1^2x_2, x_1^2x_3, x_1x_2^2, x_1x_2x_3 \rangle$. It is easy to see that $I$ is lexsegment, and $u=x_1^2x_2^2x_3^2 \in I^2$. Note that $v= x_1^3x_3^3 \not\in I^2$ and $v >_{lex}u$, so $I^2$ is not lexsegment. [3mm]{}As an application of Proposition \[operation\], we now give an alternative proof to the following: \[regular\] ([@HH Proposition 4.3.3]) Let $I{\subseteq}S$ be a monomial ideal of Borel type. Then $x_n,\ldots,x_1$ is an almost regular sequence on $S/I$. In the proof of [@HH Lemma 4.3.1], let $M=S/I$. Then the corresponding $N$ (i.e., $0:_M\mathfrak{m}^\infty$) is identical with $(I:\mathfrak{m}^\infty)/I$. Note that $M/N\cong S/(I: \mathfrak{m}^\infty )$ holds. If $M=N$, then each element of $S_1$ is almost regular on $M$ since $M$ has finite length. Now assume $M\neq N$. Since $I: \mathfrak{m}^\infty $ is monomial of Borel type and $\mathfrak{m}\not\in Ass(M/N)$, as is shown in the proof of the Lemma 4.3.1, it follows by [@HH Proposition 4.2.9(d)] that $x_n\not\in \cup Ass(M/N)$, i.e., $x_n$ is in the constructed open set $U$ and thus is almost regular on $S/I$. The result then follows by mathematical induction. Integral Closure ${\overline}{I}$ ================================= Let $I$ be any ideal of a commutative ring $R$. Recall from [@Swanson] that the integral closure ${\overline}{I}$ of an ideal $I$ consists of all elements of $R$ which are integral over $I$. Note that ${\overline}{I}$ is an ideal of $R$. For a monomial ideal $I$ of $S$, ${\overline}{I}$ is generated by all monomials $u$ such that $u^k\in I^k$ holds for some $k>0$. Thus the exponent set of all monomials in ${\overline}{I}$ is identical with the integer lattice points in the convex hull of the exponent set of all monomials in $I$, see [@Swanson Proposition 1.4.6]. In this section, we will show that ${\overline}{I}$ is a monomial ideal of Borel type (strongly stable, Borel-fixed, lexsegment, or universal lexsegment respectively), whenever $I$ has the same property. \[Closure\] Let $I$ be a monomial ideal of $S$ and let ${\overline}{I}$ be its integral closure. If $I$ is of Borel type (strongly stable, Borel-fixed, lexsegment, or universal lexsegment respectively), then ${\overline}{I}$ is also monomial of Borel type (strongly stable, Borel-fixed, lexsegment, or universal lexsegment respectively). \(1) Assume that $I$ is monomial of Borel type. Then ${\overline}{I}$ is also monomial by [@Swanson Proposition 1.4.2] (see also [@HH Theorem 1.4.2]). In order to prove that ${\overline}{I}$ is of Borel type, we need only to verify that each associated prime ideal of ${\overline}{I}$ has the form $\langle x_1,x_2,\ldots,x_j\rangle$ for some $1\le j\le n$. In fact, let $P\in Ass({\overline}{I})$ and by [@HH Corollary 1.3.10], there exists a monomial $v\in Mon(S)\setminus P$ such that $P={\overline}{I}:v$. By [@HH Theorem 1.4.2], $v\not\in {\overline}{I}$ implies that $v^r\not\in I^r$ holds for all integer $r$. For any $x_m\in P$, clearly $vx_m\in {\overline}{I}$ holds, thus there exists a positive integer $k$ such that $x_m^kv^k\in I^k$. Since $I^k$ is also monomial of Borel type and $x_m^k\mid x_m^kv^k\in I^k$ holds, thus for any $1\le j< m$, by [@HH Proposition 4.2.9(2)], there exists an integer $t\ge 0$ such that $x_j^tv^k\in I^k$. $t>0$ holds since $v^k\not\in I^k$. Then $x_j^tv^{k-1}\in I^k:v{\subseteq}{\overline}{I}:v=P$. By the choice of $v$, we have $v\not\in P$ thus $v^{k-1}\not\in P$. Then $x_j\in P$ and it shows that ${\overline}{I}$ is of Borel type. \(2) Now assume that $I$ is strongly stable. Then for any monomial $u\in {\overline}{I}$, there exists a positive integer $k$ such that $u^k\in I^k$. If $x_j\mid u$, then $x_j^k\mid u^k$. Assume $u^k=w_1w_2\cdots w_k$, in which $w_i\in Mon(S)\cap I$. Assume further that $x_j^{a_i}\mid w_i$, where $\sum_{i=1}^ka_i=k$ and $a_i\ge 0$. Then for any $i<j$, we have $x_i^{a_i}(w_1/x_j^{a_i})\in I$. Then $$[x_i(u/x_j)]^k=\prod_{i=1}^k x_i^{a_i}(w_i/x_j^{a_i})\in I^k,$$ thus by [@HH Theorem 1.4.2], $x_i(u/x_j)\in {\overline}{I}$ holds. This shows that ${\overline}{I}$ is strongly stable. \(3) Assume that $I$ is Borel-fixed. Just as in $(2)$, we assume $u^k = w_1w_2\cdots w_k \in I^k$, where $w_i\in I$. Note that $(\alpha(u))^k = \alpha(u^k)$, it will suffice to show that $\alpha(u^k) \in I^k$ for every $\alpha \in \mathcal{B}$, where $\mathcal{B}$ is the set of upper invertible $n\times n$ matrices over $K$. By Proposition \[operation\](2), it is clear since $I$ is Borel-fixed. \(4) Assume that $I$ is lexsegment. For each $u \in {\overline}{I}$, there exists a positive integer $k$, such that $u^k = \prod_{l=1}^ku_l \in I^k$. Let $u=x_i^{a_i}(\prod_{j=1}^{i-1}x_j^{a_j})(\prod_{t=i+1}^{n}x_t^{a_t})$, and let $v=x_i^{b_i}(\prod_{j=1}^{i-1}x_j^{a_j})(\prod_{t=i+1}^{n}x_t^{b_t})$ such that $b_i > a_i$ and $\sum_{t=i}^n b_{t}=\sum_{t=i}^n a_{t}$. Assume that $u_l=x_i^{a_{li}}(\prod_{j=1}^{i-1}x_j^{a_{lj}})(\prod_{t=i+1}^{n}x_t^{a_{lt}})$ for $1 \le l \le k$. It is easy to see $\sum_{l=1}^k a_{lj}=ka_j$ for $1 \le l \le n$. In the following, we will show that there exist $v_1, \ldots, v_k \in I$ such that $v^k = \prod_{l=1}^k v_l$, which implies $v \in {\overline}{I}$. In fact, we can choose $v_l$ under the following rule: If $\prod_{t=i+1}^{n}x_t^{a_{lt}}=1$, then set $v_l=u_l$ and $v_l' = 1$; If $\prod_{t=i+1}^{n}x_t^{a_{lt}} \neq 1$, then set $v_l = x_i^{a_{li}+1}(\prod_{j=1}^{i-1}x_j^{a_{lj}}) \cdot v_l'$, such that $degree(v_l') = degree(u_l) - \sum_{j=1}^i a_{lj}-1$ and $v_l' \,|\, \prod_{t=i}^n x_i^{b_i}/\prod_{t=1}^{l-1} v_t'$ with the exponent of $x_i$ as small as possible. Note that $a_i < b_i$ and $degree(v)=degree(u)$, there exist a group of $v_1, \ldots, v_k$ such that $v^k = \prod_{l=1}^k v_l$. \(5) Assume that $I$ is universal lexsegment. If the minimal generating set of $I$ is $G_{min}(I)= \{ u_1, \ldots, u_m \}$ with $u_1 > u_2 > \cdots > u_m$ by pure lexicographic order, then there exists a group of positive integers $a_1, \ldots, a_m$, such that $u_i= x_i^{a_i} \prod_{j=1}^{i-1}x_j^{a_j-1}$ for each $1 \le i \le m$. Let $\mathcal{C}(I)$ be the convex hull of the set of lattice points $\{ \alpha \,|\, x^{\alpha} \in I\}$. By Corollary 1.4.3 [@HH], ${\overline}{I}= \langle x^{\alpha} \,|\, \alpha \in \mathcal{C}(I) \rangle$. Note that the structure of $u_i$ for $1 \leq i \leq m$, it is not hard to see that ${\overline}{I} = I$. Hence ${\overline}{I}$ is universal lexsegment. [3mm]{} We remark that Theorem \[Closure\] (1) can also be proved in a similar way as is used in proving $(2)$ and $(3)$. It is known that $I{\subseteq}{\overline}{I}{\subseteq}\sqrt{I}$ holds for every ideal of a (noetherian) ring $R$. Thus $\sqrt{I}=\sqrt{{\overline}{I}}$ holds. By the primary decomposition theorem (see [@AM; @Eisenbud]), we record \[Minimal Prime ideals\] For any ideal $I$ of a noetherian ring $R$, $Min(I)=Min({\overline}{I})$ holds, where $Min(I)$ is the set of all prime ideals minimal over $I$. In particular, a squarefree monomial ideal $I$ of $S$ is integrally closed. \[integral power inclusion\] For a monomial ideal $I$ of $S$ and any integer $k \geq 1$, ${\overline}{I}^k {\subseteq}{\overline}{I^k}$ holds. First, note that $${\overline}{I^k} = \langle \{ u\in S \mid \exists\, l\,\,such\,\, that\,\, u^{l} \in I^{kl}\} \rangle$$ and $${\overline}{I}^k = \langle \{ \prod_{i=1}^k w_i \,|\, \exists l_i,\,\, such \,\,that\,\, w_i^{l_i} \in I^{l_i} \} \rangle.$$ For every $v = \prod_{i=1}^k w_i \in {\overline}{I}^k$ with $w_i^{l_i} \in I^{l_i}\,(\forall i=1, \ldots, k)$, let $l=lcm(l_1, \ldots, l_k)$. Then $w_i^{l} \in I^{l}$ holds for each $i=1, \ldots, k$. Thus $v^l = \prod_{i=1}^k w_i^l \in I^{kl}$, which implies $v \in {\overline}{I^k}$. [3mm]{} The converse inclusion does not hold even for squarefree monomial ideals. We include a counterexample below: \[not equal\] Let $u=\prod_{i=1}^6x_i$, and let $$I = \langle x_1x_2x_3, \,x_1x_4x_5,\, x_2x_4x_6,\, x_3x_5x_6\rangle$$ be a squarefree monomial ideal of $S=K[x_1, x_2, x_3, x_4, x_5, x_6]$, thus $I={\overline}{I}$. It is easy to check $u \notin I^2$, but $u^2 \in (I^2)^2$ holds and hence $u \in {\overline}{I^2}$. Thus ${\overline}{I^2}\not{\subseteq}{\overline}{I}^2.$ The $k^{th}$ symbolic power $I^{(k)}$ of an ideal $I$ ===================================================== Let $I$ be any ideal of a noetherian ring $R$. It follows from [@Eisenbud Corllary 2.19] that $Min(I)=Min(I^k)$ holds for all positive integer $k$, thus $\underset{k\ge 1} {\cup} Min(I^k)=Min(I)$. Recall that for each $P\in Min(I)$, $ker(R\mapsto (R/I)_P)$ is the $P$-primary component of $I$, and it depends only on $I$ and $P$ in an irredundant primary decomposition of $I$. If $$I^k=\underset{P\in Ass(I^k)}{\cap}Q(P)$$ is an irredundant primary decomposition of $I^k$, then $Q(P)=ker(R\to (R/I^k)_P)$ holds for each $P\in Min(I^k)$, and $\underset{P\in Min(I^k)}{\cap}Q(P)$ is independent of the primary decomposition of $I^k$. Recall that $$I^{(k)}=\underset{P\in Min(I)}{\cap}ker(R\to (R/I^k)_P)$$ is called the [*$k^{th}$ symbolic power*]{} of $I$. By [@HTT Section 3], $$I^{(k)}=I^k:(\cap_{P\in Ass^*(I)\setminus Min(I)} P)^\infty,$$ where $Ass^*(I)=\underset{k\ge 0} {\cup} Ass(I^k)$. Thus it follows from Proposition \[operation\] that if $I$ is monomial of Borel type, then so is $I^{(k)}$. In the following, we will give a direct and alternative proof to the fact. We need some preparations. \[new graded\] [*Let $B$ be a nonempty subset of $[n]$. For any monomial $u= x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$, $\underset{j\in B}{\sum}{a_{j}}$ is called [*the $B$-degree*]{} of the monomial $u$. An ideal $I$ is called $B$-graded if $f_i\in I$ holds for the $B$-graded decomposition $f=f_0 + f_1 + \cdots + f_k$ of each $f\in I$, where $f_i$ is the $B$-degree $i$ component of $f$.*]{} It is easy to check the following property. \[monomial to graded\] If $I$ is a monomial ideal, then for every subset $B$ of $[n]$, $I$ is $B$-graded. [3mm]{}Let $A$ be a subset of $[n]$. For a monomial $u=x^{{\alpha}} \in S$ with ${\alpha}= (a_1, \ldots, a_n)$, denote ${\alpha}(A)= (b_1, \cdots, b_n)$, where $$\begin{cases} b_i = a_i\,\quad if\,\, i\in A, \\ b_i = 0\,\quad if\,\, i\in [n]\setminus A. \end{cases}$$ Denote $u(A)= x^{\alpha(A)}$. We also denote $M(A)= \{u(A) \,|\, u \in M\}$ for any nonempty subset $M$ of $Mon(S)$. For a prime ideal $P$ and an ideal $I$ of $S$, denote $$J(I,P) = \{f\in S \,|\, \exists\, g\in S\setminus P,\,such\, \,that\,\, fg\in I\}.$$ Note that $J(I,P)=ker(S\to (S/I)_P).$ For a monomial ideal $I$, let $G(I)$ ($G_{min}(I)$) be its (minimal) generating set of monomials, and denote $I(A)=G_{min}(I)(A)$ for a subset $A$ of $[n]$. \[Borel evaluation\] Let $I$ and $P$ be monomial ideals of $S$. If $P$ is a prime ideal, then $J(I,P)$ is a monomial ideal. Furthermore, $$I(X_P)=\{u(X_P)\,|\,u\in G_{min}(I)\}$$ is a monomial generating set of $J(I,P)$, where $X_P=\{i\in [n] \,|\, x_i \in P\}$. In particular, $$|G_{min}(J(I,P))| \leq |G_{min}(I)|.$$ Assume $P=\langle x_{i_1}, \cdots, x_{i_t}\rangle$ and let $X_P=\{i_1,\ldots,i_t\}$. First, we will show that $J(I,P)$ is a monomial ideal. For any $f\in J(I,P)$ and any $g\in S\setminus P$ such that $fg\in I$, let $f=f_0 + f_1 + \cdots + f_m$ and $g=g_0 + g_1 + \cdots + g_l$ be their $X_P$-graded decompositions. Then $$(fg)_r= \sum_{i+j=r}f_ig_j.$$ It follows that $f_0g_0=(fg)_0\in I$ since $I$ is a $X_P$-graded ideal by Lemma \[monomial to graded\]. We will prove that $Supp(f) \subseteq J(I,P)$ holds by induction on the graded component number $m$ of $f$. Since $Supp(f_0){\subseteq}Supp(f)$, it will suffice to show that $supp(f_0) \subseteq J(I,P)$ holds. For this purpose, let $f_0 = u_1 + \cdots + u_s$ and $g_0 = v_1 + \cdots + v_c$, where $$Supp(f_0)=\{u_i\,|\,1\le i\le s\},\,\, Supp(g_0)=\{v_j\,|\, 1\le j\le c\},$$ $u_1<u_2<\cdots<u_s$ and $v_j<v_{j+1}$ under a suitable monomial order. Then it follows that $u_1v_1 \in I$ holds since $I$ is a monomial ideal. Note that $v_1$ has degree 0 under the $X_P$-grading, thus $v_1\not\in P$ and hence $u_1v_1 \in I$ implies $u_1\in J(I,P)$. Since $$u_1v_1g_0 + (u_2 + \cdots + u_s)v_1g_0=f_0g_0v_1 \in I,$$ it follows that $(u_2 + \cdots + u_s)v_1g_0 \in I$. Note that both $v_1$ and $g_0$ have degree 0 under the $X_P$-grading, so does $v_1g_0$. It follows by induction that $Supp(f_0){\subseteq}J(I,P)$ holds. This proves that $J(I,P)$ is a monomial ideal. For the second statement, for a monomial $u\in I(X_P)$, there exists a $v \in G_{min}(I)$, such that $u = v(X_P)$. Note that $v(X_P)v([n]\setminus X_P) = v \in I$, and $v([n]\setminus X_P) \in S\setminus P$, so $u=v(X_P) \in J(I,P)$. On the other hand, if a monomial $u\in J(I,P)$, then there exists a monomial $w\in S\setminus P$, such that $uw\in I$. Note that $u(X_P)=(uw)(X_P)$, there exits a monomial $v\in G_{min}(I)$, such that $v | uw$, and hence $v(X_P) | u(X_P)$. Thus $J(I,P)$ is generated by $I(X_P)$. The last statement is clear. \[disappear large variety\] Let $P = \langle x_{i_1}, \cdots, x_{i_k}\rangle$ with $x_j\notin P$ and $x_t \in P $ for every $t < j$. If a monomial ideal $I$ is of Borel type, then for every monomial $u \in G_{min}(J(I,P))$, $x_l \nmid u$ for each $l \geq j$. In particular, if $x_1 \not\in P$, then $J(I,P) = S$. Let $B = \{1, \cdots, j-1\}$. It will suffice to show that for every $u \in I$, there exists a $t \geq 0$, such that $u(B)x_j^t \in I$. But this is easy to check whenever $I$ is of Borel type. [3mm]{}Note that the above conclusion is still true when $I$ is Borel-fixed, strongly stable, lexsegment, or universal lexsegment. [3mm]{}By Proposition \[Borel evaluation\] and Corollary \[disappear large variety\], the following proposition can be checked directly, so we omit part of the proof. \[Borel transposition\] Let $P$ be a monomial prime ideal. If $I$ is of Borel type (strongly stable, Borel-fixed, lexsegment, or universal lexsegment respectively), then $J(I,P)$ is of Borel type (strongly stable, Borel-fixed, lexsegment, or universal lexsegment respectively). We only prove the case when $I$ is of Borel type. Let $P = \langle x_{i_1}, \cdots, x_{i_k}\rangle$ with $x_j\notin P$ and $x_t \in P $ for every $t < j$. Denote $X_P=\{t \,|\, x_t \in P\}$ and $B=\{1, \cdots, j-1\}$. Clearly, $B \subseteq X_P$. For a monomial $u \in G_{min}(J(I,P))$, by Corollary \[disappear large variety\] and the definition of $J(I, P)$, there exists a monomial $w \in S\setminus P$, such that $uw \in I$ and $(uw)(B)=u$. For every pair of $m<l$, if $x_l | u$, then there exists $a \geq 0$ such that $x_m^a(uw/x_l) \in I$, since $I$ is of Borel type. Let $y=x_m^a(uw/x_l)$ and note that $$y(X_P)=y(B)=x_m^a(u/x_l),$$ hence $x_m^a(u/x_l) \in J(I,P)$. [3mm]{}Note that for a universal lexsegment ideal $I$, $depth(S/I)=n-|G_{min}(I)|$, see [@MH]. By Proposition \[Borel transposition\], $J(I, P)$ is also universal lexsegment. In order to consider the depth of $S/{J(I,P)}$, we need $J(I, P)$ to be a proper ideal of $S$. \[prime inclusion\] For a monomial ideal $I$ and a monomial prime ideal $P$ of $S$, $I {\subseteq}P$ holds if and only if $I(X_P)$ generates a proper ideal of $S$, i.e., $J(I,P)$ is a proper ideal of $S$. If $P=\langle x_{i_1}, \cdots, x_{i_k}\rangle$ and is prime over $I$, then for each monomial $u \in I \subseteq P$, $u(X_P) \neq 1$, hence $\langle I(X_P)\rangle \neq S$. On the other hand, if a prime ideal $Q$ does not contain $I$, then there exists a monomial $v\in I \setminus Q$, such that $x_j \nmid v$ for every $j \in X_Q$. Thus $v(X_Q) = 1$, and hence $\langle I(X_Q)\rangle = S$. This completes the proof. [3mm]{}By Proposition \[Borel evaluation\], Proposition \[Borel transposition\] and Lemma \[prime inclusion\], the following corollary is direct to check, so we omit the proof. \[depth\] Let $I$ be a monomial ideal, and $P$ a monomial prime ideal containing $I$. If further $I$ is universal lexsegment, then $depth(S/I) \leq depth(S/{J(I,P)})$ holds. Furthermore, the identity holds true if and only if $\{ x_1, \ldots, x_{|G_{min}(I)|}\} {\subseteq}P$. [3mm]{}Now we are ready to prove the afore mentioned result: \[SymbolicPower\] Let $I$ be a monomial ideal of $S$. If $I$ is strongly stable (Borel-fixed, or of Borel type, respectively), then the $k^{th}$ symbolic power $I^{(k)}$ is also a monomial ideal which is strongly stable (Borel-fixed, or of Borel type, respectively). First we claim that $I^{(k)}$ is a monomial ideal. This can follow from the primary decomposition theorem (see e.g., [@Eisenbud Theorem 3.10]), together with [@HH Theorem 1.3.1 and Proposition 1.3.7]. For any $P\in Min(I)$, note also that $$ker(R\to (R/I^k)_P )= J(I^k,P),$$ thus gives a direct proof to the fact. Note that $I^k$ is strongly stable (Borel-fixed, or of Borel type, respectively), if $I$ is strongly stable (Borel-fixed, or of Borel type, respectively). Hence for every $P\in Min(I)$, $ker(R\to (R/I^k)_P) = J(I^k,P)$ implies that it is strongly stable (Borel-fixed, or of Borel type, respectively) by Proposition \[Borel transposition\]. Thus $$I^{(k)}=\cap_{P\in Min(I)}ker(R\to (R/I^k)_P)$$ is strongly stable (Borel-fixed, or of Borel type, respectively) by Proposition \[operation\]. [3mm]{} We remark that for a lexsegment ideal $I$, $I^{(k)}$ may be not lexsegment. \[u not p\] Let $S= K[x_1, x_2, x_3]$, and let $I= \langle x_1, x_2 \rangle$. Clearly, $I$ is universal lexsegment. But $I^{(2)} = I^2= \langle x_1^2, x_1x_2, x_2^2 \rangle$ is not lexsegment. [3mm]{} In the following, we will show that there exists a class of ideals whose symbolic powers are lexsegment. \[power lexsegment\] A monomial ideal $I$ is called stably lexsegment ideal, if $I^k$ is lexsegment for each $k > 0$. [3mm]{} Example \[u not p\] also shows that a universal lexsegment ideal may be not stably lexsegment. The following proposition shows that a stably lexsegment ideal may be not universal lexsegment. We omit the proof. \[two invariants p l\] Let $S = K[x_1, x_2]$. If $I$ is a lexsegment ideal of $S$, then $I$ is stably lexsegment. Even though $I$ being lexsegment does not imply $I^{(k)}$ being lexsegment, we have the following conclusion. \[symbolic lexsegment\] If $I$ is a stably lexsegment ideal of $S$, then $I^{(k)}$ is lexsegment for each positive integer $k$. If $I$ is stably lexsegment, then $I^k$ is lexsegment for each positive integer $k$. By Proposition \[Borel transposition\], for each monomial prime ideal $P$, $J(I^k, P)$ is lexsegment. By Proposition \[operation\], $I^{(k)}=\cap_{P\in Min(I)}J(I^k, P)$ is lexsegment. [3mm]{} We end this section with a general result on $I^{(k)}$ for an ideal $I$ in a Noetherian ring, and will improve the result for monomial ideals in section 4. \[SymbolicPower2\] Let $I$ be any ideal of a noetherian ring $R$. For each $P\in Min(I)$, let $Q(P)$ be the primary component of the isolated prime ideal $P$ of $I$. Then $(1)$ $\underset{P\in Min(I)}{\cap}Q(P)^k{\subseteq}I^{(k)}{\subseteq}\underset{P\in Min(I)}{\cap} Q(P)^{(k)}$ holds true. $(2)$ If $Q(P)^k$ is primary for each $P\in Min(I)$, then $I^{(k)}=\underset{P\in Min(I)}{\cap}Q(P)^k.$ $(3)$ For any positive integer $k$, $I^{(k)}=I^k$ if and only if $Ass(I^k){\subseteq}Ass(I)$ holds. \(1) Let $I=\underset{P\in Ass(I)}{\cap}Q(P)$ be any irredundant primary decomposition of $I$. For any $P\in Min(I)$ and any $P_1\in Ass(I)\setminus\{P\}$, there exists an element $u\in P_1\setminus P$. Thus $u^m\in Q(P_1)\setminus P$ holds for some $m\ge 1$. Thus $Q(P_1)S_P=S_P$ and hence $IS_P=Q(P)S_P$. Then $I^kS_P=Q(P)^kS_P$ and hence $Q(P)^k{\subseteq}Ker(R\to (R/I^k)_P)$. This shows $$\underset{P\in Min(I)}{\cap}Q(P)^k{\subseteq}I^{(k)}.$$ On the other hand, for any $r\in Ker(R\to (R/I^k)_P)$, there exists an element $s\not\in P$ such that $sr\in I^k$. Then $sr\in Q(P)^k {\subseteq}Q(P)^{(k)}$. Since $Q(P)^{(k)}$ is $P$-primary, it follows that $r\in Q(P)^{(k)}$. This shows $ I^{(k)}{\subseteq}\underset{P\in Min(I)}{\cap} Q(P)^{(k)}.$ \(2) This follows from (1). \(3) Consider an irredundant primary decomposition $$I^k=\underset{P\in Ass(I^k)}{\cap}Q(P)$$ of $I^k.$ Since $Min(I)=Min(I^k){\subseteq}Ass(I^k)$ always holds and $$I^{(k)}=\underset{P\in Min(I^k)}{\cap}Q(P),$$ it follows that $I^k=I^{(k)}$ holds if and only if $Min(I^k)=Ass(I^k)$, and the latter holds if and only if $Ass(I^k){\subseteq}Ass(I)$. [3mm]{}We remark that $Min(Q(P)^{k})=\{P\}$ and in fact $$Q(P)^{(k)}=ker[R\to (R/Q(P)^k)_P],$$ see [@Zariski Theorem 23, page 232]. \[Squarefree\] ([@HH Proposition 1.4.4]) Let $I$ be a squarefree monomial ideal of a polynomial ring $K[x_1,\ldots,x_n]$ over a field $K$. Then for any $k$, $$I^{(k)}=\underset{P\in Min(I)}{\cap}P^k.$$ For a squarefree monomials ideal $I$, $Ass(I)=Min(I)$ and for each $P\in Min(I)$, we have $Q(P)=P=\langle x_{i_1},\ldots, x_{i_r}\rangle$. Then it is easy to verify that $\langle x_{i_1},\ldots, x_{i_r}\rangle^k$ is $P$-primary, and the result follows from Proposition \[SymbolicPower2\](2). A simplicial complex and a decomposition of $I^{(k)}$ for a monomial ideal $I$ ============================================================================== In this section, we will use notations established before to improve Proposition \[SymbolicPower2\] (1) for monomial ideals. In doing so, we will define and study a new simplicial complex. [3mm]{} For a subset $A$ of $[n]$, $\langle I(A)\rangle = S$ if and only if $1 \in G_{min}(I)(A)$, and the latter holds if and only if there exists a monomial $x^{{\alpha}}=x_1^{a_1}\cdots x_n^{a_n}\in G_{min}(I)$ in which $a_i\neq 0$ implies $i\notin A$. Thus if $B\subseteq A$ and $\langle I(A)\rangle = S$, then clearly $\langle I(B)\rangle =S$ also holds. \[eliminating simplicial complex\] For any monomial ideal $I$ of $S$, there is the following simplicial complex $$_I\bigtriangleup = \{A\subseteq [n] \,|\, \langle I(A)\rangle = S\}.$$ It will be called [*the eliminating simplicial complex of $I$*]{}. We remark that a simplicial complex on $[n]$ usually contains all the singletons, but we do not assume this condition. By Lemma \[prime inclusion\], it is easy to prove the following proposition. \[minimal primes\] If $I$ is a monomial ideal of $S$, then $Min(I) = \{P_B \,|\, B \in \mathcal{N}(_I\bigtriangleup)\}$, where $\mathcal{N}(_I\bigtriangleup)$ consists of the minimal nonfaces of $_I\bigtriangleup$ and $P_B=\langle \{x_i\,|\,i\in B\} \rangle$. In the following, we will consider about the radical ideal $\sqrt{I}$ of $I$. Note that for a monomial ideal $u$ and a subset $B$ of $[n]$, $u(B) = 1$ if and only if $\sqrt{u}(B)=1$. So Lemma \[prime inclusion\] implies the following well known property. \[radical min\] If $I$ is a monomial ideal of $S$, then $_{\sqrt{I}}\bigtriangleup = _I \bigtriangleup$. In particular, $Min(\sqrt{I}) = Min(I)$. Let $J$ be a monomial ideal of $S$. Recall from [@Villareal; @HH] that the Stanley-Reisner ideal of the simplicial complex $_J\bigtriangleup$ is the ideal $I_{_J\bigtriangleup}$ of $S$, which is generated by the squarefree monomials $x_B= \prod_{i\in B} x_i$ with $B \notin\, _J\bigtriangleup$. The Alexander dual of $_J\bigtriangleup$, denoted by $_J\bigtriangleup^{\vee}$, is defined by $_J\bigtriangleup^{\vee} = \{[n]\setminus B \,|\, B\notin _J\bigtriangleup\}$. It is easy to see that for a subset $B$ of $[n]$, $B \in \mathcal{N}(_J\bigtriangleup)$ if and only if $[n]\setminus B \in \mathcal{F}(_J\bigtriangleup^{\vee})$. We have the following observation. \[Alexander dual\] If $J$ is a monomial ideal of $S$, then $I_{_J\bigtriangleup^{\vee}} = \sqrt{J}$. In particular, if $J$ is a squarefree monomial ideal, then $I_{_J\bigtriangleup^{\vee}} = J$. Note that $\sqrt{J}$ is squarefree, so $\sqrt{J}=\cap_{P\in Min(\sqrt{J})} P = \cap_{P\in Min(J)} P$ by Corollary \[radical min\]. By Proposition \[minimal primes\], $Min(J) = \{P_B \,|\, B \in \mathcal{N}(_J\bigtriangleup)\}$. Hence the standard primary decomposition of $I_{_J\bigtriangleup^{\vee}}$ is $$I_{_J\bigtriangleup^{\vee}}=\underset{B\in \mathcal{F}(_J\bigtriangleup^{\vee})}{\cap} P_{[n]\setminus B}=\underset{A\in \mathcal{N}(_J\bigtriangleup)}{\cap} P_{A}=\underset{P\in Min(J)}{\cap} P = \sqrt{J}.$$ Note that $\sqrt{J} = J$ while $J$ is squarefree, so the second part is clear. [3mm]{}For a monomial $u=x^{\alpha}$ with $\alpha = (a_1 , \cdots , a_n)$, denote $A(u)=\{i \,|\, a_i \neq 0\}$. In the following, we will show that the inclusions appeared in Proposition \[SymbolicPower2\](1) are actually equalities for a monomial ideal $I$ of $S$. For this purpose, we need the following Lemmas. \[evaluation identity\] Let $I$ be a monomial ideal of $S$, and $B$ a subset of $[n]$. If $A(u) {\subseteq}B {\subseteq}[n]$ holds for each $u \in G_{min}(I)$, then $J(I,P_B) = I$. By Proposition \[Borel evaluation\], $J(I,P_B) = \langle I(B)\rangle$ holds. Note also that for each $u \in G_{min}(I)$, $A(u) {\subseteq}B$ holds by assumption, hence $u(B) = u$ holds for every $u \in G_{min}(I)$. This is equivalent to saying that $J(I, P_B) = I$. [3mm]{} In the following lemma, let $G^k=\{\prod_{i=1}^ku_i\,|\,u_i\in G\}$ and set $G^k(B)=(G^k)(B)$. \[power evaluation identity\] Let $G$ be a set of monomials of $S$, and Let $B$ be a subset of $[n]$. Then for any positive integer $k$, the identity $\langle G(B)\rangle^k = \langle G^k(B)\rangle$ holds. In particular, $\langle G^k(B)\rangle =S$ holds if and only if $\langle G(B)\rangle =S$. Clearly, we only need to prove the first statement. By definition, both $\langle G(B)\rangle^k$ and $\langle G^k(B)\rangle$ are monomial ideals of $S$. On the one hand, for every monomial $u \in \langle G(B)\rangle^k$, there exists $u_1, \cdots, u_k \in G$, such that $\prod_{i=1}^k u_i(B) \,|\, u$. Note that $$\prod_{i=1}^k u_i(B) = (\prod_{i=1}^k u_i)(B),\quad \prod_{i=1}^k u_i \in G^k,$$ hold, hence $u\in \langle G^k(B)\rangle$. This shows $\langle G(B)\rangle^k {\subseteq}\langle G^k(B)\rangle$. The other inclusion follows from a similar argument. [3mm]{} The following result improved Proposition \[SymbolicPower2\] (1) for a monomial ideal $I$ of $S$. It also follows from [@HTT Lemma 3.1]. Below we include a direct and detailed proof. \[SymbolicPower3\] If $I$ is a monomial ideal of $S$, then for any positive integer $k$, the $k^{th}$ symbolic power is $$I^{(k)}=\underset{B\in \mathcal{N}(_I\bigtriangleup)}{\cap}J(I^k,P_B).$$ Furthermore, $J(I,P_B)^k= J(I,P_B)^{(k)}$ holds for each $B\in \mathcal{N}(_I\bigtriangleup)$ and thus $$I^{(k)} = \underset{B\in \mathcal{N}(_I\bigtriangleup)}{\cap}J(I,P_B)^k = \underset{P\in Min(I)}{\cap}[ker(S\to (S/I)_P)]^k$$ holds. The first equality follows from Proposition \[minimal primes\] and the definition of $I^{(k)}$. For the remaining equalities, use Proposition \[minimal primes\] again to have $$Min(I) = \{P_B \,|\, B \in \mathcal{N}(_I\bigtriangleup)\}.$$ So, the $k^{th}$ symbolic power of $I$ is nothing but $$I^{(k)}= \underset{B\in \mathcal{N}(_I\bigtriangleup)}{\cap}J(I^k,P_B).$$ Note that both $I=\langle G(I)\rangle$ and $I^k=\langle \{G(I)\}^k\rangle$ clearly holds, so that $$J(I^k,P_B) = \langle \{G(I)\}^k(B)\rangle = \langle \{G(I)(B)\}^k\rangle = J(I, P_B)^k$$ also holds by Lemma \[power evaluation identity\]. This shows $$I^{(k)} = \underset{B\in \mathcal{N}(_I\bigtriangleup)}{\cap}J(I,P_B)^k .$$ For the remaining statement, note that $J(I,P_B)$ is $P_B$-primary with $Min(J(I,P_B))=\{P_B\}.$ Hence $$J(I,P_B)^{(k)} = \underset{Q\in Min(J(I,P_B))}{\cap}J(J(I,P_B)^k,Q) = J(J(I,P_B)^k,P_B)$$ holds. Finally, note that $A(u) {\subseteq}B$ holds for every $u\in G_{min}J(I,P_B)$, it follows that $J(J(I,P_B)^k,P_B)$ $ = J(I,P_B)^k$, and hence $J(I,P_B)^{(k)} = J(I,P_B)^k$ holds. This completes the proof. [3mm]{} In the end, we include an example to illustrate Theorem \[SymbolicPower3\]: \[identity example\] Let $I = \langle x_1^2x_3^2,\, x_1x_2x_3^2\rangle$. The irredundant primary decomposition of $I$ is $I = \langle x_1^2, x_2\rangle \cap \langle x_1\rangle \cap \langle x_3^2\rangle$. Hence $Ass(I) = \{\langle x_1\rangle, \langle x_1, x_2\rangle, \langle x_3\rangle \}$ and $Min(I) = \{\langle x_1\rangle, \, \langle x_3\rangle \}$. It is easy to check that for each $k\geq 1$, $$I^k = \langle \{x_1^{k+i}x_2^{k-i}x_3^{2k} \,|\, i=0,1, \ldots, k \}\rangle = \langle x_1^k\rangle \cap \langle x_3^{2k}\rangle \cap (\cap_{i=1}^k \langle x_1^{k+i}, x_2^{k-i+1}\rangle ).$$ Hence $$I^{(k)} = \langle x_1^{k}\rangle \cap \langle x_3^{2k}\rangle = \langle x_1\rangle^k \cap \langle x_3^2\rangle^k = \underset{P\in Min(I)}{\cap}J(I,P)^k.$$ It is also clear that $$\underset{P\in Min(I)}{\cap}J(I,P)^{(k)} = \langle x_1\rangle^{(k)} \cap \langle x_3^2\rangle^{(k)} = \langle x_1\rangle^k \cap \langle x_3^2\rangle^k = I^{(k)}.$$ Polarization of universal lexsegment monomial ideal =================================================== Let $I$ be a monomial ideal of $S$, and let $<$ be a monomial order on $S= K[x_1,\ldots, x_n]$, such that $x_n < x_{n-1} < \cdots < x_1$. Let $G_{min}(I) = \{u_1, \ldots, u_m\}$ be the minimal generating set of $I$, where $u_i = \prod_{j=1}^n x_j^{a_{ij}}$ for $i=1, \ldots, m$. Let $a_i = max\{a_{1i}, \ldots, a_{mi}\}$. Recall that the polarization of $I$ is a squarefree monomial ideal $T(I) = \langle v_1, \ldots, v_m\rangle$, where $$v_i = \prod_{j=1}^n \prod_{k=1}^{a_{ij}} x_{jk}$$ for $i=1, \ldots, m$. Let $\prec$ be a monomial order on $$T = K[x_{11}, \ldots, x_{1 a_1}, \ldots, \ldots, x_{n 1}, \ldots, x_{n a_n}]$$ such that $x_{ij} \prec x_{kl}$ if $i > k$ or if $i=k$, $j>l$. If choose $<$ and $\prec$ to be the same kind of monomial order(e.g., lexicographic order, pure lexicographic order or reverse lexicographic order), which satisfies the above convention, then the polarizing process is order-preserving, i.e., for each pair of $u, v \in I$, $u<v$ if and only if $T(u) \prec T(v)$. Polarization is a powerful tool for studying quite a few important homological and combinatorial invariants, see, e.g., [@HH Corollary 1.6.3]. But unfortunately, the property of being Borel type can not be kept in almost all cases after the process of polarization, as the following example shows: \[polarization\] Consider the strongly stable monomial ideal $I=\langle x_1^3,x_1^2x_2,x_1x_2^2\rangle$. After polarization, it becomes $$J=\langle x_{11}x_{12}x_{13}, \, x_{11}x_{12}x_{21},\, x_{11}x_{21}x_{22}\rangle.$$ Since $J$ is monomial and homogeneous, it is clear that $J$ is not a Borel type monomial ideal under any monomial order. \[Four equal\] Let $I$ be a monomial ideal with $G_{min}(I)= \{u_1, \ldots, u_m\}$, where $u_i = \prod_{j=1}^n x_j^{a_{ij}}$ and $u_1 > u_2 > \cdots > u_m$ by pure lexicographic order. Let $$a_j= max\{a_{1j}, \ldots, a_{mj}\}$$ for $j=1, \ldots, n$. Then the following statement are equivalent. $(1)$ $I$ is universal lexsegment; $(2)$ $u_i = x_i^{a_i}\prod_{j=1}^{i-1} x_j^{a_j-1}$ for $i=1, \ldots, m$; $(3)$ $x_i(u/x_{m(u)}^{b_{m(u)}}) \in I$ holds for each $u= \prod_{j=1}^n x_j^{b_{j}} \in I$ and each $i< m(u)$; $(4)$ For any monomial $u= \prod_{j=1}^n x_j^{b_{j}} \in I$, $x_i(u/x_j^{b_{j}}) \in I$ holds for each pair $0\leq i<j \leq n$ with $b_j > 0$. [3mm]{} $(1) \Leftrightarrow (2)$ is well known (see [@MH]), and $(2) \Rightarrow (3)$ is clear. $(3) \Rightarrow (4)$: For any monomial $u= \prod_{k=1}^n x_k^{b_{k}} \in I$ and each pair $0\leq i<j \leq n$ with $b_j > 0$, $x_{m(u)-1}\prod_{k=1}^{m(u)-1} x_k^{b_{k}}=x_{m(u)-1}(u/x_{m(u)}^{b_{m(u)}}) \in I$ by $(3)$. By induction, $x_{j}\prod_{k=1}^{j} x_k^{b_{k}} \in I$. By $(3)$ again, $x_{i}\prod_{k=1}^{j-1} x_k^{b_{k}} \in I$. Note that $x_{i}\prod_{k=1}^{j-1} x_k^{b_{k}} \,|\, x_i(u/x_j^{b_{j}})$, $x_i(u/x_j^{b_{j}}) \in I$. $(4) \Rightarrow (2)$: It will suffice to show that for each $1\leq k\leq m$, $u_k = x_k^{a_k}\prod_{j=1}^{k-1} x_j^{a_j-1}$ and $a_{ik} = a_k-1$ for each $k+1 \leq i \leq m$. We will prove it by induction. It is clear that $u_1 = x_1^{a_1}$ under the condition $(4)$, and it is easy to see that $a_i=0$ implies that $a_j=0$ for each $j>i$. Note that $u_1 > u_2 > \cdots > u_m$ by pure lexicographic order. Hence we can assume $u_2 = x_1^{a_{21}}x_2^{a_{22}}$ with $a_{21}<a_1$ and $a_{22}>0$. Note that $x_1^{a_{21}+1} \in I$, so $a_{21}=a_1-1$. In a similar way, we get $a_{i1} = a_1-1$ for every $i = 2, \ldots, m$. Now assume that the conclusion holds true for all the integers less than $k$, and we are going to show that $u_{k} = x_{k}^{a_{k}}\prod_{j=1}^{k-1} x_j^{a_j-1}$ and $a_{ik} = a_k-1$ holds for each $k+1 \leq i \leq m$. By inductive assumption, $$u_i=(\prod_{j=1}^{k-1} x_j^{a_j-1})(\prod_{j=k}^{m} x_j^{a_{ij}})$$ holds for each $k \leq i \leq m$. If assume to the contrary that $a_{ik} < a_k-1$ holds for some $i$, then $w= x_k^{a_{ik}+1} (\prod_{j=1}^{k-1} x_j^{a_j-1}) \in I$. By the definition of $a_k$, there exists an integer $t \geq k$ such that $x_k^{a_k} \mid u_t$ holds. Hence $w $ properly divides $u_t$, contradicting $u_t \in G_{min}(I)$. Hence either $a_{ik} = a_k-1$ or $a_{ik} = a_k$ holds for each $k\leq i\leq m$. Note that $u_k > u_{k+1} > \cdots > u_m$, it is easy to check $u_k = x_k^{a_k}\prod_{j=1}^{k-1} x_j^{a_j-1}$, and that $a_{ik} = a_k-1$ holds for each $k+1 \leq i \leq m$. This completes the verification. \[universal\] Note that $(3)$, as an equivalent description of universal lexsegment ideal, explores the difference between universal lexsegment ideal, strongly stable ideal and the monomial ideal of Borel type. Actually, a universal lexsegment monomial ideal is a kind of [*super-stable*]{} monomial ideal. If $I$ is a universal lexsegment ideal of $S= K[x_1,\ldots, x_n]$ with the minimal generating set $G_{min}(I)$, it is clear that $|G_{min}(I)| \leq n$. We call a universal lexsegment ideal to be [*full*]{}, if $|G_{min}(I)| = n$. In the following, we will consider about the polarization of some class of monomial ideals with respect to $<$, and characterize the monomial ideals which become squarefree strongly stable with respect to $\prec$ after polarization. Recall that a squarefree monomial ideal $I$ is called [*squarefree strongly stable*]{}, if for each squarefree monomial $u\in I$ and each pair $j<i$ such that $x_i\mid u$ but $x_j\nmid u$, one has $x_j(u/x_i)\in I$ (see [@AHH] or [@HH page 124]). \[super-stable form\] Let $I$ be a monomial ideal with $G_{min}(I)= \{u_1, \ldots, u_m\}$, where $u_i = \prod_{j=1}^n x_j^{a_{ij}}$ and $u_1 > u_2 > \cdots > u_m$ by pure lexicographic order. Let $$a_j= max\{a_{1j}, \ldots, a_{mj}\}$$ for $j=1, \ldots, n$. Then $T(I)$ is squarefree strongly stable, if $I$ is universal lexsegment. Further more, if $a_j \neq 1$ holds for each $j=1, \ldots, n$, and for each $1\leq j < m$, there exists an integer $i$ such that $0 < a_{ij} < a_j$, then $T(I)$ is squarefree strongly stable if and only if $I$ is universal lexsegment. [3mm]{} If $I$ is universal lexsegment, then $u_k = x_k^{a_k}\prod_{j=1}^{k-1} x_j^{a_j-1}$ for $k = 1, \ldots, m$. Hence $$T(u_k) = (\prod_{i=1}^{k-1} \prod_{j=1}^{a_j-1} x_{ij})(\prod_{j=1}^{a_k} x_{kj}).$$ In the case, it is direct to check that $T(I)$ is squarefree strongly stable. This complete the proof of the first statement. For the second statement, we only need to prove the necessity part. It is easy to see that $u_1 = x_1^{a_1}$. We claim that $a_{i1} =a_1-1$ for every $i = 2, \ldots, m$. In fact, if there exists some $u_t$ such that $a_{t1}<a_1-1$, then consider $$T(u_t) = (\prod_{j=1}^{a_{t1}} x_{1j})(\prod_{i=2}^{n} \prod_{j=1}^{a_{ti}} x_{ij}) \in T(I).$$ Since $T(I)$ is squarefree strongly stable, it follows that $v = x_{1,a_{t1}+1}(T(u_t)/{x_{21}}) \in T(I)$ holds. Note that for each generating element $u$ of $T(I)$, if $x_{ij} | u$, then $x_{i,j-1} | u$, …, $x_{i,1} | u$ hold. Hence $v \in T(I)$ implies $v_1=x_{1,a_{t1}+1}(T(u_t)/{\prod_{j=1}^{a_{t2}} x_{2j}}) \in T(I)$. Note that for any $j$, $x_{2j} \nmid v_1$, thus $x_{2,a_2}(v_1/{x_{31}}) \in T(I)$ whenever $x_{31} \,|\, v_1$. Repeat the discussion above, it follows that $x_{2,a_2}(v_1/{\prod_{j=1}^{a_{t3}} x_{3j}}) \in T(I)$, and hence $v_1/{\prod_{j=1}^{a_{t3}} x_{3j}} \in T(I)$ holds since $a_2 > 1$. By induction, we have $\prod_{j=1}^{a_{t1}+1} x_{1j} \in T(I)$, contradicting $T(u_1) \in G_{min}(T(I))$. Hence $a_{t1} = a_1 - 1$ holds for each $t=2, \ldots, m$. Finally, repeat a discussion used in proving $(4) \Rightarrow (2)$ of theorem \[Four equal\], the result follows by induction. [3mm]{}Now assume that $a_j \neq 1$ holds for each $j=1, \ldots, n$. If there exists some $j$ with $a_j > 1$, such that either $a_{ij}=0$ or $a_{ij}=a_{j}$ for each $i=1, \ldots, m$, then the situation will be a little more complicated than the case in Theorem \[super-stable form\]. In this case, let $W_I = \{j \,|\, a_j \neq 0\}$, and let $A_I=\{j \in W_I \,|\,$ either $\,a_{ij}=a_j \,$ or $\, a_{ij}=0 \,$ for each $i=1, \ldots, m\}$, and let $B_I = W_I \setminus A_I$. Note that if $T(I)$ is squarefree strongly stable, then $W_I= [r]$ holds for some $r\leq n$. In this case, we can decompose $A_I$ into several mutually disjoint subsets consisting of successive integers, and denote $A_I = \cup_{t=1}^k A_t$ where $$A_t = \{j \in Z_{+} \,|\, m_{t-1}+1 \leq j\leq l_t\}.$$ Similarly, let $B_I = \cup_{t=1}^k B_t$, where $B_t = \{j \in Z_{+} \,|\, l_{t}+1 \leq j\leq m_t\}.$ For each $j\in A_I$, either $a_{ij}=0$ or $a_{ij}=a_j$ holds for any $i=1, \ldots, m$. Consider the following two cases: If $a_{ij}=0$, we claim that it implies $a_{i,j+1}=a_{i,j+2}=\cdots = a_{in}=0$. In fact, if $a_{it} > 0$ for some $t>j$, then $T(u_i)\in T(I)$ holds with $x_{t1} \mid T(u_i)$. Since $T(I)$ is squarefree strongly stable, $x_{j,a_j}(T(u_i)/x_{t1}) \in T(I)$ holds. Note also that $a_j >0$ holds, thus it implies $T(u_i)/x_{t1} \in T(I)$, contradicting $u_i \in G_{min}(I)$. On the other hand, if $a_{ij}=a_j$ holds, then the polarization of $u_i$ contains $\prod_{t=1}^{a_j} x_{jt}$ as its factor, which contains all the indeterminants related to $x_j$. Note that in each case, essentially there is little change in the problem we are working with. Due to this reason, the following proposition is routine to check, and we omit the detailed proof. \[two mixed\] If $a_j \neq 1$ for each $j=1, \ldots, n$, then $T(I)$ is squarefree strongly stable if and only if $I = \sum_{i=1}^{k} ( L_iM_i \prod_{j=1}^{i-1} (L_j M_j^{'})),$ where $$L_j= \langle \prod_{t=m_{j-1}+1}^{l_{j}} x_t^{a_t} \rangle,\,\, M_j^{'}= \langle \prod_{t=l_{j}+1}^{m_{j}} x_t^{a_t-1} \rangle,$$ for every $j=1, \ldots, k$, and $M_j$ is a monomial ideal of $S$ generated by a full universal lexsegment ideal in the polynomial ring $K[x_{l_j+1}, x_{l_j+2}, \ldots, x_{m_j}]$, where $$0=m_0 < l_1 \leq m_1 < l_2 \leq m_2 < \cdots < l_k \leq m_k \leq n.$$ \[explain\] In Proposition \[two mixed\], the equality $$I = \sum_{i=1}^{k} ( L_iM_i \prod_{j=1}^{i-1} (L_j M_j^{'}))$$ can be interpreted in the following: A universal lexsegment ideal on $B_I$ is cut into several parts by some principal ideals with respect to $A_I$. Note also that the equivalence description would be rather complicated without the assumption $a_j \neq 1$ for each $j=1, \ldots, n$, because it will be some universal lexsegment ideals, squarefree strongly stable ideals being cut into several parts by some principal ideals. \[universal lexsegment\] Let $$I= \langle x_1^3, x_1^2x_2x_3, x_1^2x_2x_4, x_1^2x_2x_5^3x_6^3, x_1^2x_2x_5^3x_6^2x_7^2, x_1^2x_2x_5^3x_6^2x_7x_8^2 \rangle.$$ It is easy to see that $T(I)$ is squarefree strongly stable. By computation, $$a_1=3, a_2=1, a_3=1, a_4=1, a_5=3, a_6=3, a_7=2, a_8=2.$$ An explanation is that $T(I)$ is going to be divided approximately into several parts: for $x_1$, it is universal lexsegment; for $x_2,x_3,x_4$, it is squarefree strongly stable; for $x_5$, it is a principal ideal; for $x_6, x_7, x_8$, again it is universal lexsegment. \[exponent vector\] Let $I$ be a monomial ideal, and let $G_{min}(I) = \{u_1, \ldots, u_m\}$ with $u_i = \prod_{j=1}^n x_j^{a_{ij}}$ for $i=1, \ldots, m$. The vector $(a_1, a_2, \ldots, a_n)$ is called the [*exponent vector*]{} of $I$, where $a_i = max\{a_{1i}, \ldots, a_{mi}\}$. For a squarefree monomial ideal $$J \subseteq K[x_{11}, \ldots, x_{1 \tau_1}, \ldots, \ldots, x_{n 1}, \ldots, x_{n \tau_n}],$$ the vector $(b_1, b_2, \ldots, b_n)$ is called the [*extension vector*]{} of $J$, where $$b_i = max\{j \,|\, \, \exists \,u \in G_{min}(J)\,\, such \,\,that \,\,x_{ij} \mid u\}.$$ By definition, the following Proposition is clear. \[exponent extension equal\] For any monomial ideal $I$, The exponent vector of $I$ is equal to the extension vector of $T(I)$. \[Borel ideals number\] Let $J$ be a polarization of some monomial ideal of $S$, with the extension vector $(b_1, \ldots, b_n)$. If $J$ is squarefree strongly stable and $b_i \neq 1$ for each $i=1, \ldots, n$, then the number of monomial $I$ such that $T(I) = J$ is $2^t$, where $t = |W_I|$. By Proposition \[two mixed\], $I$ is uniquely determined by the exponent vector $(a_1, \ldots, a_n)$ and the set $A_I$. By Proposition \[exponent extension equal\], $(a_1, \ldots, a_n) = (b_1, \ldots, b_n)$ is a constant vector. Hence $I$ is completely determined by the set $A_I$. Note that $A_I$ could be any subset of $W_I$, so the number of $I$ such that $T(I) = J$ is $2^t$, where $t = |W_I|$. [gg]{} M.F. Atiyah and I.G. MacDonald, [*Introduction to Commutative Algebra*]{}, Addison-Wesley, Reading, MA, $1969$. A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z. $228 (1998)$, $353-378.$ D. Bayer and M.A. Stillman, A criterion for detecting m-regularity. Invent. Math 87(1987), $1-11$. Mircea Cimpoeas, Some remarks on Borel type ideals, Commu. Algebra $37:2(2009)$, $724-727$. D. Eisenbud, [*Commutative Algebra with a View Toward Algebraic Geometry*]{}. Springer Science + Business Media, Inc, $2004$. J. Herzog and T. Hibi, [*Monomial Ideals*]{}. Springer-Verlag London Limited, $2011$. J. Herzog, D. Popescu and M. Vladoiu, On the Ext-Modules of ideals of Borel type, Contemporary Math. $331 (2003)$, $171-186$. J. Herzog, T. Hibi and N. V. Trung, Symbolic powers of monomial ideals and vertex cover algebras, Advances in Mathematics $210 (2007)$, $304¨C322$. S. Murai and T. Hibi, The depth of an ideal with a given Hilbert function, Proc. of AMS, $136:5(2008)$, $1533-1538$. D. Popescu, Extremal Betti Numbers and regularity of Borel type ideals, Bull. Math. Soc. Sc. Math. Roumanie, $48(96):1(2005)$, $65-72$. I. Swanson and C. Huneke, [*Integral Closures of Ideals, Rings and Modules*]{}. Cambridge University Press, $2006$. R. H. Villarreal, [*Monomial Algebra*]{}. Marcel Dekker, Inc, New York, $2001$. O. Zariski and P. Samuel, [*Commutative Algebra*]{}. Vol.1 Reprints of the $1958-60$ edition. Springer-Verlag New York, $1979$. [^1]: guojinecho@163.com [^2]: Corresponding author. tswu@sjtu.edu.cn [^3]: This research is supported by the National Natural Science Foundation of China (Grant No. 11271250).

--- abstract: 'We have used semi-numerical simulations of reionization to study the behaviour of the power spectrum of the EoR 21-cm signal in redshift space. We have considered two models of reionization, one which has homogeneous recombination (HR) and the other incorporating inhomogeneous recombination (IR). We have estimated the observable quantities — quadrupole and monopole moments of power spectrum at redshift space from our simulated data. We find that the magnitude and nature of the ratio between the quadrupole and monopole moments of the power spectrum ($P^s_2 /P^s_0$) can be a possible probe for the epoch of reionization. We observe that this ratio becomes negative at large scales for $\xb \leq 0.7$ irrespective of the reionization model, which is a direct signature of an inside-out reionization at large scales. It is possible to qualitatively interpret the results of the simulations in terms of the fluctuations in the matter distribution and the fluctuations in the neutral fraction which have power spectra and cross-correlation $P_{\Delta \Delta}(k)$, $P_{xx}(k)$ and $P_{\Delta x}(k)$ respectively. We find that at large scales the fluctuations in matter density and neutral fraction is exactly anti-correlated through all stages of reionization. This provides a simple picture where we are able to qualitatively interpret the behaviour of the redshift space power spectra at large scales with varying $\xb$ entirely in terms of a just two quantities, namely $\xb$ and the ratio $P_{xx}/P_{\Delta \Delta}$. The nature of $P_{\Delta x}$ becomes different for HR and IR scenarios at intermediate and small scales. We further find that it is possible to distinguish between an inside-out and an outside-in reionization scenario from the nature of the ratio $P^s_2 /P^s_0$ at intermediate length scales.' author: - | Suman Majumdar$^{1,2}$[^1], Somnath Bharadwaj$^1$[^2] and T. Roy Choudhury$^{3}$[^3]\ $^1$Department of Physics and Meteorology & Centre for Theoretical Studies, IIT, Kharagpur 721302, India\ $^2$ Department of Astronomy & Oskar Klein Centre, AlbaNova, Stockholm University, SE-106 91 Stockholm, Sweden\ $^3$National Centre for Radio Astrophysics, TIFR, Post Bag 3, Ganeshkhind, Pune 411007, India title: 'The effect of peculiar velocities on the epoch of reionization (EoR) 21-cm signal' --- 1[x\_[H [i]{}]{}]{} 1[P\_[[H ]{}]{}]{} 1[\_[[H ]{}]{}]{} [U]{} ¶ [[R]{}]{}[[R]{}]{} methods: data analysis - cosmology: theory: - diffuse radiation Introduction ============ The epoch when the neutral hydrogen () in the inter-galactic medium (IGM) was reionized by the first luminous sources, is one of the least known periods in the history of our universe. Observations of the CMBR [@spergel03; @page07; @komatsu11; @larson11] and absorption spectra of high redshift quasars [@becker01; @fan03; @white03; @fan06; @willott07; @goto11] suggest that the epoch of reionization (EoR) probably extended over the redshift range $6 \leq z \leq 15$ [@fan06; @choudhury06a; @alvarez06; @mitra11]. However these observations are limited in their ability to shed light on many important questions regarding EoR. What are the major sources of reionization? What are the typical sizes and the topology of the ionized regions at different stages? Observations of redshifted 21-cm radiation from neutral hydrogen hold the promise to answer some of these questions. The brightness temperature of the redshifted 21-cm radiation directly probes the distribution at the epoch where the radiation originated. It is thus possible to track the entire reionization history as it gradually proceeds with redshift. The presently functioning low frequency radio telescopes [GMRT[^4]]{} [@swarup], LOFAR[^5] and [21CMA[^6]]{}, the upcoming [MWA[^7]]{} and the future [SKA[^8]]{} all cover the frequency range relevant for the EoR 21-cm signal, and this is one of the major goals for most of these telescopes. It is therefore very important to have a good picture of the expected signal in order to make forecasts for and correctly interpret the future observations of the redshifted 21-cm radiation. There has been a considerable amount of work towards simulating the expected EoR 21-cm signal. In particular, there have been numerical simulations which use ray-tracing to follow the propagation of ionization fronts in the IGM [@gnedin00; @ciardi01; @ricotti02; @razoumov02; @maselli03; @sokasian03; @iliev06; @mellema06; @mcquinn07; @trac07; @semelin07; @shin08; @iliev08; @shapiro08; @thomas09; @baek09]. Such simulations are computationally extremely expensive, and it is difficult to simulate large volumes, and to re-run the simulations considering different values of the simulation parameters. Semi-numerical simulations which consider the average photon density in place of a detailed ray-tracing analysis provide a computationally less expensive technique to simulate the EoR 21-cm signal [@furlanetto04; @mesinger07; @geil08a; @lidz09; @choudhury09; @alvarez09; @santos10; @mesinger11; @zahn11]. The fluctuations in the brightness temperature of the redshifted 21-cm radiation essentially trace the distribution during EoR. The redshift space distortion caused by peculiar velocities also plays an important role in shaping the redshifted 21-cm signal [@bharadwaj01; @bharadwaj04]. In fact, we expect the peculiar velocities to introduce an anisotropy in the three dimensional power spectrum of the EoR 21-cm signal [@barkana05; @bharadwaj05; @wang06], very similar to the characteristic anisotropy present in the galaxy power spectrum [@kaiser87]. @barkana05 have proposed that it may be possible to use this anisotropy to separate the effect of the peculiar velocities from the other astrophysical information present in the 21-cm power spectrum. Until recently, most simulations of the EoR 21-cm signal have not considered the effect of redshift space distortions. Some of the earlier work @mellema06 and @thomas09 have considered this effect while generating maps through their simulations, but have not studied it’s implication on the statistical properties like the power spectrum of the brightness temperature fluctuations. Recently, @santos10 and @mesinger11 have included the effect of redshift space distortions in an approximate, perturbative fashion in their semi-numerical simulation and used this to study it’s implications on the redshifted 21-cm brightness temperature power spectrum at different stages of the reionization. In a very recent work @mao12 discuss the methodology to implement redshift space distortion in numerical simulations of reionization, and used this to study the 21-cm brightness temperature power spectrum during EoR. Most of the earlier semi-numerical simulations ([*e.g.*]{} @furlanetto04 [@mesinger07; @geil08a; @lidz09; @alvarez09; @santos10; @mesinger11; @zahn11]) have assumed spatially homogeneous recombination which predicts strictly inside-out reionization where the most dense regions ionize first, the ionization subsequently propagating to lower densities. However, there are observations which indicate exactly opposite picture at the end of reionization, where the high density regions remain neutral (due to self-shielding) and the low density regions are highly ionized. @choudhury09 have attempted to make their semi-numerical simulation consistent with these observations by incorporating the fact that recombination occurs faster in high density regions. In these simulations reionization is inside-out only in the early stages. However, self-shielded, high density clumps remain neutral in the later stages of reionization when inhomogeneous recombination is taken into account. In this paper we follow @choudhury09 to develop a semi-numerical code to simulate reionization, with the further improvement that we incorporate the effect of redshift space distortion due to peculiar velocities. We have used these simulations to study the effect of peculiar velocities on the EoR 21-cm signal, both with homogeneous recombination and with inhomogeneous recombination. In this paper we have used semi-numerical simulations to determine the EoR 21-cm signal at different stages of reionization, and used the power spectrum to quantify the statistical properties of this signal. We have calculated $P^r_{\HI}(k)$ the power spectrum in real space and its redshift space counterpart $P^s_{\HI}({\bf k})$, and compared these two to asses the effect of peculiar velocities. The anisotropy of the 21-cm signal, quantified through various angular multipoles of $P^s_{\HI}({\bf k})$, is a very useful tool to study the effect of redshift space distortion. In particular, we have studied the monopole and quadrupole moments of $P^s_{\HI}({\bf k})$ in order to identify the features characteristic of redshift space distortion at different stages of reionization. To our knowledge, this anisotropy has not been quantified using simulations in any of the earlier studies. Finally, we attempt to interpret the results of our simulations, and compare these against the predictions of the simple, linear model proposed by @barkana05. Unless mentioned otherwise, throughout this paper we present results for the cosmological parameters $h= 0.704$, $\Omega_m = 0.272$, $\Omega_{\Lambda} = 0.728$, $\Omega_b h^2 = 0.0226$ (all parameters from WMAP 7 year data [@komatsu11; @jarosik11]). A brief summary of the paper follows. In Section 2. we present the semi-numerical technique that we have used to simulate the EoR 21-cm signal including the effect of peculiar velocities. Section 3. contains a brief discussion of the model prediction for the effect of redshift space distortion. These were used as reference values in presenting and interpreting the results from our simulations in Section 4. Finally, we discuss our results and conclude in Section 5. Simulating redshift space distortion during Reionization {#sec:sim} ======================================================== We have used a semi-numerical simulation to generate the ionization map during reionization. The simulation essentially starts from the dark matter distribution at a given redshift, and uses this to identify the sources of ionizing photons. These sources, along with the assumption that the traces the dark matter are used to construct a snapshot of the ionization distribution. Our simulation is based on the formalism proposed by @choudhury09. This uses an excursion-set formalism as introduced by @furlanetto04. The semi-numerical simulation provides us with the ionization field in the real space [*i.e.*]{} without the redshift space distortion. We briefly discuss the semi-numerical method that we have used in this work to simulate the brightness temperature fluctuations of the redshifted 21-cm emission from EoR. We have used a Particle Mesh N-body code to generate the dark matter distribution. The spatial and mass resolution of the N-body simulation should be adequate to correctly resolve all the ionizing sources that one is going to adopt in this semi-numerical simulation. It is currently believed that the stars residing in the galaxies are the major source of photons to reionize the universe [@yan04; @stiavelli04; @bouwens05; @fan06; @choudhury06]. The presently accepted models suggest that dark matter halos having a mass $M \geq 10^9 \,h^{-1}\, M_{\odot}$ host the early galaxies that contribute to reionization. We thus include all dark matter halos of mass $M \geq 10^9 \,h^{-1}\, M_{\odot}$ in our semi-numerical simulation. Assuming that at least $10$ dark matter particles are required to constitute the smallest halo, the N-body simulation is required to have a mass resolution $\leq 10^8 \,h^{-1}\, M_{\odot}$. We have generated the dark matter distribution at $z = 8$ using the Particle Mesh N-body code. The volume of the simulation is constrained by the $16$ Gigabytes of memory available in our computer. We perform our simulation in a periodic box of size $85.12$ Mpc (comoving) with $1216^3$ grid points and $608^3$ particles, with a mass resolution $M_{part} = 7.275 \times 10^7\,h^{-1}\,M_{\odot}$. We identify halos within the simulation box using a standard Friend-of-Friend algorithm [@davis], with a fixed linking length $0.2$ (in units of mean inter particle distance) and minimum dark matter halo mass $ = 10 M_{part}$. The relation between the ionizing luminosity of a galaxy and its properties is not well known from the observations. In the semi-numerical formalism adopted here, we assume that the ionizing luminosity from a galaxy is proportional to the mass of it’s halo. The number of ionizing photons contributed by a halo of mass $M$ is given by $$N_{\gamma}(M) = N_{ion} \frac{M}{m_H} \label{eq:nion}$$ where $m_H$ is the mass of a hydrogen atom and $N_{ion}$ is a dimensionless constant. The value of $N_{ion}$ is tuned so as to achieve the desired mass-averaged neutral fraction in the simulation. The ionizing photon field is estimated on a grid which has a resolution $8$ times coarser than the N-body simulation. In our simulation we assume that the baryons follow the dark matter distribution and we also assume that each N-body particle has same hydrogen mass $M_H$. In the semi-numerical formalism, a region is said to be ionized if the average number of photons reaching there exceeds the average neutral hydrogen density at that point. Before applying the ionization condition we assume that the entire hydrogen contained in each particle is completely neutral. Using this assumption we calculate the density field in the same grid where the photon field has been generated. The photon and the density at each grid point are compared and a neutral fraction $\xh1$ is assigned to the grid point depending on the ionization conditions discussed below. In this work we consider two different models of reionization. In one model the recombination rate is assumed to be homogeneous (HR) and independent of density throughout the IGM. In the other model we have considered a density dependent inhomogeneous recombination (IR). Readers are requested to refer to the Section 2 of @choudhury09 for further details of ionization conditions (eq. \[7\] and \[15\] of @choudhury09) in these two different models of reionization. For the simulation results related to IR model presented in this paper we have set the inhomogeneous recombination parameter $\epsilon = 1.0$ (see Section 2.5 and eq. \[15\] of @choudhury09 for the definition of $\epsilon$). We next discuss how we implement the effect of peculiar velocities on the ionization maps generated. We first consider the dark matter particles to each of which we have assigned a total hydrogen mass $M_{H}$. The ionization map provides us with a neutral fraction $\xh1$ at each grid point of the simulation. For the $i$th particle in the simulation, we have interpolated the neutral fraction from its eight nearest neighbouring grid points to determine the neutral fraction $x^i_{\HI}$ at the particle’s position. We use this to calculate the particle mass $M^i_{\HI}$ as $$M^i_{\HI} = \xh1^i M_{H}\, .$$ This provides us with the distribution and the peculiar velocity associated with each element. We now consider a distant observer located along the $x$ axis, and use the $x$ component of the peculiar velocity to determine the particle positions in redshift space $$s = x + \frac{v_{x}}{a H(a)} \label{eq:rsd}$$ where $a$ and $H(a)$ are the scale factor and Hubble parameter respectively. Finally we have interpolated the distribution from the particles to the grid, and used this to generate the EoR 21-cm signal. This method of mapping the real space density in redshift space is some what similar to the PPM-RRM scheme discussed in @mao12. Modeling redshift space distortion during reionization {#sec:model} ====================================================== \[c\]\[c\]\[1\]\[0\][$\xb=0.8$]{} \[c\]\[c\]\[1\]\[0\][$\xb=0.7$]{} \[c\]\[c\]\[1\]\[0\][$\xb=0.5$]{} \[c\]\[c\]\[1\]\[0\][$\xb=0.3$]{} \[c\]\[c\]\[1\]\[0\][HR]{} \[c\]\[c\]\[1\]\[0\][HR-RS]{} \[c\]\[c\]\[1\]\[0\][IR]{} \[c\]\[c\]\[1\]\[0\][IR-RS]{} Coherent inflows into overdense regions and outflows from underdense regions appear as enhancements in the matter density fluctuations observed in redshift space. This introduces an anisotropy [@kaiser87] in $P^s (k,\mu)$ the redshift space matter power spectrum $$P^s ({k,\mu}) = \left( 1 + \mu^2 \right)^2 P^r(k) \label{eq:kaiser}$$ where $P^r(k)$ is the real space power spectrum and $\mu={\bf k} \cdot {\bf \hat{n}}/k$ is the cosine of the angle between the wave vector ${\bf k}$ and the unit vector ${\bf n}$ along the line of sight (LoS). Here we have assumed $\Omega_m =1$ throughout, which is reasonable at the high redshifts of our interest. It is convenient [@hamilton92; @hamilton98; @cole95] to decompose the anisotropy using Legendre polynomials $ {\mathcal P}_l(\mu)$ as $$P^s(k,\mu) = \sum_{l \,{\rm even}} {\mathcal P}_l(\mu) P_{l}^s(k)\, ,$$ where $P_{l}^s(k)$ are the different angular multipoles of $P^s(k,\mu)$. Under the linear approximation (eq. \[eq:kaiser\]), only the first three even moments have non-zero values $P_{0}^s(k)/P^r(k) = 28/15 $, $P_{2}^s(k)/P^r(k) = 40/21 $ and $P_{4}^s(k)/P^r(k) = 8/35$ which are constant independent of $k$. Peculiar velocities have a similar effect on the brightness temperature fluctuations $\Delta T_b$ of the 21-cm radiation from the high redshift universe [@bharadwaj01; @bharadwaj04]. Expressing the brightness temperature fluctuations as $\Delta T_b({\bf x}, z)= \bar{T}(z) \, \eta_{\HI}({\bf x}, z)$ and considering the EoR where the spin temperature is much higher than the CMBR temperature ($T_S \gg T_{\gamma}$) we have [@bharadwaj05] $$\eta_{\HI}({\bf x}, z) = \frac{\rho_{\HI}}{\bar{\rho}_{H}} \left[ 1 - \frac{(1+z)}{H(z)}\frac{\partial v_{\parallel}}{\partial r} \right] \label{eq:tb}$$ where all the quantities in the r.h.s. refer to the position and epoch where the emission originated and $ \bar{T}(z) = 4.0\, {\rm mK} (1+z)^2 \left(\frac{\Omega_b h^2}{0.02}\right) \left(\frac{0.7}{h}\right) \frac{H_0}{H(z)} $. Here $\rho_{\HI}$ refers to the density (which varies from position to position), $\bar{\rho}_{H}$ is the mean hydrogen density, $r$ is the comoving distance from the observer and $v_{\parallel}$ is the radial component of the peculiar velocity. We may express $\rho_{\HI}$ using $\rho_{\HI}/\bar{\rho}_{H}= x_{\HI} (1 + \delta) $ where $x_{\HI}$ is the hydrogen neutral fraction and $\delta$ is the matter overdensity which we have assumed to be the same as the hydrogen overdensity at the the large length-scales of our interest. Further, we may express the neutral fraction as $x_{\HI}=\langle \xh1 \rangle_{V} (1 + \delta_x)$ where $\langle \xh1 \rangle_{V}$ in the volume averaged neutral fraction and $\delta_x$ is the contrast in the $x_{\HI}$ distribution. Note that the volume averaged neutral fraction $\langle \xh1 \rangle_{V}$ and the mass averaged neutral fraction $\bar{x}_{\HI}$ refer to the average of $x_{\HI}$ and $x_{\HI}(1+\delta)$ respectively, Assuming that $\delta_x,\delta,(\partial v_{\parallel}/\partial r) \ll 1$ we drop all quadratic and higher terms involving $\delta_x,\delta$ and $(\partial v_{\parallel}/\partial r)$, to obtain $\langle \xh1 \rangle_{V}= \bar{x}_{\HI}$ whereby it is possible to express $\eta_{\HI}$ in Fourier space as $$\tilde{\eta}_{\HI}({\bf k}) = \bar{x}_{\HI} \left[ \Delta_x + (1 + \mu^2)\Delta \right] \,. \label{eq:d_tb}$$ where $\tilde{\eta}_{\HI},\Delta$ and $\Delta_x$ are the Fourier transform of $\eta_{\HI}, \delta$ and $\delta_x$ respectively. This gives the redshift space power spectrum (of $\eta_{\HI}$) to be [@barkana05] $$\begin{aligned} P^s_{\HI} (k,\mu) = \bar{x}^2_{\HI} &\left[P_{xx} (k) + 2 (1 + \mu^2) P_{\Delta x} (k)\right.\nonumber\\ &\left. + (1 + \mu^2)^2 P_{\Delta \Delta} (k) \right] \label{eq:model}\end{aligned}$$ where $P_{xx}$ and $P_{\Delta \Delta}$ are the power spectra of $\Delta_x$ and $\Delta$ respectively, and $P_{\Delta x}$ is the cross power spectrum between $\Delta$ and $\Delta_x$. We recover the real space power spectrum $P^r(k)$ if we set $\mu=0$ in eq. (\[eq:model\]). Here, and in the subsequent discussion, we drop the subscript for brevity of the symbols. Thus the power spectrum in real space will be $$P^r = \bar{x}^2_{\HI} \left( P_{\Delta \Delta} + 2 P_{\Delta x}+ P_{xx} \right) \,. \label{eq:modelr}$$ In this model only the first three even angular moments of the redshift space power spectrum have non-zero values $$P^s_0 = \bar{x}^2_{\HI} \left( \frac{28}{15} P_{\Delta \Delta} + \frac{8}{3} P_{\Delta x}+ P_{xx} \right) \,, \label{eq:model0}$$ $$P^s_2 = \bar{x}^2_{\HI} \left( \frac{40}{21} P_{\Delta \Delta} + \frac{4}{3} P_{\Delta x} \right) \,, \label{eq:model2}$$ $$P^s_{4} = \bar{x}^2_{\HI} \left( \frac{8}{35} \right) P_{\Delta \Delta}\,. \label{eq:model4}$$ To provide an interpretation of our results obtained from semi-numerical simulations in this paper we have used this linear model. Finally, we note that this model is based on the assumption $\delta_x \ll 1$, and terms of the order of $\delta_x \delta$ which appear in eq. (\[eq:tb\]) are ignored. While this is possibly a reasonable assumption in the early stages of reionization ($\bar{x}_{\HI} \sim 1$), we may expect significant deviations from this model in the late stages of the reionization where $\bar{x}_{\HI}$ is small and we expect $\delta_x$ to exhibit large fluctuation of order unity. @lidz07 and @mao12 have considered models which incorporate the non-linear effects, but these are rather complicated and we have not attempted using these models here. Method to estimate the angular multipoles of power spectrum from the simulated maps ----------------------------------------------------------------------------------- We Fourier transform the entire simulated image data cube and estimate the angular multipoles $P^s_l$ of power spectrum from the Fourier transformed data following the equation $$P^s_{l} (k) = \frac{\left( 2l + 1 \right)}{4 \pi} \int {\mathcal P}_l(\mu)\, P^s (k)\, d\Omega \,, \label{eq:p_sl}$$ where $P^s (k)$ is the power spectrum in the redshift space which has been estimated from the Fourier transform of the simulated redshift space map. The integral is done over the entire solid angle to take into account all possible orientations of the ${\bf k}$ vector with the LoS direction ${\bf \hat{n}}$. Each angular multipole is estimated at $10$ logarithmically spaced $k$ bins in the range $0.09\leq k \leq 4.80\, {\rm Mpc} ^{-1}$. Results ======= \[c\]\[c\]\[1\]\[0\][$\xb=1.0$]{} \[c\]\[c\]\[1\]\[0\][ $\,\,0.9$]{} \[c\]\[c\]\[1\]\[0\][ $\,\,0.8$]{} \[c\]\[c\]\[1\]\[0\][ $\,\,0.7$]{} \[c\]\[c\]\[1\]\[0\][ $\,\,0.5$]{} \[c\]\[c\]\[1\]\[0\][ $\,\,0.3$]{} \[c\]\[c\]\[1\]\[0\][ $\,\,0.1$]{} \[c\]\[c\]\[1\]\[0\][[IR]{}]{} \[c\]\[c\]\[1\]\[0\][[HR]{}]{} \[c\]\[c\]\[1\]\[0\][[$k$ in ${\rm Mpc}^{-1}$]{}]{} \[c\]\[c\]\[1\]\[0\][LT]{} \[c\]\[c\]\[1\]\[0\][[$P^s_0(k)/P^r(k)$]{}]{} \[c\]\[c\]\[1\]\[0\][$\xb=1.0$]{} \[c\]\[c\]\[1\]\[0\][ $\,0.9$]{} \[c\]\[c\]\[1\]\[0\][ $\,0.8$]{} \[c\]\[c\]\[1\]\[0\][ $\,0.7$]{} \[c\]\[c\]\[1\]\[0\][ $\,0.5$]{} \[c\]\[c\]\[1\]\[0\][ $\,0.3$]{} \[c\]\[c\]\[1\]\[0\][ $\,0.1$]{} \[c\]\[c\]\[1\]\[0\][[IR]{}]{} \[c\]\[c\]\[1\]\[0\][[HR]{}]{} \[c\]\[c\]\[1\]\[0\][LT]{} \[c\]\[c\]\[1\]\[0\][[$k$ in ${\rm Mpc}^{-1}$]{}]{} \[c\]\[c\]\[1\]\[0\][[$P^s_2(k)/P^s_0(k)$]{}]{} \[c\]\[c\]\[1\]\[0\][[$\xb$]{}]{} \[c\]\[c\]\[1\]\[0\][[$P^s_2/P^s_0$]{}]{} \[c\]\[r\]\[1\]\[0\][[$k=0.23\,{\rm Mpc}^{-1}$]{}]{} \[c\]\[r\]\[1\]\[0\][[$k=0.55\,{\rm Mpc}^{-1}$]{}]{} \[c\]\[r\]\[1\]\[0\][[$k=3.00\,{\rm Mpc}^{-1}$]{}]{} \[t\]\[b\]\[1\]\[0\][[IR]{}]{} \[t\]\[b\]\[1\]\[0\][[HR]{}]{} We have simulated the distribution, in both real and redshift space, for $\xb$ values starting from $1.0$ to $0.1$ with an interval of $\Delta \xb = 0.1$. For each value of $\xb$ we have used 12 independent [realizations [^9]]{} of the simulation to estimate the mean and $1\sigma$ error of the power spectrum. The simulations are all carried out at a fixed redshift $z = 8$. Ideally, one should adopt a model for the evolution of $\xb$ with $z$ and simulate each neutral fraction at the appropriate redshift. This, however, makes the results dependent on the model for the redshift evolution of $\xb$ and also makes the computations rather cumbersome. We have sidestepped this issue by keeping $z$ fixed at a value $z = 8$ which is in the redshift range that is considered favourable for observing the EoR 21-cm signal. However due to this approach we are unable account for the evolution for the dark matter density field as well the evolution of the sources of ionizing photons in our simulation. We interpret the simulations for different values of $\xb$ as representing different stages of reionization. Figure \[fig:h1\_map\] shows the distributions at different stages of reionization in both real and redshift space for a single realization of the simulation. We see that the distributions in the HR and the IR models are nearly indistinguishable in the early stages of reionization ($\xb \geq 0.7$). In the early stages reionization proceeds inside-out in both of these models with the high density regions ionizing first and the low density regions ionizing later. The maps of the two models become quite distinct at the later stages of reionization ($\xb \leq 0.5$). In the IR model we find many small isolated neutral clumps distributed within the regions which are completely ionized in the HR model. These small clumps correspond to the high density regions which become self-shielded due to the enhancement in the recombination rate in the IR model. Comparing the real and redshift space maps we see that in the initial stages the ionized and neutral regions respectively appear slightly contracted and elongated along the LoS. For the HR model the distribution exhibits the same behaviour even in the late stages. The small clumps which appear in the late stages of the IR model however, show the opposite behaviour. These neutral regions appear contracted along the LoS. All the features mentioned above can be understood based on the fact that overdense and underdense regions respectively appear contracted and elongated along the LoS. We show the ratio between the monopole of the redshift space power spectrum with its real space counterpart in Figure \[fig:pk\_ratio\]. This ratio ($P^s_0/P^r$) has been extensively used in the literature ([*e.g.*]{} @lidz07 [@mesinger11; @mao12]) to quantify the effect of redshift space distortion. The earlier works have reported that at large length scales this ratio rises to a value greater than $1.87$ in the early stages of reionization, and then falls to a value slightly less than $1$ at later stages. Our results show a similar behaviour in both the HR and IR models. We also find that at large length scales ($k \sim 0.2\, {\rm Mpc}^{-1}$) $P^s_0/P^r$ rises to a value $\approx 2$ at $\xb = 0.8-0.9$ and then abruptly falls to a value slightly less than $1$ at $\xb = 0.7$. The ratio subsequently rises gradually and approaches $1$ as $\xb$ decreases. The behaviour at intermediate length scales ($k \sim 0.5\, {\rm Mpc}^{-1}$) is somewhat similar, except that we have a sharp peak at $\xb = 0.9$. Also, the ratio again exceeds $1$ at $\xb \leq 0.2$ for the IR model. Several earlier works have highlighted the initial rise in $P^s_0/P^r$ at $\xb \sim 0.8$ as a very prominent feature of the effect of redshift space distortion on the EoR 21-cm signal. We note, that the expected signal itself drops considerably at $\xb = 0.8$ and this is possibly not very significant from the observational point of view. Further, the sudden rise in $P^s_0/P^r$ is possibly because of the rapid decline in the signal itself (due to the possible negative contribution from $P_{\Delta x}$ in eq. \[\[eq:modelr\]\] and competition between $P_{\Delta \Delta}$, $P_{\Delta x}$ and $P_{xx}$, this is further verified later in this paper in Figure \[fig:pk\_rk\] and \[fig:pk\_ak\]) at $\xb = 0.8$. Both $P^s_0$ and $P^r$ reduces at this stage, however $P^r$, which appears in the denominator of this ratio, declines much faster than $P^s_0$ (for further details see the relevant discussion in @mao12). The quantity that in principle can be directly estimated from the observational data and will quantify the strength as well as the nature of the redshift space anisotropy is the ratio between the quadrupole and the monopole of the redshift space power spectrum. Our estimations for this ratio ($P^s_2/P^s_0$) from simulations has been shown in the Figure \[fig:p2p0\_ratio\]. We find that for $\xb = 1.0$ the results from our simulations are consistent with the value $1.02$ predicted by linear theory at large length scales ($k \leq 0.5\,{\rm Mpc}^{-1}$). At smaller scales the ratio is larger than $1.02$ possibly because of the non-linear redshift space distortion. This ratio increases at all length scales for $\xb = 0.9$. The ratio increases further for $\xb = 0.8$ at large scales ($k < 0.3\,{\rm Mpc}^{-1}$) whereas it decreases at smaller length scales. The behaviour of this ratio changes significantly at $\xb \leq 0.7$ where we find that $P^s_2/P^s_0$ is negative at large scales. The value increases steadily as we move to smaller scales, crosses zero at $k \sim 1.0\,{\rm Mpc}^{-1}$ and is positive at smaller length scales. The results for the HR and IR models are very similar except that there is a bump at around $k \sim 0.5\,{\rm Mpc}^{-1}$ for the IR model. To understand the evolution of $P^s_2/P^s_0$ with $\xb$ in more detail we study the behaviour of this ratio as a function of $\xb$ (Figure \[fig:p2\_p0\_xh1\]) at three representative $k$ modes $k = 0.23, \, 0.55\, {\rm and}\, 3.00 \,{\rm Mpc}^{-1}$ hitherto referred to as large, intermediate and small scales. We could have, in principle, chosen a smaller $k$ mode to illustrate the behaviour at large scales. The errors however are rather large for $k < 0.2 \,{\rm Mpc}^{-1}$ and these scales do not provide a very reliable estimate of the behaviour, and we need larger simulations to study the behaviour at length scales as large as these. Considering the large scale first (Figure \[fig:p2\_p0\_xh1\]), the behaviour of the HR and IR models are both quite similar. The ratio rises from $\sim 1$ at $\xb = 1.0$ to $1.4$ at $\xb = 0.8-0.9$ and then abruptly falls to $\approx -0.3$ at $\xb = 0.7$. The ratio remains negative with values in the range $(-0.4) - (-0.6)$ for smaller values of $\xb$ with the exception that it rises to $\approx -0.2$ for $\xb \leq 0.2$ in the IR model. The behaviour at intermediate scales is very similar as that at the large scales except that the ratio falls to a value in the range $0.2 - 0.4$ at $\xb = 0.7$ instead of becoming negative. The ratio declines further and is negative ($\approx -0.1$) for $\xb < 0.4$ in the HR model. In the IR model the ratio is negative nowhere and is in the range $0.2 - 0.4$ for $\xb < 0.7$. At small scales the ratio is constant at $\sim 1.2$ for $\xb \geq 0.9$ where after it falls rapidly to $0.8$ at $\xb = 0.7$ and subsequently declines gradually to $\sim 0.6$ and $\sim 0.4$ in the HR and IR models respectively. The hexadecapole $P^s_4(k)$ measured from our simulations has very large error bars, and consequently we have not shown these here. Larger simulations are required for reliable estimates of $P^s_4(k)$. However it can be noted that the linear model predicts the ratio $P^s_4(k)/P^s_0(k)$ to be very small ($P^s_4/P^s_0 \simeq 0.12$) even for a completely neutral IGM, when compared with the ratio $P^s_2/P^s_0 \simeq 1.02$. Thus a successful estimation of $P^s_4(k)$ from the observational data is intrinsically a difficult task. A recent work by @shapiro13 has discussed the validity of using the $4^{{\rm th}}$ moment of the redshift space power spectrum (in $\mu$ decomposition technique), which is some what similar to the hexadecapole $P^s_4(k)$, for extracting the cosmological information from the redshifted 21 cm observations. \[c\]\[c\]\[1\]\[0\][[$\xb=0.9$]{}]{} \[c\]\[c\]\[1\]\[0\][ [$\,0.8$]{}]{} \[c\]\[c\]\[1\]\[0\][ [$\,0.7$]{}]{} \[c\]\[c\]\[1\]\[0\][ [$\,0.6$]{}]{} \[c\]\[c\]\[1\]\[0\][ [$\,0.5$]{}]{} \[c\]\[c\]\[1\]\[0\][ [$\,0.3$]{}]{} \[c\]\[c\]\[1\]\[0\][ [$\,0.1$]{}]{} \[c\]\[c\]\[1\]\[0\][[$k\, ({\rm Mpc}^{-1})$]{}]{} \[l\]\[l\]\[1\]\[0\][[$\a(k)$]{}]{} \[l\]\[l\]\[1\]\[0\][[$\r(k)$]{}]{} \[c\]\[c\]\[1\]\[0\][[IR]{}]{} \[c\]\[c\]\[1\]\[0\][[HR]{}]{} Discussion and Conclusions ========================== We first compare our results with the predictions of the model discussed in Section \[sec:model\] for which $P^s_0$ and $P^s_2$ can be calculated using eq.s (\[eq:model0\]) and (\[eq:model2\]) respectively. The model requires $P_{\Delta \Delta}(k)$ , $P_{xx}(k)$ and $P_{\Delta x}(k)$ which we have directly determined from our simulations at different stages of reionization. Several earlier authors [@lidz07; @mesinger11; @mao12] have noted that this model does not correctly reproduce the real and redshift space power spectrum $P^r$ and $P^s_0$ determined from simulations. Our results for the $\xb$ dependence of $P^s_2/P^s_0$ shown in Figure \[fig:p2\_p0\_xh1\] confirm that the model fails to quantitatively reproduce the results of the simulations. We however note that the predictions of this model are qualitatively very similar to the results of the simulations, and this provides a very useful framework for interpreting our results. In the subsequent discussion we have used the framework of this model to discuss and qualitatively interpret the $\xb$ dependence of the observable quantity $P^s_2/P^s_0$, which we have studied earlier. According to the linear model the quantities which will combinedly determine the strength as well as the nature of the redshift space distortions are the cross-correlation power spectrum ($P_{\Delta x}$) between the matter density fluctuations ($\Delta$) and the fluctuations in the neutral fraction ($\Delta_x$) and the power spectrum of fluctuations in the neutral fraction field ($P_{xx}$). The quantity $P_{xx}$ by definition will always be positive, whereas the quantity $P_{\Delta x}$ can have both positive or negative values. A negative/positive contribution from $P_{\Delta x}$ would represent an anti-correlation/correlation between the $\Delta$ and $\Delta_x$ fields. To understand their relative contribution on the simulated redshift space power spectrum we represent them in terms of two dimensionless quantities $$\a(k) = \sqrt{P_{xx}(k)/P_{\Delta\Delta}(k)} \,, \label{eq:ak}$$ and $$\r(k) = P_{\Delta x}(k)/\sqrt{P_{xx}(k) P_{\Delta \Delta}(k)}\,, \label{eq:rk}$$ from our simulations. The quantity $\a(k)$ defines the relative amplitude of $P_{xx}$ with respect to $P_{\Delta \Delta}$, whereas $\r(k)$ determines the strength of cross-correlation between $\Delta$ and $\Delta_x$. Figure \[fig:pk\_rk\] and \[fig:pk\_ak\] show $\r(k)$ and $\a(k)$ respectively at different stages of reionization. Considering the large scales first we find that $\r \simeq -1$ at all stages of reionization in both the HR and IR models. This indicate that at large scales the distribution of neutral fraction is anti-correlated with the matter distribution throughout reionization, this being a consequence of the fact that the high density regions are the locations where we expect to find the sources that drives reionization thus they are expected to get ionized first and only the low density regions remain neutral. Note that the small neutral clumps produced at the later stages in the IR model do not affect this behaviour seen at large scales. At all scales we find $\a(k)$ to monotonically increase with decreasing $\xb$. The observed sharp peak of the ratio $P^s_2/P^s_0$ (left panels of Figure \[fig:p2\_p0\_xh1\]) at the early stages of reionization ($\xb \simeq 0.9$) at large and intermediate scales thus represent the fact that at this stage the contribution from $P_{xx}$ becomes comparable to $P_{\Delta \Delta}$ and due to a complete anti-correlation between $\Delta$ and $\Delta_x$ ([*i.e.*]{} $\r \simeq -1$) the actual signal ($P^s_0$) becomes very low, which appears at the denominator of this ratio. This peak thus should not be misinterpreted as a signature of redshift space distortion. At the later stages of reionization ($0.1 < \xb \leq 0.8$ for the HR and $0.3 < \xb \leq 0.8$ for the IR model) the cross correlation still remains $\r \simeq -1$ whereas $P_{xx}>P_{\Delta \Delta}$ which leads to a negative value of $P^s_2/P^s_0$ in both models of reionization. These results points towards an inside-out reionization at large scales. We observe that the presence of small neutral clumps, produced at the later stages in the IR model, does not contribute significantly at the large scale power spectrum. Thus a negative value of $P^s_2/P^s_0$ of the 21-cm power spectrum itself will be a direct evidence that the reionization has happened and it is inside-out at large scale. The fluctuations in the neutral fraction $\Delta_x(k)$ and the matter $\Delta(k)$ are not perfectly anti-correlated ($\r > -1$) at intermediate and small scales. The value of $\r$ increases as we go to smaller scales, and the behaviour is similar in the HR and IR models at the early stages of reionization ($\xb \geq 0.8$). We see differences between the HR and IR model at intermediate and small scales at the later stages of reionization. The values of $\r$ are larger in the IR model in comparison to the HR model, and $\r$ becomes positive at small scales in the later stages in the IR model whereas it is nowhere positive in the HR model. This difference is the outcome of the small clumps seen in Figure \[fig:h1\_map\] at the later stages in the IR model. The signature of this difference is also visible in the observed power spectrum ($P^s_2/P^s_0$) at the intermediate scales (middle panels of Figure \[fig:p2\_p0\_xh1\]). At intermediate scales the ratio $P^s_2/P^s_0$ becomes negative during the later stages of reionization ($\xb \leq 0.6$) in the HR model, however in IR model it never becomes negative at these scales and more or less maintains a constant positive value of $\approx 0.2$ at the later stages of reionization. Thus the two major signatures from the redshift space 21-cm signal that can be used as the evidence of reionization as well as a characterization for the redshift space anisotropy are the following — - A negative value of the ratio $P^s_2/P^s_0$ at large scales during the intermediate and late stages of reionization. - The ratio $P^s_2/P^s_0$ stays negative even at intermediate scales for a completely inside-out reionization whereas it becomes positive for a partially outside-in reionization. The main point of concern here is how unambiguously it will be possible to detect these signatures, or in other words what level of sensitivity our measurements will require to detect these signatures. To get an idea of the accuracy level of the measurements required and to find out the possible effects of uncertainty, we have done a rough error analysis for estimations of the ratios of various angular multipoles. This analysis has been discussed in detail in Appendix \[ap:error\]. Following this analysis (eq. \[\[eq:delR\]\]) we have estimated the possible errors in our estimation of the ratio $P_2/P_0$ and shown them as error bars in Figure \[fig:p2\_p0\_xh1\]. However while estimating these errors we have not considered the possible uncertainties arising from the system noise of a specific observation with a telescope and also replaced the cosmic variance in eq. (\[eq:delR\]) with the sample variance of the zeroth moment of the power spectrum. We plan to include the effect of system noise and various other observation specific effects in our future work. From our analysis in Appendix \[ap:error\] and results in Figure \[fig:p2\_p0\_xh1\] it is evident that it will be possible to unambiguously detect both these signatures of reionization at large and intermediate scales, to which the present ([*e.g.*]{} LOFAR, see @jensen13 for more details) and upcoming telescopes will be sensitive, if the system noise can be suppressed to a sufficient level. \[c\]\[c\]\[1\]\[0\][[$k\, ({\rm Mpc}^{-1})$]{}]{} \[l\]\[l\]\[1\]\[0\][[$\a(k)$]{}]{} \[c\]\[c\]\[1\]\[0\][[IR]{}]{} \[c\]\[c\]\[1\]\[0\][[HR]{}]{} Another alternative approach to quantify the strength and the nature of redshift space distortion from the power spectrum is by decomposing the redshift space power spectrum in the various coefficients of the powers of $\mu$. These estimated coefficients can be then interpreted following a linear [@barkana05] or a quasi-linear [@mao12] model. However in this method the estimated coefficients of the powers of $\mu$ will not be completely independent of each other and the correlation between them (or the leakage of power from one component to another) may give rise to wrong interpretations as observed in a recent work by @jensen13. In comparison to this method the decomposition of the anisotropy due to peculiar velocities using Legendre polynomials, is a representation in orthonormal basis therefore the various angular multipoles estimated through this method will be independent of each other (as discussed in Appendix \[ap:error\]). Thus one does not have to be too concerned about the leakage of power between different multipoles in this method. It is anticipated that the initial observational attempts ([*e.g.*]{} GMRT, LOFAR and MWA) will probe the EoR 21-cm signal only at large scales. The multipole moments of the power spectrum of the measured signal hold the key to quantify and interpret it. We have considered the ratio between the quadrupole and monopole moments which is capable of quantifying the strength of the signal and the anisotropy which arises due to the peculiar velocities. Our simulations indicate that the prospects of detecting the EoR 21-cm signal are most favourable when the mean neutral fraction is in the range $\xb=0.4-0.5$. In this range the signal, we see, is characterized by two main features for both the HR and IR models of reionization. First, the monopole moment of the redshift space power spectrum $P^s_0(k)$ is nearly equal to the real space power spectrum $P^r(k)$, and both of these are comparable to the real space matter power spectrum. Second, the quadrupole moment of the redshift space power spectrum $P^s_2(k)$ is negative with a value which is $-0.5$ times $P^s_0(k)$. This is in contrast to the value $1.02$ predicted by linear theory for the ratio $P^s_2/P^s_0$ of the matter power spectrum. We also observe that it would be possible to distinguish between the inside-out and the outside-in reionization scenarios from the nature of the ratio between the quadrupole and monopole moments at the intermediate length scales ($k \simeq 0.5 \,{\rm Mpc}^{-1}$). This particular signature may help in ruling out the extremely outside-in reionization models using future observations. acknowledgments =============== Suman Majumdar would like to thank Prof. Sugata Pratik Khastgir for useful discussions and pointing out some errors in the initial version of the manuscript. Suman Majumdar would like to acknowledge Council of Scientific and Industrial Research (CSIR), India for providing financial assistance through a senior research fellowship (File No. 9/81 (1099)/10-EMR-I). 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We consider redshifted 21-cm observations that cover a 3-D volume $V$. In the continuum limit, the number of independent Fourier modes $d N$ corresponding to $d^3 k$ is $$dN= V \frac{d^3 k}{(2 \pi)^3}$$ We consider $\P(\k)$ which is an unbiased estimator of the power spectrum at the Fourier mode $\k$. This has the properties that $$\langle \P(\k) \rangle = P(\k) \label{eq:est1}$$ and $$\langle [\Delta \P(\k)] [\Delta \P(\k^{'})] \rangle = \frac{(2 \pi)^3}{V} \delta(\k-\k^{'}) \sigma^2(\k) \label{eq:est2}$$ where the latter equation essentially tells us that the estimates at the different Fourier modes are independent. Also the variance of power spectrum estimator has two independent contributions $$\sigma^2 (\k)=[P(\k)+\sigma_N^2]^2 \label{eq:est3}$$ which arise from the cosmic variance $[P(\k)]$ and the system noise $[\sigma_N^2]$ respectively. The system noise is inherent to the observations and it depends on the observing time, the telescope, etc. We use this to define estimators for the multipole moments of the power spectrum $$\P_{n}(k_i) = \frac{ \int d^3 k \, \Pl_n(\mu) \P(\k)}{ \int d^3 k \, \Pl_n^2(\mu) } \label{eq:mom1}$$ where $\Pl_n(\mu)$ is the Legendre polynomial of order $n$, $\mu=\k \cdot {\bf n}/k$ is the cosine of the angle between $\k$ and the line of sight ${\bf n}$, and the $d^3 \, k$ integral is over a spherical shell of radius $k_i$ and width $\Delta k_i$. We have $$\langle \P_n(k_i) \rangle = P_n(k_i) \,.$$ We assume that the bins have no overlap, whereby it is obvious that the estimators $P_n(k_i)$ and $P_m(k_j)$ in two different bins $(k_i \neq k_j)$ are uncorrelated. We now calculate the covariance between the multipoles in the same bin $$\begin{aligned} \langle [\Delta \P_n(k_i)] [\Delta \P_m(k_i)] \rangle =& \left \langle \frac{ \int d^3 k_1 \, \Pl_n(\mu_1) [\Delta \P(\k_1)]}{ \int d^3 k_1 \, \Pl_n^2(\mu_1) } \right.\nonumber\\ &\left.\frac{ \int d^3 k_2 \, \Pl_m(\mu_2) [\Delta \P(\k_2)]}{ \int d^3 k_2 \, \Pl_m^2(\mu_2) } \right \rangle\end{aligned}$$ which using eq. (\[eq:est2\]) gives $$\begin{aligned} \langle [\Delta \P_n(k_i)]& [\Delta \P_m(k_i)] \rangle =\nonumber\\ &\frac{ (2 \pi)^3 V^{-1} \int d^3 k_1 \, \Pl_n(\mu_1) \Pl_m(\mu_1) \sigma^2(\k)} { \int d^3 k_1 \, \Pl_n^2(\mu_1) \int d^3 k_2 \, \Pl_m^2(\mu_2) } \end{aligned}$$ This can be further simplified using $$\begin{aligned} \int d^3 k \Pl_n(\mu_1) &\Pl_m(\mu_1) \sigma^2(\k)= \nonumber\\ & 2 \pi k_i^2 \, \Delta k_i \, \sigma^2(k_i) \int_{-1}^{1} d \mu_1 \, \Pl_n(\mu_1) \Pl_m(\mu_1)\end{aligned}$$ and $$\int_{-1}^{1} d \mu_1 \, \Pl_n(\mu_1) \Pl_m(\mu_1) = \frac{2 \delta_{n,m}}{2 n +1 }$$ whereby $$\langle [\Delta \P_n(k_i)] [\Delta \P_m(k_i)] \rangle = \delta_{m,n} \frac{2 \pi^2 (2 n+1) \sigma^2(k_i)}{V k_i^2 \, \Delta k_i}$$ We see that the errors in the different multipole moments are uncorrelated. For the monopole $$\langle [\Delta \P_0(k_i)]^2 \rangle = \frac{2 \pi^2 [P(\k)+\sigma_N^2]^2}{V k_i^2 \, \Delta k_i}$$ and $$\langle [\Delta \P_n(k_i)]^2 \rangle = (2 n +1) \langle [\Delta \P_0(k_i)]^2 \rangle$$ for the higher multipoles. We next consider the ratio of the multipoles $$\Rh_n(k_i)=\frac{\P_n(k_i)}{\P_0(k_i)}$$ We calculate the variance of $\Rh_n(k_i)$ using $$\Delta \Rh_n(k_i) = \frac{[\Delta \P_n(k_i)]}{P_0(k_i)} - \frac{P_n(k_i)}{P_0(k_i)} \frac{[\Delta \P_0(k_i)]}{P_0(k_i)}$$ whereby $$\langle [\Delta \Rh_n(k_i)]^2 \rangle = \frac{\langle [\Delta \P_n(k_i)]^2 \rangle }{P_0^2(k_i)} + \frac{P_n^2(k_i)}{P_0^2(k_i)} \frac{\langle [\Delta \P_0(k_i)]^2 \rangle }{P_0^2(k_i)}$$ which gives $$\langle [\Delta \Rh_n(k_i)]^2 \rangle = \left[ (2n + 1) + {{\cal R}}_n^2(k_i) \right] \frac{\langle [\Delta \P_0(k_i)]^2 \rangle}{P_0^2(k_i)}$$ In summary we have a relation between $\delta {{\cal R}}_n$ which is the error in the ratio ${{\cal R}}_n=P_n/P_0$ and the fractional error $\delta P_o/P_o$ of the monopole. $$\delta {{\cal R}}_n = \sqrt{2 n + 1 + {{\cal R}}^2_n} \ \left(\frac{\delta P_0}{P_0} \right) \label{eq:delR}$$ This provides a rough estimation of possible errors that will be present in the estimations of the ratio of angular moments of 21-cm power spectrum. We plan to take up a more detailed uncertainty analysis in our future work. [^1]: E-mail: sumanm@phy.iitkgp.ernet.in [^2]: E-mail: somnath@phy.iitkgp.ernet.in [^3]: E-mail: tirth@ncra.tifr.res.in [^4]: http://www.gmrt.ncra.tifr.res.in [^5]: http://www.lofar.org/ [^6]: http://21cma.bao.ac.cn/ [^7]: http://www.haystack.mit.edu/ast/arrays/mwa/ [^8]: http://www.skatelescope.org/ [^9]: All the results shown here (Figure 2 to 6) are mean estimated from these 12 independent realizations.

--- abstract: 'Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalization of Otal’s marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal’s theorem asserts that if $X, Y$ admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map $f$ is Moebius and equivariant with respect to these actions then it extends to an isometry. In our case there are no cocompactness or equivariance assumptions, indeed the isometry groups of $X, Y$ may be trivial.' address: 'Indian Statistical Institute, Kolkata, India. Email: kingshook@isical.ac.in' author: - Kingshook Biswas bibliography: - 'moeb.bib' title: 'Moebius rigidity for simply connected, negatively curved surfaces' --- \[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Conjecture]{} \[theorem\][Remark]{} \[theorem\][Claim]{} \[theorem\][Definition-Theorem]{} Introduction ============ We continue in this article the study of Moebius maps between boundaries of CAT(-1) spaces undertaken in [@biswas3], [@biswas4], [@biswas5], [@biswas6], [@biswas7]. The principal question is whether a Moebius homeomorphism between the boundaries at infinity of two CAT(-1) spaces extends to an isometry between the spaces. We recall that the boundary $\partial X$ of a CAT(-1) space comes equipped with a positive function on the set of quadruples of distinct points in $\partial X$, called the cross-ratio, and a map $f : \partial X \to \partial Y$ between boundaries is said to be Moebius if it preserves cross-ratios. Bourdon [@bourdon2] showed that if $X$ is a rank one symmetric space of noncompact type with the metric normalized so that the maximum of the sectional curvatures equals $-1$, and $Y$ is any CAT(-1) space, then any Moebius embedding $f : \partial X \to \partial Y$ extends to an isometric embedding $F : X \to Y$. In [@biswas3] it was shown that if $X, Y$ are proper, geodesically complete CAT(-1) spaces, then any Moebius homeomorphism $f : \partial X \to \partial Y$ extends to a $(1, \log 2)$-quasi-isometry $F : X \to Y$. This extension was shown in [@biswas5] to coincide with a certain geometrically defined extension of Moebius maps called the [*circumcenter extension*]{}. For $X, Y$ complete, simply connected Riemannian manifolds of pinched negative curvature $-b^2 \leq K \leq -1$, the main result of [@biswas3] was improved in [@biswas5] to show that the circumcenter extension $F : X \to Y$ of a Moebius homeomorphism $f : \partial X \to \partial Y$ is a $(1, (1 - 1/b)\log 2)$-quasi-isometry. The case of complete, simply connected Riemannian manifolds $X, Y$ of pinched negative curvature $-b^2 \leq K \leq -1$ was further studied in [@biswas6], where it was shown that if $f : \partial X \to \partial Y$ and $g : \partial Y \to \partial X$ are mutually inverse Moebius homeomorphisms, then their circumcenter extensions $F : X \to Y$ and $G : Y \to X$ are $\sqrt{b}$-bi-Lipschitz homeomorphisms which are inverses of each other. Another case which has been considered is that of compact deformations of a negatively curved manifold [@biswas4], [@biswas7]. Here, we consider a complete, simply connected Riemannian manifold $(X, g_0)$ of pinched negative curvature $-b^2 \leq K_{g_0} \leq -1$, and a Riemannian metric $g_1$ on $X$ such that $g_1 = g_0$ outside a compact in $X$, and such that $g_1$ has sectional curvature bounded above by $-1$. The identity map $id : (X, g_0) \to (X, g_1)$ is bi-Lipschitz, and thus induces a homeomorphism $f : \partial_{g_0} X \to \partial_{g_1} X$ between the boundaries at infinity of $(X, g_0)$ and $(X, g_1)$. While some partial results were proved in [@biswas4], in [@biswas7] a complete solution to the problem in this case was obtained: if the boundary map $f : \partial_{g_0} X \to \partial_{g_1} X$ is Moebius, then its circumcenter extension $F : (X, g_0) \to (X, g_1)$ is an isometry. In the present article we obtain a complete solution to the problem of extending Moebius maps to isometries for the case of complete, simply connected Riemannian manifolds of pinched negative curvature in dimension two: \[mainthm\] Let $X, Y$ be complete, simply connected Riemannian surfaces of pinched negative curvature $-b^2 \leq K \leq -1$. If $f : \partial X \to \partial Y$ is a Moebius homeomorphism, then the circumcenter extension of $f$ is an isometry $F : X \to Y$. The above theorem may be viewed as a generalization of the well-known result of Otal on marked length spectrum rigidity for closed, negatively curved surfaces [@otal2]. This result states that if two closed, negatively curved surfaces have the same marked length spectrum, then they are isometric. It is well-known that two closed, negatively curved manifolds have the same marked length spectrum if and only if there is an equivariant Moebius map between the boundaries of their universal covers (see [@otal1] and section 5 of [@biswas3]). Thus Otal’s result is equivalent to the following: if $X, Y$ are complete, simply connected Riemannian surfaces with curvature bounded above by $-1$, admitting free, properly discontinuous, cocompact, isometric actions by a discrete group $\Gamma$, and $f : \partial X \to \partial Y$ is an equivariant Moebius map, then $f$ extends to an isometry $F : X \to Y$. We remark that the cocompactness of the actions is crucial to Otal’s proof, where a certain invariant is defined by integrating over the compact quotient $X/\Gamma$. In our case, we do not assume existence of any isometric group actions or equivariance of the Moebius map, indeed the isometry groups of $X, Y$ may well be trivial. The proof of Theorem \[mainthm\] relies on certain properties of the circumcenter extension proved in [@biswas7]. In section 2 we recall the necessary preliminaries on Moebius maps and the circumcenter extension, and then in section 3 we prove the main theorem. Preliminaries ============= For details and proofs of the assertions made in this section we refer to [@biswas3], [@biswas5], [@biswas6], [@biswas7]. Moebius metrics and visual metrics ---------------------------------- Let $(Z, \rho_0)$ be a compact metric space of diameter one. For a metric $\rho$ on $Z$, the cross-ratio with respect to the metric $\rho$ is the function of quadruples of distinct points in $Z$ defined by $$[\xi, \xi', \eta, \eta']_{\rho} := \frac{\rho(\xi, \eta)\rho(\xi', \eta')}{\rho(\xi, \eta')\rho(\xi', \eta)}$$ A metric $\rho$ on $Z$ is said to be antipodal if it has diameter one and for any $\xi \in Z$ there exists $\eta \in Z$ such that $\rho(\xi, \eta) = 1$. We assume that the metric $\rho_0$ is antipodal. We say that two metrics $\rho_1, \rho_2$ on $Z$ are Moebius equivalent if for all quadruples of distinct points in $Z$, the cross-ratios with respect to the two metrics are equal. We let $\mathcal{M}(Z, \rho_0)$ denote the set of all antipodal metrics on $Z$ which are Moebius equivalent to $\rho_0$. For any $\rho_1, \rho_2 \in \mathcal{M}(Z, \rho_0)$, there exists a positive continuous function on $Z$ called the derivative of $\rho_2$ with respect to $\rho_1$, denoted by $\frac{d\rho_2}{d\rho_1}$, such that $$\rho_2(\xi, \eta)^2 = \frac{d\rho_2}{d\rho_1}(\xi)\frac{d\rho_2}{d\rho_1}(\eta) \rho_1(\xi,\eta)^2$$ for all $\xi, \eta \in Z$, and such that $$\frac{d\rho_2}{d\rho_1}(\xi) = \lim_{\eta \to \xi} \frac{\rho_2(\xi, \eta)}{\rho_1(\xi, \eta)}$$ for all non-isolated points $\xi$ of $Z$. Moreover, $$\left( \max_{\xi \in Z} \frac{d\rho_2}{d\rho_1}(\xi) \right) \cdot \left( \min_{\xi \in Z} \frac{d\rho_2}{d\rho_1}(\xi) \right) = 1$$ The set $\mathcal{M}(Z, \rho_0)$ admits a natural metric defined by $$d_{\mathcal{M}}(\rho_1, \rho_2) = \sup_{\xi \in Z} \log \frac{d\rho_2}{d\rho_1}(\xi)$$ The metric space $(\mathcal{M}(Z, \rho_0), d_{\mathcal{M}})$ is proper. Let $X$ be a proper, geodesically complete CAT(-1) space with boundary at infinity $\partial X$. For any $x \in X$, there is a metric $\rho_x$ on $\partial X$ called the visual metric based at $x$, defined by $$\rho_x(\xi, \eta) = e^{-(\xi|\eta)_x} \ , \xi, \eta \in \partial X,$$ where $(\xi|\eta)_x$ is the Gromov inner product between the boundary points $\xi, \eta \in \partial X$ with respect to the basepoint $x$, defined by $$(\xi|\eta)_x = \lim_{y \to \xi, z \to \eta} \frac{1}{2} (d(x,y) + d(x,z) - d(y,z))$$ The metric space $(\partial X, \rho_x)$ is compact, of diameter one, and antipodal. Moreover, for any two points $x, y \in X$, the metrics $\rho_x, \rho_y$ are Moebius equivalent. Thus the metric space $(\mathcal{M}(\partial X, \rho_x), d_{\mathcal{M}})$ is independent of the choice of $x$, and we denote it by simply $\mathcal{M}(\partial X)$. The map $X \to \mathcal{M}(\partial X), x \mapsto \rho_x$ is an isometric embedding with image $1/2 \log 2$-dense in $\mathcal{M}(\partial X)$. The circumcenter extension -------------------------- Let $X, Y$ be complete, simply connected Riemannian manifolds of pinched negative curvature $-b^2 \leq K \leq -1$, and suppose there is a Moebius homeomorphism $f : \partial X \to \partial Y$. The Moebius map $f$ induces a homeomorphism between the unit tangent bundles $\phi : T^1 X \to T^1 Y$ which conjugates the geodesic flows. The map $\phi$ is defined as follows: given $v \in T^1 X$, let $\gamma : \mathbb{R} \to X$ be the unique bi-infinite geodesic such that $\gamma'(0) = v$, then let $x = \gamma(0), \xi = \gamma(+\infty), \eta = \gamma(-\infty)$. Let $(f(\xi), f(\eta)) \subset Y$ denote the unique (unparametrized) bi-infinite geodesic in $Y$ with endpoints $f(\xi), f(\eta)$. There exists a unique $y \in (f(\xi), f(\eta))$ such that $$\frac{d f_* \rho_x}{d\rho_y}(f(\xi)) = 1$$ Let $\tilde{\gamma} : \mathbb{R} \to Y$ be the unique bi-infinite geodesic such that $\tilde{\gamma}(0) = y, \tilde{\gamma}(+\infty) = f(\xi), \tilde{\gamma}(-\infty) = f(\eta)$. We then define $\phi(v) = \tilde{\gamma}'(0) \in T^1 Y$. Recall that in the CAT(-1) space $Y$, any bounded subset $B$ has a unique circumcenter $c(B) \in Y$, which is the unique point minimizing the function $y \in Y \mapsto \sup_{z \in B} d(y,z)$. Let $(\phi_t : T^1 Y \to T^1 Y)_{t \in \mathbb{R}}$ be the geodesic flow of $Y$. For any $v \in T^1 Y$, let $p(v) = \gamma(+\infty) \in \partial Y$, where $\gamma : \mathbb{R} \to Y$ is the unique geodesic such that $\gamma'(0) = v$. This defines a continuous map $p : T^1 Y \to \partial Y$. Let $\pi : T^1 Y \to Y$ denote the canonical projection. In [@biswas5], it is shown that for any compact subset $K \subset T^1 Y$ such that $p(K) \subset \partial Y$ is not a singleton, the limit of the circumcenters $c(\pi(\phi_t(K)))$ exists as $t \to +\infty$. The limit is called the asymptotic circumcenter of the compact $K$ and is denoted by $c_{\infty}(K)$. The circumcenter extension of the Moebius map $f$ is the map $F : X \to Y$ defined by $$F(x) = c_{\infty}(\phi(T^1_x X))$$ In [@biswas5], it is shown that the circumcenter extension $F$ is a $(1, (1-1/b)\log 2)$-quasi-isometry, while in [@biswas6] it is proved that the circumcenter extensions of $f$ and $f^{-1}$ are $\sqrt{b}$-bi-Lipschitz homeomorphisms which are inverses of each other. For $x \in X$ and $\xi \in \partial X$, let $\overrightarrow{x\xi} \in T^1_x X$ denote the tangent vector $\gamma'(0)$, where $\gamma$ is the unique geodesic with $\gamma(0) = 0, \gamma(+\infty) = \xi$. For $y \in Y, \eta \in \partial Y$, $\overrightarrow{y\eta} \in T^1_y Y$ is similarly defined. Let $\mu$ be a probability measure on $\partial X$. We say that $\mu$ is balanced at $x \in X$ if $$\int_{\partial X} < \overrightarrow{x\xi}, v > d\mu(\xi) = 0$$ for all $v \in T_x X$. The notion of a probability measure on $\partial Y$ being balanced at a point of $Y$ is similarly defined. Let $F : X \to Y$ be the circumcenter extension of the Moebius map $f : \partial X \to \partial Y$. For $x \in X$, define a function $u_x$ on $\partial X$ by $$u_x(\xi) = \log \frac{d\rho_x}{df^*\rho_{F(x)}}(\xi) \ , \xi \in \partial X.$$ and let $K_x \subset \partial X$ denote the set where the function $u_x$ achieves its maximum. In [@biswas6], it is shown that for any $x \in X$, there exists a probability measure $\mu_x$ on $\partial X$ with support contained in $K_x$ such that $\mu_x$ is balanced at $x \in X$ and $f_* \mu_x$ is balanced at $F(x) \in Y$. We will need the following propositions from [@biswas7]: \[rconstant\] ([@biswas7]) The function $r : X \to \mathbb{R}$ defined by $$r(x) = d_{\mathcal{M}}( \rho_x, f^* \rho_{F(x)} ) \ , x \in X$$ is constant. \[qisom\] ([@biswas7]) Let $M \geq 0$ denote the constant value of the function $r$. Then the circumcenter map $F : X \to Y$ is a $(1, 2M)$-quasi-isometry. Given $x \in X$, the flip map $T^1_x X \to T^1_x X, v \mapsto -v$ induces an involution $i_x : \partial X \to \partial X$, defined by requiring that $\overrightarrow{xi_x(\xi)} = -\overrightarrow{x\xi}$ for all $\xi \in \partial X$. \[maxmin\] ([@biswas7]) For $x \in X$, the function $u_x$ achieves its maximum at $\xi \in \partial X$ if and only if it achieves its minimum at $i_x(\xi) \in \partial X$. \[dFstar\] ([@biswas7]) Let $x \in X$ be a point of differentiability of the circumcenter map $F : X \to Y$. Then for any $\xi \in K_x$ and any $v \in T_x X$, we have $$< dF_x(v), \overrightarrow{F(x)f(\xi)} > = < v, \overrightarrow{x\xi} >$$ Equivalently, $$dF^*_x( \overrightarrow{F(x)f(\xi)} ) = \overrightarrow{x\xi}$$ for all $\xi \in K_x$. The following Lemma follows from Propositions \[qisom\] and \[maxmin\]: \[antisom\] Suppose for some $x \in X$, there exists $\xi \in \partial X$ such that $\xi, i_x(\xi) \in K_x$. Then the circumcenter map $F : X \to Y$ is an isometry. [**Proof:**]{} It follows from Proposition \[maxmin\] that the maximum and minimum values of the function $u_x$ are equal. On the other hand we know that the maximum and minimum values are negatives of each other. Since the maximum value equals the constant $M$, we have $M = - M$ and hence $M = 0$. It follows from Proposition \[qisom\] that $F : X \to Y$ is an isometry. $\diamond$ Proof of main theorem ===================== Let $X, Y$ be complete, simply connected Riemannian surfaces of pinched negative curvature $-b^2 \leq K \leq -1$, and let $f : \partial X \to \partial Y$ be a Moebius homeomorphism with circumcenter extension $F : X \to Y$. All the tools are now in hand for the proof of the main theorem: [**Proof of Theorem \[mainthm\]:**]{} As mentioned in the previous section, for any $x \in X$ there exists a probability measure $\mu_x$ on $\partial X$ with support contained in $K_x$ such that $\mu_x$ is balanced at $x \in X$, and $f_* \mu_x$ is balanced at $F(x) \in Y$. As shown in [@biswas6], this is equivalent to the fact that the convex hull in $T_x X$ of the compact $\{ \overrightarrow{x\xi} : \xi \in K_x \}$ contains the origin of $T_x X$ and the convex hull in $T_{F(x)} Y$ of the compact $\{ \overrightarrow{F(x)f(\xi)} : \xi \in K_x \}$ contains the origin of $T_{F(x)} Y$. By the classical Caratheodory theorem on convex hulls, since $X$ is of dimension two this implies that there exists $1 \leq k \leq 3$ and distinct points $\xi_1, \dots, \xi_k \in K_x$ and $\alpha_1, \dots, \alpha_k > 0$ (all depending on $x$) such that $$\alpha_1 \overrightarrow{x\xi_1} + \dots + \alpha_k \overrightarrow{x\xi_k} = 0$$ and $\alpha_1 + \dots + \alpha_k = 1$. Since the vectors $\overrightarrow{x\xi_i}$ are non-zero we must have $2 \leq k \leq 3$. Now if any two of the vectors $\overrightarrow{x\xi_i}, \overrightarrow{x\xi_j}$ for some $i \neq j$ are linearly dependent, then since they are distinct unit norm vectors we must have $\overrightarrow{x\xi_i} = -\overrightarrow{x\xi_j}$, hence $\xi_j = i_x(\xi_i)$. Thus $\xi_i, i_x(\xi_i) \in K_x$, and it follows from Lemma \[antisom\] that $F$ is an isometry and we are done. In particular if $k = 2$ then we are done. Thus we may as well assume that for any $x \in X$, there exist distinct points $\xi_1, \xi_2, \xi_3 \in K_x$ and $\alpha_1, \alpha_2, \alpha_3 > 0$ (all depending on $x$) such that any two of the vectors $\overrightarrow{x\xi_i}, \overrightarrow{x\xi_j}$ for $i \neq j$ are linearly independent. As mentioned in the previous section, the circumcenter extensions of $f$ and $f^{-1}$ are bi-Lipschitz homeomorphisms which are inverses of each other. Thus there are sets $A \subset X$ and $B \subset Y$ of full measure (with respect to the Riemannian volume measures) such that $F$ is differentiable at all points of $A$ and $F^{-1}$ is differentiable at all points of $B$. Since $F$ is bi-Lipschitz, the set $F^{-1}(B) \subset X$ has full measure, thus so does the set $C := A \cap F^{-1}(B)$. For any point $x$ of $C$, $F$ is differentiable at $x$, $F^{-1}$ is differentiable at $F(x)$, and by the Chain Rule the derivatives $dF_x, dF^{-1}_{F(x)}$ are inverses of each other, so $dF_x : T_x X \to T_{F(x)} Y$ is an isomorphism for all $x \in C$. Now let $x \in C$. As remarked earlier, we may assume that there are distinct points $\xi_1, \xi_2, \xi_3 \in K_x$ such that $$\label{no1} \alpha_1 \overrightarrow{x\xi_1} + \alpha_2 \overrightarrow{x\xi_2} + \alpha_3 \overrightarrow{x\xi_3} = 0$$ for some $\alpha_1, \alpha_2, \alpha_3 > 0$, and such that any two of the vectors $\overrightarrow{x\xi_i}, \overrightarrow{x\xi_j}$ for $i \neq j$ are linearly independent. By Proposition \[dFstar\], we have $$dF^*_x ( \alpha_1 \overrightarrow{F(x)f(\xi_1)} + \alpha_2 \overrightarrow{F(x)f(\xi_2)} + \alpha_3 \overrightarrow{F(x)f(\xi_3)} ) = \alpha_1 \overrightarrow{x\xi_1} + \alpha_2 \overrightarrow{x\xi_2} + \alpha_3 \overrightarrow{x\xi_3} = 0$$ and hence $$\label{no2} \alpha_1 \overrightarrow{F(x)f(\xi_1)} + \alpha_2 \overrightarrow{F(x)f(\xi_2)} + \alpha_3 \overrightarrow{F(x)f(\xi_3)} = 0$$ since $dF^*_x$ is an isomorphism. Now let $$\begin{aligned} a_1 = < \overrightarrow{x\xi_1}, \overrightarrow{x\xi_2} > \ , & \ a_2 = < \overrightarrow{F(x)f(\xi_1)}, \overrightarrow{F(x)f(\xi_2)} > \\ b_1 = < \overrightarrow{x\xi_1}, \overrightarrow{x\xi_3} > \ , & \ b_2 = < \overrightarrow{F(x)f(\xi_1)}, \overrightarrow{F(x)f(\xi_3)} > \\ c_1 = < \overrightarrow{x\xi_2}, \overrightarrow{x\xi_3} > \ , & \ c_2 = < \overrightarrow{F(x)f(\xi_2)}, \overrightarrow{F(x)f(\xi_3)} > \\\end{aligned}$$ Taking inner products of the left-hand side of equation (\[no1\]) above with the vectors $\overrightarrow{x\xi_i}, i = 1,2,3$, and taking inner products of the left-hand side of equation (\[no2\]) above with the vectors $\overrightarrow{F(x)f(\xi_i)}, i = 1,2,3$, we find that the vectors $\overrightarrow{u_i} := (a_i, b_i, c_i), i = 1,2$ both satisfy the same linear system of equations $$T \overrightarrow{u} = \overrightarrow{w}$$ where $T$ is the $3 \times 3$ matrix $$T = \begin{pmatrix} \alpha_2 & \alpha_3 & 0 \\ \alpha_1 & 0 & \alpha_3 \\ 0 & \alpha_1 & \alpha_2 \end{pmatrix}$$ and $\overrightarrow{w}$ is the column vector $$\overrightarrow{w} = \begin{pmatrix} -\alpha_1 \\ -\alpha_2 \\ -\alpha_3 \end{pmatrix}$$ A computation gives $\det(T) = -2\alpha_1 \alpha_2 \alpha_3 < 0$, so $T$ is nonsingular, and it follows that $\overrightarrow{u_1} = \overrightarrow{u_2}$. We thus have $$< \overrightarrow{x\xi_i}, \overrightarrow{x\xi_j} > = < \overrightarrow{F(x)f(\xi_i)}, \overrightarrow{F(x)f(\xi_j)} >$$ for all $1 \leq i,j \leq 3$. On the other hand, by Proposition \[dFstar\], we have $$< \overrightarrow{x\xi_i}, \overrightarrow{x\xi_j} > = < dF_x(\overrightarrow{x\xi_i}), \overrightarrow{F(x)f(\xi_j)} >$$ for all $1 \leq i,j \leq 3$, thus $$\label{no3} < dF_x(\overrightarrow{x\xi_i}), \overrightarrow{F(x)f(\xi_j)} > = < \overrightarrow{F(x)f(\xi_i)}, \overrightarrow{F(x)f(\xi_j)} >$$ for all $1 \leq i,j \leq 3$. Since the dimension of $X$ is two, the span of the vectors $\overrightarrow{x\xi_1}, \overrightarrow{x\xi_2}$ equals $T_x X$, thus since $dF^*_x$ is an isomorphism it follows from Proposition \[dFstar\] that the span of the vectors $\overrightarrow{F(x)f(\xi_1)}, \overrightarrow{F(x)f(\xi_2)}$ equals $T_{F(x)} Y$. Fixing $i$ and putting $j = 1,2$ in equation (\[no3\]) above, it follows that $$dF_x(\overrightarrow{x\xi_i}) = \overrightarrow{F(x)f(\xi_i)}$$ for all $1 \leq i \leq 3$. Applying $dF^*_x$ to both sides of the above equation and using Proposition \[dFstar\] it follows that $$dF^*_x dF_x ( \overrightarrow{x\xi_i} ) = \overrightarrow{x\xi_i}$$ for all $1 \leq i \leq 3$. Since the vectors $\overrightarrow{x\xi_1}, \overrightarrow{x\xi_2}$ span $T_x X$, it follows that $dF^*_x dF_x = id$. Thus $dF_x$ is an isometry for all $x \in C$, in particular $||dF_x|| = 1$ for all $x \in C$. Now it is a classical fact that if a Lipschitz map $F$ between Riemannian manifolds satisfies $||dF|| \leq L$ almost everywhere for some constant $L$, then $F$ is $L$-Lipschitz. It follows that the circumcenter extension $F : X \to Y$ of the Moebius map $f$ is $1$-Lipschitz. Now we know from [@biswas6] that $F^{-1}$ is the circumcenter extension of the Moebius map $f^{-1}$. Applying the same argument to the Moebius map $f^{-1}$, we obtain that its circumcenter extension $F^{-1}$ is also $1$-Lipschitz. Since both $F$ and $F^{-1}$ are $1$-Lipschitz, $F$ is an isometry. $\diamond$

--- abstract: 'After almost 20 years of hunting, only about a dozen hot corinos, hot regions enriched in interstellar complex organic molecules (iCOMs), are known. Of them, many are binary systems with the two components showing drastically different molecular spectra. Two obvious questions arise. Why are hot corinos so difficult to find and why do their binary components seem chemically different? The answer to both questions could be a high dust opacity that would hide the molecular lines. To test this hypothesis, we observed methanol lines at centimeter wavelengths, where dust opacity is negligible, using the Very Large Array interferometer. We targeted the NGC 1333 IRAS 4A binary system, for which one of the two components, 4A1, has a spectrum deprived of iCOMs lines when observed at millimeter wavelengths, while the other component, 4A2, is very rich in iCOMs. We found that centimeter methanol lines are similarly bright toward 4A1 and 4A2. Their non-LTE analysis indicates gas density and temperature ($\geq2\times10^6$ cm$^{-3}$ and 100–190 K), methanol column density ($\sim10^{19}$ cm$^{-2}$) and extent ($\sim$35 au in radius) similar in 4A1 and 4A2, proving that both are hot corinos. Furthermore, the comparison with previous methanol line millimeter observations allows us to estimate the optical depth of the dust in front of 4A1 and 4A2, respectively. The obtained values explain the absence of iCOMs line emission toward 4A1 at millimeter wavelengths and indicate that the abundances toward 4A2 are underestimated by $\sim$30%. Therefore, centimeter observations are crucial for the correct study of hot corinos, their census, and their molecular abundances.' author: - Marta De Simone - Cecilia Ceccarelli - Claudio Codella - 'Brian E. Svoboda' - Claire Chandler - Mathilde Bouvier - Yamamoto Satoshi - Nami Sakai - Paola Caselli - Cecile Favre - Laurent Loinard - Bertrand Lefloch - Hauyu Baobab Liu - 'Ana López-Sepulcre' - 'Jaime E. Pineda' - Vianney Taquet - Leonardo Testi bibliography: - 'IRAS4A.bib' title: 'Hot Corinos chemical diversity: myth or reality? ' --- Introduction {#sec:intro} ============ Interstellar complex organic molecules (iCOMs) are molecules detected in the interstellar medium containing carbon and at least six atoms [@herbst_complex_2009; @ceccarelli_seeds_2017]. These molecules are of particular interest because they carry a substantial fraction of carbon that can be used for prebiotic chemistry [e.g. @caselli_our_2012]. In solar-like young Class 0 protostars, iCOMs are found in relatively large quantities toward the so-called hot corinos, which are compact ($\leq$100 au), hot ($\geq$100 K) and dense ($\geq10^7$ cm$^{-3}$) regions enriched in iCOMs at the center of the envelopes accreting the future star [@ceccarelli_hot_2004; @ceccarelli_extreme_2007; @caselli_our_2012]. The first hot corino was discovered in 2003 toward the Class 0 source IRAS 16293–2422 [e.g. @cazaux_hot_2003; @jorgensen_alma_2016; @manigand_alma-pils_2020]. Since then other Class 0 hot corinos have been discovered: NGC 1333 IRAS 4A [hereafter IRAS 4A; e.g. @bottinelli_complex_2004; @taquet_constraining_2015; @lopez-sepulcre_complex_2017; @de_simone_glycolaldehyde_2017; @sahu_implications_2019], NGC 1333 IRAS 2A, NGC 1333 IRAS 4B [e.g. @jorgensen_probing_2005; @bottinelli_hot_2007; @maury_first_2014; @de_simone_glycolaldehyde_2017], HH 212 [@codella_water_2016; @bianchi_deuterated_2017; @lee_formation_2017; @lee_first_2019], B335 [@imai_discovery_2016], L483 [@oya_l483_2017; @jacobsen_organic_2018], Barnard1b-S [@marcelino_alma_2018], Ser-emb 1 [@martin-domenech_new_2019], BHR71-IRS1 [@yang_constraining_2020]. Lately, few more evolved Class I hot corinos were also discovered: NGC 1333 SVS13A [@de_simone_glycolaldehyde_2017; @bianchi_census_2019], B1a [@oberg_complex_2014] and Ser-emb 17 [@bergner_organic_2019]. Therefore, after almost 20 years, only about a dozen hot corinos are known. Recent surveys concluded that $\sim$30% of low-mass Class 0/I protostars show emission from at least three iCOMs [@de_simone_glycolaldehyde_2017; @belloche_questioning_2020]. Most of the hot corinos cited above turn out to be binary systems when imaged at high angular resolution. This is in agreement with previous surveys that found that 40-60% of protostars are multiple systems [@maury_first_2014; @tobin_vla_2016.] Interestingly, with the first hot corino maps it became clear that the two objects in a given binary system can substantially differ in molecular complexity. Illustrative examples are provided by IRAS 16293–2422 and IRAS 4A [@jorgensen_alma_2016; @lopez-sepulcre_complex_2017]. IRAS 16293–2422 is composed by two sources, A and B, separated by $5\farcs1$ ($\sim$ 720 au), where source A, weaker in millimeter continuum emission, is brighter in iCOMs lines than source B [e.g. @caux_timasss_2011; @pineda_first_2012; @jorgensen_alma_2016; @manigand_alma-pils_2020]. IRAS 4A, located in the NGC 1333 region in the Perseus cloud at $(299\pm15)$ pc of distance [@zucker_mapping_2018], is also a binary system composed by IRAS 4A1 and IRAS 4A2 (hereafter 4A1 and 4A2), separated by 1.8$''$ ($\sim$540 au): while 4A1 is brighter in the mm continuum than 4A2, only 4A2 shows bright iCOMs lines [@taquet_constraining_2015; @lopez-sepulcre_complex_2017; @de_simone_glycolaldehyde_2017]. However, not always the brightest millimeter continuum source in a binary system is the one weak in iCOMs emission [see e.g. @ospina-zamudio_first_2018]. In summary, despite two decades of hunting, only a dozen hot corinos are known so far. Of them, many are binary systems with the two components showing drastically different molecular spectra. Two related questions arise: (1) Why are hot corinos so difficult to find? While it is known that not all Class 0/I sources possess hot corinos [e.g. @sakai_warm_2013; @higuchi_chemical_2018; @bouvier_hunting_2020], observational biases might hamper their detection. (2) Why do coeval objects seem drastically differ in their chemical composition? Is this a real difference or is it only/mostly due to observational biases? A major observational bias could be caused by the dust opacity, which could be very high in Class 0/I sources, due to their high densities and, consequently, column densities [e.g. @miotello_grain_2014; @galvan-madrid_effects_2018; @galametz_low_2019]. If the effect of dust absorption is not negligible, there are three major consequences: (1) hot corinos may be difficult to detect in the millimeter (also) because of the high dust absorption of the iCOMs lines; (2) the molecular complexity diversity observed in binary systems objects may reflect a difference in the front dust column density rather than a real chemical difference of the two objects; (3) the iCOMs abundances in hot corinos could have been so far underestimated. In order to test this hypothesis, we targeted the IRAS 4A binary system, where the two objects show extremely different iCOMs line spectra at mm wavelengths (see above), and carried out observations of several methanol lines, one of the simplest iCOMS, at centimeter wavelengths where the dust is optically thin. Observations {#sec:obs} ============ The IRAS 4A system was observed at 1.3 cm using K–band receivers (18-26.5 GHz) of the Very Large Array (VLA) in C-configuration (35–3400 m) on 2018 December 10 (project ID: VLA/18B-166). We targeted 10 CH$_3$OH lines, with frequencies from 24.9 to 26.4 GHz, upper level energies $E_{up}$ from 36 to 175 K and Einstein coefficients $A_{ij}$ in the range (0.5 – 1.1) $\times 10^{-7}$ s$^{-1}$ (Table \[tab:spectral\_params&fit\_res\]). The observed spectra were divided into eight spectral windows with $\sim$0.017 MHz (0.2 km s$^{-1}$) spectral resolution and $\sim1''$ ($\sim$300 au at the distance of IRAS 4A) angular resolution. The observations were centered on 4A2, at $\rm \alpha(J2000)=03^h29^m10\fs43$, $\rm \delta(J2000)=31^\circ13'32\farcs1$. The flux calibrators were J0137+3309 and J0521+1638, while the bandpass and the gain ones were J0319+4130, and J0336+3218, respectively. The absolute flux calibration error is $\leq$15%[^1]. The data reduction and cleaning process were performed using the CASA[^2] package while data analysis and images were performed using the GILDAS[^3] package. We obtained a continuum image by averaging line-free channels from all the spectral windows (Figure \[fig:methanol\_maps+cont\]). We self-calibrated, in phase amplitude, using the line-free continuum channels and applied the solutions to both the continuum and molecular lines. The dynamic range, as defined by peak source flux over rms noise, was improved by 20% by the self-calibration. The final RMS noise in the continuum image, 3$\mu$Jy beam$^{-1}$, is consistent with that reported by the VLA Exposure Time Calculator for a line-free continuum bandwidth of 4.5GHz, 26 antennas, and an on-source integration time of 3 hr. The cube were subsequently continuum subtracted, smoothed to 1 km s$^{-1}$ ($\sim$ 0.08 MHz) and cleaned in CASA using a multiscale deconvolution[^4] (scales=\[0,5,15,18,25\]) with natural weighting. The synthesized beams for each spectral window are reported in Table \[tab:spectral\_params&fit\_res\]. The half power primary beam is $\sim 80''$. Results {#sec:results} ======= Continuum emission map ---------------------- Figure \[fig:methanol\_maps+cont\] reports the map of the continuum emission at 25 GHz. The two continuum peaks mark the two protostars, whose coordinates ($\rm \alpha(J2000)=03^h29^m10\fs536$, $\rm \delta(J2000)=31^\circ13'31\farcs07$ for 4A1, and $\rm \alpha(J2000)=03^h29^m10\fs43$, $\rm \delta(J2000)=31^\circ13'32\farcs1$ for 4A2) are consistent with those derived by @tobin_vla_2016 and @lopez-sepulcre_complex_2017 with higher angular resolution observations. Since the angular resolution of our observations ($\sim1''$) is smaller than the separation between 4A1 and 4A2 (1$'\farcs$8), they are clearly disentangled in our images, even if individually unresolved with the current resolution. At cm wavelengths, 4A1 shows a brighter continuum emission (due to dust or free-free) than 4A2. The peak fluxes are (2.1 $\pm$ 0.3) mJy beam$^{-1}$ and (0.47 $\pm$ 0.07) mJy beam$^{-1}$ toward 4A1 and 4A2, respectively. Taking into account the slightly different wavelength (1.05 cm) and angular resolution ($\mathbf{\sim 0\farcs1}$), these values are consistent with the ones measured by @tobin_vla_2016: (1.3 $\pm$ 0.2) mJy beam$^{-1}$ for 4A1 and (0.38 $\pm$ 0.04) mJy beam$^{-1}$ for 4A2. Methanol lines --------------   All the targeted methanol lines are detected with a signal-to-noise ratio larger than 3 (Table \[tab:spectral\_params&fit\_res\]). Their velocity-integrated spatial distribution is shown in Fig. \[fig:methanol\_maps+cont\]. The methanol emission peaks exactly toward the 4A1 and 4A2 continuum peaks, and it is well disentangled, even if unresolved at the current angular resolution, around the two protostars. Figure \[fig:spectra\_ch3oh\_iras4a1\] shows the 10 methanol line spectra, isolated and not contaminated by other species, extracted toward the 4A1 and 4A2 continuum peaks. The lines are slightly brighter toward 4A2 than 4A1, whereas the linewidths are very similar (see also Tab. \[tab:spectral\_params&fit\_res\]). We derived the velocity-integrated line intensities for each detected CH$_3$OH transition using a Gaussian fit, being the profile Gaussian-like. The fit results for both sources, namely the integrated emission ($ \int T_b dV$), the linewidth (FWHM), the peak velocities ($V_{peak}$) and the RMS computed for each spectral window, are reported in Table \[tab:spectral\_params&fit\_res\]. The velocity peaks are consistent with the systemic velocity of the molecular envelope surrounding IRAS 4A [$\sim6.7$ km s$^{-1}$; @choi_high-resolution_2001]. The line-widths are between 3 and 4 km s$^{-1}$ in agreement with those found by @taquet_constraining_2015 and @lopez-sepulcre_complex_2017 toward 4A2 at mm wavelength. In summary, our new VLA observations show a first clear important result: the detection of methanol emission toward 4A1, the protostar where previous mm observations showed no iCOMs emission [@lopez-sepulcre_complex_2017]. Centimeter versus millimeter observations: dust absorption derivation {#sec:cm_vs_mm} ===================================================================== We compared our new cm observations of methanol lines with previous ones at 143–146 GHz in order to understand whether the dust absorption, more important at mm than at cm wavelengths, may explain the absence of iCOMs mm line emission in 4A1 [@lopez-sepulcre_complex_2017]. We first carried out a non-LTE analysis of the cm methanol lines from which we derived the gas temperature, density and CH$_3$OH column density toward 4A1 and 4A2 (§\[subsect:T\_n\_derivation\]). Then, using the same parameters, we predicted the methanol line intensities at 143–146 GHz, the frequency of the observations by @taquet_constraining_2015 [Section \[subsect:int\_predictions\]]. Finally, we compared the predicted and measured mm line intensities and we attributed the difference to the absorption of the dust between us and the gas emitting methanol, via the usual equation: $$\label{eq:Absorption} I^{obs}_\nu=I^{pred}_\nu e^{-\tau_\nu}$$ in order to derive the dust optical depth toward 4A1 and 4A2, respectively (Section \[sec:dust-abs\]). Please note that the foreground dust opacity obtained by Eq. \[eq:Absorption\] assumes that the absorbing dust fully covers the emitting gas area, which may not be necessarily the case. Yet, the derived attenuation of the methanol line intensities is still valid, even though it is only an average over the emitting gas area. ![ Density-Temperature $\rm \chi^2$ contour plots. The contours represent 1$\sigma$ confidence level contours for 4A1 (blue) and 4A2 (green), respectively, assuming the best fit values of N$_{\rm CH_3OH}$ and $\theta$ in Table \[tab:LVG\_results\]. The best fit solutions are marked by the red (4A1) and magenta (4A2) asterisks. []{data-label="fig:LVG_results"}](figures/dens_vs_T_4a1_4a2.eps) non-LTE analysis of the cm methanol lines {#subsect:T_n_derivation} ----------------------------------------- To derive the physical properties of the gas emitting CH$_3$OH, namely gas temperature, density and methanol column density, we performed a non-LTE analysis using a Large Velocity Gradient (LVG) code @ceccarelli_theoretical_2003. CH$_3$OH can be identified in A- and E-type due to the total spin (I) state of the hydrogen nuclei in the CH$_3$ group: A-type if the total spin function is symmetric (I=3/2), E-type if asymmetric (I=1/2) [@rabli_rotational_2010]. We used the collisional coefficients of both types of CH$_3$OH with para-H$_2$, computed by @rabli_rotational_2010 between 10 and 200 K for the first 256 levels and provided by the BASECOL database [@dubernet_basecol2012:_2013]. We assumed a spherical geometry to compute the line escape probability [@de_jong_hydrostatic_1980], the CH$_3$OH-A/CH$_3$OH-E ratio equal to 1, the H$_2$ ortho-to-para ratio equal to 3, and that the levels are populated by collisions and not by the absorption of the dust background photons whose contribution is very likely negligible due to the low values of the CH$_3$OH Einstein coefficients $\rm A_{ij}$. Please note that the present LVG analysis only accounts for the line optical depth (to have also the dust $\tau$ in the methanol emitting region would require information on the structure of the region which we do not have, as the emission is unresolved). We ran a large grid of models ($\geq$10000) covering the frequency of the observed lines, a total (CH$_3$OH-A plus CH$_3$OH-E) column density $ N_{\rm CH_3OH}$ from $2\times10^{16}$ to $8\times10^{19}$ cm$^{-2}$, a gas density $ n_{\rm H_2}$ from $1\times10^{6}$ to $2\times10^{8}$ cm$^{-3}$, and a temperature T from 80 to 200 K. We then simultaneously fitted the measured CH$_3$OH-A and CH$_3$OH-E line intensities via comparison with those simulated by the LVG model, leaving $ N_{\rm CH_3OH}$, $ n_{\rm H_2}$, $T$, and the emitting size $\theta$ as free parameters. Following the observations, we assumed the linewidths equal to 3.5 km s$^{-1}$ and 3.0 km s$^{-1}$ for 4A1 and 4A2, respectively, and we included the calibration uncertainty (15%) in the observed intensities. The best fit is obtained for a total CH$_3$OH column density $\rm N_{CH_3OH}=2.8\times10^{19}$ cm$^{-2}$ with reduced chi-square $\rm \chi^2_R=0.6$ for 4A1 and $\rm N_{CH_3OH}=1\times10^{19}$ cm$^{-2}$ with $\rm \chi^2_R=0.1$ for 4A2. All the observed lines are predicted to be optically thick and emitted by a source of $0\farcs22$ for 4A1 and $0\farcs24$ for 4A2 ($\sim$70 au) in diameter. Solutions with $\rm N_{CH_3OH}\geq1\times10^{18}$ cm$^{-2}$ for 4A2 and $\geq1\times10^{19}$ cm$^{-2}$ for 4A1 are within 1$\sigma$ of confidence level. Increasing the methanol column density, the $\chi^2_R$ decreases until it reaches a constant value for $\rm N_{CH_3OH}\geq1\times10^{19}$ cm$^{-2}$ for 4A1 and $\rm N_{CH_3OH}\geq3\times10^{19}$ cm$^{-2}$ for 4A2; this is because all the observed lines become optically thick ($\tau\sim10-30$ for 4A1,$\tau\sim2-6$ for 4A2) and, consequently, the emission becomes that of a black body. The results do not change assuming a linewidth $\pm$0.5 km s$^{-1}$ with respect to the chosen one. Figure \[fig:LVG\_results\] shows, for both sources, the density-temperature $\chi^2$ surface of the $ N_{\rm CH_3OH}$ best fit. The gas temperature is (90–130) K for 4A1 and (120–190) K for 4A2, while for the gas density we obtained a lower limit of $2\times10^{6}$ cm$^{-3}$ for 4A1 and $1.5\times10^{7}$ cm$^{-3}$ for 4A2, which implies that the levels are LTE populated. The fit results are reported in Table \[tab:LVG\_results\]. The derived $ n_{\rm H_2}$ and T are consistent with those computed with the model summarised in @su_infall_2019 using our sizes. Predictions of mm methanol line intensities {#subsect:int_predictions} ------------------------------------------- Adopting the 1$\sigma$ range of gas temperature and density derived for 4A1 and 4A2 (Table \[tab:LVG\_results\]), we ran a new grid of LVG models with the CH$_3$OH column density from $1\times10^{18}$ to $8\times10^{19}$ cm$^{-2}$ at 143–146 GHz to predict the methanol line intensities observed by @taquet_constraining_2015. We then used the CH$_3$OH $3_1-2_1$ A$^+$ line at 143.866 GHz, which provides the most stringent constraint to the dust optical depth, to compare the predicted intensity with that observed by @taquet_constraining_2015. In the comparison, we took into account our LVG-derived source size and the angular resolution of the @taquet_constraining_2015 observations. While for 4A2 we considered the line intensity quoted by @taquet_constraining_2015, for 4A1, not having CH$_3$OH detection, we used the 3$\sigma$ level of the @taquet_constraining_2015 observations integrated over 3 km s$^{-1}$ (average linewidth toward 4A1: see §\[sec:results\]). The 4A1 and 4A2 CH$_3$OH predicted and observed values are reported in Table \[tab:LVG\_results\]. While the two intensities are similar toward 4A2, they differ by about a factor five toward 4A1. Dust absorption toward 4A1 and 4A2 {#sec:dust-abs} ---------------------------------- Assuming that the difference between the predicted and observed intensities is due to the (foreground) dust absorption and using Eq. \[eq:Absorption\], we derived the dust optical depth at 143 GHz ($\tau_{\rm dust}^{\rm 143GHz}$; Table \[tab:LVG\_results\]). While $\rm \tau_{dust}^{\rm 143GHz}$ toward 4A2 is small ($\sim0.3$), that toward 4A1 is large ($\geq1.6$) enough to attenuate the methanol line intensity by a factor $\geq$5. Therefore, the dust is affecting the mm line emission differently in the two sources. Discussion ========== Is IRAS 4A1 a hot corino? ------------------------- So far, only about a dozen hot corinos have been detected (§\[sec:intro\]) and the question arises whether this is because they are rare or because the searches have always been carried out at mm wavelengths, where dust could heavily absorb the line emission. Our first result is that a source that was supposed not to be a hot corino based on mm observations, IRAS 4A1 [@lopez-sepulcre_complex_2017], indeed possesses a region with temperature $\geq100$ K (§\[fig:LVG\_results\]), namely the icy mantle sublimation one, and shows methanol emission (§\[sec:results\]), the simplest of the iCOMs, when observed at cm wavelengths. According to its definition [@ceccarelli_hot_2004], thus, IRAS 4A1 is a hot corino. Although we cannot affirm that hot corinos are ubiquitous, it is clear that the searches at mm wavelengths may be heavily biased and that complementary cm observations are necessary to account for dust opacity and understand the occurrence of hot corinos. 4A2 versus 4A1: are they chemically different? ---------------------------------------------- Unlike 4A2, no sign of iCOMs mm emission was revealed toward 4A1 [@taquet_constraining_2015; @lopez-sepulcre_complex_2017]. Using ALMA observations at 250 GHz, @lopez-sepulcre_complex_2017 found that the iCOMs abundances toward 4A2 and 4A1 differ by more than a factor 17, with the largest values ($\sim$100) for HCOOCH$_3$ and CH$_3$CN. The first question to answer is whether the chemical difference between the two coeval objects is real or due to a different absorption by the surrounding dust. In Section \[sec:dust-abs\], we found that $\rm \tau_{dust}$ at 143 GHz toward 4A1 and 4A2 is $\geq$1.6 and 0.3, respectively (see Table \[tab:LVG\_results\]). Using the dependence of $\rm \tau_{dust}$ from the frequency ($\rm \tau_{\nu_2}/\tau_{\nu_1}=(\nu_2/\nu_1)^\beta$) and assuming $\beta=2$ (ISM value), the optical depth scaled at 250 GHz [frequency at which @lopez-sepulcre_complex_2017 derived the above iCOMs abundance ratios] is $\geq$4.9 for 4A1 and 0.9 for 4A2. Therefore, the different dust absorption toward 4A1 and 4A2 provides us, as lower limit, a factor 55 difference in their line intensities ($\rm I^{A2}/I^{A1}$), comparable to the 4A2/4A1 iCOMs abundance ratios derived by @lopez-sepulcre_complex_2017. A large dust absorption was also suggested by the anomalous flattened continuum spectral index at 100-230 GHz [@li_systematic_2017] and the 90$^{\circ}$ flipping of the linear polarization position angles observed at above and below 100 GHz frequencies [@ko_resolving_2020]. Although we cannot exclude that a real chemical difference exists between 4A1 and 4A2, the observations so far available cannot support that hypothesis. Centimeter observations of other iCOMs than methanol are necessary to settle this issue. This conclusion may apply to other binary systems where an apparent chemical difference is observed using mm observations. Are the iCOMs abundances in hot corinos underestimated? ------------------------------------------------------- The dust absorption also affects the iCOMs line intensities in 4A2. At 143 GHz $\rm \tau_{dust}$ is 0.3, which leads to underestimate the iCOMs abundances by about 30%. At higher frequencies, this factor becomes more important; e.g. at 250 GHz, where several hot corinos studies are carried out (see references in §\[sec:intro\]), the absorption factor would be 2.5, and at 350 GHz, frequency where the most sensitive iCOMs search has been carried out [e.g. @jorgensen_alma_2016], the absorption factor would be 6. This behaviour also agrees with what already found in massive hot cores [e.g. @rivilla_formation_2017]. Therefore, in order to derive reliable iCOMs abundances complementary cm observations are needed to estimate the dust absorption. Conclusion {#sec:conclusion} ========== We carried out observations of methanol lines at cm wavelengths with the VLA interferometer toward the binary system IRAS 4A, where previous mm observations showed a possible chemical differentiation between the two objects. Specifically, while 4A2 showed iCOMs line emission, 4A1 did not. Our new observations detected ten methanol lines in 4A1 and 4A2 with similar intensities. Using a non-LTE analysis and comparing with previous methanol mm observations, we showed that (1) 4A1 is a hot corino, (2) the lack of iCOMs detection toward 4A1 at mm wavelengths is caused by a large dust optical depth, and (3) the determination of the iCOMs abundances toward 4A2 via mm observations is slightly underestimated by the dust absorption. Therefore, the difficulty in discovering new hot corinos could be because the searches have been carried out at (sub)mm wavelengths, where the dust absorption might be not negligible. The suspected different chemical nature of coeval objects of the same binary system needs also to be verified at cm wavelengths, as well as the iCOMs abundances estimated from mm observations. We conclude that centimeter observations of hot corinos are of paramount importance for their correct study. In the future, next generation instruments in the centimeter wavelenght regime, such as ngVLA [@mcguire_science_2018] and SKA [@codella_complex_2015], could be even the most efficient way to identify hot corinos and certainly the most appropriate facilities to study them. We thank the referee P.T.P Ho for his fruitful comments and suggestions. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, for the Project “The Dawn of Organic Chemistry” (DOC), grant agreement No 741002. It was supported by the project PRIN-INAF 2016 The Cradle of Life–GENESIS-SKA (General Conditions in Early Planetary Systems for the rise of life with SKA), and partly supported by the Italian Ministero dell’Istruzione, Università e Ricerca, through the grant Progetti Premiali 2012–iALMA (CUP C52I13000140001). H.B.L. is supported by the Ministry of Science and Technology (MoST) of Taiwan (grant Nos. 108-2112-M-001-002-MY3). C.F. acknowledge support from the French National Research Agency in the framework of the Investissements d’Avenir program (ANR-15-IDEX-02), through the funding of the “Origin of Life” project of the Univ. Grenoble-Alpes. [^1]: https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/fdscale [^2]: https://casa.nrao.edu/ [^3]: http://www.iram.fr/IRAMFR/GILDAS [^4]: This technique is a scale-sensitive deconvolution algorithm efficient for images with complicated and extended spatial structures. In fact, it allows us to model the sky brightness as a linear combination of flux components of different scale sizes. The scale sizes are chosen following approximately the sizes of the dominant structures in the image and including the “0” scale to model the unresolved ones (see [casadocs-deconvolution-algorithms](https://casa.nrao.edu/casadocs/casa-5.1.0/synthesis-imaging/deconvolution-algorithms)).

[**Canonical dual theory applied to a Lennard-Jones potential minimization problem**]{}\ [Jiapu Zhang]{}\ Centre for Informatics and Applied Optimization, &\ Graduate School of Sciences, Information Technology and Engineering,\ The University of Ballarat, Mount Helen, VIC 3350, Australia.\ Emails: j.zhang@ballarat.edu.au, jiapu\_zhang@hotmail.com\ Telephones: 61-3-5327 9809 (office), 61-4 2348 7360 (mobile)\ [**Abstract**]{} The simplified Lennard-Jones (LJ) potential minimization problem is $$\mbox{minimize}~~~f(x)=4\sum_{i=1}^N \sum_{j=1,j<i}^N \left( \frac{1}{\tau_{ij}^6} -\frac{1}{\tau_{ij}^3} \right)~~~\mbox{subject to}~~~ x\in \mathbb{R}^n,$$ where $\tau_{ij}=(x_{3i-2}-x_{3j-2})^2 +(x_{3i-1}-x_{3j-1})^2 +(x_{3i} -x_{3j} )^2$, $(x_{3i-2},x_{3i-1},x_{3i})$ is the coordinates of atom $i$ in $\mathbb{R}^3$, $i,j=1,2,\dots,N(\geq 2 \quad \text{integer})$, $n=3N$ and $N$ is the whole number of atoms. The nonconvexity of the objective function and the huge number of local minima, which is growing exponentially with $N$, interest many mathematical optimization experts. In this paper, the canonical dual theory elegantly tackles this problem illuminated by the amyloid fibril molecular model building.\ [**Key words**]{} Mathematical Canonical Duality Theory $\cdot$ Mathematical Optimization $\cdot$ Lennard-Jones Potential Minimization Problem $\cdot$ Global Optimization. Introduction ============ Neutral atoms are subject to two distinct forces in the limit of large distance and short distance: a dispersion force (i.e. attractive van der Waals (vdw) force) at long ranges, and a repulsion force, the result of overlapping electron orbitals. The Lennard-Jones (L-J) potential represents this behavior ([http://en.wikipedia.org/wiki/Lennard-Jones\_potential]{}, or [@locatelli2008] and references therein). The L-J potential is of the form $$\label{LJ_r_form} V(r)=4\varepsilon \left[ (\frac{\sigma}{r})^{12} - (\frac{\sigma}{r})^6 \right],$$ where $r$ is the distance between two atoms, $\varepsilon$ is the depth of the potential well and $\sigma$ is the atom diameter; these parameters can be fitted to reproduce experimental data or deduced from results of accurate quantum chemistry calculations. The $(\frac{\sigma}{r})^{12}$ term describes repulsion and the $(\frac{\sigma}{r})^6$ term describes attraction (Fig. \[LJ\_potential\]). ![The Lennard-Jone Potential (formulas (\[LJ\_r\_form\]) and (\[LJ\_AB\_form\])) (This Figure can be found in website [http://homepage.mac.com/swain/CMC/DDResources/mol\_interactions/molecular\_interactions.html]{}](LJ_potential.eps){width="4.2in"} ). \[LJ\_potential\] In Fig. \[LJ\_potential\] we may see two points: (I) $V(r)=0$ (but the value of $V(r)$ is not the minimal value) when $r=\sigma$ (i.e. the distance between two atoms equals to the sum of [*atom radii*]{} of the atoms); and (II) when $r=2^{1/6}\sigma$ (i.e. the distance between two atoms equals to the sum of [*vdw radii*]{} of the atoms), the value of $V(r)$ reaches its minimal value $-\varepsilon$ (i.e. the bottom of the potential well; the force between the atoms is zero at this point). This paper is written based on (II). If we introduce the coordinates of the atoms whose number is denoted by $N$ and let $\varepsilon = \sigma =1$ be the reduced units, the form (\[LJ\_r\_form\]) becomes $$\label{LJ_x_form} f(x)=4\sum_{i=1}^N \sum_{j=1,j<i}^N \left( \frac{1}{\tau_{ij}^6} -\frac{1}{\tau_{ij}^3} \right),$$ where $\tau_{ij}=(x_{3i-2}-x_{3j-2})^2 +(x_{3i-1}-x_{3j-1})^2 +(x_{3i} -x_{3j} )^2 =||X_i-X_j||^2_2$ and $(x_{3i-2},x_{3i-1},x_{3i})$ is the coordinates of atom $i$, $i,j=1,2,\dots, N (\geq 2)$. The minimization of L-J potential $f(x)$ on $\mathbb{R}^n$ (where $n=3N$) is an optimization problem: $$\label{LJ_f_form} \min_{s.t. x\in \mathbb{R}^{3N}} f(x).$$ This optimization problem interests many optimization experts, for example, Pardalos [@pardalossx1994], Xue [@xuemr1992; @xue1993; @xue1994a; @xue1994b; @xue2002], Huang [@huang2002b; @huang2002a] et al..\ For (\[LJ\_f\_form\]), when its global optimization solution is reached, the value $r$ in (\[LJ\_r\_form\]) should be the sum of two [*vdw radii*]{} of the two atoms interacted. The three dimensional structure of a molecule with $N$ atoms can be described by specifying the 3-Dimensional coordinate positions $X_1, X_2, \dots, X_N \in \mathbb{R}^3$ of all its atoms. Given bond lengths $r_{ij}$ between a subset $S$ of the atom pairs, the determination of the molecular structure is $$\begin{aligned} (\mathcal{P}_0 ) \quad to \quad find \quad &X_1,X_2,\dots ,X_N \quad s.t. \quad ||X_i-X_j||=r_{ij}, (i,j)\in S, \label{orginal_problem}\end{aligned}$$ where $||\cdot ||$ denotes a norm in a real vector space and it is calculated as the Euclidean distance 2-norm in this paper. (\[orginal\_problem\]) can be reformulated as a mathematical global optimization problem (GOP) $$\begin{aligned} (\mathcal{P} ) \quad &\min P(X)=\sum_{(i,j)\in S} w_{ij} (||X_i-X_j||^2 -r_{ij}^2 )^2 \label{prime_problem}\end{aligned}$$ in the terms of finding the global minimum of the function $P(X)$, where $w_{ij}, (i,j)\in S$ are positive weights, $X = (X_1, X_2, \dots, X_N)^T \in \mathbb{R}^{N\times 3}$ [@morew1997] and usually $S$ has many fewer than $N^2/2$ elements due to the error in the theoretical or experimental data [@grossols2009; @zoubs1997]. There may even not exist any solution $X_1, X_2, \dots, X_N$ to satisfy the distance constraints in (\[orginal\_problem\]), for example when data for atoms $i, j, k \in S$ violate the triangle inequality; in this case, we may add a perturbation term $-\epsilon^TX$ to $P(X)$: $$\begin{aligned} (\mathcal{P}_{\epsilon} ) \quad &\min P_{\epsilon}(X)=\sum_{(i,j)\in S} w_{ij} (||X_i-X_j||^2 -r_{ij}^2 )^2 -\epsilon^TX, \label{prime_approx_problem}\end{aligned}$$ where $\epsilon \geq 0$. Thus, the L-J potential optimization problem (\[LJ\_f\_form\]) is rewritten into the optimization problem (\[prime\_approx\_problem\]).\ Problem (\[prime\_approx\_problem\]) is just the minimization problem of sum of fourth-order polynomials, which can be elegantly solved by the canonical dual theory (CDT) in optimization [@gao-book2000; @gaorp2010; @gaow2012]. We apply the above theory to an amyloid fibril molecular model building problem. The rest of this paper is arranged as follows. In the next section, i.e. Section 2, the CDT will be briefly introduced and its effectiveness will be illuminated by applying the CDT-based optimization approach to the famous double-well problem. In Section 3, the molecular model building works of prion AGAAAAGA amyloid fibrils will be done by the CDT. We find that just very slight refinement is needed by the optimization programs of computational chemistry package Amber 11 to get the optimal molecular models. This implies to us the effectiveness of the CDT to solve our L-J potential minimization problem. Thus, when using the time-consuming and costly X-ray crystallography or NMR spectroscopy we still cannot determine the 3D structure of a protein, we may introduce computational approaches or novel mathematical formulations and physical concepts into molecular biology to study molecular structures. These concluding remarks will be made in the last section, i.e. Section 4.\ The Canonical dual optimization approach ======================================== We briefly introduce the CDT of [@gao-book2000; @gaorp2010; @gaow2012] specially for solving the following minimization problem of the sum of fourth-order polynomials: $$\begin{aligned} &&(\mathcal{P}): \min_x \left\{ P(x) =\sum_{i=1}^m W_i (x) + \frac{1}{2}x^TQx - x^Tf: x\in \mathbb{R}^n \right\} , \label{primeP}\\ &&where \quad W_i (x)= \frac{1}{2} \alpha_i \left( \frac{1}{2} x^TA_ix +b_i^Tx +c_i \right)^2, A_i=A^T_i, Q=Q^T\in \mathbb{R}^{n\times n}, \nonumber\\ &&b_i, f\in \mathbb{R}^n, c_i, \alpha_i \in \mathbb{R}^1, i=1,2,\dots, n, x\in \mathcal{X} \subset \mathbb{R}^n. \nonumber\end{aligned}$$ The dual problem of $(\mathcal{P})$ is $$\begin{aligned} &&(\mathcal{P}^d): \max_{\varsigma} \left\{ P^d(\varsigma ) =\sum_{i=1}^m \left( c_i \varsigma_i -\frac{1}{2} \alpha_i^{-1} \varsigma^2 \right) -\frac{1}{2} F^T(\varsigma ) G^+(\varsigma ) F(\varsigma ): \varsigma \in S_a \right\} ,\label{dualPd}\\ &&where \quad F(\varsigma )=f-\sum_{i=1}^m \varsigma_i b_i, G(\varsigma )=Q+\sum_{i=1}^m \varsigma_i A_i, S_a=\{ \varsigma \in \mathbb{R}^m | F(\varsigma ) \in Col(G(\varsigma )) \}, \nonumber\end{aligned}$$ $G^+$ denotes the Moore-Penrose generalized inverse of $G$, and $Col(G(\varsigma ))$ is the column space of $G(\varsigma )$. The prime-dual Gao-Strang complementary function of CDT [@gao-book2000; @gaorp2010; @gaow2012] is $$\begin{aligned} \Xi (x,\varsigma )=\sum_{i=1}^m \left[ \left( \frac{1}{2} x^TA_ix +b_i^Tx +c_i \right) \varsigma_i -\frac{1}{2} \alpha_i^{-1} \varsigma_i^2 \right] +\frac{1}{2} x^TQx -x^Tf. \label{Gao-StrangComplementary} \end{aligned}$$ For $(\mathcal{P})$ and $(\mathcal{P}^d)$ we have the following CDT: [@gao-book2000; @gaorp2010; @gaow2012] The problem $(\mathcal{P}^d )$ is canonically dual to $(\mathcal{P} )$ in the sense that if $\bar{\varsigma}$ is a critical point of $P^d(\varsigma )$, then $\bar{x} =G^+(\bar{\varsigma} )F(\bar{\varsigma} )$ is a critical point of $P(x)$ on $\mathbb{R}^n$, and $P(\bar{x} )=P^d (\bar{\varsigma} )$. Moreover, if $\bar{\varsigma} \in S_a^+=\{ \varsigma \in S_a | G(\varsigma) \succ 0 \}$, then $\bar{\varsigma}$ is a global maximizer of $P^d(\varsigma )$ over $S_a^+$, $\bar{x}$ is a global minimizer of $P(x)$ on $\mathbb{R}^n$, and $$\begin{aligned} P(\bar{x})=\min_{x\in \mathbb{R}^n} P(x)=\Xi (\bar{x}, \bar{\varsigma}) =\max_{\varsigma \in S_a^+} P^d(\varsigma ) =P^d (\bar{\varsigma}).\end{aligned}$$ It is easy to prove that the canonical dual function $P^d(\varsigma )$ is concave on the convex dual feasible space $S_a^+$. Therefore, Theorem 1 shows that the nonconvex primal problem $(\mathcal{P} )$ is equivalent to a concave maximization problem $(\mathcal{P}^d )$ over a convex space $S_a^+$, which can be solved easily by well-developed methods. Over $S_a^-=\{ \varsigma \in S_a | G(\varsigma) \prec 0 \}$ we have the following theorem: [@gaow2012] Suppose that $\bar{\varsigma}$ is a critical point of $(\mathcal{P}^d)$ and the vector $\bar{x}$ is defined by $\bar{x} =G^+(\bar{\varsigma} )F(\bar{\varsigma} )$. If $\bar{\varsigma} \in S_a^-$, then on a neighborhood $\mathcal{X}_o \times \mathcal{S}_o \subset \mathcal{X} \times S_a^-$ of $(\bar{x}, \bar{\varsigma } )$, we have either $$\begin{aligned} P(\bar{x} ) = \min_{x \in \mathcal{X}_o} P(x) =\Xi (\bar{x}, \bar{\varsigma}) =\min_{\varsigma \in \mathcal{S}_o} P^d(\varsigma ) = P^d(\bar{\varsigma}), \label{equ-dualmin}\end{aligned}$$ or $$\begin{aligned} P(\bar{x} ) = \max_{x \in \mathcal{X}_o} P(x) =\Xi (\bar{x}, \bar{\varsigma}) =\max_{\varsigma \in \mathcal{S}_o} P^d(\varsigma ) = P^d(\bar{\varsigma}). \label{equ-dualmax}\end{aligned}$$ By the fact that the canonical dual function is a d.c. function (difference of convex functions) on $S_a^-$, the double-min duality (\[equ-dualmin\]) can be used for finding the biggest local minimizer of $(\mathcal{P} )$ and $(\mathcal{P}^d )$, while the double-max duality (\[equ-dualmax\]) can be used for finding the biggest local maximizer of $(\mathcal{P} )$ and $(\mathcal{P}^d )$. In physics and material sciences, this pair of biggest local extremal points play important roles in phase transitions.\ To illuminate the CDT above-mentioned, we minimize the well-known Double Well potential function [@gao-book2000] (blue colored in Fig. \[double\_well\]): $$P(x)=\frac{1}{2} (\frac{1}{2} x^2- 2)^2 -\frac{1}{2} x. \label{double_well_potential_prime}$$ ![The prime and dual double-well functions (Prime: blue, Dual: red).[]{data-label="double_well"}](double_well.eps){width="4.2in"} We can easily get $\Xi (x,\varsigma ) =(\frac{1}{2} x^2-2)\varsigma -\frac{1}{2} \varsigma^2 -\frac{1}{2} x$, $$P^d (\varsigma )=-\frac{1}{8\varsigma } -\frac{1}{2} \varsigma^2 -2\varsigma \label{double_well_potential_dual}$$ (red colored in Fig. \[double\_well\]) and $S_a^+=\{ \varsigma \in \mathbb{R}^1 | \varsigma >0\}$. Let $\Xi (x,\varsigma )'=0$, we get three critical points of $\Xi (x,\varsigma )$: $(\bar{x}^1, \bar{\varsigma}^1)=(2.11491, 0.236417), (\bar{x}^2, \bar{\varsigma}^2)=(-1.86081, -0.268701), (\bar{x}^3, \bar{\varsigma}^3)=(-0.254102,-1.96772)$. By Theorem 1, we know $\bar{x}^1=2.11491$ is the global minimizer of (\[double\_well\_potential\_prime\]), $\bar{\varsigma}^1=0.236417$ is the global maximizer of (\[double\_well\_potential\_dual\]) over $S_a^+$, and $P(\bar{\varsigma}^1)=\Xi (\bar{x}^1, \bar{\varsigma}^1)=P^d(\bar{\varsigma}^1)=-1.02951$. By Theorem 2, we know that the local minimizers: $\bar{x}^2=-1.86081, \bar{\varsigma}^2=-0.268701$ (over $S_a^-$), $P(\bar{\varsigma}^2)=\Xi (\bar{x}^2, \bar{\varsigma}^2)=P^d(\bar{\varsigma}^2)=0.9665031$ and the local maximizers: $\bar{x}^3=-0.254102,\bar{\varsigma}^3)=-1.96772$ (over $S_a^-$), $P(\bar{\varsigma}^3)=\Xi (\bar{x}^3, \bar{\varsigma}^3)=P^d(\bar{\varsigma}^3)=2.063$.\ Applications to a L-J potential optimization problem ==================================================== In 2007, Sawaya et al. got a breakthrough finding: the atomic structures of all amyloid fibrils revealed steric zippers, with strong vdw interactions (LJ) between $\beta$-sheets and hydrogen bonds (HBs) to maintain the $\beta$-strands [@sawaya2007]. Similarly as (\[LJ\_r\_form\]), i.e. the potential energy for the vdw interactions (Fig. \[LJ\_potential\]) between $\beta$-sheets: $$\label{LJ_AB_form} V_{LJ}(r)=\frac{A}{r^{12}} -\frac{B}{r^6},$$ the potential energy for the HBs between the $\beta$-strands has a similar formula $$\label{HB_r_form} V_{HB}(r)= \frac{C}{r^{12}} -\frac{D}{r^{10}} ,$$ where $A,B,C,D$ are constants given. Thus, the amyloid fibril molecular model building problem is reduced to well solve the optimization problem (\[LJ\_f\_form\]) or (\[prime\_approx\_problem\]) (in this paper we apply the CDT introduced in Section 2 to solve (\[prime\_approx\_problem\])).\ In this section, we will use suitable templates 3nvf.pdb, 3nvg.pdb and 3nvh.pdb from the Protein Data Bank (http://www.rcsb.org/) to build some amyloid fibril models.\ 3NVF ---- Constructions of the AGAAAAGA amyloid fibril molecular structures of prion 113–120 region are based on the most recently released experimental molecular structures of IIHFGS segment 138–143 from human prion (PDB entry 3NVF released into Protein Data Bank (http://www.rcsb.org) on 2011-03-02) [@apostolwsce2011]. The atomic-resolution structure of this peptide is a steric zipper, with strong vdw interactions between $\beta$-sheets and HBs to maintain the $\beta$-strands (Fig. \[3nvf\]). ![Protein fibril structure of IIHFGS segment 138–143 from human prion. The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf"}](3nvf_H.inpcrd.eps) In Fig. \[3nvf\] we see that H chain (i.e. $\beta$-sheet 2) of 3NVF.pdb can be obtained from A chain (i.e. $\beta$-sheet 1) by $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &-1 &0\\ 0 &0 &1 \end{array} \right) A + \left( \begin{array}{c} 27.546\\ 0\\ 0 \end{array} \right),$$ and other chains can be got by $$C(G)= A (H)+ \left( \begin{array}{c} 0\\ 0\\ 4.8\end{array} \right), B (F) = A (H) +2\left( \begin{array}{c} 0\\ 0\\ 4.8\end{array} \right), \label{3nvf_cg-bf}$$ $$D (I) = A (H) -\left( \begin{array}{c} 0\\ 0\\ 4.8\end{array} \right), E (J) = A (H) -2 \left( \begin{array}{c} 0\\ 0\\ 4.8\end{array} \right). \label{3nvf_di-ej}$$ Basing on the template 3NVF.pdb from the Protein Data Bank, three prion AGAAAAGA palindrome amyloid fibril models –- an AGAAAA model (3nvf-Model 1), a GAAAAG model (3nvf-Model 2), and an AAAAGA model (3nvf-Model 3) –- will be successfully constructed in this paper. Chain A of 3nvf-Models 1-3 were got from A Chain of 3NVF.pdb using the mutate module of the free package Swiss-PdbViewer (SPDBV Version 4.01) ([http://spdbv.vital-it.ch]{}). It is pleasant to see that almost all the hydrogen bonds are still kept after the mutations; thus we just need to consider the vdw contacts only. Making mutations for H Chain of 3NVF.pdb, we can get H Chain of 3nvf-Models 1-3. However, we find that the vdw contacts between A Chain and H Chain are too far at this moment. We know that for 3nvf-Model 1 at least the vdw interaction between A.GLY2.CA-H.GLY2.CA, A.ALA4.CB-H.GLY2.CA should be maintained, for 3nvf-Model 2 at least three vdw interactions between A.ALA4.CB-H.ALA2.CB, A.ALA2.CB-H.ALA2.CB, A.ALA2.CB-H.ALA4.CB should be maintained, and for 3nvf-Model 3 at least three vdw interactions between A.ALA2.CB-H.ALA2.CB, A.ALA2.CB-H.ALA4.CB, A.ALA4.CB-H.ALA2.CB should be maintained. Fixing the coordinates of A.GLY2.CA and A.ALA4.CB (two anchors) ((-10.919,-3.862,-1.487), (6.357,1.461,-1.905)) for 3nvf-Model 1, fixing the coordinates of A.ALA2.CB and A.ALA4.CB (two anchors) ((11.959,-2.844,-1.977), (6.357,1.461,-1.905)) for 3nvf-Models 2-3, letting $d$ equal to the twice of the vdw radius of Carbon atom (i.e. $d =3.4$ angstroms), and letting the coordinates of H.GLY2.CA of 3nvf-Model 1 (two sensors) and the coordinates of H.ALA2.CB and H.ALA4.CB of 3nvf-Models 2-3 (two sensors) be variables, we may get a simple MDGP with 3/6 variables and its dual with 2/3 variables for 3nvf-Model 1: $$\begin{aligned} P_{\epsilon}(x_1)=&&\frac{1}{2} \left\{ (x_{11} +10.919)^2+(x_{12}+3.862)^2 +(x_{13}+1.487)^2 -3.4^2 \right\}^2+\\ &&\frac{1}{2} \left\{ (x_{11} - 6.357)^2+(x_{12}-1.461)^2 +(x_{13}+1.905)^2 -3.4^2 \right\}^2-\\ &&(0.05x_{11}+0.05x_{12}+0.05x_{13}),\end{aligned}$$ $$\begin{aligned} P^d_{\epsilon}(\varsigma_1, \varsigma_2) =&&124.7908\varsigma_1 -\frac{1}{2} \varsigma_1^2 +34.615\varsigma_2 -\frac{1}{2} \varsigma_2^2 -\\ &&\frac{1}{2} \left( \begin{array}{c} 0.05-21.838\varsigma_1 +12.714\varsigma_2\\ 0.05-7.724\varsigma_1 +2.922\varsigma_2\\ 0.05-2.974\varsigma_1 -3.81\varsigma_2 \end{array} \right)^T \left( \begin{array}{cccccc} \frac{1}{2\varsigma_1 +2\varsigma_2} &0 &0\\ 0 &\frac{1}{2\varsigma_1 +2\varsigma_2} &0\\ 0 &0 &\frac{1}{2\varsigma_1 +2\varsigma_2} \end{array} \right)\\ &&\left( \begin{array}{c} 0.05-21.838\varsigma_1 +12.714\varsigma_2\\ 0.05-7.724\varsigma_1 +2.922\varsigma_2\\ 0.05-2.974\varsigma_1 -3.81\varsigma_2 \end{array} \right).\end{aligned}$$ We can get a global maximal solution (70.1836,70.1812) for $P^d_{\epsilon}(\varsigma_1, \varsigma_2)$ and its corresponding local maximal solution to $P_{\epsilon}(x_1)$:\ $\bar{x}=(-2.28097, -1.20037, -1.69582).$ By Theorem 1 we know that $\bar{x}$ is a global minimal solution of $P_{\epsilon}(x_1)$. Thus we get $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &-1 &0\\ 0 &0 &1 \end{array} \right) A + \left( \begin{array}{c} -4.5619\\ -2.4009\\ 0.0004 \end{array} \right)$$ for 3nvf-Model 1, whose other chains can be got by (\[3nvf\_cg-bf\])-(\[3nvf\_di-ej\]) (Fig. \[3nvf\_CDT\_models\]). For 3nvf-Models 2-3, similarly we may get a simple MDGP with 6 variables and its dual with 3 variables: $$\begin{aligned} P_{\epsilon}(x_1,x_2)=&&\frac{1}{2} \left\{ (x_{11} -11.959)^2+(x_{12}+2.844)^2 +(x_{13}+1.977)^2-3.4^2 \right\}^2 +\\ &&\frac{1}{2} \left\{ (x_{21} -11.959)^2+(x_{22}+2.844)^2 +(x_{23}+1.977)^2 -3.4^2 \right\}^2 +\\ &&\frac{1}{2} \left\{ (x_{11} -6.357)^2+(x_{12}-1.461)^2 +(x_{13}+1.905)^2 -3.4^2 \right\}^2 -\\ &&(0.05x_{11}+0.05x_{12}+0.05x_{13}+0.05x_{21}+0.05x_{22}+0.05x_{23}),\end{aligned}$$ $$\begin{aligned} &&P^d_{\epsilon}(\varsigma_1, \varsigma_2, \varsigma_3 )=143.4545\varsigma_1 -\frac{1}{2} \varsigma_1^2 +143.4545\varsigma_2 -\frac{1}{2} \varsigma_2^2 + 34.6150\varsigma_3 -\frac{1}{2} \varsigma_3^2 -\\ &&\frac{1}{2} \left( \begin{array}{c} 0.05+23.9180\varsigma_1 +12.7140\varsigma_3\\ 0.05- 5.6880\varsigma_1 + 2.9220\varsigma_3\\ 0.05- 3.9540\varsigma_1 - 3.8100\varsigma_3\\ 0.05+23.9180\varsigma_2\\ 0.05- 5.6880\varsigma_2\\ 0.05- 3.9540\varsigma_2 \end{array} \right)^T \left( \begin{array}{cccccc} \frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0 &0 &0\\ 0 &\frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0 &0\\ 0 &0 &\frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0\\ 0 &0 &0 &\frac{1}{2\varsigma_2} &0 &0\\ 0 &0 &0 &0 &\frac{1}{2\varsigma_2} &0\\ 0 &0 &0 &0 &0 &\frac{1}{2\varsigma_2} \end{array} \right)\\ &&\left( \begin{array}{c} 0.05+23.9180\varsigma_1 +12.7140\varsigma_3\\ 0.05- 5.6880\varsigma_1 + 2.9220\varsigma_3\\ 0.05- 3.9540\varsigma_1 - 3.8100\varsigma_3\\ 0.05+23.9180\varsigma_2\\ 0.05- 5.6880\varsigma_2\\ 0.05- 3.9540\varsigma_2 \end{array} \right) .\end{aligned}$$ We can get a global maximal solution (0.920088,0.0127286,0.921273) for $P^d_{\epsilon}(\varsigma_1, \varsigma_2, \varsigma_3)$ and its corresponding local maximal solution to $P_{\epsilon}(x_1,x_2)$:\ $\bar{x}=(9.16977, -0.676538, -1.9274, 13.9231, -0.879925, -0.0129248).$ By Theorem 1 we know that $\bar{x}$ is a global minimal solution of $P_{\epsilon}(x_1,x_2)$. Thus we get $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &-1 &0\\ 0 &0 &1 \end{array} \right) A + \left( \begin{array}{c} 20.8459\\ -2.1533\\ 0.6638 \end{array} \right).$$ for 3nvg-Models 2-3, whose other chains can be got by (\[3nvf\_cg-bf\])-(\[3nvf\_di-ej\]) (Fig. \[3nvf\_CDT\_models\]). ![Protein fibril structure of 3nvf-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf_CDT_models"}](3nvf_AG.eps "fig:") ![Protein fibril structure of 3nvf-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf_CDT_models"}](3nvf_GA.eps "fig:") ![Protein fibril structure of 3nvf-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf_CDT_models"}](3nvf_AA.eps "fig:") \ We find 3nvf-Model 1 has some atoms with bad/close contacts, 3nvf-Model 2 has 6 bad/close contacts, and 3nvf-Model 3 has no bad/close contact. This means it not necessary at all to further refine 3nvf-Model 3. We remove these bad contacts by performing energy minimization using Amber 11 [@case2010]. Even if there are no obvious bad contacts, it is still a good idea to run a short energy minimization to relax the structures a bit. We will perform the energy minimization in 2 stages. In the first stage, we’ll only minimize the water molecules and hold the protein fixed for 500 steps of steepest descent method and then 500 steps of conjugate gradient method. Our goal is just to remove bad contacts, there is no need to go overboard with minimization. In the second stage we proceed directly to minimizing the entire system as a whole for 1500 steps of steepest descent method and then 1000 steps of conjugate gradient method. RMSD (root mean square deviation) is an indicator for structural changes in a protein. It is used to measure the scalar spatial distance between atoms of the same type (for example the C$_\alpha$ atoms) for two structures in different time. The RMSDs between the last snapshot after the refinement and the snapshot illuminated in Fig. \[3nvf\_CDT\_models\] are 2.15796, 1.3089087, 1.045318 angstroms for these three 3nvf-Models respectively. The very small values of RMSD are very good measure of precision of CDT for our model building. This shows us that CDT performs well.\ 3NVG ---- In this subsection, the constructions of the AGAAAAGA amyloid fibril molecular structures of prion 113–120 region are based on the most recently released experimental molecular structures of MIHFGN segment 137–142 from mouse prion (PDB entry 3NVG released into Protein Data Bank (http://www.rcsb.org) on 2011-03-02) [@apostolwsce2011]. The atomic-resolution structure of this peptide is a steric zipper, with strong vdw interactions between $\beta$-sheets and HBs to maintain the $\beta$-strands (Fig. \[3nvg\]). ![Protein fibril structure of MIHFGN segment 137–142 from mouse prion. The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibril.[]{data-label="3nvg"}](3nvg_H.inpcrd.eps) In Fig. \[3nvg\] we see that H Chain (i.e. $\beta$-sheet 2) of 3NVG.pdb can be obtained from A Chain (i.e. $\beta$-sheet 1) by $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &1 &0\\ 0 &0 &-1 \end{array} \right) A + \left( \begin{array}{c} -27.28\\ 2.385\\ 15.738 \end{array} \right),$$ and other chains can be got by $$C(G)= A (H)+ \left( \begin{array}{c} 0\\ 4.77\\ 0 \end{array} \right), B (F) = A (H) +2\left( \begin{array}{c} 0\\ 4.77\\ 0 \end{array} \right), \label{3nvg_cg-bf}$$ $$D (I) = A (H) -\left( \begin{array}{c} 0\\ 4.77\\ 0 \end{array} \right), E (J) = A (H) -2 \left( \begin{array}{c} 0\\ 4.77\\ 0 \end{array} \right). \label{3nvg_di-ej}$$ Basing on the template 3NVG.pdb from the Protein Data Bank, three prion AGAAAAGA palindrome amyloid fibril models –- an AGAAAA model (3nvg-Model 1), a GAAAAG model (3nvg-Model 2), and an AAAAGA model (3nvg-Model 3) –- will be successfully constructed in this paper. Chain A of 3nvg-Models 1-3 were got from A Chain of 3NVG.pdb using the mutate module of the free package Swiss-PdbViewer (SPDBV Version 4.01) ([http://spdbv.vital-it.ch]{}). It is pleasant to see that almost all the hydrogen bonds are still kept after the mutations; thus we just need to consider the vdw contacts only. Making mutations for H Chain of 3NVG.pdb, we can get the H Chains of 3nvg-Models 1-3. However, the vdw contacts between A Chain and H Chain are too far at this moment. We know that for 3nvg-Model 1 at least the three vdw interaction between A.GLY2.CA-H.GLY2.CA, A.GLY2.CA-H.ALA4.CB, A.ALA4.CB-H.GLY2.CA should be maintained, for 3nvg-Model 2 at least the three vdw interactions between A.ALA2.CB-H.ALA2.CB, A.ALA2.CB-H.ALA4.CB, A.ALA4.CB-H.ALA2.CB should be maintained, and for 3nvg-Model 3 at least the three vdw interactions between A.ALA2.CB-H.ALA2.CB, A.ALA2.CB-H.ALA4.CB, A.ALA4.CB-H.ALA2.CB should be maintained. Fixing the coordinates of A.GLY2.CA and A.ALA4.CB (two anchors) ((-11.159,-2.241,4.126), (-5.865,-2.618,8.696)) for 3nvg-Model 1, fixing the coordinates of A.ALA2.CB and A.ALA4.CB (two anchors) ((-12.040,-2.675,5.307), (-5.865,-2.618,8.696)) for 3nvg-Models 1-2, letting $d$ equal to the twice of the vdw radius of Carbon atom (i.e. $d =3.4$ angstroms), and letting the coordinates of H.GLY2.CA and H.ALA4.CB of 3nvg-Model 1 (two sensors) and the coordinates of H.ALA2.CB and H.ALA4.CB of 3nvg-Models 2-3 (two sensors) be variables, we may get a simple MDGP with 6 variables and its dual with 3 variables for 3nvg-Model 1: $$\begin{aligned} P_{\epsilon}(x_1,x_2)=&&\frac{1}{2} \left\{ (x_{11} +11.159)^2+(x_{12}+2.241)^2 +(x_{13}-4.126)^2-3.4^2 \right\}^2+\\ &&\frac{1}{2} \left\{ (x_{21} +11.159)^2+(x_{22}+2.241)^2 +(x_{23}-4.126)^2 -3.4^2 \right\}^2+\\ &&\frac{1}{2} \left\{ (x_{11} + 5.865)^2+(x_{12}+2.618)^2 +(x_{13}-8.696)^2-3.4^2 \right\}^2-\\ &&(0.05x_{11}+0.05x_{12}+0.05x_{13}+0.05x_{21}+0.05x_{22}+0.05x_{23}),\end{aligned}$$ $$\begin{aligned} &&P^d_{\epsilon}(\varsigma_1, \varsigma_2, \varsigma_3) =135.009238\varsigma_1 -\frac{1}{2} \varsigma_1^2 +135.009238\varsigma_2 -\frac{1}{2} \varsigma_2^2 +105.3125\varsigma_3 -\frac{1}{2} \varsigma_3^2 -\\ &&\frac{1}{2} \left( \begin{array}{c} 0.05-22.318\varsigma_1 -11.73\varsigma_3\\ 0.05- 4.482\varsigma_1 -5.236\varsigma_3\\ 0.05+ 8.252\varsigma_1 +17.392\varsigma_3\\ 0.05-22.318\varsigma_2\\ 0.05 -4.482\varsigma_2\\ 0.05+ 8.252\varsigma_2 \end{array} \right)^T \left( \begin{array}{cccccc} \frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0 &0 &0\\ 0 &\frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0 &0\\ 0 &0 &\frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0\\ 0 &0 &0 &\frac{1}{2\varsigma_2} &0 &0\\ 0 &0 &0 &0 &\frac{1}{2\varsigma_2} &0\\ 0 &0 &0 &0 &0 &\frac{1}{2\varsigma_2} \end{array} \right)\\ &&\left( \begin{array}{c} 0.05-22.318\varsigma_1 -11.73\varsigma_3\\ 0.05- 4.482\varsigma_1 -5.236\varsigma_3\\ 0.05+ 8.252\varsigma_1 +17.392\varsigma_3\\ 0.05-22.318\varsigma_2\\ 0.05 -4.482\varsigma_2\\ 0.05+ 8.252\varsigma_2 \end{array} \right).\end{aligned}$$ We can get a global maximal solution (0.708403,0.0127287,0.699001) for $P^d_{\epsilon}(\varsigma_1, \varsigma_2, \varsigma_3)$ and its corresponding local maximal solution to $P_{\epsilon}(x_1,x_2)$:\ $\bar{x}=(-8.51192, -2.41048, 6.4135, -9.19493, -0.276929, 6.09007).$ By Theorem 1 we know that $\bar{x}$ is a global minimal solution of $P_{\epsilon}(x_1,x_2)$. Thus we get $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &1 &0\\ 0 &0 &-1 \end{array} \right) A + \left( \begin{array}{c} -18.133923\\ 0.6673703\\ 11.955023 \end{array} \right)$$ for 3nvg-Model 1, whose other chains can be got by (\[3nvg\_cg-bf\])-(\[3nvg\_di-ej\]) (Fig. \[3nvg\_CDT\_models\]). For 3nvg-Models 2-3, similarly we may get a simple MDGP with 6 variables and its dual with 3 variables: $$\begin{aligned} P_{\epsilon}(x_1,x_2)=&&\frac{1}{2} \left\{ (x_{11} +12.040)^2+(x_{12}+2.675)^2 +(x_{13}-5.307)^2-3.4^2 \right\}^2 +\\ &&\frac{1}{2} \left\{ (x_{21} +12.040)^2+(x_{22}+2.675)^2 +(x_{23}-5.307)^2 -3.4^2 \right\}^2 +\\ &&\frac{1}{2} \left\{ (x_{11} + 5.865)^2+(x_{12}+2.618)^2 +(x_{13}-8.696)^2-3.4^2 \right\}^2 -\\ &&(0.05x_{11}+0.05x_{12}+0.05x_{13}+0.05x_{21}+0.05x_{22}+0.05x_{23}),\end{aligned}$$ $$\begin{aligned} &&P^d_{\epsilon}(\varsigma_1, \varsigma_2, \varsigma_3 )= 168.721474\varsigma_1 -\frac{1}{2} \varsigma_1^2 +168.721474\varsigma_2 -\frac{1}{2} \varsigma_2^2 +105.312565\varsigma_3 -\frac{1}{2} \varsigma_3^2 -\\ &&\frac{1}{2} \left( \begin{array}{c} 0.05-24.080\varsigma_1 -11.73\varsigma_3\\ 0.05- 5.35\varsigma_1 -5.236\varsigma_3\\ 0.05+10.614\varsigma_1 +17.392\varsigma_3\\ 0.05-24.080\varsigma_2\\ 0.05 -5.35\varsigma_2\\ 0.05+10.614\varsigma_2 \end{array} \right)^T \left( \begin{array}{cccccc} \frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0 &0 &0\\ 0 &\frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0 &0\\ 0 &0 &\frac{1}{2\varsigma_1 +2\varsigma_3} &0 &0 &0\\ 0 &0 &0 &\frac{1}{2\varsigma_2} &0 &0\\ 0 &0 &0 &0 &\frac{1}{2\varsigma_2} &0\\ 0 &0 &0 &0 &0 &\frac{1}{2\varsigma_2} \end{array} \right)\\ &&\left( \begin{array}{c} 0.05-24.080\varsigma_1 -11.73\varsigma_3\\ 0.05- 5.35\varsigma_1 -5.236\varsigma_3\\ 0.05+10.614\varsigma_1 +17.392\varsigma_3\\ 0.05-24.080\varsigma_2\\ 0.05 -5.35\varsigma_2\\ 0.05+10.614\varsigma_2 \end{array} \right) .\end{aligned}$$ We can get a global maximal solution (0.849735,0.0127287,0.84036) for $P^d_{\epsilon}(\varsigma_1, \varsigma_2, \varsigma_3)$ and its corresponding local maximal solution to $P_{\epsilon}(x_1,x_2)$:\ $\bar{x}=(-8.95484, -2.63187, 7.00689, -10.0759, -0.710929, 7.27107).$ By Theorem 1 we know that $\bar{x}$ is a global minimal solution of $P_{\epsilon}(x_1,x_2)$. Thus we get $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &1 &0\\ 0 &0 &-1 \end{array} \right) A + \left( \begin{array}{c} -19.3102\\ 0.6644\\ 13.5316 \end{array} \right).$$ for 3nvg-Models 2-3, whose other chains can be got by (\[3nvg\_cg-bf\])-(\[3nvg\_di-ej\]) (Fig. \[3nvg\_CDT\_models\]). ![Protein fibril structure of 3nvg-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvg_CDT_models"}](3nvg_AG.eps "fig:") ![Protein fibril structure of 3nvg-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvg_CDT_models"}](3nvg_GA.eps "fig:") ![Protein fibril structure of 3nvg-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvg_CDT_models"}](3nvg_AA.eps "fig:") \ We did same refinements for 3nvg-Models 1-3 as for 3nvf-Models 1-3. The RMSDs between the last snapshot after the refinement and the snapshot illuminated in Fig. \[3nvg\_CDT\_models\] are 1.572438, 1.404648, 1.464767 angstroms for these three 3nvg-Models respectively. The very small values of RMSD show us that CDT performs well and precisely for 3nvg-Model building.\ 3NVH ---- Similar as the above two subsections, this subsection constructs the AGAAAAGA amyloid fibril molecular structures of prion 113–120 region basing on the most recently released experimental molecular structures of MIHFGND segment 137–143 from mouse prion (PDB entry 3NVH released into Protein Data Bank (http://www.rcsb.org) on 2011-03-02) [@apostolwsce2011]. The atomic-resolution structure of this peptide is a steric zipper, with strong vdw interactions between $\beta$-sheets and HBs to maintain the $\beta$-strands (Fig. \[3nvh\]). ![Protein fibril structure of MIHFGND segment 137–143 from mouse prion. The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibril.[]{data-label="3nvh"}](3nvh_H.inpcrd.eps) In Fig. \[3nvh\] we see that H chain (i.e. $\beta$-sheet 2) of 3NVH.pdb can be obtained from A chain (i.e. $\beta$-sheet 1) by $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &1 &0\\ 0 &0 &-1 \end{array} \right) A + \left( \begin{array}{c} 0\\ 2.437\\ -15.553 \end{array} \right),$$ and other chains can be got by $$C(G)= A (H)+ \left( \begin{array}{c} 0\\ 4.87\\ 0 \end{array} \right), B (F) = A (H) +2\left( \begin{array}{c} 0\\ 4.87\\ 0 \end{array} \right), \label{3nvh_cg-bf}$$ $$D (I) = A (H) -\left( \begin{array}{c} 0\\ 4.87\\ 0 \end{array} \right), E (J) = A (H) -2 \left( \begin{array}{c} 0\\ 4.87\\ 0 \end{array} \right). \label{3nvh_di-ej}$$ Basing on the template 3NVH.pdb from the Protein Data Bank, three prion AGAAAAGA palindrome amyloid fibril models –- an AGAAAAG model (3nvh-Model 1), a GAAAAGA model (3nvh-Model 2) –- will be successfully constructed in this paper. A chain of 3nvh-Models 1-2 were got from A chain of 3NVH.pdb using the mutate module of the free package Swiss-PdbViewer (SPDBV Version 4.01) ([http://spdbv.vital-it.ch]{}). It is pleasant to see that almost all the hydrogen bonds are still kept after the mutations; thus we just need to consider the vdw contacts only. Making mutations for H chain of 3NVH.pdb, we can get the H chains of 3nvh-Models 1-2. However, the vdw contacts between A chain and H chain are too far at this moment ($\geq$4.25 Angstroms). we may know that for 3nvh-Models 1-2 at least one vdw interaction between A.ALA4.CB-H.ALA4.CB should be maintained. Fixing the coordinates of A.ALA4.CB (the anchor) ((1.731,-1.514,-7.980)), letting $d$ equal to the twice of the vdw radius of Carbon atom (i.e. $d =3.4$ angstroms), and letting the coordinate of H.ALA4.CB (one sensor) be variables, we may get a simple MDGP with 3 variables and its dual with 1 variable: $$\begin{aligned} P_{\epsilon}(x_1)= &&\frac{1}{2} \left\{ (x_{11} -1.731)^2+(x_{12}+1.514)^2 +(x_{13}+7.980)^2 -3.4^2 \right\}^2\\ && -0.05x_{11}-0.05x_{12}-0.05x_{13},\\ P^d_{\epsilon}(\varsigma_1) = &&57.409\varsigma_1 -\frac{1}{2} \varsigma_1^2\\ &&\frac{(0.05 + 3.462\varsigma_1)^2+(0.05 - 3.028\varsigma_1)^2+(0.05 - 15.96\varsigma_1)^2}{4\varsigma_1} .\end{aligned}$$ We can easily get the global maximal solution $ 0.0127287 \in \{ \varsigma \in \mathbb{R}^1 | \varsigma_i >0, i=1\}$ for $P_{\epsilon}^d(\varsigma_1)$. Then, we get its corresponding solution for $P_{\epsilon}(x_1)$:\ $\bar{x}=(3.69507, 0.450071, -6.01593).$ By Theorem 1 we know that $\bar{x}$ is a global minimal solution of $P_{\epsilon}(x_1)$, i.e. for H.ALA4.CB. Thus we get $$H = \left( \begin{array}{ccc} -1 &0 &0\\ 0 &1 &0\\ 0 &0 &-1 \end{array} \right) A + \left( \begin{array}{c} 5.42607\\ 1.964071\\ -13.99593 \end{array} \right)$$ for 3nvh-Models 1-2, whose other chains can be got by (\[3nvh\_cg-bf\])-(\[3nvh\_di-ej\]) (Fig. \[3nvh\_CDT\_models\]). ![Protein fibril structure of 3nvh-Models 1-2 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J, K, L denote the 12 chains of the fibrils.[]{data-label="3nvh_CDT_models"}](3nvh_AG.eps "fig:") ![Protein fibril structure of 3nvh-Models 1-2 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J, K, L denote the 12 chains of the fibrils.[]{data-label="3nvh_CDT_models"}](3nvh_GA.eps "fig:") \ We carried on the same refinements for 3nvh-Models 1-2 as for 3nvf-Models 1-3 and 3nvg-Models 1-3. The RMSDs between the last snapshot after the refinement and the snapshot illuminated in Fig. \[3nvh\_CDT\_models\] are 1.534417, 1.572836 angstroms for the two 3nvh-Models respectively. The very small values of RMSD again show to us that CDT performs well and precisely for 3nvg-Model building.\ Refined 3nvf-Models 1-3, 3nvg-Models 1-3, 3nvh-Models 1-2 --------------------------------------------------------- The amyloid fibril models of prion AGAAAAGA segment refined by Amber 11 are illuminated in Fig.s \[3nvf\_CDT\_models\_min2\]-\[3nvh\_CDT\_models\_min2\]. All these models are without any bad contact now (checked by package Swiss-PdbViewer), and the vdw interactions between the two $\beta$-sheets are in a very perfect way now. ![Perfect 3nvf-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf_CDT_models_min2"}](3nvf_AG_min2.rst.pdb.eps "fig:") ![Perfect 3nvf-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf_CDT_models_min2"}](3nvf_GA_min2.rst.pdb.eps "fig:") ![Perfect 3nvf-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvf_CDT_models_min2"}](3nvf_AA_min2.rst.pdb.eps "fig:") ![Perfect 3nvg-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvg_CDT_models_min2"}](3nvg_AG_min2.rst.pdb.eps "fig:") ![Perfect 3nvg-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvg_CDT_models_min2"}](3nvg_GA_min2.rst.pdb.eps "fig:") ![Perfect 3nvg-Models 1-3 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J denote the 10 chains of the fibrils.[]{data-label="3nvg_CDT_models_min2"}](3nvg_AA_min2.rst.pdb.eps "fig:") ![Perfect 3nvh-Models 1-2 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J, K, L denote the 12 chains of the fibrils.[]{data-label="3nvh_CDT_models_min2"}](3nvh_AG_min2.rst.pdb.eps "fig:") ![Perfect 3nvh-Models 1-2 (from left to right respectively) for prion AGAAAAGA segment 113-120 . The purple dashed lines denote the hydrogen bonds. A, B, ..., I, J, K, L denote the 12 chains of the fibrils.[]{data-label="3nvh_CDT_models_min2"}](3nvh_GA_min2.rst.pdb.eps "fig:") All the initial structures before dealt by CDT approach have very far vdw contacts between the two $\beta$-sheets. CDT easily made the vdw contacts come closer and reach a state with the lowest potential energy, which has perfect vdw contacts as shown in Fig.s \[3nvf\_CDT\_models\_min2\]-\[3nvh\_CDT\_models\_min2\].\ Conclusion ========== Global optimization of Lennard-Jones clusters is a challenging problem for researchers in the field of biology, physics, chemistry, computer science, materials science, and especially for experts in mathematical optimization research field because of the nonconvexity of the L-J potential energy function and enormous local minima on the potential energy surface. In March 2008, American Mathematical Programming Society specially produced one whole issue, No. 76, to discuss this problem. In this paper through clever use of global optimization techniques of Gao’s canonical dual theory (CDT), we successfully tackle this challenging problem illuminated by the amyloid fibril molecular model building. Clearly, this paper shows to readers that CDT is very useful and powerful to tackle challenging problems in optimization area and many other areas.\ [**Acknowledgments:**]{} This research was supported by US Air Force Office of Scientific Research under the grant AFOSR FA9550-10-1-0487, by a Victorian Life Sciences Computation Initiative (VLSCI) grant number VR0063 on its Peak Computing Facility at the University of Melbourne, an initiative of the Victorian Government, and by the Head and Colleagues of Graduate School of Ballarat University.\ [99]{} Apostol, M.I., Wiltzius, J.J., Sawaya, M.R., Cascio, D., Eisenberg, D.: Atomic structures suggest determinants of transmission barriers in mammalian prion disease. Biochem. 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--- abstract: 'Material recognition can help inform robots about how to properly interact with and manipulate real-world objects. In this paper, we present a multimodal sensing technique, leveraging near-infrared spectroscopy and close-range high resolution texture imaging, that enables robots to estimate the materials of household objects. We release a dataset of high resolution texture images and spectral measurements collected from a mobile manipulator that interacted with 144 household objects. We then present a neural network architecture that learns a compact multimodal representation of spectral measurements and texture images. When generalizing material classification to new objects, we show that this multimodal representation enables a robot to recognize materials with greater performance as compared to prior state-of-the-art approaches. Finally, we present how a robot can combine this high resolution local sensing with images from the robot’s head-mounted camera to achieve accurate material classification over a scene of objects on a table.' author: - 'Zackory Erickson, Eliot Xing, Bharat Srirangam, Sonia Chernova, and Charles C. Kemp[^1][^2][^3][^4]' bibliography: - 'bibliography.bib' title: | **Multimodal Material Classification for Robots using\ Spectroscopy and High Resolution Texture Imaging** --- Introduction {#sec:intro} ============ When interacting with everyday objects, people frequently use material properties to inform their interactions [@buckingham2009cueslifting]. We make sure not to place metal in the microwave, we take caution when carrying glass or ceramic objects, we look for styrofoam or paper cups to hold hot liquids, and we sort some paper, plastic, and metal objects into recycling bins. Robots can benefit from these same skills when operating in human environments. In this work, we demonstrate how robots can use a non-contact multimodal sensing technique, based on spectroscopy and close-range texture imaging, to accurately estimate the materials of household objects prior to manipulation. This sensing approach collects near-infrared spectral measurements from a handheld micro spectrometer with a narrow field-of view camera for high resolution texture imaging. Both sensors are small and can be held by or directly integrated into a robot’s end effector. Non-contact sensing can enable a robot to determine properties and use cases of objects without the intricacies of contact physics that can affect the performance of haptic touch-based sensing. To evaluate this multimodal sensing technique, we have assembled and released a dataset of 14,400 high resolution texture images and corresponding spectral measurements. We collected this data with a PR2 mobile manipulator that interacted with 144 household objects, shown in Fig. \[fig:intro\], which spanned eight material categories: ceramic, fabric, foam, glass, metal, paper, plastic, and wood. Using this dataset, we trained a neural network that learns a shared representation of spectral and visual sensory data. By learning a compact multimodal representation, our model achieves state-of-the-art material recognition performance of 80.0% when generalizing material classification to a new set of heldout objects across eight materials (12.5% baseline with a random classifier). We further investigate the role of texture image preprocessing by comparing several ImageNet-pretrained CNN models for generating lower-dimensional visual representations. Finally, using this spectral and visual sensing approach, we demonstrate that a robot can reliably classify a scene of objects on a table without direct contact. In this work, we make the following contributions: - We introduce a near-infrared spectroscopy and high resolution texture imaging approach that surpasses prior state-of-the-art performance [@erickson2019spec] for material classification. - We release SpectroVision, a dataset of 14,400 high resolution texture images and spectral measurements collected from a PR2 mobile manipulator that interacted with 144 household objects from eight material categories. - We demonstrate that our multimodal approach surpasses the performance of models trained on each independent modality. - We show that a robot equipped only with our handheld sensors and an RGB-D camera can successfully use our approach to perform material classification on multiple objects casually arranged on a table. Related Work and Background {#sec:related_work} =========================== Material Recognition -------------------- Material recognition using haptic sensors, which require direct physical contact with objects, has been widely explored. Modalities such as force [@bhattacharjee2012tactile; @decherchi2011tactile], temperature [@kerr2013thermal; @bhattacharjee2015heat; @cho2018thermal], capacitance [@alagi2018capacitive], and vibration [@sinapov2011vibrotactile; @fang2019acoustic], have been used in haptic perception for material recognition. The BioTac fingertip, capable of sensing force, temperature, and vibration, has been studied for multimodal haptic perception [@chu2015hapticadjectives; @fishel2012biotac; @xu2013tactile; @kerr2018biotac]. Chin et al. introduced a compliant haptic sensor for robots to distinguish between plastic, metal, and paper during recycling [@chin2019automated]. Several works also use multimodal perception by combining data from multiple modalities for material recognition and outperforming single modality approaches [@chathuranga2013investigation; @sinapov2014grounding; @erickson2017semi; @bhattacharjee2018multimodal; @zhang2019multimodalcutting]. Similarly, we find that non-contact material recognition approaches also benefit from multiple sensing modalities, and we demonstrate that visual sensing couples well with spectroscopy. Several studies have evaluated visual features for material recognition [@liu2010exploring; @hu2011toward; @dimitrov2014vision; @bell2015material; @schwartz2019recognizing]. Extensive literature also exists for leveraging visual or depth imaging for vision-based tactile sensors, including the GelSight [@yuan2017shape; @li2013sensing], FingerVision [@yamaguchi2017implementing], and TacTip [@ward2018tactip; @cramphorn2018voronoi], to perform a variety of manipulation tasks [@chen2018tactile; @yamaguchi2016combining], texture recognition [@luo2018vitac; @lee2019touching], and estimation of material properties [@kampouris2016fine; @yuan2017gelsight]. Both [@yuan2017shape] and [@gao2016deep] have used visual and haptic features to estimate object properties, such as hardness or haptic adjectives. Overall, we find that multimodal approaches overcome weaknesses in the ability of any individual modality to classify materials. Spectroscopy ------------ Spectroscopy [@pasquini2018specreview] has found a number of practical applications such as for pharmaceutical manufacturing [@roggo2007pharma], food analysis [@bellon1994food], and recycled material separation [@masoumi2012plastic]. Recently, a number of handheld spectrometers have been developed for performing spectral analysis outside of lab and manufacturing settings [@crocombe2018portable; @rateni2017smartphonebased]. These portable micro spectrometers have been demonstrated for pharmaceutical quality control [@yan2018pharma] and food analysis [@das2016fruit; @lee2017nir; @kartakoullis2018meat]. Prior research has shown how a robot can use near-infrared spectroscopy with a commercial handheld SCiO spectrometer to recognize materials of household objects [@erickson2019spec]. Near-infrared spectroscopy has since been used by robots to recognize the materials of household objects for informing semantic grasp predictions and for tool construction [@liu2019cage; @nair2019autonomous; @shrivatsav2019tool]. In this paper, we demonstrate that robots can more accurately recognize common household materials by leveraging both spectroscopy and close-range texture imaging. Texture Representation ---------------------- Several techniques have been introduced for extracting or learning texture representations from visual images, including convolutional neural network (CNN) based texture analysis [@liu2019texturesurvey] and handcrafted descriptors [@paolo2017texturedescriptors]. Recent work in texture analysis has primarily investigated CNN-based texture representations [@liu2019texturesurvey]. This is due in part to a collection of works in texture and material classification tasks that have shown learned CNN feature descriptors frequently outperform alternative, handcrafted approaches [@paolo2017texturedescriptors; @liu2019bow; @schwartz2019recognizing; @bell2015material; @kalliatakis2017evaluating]. Research in texture synthesis [@gatys2015texture; @lin2016visualizing] has also provided insight into the ways in which CNNs capture and encode textures. Vision-based tactile sensing techniques for texture classification have frequently used texture features from pretrained ImageNet models [@yuan2018active]. The use of these models for extracting textural features is further supported by findings of Geirhos et al. [@geirhos2018imagenettexture] that ImageNet-trained CNNs are more biased towards recognizing and representing localized textures rather than global shape structure, similar to results by [@gatys2015texture; @long2018texturestat; @ballester2016cnnsketches]. Building on these prior findings, we leverage pretrained ImageNet CNNs to extract robust visual texture features for material classification. SpectroVision Dataset {#sec:dataset} ===================== Sensors ------- Our sensing approach consists of a micro handheld spectrometer for near-infrared spectral measurements and a narrow field-of view camera for high resolution texture imaging. Compared to haptic sensing, spectroscopy and imaging have advantageous properties for material recognition, including fast response times and no physical contact requirements. Fig. \[fig:sensors\] shows the SCiO spectrometer and camera, by themselves and when held in a PR2 robot’s end effector. The SCiO is a near-infrared spectrometer that measures light spectra in the wavelength range of $\lambda=$ 740 nm to $\lambda=$ 1,070 nm. The 35 gram spectrometer is Bluetooth enabled and has a black pigmented cover around the sensor aperture which ensures there is an $\sim$1 cm minimum air gap between an object and the sensor aperture. We capture texture images with a 2 megapixel endoscope camera. The 8.4 mm diameter camera has an optimal viewing distance of 6 cm to 10 cm and is capable of capturing images at 1600 $\times$ 1200 resolution. We placed a 12 LED light ring around the camera (see Fig. \[fig:sensors\]) to ensure consistent illumination of each object that the robot interacts with. We attached the spectrometer and camera together with a grasping mount for the PR2’s end effector. Note that there is an $\sim$3.5cm offset between the apertures of the two sensors. \ [ Y Y Y Y Y Y Y Y ]{} Ceramic & Fabric & Foam & Glass & Metal & Paper & Plastic & Wood Dataset and Data Collection --------------------------- We have collected and released SpectroVision, a dataset[^5] of 14,400 texture images and near-infrared spectral samples. This data was captured from a PR2 robot that interacted with 144 household objects from 8 material categories, as shown in Fig. \[fig:intro\]. These materials include ceramic, fabric, foam, glass, metal, paper, plastic, and wood, with 18 unique objects per material. The robot performed 100 interactions with each object, sampling at random positions and orientations along an object’s outer surface. To do this, the robot used its right end effector to hold a flat platter on which we rigidly mounted objects to, as demonstrated in Fig. \[fig:datacollection\]. For measurements collected with vertically standing upright objects, the robot would rotate the platter, then randomly sample a roll orientation for the left end effector in $[-\frac{\pi}{9}, \frac{\pi}{9}]$ and a vertical height to interact with the object at in $[0, h_i]$, where $h_i$ represents the height of object $i$. For objects that lie flat on the platter, the robot would randomly sample an end effector roll orientation in $[-\frac{\pi}{6}, \frac{\pi}{6}]$ and a point of contact in $[0, l_i]$, $[0, w_i]$ along the top surface of the object, with length $l_i$ and width $w_i$. Due to the random roll orientation of the robot’s end effector and the $\sim$3.5 cm height offset between the spectrometer and camera (seen in Fig. \[fig:sensors\]), spectral and texture images captured at the same time are not co-located and hence pairings between these measurements are not strict. In early evaluations, we found that randomizing pairings between spectral and image samples from the same object did not have considerable impact on classification performance. Video sequences of the data collection process can be found in the supplementary video. In comparison to some haptic sensing approaches that can take upwards of 15-20 seconds per measurement [@sinapov2011vibrotactile; @kerr2018biotac], spectroscopy and imaging offer consistently fast sensing times. Capturing an image takes $\sim$1.5 milliseconds, whereas the SCiO has a 1-2 second sensing time, which consists of $\sim$1 second of light exposure, reflectance data processing, and Bluetooth communication. Data processing consists of normalizing the raw spectrum reading from the SCiO’s optical head by the raw spectrum of a calibration apparatus (a high reflectance mirror material). Fig. \[fig:imagewall\] depicts sample images from each material category, captured by the camera during the interactions. Fig. \[fig:scio\_measurements\] shows the spectral measurements, which were captured alongside the images in Fig. \[fig:imagewall\]. A raw spectral measurement consists of a 331-dimensional vector with a 1 nm wavelength step between the range of $\lambda=$ 740 nm to $\lambda=$ 1,070 nm. Prior works have shown that the difference quotient (numerical first order derivative) of spectral measurements can improve learning performance [@erickson2019spec; @strother2009nir]. Given this finding, we concatenate the difference quotient to each raw spectral measurement, resulting in a 662-dimensional spectral vector. Multimodal Learning Architecture {#ssec:models} -------------------------------- ----- -- (A) (B) (C) ----- -- We construct a multimodal network that learns independent representations for each modality and fuses layers at the end for multimodal classification. We begin by building separate networks to learn low-dimensional representations for the spectral and image modalities. The spectral network, Fig. \[fig:networks\] (A), takes as input a 662-dimensional spectral sample and outputs material probability estimates from a softmax function. The model has two 64 node hidden layers followed by two 32 node layers, with batch normalization and a leaky ReLU activation applied after each layer. We apply a dropout of 0.25 after all but the last 32 node hidden layer. Prior to training a model over texture images, we first feed images through a DenseNet-201 CNN pretrained on ImageNet [@huang2017densenet]. We remove the 1000-class output layer such that the network outputs a feature vector of length 1920, resulting from global average pooling on the output of the preceding convolutional block. Models trained on ImageNet often learn strong representations for texture within an image [@geirhos2018imagenettexture]. Given this, in Section \[ssec:image\_embeddings\], we compare material recognition results across various ImageNet-trained models used for computing texture image embeddings. Fig. \[fig:networks\] (B) shows our image network, which takes as input the 1920-dimensional features from DenseNet-201. The network has three hidden layers of size 128, 64, and 32 nodes, with batch norm and leaky ReLU applied after each layer. We apply a dropout of 0.1 after the first two hidden layers. During evaluation (Section \[sec:evaluation\]), we train the spectral and image networks each for 50 epochs with a batch size of 128. Given trained spectral and texture image models, we then define our multimodal network architecture. We freeze the weights in both networks and remove the final 8 node output layer (depicted by the orange dotted lines in Fig. \[fig:networks\]). Both models output a 32-dimensional representation for their respective sensory modality. As depicted in Fig. \[fig:networks\] (C), these two outputs are concatenated and fed to a 32 node hidden layer followed by a leaky ReLU activation. We use a softmax activation after the final 8 node output layer to compute probability estimates for each of the 8 material categories. Since the spectral and texture image models are pretrained, we train only the weights for layers after the concatenation for 10 epochs. We trained all models with the Adam optimizer, using $\beta_1=$ 0.9, $\beta_2=$ 0.999, and a learning rate of 0.0005. Learning separate representations for each modality and combining into a shared representation for classification is commonly used for multimodal learning [@atrey2010multimodalsurvey; @ngiam2011multimodal; @eitel2015multimodalrgbd; @liu2018combinemodalities]. From initial tests, we found that this late fusion approach performs better than directly learning a joint representation with early fusion. Overall, our results in Section \[sec:evaluation\] show that combining modalities improves generalization to recognize materials of unseen objects, with close-range texture imaging and near-field spectroscopy providing strong individual baselines. Evaluation {#sec:evaluation} ========== Our dataset contains 14,400 spectral and image measurements from 144 distinct household objects. Prior to training and hyperparameter optimization for the models defined in Section \[ssec:models\], we split these data into a training set of measurements from 104 objects, and a heldout test set of 40 objects (5 objects per each of the 8 material categories). This heldout data was not used for optimizing our models’ hyperparameters. This heldout test set also includes the same test set objects used in [@erickson2019spec], shown in Fig. \[fig:heldout\_objects\], for a direct comparison to prior work that used only idealized spectral measurements[^6]. To reduce the influence of random weight initialization when training models, we report all results averaged over 10 random seeds. Recognizing Materials of New Objects ------------------------------------ When deployed in real-world environments, robots are likely to encounter new objects which they have not yet been exposed to. Similar to prior works in material classification, we begin by evaluating our multimodal sensing approach when recognizing the materials of new objects not found in the training data [@erickson2017semi; @erickson2019spec]. We first assess generalization across all 104 training set objects using leave-one-object-out cross-validation. To do so, we train a model on 103 objects (10,300 measurements) and evaluate material classification accuracy on the 100 samples from the one left-out object. We then repeat this process for each object and compute the average accuracy over the 104 splits. As shown in Table \[table:looo\], when using only spectral measurements with our spectral model (model A), we achieved an accuracy of 65.1% averaged over 10 random seeds. When training on visual data, our image model (model B) achieved a material classification accuracy of 70.5%. In comparison, our multimodal approach (model C) achieved an accuracy of 74.2%, a $\sim$4% improvement using low-dimensional representations of both image and spectral samples. Prior research has investigated how a robot can use near-infrared spectroscopy to recognize object materials with leave-one-object-out cross-validation over five material categories: fabric, metal, paper, plastic, and wood [@erickson2019spec]. For a direct comparison of results, we evaluate our performance given only these materials (excluding ceramic, foam, and glass objects). Table \[table:looo\] also depicts the performance of our models over these five materials (65 objects, 13 objects per material). Notably, our model trained on SCiO spectral measurements achieved 79.1% accuracy, which is identical to the 79.1% leave-one-object-out accuracy presented in prior work that used flat material objects [@erickson2019spec]. As a final assessment of how spectroscopy and texture imaging enables generalizing material classification to new objects, we evaluate results over the heldout test set consisting of five objects from each material category. Prior work has evaluated how a model trained on idealized spectral measurements from flat objects can be used to recognize the materials of 25 household objects from five material categories (fabric, metal, paper, plastic, and wood) [@erickson2019spec]. We include these same objects in our heldout test to enable a direct comparison with our multimodal sensing approach. With a spectral model trained on all 65 training set objects from the five materials, the model achieves a material recognition accuracy of 85.5% on the 25 heldout test objects, as shown in Table \[table:looo\_heldout\]. This is a $\sim$4% improvement compared to the 81.6% accuracy achieved in [@erickson2019spec], which used a nearly identical neural network architecture trained on idealized spectral measurements from flat material samples. However, by training a multimodal model on both spectral and texture images from the training set, our resulting model recognizes the material of these heldout objects with 90.6% accuracy, a 9% improvement on prior research when generalizing to new household objects. When compared to leave-one-object-out, we note that our multimodal approach performs significantly better on the paper and plastic heldout objects, 98.3% and 77.6% accuracy respectively, leading to higher overall performance on the heldout dataset. Spectral (A) Image (B) Multimodal (C) ------------- -------------- ----------- ---------------- 5 Materials **79.1** 76.8 **79.1** 8 Materials 65.1 70.5 **74.2** : \[table:looo\]Leave-one-object-out accuracy with all 8 materials and the 5 materials from [@erickson2019spec]. Spectral (A) Image (B) Multimodal (C) ------------- -------------- ----------- ---------------- 5 Materials 85.9 80.1 **90.8** 8 Materials 77.2 69.6 **80.0** : \[table:looo\_heldout\]Accuracy over the heldout test set, with all 8 materials and the 5 materials from [@erickson2019spec]. Spectral vs. Image Sensing -------------------------- In this section, we provide insight and case studies into what materials the two sensory modalities (spectral and image) perform best with and how a multimodal network architecture can leverage the strengths of each modality. Fig. \[fig:looo\_materials\] shows how our models trained on different modalities performed across material categories during leave-one-object-out cross-validation. We observe that it is easier to recognize fabrics with visual texture information, yet easier to recognize paper and glass with spectral data. Furthermore, some materials, such as plastic, remain difficult for both spectral and image data, in part due to large variation among plastic objects and difficulty distinguishing translucent plastics from glass. In addition, we observe that in many cases, a multimodal model that leverages both spectral and visual data can more accurately recognize materials than when using either modality independently. One example of this occurring is with foam objects, where the spectral and image models achieved 48.2% and 53.7% accuracy, respectively, yet our multimodal model attained 64.6% accuracy, $\sim$16% higher than the spectral model and $\sim$11% higher than the image modality. Object Spectral (A) Image (B) Multimodal (C) --------------------- -------------- ----------- ---------------- -- Fabric gray shirt 2.4 99.7 99.7 Foam plate 100.0 43.3 99.4 Wood tray 0.2 63.6 78.4 Plastic coffee-mate 0.6 81.5 50.5 Paper tissue box 100.0 15.8 48.2 : \[table:looo\_objects\] Case studies of objects with leave-one-object-out accuracy. Beyond averages over entire material categories, we also investigate examples of specific objects and how the different modalities compare, as shown in Table \[table:looo\_objects\]. A gray cotton fabric shirt was challenging for our spectral model to classify, achieving only 2.4% accuracy. In comparison, the image modality recognized this object as fabric with 99.7% accuracy. Our multimodal model also achieved 99.7% accuracy by learning to leverage the visual information to make its decision. Conversely, our image model struggled to accurately classify a foam plate with only 43.3% accuracy. By incorporating spectral data, our multimodal model correctly recognizes the foam plate with 99.4% accuracy. As indicated in the previous examples, a multimodal model frequently matches or outperforms models trained on independent modalities. Another example of this is a wood tray for which the spectral and texture image models reach 0.2% and 63.6% accuracy, respectively. Yet our multimodal model recognizes this object as wood with 78.4% accuracy, an $\sim$15% improvement over the image model. Fig. \[fig:gradients\] shows a saliency map from our multimodal model (visualization of what input features would affect the material estimate most if changed) [@simonyan2013deep] for a single measurement from the wood tray. As depicted, our multimodal model uses the entire image modality to make its classification, but also uses a small portion of the spectral data to further improve its estimate. A few limitations still remain with using a multimodal network architecture. Namely, there are instances where using a model trained on either spectral or image modalities independently performs better than a model trained on both modalities. This phenomenon occurs with both the plastic coffee-mate container and paper tissue box objects during leave-one-object-out cross-validation. For the plastic coffee-mate, texture imaging alone achieved 81.5% accuracy over 10 random seeds. Yet, our multimodal model only recognized this object as plastic 50.5% of the time. Similarly, our spectral model recognized the paper tissue box with 100.0% accuracy, yet when combined with image data, our multimodal model only classified this as paper 48.2% of the time. Upon inspection, we observed that the image model incorrectly classified 83.3% of paper tissue box images as plastic. We see these tissue box images often contain text similar to labels on commercial plastic objects in the dataset. These inaccuracies may be due in part to our dataset size, which may not fully capture the variation of materials across real-world objects. Increasing the number and variety of objects in the training set may further improve performance when generalizing material classification to new objects. Texture Image Features {#ssec:image_embeddings} ---------------------- Image Preprocessing Accuracy ---------------------------- ---------- $(320 \times 240)$ resize **70.5** $(320 \times 240)$ crop 61.6 $(640 \times 480)$ resize 69.0 $(640 \times 480)$ crop 65.7 $(1280 \times 960)$ resize 66.2 : \[table:image\_preprocessing\]8 material leave-on-object-out accuracy (model B), resizing or center cropping image input for DenseNet-201 features. In Section \[sec:dataset\], we presented a neural network architecture for material classification with visual texture data, dependent on preprocessing images with a DenseNet-201 model trained on ImageNet. Prior research has indicated that models trained on ImageNet have a high prior for recognizing texture features within an image [@geirhos2018imagenettexture]. In this section, we evaluate different texture image preprocessing techniques and compare ImageNet-trained models for texture representation. The original resolution of the captured close-range texture images is $1600 \times 1200$. CNN models are usually trained on ImageNet using center crops of $224 \times 224$. We test different resolutions for our texture images by center crop or resize, using DenseNet-201 as a feature extractor. Results are shown in Table \[table:image\_preprocessing\] for different raw image preprocessing techniques prior to computing the DenseNet-201 features, evaluated on 8 material leave-one-object-out material classification with the texture image model (model B). We find that resizing performs better than center cropping, suggesting that textural features are better captured with more visual surface area and context of the object, rather than a small but dense visual sample. Additionally, we observe that resizing to $320 \times 240$, which is near the image resolution that the CNN was trained at, performs better than resizing to higher resolutions. Network Accuracy ---------------------------------------- ---------- VGG19 [@simonyan2014vgg] 63.7 ResNet-50 [@he2016resnet] 66.2 ResNet-101 [@he2016resnet] 67.4 ResNet-152 [@he2016resnet] 66.0 DenseNet-201 [@huang2017densenet] **70.5** ResNeXt-101 [@xie2017resnext] 68.7 EfficientNet-B5 [@tan2019efficientnet] 69.4 : \[table:imagenet\_models\]Leave-on-object-out accuracy (model B), comparing features from ImageNet-trained models. $(320 \times 240)$ resized images. We generate low-dimensional visual features from common ImageNet-trained CNNs using texture images that were resized[^7] to a resolution of $320 \times 240$. Table \[table:imagenet\_models\] compares several ImageNet models for computing texture representations, which are then used to train the texture image model (model B) during leave-one-object-out cross-validation on all 8 materials. We observe that performance on ImageNet is loosely correlated with material classification accuracy with our image model. Further advances on CNN models benchmarked by ImageNet may continue to improve texture representation. Due to the architecture of our multimodal network, which learns separate and combined representations, advances in texture representation should lead to improvements on material classification with texture images. Table Scene Recognition ----------------------- We further evaluate our multimodal sensing approach by classifying materials of a scene of objects placed on a table, similar to what may be observed in a kitchen or home environment. We place one object from each material category from the heldout dataset on a table in front of the PR2. Using a 3D point cloud from its head-mounted Kinect, the PR2 segments objects from the table and defines pixel-level clusters in the 2D visual image for each distinct object found in the point cloud. The robot then classifies the material of each cluster using spectral and close-range texture image measurements from each object. To capture a measurement, the PR2 moves its left end effector to a position just in front of each object, matching a surface normal for the object computed from the point cloud. Fig. \[fig:evaluation\] shows a table setup with the PR2 and the pixel-level classification of each object using predictions from our multimodal material classification model. Our model correctly recognized the materials for seven of the eight heldout objects, missing only the foam yoga block. This demonstration can be seen in greater detail in the supplementary video. Conclusion ========== This paper introduces a multimodal sensing technique that combines near-infrared spectroscopy and close-range high resolution texture imaging for enabling robots to accurately classify the materials of household objects. We present and evaluate a new dataset of spectral measurements and high-resolution texture images for 144 household objects from 8 material categories. Compared to prior work in material classification with spectroscopy, our multimodal approach achieved 9% higher accuracy when generalizing to new, unseen household objects. In addition, we demonstrate how this sensing technique enables a robot to recognize materials across a scene of objects on a table, without physical contact with the objects. Through this work, we have shown that near-infrared spectroscopy and texturing imaging offers a reliable and accurate multimodal sensing approach for robots to estimate the materials of objects. [^1]: \*This work was supported by NSF award IIS-1514258 and AWS Cloud Credits for Research. Dr. Kemp owns equity in and works for Hello Robot, a company commercializing robotic assistance technologies. [^2]: Zackory Erickson, Eliot Xing, Bharat Srirangam, and Charles C. Kemp are with the Healthcare Robotics Lab, Georgia Institute of Technology, Atlanta, GA., USA. [^3]: Sonia Chernova is with the Robot Autonomy and Interactive Learning Lab, Georgia Institute of Technology, Atlanta, GA., USA. [^4]: Zackory Erickson is the corresponding author [zackory@gatech.edu]{}. [^5]: SpectroVisiondataset:[ https://github.com/Healthcare-Robotics/spectrovision/releases]( https://github.com/Healthcare-Robotics/spectrovision/releases) [^6]: Idealized measurements are collected with flat material objects to block out environmental light and reduce noise in spectral measurements. [^7]: Texture images were resized to $608 \times 456$ for EfficientNet-B5, near its native ImageNet input resolution.

--- abstract: | Following [@Visintin], we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we give some examples of this kind of sets, explaining how to construct them starting from well known self-similar fractals.\ In particular, we show that in the case of the von Koch snowflake $S\subset{\mathbb R}^2$ this fractal dimension coincides with the Minkowski dimension, namely $$P_s(S)<\infty\qquad\Longleftrightarrow\qquad s\in\Big(0,2-\frac{\log4}{\log3}\Big).$$ We also study the asymptotics as $s\to1^-$ of the fractional perimeter of a set having finite (classical) perimeter. author: - Luca Lombardini title: Fractional perimeter from a fractal perspective --- [Introduction and main results]{} It is well known (see e.g. [@Gamma] and [@cafenr]) that sets with a regular boundary have finite $s$-fractional perimeter for every $s\in(0,1)$. In this paper we show that also sets with an irregular, “fractal”, boundary can have finite $s$-perimeter for every $s$ below some threshold $\sigma<1$.\ Actually, the $s$-perimeter can be used to define a “fractal dimension” for the measure theoretic boundary $$\partial^-E:=\{x\in{\mathbb R}^n\,|\,0<|E\cap B_r(x)|<\omega_nr^n\textrm{ for every }r>0\},$$ of a set $E\subset{\mathbb R}^n$. Indeed, in [@Visintin] the author suggested using the index $s$ of the seminorm $[\chi_E]_{W^{s,1}}$ as a way to measure the codimension of $\partial^-E$ and he proved that the fractal dimension obtained in this way is less or equal than the (upper) Minkowski dimension. We give an example of a set, the von Koch snowflake, for which these two dimensions coincide. Moreover, exploiting the roto-translation invariance and the scaling property of the $s$-perimeter, we calculate the dimension of sets which can be defined in a recursive way similar to that of the von Koch snowflake. On the other hand, as remarked above, sets with a regular boundary have finite $s$-perimeter for every $s$ and actually their $s$-perimeter converges, as $s$ tends to 1, to the classical perimeter, both in the classical sense (see [@cafenr]) and in the $\Gamma$-convergence sense (see [@Gamma]).\ As a simple byproduct of the computations developed in this paper, we exploit Theorem 1 of [@Davila] to prove this asymptotic property for a set $E$ having finite classical perimeter in a bounded open set with Lipschitz boundary.\ This last result is probably well known to the expert, though not explicitly stated in the literature (as far as we know).\ In particular, we remark that this lowers the regularity requested in [@cafenr], where the authors asked the boundary $\partial E$ to be $C^{1,\alpha}$.\ We begin by recalling the definition of $s$-perimeter. Let $s\in(0,1)$ and let $\Omega\subset\mathbb R^n$ be an open set. The $s$-fractional perimeter of a set $E\subset\mathbb R^n$ in $\Omega$ is defined as $$P_s(E,\Omega):=\mathcal L_s(E\cap\Omega,{\mathcal C}E\cap\Omega)+ \mathcal L_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)+ \mathcal L_s(E\setminus\Omega,{\mathcal C}E\cap\Omega),$$ where $$\mathcal L_s(A,B):=\int_A\int_B\frac{1}{|x-y|^{n+s}}\,dx\,dy, $$ for every couple of disjoint sets $A,\,B\subset\mathbb R^n$. We simply write $P_s(E)$ for $P_s(E,{\mathbb R}^n)$.\ We can also write the fractional perimeter as the sum $$P_s(E,\Omega)=P_s^L(E,\Omega)+P_s^{NL}(E,\Omega),$$ where $$\begin{split} &P_s^L(E,\Omega):=\mathcal L_s(E\cap\Omega,{\mathcal C}E\cap\Omega)=\frac{1}{2}[\chi_E]_{W^{s,1}(\Omega)},\\ & P_s^{NL}(E,\Omega):={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}E\cap\Omega). \end{split}$$ We can think of $P^L_s(E,\Omega)$ as the local part of the fractional perimeter, in the sense that if $|(E\Delta F)\cap\Omega|=0$, then $P^L_s(F,\Omega)=P^L_s(E,\Omega)$. We say that a set $E$ has locally finite $s$-perimeter if it has finite $s$-perimeter in every bounded open set $\Omega\subset{\mathbb R}^n$.\ Now we give precise statements of the results obtained, starting with the fractional analysis of fractal dimensions. [Fractal boundaries]{} First of all, we prove in Section 3.1 that in some sense the measure theoretic boundary $\partial^-E$ is the “right definition” of boundary for working with the $s$-perimeter. To be more precise, we show that $$\partial^-E=\{x\in{\mathbb R}^n\,|\,P_s^L(E,B_r(x))>0,\,\forall\,r>0\},$$ and that if $\Omega$ is a connected open set, then $$P_s^L(E,\Omega)>0\quad\Longleftrightarrow\quad \partial^-E\cap\Omega\not=\emptyset.$$ This can be thought of as an analogue in the fractional framework of the fact that for a Caccioppoli set $E$ we have $\partial^-E=$ supp $|D\chi_E|$. Now the idea of the definition of the fractal dimension consists in using the index $s$ of $P_s^L(E,\Omega)$ to measure the codimension of $\partial^- E\cap\Omega$, $${\textrm{Dim}}_F(\partial^-E,\Omega):=n-\sup\{s\in(0,1)\,|\,P^L_s(E,\Omega)<\infty\}.$$ As shown in [@Visintin] (Proposition 11 and Proposition 13), the fractal dimension $\textrm{Dim}_F$ defined in this way is related to the (upper) Minkowski dimension by $$\label{intro_dim_ineq} {\textrm{Dim}}_F(\partial^-E,\Omega)\leq\overline{{\textrm{Dim}}}_\mathcal M(\partial^-E,\Omega),$$ (for the convenience of the reader we provide a proof in Proposition $\ref{vis_prop}$). If $\Omega$ is a bounded open set with Lipschitz boundary, this means that $$\label{intro_dim_ineq2} P_s(E,\Omega)<\infty\qquad\textrm{for every }s\in\big(0,n-\overline{{\textrm{Dim}}}_\mathcal M(\partial^-E,\Omega)\big),$$ since the nonlocal part of the $s$-perimeter of any set $E\subset{\mathbb R}^n$ is $$P_s^{NL}(E,\Omega)\leq2P_s(\Omega)<\infty,\qquad\textrm{for every }s\in(0,1).$$ We show that for the von Koch snowflake $(\ref{intro_dim_ineq})$ is actually an equality. ![[*The first three steps of the construction of the von Koch snowflake*]{}](fiocco){width="100mm"} Namely, we prove the following \[von\_koch\_snow\] Let $S\subset{\mathbb R}^2$ be the von Koch snowflake. Then $$\label{koch1} P_s(S)<\infty,\qquad\forall\,s\in\Big(0,2-\frac{\log4}{\log3}\Big),$$ and $$\label{koch2} P_s(S)=\infty,\qquad\forall\,s\in\Big[2-\frac{\log4}{\log3},1\Big).$$ Therefore $${\textrm{Dim}}_F(\partial S)={\textrm{Dim}}_\mathcal{M}(\partial S)=\frac{\log4}{\log3}.$$ Actually, exploiting the self-similarity of the von Koch curve, we have $${\textrm{Dim}}_F(\partial S,\Omega)=\frac{\log4}{\log3},$$ for every $\Omega$ s.t. $\partial S\cap\Omega\not=\emptyset$. In particular, this is true for every $\Omega=B_r(p)$ with $p\in S$ and $r>0$ as small as we want.\ We remark that this represents a deep difference between the classical and the fractional perimeter.\ Indeed, if a set $E$ has (locally) finite perimeter, then by De Giorgi’s structure Theorem we know that its reduced boundary $\partial^*E$ is locally $(n-1)$-rectifiable. Moreover $\overline{\partial^*E}=\partial^-E$, so the reduced boundary is, in some sense, a “big” portion of the measure theoretic boundary. On the other hand, there are (open) sets, like the von Koch snowflake, which have a “nowhere rectifiable” boundary (meaning that $\partial^-E\cap B_r(p)$ is not $(n-1)$-rectifiable for every $p\in\partial^-E$ and $r>0$) and still have finite $s$-perimeter for every $s\in(0,\sigma_0)$.\ Moreover our argument for the von Koch snowflake is quite general and can be adapted to calculate the dimension ${\textrm{Dim}}_F$ of all sets which can be constructed in a similar recursive way (see Section 3.4).\ Roughly speaking, these sets are defined by adding scaled copies of a fixed “building block” $T_0$, that is $$T:=\bigcup_{k=1}^\infty \bigcup_{i=1}^{ab^{k-1}}T_k^i,$$ where $T_k^i:=F_k^i(T_0)$ is a roto-traslation of the scaled set $\lambda^{-k}T_0$ (see Figure 2 below for an example). We also assume that $\frac{\log b}{\log\lambda}\in(n-1,n)$. Theorem $\ref{fractal_bdary_selfsim_dim}$ shows that if such a set $T$ satisfies an additional assumption, namely that “near” each set $T_k^i$ we can find a set $S_k^i=F^i_k(S_0)$ contained in ${\mathcal C}T$, then the fractal dimension of its measure theoretic boundary is $${\textrm{Dim}}_F(\partial^-T)=\frac{\log b}{\log\lambda}.$$ Many well known self-simlar fractals can be written either as (the boundary of) a set $T$ defined as above, like the von Koch snowflake, or as the difference $E=T_0\setminus T$, like the Sierpinski triangle and the Menger sponge. However sets of this second kind are often s.t. $|T\Delta T_0|=0$.\ Since the $s$-perimeters of two sets which differ only in a set of measure zero are equal, in this case the $s$-perimeter can not detect the “fractal nature” of $T$. Consider for example the Sierpinski triangle, which is defined as $E=T_0\setminus T$ with $T_0$ an equilateral triangle.\ Then $\partial^-T=\partial T_0$ and $P_s(T,\Omega)=P_s(T_0,\Omega)<\infty$ for every $s\in(0,1)$. Roughly speaking, the reason of this situation is that the fractal object is the topological boundary of $T$, while its measure theoretic boundary is regular and has finite (classical) perimeter.\ Still, we show how to modify such self-similar sets, without altering their “structure”, to obtain new sets which satisfy the hypothesis of Theorem $\ref{fractal_bdary_selfsim_dim}$. However, the measure theoretic boundary of such a new set will look quite different from the original fractal (topological) boundary and in general it will be a mix of smooth parts and unrectifiable parts. ![[*Example of a “fractal” set constructed exploiting the structure of the Sierpinski triangle (seen at the fourth iterative step), which satisfies the hypothesis of Theorem $\ref{fractal_bdary_selfsim_dim}$*]{}](Star31211color1){width="60mm"} The most interesting examples of this kind of sets are probably represented by bounded sets, like the one in Figure 2, because in this case the measure theoretic boundary does indeed have, in some sense, a “fractal nature”.\ Indeed, if $T$ is bounded, then its boundary $\partial^-T$ is compact. Nevertheless, it has infinite (classical) perimeter and actually $\partial^-T$ has Minkowski dimension strictly greater than $n-1$, thanks to $(\ref{intro_dim_ineq})$.\ However, even unbounded sets can have an interesting behavior. Indeed we obtain the following \[expl\_farc\_prop1\] Let $n\geq2$. For every $\sigma\in(0,1)$ there exists a Caccioppoli set $E\subset{\mathbb R}^n$s.t. $$P_s(E)<\infty\qquad\forall\,s\in(0,\sigma)\quad\textrm{and}\quad P_s(E)=\infty\qquad\forall\,s\in[\sigma,1).$$ Roughly speaking, the interesting thing about this Proposition is the following. Since $E$ has locally finite perimeter, $\chi_E\in BV_{loc}({\mathbb R}^n)$, it also has locally finite $s$-perimeter for every $s\in(0,1)$, but the global perimeter $P_s(E)$ is finite if and only if $s<\sigma<1$. [Asymptotics as $s\to1^-$]{} We have shown that sets with an irregular, eventually fractal, boundary can have finite $s$-perimeter. On the other hand, if the set $E$ is “regular”, then it has finite $s$-perimeter for every $s\in(0,1)$.\ Indeed, if $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary (or $\Omega={\mathbb R}^n$), then $BV(\Omega)\hookrightarrow W^{s,1}(\Omega)$. As a consequence of this embedding, we obtain $$P(E,\Omega)<\infty\qquad\Longrightarrow\qquad P_s(E,\Omega)<\infty\quad\textrm{for every }s\in(0,1).$$ Actually we can be more precise and obtain a sort of converse, using only the local part of the $s$-perimeter and adding the condition $$\liminf_{s\to1^-}(1-s)P^L_s(E,\Omega)<\infty.$$ Indeed one has the following result, which is just a combination of Theorem 3’ of [@BBM] and Theorem 1 of [@Davila], restricted to characteristic functions, \[Davila\_conv\_local\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then $E\subset{\mathbb R}^n$ has finite perimeter in $\Omega$ if and only if $P_s^L(E,\Omega)<\infty$ for every $s\in(0,1)$, and $$\label{asymptotics_fin_cond} \liminf_{s\to1}(1-s)P_s^L(E,\Omega)<\infty.$$ In this case we have $$\label{asymptotics_local_part} \lim_{s\to1}(1-s)P_s^L(E,\Omega)=\frac{n\omega_n}{2}K_{1,n}P(E,\Omega).$$ We briefly show how to get this result (and in particular why the constant looks like that) from the two Theorems cited above. We compute the constant $K_{1,n}$ in an elementary way, showing that $$\frac{n\omega_n}{2}K_{1,n}=\omega_{n-1}.$$ Moreover we show the following Condition $(\ref{asymptotics_fin_cond})$ is necessary. Indeed, there exist bounded sets (see the following Example) having finite $s$-perimeter for every $s\in(0,1)$ which do not have finite perimeter.\ This also shows that in general the inclusion $BV(\Omega)\subset\bigcap_{s\in(0,1)}W^{s,1}(\Omega)$ is strict. \[inclusion\_counterexample\] Let $0<a<1$ and consider the open intervals $I_k:=(a^{k+1},a^k)$ for every $k\in\mathbb{N}$. Define $E:=\bigcup_{k\in\mathbb{N}}I_{2k}$, which is a bounded (open) set.\ Due to the infinite number of jumps $\chi_E\not\in BV(\mathbb{R})$. However it can be proved that $E$ has finite $s$-perimeter for every $s\in(0,1)$. We postpone the proof to Appendix A. The main result of Section 2 is the following Theorem, which extends the asymptotic convergence of $(\ref{asymptotics_local_part})$ to the whole $s$-perimeter, at least when the boundary $\partial E$ intersects the boundary of $\Omega$ “transversally”. \[asymptotics\_teo\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Suppose that $E$ has finite perimeter in $\Omega_\beta$, for some $\beta\in(0,r_0)$, with $r_0>0$ small enough. Then $$\label{asymptotics_nonlocal_estimate} \limsup_{s\to1}(1-s)P_s^{NL}(E,\Omega) \leq2\omega_{n-1}\lim_{\rho\to0^+}P(E,N_\rho(\partial\Omega)).$$ In particular, if $P(E,\partial\Omega)=0$, then $$\lim_{s\to1}(1-s)P_s(E,\Omega)=\omega_{n-1}P(E,\Omega).$$ Moreover, there exists a set $S\subset(-r_0,\beta)$, at most countable, s.t. $$\label{asymptotics_ae_convergence} \lim_{s\to1}(1-s)P_s(E,\Omega_\delta)=\omega_{n-1}P(E,\Omega_\delta),$$ for every $\delta\in(-r_0,\beta)\setminus S$. Roughly speaking, the second part of this Theorem says that even if we do not have the asymptotic convergence of the $s$-perimeter in $\Omega$, we can slighltly enlarge or restrict $\Omega$ to obtain it. Actually, since $S$ has null measure, we can restrict or enlarge $\Omega$ as little as we want.\ In [@cafenr] the authors obtained a similar result for $\Omega=B_R$ a ball, but asking $C^{1,\alpha}$ regularity of $\partial E$ in $B_R$. They proved the convergence in every ball $B_r$ with $r\in(0,R)\setminus S$, with $S$ at most countable, exploiting uniform estimates.\ On the other hand, asking $E$ to have finite perimeter in a neighborhood (as small as we want) of the open set $\Omega$ is optimal.\ In [@Gamma] the authors studied the asymptotics as $s\longrightarrow1^-$ in the $\Gamma$-convergence sense. In particular, for the proof of a $\Gamma$-limsup inequality, which is typically constructive and by density, they show that if $\Pi$ is a polyhedron, then $$\limsup_{s\to1}(1-s)P_s(\Pi,\Omega) \leq\Gamma_n^*P(\Pi,\Omega)+2\Gamma_n^*\lim_{\rho\to0^+}P(\Pi,N_\rho(\partial\Omega)),$$ which is $(\ref{asymptotics_nonlocal_estimate})$, once we sum the local part of the perimeter. Their proof relies on the fact that $\Pi$ is a polyhedron to obtain the convergence of the local part of the perimeter, which is then used also in the estimate of the nonlocal part. Moreover they need an approximation result to prove that the constant is $\Gamma_n^*=\omega_{n-1}$. They also prove, in particular $$\Gamma-\liminf_{s\to1}(1-s)P_s^L(E,\Omega)\geq\omega_{n-1}P(E,\Omega),$$ which is a stronger result than the first part of Theorem $\ref{Davila_conv_local}$. [Notation and assumptions]{} - All sets and functions considered are assumed to be Lebesgue measurable. - We write $A\subset\subset B$ to mean that the closure of $A$ is compact and $\overline{A}\subset B$. - In ${\mathbb R}^n$ we will usually write $|E|=\mathcal{L}^n(E)$ for the $n$-dimensional Lebesgue measure of a set $E\subset{\mathbb R}^n$. - We write ${\mathcal H}^d$ for the $d$-dimensional Hausdorff measure, for any $d\geq0$. - We define the dimensional constants $$\omega_d:=\frac{\pi^\frac{d}{2}}{\Gamma\big(\frac{d}{2}+1\big)},\qquad d\geq0.$$ In particular, we remark that $\omega_k=\mathcal{L}^k(B_1)$ is the volume of the $k$-dimensional unit ball $B_1\subset{\mathbb R}^k$ and $k\,\omega_k={\mathcal H}^{k-1}(\mathbb{S}^{k-1})$ is the surface area of the $(k-1)$-dimensional sphere $$\mathbb{S}^{k-1}=\partial B_1=\{x\in{\mathbb R}^k\,|\,|x|=1\}.$$ - Since $$|E\Delta F|=0\quad\Longrightarrow\quad P(E,\Omega)=P(F,\Omega)\quad\textrm{and}\quad P_s(E,\Omega)=P_s(F,\Omega),$$ in Section 2 we implicitly identify sets up to sets of negligible Lebesgue measure.\ Moreover, whenever needed we can choose a particular representative for the class of $\chi_E$ in $L^1_{loc}({\mathbb R}^n)$, as in the Remark below.\ We will not make this assumption in Section 3, since the Minkowski content can be affected even by changes in sets of measure zero, that is, in general $$|\Gamma_1\Delta\Gamma_2|=0\quad\not\Rightarrow\quad \overline{\mathcal{M}}^r(\Gamma_1,\Omega)=\overline{\mathcal{M}}^r(\Gamma_2,\Omega)$$ (see Section 3 for a more detailed discussion). - We consider the open tubular $\rho$-neighborhood of $\partial\Omega$, $$N_\rho(\partial\Omega):=\{x\in{\mathbb R}^n\,|\,d(x,\partial\Omega)<\rho\}=\{|\bar{d}_\Omega|<\rho\}=\Omega_\rho\setminus\overline{\Omega_{-\rho}}$$ (see Appendix B). \[gmt\_assumption\] Let $E\subset{\mathbb R}^n$. Up to modifying $E$ on a set of measure zero, we can assume (see Appendix C) that $$\label{gmt_assumption_eq} \begin{split} &E_1\subset E,\qquad E\cap E_0=\emptyset\\ \textrm{and}\quad\partial E=\partial^-E&=\{x\in{\mathbb R}^n\,|\,0<|E\cap B_r(x)|<\omega_nr^n,\,\forall\,r>0\}. \end{split}$$ [Asymptotics as $s\to1^-$]{} We say that an open set $\Omega\subset{\mathbb R}^n$ is an extension domain if $\exists C=C(n,s,\Omega)>0$ s.t. for every $u\in W^{s,1}(\Omega)$ there exists $\tilde{u}\in W^{s,1}({\mathbb R}^n)$ with $\tilde{u}_{|\Omega}=u$ and $$\|\tilde{u}\|_{W^{s,1}({\mathbb R}^n)}\leq C\|u\|_{W^{s,1}(\Omega)}.$$ Every open set with bounded Lipschitz boundary is an extension domain (see [@HitGuide] for a proof). For simplicity we consider ${\mathbb R}^n$ itself as an extension domain. We begin with the following embedding. \[embedding\_prop\] Let $\Omega\subset{\mathbb R}^n$ be an extension domain. Then $\exists C(n,s,\Omega)\geq 1$ s.t. for every $u:\Omega\longrightarrow{\mathbb R}$ $$\label{embedding_ineq} \|u\|_{W^{s,1}(\Omega)}\leq C\|u\|_{BV(\Omega)}.$$ In particular we have the continuous embedding $$BV(\Omega)\hookrightarrow W^{s,1}(\Omega).$$ The claim is trivially satisfied if the right hand side of $(\ref{embedding_ineq})$ is infinite, so let $u\in BV(\Omega)$. Let $\{u_k\}\subset C^\infty(\Omega)\cap BV(\Omega)$ be an approximating sequence as in Theorem 1.17 of [@Giusti], that is $$\|u-u_k\|_{L^1(\Omega)}\longrightarrow0\qquad\textrm{and}\qquad\lim_{k\to\infty}\int_\Omega|\nabla u_k|\,dx=|Du|(\Omega).$$ We only need to check that the $W^{s,1}$-seminorm of $u$ is bounded by its $BV$-norm.\ Since $\Omega$ is an extension domain, we know (see Proposition 2.2 of [@HitGuide]) that $\exists C(n,s)\geq1$ s.t. $$\|v\|_{W^{s,1}(\Omega)}\leq C\|v\|_{W^{1,1}(\Omega)}.$$ Then $$[u_k]_{W^{s,1}(\Omega)}\leq\|u_k\|_{W^{s,1}(\Omega)}\leq C\|u_k\|_{W^{1,1}(\Omega)} =C\|u_k\|_{BV(\Omega)},$$ and hence, using Fatou’s Lemma, $$\begin{split} [u]_{W^{s,1}(\Omega)}&\leq\liminf_{k\to\infty}[u_k]_{W^{s,1}(\Omega)} \leq C\liminf_{k\to\infty}\|u_k\|_{BV(\Omega)}=C\lim_{k\to\infty}\|u_k\|_{BV(\Omega)}\\ & =C\|u\|_{BV(\Omega)}, \end{split}$$ proving $(\ref{embedding_ineq})$. \[embedding\_fin\_per\_coroll\] $(i)\quad$ If $E\subset{\mathbb R}^n$ has finite perimeter, i.e. $\chi_E\in BV({\mathbb R}^n)$, then $E$ has also finite $s$-perimeter for every $s\in(0,1)$.\ $(ii)\quad$ Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then there exists $r_0>0$ s.t. $$\label{unif_bound_lip_frac_per} \sup_{|r|<r_0}P_s(\Omega_r)<\infty.$$ $(iii)\quad$ If $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary, then $$P_s^{NL}(E,\Omega)\leq 2P_s(\Omega)<\infty$$ for every $E\subset{\mathbb R}^n$.\ $(iv)\quad$ Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then $$P(E,\Omega)<\infty\qquad\Longrightarrow\qquad P_s(E,\Omega)<\infty\quad\textrm{for every }s\in(0,1).$$ $(i)$ follows from $$P_s(E)=\frac{1}{2}[\chi_E]_{W^{s,1}({\mathbb R}^n)}$$ and previous Proposition with $\Omega={\mathbb R}^n$. $(ii)$ Let $r_0$ be as in Proposition $\ref{bound_perimeter_unif}$ and notice that $$P(\Omega_r)={\mathcal H}^{n-1}\big(\{\bar{d}_\Omega=r\}\big),$$ so that $$\|\chi_{\Omega_r}\|_{BV({\mathbb R}^n)}=|\Omega_r|+{\mathcal H}^{n-1}\big(\{\bar{d}_\Omega=r\}\big).$$ Thus $$\sup_{|r|<r_0}P_s(\Omega_r)\leq C\Big(|\Omega_{r_0}|+\sup_{|r|<r_0}{\mathcal H}^{n-1}\big(\{\bar{d}_\Omega=r\}\big)\Big)<\infty.$$ $(iii)$ Notice that $$\begin{split} &{\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)\leq {\mathcal L}_s(\Omega,{\mathcal C}\Omega)=P_s(\Omega),\\ & {\mathcal L}_s({\mathcal C}E\cap\Omega,E\setminus\Omega)\leq {\mathcal L}_s(\Omega,{\mathcal C}\Omega)=P_s(\Omega), \end{split}$$ and use $(\ref{unif_bound_lip_frac_per})$ (just with $\Omega_0=\Omega$). $(iv)$ The nonlocal part of the $s$-perimeter is finite thanks to $(iii)$. As for the local part, remind that $$P(E,\Omega)=|D\chi_E|(\Omega)\qquad\textrm{and}\qquad P_s^L(E,\Omega)=\frac{1}{2}[\chi_E]_{W^{s,1}(\Omega)},$$ then use previous Proposition. [Theorem $\ref{Davila_conv_local}$, asymptotics of the local part of the $s$-perimeter]{} \[bb\] Let $\Omega\subset{\mathbb R}^n$ be a smooth bounded domain. Let $u\in L^1(\Omega)$. Then $u\in BV(\Omega)$ if and only if $$\liminf_{n\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_n(x-y)\,dxdy<\infty,$$ and then $$\label{rough} \begin{split} C_1|Du|(\Omega)&\leq\liminf_{n\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_n(x-y)\,dxdy\\ & \leq\limsup_{n\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_n(x-y)\,dxdy\leq C_2|Du|(\Omega), \end{split}$$ for some constants $C_1$, $C_2$ depending only on $\Omega$. This result was refined by Davila Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Let $u\in BV(\Omega)$. Then $$\label{correct} \lim_{k\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_k(x-y)\,dxdy=K_{1,n}|Du|(\Omega),$$ where $$K_{1,n}=\frac{1}{n\omega_n}\int_{\mathbb{S}^{n-1}}|v\cdot e|\,d\sigma(v),$$ with $e\in{\mathbb R}^n$ any unit vector. In the above Theorems $\rho_k$ is any sequence of radial mollifiers i.e. of functions satisfying $$\label{rule1} \rho_k(x)\geq0,\quad\rho_k(x)=\rho_k(|x|),\quad\int_{{\mathbb R}^n}\rho_k(x)\,dx=1$$ and $$\label{rule2} \lim_{k\to\infty}\int_\delta^\infty\rho_k(r)r^{n-1}dr=0\quad\textrm{for all }\delta>0.$$ In particular, for $R$ big enough, $R>$ diam$(\Omega)$, we can consider $$\rho(x):=\chi_{[0,R]}(|x|)\frac{1}{|x|^{n-1}}$$ and define for any sequence $\{s_k\}\subset(0,1),\,s_k\nearrow1$, $$\rho_k(x):=(1-s_k)\rho(x)c_{s_k}\frac{1}{|x|^{s_k}},$$ where the $c_{s_k}$ are normalizing constants. Then $$\begin{split} \int_{{\mathbb R}^n}\rho_k(x)\,dx&=(1-s_k)c_{s_k}n\omega_n\int_0^R\frac{1}{r^{n-1+s_k}}r^{n-1}\,dr\\ & =(1-s_k)c_{s_k}n\omega_n\int_0^R\frac{1}{r^{s_k}}\,dr=c_{s_k}n\omega_nR^{1-s_k}, \end{split}$$ and hence taking $c_{s_k}:=\frac{1}{n\omega_n}R^{s_k-1}$ gives $(\ref{rule1})$; notice that $c_{s_k}\to\frac{1}{n\omega_n}$.\ Also $$\begin{split} \lim_{k\to\infty}\int_\delta^\infty\rho_k(r)r^{n-1}\,dr&= \lim_{k\to\infty}(1-s_k)c_{s_k}\int_\delta^R\frac{1}{r^{s_k}}\,dr\\ & =\lim_{k\to\infty}c_{s_k}(R^{1-s_k}-\delta^{1-s_k})=0, \end{split}$$ giving $(\ref{rule2})$.\ With this choice we get $$\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_k(x-y)\,dxdy=c_{s_k}(1-s_k)[u]_{W^{s_k,1}(\Omega)}.$$ Then, if $u\in BV(\Omega)$, Davila’s Theorem gives $$\label{limitperimeter}\begin{split} \lim_{s\to1}(1-s)[u]_{W^{s,1}(\Omega)}&=\lim_{s\to1}\frac{1}{c_s}(c_s(1-s)[u]_{W^{s,1}(\Omega)})\\ & =n\omega_nK_{1,n}|Du|(\Omega). \end{split}$$ [Proof of Theorem $\ref{asymptotics_teo}$]{} [The constant $\omega_{n-1}$]{} We need to compute the constant $K_{1,n}$. Notice that we can choose $e$ in such a way that $v\cdot e=v_n$.\ Then using spheric coordinates for ${\mathbb S}^{n-1}$ we obtain $|v\cdot e|=|\cos\theta_{n-1}|$ and $$d\sigma=\sin\theta_2(\sin\theta_3)^2\ldots(\sin\theta_{n-1})^{n-2}d\theta_1\ldots d\theta_{n-1},$$ with $\theta_1\in[0,2\pi)$ and $\theta_j\in[0,\pi)$ for $j=2,\ldots,n-1$. Notice that $$\begin{split} {\mathcal H}^k({\mathbb S}^k)&=\int_0^{2\pi}\,d\theta_1\int_0^\pi\sin\theta_2\,d\theta_2\ldots \int_0^\pi(\sin\theta_{k-1})^{k-2}\,d\theta_{k-1}\\ & ={\mathcal H}^{k-1}({\mathbb S}^{k-1})\int_0^\pi(\sin t)^{k-2}\,dt. \end{split}$$ Then we get $$\begin{split} \int_{{\mathbb S}^{n-1}}|v\cdot e|&\,d\sigma(v)={\mathcal H}^{n-2}({\mathbb S}^{n-2})\int_0^\pi(\sin t)^{n-2}|\cos t|\,dt\\ & ={\mathcal H}^{n-2}({\mathbb S}^{n-2})\Big(\int_0^\frac{\pi}{2}(\sin t)^{n-2}\cos t\,dt-\int_\frac{\pi}{2}^\pi(\sin t)^{n-2}\cos t\,dt\Big)\\ & =\frac{{\mathcal H}^{n-2}({\mathbb S}^{n-2})}{n-1}\Big(\int_0^\frac{\pi}{2}\frac{d}{dt}(\sin t)^{n-1}\,dt-\int_\frac{\pi}{2}^\pi\frac{d}{dt}(\sin t)^{n-1}\,dt\Big)\\ & =\frac{2{\mathcal H}^{n-2}({\mathbb S}^{n-2})}{n-1}. \end{split}$$ Therefore $$n\omega_nK_{1,n}=2\frac{{\mathcal H}^{n-2}({\mathbb S}^{n-2})}{n-1}=2{\mathcal L}^{n-1}(B_1(0))=2\omega_{n-1},$$ and hence $(\ref{limitperimeter})$ becomes $$\lim_{s\to1}(1-s)[u]_{W^{s,1}(\Omega)}=2\omega_{n-1}|Du|(\Omega),$$ for any $u\in BV(\Omega)$. [Estimating the nonlocal part of the $s$-perimeter]{} We prove something slightly more general than $(\ref{asymptotics_nonlocal_estimate})$. Namely, that to estimate the nonlocal part of the $s$-perimeter we do not necessarily need to use the sets $\Omega_\rho$: any “regular” approximation of $\Omega$ would do. Let $A_k,\, D_k\subset{\mathbb R}^n$ be two sequences of bounded open sets with Lipschitz boundary strictly approximating $\Omega$ respectively from the inside and from the outside, that is $(i)\quad A_k\subset A_{k+1}\subset\subset\Omega$ and $A_k\nearrow\Omega$, i.e. $\bigcup_k A_k=\Omega$, $(ii)\quad \Omega\subset\subset D_{k+1}\subset D_k$ and $D_k\searrow\overline{\Omega}$, i.e. $\bigcap_k D_k=\overline{\Omega}$.\ We define for every $k$ $$\begin{split} &\Omega_k^+:=D_k\setminus\overline{\Omega},\qquad\Omega_k^-:=\Omega\setminus\overline{A_k} \qquad T_k:=\Omega_k^+\cup\partial\Omega\cup\Omega_k^-,\\ &\qquad\qquad d_k:=\min\{d(A_k,\partial\Omega),\,d(D_k,\partial\Omega)\}>0. \end{split}$$ In particular we can consider $\Omega_\rho$ with $\rho<0$ in place of $A_k$ and with $\rho>0$ in place of $D_k$. Then $T_k$ would be $N_\rho(\partial\Omega)$ and $d_k=\rho$. Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary and let $E\subset{\mathbb R}^n$ be a set having finite perimeter in $D_1$. Then $$\limsup_{s\to1}(1-s)P_s^{NL}(E,\Omega)\leq 2\omega_{n-1}\lim_{k\to\infty}P(E,T_k).$$ In particular, if $P(E,\partial\Omega)=0$, then $$\lim_{s\to1}(1-s)P_s(E,\Omega)=\omega_{n-1}P(E,\Omega).$$ Since $\Omega$ is regular and $P(E,\Omega)<\infty$, we already know that $$\lim_{s\to1}(1-s)P_s^L(E,\Omega)=\omega_{n-1}P(E,\Omega).$$ Notice that, since $|D\chi_E|$ is a finite Radon measure on $D_1$ and $T_k\searrow\partial\Omega$ as $k\nearrow\infty$, we have $$\exists\lim_{k\to\infty}P(E,T_k)=P(E,\partial\Omega).$$ Consider the nonlocal part of the fractional perimeter, $$P_s^{NL}(E,\Omega)={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)+{\mathcal L}_s({\mathcal C}E\cap\Omega,E\setminus\Omega),$$ and take any $k$. Then $$\begin{split} {\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)&={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap\Omega_k^+)+{\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap({\mathcal C}\Omega\setminus D_k))\\ & \leq{\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap\Omega_k^+)+\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}\\ & \leq{\mathcal L}_s(E\cap\Omega_k^-,{\mathcal C}E\cap\Omega_k^+)+2\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}\\ & \leq{\mathcal L}_s(E\cap(\Omega_k^-\cup\Omega_k^+),{\mathcal C}E\cap(\Omega_k^-\cup\Omega_k^+))+2\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}\\ & =P^L_s(E,T_k)+2\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}. \end{split}$$ Since we can bound the other term in the same way, we get $$P^{NL}_s(E,\Omega)\leq2P^L_s(E,T_k)+4\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}.$$ By hypothesis we know that $T_k$ is a bounded open set with Lipschitz boundary $$\partial T_k=\partial A_k\cup\partial D_k.$$ Therefore using $(\ref{asymptotics_local_part})$ we have $$\lim_{s\to1}(1-s)P^L_s(E,T_k)=\omega_{n-1}P(E,T_k),$$ and hence $$\limsup_{s\to1}(1-s)P_s^{NL}(E,\Omega) \leq 2\omega_{n-1}P(E,T_k).$$ Since this holds true for any $k$, we get the claim. [Convergence in almost every $\Omega_\rho$]{} Having a “continuous” approximating sequence (the $\Omega_\rho$) rather than numerable ones allows us to improve the previous result and obtain the second part of Theorem $\ref{asymptotics_teo}$. We recall that De Giorgi’s structure Theorem for sets of finite perimeter (see e.g. Theorem 15.9 of [@Maggi]) guarantees in particular that $$|D\chi_E|={\mathcal H}^{n-1}\llcorner\partial^*E$$ and hence $$P(E,B)={\mathcal H}^{n-1}(\partial^*E\cap B)\qquad\textrm{for every Borel set }B\subset{\mathbb R}^n,$$ where $\partial^*E$ is the reduced boundary of $E$. Now suppose that $E$ has finite perimeter in $\Omega_\beta$. Then $$P(E,\partial\Omega_\delta)={\mathcal H}^{n-1}(\partial^*E\cap\{\bar{d}_\Omega=\delta\}),$$ for every $\delta\in(-r_0,\beta)$. Therefore, since $$M:={\mathcal H}^{n-1}(\partial^*E\cap(\Omega_\beta\setminus\overline{\Omega_{-r_0}}))\leq P(E,\Omega_\beta)<\infty,$$ the set $$S:=\left\{\delta\in(-r_0,\beta)\,|\,P(E,\partial\Omega_\delta)>0\right\}$$ is at most countable. Indeed, define $$S_k:=\Big\{\delta\in(-r_0,\beta)\,|\,{\mathcal H}^{n-1}(\partial^*E\cap\{\bar{d}_\Omega=\delta\})>\frac{1}{k}\Big\}.$$ Since $${\mathcal H}^{n-1}\Big(\bigcup_{-r_0<\delta<\beta}(\partial^*E\cap\{\bar{d}_\Omega=\delta\})\Big)=M,$$ the number of elements in each $S_k$ is at most $$\sharp S_k\leq M\,k.$$ As a consequence, $S=\bigcup_k S_k$ is at most countable.\ This concludes the proof of Theorem $\ref{asymptotics_teo}$. [Irregularity of the boundary]{} [The measure theoretic boundary as “support” of the local part of the $s$-perimeter]{} First of all we show that the (local part of the) $s$-perimeter does indeed measure a quantity related to the measure theoretic boundary. Let $E\subset{\mathbb R}^n$ be a set of locally finite $s$-perimeter. Then $$\partial^-E=\{x\in{\mathbb R}^n\,|\,P_s^L(E,B_r(x))>0\textrm{ for every }r>0\}.$$ The claim follows from the following observation. Let $A,\,B\subset{\mathbb R}^n$ s.t. $A\cap B=\emptyset$; then $${\mathcal L}_s(A,B)=0\quad\Longleftrightarrow\quad|A|=0\quad\textrm{or}\quad|B|=0.$$ Therefore $$\begin{split} x\in\partial^-E&\quad\Longleftrightarrow\quad |E\cap B_r(x)|>0\textrm{ and }|{\mathcal C}E\cap B_r(x)|>0\quad\forall\,r>0\\ & \quad\Longleftrightarrow\quad {\mathcal L}_s(E\cap B_r(x),{\mathcal C}E\cap B_r(x))>0\quad\forall\,r>0. \end{split}$$ This characterization of $\partial^-E$ can be thought of as a fractional analogue of $(\ref{support_perimeter})$. However we can not really think of $\partial^-E$ as the support of $$P_s^L(E,-):\Omega\longmapsto P_s^L(E,\Omega),$$ in the sense that, in general $$\partial^-E\cap\Omega=\emptyset\quad\not\Rightarrow\quad P_s^L(E,\Omega)=0.$$ For example, consider $E:=\{x_n\leq0\}\subset{\mathbb R}^n$ and notice that $\partial^-E=\{x_n=0\}$. Let $\Omega:=B_1(2e_n)\cup B_1(-2e_n)$. Then $\partial^-E\cap\Omega=\emptyset$, but $$P_s^L(E,\Omega)={\mathcal L}_s(B_1(2e_n),B_1(-2e_n))>0.$$ On the other hand, the only obstacle is the non connectedness of the set $\Omega$ and indeed we obtain the following Let $E\subset{\mathbb R}^n$ be a set of locally finite $s$-perimeter and let $\Omega\subset{\mathbb R}^n$ be an open set. Then $$\partial^-E\cap\Omega\not=\emptyset\quad\Longrightarrow\quad P_s^L(E,\Omega)>0.$$ Moreover, if $\Omega$ is connected $$\partial^-E\cap\Omega=\emptyset\quad\Longrightarrow\quad P_s^L(E,\Omega)=0.$$ Therefore, if $\widehat{\mathcal O}({\mathbb R}^n)$ denotes the family of bounded and connected open sets, then $\partial^-E$ is the “support” of $$\begin{split} P_s^L(E,-):\,&\widehat{\mathcal O}({\mathbb R}^n)\longrightarrow [0,\infty)\\ & \Omega\longmapsto P_s^L(E,\Omega), \end{split}$$ in the sense that, if $\Omega\in\widehat{\mathcal O}({\mathbb R}^n)$, then $$P_s^L(E,\Omega)>0\quad\Longleftrightarrow\quad\partial^-E\cap\Omega\not=\emptyset.$$ Let $x\in\partial^-E\cap\Omega$. Since $\Omega$ is open, we have $B_r(x)\subset\Omega$ for some $r>0$ and hence $$P_s^L(E,\Omega)\geq P_s^L(E,B_r(x))>0.$$ Let $\Omega$ be connected and suppose $\partial^-E\cap\Omega=\emptyset$. We have the partition of ${\mathbb R}^n$ as ${\mathbb R}^n=E_0\cup\partial^-E\cup E_1$ (see Appendix C). Thus we can write $\Omega$ as the disjoint union $$\Omega=(E_0\cap\Omega)\cup(E_1\cap\Omega).$$ However, since $\Omega$ is connected and both $E_0$ and $E_1$ are open, we must have $E_0\cap\Omega=\emptyset$ or $E_1\cap\Omega=\emptyset$. Now, if $E_0\cap\Omega=\emptyset$ (the other case is analogous), then $\Omega\subset E_1$ and hence $|{\mathcal C}E\cap\Omega|=0$. Thus $$P_s^L(E,\Omega)={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap\Omega)=0.$$ [A notion of fractal dimension]{} Let $\Omega\subset{\mathbb R}^n$ be an open set. Then $$t>s\qquad\Longrightarrow\qquad W^{t,1}(\Omega)\hookrightarrow W^{s,1}(\Omega),$$ (see e.g. Proposition 2.1 of [@HitGuide]). As a consequence, for every $u:\Omega\longrightarrow{\mathbb R}$ there exists a unique $R(u)\in[0,1]$ s.t. $$[u]_{W^{s,1}(\Omega)}\quad\left\{\begin{array}{cc} <\infty,& \forall\,s\in(0,R(u))\\ =\infty, &\forall\,s\in(R(u),1) \end{array}\right.$$ that is $$\begin{split}\label{frac_range} R(u)&=\sup\left\{s\in(0,1)\,\big|\,[u]_{W^{s,1}(\Omega)}<\infty\right\}\\ & =\inf\left\{s\in(0,1)\,\big|\,[u]_{W^{s,1}(\Omega)}=\infty\right\}. \end{split}$$ In particular, exploiting this result for characteristic functions, in [@Visintin] the author suggested the following definition of fractal dimension. Let $\Omega\subset{\mathbb R}^n$ be an open set and let $E\subset{\mathbb R}^n$. If $\partial^- E\cap\Omega\not=\emptyset$, we define $${\textrm{Dim}}_F(\partial^- E,\Omega):=n-R(\chi_E),$$ the fractal dimension of $\partial^- E$ in $\Omega$, relative to the fractional perimeter.\ If $\Omega={\mathbb R}^n$, we drop it in the formulas. Notice that in the case of sets $(\ref{frac_range})$ becomes $$\begin{split}\label{frac_range_sets} R(\chi_E)&=\sup\left\{s\in(0,1)\,\big|\,P_s^L(E,\Omega)<\infty\right\}\\ & =\inf\left\{s\in(0,1)\,\big|\,P_s^L(E,\Omega)=\infty\right\}. \end{split}$$ In particular we can take $\Omega$ to be the whole of ${\mathbb R}^n$, or a bounded open set with Lipschitz boundary.\ In the first case the local part of the fractional perimeter coincides with the whole fractional perimeter, while in the second case we know that we can bound the nonlocal part with $2P_s(\Omega)<\infty$ for every $s\in(0,1)$. Therefore in both cases in $(\ref{frac_range_sets})$ we can as well take the whole fractional perimeter $P_s(E,\Omega)$ instead of just the local part.\ Now we give a proof of the relation $(\ref{intro_dim_ineq})$ (obtained in [@Visintin]).\ For simplicity, given $\Gamma\subset{\mathbb R}^n$ we set $$\label{neigh_mink_def} \bar{N}_\rho^\Omega(\Gamma):=\overline{N_\rho(\Gamma)}\cap\Omega =\{x\in\Omega\,|\,d(x,\Gamma)\leq\rho\},$$ for any $\rho>0$. \[vis\_prop\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set. Then for every $E\subset{\mathbb R}^n$ s.t. $\partial^- E\cap\Omega\not=\emptyset$ and $\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\geq n-1$ we have $${\textrm{Dim}}_F(\partial^-E,\Omega)\leq\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega).$$ By hypothesis we have $$\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)=n-\inf\big\{r\in(0,1)\,|\,\overline{\mathcal{M}}^{n-r}(\partial^-E,\Omega)=\infty\big\},$$ and we need to show that $$\inf\big\{r\in(0,1)\,|\,\overline{\mathcal{M}}^{n-r}(\partial^-E,\Omega)=\infty\big\} \leq \sup\{s\in(0,1)\,|\,P_s^L(E,\Omega)<\infty\}.$$ Up to modifying $E$ on a set of Lebesgue measure zero we can suppose that $\partial E=\partial^-E$, as in Remark $\ref{gmt_assumption}$. Notice that this does not affect the $s$-perimeter. Now for any $s\in(0,1)$ $$\begin{split} 2P_s^L(E,\Omega)&=\int_\Omega\,dx\int_\Omega\frac{|\chi_E(x)-\chi_E(y)|}{|x-y|^{n+s}}\,dy\\ & =\int_\Omega dx\int_0^\infty d\rho\int_{\partial B_\rho(x)\cap\Omega}\frac{|\chi_E(x)-\chi_E(y)|}{|x-y|^{n+s}}\,d{\mathcal H}^{n-1}(y)\\ & =\int_\Omega dx\int_0^\infty\frac{d\rho}{\rho^{n+s}}\int_{\partial B_\rho(x)\cap\Omega}|\chi_E(x)-\chi_E(y)|\,d{\mathcal H}^{n-1}(y). \end{split}$$ Notice that $$d(x,\partial E)>\rho\quad\Longrightarrow\quad\chi_E(y)=\chi_E(x),\quad\forall\,y\in\overline{B_\rho(x)},$$ and hence $$\begin{split} \int_{\partial B_\rho(x)\cap\Omega}|\chi_E(x)-\chi_E(y)|\,d{\mathcal H}^{n-1}(y)& \leq\int_{\partial B_\rho(x)\cap\Omega}\chi_{\bar{N}_\rho(\partial E)}(x)\,d{\mathcal H}^{n-1}(y)\\ & \leq n\omega_n\rho^{n-1}\chi_{\bar{N}_\rho(\partial E)}(x). \end{split}$$ Therefore $$\label{visintin_pf} 2P_s^L(E,\Omega)\leq n\omega_n\int_0^\infty\frac{d\rho}{\rho^{1+s}}\int_\Omega \chi_{\bar{N}_\rho(\partial E)}(x) =n\omega_n\int_0^\infty\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^{1+s}}\,d\rho.$$ We prove the following\ CLAIM $$\label{visintin_proof} \overline{\mathcal{M}}^{n-r}(\partial E,\Omega)<\infty\quad\Longrightarrow\quad P_s^L(E,\Omega)<\infty,\quad\forall\,s\in(0,r).$$ Indeed $$\limsup_{\rho\to0}\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^r}<\infty\quad\Longrightarrow\quad\exists\,C>0\textrm{ s.t. } \sup_{\rho\in(0,C]}\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^r}\leq M<\infty.$$ Then $$\begin{split} 2P_s^L(E,\Omega)&\leq n\omega_n\Big\{\int_0^C\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^{1-(r-s)+r}}\,d\rho +\int_C^\infty\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^{1+s}}\,d\rho\Big\}\\ & \leq n\omega_n\Big\{ M\int_0^C\frac{1}{\rho^{1-(r-s)}}\,d\rho+|\Omega|\int_C^\infty\frac{1}{\rho^{1+s}}\,d\rho \Big\}\\ & =n\omega_n\Big\{ \frac{M}{r-s}C^{r-s}+\frac{|\Omega|}{sC^s} \Big\}<\infty, \end{split}$$ proving the claim.\ This implies $$r\leq\sup\{s\in(0,1)\,|\,P_s^L(E,\Omega)<\infty\},$$ for every $r\in(0,1)$ s.t. $\overline{\mathcal{M}}^{n-r}(\partial E,\Omega)<\infty$.\ Thus for $\epsilon>0$ very small, we have $$\inf\big\{r\in(0,1)\,|\,\overline{\mathcal{M}}^{n-r}(\partial^-E,\Omega)=\infty\big\}-\epsilon \leq\sup\{s\in(0,1)\,|\,P_s^L(E,\Omega)<\infty\}.$$ Letting $\epsilon$ tend to zero, we conclude the proof. In particular, if $\Omega$ has Lipschitz boundary we obtain \[fractal\_dim\_coroll\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Let $E\subset{\mathbb R}^n$ s.t. $\partial^-E\cap\Omega\not=\emptyset$ and $\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\in[n-1,n)$. Then $$\label{fractal_per} P_s(E,\Omega)<\infty\qquad\textrm{for every }s\in\left(0,n-\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\right).$$ \[fractal\_dim\_rmk\] Actually, previous Proposition and Corollary still work when $\Omega={\mathbb R}^n$, provided the set $E$ we are considering is bounded.\ Indeed, if $E$ is bounded, we can apply previous results with $\Omega=B_R$ s.t. $E\subset\Omega$. Moreover, since $\Omega$ has a regular boundary, as remarked above we can take the whole $s$-perimeter in $(\ref{frac_range_sets})$, instead of just the local part. But then, since $P_s(E,\Omega)=P_s(E)$, we see that $${\textrm{Dim}}_F(\partial^-E,\Omega)={\textrm{Dim}}_F(\partial^-E,{\mathbb R}^n).$$ [The measure theoretic boundary of a set of locally finite $s$-perimeter (in general) is not rectifiable]{} These results show that a set $E$ can have finite fractional perimeter even if its boundary is really irregular, unlike what happens with a Caccioppoli set and its reduced boundary, which is locally $(n-1)$-rectifiable.\ Indeed, if $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary and $E\subset{\mathbb R}^n$ is s.t. $\emptyset\not=\partial^-E\cap\Omega$ is not $(n-1)$-rectifiable, with $\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\in(n-1,n)$, thanks to previous Corollary we have $P_s(E,\Omega)<\infty$ for every $s\in(0,\sigma)$. We give some examples of this kind of sets in the following Sections.\ In particular, the von Koch snowflake $S\subset{\mathbb R}^2$ has finite $s$-perimeter for every $s\in(0,\sigma)$, but $\partial^-S=\partial S$ is not locally $(n-1)$-rectifiable.\ Actually, because of the self-similarity of the von Koch curve, there is no part of $\partial S$ which is rectifiable (see below).\ On the other hand, De Giorgi’s structure Theorem (see e.g. Theorem 15.9 and Corollary 16.1 of [@Maggi]) says that if a set $E\subset{\mathbb R}^n$ has locally finite perimeter, then its reduced boundary $\partial^*E$ is locally $(n-1)$-rectifiable.\ Moreover the reduced boundary is dense in the measure theoretic boundary, which is the support of the Radon measure $|D\chi_E|$, $$\overline{\partial^*E}=\partial^-E=\textrm{supp }|D\chi_E|.$$ This underlines a deep difference between the classical perimeter and the $s$-perimeter, which can indeed be thought of as a “fractional” perimeter.\ Namely, having (locally) finite classical perimeter implies the regularity of an “important” portion of the (measure theoretic) boundary. On the other hand, a set can have a fractal, nowhere rectifiable boundary and still have (locally) finite $s$-perimeter. [Remarks about the Minkowski content of $\partial^-E$]{} In the beginning of the proof of Proposition $\ref{vis_prop}$ we chose a particular representative for the class of $E$ in order to have $\partial E=\partial^-E$. This can be done since it does not affect the $s$-perimeter and we are already considering the Minkowski dimension of $\partial^-E$. On the other hand, if we consider a set $F$ s.t. $|E\Delta F|=0$, we can use the same proof to obtain the inequality $${\textrm{Dim}}_F(\partial^-E,\Omega)\leq\overline{{\textrm{Dim}}}_\mathcal{M}(\partial F,\Omega).$$ It is then natural to ask whether we can find a “better” representative $F$, whose (topological) boundary $\partial F$ has Minkowski dimension strictly smaller than that of $\partial^-E$. First of all, we remark that the Minkowski content can be influenced by changes in sets of measure zero. Roughly speaking, this is because the Minkowski content is not a purely measure theoretic notion, but rather a combination of metric and measure. For example, let $\Gamma\subset{\mathbb R}^n$ and define $\Gamma':=\Gamma\cup\mathbb Q^n$. Then $|\Gamma\Delta\Gamma'|=0$, but $N_\delta(\Gamma')={\mathbb R}^n$ for every $\delta>0$. In particular, considering different representatives for $E$ we will get different topological boundaries and hence different Minkowski dimensions. However, since the measure theoretic boundary minimizes the size of the topological boundary, that is $$\partial^-E=\bigcap_{|F\Delta E|=0}\partial F,$$ (see Appendix C), it minimizes also the Minkowski dimension.\ Indeed, for every $F$ s.t. $|F\Delta E|=0$ we have $$\begin{split} \partial^-E\subset\partial F&\quad\Longrightarrow\quad \bar N_\rho^\Omega(\partial^-E)\subset \bar N_\rho^\Omega(\partial F)\\ & \quad\Longrightarrow\quad \overline{\mathcal M}^r(\partial^-E,\Omega)\leq \overline{\mathcal M}^r(\partial F,\Omega)\\ & \quad\Longrightarrow\quad \overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\leq \overline{{\textrm{Dim}}}_\mathcal{M}(\partial F,\Omega). \end{split}$$ [Fractal dimension of the von Koch snowflake]{} The von Koch snowflake $S\subset{\mathbb R}^2$ is an example of bounded open set with fractal boundary, for which the Minkowski dimension and the fractal dimension introduced above coincide. Moreover its boundary is “nowhere rectifiable”, in the sense that $\partial S\cap B_r(p)$ is not $(n-1)$-rectifiable for any $r>0$ and $p\in\partial S$.\ First of all we construct the von Koch curve. Then the snowflake is made of three von Koch curves. Let $\Gamma_0$ be a line segment of unit length. The set $\Gamma_1$ consists of the four segments obtained by removing the middle third of $\Gamma_0$ and replacing it by the other two sides of the equilateral triangle based on the removed segment.\ We construct $\Gamma_2$ by applying the same procedure to each of the segments in $\Gamma_1$ and so on. Thus $\Gamma_k$ comes from replacing the middle third of each straight line segment of $\Gamma_{k-1}$ by the other two sides of an equilateral triangle. As $k$ tends to infinity, the sequence of polygonal curves $\Gamma_k$ approaches a limiting curve $\Gamma$, called the von Koch curve.\ If we start with an equilateral triangle with unit length side and perform the same construction on all three sides, we obtain the von Koch snowflake $\Sigma$.\ Let $S$ be the bounded region enclosed by $\Sigma$, so that $S$ is open and $\partial S=\Sigma$. We still call $S$ the von Koch snowflake.\ Now we calculate the (Minkowski) dimension of $\Gamma$ using the box-counting dimensions (see Appendix D).\ The idea is to exploit the self-similarity of $\Gamma$ and consider covers made of squares with side $\delta_k=3^{-k}$. The key observation is that $\Gamma$ can be covered by three squares of length $1/3$ (and cannot be covered by only two), so that $\mathcal{N}(\Gamma,1/3)=3$.\ Then consider $\Gamma_1$. We can think of $\Gamma$ as being made of four von Koch curves starting from the set $\Gamma_1$ and with initial segments of length $1/3$ instead of 1. Therefore we can cover each of these four pieces with three squares of side $1/9$, so that $\Gamma$ can be covered with $3\cdot4$ squares of length $1/9$ (and not one less) and $\mathcal{N}(\Gamma,1/9)=4\cdot3$. We can repeat the same argument starting from $\Gamma_2$ to get $\mathcal{N}(\Gamma,1/27)=4^2\cdot3$, and so on. In general we obtain $$\mathcal{N}(\Gamma,3^{-k})=4^{k-1}\cdot3.$$ Then, taking logarithms we get $$\frac{\log\mathcal{N}(\Gamma,3^{-k})}{-\log3^{-k}}=\frac{\log3+(k-1)\log4}{k\log3}\longrightarrow\frac{\log4}{\log3},$$ so that ${\textrm{Dim}}_\mathcal{M}(\Gamma)=\frac{\log4}{\log3}$. Notice that the Minkowski dimensions of the snowflake and of the curve are the same. Moreover it can be shown that the Hausdorff dimension of the von Koch curve is equal to its Minkowski dimension, so we obtain $$\label{dime_Koch_snow} {\textrm{Dim}}_{\mathcal H}(\Sigma)={\textrm{Dim}}_\mathcal M(\Sigma)=\frac{\log4}{\log3}$$ Now we explain how to construct $S$ in a recursive way and we prove that $$\partial^-S=\partial S=\Sigma.$$ As starting point for the snowflake take the equilateral triangle $T$ of side 1, with baricenter in the origin and a vertex on the $y$-axis, $P=(0,t)$ with $t>0$.\ Then $T_1$ is made of three triangles of side $1/3$, $T_2$ of $3\cdot4$ triangles of side $1/3^2$ and so on.\ In general $T_k$ is made of $3\cdot4^{k-1}$ triangles of side $1/3^k$, call them $T_k^1,\ldots,T_k^{3\cdot4^{k-1}}$. Let $x^i_k$ be the baricenter of $T_k^i$ and $P_k^i$ the vertex which does not touch $T_{k-1}$. Then $S=T\cup\bigcup T_k$. Also notice that $T_k$ and $T_{k-1}$ touch only on a set of measure zero. For each triangle $T^i_k$ there exists a rotation $\mathcal{R}_k^i\in SO(n)$ s.t. $$T_k^i=F_k^i(T):=\mathcal{R}_k^i\Big(\frac{1}{3^k}T\Big)+x_k^i.$$ We choose the rotations so that $F_k^i(P)=P_k^i$. Notice that for each triangle $T_k^i$ we can find a small ball which is contained in the complementary of the snowflake, $B_k^i\subset{\mathcal C}S$, and touches the triangle in the vertex $P_k^i$. Actually these balls can be obtained as the images of the affine transformations $F_k^i$ of a fixed ball $B$. To be more precise, fix a small ball contained in the complementary of $T$, which has the center on the $y$-axis and touches $T$ in the vertex $P$, say $B:=B_{1/1000}(0,t+1/1000)$. Then $$\label{koch3} B_k^i:=F_k^i(B)\subset{\mathcal C}S$$ for every $i,\,k$. To see this, imagine constructing the snowflake $S$ using the same affine transformations $F_k^i$ but starting with $T\cup B$ in place of $T$.\ We know that $\partial^-S\subset\partial S$ (see Appendix C).\ On the other hand, let $p\in\partial S$. Then every ball $B_\delta(p)$ contains at least a triangle $T^i_k\subset S$ and its corresponding ball $B^i_k\subset{\mathcal C}S$ (and actually infinitely many). Therefore $0<|B_\delta(p)\cap S|<\omega_n\delta^n$ for every $\delta>0$ and hence $p\in\partial^-S$. Since $S$ is bounded, its boundary is $\partial^-S=\Sigma$, and ${\textrm{Dim}}_\mathcal M(\Sigma)=\frac{\log4}{\log3}$, we obtain $(\ref{koch1})$ from Corollary $\ref{fractal_dim_coroll}$ and Remark $\ref{fractal_dim_rmk}$. Exploiting the construction of $S$ given above and $(\ref{koch3})$ we prove $(\ref{koch2})$.\ We have $$\begin{split} P_s(S)&={\mathcal L}_s(S,{\mathcal C}S)={\mathcal L}_s(T,{\mathcal C}S)+\sum_{k=1}^\infty{\mathcal L}_s(T_k,{\mathcal C}S)\\ & ={\mathcal L}_s(T,{\mathcal C}S)+\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}S) \geq\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}S)\\ & \geq\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(T_k^i,B_k^i)\qquad\textrm{(by }(\ref{koch3}))\\ & =\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(F_k^i(T),F_k^i(B))\\ & =\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}\Big(\frac{1}{3^k}\Big)^{2-s}{\mathcal L}_s(T,B)\qquad\textrm{(by Proposition }\ref{elementary_properties})\\ & =\frac{3}{3^{2-s}}{\mathcal L}_s(T,B)\sum_{k=0}^\infty\Big(\frac{4}{3^{2-s}}\Big)^k. \end{split}$$ We remark that $${\mathcal L}_s(T,B)\leq{\mathcal L}_s(T,{\mathcal C}T)=P_s(T)<\infty,$$ for every $s\in(0,1)$. To conclude, notice that the last series is divergent if $s\geq2-\frac{\log4}{\log3}$. Exploiting the self-similarity of the von Koch curve, we show that the fractal dimension of $S$ is the same in every open set which contains a point of $\partial S$. \[koch\_coroll\] Let $S\subset{\mathbb R}^2$ be the von Koch snowflake. Then $${\textrm{Dim}}_F(\partial S,\Omega)=\frac{\log4}{\log3}$$ for every open set $\Omega$ s.t. $\partial S\cap\Omega\not=\emptyset$. Since $P_s(S,\Omega)\leq P_s(S)$, we have $$P_s(S,\Omega)<\infty,\qquad\forall\,s\in\Big(0,2-\frac{\log4}{\log3}\Big).$$ On the other hand, if $p\in\partial S\cap\Omega$, then $B_r(p)\subset\Omega$ for some $r>0$. Now notice that $B_r(p)$ contains a rescaled version of the von Koch curve, including all the triangles $T_k^i$ which constitute it and the relative balls $B_k^i$. We can thus repeat the argument above to obtain $$P_s(S,\Omega)\geq P_s(S,B_r(p))=\infty,\qquad\forall\,s\in\Big[2-\frac{\log4}{\log3},1\Big).$$ [Self-similar fractal boundaries]{} The von Koch curve is a well known example of a family of rather “regular” fractal sets, the self-similar fractal sets (see e.g. Section 9 of [@Falconer] for the proper definition and the main properties). Many examples of this kind of sets can be constucted in a recursive way similar to that of the von Koch snowflake. To be more precise, we start with a bounded open set $T_0\subset{\mathbb R}^n$ with finite perimeter $P(T_0)<\infty$, which is, roughly speaking, our basic “building block”. Then we go on inductively by adding roto-translations of a scaling of the building block $T_0$, i.e. sets of the form $$T_k^i=F_k^i(T_0):=\mathcal{R}_k^i\big(\lambda^{-k}T_0\big)+x_k^i,$$ where $\lambda>1$, $k\in\mathbb N$, $1\leq i\leq ab^{k-1}$, with $a,\,b\in\mathbb N$, $\mathcal{R}_k^i\in SO(n)$ and $x_k^i\in{\mathbb R}^n$. We ask that these sets do not overlap, i.e. $$|T^i_k\cap T^j_h|=0,\qquad\textrm{if }i\not=j.$$ Then we define $$\label{frac_ind_def} T_k:=\bigcup_{i=1}^{ab^{k-1}}T_k^i\qquad\textrm{and}\qquad T:=\bigcup_{k=1}^\infty T_k.$$ The final set $E$ is either $$E:=T_0\cup\bigcup_{k\geq1}\bigcup_{i=1}^{ab^{k-1}}T^i_k,\quad\textrm{or}\quad E:=T_0\setminus\Big(\bigcup_{k\geq1}\bigcup_{i=1}^{ab^{k-1}}T^i_k\Big).$$ For example, the von-Koch snowflake is obtained by adding pieces. Examples obtained by removing the $T_k^i$’s are the middle Cantor set $E\subset{\mathbb R}$, the Sierpinski triangle $E\subset{\mathbb R}^2$ and Menger sponge $E\subset{\mathbb R}^3$.\ We will consider just the set $T$ and exploit the same argument used for the von Koch snowflake to compute the fractal dimension related to the $s$-perimeter.\ However, the Cantor set, the Sierpinski triangle and the Menger sponge are s.t. $|E|=0$, i.e. $|T_0\Delta T|=0$.\ Therefore both the perimeter and the $s$-perimeter do not notice the fractal nature of the (topological) boundary of $T$ and indeed, since $P(T)=P(T_0)<\infty$, we get $P_s(T)<\infty$ for every $s\in(0,1)$. For example, in the case of the Sierpinski triangle, $T_0$ is an equilateral triangle and $\partial^-T=\partial T_0$, even if $\partial T$ is a self-similar fractal. Roughly speaking, the problem in these cases is that there is not room enough to find a small ball $B_k^i=F_k^i(B)\subset{\mathcal C}T$ near each piece $T_k^i$. Therefore, we will make the additional assumption that $$\label{add_frac_self_hp} \exists\,S_0\subset{\mathcal C}T\quad\textrm{s.t. }|S_0|>0\quad\textrm{and }S_k^i:=F_k^i(S_0)\subset{\mathcal C}T\quad\forall\,k,\,i.$$ We remark that it is not necessary to ask that these sets do not overlap. Below we give some examples on how to construct sets which satisfy this additional hypothesis starting with sets which do not, like the Sierpinski triangle, without altering their “structure”. \[fractal\_bdary\_selfsim\_dim\] Let $T\subset{\mathbb R}^n$ be a set which can be written as in $(\ref{frac_ind_def})$. If $\frac{\log b}{\log\lambda}\in(n-1,n)$ and $(\ref{add_frac_self_hp})$ holds true, then $$P_s(T)<\infty,\qquad\forall\,s\in\Big(0,n-\frac{\log b}{\log\lambda}\Big)$$ and $$P_s(T)=\infty,\qquad\forall\,s\in\Big[n-\frac{\log b}{\log\lambda},1\Big).$$ Thus $${\textrm{Dim}}_F(\partial^-T)=\frac{\log b}{\log\lambda}.$$ Arguing as we did with the von Koch snowflake, we show that $P_s(T)$ is bounded both from above and from below by the series $$\sum_{k=0}^\infty\Big(\frac{b}{\lambda^{n-s}}\Big)^k,$$ which converges if and only if $s<n-\frac{\log b}{\log\lambda}$. Indeed $$\begin{split} P_s(T)&={\mathcal L}_s(T,{\mathcal C}T)=\sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}T)\\ & \leq \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}T_k^i) = \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(F_k^i(T_0),F_k^i({\mathcal C}T_0))\\ & =\frac{a}{\lambda^{n-s}}{\mathcal L}_s(T_0,{\mathcal C}T_0)\sum_{k=0}^\infty\Big(\frac{b}{\lambda^{n-s}}\Big)^k, \end{split}$$ and $$\begin{split} P_s(T)&={\mathcal L}_s(T,{\mathcal C}T)=\sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}T)\\ & \geq \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,S_k^i) = \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(F_k^i(T_0),F_k^i(S_0))\\ & =\frac{a}{\lambda^{n-s}}{\mathcal L}_s(T_0,S_0)\sum_{k=0}^\infty\Big(\frac{b}{\lambda^{n-s}}\Big)^k. \end{split}$$ Also notice that, since $P(T_0)<\infty$, we have $${\mathcal L}_s(T_0,S_0)\leq{\mathcal L}_s(T_0,{\mathcal C}T_0)=P_s(T_0)<\infty,$$ for every $s\in(0,1)$. Now suppose that $T$ does not satisfy $(\ref{add_frac_self_hp})$. Then we can obtain a set $T'$ which does, simply by removing a part $S_0$ of the building block $T_0$.\ To be more precise, let $S_0\subset T_0$ be s.t. $|S_0|>0$, $|T_0\setminus S_0|>0$ and $P(T_0\setminus S_0)<\infty$. Then define a new building block $T'_0:=T_0\setminus S_0$ and the set $$T':=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(T'_0).$$ This new set has exactly the same structure of $T$, since we are using the same collection $\{F_k^i\}$ of affine maps. Notice that $$S_0\subset T_0\quad\Longrightarrow\quad F_k^i(S_0)\subset F_k^i(T_0),$$ and $$F_k^i(T'_0)=F_k^i(T_0)\setminus F_k^i(S_0),$$ for every $k,\,i$. Thus $$T'=T\setminus\Big(\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(S_0)\Big)$$ satisfies $(\ref{add_frac_self_hp})$. Roughly speaking, what matters is that there exists a bounded open set $T_0$ s.t. $$|F_k^i(T_0)\cap F_h^j(T_0)|=0,\qquad\textrm{if }i\not=j.$$ This can be thought of as a compatibility criterion for the affine maps $\{F_k^i\}$.\ We also need to ask that the ratio of the logarithms of the growth factor and the scaling factor is $\frac{\log b}{\log\lambda}\in(n-1,n)$.\ Then we are free to choose as building block any set $T'_0\subset T_0$ s.t. $$|T'_0|>0,\qquad|T_0\setminus T'_0|>0\qquad\textrm{and }P(T'_0)<\infty,$$ and the set $$T':=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(T'_0).$$ satisfies the hypothesis of previous Theorem. Therefore, even if the Sierpinski triangle and the Menger sponge do not satisfy $(\ref{add_frac_self_hp})$, we can exploit their structure to construct new sets which do. However, we remark that the new boundary $\partial^-T'$ will look very different from the original fractal. Actually, in general it will be a mix of unrectifiable pieces and smooth pieces. In particular, we can not hope to get an analogue of Corollary $\ref{koch_coroll}$. Still, the following Remark shows that the new (measure theoretic) boundary retains at least some of the “fractal nature” of the original set. \[self\_sim\_frac\_bdry\_nat\_rmk\] If the set $T$ of Theorem $\ref{fractal_bdary_selfsim_dim}$ is bounded, exploiting Proposition $\ref{vis_prop}$ and Remark $\ref{fractal_dim_rmk}$ we obtain $$\overline{{\textrm{Dim}}}_{\mathcal M}(\partial^-T)\geq\frac{\log b}{\log\lambda}>n-1.$$ Moreover, notice that if $\Omega$ is a bounded open set with Lipschitz boundary, then $$P(E,\Omega)<\infty\quad\Longrightarrow\quad{\textrm{Dim}}_F(E,\Omega)=n-1.$$ Therefore, if $T\subset\subset B_R$, then $$P(T)=P(T,B_R)=\infty,$$ even if $T$ is bounded (and hence $\partial^-T$ is compact). [Sponge-like sets]{} The simplest way to construct the set $T'$ consists in simply removing a small ball $S_0:=B\subset\subset T_0$ from $T_0$. In particular, suppose that $|T_0\Delta T|=0$, as with the Sierpinski triangle.\ Define $$S:=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(B) \quad\textrm{and}\quad T':=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(T_0\setminus B)=T\setminus S.$$ Then $$\label{fractal_spazz_end} |T_0\Delta T|=0\quad\Longrightarrow\quad |T'\Delta (T_0\setminus S)|=0.$$ Now the set $E:=T_0\setminus S$ looks like a sponge, in the sense that it is a bounded open set with an infinite number of holes (each one at a positive, but non-fixed distance from the others). From $(\ref{fractal_spazz_end})$ we get $P_s(E)=P_s(T')$. Thus, since $T'$ satisfies the hypothesis of previous Theorem, we obtain $${\textrm{Dim}}_F(\partial^-E)=\frac{\log b}{\log\lambda}.$$ [Dendrite-like sets]{} Depending on the form of the set $T_0$ and on the affine maps $\{F_k^i\}$, we can define more intricated sets $T'$. As an example we consider the Sierpinski triangle $E\subset{\mathbb R}^2$.\ It is of the form $E=T_0\setminus T$, where the building block $T_0$ is an equilateral triangle, say with side length one, a vertex on the $y$-axis and baricenter in 0. The pieces $T_k^i$ are obtained with a scaling factor $\lambda=2$ and the growth factor is $b=3$ (see e.g. [@Falconer] for the construction). As usual, we consider the set $$T=\bigcup_{k=1}^\infty\bigcup_{i=1}^{3^{k-1}}T_k^i.$$ However, as remarked above, we have $|T\Delta T_0|=0$. Starting from $k=2$ each triangle $T_k^i$ touches with (at least) a vertex (at least) another triangle $T_h^j$. Moreover, each triangle $T_k^i$ gets touched in the middle point of each side (and actually it gets touched in infinitely many points). Exploiting this situation, we can remove from $T_0$ six smaller triangles, so that the new building block $T'_0$ is a star polygon centered in 0, with six vertices, one in each vertex of $T_0$ and one in each middle point of the sides of $T_0$. ![[*Removing the six triangles (in green) to obtain the new “building block” $T'_0$ (on the right)*]{}](Star_Bblock){width="90mm"} The resulting set $$T'=\bigcup_{k=1}^\infty\bigcup_{i=1}^{3^{k-1}}F_k^i(T'_0)$$ will have an infinite number of ramifications. ![[*The third and fourth steps of the iterative construction of the set $T'$*]{}](Ind_steps_star){width="110mm"} Since $T'$ satisfies the hypothesis of previous Theorem, we obtain $${\textrm{Dim}}_F(\partial^-T')=\frac{\log 3}{\log2}.$$ [“Exploded” fractals]{} In all the previous examples, the sets $T_k^i$ are accumulated in a bounded region. On the other hand, imagine making a fractal like the von Koch snowflake or the Sierpinski triangle “explode” and then rearrange the pieces $T_k^i$ in such a way that $d(T_k^i,T_h^j)\geq d$, for some fixed $d>0$. Since the shape of the building block is not important, we can consider $T_0:=B_{1/4}(0)\subset{\mathbb R}^n$, with $n\geq2$. Moreover, since the parameter $a$ does not influence the dimension, we can fix $a=1$. Then we rearrange the pieces obtaining $$\label{exploded_frac_def} E:=\bigcup_{k=1}^\infty\bigcup_{i=1}^{b^{k-1}}B_\frac{1}{4\lambda^k}(k,0,\ldots,0,i).$$ Define for simplicity $$B_k^i:=B_\frac{1}{4\lambda^k}(k,0,\ldots,0,i)\quad\textrm{and}\quad x_k^i:=k\,e_1+i\,e_n,$$ and notice that $$B_k^i=\lambda^{-k}B_\frac{1}{4}(0)+x_k^i.$$ Since for every $k,\,h$ and every $i\not=j$ we have $$d(B_k^i,B_h^j)\geq\frac{1}{2},$$ the boundary of the set $E$ is the disjoint union of $(n-1)$-dimensional spheres $$\partial^-E=\partial E=\bigcup_{k=1}^\infty\bigcup_{i=1}^{b^{k-1}}\partial B_k^i,$$ and in particular is smooth. The (global) perimeter of $E$ is $$P(E)=\sum_{k=1}^\infty\sum_{i=1}^{b^{k-1}}P(B_k^i)=\frac{1}{\lambda}P(B_{1/4}(0))\sum_{k=0}^\infty \Big(\frac{b}{\lambda^{n-1}}\Big)^k=\infty,$$ since $\frac{\log b}{\log\lambda}>n-1$. However $E$ has locally finite perimeter, since its boundary is smooth and every ball $B_R$ intersects only finitely many $B_k^i$’s, $$P(E,B_R)<\infty,\qquad\forall\,R>0.$$ Therefore it also has locally finite $s$-perimeter for every $s\in(0,1)$ $$P_s(E,B_R)<\infty,\qquad\forall\,R>0,\qquad\forall\,s\in(0,1).$$ What is interesting is that the set $E$ satisfies the hypothesis of Theorem $\ref{fractal_bdary_selfsim_dim}$ and hence it also has finite global $s$-perimeter for every $s<\sigma_0:=n-\frac{\log b}{\log\lambda}$, $$P_s(E)<\infty\qquad\forall\,s\in(0,\sigma_0)\quad\textrm{and}\quad P_s(E)=\infty\qquad\forall\,s\in[\sigma_0,1).$$ Thus we obtain Proposition $\ref{expl_farc_prop1}$. It is enough to choose a natural number $b\geq2$ and take $\lambda:=b^\frac{1}{n-\sigma}$. Notice that $\lambda>1$ and $$\frac{\log b}{\log\lambda}=n-\sigma\in(n-1,n).$$ Then we can define $E$ as in $(\ref{exploded_frac_def})$ and we are done. [Elementary properties of the $s$-perimeter]{} \[elementary\_properties\] Let $\Omega\subset{\mathbb R}^n$ be an open set. \(i) (Subadditivity)$\quad$ Let $E,\,F\subset{\mathbb R}^n$ s.t. $|E\cap F|=0$. Then $$\label{subadditive} P_s(E\cup F,\Omega)\leq P_s(E,\Omega)+P_s(F,\Omega).$$ \(ii) (Translation invariance)$\quad$ Let $E\subset{\mathbb R}^n$ and $x\in{\mathbb R}^n$. Then $$\label{translation_invariance} P_s(E+x,\Omega+x)=P_s(E,\Omega).$$ \(iii) (Rotation invariance)$\quad$ Let $E\subset{\mathbb R}^n$ and $\mathcal{R}\in SO(n)$ a rotation. Then $$\label{rotation_invariance} P_s(\mathcal{R}E,\mathcal{R}\Omega)=P_s(E,\Omega).$$ \(iv) (Scaling)$\quad$ Let $E\subset{\mathbb R}^n$ and $\lambda>0$. Then $$\label{scaling} P_s(\lambda E,\lambda\Omega)=\lambda^{n-s}P_s(E,\Omega).$$ \(i) follows from the following observations. Let $A_1,\,A_2,\,B\subset{\mathbb R}^n$. If $|A_1\cap A_2|=0$, then $${\mathcal L}_s(A_1\cup A_2,B) ={\mathcal L}_s(A_1,B)+{\mathcal L}_s(A_2,B).$$ Moreover $$A_1\subset A_2\quad\Longrightarrow\quad{\mathcal L}_s(A_1,B)\leq{\mathcal L}_s(A_2,B),$$ and $${\mathcal L}_s(A,B)={\mathcal L}_s(B,A).$$ Therefore $$\begin{split} P_s(E\cup F,\Omega)&={\mathcal L}_s((E\cup F)\cap\Omega,{\mathcal C}(E\cup F))+{\mathcal L}_s((E\cup F)\setminus\Omega,{\mathcal C}(E\cup F)\cap\Omega)\\ & ={\mathcal L}_s(E\cap\Omega,{\mathcal C}(E\cup F))+{\mathcal L}_s(F\cap\Omega,{\mathcal C}(E\cup F))\\ & \qquad+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}(E\cup F)\cap\Omega)+{\mathcal L}_s(F\setminus\Omega,{\mathcal C}(E\cup F)\cap\Omega)\\ & \leq{\mathcal L}_s(E\cap\Omega,{\mathcal C}E)+{\mathcal L}_s(F\cap\Omega,{\mathcal C}F)\\ & \qquad+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}E\cap\Omega)+{\mathcal L}_s(F\setminus\Omega,{\mathcal C}F\cap\Omega)\\ & =P_s(E,\Omega)+P_s(F,\Omega). \end{split}$$ (ii), (iii) and (iv) follow simply by changing variables in ${\mathcal L}_s$ and the following observations: $$\begin{split} &(x+A_1)\cap(x+A_2)=x+A_1\cap A_2,\qquad x+{\mathcal C}A={\mathcal C}(x+A),\\ & \mathcal{R}A_1\cap\mathcal{R}A_2=\mathcal{R}(A_1\cap A_2),\qquad\mathcal{R}({\mathcal C}A)={\mathcal C}(\mathcal{R}A),\\ & (\lambda A_1)\cap(\lambda A_2)=\lambda(A_1\cap A_2),\qquad\lambda({\mathcal C}A)={\mathcal C}(\lambda A). \end{split}$$ For example, for claim (iv) we have $$\begin{split} {\mathcal L}_s(\lambda A,\lambda B)&=\int_{\lambda A}\int_{\lambda B}\frac{dx\,dy}{|x-y|^{n+s}} =\int_A\lambda^n\,dx\int_B\frac{\lambda^n\,dy}{\lambda^{n+s}|x-y|^{n+s}}\\ & =\lambda^{n-s}{\mathcal L}_s(A,B). \end{split}$$ Then $$\begin{split} P_s(\lambda E,\lambda\Omega)&={\mathcal L}_s(\lambda E\cap\lambda\Omega,{\mathcal C}(\lambda E))+ {\mathcal L}_s(\lambda E\cap{\mathcal C}(\lambda\Omega),{\mathcal C}(\lambda E)\cap\lambda\Omega)\\ & ={\mathcal L}_s(\lambda(E\cap\Omega),\lambda{\mathcal C}E)+{\mathcal L}_s(\lambda(E\setminus\Omega),\lambda({\mathcal C}E\cap\Omega))\\ & =\lambda^{n-s}\left({\mathcal L}_s(E\cap\Omega,{\mathcal C}E)+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}E\cap\Omega)\right)\\ & =\lambda^{n-s}P_s(E,\Omega). \end{split}$$ Proof of Example $\ref{inclusion_counterexample}$ ================================================= Note that $E\subset (0,a^2]$. Let $\Omega:=(-1,1)\subset\mathbb{R}$. Then $E\subset\subset\Omega$ and $\textrm{dist}(E,\partial\Omega)=1-a^2=:d>0$. Now $$P_s(E)=\int_E\int_{{\mathcal C}E\cap\Omega}\frac{dxdy}{|x-y|^{1+s}}+ \int_E\int_{{\mathcal C}\Omega}\frac{dxdy}{|x-y|^{1+s}}.$$ As for the second term, we have $$\int_E\int_{{\mathcal C}\Omega}\frac{dxdy}{|x-y|^{1+s}}\leq\frac{2|E|}{sd^s}<\infty.$$ We split the first term into three pieces $$\begin{split} \int_E&\int_{{\mathcal C}E\cap\Omega}\frac{dxdy}{|x-y|^{1+s}}\\ & =\int_E\int_{-1}^0\frac{dxdy}{|x-y|^{1+s}} +\int_E\int_{{\mathcal C}E\cap(0,a)}\frac{dxdy}{|x-y|^{1+s}}+\int_E\int_a^1\frac{dxdy}{|x-y|^{1+s}}\\ & =\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3. \end{split}$$ Note that ${\mathcal C}E\cap(0,a)=\bigcup_{k\in\mathbb{N}}I_{2k-1}=\bigcup_{k\in\mathbb{N}}(a^{2k},a^{2k-1})$.\ A simple calculation shows that, if $a<b\leq c<d$, then $$\label{rectangle_integral}\begin{split} \int_a^b&\int_c^d\frac{dxdy}{|x-y|^{1+s}}=\\ & \frac{1}{s(1-s)}\big[(c-a)^{1-s}+(d-b)^{1-s}-(c-b)^{1-s}-(d-a)^{1-s}\big]. \end{split}$$ Also note that, if $n>m\geq1$, then $$\label{derivative_bound}\begin{split} (1-a^n)^{1-s}-(1-a^m)^{1-s}&=\int_m^n\frac{d}{dt}(1-a^t)^{1-s}\,dt\\ & =(s-1)\log a\int_m^n\frac{a^t}{(1-a^t)^s}\,dt\\ & \leq a^m (s-1)\log a\int_m^n\frac{1}{(1-a^t)^s}\,dt\\ & \leq(n-m)a^m\frac{(s-1)\log a}{(1-a)^s}. \end{split}$$ Now consider the first term $$\mathcal{I}_1=\sum_{k=1}^\infty\int_{a^{2k+1}}^{a^{2k}}\int_{-1}^0\frac{dxdy}{|x-y|^{1+s}}.$$ Use $(\ref{rectangle_integral}$) and notice that $(c-a)^{1-s}-(d-a)^{1-s}\leq0$ to get $$\int_{-1}^0\int_{a^{2k+1}}^{a^{2k}}\frac{dxdy}{|x-y|^{1+s}} \leq\frac{1}{s(1-s)}\big[(a^{2k})^{1-s}-(a^{2k+1})^{1-s}\big]\leq\frac{1}{s(1-s)}(a^{2(1-s)})^k.$$ Then, as $a^{2(1-s)}<1$ we get $$\mathcal{I}_1\leq\frac{1}{s(1-s)}\sum_{k=1}^\infty(a^{2(1-s)})^k<\infty.$$ As for the last term $$\mathcal{I}_3=\sum_{k=1}^\infty\int_{a^{2k+1}}^{a^{2k}}\int_a^1\frac{dxdy}{|x-y|^{1+s}},$$ use $(\ref{rectangle_integral}$) and notice that $(d-b)^{1-s}-(d-a)^{1-s}\leq0$ to get $$\begin{split} \int_{a^{2k+1}}^{a^{2k}}\int_a^1\frac{dxdy}{|x-y|^{1+s}}& \leq\frac{1}{s(1-s)}\big[(1-a^{2k+1})^{1-s}-(1-a^{2k})^{1-s}\big]\\ & \leq\frac{-\log a}{s(1-a)^s}a^{2k}\quad\textrm{by }(\ref{derivative_bound}). \end{split}$$ Thus $$\mathcal{I}_3\leq\frac{-\log a}{s(1-a)^s}\sum_{k=1}^\infty(a^2)^k<\infty.$$ Finally we split the second term $$\mathcal{I}_2=\sum_{k=1}^\infty\sum_{j=1}^\infty\int_{a^{2k+1}}^{a^{2k}}\int_{a^{2j}}^{a^{2j-1}} \frac{dxdy}{|x-y|^{1+s}}$$ into three pieces according to the cases $j>k$, $j=k$ and $j<k$. If $j=k$, using $(\ref{rectangle_integral})$ we get $$\begin{split} \int_{a^{2k+1}}^{a^{2k}}&\int_{a^{2k}}^{a^{2k-1}} \frac{dxdy}{|x-y|^{1+s}}=\\ & =\frac{1}{s(1-s)}\big[(a^{2k}-a^{2k+1})^{1-s}+(a^{2k-1}-a^{2k})^{1-s}-(a^{2k-1}-a^{2k+1})^{1-s}\big]\\ & =\frac{1}{s(1-s)}\big[a^{2k(1-s)}(1-a)^{1-s}+a^{(2k-1)(1-s)}(1-a)^{1-s}\\ & \quad\quad\quad\quad\quad-a^{(2k-1)(1-s)}(1-a^2)^{1-s}\big]\\ & =\frac{1}{s(1-s)}(a^{2(1-s)})^k\Big[(1-a)^{1-s}+\frac{(1-a)^{1-s}}{a^{1-s}}-\frac{(1-a^2)^{1-s}}{a^{1-s}}\Big]. \end{split}$$ Summing over $k\in\mathbb{N}$ we get $$\begin{split} \sum_{k=1}^\infty&\int_{a^{2k+1}}^{a^{2k}}\int_{a^{2k}}^{a^{2k-1}} \frac{dxdy}{|x-y|^{1+s}}=\\ & =\frac{1}{s(1-s)}\frac{a^{2(1-s)}}{1-a^{2(1-s)}}\Big[(1-a)^{1-s}+\frac{(1-a)^{1-s}}{a^{1-s}}-\frac{(1-a^2)^{1-s}}{a^{1-s}}\Big]<\infty. \end{split}$$ In particular note that $$\begin{split} (1-s)&P_s(E)\geq(1-s)\mathcal{I}_2\\ & \geq\frac{1}{s(1-a^{2(1-s)})}\big[a^{2(1-s)}(1-a)^{1-s}+a^{1-s}(1-a)^{1-s}-a^{1-s}(1-a^2)^{1-s}\big], \end{split}$$ which tends to $+\infty$ when $s\to1$. This shows that $E$ cannot have finite perimeter. To conclude let $j>k$, the case $j<k$ being similar, and consider $$\sum_{k=1}^\infty\sum_{j=k+1}^\infty\int_{a^{2j}}^{a^{2j-1}}\int_{a^{2k+1}}^{a^{2k}} \frac{dxdy}{|x-y|^{1+s}}.$$ Again, using $(\ref{rectangle_integral}$) and $(d-b)^{1-s}-(d-a)^{1-s}\leq0$, we get $$\begin{split} \int_{a^{2j}}^{a^{2j-1}}&\int_{a^{2k+1}}^{a^{2k}} \frac{dxdy}{|x-y|^{1+s}}\\ & \leq\frac{1}{s(1-s)}\big[(a^{2k+1}-a^{2j})^{1-s}-(a^{2k+1}-a^{2j-1})^{1-s}\big]\\ & =\frac{a^{1-s}}{s(1-s)}(a^{2(1-s)})^k\big[(1-a^{2(j-k)-1})^{1-s}-(1-a^{2(j-k)-2})^{1-s}\big]\\ & \leq\frac{a^{1-s}}{s(1-s)}(a^{2(1-s)})^k\frac{(s-1)\log a}{(1-a)^s}a^{2(j-k)-2}\quad\quad\textrm{by }(\ref{derivative_bound})\\ & =\frac{-\log a}{s(1-a^s)a^{s+1}}(a^{2(1-s)})^k(a^2)^{j-k}, \end{split}$$ for $j\geq k+2$. Then $$\begin{split} \sum_{k=1}^\infty&\sum_{j=k+2}^\infty\int_{a^{2j}}^{a^{2j-1}}\int_{a^{2k+1}}^{a^{2k}} \frac{dxdy}{|x-y|^{1+s}}\\ & \leq\frac{-\log a}{s(1-a^s)a^{s+1}}\sum_{k=1}^\infty(a^{2(1-s)})^k\sum_{h=2}^\infty(a^2)^h<\infty. \end{split}$$ If $j=k+1$ we get $$\begin{split} \sum_{k=1}^\infty\int_{a^{2k+2}}^{a^{2k+1}}\int_{a^{2k+1}}^{a^{2k}}\frac{dxdy}{|x-y|^{1+s}}& \leq\frac{1}{s(1-s)}\sum_{k=1}^\infty(a^{2k+1}-a^{2k+2})^{1-s}\\ & =\frac{a^{1-s}(1-a)^{1-s}}{s(1-s)}\sum_{k=1}^\infty(a^{2(1-s)})^k<\infty. \end{split}$$ This shows that also $\mathcal{I}_2<\infty$, so that $P_s(E)<\infty$ for every $s\in(0,1)$ as claimed. Signed distance function ======================== Given $\emptyset\not=E\subset{\mathbb R}^n$, the distance function from $E$ is defined as $$d_E(x)=d(x,E):=\inf_{y\in E}|x-y|,\qquad\textrm{for }x\in{\mathbb R}^n.$$ The signed distance function from $\partial E$, negative inside $E$, is then defined as $$\bar{d}_E(x)=\bar{d}(x,E):=d(x,E)-d(x,{\mathcal C}E).$$ For the details of the main properties we refer e.g. to [@Ambrosio] and [@Bellettini]. We also define the sets $$E_r:=\{x\in{\mathbb R}^n\,|\,\bar{d}_E(x)<r\}.$$ Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. By definition we can locally describe $\Omega$ near its boundary as the subgraph of appropriate Lipschitz functions. To be more precise, we can find a finite open covering $\{C_{\rho_i}\}_{i=1}^m$ of $\partial\Omega$ made of cylinders, and Lipschitz functions $\varphi_i:B'_{\rho_i}\longrightarrow{\mathbb R}$ s.t. $\Omega\cap C_{\rho_i}$ is the subgraph of $\varphi_i$. That is, up to rotations and translations, $$C_{\rho_i}=\{(x',x_n)\in{\mathbb R}^n\,|\,|x'|<\rho_i,\,|x_n|<\rho_i\},$$ and $$\begin{split} \Omega\cap C_{\rho_i}&=\{(x',x_n)\in{\mathbb R}^n\,|\,x'\in B'_{\rho_i},\,-\rho_i<x_n<\varphi_i(x')\},\\ & \partial\Omega\cap C_{\rho_i}=\{(x',\varphi_i(x'))\in{\mathbb R}^n\,|\,x'\in B_{\rho_i}'\}. \end{split}$$ Let $L$ be the sup of the Lipschitz constants of the functions $\varphi_i$. Theorem 4.1 of [@LipApprox] guarantees that also the bounded open sets $\Omega_r$ have Lipschitz boundary, when $r$ is small enough, say $|r|<r_0$.\ Moreover these sets $\Omega_r$ can locally be described, in the same cylinders $C_{\rho_i}$ used for $\Omega$, as subgraphs of Lipschitz functions $\varphi_i^r$ which approximate $\varphi_i$ (see [@LipApprox] for the precise statement) and whose Lipschitz constants are less or equal to $L$.\ Notice that $$\partial\Omega_r=\{\bar{d}_\Omega=r\}.$$ Now, since in $C_{\rho_i}$ the set $\Omega_r$ coincides with the subgraph of $\varphi_i^r$, we have $${\mathcal H}^{n-1}(\partial\Omega_r\cap C_{\rho_i})=\int_{B_{\rho_i}'}\sqrt{1+|\nabla\varphi_i^r|^2}\,dx'\leq M_i,$$ with $M_i$ depending on $\rho_i$ and $L$ but not on $r$.\ Therefore $${\mathcal H}^{n-1}(\{\bar{d}_\Omega=r\})\leq\sum_{i=1}^m{\mathcal H}^{n-1}(\partial\Omega_r\cap C_{\rho_i})\leq\sum_{i=1}^mM_i$$ independently on $r$, proving the following \[bound\_perimeter\_unif\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then there exists $r_0>0$ s.t. $\Omega_r$ is a bounded open set with Lipschitz boundary for every $r\in(-r_0,r_0)$ and $$\label{bound_perimeter_unif_eq} \sup_{|r|<r_0}{\mathcal H}^{n-1}(\{\bar{d}_\Omega=r\})<\infty.$$ Measure theoretic boundary ========================== Since $$\label{fin_spazz_basta1} |E\Delta F|=0\quad\Longrightarrow\quad P(E,\Omega)=P(F,\Omega)\quad\textrm{and}\quad P_s(E,\Omega)=P_s(F,\Omega),$$ we can modify a set making its topological boundary as big as we want, without changing its (fractional) perimeter.\ For example, let $E\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then, if we set $F:=E\cup(\mathbb Q^n\setminus E)$, we have $|E\Delta F|=0$ and hence we get $(\ref{fin_spazz_basta1})$. However $\partial F={\mathbb R}^n\setminus E$. For this reason one considers measure theoretic notions of interior, exterior and boundary, which solely depend on the class of $\chi_E$ in $L^1_{loc}({\mathbb R}^n)$.\ In some sense, by considering the measure theoretic boundary $\partial^-E$ defined below we can also minimize the size of the topological boundary (see $(\ref{ess_bdry_intersect})$). Moreover, this measure theoretic boundary is actually the topological boundary of a set which is equivalent to $E$. Thus we obtain a “good” representative for the class of $E$. We refer to Section 3.2 of [@Visintin] (see also Proposition 3.1 of [@Giusti]). For some details about the good representative of an $s$-minimal set, see the Appendix of [@graph]. Let $E\subset{\mathbb R}^n$. For every $t\in[0,1]$ define the set $$\label{density_t} E^{(t)}:=\left\{x\in{\mathbb R}^n\,\big|\,\exists\lim_{r\to0}\frac{|E\cap B_r(x)|}{\omega_nr^n}=t\right\},$$ of points density $t$ of $E$. The sets $E^{(0)}$ and $E^{(1)}$ are respectively the measure theoretic exterior and interior of the set $E$. The set $$\label{ess_bdry} \partial_eE:={\mathbb R}^n\setminus(E^{(0)}\cup E^{(1)})$$ is the essential boundary of $E$. Using the Lebesgue points Theorem for the characteristic function $\chi_E$, we see that the limit in $(\ref{density_t})$ exists for a.e. $x\in{\mathbb R}^n$ and $$\lim_{r\to0}\frac{|E\cap B_r(x)|}{\omega_nr^n}=\left\{\begin{array}{cc}1,&\textrm{a.e. }x\in E,\\ 0,&\textrm{a.e. }x\in{\mathcal C}E. \end{array} \right.$$ So $$|E\Delta E^{(1)}|=0,\qquad|{\mathcal C}E\Delta E^{(0)}|=0\qquad\textrm{and }|\partial_eE|=0.$$ In particular every set $E$ is equivalent to its measure theoretic interior.\ However, notice that $E^{(1)}$ in general is not open.\ We have another natural way to define a measure theoretic boundary. Let $E\subset{\mathbb R}^n$ and define the sets $$\begin{split} &E_1:=\{x\in{\mathbb R}^n\,|\,\exists r>0,\,|E\cap B_r(x)|=\omega_nr^n\},\\ & E_0:=\{x\in{\mathbb R}^n\,|\,\exists r>0,\,|E\cap B_r(x)|=0\}. \end{split}$$ Then we define $$\begin{split} \partial^-E&:={\mathbb R}^n\setminus(E_0\cup E_1)\\ & =\{x\in{\mathbb R}^n\,|\,0<|E\cap B_r(x)|<\omega_nr^n\textrm{ for every }r>0\}. \end{split}$$ Notice that $E_0$ and $E_1$ are open sets and hence $\partial^-E$ is closed. Moreover, since $$\label{density_subsets} E_0\subset E^{(0)}\qquad\textrm{and}\qquad E_1\subset E^{(1)},$$ we get $$\partial_eE\subset\partial^-E.$$ We have $$\label{ess_bdry_top1} F\subset{\mathbb R}^n\textrm{ s.t. }|E\Delta F|=0\quad\Longrightarrow\quad\partial^-E\subset\partial F.$$ Indeed, if $|E\Delta F|=0$, then $|F\cap B_r(x)|=|E\cap B_r(x)|$ for every $r>0$. Thus for any $x\in\partial^-E$ we have $$0<|F\cap B_r(x)|<\omega_nr^n,$$ which implies $$F\cap B_r(x)\not=\emptyset\quad\textrm{and}\quad{\mathcal C}F\cap B_r(x)\not=\emptyset\quad\textrm{for every }r>0,$$ and hence $x\in\partial F$. In particular, $\partial^-E\subset\partial E$. Moreover $$\label{ess_bdry_top2} \partial^-E=\partial E^{(1)}.$$ Indeed, since $|E\Delta E^{(1)}|=0$, we already know that $\partial^-E\subset\partial E^{(1)}$. The converse inclusion follows from $(\ref{density_subsets})$ and the fact that both $E_0$ and $E_1$ are open.\ From $(\ref{ess_bdry_top1})$ and $(\ref{ess_bdry_top2})$ we obtain $$\label{ess_bdry_intersect} \partial^-E=\bigcap_{F\sim E}\partial F,$$ where the intersection is taken over all sets $F\subset{\mathbb R}^n$ s.t. $|E\Delta F|=0$, so we can think of $\partial^-E$ as a way to minimize the size of the topological boundary of $E$. In particular $$F\subset{\mathbb R}^n\textrm{ s.t. }|E\Delta F|=0\quad\Longrightarrow\quad\partial^-F=\partial^-E.$$ From $(\ref{density_subsets})$ and $(\ref{ess_bdry_top2})$ we see that we can take $E^{(1)}$ as “good” representative for $E$, obtaining Remark $\ref{gmt_assumption}$.\ Recall that the support of a Radon measure $\mu$ on ${\mathbb R}^n$ is defined as the set $$\textrm{supp }\mu:=\{x\in{\mathbb R}^n\,|\,\mu(B_r(x))>0\textrm{ for every }r>0\}.$$ Notice that, being the complementary of the union of all open sets of measure zero, it is a closed set. In particular, if $E$ is a Caccioppoli set, we have $$\label{support_perimeter} \textrm{supp }|D\chi_E|=\{x\in{\mathbb R}^n\,|\,P(E,B_r(x))>0\textrm{ for every }r>0\},$$ and it is easy to verify that $$\partial^-E=\textrm{supp }|D\chi_E|=\overline{\partial^*E},$$ where $\partial^*E$ denotes the reduced boundary. However notice that in general the inclusions $$\partial^*E\subset\partial_eE\subset\partial^-E\subset\partial E$$ are all strict and in principle we could have $${\mathcal H}^{n-1}(\partial^-E\setminus\partial^*E)>0.$$ Minkowski dimension =================== Let $\Omega\subset{\mathbb R}^n$ be an open set. For any $\Gamma\subset{\mathbb R}^n$ and $r\in[0,n]$ we define the inferior and superior $r$-dimensional Minkowski contents of $\Gamma$ relative to the set $\Omega$ as, respectively $$\underline{\mathcal{M}}^r(\Gamma,\Omega):=\liminf_{\rho\to0}\frac{|\bar{N}_\rho^\Omega(\Gamma)|}{\rho^{n-r}},\qquad \overline{\mathcal{M}}^r(\Gamma,\Omega):=\limsup_{\rho\to0}\frac{|\bar{N}_\rho^\Omega(\Gamma)|}{\rho^{n-r}}.$$ Then we define the lower and upper Minkowski dimensions of $\Gamma$ in $\Omega$ as $$\begin{split} \underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma,\Omega)&:=\inf\big\{r\in[0,n]\,|\,\underline{\mathcal{M}}^r(\Gamma,\Omega)=0\big\}\\ & =n-\sup\big\{r\in[0,n]\,|\,\underline{\mathcal{M}}^{n-r}(\Gamma,\Omega)=0\big\}, \end{split}$$ $$\begin{split} \overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma,\Omega)&:=\sup\big\{r\in[0,n]\,|\,\overline{\mathcal{M}}^r(\Gamma,\Omega)=\infty\big\}\\ & =n-\inf\big\{r\in[0,n]\,|\,\overline{\mathcal{M}}^{n-r}(\Gamma,\Omega)=\infty\big\}. \end{split}$$ If they agree, we write $${\textrm{Dim}}_\mathcal{M}(\Gamma,\Omega)$$ for the common value and call it the Minkowski dimension of $\Gamma$ in $\Omega$.\ If $\Omega={\mathbb R}^n$ or $\Gamma\subset\subset\Omega$, we drop the $\Omega$ in the formulas. Let ${\textrm{Dim}}_\mathcal{H}$ denote the Hausdorff dimension. In general one has $${\textrm{Dim}}_\mathcal{H}(\Gamma)\leq\underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)\leq\overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma),$$ and all the inequalities might be strict. However for some sets (e.g. self-similar sets with some symmetric and regularity condition) they are all equal. We also recall some equivalent definitions of the Minkowski dimensions, usually referred to as box-counting dimensions, which are easier to compute. For the details and the relation between the Minkowski and the Hausdorff dimensions, see [@Mattila] and [@Falconer] and the references cited therein.\ For simplicity we only consider the case $\Gamma$ bounded and $\Omega={\mathbb R}^n$ (or $\Gamma\subset\subset\Omega$). Given a nonempty bounded set $\Gamma\subset{\mathbb R}^n$, define for every $\delta>0$ $$\mathcal{N}(\Gamma,\delta):=\min\Big\{k\in\mathbb{N}\,\big|\,\Gamma\subset\bigcup_{i=1}^kB_\delta(x_i),\textrm{ for some }x_i\in{\mathbb R}^n\Big\},$$ the smallest number of $\delta$-balls needed to cover $\Gamma$, and $$\mathcal{P}(\Gamma,\delta):=\max\big\{k\in\mathbb{N}\,|\,\exists\,\textrm{disjoint balls }B_\delta(x_i),\,i=1,\ldots,k\textrm{ with }x_i\in \Gamma\big\},$$ the greatest number of disjoint $\delta$-balls with centres in $\Gamma$. Then it is easy to verify that $$\label{counting} \mathcal{N}(\Gamma,2\delta)\leq\mathcal{P}(\Gamma,\delta)\leq\mathcal{N}(\Gamma,\delta/2).$$ Moreover, since any union of $\delta$-balls with centers in $\Gamma$ is contained in $N_\delta(\Gamma)$, and any union of $(2\delta)$-balls covers $N_\delta(\Gamma)$ if the union of the corresponding $\delta$-balls covers $\Gamma$, we get $$\label{counting2} \mathcal{P}(\Gamma,\delta)\omega_n\delta^n\leq|N_\delta(\Gamma)|\leq \mathcal{N}(\Gamma,\delta)\omega_n(2\delta)^n.$$ Using $(\ref{counting})$ and $(\ref{counting2})$ we see that $$\begin{split} &\underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\inf\Big\{r\in[0,n]\,\big|\,\liminf_{\delta\to0}\mathcal{N}(\Gamma,\delta)\delta^r=0\Big\},\\ & \overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\sup\Big\{r\in[0,n]\,\big|\,\limsup_{\delta\to0}\mathcal{N}(\Gamma,\delta)\delta^r=\infty\Big\}. \end{split}$$ Then it can be proved that $$\label{log_counting}\begin{split} &\underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\liminf_{\delta\to0}\frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta},\\ & \overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\limsup_{\delta\to0}\frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta}. \end{split}$$ Actually notice that, due to $(\ref{counting})$, we can take $\mathcal{P}(\Gamma,\delta)$ in place of $\mathcal{N}(\Gamma,\delta)$ in the above formulas.\ It is also easy to see that if in the definition of $\mathcal{N}(\Gamma,\delta)$ we take cubes of side $\delta$ instead of balls of radius $\delta$, then we get exactly the same dimensions. Moreover in $(\ref{log_counting})$ it is enough to consider limits as $\delta\to0$ through any decreasing sequence $\delta_k$ s.t. $\delta_{k+1}\geq c\delta_k$ for some constant $c\in(0,1)$; in particular for $\delta_k=c^k$. Indeed if $\delta_{k+1}\leq\delta<\delta_k$, then $$\begin{split} \frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta}&\leq\frac{\log\mathcal{N}(\Gamma,\delta_{k+1})}{-\log\delta_k} =\frac{\log\mathcal{N}(\Gamma,\delta_{k+1})}{-\log\delta_{k+1}+\log(\delta_{k+1}/\delta_k)}\\ & \leq\frac{\log\mathcal{N}(\Gamma,\delta_{k+1})}{-\log\delta_{k+1}+\log c}, \end{split}$$ so that $$\limsup_{\delta\to0}\frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta}\leq \limsup_{k\to\infty}\frac{\log\mathcal{N}(\Gamma,\delta_k)}{-\log\delta_k}.$$ The opposite inequality is clear and in a similar way we can treat the lower limits. [90]{} L. Ambrosio and N. Dancer, [*Calculus of variations and partial differential equations*]{}. Springer-Verlag, Berlin (2000). L. Ambrosio, G. De Philippis and L. Martinazzi, [*Gamma-convergence of nonlocal perimeter functionals*]{}. Manuscripta Math. 134, no. 3-4, 377$-$403 (2011). G. Bellettini, [*Lecture notes on mean curvature flows, barriers and singular perturbations*]{}. Edizioni della Scuola Normale 13, Pisa (2013). J. Bourgain, H. Brezis and P. Mironescu, [*Limiting embedding theorems for $W^{s,p}$ when $s\to1$ and applications*]{}. J. Anal. Math. 87, 77$-$101 (2002). L. Caffarelli, J.-M. Roquejoffre and O. Savin, [*Nonlocal minimal surfaces*]{}. Comm. pure Appl. Math. 63, no. 9, 1111$-$1144 (2010). L. Caffarelli and E. Valdinoci, [*Uniform estimates and limiting arguments for nonlocal minimal surfaces*]{}. Calc. Var. Partial Differential Equations 41, no. 1-2, 203$-$240 (2011). J. Davila, [*On an open question about functions of bounded variation*]{}. Calc. Var. Partial Differential Equations,15 no. 4, 519$-$527 (2002). E. Di Nezza, G. Palatucci and E. Valdinoci, [*Hitchhiker’s guide to the fractional Sobolev spaces*]{}. Bull. Sci. Math., 136(5):521$-$573 (2012). S. Dipierro, O. Savin and E. Valdinoci, [*Graph properties for nonlocal minimal surfaces*]{}. (2015). P. Doktor, [*Approximation of domains with Lipschitzian boundary*]{}. Cas. Pest. Mat. 101, 237$-$255 (1976). K.J. Falconer, [*Fractal geometry: mathematical foundations and applications*]{}. John Wiley and Sons (1990). E. Giusti, [*Minimal surfaces and functions of bounded variation*]{}. Monographs in Mathematics, 80. Birkhauser Verlag, Basel (1984). F. Maggi, [*Sets of finite perimeter and geometric variational problems*]{}. Cambridge Stud. Adv. Math. 135, Cambridge Univ. Press, Cambridge (2012). P. Mattila, [*Geometry of sets and measures in Euclidean spaces*]{}. Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge (1995). A. Visintin. [*Generalized coarea formula and fractal sets*]{}. Japan J. Indust. Appl. Math., 8(2):175$-$201 (1991).

--- author: - Seungchul Ryu title: 'Local Area Transform for Cross-Modality Correspondence Matching and Deep Scene Recognition' --- \ Thesis Supervisor: [**Kwanghoon Sohn**]{}\ 1.5cm\ Committee Member: [**Jaihie Kim**]{}\ 1.5cm\ Committee Member: [**Sangyoun Lee**]{}\ 1.5cm\ Committee Member: [**Bumsub Ham**]{}\ 1.5cm\ Committee Member: [**Dongbo Min**]{}\ -2cm *I would like to dedicate this dissertation to my loving family...*

--- author: - 'Nilay Bostan,' - and Vedat Nefer Şenoğuz bibliography: - 'quartic\_radiative\_v3.bib' title: 'Quartic inflation and radiative corrections with non-minimal coupling' --- Introduction {#sec:intro} ============ Inflation [@Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi; @Linde:1983gd], which is an accelerated expansion era thought to occur in the early universe, both helps explaining general properties of the universe such as its flatness and large scale homogeneity, and it leads to the primordial density perturbations that evolve into the structures in the universe. Up to now many inflationary models have been introduced with most of them depending on a scalar field called the inflaton. Predictions of these models are being tested by the cosmic microwave background radiation temperature anisotropies and polarization observations that have become even more sensitive in recent years [@Aghanim:2018eyx; @Akrami:2018odb]. The observational parameter values predicted by different potentials that the inflaton may have were calculated in many articles, see for instance ref. [@Martin:2013tda]. A vast majority of these articles assume that the inflaton is coupled to gravitation solely through the metric. On the other hand the action in general also contains a coupling term $\xi \phi^2 R$ between the Ricci scalar and the inflaton (this is required by the renormalizability of the scalar field theory in curved space-time [@Callan:1970ze; @Freedman:1974ze; @Buchbinder:1992rb]), and inflationary predictions are significantly altered depending on the coefficient of this term [@Abbott:1981rg; @Spokoiny:1984bd; @Lucchin:1985ip; @Futamase:1987ua; @Fakir:1990eg; @Salopek:1988qh; @Amendola:1990nn; @Faraoni:1996rf; @Faraoni:2004pi]. In this work, we first review in [section \[sec:inf\]]{} how to calculate the main observables, namely the spectral index $n_s$ and the tensor-to-scalar ratio $r$, for an inflaton potential in the presence of non-minimal coupling. Next, in [section \[sec:quartic\]]{} we review the $\lambda\phi^4$ quartic potential, providing analytical approximations for $n_s$ and $r$, and showing that the model agrees with current data for $\xi\gtrsim0.005$. We also briefly discuss how and to what extent can the reheating stage affect the values of observables. [Section \[sec:radiative\]]{} introduces two prescriptions that can be used to calculate radiative corrections to the inflaton potential due to inflaton couplings to bosons or fermions. In prescription I, a conformal transformation is applied to express the action in the Einstein frame; and the field dependent mass terms in the one-loop Coleman-Weinberg potential are expressed in this frame. Whereas in prescription II, the field dependent mass terms are taken into account in the original Jordan frame. The next two sections, [section \[sec:p1\]]{} and [section \[sec:p2\]]{} contain a detailed numerical investigation of how the radiative corrections due to inflaton couplings to bosons or fermions modify the predictions of the non-minimal quartic potential, for each prescription. We summarize our results in [section \[sec:conc\]]{}. The effect of radiative corrections to the predictions of the non-minimal quartic potential has been discussed mostly in the context of standard model (SM) Higgs inflation [@Bezrukov:2007ep], see for instance refs. [@Bezrukov:2013fka; @Rubio:2018ogq] and the references within. In this context, since the self coupling $\lambda$ of the inflaton is known, $\xi\gg1$ is required [@Bezrukov:2009db]. In this limit, the observational parameters are given in terms of the e-fold number $N$ by $n_s\approx 1-2/N$ and $r\approx 12/N^2$ [@Komatsu:1999mt; @Tsujikawa:2004my] as in the Starobinsky model [@Starobinsky:1980te; @Kehagias:2013mya]. Radiative corrections lead to deviations from this so called Starobinsky point in the $n_s$ and $r$ plane, however the size of these deviations differ according to the prescription used for the calculation. As discussed in refs. [@Bezrukov:2008ej; @Bezrukov:2009db; @Bezrukov:2013fka], the plateau type structure of the Einstein frame potential remains intact and the deviations in $n_s$ are rather insignificant according to prescription I. However, according to prescription II, radiative corrections lead to a linear term in the Einstein frame potential written in terms of a scalar field with a canonical kinetic term. If the inflaton is dominantly coupling to bosons the coefficient of this term is positive, and as this coefficient is increased the inflationary predictions move towards the linear potential predictions $n_s\approx1-3/(2N)$ and $r\approx4/N$ [@Martin:2013tda]. If the inflaton is dominantly coupling to fermions the coefficient of this term is negative, leading to a reduction in the values of $n_s$ and $r$ [@Okada:2010jf]. In this work we take the inflaton to be a SM singlet scalar field, and take the self-coupling $\lambda$ and $\xi$ to be free parameters, with $\xi\lesssim10^3$ as discussed in [section \[sec:quartic\]]{}.[^1] Radiative corrections for a SM singlet inflaton have been studied by refs. [@Lerner:2009xg; @Lerner:2011ge; @Kahlhoefer:2015jma]. Unlike these works, we focus on studying the effect of radiative corrections for general values of $\xi\lesssim10^3$, including the case of $\xi\ll1$. A related work which includes the case of $\xi\ll1$ is ref. [@Okada:2010jf]. In this work the inflaton is assumed to couple to fermions and prescription II is used. Ref. [@Racioppi:2018zoy] consideres a potential which coincides with the potential discussed in [section \[sec:p2\]]{} for inflaton coupling to bosons.[^2] Here, we extend previous works by considering both prescriptions I and II, and inflaton coupling to bosons or fermions. For each case we calculate the regions in the plane of coupling parameter values for which the spectral index $n_s$ and the tensor-to-scalar ratio $r$ are in agreement with the current data. We also display how $n_s$ and $r$ change due to radiative corrections in these regions. Finally, we note that the non-minimal quartic inflation model given by [eq. (\[lagrangian\])]{} is a special case of the universal attractor models discussed in ref. [@Kallosh:2013tua]. In the strong coupling limit $\xi\to\infty$, the inflationary predictions of these models coincide with those of conformal attractor models [@Kallosh:2013hoa], which correspond to the $\alpha=1$ case of the $\alpha$-attractor models [@Kallosh:2013yoa]. The relation between these types of models is elucidated in ref. [@Galante:2014ifa]. The reheating phase of Higgs and $\alpha$-attractor-type inflation models due to inflaton couplings to additional fields has been discussed in a number of works, see e.g. refs. [@Bezrukov:2008ut; @GarciaBellido:2008ab; @Bezrukov:2011gp; @Ueno:2016dim; @Drewes:2017fmn; @Dimopoulos:2017tud]. While the observational parameter values also depend on the details of the reheating phase in general, for the special case of the non-minimal quartic inflation model and for the range of $\xi$ values that we consider, the average equation of state during reheating is given by $p\approx\rho/3$ as we discuss in [section \[sec:quartic\]]{}. The number of e-folds and the observational parameter values are then to a good approximation independent of the reheat temperature. Thus, in our case the main effect of inflaton couplings to additional fields on the observational parameters is not due to the reheating phase but rather due to the radiative corrections to the potential during inflation, which we focus on this work. Inflation with non-minimal coupling {#sec:inf} =================================== Consider a non-minimally coupled scalar field $\phi$ with a canonical kinetic term and a potential $V_J(\phi)$: $$\label{vjphi} \frac{\mathcal{L}_J}{\sqrt{-g}}=\frac12F(\phi)R-\frac12g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V_J(\phi)\,,$$ where the subscript $J$ indicates that the Lagrangian is specified in Jordan frame, and $F(\phi)=1+\xi\phi^2$. We are using units where the reduced Planck scale $m_P=1/\sqrt{8\pi G}\approx2.4\times10^{18}\text{ GeV}$ is set equal to unity, so we require $F(\phi)\to1$ or $\phi\to0$ after inflation. For calculating the observational parameters given [eq. (\[vjphi\])]{}, it is convenient to switch to the Einstein ($E$) frame by applying a Weyl rescaling $g_{\mu\nu}=\tilde{g}_{\mu\nu}/F(\phi)$, so that the Lagrangian density takes the form [@Fujii:2003pa] $$\label{LE} \frac{\mathcal{L}_E}{\sqrt{-\tilde{g}}}=\frac12\tilde{R}-\frac{1}{2Z(\phi)}\tilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\,,$$ where $$\label{Zphi} \frac{1}{Z(\phi)}=\frac32\frac{F'(\phi)^2}{F(\phi)^2}+\frac{1}{F(\phi)}\,,\quad V(\phi)=\frac{V_J(\phi)}{F(\phi)^2}\,,$$ and $F'\equiv\ud F/\ud\phi$. If we make a field redefinition $$\label{redefine} \ud\sigma=\frac{\ud\phi}{\sqrt{Z(\phi)}}\,,$$ we obtain the Lagrangian density for a minimally coupled scalar field $\sigma$ with a canonical kinetic term. Once the Einstein frame potential is expressed in terms of the canonical $\sigma$ field, the observational parameters can be calculated using the slow-roll parameters (see ref. [@Lyth:2009zz] for a review and references): $$\label{slowroll1} \epsilon =\frac{1}{2}\left( \frac{V_{\sigma} }{V}\right) ^{2}\,, \quad \eta = \frac{V_{\sigma \sigma} }{V} \,, \quad $$ where $\sigma$’s in the subscript denote derivatives. The spectral index $n_s$, the tensor-to-scalar ratio and $r$ are given in the slow-roll approximation by $$\label{nsralpha1} n_s = 1 - 6 \epsilon + 2 \eta \,,\quad r = 16 \epsilon \,.\quad $$ The amplitude of the curvature perturbation $\Delta_\mathcal{R}$ is given by $$\label{perturb1} \Delta_\mathcal{R}=\frac{1}{2\sqrt{3}\pi}\frac{V^{3/2}}{|V_{\sigma}|}\,,$$ which should satisfy $\Delta_\mathcal{R}^2\approx 2.4\times10^{-9}$ from the Planck measurement [@Aghanim:2018eyx] with the pivot scale chosen at $k_* = 0.002$ Mpc$^{-1}$. The number of e-folds is given by $$\label{efold1} N_*=\int^{\sigma_*}_{\sigma_e}\frac{V\rm{d}\sigma}{V_{\sigma}}\,,$$ where the subscript “$_*$” denotes quantities when the scale corresponding to $k_*$ exited the horizon, and $\sigma_e$ is the inflaton value at the end of inflation, which can be estimated by $\epsilon(\sigma_e) = 1$. It is convenient for numerical calculations to rewrite these slow-roll expressions in terms of the original field $\phi$, following the approach in e.g. ref. [@Linde:2011nh]. Using [eq. (\[redefine\])]{}, [eq. (\[slowroll1\])]{} can be written as $$\label{slowroll2} \epsilon=Z\epsilon_{\phi}\,,\quad \eta=Z\eta_{\phi}+{\rm sgn}(V')Z'\sqrt{\frac{\epsilon_{\phi}}{2}}\,, $$ where we defined $$\epsilon_{\phi} =\frac{1}{2}\left( \frac{V^{\prime} }{V}\right) ^{2}\,, \quad \eta_{\phi} = \frac{V^{\prime \prime} }{V} \,. $$ Similarly, eqs. (\[perturb1\]) and (\[efold1\]) can be written as $$\begin{aligned} \label{perturb2} \Delta_\mathcal{R}&=&\frac{1}{2\sqrt{3}\pi}\frac{V^{3/2}}{\sqrt{Z}|V^{\prime}|}\,,\\ \label{efold2} N_*&=&\rm{sgn}(V')\int^{\phi_*}_{\phi_e}\frac{\ud\phi}{Z(\phi)\sqrt{2\epsilon_{\phi}}}\,.\end{aligned}$$ To calculate the numerical values of $n_s$ and $r$ we also need a numerical value of $N_*$. Assuming a standard thermal history after inflation, $$\label{efolds} N_*\approx64.7+\frac12\ln\frac{\rho_*}{m^4_P}-\frac{1}{3(1+\omega_r)}\ln\frac{\rho_e}{m^4_P} +\left(\frac{1}{3(1+\omega_r)}-\frac14\right)\ln\frac{\rho_r}{m^4_P}\,.$$ Here $\rho_{e}=(3/2)V(\phi_{e})$ is the energy density at the end of inflation, $\rho_{*}\approx V(\phi_{*})$ is the energy density when the scale corresponding to $k_*$ exited the horizon, $\rho_r$ is the energy density at the end of reheating and $\omega_r$ is the equation of state parameter during reheating.[^3] As discussed in [section \[sec:quartic\]]{}, $\omega_r=1/3$ is generally a good approximation for the potentials which we investigate. For this case $$\label{highN} N_*\approx 64.7+\frac{1}{2}\ln \rho_*-\frac{1}{4}\ln\rho_e\,,$$ independent of the reheat temperature. Quartic potential {#sec:quartic} ================= Inflationary predictions of non-minimal quartic inflation have been studied in detail, see e.g. refs. [@Futamase:1987ua; @Fakir:1990eg; @Kaiser:1994vs; @Tsujikawa:2004my; @Bezrukov:2008dt; @Okada:2010jf; @Bezrukov:2013fca; @Campista:2017ovq]. Here after summarizing the results following ref. [@Bezrukov:2013fca], we comment on an analytical approximation used in that work, and briefly discuss the effect of the reheating stage on the inflationary predictions. The Lagrangian of the non-minimal quartic inflation model is given by $$\label{lagrangian} \frac{\mathcal{L}_J}{\sqrt{-g}}=\frac12(1+\xi\phi^2)R-\frac12g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac14\lambda \phi^4\,.$$ In Einstein frame, the potential is $$\begin{aligned} \label{eframe} V(\phi)=\frac{(1/4)\lambda \phi^4}{\left(1+\xi \phi^2\right)^2}\,.\end{aligned}$$ Using eqs. (\[nsralpha1\]) and (\[slowroll2\]), we obtain $$\begin{aligned} \label{nsr} n_s & = & 1-\frac{24}{\phi^2}\left(\frac{1+\frac{5}{3}\psi+8\xi \psi+\frac{2}{3}\psi^2+4\xi \psi^2}{(1+(1+6\xi)\psi)^2}\right)\,, \nonumber \\ r & = & \frac{128}{\phi^2}\frac{1}{1+(1+6\xi)\psi}\,,\end{aligned}$$ where we defined $\psi\equiv\xi\phi^2$. Using [eq. (\[perturb2\])]{} we obtain $$\label{lambda1} \lambda=12\pi^2 \Delta^2_R \frac{64(1+\psi)^2\xi^3}{\psi^2(1+(1+6\xi)\psi)}\,,$$ and using [eq. (\[efold2\])]{} we obtain $$\label{e-foldd} N= \frac{3}{4s}(\psi-\psi_{\mathrm{e}})-\frac{3}{4}\ln \frac{1+\psi}{1+\psi_{\mathrm{e}}}\,,$$ where $s\equiv(6\xi)/(1+6\xi)$. Here $\psi_{\mathrm{e}}$ can be obtained from $\epsilon(\psi_{\mathrm{e}})= 1 $ as follows: $$\label{end} \psi_{\mathrm{e}}=\frac{-1+\sqrt{1+32\xi (1+6\xi)}}{2(1+6\xi)}\,.$$ For any value of $\xi$, we can calculate the observational parameters by numerically solving eqs. (\[e-foldd\]) and (\[highN\]) (with a correction for $\xi\gtrsim1$, see below for a discussion) to find the value of $\phi_*$. Formally, inverting [eq. (\[e-foldd\])]{} gives a solution in terms of the $-1$ branch of the Lambert function: $$\label{psi_e} \psi=-s W_{-1}\left(-\frac{e^{-1/s}}{s\exp\left[4N/3+\psi_{\mathrm{e}}/s-\ln(1+\psi_{\mathrm{e}})\right]}\right)-1\,.$$ As [ref. [@Bezrukov:2013fca]]{} point out, one can find reasonable approximations to the numerical solution by utilizing $N+1\approx3\psi/(4s)$. Here we point out that a slightly more complicated but better approximation can be obtained by using $W_{-1}(-x)\approx\ln x-\ln(-\ln x)$: $$\label{approx} \psi\approx\frac{4 s N}{3}+s\ln\left(1+\frac{4sN}{3}\right)\,.$$ Inserting [eq. (\[approx\])]{} in eqs. (\[nsr\]) and (\[lambda1\]), we obtain the eqs. (3.19), (3.20) and (3.22) in ref. [@Bezrukov:2013fca] only with the modification $$\label{n+1} N+1\to N'\equiv N\left[1+\frac{3}{4N}\ln\left(1+\frac{4sN}{3}\right)\right]\,.$$ Comparison of the numerical solutions and the two analytical approximations discussed here is shown in [figure \[kuartik\]]{}. As can be seen from the figure, our analytical approximation is more accurate for $\xi\gg1$, and $n_s$ ($r$) values deviate from the numerical solution by at most 2% (3%) for any $\xi$ value. ![The values of $n_s$, $r$ and $\lambda$ are shown as a function of the non-minimal coupling parameter $\xi$. “Lambert” curves show numerical solutions that are obtained using [eq. (\[psi\_e\])]{}, “Bezrukov and Gorbunov” curves show the approximate analytical expressions in ref. [@Bezrukov:2013fca], $(N+1)\to N'$ curves show the improved analytical expressions using [eq. (\[n+1\])]{}. The pink (red) contour corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx].[]{data-label="kuartik"}](quartic.pdf){width="14cm"} For the minimally coupled case, quartic potential implies an equation of state parameter $\omega_r=1/3$ after inflation [@Turner:1983he], and as a result the number of e-folds $N_*$ is given by [eq. (\[highN\])]{}, independent of the reheat temperature. This result, which removes the uncertainty in the observational parameter values due to the reheat temperature, is also valid for $\xi\lesssim1$. However, for $\xi\gtrsim1$ the reheating stage includes a phase where the Einstein frame potential for the canonical scalar field is quadratic [@Bezrukov:2008ut; @GarciaBellido:2008ab]. Following refs. [@Bezrukov:2008ut; @Gong:2015qha], we obtain the e-fold number in this case as $$\label{efolds2} N_*\approx64.7+\frac12\ln\frac{\rho_*}{m^4_P}-\frac{1}{3}\ln\frac{\rho_e}{m^4_P} +\frac{1}{12}\ln\frac{V(\phi=\sqrt2/(3^{1/4}\xi))}{m^4_P}\,.$$ For $\xi\gtrsim1$ we use this expression instead of [eq. (\[highN\])]{} in [figure \[kuartik\]]{}. The quadratic phase slightly reduces the value of $N_*$ with respect to the value calculated by [eq. (\[highN\])]{}. The difference in $N_*$ between the two expressions is approximately given by $$\label{fark} \frac{1}{12}\ln\frac{ V\left(\phi=\frac{\sqrt2}{3^{1/4}\xi}\right)}{\rho_{e}}\,.$$ Since $\xi \phi^2\lesssim1$, $V(\phi)\propto \phi^4$ after inflation. In this case $\rho_e$ is proportional to $\phi_e^4=4/(3\xi^2)$ for $\xi\gg1$. Thus, for $\xi\gg1$, the reduction in $N_*$ due to the quadratic phase is approximately given by $-(1/6)\ln\xi$ [@Gong:2015qha]. For the non-minimal quartic potential, expanding the action around the vacuum reveals a cut-off scale $\Lambda= 1/\xi$ [@Burgess:2009ea; @Barbon:2009ya; @Hertzberg:2010dc]. From eqs. (\[perturb1\]) and (\[nsralpha1\]), we obtain $V=(3/2)\pi^2 r\Delta_\mathcal{R}^2$. Thus, requiring $\Lambda$ to be higher than the energy scale during inflation corresponds to $$\xi<\left(\frac32\pi^2 r\Delta_\mathcal{R}^2\right)^{-1/4}\,,$$ leading to $\xi\lesssim10^{2.5}$. For this reason all numerical results in the following sections will be displayed for $\xi$ values up to $10^3$. With this constraint, taking into account the quadratic phase after inflation makes only a $\lesssim1$ difference in the $N_*$ value. Furthermore, preheating effects can make this difference even smaller. Thus, the uncertainty in the observational parameter values due to the reheating stage is rather small for non-minimal quartic inflation. For instance, in [figure \[kuartik\]]{} which shows $n_s$ and $r$ values for $\xi$ up to $10^3$, the effect of the quadratic phase amounts to the barely visible hook-like part at the bottom end of the $n_s$–$r$ curves. Therefore, we will use the $\omega_r=1/3$ approximation hereafter. Finally, note that since we compare the numerical $n_s$ and $r$ values with the recent Keck Array/BICEP2 and Planck data [@Ade:2018gkx], our constraints on $\xi$ are slightly more stringent compared to earlier works. Namely, non-minimal quartic inflation is compatible with the Keck Array/BICEP2 and Planck data for $\xi\gtrsim0.005$ (0.01) at $95\%$ ($68\%$) confidence level (CL). Radiative corrections {#sec:radiative} ===================== Interactions of the inflaton with other fields, required for efficient reheating, lead to radiative corrections in the inflaton potential. These corrections can be expressed at leading order as follows [@Coleman:1973jx]: $$\begin{aligned} \label{cw1loop} \Delta V(\phi)=\sum\limits_{i}\frac{(-1)^F}{64\pi^2}M_i(\phi)^4 \ln\left(\frac{M_i(\phi)^2}{\mu^2}\right)\,, \end{aligned}$$ where $F$ is $+1$ $(-1)$ for bosons (fermions), $\mu$ is a renormalization scale and $M_i(\phi)$ denote field dependent masses. First let us consider the potential terms for a minimally coupled inflaton field with a quartic potential, which couples to another scalar $\chi$ and to a Dirac fermion $\Psi$: $$\begin{aligned} \label{lmin} V(\phi,\chi,\Psi)= \frac{\lambda}{4}\phi^4+h\phi\bar{\Psi}\Psi+m_\Psi\bar{\Psi}\Psi+\frac{1}{2}g^2\phi^2\chi^2+\frac{1}{2}m_\chi^2\chi^2\,. \end{aligned}$$ Under the assumptions $$\begin{aligned} \label{varsay1} g^2\phi^2\gg m_\chi^2 \,, \quad g^2\gg\lambda \,, \quad h\phi\gg m_\Psi \,, \quad h^2\gg\lambda \,, \end{aligned}$$ the inflaton potential including the Coleman-Weinberg (CW) one-loop corrections given by [eq. (\[cw1loop\])]{} take the form: $$\begin{aligned} \label{radpot3} V(\phi)=\frac{\lambda}{4}\phi^4\pm\kappa\phi^4\ln\left(\frac{\phi}{\mu}\right)\,,\end{aligned}$$ where the $+$ ($-$) sign corresponds to the case of the inflaton dominantly coupling to bosons (fermions) and we have defined the radiative correction coupling parameter $$\begin{aligned} \label{kappatanim} \kappa\equiv\frac{1}{32\pi^2}\Big|(g^4-4h^4)\Big|\,.\end{aligned}$$ Generalizing [eq. (\[radpot3\])]{} to the non-minimally coupled case is subject to ambiguity unless the ultraviolet completion of the low-energy effective field theory is specified, as discussed in refs. [@Bezrukov:2008ej; @Bezrukov:2009db; @Bezrukov:2013fka; @Hamada:2016onh]. In the literature, typically two prescriptions for the calculation of radiative corrections are adopted. In prescription I, the field dependent masses in the one-loop CW potential are expressed in the Einstein frame. Using the transformations $$V(\phi)=\frac{V_J(\phi)}{F(\phi)^2}, \; \tilde{\phi}=\frac{\phi}{\sqrt{F(\phi)}}, \; \tilde{\Psi}=\frac{\Psi}{F(\phi)^{3/4}}, \; \tilde{m}_\Psi(\phi)=\frac{m_\Psi(\phi)}{\sqrt{F(\phi)}}, \; \tilde{m}_\chi^2=\frac{m_\chi^2}{F(\phi)}\,,$$ the one-loop corrected potential is obtained in the Einstein frame as $$\label{radpotyontem1} V(\phi)=\frac{\frac{\lambda}{4 }\phi^4\pm\kappa \phi^4 \ln\left(\frac{\phi}{\mu \sqrt{1+\xi \phi^2}}\right)}{(1+\xi\phi^2)^2}\,.$$ In prescription II, the field dependent masses in the one-loop CW potential are expressed in the Jordan frame, so that [eq. (\[radpot3\])]{} corresponds to the one-loop corrected potential in the Jordan frame. Therefore the Einstein frame potential in this case is given by $$\begin{aligned} \label{radpotyontem2} V(\phi)=\frac{\frac{\lambda}{4 }\phi^4\pm\kappa \phi^4 \ln\left(\frac{\phi}{\mu}\right)}{(1+\xi\phi^2)^2}\,. \end{aligned}$$ Note that the potentials in eqs. (\[radpotyontem1\]) and (\[radpotyontem2\]) are approximations that can be obtained from the one-loop renormalization group improved effective actions, see for instance ref. [@Okada:2010jf] for a discussion of this point. In the next two sections, we numerically investigate how the $n_s$ and $r$ values change as a function of the coupling parameters $\xi$ and $\kappa$ using prescription I and prescription II, respectively. The calculation procedure is as follows: We form a grid of points in the $\xi$ and $\kappa$ plane. For each $(\xi,\;\kappa)$ point, we start the calculation by assigning an initial $\lambda$ value. We then calculate numerical values of $\phi_e$ using $\epsilon(\phi_e) =1$, and $\phi_*$ using [eq. (\[perturb2\])]{}. The e-fold number $N_*$ is calculated using [eq. (\[efold2\])]{} and compared with [eq. (\[highN\])]{}. The initial value of $\lambda$ is then adjusted and the calculation is repeated until the two $N_*$ values match. The $\phi_*$ value obtained this way is plugged in eqs. (\[slowroll2\]) and (\[nsralpha1\]) to yield the $n_s$ and $r$ values. Finally, the calculation is repeated over the whole grid, with $\lambda$ solutions for each point used as initial values of their neighbors. For the numerical calculations, a value for $\mu$ should also be specified. However, shifting the value of $\mu$ does not change the forms of eqs. (\[radpotyontem1\]) and (\[radpotyontem2\]), corresponding only to a shift in the value of $\lambda$. Thus, for fixed values of the coupling parameters $\xi$ and $\kappa$, $n_s$ and $r$ values do not depend on $\mu$. Radiatively corrected quartic potential: Prescription I {#sec:p1} ======================================================= In this section we numerically investigate how the $n_s$ and $r$ values change as a function of the coupling parameters $\xi$ and $\kappa$, using the potential in eq. (\[radpotyontem1\]), with a $+$ ($-$) sign for the inflaton dominantly coupling to bosons (fermions). ![For prescription I and inflaton coupling to bosons, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx].[]{data-label="6.9"}](P1_boson_.pdf "fig:"){width="7.5cm"}\ ![For prescription I and inflaton coupling to bosons, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx].[]{data-label="6.9"}](P1_boson_ns_r.pdf "fig:"){width="14cm"} ![For prescription I and inflaton coupling to bosons, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx].[]{data-label="6.9"}](P1_boson_kmax_.pdf "fig:"){width="9cm"}\ ![For prescription I and inflaton coupling to bosons, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx].[]{data-label="6.9"}](kmax_ns_r2.pdf "fig:"){width="14cm"} For prescription I and inflaton coupling to bosons, [figure \[6.5\]]{} shows the region in the $\xi$ and $\kappa$ plane where $n_s$ and $r$ values are compatible with the current data. [Figure \[6.9\]]{} shows how $n_s$ and $r$ values change with $\kappa$ for chosen $\xi$ values. It is clear from the figures that $n_s$ and $r$ values depend more sensitively on the value of $\xi$ rather than $\kappa$. As $\kappa$ is increased holding $\xi$ fixed, there is a transition in $n_s$ and $r$ values for a relatively narrow range of $\kappa$. $n_s$ and $r$ no longer change at even larger $\kappa$ values, however this last result is subject to some caveats as discussed below. In contrast to the other cases covered in subsequent sections, we find that eqs. (\[perturb2\]), (\[efold2\]) and (\[highN\]) can be simultaneously satisfied for arbitrarily large values of $\kappa$. However, as mentioned in [section \[sec:radiative\]]{}, the potential we use is an approximation of the one-loop renormalization group improved effective action, and this approximation will eventually fail for large values of $\kappa$. Furthermore, higher loop corrections will eventually become also important. Even if we take the potential in [eq. (\[radpotyontem1\])]{} at face value, the inflationary solutions for large $\kappa$ values can only be obtained for fine tuned values of the coupling parameters. To show this, let us write the potential in the limit $\xi\phi^2\gg1$ and take $\mu=1$ for convenience. The potential then approximately takes the form [eq. (\[eframe\])]{}, with $\lambda/4$ replaced by $A\equiv\lambda/4-(\kappa/2)\ln\xi$. Using [eq. (\[redefine\])]{}, this potential can be written as $$\label{lambdakappa} V(\sigma)\approx \frac{A}{\xi^2}\left[1-2\exp\left(-2\sqrt{\frac s6} \sigma\right)\right]\,.$$ Using [eq. (\[efold1\])]{}, $\exp(2\sqrt{s/6}\sigma)\approx4sN/3$. Finally, using [eq. (\[perturb1\])]{}, we obtain $$\begin{aligned} \label{lambdafine} \lambda\approx \frac{72\pi^2 \Delta^2_R \xi^2}{sN^2}+2\kappa \ln\xi\,. \end{aligned}$$ The first term in the right hand side is approximately $5\times10^{-10}\xi^2$ for $\xi\gg1$. If $2\kappa\ln\xi$ is much larger than this term, [eq. (\[lambdafine\])]{} can only be satisfied if $\lambda$ almost exactly equals $2\kappa\ln\xi$. The case of inflaton having a quartic potential with radiative corrections due to coupling to fermions was discussed in ref. [@NeferSenoguz:2008nn] taking $\xi=0$. There it was pointed out that there are two solutions for every $\kappa$ value that is smaller than a maximum $\kappa_{\mathrm{max}}$ value. This is also true for $\xi\ne0$, with $\kappa_{\mathrm{max}}$ values depending on $\xi$. We label the branch of solutions with larger $\lambda$ for a given $\kappa$ as the first branch, and the other branch of solutions as the second branch. For $\kappa>\kappa_{\mathrm{max}}$ there is no solution, that is, eqs. (\[perturb2\]), (\[efold2\]) and (\[highN\]) cannot be simultaneously satisfied. ![For prescription I, inflaton coupling to fermions and second branch solutions, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="6.3"}](P1_fer.pdf "fig:"){width="7.5cm"}\ ![For prescription I, inflaton coupling to fermions and second branch solutions, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="6.3"}](P1_fer_ns_r.pdf "fig:"){width="14cm"} ![For prescription I, inflaton coupling to fermions and second branch solutions, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="6.3"}](P1_hilltop_fer.pdf "fig:"){width="7.5cm"}\ ![For prescription I, inflaton coupling to fermions and second branch solutions, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="6.3"}](P1_hilltop_fer_ns_r.pdf "fig:"){width="14cm"} For prescription I and inflaton coupling to fermions, [figure \[6.1\]]{} shows the region in the $\xi$ and $\kappa$ plane where $n_s$ and $r$ values are compatible with the current data, for the first branch of solutions. Again, $n_s$ and $r$ values depend more sensitively on the value of $\xi$ rather than $\kappa$. The observationally compatible region for the second branch of solutions is shown in [figure \[6.3\]]{}. As seen from the figure, the second branch solutions are compatible with observations for only a narrow region in the $\xi$–$\kappa$ plane. [Figure \[6.11\]]{} shows how $n_s$ and $r$ values change with $\kappa$ and the $\kappa_{\mathrm{max}}$ values for chosen $\xi$ values. The first branch solutions move from the red points towards the $\kappa=0$ curve as $\kappa$ decreases. As can be seen from the bottom panels, significant change in the $n_s$ and $r$ values only occur when $\kappa$ becomes the same order of magnitude as $\kappa_{\mathrm{max}}$. The second branch solutions, on the other hand, move towards small $n_s$ values and away from the observationally favored region in the $n_s$–$r$ plane as $\kappa$ decreases. These solutions cease to exist for $\kappa\ll\kappa_{\mathrm{max}}$ as inflation with sufficient duration cannot be obtained. ![For prescription I and inflaton coupling to fermions, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. The solid (dotted) portions of the curves correspond to first (second) branch of solutions. The red points show the maximum $\kappa$ values where the two branch of solutions meet. These values are also written in the figure. The bottom figures only show the first branch solutions.[]{data-label="6.11"}](P1_hilltop.pdf "fig:"){width="10cm"} ![For prescription I and inflaton coupling to fermions, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. The solid (dotted) portions of the curves correspond to first (second) branch of solutions. The red points show the maximum $\kappa$ values where the two branch of solutions meet. These values are also written in the figure. The bottom figures only show the first branch solutions.[]{data-label="6.11"}](kmax_ns_r_p1fer.pdf "fig:"){width="14cm"} Radiatively corrected quartic potential: Prescription II {#sec:p2} ======================================================== In this section we numerically investigate how the $n_s$ and $r$ values change as a function of the coupling parameters $\xi$ and $\kappa$, using the potential in eq. (\[radpotyontem2\]), with a $+$ ($-$) sign for the inflaton dominantly coupling to bosons (fermions). For prescription II and inflaton coupling to bosons, [figure \[7.3\]]{} shows the region in the $\xi$ and $\kappa$ plane where $n_s$ and $r$ values are compatible with the current data. [Figure \[7.12\]]{} shows how $n_s$ and $r$ values change with $\kappa$ for chosen $\xi$ values. The $\kappa_{\mathrm{max}}$ values, that is, the maximum $\kappa$ values that allow a simultaneous solution of eqs. (\[perturb2\]), (\[efold2\]) and (\[highN\]) are also shown. From the figures we see that for $\xi\gtrsim10^{-2}$, the $n_s$ and $r$ values approach the linear potential predictions $n_s\approx1-3/(2N)$ and $r\approx4/N$ as $\kappa$ approaches $\kappa_{\mathrm{max}}$. This result is not surprising since for large enough $\xi$ values $\xi\phi^2\gg1$ during inflation, in which case the Einstein frame potential written in terms of the canonical scalar field using eqs. (\[Zphi\]) and (\[redefine\]) contains a linear term which eventually dominates as the value of $\kappa$ is increased. This approach to the linear potential predictions was also noted in refs. [@Martin:2013tda; @Rinaldi:2015yoa; @Racioppi:2018zoy]. Similarly to the prescription I case for inflaton coupling to fermions, significant change in the $n_s$ and $r$ values only occur when $\kappa$ becomes the same order of magnitude as $\kappa_{\mathrm{max}}$. Similarly to the prescription I case, there are also two branch of solutions for prescription II and inflaton coupling to fermions. [Figure \[7.1\]]{} shows the region in the $\xi$ and $\kappa$ plane where $n_s$ and $r$ values are compatible with the current data, for the first branch of solutions. The second branch of solutions are not compatible with the current data at any value of $\xi$ or $\kappa$. [Figure \[7.14\]]{} shows how $n_s$ and $r$ values change with $\kappa$ and the $\kappa_{\mathrm{max}}$ values for chosen $\xi$ values. Again, the first branch solutions move from the red points towards the $\kappa=0$ curve as $\kappa$ decreases, whereas the second branch solutions move towards small $n_s$ values. Finally we note that our results for prescription II and inflaton coupling to fermions overlap and agree with those of ref. [@Okada:2010jf]. ![For prescription II and inflaton coupling to bosons, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="7.3"}](P2_boson_.pdf "fig:"){width="7.5cm"}\ ![For prescription II and inflaton coupling to bosons, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="7.3"}](P2_boson_ns_r.pdf "fig:"){width="14cm"} ![For prescription II and inflaton coupling to bosons, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. The red points show the maximum $\kappa$ values. These values, increasing with $\xi$, are also written in the figure.[]{data-label="7.12"}](P2_boson_kmax.pdf "fig:"){width="9cm"}\ ![For prescription II and inflaton coupling to bosons, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. The red points show the maximum $\kappa$ values. These values, increasing with $\xi$, are also written in the figure.[]{data-label="7.12"}](kmax_ns_r2_p2boson.pdf "fig:"){width="14cm"} ![For prescription II, inflaton coupling to fermions and first branch solutions, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="7.1"}](P2_fermion_.pdf "fig:"){width="7.5cm"}\ ![For prescription II, inflaton coupling to fermions and first branch solutions, the top figure shows in light green (green) the regions in the $\xi$–$\kappa$ plane for which $n_s$ and $r$ values are within the $95\%$ $(68\%)$ CL contours based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. Bottom figures show $n_s$ and $r$ values in these regions.[]{data-label="7.1"}](P2_fermion_ns_r.pdf "fig:"){width="14cm"} ![For prescription II and inflaton coupling to fermions, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. The solid (dotted) portions of the curves correspond to first (second) branch of solutions. The red points show the maximum $\kappa$ values where the two branch of solutions meet. These values are also written in the figure. The bottom figures only show the first branch solutions.[]{data-label="7.14"}](P2_fermion_kmax_.pdf "fig:"){width="10cm"} ![For prescription II and inflaton coupling to fermions, the change in $n_s$ and $r$ as a function of $\kappa$ is plotted for selected $\xi$ values. The pink (red) contour in the top figure corresponds to the 95% (68%) CL contour based on data taken by the Keck Array/BICEP2 and Planck collaborations [@Ade:2018gkx]. The solid (dotted) portions of the curves correspond to first (second) branch of solutions. The red points show the maximum $\kappa$ values where the two branch of solutions meet. These values are also written in the figure. The bottom figures only show the first branch solutions.[]{data-label="7.14"}](kmax_ns_r_p2fer.pdf "fig:"){width="14cm"} Conclusion {#sec:conc} ========== In this paper we revisited the non-minimal quartic inflation model consisting of a quartic potential and a coupling term $\xi \phi^2 R$ between the Ricci scalar and the inflaton, first reviewing the tree level case without any radiative corrections in [section \[sec:quartic\]]{}. We noted that the approximate analytical expressions in ref. [@Bezrukov:2013fca] can be improved by using the $W_{-1}(-x)\approx\ln x-\ln(-\ln x)$ approximation for the Lambert function. Two prescriptions used in the literature to take into account the radiative corrections to the potential were briefly discussed in [section \[sec:radiative\]]{}. We then numerically investigated the effect of the radiative corrections on the inflationary observables $n_s$ and $r$ due to inflaton coupling to bosons or fermions in [section \[sec:p1\]]{} for prescription I and in [section \[sec:p2\]]{} for prescription II. Generally, we observed that while the radiative corrections prevent inflation with a sufficient duration after a $\xi$ dependent maximum value $\kappa_{\mathrm{max}}$ of the coupling parameter $\kappa$ defined by [eq. (\[kappatanim\])]{}, they don’t change $n_s$ and $r$ values significantly unless $\kappa$ is the same order of magnitude as $\kappa_{\mathrm{max}}$. For the prescription I and coupling to bosons case, in contrast to the other cases, we found that eqs. (\[perturb2\]), (\[efold2\]) and (\[highN\]) can be simultaneously satisfied for arbitrarily large values of $\kappa$. However, as explained in [section \[sec:p1\]]{}, we regard this result as an artifact of the approximation we used for the potential. The two prescriptions for the radiative corrections lead to significantly different potentials in the limit $\xi\phi^2\gg1$, corresponding to $\xi\gg1/(8N)$. For prescription I, the plateau type structure of the potential remains intact in this limit. As a result, for the same $\kappa$ value the effect of radiative corrections is milder compared to the results obtained using prescription II. This difference is also reflected in the $\kappa_{\mathrm{max}}$ values. For example, if inflaton couples to fermions and $\xi=10$, $\kappa_{\mathrm{max}}$ is $2.2\times10^{-8}$ ($1.9\times10^{-10}$) using prescription I (prescription II). Such differences suggest the neeed for further work on the theoretical motivations of these prescriptions used in the literature to calculate the observational parameters. Acknowledgements {#acknowledgements .unnumbered} ================ This work is supported by TÜBİTAK (The Scientific and Technological Research Council of Turkey) project number 116F385. =10000 [^1]: The value of $\xi$ is ambiguous unless the inflationary part of the Lagrangian is embedded in a specific theory (see e.g. refs. [@Muta:1991mw; @Faraoni:2004pi]). [^2]: See also refs. [@Rinaldi:2015yoa; @Okada:2015lia; @Marzola:2015xbh; @Marzola:2016xgb] for related work. [^3]: For a derivation of [eq. (\[efolds\])]{} see e.g. ref. [@Liddle:2003as].

--- abstract: 'We derive the braid relations of the charged anyons interacting with a magnetic field on Riemann surfaces. The braid relations are used to calculate the quasiparticle’s spin in the fractional quantum Hall states on Riemann surfaces. The quasiparticle’s spin is found to be topological independent and satisfies physical restrictions.' address: 'International School for Advanced Studies, SISSA, I-34014 Trieste, Italy ' author: - Dingping Li title: 'Intrinsic quasiparticle’s spin and fractional quantum Hall effect on Riemann surfaces' --- The possibility of fractional statistics on two dimensional surfaces was discovered in Ref.[@lei]. When two fractional-statistics particles (anyons) are interchanged, the wave function changes by a phase $\exp(i\theta)$, where $\theta$ is neither $\theta =0$ (Bose statistics) nor $\theta =\pi$ (Fermion statistics). The quasiparticles in fractional quantum Hall systems (for a review on the fractional quantum Hall effect (FQHE), see Ref.[@girvin]) are anyons [@arovas; @halp] and this picture had been used to construct the hierarchical wave function in the FQHE [@halp]. Anyons may also find applications in some condensed matter systems [@sup]. On higher dimension spaces ($D>2$), only Fermion and Boson exist and the Fermion’s spin is an half-integer, the Boson’s spin is an integer. It will be very interesting to know what is the spin of anyons and the spin-statistics relation of anyons. The spin-statistics relation of anyons is generalized one if the spin $s$ equals to $s={\case\theta/2\pi}$. Anyons in various models, for example, in non-linear sigma models, Chern-Simons field theories and relativistic quantum field theories on $2D$ dimension spaces indeed have the generalized spin-statistics relation [@rdt; @fro]. Naturally, we will ask the question what is the spin-statistics relation of the quasiparticle in the FQHE. The recent discussion about the quasiparticle’s spin (QPS) can be found in Refs.[@lidp; @wen; @kiv]. Reference.[@lidp] calculated the QPS by analyzing the hierarchical wave function or by calculating the Berry phase of the quasiparticles moving in a closed path on the sphere. Reference.[@wen] obtained the QPS by analyzing the Ginzburg-Laudau-Chern-Simons (GLCS) theory of the FQHE on the sphere. On the other hand, Reference.[@kiv] calculated the QPS based on the GLCS theory on the disc geometry. However, the results in Refs.[@lidp; @wen; @kiv] are different from each other. The ambiguity of the QPS in the literature is due to lack of a good definition of the QPS. In this paper, we will calculate the QPS by using braid relations of anyons on Riemann surfaces [@wu; @thou; @ein; @wend; @eina; @imbo; @bla]. We will show that the QPS calculated in the following is consistent with physical restrictions and is intrinsic in the sense of its topological independence. The results of the present paper is agreed with the results of Ref.[@lidp]. Let us consider $N$ spinning anyons on an oriented compact Riemann surface with $g$ handles[@eina; @imbo; @bla]. To define the anyon’s spin, we attach an oriented local frame to every particle. When a particle moves on a curved surface (the torus is a flat surface), the attached frame is parallel transported and a path-dependent frame-rotation is associated with the particle transport. Let us denote the clockwise $2\pi$ rotation of the frame attached to the particle by $R_{2\pi}$. The action of the operator $R_{2\pi}$ on the wave function will give a phase $\exp (i2\pi S)$. We define $S$ as the spin of the particle. $S$ is equal to $1/2$ for the electron by this definition. The braiding operators are $\sigma_i ,\, \rho_{n,i} ,\, \tau_{n,i}$, where $\sigma_i$ interchanges (clockwise) particle $i$ and particle $i+1$ and $\rho_{n,i} ,\, \tau_{n,i}$ take particle $i$ around noncontractable loops on the $m^{,}th$ handle. Here we use the same definition of operators $\rho_{n,i} ,\, \tau_{n,i}$ as Ref. [@eina]. The braid relations for spinning anyons on the Riemann surface with $g$ handles are \[brarel\] $$\sigma_j\sigma_{j+1}\sigma_j =\sigma_{j+1}\sigma_j\sigma_{j+1} , \label{brarel:1}$$ $$\tau_{m,j+1}= \sigma^{-1}_j \tau_{m,j}\sigma_j , \rho_{m,j+1}=\sigma^{-1}_j\rho_{m,j}\sigma_j , \label{brarel:2}$$ $$\rho^{-1}_{m,j}\tau_{m,j+1}\sigma^{-2}_j\rho_{m,j} \sigma^2_j\tau^{-1}_{m,j+1}=\sigma^2_j , \label{brarel:3}$$ $$\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2 \sigma_1=R^{2(g-1)}_{2\pi} \prod^g_n\rho^{-1}_{n,1}\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1}. \label{relation}$$ For the spinless anyons on the sphere, Equation (\[relation\]) becomes $\sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2 \sigma_1=1 $, which had been derived in Ref. [@thou]. It expresses the fact that a closed (clockwise) loop of particle $1$ around all the other particles can be continuously shrunk to a point on the rear side of the sphere. Eq. (\[relation\]) is the generalization of the case of the spinless anyons on the sphere to the spinning anyons on general Riemann surfaces[@eina; @imbo]. When we deform the loop of the left side of Eq. (\[relation\]) to the loop of the right side of Eq. (\[relation\]), described by $\prod^g_n\rho^{-1}_{n,1}\tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1}$, the attached spin frame of is rotated ($4\pi (g-1)$ rotation), we obtain a phase $R^{2(g-1)}_{2\pi}$. We need to include this phase in the right side of the equation. However for the charged anyons in magnetic field, which is the case of quasiparticles in FQHE, the braid relation  (\[relation\]) should be changed. We also need to include the Aharonov-Bohm phase $\exp (2\pi iq\Phi )$ in the right of Eq. (\[relation\]) because the charged anyon interacts with the magnetic field, where $q$ is the anyon’s charge and $\Phi$ is the magnetic flux out of the surface. Thus in stead of Eq. (\[relation\]), for the charged anyons on the magnetic field, we have $$\begin{aligned} \sigma_1 \sigma_2 \cdots \sigma^2_{N-1}\cdots \sigma_2 \sigma_1 & = & \exp (2\pi iq\Phi ) R^{2(g-1)}_{2\pi} \nonumber \\ & \times & \prod^g_n\rho^{-1}_{n,1} \tau^{-1}_{n,1}\rho_{n,1}\tau_{n,1} . \label{relation1}\end{aligned}$$ We will only consider the Abelian fractional statistics, which means that the representation of operator $\sigma_i$ is given by $\sigma_i =\sigma =\exp{i\theta}{\bf 1}_M$, where ${\bf 1}_M$ is the $M \times M$ identity matrix. Inserting $\sigma_i =\sigma =\exp{i\theta}{\bf 1}_M$ in Eq. (\[brarel:2\]) and Eq. (\[brarel:3\]), one obtains that $\tau_{m,j}=\tau_m , \, \rho_{m,j}=\rho_m ,\, \tau_m \rho_m =\sigma^2\rho_m \tau_m $. These relations and Eq. (\[relation1\]) yield $$\begin{aligned} \exp [2i(N-1)\theta] & = & \exp [2\pi iq\Phi +4\pi(g-1)S] \nonumber \\ & \times & \exp (-2ig\theta) . \label{brarel3}\end{aligned}$$ If there are several kinds of anyons, we need to introduce mutual statistics[@mutual]. The mutual statistics $\theta_{i,j}$ means that when a particle of the $i^{,}{th}$ kind moves clockwise around a particle of the $j^{,}{th}$ kind, we get a phase $\exp (2\theta_{i,j})$. $\theta_{i,i}=\theta_i$ actually is the fractional statistics parameter of the particle of the $i^{,}{th}$ kind. The left hand of Eq. (\[brarel3\]) is a phase which is obtained by moving one particle (clockwise) around all other particles. If there exist other kinds of particles, instead of Eq. (\[brarel3\]), we have $$\begin{aligned} \exp [2i(N_i-1)\theta_i & + & 2i\sum_{j\not= i}^l N_j\theta_{i,j}] =\exp (-2ig\theta_i) \nonumber \\ & \times & \exp [2\pi iq_i\Phi +4\pi(g-1)S_i], \label{mainre}\end{aligned}$$ where there are $l$ different kinds of particles and the spin of the particle of the $i^{,}{th}$ kind is $S_i$ and the charge is $q_i$. Eq. (\[mainre\]) gives a constraint on the parameters (numbers, statistics and spin ) of anyons that $$[((N_i-1+g)\theta_i + \sum_{j\not= i}^l N_j\theta_{i,j}) / {\pi} ] -q_i\Phi - 2(g-1)S_i \label{const}$$ is an integer. Now we consider the FQHE on a surface with $g$ handles. We use the metric $ds^2=g_{z\bar z}dzd{\bar z}$ in complex coordinates. The volume form is $dv=[ig_{z\bar z} / 2]dz\wedge d{\bar z} = g_{z\bar z}dx\wedge dy$. As in the case of the FQHE on the disc, sphere and torus, we apply a constant magnetic field on the surface. The natural generalization of the constant magnetic field to high genus Riemann surfaces[@bolte] is $F=Bdv=(\partial_zA_{\bar z}- \partial_{\bar z}A_z)dz\wedge d{\bar z}$. Thus $ \partial_zA_{\bar z}- \partial_{\bar z}A_z =ig_{z\bar z}B / 2$. The flux $\Phi$ is given by $2\pi \Phi = \int F =BV$, where $V$ is the area of the surface and we assume here $B>0$ ($\Phi >0$). The Hamiltonian of an electron on the surface under the magnetic field is given by the Laplace-Beltrami operator, $$\begin{aligned} H & = & [1/ 2m \sqrt{g}] (P_{\mu}-A_{\mu})g^{\mu \nu}\sqrt{g}(P_{\nu}-A_{\nu}) \nonumber \\ & = & [ g^{z\bar z} / m] [(P_z-A_z)(P_{\bar z}-A_{\bar z}) \nonumber \\ & + & (P_{\bar z}-A_{\bar z})(P_z-A_z)] \\ \label{hamil} & = & [2g^{z\bar z}/ m] (P_z-A_z)(P_{\bar z}-A_{\bar z})+[B/ 2m] \nonumber\end{aligned}$$ where $g^{z\bar z}=[1 / g_{z\bar z}]$ and $P_{z}=-i\partial_z ,\, P_{\bar z}=-i\partial_{\bar z}$. The inner product of two wave functions is defined as $<\psi_1 | \psi_2 >=\int dv {\bar \psi_1 }\times \psi_2$. $H^{\prime}=[2g^{z\bar z} / m] (P_z-A_z)(P_{\bar z}-A_{\bar z})$ is a positive definite hermitian operator Because $<\psi | H^{\prime} |\psi > \, \geq 0$ for any $\psi$. Thus if $H^{\prime} \psi =0$, $\psi$ satisfies $(P_{\bar z}-A_{\bar z})\psi =0$. The solutions of this equation are the ground states of the Hamiltonian $H$ or $H^{\prime}$, or the lowest Landau levels (LLL). The existence of the solutions of this equation is guaranteed by Riemann-Roch theorem[@griff]. The solutions belong to the holomorphic line bundle under the gauge field. Riemann-Roch theorem tells us that $h^0(L)-h^1(L)=deg(L)-g+1$, where $h^0(L)$ is the dimension of the holomorphic line bundle or the degeneracy of the ground states of the Hamiltonian $H$, $h^1(L)$ is the dimension of the holomorphic differential and $deg(L)$ is the degree of the line bundle which is equal to the first Chern number of the gauge field, or the magnetic flux out of the surface, $\Phi$. Because $deg(L) >2g-2$ (the magnetic field is very strong in the FQHE), $h^1(L)$ is equal to zero [@griff] and $h^0(L)=\Phi-g+1$. As a consistent check, $h^0(L)$ indeed gives the right degeneracy of the ground states in the case of a particle on the sphere and torus interacting with a magnetic-monopole field. In the case of high genus surfaces, Ref. [@avron] had discussed the degeneracy of the LLL for the leaky tori (see also Ref. [@asory] for the related discussions). If the filling factor is $1 / m$ in the state of the FQHE [@laughlin], we suppose to have relation $m(N+\Delta_g)=\Phi$. We assume that $\Delta_g$ is independent of $m$, and this assumption will be justified by the explicit constructions of some examples of Laughlin wave functions on high genus surfaces [@iengoli]. If $m=1$, we have an integer quantum Hall state and expect that the LLL are completely filled. In this case, $N=\Phi-\Delta_g$, and $N$ is also the degeneracy of the LLL, which should be equal to $\Phi-g+1$ (according to the discussions above). Thus $\Delta_g$ is equal to $g-1$. The relation between the electron numbers and the flux for the FQHE at the filling factor as $1 / m$ is then $m(N+g-1)=\Phi$, which indeed gives correct results in the case of the Laughlin state on the sphere and torus [@haldane]. If $N_q$ quasiparticles are created, one has $m(N+\Delta_g)+N_q=\Phi$. The mutual statistics between the electron and quasiparticle is $2\theta_{mut}=2\pi$, the charge of the quasiparticle is $1 / m$ and the statistics parameters is $\pi / m$. We remark that, if $2\theta_{i,j}=2\pi, i\not= j$ in Eq. (\[const\]) or Eq. (\[mainre\]), we can simply omit the term $N_j\theta_{i,j}$. By applying Eq. (\[const\]) to those $N_q$ quasiparticles, one can show that $$[(N_q-1+g)/ m]+N- [\Phi / m] -2(g-1)S_q \label{spin1}$$ is an integer. From Eq. (\[spin1\]) and the equation $m(N+\Delta_g)+N_q=\Phi$, the QPS turns out to be $S_q=[1 / 2m]+[n / 2(g-1)]$ ($n$ is an integer and $g \not= 1$ is assumed). Let us discuss how to fix $n$. We consider a cluster of particles which contains $n_i$ particles of the $i^{,}{th}$ kind with the mutual statistics $\theta_{i,j}$. By using the method developed in Ref. [@thou], we get the statistics, spin and charge of the cluster, $$S_c = \sum_i [n_i(n_i-1)\theta_i / 2\pi ] +n_iS_i + \sum_{i \not= j} n_in_j\theta_{i,j} / 2\pi \label{culster:2}$$ and $q_c=\sum n_iq_i$ and $\theta_c=\sum_i n_i^2\theta_i+ \sum_{i \not= j}n_in_j\theta_{i,j}$ are the charge and statistics of the cluster respectively. If the cluster’s charge is an odd (even) integer and the cluster satisfies the fermionic (bosonic) statistics, we suppose that the cluster contains only an odd (even) number of electrons (for example, see Ref. [@frozee]), and thus the cluster’s spin is an half-integer (integer). If the cluster contains $m$ quasiparticles in the above example, the cluster’s charge is $1$ and its statistics of is fermionic. This cluster shall be the hole of the electron and the cluster’s spin is an half-integer (see also Ref. [@fro]). By using Eq. (\[culster:2\]) for this cluster, we get a restriction for the QPS, $[mn / 2(g-1)]=integer$. If $m$ and $g-1$ are coprime to each other (for example, $g$ is equal to $0,2,3$), $n$ must be equal to $2n^{\prime}(g-1)$ ($n^{\prime}$ is an integer). Thus $S_q$ is equal to $1 / 2m$ (up to an integer) and the spin-statistics relation is the standard one. However, when $m$ and $g-1$ are not coprime to each other for some high genus surfaces, there exist other solutions for the spin except $1 / 2m$. We write $m=kp$ and $g-1=kq$, where $p$ and $q$ are coprime to each other, we have solutions $n=2n^{\prime}q$. The other solutions of the QSP is $S_q=[1 / 2m]+[n^{\prime}/ k]$ where $n^{\prime}=1,\cdots , k-1$. Therefor $S_q$ can not be completely fixed by using the braid group analysis. However, we shall point out that it is highly unlikely that those [**other**]{} solutions are the [**true**]{} QPS, as we expect that the value of the spin shall be intrinsic and does not depend on the surface where quasiparticles live. To completely fix the QPS on Riemann surfaces, we can obtain the QPS by analyzing the wave function of the quasiparticles on Riemann surfaces (we plan to do so elsewhere), as it was done for the case on the sphere[@lidp]. Let us calculate the QPS in the standard hierarchical state[@halp; @lidp; @haldane; @blok; @read; @gread; @lidpt]. We remark that the above method can be used to calculate the QPS in other kinds of quantum Hall fluids, for example, the multi-layered FQHE or Jain state[@jain] etc.. The hierarchical state is described by a symmetric matrix $\Lambda_{i,j}, i,j=1,2,\cdots , l$, where $\Lambda_{i,i+1}=\Lambda_{i+1,i}=\pm 1$, $\Lambda_{1,1}$ is an odd integer and $\Lambda_{i,i}, i\not= 1$ are even integers, where $l$ is the level of the hierarchical state and $N_i$ is the number of the particles in level $i$ ($N_1$ is the number of the electrons, $N_2$ is the number of the condensed quasiparticles (or holes) of the first level (Laughlin) state, etc.). On the torus[@haldane; @lidpt], we have a relation, $\sum_j \Lambda_{i,j}N_j=\delta_{i,1}\Phi$, and on the sphere[@lidp; @haldane], the relation is $\sum_j \Lambda_{i,j}N_j -\Lambda_{i,i}=\delta_{i,1}\Phi$. Following the discussion about the Laughlin state on Riemann surfaces, we expect that the relation is $\sum_j \Lambda_{i,j}N_j + (g-1) \Lambda_{i,i}= \delta_{i,1}\Phi$ for the hierarchical state on Riemann surfaces. We define a $l$ dimension integer lattice with bases $E_i$ and the inner products $E_i\cdot E_j =\Lambda_{i,j}$ (see Ref. [@lidpt]) The above equation can be rewritten as $$\sum_{i=1}^{l} N_iE_i + (g-1) (E_i\cdot E_i)E_i^{\star}=E_1^{\star}\Phi , \label{hierar:1}$$ where $E_i^{\star}$ are the bases of the inverse lattice $E_i$ and defined by $E_i^{\star}\cdot E_j =\delta_{i,j}$. It can be verified that $E_i^{\star}\cdot E_j^{\star} =\Lambda_{i,j}^{-1}$. The quasiparticle is described by a vector ${\cal Q}_k =k_iE_i^{\star}$ ($k_i$ is an integer) on the lattice $E_i^{\star}$ (see Refs. [@blok; @read]). The statistics parameter of this quasiparticle is $\theta_k = {\cal Q}_k\cdot {\cal Q}_k \pi=\sum_{i,j}k_i\Lambda_{i,j}^{-1}k_j$ and the charge is $Q_k={\cal Q}_k \cdot E_1^{\star}=\sum_ik_i \Lambda_{i,1}^{-1}$. The mutual statistics between the quasiparticles ${\cal Q}_{k}$ and ${\cal Q}_{k^{\prime}}$ is $\theta_{k,k^{\prime}}= {\cal Q}_{k}\cdot {\cal Q}_{k^{\prime}}\pi =\sum_{i,j}k_i\Lambda_{i,j}^{-1}k^{\prime}_j$. If $N$ quasiparticles denoted by the vector ${\cal Q}_k$, is created, one has $\sum_{i=1}^{l} N_iE_i + (g-1) (E_i\cdot E_i)E_i^{\star}+ N {\cal Q}_k=E_1^{\star}\Phi$. Making inner product on two sides of this equation with ${\cal Q}_k$, one yields $${\frac {N\theta_k}{\pi}}+(g-1)\Delta \cdot {\cal Q}_k- Q_k \Phi =integer, \label{delt}$$ where $\Delta = \sum_{i=1}^{l}(E_i \cdot E_i)E_i^{\star}$. Applying Eq. (\[const\]) to those quasiparticles and comparing it with Eq. (\[delt\]), one gets $$S_k={\theta_k \over 2\pi}-{\frac {\Delta \cdot {\cal Q}_k} {2}} +{\frac {n}{2(g-1)}}, \, n=integer.\, g\not= 1. \label{result}$$ By using the argument in Ref. [@thou] (which we did for the quasiparticle of the Laughlin state), we can fix $n$ in some cases. The charge of the quasiparticle $E_i$ (denoted by the vector $E_i$) is $\delta_{i,1}$ and the statistics is $\theta = \delta_{i,1}\pi$. Thus the spin of this quasiparticle is ${\frac {\delta_{i,1}} {2}}+integer$. Because $E_i=\sum_j \Lambda_{i,j}E_j^{\star}$, the quasiparticle $E_i$ is a cluster which contains $\Lambda_{i,j}$ quasiparticles with the vector as $E_j^{\star}$. If $\det{\Lambda}$ and $g-1$ are coprime to each other, by using Eq. (\[culster:2\]) for the cluster, we find that $S_i={\frac {E_i \cdot E_i} {2}}- {\frac {\Delta \cdot E_i} {2}}-{1\over 2}$ for the quasiparticle $E_j^{\star}$ when $l-i$ is an even integer and $S_i={\frac {E_i \cdot E_i} {2}}- {\frac {\Delta \cdot E_i} {2}}$ when $l-i$ is an odd integer. We will use notation $i \in i_{e}$ if $l-i$ is an even integer. Generally, for the quasiparticle ${\cal Q}$, we find that $$S_{\cal Q}={\frac { {\cal Q}\cdot {\cal Q}} {2}}- {\frac {\Delta \cdot {\cal Q}} {2}}-{1\over 2}\Delta^{\prime} \cdot {\cal Q}, \label{result:1}$$ where $\Delta^{\prime}=\sum_{i} E_i, i\in i_e$. If the quasiparticles have the standard spin-statistics relation, it is required that ${\frac {\Delta \cdot {\cal Q}} {2}} +{1\over 2}\Delta^{\prime} \cdot {\cal Q}$ is an integer. Indeed, this number is always an integer for the Laughlin state. However, in the hierarchical state, this number may not be an integer. Thus the quasiparticles in the hierarchical state usually do not have the standard spin-statistics relation. If $\det{\Lambda}$ and $g-1$ are not coprime to each other, there exist other solutions, not only Eq. (\[result:1\]). As we argued in the case of the Laughlin state, these [**other**]{} solutions is [**unlikely**]{} the [**true**]{} QPS. Thus we suppose that Eq. (\[result:1\]) is the spin for quasiparticles and it is topological independent. The above method does not give any information about the QPS in the FQHE on the torus ($g=1$). However, the above discussion strongly suggests that the QPS in the FQHE on the torus is also given by Eq. (\[result:1\]) which suppose to be the QPS on any Riemann surfaces. The Lagrangian for the long-distance physics of the Hall fluid on Riemann surfaces is [@wen], $$\begin{aligned} {\cal L}& = & {1\over 4\pi}(\alpha_{\mu , i}\Lambda_{i,j}\epsilon^{\mu \nu \lambda}\partial_{\nu}\alpha_{\lambda , j} + 2A_{\mu}t_i\epsilon^{\mu \nu \lambda} \partial_{\nu}\alpha_{\lambda , i} \nonumber \\ & + & 2\omega s_i\epsilon^{0 \nu \lambda} \partial_{\nu}\alpha_{\lambda , i}),\end{aligned}$$ where $\omega$ is the connection one form (the curvature is given by $R=d\omega$). In the case of the hierarchical state, $t_i$ is equal to $\delta_{i,1}$ and the matrix $\Lambda$ is one we gave in the previous discussion. By using Eq. (\[hierar:1\]), we can show that $s_i=\Lambda_{i,i}$. So $s_i$ is a topological independent constant. It is reasonable to believe that $s_i$ is a topological independent constant for any kinds of quantum Hall fluids. Due to the presence of the third term in the Lagrangian, the spin-statistics relation usually is not generalized one [@wen]. 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--- abstract: 'While internal space-time symmetries of relativistic particles are dictated by the little groups of the Poincaré group, it is possible to construct representations of the little group for massive particles starting from harmonic oscillator wave functions for the quark model. The resulting oscillator wave functions are covariant and can be Lorentz-boosted. It is thus possible to obtain the parton model by boosting the quark model. A review of Wigner’s theory of the little groups is given. It is shown that the covariant oscillator wave functions become squeezed as as the system becomes boosted. It is shown also that the Lorentz-squeezed quark distribution exhibits the peculiarities of Feynman’s parton model including the lack of coherence in the calculation of cross sections. A historical review of the concept of covariance is given.' address: 'Department of Physics, University of Maryland, College Park, Maryland 20742' author: - 'Y. S. Kim' title: Covariant Model of Relativistic Extended Particles based on the Oscillator Representation of the Poincaré Group --- Introduction {#intro} ============ The present form of quantum mechanics works well in atomic systems where electrons are bound by the Coulomb force from the nucleus. Quantum mechanics works also for atomic nuclei where the nucleus is a bound state of nucleons even though it is difficult to perform exact calculations. In both atomic and nuclear physics, nucleons are regarded as point particles. However, it was found by Hofstadter in 1995 [@hofsta55] that the proton’s charge is not concentrated on a point, and its charge distribution has a non-zero space-time extension. This appears in the elastic scattering of electrons by a proton. The observed scattering cross sections deviate from those predicted by the Rutherford scattering formula based on the proton with a point charge. This deviation is commonly called the form factor. In spite of many laudable efforts to explain this form factor within the framework of quantum field theory, the workable model for the form factors did not emerge until after Gell-Mann’s formulation of the quark model, in which all hadrons are bound states of quarks and/or anti-quarks [@gell64]. The question then is whether we can use the existing models of quantum mechanics, such as the nuclear shell model, to explain hadronic mass spectra [@owg64]. For the mass spectra, one of the most effective models has been and still is the model based on harmonic oscillator wave functions [@owg64; @fkr71]. The basic advantage of the oscillator model is that its mathematics is transparent, and it does not bury physics in mathematics even though it does not always produce the most accurate numerical results. In the quark model, the charge distribution within the proton comes from the distribution of the charged particles inside the hadron. The success of the oscillator model for static or slow-moving hadrons does not necessarily mean that the model can be extended to the relativistic regime. Indeed, the calculation of the form factor with Gaussian wave functions results in an exponential decrease for large momentum-transfer variables. However, this wrong behavior comes from the use of non-relativistic wave functions for relativistic problems. Indeed, Feynman [*et al.*]{} made an attempt to construct a covariant oscillator model [@fkr71]. Even though they did not achieve their goal in their paper, Feynman [*et al.*]{} quote the work of Fujimura [*et al.*]{} [@fuji70] who calculated the nucleon form factor by taking into account the effect of the Lorentz-squeeze on the oscillator wave functions. After studying these original papers, we can raise our level of abstraction. We observe first that the spherical harmonics can represent the three-dimensional rotation group, while serving as wave functions for the angular variables. Then, we can ask whether there are wave functions which can represent the Poincaré group. We can specifically ask whether it is possible to construct a set of normalizable harmonic oscillator wave functions to represent the Poincaré group. If YES, the wave functions can be Lorentz-boosted. These wave functions then have to go through another set of tests. Are they consistent with the existing laws of quantum mechanics. If YES, they then have to be exposed to the most cruel test in physics. Do they explain what we observe in high-energy laboratories? The purpose of this paper is to show that we can use the oscillator wave functions to answer the question of whether quarks are partons. While the quark model is valid for static hadrons, Feynman’s parton picture works only in the Lorentz frame where the hadronic speed is close to that of light [@fey69]. The quark model appears to be quite different from the parton model. On the other hand, they are valid in two different Lorentz frames. The basic question is whether the quark picture and the parton picture are two different manifestations of the same covariant entity. In this paper, we shall discuss first the internal space-time symmetries of relativistic particles in terms of appropriate representations of the Poincaré group [@wig39]. We then construct the oscillator wave functions satisfying the above-mentioned theoretical criterions. This oscillator formalism will explains both the quark and the parton pictures in two separate Lorentz frames. This formalism produces all the peculiarities of Feynman’s original form of the parton picture including the incoherence of parton cross sections. In Sec. \[littleg\], we present a brief history of applications of the little groups to internal space-time symmetries of relativistic particles. In Sec. \[covham\], we construct representations of the little group using harmonic oscillator wave functions. In Sec. \[parton\], it is shown that the Lorentz-boosted oscillator wave functions exhibit the peculiarities Feynman’s parton model in the infinite-momentum limit. Much of the concept of Lorentz-squeezed wave function is derived from elliptic deformations of a sphere resulting in a mathematical technique group called contractions [@inonu53]. In Appendix \[o3e2\], we discuss the contraction of the three-dimensional rotation group to the two-dimensional Euclidean group. In Appendix \[contrac\], we discuss the little group for a massless particle as the infinite-momentum/zero-mass limit of the little group for a massive particle. In Appendix \[kant\], the author gives his confession about his educational and cultural backgrounds which led to the research program outlined in this paper. Little Groups of the Poincaré Group {#littleg} =================================== The Poincaré group is the group of inhomogeneous Lorentz transformations, namely Lorentz transformations preceded or followed by space-time translations. In order to study this group, we have to understand first the group of Lorentz transformations, the group of translations, and how these two groups are combined to form the Poincaré group. The Poincaré group is a semi-direct product of the Lorentz and translation groups. The two Casimir operators of this group correspond to the (mass)$^{2}$ and (spin)$^{2}$ of a given particle. Indeed, the particle mass and its spin magnitude are Lorentz-invariant quantities. The question then is how to construct the representations of the Lorentz group which are relevant to physics. For this purpose, Wigner in 1939 studied the subgroups of the Lorentz group whose transformations leave the four-momentum of a given free particle [@wig39]. The maximal subgroup of the Lorentz group which leaves the four-momentum invariant is called the little group. Since the little group leaves the four-momentum invariant, it governs the internal space-time symmetries of relativistic particles. Wigner shows in his paper that the internal space-time symmetries of massive and massless particles are dictated by the $O(3)$-like and $E(2)$-like little groups respectively. The $O(3)$-like little group is locally isomorphic to the three-dimensional rotation group, which is very familiar to us. For instance, the group $SU(2)$ for the electron spin is an $O(3)$-like little group. The group $E(2)$ is the Euclidean group in a two-dimensional space, consisting of translations and rotations on a flat surface. We are performing these transformations everyday on ourselves when we move from home to school. The mathematics of these Euclidean transformations are also simple. However, the group of these transformations are not well known to us. In Appendix \[o3e2\], we give a matrix representation of the $E(2)$ group. The group of Lorentz transformations consists of three boosts and three rotations. The rotations therefore constitute a subgroup of the Lorentz group. If a massive particle is at rest, its four-momentum is invariant under rotations. Thus the little group for a massive particle at rest is the three-dimensional rotation group. Then what is affected by the rotation? The answer to this question is very simple. The particle in general has its spin. The spin orientation is going to be affected by the rotation! If the rest-particle is boosted along the $z$ direction, it will pick up a non-zero momentum component. The generators of the $O(3)$ group will then be boosted. The boost will take the form of conjugation by the boost operator. This boost will not change the Lie algebra of the rotation group, and the boosted little group will still leave the boosted four-momentum invariant. We call this the $O(3)$-like little group. If we use the four-vector coordinate $(x, y, z, t)$, the four-momentum vector for the particle at rest is $(0, 0, 0, m)$, and the three-dimensional rotation group leaves this four-momentum invariant. This little group is generated by $$J_{1} = \pmatrix{0&0&0&0\cr0&0&-i&0\cr0&i&0&0\cr0&0&0&0} , \qquad J_{2} = \pmatrix{0&0&i&0\cr0&0&0&0\cr-i&0&0&0\cr0&0&0&0} ,$$ and $$\label{j3} J_{3} = \pmatrix{0 & -i & 0 & 0 \cr i & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0} ,$$ which satisfy the commutation relations: $$[J_{i}, J_{j}] = i\epsilon_{ijk} J_{k} .$$ It is not possible to bring a massless particle to its rest frame. In his 1939 paper [@wig39], Wigner observed that the little group for a massless particle moving along the $z$ axis is generated by the rotation generator around the $z$ axis, namely $J_{3}$ of Eq.(\[j3\]), and two other generators which take the form $$\label{n1n2} N_{1} = \pmatrix{0 & 0 & -i & i \cr 0 & 0 & 0 & 0 \cr i & 0 & 0 & 0 \cr i & 0 & 0 & 0} , \quad N_{2} = \pmatrix{0 & 0 & 0 & 0 \cr 0 & 0 & -i & i \cr 0 & i & 0 & 0 \cr 0 & i & 0 & 0} .$$ If we use $K_{i}$ for the boost generator along the i-th axis, these matrices can be written as $$N_{1} = K_{1} - J_{2} , \qquad N_{2} = K_{2} + J_{1} ,$$ with $$K_{1} = \pmatrix{0&0&0&i\cr0&0&0&0\cr0&0&0&0\cr i&0&0&0} , \qquad K_{2} = \pmatrix{0&0&0&0\cr0&0&0&i\cr0&0&0&0\cr0&i&0&0} .$$ The generators $J_{3}, N_{1}$ and $N_{2}$ satisfy the following set of commutation relations. $$\label{e2lcom} [N_{1}, N_{2}] = 0 , \quad [J_{3}, N_{1}] = iN_{2} , \quad [J_{3}, N_{2}] = -iN_{1} .$$ In Appendix \[o3e2\], we discuss the generators of the $E(2)$ group. They are $J_{3}$ which generates rotations around the $z$ axis, and $P_{1}$ and $P_{2}$ which generate translations along the $x$ and $y$ directions respectively. If we replace $N_{1}$ and $N_{2}$ by $P_{1}$ and $P_{2}$, the above set of commutation relations becomes the set given for the $E(2)$ group given in Eq.(\[e2com\]). This is the reason why we say the little group for massless particles is $E(2)$-like. Very clearly, the matrices $N_{1}$ and $N_{2}$ generate Lorentz transformations. It is not difficult to associate the rotation generator $J_{3}$ with the helicity degree of freedom of the massless particle. Then what physical variable is associated with the $N_{1}$ and $N_{2}$ generators? Indeed, Wigner was the one who discovered the existence of these generators, but did not give any physical interpretation to these translation-like generators. For this reason, for many years, only those representations with the zero-eigenvalues of the $N$ operators were thought to be physically meaningful representations [@wein64]. It was not until 1971 when Janner and Janssen reported that the transformations generated by these operators are gauge transformations [@janner71; @kim97poz]. The role of this translation-like transformation has also been studied for spin-1/2 particles, and it was concluded that the polarization of neutrinos is due to gauge invariance [@hks82; @kim97min]. --------------------- ---------------- ------------------------------ ----------------------- Massive, Slow COVARIANCE Massless, Fast \[4mm\] Energy- Einstein’s Momentum $E = p^{2}/2m$ $ E = [p^{2} + m^{2}]^{1/2}$ $E = cp$ \[4mm\] Internal $S_{3}$ $S_{3}$ \[-1mm\] space-time Wigner’s \[-1mm\] symmetry $S_{1}, S_{2}$ Little Group Gauge Transformations \[4mm\] Relativistic \[-1mm\] Extended Quark Model Covariant Model of Hadrons Partons \[-1mm\] Particles \[2mm\] --------------------- ---------------- ------------------------------ ----------------------- : Further contents of Einstein’s $E = mc^{2}$. Massive and massless particles have different energy-momentum relations. Einstein’s special relativity gives one relation for both. Wigner’s little group unifies the internal space-time symmetries for massive and massless particles which are locally isomorphic to $O(3)$ and $E(2)$ respectively. It is a great challenge for us to find another unification. In this note, we present a unified picture of the quark and parton models which are applicable to slow and ultra-fast hadrons respectively. Another important development along this line of research is the application of group contractions to the unifications of the two different little groups for massive and massless particles. We always associate the three-dimensional rotation group with a spherical surface. Let us consider a circular area of radius 1 kilometer centered on the north pole of the earth. Since the radius of the earth is more than 6,450 times longer, the circular region appears flat. Thus, within this region, we use the $E(2)$ symmetry group for this region. The validity of this approximation depends on the ratio of the two radii. In 1953, Inonu and Wigner formulated this problem as the contraction of $O(3)$ to $E(2)$ [@inonu53]. How about then the little groups which are isomorphic to $O(3)$ and $E(2)$? It is reasonable to expect that the $E(2)$-like little group be obtained as a limiting case for of the $O(3)$-like little group for massless particles. In 1981, it was observed by Ferrara and Savoy that this limiting process is the Lorentz boost [@ferrara82]. In 1983, using the same limiting process as that of Ferrara and Savoy, Han [*et al*]{} showed that transverse rotation generators become the generators of gauge transformations in the limit of infinite momentum and/or zero mass [@hks83pl]. In 1987, Kim and Wigner showed that the little group for massless particles is the cylindrical group which is isomorphic to the $E(2)$ group [@kiwi87jm]. This completes the second raw in Table I, where Wigner’s little group unifies the internal space-time symmetries of massive and massless particles. We are now interested in constructing the third row in Table I. As we promised in Sec. \[intro\], we will be dealing with hadrons which are bound states of quarks with space-time extensions. For this purpose, we need a set of covariant wave functions consistent with the existing laws of quantum mechanics, including of course the uncertainty principle and probability interpretation. With these wave functions, we propose to solve the following problem in high-energy physics. The quark model works well when hadrons are at rest or move slowly. However, when they move with speed close to that of light, they appear as a collection of infinite-number of partons [@fey69]. As we stated above, we need a set of wave functions which can be Lorentz-boosted. How can we then construct such a set? In constructing wave functions for any purpose in quantum mechanics, the standard procedure is to try first harmonic oscillator wave functions. In studying the Lorentz boost, the standard language is the Lorentz group. Thus the first step to construct covariant wave functions is to work out representations of the Lorentz group using harmonic oscillators [@dir45; @yuka53; @knp86]. Covariant Harmonic Oscillators {#covham} ============================== If we construct a representation of the Lorentz group using normalizable harmonic oscillator wave functions, the result is the covariant harmonic oscillator formalism [@knp86]. The formalism constitutes a representation of Wigner’s $O(3)$-like little group for a massive particle with internal space-time structure. This oscillator formalism has been shown to be effective in explaining the basic phenomenological features of relativistic extended hadrons observed in high-energy laboratories. In particular, the formalism shows that the quark model and Feynman’s parton picture are two different manifestations of one covariant entity [@knp86; @kim89]. The essential feature of the covariant harmonic oscillator formalism is that Lorentz boosts are squeeze transformations [@kn73; @knp91]. In the light-cone coordinate system, the boost transformation expands one coordinate while contracting the other so as to preserve the product of these two coordinate remains constant. We shall show that the parton picture emerges from this squeeze effect. Let us consider a bound state of two particles. For convenience, we shall call the bound state the hadron, and call its constituents quarks. Then there is a Bohr-like radius measuring the space-like separation between the quarks. There is also a time-like separation between the quarks, and this variable becomes mixed with the longitudinal spatial separation as the hadron moves with a relativistic speed. There are no quantum excitations along the time-like direction. On the other hand, there is the time-energy uncertainty relation which allows quantum transitions. It is possible to accommodate these aspect within the framework of the present form of quantum mechanics. The uncertainty relation between the time and energy variables is the c-number relation [@dir27], which does not allow excitations along the time-like coordinate. We shall see that the covariant harmonic oscillator formalism accommodates this narrow window in the present form of quantum mechanics. For a hadron consisting of two quarks, we can consider their space-time positions $x_{a}$ and $x_{b}$, and use the variables $$X = (x_{a} + x_{b})/2 , \qquad x = (x_{a} - x_{b})/2\sqrt{2} .$$ The four-vector $X$ specifies where the hadron is located in space and time, while the variable $x$ measures the space-time separation between the quarks. In the convention of Feynman [*et al.*]{} [@fkr71], the internal motion of the quarks bound by a harmonic oscillator potential of unit strength can be described by the Lorentz-invariant equation $$\label{osceq} {1\over 2}\left\{x^{2}_{\mu} - {\partial ^{2} \over \partial x_{\mu }^{2}} \right\} \psi (x)= \lambda \psi (x) .$$ It is now possible to construct a representation of the Poincaré group from the solutions of the above differential equation [@knp86]. The coordinate $X$ is associated with the overall hadronic four-momentum, and the space-time separation variable $x$ dictates the internal space-time symmetry or the $O(3)$-like little group. Thus, we should construct the representation of the little group from the solutions of the differential equation in Eq.(\[osceq\]). If the hadron is at rest, we can separate the $t$ variable from the equation. For this variable we can assign the ground-state wave function to accommodate the c-number time-energy uncertainty relation [@dir27]. For the three space-like variables, we can solve the oscillator equation in the spherical coordinate system with usual orbital and radial excitations. This will indeed constitute a representation of the $O(3)$-like little group for each value of the mass. The solution should take the form $$\psi (x,y,z,t) = \psi (x,y,z) \left({1\over \pi }\right)^{1/4} \exp \left(-t^{2}/2 \right) ,$$ where $\psi(x,y,z)$ is the wave function for the three-dimensional oscillator with appropriate angular momentum quantum numbers. Indeed, the above wave function constitutes a representation of Wigner’s $O(3)$-like little group for a massive particle [@knp86]. Since the three-dimensional oscillator differential equation is separable in both spherical and Cartesian coordinate systems, $\psi(x,y,z)$ consists of Hermite polynomials of $x, y$, and $z$. If the Lorentz boost is made along the $z$ direction, the $x$ and $y$ coordinates are not affected, and can be temporarily dropped from the wave function. The wave function of interest can be written as $$\psi^{n}(z,t) = \pmatrix{{1\over \pi }}^{1/4}\exp \pmatrix{-t^{2}/2} \psi_{n}(z) ,$$ with $$\psi ^{n}(z) = \left({1 \over \pi n!2^{n}} \right)^{1/2} H_{n}(z) \exp (-z^{2}/2) ,$$ where $\psi ^{n}(z)$ is for the $n$-th excited oscillator state. The full wave function $\psi ^{n}(z,t)$ is $$\label{2.6} \psi ^{n}_{0}(z,t) = \left({1\over \pi n! 2^{n}}\right)^{1/2} H_{n}(z) \exp \left\{-{1\over 2}\left(z^{2} + t^{2} \right) \right\} .$$ The subscript $0$ means that the wave function is for the hadron at rest. The above expression is not Lorentz-invariant, and its localization undergoes a Lorentz squeeze as the hadron moves along the $z$ direction [@knp86]. It is convenient to use the light-cone variables to describe Lorentz boosts. The light-cone coordinate variables are $$u = (z + t)/\sqrt{2} , \qquad v = (z - t)/\sqrt{2} .$$ In terms of these variables, the Lorentz boost along the $z$ direction, $$\pmatrix{z' \cr t'} = \pmatrix{\cosh \eta & \sinh \eta \cr \sinh \eta & \cosh \eta } \pmatrix{z \cr t} ,$$ takes the simple form $$\label{lorensq} u' = e^{\eta } u , \qquad v' = e^{-\eta } v ,$$ where $\eta $ is the boost parameter and is $\tanh ^{-1}(v/c)$. Indeed, the $u$ variable becomes expanded while the $v$ variable becomes contracted. This is the squeeze mechanism illustrated discussed extensively in the literature [@kn73; @knp91]. This squeeze transformation is also illustrated in Fig. 1. The wave function of Eq.(\[2.6\]) can be written as $$\label{10} \psi ^{n}_{o}(z,t) = \psi ^{n}_{0}(z,t) = \left({1 \over \pi n!2^{n}} \right)^{1/2} H_{n}\left((u + v)/\sqrt{2} \right) \exp \left\{-{1\over 2} (u^{2} + v^{2}) \right\} .$$ If the system is boosted, the wave function becomes $$\label{11} \psi ^{n}_{\eta }(z,t) = \left({1 \over \pi n!2^{n}} \right)^{1/2} H_{n} \left((e^{-\eta }u + e^{\eta }v)/\sqrt{2} \right) \times \exp \left\{-{1\over 2}\left(e^{-2\eta }u^{2} + e^{2\eta }v^{2}\right)\right\} .$$ In both Eqs. (\[10\]) and (\[11\]), the localization property of the wave function in the $u v$ plane is determined by the Gaussian factor, and it is sufficient to study the ground state only for the essential feature of the boundary condition. The wave functions in Eq.(\[10\]) and Eq.(\[11\]) then respectively become $$\label{13} \psi _{0}(z,t) = \left({1 \over \pi} \right)^{1/2} \exp \left\{-{1\over 2} (u^{2} + v^{2}) \right\} .$$ If the system is boosted, the wave function becomes $$\label{14} \psi _{\eta }(z,t) = \left({1 \over \pi }\right)^{1/2} \exp \left\{-{1\over 2}\left(e^{-2\eta }u^{2} + e^{2\eta }v^{2}\right)\right\} .$$ We note here that the transition from Eq.(\[13\]) to Eq.(\[14\]) is a squeeze transformation. The wave function of Eq.(\[13\]) is distributed within a circular region in the $u v$ plane, and thus in the $z t$ plane. On the other hand, the wave function of Eq.(\[14\]) is distributed in an elliptic region. This ellipse is a “squeezed” circle with the same area as the circle, as is illustrated in Fig. 1. Feynman’s Parton Picture {#parton} ======================== It is safe to believe that hadrons are quantum bound states of quarks having localized probability distribution. As in all bound-state cases, this localization condition is responsible for the existence of discrete mass spectra. The most convincing evidence for this bound-state picture is the hadronic mass spectra which are observed in high-energy laboratories [@fkr71; @knp86]. However, this picture of bound states is applicable only to observers in the Lorentz frame in which the hadron is at rest. How would the hadrons appear to observers in other Lorentz frames? More specifically, can we use the picture of Lorentz-squeezed hadrons discussed in Sec. \[covham\]. Proton’s radius is $10^{-5}$ of that of the hydrogen atom. Therefore, it is not unnatural to assume that the proton has a point charge in atomic physics. However, while carrying out experiments on electron scattering from proton targets, Hofstadter in 1955 observed that the proton charge is spread out. In this experiment, an electron emits a virtual photon, which then interacts with the proton. If the proton consists of quarks distributed within a finite space-time region, the virtual photon will interact with quarks which carry fractional charges. The scattering amplitude will depend on the way in which quarks are distributed within the proton. The portion of the scattering amplitude which describes the interaction between the virtual photon and the proton is called the form factor. Although there have been many attempts to explain this phenomenon within the framework of quantum field theory, it is quite natural to expect that the wave function in the quark model will describe the charge distribution. In high-energy experiments, we are dealing with the situation in which the momentum transfer in the scattering process is large. Indeed, the Lorentz-squeezed wave functions lead to the correct behavior of the hadronic form factor for large values of the momentum transfer [@fuji70]. While the form factor is the quantity which can be extracted from the elastic scattering, it is important to realize that in high-energy processes, many particles are produced in the final state. They are called inelastic processes. While the elastic process is described by the total energy and momentum transfer in the center-of-mass coordinate system, there is, in addition, the energy transfer in inelastic scattering. Therefore, we would expect that the scattering cross section would depend on the energy, momentum transfer, and energy transfer. However, one prominent feature in inelastic scattering is that the cross section remains nearly constant for a fixed value of the momentum-transfer/energy-transfer ratio. This phenomenon is called “scaling” [@bj69]. In order to explain the scaling behavior in inelastic scattering, Feynman in 1969 observed that a fast-moving hadron can be regarded as a collection of many “partons” whose properties do not appear to be identical to those of quarks [@fey69]. For example, the number of quarks inside a static proton is three, while the number of partons in a rapidly moving proton appears to be infinite. The question then is how the proton looking like a bound state of quarks to one observer can appear different to an observer in a different Lorentz frame? Feynman made the following systematic observations. a). The picture is valid only for hadrons moving with velocity close to that of light. b). The interaction time between the quarks becomes dilated, and partons\ behave as free independent particles. c). The momentum distribution of partons becomes widespread as the hadron\ moves fast. d). The number of partons seems to be infinite or much larger than that of quarks. Because the hadron is believed to be a bound state of two or three quarks, each of the above phenomena appears as a paradox, particularly b) and c) together. We would like to resolve this paradox using the covariant harmonic oscillator formalism. For this purpose, we need a momentum-energy wave function. If the quarks have the four-momenta $p_{a}$ and $p_{b}$, we can construct two independent four-momentum variables [@fkr71] $$P = p_{a} + p_{b} , \qquad q = \sqrt{2}(p_{a} - p_{b}) .$$ The four-momentum $P$ is the total four-momentum and is thus the hadronic four-momentum. $q$ measures the four-momentum separation between the quarks. We expect to get the momentum-energy wave function by taking the Fourier transformation of Eq.(\[14\]): $$\label{fourier} \phi_{\eta }(q_{z},q_{0}) = \left({1 \over 2\pi }\right) \int \psi_{\eta}(z, t) \exp{\left\{-i(q_{z}z - q_{0}t)\right\}} dx dt .$$ Let us now define the momentum-energy variables in the light-cone coordinate system as $$\label{conju} q_{u} = (q_{0} - q_{z})/\sqrt{2} , \qquad q_{v} = (q_{0} + q_{z})/\sqrt{2} .$$ In terms of these variables, the Fourier transformation of Eq.(\[fourier\]) can be written as $$\label{fourier2} \phi_{\eta }(q_{z},q_{0}) = \left({1 \over 2\pi }\right) \int \psi_{\eta}(z, t) \exp{\left\{-i(q_{u} u + q_{v} v)\right\}} du dv .$$ The resulting momentum-energy wave function is $$\label{phi} \phi_{\eta }(q_{z},q_{0}) = \left({1 \over \pi }\right)^{1/2} \exp\left\{-{1\over 2}\left(e^{-2\eta}q_{u}^{2} + e^{2\eta}q_{v}^{2}\right)\right\} .$$ Because we are using here the harmonic oscillator, the mathematical form of the above momentum-energy wave function is identical to that of the space-time wave function. The Lorentz squeeze properties of these wave functions are also the same, as are indicated in Fig. 2. When the hadron is at rest with $\eta = 0$, both wave functions behave like those for the static bound state of quarks. As $\eta$ increases, the wave functions become continuously squeezed until they become concentrated along their respective positive light-cone axes. Let us look at the z-axis projection of the space-time wave function. Indeed, the width of the quark distribution increases as the hadronic speed approaches that of the speed of light. The position of each quark appears widespread to the observer in the laboratory frame, and the quarks appear like free particles. Furthermore, interaction time of the quarks among themselves become dilated. Because the wave function becomes wide-spread, the distance between one end of the harmonic oscillator well and the other end increases as is indicated in Fig. 2. This effect, first noted by Feynman [@fey69], is universally observed in high-energy hadronic experiments. The period is oscillation is increases like $e^{\eta}$. On the other hand, the interaction time with the external signal, since it is moving in the direction opposite to the direction of the hadron, it travels along the negative light-cone axis. If the hadron contracts along the negative light-cone axis, the interaction time decreases by $e^{-\eta}$. The ratio of the interaction time to the oscillator period becomes $e^{-2\eta}$. The energy of each proton coming out of the Fermilab accelerator is $900 GeV$. This leads the ratio to $10^{-6}$. This is indeed a small number. The external signal is not able to sense the interaction of the quarks among themselves inside the hadron. The momentum-energy wave function is just like the space-time wave function. The longitudinal momentum distribution becomes wide-spread as the hadronic speed approaches the velocity of light. This is in contradiction with our expectation from nonrelativistic quantum mechanics that the width of the momentum distribution is inversely proportional to that of the position wave function. Our expectation is that if the quarks are free, they must have their sharply defined momenta, not a wide-spread distribution. This apparent contradiction presents to us the following two fundamental questions: a). If both the spatial and momentum distributions become widespread as the hadron moves, and if we insist on Heisenberg’s uncertainty relation, is Planck’s constant dependent on the hadronic velocity? b). Is this apparent contradiction related to another apparent contradiction that the number of partons is infinite while there are only two or three quarks inside the hadron? The answer to the first question is “No”, and that for the second question is “Yes”. Let us answer the first question which is related to the Lorentz invariance of Planck’s constant. If we take the product of the width of the longitudinal momentum distribution and that of the spatial distribution, we end up with the relation $$<z^{2}><q_{z}^{2}> = (1/4)[\cosh(2\eta)]^{2} .$$ The right-hand side increases as the velocity parameter increases. This could lead us to an erroneous conclusion that Planck’s constant becomes dependent on velocity. This is not correct, because the longitudinal momentum variable $q_{z}$ is no longer conjugate to the longitudinal position variable when the hadron moves. In order to maintain the Lorentz-invariance of the uncertainty product, we have to work with a conjugate pair of variables whose product does not depend on the velocity parameter. Let us go back to Eq.(\[conju\]) and Eq.(\[fourier2\]). It is quite clear that the light-cone variable $u$ and $v$ are conjugate to $q_{u}$ and $q_{v}$ respectively. It is also clear that the distribution along the $q_{u}$ axis shrinks as the $u$-axis distribution expands. The exact calculation leads to $$<u^{2}><q_{u}^{2}> = 1/4 , \qquad <v^{2}><q_{v}^{2}> = 1/4 .$$ Planck’s constant is indeed Lorentz-invariant. Let us next resolve the puzzle of why the number of partons appears to be infinite while there are only a finite number of quarks inside the hadron. As the hadronic speed approaches the speed of light, both the x and q distributions become concentrated along the positive light-cone axis. This means that the quarks also move with velocity very close to that of light. Quarks in this case behave like massless particles. We then know from statistical mechanics that the number of massless particles is not a conserved quantity. For instance, in black-body radiation, free light-like particles have a widespread momentum distribution. However, this does not contradict the known principles of quantum mechanics, because the massless photons can be divided into infinitely many massless particles with a continuous momentum distribution. Likewise, in the parton picture, massless free quarks have a wide-spread momentum distribution. They can appear as a distribution of an infinite number of free particles. These free massless particles are the partons. It is possible to measure this distribution in high-energy laboratories, and it is also possible to calculate it using the covariant harmonic oscillator formalism. We are thus forced to compare these two results. Indeed, according to Hussar’s calculation [@hussar81], the Lorentz-boosted oscillator wave function produces a reasoanbly accurate parton distribution. Concluding Remarks {#concluding-remarks .unnumbered} ================== The phenomenological aspects of the covariant oscillator formalism have been extensively discussed in the literature [@knp86]. The purpose of the present paper was to put the formalism into its proper place in the Lorentz-covariant world of physics. We have shown that the oscillator formalism constitutes a representation of Wigner’s little group governing the internal space-time symmetries of relativistic particles. For this purpose, we have given a comprehensive review of the little groups for massive and massless particles. We have discussed also the contraction procedure in which the $E(2)$-like little group for massless particles is obtained from the $O(3)$-like little group for massive particles. We have given a comprehensive review of the contents of Table I. Acknowledgments {#acknowledgments .unnumbered} =============== Since I came to the United States in 1954 right after high-school graduation, I made many new friends from many different countries. I have benefitted greatly from my association with Chinese friends and colleagues whose backgrounds are very similar to mine. When I met Professor C. N. Yang in 1958, he told me about the period before 1945 when he and other Chinese students had to move to a south-western province of China to continue their studies. Professor Yang’s story was a great encouragement to me since I had a similar experience during the Korean conflict which lasted from 1950 to 1953. In Appendix \[kant\], I discuss how my Eastern background influenced the research program which I outlined in this paper. I am indeed honored to join my Chinese colleagues in celebrating the 90th birth year of Professor T. Y. Wu. I am particularly grateful to Professor J. P Hsu for inviting me to write this review article on the covariant harmonic oscillators. Contraction of O(3) to E(2) {#o3e2} =========================== In this Appendix, we explain what the $E(2)$ group is. We then explain how we can obtain this group from the three-dimensional rotation group by making a flat-surface or cylindrical approximation. This contraction procedure will give a clue to obtaining the $E(2)$-like symmetry for massless particles from the $O(3)$-like symmetry for massive particles by making the infinite-momentum limit. The $E(2)$ transformations consist of rotation and two translations on a flat plane. Let us start with the rotation matrix applicable to the column vector $(x, y, 1)$: $$\label{rot} R(\theta) = \pmatrix{\cos\theta & -\sin\theta & 0 \cr \sin\theta & \cos\theta & 0 \cr 0 & 0 & 1} .$$ Let us then consider the translation matrix: $$T(a, b) = \pmatrix{1 & 0 & a \cr 0 & 1 & b \cr 0 & 0 & 1} .$$ If we take the product $T(a, b) R(\theta)$, $$\label{eucl} E(a, b, \theta) = T(a, b) R(\theta) = \pmatrix{\cos\theta & -\sin\theta & a \cr \sin\theta & \cos\theta & b \cr 0 & 0 & 1} .$$ This is the Euclidean transformation matrix applicable to the two-dimensional $x y$ plane. The matrices $R(\theta)$ and $T(a,b)$ represent the rotation and translation subgroups respectively. The above expression is not a direct product because $R(\theta)$ does not commute with $T(a,b)$. The translations constitute an Abelian invariant subgroup because two different $T$ matrices commute with each other, and because $$R(\theta) T(a,b) R^{-1}(\theta) = T(a',b') .$$ The rotation subgroup is not invariant because the conjugation $$T(a,b) R(\theta) T^{-1}(a,b)$$ does not lead to another rotation. We can write the above transformation matrix in terms of generators. The rotation is generated by $$J_{3} = \pmatrix{0 & -i & 0 \cr i & 0 & 0 \cr 0 & 0 & 0} .$$ The translations are generated by $$P_{1} = \pmatrix{0 & 0 & i \cr 0 & 0 & 0 \cr 0 & 0 & 0} , \qquad P_{2} = \pmatrix{0 & 0 & 0 \cr 0 & 0 & i \cr 0 & 0 & 0} .$$ These generators satisfy the commutation relations: $$\label{e2com} [P_{1}, P_{2}] = 0 , \qquad [J_{3}, P_{1}] = iP_{2} , \qquad [J_{3}, P_{2}] = -iP_{1} .$$ This $E(2)$ group is not only convenient for illustrating the groups containing an Abelian invariant subgroup, but also occupies an important place in constructing representations for the little group for massless particles, since the little group for massless particles is locally isomorphic to the above $E(2)$ group. The contraction of $O(3)$ to $E(2)$ is well known and is often called the Inonu-Wigner contraction [@inonu53]. The question is whether the $E(2)$-like little group can be obtained from the $O(3)$-like little group. In order to answer this question, let us closely look at the original form of the Inonu-Wigner contraction. We start with the generators of $O(3)$. The $J_{3}$ matrix is given in Eq.(\[j3\]), and $$\label{o3gen} J_{2} = \pmatrix{0&0&i\cr0&0&0\cr-i&0&0} , \qquad J_{3} = \pmatrix{0&-i&0\cr i &0&0\cr0&0&0} .$$ The Euclidean group $E(2)$ is generated by $J_{3}, P_{1}$ and $P_{2}$, and their Lie algebra has been discussed in Sec. \[intro\]. Let us transpose the Lie algebra of the $E(2)$ group. Then $P_{1}$ and $P_{2}$ become $Q_{1}$ and $Q_{2}$ respectively, where $$Q_{1} = \pmatrix{0&0&0\cr0&0&0\cr i &0&0} , \qquad Q_{2} = \pmatrix{0&0&0\cr0&0&0\cr0&i&0} .$$ Together with $J_{3}$, these generators satisfy the same set of commutation relations as that for $J_{3}, P_{1}$, and $P_{2}$ given in Eq.(\[e2com\]): $$[Q_{1}, Q_{2}] = 0 , \qquad [J_{3}, Q_{1}] = iQ_{2} , \qquad [J_{3}, Q_{2}] = -iQ_{1} .$$ These matrices generate transformations of a point on a circular cylinder. Rotations around the cylindrical axis are generated by $J_{3}$. The matrices $Q_{1}$ and $Q_{2}$ generate translations along the direction of $z$ axis. The group generated by these three matrices is called the [*cylindrical group*]{} [@kiwi87jm; @kiwi90jm]. We can achieve the contractions to the Euclidean and cylindrical groups by taking the large-radius limits of $$\label{inonucont} P_{1} = {1\over R} B^{-1} J_{2} B , \qquad P_{2} = -{1\over R} B^{-1} J_{1} B ,$$ and $$Q_{1} = -{1\over R}B J_{2}B^{-1} , \qquad Q_{2} = {1\over R} B J_{1} B^{-1} ,$$ where $$\label{bmatrix} B(R) = \pmatrix{1&0&0\cr0&1&0\cr0&0&R} .$$ The vector spaces to which the above generators are applicable are $(x, y, z/R)$ and $(x, y, Rz)$ for the Euclidean and cylindrical groups respectively. They can be regarded as the north-pole and equatorial-belt approximations of the spherical surface respectively [@kiwi87jm]. Contraction of O(3)-like Little Group to E(2)-like Little Group {#contrac} =============================================================== Since $P_{1} (P_{2})$ commutes with $Q_{2} (Q_{1})$, we can consider the following combination of generators. $$F_{1} = P_{1} + Q_{1} , \qquad F_{2} = P_{2} + Q_{2} .$$ Then these operators also satisfy the commutation relations: $$\label{commuf} [F_{1}, F_{2}] = 0 , \qquad [J_{3}, F_{1}] = iF_{2} , \qquad [J_{3}, F_{2}] = -iF_{1} .$$ However, we cannot make this addition using the three-by-three matrices for $P_{i}$ and $Q_{i}$ to construct three-by-three matrices for $F_{1}$ and $F_{2}$, because the vector spaces are different for the $P_{i}$ and $Q_{i}$ representations. We can accommodate this difference by creating two different $z$ coordinates, one with a contracted $z$ and the other with an expanded $z$, namely $(x, y, Rz, z/R)$. Then the generators become $$P_{1} = \pmatrix{0&0&0&i\cr0&0&0&0\cr0&0&0&0\cr0&0&0&0} , \qquad P_{2} = \pmatrix{0&0&0&0\cr0&0&0&i\cr0&0&0&0\cr0&0&0&0} .$$ $$Q_{1} = \pmatrix{0&0&0&0\cr0&0&0&0\cr i &0&0&0\cr0&0&0&0} , \qquad Q_{2} = \pmatrix{0&0&0&0\cr0&0&0&0\cr0&i&0&0\cr0&0&0&0} .$$ Then $F_{1}$ and $F_{2}$ will take the form $$\label{f1f2} F_{1} = \pmatrix{0&0&0&i\cr0&0&0&0\cr i &0&0&0\cr0&0&0&0} , \qquad F_{2} = \pmatrix{0&0&0&0\cr0&0&0&i\cr0&i&0&0\cr0&0&0&0} .$$ The rotation generator $J_{3}$ takes the form of Eq.(\[j3\]). These four-by-four matrices satisfy the E(2)-like commutation relations of Eq.(\[commuf\]). Now the $B$ matrix of Eq.(\[bmatrix\]), can be expanded to $$\label{bmatrix2} B(R) = \pmatrix{1&0&0&0\cr0&1&0&0\cr0&0&R&0\cr0&0&0&1/R} .$$ If we make a similarity transformation on the above form using the matrix $$\label{simil} \pmatrix{1&0&0&0\cr0&1&0&0\cr0&0&1/\sqrt{2} &-1/\sqrt{2} \cr0&0&1/\sqrt{2}&1/\sqrt{2}} ,$$ which performs a 45-degree rotation of the third and fourth coordinates, then this matrix becomes $$\label{simil2} \pmatrix{1&0&0&0\cr0&1&0&0\cr0&0 & \cosh\eta & \sinh\eta \cr0 & 0 & \sinh\eta & \cosh\eta} ,$$ with $R = e^\eta$. This form is the Lorentz boost matrix along the $z$ direction. If we start with the set of expanded rotation generators $J_{3}$ of Eq.(\[j3\]), and perform the same operation as the original Inonu-Wigner contraction given in Eq.(\[inonucont\]), the result is $$N_{1} = {1\over R} B^{-1} J_{2} B , \qquad N_{2} = -{1\over R} B^{-1} J_{1} B ,$$ where $N_{1}$ and $N_{2}$ are given in Eq.(\[n1n2\]). The generators $N_{1}$ and $N_{2}$ are the contracted $J_{2}$ and $J_{1}$ respectively in the infinite-momentum/zero-mass limit. Covariance and Its Historical Background {#kant} ======================================== Unlike classical physics, modern physics depends heavily on observer’s state of mind or environment. In special relativity, observers in different Lorentz frames see the same physical system differently. The importance of the observer’s subjective viewpoint was emphasized by Immanuel Kant in his book entitled [*Kritik der reinen Vernunft*]{} whose first and second editions were published in 1781 and 1787 respectively. However, using his own logic, he ended up with a conclusion that there must be the absolute inertial frame, and that we only see the frames dictated by our subjectivity. Einstein’s special relativity was developed along Kant’s line of thinking: things depend on the frame from which you make observations. However, there is one big difference. Instead of the absolute frame, Einstein introduced an extra dimension. Let us illustrate this using a CocaCola can. It appears like a circle if you look at it from the top, while it appears as a rectangle from the side. The real thing is a three-dimensional circular cylinder. While Kant was obsessed with the absoluteness of the real thing, Einstein was able to observe the importance of the extra dimension. I was fortunate enough to be close to Eugene Wigner, and enjoyed the privilege of asking him many questions. I once asked him whether he thinks like Immanuel Kant. He said Yes. I then asked him whether Einstein was a Kantianist in his opinion. Wigner said very firmly Yes. I then asked him whether he studied the philosophy of Kant while he was in college. He said No, and said that he realized he had been a Kantianist after writing so many papers in physics. He added that philosophers do not dictate people how to think, but their job is to describe systematically how people think. Wigner told me that I was the only one who asked him this question, and asked me how I knew the Kantian way of reasoning was working in his mind. I gave him the following answer. I never had any formal education in oriental philosophy, but I know that my frame of thinking is affected by my Korean background. One important aspect is that Immanuel Kant’s name is known to every high-school graduate in Korea, while he is unknown to Americans, particularly to American physicists. The question then is whether there is in Eastern culture a “natural frequency” which can resonate with one of the frequencies radiated from Kantianism developed in Europe. I would like to answer this question in the following way. Koreans absorbed a bulk of Chinese culture during the period of the Tang dynasty (618-907 AD). At that time, China was the center of the world as the United States is today. This dynasty’s intellectual life was based on Taoism which tells us, among others, that everything in this universe has to be balanced between its plus (or bright) side and its minus (or dark) side. This way of thinking forces us to look at things from two different or opposite directions. This aspect of Taoism could constitute a “natural frequency” which can be tuned to the Kantian view of the world where things depend how they are observed. I would like to point out that Hideki Yukawa was quite fond of Taoism and studied systematically the books of Laotse and Chuangtse who were the founding fathers of Taoism [@tani79]. Both Laotse and Chuangtse lived before the time of Confucius. It is interesting to note that Kantianism is also popular is Japan, and it is my assumption that Kant’s books were translated into Japanese by Japanese philosophers first, and Koreans of my father’s age learned about Kant by reading the translated versions. My publication record will indicate that I studied Yukawa’s papers before becoming seriously interested in Wignerism. Indeed, I picked up a signal of possible connection between Kantianism and Taoism while reading Yukawa’s papers carefully, and this led to my bold venture to ask Wigner whether he was a Kantianist. Kant wrote his books in German, but he was born and spent his entire life in a Baltic enclave now called Kaliningrad located between Poland and Lithuania. Historically, this place was dominated by several different countries with different ideologies [@apple94]. However, Kant’s view was that the people there may appear differently depending on who look at them, but they remain unchanged. At the same time, they had to entertain different ideologies imposed by different rulers. Kant translated this philosophy into physics when he discussed the absolute and relative frames. He was obsessed with the absolute frame, and this is the reason why Kant is not regarded as a physicist in Einstein’s world in which we live. The people of Kant’s land stayed in the same place while experiencing different ideological environments. Almost like Kant, I was exposed to two different cultural environments by moving myself from Asia to the United States. Thus, I often had to raise the question of absolute and relative values. Let us discuss this problem using one concrete example. About 4,500 years ago, there was a king named Yao in China. While he was looking for a man who could serve as the prime minister, he heard from many people that a person named Shiyu was widely respected and had a deep knowledge of the world. The king then sent his messengers to invite Shiyu to come to his palace and to run the country. After hearing the king’s message, Shiyu without saying anything went to a creek in front of his house and started washing his ears. He thought he heard the dirtiest story in his life. Shiyu is still respected in the Eastern world as one of the wisest men in history. We do not know whether this person existed or is a made-up personality. In either case, we are led to look for a similar person in the Western world. In ancient Greece, each city was run by its city council. As we experience even these days, people accomplish very little in committee meetings. Thus, it is safe to assume that the city councils in ancient Greece did not handle matters too efficiently. For this reason, there was a well-respected wiseman like Shiyu who never attended his city council meetings. His name was Idiot. Idiot was a wiseman, but he never contributed his wisdom to his community. His fellow citizens labeled him as a useless person. This was how the word idiot was developed in the Western world. Idiot and Shiyu had the same personality if they were not the same person. However, Idiot is a useless person in state-centered societies like Sparta. The same person is regarded as the ultimate wiseman in a self-centered society like Korea. I cannot say that I know everything about other Asian countries, but I have a deep knowledge of Korea where I was born and raised. The same person looks quite differently to observers in different cultural frames. While doing research in the United States with my Eastern background, I was frequently forced to find a common ground for two seemingly different views. This cultural background strongly influenced me in producing the further contents of Einstein’s $E = mc^{2}$ tabulated in Table I [@kim89]. Let us go back to the question of relative values. For Taoists, those two opposite faces of the same person is like “yang” (plus) versus “ying” (minus). Finding the harmony between these two opposite points of view is the ideal way to live in this world. We cannot always live like Shiyu, nor like Idiot. The key to happiness is to find a harmony between the individual and the society to which he/she belongs. The key word here seems to be “harmony.” To Kantianists, however, it is quite natural for the same character to appear differently in two different environments. The problem is to find the absolute value from these two different faces. Does this absolute value exist? According to Kant, it exists. To most of us, it is very difficult to find it if it exists. Let us finally visit Einstein. He avoids the question of the existence of the absolute value. Instead, he introduces a new variable. The variable is the ratio between the individual’s ability to contribute and the community’s need for his service. The best way to live in this world is to adjust this variable to the optimal value. Einstein’s approach is to a quantification of Taoism by introducing a new variable. If Taoism is so close to Einsteinism, why do we have to mention Kant at all? We have to keep in mind that Kant was the first person who formulated the idea that observers can participate in drawing the picture of the world. It is not clear whether Einstein could have formulated his relativity theory without Kant. Indeed, Kant spent many years for studying physics, namely observer-dependent physics. However, because of his obsession toward the absolute thing, he spent all of his time for finding the absolute frame. If one has a Taoist background, he/she is more likely to appreciate the concept of relativistic covariance. I would like to stress that Taoism is not confined to the ancient Eastern world. It is practiced frequently in the United States. Let us look at American football games. The offensive strategy does not rely on brute force, but is aimed at breaking the harmony of the defense. For instance, when the offensive team is near the end zone, the defense becomes very strong because it covers only a small area. Then, it is not uncommon for the offense to place four wide-receivers instead of two. This will divide the defense into two sides while creating a hole in the middle. Then the quarter-back can carry the ball to the end zone. The key word is to destroy the balance of the defense. Taoism forms the philosophical base for Sun Tzu’s classic book on military arts [@suntzu96]. When I watch the football games, I watch them as Sun-Tzu games. My maternal grandfather was fluent in the Chinese classic literature, and he was particularly fond of Sun Tzu. He told me many stories from Sun Tzu’s books. This presumably was how I inherited some of the Taoist tradition. As I said in this paper, my research life was influenced by my Asian background. Many of my Asian friends complain that they are handicapped to do original research because of the East-West cultural difference. I disagree with them. This difference could be the richest source of originality. 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--- author: - | M. Csanád$^{1}$, M. I. Nagy$^{1}$, S. Lökös$^{1}$\ $^1$Eötvös Loránd University, H-1117 Budapest, Pázmány P. s. 1/a bibliography: - '../../master.bib' title: Exact solutions of relativistic perfect fluid hydrodynamics for a QCD equation of state --- Introduction ============ The interest in relativistic hydrodynamics grew in past years mainly due to the discovery of the almost perfect fluidity of the experimentally created Quark-Gluon-Plasma [@Zajc:2007ey]. Hydrodynamical models aim to describe the space-time picture of heavy-ion collisions and infer the relation between experimental observables and the initial conditions. Besides numerical simulations there is also interest in models where exact solutions of the hydrodynamical equations are used. In this paper we generalize a previously known class of exact solutions of relativistic perfect fluid hydrodynamics [@Csorgo:2003ry] to the case of arbitrary, temperature dependent speed of sound. The mentioned class of solutions form the basis of the relativistic Buda-Lund hydrodynamical model [@Csanad:2003qa]. This model yields a successful description of hadronic observables at RHIC energies (such as the pseudorapidity and transverse momentum dependence of the azimuthal anisotropy of different hadrons as well as the HBT radii [@Csanad:2003qa]), and the reconstructed final state in this model corresponds to simple explicit scaling solutions of hydrodynamics. The same final state however can be achieved from many initial states, depending on the Equation of State [@Csanad:2009sk]. If one is given a temperature dependent speed of sound as Equation of State, the solution presented in this paper thus can be used to determine the initial state from the reconstructed final state of a heavy-ion collision. As an example, we describe the time dependence of the system if one assumes the Equation of State from lattice QCD. The solutions given in this paper are the first exact analytic solutions of 1+3 dimensional relativistic hydrodynamics, to utilize an arbitrary Equation of State.[^1] Basic equations =============== Let us adopt the following notational conventions: the fluid coordinates are $x^\mu = {\left({t, {\mathbf{r}}}\right)}$, where ${\mathbf{r}}={\left({r_x, r_y, r_z}\right)}$ is the spatial coordinate, and the metric tensor is $g_{\mu\nu}=diag{\left({1,-1,-1,-1}\right)}$. (We denote space-time indices by Greek letters, space indices by Latin letters, and assume the summation convention.) The fluid four-velocity is $u^\mu\equiv\gamma{\left({1,{\mathbf{v}}}\right)}$, where ${\mathbf{v}}$ is the three-velocity and $\gamma=1/\sqrt{1-v^2}$. The thermodynamical quantities are denoted as follows: $p$ is the pressure, $\varepsilon$ is the energy density, $\sigma$ is the entropy density, $T$ is the temperature. If the fluid consists of individual conserved particles, or if there is some conserved charge, then the conserved number density is denoted by $n$, and the corresponding chemical potential by $\mu$. (For more than one conserved number densities, we may use indices to distinguish them.) All these quantities have dependence on $x^\mu$, but mostly this will not be explicitly written out. The basic hydrodynamical equations are the continuity and energy-momentum-conservation equations: $$\begin{aligned} \partial_\mu{\left({n u^\mu}\right)} & = 0,\label{e:cont}\\ \partial_\nu T^{\mu \nu} & = 0\label{e:em}.\end{aligned}$$ The energy-momentum tensor of a perfect fluid is $$\begin{aligned} T^{\mu\nu} ={\left({\varepsilon+p}\right)}u^\mu u^\nu-pg^{\mu \nu} .\end{aligned}$$ [Eq. (\[e:em\])]{} can be then transformed to (by projecting it orthogonal and parallel to $u^\mu$, respectively): $$\begin{aligned} {\left({\varepsilon+p}\right)}u^{\nu}\partial_{\nu}u^{\mu} & ={\left({g^{\mu\nu}-u^{\mu}u^{\nu}}\right)}\partial_{\nu}p,\label{e:euler} \\ {\left({\varepsilon+p}\right)}\partial_{\nu}u^{\nu}+u^{\nu}\partial_{\nu}\varepsilon & = 0\label{e:energy}.\end{aligned}$$ [Eq. (\[e:euler\])]{} is the relativistic Euler equation, while [Eq. (\[e:energy\])]{} is the relativistic form of the energy conservation equation. In Appendix \[s:app:Teq\] we recall the well-known fact that [Eq. (\[e:energy\])]{} is equivalent to the entropy conservation equation: $$\begin{aligned} \label{e:scont} \partial_\mu{\left({\sigma u^\mu}\right)}=0 .\end{aligned}$$ An analytic hydrodynamical solution is a functional form of $\varepsilon$, $p$, $T$, $u^\mu$ (and, if dealt with, $n$), which solves [Eqs. (\[e:euler\]) and (\[e:energy\])]{}, and, if present, $n$ also solves [Eq. (\[e:cont\])]{}. The quantities $\varepsilon$, $p$, $T$, and also $\sigma$, and $n$ are subject to the Equation of State (EoS), which closes the set of equations. We investigate the following EoS: $$\begin{aligned} \label{e:eos} \varepsilon = \kappa{\left({T}\right)} p ,\end{aligned}$$ and for the case when there is a conserved $n$ number density, the additional assumption is $$\begin{aligned} \label{e:tdef} p=nT. \end{aligned}$$ For the case of $\kappa{\left({T}\right)}=\kappa$ constant, an ellipsoidally symmetric solution of the hydrodynamical equations is presented in Ref. [@Csorgo:2003ry]: $$\begin{aligned} \label{e:usol0} u^\mu = \frac{x^\mu}{\tau} ,\quad \tau=\sqrt{t^2-r^2}=\sqrt{x_\mu x^\mu} ,\end{aligned}$$ $$\begin{aligned} \label{e:tsol0} n = n_0\frac{V_0}{V}\nu{\left({s}\right)},\quad T = T_0{\left({\frac{V_0}{V}}\right)}^{{\frac{1}{\kappa}}}{\frac{1}{\nu{\left({s}\right)}}} ,\end{aligned}$$ with $\nu{\left({s}\right)}$ being an arbitrary function and $$\begin{aligned} \label{e:V0s} s = \frac{r_x^2}{X^2} + \frac{r_y^2}{Y^2} + \frac{r_z^2}{Z^2},\quad V=\tau^3 ,\end{aligned}$$ where $X$, $Y$, and $Z$ are the time ($t$) dependent principal axes of an expanding ellipsoid. They have the explicit time dependence as $$\begin{aligned} \label{e:Z0} X = \dot X_0 t,\quad Y = \dot Y_0 t, \quad Z = \dot Z_0 t\end{aligned}$$ with $\dot X_0$, $\dot Y_0$, $\dot Z_0$ constants. The quantity $s$ has ellipsoidal level surfaces, and obeys $u^\nu\partial_\nu s=0$. We call $s$ a *scaling variable*, and $V$ the effective volume of a characteristic ellipsoid[^2]. This solution is *non-accelerating*, ie. obeys $u^\nu\partial_\nu u^\mu=0$. In the next section we present a generalization of this class of solutions to more general EoS. The new solutions will be presented in Section \[s:sols\], while Section \[s:eoseqs\] details their derivation. General Equation of State {#s:eoseqs} ========================= In order to find more general solutions, where a temperature dependent EoS can be used (as in [Eq. (\[e:eos\])]{}), for a given $u^\mu$ velocity field we may *define* the $V$ and $s$ quantities by their properties that $$\begin{aligned} \label{e:V1} u^\mu\partial_\mu V = V \partial_\mu u^\mu,\quad u^\mu\partial_\mu s = 0 .\end{aligned}$$ With these quantities, [Eq. (\[e:cont\])]{} is automatically solved (for the case when there is a conserved charge present) if $$\begin{aligned} \label{e:nsol} n = n_0\frac{V_0}{V}\nu{\left({s}\right)} ,\end{aligned}$$ again, with arbitrary $\nu{\left({s}\right)}$ function. To solve the [(\[e:energy\])]{} energy equation, we must make a distinction between two possible cases. The first case is if we take a conserved $n$ into account, and use the EoS $\varepsilon=\kappa{\left({T}\right)}p$, $p=nT$ as in [Eqs. (\[e:eos\]) and (\[e:tdef\])]{}. The second case is when we do not consider any conserved $n$. In Appendix \[s:app:Teq\] we show that in both of these two cases the energy equation [Eq. (\[e:energy\])]{} can be transformed to an equation for $T$: in the first case with conserved $n$, we have $$\begin{aligned} \label{e:T1} u^\mu{\left[{{\frac{\mathrm{d}{{\left({\kappa T}\right)}}}{\mathrm{d}{T}}}\frac{\partial_\mu T}{T} + \frac{\partial_\mu V}{V}}\right]} = 0 ,\end{aligned}$$ while in the case where there is no conserved $n$, we have $$\begin{aligned} \label{e:T2} u^\mu{\left[{\frac{\partial_\mu V}{V}+{\left({{\frac{1}{\kappa+1}}{\frac{\mathrm{d}{\kappa}}{\mathrm{d}{T}}}+\frac{\kappa}{T}}\right)}\partial_\mu T}\right]}=0 .\end{aligned}$$ Remarkably, these equations are not the same (however, we may note that in the case when $\kappa=$ const., they yield the same condition). We call these equtions the temperature equations for the two cases. With the introduction of the $f{\left({T}\right)}$ function as $$\begin{aligned} \label{e:fT1} f(T)=\exp{\left\{{\int_{T_0}^T{\left({{\frac{1}{\beta}}{\frac{\mathrm{d}{}}{\mathrm{d}{\beta}}}{\left[{\kappa{\left({\beta}\right)}\beta}\right]}}\right)}{\mathrm{d}}\beta}\right\}} \end{aligned}$$ for the case of conserved $n$, and as $$\begin{aligned} \label{e:fT2} f(T)=\exp{\left\{{\int_{T_0}^T{\left({\frac{\kappa{\left({\beta}\right)}}{\beta}+{\frac{1}{\kappa{\left({\beta}\right)}+1}}{\frac{\mathrm{d}{\kappa{\left({\beta}\right)}}}{\mathrm{d}{\beta}}}}\right)}{\mathrm{d}}\beta}\right\}} \end{aligned}$$ for the case of vanishing $n$, the temperature equations can be cast in the following universal form: $$\begin{aligned} \label{e:T3} u^\mu{\left[{\frac{\partial_\mu V}{V}+\frac{\partial_\mu f(T)}{f(T)}}\right]}=0 .\end{aligned}$$ For any given $\kappa(T)$ function we can determine $f(T)$, and write up the solution of the above equation as $$\begin{aligned} \label{e:Tsol} T=f^{-1}{\left({\frac{V_0}{V}\xi{\left({s}\right)}}\right)}\end{aligned}$$ with arbitary $\xi{\left({s}\right)}$ function. (For convenience, we may normalize $\xi{\left({s}\right)}$ so that $\xi(0)=1$.) Knowing that $u^\mu\partial_\mu s=0$, it is easy to see that this indeed solves [Eq. (\[e:T3\])]{}. Note that if $\kappa=$ const., then due to [Eq. (\[e:Tsol\])]{}: $$\begin{aligned} \label{e:fkconst} f(T)={\left({\frac{T}{T_0}}\right)}^{\kappa} \;\Rightarrow\;\; T = {\left({\frac{V_0}{V}}\right)}^{1/\kappa}\xi(s)^{1/\kappa}.\end{aligned}$$ As a generalization of the solution recalled in the previous section, we assume that $u^\mu$ and thus $s$ and $V$ has the same forms as in [Eqs. (\[e:usol0\]) and (\[e:V0s\])]{}. $$\begin{aligned} \label{e:uVs} u^\mu=\frac{x^\mu}{\tau},\quad V=\tau^3,\quad s=\frac{r_x^2}{\dot{X}_0^2t^2} + \frac{r_y^2}{\dot{Y}_0^2t^2} + \frac{r_z^2}{\dot{Z}_0^2t^2} .\end{aligned}$$ For this velocity field, $u^\nu\partial_\nu u^\mu=0$, so the remaining equation, the Euler equation of [(\[e:euler\])]{} is equivalent to $$\begin{aligned} \label{e:euler0} \partial_\mu p=u_\mu u^\nu\partial_\nu p .\end{aligned}$$ In the case of vanishing $n$, using the thermodynamic relation ${\mathrm{d}}p=\sigma{\mathrm{d}}T$, [Eq. (\[e:euler0\])]{} simplifies to $$\begin{aligned} \partial_\mu T=u_\mu u^\nu\partial_\nu T ,\end{aligned}$$ and using the expression of $T$ from [Eq. (\[e:Tsol\])]{} and the definition of $u^\mu$ and $V$ from [Eq. (\[e:uVs\])]{}, we find that it is equivalent (for any $\kappa{\left({T}\right)}$, thus for any $f(T))$ to $$\begin{aligned} \label{} {f^{-1}}'{\left({\frac{V_0}{V}\xi{\left({s}\right)}}\right)}\frac{\xi'{\left({s}\right)}}{\xi{\left({s}\right)}}\partial_\mu s=0\quad\Rightarrow\quad \xi{\left({s}\right)}=1.\end{aligned}$$ In the case of non-vanishing $n$, using [Eq. (\[e:nsol\])]{} and $p=nT$, the Euler equation for our non-accelerating velocity field transforms to the following equation: $$\begin{aligned} \label{} T\partial_\mu n+n\partial_\mu T = Tu_\mu u^\nu\partial_\nu n+nu^\mu u^\nu\partial_\nu T .\end{aligned}$$ Substituting $n$ and $T$ from [Eqs. (\[e:nsol\]) and (\[e:Tsol\])]{}, and the definition of $V$, we get from this equation the following constraint: $$\begin{aligned} \label{e:euln3} {\left[{\frac{\nu'{\left({s}\right)}}{\nu{\left({s}\right)}}+ \varphi {\left({\frac{V_0}{V}\xi{\left({s}\right)}}\right)} \frac{\xi'{\left({s}\right)}}{\xi{\left({s}\right)}}}\right]} \partial_\mu s = 0 ,\end{aligned}$$ where we have introduced the following function: $$\begin{aligned} \varphi(y) = \frac{y{f^{-1}}'(y)}{f^{-1}(y)}\end{aligned}$$ Since $\partial_\mu s\neq 0$, we see from [Eq. (\[e:euln3\])]{} that there are two cases: for any EoS (ie. for any $\kappa{\left({T}\right)}$ and thus any $\varphi(T)$ function) we get a solution if $\nu{\left({s}\right)}=\xi{\left({s}\right)}=1$. The other possibility is if $\kappa=const$. It is easy to see that this case is equivalent to $\varphi=\kappa^{-1}=const$, and so [Eq. (\[e:euln3\])]{} is solved if $\xi = \nu^{-1/\kappa}$ and so from [Eq. (\[e:fkconst\])]{} we get $T=T_0{\left({V_0/V}\right)}^{1/\kappa}\nu^{-1}(s)$, i.e. the same as in [Eq. (\[e:tsol0\])]{}. In this case we indeed obtain the known solution of Ref. [@Csorgo:2003ry], recited in Eqs. [(\[e:usol0\])]{}–[(\[e:V0s\])]{}. New solutions for general Equation of State {#s:sols} =========================================== Summarizing and rewriting the results presented in the previous section, we found new solutions to the relativistic hydrodynamical equations for arbitary $\varepsilon=\kappa{\left({T}\right)}p$ Equation of State, and these are the first solutions of their kind (i.e. with a non-constant EoS). In the case where we do not consider any conserved $n$ density, the solution can be presented in the following form, in terms of $u^\mu$, $\sigma$ and $T$, with $T$ given in an implicit form: $$\begin{aligned} \sigma &= \sigma_0 \frac{\tau_0^3}{\tau^3} ,\label{e:Tsol:s:0}\\ u^\mu & = \frac{x^\mu}{\tau} ,\\ \frac{\tau_0^3}{\tau^3} & = \exp{\left\{{\int_{T_0}^T{\left({\frac{\kappa{\left({\beta}\right)}}{\beta}+{\frac{1}{\kappa{\left({\beta}\right)}+1}}{\frac{\mathrm{d}{\kappa{\left({\beta}\right)}}}{\mathrm{d}{\beta}}}}\right)}{\mathrm{d}}\beta}\right\}} . \label{e:Tsol:s}\end{aligned}$$ Also, for the case when the pressure is expressed as $p=nT$ with some conserved $n$ density, the new solution is written in terms of $u^\mu$, $T$ and $n$ as $$\begin{aligned} n &= n_0 \frac{\tau_0^3}{\tau^3} ,\\ u^\mu & = \frac{x^\mu}{\tau} ,\\ \frac{\tau_0^3}{\tau^3} & = \exp{\left\{{\int_{T_0}^T{\left({{\frac{1}{\beta}}{\frac{\mathrm{d}{}}{\mathrm{d}{\beta}}}{\left[{\kappa{\left({\beta}\right)}\beta}\right]}}\right)}{\mathrm{d}}\beta}\right\}} . \label{e:Tsol:nT}\end{aligned}$$ Note that these solutions form simple generalization of the $\nu{\left({s}\right)}=1$ case of the solutions of Ref. [@Csorgo:2003ry]. Also note that in the case when $p=nT$ and $n$ is conserved, for some choices of the $\kappa{\left({T}\right)}$ function our solution becomes ill-defined. The criterion that ${\frac{\mathrm{d}{}}{\mathrm{d}{T}}} {\left({\kappa{\left({T}\right)}T}\right)}$ should be positive limits the applicability of solutions for the case of conserved $n$ presented here. In the case when for some $T$ range ${\frac{\mathrm{d}{}}{\mathrm{d}{T}}} {\left({\kappa{\left({T}\right)}T}\right)}$ becomes negative, the implicit form of [Eq. (\[e:Tsol:nT\])]{} cannot be inverted to give a unique $T{\left({\tau}\right)}$ function. Such domains of $T$ indeed might exist in some parametrizations of the lattice QCD Equation of State around the quark-hadron transition temperature (as detailed in the next section, in particularly on Fig. \[f:validity\]). However, even for these cases, one can use the solution without conserved $n$ presented here as a physically relevant solution, since at the transition temperature a conserved density $n$ yielding pressure as $p=nT$ cannot be identified. Let us briefly mention another possibility, when $\kappa$ is a function of the pressure $p$ and not that of the temperature $T$. In this case a solution can be written up, similarly to the previous ones as $$\begin{aligned} \sigma &= \sigma_0 \frac{\tau_0^3}{\tau^3} ,\label{e:psol:s}\\ u^\mu & = \frac{x^\mu}{\tau} ,\\ \frac{\tau_0^3}{\tau^3} & = \exp{\left\{{\int_{p_0}^p{\left({\frac{\kappa{\left({\beta}\right)}}{\beta}+{\frac{\mathrm{d}{\kappa{\left({\beta}\right)}}}{\mathrm{d}{\beta}}}}\right)}\frac{{\mathrm{d}}\beta}{\kappa{\left({\beta}\right)}+1}}\right\}} , \label{e:psol:p}\end{aligned}$$ i.e. almost the same as in [Eq. (\[e:Tsol:s\])]{}, except that here the integration variable is the pressure $p$. If however the pressure can be written as a function of temperature, i.e. as $p(T)$, an integral-transformation can be made with and we get back [Eq. (\[e:Tsol:s\])]{}, so in this case these solutions are identical. This solution may be used if a $\kappa(p)$ function is given (without relation to the temperature) by an arbitrary energy density function $\varepsilon(p)=\kappa(p)p$. Application =========== Recently a QCD equation of state has been calculated by the Budapest-Wuppertal group in Ref. [@Borsanyi:2010cj]. Here (in their Eq. (3.1) and Table 2) they give a parametrization of the trace anomaly as a function of temperature. Hence the pressure, the energy density and finallly the EoS parameter $\kappa$ can be calculated, as a function of the temperature. We did this calculation, and got the $\kappa(T)$ function as shown in Fig. \[f:validity\]. Note however, that in this calculation for some $T$ range ${\frac{\mathrm{d}{}}{\mathrm{d}{T}}} {\left({\kappa{\left({T}\right)}T}\right)}$ becomes negative, as also shown in Fig. \[f:validity\]. Hence the implicit form of [Eq. (\[e:Tsol:nT\])]{} cannot be inverted to give a unique $T{\left({\tau}\right)}$ function. We can still use the solution without conserved number density $n$, presented in Eqs. [(\[e:Tsol:s:0\])]{}–[(\[e:Tsol:s\])]{}. ![The temperature dependence of the EoS parameter $\kappa$ from Ref. [@Borsanyi:2010cj] is shown with the solid black curve. Note that in the shaded $T$ range (173 MeV - 230 MeV) ${\frac{\mathrm{d}{}}{\mathrm{d}{T}}} {\left({\kappa{\left({T}\right)}T}\right)}$ (red dashed line) becomes negative, thus the implicit form of [Eq. (\[e:Tsol:nT\])]{} cannot be inverted to give a unique $T{\left({\tau}\right)}$ function. Hence we will substitute this $\kappa(T)$ in the hydrodynamic solution shown in Eqs. [(\[e:Tsol:s:0\])]{}–[(\[e:Tsol:s\])]{}.[]{data-label="f:validity"}](validity.eps){width="49.00000%"} We utilized the obtained $\kappa(T)$ and calculated the time evolution of the temperature of the fireball from this solution of relativistic hydrodynamics. The result is shown in Fig. \[f:ttau\]. Clearly, temperature falls off almost as fast as in case of a constant $\kappa=3$, an ideal relativistic gas. Hence a given freeze-out temperature yields a significantly higher initial temperature than a higher $\kappa$ (i.e. a low $c_s^2$) would. If we fix the freeze-out temperature to 170 MeV for example, then already at 30% of the freeze-out time $\tau_0$ (the value of which does not affect our results) 2.5-3$\times$ higher temperatures than at the freeze-out. To give a concrete example, if $\tau_0 = 8$ fm$/c$, and $\tau_{\rm init} = 1.5$ fm$/c$, then for $T_0 = 170$ MeV we get $T_{\rm init} \approx 550$ MeV (and even higher if $\tau_{\rm init}$ is smaller). The QCD equation of state of Ref. [@Borsanyi:2010cj] and this hydro solution yields a general $T(\tau)$ dependence. If the freeze-out temperature $T_0$ and the time evolution duration $\tau_0 / \tau_{\rm init}$ are known, the initial temperature of the fireball can be easily calculated. ![Time dependence of the temperature $T(\tau)$ (normalized with the freeze-out time $\tau_0$ and the freeze-out temperature $T_0$) is shown here. The four thin red lines show this dependence in case of constant $\kappa$ values, while the thicker blue lines show results based on the EoS of Ref. [@Borsanyi:2010cj]. The resulting curve slightly depends on the value of $T_0$. It is clear however, that the temperature fall-off is almost as fast in the QCD EoS case as in the case of fixed $\kappa=3$, which resembles a relativistic ideal gas. This means, that a fixed freeze-out temperature (which cannot vary too much due to the known quark-hadron transition temperature) results in a very high initial temperature.[]{data-label="f:ttau"}](ttau.eps){width="49.00000%"} Conclusion ========== We have presented the first analytic solutions of the equations of relativistic perfect fluid hydrodynamics for general temperature dependent speed of sound (ie. general Equation of State). They can be seen as generalizations of previously known exact solutions [@Csorgo:2003ry]. However, our new solutions are spherically symmetric, thus possible generalizations of them definitely are worth exploring: solutions for the non-accelerating case and for more general ellipsoidal symmetry would be able to analytically explore the time evolution of other hadronic observables such as the elliptic flow ($v_2$). We have shown how to use our solutions to fully utilize the lattice QCD Equation of State for exploring the initial state of heavy-ion reactions based on the reconstructed final state in the Buda-Lund hydrodynamical model. In $\sqrt{s_{NN}}=200$ GeV Au+Au collisions, our investigations reveal a very high initial temperature consistent with calculations based on the measurement spectrum of low momentum direct photons [@Adare:2008fqa]. If given a temperature-dependent direct photon emission function, then this model can be used to calculate direct photon spectra to be compared to measurements, as in Ref. [@Csanad:2011jq], but with a realistic Equation of State. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the NK-101438 OTKA grant and the Bolyai Scholarship (Hungarian Academy of Sciences) of M. Csanád. The authors also would like to thank T. Csörgő for motivating and valuable discussions. The entropy and the temperature equations {#s:app:Teq} ========================================= The fundamental thermodynamical relations connecting $\varepsilon$, $T$, $\sigma$, $p$, and any types of $n_i$ conserved charges and corresponding $\mu_i$ chemical potentials are $$\begin{aligned} \varepsilon+p & = T\sigma+\sum_i n_i\mu_i ,\label{e:thermo1}\\ {\mathrm{d}}\varepsilon & = T{\mathrm{d}}\sigma+\sum_i\mu_i{\mathrm{d}}n_i ,\label{e:thermo2}\\ {\mathrm{d}}p & = \sigma{\mathrm{d}}T+\sum_in_i{\mathrm{d}}\mu_i .\label{e:thermo3}\end{aligned}$$ In the case when there are no conserved charges, similar relations hold with all $n_i$ and $\mu_i$ variables omitted. Substituting these in the [(\[e:energy\])]{} energy conservation equation, we immediately obtain the continuity equation for the entropy density $\sigma$: $$\begin{aligned} T\sigma\partial_\mu u^\mu + Tu^\mu\partial_\mu \sigma +\sum_i\mu_i{\left({n_i\partial_\mu u^\mu+u^\mu\partial_\mu n_i}\right)}= 0 ,\end{aligned}$$ which is, for conserved (or vanishing) $n_i$s, equivalent to $$\begin{aligned} \partial_\nu{\left({\sigma u^\nu}\right)} = 0 ,\end{aligned}$$ which is the entropy conservation, [Eq. (\[e:scont\])]{}. In the case when there is no conserved $n$, we can substitute the [(\[e:thermo2\])]{} and [(\[e:thermo3\])]{} thermodynamic relations for vanishing $n$ in [Eq. (\[e:energy\])]{}. Using the Eos as $\varepsilon=\kappa{\left({T}\right)}p$, and $\varepsilon+p =T\sigma$ and ${\mathrm{d}}p=\sigma{\mathrm{d}}T$ we obtain from [Eq. (\[e:energy\])]{} the following: $$\begin{aligned} T\sigma{\left[{\partial_\mu u^\mu+{\frac{1}{\kappa+1}}{\frac{\mathrm{d}{\kappa}}{\mathrm{d}{T}}}u^\mu\partial_\mu T}\right]}+\kappa\sigma\ u^\mu\partial_\mu T = 0 ,\end{aligned}$$ which is, by using [Eq. (\[e:V1\])]{} again, equivalent to [Eq. (\[e:T2\])]{}, as was to be demonstrated. Next, we would like to obtain an equation for the temperature with our specific Equation of State as in [Eq. (\[e:eos\])]{} ($\varepsilon=\kappa{\left({T}\right)}p$), in the case when there is a conserved $n$ and $p=nT$. We can substitute these into [Eq. (\[e:energy\])]{}, and use the [(\[e:cont\])]{} continuity equation for $n$ to infer that [Eq. (\[e:energy\])]{} is equivalent to the following: $$\begin{aligned} T \partial_\mu u^\mu + {\frac{\mathrm{d}{}}{\mathrm{d}{T}}}{\left({\kappa T}\right)}u^\mu \partial_\mu T = 0 .\end{aligned}$$ Introducing $V$ by using [Eq. (\[e:V1\])]{}, we immediately see that this is equivalent to [Eq. (\[e:T1\])]{} , as stated in the text. Finally, let us show how the solution for a given $\kappa(p)$, described in Eqs. [(\[e:psol:s\])]{}–[(\[e:psol:p\])]{} can be obtained. In that case, instead of substituting the temerature to [Eq. (\[e:energy\])]{}, we write up the equation using the $\kappa(p)$ function and the relation $\varepsilon = \kappa\cdot p$, similarly to the previous cases: $$\begin{aligned} u^\nu {\left[{\frac{\partial_\nu V}{V}+{\left({\frac{\kappa}{p}+\frac{d\kappa}{dp}}\right)}\frac{\partial_\nu p}{\kappa+1}}\right]}=0.\end{aligned}$$ This equation is then solved by the implicit formula on the pressure, given in [Eq. (\[e:psol:p\])]{} [^1]: Note that it has been discussed in Ref. [@Beuf:2008vd] that the entropy flow can be calculated with an arbitrary EoS (speed of sound) from the Khalatnikov-potential, once the solution of the general Khalatnikov equation is known. [^2]: Note that with the $X{\left({t}\right)}$, $Y{\left({t}\right)}$, $Z{\left({t}\right)}$ time-dependent axes introduced as here, we can write the velocity field as $$\begin{aligned} {\mathbf{v}}={\left({\frac{\dot X}{X}r_x,\frac{\dot Y}{Y}r_y,\frac{\dot Z}{Z}r_z}\right)},\end{aligned}$$ which underlines the resemblance of this solution to certain non-relativistic exact solutions with Hubble-like expansion [@Csorgo:2001ru].

--- author: - 'E. T. Seppälä and M. J. Alava' date: 'Received: December 12, 2000 / Revised version: April 11, 2001' title: 'Energy landscapes, lowest gaps, and susceptibility of elastic manifolds at zero temperature' --- Introduction {#intro} ============ In this paper we study zero-temperature or ground state elastic manifolds that are roughened by bulk disorder, in the presence of an external field. Such objects are relevant in many contexts of condensed matter and statistical physics [@Fis86; @Emig99]. The essential point here is a competition between the elasticity, which prefers flat manifolds, and the disorder, which induces wandering in order to take advantage of the low energy regions in the system. This leads to a glassy, complicated energy landscape and the fact that the quenched randomness dominates thermal effects at low temperatures. In two embedding dimensions (2D) such manifolds are under the name directed polymers (DP) [@KPZ; @HaH95; @Las98] particularly interesting through their connection to the celebrated Kardar-Parisi-Zhang (KPZ) equation of nonlinear surface growth. Directed polymers have an experimental realization as vortex-lines in granular superconductors [@blatter]. In higher dimensions elastic manifolds may be best considered as domain walls (DW) in ferromagnets with quenched impurities. Elastic manifolds have also other connections to charged density waves, the sine-Gordon model with disorder, random substrate problems, and vortex lattices, to name but a few [@blatter; @Cardy82; @Toner90]. Let us start by introducing the classical spin-half random Ising Hamiltonian $${\mathcal H} = - \sum_{\langle ij \rangle} J_{ij} S_i S_j - \sum_i H_i S_i, \label{RHamilton}$$ where $J_{ij}$ is the coupling constant between the nearest-neighbor spins $S_i$ and $S_j$, and $H_i$ is a field assigned to each spin. To this system we apply antiperiodic or domain wall -enforcing boundary conditions in one direction. The spins in the opposite boundaries, let us define in $z$-direction, $z=0$ and $z=L_z$ are forced to be up and down, respectively. In the case of ferromagnetic random bond (RB) Ising systems one has $J_{ij} \geq0$ and $H_i=0$. In the minimum energy state the spins prefer to align on each side of the induced domain wall. When $J_{ij} \lessgtr 0$ the spins become frustrated and the task to find the ground state (GS) structure is most often related to spin glass physics [@review; @droplet; @replica]. On the other hand, when for simplicity $J_{ij} = {\rm const.} = J > 0$, and $H_i \lessgtr 0$ one arrives at random field (RF) Ising systems. The random field Ising model (RFIM) has an experimental realization as a diluted antiferromagnet in a field. In the RFIM the ferromagnetic couplings compete with the random field contribution, which prefers in the ground state to have the spins to be oriented towards the field assigned to them. Here we concentrate mainly on the random bond Ising Hamiltonian, i.e., $J_{ij}>0$ and $H=0$, and in some special cases extend the discussion to RF domain walls as well. In the simplest case the spins are located in a square lattice, in $d=2$, or a cube in $d=3$, so that the lattice orientation is in the {10} and {100} directions, respectively. The elastic manifold is the interface, with the dimension $D=d-1$, which divides the system in two parts of up and down spins. At $T=0$ the problem of finding the ground state domain wall, which minimizes the path consisting of unsatisfied bonds between the spins on opposite sides of the domain wall becomes “global optimization”. In our case the displacement field is one-dimensional, $n=1$, but one can certainly think about generalizations so that the total dimension of a system $d=(D+n)$, where $n \geq 1$ is the dimension in which the manifold is able to fluctuate. The continuum Hamiltonian of elastic manifolds with an external field and $n=1$ may be written as $${\mathcal H} = \int \left[ \frac{\Gamma}{2} \{ \nabla z({\bf x}) \}^2 + V_r \{ {\bf x},z({\bf x})\} + h\{z({\bf x})\} \right] \, {\rm d}^D{\bf x}. \label{H}$$ The elastic energy is proportional to the area of the interface given by the first term, and $\Gamma$ is the surface stiffness of the interface. The second term of the integrand comes from the random potential, and the last term accounts for the potential caused by the external field. The use of random bond disorder means that the random potential is delta-point correlated, i.e., $\langle V_r ({\bf x},z)V_r ({\bf x'},z') \rangle =2 {\mathcal D} \delta ({\bf x}- {\bf x'})\delta (z-z')$. In the random field case $\langle V_r ({\bf x},z)V_r ({\bf x'},z') \rangle \sim \delta ({\bf x}- {\bf x'}) (z-z')$. The Hamiltonian (\[H\]) is also applicable to wetting in a three phase system, where two of the phases are separated by an interface in a random bulk [@Lip86; @Wuttke91; @Dole91]. In that case $h(z)$ is equivalent to the chemical potential, which tries to bind the interface to the wall, and competes with the random potential, in the presence of which the interface tends to wander in the low energy regions of the system. Below the upper critical dimension the geometric behavior of elastic manifolds is characterized by the spatial fluctuations. For the mean-square fluctuations one has $$w^2 = \left \langle \left[z({\bf x}) - \overline{z({\bf x})} \right ]^2 \right \rangle \sim L^{2 \zeta}, \label{w}$$ where $z({\bf x})$ is the height of the interface with the mean $\overline{z}$, ${\bf x}$ is the $D=d-n$ dimensional internal coordinate of the manifold, $L$ is the linear size of the system, and $\zeta$ is the corresponding roughness exponent. At low temperatures in $(D+n)=(1+1)$ dimensions with RB disorder, i.e., when one actually considers a directed polymer in random media [@KPZ; @HaH95], the roughness exponent is calculated exactly via the KPZ formalism to be $\zeta=2/3$. In higher manifold dimensions $D$ with $n=1$ the functional renormalization group (FRG) calculations give the approximatively values $\zeta \simeq 0.208(4-D)$ for RB disorder and $\zeta = (4-D)/3$ for RF disorder [@Fis86; @new]. The expression for $\zeta$ tells also that the upper critical dimension for the elastic manifold is $D_u=4$. For manifolds with varying $n$ and $D$ Balents and D. Fisher have derived using FRG $\zeta \simeq [(4-D)/(n+4)]\{1+(1/4e)2^{-[(n+2)/2]}[(n+2)^2/(n+4)] [1-\ldots]\}$ [@Balents]. At zero temperature the total average energy $\overline{E}$ of an elastic manifold equals its free energy and grows linearly with the system size $L^D$ and its fluctuations scale for all $n$ as $\Delta E = \left \langle ( E- \overline{E} )^2 \right \rangle^{1/2} \sim L^\theta$, where [@HuHe] $$\theta = 2 \zeta +D -2 \label{HuHe}$$ is the first non-analytic correction to the energy. The same hyperscaling law holds for RF manifolds, too, in $D>1$ [@Seppala98]. Having a positive $\theta$ implies that the temperature is an irrelevant variable in the renormalization group (RG) sense and the $T=0$ fixed point dominates. For $D=1$, $n > 2$ there exists a $T_c$, and $T=0$ fixed point dominates only below $T_c$, likewise always for $n\leq 2$ [@Fish91]. At the randomness dominated pinned phase the temperature is “dangerously” irrelevant, which means that in RG calculations the interesting correlation functions cannot be obtained by setting $T$ to zero. Above $T_c$ the fluctuations become random walk -like with $\zeta=1/2$ and $\theta =0$. For $D=1$ with $n>1$ there is no exact result existing for the roughness exponent below $T_c$, and hence whether $\zeta \to 1/2$ and $\theta \to 0$ for a finite $n_c$ is still an open question – that is, what is the upper critical dimension of the KPZ equation. Elastic manifolds self-average in the sense that the intensive fluctuations of the roughness and the energy decrease with system size. However, in this paper we will show that, due to the glassiness of the energy landscape, for example the behavior of the mean position of a typical example of a manifold $\overline{z({\bf x})}$ does not coincide with the disorder average, here denoted by $\langle \, \rangle$, over many realizations with different random configurations. Introducing the external field to a random system induces often drastic changes and one experimental possibility is to study the ground state behavior e.g. by measuring the susceptibility. In disordered superconductors the external field is due to the current density $j$, which drives the vortex lines. Here we study the elastic manifolds in an external field while being especially interested in the information it gives about the energy landscape of the random system [@Hou1], which is closely related to the susceptibility. The external field is applied to the coupling constants \[see Eq. (\[H\])\] so that $J_\perp(z) = J_{random} + h(z)$, where $J_\perp$ are ones in the $z$-direction and $h$ is the amplitude of $h(z)$, the strength of the external field. Since we have the field potential linear in height, $h(z)= hz$, in ferromagnetic random systems with a domain wall the external field $h(z)$ may be transformed to a constant external field term $-H\sum_i S_i$ in the Hamiltonian (\[RHamilton\]). Our results are a systematic extension of early numerical work by Mezárd and relate to more recent discussion of the energy landscapes of directed polymers by Hwa and D. Fisher [@Mez90; @Hwa94]. Due to the glassiness of the energy landscape the position changes in “first order type” large jumps, at sample-dependent values of the external field. The second scenario in which the perturbations take place locally via “droplet”-like excitations is ruled out by a scaling argument and numerical results. We start by studying a specific case where the number of valleys is fixed to a constant. We first consider the case without the external field, and find the scaling behavior of the energy difference of the global energy minimum and the next lowest minimum of a manifold numerically and derive it also from an extreme statistics argument. The scaling of the gap does not only depend on the energy fluctuation scale, but has in addition a logarithmic factor dependent on the number of the low energy minima. The gap scaling is extended to the finite external field case, to derive the susceptibility of the manifolds under the assumption that the zero-field energy landscape is still relevant. The corresponding numerics allows us also to deduce the effective gap probability distribution without any a priori assumptions. We then proceed by constructing a mean field argument for the finite size scaling of the first jump field in a more general case. This agrees with “grand-canonical” numerics in which the manifold is allowed to have an arbitrary ground state position in the system: the number of valleys in the energy landscape fluctuates and the important physics is caught by the simple scaling considerations. Finally, we discuss wetting in random systems in the light of our results, and compare with the necklace theory of M. Fisher and Lipowsky that applies in the limit of large fields. We also consider finite temperatures, and relate the physics at finite external fields to the Kardar-Parisi-Zhang growth problem through the arrival time mapping. The implications of our results in that case concern the [*two first*]{} arrival times and their difference of an interface to a prefixed height. Note that we have published short accounts of some of the work contained here earlier[@Seppala00; @unpublished]. The paper is organized so that it starts with a short review of the thermodynamic behavior of the interfaces in random media at zero temperature in the presence of an applied field, and of the folklore concerning energy landscapes, in Section \[thermo\]. Section \[glassy\] introduces the mean-field level behavior of the interface when the external field is applied. The analytical probability distribution of the first [*jump*]{} field is derived and its relation to the susceptibility is discussed in Section \[argumentti\]. In the section also the lowest energy gaps between the local energy minima and the global minimum are derived from extreme statistics arguments. The numerical method of calculating the first jump field $h_1$ of an interface in a fixed height is introduced in Section \[numerics\]. Section \[results\] contains the numerical results of the first jump field in $(1+1)$ and $(2+1)$ dimensions with the corresponding jump geometry statistics as well as the energy gaps of the first lowest energy minima. The results are compared with the analytical arguments presented in the previous section. In Section \[first\_jump\] a mean field argument for the first jump field of an interface which lies originally in an arbitrary height is derived and compared with the numerical data and the evolution of the jump size distributions is studied. As an application for the finite field case the wetting phenomenon is considered and the corresponding wetting exponents are numerically studied in Section \[wetting\]. The arguments presented in the paper are discussed from the viewpoint of interface behavior at low but finite temperatures in Section \[creep\] and in the context of first arrival times in KPZ growth in Section \[secKPZ\]. The paper is finished with conclusions in Section \[concl\]. Thermodynamic considerations {#thermo} ============================ The interesting behavior arises since in the low temperature phase e.g. directed polymers are found to have [*anomalous*]{} fluctuations resulting from the regions of the random potential with almost degenerate energy minima, which are separated by large energy barriers [@Fish91; @Mez90; @Hwa94]. These spatially large-scale low-energy excitations are rare, but are expected to dominate the thermodynamic properties and cause large variations in the structural properties at low temperatures. We will use the arguments in the next section in order to study the movement of the manifolds in equilibrium when an external field is applied. Hwa and D. Fisher [@Hwa94] derived that for a polymer fixed at one end a small energy excitation with a transverse scale $\Delta$ ($\simeq l^\zeta$, where $l$ is the linear length of the excitation), scales as $\Delta^{\theta/\zeta}$ and gives rise to large sample-to-sample variations in the two-point correlation function and dominates the disorder averages. The probability distribution for the sizes of the excitations was found to be a power-law from the normalization of “density of states” of small energy excitations $W \sim \Delta^{-n} l^{-\theta} \sim 1/\Delta^{n+2-1/\zeta}$, since $\theta =2\zeta -1$, Eq. (\[HuHe\]). This is in contrast to the high temperature phase one, where $W$ is Gaussian. For the finite field case Hwa and D. Fisher argued that the power-law distribution for the excitations is due to the statistical tilt symmetry, which means that the random part of the new potential of the polymer is statistically the same as the old one, when the polymer is excited by an applied field. In Ref. [@Mez90] Mézard showed numerically that the energy difference of two copies of the polymer in the same realization of the random potential scales as $L^\theta$, where $\theta=1/3$ in $d=2$. He also showed that the ground state configuration of a polymer, which is fixed at one end and applied with an external field $h$ to the end point of the polymer, i.e., $h\{z({\bf x})\} = hz\delta(x-L)$ in Hamiltonian (\[H\]), changes abruptly with a distance $L^\zeta, \zeta=2/3$, when the field is increased by $L^{\theta}$. Calculating the variance of the end point of the polymer $var(z) \sim \langle z^2 \rangle - \langle z \rangle^2$ Mézard found that typically it is zero, i.e., the ground state configuration does not change. With the probability $L^{-\theta}$, there is a sample for which $h=0$ is the critical value of $h$, and the configuration changes with a large difference compared to the original state $L^{2\zeta} \sim L^{1+\theta}$, Eq. (\[HuHe\]). Hence the total average variance becomes $var(z)_{ave} \sim L$. In this paper we generalize such studies as discussed later. The physics of DP’s have been shown by Parisi to obey weakly broken replica symmetry [@Parisi90], a “baby-spin glass” phenomenon. This means that the replica symmetric solution of a DP is degenerate with the solution with the broken replica symmetry. This is also evidence for an energy landscape with several, far away from each other, nearly degenerate local minima. Mézard and Bouchaud studied later the connection between extreme statistics and one step replica symmetry breaking in the random energy model (REM) [@derrida], in a finite, one-dimensional form [@boumez]. They computed using extreme statistics the probability distribution of the minimum of all the energies $\frac{\gamma}{2}x^2 + E(x)$, where $E(x)$ are random and follow a suitable probability distribution so that its tails decay faster than any power-law, e.g. Gaussian. This can be seen as a toy model for an interface in a random media, in which case the quadratic part plays the role of the elasticity. The minimum of the total energy is easily seen to be distributed according to the Gumbel distribution [@Gumbel] of extreme statistics, and likewise for the position of the interface the shape of the probability distribution to be approximately Gaussian. In Section \[argumentti\] we will use extreme statistics to study the scaling of the lowest minima and energy gaps in elastic manifolds in a situation analogous to that of Bouchaud and Mézard. The response of the manifolds pinned by quenched impurities to perturbations was discussed in terms of thermodynamic functions by Shapir [@Shapir91], including the susceptibility of the manifolds. The formal definition of the susceptibility for a $D$ dimensional manifold in a $d$ dimensional embedding space reads: $$\chi = \lim_{h \to 0+} \left \langle \frac{\partial m}{\partial h} \right \rangle, \label{Eqsuskis}$$ where the change in the magnetization of the whole $d$ dimensional system is calculated in the limit of the vanishing external field from the positive side and the brackets imply disorder average. In Section \[results\] we derive the susceptibility based on energy landscape arguments. By assuming smooth, analytic behavior in the manifolds’ thermodynamic functions, Shapir found that the susceptibility (for a surface of dimension $D$) is proportional to the displacement of the manifold in the limit of small field, which is applied uniformly to the whole manifold, and $\chi_{D} \sim d^2 E / dh^2 \sim L^{\theta +2 \tilde{\alpha}}$. $\tilde{\alpha}$ results from an argument concerning the energy gap and the external field: $\Delta E \sim h L^\zeta L^D$, where the external field couples to a droplet of size $L^\zeta$ and the area of the manifold is $L^D$. This should be equal to the energy difference: $\Delta E \sim L^{\theta+\tilde{\alpha}}$, and thus $\tilde{\alpha} = 2 - \zeta$, since $\theta =2\zeta+D-2$. Hence the susceptibility was derived to be $\chi_D \sim L^{D+2}$ and the susceptibility per unit hyper-surface vary as $L^2$ for a manifold of scale $L$. This surprisingly is [*independent*]{} of the type of the pinning randomness. The landscapes of DP’s have also been studied by Jögi and Sornette [@Joe98] by moving both end points step by step (with periodic boundary conditions) from $(0,z)$ and $(L,z)$ to $(0,z+1)$ and $(L,z+1)$, and finding the ground state at each step. They studied the sizes of the [*avalanches*]{} as in self-organized systems, i.e., the areas the polymer covers when moving to the next position. They found a power-law for the avalanche sizes $P(S)\, dS \sim S^{-(1+m)} \, dS$, where $m =2/5$, up to the maximum size $S_{max} \sim L^{5/3}$. $S_{max}$ is actually the size of the largest fluctuation of the polymer in such a controlled movement, i.e., $L \times L^{\zeta}$. The authors did not however study the energy variations of the manifold during the “dragging process”. In the limit of a large $h$ the Hamiltonian of Eq. (\[H\]) is applicable to (complete) wetting interface in random systems with three phases. In this case the manifold is the interface between i) non-wet and ii) wet phases near iii) a hard wall at $z=0$. This problem was studied by mean-field arguments by first Lipowsky and M.Fisher and later elaborated numerically [@Lip86; @Wuttke91]. The choice of wetting potential that corresponds to the linear field used here is called in the wetting literature the weak-fluctuation regime (WFL). In the WFL regime one can derive the wetting exponent $\psi$ by a Flory argument, since the mean distance of the interface from the inert wall $\overline{z}$ is of the order of the interface transverse fluctuations $\xi_\perp$, see Fig. \[fig1\], i.e., the field is strong enough to bind the interface to the wall. The Hamiltonian can now be minimized by estimating the total potential to be $V_{tot} = V_{fl}(\overline{z})+ V_W(\overline{z})$. $V_W \sim h z$ is the wetting potential induced by the external field $h$. The fluctuation-induced potential $V_{fl}$ is the potential of the free interface $V_{el} + V_r = \frac{\Gamma}{2} \{ \nabla z({\bf x}) \}^2 + V_r ({\bf x},z)$, which gives, when minimized, $\xi_\perp \simeq \xi_\|^\zeta$. The elastic energy term $V_{el}$ gives the scaling of $V_{fl}$ with respect to the correlation lengths, and expanding the square-term of the gradient of the height gives $\rm{const}_1 + \rm{const}_2 \left| \frac{\xi_\perp}{\xi_\|} \right |^2 +$ higher order terms. So, $V_{fl} \sim \xi_\perp^{-\tau}$, where the decay exponent $\tau = (2-2\zeta)/\zeta$ defines the scaling between the pinning and elastic energies. Minimizing $V_{tot}$ in WFL, where $\overline{z} \simeq \xi_\perp$ and is much larger than a lattice constant, gives $$\overline{z} \sim h^{-1/(1 +\tau)} \sim h^{-\psi} . \label{cwexp}$$ Thus the wetting-exponent $\psi = 1/(1+\tau)$ becomes $$\psi = \frac{\zeta}{2-\zeta}. \label{psi}$$ Huang [*et al.*]{} [@Hua89; @Wuttke91] did numerical calculations for directed polymers at zero temperature using transfer-matrix techniques for slab geometries, $L \gg L_z$, where $L$ is the length of the polymer, and $L_z$ is the height of the system, confirming roughly the expected exponent $\psi \simeq 0.5$. Our results, presented in Section \[wetting\] are in line with the expectations in (1+1) and we also present the first studies in (2+1) dimensions. Glassy energy landscapes: length scale of perturbations {#glassy} ======================================================= We next employ a version of the scaling arguments by Hwa and Fisher, and Mézard for the next optimal position of the directed polymer. The goal is to investigate the preferred length scale of the change that takes place, i.e., the size of the “droplet” created with the field. The numerical calculations presented in this paper are done such a way, that first the ground state is searched and the configuration stored. After that the external field is applied and the energy is minimized again. If there is a change in the configuration, the difference in the mean height of the manifold, the so called “jump size” is analyzed. Most of the studies are done in isotropic systems, i.e., the height $L_z$ of the system is of the order of the linear length of the manifold, $L_z \propto L$. One finds that the behavior of the polymer has a [*first order character*]{}: the optimal length scale is such that the whole configuration changes, and this holds for higher dimensional manifolds, too. Hwa and Fisher defined the next optimal position of the DP fixed at one end and with displacement $\Delta$ to scale as $\Delta \sim L^{\zeta}$, where $L$ is the length of the polymer. Now one assumes that the energy [*gaps*]{} between two copies of the polymers with overlap, and energies $E_0$ and $E_1$, grows as $E_1 -E_0 \sim L^\theta$, i.e., is just energy fluctuation exponent of a polymer (as demonstrated by Mézard numerically). Then as a generalization with $n=1$ it follows that $E_1 -E_0 \sim \Delta^{\theta/\zeta}$. The external field has a contribution for the energy differences of interfaces with the dimension $D$ $E_1 -E_0 \sim h L^D \Delta \sim h \Delta^{1+D/\zeta}$. Assuming that this difference balances the gap, and using the hyperscaling law for the roughness and energy fluctuation exponents $\theta = 2\zeta+D-2$, we get $$h \sim \Delta^{\overline{\alpha}} = \Delta^{(\zeta-2)/\zeta}. \label{excitation}$$ The exponent $\overline{\alpha}$ is negative assuming that the roughness exponent is below two, which is of course satisfied in the case of both RF and RB domain walls. Hence, the smaller the field the larger the excitations and thus large excitations are the preferred ones, at least below the upper critical dimension. Besides RB systems Eq. (\[excitation\]) works only for RF systems with a ferromagnetic bulk (among others, the relation $E_1 -E_0 \sim L^\theta$ has not been demonstrated to hold for RF interfaces). If the bulk of a RF magnet is paramagnetic, the stiffness of DW’s is expected vanish on large enough length scales, translational invariance is broken, and $\Delta E = E_1-E_0$ does not scale. The scaling relation (\[excitation\]) should hold also for general $n$. One should note in the case $h$ is applied in one direction but the droplet may extend in $n$ dimensions, in order to Eq. (\[excitation\]) to hold, also the [*projection*]{} of the droplet in the applied field direction has to have the scaling as $\Delta^{\theta/\zeta}$. In Fig. \[fig2\] it is shown what happens for two different random realizations of DPs, when a perturbing external field is applied. Fig. \[fig2\](a) shows the mean height of the polymer normalized by its original height $\overline{z}/\overline{z}_0$, when the external field is increased. The first realization $1^\circ$ shows a large [*jump*]{} of a size half of the height of its original position at [*first jump field*]{} $h_1 = 8 \times 10^{-5}$. In Fig. \[fig2\](b) the two positions of $1^\circ$, before and after the first jump, $z_0(x)$ and $z_1(x)$, are shown. The other scenario would be that the directed polymer would continuously undergo small geometric adjustments and get meanwhile bound to the wall, $z=0$. An example of such a droplet, is shown in Fig. \[fig2\](b) as the second realization $2^\circ$, see also Fig. \[fig1\]. However, a further field increase, demonstrated in Fig. \[fig2\](a) for $2^\circ$, leads to a global jump too. This leads to a picture in which assuming a starting position far enough from the system boundary, a finite number of large jumps exists from the original position $z_0({\bf x})$ to the positions $z_1({\bf x}), z_2({\bf x}), \ldots, z_n({\bf x}), \ldots $, closer and closer to the wall, i.e., $z_0({\bf x}) > z_1({\bf x}) > z_2({\bf x}) > \ldots > z_n({\bf x}) > 0$. The mean jump length is defined to be $\Delta z_n = \overline{z}_{n-1} - \overline{z}_{n}$, and the jumps take place at the fields $h_1, h_2, \ldots, h_n, \ldots$. The field value $h_1$ corresponds the jump from $z_0({\bf x})$ to $z_1({\bf x})$. Finally, after a finite number of jumps, with the interface being in the proximity of the wall, around $h \simeq 2 \times 10^{-2}$ in Fig. \[fig2\](a), see also Fig. \[fig1\], the interface evolves inside the last valley next to the wall continuously, and the wetting behavior, Eqs. (\[cwexp\]) and (\[psi\]), applies. We discuss this picture of consequent jumps before the wetting regime in Section \[first\_jump\] together with the numerical results. The global changes (large jumps) induce finite changes in the magnetization ($m$), and are reminiscent of first order phase transitions. Note that the field $h_1$ of the first change or jump obeys for any particular ensemble a probability distribution, and thus a co-existence follows between systems that have undergone a jump or change in $m$ and those that are still in the original state. This “transition” is a result of level-crossing between the valleys in the energy landscape for the interface, sketched in Fig. \[fig3\]. To change between the geometrically separated minima, an external field is applied, which at jumps plays the role of a latent heat. Originally the interface lies at height $\overline{z}_0$ and has an energy $E_0$. When the field is applied interface’s energy increases by $h L^D \overline{z}_0$ and at $h_1$ the interface jumps to $\overline{z}_1 < \overline{z}_0$ having energy $E(h_1)=E_1 + h_1 L^D \overline{z}_1 = E_0 + h_1 L^D \overline{z}_0$, where $E_1$ is the energy of the interface at $z_1$ without the field. Similar behavior takes place at $h_n$, $n=2,3,\dots $, when the interface moves from $\overline{z}_{n-1}$ to $\overline{z}_n < \overline{z}_{n-1}$, until the cross-over to the wetting regime. In the thermodynamic (TD) limit, i.e., for very large systems, when $L, L_z \to \infty$, the interface may originally be located anywhere in the system. Assuming that the roughness exponent $\zeta$ does not define only the width of the manifold, but also the width of the minimum energy valley, i.e., $L^\zeta$ is the only relevant length scale in the transverse direction, gives that the manifold should have $N_z \sim L_z/L^\zeta$ minima from which to choose the global minimum position. For $\zeta <1$ the number of the minima grows with system size, if the geometry is kept isomorphic, $L_z \propto L$. The requirement $\zeta <1$ holds for all RB interfaces; and for RF ones with ferromagnetic bulk, when $D>1$. In the special case that the local minimum closest to the wall coincides with the global minimum, the wetting regime is entered at once without any jumps. The prediction of Eq. (\[excitation\]) that large scale excitations should dominate is an asymptotic one, and for finite system sizes one may find also small excitations, which are less costly in energy (Fig. \[fig2\]). However, the fraction of these cases decreases with system size, see Fig. \[fig4\]. It depicts in $d=2$ up to $L^2 = 1000^2$, with at least $N=3000$ realizations per data point, the probability of finding a non-zero overlap $q$ between the interfaces before and after the first jump. $q$ is the disorder average of the fraction of the samples, which have overlap between the cases before and after the jump, i.e., for at least one $x$ $z_0(x) = z_1(x)$. The probability goes as $q \simeq 0.56L^{-0.23 \pm 0.01}$, and confirms the expected behavior that for isotropic systems in the $L \to \infty$ limit the first jump is always without an overlap, i.e., a macroscopic one. A first guess would give that $q \sim 1/N_z \sim L^{\zeta-1}$. A similar trend seems to exist in $(2+1)$ dimensions, too, but our statistics is not good enough to determine the exponent of $q$ reliably. This is since $q$ is averaged over a binary distribution (overlapping and non-overlapping jumps), and thus very large ensembles would be needed. Extreme statistics arguments {#argumentti} ============================ Next we compute the field $h_1$ and the susceptibility, Eq. (\[Eqsuskis\]), by taking into accont the “co-existence” of the original ground state and the state after the jump. Assuming that the relevant process in the response of a domain wall is the large-scale jump, the susceptibility per spin of a system with a domain wall, Eq. (\[Eqsuskis\]), may be written in the form $$\chi = \lim_{h \to 0+} \left \langle \frac{\Delta m(h)}{\Delta h} \right \rangle \simeq \left \langle \frac{\Delta z_1}{L_z} \right \rangle \lim_{h \to 0+} P(h_1), \label{Eqsuskis2}$$ where $P(h_1)$ is the probability distribution of the first jump fields with the corresponding first jump size $\Delta z_1$, and the magnetization $m(h) \sim z(h)/L_z$. The limit $h \to 0+$ is taken over the probability of having a jump, and thus the susceptibility will reflect the co-existence phenomenon discussed above. One uses also a plausible assumption, found to be true in the numerics, that the change in the interface does not depend on the threshold field $h_1$. A further simplification is obtained by assuming (this will be shown numerically later) that the jumps have an invariant size distribution, independent of $L$ and $L_z$, after a normalization with $L_z$. This gives $\langle \Delta z_1 \rangle \sim L_z$. Hence the finite size scaling of the susceptibility per spin depends only on the probability distribution of the first jump field in the limit of the vanishing external field, and in particular on the “rare events” measured by such a distribution. The dependence of $h_1$ on the anomalous scaling is found to be true not only for directed polymers but also for the higher dimensional case. Such behavior is contrary to Shapir’s work that assumed smooth, analytic thermodynamic functions and it is thus no surprise that the scaling of the susceptibility, derived in detail in Section \[results\], differs from that. Next we first derive the distribution for the first jump field and its relation to the lowest energy gap probability distributions. Then the probability distribution and the finite size scaling of the lowest gaps and also the lowest energy level are derived using extreme statistics arguments (see also [@Galambos; @DBL] and our shorter account of parts of this work [@unpublished]). These gaps have a logarithmic dependence on the number of local minima. In order to derive the probability distribution of the first jump field $P(h_1)$, let us first calculate the probability $P_0$, that with a certain test field $h'<h_1$, the interface has not changed yet, i.e. $$P_0(h_1 >h')= 1-P_0(h_1 \leq h') = 1-\int_0^{h'} P(h) \, dh. \label{P0}$$ By differentiating $P_0(h_1 >h')$, one gets the probability distribution of the first jump field, since $$P(h_1) = \frac{\partial}{\partial h} \left. P_0(h_1 \leq h') \right|_{h'=h_1}. \label{P0derv}$$ We assume all the minima to be non-correlated and well separated from each other, see Fig. \[fig5\](a). There we have a global energy minimum $E_0$ at $z_0$ (note, that here the energy values as well as the field contributions are normalized by $L^D$, constant with a fixed system size), and energy gaps $\Delta E_1 = E_1 - E_0$ and $\Delta E_{1^*} = E_{1^*} - E_0> \Delta E_1$ with energies $E_1$ at $z_1$ and $E_{1^*}$ at $z_{1^*}$, respectively. Then we apply the field $h$, i.e., we tilt the energy landscape, see Fig. \[fig5\](b). Due to the statistical tilt symmetry the external field $h$ is assumed not to change the shape of the random landscape. At the lowest tilt to move the interface, i.e., at field $h_1$, it moves to $z_{1^*}$, since with the field $h_1$ $E_0(h_1) = E_0 +h_1z_0 = E_{1^*}(h_1) =E_{1^*}+h_1z_{1^*} < E_1(h_1) = E_1+h_1z_1$. However, sometimes $E_1$ and $E_{1^*}$ coincide, so that the interface jumps to the true second lowest minimum. Now we index the positions of the $N_z$ minima as $z_i$ (need not to be in order), so that $h(z_0-z_i) = h \Delta z_i$, and we have $$\begin{aligned} P_0(h_1>h') = \prod_{i=1}^{N_z} {\rm prob}_i(\Delta E_i > h\Delta z_i) \nonumber \\ = \prod_{i=1}^{N_z} \left\{ 1- \int_0^{\Delta z_ih} \hat{P}_i(\Delta E_i) \, d(\Delta E_i) \right\}. \label{P0prod}\end{aligned}$$ By assuming all the gap energies from the same probability distribution $\hat{P} (\Delta E)$, setting $\Delta z_k \hat{=} \frac{k}{N_z}$ and taking the continuum-limit we get for the probability distribution of the first jump field using Eqs. (\[P0\]), (\[P0derv\]) and (\[P0prod\]), $$\begin{aligned} P(h_1) = \exp\left[-\int_1^{N_z} \int_0^{kh_1/N_z} \hat{P}(\Delta E) \, d(\Delta E) \, dk \right] \times \nonumber \\ \int_1^{N_z} \frac{ \hat{P}(kh_1/N_z)}{1 -\int_0^{kh_1/N_z} \hat{P}(\Delta E) \, d(\Delta E)} \, dk. \label{Ph1}\end{aligned}$$ There are two approaches to go on from the distribution of Eq. (\[Ph1\]): one may either compare its prediction with numerics with a trial distribution for $\hat{P}$, or alternatively proceed by a complete Ansatz for the valley energies, from which the gap naturally follows. We now attempt the latter, and comment on the first one together with the numerics. In order to calculate the probability distribution for the gap energies $\hat{P}(\Delta E)$, we use a probability distribution for the energy minima suitable for directed polymers, which has the advantage of decaying faster than any power-law, ${\mathcal P}(E) \sim \exp(-BE^\eta)$, $\eta>0$. For directed polymers with the “single valley” boundary condition case, i.e., one end of the manifold fixed and thus no room for other minima, it is known mostly numerically that the bulk of the distribution of the energy of a directed polymer ${\mathcal P}(E)$ is Gaussian (the exponent $\eta =2$ in the exponential), except for the tails, ($E <E_{min}$ and $E > E_{max}$) in which different cut-offs, ($\eta_- =1.6$ and $\eta_+=2.4$) take over [@KBM; @HaH95]. For the other manifolds that one might be interested in the actual distributions have not generally been studied. In the following we usually approximate the distribution with Gaussian distribution (this is valid for $1/N_z \geq E_{min}$ and $< E_{max}$). The validity of the approximation is discussed in Section \[results\], when the analytical arguments are compared with the numerics. The distribution ($\eta =2$ for Gaussian) for ${\mathcal P} (E)$ is written in the form: $${\mathcal P}(E) = k \exp\left \{-\left( {|E - \langle E \rangle |\over \Delta E} \right)^\eta \right \}, \label{distr}$$ where $\langle E \rangle \sim L^D$ is the average energy of the manifold and $k \sim (\Delta E)^{-1} \sim L^{-\theta}$ normalizes the distribution. The extreme statistics argument goes so that in a system with $N_z \sim L_z/L^\zeta$ minima the probability for the lowest energy to be $E$ is $$L_{N_z}(E) = N_z {\mathcal P}(E) \left \{1-C_1(E) \right \}^{N_z-1}, \label{LNdistr}$$ where $C_1(E) = \int_{-\infty}^E {\mathcal P}(e) \, de$ is called the error-function when $\eta =2$. For Gaussian ${\mathcal P}(E)$, likewise for general $\eta>0$, $L_{N_z}(E)$ is known to be Gumbel distributed, $\exp(u-\exp u)$ [@boumez; @Gumbel; @Rowlands]. The gap $\Delta E_1$ follows similarly as $L_{N_z}(E)$. Its distribution, $G_{N_z}(\Delta E_1,E)$ is given by $$\begin{aligned} G_{N_z}(\Delta E_1,E) = {N_z(N_z-1)\over 2} {\mathcal P}(E) {\mathcal P}(E+\Delta E_1) \nonumber\\ \{1-C_1(E+\Delta E_1)\}^{N_z-2}. \label{DeE}\end{aligned}$$ $G_{N_z}(\Delta E_1,E)$ is the probability that if the lowest energy manifold has an energy $E$, then the gap to the next lowest energy level is $\Delta E_1$. $G_{N_z}(\Delta E_1,E)$ is actually a generalization of Gumbel distribution. Integrating Eq. (\[DeE\]) over all energies gives the probability distribution for the $\Delta E_1$ $$\begin{aligned} \hat{P}(\Delta E_1) = \int_{-\infty}^{\infty} G_{N_z}(\Delta E_1,E) \, dE. \label{DE1}\end{aligned}$$ This becomes for Gaussian ($\eta=2$) and $\Delta E_1 \ll \langle E \rangle$ $$\begin{aligned} \hat{P}(\Delta E_1) = \int_{-\infty}^{\infty} {N_z(N_z-1)\over 2} k^2 \exp \left \{ {- (E- \langle E \rangle)^2 -2\Delta E_1(E- \langle E \rangle) \over \Delta E^2 } \right \} \left \{ k\, {\rm erfc} \left( E +\Delta E_1- \langle E \rangle \over \Delta E \right) \right\}^{N_z-2} \, dE, \label{DE1_2}\end{aligned}$$ where erf denotes the error-function and erfc $=1-$ erf. Neglecting all ${\mathcal O} \left[ \left\{ k\, {\rm erfc} \left( E +\Delta E_1 - \langle E \rangle \over \Delta E \right) \right\}^2 \right]$ and higher order terms and using a Taylor expansion around $\Delta E_1=0$, i.e., Maclaurin-series, gives $$\begin{aligned} \hat{P}(\Delta E_1) = \int_{-\infty}^{\infty} {N_z(N_z-1)\over 2} k^2 \exp \left \{ {- (E- \langle E \rangle)^2 -2\Delta E_1(E- \langle E \rangle) \over \Delta E^2 } \right \} \nonumber \\ \left[ 1 -(N_z-2) k\, {\rm erf} \left( E - \langle E \rangle \over \Delta E \right) +(N_z-2) \Delta E_1 k \exp \left \{ {- (E- \langle E \rangle)^2 \over \Delta E^2 } \right \} \right ] \, dE. \label{DE1_3}\end{aligned}$$ The first two terms within the second parenthesis dominate for $\Delta E_1$ small, and thus we see that to first order of $\hat{P}(\Delta E_1) \sim$ const (compare with the numerics presented below). In particular, one should notice that the probability distribution does not vanish for $\Delta E_1 \simeq 0$. This approach is very similar to the extreme statistics calculation of the one-dimensional version of the random energy model by Bouchaud and Mézard [@boumez] (see Section \[thermo\]). They derived the probability distribution for the location that gives minimum energy for the system, with the difference to our case that Hamiltonian reads $\frac{\gamma}{2}x^2 + E(x)$. The result becomes such that the distribution for the position is approximately Gaussian, too. The one-dimensional system is close to our example except for the functional form of the external potential which is quadratic instead of a linear one. There is however one essential difference in that in their analysis the “field value” is fixed, whereas in our case we are interested in what happens in any particular sample as the field is varied. Nevertheless such a calculation could be compared to the distribution of the interface locations at a fixed field $h$. More important than the actual distributions is that the finite size scaling of the gap energies may be computed. Let us start by calculating the typical lowest energy level. The average of it is given by $$\langle E_0 \rangle = \int_{-\infty}^{\infty} E L_{N_z}(E) \,dE, \label{E0true}$$ and the typical value of the lowest energy is estimated from $$N_z \frac{1}{k} {\mathcal P}(\langle E_0 \rangle) \approx 1. \label{E0estimate}$$ Note, that in approximating the integral, Eq. (\[E0true\]), with the aid of the distribution, Eq. (\[E0estimate\]) (in the limit $C_1(E)$ in Eq. (\[LNdistr\]) is small) the normalization $1/k$ should be taken into account. Eq. (\[E0estimate\]) gives $$\langle E_0 \rangle \sim \langle E \rangle - \Delta E \left [ \ln (N_z) \right ]^{1/\eta}, \label{typicalene}$$ where $\Delta E \sim L^\theta$ and for Gaussian $\eta =2$. To estimate the typical value of the gap, we make a similar approximation as in Eq. (\[E0estimate\]) for $\langle E_0 \rangle$, $${N_z(N_z-1)\over 2 k^2} {\mathcal P}(\langle E_0 \rangle ) {\mathcal P}(\langle E_0\rangle + \langle \Delta E_1 \rangle ) \approx 1,$$ which with (\[typicalene\]) and the fact that $| \langle \Delta E_1 \rangle | \ll | \langle E_0 \rangle| $ yields, $$\langle \Delta E_1 \rangle \approx { \Delta E^\eta \over \eta (\langle E \rangle - \langle E_0\rangle)^{\eta-1}} \approx { \Delta E \over \eta \left [ \ln(N_z) \right ]^{(\eta-1)/\eta}}. \label{gap}$$ We thus find that the gap scales as $\Delta E_1 \sim \Delta E [\ln (N_z)]^{-1/2} \sim L^{\theta} [\ln (L_z L^{-\zeta})]^{-1/2}$, when $\eta =2$ and assuming as in the previous section, that $N_z \sim L_z/L^\zeta$. If the interfaces are flat, i.e., $\zeta =0$, which is true for $\{100\}$ RB interfaces below the cross-over roughening scale $L_c$ [@Bouchaud92; @Alava96; @Raisanen98; @unpublished2] if randomness is weak, the same arguments should hold. However, then the energy distribution is pure Gaussian, $\eta =2$, and $\theta =D/2$ due to Poissonian statistics. Susceptibility of manifolds {#suskis} =========================== Numerical method {#numerics} ---------------- For the numerical calculations the RB Hamiltonian (\[RHamilton\]) with $J_{ij} > 0$, $H=0$, has been transformed to a random flow graph. The graph is formed by the Ising lattice and two extra sites: the source and the sink; and the coupling constants $2J_{ij} \equiv c_{ij}$ between the spins correspond to flow capacities $c_{ij} \equiv c_{ji}$ from a site $S_i$ to its neighboring one $S_j$ [@Alavaetal]. The graph-theoretical optimization algorithm, a maximum-flow minimum-cut algorithm, enables us to find the bottleneck, which restricts the amount of the flow that can get from the source to the sink given the capacities in such a random graph. This bottleneck, a path $P$ which divides the system into two parts (sites connected to the sink and sites connected to the source) is the minimum cut of the graph and the sum of the capacities belonging to the cut $\sum_P c_{ij}$ equals the maximum flow, the smallest of all cut-paths in the system. The source is connected to the sites in the Ising lattice which are forced to be up and the sink is connected to the sites which are forced to be down. The value of the maximum flow is the total minimum energy of the domain wall, equivalent to the minimum cut. The maximum flow algorithms can be proven to give the exact minimum cut for all the random graphs, in which the capacities are positive semidefinite and with a single source and sink [@network]. The best known maximum flow method is by Ford and Fulkerson and called the augmenting path method [@FordF]. We have used a more sophisticated method called push-and-relabel by Goldberg and Tarjan [@Goltar88], which we have optimized for our purposes. It scales almost linearly, ${\mathcal O}(n^{1.2})$, with the number of spins and gives the ground state DW in about minute for a million of spins in a workstation. Notice that one could use for DP’s the usual transfer matrix method as well, but the max-flow implementation is convenient in the case of systematic perturbations to each sample. In this study we have done simulations for $(1+1)$ and $(2+1)$ dimensional manifolds. The system sizes extend to $L \times L_z =1000^2$ and $L^2 \times L_z = 400^2 \times 50$. The number of realizations $N$ ranges from $200$ to $2 \times 10^4$. The random bonds are such that in 2D $J_{ij,z} \in [0,1]$ uniform distribution and $J_{ij,x} = 0.5$ unless otherwise mentioned. In three dimensions we have either $J_{ij} \in [0,1]$ uniform distribution in all $x,y,$ and $z$ directions or dilution type disorder, $P(J_{ij}) = p \, \delta(J_{ij}-1) + (1-p) \, \delta(J_{ij})$, i.e., a bond has a value of unity or zero with the probability $p$. We have used $p=0.5$ and $p=0.95$. The linear field contribution is applied in the $z$-directional bonds as $J_\perp(z) = J_{random} + hz$. When the jump field values with the corresponding jump distances are searched, the precision is as small as $\Delta h =10^{-5}$ in order to be sure that no smaller changes would occur between the jumps, which could be the case if the precision was much larger. Periodic boundary conditions are used in $x$ and $y$ directions. When studying the susceptibility, we have controlled the number of the minima in the systems to avoid fluctuations in $N_z$ (and the grand-canonical ensemble). To fix the number of the minima $\langle N_z \rangle \sim L_z/L^\zeta$ in a certain system size, we have set the initial position of the interface $\overline{z}_0$ in a fixed size window at height $\overline{z}_0/L_z \simeq \rm{const}$. If the ground state interface is originally outside the window, large enough only for a single valley, it is discarded so that the statistics quoted are based on the successful attempts. If the window is well-separated in space from the $z$-directional boundaries it is obvious that this sampling has no effect on the statistical properties as the average energy. After an original ground state is found, with an energy $E_0$, the external field $h$ is applied by increasing the couplings by constant steps until the first jump is observed [@resgraph]. We have also calculated the energy gap between the global minimum and the next lowest minimum. In that case the initial position of the interface $\overline{z}_0$ is also set to be in a fixed size window at height $\overline{z}_0/L_z \simeq \rm{const}$ by discarding all the samples with the global minimum outside the window. After that the lattice is reduced so that bonds in and above the window are neglected and the new ground state, its energy $E_1$, and the corresponding gap energy $\Delta E_1$ are found. Although the discarding procedure is slow we have at least $N=500$ realizations up to system sizes $L=300$, $L_z=500$. Results ------- In order to compare the analytical arguments presented in Section \[argumentti\] and in the end to compute the total susceptibility of the manifolds we will first study numerically in $(1+1)$ dimensions the finite size scaling of the average lowest energy level. Then the shape of the energy gap distribution as well as the finite size scaling of the average gap energies are considered. After that the numerical results of the first jump field distribution are presented together with the finite size scaling of the average first jump field. The shape of the first jump field distribution and the finite size scaling of its average are shortly reported for $(2+1)$ dimensional interfaces, too. The susceptibility is derived from the distribution and the finite size scaling of the first jump field. In the end of the section the jump distance distributions are considered. In order to see the logarithmic correction of the lowest energy level $\langle E_0 \rangle$, Eq. (\[typicalene\]), when the height of a system, and thus the number of the minima, are increased, we have plotted in Fig. \[fig6\] for three different lengths of the directed polymers $L=100,200,$ and $300$ $$\frac{\langle E \rangle - \langle E_0 \rangle}{L^\theta} \sim \left [ \ln \left( N_z \right) \right ]^{1/2}, \label{typicalene2}$$ where $N_z \sim L_z/L^{\zeta}$, and $\langle E \rangle \simeq 0.365L+1$. The prefactor, 0.365, of the average energy of a polymer with only a single valley in a system, $\langle E \rangle$, we have estimated by calculating systems of sizes $L_z \times L$, where $L_z = 6.5\times w$, $w$ is the average roughness of a polymer, up to system sizes $50 \times 1000$ with $2000$ realizations. One should note, that the prefactor is highly sensitive to the disorder type and boundary conditions, c.f. Ref. [@Hansenetal], and the constant factor (unity) also to the estimate of the size of the single valley. We have used it as a free parameter. Although for the following results the relevant part of the tail of $\langle E \rangle$ should not be a pure Gaussian (at least if we have enough many minima $N_z$, so that the bulk of the distribution is avoided), we have used for $1/\eta = 1/2$ instead of $1/\eta = 1/1.6=0.625$ in the fit of the logarithmic correction. In practice one can not tell these two choices apart in the range of the system sizes used. The probability distributions for the energy gaps of directed polymers in system sizes $L^2 =100^2$ and $200^2$ are shown in Fig. \[fig7\](a). The distribution has a finite value at $\Delta E_1 =0$ and the tail has approximately a stretched exponential behavior, $\exp(-ax^b)$, with an exponent $b \simeq 1.3$. In the figure there is plotted as a comparison an exponential $\exp(-x)$ line, from which the deviation of the probability distribution is more clearly seen in the inset, where the distribution is in a natural-log scale. The finite value at $\Delta E_1 =0$ is consistent with the weak replica symmetry breaking picture. To derive the scaling function for the $\Delta E_1$ it is expected that in systems with height $L_z$ small enough to restrict the number of minima $\Delta E_1$ mainly depends on the height of the system $L_z$. On the other hand, when $L_z$ is large enough, there are enough valleys of which to choose the two minima, and one has $\Delta E_1 \sim \Delta E \sim L^\theta$, hence $$\langle \Delta E_1(L,L_z) \rangle \sim \left\{ \begin{array}{lll} \tilde {f}(L_z),&\mbox{\hspace{5mm}}&L_z \ll L, \\ L^{\theta},& &L_z \gg L, \end{array} \right. \label{limits}$$ when the smaller parameter being varied. Since it is assumed, that $L_z \sim L^\zeta$, a natural scaling form based on these limiting behaviors is, $$\langle \Delta E_1(L,L_z) \rangle\sim L^{\theta} f\left(\frac{L_z} {L^{\zeta}}\right). \label{scalingDE}$$ The argument $y = L_z/L^{\zeta}$ for the scaling function $f(y)$ is just a function of the number of the minima, i.e., $L_z/L^\zeta \sim N_z$, and the scaling function has the form from Eq. (\[gap\]), when $\eta=2$, $$f(y) \sim [\ln y]^{-1/2}. \label{scalingf} \label{Nscaling}$$ In Fig. \[fig7\](b) we have plotted the scaling function (\[scalingf\]) by collapsing $\langle \Delta E_1(L,L_z) \rangle/L^\theta$ versus $L_z/L^{\zeta}$ for various $L$ and $L_z$, and find a nice agreement confirming the scaling behavior Eqs. (\[scalingDE\]), (\[scalingf\]) as well as the analytic form Eq. (\[gap\]) again assuming a Gaussian distribution ($\eta =2$). Next we explore the first jump fields. Consider the relation of the gap distribution and that of the jump fields given by Eq. (\[Ph1\]). If we approximate $\hat{P}(\Delta E_1)$, Eq. (\[DE1\_3\]), with a uniform distribution, we get for the probability distribution of the first jump field $$P(h_1) = \exp \left[-\frac{N_z h_1}{2}\right] \frac{N_z}{h_1} \ln \left[ \frac{1- \frac{h_1}{N_z}} {1-h_1}\right] \sim \exp (-h_1). \label{Ph}$$ The form of $P(h_1)$ implies that it has a finite value at $h_1=0$, which is again consistent with the weakly broken replica symmetry picture [@Parisi90] and also with the discussion in Section \[argumentti\]. We have also tried exponential and power-law type of probability distributions for $\hat{P}(\Delta E_1)$ in Eq. (\[Ph1\]), but all trials with negative exponents vanishes too fast with $h_1$ compared to the numerical data, and all positive exponents have behaviors with $P(h_1 \to 0) \to 0$ and diverge for larger $h_1$. In Fig. \[fig8\](a) we have plotted the probability distribution of the first jump field for the system sizes $L^2 =100^2$ and $200^2$. The probability distribution of the first jump field is similar to the probability distribution of the gap energies. The analytic formula, Eq. (\[Ph\]), is drawn as a line in the figure. One clearly sees that the line is a pure exponential, $\exp(-x)$, and the deviation of the numerical data from the exponential is similar to Fig. \[fig7\](a) of the energy gap distributions. Hence the shape of the numerical first jump field distribution is approximately a stretched exponential, $\exp(-ax^b)$, with an exponent $b \simeq 1.3$. Based on Figs. \[fig7\](a) and \[fig8\](a) one sees, that the probability distributions of the energy gaps and first jump fields are the same. Thus with the correct $\hat{P}(\Delta E)$ one gets from Eq. (\[Ph1\]) the same $P(h_1)=\hat{P}(\Delta E)$. This distribution may be for small $h_1$ flat, but obviously starts to vanish for larger $h_1$ since $P$ decays faster than exponentially. Another reason can be the fact that by discarding samples with the GS outside of a predefined window we just constrain the number of valleys in the sample so that the expectation value is the same in each one. $N_z$ can however fluctuate from sample to sample. The most important consequence is in any case that $\hat{P}(\Delta E_1=0)=P(h_1=0) \not=0$. The finite size scaling and the normalization of the probability distribution of $\Delta E_1$ give $\hat{P}(\Delta E_1=0) \sim L^{-\theta} [\ln(L_z L^{-\zeta})]^{1/\eta}$. In order to find the scaling relation for the first jump field $h_1$, we make the Ansatz $\langle \Delta E_1 \rangle = \langle h_1 \rangle L L_z$, since the field contributes to the manifold energy proportional to $L^D$ ($D=1$) and $L_z \sim \langle \Delta z_1 \rangle$ is the difference in the field contributions $hz$ to the energy at finite $h$ at different average valley heights $z_0$, $z_1$. Hence $$\langle h_1(L,L_z) \rangle L L_z \sim L^{\theta} f\left(\frac{L_z}{L^{\zeta}}\right), \label{scalingh1}$$ where the scaling function $f(y)$ for the number of the minima $N_z \sim L_z/L^\zeta \sim y$ has the same scaling function Eq. (\[Nscaling\]). Fig. \[fig8\](b) shows the scaling function (\[scalingf\]) with a collapse of $\langle h_1(L,L_z) \rangle L^{1-\theta}L_z$ versus $L_z/L^{\zeta}$ for various $L$ and $L_z$ with a good agreement, again. Next we move over to the $(2+1)$ dimensional manifolds. The inset of Fig. \[fig9\] shows the tail of the distribution for the system size $L^3 =50^3$ with dilution type of disorder and bond occupation probability $p=0.5$. The first non-overlapping jumps are included in the distribution, since due to the anomaly of the dilution disorder (lots of small scale degeneracy), there are typically two adjustments in the interface before the large jump. As a comparison, we plot again the $\exp(-x)$ line, too. One sees that the deviation of the tail of the distribution from the exponential behavior is similar to the $(1+1)$ dimensional case. The finite scaling of the first jump field in Fig. \[fig9\] is shown for interfaces of size $L_x \times L_y = 50^2$ and systems of height $L_z = 30\--90$. We have fitted for a constant $L$ the formula (\[scalingh1\]), i.e., $\langle h_1(L_z) \rangle \sim L_z^{-1} [\ln(L_z)]^{-1/2}$ and it works within error bars. Generalizing the numerical results of $(1+1)$ and $(2+1)$ dimensional calculations and the analytic arguments from the previous section to arbitrary dimensions gives the behavior of $\langle h_1(L,L_z) \rangle \sim L^{\theta-D} L_z^{-1} [\ln(L_z/ L^{\zeta})]^{-1/2}$. Since the probability distribution has a finite value at $P(h_1 = 0)$ and $\langle h_1(L,L_z) \rangle$ vanishes with increasing system size, one obtains from the normalization factor at $P(h_1 = 0)$ for the scaling of the susceptibility, Eq. (\[Eqsuskis2\]) $$\chi \sim L^{D-\theta} L_z [\ln(L_z/L^{\zeta})]^{1/2}, \label{Eqsuskis3}$$ and in the isotropic limit, $L \propto L_z$, the total susceptibility $\chi_{tot} = L^d \chi$ becomes $$\chi_{tot} \sim L^{2D+1-\theta} [(1-\zeta)\ln(L)]^{1/2}. \label{Eqsuskistot}$$ Thus the extreme statistics of energy landscapes shows up in the susceptibility of random manifolds or domain walls in a form that Eq. (\[Eqsuskistot\]) has a logarithmic multiplier and the scaling behavior of the first jump field is due to the scaling of the energy minima differences. This is in contrast to Shapir’s result [@Shapir91] and the logarithmic contribution is also missing from our earlier paper [@Seppala00] where the valley energy scale was taken to follow the standard $L^\theta$ assumption. Therefore one can conclude that the effect of extreme statistics is important in this problem: since we look at the finer details of the landscape “typical” differences are not sufficient. Notice that for two dimensional random field Ising magnets $\zeta \simeq 1$ at large scales [@Seppala98] and the susceptibility does not diverge [@Aizenman]: the premise that $N_z > 1$ is broken in that case. If the condition $N_z > 1$ is violated the extreme statistics correction disappears. For flat interfaces so that the effective roughness exponent is zero ($\zeta_{eff}=0$), e.g. {100} oriented RB interfaces with weak disorder and small system sizes $L<L_c$, when $\zeta=0$ [@Bouchaud92; @Alava96; @Raisanen98; @unpublished2], $\theta =D/2$ and Eq. (\[Eqsuskistot\]) becomes $\chi_{tot} \sim L^{2D+1-D/2} [\ln(L)]^{1/2} \sim L^4 [\ln(L)]^{1/2}$, when $D=2$. Finally we report the jump distance distributions of the first jumps. Fig. \[fig10\](a) shows $P(\Delta z_1/\overline{z}_0)$ of the first jump with the field in $(1+1)$ dimensions. The distribution is a superposition of two behaviors: the interface jumps to the lowest minimum as such, and the external field favors the minima close to the wall (remember the differences of $z_1$ and $z_{1^*}$ in Fig. \[fig5\]). Since the non-overlapping cases are excluded and the wall has a repulsive effect, there are cut-offs in the both ends of the distributions. The shape of the distribution does not change with the system size, which is consistent with the assumptions in Eqs. (\[Eqsuskis2\]) and (\[scalingh1\]), $\langle \Delta z_1 \rangle \sim L_z$. Fig. \[fig10\](b) shows the same distribution of $P(\Delta z_1/\overline{z}_0)$ with a field for $(2+1)$ dimensional manifolds and it is clear that the shape of the distribution does not depend on the dimension. In Fig. \[fig10\](c) we have plotted $P(\Delta z_1/\overline{z}_0)$ without a field, i.e., the distance of the true lowest energy minima in $(1+1)$ dimensions. Now there is no field, and thus the shape of the distribution is just a uniform one, again consistent with the predictions made in Section \[argumentti\] in Eq. (\[P0prod\]). Scaling of the first jumps: field and distance {#first_jump} ============================================== Following similar arguments as Eq. (\[excitation\]) in Section \[glassy\] for an excitation, a mean field result for the finite size scaling of the first jump field is next derived. Let us have an interface at an arbitrary height $z_0$ with an energy $E_0$ and an isotropic system $L \propto L_z$. The energy gap between the two lowest energy minima, Eq. (\[gap\]), scales as $\Delta E_1 \sim L^{\theta} [ (1-\zeta) \ln (L)]^{-1/2}$, when the $z_0 \simeq L_z$. However, when the interface is at an arbitrary height the number of the available minima depends on the original position of the interface, and we use only the dominating algebraic behavior so that $\Delta E_1 \sim L^{\theta}$. On the other hand the energy difference of elastic manifolds at different heights due to the field contribution is $\Delta E \simeq h \Delta z L^D$. Assuming that $\langle \Delta z \rangle \sim L$ as in the earlier arguments, the field contribution becomes $\Delta E \sim h L^d$. We expect that the first jump happens, when the gap equals the field energy, and thus the first jump field has a scaling $$\langle h_1 \rangle \sim L^{\alpha} = L^{\theta-d}. \label{MFh1}$$ In Figs. \[fig11\](a) and \[fig11\](b) we have plotted the average first jump field $\langle h_1 \rangle$ in isotropic systems for $(1+1)$ and $(2+1)$ dimensional manifolds, respectively. The data is taken from non-overlapping jumps, to minimize finite size effects since they are smaller than with the overlapping jumps included. However, the fraction of overlapping jumps is small as seen in Fig. \[fig4\] for $(1+1)$ dimensional case. The data in Figs. \[fig11\] confirms within error bars the expected behavior of $\langle h_1 \rangle \sim L^{-5/3}$ and $\langle h_1 \rangle \sim L^{-2.18}$ for $(1+1)$ and $(2+1)$ dimensional manifolds, respectively. The logarithmic term $[\ln (N_z)]^{-1/2}$ might contribute a downward trend since it decreases with the number of the available minima $N_z$, here disorder averaged due to the arbitrary GS location. This is however not noticeable in the data: the finite size effects in the calculated data are towards smaller absolute value of the exponent. In the insets of Figs. \[fig11\](a) and (b) the linear scaling of jump sizes $\langle \Delta z_1 \rangle \sim L$ is confirmed, which was assumed in Eqs. (\[Eqsuskis2\]) and (\[scalingh1\]). The evolution of the jump size distribution for succeeding jumps is demonstrated in Fig. \[fig12\]. As a comparison for Fig. \[fig10\] where only the non-overlapping jumps were considered, here we have shown the overlapping ones, too. It is seen as a peak near $\Delta z_{n}/ \overline{z}_{n-1} = 0$. The first jumps have clearly the most weight in the large jump end of the distribution, but for the following jumps the whole distribution shifts towards $\Delta z_{n}/\overline{z}_{n-1} = 0$. After a small number of jumps the interface is already in the minimum closest to the wall and then the random-bulk wetting behavior takes over. In order to estimate the number of the jumps the manifold does before binding to the wall, it is assumed that after a jump the system looks statistically the same as before it. From Fig. \[fig10\](a) we infer that the probability distribution for the jump size has a form $P(\Delta z) \sim \Delta z^2$ (note that we did not attempt to compute this from the analytical valley arguments) and that is then taken to be the same for all (large) jumps $\Delta z_n$. To calculate the average jump size of jumps which [*do not*]{} jump to the closest valley to the wall we have $$\begin{aligned} \langle \Delta z_n \rangle = \int_0^{z_{n-1}-A_1w} B_{n-1} \Delta z (\Delta z)^2 \, d(\Delta z) \nonumber \\ = \frac{3(z_{n-1}-A_1w)^4}{4z_{n-1}^3}, \label{Eqjumpsize}\end{aligned}$$ where the upper bound of the integral is to neglect the cases that jump to the closest valley to the wall, $A_1$ is the prefactor to multiply the roughness value to get the valley width, and $B_{n-1} = 3/z_{n-1}^3$ normalizes the probability distribution. Using $\langle z_n \rangle = \langle z_{n-1} \rangle - \langle \Delta z_n \rangle$ we get the next position of the interface. In order to calculate the probability of an interface to jump with the first jump to the closest valley to the wall, $p_1$, we use the same probability distribution as in Eq. (\[Eqjumpsize\]), $$p_1 = \int_{z_0-A_1w}^{z_0} B_0 (\Delta z)^2 \, d(\Delta z) = 1-\left[1-\frac{A_1w}{z_0}\right]^3 = {\tilde p}_1, \label{jumpp1}$$ and similarly for the probability of an interface to jump with the second jump to the closest valley to the wall $$\begin{aligned} p_2 = (1-{\tilde p}_1) {\tilde p}_2 = (1-{\tilde p}_1) \int_{z_1-A_1w}^{z_1} B_1 (\Delta z)^2 \, d(\Delta z) \nonumber \\ = \left[1-\frac{A_1w}{z_0}\right]^3 \left\{1-\left[1-\frac{A_1w}{z_1 }\right]^3 \right\} \label{jumpp2}\end{aligned}$$ Due to the hierarchy we finally get for the $n^{th}$ jump $$p_n = {\tilde p}_n \prod_{k=1}^{n-1} (1-{\tilde p}_k) = \left[1-\left(1-q_{n}\right)^3 \right] \prod_{k=1}^{n-1} (1-q_k)^3, \label{jumppn}$$ where $$q_n = \frac{A_1w}{z_{n-1}}. \label{qfrac}$$ To get an estimate for the number of jumps, we write $$\langle {\mathcal N} \rangle = \sum_{n=1}^{\infty} np_n. \label{n_estim}$$ This can be evaluated, but only approximately since among others the estimate of Eq. (\[jumppn\]) breaks down beyond $n$ small. Using $A_1w \simeq AL^\zeta= 50$, $L = 1000$, which is the case with $A_1 \simeq 6.5$, (see the numerics in Section \[results\]), and taking $\overline{z}_0 =1000$, we get $\langle {\mathcal N} \rangle \simeq 3$. Eq. (\[Eqjumpsize\]) gives for the first jump size, with the above numerical values, $\langle \Delta z_1 \rangle \simeq 0.6 \overline{z}_0$, which is not far from the behavior in the inset of Fig. \[fig11\](a), where $\langle \Delta z_1 \rangle \sim 0.4 L_z \sim 0.8 \overline{z}_0$, since $\langle \overline{z}_0 \rangle \sim L_z/2$. Application of a non-zero field: random-bulk wetting {#wetting} ==================================================== In order to see the wetting behavior one just studies the effect of a large external field on the average interface to wall distance. Notice that the low-field physics discussed extensively above can be considered as the eventual outcome in any sample with $L_z > L^\zeta$ so that more than one valley is available. This means that when the field is decreased from a large field value, the interface finally jumps into the bulk. Fig. \[fig13\](a) we show the average mean height $\langle \overline{z}(h) \rangle$ versus the field $h$ for $(1+1)$ and $(2+1)$ dimensional manifolds. The system sizes are $L_z \times L = 100 \times 3000$ and $L_z \times L^2 =50\times 300^2$. To maximize the prefactor, $A_2$, in the scaling of roughness, $w \sim A_2L^\zeta$, and hence the width of the minimum energy valley, the disorder has been chosen to be strong, i.e., dilution type of disorder with small $p$, but above the bond-percolation threshold. However, there are still some deviations in the form of greater exponents than the expected from Eq. (\[psi\]), which gives the values $\psi =1/2$ and $\psi \simeq 0.26$ in $(1+1)$ and $(2+1)$ dimensions from $\zeta_{(1+1)}=2/3$ and $\zeta_{(2+1)}=0.41\pm 0.01$, respectively. We found that the trend is nevertheless clear, with greater $L$ and fixed $L_z$ the exponents become closer to the expected one. The effective exponent $\psi_{eff}(L)$ can be used to extract the asymptotic, $L$-independent exponent in particular in $(2+1)$ dimensions since this case is hampered most by finite-size effects. $\psi_{eff}(L)$ as a function of $1/L$ indicates that the asymptotic value is indeed $0.26 \pm 0.02$ and that the system sizes at which $\psi_{eff}(L)$ approaches that are of the order of $L = 10^4$. That is, only with $10^{10}$ sites it becomes possible to reach the asymptotic regime. When one has $L \propto L_z$ calculating the average mean height $\langle \overline{z}(h) \rangle$ with a fixed field $h$ is nothing but averaging over the jumped and not jumped interfaces together with their location, see Fig. \[fig2\](a). In Fig. \[fig13\](b) we have plotted as a comparison the average mean height $\langle \overline{z}(h) \rangle$ versus the field $h$ for $(2+1)$ dimensional manifolds with a weak disorder. In this case the [*weak*]{} means for system sizes used, that the roughness of the manifolds are not in the asymptotic roughness limit yet, $L<L_c$ [@Bouchaud92; @Alava96; @Raisanen98; @unpublished2]. The system sizes used are $L_z \times L^2 =50\times 80^2\--400^2$ and the behavior is simple: either the interface stays in its original position or jumps directly to the wall. Taking into account the jumped and original interfaces as $\langle z(h) \rangle = \langle z [1-P(\overline{z}_0,h)] \overline{z}_0 + P(\overline{z}_0,h) \times 0 \rangle = \langle [1-P(\overline{z}_0,h)] \overline{z}_0 \rangle = \int_0^{L_z} [1-P\{\overline{z}_0(h)\}] \overline{z}_0 \, d\overline{z}_0$, gives the behavior $\langle z(h) \rangle \sim h^{-1}$, i.e., the exponent $\psi \simeq 1$, if $P\{\overline{z}_0(h)\} \sim h^{-1}$. The larger manifolds jump faster to the wall, i.e., they feel the perturbation earlier, since Eq. (\[scalingh1\]) for flat interfaces ($\zeta =0$) becomes $\langle h_1(L,L_z) \rangle \sim L^{\theta-D} L_z^{-1} [\ln(L_z)]^{-1/2}$. With fixed $L_z$ and $\theta =D/2$ from Poissonian statistics, we get in $D=2$ $\langle h_1(L) \rangle \sim L^{-1}$ consistent with the numerical data. This leads to the behavior of the wetting scaling, $\langle \bar{z}(h) \rangle \sim c(L) h^{-\psi}$ where $c(L) \simeq L^{-1}$ and $\psi=1$. The finite size scaling of the prefactor indicates that at large $L$-limit with fixed $L_z$ the flat interfaces are immediately at the wall, and thus the systems are non-wet. This implies that there is an interesting cross-over around $L_c$ between such a “dry” regime and the bulk wetting that takes over for still larger $L$. In $D>2$ this kind of behavior is relevant even in the thermodynamic limit, if the disorder is weak due to the presence of a bulk roughening transition for bond disorder. Discussion {#disc} ========== Finite temperature behavior {#creep} --------------------------- The movement of the elastic manifolds in random media at low temperatures, when an applied force is much below the depinning threshold $F_c$, is characterized as creep. The dynamics is controlled by thermally activated jumps over pinning energy barriers, which separate the metastable states. D. Fisher and Huse [@Fish91] showed that for a DP at finite temperature $T>0$ the fluctuations of the entropy $(\Delta S)^2$ and the internal energy $(\Delta E_{int})^2$ scale linearly with the length of the polymer and cancel each other. Hence there are only the fluctuations of the free energy $(\Delta F)^2$, which scale with the zero temperature energy fluctuation exponent $2\theta=2/3$. Since the free energy is the one which should be minimized at finite temperature, it is the one which defines statistically the shape of the energy landscape, although the energy valleys and minima need not to have exactly the same real space structure as at $T=0$. Thus the $\theta=1/3$ exponent should define the energy gaps also when $T>0$, expect in the cases there is a critical temperature $T_c$. Hence, our derivation of the susceptibility and also the first order character in the reorganization of valleys should be relevant also at $T>0$. First arrival times in nonlinear surface growth {#secKPZ} ----------------------------------------------- The $(1+n)$ dimensional directed polymers map, in the continuum limit, to the KPZ [@KPZ; @Barab] equation by associating the minimum energy of a DP-configuration with the minimum [*arrival time*]{} $t_1 \equiv E_0$ of a KPZ-surface to height $H$. The connection is illustrated in Fig. \[fig14\]. The minimal path of the DP with the end point ${\bf x}_1(t_1)$ equals the path by which the interface reaches $H$, at location ${\bf x}_1$ and at time $t_1$. Thus $t_1$ attains a logarithmic correction, from Eq. (\[typicalene\]), of size $-H^{\beta} \{\ln(L/H^{1/z}) \}^{1/\eta}$, where $L$ is the linear size of the system, $\beta = \theta > 0$ and $z=1/\zeta$ are now the roughening exponent and dynamical exponent of the KPZ universality class and the values of $\theta = 2\zeta-1$ and $\zeta >1/2$ depend on $n$. Notice that if there is a upper critical dimension $n_c$ in the KPZ growth, then the logarithmic correction is not there anymore, and $\theta=0$, $\zeta=1/2$, i.e., a random walk ensues. Consider now the second smallest arrival time $t_2$. In the language of directed polymers, if the path ${\bf x}_2 (t')$ corresponding to $t_2$ is independent of ${\bf x}_1 (t')$ that corresponds to $t_1$ the time and the path are found inside a separate, independent valley. The [*difference*]{} $\Delta t = t_2 - t_1$ then equals $\Delta E_1$, and likewise obeys extreme statistics, so that $\Delta t \sim H^\beta [\ln (L/H^{1/z})]^{-(\eta-1)/\eta}$. For growing surfaces this limit is the [*early stages*]{} of growth, in which the correlation length $\xi \ll L$, and therefore the arrival times, or directed polymer energies, are independent. On the other hand if we disturb the growing process such a way that it depends on the ${\bf x}$, e.g. linearly with a factor $h$ [@spohn], the polymer ending at ${\bf x}_2$ becomes faster if $h>h_1$, see right hand side of Fig. \[fig14\]. Similarly now the factor $h_1$ has a scaling behavior from Eq. (\[scalingh1\]), $h_1 \sim H^{\beta-1} L^{-1} [\ln (L/H^{1/z})]^{-(\eta-1)/\eta}$. Conclusions {#concl} =========== To conclude we have studied the $(1+1)$ and $(2+1)$ dimensional elastic manifolds at zero temperature, when an external field is applied. We have demonstrated that the response of manifolds shows a first order character (“jump”) in the sense that the manifolds change their configuration in the large system size limit completely. This persists in a finite system over a small number of such jumps. The distance that the center-of-mass moves is extensive. The whole picture is based on a level-crossing between two low-energy valleys in the energy landscape. Averaging over the total magnetization of such random magnets with a domain wall or over the positions of manifolds when an external field is applied becomes dependent on whether the DW or manifold has jumped or not. This leads to the problem of self-averaging in random systems. Here a disorder average smooths over the “coexistence” between systems that are not affected by a finite field $h$ and those that have responded. In order to study the susceptibility of the DW in random media, one has to take into account the probability distribution of the sample-dependent field associated with the global change of the configuration, and take the limit of vanishing fields. This probability distribution has a finite density at $h=0$. The finite size scaling of the density is dependent on the finite size scaling of the number of the low lying nearly degenerate energy minima in the system. The scaling of the number of the energy minima leads to a logarithmic factor in the susceptibility, and can be accounted for by using extreme statistics. Such effects are difficult to study by usual field theoretical means, since one has to have access to the whole probability distribution and not only the few first moments thereof. Notice that the crucial difference to much previous work is the simple fact that we allow for multiple minima in the energy landscape, which is often most excluded by the boundary conditions applied to the problem. Although the derivations and the numerical calculations done here have concentrated on random bond type of randomness similar behavior should be seen in random field cases, too. The discrete character in the movement of elastic manifolds with an external field results also in that the continuum theory for wetting in random systems works only in slab geometries, where there is room only for a single valley, or in the large external field limit, when the interface is close to the wall. On the other hand the flat interfaces are shown to jump directly to the wall, i.e., to be non-wet. It would be interesting to see if the dynamics of the manifolds at finite temperature reflects the first order character seen here at $T=0$. Through the connection of (1+1) dimensional DW’s to the directed polymers of the KPZ surface growth, we have shown that the logarithmic factor is also present in the statistics of growth times in nonlinear surface growth. The authors would like to thank the Academy of Finland’s Centre of Excellence Programme for financial support and the Center for Scientific Computing, Espoo, Finland for computing resources. 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J. C. Anglès d’Auriac and N. Sourlas, Europhys. Lett. **39** (1997), 473. ![A schematic figure of two interfaces in a system of parallel length $L$ and perpendicular one $L_z > L^\zeta$ at heights $\bar{z}_1$ and $\bar{z}_2 \simeq w_2$, with the corresponding roughness values $w_1$ and $w_2$. $\xi_\perp$ denotes the transverse correlation length of a part of the interface of length $\xi_\|$. See the text. A droplet is seen in the interface of height $\bar{z}_1$ as a dotted line.[]{data-label="fig1"}](./fig1.eps){width="9cm"} ![The level-crossing phenomenon for interfaces in random systems in the presence of an external field. Originally the interface lies at height $z_0$ and has an energy $E_0$. When the field is applied its energy increases linearly and at $h_1$ the interfaces jumps to $z_1 < z_0$, with the energy $E_1(h_1)=E_1 + h_1 L^D z_1$, where $E_1$ is the energy of the interface at $z_1$ without the field. Similar behavior takes place at $h_n$, $n=2,3,\dots $, when the interface moves from $z_{n-1}$ to $z_n < z_{n-1}$. The thick line represents the minimum energy of the interface $E(h)$ as a function of the field $h$, with discontinuities of the derivative at $h_n$.[]{data-label="fig3"}](./fig3.eps){width="7cm"} ![(a) A simplified view of the minima in the energy landscape of a random system. $\Delta E_1 = E_1 -E_0$ is the energy gap between the ground state at $z_0$, denoted with a black circle, and the second lowest minimum at $z_1$ denoted with a gray circle. $\Delta E_{1^*} = E_{1^*} -E_0$ is the energy gap (normalized with $L^D$) between the ground state and the minimum at $z_{1^*}$. (b) The view of the minima in the random system when the field is applied. At $h_1$ the interface moves from $z_0$ to $z_{1^*}$, indicated with a gray circle, since the energy difference $\Delta E_{1^*} = E_{1^*} -E_0 = h_1 z_0 - h_1 z_{1^*} = 0$ while all the other $\Delta E$’s are greater. However, often $E_{1^*}$ and $E_1$ coincide.[]{data-label="fig5"}](./fig5a.eps){width="8cm"} ![(a) A simplified view of the minima in the energy landscape of a random system. $\Delta E_1 = E_1 -E_0$ is the energy gap between the ground state at $z_0$, denoted with a black circle, and the second lowest minimum at $z_1$ denoted with a gray circle. $\Delta E_{1^*} = E_{1^*} -E_0$ is the energy gap (normalized with $L^D$) between the ground state and the minimum at $z_{1^*}$. (b) The view of the minima in the random system when the field is applied. At $h_1$ the interface moves from $z_0$ to $z_{1^*}$, indicated with a gray circle, since the energy difference $\Delta E_{1^*} = E_{1^*} -E_0 = h_1 z_0 - h_1 z_{1^*} = 0$ while all the other $\Delta E$’s are greater. However, often $E_{1^*}$ and $E_1$ coincide.[]{data-label="fig5"}](./fig5b.eps){width="8cm"} ![The relation between directed polymers and growing interfaces. Two directed polymers in independent valleys equal the fastest arrival time $t_1$ at $\vec{x}_1$, solid line, and the second fastest at $\vec{x}_2$ with time $t_2$, dotted line, of a KPZ interface to a fixed height $H$, when the external field $h=0$. In the right hand side figure an external field is added, which increases the growth time depending on the position in the direction of the arrow and the polymer at $\vec{x}_2$ becomes the one corresponding to the fastest time to reach $H$.[]{data-label="fig14"}](./fig14.eps){width="9cm"}

--- abstract: 'In the present contribution, we investigate first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and non-conservative first order terms. Whereas the theory of first-order systems of conservation laws is well established and the conditions for the existence of supplementary conservation laws, and more specifically of an entropy supplementary conservation law for smooth solutions, well known, there exists so far no general extension to obtain such supplementary conservation laws when non-conservative terms are present. We propose a framework in order to extend the existing theory and show that the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to a supplementary conservation law. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms with the objective of obtaining specifically an entropy supplementary conservation law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziato two-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluid model, with non-conservative terms impacting on the linearly degenerate field and requires a closure since there is no way to derive interfacial quantities from averaging principles and we need guidance in order to close the pressure and velocity of the interface and the thermodynamics of the mixture. The second one involves first order terms for the heavy species coupled to second order terms for the electrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rigorously derived from kinetic theory. We show how the theory allows to recover the whole spectrum of closures obtained so far in the literature for the two-phase flow system as well as conditions when one aims at extending the thermodynamics and also applies to the plasma case, where we recover the usual entropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory.' author: - 'Pierre CORDESSE[^1]' - 'Marc MASSOT[^2]' bibliography: - '../biblatex/jabref\_bdd.bib' title: 'Entropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: application to the modelling and analysis of complex fluid flows using computer algebra[^3]' --- Nonlinear PDEs with non-conservative terms, supplementary conservation law, entropy, computer algebra, two-phase flow, Baer-Nunziato model, multicomponent plasma fluid model 35L60; 68W30; 76N15; 76T10; 82D10 Introduction ============ First-order nonlinear systems of partial differential equations and more specifically systems of conservation laws have been the subject of a vast literature since the second half of the twentieth century because they are ubiquitous in mathematical modelling of fluid flows and are used extensively for numerical simulation in applications and industrial context [[@Bissuel_2018; @Gaillard_2016]]{}. Such systems of equation can either be rigorously derived from kinetic theory of gases through various expansion techniques [[@Ferziger_1972; @Woods_1975]]{}, or can be derived using rational thermodynamics and fluid mechanics including stationary action principle (SAP) [[@Serrin_1959; @Landau_1976; @Truesdell_1969]]{}. As far as Euler or Navier-Stokes equations are concerned for a gaseous flow field, the outcome of both approaches are similar and the mathematical properties of these systems have been thoroughly investigated for the past decades. An interesting related problem is the quest for supplementary conservation laws. Noether’s theorem [[@Olver_1986]]{} leads, within the framework of SAP, to the derivation of supplementary conservation laws based on symmetry transformations of the variational problem under investigation[^4]. Examples of such derivations on two-phase flow modelling can be found in [[@Gavrilyuk_Saurel_2002; @Drui_JFM_2019]]{}. However, to the authors knowledge, no symmetry transformations have been identified yielding a conservative law on the entropy of the system. In fact, SAP does not allow to reach a closed system of equations, and one has to provide a closure for the entropy (see [[@Gouin_2009]]{} for example). A specific type of supplementary conservation equation for smooth solution is especially important, namely the *entropy equation*, derived through the theory developed in [[@Godunov_1961; @Friedrichs_1971]]{} for systems of conservation laws. Such systems of PDEs are hyperbolic at any point where a locally convex entropy function exists [[@Mock_1980]]{}, and when they are equipped with a strictly convex entropy, they can be symmetrized [[@Friedrichs_1971]]{} [[@Harten_Hyman_1983]]{} and thus are hyperbolic. These properties have been at the heart of the mathematical theory of existence and uniqueness of smooth solutions [[@Kawashima_1988]]{} [[@Giovangigli_1998]]{}, but they are also a corner stone for the study of weak solutions for which the work of [[@Kruzkov_1970]]{} proves the well-posedness of Cauchy problem for one-dimensional systems. Nonetheless, for a number of applications, where reduced-order fluid models have to be used for tractable mathematical modelling and numerical simulations, be it in the industry or in other disciplines, micro-macro kinetic-theory-like approaches as well as rational thermodynamics and SAP approaches often lead to system of conservation laws involving *non-conservative terms*. Among the large spectrum of applications, we focus on two types of models, which exemplify the two approaches: 1- two phase flows models which rely on a hierarchy of diffuse interface models among which stands the Baer-Nunziato [[@Baer_Nunziato_1986]]{} model used when full disequilibrium of the phases must be taken into account. Since this model is derived through rational thermodynamics, the macroscopic set of equations can not be derived from physics at small scale of interface dynamics and thus require closure of interfacial pressure and velocity, 2- multicomponent fluid modelling of plasmas flows out of thermal equilibrium, where the equations can be derived rigorously from kinetic theory using a multi-scale Chapman-Enskog expansion mixing a hyperbolic scaling for the heavy species and a parabolic scaling for the electrons [[@Graille_2007]]{}. Concerning the thermodynamics, whereas for the first model it has to be postulated and requires assumptions, it can be obtained from kinetic theory in the second model. In both cases, the models involve non-conservative terms, but these terms do not act on the same fields; linearly degenerate field is impacted for the two-phase flow model, whereas it acts on the genuinely nonlinear fields in the second [[@Wargnier_2018]]{}. Whereas hyperbolicity depends on the closure and is not guaranteed for the first class of models [[@Gallouet_2004]]{}, the second is naturally hyperbolic [[@Graille_2007]]{} and also involves second-order terms and eventually source terms [[@Magin_2009]]{}. Thus, the presence of *non-conservative terms* encompasses several situations and requires a general theoretical framework. While Noether’s theorem can still applied to obtain some supplementary conservation laws, it does not permit to exhibit all of them and especially not an entropy supplementary conservation law. A unifying theory extending the standard approach for systems of conservations laws (entropy supplementary conservation law, entropic symmetrization, Godunov-Mock theorem, hyperbolicity) is still missing for such systems even if some key advances exist. The system has been shown to be symmetrizable by [[@Coquel_2013]]{} – not in the sens of Godunov-Mock – far from the resonance condition for which hyperbolicity degenerates. In [[@Forestier_2011]]{}, the model is proved to be partially symmetrizable in the sense of Godunov-Mock. The present paper first proposes an extension of the theory of supplementary conservation laws for system of conservation laws to first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and *non-conservative first order terms*.We emphasize how the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to supplementary conservation laws. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms to obtain an entropy supplementary conservation law. The proposed theoretical approach is then applied to the two systems identified so far for their diversity of behaviour. Even if the whole theory is valid for any supplementary conservation law, we focus on obtaining an *entropy* supplementary conservation law. For the two-phase flow model, assuming a thermodynamics of non-miscible phases, we derive conditions to obtain an entropy supplementary conservative equation together with a compatible thermodynamics and closures for the non-conservative terms. Interestingly enough, all the closures proposed so far in the literature are recovered [[@Baer_Nunziato_1986; @Kapila_1997; @Bdzil_1999; @Lochon_PhdThesis_2016; @Saurel_Gavrilyuk_2003]]{}. The strength of the formalism lies also in the capacity to derive such conditions for some level of mixing of the phases. By introducing a mixing term in the definition of the entropy, the new theory brings out constraints on the form of the added mixing term. We recover not only the closure proposed to account for a configuration energy as in the context of deflagration-to-detonation [[@Baer_Nunziato_1986]]{} or in [[@Coquel_2002]]{}, but we also rigorously find new closures leading to a conservative system of equations[^5]. We also prove that the theory encompasses the plasma case, where we recover the usual *entropy* supplementary equation assessing the effectiveness and scope of the proposed theory. The paper is organized as follows. The extension of the theory for system of conservation laws to first-order nonlinear systems of partial differential equations including non-conservative terms, as well as the framework to apply the theory by means of computer algebra are introduced in Section \[sec:theory\]. These results are then applied first to the Baer-Nunziato model in Section \[sec:BNZ\] and then to the plasma model in Section \[sec:plasma\] to obtain an entropy supplementary conservation law compatible with the model closure. **Notations:** Let ${\boldsymbol{a}} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$, ${\boldsymbol{b}} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$, $\mathcal{B} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p\times p}$, $\mathcal{C} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p\times p}$, ${{\boldsymbol{\mathcal{D}}}} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p\times p \times p}$ be a $p$-component line first-order tensor, a $p$-component column first-order tensor, two $p$-square second-order tensor and a third-order tensor respectively. We introduce the following notations: - ${\boldsymbol{a}} \mathcal{B}$ is a line first-order tensor in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ whose $i$ component are defined by $$\begin{aligned} \left({\boldsymbol{a}} \mathcal{B}\right)_{i} = \sum_{j=1,p} {\boldsymbol{a}}_{j} \mathcal{B}_{j,i}, \end{aligned}$$ - $\mathcal{B} {\boldsymbol{b}}$ is a column first-order tensor in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ whose $i$ component is defined by $$\begin{aligned} \left(\mathcal{B} {\boldsymbol{b}}\right)_{i} = \sum_{j=1,p} \mathcal{B}_{i,j}{\boldsymbol{b}}_{j}, \end{aligned}$$ - $\mathcal{B} \times \mathcal{C}$ is $p$-square second-order tensor whose $(i,j)$ component is defined by $$\begin{aligned} \left(\mathcal{B} \times \mathcal{C}\right)_{i,j} = \sum_{k=1,p} \mathcal{B}_{i,k} \mathcal{C}_{k,j}, \end{aligned}$$ - ${\boldsymbol{a}} \otimes \mathbb{D}$ is a $p$-square second-order tensor whose $(i,j)$ component is defined by $$\begin{aligned} \left( {\boldsymbol{a}} \otimes \mathbb{D} \right)_{(i,j)} = \sum_{k=1,p} {\boldsymbol{a}}_{k} \times \mathbb{D}_{k,i,j}. \end{aligned}$$ Hereafter, we will name zero- first- and second-order tensors by scalar, vector and matrix respectively and for convenience we will use vector and matrix representations of functions. Moreover, given a scalar function $S$, the partial differentiation of $S$ by a column vector ${\boldsymbol{a}}$, $\partial_{{\boldsymbol{a}}}S$ is a line vector in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$. Finally, $\cdot$ denotes the Euclidean scalar product in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$. Supplementary conservation law {#sec:theory} ============================== First we recall the theory of the existence of a supplementary conservative equation for first-order nonlinear systems of conservation laws. Second, this notion is extended to systems containing first order non-conservative terms. Third, we introduce a framework to apply this new theory to design and analyze physical models using computer algebra. A one-dimensional framework is adopted from now on, $x \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$, in order to simplify the derivation. Nonetheless, the results can easily be extended to the multi-dimensional approach as presented in [[@Godlewski_1996]]{} for systems of conservation laws. First-order nonlinear conservative systems. {#ssec:CS_theory} ------------------------------------------- The homogeneous form of a first-order nonlinear system of $p$ conservation laws writes $$\begin{aligned} \label{eq:cons_syst_non_linear} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \partial_{x} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}},\end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ denotes the conservative variables with $\Omega$ an open convex of ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ and ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ the conservative fluxes. Focusing on smooth solution of the system , its quasi-linear form is given by $$\begin{aligned} \label{eq:cons_syst_quasi_linear} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ \[theo:cons\_syst\_SCE\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. The following statements are equivalent: 1. System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:cons_syst_entropy_eq} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function. 2. There exists a scalar function ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ such that $$\begin{aligned} \label{eq:cons_syst_compatibility_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ 3. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ is a $p$-square symmetric matrix. The proofs of the theorem can be found in the literature. We would like to recall how the last statement is obtained. Assuming $(C_{2})$, differentiating Equation  leads to $$\begin{aligned} \label{eq:cons_syst_sym_cond} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}),\end{aligned}$$ where $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ is a $p$-square matrix defined as $\sum_{i} \partial_{{{{\boldsymbol{\mathrm{u}}}}}_{i}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}_{i}({{{\boldsymbol{\mathrm{u}}}}})$ which is a linear combination of Hessian matrices and hence symmetric. Moreover, the RHS of Equation  $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}})$ is symmetric. Therefore $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ is symmetric. Theorem \[theo:cons\_syst\_SCE\] applies for any type of supplementary conservative equations and other formulations of Theorem \[theo:cons\_syst\_SCE\] can be found in the literature [[@Harten_Hyman_1983; @Godlewski_1996; @Despres_2005]]{}. Extension to systems involving non-conservative terms. {#ssec:NC_theory} ------------------------------------------------------ Let us now consider the homogeneous form of a first-order nonlinear system of partial differential equations constituted of two parts: conservations laws and first-order non-conservative terms. Its quasi-linear form can be written as $$\begin{aligned} \label{eq:NC_syst} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= {{\boldsymbol{\mathrm{0}}}},\end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution with $\Omega$ an open convex of ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$, ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ the conservative fluxes, ${{{\boldsymbol{\mathcal{N}}}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ the $p$-square matrix containing the first-order non-conservative terms. In the following we extend the theory introduced in Section \[ssec:CS\_theory\] to system . Given a scalar function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$, multiplying system  by the line vector $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ yields $$\begin{aligned} \label{eq:NC_syst_SCE} \partial_{t} {{\mathsf{H}}}+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}= 0.\end{aligned}$$ Compared to Equation , the presence of the non-conservative terms in Equation  complexifies the question of the existence of a supplementary conservative equation. Therefore we propose to decompose in a specific way the conservative and non-conservative terms in Definition \[def:decomposition\]. \[def:decomposition\] Given a scalar function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ and a first-order nonlinear non-conservative system , let us define the four $p$-square matrices, ${{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ in ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p \times p}$ such that $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) & = {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}), \\ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) & = {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})+{{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}),\end{aligned}$$ with the condition $$\begin{aligned} \label{eq:decomposition_condition} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] = {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ In light of Definition \[def:decomposition\], Theorem \[theo:cons\_syst\_SCE\] can be extended as follows: \[theo:NC\_syst\_SCE\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Given a first-order nonlinear system of non-conservation laws , if we introduce the decomposition as in Definition \[def:decomposition\], then the following statements are equivalent: 1. System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:NC_syst_entropy_eq} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function. 2. There exists a scalar function ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ such that $$\begin{aligned} \label{eq:NC_syst_compatibility_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ 3. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \left[{{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] $ is a $p$-square symmetric matrix. Rewriting Equation  using the decomposition of the conservative and non-conservative terms as $$\begin{aligned} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}= - \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}\end{aligned}$$ outlines the result. \[remark:symmetry\_condition\]Theorem \[theo:NC\_syst\_SCE\] applies for any type of supplementary conservative equations. The usual symmetry condition on which relies the existence of a supplementary conservation equation is strongly modified when non-conservation terms are present. From Theorem \[theo:cons\_syst\_SCE\] to Theorem \[theo:NC\_syst\_SCE\] the condition $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric},\end{aligned}$$ is modified into $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \left[{{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \text{ symmetric}.\end{aligned}$$ In the context of systems of conservation laws, an interesting algebraic approach is proposed in [[@Barros_2006]]{} based on the reinterpretation of the symmetric Condition $(C_{3})$ in Theorem \[theo:cons\_syst\_SCE\] as a Frobenuis problem. Nevertheless, when dealing with additional non-conservative terms, the above new symmetry condition prevents us from applying efficiently such an approach. In Definition \[def:decomposition\], the condition  implies that the conservative and non-conservative terms depend only on the variables ${{{\boldsymbol{\mathrm{u}}}}}$, and not on their gradient. Some authors have allowed the matrices ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ to depend also on the gradients of the variables ${{{\boldsymbol{\mathrm{u}}}}}$, then a more general condition for the decomposition can be written $$\begin{aligned} \label{eq:theo_b1b2_extended} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}\leq 0.\end{aligned}$$ In Section \[sec:BNZ\], we will see that such a condition has been chosen to close the Baer-Nunziato model [[@Saurel_Gavrilyuk_2003]]{}. However, since it changes the mathematical nature of the PDE under investigation, we will not include it in our study. From a modelling perspective, System  under consideration is not necessary closed. Therefore, the following corollary yields conditions on the model to obtain a supplementary conservative equation once we have postulated the thermodynamics. Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Given a first-order nonlinear system of non-conservation laws where ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ and ${{{\boldsymbol{\mathcal{N}}}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ are unknown functions to be modelled. If we introduce the decomposition as in Definition \[def:decomposition\], then System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:NC_syst_entropy_eq_corro} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ a scalar function, if and only if the following conditions hold 1. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \left[{{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] $ is a $p$-square symmetric matrix. 2. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] = {{\boldsymbol{\mathrm{0}}}}$. Design or analysis of physical models using computer algebra. {#ssec:NC_theory_applied} ------------------------------------------------------------- We would like to apply the theory on first-order nonlinear non-conservative systems introduced in Section \[ssec:NC\_theory\] to physical models such as the Baer-Nunziato model and the plasma model in order to design and analyze them. We recall that our prior interest is to obtain an *entropy* supplementary conservation law. However, the difficulty is manifold: - The combination of the non-conservative terms and conservative terms proposed in Definition \[def:decomposition\] to build a supplementary conservative equation is not unique and thus many degrees of freedom exist in defining the matrices ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$. - When the model is derived trough rational thermodynamics, terms in the system of equations might need closure and the thermodynamics has to be postulated. Therefore, the matrices ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ can contain unknowns related to the system and the definition of ${{\mathsf{H}}}$. - The calculations needed to derive a supplementary conservative equation are heavy and choice-based. Any change of ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ that respects Definition \[def:decomposition\], or any new postulated thermodynamics would require to derive again all the equations, and eventually a very limited range of possibilities would be examined. These difficulties to apply the theory and examine all the possibilities makes computer algebra very appealing since it allows symbolic operations to be implemented and thus can derive equations systematically and quasi-instantaneously for any combinations of conservative and non-conservative terms as well as model closure and ${{\mathsf{H}}}$ definition. Furthermore, the generic level handled by computer algebra is not unlimited and therefore Definition \[def:decomposition\] requires further assumptions to circumscribe the number of degrees of freedom that can be accounted for. Even if the theory proposed hereinbefore is valid to obtain any kind of supplementary conservation laws, we are mainly interested in obtaining an entropy supplementary conservation law. We thus need to define the notions of *entropy* and *entropic variables* in the following two definitions. ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is said to be an *entropy* of the system  if ${{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ is a convex scalar function of the variables ${{{\boldsymbol{\mathrm{u}}}}}$ which fulfills Theorem \[theo:cons\_syst\_SCE\]. The supplementary conservative equation  is then named the *entropy equation* and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is the associated *entropy flux*. \[def:entropic\_variable\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Given a first-order nonlinear conservative system , let us define the *entropic variables* ${{{\boldsymbol{\mathrm{v}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ such that $$\begin{aligned} {{{\boldsymbol{\mathrm{v}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\right)^{t}.\end{aligned}$$ The entropic variables have been studied in [[@Giovangigli_1998]]{} in order to obtain symmetric and normal forms of the system of equation and used in the framework of gaseous mixtures, where the mathematical entropy ${{\mathsf{H}}}$ is usually defined as the opposite of a physical entropy density per unit volume of the system [[@Giovangigli_1998]]{}. \[def:decomposition\_applied\] Given a scalar function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$, a first-order nonlinear non-conservative system , and the four $p$-square matrices ${{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ in ${ \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ defined in Definition \[def:decomposition\], we introduce the unknown line vector $\transfer{v}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ such that $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}),\\ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ The condition of Equation  rewrites into $$\begin{aligned} \label{eq:decomposition_condition_applied} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ Since Definition \[def:decomposition\_applied\] is a projection of the matrix equations of Definition \[def:decomposition\] on the vector $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$, it may be interesting to introduce an unknown matrix ${{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p \times p}$ associated to the unknown line vector $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ such that $$\begin{aligned} \label{def:gamma_matrix} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}).\end{aligned}$$ Thus, Definition \[def:decomposition\_applied\] can be formulated as follows $$\begin{aligned} {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}), \\ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) &= {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}),\end{aligned}$$ with the condition $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] = {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ The unknown functional line vector $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ represents the transfer of non-conservative terms to the conservative terms. In the degenerate case where $\transfer{v}={{\boldsymbol{\mathrm{0}}}}$, ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ receives all the conservative terms and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ all the non-conservative terms. Condition  forces all the non-conservative terms to vanish and System  is fully conservative, hence the theory of conservative system can be applied. Definition \[def:decomposition\_applied\] being more restrictive than Definition \[def:decomposition\], computer algebra is now applicable to analyze the properties of a first-order nonlinear non-conservative system leading to a reformulation of Theorem \[theo:NC\_syst\_SCE\]. \[theo:NC\_SCE\_applied\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Consider a first-order nonlinear system of non-conservation laws . If we introduce the decomposition as in Definition \[def:decomposition\_applied\], then the following statements are equivalent: 1. System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:entropy_eq} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function. 2. There exists a scalar function ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ such that $$\begin{aligned} \label{eq:compatibility_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ 3. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ is a $p$-square symmetric matrix. Injecting Definition \[def:decomposition\_applied\] into Theorem \[theo:NC\_syst\_SCE\] leads to these results. When ${{\mathsf{H}}}$ is the entropy of the system, Theorem \[theo:NC\_SCE\_applied\] provides equations that relate the thermodynamics of the model through ${{\mathsf{H}}}$, the model itself with possible terms to be closed in ${{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}})$, and the unknown line vector $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$. Combined with the Definition \[def:decomposition\_applied\], Theorem \[theo:NC\_SCE\_applied\] brings out conditions on the model to obtain a supplementary conservative equation given a postulated thermodynamics and it leads to the following corollary. \[coro:NC\_syst\_applied\_metho\] Consider a first-order nonlinear system of non-conservation laws where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution with $\Omega$ an open convex of ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ but ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ and ${{{\boldsymbol{\mathcal{N}}}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ are unknown functions to be modelled. Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. If we introduce the decomposition as in Definition \[def:decomposition\_applied\], then System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:NC_syst_entropy_eq_corro_applied} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function if and only if the following conditions hold 1. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ is symmetric. 2. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}$. The previous framework can be extended to the multi-dimensional case in a straightforward manner. If the original system is isotropic, such as for the applications we have in mind, then the previous conditions will be the same in the various directions. In the framework of more general non-isotropic systems, which satisfy Galilean and rotational invariances for example, we will obtain different conditions and we have to check that the decomposition we perform in the various directions satisfies some compatibility relations so that the obtained conservation law satisfies the original invariance properties of the system. Methodology. {#ssec:NC_metho} ------------ Corollary \[coro:NC\_syst\_applied\_metho\] draws the methodology we have implemented in the Maple computer algebra software[^6] in order to obtain an *entropy* supplementary conservation law. Our methodology is the following: 1. We define the thermodynamics by postulating - if need be - an entropy function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$. 2. We then use Condition $(C_{1})$ and $(C_{2})$ of Corollary \[coro:NC\_syst\_applied\_metho\] to ensure the existence of an entropy flux ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ and solve $$\begin{aligned} \label{eq:NC_SCE_applied_cond} \left\{ \, \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{l} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric}, \\ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}. \end{IEEEeqnarraybox} \right. \end{aligned}$$ In System , $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ is systematically an unknown, ${{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ as well as ${{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ can include unknown terms for which the variable dependency is specified. Maple generates then an exhaustive solution for $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ and constraints on all the other unknown terms. 3. From that, the software derives the admissible entropy flux ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ which gives then the supplementary conservative equation. Application to the Baer-Nunziato model {#sec:BNZ} ====================================== Context and presentation of the model. -------------------------------------- The Baer-Nunziato model has been derived through rational thermodynamics in [[@Baer_Nunziato_1986]]{} and describes a two-phase flow out of equilibrium. Extended by the work of [[@Saurel_1999]]{} thanks to the introduction of interfacial quantities, the homogeneous form of the Baer-Nunziato model is $$\begin{aligned} \label{sys:BNZ_eq} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{c} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= {{\boldsymbol{\mathrm{0}}}},\\ \\ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \begin{pmatrix} 0 & {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}} \\ {{\boldsymbol{\mathrm{0}}}} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}_{2}} {{{\boldsymbol{\mathrm{f}}}}}_{2}({{{\boldsymbol{\mathrm{u}}}}}_{2}) & {{\boldsymbol{\mathrm{0}}}} \\ {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}_{1}} {{{\boldsymbol{\mathrm{f}}}}}_{1}({{{\boldsymbol{\mathrm{u}}}}}_{1}) \end{pmatrix}, \ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \begin{pmatrix} \speed{sint} & {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}}\\ {{{\boldsymbol{\mathrm{n}}}}_{2}}& {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}}\\ {{{\boldsymbol{\mathrm{n}}}}_{1}}& {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}} \end{pmatrix}, \end{IEEEeqnarraybox}\end{aligned}$$ where the column vector ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{7}$ is defined by ${{{\boldsymbol{\mathrm{u}}}}}^{{T}} = \left(\alpha_{2},\, {{{\boldsymbol{\mathrm{u}}}}}_{2}^{{T}},\, {{{\boldsymbol{\mathrm{u}}}}}_{1}^{{T}} \right)$, ${{{\boldsymbol{\mathrm{u}}}}}_{k}^{{T}} = ( \alpha_{k} \rho_{k},\, \allowbreak \allowbreak \alpha_{k} \rho_{k} \speed{sk},\, \allowbreak \alpha_{k} \rho_{k} E_{k} )$. The conservative flux ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R7}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R7}{}]}$ reads ${{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = (0,\, {{{\boldsymbol{\mathrm{f}}}}}_{2}({{{\boldsymbol{\mathrm{u}}}}}_{2})^{{T}},\, {{{\boldsymbol{\mathrm{f}}}}}_{1}({{{\boldsymbol{\mathrm{u}}}}}_{1})^{{T}})$ with ${{{\boldsymbol{\mathrm{f}}}}}_{k}({{{\boldsymbol{\mathrm{u}}}}}_{k})^{{T}} = (\alpha_{k} \rho_{k} \speed{sk},\, \allowbreak \alpha_{k}(\rho_{k} \speed{sk}^{2}+p_{k}),\, \allowbreak \alpha_{k} ( \rho_{k} E_{k}+p_{k})\speed{sk} )$. ${{{\boldsymbol{\mathcal{N}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R77}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R77}{}]}$ is the matrix containing the non-conservative terms with ${{{\boldsymbol{\mathrm{n}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = - {{{\boldsymbol{\mathrm{n}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = (0,\, \allowbreak -{p_{I}},\, \allowbreak -{p_{I}}\speed{sint})$. Then, $\alpha_{k}$ is the volume fraction of phase $k \in \left[ 1,2 \right]$, $\rho_{k}$ the partial density, $\speed{sk}$ the phase velocity, $p_{k}$ the phase pressure, $E_{k}=\epsilon_{k} + \allowbreak \speed{sk}^{2}/2$ the total energy per unit of mass, $\epsilon_{k}$ the internal energy, $\speed{sint}$ the interfacial velocity and ${p_{I}}$ the interfacial pressure. Two levels of ingredients are still missing for this model. First, the macroscopic set of equations includes the interface dynamics through the interfacial terms $\speed{sint}$ and ${p_{I}}$ and thus needs closure on these terms. Second the thermodynamics has to be postulated. The mathematical properties of the model have been studied by [[@Embid_Baer_1992; @Coquel_2002; @Gallouet_2004]]{} among others and many closure have been proposed for the interfacial terms based on wave-type considerations and the entropy inequality. Regarding the thermodynamics, for non-miscible phases, the entropy ${{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ is commonly defined by Equation  as in [[@Coquel_2002; @Lochon_PhdThesis_2016]]{}, $$\begin{aligned} \label{eq:mixture_entropy_immiscible} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) = - \sum_{k=1,2} \alpha_{k} \rho_{k} s_{k},\end{aligned}$$ with $s_{k}=s_{k}(\rho_{k},p_{k})$ the phase entropy which takes for the Ideal Gas equation of state the form $$\begin{aligned} s_{k}= c_{v,k} \text{ln}\left(\frac{p_{k}}{\rho_{k}^{\gamma_{k}}}\right),\end{aligned}$$ with $c_{v,k}$ the heat capacity, $p_{k}$ the pressure, $\rho_{k}$ the density and $\gamma_{k}$ the isentropic coefficient of phase $k$. If we were to account for partial miscibility between the two phases, we would have to add a mixing term to the definition of the non-miscible entropy. The mixing term could take the form proposed in [[@Gallouet_2004]]{}, so that the entropy rewrites $$\begin{aligned} \label{eq:mixture_entropy_extended} {{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} \left[ {s}_{k}(\rho_{k},p_{k}) - \psi_{k}(\alpha_{k}) \right],\end{aligned}$$ with $\psi_{k}$, $k=\left[ 1,2 \right]$, two strictly convex nonlinear arbitrary functions depending on the volume fraction. Nevertheless, so far in the literature, no explicit expressions of these functions have been proposed. In [[@Gallouet_2004]]{}, in order to obtain a supplementary conservative equation using the entropy defined in Equation , the authors show that the following condition has to be fulfilled $$\begin{aligned} \label{eq:mixing_term_condition_BNZ} \psi_{k}(\alpha_{k}) = \psi_{k^{\prime}}(\alpha_{k^{\prime}}).\end{aligned}$$ In this section, we apply to the Baer-Nunziato model the framework introduced in Section \[sec:theory\] by means of computer algebra. We will firstly assume the phases are non-miscible and derive an entropy supplementary conservative equation along with conditions on the interfacial terms. All the closures proposed in the literature will be recovered. Secondly, we will also apply the methodology in the case of a thermodynamics with partial miscibility and derive an entropy supplementary conservative equation together with conditions on both the interfacial terms and the mixing terms of the entropy. Not only all the closures proposed in the literature are recovered but also new ones and we also propose explicit formulations of the mixing terms and show that depending on their expression, the condition expressed in [[@Gallouet_2004]]{} is not necessary. Methodology and decomposition. ------------------------------ We start without any condition on $(\speed{sint}, {p_{I}})$. We need initially to fix a decomposition of $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ including a certain degree of freedom as explained in Section \[ssec:NC\_theory\_applied\]. Given an entropy ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ of System , by expressing the entropic variables as ${{{\boldsymbol{\mathrm{v}}}}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = \left( {\mathrm{v_{\alpha}}}, {{{\boldsymbol{\mathrm{v}}}}}_{2}^{{T}}, {{{\boldsymbol{\mathrm{v}}}}}_{1}^{{T}}\right)$, we use the decomposition proposed in Definition . Since we do not want to generate other non-conservative terms, we choose to define the line vector $\transfer{v}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ by $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = \left( \transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}}), {{\boldsymbol{\mathrm{0}}}}, {{\boldsymbol{\mathrm{0}}}} \right)$ where $\transfer{salpha}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is the unknown scalar function a priori of all the variables ${{{\boldsymbol{\mathrm{u}}}}}$. We obtain the following decompositions [rClrClrClrCl]{}\[eq:final\_entropic\_condition\_decomp\_BNZ\] ( \_ )\^[[T]{}]{} &=& ()\ \_[2]{} \_[\_[2]{}]{}\_[2]{}(\_[2]{})\ \_[1]{} \_[\_[1]{}]{}\_[1]{}(\_[1]{}) ,\ (\_ )\^[[T]{}]{} &=& -() + + \_\_[k]{} [\_[k]{}]{}\ \ . $\transfer{salpha}$ allows fractions of the non-conservative terms to feed the matrix ${{{\boldsymbol{\mathcal{C}}}}_{k}}$. Given this decomposition, we use the methodology proposed in Section \[ssec:NC\_metho\]. ($Step$ 2) will be split here into two sub-steps. 1. Condition $(C_{1})$ on the symmetry of the matrix $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ ensures the existence of an entropy flux ${{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}})$. It will determine $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$. 2. Knowing $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$, Condition $(C_{2})$, $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}$, will return an equation linking $(\speed{sint},{p_{I}})$ and also $\psi_{k}$ when miscibility is accounted for. Non-miscible phases entropy. ---------------------------- We start applying our method $(Step \ 1)$ by postulating ${{\mathsf{H}}}$ as in Equation . The thermodynamics is entirely known and we use the Ideal Gas EOS. The entropic variables ${{{\boldsymbol{\mathrm{v}}}}}$ are then $$\begin{aligned} {{{\boldsymbol{\mathrm{v}}}}}= \begin{pmatrix} {\mathrm{v_{\alpha}}}\\ {{{\boldsymbol{\mathrm{v}}}}}_{2} \\ {{{\boldsymbol{\mathrm{v}}}}}_{1} \end{pmatrix} \text{ with } {\mathrm{v_{\alpha}}}= \frac{p_{1}}{T_{1}} - \frac{p_{2}}{T_{2}} \text{ and } {{{\boldsymbol{\mathrm{v}}}}}_{k} = \frac{1}{T_{k}}\begin{pmatrix} g_{k} - \speed{sk}^{2}/2 \\ \speed{sk} \\ -1 \end{pmatrix},\end{aligned}$$ with $g_{k}$ the Gibbs free energy, $g_{k} =\epsilon_{k} + p_{k}/\rho_{k} - T_{k}s_{k}$. We now apply the conditions to determine $\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}})$ and derive the equation that links the interfacial quantities $\speed{sint}$ and ${p_{I}}$. \[theo:BNZ\_ab\_classic\] Consider System . If the mixture entropy is defined as ${{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} {s}_{k}$ then with the decomposition proposed in Equations  $$\begin{aligned} \label{eq:classic_cond_a} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric } \Leftrightarrow \transfer{salpha} ({{{\boldsymbol{\mathrm{u}}}}}) = {F}(\alpha_{2}) + \frac{p_{1}}{T_{1}}u_{1} - \frac{p_{2}}{T_{2}}u_{2},\end{aligned}$$ with ${F}$ a strictly convex arbitrary function depending on the volume fraction $\alpha_{2}$. As a consequence the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]$ gives $$\begin{aligned} \label{eq:classic_cond_b} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rl} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= {{\boldsymbol{\mathrm{0}}}} \\ \Leftrightarrow \ - {F}(\alpha_{2}) + \sum_{k=1,2} \frac{(-1)^{k}}{T_{k}} ({p_{I}}- p_{k} )( \speed{sk} - \speed{sint}) &= 0. \end{IEEEeqnarraybox}\end{aligned}$$ The function $\transfer{salpha}$ is found relying on symbolic computation and it holds as a proof. As explained in $(Step\ 2.a)$, Equation  guarantees the existence of an entropy flux ${{\mathsf{G}}}$ associated with the mixture entropy ${{\mathsf{H}}}$ chosen as in Equation  by defining the unknown function $\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}})$. Then as described in $(Step\ 2.b)$, Equation  relates the interfacial terms $(\speed{sint}, {p_{I}})$. By choosing ${F}(\alpha_{2}) =0$, the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\times \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ writes $$\begin{aligned} \label{eq:classic_cond_b_reduced} \sum_{k = 1,2} \frac{1}{T_{k}} \left(p_{k}-{p_{I}}\right) \left(\speed{sint}-\speed{sk}\right) = 0.\end{aligned}$$ So now, to obtain a closed model along with a supplementary conservative equation, we can postulate an interfacial velocity $\speed{sint}$ and derive the corresponding ${p_{I}}$. We will limit ourselves to defining $\speed{sint}$ such that the field associated to $\speed{sint}$ is linearly degenerate. In that case, the only admissible interfacial velocities are $\speed{sint} = \beta u_{1} + (1-\beta) u_{2}$ with $\beta \in \left[ 0,1, \alpha_{1} \rho_{1}/\rho \right]$ [[@Coquel_2002]]{}, [[@Lochon_PhdThesis_2016]]{}. We will focus on the particular case where ${F}(\alpha_{2}) =0$. We obtain the following results: - If $\speed{sint} = \speed{sk}$, then Equation  returns ${p_{I}}= p_{k^{\prime}}$. $(\speed{sk},p_{k^{\prime}})$ is the closure proposed first by [[@Baer_Nunziato_1986]]{}, [[@Kapila_1997]]{}, [[@Bdzil_1999]]{}, in the context of deflagration-to-detonation. - If $\speed{sint} = \beta u_{1} + (1-\beta) u_{2}$ with $\beta = \alpha_{1} \rho_{1}/\rho$, then Equation  returns ${p_{I}}=\mu p_{1} + (1- \mu ) p_{2}$ with $\mu\left(\beta\right) =(1-\beta) T_{2} / ( \beta T_{1} + (1-\beta) T_{2})$. It is the closure found in [[@Lochon_PhdThesis_2016]]{} among others. We see that first these closures are a specific case where $F(\alpha_{2})$ is chosen to be zero in Equation . Second, one could have chosen another interfacial velocity $\speed{sint}$ and it would have led to another interfacial pressure ${p_{I}}$ compatible with an entropy pair. If we had used the extended condition expressed in Equation , then the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ would be $$\begin{aligned} \label{eq:classic_cond_b_extended} & \sum_{k = 1,2} \frac{1}{T_{k}} \left[p_{k}-{p_{I}}\left({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}\right) \right] \left[\speed{sint}\left({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}\right)-\speed{sk}\right] \partial_{x} \alpha_{k} \leq 0 \\ \Leftrightarrow \ & - \sum_{k = 1,2} \frac{1}{T_{k}} \frac{Z_{k}}{(Z_{1}+Z_{2})^{2}}\left[ p_{k^{\prime}}-p_{k} + sgn\left(\partial_{x} \alpha_{1} \right) (u_{k^{\prime}}-\speed{sk}) Z_{k^{\prime}} \right]^{2} \leq 0,\end{aligned}$$ where $Z_{k}$ is defined by $Z_{k} = \rho_{k} a_{k}$ with the phase sound speed $a^{2}_{k} = \left. \partial p_{k} / \partial \rho_{k} \right|_{\scriptstyle s_{k}}$. From Equation , one sees that the dependency on $\partial_{x} {{{\boldsymbol{\mathrm{u}}}}}$ reduces to $\partial_{x} \alpha_{2}$ otherwise some terms would not be signable. Then closures such as the one found through Discrete Element Method (DEM) [[@Saurel_Gavrilyuk_2003]]{} are obtained $$\begin{aligned} \speed{sint} &= \frac{Z_{1} u_{1} + Z_{2} u_{2}}{Z_{1} + Z_{2}} + sgn\left(\partial_{x} \alpha_{1} \right) \frac{p_{2}-p_{1}}{Z_{1}+Z_{2}}, \\ {p_{I}}&= \frac{Z_{2} p_{1} + Z_{1} p_{2}}{Z_{1} + Z_{2}} + sgn\left(\partial_{x} \alpha_{1} \right) \frac{Z_{1}Z_{2}}{Z_{1}+Z_{2}} \left(u_{2}-u_{1}\right).\end{aligned}$$ Partially miscible phases entropy. ---------------------------------- Now, let us add a degree of freedom in the thermodynamics by introducing mixing terms in the definition of the entropy ${{\mathsf{H}}}$ as in Equation  to account for partial miscibility of the phases. The added terms, $\psi_{k}$, functions of the volume fraction $\alpha_{k}$ only, are to be determined. The entropic variables ${{{\boldsymbol{\mathrm{v}}}}}$ are $$\begin{aligned} \label{eq:v_r_ln_alpha} {{{\boldsymbol{\mathrm{v}}}}}= \begin{pmatrix} \sum\limits_{\text{\tiny $k{=}1,2$}} (-1)^{k+1} \dfrac{p_{k}}{T_{k}} \left[ 1 - \dfrac{\alpha_{k}}{r_{k}} \psi_{k}^{\prime}(\alpha_{k}) \right]\\ {{{\boldsymbol{\mathrm{v}}}}}_{2} \\ {{{\boldsymbol{\mathrm{v}}}}}_{1} \end{pmatrix} \text{ with } {{{\boldsymbol{\mathrm{v}}}}}_{k} = \frac{1}{T_{k}}\begin{pmatrix} g_{k} - \speed{sk}^{2}/2\\ \speed{sk} \\ -1 \end{pmatrix}\end{aligned}$$ \[theo:BNZ\_ab\_mixing\] Consider System . If the mixture entropy is defined as ${{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} \left[ {s}_{k} - \psi_{k}(\alpha_{k}) \right]$ with $\psi_{k}$, $k=\left[ 1,2 \right]$, two strictly convex arbitrary functions depending on the volume fraction, then with the decomposition proposed in Equations , we have $$\begin{aligned} \label{eq:mixture_cond_a} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rl} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v} \text{ symmetric } \\ \Leftrightarrow \, &\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}}) = {F}(\alpha_{2}) + \frac{p_{1}}{T_{1}}u_{1} \left[ 1 - \frac{\alpha_{1}}{r_{1}} \psi_{1}^{\prime}(\alpha_{1}) \right] - \frac{p_{2}}{T_{2}}u_{2} \left[ 1 - \frac{\alpha_{2}}{r_{2}} \psi_{2}^{\prime}(\alpha_{2}) \right] \end{IEEEeqnarraybox}\end{aligned}$$ with ${F}$ a strictly convex arbitrary function depending on the volume fraction. As a consequence the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ gives $$\begin{aligned} \label{eq:mixture_cond_b} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rcl} & {{\boldsymbol{\mathrm{0}}}} &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \\ \Leftrightarrow \ & 0 &= - {F}(\alpha_{2}) + \sum_{k=1,2} (-1)^{k+1} \alpha_{k}\rho_{k} \psi_{k}^{\prime}(\alpha_{k}) (u_{k}-\speed{sint})\\ & &+ \sum_{k=1,2} \frac{(-1)^{k}}{T_{k}} ({p_{I}}- p_{k} )( \speed{sk} - \speed{sint}) \end{IEEEeqnarraybox}\end{aligned}$$ Again, Equation  guarantees the existence of an entropy flux ${{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}})$ conditioning the function $\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}})$ $(Step\ 2.a)$. The interfacial quantities $(\speed{sint},{p_{I}})$ and $\psi_{k}$ are linked by Equation  $(Step \ 2.b)$. The difference with the previous case for immiscible phases is that there are two supplementary unknowns $\psi_{k}$, $k=1,2$. We thus are free to either postulate first an interfacial velocity $\speed{sint}$ and then derive the corresponding ${p_{I}}$ and $\psi_{k}$ or postulate first the functions $\psi_{k}$ and see what choices we have for the interfacial terms. In the following we investigate the two approaches. ### Interfacial closures impacting thermodynamics. Let us postulate $\speed{sint}$ and limit ourselves to the case ${F}(\alpha_{2}) =0$. We will again seek a linearly degenerate field for $\speed{sint}$. In such case, the results in Table \[table:mixing\_entropy\_cond\_case12\] are obtained. [c|c|c|c]{} & $\speed{sint}$ & ${p_{I}}$ & $\left( \psi_{k} , \psi_{k^{\prime}}\right)$\ Case 1 &$\speed{sk}$ & $p_{k^{\prime}}$ & $\left( \psi_{k}, 0 \right)$\ Case 2 & -------------------------------------- $\beta u_{1} + (1-\beta) u_{2}$ $\beta = \alpha_{1} \rho_{1} / \rho$ -------------------------------------- : Admissiblethermodynamics and model closures obtained by postulating$\speed{sint}$[]{data-label="table:mixing_entropy_cond_case12"} & --------------------------------------------------------------------------------- $\mu p_{1} + (1- \mu ) p_{2}$ $\mu\left(\beta\right) = \frac{(1-\beta) T_{2}}{\beta T_{1} + (1-\beta) T_{2}}$ --------------------------------------------------------------------------------- : Admissiblethermodynamics and model closures obtained by postulating$\speed{sint}$[]{data-label="table:mixing_entropy_cond_case12"} & $\psi_{k}(\alpha_{k})=\psi_{k^{\prime}}(\alpha_{k^{\prime}})$ In Case 1 of Table \[table:mixing\_entropy\_cond\_case12\], $\psi_{k}$ can be interpreted as a configuration energy of phase $k$ as in [[@Baer_Nunziato_1986]]{}, [[@Kapila_1997]]{} [[@Bdzil_1999]]{}, in the context of deflagration-to-detonation. It is a term defining an interaction of one phase with itself only. More importantly, Equation  shows that it is not possible to include a configuration energy for each phase when choosing the closure $(\speed{sint},{p_{I}})=(\speed{sk},p_{k^{\prime}})$. In Case 2 of Table \[table:mixing\_entropy\_cond\_case12\], the condition on the mixing term introduced in Equation  by [[@Gallouet_2004]]{} is recovered and the closures are the one stated in [[@Coquel_2002]]{}. However, the condition on the mixing terms imposes a constraint on the volume fraction and thus on the flow topology. Since mixing of the phases should be able to occur disregarding the flow topology, these terms fail to introduce free mixing among the phases. ### Thermodynamics impacting interfacial term closures. Since Case 1 and Case 2 of Table \[table:mixing\_entropy\_cond\_case12\] do not allow the phases to mix, let us choose first the thermodynamics of the system and induce the admissible interfacial terms. It has been shown that the mixing entropy of an ideal compressible binary mixture is of the form $\sum_{k=1,2} \alpha_{k} \text{ln}(\alpha_{k})$. Therefore, we choose to define the functions $\psi_{k}$ by $\psi_{k}(\alpha_{k}) =r_{k} \text{ln}(\alpha_{k})$. In this case, the entropy writes $$\begin{aligned} {{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} \left[ {s}_{k} - r_{k}\text{ln}(\alpha_{k}) \right],\end{aligned}$$ with $r_{k}$ the specific gas constant of phase $k$, we now account for quasi-miscibility between the phases. The condition on $\transfer{salpha}$ degenerates, $\transfer{salpha} = {F}(\alpha_{2})$ and the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ is now $$\begin{aligned} \label{eq:mixture_cond_b_rln} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{c} - {F}(\alpha_{2}) + {p_{I}}\left( \frac{u_{1}-\speed{sint}}{T_{1}}-\frac{u_{2}-\speed{sint}}{T_{2}} \right) = 0. \end{IEEEeqnarraybox}\end{aligned}$$ It is no more possible to obtain the classic definition on $\speed{sint}$ and ${p_{I}}$. In the case ${F}(\alpha_{2})=0$ two choices are possible to verify Equation  and summarized in Table \[table:mixing\_entropy\_cond\]. $\speed{sint}$ ${p_{I}}$ -------- ---------------------------------------------------------------------- --------------- Case 3 $\beta u_{1} + (1-\beta) u_{2}$ with $\beta = T_{2} / (T_{2}-T_{1})$ no constraint Case 4 no constraint 0 : Admissiblethermodynamics and model closures obtained by postulating$\psi_{k}$[]{data-label="table:mixing_entropy_cond"} Case 3 of Table \[table:mixing\_entropy\_cond\] proposes a temperature-based averaged velocity for $\speed{sint}$, which does not seem to be physically reasonable. In Case 4, the interfacial pressure must vanish for the system to admit an entropy supplementary conservation equation and the Baer-Nunziato model becomes a conservative system if one assumes the field associated to $\speed{sint}$ to be linearly degenerate. One knows how much it simplifies the problem in terms of numerical implementation. This result can be interpreted as an incompatibility between the existence of a mixing process in the thermodynamics of the mixture and an interfacial pressure, that stays meaningful as long as there is an interface between the two phases. ### Link with dispersed phase flow. When the thermodynamics accounts for mixing (Case 4 Table \[table:mixing\_entropy\_cond\]), the existence of an entropy supplementary conservative equation is incompatible with the interfacial pressure, and thus the nozzling terms ${p_{I}}\partial_{x} \volfrac{k}$ vanish. In separated two-phase flows, these terms are known to be necessary to preserve uniformity in velocity and pressure of the flow during its temporal evolution [[@Andrianov_2003]]{} and are usually compared to the terms obtained in a single gas with a variable section [[@Saurel_2001]]{}. Whereas these arguments seem valid for separated two-phase flows, one may question the role these terms play in a dispersed phase flows. Taking the particular case ${p_{I}}= 0$ and $p_{2}=0$ in the Baer-Nunziato model seems to lead to a system of equations similar to one that would describe a flow of incompressible suspended particles, where 1 would denote the carrier phase and 2 the dispersed phase. Doing so, one recovers not only the Marble model [[@Marble_1963]]{}, which proposes a pressureless gas dynamic equations for the particle phase, valid in the limit where $\alpha_{2} < 10^{-3}$, but also the model obtained by Sainsaulieu [[@Sainsaulieu_1995]]{} in the asymptotic limit where the volume fraction of the particles $\alpha_{2} \rightarrow 0$. Nevertheless, even if the partial differential equations are alike, the thermodynamics associated to Marble and Sainsaulieu models differ from the one we propose for the Baer-Nunziato model. The latter accounts for compressibility of the two phases and partial miscibility whereas the thermodynamics of the Marble model assumes incompressibility of the particles and non-miscibility between the two phases. To conclude, if one aims at unifying the description of both separated phases and dispersed flow through a unique model, the thermodynamics must be treated together with the system modelling. Application to the plasma model {#sec:plasma} =============================== The multicomponent fluid modelling of plasma flows out of thermal equilibrium has been derived rigorously from kinetic theory using a multi-scale Chapman-Enskog expansion mixing a hyperbolic scaling for the heavy species with a parabolic scaling for the electrons [[@Graille_2007]]{}. The system takes the form $$\begin{aligned} \label{sys:plasma_eq_full} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= \partial_{x} \left( {{{\boldsymbol{\mathcal{D}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}\right),\end{aligned}$$ with $$\begin{aligned} \label{eq:plasma_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) &= \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ (\kappa/2-1)\speed{s}^2 & (2-\kappa)\speed{s} & \kappa & 0 & 0\\ (\kappa/2 \speed{s}^2 - \frac{h^{tot}}{\rho_{h}})\speed{s} & \frac{h^{tot}}{\rho_{h}} - \kappa \speed{s}^2 & (1+\kappa)\speed{s} & 0 & 0\\ -\frac{\rho_{e}}{\rho_{h}} \speed{s} & \frac{\rho_{e}}{\rho_{h}} & 0 & \speed{s} & 0 \\ - \frac{\rho_{e} \epsilon_{e}}{\rho_{h}} \speed{s} & \frac{\rho_{e} \epsilon_{e}}{\rho_{h}} & 0 & 0 & \speed{s} \end{pmatrix}, \\ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) &= \begin{pmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ -\frac{\rho_{e} \epsilon_{e}}{\rho_{h}} \kappa \speed{s} & \frac{\rho_{e} \epsilon_{e}}{\rho_{h}} \kappa & 0 & 0 & 0 \end{pmatrix}, \\ {{{\boldsymbol{\mathcal{D}}}}}({{{\boldsymbol{\mathrm{u}}}}}) &= \begin{pmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} & \frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} + \gamma D\\ 0 & 0 & 0 & 0 & \frac{D\kappa}{T_{e}}\\ 0 & 0 & 0 & -\frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} & \frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} + \gamma D \end{pmatrix},\end{aligned}$$ where the column vector ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{5}$ is defined by ${{{\boldsymbol{\mathrm{u}}}}}^{{T}} = \left( \rho_{h}, \rho_{h} \speed{s}, E, \rho_{e}, \rho_{e} \epsilon_{e}\right)$ with $\rho_{h}$ is the density of the heavy particles, $\speed{s}$ the hydrodynamic velocity, $E$ the total energy defined by $E= \rho_{h} \speed{s}^2/2 + \rho_{h} \epsilon_{h} + \rho_{e} \epsilon_{e}$, $ \epsilon_{h}$ the internal energy of the heavy particles, $\rho_{e}$ the density of the electrons, $\epsilon_{e}$ the internal energy of the electrons, $h^{tot}$ the total enthalpy defined by $h^{tot}= E + p$ with $p=p_{h}+p_{e}$, $T_{e}$ the temperature of the electrons, the constant $\kappa$ defined by $\kappa = \gamma-1$ with $\gamma$ the isentropic coefficient, $p_{h}$ is the pressure of the heavy particles and $p_{e}$ is the pressure of the electrons. In the diffusive terms, $\lambda$ is the electron thermal conductivity, D the electron diffusion coefficient. Concerning the thermodynamics, it can be obtained from kinetic theory. The electrons and the heavy particles thermodynamics are defined by an ideal gas equation of state, and they share both the same isentropic coefficient: $p_{h} = \kappa \rho_{h} \epsilon_{h}$, $p_{e} = \kappa \rho_{e} \epsilon_{e}$ where $p_{h}$ is the pressure of the heavy particles and $p_{e}$ is the pressure of the electrons, $r$ is the constant of the gas $r=c_{v} \kappa$ with $c_{v}$ the calorific heat at constant volume, the model being adimensionalized $r=c_{v}(\gamma-1)=1$. The model is naturally hyperbolic [[@Graille_2007]]{} and also involves second-order terms and eventually source terms [[@Magin_2009]]{}. Here we considered the homogeneous form. In this section, we would like to derive the usual entropy supplementary conservative equation found by [[@Graille_2007]]{} and show that it is unique, to attest the effectiveness of the theory. Decomposition. -------------- We need to proceed to the decomposition of the conservative and non conservative terms of System . We restrict ourselves again to the decomposition proposed in Definition  and we add a degree of liberty to each non-null non-conservative components by defining $\transfer{v}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R5}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R5}{}]}$ as $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = ( \transfer{s1}({{{\boldsymbol{\mathrm{u}}}}}), \transfer{s2}({{{\boldsymbol{\mathrm{u}}}}}), 0, 0, 0)$ such that the following decompositions are obtained $$\begin{aligned} \label{eq:final_entropic_condition_decomp_plasma} \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \right)^{{T}} = {{{\boldsymbol{\mathrm{v}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \cdot \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \begin{pmatrix} \transfer{s1}({{{\boldsymbol{\mathrm{u}}}}})\\ \transfer{s2}({{{\boldsymbol{\mathrm{u}}}}})\\ 0 \\ 0 \\ 0 \end{pmatrix}, \\ \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \right)^{{T}} =\begin{pmatrix} - \transfer{s1}({{{\boldsymbol{\mathrm{u}}}}}) - \frac{\rho_{e}}{\rho_{h}} \left(1-\frac{T_{e}}{T_{h}}\right) \speed{s} \\ - \transfer{s2}({{{\boldsymbol{\mathrm{u}}}}}) + \frac{\rho_{e}}{\rho_{h}} \left(1-\frac{T_{e}}{T_{h}}\right) \\ 0 \\ 0 \\ 0 \end{pmatrix}.\end{aligned}$$ The unknown scalar functions $\transfer{v}_{k}({{{\boldsymbol{\mathrm{u}}}}})$ give the possibility to fractions of the non-conservative terms to be given to the matrix ${{{\boldsymbol{\mathcal{C}}}}_{k}}$. Ideal Gas entropy. ------------------ The entropy ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ for two perfect gases is defined as $$\begin{aligned} \label{eq:mixture_entropy_plasma} {{\mathsf{H}}}= - \rho_{h} {s}_{h} - \rho_{e} {s}_{e},\end{aligned}$$ with the partial entropies defined by $$\begin{aligned} {s}_{h}= c_{v} \, \text{ln}\left(\frac{p_{h}}{\kappa \rho_{h}^{\transfer{v}}}\right), && {s}_{e}= c_{v} \, \text{ln}\left(\frac{p_{e}}{\kappa \rho_{e}^{\transfer{v}}}\right).\end{aligned}$$ This entropy includes mixing between the electrons and the heavy particles. Thus, we start applying our method $(Step \ 1)$ by postulating ${{\mathsf{H}}}$ as in Equation . The entropic variables ${{{\boldsymbol{\mathrm{v}}}}}$ are then $$\begin{aligned} {{{\boldsymbol{\mathrm{v}}}}}= \begin{pmatrix} \dfrac{1}{T_{h}}\left(g_{h} - \speed{s}^{2}/2\right)\\ \dfrac{1}{T_{h}}\speed{s}\\ -\dfrac{1}{T_{h}}\\ \dfrac{1}{T_{e}} g_{e}\\ \dfrac{1}{T_{h}}-\dfrac{1}{T_{e}} \end{pmatrix},\end{aligned}$$ with $g_{k}$ the Gibbs free energy, $g_{k} =\epsilon_{k} + p_{k}/\rho_{k} - T_{k}s_{k}$. In the fourth component of the entropic variable, the kinetic energy of the electrons has vanished. This is due to the low-Mach assumption made for the electrons. We now apply the conditions to determine $\transfer{sk}({{{\boldsymbol{\mathrm{u}}}}})$. \[theo:plasma\_ab\_classic\] Consider System . If the mixture entropy is defined as ${{\mathsf{H}}}= - \rho_{h} {s}_{h} - \rho_{e} {s}_{e}$, then with the decomposition proposed in Equations , we have $$\begin{aligned} \label{eq:plasma_classic_cond_a} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rl} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric } \\ \Leftrightarrow \ & \transfer{s1} ({{{\boldsymbol{\mathrm{u}}}}}) = \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)\speed{s} \text{ and } \transfer{s2} ({{{\boldsymbol{\mathrm{u}}}}}) = -\frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right) \end{IEEEeqnarraybox},\end{aligned}$$ and the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]$ is $$\begin{aligned} \label{eq:plasma_classic_cond_b} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] & = (0,\, 0,\, 0,\, 0,\, 0).\end{aligned}$$ Using Maple, we find $$\begin{aligned} &\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric } \nonumber \\ \Leftrightarrow & \transfer{s1} ({{{\boldsymbol{\mathrm{u}}}}}) = \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)\speed{s} + \int \left[ -\speed{s} \partial_{\speed{s}}{F}_{1}(\rho_{h},\speed{s}) +\rho_{h} \partial_{\rho_{h}}{F}_{1}(\rho_{h},\speed{s}) \right] d\speed{s} + {F}_{2}(\rho_{h}) \nonumber \\ & \text{and } \transfer{s2} ({{{\boldsymbol{\mathrm{u}}}}}) = -\frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right) + {F}_{1}(\rho_{h},\speed{s}),\end{aligned}$$ with $\mathsf{F_{1}}$, $\mathsf{F_{2}}$ two arbitrary functions and the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]$ is $$\begin{aligned} \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \right)^{{T}} & = \begin{pmatrix} - \int \left[ -\speed{s} \partial_{\speed{s}}{F}_{1}(\rho_{h},\speed{s}) +\rho_{h} \partial_{\rho_{h}}{F}_{1}(\rho_{h},\speed{s}) \right] d\speed{s} - {F}_{2}(\rho_{h}) \\ -{F}_{1}(\rho_{h},\speed{s}) \\ 0\\ 0\\ 0 \end{pmatrix}\\ &= {\boldsymbol{0}}.\end{aligned}$$ One sees that the last equation imposes first ${F}_{1}=0$ and thus ${F}_{2}=0$. Reinjecting these terms into the first equation gives the result. As explained in $(Step\ 2.a)$, the Equation  guarantees the existence of an entropy flux ${{\mathsf{G}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ associated with the entropy ${{\mathsf{H}}}$ defined in Equation  by solving the unknown functions $\transfer{s1}({{{\boldsymbol{\mathrm{u}}}}})$ and $\transfer{s2}({{{\boldsymbol{\mathrm{u}}}}})$. Therefore, for the entropy ${{\mathsf{H}}}$ defined in Equation , there is a unique decomposition which ensures the existence of a supplementary conservative equation which is given by $$\begin{aligned} \label{eq:final_entropic_condition_decomp_plasma_determined} \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}+ {{{\boldsymbol{\mathcal{C}}}}_{2}}\right] \right)^{{T}}= {{{\boldsymbol{\mathrm{v}}}}}^{{T}} \cdot \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \begin{pmatrix} \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)v\\ \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)\\ 0 \\ 0 \\ 0 \end{pmatrix},\end{aligned}$$ $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right] = {\boldsymbol{0}}.\end{aligned}$$ It leads to the following entropy flux couple $$\begin{aligned} {{\mathsf{H}}}&= - \rho_{h} {s}_{h} - \rho_{e} {s}_{e}, \\ {{\mathsf{G}}}&= - \left( \rho_{h} {s}_{h} + \rho_{e} {s}_{e} \right) \speed{s}.\end{aligned}$$ The theory recovers the supplementary conservative equation already found in the literature from the kinetic theory [[@Graille_2007]]{}. Conclusion ========== In the present contribution, we have proposed a theoretical framework for the derivation of supplementary conservation laws for systems of partial differential equation including first-order non-conservative terms - commonly encountered in modeling of complex flows - thus extending the standard approach for systems of conservation laws. Since our main objective is deriving an entropy supplementary conservation law, we have used this framework to make a first step to extend the theory of Godunov-Mock to such non-conservative systems. Given a reasonable choice in the combination of the conservative and non-conservative terms, we have been able to show how to use the theory to design or analyze systems by means of computer algebra on two applications chosen for their numerous differences in terms of model and thermodynamics closure as well as the nature of the waves impacted by the non-conservative terms. Firstly, applied to the Baer-Nunziato two-phase flow model derived from rational thermodynamics, the theory has brought about entropy supplementary conservative equations together with constraints on the interfacial quantities and the definition of the thermodynamics for non-miscible fluids and also when accounting for some level of mixing of the two phases. A new closure for the interfacial quantities has been proposed and leads to a conservative system. Secondly, for a plasma model obtained rigorously from the kinetic theory of gases, where the thermodynamics is also provided, the approach allows to recover as unique the supplementary conservation equation related to the kinetic entropy and is thus assessed. The content of the paper is a first step into studying the entropic symmetrization in the sense of Godunov-Mock and relation to source terms for two-phase flow modeling. Some partial symmetrization of the Baer-Nunziato model has been obtained in the classical framework by [@Forestier_2011]. Combining such symmetrization theory with source terms can then be envisioned such as in the case of plasma flows [@Magin_2009], even if the symmetrization is only partial in the framework of [@Graille_2007] where the electron are considered in a low-Mach limit. Nevertheless, for such a study to be complete, several other steps have to be handled first: the question of the strict convexity of the entropy for the change of variable to be admissible and its relation to thermodynamics (a difficult question [@Coquel_2002; @Gallouet_2004]); it is a part of Pierre Cordesse’s PhD Thesis [@Cordesse_PhD]. This loss of strict convexity in the framework of non-interacting thermodynamics has been investigated in [[@Cordesse_CMT_2019]]{} where a mixing thermodynamics for multi-fluids has been developed. Based on this new developments, we hope that equipping the Baer-Nunziato system with an extended thermodynamics closure will lead to a strictly convex entropy and thus allow the study of entropic full symmetrization and source terms, in the spirit of [@Giovangigli_1998; @Massot_2002; @Giovangigli_2004; @Magin_2009]. This is the subject of our current research. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to acknowledge the support of a CNES/ONERA PhD Grant for P. Cordesse and the help of M. Théron (CNES). They would like to express their special thanks to F. Coquel, S. Kokh, V. Giovangigli and A. Murrone for their invaluable help and numerous pieces of advice during the writing of the paper. We also would like to thank discussions with J.M. Hérard, which prompted this research path. Part of this work was conducted during the Summer Program 2018 at NASA Ames Research Center and the support and help of Nagi N. Mansour is also gratefully acknowledged. [^1]: ONERA, DMPE, 8 Chemin de la Hunière, 91120 Palaiseau, France, and CMAP, Ecole polytechnique, Route de Saclay 91128 Palaiseau Cedex, France, (pierre.cordesse@polytechnique.edu) [^2]: CMAP, Ecole polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, (marc.massot@polytechnique.edu) [^3]: Received date, and accepted date (The correct dates will be entered by the editor). [^4]: Among the most well-known symmetry transformations, the time translation yields the conservation of the total energy of the system if the associated Lagrangian is invariant to time-shift and the space translation yields the conservation of the total momentum of the system if the Lagrangian is invariant to space-shift [^5]: Such closure is similar to the one used in [[@Powers_1988; @Powers_1990]]{} which led to a controversy [[@Drew_1983; @Bdzil_1999; @Andrianov_2003]]{} [^6]: Maple is a trademark of Waterloo Maple Inc.

--- abstract: 'Let $G$ be a finite group, and $\a$ a nontrivial character of $G$. The McKay graph $\MC(G,\a)$ has the irreducible characters of $G$ as vertices, with an edge from $\c_1$ to $\c_2$ if $\c_2$ is a constituent of $\a\c_1$. We study the diameters of McKay graphs for finite simple groups $G$. For alternating groups, we prove a conjecture made in [@LST]: there is an absolute constant $C$ such that $\hbox{diam}\,{\mathcal M}(G,\a) \le C\frac{\log |\AAA_n|}{\log \a(1)}$ for all nontrivial irreducible characters $\a$ of $\AAA_n$. Also for classsical groups of symplectic or orthogonal type of rank $r$, we establish a linear upper bound $Cr$ on the diameters of all nontrivial McKay graphs.' address: - 'M.W. Liebeck, Department of Mathematics, Imperial College, London SW7 2BZ, UK' - 'A. Shalev, Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel' - 'P.H. Tiep, Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA' author: - 'Martin W. Liebeck' - Aner Shalev - Pham Huu Tiep title: McKay graphs for alternating and classical groups --- [^1] Introduction ============ For a finite group $G$, and a (complex) character $\a$ of $G$, the [*McKay graph*]{} $\MC(G,\a)$ is defined to be the directed graph with vertex set ${\rm Irr}(G)$, there being an edge from $\c_1$ to $\c_2$ if and only if $\c_2$ is a constituent of $\a\c_1$. A classical result of Burnside and Brauer [@Br] shows that $\MC(G,\a)$ is connected if and only if $\a$ is faithful. The study of McKay graphs for finite simple groups $G$ was initiated in [@LST], with a particular focus on the diameters of these graphs. Theorem 2 of [@LST] establishes a quadratic upper bound $\hbox{diam}\,{\mathcal M}(G,\a) \le Cr^2$ for any simple group $G$ of Lie type or rank $r$ and any nontrivial $\a \in {\rm Irr}(G)$. Notice that the smallest (resp. largest) nontrivial irreducible character degrees of $G$ are at most $q^{cr}$ (resp. at least $q^{c'r^2}$), where $c,c'$ are constants, and hence the maximal diameter of a McKay graph ${\mathcal M}(G,\a)$ is at least a linear function of $r$. Theorem 3 of [@LST] implies a linear upper bound on these diameters for the classical groups $G=\PSL_n^\e(q)$, provided $q$ is large compared to $n$. Our first main result establishes a linear upper bound for the remaining classical groups. \[main1\] Let $G$ be a quasisimple classical group $Sp_n(q)$ or $\O_n^\e(q)$, and let $\a$ be a nontrivial irreducible character of $G$. Then $\hbox{diam}\,{\mathcal M}(G,\a) \le Cn$, where $C=16$ or $32$, respectively. An obvious lower bound for $\hbox{diam}\,{\mathcal M}(G,\a)$ (when $\a(1)>1$) is given by $\frac{\log \bmax(G)}{\log \a(1)}$, where $\bmax(G)$ is the largest degree of an irreducible character of $G$. In [@LST Conjecture 1] we conjectured that for simple groups $G$, this bound is tight up to a multiplicative constant. This conjecture was proved in [@LST Theorem 3] for the simple groups $\PSL_n^\e(q)$, provided $q$ is large compared to $n$. Recently it has also been established for the symmetric groups in [@S]. Deducing it for the alternating groups is not entirely trivial, and this is the content of our next result. \[main2\] There is an effective absolute constant $C$ such that, for all $n \geq 5$ and for all nontrivial irreducible characters $\a$ of $G:=\AAA_n$, $$\hbox{diam}\,{\mathcal M}(G,\a) \le C\frac{\log |G|}{\log \a(1)}.$$ In our final result, we consider covering ${\rm Irr}(G)$ by products of arbitrary irreducible characters, instead of powers of a fixed character. This idea was suggested by Gill [@G], inspired by an analogous result of Rodgers and Saxl [@RS] for conjugacy classes in $G=\SL_n(q)$: this states that if a collection of conjugacy classes of $G$ satisfies the condition that the product of the class sizes is at least $|G|^{12}$, then the product of the classes is equal to $G$. As a piece of notation, for characters $\c_1,\ldots,\c_l$ of $G$, we write $\c_1\c_2\cdots \c_l \supseteq \Irr(G)$ to mean that every irreducible character of $G$ appears as a constituent of $\c_1\c_2\cdots \c_l$. Also, let $g: \N\to \N$ be the function appearing in [@LST Theorem 3]. \[rodsax\] - Let $G$ be a simple group of Lie type of rank $r$, let $l \ge 489r^2$, and let $\c_1,\ldots,\c_l \in \Irr(G) \setminus 1_G$. Then $\c_1\c_2\cdots \c_l \supseteq \Irr(G)$. - Let $G = \PSL_n^\e(q)$ with $q>g(n)$, let $l \in \N$, and let $\c_1,\ldots \c_l \in \Irr(G)$ satisfy $\prod_1^l \c_i(1) > |G|^{10}$. Then $\c_1\c_2\cdots \c_l \supseteq \Irr(G)$. Gill [@G] has conjectured that part (ii) of the theorem holds for all simple groups (with the constant 10 possibly replaced by a different constant). As a stepping stone in the spirit of the linear bound given by Theorem \[main1\], let us pose the following more modest conjecture. \[rsax\] There is an absolute constant $C>0$ such that the following holds. Let $G=\Cl_n(q)$, a classical simple group of dimension $n$, or $\AAA_n$, an alternating group of degree $n\ge 5$. Let $l \ge Cn$, and let $\c_1,\ldots,\c_l \in \Irr(G) \setminus 1_G$. Then $\c_1\c_2\cdots \c_l \supseteq \Irr(G)$. See Proposition \[rs2-an\] for some partial result on Conjecture \[rsax\] in the cae of $\AAA_n$. The layout of the paper is as follows. Section \[prel1\] contains a substantial amount of character theory for symplectic and orthogonal groups that is required for the proof of Theorem \[main1\], which is completed in Section \[pfth1\]. The remaining sections \[pfth2\] and \[pfth3\] contain the proofs of Theorems \[main2\] and \[rodsax\], respectively. Some character theory for symplectic and orthogonal groups {#prel1} ========================================================== Let $V = \F_q^d$ be endowed with a non-degenerate, alternating or quadratic of type $\e = \pm$, form and let $G$ denote the derived subgroup of the full isometry group of the form. Assume that $G$ is quasisimple, so that $G = \Sp(V) = \Sp_d(q)$ or $\O(V) = \O^\e_d(q)$. This section contains a detailed study of some specific irreducible characters $\c$ of $G$ – namely, the constituents of the permutation character $\Ind^G_{[P,P]}(1_{[P,P]})$, where $P$ is the maximal parabolic subgroup of $G$ stabilizing a singular 1-space. Two of the main results of the section are Propositions \[rat-so21\] and \[rat-sp-so22\], which give upper bounds for the character ratios $|\c(g)/\c(1)|$ for $g\in G$. These will be used in Section \[pfth1\] to prove Theorem \[main1\]. Reduction lemmas {#red} ---------------- It is well known that the permutation action of $G$ on the set of singular $1$-spaces of $V$ is primitive of rank $3$, and thus its character is $\rho = 1_G + \a + \b$, with $\a, \b \in \Irr(G)$. Let (the parabolic subgroup) $P=QL$ denote a point stabilizer in this action, with $Q$ the unipotent radical and $L$ a Levi subgroup. Aside from $\a,\b$, we also need to consider the remaining non-principal irreducible constituents $\g_i$ of $\Ind^G_{[P,P]}(1_{[P,P]})$. Let $\St$ denote the Steinberg character of $G$. \[mc-r1\] The following statements hold. 1. Suppose that every semisimple element $s \in G$ is real. Then for any $\chi \in \Irr(G)$ and $k \in \N$, $\chi^{2k}$ contains $\St$ if and only if $(\c\overline\c)^k$ contains $\St$. 2. All semisimple elements in $G$ are real, if $G = \Sp_{2n}(q)$, $\O_{2n+1}(q)$, or $\O^\e_{4n}(q)$. \(i) Recall that $\St(g) = 0$ if $g \in G$ is not semisimple. Furthermore, $\c(g) = \overline\c(g)$ if $g \in G$ is semisimple, by hypothesis. Hence $$\begin{aligned} ~[\chi^{2k},\St]_G & = \frac{1}{|G|}\sum_{g \in G}\chi(g)^{2k}\overline\St(g)\\ & = \frac{1}{|G|}\sum_{g \in G,~g\mbox{ {\tiny semisimple}}}\chi(g)^{2k}\overline\St(g)\\ & = \frac{1}{|G|}\sum_{g \in G,~g\mbox{ {\tiny semisimple}}}\chi(g)^{k}\overline\c(g)^k\overline\St(g)\\ & = \frac{1}{|G|}\sum_{g \in G}\c(g)^k\overline\c(g)^{k}\overline\St(g) = [(\c\overline\c)^k,\St]_G, \end{aligned}$$ and the claim follows. \(ii) This is well known, see e.g. [@TZ2 Proposition 3.1]. \[mc-r2\] Let $G = \Sp(V) = \Sp_{2n}(q)$ with $n \geq 3$. Suppose $C \in \N$ is such that both $\a^C$ and $\b^C$ contain $\St$. Then for any $1_G \neq \chi \in \Irr(G)$, $\c^{2C}$ contains $\St$. In the aforementioned rank $3$ permutation action of $G$ with character $\rho = 1_G+\a+\b$, a point stabilizer $P$ is the normalizer $\NB_G(Z)$ of some long-root subgroup $Z$. Since $n \geq 3$, $Z$ has a nonzero fixed point on any $\C G$-module affording $\c$ by [@T Theorem 1.6]. It follows that $\c|_P$ is reducible, and so $$\label{eq:mc1} 2 \leq [\c|_P,\c|_P]_P = [\c\overline\c,\Ind^G_P(1_P)]_G = [\c\overline\c,\rho]_G.$$ As $[\c\overline\c,1_G]_G = 1$, $\c\overline\c$ contains either $\a$ or $\b$, whence $(\c\overline\c)^C$ contains $\St$. Applying Lemma \[mc-r1\], we conclude that $\c^{2C}$ contains $\St$. \[mc-r3\] Let $G = \O(V) = \O^\e_{n}(q)$ with $n \geq 5$. Suppose $C \in \N$ is such that both $\a^C$ and $\b^C$ contain $\St$. Consider any $1_G \neq \chi \in \Irr(G)$, and suppose in addition that either $n \not\equiv 2 (\bmod\ 4)$, or $\c = \overline\c$. Then $\c^{4C}$ contains $\St$. Again we consider a point stabilizer $P=QL$ in the aforementioned rank $3$ permutation action of $G$ with character $\rho = 1_G+\a+\b$. Note that $Q$ is elementary abelian, $[L,L] \cong \O^\e_{n-2}(q)$, and we can identify $\Irr(Q)$ with the natural module $\F_q^{n-2}$ for $[L,L]$. In particular, any $[L,L]$-orbit on $\Irr(Q) \smallsetminus \{1_Q\}$ has length at least $2$. It is also clear that some irreducible constituent of $\c|_Q$ is non-principal, since $\Ker(\c) \leq \ZB(G)$ and $Q \not\leq \ZB(G)$. It follows that $\c|_Q$ is reducible, and so $$2 \leq [\c|_Q,\c|_Q]_Q = [(\c\overline\c)|_Q,1_Q]_Q.$$ Since $[\c\overline\c,1_G]_G = 1$, at least one non-principal irreducible constituent $\theta$ of $\c\overline\c$ contains $1_Q$ on restriction to $Q$. But $P$ normalizes $Q$, so the latter implies that $\theta|_P$ is reducible. Thus holds for $\theta$ instead of $\c$. Arguing as in the proof of Lemma \[mc-r1\], we obtain that $\theta\overline\theta$ contains either $\a$ or $\b$, whence $(\c\overline\c)^2$ contains either $\a$ or $\b$. It follows that $(\c\overline\c)^{2C}$ contains $\St$, and we are done if $\c = \overline\c$. Applying Lemma \[mc-r1\], we also have that $\c^{4C}$ contains $\St$ in the case $n \not\equiv 2 (\bmod\ 4)$. \[mc-r4\] Let $G = \O(V) = \O^\e_{n}(q)$ with $n \geq 10$ and $n \equiv 2 (\bmod\ 4)$. Suppose $C \in \N$ is such that each of $\a^C$, $\b^C$, and $\g_i^C$ contains $\St$. Then for any $\chi \in \Irr(G)$ with $\c \neq \overline\c$, $\c^{4C}$ contains $\St$. \(i) As noted in the proof of Lemma \[mc-r3\], $Q$ is elementary abelian, $[L,L] \cong \O^\e_{n-2}(q)$, and we can identify $\Irr(Q)$ with the natural module $\F_q^{n-2}$ for $[L,L]$. Since $n-2 \geq 8$, it is straightforward to check that any $[L,L]$-orbit on nonzero vectors of $\F_q^{n-2}$ contains a vector $v$ and also $-v$. Thus, any $[L,L]$-orbit on $\Irr(Q) \smallsetminus \{1_Q\}$ contains a characters $\l$ and also its complex conjugate $\overline\l$. As noted in the proof of Lemma \[mc-r3\], $Q \not\leq \Ker(\c)$. Thus we may assume that $\c|_Q$ contains $\l$ and also $\overline\l$. It follows that $1 \leq [\c^2|_Q,1_Q]_Q$. Since $[\c^2,1_G]_G = [\c,\overline\c]_G = 0$, at least one non-principal irreducible constituent $\theta$ of $\c^2$ contains $1_Q$ on restriction to $Q$. In particular, $\theta|_P$ is reducible, since $P$ normalizes $Q$, and holds for $\theta$ instead of $\c$, and so the arguments in the proof of Lemma \[mc-r2\] shows that $\theta\overline\theta$ contains $\a$ or $\b$. If, moreover, $\theta = \overline\theta$, then we conclude that $\theta^2$ contains $\a$ or $\b$. \(ii) Now consider the case $\theta \neq \overline\theta$, and let $\theta$ be afforded by a $\C G$-module $U$. As shown in (i), the $Q$-fixed point subspace $U^Q$ on $U$ is nonzero, and $L$ acts on $U^Q$. Recall that $4|(n-2)$ and $n-2 \geq 8$. Now, if $(\e,q) \neq (+, \equiv 3(\bmod\ 4))$, then all irreducible characters of $[L,L] \cong \O^e_{n-2}(q)$ are real-valued, and so the $[L,L]$-module $U^Q$ contains an irreducible submodule $W \cong W^*$. Consider the case $(\e,q) = (+,\equiv 3(\bmod\ 4))$ and let $P = \Stab_G(\langle u \rangle_{\F_q})$ for a singular vector $0 \neq u \in V$. We can consider $P$ inside $\tilde P:=\Stab_{\SO(V)}(\langle u \rangle_{\F_q})=Q\tilde L$, and find another singular vector $u' \in V$ such that $V = V_1 \oplus V_2$, with $V_1 = \langle u,u' \rangle_{\F_q}$, $V_2 = V_1^{\perp}$, and $[L,L] = \O(V_2)$. Since $q \equiv 3 (\bmod\ 4)$, $t:=-1_{V_1} \in \SO(V_1) \smallsetminus \O(V_1)$. Choosing some $t' \in \SO(V_2) \smallsetminus \O(V_2)$, we see that $tt' \in \tilde L \cap \O(V) = L$, and $L_1 := \langle [L,L],tt' \rangle \cong \SO^+_{n-2}(q)$. By [@Gow], all irreducible characters of $L_1$ are real-valued, and so the $L_1$-module $U^Q$ contains an irreducible submodule $W \cong W^*$. We have shown that the $[L,L]$-module $U^Q$ contains a nonzero submodule $W \cong W^*$. We can also inflate $W$ to a nonzero self-dual module over $[P,P] = Q[L,L]$. It follows that $(U \otimes_{\C} U)|_{[P,P]}$ contains $W \otimes_{\C} W^*$, which certainly contains the trivial submodule. Thus, $\theta^2|_{[P,P]}$ contains the principal character $1_{[P,P]}$, and so $$\label{eq:mc2} 1 \leq [\theta^2,\Ind^G_{[P,P]}(1_{[P,P]})]_G.$$ Recall we are assuming that $0 = [\theta,\overline\theta]_G = [\theta^2,1_G]_G$. Hence implies that $\theta^2$ contains at least one of $\a$, $\b$, or $\g_i$. \(iii) We have shown that, in all cases, $\theta^2$ contains at least one of $\a$, $\b$, or $\g_i$. As $\chi^2$ contains $\theta$, we see that $\c^4$ contains at least one of $\a$, $\b$, or $\g_i$, and so $\c^{4C}$ contains $\St^2$. Classical groups in characteristic $2$ -------------------------------------- In this subsection we study certain characters of $\tilde G = \Sp(V) = \Sp_{2n}(q)$ and $G = \O(V)=\O^\e_{2n}(q)$, where $n \geq 5$ and $2|q$. These results will be used subsequently and are also of independent interest. First we endow $V$ with a non-degenerate alternating form $(\cdot,\cdot)$, and work with its isometry group $\tilde G = \Sp(V)$. We will consider the following irreducible characters of $\tilde G$: $\bullet$ the $q/2+1$ [*linear-Weil*]{} characters: $\rho^1_n$ of degree $(q^n+1)(q^n-q)/2(q-1)$, $\rho^2_n$ of degree $(q^n-1)(q^n+q)/2(q-1)$, and $\tau^i_n$ of degree $(q^{2n}-1)/(q-1)$, $1 \leq i \leq (q-2)/2$, and $\bullet$ the $q/2+2$ [*unitary-Weil*]{} characters: $\a_n$ of degree $(q^n-1)(q^n-q)/2(q+1)$, $\b_n$ of degree $(q^n+1)(q^n+q)/2(q+1)$, and $\zeta^i_n$ of degree $(q^{2n}-1)/(q+1)$, $1 \leq i \leq q/2$;\ see [@GT Table 1]. Then $$\label{eq:dec11} \rho:=1_{\tilde G}+\rho^1_n+\rho^2_n$$ is the rank $3$ permutation character of $\tilde G$ acting on the set of $1$-spaces of $V$. The following statement is well known, see e.g. formula (1) of [@FST]: \[quad1\] For $\e = \pm$, the character $\pi^\e$ of the permutation action of $\tilde G$ on quadratic forms of type $\e$ associated to $(\cdot,\cdot)$ is given as follows: $$\pi^+ = 1_{\tilde G} + \rho^2_n + \sum^{(q-2)/2}_{i=1}\tau^i_n,~~~ \pi^- = 1_{\tilde G} + \rho^1_n + \sum^{(q-2)/2}_{i=1}\tau^i_n.$$ Given any $g \in \GL(V)$, let $$d(x,g):= \dim_{\overline{\F}_q}\Ker(g-x \cdot 1_{V \otimes_{\F_q}}\overline{\F}_q)$$ for any $x \in \overline{\F}_q^\times$, and define the [*support*]{} of $g$ to be $$\supp(g) := \dim(V)-\max_{x \in \overline{\F}_q^\times}d(x,g).$$ Set $$d(g):= \dim(V)-\supp(g).$$ \[rat-sp2\] Let $\tilde G = \Sp_{2n}(q)$ with $n \geq 3$ and $2|q$, and let $g \in \tilde G$ have support $s=\supp(g)$. If $\chi \in \{\rho^1_n,\rho^2_n\}$, then $$\frac{|\c(g)|}{\c(1)} \leq \frac{1}{q^{s/3}}.$$ The statement is obvious if $s=0$. Suppose $s=1$. It is easy to see that in this case $g$ is a transvection, and so $$\rho^1_n(g) = \rho^2_n(g) = \frac{q^{2n-1}-q}{2(q-1)}$$ by [@GT Corollary 7.8], and the statement follows. From now on we may assume $s \geq 2$. Observe that $d:=\max_{x \in \F_q^\times}d(x,g) \leq d(g) = 2n-s$. Hence, $$0 \leq \rho(g) = \sum_{x \in \F_q^\times}\frac{q^{d(x)}-1}{q-1} \leq q^d-1,$$ and so implies $$|\rho^1_n(g)+\rho^2_n(g)| \leq q^d-1.$$ On the other hand, since $\pi^\pm(g) \geq 0$ and $\pi^++\pi^-$ is just the permutation character of $\tilde G$ acting on $V$, Lemma \[quad1\] implies that $$|\rho^1_n(g)-\rho^2_n(g)| = |\pi^+(g)-\pi^-(g)| \leq \pi^+(g)+\pi^-(g) = q^{d(1,g)} \leq q^d.$$ It follows for any $i \in \{1,2\}$ that $$|\rho^i_n(g)| \leq \bigl(|\rho^1_n(g)+\rho^2_n(g)|+|\rho^1_n(g)+\rho^2_n(g)|\bigr)/2 < q^d \leq q^{2n-s}.$$ Since $n \geq 3$, we can also check that $$\rho^i_n(1) \geq \frac{(q^n+1)(q^n-q)}{2(q-1)} > q^{2n-4/3}.$$ Thus $|\c(g)|/\c(1)| < q^{4/3-s} \leq q^{-s/3}$, as stated. Next we endow $V = \F_q^{2n}$ with a non-degenerate quadratic form $\QF$ of type $\e = \pm$ associated to the alternating form $(\cdot,\cdot)$. Choose a Witt basis $(e_1,\ldots,e_n,f_1, \ldots, f_n)$ for $(\cdot,\cdot)$, such that $\QF(e_1)=\QF(f_1)=0$. We may assume that $P = \Stab_G(\langle e_1 \rangle_{\F_q}) = QL$, where $Q$ is elementary abelian of order $q^{2n-2}$, $L \cong \O^\e_{2n-2}(q) \times C_{q-1}$, and $$[P,P] = \Stab_G(e_1)=Q \rtimes [L,L]$$ has index $(q^n-\e)(q^{n-1}+\e)$ in $G$. Also consider $H := \Stab_G(e_1+f_1)$. According to [@N Theorem 1.3], $G$ has $q+1$ non-principal complex irreducible characters of degree at most $(q^n-\e)(q^{n-1}+\e)$, namely, $\a$ of degree $(q^n-\e)(q^{n-1}+\e q)/(q^2-1)$, $\b$ of degree $(q^{2n}-q^2)/(q^2-1)$, $\g_i$ of degree $(q^n-\e)(q^{n-1}+\e)/(q-1)$, $1 \leq i \leq (q-2)/2$, and $\d_j$ of degree $(q^n-\e)(q^{n-1}-\e)/(q+1)$, $1 \leq j \leq q/2$. \[dec-so2\] Let $G = \O^\e_{2n}(q)$ with $n \geq 5$ and $2|q$, and consider $P = \Stab_G(e_1)$ and $H = \Stab_G(e_1+f_1)$ as above. Then the following statements hold. 1. $\Ind^G_P(1_P) = 1_G + \a + \b$. 2. $\Ind^G_{[P,P]}(1_{[P,P]}) = 1_G +\a+\b + 2\sum^{(q-2)/2}_{i=1}\g_i$. 3. $\Ind^G_H(1_H) = 1_G +\b + \sum^{(q-2)/2}_{i=1}\g_i+\sum^{q/2}_{j=1}\d_j$. \(i) is well known. Next, $P/[P,P] \cong C_{q-1}$ has $q-1$ irreducible characters: $1_P$ and $(q-2)/2$ pairs of $\{\nu_i,\overline\nu_i\}$, $1 \leq i \leq (q-2)/2$. An application of Mackey’s formula shows that $\Ind^G_P(\nu_i) = \Ind^G_P(\overline\nu_i)$ is irreducible for all $i$. Now using (i) we can write $$\label{eq:dec1} \Ind^G_{[P,P]}(1_{[P,P]}) = \Ind^G_P\bigl( \Ind^P_{[P,P]}(1_{[P,P]}) \bigr) = 1_G+\a+\b + 2\sum^{(q-2)/2}_{i=1}\Ind^G_P(\nu_i).$$ On the other hand, note that $[P,P]$ has exactly $2q-1$ orbits on the set of nonzero singular vectors in $V$: $q-1$ orbits $\{xe_1\}$ with $x \in \F_q^\times$, one orbit $\{v \in e_1^\perp \smallsetminus \langle e_1 \rangle_{\F_q} \mid \QF(v)=0\}$, and $(q-1)$ orbits $\{yf_1 + v \mid v \in e_1^\perp, \QF(yf_1+v) =0\}$ with $y \in \F_q^\times$. Together with , this implies that all summands in the last decomposition in are pairwise distinct. Since $\g_i = (q^n-\e)(q^{n-1}+\e)/(q-1) = \Ind^G_P(\nu_{i'})$, renumbering the $\nu_i$ if necessary, we may assume that $\Ind^G_P(\nu_i)=\g_i$, and (ii) follows. For (iii), first note that $P$ has two orbits on the set $\XC := \{ v \in V \mid \QF(v)=1\}$, namely, $\XC \cap e_1^\perp$ and $\XC \smallsetminus e_1^\perp$. Since $\Ind^G_H(1_H)$ is the character of the permutation action of $G$ on $\XC$, we get $$\label{eq:dec2} [\Ind^G_P(1_P),\Ind^G_H(1_H)]_G = 2.$$ Next, $[P,P]$ has $q$ orbits on $\XC$, namely, $\XC \cap e_1^\perp$, and $\{yf_1+w \in \XC \mid w \in e_1^\perp\}$ with $y \in \F_q^\times$. Thus $$\label{eq:dec3} [\Ind^G_{[P,P]}(1_{[P,P]}),\Ind^G_H(1_H)]_G = q.$$ Combining the results of (i), (ii), with , , and again using [@N Theorem 1.3], we can write $$\label{eq:dec4} \Ind^G_H(1_H) = 1_G + (a\a + b\b) +\sum^{(q-2)/2}_{i=1}c_i\g_i + \sum^{q/2}_{j=1}d_j\d_j,$$ where $a,b,c_i,d_j \in \Z_{\geq 0}$, $a+b=1$, $\sum_ic_i = (q-2)/2$. Let $\tau$ denote the character of the permutation action of $G$ on $V \smallsetminus \{0\}$, so that $$\tau = \Ind^G_{[P,P]}(1_{[P,P]}) + (q-1)\Ind^G_H(1_H).$$ Note that $G$ has $q^3+q^2-q$ orbits on $(V \smallsetminus \{0\}) \times (V \smallsetminus \{0\})$, namely, $q(q-1)$ orbits of $(u,xu)$, where $x \in \F_q^\times$ and $\QF(u) = y \in \F_q$, and $q^3$ orbits of $(u,v)$, where $u,v$ are linearly independent and $(\QF(u),(u,v),\QF(v)) = (x,y,z) \in \F_q^3$. In other words, $[\tau,\tau]_G = q^3+q^2-q$. Using (ii) and , we deduce that $$\label{eq:dec5} [\Ind^G_H(1_H),\Ind^G_H(1_H)]_G = q+1.$$ In particular, if $q=2$ then $\Ind^G_H(1_H)$ is the sum of $3$ pairwise distinct irreducible characters. By checking the degrees of $\a,\b$ and $\d_1$, (iii) immediately follows from . Now we may assume $q=2^e \geq 4$. Let $\ell_+ = \ell(2^{ne}-1)$ denote a primitive prime divisor of $2^{ne}-1$, which exists by [@Zs]. Likewise, let $\ell_- = \ell(2^{2ne}-1)$ denote a primitive prime divisor of $2^{2ne}-1$. Then note that $\ell_\e$ divides the degree of each of $\a$, $\g_i$, $d_j$, but neither $[G:H]-1$ nor $\b(1)$. Hence implies that $(a,b)=(0,1)$. Comparing the degrees in , we also see that $\sum_jd_j = q/2$. Now $$q+1 = [\Ind^G_H(1_H),\Ind^G_H(1_H)]_G = 2 + \sum^{(q-2)/2}_{i=1}c_i^2 + \sum^{q/2}_{j=1}d_j^2 \geq 2 + \sum^{(q-2)/2}_{i=1}c_i + \sum^{q/2}_{j=1}d_j = 2 +\frac{q-2}{2}+\frac{q}{2},$$ yielding $c_i^2=c_i$, $d_j^2=d_j$, $c_i,d_j \in \{0,1\}$, and so $c_i = d_j = 1$, as desired. In the next statement, we embed $G = \O(V)$ in $\tilde G := \Sp(V)$ (the isometry group of the form $(\cdot,\cdot)$ on $V$). \[sp-so1\] Let $n \geq 5$, $2|q$, and $\e = \pm$. Then the characters $\rho^1_n$ and $\rho^2_n$ of $\Sp(V) \cong \Sp_{2n}(q)$ restrict to $G = \O(V) \cong \O^\e_{2n}(q)$ as follows: $$\begin{array}{ll}(\rho^1_n)|_{\O^+_{2n}(q)} = \b + \sum^{q/2}_{j=1}\d_j, & (\rho^2_n)|_{\O^+_{2n}(q)} = 1+\a+\b + \sum^{(q-2)/2}_{i=1}\g_i,\\ (\rho^1_n)|_{\O^-_{2n}(q)} = 1+\a+\b + \sum^{(q-2)/2}_{i=1}\g_i, & (\rho^2_n)|_{\O^-_{2n}(q)} = \b + \sum^{q/2}_{j=1}\d_j. \end{array}$$ Note by that $1_G + (\rho^1_n+\rho^2_n)|_G$ is just the character of the permutation action on the set of $1$-spaces of $V$. Hence, by Proposition \[dec-so2\] we have $$\label{eq:dec21} \bigl( \rho^1_n+\rho^2_n \bigr)|_G = \Ind^G_P(1_P) + \Ind^G_H(1_H) -1_G = 1_G +\a+2\b + \sum^{(q-2)/2}_{i=1}\g_i + \sum^{q/2}_{j=1}\d_j.$$ Furthermore, Lemma \[quad1\] implies by Frobenius’ reciprocity that $$\label{eq:dec22} \bigl(\rho^2_n\bigr)|_G \mbox { contains }1_G \mbox { when }\e=+, \mbox{ and }\bigl(\rho^1_n\bigr)|_G \mbox { contains }1_G \mbox { when }\e=-.$$ \(i) First we consider the case $\e = +$. If $(n,q) \neq (6,2)$, one can find a primitive prime divisor $\ell = \ell(2^{ne}-1)$, where $q = 2^e$. If $(n,q) = (6,2)$, then set $\ell = 7$. By its choice, $\ell$ divides the degrees of $\rho^2_n$, $\a$, $\g_i$, and $\d_j$, but $\b(1) \equiv \rho^1_n(1) \equiv -1 (\bmod\ \ell)$. Hence, and imply that $$\bigl(\rho^2_n\bigr)|_G = 1_G +\b +x\a + \sum^{(q-2)/2}_{i=1}y_i\g_i + \sum^{q/2}_{j=1}z_j\d_j,$$ where $x,y_i,z_j \in \{0,1\}$. Setting $y:=\sum^{(q-2)/2}_{i=1}y_i$ and $z:=\sum^{q/2}_{j=1}z_j$ and comparing the degrees, we get $$(1-x)(q^{n-1}+q)+(q^{n-1}+1)(q+1)((q-2)/2-y) = z(q^{n-1}-1)(q-1),$$ and so $q^{n-1}+1$ divides $(1-x+2z)(q-1)$. Note that $\gcd(q-1,q^{n-1}+1)=1$ and $0 \leq (1-x+2z)(q-1) \leq q^2-1 < q^{n-1}+1$. It follows that $x=1$, $z=0$, $y=(q-2)/2$, whence $y_i=1$ and $z_j=1$, as stated. \(ii) Now let $\e = -$, and choose $\ell$ to be a primitive prime divisor $\ell(2^{2ne}-1)$. By its choice, $\ell$ divides the degrees of $\rho^1_n$, $\a$, $\g_i$, and $\d_j$, but $\b(1) \equiv \rho^2_n(1) \equiv -1 (\bmod\ \ell)$. Hence, and imply that $$\bigl(\rho^1_n\bigr)|_G = 1_G +\b +x\a + \sum^{(q-2)/2}_{i=1}y_i\g_i + \sum^{q/2}_{j=1}z_j\d_j,$$ where $x,y_i,z_j \in \{0,1\}$. Setting $y:=\sum^{(q-2)/2}_{i=1}y_i$ and $z:=\sum^{q/2}_{j=1}z_j$ and comparing the degrees, we get $$(1-x)(q^{n-1}-q)+(q^{n-1}-1)(q+1)((q-2)/2-y) = z(q^{n-1}+1)(q-1),$$ and so $(q^{n-1}-1)/(q-1)$ divides $1-x+2z$. Since $0 \leq 1-x+2z \leq q+1 < (q^{n-1}-1)/(q-1)$, it follows that $x=1$, $z=0$, $y=(q-2)/2$, whence $y_i=1$ and $z_j=1$, as stated. For the subsequent discussion, we recall the [*quasi-determinant*]{} $\kappa_\e: \GO_\e \to \{-1,1\}$, where $\GO_\e:= \mathrm{GO}(V) \cong \mathrm{GO}^\e_{2n}(q)$, defined via $$\kappa_\e(g) := (-1)^{\dim_{\F_q}\Ker(g-1_V)}.$$ It is known, see e.g. [@GT Lemma 5.8(i)], that $\kappa$ is a group homomorphism, with $$\label{eq:kappa1} \Ker(\kappa_\e) = \O_\e:= \O(V) \cong \O^\e_{2n}(q).$$ Now we prove the “unitary” analogue of Lemma \[quad1\]: \[quad2\] For $n \geq 3$ and $2|q$, the following decompositions hold: $$\Ind^{\tilde G}_{\GO_+}(\kappa_+) = \b_n + \sum^{q/2}_{i=1}\zeta^i_n,~~~ \Ind^{\tilde G}_{\GO_-}(\kappa_-) = \a_n + \sum^{q/2}_{i=1}\zeta^i_n.$$ According to formulae (10) and (4)–(6) of [@GT], $$\label{eq:dec31} \Ind^{\tilde G}_{\O_+}(\kappa_+)+ \Ind^{\tilde G}_{\O_-}(\kappa_-) = \a_n+\b_n + 2\sum^{q/2}_{i=1}\zeta^i_n.$$ Hence we can write $$\label{eq:dec32} \Ind^{\tilde G}_{\O_+}(\kappa_+) = x\a_n+y\b_n + \sum^{q/2}_{i=1}z_i\zeta^i_n,$$ where $x,y,z_i \in \Z_{\geq 0}$, $x,y \leq 1$ and $z_i \leq 2$. Note that, since $\pi^+= \Ind^{\tilde G}_{\GO_+}(1_{\GO_+})$, Lemma \[quad1\] implies that $$|\GO_+ \backslash \tilde G/\GO_+| = \frac{q}{2}+1.$$ Next, by Mackey’s formula we have $$[\Ind^{\tilde G}_{\O_+}(\kappa_+),\Ind^{\tilde G}_{\O_+}(\kappa_+)]_G = \sum_{\GO_+t\GO_+ \in \GO_+ \backslash \tilde G/\GO_+} [(\kappa_+)|_{\GO_+ \cap t\GO_+t^{-1}},(\kappa^t_+)|_{\GO_+ \cap t\GO_+t^{-1}}]_{\GO_+ \cap t\GO_+t^{-1}},$$ where $\kappa^t_+(x) = \kappa(x^t) := \kappa(t^{-1}xt)$ for any $x \in \GO_+ \cap t\GO_+t^{-1}$. For such an $x$, note that $$\label{eq:dec321} \kappa_+(x) = 1 \Leftrightarrow 2 | \dim_{\F_q}\Ker(x-1_V) \Leftrightarrow 2 | \dim_{\F_q}\Ker(x^{t-1}-1_V) \Leftrightarrow (\kappa_+)^t(x) = 1,$$ i.e. $\kappa_+(x) = \kappa^t_+(x)$. It follows that $$\label{eq:dec33} x^2+y^2+\sum^{q/2}_{i=1}z_i^2= [\Ind^{\tilde G}_{\O_+}(\kappa_+),\Ind^{\tilde G}_{\O_+}(\kappa_+)]_G = |\GO_+ \backslash \tilde G/\GO_+| = \frac{q}{2}+1.$$ On the other hand, equating the character degrees in we obtain $$\label{eq:dec34} \frac{q^n(q^n+1)}{2} = x\frac{(q^n-1)(q^n-q)}{2(q+1)}+y\frac{(q^n+1)(q^n+q)}{2(q+1)}+\sum^{q/2}_{i=1}z_i \cdot \frac{q^{2n}-1}{q+1}.$$ We claim that $x=0$. Indeed, if $(n,q) = (3,2)$, then implies that $3|x$, and so $x=0$ as $0 \leq x \leq 1$. Assume $(n,q) \neq (3,2)$. Then we can find a primitive prime divisor $\ell = \ell(2^{2ne}-1)$ for $q = 2^e$, and note from that $\ell|x$. Since $\ell > 2$ and $x \in \{0,1\}$, we again have $x=0$. Now if $y=0$, then implies that $q^n(q^n+1)/2$ is divisible by $(q^{2n}-1)/(q+1)$, a contradiction. Hence $y=1$, and from we obtain that $\sum^{q/2}_{i=1}z_i = q/2$. On the other hand, $\sum^{q/2}_{i=1}z_i^2 = q/2$ by . Thus $\sum^{q/2}_{i=1}(z_i-1)^2 = 0$, and so $z_i = 1$ for all $i$. Together with , this yields the two stated decompositions. \[sp-so2\] Let $n \geq 5$, $2|q$, and $\e = \pm$. Then the characters $\a_n$ and $\b_n$ of $\Sp(V) \cong \Sp_{2n}(q)$ restrict to $G = \O(V) \cong \O^\e_{2n}(q)$ as follows: $$\begin{array}{ll}(\a_n)|_{\O^+_{2n}(q)} = \sum^{q/2}_{j=1}\d_j, & (\b_n)|_{\O^+_{2n}(q)} = 1+\a + \sum^{(q-2)/2}_{i=1}\g_i,\\ (\a_n)|_{\O^-_{2n}(q)} = 1+\a + \sum^{(q-2)/2}_{i=1}\g_i, & (\b_n)|_{\O^-_{2n}(q)} = \sum^{q/2}_{j=1}\d_j. \end{array}$$ In particular, the following formula holds for the irreducible character $\b$ of $G$ of degree $(q^{2n}-q^2)/(q^2-1)$: $$\bigl( (\rho^1_n+\rho^2_n)-(\a_n+\b_n)\bigr)|_{\O^\e_{2n}(q)} = 2\b.$$ By Mackey’s formula, $$\bigl(\Ind^{\tilde G}_{\O_+}(\kappa_+)\bigr)|_G = \sum_{Gt\GO_+\in G \backslash \tilde G/\GO_+} \Ind^G_{G \cap t\GO_+t^{-1}}\bigl((\kappa^t_+)|_{G \cap t\GO_+t^{-1}}\bigr),$$ and similarly for $\pi^+=\Ind^{\tilde G}_{\O_+}(1_{\GO_+})$. The argument in shows that $\kappa^t_+(x)=1$ for all $x \in G \cap t\GO_+t^{-1}$, and so $\pi^+$ and $\Ind^{\tilde G}_{\O_+}(\kappa_+)$ agree on $G$. Similarly, $\pi^-$ and $\Ind^{\tilde G}_{\O_-}(\kappa_-)$ agree on $G$. It then follows from Lemmas \[quad1\] and \[quad2\] that $$\label{eq:dec41} \bigl( \rho^2_n-\rho^1_n\bigr)|_G = \bigl( \pi^+-\pi^-\bigr)|_G = \bigl( \Ind^{\tilde G}_{\O_+}(\kappa_+)- \Ind^{\tilde G}_{\O_-}(\kappa_-) \bigr)|_G = \bigl( \b_n-\a_n\bigr)|_G.$$ First assume that $\e=+$. Then using Proposition \[sp-so1\] we get $$\bigl( \b_n-\a_n\bigr)|_G= 1_G+\a+\sum^{(q-2)/2}_{i=1}\g_i-\sum^{q/2}_{j=1}\d_j,$$ i.e. $$\sum^{q/2}_{j=1}\d_j+(\b_n)|_G = 1_G +\a+\sum^{(q-2)/2}_{i=1}\g_i + (\a_n)|_G.$$ Aside from $(\a_n)|_G$ and $(\b_n)|_G$, all the other characters in the above equality are irreducible and pairwise distinct. It follows that $(\a_n)|_G$ contains $\sum^{q/2}_{j=1}\d_j$. Comparing the degrees, we see that $$(\a_n)|_G = \sum^{q/2}_{j=1}\d_j,$$ which then implies that $$(\b_n)|_G = 1_G +\a+\sum^{(q-2)/2}_{i=1}\g_i.$$ Now assume that $\e=-$. Then again using Proposition \[sp-so1\] and we get $$\bigl( \a_n-\b_n\bigr)|_G= 1_G+\a+\sum^{(q-2)/2}_{i=1}\g_i-\sum^{q/2}_{j=1}\d_j,$$ i.e. $$\sum^{q/2}_{j=1}\d_j+(\a_n)|_G = 1_G +\a+\sum^{(q-2)/2}_{i=1}\g_i + (\b_n)|_G.$$ Aside from $(\a_n)|_G$ and $(\b_n)|_G$, all the other characters in the above equality are irreducible and pairwise distinct. It follows that $(\b_n)|_G$ contains $\sum^{q/2}_{j=1}\d_j$. Comparing the degrees, we see that $$(\b_n)|_G = \sum^{q/2}_{j=1}\d_j,$$ which then implies that $$(\a_n)|_G = 1_G +\a+\sum^{(q-2)/2}_{i=1}\g_i.$$ For both $\e = \pm$, the last statement now follows from . Proposition \[sp-so2\] leads to the following explicit formula for $\b$, which we will show to hold for all special orthogonal groups in all characteristics and all dimensions, and which is of independent interest. In this result, we let $V = \F_q^n$ be a quadratic space, $L := \SO(V)$ if $2 \nmid q$, $L := \O(V)$ if $2|q$, and extend the action of $L$ on $V$ to $\tilde V := V \otimes_{\F_q}\F_{q^2}$, and we assume $2 \nmid q$ if $2 \nmid n$. Also, set $$\mu_{q-1}:= \F_q^\times,~~\mu_{q+1} := \{ x \in \F_{q^2}^\times \mid x^{q+1} = 1 \}.$$ If $2 \nmid q$, let $\chi_2^+$ be the unique linear character of order $2$ of $\mu_{q-1}$, and let $\chi_2^-$ be the unique linear character of order $2$ of $\mu_{q+1}$. \[beta-so2\] Let $n \geq 10$, $\e = \pm$, and let $q$ be any prime power. If $2|n$, let $\psi = \b$ be the irreducible constituent $\b$ of degree $(q^{n}-q^2)/(q^2-1)$ of the rank $3$ permutation character of $L = \O(V)$ when $2|q$, and of $L = \SO(V)$ when $2 \nmid q$, on the set of singular $1$-spaces of its natural module $V=\F_q^{n}$. If $2 \nmid qn$, let $\psi$ be the irreducible character of $L = \SO(V)$ of degree $(q^n-q)/(q^2-1)$ denoted by $D_{\St}$ in [@LBST Proposition 5.7]. Then for any $g \in L$ we have $$\psi(g) = \frac{1}{2(q-1)}\sum_{\l \in \mu_{q-1}}q^{\dim_{\F_q}\Ker(g - \l \cdot 1_V)} - \frac{1}{2(q+1)}\sum_{\l \in \mu_{q+1}}(-q)^{\dim_{\F_{q^2}}\Ker(g - \l \cdot 1_{\tilde V})} -1$$ when $2|n$, and by $$\psi(g) = \frac{1}{2(q-1)}\sum_{\l \in \mu_{q-1}}\chi_2^+(\l)q^{\dim_{\F_q}\Ker(g - \l \cdot 1_V)} + \frac{1}{2(q+1)}\sum_{\l \in \mu_{q+1}}\chi^-_2(\l)(-q)^{\dim_{\F_{q^2}}\Ker(g - \l \cdot 1_{\tilde V})}$$ when $2 \nmid qn$. In the case $2|q$, the statement follows from the last formula in Proposition \[sp-so2\], together with formulae (3) and (6) of [@GT]. Assume now that $2 \nmid q$, and set $\kappa := 1$ if $2|n$ and $\kappa := 0$ if $2 \nmid n$. By [@LBST Proposition 5.7] (and in the notation of [@LBST §5.1]), $$\psi(g)=\frac{1}{|\Sp_2(q)|}\sum_{x \in \Sp_2(q)}\omega_{n}(xg)\St(x)-\kappa,$$ where $\omega_{2n}$ denotes a reducible Weil character of $\Sp_{2n}(q)$ and $\St$ denotes the Steinberg character of $S:= \Sp_2(q)$. If $x \in S$ is not semisimple, then $\St(x) = 0$. Suppose $x = \mathrm{diag}(\l,\l^{-1}) \in T_1 <S$, where $T_1 \cong C_{q-1}$ is a split torus and $\l \in \mu_{q-1}$. In this case, we can view $T_1$ as $\GL_1(q)$, embed $G$ in $\GL_{n}(q)$, and view $xg$ as an element $h=\l g$ in a Levi subgroup $\GL_{n}(q)$ of $\Sp_{2n}(q)$, with $\det(h) = \l^{n}$. It follows from [@Ge Theorem 2.4(c)] that $$\omega_{n}(xg) = \chi_2^+(\l^n) q^{\dim_{\F_q}\Ker(h-1)} = \chi_2^+(\l^n) q^{\dim_{\F_q}\Ker(g-\l^{-1})}.$$ If $\l \neq \pm 1$, then $|x^S| = q(q+1)$ and $\St(x) = 1$. If $\l = \pm 1$, then $|x^S| = 1$ and $\St(x)=q$. Note that since $g \in \mathrm{GO}(V)$, $$\dim_{\F_q}\Ker(g-\l^{-1}) = \dim_{\F_q}\Ker({{}^t\!} g-\l^{-1}) = \dim_{\F_q}\Ker(g^{-1}-\l^{-1}) = \dim_{\F_q}\Ker(g-\l).$$ We also note that since $g \in \SO(V)$, $$\label{eq:kappa2} \dim_{\F_q}\Ker(g-1_V) \equiv n ({\bmod \,}2),~~ \dim_{\F_q}\Ker(g+1_V) \equiv 0 ({\bmod \,}2).$$ (Indeed, since $\det(g)=1$, each of $\Ker(g_s-1_V)$ and $\Ker(g_s+1_V)$ is a non-degenerate subspace of $V$ if nonzero, where $g=g_sg_u$ is the Jordan decomposition; furthermore, $2|\dim_{\F_q}\Ker(g_s+1_V)$ and $\dim\Ker_{\F_q}\Ker(g_s-1_V) \equiv n ({\bmod \,}2)$. Hence the claim reduces to the unipotent case $g=g_u$. In the latter case, the number of Jordan blocks of $g_u$ of each even size is even, see [@Car §13.1], and the claim follows.) Suppose $x = \mathrm{diag}(\mu,\mu^{-1}) \in T_2 <S$, where $T_2 \cong C_{q+1}$ is a non-split torus and $\mu \in \mu_{q+1}$ with $\mu \neq \pm 1$. Then $\St(x) = -1$ and $|x^S| = q(q-1)$. In this case, we can view $T_2$ as $\GU_1(q)$, embed $G$ in $\GU_{n}(q)$, and view $xg$ as an element $h=\mu g$ in a subgroup $\GU_{n}(q)$ of $\Sp_{2n}(q)$, with $\det(h) = \mu^{n}$. It follows from [@Ge Theorem 3.3] that $$\omega_{n}(xg) = (-1)^n\chi_2^-(\mu^n)(-q)^{\dim_{\F_{q^2}}\Ker(h-1)} = (-1)^n\chi_2^-(\mu^n)(-q)^{\dim_{\F_{q^2}}\Ker(g-\mu^{-1})}.$$ Altogether, we have shown that $$\label{eq:dec51} \begin{aligned}\psi(g) & = \frac{1}{q^2-1}\bigl(q^{\dim_{\F_q}\Ker(g-1)}+\chi^+_2((-1)^n)q^{\dim_{\F_q}\Ker(g+1)}\bigr)\\ & +\frac{1}{2(q-1)}\sum_{\l \in \mu_{q-1} \smallsetminus \{\pm 1\}}\chi^+_2(\l^n)q^{\dim_{\F_q}\Ker(g -\l)}\\ & - \frac{(-1)^n}{2(q+1)}\sum_{\mu \in \mu_{q+1} \smallsetminus \{\pm 1\}}\chi^-_2(\mu^n)(-q)^{\dim_{\F_{q^2}}\Ker(g-\mu)} -\kappa,\end{aligned}$$ and the statement now follows if we use . Some character estimates ------------------------ \[rat-so21\] Let $q$ be any prime power, $G = \O^\e_{2n}(q)$ with $n \geq 5$, $\e=\pm$, and let $g \in G$ have support $s=\supp(g)$. Assume that $\chi \in \{\a,\b\}$ if $2 \nmid q$, and $\chi \in \{\a,\b,\g_i\}$ if $2|q$. Then $$\frac{|\c(g)|}{\c(1)} \leq \frac{1}{q^{s/3}}.$$ \(i) First we consider the case $s \geq n \geq 5$. Then $$\label{eq:rb1} d(x,g) \leq 2n-s$$ for any $x \in \overline{\F}_q^\times$. In particular, $$\label{eq:rb2} 0 \leq \rho(g) \leq \sum_{x \in \F_q^\times}\frac{q^{d(x,g)}-1}{q-1} \leq q^{2n-s}-1.$$ Now, (when $2|q$) part (i) of the proof of Proposition \[dec-so2\] shows that $\g_i = \Ind^G_P(\nu_j)$ for some linear character $\nu_j$ of $P$, and recall that $\rho = \Ind^G_P(1_P)$. It follows that $$|\g_i(g)| \leq |\rho(g)| \leq q^{2n-s}-1,$$ and so $|\g_i(g)/\g_i(1)| < 1/q^{s-2} \leq q^{-3s/5}$ as $\g_i(1) = [G:P] > q^{2n-2}$. Next, using Theorem \[beta-so2\] and we also see that $$\label{eq:rb3} |\b(g)+1| \leq \frac{1}{2(q-1)}\sum_{x \in \F_q^\times}q^{d(x,g)} + \frac{1}{2(q+1)}\sum_{x \in \overline{\F}_q^\times,x^{q+1}=1}q^{d(x,g)} \leq q^{2n-s}.$$ In particular, $|\b(g)| \leq q^{2n-s}+1$. Since $\b(1) = (q^{2n}-q^2)/(q^2-1)$, we deduce that $|\b(g)/\b(1)| < q^{-3s/5}$. Furthermore, as $\a(g) = \rho(g)-(\b(g)+1)$, we obtain from – that $$|\a(g)| \leq 2q^{2n-s}-1.$$ If $s \geq 6$, then it follows that $|\a(g)/\a(1)| < q^{4-s} \leq q^{-s/3}$, since $\a(1) > q^{2n-3}$. Suppose that $s=n=5$. Then we can strengthen to $$\frac{-2q^5-(q-1)q^3}{2(q+1)} \leq \b(g)+1 \leq q^5.$$ Together with , this implies that $$|\a(g)| = |\rho(g)-(\b(g)+1)| < q^5+q^4 < \a(1)/q^{s/3}$$ since $\a(1) \geq (q^5+1)(q^4-q)/(q^2-1)$. \(ii) From now on we may assume that $s \leq n-1$. As $g \in G=\O^\e_{2n}(q)$, it follows that $d(z,g) = 2n-s$ for a unique $z \in \{1,-1\}$. Furthermore, $2|s$. (Indeed, this has been recorded in when $2|q$, and in when $2 \nmid q$.) We also have that $$\label{eq:rb4} d(x,g) \leq 2n-d(z,g) =s$$ for all $x \in \overline\F_q^\times \smallsetminus \{z\}$, Assume in addition that $s \geq 4$. Using we obtain $$\label{eq:rb5} 0 \leq \rho(g) \leq \frac{q^{2n-s}-1+(q-2)(q^s-1)}{q-1}.$$ As $\rho(1)=(q^n-\e)(q^{n-1}+\e)/(q-1)$, it follows that $|\rho(g)/\rho(1)| < q^{-3s/5}$. As above, the same bound also applies to $\chi=\g_i$ when $2|q$. Next, since $2|s$, using Theorem \[beta-so2\] and applying to $x^{q \pm 1} = 1$ and $x \neq z$, we have that $$\label{eq:rb6} \frac{q^{2n-s}}{q^2-1}-q^s \cdot \frac{q}{2(q+1)} \leq \beta(g)+1 \leq \frac{q^{2n-s}}{q^2-1}+ q^s \cdot \biggl( \frac{q-2}{2(q-1)} + \frac{q}{2(q+1)} \biggr);$$ in particular, $$|\beta(g)| < \frac{q^{2n-s}+q^s(q^2-q-1)}{q^2-1}.$$ Since $\b(1) = (q^{2n}-q^2)/(q^2-1)$, we obtain that $|\b(g)/\b(1)| < q^{-4s/5}$. Furthermore, using –, we can bound $$|\a(g)| = |\rho(g)-(\b(g)+1)| < \frac{q^{2n-s+1}+q^s(3q^2-3q-4)/2}{q^2-1} <\frac{\a(1)}{q^{2s/5}}$$ since $\a(1) \geq (q^n+1)(q^{n-1}-q)/(q^2-1)$. \(iii) Since the statement is obvious for $s=0$, it remains to consider the case $s=2$, i.e. $d(1,zg) = 2n-2$. Using [@TZ1 Lemma 4.9], one can readily show that $g$ fixes an orthogonal decomposition $V = U \oplus U^\perp$, with $U \subset \Ker(g-z \cdot 1_V)$ being non-degenerate of dimension $2n-4$, and $$\label{eq:rb7} \dim_{\F_q}(U^\perp)^{zg} = 2.$$ First we estimate $\rho(g)$. Suppose $g(v) = tv$ for some singular $0 \neq v \in V$ and $t \in \F_q^\times$. If $t \neq z$, then $v \in U^\perp$, and implies that $g$ can fixes at most $q+1$ such singular $1$-spaces $\langle v \rangle_{\F_q}$. Likewise, $g$ fixes at most $q+1$ singular $1$-spaces $\langle v \rangle_{\F_q} \subset U^\perp$ with $g(v) = zv$. Assume now that $g(v) = zv$ with $v = u+u'$, $0 \neq u \in U$ and $u' \in U^\perp$. As $0 = \QF(v) = \QF(u)+\QF(u')$, the total number of such $v$ is $$N:=\sum_{x \in \F_q}|\{ 0 \neq w \in U \mid \QF(w) = x \}| \cdot |\{ w' \in U^\perp \mid g(w') = zw',\QF(w') = -x \}|.$$ Note that, since $U$ is a non-degenerate quadratic space of dimension $2n-4$, $$(q^{n-2}+1)(q^{n-3}-1) \leq |\{ 0 \neq w \in U \mid \QF(w) = x \}| \leq (q^{n-2}-1)(q^{n-3}+1)$$ for any $x \in \F_q$. On the other hand, implies that $$\sum_{x \in \F_q}|\{ w' \in U^\perp \mid g(w') = zw',\QF(w') = -x \}| = |(U^\perp)^{zg}| = q^2.$$ It follows that $$q^2(q^{n-2}+1)(q^{n-3}-1) \leq N \leq q^2(q^{n-2}-1)(q^{n-3}+1),$$ and so $$\label{eq:rb8} \frac{q^2(q^{n-2}+1)(q^{n-3}-1)}{q-1} \leq \rho(g) \leq 2q+2+\frac{q^2(q^{n-2}-1)(q^{n-3}+1)}{q-1}.$$ In particular, when $2|q$ we have $|\g_i(g)| \leq |\rho(g)| < \rho(1)/q^{4s/5}$. Next, applying to $s=2$ we have $$|\b(g)| \leq \frac{q^{2n-2}+q^2(q^2-q-1)}{q^2-1} < \frac{\b(1)}{q^{4s/5}}.$$ Finally, using with $s=2$ and , we obtain $$|\a(g)| = |\rho(g)-(\b(g)+1)| < \frac{q^{2n-3}+q^{n+1}-q^{n-1}}{q^2-1}+(q+1) <\frac{\a(1)}{q^{3s/5}}.$$ \[rat-sp-so22\] Let $q$ be any odd prime power, $n \geq 5$, and $\e=\pm$. Assume that $\chi \in \Irr(G)$, where either $G \in \{\Sp_{2n}(q), \O_{2n+1}(q)\}$ and $\chi \in \{\a,\b\}$, or $G = \O^\e_{2n}(q)$ and $\chi \in \{\a,\b,\g_i\}$. If $g \in G$ has support $s=\supp(g)$, then $$\frac{|\c(g)|}{\c(1)} \leq \frac{1}{q^{s/3}}.$$ \(i) As usual, we may assume $s \geq 1$. First we consider the case $G = \O^\e_{2n}(q)$. Then [@NT Corollary 5.14] and [@LBST Proposition 5.7] show (in their notation) that $\a=D_{1}-1_G$, $\b = D_{\St}-1_G$. Furthermore if $\nu \neq 1_P$ is a linear character of $P$, then $\Ind^G_P(\nu) = D_{\chi_j}$ if $\nu$ has order $>2$, and $\Ind^G_P(\nu) = D_{\xi_1}+D_{\xi_2}$ if $\nu$ has order $2$. If $\chi = \a$ or $\b$, then the statement is already proved in Proposition \[rat-so21\], whose proof also applies to the case $\chi=\g_i = D_{\chi_j}$ (using the estimate $|\Ind^G_P(\nu)(g)| \leq \rho(g)$). It remains to consider the case $\chi = \g_i = D_{\xi_j}$ for $j = 1,2$. Again the previous argument applied to $\nu$ of order $2$ shows that $$|D_{\xi_1}(g)+D_{\xi_2}(g)| \leq \frac{[G:P]}{q^{3s/5}} = \frac{2\chi(1)}{q^{3s/5}}.$$ On the other hand, the formula for $D_\a$ in [@LBST Lemma 5.5], the character table of $\SL_2(q)$ [@D Theorem 38.1], and part 1) of the proof of [@LBST Proposition 5.11] imply that $$\label{eq:rb21} |D_{\xi_1}(g)-D_{\xi_2}(g)| \leq \frac{2(q^2-1)q^n \cdot\sqrt{q}}{q(q^2-1)} = 2q^{n-1/2}.$$ If $4 \leq s \leq 2n-2$, then since $\chi(1) \geq (q^n+1)(q^{n-1}-1)/2(q-1)> q^{2n-3}(q+1)$ it follows that $$\begin{aligned}|\chi(g)| & \leq \bigl(|D_{\xi_1}(g)+D_{\xi_2}(g)|+|D_{\xi_1}(g)+D_{\xi_2}(g)|\bigr)/2 \\ & \leq \frac{\chi(1)}{q^{3s/ 5}}+q^{n-1/2} < \frac{\chi(1)}{q^{3s/5}} + \frac{2\chi(1)}{q^{s/3-1/6}(q+1)} < \frac{\chi(1)}{q^{s/3}}.\end{aligned}$$ If $1 \leq s \leq 4$, then $s < n$, and so $2|s$ as shown in part (ii) of the proof of Proposition \[rat-so21\]. Hence $s=2$, and we again have $$|\chi(g)| \leq \frac{\chi(1)}{q^{3s/ 5}}+q^{n-1/2} < \frac{\chi(1)}{q^{3s/5}} + \frac{2\chi(1)}{q^{s/3+17/6}} < \frac{\chi(1)}{q^{s/3}}.$$ Finally, if $s=2n-1$, then $d(x,g) \leq 1$ for all $x \in \overline\F_q^\times$ by ; moreover, $d(\pm 1,g) = 0$. Hence, instead of we now have the stronger bound $$|D_{\xi_1}(g)-D_{\xi_2}(g)| \leq \frac{2(q^2-1) \cdot\sqrt{q}}{q(q^2-1)} = 2q^{-1/2},$$ whence $|\chi(g)| \leq \chi(1)q^{-3s/5}+q^{-1/2} < \chi(1)q^{-s/3}$. \(ii) Next we consider the case $G = \O^\e_{2n+1}(q)$. Then [@NT Corollary 5.15] and [@LBST Proposition 5.7] show (in their notation) that $\a=D_{\xi_1}-1_G$, $\b = D_{\xi_2}-1_G$. Again using the formula for $D_\a$ in [@LBST Lemma 5.5], the character table of $\SL_2(q)$ [@D Theorem 38.1], and part 1) of the proof of [@LBST Proposition 5.11], we obtain that $$\label{eq:rb22} |\a(g)-\b(g)| = |D_{\xi_1}(g)-D_{\xi_2}(g)| \leq \frac{2(q^2-1)q^{n+1/2} \cdot\sqrt{q}}{q(q^2-1)} = 2q^n.$$ Suppose in addition that $3 \leq s \leq 2n-2$. Since $d(x,g) \leq 2n+1-s$ by , we have that $$0 \leq \rho(g)=1+\a(g)+\b(g) \leq \sum_{x \in \mu_{q-1}}\frac{q^{d(x,g)}-1}{q-1} \leq q^{2n+1-s}.$$ As $\chi(1) \geq (q^n+1)(q^n-q)/2(q-1)$, it follows that $$|\a(g)+\b(g)| \leq q^{2n+1-s}-1 < \frac{2(1-1/q)q^{2-s}\chi(1)}{(1+1/q^n)(1-1/q^{n-1})} < \frac{2(1-1/q)\chi(1)}{q^{s/3}(1-1/q^{n-1})}.$$ On the other hand, implies that $$|\a(g)-\b(g)| \leq \frac{4(1-1/q)\chi(1)}{q^{(s+4)/3}(1-1/q^{n-1})},$$ and so $$\frac{|\chi(g)|}{\chi(1)} < \frac{(1-1/q)}{q^{s/3}(1-1/q^{n-1})}+ \frac{2(1-1/q)}{q^{(s+4)/3}(1-1/q^{n-1})} < \frac{1}{q^{s/3}}.$$ If $s=2n-1$ or $2n$, then $d(x,g) \leq 2$ for all $x \in \overline\F_q^\times$ by . Hence, instead of we now have the stronger bound $$|\a(g)-\b(g)|=|D_{\xi_1}(g)-D_{\xi_2}(g)| \leq \frac{2(q^2-1)q^2 \cdot\sqrt{q}}{q(q^2-1)} = 2q^{3/2},$$ whence $$|\chi(g)| < \frac{(1-1/q)q^{2-s}\chi(1)}{(1-1/q^{n-1})} +q^{3/2} < \chi(1)q^{-s/3}.$$ It remains to consider the case $s=1,2$, i.e. $d(1,zg) = 2n$ or $2n-1$ for some $z \in \{1,-1\}$. Using [@TZ1 Lemma 4.9], one can readily show that $g$ fixes an orthogonal decomposition $V = U \oplus U^\perp$, with $U \subset \Ker(g-z \cdot 1_V)$ being non-degenerate of dimension $2n-3$, and $$\label{eq:rb23} \dim_{\F_q}(U^\perp)^{zg} = 4-s.$$ First we estimate $\rho(g)$. Suppose $g(v) = tv$ for some singular $0 \neq v \in V$ and $t \in \F_q^\times$. If $t \neq z$, then $v \in U^\perp$, and implies that $g$ can fixes at most $(q^s-1)/(q-1) \leq q+1$ such singular $1$-spaces $\langle v \rangle_{\F_q}$. Likewise, $g$ fixes at most $(q+1)^2$ singular $1$-spaces $\langle v \rangle_{\F_q} \subset U^\perp$ with $g(v) = zv$, since $\dim U^\perp = 4$. Assume now that $g(v) = zv$ with $v = u+u'$, $0 \neq u \in U$ and $u' \in U^\perp$. As $0 = \QF(v) = \QF(u)+\QF(u')$, the total number of such $v$ is $$N:=\sum_{x \in \F_q}|\{ 0 \neq w \in U \mid \QF(w) = x \}| \cdot |\{ w' \in U^\perp \mid g(w') = zw',\QF(w') = -x \}|.$$ Note that, since $U$ is a non-degenerate quadratic space of dimension $2n-3$, $$q^{n-2}(q^{n-2}-1) \leq |\{ 0 \neq w \in U \mid \QF(w) = x \}| \leq q^{n-2}(q^{n-2}+1)$$ for any $x \in \F_q$. On the other hand, implies that $$\sum_{x \in \F_q}|\{ w' \in U^\perp \mid g(w') = zw',\QF(w') = -x \}| = |(U^\perp)^{zg}| = q^{4-s}.$$ It follows that $$q^{n+2-s}(q^{n-2}-1) \leq N \leq q^{n+2-s}(q^{n-2}+1),$$ and so $$\frac{q^{n+2-s}(q^{n-2}-1)}{q-1} \leq \rho(g)=1+\a(g)+\b(g) \leq q^2+3q+2+\frac{q^{n+2-s}(q^{n-2}+1)}{q-1}.$$ Together with , this implies that $$\frac{|\chi(g)|}{\chi(1)} \leq \frac{(q^2-1)(q+2)+q^{n+2-s}(q^{n-2}+1)+2q^n(q-1)}{(q^n+1)(q^n-q)} < \frac{1}{q^{s/2}}.$$ \(iii) Finally, we consider the case $G = \Sp_{2n}(q)$. In this case, arguing similarly to the proof of [@LBST Proposition 5.7], one can show that $\{\a,\b\} = \{D^\circ_{\l_0},D^\circ_{\l_1}\}$, where $S = \GO^+_2(q) \cong D_{2(q-1)}$, with $\l_0$, $\l_1$ being the two linear characters trivial at $\SO^+_2(q)$, and we consider the dual pairs $G \times S \to \Sp_{4n}(q)$. In particular, $\chi(1) \geq (q^n+1)(q^n-q)/2(q-1) > q^{2n-4/3}$. Now, the formula for $D_\a$ in [@LBST Lemma 5.5], the character table of $S$, and part 1) of the proof of [@LBST Proposition 5.11] imply that $$\label{eq:rb24} |\a(g)-\b(g)| \leq q^{(d(1,g)+d(-1,g))/2} \leq q^{2n-s}.$$ On the other hand, using we have $0 \leq \rho(g) = \a(g)+\b(g)+1 \leq q^{2n-s}-1$. In particular, when $s \geq 2$ we have $$|\chi(g)| \leq \bigl(|\a(g)+\b(g)|+|\a(g)-\b(g)|\bigr)/2 \leq q^{2n-s} < \chi(1)q^{-s/3}.$$ Assume now that $s=1$. Then $g = zu$ for some $z = \pm 1$ and unipotent $u \in G$; furthermore, $\rho(g) = (q^{2n-1}-1)/(q-1)$. Applying also , we obtain $$|\chi(g)| \leq \biggl(|\a(g)+\b(g)|+|\a(g)-\b(g)|\biggr)/2 \leq \biggl(\frac{q^{2n-1}-q}{q-1}+q^{n-1/2}\biggr)/2 < \chi(1)q^{-4s/5},$$ and the proof is complete. Classical groups: Proof of Theorem \[main1\] {#pfth1} ============================================ Let $G = \Sp(V)$ or $\O(V)$, where $V = V_n(q)$. Write $G = \Cl_n(q)$ to cover both cases. As before, for a semisimple element $g \in G$, define $\nu(s) = \hbox{supp}(g)$, the codimension of the largest eigenspace of $g$ over $\bar \F_q$. For $n<10$, Theorem \[main1\] can be easily proved by exactly the same method of proof of [@LST Theorem 2] (improving the constant $D$ in Lemma 2.3 of [@LST] by using better bounds for $|G|$ and $|C_G(g)|_p$). So assume from now on that $n\ge 10$, so that the character ratio bounds in Propositions \[rat-so21\] and \[rat-sp-so22\] apply. We begin with a lemma analogous to [@LST Lemma 3.2]. \[sest\] For $1\le s<n$, define $$N_s(G) := \{g\in \GSS : \nu(g)=s\}$$ and let $n_s(G):=|N_s(G)|$. - If $g \in N_s(g)$ and $s<\frac{n}{2}$ then $|\CB_G(g)|_p < q^{\frac{1}{4}((n-s)^2+s^2) - v\frac{n-1}{2}}$, where $v=0$ or $1$ according as $G$ is symplectic or orthogonal. - If $g \in N_s(g)$ and $s\ge \frac{n}{2}$ then $|\CB_G(g)|_p < q^{\frac{1}{4}(n^2-ns)}$. - $\sum_{n-1 \geq s \geq n/2}n_s(G) < |G| < q^{\frac{1}{2}(n^2+n)-vn}$, where $v$ is as in $(ii)$. - If $s < n/2$, then $n_s(G) < cq^{\frac{1}{2}s(2n-s+1)+\frac{n}{2}}$, where $c$ is an absolute constant that can be taken to be $15.2$. \(i) If $\nu(g)=s<\frac{n}{2}$, then the largest eigenspace of $g$ has dimension $n-s>\frac{n}{2}$, so has eigenvalue $\pm 1$, and so $\CB_G(g) \le \Cl_{n-s}(q) \times \Cl_s(q)$. Part (i) follows. \(ii) Now suppose $\nu(g) = s \ge \frac{n}{2}$, and let $E_\l$ ($\l \in \bar \F_q$) be an eigenspace of maximal dimension $n-s$. Assume first that $\l \ne \pm 1$. Then letting $a$ and $b$ denote the dimensions of the $+1$- and $-1$-eigenspaces, we have $$\label{cent} \CB_G(g) \le \prod_{i=1}^t \GL_{d_i}(q^{k_i}) \times \Cl_a(q) \times \Cl_b(q),$$ where $n-s = d_1 \ge d_2\ge \cdots \ge d_t$ and also $d_1 \ge a\ge b$ and $2\sum_1^t k_id_i+a+b = n$. Hence $|\CB_G(g)|_p \le q^D$, where $$\label{expd} D = \frac{1}{2}\sum_{i=1}^t k_id_i(d_i-1) + \frac{1}{4}(a^2+b^2).$$ If $n\ge 4d_1$, this expression is maximised when $a=b=d_1$ and $(d_1,\ldots ,d_t) = (d_1,\ldots ,d_1,r)$ with $r\le d_1$ and $k_i=1$ for all $i$.. Hence in this case, $$D \le \frac{1}{2}(t-1)d_1(d_1-1) + \frac{1}{2}r(r-1) + \frac{1}{2}d_1^2 = \frac{1}{2}td_1^2-\frac{1}{2}(t-1)d_1+\frac{1}{2}r(r-1),$$ and this is easily seen to be less than $\frac{1}{4}nd_1$, as required for part (ii). Similarly, if $4d_1>n\ge 3d_1$, the expression (\[expd\]) is maximised when $t=1$, $k_1=1$, $a=d_1$ and $b=r < d_1$; and when $3d_1>n \ge 2d_1$ (note that $n\ge 2d_1 = 2(n-s)$ by our assumption that $\nu(g) = s \ge \frac{n}{2}$), the expression (\[expd\]) is maximised when $t=1$ and $a=r< d_1$. In each case, we see that $D< \frac{1}{4}nd_1$ as above. Assume finally that the eigenvalue $\l = \pm 1$. In this case the centralizer $\CB(g)$ is as in (\[cent\]), with $n-s=a \ge d_1\ge \cdots \ge d_t$ and also $a\ge b$ and $2\sum_1^t k_id_i+a+b = n$. Again we have $|\CB_G(g)|_p \le q^D$, with $D$ as in (\[expd\]), and we argue as above that $D < \frac{1}{4}na = \frac{1}{4}n(n-s)$. This completes the proof of (ii). \(iii) This is clear. \(iv) If $\nu(g) = s < \frac{n}{2}$ then as in (i), the largest eigenspace of $g$ has eigenvalue $\pm 1$, so we have $\CB_G(g) \ge \Cl_{n-s}(q) \times T_s$, where $T_s$ is a maximal torus of $\Cl_s(q)$. Hence $|g^G| \le |G:\Cl_{n-s}(q)T_s| \le q^{\frac{1}{2}s(2n-s+1)}$. Also the number of conjugacy classes in $G$ is at most $15.2q^{n/2}$ by [@FG], and (iv) follows. \[stein\] Let $\c \in \{\a,\b,\g_i\}$, where $\a,\b,\g_i$ are the irreducible characters of $G$ defined in Section \[red\]. Then $\St \subseteq \c^{4n}$. As in the proof of [@LST Lemma 2.3], there are signs $\e_g=\pm 1$ such that $$\label{useag} \begin{array}{ll} [\c^l,\St]_G & = \dfrac{1}{|G|}\sum_{g\in \GSS} \e_g \c^l(g)|\CB_G(g)|_p \\ & = \dfrac{\chi^l(1)}{|G|}\left(|G|_p + \sum_{1 \neq g \in \GSS} \e_g \left(\frac{\chi(g)}{\chi(1)}\right)^l|\CB_G(g)|_p\right). \end{array}$$ Hence $[\c^l,\St]_G \ne 0$ provided $\Sigma_l < |G|_p$, where $$\Sigma_l := \sum_{1 \neq g\in \GSS} \left|\frac{\chi(g)}{\chi(1)}\right|^l|\CB_G(g)|_p.$$ By Propositions \[rat-so21\] and \[rat-sp-so22\], if $s = \nu(g)$ we have $$\frac{|\c(g)|}{\c(1)} \le \frac{1}{q^{s/3}}.$$ Hence applying Lemma \[sest\], we have $\Sigma_l \le \Sigma_1+\Sigma_2$, where $$\begin{array}{l} \Sigma_1 = \sum_{1\le s<\frac{n}{2}} cq^{\frac{1}{2}s(2n-s+1)+\frac{n}{2}}. \frac{1}{q^{ls/3}} . q^{\frac{1}{4}((n-s)^2+s^2) - v\frac{n-1}{2}}, \\ \Sigma_2 = \sum_{\frac{n}{2}\le s < n} q^{\frac{1}{2}(n^2+n)-vn}. \frac{1}{q^{ls/3}} . q^{\frac{1}{4}(n^2-ns)}. \end{array}$$ For a term in $\Sigma_1$, the exponent of $q$ is $$\frac{1}{4}n^2-v\frac{n-1}{2} + \frac{1}{2}s(n+1)+\frac{1}{2}n-\frac{ls}{3}.$$ As $|G|_p \le q^{\frac{1}{4}n^2-v\frac{n-1}{2}}$, taking $l=4n$ this gives $$\begin{array}{ll} \frac{\Sigma_1}{|G|_p} & \le \sum_{1\le s<\frac{n}{2}} cq^{\frac{1}{2}s(n+1)+\frac{n}{2}-\frac{ls}{3}} \\ & \le \sum_{1\le s<\frac{n}{2}} cq^{\frac{1}{2}n(1-\frac{5s}{3})+\frac{s}{2}}. \end{array}$$ Recalling that $c=15.2$, it follows that $\frac{\Sigma_1}{|G|_p} < \frac{1}{2}$ (except for $q=2, n\le 20$, in which case we obtain the same conclusion using slightly more refined estimates instead of Lemma \[sest\](iv)). For a term in $\Sigma_2$, the exponent of $q$ is $$\frac{1}{2}(n^2+n)-vn +\frac{1}{4}n(n-s) - \frac{ls}{3},$$ and leads similarly to the inequality $\frac{\Sigma_2}{|G|_p} < \frac{1}{2}$ when $l=4n$. We conclude that $\Sigma_l < |G|_p$ for $l=4n$, proving the lemma. Let $1\ne \psi \in \Irr(G)$. By Lemma \[stein\] together with Lemmas \[mc-r2\] and \[mc-r3\], we have $\St \subseteq \psi^{8n}$ for $G = Sp_n(q)$, and $\St \subseteq \psi^{16n}$ for $G = \O^\e_n(q)$. Since $\St^2$ contains all irreducible characters by [@HSTZ], the conclusion of Theorem \[main1\] follows. Alternating groups: Proof of Theorem \[main2\] {#pfth2} ============================================== In this section we prove Theorem \[main2\]. \[staircase\] Let $n := m(m+1)/2$ with $m \in \Z_{\geq 6}$, and let $\chi_m := \chi^{(m,m-1,\ldots,1)}$ be the staircase character of $\SSS_n$. Then $$\chi_m(1) \geq |\SSS_n|^{5/11}.$$ We will proceed by induction on $m \geq 6$. The induction base $m=6,7$ can be checked directly. For the induction step going from $m$ to $m+2$, note by the hook length formula that $\chi_m(1)= n!/H_m$, where $H_m$ is the product of all the hook lengths in the Young diagram of the staircase partition $(m,m-1, \ldots,1)$. Hence it is equivalent to to prove that $$(m(m+1)/2)! > H_m^{11/6}.$$ Since the statement holds for $m$ and $H_{m+2}/H_m = (2m+3)!!(2m+1)!!$, it suffices to prove that $$\label{eq:st1} \prod^{2m+3}_{i=1}(m(m+1)/2+i) > \bigl((2m+3)!!(2m+1)!!\bigr)^{11/6}$$ for any $m \geq 6$. Direct computation shows that holds when $3 \leq m \leq 40$. When $m \geq 40$, note that $$\begin{array}{ll} \prod^{2m+3}_{i=1}(m(m+1)/2+i) & > \bigl(m(m+1)/2+1\bigr)^{2m+3}\\ & > \bigl((m+3)^{m+1}(m+2)^m\bigr)^{11/6}\\ & > \bigl((2m+3)!!(2m+1)!!\bigr)^{11/6},\end{array}$$ proving and completing the induction step. We will make use of [@S Theorem 1.4] which states that there exists an effective absolute constant $C_1 \geq 2$ such that $$\label{eq:s} \chi^{t} \mbox{ contains }\Irr(\SSS_n) \mbox{ whenever }t \geq C_1n\log(n)/\log(\chi(1))$$ for every non-linear $\chi \in \Irr(\SSS_n)$. With this, we will prove that when $n$ is sufficiently large we have $$\label{eq:main2} \varphi^{k} \mbox{ contains }\Irr(\AAA_n) \mbox{ whenever }k \geq Cn\log(n)/\log(\varphi(1))$$ for every nontrivial $\varphi \in \Irr(\AAA_n)$, with $C=5C_1^2$. \(i) Consider any $n \geq 5$ and any nontrivial $\varphi \in \Irr(\AAA_n)$. If $\varphi$ extends to $\SSS_n$, then we are done by . Hence we may assume that $\varphi$ lies under some $\chi^\l \in \Irr(\SSS_n)$, where $\l \vdash n$ is self-associate, and that $n$ is sufficiently large. By [@KST Proposition 4.3], the latter implies that $$\label{eq:a1} \varphi(1) \geq 2^{(n-5)/4}.$$ Consider the Young diagram $Y(\l)$ of $\l$, and let $A$ denote the removable node in the last row of $Y(\l)$. Also let $\rho:=\chi^{\l \smallsetminus A} \in \Irr(\SSS_{n-1})$. Since $\l \smallsetminus A$ is not self-associate, $\rho$ is also irreducible over $\AAA_{n-1}$. Furthermore, by Frobenius’ reciprocity, $$1 \leq [(\chi^\l)|_{\SSS_{n-1}},\rho]_{\SSS_{n-1}} = [\chi^\l,\Ind^{\SSS_n}_{\SSS_{n-1}}(\rho)]_{\SSS_n},$$ whence $2\varphi(1) = \chi^\l(1) \leq \Ind^{\SSS_n}_{\SSS_{n-1}}(\rho)(1) = n\rho(1)$, and so $$\label{eq:a2} \rho(1) \geq (2/n)\varphi(1).$$ It follows from and that when $n$ is large enough, $$\log(\rho(1)) \geq \log(\varphi(1))-\log(n/2) \geq (9/10)\log(\varphi(1)).$$ Now we consider any integer $$\label{eq:a3} s \geq \frac{10C_1}{9} \cdot \frac{n\log(n)}{\log(\varphi(1))}.$$ This ensures that $s \geq C_1(n-1)\log(n-1)/\log(\rho(1))$, and so, by applied to $\rho$, $\rho^s$ contains $\Irr(\SSS_{n-1})$. \(ii) Next, we can find a unique $m \in \Z_{\geq 3}$ such that $$\label{eq:a4} n_0:=m(m+1)/2 \leq n-3 < (m+1)(m+2)/2,$$ and consider the following partition $$\label{eq:a5} \mu:= (n-1-m(m-1)/2,m-1,m-2, \ldots,2,1)$$ of $n-1$. Note that $\mu$ has $m$ rows, with the first (longest) row $$\mu_1=n-1-m(m-1)/2 \geq m+2$$ by . Hence, if $B$ is any addable node for the Young diagram $Y(\mu)$ of $\mu$, $Y(\mu \sqcup B)$ has at most $m+1$ rows and at least $m+2$ columns, and so is not self-associate. It follows that, for any such $B$, the character $\chi^{\mu \sqcup B}$ of $\SSS_n$ is irreducible over $\AAA_n$. \(iii) Recall that $\chi^\l|_{\AAA_n} = \varphi+\varphi^\star$ with $\varphi^\star$ being $\SSS_n$-conjugate to $\varphi$. It suffices to prove for an $\SSS_n$-conjugate of $\varphi$. As $\chi^\l|_{\SSS_{n-1}}$ contains $\rho=\chi^{\l \smallsetminus A}$ which is irreducible over $\AAA_{n-1}$, without loss we may assume that $\varphi|_{\AAA_{n-1}}$ contains $\rho|_{\AAA_{n-1}}$. By the result of (i), $\rho^s$ contains $\chi^\mu$, with $\mu$ defined in Thus $$\label{eq:a6} 1 \leq [\varphi^s|_{\AAA_{n-1}},(\chi^\mu)|_{\AAA_{n-1}}]_{\AAA_{n-1}}=\bigl[\varphi^s, \Ind^{\AAA_n}_{\AAA_{n-1}}\bigl((\chi^\mu)|_{\AAA_{n-1}}\bigr)\bigr]_{\AAA_n}.$$ Also recall that $\chi^\mu$ is an $\SSS_{n-1}$-character and $\SSS_n = \AAA_n\SSS_{n-1}$. Hence $$\Ind^{\AAA_n}_{\AAA_{n-1}}\bigl((\chi^\mu)|_{\AAA_{n-1}}\bigr) = \bigl(\Ind^{\SSS_n}_{\SSS_{n-1}}(\chi^\mu)\bigr)|_{\AAA_{n}}.$$ Next, $$\Ind^{\SSS_n}_{\SSS_{n-1}}(\chi^\mu) = \sum_{B~{\rm \tiny{addable}}}\chi^{\mu \sqcup B},$$ where, as shown in (ii), each such $\chi^{\mu \sqcup B}$ is irreducible over $\AAA_n$. Hence, it now follows from that there is an addable node $B_0$ for $Y(\mu)$ that $\varphi^s$ contains $\psi|_{\AAA_n}$, with $\psi:=\chi^{\mu \sqcup B_0}$. \(iv) By the choice of $B_0$, $\psi|_{\SSS_{n-1}}$ contains $\chi^\mu$, whence $\psi(1) \geq \chi^\mu(1)$. Next, by , we can remove $n-1-n_0 \geq 2$ nodes from the first row to arrive at the staircase partition $(m,m-1, \ldots,1) \vdash n_0$. In particular, $\psi|_{\SSS_{n_0}}$ contains the character $\chi_m$ of $\SSS_{n_0}$. By Lemma \[staircase\], for $n$ sufficiently large we have $$\label{eq:a7} \log(\psi(1)) \geq \log(\chi_m(1)) \geq (5/11)\log(n_0!) \geq (2/5)n\log(n),$$ since $$n_0 = m(m+1)/2 \geq n-(m+2) \geq n-(3/2+\sqrt{2n-4})$$ by the choice of $m$. Now we consider the integer $t := \lceil (5/2)C_1 \rceil \leq 3C_1$ (since $C_1 \geq 2$). Then $$C_1n\log(n)/\log(\psi(1)) \leq (5/2)C_1 \leq t$$ by , and so $\psi^t$ contains $\Irr(\SSS_n)$ by applied to $\psi$. In particular, $(\psi^t)|_{\AAA_n}$ contains $\Irr(\AAA_n)$. Recall from (iii) that $\varphi^s$ contains the irreducible character $\psi|_{\AAA_n}$. It follows that $\varphi^{st}$ contains $(\psi^t)|_{\AAA_n}$, and so $\varphi^{st}$ contains $\Irr(\AAA_n)$. \(v) Finally, consider any integer $k \geq Cn\log(n)/\varphi(1)$ with $C=5C_1^2$. Then $$k/t \geq k/3C_1 \geq (5/3)C_1n\log(n)/\log(\varphi(1)).$$ As $C_1\geq 1$ and $n\log(n)/\log(\varphi(1) \geq 2$, we have that $$(5/3-10/9)C_1n\log(n)/\log(\varphi(1)) \geq 10/9.$$ In particular, we can find an integer $s_0$ such that $$k/t \geq s_0 \geq (10/9)C_1n\log(n)/\log(\varphi(1)).$$ As $s$ satisfies , the result of (iv) shows that $\varphi^{s_0t}$ contains $\Irr(\AAA_n)$. Now, given any $\gamma \in \Irr(\AAA_n)$, we can find an irreducible constituent $\delta$ of $\varphi^{k-s_0t}\overline\gamma$. By the previous result, $\varphi^{s_0t}$ contains $\overline\delta$. It follows that $\varphi^k$ contains $\varphi^{k-s_0t}\overline\delta$, and $$[\varphi^{k-s_0t}\overline\delta,\gamma]_{\AAA_n}= [\varphi^{k-s_0t}\overline\gamma,\delta]_{\AAA_n} \geq 1,$$ i.e. $\varphi^k$ contains $\gamma$, and the proof of is completed. Products of characters {#pfth3} ====================== Products of characters in classical groups ------------------------------------------ This is very similar to the proof of Theorem 2 of [@LST]. Let $G = G_r(q)$ be a simple group of Lie type of rank $r$ over $\F_q$. \[stdiam\] There is an absolute constant $D$ such that for any $m\ge Dr^2$ and any $\c_1,\ldots,\c_m \in {\rm Irr}(G)$, we have $[\prod_1^m\c_i,\St]_G\ne 0$. Indeed, $D=163$ suffices. This is proved exactly as for [@LST Lemma 2.3], replacing the power $\c^m$ by the product $\prod_1^m\c_i$. Take $c_1=3D$ with $D$ as in the lemma, and let $\c_1,\ldots ,\c_l \in {\rm Irr}(G)$ with $l=c_1r^2$. Writing $m=l/3 = Dr^2$, Lemma \[stdiam\] shows that each of the products $\prod_1^m\c_i$, $\prod_{m+1}^{2m}\c_i$ and $\prod_{2m+1}^{3m}\c_i$ contains $\St$. Hence $\prod_1^l\c_i$ contains $\St^3$, and this contains ${\rm Irr}(G)$ by [@LST Prop. 2.1]. This completes the proof. Products of characters in linear and unitary groups --------------------------------------------------- This is similar to the proof of Theorem 3 of [@LST]. Let $G = \PSL_n^\e(q)$. We shall need [@LST Theorem 3.1], which states that there is a function $f:\N\to \N$ such that for any $g \in \GSS$ with $s = \nu(g)$, and any $\c \in {\rm Irr}(G)$, we have $$\label{31lst} |\c(g)| < f(n)\c(1)^{1-\frac{s}{n}}.$$ Again we begin with a lemma involving the Steinberg character. \[ste\] Let $m\in \N$ and let $\c_1,\ldots,\c_m \in {\rm Irr}(G)$. Set $c=44.1$, and define $$\begin{array}{l} \D_{1m} = cf(n)^m \sum_{1\le s <n/2} q^{ns+\frac{3n}{2}-1}\left(\prod_1^m\c_i(1)\right)^{-s/n},\\ \D_{2m} = f(n)^m \sum_{n/2\le s<n}q^{n^2-\frac{1}{2}n(s-1)-1}\left(\prod_1^m\c_i(1)\right)^{-s/n}. \end{array}$$ If $\D_{1m}+\D_{2m}<1$, then $[\prod_1^m\c_i,\,\St]_G \ne 0$. Arguing as in the proof of [@LST Lemma 3.3], we see that $[\prod_1^m\c_i,\,\St]_G \ne 0$ provided $\D_m <1$, where $$\D_m := \sum_{1 \leq s < n/2} cq^{ns+\frac{3n}{2}-1}\left|\prod_1^m\frac{\chi_i(g_{i,s})}{\chi(1)}\right| + \sum_{n/2 \leq s < n} q^{n^2-\frac{1}{2}n(s-1)-1}\left|\prod_1^m\frac{\chi(g_{i,s})}{\chi(1)}\right|,$$ where $g_{i,s} \in \GSS$ is chosen such that $\nu(g_{i,s})=s$ and $|\c_i(g_{i,s})|$ is maximal. Now application of (\[31lst\]) gives the conclusion. \[better\] There is a function $g:\N\to \N$ such that the following holds. Suppose that $\c_1,\ldots,\c_m \in {\rm Irr}(G)$ satisfy $\prod_1^m \c_i(1) > |G|^3$. Then provided $q>g(n)$, we have $[\prod_1^m\c_i,\,\St]_G \ne 0$. We have $|G|>\frac{1}{2}q^{n^2-2}$, so for $s<n$, $$\left(\prod_1^m\c_i(1)\right)^{-s/n} < 8q^{-3ns+\frac{6s}{n}}.$$ Hence $$\D_{1m} \le 8cf(n)^m \sum_{1\le s <n/2} q^{-2ns+\frac{3n}{2}+2},$$ and $$\begin{array}{ll} \D_{2m} & \le 8f(n)^m \sum_{n/2\le s<n}q^{n^2-\frac{1}{2}n(s-1)-1} q^{-3ns+6} \\ & \le 8f(n)^m \sum_{n/2\le s<n}q^{-\frac{3n^2}{4}+\frac{1}{2}n+5}. \end{array}$$ Now the conclusion follows from Lemma \[ste\] (using some slight refinements of the above inequalities for $n\le 4$). Assume $\c_1,\ldots,\c_l \in {\rm Irr}(G)$ satisfy $\prod_1^l \c_i(1) > |G|^{10}$. Since $\c_i(1) < |G|^{1/2}$ for all $i$, there are disjoint subsets $I_1,I_2,I_3$ of $\{1,\ldots ,m\}$ such that $\prod_{i\in I_k} \c_i(1) > |G|^3$ for $k=1,2,3$. Then $\prod_{i\in I_k} \c_i$ contains $\St$ for each $k$, by Lemma \[better\], and so $\prod_1^l\c_i$ contains $\St^3$, hence contains ${\rm Irr}(G)$, completing the proof. Products of characters in symmetric and alternating groups ---------------------------------------------------------- \[rs2-an\] Let $G \in \{\SSS_n,\AAA_n\}$, $l \in \Z_{\geq 1}$, and let $\chi_1,\chi_2, \ldots,\chi_l \in \Irr(G)$ with $\chi_i(1) > 1$ for all $i$. 1. If $l \geq 8n-11$, then $\bigl(\prod^l_{i=1}\chi_i\bigr)^{2}$ contains $\Irr(G)$. 2. Suppose that, for each $1 \leq i \leq l$, there exists some $j \neq i$ such that $\chi_j = \chi_i$. If $l \geq 24n-33$ then $\prod^l_{i=1}\chi_i$ contains $\Irr(G)$. \(i) Let $\chi^\l$ denote the irreducible character of $\SSS_n$ labeled by the partition $\l \vdash n$. A key result established in the proof of [@LST Theorem 5] is that, for any $i$ there exists $$\a_i \in \left\{\chi^{(n-1,1)},\chi^{(n-2,2)},\chi^{(n-2,1^2)},\chi^{(n-3,3)}\right\}$$ such that $\chi_i^2$ contains $(\a_i)|_G$. Since $l \geq 8n-11$, there must be some $$\b \in \left\{\chi^{(n-1,1)},\chi^{(n-2,2)},\chi^{(n-2,1^2)},\chi^{(n-3,3)}\right\}$$ such that $\b=\a_i$ for at least $2n-2$ distinct values of $i$. It follows that $\bigl(\prod^l_{i=1}\chi_i\bigr)^{2}=\g\d$, where $\g := \b^{2n-2}|_G$, and $\d$ is a character of $G$. By [@LST Theorem 5], $\b^{2n-2}$ contains $\Irr(\SSS_n)$, whence $\g$ contains $\Irr(G)$. Now the arguments in the last paragraph of the proof of Theorem \[main2\] show that $\g\d$ contains $\Irr(G)$ as well. \(ii) Note that the assumptions imply, after a suitable relabeling, that $\prod^l_{i=1}\chi_i$ contains $\sigma\l$, where $\l$ is a character of $G$ and $$\s= \prod^{8n-11}_{i=1}\chi_i^2.$$ (Indeed, any subproduct $\chi_{i_1}\ldots\chi_{i_t}$ with $t> 1$ and $\chi_{i_1}=\ldots =\chi_{i_t}$ yields a term $(\chi_{i_1}^2)^{\lfloor t/2 \rfloor}$ in $\s$.) By (i), $\s$ contains $\Irr(G)$, and so we are done as above. 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[^1]: The second author acknowledges the support of ISF grant 686/17, and the Vinik chair of mathematics which he holds. The third author gratefully acknowledges the support of the NSF (grant DMS-1840702), and the Joshua Barlaz Chair in Mathematics. The second and the third authors were partially supported by BSF grant 2016072. The authors also acknowledge the support of the National Science Foundation under Grant No. DMS-1440140 while they were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. Part of this work was done when the authors were in residence at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, in Spring 2020, and partially supported by a grant from the Simons Foundation.

--- abstract: 'We extend the Higgs triplet model so as to include dark matter candidates and a simple suppression mechanism for the vacuum expectation value ($v_\Delta$) of the triplet scalar field. The smallness of neutrino masses can be naturally explained with the suppressed value of $v_\Delta$ even when the triplet fields are at the TeV scale. The Higgs sector is extended by introducing $Z_2$-odd scalars (an ${{\text{SU}}}(2)_L$ doublet $\eta$ and a real singlet $s_2^0$) in addition to a $Z_2$-even complex singlet scalar $s_1^0$ whose vacuum expectation value violates the lepton number conservation by a unit. In our model, $v_\Delta$ is generated by the one-loop diagram to which $Z_2$-odd particles contribute. The lightest $Z_2$-odd scalar boson can be a candidate for the dark matter. We briefly discuss a characteristic signal of our model at the LHC.' author: - Shinya Kanemura - Hiroaki Sugiyama title: | Dark matter and a suppression mechanism for neutrino masses\ in the Higgs triplet model --- introduction {#sec:intro} ============ Existence of dark matter (DM) has been established, and its thermal relic abundance has been determined by the WMAP experiment [@WMAP; @Komatsu:2008hk]. If the essence of DM is an elementary particle, the weakly interacting massive particle (WIMP) would be a promising candidate. It is desired to have a viable candidate for the dark matter in models beyond the standard model (SM). The WIMP dark matter candidate can be accommodated economically by introducing only an inert scalar field [@Silveira:1985rk; @Deshpande:1977rw; @i-nplet], where we use “inert” for the $Z_2$-odd property. The imposed $Z_2$ parity ensures the stability of the DM candidate. Phenomenology in such models have been studied in, e.g., Refs. [@c-i-singlet; @r-i-singlet; @i-doublet; @Lundstrom:2008ai; @Araki:2011hm; @THDM-iSDM; @HTM-iSDM]. On the other hand, it has been confirmed by neutrino oscillation measurements that neutrinos have nonzero but tiny masses as compared to the electroweak scale [@solar; @atm; @acc; @reactor-S; @reactor-L]. The different flavor structure of neutrinos from that of quarks and leptons may indicate that neutrino masses are of Majorana type. In order to explain tiny neutrino masses, many models have been proposed. The seesaw mechanism is the simplest way to explain tiny neutrino masses, in which right-handed neutrinos are introduced with large Majorana masses [@seesaw; @Mohapatra:1979ia]. Another simple model for generating neutrino masses is the Higgs Triplet Model (HTM) [@Mohapatra:1979ia; @HTM]. However, these scenarios do not contain dark matter candidate in themselves. In a class of models where tiny neutrino masses are generated by higher orders of perturbation, the DM candidate can be naturally contained [@KNT; @Ma; @AKX; @Aoki:2011yk; @Kanemura:2010bq; @Gu:2007ug]. In models in Refs. [@KNT; @Ma; @AKX; @Aoki:2011yk; @Kanemura:2010bq], the Yukawa couplings of neutrinos with the SM Higgs boson are forbidden at the tree level by imposing a $Z_2$ parity. The same $Z_2$ parity also guarantees the stability of the lightest $Z_2$-odd particle in the model which can be the candidate of the DM as long as it is electrically neutral. In this paper, we consider an extension of the HTM in which by introducing the $Z_2$ parity $m_\nu$ is generated at the one-loop level and the DM candidate appears. In the HTM, Majorana masses for neutrinos are generated via the Yukawa interaction $h_{\ell{{\ell^\prime}}} \overline{L_\ell^c}\, i\sigma_2 \Delta L_{{\ell^\prime}}$ with a nonzero vacuum expectation value (VEV) of an ${{\text{SU}}}(2)_L$ triplet scalar field $\Delta$ with the hypercharge of $Y=1$. The VEV of $\Delta$ is described by $v_\Delta \sim \sqrt{2} \mu v^2/(2 M_\Delta^2)$, where $v$ is the VEV of the Higgs doublet field $\Phi$ and $M_\Delta$ is the typical mass scale of the triplet field; the dimensionful parameter $\mu$ breaks lepton number conservation at the trilinear term $\mu\,\Phi^T i\sigma_2 \Delta^\dagger \Phi$ which we refer to as the $\mu$-term. As the simplest explanation for the smallness of neutrino masses, the mass of the triplet field is assumed to be much larger than the electroweak scale. On the other hand a characteristic feature of the HTM is the fact that the structure of the neutrino mass matrix $(m_\nu)_{\ell{{\ell^\prime}}}$ is given by that of the Yukawa matrix, $h_{\ell{{\ell^\prime}}} \propto (m_\nu)_{\ell{{\ell^\prime}}}$. The direct information on $(m_\nu)_{\ell{{\ell^\prime}}}$ would be extracted from the decay $H^{\pm\pm} \to \ell^\pm{{\ell^\prime}}^{\prime\pm}$ [@nu-LHC] if $H^{++}$ is light enough to be produced at collider experiments, where $H^{++}$ is the doubly charged component of the triplet field $\Delta$. At hadron colliders, the $H^{\pm\pm}$ can be produced via ${{q\overline{q} \to Z^\ast (\gamma^\ast) \to H^{++}H^{--}}}$ [@HppHmm] and ${{q^\prime\overline{q} \to W^{\pm\ast} \to H^{\pm\pm}H^\mp}}$ [@HppHm]. The $H^{\pm\pm}$ searches at the LHC put lower bound on its mass as $m_{H^{\pm\pm}}^{}\gtrsim 300\,{{\text{GeV}}}$ [@Hpp-CMS; @Hpp-ATLAS], assuming that the main decay mode is $H^{\pm\pm} \to \ell^\pm \ell^{\prime\pm}$. Phenomenological analyses for $H^{\pm\pm}$ in the HTM at the LHC have also been performed in Ref. [@Hpp]. Triplet scalars can contribute to lepton flavor violation (LFV) in decays of charged leptons, e.g., $\mu \to \bar{e}ee$ and $\tau \to \bar{\ell}{{\ell^\prime}}\ell^{{{\prime\prime}}}$ at the tree level and $\ell \to {{\ell^\prime}}\gamma$ at the one-loop level. Relation between these LFV decays and neutrino mass matrix constrained by oscillation data was discussed in Refs. [@Chun:2003ej; @LFV-HTM]. In order to explain the small $v_\Delta$ with such a detectable light $H^{++}$, the $\mu$ parameter has to be taken to be unnaturally much lower than the electroweak scale. Therefore, it would be interesting to extend the HTM in order to include a natural suppression mechanism of the $\mu$ parameter (therefore $v_\Delta$) in addition to the DM candidate. In our model, lepton number conservation is imposed to the Lagrangian in order to forbid the $\mu$-term in the HTM at the tree level while the triplet Yukawa term $h_{\ell{{\ell^\prime}}}^{} \overline{L_\ell^c}\, i\sigma_2 \Delta L$ exists. The VEV of a $Z_2$-even complex singlet scalar $s_1^0$ breaks the lepton number conservation by a unit. An ${{\text{SU}}}(2)_L$ doublet $\eta$ and a real singlet $s_2^0$ are also introduced as $Z_2$-odd scalars in order to accommodate the DM candidate. Then, the $\mu$-term is generated at the one-loop level by the diagram in which the $Z_2$-odd scalars are in the loop. By this mechanism, the smallness of $v_\Delta \ll v$ is realized, and the tiny neutrino masses are naturally explained without assuming the triplet fields to be heavy. The Yukawa sector is then the same as the one in the HTM, so that its predictions for the LFV processes are not changed. See Refs. [@Babu:2001ex; @Chun:2003ej] for some discussions about two-loop realization of the $\mu$-term[^1]. This paper is organized as follows. In Sec. \[sec:HTM\], we give a quick review for the HTM to define notation. In Sec. \[sec:1-loop\], the model for radiatively generating the $\mu$ parameter with the dark matter candidate is presented. Some phenomenological implications are discussed in Sec. \[sec:pheno\], and the conclusion is given in Sec. \[sec:concl\]. The full expressions of the Higgs potential and mass formulae for scalar bosons in our model are given in Appendix. Higgs Triplet Model {#sec:HTM} =================== In the HTM, an $\text{SU}(2)_L$ triplet of complex scalar fields with hyperchage $Y=1$ is introduced to the SM. The triplet $\Delta$ can be expressed as $$\begin{aligned} \Delta &=& \begin{pmatrix} \Delta^+/\sqrt{2} & \Delta^{++}\\ \Delta^0 & -\Delta^+/\sqrt{2} \end{pmatrix} ,\end{aligned}$$ where $\Delta^0 = (\Delta^0_r + i \Delta^0_i)/\sqrt{2}$. The triplet has a new Yukawa interaction term with leptons as $$\begin{aligned} {\mathcal L}_{\text{triplet-Yukawa}} &=& h_{\ell{{\ell^\prime}}}\, \overline{L_\ell^c}\, i\sigma_2\, \Delta\, L_{{\ell^\prime}}+ \text{h.c.} ,\end{aligned}$$ where $h_{\ell{{\ell^\prime}}}$ ($\ell, {{\ell^\prime}}= e, \mu, \tau$) are the new Yukawa coupling constants, $L_\ell$ \[$= (\nu_{\ell L}, \ell)^T$\] are lepton doublet fields, a superscript $c$ means the charge conjugation, and $\sigma_i$ ($i = 1\text{-}3$) denote the Pauli matrices. Lepton number ($L\#$) of $\Delta$ is assigned to be $-2$ as a convention such that the Yukawa term does not break the conservation. A vacuum expectation value $v_\Delta^{}$ \[$=\sqrt{2}\,\langle\Delta^0\rangle$\] breaks lepton number conservation by two units. The new Yukawa interaction then yields the Majorana neutrino mass term $(m_\nu)_{\ell{{\ell^\prime}}}\,\overline{(\nu_{\ell L}^{})^c}\,\nu_{{{\ell^\prime}}L}^{}/2$ where $(m_\nu)_{\ell{{\ell^\prime}}} = \sqrt{2}\, v_\Delta\, h_{\ell{{\ell^\prime}}}$. The scalar potential in the HTM can be written as $$\begin{aligned} V_{{\text{HTM}}}&=& -m_\Phi^2\, \Phi^\dagger \Phi + m_\Delta^2 {{\text{tr}}}(\Delta^\dagger \Delta) + \left\{ \mu\, \Phi^T i\sigma_2 \Delta^\dagger \Phi + \text{h.c.} \right\} \nonumber\\ &&{} + \lambda_1 (\Phi^\dagger \Phi)^2 + \lambda_2 \left[ {{\text{tr}}}(\Delta^\dagger \Delta) \right]^2 + \lambda_3\, {{\text{tr}}}[(\Delta^\dagger \Delta)^2] \nonumber\\ &&{} + \lambda_4\, (\Phi^\dagger \Phi) {{\text{tr}}}(\Delta^\dagger \Delta) + \lambda_5\, \Phi^\dagger \Delta \Delta^\dagger \Phi ,\end{aligned}$$ where $\Phi=(\phi^+, \phi^0)^T$ \[$\phi^0=(\phi^0_r + i\phi^0_i)/\sqrt{2}$\] is the Higgs doublet field in the SM. The $\mu$ parameter can be real by using rephasing of $\Delta$. Because we take $m_\Delta^2 > 0$, there is no Nambu-Goldstone boson for spontaneous breaking of lepton number conservation. The small triplet VEV $v_\Delta^{}$ is generated by an explicit breaking parameter $\mu$ of the lepton number conservation as $$\begin{aligned} v_\Delta^{} &\simeq& \frac{ \sqrt{2}\, \mu v^2 } { 2m_\Delta^2 + (\lambda_4+\lambda_5) v^2 } ,\end{aligned}$$ where $v$ ($\simeq 246\,{{\text{GeV}}}$) is the doublet VEV defined by $v=\sqrt{2}\,\langle\phi^0\rangle$. In order to obtain small neutrino masses in the HTM, at least one of $v^2/m_\Delta^2$, $h_{\ell{{\ell^\prime}}}$, $\mu/v$ should be tiny. A small $\mu$ is an attractive option because $m_\Delta^{}$ can be small ($\lesssim 1\,{{\text{TeV}}}$) so that triplet scalars can be produced at the LHC. Furthermore, large $h_{\ell{{\ell^\prime}}}$ can be taken, which have direct information on the flavor structure of $(m_\nu)_{\ell{{\ell^\prime}}}$. There is, however, no reason why the $\mu$ parameter is tiny in the HTM. In our model presented below, the $\mu$ parameter is naturally small because it arises at the one-loop level. An extension of the Higgs Triplet Model {#sec:1-loop} ======================================= Since we try to generate the $\mu$-term in the HTM radiatively, the term must be forbidden at the tree level. The simplest way would be to impose lepton number conservation to the Lagrangian. The conservation is assumed to be broken by the VEV of a new scalar field $s_1^0$ which is singlet under the SM gauge symmetry. Notice that $s_1^0$ \[$= (s_{1r}^0 + i s_{1i}^0)/\sqrt{2}$\] is a complex (“charged”) field with non-zero lepton number although it is electrically neutral. One might think that the VEV of $s_1^0$ could be generated by using soft breaking terms of $L\#$. However, the $\mu$-term is also a soft breaking term. Therefore lepton number must be broken spontaneously in our scenario. One may worry about Nambu-Goldstone boson corresponds to the spontaneous breaking of the lepton number conservation (the so-called Majoron, $J^0$). However the Majoron which comes from gauge singlet field can evade experimental searches (constraints) because it interacts very weakly with matter fields [@majoron]. It is also possible to make it absorbed by a gauge boson by introducing the $\text{U}(1)_{{\text{B}-\text{L}}}$ gauge symmetry to the model (See, e.g., Ref. [@BL]). In this paper we just accept the Majoron without assuming the $\text{U}(1)_{{\text{B}-\text{L}}}$ gauge symmetry for simplicity. If $s_1^0$ has $L\# = -2$, we can have a dimension-4 operator $\lambda\, s_1^0\, \Phi^T i\sigma_2 \Delta^\dagger \Phi$. This gives a trivial result $\mu = \lambda \langle s_1^0 \rangle$ at the tree level. Although the dim.-4 operator could be forbidden by some extra global symmetries with extra scalars to break them, we do not take such a possibility in this paper. We just assume the $s_1^0$ has $L\#=-1$. Then the lepton number conserving operator which results in the $\mu$-term is of dimension-5 as $$\begin{aligned} (s_1^0)^2 \Phi^T i\sigma_2 \Delta^\dagger \Phi . \label{eq:dim5op}\end{aligned}$$ We consider below how to obtain the dim.-5 operator at the loop level by using renormalizable interactions[^2]. We restrict ourselves to extend only the $\text{SU}(3)_c$-singlet scalar sector in the HTM because it seems a kind of beauty that the HTM does not extend the fermion sector and colored sector in the SM. An unbroken $Z_2$ symmetry is introduced in order to obtain dark matter candidates, and new scalars which appear in the loop diagram for the $\mu$-term are aligned to be $Z_2$-odd particles. We emphasize that the unbroken $A_2$ symmetry is not for a single purpose to introduce dark matter candidates but utilized also for our radiative mechanism for the $\mu$-term.  $L$  $\Phi$  $\Delta$  $s_1^0$  $s_2^0$  $\eta$ ------------------- ----------- ----------- ----------- ----------- ----------- -----------  $\text{SU}(2)_L$  [****]{}  [****]{}  [****]{}  [****]{}  [****]{}  [****]{}  $\text{U}(1)_Y$  $1/2$  $1/2$  $1$  $0$  $0$  $1/2$  $L\#$  $1$  $0$  $-2$  $-1$  $0$  $-1$  $Z_2$  $+$  $+$  $+$  $+$  $-$  $-$ : List of particle contents of our one-loop model. []{data-label="tab:1-loop"} We present the minimal model where the dim.-5 operator in eq. (\[eq:dim5op\]) is generated by a one-loop diagram with dark matter candidates. Table \[tab:1-loop\] shows the particle contents. A real singlet scalar field $s_2^0$ and the second doublet scalar field $\eta$ \[$=(\eta^+, \eta^0)^T, \eta^0=(\eta^0_r+i\eta^0_i)/\sqrt{2}$\] are introduced to the HTM in addition to $s_1^0$. Lepton numbers of $s_2^0$ and $\eta$ are 0 and $-1$, respectively. Then $\eta^T i\sigma_2 \Delta^\dagger \eta$ conserves lepton number. In order to forbid the VEV of $\eta$, we introduce an unbroken $Z_2$ symmetry for which $s_2^0$ and $\eta$ are odd. Other fields are even under the $Z_2$. The Yukawa interactions are the same as those in the HTM. The Higgs potential is given as $$\begin{aligned} V &=& \frac{1}{\,2\,} m_{s_2^0}^2 (s_2^0)^2 + \left\{ \mu_\eta^{}\, \eta^T\, i\sigma_2\, \Delta^\dagger\, \eta + \text{h.c.} \right\} + \left\{ \lambda_{s\Phi\eta}\, s_1^0\, s_2^0\, (\eta^\dagger\, \Phi) + \text{h.c.} \right\} + \cdots .\end{aligned}$$ Here we show only relevant parts for radiative generation of the $\mu$-term. See Appendix for the other terms. Vacuum expectation values $v$ and $v_s$ \[$= \sqrt{2}\,\langle s_1^0 \rangle$\] are given by $$\begin{aligned} \begin{pmatrix} v^2\\ v_s^2 \end{pmatrix} = \frac{2}{ 4 \lambda_{1\Phi} \lambda_{s1} - \lambda_{s\Phi 1}^2 } \begin{pmatrix} 2 \lambda_{s1} & -\lambda_{s\Phi 1}\\ -\lambda_{s\Phi 1} & 2 \lambda_{1\Phi} \end{pmatrix} \begin{pmatrix} m_\Phi^2\\ m_{s_1}^2 \end{pmatrix} .\end{aligned}$$ The $Z_2$-odd scalars in this model are two CP-even neutral ones (${\mathcal H}_1^0$ and ${\mathcal H}_2^0$), a CP-odd neutral one (${\mathcal A}^0 = \eta_i^0$), and a charged pair (${\mathcal H}^\pm = \eta^\pm$). The CP-even scalars are defined as $$\begin{aligned} \begin{pmatrix} {\mathcal H}_1^0\\ {\mathcal H}_2^0 \end{pmatrix} = \begin{pmatrix} \cos\theta_0^\prime & -\sin\theta_0^\prime\\ \sin\theta_0^\prime & \cos\theta_0^\prime \end{pmatrix} \begin{pmatrix} \eta_r^0\\ s_2^0 \end{pmatrix} , \quad \tan{2\theta_0^\prime} = \frac{ \sqrt{2}\, \lambda_{s\Phi\eta}\, v\, v_s } { ({\mathcal M}_0)_{ss}^2 - ({\mathcal M}_0)_{\eta\eta}^2 } ,\end{aligned}$$ where $({\mathcal M}_0)_{\eta\eta}^2 \equiv m_\eta^2 + ( \lambda_{1\Phi\Phi} + \lambda_{1\Phi\eta} )\, v^2/2 + \lambda_{s\eta 1}\, v_s^2/2$ and $({\mathcal M}_0)_{ss}^2 \equiv m_{s_2^0}^2 + \lambda_{s3}\, v_s^2 + \lambda_{s\Phi 2}\, v^2$. Squared masses of these scalars are given by $$\begin{aligned} m_{{\mathcal H}_1^0}^2 &=& \frac{1}{\,2\,} \left\{ ({\mathcal M}_0)_{\eta\eta}^2 + ({\mathcal M}_0)_{ss}^2 - \sqrt{ \bigl\{ ({\mathcal M}_0)_{\eta\eta}^2 - ({\mathcal M}_0)_{ss}^2 \bigr\}^2 + 2\, \lambda_{s\Phi\eta}^2\, v^2\, v_s^2 \ } \right\} , \label{eq:mH1}\\ m_{{\mathcal H}_2^0}^2 &=& \frac{1}{\,2\,} \left\{ ({\mathcal M}_0)_{\eta\eta}^2 + ({\mathcal M}_0)_{ss}^2 + \sqrt{ \bigl\{ ({\mathcal M}_0)_{\eta\eta}^2 - ({\mathcal M}_0)_{ss}^2 \bigr\}^2 + 2\, \lambda_{s\Phi\eta}^2\, v^2\, v_s^2 \ } \right\} , \label{eq:mH2}\\ m_{{\mathcal A}^0}^2 &=& ({\mathcal M}_0)_{\eta\eta}^2 , \label{eq:mA}\\ m_{{\mathcal H}^\pm}^2 &=& ({\mathcal M}_0)_{\eta\eta}^2 - \frac{1}{\,2\,} \lambda_{1\Phi\eta}\, v^2 . \label{eq:mHpm}\end{aligned}$$ Notice that $m_{{\mathcal H}_1^0} \leq m_{{\mathcal A}^0} \leq m_{{\mathcal H}_2^0}$. We assume $m_{{\mathcal H}_1^0} < m_{{\mathcal H}^\pm}$ and then ${\mathcal H}_1^0$ becomes the dark matter candidate. Hereafter it is assumed that the mixing $\theta_0^\prime$ is small. The $\mu$-term is generated by the one-loop diagram. Figure \[fig:1-loop\] is the dominant one in the case of small $\theta_0^\prime$. Then, the parameter $\mu$ is calculated as $$\begin{aligned} \mu &=& \frac{ \lambda_{s\Phi\eta}^2\, \mu_\eta^{} v_s^2 } { 64\pi^2 \bigl\{ ({\mathcal M}_0)_{ss}^2 - ({\mathcal M}_0)_{\eta\eta}^2 \bigr\} } \left\{ 1 - \frac{ ({\mathcal M}_0)_{ss}^2 } { ({\mathcal M}_0)_{ss}^2 - ({\mathcal M}_0)_{\eta\eta}^2 } \ln\frac{({\mathcal M}_0)_{ss}^2}{({\mathcal M}_0)_{\eta\eta}^2} \right\} . \label{eq:mu-loop}\end{aligned}$$ The one-loop induced $\mu$ parameter can be expected to be much smaller than $\mu_\eta$. The suppression factor $|\mu/\mu_\eta|$ is estimated in Sec. \[subsec:DM\]. ![ One-loop diagram for the $\mu$-term. We call it “A. oryzae diagram” [@moyasimon]. []{data-label="fig:1-loop"}](1-loop.eps "fig:") ![ One-loop diagram for the $\mu$-term. We call it “A. oryzae diagram” [@moyasimon]. []{data-label="fig:1-loop"}](a-oryzae.eps "fig:") phenomenology {#sec:pheno} ============= Dark matter {#subsec:DM} ----------- If $({\mathcal M}_0)_{\eta\eta} < ({\mathcal M}_0)_{ss}$, the dark matter candidate ${\mathcal H}_1^0$ is given by $\eta_r^0$ approximately because we assume small mixing. See, e.g., Ref. [@i-doublet] for studies about the inert doublet scalar. Let us assume $m_{{\mathcal H}_1^0}^{}\simeq 75\,{{\text{GeV}}}$ and $m_{{\mathcal A}^0}^{} \gtrsim 125\,{{\text{GeV}}}$. As shown in Ref. [@Lundstrom:2008ai], these values satisfy constraints from the LEP experiments [@:2005ema; @EspiritoSanto:2003by] and the WMAP experiment [@Komatsu:2008hk]. The mass splitting ($m_{{\mathcal A}^0}^{} - m_{{\mathcal H}_1^0}^{} \gtrsim 50\,{{\text{GeV}}}$) suppresses quasi-elastic scattering on nuclei (${\mathcal H}_1^0 N \to {\mathcal A}^0 N$ mediated by the $Z$ boson) enough to satisfy constraints from direct search experiments of the DM [@Aprile:2011hi]. By using eqs.  and , we obtain $$\begin{aligned} \frac{ \lambda_{s\Phi\eta}^2 v_s^2 }{ ({\mathcal M}_0)_{ss}^2 } \simeq \frac{ 2}{ v^2 } \left( m_{{\mathcal A}^0}^2 - m_{{\mathcal H}_1^0}^2 \right) \gtrsim 0.3 . \label{eq:lambda}\end{aligned}$$ In order to be consistent with our assumption of small $\theta_0^\prime$ (e.g., $\simeq 0.1$), $({\mathcal M}_0)_{ss} \gtrsim 3\,{{\text{TeV}}}$ is required. The value in eq.  results in $$\begin{aligned} \frac{\mu}{\mu_\eta} \gtrsim 10^{-4} . \label{eq:mu-50GeV}\end{aligned}$$ For the greater value of $m_{{\mathcal A}^0}^{}$, the larger $\mu/\mu_\eta$ is predicted. In particular, by taking $m_{{\mathcal A}^0}^{}$ to be the TeV scale, we obtain $\mu/\mu_\eta \sim 10^{-2}$, which yields $v_\Delta^{} \sim 1\,{{\text{GeV}}}$ for $\mu_\eta^{}$ and $m_\Delta^{}$ to be at the electroweak scale. Such a value for $v_\Delta^{}$ is suggested in the recent study of radiative corrections to the electroweak parameters [@Kanemura:2012rs]. On the contrary, if we take $m_{{\mathcal A}^0}^{} \simeq 83\,{{\text{GeV}}}$ which is allowed in a tiny region [@Lundstrom:2008ai], values in eqs.  and become 10 times smaller. We mention that the WMAP constraint might be changed by a characteristic annihilation process ${\mathcal H}_1^0 {\mathcal H}_1^0 \to \Delta_r^0 \to \overline{\nu}\,\overline{\nu}$ where ${\mathcal H}_1^0 {\mathcal H}_1^0 (\Delta_r^0)^\ast$ interaction is governed by $\mu_\eta^{}$ (not by a tiny $\mu$). This additional process could sift allowed value of $m_{{\mathcal H}_1^0}^{}$ to lower one while $m_{{\mathcal A}^0}^{} \gtrsim 100\,{{\text{GeV}}}$ due to the LEP constraint. Then, $\mu/\mu_\eta$ might become larger than the value in eq.  because of larger $m_{{\mathcal A}^0}^{} - m_{{\mathcal H}_1^0}^{}$. This undesired effect would be easily avoided if $m_{\Delta_r^0}^{}$ is away enough from $2 m_{{\mathcal H}_1^0}^{}$. On the other hand, ${\mathcal H}_1^0$ comes dominantly from $s_2^0$ if $({\mathcal M}_0)_{\eta\eta} > ({\mathcal M}_0)_{ss}$. See, e.g., Ref. [@r-i-singlet] for studies about the real inert singlet scalar. Coupling $\sqrt{2}\, \lambda_{s\Phi 1}\, v$ of the ${\mathcal H}_1^0 {\mathcal H}_1^0 h^0$ interaction ($h^0$ is the SM Higgs boson) determines annihilation cross section of ${\mathcal H}_1^0$ and scattering cross section on nuclei. If we introduce the $\text{U}(1)_{{\text{B}-\text{L}}}$ gauge symmetry, the scattering of $s_2^0$ on nuclei can be mediated also by the gauge boson $Z^\prime$. Notice that the parameter $\lambda_{s\Phi 1}$ (and also the $\text{U}(1)_{{\text{B}-\text{L}}}$ gauge coupling constant) does not affect on $\mu$ parameter in eq. . Let us estimate the magnitude of $\mu/\mu_\eta$. In the usual HTM, $h_{\ell{{\ell^\prime}}}$ is expected to be $\lesssim 10^{-2}$ for $m_{H^{\pm\pm}} \sim 100\,{{\text{GeV}}}$ in order to suppress LFV processes. Thus, we may accept $\lambda_{s\Phi\eta} \sim 1 \text{-} 10^{-2}$ as a value which is not too small. Assuming $({\mathcal M}_0)_{ss} \ll ({\mathcal M}_0)_{\eta\eta} \sim v_s \sim 1\,{{\text{TeV}}}$ for example[^3], we have a suppression factor as $$\begin{aligned} \left| \frac{\mu}{\mu_\eta} \right| \sim \frac{ \lambda_{s\Phi\eta}^2 v_s^2 } { 64 \pi^2 ({\mathcal M}_0)_{\eta\eta}^2 } \sim 10^{-3} \text{-} 10^{-7} .\end{aligned}$$ Thus, even if the value of $\mu_\eta$ is in the TeV scale, we can obtain $\mu \sim 0.1\,{{\text{MeV}}}$ although we need further suppression with $h_{\ell{{\ell^\prime}}} \lesssim 10^{-5}$ to have $m_\nu \lesssim 1\,{{\text{eV}}}$. If we use $h_{\ell{{\ell^\prime}}} \sim \lambda_{s\Phi\eta} \sim 10^{-3}$, we obtain $|\mu/\mu_\eta| h_{\ell{{\ell^\prime}}} \sim 10^{-12}$ which can connect the TeV scale $\mu_\eta$ to the eV scale $m_\nu$. Collider {#subsec:col} -------- ![ The unique process in our model for ${\mathcal H}_1^0 \simeq \eta^0_r$. The bosonic decay of $H^{+}$ contains information of $\mu_\eta$ indicated by a red blob. []{data-label="fig:LHC"}](loop_muterm_LHC.eps) The characteristic feature of our model is that $\mu_\eta$ is much larger than $\mu$. Let us consider possibility to probe the large $\mu_\eta$ in collider experiments. A favorable process is shown in Fig. \[fig:LHC\] for ${\mathcal H}_1^0 \simeq \eta^0_r$. For simplicity, we take $\lambda_5=0$ which results in $m_{H^{\pm\pm}}^{} \simeq m_{H^\pm}^{} \simeq m_{H^0,A^0}^{}$. Recently, it was found in Ref. [@Kanemura:2012rs] that the electroweak precision test prefers $\lambda_5 > 0$ in the HTM where the electroweak sector is described by four input parameters. However, results in Ref. [@Kanemura:2012rs] might not be applied directly to our model[^4] because the scalar sector is extended. Since $H^{\pm\pm} \to \ell^\pm {\ell^\prime}^\pm$ is the most interesting decay in the HTM, we assume $2 m_{{\mathcal H}^\pm}^{} > m_{H^{\pm\pm}}^{}$ in order to forbid $H^{\pm\pm} \to {\mathcal H}^\pm {\mathcal H}^\pm$. Even in this case, the DM ${\mathcal H}_1^0$ can be light enough ($m_{H^\pm}^{} > m_{{\mathcal H}^\pm}^{} + m_{{\mathcal H}_1^0}^{}$) so that $Z_2$-even charged scalar $H^\pm$ ($\simeq \Delta^\pm$) can decay into ${\mathcal H}^\pm {\mathcal H}_1^0$ via $\mu_\eta$-term which is indicated by a red blob in Fig. \[fig:LHC\]. The partial decay width of $H^\pm \to {\mathcal H}^\pm {\mathcal H}_1^0$ is determined by $(\mu_\eta/\mu)^2 v_\Delta^2/m_{H^\pm}^{}$ while the width of $H^\pm \to \ell^\pm \nu$ is proportional to $m_{H^\pm}^{} m_\nu^2/v_\Delta^2$. Taking $\mu_\eta/\mu \sim 10^4$, $v_\Delta \sim 10\,{{\text{keV}}}$, $m_{H^\pm}^{} \sim 100\,{{\text{GeV}}}$, and $m_\nu^{} \sim 0.1\,{{\text{eV}}}$ for example, we have $(\mu_\eta/\mu)^2 v_\Delta^2/m_{H^\pm}^{} \sim 10^5\,{{\text{eV}}}$ and $m_{H^\pm}^{} m_\nu^2/v_\Delta^2 \sim 10\,{{\text{eV}}}$. Then, $H^\pm$ dominantly decays into ${\mathcal H}^\pm {\mathcal H}_1^0$. Finally, ${\mathcal H}^\pm$ decays into $(W^\pm)^\ast {\mathcal H}_1^0$. Therefore, from a production mechanism $pp \to (W^\pm)^\ast \to H^{\pm\pm} H^\mp$, we would have $\ell\ell j j \cancel{E}_T$ as a final state[^5] for which $\ell\ell$ has the invariant mass $m(\ell\ell)$ at $m_{H^{\pm\pm}}^{}$ assuming that the value of $m_{H^{\pm\pm}}^{}$ has been known already. If ${\mathcal H}_1^0 \simeq s_2^0$, then $H^\pm$ decays via $\mu_\eta$-term into ${\mathcal H}^\pm {\mathcal A}^0$ or ${\mathcal H}^\pm {\mathcal H}_2^0$ followed by ${\mathcal H}_2^0 \to {\mathcal A}^0 J^0$ where a sizable $\lambda_{s\eta 1}$ is assumed[^6]. Because of ${\mathcal A}^0 \to {\mathcal H}_1^0 J^0$ through $\lambda_{s\Phi\eta}$, we have again $\ell\ell j j \cancel{E}_T$ with $m(\ell\ell) = m_{H^{\pm\pm}}^{}$ from $pp \to (W^\pm)^\ast \to H^{\pm\pm} H^\mp$. In the usual HTM in contrast, the final state with such $\ell\ell$ is likely to include additional charged leptons ($\ell\ell\ell\ell$ from $H^{++} H^{--}$, $\ell\ell\ell\cancel{E}_T$ from $H^{\pm\pm} H^\mp$, etc.) if $H^{\pm\pm}$ decay dominantly into $\ell^\pm\ell^{\prime\pm}$. Therefore, our model would be supported if experiments observe final states which include jets and only two $\ell$ whose invariant mass gives $m(\ell\ell) = m_{H^{\pm\pm}}^{}$. This potential signature might be disturbed by hadronic decays of $\tau$ because $H^{++}H^{--} \to \ell\ell\tau\tau$ can result in $\ell\ell j j \cancel{E}_T$ with $m(\ell\ell) = m_{H^{\pm\pm}}^{}$. Realistic simulation is necessary to see the feasibility. conclusions and discussion {#sec:concl} ========================== We have presented the simple extension of the HTM by introducing a $Z_2$-even neutral scalar $s_1^0$ of $L\#=-1$, a $Z_2$-odd neutral real scalar $s_2^0$ of $L\#=0$, and a $Z_2$-odd doublet scalar field $\eta$ of $L\#=-1$. The DM candidate ${\mathcal H}^0_1$ in our model is made from $s_2^0$ and $\eta_r$. The $\mu \Phi^T i\sigma_2 \Delta^\dagger \Phi$ interaction which is the origin of $v_\Delta$ (and neutrino masses) is induced at the one-loop level while the $\mu_\eta^{} \eta^T i\sigma_2 \Delta^\dagger \eta$ interaction exists at the tree level. Because of the loop suppression for $\mu$ parameter, the model gives small neutrino masses naturally without using very heavy particles. For ${\mathcal H}^0_1 \simeq \eta^0_r$, the suppression factor $|\mu/\mu_\eta^{}|$ is constrained by the DM relic abundance measured by the WMAP experiment. We have shown that $|\mu/\mu_\eta^{}| \sim 10^{-4}\,\text{-}\,10^{-5}$ is possible. On the other hand, for ${\mathcal H}^0_1 \simeq s_2^0$, the suppression factor is somewhat free from experimental constraints on the DM. In our estimate, $|\mu/\mu_\eta^{}| \sim 10^{-3} \text{-} 10^{-7}$ can be obtained as an example with $\lambda_{s\Phi\eta} \sim 1 \text{-} 10^{-2}$. The characteristic feature of the model is that $\mu_\eta$ is not small while $\mu$ can be small. A possible collider signature which depends on $\mu_\eta$ would be $\ell\ell j j \cancel{E}_T$ with the invariant mass $m(\ell\ell) = m_{H^{\pm\pm}}^{}$ because more charged leptons are likely to exist in such final states in the usual HTM. The work of S.K. was supported by Grant-in-Aid for Scientific Research Nos. 22244031 and 23104006. The work of H.S. was supported by the Sasakawa Scientific Research Grant from the Japan Science Society and Grant-in-Aid for Young Scientists (B) No. 23740210. appendix {#appendix .unnumbered} ======== The Higgs potential of our model is given by $V=V_2+V_3+V_4$ where $$\begin{aligned} V_2 &\equiv& - m_{s_1}^2 |s_1^0|^2 + \frac{1}{\,2\,} m_{s_2}^2 (s_2^0)^2 - m_\Phi^2\, \Phi^\dagger \Phi + m_\eta^2\, \eta^\dagger \eta + m_\Delta^2\, {{\text{tr}}}(\Delta^\dagger \Delta) ,\end{aligned}$$ $$\begin{aligned} V_3 &\equiv& ( \mu_\eta^{}\, \eta^T\, i\sigma_2\, \Delta^\dagger\, \eta) + \text{h.c.} ,\end{aligned}$$ $$\begin{aligned} V_4 &\equiv& \lambda_{1\Phi}\, (\Phi^\dagger \Phi)^2 + \lambda_{1\eta}\, (\eta^\dagger \eta)^2 + \lambda_{1\Phi\Phi}\, (\Phi^\dagger \Phi) (\eta^\dagger \eta) + \lambda_{1\Phi\eta}\, (\Phi^\dagger \eta) (\eta^\dagger \Phi) \nonumber\\ &&{} + \lambda_2\, [{{\text{tr}}}(\Delta^\dagger \Delta)]^2 + \lambda_3\, {{\text{tr}}}[(\Delta^\dagger \Delta)^2] \nonumber\\ &&{} + \lambda_{4\Phi}\, (\Phi^\dagger \Phi)\, {{\text{tr}}}(\Delta^\dagger \Delta) + \lambda_{4\eta}\, (\eta^\dagger \eta)\, {{\text{tr}}}(\Delta^\dagger \Delta) \nonumber\\ &&{} + \lambda_{5\Phi}\, (\Phi^\dagger \Delta \Delta^\dagger \Phi) + \lambda_{5\eta}\, (\eta^\dagger \Delta \Delta^\dagger \eta) \nonumber\\ &&{} + \lambda_{s1}\, |s_1^0|^4 + \lambda_{s2}\, (s_2^0)^4 + \lambda_{s3}\, |s_1^0|^2 (s_2^0)^2 \nonumber\\ &&{} + \lambda_{s\Phi 1}\, |s_1^0|^2\, (\Phi^\dagger \Phi) + \lambda_{s\Phi 2}\, (s_2^0)^2\, (\Phi^\dagger \Phi) \nonumber\\ &&{} + \lambda_{s\eta 1}\, |s_1^0|^2\, (\eta^\dagger \eta) + \lambda_{s\eta 2}\, (s_2^0)^2\, (\eta^\dagger \eta) + \left\{ \lambda_{s\Phi\eta}\, s_1^0\, s_2^0\, (\eta^\dagger \Phi) + \text{h.c.} \right\} \nonumber\\ &&{} + \lambda_{s\Delta 1}\, |s_1^0|^2 {{\text{tr}}}(\Delta^\dagger \Delta) + \lambda_{s\Delta 2}\, (s_2^0)^2 {{\text{tr}}}(\Delta^\dagger \Delta) .\end{aligned}$$ All coupling constants are real because the phases of $\mu_\eta$ and $\lambda_{s\Phi\eta}$ can be absorbed by $\Delta$ and $s_1^0$, respectively. Mass eigenstates of two $Z_2$-even CP-even neutral scalars which are composed of $s_{1r}^0$ and $\phi_r^0$ are obtained as $$\begin{aligned} \begin{pmatrix} h^0\\ H^0 \end{pmatrix} = \begin{pmatrix} \cos\theta_0 & -\sin\theta_0\\ \sin\theta_0 & \cos\theta_0 \end{pmatrix} \begin{pmatrix} \phi^0_r\\ s_{1r}^0 \end{pmatrix} , \quad \tan{2\theta_0} = \frac{ \lambda_{s\Phi 1}\, v\, v_s } { \lambda_{s1} v_s^2 - \lambda_{1\Phi} v^2 } .\end{aligned}$$ Their masses eigenvalues are given by $$\begin{aligned} m_{h^0}^2 &\simeq& \lambda_{1\Phi} v^2 + \lambda_{s1} v_s^2 - \sqrt{ \left( \lambda_{1\Phi} v^2 - \lambda_{s1} v_s^2 \right)^2 + \lambda_{s\Phi 1}^2 v^2\, v_s^2 } \ ,\\ m_{H^0}^2 &\simeq& \lambda_{1\Phi} v^2 + \lambda_{s1} v_s^2 + \sqrt{ \left( \lambda_{1\Phi} v^2 - \lambda_{s1} v_s^2 \right)^2 + \lambda_{s\Phi 1}^2 v^2\, v_s^2 } \ ,\end{aligned}$$ where small contributions from $v_\Delta^{}$ are neglected. 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--- abstract: 'We demonstrate rotational and vibrational cooling of cesium dimers by optical pumping techniques. We use two laser sources exciting all the populated rovibrational states, except a target state that thus behaves like a dark state where molecules pile up thanks to absorption-spontaneous emission cycles. We are able to accumulate photoassociated cold Cs$_{2}$ molecules in their absolute ground state ($v=0,J=0$) with up to $40\%$ efficiency. Given its simplicity, the method could be extended to other molecules and molecular beams. It also opens up general perspectives in laser cooling the external degrees of freedom of molecules.' author: - 'I. Manai' - 'R. Horchani' - 'A. Fioretti' - 'M. Allegrini' - 'H. Lignier' - 'P. Pillet' - 'D. Comparat' title: Rovibrational cooling of molecules by optical pumping --- It is commonly admitted that optical manipulation of molecules comes up against strong limitations. The difficulty originates from the large number of internal states accessible by spontaneous emission events [@2008_PRL_Ye_direct_cooling]. However, a few recent experiments succeeded to implement molecular optical pumping: for example, the vibrational cooling of Cs$_2$ [@MatthieuViteau07112008], the optical cooling of SrF [@shuman2010laser], or the rotational cooling of molecular ions [@2010NatPhDrewsen_rotational_cooling]. These methods, like the one we present in this article, are fundamentally different from coherent optical manipulations, such as STIRAP, that affect the population of a single quantum state [@JohannDanzl07102008; @2010JinPRL_HyperfinePreparation]. Optical cooling of both the vibration and the rotation of Cs$_2$ molecules is complicated because these two degrees of freedom can not be manipulated independently. The pumping of one of them tends to impair the other one, i.e. the vibrational pumping is likely to modify the rotational quantum number, just as the rotational pumping is likely to modify the vibrational quantum number. As a consequence, a global rovibrational cooling can only be achieved through an interplay between both processes. Our vibrational pumping, already demonstrated in [@MatthieuViteau07112008; @2009_JMOp_Cooling; @2009NJPh...11e5037S; @2009PhRvA..80e1401S; @2012_PRA_Horchani_Conversion], makes use of a broadband laser whose spectrum has been specifically shaped to excite all the vibrational levels but one where molecules accumulate. The frequency resolution is limited to $\sim 0.1$ cm$^{-1}$. For rovibrational pumping, a requirement is the control of the light spectrum with a resolution on the order of magnitude of the rotational constants [@2009_JMOp_Cooling; @2010MolPh.108..795S]. As the Cs$_2$ molecules have a rotational constant of about $0.01$ cm$^{-1}$, which is exceptionally small compared with the great majority of diatomic molecules, we have designed a method improving the spectral resolution of molecular optical pumping with respect to the grating technique. It consists in scanning a narrow-band laser diode in appropriate regions of the rotational spectrum, which allows one to modify the population distribution among the rotational levels characterized by their $J$ quantum numbers. Our experimental setup is based on a caesium magneto-optical trap (MOT). All the manipulations achieved on this source last a total of $100$ ms. During the first 20 ms, a cw Ti:Sa laser is focused on the MOT to form molecules in the electronic ground state by photoassociation (PA). The PA scheme, detailed in Ref. [@2011PCCP...1318910L], consists in exciting a $0^-_g (6s + 6p_{1/2})(v'=26,J')$ rovibrational level, where $v'$ is the vibrational quantum number counted from the dissociation limit, and $J'$ the rotational quantum number. The rotational and vibrational cooling lasers are switched on at the same time as the PA laser, but are respectively applied for $25$ ms and $28$ ms. The vibrational populations are then probed by converting molecules into ions via a resonant enhanced 2 photon ionization scheme (RE2PI). For this purpose, a pulsed dye laser is turned on $1$ ms after switching off the vibrational cooling laser. It allows us to scan the vibrational transitions between the X$^1\Upsigma_{\rm g}^{+}$ state and the C$^1\Uppi_u$ or D$^1\Upsigma^+_u$ intermediate states [@2011PCCP...1318910L]. Cs$_2^+$ ions are finally detected by micro-channel plates. To access the rotational distributions, we insert a 50 $\mu$s pulse of a narrow band laser $0.5$ ms before the application of the ionizing pulsed laser. . Vibrational cooling is performed according to Ref. [@MatthieuViteau07112008], i.e. with a femtosecond laser ($200$ mW final power, $120$ fs pulse duration, $1$ mm$^2$ beam size) tuned to rovibrational transitions between the ground state and the B$^1\Uppi_u$ excited state as shown in Fig. \[fig:figure1\]. This allows us to accumulate molecules in $v=0$. It is conceivable to extend the shaping technique of the femtosecond laser to rotational cooling provided that the molecular species under study has larger rotational constants [@2011PCCP...1318825L; @2009_JMOp_Cooling] than the laser shaping resolution. We stress the fact that this requirement is not fulfilled in the case of Cs$_2$. To manipulate the rotational populations of Cs$_2$, we employ a cw DFB diode laser ($\sim 2$ MHz linewidth, $\sim 6$ mW power, $3.5$ mm$^2$ beam size) scanned across the B$^{1}\Uppi_{\rm u}(v_B=3,J_B)\leftarrow $X$^{1}\Upsigma_{\rm g}^{+}(v=0,J)$ transitions though the diode current (see right part of Fig. \[fig:figure1\]). The choice of these transitions originates from the wavelength accessibility of the diode laser. We can explore the $P$, $Q$, or $R$ rotational branches, schematically shown in Fig. \[fig:figure3\], and thus change the $J$ value as proposed in [@1996_JCP_BahnsLaserCooling]. The trend in the evolution of $J$ is governed by the branch selected at the excitation. The rotational spectroscopy of the population in $v=0$ cannot be achieved with the pulsed dye laser alone. Its resolution ($0.1$ cm$^{-1}$) is not good enough to probe the rotational levels. Therefore, we use an additional narrow band cw DFB laser scanning the B$^{1}\Uppi_{\rm u}(v_B=3,J_B)\leftarrow$X$^{1}\Upsigma_{\rm g}^{+}(v=0,J)$ transitions. After excitation and spontaneous emission, the rovibrational populations in $v=0$ are transferred to several vibrational levels, and mostly to those having a good Franck-Condon (FC) factor with the vibrational excited level [@2010PhRvL.105t3001A]. For our experiment, the FC factor between $v=7$ and $v_B=3$ is the largest available one ($\sim0.15$). The $v=7$ population is probed by the pulsed dye laser through the $9\leftarrow 7$ vibrational transition shown in Fig. \[fig:figure2\]. Examples of rotational spectra obtained by scanning by the DFB laser diode frequency are given in Fig. \[fig:figure3\] and \[fig:figure4\]. In these figures, there is a systematic, irrelevant offset of $\sim7$ ions due to a measurement bias, and the real number of molecules in a specific rotational level is found by dividing the number of detected ions by the FC factor between $v_B=3$ and $v=7$ and the MCP efficiency ($\sim 30\%$). In a first experiment, we perform a simple PA on $0^-_g (6s + 6p_{1/2})(v'=26,J')$. We then find vibrational spectra that hardly depend on the $J'$ value and look like the one shown in Fig. \[fig:figure2\](a). In accordance with the experimental and theoretical study related in Ref. [@2011PCCP...1318910L], we find a distribution spread over more than 60 vibrational levels of the ground state. At this stage, as very few molecules are produced in $v=0$, we are unable to record any valuable rotational spectrum. Next, by adding the femtosecond laser, we obtain the typical vibrational spectrum displayed in Fig. \[fig:figure2\](b) where we see that a substantial fraction ($\sim 25$%) of molecules pumped into $v=0$ [@2011PCCP...1318910L]. The large population in $v=0$ enables us to study its rotational distribution. We find that rotational spectra depend on the $J'$ value: for $J'=1$, we find the upper spectrum shown in Fig. \[fig:figure3\], and for $J'=2$, the upper spectrum shown in Fig. \[fig:figure4\](a). In both cases, the rotational distributions are spread over less than four values of $J$, which indicates that vibrational optical pumping slightly broadens the rotational distribution [@2012_OptExpr_Bigelow_NaCs_Cooling]. We see that when PA is tuned on the $J'=1$($2$) rotational level, only even (odd) $J$ values are populated. This is due to the parity conversion occurring at any stage of the all procedure. ![(color online) Three vibrational spectra obtained by photoionization (RE2PI). (a) When only PA $0^-_g (6s + 6p_{1/2})(v'=26,J'=2)$ is applied, the spectrum reveals that molecules are mainly stabilized in the X state [@2011PCCP...1318910L]. (b) the application of the vibrational cooling provokes the emergence of a few intense lines while the intensity of the other lines is reduced. This proves the accumulation of molecules in $v=0$. (c) rotational and vibrational cooling to ($v=0,J=1$) leads to a spectrum similar to (b), but some vibrational levels $v\neq0$ are more populated than in (b). The number pairs above indicate the wavenumber of some $v_C-v$ transitions. The wavenumber of the $9-7$ transition, indicated by an arrow, is used for rotational spectroscopy. []{data-label="fig:figure2"}](Photoionization_spectra){width="1\columnwidth"} ![(color online) Upper Panel: Fortrat diagram showing the transition wavenumbers from the $(v=0,0\leq J \leq 9)$ rotational levels of the X state. The $P$, $Q$, $R$ branches respectively correspond to the $J_B-J=-1,0,1$ allowed transitions. The parity of the $J$ state is also indicated on the right hand side. The spectrum covered by the rotational cooling laser is shown by the hatched area. Lower panel: Comparison of rotational spectra obtained with vibrational pumping (upper black dotted line) and rovibrational pumping (lower red solid line). The narrow band laser used for rotational cooling is swept over the P(6), P(4) and P(2) transitions with repetitive $100$ $\mu$s frequency ramps. The rotational cooling efficiency is $\sim 40$%.[]{data-label="fig:figure3"}](rotational_diagram "fig:"){width="1\columnwidth"} ![(color online) Upper Panel: Fortrat diagram showing the transition wavenumbers from the $(v=0,0\leq J \leq 9)$ rotational levels of the X state. The $P$, $Q$, $R$ branches respectively correspond to the $J_B-J=-1,0,1$ allowed transitions. The parity of the $J$ state is also indicated on the right hand side. The spectrum covered by the rotational cooling laser is shown by the hatched area. Lower panel: Comparison of rotational spectra obtained with vibrational pumping (upper black dotted line) and rovibrational pumping (lower red solid line). The narrow band laser used for rotational cooling is swept over the P(6), P(4) and P(2) transitions with repetitive $100$ $\mu$s frequency ramps. The rotational cooling efficiency is $\sim 40$%.[]{data-label="fig:figure3"}](ref_vers_J=0 "fig:"){width="1\columnwidth"} The key result of this letter is obtained when rovibrational pumping is achieved. The corresponding vibrational spectrum in Fig. \[fig:figure2\](c) still shows an accumulation in $v=0$. However, with respect to the spectrum in Fig. \[fig:figure2\](b), we find more molecules in $v\neq0$. This is caused by the fact that the DFB laser has a slightly larger interaction zone than the femtosecond laser: some molecules, pumped out of $v=0$ by the DFB laser, remain in $v\neq 0$. The lower panel in Fig. \[fig:figure3\] compares two rotational spectra obtained by rovibrational cooling and vibrational cooling in the case of a PA on $J'=1$. It undeniably indicates that the population of the absolute rotational ground state ($v=0,J=0$) has increased. The cooling results from an accumulation of absorption-spontaneous emission cycles using $J$ lowering absorption transitions, i.e. a laser spectrum covering mainly the $P$ branch transitions: B$^{1}\Uppi_{\rm u},(v_B=3,J-1)\leftarrow $X$^{1}\Upsigma_{\rm g}^{+}(v=0,2\leq J\leq 6)$. Due to the transition rules regarding the parity and the rotational quantum number, a single absorption-spontaneous emission cycle leads to either recover the same $J$ value or decrease it by two units. The frequency sweep, a sawtooth shape going from P(6) to P(2), may facilitate the cooling: if a molecule is excited from a given $J$ and immediately decays to $J-2$, it can be excited again by the same ramp. A few cycles are thus sufficient to transfer all the initial populations to $J=0$. Our procedure also affects the polarization of the molecular sample since it reduces the number of the $2J+1$ projections of the total angular momentum $\bm J$ on a given quantization axes. Indeed, the sample of unpolarized molecules is eventually transferred into a single well defined ($v=0,J=0,M_J=0$) state. This polarization is not straightforward because of the possible existence of dark states. That is why the two lasers used for vibrational and rotational cooling have different propagation axis and polarization states. To illustrate the capabilities of the method, we also pump molecules toward other rotational levels. For instance, when PA is performed on $J'=2$, we obtain the results summarized in Fig. \[fig:figure4\](a): The accumulation occurs in the lowest odd-parity rotational level, i.e. ($v=0,J=1$). Finally, Fig. \[fig:figure4\](b) shows the case where populations are optically pumped to a higher $J$ value, here $J=4$, by scanning the $R$ branch. The efficiency of these rotational pumping processes is defined as the number of molecules transferred into the intended rotational level divided by the number of molecules initially present in the other rotational levels when only vibrational pumping is applied. The results change a bit according to the type of pumping: respectively $\sim 40$%, $\sim 30$% and $\sim 25$% toward $J=0$, $J=1$ and $J=4$. In all the cases, a proportion of $\sim 30$% remain in the other rotational levels. The residual part is lost in the $v\neq0$ levels as revealed by the comparison of Fig. \[fig:figure2\](b) and \[fig:figure2\](c). The weak number of photons involved in the process necessarily implies a negligible increase of temperature of $\sim1$ $\mu$K at most. ![(color online) Rotational spectra demonstrating the rovibrational pumping toward $J=1$ (a) and $J=4$ (b) when the narrow band laser is swept in the frequency range indicated by the hatched areas. For comparison, the spectra resulting from vibrational pumping (dotted line) are displayed above the spectra resulting from rovibrational pumping (solid line).[]{data-label="fig:figure4"}](ref_vers_J=1 "fig:"){width="1\columnwidth"} ![(color online) Rotational spectra demonstrating the rovibrational pumping toward $J=1$ (a) and $J=4$ (b) when the narrow band laser is swept in the frequency range indicated by the hatched areas. For comparison, the spectra resulting from vibrational pumping (dotted line) are displayed above the spectra resulting from rovibrational pumping (solid line).[]{data-label="fig:figure4"}](chauffage_vers_J=4 "fig:"){width="1\columnwidth"} ![(color online) Increase of the number of molecules in $J=1$ with its typical error bar versus the period of the frequency ramp applied on the narrow band laser used for rotational pumping. Because molecules escape from the interaction zone of the vibrational pumping laser, the transfer of molecules in $J=1$ decreases for a period longer than $100$ $\mu$s.[]{data-label="fig:figure5"}](evolution){width="1\columnwidth"} To fully understand the mechanisms and limitations of our cooling method, we now detail the action of the pumping lasers. First, it is important to note that the vibrational cooling causes a little modification of $J$. For low $J$ values, typically only $1/5$ of the population ends up in an increased $J$ value after a single absorption-spontaneous emission cycle induced by the vibrational pumping laser. As the complete vibrational pumping to $v=0$ requires less than $5$ such cycles [@MatthieuViteau07112008], about $50$% of the molecules undergo an increase of $J$. We roughly estimate that it takes about $100$ $\mu$s to bring back molecules to $v=0$ from any $v$ when the saturation intensity of a typical rovibrational transition is $\sim 100$ mW/cm$^2$ (considering typical FC factors of $0.1$ and Hönl-London factor of 1/4) and the vibrational cooling bandwidth is $\sim 100$ cm$^{-1}$. The minimum scan period ensuring a rotational pumping with a single ramp probability of $100$% is also about $100$ $\mu$s, a time that is estimated through the maximum scan velocity roughly given by $\Gamma^2\sqrt{I/I_{sat}}$ where $\Gamma\approx 15\times 10^6$ s$^{-1}$ is the linewidth, $I$ the laser intensity, and $I_{sat}$ is the saturation intensity of some rovibrational transition. These considerations allow us to interpret the results in Fig. \[fig:figure5\] where the relative transfer of molecules in $J=1$ is plotted versus the scan period. In principle, below $100$ $\mu$s, a single ramp does not allow a complete rotational pumping, but several ramps can be accumulated, which gives a pumping efficiency equivalent to that obtained with a $100$ $\mu$s scan period. On the contrary, once the scan period exceeds a few $100$ $\mu$s, the cooling efficiency decreases. This is explained by the molecular diffusion out of the laser interaction zone: Thus, the scan period limits the effective number of optical transitions. We have demonstrated that rovibrational pumping of molecules can be achieved by using different and complementary laser sources. Combined with the possibility of transferring populations from a given electronic state to another one [@2012_PRA_Horchani_Conversion], we believe that such a method can be a very efficient tool to manipulate the rovibronic population of ultracold molecular samples or molecular beams. Besides, the efficiency of the rovibrational pumping should be easily enhanced by an optimized frequency shaping [@2010MolPh.108..795S] or with a DFB laser tuned on the $(v_B =0\leftarrow v=0)$ transitions rather than the $(v_B =3\leftarrow v=0)$ ones. Also, by using a more powerful femtosecond source, it becomes conceivable to realize an optical pumping of the external degrees of freedom based on a closed system made up of several states. In other words, the method can be a decisive help for (standard or Sisyphus) laser cooling of molecules [@2009PhRvA..80d1401Z; @2009JPhB...42s5301R; @shuman2010laser; @2011_CRPhy_Perrin_laser]. Laboratoire Aimé Cotton is a member of Institut Francilien de Recherche sur les Atomes Froids (IFRAF) and of the LABEX PALM initiative. The exchange project between the University of Pisa and the University of Paris-Sud is acknowledged. A. F. and I. M. have been supported by the “Triangle de la Physique” under contracts No. 2007-n.74T and No. 2009-035T “GULFSTREAM” and No. 2010-097T-COCO2. We thank O. Dulieu and N. Bouloufa-Maafa for fruitful discussions as well as E. Dimova, L. Wang, L. Couturier, B. Le Crom and E. Mangaud for their help with the experiment. [10]{} B. K. [Stuhl]{}, B. C. 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--- abstract: 'We investigate theoretically the Seebeck effect in materials close to a ferromagnetic quantum critical point to explain anomalous behaviour at low temperatures. It is found that the main effect of spin fluctuations is to enhance the coefficient of the leading $T$-linear term, and a quantum critical behaviour characterized by a spin-fluctuation temperature appears in the temperature dependence of correction terms as in the specific heat.' address: ' Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561,Japan' author: - Takuya Okabe bibliography: - 'main.bib' title: 'Spin-fluctuation drag thermopower of nearly ferromagnetic metals ' --- Introduction ============ Experiments on clean materials near ferromagnetic quantum critical point (QCP) have revealed unusual properties, including non Fermi liquid transport and unconventional superconductivity.[@saxena00; @nblkstbbf05] The effects caused by quantum critical dynamics of spin fluctuations on the specific heat coefficient, the spin susceptibility, the resistivity, and so on, have been elucidated analytically at low temperatures.[@de66; @bmf68; @mathon68; @moriya85; @lrvw07] In most of such theoretical analyses made so far, critical spin fluctuations are regarded to stay in thermal equilibrium. On the other hand, one may conceive of its inequilibrium counterpart of anomalous behaviours as well, which would be of fundamental interest too and should be paid due attention theoretically. As a representative of such phenomena, there are observations suggesting spin-fluctuation (or paramagnon) drag thermopower. In the Seebeck coefficient $S(T)$ of ${\rm UAl_2}$, for example, there have remained a structure at low temperature, which is observed experimentally,[@afss79; @po97] but left unexplained theoretically.[@ijc78; @cij78] Among others, the most typical clear-cut experimental evidence would be those reported by Gratz $et$ $al$.,[@grbbmg95; @gratz97; @gm01] where the pronounced low-temperature minimum in $S(T)$ of strong paramagnet $\rm RCo_2$ (R$=$Sc, Y and Lu) was attributed to the paramagnon drag effect. Recently, Matsuoka $et$ $al$.[@mhittm05] found a similar structure for $\rm A Fe_4Sb_{12}$ (A$=$Ca, Sr and Ba). In effect, Takabatake $et$ $al$.[@tmnhmsus06] made it clear that the anomaly in $S(T)$ is indeed caused by the ferromagnetic spin fluctuations prevalent in the materials by showing that those structure is completely suppressed by applying a uniform magnetic field. In contrast with the accumulating experimental evidence, there seems no theory to compare with the experiments available so far, with the exception of a brief account on a qualitative effect expected for [*localized*]{} spin fluctuations around impurity sites of alloys.[@kaiser76] In this paper, we discuss an effect of uniform spin fluctuations in a translationally invariant system, and intend to provide a more solid footing on which to discuss the phenomenon. In section \[sec:model\], we give an outline of a two-band model, which we adopt as a relevant model, along with approximations and assumptions conventionally made. In section \[sec:sfd\], we introduce a function $\Phi_k^d$ to represent inequilibrium displacement of spin fluctuations. In section \[sec:ltc\], we discuss that the leading effect of spin fluctuations appears on the $T$-linear term of $S(T)$. In effect, in section \[subsec:ee\], we discuss that the leading term contribution follows a universal relation to the specific heat, that is, $ q\equiv e S/ C \simeq \pm 1$ revealed by Behnia $et$ $al$.[@BJF04] In the higher order terms, we have to consider not only a critical effect originating from equilibrium quantities, but also a genuinely non-equilibrium effect which has not been investigated before. In section \[sec:std\], we investigate the latter contributions to find a characteristic temperature dependence, and the results are summarized in the last subsection \[sec:summary\]. In section \[sec:disscs\], we discuss the results and comparison is made with experiment. \[sec:model\]Model ================== Let us introduce a two-band paramagnon model, which is conventionally employed to explain an enhanced resistivity of transition metals at low temperature.[@mathon68; @ml66; @rice68] The model has been applied successfully to explain, e.g., a saturation behaviour at elevated temperatures by taking into account a proper temperature dependence of spin susceptibility.[@jbc74; @um75] The model is comprised of two types of electrons, i.e., wide-band conduction electrons and narrow-band itinerant electrons on the border of ferromagnetism. We denote the former as the $s$ electron and the latter as the $d$ electrons, representatively. The Hamiltonian consists of three parts, $$H=H_s + H_{sd} + H_d.$$ The free Hamiltonian of the $s$ electron is given by $$H_s =\sum_{k\sigma} \varepsilon_s (k) c^\dagger_{k\sigma} c_{k\sigma},$$ where $c^\dagger_{k\sigma}$ and $c_{k\sigma}$ are the creation and annihilation operators for the electron with momentum $k$ and spin $\sigma$. For simplicity, it is often assumed that the $s$ electrons make a parabolic band with mass $m_s$, i.e., $$\varepsilon_s (k) =\frac{k^2}{2m_s}. \label{varepsk}$$ At each site $i$, they are scattered by the spin ${\bi S}_i$ of the $d$ electron at the same site through the Kondo $s$-$d$ coupling, $$H_{sd} = J \sum_i {\bi s}_i\cdot {\bi S}_i, \label{Hsd}$$ where $J$ denotes a coupling constant, and ${\bi s}_i = \frac{1}{2}\sum_{\sigma \sigma'} c^{\dagger}_{i\sigma} {\bi \tau}_{\sigma\sigma'}c_{i\sigma}$ is the spin of the $s$ electron at the site $i$ expressed in terms of the Pauli matrix vector ${\bi \tau}_{\sigma\sigma'}$. Similarly, the $d$ electron spin at the site $i$ is given by ${\bi S}_i = \frac{1}{2}\sum_{\sigma \sigma'} d^{\dagger}_{i\sigma} {\bi \tau}_{\sigma\sigma'}d_{i\sigma}$ in terms of the creation and annihilation operators $d^\dagger_{i\sigma}$ and $d_{i\sigma}$ for the $d$ electron. Spin dynamics of the $d$ electrons is described by the Hubbard Hamiltonian, $$H_d =\sum_{k\sigma} \varepsilon_d (k) d^\dagger_{k\sigma} d_{k\sigma} + U \sum_i n_{i\uparrow}n_{i\downarrow}, \label{Hd}$$ where $n_{i\sigma}=d^\dagger_{i\sigma} d_{i\sigma}$ $(\sigma =\uparrow,\downarrow)$ is the number operator of the $d$ electron at the site $i$. The on-site repulsion $U$ is fixed such that the $d$ band is nearly ferromagnetic. To make analytical evaluation feasible, it is often assumed further that the $d$ electrons are also parabolic with a different mass $m_d$ heavier than $m_s$, i.e., $$\varepsilon_d (k) =\frac{k^2}{2m_d}, \label{varepdk}$$ and $m_d \gg m_s$. The latter inequality is regarded as the basic ingredient of the model. Hence the $d$ electrons act as heavy and fluctuating scatterers against the $s$ electrons through the coupling of (\[Hsd\]). In effect, this is taken into account as the second order effect with respect to the coupling $J$, i.e., through the Born approximation.[@ml66] Then, the $d$ electron comes into play through the (transverse) spin susceptibility $\chi(q, \omega)$. In the random phase approximation, it is given by $$% \chi^{-+} \chi(q,\omega) =\frac{ \chi_{0}(q,\omega) }{ 1-U \chi_{0}(q,\omega)}, \label{chi+-qom}$$ where $$\chi_{0}(q,\omega) = \sum_k \frac{f^0_k -f^0_{k+q}}{\varepsilon_d(k+q) -\varepsilon_d(k) - \omega -{\rm i}\delta }. \label{chi0qomega}$$ Here, $f_k^0 \equiv f^0(\varepsilon_d (k)) %=1/(\exp((f^0(\varepsilon_d (k)-\mu)/T+1) =1/({\rm e}^{(\varepsilon_d (k)-\mu)/T}+1) $ is the Fermi distribution function, and $\delta$ is a positive infinitesimal. To investigate critical properties at low temperatures,[@ikk63] (\[chi0qomega\]) is expanded for small $q$ and $\omega/q$ as $$\chi_{0}(q,\omega) = %N(0) \rho_{F,d} \left( 1-\frac{1}{12} \bar{q}^2 +{\rm i} \frac{\pi}{4} \frac{\bar{\omega}}{\bar{q}} \right), \label{chi0qom}$$ for $\bar{\omega}< 2 \bar{q} $, where $\bar{q} = q/k_{F,d}$ and $\bar{\omega} = \omega/\varepsilon_{F,d}$ are the momentum and energy normalized by the Fermi momentum $k_{F,d}$ and the Fermi energy $\varepsilon_{F,d}$ of the $d$ electron. $\rho_{F,d} %N(0) =m_d k_{F,d}/2\pi^2$ is the density of states (DOS) at the Fermi level of the $d$ electron per spin. Substituting (\[chi0qom\]) into (\[chi+-qom\]), we obtain $$%^{-+} \chi (q,\omega) = \frac{\rho_{F,d}}{K_0^2 + \frac{\bar{U}}{12}\bar{q}^2 -{\rm i} \frac{\pi\bar{U}}{4} \frac{\bar{\omega}}{\bar{q}}}, \label{chi+-qomega}$$ for $\bar{\omega}< 2 \bar{q} $, where $\bar{U}=\rho_{F,d}U$, and $K_0^2= 1-\bar{U} (\ll 1)$ represents the distance to the QCP. The intrinsic transition probability $ {\cal Q}_{k,q}^{k+q}$ that an $s$ electron with momentum $k$ is scattered to $k+q$ by absorbing a spin fluctuation with $q$ and $\omega$ via the coupling in (\[Hsd\]) is given by $${\cal Q}_{k,q}^{k+q}(\omega) = \frac{3J^2}{4} S(q,\omega),$$ where $S(q,\omega)$ denotes the Fourier transform of the spin density correlation function, which is related to the dynamical susceptibility by the fluctuation dissipation theorem.[@ikk63] $$S(q,\omega) %= \langle S^+_{-q} S^-_{q}(\omega)\rangle=\int \langle S^+_{-q} S^-_{q}(t)\rangle {\rm e}^{-{\rm i}\omega t}dt =\frac{2 }{1-{\rm e}^{-\omega/T}}{\rm Im}\chi %^{+-} (q,\omega).$$ The equilibrium transition rate is given by $$\fl {\cal P}_{k,q}^{k+q}= \int {\rm d}\omega (1-f^0(\varepsilon_s({k+q}))) f^0(\varepsilon_s(k)) n^0(\omega) %-f_{k+q} (1-f_k) (n(\omega) +1) {\cal Q}_{k,q}^{k+q}(\omega) \delta(\omega +\varepsilon_s(k)-\varepsilon_s({k+q})), \label{calP=intdomega1-f0fn0Q}$$ where $f^0(\varepsilon_s(k))$ is the Fermi factor for the $s$ electron, and $n^0(\omega) =1/({\rm e}^{\omega/T}-1)$ is the Bose function. With this ${\cal P}_{k,q}^{k+q}$, transport coefficients are derived by following the formal transport theory of Ziman[@ziman60] (cf. \[sec:formaltransporttheory\]). Transport properties of the $s$ electrons in an electric field ${\bi E}$ and a gradient of temperature $\nabla T$ are described by the Boltzmann transport equation, $$-{\bi v}_s(k) \cdot \nabla T \frac{\partial f^0(\varepsilon_s(k))}{\partial T} -e {\bi v}_s(k) \cdot {\bi E} \frac{\partial f^0(\varepsilon_s(k))}{\partial \varepsilon_s(k)} = %-\left. \dot{f}_k \right]_{\rm scatt}, - \dot{f}_k, \label{boltzeq0}$$ where $ {\bi v}_s(k) =\nabla_k \varepsilon_s(k) $ is the velocity of the $s$ electron, and $e (<0)$ is the electronic charge. The right-hand side in (\[boltzeq0\]) is the collision integral for the $s$ electron. To linearize the transport equation for the conduction electrons, a function $\Phi_k^s $ to represent the displacement of the distribution function $f(\varepsilon_s(k))$ from the equilibrium one $f^0(\varepsilon_s(k))$ is introduced, i.e., by $$f(\varepsilon_s(k)) =f^0(\varepsilon_s(k)) -\frac{\partial f^0(\varepsilon_s(k)) }{\partial \varepsilon_s(k)} \Phi_k^s. \label{f=f0-Phik}$$ On the contrary, the $d$ electrons are commonly assumed to stay in equilibrium, despite the applied fields. Then, for the collision integral in (\[boltzeq0\]), we obtain $$%\left. \dot{f}_k \right]_{\rm scatt} = \dot{f}_k = -\frac{1}{T}\sum_{q} \left( \Phi_k^s - \Phi_{k+q}^s\right) {\cal P}_{k,q}^{k+q}. \label{fkscattLinear}$$ For definiteness, let the fields ${\bi E}$ and $\nabla T$ be in the direction parallel to a unit vector ${\bi u}$. For the isotropic model, the magnitudes of the electric and heat currents due to the $s$ electrons are given by $$% {\bi J}_s[\Phi^s] = e \sum_k {\bi v}_s(k) {J}_s[\Phi^s] = 2e \sum_k {\bi u}\cdot {\bi v}_s(k) %{\bi v}_s(k) \left( -\frac{\partial f^0(\varepsilon_s(k)) }{\partial \varepsilon_s(k)} \right) %- \frac{\partial f_k^0}{\partial \varepsilon_k} \Phi_k^s, \label{J=esumkvck}$$ and $$% {\bi U}_s[\Phi^s] = \sum_k {\bi v}_s(k) {U}_s[\Phi^s] = 2\sum_k {\bi u}\cdot {\bi v}_s(k) (\varepsilon_s(k) -\mu) \left( -\frac{\partial f^0(\varepsilon_s(k)) }{\partial \varepsilon_s(k)} \right) %\left( %- \frac{\partial f_k^0}{\partial \varepsilon_k} %\right) \Phi_k^s. \label{U0=sumkvck}$$ The factor 2 in front of the $k$ sum accounts for the two spin components. As noted below (\[f=f0-Phik\]), it is conventionally assumed that the corresponding currents due to the $d$ electrons are neglected against the $s$ electron currents. To obtain a solution $\Phi_k^s$, one may set $\Phi_k^s =\tau {\bi u}\cdot {\bi v}_s(k)$, while the constant $\tau$ is fixed by the equation. Consequently, for the electric resistivity $R=R_0(T)$ and the diffusion thermopower coefficient $S=S_0^s(T)$, we obtain $$R_0 (T)= \frac{P_{ss}}{({J_s}[\Phi^s])^2}, \label{R0T=fracpss}$$ and $$S_0^s(T) = \frac{1}{T} \frac{{ U_s}[\Phi^s]}{{J_s}[\Phi^s]}, \label{S0sT}$$ where $$P_{ss} =\frac{1}{T}\sum_{k,q} {\cal P}^{k+q}_{kq} \left( \Phi_k^s - \Phi_{k+q}^s \right)^2. %\label{Pss}$$ The ordinary diffusion thermopower in (\[S0sT\]) is linear in $T$ at low temperature, and is often expressed as $$S_0^s (T) %= S_{\rm diffuse}(T) =\frac{\pi^2 T}{3e} \frac{\partial \log \sigma_s(\varepsilon_{F,s}) }{\partial \varepsilon},$$ in terms of the spectral conductivity $\sigma_s(\varepsilon)$ of the conduction electron. \[sec:sfd\] Spin-fluctuation drag ================================= As remarked above, the $d$ electrons are customarily assumed to stay in equilibrium regardless of the applied fields. To generalize the above framework to describe spin fluctuations with a shifted distribution theoretically, let us consider a bare dragged susceptibility $\chi^{q_0}_0(q,\omega)$, which is obtained by shifting uniformly the equilibrium bare susceptibility $\chi_0(q,\omega)$ in (\[chi0qomega\]) by a small but finite amount ${\bi q}_0$ in momentum space. Similarly, we may define $\chi^{q_0}(q,\omega)$ for the full susceptibility $\chi(q,\omega)$ as well. Hence, $\chi^{q_0}(q,\omega)$ is strongly peaked at ${\bi q}={\bi q}_0$. First we derive a simple relation between $\chi^{q_0}_0(q,\omega)$ and $\chi_0(q,\omega)$. According to (\[chi+-qom\]), we will obtain a similar relation for the full susceptibility. For the derivation, we introduce a shifted energy of the $d$ electron, $$\varepsilon_d^{q_0}(k) = \varepsilon_d (k-q_0) \simeq \varepsilon_d(k) - {\bi q}_0\cdot{\bi v}_{d}(k), \label{espdq0k}$$ where ${\bi v}_{d}(k) = \nabla_k \varepsilon_d$. Then, $\chi^{q_0}_0(q,\omega)$ is obtained by distributing the $d$ electron with momentum $k$ according to the shifted distribution $f^0(\varepsilon^{q_0}_{k})$[^1], that is to say, by $$\begin{aligned} \chi^{q_0}_0(q,\omega) &=& \sum_k \frac{ f^0(\varepsilon_d^{q_0}({k}))-f^0(\varepsilon_d^{q_0}({k+q})) }{\varepsilon_d(k+q) -\varepsilon_d({k})-\omega} \label{chiq00} \\ &=&\sum_k \frac{ f^0(\varepsilon_d({k}))-f^0(\varepsilon_d({k+q})) %f^0_k -f^0_{k+q} }{\varepsilon^{-q_0}_d(k+q) -\varepsilon^{-q_0}_d(k) - \omega }.\end{aligned}$$ Thus, by (\[espdq0k\]), we obtain the relation $$\chi_0^{q_0} (q,\omega) \simeq \chi_0(q,\omega + {\bi q}_0\cdot{\bi v}_d(q)). %{\bi V}_0\cdot{\bi q}), \label{chi0q0qom=chi0qom-q0}$$ This is the result on which we base ourselves in the following. According to (\[chi0q0qom=chi0qom-q0\]), the drag effect is described by a function $\Phi_q^d \equiv{\bi q}_0\cdot{\bi v}_{d}(q)$. To understand what this represents, it is instructive to consider the isotropic case of (\[varepdk\]), where ${\bi v}_{d}(q)= {\bi q}/m_d$. In this case, we obtain $\Phi_q^d={\bi V}\cdot{\bi q}$ where ${\bi V}={\bi q}_0/m_{d}$ denotes a uniform drift velocity of the $d$ electrons, or the spin fluctuations. In effect, the energy $\varepsilon^{-q_0}_d(k)$ represents the excitation energy of the $d$ electron in the moving frame drifting with the velocity ${\bi V}$. This is just a Galilean transformation. Indeed, noting that we can write $$f^0(\varepsilon_d^{q_0}({k})) =f^0(\varepsilon_d({k})) -\frac{\partial f^0(\varepsilon_d(k)) }{\partial \varepsilon_d(k)} \Phi_k^d,$$ and comparing this with (\[f=f0-Phik\]), it would be clear that the new function $\Phi_k^d$ represents the distribution shift of the $d$ electrons, just as $\Phi_k^s$ does for the $s$ electrons. Thus, we argue that the drag effect of spin fluctuations is described in terms of $\Phi_q^d$ in the way that $\chi_{\rm drag} (q,\omega)$ of the dragged fluctuations is represented as $$\chi_{\rm drag} (q,\omega) = \chi(q,\omega+\Phi_q^d), \label{chiphiq=}$$ in terms of the equilibrium susceptibility $\chi(q,\omega)$. Given the above argument, we have next to investigate how the formalism in the last section should be affected by a non-vanishing $\Phi_q^d$. The first effect is to modify the collision integral in (\[fkscattLinear\]). To see this, here we follow how (\[fkscattLinear\]) is derived. The collision term in the right-hand side of (\[boltzeq0\]) is explicitly given by $$\begin{aligned} \fl \dot{f}_k= -\sum_{q} \int {\rm d}\omega \left[ (1-f_{k+q}) f_k %n(\omega) n^0(\omega) - f_{k+q}(1- f_k) (n^0(\omega) +1) \right] \\ \times {\cal Q}_{k,q}^{k+q}(\omega) \delta(\omega +\varepsilon_s(k)-\varepsilon_s({k+q})), \label{fkscattFull}\end{aligned}$$ where we denoted $f_k=f(\varepsilon_s(k))$ for the distribution function. According to the condition of detailed balance, the equilibrium distribution functions $f^0_k$ and $n^0(\omega)$ satisfy the relation $$(1-f^0_{k+q}) f^0_k n^0(\omega) - f^0_{k+q}(1- f^0_k)(n^0(\omega) +1)=0. \label{detailedbalance}$$ Accordingly, by substituting (\[f=f0-Phik\]) into (\[fkscattFull\]), we obtain (\[fkscattLinear\]) to the linear order in $\Phi_k^s$. To go further to take into account the inequilibrium shift of the $d$ electrons, we regard that ${\cal Q}_{k,q}^{k+q}(\omega)$ in (\[fkscattFull\]), or ${\cal P}_{k,q}^{k+q}$ of (\[calP=intdomega1-f0fn0Q\]), depends on $\chi_{\rm drag}(q,\omega)$ in place of $\chi(q,\omega)$. Then we can make use of (\[chiphiq=\]). The first effect of $\Phi_q^d$ is to change the scattering probability ${\cal P}_{k,q}^{k+q}$, which eventually has no effect owing to (\[detailedbalance\]). The second is to replace $n^0(\omega)$ in (\[fkscattFull\]) by $$n^0(\omega-\Phi_q^d) \simeq % n(\omega) = n^0(\omega) -\frac{\partial n^0}{\partial \omega} \Phi_q^d. \label{nomega=n0omega-}$$ As a result, we obtain $$%\left. \dot{f}_k \right]_{\rm scatt} = \dot{f}_k = -\frac{1}{T}\sum_{q} \left( \Phi_k^s + \Phi_q^d - \Phi_{k+q}^s \right) {\cal P}_{k,q}^{k+q}. \label{fkscattLinearD}$$ At this point, (\[fkscattLinearD\]) clearly indicates a close analogy to the similar problem of phonon drag.[@ziman60] On the one hand, we can reproduce the previous results under the assumption $\Phi_q^d=0$ of no drag. On the other hand, owing to $\Phi_q^d$ in (\[fkscattLinearD\]), we can recover the correct identity $\dot{f}_k =0$ when the model is genuinely isotropic as implied by (\[varepsk\]) and (\[varepdk\]). In fact, in this case, we may set $$%\Phi_k^s ={\bi u}\cdot {\bi k}, \quad \Phi_q^d ={\bi u}\cdot {\bi q}, \Phi_k^s =\Phi_k^d= {\bi u}\cdot {\bi k}, % \quad \Phi_q^d ={\bi u}\cdot {\bi q}, \label{phiks=ukphiqd=uq}$$ where we put ${\bi V}={\bi u}$ without loss of generality. Then the null result for (\[fkscattLinearD\]) obtains from the total momentum conservation. This means that, if properly treated, the model should give no resistivity at all, irrespective of strong scatterings with spin fluctuations. In effect, the spin fluctuations in the inequilibrium state represented by (\[phiks=ukphiqd=uq\]) are completely dragged along with the conduction electron currents. It is the fully dragged state in which all the $s$ and $d$ electrons drift with the same uniform velocity ${\bi V}$, independently of the electric field ${\bi E}$. This is the opposite limit to the case $\Phi_q^d=0$ without drag. In practice, in any case, we should have a finite rate $\dot{f}_k $ by some mechanism neglected in the simple model, e.g., by Umklapp scatterings or by scatterings with extraneous agents. Moreover, generally, in order to investigate the degree of drag quantitatively, e.g., the temperature dependence through a wide range over a characteristic spin fluctuation temperature, $\Phi_q^d$ should be determined consistently on the basis of its own transport equation. In general, the $k$ dependence of $\Phi_k^s$ and $\Phi_k^d$ may not be as simple as in (\[phiks=ukphiqd=uq\]). For definiteness, however, we restrict ourselves to the low temperature regime, where we make use of the full drag assumption, (\[phiks=ukphiqd=uq\]), to elucidate non-trivial effects arising from our extra degree of freedom, $\Phi_q^d$. A formal theory to discuss a general case is given in \[sec:formaltransporttheory\]. \[sec:ltc\]Leading effect ========================= Limiting cases -------------- In the original model, the $d$ electron currents are neglected on the basis of the basic inequality $|{\bi v}_s(k)| \gg |{\bi v}_d(k)|$, or $m_d \gg m_s$.[@ziman60; @mott35] Close inspection indicates that this is concluded through the additional implicit assumption $\Phi_k^i ={\bi u}\cdot {\bi v}_i(k)$ $(i=s,d)$ on the solutions of the transport equations, namely, by $\Phi_k^s\gg \Phi_k^d\simeq 0$. As we saw above in (\[phiks=ukphiqd=uq\]), this does not hold true in the presence of the $d$ electron drag. In effect, the leading term contribution to the thermopower will arise from those dragged $d$ electron currents, which would outweigh the normal diffusion term $S_0^s(T)$ in (\[S0sT\]) due to the conduction electrons by a factor of $m_d/m_s\gg 1$. We obtain from (\[varepsk\]), (\[J=esumkvck\]), and $\Phi_k^s ={\bi u}\cdot {\bi k}$, $$J_s^s\equiv {J}_s [ \Phi^s] = \frac{2 e }{3} v_{F,s}k_{F,s}\rho_{F,s}, \label{JssequivJsphis}$$ where $v_{F,s}=k_{F,s}/m_s$ is the Fermi velocity. Similarly, (\[U0=sumkvck\]) may be written as $$U_s^s\equiv {U}_s [ \Phi^s] = \frac{\pi^2}{3e} T^2 \frac{\partial {J}_s^s }{\partial \varepsilon_{F,s}}.$$ where $\varepsilon_{F,s}$ is the Fermi energy. The latter is obtained by expanding the integrand in (\[U0=sumkvck\]) with respect to the excitation energy $\varepsilon_s(k) -\mu$. The factor of ${\pi^2}T^2/3$ derives from the energy integral over $\varepsilon_s(k)$ to replace the $k$ sum. Hence, from (\[S0sT\]) we obtain the ordinary $T$-linear Seebeck coefficient $$S_0^s= %\frac{U_s^s}{TJ_s^s}= \frac{\pi^2}{3e } \frac{\partial \log {J}_s^s }{\partial \varepsilon_{F,s}} T. \label{S0^s}$$ In the same manner, the $d$ electron currents are evaluated. We may use $$J_d^d\equiv {J}_d[\Phi^d] = 2e \sum_k {\bi u}\cdot {\bi v}_d(k) \left( -\frac{\partial f^0(\varepsilon_d(k)) }{\partial \varepsilon_d(k)} \right) \Phi_k^d, \label{J=esumkvdk}$$ in place of (\[J=esumkvck\]), and $U_d^d\equiv {U}_d[\Phi^d]$ as in (\[U0=sumkvck\]), with which we obtain $$S_0^d\equiv \frac{ U_d^d}{T{J_d^d}}= \frac{\pi^2}{3e } \frac{\partial \log {J}_d^d }{\partial \varepsilon_{F,d}} T, \label{S0^d}$$ as in (\[S0\^s\]). Formally, this represents the diffusion thermopower due to the $d$ electrons, as $S_0^s$ does for the $s$ electrons. Therefore, we should expect $$|S_0^d| \gg |S_0^s|, \label{s0dggs0s}$$ for $S_0^i$ is proportional to the mass $m_i$. Still, it is remarked that $S_0^d$ in (\[S0\^d\]) is not a directly observable quantity in general. In fact, from (\[S=frac1TUs+Ud/Js+Jd\]), the total thermopower is given by $$S_0 = \frac{ U_s^s+U_d^d }{T \left( J_s^s + J_d^d \right) }. \label{S=fracUss+Udd/T}$$ Therefore, on the one hand, in the conventional case without $d$ electron drag, where $|J_s^s|\gg |J_d^d|$ and $|U_s^s|\gg |U_d^d|$, we recover the normal result $S_0\simeq S_0^s$. On the other hand, in the opposite limiting case of the full drag, as the two currents $J_s^s$ and $J_d^d$ become comparable with each other, we expect a sizable modification from the normal result. To make this explicit, we remark that the currents are conveniently expressed in terms of their electron numbers $n_s$ and $n_d$. In effect, it is straightforward to show $ J_s^s =2 n_s e $ from (\[JssequivJsphis\]), or more generally, we get it by a partial integration as follows. $$\begin{aligned} J_s^s &=& 2e \sum_k {\bi u}\cdot {\bi v}_s(k) \left( -\frac{\partial f^0(\varepsilon_s(k)) }{\partial \varepsilon_s(k)} \right) {\bi u}\cdot {\bi k} \label{Jss=-2e} %\nonumber \\ &=& - 2e \sum_k {\bi u}\cdot \nabla_{\bi k} %\frac{\rm d}{{\rm d}k_x} \left( {\bi u}\cdot {\bi k} f^0(\varepsilon_s(k))\right) + 2e \sum_k f^0(\varepsilon_s(k)). \nonumber %\nonumber\end{aligned}$$ The first term represents the contribution from the Brillouin zone boundary of the $k$ sum, which vanishes when the states there are unfilled. The second sum gives the result of the total number times $e$. Similarly, we obtain $ J_d^d =2 n_d e $ for the $d$ electron. These results simply represent that the whole electrons are drifting all together, as noted in the last section. Hence, from (\[S=fracUss+Udd/T\]) we get $$S_0= \frac{ \displaystyle {n_s} S_0^s+ {n_d} S_0^d }{ \displaystyle{n_s} +{n_d} }. \label{S_0=diffuse}$$ Especially, in the limit $n_d\gg n_s$, we obtain the enhanced diffusion thermopower $S_0\simeq S_0^d $ given in (\[S0\^d\]), which is wholly due to the $d$ electrons carrying the spin fluctuations. Equilibrium effect {#subsec:ee} ------------------ To the extent that we make use of an approximate expression $ J_d^d \simeq 2 n_d e$ as above, one may obtain $U_d^d \simeq 2 e_d$ correspondingly similarly, where $e_d$ generally represents free energy of the $d$ electrons. Then we obtain $$S_0^d \simeq {e_d}/(T n_d e). \label{S0edTnde}$$ This expression may be valuable as it is expressed in terms of the [*equilibrium*]{} quantities, which have been vigorously investigated. For example, one may have recourse to scaling argument for $e_d$.[@lrvw07] We obtain $S_0^d \propto T$ by $e_d \propto T^2$ normally, while at the QCP, $S_0^d \propto T\log T^{-1}$ according to $e_d \propto T^2\log T^{-1}$. In terms of the electronic heat capacity $C$, one may substitute $e_d=CT $ to obtain $S_0^d \simeq C /(n_d e)$, or $$q\equiv \frac{e S}{ C } = \frac{1}{n}, %= \frac{1}{n_s+ n_d}, \label{qequiv es/c}$$ where $n\equiv n_s+ n_d$ and $S\simeq S_0 \simeq n_d S_0^d/ n $ under (\[s0dggs0s\]). For hole like carriers, following as in (\[Jss=-2e\]), we find that the number $n_d$ becomes negative with the absolute value $|n_d|$ representing the hole number. Thus our drag mechanism supports the material-independent universality in $q$ as revealed by Behnia $et$ $al$.[@BJF04] This is contrasted with the explanation by resorting to dominant impurity scatterings.[@MK05] To go further to investigate the next order contributions, we have to consider not only those originating from the equilibrium quantities, which may be related to singular behaviour of the specific heat, but also the [*non-equilibrium*]{} effect which manifest itself in linear response to an applied field. The latter, though potentially important, has not been investigated before. In the next section, we focus ourselves to such singular contributions which vanish at zero field ${\bi E}=0$. We find similar temperature dependences as that expected from the equilibrium effect through (\[S0edTnde\]). \[sec:std\]Sub-leading corrections ================================== Extra currents -------------- The effect of spin fluctuations on the single particle excitation of conduction electron is described by a particle self-energy $\Sigma({\bi k}, \varepsilon)$. The dragged spin fluctuations bring about a similar effect as those in equilibrium affect the thermodynamical properties.[@de66; @bmf68] We pay attention to the extra quasiparticle currents induced by the change of states at the Fermi level, as they are expected to make dominant contributions. We write an energy shift caused by a non-vanishing factor $\Phi^d_k$ as $\delta \varepsilon_s(k)$. Then the extra currents are given by $${J}_s [ \Phi^d] = 2e \sum_k {\bi u}\cdot {\bi v}_{s}(k) \frac{\partial f^0}{\partial \varepsilon_{s}(k)} \delta \varepsilon_{s}(k), \label{Jsphid}$$ and $${U}_s [ \Phi^d] = 2 \sum_k {\bi u}\cdot {\bi v}_{s}(k) (\varepsilon_{s}(k) -\mu) \frac{\partial f^0}{\partial \varepsilon_{s}(k)} \delta \varepsilon_{s}(k). \label{Usphid}$$ The effective energy of the conduction electron at the Fermi level is given in terms of the real part of the self-energy ${\rm Re } \Sigma(k, \varepsilon)$ by $$\varepsilon^*_{s}(k) = %\left. \frac{\varepsilon_{s}(k) + {\rm Re } \Sigma (k,0)}{ \displaystyle 1 -\frac{\partial }{\partial \varepsilon}{\rm Re } \Sigma(k, 0) }. %\right|_{\varepsilon=0}$$ For the self-energy, we are interested in those part induced by the dragged spin fluctuations, which we denote as $\delta\left({\rm Re } \Sigma(k, 0)\right)$. Thus we have $$\delta \varepsilon_{s}(k) \simeq \delta\left( {\rm Re } \Sigma (k,0) \right) +(\varepsilon_{s}(k)-\mu) \frac{\partial }{\partial \varepsilon} \delta\left({\rm Re } \Sigma(k, 0)\right), \label{deltavarepsksimeq}$$ as we need $\delta \varepsilon_{s}(k)$ and $\delta\left({\rm Re } \Sigma(k, 0)\right)$ only to the linear order in $\Phi^d_k$. The first and the second terms in (\[deltavarepsksimeq\]) contribute mainly to ${J}_s [ \Phi^d]$ and ${U}_s [ \Phi^d]$, respectively. In effect, we find $$\begin{aligned} \fl { U}_s [ \Phi^d] = % \sum_k {\bi u}\cdot {\bi v}_{s}(k) %(\varepsilon_{s}(k) -\mu) % \frac{\partial f^0}{\partial % \varepsilon_{s}(k)} %\delta \varepsilon_{s,k}[\Phi^d] = 2 \sum_k {\bi u}\cdot {\bi v}_{s}(k) (\varepsilon_{s}(k) -\mu)^2 \frac{\partial f^0}{\partial \varepsilon_{s}(k)} \frac{\partial }{\partial \varepsilon} \delta\left( {\rm Re } \Sigma(k, 0) \right) \nonumber\\ \simeq 2\langle \left({\bi v}_{s}(k)\cdot {\bi u}\right) \frac{\partial }{\partial \varepsilon} \delta\left( {\rm Re } \Sigma(k, 0) \right) \rangle_{k_{F,s}} \int_0^\infty \rho_s(\varepsilon) {\rm d}\varepsilon (\varepsilon_{s,k} -\mu)^2 \frac{\partial f^0}{\partial \varepsilon_{s,k}} \label{simeqleftvskuright}\\ = - \frac{2\pi^2}{3}\rho_{F,s} I'(0) T^2, \label{UsPhid}\end{aligned}$$ where $I'(0)$ is the derivative at $\varepsilon=0$ of $$I(\varepsilon)= \langle {\bi u}\cdot{\bi v}_{s}(k) \delta \left( {\rm Re } \Sigma({ k}, \varepsilon) \right) \rangle_{k_{F,s}}. \label{Ivarepsilondef}$$ The angular bracket in (\[Ivarepsilondef\]) represents the average over the Fermi surface. In (\[simeqleftvskuright\]), $\rho_s(\varepsilon)$ is the DOS per spin of the $s$ electron, and $\rho_{F,s}=\rho_s(\varepsilon_{F,s})$. Furthermore, we used $$\int_0^\infty \rho_s(\varepsilon) {\rm d}\varepsilon (\varepsilon_{s,k} -\mu)^2 \frac{\partial f^0}{\partial \varepsilon_{s,k}} =- \frac{\pi^2}{3}\rho_{F,s} T^2.$$ Similarly as (\[UsPhid\]), we obtain $${J}_s [ \Phi^d]= -2 \rho_{F,s}I(0), \label{JsPhid}$$ using $I(\varepsilon)$ in (\[Ivarepsilondef\]). As we find $I(0)$ is insignificant, a correction to the thermopower due to the $s$ electrons affected by the spin fluctuations is given by $$\Delta S_s =\frac{{U}_s[\Phi^d]}{T({ J}_s^s+{ J}_d^d)} = %- \frac{2\pi^2}{3 e } - \frac{\pi^2}{3 e } \frac{ \rho_{F,s} I'(0) }{ n_s+n_d}T. \label{DeltaS}$$ Self-energy ----------- We employ the self-energy in which a spin fluctuation excitation is emitted at one vertex and absorbed at the other one. It is given by $$\Sigma({\bi k}, \varepsilon_n) =-\frac{3}{2}J^2 T \sum_{\varepsilon_n'} \sum_{k'} {G}({\bi k}',\varepsilon_n') %^{-+} \chi ( {\bi k}-{\bi k}', \varepsilon_n-\varepsilon_n') %\delta_{q+k'-k}$$ where $\varepsilon_n=(2n+1)\pi T$ and $\varepsilon_n'=(2n'+1)\pi T$ are the fermion Matsubara frequencies, ${G}({\bi k},\varepsilon_n)$ is the temperature Green’s function for the $s$ electron, and $%^{-+} \chi ({\bi q},\omega_n)$ is related to the $d$ electron susceptibility $%^{-+} \chi ({\bi q},\omega)$ at the imaginary frequency $\omega = {\rm i}\omega_n$, where $\omega_n=2n\pi T$ is the boson Matsubara frequency. By an analytic continuation, we obtain the following relation for the retarded functions, denoted below with the subscript $R$, which are analytic in the upper half plane of the complex frequencies; $$\begin{aligned} \fl {\rm Re} \Sigma_R({\bi k}, \varepsilon)= - \frac{3}{2}J^2 \sum_{k'} \int^\infty_{-\infty} \frac{{\rm d}\omega}{2\pi} {\rm Im } G_R({\bi k}',\omega) {\rm Re} \chi_R( %^{-+} {\bi k}-{\bi k}', \varepsilon-\omega) %\left(1- 2f^0(\omega)\right) \tanh \frac{\omega}{2T} \nonumber\\ - \frac{3}{2}J^2 \sum_{k'} %\int d{\bi k}' \int^\infty_{-\infty} \frac{{\rm d}\omega}{2\pi} {\rm Re} G_R({\bi k}',\varepsilon-\omega) {\rm Im } %\chi^{-+}_R \chi_R ( {\bi k}-{\bi k}', \omega) %( 2n^0(\omega) +1) \coth \frac{\omega}{2T}. \label{SigmaR}\end{aligned}$$ To obtain the effect of $\Phi^d_k$, we substitute $ %\chi^{-+}_R(q,\omega) =\chi^{-+}_{\rm drag}(q,\omega +{\rm i}\delta) \chi_R(q,\omega) =\chi_{\rm drag}(q,\omega +{\rm i}\delta) $ from (\[chiphiq=\]). Hence the shift $\delta \left( {\rm Re} \Sigma_R({\bi k}, \varepsilon) \right)$ is obtained from (\[SigmaR\]) by substituting $ \frac{\partial %\chi^{-+}_R(q,\omega ) \chi_R(q,\omega ) }{\partial \omega} \Phi_q^d $ in place of $ %\chi^{-+}_R(q,\omega) \chi_R(q,\omega) $. For $G_R({\bi k}',\omega)$, we use a free propagator $ G_R({\bi k},\omega) ={1}/({\omega - \xi_k + {\rm i}\delta}), $ where $ \xi_k =\varepsilon_s(k)-\mu$. Owing to $ {\rm Im} G_R({\bi k}',\omega) = - \pi \delta (\omega - \xi_{k'})$ and (\[chi+-qomega\]) for $ %\chi^{-+}_R({\bi q},\omega) \chi_R({\bi q},\omega) $, the first term of (\[SigmaR\]) gives $$\begin{aligned} \fl { \sum_{k'} \int^\infty_{-\infty} \frac{{\rm d}\omega}{2\pi} {\rm Im } G_R({\bi k}',\omega) \frac{\partial}{\partial \varepsilon} %{\rm Re}\chi^{-+}_R({\bi k}-{\bi k}',\varepsilon-\omega) {\rm Re}\chi_R({\bi k}-{\bi k}',\varepsilon-\omega) \tanh \frac{\omega}{2T} %\left(1- 2f^0(\omega)\right) \Phi_{k-k'}^d } \nonumber\\ = -\frac{1}{2} \sum_{q} \frac{\partial}{\partial \varepsilon} {\rm Re} %\chi^{-+}_R({\bi q},\varepsilon-\xi_{k-q}) \chi_R({\bi q},\varepsilon-\xi_{k-q}) % \chi_{\rm drag}({\bi q},\varepsilon-\xi_{k-q}) \left(1- 2f^0(\xi_{k-q})\right)\Phi_q^d. \nonumber %\\ %&=&\sum_{q} %%\frac{\partial}{\partial \omega} \chi_R^{-+}({\bi q},\varepsilon-\xi_{k-q}) %\left( %\frac{\pi \bar{I}}{4 \bar{q}} %\right)^2 %\frac{{\varepsilon-\xi_{k-q}}}{\varepsilon_{F,d}} %\frac{\rho_{F,d}\left( %K_0^2 + \frac{\bar{I}}{12}\bar{q}^2 %\right)}{ %\left( %\left( %K_0^2 + \frac{\bar{I}}{12}\bar{q}^2 %\right)^2 %+\left( %\frac{\pi}{4}\bar{I} \frac{{\varepsilon-\xi_{k-q}}}{\bar{q} \varepsilon_{F,d}} %\right)^2 %\right)^2 %} %\left(1- 2f^0(\xi_{k-q})\right)\Phi_{q}'.\end{aligned}$$ As this give only a convergent result, we neglect this part. Using (\[phiks=ukphiqd=uq\]) for $\Phi_q^d$, for (\[Ivarepsilondef\]) we find $$\fl I(\varepsilon)= - \frac{J^2} {2(2\pi)^4}\int {\rm d}{\bi q} \int^\infty_{-\infty} \frac{{\rm d}\omega }{\varepsilon-\omega + {\bi v}_{s}(k)\cdot {\bi q} %- \xi_{k-q} } \frac{\partial %{\rm Im } \chi^{-+}_R({\bi q},\omega) {\rm Im } \chi_R({\bi q},\omega) }{\partial \omega}\coth \frac{\omega}{2T} \langle{\bi v}_s(k_{F,s})\cdot{\bi q}\rangle_{k_{F,s}},$$ where we substituted $\xi_{k-q} \simeq - {\bi v}_{s}(k_{F,s})\cdot {\bi q}$, which holds in the important integral region of small $|{\bi q}|$. Integrating over the angle between ${\bi v}_{s}(k_{F,s})$ and ${\bi q}$, we obtain $$\begin{aligned} \fl I(\varepsilon) %&=& %- %\frac{J^2}{2(2\pi)^3} %%\int {\rm d}{\bi q} %%2\pi %\int_0^{2k_{F,s}} q^2{\rm d}q % \int_{-1}^1 {\rm d}x %\int^\infty_{-\infty} %\frac{{\rm d}\omega }{\varepsilon-\omega +{v}_{k,s}q x} %\frac{\partial % {\rm Im } \chi^{-+}_R({q},\omega) %}{\partial \omega}\coth \frac{\omega}{2T} %v_{k,s} q x %\nonumber\\ = - \frac{J^2}{2(2\pi)^3} \int_0^{2k_{F,s}} q^2{\rm d}q % \int_{-1}^1 {\rm d}x \int^\infty_{-\infty}{\rm d}\omega \left( %\frac{2}{{v}_{k,s}q} - 2- \frac{\varepsilon-\omega}{{v}_{F,s}q} \log \left| \frac{ \varepsilon-\omega+{v}_{F,s}q } { \varepsilon-\omega-{v}_{F,s}q } \right| \right) \nonumber\\ %\frac{{\rm d}\omega }{\varepsilon-\omega +{v}_{k,s}q x} \times \frac{\partial % {\rm Im } \chi^{-+}_R({q},\omega) {\rm Im } \chi_R({q},\omega) }{\partial \omega}\coth \frac{\omega}{2T}. %v_{k,s} q %x \label{defIeps}\end{aligned}$$ In the parenthesis, only those terms odd in $\omega$ contribute to the integral over $\omega$. Hence we find $I(0) =0,$ and the leading term in $|\omega/(v_{F,s} q)|$ gives $$\begin{aligned} \fl I'(0)\simeq - \frac{J^2}{2\pi^3{v}_{F,s}^2} \int_0^{2k_{F,s}} {\rm d}q \int^\infty_0 \omega {{\rm d}\omega } \frac{\partial % {\rm Im } \chi^{-+}_R({q},\omega) {\rm Im } \chi_R({q},\omega) }{\partial \omega}\coth \frac{\omega}{2T} \\ = - \frac{J^2 k_{F,d}\varepsilon_{F,d}}{{2\pi^3}{{v}_{F,s}^2}} \int_0^{2k_{F,s}/k_{F,d}} {\rm d}\bar{q} \int^\infty_0 \bar{\omega} {{\rm d}\bar{\omega} } \frac{\partial % {\rm Im } \chi^{-+}_R(\bar{q},\bar{\omega}) {\rm Im } \chi_R(\bar{q},\bar{\omega}) }{\partial \bar{\omega}}\coth \frac{ \varepsilon_{F,d}\bar{\omega} }{2T}.\end{aligned}$$ To put in this expression, we may write the susceptibility in (\[chi+-qomega\]) as $$\frac{\partial}{\partial \bar{\omega}} %{\rm Im } \chi_R^{-+}({q},{\omega}) = {\rm Im } \chi_R({q},{\omega}) = \frac{\partial}{\partial \bar{\omega}} \left( \frac{\bar{\omega}}{\bar{q}} \frac{A}{\left( \bar{\kappa}^2+ \bar{q}^2 \right)^2 + \left( C \bar{\omega}/\bar{q} \right)^2 } \right)$$ for $ \bar{\omega} < 2\bar{q}, $ where $ A={36 \pi \rho_{F,d}}/{\bar{U}^2}$, $C =3 {\pi}$, and $$%$ \bar{\kappa}^2= 12 K_0^2/{\bar{U}} = 12 (1-{\bar{U}})/{\bar{U}}. %$$$ We find $$\begin{aligned} \fl I'(0) = - \frac{J^2 k_{F,d}\varepsilon_{F,d} A}{{\pi^3}{{v}_{F,s}^2}} \left( {\cal I}_0 +{\cal I}(T) \right) %\\&=& = - 36 \left( \frac{\bar{J}}{\bar{U}} \right)^2 \left( \frac{k_{F,d}}{k_{F,s}} \right)^4 \left( {\cal I}_0 +{\cal I}(T) \right), \label{I'0=calI0IT}\end{aligned}$$ where $ \bar{J}\equiv \rho_{F,s} J, $ $$\fl {\cal I}_0= \frac{1}{2} \int_0^{2k_{F,s}/k_{F,d}} \frac{{\rm d}\bar{q}}{{\bar{q}}} \int^{2\bar{q}}_0 \bar{\omega} {{\rm d}\bar{\omega} } \frac{\partial}{\partial \bar{\omega}} \left( \frac{ {\bar{\omega}} }{\left( \bar{\kappa}^2+ \bar{q}^2 \right)^2 + \left( C \bar{\omega}/\bar{q} \right)^2 } \right),$$ and $$\fl {\cal I}(T)=\int_0^{2k_{F,s}/k_{F,d}} \frac{{\rm d}\bar{q}}{{\bar{q}}} \int^{2\bar{q}}_0 \bar{\omega} {{\rm d}\bar{\omega} } \frac{\partial}{\partial \bar{\omega}} \left( \frac{ {\bar{\omega}} }{\left( \bar{\kappa}^2+ \bar{q}^2 \right)^2 + \left( C \bar{\omega}/\bar{q} \right)^2 } \right) n^0(\varepsilon_{F,d}\bar{\omega}). \label{calI}$$ The former ${\cal I}_0$ is the part independent of temperature $T$, while the temperature dependence in the latter ${\cal I}(T)$ arises from the Bose factor $n^0(\omega)$. In particular, for $\bar{\kappa}=0$, we obtain $$\begin{aligned} {\cal I}_0 &=& \frac{1}{4} \int_0^{(2k_{F,s}/k_{F,d})^2} {{\rm d}\bar{q}^2} \int^{2\bar{q}}_0 \bar{\omega} {{\rm d}\bar{\omega} } \frac{\partial}{\partial \bar{\omega}} \left( \frac{ {\bar{\omega}} }{ \bar{q}^6 + \left( C\bar{\omega} \right)^2 } \right) \nonumber\\&=& % \frac{1}{8C^2} \int_0^{(2k_{F,s}/k_{F,d})^2}{{\rm d}\bar{q}^2} \left( \frac{1 }{\bar{q}^4+(2C)^2 } + \frac{1}{8C^2} \log \frac{\bar{q}^4}{\bar{q}^4 + (2C)^2 } \right) \nonumber\\ &=& \frac{(k_{F,s}/k_{F,d})}{4C^2} \log \frac{( k_{F,s}/k_{F,d})^2}{ (k_{F,s}/k_{F,d})^2+ C^2}.\end{aligned}$$ \[sec:calIT\]Temperature dependence: ${\cal I}(T)$ -------------------------------------------------- To obtain an explicit expression for the temperature dependent part ${\cal I}(T)$, we adopt an approximation to set $$n^0(\omega)= \left\{ \begin{array}{ll} \displaystyle {T}/{\omega}, \quad& \omega< cT\\ 0, & \omega>cT\\ \end{array} \right. \label{n0omega=om<cT}$$ where $c$ is a constant of order unity (cf. (\[constc0.928\])). Consequently, we obtain $${\cal I}(T)= {\cal I}_a(T)+{\cal I}_b(T),$$ where $$\fl {\cal I}_a(T) = \frac{1}{2} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 \int_0^{\bar{q}_0^2} {{\rm d}\bar{q}^2} \int^{2\bar{q} \varepsilon_{F,d}/T}_0 {{\rm d}u } \frac{\partial}{\partial u} \left( \frac{{u}}{ \bar{q}^2\left( \bar{\kappa}^2+ \bar{q}^2 \right)^2 + \left( C Tu /\varepsilon_{F,d} \right)^2 } \right), \label{IaTdef}$$ and $$\fl {\cal I}_b(T) = \frac{1}{2} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 \int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} {{\rm d}\bar{q}^2} \int^{c }_0 {{\rm d}u } \frac{\partial}{\partial u} \left( \frac{{u}}{ \bar{q}^2\left( \bar{\kappa}^2+ \bar{q}^2 \right)^2 + \left( C Tu /\varepsilon_{F,d} \right)^2 } \right). \label{IbTdef}$$ Here we introduced a characteristic scale for the normalized momentum, $$\bar{q}_0 \equiv \frac{c T}{2\varepsilon_{F,d}}. \label{q0def}$$ We may take the limit $\bar{\kappa}=0$ for (\[IaTdef\]) to obtain $$\begin{aligned} {\cal I}_a(T) %&= & (to omit) %\frac{1}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\int_0^{\bar{q}_0^2}{{\rm d}\bar{q}^2} %%% %% \int^{2\bar{q} \varepsilon_{F,d}/T}_0{{\rm d}u } %%\frac{\partial}{\partial u} %\left( %\frac{{2\bar{q} \varepsilon_{F,d}/T}}{ %\bar{q}^2\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( % C T %2\bar{q} \varepsilon_{F,d}/T %/\varepsilon_{F,d} %\right)^2 %} %\right) %\nonumber\\ &=& (to omit) %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\int_0^{\bar{q}_0^2}{{\rm d}\bar{q}} %\left( %\frac{{2 %%\bar{q} %\varepsilon_{F,d}/T}}{ %%\bar{q}^2 %\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( %2C %%\bar{q} %\right)^2 %} %\right) %\nonumber\\ %&=& %\frac{2T}{\varepsilon_{F,d}} %\int_0^{\bar{q}_0} %\frac{{{{\rm d}\bar{q}} %%\bar{q} %%\varepsilon_{F,d}/T %}}{ %%\bar{q}^2 %\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( %2C %%\bar{q} %\right)^2 %} %\nonumber\\ \simeq \frac{2T}{\varepsilon_{F,d}} \int_0^{\bar{q}_0} \frac{{{{\rm d}\bar{q}} %\bar{q} %\varepsilon_{F,d}/T }}{ %\bar{q}^2 \bar{q}^4 + \left( 2C %\bar{q} \right)^2 } \simeq %2\left(\frac{T}{\varepsilon_{F,d}}\right) \frac{ {\bar{q}_0}T}{ 2C^2\varepsilon_{F,d}} %\nonumber\\ %= %\frac{c}{ \left(2C\right)^2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 = c \left(\frac{T}{2C\varepsilon_{F,d}}\right)^2.\end{aligned}$$ On the other hand, for (\[IbTdef\]), we obtain $${\cal I}_b(T) = %(to omit) %\frac{1}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} %{{\rm d}\bar{q}^2} % \int^{c %}_0 %{{\rm d}u } %\frac{\partial}{\partial u} %\left( %\frac{{u}}{ %\bar{q}^2\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( % C Tu %/\varepsilon_{F,d} %\right)^2 %} %\right) %\nonumber\\&=&(to omit) %\frac{c}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} %%{{\rm d}\bar{q}^2} %%\left( %\frac{ %{{\rm d}\bar{q}^2} %}{ %\bar{q}^2\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( %c C T %/\varepsilon_{F,d} %\right)^2 %} %%\right) %\nonumber\\&=& \frac{c}{2} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 \int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} %{{\rm d}\bar{q}^2} %\left( \frac{ {{\rm d}\bar{q}^2} }{ \bar{q}^2\left( \bar{\kappa}^2+ \bar{q}^2 \right)^2 + \left( 2C \bar{q}_0 \right)^2 }, \label{Ib(T)} %\right)$$ for which the main contribution comes from around the lower limit of the integral. Let us discuss two cases depending on the relative size of $\bar{\kappa}$ and $\bar{q}_0$, separately. First we consider the case $\bar{\kappa}/\bar{q}_0 \gg 1$, which is the low temperature limit for $\bar{\kappa}>0$. In this case, we obtain $$\begin{aligned} {\cal I}_b(T) %\frac{c}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} %\frac{ %{{\rm d}\bar{q}^2} %}{ %\bar{q}^2\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( %2C \bar{q}_0 %\right)^2 %} &\simeq & \frac{c}{2} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 \frac{1}{\bar{\kappa}^4} \int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} \frac{ {{\rm d}\bar{q}^2} }{ \bar{q}^2 + \left( 2C \bar{q}_0 /\bar{\kappa}^2 \right)^2 } %\nonumber\\ %&=& %(to omit) %\frac{c}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\frac{1}{\bar{\kappa}^4} %\log \frac{ %{(2k_{F,s}/k_{F,d})^2} %+ % \left( %2C \bar{q}_0 %/\bar{\kappa}^2 %\right)^2 %}{ %\bar{q}_0^2 %+ % \left( %2C \bar{q}_0 %/\bar{\kappa}^2 %\right)^2 %} %\nonumber\\&\simeq %&(to omit) %\frac{c}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\frac{1}{\bar{\kappa}^4} %\log \frac{ %{(2k_{F,s}/k_{F,d})^2} %+ % \left( %2C \bar{q}_0 %/\bar{\kappa}^2 %\right)^2 %}{ %%\bar{q}_0^2+ % \left( %2C \bar{q}_0 %/\bar{\kappa}^2 %\right)^2 %} \nonumber\\ &\simeq& %\simeq \frac{c}{2} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 \frac{1}{\bar{\kappa}^4} \log \frac{ {(k_{F,s}/k_{F,d})^2} + \left( C \bar{q}_0 /\bar{\kappa}^2 \right)^2 }{ %\bar{q}_0^2+ \left( C \bar{q}_0 /\bar{\kappa}^2 \right)^2 }.\end{aligned}$$ In terms of a characteristic temperature of spin fluctuations defined by $$\bar{T} \equiv \frac{\varepsilon_{F,d} \bar{\kappa}^2}{C} %=\varepsilon_{F,d} (12 K_0^2/{\bar{U}})/(3\pi) =\frac{4 \varepsilon_{F,d} K_0^2}{\pi{\bar{U}}}, \label{Tsfdef}$$ we find $${\cal I}_b(T) \simeq \frac{c}{2C^2} \left(\frac{T}{ \bar{T}}\right)^2 %\frac{1}{\bar{\kappa}^4} \log \frac{ {(2 k_{F,s}/k_{F,d})^2} + \left(c T/\bar{T}\right)^2 }{ \left(c T/\bar{T}\right)^2 % \left(C \bar{q}_0/\bar{\kappa}^2\right)^2 }. \label{IbT=c2C2log}$$ In the literature, a spin fluctuation temperature, $${T}_{\rm sf} =\varepsilon_{F,d} K_0^2 ={\varepsilon_{F,d}}({1-\bar{U}}),$$ is commonly used as well. Indeed we have $\bar{T} \simeq {T}_{\rm sf} $ for $\bar{U}\simeq 1$. Lastly, in the quantum critical limit $\bar{\kappa}/\bar{q}_0 \ll 1$, we obtain $$\begin{aligned} {\cal I}_b(T) %&= & %(to omit)\frac{1}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %..\int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} %\frac{ %{{\rm d}\bar{q}^2} %}{ %\bar{q}^2\left( %\bar{\kappa}^2+ %\bar{q}^2 %\right)^2 %+ % \left( %2C \bar{q}_0 %\right)^2 %} %%\nonumber\\ % %&\simeq & %\simeq %\frac{1}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 %\int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} %\frac{ %{{\rm d}\bar{q}^2} %}{ %\bar{q}^6 %+ % \left( %2C \bar{q}_0 %\right)^2 %} &\simeq& \frac{1}{2} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 \int_{\bar{q}_0^2}^{(2k_{F,s}/k_{F,d})^2} \frac{ {{\rm d}\bar{q}^2} }{ \bar{q}^6 + \left( 2C \bar{q}_0 \right)^2 } %\nonumber\\&\simeq& %\simeq %\frac{1}{2} %\left(\frac{T}{\varepsilon_{F,d}}\right)^2 % \frac{2\sqrt{3}{\pi}}{9 (2C\bar{q}_0)^{4/3}} \nonumber\\&\simeq& \frac{\sqrt{3}{\pi}}{9 (2C\bar{q}_0)^{4/3}} \left(\frac{T}{\varepsilon_{F,d}}\right)^2 = %= \frac{{\pi}}{3\sqrt{3} (C {c })^{4/3}} \left(\frac{T}{\varepsilon_{F,d}}\right)^{2/3}. \label{Ib(T)c}\end{aligned}$$ \[sec:summary\]Results ---------------------- We may neglect ${\cal I}_a(T)$ against ${\cal I}_b(T)$, for $\bar{T}\ll \varepsilon_{F,d}$. For (\[DeltaS\]), we obtain $$\Delta S_s %&=&(to omit)\frac{{U}_s[\Phi^d]}{T({ J}_s^s+{ J}_d^d)} %= %- \frac{2\pi^2}{3 e } %\frac{ %\rho_{F,s} I'(0) T %}{ n_s+n_d} %\nonumber\\ %&=&(to omit) % \frac{2\pi^2}{3 e } %36 %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^4 %\frac{ %\rho_{F,s} %( %\left( %{\cal I}_0 +{\cal I}(T) %\right), %) % T %}{ n_s+n_d} %\nonumber\\ \simeq \Delta S_0^s + \Delta S_s(T),$$ where $$\Delta S_0^s %&=& % \frac{24\pi^2}{ e( n_s+n_d) } %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^4 %\rho_{F,s} {\cal I}_0 % T %\nonumber\\ %&=& % \frac{24\pi^2}{ e( n_s+n_d) } %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^4 %\rho_{F,s} % \frac{(k_{F,s}/k_{F,d})}{4C^2} % T % \log \frac{( k_{F,s}/k_{F,d})^2}{ %(k_{F,s}/k_{F,d})^2+ C^2} %\nonumber\\ %&=& % \frac{24\pi^2}{ e( n_s+n_d) } %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^3 %\rho_{F,s} % \frac{1}{4(3\pi)^2} % T % \log \frac{( k_{F,s}/k_{F,d})^2}{ %(k_{F,s}/k_{F,d})^2+ C^2} %\nonumber\\ = % \frac{2 \rho_{F,s} T \frac{ \rho_{F,s} T }{3 e( n_s+n_d) } \left( \frac{\bar{J}}{\bar{U}} \right)^2 \left( \frac{k_{F,d}}{k_{F,s}} \right)^3 \log \frac{( k_{F,s}/k_{F,d})^2}{ (k_{F,s}/k_{F,d})^2+ (3\pi)^2},$$ and $$\Delta S_s(T) = % \frac{24\pi^2}{ e( n_s+n_d) } \frac{12\pi^2}{ e( n_s+n_d) } \left( \frac{\bar{J}}{\bar{U}} \right)^2 \left( \frac{k_{F,d}}{k_{F,s}} \right)^4 \rho_{F,s}{\cal I}_b(T) T. \label{DeltaSs=24pi2}$$ The former $\Delta S_0^s$ to modify the $T$ linear term may be effectively neglected, while the latter $ \Delta S_s(T)$ gives a sub-leading correction. At the low temperature $T\ll \bar{T}$, with (\[IbT=c2C2log\]), we get $$\begin{aligned} \fl \Delta S_s(T) %&=& % \frac{24\pi^2}{ e( n_s+n_d) } %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^4 %\rho_{F,s} % T %\frac{c}{2C^2} %\left(\frac{T}{ % \bar{T}}\right)^2 %\log \frac{ %{(2 k_{F,s}/k_{F,d})^2} %+ % \left(c T/\bar{T}\right)^2 %}{ % \left(c T/\bar{T}\right)^2 %} %\nonumber\\ %&=& %% \frac{4c }{3 e( n_s+n_d) } % \frac{2c }{3 e( n_s+n_d) } %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^4 %\rho_{F,s} % T %\left(\frac{T}{ % \bar{T}}\right)^2 %\log \frac{ %{(2 k_{F,s}/k_{F,d})^2} %+ % \left(c T/\bar{T}\right)^2 %}{ % \left(c T/\bar{T}\right)^2 %} %\nonumber\\ &\simeq & % \frac{4 }{3 e( n_s+n_d) } \frac{2 }{3 e( n_s+n_d) } \left( \frac{\bar{J}}{\bar{U}} \right)^2 \left( \frac{k_{F,d}}{k_{F,s}} \right)^4 \rho_{F,s} T \left(\frac{T}{ \bar{T}}\right)^2 \log \frac{ {(2 k_{F,s}/k_{F,d})^2} + \left( T/\bar{T}\right)^2 }{ \left( T/\bar{T}\right)^2 }, \label{Delta S_s(T)}\end{aligned}$$ where we set $c\simeq 1$ for simplicity (instead of (\[constc0.928\])). In the opposite limit, from (\[Ib(T)c\]), we obtain $$\begin{aligned} \Delta S_s(T) %&=& %% \frac{24\pi^2}{ e( n_s+n_d) } % \frac{12\pi^2}{ e( n_s+n_d) } %\left( %\frac{\bar{J}}{\bar{U}} %\right)^2 %\left( %\frac{k_{F,d}}{k_{F,s}} %\right)^4 %\rho_{F,s} % T % \frac{{\pi}}{3\sqrt{3} (C {c })^{4/3}} %\left(\frac{T}{\varepsilon_{F,d}}\right)^{2/3} %\nonumber\\ &\simeq& % \frac{8\pi^{5/3}}{{3}^{11/6} e( n_s+n_d) } \frac{4\pi^{5/3}}{{3}^{11/6} e( n_s+n_d) } \left( \frac{\bar{J}}{\bar{U}} \right)^2 \left( \frac{k_{F,d}}{k_{F,s}} \right)^4 \rho_{F,s} T \frac{}{ } \left(\frac{T}{\varepsilon_{F,d}}\right)^{2/3}. \label{Deltasst.QC}\end{aligned}$$ In the same manner as $U_s[\Phi^d]$ discussed above, one can think of an additional heat current $\Delta U_d[\Phi^d]$ caused by the intraband many-body effect due to the on-site repulsion $U$ in the $d$ band. Formally, the corresponding results are obtained straightforwardly by replacing $k_{F,s}$, $\rho_{F,s}$, and $3J^2/2$ in the above results by $k_{F,d}$, $\rho_{F,d}$, and $U^2$, respectively, i.e., $$\Delta S_d(T) \simeq % \frac{8 }{9 e( n_s+n_d) } \frac{4 }{9 e( n_s+n_d) } \rho_{F,d} T \left(\frac{T}{ \bar{T}}\right)^2 \log \frac{ 4 + \left( T/\bar{T}\right)^2 }{ \left( T/\bar{T}\right)^2 }, \label{DeltaSdT}$$ in place of (\[Delta S\_s(T)\]). The results are modified in some ways in generalizing the model. The constant of 4 in the logarithm of (\[DeltaSdT\]) stems from $(2k_{F,d}/k_{F,d})^2$, where $2k_{F,d}$ sets the upper cutoff for the momentum $q$ of spin fluctuations. If we should have set a cutoff parameter $q_c$ differently, the factor should be replaced by $\bar{q}_c^2$, where $\bar{q}_c \equiv q_c/k_{F,d}$. Moreover, if we had assumed a phenomenological coupling $g$ between electrons and spin fluctuations instead of $U$, the results will be reduced by a factor of $(g/U)^2$. \[sec:disscs\]Discussion: comparison with experiment ==================================================== To compare the theoretical result $S(T)$ with experiment, some assumptions like the free dispersions in (\[varepsk\]) and (\[varepdk\]) should not be taken literally. In particular, the $T$-linear terms $S_0^i$ ($i=s,d$) for $S_0$ would be able to have either positive or negative sign, depending on the energy dependence of the respective DOS at the Fermi level, while the relation $|S_0^d|\gg |S_0^s|$ will always hold true for their relative magnitudes. Therefore, as the leading effect at low temperature, we generally expect an enhanced $T$-linear term, $$S_0 \simeq \bar{S}_0^d\equiv \frac{ \displaystyle {n_d} }{ \displaystyle{n_s} +{n_d} } S_0^d, \label{Sdrag simeq nd}$$ unless $n_s\gg n_d$. Effectively, this term is indistinguishable from the diffusion term contribution, as discussed below (\[S0\^d\]). It is indeed due to the drag current of the heavy $d$ electrons. Without drag, we recover the conventional result $S\simeq S_0\simeq S_0^s$ of the diffusion thermopower due to the conduction electrons. We expect that the latter holds true at high temperature $T\agt \bar{T}$ where the $s$-$d$ scatterings become too weak to sustain the $d$ electron drag. Therefore, it is reasonably expected that we should find some structure in the temperature dependence of the thermopower $S(T)$ around $T\alt \bar{T}$, which is brought about by the crossover between the $T$-linear terms with different magnitudes of coefficients. This is schematically shown in figure \[fig1\]. ![\[fig1\] The effect of spin-fluctuation drag on the thermopower $S(T)$ is schematically shown. The bold lines are drawn to interpolate the two linear relations, $S(T)=\bar{S}_0^d$ and $\bar{S}_0^s$. The low temperature $\bar{S}_0^d$ in (\[Sdrag simeq nd\]) is due to drag of those heavy electrons pertaining to the spin fluctuations, while the normal diffusion term $S\simeq {S}_0^s$ due to light conduction electrons appear at high temperature $T\gg \bar{T}\simeq T_{\rm sf}$, a spin-fluctuation temperature. The left (a) is for $\bar{S}_0^d>0$, while (b) for $\bar{S}_0^d<0$. The latter is compared with ${\rm RCo_2}$ (R=Sc, Y and Lu) by Gratz $et$ $al$.[@grbbmg95; @gratz97; @gm01] ](fig1.eps) Takabatake $et$ $al$.[@tmnhmsus06] have shown experimentally by applying the magnetic field of 15T that an S-shaped structure in $S(T)$ of ${\rm CaFe_4Sb_{12}}$ observed at low $T< \bar{T}\simeq 50$K is suppressed to yield a normal $T$-linear diffusion term. This is consistent with our result for $\bar{S}_0^d<0$, $S_0^s>0$ and $|\bar{S}_0^d/S_0^s|\simeq 8$. In this case, the conduction band for $S_0^s$ consists mainly of 5$p$ states of antimony. Moreover, they have shown that the temperature dependence of the spin-fluctuation contribution $\Delta S=S-S_0^s$ is not monotonic. To explain this theoretically goes beyond the scope of this paper, as it requires us to solve the transport equations concretely. Similarly known before were the low temperature minima in the thermopower of ${\rm RCo_2}$ (R=Sc, Y and Lu), which had been stressed by Gratz $et$ $al$.[@grbbmg95; @gratz97; @gm01] as the experimental evidence of paramagnon drag. Their results can be compared with our result for $\bar{S}_0^d \ll S_0^s< 0$ in figure \[fig1\] (b). On the correction terms, it is generally expected that the $d$ electron contribution $\Delta S_d$ will become more important than $\Delta S_s$ when the $d$ electron current becomes relevant indeed. As discussed in section \[subsec:ee\], we have to consider two sources of contributions, one due to the equilibrium effect and the other due to the non-equilibrium effect in section \[sec:std\]. Interestingly, we find that both give the same temperature dependence, $ S(T)= \alpha T + \beta T^3 \log T^{-1} %\label{STLSF} $ away from the QCP. Nevertheless, we notice an important difference. While we observe $\beta\propto 1/K_0^4$ from the results of the last section, $\beta$ expected from a correction term in (\[S0edTnde\]) has an extra factor of $1/K_0^2$.[@de66] This means that the equilibrium effect becomes more important. We suspect that this would hold true at the QCP too, although there has been no definite calculation deriving the corresponding free energy correction $\propto T^{5/3}$ in accordance with our result. In any case, we remark that the relative magnitude of the electron numbers $n_s$ and $n_d$ may have an effect on the correction terms, the sign of which will depend on the factor $ en= e(n_s+ n_d)$, that is, the direction of the net current. In most cases where the model applies, the current carrier in the heavy-electron band will be hole like. Moreover, we generally expect that $|n_s|$ will not exceed $|n_d|$, or the net current would be hole-like, $e(n_s+ n_d) >0$. Accordingly, $\Delta S_i >0$ (cf. (\[qequiv es/c\])). This is consistent with a model calculation of the spin fluctuation effect on the resistivity, where Jullien $et$ $al.$[@jbc74; @cij78] pointed out the important role of the parameter $\xi = k_{F,c}/k_{F,d}$ on the transport properties of spin fluctuations systems. We observe the dependence in our results of (\[Delta S\_s(T)\]) and (\[Deltasst.QC\]). To compare their numerical results with experiments, they should choose $\xi\le 1$ generically, that is, $|n_s|\alt |n_d|$. To conclude, let us fit the low-temperature experimental data for $S(T)$ of $\rm AFe_4Sb_{12}$ $\rm (A=Ba, Sr, Ca)$ reported by Matsuoka $et$ $al.$[@mhittm05] with $$S(T)= \alpha T + \beta T \left(\frac{T}{ \bar{T}}\right)^2 \log \frac{ \delta + \left( T/\bar{T}\right)^2 }{ \left( T/\bar{T}\right)^2 }, \label{STLSF}$$ where $\alpha$, $\beta$, $\bar{T}$ and $\delta$ are regarded as parameters. In table \[tab:table\], we present the fitting parameters obtained for $\delta =4$ by the least squares fits of the low temperature part of the data for $T\alt 17$K $(< \bar{T})$. The results are shown in figure \[fig\], along with the experimental data points. We find that $\alpha$’s do not depend much on the other parameters, and the ratios of $\beta$ and $\bar{T}$ between materials are nearly independent of $\delta$. The relatively large values for $\alpha$ and $\beta$ will be more properly ascribed to the heavier $d$ band than to the conduction band, in accordance with our result. Note that these coefficients are susceptible to the equilibrium effect of mass enhancement,[@de66; @bmf68] which we did not take into account explicitly (cf. section \[subsec:ee\]).[^2] The positive $\beta $ implies that the net current is in the hole-like direction. The relative material dependence of $\beta$ in table \[tab:table\] may be qualitatively compared with the observed static uniform susceptibility $\chi_0 \propto \rho_d/K_0^2$, that is, $\chi_0({\rm BaFe_4Sb_{12}}): \chi_0({\rm SrFe_4Sb_{12}}): \chi_0({\rm CaFe_4Sb_{12}})\simeq 1:1.6:2.5 $. ![\[fig\] The points are the experimental data of $S(T)$ for $\rm AFe_4Sb_{12}$ $\rm (A=Ba, Sr, Ca)$.[@mhittm05] The lines are the least squares fits by the theoretical expression in (\[STLSF\]) with the parameters given in table \[tab:table\]. ](fig2.eps) [@lccc]{} & $\alpha$ [\[$\rm \mu V/K^2$\]]{} & $\beta$ [\[$\rm \mu V/K^2$\]]{} & $\bar{T}$ [\[K\]]{}\ $\rm BaFe_4 Sb_{12}$ & $-0.35$ & 1.5 & 48\ $\rm SrFe_4 Sb_{12}$ & $-0.74$ & 1.6 & 37\ $\rm CaFe_4 Sb_{12}$ & $-1.4\0$ & 3.7 & 58\ The author is very grateful to Eiichi Matsuoka for the original data of [@mhittm05]. \[sec:formaltransporttheory\]Formal transport theory ==================================================== The formal expressions for resistivity and thermopower referred to in the main text are derived by adapting a general variational method of Ziman,[@ziman60] according to which $\Phi^s$ in (\[f=f0-Phik\]) and $\Phi^d$ in (\[chiphiq=\]) are regarded as variational trial functions. Below we substitute $\eta_i \Phi^i$ for $\Phi^i$ ($i=s,d$), and take variation with respect to the arbitrary parameters $\eta_i$. On the one hand, the microscopic entropy production rate corresponding to (\[fkscattLinearD\]) is given by $$\begin{aligned} \dot{S}_{\rm scatt}&=& \frac{1}{T^2} \sum_{k,q} \left( %\sum_{i=0,1} \eta_s \Phi_k^s - \eta_s\Phi_{k+q}^s + \eta_d \Phi_q^d \right)^2 {\cal P}^{k+q}_{kq} \nonumber \\ &&\equiv \frac{1}{T} \sum_{i,j=s,d} P_{ij} \eta_i \eta_j. \label{Sscatt=}\end{aligned}$$ The components of the matrix $P_{ij}$ defined in (\[Sscatt=\]) are explicitly given by $$\begin{aligned} P_{ss} &=&\frac{1}{T}\sum_{k,q} {\cal P}^{k+q}_{kq} \left( \Phi_k^s - \Phi_{k+q}^s \right)^2, \label{Pss} \\ P_{sd}=P_{ds} &=&\frac{1}{T}\sum_{k,q} {\cal P}^{k+q}_{kq} \left( \Phi_k^s - \Phi_{k+q}^s \right) \Phi_q^d , \label{Psd} \\ P_{dd} &=&\frac{1}{T}\sum_{k,q} {\cal P}^{k+q}_{kq} (\Phi_q^d) ^2. \label{Pdd}\end{aligned}$$ In (\[Sscatt=\]), not only emission of a paramagnon corresponding to (\[fkscattLinearD\]), but the reverse absorption process is also taken into account. In the special case of the full drag without Umklapp processes, there holds the relation $\Phi_{k+q}^s - \Phi_k^s =\Phi_q^d$ by (\[phiks=ukphiqd=uq\]), so that we get the following identities, $$P_{ss}=P_{dd} =-P_{sd}. \label{Pss=Pdd=-Psd}$$ On the other hand, the macroscopic entropy production is given by $$\dot{S}_{\rm macro}= \frac{{\bi J} \cdot {\bi E}}{T} + {\bi U} \cdot \nabla \frac{1}{T}.$$ In the linear response regime, the electric current ${\bi J}$ and the heat current ${\bi U}$ are written as $$\begin{aligned} {\bi J} &=& \eta_s {\bi J}[\Phi^s] +\eta_d {\bi J}[\Phi^d], \label{J=etasd} \\ {\bi U} &=& \eta_s {\bi U}[\Phi^s] +\eta_d {\bi U}[\Phi^d], \label{U=etasd}\end{aligned}$$ where ${\bi J}[\Phi^i]$ denotes the current flow caused by $\Phi^i$ ($i=s,d$), i.e., ${\bi J}[\Phi^i]$ formally represents the part of the total current which depends linearly on $\Phi^i$. ${\bi U}[\Phi^i]$ is similarly defined. In general, these currents have different functional forms. It is remarked that ${\bi J}[\Phi^i]$ is not to be identified with the current in the $i$ band. Owing to the interband interaction, the distribution shift $\Phi^i$ in the $i$ band can induce a current in the other band. The variational parameters $\eta_i$ are determined so as to maximize $\dot{S}_{\rm scatt}$ after equating $\dot{S}_{\rm scatt}$ and $\dot{S}_{\rm macro}$.[@ziman60] Substituting the solutions into (\[J=etasd\]) and (\[U=etasd\]), we obtain $${\bi J} = \sum_{i,j=s,d} {\bi J}[\Phi^i] (P^{-1})_{ij} \left( {{\bi J}[\Phi^j] \cdot {\bi E}} -\frac{1}{T} {\bi U}[\Phi^j] \cdot \nabla T \right), \label{J=sumijsd=J}$$ and $${\bi U} =\sum_{i,j=s,d} {\bi U}[\Phi^i] (P^{-1})_{ij} \left( { {\bi J}[\Phi^j] \cdot {\bi E}} -\frac{1}{T} {\bi U}[\Phi^j] \cdot \nabla T \right), \label{U=sumijsd=U}$$ where $(P^{-1})_{ij}$ is the inverse matrix of $P_{ij}$. For definiteness, let the applied field ${\bi E}$ and $\nabla T$ be in the direction of a unit vector ${\bi u}$. In an isotropic system, or in cubic symmetry, the results are expressed with the magnitudes $J[\Phi^i] ={\bi J}[\Phi^i] \cdot {\bi u}$ and $U[\Phi^i] ={\bi U}[\Phi^i] \cdot {\bi u}$. From (\[J=sumijsd=J\]), we obtain the electrical conductivity, $$\sigma = \sum_{l,m=s,d} { J}[\Phi^l] (P^{-1})_{lm} { J}[\Phi^m].$$ The resistivity $R=\sigma^{-1}$ is given by $$R= R_0(T)\frac{ \displaystyle {1 - \frac{P_{sd}P_{ds}}{P_{ss} P_{dd}}} }{ \displaystyle 1+ \left( \frac{{ J}[\Phi^d]}{{ J}[\Phi^s]} \right)^2 \frac{P_{ss}}{P_{dd}} }, \label{R=R01-frapsd}$$ where $$R_0 (T)= \frac{P_{ss}}{({J}[\Phi^s])^2}. \label{R0T=fracpssA}$$ The latter, given in (\[R0T=fracpss\]), is the resistivity that we obtain when we have no spin-fluctuation drag. In fact, this is the central formula to explain an enhanced resistivity of a spin fluctuation system due to normal scattering processes with long-lived spin fluctuations.[@ml66; @rice68; @mathon68] According to (\[R=R01-frapsd\]), the $d$ electron drag modifies the resistivity in two ways. First, we note that the numerator in (\[R=R01-frapsd\]) vanishes in the full drag case, (\[Pss=Pdd=-Psd\]). This represents physically that a finite resistivity is brought about only with those scattering processes which can degrade the total net current. On the basis of a more realistic model, a proper treatment of Umklapp scattering processes could make the numerator a non-vanishing factor of order unity. Secondly, the positive factor in the denominator has the effect of suppressing the resistivity. This is due to an additional drag current of the $d$ electrons. When fully dragged, the $d$ electrons carry $n_d/n_s$ times as large current as the $s$ electrons, where $n_d/n_s$ is the ratio of the electron densities. In general, this would not be negligible quantitatively, and it might be so even qualitatively. From the condition of no heat flow ${\bi U}=0$ for (\[U=sumijsd=U\]), we obtain the Seebeck coefficient, $$S = \frac{1}{T}\frac{ \displaystyle \sum_{l,m=s,d} { J}[\Phi^l] (P^{-1})_{lm} { U}[\Phi^m] }{ \displaystyle \sum_{l,m=s,d} { J}[\Phi^l] (P^{-1})_{lm} { J}[\Phi^m] }.$$ From this we can obtain the result for the the full drag case of (\[Pss=Pdd=-Psd\]) formally as a special limit. It is expressed simply by the ratio of the total energy current to the total momentum current as $$S = \frac{1}{T} \frac{ { U}[\Phi^s] +{U}[\Phi^d] }{ { J}[\Phi^s] +{ J}[\Phi^d] }. \label{S=frac1TUs+Ud/Js+Jd}$$ Indeed, the simple result in this limit is straightforwardly generalized to many-band models. It is owing to this simple property that we investigated this limit devotedly in the main text. On the other side, the case without drag is obtained for ${U}[\Phi^d]={ J}[\Phi^d]=0$ as $$S_0^s = \frac{1}{T} \frac{ { U}[\Phi^s] }{ { J}[\Phi^s] }, \label{S0s=1/Tfrac}$$ as presented in (\[S0sT\]). As a matter of fact, the no-drag results of (\[R0T=fracpssA\]) and (\[S0s=1/Tfrac\]) are directly derived without taking $\Phi^d$ into account from the beginning. \[sec:IbT\] Temperature dependence of ${\cal I}(T)$ at $\bar{\kappa}=0$ ======================================================================= To evaluate ${\cal I}(T)$ in (\[calI\]), we made the approximation as given in (\[n0omega=om&lt;cT\]). We obtained (\[Ib(T)c\]) for ${\cal I}_b(T)$ in (\[IbTdef\]), which signifies the main correction term of $\Delta S\propto T^{5/3}$ in the quantum critical regime. The exponent 5/3 is the same as for the resistivity.[@mathon68] The derivation in section \[sec:calIT\] indicates that the important contributions come from $\omega \simeq T$. In effect, this is the upper limit of the $\omega$ integral for $\bar{q}\agt \bar{q}_0$, and the high-energy cutoff is naturally provided by the Bose factor $n^0(\omega)$ in the integrand, without employing the approximation in (\[n0omega=om&lt;cT\]). With this in mind, we can obtain the result for $\bar{\kappa}=0$ directly by transforming the integral and taking the limits for the bounds of integration as follows. $$\begin{aligned} \fl {\cal I}_b(T) = \frac{1}{2 \varepsilon_{F,d}^2} \int_{\bar{q}_0^2}^{2k_{F,s}/k_{F,d}} {{\rm d}\bar{q}^2} \int^{2\bar{q}}_0 {\omega} {{\rm d}{\omega} } \frac{\partial}{\partial{\omega}} \left( \frac{ {{\omega}} }{ \bar{q}^6 + \left( C {\omega}/\varepsilon_{F,d} \right)^2 } \right) n^0(\omega) \nonumber\\ \simeq \frac{1}{2 \varepsilon_{F,d}^2} \int_{\bar{q}_0^2}^{2k_{F,s}/k_{F,d}} {{\rm d}\bar{q}^2} \int^{\infty}_0 {\omega} {{\rm d}{\omega} } \frac{\partial}{\partial{\omega}} \left( \frac{ {{\omega}} }{ \bar{q}^6 + \left( C {\omega}/\varepsilon_{F,d} \right)^2 } \right) n^0(\omega) \nonumber\\ %&= & %-\frac{1}{2 \varepsilon_{F,d}^2} %\int_{\bar{q}_0^2}^{2k_{F,s}/k_{F,d}} %{{\rm d}\bar{q}^2} % \int^{\infty}_0 % \frac{ %{{\omega}}{{\rm d}{\omega} } %}{ %\bar{q}^6 %+ % \left( %C %{\omega}/\varepsilon_{F,d} %\right)^2 %} %\frac{\partial\left({\omega}n^0(\omega)\right) %}{\partial{\omega}} %\nonumber\\ = -\frac{T^{2/3}}{2 \varepsilon_{F,d}^2} \int_{\bar{q}_0^2/T^{2/3}}^{2k_{F,s}/k_{F,d}/T^{2/3}} {{\rm d}v %\bar{q}^2 } \int^{\infty}_0 \frac{ {u} {{\rm d}u } }{ %\bar{q}^6 v^3 + \left( C u /\varepsilon_{F,d} \right)^2 } \frac{\partial}{\partial u} \left(\frac{u}{{\rm e}^u-1} \right) \nonumber\\ \simeq -\frac{T^{2/3}}{2 \varepsilon_{F,d}^2} \int_0^\infty {{\rm d} v %\bar{q}^2 } \int^{\infty}_0 \frac{ {u}{{\rm d}u } }{ %\bar{q}^6 v^3 + \left(C u/\varepsilon_{F,d}\right)^2 } \frac{\partial}{\partial u} \left(\frac{u}{{\rm e}^u-1} \right) \nonumber\\ %&= & %-\frac{T^{2/3}}{2 \varepsilon_{F,d}^2} %\frac{2\pi}{3\sqrt{3} \left(C /\varepsilon_{F,d}\right)^{4/3}} % \int^{\infty}_0 % {{\rm d}u } %u^{-1/3} %\frac{\partial}{\partial u} %\left(\frac{u}{{\rm e}^u-1}\right) %\nonumber\\ = - %\frac{T^{2/3}}{ \varepsilon_{F,d}^2} \frac{\pi}{3\sqrt{3} C^{4/3}} \left(\frac{T}{\varepsilon_{F,d}}\right)^{2/3} \int^{\infty}_0 {{\rm d}u } u^{-1/3} \frac{\partial}{\partial u} \left(\frac{u}{{\rm e}^u-1}\right) \nonumber\\ = 1.10 %477 \frac{\pi}{3\sqrt{3}C^{4/3}} \left(\frac{T}{\varepsilon_{F,d}}\right)^{2/3}. \label{Ib(T)ex}\end{aligned}$$ By comparing (\[Ib(T)ex\]) and (\[Ib(T)c\]), we obtain $ {c }^{-4/3} \simeq 1.10 %477 $, or $${c} %=1.10477^{-3/4} =1.10477^{-3/4}==0.927996 \simeq 0.928. \label{constc0.928}$$ References {#references .unnumbered} ========== [^1]: A similar consideration was taken to derive the Drude weight of a Fermi liquid.[@cond1] [^2]: Owing to prevalent anharmonic phonons in this skutterudite system, it may not be a simple matter to extract the electronic contribution from the observed specific heat coefficient $\gamma$, which does not depend sensitively on the divalent ion A.[@mhittm05]

--- abstract: 'Quantum feedback control is a technology which can be used to drive a quantum system into a predetermined eigenstate. In this article, sufficient conditions for the experiment parameters of a quantum feedback control process of a homodyne QND measurement are given to guarantee feedback control of a spin-1/2 quantum system in case of imperfect detection efficiency. For the case of pure states and perfect detection efficiency, time scales of feedback control processes are calculated.' author: - Andreas de Vries title: '[Global stability criterion for a quantum feedback control process on a single qubit and exponential stability in case of perfect detection efficiency]{}' --- Introduction ============ In classical control theory, feedback control describes processes in which a closed-loop controller is used to steer the states or outputs of a dynamical system, which in turn effect the inputs of the controller into the system. A remarkable approach to feedback control of quantum spin systems has recently been elaborated in [@van-Handel-et-al-2005]. Here QND measurements are utilized to let a quantum system collapse deterministically onto a predetermined eigenstate. In the present article, the stability and the time scale of quantum feedback control processes are studied. As a result (Theorem \[satz-asymptotically-stable\]), sufficient limits for the experiment control parameters are derived to guarantee asymptotically stable quantum feedback control processes on a spin-$\frac12$ quantum system. It is proved by applying Lyapunov’s method to the stochastic differential equation governing the quantum state evolution, and thus differs from the similar result in [@van-Handel-et-al-2005] proposing numerical methods of semialgebraic geometry and aiming at applicability for higher spin systems where efficient search for Lyapunov functions is practically impossible. For the special case of pure states and perfect detection efficiency, the quantum feedback control process is proved to terminate even exponentially fast in time. The article is organized as follows. First, the notions of QND measurements and quantum feedback control for a spin-$\frac12$ system are shortly reviewed, before the stochastic stability of quantum feedback control processes with imperfect and perfect detection efficiency are studied, and the results are shortly discussed. QND measurements ================ In contrast to a measurement in classical physics, a quantum measurement inevitably changes, or even destroys, the measured quantum system itself [@Goswami-1997; @Nielsen-Chuang-2000]. Theoretical as well as experimental investigation have been intensively made dealing with processes where quantum measurements are utilized constructively, for instance theoretical considerations of measurement determination by the quantum register [@Dusek-Buzek-2002], quantum feedback control by continuous measurements [@Belavkin-1992; @Belavkin-1992b; @Belavkin-1994; @Wiseman-1994], especially in quantum optics [@Armen-et-al-2002; @Stockton-et-al-2002; @Stockton-et-al-2004; @van-Handel-et-al-2005], stabilization and purification of two-level systems [@Wiseman-et-al-2002; @Wiseman-Ralph-2006], conditional measurements of coupled quantum dots by a point contact detector [@Goan-Milburn-2001; @Fujisawa-et-al-2004] or by a SET [@Shnirman-Schoen-1998; @Gurvitz-2003; @Gurvitz-Berman-2005], and the conditional measurement approach due to Sherman and Kurizki [@Sherman-Kurizki-1992] to prepare predetermined field states of atoms trapped in optical QED cavities [@Harel-et-al-1996; @Fortunato-et-al-1996; @Fortunato-et-al-1999], as well as a similar approach analyzed for spin squeezing in Cs clocks [@Oblak-et-al-2005]. Although these approaches differ considerably in detail, most of them utilize repeated quantum nondemolition (QND) measurements [@Joos-et-al-2003 §3.3], i.e., measurements of an observable $Y$ satisfying the *self-nondemolition condition* $ [Y(t), Y(t')] = 0 $ for all times $t$, $t'$, as well as the *back action evasion condition* $ [Y,H_{\mathrm{int}}] = 0, $ where $H_{\mathrm{int}} = \sum_j |j\rangle \langle j| \otimes B_j$ denotes the interaction Hamiltonian between the considered system (the projections $|j\rangle \langle j|$) and the measuring apparatus ($B_j$). The QND observable $Y$ may correspond, for instance, to a Hermitian Lindblad operator $L$, or to a conserved quantity, such as a constant of motion of the considered system like polarization or momentum. Quantum feedback control ======================== Due to ideas of Belavkin [@Belavkin-1992; @Belavkin-1992b; @Belavkin-1994] as well as Wiseman and coworkers [@Wiseman-1994; @Thomsen-et-al-2002], an approach to quantum feedback control of spin systems has been recently developed by van Handel, Stockton, and Mabuchi [@van-Handel-et-al-2005]. In this approach repeated quantum nondemolition measurements are engineered to let quantum spin systems collapse deterministically onto a previously chosen eigenstate. This idea, surprising from a traditional physics perspective, bases on the fact that realistic measurements are not instantaneous but take some finite time. If these reduction time scales are of an order attainable by modern digital electronics, a quantum filter and a controller can respond on the spin system state, feeding back the intermediate nondemolition measurement results to a Hamiltonian parameter. In this way it is possible, for instance, to deterministically prepare highly entangled Dicke states [@Stockton-et-al-2004], to generate and utilize squeezed quantum states of trapped atoms in an optical cavity [@Oblak-et-al-2005], or to improve quantum error correction [@Ahn-et-al-2002]. The quantum stochastic control formalism of van Handel, Stockton, and Mabuchi [@van-Handel-et-al-2005] can be considered as an extension of probability theory, and the traditional formulation of quantum mechanics can be directly recovered from it. In [@van-Handel-et-al-2005 §IV.C] a stabilizing controller is given for a quantum system of spin $j=\frac12$, schematically depicted in Figure \[fig-control\]. 1ex (49,16) ( 0,-0.3)[![\[fig-control\] (Color online) Schema of a quantum feedback control process. The QND measurement output $y_t$ from the quantum system is used to propagate the conditional state of the filter, via the feedback signal $H(t)$. The dashed line indicates (classical) digital processing, the filter is determined by Eq. (\[eq-SME\]). ](./control "fig:"){width="50ex"}]{} (25.0,14.3)[(0,0)[Quantum system]{}]{} (40.0,14.5)[(0,0)\[b\][$y_t$]{}]{} (25.0, 9.5)[(0,0)]{} ( 7.0, 7.5)[(0,0)\[r\][$H(t)$]{}]{} ( 8.5, 3.2)[(0,0)[Controller]{}]{} (25.0, 4.0)[(0,0)[$\rho_t$]{}]{} (41.5, 3.2)[(0,0)[Filter]{}]{} The conditional evolution of the density operator $\rho$ describing the quantum system depends on the probe parameter measurement rate $M>0$ in Hz, and the detection efficiency $\eta$ $\in$ $[0,1]$, a pure number. More precisely, the conditional evolution of $\rho$ is determined by the stochastic master equation $${\,\mathrm{d}}\rho_t = \mathscr{G}^*[H(t), L] \rho_t {\,\mathrm{d}}t +\sqrt\eta\, \mathscr{H}[L] \rho_t {\,\mathrm{d}}W_t , \label{eq-SME}$$ where $H(t)$ is the control Hamiltonian (with $H(t)=0$ in case of no feedback), $L$ is an observable one of whose eigenstates is the desired final state of the system, $\mathscr{G}^* = \mathscr{G}^*[H(t),L]$ is the adjoint generator $$\mathscr{G}^* \rho_t = -{\mathrm{i}}[H(t), \rho_t] + L \rho_t L^* - {\textstyle \frac12} (L^* L \rho_t + \rho_t L^* L) ,$$ $\mathscr{H}$ is the superoperator $$\begin{aligned} \mathscr{H}[L] \rho_t & \hspace*{-1.0ex} = \hspace*{-1.0ex} & L \rho_t + \rho_t L^* - \mathrm{Tr}[\rho_t (L+L^*)]\,\rho_t,\end{aligned}$$ and ${\,\mathrm{d}}W_t$ is the innovations process $${\,\mathrm{d}}W_t = 2 \sqrt{M\eta}\, y_t {\,\mathrm{d}}t - \sqrt\eta \, \mathrm{Tr}[\rho_t (L+L^*)]\, {\,\mathrm{d}}t$$ depending on the QND measurement record $y_t$ of the output corresponding to the observable $Y(t)$ [@Stockton-et-al-2004 §II]. Here $Y$ is normalized (i.e., ${\,\mathrm{d}}Y_t^2 = {\,\mathrm{d}}t$) and related to $L$ and the standard noises $A$ and $A^*$ by the Hudson-Parthasarathy equation. The innovations ${\,\mathrm{d}}W_t$ is a Wiener increment and ${\,\mathrm{d}}W_t/{\,\mathrm{d}}t$ is a Gaussian white noise. Note that ${\,\mathrm{d}}W_t$ is one-dimensional, whereas Eq. (\[eq-SME\]) is operator-valued. Usually, the controller Hamiltonian $H(t)$ is determined by a few control parameters. It is most desirable if they could be adjusted in a way such that the master equation has an asymptotically stable fixed point. Density operator space of a spin-$\frac12$ system ------------------------------------------------- In the prototypical physical model of homodyne measurement of a spin system [@van-Handel-et-al-2005], where $y_t$ denotes the homodyne measurement record of the output, the observable $L$ and the controller Hamiltonian $H(t)$ in Eq. (\[eq-SME\]) are given by $$L=\sqrt{M} J_z \qquad \mbox{and} \qquad H(t)=B(t) J_y$$ with the usual angular momentum observables $J_y$, $J_z$, and ${\,\mathrm{d}}W_t/{\,\mathrm{d}}t$ can be identified with the shot-noise of the homodyne local oscillator. Here $M > 0$ is the strength of the interaction between the light and the atoms and is regulated experimentally by the optical cavity, and the control input $B(t)$ is the applied magnetic field. The time scale then only depends on the sensitivity $1/(2\sqrt{M\eta})$ of the photodetection per $\sqrt{\mathrm{Hz}}$, and the feedback gain parameters of the controller $B(t)$. Hence the time scale only depends on experimentally controlled parameters. For perfect detection efficiency, $\eta=1$, the stochastic master equation governing a pure quantum system under feedback control is one-dimensional and will be tackled analytically below. For the special case of a quantum system of spin $\frac12$, i.e., a qubit, the space of all density operators $\rho$ of the system is represented by the two-dimensional disc $$D^2 = \{ (\lambda, \nu) \in \mathbb{R}^2: \lambda^2 + \nu(\nu-1) \leqq 0\}$$ with center $(0,\frac12)$ and radius $\frac12$, [@van-Handel-et-al-2005 §IV.C]. Here the density matrix entries are given by $\rho_{11} = \nu$, $\rho_{22} = 1 - \nu$, $\rho_{21} = \rho_{12}^* = \lambda$, and the state $(\lambda,\nu) = (0,0)$ to be stabilized corresponds to $\rho$ $=$ $|1\rangle \langle 1|$ $=$ diag$\,(0,1)$. 1ex (20,16.5) ( 0,-0.5)[![\[fig-D-2\] (Color online) The density operator space $D^2$ of a spin-$\frac12$ quantum system. The origin $(\lambda, \nu) = (0,0)$ corresponds to the quantum state $|1\rangle \langle 1|$, the point $(0,1)$ to $|0\rangle \langle 0|$. ](./D-2 "fig:"){width="20ex"}]{} (10,16.3)[(0,0)\[b\][$\nu$]{}]{} (13.0,10.0)[(0,0)[$D^2$]{}]{} (9.0,7.8)[(0,0)\[r\][$\frac12$]{}]{} (9.3,0.0)[(0,0)\[ur\][$0$]{}]{} (20.3,1.3)[(0,0)\[l\][$\lambda$]{}]{} Note that in this case the imaginary parts of the off-diagonal entries of $\rho$ decouple and may be neglected. The stochastic master equation (\[eq-SME\]) describing the conditional evolution of a single qubit thus is reducible to the two-dimensional Itô equation [@van-Handel-et-al-2005] $$\begin{aligned} {\,\mathrm{d}}\lambda_t & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \textstyle [B(t) (\nu_t - \frac12) - \frac M2 \lambda_t] {\,\mathrm{d}}t + \sqrt{M\eta}\, \lambda_t (1-2\nu_t){\,\mathrm{d}}W_t , \nonumber \\ {\,\mathrm{d}}\nu_t & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \textstyle - B(t) \lambda_t {\,\mathrm{d}}t - 2 \sqrt{M\eta}\, \nu_t (\nu_t-1){\,\mathrm{d}}W_t . \label{eq-SME-j=1/2}\end{aligned}$$ Its infinitesimal generator [@Oeksendal-1998 §7.3] is given as $$\begin{aligned} \mathscr{L} & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \textstyle [B(\lambda,\nu) (\nu - \frac12) - \frac M2 \lambda] \frac{\partial}{\partial \lambda} - B(\lambda,\nu) \lambda \frac{\partial}{\partial \nu} \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}+ 2M\eta \left[\lambda^2 (\nu - \frac12)^2 \frac{\partial^2}{\partial \lambda^2} + \nu^2 (\nu - 1)^2 \frac{\partial^2}{\partial \nu^2} \right] ,\end{aligned}$$ so $\frac{{\,\mathrm{d}}\mathbb{E}[f(x_t)]}{{\,\mathrm{d}}t} = \mathbb{E}[\mathscr{L}f(x_t)]$. Stochastic stability of quantum feedback control processes ========================================================== Stochastic stability -------------------- In control theory stabilization of nonlinear systems is usually investigated using Lyapunov theory. In the 1960s, the stochastic counterpart of Lyapunov theory [@Hasminskii-1980] was developed by Has’minskiǐ and others. To prove the central results of this article, we first have to define asymptotic stability of stochastic processes. Let $W_t$ be a Wiener process on the canonical Wiener space $(\Omega, \mathscr{F}, \mathbb{P})$, and let $x$ obey the Itô equation on $\mathbb{R}^n$, $${\,\mathrm{d}}x_t = b(x_t) {\,\mathrm{d}}t + \sigma(x_t) {\,\mathrm{d}}W_t, \label{eq-Ito}$$ where $b$, $\sigma: \mathbb{R}^n \to \mathbb{R}^n$ satisfy the usual growth and Lipschitz conditions for existence and uniqueness of solutions [@Oeksendal-1998]. Then an equlibrium solution $x_*$ of Eq. (\[eq-Ito\]), i.e., a solution satisfying $b(x_*) = \sigma(x_*) = 0$, is called *stable in probability* if $$\lim_{x_0 \to x_*} \mathbb{P} \Big[ \sup_{t\geqq 0} |x_t - x_*| > \varepsilon \Big] = 0 \qquad \forall \varepsilon > 0.$$ It is called *asymptotically stable* if it is stable in probability and $$\lim_{x_0 \to x_*} \mathbb{P} \left[ \lim_{t\to \infty} |x_t - x_*| = 0 \right] = 1 .$$ It is called *globally stable* if it is stable in probability and $$\mathbb{P} \left[ \lim_{t\to \infty} |x_t - x_*| = 0 \right] = 1 .$$ $x_*$ is called *exponentially stable in $p$-th moment*, $p \in \mathbb{N}$, [@Higham-et-al-2003] if there exists a pair of constants $a$, $\alpha>0$ such that $$\mathbb{E} \big[ |x_t - x_*|^p \big] \leqq a \, \mathbb{E}\big[ |x_0 - x_*|^p \big] \, {\,\mathrm{e}}^{-\alpha t}$$ for all $t\geqq 0$. Especially for $p=1$, $x_*$ is then called *exponentially stable in mean*, and for $p=2$ *exponentially stable in mean square* . The smallest possible value of the constant $a$ is referred to as the *growth constant*, and the largest possible value of $\alpha$ as the *rate constant* or *rate of convergence*. The first two notions are local properties, whereas the third one is a global property of the system. Imperfect detection efficiency ------------------------------ If quantum feedback control is performed with only imperfect detection efficiency, i.e., $0 < \eta < 1$, the following theorem yields a sufficient condition for the global asymptotical stability of its final state. Although a similar result for a special controller has been shown already in [@van-Handel-et-al-2005 §IV.F] by numerical semialgebraic methods, here a general analytical criterion relating the controller parameters is given. \[satz-asymptotically-stable\] Consider a quantum feedback control process of a spin-$\frac{1}{2}$ quantum system, described by the density operators in the state space $D^2$ and by the stochastic master equation (\[eq-SME-j=1/2\]), with the probe parameter measurement rate $M>0$, the detection efficiency $\eta$ $\in$ $(0,1]$ and the controller $$B(\lambda, \nu) = g_1 \lambda + g_2 \nu . \label{eq-controller}$$ Assume that for the feedback gain parameters $g_1$ and $g_2$ there exist real constants $c>1$, $0<d<2(c-1)$ such that the maximum $f_{\max}$ of the auxiliary function $f:[0,\frac12] \times [-\pi,\pi] \to \mathbb{R}$, $$\begin{aligned} f(r,\theta) & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \textstyle [M - (c-1)g_1] (1-\cos\theta) {}- dg_1 r \sin \theta (1-\cos\theta) \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}+ [d g_1 (1+\cos\theta)\, r - (c-1) g_2 - \frac {d(g_1+M)}{2}] \sin \theta \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}+ 2 d g_2 (1+\cos\theta) ( 2 r \cos\theta - \frac{1}{2}) \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}- 4M\eta (1-\cos\theta) ((1+\cos\theta)\,r - \frac12)^2 \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}- 4M\eta (1+\cos\theta) ( (1+\cos\theta)\, r - 1)^2 , \label{condition-g1-g2} \end{aligned}$$ is negative, i.e., $f_{\max} < 0$. Then the state $(0,0)\in D^2$, corresponding to $\rho_0$ $=$ $|1\rangle\langle 1|$, of the quantum system undergoing a quantum feedback control process is globally stable. Moreover, condition (\[condition-g1-g2\]) implies $$g_1 > \frac{(1-\eta)M}{c-1} \geqq 0, \qquad -\frac{4M\eta}{d} < g_2 < 0. \label{condition-g1-g2-2}$$ Defining the function $$V(\lambda,\nu) = c \nu + d \lambda \nu - \lambda^2 - \nu^2, \label{eq-V}$$ we have $V(0,0)=0$ and $V(\lambda,\nu) > 0$ for $(\lambda,\nu)\in D^2\setminus\{{\mbox{\mathversion{bold}$0$}}\}$. Since $0\leqq \lambda^2 \leqq \nu(1-\nu)$ and $\lambda \geqq -\frac12$, we have $ V=(c-1) \nu + d \lambda \nu - (\lambda^2 + \nu(\nu-1) ) \geqq (c-1)\nu - \frac{d}{2} \nu $, i.e., $$\textstyle V(\lambda,\nu) \geqq (c-\frac{d}{2} -1)\, \nu, \label{eq-V>cnu}$$ and especially $V>0$ for $\nu>0$. Moreover by (\[eq-controller\]), $$\begin{aligned} \mathscr{L}V & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \textstyle (g_1 \lambda + g_2 \nu) [(d(\nu^2 - \frac{\nu}{2} - \lambda^2) - (c-1)\lambda] \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}- \frac M2 \lambda (d \nu - 2\lambda) \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}- 4M\eta \left[\lambda^2 (\nu - \frac12)^2 + \nu^2 (\nu - 1)^2 \right] \nonumber \\* & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \textstyle [M - d (g_1 \lambda + g_2 \nu) - (c-1) g_1] \lambda^2 {}+ d g_2 \nu^2 (\nu - \frac12) \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}+ [d g_1 (\nu - \frac12) - (c-1) g_2 - \frac {dM}{2}] \lambda\nu \nonumber \\* & \hspace*{-1.0ex} \hspace*{-1.0ex} & \textstyle {}- 4M\eta \left[\lambda^2 (\nu - \frac12)^2 + \nu^2 (\nu - 1)^2 \right] . \label{eq-LV-1} \end{aligned}$$ We see immediately that $\mathscr{L}V(0,0) = 0$. We will prove next that $\mathscr{L}V(\lambda,\nu) < 0$ for $(\lambda,\nu) \in D^2\setminus \{{\mbox{\mathversion{bold}$0$}}\}$. Using the coordinates $(r,\theta)$, where $r\in(0,\frac12]$ and $\theta \in (-\pi, \pi)$, given by $r=\frac{\lambda^2+\nu^2}{2\nu}$, $\tan\frac{\theta}{2} = \frac{\lambda}{\nu}$, we have $(\lambda,\nu)$ $=$ $r(\sin\theta, 1+\cos\theta)$. Thus we obtain $$f(r,\theta) = \frac{\mathscr{L}V\big(\lambda(r,\theta), \nu(r,\theta)\big)}{r^2 (1+\cos\theta)} . \label{eq-LV-f}$$ By the assumption of the Theorem, $f(r,\theta) < 0$ for $(r,\theta) \in [0,\frac12] \times (-\pi,\pi)$, hence $\mathscr{L}V(\lambda,\nu) < 0$ on $D^2\setminus \{{\mbox{\mathversion{bold}$0$}}\}$. Therefore, $V$ is a strict Lyapunov function on $D^2$ with the only asymptotically stable state $(\lambda,\nu)=0$. Since $f(r,\pm\pi) = 2( (1-\eta)M - (c-1) g_1)$, as well as $f(r,0) = -32M\eta (r^2 - (1+\frac{dg_2}{4M\eta})r + \frac{4M\eta + dg_2}{16 M\eta})$, assumption (\[condition-g1-g2\]) implies (\[condition-g1-g2-2\]). Therefore, a quantum feedback control process satisfying the assumptions of Theorem \[satz-asymptotically-stable\] drives a spin-$\frac12$ quantum system to the state $\binom{0}{0} \in D^2$, [no matter in which quantum state the system is initially.]{} This holds true even for the worst case, when the initial state is $\rho_i = |0\rangle \langle 0| = \binom{0}{1} \in D^2$. Note that a standard state reduction measurement would leave the quantum system in this state with certainty. The next example shows that there indeed *exist* parameter constellations satisfying Theorem \[satz-asymptotically-stable\]. For the experimentally controlled parameters $g_1=\frac{3M}{4}$, $g_2=-\frac{M}{4}$, and $\eta=\frac{1}{2}$, and =1ex (32,27) (-5,-.5)[![\[fig-Lyapunov\] (Color online) The auxiliary function $h$ in (\[condition-g1-g2\]) for $g_1=\frac{3M}{4}$, $g_2=-\frac{M}{4}$, $\eta=\frac{M}{2}$, $c=4$, $d=2$. Left figure: The graph of $f(r,\theta)$. Right figure: The corresponding 2-manifold imbedded in 3-space $(\lambda,\nu, z)$, parametrized by $\lambda=r\sin\theta$, $\nu=(1+\cos\theta)r$, $z=\frac{f(r,\theta)}{5}$ for $0<r\leqq\frac12$, $|\theta| < \pi$. ](./Lyapunov1 "fig:"){width="45ex"}]{} ( 0,12)[(0,0)\[r\][$f$]{}]{} (10,3)[(0,0)\[t\][$r$]{}]{} (30,7)[(0,0)\[l\][$\theta$]{}]{} (28,27) (-8,-.5)[![\[fig-Lyapunov\] (Color online) The auxiliary function $h$ in (\[condition-g1-g2\]) for $g_1=\frac{3M}{4}$, $g_2=-\frac{M}{4}$, $\eta=\frac{M}{2}$, $c=4$, $d=2$. Left figure: The graph of $f(r,\theta)$. Right figure: The corresponding 2-manifold imbedded in 3-space $(\lambda,\nu, z)$, parametrized by $\lambda=r\sin\theta$, $\nu=(1+\cos\theta)r$, $z=\frac{f(r,\theta)}{5}$ for $0<r\leqq\frac12$, $|\theta| < \pi$. ](./Lyapunov2 "fig:"){width="45ex"}]{} ( 6,25)[(0,0)\[r\][$\nu$]{}]{} ( 1,12)[(0,0)\[r\][$z$]{}]{} (10,1)[(0,0)\[t\][$\lambda$]{}]{} the constants $c=4$, $d=2$, the function $h$ in (\[condition-g1-g2\]) is negative on $D^2\setminus \{{\mbox{\mathversion{bold}$0$}}\}$, which may be seen graphically (Fig. \[fig-Lyapunov\]). Pure states and perfect detection efficiency -------------------------------------------- For pure quantum states and perfect detection efficiency, we are able to estimate the expected running time for a quantum feedback control process more precisely, as is shown in the following theorem. \[satz-T-qfc\] Consider a quantum feedback control process of a spin-$\frac{1}{2}$ quantum system of pure quantum states, described by the stochastic master equation (\[eq-SME-j=1/2\]) with the probe parameter measurement rate $M>0$, the perfect detection efficiency $\eta=1$, and the controller $B(t)$ of (\[eq-controller\]) with feedback gain parameters $$g_1 > 0, \quad g_2<0 . \label{eq-cond-g1-g2}$$ Then this process has an expected running time $$ T_{\mathrm{qfc}}(M,g_1) = O\big({\,\mathrm{e}}^{- \left(g_1 + M \right) t/2}\big) , \label{eq-T-qfc}$$ where $t$ denotes the time duration since the process start. In particular, ${\mbox{\mathversion{bold}$0$}} \in D^2$ is globally exponentially stable in mean. For a quantum feedback control process of a spin-$\frac12$ system, performed with perfect efficiency, the space of all density operators $\rho$ of pure states of the system can be reduced to the circle $$S^1 = \{ (\sin \theta, 1+\cos\theta) \in \mathbb{R}^2: \theta \in (-\pi,\pi]\}$$ with center $(0,\frac12)$ and radius $\frac12$, [@van-Handel-et-al-2005 §IV.C], and the stochastic master equation (\[eq-SME\]) describing the conditional evolution of a single qubit, with perfect efficiency $\eta=1$, then is reducible to the one-dimensional It[ô]{} equation $$\textstyle {\,\mathrm{d}}\theta_t = ( B(t) - \frac M2 \sin \theta_t \cos\theta_t) {\,\mathrm{d}}t - \sqrt M \sin\theta_t {\,\mathrm{d}}W_t . \label{eq-SME-j=1/2-pure}$$ Choosing the controller as $$B(t) = \frac{g_1}{2} \sin \theta_t + \frac{g_2}{2} (1+\cos\theta_t) ,$$ the system stabilizes the state $\theta=\pm\pi$, which we mark as $|1\rangle \langle 1|$. Eq. (\[eq-SME-j=1/2-pure\]) has a unique solution on the interval $[-\pi,\pi]$ because its coefficients satisfy the sufficient Lipschitz and growth conditions [@Oeksendal-1998 Theor.5.2.1]. Since the diffusion coefficient $$\sigma(\theta) = \sqrt{M} \sin\theta$$ vanishes at $\theta=0$, we have to consider the two intervals $J^- = (-\pi,0)$ and $J^+=(0,\pi)$ separately. Since moreover the drift coefficient $$b(\theta) = B(\theta) - \frac{M}{2} \sin\theta \cos\theta$$ satisfies $b(-\pi) = b(\pi) = 0$ and $b(0)=g_2<0$, the state $\theta=0$ is a reflecting barrier (‘entrance boundary’) for states in $J^-$, but an absorbing barrier (‘exit boundary’) for states in $J^+$, whereas $\theta=\pm\pi$ both are absorbing barriers [@Gardiner-1990 §5.2.1]. To estimate the expected time that a given pure state requires to reach the desired state $|1\rangle\langle1|$, represented by $\theta=\pm\pi$, we have to compute the expected first exit time $T$ for the random variable $\theta$ to leave the interval $J^-$ or $J^+$, respectively. $T$ is the solution of the inhomogeneous linear differential equation [@Gardiner-1990 §5.2.7], [@Wilmott-1998 §10.9] $$\frac{\partial T}{\partial t} + \frac{1}{2}\, \sigma^2(\theta)\, \frac{\partial^2 T}{\partial \theta^2} + b(\theta)\, \frac{\partial T}{\partial \theta} = -1 \label{eq-first-exit-time}$$ on $J^\pm$ under the boundary conditions $$T(\pm\pi,t) = T(0+,t) = \frac{\partial}{\partial \theta} T(0-,t) = 0, \quad$$ and $T(\theta,0) = f(\theta)$. With the change of variable $$x = \frac{1 + \cos\theta}{\sin\theta} = \cot\frac{\theta}{2} \label{eq-x-theta}$$ we obtain the relations $\sin\theta = \frac{2x}{1+x^2}$ and $\cos \theta = \frac{x^2 - 1}{x^2 + 1}$, i.e., $\frac{\sigma^2}{2} = \frac{2Mx^2}{(1+x^2)^2}$ and $b =\frac{2g_1 x}{1+x^2} + \frac{2 g_2 x^2}{1+x^2} - \frac{M(x-x^2)}{(1+x^2)^2}$. Moreover, ${\,\mathrm{d}}\theta = -\frac{2 {\,\mathrm{d}}x}{1 + x^2}$, i.e., $$\textstyle \frac{\partial}{\partial \theta} = - \frac{1+x^2}{2}\frac{\partial}{\partial x}, \quad \frac{\partial^2}{\partial \theta^2} = \frac{(1+x^2)^2}{4}\frac{\partial^2}{\partial x^2} - \frac{x(1+x^2)}{2}\frac{\partial}{\partial x} .$$ Hence Eq. (\[eq-first-exit-time\]) is rewritten as $\frac{\partial T}{\partial t} - \frac{M}{2} L T = -1$, where $$L = - x^2 \frac{\partial^2}{\partial x^2} + h(x) \frac{\partial}{\partial x}$$ with $$h(x) = \frac{g_1}{M} x + \frac{g_2}{M} x^2 + \frac{x - 3x^3}{1+x^2} ,$$ and where the boundary conditions $T(0,t) = 0$, $T(x,t) \to 0$ as $x \to \infty$, and $\frac{\partial}{\partial x} T(x,t) \to 0$ as $x \to -\infty$ hold. Since for $|x|<1$ we have $\frac{1}{1+x^2} = \sum_{0}^{\infty} (-1)^\nu x^{2\nu}$, we can write $h$ as $$h(x) = \frac{g_1}{M} x + \frac{g_2}{M} x^2 + (x - 3x^3) \sum_{\nu=0}^\infty (-1)^\nu x^{2\nu} \label{eq-h-2}$$ for $|x|<1$. An eigenvalue $\lambda$ of $L$ is given by the equation $Ly = \lambda y$. We use the ansatz $y(x) = \sum_0^\infty a_k x^k$. By the boundary conditions, $y(0)=0$, hence $a_0=0$. Moreover, $ y'(x) = \sum_{1}^\infty k a_k x^{k-1}, $ and $ y''(x) = \sum_{2}^\infty k(k-1) a_k x^{k-2}, $ hence $$Ly(x) = \sum_{k=2}^\infty k(k-1) a_k x^{k} + h(x) \sum_{k=1}^\infty k a_k x^{k-1} .$$ For $|x|<1$, the series $\sum_{0}^\infty (-1)^\nu x^{2\nu}$ is absolutely convergent and we have $ (\sum_{1}^N k a_k x^{k-1}) (\sum_{0}^\infty (-1)^\nu x^{2\nu}) = \sum_{1}^N c_k $ with $ c_k = k a_k \sum_{\nu=0}^k (-1)^{k-\nu} x^{3k-2\nu-1} $, i.e., $c_1=a_1(1 - x^2)$, $c_2 = 2a_2 (x - x^3 + x^5)$, $c_3 = 3a_3 (x^2 - x^4 + x^6 - x^8)$. Hence $Ly(x)$ has the following coefficients for the first powers of $x$, $$\begin{aligned} x & \hspace*{-1ex} : & \left(\frac{g_1}{M} + 1\right) a_1 \\ x^2 & \hspace*{-1ex} : & 2 \left(\frac{g_1}{M} + 2\right) a_2 + \frac{g_2}{M} a_1 \\ x^3 & \hspace*{-1ex} : & 3 \left(\frac{g_1}{M} + 3\right) a_3 + 2 \cdot \frac{g_2}{M} a_2 - 3 a_1 \end{aligned}$$ if $|x|<1$. Especially, the lowest power of $h(x)$ is the term $\left( \frac{g_1}{M} + 1 \right)$. Let $y_{n}(x) = \sum_{n}^\infty a_k x^k$ for $n\in\mathbb{N}$, i.e., $a_k=0$ for $k \leqq n$. Then the lowest power of $Ly_n(x)$ is the term $\left( \frac{g_1}{M} + n \right) n a_{n} x^{n}$. Setting $Ly_n(x) = \lambda_n y_n(x)$ and comparing the coefficients, we then get the eigenvalue $$\lambda_n = \left( \frac{g_1}{M} + n \right) n \label{eq-eigenvalues}$$ corresponding to the function $y_n$. In turn, once the eigenvalue is specified, the coefficients $a_{n+1}$, $a_{n+2}$, …, are determined recursively by comparing the coefficients of $Ly_n$ and $\lambda_n y_n$. Although the above arguments hold true only for $|x|<1$, $\lambda_n$ is the eigenvalue corresponding to $y_n$ for the entire domains of definition, $x\in (-\infty,0)$ and $(0,\infty)$, respectively. Thus the smallest eigenvalue of $L$ is $\lambda_1 = \frac{g_1}{M} + 1$. Now, $L$ can be expressed as $Ly =-\frac{1}{r}(py')'$ with $p(x) = \exp(-\int \frac{h(x)}{x^2}{\,\mathrm{d}}x)$, i.e., $$p(x) = \frac{1+x^2}{|x|^{\frac{g_1}{M}}} {\,\mathrm{e}}^{-\frac{g_2}{M}x} ,$$ and $r(x) = \frac{p(x)}{x^2}$, satisfying the boundary conditions $y(0,t) = 0$, $y(x,t) \to 0$ as $x \to \infty$, $\frac{\partial}{\partial x} y(x,t) \to 0$ as $x \to -\infty$, and $y(x,0) = \frac{M}{2} f(x)$. Thus, we have a Sturm-Liouville problem on each interval $(-\infty,0)$ and $(0,\infty)$ separately, possessing the eigensolutions $y_n$ corresponding to the eigenvalues $\lambda_n$ of (\[eq-eigenvalues\]), and our initial problem (\[eq-first-exit-time\]) has the solutions $$\begin{aligned} T(\theta,t) & \hspace*{-1.0ex} = \hspace*{-1.0ex} & \int_{J^\pm} G(\theta,\alpha,t)\, r(\alpha) f(\alpha) {\,\mathrm{d}}\alpha \nonumber \\ & \hspace*{-1.0ex} \hspace*{-1.0ex} & -\int_0^t \int_{J^\pm} G(\theta,\alpha, t-\tau)\, r(\alpha) {\,\mathrm{d}}\alpha {\,\mathrm{d}}\tau \end{aligned}$$ with $$G(\theta,\alpha,t) = \sum_{n=1}^\infty \frac{y_n(\theta)y_n(\alpha)}{\|y_n\|^2} \textstyle {\,\mathrm{e}}^{- \left(g_1 + Mn \right) n t/2}$$ where $\theta=2 \arctan x$ and $\alpha = 2 \arctan x'$ according to (\[eq-x-theta\]), and $\|y_n\|^2 = \int_{J^\pm} r y_n^2 {\,\mathrm{d}}x$ . Hence an arbitrarily given initial state $\theta_0$, i.e., $f(\theta) = \delta_{\theta_0}(\theta)$, is pushed exponentially fast in time $t$ to one of the final states $\theta=\pm \pi$, or $x=0$, because $$\begin{aligned} T(\theta, t) & \hspace*{-.75ex} = \hspace*{-.75ex} & r(\theta_0) \sum_{n=1}^\infty \frac{y_n(\theta)y_n(\theta_0)}{\|y_n\|^2} \left(\frac{1+\lambda_n}{\lambda_n}\, {\,\mathrm{e}}^{-\lambda_n t} - \frac{1}{\lambda_n}\right) \nonumber \\ & \hspace*{-.75ex} \leqq \hspace*{-.75ex} & 2 \, {\,\mathrm{e}}^{-\lambda_1 t} \, r(\theta_0) \sum_{n=1}^\infty \frac{y_n(\theta)y_n(\theta_0)}{\|y_n\|^2} \end{aligned}$$ i.e., Eq. (\[eq-T-qfc\]), for the smallest eigenvalue $\lambda_1$ in (\[eq-eigenvalues\]). Consider the the case $g_1=M$, $g_2=-\frac{M}{2}$ given in [@van-Handel-et-al-2005 Fig. 3(b)]. Then the eigenvalues $\mu$ of the expected time $T$ to set the quantum system into the state $|1\rangle \langle 1|$ given by (\[eq-eigenvalues\]) are bounded by $\mu \geqq \frac{M}{2} \lambda_1 = M$. Therefore, the greater the measurement rate $M$, the greater is the smallest possible eigenvalue of $T$. Discussion ========== In this article, a stability criterion for a quantum feedback control process has been introduced, as well as its expected running time in case of perfect detection efficiency. In Theorem \[satz-asymptotically-stable\], a sufficient limit for the experimental control parameters leading to globally stable quantum feedback control processes acting on a spin-$\frac12$ quantum system are given. The proof consists of the application of Lyapunov’s method to the stochastic differential equation governing the quantum state evolution under feedback control. In Theorem \[satz-T-qfc\] it is shown that, for perfect detection efficiency, the quantum feedback control process terminates even exponentially fast in time. The proof bases on the power series ansatz $y=\sum_n^\infty a_k x^k$ for the derived equation $\frac{\partial T}{\partial t} - \frac{M}{2} LT = -1$ determining the expected time $T$, yielding eigenfunctions $y_n$ with corresponding positive eigenvalues $\lambda_n$ by Eq. (\[eq-eigenvalues\]). Mathematically, $T$ is the expected first exit time. Theorem \[satz-T-qfc\] implies that the expected running time $T_{\mathrm{qfc}}$ of a quantum feedback control process does not depend on the probability neither of the desired state, nor of the initial state. Former numerical investigations indicate that the running time of quantum feedback control algorithms, and thus $T_{\mathrm{qfc}}$, is about a tenth of the decoherence time [@Ahn-et-al-2002] up to the order of the decoherence time [@Atkins-et-al-2005]. Thus we are left with the unsatisfactory situation that the general case of quantum feedback control with imperfect detection efficiency could not yet be proved to be exponentially stable, in contrast to the marginal case of pure states and perfect detection efficiency. Of course, the fact that a Lyapunov function proving exponential stability could not be found does not mean that there does not exist any at all. However, by the proof of Theorem \[satz-asymptotically-stable\] such a Lyapunov function cannot be of the form (\[eq-V\]), since with ${\mbox{\mathversion{bold}$x$}} = {\lambda \choose \nu}\in D^2$, $|{\mbox{\mathversion{bold}$x$}}|^2 = \lambda^2 + \nu^2 = 2r\nu = 2r^2(1+\cos \theta),$ i.e., with Eq. (\[eq-LV-f\]), $$\textstyle \frac{f_{\min}}{2}\, |{\mbox{\mathversion{bold}$x$}}|^2 \leqq \mathscr{L}V({\mbox{\mathversion{bold}$x$}}) \leqq \frac{f_{\max}}{2}\, |{\mbox{\mathversion{bold}$x$}}|^2 \leqq 0 , \label{eq-V-LV-1}$$ but $V$ has a linear term such that there does not exist a constant $\alpha>0$ satisfying $\mathscr{L}V({\mbox{\mathversion{bold}$x$}}) \leqq -\alpha V({\mbox{\mathversion{bold}$x$}})$. By Theorem \[satz-T-qfc\], however, a Lyapunov function may exist satisfying this criterion at least for pure states. Thus for future mathematical investigation the question remains: Can quantum feedback control, including purification of a mixed state, be exponentially stable in general? [32]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (), . , ** (, , ), ed. , ** (, , ). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , , , ****, (), . , , , ****, (). , , , ****, (), . , , , ****, (), . , ****, (), . , ****, (), . , , , ****, (). , ****, (). , ****, (), . , ****, (), . , ****, (). , , , , ****, (). , , , ****, (). , , , ****, (), . , , , , , , , , ****, (), . , , , , , , ** (, , ), ed. , , , ****, (), . , , , ****, (), . , ** (, , ), ed. , ** (, , ). , , , ****, (), . , ** (, , ), ed. , ** (, , ). , , , , ****, (), .

--- abstract: 'In recent years, on-policy reinforcement learning (RL) has been successfully applied to many different continuous control tasks. While RL algorithms are often conceptually simple, their state-of-the-art implementations take numerous low- and high-level design decisions that strongly affect the performance of the resulting agents. Those choices are usually not extensively discussed in the literature, leading to discrepancy between published descriptions of algorithms and their implementations. This makes it hard to attribute progress in RL and slows down overall progress [@implementation_matters]. As a step towards filling that gap, we implement &gt;50 such “choices” in a unified on-policy RL framework, allowing us to investigate their impact in a large-scale empirical study. We train over 250’000 agents in five continuous control environments of different complexity and provide insights and practical recommendations for on-policy training of RL agents.' author: - | Marcin Andrychowicz, Anton Raichuk, Piotr Stańczyk, Manu Orsini,\ **Sertan Girgin, Raphael Marinier, Léonard Hussenot, Matthieu Geist,**\ **Olivier Pietquin, Marcin Michalski, Sylvain Gelly, Olivier Bachem**\ \ Google Research, Brain Team bibliography: - 'main.bib' title: 'What Matters In On-Policy Reinforcement Learning? A Large-Scale Empirical Study' --- Introduction ============ Deep reinforcement learning (RL) has seen increased interest in recent years due to its ability to have neural-network-based agents learn to act in environments through interactions. For continuous control tasks, on-policy algorithms such as REINFORCE [@pg], TRPO [@trpo], A3C [@a3c], PPO [@ppo] and off-policy algorithms such as DDPG [@ddpg] and SAC [@sac] have enabled successful applications such as quadrupedal locomotion [@sac2], self-driving [@kendall2019learning] or dexterous in-hand manipulation [@sac2; @openai2018learning; @openai2019solving]. Many of these papers investigate in depth different loss functions and learning paradigms. Yet, it is less visible that behind successful experiments in deep RL there are complicated code bases that contain a large number of low- and high-level design decisions that are usually not discussed in research papers. While one may assume that such “choices” do not matter, there is some evidence that they are in fact crucial for or even driving good performance [@implementation_matters]. While there are open-source implementations available that can be used by practitioners, this is still unsatisfactory: In research publications, often different algorithms implemented in different code bases are compared one-to-one. This makes it impossible to assess whether improvements are due to the algorithms or due to their implementations. Furthermore, without an understanding of lower-level choices, it is hard to assess the performance of high-level algorithmic choices as performance may strongly depend on the tuning of hyperparameters and implementation-level details. Overall, this makes it hard to attribute progress in RL and slows down further research [@henderson2018deep; @implementation_matters; @islam2017reproducibility]. **Our contributions.** Our key goal in this paper is to investigate such lower level choices in depth and to understand their impact on final agent performance. Hence, as our key contributions, we (1) implement &gt;50 choices in a unified on-policy algorithm implementation, (2) conducted a large-scale (more than 250’000 agents trained) experimental study that covers different aspects of the training process, and (3) analyze the experimental results to provide practical insights and recommendations for the on-policy training of RL agents. **Most surprising finding.** While many of our experimental findings confirm common RL practices, some of them are quite surprising, e.g. the policy initialization scheme significantly influences the performance while it is rarely even mentioned in RL publications. In particular, we have found that initializing the network so that the initial action distribution has zero mean, a rather low standard deviation and is independent of the observation significantly improves the training speed (Sec. \[sec:results-arch\]). The rest of of this paper is structured as follows: We describe our experimental setup and performance metrics used in Sec. \[sec:performance\]. Then, in Sec. \[sec:results\] we present and analyse the experimental results and finish with related work in Sec. \[sec:related\] and conclusions in Sec. \[sec:conclusions\]. The appendices contain the detailed description of all design choices we experiment with (App. \[sec:choices\]), default hyperparameters (App. \[sec:default-settings\]) and the raw experimental results (App. \[exp\_final\_losses\] - \[exp\_final\_regularizer\]). Study design {#sec:performance} ============ #### Considered setting. In this paper, we consider the setting of *on-policy reinforcement learning for continuous control*. We define on-policy learning in the following loose sense: We consider policy iteration algorithms that iterate between generating experience using the current policy and using the experience to improve the policy. This is the standard *modus operandi* of algorithms usually considered on-policy such as PPO [@ppo]. However, we note that algorithms often perform several model updates and thus may operate technically on off-policy data within a single policy improvement iteration. As benchmark environments, we consider five widely used continuous control environments from OpenAI Gym [@gym] of varying complexity: Hopper-v1, Walker2d-v1, HalfCheetah-v1, Ant-v1, and Humanoid-v1 [^1]. #### Unified on-policy learning algorithm. We took the following approach to create a highly configurable unified on-policy learning algorithm with as many choices as possible: 1. We researched prior work and popular code bases to make a list of commonly used choices, i.e., different loss functions (both for value functions and policies), architectural choices such as initialization methods, heuristic tricks such as gradient clipping and all their corresponding hyperparameters. 2. Based on this, we implemented a single, unified on-policy agent and corresponding training protocol starting from the SEED RL code base [@seed]. Whenever we were faced with implementation decisions that required us to take decisions that could not be clearly motivated or had alternative solutions, we further added such decisions as additional choices. 3. We verified that when all choices are selected as in the PPO implementation from OpenAI baselines, we obtain similar performance as reported in the PPO paper [@ppo]. We chose PPO because it is probably the most commonly used on-policy RL algorithm at the moment. The resulting agent implementation is detailed in Appendix \[sec:choices\]. The key property is that the implementation exposes all choices as configuration options in an unified manner. For convenience, we mark each of the choice in this paper with a number (e.g., ) and a fixed name (e.g. ) that can be easily used to find a description of the choice in Appendix \[sec:choices\]. #### Difficulty of investigating choices. The primary goal of this paper is to understand how the different choices affect the final performance of an agent and to derive recommendations for these choices. There are two key reasons why this is challenging: First, we are mainly interested in insights on choices for good hyperparameter configurations. Yet, if all choices are sampled randomly, the performance is very bad and little (if any) training progress is made. This may be explained by the presence of sub-optimal settings (e.g., hyperparameters of the wrong scale) that prohibit learning at all. If there are many choices, the probability of such failure increases exponentially. Second, many choices may have strong interactions with other related choices, for example the learning rate and the minibatch size. This means that such choices need to be tuned together and experiments where only a single choice is varied but interacting choices are kept fixed may be misleading. #### Basic experimental design. To address these issues, we design a series of experiments as follows: We create groups of choices around thematic groups where we suspect interactions between different choices, for example we group together all choices related to neural network architecture. We also include in all of the groups as we suspect that it may interact with many other choices. Then, in each experiment, we train a large number of models where we randomly sample the choices within the corresponding group [^2]. All other settings (for choices not in the group) are set to settings of a competitive base configuration (detailed in Appendix \[sec:default-settings\]) that is close to the default PPOv2 configuration[^3] scaled up to $256$ parallel environments. This has two effects: First, it ensures that our set of trained models contains good models (as verified by performance statistics in the corresponding results). Second, it guarantees that we have models that have different combinations of potentially interacting choices. We then consider two different analyses for each choice (e.g, for ): *Conditional 95th percentile*: For each potential value of that choice (e.g., = `N-Step`), we look at the performance distribution of sampled configurations with that value. We report the 95th percentile of the performance as well as a confidence interval based on a binomial approximation [^4]. Intuitively, this corresponds to a robust estimate of the performance one can expect if all other choices in the group were tuned with random search and a limited budget of roughly 20 hyperparameter configurations. *Distribution of choice within top 5% configurations.* We further consider for each choice the distribution of values among the top 5% configurations trained in that experiment. The reasoning is as follows: By design of the experiment, values for each choice are distributed uniformly at random. Thus, if certain values are over-represented in the top models, this indicates that the specific choice is important in guaranteeing good performance. #### Performance measures. We employ the following way to compute performance: For each hyperparameter configuration, we train $3$ models with independent random seeds where each model is trained for one million (Hopper, HalfCheetah, Walker2d) or two million environment steps (Ant, Humanoid). We evaluate trained policies every hundred thousand steps by freezing the policy and computing the average undiscounted episode return of 100 episodes (with the stochastic policy). We then average these score to obtain a single performance score of the seed which is proportional to the area under the learning curve. This ensures we assign higher scores to agents that learn quickly. The performance score of a hyperparameter configuration is finally set to the median performance score across the 3 seeds. This reduces the impact of training noise, i.e., that certain seeds of the same configuration may train much better than others. Related Work {#sec:related} ============ Islam et al. [@islam2017reproducibility] and Henderson et al. [@henderson2018deep] point out the reproducibility issues in RL including the performance differences between different code bases, the importance of hyperparameter tuning and the high level of stochasticity due to random seeds. Tucker et al. [@tucker2018mirage] showed that the gains, which had been attributed to one of the recently proposed policy gradients improvements, were, in fact, caused by the implementation details. The most closely related work to ours is probably Engstrom et al. [@implementation_matters] where the authors investigate code-level improvements in the PPO [@ppo] code base and conclude that they are responsible for the most of the performance difference between PPO and TRPO [@trpo]. Our work is also similar to other large-scale studies done in other fields of Deep Learning, e.g. model-based RL [@langlois2019benchmarking], GANs [@lucic2018gans], NLP [@kaplan2020scaling], disentangled representations [@locatello2018challenging] and convolution network architectures [@radosavovic2020designing]. Conclusions {#sec:conclusions} =========== In this paper, we investigated the importance of a broad set of high- and low-level choices that need to be made when designing and implementing on-policy learning algorithms. Based on more than 250’000 experiments in five continuous control environments, we evaluate the impact of different choices and provide practical recommendations. One of the surprising insights is that the initial action distribution plays an important role in agent performance. We expect this to be a fruitful avenue for future research. [^1]: It has been noticed that the version of the Mujoco physics simulator [@mujoco] can slightly influence the behaviour of some of the environments — <https://github.com/openai/gym/issues/1541>. We used Mujoco 2.0 in our experiments. [^2]: Exact details for the different experiments are provided in Appendices \[exp\_final\_losses\] - \[exp\_final\_regularizer\]. [^3]: <https://github.com/openai/baselines/blob/master/baselines/ppo2/defaults.py> [^4]: We compute confidence intervals with a significance level of $\alpha=5\%$ as follows: We find $i_l = icdf\left(\frac\alpha2\right)$ and $i_h = icdf\left(1-\frac\alpha2\right)$ where $icdf$ is the inverse cumulative density function of a binomial distribution with $p=0.95$ (as we consider the 95th percentile) and the number of draws equals the number of samples. We then report the $i_l$th and $i_h$th highest scores as the confidence interval.

--- author: - '**[Sarbari Guha and Samarjit Chakraborty]{}**' title: '[**On the gravitational entropy of accelerating black holes** ]{}' --- Abstract {#abstract .unnumbered} ======== In this paper we have examined the validity of a proposed definition of gravitational entropy in the context of accelerating black hole solutions of the Einstein field equations, which represent the realistic black hole solutions. We have adopted a phenomenological approach proposed in Rudjord et al \[20\] and expanded by Romero et al \[21\], in which the Weyl curvature hypothesis is tested against the expressions for the gravitational entropy. Considering the $C$-metric for the accelerating black holes, we have evaluated the gravitational entropy and the corresponding entropy density for four different types of black holes, namely, non-rotating black hole, non-rotating charged black hole, rotating black hole and rotating charged black hole. We end up by discussing the merits of such an analysis and the possible reason of failure in the particular case of rotating charged black hole and comment on the possible resolution of the problem. KEYWORDS: Gravitational entropy, Accelerating Black holes. Introduction ============ The $C$-metric was independently discovered by Levi-Civita [@Levi] and Weyl [@Weyl] in 1917. Ehlers and Kundt [@EK] while working on the classification of the degenerated static vacuum fields, constructed a table in which this metric was placed in the slot “$C$”, leading to the name ‘$C$-metric’. Kinnersley and Walker [@KW] pointed out that this metric is an exact solution of Einstein’s equations which describes the combined electromagnetic and gravitational field of a uniformly accelerating object having mass $m$ and charge $e$, and is an example of “almost everything”. It is for this reason that the $C$-metric is the focus of our attention in this paper. Dray and Walker [@DW] showed that this spacetime represents the gravitational field of a pair of uniformly accelerating black holes. Letelier and Oliveira [@LO], studied the static and stationary $C$-metric and sought its interpretation in details, in particular those cases charaterized by two event horizons, one for the black hole and another for the acceleration. For spacetimes with vanishing or positive cosmological constant, the $C$-metric represents two accelerated black holes in asymptotically flat or de Sitter (dS) spacetime, and for a negative $\Lambda$ term, depending on the magnitude of acceleration [@DL], it may represent a single accelerated black hole or a pair of causally separated black holes which accelerate away from each other [@Krtous]. The acceleration $A$ is due to forces represented by conical singularities arising out of a strut between the two black holes or because of two semi-infinite strings connecting them to infinity [@Podolsky; @GKP]. The second law of thermodynamics is one of the most fundamental laws of physics. We know that for an ensemble of ideal gas molecules confined to a closed chamber, the gas spreads out to fill the entire space once the chamber is opened, thereby reaching a state of maximum entropy. However, in the case of the universe with its matter content modelled as a fluid (or gas), this is not exactly true. The universe was born from a very homogeneous state and later on, small density fluctuations appeared due to the effect of gravity, that ultimately led to the formation of structures in the universe. This evolution is contrary to our expectations from the thermodynamic point of view, since the “gas” condenses into clumps of matter, instead of spreading out. Moreover in the past, the universe was much hotter and at some point of time, matter and radiation were in thermal equilibrium, and the entropy was maximum. So, how can the entropy increase if it was maximum in the past? It appears that if the evolution of the universe is dominated solely by gravity, then we may encounter a violation of the second law of thermodynamics, if we are considering the contribution of the thermodynamic entropy only. To resolve this problem and to provide a proper sequence to the occurrence of gravitational processes, Penrose [@Penrose1] proposed that we must assign an entropy function to the gravitational field itself. He suggested that the Weyl curvature tensor could be used as a measure of the gravitational entropy. The Weyl tensor $C_{\alpha\beta\gamma\delta}$ in $ n $ dimensions is expressed as [@Chandra] $$\label{decom} C_{\alpha\beta\gamma\delta}= R_{\alpha\beta\gamma\delta} - \dfrac{1}{(n-2)}(g_{\alpha\gamma}R_{\beta\delta} + g_{\beta\delta}R_{\alpha\gamma} - g_{\beta\gamma}R_{\alpha\delta} - g_{\alpha\delta}R_{\beta\gamma}) + \dfrac{1}{(n-1)(n-2)}R(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma}),$$ where $R_{\alpha\beta\gamma\delta}$ is the covariant Riemann tensor, $R_{\alpha\beta}$ is the Ricci tensor and $R$ is the Ricciscalar. According to Penrose, initially after the ‘big bang’, when the universe started evolving, the Weyl tensor component was much smaller than the Ricci tensor component of the spacetime curvature. This hypothesis sounds credible because the Weyl tensor is independent of the local energy–momentum tensor. Moreover, the universe was in a nearly homogeneous state before structure formation began, and the FRW models successfully describe this homogeneous phase of the evolution. Further, the Weyl curvature is zero in the FRW models. However, the Weyl is large in the Schwarzschild spacetime. Thus we need a description of gravitational entropy, which should increase throughout the history of the universe on account of formation of more and more structures leading to the growth of inhomogeneity [@Penrose2; @Bolejko], and thus preserve the second law of thermodynamics. But there is still doubt regarding the definition of gravitational entropy in a way analogous to the thermodynamic entropy, which would be applicable to all gravitational systems [@CET]. The definition of gravitational entropy as the ratio of the Weyl curvature and the Ricci curvature faces problems with radiation [@Bonnor]. Once Senovilla showed that the Bel-Robinson tensor is suitable for constructing a measure of the “energy” of the gravitational field [@Senovilla], several attempts were made to define the gravitational entropy based on the Bel-Robinson tensor and also in terms of the Riemann tensor and its covariant derivatives [@PL; @PC]. Many efforts has been made to explain the entropy of black holes using the quantized theories of gravity, such as the string theory and loop quantum gravity. However, in this paper we will handle the problem from a phenomenological approach proposed in [@entropy1] and expanded in [@entropy2], in which the Weyl curvature hypothesis is tested against the expressions for the entropy of cosmological models and black holes. They considered a measure of gravitational entropy in terms of a scalar derived from the contraction of the Weyl tensor and the Riemann tensor, and matched it with the Bekenstein-Hawking entropy [@SWH1; @Bekenstein]. In our current work we will consider the accelerating black holes only, which represent more realistic black holes for several reasons. For instance, collision of galaxies is a rather common phenomenon occurring in the universe, and it inevitably leads to black hole mergers with the associated production of gravitational waves [@POK]. In such situations, we may imagine that the black holes at the centre of these galaxies are accelerating towards each other, although we can always think of any black hole as accelerating since no black hole is gravitationally isolated from the neighboring massive systems. Moreover, a static black hole may be considered as the limiting case of an accelerating black hole. Thus the study of accelerating black holes is very important. Here we will investigate whether the calculations for gravitational entropy proposed in [@entropy1] and [@entropy2] can be applied in this context. The organization of our paper is as follows: Sec. II deals with the definition of gravitational entropy and Sec. III enlists the metrics of accelerating black holes considered by us. Sec. IV provides the main analysis of our paper where we evaluate the gravitational entropy and the corresponding entropy density for these black holes. We discuss our results in Sec. V and present the conclusions in Sec. VI. Gravitational Entropy ===================== The entropy of a black hole can be described by the surface integral [@entropy1] $$S_{\sigma}=k_{s}\int_{\sigma}\mathbf{\Psi}.\mathbf{d\sigma},$$ where $ \sigma $ is the surface of the horizon of the black hole and the vector field $\mathbf{\Psi}$ is given by $$\mathbf{\Psi}=P \mathbf{e_{r}},$$ with $ \mathbf{e_{r}} $ as a unit radial vector. The scalar $ P $ is defined in terms of the Weyl scalar ($ W $) and the Krestchmann scalar ($ K $) in the form $$\label{P_sq} P^2=\dfrac{W}{K}=\dfrac{C_{abcd}C^{abcd}}{R_{abcd}R^{abcd}}.$$ In order to find the gravitational entropy, we need to do our computations in a 3-space. Therefore, we consider the spatial metric which is defined as $$\label{sm} h_{ij}=g_{ij}-\dfrac{g_{i0}g_{j0}}{g_{00}},$$ where $ g_{\mu\nu} $ is the concerned 4-dimensional space-time metric and the Latin indices denote spatial components, $i, j = 1, 2, 3$. So the infinitesimal surface element is given by $$d\sigma=\dfrac{\sqrt{h}}{\sqrt{h_{rr}}}d\theta d\phi.$$ Using Gauss’s divergence theorem, we can easily find out the entropy density [@entropy1] as $$s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|.$$ Accelerating Black holes ======================== Non-rotating black hole ----------------------- The $C$-metric in spherical type coordinates is given by $$\label{cmetric} ds^2=\dfrac{1}{(1-\alpha r cos\theta)^2}\left(-Qdt^2+\dfrac{dr^2}{Q}+\dfrac{r^2d\theta^2}{P}+Pr^2sin^2\theta d\phi^2\right),$$ where $ P=(1-2\alpha m cos\theta)$, and $ Q=\left(1-\dfrac{2m}{r}\right)(1-\alpha^2r^2) $. This metric represents an accelerating massive black hole, which has two coordinate singularities, one is at $ r_{a}=\dfrac{1}{\alpha} $ and the other is at $ r_{h}=2m $. The $ r_{h}=2m $ singularity stands for the familiar *event horizon*, but the $ r_{a}=\dfrac{1}{\alpha} $ singularity is the *acceleration horizon* formed due to the acceleration of the black hole [@GP]. Here $m$ is the mass of the black hole and $ \alpha $ is the acceleration parameter. The important feature of this metric is that $ \phi\in [0,2\pi C) $ unlike the full $ 2\pi $ range for stationary black holes because of the conical singularity arising due to acceleration. Here $ C=\dfrac{1}{(1+2\alpha m)} $ is the deficiency factor in the range of $ \phi $. If the acceleration of the black hole vanishes, i.e., $ \alpha=0 $, then the deficiency factor $ C $ becomes unity and $ \phi $ reduces to the conventional polar coordinate running from $ 0 $ to $ 2\pi $. All the accelerated black hole metrics discussed below also have this property. Non-rotating charged black hole ------------------------------- We now consider the metric representing charged accelerating black holes. The charged $C$-metric in spherical type coordinate is [@GP] $$\label{cmetric1} ds^2=\dfrac{1}{(1-\alpha r cos\theta)^2}\left(-Qdt^2+\dfrac{dr^2}{Q}+\dfrac{r^2d\theta^2}{P}+Pr^2sin^2\theta d\phi^2\right),$$ where $ P=(1-2\alpha m cos\theta+\alpha^2e^2cos^2\theta)$, and $ Q=\left(1-\dfrac{2m}{r}+\dfrac{e^2}{r^2}\right)(1-\alpha^2r^2) $. This is just the charged version of the previous metric with the parameter $ e $ representing the charge of the black hole. We can also think of it as an accelerated Reissner–Nordstrom (RN) black hole, because as $ \alpha\rightarrow 0 $, the metric reduces to the familiar RN metric. In the case of this metric also, we have $ r=\dfrac{1}{\alpha} $ as the acceleration horizon, and because of the introduction of charge, we have the outer and inner horizons at $ r_{\pm}=m\pm \sqrt{m^{2}-e^{2}}.$ Here the corresponding deficiency factor is given by $ C=\dfrac{1}{(1+2\alpha m+\alpha^2 e^2)} $. Rotating black hole ------------------- The general line element for **an accelerating** rotating black hole is given by $$\label{htn} ds^2=\dfrac{1}{\Omega^2}\left(-\dfrac{Q}{R}(dt-asin^2\theta d\phi)^2 + \dfrac{R}{Q}dr^2+\dfrac{R}{P}d\theta^2+\dfrac{P}{R}sin^2\theta[adt-(r^2+a^2)d\phi]^2\right),$$ where $ \Omega=1-\alpha rcos\theta $, $ R=r^2+a^2cos^2\theta $, $ P=(1-2\alpha m cos\theta +\alpha^2a^2cos^2\theta) $, and $ Q=(a^2-2mr+r^2)(1-\alpha^2r^2) $. This metric represents the rotating version of the $ C$-metric, and contains three coordinate singularities, namely $ r_{\pm}=m\pm \sqrt{m^2-a^2} ,$ representing the outer and inner horizons, and $ r=\dfrac{1}{\alpha} $ representing the acceleration horizon [@GP] with the deficiency factor $C=\dfrac{1}{(1+2\alpha m+\alpha^2 a^2)}$ . Rotating charged black hole --------------------------- This is the charged version of the previous accelerating rotating metric. It may be regarded as the most general case among all the black holes considered by us, and is given by $$\label{ht} ds^2=\dfrac{1}{\Omega^2}\left(-\dfrac{Q}{R}(dt-asin^2\theta d\phi)^2 + \dfrac{R}{Q}dr^2+\dfrac{R}{P}d\theta^2+\dfrac{P}{R}sin^2\theta[adt-(r^2+a^2)d\phi]^2\right),$$ where $ \Omega=1-\alpha rcos\theta $, $ R=r^2+a^2cos^2\theta $, $ P=(1-2\alpha m cos\theta +\alpha^2(a^2+e^2)cos^2\theta) $, and $ Q=(a^2+e^2-2mr+r^2)(1-\alpha^2r^2) $. In this case the deficiency factor is given by $ C=\dfrac{1}{(1+2\alpha m+\alpha^2 (a^2+e^2))}. $ As in the previous case, the acceleration horizon is at $ r=\dfrac{1}{\alpha} $, however, the outer (or inner) horizons are located at $ r_{\pm}=m\pm \sqrt{m^2-a^2-e^2}. $ Analysis ======== Non-rotating accelerating black hole ------------------------------------ We know that the Kretschmann scalar for a given spacetime geometry is defined by the relation $$K=R_{abcd}R^{abcd},$$ where $R_{abcd} $ is the covariant Riemann curvature tensor. For the $C$-metric (\[cmetric\]), the Kretschmann scalar turns out to be $$\label{non-rot_Kscalar} K_{c}=\dfrac{48m^2(\alpha r cos\theta-1)^6}{r^6}.$$ The Weyl scalar is defined by $$W=C_{abcd}C^{abcd},$$ where the $C_{abcd} $ is the Weyl curvature tensor. For the $C$-metric (non-rotating black hole) the Weyl scalar is evaluated as $$\label{non-rot_Wscalar} W_{c}=\dfrac{48m^2(\alpha r cos\theta-1)^6}{r^6}.$$ This result is expected since the Ricci tensor for this metric turns out to be zero. As the Riemann tensor can be decomposed into the Ricci and the Weyl parts according to equation (\[decom\]), the vanishing Ricci component renders the Riemann and Weyl tensors identical as evident from equations (\[non-rot\_Kscalar\]) and (\[non-rot\_Wscalar\]). The scalar function $P$ is defined by the relation (\[P\_sq\]) as $$P^2=\dfrac{C_{abcd}C^{abcd}}{R_{abcd}R^{abcd}}.$$ For this $C$-metric, we get $ P^2=1 $. Therefore we assume that $ P=+1 $ for our entropy calculations, since the entropy must be non-negative. Now the *spatial section* corresponding to this metric is $$h_{ij}=diag\left[\dfrac{1}{( 1 - \alpha r cos\theta )^2 (1-2m/r) (1-\alpha^2 r^2)}, \dfrac{r^2}{(( 1 - \alpha r cos\theta )^2 ( 1 - 2\alpha m cos\theta ))}, \dfrac{r^2 sin^2 \theta ( 1 - 2 \alpha m cos\theta )}{(1 - \alpha r cos\theta )^2}\right],$$ with the determinant given by $$h=\dfrac{sin^2(\theta)r^5}{(\alpha^2 r^2-1)(-r+2m)(\alpha r cos(\theta)-1)^6}.$$ Therefore, the infinitesimal surface element has the form $$d\sigma=\dfrac{\sqrt{h}}{\sqrt{h_{rr}}}d\theta d\phi=\dfrac{r^2 sin\theta}{(\alpha r cos\theta-1)^2} d\theta d\phi.$$ We are now in a position to calculate the magnitude of the gravitational entropy on the event horizon $H_{0}$ at the location $ r_{h}=2m $ for this metric, which is $$\label{s_grav_nonrot} S_{grav}=k_{s}r_{h}^2\int_{\theta=0}^{\pi}\dfrac{sin\theta}{(\alpha r_{h} cos\theta-1)^2}d\theta \int_{\phi=0}^{2\pi C} d\phi=k_{s}\dfrac{4 \pi C r_{h}^2}{(1-r_{h}^2\alpha^2)}=k_{s}\dfrac{4 \pi r_{h}^2}{(1-r_{h}^2\alpha^2)(1+2\alpha m)}.$$ From equation (\[s\_grav\_nonrot\]) it is evident that the gravitational entropy is proportional to the area of the event horizon of the black hole, as in the case of the Bekenstein-Hawking entropy [@SWH1; @Bekenstein]. Here $ C=\dfrac{1}{(1+2\alpha m)} $ is the deficiency factor in the limit of $ \phi $ as it runs from $ 0\rightarrow 2\pi C $ (as mentioned earlier). In FIG. \[plot3\], we have shown the variation of the total entropy on the horizon with the acceleration parameter $ \alpha $. Similarly we can compute the entropy density as $$\label{s_nonrot} s=k_{s}\frac{1}{\sqrt{h}}\frac{\partial}{\partial r}\left(\sqrt{h}\dfrac{P}{\sqrt{h_{rr}}} \right)=\dfrac{2k_{s}}{r}\sqrt{\left(1-\alpha^{2}r^{2}\right)\left(1-\frac{r_{h}}{r}\right)}.$$ In the above equation (\[s\_nonrot\]), inserting $ \alpha=0 $, we get the entropy density for the Schwarzschild black hole. In FIG. \[fig2\], the dependence of the gravitational entropy density corresponding to this metric on other relevant parameters have been indicated. From equation (\[s\_nonrot\]) we can see that the zeroes of the gravitational entropy density function are located at the acceleration horizon $ r=\dfrac{1}{\alpha} $, and at the event horizon $ r=2m $, which is clearly evident from FIG. \[fig2\]. Specifically, FIG. \[fig2\](a) shows that for $ \alpha=0 $, the acceleration horizon goes to infinity where the entropy density reduces to zero, and at the event horizon $r=2$, the entropy density becomes zero. Similarly for $ \alpha=0.5 $, the acceleration horizon and the event horizon coincide at $ r=2 $, where the entropy density becomes zero. FIG. \[fig2\](b) indicates that for $\alpha=0.25$, the acceleration horizon lies at $r=4$ and the event horizon is at $ r=2 $, the entropy density going to zero at both these places, and diverges at the singularity $ r=0 $ which is in agreement with equation (\[s\_nonrot\]). ![Plot showing the variation of the total gravitational entropy for the accelerating non-rotating BH with respect to the acceleration parameter $ \alpha $, where we have taken $m=1 \: \textrm{and} \: k_{s}=1$.[]{data-label="plot3"}](Salphanew1.png){width="34.00000%"} Non-rotating charged accelerating black hole -------------------------------------------- The Kretschmann scalar for the non-rotating charged black hole given by the metric (\[cmetric1\]) is evaluated to be $$K=\dfrac{56\left(\alpha rcos\theta-1\right)^6 \left(cos^2\theta \alpha^2 e^4 r^2+\dfrac{10}{7} \left(e^2-\dfrac{6}{5}mr\right)re^2 \alpha cos\theta+e^4-\dfrac{12}{7} e^2mr+\dfrac{6}{7}m^2r^2 \right)}{r^8},$$ and the corresponding Weyl scalar is $$W=\dfrac{4}{3}\dfrac{\left(\alpha r cos\theta -1\right)^4 \left(5cos^2\theta \alpha^2 e^2 r^2-sin^2\theta\alpha^2 e^2 r^2+\alpha^2 e^2 r^2-6 m \alpha cos\theta r^2-6e^2+6mr\right)^2}{r^8}.$$ Therefore the quantity $P$ is given by the expression $$P^2=\dfrac{6(e^2\alpha cos\theta r+e^2-mr)^2}{(7 cos^2\theta\alpha^2e^4r^2+10\alpha r cos\theta e^4-12\alpha r^2 cos\theta e^2 m+7e^4-12e^2 mr+6 m^2 r^2)}.$$ The spatial metric for this case is $$\begin{aligned} % \nonumber to remove numbering (before each equation) h_{ij} &=& \dfrac{1}{(1-\alpha r cos\theta)^2} diag\left[\dfrac{1}{\left(1-\dfrac{2m}{r}+\dfrac{e^2}{r^2}\right)\left(-\alpha^2 r^2+1 \right)}, \dfrac{r^2}{\beta}, \beta r^2 sin^2\theta \right],\end{aligned}$$ where $$\beta=\left(1-2\alpha m cos\theta +\alpha^2 e^2 cos^2\theta\right). \nonumber$$ Consequently the determinant of the spatial $ h_{ij} $ metric is given by $$h=-\dfrac{sin^2\theta r^6}{(\alpha^2 r^2-1)(e^2-2mr+r^2)(\alpha r cos\theta-1)^6},$$ and the infinitesimal surface element is $$d\sigma=\dfrac{\sqrt{h}}{\sqrt{h_{rr}}}d\theta d\phi=\dfrac{r^2 sin\theta}{(\alpha r cos\theta-1)^2} d\theta d\phi.$$ Next we calculate the gravitational entropy on the horizon $H_{0}$ at $ r_{h}=r_{\pm}=m \pm \sqrt{m^2-e^2} ,$ which turns out to be $$\label{s_grav_nonrot_chrg} S_{grav}=k_{s}r_{h}^2\int_{\theta=0}^{\pi}\dfrac{P(r_{h},\theta) sin\theta}{(\alpha r_{h} cos\theta-1)^2}d\theta \int_{\phi=0}^{2\pi C} d\phi=\dfrac{k_{s}(4\pi r_{\pm}^2)}{(1+2\alpha m+\alpha^2 e^2)}\int_{\theta}\dfrac{P_{\pm}(\theta)sin\theta d\theta}{2(\alpha r_{\pm} cos\theta-1)^2},$$ where the quantity $P_{\pm}$ corresponds to the value calculated for $r_{\pm}$. From equation (\[s\_grav\_nonrot\_chrg\]) we find that the gravitational entropy is proportional to the area of the event horizon of the black hole, just as in the case of the Bekenstein-Hawking entropy. We can further check the validity of our result by setting $\alpha=0$ in (\[s\_grav\_nonrot\_chrg\]), to see whether it leads us to the desired expression for the entropy of the Reissner–Nordstrom (RN) black hole. This exercise yields the result $$S_{grav}^{RN}=k_{s}(4\pi r_{\pm}^{2})\int_{\theta}P^{RN}_{\pm}(\theta)\dfrac{\sin\theta}{2} d\theta.$$ We can easily see that $P^{RN}_{\pm}(\theta)=P_{\pm}(\alpha=0)=\dfrac{6e^{4}-12e^{2}mr+6m^{2}r^{2}}{7e^{4}-12e^{2}mr+6m^{2}r^{2}} , $ and therefore the gravitational entropy for the RN black hole is $$S_{grav}^{RN}=k_{s}(4\pi r_{\pm}^{2})\sqrt{\dfrac{6e^{4}-12e^{2}mr+6m^{2}r^{2}}{7e^{4}-12e^{2}mr+6m^{2}r^{2}}}\int_{\theta}\dfrac{\sin\theta}{2}d\theta=k_{s}(4\pi r_{\pm}^{2})\sqrt{\dfrac{6e^{4}-12e^{2}mr+6m^{2}r^{2}}{7e^{4}-12e^{2}mr+6m^{2}r^{2}}}.$$ This result matches with the expression of gravitational entropy for the RN black hole derived in [@entropy2] by Romero et al. The entropy density for the non-rotating charged black hole is obtained as $$\begin{aligned} \label{s_nonrot_chrg} \left.s\right. & = \frac{16\sqrt {6}k_{s}\sqrt { \left( -{\alpha}^{2}{r}^{2}+1 \right) \left( {e}^{2}-2mr+{r}^{2} \right) }}{{{r}^{2} \left( 7{e}^{4}{\alpha}^{2} \cos^{2}\theta{r}^{2} + 10 \left( {e}^{2}-\dfrac{6mr}{5} \right) r\alpha{e}^{2}\cos\theta + 7{e}^{4}-12{e}^{2}mr+6{m}^{2}{r}^{2} \right) ^{3/2}}} \nonumber \\ & \times \left[ \cos^{3}\theta{\alpha}^{3}{e}^{6}{r}^{3} + {\frac {15{e}^{4}{\alpha}^{2} \cos^{2}\theta{r}^{2}}{8} \left( {e}^{2} - {\frac{13mr}{10}} \right) } + \dfrac{9 r\alpha{e}^{2} \cos\theta }{4} \left( {e}^{4}-{\frac {11{e}^{2}mr}{6}}+{m}^{2}{r}^{2} \right) \right. \nonumber \\ & \qquad\qquad\qquad + \left. {\frac {7{e}^{6}}{8}} - \frac{3mr}{4} \left({ \frac {13{e}^{4}}{4}} - 3{e}^{2}mr + {m}^{2}{r}^{2} \right) \right] . \end{aligned}$$ If in this expression we substitute $ e=0 $, and consider the absolute value of this quantity, then we get back the expression (\[s\_nonrot\]) for the entropy density of the accelerating black hole. In FIG. \[fig3\] we have shown the dependence of the gravitational entropy density of the non-rotating charged black hole on different parameters appearing in (\[s\_nonrot\_chrg\]). From FIG. \[fig3\](a) we can determine the zeroes of the gravitational entropy function (\[s\_nonrot\_chrg\]), e.g., for $ \alpha=0, \theta=\frac{\pi}{2} $, the acceleration horizon goes to infinity and by solving the entropy density function, we obtain the zeroes at $ r=0.13, 1.87 $, and also at $ r=0.18 $, where $ r=0.13, 1.87 $ are the horizons. Again from (\[s\_nonrot\_chrg\]), using $ \alpha=0.45, \theta=\frac{\pi}{2} $, we find that the zeroes of the entropy density function are located at the acceleration horizon $ r=\frac{1}{\alpha}=2.22 $, and at the event horizon $ r=m\pm\sqrt{m^2 - e^2}=1 $, which is evident from FIG. \[fig3\](b). The additional zero can be found by solving for the roots of the second factor in (\[s\_nonrot\_chrg\]) which gives us the only real root at $ r=0.74 $. We have also analyzed the case for $ \alpha=0.25 $ (shown in FIG. \[plot4a\]) from which we can identify the zeroes clearly, i.e., at the acceleration horizon $ r=\frac{1}{\alpha}=4 $, and at the horizons $ r=0.13, 1.87 $. Further, another zero arises from the second term of the entropy density function at $ r=0.19 $. The overall behavior is also as we expected, that is, the entropy density diverges near the $r=0$ singularity and it increases inside the horizon, encountering some zeroes in between. Rotating accelerating black hole -------------------------------- Using the metric for the rotating accelerating black hole we have calculated the Ricci tensor, which turns out to be zero. Thus the Kretschmann scalar $K$ and the Weyl scalar $W$ are identical. Thus we have $$\begin{aligned} \nonumber K= W = 48{m}^{2} \left( \alpha r\cos \left( \theta \right) -1 \right) ^{6} \left( \left({a}^{4}\alpha+{a}^{3} \right) \cos^{3}\theta + 3{a}^{2}r \left( a \alpha-1 \right) \cos^{2}\theta - 3a {r}^{2} \left( a\alpha+1 \right) \cos \theta -{r}^{3} \left( a\alpha-1 \right) \right) \\ \times \dfrac{ \left( \left( {a}^{4}\alpha - {a}^{3} \right) \cos^{3}\theta - 3{a}^{2}r \left( a \alpha+1 \right) \cos^{2}\theta - 3a {r}^{2}\left(a\alpha-1 \right) \cos \theta + {r}^{3} \left( a\alpha+1 \right) \right) } { \left( {r}^{2} + {a}^{2} \cos^{2}\theta \right) ^{6} }.\end{aligned}$$ Therefore $ P^{2}=\dfrac{W}{K}=1 $, i.e. $ P=+1 $. Hence the total gravitational entropy in this case is given by $$\label{s_grav_rot} S_{grav}=k_{s}\int_{\sigma}\mathbf{\Psi}.\mathbf{d\sigma}=k_{s}\int_{\sigma}d\sigma=k_{s}\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi C}\sqrt{g_{\theta\theta}g_{\phi\phi}}d\theta d\phi.$$ The entropy evaluated at $r_{\pm}$ is obtained as $$\label{s_grav_pm} S_{grav_{\pm}}=k_{s}\dfrac{4\pi C(r^{2}_{\pm}+a^{2})}{(1-\alpha^{2}r_{\pm}^{2})}=k_{s}\dfrac{4\pi (r^{2}_{\pm}+a^{2})}{(1-\alpha^{2}r_{\pm}^{2})(1+2\alpha m+\alpha^2a^2)}.$$ If we substitute $ a=0 $ in (\[s\_grav\_pm\]), then we get back the expression (\[s\_grav\_nonrot\]) for the entropy of the non-rotating accelerating black holes. We see that as the acceleration parameter vanishes, i.e., $ \alpha\rightarrow 0 $, the equation (\[s\_grav\_pm\]) reduces to the expression of gravitational entropy for Kerr black holes derived in [@entropy2]. However for this axisymmetric metric, it is not possible to evaluate the spatial metric using equation (\[sm\]) because the object is rotating, and so there is a nonzero contribution from the component of $ g_{t\phi} $, which changes the spatial positions of events in course of time. Therefore the entropy density is calculated by using the full four-dimensional metric determinant $ g $ in the expression involving the covariant derivative [@entropy2], and we get $$\label{enden1} s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left(\dfrac{\partial}{\partial r}\sqrt{-g}P\right)=2k_{s}\dfrac{(2\cos^{3}\theta a^{2}\alpha+\cos\theta \alpha r^{2}+r)}{(1-\alpha r \cos\theta)(r^{2}+a^{2}cos^{2}\theta)},$$ where $ g=-\sin^{2}\theta \dfrac{(a^{2}\cos^{2}\theta+r^{2})^{2}}{(\alpha r \cos\theta -1)^{8}} $. From equation (\[enden1\]) we see that the entropy density diverges at the ring singularity and at $ r=\dfrac{1}{\alpha\cos\theta}, $ which is the conformal infinity in this spherical type coordinate system, as is evident from the metric (\[htn\]). This can also be further verified from the expressions of the Kretschmann scalar and the Weyl scalar in this case, since they vanish at the conformal infinity but diverge at the ring singularity. To compute the zeroes of the entropy density function we only need to find the roots of the numerator in (\[enden1\]), which is a quadratic function in $ r $. Substituting $ \alpha=0 $ in the above expression of entropy density, we get the entropy density for the Kerr black hole: $$s_{kerr}=\dfrac{2k_{s}r}{(r^{2}+a^{2}\cos^{2}\theta)}.$$ ![Plot showing the variation of the gravitational entropy density for the accelerating non-rotating charged BH with respect to the radial coordinate $ r $, where $ m=1, \; k_{s}=1, \; \alpha=0.25, \; e=0.5, \; \textrm{and} \; \theta=\dfrac{\pi}{2} $.[]{data-label="plot4a"}](chcsralphav.png){width="55.00000%"} ![Plot showing the variation of the gravitational entropy density for the accelerating rotating BH with respect to the radial coordinate $ r $ and the angular coordinate $ \theta $, where $ \alpha=0.45, \: m=1, \: {\color{blue}{a=0.5,}} \: \textrm{and} \: k_{s}=1 $. This figure clearly indicates that at the ring singularity $\left( r=0, \: \theta=\dfrac{\pi}{2} \right) $ the gravitational entropy density diverges.[]{data-label="plot4b"}](htnsrthetanew1.png){width="40.00000%"} ![Plot showing the variation of the gravitational entropy density for the accelerating rotating BH with respect to the radial coordinate $ r $ for different values of $ \theta $, where we have taken $m=1 \: a=0.25, \: \alpha=0.5, \textrm{and} \: k_{s}=1 $.[]{data-label="plottn"}](htnconfor.png){width="60.00000%"} FIG. \[plot4b\] clearly shows that the measure of entropy density is well behaved everywhere except at the ring singularity. In figures FIG. \[fig4\](a) and FIG. \[fig4\](b), we have shown that for different values of $\theta$ we can have a diverging or finite entropy density at $r=0$. When $ \theta=\frac{\pi}{2} $, the expression of entropy density in (\[enden1\]) simply becomes $ \dfrac{2k_{s}}{r} $, which can be seen clearly in FIG. \[fig4\](a) and it also diverges at the ring singularity at $ r=0 $ for this case. Whereas in FIG. \[fig4\](b), we can see that for $ \theta=\frac{\pi}{6} $ the entropy density is finite at $ r=0 $ for nonzero values of acceleration parameter whereas for $ \alpha=0 $ the entropy density becomes zero at the central singularity at $ r=0 $. FIG. \[fig5\](a) shows us that the entropy density simply behaves like inverse squared in $ r $ when $ \theta=\frac{\pi}{2} $, for different values of $ a $, as it becomes independent of the rotation parameter and the acceleration parameter, which can be easily seen from the expression (\[enden1\]). In FIG. \[fig5\](b), the entropy density diverges for the condition $ r=0, \: \textrm{and} \: a=0 $, because it corresponds to the central singularity of an accelerating non-rotating BH. These behaviors of the entropy density are in conformity with our expectations and so we can say that this definition of entropy density is quite suitable for these kinds of black holes. In FIG. \[plottn\], the nature of the gravitational entropy density for accelerating rotating black hole is studied for different values of $ \theta $, where we have fixed the values of the black hole parameters as the following: $m=1, \: a=0.25, \: \alpha=0.5, \textrm{and} \: k_{s}=1 $. The conformal infinity lies at $ r=\dfrac{1}{\alpha\cos\theta} $. From (\[enden1\]) it is clear that as the value of $ \theta $ goes from $ 0 $ to $ \dfrac{\pi}{2} $, the conformal infinity shifts towards infinity, giving rise to the $ \sim \dfrac{1}{r^2} $ behavior which diverges only at the ring singularity at $ r=0, \theta=\dfrac{\pi}{2} $. Moreover, we observe that except for $ \theta=\dfrac{\pi}{2} $, the gravitational entropy density remains finite at $ r=0 $ for all other values of $\theta$. Rotating charged accelerating black hole ---------------------------------------- For the rotating charged black hole, the Weyl scalar $W$ is $$\begin{aligned} \left.W\right. &=48\frac { \left( \alpha r\cos \left( \theta \right) -1 \right) ^{6}} { \left( {r}^{2}+{a}^{2} \left( \cos \left(\theta \right) \right) ^{2} \right) ^{6}} \times\nonumber\\ &( \left( {e}^{2}r\alpha+am \left( a\alpha+1 \right) \right) { a}^{2}\cos^{3}\theta+ \left( 2a{e} ^{2}{r}^{2}\alpha+3{a}^{2}m \left( a\alpha-1 \right) r+{a}^{2}{e}^{2 } \right) \cos^{2}\theta \nonumber\\ &+ \left( -{e }^{2}\alpha\,{r}^{3}-3am \left( a\alpha+1 \right) {r}^{2}+2 a{e}^{2 }r \right) \cos \left( \theta \right) -{r}^{2} \left( m \left( a\alpha -1 \right) r+{e}^{2} \right) ) \nonumber\\ & ( \left( {e}^{2}r\alpha+a m \left( a\alpha-1 \right) \right) {a}^{2} \left( \cos \left( \theta \right) \right) ^{3}+ \left( -2a{e}^{2}{r}^{2}\alpha-3{a}^{2}m \left( a\alpha+1 \right) r+{a}^{2}{e}^{2} \right) \left( \cos \left( \theta \right) \right) ^{2} \nonumber\\ &+ \left( -{e}^{2}\alpha{r}^{3}-3 am \left( a\alpha-1 \right) {r}^{2}-2a{e}^{2}r \right) \cos \left( \theta \right) +{r}^{2} \left( m \left( a\alpha+1 \right) r-{e }^{2} \right) ),\end{aligned}$$ and the Kretschmann scalar $K$ is $$\begin{aligned} \left.K\right. &= 48\frac {\left( \alpha r \cos \left( \theta \right)-1 \right)^{6}}{\left({r}^{2}+{a}^{2}\left(\cos\left(\theta \right)\right)^{2}\right)^{6}}({a}^{4}\left({a}^{4}{\alpha}^{2}{m}^{2}+2{a}^{2}{\alpha}^{2}{e}^{2}mr+7/6{\alpha}^{2}{e}^{4}{r}^{2} -{a}^{2}{m}^{2} \right) \left( \cos \left( \theta \right) \right) ^{6} \nonumber\\ &+2 \left( {a}^{2}m+5/6{e}^{2}r \right) \left( {e}^{2}-6mr\right) {a}^{4}\alpha \left(\cos\left(\theta \right)\right)^{5} \nonumber\\ & +\left(15{a}^{4}{m}^{2}{r}^{2}-20{a}^{4}{\alpha}^{2}{e}^{2}m{r}^{3}-15{a}^{6}{\alpha}^{2}{m}^{2}{r}^{2}-{\frac {17{a}^{2}{\alpha}^{2}{e}^{4}{r}^{4}}{3}}-10{a}^{4}{e}^{2}mr+7/6{a}^{4}{e}^{4}\right) \left( \cos \left( \theta \right) \right) ^{4} \nonumber\\ &-20 \left(-{e}^{2}m{r}^{2}+ \left( -2{a}^{2}{m}^{2}+{\frac {19{e}^{4}}{30}}\right) r+{a}^{2}{e}^{2}m \right) {r}^{2}{a}^{2}\alpha \left( \cos\left( \theta \right) \right) ^{3}+ \nonumber\\ & \left( 7/6{\alpha}^{2}{e}^{4}{r}^{6}-{\frac {17{a}^{2}{e}^{4}{r}^{2}}{3}}-15{a}^{2}{m}^{2}{r}^{4 }+10{a}^{2}{\alpha}^{2}{e}^{2}m{r}^{5}+15{a}^{4}{\alpha}^{2}{m}^{2}{r}^{4}+20{a}^{2}{e}^{2}m{r}^{3} \right) \left( \cos \left( \theta\right) \right) ^{2} \nonumber\\ &+ 10 \left( {e}^{2}-6/5mr \right) {r}^{4}\alpha \left( {a}^{2}m+1/6{e}^{2}r \right) \cos \left( \theta \right) + \left( -{a}^{2}{\alpha}^{2}{m}^{2}+{m}^{2} \right) {r}^{6}-2{e}^{2}m{r}^{5}+7/6{e}^{4}{r}^{4} ).\end{aligned}$$ From the above scalars we can calculate the ratio $ P=\sqrt{\dfrac{W}{K}} $, defined in [@entropy1], which serves as the measure of gravitational entropy, $S_{grav}$ of black holes. The four-dimensional determinant of the metric is $$g=-{\frac { \left( \sin \left( \theta \right)\right) ^{2} \left( {r}^{2}+{a}^{2} \left( \cos \left( \theta \right) \right) ^{2} \right) ^{2}}{ \left( \alpha\,r\cos \left( \theta \right) -1 \right) ^{8}}}.$$ Here again the axisymmetric metric denies us the calculation of the spatial metric due to the nonzero metric component $ g_{t\phi} $. Therefore as in the previous calculation for rotating black holes, the entropy density is calculated by using the metric determinant $ g $ in the covariant derivative. We thus have $$\label{enden2} s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left(\dfrac{\partial}{\partial r}\sqrt{-g}P\right).$$ Here we have intentionally avoided writing the exact expression of entropy density as it is lengthy and too much complicated, but we can easily check the validity of the result. We have checked that if we substitute $ e=0 $ in these calculations, then we get back the result for the accelerating rotating black hole. In FIG. \[fig6\] we find that the gravitational entropy density is not smooth, but contains several singularities. The above analysis clearly shows that the measure of the gravitational entropy used above is not adequate to explain the case of the accelerating rotating charged black holes. Therefore we have to use the measure proposed in [@entropy2] for the expression of $ P $, which is $$\label{mod_P} P=C_{abcd}C^{abcd}.$$ Using the definition (\[mod\_P\]) of $ P $, we have calculated the gravitational entropy density, which is given in equation (\[new3\]): $$\begin{aligned} \label{new3} \left.s\right.&=\dfrac{k_{s}}{(a^2\cos^2(\theta)+r^2)^7}\bigg(96\Big(a^6\alpha(a^4 \alpha^2 m^2+3a^2\alpha^2 e^2 mr+2\alpha^2 e^4 r^2 -a^2m^2)\cos^9(\theta)\nonumber\\ &+(-20e^2mr^2+(-18a^2m^2+2e^4)r+a^2e^2m)\alpha^2a^6\cos^8(\theta)- \nonumber\\ &34\alpha a^4(21/34e^4r^4\alpha^2+57/34a^2e^2mr^3\alpha^2+(a^4\alpha^2m^2-a^2m^2)r^2+5/34a^2e^2mr-3/17a^4m^2)\cos^7(\theta)+\nonumber\\ &(90a^4e^2mr^4\alpha^2+(142a^6\alpha^2m^2-21a^4\alpha^2e^4)r^3-9a^6e^2m\alpha^2r^2+(20a^8\alpha^2 m^2-20a^6m^2)r+5a^6e^2m)\cos^6(\theta)\nonumber\\ &+30\alpha a^2(8/15e^4r^5\alpha^2+17/6a^2e^2 m r^4\alpha^2+(3a^4\alpha^2 m^2-3a^2m^2)r^3+1/6a^2e^2mr^2\nonumber\\ &+(-19/5a^4m^2+11/30e^4a^2)r+a^4e^2m)r\cos^5(\theta)+(-48a^2e^2m\alpha^2r^6+(-142a^4\alpha^2m^2+16a^2\alpha^2e^4)r^5 \nonumber\\ &-5a^4e^2mr^4\alpha^2+(-90a^6\alpha^2m^2+90a^4m^2)r^3-75a^4e^2mr^2+11a^4e^4r)\cos^4(\theta)-100(1/100e^4r^5\alpha^2+\nonumber\\ &3/20a^2e^2mr^4\alpha^2+(17/50a^4\alpha^2m^2-17/50a^2m^2)r^3-9/100a^2e^2mr^2+(-17/10a^4m^2+13/50e^4a^2)r+ \nonumber\\ &a^4e^2m)\alpha r^3\cos^3(\theta)+(2e^2m\alpha^2r^8+(18a^2\alpha^2 m^2-\alpha^2e^4)r^7+5a^2e^2m\alpha^2 r^6+ \nonumber\\ &(48a^4\alpha^2m^2-48a^2m^2)r^5+75a^2e^2mr^4-26e^4a^2r^3)\cos^2(\theta)+30((1/30a^2\alpha^2m^2-1/30m^2)r^3-1/30e^2mr^2+\nonumber\\ &(-a^2m^2+1/10e^4)r+a^2e^2m)\alpha r^5\cos(\theta)+(-2a^2\alpha^2m^2+2m^2)r^7-5e^2 mr^6+3e^4r^5\Big)(r\alpha \cos(\theta)-1)^5\bigg).\end{aligned}$$ In FIG. \[fig7\] we have shown the variation of the gravitational entropy density with the radial distance and the acceleration parameter using this new definition (\[mod\_P\]) of the scalar $ P $. The entropy density function is now well-behaved and all the singularities vanish, except the ring singularity, on account of the introduction of this new definition. In FIG. \[fig7\](a), the entropy density function diverges at $ r=0 $ and $ \theta=\frac{\pi}{2} $, as it encounters the ring singularity, whereas in FIG. \[fig7\](b) the entropy density stays finite at $ r=0 $ and $ \theta=\frac{\pi}{4} $. Although the entropy density function (\[new3\]) vanishes at the conformal infinity $ r=\dfrac{1}{\alpha\cos(\theta)} $, we cannot simply associate the zeroes of the entropy density function with the horizons, because according to this modified definition, the expression (\[new3\]) does not have such factors, and so we have to solve the function explicitly in order to determine the zeroes of the entropy density. Discussions =========== We now discuss the possibility of having an angular component in the vector field $ \mathbf{\Psi} $ for axisymmetric spacetimes as proposed in [@entropy2]. Using this modified definition of $ \mathbf{\Psi} $, and the modified expression (\[mod\_P\]), we now calculate the gravitational entropy density for axisymmetric space-times, using the following expression: $$\label{news} s=k_{s}|\mathbf{\nabla}.\mathbf{\Psi}|=\dfrac{k_{s}}{\sqrt{-g}}\left|\left(\dfrac{\partial}{\partial r}(\sqrt{-g}P)+\dfrac{\partial}{\partial \theta}(\sqrt{-g}P) \right)\right|.$$ The gravitational entropy density for the uncharged rotating accelerating black hole is given by $$\begin{aligned} \label{new1} \left.s\right. &= \dfrac{k_{s}}{\sqrt{\dfrac{\sin^2(\theta)(a^2\cos^2(\theta)+r^2)^2}{(r\alpha\cos(\theta)-1)^8}}}\Bigg(\Bigg|\dfrac{48}{\sqrt{\dfrac{\sin^2(\theta)(a^2\cos^2(\theta)+r^2)^2}{(r\alpha\cos(\theta)-1)^8}}(a^2\cos^2(\theta)+r^2)^5(r\alpha \cos(\theta)-1)^3} \nonumber\\ &\bigg(\sin(\theta)\Big(((2a^{10}\alpha^{3} m^{2} r-2a^{8}\alpha m^{2}r)\cos^{8}(\theta)-4(a^{4}m^{2}\alpha^{2}+(9\alpha^{2}r^{2}-1)m^{2}a^{2})a^{6}\cos^{7}(\theta)-10(m^{2}(\frac{34}{5}r^{3}\alpha^{2} \nonumber\\ &-6r)a^{4}-\frac{34}{5}r^{3}a^{2}m^{2})\alpha a^{4}\cos^{6}(\theta) +96 a^{4}(a^{4}m^{2}r^{2}\alpha^{2}+(\frac{71}{24}\alpha^{2}r^{4}-r^{2})m^{2}a^{2})\cos^{5}(\theta)+150((\frac{6}{5}r^{3}\alpha^{2}-\frac{34}{15}r) \nonumber\\ &m^{2}a^{4}-\frac{6}{5}r^{3}a^{2}m^{2}) \alpha a^{2}r^{2}\cos^{4}(\theta)-180a^{2}(a^{4}m^{2}r^{2}\alpha^{2}+(\frac{71}{45}\alpha^{2}r^{4}-r^2)m^2a^2)r^2\cos^{3}(\theta)-150\alpha((\frac{34}{75} r^3\alpha^{2} \nonumber\\ &-\frac{38}{25} r)m^2a^4-\frac{34}{75} r^3 a^2 m^2)r^4\cos^{2}(\theta)+40(a^4 m^2 r^2 \alpha^2+(\frac{9}{10} \alpha^2 r^4-r^2)m^2a^2)r^4\cos(\theta)+10\alpha((\frac{1}{5}r^3\alpha^2-\frac{6}{5}r) \nonumber\\ &m a^2-\frac{1}{5}r^3 m)mr^6)\sin^2(\theta)+(2\alpha a^6 (a^4\alpha^2m^2-a^2m^2)\cos^{9}(\theta)-36\cos^8(\theta) a^8 \alpha^2 m^2 r-68 \alpha a^4(m^2(\alpha^2 r^2-\frac{3}{17})a^4 \nonumber\\ & -a^2 m^2 r^2)\cos^7(\theta)+40 a^4(a^4 m^2 r\alpha^2-\frac{9}{20}(-\frac{142}{9} r^3\alpha^2+\frac{20}{9}r)m^2a^2)\cos^6(\theta)+60\alpha a^2((3 r^3 \alpha^2-\frac{19}{5}r)m^2 a^4 \nonumber\\ &-3 r^3 a^2 m^2)r\cos^5(\theta)-180 a^2(a^4 m^2 r^2\alpha^2+(\frac{71}{45} \alpha^2r^4-r^2)m^2a^2)r\cos^{4}(\theta)-200((\frac{17}{50}r^3\alpha^2-\frac{17}{10}r)m^2a^4- \nonumber\\ &\frac{17}{50}r^3 a^2 m^2)\alpha r^3 \cos^3(\theta)+96 r^3(a^4 m^2 r^2\alpha^2+(\frac{3}{8}\alpha^2 r^4-r^2)m^2a^2)\cos^2(\theta)+60((\frac{1}{30} r^3\alpha^2-r)m^2 a^2-\frac{1}{30}r^3 m^2)\nonumber\\ &\alpha r^5\cos(\theta)-4a^2\alpha^2 m^2 r^7+4 m^2 r^7)\sin(\theta)+(a^2\cos^{2}(\theta)+r^2)(a^2(a^2\alpha m -a m)\cos^3(\theta)+(-3 a^3\alpha m r-3 a^2 m r) \nonumber\\ & \cos^2(\theta)+ (-3 a^2 \alpha m r^2+3 a m r^2)\cos(\theta)+r^2(a\alpha m r+m r))(r\alpha\cos(\theta)-1)(a^2(a^2\alpha m+a m)\cos^3(\theta)+ \nonumber\\ &(3a^3\alpha m r-3a^2mr)\cos^{2}(\theta)+(-3 a^2\alpha m r^2-3 a m r^2)\cos(\theta)-r^3 a m\alpha+r^3 m)\cos(\theta)\Big)\bigg)\Bigg|\Bigg)\end{aligned}$$ In FIG. \[fig8\](a), we indicate the variation of the gravitational entropy density with radial distance and the angular coordinate, using the modified expression in (\[new1\]). Similarly the gravitational entropy density for the charged rotating accelerating black hole is given by equation (\[new2\]), which is quoted below. We can always check that if we substitute $ e=0 $ in (\[new2\]), we get back the expression in (\[new1\]). In FIG. \[fig8\](b), we have shown the corresponding variation of the gravitational entropy density with radial distance and angular coordinate. The expressions of gravitational entropy density in both (\[new1\]) and (\[new2\]) diverges at the ring singularity i.e. at $ (r^2 + a^2 \cos^2(\theta))=0 $ and vanishes at the conformal infinity i.e. at $ r=\dfrac{1}{\alpha \cos(\theta)} $. $$\begin{aligned} \label{new2} \left.s\right. &= \dfrac{k_{s}}{\sqrt{\dfrac{\sin^2(\theta)(a^2\cos^2(\theta)+r^2)^2}{(r\alpha\cos(\theta)-1)^8}}}\Bigg(\Bigg|\dfrac{48}{\sqrt{\dfrac{\sin^2(\theta)(a^2\cos^2(\theta)+r^2)^2}{(r\alpha\cos(\theta)-1)^8}}(a^2\cos^2(\theta)+r^2)^5(r\alpha \cos(\theta)-1)^3} \nonumber\\ &\bigg(\sin(\theta)\Big(((2a^{10} m^2r\alpha^3+(4\alpha^3e^2m r^2-2\alpha m^2 r)a^8+2a^6e^4r^3\alpha^3)\cos^8(\theta)- 4(a^4m^2\alpha^2+((9\alpha^2r^2-1)m^2+\frac{1}{2}e^2mr\alpha^2) \nonumber\\ &a^2 -\frac{1}{2}e^2r^2\alpha^2(e^2-15mr))a^6\cos^7(\theta)-10(m((\frac{34}{5}r^3\alpha^2-6r)m+e^2)a^4+\frac{3}{5}(-\frac{34}{3}m^2r^2+16e^2r(\alpha^2r^2-\frac{5}{48})m+e^4) \nonumber\\ &ra^2 +\frac{16}{5}e^4r^5\alpha^2)\alpha a^4\cos^6(\theta)+96a^4(a^4m^2r^2\alpha^2+((\frac{71}{24}\alpha^2r^4-r^2)m^2+\frac{3}{16}e^2(\alpha^2r^2+\frac{10}{3})rm-\frac{1}{16}e^4)a^2-\frac{1}{3}e^2\alpha^2 \nonumber\\ &(e^2-\frac{85}{16}mr)r^4)\cos^5(\theta)+150(((\frac{6}{5}r^3\alpha^2-\frac{34}{15}r)m+e^2)ma^4+\frac{26}{75}(-\frac{45}{13}m^2r^2+\frac{45}{13}(\alpha^2r^2-\frac{1}{18})e^2rm+e^4)ra^2 \nonumber\\ &+\frac{7}{25}e^4r^5\alpha^2)\alpha a^2r^2\cos^4(\theta)-180a^2(a^4m^2r^2\alpha^2+((\frac{71}{45}\alpha^2r^4-r^2)m^2-\frac{1}{18}e^2r(\alpha^2r^2-20)m-\frac{13}{45}e^4)a^2-\frac{7}{30}e^2\alpha^2r^4 \nonumber\\ &(e^2-\frac{19}{7}mr))r^2\cos^3(\theta)-150\alpha(((\frac{34}{75}r^3\alpha^2-\frac{38}{25}r)m+e^2)ma^4+\frac{11}{75}r(-\frac{34}{11}m^2r^2+\frac{20}{11}e^2(\alpha^2r^2+\frac{9}{20})rm \nonumber\\ &+e^4)a^2+\frac{2}{75}e^4r^5\alpha^2)r^4\cos^2(\theta)+40(a^4m^2r^2\alpha^2+((\frac{9}{10}\alpha^2r^4-r^2)m^2-\frac{1}{4}e^2r(\alpha^2r^2-6)m-\frac{11}{20}e^4)a^2-\frac{1}{10}(e^2 \nonumber\\ &-\frac{3}{2}mr)e^2\alpha^2r^4)r^4\cos(\theta)+10\alpha(((\frac{1}{5}r^3\alpha^2-\frac{6}{5}r)m+e^2)a^2+\frac{1}{5}r^2(e^2-mr))mr^6)\sin^2(\theta)+(2\alpha a^6(a^4m^2\alpha^2+ \nonumber\\ &(3\alpha^2e^2mr-m^2)a^2+2\alpha^2e^4r^2)\cos^9(\theta)+2\alpha^2a^6((e^2m-18m^2r)a^2+2re^4-20e^2mr^2)\cos^8(\theta)-68\alpha a^4(m^2(\alpha^2r^2 \nonumber\\ &-\frac{3}{17})a^4+(-m^2r^2 +\frac{57}{34}e^2r(\alpha^2r^2+\frac{5}{57})m)a^2+\frac{21}{34}e^4r^4\alpha^2)\cos^7(\theta)+40a^4(a^4m^2r\alpha^2-\frac{9}{20}((-\frac{142}{9}r^3\alpha^2+\frac{20}{9}r)m \nonumber\\ &+e^2(\alpha^2r^2-\frac{5}{9}))ma^2-\frac{21}{20}e^2\alpha^2(e^2-\frac{30}{7}mr)r^3)\cos^6(\theta)+60\alpha a^2(((3r^3\alpha^2-\frac{19}{5}r)m+e^2)ma^4+(-3r^3m^2+\frac{17}{6}e^2 \nonumber\\ &(\alpha^2r^2+\frac{1}{17})r^2m+\frac{11}{30}re^4)a^2+\frac{8}{15}e^4r^5\alpha^2)r\cos^5(\theta)-180a^2(a^4m^2r^2\alpha^2+((\frac{71}{45}\alpha^2r^4-r^2)m^2+\frac{1}{18}e^2r(\alpha^2r^2+15)m \nonumber\\ &-\frac{11}{90}e^4)a^2 -\frac{8}{45}e^2r^4\alpha^2(e^2-3mr))r\cos^4(\theta)-200(((\frac{17}{50}r^3\alpha^2-\frac{17}{10}r)m+e^2)ma^4+\frac{13}{50}(-\frac{17}{13}m^2r^2+\frac{15}{26}e^2(\alpha^2r^2 \nonumber\\ &-\frac{3}{5}) rm+e^4)ra^2+\frac{1}{100}e^4r^5\alpha^2)\alpha r^3\cos^3(\theta)+96r^3(a^4m^2r^2\alpha^2+((\frac{3}{8}\alpha^2r^4-r^2)m^2+\frac{5}{48}e^2r(\alpha^2r^2+15)m \nonumber\\ &-\frac{13}{24}e^4)a^2 -\frac{1}{48}e^2r^4\alpha^2(e^2-2mr))\cos^2(\theta)+60(((\frac{1}{30}r^3\alpha^2-r)m+e^2)ma^2+\frac{1}{10}r(e^4-\frac{1}{3}e^2mr-\frac{1}{3}m^2r^2)) \times \nonumber\\ &\alpha r^5\cos(\theta) -4a^2\alpha^2m^2r^7+6e^4r^5-10e^2mr^6+4m^2r^7)\sin(\theta)+(a^2\cos^2(\theta)+r^2)(a^2(a^2\alpha m+\alpha e^2r-am)\cos^3(\theta)+ \nonumber\\ &(-3a^3mr\alpha+(e^2-3mr)a^2-2ae^2\alpha r^2)\cos^2(\theta)+(-3a^2mr^2\alpha+(-2e^2r+3mr^2)a-e^2\alpha r^3)\cos(\theta)+r^2(a\alpha mr-e^2 \nonumber\\ &+mr))(r\alpha \cos(\theta)-1)(a^2(a^2\alpha m+\alpha e^2 r+am)\cos^3(\theta)+(3a^3mr\alpha+(e^2-3mr)a^2+2ae^2\alpha r^2)\cos^2(\theta) \nonumber\\ &+(-3a^2mr^2\alpha+(2e^2r-3mr^2)a-e^2\alpha r^3)\cos(\theta)-r^3am\alpha-e^2r^2+r^3m)\cos(\theta)\Big)\bigg)\Bigg|\Bigg)\end{aligned}$$ Therefore from FIG. \[fig8\], we find that the entropy density measure diverges not only at the ring singularity but also at $ \theta=\pi $, which renders this measure inappropriate for determining the gravitational entropy in these cases. There is another singularity at $ \theta=0 $, though not visible in FIG. \[fig8\], but can be inferred from the mathematical analysis. This is in agreement with the observations in [@entropy2] for non-accelerating axisymmetric black holes. This is a disturbing feature of this method of analysis. For a possible resolution of this problem we want to point out that in the case of rotating black holes, the existence of stationary observer is not well defined because of the effect of frame dragging. Nevertheless, we have worked with the chosen definition of gravitational entropy density to get an overall idea of the way things work out. For such cases of axisymmetric space-times, it is not possible to determine the spatial metric $h_{ij} $ because of the presence of the metric coefficient $g_{t\phi}$ in the metric (\[ht\]) and in metric (\[htn\]). This is because the object is rotating and the spatial position of each event in the space-time depends on time. Therefore the covariant divergence is calculated from the determinant of the full metric and is given in equations (\[enden1\]) and (\[enden2\]). Conclusions =========== In this paper we have adopted a phenomenological approach of determining the gravitational entropy of accelerating black holes as done in [@entropy1] and [@entropy2]. We find that the gravitational entropy proposal [@entropy1] for the accelerating black holes and charged accelerating black holes works pretty well, except for the rotating charged metric where we faced difficulties in this regard. We then considered the alternative definition of $ P $ given in [@entropy2] to compute the entropy density and showed that the gravitational entropy is well defined in this case. In the end we considered the vector $ \mathbf{\Psi} $ to have additional angular components for axisymmetric spacetimes, as proposed in [@entropy2], to compute the entropy density for accelerating rotating and accelerating charged rotating black holes. From our calculations and the corresponding plots, we can conclude that for the rotating black holes the entropy density will be well-defined if we change our definition of the vector field $ \mathbf{\Psi} $, be it in the magnitude ($ P $) of it, or in the vector directions (having additional angular components). Acknowledgments {#acknowledgments .unnumbered} =============== The authors are thankful to the anonymous reviewers for their valuable comments and suggestions. SC is grateful to CSIR, Government of India for providing junior research fellowship. SG gratefully acknowledges IUCAA, India for an associateship and CSIR, Government of India for approving the major research project No. 03(1446)/18/EMR-II. T. Levi-Civita, Atti Accad. 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--- bibliography: - 'reion.bib' --- Introduction ============ The argument for an extended period of reionization is as follows. The WMAP has detected the correlation between temperature and polarization on large angular scales [@kogut03] that has an amplitude proportional to the total optical depth of CMB photons to Thomson scattering, $\tau$ [@sunyaev80; @zaldarriaga97a; @kaplinghat02]. Modeling reionization with a single sharp transition at $z_{ri}$, a multi–parameter fit to the WMAP data gives $z_{ri} = 17 \pm 5$ [@spergel03]. On the other hand, the evolution of quasar spectra from $z=6.3$ and $z=6.4$ to $z = 6$ shows a rapid decrease in the amount of neutral Hydrogen, indicating the end of reionization [@fan03]. A simple interpretation to explain these two very different datasets is that reionization started early, $z_{ri} \sim 20$, but did not conclude until much later ($z \sim 6$). An extended period of reionization has its effect at low $\ell$ in the polarization (as detected by WMAP), and also at higher $\ell$ in temperature. In Section II we disucss the low $\ell$ effect and in Section III the high $\ell$ effect. In section IV we discuss them in combination. All the work I present here is described in more detail in @haiman03 (reionization models), @kaplinghat03a [@holder03] (low $\ell$ signal) and @santos03 (high $\ell$). The Reionization Bumps ====================== Quadrupole radiation incident upon an electron leads to linear polarization. For a review of CMB polarization, see [@hu97a]. As drawn in Fig. \[fig:bump\] a free electron at high redshift sees a quadrupole from free–streaming of the monopole on its last–scattering surface. A monopole with wavenumber $k$ free streams primarily into $l=k(\eta_{ri} - \eta_{lss})$ so the quadrupole is given primarily by $k=2/(\eta_{ri} - \eta_{lss})$. The linear polarization thus generated at $\eta_{ri}$ projects to $l = k(\eta_0-\eta_{ri}) = 2(\eta_0-\eta_{ri})/(\eta_{ri} - \eta_{lss})$ today. Thus, the feature appears at low $\ell$. The polarization signature is proportional to the amount of scattering and hence the optical depth $\tau$. Therefore $C_l^{EE} \propto \tau^2$ and $C_l^{TE} \propto \tau$. See @zaldarriaga97a and @kaplinghat03a. The same considerations lead to a reionization bump in the tensor B mode with $C_l^{BB} \propto \tau^2$ so higher $\tau$ can improve the detectability of gravitational waves as quantified in @knox02 [@kaplinghat03b]. The polarization angular power spectra are shown in Fig. \[fig:foregrounds\]. The low l polarization has already been detected by WMAP through the correlation of this effect with the temperature map [@kogut03]. From the WMAP measurements only one number can be inferred: a joint fit of the TT and TE power spectra to a six-parameter model results in $\tau = 0.17 \pm 0.06$ [@spergel03]. If the EE power spectrum is measured with cosmic variance precision, there are 5 uncorrelated numbers (the amplitudes of 5 eigenmodes of the ionization history), that can be measured with signal-to-noise greater than 1 [@hu03]. These 5 numbers will provide strong constraints on models of the first generation of stars, since these are presumably what cause the reionization of the inter-galactic medium. Models of foreground polarization indicate that this low l signal can, in principle, be measured with near cosmic variance precision, as seen in Fig. \[fig:foregrounds\]. Although these models are highly uncertain. We will know more soon from further releases of WMAP data. To quantify how much improvement is possible beyond WMAP we show forecasted constraints on $\tau$ and the initial reionization redshift $z_e$ assuming statistical errors only for WMAP, Planck/LFI and Planck/HFI. Model B has sudden and complete reionization at $z_e=15$. Model A has a two–stage reionization, first increasing $X_e$ suddently to 0.42 at $z_e=25$ and then suddenly completely ionizing at $z=6.3$. The ionization fraction, $X_e\equiv n_e/n_p$ cannot be greater than 1.08 if Helium does not doubly ionize. Thus for a given $z_e$ there is a minimum value of $\tau$ that gives rise to the straight lines in the lower portions of the contours around model A. The kSZ power spectrum ====================== An extended period of reionization (as suggested by combination of WMAP data and high-redshift quasar spectra) is likely to be a period with a highly inhomogeneous reionization fraction. Prior to percolation of the ionized regions, the ionization fraction will be near unity in the vicinity of the sources of the reioinizing radiation and zero far away from any sources. If the sources are in high-mass halos, as semi-analytic models suggest, then the patches of high ionization fraction will be highly correlated. These correlated patches can lead to a kinetic SZ power spectrum with an amplitude of about $10^{-12}$ at $l$ values of 1500 and higher. At $l > 3000$ this kinetic SZ power spectrum may be the dominant source of flucutation power on the sky, for components with the same spectral signature as thermal fluctuations about a 2.7 K black body. Here we reproduce a figure from @santos03 showing a range of the kSZ power spectra that come from the reionization models of @haiman03. The error bars on one of the curves are for an observation of 1% of the sky with 0.9’ resolution. They are due to the “noise” from primary + lensing CMB and residual point sources. Instrument noise is assumed to be subdominant. We assume thermal SZ has been removed by taking advantage of its unique spectral dependence. We see that the models do not show much variation in shape, but only in amplitude. There is a degeneracy between the clustering bias of the dark matter halos hosting the reionizing sources and the optical depth. The low $\ell$ measurements can break this degeneracy, allowing us to place some constraint on the clustering bias, and therefore the halo masses. Although predicting the amplitude of the signal is difficult, the shape appears to be robust, particularly at $l < 3000$ where the kSZ power spectrum might contaminate attempts to measure cosmological parameters and reconstruct the gravitational lensing potential. We can therefore model this contaminant with one free parameter: an amplitude. Doing parameter estimation without such modeling (i.e., ignoring the kSZ power spectrum) can lead to significant biases for parameter estimation from Planck and higher–resolution observations [@knox98; @santos03]. But @santos03 find that modeling the kSZ power spectrum as this robust shape times a floating amplitude removes all significant parameter estimation biases even for an experiment that is cosmic–variance limited out to $l=3,000$. The simple prescription for removing the bias works because there is so little variation in the shape of the kSZ power spectrum in our reionization models. This shape is merely a projection of the matter power spectrum from high redshift. The small dependence on shape that there is comes from the different mean angular diameter distances in the different models. If necessary, the modeling could be extended to include this as a free parameter also. If the real kSZ power spectrum shape (at $l < 3,000$) were significantly different from the shapes we calculate then our simple modeling (with one or maybe two parameters) may not be sufficient. However, this could only be the case if the density of free electrons does not trace the density of matter on scales larger than about 3 Mpc. Note that even if $X_e$ were completely homogeneous, the density of free electrons traces the density of matter. Even in this homogeneous $X_e$ case the shape would still be a projection of the matter power spectrum, although this time from a smaller mean angular-diameter distance. Our calculations have relied on semi–analytic models of reionization. For a more numerical approach, see the contribution from Naoshi Sugiyama. For more on patchy reionization, see @aghanim96 [@gruzinov98; @knox98; @valageas01] and other papers already mentioned above.

--- abstract: 'This paper provides a sample of a LaTeX document which conforms, somewhat loosely, to the formatting guidelines for ACM SIG Proceedings.[^1]' author: - Ben Trovato - 'G.K.M. Tobin' - 'Lars Th[ø]{}rv[ä]{}ld' - Valerie Béranger - Aparna Patel - Huifen Chan - Charles Palmer - John Smith - 'Julius P. Kumquat' bibliography: - 'sample-bibliography.bib' subtitle: Extended Abstract title: SIG Proceedings Paper in LaTeX Format --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010562&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Embedded systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010575.10010755&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Redundancy&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010554&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Robotics&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003033.10003083.10003095&lt;/concept\_id&gt; &lt;concept\_desc&gt;Networks Network reliability&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; [^1]: This is an abstract footnote

--- author: - 'V. M. Passegger$^{1}$, A. Reiners$^{1}$, S. V. Jeffers$^{1}$, S. Wende$^{1}$, P. Schöfer$^{1}$, P. J. Amado$^{2}$, J. A. Caballero$^{3}$, D. Montes$^{4}$, R. Mundt$^{5}$, I. Ribas$^{6}$, A. Quirrenbach$^{3}$, and the CARMENES Consortium' bibliography: - 'passegger.bib' title: 'Spectroscopic characterisation of CARMENES target candidates from FEROS, CAFE and HRS high-resolution spectra' --- Introduction ============ The new CARMENES instrument is mounted at the 3.5 m telescope at Calar Alto Observatory, located in the Sierra de los Filabres in southern Spain. It consists of two fibre-fed high-resolution spectrographs, operating in the visible wavelength range from 0.52 to 0.96 $\mu$m and in the near-infrared from 0.96 to 1.71 $\mu$m, having a spectral resolution of R &gt; 80,000. [@Quirrenbach2010; @Quirrenbach2012; @Quirrenbach2014] Both spectrographs will simultaneously perform high-accuracy radial-velocity measurements of about 300 M dwarfs during three years of guaranteed observing time. The aim is to detect low-mass planets within the habitable zones of these stars.\ For science preparation over 1500 high-resolution spectra have been observed with FEROS, CAFE and HRS to determine effective temperature, surface gravity and metallicity. These parameters are fundamental for characterising star-planet systems. The spectra of M dwarfs are very complex, with molecular lines forming due to the low temperatures. This makes it difficult to use a line-by-line approach and requires a full spectral synthesis, which in turn necessitates for accurate models that take into account the formation of molecules. We use the latest generation PHOENIX model grid, the PHOENIX ACES models [@Husser2013]. These models are especially designed for low temperature stellar atmospheres and use a new equation of state to accurately reproduce molecular lines. Methods and Data ================ Name Resolution Coverage \[nm\] No. Spectra No. Stars Observing Period ------- ------------ ----------------- ------------- ----------- -------------------------- CAFE \~65,000 396-950 623 236 2013-01-21 to 2014-09-26 FEROS 48,000 350-920 455 217 2012-12-31 to 2014-07-11 HRS 60,000 420-1100 93 29 2011-09-29 to 2013-06-18 Table \[tab:obs\] summarizes the properties of the spectrographs used for observation and the data taken. Some observed spectra could not be used for analysis because of different issues, e.g. very low signal-to-noise, observation of wrong target, polluting light from close companions. The method we use was described in detail in [@Passegger2016a]. We fit PHOENIX ACES model spectra to our observed spectra. This is done for different spectral ranges, including the $\gamma$- and $\epsilon$-TiO bands (sensitive to temperature and metallicity), the K- and Na-doublets around 768 nm and 819 nm (sensitive to surface gravity and metallicity) and two CaII-lines. Rotational velocities determined by [@Jeffers2016] are included to account for line broadening due to stellar rotation. Other than [@Passegger2016a] a downhill simplex is implemented for linear interpolation between the model grid points and a $\chi2$ -minimization determines the best fit to the data. Figure \[fig:fit\] shows an example fit to CARMENES data. {width="0.85\linewidth"} Results and Discussion ====================== We obtained stellar parameters for 351 stars from 977 spectra. We find that most stars lie within 3200-3900 K, corresponding to spectral types M1V-M5V, as shown in the upper left panel of Figure \[fig:results\]. The higher the metallicity the higher the temperature for each spectral type (Figure \[fig:results\], lower left panel). This is consistent with results by [@Mann2015]. They showed that with increasing metallicity the radius increases, for fixed temperature. The spectral types have been calculated using spectral indices [@Schoefer2015]. The green squares correspond to a literature computation by [@PecautMamajek2013] for solar metallicity. A literature comparison with [@RojasAyala2012], [@GaidosMann2014] and [@Maldonado2015] shows that our values for metallicity turn out to be higher than published ones. (Figure \[fig:results\], upper right). One possible explanation for this is that PHOENIX ACES models still cannot reproduce the full depths of some lines (see Figure \[fig:fit\], 4th wavelength range), which might cause the algorithm to choose higher metallicity models to fit the lines. On the other hand it seems that the signal-to-noise ratio is also very important for parameter determination. 75 percent of the stars with \[Fe/H\] higher than 0.6 have SNRs lower than 50. We find good agreement with expected \[Fe/H\] values for SNR&gt;50 (Figure \[fig:results\], lower right). For the first four months of CARMENES data we find that the parameters show better agreement with literature, having better SNRs. {width="0.85\linewidth"} Acknowledgments {#acknowledgments .unnumbered} =============== [CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA). CARMENES was funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Cientícas (CSIC), the European Union through European Regional Fund (FEDER/ERF), Spanish Ministry of Economy and Competitiveness, the state of Baden-Württemberg, the German Science Foundation (DFG), the Junta de Andalucía, and by the Klaus Tschira Stiftung, with additional contributions by the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l’Espai, Institut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología, and the Centro Astronómico Hispano-Alemán).]{}

--- abstract: | Given a flat injective ring epimorphism $u\colon R\to U$ between commutative rings, we consider the Gabriel topology ${\mathcal{G}}$ associated to $u$ and the class ${\mathcal{D}}_{\mathcal{G}}$ of ${\mathcal{G}}$-divisible modules. We prove that ${\mathcal{D}}_{\mathcal{G}}$ is an enveloping class if and only if it is the tilting class corresponding to the $1$-tilting module $U\oplus U/R$ and for every ideal $J\in {\mathcal{G}}$ the quotient rings $R/J$ are perfect rings. Equivalently, ${\operatorname{p.dim}}U\leq 1$ and the discrete quotient rings $\mathfrak R/\mathfrak RJ$ of the topological ring $\mathfrak R={\operatorname{End}}(U/R)$ are perfect rings. Moreover, we show that every enveloping $1$-tilting class over a commutative ring arises from a flat injective ring epimorphism. address: - | Dipartimento di Matematica “Tullio Levi-Civita”\ Università di Padova\ Via Trieste 63, 35121 Padova (Italy) - | Dipartimento di Matematica “Tullio Levi-Civita”\ Università di Padova\ Via Trieste 63, 35121 Padova (Italy) author: - Silvana Bazzoni - Giovanna Le Gros title: Enveloping classes over commutative rings --- Introduction ============ The classification problem for classes of modules over arbitrary rings is in general very difficult, or even hopeless. Nonetheless, approximation theory was developed as a tool to approximate arbitrary modules by modules in classes where the classification is more manageable. Left and right approximations were first defined in the case of modules over finite dimensional algebras by work of Auslander, Reiten, and Smalø and generalised later by Enochs and Xu for modules over arbitrary rings using the terminology of preenvelopes and precovers. An important problem in approximation theory is when minimal approximations, that is covers or envelopes, over certain classes exist. In other words, for a certain class ${\mathcal{C}}$, the aim is to characterise the rings over which every module has a minimal approximation in ${\mathcal{C}}$ and furthermore to characterise the class ${\mathcal{C}}$ itself. The most famous positive result of when minimal approximations exist is the construction of an injective envelope for every module. Instead, Bass proved in [@Bass] that projective covers rarely exist. In his paper, Bass introduced and characterised the class of perfect rings which are exactly the rings over which every module admits a projective cover. Among the many characterisations of perfect rings, the most important from the homological point of view is the closure under direct limits of the class of projective modules. A comparison between the existence of injective envelopes and projective covers shows that the existence of minimal left or right approximations is not a symmetric phenomenon in general. A class ${\mathcal{C}}$ of modules is called covering, respectively enveloping, if every module admits a ${\mathcal{C}}$-cover, respectively a ${\mathcal{C}}$-envelope. A cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ admits (special) ${\mathcal{A}}$-precovers if and only if it admits (special) ${\mathcal{B}}$-preenvelopes. This observation lead to the notion of complete cotorsion pairs, that is cotorsion pairs admitting approximations. Results by Enochs and Xu ([@Xu Theorem 2.2.6 and 2.2.8]) show that a complete cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ such that ${\mathcal{A}}$ is closed under direct limits admits both ${\mathcal{A}}$-covers and ${\mathcal{B}}$-envelopes. Note that in the case of the cotorsion pair $({\mathcal{P}}_0, {\mathrm{Mod}\textrm{-}{R}})$, where ${\mathcal{P}}_0$ is the class of projective modules, Bass’s results state that ${\mathcal{P}}_0$ is a covering class if and only if ${\mathcal{P}}_0$ is closed under direct limits. In this paper we are interested in the conditions under which a class ${\mathcal{C}}$ is enveloping. We will deal with classes of modules over commutative rings and in particular with $1$-tilting classes. An important starting point is the bijective correspondence between faithful finitely generated Gabriel topologies ${\mathcal{G}}$ and $1$-tilting classes over commutative rings established by Hrbek in [@H]. The tilting class can then be characterised as the class ${\mathcal{D}}_{\mathcal{G}}$ of ${\mathcal{G}}$-divisible modules, that is, the modules $M$ such that $JM=M$ for every $J\in {\mathcal{G}}$. We prove that if a $1$-tilting class is enveloping, then $R_{\mathcal{G}}$, the ring of quotients with respect to the Gabriel topology ${\mathcal{G}}$, is ${\mathcal{G}}$-divisible, so that $R\to R_{\mathcal{G}}$ is a flat injective ring epimorphism. It is well known that every flat ring epimorphism $u\colon R\to U$ gives rise to a finitely generated Gabriel topology. We will consider the case of a flat injective ring epimorphism $u\colon R\to U$ between commutative rings and show that if the module $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope, then $U$ has projective dimension at most one. From results by Angeleri Hügel and S[á]{}nchez [@AS], we infer that the module $U\oplus K$, where $K$ is the cokernel of $u$, is a $1$-tilting module with ${\mathcal{D}}_{\mathcal{G}}$ as associated tilting class. In other words, ${\mathcal{D}}_{\mathcal{G}}$ coincides with the class of modules generated by $U$, that is epimorphic images of direct sums of copies of $U$ or also with $K^\perp$, the right Ext-orthogonal of $K$. Assuming furthermore that the class ${\mathcal{D}}_{\mathcal{G}}$ is enveloping, we prove that all the quotient rings $R/J$, for $J\in {\mathcal{G}}$ are perfect rings and so are all the discrete quotient rings of the topological ring $\mathfrak R={\operatorname{End}}(K)$ (Theorems  \[T:rjperfect\] and  \[T:EndK-properfect\]). In the terminology of [@BP2] this means that $\mathfrak R$ is a pro-perfect topological ring. Moreover, the converse holds, that is if $\mathfrak R={\operatorname{End}}(K)$ is a pro-perfect topological ring and the projective dimension of $U$ is at most one, then the class of ${\mathcal{G}}$-divisible modules is enveloping (Theorem \[T:characterisation\]). Consequently, applying results from [@BP2 Section 19], we obtain that ${\mathrm{Add}}(K)$, the class of direct summands of direct sums of copies of $K$, is closed under direct limits. Since ${\mathcal{D}}_{\mathcal{G}}$ coincides with the right Ext-orthogonal of ${\mathrm{Add}}(K)$, we have an instance of the necessity of the closure under direct limits of a class whose right Ext-orthogonal admits envelopes. Therefore in our situation we prove a converse of the result by Enochs and Xu ([@Xu Theorem 2.2.6]) which states that if a class ${\mathcal{A}}$ of modules is closed under direct limits and extensions and whose right Ext-orthogonal ${\mathcal{A}}^\perp$ admits special preenvelopes with cokernel in ${\mathcal{A}}$, then ${\mathcal{A}}^\perp$ is enveloping. The case of a non-injective flat ring epimorphism $u\colon R\to U$ is easily reduced to the injective case, since the class of ${\mathcal{G}}$-divisible modules is annihilated by the kernel $I$ of $u$, so all the results proved for $R$ apply to the ring $R/I$ and to the cokernel $K$ of $u$. As a byproduct we obtain that a $1$-tilting torsion class over a commutative ring is enveloping if and only if it arises from a flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$ such that the factor rings $R/J$ are perfect rings for every $J\in {\mathcal{G}}$ (Theorem \[T: tilting-envelope\]). This provides a partial answer to Problem 1 of [@GT12 Section 13.5] and generalises the result proved in [@B] for the case of commutative domains and divisible modules. The paper is organised as follows. After the necessary preliminaries, in Section \[S:envelope\] we state some general results concerning properties of envelopes with respect to classes of modules. In Section \[S:gab-top\] we recall the notion of a Gabriel topology and outline the properties of the related ring of quotients. In Section \[S:tilting-enveloping\], we consider a $1$-tilting class over a commutative ring and its associated Gabriel topology via Hrbek’s results  [@H]. We prove that if the $1$-tilting class is enveloping, then the ring of quotients with respect to the Gabriel topology ${\mathcal{G}}$ is ${\mathcal{G}}$-divisible, hence flat. In Section \[S:compl-endK\] we introduce the completion of a ring with respect to a Gabriel topology and the endomorphism ring of a module as a topological ring. Considering the particular case of a perfect localisation corresponding to a flat injective ring epimorphism $u\colon R\to U$ between commutative rings, we show the isomorphism between the completion of $R$ with respect to the associated Gabriel topology and the topological ring $\mathfrak R={\operatorname{End}}(K)$. In the main Sections \[S:enveloping\] and  \[S:properfect\], we prove a ring theoretic and topological characterisation of commutative rings for which the class of ${\mathcal{G}}$-divisible modules is enveloping where ${\mathcal{G}}$ is the Gabriel topology associated to a flat injective ring epimorphism. Namely, the characterisation in terms of perfectness of the factor rings $R/J$, for every $J\in {\mathcal{G}}$ and the pro-perfectness of the topological ring $\mathfrak R={\operatorname{End}}(K)$. In Section \[S:notmono\] we extend the results proved in Sections \[S:enveloping\] and  \[S:properfect\] to the case of a non-injective flat ring epimorphism Preliminaries ============= The ring $R$ will always be associative with a unit and ${\mathrm{Mod}\textrm{-}{R}}$ the category of right $R$-modules. Let ${\mathcal{C}}$ be a class of right $R$-modules. The right ${\operatorname{Ext}}^1$-orthogonal and right ${\operatorname{Ext}}^\infty$-orthogonal classes of ${\mathcal{C}}$ are defined as follows. $${\mathcal{C}}^{\perp_1} =\{M\in {\mathrm{Mod}\textrm{-}{R}} \ | \ {\operatorname{Ext}}_R^1(C,M)=0 \ {\rm for \ all\ } C\in {\mathcal{C}}\}$$ $${\mathcal{C}}^\perp = \{M\in {\mathrm{Mod}\textrm{-}{R}} \ | \ {\operatorname{Ext}}_R^i(C,M)=0 \ {\rm for \ all\ } C\in {\mathcal{C}}, \ {\rm for \ all\ } i\geq 1 \}$$ The left Ext-orthogonal classes ${}^{\perp_1} {\mathcal{C}}$ and ${}^\perp {\mathcal{C}}$ are defined symmetrically. If the class ${\mathcal{C}}$ has only one element, say ${\mathcal{C}}= \{X\}$, we write $X^{\perp_1}$ instead of $\{X\}^{\perp_1}$, and similarly for the other ${\operatorname{Ext}}$-orthogonal classes. We will now recall the notions of ${\mathcal{C}}$-preenvelope, special ${\mathcal{C}}$-preenvelope and ${\mathcal{C}}$-envelope for a class ${\mathcal{C}}$ of $R$-modules. Let ${\mathcal{C}}$ be a class of modules, $N$ a right $R$-module and $C\in {\mathcal{C}}$. A homomorphism $\mu\in {\operatorname{Hom}}_R(N, C)$ is called a ${\mathcal{C}}$-[*preenvelope*]{} (or left approximation) of $N$ if for every homomorphism $f' \in {\operatorname{Hom}}_R(N, C')$ with $C'\in {\mathcal{C}}$ there exists a homomorphism $f\colon C\to C'$ such that $f '= f \mu$. A ${\mathcal{C}}$-preenvelope $\mu\in {\operatorname{Hom}}_R(N, C)$ is called a ${\mathcal{C}}$-[*envelope*]{} (or a minimal left approximation) of $N$ if for every endomorphism $f$ of $C$ such that $f \mu= \mu$, $f$ is an automorphism of $C$. A ${\mathcal{C}}$-preenvelope $\mu$ of $N$ is said to be [*special*]{} if $\mu$ it is a monomorphism and ${\operatorname{Coker}}\mu\in {}^\perp {\mathcal{C}}$. The notions of ${\mathcal{C}}$-[*precover*]{} (right approximation), [*special*]{} ${\mathcal{C}}$-[*precover*]{} and of ${\mathcal{C}}$-[*cover*]{} (minimal right approximation) (see [@Xu]) are defined dually. A class ${\mathcal{C}}$ of $R$-modules is called *enveloping* (*covering*) if every module admits a ${\mathcal{C}}$-envelope (${\mathcal{C}}$-cover). A pair of classes of modules $({\mathcal{A}}, {\mathcal{B}})$ is a *cotorsion pair* provided that $\mbox{${\mathcal{A}}$} = {}^{\perp_1} \mbox{${\mathcal{B}}$} $ and $\mbox{${\mathcal{B}}$} = \mbox{${\mathcal{A}}$} ^{\perp_1}$. We consider preenvelopes and envelopes for particular classes of modules, that is classes which form the right-hand class of a cotorsion pair. A cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ is *complete* provided that every $R$-module $M$ admits a [special ${\mathcal{B}}$-preenvelope]{} or equivalently, every $R$-module $M$ admits a [special ${\mathcal{A}}$-precover]{}. Results by Enochs and Xu ([@Xu Theorem 2.2.6 and 2.2.8]) show that a complete cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ such that ${\mathcal{A}}$ is closed under direct limits admits both ${\mathcal{B}}$-envelopes and ${\mathcal{A}}$-covers. A cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$ is [*hereditary*]{} if for every $A \in {\mathcal{A}}$ and $B \in {\mathcal{B}}$, ${\operatorname{Ext}}^i_R(A, B)=0$ for all $i \geq 1$.\ Given a class $\mathcal{C}$ of modules, the pair $(^{\perp }({\mathcal{C}}^{\perp}),{\mathcal{C}}^{\perp})$ is a (hereditary) cotorsion pair called the cotorsion pair *generated* by ${\mathcal{C}}$, while $(^{\perp }{\mathcal{C}}, (^{\perp }{\mathcal{C}}) ^{\perp})$ is a (hereditary) cotorsion pair called the cotorsion pair *cogenerated* by ${\mathcal{C}}$. Examples of complete cotorsion pairs are abundant. In fact, by [@ET01 Theorem 10] a cotorsion pair generated by a set of modules is complete.\ For an $R$-module $C$, we let ${\mathrm{Add}}(C)$ denote the class of $R$-modules which are direct summands of direct sums of copies of $C$, and ${\operatorname{Gen}}(C)$ denote the class of $R$-modules which are homomorphic images of direct sums of copies of $C$.\ We now define $1$-tilting and silting modules.\ A right $R$-module $T$ is [*$1$-tilting*]{} if the following conditions hold. 1. ${\operatorname{p.dim}}T \leq1$. 2. ${\operatorname{Ext}}_R^i (T, T^{(\kappa)}) =0$ for every cardinal $\kappa$ and every $i >0$. 3. There exists an exact sequence of the following form where each $T_i$ is in ${\mathrm{Add}}(T)$. $$0 \to R \to T_0 \to T_1 \to 0$$ Equivalently, $T$ is $1$-tilting if and only if $T^\perp = {\operatorname{Gen}}(T)$. The cotorsion pair $({}^\perp(T^{\perp}), T^\perp)$ is called a [*$1$-tilting cotorsion pair*]{} and the torsion class $T^\perp$ is called [*$1$-tilting class*]{}. Two $1$-tilting modules are [*equivalent*]{} if they define the same $1$-tilting class (equivalently, if ${\mathrm{Add}}(T)={\mathrm{Add}}(T')$). A $1$-tilting class can be generalised in the following way. For a homomorphism $\sigma:P_{-1} \to P_0$ between projective modules in ${\mathrm{Mod}\textrm{-}R}$, consider the following class of modules. $$D_\sigma := \{ X \in {\mathrm{Mod}\textrm{-}R}: {\operatorname{Hom}}_R( \sigma,X) \text{ is surjective}\}$$ An $R$-module $T$ is said to be *silting* if it admits a projective presentation $$P_{-1} \overset{\sigma}\to P_0 \to T \to 0$$ such that ${\operatorname{Gen}}(T) = D_\sigma$. In the case that $\sigma$ is a monomorphism, ${\operatorname{Gen}}(T)$ is a $1$-tilting class. A ring $R$ is [*left perfect*]{} if every module in ${R\textrm{-}\mathrm{Mod}}$ has a projective cover. By [@Bass], $R$ is left perfect if and only if all flat modules in ${R\textrm{-}\mathrm{Mod}}$ are projective. An ideal $I$ of $R$ is said to be [*left T-nilpotent*]{} if for every sequence of elements $a_1, a_2, ..., a_i, ...$ in $I$, there exists an $n >0$ such that $a_1 a_2 \cdots a_n =0$. The following proposition for the case of commutative perfect rings is well known. \[P:perfect\] Suppose $R$ is a commutative ring. The following statements are equivalent for $R$. - $R$ is perfect - The Jacobson radical $J(R)$ of $R$, is T-nilpotent and $R/J(R)$ is semi-simple. - $R$ is a finite product of local rings, each one with a T-nilpotent maximal ideal. The following fact will be useful. Let $_RF$ be a left $R$-module $_SG_R$ be an $S$-$R$-bimodule such that ${\operatorname{Tor}}_1^R(G, F)=0$. Then, for every left $S$-module $M$ there is an injective map of abelian groups $${\operatorname{Ext}}^1_R(F, {\operatorname{Hom}}_S(G, M))\hookrightarrow{\operatorname{Ext}}^1_S(G\otimes_RF, M)).$$ Envelopes {#S:envelope} ========= In this section we discuss some useful results in relation to envelopes. The following result is crucial in connection with the existence of envelopes. \[P:Xu-env\] [@Xu Proposition 1.2.2] Let ${\mathcal{C}}$ be a class of modules and assume that a module $N$ admits a ${\mathcal{C}}$-envelope. If $\mu\colon N\to C$ is a ${\mathcal{C}}$-preenvelope of $N$, then $C=C'\oplus H$ for some submodules $C'$ and $H$ such that the composition $N\to C\to C'$ is a ${\mathcal{C}}$-envelope of $N$. We will consider ${\mathcal{C}}$-envelopes where ${\mathcal{C}}$ is a class closed under direct sums and therefore we will make use of the following result which is strongly connected with the notion of T-nilpotency of a ring. \[T:Xu-sums\][@Xu Theorem 1.4.4 and 1.4.6] 1. Let ${\mathcal{C}}$ be a class closed under countable direct sums. Assume that for every $n\geq 1$, $\mu_n\colon M_n\to C_n$ are ${\mathcal{C}}$-envelopes of $M_n$ and that $\oplus_nM_n$ admits a ${\mathcal{C}}$-envelope. Then $\oplus \mu_n\colon \oplus_nM_n\to \oplus_nC_n$ is a ${\mathcal{C}}$-envelope of $\oplus_nM_n$. 2. Assume that $\oplus \mu_n\colon \oplus_nM_n\to \oplus_nC_n$ is a ${\mathcal{C}}$-envelope of $\oplus_nM_n$ with $M_n\leq C_n$ and let $f_n\colon C_n\to C_{n+1}$ be a family of homomorphisms such that $f_n(M_n)=0$. Then, for every $x\in C_1$ there is an integer $m$ such that $f_m f_{m-1} \dots f_1(x)=0$. For a complete cotorsion pair $({\mathcal{A}}, {\mathcal{B}})$, we investigate the properties of ${\mathcal{B}}$-envelopes of arbitrary $R$-modules. First of all we state two straightforward lemmas. \[L:endomorph-env\] Let $0\to N\overset{\mu}\to B\overset{\pi}\to A\to 0$ be an exact sequence. Let $f$ be an endomorphism of $B$ such that $\mu = f \mu$. Then $f(B)\supseteq \mu(N)$ and ${\operatorname{Ker}}f\cap \mu(N)=0$. \[L:identity-env\] Let $0\to N\overset{\mu}\rightarrow B\overset{\pi}\to A\to 0$ be an exact sequence. For every endomorphism $f$ of $B$, the following are equivalent 1. $\mu = f \mu$. 2. The restriction of $f$ to $\mu(N)$ is the identity of $\mu(N)$. 3. There is a homomorphism $g\in {\operatorname{Hom}}_R(A,B)$ such that $f=id_B-g \pi$. \(1) $\Leftrightarrow$ (2) This is clear. \(1) $\Leftrightarrow$ (3) $\mu = f \mu$ if and only if $ (id_B-f) \mu=0$, that is if and only if $\mu(N)$ is contained in ${\operatorname{Ker}}(id_B-f)$. Equivalently, there exists $g\in {\operatorname{Hom}}_R(A,B)$ such that $id_B-f=g \pi $. \[P:B-envelopes\] Let $({\mathcal{A}}, {\mathcal{B}})$ be a complete cotorsion pair over a ring $R$. Assume that $0\to N\overset{\mu}\to B$ is a ${\mathcal{B}}$-envelope of the $R$-module $N$. Let $\alpha$ be an automorphism of $N$ and let $\beta $ be any endomorphism of $B$ such that $\beta\mu=\mu\alpha$. Then $\beta$ is an automorphism of $B$. By Wakamatsu’s Lemma (see [@Xu Lemma 2.1.2]), $\mu$ induces an exact sequence $$0\to N\overset{\mu}\to B\overset{\pi}\to A\to 0$$ with $A\in {\mathcal{A}}$. Since $\alpha$ is an automorphism of $N$, it is easy to show that $$0\to N\overset{\mu\alpha}\to B\to A\to 0$$ is a ${\mathcal{B}}$-envelope of $N$. Let $\beta$ be as assumed and consider an endomorphism $g$ of $B$ such that $g\mu\alpha=\mu$. Then $g\beta\mu=\mu$ and thus $g\beta$ is an automorphism of $B$, since $\mu$ is a ${\mathcal{B}}$-envelope. This implies that $\beta$ is a monomorphism so that $\beta(B)\in {\mathcal{B}}$. Since $\mu(N)\subseteq \beta(B)$ there is an epimorphism $\tau\colon B/\mu(N)\to B/\beta(B)$, where $B/\mu(N)$ can be identified with $A$. Consider the diagram: $$\xymatrix{ 0\ar[r]& {\beta(B)}\ar[r]&B\ar[r]^{\rho} & B/\beta(B)\ar[r]&0\\ && A\ar@{-->}[u]^{h} \ar[ur]_{\tau}}$$ where $\rho$ is the canonical projection and $\tau\pi=\rho$. It can be completed by $h$, since ${\operatorname{Ext}}_R^1(A, \beta(B))=0$. Consider the homomorphism $f=id_B-h\pi$. $f$ is an endomorphism of $B$ satisfying $f\mu=\mu$. By assumption $f$ is an isomorphism, hence, in particular$ f(B)=B$. Now, $\rho f=\rho-\rho h \pi= \rho-\tau\pi=0$. Hence $ f(B)\subseteq {\operatorname{Ker}}\rho= \beta(B)$; so $\beta(B)=B$ and $\beta$ is an isomorphism. Gabriel topologies {#S:gab-top} ================== In this section we briefly introduce Gabriel topologies and discuss some advancements that relate Gabriel topologies to $1$-tilting classes and silting classes over commutative rings as done in [@H] and [@AHHr]. For more detailed discussion on torsion pairs and Gabriel topologies, refer to [@Ste75 Chapters VI and IX]. We will start by giving definitions in the case of a general ring with unit (not necessarily commutative). Recall that a [*torsion pair*]{} $({\mathcal{E}}, {\mathcal{F}})$ in ${\mathrm{Mod}\textrm{-}R}$ is a pair of classes of modules in ${\mathrm{Mod}\textrm{-}R}$ which are mutually orthogonal with respect to the ${\operatorname{Hom}}$-functor and maximal with respect to this property. The class ${\mathcal{E}}$ is called a [*torsion class*]{} and ${\mathcal{F}}$ a [*torsion-free class*]{}. A class ${\mathcal{C}}$ of modules is a [ torsion class]{} if and only if it is closed under extensions, direct sums, and epimorphic images. A torsion pair $({\mathcal{E}}, {\mathcal{F}})$ is called [*hereditary*]{} if ${\mathcal{E}}$ is also closed under submodules. A torsion pair $({\mathcal{E}}, {\mathcal{F}})$ is [*generated*]{} by a class ${\mathcal{C}}$ if ${\mathcal{F}}$ consists of all the modules $F$ such that ${\operatorname{Hom}}_R(C, F)=0$ for every $C\in {\mathcal{C}}$. A [*(right) Gabriel topology*]{} on $R$ is a filter of right ideals of $R$, denoted ${\mathcal{G}}$, such that the following conditions hold. Recall that for a right ideal $I$ in $R$ and an element $t \in R$, $(I:t) := \{r \in R: tr \in I\}$. - If $I \in {\mathcal{G}}$ and $r \in R$ then $(I:r) \in {\mathcal{G}}$. - If $J$ is a right ideal of $R$ and there exists a $I \in {\mathcal{G}}$ such that $(J:t) \in {\mathcal{G}}$ for every $t \in I$, then $J \in {\mathcal{G}}$. Right Gabriel topologies on $R$ are in bijective correspondence with hereditary torsion pairs in ${\mathrm{Mod}\textrm{-}R}$. Indeed, to each right Gabriel topology ${\mathcal{G}}$, one can associate the following hereditary torsion class. $${\mathcal{E}}_{\mathcal{G}}= \{ M \mid {\mathrm{Ann}}_R (x) \in {\mathcal{G}}\text{ for every } x \in M\}$$ Then, the corresponding torsion pair $({\mathcal{E}}_{\mathcal{G}}, {\mathcal{F}}_{\mathcal{G}})$ is generated by the cyclic modules $R/J$ where $J \in {\mathcal{G}}$. The classes ${\mathcal{E}}_{\mathcal{G}}$ and ${\mathcal{F}}_{\mathcal{G}}$ are referred to as the [*${\mathcal{G}}$-torsion*]{} and [*${\mathcal{G}}$-torsion-free*]{} classes, respectively. Conversely, if $({\mathcal{E}}, {\mathcal{F}})$ is a hereditary torsion pair in ${\mathrm{Mod}\textrm{-}R}$, the set $$\{J\leq R\mid R/J\in {\mathcal{E}}\}$$ is a right Gabriel topology. For a right $R$-module $M$ let $t_{\mathcal{G}}(M)$ denote the ${\mathcal{G}}$-torsion submodule of $M$, or sometimes $t(M)$ when the Gabriel topology is clear from context. The [*module of quotients*]{} of the Gabriel topology ${\mathcal{G}}$ of a right $R$-module $M$ is the module $$M_{\mathcal{G}}:= \varinjlim_{\substack{J \in {\mathcal{G}}}} {\operatorname{Hom}}_R(J, M/t_{\mathcal{G}}(M)).$$ Furthermore, there is a canonical homomorphism $$\psi_M: M\cong {\operatorname{Hom}}_R(R, M) \to M_{\mathcal{G}}.$$ By substituting $M=R$, the assignment gives a ring homomorphism $\psi_R:R \to R_{\mathcal{G}}$ and furthermore, for each $R$-module $M$ the module $M_{\mathcal{G}}$ is both an $R$-module and an $R_{\mathcal{G}}$-module. Both the kernel and cokernel of the map $\psi_M$ are ${\mathcal{G}}$-torsion $R$-modules, and in fact ${\operatorname{Ker}}(\psi_M) = t_{\mathcal{G}}(M)$. Let $q: {\mathrm{Mod}\textrm{-}R}\to {\mathrm{Mod}\textrm{-}R}_{\mathcal{G}}$ denote the functor that maps each $M$ to its module of quotients. Let $\psi^\ast$ denote the right exact functor ${\mathrm{Mod}\textrm{-}R}\to {\mathrm{Mod}\textrm{-}R}_{\mathcal{G}}$ where $\psi^\ast(M):= M \otimes R_{\mathcal{G}}$. In general, there is a natural transformation $\Theta: \psi^\ast \to q$ with $\Theta_M:M\otimes R_{\mathcal{G}}\to M_{\mathcal{G}}$ which is defined as $m \otimes \eta \mapsto \psi_M(m) \cdot \eta$. That is, for every $M$ the following triangle commutes. $$(\star)\qquad \qquad \xymatrix{ M \ar[rr]^{\psi^\ast(M)} \ar[dr]_{\psi_M}&&M \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_M}\\ &M_{\mathcal{G}}&}$$ A right $R$-module is [*${\mathcal{G}}$-closed*]{} if the following natural homomorphisms are all isomorphisms for every $J \in {\mathcal{G}}$. $$M \cong {\operatorname{Hom}}_R(R, M) \to {\operatorname{Hom}}_R (J, M)$$ This amounts to saying that ${\operatorname{Hom}}_R(R/J,M) =0$ for every $J \in {\mathcal{G}}$ (i.e. $M$ is [*${\mathcal{G}}$-torsion-free*]{}) and ${\operatorname{Ext}}^1_R(R/J,M) =0$ for every $J \in {\mathcal{G}}$ (i.e. $M$ is [*${\mathcal{G}}$-injective*]{}). Thus if $M$ is ${\mathcal{G}}$-closed then $M$ is isomorphic to its module of quotients $M_{\mathcal{G}}$. Conversely, every $R$-module of the form $M_{\mathcal{G}}$ is ${\mathcal{G}}$-closed. The ${\mathcal{G}}$-closed modules form a full subcategory of both ${\mathrm{Mod}\textrm{-}R}$ and ${\mathrm{Mod}\textrm{-}R}_{\mathcal{G}}$. A left $R$-module $N$ is called [*${\mathcal{G}}$-divisible*]{} if $JN = N$ for every $J\in {\mathcal{G}}$. Equivalently, $N$ is ${\mathcal{G}}$-divisible if and only if $R/J \otimes_R N =0$ for each $J \in {\mathcal{G}}$. We denote the class of ${\mathcal{G}}$-divisible modules by ${\mathcal{D}}_{\mathcal{G}}$. It is straightforward to see that ${\mathcal{D}}_{\mathcal{G}}$ is a torsion class in ${R\textrm{-}\mathrm{Mod}}$. A right Gabriel topology is [*faithful*]{} if ${\operatorname{Hom}}_R(R/J, R) =0$ for every $J \in {\mathcal{G}}$, or equivalently if $R$ is ${\mathcal{G}}$-torsion-free, that is the natural map $\psi_R\colon R \to R_{\mathcal{G}}$ is injective. A right Gabriel topology is [*finitely generated*]{} if it has a basis consisting of finitely generated right ideals, or equivalently if the torsion-free class ${\mathcal{F}}_{\mathcal{G}}$ is closed under direct limits. In this paper, we will only be concerned with Gabriel topologies over commutative rings. In this setting, much useful research has already done in this direction. Specifically, in [@H], Hrbek showed that over commutative rings the faithful finitely generated Gabriel topologies are in bijective correspondence with $1$-tilting classes, and that the latter are exactly the classes of ${\mathcal{G}}$-divisible modules for some faithful finitely generated Gabriel topology ${\mathcal{G}}$, as stated in the following theorem. [@H Theorem 3.16] \[T:Hrb-tilting\] Let R be a commutative ring. There are bijections between the following collections. 1. 1-tilting classes ${\mathcal{T}}$. 2. faithful finitely generated Gabriel topologies ${\mathcal{G}}$. 3. faithful hereditary torsion pairs $({\mathcal{E}},{\mathcal{F}})$ of finite type in ${\mathrm{Mod}\textrm{-}R}$. Moreover, the tilting class ${\mathcal{T}}$ is the class of ${\mathcal{G}}$-divisible modules with respect to the Gabriel topology ${\mathcal{G}}$. When we refer to the [*Gabriel topology associated to the $1$-tilting class ${\mathcal{T}}$*]{} we will always mean the Gabriel topology in the sense of the above theorem. In addition we will often denote ${\mathcal{A}}$ to be the right ${\operatorname{Ext}}$-orthogonal class to ${\mathcal{D}}_{\mathcal{G}}={\mathcal{T}}$ in the situation just described, so $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ will denote the $1$-tilting cotorsion pair. In [@AHHr] the correspondence between faithfully finitely generated Gabriel topologies and $1$-tilting classes over commutative rings was extended to finitely generated Gabriel topologies which were shown to be in bijective correspondence with silting classes. Thus in this case the class ${\mathcal{D}}_{\mathcal{G}}$ of ${\mathcal{G}}$-divisible modules coincides with the class ${\operatorname{Gen}}T$ for some silting module $T$. Homological ring epimorphisms {#S:homological} ----------------------------- There is a special class of Gabriel topologies which behave particularly well and are related to ring epimorphisms. The majority of this paper will be restricted to looking at these Gabriel topologies. The standard examples of these Gabriel topologies over $R$ are localisations of $R$ with respect to a multiplicative subset. A [*ring epimorphism*]{} is a ring homomorphism $R \overset{u}\to U$ such that $u$ is an epimorphism in the category of unital rings. This is equivalent to the natural map $ U \otimes_R U \to U$ induced by the multiplication in $U$ being an isomorphism, or equivalently that $U \otimes_R (U / u(R)) =0$ (see [@Ste75 Chapter XI.1]. Two ring epimorphisms $R \overset{u}\to U$ and $R \overset{u'}\to U'$ are equivalent if there is a ring isomorphism $\sigma\colon U\to U'$ such that $\sigma u=u'$. A ring epimorphism is [*homological*]{} if ${\operatorname{Tor}}^R_n(U_R,{}_RU) = 0$ for all $n >0$. A ring epimorphism is called [*(left) flat*]{} if $u$ makes $U$ into a flat left $R$-module. Clearly all flat ring epimorphisms are homological. We will denote the cokernel of $u$ by $K$ and sometimes by $U/R$ or $U/u(R)$. A left flat ring epimorphism $R \overset{u}\to U$ is called a [*perfect right localisation*]{} of $R$. In this case, by [@Ste75 Chapter XI.2, Theorem 2.1] the family of right ideals $${\mathcal{G}}= \{ J \leq R \mid J U = U \}$$ forms a right Gabriel topology. Moreover, there is a ring isomorphism $\sigma:U \to R_{\mathcal{G}}$ such that $\sigma u: R \to R_{\mathcal{G}}$ is the canonical isomorphism $\psi_R: R \to R_{\mathcal{G}}$, or, in other words, $u$ and $\psi_R$ are equivalent ring epimorphisms. Note also that a right ideal $J$ of $R$ is in ${\mathcal{G}}$ if and only if $R/J \otimes_R U =0$. We will make use of the characterisations of perfect right localisations from Proposition 3.4 in Chapter XI.3 of Stenström’s book [@Ste75]. In particular, Proposition 3.4 states that the right Gabriel topology ${\mathcal{G}}$ associated to a flat ring epimorphism $R \overset{u}\to U$ is finitely generated and the ${\mathcal{G}}$-torsion submodule $t_{\mathcal{G}}(M)$ of a right $R$-module $M$ is the kernel of the canonical homomorphism $M\to M \otimes_R U $. Thus, $K=U/u(R)$ is ${\mathcal{G}}$-torsion, hence ${\operatorname{Hom}}_R(K, U)=0$. If moreover the flat ring epimorphism $R \overset{u}\to U$ is injective, then $ {\operatorname{Tor}}^R_1(M, K) \cong t_{\mathcal{G}}(M)$ and ${\mathcal{G}}$ is faithful. \[R:pdU=1\] *From the above observations and results in [@H], when $R$ is commutative and $R \overset{u}\to U$ is a flat injective epimorphism one can associate a $1$-tilting class which is exactly the class of ${\mathcal{G}}$-divisible modules. In the case that additionally ${\operatorname{p.dim}}_R U \leq 1$, one can apply a result from [@AS] which states that $U \oplus K$ is a $1$-tilting module, so there is a $1$-tilting class denoted ${\mathcal{T}}: =(U \oplus K)^\perp = {\operatorname{Gen}}(U)$. In fact, we claim that this is exactly the $1$-tilting class of ${\mathcal{G}}$-divisible modules. Explicitly, the Gabriel topology associated to ${\mathcal{T}}$ in the sense of Theorem \[T:Hrb-tilting\] is exactly the collection of ideals $\{J \mid JM = M \text{ for every } M \in {\mathcal{T}}\}$. The Gabriel topology that arises from the perfect localisation is the collection $\{J \mid JU = U \}$ and since $U \in {\mathcal{T}}= {\operatorname{Gen}}U$, the Gabriel topologies associated to these two $1$-tilting classes are the same. We conclude that the two $1$-tilting classes coincide: ${\operatorname{Gen}}_R(U) = {\mathcal{D}}_{\mathcal{G}}$.\ In [@H Proposition 5.4] the converse is proved: If one starts with a $1$-tilting class ${\mathcal{T}}$ with associated Gabriel topology ${\mathcal{G}}$, so that ${\mathcal{T}}={\mathcal{D}}_{\mathcal{G}}$, then $R_{\mathcal{G}}$ is a perfect localisation and ${\operatorname{p.dim}}R_{\mathcal{G}}\leq 1$ if and only if ${\operatorname{Gen}}R_{\mathcal{G}}= {\mathcal{D}}_{\mathcal{G}}$.* The following lemma will be useful when working with a Gabriel topology over a commutative ring that arises from a perfect localisation. \[L:finmanyann\] Let $R$ be a commutative ring, $u:R \to U$ a flat injective ring epimorphism, and ${\mathcal{G}}$ the associated Gabriel topology. Then the annihilators of the elements of $U/R$ form a sub-basis for the Gabriel topology ${\mathcal{G}}$. That is, for every $J\in {\mathcal{G}}$ there exist $z_1, z_2, \dots , z_n \in U$ such that $$\bigcap_{\substack{ 0 \leq i \leq n}} {\mathrm{Ann}}_R(z_i +R) \subseteq J.$$ Every ideal of the form ${\mathrm{Ann}}_R(z+R)$ is an ideal in ${\mathcal{G}}$ since $K=U/R$ is ${\mathcal{G}}$-torsion. Fix an ideal $J \in {\mathcal{G}}$. Then, $U = JU$, so $1_U = \sum_{0 \leq i \leq n} a_i z_i$ where $a_i \in J$ and $z_i \in U$. We claim that $$\bigcap_{\substack{ 0 \leq i \leq n}} {\mathrm{Ann}}_R(z_i +R) \subseteq J.$$ Take $b \in \bigcap_{\substack{ 0 \leq i \leq n}} {\mathrm{Ann}}_R(z_i +R)$. Then $$b = \sum_{0 \leq i \leq n} b a_i z_i \in J$$ since each $b z_i \in R$, hence $b a_i z_i \in J$, and it follows that $b \in J$. Enveloping $1$-tilting classes over commutative rings {#S:tilting-enveloping} ===================================================== For this section, $R$ will always be a commutative ring and ${\mathcal{T}}$ a $1$-tilting class. By Theorem \[T:Hrb-tilting\] there is a faithful finitely generated Gabriel topology ${\mathcal{G}}$ such that ${\mathcal{T}}$ is the class of ${\mathcal{G}}$-divisible modules. We denote again by $({\mathcal{E}}_{\mathcal{G}}, {\mathcal{F}}_{\mathcal{G}})$ the associated faithful hereditary torsion pair of finite type. We use ${\mathcal{D}}_{\mathcal{G}}$ and ${\mathcal{T}}= {\operatorname{Gen}}T = T^\perp$ interchangeably to denote the $1$-tilting class, and ${\mathcal{A}}$ to denote the right orthogonal class ${}^\perp {\mathcal{D}}_{\mathcal{G}}$.\ The aim of this section is to show that if ${\mathcal{T}}$ is enveloping, then $R_{\mathcal{G}}$, the ring of quotients with respect to ${\mathcal{G}}$, is ${\mathcal{G}}$-divisible and therefore $\psi_R:R \to R_{\mathcal{G}}$ is a perfect localisation of $R$. In Section \[S:enveloping\], we will moreover, show that $R_{\mathcal{G}}$ has projective dimension at most one, thus the $1$-tilting class arises from the flat injective epimorphism $R\to R_{\mathcal{G}}$ (see Proposition \[P:pd1\], Corollary \[C:U-envelope\]). Recall that if ${\mathcal{T}}$ is $1$-tilting, ${\mathcal{T}}\cap {}^\perp{\mathcal{T}}= {\mathrm{Add}}(T)$ (see [@GT12 Lemma 13.10]). By (T3) of the definition of a $1$-tilting module we have the following short exact sequence $$\text{(T3) }\quad 0 \to R \overset{\varepsilon}\to T_0 \to T_1 \to 0$$ where $T_0, T_1 \in {\mathrm{Add}}(T)$. In fact, this short exact sequence is a special ${\mathcal{D}}_{\mathcal{G}}$-preenvelope of $R$, and $T_0 \oplus T_1$ is a $1$-tilting module which generates ${\mathcal{T}}$ by [@GT12 Theorem 13.18 and Remark 13.19]. Furthermore, assuming that $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope, we can suppose without loss of generality that the sequence (T3) is the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$, since an envelope is extracted from a special preenvelope by passing to direct summands (Proposition \[P:Xu-env\]). For the rest of the section we will denote the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$ by $\varepsilon$. Recall from Section \[S:gab-top\] that for every $M \in {\mathrm{Mod}\textrm{-}R}$ there is the commuting diagram $(\star)$. Since ${\mathcal{G}}$ is faithful we have the following short exact sequence where $\psi_R$ is a ring homomorphism and $R_{\mathcal{G}}/R$ is ${\mathcal{G}}$-torsion. $$(\dag)\quad 0 \to R \overset{\psi_R} \to R_{\mathcal{G}}\to R_{\mathcal{G}}/R \to 0$$ We begin with some preliminary facts that hold for a general $1$-tilting class ${\mathcal{D}}_{\mathcal{G}}$ and which use only properties of the associated Gabriel topology. Recall that $D$ is ${\mathcal{G}}$-divisible if and only if $R/J \otimes_R D=0$ for every $J \in {\mathcal{G}}$ if and only if $M\otimes_RD=0$ for every ${\mathcal{G}}$-torsion module $M$. \[L:tor-Rg\] Let ${\mathcal{D}}_{\mathcal{G}}$ be a $1$-tilting class. Then the following statements hold. 1. If $N$ is a ${\mathcal{G}}$-torsion-free module then the natural map\ $\psi^\ast(N): N \to N \otimes_R R_{\mathcal{G}}$ is a monomorphism. 2. If $D$ is both ${\mathcal{G}}$-divisible and ${\mathcal{G}}$-torsion-free, then $D$ is a $R_{\mathcal{G}}$-module and $D \cong D\otimes_R R_{\mathcal{G}}$ via the natural map\ $\psi^\ast(D)={\operatorname{id}}_D \otimes_R \psi_R:D \otimes_R R \to D \otimes_R R_{\mathcal{G}}$. 3. If ${\operatorname{p.dim}}M \leq 1$, then ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0$. 4. If ${\operatorname{p.dim}}M \leq 1$ and $M$ is ${\mathcal{G}}$-torsion-free, then\ ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0={\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$. <!-- --> 1. Consider the following commuting triangle where $N$ is ${\mathcal{G}}$-torsion-free. $$\xymatrix{ 0 \ar[d]& \\ N \ar[r]^{\psi^\ast(N) \hspace{10pt}} \ar[d]_{\psi_N}&N \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_N}\\ N_{\mathcal{G}}&}$$ Then $\psi_N$ is a monomorphism and since $\psi_N = \Theta_N \circ \psi^\ast(N)$, also $\psi^\ast(N)$ is a monomorphism. 2. Consider the following commuting diagram where the horizontal sequence is exact by (1) as $D$ is ${\mathcal{G}}$-torsion-free. $$\xymatrix{ &0 \ar[d]& \\ 0 \ar[r] &D \ar[r]^{\psi^\ast(D) \hspace{10pt}} \ar[d]_{\psi_D}&D \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_D} \ar[r] & D \otimes_R R_{\mathcal{G}}/R \ar[r] &0\\ &D_{\mathcal{G}}&}$$ Additionally, $D\otimes_R R_{\mathcal{G}}/R=0$, since $R_{\mathcal{G}}/R$ is ${\mathcal{G}}$-torsion. Therefore $$\psi^\ast(D):D \to D \otimes_R R_{\mathcal{G}}$$ is an isomorphism. 3. Consider the following exact sequence formed by taking the tensor product of $M$ with the short exact sequence $(\dag)$. $$0 = {\operatorname{Tor}}^R_1(M, R) \to {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}) \to {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$$ By assumption ${\operatorname{p.dim}}M \leq 1$, so there is a projective resolution of $M$,$$0 \to P_1 \overset{\gamma}\to P_0 \to M \to 0$$ where $P_0, P_1$ are projective modules. It follows that ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is isomorphic to the kernel of the map $\gamma \otimes_R {\operatorname{id}}_{R_{\mathcal{G}}}$. $$P_1 \otimes_R R_{\mathcal{G}}\xrightarrow{\gamma \otimes_R {\operatorname{id}}_{R_{\mathcal{G}}}} P_0 \otimes_R R_{\mathcal{G}}$$ As $P_1$ is a submodule of $R^{(\alpha)}$ for some cardinal $\alpha$, also $P_1 \otimes_R R_{\mathcal{G}}$ is a submodule of the ${\mathcal{G}}$-torsion-free module $R_{\mathcal{G}}^{(\alpha)}$. Thus ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is itself a ${\mathcal{G}}$-torsion-free module.\ However, by applying the tensor product $(- \otimes_R R_{\mathcal{G}}/R)$ to the above projective presentation of $M$, we find that ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$ is a submodule of $P_1 \otimes_R R_{\mathcal{G}}/R$ which is ${\mathcal{G}}$-torsion. Since ${\mathcal{E}}_{\mathcal{G}}$ is a hereditary torsion class also ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$ is ${\mathcal{G}}$-torsion. Therefore, also $ {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is ${\mathcal{G}}$-torsion since it is a submodule of ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)$. We conclude that $ {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})$ is both ${\mathcal{G}}$-torsion and ${\mathcal{G}}$-torsion-free, hence $ {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0$. 4. Consider the following commuting triangle where $\psi^\ast(M)$ is a monomorphism from (1). $$\xymatrix{ &0 \ar[d]& \\ 0 \ar[r]&M \ar[r]^{\psi^\ast(M) \hspace{10pt}} \ar[d]_{\psi_M}&M \otimes_R R_{\mathcal{G}}\ar[dl]^{\Theta_M}\\ &M_{\mathcal{G}}&}$$ By applying the functor $(M \otimes_R -)$ to the short exact sequence $(\dag)$, we have the following exact sequences. $$\xymatrix{ 0 \ar[r] & {\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}) \ar[r] & {\operatorname{Tor}}^R_1( M, R_{\mathcal{G}}/R) \ar[r]&0&}$$ $$\xymatrix{ &0 \ar[r] & M \ar[r]^{\psi^\ast(M) \hspace{10pt}} & M \otimes_R R_{\mathcal{G}}\ar[r] & M \otimes_R R_{\mathcal{G}}/R \ar[r] & 0}$$ By (3), ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}})=0$, thus also ${\operatorname{Tor}}^R_1(M, R_{\mathcal{G}}/R)=0$ as these two modules are isomorphic from the above short exact sequence. We now show two lemmas about the $1$-tilting module $T_0 \oplus T_1$ and the class ${\mathrm{Add}}(T_0 \oplus T_1)$ assuming that $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope. \[L:R-env\] Let the following short exact sequence be a ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$. $$0 \to R \overset{\varepsilon}\to T_0 \to T_1 \to 0$$ Then $T_0$ is ${\mathcal{G}}$-torsion-free and $T_0 \cong T_0 \otimes_R R_{\mathcal{G}}$. We will show that for every $J \in {\mathcal{G}}$, $T_0 ]$, the annihilator of $J$ in $T_0$ is zero. Set $w:=\varepsilon(1_R)$ and fix a $J \in {\mathcal{G}}$. As $T_0 = JT_0$, $w = \sum_{1 \leq i \leq n} a_i z_i$ where $a_i \in J$ and $z_i \in T_0$. This sum is finite, so we can define the following maps. $$\xymatrix@R=.1cm{ {\mathbf{z}}:R \ar[r] & \bigoplus_{1 \leq i \leq n} T_0 & {\mathbf{a}}:\bigoplus_{1 \leq i \leq n} T_0 \ar[r] & T_0\\ \hspace{15pt}1_R \ar@{|-_{>}}[r] &(z_1, ..., z_n) & \hspace{5pt}(x_1, ..., x_n) \ar@{|-_{>}}[r] & \sum_i a_ix_i}$$ As $\bigoplus_nT_0$ is also ${\mathcal{G}}$-divisible, by the preenvelope property of $\varepsilon$ there exists a map $f:T_0 \to \bigoplus_nT_0$ such that $f \varepsilon = {\mathbf{z}}$. Also, ${\mathbf{a}}{\mathbf{z}}(1_R) = \sum_{1 \leq i \leq n} a_i z_i = w$, so ${\mathbf{a}}{\mathbf{z}} = \varepsilon$ and the following diagram commutes. $$\xymatrix@C=1.8cm@R=1.3cm{ 0\ar[r]& {R}\ar[dr]^{{\mathbf{z}}} \ar[ddr]_\varepsilon \ar[r]^\varepsilon&T_0\ar[r]^{\beta} \ar[d]^{f}& T_1 \ar[r]&0\\ & & \bigoplus_n T_0 \ar^{{\mathbf{a}}}[d]\\ & & T_0}$$ By the envelope property of $\varepsilon$, ${\mathbf{a}}f$ is an automorphism of $T_0$. The restriction of the automorphism ${\mathbf{a}}f$ to $T_0[J]$ is an automorphism of $T_0[J]$, and factors through the module $\bigoplus_nT_0[J]$. However $\mathbf{a}( \bigoplus_nT_0[J]) =0$, so ${\mathbf{a}}f(T_0[J]) =0$, but $\mathbf{a}f$ restricted to $T_0[J]$ is an automorphism, thus $T_0[J] =0$.\ It follows from (3) of Lemma \[L:tor-Rg\] $T_0 \cong T_0 \otimes_R R_{\mathcal{G}}$ since $T_0$ is ${\mathcal{G}}$-divisible. \[L:addT-tens-tf\] Suppose $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope in ${\mathrm{Mod}\textrm{-}R}$. Then for every $M \in {\mathrm{Add}}(T_0 \oplus T_1)$, $M \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free. From Lemma \[L:R-env\], $T_0 \cong T_0 \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free. We first show that $T_1 \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free. Consider the following short exact sequence obtained by applying $(- \otimes_R R_{\mathcal{G}})$ to the envelope of $R$, and note that ${\operatorname{Tor}}^R_1(T_1, R_{\mathcal{G}})=0$ by Lemma \[L:tor-Rg\] (3). $$0 \to R_{\mathcal{G}}\to T_0 \otimes_R R_{\mathcal{G}}\to T_1 \otimes_R R_{\mathcal{G}}\to 0$$ As $R_{\mathcal{G}}$ is ${\mathcal{G}}$-closed and $T_0 \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free, by applying the covariant functor ${\operatorname{Hom}}_R(R/J,-)$ to the above sequence for every $J \in {\mathcal{G}}$, we obtain that $T_1 \otimes_R R_{\mathcal{G}}$ must be ${\mathcal{G}}$-torsion-free.\ It is now straightforward to see that the statement holds for any direct summand of $ (T_0 \oplus T_1)^{(\alpha)}$. We look at ${\mathcal{D}}_{\mathcal{G}}$-envelopes of ${\mathcal{G}}$-torsion modules in ${\mathrm{Mod}\textrm{-}R}$, and find that they are also ${\mathcal{G}}$-torsion. \[L:torsion-env\] Suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ and $M$ is a ${\mathcal{G}}$-torsion $R$-module. Then the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $M$ is ${\mathcal{G}}$-torsion. To begin with, fix a finitely generated $J \in {\mathcal{G}}$ with a set $\{a_1, \dots, a_t\}$ of generators and consider a ${\mathcal{D}}_{\mathcal{G}}$-envelope $D(J)$ of the cyclic ${\mathcal{G}}$-torsion module $R/J$, denoted as follows. $$0 \to R/J \hookrightarrow D(J) \to A(J) \to 0$$ We will use the T-nilpotency of direct sums of envelopes as in Theorem \[T:Xu-sums\] (2). Consider the following countable direct sum of envelopes of $R/J$ which is itself an envelope, by Theorem \[T:Xu-sums\] (1): $$0 \to \bigoplus_{n} (R/J)_{n}\hookrightarrow \bigoplus_{n} D(J)_{n} \to \bigoplus_{n}A(J)_{n} \to 0 .$$ Choose an element $a \in J$ and for each $n$ set $f_n\colon D(J)_n\to D(J)_{n+1}$ to be the multiplication by $a$. Then clearly $(R/J)_n$ vanishes under the action of $f_n $, hence we can apply Theorem \[T:Xu-sums\] (2). For every $d \in D(J)$, there exists an $m$ such that $$f_m \circ \cdots \circ f_2 \circ f_1 (d) = 0 \in D(J)_{(m+1)}.$$ Hence for every $d \in D$ there is an integer $m$ for which $a^m d = 0$.\ Fix $d \in D$ and let $m_i$ be the minimal natural number for which $(a_i)^{m_i}d=0$ and set $m:= \sup\{m_i: 1 \leq i \leq t\}$. Then for a large enough integer $k$ we have that $J^k d =0$ (for example set $k=tm$), and $J^k \in {\mathcal{G}}$. Thus every element of $D(J)$ is annihilated by an ideal contained in ${\mathcal{G}}$, therefore $D(J)$ is ${\mathcal{G}}$-torsion. Now consider an arbitrary ${\mathcal{G}}$-torsion module $M$. Then $M$ has a presentation $\bigoplus_{\alpha\in \Lambda} R/J_\alpha \overset{p}\to M \to 0$ for a family $\{J_\alpha\}_{\alpha \in \Lambda}$ of ideals of ${\mathcal{G}}$. Since ${\mathcal{G}}$ is of finite type, we may assume that each $J_{\alpha}$ is finitely generated. Take the push-out of this map with the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $\bigoplus_\alpha R/J_\alpha$. $$\xymatrix{ 0 \ar[r]&\bigoplus_{\substack{\alpha \in \Lambda}} R/J_\alpha \ar[r] \ar[d]^p& \bigoplus_{\substack{\alpha \in \Lambda}}D(J_\alpha) \ar[r] \ar[d]& \bigoplus_{\substack{\alpha \in \Lambda}}A(J_\alpha) \ar@{=}[d] \ar[r] & 0\\ 0 \ar[r]&M\ar[r] \ar[d] & Z \ar[r] \ar[d]& \bigoplus_{\substack{\alpha \in \Lambda}}A(J_\alpha) \ar[r] & 0\\ &0&0 &&}$$ The bottom short exact sequence forms a preenvelope of $M$. We have shown above that for every $\alpha$ in $A$, $D(J_\alpha)$ is ${\mathcal{G}}$-torsion, so also $Z$ is ${\mathcal{G}}$-torsion. Therefore, as the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $M$ must be a direct summand of $Z$ by Proposition \[P:Xu-env\], also the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $M$ is ${\mathcal{G}}$-torsion. The following is a corollary to Lemma \[L:addT-tens-tf\] and Lemma \[L:torsion-env\]. \[C:tor-tens-div\] Suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ and suppose $M$ is a ${\mathcal{G}}$-torsion $R$-module. Then $M \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible. Let the following be a ${\mathcal{D}}_{\mathcal{G}}$-envelope of a ${\mathcal{G}}$-torsion module $M$, where both $D$ and $A$ are ${\mathcal{G}}$-torsion by Lemma \[L:torsion-env\]. $$0 \to M \to D \to A \to 0$$ The module $A$ is ${\mathcal{G}}$-divisible and $R_{\mathcal{G}}/R$ is ${\mathcal{G}}$-torsion so $A \otimes_R R_{\mathcal{G}}/R =0$, hence $A \to A \otimes_R R_{\mathcal{G}}$ is surjective. In particular, $A \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion. Also as $A \in {\mathrm{Add}}(T_0 \oplus T_1)$, $A \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-torsion-free by Lemma \[L:addT-tens-tf\] (2). It follows that $A\otimes_R R_{\mathcal{G}}$ is both ${\mathcal{G}}$-torsion and ${\mathcal{G}}$-torsion-free so $A\otimes_R R_{\mathcal{G}}=0$. Additionally as ${\operatorname{p.dim}}A \leq 1$, ${\operatorname{Tor}}^R_1(A, R_{\mathcal{G}})=0$, so the functor $(- \otimes_R R_{\mathcal{G}})$ applied to the envelope of $M$ reduces to the following isomorphism. $$0={\operatorname{Tor}}^R_1(A, R_{\mathcal{G}}) \to M \otimes_R R_{\mathcal{G}}\overset{\cong}\to D \otimes_R R_{\mathcal{G}}\to A \otimes_R R_{\mathcal{G}}=0$$ Hence as $D \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, also $M \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, as required. \[P:R\_G-divisible\] Suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$. Then $R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible. We will show that for each $J \in {\mathcal{G}}$, $R/J \otimes_R R_{\mathcal{G}}=0$. Fix a $J \in {\mathcal{G}}$. By Corollary \[C:tor-tens-div\], $R/J \otimes_R R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, Thus we have $R/J \otimes_R (R/J \otimes_R R_{\mathcal{G}}) =0$. However $$0=R/J \otimes_R (R/J \otimes_R R_{\mathcal{G}}) \cong (R/J \otimes_R R/J) \otimes_R R_{\mathcal{G}}\cong R/J\otimes_RR_{\mathcal{G}},$$ since $R\to R/J$ is a ring epimorphism, thus $R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible. Using the characterisation of a perfect localisation of [@Ste75 Chapter XI.3, Proposition 3.4], we can state the main result of this section. \[P:tilting-env\] Assume that ${\mathcal{T}}$ is a $1$-tilting class over a commutative ring $R$ such that the class ${\mathcal{T}}$ is enveloping. Then the associated Gabriel topology ${\mathcal{G}}$ of ${\mathcal{T}}$ arises from a perfect localisation. By Proposition \[P:R\_G-divisible\], $R_{\mathcal{G}}$ is ${\mathcal{G}}$-divisible, hence by [@Ste75 Proposition 3.4 (g)], $\psi\colon R\to R_{\mathcal{G}}$ is flat ring epimorphism and moreover it is injective. The ${\mathcal{G}}$-completion of $R$ and the endomorphism ring of $K$ {#S:compl-endK} ====================================================================== The aim of this section is to prove that if $R \overset{u}\to U$ is a commutative flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$, then there is a natural ring isomorphism between the following two rings. $$\Lambda(R) = \varprojlim_{\substack{J \in {\mathcal{G}}}} R/J {\rm \ and\ } {\operatorname{End}}_R(K)=\mathfrak{R}$$ This was mentioned in [@BP2 Remark 19.4], and a much stronger equivalence was shown in [@Pos3]. Also, it follows from this ring isomorphism that $\mathfrak{R}$ is a commutative ring.\ For completeness, we will give an explicit description of the isomorphism between the two rings.\ We will begin by briefly recalling some useful definitions about topological rings specifically referring to Gabriel topologies. Our reference is [@Ste75 Chapter VI.4]. Next we will continue by introducing $u$-contramodules in an analogous way to Positselski in [@Pos]. To finish, we show the ring isomorphism as well as a lemma and a proposition which relate the ${\mathcal{G}}$-torsion $R$-modules $R/J$ to the discrete quotient rings of $\mathfrak{R}$. Topological rings ----------------- A ring $R$ is a [*topological ring*]{} if it has a topology such that the ring operations are continuous. A topological ring $R$ is [*right linearly topological*]{} if it has a topology with a basis of neighbourhoods of zero consisting of right ideals of $R$. The ring $R$ with a right Gabriel topology is an example of a right linearly topological ring. If $R$ is a right linearly topological ring, then the set of right ideals $J$ in a basis $\mathfrak{ B}$ of the topology form a directed set, hence $\{R/J\mid J\in \mathfrak B\}$ is an inverse system. The [*completion*]{} of $R$ is the module $$\Lambda_{\mathfrak{ B}}(R) := \varprojlim_{\substack{J \in \mathfrak B}} R/J.$$ There is a canonical map $\lambda:R \to \Lambda_\mathfrak{B}(R)$ which sends the element $r\in R$ to $(r +J)_{J\in \mathfrak{ B}}$. If the homomorphism $\lambda_R$ is injective, then $R$ is called [*separated*]{}, which is equivalent to $\bigcap_{J \in \mathfrak{ B}}J =0$. If the map $\lambda$ is surjective, $R$ is called [*complete*]{}.\ The [*projective limit topology*]{} on $\Lambda_{\mathfrak{ B}}(R)$ is the topology where a sub-basis of neighbourhoods of zero is given by the the kernels of the projection maps $\Lambda_{\mathfrak{ B}}(R) \to R/J$. That is, it is the topology induced by the product of the discrete topology on $\prod_{J \in \mathfrak{ B}} R/J$. If the ideals in $\mathfrak{ B}$ are two-sided in $R$, then the module $\Lambda_{\mathfrak{ B}}(R)$ is a ring. Furthermore, it is a linearly topological ring with respect to the projective limit topology. In this case, the ring $\Lambda_{\mathfrak{ B}}(R)$ is both separated and complete with this topology. Each element in $\Lambda_{\mathfrak{ B}}(R)$ is of the form $(r_J +J)_{J\in \mathfrak{ B}}$ with the relation that for $J \subseteq J'$, $r_J - r_{J'} \in J'$. We will simply write $\Lambda(R)$ when the basis $\mathfrak{ B}$ is clear from the context. \[R:topologies\] *If $W(J)$ is the kernel of the projection $\pi_J\colon\Lambda_{\mathfrak{ B}}(R)\to R/J$, then clearly $W(J)\supseteq \Lambda(R)J$.* Let $R$ be a linearly topological ring. A right $R$-module $N$ is [*discrete*]{} if for every $x \in N$, the annihilator ideal ${\mathrm{Ann}}_R(x) = \{r \in R: xr =0\}$ is open in the topology of $R$. In case the topology on $R$ is a Gabriel topology ${\mathcal{G}}$ on $R$, then $N$ is discrete if and only if it is ${\mathcal{G}}$-torsion. A linearly topological ring is [*left pro-perfect*]{} ([@BP2]) if it is separated, complete, and with a base of neighbourhoods of zero formed by two-sided ideals such that all of its discrete quotient rings are perfect. *For the rest of this subsection, we will be considering a flat injective ring epimorphism of commutative rings denoted $0 \to R \overset{u} \to U$, and we will denote by $K$ the cokernel $U/R$ of $u$.* Let $\mathfrak{R}$ denote the endomorphism ring ${\operatorname{End}}_R(K)$. Take a finitely generated submodule $F$ of $K$, and consider the ideal formed by the elements of $\mathfrak{R}$ which annihilate $F$. The ideals of this form form a base of neighbourhoods of zero of $\mathfrak{R}$. Note that this is the same as considering ${\operatorname{End}}_R(K)$ with the subspace topology of the product topology on $K^K$ where the topology on $K$ is the discrete topology. We will consider $\mathfrak{R}$ endowed with this topology, which is also called the [*finite topology*]{}. We will now state the above in terms of a Gabriel topology that arises from a perfect localisation. Let ${\mathcal{G}}$ be the Gabriel topology associated to the flat ring epimorphism $u$. As $K\otimes_RU=0$, $K$ is ${\mathcal{G}}$-torsion, or equivalently a discrete module. Thus there is a natural well-defined action of $\Lambda(R)$ on $K$. In other words, $K$ is a $\Lambda(R)$-module where for every element $(r_J +J)_{J \in {\mathcal{G}}}\in \Lambda (R)$ and every element $z\in U$, the scalar multiplication is defined by $(r_J +J)_{J \in {\mathcal{G}}} \cdot (z+R):= r_{J_z} z +R$ where $J_z:= {\mathrm{Ann}}_R(z+R)$. As well as the natural map $\lambda:R \to \Lambda(R)$, there is also a natural map $\nu:R \to \mathfrak{R}$ where each element of $R$ is mapped to the endomorphism of $K$ which is multiplication by that element. If $R \overset{u}\to U$ is a flat injective ring epimorphism, then there is a homomorphism $$\alpha: \Lambda(R) = \varprojlim_{\substack{J \in {\mathcal{G}}}} R/J \to \mathfrak{R},$$ where $\alpha$ is induced by the action of $\Lambda(R)$ on $K$. It follows that the following triangle commutes. $$(\ast)\quad \xymatrix{ R \ar[d]_{\nu} \ar[r]^\lambda& \Lambda(R) \ar[dl]^{\alpha} \\ \mathfrak{R} & }$$ The rest of this section is dedicated to showing that $\alpha$ is a ring isomorphism. We will first show that $\alpha$ is injective, but before that we have to recall some terminology. A module $M$ is [*$U$-h-divisible*]{} if $M$ is an epimorphic image of $U^{(\alpha)}$ for some cardinal $\alpha$. An $R$-module $M$ has a unique $U$-h-divisible submodule denoted $h_U(M)$, and it is the image of the map ${\operatorname{Hom}}_R(U,M) \to {\operatorname{Hom}}_R(R,M) \cong M$. Hence for an $R$-module $M$, by applying the contravariant functor ${\operatorname{Hom}}_R(-,M)$ to the short exact sequence $0 \to R \overset{u}\to U \to K \to 0$ we have the following short exact sequences. $$\label{eq:1contra} 0 \to {\operatorname{Hom}}_R(K, M) \to {\operatorname{Hom}}_R(U,M) \to h_U(M) \to 0$$ $$\label{eq:2contra} 0 \to M / h_U(M) \to {\operatorname{Ext}}^1_R(K,M) \to {\operatorname{Ext}}^1_R(U,M) \to 0$$ By applying the covariant functor ${\operatorname{Hom}}_R(K,-)$ to the same short exact sequence we have the following. $$\label{eq:3contra} 0= {\operatorname{Hom}}_R(K, U) \to {\operatorname{Hom}}_R(K,K) \overset{\delta}\to {\operatorname{Ext}}^1_R(K, R) \to {\operatorname{Ext}}^1_R(K,U) =0, $$ where the last term vanishes since by the flatness of the ring $U$, there is an isomorphism ${\operatorname{Ext}}^1_R(K,U)\cong {\operatorname{Ext}}^1_U(K\otimes_RU,U) =0$. Thus note that ${\operatorname{Hom}}_R(K, K)$ is isomorphic to ${\operatorname{Ext}}^1_R(K, R)$ via $\delta$. Recall from Lemma \[L:finmanyann\] that the ideals ${\mathrm{Ann}}_R(z +R)$ for $z+R \in K$ form a sub-basis of the topology ${\mathcal{G}}$. Let ${\mathcal{S}}\subset {\mathcal{G}}$ denote denote the ideals of ${\mathcal{G}}$ of the form ${\mathrm{Ann}}_R(z +R)$ for $z+R \in K$. Clearly, the following two intersections of ideals coincide. $$\bigcap_{\substack{J \in {\mathcal{G}}}} J = \bigcap_{\substack{J \in {\mathcal{S}}}} J$$ We begin with some facts about $\Lambda(R)$ and $\mathfrak{R}$. \[L:mapfacts\] Let $u:R \to U$ be a flat injective ring epimorphism. Then the following hold. 1. The kernel of $\nu: R \to \mathfrak{R}$ is the intersection $\bigcap_{J \in {\mathcal{S}}} J$. 2. The kernel of $\lambda: R \to \Lambda(R)$ is the intersection $\bigcap_{J \in {\mathcal{G}}} J$. 3. The ideal $\bigcap_{J \in {\mathcal{G}}} J$ is the maximal $U$-h-divisible submodule of $R$. 4. The homomorphism $\alpha: \Lambda(R) \to \mathfrak{R}$ is injective. <!-- --> 1. For $r \in R$, $\nu (r)=0$ if and only if $rK=0$ if and only if $rz \in R$ for every $z \in U$. This amounts to $r \in {\mathrm{Ann}}_R(z +R)$ for every $z\in U$, hence $r \in \bigcap_{J \in {\mathcal{S}}} J$. 2. By the definition of $\lambda$ it is clear that $\lambda (r) =0$ if and only if $r \in J$ for every $J \in {\mathcal{G}}$. 3. First we show that $\bigcap_{J \in {\mathcal{G}}} J \subseteq h_U(R)$. Take $a \in \bigcap_{J \in {\mathcal{G}}} J$. We want to see that multiplication by $a$, $\dot{a}: R \to R$ extends to a map $f:U \to R$ (that is $\dot{a}$ is in the image of the map $u^\ast: {\operatorname{Hom}}_R(U, R) \to {\operatorname{Hom}}_R(R,R)$). By part (1) and its proof, $az \in R$ for every $z \in U$, so we have a well-defined map $\dot{a}: U \to R$, which makes the following triangle commute as desired. $$\xymatrix{ R \ar[d]_{\dot{a}} \ar[r]^u&U \ar[ld]^{\dot{a}} \\ R &}$$ Now take $a \in h_U(R)$. Since $h_U(R)$ is a ${\mathcal{G}}$-divisible submodule of $R$, $a \in J( h_U(R))\leq J$ for each $J \in {\mathcal{G}}$, as required. 4. Take $\eta=(r_J + J)_{J \in {\mathcal{G}}} \in \Lambda(R)$ such that $\alpha(\eta)=0$ or $\eta(z+R)=0$ for each $z \in U$. Then $r_Iz \in R$ where $I = {\mathrm{Ann}}_R(z+R)$. This implies $r_J \in J$ for each $J \in {\mathcal{S}}$, so $\eta=0$. $u$-contramodules {#S:U-contra} ----------------- We will begin by discussing a general commutative ring epimorphism $u$ before moving onto a flat injective ring epimorphisms. Let $u\colon R\to U$ be a ring epimorphism. A [*$u$-contramodule*]{} is an $R$-module $M$ such that $${\operatorname{Hom}}_R(U, M) = 0 = {\operatorname{Ext}}^1_R(U,M).$$ \[L:geiglenz\][@GL91 Proposition 1.1] The category of $u$-contramodules is closed under kernels of morphisms, extensions, infinite products and projective limits in ${R\textrm{-}\mathrm{Mod}}$. The following two lemmas are proved in [@Pos] for the case of the localisation of $R$ at a multiplicative system. For completeness we include their proofs in our setting. \[L:pos1.2\][@Pos Lemma 1.2] Let $u\colon R\to U$ be a ring epimorphism and let $M$ be an $R$-module. 1. If ${\operatorname{Hom}}_R(U,M)=0$, then ${\operatorname{Hom}}_R(Z,M)=0$ for any $U$-h-divisible module $Z$. 2. If $M$ is a $u$-contramodule, then ${\operatorname{Ext}}_R^1(Z,M)=0={\operatorname{Hom}}_R(Z,M)$ for any $U$-module $Z$. <!-- --> 1. By the $U$-h-divisibility of $Z$ there exists a map $U^{(\alpha)} \to Z \to 0$. As ${\operatorname{Hom}}_R(U^{(\alpha)}, M)=0$, it follows that ${\operatorname{Hom}}_R(Z, M)=0$. 2. First note that if $Z \in {\mathrm{Mod}\textrm{-}U}$, there is a short exact sequence $0 \to H \to U^{(\alpha)} \to Z \to 0$ of $U$-modules in ${\mathrm{Mod}\textrm{-}R}$. As ${\operatorname{Ext}}^1_R(U^{(\alpha)}, M)=0$ then ${\operatorname{Hom}}_R(H, M) \cong {\operatorname{Ext}}^1_R(Z,M) $. However, from (1) ${\operatorname{Hom}}_R(H, M)=0$, so also ${\operatorname{Ext}}^1_R(Z,M)=0$. \[L:oneten\] [@Pos Lemma 1.10] Let $b:A \to B$ and $c:A \to C$ be two $R$-module homomorphisms such that $C$ is a $u$-contramodule while ${\operatorname{Ker}}(b)$ is a $U$-h-divisible $R$-module and ${\operatorname{Coker}}(b)$ is a $U$-module. Then there exists a unique homomorphism $f:B \to C$ such that $c =fb$. First we show the existence of a homomorphism $f: B \to C$ such that $c =fb$. ${\operatorname{Ker}}b$ is a $U$-h-divisible module, so the composition $c \circ \ker b =0$ by Lemma \[L:pos1.2\] (1), hence the map $c$ factors through $\bar{c}:A/{\operatorname{Ker}}b \to C$ as in the following diagram. $$\xymatrix@C=1.2cm{ {\operatorname{Ker}}b \ar[r]^{\ker b} & A \ar@{->>}[dr] \ar[rdd]_c \ar[rr]^b & & B \ar[r]^{{\operatorname{coker}}b}& {\operatorname{Coker}}b \\ &&A/{\operatorname{Ker}}b \ar[d]^{\bar{c}} \ar@{^{(}->}[ru]\\ & &C}$$ By applying the functor ${\operatorname{Hom}}_R(-,C)$ to the right short exact sequence above we get the following exact sequence. $${\operatorname{Hom}}_R(B, C) \to {\operatorname{Hom}}_R(A/{\operatorname{Ker}}b, C) \to {\operatorname{Ext}}^1_R({\operatorname{Coker}}b, C)$$ By Lemma \[L:pos1.2\] (2), ${\operatorname{Ext}}^1_R({\operatorname{Coker}}b, C)=0$ as ${\operatorname{Coker}}b$ is a $U$-module. Now we show the uniqueness of such a homomorphism. Suppose $h = f -g$ is such that $hb=0$. Then there exists a homomorphism $\bar{h}: {\operatorname{Coker}}b \to C$ such that $\bar{h} \circ {\operatorname{coker}}b = h$. $$\xymatrix@C=1.2cm{ A \ar[r]^b \ar[rd]& B \ar[d]^h \ar[r]^{{\operatorname{coker}}b} & {\operatorname{Coker}}b \ar[ld]^{\bar{h}}\\ &C&}$$ By assumption, ${\operatorname{Coker}}b$ is a $U$-module, and $C$ is a $u$-contramodule, so ${\operatorname{Hom}}_R({\operatorname{Coker}}b,C) =0$ by Lemma \[L:pos1.2\] (2). Thus $h$ must be the zero homomorphism, so $f =g$. From now on, $u: R \to U$ will always be a flat injective ring epimorphism. \[L:endcontra\][@BP2 Lemma 16.2] Let $u:R \to U$ be a flat injective ring epimorphism. Then $\mathfrak{R}$ is a $u$-contramodule and is ${\mathcal{G}}$-torsion-free. To see that $\mathfrak{R}$ is ${\mathcal{G}}$-torsion-free we note that it is contained in a $U$-module which is always ${\mathcal{G}}$-torsion-free, as follows. $$0 \to {\operatorname{Hom}}_R(K,K) \to {\operatorname{Hom}}_R(U,K)$$ Now we will show that $\mathfrak{R}$ is a $u$-contramodule. By the tensor-hom adjunction, we have the following isomorphism. $${\operatorname{Hom}}_R(U, {\operatorname{Hom}}_R(K,K)) \cong {\operatorname{Hom}}_R(U \otimes_R K, K) =0$$ Similarly, to see that ${\operatorname{Ext}}_R^1(U, \mathfrak{R})=0$, we use the flatness of $U$ so ${\operatorname{Tor}}^R_1(U,K)=0$. Hence there is the following inclusion. $$0 \to {\operatorname{Ext}}_R^1(U, {\operatorname{Hom}}_R(K,K)) \to {\operatorname{Ext}}_{R}^1(U \otimes_R K, K)=0$$ \[L:rjcontra\] Let $u:R \to U$ be a flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$. Then for every $J \in {\mathcal{G}}$, every $R/J$-module $M$ is a $u$-contramodule. To see that ${\operatorname{Hom}}_R(U,M)=0$, take $f:U \to M$. Then $f(U) = f(JU) = Jf(U) =0$ as $J$ annihilates $M$. As ${\operatorname{Tor}}^R_i(R/J, U)=0$ and $R \to R/J$ is a ring epimorphism, one has that the following isomorphism. $${\operatorname{Ext}}_R^1(U, M) \cong {\operatorname{Ext}}_{R/J}^1 (R/J \otimes_R U, M) =0$$ \[C:compl-contra\] Let $u:R \to U$ be a flat injective ring epimorphism. Then $\Lambda(R)$ is a $u$-contramodule. This follows immediately by Lemma \[L:rjcontra\] and by the closure properties of $u$-contramodules in Lemma \[L:geiglenz\]. \[L:coker-nu\] Let $u:R \to U$ be a flat injective ring epimorphism. Then the cokernel of $\nu:R \to \mathfrak{R}$ is a $U$-module. Recall that $h_U(R)$ is the $U$-h-divisible submodule of $R$ and $\delta$ is as in sequence (\[eq:3contra\]). Consider the following commuting diagram. $$\xymatrix{ 0 \ar[r] &R/h_U(R) \ar[r]^\nu \ar@{=}[d] & \mathfrak{R} \ar[r] \ar[d]^\cong_\delta & {\operatorname{Coker}}(\nu) \ar[r] \ar[d] &0\\ 0 \ar[r] &R/h_U(R) \ar[r] & {\operatorname{Ext}}^1_R(K,R) \ar[r] & {\operatorname{Ext}}^1_R(U,R) \ar[r] &0}$$ By the five-lemma, the last vertical arrow is an isomorphism, so ${\operatorname{Coker}}(\nu) \cong {\operatorname{Ext}}^1_R(U,R)$ which is a $U$-module, as required. The isomorphism between the ${\mathcal{G}}$-completion of $R$ and ${\operatorname{End}}(K)$ ------------------------------------------------------------------------------------------- We now prove the main result of this section. \[P:ringiso\] Let $u:R \to U$ be a flat injective ring epimorphism. Using the notation of subsection 6.1 the morphism $\alpha: \Lambda(R) \to \mathfrak{R}$ is a ring isomorphism. From $(\ast)$ we have the following commuting triangle: $$\xymatrix{ R \ar[d]_{\nu} \ar[r]^\lambda& \Lambda(R) \ar[dl]^{\alpha} \\ \mathfrak{R} & }$$ From sequences (2) and (3) we have the following exact sequence. $$0 \to h_U(R) \to R \overset{\nu} \to \mathfrak{R} \to {\operatorname{Coker}}(\nu) \to 0$$ where $h_U(R)$ is $U$-h-divisible and ${\operatorname{Coker}}(\nu)$ is a $U$-module by Lemma \[L:coker-nu\]. Both $\Lambda(R)$ and $\mathfrak{R}$ are $u$-contramodules so one can apply Lemma \[L:oneten\] to the two triangles below. That is, firstly, there exists a unique map $\beta$ such that $\beta \nu = \lambda$, and secondly by uniqueness, the identity on $\mathfrak{R}$ is the only homomorphism that makes the triangle on the right below commute. $$\xymatrix{ R \ar[d]_{\lambda} \ar[r]^\nu& \mathfrak{R} \ar[dl]^{\beta} & R \ar[d]_{\nu} \ar[r]^\nu& \mathfrak{R} \ar[dl]^{\text{id}_\mathfrak{R}} \\ \Lambda(R) && \mathfrak{R} & }$$ It follows that since $ \alpha \beta \nu = \alpha \lambda= \nu $, by uniqueness $\alpha \beta = \text{id}_\mathfrak{R}$. Therefore, $\alpha$ is surjective. It was shown in Lemma \[L:mapfacts\] that $\alpha$ is injective, hence $\alpha$ is an isomorphism. It remains to see that $\alpha$ is a ring homomorphism. First note that if $z\in U$, $s\in R$ and $Jz \subseteq R $, then also $J(sz) \subseteq R$, that is $J \subseteq {\mathrm{Ann}}_R(sz +R)$. Let $\tilde{r}= (r_J+J)_{J \in {\mathcal{G}}}$ and $\tilde{s}= (s_J+J)_{J \in {\mathcal{G}}}$ denote elements of $\Lambda(R)$. Let $L$ denote ${\mathrm{Ann}}_R(z +R)$ and $L_s$ denote ${\mathrm{Ann}}_R(sz +R)$ for a fixed $z+R$ and note that $L \subseteq L_s$. $$\alpha(\tilde{r} \cdot \tilde{s} ):K \to K: z+R \mapsto r_L s_L z +R$$ $$\alpha(\tilde{r}) \alpha(\tilde{s}) = (K \overset{\tilde{r}}\to K) (K \overset{\tilde{s}}\to K) : z +R \mapsto s_L z +R \mapsto r_{L_s}s_Lz+R$$ Then clearly $r_{L_s} - r_L \in L_s$, so the endomorphisms $\alpha(\tilde{r} \cdot \tilde{s} )$ and $\alpha(\tilde{r}) \alpha(\tilde{s})$ are equal. The following lemma will be useful when passing from the ring $R$ to the complete and separated topological ring $\mathfrak{R}$. \[L:discreteiso\] Let $u:R \to U$ be a flat injective ring epimorphism with associated Gabriel topology ${\mathcal{G}}$. The $R$-module $R/J$ is isomorphic to $\mathfrak{R}/ J\mathfrak{R}$ and to $\Lambda(R)/J\Lambda(R)$, for every $J\in {\mathcal{G}}$. $\mathfrak{R}/ J\mathfrak{R}$ and $\Lambda(R)/J\Lambda(R)$ are isomorphic by Proposition \[P:ringiso\]. Both $R/J$ and $\mathfrak{R} / J \mathfrak{R}$ are $R/J$-modules, hence by Lemma \[L:rjcontra\] and Lemma \[L:oneten\], there exists a unique $f$ such that the left triangle below commutes. The map $f$ induces $\bar{f}$ since $J\mathfrak{R}\subseteq {\operatorname{Ker}}f$, so the right triangle below also commutes. $$\xymatrix{ R \ar[d]_{p} \ar[r]^\nu& \mathfrak{R} \ar[dl]^{f} & \mathfrak{R} \ar[d]_{f} \ar[r]^{\pi \hspace{15pt}}& \mathfrak{R} / J\mathfrak{R} \ar[dl]^{\bar{f} }\\ R/J & &R/J & }$$ Let $\bar{\nu}$ be the map induced by $\nu$ as in the following commuting diagram. We will show that $\bar{f}$ and $\bar{\nu}$ are mutually inverse. $$\xymatrix{ R \ar[r]^\nu \ar[d]_{p} & \mathfrak{R} \ar[d]_{\pi}\\ R/J \ar[r]^{\bar{\nu}} & \mathfrak{R} / J\mathfrak{R}}$$ Then, we have that $\pi \nu = \bar{\nu} p$, and so using the above commuting triangles it follows that $\bar{f} \bar{\nu} p = \bar{f} \pi \nu =f \nu =p $. As $p$ is surjective, $\bar{f} \bar{\nu} = \text{id}_{R/J}$. We now show that $\bar{\nu} \bar{f} = \text{id}_{{\mathfrak{R}}/J\mathfrak{R}}$. $$\xymatrix{ R \ar[d]_{\pi \nu} \ar[r]^\nu&\mathfrak{R} \ar[dl]^{h} \\ \mathfrak{R} / J\mathfrak{R} & }$$ By uniqueness, $\pi$ is the unique map that fits into the triangle above, that is $\pi \nu = h \nu$ implies that $h = \pi$. So, $$\pi \nu = \bar{\nu} p = \bar{\nu} f \nu = \bar{\nu} \bar{f} \pi \nu$$ Therefore $\pi = \bar{\nu} \bar{f} \pi$, and as $\pi$ is surjective, $\bar{\nu} \bar{f} = \text{id}_{{\mathfrak{R}}/J\mathfrak{R}}$ as required. \[P:topologies-2\] If $V$ is an open ideal in the topology of $\mathfrak{R}={\operatorname{End}}_R(K)$, then there is $J\in {\mathcal{G}}$ and a surjective ring homomorphism $R/J\to \mathfrak{R}/V$. By the definition of the topology on $\mathfrak{R}$, if $V$ is an open ideal, then by Proposition \[P:ringiso\], $W=\alpha^{-1}(V)$ is an open ideal in the projective limit topology of $\Lambda(R)$. Hence by Remark \[R:topologies\], there is $J\in {\mathcal{G}}$ such that $W\supseteq \Lambda(R)J$. By Lemma \[L:discreteiso\] there is a surjective ring homomorphism $R/J\to \mathfrak{R}/V.$ When a ${\mathcal{G}}$-divisible class is enveloping {#S:enveloping} ==================================================== For this section, $R$ will always be a commutative ring. Fix a flat injective ring epimorphism $u$ and an exact sequence $$0 \to R \overset{u}\to U \to K \to 0.$$ Denote by ${\mathcal{G}}$ the corresponding Gabriel topology. The aim of this section is to show that if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping then for each $J \in {\mathcal{G}}$ the ring $R/J$ is perfect. It will follow from Section \[S:properfect\] that also $\mathfrak{R}$ is pro-perfect. We begin by showing that for a local ring $R$ the rings $R/J$ are perfect, before extending the result to all commutative rings by showing that all ${\mathcal{G}}$-torsion modules (specifically the $R/J$ for $J \in {\mathcal{G}}$) are isomorphic to the direct sum of their localisations.\ In Lemma \[L:R-env\], it was shown that if $\varepsilon:R \to D$ is a ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$ in ${\mathrm{Mod}\textrm{-}R}$, then $D$ must be ${\mathcal{G}}$-torsion-free. Furthermore, if ${\mathcal{G}}$ arises from a perfect localisation $u:R \to U$ and $R$ has a ${\mathcal{D}}_{\mathcal{G}}$ envelope, then the following proposition allows us to work in the setting that ${\mathcal{D}}_{\mathcal{G}}= {\operatorname{Gen}}U$, thus $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ is the $1$-tilting cotorsion pair associated to the $1$-tilting module $U \oplus K$ (see Remark \[R:pdU=1\]). \[P:pd1\] Let $u:R \to U$ be a (non-trivial) flat injective ring epimorphism and suppose $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope. Then ${\operatorname{p.dim}}_R U \leq 1$. Let $$0 \to R \overset{\varepsilon}\to D \to D/R \to 0 \eqno(** )$$ denote the ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$. First we claim that $D$ is a $U$-module by showing that $D$ is ${\mathcal{G}}$-closed, or that $D \cong U \otimes_R D$. Consider the following exact sequence. $$0 \to {\operatorname{Tor}}^R_1( D, K) \to D \to D \otimes_R U \to D \otimes_R K \to 0$$ Therefore we must show that ${\operatorname{Tor}}^R_1(D, K) = 0 = D\otimes_R K$. As $D$ is ${\mathcal{G}}$-divisible and $K$ is ${\mathcal{G}}$-torsion it follows that $D \otimes_R K =0$. By Lemma \[L:R-env\] $D$ is ${\mathcal{G}}$-torsion-free, hence $D\cong D\otimes_R U$ and $D$ is a $U$-module. The cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ is complete, which implies $R$-module $D/R$ is in ${\mathcal{A}}$, so ${\operatorname{p.dim}}_R D/R \leq 1$. From the short exact sequence [$(**)$]{} it follows that also ${\operatorname{p.dim}}_R D \leq 1$. Consider the following short exact sequence of $U$-modules $$0 \to U \to D \otimes_R U \cong D \to D/R \otimes_R U \to 0$$ We now claim that $D/R \otimes_R U$ is $U$-projective. Indeed, take any $Z \in {U\textrm{-}\mathrm{Mod}}$ and note that $Z \in {\mathcal{D}}_{\mathcal{G}}$. Then $0={\operatorname{Ext}}^1_R(D/R, Z)\cong{\operatorname{Ext}}^1_U(D/R\otimes_R U, Z)$. Therefore the short exact sequence above splits in ${\mathrm{Mod}\textrm{-}U}$ and so $U$ is a direct summand of $D$ also as an $R$-module, and the conclusion follows. \[C:U-envelope\] Let $u:R \to U$ be a (non-trivial) flat injective ring epimorphism and suppose $R$ has a ${\mathcal{D}}_{\mathcal{G}}$-envelope. Then $$0 \to R \overset{u} \to U \to K \to 0$$ is a ${\mathcal{D}}_{\mathcal{G}}$-envelope of $R$. By Proposition \[P:pd1\] ${\operatorname{p.dim}}U \leq 1$, so from the discussion in Section \[S:gab-top\], $U \oplus K$ is a $1$-tilting module such that $(U \oplus K)^\perp = {\mathcal{D}}_{\mathcal{G}}$. Thus $K \in {\mathcal{A}}$ and so $u$ is a ${\mathcal{D}}_{\mathcal{G}}$-preenvelope. To see that $u$ is an envelope, note that ${\operatorname{Hom}}_R(K,U)=0$, so by Lemma \[L:identity-env\], if $u = f u$, then $f = \text{id}_U$ is an automorphism of $U$, thus $u$ is a ${\mathcal{D}}_{\mathcal{G}}$-envelope as required. We now begin by showing that when $R$ is a commutative local ring, if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ then for each $J \in {\mathcal{G}}$, $R/J$ is a perfect ring. We will use the ring isomorphism $\alpha: \Lambda(R) \cong \mathfrak{R}$ of Proposition \[P:ringiso\]. \[L:Kindecomp\] Let $R$ be a commutative local ring and $u:R \to U$ a flat injective ring epimorphism and let $K$ denote $U/R$. Then $K$ is indecomposable. It is enough to show that every idempotent of ${\operatorname{End}}_R(K)$ is either the zero homomorphism or the identity on $K$. Let ${\mathfrak{m}}$ denote the maximal ideal of $R$. Take a non-zero idempotent $e \in {\operatorname{End}}_R(K)$. Then there is an associated element $\alpha^{-1}(e)=\tilde{r}:=(r_J +J)_{J \in {\mathcal{G}}} \in \Lambda(R)$ via the ring isomorphism $\alpha:\Lambda(R) \cong \mathfrak{R}$ of Proposition \[P:ringiso\]. Clearly $\tilde{r}$ is also non-zero and an idempotent in $\Lambda(R)$. We will show this element is the identity in $\Lambda(R)$. As $\tilde{r}$ is non-zero, there exists a $J_0 \in {\mathcal{G}}$ such that $r_{J_0} \notin J_0$. Also, $\tilde{r} \cdot \tilde{r} - \tilde{r} =0$, hence $$r_{J_0}r_{J_0} - r_{J_0} = r_{J_0}(r_{J_0} -1_R) \in J_0.$$ We claim that $r_{J_0}$ is a unit in $R$. Suppose not, then $r_{J_0} \in {\mathfrak{m}}$, hence $r_{J_0} -1_R$ is a unit, which implies that $r_{J_0} \in J_0$, a contradiction. Consider some other $J \in {\mathcal{G}}$ such that $J \neq R$. $r_{J \cap J_0} - r_{J_0} \in J_0$, hence $r_{J \cap J_0} \notin J_0$. Therefore, by a similar argument as above, $r_{J \cap J_0}$ is a unit in $R$. As $r_{J \cap J_0} - r_{J} \in J$ and $r_{J \cap J_0}$ is a unit, $r_J \notin J$. Therefore by a similar argument as above $r_J$ is a unit in $R$ for each $J \in {\mathcal{G}}$ and we conclude that $\tilde{r}$ is a unit in $\Lambda(R)$. Finally, as $r_J(r_J - 1_R) \in J$ for every $J$, and $\tilde{r}:=(r_J +J)_{J \in {\mathcal{G}}} $ is a unit, it follows that $r_J - 1_R \in J$ for each $J$, implying that $\tilde{r}$ is the identity in $\Lambda(R)$. \[P:localperfect\] Let $R$ be a commutative local ring and consider the $1$-tilting cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ induced by the flat injective ring epimorphism $u:R \to U$. If $ {\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$, then $R/J$ is a perfect ring for every $J \in {\mathcal{G}}$. Let ${\mathfrak{m}}$ denote the maximal ideal of $R$. As $R$ is local, to show that $R/J$ is perfect it is enough to show that for every sequence of elements $\{a_1, a_2, \dots, a_i, \dots \}$ with $a_i \in {\mathfrak{m}}\setminus J$, there exists an $m >0$ such that the product $a_1 a_2 \cdots a_m \in J$ (that is ${\mathfrak{m}}/J$ is T-nilpotent) by Proposition \[P:perfect\]. Fix a $J\in {\mathcal{G}}$ and take $\{a_1, a_2, \dots, a_i, \dots \}$ as above. Consider the following preenvelope of $R/a_iR$. $$0 \to R/a_iR \hookrightarrow U/a_iR \to K \to 0$$ As $R$ is local, by Lemma \[L:Kindecomp\], $K$ is indecomposable, and as $R/a_iR$ is not ${\mathcal{G}}$-divisible this is an envelope of $R/a_iR$. We will use the T-nilpotency of direct sums of envelopes from Theorem \[T:Xu-sums\]. Consider the following countable direct sum of envelopes of $R/a_iR$ which is itself an envelope by Theorem \[T:Xu-sums\] (1). $$0 \to \bigoplus_{\substack{ i>0 }} R/a_iR \hookrightarrow \bigoplus_{\substack{ i>0 }}U/a_iR \to \bigoplus_{\substack{ i>0 }}K \to 0$$ For each $i>0$, we define a homomorphism $f_i:U/a_iR \to U/a_{i+1}R$ between the direct summands to be the multiplication by the element $a_{i+1}$. Then clearly $R/a_iR \subseteq U/a_iR$ vanishes under the action of $f_i = \dot{a}_{i+1}$, hence we can apply Theorem \[T:Xu-sums\] (2) to the homomorphisms $\{f_i\}_{i>0}$. So, for every $z + a_1R \in U/a_1R$, there exists an $n>0$ such that $$f_n \cdots f_2 f_1 (z+a_1R) = 0 \in U/a_{n+1}R,$$ which can be rewritten as $$a_{n+1} \cdots a_3 a_2 (z) \in a_{n+1}R.$$ By Lemma \[L:finmanyann\], there exist $z_1, z_2, \dots , z_n \in U$ such that $$\bigcap_{\substack{ 0 \leq j \leq n}} {\mathrm{Ann}}_R(z_j +R) \subseteq J.$$ Let $\Omega = \{z_1, z_2, \dots , z_n\}$. For each $z_j$, there exists an $n_j$ such that $a_{n_j+1} \cdots a_3 a_2$ annihilates $z_j$. That is, $$a_{n_j+1} \cdots a_3 a_2 (z_j) \in a_{n_j+1}R\subseteq R.$$ We now choose an integer $m$ such that $a_{m} \cdots a_3 a_2$ annihilates all the $z_j$ for $a \leq j \leq n$. Set $m = max\{n_j\mid j=1,2\dots, n\}$. Then this $m$ satisfies the following, which finishes the proof. $$a_m a_{m-1} \cdots a_3 a_2 \in \bigcap_{\substack{ 0 \leq j \leq n}} {\mathrm{Ann}}_R(z_j +R) \subseteq J$$ Now we extend the result to general commutative rings. Our assumption is that the Gabriel topology ${\mathcal{G}}$ is arises from a perfect localisation $u:R \to U$ and that the associated $1$-tilting class ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$. \[N:simple-env\] There is a preenvelope of the following form induced by the map $u$. $$0 \to R/{\mathfrak{m}}\to U / {\mathfrak{m}}\to K \to 0$$ Let the following sequence denote an envelope of $R/{\mathfrak{m}}$. $$0 \to R/{\mathfrak{m}}\to D({\mathfrak{m}}) \to X({\mathfrak{m}}) \to 0$$ By Proposition \[P:Xu-env\], $D({\mathfrak{m}})$ and $X({\mathfrak{m}})$ are direct summands of $U/{\mathfrak{m}}$ and $K =U/R$ respectively. For convenience we will consider $R/{\mathfrak{m}}$ as a submodule of $D({\mathfrak{m}})$ and $X({\mathfrak{m}})$ as a submodule of $K$. \[R:tors-facts\] The following lemma allows us to use Proposition \[P:localperfect\] to say that if $D_{\mathcal{G}}$ is enveloping in $R$, all localisations $R_{\mathfrak{m}}/J_{\mathfrak{m}}$ are perfect rings where ${\mathfrak{m}}$ is a maximal ideal in ${\mathcal{G}}$ and $J \in {\mathcal{G}}$. \[L:localfacts\] Let $R$ be a commutative ring and consider the $1$-tilting cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ induced from the flat injective ring epimorphism $u:R \to U$. Fix a maximal ideal ${\mathfrak{m}}$ of $R$ and let $u_{\mathfrak{m}}: R_{\mathfrak{m}}\to U_{\mathfrak{m}}$ be the corresponding flat injective ring epimorphism in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. Then the following hold. 1. $K_{\mathfrak{m}}= 0$ if and only if ${\mathfrak{m}}\notin {\mathcal{G}}$. 2. The induced Gabriel topology of $u_{\mathfrak{m}}$ denoted $${\mathcal{G}}({\mathfrak{m}}) = \{L \leq R_{\mathfrak{m}}: L U_{\mathfrak{m}}= U_{\mathfrak{m}}\}$$ contains the localisations ${\mathcal{G}}_{\mathfrak{m}}= \{J_{\mathfrak{m}}: J \in {\mathcal{G}}\}$. 3. Suppose ${\operatorname{p.dim}}U \leq 1$. Then $({\mathcal{A}}_{\mathfrak{m}}, ({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}})$ is the $1$-tilting cotorsion pair associated to the flat injective ring epimorphism $u_{\mathfrak{m}}: R_{\mathfrak{m}}\to U_{\mathfrak{m}}$. That is, $({\mathcal{D}}_{{\mathcal{G}}})_{\mathfrak{m}}= {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ and ${\mathcal{A}}_{\mathfrak{m}}= {}^\perp {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$. 4. If ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$, then ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. <!-- --> 1. Since $K$ is ${\mathcal{G}}$-torsion, this follows by Remark \[R:tors-facts\] (3). Note that if ${\mathfrak{m}}\notin {\mathcal{G}}$ the rest of the lemma follows trivially. 2. Take $J_{\mathfrak{m}}\in {\mathcal{G}}_{\mathfrak{m}}$. Then $ R_{\mathfrak{m}}/ J_{\mathfrak{m}}\otimes_R U_{\mathfrak{m}}\cong (R/J \otimes_R U) \otimes_R R_{\mathfrak{m}}= 0 $, so $J_{\mathfrak{m}}\in {\mathcal{G}}({\mathfrak{m}})$. 3. That $({\mathcal{A}}_{\mathfrak{m}}, ({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}})$ is the $1$-tilting cotorsion pair associated to the $1$-tilting module $(U \oplus K)_{\mathfrak{m}}$ is [@GT12 Proposition 13.50], therefore ${\operatorname{Gen}}(U_{\mathfrak{m}}) = ({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}}$ in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. As $u_{\mathfrak{m}}:R_{\mathfrak{m}}\to U_{\mathfrak{m}}$ is a flat injective ring epimorphism and ${\operatorname{p.dim}}_{R_{\mathfrak{m}}}U_{\mathfrak{m}}\leq 1$ the $1$-tilting classes ${\operatorname{Gen}}(U_{\mathfrak{m}})$ and ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})} $ coincide in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ by [@H Theorem 5.4]. Thus $({\mathcal{D}}_{{\mathcal{G}}})_{\mathfrak{m}}= {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ and it follows that ${\mathcal{A}}_{\mathfrak{m}}= {}^\perp {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$. 4. Assume that ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$ and take some $M \in {\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ with the following ${\mathcal{D}}_{\mathcal{G}}$-envelope. $$0 \to M \to D \to X \to 0$$ We claim that $M$ has a ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$-envelope in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. Since $M \in {\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$, $D$ and $X$ are $R_{\mathfrak{m}}$-modules by Proposition \[P:B-envelopes\]. By Proposition \[P:pd1\] ${\operatorname{p.dim}}U \leq 1$. By (3), $({\mathcal{D}}_{{\mathcal{G}}})_{\mathfrak{m}}= {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ so $D \in {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ and $X \in {}^\perp {\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$. Since $R \to R_{\mathfrak{m}}$ is a ring epimorphism, any direct summand of $D$ which contains $M$ in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ would also be a direct summand in ${\mathrm{Mod}\textrm{-}R}$. Thus we conclude that $0\to M\to D\to X\to 0$ is a ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$-envelope of $M$ in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. By the above lemma, if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$, then ${\mathcal{D}}_{{\mathcal{G}}({\mathfrak{m}})}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$. Next we show that, under our enveloping assumption, all ${\mathcal{G}}$-torsion modules are isomorphic to the direct sums of their localisations at maximal ideals.\ The proof of the following lemma uses an almost identical argument to the proof of Lemma \[L:torsion-env\]. \[L:Xmnilpotent\] Let $u:R \to U$ be a flat injective ring epimorphism, ${\mathcal{G}}$ the associated Gabriel topology and suppose that ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Let $D({\mathfrak{m}})$ and $X({\mathfrak{m}})$ be as in Notation \[N:simple-env\] and fix a maximal ideal ${\mathfrak{m}}\in {\mathcal{G}}$. For every element $d \in D({\mathfrak{m}})$ and every element $a \in {\mathfrak{m}}$, there is a natural number $n > 0$ such that $a^n d = 0$. Moreover, for every element $x \in X({\mathfrak{m}})$ and every element $a \in {\mathfrak{m}}$, there is a natural number $n > 0$ such that $a^n x = 0$. We will use the T-nilpotency of direct sums of envelopes as in Theorem \[T:Xu-sums\] (2). Consider the following countable direct sum of envelopes of $R/{\mathfrak{m}}$ which is itself an envelope by Theorem \[T:Xu-sums\] (1). $$0 \to \bigoplus_{\substack{ 0<i }} (R/{\mathfrak{m}})_{(i)} \to \bigoplus_{\substack{ 0<i }}D({\mathfrak{m}})_{(i)} \to \bigoplus_{\substack{ 0<i }}X({\mathfrak{m}})_{(i)} \to 0$$ For a fixed element $a \in {\mathfrak{m}}$, we choose the homomorphisms $f_i:D({\mathfrak{m}})_{(i)} \to D({\mathfrak{m}})_{(i+1)}$ between the direct summands to all be multiplication by $a$. Then clearly $R/{\mathfrak{m}}\subseteq D({\mathfrak{m}})$ vanishes under the action of $f_i = \dot{a}$, hence we can apply Theorem \[T:Xu-sums\] (2): for every $d \in D({\mathfrak{m}})$, there exists an $n$ such that $$f_n \cdots f_2 f_1 (d) = 0 \in D({\mathfrak{m}})_{(n+1)}.$$ Since each $f_i$ acts as multiplication by $a$, for every $d \in D$ there is an integer $n$ for which $a^n d = 0$, as required. It is straightforward to see that $X({\mathfrak{m}})$ has the same property as $X({\mathfrak{m}})$ is an epimorphic image of $D({\mathfrak{m}})$. \[L:suppXm\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Let ${\mathfrak{m}}\in {\mathcal{G}}$ and let $X({\mathfrak{m}})$ be as in Notation \[N:simple-env\]. The support of $X({\mathfrak{m}})$ is exactly $\{ {\mathfrak{m}}\}$, and each $X({\mathfrak{m}}) \cong X({\mathfrak{m}})_{\mathfrak{m}}$ is $K_{\mathfrak{m}}$. We claim that $X({\mathfrak{m}})$ is non-zero. Otherwise, $X({\mathfrak{m}})=0$ would imply that $R/{\mathfrak{m}}$ is ${\mathcal{G}}$-divisible, so $R/{\mathfrak{m}}= {\mathfrak{m}}(R/{\mathfrak{m}}) = 0$, a contradiction. Consider a maximal ideal ${\mathfrak{n}}\neq {\mathfrak{m}}$. Take an element $a \in {\mathfrak{m}}\setminus {\mathfrak{n}}$. Then for any $x \in X({\mathfrak{m}})$, $a^n x = 0$ for some $n > 0$, by Lemma \[L:Xmnilpotent\] and since $a$ is an invertible element in $R_{\mathfrak{n}}$, $x$ is zero in the localisation with respect to ${\mathfrak{n}}$. This holds for any element $x \in X({\mathfrak{m}})$, hence $X({\mathfrak{m}})_{\mathfrak{n}}= 0$. It follows that since $X({\mathfrak{m}})$ is non-zero, $X({\mathfrak{m}})_{\mathfrak{m}}\neq 0$. As mentioned in Remark \[R:tors-facts\], $X({\mathfrak{m}})$ is an $R_{\mathfrak{m}}$-module and since $X({\mathfrak{m}})$ is a direct summand of $K$, $X({\mathfrak{m}})$ is a direct summand of $K_{\mathfrak{m}}$ which is indecomposable, by Lemma \[L:Kindecomp\]. Therefore $X({\mathfrak{m}})$ is non-zero and is isomorphic to $K_{\mathfrak{m}}$. \[L:directsumxm\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Then the sum of the submodules $X({\mathfrak{m}})$ in $K$ is a direct sum.$$\sum_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}) = \bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}})$$ Recall that $X({\mathfrak{m}})$ is non-zero only for ${\mathfrak{m}}\in {\mathcal{G}}$ by Remark \[R:tors-facts\]. Consider an element $$x \in X({\mathfrak{m}}) \cap \sum_{\substack{ {\mathfrak{n}}\neq {\mathfrak{m}}\\ {\mathfrak{n}}\in {\mathcal{G}}}} X({\mathfrak{n}}).$$ We will show that this element must be zero. By Lemma \[L:suppXm\], since $x \in X({\mathfrak{m}})$, $x$ is zero in the localisation with respect to all maximal ideals ${\mathfrak{n}}\neq {\mathfrak{m}}$. But $x$ can also be written as a finite sum of elements $x_i \in X({\mathfrak{n}}_i)$, each of which is zero in the localisation with respect to ${\mathfrak{m}}$, by Lemma \[L:suppXm\]. Therefore, $(x)_{\mathfrak{n}}=0$ for all maximal ideals ${\mathfrak{n}}$, hence $x = 0$ . \[P:directsumk\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. The module $K$ can be written as a direct sum of its localisations $K_{\mathfrak{m}}$, as follows. $$K \cong \bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} K_{\mathfrak{m}}= \bigoplus_{\substack{ {\mathfrak{m}}\in {\operatorname{Max}}{R} }} K_{\mathfrak{m}}$$ From Lemma \[L:directsumxm\], we have the following inclusion. $$\bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}) \leq K$$ To see that this is an equality we show that these two modules have the same localisation with respect to every ${\mathfrak{m}}$ maximal in $R$. Recall that by Lemma \[L:localfacts\](1) if ${\mathfrak{n}}$ is maximal, then $K_{\mathfrak{n}}= 0$ if and only if ${\mathfrak{n}}\notin {\mathcal{G}}$ and by Lemma \[L:suppXm\], $\text{Supp}(X({\mathfrak{m}})) = \{ {\mathfrak{m}}\}$. Using these lemmas, it follows that for ${\mathfrak{n}}\notin {\mathcal{G}}$, $K_{\mathfrak{n}}= 0 = ( \bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}))_{\mathfrak{n}}$. Similarly, if ${\mathfrak{m}}\in {\mathcal{G}}$, then $K_{\mathfrak{m}}= X({\mathfrak{m}})_{\mathfrak{m}}$. Hence we have shown the following. $$\bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} X({\mathfrak{m}}) = K$$ Since $K_{\mathfrak{m}}= X({\mathfrak{m}})_{\mathfrak{m}}$, it only remains to see that $X({\mathfrak{m}}) \cong X({\mathfrak{m}})_{\mathfrak{m}}$, which follows from Remark \[R:tors-facts\]. \[C:torsion-decomposes\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Then for every ${\mathcal{G}}$-torsion module $M$, the following isomorphism holds. $$M \cong \bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} M_{\mathfrak{m}}= \bigoplus_{\substack{{\mathfrak{m}}\in {\operatorname{Max}}R}} M_{\mathfrak{m}}$$ Furthermore, it follows that for every $J \in {\mathcal{G}}$, $J$ is contained only in finitely many maximal ideals of $R$. For the first isomorphism, recall that if an $R$-module $M$ is ${\mathcal{G}}$-torsion, then $M \cong {\operatorname{Tor}}^R_1(M, K)$. Also, note that in this case, $M_{\mathfrak{m}}\cong {\operatorname{Tor}}^R_1(M, K)_{\mathfrak{m}}\cong {\operatorname{Tor}}^{R_{\mathfrak{m}}}_1(M_{\mathfrak{m}}, K_{\mathfrak{m}}) \cong {\operatorname{Tor}}^{R}_1(M, K_{\mathfrak{m}})$. Hence we have the following isomorphisms. $$M \cong {\operatorname{Tor}}^R_1(M, K) \cong {\operatorname{Tor}}^R_1(M, \bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} K_{\mathfrak{m}}) \cong \bigoplus_{\substack{{\mathfrak{m}}\in {\mathcal{G}}}} {\operatorname{Tor}}^R_1(M, K_{\mathfrak{m}}) \cong \bigoplus_{\substack{{\mathfrak{m}}\in {\mathcal{G}}}} M_{\mathfrak{m}}$$ The fact that $$\bigoplus_{\substack{ {\mathfrak{m}}\in {\mathcal{G}}}} M_{\mathfrak{m}}= \bigoplus_{\substack{{\mathfrak{m}}\in {\operatorname{Max}}R}} M_{\mathfrak{m}}$$ follows from Remark \[R:tors-facts\] (3).\ For the final statement of the proposition, one only has to replace $M$ with the ${\mathcal{G}}$-torsion module $R/J$ where $J \in {\mathcal{G}}$. Hence as $R/J$ is cyclic, it cannot be isomorphic to an infinite direct sum. Therefore, $(R/J)_{\mathfrak{m}}$ is non-zero only for finitely many maximal ideals and the conclusion follows. We are now in the position to show the main results of this section. \[T:rjperfect\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. Then $R/J$ is a perfect ring for every $J \in {\mathcal{G}}$. By Corollary \[C:torsion-decomposes\], every $R/J$ is a finite product of local rings $R_{\mathfrak{m}}/J_{\mathfrak{m}}$. Additionally as $({\mathcal{D}}_{\mathcal{G}})_{\mathfrak{m}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}_{\mathfrak{m}}$ by Lemma \[L:localfacts\] each $R_{\mathfrak{m}}/J_{\mathfrak{m}}$ is a perfect ring by Proposition \[P:localperfect\]. Therefore, by Proposition \[P:perfect\], $R/J$ itself is perfect. \[T:EndK-properfect\] Let $u:R \to U$ be a flat injective ring epimorphism and suppose $ {\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$. Then the topological ring $\mathfrak{R}={\operatorname{End}}(K)$ is pro-perfect. Recall that the topology of $\mathfrak{R}$ is given by the annihilators of finitely generated submodules of $K$, so that $\mathfrak{R}={\operatorname{End}}_R(K)$ is separated and complete in its topology. Let $V$ be an open ideal in the topology of $\mathfrak{R}$. By Proposition \[P:topologies-2\] there is $J\in {\mathcal{G}}$ and a surjective ring homomorphism $R/J\to \mathfrak{R}/V$. By Theorem \[T:rjperfect\] $R/J$ is a perfect ring and thus so are the quotient rings $\mathfrak{R}/V$. ${\mathcal{D}}_{\mathcal{G}}$ is enveloping if and only if $\mathfrak{R}$ is pro-perfect {#S:properfect} ======================================================================================== Suppose that $u:R \to U$ is a commutative flat injective ring epimorphism where ${\operatorname{p.dim}}_R U \leq 1$ and denote $K = U/R$. In this section we show that if the endomorphism ring $\mathfrak{R} = {\operatorname{End}}_R(K)$ is pro-perfect, then ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${\mathrm{Mod}\textrm{-}R}$. So combining with the results in the Section \[S:enveloping\] we obtain that ${\mathcal{D}}_{\mathcal{G}}$ is enveloping if and only if ${\operatorname{p.dim}}U \leq 1$ and $\mathfrak{R}$ is pro-perfect. Recall that if ${\operatorname{p.dim}}U \leq 1$, $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ denotes the $1$-tilting cotorsion pair associated to the $1$-tilting module $U \oplus K$. The following theorem of Positselski is vital for this section. \[T:pos-addK\]([@BP2 Theorem 19.6]) Suppose $R$ is a commutative ring and $u:R \to U$ a flat injective ring epimorphism with ${\operatorname{p.dim}}_R U \leq 1$. Then the topological ring $\mathfrak{R}={\operatorname{End}}(K)$ is pro-perfect if and only if $\varinjlim {\mathrm{Add}}(K) = {\mathrm{Add}}(K)$. A second crucial result that we will use is the following. \[T:Enochs2\] ([@Xu Theorem 2.2.6]) Assume that ${\mathcal{C}}$ is a class of modules closed under direct limits and extensions. If a module $M$ admits a special ${\mathcal{C}}^{\perp_1}$-preenvelope with cokernel in ${\mathcal{C}}$, then $M$ admits a ${\mathcal{C}}^{\perp_1}$-envelope. We now show that if $\mathfrak{R}$ is pro-perfect, then ${\mathrm{Add}}(K)$ does in fact satisfy the conditions of Theorem \[T:Enochs2\]. From Theorem \[T:pos-addK\] ${\mathrm{Add}}(K)$ is closed under direct limits. Moreover, ${\mathrm{Add}}(K)$ is closed under extensions as any short exact sequence $0 \to L \to M \to N \to 0$ with $L, N \in {\mathrm{Add}}(K)$ splits. As the cotorsion pair $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ is complete, every $R$-module $M$ has an injective ${\mathcal{D}}_{\mathcal{G}}$-preenvelope, and as ${\mathcal{D}}_{\mathcal{G}}= K^\perp = ({\mathrm{Add}}(K))^\perp$, $M$ has a $({\mathrm{Add}}(K))^\perp$-preenvelope. It remains to be seen that every $M$ has a special preenvelope $\nu$ such that ${\operatorname{Coker}}\nu \in {\mathrm{Add}}(K)$, which we will now show. \[L:coker-in-AddK\] Suppose $u:R \to U$ is a flat injective ring epimorphism where ${\operatorname{p.dim}}_R U \leq 1$. Let $({\mathcal{A}}, {\mathcal{D}}_{\mathcal{G}})$ be the $1$-tilting cotorsion pair associated to the $1$-tilting module $U \oplus K$. Then every module has a special ${\mathcal{D}}_{\mathcal{G}}$-preenvelope $\nu$ such that ${\operatorname{Coker}}\nu \in {\mathrm{Add}}(K)$. For every cardinal $\alpha$ the short exact sequence $0 \to R^{(\alpha)} \to U^{(\alpha)} \to K^{(\alpha)} \to 0$ is a ${\mathcal{D}}_{\mathcal{G}}$-preenvelope and is of the desired form. Take an $R$-module $M$ and consider the canonical surjection $R^{(\alpha)} \overset{p}\to M \to 0$. Consider the following pushout $Z$ of $M \gets R^{(\alpha)} \to U^{(\alpha)}$. $$\xymatrix{ &0 \ar[d]& 0 \ar[d]& & \\ &\ker p \ar[d] \ar@{=}[r]& \ker p \ar[d] & & \\ 0 \ar[r]&R^{(\alpha)} \ar[r] \ar[d]^p& U^{(\alpha)} \ar[r] \ar[d]& K^{(\alpha)} \ar@{=}[d] \ar[r] & 0\\ 0 \ar[r]&M\ar[r] \ar[d] & Z \ar[r] \ar[d]& K^{(\alpha)} \ar[r] & 0\\ &0&0 &&}$$ The module $Z$ is in ${\operatorname{Gen}}U={\mathcal{D}}_{\mathcal{G}}$, and so the bottom row of the above diagram is a ${\mathcal{D}}_{\mathcal{G}}$-preenvelope of $M$ of the desired form.\ The following theorem follows easily from the above discussion. \[T:divenv\] Suppose $u:R \to U$ is a flat injective ring epimorphism with ${\operatorname{p.dim}}_R U \leq 1$. If the topological ring $\mathfrak{R}$ is pro-perfect, then ${\mathcal{D}}_{\mathcal{G}}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$. From Theorem \[T:pos-addK\] and Lemma \[L:coker-in-AddK\], ${\mathrm{Add}}(K)$ does satisfy the conditions of Theorem \[T:Enochs2\]. Thus the conclusion follows, since ${\mathcal{D}}_{\mathcal{G}}={\mathrm{Add}}(K)^\perp$. Finally combining the above theorem with the results in Section \[S:tilting-enveloping\] and Section \[S:enveloping\] we obtain the two main results of this paper. \[T:characterisation\] Suppose $u:R \to U$ is a commutative flat injective ring epimorphism, ${\mathcal{G}}$ the associated Gabriel topology and $\mathfrak R$ the topological ring ${\operatorname{End}}_R(K)$. The following are equivalent. 1. ${\mathcal{D}}_{\mathcal{G}}$ is enveloping. 2. ${\operatorname{p.dim}}U \leq 1$ and $R/J$ is a perfect ring for every $J\in {\mathcal{G}}$. 3. ${\operatorname{p.dim}}U \leq 1$ and $\mathfrak{R}$ is pro-perfect. In particular, if ${\mathcal{D}}_{\mathcal{G}}$ is enveloping then the class ${\mathrm{Add}}(K)$ is closed under direct limits. (1)$\Rightarrow$(2) Follows by Proposition \[P:pd1\] and Theorem \[T:rjperfect\]. (2)$\Rightarrow$(3) Follows from Lemma \[L:discreteiso\] and Proposition \[P:topologies-2\]. (3)$\Rightarrow$(1) Follows from Theorem \[T:divenv\]. \[T: tilting-envelope\] Assume that $T$ is a $1$-tilting module over a commutative ring $R$ such that the class $T^\perp$ is enveloping. Then there is a flat injective ring epimorphism $u\colon R\to U$ such that $U\oplus U/R$ is equivalent to $T$ and the topological ring $\mathfrak R={\operatorname{End}}(U/R)$ is a pro-perfect ring. Moreover, if ${\mathcal{G}}$ is the associated Gabriel topology, then $R/J$ is perfect ring for every $J\in {\mathcal{G}}$. By Proposition \[P:tilting-env\], the Gabriel topology ${\mathcal{G}}$ associated to $T^\perp$ arises from a perfect localisation. Moreover, $\psi:R \to R_{\mathcal{G}}$ is injective so by setting $U=R_{\mathcal{G}}$ we can apply Theorem \[T:characterisation\] to conclude. The case of a non-injective flat ring epimorphism {#S:notmono} ================================================= Now we extend the results of the previous section to the case of a non-injective flat ring epimorphism $u\colon R \to U$ with $K={\operatorname{Coker}}u$. As before, the Gabriel topology ${\mathcal{G}}_u=\{J\leq R\mid JU=U\}$ associated to $u$ is finitely generated and the class $${\mathcal{D}}_{{\mathcal{G}}_u}=\{M\in R{\textrm{-}\mathrm{Mod}}\mid JM=M \text{ for every } J\in{\mathcal{G}}_u\}$$ of ${\mathcal{G}}_u$-divisible modules is a torsion class. Moreover, by [@AHHr] it is a silting class, that is there is a silting module $T$ such that ${\operatorname{Gen}}T={\mathcal{D}}_{{\mathcal{G}}_u}$. The ideal $I$ will denote the kernel of $u$ and $\bar R$ the ring $R/I$ so that there is a flat injective ring epimorphism $\bar u\colon \bar R\to U$. To the is the associated Gabriel topology ${\mathcal{G}}_{\bar u}=\{L/I\leq \bar R\mid LU=U\}$ in $\bar{R}$ and the following class of $\bar{R}$-modules. $${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}=\{M\in {\bar{R}\textrm{-}\mathrm{Mod}}\mid (L/I)M=M, \text{ for every } L/I\in{\mathcal{G}}_{\bar u}\}.$$ We first note the following \[L: I-annih-div\] Every module in ${\mathcal{D}}_{{\mathcal{G}}_u}$ is annihilated by $I$, thus ${\mathcal{D}}_{{\mathcal{G}}_u}={\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$. Note that ${\operatorname{Ker}}u=I$ is the ${\mathcal{G}}_u$-torsion submodule of $R$. Hence for every $b\in I$ there is $J\in {\mathcal{G}}_u$ such that $bJ=0$. Let $M\in {\mathcal{D}}_{{\mathcal{G}}_u}$, then $bM =bJM=0$, thus $IM=0$. We conclude that ${\mathcal{D}}_{{\mathcal{G}}_u}$ can be considered a class in ${\bar{R}\textrm{-}\mathrm{Mod}}$ and coincides with ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$. \[P:same-env\] The class ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$ if and only if ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$ is enveloping in ${\bar{R}\textrm{-}\mathrm{Mod}}$. Assume that ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$ and let $\bar M\in {\bar{R}\textrm{-}\mathrm{Mod}}$. Consider a ${\mathcal{D}}_{{\mathcal{G}}_u}$-envelope $\bar{\psi}\colon \bar M\to D$ in ${R\textrm{-}\mathrm{Mod}}$. Since $R\to R/I$ is a ring epimorphism and $D$ is annihilated by $I$ by Lemma \[L: I-annih-div\], it is immediate to conclude that $\bar{\psi}$ is also a ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$-envelope of $\bar M$. Conversely, assume that ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$ is enveloping in ${\bar{R}\textrm{-}\mathrm{Mod}}$. Take $M\in{R\textrm{-}\mathrm{Mod}}$ and let $\bar{\psi}\colon M/IM\to D$ be a ${\mathcal{D}}_{{\mathcal{G}}_{\bar u}}$-envelope of $ M/IM$ in ${\bar{R}\textrm{-}\mathrm{Mod}}$. Let $\pi\colon M\to M/IM$ be the canonical projection. We claim that $\psi=\bar{\psi} \pi$ is a ${\mathcal{D}}_{{\mathcal{G}}_u}$-envelope of $M$ in ${R\textrm{-}\mathrm{Mod}}$. Indeed, if $f\colon D\to D$ satisfies $f\psi=\psi$, then $f\bar{\psi}\pi =\bar{\psi} \pi$. As $\pi$ is a surjection, $f \bar{\psi} = \bar{\psi}$ and so $f$ is an automorphism of $D$. Note that ${\operatorname{End}}_R(K)$ is the same as ${\operatorname{End}}_{\bar R}(K)$ both as a ring and as a topological ring. It will be still denoted by $\mathfrak R$. Thus if ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping in ${R\textrm{-}\mathrm{Mod}}$ we can apply the results of the previous sections to the ring $\bar R$, in particular Theorem \[T:characterisation\]. \[T:non-mono\] Let $u\colon R\to U$ be a commutative flat ring epimorphism with kernel $I$. Let ${\mathcal{G}}$ be the associated Gabriel topology and $\mathfrak R$ the topological ring ${\operatorname{End}}_R(K)$. The following are equivalent. 1. ${\mathcal{D}}_{{\mathcal{G}}_u}$ is enveloping. 2. ${\operatorname{p.dim}}_{\bar{R}} U \leq 1$ and $R/L$ is a perfect ring for every $L\in {\mathcal{G}}$ such that $L\supseteq I$. 3. ${\operatorname{p.dim}}_{\bar R} U \leq 1$ and $\mathfrak{R}$ is pro-perfect. In particular, $U\oplus K$ is a $1$-tilting module over the ring $\bar R$ and since ${\operatorname{Gen}}U$ is contained in ${\bar{R}\textrm{-}\mathrm{Mod}}$, ${\mathcal{D}}_{{\mathcal{G}}_u}={\operatorname{Gen}}U$. As already noted, results from [@AHHr] imply that ${\operatorname{Gen}}U$ is a silting class. Since we have that $U\oplus K$ is a $1$-tilting module in ${\bar{R}\textrm{-}\mathrm{Mod}}$ inducing the silting class ${\operatorname{Gen}}U$, it is natural to ask the following question. Is $U\oplus K$ a silting module in ${R\textrm{-}\mathrm{Mod}}$? [AHH17]{} Lidia Angeleri Hügel and Michal Hrbek. Silting modules over commutative rings. , (13):4131–4151, 2017. Hyman Bass. Finitistic dimension and a homological generalization of semi-primary rings. , 95:466–488, 1960. Silvana Bazzoni. Divisible envelopes, p-1 covers and weak-injective modules. , 9(4):531–542, 2010. Silvana Bazzoni and Leonid Positselski. Contramodules over pro-perfect topological rings, the covering property in categorical tilting theory, and homological ring epimorphisms. Preprint, arXiv:1808.00937, 2018. Paul C. Eklof and Jan Trlifaj. How to make [E]{}xt vanish. , 33(1):41–51, 2001. Werner Geigle and Helmut Lenzing. Perpendicular categories with applications to representations and sheaves. , 144(2):273–343, 1991. R[ü]{}diger G[ö]{}bel and Jan Trlifaj. , volume 41 of [*de Gruyter Expositions in Mathematics*]{}. Walter de Gruyter GmbH & Co. KG, Berlin, extended edition, 2012. Approximations. Michal Hrbek. One-tilting classes and modules over commutative rings. , 462:1–22, 2016. Lidia Angeleri H[ü]{}gel and Javier S[á]{}nchez. Tilting modules arising from ring epimorphisms. , 14(2):217–246, 2011. Leonid Positselski. Flat ring epimorphisms of countable type. Preprint, arXiv:1808.00937, 2018. Leonid Positselski. Triangulated [M]{}atlis equivalence. , 17(4):1850067, 44, 2018. Bo Stenstr[ö]{}m. . Springer-Verlag, New York, 1975. Die Grundlehren der Mathematischen Wissenschaften, Band 217, An introduction to methods of ring theory. Jinzhong Xu. , volume 1634 of [*Lecture Notes in Mathematics*]{}. Springer-Verlag, Berlin, 1996.

--- author: - 'Chee Sheng Fong$^{1}$' - 'Hisakazu Minakata$^{2,3}$' - 'Hiroshi Nunokawa$^{4}$' title: 'Non-unitary evolution of neutrinos in matter and the leptonic unitarity test\' --- IFT-UAM/CSIC-17-117 Introduction {#sec:introduction} ============ Studies of neutrino oscillation entered into a “matured phase” after the structure of the three-flavor lepton mixing [@Maki:1962mu] is elucidated. The long-lasted discovery phase of neutrino oscillation has been unambiguously concluded by the Super-Kamiokande (Super-K) atmospheric neutrino observation which discovered neutrino oscillation and hence neutrino mass [@Fukuda:1998mi]. It was followed by the KamLAND reactor and the solar neutrino experiments which uncovered the three-flavor nature of the mixing by observing oscillation and/or nonadiabatic flavor conversion of neutrinos in matter [@Mikheev:1986gs; @Wolfenstein:1977ue] in the 1-2 sector [@Eguchi:2002dm; @Ahmad:2002jz].[^1] The last step of understanding the three-flavor structure of neutrino oscillation was carried out by the reactor [@An:2016ses; @RENO:2015ksa; @Schoppmann:2016iww] and the accelerator [@Abe:2017vif; @Adamson:2016tbq] measurement of $\theta_{13}$. It lefts only the two remaining unknowns in the standard three-flavor mixing paradigm, that is, measurement of CP violating phase and determination of neutrino mass ordering.[^2] The completion of the theory of the three-flavor neutrino mixing, however, necessitates the paradigm test. A well-known example of such efforts is to verify unitarity of the quark CKM matrix [@Olive:2016xmw]. We have argued in ref. [@Fong:2016yyh] that we may need a different strategy to test leptonic unitarity. That is, first prepare a generic framework which describes unitarity violation at certain energy scale, and then confront it to experimental data. We contrasted the two typical alternatives, unitarity violation by new physics at high ($E \gg m_{W}$) and low ($E \ll m_{W}$) energy scales, which are dubbed as high-scale and low-scale unitarity violation, respectively.[^3] They differ in certain characteristic features, such as absence (low-scale) or presence (high-scale) of violation of flavor universality and zero-distance flavor transition. In a previous paper, we have proposed a model-independent framework for testing leptonic unitarity assuming low-scale unitarity violation [@Fong:2016yyh]. Our framework is based on the three active plus $N$ sterile lepton (called neutrino) system, which is unitary in the whole $(3+N)$ dimensional state space but restriction to observables in the active neutrino subspace renders the theory non-unitary in that subspace. It is referred to as the “$(3+N)$ space unitary model”. We have shown that the restriction of sterile state masses to $0.1 \,\text{eV}^2 \lsim m^2_{J} \lsim 1 \,\text{MeV}^2$ is sufficient to make the observables sterile-sector model independent. That is, the neutrino oscillation probability can be written in such a way that it is independent of details of the sterile mass spectrum and mixing with active neutrinos. The model-independent nature of the framework will be translated into that of the constraints obtained, thereby making leptonic unitarity test more powerful. See ref. [@Parke:2015goa; @Blennow:2016jkn] for a comprehensive analysis of the currently available neutrino data with the active plus sterile framework, and [@Blennow:2016jkn; @Tang:2017khg] for analyses of the future experiments. Can one distinguish between high-scale and low-scale unitarity violation? The answer is in principle yes, given the above difference between them, which can only be done by detecting violation of flavor universality or zero-distance flavor transition. In ref. [@Fong:2016yyh] we have pointed out another way of distinguishing them by observing the probability leaking term in the oscillation probability, which signals existence of energetically accessible sterile states. Unfortunately, this important trace of the hidden sterile sector has neither been mentioned in the context of unitarity test, nor included into the foregoing analyses before our analysis for JUNO [@JUNO]-like setting. In this paper we will give a comprehensive treatment of the $(3+N)$ space unitary model. Our particular emphasis in this paper is twofold: - We formulate a novel perturbative framework for the $(3+N)$ space unitary model, which we call “[*small unitarity-violation perturbation theory*]{}”. It allows us to calculate the oscillation probability in the presence of matter effect comparable in size to the vacuum effect. - We show that the framework can be used in dual modes: It serves (1) as a suitable framework for leptonic unitarity test in neutrino oscillation experiments, and (2) as a hunting tool for unitarity violation effects, which could serve for another way of distinguishing low-scale unitarity violation from high-scale one. It must be remarked that the sterile sector model-independent nature of the $(3+N)$ space unitary model is demonstrated in ref. [@Fong:2016yyh] only in vacuum and in matter to first order in matter perturbation theory. It is the first goal of this paper to formulate the appropriate framework to address this question, and show that the model-independence holds after inclusion of sizeable matter effect. In fact, we will observe that the same condition on the sterile neutrino masses guarantees this property. It may be useful in the application of our framework to some of the ongoing and next generation neutrino oscillation experiments [@Abe:2017vif; @Adamson:2016tbq; @Abe:2017aap; @Collaboration:2011ym; @Abe:2015zbg; @Acciarri:2015uup; @TheIceCube-Gen2:2016cap; @Adrian-Martinez:2016zzs]. The discussion of the second point above, our second goal, will follow once the formulation of perturbation theory is completed. It may be worth to note that there exists difference between low- and high-scale unitarity violation with regard to need for and implication to future neutrino experiments. The scenario of high-scale unitarity violation has been studied extensively in the literature [@Antusch:2006vwa; @FernandezMartinez:2007ms; @Antusch:2009pm; @Antusch:2009gn; @Antusch:2014woa; @Escrihuela:2015wra; @Fernandez-Martinez:2016lgt; @Blennow:2016jkn; @Escrihuela:2016ube].[^4] In this case, due to preserved $SU(2) \times U(1)$ symmetry at high scales, it is conceivable that the constraints from measurements using probes in the charged lepton sector play a dominant role. On the other hand, in the case of low-scale unitarity violation, neutrino oscillation experiments will play key role in constraining unitarity violation. Essence of the present and the previous papers {#sec:essence} ============================================== In this section, we present essence of the present and the previous [@Fong:2016yyh] papers in which we try to construct an adequate formulation to describe unitarity violation at low energies, $E \ll m_{W}$. Unitary 3 active + $N$ sterile neutrino system {#sec:3+N-system} ---------------------------------------------- We have shown in the $(3+N)$ space unitary model that the active neutrino oscillation probability can be written in a sterile sector model-independent way under the constraint on sterile state masses to $0.1\, \text{eV}^2 \lsim m^2_{J} \lsim 1\, \text{MeV}^2$ both in vacuum and to first order in matter effect [@Fong:2016yyh]. The lower limit of the sterile mass range comes from the requirement that fast oscillations due to active-sterile and sterile-sterile mass differences are averaged out due to decoherence. Whereas the upper limit is for sterile neutrinos to take part in reactor neutrino experiments, which may be relaxed to $m^2_{J} \lsim (1 - 10) \,\text{GeV}^2$ for accelerator neutrinos. Then, the obvious question is whether the conclusion remains the same when all order effect of matter is taken into account. We will answer the question in the positive in this paper. Non-unitary evolution of neutrinos in vacuum {#sec:nonunitarity-vacuum} -------------------------------------------- Despite large state space of $(3+N)$ dimensions the resultant expression of the oscillation probability has a simple form under the above stated restriction on the sterile neutrino mass spectrum. In vacuum it has the form $$\begin{aligned} P(\nu_\beta \rightarrow \nu_\alpha) &=& \mathcal{C}_{\alpha \beta} + \left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 - 2 \sum_{j \neq k} \mbox{Re} \left( U_{\alpha j} U_{\beta j}^* U_{\alpha k}^* U_{\beta k} \right) \sin^2 \frac{ ( \Delta_{k} - \Delta_{j} ) x }{ 2 } \nonumber\\ &-& \sum_{j \neq k} \mbox{Im} \left( U_{\alpha j} U_{\beta j}^* U_{\alpha k}^* U_{\beta k} \right) \sin ( \Delta_{k} - \Delta_{j} ) x, \label{P-beta-alpha-ave-vac}\end{aligned}$$ where $$\begin{aligned} \mathcal{C}_{\alpha \beta} \equiv \sum_{J=4}^{3+N} \vert W_{\alpha J} \vert^2 \vert W_{\beta J} \vert^2, \label{Cab} \end{aligned}$$ in the appearance ($\alpha \neq \beta$) as well as in the disappearance channels ($\alpha = \beta$).[^5] The $(3 \times N)$ $W$ matrix elements bridge between active and sterile state space, see eq. (\[U-parametrize\]) for definition. The characteristic features of (\[P-beta-alpha-ave-vac\]) are: 1. The non-unitary matrix $U$ replaces the standard unitary three-flavor mixing matrix often parametrized with Particle Data Group convention $U_{\text{\tiny PDG}}$ [@Olive:2016xmw]. 2. Probability leakage term $\mathcal{C}_{\alpha \beta}$ appears reflecting the nature of low-energy unitarity violation in which the probability can flow out into the sterile state space from active neutrino space. 3. Due to non-unitarity, $\delta_{\alpha \beta}$ term in the unitary case is modified to $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2$. The points 2 and 3 above are important ones and the clarifying remarks about them are in order: - Presence or absence of the probability leakage term $\mathcal{C}_{\alpha \beta}$ distinguishes between low-energy and high-energy unitarity violation [@Fong:2016yyh]. Unfortunately, $\mathcal{C}_{\alpha \beta}$ may be small because it is of fourth order in $W$. - Difference in normalization factor, the second term in (\[P-beta-alpha-ave-vac\]), between unitary and non-unitary cases is of order $\sim W^4$ ($\sim W^2$) in the appearance (disappearance) channels. To understand the latter point, we notice that unitarity in the $(3+N)$ space unitary model can be written as $$\begin{aligned} \delta_{\alpha \beta} = \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} + \sum_{J=4}^{N+3} W_{\alpha J} W^{*}_{\beta J}. \label{unitarity0}\end{aligned}$$ Then, $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 = \left| \sum_{J=4}^{N+3} W_{\alpha J} W^{*}_{\beta J} \right|^2$ in the appearance channel ($\alpha \neq \beta$), and $\left( \sum_{j=1}^{3} \vert U_{\alpha j} \vert^2 \right)^2 = \left( 1 - \sum_{J=4}^{N+3} \vert W_{\alpha J} \vert^2 \right)^2 = 1 - \mathcal{O} (W^2)$ in the disappearance channel ($\alpha = \beta$), which justifies the above statement. We emphasize, therefore, that the probability leaking term $\mathcal{C}_{\alpha \beta}$ and the another constant term $\left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2$ in the oscillation probabilities are the same order, $\mathcal{O} (W^4)$, in the appearance channels. Hence, we do not see any good reasons why the former can be ignored, as was done in the existing literatures. It is also worth to note that $\mathcal{O} (W^2)$ difference in normalization in the disappearance channel would make detection of unitarity violation more feasible. It is one of the reasons for high sensitivity to unitarity violation that could be reached in disappearance measurement in the JUNO-like setting [@Fong:2016yyh]. Non-unitary evolution of neutrinos in matter {#sec:nonunitarity-matter} -------------------------------------------- We have also examined in ref. [@Fong:2016yyh] the question of whether inclusion of the matter effect alters the above features of non-unitary evolution of neutrino states in vacuum. We have found that as far as we remain in the region of unitarity violating element $\vert W \vert \simeq 0.1$[^6] or larger, the matter effect does not alter the above features of the oscillation probability in (\[P-beta-alpha-ave-vac\]) under the same restriction on sterile masses. Notice that $\vert W \vert \simeq 0.1$ implies that the unitarity violating effect in the probability is of the order of $\vert W \vert^4 \sim 10^{-4}$, except for the $\mathcal{O} (W^2)$ difference in normalization constant in the disappearance probability. It is practically the limit of order of magnitude that can be explored by the next generation neutrino oscillation experiments. We must note, however, that this conclusion is based on our treatment to first order in matter perturbation theory. Hence, the statement is not a very convincing one. Given the fact that setup of some of the next generation long-baseline (LBL) experiments require consideration of matter effect of comparable size with the vacuum effect, it is clear that a better treatment is necessary to understand the influence of the matter effect in the $(3+N)$ model. In particular, the conditions that allows us to make the active space oscillation probabilities sterile-sector model independent have to be worked out. This task will be carried out in section \[sec:formulation\]. As an outcome of the honest computation using the small unitarity-violation perturbation theory in the $(3+N)$ space unitary model to order $W^4$, we postulate the following theorem: [**Uniqueness theorem**]{} - All the matter dependent perturbative corrections in $W$ in the oscillation probability vanish or can be ignored after averaging over the fast oscillations and using the suppression due to the large energy denominators, leaving only the probability leakage term $\mathcal{C}_{\alpha \beta}$, the first term in (\[P-beta-alpha-ave-vac\]). It must be remarked here that unitarity violation effects which are hidden in non-unitary active space mixing matrix $U$ produces zeroth- to higher order effects of $W$. The above theorem is only about the terms generated by explicit perturbative corrections in $W$.[^7] As a result of the explicit computation, we show that the oscillation probability in matter between active flavor neutrinos in the $(3+N)$ space unitary model to fourth order in $W$ in our small unitarity-violation perturbation theory can be written as $$\begin{aligned} P(\nu_\beta \rightarrow \nu_\alpha) &=& \mathcal{C}_{\alpha \beta} + \left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 \nonumber\\ &-& 2 \sum_{j \neq k} \mbox{Re} \left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right] \sin^2 \frac{ ( h_{k} - h_{j} ) x }{ 2 } \nonumber\\ &-& \sum_{j \neq k} \mbox{Im} \left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right] \sin ( h_{k} - h_{j} ) x, \label{P-beta-alpha-final}\end{aligned}$$ where $h_{i}$ $(i=1,2,3)$ denote the energy eigenvalues of zeroth-order states of active neutrinos in matter, and $X$ is the unitary matrix which diagonalize the zeroth-order Hamiltonian used to formulate our perturbation theory. $\mathcal{C}_{\alpha \beta}$ is given in (\[Cab\]). The expression is valid under the restriction on sterile neutrino masses $0.1\, \text{eV}^2 \lsim m^2_{J} \lsim 1\, \text{MeV}^2$ for $\vert W \vert^4 \gsim 10^{-4}$. For the restriction needed on the sterile state masses for smaller $W$, see section \[sec:probability-2nd\]. Notice that the normalization term, the second term both in (\[P-beta-alpha-ave-vac\]) and (\[P-beta-alpha-final\]), comes out from the contribution of zeroth-order Hamiltonian which contains all orders effect of the matter potential. It is true despite the fact that it is a vacuum effect, which indeed exists in vacuum.[^8] The expression (\[P-beta-alpha-final\]) is a very transparent result in the sense that (1) the vacuum non-unitary mixing matrix $U$ is “dressed” in a simple way by the matter effect represented by $X$, the unitary matrix which diagonalizes the zeroth-order Hamiltonian, and (2) the probability leaking term and the normalization term stay as they are in vacuum. Enhanced higher-order $W$ corrections and nature of unitarity violation {#sec:higher-order-corrections} ------------------------------------------------------------------------ We will point out that, despite our above theorem, there are regions of neutrino energy and baseline that condition for suppression due to the large energy denominators is not fully effective. In this region outside validity of the theorem we show that second order correction terms in $W$, together with the leaking term $\mathcal{C}_{\alpha \beta}$, may not be totally negligible, and it could be detectable. If it were the case, it could distinguish low-scale unitarity violation from high-scale one. Small unitarity-violation perturbation theory of neutrino oscillation in matter {#sec:formulation} ================================================================================= We formulate a perturbation theory of the $(3+N)$ state unitary model using an expansion parameter of matrix elements of $W$ signifying unitarity violation effects, assuming it small. In the main text we mostly confine ourselves to the formulas to second order in $W$, but include fourth order terms whenever it is necessary. Our formulation of the perturbative framework in this section will be done aiming at constructing a model-independent framework for leptonic unitarity test. Usage of the same probability formula as a hunting tool of unitarity violation and discriminator between low-scale and high-scale unitarity violation will be discussed in section \[sec:UV-low-high-E\]. For simplicity, we take the uniform number density approximation for electrons and neutrons in matter. However, extension to the varying density case is, in principle, straightforward as far as adiabaticity holds. 3 active plus $N$ sterile neutrino system in the flavor basis {#sec:flavor-basis} ------------------------------------------------------------- The $S$ matrix describes possible flavor changes after traversing a distance $x$ $$\begin{aligned} \nu_{\alpha} (x) = S_{\alpha \beta} \nu_{\beta} (0), \label{def-S}\end{aligned}$$ and the oscillation probability is given by $$\begin{aligned} P(\nu_{\beta} \rightarrow \nu_{\alpha}; x)= \vert S_{\alpha \beta} \vert^2. \label{def-P}\end{aligned}$$ The neutrino evolution in flavor basis in the $(3+N)$ space unitary model is governed by the Schrödinger equation $$\begin{aligned} i \frac{d}{dx} \nu = H \nu. \label{evolution}\end{aligned}$$ Given the flavor basis Hamiltonian $H$, the $S$ matrix is given by $$\begin{aligned} S = T \text{exp} \left[ -i \int^{x}_{0} dx^{\prime} H(x^{\prime}) \right], \label{S-matrix-def}\end{aligned}$$ where $T$ symbol indicates the “time ordering” (in fact “space ordering” here). The right-hand side of (\[evolution\]) may be written as $e^{-i H x}$ for the case of constant matter density. The flavor basis Hamiltonian $H$ is $(3+N) \times (3+N)$ matrix: $$\begin{aligned} H = {\bf U} \left[ \begin{array}{cccccc} \Delta_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & \Delta_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \Delta_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \Delta_{4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & 0 \\ 0 & 0 & 0 & 0 & 0 & \Delta_{3+N} \\ \end{array} \right] {\bf U}^{\dagger} + \left[ \begin{array}{cccccc} \Delta_{A} - \Delta_{B} & 0 & 0 & 0 & 0 & 0 \\ 0 & - \Delta_{B} & 0 & 0 & 0 & 0 \\ 0 & 0 & - \Delta_{B} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right], \label{hamiltonian}\end{aligned}$$ where $$\begin{aligned} \Delta_{i} \equiv \frac{ m^2_{i} }{2E} \hspace{4mm} (i = 1,2,3), \hspace{8mm} \Delta_{J} \equiv \frac{ m^2_{J} }{2E} \hspace{4mm} (J = 4, \cdot \cdot \cdot, 3+N). \label{Delta-def}\end{aligned}$$ Here, $m_{i}$ ($m_{J}$) denote the mass of mostly active (sterile) neutrinos and $E$ is the neutrino energy. $\Delta_{A}$ and $\Delta_{B}$ are related to Wolfenstein’s matter potential [@Wolfenstein:1977ue] due to CC and NC reactions, $a$ and $b$, as $$\begin{aligned} \Delta_{A} \equiv \frac{ a }{2E}, \hspace{10mm} \Delta_{B} \equiv \frac{ b }{2E}, \label{Delta-ab-def}\end{aligned}$$ where $$\begin{aligned} a &=& 2 \sqrt{2} G_F N_e E \approx 1.52 \times 10^{-4} \left( \frac{Y_e \rho}{\rm g\,cm^{-3}} \right) \left( \frac{E}{\rm GeV} \right) {\rm eV}^2, \nonumber \\ b &=& \sqrt{2} G_F N_n E = \frac{1}{2} \left( \frac{N_n}{N_e} \right) a. \label{matt-potential}\end{aligned}$$ Here, $G_F$ is the Fermi constant, $N_e$ and $N_n$ are the electron and neutron number densities in matter. $\rho$ and $Y_e$ denote, respectively, the matter density and number of electron per nucleon in matter. In (\[hamiltonian\]), ${\bf U}$ denotes the flavor mixing matrix which relates $(3+N)$ dimensional flavor neutrino states to the vacuum mass eigenstate basis as $\nu_{\zeta} = {\bf U}_{\zeta z} \tilde{\nu}_{z}$, where $\zeta$ runs over active flavor $\alpha = e,\mu,\tau$ and sterile flavor $s = s_1,\cdot \cdot \cdot,s_N$ indices, $z$ runs over mostly active $i=1,2,3$ and mostly sterile mass eigenstate indices $J=4,5,\cdot \cdot \cdot, N+3$. For simplicity, we introduce a compact notation which writes $(3+N) \times (3+N)$ matrix in a form of $2 \times 2$ matrix. By defining the active $3 \times 3$ matter potential matrix $$\begin{aligned} A = \left[ \begin{array}{ccc} \Delta_{A} - \Delta_{B} & 0 & 0 \\ 0 & - \Delta_{B} & 0 \\ 0 & 0 & - \Delta_{B} \\ \end{array} \right] \label{matter-pot2}\end{aligned}$$ the flavor basis Hamiltonian is written as $$\begin{aligned} H = {\bf U} \left[ \begin{array}{cc} {\bf \Delta_{a} } & 0 \\ 0 & {\bf \Delta_{s} } \\ \end{array} \right] {\bf U}^{\dagger} + \left[ \begin{array}{cc} A & 0 \\ 0 & 0 \\ \end{array} \right] \equiv H_{ \text{vac} } + H_{ \text{matt} } \label{flavor-hamiltonian}\end{aligned}$$ where ${\bf \Delta_{a} } = \text{diag} ( \Delta_{1}, \Delta_{2}, \Delta_{3})$ and ${\bf \Delta_{s} } = \text{diag} (\Delta_{4}, \Delta_{5}, \cdot \cdot \cdot, \Delta_{N+3})$. As an application of our framework, we anticipate leptonic unitarity test in the LBL accelerator neutrino experiments which utilize atmospheric-scale neutrino oscillations. Therefore, we assume that the system satisfies the following conditions in formulating our perturbation theory $$\begin{aligned} \frac{ \Delta m^2_{31} L }{2E} \sim \frac{ \Delta m^2_{32} L }{2E} \sim \mathcal{O} (1), \hspace{10mm} \frac{ a L }{2E} \sim \frac{ b L }{2E} \sim \mathcal{O} (1), \label{assumption}\end{aligned}$$ where $L$ denotes baseline, $\Delta m^2_{ji} \equiv m_j^2 - m_i^2$, and $$\begin{aligned} \frac{ a L }{2E} &=& \sqrt{2} G_F N_e L = 0.58 \left(\frac{\rho}{3 \text{g/cm}^3}\right) \left(\frac{L}{1000 \mbox{km}}\right). \label{aL-2E}\end{aligned}$$ They probably ensure that our oscillation probability formulas have applicability to the terrestrial LBL and atmospheric neutrino experiments with baseline up to $\sim 10^4$ km and energies from low to high, up to $E \sim 100$ GeV. More precise discussions on where our formulas are valid will be given in sections \[sec:energy-denominator\] and \[sec:higher-order\]. Vacuum mass eigenstate basis, or tilde basis {#sec:tilde-basis} -------------------------------------------- To formulate perturbative treatment it is convenient to consider the vacuum mass eigenstate basis, the tilde basis, introduced in the previous section $$\begin{aligned} \tilde{\nu}_{z} = ({\bf U}^{\dagger})_{z \zeta} \nu_{\zeta}. \label{tilde-basis}\end{aligned}$$ The tilde basis Hamiltonian is related to the flavor basis one as $$\begin{aligned} \tilde{H} = {\bf U}^{\dagger} H {\bf U}. \label{tilde-hamiltonian}\end{aligned}$$ The explicit form of $\tilde{H}$ is given by $$\begin{aligned} \tilde{H} &=& \tilde{H}_{ \text{vac} } + \tilde{H}_{ \text{matt} } = \left[ \begin{array}{cc} {\bf \Delta_{a} } & 0 \\ 0 & {\bf \Delta_{s} } \\ \end{array} \right] + {\bf U}^{\dagger} \left[ \begin{array}{cc} A & 0 \\ 0 & 0 \\ \end{array} \right] {\bf U}. \label{H-tilde-3+N}\end{aligned}$$ We parametrize the $(3+N) \times (3+N)$ dimensional flavor mixing matrix ${\bf U}$ as $$\begin{aligned} {\bf U} = \left[ \begin{array}{cc} U & W \\ Z & V \\ \end{array} \right]. \label{U-parametrize}\end{aligned}$$ The matrix $U$ and $V$ are $3 \times 3$ and $N \times N$ matrices, respectively, and $W$ and $Z$ have sizes that just fill in the space. In our $(3+N)$ model, unitarity is obeyed in the whole $(3+N)$ state space: $$\begin{aligned} {\bf U} {\bf U}^{\dagger} &=& \left[ \begin{array}{cc} U U^{\dagger} + W W^{\dagger} & U Z^{\dagger} + W V^{\dagger} \\ Z U^{\dagger} + V W^{\dagger} & Z Z^{\dagger} + V V^{\dagger} \\ \end{array} \right] = \left[ \begin{array}{cc} {\bf 1}_{3 \times 3} & 0 \\ 0 & {\bf 1}_{N \times N} \\ \end{array} \right], \nonumber \\ {\bf U}^{\dagger} {\bf U} &=& \left[ \begin{array}{cc} U^{\dagger} U + Z^{\dagger} Z & U^{\dagger} W + Z^{\dagger} V \\ W^{\dagger} U + V^{\dagger} Z & W^{\dagger} W + V^{\dagger} V \\ \end{array} \right] = \left[ \begin{array}{cc} {\bf 1}_{3 \times 3} & 0 \\ 0 & {\bf 1}_{N \times N} \\ \end{array} \right]. \label{unitarity}\end{aligned}$$ Then, the Hamiltonian $\tilde{H}$ in vacuum mass eigenstate basis is given by $$\begin{aligned} \tilde{H} &=& \left[ \begin{array}{cc} {\bf \Delta_{a} } & 0 \\ 0 & {\bf \Delta_{s} } \\ \end{array} \right] + \left[ \begin{array}{cc} U^{\dagger} A U & U^{\dagger} A W \\ W^{\dagger} A U & W^{\dagger} A W \\ \end{array} \right]. \label{tilde-H}\end{aligned}$$ As in vacuum, the neutrino oscillation is governed only by the $U$ and $W$ matrices, and is independent of $Z$ and $V$ matrices. It is natural that $V$ matrix does not show up in physical Hamiltonian matrix because the rotations inside sterile basis does not have any physical meaning, if we observe the system only by the Standard Model interactions. Formulating small unitarity-violation perturbation theory {#sec:UV-perturb} --------------------------------------------------------- We now construct the small unitarity-violation perturbation theory. It is natural to consider the framework in which the tilde-basis Hamiltonian $\tilde{H}$ is decomposed into the un-perturbed and perturbed parts, $\tilde{H}_{0} + \tilde{H}_{1}$, as follows: $$\begin{aligned} \tilde{H}_{0} = \left[ \begin{array}{cc} {\bf \Delta_{a} } + U^{\dagger} A U & 0 \\ 0 & {\bf \Delta_{s} } \\ \end{array} \right], \hspace{10mm} \tilde{H}_{1} = \left[ \begin{array}{cc} 0 & U^{\dagger} A W \\ W^{\dagger} A U & W^{\dagger} A W \\ \end{array} \right]. \label{tilde-H0+H1} \end{aligned}$$ Therefore, what we mean by “expansion by unitarity violation effect” is an expansion by the $W$ matrix elements.[^9] We assume, for simplicity, that all the $W$ matrix elements are small and have the same order $\epsilon_{s}$. Then, $3 \times N$ ($N \times 3$) sub-matrix elements in $\tilde{H}_{ \text{matt} }$ are of order $\epsilon_{s}$, while the pure sterile space $N \times N$ sub-matrix elements are of order $\epsilon_{s}^2$. For simplicity, we often use the expression “expanding to order $W^n$” which means to order $\epsilon_{s}^n$ in this paper. ### Hat basis To formulate perturbation theory with $\tilde{H}_{0}$ and $\tilde{H}_{1}$ given above we transform to a basis in which the un-perturbed part of the Hamiltonian is diagonal, which we call the “hat basis”. Since the $3 \times 3$ sub-matrix ${\bf \Delta_{a} } + U^{\dagger} A U$ in $\tilde{H}_{0}$ is hermitian, it can be diagonalized by the unitary transformation $$\begin{aligned} X^{\dagger} \left( {\bf \Delta_{a} } + U^{\dagger} A U \right) X = \left[ \begin{array}{ccc} h_{1} & 0 & 0 \\ 0 & h_{2} & 0 \\ 0 & 0 & h_{3} \\ \end{array} \right] \equiv {\bf h} \label{H0-diag}\end{aligned}$$ with $X$ being the $3 \times 3$ unitary matrix. Then, $\tilde{H}_{0}$ can be diagonalized by using $$\begin{aligned} {\bf X} \equiv \left[ \begin{array}{cc} X & 0 \\ 0 & 1 \\ \end{array} \right] \label{bfX-def}\end{aligned}$$ as $$\begin{aligned} && {\bf X}^{\dagger} \tilde{H}_{0} {\bf X} = \left[ \begin{array}{cc} X^{\dagger} \left( {\bf \Delta_{a} } + U^{\dagger} A U \right) X & 0 \\ 0 & {\bf \Delta_{s} } \\ \end{array} \right] = \left[ \begin{array}{cc} {\bf h} & 0 \\ 0 & {\bf \Delta_{s} } \\ \end{array} \right] \equiv \hat{H}_{0}, \label{hat-H0}\end{aligned}$$ the zeroth-order Hamiltonian in the hat basis. Since $\hat{H}_{0}$ is diagonal it is easy to compute $e^{ \pm i \hat{H}_{0} x}$: $$\begin{aligned} e^{ \pm i \hat{H}_{0} x} = \left[ \begin{array}{cc} e^{ \pm i {\bf h} x } & 0 \\ 0 & e^{ \pm i {\bf \Delta_{s} x } } \\ \end{array} \right]. \label{exp-H0}\end{aligned}$$ Then, the perturbed Hamiltonian is given by $$\begin{aligned} \hat{H}_{1} &=& {\bf X}^{\dagger} \tilde{H}_{1} {\bf X} = \left[ \begin{array}{cc} 0 & (UX)^{\dagger} A W \\ W^{\dagger} A (UX) & W^{\dagger} A W \\ \end{array} \right]. \label{hat-H1}\end{aligned}$$ The eigenvalues of $\tilde{H}_{0}$ is therefore $h_{1}$, $h_{2}$, $h_{3}$, and $\Delta_{J}$ ($J=4, \cdot \cdot \cdot, 3+N$). Therefore, the sterile neutrino masses are affected neither by the active states nor the matter potential in our zeroth-order unperturbed basis. It must be a good approximation because we have assumed that the sterile masses are much heavier than the active ones, and we are interested in the energy region $a \sim \Delta m^2_{31}$. To do real calculations of the $S$ matrix elements we must solve the zeroth order Hamiltonian $\tilde{H}_{0}$. This task will be carried out in section \[sec:exact-solution-zeroth\], in which we derive explicit expressions of the eigenvalues $h_{i}$ and the unitary matrix $X$. Now, we formulate perturbation theory with the hat basis Hamiltonian, $\hat{H}_{0}$ in (\[hat-H0\]) and $\hat{H}_{1}$ in (\[hat-H1\]) after a clarifying note in the next subsection. ### The relationship between quantities in various bases So far we have introduced the tilde- and the hat-basis: $$\begin{aligned} \tilde{H} = {\bf U}^{\dagger} H {\bf U}, \hspace{10mm} \hat{H} = {\bf X}^{\dagger} \tilde{H} {\bf X}, \label{tilde-hat-relation}\end{aligned}$$ where ${\bf X}$ is given by eq. (\[bfX-def\]). Therefore, $$\begin{aligned} \hat{H} = \left( {\bf U} {\bf X} \right)^{\dagger} H \left( {\bf U} {\bf X} \right). \label{flavor-hat-relation}\end{aligned}$$ Or $$\begin{aligned} H = \left( {\bf U} {\bf X} \right) \hat{H} \left( {\bf U} {\bf X} \right)^{\dagger}, \hspace{10mm} S = \left( {\bf U} {\bf X} \right) \hat{S} \left( {\bf U} {\bf X} \right)^{\dagger}. \label{flavor-hat-relation2}\end{aligned}$$ Notice that both ${\bf U}$ and ${\bf X}$ are unitary, and hence ${\bf U} {\bf X}$ is unitary too. The relationship between wave functions of various basis are given by $$\begin{aligned} \hat{\nu}_{y} &=& {\bf X}^{\dagger}_{y z} \tilde{\nu}_{z} = \left( {\bf U} {\bf X} \right)^{\dagger}_{y \zeta} \nu_{\zeta}, \nonumber \\ \nu_{\zeta} &=& \left( {\bf U} {\bf X} \right)_{\zeta y} \hat{\nu}_{y}. \label{hat-basis}\end{aligned}$$ where $y$ denote the hat-basis indices. Using the explicit parametrization of the ${\bf U}$ matrix we have $$\begin{aligned} {\bf U} {\bf X} &=& \left[ \begin{array}{cc} U & W \\ Z & V \\ \end{array} \right] \left[ \begin{array}{cc} X & 0 \\ 0 & 1 \\ \end{array} \right] = \left[ \begin{array}{cc} U X & W \\ Z X & V \\ \end{array} \right], \nonumber \\ \left( {\bf U} {\bf X} \right)^{\dagger} &=& \left[ \begin{array}{cc} (UX)^{\dagger} & \left( Z X \right)^{\dagger} \\ W^{\dagger} & V^{\dagger} \\ \end{array} \right]. \label{UX-parametrize}\end{aligned}$$ It may be helpful for our discussions later to understand the relationship between $S$ and $\hat{S}$ matrix elements. For this purpose, we denote them in the block form $$\begin{aligned} S = \left[ \begin{array}{cc} S_{aa} & S_{aS} \\ S_{Sa} & S_{SS} \\ \end{array} \right], \hspace{10mm} \hat{S} = \left[ \begin{array}{cc} \hat{S}_{aa} & \hat{S}_{aS} \\ \hat{S}_{Sa} & \hat{S}_{SS} \\ \end{array} \right], \label{S-hatS}\end{aligned}$$ where the subscripts $a$ and $S$ indicate that they act (for the right index) to the active or the sterile subspaces. Notice that $S_{aa}$ and $S_{aS}$, for example, are $3 \times 3$ and $3 \times N$ matrices, respectively. Then, the relationship between $S$ and $\hat{S}$ matrix elements can be written explicitly as $$\begin{aligned} S_{aa} &=& (U X) \hat{S}_{aa} (UX)^{\dagger} + (U X) \hat{S}_{aS} W^{\dagger} + W \hat{S}_{Sa} (UX)^{\dagger} + W \hat{S}_{SS} W^{\dagger}, \nonumber \\ S_{aS} &=& (U X) \hat{S}_{aa} \left( Z X \right)^{\dagger} + (U X) \hat{S}_{aS} V^{\dagger} + W \hat{S}_{Sa} \left( Z X \right)^{\dagger} + W \hat{S}_{SS} V^{\dagger}, \nonumber \\ S_{Sa} &=& \left( Z X \right) \hat{S}_{aa} (U X)^{\dagger} + \left( Z X \right) \hat{S}_{aS} W^{\dagger} + V \hat{S}_{Sa} (U X)^{\dagger} + V \hat{S}_{SS} W^{\dagger}, \nonumber \\ S_{SS} &=& \left( Z X \right) \hat{S}_{aa} \left( Z X \right)^{\dagger} + \left( Z X \right) \hat{S}_{aS} V^{\dagger} + V \hat{S}_{Sa} \left( Z X \right)^{\dagger} + V \hat{S}_{SS} V^{\dagger}. \label{S-Shat-relation}\end{aligned}$$ ### Computation of $\hat{S}$ matrix elements {#sec:hatS-matrix} To calculate $\hat {S} (x) = \exp \left[ -i \int^{x}_{0} dx \hat{H} (x) \right] $ we define $\Omega(x)$ as $$\begin{aligned} \Omega(x) = e^{i \hat{H}_{0} x} \hat{S} (x). \label{def-omega}\end{aligned}$$ $\Omega(x)$ obeys the evolution equation $$\begin{aligned} i \frac{d}{dx} \Omega(x) = H_{1} \Omega(x), \label{omega-evolution}\end{aligned}$$ where $$\begin{aligned} H_{1} \equiv e^{i \hat{H}_{0} x} \hat{H}_{1} e^{-i \hat{H}_{0} x}. \label{def-H1}\end{aligned}$$ Then, $\Omega(x)$ can be computed perturbatively as $$\begin{aligned} \Omega(x) &=& 1 + (-i) \int^{x}_{0} dx' H_{1} (x') + (-i)^2 \int^{x}_{0} dx' H_{1} (x') \int^{x'}_{0} dx'' H_{1} (x'') \nonumber \\ &+& (-i)^3 \int^{x}_{0} dx' H_{1} (x') \int^{x'}_{0} dx'' H_{1} (x'') \int^{x''}_{0} dx''' H_{1} (x''') \nonumber \\ &+& (-i)^4 \int^{x}_{0} dx' H_{1} (x') \int^{x'}_{0} dx'' H_{1} (x'') \int^{x''}_{0} dx''' H_{1} (x''') \int^{x'''}_{0} dx'''' H_{1} (x'''') + \cdot \cdot \cdot, \nonumber \\ \label{Omega-expand}\end{aligned}$$ where the “space-ordered” form in (\[Omega-expand\]) is essential because of the highly nontrivial spatial dependence in $H_{1}$. Upon obtaining $\Omega(x)$, $\hat{S}$ matrix can be obtained as $$\begin{aligned} \hat{S} (x) = e^{- i \hat{H}_{0} x} \Omega(x). \label{hatS-Omega}\end{aligned}$$ By knowing $\hat{S}$ matrix elements, the $S$ matrix is obtained by using (\[flavor-hat-relation2\]). The perturbing Hamiltonian $H_{1}$ defined in (\[def-H1\]) has a structure $$\begin{aligned} H_{1} = \left[ \begin{array}{cc} 0 & e^{ i {\bf h} x} (UX)^{\dagger} A W e^{ - i {\bf \Delta_{s} } x} \\ e^{ i {\bf \Delta_{s} } x} W^{\dagger} A (UX) e^{ - i {\bf h} x} & e^{ i {\bf \Delta_{s} } x} W^{\dagger} A W e^{ - i {\bf \Delta_{s} } x} \\ \end{array} \right]. \label{H1-matrix}\end{aligned}$$ That is, $(H_{1})_{i j} = 0$ in the whole active neutrino subspace. The non-vanishing elements of $H_{1}$ are as follows: $$\begin{aligned} (H_{1})_{i J} &=& e^{- i ( \Delta_{J} - h_{i} ) x} \left\{ (UX)^{\dagger} A W \right\}_{i J}, \nonumber \\ (H_{1})_{J i} &=& e^{ - i ( h_{i} - \Delta_{J} ) x} \left\{ W ^{\dagger} A (UX) \right\}_{J i}, \nonumber \\ (H_{1})_{J K} &=& e^{- i ( \Delta_{K} - \Delta_{J} ) x} \left\{ W ^{\dagger} A W \right\}_{J K}. \label{H1-elements}\end{aligned}$$ Inserting eq. (\[H1-elements\]) into (\[Omega-expand\]), we can compute all the $\Omega$ matrix elements. The simplest ones in first order in $H_{1}$, the second term in (\[Omega-expand\]), are given by $$\begin{aligned} \Omega_{i j} [1] &=& 0, \nonumber \\ \Omega_{i J} [1] &=& \frac{e^{- i ( \Delta_{J} - h_{i} ) x} - 1 }{ ( \Delta_{J} - h_{i} ) } \left\{ (UX)^{\dagger} A W \right\}_{i J}, \nonumber \\ \Omega_{J i} [1] &=& - \frac{e^{ i ( \Delta_{J} - h_{i} ) x} - 1 }{ ( \Delta_{J} - h_{i} ) } \left\{ W ^{\dagger} A (UX) \right\}_{J i}, \nonumber \\ \Omega_{J K} \vert_{J \neq K} [1] &=& \frac{e^{- i ( \Delta_{K} - \Delta_{J} ) x} - 1 }{ ( \Delta_{K} - \Delta_{J} ) } \left\{ W ^{\dagger} A W \right\}_{J K}, \nonumber \\ \Omega_{J J} [1] &=& (-i x) \left\{ W ^{\dagger} A W \right\}_{J J}, \label{Omega-1st-order}\end{aligned}$$ which serve as a building block of the perturbation series because of the structure in (\[Omega-expand\]). The notation “\[1\]” implies that the terms come from first order perturbation with $H_{1}$. For more about notations, see appendix \[sec:hatS-elements\]. We need to compute up to fourth order in $H_{1}$ because we want to keep all the order $W^4$ terms. The requirement arises because the probability leaking term, whose observation is crucial to distinguish between low-energy and high-energy unitarity violation, is of order $W^4$. The other normalization term, the second term in (\[P-beta-alpha-ave-vac\]), also deviates from the one in unitary case by a quantity of order $W^4$ in the appearance channels, but in an implicit way. The resulting expressions of $\hat{S}$ matrix elements to order $W^4$ are summarized in appendix \[sec:hatS-elements\]. There exists important consistency check in the calculation. That is, the identity relation between $\hat{S}$ matrix elements that follows from generalized T invariance:[^10] $$\begin{aligned} \hat{S}_{i j} (U, W, X, A) &=& \hat{S}_{j i} (U^*, W^*, X^*, A^*), \nonumber \\ \hat{S}_{i J} (U, W, X, A) &=& \hat{S}_{J i} (U^*, W^*, X^*, A^*), \nonumber \\\hat{S}_{I J} (U, W, X, A) &=& \hat{S}_{J I} (U^*, W^*, X^*, A^*), \label{T-invariance}\end{aligned}$$ where $\hat{S}_{J i}$ is obtained by performing the exchange $h_{i} \leftrightarrow \Delta_{J}$ in $\hat{S}_{i J}$. The generalized T invariance relation will be explicitly verified by the computed results of $\hat{S}$ matrix elements in appendix \[sec:hatS-elements\].[^11] To carry out a complete consistency check, we have obtained all the $\hat{S}$ matrix elements (including $\hat{S}_{I J}$) to fourth order in $W$, and verified their generalized T invariance. However, only the ones which are required to obtain $S$ matrix elements to fourth order are exhibited in appendix \[sec:hatS-elements\]. ### Computation of $S$ matrix elements {#sec:S-matrix} Given the results of $\hat{S}$ matrix elements it is straightforward to calculate $S$ matrix elements by using the formulas in eq. (\[flavor-hat-relation2\]). The active neutrino space $S$ matrix elements can be written in perturbative forms, $S_{\alpha \beta} = S_{\alpha \beta}^{(0)} + S_{\alpha \beta}^{(2)} + S_{\alpha \beta}^{(4)}$, where $$\begin{aligned} S_{\alpha \beta}^{(0)} &=& \sum_{k l} (UX)_{\alpha k} (UX)^*_{\beta l} \hat{S}_{kl}^{(0)}, \nonumber \\ S_{\alpha \beta}^{(2)} &=& \sum_{k l} (UX)_{\alpha k} (UX)^*_{\beta l} \hat{S}_{kl}^{(2)} + \sum_{k L} (UX)_{\alpha k} W^*_{\beta L} \hat{S}_{kL}^{(1)} \nonumber \\ &+& \sum_{K l} W_{\alpha K} (UX)^*_{\beta l} \hat{S}_{K l}^{(1)} + \sum_{K L} W_{\alpha K} W^*_{\beta L} \hat{S}_{KL}^{(0)}, \nonumber \\ S_{\alpha \beta}^{(4)} &=& \sum_{k l} (UX)_{\alpha k} (UX)^*_{\beta l} \hat{S}_{kl}^{(4)} + \sum_{k L} (UX)_{\alpha k} W^*_{\beta L} \hat{S}_{kL}^{(3)} \nonumber \\ &+& \sum_{K l} W_{\alpha K} (UX)^*_{\beta l} \hat{S}_{K l}^{(3)} + \sum_{K L} W_{\alpha K} W^*_{\beta L} \hat{S}_{KL}^{(2)}. \label{Sab-hatSab}\end{aligned}$$ Using (\[Sab-hatSab\]) the explicit expressions of $S$ matrix elements can be easily obtained with use of $\hat{S}$ matrix elements given in appendix \[sec:hatS-elements\]. For example, $S_{\alpha \beta}$ in zeroth and second orders in $W$ are given, respectively, by $$\begin{aligned} S_{\alpha \beta}^{(0)} &=& \sum_{k} (UX)_{\alpha k} (UX)^*_{\beta k} e^{-i h_{k} x}, \label{S-alpha-beta-0th}\end{aligned}$$ and $$\begin{aligned} && S_{\alpha \beta}^{(2)} = \sum_{k, K} \frac{ 1 }{ \Delta_{K} - h_{k} } \left[ (ix) e^{- i h_{k} x} + \frac{e^{- i \Delta_{K} x} - e^{- i h_{k} x} }{ ( \Delta_{K} - h_{k} ) } \right] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &-& \sum_{k \neq l} \sum_{K} \frac{ 1 }{ ( h_{l} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{l}) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i h_{l} x} - \left( \Delta_{K} - h_{l} \right) e^{- i h_{k} x} - ( h_{l} - h_{k} ) e^{- i \Delta_{K} x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \nonumber \\ &+& \sum_{k, K} \frac{e^{- i \Delta_{K} x} - e^{- i h_{k} x} }{ ( \Delta_{K} - h_{k} ) } \biggl[ (UX)_{\alpha k} W^*_{\beta K} \left\{ (UX)^{\dagger} A W \right\}_{k K} + W_{\alpha K} (UX)^*_{\beta k} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \biggr] \nonumber \\ &+& \sum_{K} e^{- i \Delta_{K} x} W_{\alpha K} W^*_{\beta K}. \label{S-alpha-beta-2nd}\end{aligned}$$ The oscillation probability to second order in $W$ {#sec:probability-2nd} -------------------------------------------------- In this section, we discuss the oscillation probability to second order in $W$. It is to illuminate the principle of calculation, how averaging over the fast oscillation works, and to show which constraints are obtained on the sterile state masses by the requirement of suppression by the energy denominator to make these sterile-sector model dependent terms negligible. Of course, we will calculate in this paper all the oscillation probabilities $P(\nu_\beta \rightarrow \nu_\alpha)$ in matter to fourth order in $W$ to keep the necessary term, the probability leaking term $\mathcal{C}_{\alpha \beta}$, as mentioned earlier. The key features of the fourth-order terms will be described in the next section \[sec:probability-4th\]. The following formulas include the cases of both disappearance ($\alpha=\beta$) and appearance ($\alpha \neq \beta$) channels. The oscillation probability $P(\nu_\beta \rightarrow \nu_\alpha)$ is given to second order in $W$ as $$\begin{aligned} &&P(\nu_\beta \rightarrow \nu_\alpha)^{(0+2)} = \left| S^{(0)}_{\alpha \beta} \right|^2 + 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S^{(2)}_{\alpha \beta} \right] \nonumber \\ &=& \sum_{k} (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha k} (UX)_{\beta k} + \sum_{k \neq l} (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha l} (UX)_{\beta l} e^{-i ( h_{k} - h_{l} ) x} \nonumber \\ &+& 2 \mbox{Re} \biggl\{ \sum_{m} \sum_{k, K} \frac{ 1 }{ \Delta_{K} - h_{k} } \left[ (ix) e^{- i ( h_{k} - h_{m} ) x} + \frac{e^{- i ( \Delta_{K} - h_{m} ) x} - e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) } \right] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &-& \sum_{m} \sum_{k \neq l} \sum_{K} \frac{ 1 }{ ( h_{l} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{l}) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i ( h_{l} - h_{m} ) x} - \left( \Delta_{K} - h_{l} \right) e^{- i ( h_{k} - h_{m} ) x} - ( h_{l} - h_{k} ) e^{- i ( \Delta_{K} - h_{m} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \nonumber \\ &+& \sum_{m} \sum_{k, K} \frac{e^{- i (\Delta_{K} - h_{m} ) x} - e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) } \biggl[ (UX)_{\alpha k} W^*_{\beta K} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \nonumber \\ &+& W_{\alpha K} (UX)^*_{\beta k} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \biggr] \nonumber \\ &+& \sum_{m} \sum_{K} e^{- i (\Delta_{K} - h_{m} ) x} W_{\alpha K} W^*_{\beta K} (UX)^*_{\alpha m} (UX)_{\beta m} \biggr\}. \label{P-beta-alpha-0th+2nd}\end{aligned}$$ Notice that there is no matter dependent terms without suppression either by high-frequency oscillations $\propto \cos (\Delta_{K} - h_{m}) x$ (or $\sin$), or by large energy denominators $\propto \frac{ 1 }{ \Delta_{K} - h_{k} }$. We take averaging over fast oscillations due to active-sterile and sterile-sterile mass squared differences which leads to $$\begin{aligned} \left\langle \sin \Delta_{J i} x \right\rangle &\approx& \left\langle \sin \Delta_{J K} x \right\rangle \approx 0, \nonumber \\ \left\langle \cos \Delta_{J i} x \right\rangle &\approx& \left\langle \sin \Delta_{J K} x \right\rangle \approx 0, \label{average-out}\end{aligned}$$ where $\langle ... \rangle$ stands for averaging over neutrino energy within the uncertainty of energy resolution, as well as averaging over uncertainty of distance between production and detection points of neutrinos.[^12] The second approximate equalities in assume that there is no accidental degeneracy among the sterile state masses. That is, we assume that the relation $|\Delta m^2_{JK}| \gg |\Delta m^2_{31}|$ always holds. After averaging out the fast oscillations, $P(\nu_\beta \rightarrow \nu_\alpha)$ is given to second order in $W$ by $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(0+2)} \nonumber \\ &=& \left| \sum_{j=1}^{3} U_{\alpha j} U^{*}_{\beta j} \right|^2 - 2 \sum_{j \neq k} \mbox{Re} \left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right] \sin^2 \frac{ ( h_{k} - h_{j} ) x }{ 2 } \nonumber\\ &-& \sum_{j \neq k} \mbox{Im} \left[ (UX)_{\alpha j} (UX)_{\beta j}^* (UX)_{\alpha k}^* (UX)_{\beta k} \right] \sin ( h_{k} - h_{j} ) x \nonumber \\ &+& 2 \mbox{Re} \biggl\{ \sum_{m} \sum_{k, K} \frac{ 1 }{ \Delta_{K} - h_{k} } \left[ (ix) e^{- i ( h_{k} - h_{m} ) x} - \frac{ e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) } \right] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &-& \sum_{m} \sum_{k \neq l} \sum_{K} \frac{ 1 }{ ( h_{l} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{l}) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i ( h_{l} - h_{m} ) x} - \left( \Delta_{K} - h_{l} \right) e^{- i ( h_{k} - h_{m} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \nonumber \\ &-& \sum_{m} \sum_{k, K} \frac{ e^{- i ( h_{k} - h_{m} ) x} }{ ( \Delta_{K} - h_{k} ) } \biggl[ (UX)_{\alpha k} W^*_{\beta K} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \nonumber \\ &+& W_{\alpha K} (UX)^*_{\beta k} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \biggr] \biggr\}, \label{P-beta-alpha-2nd-averaged}\end{aligned}$$ where the zeroth-order terms are rewritten in a familiar way. However, the expression in (\[P-beta-alpha-2nd-averaged\]) does not mean a great simplification. That is, the matter dependent terms with sterile sector model-dependence remain. Suppression by the energy denominator {#sec:energy-denominator} -------------------------------------- We, therefore, have to examine the effect of suppression by the large energy denominator which characterizes transition between active-sterile sates, $1/ ( \Delta_{K} - h_{k} )$. We demand that the matter dependent terms in (\[P-beta-alpha-2nd-averaged\]) be smaller than the probability leaking and the normalization terms of order $\sim W^4$. It leads to $$\begin{aligned} \biggl | \frac{ AA L }{ ( \Delta_{J} - h_{i} ) } \biggr | < W^2, \hspace{4mm} \biggl | \frac{ AA }{ ( h_{k} - h_{j} ) ( \Delta_{J} - h_{i} ) } \biggr | < W^2, \hspace{4mm} \text{and} \hspace{4mm} \biggl | \frac{ A }{ ( \Delta_{J} - h_{i} ) } \biggr | < W^2, \label{denominator-size}\end{aligned}$$ where $L$ is the baseline distance and $i$ and $J$ denote, respectively, generic indices for active and sterile states. For notational convenience, we define $\lambda_{i}$ $(i=1,2,3)$ to be the eigenvalues of $3 \times 3$ submatrix $2E \tilde{H}_0$ in (\[tilde-H0+H1\]) corresponding to the active neutrino mass squared in matter. Then, $h_{i} = \frac{ \lambda_{i} }{2E}$. As one can easily see[^13] the last one in (\[denominator-size\]) gives the severest constraint (taking the matter potential due to CC in $A$ and removing the factor $\frac{1}{2E}$) $$\begin{aligned} \biggl | \frac{ a }{ m^2_{J} - \lambda_{i} } \biggr | \approx \biggl | \frac{ a }{ \Delta m^2_{J i} } \biggr | < W^2. \label{suppression-cond}\end{aligned}$$ Notice that, in order for the first inequality in (\[suppression-cond\]) to be valid, we have restricted the energy region for a given matter density such that $\lambda_{i}$ remain in the order of active neutrino masses. Roughly speaking, it corresponds to $- 50 \,\text{ (g/cm}^3) \text{GeV} \lsim Y_{e} \rho E \lsim 50 \, \text{ (g/cm}^3) \text{GeV}$. See e.g., figure 3 of ref. [@Minakata:2015gra]. Clearly, it excludes the interesting region of “IceCube resonance” due to sterile neutrino mass of eV scales [@Nunokawa:2003ep], for which an entirely different theoretical framework would be necessary. Then, we notice that in a regime $W^2 \sim 10^{-2}$, the condition in (\[suppression-cond\]) is valid given the estimation (assuming $Y_{e} = 0.5$) $$\begin{aligned} \biggl | \frac{ a }{ \Delta m^2_{J i} } \biggr | = 2.13 \times 10^{-3} \left(\frac{ \Delta m^2_{J i} }{ 0.1~\mbox{eV}^2}\right)^{-1} \left(\frac{\rho}{2.8 \,\text{g/cm}^3}\right) \left(\frac{E}{1~\mbox{GeV}}\right), \label{rA-def-value}\end{aligned}$$ unless $\rho E \gsim 10 \, \text{ (g/cm}^3) \text{GeV}$. That is, the second-order matter dependent correction terms can be ignored in comparison with $\mathcal{O} (W^4)$ terms if $\Delta m^2_{J k} \gsim 0.1$ eV$^2$, which is already required in vacuum. If we want to treat the regime $W^2 \gsim 10^{-n}$, we need to limit the sterile masses to $\Delta m^2_{J k} \simeq m^2_{J} \gsim 10^{(n-3)}$ eV$^2$ to keep our $(3+N)$ space unitary model insensitive to details of the sterile sector [@Fong:2016yyh]. We note, however, that terms of order $W^4 \sim 10^{-4}$ may be the limit of exploration for near future neutrino oscillation experiments. The condition (\[suppression-cond\]) is identical with the one obtained using the first order matter perturbation theory [@Fong:2016yyh], which may look strange to the readers. Let us understand the reason why taking care of all order matter effect does not alter the condition obtained by first-order treatment in matter perturbation theory. The matter-dependent term in the zeroth-order Hamiltonian $\tilde{H}_{0}$ only involves $U$ matrix, but no $W$ matrix. Since we treat $\tilde{H}_{0}$ in an unperturbed fashion it produces all-order effect of the matter potential which is however independent of $W$ matrix elements. On the other hand, perturbative effects that come from single or double powers in $W$ in $\hat{H}_{1}$ are always accompanied by the matter potential in the form of $WA$ or $W^{\dagger} A$, as in eq. (\[H1-matrix\]). That is, perturbative effect of $W$ is always accompanied by matter potential, and hence can always be dealt with matter perturbation theory. It is the reason why the matter perturbation theory is able to yield the same condition on sterile masses as obtained in a fuller treatment of matter effect done in this paper.[^14] The oscillation probability in fourth order in $W$ {#sec:probability-4th} -------------------------------------------------- The oscillation probability in fourth order in $W$ contains the two terms $$\begin{aligned} &&P(\nu_\beta \rightarrow \nu_\alpha)^{(4)} = \left| S^{(2)}_{\alpha \beta} \right|^2 + 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S^{(4)}_{\alpha \beta} \right]. \label{P-beta-alpha-4th-def}\end{aligned}$$ We will show in appendix \[sec:second-order-square\] that the first term in (\[P-beta-alpha-4th-def\]), after averaging over the fast oscillations and using the suppression by energy denominator as discussed in the previous section, leaves the unique term, the probability leaking term $\mathcal{C}_{\alpha \beta}$ in (\[P-beta-alpha-ave-vac\]). We will also show in appendix \[sec:interference\] that the second term in (\[P-beta-alpha-4th-def\]), under the same treatment for the first term, gives vanishing contribution. Therefore, no matter-dependent term remains after using energy denominator suppression and averaging over the fast oscillations. In conclusion, the oscillation probability in matter between active flavor neutrinos in the $(3+N)$ space unitary model to fourth order in $W$ in our small unitarity-violation perturbation theory can be written as in eq. (\[P-beta-alpha-final\]) in section \[sec:essence\]. We hope that it serves as a useful tool to test leptonic unitarity in various ongoing and future neutrino oscillation experiments. Analytical and numerical methods for solving non-unitary evolution in matter {#sec:analytical-numerical} ============================================================================== In this section, we describe the numerical and analytical methods for calculating the neutrino oscillation probability by solving non-unitary evolution in matter. Numerical method for calculating neutrino oscillation probability {#sec:numerical} ------------------------------------------------------------------- We describe a numerical method for computing the oscillation probability in matter. This method can be used, assuming adiabaticity, in cases with varying matter density. We show that in zeroth order in $W$ the system simplifies to an evolution equation in the $3 \times 3$ active subspace. We solve the Schrödinger equation in the vacuum mass eigenstate basis (“tilde basis”), $\tilde{\nu}_{z} = ({\bf U}^{\dagger})_{z \zeta} \nu_{\zeta}$ with Hamiltonian $\tilde{H}$ in (\[tilde-H\]): $$\begin{aligned} i \frac{d}{dx} \left[ \begin{array}{c} \tilde{\nu}_{i} \\ \tilde{\nu}_{J} \\ \end{array} \right] = \left[ \begin{array}{cc} {\bf \Delta_{a} } + U^{\dagger} A U & U^{\dagger} A W \\ W^{\dagger} A U & {\bf \Delta_{s} } + W^{\dagger} A W \\ \end{array} \right] \left[ \begin{array}{c} \tilde{\nu}_{i} \\ \tilde{\nu}_{J} \\ \end{array} \right], \label{Schroedinger-eq}\end{aligned}$$ where $i = 1,2,3$ and $J= 4,5,\cdot \cdot \cdot,3+N$ denote mostly active and mostly sterile neutrino mass eigenstate labels, respectively. The initial condition with only active component implies $$\begin{aligned} \tilde{\nu}_{i} (0) &=& \sum_{\alpha} (U^{\dagger})_{i \alpha} \nu_{\alpha} (0), \nonumber \\ \tilde{\nu}_{J} (0) &=& \sum_{\alpha} (W^{\dagger})_{J \alpha} \nu_{\alpha} (0). \label{initial-condition}\end{aligned}$$ Using the solution of equation (\[Schroedinger-eq\]), we need the wave function of active flavor component to calculate the probability at baseline $x=L$. $$\begin{aligned} \nu_{\alpha} (L) &=& \sum_{i} U_{\alpha i} \tilde{\nu}_{i} (L) + \sum_{J} W_{\alpha J} \tilde{\nu}_{J} (L). \label{final-condition}\end{aligned}$$ Therefore, in the mass-basis formulation only $U$ and $W$ are involved, which is consistent with our experience in $W$ perturbation theory. An apparent contradiction to this property that one faces in the evolution equation in the flavor basis is resolved in appendix \[sec:flavor-basis-evolution\]. A drawback of this method is that we have to solve explicitly the evolution of the sterile states which are coupled to the active states. Then, we need to specify the sterile sector model, and have to know how to deal with averaging over the fast modes. We notice, however, that in the zeroth-order in $W$ the system simplifies. Since the Hamiltonian $\tilde{H}$ is block-diagonal it suffices to solve the equation only in the $3 \times 3$ active neutrino subspace: $$\begin{aligned} i \frac{d}{dx} \nu_{i} = \sum_{j} \left( {\bf \Delta_{a} } + U^{\dagger} A U \right)_{ij} \nu_{j}. \label{Schroedinger-eq-0th}\end{aligned}$$ The initial condition (\[initial-condition\]) and final reverse-back formula (\[final-condition\]) involve only $U$ matrix elements. Therefore, the oscillation probability in the zeroth-order in $W$ can be calculable in a manner independent of sterile sector models.[^15] An exact solution of zeroth-order oscillation probability {#sec:exact-solution-zeroth} ----------------------------------------------------------- Here, we describe a method for obtaining the analytical solution of the zeroth-order Hamiltonian. The exact solution, as well as the numerical one described in the previous section, provides the basis for computing the higher order corrections in $W$. We calculate an exact form of the oscillation probability $P(\nu_\beta \rightarrow \nu_\alpha)$ in leading order in our perturbative framework, the one in (\[P-beta-alpha-final\]) except for $\mathcal{C}_{\alpha \beta}$, in the case of uniform matter density. The zeroth-order $S$ matrix element $S_{\alpha \beta}^{(0)}$ in (\[S-alpha-beta-0th\]) can be written as $$\begin{aligned} S_{\alpha \beta}^{(0)} &=& \sum_{i, j} U_{\alpha i} U^*_{\beta j} \left( \sum_{k} X_{i k} X^*_{j k} e^{-i h_{k} x} \right), \label{S-alpha-beta-0th-KTY}\end{aligned}$$ and the factor in parenthesis can be calculated by the KTY technique [@Kimura:2002wd]. We want to diagonalize the Hamiltonian $$\begin{aligned} H_{0} \equiv \frac{1}{2E} \left\{ \left[ \begin{array}{ccc} m^2_{1} & 0 & 0 \\ 0 & m^2_{2} & 0 \\ 0 & 0 & m^2_{3} \\ \end{array} \right] + U^{\dagger} \left[ \begin{array}{ccc} a - b & 0 & 0 \\ 0 & -b & 0 \\ 0 & 0 & -b \\ \end{array} \right] U \right\} , \label{H0-def}\end{aligned}$$ the active $3 \times 3$ block of $\tilde{H}_0$ in (\[tilde-H0+H1\]). We have defined in eq. (\[H0-diag\]) the unitary matrix $X$ which diagonalize $H_{0}$ as $$\begin{aligned} H_{0} = \frac{1}{2E} X \left[ \begin{array}{ccc} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \\ \end{array} \right] X^{\dagger} \equiv H_{d}. \label{H0-diagonal}\end{aligned}$$ For our notational convenience we call this form of $H_{0}$ as $H_{d}$. Note that $h_{i} = \frac{ \lambda_{i} }{2E}$ where $$\begin{aligned} \lambda_{1} & = & \frac{T}{3}-\frac{1}{3}\sqrt{T^{2}-3A}\cos\left[\frac{1}{3}\arccos\left(\frac{2T^{3}-9AT+27D}{2\left(T^{2}-3A\right)^{3/2}}\right)\right]\nonumber \\ & & -\frac{1}{\sqrt{3}}\sqrt{T^{2}-3A}\sin\left[\frac{1}{3}\arccos\left(\frac{2T^{3}-9AT+27D}{2\left(T^{2}-3A\right)^{3/2}}\right)\right],\\ \lambda_{2} & = & \frac{T}{3}-\frac{1}{3}\sqrt{T^{2}-3A}\cos\left[\frac{1}{3}\arccos\left(\frac{2T^{3}-9AT+27D}{2\left(T^{2}-3A\right)^{3/2}}\right)\right]\nonumber \\ & & +\frac{1}{\sqrt{3}}\sqrt{T^{2}-3A}\sin\left[\frac{1}{3}\arccos\left(\frac{2T^{3}-9AT+27D}{2\left(T^{2}-3A\right)^{3/2}}\right)\right],\\ \lambda_{3} & = & \frac{T}{3}+\frac{2}{3}\sqrt{T^{2}-3A}\cos\left[\frac{1}{3}\arccos\left(\frac{2T^{3}-9AT+27D}{2\left(T^{2}-3A\right)^{3/2}}\right)\right],\end{aligned}$$ with $$\begin{aligned} T & = & (2E)\, {\rm Tr} H_0,\\ A & = & (2E)^2\, {\rm Tr}\left({\rm Adj} H_0 \right),\\ D & = & (2E)^3 \det H_0.\end{aligned}$$ The adjugate of $H_{0}$ is defined as $\text{Adj} H_{0} \equiv (H_{0})^{-1} \text{det} H_{0}$. Notice that $T$, $A$ and $D$ are invariant under unitary transformation of $ H_0\to K H_0 K^{\dagger}$ with $K$ any unitary matrix and so are $\lambda_i$. Following the notation in [@Kimura:2002wd] we define $p_{ij}$ and $q_{ij}$ as ($i,j=1,2,3$) $$\begin{aligned} \frac{ p_{ij} }{ 2E } &\equiv& \left( H_{0} \right)_{ij}, \nonumber \\ \frac{ q_{ij} }{ (2E)^2 } &\equiv& \left( \text{Adj} H_{0} \right)_{ij}. \label{pq-def}\end{aligned}$$ Notice that $p_{ij}$ and $q_{ij}$ are written only by the known (or given) quantities. Then, the equations $$\begin{aligned} \left( H_{d} \right)_{ij} &=& \frac{ p_{ij} }{ 2E }, \nonumber \\ \left( \text{Adj} H_{d} \right)_{ij} &=& \frac{ q_{ij} }{ (2E)^2 }, \label{KTY-eq}\end{aligned}$$ together with unitarity of $X$, become the equations to determine $X X^{\dagger}$: $$\begin{aligned} X_{i 1} X_{j 1}^* + X_{i 2} X_{j 2}^* + X_{i 3} X_{j 3}^* &=& \delta_{ij}, \nonumber \\ \lambda_{1} X_{i 1} X_{j 1}^* + \lambda_{2} X_{i 2} X_{j 2}^* + \lambda_{3}X_{i 3} X_{j 3}^* &=& p_{ij}, \nonumber \\ \lambda_{2} \lambda_{3} X_{i 1} X_{j 1}^* + \lambda_{3} \lambda_{1} X_{i 2} X_{j 2}^* + \lambda_{1} \lambda_{2} X_{i 3} X_{j 3}^* &=& q_{ij}. \label{KTY-eq-explicit}\end{aligned}$$ They lead to the solution ($k=1,2,3$) $$\begin{aligned} X_{i k} X_{j k}^* = \frac{ q_{ij} + p_{ij} \lambda_{k} - \delta_{ij} \lambda_{k} ( \lambda_{l} + \lambda_{m} ) }{ (\lambda_{l} -\lambda_{k} ) (\lambda_{m} -\lambda_{k} ) }, \label{KTY-eq-solution}\end{aligned}$$ where $k,l,m$ is cyclic, and sum over $k$ is not implied in (\[KTY-eq-solution\]). Therefore, to zeroth-order in $W$ expansion, the $S$ matrix elements are given by $$\begin{aligned} S_{\alpha \beta}^{(0)} &=& \sum_{k} \left( \sum_{i, j} U_{\alpha i} \left[ q_{ij} + p_{ij} \lambda_{k} - \delta_{ij} \lambda_{k} ( \lambda_{l} + \lambda_{m} ) \right] U^*_{\beta j} \right) \frac{ e^{-i h_{k} x} }{ (\lambda_{l} -\lambda_{k} ) (\lambda_{m} -\lambda_{k} ) }, \label{S-alpha-beta-0th-final}\end{aligned}$$ and the oscillation probability by $P(\nu_\beta \rightarrow \nu_\alpha) = \vert S_{\alpha \beta}^{(0)} \vert^2$. Finally, armed with the solution , we can also calculate all higher order terms in oscillation probability for e.g. those in eq.  since only such combination $X_{ik} X_{jk}^*$ (no sum over $k$ implied) can appear. Low-scale vs. high-scale unitarity violation and $W$ corrections {#sec:UV-low-high-E} ================================================================= Low-scale versus high-scale unitarity violation {#sec:low-vs-high} ------------------------------------------------- In leptonic unitarity test, a clear understanding of the relationship between low-scale and high-scale unitarity violation may be one of the key issues. While the presence of probability leaking term $\mathcal{C}_{\alpha \beta}$, if detected, clearly testifies for low-scale unitarity violation [@Fong:2016yyh], it would be better to provide a global view of the difference between them, looking for alternative ways to reach the goal. In this section, we would like to give a preliminary discussion on this topics. One would guess, on intuitive ground, that at zeroth order in $W$ our system describes high-scale unitarity violation. There is no “$W$ corrections” in high-scale unitarity violation because the energy scale is so high that the high-mass sector is truncated. It is in agreement with the formulations in ref. [@Blennow:2016jkn] with which we share the same evolution equation (\[Schroedinger-eq-0th\]) in the vacuum mass eigenstate basis. See also [@Antusch:2006vwa].[^16] Based on this perspective, we just conclude this subsection by stating the generic characteristic differences between high-scale and low-scale unitarity violation: - Our system calculated with evolution equations (\[Schroedinger-eq-0th\]) in leading (zeroth) order in $W$ expansion, which is common to both high- and low-scale unitarity violation, would serve as a first indicator to the existence of leptonic unitarity violation. Existence of second and higher order corrections in $W$, if detected, uniquely identifies the case for low-scale unitarity violation. - Presence (low-$E$) or absence (high-$E$) of the probability leaking term $\mathcal{C}_{\alpha \beta}$ in (\[Cab\]) remains to be the best possible way to distinguish between high- and low-scale unitarity violation. Oscillation probabilities with and without unitarity violation {#sec:probabilities} ---------------------------------------------------------------- Under the hope to uncover in which region of parameter space the effect of non-unitarity is most prominent, we first compare the oscillation probabilities with and without unitarity violation. We examine the three channels $\nu_{\mu} \rightarrow \nu_{e}$, $\nu_{\mu} \rightarrow \nu_{\tau}$, and $\nu_{\mu} \rightarrow \nu_{\mu}$. We do not intend to do quantitative analyses, nor attempt to cover the whole parameter space. Yet, we try to give the readers a feeling on how large and where are the effects of unitarity violation. Therefore, we present the results just for a specific choice of the parameters. Also, the use of uniform matter density $\rho = 3.2~{\rm g\,cm}^{-3} $ over the entire baseline is not realistic.[^17] Notice that the results given in section \[sec:probabilities\] apply to both high-scale unitarity violation as well as low-scale one in its leading order in $W$. The probability leaking term is not included in the analysis in this section \[sec:probabilities\], but the effect is discussed in section \[sec:correction-terms\]. ### A brief note on the parameter choice {#sec:parameter-choice} Let us start by explaining briefly how the unitarity violation parameters are chosen. We go to the $(3+1)$ model in which the constraints on the parameters are best understood [@Kopp:2013vaa; @deGouvea:2015euy; @Fernandez-Martinez:2016lgt; @TheIceCube:2016oqi]. In consistent with the current ones we have chosen: $\sin^2\theta_{14} = 0.02$, $\sin^2\theta_{24} = 0.01$, and $\sin^2\theta_{34} = 0.1$ for $\Delta m^2_{41} = 0.1$ eV$^2$, and set all the CP phases to zero. Then, we cut out the $3\times3$ active neutrino mixing matrix, which is non-unitary.[^18] For the Standard Model mixing parameters in $U_{\text{\tiny PDG}}$, we take $\sin^2\theta_{12} = 0.3$, $\sin^2\theta_{23} = 0.5$, $\sin^2 (2\theta_{13}) = 0.09$, and the mass squared differences $\Delta m_{21}^2 = 7.4 \times 10^{-5}$ eV$^2$ and $\Delta m_{31}^2 = 2.4 \times 10^{-3}$ eV$^2$, and set the CP phase $\delta_{\text{CP}}$ to zero. ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in space spanned by neutrino energy $E$ and baseline $L$. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ is presented. For the values of unitarity-violating as well as the standard mixing parameters taken, see the text. []{data-label="fig:Pmue_energy_dist"}](Pmue_energy_dist_non_unitary_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in space spanned by neutrino energy $E$ and baseline $L$. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ is presented. For the values of unitarity-violating as well as the standard mixing parameters taken, see the text. []{data-label="fig:Pmue_energy_dist"}](Pmue_energy_dist_difference_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\tau}) \equiv P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmutau_energy_dist"}](Pmutau_energy_dist_non_unitary_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\tau}) \equiv P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\tau})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmutau_energy_dist"}](Pmutau_energy_dist_difference_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu}) \equiv P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmumu_energy_dist"}](Pmumu_energy_dist_non_unitary_small_size.jpeg "fig:"){width="85.00000%"} ![ In the upper panel (a), presented is the iso-contour of $P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ in $E-L$ space. In the lower panel (b), the iso-contour of the difference $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu}) \equiv P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$ is presented. The parameters used are the same as in figure \[fig:Pmue\_energy\_dist\]. []{data-label="fig:Pmumu_energy_dist"}](Pmumu_energy_dist_difference_small_size.jpeg "fig:"){width="85.00000%"} ### $P(\nu_{\mu} \rightarrow \nu_{e})$ In figure \[fig:Pmue\_energy\_dist\]-(a) (upper panel) and (b) (lower panel), presented are the iso-contours of $P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ and $\Delta P (\nu_{\mu} \rightarrow \nu_{e}) \equiv P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{standard} }- P(\nu_{\mu} \rightarrow \nu_{e})_{ \text{non-unitary} }^{(0)}$ in $E - L$ space. Here, the superscript $(0)$ implies that it is calculated in zeroth-order in $W$ using (\[Schroedinger-eq-0th\]) with appropriate initial condition and final projection to flavor eigenstate. In most of the $E - L$ space $P(\nu_{\mu} \rightarrow \nu_{e})$ is small. However, we identify the two regions where $P(\nu_{\mu} \rightarrow \nu_{e})$ is relatively large, $\gsim 0.3$. One of them is at low energy, $E \lsim \text{a few hundred}$ MeV, and baseline $L \gsim$1000 km. The other one is a region $E \sim 10$ GeV and $L \sim$ 10000 km. The former may be understood as due to the solar MSW enhancement, and the latter as the atmospheric MSW enhancement [@Mikheev:1986gs; @Wolfenstein:1977ue]. Roughly speaking, the regions with relatively large $| \Delta P (\nu_{\mu} \rightarrow \nu_{e}) |$ overlap with these regions. ### $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ In figures \[fig:Pmutau\_energy\_dist\] and \[fig:Pmumu\_energy\_dist\], the same quantities (in each upper (a) and lower (b) panel) are presented but in $\nu_{\mu} \rightarrow \nu_{\tau}$ and $\nu_{\mu} \rightarrow \nu_{\mu}$ channels, respectively. In contrast to $\nu_{\mu} \rightarrow \nu_{e}$ channel, $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ contours are globally “vacuum effect dominated”, apart from the solar MSW region, both in the standard (not shown) and the non-unitary cases. The first oscillation peak of $P(\nu_{\mu} \rightarrow \nu_{\tau})$ scales roughly as the vacuum oscillation peak does, $L / 10^3 \,\text{km} = 0.33 E / 1 \, \text{GeV}$. This feature is more or less seen in $P(\nu_{\mu} \rightarrow \nu_{e})$, but $P(\nu_{\mu} \rightarrow \nu_{\tau})$ has a higher peak height $\simeq 0.7-0.8$, and the effect of atmospheric MSW enhancement is less prominent. For $P(\nu_{\mu} \rightarrow \nu_{\mu})_{ \text{non-unitary} }^{(0)}$, roughly speaking, the relation $P(\nu_{\mu} \rightarrow \nu_{\mu}) \approx 1 - P(\nu_{\mu} \rightarrow \nu_{\tau}) $ holds in region where $P(\nu_{\mu} \rightarrow \nu_{e})$ is small. It must be the case in the unitary case, but even in non-unitary case the relation holds approximately because unitarity violation is small in our choice of the parameters. Therefore, $P(\nu_{\mu} \rightarrow \nu_{\mu})$ is large in region where $P(\nu_{\mu} \rightarrow \nu_{\tau})$ is small, and vice versa, as seen in figure \[fig:Pmumu\_energy\_dist\]. It appears that the anticorrelation is inherited to the relationship between $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu})$ and $\Delta P (\nu_{\mu} \rightarrow \nu_{\tau})$. Due to relatively large $\Delta P (\nu_{\mu} \rightarrow \nu_{\tau})$ in the first and second oscillation maxima over the entire baseline, or similarly because of relatively large depletion of $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu})$ there, it should be possible to detect the effect of non-unitarity if the detector has a good $\tau$ (in the former channel) and $\mu$ (in the latter channel) detection capabilities. It would be ideal that the detector has sensitivities to the both channels because then one can take anticorrelation between $\mu$ and $\tau$ yields to enhance the sensitivity to unitarity violation. Some comments on observational aspects: In the two regions where $| \Delta P (\nu_{\mu} \rightarrow \nu_{e}) |$ is large, and $\Delta P (\nu_{\mu} \rightarrow \nu_{\mu})$ in energy region $E \lsim 10$ GeV may be explored by high-statistics atmospheric neutrino observation by Super-K/Hyper-K, or DUNE [@Abe:2017aap; @Abe:2015zbg; @Acciarri:2015uup]. The atmospheric MSW enhanced region of $P(\nu_{\mu} \rightarrow \nu_{e})$ would be a good target for PINGU extensions of IceCube and KM3NeT-ORCA [@TheIceCube-Gen2:2016cap; @Adrian-Martinez:2016zzs]. $P(\nu_{\mu} \rightarrow \nu_{\tau})$ and $P(\nu_{\mu} \rightarrow \nu_{\mu})$ would be explored by them, with possibility of seeing anticorrelation between $\mu$ and $\tau$ yields. Although it is very interesting to investigate these experimental prospects, a detailed examination of these questions is beyond the scope of this paper. The probability leaking and $W$ correction terms {#sec:correction-terms} -------------------------------------------------- As we stated in section \[sec:low-vs-high\], uncovering the effect of $W$ correction or the probability leaking term $\mathcal{C}_{\alpha \beta}$ in eq. (\[Cab\]) would offer another way of distinguishing low-scale unitarity violation from high-scale one. Let us give a brief sketch of how and where we might see visible effect. ### A further note on the parameter choice {#sec:parameter-choice2} To discuss $W$ correction and the probability leaking term we have to determine the $W$ matrix. Given the non-unitary $U$ matrix there is a way to construct the $W$ matrix. In general, it is given by $$W = S \sqrt{w} R, \label{W-construction}$$ where $S$ is a $3\times 3$ matrix which diagonalizes ${\bf 1}_{3\times 3} - UU^\dagger$, $w$ is diagonal matrix which consists of eigenvalues of ${\bf 1}_{3\times 3} - UU^\dagger$, and $R$ is an arbitrary $3\times N$ complex matrix obeying $RR^\dagger = \bf{1}_{3 \times 3}$. The construction makes sense for $N \geq 3$. Therefore, for a given $N$ there is a large arbitrariness on the choice of the $W$ matrix, and hence on the sizes of the $W$ corrections and $\mathcal{C}_{\alpha \beta}$. Lacking a guiding principle of how to choose the $R$ matrix in (\[W-construction\]), we examine the cases with largest and smallest possible values of $\mathcal{C}_{\alpha \beta}$ for given values of unitarity violation $1 - \sum_{j=1}^3 |U_{\alpha j}|^2$ ($\alpha=e,\mu,\tau$). It is shown that in the $(3+N)$ model $\mathcal{C}_{\alpha \beta}$ is bounded from above and below as [@Fong:2016yyh] $$\frac{1}{N} \biggl( 1 - \sum_{j=1}^3 |U_{\alpha j}|^2 \biggr) \biggl( 1 - \sum_{j=1}^3 |U_{\beta j}|^2 \biggr) \leq \mathcal{C}_{\alpha \beta} \leq \biggl( 1 - \sum_{j=1}^3 |U_{\alpha j}|^2 \biggr) \biggl( 1 - \sum_{j=1}^3 |U_{\beta j}|^2 \biggr). \label{Cab-bound}$$ In the $(3+1)$ model, the $W$ matrix elements are unique, with the upper and lower bound being equal. For the numbers given in section \[sec:parameter-choice\], we have $W_{e4} = 0.141$, $W_{\mu 4} = 0.099$, and $W_{\tau 4} = 0.141$ assuming that they are real. Then, the leaking constants have the unique values, $\mathcal{C}_{e \mu}^{(N=1)} = 2 \times 10^{-4}$, $\mathcal{C}_{\mu \mu}^{(N=1)} = 9.6 \times 10^{-5}$, and $\mathcal{C}_{\tau \mu}^{(N=1)} = 9.5 \times 10^{-4}$. The lower bound is realized in the “universal scaling” model described in appendix \[sec:scaling-model\], which predicts $W_{\alpha J} = \frac{ 1 }{ \sqrt{N} } W_{\alpha 4}^{(N=1)}$ ($J=4,5, \cdot \cdot \cdot, 3+N$).[^19] It is shown in appendix \[sec:scaling-model\] that under the assumption of equal sterile state masses the universal scaling model predicts the same $W^2$ correction terms as those of the $(3+1)$ model. ### How large are the $W$ corrections and $\mathcal{C}_{\alpha \beta}$? {#sec:how-large} Let us go back to the expression of the oscillation probability to second order in $W$, eq. (\[P-beta-alpha-2nd-averaged\]), in section \[sec:probability-2nd\] to know where we might see visible effects. If we enter into the region $\rho E \gg 10 \, \text{ (g/cm}^3) \text{GeV}$ at around the first oscillation maximum, the first two terms in $2 \mbox{Re} \{ \cdot \cdot \cdot \}$ in (\[P-beta-alpha-2nd-averaged\]) can become large apart from $W^2$ suppression, $$\begin{aligned} \biggl | \frac{ AA L }{ ( \Delta_{J} - h_{i} ) } \biggr | &\sim& \biggl | \frac{ AA }{ ( h_{k} - h_{j} ) ( \Delta_{J} - h_{i} ) } \biggr | = 0.27 \left(\frac{ \Delta m^2_{J i} }{ 0.1 \mbox{eV}^2}\right)^{-1} \left( \frac{ \rho E }{ 100 (\text{g/cm}^3) \mbox{GeV} }\right)^2. \label{enhanced-case}\end{aligned}$$ It should be compared to (\[denominator-size\]). After taking account of $W^2$ suppression of $\sim 0.01$ (assuming $W \simeq 0.1$), $| \frac{ AA L }{ ( \Delta_{J} - h_{i} ) } W^2 | \sim 3 \times 10^{-2}$ at $E \sim 100$ GeV, assuming $\Delta m^2_{J i} =0.1$ eV$^2$. ![ The sum of the order $W^2$ correction terms (see eq. (\[P-beta-alpha-2nd-averaged\])) plus the probability leaking term $\mathcal{C}_{\mu \alpha}$ (see eq. (\[Cab\]) for definition) in $P(\nu_{\mu} \rightarrow \nu_{\alpha})$, namely, $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha}) \equiv P(\nu_{\mu} \rightarrow \nu_{\alpha}) - P(\nu_{\mu} \rightarrow \nu_{\alpha})^{(0)}$, are plotted assuming a common $m_J^2 = 0.1$ eV$^2$. The top, middle and bottom panels are for $\alpha = e, \tau$, and $\mu$, respectively. In each panel the three cases are shown: $N=1$ case with maximal $\mathcal{C}_{\mu \alpha}$ (solid line), the universal scaling model with $N=3$ (dotted line), and the order $W^2$ correction only (dashed line). The last case corresponds to the universal scaling model with $N=\infty$. The blue lines are for $E=10$ GeV, and the red for $E=100$ GeV. The leaking constants in the $N=1$ model (shown without superscript $(N=1)$ in the legend) have values $\mathcal{C}_{e \mu} = 2 \times 10^{-4}$, $\mathcal{C}_{\tau \mu} = 9.5 \times 10^{-4}$, and $\mathcal{C}_{\mu \mu} = 9.6 \times 10^{-5}$. []{data-label="fig:W-correction"}](Delta_Prob_W_corrections.pdf){width="120.00000%"} To know more quantitatively their sizes, we fix the parameters as discussed in section \[sec:parameter-choice2\] with a common $m_J^2 = 0.1$ eV$^2$,[^20] and plot in figure \[fig:W-correction\] the order $W^2$ correction terms in $P(\nu_{\mu} \rightarrow \nu_{\alpha})$ in eq. (\[P-beta-alpha-2nd-averaged\]) plus the probability leaking term $\mathcal{C}_{\mu \alpha}$, $\alpha = e$ (top panel), $\alpha = \tau$ (middle panel), and $\alpha = \mu$ (bottom panel). Under the approximation of ignoring the fourth (and higher-order) corrections, they are identical to $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha}) \equiv P(\nu_{\mu} \rightarrow \nu_{\alpha}) - P(\nu_{\mu} \rightarrow \nu_{\alpha})^{(0)}$. In each panel the three cases are examined. $N=1$ case with maximal $\mathcal{C}_{\mu \alpha}$ (solid line), the universal scaling model with $N=3$ (dotted line), and the order $W^2$ correction only (dashed line). The last case corresponds to the universal scaling model with $N=\infty$. The blue lines are for $E=10$ GeV, and the red for $E=100$ GeV. We will first focus on the appearance channels $\nu_\mu \to \nu_e$ and $\nu_\mu \to \nu_\tau$. At $E=10$ GeV (100 GeV) $\delta P$ depends very much on the above three cases, $N=1$, $N=3$, and $N=\infty$ for baseline $L$ of several 100 km ($L \gsim 1000$ km). The maximum value of $| \delta P |$ is always given by the case of maximal (minimal) $\mathcal{C}_{\mu \alpha}$ for positive (negative) $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha})$ shown by the solid (dashed) lines. These maximal values of $| \delta P(\nu_{\mu} \rightarrow \nu_{\alpha}) |$ are, roughly speaking, $\simeq 10^{-3}$ for $\nu_{\mu} \rightarrow \nu_{\tau}$, and $\simeq \text{a few} \times 10^{-4}$ for $\nu_{\mu} \rightarrow \nu_{e}$. The effect might be visible for the former, though it might be challenging for the latter channel.[^21] At longer distance, however, we see enhancement. At $E=10$ GeV, we observe a factor of several enhancement both in $| \delta P (\nu_{\mu} \rightarrow \nu_{\alpha})|$ for $\nu_{\mu} \rightarrow \nu_{e}$ and $\nu_{\mu} \rightarrow \nu_{\tau}$ in region $L \gsim 3000$ km, which may provide a clearer signature. The similar tendency exists at $E=100$ GeV, but in a less pronounced way. On the other hand, $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha})$ flips sign at around $1000-3000$ km for $\nu_{\mu} \rightarrow \nu_{e}$, and $3000-6000$ km for $\nu_{\mu} \rightarrow \nu_{\tau}$ channels. It produces, assuming detector’s sensitivity, a peculiar zenith angle dependence. The relevant energy region of $\rho E = 50 - 1000 \text{ (g/cm}^3) \text{GeV}$ may be explored, for example, by atmospheric neutrino observation by Deep Core, PINGU, or KM3NeT-ORCA [@Collaboration:2011ym; @TheIceCube-Gen2:2016cap; @Adrian-Martinez:2016zzs]. For the disappearance channel $\nu_\mu \to \nu_\mu$, $|\delta P(\nu_{\mu} \rightarrow \nu_{\mu})| \sim 10^{-3} (10^{-2})$ for $E = 10$ GeV (100 GeV). In this case, the contribution from $\mathcal{C}_{\mu \mu}$ is subdominant compared to $W^2$ correction terms. Since $\mathcal{C}_{\alpha \beta}$ is a constant term in the oscillation probability, it can in principle be distinguished from the other normalization term which shares $U$ matrix element dependences with the oscillation terms. In particular, they can dominate for large $m_J^2$ since the $W^2$ correction terms are suppressed by at least $\sim 1/m_J^2$. In this case, they will be the sole indicator of low-scale unitarity violation. In general (though not in the $N=1$ model), the order $W^2$ terms depend upon details of the sterile sector, e.g., matrix structure of $W$. Therefore, once the effect is seen it would give us useful information on the structure of low-scale leptonic unitarity violation. Some remaining theoretical issues {#sec:theoretical} =================================== In this section, we will give some remarks on the theoretical basis in our framework, basic one as well as on its perturbative aspects. They include our treatment of decoherence, generic structure of higher-order corrections and its relation to the “Uniqueness theorem” (see section \[sec:nonunitarity-matter\]), and absence of enhancement due to small solar mass splitting denominator. Decoherence imposed onto coherent evolution system {#sec:decoherence} ---------------------------------------------------- We have started with the Schrödinger equation (\[evolution\]) with Hamiltonian (\[hamiltonian\]) assuming that all the neutrino states remain coherent. We have shown in this and the previous papers that the coherence between active and sterile, and sterile and sterile states are not maintained for sterile mass differences larger than $0.1$ eV$^2$. The effect of decoherence is taken into account by making average over the fast oscillations. We feel it desirable for the current treatment be replaced by the real quantum mechanical one using wave packets, in which the effect of decoherence would automatically come in. But, we do believe that our present framework is able to describe effectively the right physics derived from such improved treatment. Smallness of expansion parameters and higher order corrections {#sec:higher-order} ---------------------------------------------------------------- Here, we discuss general structure of the perturbation series without recourse to averaging out the fast oscillations. The effective expansion parameters in our perturbative framework are the following four, $$\begin{aligned} \frac{A W }{ \Delta_{J} - h_{i} }, \hspace{10mm} \frac{A W }{ h_{j} - h_{i} }, \hspace{10mm} A L W, \hspace{6mm} \text{and} \hspace{6mm} W. \label{expansion-parameters}\end{aligned}$$ We already saw them, except for the last one, in the discussion in section \[sec:energy-denominator\], and it can be seen by inspecting the expressions of the oscillation probabilities up to the fourth orders given in section \[sec:probability-2nd\] and appendix \[sec:expression-probability-4th\]. Formally, the expansion parameter is the first one in (\[expansion-parameters\]) in view of (\[Omega-expand\]) with $\Omega [1]$, the kernel, in (\[Omega-1st-order\]). But, the spacial integration in (\[Omega-expand\]) produces different effective expansion parameters, the second and the third ones in (\[expansion-parameters\]). The extra factor of $W$’s without the kinematical factors is provided when transforming from the $\hat{S}$ to $S$ matrices, as seen in section \[sec:S-matrix\]. For simplicity of the discussion in this section, we limit ourselves to the case of $|W| \sim 0.1$. Under the same conditions we have imposed in section \[sec:energy-denominator\] the first one in (\[expansion-parameters\]) is $\simeq 7.6 \times 10^{-4}$ for $\Delta m^2_{J i} = 0.1$ eV$^2$ and $\rho E = 10 \text{ (g/cm}^3) \text{GeV}$, while the second and the third, which are comparable at around the first oscillation maximum, are estimated as $2.3 \times 10^{-2}$ for the same condition. Therefore, the smallness of the expansion parameter is ensured unless $\rho E \gg 10 \text{ (g/cm}^3) \text{GeV}$. In fact, a close examination of the order $W^4$ terms in the oscillation probability (see appendix \[sec:expression-probability-4th\]) shows that all the formally $W^4$ terms are actually further suppressed. The largest term in the fourth-order oscillation probabilities is of the one suppressed by a factor $\left\vert \left( \frac{A W }{ \Delta_{J} - h_{i} } \right) \left( A L W \right) W^2 \right\vert \lsim 1.7 \times 10^{-7}$, which is as small as $\sim10^{-4}$ even in the case $|W|=0.5$. Therefore, we expect that the formula for the oscillation probability in (\[P-beta-alpha-final\]) works under much relaxed conditions than the one in (\[suppression-cond\]). On Uniqueness theorem and matter-dependent dynamical phase {#sec:U-theorem} ------------------------------------------------------------ We have shown in sections \[sec:probability-2nd\] and \[sec:probability-4th\] that there is no surviving matter dependent correction term in the oscillation probability up to order $W^4$ after averaging out fast oscillations and using the suppression by energy denominators. Should we expect that this feature is stable against higher order corrections beyond order $W^4$, as postulated by the Uniqueness theorem, in our perturbative framework? We argue that the answer is [*Yes*]{}. We first note that higher-order corrections in terms of $W$ are computed by using $\Omega [1]$ as the kernel, as indicated in eq. (\[Omega-expand\]). Notice also that all the elements of $\Omega [1]$, except for $\Omega [1]_{JJ}$, carry the energy denominator, as shown in (\[Omega-1st-order\]). Then, higher order correction terms are always accompanied by the energy denominators which are composed of some of the first three in (\[expansion-parameters\]), and therefore they are suppressed. The unique exception for it is the terms generated only by $\Omega [1]_{JJ}$ which lacks the energy denominator. Therefore, apart from this special case, we have shown that higher-order corrections in $W$ does not produce the surviving terms after averaging over fast oscillation and using the energy denominator suppression. It is consistent with what we saw in our explicit computation to order $W^4$. This concludes our justification of the Uniqueness theorem. We need to clear up the issue of special type of perturbative correction terms which involve only $\Omega [1]_{JJ}$ as the kernel in (\[Omega-expand\]). It produces the unique form of $\hat{S}_{JJ}$ as $$\begin{aligned} \hat{S}_{JJ} = e^{ - i \Delta_{J} x } \sum_{n} \frac{ (-ix)^n }{n !} \left\{ (W^{\dagger} A W)_{JJ} \right\}^n, \label{hatS-JJ-term}\end{aligned}$$ a collection of terms of matter-dependent higher order renormalization to $\sum_{J} W_{\alpha J} W_{\beta J}^*$, the probability leaking term at the amplitude level. However, it exponentiates and has contribution to the $S$ matrix element as[^22] $$\begin{aligned} S_{\alpha \beta} = \sum_{J} W_{\alpha J} W_{\beta J}^* \exp { \left[ -i \left\{ \Delta_{J} + (W^{\dagger} A W )_{JJ} \right\} x \right] }. \label{S-alpha-beta-WW-term}\end{aligned}$$ The unique form of $S$ matrix, in principle, raises an interesting issue of dynamically generated phase produced jointly by unitarity violation and matter effect.[^23] In our setting, however, it either disappears from the amplitude squared, or has vanishing effect when the high frequency oscillation is averaged out. Absence of enhancement due to small solar mass splitting denominator {#sec:no-enhancement} ---------------------------------------------------------------------- In perturbation theory one has to sum up intermediate states including off mass shell states. Therefore, even though we sit in the kinematic region where atmospheric-scale oscillations are large the energy denominator can become small, to the order of solar $\Delta m^2$ mass splitting. Then, one might question whether the correction terms blow up at the small denominator, which would invalidate our perturbative treatment. Fortunately, one can show that the “singularity” which could be produced in the limit of small solar mass splitting always cancels against the small numerator of the similar size. This problem exists already in the second-order expression of the oscillation probability (\[P-beta-alpha-0th+2nd\]). See the second term in second order (in $W$) term. If we denote $h_{l} - h_{k} \equiv \epsilon$ the term would have $1/\epsilon$ singularity in the limit of $\epsilon \rightarrow 0$. However, one can see by inspection by eye that the expression inside the square parenthesis is antisymmetric under $l \leftrightarrow k$, and hence it is of order $\epsilon$ or higher. Therefore, the singularity cancels. Notice that the antisymmetry under $l \leftrightarrow k$ is not required for the whole expression including the matrix element factor. The situation is a little bit more complicated in the fourth-order expression of the oscillation probability given in appendix \[sec:expression-probability-4th\]. In addition to $1/\epsilon$ singularity similar to the one we already saw, there exist apparent singularity of $1/\epsilon^2$ type. See, for example, the second term in (\[P-beta-alpha-W4-H4-diag\]) and the last term in (\[P-beta-alpha-W4-H4-single\]). But, an explicit calculation shows that the $1/\epsilon^2$ singularity always cancels against order $\epsilon^2$ numerator in the limit of small solar splitting. This phenomenon is reminiscent of the finiteness of the oscillation probability at the small solar mass splitting limit in helio-perturbation theory with the unique expansion parameter $\frac{\Delta m^2_{21}}{\Delta m^2_{31}}$ (or a renormalized one), see e.g., [@Minakata:2015gra] and the references therein. Possible interpretation of applicability of the perturbative framework to the region of solar level crossing has been discussed [@Xu:2015kma; @Ge:2016dlx]. Another example for the similar phenomena is the one at the small atmospheric mass splitting limit with additional expansion parameter $\sin \theta_{13}$. In this case it is observed that near the atmospheric resonance region not only the oscillation probability is finite but also its accuracy improves when the higher order terms to fourth order in $\sin \theta_{13}$ is added [@Asano:2011nj]. Then, one might ask if our small unitarity violation perturbation theory gives quantitatively accurate result at around the denominator with small solar mass splitting. However, we note that this problem is not relevant in our case because all these terms with apparent singularities vanish after averaging over the high-frequency oscillations and using the suppression by the sterile mass denominators. Yet, we must remark that if we investigate possible enhancement of the correction terms outside the condition (\[suppression-cond\]), as done in section \[sec:correction-terms\], the quantitative accuracy of the expression may become an issue. Concluding remarks {#sec:conclusion} ================== In this paper, we have presented a comprehensive treatment of the three active plus $N$ sterile neutrino model in the context of leptonic unitarity test. We have formulated an appropriate perturbative framework with expansion in small unitarity violating $W$ matrix elements, while keeping (non-$W$ suppressed) matter effect to all orders. What we have done in this paper is mainly threefold: - We have shown that the same condition on sterile state masses $0.1\, \text{eV}^2 \lsim m^2_{J} \lsim 1\, \text{MeV}^2$, as we imposed in vacuum, is sufficient to make the $(3+N)$ model sterile-sector model independent, apart from $N$ dependence in the lower bound on probability leaking constant $\mathcal{C}_{\alpha \beta}$. We have shown in an explicit computation to fourth order in $W$ that these higher order correction terms either vanish or are negligible after averaging over fast oscillations and using energy denominator suppression, leaving only the leaking term $\mathcal{C}_{\alpha \beta}$. We have argued by postulating the “Uniqueness theorem” that this feature prevails to all orders in $W$ perturbation theory. - As an outcome of our framework, we have derived very simple formulas for oscillation probabilities between active neutrinos in the $(3+N)$ model. We have analyzed them in $\nu_{\mu} \rightarrow \nu_{\alpha}$ channel, $\alpha=e,\mu,\tau$, to know in which region of energy and baseline the effects of unitarity violation are large. We have observed relatively large effects in $\nu_{\mu} \rightarrow \nu_{\mu}$ and $\nu_{\mu} \rightarrow \nu_{\tau}$ channels and pointed out, though qualitatively, that anticorrelation of signals between them would enhance the sensitivity to unitarity violating effects. - We have discussed the question of how to distinguish low-scale unitarity violation from high-scale one. We have pointed out that the second order $W$ correction to the leading order, if detected, would signal low-scale unitarity violation. This is to add to the way we have discussed in [@Fong:2016yyh], i.e., to detect the probability leaking term $\mathcal{C}_{\alpha \beta}$ which testifies for the existence of “sterile”, or undetectable but communicable, sector at low energy scales. Notice that in “constraining mode” of unitarity violation, the model-independence of the framework translates into a universal nature of the bounds, thereby making them more powerful. Whereas in “discovery mode” of unitarity violation, the model dependence, in particular through the $W$ dependent correction terms, is welcome because it serves for identifying the structure of the sterile sector. During the course of this work, we have obtained the new results and had some interesting observations including: - We have obtained an exact solution of the Hamiltonian (\[H0-def\]) for uniform matter density. It describes neutrino evolution in low-scale unitarity violation in zeroth-order in $W$, which applies also to the case of high-scale unitarity violation. The key parts of the solution can also be utilized to calculate the higher order corrections in $W$. When applied to each shell inside the earth, it could provide a semi-quantitative way of simulating non-unitary neutrino evolution for the terrestrial experiments. - We have observed outside of the region of validity of our Uniqueness theorem there is a limited phase space, $\rho E \gg 10 \text{ (g/cm}^3) \text{GeV}$, in which the second order $W$ correction terms can be sizeable. If detected, it may allow us to probe structure of $W$ matrix elements which bridges between active and sterile sectors. - We have examined a toy model of equally distributed $W$ matrix elements within each flavor, in which the probability leaking term scales as $1/N$. Likewise, possible detection of $\mathcal{C}_{\alpha \beta}$ may provide information of the hidden sterile sector. We emphasize that, unlike in the case of high-scale unitarity violation, neutrino experiment is the most powerful way to execute leptonic unitarity test in the scenarios of low-scale unitarity violation. Nonetheless, we have to admit that our observations on what we could do for experimental detection of possible non-unitarity effects are too qualitative to make any definitive claim for possible detection in the future. Clearly, the better analyses are called for. While we worked exclusively on the $(3+N)$ state unitary model as a model of low-scale unitarity violation, we do not know if it is the unique choice, or it merely reflects our ignorance. Even in the case there exist more generic class of models for low-scale unitarity violation, the phenomenon of probability leaking is likely to survive. It is because the probability leaking must take place whenever the extra light sector exists and communicates with the three active neutrinos. One of the authors (H.M.) thanks Enrique Fernandez-Martinez for interesting discussions about the relationship between high-scale and low-scale unitarity violation. He expresses a deep gratitude to Instituto Física Teórica, UAM/CSIC in Madrid, for its support via “Theoretical challenges of new high energy, astro and cosmo experimental data” project, Ref: 201650E082. He had been a member of Yachay Tech for 14 months, at that time the first research-oriented university in Ecuador [@Yachay-story], during which he was warmly supported by Ecuadorian people. He thanks kind supports by ICTP-SAIFR (FAPESP grant 2016/01343-7), UNICAMP (FAPESP grant 2014/19164-6), and PUC-Rio (CNPq) which enabled him to visit these institutions where part of this work was done. C.S.F. is supported by FAPESP under grants 2013/01792-8 and 2012/10995-7. H.N. is supported by CNPq. This work was supported in part by the Fermilab Neutrino Physics Center. $\hat{S}$ matrix elements {#sec:hatS-elements} ========================= The method of computing $\hat{S}$ matrix elements is outlined in section \[sec:hatS-matrix\]. In this appendix \[sec:hatS-elements\] we carry out this task. With the expression of $H_{1}$ in (\[H1-matrix\]) we compute perturbatively the matrix elements of $\Omega (x)$. Then, $\hat{S} (x) = e^{- i \hat{H}_{0} x} \Omega(x)$. We denote computated results of the matrix elements of $\hat{S}$ as $\hat{S} [n]$ to indicate that it is the one that comes from $n$-th order contribution in $H_{1}$. Since the elements of $H_{1}$ are of order either $W$ or $W^2$, $\hat{S} [n]$ generally has order $W^n$ or higher. To show that a particular contribution is of order $W^m$ we use the superscript $``(m)''$. That is, $\hat{S}^{(m)} [n]$ denotes contribution to $\hat{S}$ that arizes from $n$-th order perturbative contribution in $H_{1}$ and is of order $W^m$. Contribution to $\hat{S}$ matrix elements from zeroth and first order in $H_{1}$ {#sec:hatS-0th-1st} -------------------------------------------------------------------------------- The zeroth and first order $\hat{S}$ matrix elements can be calculated as follows: $$\begin{aligned} \hat{S}_{ii}^{(0+1)} &=& \left( e^{-i \hat{H}_{0} x} \right)_{i k} (\Omega_{k i}) + \left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K i}) = e^{-i h_{i} x} (\Omega_{i i}) = e^{-i h_{i} x}, \nonumber \\ \hat{S}_{i j}^{(1)} \vert_{i \neq j} &=& \left( e^{-i \hat{H}_{0} x} \right)_{i k} (\Omega_{k j}) + \left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K j}) = e^{-i h_{i} x} (\Omega_{i j}) = 0, \nonumber \\ \hat{S}_{i J}^{(1)} &=& \left( e^{-i \hat{H}_{0} x} \right)_{i j} (\Omega_{j J}) + \left( e^{-i \hat{H}_{0} x} \right)_{i K} (\Omega_{K J}) = e^{- i h_{i} x} (\Omega_{i J}) = \frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \left\{ (UX)^{\dagger} A W \right\}_{i J}, \nonumber \\ \hat{S}_{J i}^{(1)} &=& \left( e^{-i \hat{H}_{0} x} \right)_{J k} (\Omega_{k i}) + \left( e^{-i \hat{H}_{0} x} \right)_{J K} (\Omega_{K i}) = \frac{e^{- i \Delta_{J} x} - e^{ - i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \left\{ W ^{\dagger} A (UX) \right\}_{J i}, \nonumber \\ \hat{S}_{J K}^{(1)} \vert_{J \neq K} &=& \left( e^{-i \hat{H}_{0} x} \right)_{J i} (\Omega_{i K}) + \sum_{I} \left( e^{-i \hat{H}_{0} x} \right)_{J I} (\Omega_{I K}) = \frac{e^{- i \Delta_{K} x } - e^{- i \Delta_{J} x} }{ ( \Delta_{K} - \Delta_{J} ) } \left( W ^{\dagger} A W \right)_{J K}, \nonumber \\ \hat{S}_{J J}^{(0+1)} &=& \sum_{i} \left( e^{-i \hat{H}_{0} x} \right)_{J i} (\Omega_{i J}) + \sum_{I} \left( e^{-i \hat{H}_{0} x} \right)_{J I} (\Omega_{I J}) = e^{-i \Delta_{J} x} \left[ 1 - (i x) \left( W ^{\dagger} A W \right)_{J J} \right]. \label{hat-S-elements-1st}\end{aligned}$$ Contribution to $\hat{S}$ matrix elements from second order in $H_{1}$ {#sec:hatS-2nd} ---------------------------------------------------------------------- Likewise, $\hat{S}$ matrix elements can be calculated in second order in $\hat{H}_{1}$ by using the formula for $\Omega$ in (\[Omega-expand\]) and $\hat{S}$-$\Omega$ relation in (\[hatS-Omega\]) as $$\begin{aligned} \hat{S}_{i i}^{(2)} [2] &=& \sum_{K} \left[ (ix) e^{- i h_{i} x} + \frac{e^{- i \Delta_{K} x} - e^{- i h_{i} x} }{ ( \Delta_{K} - h_{i} ) } \right] \left\{ (UX)^{\dagger} A W \right\}_{i K} \frac{ 1 }{ \Delta_{K} - h_{i} } \left\{ W ^{\dagger} A (UX) \right\}_{K i}, \nonumber \\ \hat{S}_{i j}^{(2)} \vert_{i \neq j} [2] &=& - \sum_{K} \left[ \frac{e^{- i h_{j} x} - e^{- i h_{i} x} }{ ( h_{j} - h_{i} ) } - \frac{e^{- i \Delta_{K} x} - e^{- i h_{i} x} }{ ( \Delta_{K} - h_{i} ) } \right] \left\{ (UX)^{\dagger} A W \right\}_{i K} \frac{ 1 }{ \Delta_{K} - h_{j} } \left\{ W ^{\dagger} A (UX) \right\}_{K j}, \nonumber \\ \hat{S}_{i J}^{(3)} [2] &=& - \left[ (ix) e^{- i \Delta_{J} x} + \frac{ e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \right] \left\{ (UX)^{\dagger} A W \right\}_{i J} \frac{ 1 }{ \Delta_{J} - h_{i} } \left\{ W^{\dagger} A W \right\}_{J J} \nonumber \\ &+& \sum_{K \neq J} \left[ \frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } - \frac{e^{- i \Delta_{K} x} - e^{- i h_{i} x} }{ ( \Delta_{K} - h_{i} ) } \right] \left\{ (UX)^{\dagger} A W \right\}_{i K} \frac{ 1 }{ \Delta_{J} - \Delta_{K} } \left\{ W^{\dagger} A W \right\}_{K J}, \nonumber \\ \hat{S}_{J i}^{(3)} [2] &=& - \left[ (ix) e^{- i \Delta_{J} x} + \frac{ e^{- i \Delta_{J} x} - e^{ - i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \right] \left\{ W^{\dagger} A W \right\}_{J J} \frac{ 1 }{ \Delta_{J} - h_{i} } \left\{ W^{\dagger} A (UX) \right\}_{J i} \nonumber \\ &-& \sum_{K \neq J} \left[ \frac{e^{- i h_{i} x} - e^{- i \Delta_{J} x} }{ h_{i} - \Delta_{J} } - \frac{e^{- i \Delta_{K} x} - e^{- i \Delta_{J} x} }{ ( \Delta_{K} - \Delta_{J} ) } \right] \left\{ W^{\dagger} A W \right\}_{J K} \frac{ 1 }{ \Delta_{K} - h_{i} } \left\{ W^{\dagger} A (UX) \right\}_{K i}, \nonumber \\ \hat{S}_{I J}^{(2+4)} \vert_{I \neq J} [2] &=& \sum_{k} \left[\frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) } - \frac{e^{- i h_{k} x} - e^{- i \Delta_{I} x} }{ ( h_{k} - \Delta_{I} ) } \right] \left\{ W^{\dagger} A (UX) \right\}_{I k} \frac{ 1 }{ \Delta_{J} - h_{k} } \left\{ (UX)^{\dagger} A W \right\}_{k J} \nonumber \\ &+& \left[ (ix) e^{- i \Delta_{I} x} + \frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) } \right] \left\{ W^{\dagger} A W \right\}_{I I} \frac{ 1 }{ \Delta_{J} - \Delta_{I} } \left\{ W^{\dagger} A W \right\}_{I J} \nonumber \\ &-& \left[ (ix) e^{- i \Delta_{J} x} + \frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) } \right] \left\{ W^{\dagger} A W \right\}_{I J} \frac{ 1 }{ \Delta_{J} - \Delta_{I} } \left\{ W^{\dagger} A W \right\}_{J J} \nonumber \\ &+& \sum_{K \neq I, J} \left[ \frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} ) } - \frac{e^{- i \Delta_{K} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{K} - \Delta_{I} ) } \right] \left\{ W^{\dagger} A W \right\}_{I K} \frac{ 1 }{ \Delta_{J} - \Delta_{K} } \left\{ W^{\dagger} A W \right\}_{K J}, \nonumber \\ \hat{S}_{I I}^{(2+4)} [2] &=& - \sum_{k} \left[ (ix) e^{- i \Delta_{I} x} + \frac{e^{- i h_{k} x} - e^{- i \Delta_{I} x} }{ h_{k} - \Delta_{I} } \right] \left\{ W^{\dagger} A (UX) \right\}_{I k} \frac{ 1 }{ \Delta_{I} - h_{k} } \left\{ (UX)^{\dagger} A W \right\}_{k I} \nonumber \\ &+& \frac{ (- ix)^2 }{ 2 } e^{- i \Delta_{I} x} \left( \left\{ W^{\dagger} A W \right\}_{I I} \right)^2 \nonumber \\ &-& \sum_{K \neq I} \left[ (ix) e^{- i \Delta_{I} x} + \frac{e^{- i \Delta_{K} x} - e^{- i \Delta_{I} x} }{ \Delta_{K} - \Delta_{I} } \right] \left\{ W^{\dagger} A W \right\}_{I K} \frac{ 1 }{ \Delta_{I} - \Delta_{K} } \left\{ W^{\dagger} A W \right\}_{K I}. \label{hatS-2nd-order}\end{aligned}$$ We now discuss generalized T invariance of second order $\hat{S}$ matrix elements. Let us first examine $\hat{S}_{i j} \vert_{i \neq j} [2]$. We first note that the matrix element transforms under generalized T transformation $i \leftrightarrow j$, $U \rightarrow U^*$, $W \rightarrow W^*$, etc. as $$\begin{aligned} \left\{ (UX)^{\dagger} A W \right\}_{i K} &\rightarrow & \left\{ W^{\dagger} A (UX) \right\}_{K j}, \nonumber \\ \left\{ W^{\dagger} A (UX) \right\}_{K j} &\rightarrow & \left\{ (UX)^{\dagger} A W \right\}_{i K}. \label{matrix-T-transf}\end{aligned}$$ Therefore, the matrix element part is consistent with generalized T invariance. However, the rest of $\hat{S}_{i j} [2] \vert_{i \neq j}$ does not appear to be manifestly generalized T invariant. But, one notices that $\hat{S}_{i j} \vert_{i \neq j} [2]$ can be written as $$\begin{aligned} && \hat{S}_{i j}^{(2)} \vert_{i \neq j} [2] = - \sum_{K} \frac{ 1 }{ ( h_{j} - h_{i} ) (\Delta_{K} - h_{i}) (\Delta_{K} - h_{j}) } \nonumber \\ &\times& \left[ \left( e^{- i h_{j} x} - e^{- i h_{i} x} \right) \Delta_{K} - ( h_{j} - h_{i} ) e^{- i \Delta_{K} x} + h_{j} e^{- i h_{i} x} - h_{i} e^{- i h_{j} x} \right] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K j}. \label{hatS-ij[2]}\end{aligned}$$ Then, $\hat{S}_{i j} \vert_{i \neq j} [2] (U, W, \text{etc.} )= \hat{S}_{j i} \vert_{i \neq j} [2] (U^*, W^*, \text{etc.} )$ holds, that is, generalized T invariance is maintained. Similarly, the first and the fourth terms in $\hat{S}_{I J} \vert_{I \neq J} [2]$ are not manifestly generalized T invariant. But, they can also be written as manifestly generalized T invariant forms as $$\begin{aligned} && \hat{S}_{I J}^{(2+4)} \vert_{I \neq J} [2] = \sum_{k} \frac{ 1 }{ ( \Delta_{J} - \Delta_{I} ) (\Delta_{J} - h_{k}) (\Delta_{I} - h_{k}) } \nonumber \\ &\times& \left[ \Delta_{J} e^{- i \Delta_{I} x} - \Delta_{I} e^{- i \Delta_{J} x} + \left( e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} \right) h_{k} - ( \Delta_{J} - \Delta_{I} ) e^{- i h_{k} x} \right] \nonumber \\ &\times& \left\{ W^{\dagger} A (UX) \right\}_{I k} \left\{ (UX)^{\dagger} A W \right\}_{k J} \nonumber \\ &+& (ix) e^{- i \Delta_{I} x} \frac{ 1 }{ \Delta_{J} - \Delta_{I} } \left\{ W^{\dagger} A W \right\}_{I I} \left\{ W^{\dagger} A W \right\}_{I J} \nonumber \\ &-& (ix) e^{- i \Delta_{J} x} \frac{ 1 }{ \Delta_{J} - \Delta_{I} } \left\{ W^{\dagger} A W \right\}_{I J} \left\{ W^{\dagger} A W \right\}_{J J} \nonumber \\ &+& \frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} )^2 } \left\{ W^{\dagger} A W \right\}_{I I} \left\{ W^{\dagger} A W \right\}_{I J} \nonumber \\ &-& \frac{e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} }{ ( \Delta_{J} - \Delta_{I} )^2 } \left\{ W^{\dagger} A W \right\}_{I J} \left\{ W^{\dagger} A W \right\}_{J J} \nonumber \\ &-& \sum_{K \neq I, J} \frac{ 1 }{ ( \Delta_{J} - \Delta_{I} ) ( \Delta_{K} - \Delta_{I} ) ( \Delta_{K} - \Delta_{J} ) } \nonumber \\ &\times& \left[ \Delta_{J} e^{- i \Delta_{I} x} - \Delta_{I} e^{- i \Delta_{J} x} + \left( e^{- i \Delta_{J} x} - e^{- i \Delta_{I} x} \right) \Delta_{K} - ( \Delta_{J} - \Delta_{I} ) e^{- i \Delta_{K} x} \right] \nonumber \\ &\times& \left\{ W^{\dagger} A W \right\}_{I K} \left\{ W^{\dagger} A W \right\}_{K J}. \label{hatS-IJ(2+4)}\end{aligned}$$ Under generalized T transformation, the second term goes to the third term (both $ix$ terms), and vice versa, and hence they are invariant. The situation is completely the same as in the fourth and the fifth terms. The first and the last terms are invariant in an analogous way as $\hat{S}_{i j} \vert_{i \neq j} [2]$. For $\hat{S}_{i J} [2]$ and $\hat{S}_{J i} [2]$ generalized T invariance holds as they are. For $\hat{S}_{i i} [2]$ and $\hat{S}_{I I} [2]$ the matrix factor is self-invariant, and therefore generalized T invariance holds. What we should do in the rest of appendix is to compute $\hat{S}$ matrix elements perturbatively to fourth order in $H_{1}$. In presenting the results of computation of $\hat{S}$ matrix elements, however, we change strategy of our description. That is, - We present $\hat{S}$ matrix elements at fixed order in $W$. To do this we, of course, have to sum up contributions that come from different order in $H_{1}$. For example, $\hat{S}_{i J}^{(3)} = \hat{S}_{i J}^{(3)} [3] + \hat{S}_{i J}^{(3)} [2]$. - We present only the terms which are required to compute $S$ matrix elements to order $W^4$. In view of the relations between $\hat{S}$ and $S$ matrix elements given in eq. (\[Sab-hatSab\]), $\hat{S}_{I J}^{(3)}$, $\hat{S}_{I J}^{(4)}$, and $\hat{S}_{i J}^{(4)}$ (and $\hat{S}_{J i}^{(4)}$) are all unnecessary. - We only give the results of manifestly generalized T invariant form of $\hat{S}$ matrix elements with which it must be straightforward to prove generalized T invariance. Order $W^2$ $\hat{S}$ matrix elements from Second-order in $H_{1}$ {#sec:hatSIJ-2nd} ------------------------------------------------------------------- The second order in $H_{1}$ contribution $\hat{S}_{I J} [2]$ (and $\hat{S}_{I I} [2]$) has both order $W^2$ and $W^4$ terms. To compute $S$ matrix elements to order $W^4$ we need only the former term because the latter produces order $W^6$ terms, as one can see in (\[Sab-hatSab\]). Then, it is necessary to remind the readers that the expressions of order $W^2$ terms in $\hat{S}_{I J} [2]$ and $\hat{S}_{I I} [2]$ already exist as the first terms in (\[hatS-IJ(2+4)\]) and (\[hatS-2nd-order\]), respectively. Order $W^3$ $\hat{S}$ matrix elements $\hat{S}_{i J}^{(3)}$ and $\hat{S}_{J i}^{(3)}$ {#sec:hatSiJ-2nd-]} -------------------------------------------------------------------------------------- By adding the contribution of the third and second order in $H_{1}$ we obtain for $\hat{S}$ matrix elements $\hat{S}_{i J}^{(3)}$ at order $W^3$ $$\begin{aligned} && \hat{S}_{i J}^{(3)} [2+3] \nonumber \\ &=& - \frac{ 1 }{ \Delta_{J} - h_{i} } \left[ (ix) e^{- i \Delta_{J} x} + \frac{ e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \right] \left\{ (UX)^{\dagger} A W \right\}_{i J} \left\{ W^{\dagger} A W \right\}_{J J} \nonumber \\ &+& \sum_{K \neq J} \frac{ 1 }{ ( \Delta_{J} - \Delta_{K} ) ( \Delta_{J} - h_{i} ) ( \Delta_{K} - h_{i} ) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{i} \right) e^{- i \Delta_{J} x} - \left( \Delta_{J} - h_{i} \right) e^{- i \Delta_{K} x} - \left( \Delta_{K} - \Delta_{J} \right) e^{- i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W^{\dagger} A W \right\}_{K J} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{J} - h_{i} )^2 } \biggl[ (ix) \left( e^{- i h_{i} x} + e^{- i \Delta_{J} x} \right) + 2 \frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i J} \left\{ W^{\dagger} A (UX) \right\}_{J i} \left\{ (UX)^{\dagger} A W \right\}_{i J} \nonumber \\ &+& \sum_{k \neq i} \biggl[ - \frac{ (ix) e^{- i \Delta_{J} x} }{ ( \Delta_{J} - h_{i} )( \Delta_{J} - h_{k} ) } + \frac{ 1 }{ ( h_{i} - h_{k} ) (\Delta_{J} - h_{i} )^2 (\Delta_{J} - h_{k} )^2 } \nonumber \\ &\times& \biggl\{ (\Delta_{J} - h_{k} )^2 e^{- i h_{i} x} - (\Delta_{J} - h_{i} )^2 e^{- i h_{k} x} + ( h_{i} - h_{k} )( h_{i} + h_{k} - 2 \Delta_{J} ) e^{- i \Delta_{J} x} \biggr\} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i J} \left\{ W^{\dagger} A (UX) \right\}_{J k} \left\{ (UX)^{\dagger} A W \right\}_{k J} \nonumber \\ &+& \sum_{K \neq J} \biggl[ - \frac{ (ix) e^{- i h_{i} x} }{ (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) } + \frac{ 1 }{ (\Delta_{J} - \Delta_{K}) (\Delta_{J} - h_{i})^2 ( \Delta_{K} - h_{i} )^2 } \nonumber \\ &\times& \biggl\{ ( \Delta_{K} - h_{i} )^2 e^{- i \Delta_{J} x} - (\Delta_{J} - h_{i})^2 e^{- i \Delta_{K} x} + e^{- i h_{i} x} (\Delta_{J} - \Delta_{K}) (\Delta_{J} + \Delta_{K} - 2 h_{i} ) \biggr\} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W^{\dagger} A (UX) \right\}_{K i} \left\{ (UX)^{\dagger} A W \right\}_{i J} \nonumber \\ &+& \sum_{K \neq J} \sum_{k \neq i} \frac{ 1 }{ (\Delta_{J} - \Delta_{K}) ( h_{k} - h_{i} ) (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) (\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) } \nonumber \\ &\times& \biggl[ ( h_{k} - h_{i} ) \biggl\{(\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) e^{- i \Delta_{J} x} - (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) e^{- i \Delta_{K} x} \biggr\} \nonumber \\ &-& (\Delta_{J} - \Delta_{K}) \biggl\{ (\Delta_{J} - h_{k}) (\Delta_{K} - h_{k}) e^{- i h_{i} x} - (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) e^{- i h_{k} x} \biggr\} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k J}, \label{hatS-iJ-W3-H2+3}\end{aligned}$$ and similarly, for $\hat{S}_{J i}^{(3)}$ $$\begin{aligned} && \hat{S}_{J i}^{(3)} [2+3] \nonumber \\ &=& - \frac{ 1 }{ \Delta_{J} - h_{i} } \left[ (ix) e^{- i \Delta_{J} x} + \frac{ e^{- i \Delta_{J} x} - e^{ - i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \right] \left\{ W^{\dagger} A W \right\}_{J J} \left\{ W^{\dagger} A (UX) \right\}_{J i} \nonumber \\ &+& \sum_{K \neq J} \frac{ 1 }{ ( \Delta_{J} - \Delta_{K} ) ( \Delta_{J} - h_{i} ) ( \Delta_{K} - h_{i} ) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{i} \right) e^{- i \Delta_{J} x} - \left( \Delta_{J} - h_{i} \right) e^{- i \Delta_{K} x} - \left( \Delta_{K} - \Delta_{J} \right) e^{- i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ W^{\dagger} A W \right\}_{J K} \left\{ W^{\dagger} A (UX) \right\}_{K i} \nonumber \\ &-& \frac{ 1 }{ (\Delta_{J} - h_{i})^2 } \biggl[ (ix) \left( e^{- i h_{i} x} + e^{- i \Delta_{J} x} \right) + 2 \frac{e^{- i \Delta_{J} x} - e^{- i h_{i} x} }{ ( \Delta_{J} - h_{i} ) } \biggr] \nonumber \\ &\times& \left\{ W^{\dagger} A (UX) \right\}_{J i} \left\{ (UX)^{\dagger} A W \right\}_{i J} \left\{ W^{\dagger} A (UX) \right\}_{J i} \nonumber \\ &+& \sum_{k \neq i} \biggl[ - \frac{ (ix) e^{- i \Delta_{J} x} }{ ( \Delta_{J} - h_{i} )( \Delta_{J} - h_{k} ) } + \frac{ 1 }{ ( h_{i} - h_{k} ) (\Delta_{J} - h_{i} )^2 (\Delta_{J} - h_{k} )^2 } \nonumber \\ &\times& \biggl\{ (\Delta_{J} - h_{k} )^2 e^{- i h_{i} x} - (\Delta_{J} - h_{i} )^2 e^{- i h_{k} x} + ( h_{i} - h_{k} )( h_{i} + h_{k} - 2 \Delta_{J} ) e^{- i \Delta_{J} x} \biggr\} \biggr] \nonumber \\ &\times& \left\{ W^{\dagger} A (UX) \right\}_{J k} \left\{ (UX)^{\dagger} A W \right\}_{k J} \left\{ W^{\dagger} A (UX) \right\}_{J i} \nonumber \\ &+& \sum_{K \neq J} \biggl[ - \frac{ (ix) e^{- i h_{i} x} }{ (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) } + \frac{ 1 }{ (\Delta_{J} - \Delta_{K}) (\Delta_{J} - h_{i})^2 ( \Delta_{K} - h_{i} )^2 } \nonumber \\ &\times& \biggl\{ ( \Delta_{K} - h_{i} )^2 e^{- i \Delta_{J} x} - (\Delta_{J} - h_{i})^2 e^{- i \Delta_{K} x} + e^{- i h_{i} x} (\Delta_{J} - \Delta_{K}) (\Delta_{J} + \Delta_{K} - 2 h_{i} ) \biggr\} \biggr] \nonumber \\ &\times& \left\{ W^{\dagger} A (UX) \right\}_{J i} \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W^{\dagger} A (UX) \right\}_{K i} \nonumber \\ &+& \sum_{K \neq J} \sum_{k \neq i} \frac{ 1 }{ (\Delta_{J} - \Delta_{K}) ( h_{k} - h_{i} ) (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) (\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) } \nonumber \\ &\times& \biggl[ ( h_{k} - h_{i} ) \biggl\{ (\Delta_{K} - h_{i}) (\Delta_{K} - h_{k}) e^{- i \Delta_{J} x} - (\Delta_{J} - h_{i}) (\Delta_{J} - h_{k}) e^{- i \Delta_{K} x} \biggr\} \nonumber \\ &-& (\Delta_{J} - \Delta_{K}) \biggl\{ (\Delta_{J} - h_{k}) (\Delta_{K} - h_{k}) e^{- i h_{i} x} - (\Delta_{J} - h_{i}) (\Delta_{K} - h_{i}) e^{- i h_{k} x} \biggr\} \biggr] \nonumber \\ &\times& \left\{ W^{\dagger} A (UX) \right\}_{J k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W^{\dagger} A (UX) \right\}_{K i}. \label{hatS-Ji-W3-H2+3}\end{aligned}$$ Using (\[hatS-iJ-W3-H2+3\]) and (\[hatS-Ji-W3-H2+3\]), one can easily confirm generalized T invariance $\hat{S}_{i J}^{(3)} (U, W, \text{etc}) = \hat{S}_{J i}^{(3)} (U^*, W^*, \text{etc})$. Order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i j}^{(4)} \vert_{i \neq j} [3]$ and $\hat{S}_{i j}^{(4)} \vert_{i \neq j} [4]$ {#sec:hatSij-3nd-4th} -------------------------------------------------------------------------------------------------------------------------------- The third order in $H_{1}$ contribution to order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i j}^{(4)} \vert_{i \neq j}$ is given by $$\begin{aligned} && \hat{S}_{ij}^{(4)} \vert_{i \neq j} [3] \nonumber \\ &=& - \sum_{K} (ix) e^{- i \Delta_{K} x} \frac{ 1 }{ \Delta_{K} - h_{i} } \frac{ 1 }{ \Delta_{K} - h_{j} } \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A W \right\}_{K K} \left\{ W ^{\dagger} A (UX) \right\}_{K j} \nonumber \\ &+& \sum_{K} \frac{ 1 }{ ( h_{j} - h_{i} ) ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} )^2 } \nonumber \\ &\times& \biggl[ ( h_{j} - h_{i} ) ( h_{j} + h_{i} - 2 \Delta_{K} ) e^{- i \Delta_{K} x} + ( \Delta_{K} - h_{i} )^2 e^{ - i h_{j} x} - ( \Delta_{K} - h_{j} )^2 e^{ - i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A W \right\}_{K K} \left\{ W ^{\dagger} A (UX) \right\}_{K j} \nonumber \\ &+& \sum_{K \neq L} \frac{ 1 }{ ( h_{j} - h_{i} ) ( \Delta_{L} - \Delta_{K} ) ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) } \nonumber \\ &\times& \biggl[ ( h_{j} - h_{i} ) \biggl\{ ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) e^{- i \Delta_{L} x} - ( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) e^{- i \Delta_{K} x} \biggr\} \nonumber \\ &+& ( \Delta_{L} - \Delta_{K} ) \biggl\{ ( \Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) e^{- i h_{j} x} - ( \Delta_{K} - h_{j} ) ( \Delta_{L} - h_{j} ) e^{- i h_{i} x} \biggr\} \biggl] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A W \right\}_{K L} \left\{ W ^{\dagger} A (UX) \right\}_{L j}. \label{hatS-3rd-order-ij-3}\end{aligned}$$ For bookkeeping purpose we decompose the fourth order in $H_{1}$ contribution to order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i j}^{(4)} \vert_{i \neq j}$ into the two terms $$\begin{aligned} \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j} = \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{single sum}) + \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{double sum}) \label{hatS-4th-order-ij-T-transf}\end{aligned}$$ where $$\begin{aligned} && \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{single sum}) \nonumber \\ &=& \sum_{K} \biggl[ (ix) e^{ - i h_{i} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( h_{j} - h_{i} ) } - (ix) e^{ - i \Delta_{K} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} ) } \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( h_{j} - h_{i} )^2 } e^{ - i h_{j} x} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( h_{j} - h_{i} )^2 } \left( \Delta_{K} + 2 h_{j} - 3 h_{i} \right) e^{ - i h_{i} x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{K} - h_{j} )^2 } \left( h_{i} + 2 h_{j} - 3 \Delta_{K} \right) e^{ - i \Delta_{K} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K j} \nonumber \\ &+& \sum_{K} \biggl[ - (ix) e^{ - i h_{j} x} \frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( h_{j} - h_{i} ) } - (ix) e^{ - i \Delta_{K} x} \frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( \Delta_{K} - h_{i} ) } \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} )^3 } \left( h_{j} + 2 h_{i} - 3 \Delta_{K} \right) e^{ - i \Delta_{K} x} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{j} )^3 ( h_{j} - h_{i} )^2 } \left( \Delta_{K} + 2 h_{i} - 3 h_{j} \right) e^{ - i h_{j} x} + \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( h_{j} - h_{i} )^2 } e^{ - i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K j} \left\{ (UX)^{\dagger} A W \right\}_{j K} \left\{ W ^{\dagger} A (UX) \right\}_{K j} \nonumber \\ &+& \sum_{K} \sum_{k \neq i, j} \biggl[ - (ix) e^{- i \Delta_{K} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{K} - h_{k} ) } \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{j} )^2 ( \Delta_{K} - h_{k} ) } \left( h_{i} + h_{j} - 2 \Delta_{K} \right) e^{- i \Delta_{K} x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{j} )^2 ( \Delta_{K} - h_{k} ) ( h_{j} - h_{i} ) } e^{ - i h_{j} x} - \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{k} ) ( h_{j} - h_{i} ) } e^{ - i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K j}, \label{hatS-ij-W4-H4-single}\end{aligned}$$ and $$\begin{aligned} && \hat{S}_{ij}^{(4)} [4] \vert_{i \neq j}~(\text{double sum}) \nonumber \\ &=& \sum_{K \neq L} \biggl[ (ix) e^{- i h_{i} x} \frac{ 1 }{ (\Delta_{K} - h_{i} ) (\Delta_{L} - h_{i} ) ( h_{j} - h_{i} ) } \nonumber \\ &-& \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{i} )^2 (\Delta_{L} - h_{j} ) } e^{- i \Delta_{L} x} + \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{i} )^2 (\Delta_{K} - h_{j} ) } e^{- i \Delta_{K} x} \nonumber \\ &+& \frac{ 1 }{ (\Delta_{K} - h_{j} ) (\Delta_{L} - h_{j} ) ( h_{j} - h_{i} )^2 } e^{- i h_{j} x} \nonumber \\ &-& \frac{ 1 }{ (\Delta_{K} - h_{i} )^2 (\Delta_{L} - h_{i} )^2 ( h_{j} - h_{i} )^2 } \biggl\{ 3 h_{i}^2 - 2 h_{i} h_{j} + \left( h_{j} - 2 h_{i} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L} \biggr\} e^{- i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \left\{ (UX)^{\dagger} A W \right\}_{i L} \left\{ W ^{\dagger} A (UX) \right\}_{L j} \nonumber \\ &+& \sum_{K \neq L} \biggl[ - (ix) e^{- i h_{j} x} \frac{ 1 }{ (\Delta_{K} - h_{j} ) (\Delta_{L} - h_{j} ) ( h_{j} - h_{i} ) } \nonumber \\ &+& \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{i} ) (\Delta_{K} - h_{j} )^2 } e^{- i \Delta_{K} x} - \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{i} ) (\Delta_{L} - h_{j} )^2 } e^{- i \Delta_{L} x} \nonumber \\ &+& \frac{ 1 }{ (\Delta_{K} - h_{i} ) (\Delta_{L} - h_{i} ) ( h_{j} - h_{i} )^2 } e^{- i h_{i} x} \nonumber \\ &-& \frac{ 1 }{ (\Delta_{K} - h_{j} )^2 (\Delta_{L} - h_{j} )^2 ( h_{j} - h_{i} )^2 } \biggl\{ 3 h_{j}^2 - 2 h_{i} h_{j} - \left( 2 h_{j} - h_{i} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L} \biggr\} e^{- i h_{j} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K j} \left\{ (UX)^{\dagger} A W \right\}_{j L} \left\{ W ^{\dagger} A (UX) \right\}_{L j} \nonumber \\ &+& \sum_{K \neq L} \sum_{k \neq i, j} \biggl[ \frac{1}{ ( h_{k} - h_{i} ) ( h_{k} - h_{j} ) ( h_{j} - h_{i} ) ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) } \nonumber \\ &+& \biggl\{ ( h_{k} - h_{j} ) (\Delta_{K} - h_{j} ) ( \Delta_{L} - h_{j} ) e^{ - i h_{i} x} - ( h_{k} - h_{i} ) (\Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) e^{ - i h_{j} x} \biggr\} \nonumber \\ &+& \frac{1}{ ( h_{k} - h_{i} ) ( h_{k} - h_{j} ) ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) } e^{ - i h_{k} x} \nonumber \\ &+& \frac{1}{ ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - \Delta_{K} ) } \nonumber \\ &\times& \biggl\{ ( \Delta_{K} - h_{i} ) ( \Delta_{K} - h_{j} ) ( \Delta_{K} - h_{k} ) e^{- i \Delta_{L} x} - ( \Delta_{L} - h_{i} ) ( \Delta_{L} - h_{j} ) ( \Delta_{L} - h_{k} ) e^{- i \Delta_{K} x} \biggr\} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W ^{\dagger} A (UX) \right\}_{L j}. \label{hatS-ij-W4-H4-double}\end{aligned}$$ Order $W^4$ $\hat{S}$ matrix elements $\hat{S}_{i i}^{(4)} [3]$ and $\hat{S}_{i i}^{(4)} [4]$ {#sec:hatSii-3nd-4th} ---------------------------------------------------------------------------------------------- $\hat{S} _{i i}^{(4)} [3]$ is given by $$\begin{aligned} && \hat{S}_{ii}^{(4)} [3] \nonumber \\ &=& - \sum_{K} \biggl[ \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 } \{ (ix) e^{- i \Delta_{K} x} + (ix) e^{- i h_{i} x} \} + \frac{ 2 }{ ( \Delta_{K} - h_{i} )^3 } \left( e^{- i \Delta_{K} x} - e^{- i h_{i} x} \right) \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A W \right\}_{K K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \nonumber \\ &+& \sum_{K \neq L} \biggl[ \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - \Delta_{L} )} e^{- i \Delta_{K} x} - \frac{ 1 }{ ( \Delta_{L} - h_{i} )^2 ( \Delta_{K} - \Delta_{L} )} e^{- i \Delta_{L} x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{L} - h_{i} )^2 } \left( \Delta_{K} + \Delta_{L} - 2 h_{i} \right) e^{- i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A W \right\}_{K L} \left\{ W ^{\dagger} A (UX) \right\}_{L i}. \label{hatS-3rd-order-ii-T-transf}\end{aligned}$$ $\hat{S} _{i i}^{(4)} [4]$ is given by $$\begin{aligned} && \hat{S} _{i i}^{(4)} [4] \nonumber \\ &=& \sum_{K} \biggl[ - \frac{x^2}{2} e^{- i h_{i} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 } - (ix) e^{- i h_{i} x} \frac{ 2 }{ ( \Delta_{K} - h_{i} )^3 } - (ix) e^{- i \Delta_{K} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 } \nonumber \\ &-& \frac{ 3 }{ ( \Delta_{K} - h_{i} )^4 } \left( e^{- i \Delta_{K} x} - e^{- i h_{i} x} \right) \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \nonumber \\ &+& \sum_{K} \sum_{k \neq i} \biggl[ (ix) e^{- i h_{i} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( h_{k} - h_{i} ) } - (ix) e^{- i \Delta_{K} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{k} ) } \nonumber \\ &+& \frac{ ( h_{i} + 2 h_{k} - 3 \Delta_{K} ) }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{K} - h_{k} )^2 } e^{- i \Delta_{K} x} + \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( h_{k} - h_{i} )^2 } e^{- i h_{k} x} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( h_{k} - h_{i} )^2 } \left( \Delta_{K} + 2 h_{k} - 3 h_{i} \right) e^{- i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \nonumber \\ &+& \sum_{K \neq L} \biggl[ - \frac{x^2}{2} e^{- i h_{i} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) } \nonumber \\ &-& (ix) e^{- i h_{i} x} \frac{ 1 }{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{L} - h_{i} )^2 } \left( \Delta_{K} + \Delta_{L} - 2 h_{i} \right) \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{L} - \Delta_{K} ) } e^{- i \Delta_{K} x} + \frac{ 1 }{ ( \Delta_{L} - h_{i} )^3 ( \Delta_{L} - \Delta_{K} ) } e^{- i \Delta_{L} x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{i} )^3 ( \Delta_{L} - h_{i} )^3 } \biggl\{ \Delta_{L}^2 + \Delta_{L} \Delta_{K} + \Delta_{K}^2 - 3 h_{i} ( \Delta_{L} + \Delta_{K} ) + 3 h_{i}^2 \biggr\} e^{- i h_{i} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K i} \left\{ (UX)^{\dagger} A W \right\}_{i L} \left\{ W ^{\dagger} A (UX) \right\}_{L i} \nonumber \\ &+& \sum_{K \neq L} \sum_{k \neq i} \biggl[ \frac{1}{ ( \Delta_{K} - h_{i} ) ( \Delta_{L} - h_{i} ) ( h_{k} - h_{i} ) } (ix) e^{- i h_{i} x} \nonumber \\ &-& \frac{1}{ ( \Delta_{K} - h_{i} )^2 ( \Delta_{L} - h_{i} )^2 ( h_{k} - h_{i} )^2 } \biggl\{ \Delta_{K} \Delta_{L} + ( h_{k} - 2 h_{i} ) ( \Delta_{K} + \Delta_{L} ) + 3 h_{i}^2 - 2 h_{k} h_{i} \biggr\} e^{- i h_{i} x} \nonumber \\ &+& \frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) ( h_{k} - h_{i} )^2 } e^{- i h_{k} x} \nonumber \\ &+& \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{K} - h_{i} )^2 ( \Delta_{K} - h_{k} ) } e^{- i \Delta_{K} x} - \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{L} - h_{i} )^2 ( \Delta_{L} - h_{k} ) } e^{- i \Delta_{L} x} \biggr] \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{i K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W ^{\dagger} A (UX) \right\}_{L i}. \label{hatS-4th-order-ii-4th}\end{aligned}$$ Structure of $S$ matrix elements {#sec:structure-S-matrix} ================================ Here we present details in computation of the $S$ matrix elements. The $S$ matrix elements $S_{\alpha \beta}^{(4)}$ ------------------------------------------------ We decompose $S_{\alpha \beta}^{(4)}$ into the following three pieces (include both $\alpha \neq \beta$ and $\alpha = \beta$) $$\begin{aligned} S_{\alpha \beta}^{(4)} &=& S_{\alpha \beta}^{(4)} [3+4] + S_{\alpha \beta}^{(4)} [3] + S_{\alpha \beta}^{(4)} [2] \label{Sab-4th}\end{aligned}$$ where $[n]$ implies that the term comes from $n$-th order perturbation of $H_{1}$. To prevent too long expression, we decompose the first term in (\[Sab-4th\]) as $$\begin{aligned} S_{\alpha \beta}^{(4)} [3+4] &=& S_{\alpha \beta}^{(4)} [3]_{ \text{ diag } } + S_{\alpha \beta}^{(4)} [4]_{ \text{ diag } } + S_{\alpha \beta}^{(4)} [3]_{ \text{ offdiag } } + S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } \label{Sab-4th-3+4}\end{aligned}$$ where ($n = 3, 4$) “diag” and “offdiag”, respectively, implies $$\begin{aligned} S_{\alpha \beta}^{(4)} [n]_{ \text{ diag } } &=& \sum_{k} (UX)_{ik} \left( \hat{S}_{kk}^{(4)} [n] \right) \left\{ (UX)^{\dagger} \right\}_{kj}, \nonumber \\ S_{\alpha \beta}^{(4)} [n]_{ \text{ offdiag } } &=& \sum_{k \neq l} (UX)_{ik} \left( \hat{S}_{kl}^{(4)} [n] \right) \left\{ (UX)^{\dagger} \right\}_{lj}. \end{aligned}$$ The latter two terms in (\[Sab-4th\]) are given, respectively, by $$\begin{aligned} S_{\alpha \beta}^{(4)} [3] &=& \sum_{k L} (UX)_{ik} \hat{S}_{kL}^{(3)} \left\{ (W^{\dagger}) \right\}_{L j} + \sum_{K l} W_{iK} \hat{S}_{Kl}^{(3)} \left\{ (UX)^{\dagger} \right\}_{lj}, \nonumber \\ S_{\alpha \beta}^{(4)} [2] &=& \sum_{K} W_{iK} \hat{S}_{KK}^{(2)} \left\{ (W^{\dagger}) \right\}_{K j} + \sum_{K \neq L} W_{iK} \hat{S}_{KL}^{(2)} \left\{ (W^{\dagger}) \right\}_{L j}. \label{S-alpha-beta-4th-[3]}\end{aligned}$$ We do not display explicitly the expression of each term in (\[Sab-4th\]). But, the notation of $S_{\alpha \beta}^{(4)} [n]_{ \text{ diag } }$ and $S_{\alpha \beta}^{(4)} [n]_{ \text{ offdiag } }$ will be transported to the notation for the oscillation probability such that $2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [n]_{ \text{ diag } } \right]$. Similarly, to make the equation fit to a single page we present the first and the second terms of $S_{\alpha \beta}^{(4)} [3]$ in (\[S-alpha-beta-4th-\[3\]\]) separately, $S_{\alpha \beta}^{(4)} [3]_\text{First} = \sum_{k L} (UX)_{\alpha k} W^*_{\beta L} \hat{S}_{kL}^{(3)}$ and $S_{\alpha \beta}^{(4)} [3]_\text{Second} = \sum_{L k} W_{\alpha L} (UX)^*_{\beta k} \hat{S}_{L k}^{(3)}$, whose notations are also transported to the oscillation probability. Expression of the oscillation probability in fourth order in $W$ {#sec:expression-probability-4th} ================================================================ The oscillation probability to second order in $W$ is given in eq. (\[P-beta-alpha-0th+2nd\]) in section \[sec:probability-2nd\]. What is left is, therefore, the expressions of the oscillation probability in fourth order in $W$, the explicit form of the two terms in (\[P-beta-alpha-4th-def\]), $P(\nu_\beta \rightarrow \nu_\alpha) = \left| S^{(2)}_{\alpha \beta} \right|^2 + 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S^{(4)}_{\alpha \beta} \right] $. Second order $S$ matrix squared term: $\left| S^{(2)}_{\alpha \beta} \right|^2$ {#sec:second-order-square} ------------------------------------------------------------------------------- The $S$ matrix element $S^{(2)}_{\alpha \beta}$ in eq. (\[S-alpha-beta-2nd\]) contains four terms. To prevent too long expressions, we divide $\left| S^{(2)}_{\alpha \beta} \right|^2$ into the two terms, one sum of each term squared and the other one composed of cross terms. The first one is given by $$\begin{aligned} && \left| S^{(2)}_{\alpha \beta} \right|^2_{\text{1st}} = \sum_{k, K} \sum_{l, L} \frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } \nonumber \\ &\times& \biggl[ x^2 e^{- i ( h_{k} - h_{l} ) x} - (ix) \frac{e^{- i ( \Delta_{K} - h_{l} ) x} - e^{- i ( h_{k} - h_{l} ) x} }{ ( \Delta_{K} - h_{k} ) } + (ix) \frac{e^{- i ( h_{k} - \Delta_{L} ) x} - e^{- i ( h_{k} - h_{l} ) x} }{ ( \Delta_{L} - h_{l} ) } \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } \biggl\{ e^{- i ( \Delta_{K} - \Delta_{L} ) x} + e^{- i ( h_{k} - h_{l} ) x} - e^{- i ( \Delta_{K} - h_{l} ) x} - e^{- i ( h_{k} - \Delta_{L} ) x} \biggr\} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &\times& (UX)_{\alpha l}^* (UX)_{\beta l} \left\{ (UX)^{\dagger} A W \right\}_{l L} \left\{ W^{\dagger} A (UX) \right\}_{L l} \nonumber \\ &+& \sum_{k \neq m} \sum_{K} \sum_{l \neq n} \sum_{L} \frac{ 1 }{ ( h_{m} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) } \frac{ 1 }{ ( h_{n} - h_{l} ) (\Delta_{L} - h_{l}) (\Delta_{L} - h_{n}) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i h_{m} x} - \left( \Delta_{K} - h_{m} \right) e^{- i h_{k} x} - ( h_{m} - h_{k} ) e^{- i \Delta_{K} x} \biggr] \nonumber \\ &\times& \biggl[ \left( \Delta_{L} - h_{l} \right) e^{+ i h_{n} x} - \left( \Delta_{L} - h_{n} \right) e^{+ i h_{l} x} - ( h_{n} - h_{l} ) e^{+ i \Delta_{L} x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K m} \nonumber \\ &\times& (UX)_{\alpha l}^* (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{n L} \left\{ W^{\dagger} A (UX) \right\}_{L l} \nonumber \\ &+& \sum_{k, K} \sum_{l, L} \frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } \left( e^{- i \Delta_{K} x} - e^{- i h_{k} x} \right) \left( e^{+ i \Delta_{L} x} - e^{+ i h_{l} x} \right) \nonumber \\ &\times& \biggl[ (UX)_{\alpha k} W^*_{\beta K} \left\{ (UX)^{\dagger} A W \right\}_{k K} + W_{\alpha K} (UX)^*_{\beta k} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \biggr] \nonumber \\ &\times& \biggl[ (UX)_{\alpha l}^* W_{\beta L} \left\{ W ^{\dagger} A (UX) \right\}_{L l} + W_{\alpha L}^* (UX)_{\beta l} \left\{ (UX)^{\dagger} A W \right\}_{l L} \biggr] \nonumber \\ &+& \sum_{K} \vert W_{\alpha K} \vert^2 \vert W_{\beta K} \vert^2 + \sum_{K \neq L} e^{- i ( \Delta_{K} - \Delta_{L} ) x} W_{\alpha K} W^*_{\beta K} W_{\alpha L}^* W_{\beta L}. \label{S(2)-squared-1}\end{aligned}$$ Except for the first term in (\[S(2)-squared-1\]) we did not try to unify the two exponential factors because the expressions become cumbersome. Apart from the last line in (\[S(2)-squared-1\]) all the terms are suppressed by the two energy denominators with $\Delta m^2_{J k}$ which doubly suppress the active-sterile state transition. The first term in the last line is the probability leaking term mentioned in section \[sec:nonunitarity-vacuum\]. The second term of $\left| S^{(2)}_{\alpha \beta} \right|^2$ (interference terms) is given by $$\begin{aligned} && \left| S^{(2)}_{\alpha \beta} \right|^2_{\text{2nd}} = - 2 \mbox{Re} \biggl\{ \sum_{k, K} \sum_{l \neq m} \sum_{L} \frac{ 1 }{ ( \Delta_{K} - h_{k} ) (\Delta_{L} - h_{l}) (\Delta_{L} - h_{m}) ( h_{m} - h_{l} ) } \nonumber \\ &\times& \biggl[ \left( \Delta_{L} - h_{l} \right) e^{- i h_{m} x} - \left( \Delta_{L} - h_{m} \right) e^{- i h_{l} x} - ( h_{m} - h_{l} ) e^{- i \Delta_{L} x} \biggr] \left[ - (ix) e^{+ i h_{k} x} + \frac{e^{+ i \Delta_{K} x} - e^{+ i h_{k} x} }{ ( \Delta_{K} - h_{k} ) } \right] \nonumber \\ &\times& (UX)_{\alpha l} (UX)^*_{\beta m} (UX)_{\alpha k}^* (UX)_{\beta k} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{l L} \left\{ W ^{\dagger} A (UX) \right\}_{L m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &+& 2 \mbox{Re} \biggl\{ \sum_{k, K} \sum_{l, L} \left[ - (ix) e^{+ i h_{k} x} + \frac{e^{+ i \Delta_{K} x} - e^{+ i h_{k} x} }{ ( \Delta_{K} - h_{k} ) } \right] \frac{e^{- i \Delta_{L} x} - e^{- i h_{l} x} }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } \nonumber \\ &\times& (UX)_{\alpha k}^* (UX)_{\beta k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &\times& \biggl[ (UX)_{\alpha l} W^*_{\beta L} \left\{ (UX)^{\dagger} A W \right\}_{l L} + W_{\alpha L} (UX)^*_{\beta l} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \biggr] \biggr\} \nonumber \\ &+& 2 \mbox{Re} \biggl\{ \sum_{k, K} \sum_{L} \left[ - (ix) e^{- i ( \Delta_{L} - h_{k} ) x} + \frac{ e^{- i ( \Delta_{L} - \Delta_{K} ) x} - e^{- i ( \Delta_{L} - h_{k} ) x} }{ ( \Delta_{K} - h_{k} ) } \right] \frac{ 1 }{ \Delta_{K} - h_{k} } \nonumber \\ &\times& (UX)_{\alpha k}^* (UX)_{\beta k} W_{\alpha L} W^*_{\beta L} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \biggr\} \nonumber \\ &-& 2 \mbox{Re} \biggl\{ \sum_{k \neq m} \sum_{K} \sum_{l, L} \frac{ 1 }{ ( h_{m} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{+ i h_{m} x} - \left( \Delta_{K} - h_{m} \right) e^{+ i h_{k} x} - ( h_{m} - h_{k} ) e^{+ i \Delta_{K} x} \biggr] \frac{e^{- i \Delta_{L} x} - e^{- i h_{l} x} }{ ( \Delta_{L} - h_{l} ) } \nonumber \\ &\times& (UX)^*_{\alpha k} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{m K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &\times& \biggl[ (UX)_{\alpha l} W^*_{\beta L} \left\{ (UX)^{\dagger} A W \right\}_{l L} + W_{\alpha L} (UX)^*_{\beta l} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \biggr] \nonumber \\ &-& 2 \mbox{Re} \biggl\{ \sum_{k \neq m} \sum_{K} \sum_{L} \frac{ 1 }{ ( h_{m} - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i ( \Delta_{L} - h_{m} ) x} - \left( \Delta_{K} - h_{m} \right) e^{- i ( \Delta_{L} - h_{k} ) x} - ( h_{m} - h_{k} ) e^{- i ( \Delta_{L} - \Delta_{K} ) x} \biggr] \nonumber \\ &\times& (UX)^*_{\alpha k} (UX)_{\beta m} W_{\alpha L} W^*_{\beta L} \left\{ (UX)^{\dagger} A W \right\}_{m K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \biggr\} \nonumber \\ &+& 2 \mbox{Re} \biggl\{ \sum_{K} \sum_{l, L} \frac{e^{- i ( \Delta_{L} - \Delta_{K} ) x} - e^{- i ( h_{l} - \Delta_{K} ) x} }{ ( \Delta_{L} - h_{l} ) } \nonumber \\ &\times& \biggl[ W^*_{\alpha K} W_{\beta K} (UX)_{\alpha l} W^*_{\beta L} \left\{ (UX)^{\dagger} A W \right\}_{l L} + W^*_{\alpha K} W_{\beta K} W_{\alpha L} (UX)^*_{\beta l} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \biggr] \biggr\}. \label{S(2)-squared-2}\end{aligned}$$ Interference terms of the type $2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S^{(4)}_{\alpha \beta} \right]$ {#sec:interference} ---------------------------------------------------------------------------------------------------------------------------- We classify the fourth order in $W$ contribution of the interference terms into 8 terms: $$\begin{aligned} P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{ \text{interference} } &=& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{1st} + P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{2nd} + P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{3rd} \nonumber \\ &+& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-s} + P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-d} \nonumber \\ &+& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-1st} + P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-2nd} + P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{6th}. \nonumber \\\end{aligned}$$ The nature of each term is explicitly indicated as follows: $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{1st} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [3]_{ \text{ diag } } \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ - \sum_{K} \sum_{k} \sum_{m} \biggl[ \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 } \left\{ (ix) + \frac{ 2 }{ ( \Delta_{K} - h_{k} ) } \right\} \left( e^{- i ( \Delta_{K} - h_{m} ) x} + e^{- i ( h_{k} - h_{m} ) x} \right) \biggr] \nonumber \\ &\times& (UX)^*_{\alpha m} (UX)_{\beta m} (UX)_{\alpha k} (UX)^*_{\beta k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A W \right\}_{K K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &+& \sum_{K \neq L} \sum_{k} \sum_{m} \biggl[ \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - \Delta_{L} )} e^{- i ( \Delta_{K} - h_{m} ) x} - \frac{ 1 }{ ( \Delta_{L} - h_{k} )^2 ( \Delta_{K} - \Delta_{L} )} e^{- i ( \Delta_{L} - h_{m} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{L} - h_{k} )^2 } \left( \Delta_{K} + \Delta_{L} - 2 h_{k} \right) e^{- i ( h_{k} - h_{m} ) x} \biggr] \nonumber \\ &\times& (UX)^*_{\alpha m} (UX)_{\beta m} (UX)_{\alpha k} (UX)^*_{\beta k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A W \right\}_{K L} \left\{ W ^{\dagger} A (UX) \right\}_{L k} \biggr\}. \nonumber \\ \label{P-beta-alpha-W4-H3-diag}\end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{2nd} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [4]_{ \text{ diag } } \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ \sum_{n} \sum_{k} \sum_{K} \biggl[ - \frac{x^2}{2} \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 } e^{- i ( h_{k} - h_{n} ) x} - \frac{ 2 (ix) }{ ( \Delta_{K} - h_{k} )^3 } e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &-& \frac{ (ix) }{ ( \Delta_{K} - h_{k} )^3 } e^{- i ( \Delta_{K} - h_{n} ) x} - \frac{ 3 }{ ( \Delta_{K} - h_{k} )^4 } \left( e^{- i ( \Delta_{K} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} \right) \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &+& \sum_{n} \sum_{k} \sum_{K} \sum_{m \neq k} \biggl[ \frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( h_{m} - h_{k} ) } e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &-& \frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{m} ) } e^{- i ( \Delta_{K} - h_{n} ) x} + \frac{ ( h_{k} + 2 h_{m} - 3 \Delta_{K} ) }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{K} - h_{m} )^2 } e^{- i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{m} )^2 ( h_{m} - h_{k} )^2 } e^{- i ( h_{m} - h_{n} ) x} - \frac{ \left( \Delta_{K} + 2 h_{m} - 3 h_{k} \right) }{ ( \Delta_{K} - h_{k} )^3 ( h_{m} - h_{k} )^2 } e^{- i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K m} \left\{ (UX)^{\dagger} A W \right\}_{m K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &+& \sum_{n} \sum_{k} \sum_{K \neq L} \biggl[ - \frac{x^2}{2} \frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) } e^{- i ( h_{k} - h_{n} ) x} - (ix) \frac{ \left( \Delta_{K} + \Delta_{L} - 2 h_{k} \right) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{L} - h_{k} )^2 } e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{L} - \Delta_{K} ) } e^{- i ( \Delta_{K} - h_{n} ) x} + \frac{ 1 }{ ( \Delta_{L} - h_{k} )^3 ( \Delta_{L} - \Delta_{K} ) } e^{- i ( \Delta_{L} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{L} - h_{k} )^3 } \biggl\{ \Delta_{L}^2 + \Delta_{L} \Delta_{K} + \Delta_{K}^2 - 3 h_{k} ( \Delta_{L} + \Delta_{K} ) + 3 h_{k}^2 \biggr\} e^{- i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W ^{\dagger} A (UX) \right\}_{L k} \nonumber \\ &+& \sum_{n} \sum_{k} \sum_{K \neq L} \sum_{m \neq k} \biggl[ \frac{ (ix) }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) ( h_{m} - h_{k} ) } e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &-& \frac{1}{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{L} - h_{k} )^2 ( h_{m} - h_{k} )^2 } \biggl\{ \Delta_{K} \Delta_{L} + ( h_{m} - 2 h_{k} ) ( \Delta_{K} + \Delta_{L} ) + 3 h_{k}^2 - 2 h_{m} h_{k} \biggr\} e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &+& \frac{1}{ ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{m} ) ( h_{m} - h_{k} )^2 } e^{- i ( h_{m} - h_{n} ) x} \nonumber \\ &+& \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{m} ) } e^{- i ( \Delta_{K} - h_{n} ) x} - \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{L} - h_{k} )^2 ( \Delta_{L} - h_{m} ) } e^{- i ( \Delta_{L} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K m} \left\{ (UX)^{\dagger} A W \right\}_{m L} \left\{ W ^{\dagger} A (UX) \right\}_{L k} \biggr\}. \label{P-beta-alpha-W4-H4-diag}\end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{3rd} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ \sum_{m} \sum_{k \neq l } \sum_{K} \biggl[ - \frac{ (ix) }{ ( \Delta_{K} - h_{k} ) (\Delta_{K} - h_{l} ) } e^{- i ( \Delta_{K} - h_{m} ) x} + \frac{ 1 }{ ( h_{l} - h_{k} ) ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} )^2 } \nonumber \\ &\times& \biggl\{ ( h_{l} - h_{k} ) ( h_{l} + h_{k} - 2 \Delta_{K} ) e^{- i ( \Delta_{K} - h_{m} ) x} + ( \Delta_{K} - h_{k} )^2 e^{ - i ( h_{l} - h_{m} ) x} - ( \Delta_{K} - h_{l} )^2 e^{ - i ( h_{k} - h_{m} ) x} \biggr\} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A W \right\}_{K K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \nonumber \\ &+& \sum_{m} \sum_{k \neq l } \sum_{K \neq L } \frac{ 1 }{ ( h_{l} - h_{k} ) ( \Delta_{L} - \Delta_{K} ) ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) } \nonumber \\ &\times& \biggl[ ( h_{l} - h_{k} ) \biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) e^{- i ( \Delta_{L} - h_{m} ) x} - ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) e^{- i ( \Delta_{K} - h_{m} ) x} \biggr\} \nonumber \\ &+& ( \Delta_{L} - \Delta_{K} ) \biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) e^{- i ( h_{l} - h_{m} ) x} - ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{l} ) e^{- i ( h_{k} - h_{m} ) x} \biggr\} \biggl] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha m} (UX)_{\beta m} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A W \right\}_{K L} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \biggr\}. \label{P-beta-alpha-W4-H3-offdiag}\end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-s} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } (\text{single}) \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ \sum_{n} \sum_{k \neq l } \sum_{K} \biggl[ \frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( h_{l} - h_{k} ) } e^{ - i ( h_{k} - h_{n} ) x} - \frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} ) } e^{ - i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{k} )^2 } e^{ - i ( h_{l} - h_{n} ) x} - \frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( h_{l} - h_{k} )^2 } \left( \Delta_{K} + 2 h_{l} - 3 h_{k} \right) e^{ - i ( h_{k} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{K} - h_{l} )^2 } \left( h_{k} + 2 h_{l} - 3 \Delta_{K} \right) e^{ - i ( \Delta_{K} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \nonumber \\ &+& \sum_{n} \sum_{k \neq l } \sum_{K} \biggl[ - \frac{ (ix) }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{k} ) } e^{ - i ( h_{l} - h_{n} ) x} - \frac{ (ix) }{ ( \Delta_{K} - h_{l} )^2 ( \Delta_{K} - h_{k} ) } e^{ - i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} )^3 } \left( h_{l} + 2 h_{k} - 3 \Delta_{K} \right) e^{ - i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{l} )^3 ( h_{l} - h_{k} )^2 } \left( \Delta_{K} + 2 h_{k} - 3 h_{l} \right) e^{ - i ( h_{l} - h_{n} ) x} + \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( h_{l} - h_{k} )^2 } e^{ - i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \left\{ (UX)^{\dagger} A W \right\}_{l K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \nonumber \\ &+& \sum_{n} \sum_{k \neq l } \sum_{K} \sum_{m \neq k, l} \biggl[ \frac{ (ix) }{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{K} - h_{m} ) } e^{ - i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &-& \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} )^2 ( \Delta_{K} - h_{m} )^2 } \nonumber \\ &\times& \left\{ 3 \Delta_{K}^2 - 2 \Delta_{K} \left( h_{k} + h_{l} + h_{m} \right) + \left( h_{k} h_{l}+ h_{l} h_{m} + h_{m} h_{k} \right) \right\} e^{ - i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{m} )^2 ( h_{m} - h_{k} ) ( h_{m} - h_{l} ) } e^{ - i ( h_{m} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{m} ) ( h_{l} - h_{k} ) } e^{ - i ( h_{l} - h_{n} ) x} - \frac{ 1 }{ ( \Delta_{K} - h_{k} )^2 ( h_{k} - h_{m} ) ( h_{l} - h_{k} ) } e^{ - i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K m} \left\{ (UX)^{\dagger} A W \right\}_{m K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \biggr\}. \label{P-beta-alpha-W4-H4-single}\end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-d} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } (\text{double}) \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ \sum_{n} \sum_{k \neq l } \sum_{K \neq L} \biggl[ \frac{ (ix) }{ (\Delta_{K} - h_{k} ) (\Delta_{L} - h_{k} ) ( h_{l} - h_{k} ) } e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &-& \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{k} )^2 (\Delta_{L} - h_{l} ) } e^{- i ( \Delta_{L} - h_{n} ) x} + \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{k} )^2 (\Delta_{K} - h_{l} ) } e^{- i ( \Delta_{K} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ (\Delta_{K} - h_{l} ) (\Delta_{L} - h_{l} ) ( h_{l} - h_{k} )^2 } e^{- i ( h_{l} - h_{n} ) x} \nonumber \\ &-& \frac{ 1 }{ (\Delta_{K} - h_{k} )^2 (\Delta_{L} - h_{k} )^2 ( h_{l} - h_{k} )^2 } \biggl\{ 3 h_{k}^2 - 2 h_{k} h_{l} + \left( h_{l} - 2 h_{k} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L} \biggr\} e^{- i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \nonumber \\ &+& \sum_{n} \sum_{k \neq l } \sum_{K \neq L} \biggl[ - \frac{ (ix) }{ (\Delta_{K} - h_{l} ) (\Delta_{L} - h_{l} ) ( h_{l} - h_{k} ) } e^{- i ( h_{l} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{K} - h_{k} ) (\Delta_{K} - h_{l} )^2 } e^{- i ( \Delta_{K} - h_{n} ) x} - \frac{ 1 }{ (\Delta_{K} - \Delta_{L} ) (\Delta_{L} - h_{k} ) (\Delta_{L} - h_{l} )^2 } e^{- i ( \Delta_{L} - h_{n} ) x} \nonumber \\ &+& \frac{ 1 }{ (\Delta_{K} - h_{k} ) (\Delta_{L} - h_{k} ) ( h_{l} - h_{k} )^2 } e^{- i ( h_{k} - h_{n} ) x} \nonumber \\ &-& \frac{ 1 }{ (\Delta_{K} - h_{l} )^2 (\Delta_{L} - h_{l} )^2 ( h_{l} - h_{k} )^2 } \biggl\{ 3 h_{l}^2 - 2 h_{k} h_{l} - \left( 2 h_{l} - h_{k} \right) (\Delta_{K} + \Delta_{L} ) + \Delta_{K} \Delta_{L} \biggr\} e^{- i ( h_{l} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K l} \left\{ (UX)^{\dagger} A W \right\}_{l L} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \nonumber \\ &+& \sum_{n} \sum_{k \neq l } \sum_{K \neq L} \sum_{m \neq k, l} \biggl[ \frac{1}{ ( h_{m} - h_{k} ) ( h_{m} - h_{l} ) ( h_{l} - h_{k} ) ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) } \nonumber \\ &+& \biggl\{ ( h_{m} - h_{l} ) (\Delta_{K} - h_{l} ) ( \Delta_{L} - h_{l} ) e^{- i ( h_{k} - h_{n} ) x} - ( h_{m} - h_{k} ) (\Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) e^{- i ( h_{l} - h_{n} ) x} \biggr\} \nonumber \\ &+& \frac{1}{ ( h_{m} - h_{k} ) ( h_{m} - h_{l} ) ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{m} ) } e^{- i ( h_{m} - h_{n} ) x} \nonumber \\ &+& \frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) ( \Delta_{L} - h_{m} ) ( \Delta_{L} - \Delta_{K} ) } \nonumber \\ &\times& \biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{K} - h_{m} ) e^{- i ( \Delta_{L} - h_{n} ) x} - ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) ( \Delta_{L} - h_{m} ) e^{- i ( \Delta_{K} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} (UX)^*_{\beta l} (UX)^*_{\alpha n} (UX)_{\beta n} \nonumber \\ &\times& \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W ^{\dagger} A (UX) \right\}_{K m} \left\{ (UX)^{\dagger} A W \right\}_{m L} \left\{ W ^{\dagger} A (UX) \right\}_{L l} \biggr\}. \label{P-beta-alpha-W4-H4-double}\end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-1st} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [3]_\text{First} \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ - \sum_{n} \sum_{k L} \frac{ 1 }{ \Delta_{L} - h_{k} } \left[ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} + \frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{K} - h_{k} ) } \right] \nonumber \\ &\times& (UX)_{\alpha k} W^*_{\beta L} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W^{\dagger} A W \right\}_{L L} \nonumber \\ &+& \sum_{n} \sum_{k L} \sum_{K \neq L} \frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) ( \Delta_{L} - h_{k} ) ( \Delta_{K} - h_{k} ) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i ( \Delta_{L} - h_{n} ) x} - \left( \Delta_{L} - h_{k} \right) e^{- i ( \Delta_{K} - h_{n} ) x} - \left( \Delta_{K} - \Delta_{L} \right) e^{- i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} W^*_{\beta L} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W^{\dagger} A W \right\}_{K L} \nonumber \\ &-& \sum_{n} \sum_{k L} \frac{ 1 }{ ( \Delta_{L} - h_{k} )^2 } \biggl[ (ix) \left( e^{- i ( h_{k} - h_{n} ) x} + e^{- i ( \Delta_{L} - h_{n} ) x} \right) + 2 \frac{e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) } \biggr] \nonumber \\ &\times& (UX)_{\alpha k} W^*_{\beta L} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W^{\dagger} A (UX) \right\}_{L k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \nonumber \\ &+& \sum_{n} \sum_{k L} \sum_{m \neq k} \biggl[ - \frac{ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} )( \Delta_{L} - h_{m} ) } + \frac{ 1 }{ ( h_{k} - h_{m} ) (\Delta_{L} - h_{k} )^2 (\Delta_{L} - h_{m} )^2 } \nonumber \\ &\times& \biggl\{ (\Delta_{L} - h_{m} )^2 e^{- i ( h_{k} - h_{n} ) x} - (\Delta_{L} - h_{k} )^2 e^{- i ( h_{m} - h_{n} ) x} + ( h_{k} - h_{m} )( h_{k} + h_{m} - 2 \Delta_{L} ) e^{- i ( \Delta_{L} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} W^*_{\beta L} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W^{\dagger} A (UX) \right\}_{L m} \left\{ (UX)^{\dagger} A W \right\}_{m L} \nonumber \\ &+& \sum_{n} \sum_{k L} \sum_{K \neq L} \biggl[ - \frac{ (ix) e^{- i ( h_{k} - h_{n} ) x} }{ (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) } + \frac{ 1 }{ (\Delta_{L} - \Delta_{K}) (\Delta_{L} - h_{k})^2 ( \Delta_{K} - h_{k} )^2 } \nonumber \\ &\times& \biggl\{ ( \Delta_{K} - h_{k} )^2 e^{- i ( \Delta_{L} - h_{n} ) x} - (\Delta_{L} - h_{k})^2 e^{- i ( \Delta_{K} - h_{n} ) x} + (\Delta_{L} - \Delta_{K}) (\Delta_{L} + \Delta_{K} - 2 h_{k} ) e^{- i ( h_{k} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} W^*_{\beta L} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \nonumber \\ &+& \sum_{n} \sum_{k L} \sum_{K \neq L} \sum_{m \neq k} \frac{ 1 }{ (\Delta_{L} - \Delta_{K}) ( h_{m} - h_{k} ) (\Delta_{L} - h_{k}) (\Delta_{L} - h_{m}) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) } \nonumber \\ &\times& \biggl[ ( h_{m} - h_{k} ) \biggl\{(\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) e^{- i ( \Delta_{L} - h_{n} ) x} - (\Delta_{L} - h_{k}) (\Delta_{L} - h_{m}) e^{- i ( \Delta_{K} - h_{n} ) x} \biggr\} \nonumber \\ &-& (\Delta_{L} - \Delta_{K}) \biggl\{ (\Delta_{L} - h_{m}) (\Delta_{K} - h_{m}) e^{- i ( h_{k} - h_{n} ) x} - (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) e^{- i ( h_{m} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& (UX)_{\alpha k} W^*_{\beta L} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W^{\dagger} A (UX) \right\}_{K m} \left\{ (UX)^{\dagger} A W \right\}_{m L} \biggr\}. \label{P-beta-alpha-W4-H3-First} \end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-2nd} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [3]_\text{Second} \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ - \sum_{n} \sum_{L k} \frac{ 1 }{ \Delta_{L} - h_{k} } \left[ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} + \frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) } \right] \nonumber \\ &\times& W_{\alpha L} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ W^{\dagger} A W \right\}_{L L} \left\{ W^{\dagger} A (UX) \right\}_{L k} \nonumber \\ &+& \sum_{n} \sum_{L k} \sum_{K \neq L} \frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) ( \Delta_{L} - h_{k} ) ( \Delta_{K} - h_{k} ) } \nonumber \\ &\times& \biggl[ \left( \Delta_{K} - h_{k} \right) e^{- i ( \Delta_{L} - h_{n} ) x} - \left( \Delta_{L} - h_{k} \right) e^{- i ( \Delta_{K} - h_{n} ) x} - \left( \Delta_{K} - \Delta_{L} \right) e^{- i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& W_{\alpha L} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ W^{\dagger} A W \right\}_{L K} \left\{ W^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &-& \sum_{n} \sum_{L k} \frac{ 1 }{ (\Delta_{L} - h_{k})^2 } \biggl[ (ix) \left( e^{- i ( h_{k} - h_{n} ) x} + e^{- i ( \Delta_{L} - h_{n} ) x} \right) + 2 \frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) } \biggr] \nonumber \\ &\times& W_{\alpha L} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ W^{\dagger} A (UX) \right\}_{L k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \left\{ W^{\dagger} A (UX) \right\}_{L k} \nonumber \\ &+& \sum_{n} \sum_{L k} \sum_{m \neq k} \biggl[ - \frac{ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} )( \Delta_{L} - h_{m} ) } + \frac{ 1 }{ ( h_{k} - h_{m} ) (\Delta_{L} - h_{k} )^2 (\Delta_{L} - h_{m} )^2 } \nonumber \\ &\times& \biggl\{ (\Delta_{L} - h_{m} )^2 e^{- i ( h_{k} - h_{n} ) x} - (\Delta_{L} - h_{k} )^2 e^{- i ( h_{m} - h_{n} ) x} + ( h_{k} - h_{m} )( h_{k} + h_{m} - 2 \Delta_{L} ) e^{- i ( \Delta_{L} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& W_{\alpha L} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ W^{\dagger} A (UX) \right\}_{L m} \left\{ (UX)^{\dagger} A W \right\}_{m L} \left\{ W^{\dagger} A (UX) \right\}_{L k} \nonumber \\ &+& \sum_{n} \sum_{L k} \sum_{K \neq L} \biggl[ - \frac{ (ix) e^{- i ( h_{k} - h_{n} ) x} }{ (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) } + \frac{ 1 }{ (\Delta_{L} - \Delta_{K}) (\Delta_{L} - h_{k})^2 ( \Delta_{K} - h_{k} )^2 } \nonumber \\ &\times& \biggl\{ ( \Delta_{K} - h_{k} )^2 e^{- i ( \Delta_{L} - h_{n} ) x} - (\Delta_{L} - h_{k})^2 e^{- i ( \Delta_{K} - h_{n} ) x} + (\Delta_{L} - \Delta_{K}) (\Delta_{L} + \Delta_{K} - 2 h_{k} ) e^{- i ( h_{k} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& W_{\alpha L} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ W^{\dagger} A (UX) \right\}_{L k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \left\{ W^{\dagger} A (UX) \right\}_{K k} \nonumber \\ &+& \sum_{n} \sum_{L k} \sum_{K \neq L} \sum_{m \neq k} \frac{ 1 }{ (\Delta_{L} - \Delta_{K}) ( h_{m} - h_{k} ) (\Delta_{L} - h_{k}) (\Delta_{L} - h_{m}) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) } \nonumber \\ &\times& \biggl[ ( h_{m} - h_{k} ) \biggl\{ (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) e^{- i ( \Delta_{L} - h_{n} ) x} - (\Delta_{L} - h_{k}) (\Delta_{L} - h_{m}) e^{- i ( \Delta_{K} - h_{n} ) x} \biggr\} \nonumber \\ &-& (\Delta_{L} - \Delta_{K}) \biggl\{ (\Delta_{L} - h_{m}) (\Delta_{K} - h_{m}) e^{- i ( h_{k} - h_{n} ) x} - (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) e^{- i ( h_{m} - h_{n} ) x} \biggr\} \biggr] \nonumber \\ &\times& W_{\alpha L} (UX)^*_{\beta k} (UX)^*_{\alpha n} (UX)_{\beta n} \left\{ W^{\dagger} A (UX) \right\}_{L m} \left\{ (UX)^{\dagger} A W \right\}_{m K} \left\{ W^{\dagger} A (UX) \right\}_{K k} \biggr\}. \label{P-beta-alpha-W4-H3-Second} \end{aligned}$$ $$\begin{aligned} && P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{6th} \equiv 2 \mbox{Re} \left[ \left( S^{(0)}_{\alpha \beta} \right)^{*} S_{\alpha \beta}^{(4)} [2] \right] \nonumber \\ &=& 2 \mbox{Re} \biggl\{ - \sum_{n} \sum_{K} \sum_{k} \frac{ 1 }{ \Delta_{K} - h_{k} } \left[ (ix) e^{- i ( \Delta_{K} - h_{n} ) x} + \frac{e^{- i ( h_{k} - h_{n} ) x} - e^{- i ( \Delta_{K} - h_{n} ) x} }{ h_{k} - \Delta_{K} } \right] \nonumber \\ &\times& (UX)^*_{\alpha n} (UX)_{\beta n} W_{\alpha K} W^*_{\beta K} \left\{ W^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k K} \nonumber \\ &+& \sum_{n} \sum_{K \neq L} \sum_{k} \frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) (\Delta_{L} - h_{k}) (\Delta_{K} - h_{k}) } \nonumber \\ &\times& \biggl[ \Delta_{L} e^{- i ( \Delta_{K} - h_{n} ) x} - \Delta_{K} e^{- i ( \Delta_{L} - h_{n} ) x} + \left( e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( \Delta_{K} - h_{n} ) x} \right) h_{k} - ( \Delta_{L} - \Delta_{K} ) e^{- i ( h_{k} - h_{n} ) x} \biggr] \nonumber \\ &\times& (UX)^*_{\alpha n} (UX)_{\beta n} W_{\alpha K} W^*_{\beta L} \left\{ W^{\dagger} A (UX) \right\}_{K k} \left\{ (UX)^{\dagger} A W \right\}_{k L} \biggr\}. \label{P-beta-alpha-W4-H2} \end{aligned}$$ Neutrino evolution equation in flavor basis {#sec:flavor-basis-evolution} =========================================== The Schrödinger equation takes the form with flavor basis Hamiltonian $H$ in (\[flavor-hamiltonian\]) $$\begin{aligned} i \frac{d}{dx} \left[ \begin{array}{c} \nu_{a} \\ \nu_{s} \\ \end{array} \right] = \left[ \begin{array}{cc} U {\bf \Delta_{a} } U^{\dagger} + W {\bf \Delta_{s} } W^{\dagger} + A & U {\bf \Delta_{a} } Z^{\dagger} + W {\bf \Delta_{s} } V^{\dagger} \\ Z {\bf \Delta_{a} } U^{\dagger} + V {\bf \Delta_{s} } W^{\dagger} & Z {\bf \Delta_{a} } Z^{\dagger} + V {\bf \Delta_{s} } V^{\dagger} \\ \end{array} \right] \left[ \begin{array}{c} \nu_{a} \\ \nu_{s} \\ \end{array} \right] \label{Schroedinger-eq-flavor-basis}\end{aligned}$$ where $\nu_{a}$ ($\nu_{s}$) denotes $3$ ($N$) component vector in active (sterile) space. Apparently, the system depends not only on $U$ and $W$, but also on $Z$ and $V$ matrix elements, which is not the case in our treatment using the mass eigenstate basis. Here, we show that the dependence on $Z$ and $V$ is superficial. Since there is no physical meaning of the particular basis for the sterile sector fermions we can redefine it by doing the transformation $$\begin{aligned} \left[ \begin{array}{c} \nu_{a} \\ \nu_{s} \\ \end{array} \right] \rightarrow \left[ \begin{array}{cc} 1 & 0 \\ 0 & Y \\ \end{array} \right] \left[ \begin{array}{c} \nu_{a} \\ \nu_{s} \\ \end{array} \right] \equiv \left[ \begin{array}{c} \nu_{a}' \\ \nu_{s}' \\ \end{array} \right] $$ where $Y$ is a $N \times N$ unitary matrix. In the primed basis the Hamiltonian becomes $$\begin{aligned} H' \equiv \left[ \begin{array}{cc} 1 & 0 \\ 0 & Y \\ \end{array} \right] H \left[ \begin{array}{cc} 1 & 0 \\ 0 & Y^{\dagger} \\ \end{array} \right] = \left[ \begin{array}{cc} ( U {\bf \Delta_{a} } U^{\dagger} + W {\bf \Delta_{s} } W^{\dagger} + A ) & ( U {\bf \Delta_{a} } Z^{\dagger} + W {\bf \Delta_{s} } V^{\dagger} ) Y^{\dagger} \\ Y ( Z {\bf \Delta_{a} } U^{\dagger} + V {\bf \Delta_{s} } W^{\dagger} ) & Y ( Z {\bf \Delta_{a} } Z^{\dagger} + V {\bf \Delta_{s} } V^{\dagger} ) Y^{\dagger} \\ \end{array} \right]. $$ We can arbitrarily choose $Y=V^{\dagger}$. Then, one can show by using unitarity (\[unitarity\]) that $$\begin{aligned} H' = \left[ \begin{array}{cc} ( U {\bf \Delta_{a} } U^{\dagger} + W {\bf \Delta_{s} } W^{\dagger} + A ) & - U {\bf \Delta_{a} } U^{\dagger} W + W {\bf \Delta_{s} } ( {\bf 1} - W^{\dagger} W )\\ - W^{\dagger} U {\bf \Delta_{a} } U^{\dagger} + ( {\bf 1} - W^{\dagger} W ) {\bf \Delta_{s} } W^{\dagger} & W^{\dagger} U {\bf \Delta_{a} } U^{\dagger} W + ( {\bf 1} - W^{\dagger} W ) {\bf \Delta_{s} } ( {\bf 1} - W^{\dagger} W ) \end{array} \right]. \nonumber \\ \end{aligned}$$ Therefore, our system depends only on $U$ and $W$, and the dependence on $Z$ and $V$ is superficial. Universal scaling model of $N$ sterile sector {#sec:scaling-model} ============================================== Suppose that we obtain a particular parametrization of $U$ matrix by taking $N=1$ sterile sector, as we did in section \[sec:probabilities\]. In this $(3+1)$ model, the $W$ matrix elements are completely determined, up to phase, by unitarity $$\begin{aligned} \vert W_{\alpha 4} \vert^2 = 1 - \sum_{j=1}^3 |U_{\alpha j}|^2. \end{aligned}$$ Now, we attempt to create a toy model of $N$ sterile sector by “universal scaling”. We postulate that all the $W$ matrix elements are real and equal: $$\begin{aligned} W_{\alpha 4} = W_{\alpha 5} = \cdot \cdot \cdot W_{\alpha N+3} = \frac{ 1 }{ \sqrt{N} } \biggl( 1 - \sum_{j=1}^3 |U_{\alpha j}|^2 \biggr)^{1/2}\end{aligned}$$ which is consistent with $(3+N)$ space unitarity. In this universal scaling model, the order $W^2$ correction terms in (\[P-beta-alpha-2nd-averaged\]) remains unchanged provided that we further assume that all the sterile masses are equal.[^24] It is because the $W$ matrix elements enter into the $W^2$ terms in the form $$\begin{aligned} \sum_{K} W_{\alpha K} W^{\dagger}_{K \beta} \frac{ 1 }{ ( \Delta_{K} - h_{k} )^n }, \label{W2-scaling} \end{aligned}$$ where $n=1$ or 2. However, the leaking constant $\mathcal{C}_{\alpha \beta}$ becomes smaller by a factor of $N$ in the universal scaling model. In the $(3+1)$ model, $\mathcal{C}_{\alpha \beta}$ takes the largest value, the upper limit in eq. (\[Cab-bound\]). Because $\mathcal{C}_{\alpha \beta}$ is fourth order in $W$ it is evident that in the universal scaling model, $$\mathcal{C}_{\alpha \beta} = \frac{1}{N} \biggl( 1 - \sum_{j=1}^3 |U_{\alpha j}|^2 \biggr) \biggl( 1 - \sum_{j=1}^3 |U_{\beta j}|^2 \biggr), \label{Cab-USM}$$ which is the lower limit of (\[Cab-bound\]). 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Emiliano Rodríguez Mega, “Plans for a research powerhouse in the Andes begin to unravel”, Science (2017) doi:10.1126/science.aan7140. [^1]: The unique citation of the solar neutrino measurement here must be understood as the representative of all the foregoing solar neutrino experiments [@Cleveland:1998nv; @Hirata:1991ub; @Hampel:1998xg; @Abdurashitov:2002nt; @Fukuda:2001nj; @Aharmim:2011vm]. [^2]: Recently, however, there exists accumulating indication that CP phase $\delta$ takes value around $\sim\frac{3\pi}{2}$ [@Hartz-KEK-colloquium]. [^3]: In recent years there exist a good amount of activities addressing the possibilities of new physics at low energies. For scenarios which involve light sterile neutrino(s), see, e.g., [@Nelson:2007yq; @Pospelov:2012gm; @Harnik:2012ni], and the references therein. [^4]: Works have also been done on unitarity violation by sterile sector from somewhat different point of view, e.g., if it exists, how it could disturb measurement of lepton Kobayashi-Maskawa phase $\delta_{\rm CP}$, or mass ordering. See for example, [@Klop:2014ima; @Gandhi:2015xza; @Agarwalla:2016mrc; @Miranda:2016wdr; @Ge:2016xya; @Dutta:2016vcc; @Dutta:2016czj; @Rout:2017udo]. [^5]: To understand the formulas presented in this section, minimal explanation of definitions of the quantities may be helpful. Let the flavor mixing matrix $\bf{U}$ in $(3+N) \times (3+N)$ space that connect the flavor and mass eigenstates as $\nu_{\zeta} = {\bf U}_{\zeta z} \nu_{z}$, where $\zeta$ runs over active $\alpha = e,\mu,\tau$ and sterile flavor $s=s_1,\cdot \cdot \cdot,s_N$ indices, and $z$ over mostly active $i=1,2,3$ and mostly sterile mass eigenstate $J=4,5,\cdot \cdot \cdot, N+3$ indices. Let $U$ and $W$ be a part of $\bf{U}$ so that they connect the mass eigenstates to active neutrino flavor states, $\nu_{\alpha} = \sum_{i} (U)_{\alpha i} \nu_{i} + \sum_{J} (W)_{\alpha J} \nu_{J}$. The kinematical phase factors $\Delta_{j}$ and $\Delta_{J}$ are defined as $\Delta_{j} \equiv \frac{m^2_{j}}{2E}$ and $\Delta_{J} \equiv \frac{m^2_{J}}{2E}$, respectively. $m_{j}$ and $m_{J}$ denote the active and sterile neutrino masses, respectively. [^6]: Speaking more precisely, we mean that all the $W$ matrix elements are assumed to be small, of the order of $\simeq 0.1$. [^7]: Our result also means that taking care of all order matter effect does not change the feature obtained by first-order treatment in matter perturbation theory. That is, the same condition on sterile states masses derived by using first-order matter perturbation theory suffices to guarantee the absence of matter-dependent higher order correction terms in $W$. The reasons for this feature will be explained at the end of section \[sec:energy-denominator\]. [^8]: In matter, we have $\sum_{j=1}^{3} (UX)_{\alpha j} (UX)^{*}_{\beta j} = \sum_{j,k,l=1}^{3} (U)_{\alpha k} (U)^{*}_{\beta l} X_{kj} X^*_{lj} = \sum_{k=1}^{3} (U)_{\alpha k} (U)^{*}_{\beta k}$ where in the last step, we have used the unitarity relation $\sum_{j=1}^3 X_{kj} X^*_{lj} = \delta_{kl}$. [^9]: Through unitarity (\[unitarity\]), $U$ matrix elements have some dependence on $W$ matrix elements. We choose not to expand $U$ matrix elements by this $W$ dependence. In this sense, we use a “renormalized basis” (in the same sense as in ref. [@Minakata:2015gra]) in which some higher order effects are absorbed into the zeroth-order state. [^10]: As in the Standard Model in particle physics T invariance is broken in our system only by complex numbers in the mixing matrix. [^11]: Since $\hat{H}$ system is a consistent dynamical system it is legitimate and easier to verify generalized T invariance in the $\hat{S}$ level, though it can be done in the $S$ matrix level as well. [^12]: To check the point of how the “averaging out the fast oscillation” procedure works, we numerically solved the $3+1$ system explicitly and confirmed that it does, as it should be. [^13]: One can show that $L \sim \frac{ 2E }{ \Delta m^2_{31} } \sim \frac{1}{( h_{k} - h_{j} ) }$ holds at around the oscillation maximum of atmospheric-scale oscillations under the restriction we make for (\[suppression-cond\]). Then, the left-hand side of the first two inequalities receive an extra factor $LA \sim \frac{A}{( h_{k} - h_{j} ) } \sim \frac{ a }{ \Delta m^2_{31} } \simeq 0.1 \left(\frac{\rho}{2.8 \,\text{g/cm}^3}\right) \left(\frac{E}{1~\mbox{GeV}}\right)$, which further suppress it unless $\rho E \gsim 10\, \text{ (g/cm}^3) \text{GeV}$. [^14]: We must remark, however, that this reasoning does not prove that the first order in matter perturbation theory is sufficient to obtain all the necessary conditions on the sterile state masses. [^15]: As we remarked in footnote 7 the non-unitary mixing matrix $U$ has some $W$ dependence through unitarity of the ${\bf U}$ matrix in the whole $(3+N)$ space. Therefore, the nature of the eq. (\[Schroedinger-eq-0th\]) as the zeroth-order in $W$ is ambiguous. However, following [@Fong:2016yyh], we remain in the treatment with this “$W$ effect renormalized basis” in this paper. [^16]: However, it appears that the flavor basis formulation of neutrino evolution in matter in high-scale unitarity violation poses highly nontrivial features such as non-Hermitian Hamiltonian [@Antusch:2006vwa], or the evolution equation $i \frac{d}{dx} \nu_{\alpha} = \sum_{j} \left[ U \left( {\bf \Delta_{a} } + U^{\dagger} A U \right) U^{\dagger} \right]_{\alpha \beta} \nu_{\beta}$ [@Escrihuela:2016ube]. The latter is not equivalent to (\[Schroedinger-eq-0th\]) in the vacuum mass eigenstate basis due to non-unitarity of the $U$ matrix. [^17]: One can apply our formulas of $S$ matrix obtained under the constant matter density approximation to semi-realistic calculation for earth crossing neutrinos by using them in each shell (core, mantle, and crust regions, etc.) with proper connecting conditions at the boundaries. [^18]: It can be re-parametrized in terms of the “$\alpha$ matrix parametrization” defined in ref. [@Escrihuela:2015wra]. The resultant values of $\alpha$ parameters are given as follows: $\alpha_{11} = 0.990$, $\alpha_{21} = - 0.0141$, $\alpha_{22} = 0.995$, $\alpha_{31} = -0.0445$, $\alpha_{32} = -0.0316$, $\alpha_{33} = 0.949$. [^19]: This feature must be obvious if one goes back to the derivation of bound on $\mathcal{C}_{\alpha \beta}$ in [@Fong:2016yyh]. [^20]: We are aware that the assumption of equal sterile neutrino masses is contradictory to the assumption of no accidental degeneracy in the sterile mass spectrum we made in section \[sec:probability-2nd\]. It was done not to complicate term by term evaluation of the perturbative series, and to avoid using degenerate perturbation theory. Fortunately, we can remove this assumption to second order in $W$ in which no purely sterile sector energy denominator is involved. [^21]: Of course, there is an issues of how to separate effects of $W^2$ correction terms from unitarity violation through $U$ matrix in leading order. [^22]: It might be easier to obtain the phase factor if we use a different decomposition of $\tilde{H}$ from (\[tilde-H0+H1\]) by absorbing $W^{\dagger} A W$ into $\tilde{H}_{0}$. [^23]: The phase itself needs not be small. Taking the matter potential of CC reaction and the earth diameter, $AL = 6.2 \left(\frac{\rho}{5 \text{g/cm}^3}\right) \left(\frac{L}{6,400 \mbox{km}}\right)$. Therefore, $ALW^2$ can be order unity for $|W| \simeq 0.4$. [^24]: This statement applies also to the original expression (\[P-beta-alpha-0th+2nd\]).

--- abstract: 'The simple linear model $$Y_i = \alpha + \beta \, x_i + \epsilon_i \qquad i=1,2, \ldots,N \geq 2$$ is considered, where the $x_i$’s are given constants and $\epsilon_1, \epsilon_2 , \ldots, \epsilon_N$ are iid with continuous distribution function $F$. An estimator of $\beta$ is proposed, based on the stochastic process in (\[due\]) and defined as $\tilde{\beta} = \frac 12 \, \left\{ \sup (b: G(\underline y;b) >0) + \right. $ $ \left. \inf (b: G(\underline y;b) <0) \right\}.$ The properties of $\tilde{\beta}$ and of the related confidence interval are studied. Some comparisons are given, in terms of asymptotic relative efficiency, with other estimators of $\beta$ including that obtained with the method of least squares.' author: - 'D. Michele Cifarelli' date: '[**Translation from Italian of the paper:**]{} Cifarelli, D.M. (1978). “La stima del coefficiente di regressione mediante l’indice di cograduazione di Gini", [*Rivista di matematica per le scienze economiche e sociali*]{} (now: [*Decisions in economics and finance*]{}), 1, 7–38.' title: 'Estimation of the regression slope by means of Gini’s cograduation index' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore Introduction and summary ======================== Consider the simple linear model $$Y(x_i) = Y_i = \alpha + \beta \, x_i + \epsilon_i \qquad i=1,2, \ldots,N,$$ where 1. $x_1, x_2, \ldots , x_N$ are known constants, supposed to be all distinct and increasingly ordered 2. $\epsilon_1, \epsilon_2, \ldots, \epsilon_N$ are mutually independent random variables with the same distribution function $F$ 3. $\alpha$ and $\beta$ are unknown parameters. The usual estimators of $\alpha$ and $\beta$ are those derived from the least squares method. As known, if the $\epsilon_i$’s have finite variance, such estimators possess some good properties. More specifically, they are unbiased and have minimum variance in the class of linear estimators (BLUE). When, in addition, the $\epsilon_i$’s are assumed to be normal, the above estimators coincide with the ones obtained by the maximum likelihood method and, besides being unbiased, they have minimum variance in the class of all unbiased estimators (MVUE) and they are normally distributed. Consider then the least squares estimator of $\beta$: $$\label{uno} \hat{\beta} = \frac{\displaystyle{\sum_{1\leq i \leq N}} (x_i - \bar x) (Y_i - \bar Y)}{\displaystyle{\sum_{1\leq i \leq N}} (x_i - \bar x)^2} \qquad \bar Y = \frac 1N \sum_{1\leq i \leq N} Y_i, \quad \bar x = \frac 1N \sum_{1\leq i \leq N} x_i.$$ As the corresponding estimate of $\beta$ strongly depends on the observed values $y_1, y_2, \ldots , y_N,$ the occurrence of outliers, that is of observations deviating from the main core of data, will likely influence such a procedure. This chance will often arise when the distribution of the disturbances $\epsilon_i$ has heavy tails, like in the case of the Cauchy, the double-exponential and other distributions. It is quite a serious drawback of the estimator $\hat{\beta}$ and attempts are occasionally made to remedy it by unconventionally deleting the most extreme observations. Another completely different problem of least squares concerns the interval estimation of $\beta.$ The possibility of producing a confidence interval for $\beta,$ or equivalently of testing the hypothesis $\beta = \beta_0,$ rests indeed on the assumption of normality for the variables $\epsilon_i$’s, so that, at least for limited values of $N,$ the whole procedure proves to be fairly “unrobust" when such an assumption is not met (even if the asymptotic normality of $\hat{\beta}$ is assumed). The asymptotic theory for such intervals cannot be always invoked, besides, for such a theory rests on the asymptotic normality of $\hat{\beta}$ which is not always ensured (\[1\]). Two distinct methods can be used to solve the first of the problems above: two distinct ways can be tried: one can decide to delete outliers or, alternatively, to base the estimation of $\beta$ on suitable functions of ranks, which are possibly unaffected by the extreme observations. Common thinking is that the deletion of outliers must follow rules that are clearly stated before, and not after, data are available; this task cannot then rely on a subjective judgment, which will deprive the researcher of any foundation to study the related procedure. The papers by Brown and Mood (\[2\]), Adichie (\[3\]), Theil (\[4\]) and Sen (\[5\]) are framed, instead, in the logic of ranks, which proved to be able to overcome both the drawbacks outlined above. To introduce such kinds of procedures, notice that the estimator (\[uno\]) can be rewritten so that the slopes $$P_{ij} = \frac{Y_j-Y_i}{x_j-x_i}, \qquad i<j,$$ are explicitly shown. Indeed, $$\hat{\beta} = \frac{\displaystyle{\sum_{i<j}} P_{ij} \, (x_j - x_i)^2}{\displaystyle{\sum_{i<j}} (x_j-x_i)^2} = \frac{\displaystyle{\sum_{i<j}} (Y_j-Y_i) \, (x_j - x_i)}{\displaystyle{\sum_{i<j}} (x_j-x_i)^2}.$$ The above equality shows that $\hat{\beta}$ can be regarded as a mean of the $P_{ij}$’s with weights $(x_j-x_i)^2.$ To solve the problem of outliers, one can then obviously substitute such a weighted mean with a suitable function of the slopes $P_{ij},$ so as to result unaffected (at least less affected) by the extreme observations. This approach is substantially the one used by Theil, who proposed, as an estimator of $\beta,$ the median of the slopes $P_{ij}$, or the central value of the median interval when dealing with an even number of slopes. Theil’s procedure is related to the one by Sen, who derived an estimator of $\beta$ by using a measure of concordance, which is essentially Kendall’s $\tau,$ between the ranks of $Y_i-bx_i$ and those of $x_i,$ $i=1, 2, \ldots , N.$ The obtained estimator is the same proposed by Theil, but it can be applied under the general assumption that the $x_i$’s are not all distinct. It is interesting to note that the same result can be obtained by starting from a completely different point of view, namely by using the minimax estimator with a non-quadratic loss function (\[6\]). The study of the asymptotic properties of both the point and the interval estimators is due to Sen as well, along with the determination of the asymptotic relative efficiency of the proposed estimator with respect to the one of least squares and to other estimators, proposed by Adichie (\[3\]), which were generalized, somehow under a more general framework, by Koul (\[7\]). To have an idea of the efficiency gained by the Theil-Sen estimator, $\beta^*,$ with respect to that of least squares, $\hat{\beta},$ it suffices to notice that there are cases where $$\lim_{N \rightarrow +\infty} \frac{{{\rm Var}}(\hat{\beta})}{{{\rm Var}}(\beta^*)} = \alpha > 1$$ and that, even in the normal case, if the constants $x_i$’s are conveniently chosen, $$\lim_{N \rightarrow +\infty} \frac{{{\rm Var}}(\hat{\beta})}{{{\rm Var}}(\beta^*)} = \frac{3}{\pi} \simeq 0.95.$$ Instead of measuring the concordance between the residuals $Y_i-bx_i$ and $x_i,$ $i=1,2,\ldots, N,$ by means of $\tau$ or other indices, as later proposed (\[8\]), one can obviously consider Gini’s cograduation index $G.$ This procedure is quite different from the one proposed by Adichie, who used a class of indices which are functions of the ranks of residuals $Y_i-bx_i$ and of the values $x_i,$ while $G$ is based, as known, on the ranks of $Y_i-bx_i$ and on the [*ranks*]{} of $x_i,$ $i=1, 2, \ldots, N.$ In addition, the results gained using $G$ are likely to be structurally different from the ones obtained from $\tau$ or Spearman’s $R,$ because $G$ is believed to locate some aspects of cograduation which neither $\tau$ nor $R$ can account for. This statement, in effect, is also confirmed by the fact that the correlation coefficient between $G$ and $\tau$ (or between $G$ and $R$), as shown in (\[9\]) and in (\[10\]), even though quite large for a limited value of $N$ (in absence of cograduation), never reaches one, not even asymptotically. Let $\underline y$ be a realization of $\underline Y,$ $G(\underline y,b)$ be Gini’s cograduation index computed from the residuals $y_i-bx_i$ and $x_i,$ $i=1,2,\ldots, N,$ and $\tilde{\beta} (\underline y)$ be the function of data obtained by making $G(\underline y;b)$ as close to zero as possible.[^1] $\tilde{\beta}(\underline y)$ can then be regarded as a minimum $G-$dependence estimate or, more correctly, as a maximum $G-$indifference estimate of $\beta.$ The same notation can be used for the estimator $\tilde{\beta}(\underline Y).$ This terminology is coherent with the term “indifference” proposed by Gini (\[11\], p. 330) to indicate the lack of concordance or discordance between two rankings, in comparison with the term “(stochastic) independence" which should instead be used to indicate lack of connection. Indeed, the two conditions (independence and indifference) are not equivalent, though W. Hoeffding (\[12\], p. 555) showed that, under suitable assumptions, they imply each other. This paper aims at proposing the estimator $\tilde{\beta}$ and at analyzing its properties. In section 2, the main problem is framed and the stochastic process $G(\underline Y;b),$ whose properties are studied in section 3, is introduced. In section 4 the estimator $\tilde{\beta}$ is formally defined and some properties of its distribution are analyzed. In section 5 the task of building a confidence interval for $\beta$ is faced, for every sample size and independently of the distribution function of disturbances, $F,$ which will be exclusively assumed to be continuous. As the proposed estimator does not possess a closed form as a function of data, section 6 gives some hints to fasten its computation for a given sample realization. Section 7 deals with the asymptotic distribution of $\tilde{\beta}.$ Such distribution is closely related to the one of $G(\underline Y;b)$ whose analysis is rather long and hence is developed in the Appendix, to simplify the structure of the paper. Finally, section 8 focuses on the comparison, based on the asymptotic relative efficiency (ARE), of the estimator $\tilde{\beta}$ with the one of least squares and the one by Theil and Sen. The drawn conclusions are quite interesting, as the asymptotic efficiency of $\tilde{\beta} $ relative to the other two estimators is shown to be (for the chosen values of the $x_i$’s) greater than 1 when the distribution has tails heavier than the normal case; this fact recommends a wide use of $\tilde{\beta}$, even if its computation might seem somehow unpractical. Problem settings ================ Let $Y_1, Y_2, \ldots, Y_n$ be $N$ mutually independent random variables with distribution functions $$P\left\{ Y_i \leq y \right\} = F_i(y) = F(y- \alpha - \beta \, x_i) \qquad i=1,2, \ldots, N; \, N \geq 2$$ where $F$ is any continuous distribution function and $x_1 < x_2 < \cdots < x_N$ are known constants. As the main interest is the estimation of $\beta,$ in the following $\alpha = 0$ will be supposed, without loss of generality. For every real $b,$ consider the new variables $$Z_i(b) = Y_i - b \, x_i \qquad i =1,2, \ldots , N$$ and use them to build the function (of $b$) $$\label{due} G(\underline Y;b) = \frac 2D \sum_{1 \leq i \leq N} \left\{ \left| N+1-i -R(Z_i(b)) \right| - \left| i - R(Z_i(b)) \right| \right\} \: \: b \in \Re - B$$ with $$B = \left\{ b: \: \: Y_i - b \, x_i = Y_j - b \, x_j \: \: \mbox{ for at least a couple } i \neq j \right\},$$ where $R(Z_i(b))$ denotes the rank of $Z_i(b)$ in the sorting of $\{ Z_1(b), \ldots , Z_N(b) \}$ and $D=N^2$ if $N$ is even or $D=N^2-1$ if $N$ is odd. The function $G(\underline Y;b)$ is not defined in the set $B,$ which is clearly finite. For every chosen $b \notin B,$ the function (\[due\]) is the known Gini’s cograduation index between $Y_1- b \, x_1, \ldots , Y_N - b \, x_N$ and $x_1, \ldots , x_N;$ conversely, as a function of $b,$ it is a stochastic process whose realizations correspond to the events $\omega \equiv (y_1, \ldots , y_N) \in A \subseteq \Re^N.$ As the random variables $Y_i - \beta \, x_i ,$ $i=1,2, \ldots, N,$ are iid, $G(\underline Y;\beta)$ has the known distribution of Gini’s index in case of indifference and hence $${{\rm E}}\left\{ G(\underline Y; \beta) \right\} = 0.$$ One can then naturally estimate the parameter $\beta$ by making $G(\underline Y;b)$ as close to zero as possible, that is by letting $$G (\underline Y; b) \simeq 0 .$$ Equivalently, the estimator proposed in this paper is a function $\tilde{\beta} = \tilde{\beta}(\underline Y)$ so that the sequence $Y_i-\tilde{\beta} \, x_i$ will result as indifferent as possible to $x_i,$ $i=1, 2,\ldots , N.$ In effect, this is a natural requirement when considering that the least squares estimator can be regarded as a function $\hat{\beta} = \hat{\beta} (\underline Y)$ which makes the usual sample covariance between $Y_i - \hat{\beta} \, x_i$ and $x_i$ $$\frac 1N \sum_{1 \leq i \leq N} \left( Y_i - \bar Y - \hat{\beta} \, (x_i - \bar x) \right) \, (x_i - \bar x)$$ vanish. Such a covariance plays then, in another framework, the same role of $G(\underline Y;\tilde{\beta}).$ Of course, to implement the proposed procedure one must be sure that the obtained estimator is, in some sense, unique. This could be the case if the realizations of the process $G(\underline Y;b)$ resulted strictly monotonic functions of $b.$ In the following section, such realizations are shown to be non increasing functions of $b.$ This fact implies that a whole interval of values of $b$ may exist where $G(\underline Y; b)=0$ or, alternatively, two consecutive intervals $I_1$ and $I_2$ so that $$G(\underline Y;b)>0 \quad b \in I_1 \qquad \mbox{and} \qquad G(\underline Y;b)<0 \quad b \in I_2 .$$ Properties of $G(\underline Y;b)$ ================================= As claimed in the previous section, $G(\underline Y;b)$ is not defined for every real $b;$ more specifically, if $\underline y = (y_1, \ldots, y_N)$ is a realization of $\underline Y,$ $G(\underline y;b)$ turns out to be undefined in the set $$B = \left\{ b: \: \: y_i - b \, x_i = y_j - b \, x_j \: \: \mbox{ for at least a couple } i \neq j \right\},$$ which will be referred to, in the following, as $$B = \left\{ b^{(1)}, b^{(2)}, \cdots , b^{(r)} \right\} \qquad 1 \leq r \leq {N \choose 2} \qquad b^{(1)} < b^{(2)} < \cdots < b^{(r)}.$$ For any $n-$tuple $\underline y,$ the function $G(\underline y;b)$ is constant inside each interval $$(-\infty,b^{(1)}), \: (b^{(1)},b^{(2)}), \: \cdots \:, \: (b^{(r)}, +\infty) .$$ To prove such a claim, it suffices to show that, inside each of the intervals above, $\{ R(Z_1(b)), \ldots , R_N(Z(b)) \}$ is the same permutation of the set of integers $\{ 1, 2,$ $\ldots , N \}.$ Let $b^{(i)} < b < b^{(i+1)},$ $i=1,\ldots, r-1.$ There are at least two couples of indices $(u_1,v_1)$ and $(u_2,v_2),$ with $u_1<v_1,$ $u_2<v_2,$ such that $$b^{(i)} = \frac{y_{v_1} - y_{u_1}}{x_{v_1} - x_{u_1}} \, < \, b \, < \, \frac{y_{v_2} - y_{u_2}}{x_{v_2} - x_{u_2}} = b^{(i+1)} .$$ This fact implies that, for every $b$ belonging to the interval $\left( b^{(i)}, b^{(i+1)} \right),$ $$R(Z_{v_1}(b)) < R(Z_{u_1}(b)) \qquad R(Z_{v_2}(b)) > R(Z_{u_2}(b)).$$ The former of the above inequalities holds equivalently for every couple whose slope is less than or equal to $b^{(i)}$; the latter inequality holds for those couples whose slope is greater than or equal to $b^{(i+1)}.$ This remark shows that, for every $b$ belonging to the considered interval, the permutation taken by $\{ R(Z_1(b)), \ldots ,$ $R_N(Z(b)) \}$ does not change, which suffices to state that $G(\underline y;b)$ does not change its value. The same conclusions can be drawn when considering the first and the last intervals for $b.$ Specifically, as $$\begin{array}{rcll} \{ R(Z_1(b)), \ldots , R_N(Z(b)) \} &=& \{ 1,2, \ldots , N \} & \quad \forall b < b^{(1)} \\ \{ R(Z_1(b)), \ldots , R_N(Z(b)) \} &=& \{ N, N-1,\ldots , 1 \} & \quad \forall b > b^{(r)} , \end{array}$$ one gets $$G(\underline y;b) = 1 \: \: \forall b < b^{(1)} \qquad \mbox{and} \qquad G(\underline y;b) = -1 \: \: \forall b > b^{(r)} .$$ Obviously the definition of $G(\underline Y;b)$ can be supplemented by setting $$G(\underline y; b^{(i)}) = \lim_{b \rightarrow b^{(i)+}} \, \, G(\underline y;b) ,$$ so as to let every realization of the process be right continuous. In the following, $G(\underline Y;b)$ will be supposed to be defined for every real $b.$ Consider now two adjacent intervals ${\cal I}_1 = [ b^{(s)}, b^{(s+1)} )$ and ${\cal I}_2 = [ b^{(s+1)},$ $ b^{(s+2)} )$ and let $b \in {\cal I}_1,$ $b_1 \in {\cal I}_2.$ When shifting from $b$ to $b_1,$ the above discussion shows that the ranks of $Z_i(b)$ will be only partially modified. Specifically, suppose that $$b^{(s+1)} = \frac{y_{v_1} - y_{v_0}}{x_{v_1} - x_{v_0}} = \frac{y_{v_2} - y_{v_0}}{x_{v_2} - x_{v_0}} = \cdots = \frac{y_{v_m} - y_{v_0}}{x_{v_m} - x_{v_0}}$$ with $$v_0 < v_1 < \cdots <v_m \qquad \mbox{and} \qquad m \geq 1 ,$$ which means that the observations $y_{v_0}, y_{v_1} , \ldots , y_{v_m}$ lie on the same straight line. When shifting from $b$ to $b_1,$ only the ranks of $Z_{v_0}, \ldots , Z_{v_m}$ will be modified, that is $$\label{tre} R(Z_{v_i}(b_1)) = R(Z_{v_i}(b)) + m -2i \qquad i =0,1,\ldots, m;$$ furthermore $$\label{quattro} R(Z_{v_i}(b)) = R(Z_{v_0}(b)) +i \qquad i =0,1,\ldots, m.$$ The above equalities derive immediately after considering that the rank of the generic $Z_j(b)$ equals the number of observations $y_j$ which lie under or on the straight line with slope $b$ passing through $(x_j,y_j).$ If $1 \leq v_0 < v_1 < \cdots < v_m,$ $m \geq 0,$ then $$\sum_{ 1 \leq i \leq m} \left| v_i - m +i - \xi \right| \: \geq \: \sum_{1 \leq i \leq m} \left| v_i - i - \xi \right| \qquad \forall \xi \in \Re.$$ Consider the functions $$\phi_i (\xi) = \left| v_i - m + i -\xi \right| - \left| v_i - i - \xi \right| \qquad i =0,1, \ldots , m$$ and define $$\phi (\xi) = \sum_{0 \leq i \leq m} \phi_i(\xi) = \sum_{0 \leq i \leq [m/2]} \left( \phi_i(\xi) + \phi_{m-i} (\xi) \right).$$ For every $0 \leq i \leq [m/2],$ one gets $$\begin{array}{rcl} \phi_i(\xi) &=& \left\{ \begin{array}{l@{\qquad}l} -(m-2i) \leq 0 & \xi< v_i -m+i \\ 2\xi-2v_i+m & v_i -m +i \leq \xi <v_i -i \\ m-2i \geq 0 & \xi \geq v_i -i \end{array} \right. \\ \\ \phi_{m-i} (\xi) &=& \left\{ \begin{array}{l@{\qquad}l} m -2i \geq 0 & \xi < v_{m-i} - m + i \\ 2v_{m-i} - m - 2 \xi & v_{m-i} - m + i \leq \xi < v_{m-i} - i \\ -(m-2i) \leq 0 & \xi \geq v_{m-i} -i \end{array} \right. \end{array}$$ hence $$\phi_i(\xi) + \phi_{m-i} (\xi) \geq 0 \qquad \forall \xi \in \Re,$$ from which the proof follows. $\qed$ The following theorem can now be stated. For every $n-$tuple $\underline y,$ the function $G(\underline y;b)$ is non increasing. It suffices to prove that the function $${\cal U}(b) = \sum_{1 \leq i \leq N} \left| i - R(Z_i(b)) \right|$$ is non decreasing and that the function $${\cal V}(b) = \sum_{1 \leq i \leq N} \left| N+1 - i - R(Z_i(b)) \right|$$ is non increasing. Only the statement for ${\cal U}(b)$ will be proved; the one for ${\cal V}(b)$ follows similarly. Suppose that $b^{(s)} < b < b^{(s+1)},$ $b^{(s+1)} < b_1 < b^{(s+2)},$ with $b^{(0)} = -\infty,$ $b^{(r+1)} = + \infty,$ $$b^{(s+1)} = \frac{y_{v_{i,t}} - y_{v_{0,t}}}{x_{v_{i,t}} - x_{v_{0,t}}} \qquad i = 1,2,\ldots, m_t \qquad t=1,2,\ldots ,k \geq 1$$ and $v_{0,t} < v_{1,t} < \cdots < v_{m,t}.$ When shifting from $b$ to $b_1,$ only the ranks of $$\begin{aligned} \left\{ Z_{v_{0,1}}(b), \ldots , Z_{v_{m_1,1}}(b) \right\} \: , \: \left\{ Z_{v_{0,2}}(b), \ldots , \right. & \left. Z_{v_{m_2,2}}(b) \right\}, \: \ldots \: , \\ & \left\{ Z_{v_{0,k}}(b), \ldots , Z_{v_{m_k,k}}(b) \right\} \end{aligned}$$ will change; hence $$\begin{aligned} \lefteqn{{\cal U}(b_1) - {\cal U}(b) = } \\ && = \sum_{1 \leq t \leq k} \left\{ \sum_{0 \leq i \leq m_t} \left| v_{i,t} - R(Z_{v_{i,t}} (b_1)) \right| - \sum_{0 \leq i \leq m_t} \left| v_{i,t} - R(Z_{v_{i,t}} (b)) \right| \right\} .\end{aligned}$$ By (\[tre\]) and (\[quattro\]), $$\begin{aligned} {\cal U}(b_1) - {\cal U}(b) & = \sum_{1 \leq t \leq k} \left\{ \sum_{0 \leq i \leq m_t} \left| v_{i,t} - m_t + i - R(Z_{v_{0,t}} (b)) \right| + \right. \\ & \quad \left. - \sum_{0 \leq i \leq m_t} \left| v_{i,t} - i - R(Z_{v_{0,t}} (b)) \right| \right\} .\end{aligned}$$ According to the Lemma stated above, the quantity in brackets is non negative for every value of $R(Z_{v_{0,t}}(b))$ and hence $${\cal U}(b_1) \geq {\cal U}(b),$$ which holds for all intervals and thus gives the proof. $\qed$ Definition of the point estimator and related properties ======================================================== Section 3 showed that all trajectories of the stochastic process $G(\underline Y;b)$ are non increasing functions of $b$ and that $$G(\underline y;b) = 1 \quad \forall b < b^{(1)}; \qquad \qquad G(\underline y;b) = -1 \quad \forall b > b^{(r)} .$$ For any observed $n-$tuple $\underline y,$ the following two cases will then arise: - a whole interval of $b$ exists where $G(\underline y;b)=0$ pointwise - two adjacent intervals exist such that $$\begin{array}{rl} G(\underline y;b) >0 & \qquad \mbox{in the first interval} \\ G(\underline y;b) <0 & \qquad \mbox{in the second interval.} \end{array}$$ When in case a), one could choose the central value of the interval as an estimate of $\beta;$ in case b), the value $$\sup \left\{ b: \: \: G(\underline y;b)>0 \right\} = \inf \left\{ b: \: \: G(\underline y;b)<0 \right\}$$ could instead be chosen. The two cases can then be unified by defining the following maximum $G-$indifference estimator: $$\label{cinque} \tilde{\beta} = \tilde{\beta}(\underline y) = \frac 12 \left[ \sup \left\{ b: \: G(\underline Y;b)>0 \right\} + \inf \left\{ b: \: G(\underline Y;b)<0 \right\} \right]$$ which is similar to the estimator proposed in (\[13\]) for the location parameter. One of the following sections will show how to get a fast computation of $\tilde{\beta}(\underline y).$ First of all, the next propositions will give three quite immediate properties of the distribution of $\tilde{\beta}.$ The distribution of $\tilde{\beta} - \beta$ does not depend on the parameter $\beta.$ Let $$\tilde{\beta}_1 (Y_1-\beta \, x_1, \ldots , Y_N-\beta \, x_N) = \sup \left\{ b: \: G(Y_1 - \beta \, x_1, \ldots , Y_N - \beta \, x_N; \, b) >0 \right\} .$$ From the definition of $G(\underline Y;b),$ $$\tilde{\beta}_1 (Y_1-\beta \, x_1, \ldots , Y_N-\beta \, x_N) = \tilde{\beta}_1(\underline Y) - \beta .$$ Similarly, if $$\tilde{\beta}_2 (Y_1-\beta \, x_1, \ldots , Y_N-\beta \, x_N) = \inf \left\{ b: \: G(Y_1 - \beta \, x_1, \ldots , Y_N - \beta \, x_N; \, b) <0 \right\} ,$$ one gets $$\tilde{\beta}_2 (Y_1-\beta \, x_1, \ldots , Y_N-\beta \, x_N) = \tilde{\beta}_2(\underline Y) - \beta$$ and, by definition (\[cinque\]), $$\tilde{\beta}(Y_1-\beta \, x_1, \ldots , Y_N-\beta \, x_N) = \tilde{\beta}(\underline Y) - \beta .$$ However, the lhs of the above equality is a function of the variables $Y_1-\beta \, x_1, \ldots , Y_N-\beta \, x_N,$ whose distributions, by hypothesis, do not depend on $\beta.$ $\qed$ Proposition 1 equivalently states that, if $P\left( \tilde{\beta} \leq b \right) = \psi(b; \beta),$ then $\psi(b; \beta)=\varphi(b-\beta),$ namely $\beta$ is a location parameter of the distribution of $\tilde{\beta}.$ This fact will allows setting $\beta = 0$ in the following, without loss of generality. $\tilde{\beta}$ has a continuous distribution. It suffices to prove that the two variables $$\begin{aligned} \tilde{\beta}_1 (\underline Y) = & \sup \{ b: G(\underline Y;b) >0 \} \\ \tilde{\beta}_2 (\underline Y) = & \inf \{ b: G(\underline Y;b) <0 \} \end{aligned}$$ are both continuous. The continuity of $\tilde{\beta}_1$ and $\tilde{\beta}_2,$ indeed, implies that the joint distribution of $(\tilde{\beta}_1, \tilde{\beta}_2)$ is continuous and similarly for $\tilde{\beta} = \frac 12 \left( \tilde{\beta}_1 + \tilde{\beta}_2 \right).$ As every realizations of $G(\underline Y;b)$ is non-increasing with at least a jump at a point of the form $(Y_j-Y_i)/(x_j-x_i),$ $i \neq j,$ the event $\tilde{\beta} = a \in \Re$ implies that $G(\underline Y;b)$ has a jump at $a.$ Hence $$\begin{aligned} P \left\{ \tilde{\beta}_1 = a \right\} & \leq P \left\{ \frac{Y_j-Y_i}{x_j-x_i} = a \quad \mbox{ for at least a couple } (i<j) \right\} \\ & \leq \sum_{i<j} \, P \left\{ \frac{Y_j-Y_i}{x_j-x_i} = a \right\} \end{aligned}$$ As, by hypothesis, the variables $Y_k$’s are continuous (and independent), the same is true for the variables $(Y_j-Y_i)/(x_j-x_i);$ hence $P(\tilde{\beta}_1 = a) =0.$ Similarly, one can show that $P(\tilde{\beta}_2 = a) =0.$ $\qed$ Notice that, by following the same steps as for the proof of Proposition 2, a similar result can be obtained for the variables $$\begin{aligned} \tilde{\beta}_I (\underline Y) = & \inf \{ b: G(\underline Y;b) <G^* \} \\ \tilde{\beta}_S (\underline Y) = & \sup \{ b: G(\underline Y;b) >-G^*\} \end{aligned}$$ where $G^*>0$ is a given constant. Before stating another property concerning the distribution of $\tilde{\beta},$ the following equality should be considered: $$\label{sei} G(-\underline y;b) = - G(\underline y;-b) \qquad \forall \, \underline y \subseteq \Re^N \mbox{ and } \forall \, b \in \Re.$$ Indeed, $$\begin{aligned} G(- \underline y;b) & = \frac 2D \, \left\{ \sum_i \left| N+1-i-R(-y_i+b\, x_i) \right| - \sum_i \left| i - R(-y_i+b\, x_i) \right| \right\} \\ & = \frac 2D \, \left\{ \sum_i \left| N+1-i-R[-(y_i-b\, x_i)] \right| - \sum_i \left| i - R[-(y_i-b\, x_i)] \right| \right\} \end{aligned}$$ from which the above result follows, as it is obviously $$R[-(y_i-b\, x_i)] = N+1 - R(y_i-b\, x_i).$$ If $Y_1, Y_2, \ldots , Y_n$ are symmetrically distributed, $${{\rm E}}(\tilde{\beta}) = \beta \qquad \forall \, N \geq 2 \quad \forall \, \beta \in \Re .$$ According to Proposition 1, it can be assumed that $\beta =0.$ Now notice that $$\label{sette} \tilde{\beta} (- \underline Y ) = - \tilde{\beta} (\underline Y)$$ Indeed, $$\tilde{\beta} (- \underline Y ) = \frac 12 \left[ \sup \left\{ b: \, G(-\underline Y;b) >0 \right\} + \inf \left\{ b: \, G(-\underline Y;b) <0 \right\} \right]$$ and, by (\[sei\]), $$\begin{aligned} \tilde{\beta} (- \underline Y ) &= \frac 12 \left[ \sup \left\{ b: \, G(\underline Y;-b) <0 \right\} + \inf \left\{ b: \, G(\underline Y;-b) >0 \right\} \right] = \\ &= \frac 12 \left[ -\inf \left\{ b: \, G(\underline Y;b) <0 \right\} - \sup \left\{ b: \, G(\underline Y;b) >0 \right\} \right] = \\ & = -\tilde{\beta} ( \underline Y ) \end{aligned}$$ By the symmetry of $Y_1, Y_2, \ldots , Y_N,$ the variables $\tilde{\beta} ( \underline Y )$ and $\tilde{\beta} ( -\underline Y )$ share the same distribution. By using (\[sette\]) and this latter property, one can then claim that $\tilde{\beta} ( \underline Y )$ has a distribution symmetric around zero (which is the value of $\beta$); this fact completes the proof. $\qed$ Confidence intervals for $\beta$ ================================ As the variables $Y_i - \beta \, x_i,$ $i=1,2, \ldots , N$ are iid, $G(\underline Y ; \beta)$ has the known distribution of Gini’s cograduation index under indifference. There exist quite complete tables of such a distribution. By using these tables, a constant $G^*>0$ such that, for a suitable $\alpha,$ $$P \{ - G^* < G(\underline Y;\beta) < G^* \} = 1-\alpha \qquad 0 < \alpha < 1$$ can be easily determined. Consider now the variables $$\begin{aligned} \tilde{\beta}_I (\underline Y) = & \inf \{ b: G(\underline Y;b) <G^* \} \label{otto} \\ \tilde{\beta}_S (\underline Y) = & \sup \{ b: G(\underline Y;b) >-G^* \label{nove} \} .\end{aligned}$$ From (\[otto\]), as $G(\underline y;b)$ is non increasing, $$\inf \{ b: \, G(\underline y;b) < G^* \} < \beta \, \, \Rightarrow \, \, G(\underline y;\beta) < G^* \,\, \Rightarrow \, \, \inf \{ b: \, G(\underline y;b) < G^* \} \leq \beta$$ Similarly, from (\[nove\]), $$\sup \{ b: G(\underline y;b) > -G^* \} > \beta \Rightarrow \, G(\underline y;\beta) > - G^* \Rightarrow \, \sup \{ b: G(\underline y;b) > -G^* \} \geq \beta$$ It follows that $$\label{dieci} \{ \tilde{\beta}_I < \beta < \tilde{\beta}_S \} \,\, \Rightarrow \, \, \{ - G^* < G(\underline Y;\beta) < G^* \} \, \, \Rightarrow \, \, \{ \tilde{\beta}_I \leq \beta \leq \tilde{\beta}_S \} .$$ Hence, from (\[dieci\]), $$P\{ \tilde{\beta}_I < \beta < \tilde{\beta}_S \} \,\leq \, P \{ - G^* < G(\underline Y;\beta) < G^* \} \, \leq \, P \{ \tilde{\beta}_I \leq \beta \leq \tilde{\beta}_S \}$$ and, by the continuity of $\tilde{\beta}_I$ and $\tilde{\beta}_S,$ $$P\{ \tilde{\beta}_I < \beta < \tilde{\beta}_S \} \,= \, P \{ - G^* < G(\underline Y;\beta) < G^* \} \, = \, 1-\alpha .$$ The variables (\[otto\]) and (\[nove\]) are then respectively the lower and the upper bounds of the confidence interval for $\beta,$ for any continuous distribution function $F.$ Computation of $\tilde{\beta}$ ============================== In the previous sections, the point estimator for $\beta$ and the bounds of the confidence interval for the same parameter were defined. However, a closed expression for such statistics as functions of the elements of the sample, was not provided. This fact makes it difficult to study further properties of the considered statistics for a finite value of $N.$ The following section will then deal with the asymptotic distribution of $\tilde{\beta}.$ Before doing that, this section aims at providing an easy scheme to determine the values taken by $\tilde{\beta},$ $\tilde{\beta}_I$ and $\tilde{\beta}_S$ for any given sample realization. Let $\underline y = (y_1, \ldots , y_N)$ denote the observations on the response variable corresponding to $x_1, \ldots , x_N$ and suppose computing the ${N \choose 2}$ slopes $P_{ij} = \frac{y_j-y_i}{x_j-x_i},$ $i<j$ which are not necessarily all distinct. Denote with ${}_{(k)}P_{ij}$ the distinct sorted values of such slopes, $k=1, 2, \ldots, r.$ Of course the same slope can correspond to more than a couple of indices $(i,j).$ With the aid of equality (\[tre\]) in section 3, one can then produce a table displaying, for each row $i=1,2,\ldots, N,$ the ranks of $y_i-b\, x_i$ when $b$ belongs to the possible intervals determined by the slopes ${}_{(k)}P_{ij}.$ To illustrate the construction of such a table, suppose that the observed values are $$\begin{array}{c|@{\quad}c@{\quad}c@{\quad}c@{\quad \:}c} x_i & 1 & 2 & 3 & 4 \\ \hline y_i & 2 & 2.5 & 4 & 5 \end{array}$$ The six possible slopes are $$P_{12}=0.5 \quad P_{13}=1 \quad P_{14}=1 \quad P_{23}=1.5 \quad P_{24}=1.25 \quad P_{34}=1,$$ so that the sorted distinct slopes are $${}_{(1)}P_{12} = 0.5 \quad {}_{(2)}P_{13} = {}_{(2)}P_{14} = {}_{(2)}P_{34} = 1 \quad {}_{(3)}P_{24} = 1.25 \quad {}_{(4)}P_{23} = 1.5.$$ A table with $N=4$ rows can now be produced as follows. First of all a vertical line is built for every slope ${}_{(k)}P_{ij}$ and, on this line, the $i-$th row is marked with a circle and the $j-$th row is marked with a square. For every $h-$th row, one can then put suitable integer values, starting from $h,$ by adding a unit if a circle is met and by subtracting a unit if a square is met. If more than a single circle or square is met, the number in the previous column will be simply increased by the number of circles and decreased by the number of squares. As an example, on the third row, when passing from the second to the third column, a circle and a square are met; the number on the second column (3) should then be increased by 1 and decreased by 1, so that the same value (3) is reported in the third column. (8,8)(0,0) (1,4.1)[(1,0)[5]{}]{}(1,5.1)[(1,0)[5]{}]{} (1,6.1)[(1,0)[5]{}]{} (1,7.1)[(1,0)[5]{}]{} (2,3.5)[(0,1)[4.5]{}]{}(3,3.5)[(0,1)[4.5]{}]{} (4,3.5)[(0,1)[4.5]{}]{} (5,3.5)[(0,1)[4.5]{}]{} (0.5,4)[4]{}(0.5,5)[3]{} (0.5,6)[2]{} (0.5,7)[1]{} (6.5,4.5)[$R(Z_4(b))$]{}(6.5,5.5)[$R(Z_3(b))$]{} (6.5,6.5)[$R(Z_2(b))$]{} (6.5,7.5)[$R(Z_1(b))$]{} (1.5,4.5)[4]{}(1.5,5.5)[3]{} (1.5,6.5)[2]{} (1.5,7.5)[1]{} (2.5,4.5)[4]{} (2.5,5.5)[3]{} (2.5,6.5)[1]{} (2.5,7.5)[2]{} (3.5,4.5)[2]{} (3.5,5.5)[3]{} (3.5,6.5)[1]{} (3.5,7.5)[4]{} (4.5,4.5)[1]{} (4.5,5.5)[3]{} (4.5,6.5)[2]{} (4.5,7.5)[4]{} (5.5,4.5)[1]{} (5.5,5.5)[2]{} (5.5,6.5)[3]{} (5.5,7.5)[4]{} (1.5,3)[$\begin{array}{c} {}_{(1)}P_{12} \\ \scriptstyle{\bf (0.5)} \end{array}$]{}(2.5,2.6)[$\begin{array}{c} {}_{(2)}P_{13} \\ {}_{(2)}P_{14} \\ {}_{(2)}P_{34} \\ \scriptstyle{\bf (1)} \end{array}$]{} (3.5,3)[$\begin{array}{c} {}_{(3)}P_{24} \\ \scriptstyle{\bf (1.25)} \end{array}$]{} (4.5,3)[$\begin{array}{c} {}_{(4)}P_{23} \\ \scriptstyle{\bf (1.5)} \end{array}$]{} (2,7.1)(3,7.1) (3,7.1) (4,6.1) (5,6.1) (3,5.1) (1.9,6.0)[(0.2,0.2)]{} (2.9,5.0)[(0.2,0.2)]{} (4.9,5.0)[(0.2,0.2)]{} (2.9,4.0)[(0.2,0.2)]{} (2.825,3.925)[(0.35,0.35)]{} (3.9,4.0)[(0.2,0.2)]{} The figures on the $h-$th row of the above table are the ranks of $y_h-b\, x_h$ as long as $b$ ranges in the intervals $(-\infty, {}_{(1)}P),$ $({}_{(1)}P, {}_{(2)}P),$ $({}_{(2)}P, {}_{(3)}P),$ $({}_{(3)}P, {}_{(4)}P),$ $({}_{(4)}P, + \infty).$ For example, $$\begin{array}{ll@{\qquad}rl@{\,}l} R(Z_2(b)) & = 2 & &b < {}_{(1)}P & = 0.5 \\ & = 1 & 0.5 \leq &b < {}_{(3)}P & = 1.25 \\ & =2 & 1.25 \leq &b < {}_{(4)}P & = 1.5 \\ & =3 &&b \geq 1.5 \end{array}$$ After the above table, two further tables can be produced by computing, for the $i-$th row, the quantities $$|N+1-i-R(Z_i(b))| \qquad \mbox{ and } \qquad |i-R(Z_i(b))|$$ One then gets (12,3.5)(0,0) (1,0.7)[(1,0)[5]{}]{}(1,1.4)[(1,0)[5]{}]{} (1,2.1)[(1,0)[5]{}]{} (1,2.8)[(1,0)[5]{}]{} (2,0)[(0,1)[3.5]{}]{}(3,0)[(0,1)[3.5]{}]{} (4,0)[(0,1)[3.5]{}]{} (5,0)[(0,1)[3.5]{}]{} (1.5,0.2)[**8**]{} (1.5,0.9)[3]{} (1.5,1.6)[1]{} (1.5,2.3)[1]{} (1.5,3.0)[3]{} (2.5,0.2)[**8**]{} (2.5,0.9)[3]{} (2.5,1.6)[1]{} (2.5,2.3)[2]{} (2.5,3.0)[2]{} (3.5,0.2)[**4**]{} (3.5,0.9)[1]{} (3.5,1.6)[1]{} (3.5,2.3)[2]{} (3.5,3.0)[0]{} (4.5,0.2)[**2**]{} (4.5,0.9)[0]{} (4.5,1.6)[1]{} (4.5,2.3)[1]{} (4.5,3.0)[0]{} (5.5,0.2)[**0**]{} (5.5,0.9)[0]{} (5.5,1.6)[0]{} (5.5,2.3)[0]{} (5.5,3.0)[0]{} (0.5,0.2)[**Tot.**]{} (7,0.7)[(1,0)[5]{}]{}(7,1.4)[(1,0)[5]{}]{} (7,2.1)[(1,0)[5]{}]{} (7,2.8)[(1,0)[5]{}]{} (8,0)[(0,1)[3.5]{}]{}(9,0)[(0,1)[3.5]{}]{} (10,0)[(0,1)[3.5]{}]{} (11,0)[(0,1)[3.5]{}]{} (7.5,0.2)[**0**]{} (7.5,0.9)[0]{} (7.5,1.6)[0]{} (7.5,2.3)[0]{} (7.5,3.0)[0]{} (8.5,0.2)[**2**]{} (8.5,0.9)[0]{} (8.5,1.6)[0]{} (8.5,2.3)[1]{} (8.5,3.0)[1]{} (9.5,0.2)[**6**]{} (9.5,0.9)[2]{} (9.5,1.6)[0]{} (9.5,2.3)[1]{} (9.5,3.0)[3]{} (10.5,0.2)[**6**]{} (10.5,0.9)[3]{} (10.5,1.6)[0]{} (10.5,2.3)[0]{} (10.5,3.0)[3]{} (11.5,0.2)[**8**]{} (11.5,0.9)[3]{} (11.5,1.6)[1]{} (11.5,2.3)[1]{} (11.5,3.0)[3]{} (6.5,0.2)[**Tot.**]{} The total of every column in the left table above gives the value of $$\sum_{1 \leq i \leq N}|N+1-i-R(Z_i(b))|$$ when $b$ ranges in each interval; similarly, the totals in the right table give the values of $$\sum_{1 \leq i \leq N}|i-R(Z_i(b))|.$$ Such totals provide an easy computation of the value taken by $\tilde{\beta}.$ Indeed, after noticing that in the considered example $D=8,$ one gets $$\label{undici} \begin{array}{ll@{\qquad}rl@{\,}ll} G(\underline y;b) & = \frac{8-0}{8} = 1 & &b < 0.5 &\, =\, b^{(1)} \vspace{0.2cm}\\ & = \frac{8-2}{8} = \frac 34 & 0.5 \leq &b < 1 & \, =\, b^{(2)} \vspace{0.2cm} \\ & = \frac{4-6}{8} = -\frac 14 & 1 \leq &b < 1.5 & \, =\, b^{(3)} \vspace{0.2cm} \\ & = \frac{2-6}{8} = -\frac 12 & 1.25 \leq &b < 1.5 & \, =\, b^{(4)} \vspace{0.2cm} \\ & = \frac{0-8}{8} = -1 & &b \geq 1.5 \end{array}$$ so that $\tilde{\beta}=1.$ Notice that $\tilde{\beta}=1$ is also the median of the possible slopes, even if this coincidence is not a general rule. The least-squares estimate is $\hat{\beta}=1.05$ instead. Concerning the determination of the confidence interval for $\beta$ and thus of the bounds $\tilde{\beta}_I$ and $\tilde{\beta}_S,$ notice that the tables of the distribution of $G$ under indifference provide $$P \{ -1 <G(\underline Y;\beta) < 1 \} = P \left\{ -\frac 34 \leq G(\underline Y;\beta) \leq \frac 34 \right\} = \frac{11}{12} \simeq 0.92.$$ By (\[undici\]), one can then deduce that $$\begin{array}{rll} \tilde{\beta}_I &= \inf \{ b: \, G(\underline y;b) < 1 \} &= 0.5 = \inf \left\{ b: \, G(\underline y;b) \leq \frac 34 \right\} \vspace{0.2cm}\\ \tilde{\beta}_S &= \sup \{ b: \, G(\underline y;b) >- 1 \} &= 1.5 = \sup \left\{ b: \, G(\underline y;b) \geq - \frac 34 \right\} \end{array}$$ so that the confidence interval for $\beta$ with level $1-\alpha =92 \%,$ whatever the distribution function $F,$ is $$0.5 < \beta < 1.5.$$ Notice that the least squares method cannot provide a similar result, without any further assumptions. The asymptotic distribution of $\tilde{\beta}$ {#s7} ============================================== In order to compare the estimator $\tilde{\beta}$ with the other cited estimators for $\beta,$ some information about its asymptotic distribution is needed. The following theorem, whose proof is found in the Appendix, will be of use \[t2\] Let $Y_1, Y_2, \ldots, Y_n $ be independent variables with a common distribution function $F$ and absolutely continuous density $f,$ whose support is $\Re,$ and suppose that 1. $I(f) = \int_{-\infty}^{+\infty} \left( \frac{f'}{f} \right) \, f \, dy < \infty $ 2. $\int_{-\infty}^{+\infty} |f'| \, dy < \infty $ and that $\displaystyle{\frac{T^2}{M} = \frac 1M \sum_{i=1}^N (x_i - \bar x)^2 \rightarrow +\infty,}$ with $M = \displaystyle{\max_{1 \leq i \leq N} (x_i -\bar x)^2,}$ then $$\lim_{N \rightarrow + \infty} P \left\{ \sqrt N \, G \left( \underline Y; \frac bT \right) \leq z \right\} = \phi \left( \frac{z-\sigma_{12}}{\sqrt{2/3}} \right)$$ where $\phi$ denotes the normal cdf with zero mean and unit variance and $$\begin{aligned} \sigma_{12} &= 4b \, \int_{-\infty}^{+\infty} \left[ \psi(1-F(y)) - \psi(F(y)) \right] \, f'(y) \, dy \\ \psi(y) & = \lim_{N \rightarrow + \infty} \, \frac{1}{N^{3/2} T} \, \sum_{i=1}^{[Ny]} (x_i - \bar x) (Ny-i) \qquad 0 <y \leq 1\end{aligned}$$ [*Remark.*]{} The quantity $$\label{dodici} C = \int_{-\infty}^{+\infty} \left[ \psi(1-F(y)) - \psi(F(y)) \right] \, f'(y) \, dy$$ is negative or null. It suffices to notice that the function $$g(y) = \psi(1-y) - \psi(y) \qquad 0 \leq y \leq 1$$ is non-decreasing and bounded with $$g \left( \frac 12 + \xi \right) = - g \left( \frac 12 - \xi \right) \qquad -\frac 12 \leq \xi \leq \frac 12$$ and that $${{\rm Cov}}\left\{ F(Y) \, , \, \frac{f'(Y)}{f(Y)} \right\} = \int_{-\infty}^{+\infty} F(y) \, f'(y) \, dy = - \int_{-\infty}^{+\infty} f^2 < 0.$$ One then gets $$C = \int_{-\infty}^{+\infty} g ( F(y) ) \, f'(y) \, dy = {{\rm Cov}}\left\{ g(F(Y)) \, , \, \frac{f'(Y)}{f(y)} \right\} \leq 0 .$$ When $f$ is an even function, in addition, it immediately follows that $$\label{tredici} C = -2 \int_{-\infty}^{+\infty} \psi(F(y)) \, (f'(y)) \, dy .$$ The function $\psi$ may also happen to be identically null for peculiar sequences of the $x_i$’s, so that it is trivially $C=0.$ This chance may arise when the sequence of the $x_i$’s grows “too fast" wrt $i,$ for example when $x_i=\alpha^i$ with $\alpha >1.$ If the assumptions of Theorem 2 are met and if $C \neq 0,$ then $$\lim_{N \rightarrow + \infty} P \left\{ T (\tilde{\beta} - \beta) \leq b \right\} = \phi \left( - \frac{b}{\sqrt{\frac{1}{24 \, C^2}}} \right)$$ According to Proposition 1, $\beta=0$ can be assumed. As the realizations of $G(\underline Y;b)$ are non-increasing and by (\[cinque\]), $$\left\{ G(\underline Y; \frac bT) < 0 \right\} \quad \Rightarrow \quad \left\{ \tilde{\beta} < \frac bT \right\} \quad \Rightarrow \quad \left\{ G\left(\underline Y; \frac bT \right) \leq 0 \right\} ,$$ so that, by Theorem 2, $$\lim_{N \Rightarrow + \infty} P \left\{ T \tilde{\beta} < b \right\} = \phi \left( -\frac{\sigma_{12}}{\sqrt{2/3}} \right) = \phi \left( - \frac{b}{\sqrt{\frac{1}{24\, C^2}}} \right) . \quad \qed$$ Theorem 3 assures that, under the stated assumptions, the estimator $\tilde{\beta}$ is asymptotically normally distributed with mean $\beta$ and variance $$\label{quattordici} {{\rm Var}}(\tilde{\beta}) \simeq \frac{1}{24 \, T^2 \, C^2} .$$ Asymptotic relative efficiency of $\tilde{\beta}$ ================================================= Some comparisons of the proposed estimator $\tilde{\beta}$ with other known estimators will now be conducted in the very important case $x_i=i,$ $i=1,2,\ldots , N.$ Comparisons with other kinds of sequences can be produced analogously. First of all, notice that, in the considered case, $$\psi(y) = \frac{1}{\sqrt{12}} (2y^3-3y^2) \qquad 0 \leq y \leq 1.$$ To develop suitable comparisons, the asymptotic relative efficiency (ARE) can be used. As known, this technique compares the sample sizes corresponding to two unbiased estimators having the same asymptotic variance. More specifically, if two estimators $T_1$ and $T_2,$ both asymptotically unbiased for the same parameter $\theta$ and with variances ${{\rm Var}}(T_1)$ and ${{\rm Var}}(T_2),$ need $n_1$ and $n_2$ observations respectively to obtain the same variance, then $$ARE (T_1, T_2) = \lim_{N \rightarrow + \infty} \frac{n_1}{n_2} = \lim_{N \rightarrow + \infty} \frac{{{\rm Var}}(T_2)}{{{\rm Var}}(T_1)}$$ For the considered sequence of the $x_i$’s, the least squares estimator $\hat{\beta}$ is known to be asymptotically normally distributed with mean $\beta$ and variance $$\label{quindici} {{\rm Var}}(\hat{\beta}) \simeq \frac{\sigma^2(F)}{T^2}$$ where $\sigma^2(F)$ denotes the population variance depending on $F.$ Hence $$\label{sedici} ARE(\tilde{\beta}, \hat{\beta} ) = 24 \, \sigma^2(F) \, C^2.$$ The asymptotic efficiency of $\tilde{\beta}$ relative to the Theil’s estimator $\beta^*$ can be obtained using Theorem 6.1 in \[5\] (p. 1385) which, for the considered sequence of the $x_i$’s, states that $\beta^*$ is asymptotically normally distributed with mean $\beta$ and variance $$\label{diciassette} \frac{1}{12 \, T^2 \, B^2}$$ where $B = \int f^2.$ One then gets $$\label{diciotto} ARE (\tilde{\beta}, \beta^*) = 2 \, \frac{C^2}{B^2}.$$ It is easy to prove that (\[sedici\]) and (\[diciotto\]) are invariant under location and scale shifts. The following propositions are of interest: For any $F$ possessing finite and positive variance, $$ARE (\tilde{\beta} , \hat{\beta} ) > \frac 89 \left( \frac{\sigma(F)}{\Delta(F)} \right)^2 \geq \frac 23$$ where $\Delta(F)$ denotes the population mean difference depending on $F.$ After integrating by parts, (\[dodici\]) gives $$C^2 = 12 \, \left( \int_{-\infty}^{+\infty} F(y) \, (1-F(y)) \, f^2(y) \, dy \right)^2 .$$ Now notice that $$\int_{-\infty}^{+\infty} F \, (1-F) \, f^2 \, dy = \frac{\Delta(F)}{2} \, \int f^2 \, \frac{2\, F\, (1-F)}{\Delta(F)} .$$ The function $$\varphi (y) = \frac{2\, F(y)\, (1-F(y))}{\Delta(F)} \geq 0$$ is such that, by the definition of $\Delta (F),$ $$\int_{-\infty}^{+\infty} \varphi(y) \, dy = 1$$ so that it can be considered as a density function. Hence $$\int_{-\infty}^{+\infty} F \, (1-F) \, f^2 \, dy = \frac{\Delta(F)}{2} \, {{\rm E}}\{ f^2(Y) \}$$ where $Y$ is a random variable with density $\varphi(y).$ By a trivial inequality, one has then $${{\rm E}}\{ f^2(Y) \} > {{\rm E}}^2 (f(Y)) = \frac{1}{9 \, \Delta^2(F)}$$ and hence $$C^2 > \frac{1}{27 \, \Delta^2(F)} .$$ Formula (\[sedici\]) gives $$ARE(\tilde{\beta}, \beta^*) > \frac 89 \, \left( \frac{\sigma(F)}{\Delta(F)} \right)^2$$ and the proof follows by remembering (\[14)) that, for any distribution, $$\frac{\sigma(F)}{\Delta(F)} \geq \frac{\sqrt 3}{2} .\quad \qed$$ For any $F,$ $$ARE (\tilde{\beta}, \beta^*) < \frac 32 .$$ One can obtain $$\begin{aligned} C^2 &= 12 \left( \int_{-\infty}^{+\infty} (F-F^2) \, f^2 \, dy \right)^2 = \\ &= 12 \left( \frac 14 \, \int f^2 - \int \left( F - \frac 12 \right) ^2 \, f^2 \right)^2 = \\ &= \frac 34 \left( \int f^2 - \int (2F-1)^2 \, f^2 \right)^2 . \end{aligned}$$ By using (\[diciotto\]), $$ARE(\tilde{\beta}, \beta^*) = \frac 32 \left( 1- \frac{\int (2F-1)^2 \, f^2}{\int f^2} \right)^2 < \frac 32 . \quad \qed$$ In the following, the values taken by (\[sedici\]) and (\[diciotto\]) will be computed for three specific distributions: 1. normal 2. double exponential or Laplace 3. Cauchy which are characterized by a different tail behavior. More specifically, when $|x| \rightarrow + \infty,$ the Cauchy density tends to zero very slowly, in the same manner as $1/x^2;$ the double exponential distribution, instead, has a density tending to zero rather faster than the Cauchy, but more slowly than the normal density. 1\) [*normal with zero mean and unit variance*]{} By applying (\[tredici\]), one gets $$C = \frac{1}{\sqrt 3} \, \int_0^1 \, (2y^3-3y^2) \, \phi^{-1} (y) \, dy = - \frac{0.3317}{\sqrt 3}$$ being that $$\begin{aligned} \int_0^1 y^3 \, \phi^{-1}(y) \, dy &= \int_{-\infty}^{+\infty} z \, \phi^3(z) \, d\phi(z) = 0.2573 \\ \int_0^1 y^2 \, \phi^{-1}(y) \, dy &= \int_{-\infty}^{+\infty} z \, \phi^2(z) \, d\phi(z) = 0.2821 . \end{aligned}$$ One has also $$B = \int f^2 = \frac{1}{2 \sqrt{\pi}},$$ so that (\[sedici\]) gives $$ARE (\tilde{\beta}, \hat{\beta}) = 8 (0.3317)^2 \simeq 0.88$$ and (\[diciotto\]) gives $$ARE (\tilde{\beta}, \beta^*) = \frac{8\, \pi}{3} \, (0.3317)^2 \simeq 0.93.$$ Hence, in the normal case the least squares estimator is better than both $\tilde{\beta}$ and $\beta^*,$ even if none of the latter two estimators shows a substantial loss of efficiency. 2\) [*Double exponential*]{} In this case, $$f(y) = \frac 12 \exp \{ - |y| \} \qquad -\infty < y < + \infty,$$ so that $\sigma^2(F)=2.$ After some more computations, one gets $$C = - \frac{5}{\sqrt 3 \, 2^4} \qquad \mbox{ and } \qquad B = \frac 14.$$ Hence $$ARE (\tilde{\beta}, \hat{\beta}) = \frac{25}{16} \simeq 1.56; \qquad \qquad ARE (\tilde{\beta}, \beta^*) = \frac{25}{24} \simeq 1.05.$$ 2\) [*Cauchy*]{} Obviously this is an extreme case, because the density $$f(y) = \frac{1}{\pi} \, \frac{1}{1-y^2} \qquad -\infty < y < + \infty$$ does not possess finite variance, so the least squares estimator is not consistent. By definition, one has then $$ARE (\tilde{\beta}, \hat{\beta}) = + \infty .$$ However, it makes sense to compare $\tilde{\beta}$ and $\beta^*$. This task results again in favor of $\tilde{\beta}$. Some tedious but trivial computations indeed give $$C = - \frac{\sqrt 3}{2 \, \pi} \left( \frac 13 + \frac{1}{\pi^2} \right); \qquad B = \frac{1}{2 \, \pi}$$ and $$ARE (\tilde{\beta}, \beta^*) = 6 \, \left( \frac 13 + \frac{1}{\pi^2} \right)^2 \simeq 1.13.$$ The above results clearly show that the asymptotic efficiency of $\tilde{\beta}$ relative to $\hat{\beta},$ but also to $\beta^*,$ tend to grow as distributions with more and more heavy tails are considered. Appendix {#appendix .unnumbered} ======== To prove Theorem \[t2\] of section \[s7\], some preliminary results will be considered. Let $f$ be a probability density function with support in $\Re$ and define the two probability measures $$P_N(A) = \int_A \, \prod_{1 \leq i \leq N} f(y_i) \quad \mbox{ and } \quad Q_N(A) = \int_A \, \prod_{1 \leq i \leq N} f \left( y_i + \frac{b}{T} (x_i-\bar x) \right)$$ where $x_1 < x_2 < \ldots < x_N$ as usual, $b \neq 0$ is finite, $A$ is any event and $$T^2 = \sum_{1 \leq i \leq N} (x_i - \bar x)^2 \qquad M = \max_{1 \leq i \leq N} (x_i - \bar x)^2 .$$ \[lemma1app\] [(\[15\], p. 208) (\[16, p. 1134\])]{}\ If the vector $$\left( \sqrt N \, G(\underline Y;0) \: , \: \log \frac{\displaystyle{\prod_{1 \leq i \leq N}} \, f \left( Y_i + \frac{b}{T} (x_i - \bar x) \right) }{\displaystyle{\prod_{1 \leq i \leq N}} f(Y_i)} \right)$$ converges, with the measure $P_N$, to the normal distribution with parameters $$\left( \mu \, , \, - \frac 12 \sigma_2^2 \, , \, \sigma_1^2 \, , \, \sigma_2^2 \, , \, \sigma_{12} \right)$$ then the variable $$\sqrt N \, G(\underline Y;0)$$ converges, with the measure $Q_N$, to the normal distribution with mean $\mu + \sigma_{12}$ and variance $\sigma_1^2.$ \[lemma2app\] [(\[15\], p. 213), (\[16\], p. 1136).]{}\ If $ \displaystyle{I(f) = \int \left( \frac{f'}{f} \right)^2 f < \infty}$ and if $\displaystyle{\frac{T^2}{M} \rightarrow \infty },$ then $$\begin{aligned} \lefteqn{ {P_N\lim}_{N \rightarrow \infty} \Bigg( \log \frac{\displaystyle{\prod_{1 \leq i \leq N}} f \left( Y_i + \frac{b}{T} (x_i-\bar x) \right)}{\displaystyle{\prod_{1 \leq i \leq N}} \, f(Y_i)} - \frac bT \sum_{1 \leq i \leq N} (x_i - \bar x) \frac{f'(Y_i)}{f(Y_i)} } && \\ && \hspace{8.5cm} + \frac b2 \, I(f) \Bigg) =0 \end{aligned}$$ where $P_N \lim$ denotes the limit in $P_N$–probability. [*Remark to Lemma \[lemma2app\]*]{}\ According to the measure $P_N,$ the variable $$\frac bT \sum_{1 \leq i \leq N} (x_i - \bar x) \, \frac{f'(Y_i)}{f(Y_i)}$$ has the following variance $$\mbox{Var} \left( \frac bT \sum_{1 \leq i \leq N} (x_i - \bar x) \, \frac{f'(Y_i)}{f(Y_i)} \right) = b^2 \, I(f) < \infty$$ and expectation (\[16\], p. 1125) $$\mbox{E} \left( \frac bT \sum_{1 \leq i \leq N} (x_i - \bar x) \, \frac{f'(Y_i)}{f(Y_i)} \right) = 0 .$$ Moreover, the variable $$\sum_{1 \leq i \leq N} Z_{2i} = \sum_{1 \leq i \leq N} \frac{\displaystyle{\frac bT \, (x_i - \bar x) \, \frac{f'(Y_i)}{f(Y_i)}}}{\sqrt{b^2 I (f)}}$$ satisfies the Lindeberg-Feller condition. Indeed, after defining $$Z_{2i} (\delta) = Z_{2i} \, \, s \left( |Z_{2i}| - \delta \right)$$ where $\delta >0$ and $s(x)$ equals 1 if $x\geq 0$ and $0$ elsewhere, such a condition can be written as $$\sum_{1 \leq i \leq N} \mbox{E} \left( Z_{2i}^2 (\delta) \right) = \sum_{1 \leq i \leq N} \int_{|z| \geq \delta} z^2 \, d P_N \left\{ \frac bT (x_i - \bar x) \frac{f'(Y_i)}{f(Y_i} \leq z \, \sqrt{b^2 \, I(f)} \right\} \rightarrow 0 .$$ However, by putting $t = z \, \sqrt{b^2 \, I(f)},$ one gets $$\begin{aligned} \lefteqn{\sum_{1 \leq i \leq N} \frac{1}{b^2 \, I(f)} \, \int_{|t| \geq \delta \sqrt{b^2 \, I(f)}} \, t^2 \, d P_N \left\{ \frac{f'(Y_i)}{f(Y_i} \leq t \right\} = } && \\ && = \frac{1}{T^2} \, \sum_{1 \leq i \leq N} \frac{(x_i- \bar x)^2}{I(f)} \int_{|y| \geq \delta \sqrt{I(f)} \left| \frac{T}{x_i - \bar x} \right|} \, y^2 \, d P_N \left\{ \frac{f'(Y_i)}{f(Y_i} \leq y \right\} \rightarrow 0\end{aligned}$$ because, by hypothesis, $$\frac{T^2}{M} \rightarrow + \infty \quad \Rightarrow \quad \left| \frac{T}{x_i- \bar x} \right| \Rightarrow + \infty.$$ \[lemma3app\] If $F'=f$ and if $$\hat G(\underline Y;0) = \frac{2N}{D} \sum_{1 \leq i \leq N} \left( \left| 1 - \frac iN - F(Y_i) \right| - \left| \frac iN - F(Y_i) \right| \right)$$ then $${P_N \lim}_{N \rightarrow \infty} \, \sqrt N \, (G(\underline Y;0) - \hat G(\underline Y;0)) = 0.$$ By using the identity $ |x| = 2x \, s(x) - x $ $(x \in \Re),$ the definition in (\[due\]) and the expression of $\hat G(\underline Y; 0),$ one gets $$\sqrt N (G(\underline Y;0) - \hat G (\underline Y;0)) = A_N + B_N + C_N + D_N$$ where $$\begin{aligned} A_N & =\frac{4 \, N^{3/2}}{D} \, \sum_{1 \leq i \leq N} \left( 1- \frac iN - F(Y_i) \right) \left[ s(N+1-i-R(Y_i)) \right. \\ & \hspace{8cm} \left. - s(N-i-N\, F(Y_i)) \right] \\ B_N & =- \frac{4 \, N^{3/2}}{D} \, \sum_{1 \leq i \leq N} \left(\frac iN - F(Y_i) \right) \left[ s(i-R(Y_i)) - s(i-N\, F(Y_i)) \right] \\ C_N & =\frac{4 \, N^{3/2}}{D} \, \sum_{1 \leq i \leq N} \left( \frac{R(Y_i)}{N} - F(Y_i) \right) \left[ s(i - R(Y_i)) - s(N+1-i-R(Y_i)) \right] \\ D_N & =\frac{2 \, N^{1/2}}{D} \, \sum_{1 \leq i \leq N} s(N+1-i-R(Y_i)) . \end{aligned}$$ It follows that $$\begin{aligned} \mbox{E} \{ |D_N| \} & \leq \frac{2 \, N^{3/2}}{D} \, \rightarrow 0 \\ \mbox{E} \{ |B_N| \} & \leq \frac{4 \, N^{3/2}}{D} \sum_{1 \leq i \leq N} \mbox{E} \left\{ \left| \frac iN - F(Y_i) \right| \, \, \left| s(i-R(Y_i)) - s(i-NF(Y_i)) \right| \right\} \end{aligned}$$ By using the joint distribution of $(Y_i, R(Y_i)),$ that is $$\begin{aligned} \lefteqn{ \mbox{Pr} \{ R(Y_i) = r \, ; \, y < Y_i < y+dy \} = } &&\\ && = \frac 1N \, g_{Y_{(r)}} (y) \, dy = \\ && = \frac 1N \frac{N!}{(N-r)! \, (r-1)!} \, [F(y)]^{r-1} \, [1-F(y)]^{N-r} \, f(y) \, dy \quad r=1,2, \ldots, N; y \in \Re , \end{aligned}$$ one gets $$\begin{aligned} \lefteqn{ \mbox{E} \left\{ \left| \frac iN - F(Y_i) \right| \, \, \left| s(i-R(Y_i) - s(i-NF(Y_i)) \right| \right\} = } &&\\ && = \frac 1N \sum_{1 \leq r \leq N} \int_{- \infty}^{+ \infty} \left| \frac iN - F(y) \right| \, \, \left| s(i-r) - s(i-NF(y)) \right| \, g_{Y_{(r)}} (y) \, dy = \\ && = \sum_{1 \leq r \leq N} \int_0^1 \left| \frac iN - v \right| \, \, \left| s(i-r) - s(i-Nv) \right| {N-1 \choose r-1} \, v^{r-1} \, (1-v)^{N-r} \, dv = \\ && =\int_0^1 \left| \frac iN - v \right| \left[ \left( 1- s(i-Nv) \right) \, {{\rm Pr}}\{ U_{(i)} > v \} + s(1-Nv) \, {{\rm Pr}}\{ U_{(i)} \leq r \} \right] \, dv \end{aligned}$$ where $U_{(i)}$ is the $i-$th order statistic of a $(N-1)-$sized random sample drawn from a uniform population in $(0,1).$ By partitioning the integration interval, after some trivial passages, one gets $$\begin{aligned} \lefteqn{ \mbox{E} \left\{ \left| \frac iN - F(Y_i) \right| \, \, \left| s(i-R(Y_i) - s(i-NF(Y_i)) \right| \right\} = } &&\\ && = \frac{i^2}{2 \, N^2} - \frac iN \, \int_0^1 {{\rm Pr}}\{ U_{(i)} >v \} \, dv + \int_0^1 v \, \, {{\rm Pr}}\{ U_{(i)} >v \} \, dv = \\ && = \frac{i^2}{2 \, N^2} - \frac i N \, \mbox{E} \left( U_{(i)} \right) + \frac 12 \, \mbox{E} \left( U_{(i)}^2 \right) = \\ && = \frac{i \, (N-i)}{2 \, N^2 (N+1)} \qquad \qquad 1 \leq i \leq N \end{aligned}$$ so that $$\mbox{E} \{ |B_N| \} \leq \frac{2 \, N^{3/2}}{D} \, \sum_{1 \leq i \leq N} \frac{i(N-i)}{N^2(N+1)} \quad\rightarrow 0 .$$ By following similar steps, one can prove that $$\mbox{E} \{ |A_N| \} \rightarrow 0 .$$ Now let $$S_{i,N} = \left( \frac{R(Y_i)}{N} - F(Y_i) \right) \left[ s ( i - R(Y_i) ) - s(N+1-i-R(Y_i)) \right] \qquad i =1, \ldots , N .$$ and simply consider that $$\mbox{E} (C_N) = \frac{4 \, N^{3/2}}{D} \, \sum_{1 \leq i \leq N} \mbox{E} \left( S_{i,N} \right)=0$$ Moreover, $$\label{asterisco} \mbox{Var} (C_N) = \frac{16 \, N^3}{D^2} \sum_{1 \leq i \leq N} \mbox{E} \left( S_{i,N}^2 \right) + \frac{16 \, N^3}{D^2} \sum_{i \neq j} \, \mbox{E} \left( S_{i,N} \, S_{j,N} \right)$$ and $$\begin{aligned} \sum_{1 \leq i \leq N} \mbox{E} (S_{i,N}^2) &=& \sum_{1 \leq i \leq N} \frac 1N \sum_{1 \leq r \leq N} \int_{-\infty}^{+ \infty} \left( \frac rN - F(y) \right)^2 \cdot \\ && \hspace{3cm} \cdot \, \left[ s(i-r) -s(N+1-i-r) \right]^2 \, g_{Y_{(r)}} (y) \, dy \\ & \leq & \sum_{1 \leq r \leq N} \int_{-\infty}^{+\infty} \left( \frac rN - F(y) \right)^2 \, g_{Y_{(r)}}(y)\,dy \\ & \leq & \sum_{1 \leq r \leq N} \mbox E \left\{ \left( F(Y_{(r)}) -\frac rN \right)^2 \right\} \, \rightarrow \, A < +\infty \end{aligned}$$ being that $$\mbox E \left\{ \left( F(Y_{(r)}) - \frac rN \right)^2 \right\} = \frac{r(N-r+1)}{(N+2)\, (N+1)^2} + \frac{r^2}{N^2 \, (N+1)^2} \qquad \forall r = 1,2, \ldots, N$$ The first summand in the rhs of (\[asterisco\]) thus tends to zero. Moreover, $$\begin{aligned} \lefteqn{ \left| \sum_{i \neq j} \mbox E (S_{i,N} \, S_{j,N}) \right| = } && \\ && = \left| \frac{1}{N(N-1)} \, \sum_{i \neq j} \, \sum_{r \neq k} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \left( \frac rN - F(x) \right) \, \left( \frac kN - F(y) \right) \right. \\ && \hspace{2cm} \left( s(i-r) - s(N+1-i-r) \right) \, \left( s(j-k) - s (N+1-j-k) \right) \\ && \hspace{7.5cm} g_{Y_{(r)},Y_{(k)}} (x,y) \, dx \, dy \Big| = \\ && = \left| \frac{1}{N \, (N-1)} \sum_{r \neq k} \left[ \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \left( \frac rN - F(x) \right) \, \left( \frac kN - F(y) \right) \, g_{Y_{(r)}, Y_{(k)}} (x,y) \right. \right. \\ && \hspace{0.5cm} dx \, dy \Big] \sum_{i \leq i \leq N} \left( s(i-r) -s(N+1-i-r) \right) \, \left( s(i-k) - s(N+1-i-k) \right) \big| . \end{aligned}$$ Now, as $$\begin{aligned} \lefteqn{ \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \left( \frac rN - F(x) \right) \, \left( \frac kN - F(y) \right) \, g_{Y_{(r)}, Y_{(k)}} (x,y) \, dx \, dy = } \\ && = \mbox{Cov} \left\{ F(Y_{(r)}) \, , \, F(Y_{(k)}) \right\} + \frac{rk}{N^2 \, (N+1)^2} = \\ && = \left\{ \begin{array}{ll} \dfrac{r(N+1-k)}{(N+2)\, (N-1)^2} + \dfrac{rk}{N^2 \, (N+1)^2} & \hspace{1.5cm} r<k \\ \dfrac{k(N+1-r)}{(N+2)\, (N+1)^2} + \dfrac{rk}{N^2 \, (N+1)^2} & \hspace{1.5cm} r>k \end{array} \right. \end{aligned}$$ one obtains $$\begin{aligned} \lefteqn{ \left| \sum_{i \neq j} \mbox{E} (S_{i,N} \, S_{j,N}) \right| \leq} \\ && \leq \frac{1}{N-1} \sum_{r \neq k} \left\{ \mbox{Cov} (F(Y_{(r)}) \, , \, F(Y_{(k)}) + \frac{rk}{N^2 \, (N+1)^2} \right\} \, \rightarrow \, B < + \infty \end{aligned}$$ so that the second summand in (\[asterisco\]) tends to zero as well. The proof follows then by applying Thchebycheff’s inequality in its suitable form to the four variables. $\qed$ [*Remark to Lemma \[lemma3app\]*]{}\ Lemma \[lemma3app\] makes it possible to obtain the asymptotic distribution of Gini’s cograduation index under indifference in an alternative way with respect to a former paper (\[17\]). Indeed, Lemma \[lemma3app\] assures that $\sqrt N \, G(\underline Y;0)$ is asymptotically equally distributed as $\sqrt N \, \hat G (\underline Y; 0),$ for which the classical limit theorems can be applied, because it can be regarded as a sum of independent variables. As a matter of fact, the variable $$\sqrt N \, \hat G(\underline Y; 0) = \frac{2 \, N^{3/2}}{D} \, \sum_{1 \leq i \leq N} \left( \left| 1-\frac iN - F(Y_i) \right| - \left| \frac iN - F(Y_i) \right| \right)$$ has the following mean and variance $$\mbox{E} (\sqrt N \, \hat G(\underline Y;0)) = 0 \qquad \mbox{Var} (\sqrt N \, \hat G(\underline Y;0)) \simeq \frac 23$$ and, by letting $$\sum_{1 \leq i \leq N} \dfrac{\frac{2 \, N^{3/2}}{D} \, \left( | 1 - \frac iN - F(Y_i) | - | \frac iN - F(Y_i) | \right)}{\sqrt{\frac 23}} = \sum_{1 \leq i \leq N} Z_{1i}$$ and, for every $\delta >0,$ $$Z_{1i} (\delta) = Z_{1i} \: s(|Z_{1i} - \delta|),$$ the Lindeberg condition is satisfied: $$\sum_{1 \leq i \leq N} \mbox{E} (Z_{1i}^2 (\delta)) \, \rightarrow \, 0 \qquad \forall \delta >0$$ \[lemma4app\] If $F' = f,$ $I(f) < +\infty,$ $\int |f'| < + \infty$ and $\frac{T^2}{M} \rightarrow + \infty,$ then the vector $$\left( \sqrt N \, G (\underline Y;0) \, , \, \log \frac{\displaystyle{\prod_{1 \leq i \leq N}} \, f \left( Y_i + \frac bT (x_i - \bar x) \right) }{\displaystyle{\prod_{1 \leq i \leq N}} f(Y_i)} \right)$$ converges in distribution, with the measure $P_N,$ to the bivariate normal with parameters $$\left( 0, \quad -\frac{b^2}{2} I(f), \quad \frac 23 , \quad b^2 I(f), \quad \sigma_{12} \right)$$ where $$\begin{aligned} \sigma_{12} & = 4b \, \int_{-\infty}^{+\infty} \left[ \psi(1-F(y)) - \psi(F(y)) \right] \, f'(y) \, dy \\ \psi(y) & = \lim_{N \rightarrow + \infty} \, \frac{1}{N^{3/2} \, T} \, \sum_{i=1}^{[Ny]} (x_i - \bar x) (Ny-i) \qquad \qquad 0 <y \leq 1 \end{aligned}$$ By following lemmas \[lemma2app\] and \[lemma3app\], it suffices to show that the vector $$\left( \sqrt{N} \, \hat G (\underline Y;0) \, , \, \frac bT \sum_{1 \leq i \leq N} (x_i -\bar x) \dfrac{f'(Y_i)}{f(Y_i)} \right)$$ converges in distribution to the bivariate normal with parameters $$\left( 0, \quad 0, \quad \frac 23 , \quad b^2 I(f), \quad \sigma_{12} \right) .$$ By the remarks following Lemma \[lemma2app\] and Lemma \[lemma3app\], the limiting distribution surely takes the first four parameters listed above. Moreover, consider that $$\begin{aligned} \lefteqn{\mbox{Cov} \left\{ \sqrt N \hat G (\underline Y;0)\, , \, \frac bT \sum_{1 \leq i \leq N} (x_i - \bar x) \dfrac{f'(Y_i)}{f(Y_i)} \right\} = } \\ && = \frac{2 \, N^{3/2}}{D \, T} \, b \, \int_{-\infty}^{+\infty} f'(y) \, \left[ \sum_{1 \leq i \leq N} (x_i - \bar x) \left( \left| 1 - \frac iN - F(y) \right| - \left| \frac iN - F(y) \right| \right) \right] \, dy = \\ && = 4b \, \int_{- \infty}^{+ \infty} f'(y) \left[ \frac{N^{3/2}}{D \, T} \left( \sum_{i=1}^{[N\, (1-F(y))]} (x_i - \bar x) \left( 1-F(y) - \frac iN \right) + \right. \right. \\ && \hspace{6cm} - \left. \left. \sum_{i=1}^{[N\, F(y)]} (x_i - \bar x) \left( F(y) - \frac iN \right) \right) \right] \, dy \end{aligned}$$ By passing to the limit (with $N$) under the integral sign, one gets $$\begin{aligned} \sigma_{12} & = \lim_{N \rightarrow + \infty} \mbox{Cov} \left\{ \sqrt N \hat G (\underline Y;0)\, , \, \frac bT \sum_{1 \leq i \leq N} (x_i - \bar x) \dfrac{f'(Y_i)}{f(Y_i)} \right\} = \\ & = 4b \, \int_{- \infty}^{+ \infty} \left[ \psi(1-F(y)) - \psi(F(y)) \right] \, f'(y) \, dy = \\ & = 4b \, \int_0^1 \left[ \psi(1-v) - \psi(v) \right] \, \dfrac{f'(F^{-1}(v))}{f(F^{-1}(v))} \, dv\end{aligned}$$ To prove that the limiting distribution is normal, one can then show that, for every real $\lambda_1$ and $\lambda_2,$ the following variable is asymptotically normally distributed: $$\lambda_1 \, \sqrt{N} \, \hat G (\underline Y;0)+ \lambda_2 \, \frac bT \sum_{1 \leq i \leq N} (x_i -\bar x) \dfrac{f'(Y_i)}{f(Y_i)} .$$ However, as both the variables $$\lambda_1 \, \sqrt{N} \, \hat G (\underline Y;0) \quad \mbox{ and } \quad \lambda_2 \, \frac bT \sum_{1 \leq i \leq N} (x_i -\bar x) \dfrac{f'(Y_i)}{f(Y_i)}$$ satisfy the Lindeberg condition, one can get the aimed result as in \[15\], page 218. $\qed$ The proof of Theorem \[t2\] in section \[s7\] now immediately follows from lemmas \[lemma1app\] and \[lemma4app\]. Indeed, for every real $z,$ $$\begin{aligned} Q_N \left\{ \sqrt N \, G(\underline Y;0) \leq z \right\} & = \int_{\left\{ \sqrt N \, G(\underline y ; 0) \leq z \right\}} \prod_{1 \leq i \leq N} \, f \left( y_i + \frac bT (x_i - \bar x) \right) = \\ & = \int_{\left\{ \sqrt N \, G \left(\underline y ; \frac bT \right) \leq z \right\}} \prod_{1 \leq i \leq N} \, f(y_i) = \\ & = P_N \left\{ \sqrt N \, G \left(\underline Y ; \frac bT \right) \leq z \right\} \end{aligned}$$ and, by lemmas \[lemma1app\] and \[lemma4app\], $$\lim_{N \rightarrow + \infty} Q_N \left\{ \sqrt N \, G(\underline Y;0) \leq z \right\} = \phi \left( \frac{z-\sigma_{12}}{\sqrt{2/3}} \right) .$$ References {#references .unnumbered} ========== \[1\] <span style="font-variant:small-caps;">F. Eicker</span>, *Asymptotic normality and consistency of least squares estimators for families of linear regressions.* Ann. of Math. Stat., *34*, 1963, pp. 447–456. \[2\] <span style="font-variant:small-caps;">A. M. 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Gini</span>, *Sul criterio di concordanza tra due caratteri.* Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, 1915-16, pp. 309–331. \[12\] <span style="font-variant:small-caps;">W. Hoeffding</span>, *A non parametric test of independence.* Ann. of Math. Stat., *19*, 1948, pp. 546–557. \[13\] <span style="font-variant:small-caps;">J. L. Hodges</span> and <span style="font-variant:small-caps;">E. L. Lehmann</span>, *Estimates of location based on rank tests.* Ann. of Math. Stat., *34*, 1963, pp. 598–611. \[14\] <span style="font-variant:small-caps;">B. Michetti</span> and <span style="font-variant:small-caps;">G. Dall’Aglio</span>, *La differenza semplice media.* Statistica, *17*, 1957, pp. 159–255. \[15\] <span style="font-variant:small-caps;">J. Hájek</span> and <span style="font-variant:small-caps;">Z. Šidák</span>, *Theory of rank tests.* Academic Press, London, 1962. \[16\] <span style="font-variant:small-caps;">J. Hájek</span>, *Asymptotically most powerful rank-order tests.* Ann. of Math. Stat., *33*, 1962, pp. 1124–1147. \[17\] <span style="font-variant:small-caps;">D. M. Cifarelli</span> and <span style="font-variant:small-caps;">E. Regazzini</span>, *On a distribution-free test of independence based on Gini’s rank association coefficient.* Recent developments in Statistics. Proceedings of the European Meeting of Statisticians (Grenoble, 6-11 sept. 1976), North-Holland, Amsterdam, 1977, pp. 375–385. [^1]: As later shown, $G(\underline y;b)$ is actually a non increasing step function; hence the stated condition does not imply that $\tilde{\beta}$ is a root of the equation $G(\underline y;b)=0,$ which might not admit any root.

--- abstract: 'We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric) we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.' address: - | Mathematisches Institut\ Friedrich Schiller Universit[ä]{}t Jena\ 07743 Jena, Germany - | Mathematisches Institut\ Friedrich Schiller Universit[ä]{}t Jena\ 07743 Jena, Germany - | York College of the City University of New York\ Jamaica, NY 11451\ USA author: - Sebastian Haeseler - Matthias Keller - 'Rados[ł]{}aw K. Wojciechowski' title: Volume growth and bounds for the essential spectrum for Dirichlet forms --- Introduction and Main Results ============================= In 1981 Brooks proved that the bottom of the essential spectrum of the Laplace Beltrami operator on a complete non compact Riemannian manifold with infinite measure can be bounded by the exponential volume growth rate of the manifold [@Br]. Following this, similar results were proven in various contexts, see [@DK; @Fuj; @Hi; @Hi2; @OU; @Stu]. Very recently it was shown in [@KLW] that such a result fails to be true in the case of the (non-normalized) graph Laplacian when the volume is measured with respect to the natural graph distance. Indeed, there are graphs of cubic polynomial volume growth that have positive bottom of the spectrum and slightly more than cubic growth already allows for purely discrete spectrum. This suggests that one should look for other candidates for a metric on a graph. In this work we use the context of regular Dirichlet forms (without killing term) and the corresponding concept of intrinsic metrics, see [@Stu] and [@FLW], to prove a Brooks-type theorem. The purpose of this approach is threefold. First, we provide a set up which includes all known examples (and various others, e.g., quantum graphs) and give a unified treatment. Additionally, our estimates are slightly better than most of the previous results. Secondly, our method of proof seems to be much clearer and simpler than most of the previous works. Finally, graph Laplacians are now included and the disparity discussed above is resolved by considering suitable metrics. As an application, we can now prove that the examples found in [@KLW] for Laplacians on graphs do indeed give the borderline for positive bottom of the spectrum. In particular, for the natural graph distance the threshold for zero bottom of the essential spectrum and the discreteness of the spectrum lies at cubic growth. Let $X$ be a locally compact separable metric space and $m$ a positive Radon measure of full support. Let ${{\mathcal E}}$ be a closed, symmetric, non-negative form on the Hilbert space $L^{2}(X,m)$ of real-valued square integrable functions with domain $D$. We assume that ${{\mathcal E}}$ is a regular Dirichlet form without killing term (for background on Dirichlet forms see [@Fuk], more details are given in Section \[s:DF\]). Let $L$ be the positive self adjoint operator arising from ${{\mathcal E}}$. Define $$\begin{aligned} {{\lambda}}_{0}(L):=\inf{{\sigma}}(L)\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L):=\inf{\sigma_{\!\mathrm{ess}}}(L)\end{aligned}$$ where ${\sigma_{\!\mathrm{ess}}}(L)$ denotes the essential spectrum of $L$. We let $\rho$ be an intrinsic pseudo metric in the sense of [@FLW]. For $x_{0}\in X$ and $r\geq 0$, we define the distance ball $B_{r}=B_{r}(x_{0})=\{x\in X\mid \rho(x,x_{0})\leq r\}$. Let the *exponential volume growth* be defined as $$\begin{aligned} \mu=\liminf_{r\to\infty}\frac{1}{r}\log m(B_{r}(x_{0})).\end{aligned}$$ Note that, in contrast to previous works on manifolds [@Br], graphs [@Fuj] and strongly local forms [@Stu], we consider a $\liminf$ here, rather than a $\limsup$. If $\rho$ takes values in $[0,\infty)$, then $X=\bigcup_{r} B_{r}(x_{0})$. In this case $\mu$ does not depend on the particular choice of $x_{0}$. There is another constant first introduced in [@Stu] which we call the *minimal exponential volume growth* and which is defined as $$\begin{aligned} {\widetilde{ \mu}}=\liminf_{r\to\infty}\frac{1}{r}\inf_{x\in X}\log \frac{m(B_{r}(x))}{m(B_{1}(x))}.\end{aligned}$$ In this paper we prove the following theorem. \[t:main\] Let $L$ be the positive self adjoint operator arising from a regular Dirichlet form ${{\mathcal E}}$ without killing term and let $\rho$ be an intrinsic metric such that all distance balls are compact. Then, $$\begin{aligned} {{\lambda}}_{0}(L)\leq \frac{{\widetilde{ \mu}}^{2}}{4}.\end{aligned}$$ If additionally $m(\bigcup_{r}B_{r}(x_{0}))=\infty$ for some $x_{0}$, then $$\begin{aligned} {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq \frac{\mu^{2}}{4}.\end{aligned}$$ This has the following immediate corollary. The corollary has various consequences, for example, the exponential instability of the semigroup $(e^{-tL})_{t\geq0}$ on $L^{p}(X,m)$, $p\in [1,\infty]$, see [@Stu Corollary 2]. Suppose that $(X,d)$ is of subexponential growth, i.e., ${\widetilde{ \mu}}=0$ (respectively, $\mu=0$). Then, ${{\lambda}}_{0}(L)=0$ (respectively, ${{\lambda}}_{0}^{\mathrm{ess}}(L)=0$). \(a) Let us discuss Theorem \[t:main\] in the perspective of the present literature: For the Laplace Beltrami operator on a Riemannian manifolds an estimate for ${{\lambda}}_{0}^{\mathrm{ess}}$ can be found in [@Br], see also [@Hi2]. In [@Stu] the statement for ${{\lambda}}_{0}$ is proven for strongly local Dirichlet forms. For non-local operators such results were known only for normalized Laplacians on graphs, see [@DK; @Fuj; @Hi; @OU]. These operators are of a very special form, in particular, they are always bounded. For unbounded Laplacians on graphs the conclusions of the theorem do not hold if one considers volume with respect to the natural graph metric, see [@KLW]. However, by [@FLW] (see also [@GHM]), there is now a suitable notion of intrinsic metric for non-local forms. Let us stress that our result covers the results in [@Br; @DK; @Fuj; @OU; @Stu]. The results of type [@Hi; @Hi2] could certainly also be obtained with slightly more technical effort which we avoid here for clarity of presentation. \(b) Despite the fact that our result is much more general, we have a unified method of proof for the bounds on the spectrum and the essential spectrum. Moreover, for the essential spectrum, the proof is significantly simpler than the one of [@Br; @Fuj] as we use test functions that converge weakly to zero and, therefore, avoid a cut-off procedure. \(c) Indeed, we prove a slightly more general result than above for non-local forms in Section \[s:nonlocal\]. In particular, for some special cases we prove much better estimates and recover the results of [@DK; @Fuj; @OU] in Corollary \[c:normalized\] in Section \[s:graph\]. \(d) If we assume that $\rho$ takes values in $[0,\infty)$, then we can clearly replace the assumption that $m(\bigcup_{r}B_{r}(x_{0}))=\infty$ with $m(X)=\infty$. The case when $m(X) < \infty$ is notably different, see [@HKLW2] for more details. \(e) If $\inf_{x\in X}m(B_{1}(x))>0$, then one can also show that $ {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq {{\widetilde{ \mu}}^{2}}/{4}$. \(f) Our result deals exclusively with Dirichlet forms with vanishing killing term. The major challenge in the case of non vanishing killing term is to give a proper definition of volume which incorporates the killing term. We shortly discuss a strategy of how one could approach this case: We need an positive generalized harmonic function $u$, i.e., ${{\mathcal E}}(u,{{\varphi}})=0$ for all ${{\varphi}}\in D$, where $u$ is assumed to be locally in the domain of ${{\mathcal E}}$ (this space is introduced in [@FLW] as $\mathcal{D}_{\mathrm{loc}}^{*}$). Such a function exists in many settings, see e.g. [@DK; @HK; @LSV], and the result which guarantees the existence of such a function is often referred to as a Allegretto-Piepenbrink type theorem. Then, by a ground state representation, see Theorem 10.1 [@FLW], one obtains a form ${{\mathcal E}}_{u}$ with vanishing killing term such that ${{\mathcal E}}={{\mathcal E}}_{u}$ on the intersection of their domains. Now, we can apply the methods above for ${{\mathcal E}}_{u}$ to derive the result for ${{\mathcal E}}$. However, as shown in [@HK], there are examples of non-locally finite weighted graphs that do not have such a generalized harmonic function. Therefore, it would be interesting to find sufficient conditions under which the approach above can be carried out. Let us highlight one of the applications of our results for graphs. Let $\Delta $ be the graph Laplacian on $\ell^{2}(X)$ acting as $$\begin{aligned} \Delta{{\varphi}}(x)= \sum_{y \sim x}({{\varphi}}(x)-{{\varphi}}(y))\end{aligned}$$ (for more details, see Sections \[s:graph\] and \[s:graph2\]). Moreover, let $B_{r}^{d}$, for $r\geq0$, be balls with respect to the natural graph distance $d$ defined as the length of the shortest path of edges between two vertices. It has to be stressed that this metric is not an intrinsic metric for $\Delta$. However, we will show in Theorem \[t:graph2\] that, if the growth of the balls $B_{r}^{d}$ is $r^{3-{{\varepsilon}}}$ for any ${{\varepsilon}}>0$, then ${{\lambda}}_{0}(\Delta)={{\lambda}}_{0}^{\mathrm{ess}}(\Delta)=0$ and if it is less than $r^{3}$, then ${{\lambda}}_{0}^{\mathrm{ess}}(\Delta)<\infty$. We demonstrate by examples that this result is sharp, see Section \[s:graph2\]. The paper is structured as follows. In Section \[s:Preliminaries\] we recall some basic facts about Dirichlet forms and intrinsic metrics. Moreover, we give a bound on the bottom of the essential spectrum via weak null sequences and introduce the test functions. In Section \[s:proof\] we prove the crucial estimate for the strongly local and the non-local parts of the Dirichlet form and prove the main theorem. In Section \[s:applications\] we discuss the result for weighted graphs and prove the polynomial growth bound discussed above. Note added: After this work was completed we learned about the very recent preprint of Matthew Folz “Volume growth and spectrum for general graph Laplacians" which contains related material in the special case of graphs. Preliminaries {#s:Preliminaries} ============= In this section we introduce the basic notions and concepts. The first subsection is devoted to recalling the setting of Dirichlet forms. In the second subsection we prove an estimate for the bottom of the essential spectrum and in the third subsection we discuss the basic properties of the test functions that are used to prove our result. Dirichlet forms {#s:DF} --------------- In this section we recall some elementary facts about Dirichlet forms, see e.g. [@Fuk] and, for recent work on non-local forms, [@FLW]. As above let $X$ be a locally compact separable metric space and let $m$ be a positive Radon measure of full support. We consider all functions on $X$ to be real-valued, but, by complexifying the corresponding Hilbert spaces and forms, we could also consider complex-valued functions. A closed non-negative form on $L^{2}(X,m)$ consists of a dense subspace $D\subseteq L^{2}(X,m)$ and a sesqui-linear non-negative map ${{\mathcal E}}:D\times D\to{{\mathbb R}}$ such that $D$ is complete with respect to the form norm $\|\cdot\|_{{{\mathcal E}}}=\sqrt{{{\mathcal E}}(\cdot,\cdot)+\|\cdot\|^{2}}$ where $\|\cdot\|$ always denotes the $L^{2}$ norm. We write ${{\mathcal E}}(u):={{\mathcal E}}(u,u)$ for $u\in D$. A closed non-negative form $({{\mathcal E}},D)$ is called a *Dirichlet form* if for any $u\in D$ and any normal contraction $c:{{\mathbb R}}\to{{\mathbb R}}$ we have $c\circ u\in D$ and ${{\mathcal E}}(c\circ u)\leq {{\mathcal E}}(u)$. Here, $c$ is a normal contraction if $c(0)=0$ and $|c(x)-c(y)|\leq|x-y|$ for $x,y\in{{\mathbb R}}$. A Dirichlet form is called *regular* if $D\cap C_{c}(X)$ is dense both in $(D,\|\cdot\|_{{{\mathcal E}}})$ and $(C_{c}(X),\|\cdot\|_{\infty})$ where $C_{c}(X)$ is the space of continuous compactly supported functions. A function $f:X\to{{\mathbb R}}$ is said to be *quasi continuous* if for every ${{\varepsilon}}>0$ there is an open set $U\subseteq X$ with $$\begin{aligned} \mathrm{cap}(U):=\inf\{\|v\|_{{{\mathcal E}}}\mid v\in D,\, 1_{U}\leq v\}\leq {{\varepsilon}},\end{aligned}$$ such that $f\vert _{X\setminus U}$ is continuous (where $\inf\emptyset=\infty$ and $1_{U}$ is the characteristic function of $U$). For a regular Dirichlet form $({{\mathcal E}},D)$ every $u\in D$ admits a quasi continuous representative, see [@Fuk Theorem 2.1.3]. In the following we assume that when considering $u$ as a function we always choose a quasi continuous representative. There is a fundamental representation theorem for regular Dirichlet forms called the Beurling-Deny formula, see [@Fuk Theorem 3.2.1.]. It states that there is a non-negative Radon measure $k$ on $X$, a non-negative Radon measure $J$ on $X\times X\setminus d$ which is $X\times X$ without the diagonal $d:=\{(x,x)\mid x\in X\}$ and a positive semi-definite bilinear form ${{\Gamma}}^{(c)}$ on $D\times D$ with values in the signed Radon measures on $X$ which is *strongly local*, i.e., satisfies ${{\Gamma}}^{(c)}(u,v)=0$ if $u$ is constant on the support of $v$, such that $$\begin{aligned} {{\mathcal E}}(u)=\int_{X}d{{\Gamma}}^{(c)}(u)+\int_{X\times X\setminus d} (u(x)-u(y))^{2}dJ(x,y)+\int_{X}u(x)^{2}dk(x),\end{aligned}$$ where we choose a quasi continuous representative of $u$ in the second and third integral. The first term on the right hand side is called the *strongly local part* of ${{\mathcal E}}$, the second term is called the *jump part* and the third term is called the *killing term*. The measure $J$ gives rise to a Radon measure ${{\Gamma}}^{(j)}$ (where the $j$ refers to ‘jump’) which is characterized by $$\begin{aligned} \int_{K}d{{\Gamma}}^{(j)}(u)=\int_{K\times X\setminus d}(u(x)-u(y))^{2}dJ(x,y)\end{aligned}$$ for $K\subseteq X$ compact and $u\in D$. The focus of this paper is on regular Dirichlet forms ${{\mathcal E}}$ without killing term, i.e., $k\equiv0$. Thus, we denote $$\begin{aligned} {{\Gamma}}={{\Gamma}}^{(c)}+{{\Gamma}}^{(j)}.\end{aligned}$$ The space $D_{\mathrm{loc}}^{*}$ of *functions locally in the domain* of ${{\mathcal E}}$ was introduced in [@FLW] and is important for the definition of intrinsic metrics. It is defined as the set of functions $u\in L^{2}_{\mathrm{loc}}(X,m)$ such that for all open and relatively compact sets $G$ there is a function $v\in D$ such that $u$ and $v$ agree on $G$ and for all compact $K\subseteq X$ $$\begin{aligned} \int_{K\times X\setminus d}(u(x)-u(y))^{2}dJ(x,y)<\infty.\end{aligned}$$ We can extend ${{\Gamma}}^{(c)}$ and ${{\Gamma}}^{(j)}$ to $D_{\mathrm{loc}}^{*}$, see [@Fuk Remarks after the proof of Theorem 3.2.1.] and [@FLW Proposition 3.3]. For the strongly local part we have a *chain rule* (see [@Fuk Theorem 3.2.2.]) as follows: for ${{\varphi}}:{{\mathbb R}}\to{{\mathbb R}}$ continuously differentiable with bounded derivative ${{\varphi}}'$, $$\begin{aligned} {{\Gamma}}^{(c)}({{\varphi}}(u),v) = {{\varphi}}'(u){{\Gamma}}^{(c)}(u,v),\quad u,v\in D_{\mathrm{loc}}^{*}\cap L^{\infty}(X,m).\end{aligned}$$ A *pseudo metric* is a map $\rho:X\times X\to[0,\infty]$ which is symmetric, satisfies the triangle inequality and $\rho(x,x)=0$ for all $x\in X$. For $A\subseteq X$ we define the map $\rho_{A}:X\to[0,\infty]$ by $$\begin{aligned} \rho_{A}(x)=\inf_{y\in A}\rho(x,y).\end{aligned}$$ If $\rho$ is a pseudo metric and $T>0$, then $\rho\wedge T$ is a pseudo metric and we have that $(\rho\wedge T)_{A}=\rho_{A} \wedge T$ and $|\rho_{A}(x)\wedge T-\rho_{A}(y)\wedge T|\leq\rho(x,y)$. By [@FLW Definition 4.1.] a pseudo metric $\rho$ is called an *intrinsic metric* for the Dirichlet form ${{\mathcal E}}$ if there are Radon measures $m^{(c)}$ and $m^{(j)}$ with $m^{(c)}+m^{(j)}\leq m$ such that for all $A\subseteq X$ and all $T>0$ the functions $\rho_{A}\wedge T$ are in $D_{\mathrm{loc}}^{*}\cap C(X)$ and satisfy $$\begin{aligned} {{\Gamma}}^{(c)}(\rho_{A}\wedge T)\leq m^{(c)}\quad\mbox{and}\quad{{\Gamma}}^{(j)}(\rho_{A}\wedge T)\leq m^{(j)}.\end{aligned}$$ This implies that if $A\subseteq X$ is such that $\rho_{A}(x)<\infty$ for all $x\in X$, then $\rho_{A}\in D_{\mathrm{loc}}^{*}\cap C(X)$ and ${{\Gamma}}(\rho_{A})\leq m$. We assume that $\rho$ is continuous with respect to the original topology. An estimate for the bottom of the essential spectrum ---------------------------------------------------- The following Persson-type theorem seems to be standard in some settings, see [@Per; @Gri]. However, since we are not able to find a proper reference in the literature which covers our case, we include a short proof. \[p:h\] Let $h$ be a closed quadratic form on $L^{2}(X,m)$ that is bounded from below and let $H$ be the corresponding self adjoint operator. Assume that there is a normalized sequence $(f_{n})$ in $D(h)$ that converges weakly to zero. Then, $$\begin{aligned} {{\lambda}}_{0}^{\mathrm{ess}}(H)\leq\liminf_{n\to\infty} h(f_{n}).\end{aligned}$$ Without loss of generality assume that $h\geq0$ and that ${{\lambda}}_0^{\mathrm{ess}}(H)>0$. Let $0< {{\lambda}}<{{\lambda}}_{0}^{\mathrm{ess}}(H)$. We will show that there is an $N\geq0$ such that $h(f_{n})>{{\lambda}}$ for all $n \geq N$. Let ${{\lambda}}_{1}$ be such that ${{\lambda}}<{{\lambda}}_{1}<{{\lambda}}_{0}^{\mathrm{ess}}(H)$ and let ${{\varepsilon}}>0$ be arbitrary. Since $D(H)$ is a core for $D(h)$ there exist $g_{n}\in D(H)$ for all $n\geq0$ such that ${\left\Vert f_{n}-g_{n}\right\Vert}_{h}^{2}=h(f_{n}-g_{n})+{\left\Vert f_{n}-g_{n}\right\Vert}^{2}\leq {{\varepsilon}}$ and $(g_{n})$ converges weakly to zero as well. As ${{\lambda}}_{1}<{{\lambda}}_{0}^{\mathrm{ess}}(H)$, the spectral projection $E_{(-\infty,{{\lambda}}_{1}]}$ of $H$ and the interval $(-\infty,{{\lambda}}_{1}]$ is a finite rank operator. Therefore, as $(g_{n})$ converges weakly to zero, there is an $N\geq0$ such that ${\left\Vert E_{(-\infty,{{\lambda}}_{1}]}g_{n}\right\Vert}^{2}< {{\varepsilon}}$ for $n\geq N$. Letting $\nu_{n}$ be the spectral measure of $H$ with respect to $g_{n}$, we estimate for $n\ge N$ $$\begin{aligned} h(g_{n})\geq\int_{{{\lambda}}_{1}}^{\infty}td\nu_{n}(t)\geq {{\lambda}}_{1} \int_{{{\lambda}}_{1}}^{\infty}d\nu_{n}(t)={{\lambda}}_{1}(\|g_{n}\|^{2} -\|E_{(-\infty,{{\lambda}}_{1}]}g_{n}\|^{2})> {{\lambda}}_{1}(1-{{\varepsilon}}),\end{aligned}$$ where we used ${{\lambda}}_{1}\ge0$ as $h\geq0$. Since $h(f_{n})\geq h(g_{n})-{{\varepsilon}}$ by the choice of $g_{n}$, we conclude the asserted inequality by choosing ${{\varepsilon}}=({{\lambda}}_1-{{\lambda}})/(1+{{\lambda}}_{1})>0$. The test functions ------------------ In this section we introduce the sequence of test functions which we will use to estimate the bottom of the (essential) spectrum. If $\mu = \infty$ or ${\widetilde{ \mu}} =\infty$ the statements of our theorem become obvious, therefore, from now on, we assume that $\mu, {\widetilde{ \mu}} < \infty$. For $r\in{{\mathbb N}},x_{0}\in X,{{\alpha}}>0$, define $$\begin{aligned} f_{r,x_{0},{{\alpha}}}&:X\to[0,\infty),\quad x\mapsto \big( (e^{{{\alpha}}r}\wedge e^{{{\alpha}}(2r-\rho(x_{0},x))}) - 1 \big)\vee 0.\end{aligned}$$ Then, for fixed $r$, ${{\alpha}}$, $x_{0}$, we have $f\vert_{B_{r}}\equiv e^{{{\alpha}}r}-1$, $f\vert_{B_{2r}\setminus B_{r}}= e^{ {{\alpha}}(2r- \rho(x_{0},\cdot))}-1$ and $f\vert_{X\setminus B_{2r}}\equiv0$. Clearly, $f$ is spherically homogeneous, i.e., there exists $h:[0,\infty)\to[0,\infty)$ such that $f(x)=h(\rho(x_{0},x))$. The definition of $f$ combines ideas from [@Br], [@Fuj] and [@Stu]. Moreover, for $r\in{{\mathbb N}},x_{0}\in X,{{\alpha}}>0$, let $g_{r,x_{0},{{\alpha}}}:X\to[0,\infty)$, be given by $$\begin{aligned} g_{r,x_{0},{{\alpha}}}=(f_{r,x_{0},{{\alpha}}}+2)1_{B_{2r}}\end{aligned}$$ \[l:f\] Let ${{\alpha}}>\mu/2$, $x_{0}\in X $ and $f_{r}=f_{r,x_{0},{{\alpha}}}$ and $g_{r}=g_{r,x_{0},{{\alpha}}}$ for $r\geq0$. Then, - $f_{r},g_{r}\in L^{2}(X,m)$ for all $r\ge0$. - If $m(\bigcup_{r}B_{r})=\infty$, then $f_{r}/\|f_{r}\|$ converges weakly to $0$ as $r\to\infty$. - There is a sequence $(r_{k})$ such that $\|g_{r_{k}}\|/\|f_{r_{k}}\|\to 1$ as $k\to\infty$. If ${{\alpha}}>{\widetilde{ \mu}}/2$, then - There are a sequences $(x_{k})$ in $X$ and $(r_{k})$ such that $f_{k}=f_{r_{k},x_{k},{{\alpha}}},g_{k}=g_{r_{k},x_{k},{{\alpha}}}\in L^{2}(X,m)$ and we have that $\|g_{k}\|/\|f_{k}\|\to 1$ as $k\to\infty$. \(a) As $\mu<\infty$ it follows that $m(B_{r}(x_{0}))<\infty$ for all $r\geq0$. Therefore, $f_{r},g_{r}\in L^{2}(X,m)$ for all $r\ge0$ since $f_{r},g_{r}$ are supported in $B_{2r}$ and bounded.\ (b) Let $\psi\in L^{2}(X,m)$ with ${\left\Vert \psi\right\Vert}=1$, ${{\varepsilon}}>0$ and set ${{\varphi}}=\psi1_{\bigcup B_{r}}$. There exists $R>0$ such that ${\left\Vert {{\varphi}}1_{X\setminus B_{R}}\right\Vert}\leq {{\varepsilon}}/2$. Moreover, let $r\geq R$ be such that $m(B_{R})\leq{{\varepsilon}}^{2} m(B_{r})/4$ (this choice is possible since $m(\bigcup B_{r})=\infty$). We conclude by the Cauchy-Schwarz inequality and $\|f_{r}1_{B_{R}}\|\leq\frac{{{\varepsilon}}}{2}\|f_{r}\|$ that $$\begin{aligned} {\left\langle {{\varphi}},f_{r}\right\rangle} ={\left\langle {{\varphi}}1_{B_{R}},f_{r}\right\rangle} + {\left\langle {{\varphi}}1_{X\setminus B_{R}},f_{r}\right\rangle}\leq {\left\Vert {{\varphi}}\right\Vert}{\left\Vert f_{r}1_{B_{R}}\right\Vert}+{\left\Vert {{\varphi}}1_{X\setminus B_{R}}\right\Vert} {\left\Vert f_{r}\right\Vert}\leq {{\varepsilon}}{\left\Vert f_{r}\right\Vert}.\end{aligned}$$ As ${{\mathrm {supp}\,}}f_{r}\subseteq \bigcup_{s} B_{s}$, it follows that ${\left\langle \psi,f_{r}\right\rangle}={\left\langle {{\varphi}},f_{r}\right\rangle}$ for $r\geq0$ which proves (b).\ Before we prove (c) we show (d) and indicate how to adapt the proof to (c) afterwards. Let $0<{{\varepsilon}}<{{\alpha}}-{\widetilde{ \mu}}/2$. By the definition of ${\widetilde{ \mu}}$ there are sequences $(r_{k})$ of increasing positive numbers and $(x_{k})$ of elements in $X$ such that $$\begin{aligned} \frac{ m(B_{2r_{k}}(x_{k}))}{m(B_{1}(x_{k}))}&\leq e^{(2{\widetilde{ \mu}}+{{\varepsilon}})r_{k}},\quad k\geq0.\end{aligned}$$ We set $f_{k}=f_{r_{k},x_{k},{{\alpha}}}$, $g_{k}=g_{r_{k},x_{k},{{\alpha}}}$. As $m(B_{2r_{k}}(x_{k}))<\infty$ and the functions $f_{k},g_{k}$ are supported in $B_{2r_{k}}(x_{k})$ and bounded, they are in $L^{2}(X,m)$. By definition we have $g_{k}=g_{k}1_{B_{2r_{k}}}=(f_{k}+2)1_{B_{2r_{k}}}$, $k\geq0$. Using the inequalities $(a+b)^{2}\leq \frac{1}{(1-{{\varepsilon}})}a^{2}+\frac{1}{{{\varepsilon}}}b^{2}$ and $\|f_{k}\|^{2}\geq m({B_{r_{k}}}(x_{k}))(e^{{{\alpha}}r_{k}}-1)^{2}\geq m({B_{r_{k}}}(x_{k}))e^{2{{\alpha}}r_{k}}/c$ for some $c>0$ we get $$\begin{aligned} \frac{\|g_{k}\|^{2}}{\|f_{k}\|^{2}} &\leq\frac{(\|f_{k}\|+2\sqrt{m(B_{2r_{k}}(x_{k}))})^{2}}{\|f_{k}\|^{2}} \leq\frac{\frac{1}{(1-{{\varepsilon}})}\|f_{k}\|^{2}+\frac{4}{{{\varepsilon}}}{m({B_{2r_{k}}(x_{k})})}} {\|f_{k}\|^{2}}\\ &\leq\frac{1}{(1-{{\varepsilon}})}+\frac{4c}{{{\varepsilon}}} {\frac{{m({B_{2r_{k}}(x_{k})})}}{m({B_{r_{k}}}(x_{k}))}} e^{-2{{\alpha}}r_{k}}.\end{aligned}$$ For $r_{k}$ large enough we have $$\begin{aligned} \frac{ m(B_{r_{k}}(x_{k}))}{m(B_{1}(x_{k}))}\geq\inf_{x\in X}\frac{ m(B_{r_{k}}(x))}{m(B_{1}(x))}\geq e^{({\widetilde{ \mu}}-{{\varepsilon}})r_{k}}.\end{aligned}$$ Thus, by the choice of $(r_{k})$ and $(x_{k})$, we have ${\frac{{m({B_{2r_{k}}})}}{m({B_{r_{k}}})}} \leq e^{({\widetilde{ \mu}}+2{{\varepsilon}})r_{k}}$. As $0<{{\varepsilon}}<{{\alpha}}-{\widetilde{ \mu}}/2$ $$\begin{aligned} \frac{\|g_{{k}}\|^{2}}{\|f_{{k}}\|^{2}}\leq\frac{1}{(1-{{\varepsilon}})}+\frac{4c}{{{\varepsilon}}} e^{({\widetilde{ \mu}}+2{{\varepsilon}}-2{{\alpha}}) r_{k}}\to \frac{1}{(1-{{\varepsilon}})}\quad\mbox{as $k\to\infty$}.\end{aligned}$$ Since ${{\varepsilon}}$ can be chosen to be arbitrarily small and ${\|g_{{k}}\|}\geq{\|f_{{k}}\|}$ we deduce the statement.\ For (c) we choose $(x_{k})$ to be $x_{0}$ and follow the lines of the proof replacing ${\widetilde{ \mu}}$ by $\mu$. If $\inf_{x\in X}m(B_{1}(x))>0$, then $f_{k}/\|f_{k}\|$ of (d) also converges weakly to zero as $k\to\infty$. The following auxiliary estimates will later give us bounds for the Lipshitz constants of $f_{r,x,{{\alpha}}}$. \[l:e\] Let ${{\alpha}}>0$. For all $R\geq 0$ one has $$\begin{aligned} \frac{{\left({e}^{\alpha R} -1\right)}^{2}}{\left({e}^{2\alpha R} +1\right)} \leq\frac{{\alpha }^{2}{R}^{2}}{2}.\end{aligned}$$ Moreover, for $ R\in [0, 1] $ one has $$\begin{aligned} \frac{{\left({e}^{\alpha R} -1\right)}^{2}}{\left({e}^{2\alpha R} +1\right)} \leq\frac{{R}^{2} {\left({e}^{\alpha}-1\right)}^{2}}{\left( R^2{e}^{2\alpha}+1\right)}.\end{aligned}$$ For the first statement let $s={{\alpha}}R$ and check via a series expansion that $s\mapsto{s}^{2}\left({e}^{2s} +1\right)- 2{\left({e}^{s}-1\right)}^{2}$ is non-negative. The second statement follows by direct calculation since we have $e^{{{\alpha}}R}-1\leq R(e^{{{\alpha}}}-1)$ for $R\in[0,1]$ and ${{\alpha}}> 0$. \[l:f\_Lip\] Let $r\in{{\mathbb N}}$, $x_{0}\in X$, ${{\alpha}}>0$ and set $f:=f_{r,x_{0},{{\alpha}}}$, $g:=g_{r,x_{0},{{\alpha}}}$. Then, for all $ x,y\in X$ $$\begin{aligned} (f(x)-f(y))^{2}&\leq c({{\alpha}}) (g(x)^{2}+g(y)^2) \rho(x,y)^{2}\end{aligned}$$ where $c({{\alpha}})= \frac{{{{\alpha}}^{2}}}{2} $. If additionally $\rho(x,y)\leq 1$, then $c({{\alpha}})$ can be chosen to be $c({{\alpha}},\rho(x,y))=\frac{(e^{{{\alpha}}}-1)^{2}}{\rho(x,y)^{2}e^{2{{\alpha}}}+1}$. In particular, $f$ is Lipshitz continuous with Lipshitz constant ${{\alpha}}( e^{{{\alpha}}r}+1)$. We fix $r$, ${{\alpha}}$ and $x_{0}$ for the proof. Let $x,y\in X$ be given and let $s=\rho(x_{0},x)$ and $t=\rho(x_{0},y)$. We define $ D_{s,t}:=(f(x)- f(y))^{2}$. Moreover, we use the estimate on $F(R):=\frac{(e^{{{\alpha}}R}-1)^{2}}{e^{2{{\alpha}}R}+1}$, $R\geq0,$ by $c({{\alpha}})R^{2}$ (and by $c({{\alpha}},R)R^{2}$ for $R\leq1$) from Lemma \[l:e\]. By symmetry we may assume, without loss of generality, that $s\leq t$ so that we have six cases to check.\ Case 1: If $s\leq t\leq r$, then $D_{s,t}=0$.\ Case 2: If $s\leq r\leq t\leq 2r$, then since $t-r\leq t-s=\rho(x_{0},y)-\rho(x_{0},x)\leq \rho(x,y)$ and $g(x)=e^{{{\alpha}}r}+1$, $g(y)=e^{{{\alpha}}(2r-t)}+1$, $$\begin{aligned} D_{s,t}&=(e^{{{\alpha}}r} -e^{{{\alpha}}(2r-t)})^{2} =( e^{2{{\alpha}}r} +e^{2{{\alpha}}(2r-t)})F(t-r) \leq ( e^{2{{\alpha}}r} +e^{2{{\alpha}}(2r-t)})c({{\alpha}})(t-r)^{2}\\ &\leq c({{\alpha}})(g(x)^{2}+ g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 3: If $s\leq r\leq 2r\leq t$, then since $r\leq t-s\leq \rho(x,y)$, $g(x)=e^{{{\alpha}}r}+1$ and $g({y})=0$, $$\begin{aligned} D_{s,t}=(e^{{{\alpha}}r} -1)^{2} = (e^{2{{\alpha}}r}+1)F(r) \leq (e^{2{{\alpha}}r}+1)c({{\alpha}}) r^{2}\leq2c({{\alpha}}) (g(x)^{2}+g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 4: If $ r\leq s\leq t\leq 2r$, then since $t-s\leq \rho(x,y)$ and $g(x)=e^{{{\alpha}}(2r- s)}+1$, $g(y)=e^{{{\alpha}}(2r-t)}+1$, $$\begin{aligned} D_{s,t}&=(e^{{{\alpha}}(2r-s)} -e^{{{\alpha}}(2r-t)})^{2}= ( e^{2{{\alpha}}(2r- s)}+ e^{2{{\alpha}}(2r -t)}) F(t-s)\\ &\leq c({{\alpha}})(g(x)^{2}+ g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 5: If $ r\leq s\leq 2r\leq t$, then since $2r-s\leq t-s\leq \rho(x,y)$, $g(x)=e^{{{\alpha}}(2r- s)}+1$ and $g(y)=0$, $$\begin{aligned} D_{s,t}=(e^{{{\alpha}}(2r-s)}-1)^{2} = ( e^{2{{\alpha}}(2r-s)}+1)F(2r-s)\leq c({{\alpha}})(g(x)^{2}+g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 6: If $2r\leq s\leq t$, then $D_{s,t}=0$.\ The Lipshitz bound follows since $g$ is bounded by $e^{{{\alpha}}r}+1$. \[l:f\_in\_D\] Let $({{\mathcal E}},D)$ be a regular Dirichlet form and $\rho$ an intrinsic metric. For all $r>0$, $x_{0}\in X$ and ${{\alpha}}>0$ we have $f:=f_{r,x_{0},{{\alpha}}}\in D_{\mathrm{loc}}^{*}$. Moreover, if $B_{2r}(x_{0})$ is compact, then $f\in D$. By Lemma \[l:f\_Lip\] the functions $f:=f_{r,x_{0},{{\alpha}}}$ are Lipshitz continuous for all $r>0$, $x_{0}$ and ${{\alpha}}>0$. Thus, by a Rademacher type theorem, see e.g. [@Sto Theorem 5.1] for strongly local forms or [@FLW Theorem 4.8] for general Dirichlet forms, we have $f\in D_{\mathrm{loc}}^{*}$ and ${{\Gamma}}(f)\leq m$. If $B_{2r}(x_{0})$ is compact, then the function $f$ is compactly supported which implies that $f\in D$. Proof of the main theorem {#s:proof} ========================= The strongly local estimate --------------------------- In this subsection we give an estimate which will be used to prove the theorem for the strongly local part of the Dirichlet form. For given $r\in{{\mathbb N}}$, $x_{0}\in X$ and ${{\alpha}}>0$ we denote $f:=f_{r,x_{0},{{\alpha}}}$ and $g:=g_{r,x_{0},{{\alpha}}}$. \[l:SL\] Let $\rho$ be an intrinsic metric for a regular strongly local Dirichlet form $\mathcal{E}$. Then, for all $r>0$, $x_{0}\in X$ and ${{\alpha}}>0$ such that $f\in D$ we have $$\begin{aligned} {{\mathcal E}}(f)\leq{{\alpha}}^{2}\int_{X}g^{2}dm^{(c)}.\end{aligned}$$ As ${{\mathcal E}}$ is strongly local, we get by the chain rule and the fact that $\rho $ is an intrinsic metric that $$\begin{aligned} {{\mathcal E}}(f)&=\int_{B_{2r}\setminus B_{r}}d{{\Gamma}}^{(c)}(f)= \int_{B_{2r}\setminus B_{r}}d{{\Gamma}}^{(c)}(e^{{{\alpha}}(2r-\rho(x_{0},\cdot))} -1)\\ &={{\alpha}}^{2}\int_{B_{2r}\setminus B_{r}} e^{2{{\alpha}}(2r-\rho(x_{0},\cdot))}d{{\Gamma}}^{(c)}(\rho(x_{0},\cdot))\\ &\leq{{\alpha}}^{2}\int_{B_{2r}\setminus B_{r}} e^{2{{\alpha}}(2r-\rho(x_{0},\cdot))}dm^{(c)}\leq {{\alpha}}^{2}\int_{X}g_{r,x_{0},{{\alpha}}}^{2}dm^{(c)}.\end{aligned}$$ The non-local estimate {#s:nonlocal} ---------------------- Next, we treat the non-local case. With applications to graphs in the next section in mind, we do not assume that the jump part is a regular Dirichlet form for now. For this subsection, let $m$ be a Radon measure on $X$ and let $J$ be a symmetric Radon measure on $X\times X\setminus d$ such that for every $m$-measurable $A\subseteq X$ the set $A\times X\setminus d$ is $J$ measurable and vice versa. Let $\rho$ be a pseudo metric on $X$ which is $J$ measurable and assume that for all measurable $A\subseteq X$ $$\begin{aligned} \label{e:adapted}\tag{$\clubsuit$} \int_{A\times X\setminus d}\rho(x,y)^{2}dJ(x,y)\leq m(A)\end{aligned}$$ which immediately implies that for all measurable functions ${{\varphi}}$ $$\begin{aligned} \int_{X\times X\setminus d}{{\varphi}}(x)^{2}\rho(x,y)^{2}dJ(x,y)\leq \int_{X} {{\varphi}}^{2} dm.\end{aligned}$$ We say that the pseudo metric $\rho$ has *jump size in* $[a,b]$, $0\leq a\leq b$, if for the set $A_{a,b}:=\{(x,y)\in X \times X \mid\rho(x,y)\in[a,b]\}\setminus d$ $$\begin{aligned} \int_{X\times X\setminus d}\rho(x,y)^{2}dJ(x,y)=\int_{A_{a,b}} \rho(x,y)^{2}dJ(x,y).\end{aligned}$$ For given $r\in{{\mathbb N}}$, $x_{0}\in X$ and ${{\alpha}}>0$ we denote $f:=f_{r,x_{0},{{\alpha}}}$ and $g:=g_{r,x_{0},{{\alpha}}}$. \[l:NL\]Assume that $\rho$ satisfies . For all $r\in{{\mathbb N}}$, $x_{0}\in X$ and ${{\alpha}}>0$ $$\begin{aligned} \int_{X\times X\setminus d}(f(x)-f(y))^{2}dJ(x,y)\leq\ 2c({{{\alpha}}}) \int_{X} g^2dm,\end{aligned}$$ where $c({{\alpha}})= \frac{{{\alpha}}^{2}}{2}$. If $\rho$ has jump size in $[{{\delta}},1]$ for some $0\leq{{\delta}}\leq1$, then $c({{\alpha}})$ can be chosen to be $c({{\alpha}},{{\delta}})=\frac{(e^{{{\alpha}}}-1)^{2}}{1+{{\delta}}^{2}e^{2{{\alpha}}}}$. By Lemma \[l:f\_Lip\] and since $\rho$ satisfies $$\begin{aligned} \int_{X\times X\setminus d} (f(x)-f(y))^{2}dJ(x,y)\leq {{{\alpha}}^{2}}\int_{X\times X\setminus d} g(x)^{2}\rho(x,y)^{2}dJ(x,y)\leq{{{\alpha}}^{2}} \int_{X} g^2dm.\end{aligned}$$ Let ${{\delta}}>0$. If the jump size is in $[{{\delta}},1]$, then $$\begin{aligned} \int_{X\times X\setminus d} &(f(x)-f(y))^{2}dJ(x,y)= \int_{{A_{{{\delta}},1}}} (f(x)-f(y))^{2} dJ(x,y)\\ \leq& \int_{{A_{{{\delta}},1}}} |g(x)|^{2}\frac{2(e^{{{\alpha}}}-1)^{2}}{(1+\rho(x,y)^{2}e^{2{{\alpha}}})} \rho(x,y)^{2}dJ(x,y) \\ \leq& \frac{2(e^{{{\alpha}}}-1)^{2}}{(1+{{\delta}}^{2}e^{2{{\alpha}}})} \int_{X\times X\setminus d} g(x)^{2}\rho(x,y)^{2}dJ(x,y)\\ \leq& \frac{2(e^{{{\alpha}}}-1)^{2}}{(1+{{\delta}}^{2}e^{2{{\alpha}}})} \int_{X}g^{2}dm.\end{aligned}$$ Proof of Theorem \[t:main\] --------------------------- We now have all of the ingredients to prove our main result. By [@FLW Lemma 4.7] an intrinsic metric satisfies . Moreover, under the assumption that the distance balls are compact we have that $f_{r,x,{{\alpha}}}\in D$ for all $r>0$, $x\in X$, ${{\alpha}}>0$ by Lemma \[l:f\_in\_D\]. By Lemma \[l:f\] (d) there are a sequences $(x_{k})$ and $r_{k}$ such that for $f_{k}=f_{r_{k},x_{k},{{\alpha}}}$, $g_{k}=g_{r_{k},x_{k},{{\alpha}}}$ with ${{\alpha}}>{\widetilde{ \mu}}/2$ $$\begin{aligned} {{\lambda}}_{0}(L)\leq \lim_{k\to\infty}\frac{{{\mathcal E}}(f_{k})}{\|f_{k}\|^{2}}\leq {{\alpha}}^{2}\lim_{k\to\infty}\frac{\|g_{k}\|^{2}}{\|f_{k}\|^{2}}={{\alpha}}^{2}, \end{aligned}$$ where the second inequality follows from Lemmas \[l:SL\] and \[l:NL\] and the equality follows from Lemma \[l:f\] (d). Hence, ${{\lambda}}_{0}(L)\leq {\widetilde{ \mu}}^{2}/4$. Let now $(r_{k})$ be the sequence given by Lemma \[l:f\] (c) for some fixed $x_{0}\in X$ and let $x_{k}=x_{0}$ for all $k\geq0$. By Lemma \[l:f\] (b) the sequence $(f_{k}/\|f_{k}\|)$ converges weakly to zero and, therefore, we get by Proposition \[p:h\] and Lemma \[l:f\] (c), that $$\begin{aligned} {{\lambda}}_{0}^{\mathrm{ess}}(L) \leq\lim_{k\to\infty}\frac{{{\mathcal E}}(f_{k})}{\|f_{k}\|^{2}} \leq {{\alpha}}^{2}\lim_{k\to\infty}\frac{\|g_{k}\|^{2}}{\|f_{k}\|^{2}}={{\alpha}}^{2}. \end{aligned}$$ Therefore, ${{\lambda}}_{0}^{\mathrm{ess}}(L) \leq\mu^{2}/4$. A more general non-local estimate --------------------------------- Let $L$ be the positive selfadjoint operator associated to ${{\mathcal E}}$. \[t:jump\] Assume that $\rho$ satisfies and $f_{r,x,{{\alpha}}}\in D$ for all $r\geq0$, $x\in X$ and ${{\alpha}}>{\widetilde{ \mu}}/2$. Then, $$\begin{aligned} {{\lambda}}_{0}(L)\leq\frac{{\widetilde{ \mu}}^{2}}{4}\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq \frac{\mu^{2}}{4}\end{aligned}$$ if $m(\bigcup B_{r}(x_{0}))=\infty$ for $x_{0}$ used to define $\mu$.\ If the jump size is bounded in $[{{\delta}},1]$ for some $0\leq{{\delta}}\leq1$, then $$\begin{aligned} {{\lambda}}_{0}(L)\leq\frac{2{(e^{{\widetilde{ \mu}}/2}-1)^{2}}}{{{{\delta}}^{2}e^{{\widetilde{ \mu}}}}+1} \quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq \frac{2{(e^{\mu/2}-1)^{2}}}{{{{\delta}}^{2}e^{\mu}+1}}\end{aligned}$$ if $m(\bigcup B_{r}(x_{0}))=\infty$ for $x_{0}$ used to define $\mu$. The proof follows analogously to the proof of the main theorem from Proposition \[p:h\], Lemma \[l:f\] and Lemma \[l:NL\]. Applications {#s:applications} ============ Weighted graphs {#s:graph} --------------- In this section we derive consequences of Theorem \[t:main\] and Theorem \[t:jump\] for graphs. We briefly introduce the setting and refer for more background to [@KL1]. Let $X$ be a countable discrete set. Every Radon measure of full support on $X$ is given by a function $m:X\to(0,\infty)$. Then, $L^{2}(X,m)$ is the space $\ell^{2}(X,m)$ of $m$-square summable functions with norm ${\left\Vert u\right\Vert}=(\sum_{x}u(x)^{2}m(x))^{\frac{1}{2}}$, $u\in \ell^{2}(X,m)$. From [@KL1 Theorem 7] it can be seen that all regular Dirichlet forms without killing term are determined by a symmetric map $b:X\times X\to[0,\infty)$ with vanishing diagonal that satisfies $$\begin{aligned} \sum_{y\in X}b(x,y)<\infty,\qquad\mbox{for all } x\in X,\end{aligned}$$ which gives rise to a measure $J$ on $X\times X\setminus d$ by $J=\frac{1}{2}b$. The one half stems from the convention that in the form we consider each edge only once. The map $b$ can then be interpreted as a weighted graph with vertex set $X$. Namely, the vertices $x,y\in X$ are connected by an edge with weight $b(x,y)$ if $b(x,y)>0$. In this case, we write $x\sim y$. A graph is called *connected* if for all $x,y\in X$ there are vertices $x_i \in X$ such that $x=x_{0}\sim x_{1}\sim\ldots \sim x_{n}=y$. Let a map ${\widetilde{ {{\mathcal E}}}}:\ell^{2}(X,m)\to[0,\infty]$ be given by $$\begin{aligned} {\widetilde{ {{\mathcal E}}}}(u)=\frac{1}{2}\sum_{x,y\in X}b(x,y)(u(x)-u(y))^{2}.\end{aligned}$$ The regular Dirichlet form ${{\mathcal E}}$ associated to $J$ is the restriction of ${\widetilde{ {{\mathcal E}}}}$ to $\overline{C_{c}(X)}^{\|\cdot\|_{\mathcal{E}}}$. Moreover, let $$\begin{aligned} \mbox{ ${{\mathcal E}}^{\max}={\widetilde{ {{\mathcal E}}}}\vert_{D^{\max}},\quad D^{\max}=\{u\in \ell^{2}(X,m)\mid {\widetilde{ {{\mathcal E}}}}(u)<\infty\}$}\end{aligned}$$ which is also a Dirichlet form. We denote the operator arising from ${{\mathcal E}}$ by $L$ and the operator arising from ${{\mathcal E}}^{\max}$ by $L^{\max}$. Let $\rho$ be an intrinsic pseudo metric on $X$. In this context this is equivalent to (see [@FLW Lemma 4.7, Theorem 7.3]) which reads as $$\begin{aligned} \frac{1}{2}\sum_{y\in X}b(x,y)\rho(x,y)^{2}\leq m(x),\quad x\in X.\end{aligned}$$ For simplicity we restrict ourselves to the case when $\rho$ takes values in $[0,\infty)$. (Otherwise, we can easily consider the graph componentwise.) \[r:rho\] Very often it is convenient to consider intrinsic metrics which satisfy $\sum_{y\in X}b(x,y)\rho(x,y)^{2}\leq m(x)$ for all $ x\in X$ (i.e., we drop the $\frac{1}{2}$ on the left hand side). For example, in [@Hu] an explicit example of such a metric $\rho$ is given, for $x,y \in X$, by $$\begin{aligned} \rho(x,y):=\inf\{l(x_{0},\ldots,x_{n})\mid n\geq 1,x_{0}=x, x_{n}=y, x_{i}\sim x_{i-1}, i=1,\ldots,n\}\end{aligned}$$ where the length $l$ is given by $l(x_{0},\ldots,x_{n})= \sum_{i=1}^{n}\min\{{{\mathrm{Deg}}}(x_{i})^{-\frac{1}{2}}, {{\mathrm{Deg}}}(x_{i-1})^{-\frac{1}{2}}\}$ and ${{\mathrm{Deg}}}(z)=\sum_{w}b(z,w)/m(z)$ is a generalized vertex degree. In this case all estimates in the theorem above can be divided by $2$. In general, it is hard to determine whether distance balls with respect to a certain metric are compact, which means finite in the original topology, in the situation of graphs. However, we always have a statement for the operator $L^{\max}$ related to ${{\mathcal E}}^{\max}$. \[t:graph\]Assume that $b$ is connected and $m(X) = \infty$. Then, $$\begin{aligned} {{\lambda}}_{0}(L^{\max})\leq\frac{{\widetilde{ \mu}}^{2}}{4}\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L^{\max})\leq \frac{\mu^{2}}{4}.\end{aligned}$$ If $\rho(x,y)\in[{{\delta}},1]$ for all $x\sim y$, then $$\begin{aligned} {{\lambda}}_{0}(L^{\max})\leq\frac{{2(e^{{\widetilde{ \mu}}/2}-1)^{2}}}{{{{\delta}}^{2}e^{{\widetilde{ \mu}}}}+1} \quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L^{\max})\leq \frac{{2(e^{\mu/2}-1)^{2}}}{{{{\delta}}^{2}e^{\mu}+1}}.\end{aligned}$$ In this case where the assumption on the adapted metric above is posed without the $1/2$ on the left hand side, all estimates in the theorem above can be divided by $2$. Let $B_{r_{k}}(x_{0})$, (respectively $B_{{\widetilde{ r}}_{k}}(x_{k})$) be a sequence of distance balls that realizes $\mu$ (respectively ${\widetilde{ \mu}}$), i.e., $\mu=\lim_{k\to\infty} r_{k}^{-1}\log m(B_{r_{k}}(x_{0}))$ (respectively ${\widetilde{ \mu}}=\lim_{k\to\infty} r_{k}^{-1}\log m(B_{{\widetilde{ r}}_{k}}(x_{k}))$). If the measure of $B_{r_{k}}(x_{0})$, (respectively $B_{{\widetilde{ r}}_{k}}(x_{k})$) is infinite for some $k$, then $\mu=\infty$ (respectively ${\widetilde{ \mu}}=\infty$) and we are done. Otherwise, $f_{r_{k},x_{0},{{\alpha}}},g_{r_{k},x_{0},{{\alpha}}} \in \ell^{2}(X,m)$ (respectively $f_{{\widetilde{ r}}_{k},x_{k},{{\alpha}}}$, $g_{{\widetilde{ r}}_{k},x_{k},{{\alpha}}}\in \ell^{2}(X,m)$) and $f_{r_{k},x_{0},{{\alpha}}}\in D^{\max}$ (respectively $f_{{\widetilde{ r}}_{k},x_{k},{{\alpha}}}\in D^{\max}$) by Lemma \[l:NL\]. Thus, the statement follows directly from Theorem \[t:jump\]. In the case when we know more about the measure or the metric structure we can say something about the operator $L$. This is the case under either of the following additional assumptions: - Every infinite path of vertices has infinite measure. - $\rho$ is any adapted path metric on a locally finite graph such that $(X, \rho)$ is metrically complete. In particular, (A) is satisfied if $\inf_{x\in X}m(x)>0$ and (B) is satisfied if all infinite geodesics have infinite length. \[c:graph\]Assume that either (A) or (B) is satisfied. Then, the statement of Theorem \[t:graph\] holds for $L=L^{\max}$. By [@KL1 Theorem 6], respectively [@HKMW Theorem 2], (A), respectively (B), imply that ${{\mathcal E}}={{\mathcal E}}^{\max}$ and $L=L^{\max}$. Under the slightly stronger assumption that connected infinite sets have infinite measure we can prove the corollary directly. Namely, if one of the relevant distance balls is infinite, then it has infinite measure and the exponential volume growth is infinite. In the other case the corollary follows from Theorem \[t:jump\]. We also recover the result of [@Fuj] which already covers [@DK; @OU]. In their very particular situation, $m$ is the vertex degree and $b$ takes values in $\{0,1\}$. The natural graph distance $d$ is given as the minimum length of a path of edges connecting two vertices where the length is the number of edges contained in the path. (Normalized Laplacians)\[c:normalized\] Let $b$ be a connected weighted graph over $(X,n)$, with $n(x)=\sum_{y\in X}b(x,y)$, $x\in X$ and let $d$ be the natural graph metric. Then, ${{\lambda}}_{0}^{\mathrm{ess}}(L)\leq 1- 2e^{{\widetilde{ \mu}}/2}/({1+e^{{\widetilde{ \mu}}}})$ and ${{\lambda}}_{0}(L)\leq 1-2e^{\mu/2}/({1+e^{\mu}})$. Clearly, $L$ is a bounded operator and thus $L=L^{\max}$. Moreover, the natural graph metric is an intrinsic metric for $2L$ and its jump size in exactly $1$. Thus, the statement follows from the previous theorem. Unweighted graphs and the natural graph distance {#s:graph2} ------------------------------------------------ Let $b:X\times X\to\{0,1\}$ and $m\equiv 1$. Then, the operator $L$ becomes the graph Laplacian $\Delta $ acting on $D(\Delta)=\{{{\varphi}}\in \ell^{2}(X)\mid (x\mapsto\sum_{y\sim x}({{\varphi}}(x)-{{\varphi}}(y)))\in\ell^{2}(X)\}$, see [@KL1; @Woj1], as $$\begin{aligned} \Delta{{\varphi}}(x)=\sum_{y\sim x}({{\varphi}}(x)-{{\varphi}}(y)),\end{aligned}$$ where $x\sim y$ means that $b(x,y)=1$. By $m\equiv 1$ we have that $m(A)=|A|$ for all $A\subseteq X$. For simplicity we assume that the graph is connected. \[t:graph2\] Let the $d$ be the natural graph distance on an infinite graph and $B_{r}^{d}=\{x\in X\mid d(x,x_{0})\leq r\}$ for some $x_{0}\in X$ and $r\ge0$. If $$\begin{aligned} \liminf_{r\to\infty}\frac{\log |B_{r}^{d}(x_{0})|}{\log r}<3,\end{aligned}$$ then, ${{\lambda}}_{0}(\Delta)={{\lambda}}_{0}^{\mathrm{ess}}(\Delta)=0$. Moreover, if $$\begin{aligned} \limsup_{r\to\infty}\frac{ |B_{r}^{d}(x_{0})|}{ r^{3}}<\infty,\end{aligned}$$ then ${{\lambda}}_{0}^{\mathrm{ess}}(\Delta)<\infty$ and, in particular, ${{\sigma}}_{\mathrm{ess}}(\Delta)\neq \emptyset$. \(a) The result above is sharp. This can be seen by the examples of antitrees discussed below the proof. \(b) In [@GHM Theorem 1.4] it is shown that less than cubic growth implies stochastic completeness. \(c) In the case where the vertex degree is bounded by some $K$, the situation is very different: the $n$ in Corollary \[c:normalized\] becomes $\deg$ in our situation, where $\deg:X\to{{\mathbb N}}$ is the function assigning to a vertex the number of adjacent vertices, and the corresponding normalized operator is ${\widetilde{ \Delta}}$ acting on $\ell^{2}(X,\deg)$ as ${\widetilde{ \Delta}}{{\varphi}}(x)=\frac{1}{\deg(x)}\sum_{y\sim x}({{\varphi}}(x)-{{\varphi}}(y))$. Then, $$\begin{aligned} {{\lambda}}_{0}({\widetilde{ \Delta}})\leq{{\lambda}}_{0}(\Delta)\leq K{{\lambda}}({\widetilde{ \Delta}})\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}({\widetilde{ \Delta}})\leq{{\lambda}}_{0}^{\mathrm{ess}}(\Delta)\leq K{{\lambda}}^{\mathrm{ess}}({\widetilde{ \Delta}}),\end{aligned}$$ see, e.g., [@K]. Thus, in the bounded situation, the threshold lies again at subexponential growth by Corollary \[c:normalized\] (as the measures $m\equiv 1$ and $n=\deg$ also give the same exponential volume growth.) Explicit estimates for the exponential volume growth of planar tessellations in terms of curvature can be found in [@KP]. \(d) In the case of bounded vertex degree we also have a threshold for recurrence of the corresponding random walk at quadratic volume growth, see [@Woe Lemma 3.12]. Let $\rho$ be the intrinsic metric from [@Hu] introduced above in Remark \[r:rho\] which, in the case of unweighted graphs, is given by $$\rho(x,y)=\inf\{ \sum_{i=0}^{n-1}\min\{\deg(x_{i})^{-\frac{1}{2}}, \deg(x_{i+1})^{-\frac{1}{2}}\}\mid (x_{0},\ldots, x_{n})\mbox{ is a path from $x$ to $y$}\}.$$ Let $ B_{r}^{\rho}=\{x\in X\mid \rho(x,x_{0})\leq r\}$, while $B_{r}^{d}$ are the balls with respect to the natural graph distance $d$. The proof of the theorem is based on the following lemma which is inspired by the proof of [@GHM Theorem 1.4]. Indeed, the second statement is taken directly from there. If $\liminf\limits_{r\to\infty}{\log |B_{r}^{d}|}/{\log r}=\beta\in[1,3)$, then $\liminf\limits_{r\to\infty}{\log|B_{r}^{\rho}|}/{\log r}\leq {\frac{2\beta}{3-\beta}}$. Moreover, if $\limsup\limits_{r\to\infty}{ |B_{r}^{d}|}/{ r^{3}}<\infty$, then $\limsup\limits_{r\to\infty}\frac{1}{r}\log|B_{r}^{\rho}|<\infty$. Let $S_{r}^{d}=B_{r}^{d}\setminus B_{r-1}^{d}$, $r\geq0$, and for convenience set $S_{-r}^{d}=B_{-r}^{d}=\emptyset$ for $r>0$. Let $1\leq{{\alpha}}<3$ and $(r_{k})$ be an increasing sequence such that ${\log |B_{r_{k}}^{d}(x_{0})|}/{\log r_{k}}<{{\alpha}}$ for all $k\ge0$. Then, $$\begin{aligned} |B_{r_{k}}^{d}|=\sum_{r=0}^{r_{k}}|S_{r}^{d}|< r^{{{\alpha}}}_{k}\end{aligned}$$ for large $k\ge0$. For ${{\varepsilon}}>0$ and $k\geq0$ set $$\begin{aligned} A_{k}:=\{r\in[0,r_{k}]\cap {{\mathbb N}}_{0}\mid |S_{r}^{d}|> \frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\}.\end{aligned}$$ We can estimate $|A_{k}|\leq{{\varepsilon}}r_{k}$ via $$\begin{aligned} r_{k}^{{{\alpha}}}>|B_{r_{k}}^{d}| \geq\frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}}\sum_{r\in A_{k}}r^{{{\alpha}}-1} \geq\frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}}\sum_{r=0}^{|A_{k}|}r^{{{\alpha}}-1} \geq\frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}}\int_{0}^{|A_{k}|}r^{{{\alpha}}-1}dr =\frac{|A_{k}|^{{{\alpha}}}}{{{\varepsilon}}^{{{\alpha}}}}.\end{aligned}$$ Thus, $$\begin{aligned} |\{r\in[1,r_{k}]\cap {{\mathbb N}}_{0}\mid \max_{i=0,1,2,3}|S_{r-i}^{d}|> \frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\}|\leq 4{{\varepsilon}}r_{k}\end{aligned}$$ and $$\begin{aligned} |\{r\in[1,r_{k}]\cap {{\mathbb N}}_{0}\mid \max_{i=0,1,2,3}|S_{r-i}^{d}|\leq \frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\}|\geq (1-4{{\varepsilon}})r_{k}.\end{aligned}$$ As we have $\deg \leq |S_{r-1}^{d}\cup S_{r}^{d}\cup S_{r+1}^{d}|$ on $S_{r}^{d}$, we get $|D_{k}|\geq (1-4{{\varepsilon}})r_{k}$, where $$\begin{aligned} D_{k}:=\{(r+1)\in[0,r_{k}-1]\cap {{\mathbb N}}_{0}\mid \deg\leq \frac{3{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\mbox{ on } S_{r-1}^{d}\cup S_{r}^{d}\}.\end{aligned}$$ Hence, for $(r+1)\in D_{k}$ we have for $x\in S_{r-1}^{d}$, $y\in S_{r}^{d}$ $$\rho(x,y)\geq cr^{-\frac{{{\alpha}}-1}{2}},\quad\mbox{ with } c=\sqrt{{{{\varepsilon}}^{{{\alpha}}}}/{3{{\alpha}}}}.$$ Since any path from $x_{0}$ to $S_{r_{k}}^{d}$ contains such edges we have for any $x\in S_{r_{k}}^{d}$ $$\begin{aligned} \rho(x_{0},x)\geq c\sum_{(r+1)\in D_{k}}r^{-\frac{{{\alpha}}-1}{2}} \geq c \sum_{r=4{{\varepsilon}}r_{k}}^{r_{k}-1}r^{-\frac{{{\alpha}}-1}{2}} \geq c\int_{4{{\varepsilon}}r_{k}} ^{r_{k}-1}r^{-\frac{{{\alpha}}-1}{2}} dr\geq C_{0} r_{k}^{{\frac{3-{{\alpha}}}{2}}}\end{aligned}$$ with $C_{0}>0$ for ${{\varepsilon}}>0$ chosen sufficiently small and $r_{k}$ large. Let $R_{k}:=C_{0} r_{k}^{{\frac{3-{{\alpha}}}{2}}}$ and $C:=C_{0}^{-\frac{2{{\alpha}}}{3-{{\alpha}}}}$. Then, $B_{R_{k}}^{\rho}\subseteq B_{r_{k}}^{d}$ and since $|B_{r_{k}}^{d}|=\sum_{r=0}^{r_{k}}|S_{r}^{d}|< r^{{{\alpha}}}_{k}$, we conclude $$\begin{aligned} |B_{R_{k}}^{\rho}|\leq |B_{r_{k}}^{d}|<r^{{{\alpha}}}_{k}\leq CR_{k}^{\frac{2{{\alpha}}}{3-{{\alpha}}}}.\end{aligned}$$ Thus, the first statement follows. The second statement is shown in the proof of [@GHM Theorem 1.4]. In the case where the polynomial growth is strictly less than cubic we get by the lemma above that $\mu=0$ with respect to the intrinsic metric $\rho$ and in the case where it is less than cubic we still have $\mu<\infty$. Thus, the statement follows from Corollary \[c:graph\], where (A) is clearly satisfied as $m\equiv 1$. Let us discuss the example of antitrees which show the sharpness of the result. They were first introduced in [@Woj3] and further studied in [@BK; @KLW]. An antitree is a spherically symmetric graph, where a vertex in the $r$-th sphere is connected to all vertices in the $(r+1)$-th sphere for $r\ge0$, and there are no horizontal edges. Thus, an antitree is characterized by a sequence $(s_{r})$ taking values in ${{\mathbb N}}$ which encodes the number of vertices in the sphere $S_{r}^{d}=B_{r}^{d}\setminus B_{r-1}^{d}$. *Stronger growth than cubic:* In [@KLW Corollary 6.6] it is shown that if the polynomial volume growth of an antitree is more than cubic, i.e., as $r^{3+{{\varepsilon}}}$ for ${{\varepsilon}}>0$, then ${{\lambda}}_{0}(\Delta)>0$ and ${{\sigma}}_{\mathrm{ess}}(\Delta)=\emptyset$. Indeed, in the intrinsic metric $\rho$, these antitrees have finite diameter and thus $\mu=\infty$, see [@Hu]. *Cubic growth:* If the distance spheres of an antitree satisfy $|S_{r}^{d}|=(r+1)^{2}$, then $|B_{r}^{d}|\sim (r+1)^{3}$. Moreover, the function which takes the value $r^{-2}$ on vertices of the $(r-1)$-th sphere, $r\geq1$, is a positive generalized super-solution for $\Delta$ to the value $2$, that is, $\Delta {{\varphi}}\geq 2 {{\varphi}}$. Thus, by a discrete Allegretto-Piepenbrink theorem (see [@Woj2 Theorem 4.1] or [@HK Theorem 3.1]) it follows that ${{\lambda}}_{0}(\Delta)\geq2$. By Theorem \[t:graph2\] we thus have $2\leq {{\lambda}}_{0}^{\mathrm{ess}}(\Delta)<\infty$. *Weaker growth than cubic:* In this case Theorem \[t:graph2\] shows that ${{\lambda}}_{0}(\Delta)={{\lambda}}_{0}^{\mathrm{ess}}(\Delta)=0$. **Acknowledgements.** The authors are grateful to J[ó]{}zef Dodziuk and Daniel Lenz for their continued support and for generously sharing their knowledge. The research of RKW was partially sponsored by the Fundação para a Ciência e a Tecnologia through project PTDC/MAT/101007/2008 and by the Research Foundation of CUNY through the PSC-CUNY Research Award 42. [10]{} J. Breuer, M. 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--- author: - Ulrich Langer - 'Stephen E. Moore' bibliography: - 'DGIGASurface\_MooreLanger.bib' title: Discontinuous Galerkin Isogeometric Analysis of Elliptic PDEs on Surfaces --- Introduction {#sec1:Introduction} ============ The Isogeometric Analysis (IGA) was introduced by [@HughesCottrellBazilevs:2005a] and has since been developed intensively, see also monograph [@CottrellHughesBazilevs:2009a], is a very suitable framework for representing and discretizing Partial Differential Equations (PDEs) on surfaces. We refer the reader to the survey paper by [@DziukElliot:2013a] where different finite element approaches to the numerical solution of PDEs on surfaces are discussed. Very recently, [@DednerMadhavanStinner:2013a] have used and analyzed the Discontinuous Galerkin (DG) finite element method for solving elliptic problems on surfaces. The IGA of second-order PDEs on surfaces that avoid errors arising from the approximation of the surface, has been introduced and numerically studied by [@DedeQuarteroni:2012a]. [@Brunero:2012a] presented some discretization error analysis of the DG-IGA applied to plane (2d) diffusion problems that carries over to plane linear elasticity problems which have recently been studied numerically in [@ApostolatosSchmidtWuencherBletzinger:2013a]. The efficient generation of the IGA equations, their fast solution, and the implementation of adaptive IGA schemes are currently hot research topics. The use of DG technologies will certainly facilitate the handling of the multi-patch case. In this paper, we use the DG method to handle the IGA of diffusion problems on closed or open, multi-patch NURBS surfaces. The DG technology easily allows us to handle non-homogeneous Dirichlet boundary condition as in the Nitsche method and the multi-patch NURBS spaces which can be discontinuous across the patch boundaries. We also derive discretization error estimates in the DG- and $L_{2}$-norms. Finally, we present some numerical results confirming our theoretical estimates. Surface Diffusion Model Problem {#sec2:SurfaceDiffusionModelProblem} =============================== Let us assume that the physical (computational) domain $\Omega$, where we are going to solve our diffusion problem, is a sufficiently smooth, two-dimensional generic (Riemannian) manifold (surface) defined in the physical space $\mathbb{R}^{3}$ by means of a smooth multi-patch NURBS mapping that is defined as follows. Let $\mathcal{T}_{H}= \{\Omega^{(i)}\}_{i=1}^{N}$ be a partition of our physical computational domain $\Omega$ into non-overlapping patches (subdomains) $\Omega^{(i)}$ such that $\overline{\Omega}= \bigcup_{i=1}^{N} \overline{\Omega}^{(i)} $ and $\Omega^{(i)} \cap \Omega^{(j)}= \emptyset $ for $i \neq j$, and let each patch $\Omega^{(i)}$ be the image of the parameter domain $\widehat{\Omega} = (0,1)^2 \subset \mathbb{R}^{2}$ by some NURBS mapping $G^{(i)} : \widehat{\Omega} \rightarrow \Omega^{(i)} \subset \mathbb{R}^{3}, \mathbf{\xi} = (\mathbf{\xi}_1,\mathbf{\xi}_2) \mapsto \mathbf{x} = (\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3)=G^{(i)}(\mathbf{\xi})$, which can be represented in the form $$\label{sec2:GeometricalMappingRepresentation} G^{(i)}(\xi_{1},\xi_{2}) = \sum_{k_{1}=1}^{n_{1}} \sum_{k_{2}=1}^{n_{2}} \mathbf{P}^{(i)}_{(k_{1},k_{2})} \widehat{R}^{(i)}_{(k_{1},k_{2})}(\xi_{1},\xi_{2})$$ where $\{ \widehat{R}^{(i)}_{(k_{1},k_{2})} \}$ are the bivariate NURBS basis functions, and $\{\mathbf{P}^{(i)}_{(k_{1},k_{2})} \}$ are the control points, see [@CottrellHughesBazilevs:2009a] for a detailed description. Let us now consider a diffusion problem on the surface $\Omega$ the weak formulation of which can be written as follows: find $u \in V_g$ such that $$\label{sec2:VariationalFormulation} a(u,v) = \langle F,v \rangle \quad \forall v \in V_0,$$ with the bilinear and linear forms are given by the relations $$a(u,v) = \int_\Omega \alpha \, \nabla_\Omega u \cdot \nabla_\Omega v \, d \Omega \quad \mbox{and} \quad \langle F,v \rangle = \int_\Omega f v \, d \Omega + \int_{\Gamma_N} g_N v \,d \Gamma,$$ respectively, where $\nabla_\Omega$ denotes the so-called tangential or surface gradient, see e.g. Definition 2.3 in [@DziukElliot:2013a] for its precise description. The hyperplane $V_g$ and the test space $V_0$ are given by $V_g=\{v \in V = H^1(\Omega): v=g_D \;\mbox{on}\; \Gamma_D\}$ and $V_0=\{v \in V: v=0 \;\mbox{on}\; \Gamma_D\}$ for the case of an open surface $\Omega$ with the boundary $\Gamma = \overline{\Gamma}_D \cup \overline{\Gamma}_N$ such that $\mbox{meas}_1(\Gamma_D) > 0$, whereas $V_g=V_0=\{v \in V: \int_\Omega v \, d \Omega =0\}$ in the case of a pure Neumann problem ($\Gamma_N = \Gamma$) as well as in the case of closed surfaces unless there is a reaction term. In case of closed surfaces there is of course no integral over $\Gamma_N$ in the linear functional on the right-hand side of (\[sec2:VariationalFormulation\]). In the remainder of the paper, we will mainly discuss the case of mixed boundary value problems on an open surface under appropriate assumptions (e.g., $\mbox{meas}_1(\Gamma_D) > 0$, $\alpha$ - uniformly positive and bounded, $f\in L_2(\Omega)$, $g_D \in H^{\frac{1}{2}}(\Gamma_{D})$ and $g_{N} \in L_{2}(\Gamma_{N})$ ) ensuring existence and uniqueness of the solution of (\[sec2:VariationalFormulation\]). For simplicity, we assume that the diffusion coefficient $\alpha$ is patch-wise constant, i.e. $\alpha = \alpha_i$ on $\Omega^{(i)}$ for $i=1,2,\ldots,N$. The other cases including the reaction-diffusion case can be treated in the same way and yield the same results like presented below. DG-IGA Schemes and their Properties {#sec3:DGIGASchemesAndProperties} =================================== The DG-IGA variational identity $$\label{sec3:DG-VariationalIdentity} a_{DG}(u,v) = \langle F_{DG},v \rangle \quad \forall v \in \mathcal{V} = H^{1+s}(\mathcal{T}_{H}),$$ which corresponds to (\[sec2:VariationalFormulation\]), can be derived in the same way as their FE counterpart, where $H^{1+s}(\mathcal{T}_{H}) =\{v \in L_{2}(\Omega): v|_{\Omega^{(i)}} \in H^{1+s}(\Omega^{(i)}), \; \forall \, i = 1,\ldots,N\}$ with some $s > 1/2$. The DG bilinear and linear forms in the *Symmetric Interior Penalty Galerkin* (SIPG) version, that is considered throughout this paper for definiteness, are defined by the relationships $$\begin{aligned} \label{sec3:DG-BilinearForm} \nonumber a_{DG}(u,v) &=&\sum_{i=1}^{N} \int_{\Omega^{(i)}} \alpha_{i} \nabla_{\Omega} u \cdot \nabla_{\Omega} v \, d\Omega\\ \nonumber && -\sum_{\gamma \in \mathcal{E}_{I} \cup \mathcal{E}_{D}} \int_{\gamma} \left( \{ \alpha \nabla_{\Omega} u \cdot \mathbf{n}\} [v] + \{\alpha \nabla_{\Omega} v \cdot \mathbf{n}\} [u] \right)\,d\Gamma\\ && + \sum_{ \gamma \in \mathcal{E}_{I} \cup \mathcal{E}_{D}} \frac{\delta}{ h_{\gamma} } \int_{\gamma} \alpha_{\gamma} [u][v]\,d\Gamma\end{aligned}$$ and $$\begin{aligned} \label{sec3:DG-LinearForm} \nonumber \langle F_{DG},v \rangle &=& \int_{\Omega} f v d\,\Omega + \sum_{\gamma \in \mathcal{E}_{N}} \int_{\gamma} g_{N}v\, d\Gamma\\ && + \sum_{\gamma \in \mathcal{E}_{D}} \int_{\gamma} \alpha_{\gamma} \left( - \nabla_{\Omega} v \cdot \mathbf{n} + \frac{\delta}{ h_{\gamma} } v \right) g_{D}\,d\Gamma,\end{aligned}$$ respectively, where the usual DG notations for the averages $\{ \cdot \}$ and jumps $[\cdot ]$ are used, see, e.g., [@Riviere:2008a]. The sets $\mathcal{E}_{I}$, $\mathcal{E}_{D}$ and $\mathcal{E}_{N}$ denote the sets of edges $\gamma$ of the patches belonging to $\Gamma_I = \cup \,\partial \Omega^{(i)} \setminus \{\Gamma_D \cup \Gamma_N\}$, $\Gamma_D$ and $\Gamma_N$, respectively whereas $h_{\gamma}$ is the mesh-size on $\gamma$. The penalty parameter $\delta$ must be chosen such that the ellipticity of the DG bilinear on $\mathcal{V}_{h}$ can be ensured. The relationship between our model problem (\[sec2:VariationalFormulation\]) and the DG variational identity (\[sec3:DG-VariationalIdentity\]) is given by the consistency theorem that can easily be verified. \[Thm:Sec3:Consistency\] If the solution $u$ of the variational problem (\[sec2:VariationalFormulation\]) belongs to $V_g \cap H^{1+s}(\mathcal{T}_{H })$ with some $s > 1/2$, then $u$ satisfies the DG variational identity (\[sec3:DG-VariationalIdentity\]). Conversely, if $u \in H^{1+s}(\mathcal{T}_{H})$ satisfies (\[sec3:DG-VariationalIdentity\]), then $u$ is the solution of our original variational problem (\[sec2:VariationalFormulation\]). Now we consider the finite-dimensional Multi-Patch NURBS subspace $$\mathcal{V}_{h}= \{v \in L_{2}(\Omega): \; v|_{\Omega^{(i)}}\in V^{i}_{h}(\Omega^{(i)}), \; i= 1,\ldots,N \}$$ of our DG space $\mathcal{V}$, where $ V^{i}_{h}(\Omega^{(i)}) = \text{span}\{R_{\textbf{k}}^{(i)} \} $ denotes the space of NURBS functions on each single-patch $ \Omega^{(i)}, \; i= 1,\ldots,N$, and the NURBS basis functions $ R_{\textbf{k}}^{(i)} = \widehat{R}^{(i)}_{\textbf{k}} \circ G^{(i)^{-1}} $ are given by the push-forward of the NURBS functions $\widehat{R}^{(i)}_{\textbf{k}}$ to their corresponding physical sub-domains $ \Omega^{(i)}$ on the surface $\Omega$. Finally, the DG scheme for our model problem reads as follows: find $u_{h} \in \mathcal{V}_{h}$ such that $$\label{sec3:DiscreteDGVariationalFormulation} a_{DG}(u_{h},v_{h}) = \langle F_{DG},v_{h} \rangle , \quad \forall v_{h} \in \mathcal{V}_{h}.$$ For simplicity of our analysis, we assume matching meshes in the IGA sense, where the discretization parameter $h_i$ characterizes the mesh-size in the patch $\Omega^{(i)}$ whereas $p$ always denotes the underlying polynomial degree of the NURBS. Using special trace and inverse inequalities in the NURBS spaces $\mathcal{V}_{h}$ and Young’s inequality, for sufficiently large DG penalty parameter $\delta$, we can easily establish $\mathcal{V}_{h}$ coercivity and boundedness of the DG bilinear form with respect to the DG energy norm $$\label{sec3:DG-Norm} \|v \|^{2}_{DG} = \sum_{i=1}^{N} \alpha_{i} \|\nabla_{\Omega} v_{i} \|_{L^{2}(\Omega^{i})}^{2} + \sum_{ \gamma \in \mathcal{E}_{I} \cup \mathcal{E}_{D}} \alpha_{\gamma} \frac{\delta}{h_{\gamma}} \| [v] \|_{L^{2}(\gamma)}^{2},$$ yielding existence and uniqueness of the DG solution $u_{h} \in \mathcal{V}_{h}$ of (\[sec3:DiscreteDGVariationalFormulation\]) that can be determined by the solution of a linear system of algebraic equations. Discretization Error Estimates {#sec4:DiscretizationErrorEstimates} ============================== \[sec4:DG-Norm-ErrorEstimate\] Let $u \in V_g \cap H^{1+s}(\mathcal{T}_{H})$ with some $s > 1/2$ be the solution of (\[sec2:VariationalFormulation\]), $u_{h} \in \mathcal{V}_{h}$ be the solution of (\[sec3:DiscreteDGVariationalFormulation\]), and the penalty parameter $\delta$ be chosen large enough . Then there exists a positive constant $c$ that is independent of $u$, the discretization parameters and the jumps in the diffusion coefficients such that the DG-norm error estimate $$\label{sec4:DG-normError Estimate} \|u-u_{h} \|_{DG}^{2} \leq c \left(\sum_{i=1}^{N} \alpha_{i} h_{i}^{2t}\|u\|^{2}_{H^{1+t}(\Omega^{(i)})}\right)^{1/2},$$ holds with $t:= \min\{s,p\} $. Let us give a sketch of the proof. By the triangle inequality, we have $$\label{sec4:DGTriangleInequality} \|u-u_{h} \|_{DG} \leq \|u- \Pi_{h} u\|_{DG} + \|\Pi_{h} u - u_{h}\|_{DG}$$ with some quasi-interpolation operator $\Pi_{h}: \mathcal{V} \mapsto \mathcal{V}_{h}$ such that the first term can be estimated with optimal order, i.e. by the term on the right-hand side of (\[sec4:DG-normError Estimate\]) with some other constant $c$. This is possible due to the approximation results known for NURBS, see, e.g., [@BazilevsBeiraoCottrellHughesSangalli:2006a] and [@CottrellHughesBazilevs:2009a]. Now it remains to estimate the second term in the same way. Using the Galerkin orthogonality $a_{DG}(u-u_h,v_h)=0$ for all $v_h \in \mathcal{V}_{h}$, the $\mathcal{V}_{h}$ coercitivity of the bilinear form $a_{DG}(\cdot,\cdot)$, the scaled trace inequality $$\label{sec4:EpsilonTraceInequality} \|v \|_{L^{2}(e)} \leq C h^{-1/2}_{E} \left( \|v \|_{L^{2}(E)} + h^{1/2+\epsilon}_{E} |v|_{H^{1/2+\epsilon}(E)} \right),$$ that holds for all $v \in H^{1/2+\epsilon}(E)$, for all IGA mesh elements $E$, for all edges $e \subset \partial E$, and for $\epsilon > 0$, where $ h_{E}$ denotes the mesh-size of $E$ or the length of $e$, Young’s inequality, and again the approximation properties of the quasi-interpolation operator $\Pi_{h}$, we can estimate the second term by the same term $ c \left(\sum_{i=1}^{N} \alpha_{i} h_{i}^{2t}\|u\|^{2}_{H^{1+t}(\Omega^{(i)})}\right)^{1/2}$ with some (other) constant $c$. This completes the proof of the theorem. Using duality arguments, we can also derive $L_2$-norm error estimates that depend on the elliptic regularity. Under the assumption of full elliptic regularity, we get $\|u-u_{h} \|_{L_2(\Omega)} \le c\, h^{p+1} \|u\|_{H^{p+1}(\Omega)}$ that is nicely confirmed by our numerical experiments presented in the next section for $p=1,2,3,4$. Numerical Results {#sec5:NumericalResults} ================= The DG IGA method presented in this paper as well as its continuous Galerkin counterpart have been implemented in the object oriented C++ IGA library ”Geometry + Simulation Modules” (G+SMO) [^1]. We present some first numerical results for testing the numerical behavior of the discretization error with respect to the mesh parameter $h$ and the polynomial degree $p$ Concerning the choice of the penalty parameter, we used $\delta = 2(p+2)(p+1).$ As a first example, we consider a non-homogeneous Dirichlet problem for the Poisson equation in the 2d computational domain $\Omega \subset \mathbb{R}^2$ called Yeti’s footprint, see also [@KleissPechsteinJuttlerTomar:2012a], where the right-hand side $f$ and the Dirichlet data $g_D$ are chosen such that $u(x_1,x_2) = \sin(\pi x_1)\sin(\pi x_2)$ is the solution of the boundary value problem. The computational domain (left) and the solution (right) can be seen in Fig. \[sec5:fig1:YetiFoot\]. The Yeti footprint consists of 21 patches with varying open knot vectors $\Xi$ describing the NURBS discretization in a short and precise way, see, e.g., [@CottrellHughesBazilevs:2009a] for a detailed definition. The knot vector for patches 1 to 16 and 21 is given as $\Xi = (0,\ldots,0,0.5,1,\ldots,1)$ in both directions whereas the knot vectors for the patches 17 to 20 are given as $\Xi_1 = ( 0,\ldots,0,0.5,1,\ldots,1)$ and $\Xi_2 = ( 0,\ldots,0,0.25,0.5,0.75,1,\ldots,1).$ In Fig. \[sec5:fig2:YetiFootL2Error\] and \[sec5:fig3:YetiFootDGError\], the errors in the $L_2$-norm and in the DG energy norm (\[sec3:DG-Norm\]) are plotted against the degree of freedom (DOFs) with polynomial degrees from 1 to 4. It can be observed that we have convergence rates of $\mathcal{O}(h^{p+1})$ and $\mathcal{O}(h^{p})$ respectively. This corresponds to our theory in Section \[sec4:DiscretizationErrorEstimates\]. In the second example, we apply the DG-IGA to the same Laplace-Beltrami problem on an open surface as described in [@DedeQuarteroni:2012a], section 5.1, where $\Omega$ is a quarter cylinder represented by four patches in our computations, see Fig. \[sec5:fig4:MultiPatchCylinder\] (left). The $L_{2}-$norm errors plotted on the right side of Fig. \[sec5:fig4:MultiPatchCylinder\] exhibit the same numerical behavior as in the plane case of the Yeti foot. The same is true for the DG-norm. Conclusions {#sec6:Conclusion} =========== We have developed and analyzed a new method for the numerical approximation of diffusion problems on open and closed surfaces by combining the discontinuous Galerkin technique with isogeometric analysis. We refer to our approach as the Discontinuous Galerkin Isogeometric Analysis (DG-IGA). In our DG approach we allow discontinuities only across the boundaries of the patches, into which the computational domain is decomposed, and enforce the interface conditions in the DG framework. For simplicity of presentation, we assume that the meshes are matching across the patches, and the solution $u$ is at least patch-wise in $H^{1+s}$, i.e. $u \in H^{1+s}(\mathcal{T}_{H})$, with some $s > 1/2$. The cases of non-matching meshes and low-regularity solution, that are technically more involved and that were investigated, e.g., by [@Dryja:2003a] and [@DiPietroErn:2012a], will be considered in a forthcoming paper. The parallel solution of the DG-IGA equations can efficiently be performed by Domain Decomposition (DD) solvers like the IETI technique proposed by [@KleissPechsteinJuttlerTomar:2012a], see also [@ApostolatosSchmidtWuencherBletzinger:2013a] for other DD solvers. The construction and analysis of efficient solution strategies is currently a hot research topic since, beside efficient generation techniques, the solvers are the efficiency bottleneck in large-scale IGA computations. Acknowledgement {#acknowledgement .unnumbered} =============== The authors gratefully acknowledge the financial support of the research project NFN S117-03 by the Austrian Science Fund (FWF). Furthermore, the authors want to thank their colleagues Angelos Mantzaflaris, Satyendra Tomar, Ioannis Toulopoulos and Walter Zulehner for fruitful and enlighting discussions as well as for their help in the implementation in GISMO. [^1]: G+SMO : https://ricamsvn.ricam.oeaw.ac.at/trac/gismo

--- abstract: 'The response of ultracold atomic Bose gases in time-dependent optical lattices is discussed based on direct simulations of the time-evolution of the many-body state in the framework of the Bose-Hubbard model. We focus on small-amplitude modulations of the lattice potential as implemented in several recent experiment and study different observables in the region of the first resonance in the Mott-insulator phase. In addition to the energy transfer we investigate the quasimomentum structure of the system which is accessible via the matter-wave interference pattern after a prompt release. We identify characteristic correlations between the excitation frequency and the quasimomentum distribution and study their structure in the presence of a superlattice potential.' author: - Markus Hild - Felix Schmitt - Ilona Türschmann - Robert Roth title: 'Ultracold Bose gases in time-dependent 1D superlattices: response and quasimomentum structure' --- Ultracold atomic gases in optical lattices are a versatile laboratory for the study of fundamental quantum phenomena. The accurate control of the important physical parameters over a wide range has been utilized for detailed experimental investigations of quantum phase transitions, e.g. the superfluid to Mott insulator phase transition (SF-MI) [@JaBr98; @GrMa02a; @GrMa02b]. The primary experimental observable is the matter-wave interference pattern of the atoms after release from the confining potentials and ballistic expansion. The interference pattern obtained after a sudden release of the atoms provides direct information on the (quasi)momentum distribution of the system before the release. The interference pattern thus helps with the identification of different quantum phases, such as the SF and MI phases [@GrMa02a; @GrMa02b]. In recent experiments it was also used to study the response of the system in a two-photon Bragg-spectroscopy scheme based on a temporal modulation of the lattice potential. The broadening of the central interference peak after re-thermalization in a shallow lattice was used as a measure for the response [@StMo04; @FaLy07]. These techniques can also be employed in the context of more complicated irregular lattices as they can be produced, e.g., by using speckle patterns [@LyFa05; @ClVa05; @FoFa05; @ScDr05] or two-color superlattice potentials [@RoBu03b; @RoBu03c; @FaLy07]. In pioneering experiments on the response in two-color superlattices the impact of a superlattice has been investigated [@FaLy07]. In agreement with theoretical predictions a broadening of the response as function of modulation frequency was observed with increasing superlattice amplitude. Motivated by these experiments we study the response of a bosonic system to the modulation of the superlattice potential deep in the Mott-insulating regime. Going beyond the observables used in experiments, we study the quasimomentum structure of the system as it is revealed by the interference pattern after a prompt release of the atoms without any re-thermalization phase. To this end we perform an explicit time-evolution of the many-body state in the presence of a time-dependent lattice based on the Bose-Hubbard Hamiltonian [@HiSc06]. Similar studies have been done using the time-dependent density matrix renormalization group method (tDMRG) [@KoIu06]. Our simulations reveal subtle correlations between the modulation frequency within the resonance region and the quasimomentum distribution, which should be accessible to experiment. A system of $N$ bosons in an one-dimensional superlattice potential with $I$ sites at zero temperature is well described by the single-band Bose-Hubbard model (BHM) [@JaBr98]. The Bose-Hubbard Hamiltonian, formulated in second quantization in the basis of localized Wannier states of the lowest bands, reads $$\begin{aligned} \label{eq_sec2_hamiltonian} {{\hat{\mathrm{H}}}}_0= -J_0\!\sum_{\langle{}i,j\rangle}{{\hat{\mathrm{a}}}^{\dagger}}_i{{\hat{\mathrm{a}}}^{\phantom{\dagger}}}_j +\frac{U_0}{2}\!\sum_{i}{\hat{\mathrm{n}}}_i({\hat{\mathrm{n}}}_i\!-\!1) +\Delta_0\!\!\sum_{i}\epsilon_i{\hat{\mathrm{n}}}_i,\end{aligned}$$ where the first sum runs over adjacent sites including a term connecting the first and last site of the lattice (cyclic boundary conditions). The Hamiltonian with the bosonic creation (annihilation) operators ${{\hat{\mathrm{a}}}^{\dagger}}_i$ (${{\hat{\mathrm{a}}}^{\phantom{\dagger}}}_i$) and the occupation number operator ${\hat{\mathrm{n}}}_i$ consists of three terms: The first term describes the tunneling between adjacent sites, the second term accounts for the on-site interaction of the atoms, and the third term represents a site-dependent external potential. We introduce the superlattice potential via the latter term and describe its spatial structure by the reduced on-site energies $\epsilon_i \in [-1,0]$. The physics of the BHM is governed by the competition between the relative strengths of these three terms, i.e. the tunneling strength $J_0$, the interaction strength $U_0$, and the superlattice amplitude $\Delta_0$. The basis of the BHM for $N$ bosons and $I$ sites is spanned by the occupation number states ${\,\big|{\{n_1,n_2,\dots,n_I\}_{\alpha}}\big> }$ for all possible sets of occupation numbers $n_i$ with $\sum_{i} n_i = N$. An arbitrary state can be expanded in this number-state basis leading to ${\,\big|{\psi}\big> }=\sum_{\alpha=1}^D C_{\alpha}{\,\big|{\{n_1,n_2,\dots,n_I\}_{\alpha}}\big> }\label{eq_sec2_state}$ with complex coefficients $C_{\alpha}$. The coefficients of the eigenstates ${\,\big|{\nu}\big> }$ can be obtained by the numerical solution of the eigenvalue problem of the Hamilton matrix. {width="80.00000%"} The basis dimension $D$ grows factorially with $I$ and $N$, thus limiting any calculation using the full basis to small systems. However, in the strongly interacting regime ($U_0\gg{}J_0$) only a few basis states contribute to the low-lying eigenstates. This allows for a physically motivated truncation [@HiSc06; @ScHi07] of the many-body basis. The relevant basis states are identified using the expectation value of the Hamiltonian ${\big<{\{n_1,n_2,\dots,n_I\}}\big|\,{{{\hat{\mathrm{H}}}}_0}\big|\,{\{n_1,n_2,\dots,n_I\}}\big> } \leq E_\text{trunc}$, and only states below the truncation energy $E_\text{trunc}$ are included. By varying $E_\text{trunc}$ one can explicitly assess and control the impact of the truncation on observables. For regular lattices ($\Delta_0\!=\!0$) and filling factor $N/I\!=\!1$ this truncation allows for all relevant $n$-particle–$n$-hole states with $n \leq E_\text{trunc}/U_0$ with respect to the reference state ${\,\big|{1,1,\dots,1}\big> }$. We investigate the dynamics and response of the system induced by external time-dependent perturbations based on the explicit time evolution of the many-body state. Our calculations are motivated by recent experiments [@StMo04; @FaLy07] using a sinusoidal modulation of the lattice depth to perform two-photon Bragg spectroscopy. Unlike our calculations, these experiments include a rethermalization phase in the superfluid regime before the time-of-flight measurement in order to assess the energy transfer to the system. We assume a prompt release without rethermalization to directly access the quasimomentum distribution after a certain modulation time. Formally, the time-dependent lattice potential is written as $V(x,t)=[1+F\sin(\omega t)]\,V(x)$, where $V(x)$ is the static spatial lattice, $\omega$ the frequency, and $F$ the relative amplitude of the modulation. All simulations are performed with a small amplitude $F=0.1$ in accord with experiment. The time-dependence enters the Bose-Hubbard Hamiltonian ${{\hat{\mathrm{H}}}}(t)$ via the time-dependent parameters $J(t)$, $U(t)$, and $\Delta(t)$, which are obtained within a Gaussian approximation for the localized Wannier functions [@KBM05; @HiSc06]. The parameters oscillate around their initial values $J_0$, $U_0$, and $\Delta_0$ at $t\!=\!0$. We investigate the response of the system deep inside the Mott-insulating regime for fixed $U_0/J_0=40$. The modulation frequency $\omega$ is varied in the range $\omega/J_0=32$ to $52$, which covers the so-called 1U resonance at $\omega/J_0\approx U_0/J_0 = 40$. The ground state obtained for the initial Bose-Hubbard Hamiltonian ${{\hat{\mathrm{H}}}}_0$ is used as initial state ${\,\big|{\psi,0}\big> }$ and evolved up to $t J_0=10$ for each frequency $\omega$. The exact time-evolution is performed using a Crank-Nicholson scheme [@ScGa00; @ScGa04] with typically $3000$ time steps. The relevant observables are evaluated at every 30th step. Following an initial phase with large changes, the observables saturate [@KoIu06] and show only minor fluctuations [@HiSc06]. Since this residual time-dependence will not be resolved in experiment, we average the observables over an interval of evolution times within the saturated regime from $t J_0=6$ to $10$ (time-averaged quantities indicated by a bar). The simplest theoretical quantity to characterize the response of the system to the lattice modulation is the energy transfer $\Delta E(t)\!=\!{\big<{\psi,t}\big|\,{\!{{\hat{\mathrm{H}}}}_0}\big|\,{\psi,t}\big> }\!-\!{\big<{\psi,0}\big|\,{\!{{\hat{\mathrm{H}}}}_0}\big|\,{\psi,0}\big> }$. As an example, Fig. \[fig:etrans\_int\](a) shows the energy transfer for a system with $I=N=10$ in the Mott-insulator phase ($U_0/J_0=40$) for a regular optical lattice ($\Delta_0=0$) in the vicinity of the 1U resonance at the modulation frequency $\omega=U_0$. A detailed analysis of the energy transfer is given in Ref. [@HiSc06]. Additional information can be obtained by a linear response analysis [@ClJa06; @IuCa06; @HiSc06]. To this end, the Hamiltonian is linearized with respect to the modulation amplitude $F$, leading to ${{\hat{\mathrm{H}}}}_{\text{lin}}(t) = {{\hat{\mathrm{H}}}}_0+FV_0\sin(\omega t) [\mu{{\hat{\mathrm{H}}}}_0-\kappa{{\hat{\mathrm{H}}}}_J ]$ with the amplitude $V_0$ of the lattice potential and the couplings $\mu$ and $\kappa$. The first part of the linear term is irrelevant, because it only generates an energy shift. The second part couples the ground state to excited states via the tunneling operator ${{\hat{\mathrm{H}}}}_J=-J\sum_{\langle i,j\rangle}{{\hat{\mathrm{a}}}^{\dagger}}_i{{\hat{\mathrm{a}}}^{\phantom{\dagger}}}_j$. In a linear response picture, resonant transitions from the ground state ${\,\big|{0}\big> }$ are expected whenever the frequency $\omega$ coincides with the energy $E_{\nu}$ of an excited state ${\,\big|{\nu}\big> }$ with a sizable matrix element ${\big<{0}\big|\,{{{\hat{\mathrm{H}}}}_J}\big|\,{\nu}\big> }$. This interrelation is illustrated in Fig. \[fig:etrans\_int\](c), where the energy eigenvalues for the first Hubbard band are shown. The vertical lines indicate excited states with sizable matrix elements ${\big<{0}\big|\,{{{\hat{\mathrm{H}}}}_J}\big|\,{\nu}\big> }$. According to the linear response analysis, the energies of those eigenstates should correspond to the resonance frequencies in the response. The comparison of the resonance energies emerging in the energy transfer in Fig. \[fig:etrans\_int\](a) with these excitation energies in \[fig:etrans\_int\](c) confirms this interpretation. In addition to the energy transfer we investigate the evolution of the interference pattern. The matter-wave interference pattern after ballistic expansion is used to gain experimental information on the many-body state. In recent experiments [@StMo04; @FaLy07] the central peak of the interference pattern was employed as a measure for the response of the gas to a lattice modulation. The intensity distribution $\mathcal{I}(\delta)$ of the interference pattern as function of the relative phase $\delta$ is given by $\mathcal{I}(\delta)=\frac{1}{I}\sum_{i,j}\exp[{\mathrm{i}}(i\!-\!j)\delta]\;{\big<{\psi}\big|\,{{{\hat{\mathrm{a}}}^{\dagger}}_i{{\hat{\mathrm{a}}}^{\phantom{\dagger}}}_j}\big|\,{\psi}\big> }$ for an arbitrary state ${\,\big|{\psi}\big> }$ [@RoBu03a]. For $\delta=\!2\pi q/I$ the intensity $\mathcal{I}(\delta)$ corresponds to the occupation numbers $n_q\!=\!{\big<{\psi}\big|\,{{{\hat{\mathrm{c}}}^{\dagger}}_q{{\hat{\mathrm{c}}}^{\phantom{\dagger}}}_q}\big|\,{\psi}\big> }\!=\!\mathcal{I}(\delta\!=\!2\pi q/I)$ of quasimomentum eigenstates with $q\!=\!\delta \,I/2\pi$. Here, ${{\hat{\mathrm{c}}}^{\dagger}}_q$(${{\hat{\mathrm{c}}}^{\phantom{\dagger}}}_q$) are the bosonic creation (annihilation) operators with respect to the quasimomentum basis, which can be written as ${{\hat{\mathrm{c}}}^{\dagger}}_q=\frac{1}{\sqrt{I}}\sum_{i}\exp[{\mathrm{i}}\tfrac{2\pi}{I}qi]\;{{\hat{\mathrm{a}}}^{\dagger}}_i$. Figure \[fig:etrans\_int\](b) illustrates the frequency-dependence of the interference pattern in the region of the 1U resonance. Each horizontal cut through the density plot represents the time-averaged interference pattern for a certain frequency $\omega$. We assume that the lattice is switched off instantaneously after a certain evolution time in the modulated lattice—different from recent experiments which involve a re-thermalization period in the superfluid regime [@StMo04; @FaLy07]. The general interference structure reveals a specific correlation between the frequency $\omega$ relative to the centroid of the 1U resonance and the quasimomentum distribution, i.e. the peaks of the interference pattern. Away from the resonance region, the intensity $\mathcal{I}(\delta)$ exhibits a broad background distribution characteristic for the Mott-insulating phase. For frequencies $\omega$ at the low-frequency end of the resonance a sharp interference peak emerges at $\delta=0$ indicating the resonant transition to the $q=0$ state. With increasing frequency $\omega$ this population moves to successively higher quasimomenta $|q|$, i.e. the interference peak splits and shifts towards larger $|\delta|$. The fine-structure of the resonance is thus mapped onto the interference pattern in an experimentally accessible way. ![(color online) Interference pattern as function of $\omega$ in the region of the 1U resonance for a system with $I\!=N\!=10$ at $U_0/J_0\!=\!40$ and $\Delta_0/J_0\!=\!2$ for truncation energies $E_\text{trunc}/J_0\!=\!120$ (a), $E_\text{trunc}/J_0\!=\!80$ (b), and $E_\text{trunc}/J_0\!=\!40$ (c). The color coding is the same as in Fig.\[fig:results\_ic\].[]{data-label="fig:benchmark"}](fig2.eps){width="0.95\columnwidth"} ![Distribution of the $\epsilon_i$ for $I\!=\!20$ lattice sites of the incommensurate superlattice used in the calculations.[]{data-label="fig:lattice_struct"}](fig3.eps){width="0.8\columnwidth"} So far, the simulations were restricted to small systems with large bases. To treat larger lattices we reduce the truncation energy $E_{\text{trunc}}$ controlling the basis size. Figure \[fig:benchmark\] illustrates the insensitivity of the interference pattern on changes of $E_{\text{trunc}}$. There is practically no difference when reducing the truncation energy from $E_\text{trunc}/J_0=120$ to $80$. Even for $E_\text{trunc}/J_0=40$ all relevant features are reproduced, although the intensity of the interference peaks shows slight deviations. All qualitative conclusions regarding the correlations between frequency and quasimomentum distribution remain unaffected. Using the lowest truncation energy we investigate the response of a system with $I\!=\!N\!=\!20$ at $U_0/\!J_0\!=\!40$. The major focus is on the change of the response and the interference pattern if a two-color superlattice potential of increasing amplitude $\Delta_0$ is added. The distribution of the relative strengths $\epsilon_i$ for the incommensurate superlattice used in the following are shown in Fig. \[fig:lattice\_struct\]. The the ratio of the wavelengths of the two standing waves is $\lambda_1/\lambda_2\!\approx\!0.81$, similar to a recent experiment [@FaLy07]. Figure \[fig:results\_ic\] depicts the evolution of the interference structure as function of the superlattice amplitude $\Delta_0$ for the incommensurate case. The right-hand panels show the energy spectrum with vertical lines marking sizeable matrix elements ${\big<{0}\big|\,{{{\hat{\mathrm{H}}}}_J}\big|\,{\nu}\big> }$. The left-hand panels depict the energy transfer as function of frequency. The result for $\Delta_0/J_0\!=\!0$ shown in Fig. \[fig:results\_ic\](a) confirms our previous discussion of the smaller regular lattice in Fig. \[fig:etrans\_int\]. The linear response analysis provides a good estimate for the resonance energies via the energies of those excited states that exhibit strong transition matrix elements to the ground state. Furthermore, the correlation between frequency $\omega$ and the quasimomentum distribution is even more pronounced. At the low-frequency end of the resonance quasimomentum states around $q\approx0$ are populated, whereas for larger modulation frequencies successively higher quasi-momenta are occupied. The results for a small superlattice amplitude $\Delta_0/J_0\!=\!1$ in Fig. \[fig:results\_ic\](b) show minor changes in the quasimomentum structure and the energy transfer as compared to (a). Nevertheless, the number of possible excitations from the ground state increases. A further increase of the superlattice amplitude to $\Delta_0/J_0\!=\!2$ leads to a weak suppression of the interference structure as shown in Fig. \[fig:results\_ic\](c). In comparison to the energy spectrum in (b) there are many more large matrix elements which are not localized at distinct energies but spread over the whole range. The occurrence of the small gaps at both ends of the energy band is also visible in the density plots as small shifts in the interference structure along the energy (vertical) axis. This also indicates the broadening of the resonance due to the superlattice [@FaLy07; @HiSc06]. ![(color online) Energy transfer (left-hand panels), interference pattern (middle panels), and excitation spectrum (right-hand panels) of a system with $I\!\!=\!\!N\!\!=\!\!20$ and $U_0/J_0\!\!=\!\!40$ for several superlattice amplitudes $\Delta_0$ as indicated in the plots. The density plots illustrate the correlations between quasimomentum distribution and excitation frequency in the region of the 1U-resonance. The vertical lines in the energy spectra (right-hand panels) point out strong matrix elements ${\big<{0}\big|\,{{{\hat{\mathrm{H}}}}_J}\big|\,{\nu}\big> }$.[]{data-label="fig:results_ic"}](fig4a.eps "fig:"){width="0.95\columnwidth"}\ ![(color online) Energy transfer (left-hand panels), interference pattern (middle panels), and excitation spectrum (right-hand panels) of a system with $I\!\!=\!\!N\!\!=\!\!20$ and $U_0/J_0\!\!=\!\!40$ for several superlattice amplitudes $\Delta_0$ as indicated in the plots. The density plots illustrate the correlations between quasimomentum distribution and excitation frequency in the region of the 1U-resonance. The vertical lines in the energy spectra (right-hand panels) point out strong matrix elements ${\big<{0}\big|\,{{{\hat{\mathrm{H}}}}_J}\big|\,{\nu}\big> }$.[]{data-label="fig:results_ic"}](fig4b.eps "fig:"){width="0.95\columnwidth"} Further increase of the superlattice amplitude to $\Delta_0/J_0\!=\!4$ leads to the disappearance of the interference peaks as depicted in Fig. \[fig:results\_ic\](d). The number of strong matrix elements which couple ground and excited states is further increased. Although the interference pattern does not show sharp peaks anymore, the energy transfer in these cases still exhibits a strong resonance behavior [@HiSc06]. Only the fine-structure of the resonance in the energy transfer is affected by the superlattice. However, the distinct correlations between excitation frequency and quasimomentum distribution vanish far below the transition from the homogeneous Mott insulator to the Bose-glass phase at $\Delta_0\approx{}U_0$. These general features are robust against changes of the superlattice structure. We have performed explicit simulations using a commensurate superlattice showing the same behavior. The rapid change of the interference pattern can be explained in the linear response picture: In the absence of a superlattice, resonant transitions connect the ground state to a few excited states only—those characterized by large transition matrix elements ${\big<{0}\big|\,{{{\hat{\mathrm{H}}}}_J}\big|\,{\nu}\big> }$ \[Fig. 4(a) and (b)\]. The many-body state ${\,\big|{\psi,t}\big> }$ during the time evolution is dominated by these few states and exhibits well-defined interference peaks. With increasing superlattice amplitude $\Delta_0$ more and more sizable transition matrix elements emerge and the time-evolved state is a superposition of a large number of eigenstates \[Fig. 4(c) and (d)\]. The fragmentation of the state causes a fragmentation of the interference pattern and in effect a suppression of the distinct peaks. This mechanism can be confirmed within a simple toy-model by comparing the interference pattern of an excited eigenstate with the one resulting from a coherent superposition of a few neighboring eigenstates. Note that this phenomenon is quite different from the Mott-insulator to Bose-glass transition, which appears at much larger superlattice amplitudes. In summary, we have discussed the response of Bose gases in modulated lattice and superlattice potentials with an emphasize on the quasimomentum distribution and the interference pattern after prompt release from the lattice. For modulation frequencies in the region of the 1U resonance distinct peaks appear in the interference pattern. Their position is correlated with the modulation frequency relative to the centroid of the resonance: with increasing frequency they shift to larger quasimomenta. In the presence of a superlattice this characteristic correlation vanishes already for small superlattice amplitudes, much faster than the resonance observed in the energy transfer. [10]{} url \#1[`#1`]{}urlprefix D. Jaksch *et al.*, Phys. Rev. Lett. **81** (1998) 3108. M. Greiner *et al.*, Nature **415** (2002) 39. M. Greiner *et al.*, Nature **419** (2002) 51. T. Stöferle *et al.*, Phys. Rev. Lett. **92** (2004) 130403. L. Fallani *et al.*, Phys. Rev. Lett. **98** (2007) 130404. J. E. Lye *et al.*, Phys. Rev. Lett. **95** (2005) 070401. D. Clement *et al.*, Phys. Rev. Lett. **95** (2005) 170409. C. Fort *et al.*, Phys. Rev. Lett. **95** (2005) 170410. T. Schulte *et al.*, Phys. Rev. Lett. **95** (2005) 170411. R. Roth, K. Burnett, J. Opt. B: Quantum Semiclass. Opt. **5** (2003) S50. R. Roth, K. Burnett, Phys. Rev. A **68** (2003) 023604. M. Hild, F. Schmitt, R. Roth, J. Phys. B **39** (2006) 4547. C. Kollath *et al.*, Phys. Rev. Lett. **97** (2006) 050402. F. Schmitt, M. Hild, R. Roth, J. Phys. B **40** (2007) 371. K. Braun-Munzinger, Ph.D. thesis; Oxford (2005). O. Schenk *et al.*, BIT **40** (2000) 158. O. Schenk, K. Gärtner, FGCS **20** (2004) 475. S. R. Clark, D. Jaksch, New J. Phys. **8** (2006) 160. A. Iucci *et al.*, Phys. Rev. A **73** (2006) 041608(R). R. Roth, K. Burnett, Phys. Rev. A **67** (2003) 031602(R).

--- abstract: 'Simulations of vortex tube dynamics reveal that the non-Gaussian nature of turbulent fluctuation originates in the effect of random advection. A similar non-Gaussian distribution is found numerically in a simplified statistical model of random advection. An analytical solution is obtained in the mean-field case.' author: - 'Y-h. Taguchi' - Hideki Takayasu --- v Non-Gaussian distribution in Random advection dynamics Institut für Festkörperforschung, Forschungszentrum Jülich,\ D-5170 Jülich, Germany\ and\ Department of Physics, Tokyo Institute of Technology,\ Oh-okayama, Meguro-ku, Tokyo 152, Japan[@present] Department of Earth Science, Kobe University, Kobe 657, Japan. In the study of statistical physics for non-equilibrium systems the deviation from the Gaussian distribution has been the central issue. We are expecting the existence of universal mechanisms of producing non-Gaussian distributions as in the case of thermal equilibrium systems, however, our knowledge has been still in the elementary level. Non-Gaussian probability distribution function (NGPDF) is especially important in fluid turbulence. Turbulence is a typical far-from-equilibrium phenomena as it is characterized by the energy cascade which violates both the detailed balance and equi-partition. The appearance of NGPDF is so common and we can not construct any theory without taking the non-Gaussian nature into account[@Batchelor; @Monin]. Latest technique of direct observation of turbulence[@Goldburg] and also direct numerical integration of Navier-Stokes equation[@Kida] are clarifying the details NGPDF from experimental viewpoints. Recently, theoretical investigation to NGPDF in turbulence is attracting much attention. An approach is to assume the multifractality in the geometry of turbulent velocity field as an ideal limit[@Benzi], and another one called the mapping closure conjectures the existence of a kind of smooth map which describes the time evolution of statistical quantities[@She91; @Kraichnan]. Yakhot et al.[@Yakhot] also present a phenomenological excellent approach in order to derive a NGPDF for vorticities in the fluid turbulence. Although it is fairly reasonable, physical meanings in their assumptions are not clear. In this paper we first introduce a numerical model of vortex dynamics on lattice and show that the vorticity distribution in a turbulent steady state is actually far from Gaussian but closer to a symmetric exponential distribution. In order to clarify the origin of this NGPDF we modify the dynamics so that vortex tubes move randomly by passive advection. It is shown that this modification does not seem to affect the vorticity distribution, which implies that the random advection is playing an important role. Then, we focus our attention to the effect of random advection and introduce a random scalar advection model on lattice. Surprisingly the scalar also follows a distribution which is almost identical to the former NGPDFs. A mean-field version of the scalar model is then solved analytically showing a clear evidence of NGPDF. Let us first introduce our lattice vortex model[@Tag] which is equivalent to the following set of vorticity equations of incompressible fluid in the continuum limit $$\begin{aligned} \frac{\partial \w}{\partial t} + (\v \cdot \nabla ) \w &=& (\w \cdot \nabla)\v + \nu \Delta \w, \label{vor}\\ \rot \v &=& \w, \label{rot}\\ \div \v &=& 0, \label{div}\\ \nonumber\end{aligned}$$ where $\v, \w$ and $\nu$ are the velocity, vorticity and viscosity, respectively. We assign vorticities on a simple cubic lattice. Every bond has only one vorticity component along its direction. This means that an $x$-bond has only $x$ component of vorticity, i.e., a bond can be viewed as a vortex tube. For a given configuration of vorticities the vorticity field is calculated by using the Biot-Savier law (equivalent to eqs.(\[rot\]) and (\[div\]). The fluid velocity of a bond at $\r,\v(\r)$, is estimated at the mid-point of the bond. Dynamics is defined so as to satisfy the vorticity equation(\[vor\]). Due to the advection term $(\v \cdot \nabla) \w$ in eq. (\[vor\]) the vorticity flows with the fluid as shown by the name of Kelvin’s theorem[@Landau], the vorticity $\omega_z$ at a $z$-bond is transported to its six neighboring $z$-bonds. By this effect the value of $\omega_z$ at $\r \pm \hi$ is increased by $\pm J_{iz} \Delta t / 2$, where $\hi$ is a unit vector directing either $x$, $y$ or $z$ direction, and $J_{ij}$ is the flux of vorticity, $\tvi \wj$. The coefficient $1/2$ of the vorticity flux and the signs $\pm$ are introduced to keep the spatial symmetry. The first term on the right hand side of eq.(\[vor\]) shows the vortex stretching term, so we have to modify the vorticities around the advectively transported bonds so that modified vorticities make loops as shown in Fig.\[dyna\]. In this procedure we add $\mp J_{iz} \Delta t/2$ to the bonds at $\r \pm \hi/2 \pm \hz/2$, namely, $x$ and $y$ component appear due to the elongation of vortex tube. The diffusion term in eq. (\[vor\]) is discretized by the usual finite difference method. It has been shown that this dynamics is equivalent to eq.(\[vor\]) in the continuum limit and numerical simulation of this vortex tube dynamics on a periodic $24 \times 24 \times 24$ lattice has reproduced many basic properties of fluid turbulence[@Tag]. As for the probability distributions we obtained the following results: 1.The velocity components (for example $v_x$) follows nearly Gaussian. 2\. The distribution of differentiated quantities, such as $\partial v_x /\partial x, \partial v_x / \partial y $ and $\omega_x$, are non-Gaussian and close to exponential (see Fig.\[wx\]). 3\. By removing lower wave number components, the distribution of $v_x$ also becomes closer to an exponential as firstly found by She et al[@She88]. We also observe a kind of conditional distributions for velocities on bonds whose vorticities are in a fixed range. The distribution are nearly identical both for large $\omega$ and for small $\omega$, indicating that the velocity on a bond is nearly independent of its vorticity strength. We now modify our model to seek the origin of the NGPDF. The modification is to randomize the velocity field at each time step keeping its distribution. Namely, we do not use Biot-Savier law, but the vorticities are transported and elongated by random advection. The probability distribution of $\omega$ after several hundreds time steps is quite similar to that of original lattice vortex model (see Fig.\[wx\]). This result clearly shows that the appearance of NGPDF is independent of the details of velocity field. As suggested by Sinai and Yakhot[@Sinai; @Yakhot] NGPDF may be caused by random advection, and if so, we can expect a large universality class including another exponential-like distribution in thermal convection flow[@Siggia; @Yanagita]. In order to see the effect of random advection more clearly we introduce a scalar model on lattice. The model is governed by the following stochastic equation $$\omega (r_0,t + \Delta t) = \omega (r_0,t) - v(r_0,t) \omega (r_0,t) \Delta t + \sum_{r} P(r_0,r,t) v(r,t) \omega (r,t) \Delta t, \label{eq:adv}$$ where $v(r,t)$ and $P(r_0,r,t)$ are independent random variables. $v(r,t)$ takes non-negative values, and $P(r_0,r,t)=1$ when a flow from site $r$ to $r_0$ exists at time $t$, otherwise $P(r_0,r,t)=0$ and it is normalized as $\sum_r P(r_0,r,t)=1$. The one-dimensional version of eq.(\[eq:adv\]) is defined by the special case that either $P(r_0,r_0-1,t)$ or $P(r_0,r_0+1,t)$ is equal to 1 with probability $1/2$. We perform the simulation on a 1-dimensional lattice of size $10000$ with the periodic boundary condition. The random number $v(r,t)$ is in the range of $[0,0.5]$ and $\Delta t =0.5$. For a given random initial condition of ${\omega (r,0)}$ we observe the distribution of $\omega(r,t)$ at sufficiently large $t$. The result is also plotted in Fig.\[wx\]. The distribution is far from Gaussian and is very close to those of former cases. Now it seems obvious that the origin of NGPDF is in the simple random advection transport. A mean-field version of eq.(\[eq:adv\]) can be solved analytically. We consider the situation that a site is interacting with a mean-field site. To avoid unimportant complexity we assume the case that the random number $v$ takes either 0 or $j_0/\Delta t$ with probability $1/2$ ($j_0 \in [0,1]$). In this situation the stochastic eq.(\[eq:adv\]) becomes $$\omega (t+t_0) = \left \{ \begin{array}{ll} \omega (t), & {\rm Prob.} 1/4 \\ (1-j_0) \omega (t), & {\rm Prob.} 1/4 \\ \omega (t) + j_0 \omega_M, & {\rm Prob.} 1/4 \\ (1-j_0) \omega (t) + j_0 \omega_M, & {\rm Prob.} 1/4 \end{array}, \right. \label{eq:mean}$$ where $\omega_M$ is an independent random number having the same distribution as $\omega (t)$. By introducing the characteristic function $$Y(\rho,t)=\int^{\infty}_{-\infty} e^{i\rho \omega} P(\omega,t) d \omega,$$ eq.(\[eq:mean\]) becomes $$Y(\rho, t+\Delta t) = \frac{1}{4} \{ Y(\rho,t) + Y(\rho-\rho j_0,t)\} \{ 1 + Y ( \rho j_0,t) \}. \label{eq:mean2}$$ Taylor expansion of eq.(\[eq:mean2\]) in terms of $\rho$ gives a set of equations for the moment functions $\{ < \omega^n > \}$, and it can be easily shown that a steady state exists when $< \omega > \neq 0$. The steady state solution is obtained with the aid of algebraic calculation by computer. In Fig.\[fig:j0\] we plot the cumulants for orders up to $7$ together with those for one-sided exponential distribution for comparison. Higher order cumulants are not zero in any case and showing a tendency to diverge as the order goes to infinity, which clearly demonstrates that the steady state distributions are not Gaussian. In the special case of $j_0 = 1/2$ we can estimate the functional form of $Y(\rho)$ as follows. By denoting $Y_n = Y(\rho 2^n)$ eq.(\[eq:mean2\]) becomes $$Y_{n+1}-Y_n = 2 \cdot \frac{Y_n^2 -Y_n}{3 - Y_n}. \label{eq:j2}$$ Supposing that $n$ can be a continuous number and we approximate the left hand side by a derivative $dY/dn$. Then eq.(\[eq:j2\]) can easily be integrates and we have the following cubic equation for $Y(\rho)$ with $< \omega > = 1$, $$\rho^2 Y(\rho)^3 + (Y(\rho)-1)^2 =0,$$ which can be solved explicitly by Cardano’s formula. The asymptotic behavior for $\rho \rightarrow \infty$ is given as $$Y(\rho) \propto \rho^{-2/3}.$$ This power law decay is very different from the case of a Gaussian distribution where the characteristic function decays faster than any power of $\rho$. In the case of exponential distribution the characteristic function is given by $1/(1 + i \rho)$ (asymmetric case) or $1/(1+\rho^2)$ (symmetric case), which also shows a power law decay as $\rho \rightarrow \infty$. In this sense our distribution is far from Gaussian, but has a similarity to the exponential distribution. For smaller $j_0$ the cumulants are smaller, and in the limit of $j_0 \rightarrow 0$ it can be shown that the normalized cumulant of order $n$, $< \omega^n >_c / < \omega^2 >_c^{n/2}$, vanishes for $n \geq 3$. Namely, the distribution converges to a Gaussian in this limit, which agrees with the well-known fact that the distribution function satisfying the usual diffusion equation in the continuum limit is a Gaussian since eq.(\[eq:adv\]) becomes a usual diffusion equation in the limit of $\Delta t \rightarrow 0$. This indicates that the finiteness of $\Delta t $ is very important for the appearance of NGPDF in our random advection dynamics. One may think that these results for $\Delta t \neq 0$ are meaningless because we must take the limit of $\Delta t \rightarrow 0$ in order to recover eq.(\[vor\]). This is a wrong argument neglecting the finiteness of correlation time in the real dynamics with $\Delta t =0$. It is known that the fastest mode in real turbulent flows has a characteristic time approximately given by $\sqrt{\nu / \varepsilon}$, where $\varepsilon$ is the mean energy dissipation rate [@visco]. For time scales less than this value advections can not be treated as random. In our random advection models $\Delta t$ should be viewed as the correlation time in real dynamics, therefore, it should take a finite value. Our theoretical approach is successful only in the case of $< \omega > \neq 0$, while observed distributions are symmetric with $< \omega > = 0$. At present we have no rigorous way of connecting these two cases, but we are now considering the effect of spatial fluctuation as the key of solving this difficulty. Even in the case of $< \omega > =0$ averaged over the whole space, $<\omega >$ can be nonzero if the average is taken over a finite area due to the spatial fluctuation. So, if the distribution of $\omega$ is determined rather locally not using the information from the whole space, then the distribution for $< \omega > \neq 0$ may have direct consequence to the real distributions. In summary, we showed that the random advection dynamics creates far-from Gaussian fluctuations whose distribution functions are closer to the exponential. As shown in the discussions of the statistical scalar model on lattice and its mean-field analysis, the mechanism of NGPDF is so simple that a wide application may be expected. H.T. thanks M. Takayasu for useful discussions. NEC software (c) is also acknowledged for allowing Y.T. to use EWS-4800/220, with which all calculations were performed. This work is supported by Grant-Aid for Scientific Research on Priority Areas, “Computational Physics as a New Frontier in Condensed Matter Research”, from the Ministry of Education, Science and Culture, Japan. Present and permanent address G. K. Batchelor, Theory of Homogeneous Turbulence (Cambridge Univ. Press,Cambridge, 1953). A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT, Cambridge,1975). H.K.Pak and W.I.Goldburg, Phys. Rev. Lett. [**68**]{}, 938(1992). R. Benzi, L. Biferate, G. Pladin, A. Vulpiani and M. Vergassola, Phys. Rev. Lett. [**67**]{}, 2299 (1991). Z-S. She, Phys. Rev. Lett. [**66**]{}, 600 (1991). R.H. Kraichnan, Phys. Rev. Lett. [**65**]{} 575 (1990) V. Yakhot, S.A. Orszag, S. Balachander, E. Jackson, Z-S. She and L. Sirovich, J. Sci. Comp., [**5**]{}, (1990) 199. S. Kida and Y. Murakami, Fluid Dynamics Research [**4**]{}, 347 (1989). Y-h. Taguchi and H. Takayasu, submitted to Physica D. L. D. Landau and E.M.Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1979) p14. Z-S. She, E. Jackson and S.A. Orszag, J. Sci. Comp., [**3**]{}, 407 (1988). A. Pumir, B. Shraiman and E.D. Siggia, Phys. Rev. Lett. [**66**]{}, 2984 (1991), and references therein. T.Yanagita and K.Kaneko, preprint. E.D. Sinai and V. Yakhot, Phys. Rev. Lett. [**63**]{}, 1962 (1989). According to the Kolmogorov’s theory[@Batchelor], the smallest scale which appears in the homogeneous isotropic turbulence is estimated by $( \nu^3 / \varepsilon )^{1/4}$. The characteristic time scale corresponding to this scale is $\sqrt{\nu/\varepsilon}$.

--- abstract: | The high-frequency Raman-active phonon modes of metallic single-walled carbon nanotubes (SWNTs) are thought to be characterized by Kohn anomalies (KAs) resulting from the combination of SWNTs intrinsic one-dimensional nature and a significant electron-phonon coupling (EPC). KAs are expected to be modified by the doping-induced tuning of the Fermi energy level $\epsilon_F$, obtained through the intercalation of SWNTs with alkali atoms or by the application of a gate potential. We present a Density-Functional Theory (DFT) study of the phonon properties of a (9,9) metallic SWNT as a function of electronic doping. For such study, we use, as in standard DFT calculations of vibrational properties, the Born-Oppenheimer (BO) approximation. We also develop an analytical model capable of reproducing and interpreting our DFT results. Both DFT calculations and this model predict, for increasing doping levels, a series of EPC-induced KAs in the vibrational mode parallel to the tube axis at the $\mathbf\Gamma$ point of the Brillouin zone, usually indicated in Raman spectroscopy as the $G^-$ peak. Such KAs would arise each time a new conduction band is populated. However, we show that they are an artifact of the BO approximation. The inclusion of non-adiabatic (NA) effects dramatically affects the results, predicting KAs at $\mathbf\Gamma$ only when $\epsilon_F$ is close to a band crossing $E_{X}$. For each band crossing a double KA occurs for $\epsilon_F=E_{X}\pm \hbar\omega/2$, where $\hbar\omega$ is the phonon energy. In particular, for a 1.2 $nm$ metallic nanotube, we predict a KA to occur in the so-called $G^-$ peak at a doping level of about $N_{el}/C=\pm 0.0015$ atom ($\epsilon_F\approx \pm 0.1 ~eV$) and, possibly, close to the saturation doping level ($N_{el}$/$C$$\sim$ 0.125), where an interlayer band crosses the $\pi^*$ nanotube bands. Furthermore, we predict that the Raman linewidth of the $G^-$ peak significantly decreases for $|\epsilon_F| \geq \hbar\omega/2$. Thus our results provide a tool to determine experimentally the doping level from the value of the KA-induced frequency shift and from the linewidth of the $G^-$ peak. Finally, we predict KAs to occur in phonons with finite momentum **q** not only in proximity of a band crossing, but also each time a new band is populated. Such KAs should be observable in the double-resonant Raman peaks, such as the defect-activated $D$ and peak, and the second-order peaks $2D$ and $2G$. author: - Nicolas Caudal - 'A. Marco Saitta' - Michele Lazzeri - Francesco Mauri title: 'Kohn anomalies and non-adiabaticity in doped carbon nanotubes' --- Introduction ============ Since their discovery in 1991 [@Iijima91], carbon nanotubes have raised an enormous interest both from the academic and the technological points of view. They exhibit in fact a variety of exciting features: their quasi-one-dimensional nature, due to a diameter of 1-2 $nm$ and a length of up to several micrometers, makes them sharp probes for scanning tunneling microscopes and an excellent model for one-dimensional physics. Their mechanical and tensile strength make them of great interest in composite materials, and, most significantly of all, they have very unusual and extremely promising electronic properties, displaying metallic or semiconducting behavior according to their structure and helicity [@Saito; @Reich]. The electronic properties of SWNTs, already particularly interesting from the technological point of view, promise to be the future of nano-electronics due to their *tunability*, achieved by doping the nanotubes through intercalation with alkali atoms [@Ye03; @Bendiab01; @Bendiab01b; @Bendiab01c; @BendiabThesis; @Furtado05; @Meunier02; @Liu03; @Rauf04; @Bantignies05] or application of a gate potential [@Corio04; @Cronin04; @Rafailov05; @Wang06]. However, there are still a number of experimental challenges to be solved in order to fully develop SWNT-based nano-electronic technology, in particular the low-cost industrial-scale synthesis of nanotubes of good chemical purity, crystalline quality and given helicity. An experimental tool largely used to characterize SWNTs is Raman spectroscopy [@Bendiab01; @Bendiab01b; @Bendiab01c; @BendiabThesis; @Ye03; @Furtado05; @Maultzsch02; @Maultzsch03; @Cronin04; @Rafailov05; @Maultzsch05; @Son06; @Wang06; @Jorio02; @Jorio05]. Typical Raman spectra of carbon nanotubes display a peak around 150-300 $cm^{-1}$, due to the radial breathing mode (RBM), which has been recently used as a tool to infer the size and chirality of the nanotubes [@Maultzsch05; @Jorio05]. Other important features of SWNT Raman spectra include a peak around 1350 $cm^{-1}$, activated by defects and impurities, and known in the literature as $D$ peak, and a large structure around 1570 $cm^{-1}$, due to modes tangential to the nanotube and known as $G$ peak. This last feature is thought to have two components, usually referred to as $G^+$ and $G^-$, originating from the $E_{2g}$ in-plane modes of graphite. In refs. [@Lazzeri06; @Piscanec06; @Popov06] it has been shown that in metallic SWNTs the $G^+$ component corresponds to the tangential vibrational mode *perpendicular* to the nanotube axial direction, while the $G^-$ component corresponds to the tangential vibrational mode *parallel* to the nanotube axial direction. In the following, we will refer to the former as “the nanotube TO tangential mode”, and to the latter as “the nanotube LO axial mode”. Some Raman studies [@Bendiab01; @Bendiab01b; @Bendiab01c; @BendiabThesis] show that the frequency of the $G$ peak increases up to 1600 $cm^{-1}$ at low doping levels, and then suddenly drops to about 1550 $cm^{-1}$ at the saturation threshold of alkali intercalation, estimated around a number of electrons per carbon atom $N_{el}/C=0.12$ (MC$_{\rm 8}$). Other experimental [@Maultzsch02; @Rafailov05; @Furtado05] and theoretical [@Dubay02; @Dubay03; @Akdim05] works on the effect of doping on SWNTs report a similar $G$-peak softening or even a Luttinger-Fermi liquid transition [@Rauf04]. Since the LO axial and TO tangential modes are particularly sensitive to the electronic structure of SWNTs around the Fermi energy  [@Dubay02; @bohnen04; @connetable05; @Lazzeri06; @Popov06] $\epsilon_F$, and $\epsilon_F$ directly depends on the charge doping level, a profound understanding of the interplay between the vibrational and the electronic properties of nanotubes looks to be crucial for technological development. In this work we report our theoretical study of the electronic and vibrational properties of doped SWNTs, based on DFT first-principles calculations and analytical results. We will show in the following that: *i)* the vibrational properties of SWNTs can be obtained from the so-called *electronic zone-folding* of a graphene sheet; *ii)* their behavior as a function of the charge doping level can be determined by the knowledge of electron-phonon coupling (EPC) in graphene; *iii)* ordinary quantum-mechanics calculations relying on the adiabatic Born-Oppenheimer approximation fail when applied to SWNTs, where non-adiabatic effects are enhanced by their intrinsic one-dimensional nature. Our paper is organized as follows: in section \[theo\] we will describe our theoretical framework, and in particular how the Raman-active modes of a nanotube can be accurately obtained through an appropriate electronic sampling of the graphene Brillouin Zone (BZ). We will then show in section \[model\] that the DFT results can be almost perfectly reproduced by an integral model, that uses the graphite EPC and the slope of the electronic bands around the (undoped) Fermi level $\epsilon^0_F$ (usually referred to as the $\pi$ and $\pi^*$ bands) as external inputs, and that becomes analytical in the limit of vanishing temperature. The results on the DFT and the model-derived LO axial and TO tangential modes of a SWNT will be presented, showing that Kohn anomalies (KA) would occur, within the adiabatic approximation, each time a new electronic band is populated by the electrons at $\epsilon_F$. In section \[nonadia\] we show that when the Born-Oppenheimer approximation is lifted the outcome is dramatically different, and that a drop in the frequency of the $G^-$ peak occurs at a doping level such that $\epsilon_F-\epsilon^0_F\approx 0.1 ~eV$ or close to the saturation level. Section \[conclusions\] will be devoted to the discussion of the physical properties of metallic SWNTs that are experimentally accessible, and to the conclusions. Theoretical background {#theo} ====================== General description of graphene and SWNTs {#graphene} ----------------------------------------- Graphene is a semimetal, and its highest valence and lowest conduction bands have a circular conical surface shape around the points **K** and **K$^\prime$** of the hexagonal first Brillouin zone at zero doping; its Fermi surface is reduced to the vertexes of the cones at those two points. The structure of a SWNT is obtained by rolling a sheet of 2D graphene, and closing both ends with fullerene-like semi-spheres. However, nanotubes are commonly studied as infinite 1D crystals, [*i.e.*]{} neglecting the details of those ends. Nanotubes are then uniquely determined by the knowledge of the chiral vector ${\bf C_h}=n{\bf a_1}+m{\bf a_2}$, where ${\bf a_1}$ and ${\bf a_2}$ are the basis vectors of graphene, $n$ and $m$ are integers [@Saito; @Reich]. The translational vector determines the periodicity along the tube axis and is ${\bf T}=\frac{(2m+n)}{d_R}{\bf a_1}-\frac{(2n+m)}{d_R}{\bf a_2}$, where $d_R$ is the greatest common divisor of $(2m+n)$ and $(2n+m)$. Nanotubes with indexes such that $n-m=3k$, where $k$ is an integer, are metallic, otherwise they are semiconductors. Nanotubes with identical indexes, *i.e. n=m*, are thus always metallic, and are usually called “armchair” tube. Electronic zone folding (EZF) is a common approximation for the electronic band structure of a SWNT, which consists in neglecting the curvature of the SWNT. Within the EZF the electronic states of a SWNT are approximated by the electronic states of a graphene sheet having wavevector ${\bf k}$ such that ${\bf k}\cdot{\bf C_h}=2\pi\nu$, being $\nu$ an integer. The wave-vectors allowed for the SWNT satisfy $$\label{nano_nu} \mathbf{k}=\nu \mathbf{k}_\perp + k \mathbf{k}_\parallel / \| \mathbf{k}_\parallel \|$$ where $k$ is a real, $\mathbf{k_\parallel}$ is parallel to the tube axis, $\mathbf{k_\perp}$ is perpendicular to the tube axis and satisfies $\|\mathbf{k}_\perp\|=2 \pi/ \|\mathbf{C}_\mathrm{h}\|=2/d$, $d$ being the tube diameter. This way of constructing the electronic band structure accounts for the fact that SWNTs can be either metallic or semiconducting. In fact, if in SWNTs a line of allowed wave vectors **k** crosses the **K** or the **K$^\prime$** point, the nanotube is metallic, otherwise it is a semiconductor. In particular, in the BZ of armchair metallic SWNTs the section of the vertical plane containing the $\nu=0$ line of **k** vectors and the conical bands consists of two straight lines of slope $\pm\beta$, while the conic sections for $|\nu|\ge 1$ consist of arms of hyperbolae. The energy difference between the maximum of the lower arm and the minimum of the upper arm are usually indicated as $E_{\nu\nu}^M$ in the metallic case (see also Fig. \[fermi\]). Previous works [@Saito; @Reich; @Zolyomi04; @Piscanec06; @Lazzeri06] have shown that the effects of the nanotube curvature can be neglected as a first approximation for SWNTs whose diameter is larger than 1 $nm$ (the most common nanotubes present in experimental samples) and the electronic properties of a nanotube can be quite accurately obtained from EZF. Electronic zone-folding for phonon calculations ----------------------------------------------- The explicit calculation of the full dynamical matrix of an infinite nanotube, even within an efficient scheme such as Density-Functional Perturbation Theory (DFPT) [@Baroni01], is a quite demanding task from the computational point of view. On the other hand, the SWNT vibrational modes typically observed in Raman spectra can be traced back to the phonon and elastic modes of an isolated graphene sheet. However, it has been shown that the phonon dispersions of SWNTs cannot be as accurately reconstructed from the graphene ones as the electronic bands, because of the presence of KAs which behave differently in graphene [@Piscanec04] and metallic SWNTs  [@Dubay02; @bohnen04; @connetable05; @Lazzeri06; @Piscanec06; @Popov06]. The KAs are determined by singularities in the electron screening and such singularities depends on the dimensionality of the electron Brillouin zone. The dimensionality is two for graphene and one for SWNTs. This reduced SWNT dimensionality is due to the quantization ([*confinement*]{}) of the electronic wavevector around the nanotube circumference, which is actually described by the EZF. This last consideration suggests a practical scheme for the calculation of the phonon dispersion in SWNTs which neglects curvature, but fully takes into account the more important confinement effects. Phonons in nanotubes are well approximated by the phonons of a flat graphene sheet, if the calculation is done performing the electronic Brillouin-zone integration on the lines of the electronic zone-folding (Eq. \[nano\_nu\]). Such phonon calculation method (phonon-EZF) was introduced in refs. [@Lazzeri06; @Piscanec06]. A phonon-EZF calculation requires the use of a unit cell containing two atoms and thus, is clearly much less computationally demanding than full calculations on an actual SWNT, which requires a unit-cell with tens or hundreds of atoms. In Subsect. \[accuracy\] we will demonstrate that, for the preset study, curvature effects are negligible and the phonon-EZF provides an accurate description of the high-frequency phonon modes. Computational details --------------------- Our first-principles calculations are based on DFT, within the plane-wave (PW)/pseudopotential scheme implemented in the Quantum-ESPRESSO code [@PWSCF]. We adopt a Perdew-Burke-Erzherhof gradient corrected functional, and an ultrasoft pseudopotential to describe the $C$ atom. A PW kinetic energy cutoff of 30 Ry is sufficient to ensure convergence on the structural, electronic, and vibrational properties. We choose as a case-study the (9,9) metallic armchair SWNT, containing 36 distinct carbon atoms, and having a tube diameter of 1.24 $nm$. The full nanotube calculations are performed by setting the nanotube in an infinite lattice of hexagonal symmetry in the plane perpendicular to the tube axis, mimicking thus the bulk effect of a real nanotube bundle or an isolated tube by tuning the lattice parameter between 1.56 $nm$ and 1.72 $nm$. The $c$ axis is maintained constant. Integrations in the nanotube one-dimensional Brillouin zone have been performed by using regular grids of **k** points along the $c$ reciprocal axis. We use a Fermi-Dirac electronic smearing of 0.01 Ry (corresponding to an electronic temperature of 1578 K) for SWNT full phonon calculations, and of 0.002 Ry (315 K) for structural calculations and frozen-phonon tests (see below). Grids of 8 and 40 **k** points were sufficient, respectively, to ensure a good convergence ($\sim$ 5 $cm^{-1}$) on the LO axial frequencies in the two cases. In the phonon-EZF calculation we use a graphene hexagonal unit cell of lattice parameter $a_{\mathrm{exp}}=2.47$ Å and $c=5$ Å. The number of points is chosen as to ensure phonon frequencies converged within about 0.3 $cm^{-1}$, and is inversely proportional to temperature. At a temperature of 315 K, the total number of points is 3240, equivalent to 180 points along the $c$ direction of the reciprocal space of the (9,9) nanotube, while at 1578 K this number reduces to 720, equivalent to 40 points in the nanotube case. For both the full nanotube and the phonon-EZF case, the effect of (low) charge doping is simulated by adding an excess electronic charge which is compensated by a uniformly charged backgournd. This is done using the standard implementation of the ESPRESSO-package [@PWSCF]. This approach is known to describe very well the properties of doped SWNTs [@Margine06]. This is justified even in the case of alkali intercalation by the report that at low doping levels, [*i.e.*]{} up to MC$_{\rm 15}$, the electronic charge of the metal atoms is thought to be completely transferred [@Lu04], and that the effect of the intercalation is essentially described by taking into account the sole effect of the charge transfer. ![\[compare\] Variation of the LO axial frequencies with electron doping at 315 K, calculated from phonon-EZF (full diamonds) or through full-nanotube frozen-phonon (open circles). The solid line is a guide to the eye. The metal/carbon composition corresponding to the doping level is indicated on the top of the figure. Inset: variation with electronic doping of the phonon-EZF LO axial frequency (full diamonds) compared to a SWNT full DFPT dynamical matrix calculation (open squares), at 1578 K.](frozen.eps){width="8.25cm"} Accuracy of the method of phonon calculation {#accuracy} -------------------------------------------- The first step of our study is to assess the accuracy of the phonon-EZF method with respect to the SWNT phonon calculations. In Fig. \[compare\] we show the frequency of the LO axial mode as a function of electronic doping at 315 K, obtained both through the phonon-EZF method or by an explicit nanotube frozen-phonon calculation. In the latter case, the atoms are displaced along the axial direction of the nanotube, and the frequency is obtained by the curvature of the displacement energy. The convergence of this single-mode frozen-phonon frequency has been confirmed by testing the result with a more accurate sampling of the BZ, that is up to 200 uniformly-spaced **k** points in the one-dimensional BZ grid. In the inset we analogously compare the phonon-EZF LO axial frequency with a full DFPT nanotube phonon calculation at 1578 K. In this latter case, all the 108 $\Gamma$ vibrational modes of the nanotube are explicitly calculated; a lower temperature, and thus a correspondingly larger grid of at least 40 **k** points, would be computationally too demanding for a complete and accurate study of the problem. Both graphs show that phonon-EZF not only determines the nanotube phonon frequencies in good agreement with explicit SWNT calculations, but it also captures very well the strikingly non-monotonic dependence of the LO axial mode with respect to electronic doping, to be discussed later on, confirming that the effects of the SWNT curvature can be confidently neglected for a (9,9) armchair nanotube, and that phonon-EZF is a reliable and accurate method. ![\[fermi\] Variation of the electronic doping per $C$ atom as a function of the Fermi energy $\epsilon_F$. Solid line: analytical expression (Eq.\[eq:Fermi-dop\]) with $E_{11}^M/2=0.83~eV$; diamonds: DFT calculations, within phonon-EZF, of the Fermi energy in a (9,9) SWNT at 315 K. The $E_{11}^M/2$ and $E_{22}^M/2$ levels are indicated by vertical dotted lines. Inset: Calculated electronic bands of the (9,9) SWNT around the **K** point. $\epsilon_F^0$ is taken as the zero energy, $E_{11}^M/2$ and $E_{22}^M/2$ are the minima of the second and third (hyperbolic) conduction bands. The valence bands (below $\epsilon_F^0$) and the conduction bands (above $\epsilon_F^0$) are usually referred to as the $\pi$ and $\pi^*$ bands, respectively. The bands crossing at $\epsilon_F^0$ have the index $\nu=0$, while the valence and conduction bands immediately below and above have indexes $\nu=\pm 1$ (see Eq. \[nano\_nu\]).](Fermi_bands.eps){width="8.25cm"} Fermi level in doped SWNTs {#subsectionFermi} -------------------------- The key quantity intervening in the anomalous vibrational properties of SWNTs is the Fermi energy level $\epsilon_F$. In undoped graphene and in metallic armchair nanotubes, its value ($\epsilon_F^0$) coincides with the band energy at the crossing of the $\pi$ and $\pi^*$ bands at the **K** and **K$^\prime$** points of the graphene BZ, and is chosen as the zero band energy hereafter. Since the results are independent on the sign of $\epsilon_F$, we consider here and in the following that $\epsilon_F>0$, without loss of generality. Following the description of the graphene electronic bands in terms of conic sections as in subsection \[graphene\], the increase of $\epsilon_F$ due to electronic doping per carbon atom $N_{el}/C$ in a ($n,n$) SWNT can be analytically expressed, at low doping levels and a temperature T=0 K, by the following formula: $$\label{fermieq} N_{\mathrm{el}}/C(\epsilon_F)=\frac{a_0^2\sqrt{3}}{\pi \beta d}\epsilon_F+ \theta(\epsilon_F - E_{11}^M/2) \frac{2 a_0^2\sqrt{3}}{\pi \beta d} \sqrt{\epsilon_F^2-(E_{11}^M/2)^2}, \label{eq:Fermi-dop}$$ where $a_0$ is the lattice parameter, $\beta$ is the slope of the conical bands and equals 14.1 $eV\frac{a_0}{2\pi}$, and $\theta(x)$ is the step function. As previously mentioned, the energies of electronic transitions in SWNTs are usually indicated as $E_{ii}^{M/S}$, where, for armchair SWNTs, $i$ coicides with the $\nu$ band index of the initial and final states, and the superscript refers to metallic or semiconducting SWNTs. In the case of a ($n,n$) nanotube, assuming perfect conical bands, $E_{11}^M/2=2\beta/d$. In particular, for the (9,9) nanotube $E_{11}^M/2=0.89~eV$. We report in Fig. \[fermi\] the corresponding curve relating $\epsilon_F$ to $N_{el}/C$, along with our DFT calculations at 315 K. The most significant discrepancy being between the ideal value of $E_{11}^M/2$ and the calculated one, which is 0.83 $eV$. If however we treat $E_{11}^M/2$ as an independent parameter, and we red-shift it by 0.06 $eV$, the agreement is very good. This indicates that the analytical expressions can be safely used to develop an integral and analytical model, based on the EPC and the electronic bands, aimed at describing and predicting the behavior of the LO axial and TO tangential modes in metallic nanotubes. The electronic bands of the nanotube, explicitly calculated by DFT, are shown in the inset around the **K** point of the BZ, along with the band nomenclature $\pi/\pi^*,\nu$ used in the literature and adopted in this work. One can notice that the minimum of the $\nu=\pm 1$ conduction band is slightly displaced to the right, and that the bands deviate from the conical shape far from **K**. In the following we will use a polynomial fit of the calculated bands in numerical evaluation of integral expressions, while the ideal conical shape of the bands will be used to determine the analytical limits at low temperature. Integral and analytical model {#model} ============================= Electron-phonon coupling contribution to the dynamical matrix {#general} ------------------------------------------------------------- The frequency of a vibrational mode at a **q** wavevector, is obtained from the dynamical matrix $D_{\mathbf{q}}$ $$\label{dynmat} \omega_{\mathbf{q}}= \sqrt{\frac{D_{\mathbf{q}}}{M}},$$ where $M$ is the atomic mass of carbon. As shown in refs. [@Piscanec04; @Piscanec06], the dynamical matrix can be written as the sum of an EPC direct contribution $\tilde D_{\mathbf{q}}$, which contains non-analytical terms giving rise to Kohn anomalies, and a term containing all the other contributions: $$D_{\mathbf{q}}= \tilde D_{\mathbf{q}}+D_{\mathbf{q}}^{\mathrm{other}}$$ The direct contribution of EPC to the dynamical matrix can be written as $$\tilde D_{\mathbf{q}}=\frac{2}{N_{\mathrm{k}}} \sum_{\mathbf{k}, i, f} \frac{f(\epsilon_{\mathbf{k},i})-f(\epsilon_{\mathbf{k}+\mathbf{q},f})}{\epsilon_{\mathbf{k}, i}- \epsilon_{\mathbf{k}+\mathbf{q}, f}} \cdot |G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}|^2$$ where $N_{\mathrm{k}}$ is the number of points in the SWNT Brillouin zone, $i$ and $f$ are the band indexes indicating the two states involved in the the electronic transition. Since we consider only the contribution of the $\pi$ and $\pi^*$ bands, $i,f=\pi,\pi^*$. The function $f(\epsilon)=\{\exp[(\epsilon-\epsilon_F)/kT]+1\}^{-1}$, is the Fermi-Dirac distribution; $G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}$ is the EPC matrix element, defined as $$G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}=\langle \mathbf{k}+\mathbf{q}, f |\Delta V_{\mathbf{q}}| \mathbf{k}, i \rangle$$ where $\Delta V_{\mathbf{q}}$ is the first-order derivative of the Kohn-Sham self-consistent potential with respect to the atomic displacements corresponding to a **q**-vector phonon; $| \mathbf{k}, i \rangle$ is the Bloch electronic wavefunction. Hereafter, we develop our model on the basis of the EZF sampling, and the **k** vectors are those allowed by Eq. \[nano\_nu\]. Since we study phonons close to $\mathbf\Gamma$ we limit ourselves to wave vectors entirely along $\mathbf{k_\parallel}$ writing $\mathbf{q}=q \mathbf{k}_\parallel/\| \mathbf{k}_\parallel \|$ and $\|\mathbf{q}\|\ll\|\mathbf{K}\|$. Therefore $\nu$ is conserved through the transition $i \rightarrow f$. In the limit of a nanotube of infinite length we replace $\frac{1}{N_{\mathrm{k}}} \sum_{\mathbf{k}}$ by $\sum_\nu \frac{T}{2 \pi} \int \mathrm{d}k$, which yields for the non-analytical part of the dynamical matrix $$\label{eq:start} \tilde D_{\mathbf{q}}= \frac{T}{\pi} \sum_{\nu, i, f} \int \mathrm{d}k \frac{f \left[\epsilon_{\nu, i}(k)\right]-f\left[\epsilon_{\nu,f}(k+q)\right] } {\epsilon_{\nu, i}(k)-\epsilon_{\nu, f}(k+q)} \cdot |G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}|^2 \label{eq:Dq}$$ Neglecting the effects of curvature, the EPC matrix element $G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}$ of a nanotube of diameter $d$ and longitudinal period $T$ is related to the EPC $\tilde G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}$ of graphene by the ratio of the unit cells areas (Eq.4 of [@Lazzeri05]): $$d \pi T |G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}|^2 =\frac{a_0^2\sqrt{3}}{2} |\tilde G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}|^2$$ The main contribution to the EPC term originates from the **K** point of the graphene BZ, so we can define the wavevector $\mathbf{k'}=\mathbf{k}-\mathbf{K}$. Furthermore, since KAs originate when the denominator in Eq. \[eq:start\] vanishes, we can restrict the integral in the BZ to a small interval of width $2\overline{k}$ around **K** (and **K’**). We can also define the angles $\theta$, between **k$^\prime$** and **q**, and $\theta'$, between $\mathbf{k^\prime}+\mathbf{q}$ and **q**. In refs. [@Piscanec04; @Lazzeri06] it was shown that for phonons close to $\mathbf\Gamma$, the nanotube EPC can be in principle expressed in terms of the graphene EPC $\langle G_\mathbf{\Gamma}^2\rangle_\mathrm{F}= 45.60$ ($eV$/Å)$^2$ [@Piscanec04], modulated by a geometric factor: $$|\tilde G_{(\mathbf{k}+\mathbf{q}), f; \mathbf{k}, i}^{\rm TO/LO}|^2=\langle G_\mathbf{\Gamma}^2\rangle_\mathrm{F}\left[1\pm sign(\epsilon_{\nu, f}\cdot\epsilon_{\nu, i})\cos(\theta+\theta')\right]\label{eq:G}$$ where $$\cos (\theta+\theta')=\frac{k' (k'+q)-(\nu k_\perp)^2}{\sqrt{k'^2+(\nu k_\perp)^2} \sqrt{(k'+q)^2+(\nu k_\perp)^2}} \label{eq:cos}$$ and the $+$($-$) sign has to be considered for TO tangential(LO axial) modes. This expression, developed through a first-neighbor tight-binding model was then quantitatively confirmed by direct DFT calculations [@Piscanec04]. The TO tangential/LO axial terms of the non-analytical part of the dynamical matrix can thus be written as: $$\tilde D_{\mathbf{q}}^{\rm TO/LO}=\frac{a_0^2 \sqrt{3}\langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}}{\pi^2 d} \sum_{\nu, f, i}\int_{-\overline{k}}^{\overline{k}} \mathrm{d}k' \;\frac{f\left[\epsilon_{\nu, i}(k')\right]- f\left[\epsilon_{\nu, f}(k'+q)\right]}{\epsilon_{\nu, i}(k')-\epsilon_{\nu, f}(k'+q)} \cdot \left[1\pm sign(\epsilon_{\nu, f}\cdot\epsilon_{\nu, i})\cos(\theta+\theta')\right] \label{eq:stat}$$ where a factor 2 is included to take into account the contribution of the two equivalent points **K** and **K$^\prime$**. Since we are interested in phonons close to $\Gamma$, we should consider the $\mathbf{q}\rightarrow 0$ limit in Eq. \[eq:stat\]: $$\begin{aligned} \label{eq:statgamma} \tilde D_{\mathbf{\Gamma}}^{\rm TO/LO}&=& \frac{a_0^2 \sqrt{3}\langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}}{\pi^2 d} \int_{-\overline{k}}^{\overline{k}} \mathrm{d}k' \; \left\{ \sum_{\nu, f\ne i}\frac{f\left[\epsilon_{\nu, i}(k')\right]- f\left[\epsilon_{\nu, f}(k')\right]}{\epsilon_{\nu, i}(k')-\epsilon_{\nu, f}(k')} \cdot \left[1\pm sign(\epsilon_{\nu, f}\cdot\epsilon_{\nu, i})\cos(2\theta)\right] + \right. \nonumber \\ %\left. &+& \left. \sum_{\nu, f}\frac{\partial f} {\partial \epsilon} \left[\epsilon_{\nu, f}(k')\right]\cdot \left[1\pm \cos(2\theta)\right] \right\}\end{aligned}$$ where we distinguish between *interband* ($i\ne f$, first line) and *intraband* ($i=f$, second line) transitions, for which we used the following limit: $$\label{DOS1} \lim_{\mathbf{q}\rightarrow 0}\frac{f\left[\epsilon_{\nu, f}(k')\right]-f\left[\epsilon_{\nu, f}(k'+q)\right]} {\epsilon_{\nu, f}(k')-\epsilon_{\nu, f}(k'+q)} =\frac{\partial f} {\partial \epsilon} \left[\epsilon_{\nu, f}(k')\right].$$ Then if the temperature $T\rightarrow 0$, $$\label{DOS2} \frac{\partial f}{\partial \epsilon} \left[\epsilon_{\nu, f}(k')\right]= -\frac{4\pi^2 d}{\sqrt{3}a_0^2}\sum_{\sigma=\pm}DOS_{\nu, f,\sigma}(\epsilon_F)\delta(k'-k_{\nu, f,\sigma}^{F})$$ where $DOS_{\nu, f,\sigma}(\epsilon_F)$ and $k_{\nu, f,\sigma}^{F}$ are the electronic *density of states* (DOS) per $C$ atom and the Fermi wavevector of the $f$ band of $\nu$ index and slope of sign $\sigma$. The intraband EPC contribution to the dynamical matrix at $\Gamma$ is thus proportional to the density of states at the Fermi level. The dynamical matrix at finite **q** can be calculated by using the general expression (\[eq:stat\]); in the following we will focus on the contribution at **q**=0 of the lowest conduction band $\nu$=0, $\tilde D_{\mathbf{\Gamma}}^{0}$, and the second lowest bands $\nu=\pm 1$, $D_{\mathbf{\Gamma}}^{\pm 1}$, as functions of the Fermi energy $$\label{w_first_second} \tilde D_{\mathbf{\Gamma}}(\epsilon_F)= \tilde D_{\mathbf{\Gamma}}^{0}(\epsilon_F)+ \tilde D_{\mathbf{\Gamma}}^{\pm 1}(\epsilon_F)$$ The contribution of higher conduction bands could be in principle included with no difficulty in the same form as the $\nu=\pm 1$ bands, but their effect is negligible for the doping levels considered in this work, up to $N_{el}/C\sim 0.06$, $i.e.$ the value of estimated total charge transfer in the case of alkali intercalation [@Lu04]. Contribution of the $\nu=0$ bands --------------------------------- The bands intersecting at the **K** point of the graphene BZ are at a good approximation linear around **K**, as shown in the previous section, with slopes $\pm\beta= 14.1\cdot \frac{a_0}{2\pi}$ $eV$. For these branches, $\nu=0$, so that the geometric contribution $1\pm\cos(\theta+\theta')$ equals 0 or 2. In practice, in the LO axial case it equals 2 when the transition involves *interband* states with opposite slope, and vanishes for the *intraband* transitions; the opposite behavior occurs in the TO tangential case [@Lazzeri05]. For LO axial modes Eq. \[eq:statgamma\] becomes $$\label{eq:first} \tilde D_{\mathbf{\Gamma}}^{\mathrm{LO},0}=\frac{2 a_0^2 \sqrt{3}}{\pi^2 d} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}\int_{-\overline{k}}^{\overline{k}} \mathrm{d}k' \; \left[ \frac{f(\beta k') -f(-\beta k')}{\beta k'} \right]$$ In the limit $T\rightarrow 0$, one obtains $$\label{eq:statfirst} \tilde D_{\mathbf{\Gamma}}^{\mathrm{LO},0}=\frac{4 a_0^2 \sqrt{3}}{\pi^2 d \beta} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F} \ln\frac{|\epsilon_F|}{|\beta \overline{k}|}$$ The quantitative result depends on the value of $\overline{k}$, which enters the additive constant defined for $\omega_{LO}$ at the end of the next subsection. Since the density of states is independent of energy for the $\nu=0$ band, the non-analytical direct EPC contribution to the TO tangential mode is also a constant, independent of $\epsilon_F$ $$\label{eq:statfirst3} \tilde D_{\mathbf{\Gamma}}^{\mathrm{TO},0}=-\frac{4 a_0^2 \sqrt{3}}{\pi^2 d \beta} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}$$ which we include in the additive constant defined for $\omega_{TO}$. In other words, our model predicts, in the limit of low temperatures, a logarithmic divergence of the dynamical matrix at $\Gamma$ for zero doping in the LO axial case, while the TO tangential modes are simply red-shifted by the non-analytical EPC contribution. Contribution of the $\nu=\pm 1$ bands ------------------------------------- Let us now consider the second lowest $\nu=\pm 1$ bands. Since there is a large gap $E_{11}^M$ between the $\pi$ and $\pi^*$ bands, the interband transitions cannot give rise to KAs, and are expected to contribute negligibly to the Fermi level dependence of the phonon frequencies. On the other hand, in the case of doped nanotubes where the $\nu=\pm 1$ band is partly populated, *intraband* transitions involve a density of states which diverges in one dimension, leading to possible Kohn anomalies. We develop the following analytical expressions based on the ideal hyperbolic $\nu=\pm 1$ bands. According to Eqs. \[eq:statgamma\], \[DOS1\], and \[DOS2\], the contribution of intraband transitions to $\tilde D_{\mathbf{\Gamma}}$ amounts to $$\tilde D_{\mathbf{\Gamma}}^{\mathrm{TO/LO},1}=-8\langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F} DOS_{ 1,\pi^*}(\epsilon_F) \cdot [1\pm\cos(2\theta_F)] \label{eq:nextbands}$$ where the $\nu\pm 1$ degeneracy is taken into account by a factor of 2, and $$DOS_{ 1,\pi^*}(\epsilon_F)=\theta(\epsilon_F - E_{11}^M/2) \frac{2 a_0^2 \sqrt{3}}{\pi \beta d}\frac{\epsilon_F}{\sqrt{\epsilon_F^2-(E_{11}^M/2)^2}}$$ is the total DOS of the $\nu=1, \pi^*$ band (see Eq. \[fermieq\]). The angular factor at the Fermi wavevector is $$\cos(2\theta_F)=\frac{(k^F_{1,\pi^*})^2-k_\perp^2}{(k^F_{1,\pi^*})^2+k_\perp^2}$$ where $k^F_{1,\pi^*}=\pm\frac{1}{\beta}\sqrt{\epsilon_F^2-(E_{11}^M/2)^2}$ is the Fermi momentum of the $\nu=1, \pi^*$ band. We can notice that the EPC direct contribution $\tilde D_{\mathbf{\Gamma}}^{\mathrm{LO},1}$ is proportional to the density of states of the $\nu=\pm 1$ conduction bands at the Fermi energy, modulated by the cosine factor. Kohn anomalies would thus occur *where the density of states diverges*. With a few algebraic calculations one obtains, for LO axial phonons, the compact form: $$\tilde D_{\mathbf{\Gamma}}^{\mathrm{LO},1}=-\frac{8 a_0^2 \sqrt{3}}{\pi^2 d \beta} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F} \frac{(E_{11}^M/2)^2} {\epsilon_F\sqrt{\epsilon_F^2-(E_{11}^M/2)^2}}\cdot\theta(\epsilon_F-E_{11}^M/2) \label{eq:statnextLO}$$ For TO tangential modes through similar calculations one finally obtains: $$\tilde D_{\mathbf{\Gamma}}^{\mathrm{TO},1}=-\frac{8 a_0^2 \sqrt{3}} {\pi^2 d \beta} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F} \frac{\sqrt{\epsilon_F^2-(E_{11}^M/2)^2}}{\epsilon_F}\cdot\theta(\epsilon_F-E_{11}^M/2) \label{eq:statnextTO}$$ In this latter case, the divergence of the DOS is canceled by the cosine factor, which equals 0 for $\epsilon_F=E_{11}^M/2$. In the following subsection, we will thus calculate through this model the $\Gamma$ frequency of the TO/LO modes as: $$\label{omegasum} \omega^\mathrm{TO/LO}_\mathbf{\Gamma}= \sqrt{(\omega_{\mathbf{\Gamma}}^{\mathrm{TO/LO},\mathrm{other}})^2 + \frac{\tilde D_{\mathbf{\Gamma}}^{\mathrm{TO/LO},0} + \tilde D_{\mathbf{\Gamma}}^{\mathrm{TO/LO},1}}{M}},$$ where $\omega_{\mathbf{\Gamma}}^{\mathrm{TO/LO},\mathrm{other}}=\omega_{\mathbf{\Gamma}}+C^{\mathrm{TO/LO}}$ contains the unperturbed value $\omega_{\mathbf{\Gamma}}=1581~cm^{-1}$, plus an additive constant $C^{\mathrm{TO/LO}}$, which also includes the terms discussed at the end of the previous subsection, and is chosen as to match the DFT phonon-EZF results at zero doping; its numerical value, for $\overline{k}=0.1 \frac{2\pi}{a_0}$; is 15.1/39.1 $cm^{-1}$ in the TO/LO case respectively. Results: Kohn anomalies within the adiabatic approximations ----------------------------------------------------------- At this point, the actual determination of the effect of EPC in the vibrational properties of doped nanotubes can be in principle carried out by adopting approximated calculation schemes. The DFT phonon calculation of a SWNT is very demanding and is doable only at a relatively high electronic temperature, where the potentially interesting behaviors are smeared out. However, the comparison of the calculations done on a real SWNT with those done using the phonon-EZF (Fig. \[compare\]) shows that a quantitative determination of the phonon frequency can be done neglecting the nanotube curvature. Now, we will analize an integral model which takes another step in identifying the key role played by graphene-derived EPC in the vibrational properties of SWNTs, and its low-temperature limit can be put into an analytical expression. We report in Fig. \[adiabatic\] the frequencies of the LO axial and the TO tangential modes of the $G$ peak at an electronic temperature of 315 K as a function of the Fermi energy, and thus of the electronic doping, as calculated $i)$ from the phonon-EZF method; $ii)$ from the numerical integration of Eq. \[eq:stat\] at T=315 K (polynomial fits are used to describe the electronic bands); $iii)$ from its analytical limit at T=0 K, where we use the ideal conical bands (Eqs. \[eq:statfirst\],  \[eq:statfirst3\], \[eq:statnextLO\], \[eq:statnextTO\], and \[omegasum\]), with $E_{11}^M/2$ red-shifted by 0.06 $eV$ as explained in subsection \[subsectionFermi\]. ![\[adiabatic\] (Color online) Variation, within the adiabatic approximation, of the LO axial (black) and TO tangential (red) $\Gamma$ frequencies with the Fermi level at T=315 K, calculated from DFT phonon-EZF (full diamonds), from the integral model (solid line, Eq. \[eq:stat\]), and from the analytical model at T=0 K (dashed line, Eqs. \[eq:statfirst\], \[eq:statnextLO\], and \[eq:statnextTO\]). The metal/carbon composition corresponding to the Fermi level is indicated on the top of the figure.](adiabatic.eps){width="8.25cm"} A first inspection of the graphs indicates that the agreement within the different approaches, and in particular between the DFT calculations and the integral model, is excellent. The LO axial frequency drops at $\epsilon_F=0$ due to a Kohn anomaly arising from the first conduction band, and $\tilde D_{\mathbf{\Gamma}}^{\mathrm{LO},0}$ varies as $\ln \epsilon_F$ according to Eq. \[eq:statfirst\]. Another KA occurs when the Fermi energy reaches the minimum of the second bands $E_{11}^M/2=0.83$ $eV$. This second KA is much stronger than that at $\epsilon_F=0$, in agreement with Eq. \[eq:statnextLO\] which predicts a variation going as $-1/\sqrt{\epsilon_F^2-(E_{11}^M/2)^2}$. A frequency drop of about 40 $cm^{-1}$, with respect to zero doping, was reported in a theoretical calculation of alkali-doped SWNTs [@Akdim05] at a doping level of about $N_{el}/C\sim 0.02$, in good agreement with our results (see Fig. \[compare\]). Other KAs occur each time the Fermi level increases such that a new conduction band is populated. In TO tangential modes, due to the vanishing of the EPC interband matrix element, the first band does not give rise to a KA; the curve is practically flat until $E_{11}^M/2$. The effect of the second conduction band induces a variation going as -$\sqrt{\epsilon_F^2-(E_{11}^M/2)^2}$ (Eq. \[eq:statnextTO\]). However, as we will see in the following section, although the physics described by these results seems very intriguing, the main outcome of this comparison is the accuracy of the integral model, with respect to full DFT calculations, in predicting the Raman properties of SWNTs, and in particular the anomalous doping dependence of the vibrational axial mode. Non-adiabaticity {#nonadia} ================ Time-dependent perturbation theory ---------------------------------- The vast majority, if not the totality, of *ab initio* phonon-calculations rely on the so-called Born-Oppenheimer adiabatic approximation. In this framework one can decouple the electron motion from the ion dynamics, on the basis of their large mass and velocity difference, and thus treat the electronic properties as they were completely independent of the ionic motion. This approximation for phonon calculation is equivalent to first-order time-[*independent*]{} perturbation theory and is wholly justified for insulators and semiconductors. Even in ordinary metals a proper, specific treatment of the electronic degrees of freedom, such the inclusion of a finite Fermi-Dirac electronic temperature, usually suffices to avoid a failure of the BO approximation. Metallic SWNTs do however represent an “extraordinary” case, due to their intrinsic one-dimensional nature, and the use of the adiabatic BO approximation leads to significant deviations from experimental results. This could be particularly true in the case of Raman scattering where a sinusoidal excitation induces the oscillatory motion of ions. In order to check whether a proper inclusion of non-adiabaticity would affect our previous results, we introduce a non-adiabatic model, based on time-[*dependent*]{} perturbation theory. In the process of absorption and emission of a phonon by the electrons, the phonon energy is no longer neglected: we replace, in Eq. \[eq:stat\], the energy difference between the two electronic scattering states,$(\epsilon_{\mathbf{k}, i}-\epsilon_{\mathbf{k}+\mathbf{q}, f})$, by $(\epsilon_{\mathbf{k}, i}-\epsilon_{\mathbf{k}+\mathbf{q}, f}+\hbar \omega_\mathbf{q}+i\delta)$. Here we add a small imaginary part $\delta$ to the energy to control the divergences. Eq. \[eq:stat\] then becomes: $$\tilde D_{\mathbf{q}}^{\rm TO/LO}=\frac{a_0^2 \sqrt{3}\langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}}{\pi^2 d} \sum_{\nu, f, i}\int_{-\overline{k}}^{\overline{k}} \mathrm{d}k' \;\frac{f\left[\epsilon_{\nu, i}(k')\right]- f\left[\epsilon_{\nu, f}(k'+q)\right]}{\epsilon_{\nu, i}(k')-\epsilon_{\nu, f}(k'+q)+\hbar\omega_\mathbf{q}+i\delta} \cdot \left[1\pm sign(\epsilon_{\nu, f}\cdot\epsilon_{\nu, i})\cos(\theta+\theta')\right] \label{eq:dyn}$$ The frequency is obtained as $\omega_\mathbf{q}= \sqrt{\frac{\Re\mathrm{e} (D_{\mathbf{q}})}{M}}$, that is, we take the principal part of the above integral by letting $\delta\rightarrow 0$. Since $D_{\mathbf{q}}$ depends on the frequency, this equation should be solved self-consistently. ![\[transitions\]. Schematic representation of electronic transitions allowed by conservation of energy and momentum in electron-phonon scattering processes involving a phonon with momentum **q** and energy $\hbar\omega_\mathbf{q}$. An abrupt change in the phonon dispersion, [*i.e.*]{} a Kohn anomaly, occurs when, by change of the Fermi energy, one of this transitions becomes allowed or forbidden by the Pauli exclusion principle. In practice, this occurs when the Fermi level crosses a tip of one of the arrows corresponding to allowed transitions. Top panel: transitions at **q**=**0** ($\mathbf\Gamma$). Bottom panel: transitions at a finite **q**-point.](transitions.eps){width="8.25cm"} In the $\mathbf{q}\rightarrow \mathbf{0}$ limit, Eq. \[eq:statgamma\] then becomes, in the NA case, $$\label{eq:dyngamma} \tilde D_{\mathbf{\Gamma}}^{\rm TO/LO}= \frac{a_0^2 \sqrt{3}\langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}}{\pi^2 d} \int_{-\overline{k}}^{\overline{k}} \mathrm{d}k' \; \left\{ \sum_{\nu, f\ne i}\frac{f\left[\epsilon_{\nu, i}(k')\right]- f\left[\epsilon_{\nu, f}(k')\right]}{\epsilon_{\nu, i}(k')-\epsilon_{\nu, f}(k')+\hbar\omega_\mathbf{\Gamma}+i\delta} \cdot \left[1\pm sign(\epsilon_{\nu, f}\cdot\epsilon_{\nu, i})\cos(2\theta)\right]\right\}$$ Here only the interband transitions contribute to the dynamical matrix, since, for an optical phonon, the limit $$\label{DOS3} \lim_{\mathbf{q}\rightarrow 0}\frac{f\left[\epsilon_{\nu, f}(k')\right]-f\left[\epsilon_{\nu, f}(k'+q)\right]} {\epsilon_{\nu, f}(k')-\epsilon_{\nu, f}(k'+q)+\hbar\omega_\mathbf{q}+i\delta} =0$$ We can anticipate that the absence of the intraband contributions dramatically affects the results when such terms are important, *i.e.* when the Fermi level is close to a $E_{\nu\nu}^M/2$ band minimum. As in the previous section, we study the contribution of the $\nu=0,\pm 1$ bands to the non-analytical part of the dynamical matrix. The mode frequencies are obtained from the dynamical matrix as in Eq. \[omegasum\]. Contribution of the $\nu=0$ bands --------------------------------- Eq. \[eq:dyngamma\] can be integrated, by using linear bands and $T=0$, analogously to the calculations developed in the static case; we thus obtain for the non-analytical part of $D_\mathbf{\Gamma}$ in the case of LO axial phonons $$\tilde D_{\mathbf{\Gamma}}^{\mathrm{LO},0}=\frac{2 a_0^2 \sqrt{3}}{\pi^2 d \beta} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F} \ln\frac{|2\epsilon_F+\hbar\omega^\mathrm{LO}_\mathbf{\Gamma}||2\epsilon_F- \hbar\omega^\mathrm{LO}_\mathbf{\Gamma}|}{|2\beta \overline{k}|^2} \label{eq:dynfirst}$$ while the TO tangential term $\tilde D_{\mathbf{\Gamma}}^{\mathrm{TO},0}$ is zero since it involves the intraband terms only. We note that the KA observed in the static case for zero doping is replaced by two logarithmic divergences at a doping level $\epsilon_F=\pm\hbar \omega^\mathbf{LO}_\mathbf{\Gamma}/2$, which is $N_{el}/C\approx 0.0015$ ($\sim$ MC$_{\rm 650}$). As shown schematically in the top panel of Fig. \[transitions\], $\Gamma$ transitions conserving energy and momentum in the scattering process can indeed occur when connecting two bands separated by an energy equal to $\hbar\omega_\mathbf{\Gamma}$, which is in practice the case only in correspondence of the electronic vertical transitions (**q**=**0**) between the $\pi$ and the $\pi^*$ bands. Contribution of the $|\nu|\ge1$ bands ------------------------------------- The effect of the non-adiabatic terms in the $\nu=0$ bands contribution merely consists in a splitting of the zero-doping Kohn anomaly observed in the static case. The picture changes completely for the $|\nu|\ge1$ bands. In fact, given the energy gap between the $\pi$ and $\pi^*$ bands having $\nu\neq 0$, only the intraband terms are relevant to Kohn anomalies. These intraband terms are suppressed by the dynamical effects (Eq. \[eq:dyngamma\]) in both LO and TO modes and thus $\tilde D_{\mathbf{\Gamma}}^{\mathrm{LO,TO},\nu\neq 0}=0$. Indeed in the hyperbolic band it is not possible to conserve both energy and momentum in an electron-phonon scattering process with a ${\rm q}=0$ optical phonon, see Fig. \[transitions\]. In addition to the analytical calculation at T=0, we performed numerical integrations of Eq. \[eq:dyngamma\] at different temperatures (77 K, 315 K). As in the static case, we used polynomial fits to describe the band dispersions. We report in Fig. \[nonadiaGamma\] the variation of the LO axial mode as function of the Fermi level in the non-adiabatic case at those two temperatures. The results are consistent with the analytical ones obtained at vanishing temperature; the only Kohn anomaly is observed, as explained in the previous subsection, at $\hbar\omega_\mathbf{\Gamma}/2$. All the other Kohn anomalies, that in the adiabatic approximation and thus in all standard DFT calculations are due to the population of new energy levels, completely disappear. This is a general behavior for vertical transitions. LO phonon linewidth ------------------- An important measurable quantity is the phonon linewidth. In ref. [@Lazzeri06] it is shown that it can be split into a EPC direct contribution, only relevant in metallic nanotubes, and a term due to inhomogeneous broadening and anharmonicites, common to all nanotubes and estimated around 10$~cm^{-1}$ from experimental data. The LO axial phonon linewidth (full width at half maximum) $\gamma^{EPC}_{LO}$ as function of the electronic doping can instead be calculated [@JonesMarch] from the imaginary part of the non-adiabatic dynamical matrix, Eq. \[eq:dyn\], as $\gamma_{\mathbf{q}}=|\Im(\tilde D_{\mathbf{q}})|/(\omega_{\mathbf{q}} M)$. An identical result is obtained by using the Fermi golden rule, as in ref. [@Lazzeri06]. The result of ref. [@Lazzeri06] derived in nanotubes for zero doping can be easily generalized for any doping as: $$\begin{aligned} \gamma^\mathrm{EPC}_\mathrm{LO}(\epsilon_F) &=& \frac{2 a_0^2 \sqrt{3}}{\pi d \beta M \omega^\mathrm{LO}_\mathbf{\Gamma}} \langle G_\mathbf{\Gamma}^2 \rangle_\mathrm{F}\left[\frac{1}{e^{(\epsilon_F-\hbar\omega^\mathrm{LO}_\mathbf{\Gamma}/2)/k_B T}+1}-\frac{1}{e^{(\epsilon_F+\hbar\omega^\mathrm{LO}_\mathbf{\Gamma}/2)/k_B T}+1}\right] \\ \nonumber &=& \frac{79~cm^{-1} nm}{d} \left[\frac{1}{e^{(\epsilon_F-\hbar\omega^\mathrm{LO}_\mathbf{\Gamma}/2)/k_B T}+1}-\frac{1}{e^{(\epsilon_F+\hbar\omega^\mathrm{LO}_\mathbf{\Gamma}/2)/k_B T}+1}\right]\end{aligned}$$ At zero doping this FWHM equals, at T=315 K, about 60 $cm^{-1}$ for a (9,9) SWNT. We show the EPC LO linewidth in the top panel of Fig. \[nonadiaGamma\]. At $T\rightarrow 0$ this term is constant up to the Kohn anomaly at a $N_{el}/C\approx 0.0015$ (MC$_{\rm 650}$) doping level, and vanishes abruptly for higher doping levels, while a smoother behavior is observed at finite temperature. Since all the terms contributing to the EPC part of the dynamical matrix vanish for the TO tangential modes, the corresponding phonon linewidth will vanish as well. ![\[nonadiaGamma\] (Color online) Lower panel: variation of the LO axial ($G^-$ peak in Raman experiments, black) and TO tangential ($G^+$ peak, red) $\Gamma$ frequencies with the Fermi level at T=315 K (solid line) and at 77 K (LO axial, dashed line), calculated from the **non-adiabatic** integral model (Eq. \[eq:dyn\]). Top panel: variation, at the same temperatures, of the EPC contribution to the FWHM of the LO axial mode with the doping level. The metal/carbon composition corresponding to the Fermi level is indicated on the top of the figure.](nonadiabatic_Gamma.eps){width="8.25cm"} Finite-q results ---------------- In the previous subsections, we have developed our analytical model at the $\Gamma$ point of the Brillouin zone, and we have shown that the proper treatment of non-adiabatic terms lifts all the divergences resulting from intraband, vertical transitions. However, as shown schematically in the lower panel of Fig. \[transitions\], at finite **q** the outcome is different from the $\Gamma$ case. Indeed, non-vertical intraband transitions conserving energy and momentum can occur when **q** is approximately larger than $\omega_q/\mathrm{v_{M}}$, $\mathrm{v_{M}}$ being the maximum electronic band velocity. We report in Fig. \[nonadia\_q\] the dependence on electronic doping of the LO axial frequency at **q**=$0.05 \cdot \frac{2\pi}{a_0}$ and $0.1 \cdot \frac{2\pi}{a_0}$, calculated numerically through Eq. \[eq:dyn\] at T=77 K and 315 K. In this case the intraband Kohn anomalies observed in the adiabatic approximation are not suppressed, as it is the case for vertical transitions. The interband KA is shifted towards higher doping levels with respect to the $\Gamma$ case. The intraband KAs are shifted in the same direction with respect to the energy levels $E_{\nu\nu}^M/2$. Both interband and intraband Kohn anomalies appear in doublets, separated by $\hbar \omega_\mathbf{q}$. By using the expressions for EPC derived in Refs. [@Piscanec04; @Lazzeri05] for the scattering involving phonons near the **K** point, it can be shown that the behavior of the highest optical phonon at a wavevector **q**+**K** is very similar to the one reported in Fig. \[nonadia\_q\] for the LO axial frequency at **q**. This phonon near the **K** point is responsible for the $D$ band observed experimentally in Raman spectra. ![\[nonadia\_q\] (Color online) Lower panel: variation, with respect to zero doping, of the LO axial frequencies with the Fermi level at T=315 K (black) and T=77 K (red), calculated from the **non-adiabatic** integral model (Eq. \[eq:dyn\]). The phonon momentum is equal to **q**=0.1 $\cdot \frac{2\pi}{a_0}$. Top panel: same, at **q**=0.05 $\cdot \frac{2\pi}{a_0}$. The dotted vertical line indicates the minimum of the second conduction band. The energy values $\mu_{1-}$, $\mu_{1+}$, $\mu_{2-}$, and $\mu_{2+}$ correspond to the levels schematically shown in Fig. \[transitions\]. A similar behavior is expected for the highest optical phonon near the **K** point, which is responsible for the Raman $D$ band. The metal/carbon composition corresponding to the Fermi level is indicated on the top of the figure.](nonadiabatic_q.eps){width="8.25cm"} Discussion {#conclusions} ========== Physical properties of metallic nanotubes ----------------------------------------- In this work we presented a combination of *ab initio* DFT calculations with an integral and analytical model developed to take into account in a simple way the direct EPC contribution to the dynamical matrix, responsible for the divergences originating the KAs. Since however the mathematics behind our models might be cumbersome, here we summarize the most interesting *physical* properties of metallic SWNTs we observe, as well as the quantitative results on experimentally measurable quantities. We showed that KAs in SWNTs can be identified by considering electron-phonon scattering processes that conserve both energy and momentum (see Fig. \[transitions\] and relative discussion). Since there are strong disagreements between the adiabatic and the non-adiabatic case, only the results obtained through the latter model are reliable. The failure of the adiabatic calculation originates from the fact that, in this approximation, the phonon energy is neglected and the energy conservation in the electron-phonon scattering processes is thus violated. Since the curves shown in Figs. \[compare\] and \[adiabatic\] are obtained through the adiabatic model and through DFT frozen-phonon and DFPT calculations, those results should not be used for comparison to experimental data. On the other hand, all the results presented in Figs. \[nonadiaGamma\] and  \[nonadia\_q\], which conflict with any existing DFT calculation based on the BO approximation, are in principle *correct*, and are thus meant to be directly compared to experiments. The results shown in Fig. \[nonadiaGamma\] promise to be useful tools to determine experimentally the doping level of individual metallic SWNTs. In particular, by measuring the shift of the $G^-$ peak with respect to zero doping and its linewidth, one can determine the variation of the Fermi energy and thus the doping level. Those results refer to a (9,9) nanotube of diameter 1.23 $nm$; however, they can be easily generalized to any $d$-diameter metallic SWNT by observing that the dependence of both the linewidth of the $G^-$ peak and its frequency shift with respect to zero doping case are proportional to $1/d$ and independent of the chirality. We note here that we are assuming that the $G$ peak originates from single resonance. It was showed that carbon nanotubes [@Lu04; @Margine06], as graphite [@Posternak83; @Csanyi05], present a free-electron-like interlayer state whose energy level decreases rapidly with increasing doping, and reaches the Fermi energy at about the saturation doping of $\sim$MC$_{\rm 8}$. We showed that KAs occur at $\mathbf\Gamma$ only when the Fermi energy is in proximity of a band crossing with non-zero interband electron-phonon coupling. Besides the crossing of the $\pi$-bands at zero doping, we estimate that the first band-crossing with symmetry-allowed transitions is that between the interlayer level and the $\pi^*$ ($\nu=0$) bands. Such crossing is a candidate for a Kohn anomaly. This could explain the strong softening of about $50~cm^{-1}$ observed at saturation doping in experimental Raman studies on the $G$ peak in alkali-doped SWNTs [@Bendiab01; @Sauvajol03]. Finally, we predicted KAs to occur in phonons with *finite* momentum **q** not only in proximity of a band crossing, but also each time a new band is populated (see Fig. \[nonadia\_q\]). Therefore, by experimentally probing such KAs, the filling of the hyperbolic bands could be detected. Phonons at finite-**q** vectors are experimentally accessible to double-resonant Raman scattering [@Thomsen00], and correspond in the Raman spectra, *e.g.*, to the defect-activated $D$ peak at about 1350 $cm^{-1}$, and to the second-order (two-phonon) $2D$ and $2G$ peaks at approximately 2700 and 3200 $cm^{-1}$. We showed in Fig. \[nonadia\_q\] the variation of the phonon frequency with respect to doping at **q**=$0.05 \cdot \frac{2\pi}{a_0}$ and $0.1 \cdot \frac{2\pi}{a_0}$. These phonon momenta are experimentally accessible to double-resonant Raman scattering [@Maultzsch03]. Although our computational study has been limited to a metallic SWNT, our analytical results allow to predict the behavior of the high-energy Raman peaks in semiconducting nanotubes. Since in this case only intraband terms contribute to the EPC term of the dynamical matrix, we expect the $G$ peak to be unaffected by electron doping. On the other hand, analogously to the metallic case, the defect-activated $D$ peak should be sensitive to doping, provided that the excitation energy is sufficient to populate the first excited electronic level in the conduction band. Conclusions ----------- In this work we presented a theoretical study on the vibrational/Raman properties of electron-doped SWNTs using Density-Functional Theory and analytical models. We performed our calculations within the adiabatic Born-Oppenheimer approximation, but we also used time-dependent perturbation theory to explore non-adiabatic effects beyond this approximation. We showed that the Born-Oppenheimer approximation, predicts, for increasing doping levels, a series of EPC-induced KAs in the vibrational mode parallel to the tube axis at the $\mathbf\Gamma$ point of the Brillouin zone, usually indicated in Raman spectroscopy as the $G^-$ peak. Such Kohn anomalies would arise each time a new conduction band is populated. However, we showed that they are an artifact of the adiabatic approximation, which is the standard approach for ab-initio phonon calculations. The inclusion of non-adiabatic effects dramatically affects the results, predicting KAs at $\mathbf\Gamma$ only when $\epsilon_F$ is close to a band crossing $E_{X}$. For each band crossing a double KA occurs for $\epsilon_F=E_{X}\pm \hbar\omega/2$, where $\hbar\omega$ is the phonon energy. In particular, for a 1.2 $nm$ metallic nanotube, we predicted a KA to occur in the so-called $G^-$ peak at a doping level of about $N_{el}/C=\pm 0.0015$ atom ($\epsilon_F\approx \pm 0.1 ~eV$) and, possibly, close to the saturation doping level ($N_{el}$/$C$$\sim$ 0.125), where an interlayer band crosses the $\pi^*$ nanotube bands. Furthermore, we predicted that the Raman linewidth of the $G^-$ peak significantly decreases for $|\epsilon_F| \geq \hbar\omega/2$. Thus our results provide a tool to determine experimentally the doping level from the value of the frequency shift and from the linewidth of the $G^-$ peak. Finally, we predicted Kohn anomalies to occur in phonons with finite momentum **q** not only in proximity of a band crossing, but also each time a new band is populated. Such Kohn anomalies should be observable in the double-resonant Raman peaks, such as the defect-activated $D$ and peak, and the second-order peaks $2D$ and $2G$. We also predict that in semiconducting nanotubes the $G$ peak should be insensitive to doping, while the $D$ peak should be affected analogously to the metallic case. We thank N. Bendiab, S. Piscanec and A. C. Ferrari for useful discussions. DFT calculations have been performed at the IDRIS French National Computational Facility under the projet CP9-61387. [41]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , , , ** (, ). , , , ** (, ). , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , ****, (). , ** (), . , , , , ****, (). , , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , , , ****, (). , , , , ****, (). . , , , ****, (). , , , , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). . . . , , , , , ****, (). , , , , (), . , ****, (). , , , ****, (). , ****, (). , , , , ****, (). K.-P. Bohnen, R. Heid, H.J. Liu, and C.T. Chan, Phys. Rev. Lett. [**93**]{}, 245501 (2004). D. Connétable, G.-M. Rignanese, J.-C. Charlier, and X. Blase, Phys. Rev. Lett. [**94**]{}, 015503 (2005). , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , , , , , , , . , ****, (). , , , , ****, (). , , , , , ****, (). , ** (, ). , , , , , ****, (). , , , , , ****, (). , , , , ****, (). , ****, ().

--- abstract: | Photoproduction of $ K\Sigma^{*}(1385)$ on the nucleon is investigated within the Regge framework and the reaction mechanism is analyzed based on the data existing in the channels $\gamma p\to K^+\Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$. The Reggeization of the $t$-channel meson exchanges $K(494)+K^*(892)+K_2^*(1430)$ is employed to construct the photoproduction amplitude. The Rarita-Schwinger formalism is applied for the spin-3/2$^+$ strangeness-baryon $\Sigma^*$ with a special gauge prescription utilized for the convergence of these reaction processes. Within a set of coupling constants determined from the symmetry arguement for the $K$ and $K^*$ and from the duality and vector dominance for the $K_2^*$, the data of the both processes are reproduced to a good degree. The production mechanism of these processes are featured by the dominance of the contact term plus the $K$ exchange with the role of the $K_2^*$ following rather than the $K^*$. author: - 'Byung-Geel Yu' - 'Kook-Jin Kong' title: ' Photoproduction of $\gamma N\to K^+ \Sigma^*(1385)$ in the Reggeized framework ' --- Introduction ============ Kaon photoproduction off the nucleon target has been a useful tool to investigate strangeness production with data on a clean background from the electromagnetic probe. The experimental studies of the reactions involving $\Lambda(1116)$, and $\Sigma(1190)$ hyperons, or their resonances in the final state have been extensively conducted up to recent at the electron/photon accelerator for hadron facilities [@mc; @glander; @bradford; @moriya]. Of recent experimental achievements on these reactions the measurements of reaction cross sections for the $\gamma p\to K^+\Sigma^{*0}(1385)$ process from the CLAS [@moriya; @mattione] and LEPS [@niiyama], and the $\gamma n\to K^+\Sigma^{*-}(1385)$ process from the LEPS [@hicks] Collaboration draw our attention. In these reactions One reason for our interest is an advantage of studying baryon resonances whose existences have been predicted by the quark model, but are still missing, or remain an indefinite state. On the other hand, these reactions have their own issues of how to deal with the spin-3/2 baryon resonance in describing the reaction, because the propagation of the spin-3/2 resonance would give rise to a divergence as the reaction energy increases [@bgyu-pi-delta; @bgyu-rho-delta]. Theoretical investigation of baryon resonances in the $\gamma p\to K^+\Sigma^{*0}$ process was carried out in Ref. [@ysoh], where a set of $\Delta$ and $N^*$ resonances was considered in the effective Lagrangian approach. In this pioneering work the role of the baryon resonances was analyzed up to spin-5/2 state in the $s$- and $u$-channel contributions to the reaction process. Meanwhile, as an extension to the high energy realm a Regge plus resonance approach was applied for the $\gamma p\to K^+\Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$ processes in Refs. [@junhe; @wang] with the empirical data up-dated by the recent experiments. However, in these works, the description of the reactions was complicated by using a hybrid-type propagation which mixed the pure Regge-pole and the Feynman propagator in the $t$-channel, apart from the cutoff functions to suppress the divergence at high energies, as in Ref. [@ysoh]. In this paper, we investigate photoproduction of $K\Sigma^*$ in two different isospin channels, $\gamma p \to K^+ \Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$, where the Reggeization of the $t$-channel meson exchange is exploited for the photoproduction amplitude at forward angles and high energies. Our focus here is to describe these reaction processes up to high energy without fit-parameters rather than to search for baryon resonances, because their roles in these reactions are found to be less important as discussed in Ref. [@ysoh]. Avoiding such complications as mentioned above, we will utilize the model of the $\gamma N\to \pi^\pm\Delta$ in Ref. [@bgyu-pi-delta] to apply to the present processes with the coupling constant $f_{KN\Sigma^*}$ considered from the SU(3) symmetry. Since the $\Sigma^*$ of $3/2^+$ is the lowest mass hyperon in the baryon decuplet, this will be a valuable test of the flavor SU(3) symmetry with an expectation that the production mechanism of $K\Sigma^*$ is essentially identical to the $\pi\Delta$ case. For the analysis of the process involving the spin-3/2 baryon resonance, in particular, it is worth asking how to describe the process without cutoff functions because they could sometimes hide the pieces of the reaction mechanism that are missing, or malfunctioning through the adjustment of the cutoff masses. From the previous studies on photoproduction of $\pi\Delta$ [@bgyu-pi-delta] we have learned two important things as to the dynamical feature of the spin-3/2 baryon photoproduction: The minimal gauge prescription is the one requisite for a convergence of the reaction cross section and the other is the role of the tensor meson $a_2(1320)$ significant in the high energy region. Therefore, as a natural extension of the model in Ref. [@bgyu-pi-delta] to strangeness sector, we here consider the $K(494)+K^*(892)+K_2^*(1430)$  exchanges in the $t$-channel to analyze the production mechanism of the $\gamma p\to K^+\Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$ processes. This paper is organized as follows. In Sec. II, we discuss the construction of the photoproduction amplitude in association with the gauge-invariant $K$ exchange in the $t$-channel. This will include a brief introduction of the minimal gauge, and the new coupling vertex for the tensor meson interaction $K_2^* N \Sigma^*$ which has been missed in previous works. Numerical results in the total and differential cross sections as well as the beam polarization asymmetry are presented for both reactions in Sec. III. We give a summary and discussion in Sec. IV. The SU(3) coefficients for the octet and decuplet baryons coupling to octet mesons are given in the Appendix. formalism ========= For a description of the reaction, $$\begin{aligned} &&\gamma(k)+N(p)\to K(q)+\Sigma^{*}(p'),\end{aligned}$$ with the momenta of the initial photon, nucleon and the final $K$ and $\Sigma^*$ denoted by $k$, $p$, $q$, and $p'$, respectively, we first construct the photoproduction amplitude which is gauge invariant as to the coupling of photon with particles in the reaction process. Then, the Reggeization of the $t$-channel meson-pole follows as has been done before. ![Feynman diagrams for $\gamma N \to K^+ \Sigma^{*}$. The exchange of $K$ in the $t$-channel $(a)$, the proton-pole in the $s$-channel $(b)$, the $\Sigma^*$-pole in the $u$-channel $(c)$, and the contact term $(d)$ are the basic ingredients for gauge invariance of the reaction. The $K^*$ and $K_2^*$ exchanged in the $t$-channel $(a)$ are themselves gauge-invariant.[]{data-label="fig1"}](fig1.eps){width="0.6\hsize"} Photoproduction amplitude ------------------------- Viewed from the $t$-channel meson exchange the Born amplitudes in four different isospin channels are read as $$\begin{aligned} &&{M}_{\gamma p\to K^{+}\Sigma^{*0}}={M}_K+{M}_{K^*}+{M}_{K_2^*},\label{p+}\\ % &&{M}_{\gamma n\to K^{+}\Sigma^{*-}}=\sqrt{2}\left({M}_K+{M}_{K^*}+{M}_{K_2^*}\right),\label{n+}\\ % &&{M}_{\gamma p\to K^{0}\Sigma^{*+}}=-\sqrt{2}\left({ M}_{K^*}+{M}_{K_2^*}\right),\label{p0}\\ % &&{M}_{\gamma n\to K^{0}\Sigma^{*0}}=-\left({M}_{K^*}+{M}_{K_2^*}\right),\label{n0}\end{aligned}$$ where the $\sqrt{2}$ factors and signs result from our convention of the meson-baryon-decuplet coupling of the $10-8-8$ type presented in the Appendix. Hereafter, we call the reaction process in Eq. (\[p+\]), the $\gamma p$ process, and the process in Eq. (\[n+\]), the $\gamma n$ process, respectively. In experimental sides, the cross sections for total and differential were measured recently for the charged state Eq. (\[p+\]) at the CLAS [@moriya] and LEPS [@niiyama] Collaborations, and the differential cross section and the beam asymmetry were measured for the process in Eq. (\[n+\]) at the LEPS Collaboration [@hicks]. There exist data from the CBCG [@crouch] and ABBHHM Collaboration [@erbe-nc; @erbe-pr] in the pre-1970’s where the total cross section for the charged process in Eq. (\[p+\]) as well as the total and differential cross sections for the process in Eq. (\[n+\]) reported by the ABHHM Collaboration [@benz]. Therefore, these data will be of use to constrain the physical quantities such as the coupling constants in the reaction once the trajectories of the Regge-poles for $K$, $K^*$, and $K_2^*$ are chosen. $K(494)$ exchange ----------------- For nucleon, kaon, and $\Sigma^*$ charges, the current conservation following the charge conservation, $e_N-e_{K}-e_{\Sigma^*}=0$, requires that the $\gamma p$ process includes the proton-pole in the $s$-channel and the contact term for gauge-invariance of the $t$-channel $K$ exchange. Similarly the $\gamma n$ process includes the $u$-channel $\Sigma^*$-pole and the contact term in addition to the $K$ exchange, respectively. These are depicted in Fig. \[fig1\]. Thus, the gauge-invariant $K$ exchange in the $t$-channel for these reactions are given by $$\begin{aligned} \label{pk+} &&iM_K^{\gamma p}=\overline{u}_{\nu}(p') i\left[{M}^{\nu\mu}_{t(K)}+M^{\nu\mu}_{s(N)}+ {M}^{\nu\mu}_c\right] \epsilon_\mu(k)u(p),\\ % &&iM_K^{\gamma n}=\overline{u}_{\nu}(p')i\left[{M}^{\nu\mu}_{t(K)}+ M^{\nu\mu}_{u(\Sigma^*)}+ {M}^{\nu\mu}_c\right] \epsilon_\mu(k)u(p),\label{nk+}\ \ \\end{aligned}$$ where $$\begin{aligned} &&i{M}^{\nu\mu}_{s(N)}= \Gamma_{K N\Sigma^*}^{\nu}(q) \frac{\rlap{/}p+\rlap{/}k+M_N}{s-M^{2}_{N}}\Gamma^\mu_{\gamma NN}(k),\label{nucleon}\\ % && i{M}^{\nu\mu}_{t(K)} =\Gamma^{\mu}_{\gamma KK}(q,Q) \frac{1}{t-m^{2}_{K}} \Gamma_{K N\Sigma^*}^{\nu}(Q), \label{kaon}\\ % &&i{M}^{\nu\mu}_{u(\Sigma^*)}= \Gamma_{\gamma\Sigma^*\Sigma^*}^{\nu\mu\sigma}(k)\frac{\rlap{/}p'-\rlap{/}k+M_{\Sigma^*}} {u-M^2_{\Sigma^*}}\Pi^{\Sigma^*}_{\sigma\beta}(p'-k)\nonumber\\&&\hspace{1.5cm}\times \Gamma_{K N\Sigma^*}^{\beta}(q), \label{sigma33}\hspace{0.5cm} %\end{aligned}$$ with $Q^\mu=(q-k)^\mu$, the $t$-channel momentum transfer, and the spin-3/2 projection which is given by $$\begin{aligned} \label{sigma-propagator} \Pi_{\Sigma^*}^{\mu\nu}(p)=-g^{\mu\nu}+\frac{\gamma^{\mu}\gamma^{\nu}}{3} +\frac{\gamma^{\mu}p^{\nu}-\gamma^{\nu}p^{\mu}}{3M_{\Sigma^*}} +\frac{2p^{\mu}p^{\nu}}{3M^{2}_{\Sigma^*}}\,.\ \\end{aligned}$$ Here $u_\nu(p')$, $u(p)$, and $\epsilon_\mu(k)$ are the spin-3/2 Rarita-Schwinger field for the $\Sigma^*(1385)$, Dirac spinor for nucleon, and the spin polarization of photon, respectively. The charge-coupling vertices $\gamma NN$, $\gamma\Sigma^*\Sigma^*$ and $\gamma KK$ [@bgyu-pi-delta] are given as follows, $$\begin{aligned} &&\epsilon_\mu\Gamma^\mu_{\gamma NN}=e_N\rlap{/}\epsilon,\label{gammann}\\ % &&\epsilon_\mu\Gamma_{\gamma\Sigma^*\Sigma^*}^{\nu\mu\sigma} =-e_{\Sigma^*}\left(g^{\nu\sigma}\rlap{/}\epsilon-\epsilon^\nu\gamma^\sigma -\gamma^\nu \epsilon^\sigma +\gamma^\nu\rlap{/}\epsilon \gamma^\sigma\right),\label{gamma33}\\ % &&\epsilon_\mu\Gamma^{\mu}_{\gamma KK}(q,Q)=e_K (q+Q)^\mu \epsilon_\mu\,, \label{gkk}\end{aligned}$$ where $e_N$, $e_{\Sigma^*}$, and $e_K$ are the nucleon, $\Sigma^*$ and kaon charges, respectively. For the strong coupling vertex $K N\Sigma^*$ we use $$\begin{aligned} \Gamma^\nu_{K N\Sigma^*}(q)={f_{K N\Sigma^*}\over m_K}q^\nu,\end{aligned}$$ and neglect the off-shell effect of the spin-3/2 Rarita-Schwinger field for simplicity. Then, the contact term is given by $$\begin{aligned} \label{gau3} i{M}_c^{\nu\mu}=-e_K\frac{f_{K N\Sigma^*}}{m_K}\ g^{\nu\mu}\,.\end{aligned}$$ Note that the charge-coupling terms in Eqs. (\[gammann\]), (\[gamma33\]), and (\[gkk\]) satisfy the Ward identities in their respective vertices [@bgyu-pi-delta], and the full expressions for the spin-3/2 baryon electromagnetic form factors will be found in Ref. [@bgyu-rho-delta]. Since the mass of $\Sigma^*$ lies below $\bar{K}N$ threshold the empirical decay channel $\Sigma\to \bar{K}N$ is not available for the estimate of the $KN\Sigma^*$ coupling constant, and we follow the SU(3) symmetry which predicts, $$\label{su3-rel} \frac{f_{\pi^- p \Delta^{++}}}{m_\pi} = -\sqrt6 \frac{f_{K^+p\Sigma^{*0}}}{m_K}\,,$$ and determine the coupling constant $f_{K^+p\Sigma^{*0}}$ from the empirically known coupling constant $f_{\pi^- p \Delta^{++}}$. (See the Clebsch-Gordan coefficients with phase for the SU(3) baryon decuplet in the Appendix.) Hereafter, we will write $f_{K^+p\Sigma^{*0}}$ as $f_{KN\Sigma}$ for brevity. In our previous work [@bgyu-pi-delta] we considered the coupling constant in the range from $f_{\pi^- p \Delta^{++}}= 1.7$ to $2$. From these we estimate $f_{KN\Sigma^{*}} = -2.46$ and $-2.83$, respectively. In other model calculations, however, the determination of $f_{KN\Sigma^{*}}$ is found to be rather scattered, e. g., $f_{KN\Sigma^{*}}=-3.22$ for the $\gamma p$ process in the effective Lagrangian approach by applying $f_{\pi^- p\Delta^{++}}=2.23$ to the symmetry relation above [@ysoh]. The coupling constant $f_{KN\Sigma^*}=-4.74$ for the $\gamma p$ process [@junhe], and $-1.22$ for the $\gamma n$ process [@wang] were obtained from the $\chi^2$-fit of data in the Regge plus resonance approach. In this work we take the coupling constant $f_{KN\Sigma^{*}}=-2.2$ within the range discussed above for a better agreement with experiment. ### Minimal gauge {#minimal-gauge .unnumbered} It is well known that the propagation of spin-3/2 $\Sigma^*$ baryon in Eq. (\[sigma33\]) causes divergence of the reaction at high energy. However, if we expect that only the peripheral $K$ exchange in the $t$-channel should dominate at high energies and small angles, then the particle exchanges in the reaction should contribute only to the Coulomb component of the photoproduction currents in Eqs. (\[pk+\]) and (\[nk+\]). This we call the minimal gauge prescription for the $K$ exchange advocated in Refs. [@stichel; @clark; @bgyu-pi-delta], and this is physically sensible because the higher multipoles of the $\Sigma^*$ as a resonance are defined uniquely in the static limit and such a uniqueness can no longer be valid at high energy. In the Reggeized model we recall that the $u$-channel $\Sigma^*$-pole in Eq. (\[nk+\]) as well as the $s$-channel proton-pole term in Eq. (\[pk+\]) is introduced merely to preserve gauge invariance for the $t$-channel $K$-pole exchange, respectively. By the above speculation at high energies we consider only the Coulomb components of the $s$-, and $u$-channel amplitudes that are indispensable to restore gauge invariance of the $K$ exchange. Technically speaking, these correspond to the non-gauge invariant terms in the $s$- and $u$-channels after we remove the transverse component of the production current by redundancy with respect to gauge invariance. In the $u$-channel amplitude in Eq. (\[sigma33\]), for instance, the full expression is now written as [@bgyu-pi-delta], $$\begin{aligned} \label{uch1} &&i{M}_{u(\Sigma^*)} =e_{\Sigma^*} \frac{f_{K N\Sigma^*}}{m_{K}} \bar{u}^{\nu}(p')\biggl[{ 2\epsilon \cdot p'\over u-M^2_{\Sigma^*}}g_{\nu\alpha}\nonumber\\&&\hspace{2cm}+G_{\nu\alpha}(p',k)\biggr]q^\alpha u(p),\end{aligned}$$ where $G_{\nu\alpha}(p',k)$ is the part of the amplitude which collects all the terms that are gauge-invariant themselves. Thus, in this minimal gauge the production amplitude simply consists of the non-invariant terms in three channels, i.e., $$\begin{aligned} \label{min} &&i{M}_{K}=\frac{f_{K N\Sigma^*}}{m_{K}} \bar{u}_{\nu}(p')\biggl[q^\nu\frac{2p\cdot\epsilon}{s-M^{2}_{N}}e_N \nonumber\\&&\hspace{0cm} % +e_{\Sigma^*} \frac{2p'\cdot\epsilon} {u-M^{2}_{\Sigma^*}}q^{\nu} %\nonumber\\&&\hspace{3cm} % + e_{K}\frac{2q\cdot \epsilon }{t-m^{2}_{K}}(q-k)^\nu\biggr] u+iM_c\,.\hspace{0.3cm}\end{aligned}$$ With the $K$ exchange given in Eq. (\[min\]), we now make it Reggeized by the following procedure, $$\begin{aligned} \label{regge} &&{\cal M}_K=M_{K}\times(t-m_K^2){\cal R}^{K}(s,t),\end{aligned}$$ where $$\begin{aligned} \label{regge3} {\cal R}^\varphi =\frac{\pi\alpha'_\varphi}{\Gamma(\alpha_\varphi(t)+1-J)} \frac{\mbox{phase}}{\sin\pi\alpha_\varphi(t)} \left(\frac{s}{s_0}\right)^{\alpha_\varphi(t)-J},\hspace{0.3cm}\end{aligned}$$ is the Regge pole written collectively for the meson $\varphi(=K,\,K^*,\,K^*_2)$ of spin-$J$ with the canonical phase ${1\over2}((-1)^J+e^{-i\pi\alpha_J(t)})$ taken for the exchange-nondegenerate meson in general. For the trajectory of $K$ we use $$\begin{aligned} \label{regge2} &&\alpha_K(t)=0.7\,(t-m_K^2)\,.\end{aligned}$$ The phases of the $K$ exchange is taken from the reaction $\gamma p\to K^+\Lambda$ [@bgyu-kaon] as a natural extension. As for the $\gamma n\to K^+\Sigma^-$ process, however, we favor to choose the phase of the $K$ exchange for a better description of the reaction processes, as will be discussed later. $K^*(892)$ exchange ------------------- The $K^*$ exchange in the $t$-channel is one of the ingredients to consider for the analysis of the production mechanism. The production amplitude is give by [@bgyu-rho-delta], $$\begin{aligned} \label{amp2} &&i{\cal M}_{K^*}=-i{g_{\gamma KK^*}\over m_0} \,\epsilon^{\mu\rho\lambda\alpha} \epsilon_{\mu}k_{\rho}q_{\lambda} \bar{u}_{\nu}(p')\nonumber\\&&\hspace{1cm}\times\left(-g_{\alpha\beta}+Q_\alpha Q_\beta/m^2_{K^*}\right)\Gamma^{\beta\nu}_{K^* N\Sigma^*}(Q,p',p) %\left(\rlap{/}Q g_{\nu\sigma}-Q_{\nu}\gamma_{\sigma}\right) \gamma_5 u(p)\nonumber\\&&\hspace{1cm}\times{\cal R}^{K^*}(s,t).\end{aligned}$$ For the $K^*N\Sigma^*$ coupling we consider only the following form, $$\begin{aligned} \label{kstar} &&\Gamma^{\beta\nu}_{K^* N\Sigma^*}(q,p',p)={f_{K^*N\Sigma^*}\over m_{K^*}}\left( q^\beta\gamma^\nu - \rlap{/}q g^{\beta\nu} \right)\gamma_5,\end{aligned}$$ and disregard the other nonleading terms simply because the leading contribution of the $K^*$ exchange in Eq. (\[kstar\]) itself is not significant. In our previous work on the $\gamma p\to\pi^\pm\Delta$ process we used $f_{\rho N\Delta}=5.5$ for the Model I, and 8.57 for the Model II [@bgyu-rho-delta]. These values lead to $f_{K^*N\Sigma^*}=-2.58$ and $-4.03$, respectively, according to the SU(3) relation $$\frac{f_{\rho N \Delta}}{m_\rho} = -\sqrt6 \frac{f_{K^* N \Sigma^*}}{m_{K^*}}\,.$$ With these values we try to find which one yields the better result in the numerical analysis. From the decay width $\Gamma_{K^*\to K\gamma}=50$ keV for the charged state, we estimate $g_{\gamma K^*K}=\pm0.254$ and the take the negative sign for an agreement with data. The trajectory for $K^*$ is taken to be $$\begin{aligned} \label{regge2} &&\alpha_{K^*}(t)=0.83\,t+0.25 \,,\end{aligned}$$ which is consistent with the previous works [@glv; @bgyu-kaon]. The complex phase for the $\gamma p$  and the constant phase for $\gamma n$ processes are considered for the exchange-degenerate (EXD) pair $K^*$-$K_2^*$. $K^*_2(1430)$ exchange ---------------------- It is found that the tensor-meson $a_2(1320)$ of spin-2 exchange plays the role at high energy from the previous studies of the reactions $\gamma N\to\pi^\pm N$ [@bgyu-pion] and $\gamma N\to\pi^\pm\Delta$ [@bgyu-pi-delta]. Furthermore the role of the $K_2^*$ in the strangeness sector is also noticeable in the $\gamma p\to K^+\Lambda$ [@bgyu-kaon]. Therefore, it is quite reasonable to consider the tensor meson $K_2^*$ exchange in these reaction processes. As an application of the $a_2N\Delta$ coupling in Ref. [@bgyu-pi-delta] to the strangeness sector, we write the Lagrangian for the $K_2^*N\Sigma^*$ as, $$\begin{aligned} {\cal L}_{K^*_2N\Sigma^*}=i{f_{K_2^*N\Sigma^*}\over m_{K_2^*}}\overline{\Sigma^*}^\lambda(g_{\lambda\mu}\partial_\nu +g_{\lambda\nu}\partial_\mu) \gamma_5 N {K_2^*}^{\mu\nu}.\hspace{0.4cm}\end{aligned}$$ Here ${K_2^*}^{\mu\nu}$ is the tensor field of spin-2 with the coupling constant assumed to be $$\begin{aligned} \label{tensorcc} {f_{K^*_2 N\Sigma^*}\over m_{K^*_2}}\approx-3{f_{K^* N\Sigma^*}\over m_{K^*}}\end{aligned}$$ by a simple extension to the strangeness sector from the $\rho$ and $a_2$ meson case which is based on the duality and vector dominance [@goldstein; @thews]. In the $\pi\Delta$ photoproduction the tensor meson-$\Delta$ baryon coupling constant determined by such a relation above yielded a reasonable result in the high energy region, as illustrated in Ref. [@bgyu-pi-delta]. The Lagrangian for the $\gamma KK^*_2$ coupling was investigated in Ref. [@giacosa] and given by $$\begin{aligned} {\cal L}_{\gamma KK^*_2}=-i\frac{g_{\gamma KK^*_2}}{m^2_0}\tilde{F}_{\alpha\beta}(\partial^\alpha {K_2^*}^{\beta\rho}-\partial^\beta {K_2^*}^{\alpha\rho})%K_2^{[\alpha\beta]\rho} \partial_\rho K \,,\\end{aligned}$$ where $\tilde{F}_{\alpha\beta}={1\over2}\epsilon_{\mu\nu\alpha\beta}F^{\mu\nu}$ is the pseudotensor field of photon. The decay of the tensor meson $K^*_2$ to $K\gamma$ is reported to be $\Gamma_{K^*_2\to K\gamma}=(0.24\pm0.05)$ MeV in the Particle Data Group (PDG) and we estimate $g_{\gamma KK^*_2}=0.276$ [@bgyu-kaon] with the sign determined to agree with existing data. The Reggeized amplitude for the $K_2^*$ exchange is thus written as $$\begin{aligned} \label{amp4} && i{\cal M}_{K_2^*}= -i\frac{2g_{\gamma KK^*_2}}{m_0^2}{f_{K_2^* N\Sigma^*}\over m_{K_2^*}} \,\epsilon^{\alpha\beta\mu\lambda}\epsilon_\mu k_\lambda Q_\alpha q_\rho \nonumber\\ &&\hspace{1cm}\times\Pi_{K_2^*}^{\beta\rho;\sigma\xi}(Q) \bar{u}^{\nu}(p')(g_{\nu\sigma}P_\xi+g_{\nu\xi}P_\sigma)\gamma_5 u(p)\nonumber\\ &&\hspace{1cm}\times {\cal R}^{K_2^*}(s,t) \,, %\nonumber\\ %&&\hspace{-1cm}=-i\frac{2g_{\gamma\pi a_2}}{m_0^2}{f_{a_2 %N\Delta}\over m_{a_2}} {1\over %t-m^2_{a_2}}\,\epsilon^{\mu\lambda\alpha\beta}\epsilon_\mu %k_\lambda %q_\alpha\bar{\Delta}^\nu(p')\left[g_{\nu\beta}\left(q\cdot %P-q\cdot Q {P\cdot Q\over m_T^2}\right)+P_\beta\left(q_\nu-Q_\nu %{q\cdot Q\over m_T^2} \right)\right]\gamma_5 u(p)\end{aligned}$$ where $P=(p+p')/2$ and the spin-2 projection is given by $$\begin{aligned} \Pi^{\beta\rho;\sigma\xi}_{K_2^*}(Q)={1\over2}(\eta^{\beta\sigma}\eta^{\rho\xi} +\eta^{\beta\xi}\eta^{\rho\sigma})-{1\over3}\eta^{\beta\rho}\eta^{\sigma\xi}\end{aligned}$$ with $\eta^{\beta\rho}=-g^{\beta\rho}+Q^\beta Q^\rho/m^2_{a_2}$. For the $K_2^*$ Regge-pole exchange we take the EXD phase $e^{-i\pi\alpha_{K_2^*}}$ for the $\gamma p$  and the constant phase for the $\gamma n$  processes, respectively, as discussed above, and choose the trajectory $$\begin{aligned} \label{regge2} &&\alpha_{K^*_2}(t)=0.83\,(t-m^2_{K^*_2})+2 \,,\end{aligned}$$ to be consistent with Ref. [@bgyu-kaon]. Meson $^{(a)}$Phase $^{(b)}$Phase Cpl. const. --------- -------------------- -------------------------- ------------------------------- $K$ $e^{-i\pi \alpha}$ $(1+e^{-i\pi \alpha})/2$ $f_{KN\Sigma^*}=-2.2$ $K^*$ $e^{-i\pi \alpha}$ $1$ $f_{K^*N\Sigma^*}=-4.03$ $K_2^*$ $e^{-i\pi \alpha}$ $1$ ${\rm Eq}.~ (\ref{tensorcc})$ : Physical constants from the SU(3) symmetry, Regge trajectories and phases for $^{(a)}\gamma p\to K^+\Sigma^{*0}$ and $^{(b)}\gamma n\to K^+\Sigma^{*-}$. The radiative decay constants are $g_{\gamma KK^*}=-0.254$ and $g_{\gamma KK^*_2}=0.276$. \[tb1\] In model calculations where the Regge-poles are employed to estimate physical observables the results in shape and magnitude are, in general, very sensitive to a change of the phase as well as the trajectory. Therefore, it is of importance to choose the phase of $K$ exchange which dominates over other meson exchanges. We take the complex phase for the $K$ exchange in the $\gamma p$ process, as before. In the case of $\gamma n$ process, however, the choice of the constant phase leads to an overestimation of the total cross section in the resonance peak, while fixing the coupling constant $f_{K^+n\Sigma^{*-}}=\sqrt{2}f_{K^+p\Sigma^{*0}}$. Without altering the coupling constant, thus, we take the canonical phase which is more adaptive to describe the reaction processes. In Table \[tb1\] we list the coupling constants and phases used for the calculation of the $\gamma p\to K^+\Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$ reactions. numerical Results ================= In this section we present numerical consequences in the cross sections for the total, differential and beam polarization for the reactions $\gamma p\to K^+\Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$. $\gamma p\to K^+\Sigma^{*0}$ ---------------------------- Given the production amplitudes in Eq. (\[p+\]) with the coupling constants in Table \[tb1\] determined from the symmetry consideration, we calculate total and differential cross sections for $\gamma p \to K^+ \Sigma^{*0}$ and present the result to compare with existing data. There is a discrepancy between the recent CLAS data and old ones measured in the pre-1970’s by the CBCG [@crouch] and ABBHHM Collaboration [@erbe-nc; @erbe-pr]. The solid curve in Fig. \[fig2\] corresponds to the full calculation of the cross section with the coupling constants chosen to agree with the CLAS data, and the respective contributions of meson exchanges are displayed. As shown in the figure the production mechanism is solely understood as the dominating role of the contact term in Eq. (\[gau3\]) plus the pseudoscalar $K$ exchange in Eq. (\[kaon\]), while the tensor meson $K_2^*$ exchange in Eq. (\[amp4\]) gives a contribution gradually growing as the energy increases. The contribution of the vector meson $K^*$ exchange in Eq. (\[amp2\]) is small and less significant than that of the tensor-meson $K_2^*$. That the $K^*$ contribution is small and thus insignificant is consistent with the observation in other model calculations of the process [@ysoh], and confirms the validity of the leading $K^*N\Sigma^*$ interaction considered only for the $K^*$ exchange. ![Total cross section for $\gamma p \to K^+ \Sigma^{*0}(1385)$. Contributions of the contact term and meson exchanges are shown in the dotted, dashed, dash-dotted and dash-dot-dotted curves, respectively. The dominance of the contact term is shown. The CLAS data are taken from Ref. [@moriya]. The data of the CBCG and ABBHHM are from Refs. [@crouch; @erbe-nc; @erbe-pr]. []{data-label="fig2"}](fig2.eps){width="0.9\hsize"} ![Differential cross sections for $\gamma p \to K^+ \Sigma^{*0}(1385)$. The angle dependence of the cross sections are shown in the first three panels (a), (b), (c) with the data taken from the CLAS Collaboration [@moriya]. The energy dependence is shown in the panel (d) with data from the LEPS Collaboration [@niiyama]. The contributions of the contact terms and the respective meson exchanges are displayed in (b) and (d) with same nations as in Fig. \[fig2\]. []{data-label="fig3"}](fig3.eps){width="0.9\hsize"} The dependences of differential cross sections on the angle and energy are presented in Figs. \[fig3\]. The slope of the CLAS data in the forward direction is reproduced to a degree in the panels (a), (b), and (c). The rise of the cross section data in the backward angle in (c) may signify the contributions of the baryon resonances. For the energy dependence of the differential cross section in (d) our prediction also agrees with the LEPS data as well. The contributions of the contact term and the respective meson exchanges are analyzed in the panels (b) and (d). $\gamma n\to K^+\Sigma^{*-}$ ---------------------------- ![Total cross section for $\gamma n \to K^+ \Sigma^{*-}(1385)$. Contributions of the contact term and meson exchanges are displayed with the same notations as in Fig. \[fig2\]. The dominance of the contact term is shown. Data are taken from Ref. [@benz]. []{data-label="fig4"}](fig4.eps){width="0.9\hsize"} ![Differential cross section $(s-M^2)^2{d\sigma/dt}$ for $\gamma n \to K^+ \Sigma^{*-}(1385)$ at $E_\gamma=3,\,4$, and 5 GeV. The solid and dotted curves are the cross sections at $E_\gamma=3$ GeV with and without $K_2^*$, showing the role of the tensor meson. Data are taken from Ref. [@benz]. []{data-label="fig5"}](fig5.eps){width="0.9\hsize"} There are various sorts of data on the $\gamma n\to K^+\Sigma^{*-}$ process in comparison to the former $\gamma p$ process. The total and differential cross sections are found in the experiment at the ABHHM Collaboration in the mid-1970’s [@benz]. Very recently the angular distribution and beam polarization asymmetry were measured in the LEPS experiment [@hicks]. We calculate the energy dependence of the cross section and present the result in Fig. \[fig4\]. There might be a room for improving the accuracy in future experiment as can be seen in Fig. \[fig2\]. But the data of the ABHHM are enough to test our model prediction at the present stage, exhibiting the maximum peak and the slope of the decrease along with the increase of photon energy. We note that the $K_2^*$ exchange gives an equal amount of contribution to the $K$ over $E_\gamma\approx 3$ GeV. Figure \[fig5\] shows the differential cross section scaled by the factor $(s-M_n^2)^2$ so that the $-t$ distribution of the cross section is energy independent. We reproduced the cross section at the photon energies, 3, 4, and 5 GeV up to the limit of the experiment, $E_\gamma=5.3$ GeV. It should be pointed out that the role of the $K_2^*$ is crucial to meet with the data in the region $-t>0.5$ GeV$^2$/$c^2$. ![Dependence of differential cross sections for $\gamma n \to K^+ \Sigma^{*-}(1385)$ on the energy (upper panels) and angle (lower panels). Model predictions are given by the grey band to cover the range of angle denoted. The contributions of the contact terms and the respective meson exchanges estimated at $\cos\theta=0.85$ are displayed in upper right and at $E_\gamma=2.4$ GeV in the lower right panels with same nations as in Fig. \[fig4\]. The dip structure appears there due to the canonical phase of the $K$ exchange. Data of the LEPS (black squares) are taken from Ref. [@hicks] and those of the CLAS (empty circles) are from Ref. [@mattione].[]{data-label="fig6"}](fig6.eps){width="0.9\hsize"} Shown in Fig. \[fig6\] is the energy dependence of the differential cross section at forward angles and its angle dependence in two energy bins. The energy dependence of the $d\sigma/d\cos\theta$ is shown in the range calculated between two boundaries $\cos\theta=0.9$ and $0.99$ in the first panel, for instance. The angle dependence of $d\sigma/d\cos\theta$ is calculated at the $E_\gamma=2.2$ and 2.4 GeV, respectively. These results reproduce quite well the overall feature of the cross section data. The contributions of the contact terms and the respective meson exchanges are analyzed in upper right and lower right panels, where the dip structure of the $K$ exchange, and of the contact term, as a result, are shown at the $-t\approx 0.3$ GeV$^2$ due to the zero of the trajectory $\alpha_K(t)=0$ in the canonical phase of the $K$ exchange. ![Energy-dependence of beam polarization asymmetries for $\gamma n \to K^+ \Sigma^-(1190)$ from $\Sigma={d\sigma_y-d\sigma_x\over d\sigma_y+d\sigma_x}\,$(a) and $\gamma n \to K^+ \Sigma^{*-}(1385)$ from $\Sigma=-{d\sigma_y-d\sigma_x\over d\sigma_y+d\sigma_x}\,$ (b). The beam polarization asymmetry in (a) is calculated by the model in Ref. [@bgyu-kaon]. Model predictions are given by the grey band to cover the range of the angle denoted. Data are taken from Ref. [@hicks].[]{data-label="fig7"}](fig7.eps){width="0.75\hsize"} The energy dependence of the beam polarization asymmetry $\Sigma$ was measured in the LEPS experiment of the reaction $\gamma n \to K^+ \Sigma^{*-}(1385)$ and the result is compared with the case of the $\gamma n\to K^+\Sigma^-(1190)$ in Fig. \[fig7\]  in the same range of the angle, $0.6<\cos\theta<1$. With the $\Sigma$ defined as $$\label{beam} \Sigma = \frac{d\sigma_y - d\sigma_x}{d\sigma_y + d\sigma_x}\,,$$ where $d\sigma_{x(y)}$=${d\sigma_{x(y)}\over d\Omega}$ is the component of the differential cross section in the $xyz$-system spanned by the photon momentum ($z$-direction) and two other axes orthogonal to it in the production plane, we calculated the $\Sigma$ of the $\gamma p\to K^+\Sigma$ process in Fig. \[fig7\] (a) by using the model of Ref. [@bgyu-kaon] where the production amplitude consists of the $K+K^*+K_2^*$ similar to Eq. (\[n+\]), but the phases for all the exchanged mesons are taken to be constant, i.e., 1. As for the case of the $\gamma n\to K^+\Sigma^{*-}$ process, however, the beam polarization $\Sigma$ as shown by the grey band in Fig. \[fig7\] (b) is predicted in the present framework with the sign of the $\Sigma$ in Eq. (\[beam\]) reversed. At the present stage, we leave it a problem how to reconcile the sign of the $\Sigma$ between theory and experiment, and suggest that such an uncertainty in measuring the $\Sigma$ in the $\gamma n$ reaction needs to be more analyzed in future experiments. Summary and Discussion ====================== In this work, we have investigated the reaction processes $\gamma p\to K^+\Sigma^{*0}$ and $\gamma n\to K^+\Sigma^{*-}$ to analyze the production mechanism based on the data provided by the CLAS and LEPS Collaborations as well as those by the CBCG, ABBHHM, and ABHHM Collaborations. By using a set of coupling constants common in both reactions total and differential cross sections as well as the beam polarization asymmetry are analyzed and the results in these reactions are quite reasonable to account for the experimental data. Nevertheless, we need to work further on the beam polarization $\Sigma$ to resolve the inconsistency between the model calculation and measurement. The results obtained in this work show that the most important contribution comes from the contact term which is a feature of the spin-3/2 baryon photoproduction. Then, the contribution of the pseudoscalar $K$ exchange follows as the dominant one among the $t$-channel meson exchanges. The role of the $K^*$ exchange from the present analysis turned out to be of secondary importance, as concluded in previous works. Nevertheless, it cannot be neglected in these processes because of its relation with the $K_2^*$ which plays the role crucial to explain the data at high energy, as demonstrated in the scaled differential cross section of the $\gamma n$ reaction. A few remarks are in order. First, we note that the size of the total cross section for the $\gamma p$ process is about the same as that of the $\gamma n$, though the amplitude of the latter process differs by a factor of $\sqrt{2}$ from the former, i.e. $$\begin{aligned} {\sigma_{\gamma n}\over\sigma_{\gamma p}}\sim {|\sqrt{2}({\rm contact}+ t{\rm-ch.}\ K+\cdots)|^2\over |{\rm contact}+t{\rm-ch.}\ K+\cdots|^2}\sim1.\end{aligned}$$ This could be understood as the similar size of the contact term contribution which is dominant in both reactions, as shown in Figs. \[fig2\] and \[fig4\]. By comparing the maximum size of the cross section $\sigma\approx 10$ $\mu$b for the $\gamma p\to \pi^+\Delta^{0}$ process [@wu] with that of $\sigma\approx 1$ for the $\gamma p\to K^+\Sigma^{*0}$ process, their ratio is basically consistent with the reduction of the leading coupling constant $f_{KN\Sigma^*}$ by a factor of 36 $\%$ as compared to the $f_{\pi N\Delta}=1.7$ in the same mass unit, i.e., $$\begin{aligned} {\sigma(\gamma p\to K^+\Sigma^{*0})\over\sigma(\gamma p\to \pi^+\Delta^{0})}\approx {\left|f_{KN\Sigma^*}\over f_{\pi N\Delta}\right|^2}.\end{aligned}$$ Therefore, it is reasonable to assume that both reactions share the same production mechanism as the members of the baryon-decuplet within the present framework. Finally, we give a comment on the study of $N^*$ resonances, though it is beyond the scope of the present work. For future work it is desirable to investigate the role of $N^*$ in the neutral processes such as in Eqs. (\[p0\]) and (\[n0\]), because they have only $K^*+K_2^*$ exchanges in the $t$-channel which are expected to be small as can be seen in Figs. \[fig2\] and \[fig4\]. In this sense, the reaction $\gamma p\to K^0\Sigma^{*+}$ in Eq. (\[p0\]), in particular, could provide a ground more advantageous to identify $N^*$ resonances in the measured cross section of $\sigma=0.68\pm0.48\, (\mu b)$ at $E_\gamma=1.42\sim2$ GeV and $\sigma=0.13\pm0.09\, (\mu b)$ at $E_\gamma=2\sim5.8$ GeV [@erbe-pr], which is of the same order of magnitude as the charged ones we have presented in this work. We are grateful to Hungchong Kim for fruitful discussions. This work was supported by the grant NRF-2013R1A1A2010504 from National Research Foundation (NRF) of Korea. SU(3) relation of the meson-baryon coupling constants for the interactions of the ${\bf 8-8-8}$ and ${\bf 10-8-8}$ types ======================================================================================================================== We use the phase and coupling constants of the meson-baryon interaction ($MBB$) of the type ${\bf 8-8-8}$ which is defined by the following tensor operators, $$\begin{aligned} B_i^j=\left(\begin{array}{ccc} {1\over \sqrt{2}}\Sigma^0+{1\over \sqrt{6}}\Lambda & \Sigma^+ & p\\ \Sigma^- & -{1\over \sqrt{2}}\Sigma^0+{1\over \sqrt{6}}\Lambda & n\\ -\Xi^- & \Xi^0 & -{2\over \sqrt{6}}\Lambda\\ \end{array} \right)\end{aligned}$$ for the $J^P={1\over 2}^+$ baryon octet, and $$\begin{aligned} M_i^j=\left(\begin{array}{ccc} {1\over \sqrt{2}}\pi^0+{1\over \sqrt{6}}\eta & \pi^+ & K^+\\ \pi^- & -{1\over \sqrt{2}}\pi^0+{1\over \sqrt{6}}\eta & K^0\\ K^- & \bar{K}^0 & -{2\over \sqrt{6}}\eta\\ \end{array} \right)\end{aligned}$$ for the $J^P=0^-$ pseudoscalar meson octet. The meson-baryon-baryon ($MBB$) interaction of the ${\bf 8-8-8}$ type can be constructed from fully contracting the indices as $$\begin{aligned} \label{app1} a\bar{B}^i_j B^j_k M^k_i + b\bar{B}^i_j B^k_i M^j_k + {\rm h.c.},\end{aligned}$$ Therefore, two types of coupling are possible in the SU(3) limit, which are equivalent to the conventional $F$ and $D$ types. For the $J^P={3\over 2}^+$ baryon decuplet, totally symmetric tensor $D^{ijk}$ can be identified with the baryon resonances. $$\begin{aligned} D^{111}=\Delta^{++} ,\,\,\, D^{112}={1\over\sqrt{3}}\Delta^+, \,\,\, D^{122}={1\over\sqrt{3}}\Delta^0, \,\,\, D^{222}=\Delta^- ,\\ D^{113}={1\over\sqrt{3}}\Sigma^{*+}, \,\,\, D^{123}={1\over\sqrt{6}}\Sigma^{*0}, \,\,\, D^{223}={1\over\sqrt{3}}\Sigma^{*-} \label{223},\\ D^{133}={1\over\sqrt{3}}\Xi^{*0}, \,\,\, D^{233}={1\over\sqrt{3}}\Xi^{*-}, \\ D^{333}=\Omega^{*-}.\end{aligned}$$ The meson-baryon-decuplet baryon ($MBD$) interaction of the ${\bf 10-8-8}$ type in SU(3) limit can be again from fully contracting the indices as $$\begin{aligned} \label{app2} g\bar{D}^{ijk}B^l_j M^m_k \epsilon_{ilm} + {\rm h.c.},\end{aligned}$$ where the Levi-Civita tensor $\epsilon_{ilm}$ is needed because the total number of index is odd. Therefore, only one type of coupling is possible in the SU(3) limit as in Eq. (\[app2\]). 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--- abstract: 'We report in this paper the proofs that the pulse shape analysis can be used in some bolometers to identify the nature of the interacting particle. Indeed, while detailed analyses of the signal time development in purely thermal detectors have not produced so far interesting results, similar analyses on bolometers built with scintillating crystals seem to show that it is possible to distinguish between an electron or $\gamma$-ray and an $\alpha$ particle interaction. This information can be used to eliminate background events from the recorded data in many rare process studies, especially Neutrinoless Double Beta decay search. Results of pulse shape analysis of signals from a number of bolometers with absorbers of different composition (CaMoO$_4$, ZnMoO$_4$, MgMoO$_4$ and ZnSe) are presented and the pulse shape discrimination capability of such detectors is discussed.' address: - 'INFN - Milano Bicocca, Italy' - 'Dipartimento di Fisica - Università di Milano Bicocca, Italy' author: - 'C.Arnaboldi' - 'C.Brofferio' - 'O.Cremonesi' - 'L.Gironi' - 'M.Pavan' - 'G.Pessina' - 'S.Pirro' - 'E.Previtali' title: A novel technique of particle identification with bolometric detectors --- Bolometers ,Scintillators ,Pulse shape discrimination (PSD) 23.40B ,95.35.+d ,07.57.K ,29.40M ,66.70.-f Rare event searches {#RES} =================== Rare event studies, such as the search for Neutrinoless Double Beta decay () [@BBreview] or the identification of Weakly Interacting Massive Particle (WIMP) interactions with ordinary matter [@DMreview], are of extreme interest in astroparticle physics, since they would imply new physics beyond the Standard Model. In both cases, as in all the rare event studies, spurious events are a limiting factor to the reachable sensitivity of the experiment. Unfortunately natural radioactive background is often present in the detector itself or in the materials surrounding it, no matter how much one can try to reduce it with shieldings, selection of materials and complicated purification techniques. In order to handle the residual unavoidable background, all the envisaged approaches require both a good energy resolution (which always helps in the comprehension of the different structures of an energy spectrum) and the capability to identify the nature of the projectile that interacted with the detector. Indeed, the searched event has always well defined signatures helping to distinguish it from background, for instance two electrons with a fixed sum energy in the case of the . Bolometers [@bolometers] are based on the detection of phonons produced after an energy release by an interacting particle and can have both an excellent energy resolution and extremely low energy threshold with respect to conventional detectors. They can be fabricated from a wide variety of materials, provided they have a low enough heat capacity at low temperatures, which is the only requirement really unavoidable to build a working bolometer. The latter is a priceless feature for experiments that aim at detectors containing particular atomic or nuclear species to optimize the detection efficiency. If other excitations (such as ionization charge carriers or scintillation photons) are collected in addition to phonons, bolometers have already shown to be able to discriminate nuclear recoils from electron recoils, or $\alpha$ particles from $\beta$ particles and $\gamma$-rays. In this paper we will report on the possibility to obtain similar results just by pulse shape analysis, without the requirement of a double readout for phonons and ionization or scintillation light. Bolometric Technique and Scintillating Bolometers {#BolTec} ================================================= Bolometers can be essentially sketched as a two-component object: an energy absorber in which the energy deposited by a particle is converted into phonons, and a sensor that converts thermal excitations into a readable signal. The absorber must be coupled to a constant temperature bath by means of a weak thermal conductance. Denoting by C the heat capacity of the bolometer, the temperature variation induced by an energy release E in the absorber can be written as $$\label{eq:temperature} \Delta T = \frac{E}{C}$$ The accumulated heat flows then to the heat sink through the thermal link and the absorber returns to the base temperature with a time constant $\tau$ = C/G, where G is the thermal conductance of the link: $$\label{eq:signal} \Delta T(t) = \frac{E}{C} e^{ - \frac{t}{\tau}}$$ In order to obtain a measurable temperature rise the heat capacity of the absorber must be very small: this is the reason why bolometers need to be operated at cryogenic temperatures (of the order of 10-100 mK). A real bolometer is somewhat more complicated than the naive description presented above. It is made of different elements and it is therefore represented by more than one heat capacity and heat conductance. As such, the time development of the thermal pulse is characterized by various time constants. In principle, if the bolometer performs as an ideal calorimeter and if the conversion of the energy into heat deposited by the particle is instantaneous (as assumed in equation \[eq:signal\]), then the device is insensitive to the nature of the interacting particle. Although this situation is generally very far from reality, it is however true that the small differences are difficult to detect and the goal has been so far achieved only relying on complicated solutions. Among these are scintillating bolometers. The concept of a scintillating bolometer is very simple: a bolometer coupled to a light detector [@CaF2]. The first must consist of a scintillating absorber thermally linked to a phonon sensor while the latter can be any device able to measure the emitted photons. The driving idea of this hybrid detector is to combine the two available informations, the heat and the scintillation light, to distinguish the nature of the interacting particles, exploiting the different scintillation yield of $\beta$/$\gamma$, $\alpha$ and neutrons. Dark Matter as well as searches can benefit of this capability of tagging the different particles, and more generally this technique can be exploited in any research where background suppression or identification is important. Dark matter experiments look for a very rare signal generally hidden in a huge background. The signal is a nuclear recoil with an energy of few keV (or less) induced by the scattering of a WIMP off a target nucleus. Experiments like CDMS [@CDMS], Edelweiss [@Edelweiss] or CRESST [@CRESST] clearly show that in such energy region the background is dominated by $\beta$/$\gamma$ interactions. A second source of background are $\alpha$ decays, contributing through energy degraded $\alpha$’s and nuclear recoils. The capability to distinguish a nuclear recoil - candidate for a WIMP interaction - from $\alpha$ or $\beta$/$\gamma$ clearly allows to improve drastically the experimental sensitivity. A similar approach was proposed also for applications in searches [@CaF2]. More recently, such a possibility has been demonstrated to be viable for a number of candidate nuclei [@Pirr06]. In this case the major interest is the identification of $\alpha$ interactions. Indeed the other important source of background, namely $\gamma$-rays, is virtually indistinguishable from the signal. The suggested way to eliminate the problem of $\gamma$-rays contribution is to study isotopes with a transition energy above 2615 keV. This corresponds in fact to the highest energy $\gamma$-ray line from natural radioactivity and is due to $^{208}$Tl. Above this energy there are only extremely rare high energy $\gamma$’s from $^{214}$Bi (all the active isotopes with Q$_{\beta\beta}>$2615 keV are listed in Tab. \[tab:isotopes\]). Once $\gamma$-rays are no more a worrisome source of background, what is left - on the side of radioactivity- are $\alpha$ emissions. Indeed $\alpha$ surface contaminations not only can represent the dominant background source for searches based on high transition energy isotopes, but they are already recognized as the most relevant background source in the bolometric experiment CUORICINO [@CUORICINO; @CUOpotential; @ArtChambery] and as a limiting factor for the experiment CUORE [@CUOpotential; @CUORE]. Both the experiments search for the of $^{130}$Te whose transition energy is at 2527 keV, therefore in a region where $\gamma$ background (mainly due to Compton events produced by 2615 keV photons) can be still important. [ccc]{} Isotope & Q$_{\beta\beta}$ \[MeV\] & natural abundance\ $^{116}$Cd & 2.80 & 7.5 %\ $^{82}$Se & 3.00 & 9.2 %\ $^{100}$Mo & 3.03 & 9.6 %\ $^{96}$Zr & 3.35 & 2.8 %\ $^{150}$Nd & 3.37 & 5.6 %\ $^{48}$Ca & 4.27 & 0.19 %\ \[tab:isotopes\] The $\alpha$ contribution to the background in the region (i.e. at about 3 MeV) is the following. In the natural chains we have various nuclei that decay emitting an $\alpha$ particle with an energy between 4 and 8 MeV, their energy is quite higher than most Q-values. However, if the radioactive nucleus is located at a depth of a few $\mu$m inside a material facing the detector, the $\alpha$ particle looses a fraction of its energy before reaching the detector and its energy spectrum looks as a continuum between 0 and 4-8 MeV [@EPJA]. A similar mechanism holds in the case of surface contaminations on the bolometer. This radioactive source plays a role in almost all detectors but it turns out to be particularly dangerous for fully active detectors, as in the case of bolometers. It can be efficiently identified and removed with active background suppression technique such as that conceived with scintillating bolometers. The development of a hybrid detector, able to discriminate $\alpha$ particles and optimized for searches, was the main purpose of our studies on scintillating bolometers. We tested several devices, differing mainly in the scintillating crystal material and size, to study their thermal response, light yield and radio purity. The results obtained so far are reported in [@CDWO4; @CAMOO4; @ZNSE; @ZNMOO4]. On our way, we discovered an extremely interesting feature of some of the tested crystals: the different pulse shape of the thermal signals produced by $\alpha$’s and by $\beta$/$\gamma$’s. This feature opens the possibility of realizing a bolometric experiment that can discriminate among different particles, without the need of a light detector coupled to each bolometer. In the case of a huge, multi-detector array, such as CUORE [@CUORE] and EURECA [@EURECA], the benefits of employing this technique would be impressive: - more ease during assembly because the single element of the array would be a quite simpler device. - fewer readout channels, with not only an evident reduction of cost and work, but also a cryogenic benefit (in a cryogenic experiment particular care should be devoted to reduce any thermal link between room temperature and the bolometers working at few mK: the heat load of the readout channels must be taken into account and their reduction is always a good solution). - a significant cost reduction, saving money and work that would be necessary for the light detectors procurement and optimization. - no need of light collectors, this would simplify the structure of the assembly and it would allow the use of coincidences between facing crystals to further reduce the background. As a final remark, it is worth to be mentioned that these devices could be used also for the measurement of $\alpha$ emissions from surfaces, when extremely low counting rates are needed. Indeed, due to their lack of a dead layer and their high energy resolution, bolometers have an extraordinary sensitivity to low range particles like $\alpha$’s. However, a conventional bolometer cannot distinguish the nature of the interacting particle. It provides therefore only a limited diagnostic power (especially for $\alpha$ particles with energies lower than 2615 keV where the $\beta$/$\gamma$ induced background dominates the detector counting rate). On the other hand, a scintillating bolometer has to be surrounded by a reflector to properly collect the scintillation light (therefore cannot be faced to a sample whose radioactive emission has to be identified). The devices here discussed overcome these two difficulties. Traditionally the devices used in this field are Si surface barrier detectors. For low counting rates, large area low background detectors are needed. Today Si surface barriers detectors with an active area of about 10 cm$^2$, a typical energy resolution of about 25-30 keV FWHM, and counting rates of the order 0.05 count/h/cm$^2$ between 3 and 8 MeV are available [@CANBERRA]. A bolometer like those here discussed can easily reach a much larger active area, has a typical energy resolution of 10 keV and a background counting rate in the 3-8 MeV region that can be as low as 0.001 count/h/cm$^2$. Thanks to the particle identification technique discussed in this paper, it can distinguish an $\alpha$ emission from a $\beta/\gamma$ one[^1] and finally can reject the $\beta$/$\gamma$ background extending its measurement field to energies by far lower than 3 MeV. In the following section we report the results obtained with the pulse shape analysis on some of the tested crystals. Detectors, set-up and data analysis =================================== The results discussed in this paper have been obtained operating different scintillating bolometers in an Oxford 200 $^{3}$He/$^{4}$He dilution refrigerator located deep underground, in the National Laboratory of Gran Sasso (L’Aquila, Italy). The rock overburden (average depth $\sim$3650 m.w.e. [@Hime]) ensures a strong suppression of cosmic rays that in our case is mandatory to be able to operate the detectors without an overwhelming pile-up. A detailed description of the experimental setup can be found in [@SETUP]. In order to study the pulse shape characteristics of different materials (in particular those of interest for ), we operated a number of scintillating bolometers differing for size, geometry and, of course, the absorbing material. As light detector we have used a second bolometer able to absorb scintillation photons converting their energy into heat. This was realised using as absorber Ge wafers of about 5 g, covered - on the side facing the scintillating crystal - with a 600 Å thick layer of SiO$_2$ in order to increase the light absorption. In this way they provided measurable thermal signals over an extremely large band of scintillation wavelengths. Both the scintillating crystal and the Ge wafer were equipped with a Neutron Transmutation Doped Ge thermistor (NTD) [@NTD], glued on the crystal surface and used as a thermometer to measure the heat or light signal produced by particles traversing the scintillating crystal. A silicon resistance, glued on the crystals, was used to produce a calibrated heat pulse in order to monitor the thermal gain of the bolometer. This is indeed subject to variation upon temperature drifts of the cryostat that can spoil the energy resolution. In most cases this temperature drift could be re-corrected off-line on the basis of the measured thermal gain variation [@ALES98; @ARNA03]. Read-out and DAQ ---------------- The read-out [@ELE] of the thermistor was performed via a preamplifier stage, a second stage of amplification and an antialiasing filter (a 6 pole roll-off active Bessel filter 120 db/decade [@Bessel]) located in a small Faraday cage. The ADC was a NI USB-6225 device (16 bit 40 differential input channels). For each triggered signal the entire waveform (*raw-pulse*) is sampled, digitized and acquired for the off-line analysis. Since all the relevant parameters (including the amplitude) of the triggered signals are evaluated off-line, a particular care has to be dedicated to the optimization of the signal filtering and digitization. In the case of the scintillating bolometer the large heat capacity of the absorber, coupled to the finite conductance of the crystal-glue-thermistor interface results in quite slow signals, characterised by a rise-time of the order of few ms[^2] and a decay-time of hundreds of ms (determined by the crystal heat capacity and by its thermal conductance toward the heat sink). Consequently the sampling rate typically used for the signal is 1-4 kHz, over a time window of 200-2000 ms. The Bessel filter acts mainly as antialising, to avoid spurious contributions in the sampled signal. Generally it is preferred to fix its cut-off frequency at the lowest value that does not deteriorate the signal to noise ratio (i.e. to obtain the best results in terms of energy resolution). This results to be a frequency of the order of 10 Hz, which is by far lower than what needed for antialiasing purposes. In the studies here presented the Bessel cut-off frequency was fixed at 120 Hz in order to exploit the maximum available information in the signal bandwidth. Analysis techniques {#AnaTech} ------------------- Off-line analysis aims at determining the pulse amplitude and energy together with several pulse shape parameters associated with each raw-pulse waveform recorded by the data acquisition system. Starting from these quantities the physical informations that are relevant for the scientific goals can be extracted. The first step of the analysis consists in the correct evaluation of the pulse amplitude. Since thermal pulses are superimposed to stochastic noise, a simple maximum-minimum algorithm would not give the better achievable resolution. We therefore use the Optimum Filter (OF) approach [@GATTI86]. This algorithm has proven to provide the best estimate of the pulse amplitude and, as a consequence, the best energy resolution. The basic concept is to build a filter that, when applied to the raw-pulse, produces - as output - a pulse with the best signal to noise ratio. The filtered pulse is then used to evaluate the signal amplitude. It can be proven that in the frequency domain the OF transfer function H( $\omega$ ) is given by $$\label{eq:OF} H(\omega) = K ~ \frac{S^*(\omega)}{N(\omega)} ~ e^{-j\omega t_M}$$ where S($\omega$) is the Fourier transform of the ideal thermal signal (reference pulse in the absence of noise), N($\omega$) is the noise power spectrum, $t_M$ is the delay of the current pulse with respect to the reference pulse and K is a proper normalising factor usually chosen in order to obtain the correct event energy. The role of the optimum filter is to weight the frequency components of the signal in order to suppress those frequencies that are more affected by noise. It can be seen from eq. \[eq:OF\] that, in order to build the filter, the shape of the reference pulse S($\omega$) and the noise power spectrum N($\omega$) must be known. S($\omega$) is usually estimated by averaging a large number of recorded raw-pulses, so that the noise associated with each of them averages to zero. N($\omega$) is obtained according to the Wiener-Khintchine theorem by acquiring many detector baselines in absence of thermal pulses and averaging the corresponding noise power spectra. Once the pulse amplitude has been evaluated, gain instability corrections are applied to data. Due to the dependence of the detector response on the working temperature, the same amount of released energy can produce thermal pulses of different amplitudes. Gain instabilities are corrected monitoring the time behaviour of thermal pulses of fixed energy, generated every few minutes across a Si heater resistor attached on the crystal absorber [@ALES98; @ARNA03]. Finally the amplitude to energy conversion (calibration) is determined by measuring the pulse amplitudes corresponding to fixed calibration lines. In the measurements here reported the signal of the scintillating bolometer (we will refer to this signal as to the heat or thermal signal) has been calibrated on the basis of the full energy peaks visible in the spectrum collected when the detector was exposed to an (external to the cryostat) source. These peaks have a nominal energy of: 511, 583, 911, 968 and 2615 keV. Below 511 keV and above 2615 keV the energy calibration is extrapolated. However, the heat response for $\alpha$ particles is slightly different from the $\beta$/$\gamma$ response in scintillating bolometers [@CDWO4]. For the molybdates that are here reported this heat quenching factor is lower than few percent while for ZnSe it is a little bit higher, about 10 percent for $\sim$6 MeV alpha particles [@ZNSE]. This miscalibration of the $\alpha$ band however does not imply an appreciable change in the discrimination confidence level described below. In the case of the light signal the energy calibration is not needed and we therefore present its value in arbitrary units. Besides the amplitude, few other characteristic parameters of the pulse are computed by the off-line analysis. Some of them are: $\tau_{rise}$ and $\tau_{decay}$, TVL and TVR. The rise-time ($\tau_{rise}$) and the decay-time ($\tau_{decay}$) are determined on the recorded raw-pulse as (t$_{90\%}$-t$_{10\%}$) and (t$_{30\%}$-t$_{90\%}$) respectively. TVR (Test Value Right) and TVL (Test Value Left) are computed on the optimally filtered pulse A(t). They are the root mean square differences between the current signal A(t) and the reference pulse after OF filtering A$_{0}$(t)= H(t) $\otimes$ S(t). In more detail, the filtered response function A$_{0}$(t) is synchronized with the filtered signal A(t), making their maxima to coincide, then the least square differences of the two functions are evaluated on the right (TVR) and left (TVL) side of the maximum on a proper time interval which is usually chosen depending on the shape of the OF signals. Although these two parameters do not have a direct physical meaning, however they are very sensitive (even in noisy conditions) to any difference between the shape of the analyzed pulse and the response function. Consequently, they are used either to reject fake triggered signals (e.g. spikes) or to identify variations in the pulse shape with respect to the reference response function (and this will be our case). {width="1.\linewidth"} Pulse shape signature in the heat pulse {#PSA} ======================================= A series of measurements was carried out, in which different scintillating bolometers, each coupled to a light detector (but in one case), were exposed to $\gamma$ and $\alpha$ sources. This allowed us to study the response of our devices to different radiations. The light signal was used to identify - on the basis of the heat to light ratio - the particle producing the event under study. As mentioned in section \[AnaTech\], for each triggered signal different pulse shape parameters are computed by the off-line analysis, generally to isolate spurious and pile-up events. In the case of scintillating bolometers, looking at the distribution of the pulse shape parameters for the heat signals, we realized that it was possible to distinguish $\beta$/$\gamma$ from $\alpha$ events. This is clearly evident in Fig. \[fig:camoo4\] where the scatter plot of the amplitudes measured for the light and heat signals (Light vs. Heat) is compared with the scatter plot (obtained for the same events) of the linearized rise-time (see later in the text) vs. amplitude for the heat signal ($\tau_{rise}^{lin}$ vs. Heat). In this detector, $\alpha$ and $\beta$/$\gamma$ interactions draw different distributions in both the scatter plots, definitely proving that the shape of the thermal pulse induced by an $\alpha$ particle is different from that of a $\beta$/$\gamma$ interaction. This behavior can be explained by the dependence of light yield on the nature of the interacting particle. The high ionization density of $\alpha$ particles implies that all the scintillation states along their path are occupied. This saturation effect does not occur or at least is much less for $\beta$/$\gamma$ particles. Therefore, in $\alpha$ interactions a larger fraction of energy flow in the heat channel with respect to $\beta$/$\gamma$ events. This leads not only to a different light and heat yield but also to a different time evolution of both signals. The pulse shape of the thermal signal then can be explained by the partition of energy in the two channels with different decay constants. In particular, as shown by [@LightDep1; @LightDep2] the scintillation produced in molybdates by $\alpha$ and $\beta$/$\gamma$ particles presents few decay-time constants with different relative intensities. Also a strong temperature dependence of the averaged decay-time of the light pulses in these same crystals was reported [@TimeDep1; @TimeDep2]. For this reason - while at room temperature the averaged decay-time of molybdate or tungstate scintillators is of the order of tens of $\mu$s (thus almost instantaneous in the heat pulse timescale) - at low temperatures it increases to hundreds of $\mu$s and is therefore comparable to the typical rise-time of the heat signal of our scintillating bolometers. In the following sections we analyse the results obtained with different crystals, each being a possible candidate for a experiment. We quote for each crystal the discrimination power achieved - in the region - between $\alpha$ and $\beta$/$\gamma$ particles on the basis of the light/heat ratio or simply on the basis of the pulse shape of the heat signal. We discuss in more detail the case of CaMoO$_4$, briefly summarizing the results obtained for other crystals. CaMoO$_{4}$ {#CaMoO4} =========== Recently CaMoO$_{4}$ has been intensively studied, for its possible application as a scintillating bolometer for and Dark Matter experiments [@Pirr06; @Seny06; @Mikh06-JPDAP]. This crystal contains two isotopes that could undergo : $^{48}$Ca (Q$_{\beta\beta}$=4.27 MeV) and $^{100}$Mo (Q$_{\beta\beta}$=3.03 MeV). Actually, while the large content of $^{100}$Mo makes this crystal very attractive, the presence of $^{48}$Ca is a problem. Indeed, the natural isotopic abundance of $^{48}$Ca (a.i.=0.19%) is too low to study the without enrichment, which is extremely difficult, and at the same time, it is too high to study the of $^{100}$Mo, since the background due to the of $^{48}$Ca in the region of $^{100}$Mo will limit the reachable sensitivity for the latter isotope. A possible solution to this problem was suggested by Annenkov et al. [@CAMOO4_DEP] who proposed an experiment with CaMoO$_{4}$ depleted in $^{48}$Ca. Despite these possible problems, we discovered that CaMoO$_{4}$ is an extremely interesting crystal because of its capability to discriminate $\beta$/$\gamma$ from $\alpha$, thanks to the different shape of the thermal pulses. The sample we used was a cylindric CaMoO$_{4}$ crystal with a mass of 158 g (h = 40mm, $\varnothing$ = 35mm). The crystal was faced to two $\alpha$ sources. Source A was obtained by implantation of $^{224}$Ra in an Al reflecting stripe. The shallow implantation depth allows to reduce to a minimum the energy released by the $\alpha$’s in the Al substrate so that monochromatic $\alpha$ lines (those produced in the decay chain of $^{224}$Ra to the stable $^{208}$Pb isotope) can be observed in the scintillating bolometer. Besides these $\alpha$ particles - all with energies above 5 MeV - the source emits a $\beta$ with a maximum energy of 5 MeV, due to the decay of . Since our main goal was to study the efficiency of $\alpha$ particle rejection in the region (i.e. at about 3 MeV), a second source (B) was also used. This was obtained contaminating an Al stripe with a liquid solution and later covering the stripe with an alluminated Mylar film (6 $\mu$m thick). Thus the source produced a continuous spectrum of $\alpha$ particles, extending from about 3 MeV down to 0. In Fig. \[fig:camoo4\] we show the Light vs. Heat scatter plot, collected while the crystal was exposed to an external source. The total live time of this measurement was $\sim$43 h. The FWHM energy resolution, on the heat signal, ranges from 2.7 keV at 243 keV to 8.7 keV at 2615 keV on $\beta$/$\gamma$ events and is about 10 keV on 5 MeV $\alpha$’s. The light yield of the crystal, evaluated calibrating the light detector with a $^{55}$Fe source, is 1.87 keV/MeV (for more details on the procedure used for the evaluation of the light yield see references [@CDWO4; @ZNSE]). The two separate bands - clearly visible in the Light vs. Heat scatter plot - are ascribed to $\beta$/$\gamma$’s (upper band) and $\alpha$’s (lower band). The upper band is dominated by the source $\gamma$’s (plus the environmental $\gamma$’s). The lower band is due to $\alpha$’s from source A and from Uranium and Thorium internal contamination of the crystal (these are the monochromatic lines above 4 MeV) and source B (the continuum counting rate below 4 MeV). Above 8 MeV we observe a group of events ascribed to $\alpha$+$\beta$ summed signals due to internal contamination in Bismuth and Polonium. Indeed, due to the long rise-time of this device (5 ms), the beta decay of $^{214}$Bi or $^{212}$Bi (respectively of $^{238}$U and $^{232}$Th chains) followed immediately by $\alpha$ decay of $^{214}$Po ($\tau$=164 $\mu$s) and $^{212}$Po ($\tau$=298 $\mu$s ), may lead to a pile up on the rise-time of the thermal pulses that can hardly be recognized as a double signal. The two decays produce therefore a single pulse, with an energy that is the sum of the two. The discrimination between the $\alpha$ and the $\beta$/$\gamma$ populations, provided by the scintillation signal, can be evaluated by measuring the difference between the average amplitude of the light signal produced by the two kinds of particles (Light$_{\beta/\gamma}$ and Light$_{\alpha}$), considering a group of events releasing the same energy in the scintillating crystal. This difference is then compared with the width of the two distributions ($\sigma_{\beta/\gamma}$ and $\sigma_{\alpha}$). The discrimination confidence level D$_{Light}$ can be then defined as: $$D_{Light} = \frac{Light_{\beta/\gamma}-Light_{\alpha}}{\sqrt{\sigma_{\beta/\gamma}^2+\sigma_{\alpha}^2}}$$ To evaluate this discrimination power in the region we selected events belonging to the 2615 keV $\gamma$-line full energy peak and compared their light pulse distribution with that of events of similar energy in the $\alpha$ band (Fig. \[fig:camoo4\_light\] left panel). D$_{light}$ results to be 12.6 sigma. The reason for using events with the same energy to evaluate D$_{light}$ is in the energy dependence of the distance (and width) of the $\alpha$ and $\beta$/$\gamma$ bands, that induce an energy dependence of the discrimination confidence level. In the energy range where both the bands are populated (i.e. between 1 and 3 MeV), D$_{Light}$(E) appears to be linearly decreasing with energy. Its extrapolated value at E=0 being 3 sigma. As already anticipated, we discovered that the heat signal shape is enough to discriminate $\beta$/$\gamma$’s from $\alpha$’s. This is shown in the right panel of Fig. \[fig:camoo4\] where we report the (thermal pulse) $\tau_{rise}^{lin}$ vs. Heat scatter plot for the same events shown in the left panel. Two separate bands, ascribed to $\beta$/$\gamma$ and $\alpha$ events can be identified. The former with an average rise-time of 5.8 ms, the latter with 5.6 ms. In this plot the weak energy (Heat) dependence of $\tau_{rise}$ observed for the two populations was corrected by fitting their rise-time distributions with two lines having the same slope. To do this we used $\beta$/$\gamma$ events in the 0.5-2.6 MeV range and $\alpha$ events in the 1.5-7 MeV range. The $\tau_{rise}$ is then linearized re-defining it as $\tau_{rise}^{lin}=\tau_{rise} - slope \times E$ with $slope$=0.0075 ms/MeV. In order to emphasize the correspondence in the identification of $\beta$/$\gamma$ and $\alpha$ events, signals selected in the Light vs. Heat scatter plot as having an energy between 2 and 4 MeV and belonging to the $\beta$/$\gamma$ or $\alpha$ bands, are reported in different colors. We will refer in the following to these two groups of events as to 2-4 MeV $\gamma$’s and 2-4 MeV $\alpha$’s, although - above the 2615 keV line - the $\gamma$ group is empty. These two samples will be used to evaluate the difference in shape among $\beta$/$\gamma$ and $\alpha$ signals. In Fig. \[fig:Impulso\] we compare two pulses obtained by averaging - separately - signals belonging to the 2615 keV $\gamma$-line and signals due to $\alpha$ particles that release a similar energy. The average is here needed to get rid of the noise that can mask the small differences among the pulses. The difference in shape can be appreciated in the bottom panel of Fig. \[fig:Impulso\]. The discriminating power D$_{RiseTime}$ of the heat pulse shape method can be defined exactly as done for D$_{Light}$: $$D_{RiseTime}(E) = \frac{RiseTime_{\beta/\gamma}-RiseTime_{\alpha}}{\sqrt{\sigma_{\beta/\gamma}^2+\sigma_{\alpha}^2}}$$ where, however, the dependence on energy is here due only to the variation of the width of the rise-time distributions since the difference in rise-time appears to be independent on energy. The distribution obtained projecting the rise-time of 2-4 MeV $\gamma$’s and 2-4 MeV $\alpha$’s is shown in Fig. \[fig:camoo4\_light\] (right panel). A gaussian fit of the two peaks yields a rise-time of (5.788$\pm$0.017) ms for $\gamma$’s and of (5.649$\pm$0.013) ms for $\alpha$’s. D$_{RiseTime}$ results equal to 6.5 sigma. As for the D$_{Light}$ case, we can extrapolate the D$_{RiseTime}$ value also in the energy region where the $\alpha$ band is poorly populated, simply assuming that the distance between the two bands remains constant (as we observe) and D$_{RiseTime}$ changes because the width of the two distributions becomes larger, at low energies, due to noise. The result is that D$_{RiseTime}$ becomes lower than 2 sigma at 500 keV. In other words the 150 $\mu$s difference in rise-time of $\alpha$ and $\beta$/$\gamma$ events cannot be appreciated when the pulse amplitude is too low and, therefore, the noise modifies appreciably the signal shape. Finally, although in this measurement the rise-time is the most efficient shape parameter for $\alpha$ event discrimination, good discrimination levels were observed also in other parameters such as the decay-time and the TVR. The possibility to increase the discrimination power by combining the information from different shape parameters or the fit on the rise-time is under study. We note that either in the Light vs. Heat scatter plot as in the $\tau_{rise}^{lin}$ vs. Heat one we can see some outliers that could indicate a possible failure of the particle identification technique. However such events can be accounted for if one considers the following two effects. If a $\gamma$-ray interacts both in the light detector and in the scintillator the light signal that is read-out has a wrong amplitude since it is not only ascribed to scintillating photons but also to a direct $\gamma$ interaction in the Ge wafer. The measurement is performed in a high rate condition, therefore we expect to have a number of pile-up events in each of the two detectors, this leads to an erroneous evaluation of the pulse amplitude and pulse rise-time. To conclude, in the case of decay the discrimination power provided by the use of the scintillation signal is comparable with that provided by the pulse shape analysis. For what concerns the use of this device for the measurements of $\alpha$ particle emissions from an external sample, a D$_{RiseTime}$ better than 2 sigma above 500 keV means that this detector has a good sensitivity even in the region where Si surface barrier detectors start to be dominated by $\gamma$ background. On the contrary the applicability of this technique to Dark Matter searches cannot be proved directly. Other crystals -------------- Other scintillating crystals have shown the possibility to discriminate interacting particles through the thermal pulse shape differences. Among the tested crystals, other molybdates (ZnMoO$_{4}$, MgMoO$_{4}$) and also other crystals such as ZnSe showed a good discrimination power. ZnMoO$_{4}$ ----------- We tested a 19.8 g ZnMoO$_{4}$ crystal, having the shape of a prism with height of 11mm and a regular hexagonal base, with a diagonal of 25 mm [@ZNMOO4]. The FWHM energy resolution, on the heat signal, is 4.2 keV FWHM on $\beta$/$\gamma$ events at 2615 keV, and about 6 keV on the 5.4 MeV $\alpha$ line. The light yield of the crystal is 1.1 keV/MeV. The total live time of this measurement was $\sim$195 h. Also for this analysis we have used the Light vs. Heat scatter plot to select $\beta$/$\gamma$ and $\alpha$ between 2 and 4 MeV, in order to evaluate the discrimination power provided by pulse shape studies. Unlike for the CaMoO$_4$ case where the most efficient parameter for the discrimination is the rise-time, this measurement showed a higher discrimination power on the decay-time of the thermal pulse. In order to emphasize the discrimination power, all signals have been fitted with a function obtained as the sum of two exponentials: $$\label{eq:vs. energy} \Delta V(t) = (e^{ - t/ \tau_1 + A_1} + e^{ - t/ \tau_2 + A_2})$$ where the $\tau_1$ and $\tau_2$ parameters are obtained by fitting the decay of the thermal signals in the raw-pulse acquired for each event. It was observed that the best discrimination power is obtained by the ratio of the two decay constants (RDC): $$\label{RDC_def} RDC = \frac{\tau_1}{\tau_2}$$ The scatter plot of RDC vs. Heat is reported in the right panel of Fig. \[fig:znmoo4\] (in the left panel the corresponding Light vs. Heat scatter plot). The discrimination between $\alpha$ and $\beta$/$\gamma$ populations provided by RDC is evaluated linearizing the RDC vs. Heat relationship (fitting the $\alpha$ band in scatter plot of Fig. \[fig:znmoo4\] with a polynomial as done for the $\tau_{rise}$ of CaMoO$_4$) and then projecting the distribution of RDC$^{lin}$ for events in the 2-4 MeV range. The two peaks (${\beta/\gamma}$ and ${\alpha}$) are then fitted with a gaussian (Fig. \[fig:ZnMoO4\_RDC\_Proj\]). The discrimination power D$_{RDC}$ is 6.4 sigma. MgMoO$_{4}$ ----------- This compound contains, as in the two previous cases, the active isotope $^{100}$Mo which is here present in a larger concentration (52% in mass). The crystal tested is 32x31x24 mm$^3$ with a weight of 89.1 g. The total live time of this measurements was $\sim$22 h. In this run it was not possible to face the crystal to a light detector because of the assembly structure so we can’t use the Light vs. Heat scatter plot in order to tag $\alpha$ events. The performances of the bolometer were quite poor, most probably this was due to a problem with the gluing of the NTD thermistor: at the end of the measurement, when the crystal was back to room temperature, we discovered that under the thermistor the crystal showed a crack. This could explain why the signal to noise ratio was so bad (the energy resolution measured on $\alpha$ lines was 150 keV FWHM) and consequently also the resolution in the evaluation of the signal shape parameters was limited. Despite these problems, we were able to observe in the scatter plot of RDC$^{lin}$ vs. Heat (fig. \[fig:MgMoO4\_RT\]) clear difference between events due to the interaction of $\alpha$ particles and events due to $\beta$/$\gamma$ events. The discrimination between the two populations provided by the RDC parameter can be evaluated by means of a gaussian fit of the two peaks obtained by projecting selected events after linearization. The resulting discrimination power D$_{RDC}$ is 1.8 sigma. These preliminary results, even if limited because of the problems reported above, lead us to program new more detailed measurements to better study this promising crystal. ZnSe ---- $^{82}Se$ is a emitter with an isotopic abundance of 9.2% and a Q-value of (2995.5 $\pm$ 2.7) keV. It has always been considered a good candidate for studies because of its high transition energy and the favorable nuclear factor of merit. For these reasons in the last years an R&D work was carried out in which we have extensively studied the performances of ZnSe detectors in different conditions. For our studies we have used different ZnSe crystals. Characteristics of measurements done and obtained results are reported in detail in [@ZNSE]. Here we report the discrimination power on the RDC obtained with the Huge ZnSe crystal (h = 50mm, $\varnothing$ = 40mm, 337g). In Fig. \[fig:znse\] the Light vs. Heat scatter plot of a measurement of $\sim$70 h of live time is shown. In order to have a high number of $\alpha$ counts in the 2-3 MeV region also in these measurements a degraded $^{238}$U source was placed in front of the crystal. An explanation of the Quenching Factor larger than one (i.e. the $\alpha$ band lies above the $\beta$/$\gamma$ band) and other anomalies observed in this crystal can be found in [@ZNSE]. Also in this case, the RDC$^{lin}$ vs. Heat (fig. \[fig:znse\_RDC\]) showed a difference between events due to the interaction of $\alpha$ particles and events due to $\beta$/$\gamma$ events. The discrimination power D$_{RDC}$ is 2.2 sigma. Conclusion ========== The possibility to discriminate the nature of the particle interacting in a bolometric detector, simply on the base of the shape of the thermal pulse, is now definitely proved and opens new possibilities for the application of these devices in the field of rare events searches. In particular, the high rejection capability that could allow to completely rule out the $\alpha$ background in experiments was demonstrated. Unfortunately, based on the results obtained so far, the applicability of this technique to Dark Matter searches cannot yet be proved directly. This feature was observed in different scintillating crystals (CaMoO$_4$, ZnMoO$_4$, MgMoO$_4$ and ZnSe) and new tests are under preparation in order to investigate if a similar behavior can be observed also in other compounds. A discrimination confidence level of $\sim$6.5 sigma was obtained both for CaMoO$_4$ and ZnMoO$_4$ crystals in the 2-4 MeV energy region. Discrimination confidence levels reached with MgMoO$_4$ (D$_{RDC}$=1.8$\sigma$) and ZnSe (D$_{RDC}$=2.2$\sigma$) are not so high but they indicate that even with these crystals the discrimination based on pulse shape analysis is possible. New techniques aiming at improving the discrimination power of the pulse shape analysis are being studied. 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Sci **57 (2010) 1475************ [^1]: This feature is extremely important when the sample which is investigated produces a continuous counting rate which could be due either to $\alpha$ emissions from a thick contamination or to a $\beta/\gamma$ continuum. [^2]: The minimum observable signal rise-time is limited by the integration on the parasitic capacitance of the signal wires that connect the NTD thermistor to the front-end electronics.

-1truecm 2truecm 1truecm ****\ 0.3truecm [Nordita, KTH Royal Institute of Technology and Stockholm University,\ Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden]{}\ [Email: yenchin.ong@nordita.org]{}\ 1truecm Christodoulou and Rovelli have shown that black holes have large interiors that grow asymptotically linearly in advanced time, and speculated that this may be relevant to the information loss paradox. We show that there is no simple relation between the interior volume of an arbitrary black hole and its horizon area. That is, the volume enclosed is not necessarily a monotonically increasing function of the surface area. {#section .unnumbered} An asymptotically flat Schwarzschild black hole has a spherical event horizon. The usual metric of this geometry in 4-dimensions, in the units $G=c=1$, is $$g[\text{Sch}]=-\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1} dr^2 + r^2\left(d\theta^2 + \sin^2\theta d\phi^2 \right),$$ where $M$ is the ADM mass, and $r$ is the areal radius. That is, the area of the event horizon $r_h$ is the area of the 2-sphere: $4\pi r_h^2$. Unlike its surface area, the concept of “the volume” of a black hole is a tricky one. The reason is that volume depends on the choice of 3-dimensional spacelike hypersurface. As such, no unique volume can be prescribed to a black hole. Furthermore, the interior of a static black hole is nevertheless dynamical, so one should definitely *not* think of a black hole as a black box that bounds a certain amount of volume that can be easily estimated from knowing the size of its area. This is well-known: a maximally extended Schwarzschild \[Kruskal-Szekeres\] geometry has an infinitely large asymptotically flat region on the “other side”, connected via the Einstein-Rosen bridge. Similarly, one could attach a closed FLRW universe to the interior of a black hole via the Einstein-Rosen bridge, resulting in the so-called Wheeler’s “bag-of-gold” geometry [@Wheeler]. Even non-black hole configurations can have arbitrarily large interiors than their areas might suggest [@monsters]. What about black holes that were formed from gravitational collapse and have no second asymptotic region? One important motivation to study the interiors of *generic* black holes is of course the information loss paradox. As matter falls into a black hole and the black hole gradually Hawking evaporates away, it seems that when the black hole disappears, the information of the in-fallen matter will be lost forever, thereby threatening the unitarity of quantum mechanics. One proposal to resolve this paradox is the idea that black holes do not completely evaporate away, but instead settle down to a \[meta-\]stable remnant. \[See [@1412.8366] for a recent review of black hole remnants.\] An obvious shortcoming of such a proposal is that, as the black hole shrinks in size it would seem that room is running out for storing a large amount of information. This objection presumably does not arise if the interior of a rather small black hole can remain large[^1]. However, since “bag-of-gold” geometry is unlikely to be generic, and Kruskal-Szekeres geometry only holds for eternal black holes, we have to look elsewhere to resolve the information loss paradox, which is most important in the case of black holes formed from gravitational collapse. Fortunately, it turns out that even such black holes have large interiors. \[Whether or not such interior volumes remain sufficiently large even as the black hole shrinks to Planck scale is, of course, a relevant and important question.\] In a recent paper, Christodoulou and Rovelli [@1411.2854] pointed out that, while there is no unique volume that can be prescribed to the interior of a black hole, one could look at the volume of the *largest* spacelike spherically-symmetric surface bounded by the event horizon of the black hole. Such a volume is a geometrical property that is coordinate-independent. Christodoulou and Rovelli \[hereinafter, CR\] showed that most of the volume contribution comes from a region which is not causally connected with matter that has fallen far into the black hole \[see the work of Bengtsson and Jakobsson [@1502.01907] for an explicit and nice illustration of this fact, as well as their generalization of the work of CR to the case of asymptotically flat Kerr black holes.\] We will henceforth refer to such a volume measure as the “CR-volume”. For an asymptotically flat Schwarzschild black hole in 4-dimensions, most of the CR-volume contribution is given by the integral [@1411.2854; @1502.01907] $$\label{int1} \text{Vol.} \sim \int^v \int_{S^2} \max\left[r^2 \sqrt{\frac{2M}{r}-1}\right] \sin\theta ~d\theta d\phi dv,$$ where $v$ is the advanced time defined by $$v := t + \int \frac{dr}{f(r)} = t + r + 2M \ln |\frac{r}{2M}-1|; ~~f(r):=1-\frac{2M}{r}.$$ Following [@1502.01907], we omitted the lower limit in the integral with respect to $v$, which only contributes to a negligible finite value, whereas the integral will be dominated by its upper limit $v$. The coefficient of $v$ can be maximized by maximizing the function $$\mathcal{F}(r) := r^2 \sqrt{\frac{2M}{r}-1}.$$ Elementary calculus shows that $r=3M/2$ maximizes $\mathcal{F}(r)$. Indeed, most of the volume comes from the contribution of this constant $r$ slice. \[Note that this slice is rather “close” to the event horizon $r=2M$.\] This leads to $$\text{Vol.} \sim 3\sqrt{3} \pi M^2 v.$$ That is, the CR-volume grows asymptotically linearly in $v$. In other words, even though a static black hole looks the same to the exterior observer no matter how long one waits \[this is a classical statement without taking into account Hawking radiation\], its interior gets larger with time \[c.f. another proposal of black hole volume in [@0508108]\]. The estimate given by CR is that the supermassive black hole at the center of our galaxy, Sagittarius A$^*$, contains sufficient space to fit a *million* solar systems, despite its areal radius is only a factor of 10 or so larger than the Earth-Moon distance. Taking into account the rotation of the black hole does not change this result by much, despite the rotation rate of Sagittarius A$^*$ is about 90% of the extremal limit. In other words, the CR-volume for asymptotically flat black holes seem to be robust against rotational effect, as long as it stays below $\sim 99\%$ of the extremal limit [@1502.01907]. In view of the potentially important role the CR-volume may play in the resolution of the information loss paradox \[as suggested by CR [@1411.2854]\], the properties of the CR-volume of other black hole solutions should be investigated. For example, one may wish to consider black holes with other asymptotic geometries. One especially interesting arena to explore is the anti-de Sitter \[AdS\] space — in the presence of a negative cosmological constant, a large class of black hole solutions are allowed. Unlike their asymptotically flat cousins, these black holes can have non-trivial horizon topologies, hence are often referred to as “topological black holes” [@9705004; @9705012; @9709039; @9808032]. The metric tensor of a static topological black hole in $(n+2)$-dimensions \[$n\geqslant 2$\] is given by: \[metric\] g\[\^k\_[n+2]{}\] =& -dt\^2\ & + \^[-1]{} dr\^2 + r\^2 d\^2 \[X\^k\_n\], where $L$ is the AdS curvature length scale, and $d\Omega^2[X^k_n]$ is a metric of constant curvature $k=\left\{-1,0,+1\right\}$ on the $n$-dimensional Riemannian manifold $X^k_n$, with \[dimensionless\] area $\Gamma[X^k_n]$. For example, if $X^k_n$ is a 2-sphere, then $\Gamma[X^k_n]=\Gamma[S^1_2] = 4\pi$. Note that in general, the geometry is only asymptotically *locally* AdS. {#section-1 .unnumbered} In the $k=0$ case, the metric Eq.(\[metric\]) reduces to the simpler form: \[flatmetric\] g\[\^0\_[n+2]{}\] =& - dt\^2\ &+\^[-1]{}dr\^2 +r\^2, where the $\zeta_i$’s are dimensionless coordinates on a flat space. If the black hole has a compact horizon with toral topology \[$T^n = \Bbb{R}^n/\Bbb{Z}^n$\], the mass parameter $M^*$ is simply $M/(2\pi K)^n$, where $K$ is a dimensionless “compactification parameter”. Similarly $Q^* = Q/(2\pi K)^n$. For example, if the horizon is a flat 2-torus, one may think of it as a product of two circles: $T^2 = S^1 \times S^1$, each of which has period $2\pi K$, so the dimensionless area corresponds to the event horizon is $\Gamma[X^k_n=T^0_2]=4\pi^2 K^2$. The dimensionful area is $4\pi^2K^2 r_h^2$, where $r_h$ denotes the horizon. This generalizes straightforwardly to $(2\pi K)^n r_h^n$ for general dimensions $n \geqslant 2$. Of course, even if we restrict to the case in which the horizon is compact, there are still many other possible topologies. In 5-dimensions, i.e., if the event horizon is a 3-dimensional manifold, there are 6 orientable compact topologies with constant zero curvature [@0311476], namely the torus $T^3$ and its various quotient topologies: the dicosm $T^3/\Bbb{Z}_2$, the tricosm $T^3/\Bbb{Z}_3$, the tetracosm $T^3/\Bbb{Z}_4$, the hexacosm $T^3/\Bbb{Z}_6$, and the didicosm \[also called the Hantzsche-Wendt space\] $T^3/(\Bbb{Z}_2 \times \Bbb{Z}_2)$. In such cases, we still *define* the area as $8\pi^3K^3$. That is, we use $K$ to measure the relative size of these spaces. We can also consider a non-compact \[planar\] event horizon. To do this, we take $M$, $Q$ and $K$ to infinity, but in a way such that the parameters $M^*$ and $Q^*$ remain finite. Let us consider for concreteness, toral black holes in 4-dimensional spacetime. These flat black holes exhibit many surprisingly elegant properties [@1403.4886], such as: - The maximal in-falling time $\tau_{\text{max}}$ from the horizon to the singularity for a neutral toral black hole[^2] is independent of the black hole mass $M$: $$\label{wow} \tau_{\text{max}} = \int_0^{r_h} \left(\frac{2M}{\pi K^2r} -\frac{r^2}{L^2}\right)^{-\frac{1}{2}} dr = \frac{\pi L}{3}, ~~r_h = \left(\frac{2ML^2}{\pi K^2}\right)^{\frac{1}{3}}.$$ - In the charged case, the Kretschmann scalar $R_{abcd}R^{abcd}$ at the event horizon of the extremal black hole is precisely $144/L^4$. \[It is also the square of the scalar curvature $R=-12/L^2$.\] In the neutral case, the Kretschmann scalar at the event horizon is $36/L^4$. In both of these cases, the Kretschmann scalar at the horizon is independent of the black hole mass, and only depends on $L$. In some sense, these black holes are thus behaving more like pure AdS space than black holes \[property (1) should be compared to the fact that in pure AdS, the time to fall from anywhere to the “center” of AdS only depends on the curvature radius\]. We thus expect some surprises in the properties of the CR-volume of these black holes. As we shall see below, this is indeed so. As pointed out by CR [@1411.2854], their analysis generalizes in a straight-forward manner to other spherically symmetric black holes. In the case of neutral toral black holes in 4-dimensions, the metric is $$g[\text{AdSRN}^0_{4}] = -\left(\frac{r^2}{L^2}-\frac{2M}{\pi K^2 r}\right)dt^2 + \left(\frac{r^2}{L^2}-\frac{2M}{\pi K^2 r}\right)^{-1}dr^2 + r^2(d\zeta^2 + d\xi^2); ~~\zeta, \xi \in [0, 2\pi K).$$ Its CR-volume is $$\label{v1} \text{Vol.} \sim \int^v \int_{T^2} \max\left[r^2 \sqrt{\frac{2M}{\pi K^2 r} -\frac{r^2}{L^2}}\right]~ d\zeta d\xi dv.$$ The coefficients in front of $v$ is maximized when $$r=r_*=\left(\frac{ML^2}{\pi K^2}\right)^{\frac{1}{3}}.$$ Note that $r_* < r_h$. Substituting the value of $r_*$ into the function $$\mathcal{F}(r) = r^2 \sqrt{\frac{2M}{\pi K^2 r} -\frac{r^2}{L^2}},$$ we get $\mathcal{F}(r_*)=ML/(\pi K^2)$. We therefore find that the CR-volume is $$\text{Vol.} \sim 4\pi^2 K^2 \left[\frac{ML}{\pi K^2}\right] v = 4\pi ML v,$$ which is independent of the compactification parameter $K$. Alternatively, we can show this by checking that $$\frac{\partial}{\partial K} \left[4\pi K^2 \mathcal{F}(r_*) \right]= 0.$$ The result generalizes to arbitrary dimension. The value of $r$ that maximizes the volume is $$r_* = \left[\frac{8\pi ML^2}{n (2\pi K)^n}\right]^{\frac{1}{n+1}},$$ The CR-volume can be shown to be $$\label{CR1} \text{Vol.} \sim (2\pi K)^n \mathcal{F}(r_*) v = \frac{8\pi ML}{n}v.$$ Thus, in arbitrary dimension, the CR-volume of a neutral AdS black hole with toral event horizon is *independent* of the compactification parameter. This is surprising since our intuition of volume would be that larger surface area can bound a larger interior volume. This is certainly still true for the CR-volume of an asymptotically flat Schwarzschild black hole — for 4-dimensional case, we have $\text{Vol.}\sim 3\sqrt{3} \pi M^2 v = 3\sqrt{3}Av/16$, where $A=4\pi r_h^2 = 16 \pi M^2$ is the area of its horizon. Thus a larger Schwarzschild black hole *does* have a larger interior volume[^3], which agrees with our flat space expectation. This is perhaps only a coincidence. As we have seen, the flat AdS black holes behave in an entirely different way — for fixed mass $M$, the CR-volume of a black hole with a larger horizon[^4] \[i.e. with larger $K$\] grows asymptotically the same way as a black hole with a smaller horizon \[i.e. with smaller $K$\]. In other words, at sufficiently late time, the maximum volume in any flat AdS black hole with compact event horizon is identical \[up to some finite pieces which are negligible in the volume integral\]. This provides an example in which a large black hole can nevertheless have small interior volume. Note that the planar case is different since there we have to take both $M$ and $K$ to infinity, and the expression given in Eq.(\[CR1\]) does in fact diverge. This remarkable behavior of $K$-independence of the CR-volume nevertheless only holds if the black holes are neutral[^5]. The proof is straightforward but tedious. For simplicity, let us consider only the 4-dimensional case. One starts with the function to be maximized: $$\mathcal{F}(r) = r^2 \sqrt{\frac{2M}{\pi K^2 r}-\frac{r^2}{L^2} - \frac{Q^2}{\pi K^2 r^2}}.$$ Let us denote the value of $r$ that maximizes $\mathcal{F}(r)$ by $r_*$. This can be solved analytically but it is extremely complicated. Fortunately we do not need the exact expression of $r_*$. It can be shown by elementary calculus that $r_*$ satisfies the equation $$\label{condition} 3\pi K^2 r_*^4 - 3ML^2 r_* + L^2Q^2 = 0.$$ Using this condition, we can re-write $$\mathcal{F}(r_*) = \frac{r_*}{\sqrt{\pi}KL}\left[\pi K^2 r_*^4 - \frac{Q^2L^2}{3}\right]^{\frac{1}{2}}.$$ The CR-volume is $$\text{Vol.} \sim 4\pi^2 K^2 \mathcal{F}(r_*) v.$$ Therefore to check for $K$-dependence, it suffices to check whether $$\frac{\partial}{\partial K} \left[K^2 \mathcal{F}(r_*)\right] = 0.$$ A straightforward algebraic manipulation, using the fact that from Eq.(\[condition\]), $$\frac{\partial r_*}{\partial K} = \frac{6\pi K r_*^5}{L^2Q^2 - 9\pi K^2 r_*^4},$$ yields the result that $$\frac{\partial}{\partial K} \left[K^2 \mathcal{F}(r_*)\right] = 0 \Longleftrightarrow Q=0.$$ Therefore, only neutral toral black holes can have $K$-independent CR-volumes. {#section-2 .unnumbered} We have discussed the intriguing property of the CR-volume for neutral toral black holes in AdS, namely that it is independent of the compactification parameter $K$. It is therefore of paramount importance to compare this result with AdS black holes with positively curved event horizons. There is an anologous “compactification parameter” one can consider here, namely by taking quotients of the spherical horizons. More specifically, one constructs the quotient space $S^3/G$ of the 3-sphere by a finite subgroup $G$ of $\text{SO}(4)$ acting freely on $S^3$. By the classification theorem of closed surfaces, there is no other orientable topology other than $S^2$ for a positively curved 2-dimensional compact horizon. So let us consider horizons with $S^3$ topology in 5-dimensional spacetime instead. We can then construct the so-called “black lens” [@ida], which is also allowed in asymptotically flat spacetime \[a rotating black lens was constructed in [@0808.0587]\]. A black lens is a black hole with lens space topology $L(p,1):=S^3/\Bbb{Z}_p$, where $p$ is a positive integer[^6]. For example, $S^3/\Bbb{Z}_1$ is just $S^3$ itself. The first non-trivial example is $S^3/\Bbb{Z}_2 \cong \Bbb{R}\text{P}^3$, the real projective 3-space. Note that $\Bbb{R}\text{P}^n$ is orientable when $n$ is odd. The parameter $p$ is, in some sense, analogous to the compactification parameter $K$ for the flat AdS black holes[^7]. Of course, $p$ is discrete while $K$ is continuous. Nevertheless, it would be interesting to see what happens to the CR-volume of black lenses as we take $p$ to infinity. The metric of a black lens is given by Eq.(\[metric\]), with $k=1$ and $X^1_3 = S^3/\Bbb{Z}_p$. The area of $S^3$ is $2\pi^2$, so the area of $S^3/\Bbb{Z}_p$ is simply $2\pi^2/p$. The area of the horizon is therefore given by $$\frac{2\pi^2}{p} r_h^3 = \frac{12^{1/4}\pi^{5/4}L^{3/2} (\sqrt{3\pi L^2 + 32pM}-L\sqrt{3\pi})^{3/2} }{6p}.$$ For fixed $M$ such that $M$ is small[^8] compared to $L^2$, this function initially *increases* with $p$ but eventually reaches a maximum, and then starts to decrease with increasing $p$. That is to say, for $M$ small compared to $L^2$, taking quotient with a sufficiently small group $\Bbb{Z}_p$ actually causes the area of the event horizon to *increase*. However, for fixed $M$ such that $M$ is sufficiently large compared to $L^2$, the horizon area always decreases upon taking quotients. \[See also the discussion in [@0806.3818] for an expression of the horizon area in terms of the dimensionless area $\Gamma[L(p,1)]$.\] Thus in either case, for sufficiently large $p$, the black hole horizon is small. To compute the CR-volume, the function that we need to maximize is now $$\mathcal{F}(r) = r^3 \sqrt{\frac{16\pi M}{3r^2\left(\frac{2\pi^2}{n}\right)}-\frac{r^2}{L^2}-1}=r^3\sqrt{\frac{8nM}{3\pi r^2}-\frac{r^2}{L^2}-1}.$$ It turns out that $\mathcal{F}(r)$ is maximized by $$r=r_* = \sqrt{\frac{3L^2}{8}\left(-1+\sqrt{1+\frac{256 p M}{27 \pi L^2}}\right)},$$ and the CR-volume is \[v2\] &\~ (r\_\*) v\ & = (-3L)\ &      . It can be checked that for all values of fixed $M$ and $L$, this expression is monotonically *increasing* in $p$, and in the limit $p \to \infty$, the expression in front of $v$ in the CR-volume tends toward a constant $8\pi MLv/3$. Note that this is the same value as the CR-volume of a 5-dimensional neutral toral black hole \[independent of $K$\]. It would be interesting to investigate if there is a deeper reason to this remarkable fact. As we have mentioned, for $M$ that is smaller than $ L^2$, the effect of taking quotient by a small cyclic group $\Bbb{Z}_p$ is to increase the horizon area. Since the CR-volume is monotonically increasing, these black holes conform to our flat space intuition that the volume is a monotonically increasing function of its area. However if $p$ is sufficiently large, then regardless of the value of the ratio $M/L^2$, the horizon area is monotonically decreasing in $p$, whereas the CR-volume is monotonically increasing in $p$. Here we have an example in which a very small black hole could nevertheless possess much larger interior than a bigger black hole of the same mass. {#section-3 .unnumbered} Despite the fact that no unique volume can be prescribed to a black hole, Christodoulou and Rovelli [@1411.2854] showed that it makes sense to talk about the volume of the largest hypersurface bounded by the event horizon \[“CR-volume”\]. Specifically they investigated the CR-volumes of asymptotically flat Schwarzschild and Reissner-Nordström black holes. Bengtsson and Jakobsson [@1502.01907] recently extended the result to asymptotically flat Kerr black holes. Since interior volumes of black holes may play some roles in the context of information loss paradox, it is crucial to understand the properties of CR-volumes for various other black holes. Our flat space intuition is that the volume of a closed surface should be a monotonically increasing function of its area. In other words, a smaller surface area means the volume enclosed is also small. For example, the volume of a 2-sphere is proportional to $A^{3/2}$, where $A$ is its surface area. The interior volume proposed by Christodoulou and Rovelli does have such a property in the case of asymptotically flat Schwarzschild black hole \[there the volume is proportional to $A$ $(\times v)$\]. That the same remains true in the case of asymptotically flat Kerr black holes can be seen in [@1502.01907] — for a fixed mass $M$, increasing the angular momentum $J=aM$ would decrease the horizon area $8\pi M (M + \sqrt{M^2-a^2})$ and likewise its CR-volume also decreases. \[However no such volume arises in the extremal case despite of the nonzero horizon area. The reason is that the region in which the calculation is valid is not present in the extremal case.\] The case for asymptotically flat Reissner-Nordström black hole is very similar. In this work we study topological black holes with toral and lens space event horizons and found that the CR-volume of a black hole is *not* always a monotonically increasing function of its horizon area. In the toral case, if the black hole is neutral, then the CR-volume is independent of the compactification parameter. In other words, for fixed black hole mass $M$ and fixed AdS length scale $L$, the CR-volume is independent of the size of the horizon area. The CR-volume for an AdS black hole in 5-dimensional spacetime with lens space topology $S^3/\Bbb{Z}_p$ \[“black lens”\] is even more remarkable — for any fixed $M$ and $L$, it is monotonically increasing with $p$ and tends to a constant $8\pi MLv/3$ in the limit $p \to \infty$, whereas in the same limit the horizon area shrinks toward 0. Thus a smaller black hole can have a larger CR-volume than a bigger hole of the same mass. \[We remind the readers that this statement is not exact since the CR-volumes we calculated are asymptotic expressions — they ignore the lower limit of the integral in, e.g, Eq.(\[int1\]). \] Therefore, there appears to be no simple relation between the area of the event horizon and the CR-volume. In particular, the CR-volume certainly is *not* always a monotonically increasing function of its horizon area. It may be interesting to investigate the behavior of the CR-volumes for other black holes, such as black ring, and black hole solutions of modified gravity theories. We now speculate on the implication of this result to the information loss paradox. As discussed by Christodoulou and Rovelli, the large interior of a black hole may store information even if its area is shrinking. Of course, this would mean that the Bekenstein-Hawking entropy does not measure the entire information content \[i.e., we have to subscribe to the “weak-form interpretation” of the Bekenstein-Hawking entropy [@1412.8366; @0901.3156; @0003056; @0501103]\]. For a generic topological black hole in AdS, its horizon area \[and hence its Bekenstein-Hawking entropy\] is typically not completely fixed by its mass. For example, for fixed mass $M$ which is small compared to the square of the AdS length scale $L^2$, two AdS black lenses with different values of $p$ can have the same horizon area. If the Bekenstein-Hawking entropy measures the entire information content of a black hole, this implies that these black holes have the same information storage “capacity”. However, if the weak form interpretation is correct, and the CR-volume is responsible for information storage, then these two black holes would have different capacities. On the other hand, this would mean that for fixed $L$, AdS neutral toral black holes with different horizon areas \[due to different values of the compactification parameter $K$\] all have the same capacity for information storage, as long as they have the same mass. The fact that a neutral AdS *planar* black hole possesses infinite CR-volume is intriguing. Although the Bekenstein-Hawking entropy of a planar black hole is formally infinite, from the perspective of AdS/CFT correspondence, the relevant quantity is the entropy *density* \[see, e.g., [@kovtun]\]. If the entropy density measures the information content of the planar black hole, then under Hawking evaporation[^9] it would eventually reach the Page time [@page1; @page2], at which point firewall [@amps1; @amps2] is argued to set in. \[See also [@BPZ]\]. However, if the true information content is given by the CR-volume, which is infinite, then presumably the Page time will never be reached for a planar black hole. See also, Sec.(3.3) of [@1412.8366] for related discussions. \[Of course, having an infinite capacity does not necessarily mean that it has that much information, but only that *in principle, in could*.\] Likewise, even for other black holes, depending on whether the area or the CR-volume encodes information, the Page time will be different. The role of the CR-volume may therefore have profound implication on the onset of the Page time, as well as the firewall paradox. We would like to emphasize that as far as information loss is concerned, we must know what happens to the singularity in a theory of quantum gravity \[see [@1412.8366] for further discussion on this emphasis\]. This is because despite the large interiors, the worldlines of in-falling objects always hit the curvature singularities in finite proper time. In 4-dimensions, for asymptotically flat Schwarzschild black hole the maximal in-fall time is $\pi M$; for AdS neutral flat toral black hole, it is $\pi L/3$, as we have shown in Eq.(\[wow\]). That is, classically the information crashes into the singularity and may still be lost. In addition, we did not discuss possible instabilities of the black hole solutions in this work. If we are really interested in information loss paradox, the final fate of a given black hole is an important issue — if the black hole becomes unstable and evolves into another geometry, its large interior may be irrelevant to information loss. Lastly, we note that AdS black holes with flat horizons, especially the planar ones, are especially important in the various applications of AdS/CFT correspondence, e.g., to condensed matter and heavy ion collider physics \[see [@1403.4886] and the references therein\]. Since the field theory physics is unitary, this provides a strong reason to believe that the bulk physics is also unitary. Understanding how information loss paradox is solved in the bulk however remains an open problem. There are recent attempts to probe the interiors of black holes in string theory [@1310.6334; @1412.1084]. If the CR-volume stores information, it would be important to understand its role in the AdS/CFT context. {#section-4 .unnumbered} The author is grateful to Ingemar Bengtsson and Brett McInnes for comments and discussions. [99]{} John A. Wheeler, *Relativity, Groups, and Topology*, edited by B. S. DeWitt and C. M. DeWitt, Gordon and Breach, New York, 1964. Stephen D.H. Hsu, David Reeb, “Black Hole Entropy, Curved Space and Monsters”, Phys. Lett. B **658** (2008) 244, [\[arXiv:0706.3239 \[hep-th\]\]](http://arxiv.org/abs/0706.3239v2). 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Quantum Grav. **32** (2015) 055014, [\[arXiv:1412.1084 \[hep-th\]\]](http://arxiv.org/abs/1412.1084). [^1]: The usual objection against black hole remnants is that they will be copiously produced \[“infinite production problem”\]. Large interiors may ameliorate this [@0901.3156]. [^2]: For an asymptotically flat Schwarzschild black hole, we have instead $$\tau_{\text{max}}= \int_0^{2M} \left(\frac{2M}{r} -1 \right)^{-\frac{1}{2}}dr = \pi M. \notag$$ [^3]: Here, as well as later, we are comparing the CR-volumes at the same \[sufficiently large\] constant advanced time $\nu$. [^4]: In 4-dimensions, the exact expression for the horizon area is $\displaystyle 4\pi^2K^2\left(\frac{2ML^2}{\pi K^2}\right)^{2/3} \propto K^{2/3}$. [^5]: CR discussed the interior volume of asymptotically flat Reissner-Nordström black holes[@1411.2854]. [^6]: More generally, a lens space $L(p,q)$ is characterized by two parameters. It is the 3-manifold obtained by gluing the boundaries of two solid tori together such that the meridian of the first goes to a $(p,q)$-curve on the second. By $(p,q)$-curve we mean a smooth curve that wraps around the meridian $q$ times and around the longitude $p$ times. A lens space can also be constructed by making appropriate identifications on the boundary 2-sphere of a solid ball. See, e.g., [@maunder]. [^7]: Of course, the lens space is not the only possible topology; one could take quotients by other groups. If we assume the geometry is homogeneous, then in addition to the cyclic groups $\Bbb{Z}_p$ of order $p$, one can take $G$ to be the binary dihedral groups of order $4n$, where $n \geqslant 2$; the binary tetrahedral group of order 24; the binary octahedral group of order 48; and the binary icosahedral group of order 120 [@wolf]. [^8]: These are “small” black holes that are subjected to Hawking evaporation, unlike the “large” black holes that attain thermal equilibrium with their own Hawking radiation. [^9]: Large AdS black holes tend to achieve thermal equilibrium with their own Hawking radiation. Nevertheless they can evaporate if we change the boundary conditions. This can be achieved by coupling the boundary CFT with an auxiliary system [@aux].

8.5in -30pt ł c [FERMILAB–Pub–94/XXX-A\ March 1994]{} [**Quantum Cosmology and Higher-Order Lagrangian Theories**]{}\ Henk van Elst$^{1a}$, James E. Lidsey$^{2b}$ & Reza Tavakol$^{1c}$\ $^1$[*School of Mathematical Sciences\ Queen Mary & Westfield College\ Mile End Road\ London E1 4NS, UK\ *]{} $^2$[*NASA/Fermilab Astrophysics Center\ Fermi National Accelerator Laboratory\ Batavia IL 60510, USA*]{} > In this paper the quantum cosmological consequences of introducing a term cubic in the Ricci curvature scalar $R$ into the Einstein–Hilbert action are investigated. It is argued that this term represents a more generic perturbation to the action than the quadratic correction usually considered. A qualitative argument suggests that there exists a region of parameter space in which neither the tunneling nor the no-boundary boundary conditions predict an epoch of inflation that can solve the horizon and flatness problems of the big bang model. This is in contrast to the $R^2$–theory. > > e-mail: $^a$hve@maths.qmw.ac.uk; $^b$jim@fnas09.fnal.gov;  $^c$reza@maths.qmw.ac.uk > > PACS number(s): 98.80.Hw; 04.50.+h; 04.60.Kz; 98.80.Cq Introduction ============ An important motivation for the development of the quantum cosmology programme has been to explain the initial conditions for the emergence of the Universe as a classical outcome. In principle one must find the form of the wave function $\Psi$ satisfying the Wheeler–DeWitt equation [@wdw]. This equation describes the annihilation of the wave function by the Hamiltonian operator and since it admits an infinite number of solutions, one must also choose the boundary conditions in order to specify the wave function uniquely. Such boundary conditions must be viewed as an additional physical law since, by definition, there is nothing external to the Universe. In practice one assumes, at least implicitly, that a finite subset of all possible boundary conditions is favoured by cosmological observations, in the sense that the wave functions corresponding to such boundary conditions predict outcomes which are compatible with observations. For example, if one believes in the inflationary scenario, the requirement that sufficient inflation occurred, in order to solve the assorted problems of the standard big bang model can, in principle, restrict the number of plausible boundary conditions. Among the set of all possible choices the Vilenkin, or [*tunneling from nothing*]{}, boundary condition [@v1; @v2] and the Hartle–Hawking, or [*no-boundary*]{}, boundary condition [@HH83] have been the subject of intense discussion. Given the non-uniqueness of such conditions, the question arises as to the consequences of choosing different boundary conditions for the resulting wave function of the Universe and its corresponding probability measures. An important study in this regard is due to Vilenkin [@v2], who considered the effects of the above boundary conditions within the context of Einstein gravity minimally coupled to a self-interacting scalar field. He restricted his analysis to the minisuperspace corresponding to the spatially closed, isotropic and homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) Universe and showed that the tunneling wave function predicts initial states that are likely to lead to sufficient inflation, whereas the Hartle–Hawking wave function does not. It is sometimes argued that this result indicates that observations favour the tunneling as opposed to the no-boundary boundary condition. However, the precise relation between the boundary conditions and the observations is determined by the specific models employed and since such models always involve idealisations in the form of a set of simplifying assumptions, it follows that the above conclusion can not be made [*a priori*]{}. Indeed it only makes sense in general if the correspondence between the observations and the boundary conditions is robust under physically motivated perturbations to the underlying quantum cosmological model. Consequently, it is important to consider the ‘stability’ of the above conclusions. In particular, are the conclusions robust under higher-order perturbations to the Einstein–Hilbert action? Quadratic and higher-order terms in the Riemann curvature tensor and its traces appear in the low-energy limit of superstrings [@canetal85] and they also arise when the usual perturbation expansion is applied to General Relativity [@barchr83; @anttom86]. Such terms diverge as the initial singularity is approached, but can in principle be eliminated if higher-order corrections are included in the action. In four-dimensional space-times the Hirzebrucht signature and Euler number imply that the most general, four-dimensional gravitational action to quadratic order is S = d\^4x    , where $R$ is the Ricci curvature scalar of the space-time with metric tensor $g_{\mu\nu}$, $g={\rm det}\,g_{\mu\nu}$, $C_{\alpha\beta\gamma\delta}$ is the Weyl conformal curvature tensor, $\kappa^{2}$ is the gravitational coupling constant and $\epsilon_1$ and $\gamma$ are coupling constants of dimension $(\mbox{length})^{2}$. The action simplifies further for spatially homogeneous and isotropic four-geometries, since the conformal flatness of these space-times implies that the Weyl tensor vanishes. The effects of including quadratic terms have been investigated in Refs. [@bg93; @hawlut84; @mijetal89]. In particular Mijić et al [@mijetal89] studied the effects of such perturbations on Vilenkin’s result [@v2] and found that those results remain robust in the sense that the inflationary scenario still favours the tunneling boundary condition in the presence of quadratic terms in the action. On the other hand Biswas and Guha have recently arrived at the opposite conclusion [@bg93]. The renormalisation of higher loop contributions introduces terms into the effective action that are higher than quadratic order. Consequently it is important to also study the effects of these additional terms. In this paper we shall investigate what happens to the wave function if an $R^3$-contribution is present. By employing the conformal equivalence of higher-order gravity theories with Einstein gravity coupled to matter fields, we argue that this term represents a more general perturbation to the Einstein–Hilbert action than the $R^2$-correction, at least within the context of four-dimensional FLRW space-times. We then consider the conditional probability that an inflationary epoch of sufficient duration can occur. We estimate how the qualitative behaviour of this quantity changes when higher-order perturbations to the action are included. Our main result is that for the $R^3$–theory there exists a finite region of parameter space in which neither of the boundary conditions discussed above predict an epoch of inflationary expansion that leads to the observed Universe. We use (dimensionless) Planckian units defined by $\hbar = c = G = 1$ throughout and define $\kappa^2 = 8\pi$. Higher-Order Lagrangians as Einstein Gravity plus Matter {#Lagrange} ======================================================== The wave function of the Universe in higher-order Lagrangian theories can be determined in one of two ways. It is well known that theories with a Lagrangian given by a differentiable function of the Ricci curvature scalar are conformally equivalent to Einstein gravity with a matter sector containing a minimally coupled, self-interacting scalar field [@whitt84; @m]. The precise form of the self-interaction is uniquely determined by the higher-derivative metric terms in the field equations. It follows that one can start either from the original action or the conformal action and derive the corresponding Wheeler–DeWitt equation [@hall91]. One takes the related Lagrangian as the defining feature of the theory and then applies the canonical quantisation rules. The advantage of the conformal transformation is that it allows the known results from Einstein gravity to be carried over to the higher-order examples and we shall follow such an approach in this paper. Consider the general, $D$-dimensional, vacuum theory S = d\^Dx   , where the Lagrangian $f(R)$ is some arbitrary differentiable function of the Ricci curvature scalar satisfying $\{ f(R), df(R)/dR \} > 0$ and $g_D$ is the determinant of the $D$-dimensional space-time metric $g_{D\,\mu\nu}$. If we perform the conformal transformation [@m] ł[conf]{} \_[D]{} = \^2g\_[D]{} \^2 = ( 2\^2)\^[2/(D-2)]{} , and define a new scalar field ł[phi]{} | ( )\^[1/2]{}  , the conformally transformed action takes the Einstein–Hilbert form ł[conformal]{} S = d\^D x   , where the self-interaction potential is given by ł[\*]{} U(|) ( 2\^2 )\^[-D/(D-2)]{}( R(|) -f\[R(|)\] ) . Definition (\[phi\]) yields a correspondence between the values of the Ricci curvature scalar $R$ and the values of the scalar field $\bar{\phi}$. We shall consider the quadratic and cubic Lagrangians \[lagr\] f\_2 (R) & = & (R + \_1R\^2)\ f\_3 (R) & = & (R + \_1R\^2 +\_2R\^[3]{}) , in four dimensions, where the parameters $\epsilon_{1}$ and $\epsilon_{2}$ have dimensions $(\mbox{length})^{2}$ and $(\mbox{length})^{4}$ respectively before the introduction of Planckian units. The corresponding potentials for positive $\epsilon_1$ and $\epsilon_2$ are given by [@mijetal89; @b]: \[pot2\] U\_[f\_[2]{}]{}(|) & = & \^2\ \[pot3\] U\_[f\_[3]{}]{}(|) & = & (-2|)  , and are semi-positive definite for all values of $\bar{\phi}$. **Figures 1a & 1b** In the classical $R^{3}$–theory the requirement that the inflationary epoch lasts sufficiently long implies that the coupling constants must satisfy $|\epsilon_2| \ll \epsilon_1\!^2$ [@b]. Moreover, the observed isotropy of the cosmic microwave background radiation requires that $\epsilon_{1}\approx 10^{11}$ [@mijetal89]. In view of these constraints we specify $\epsilon_1=10^{11}$ in the subsequent numerical calculations. Figures 1a and 1b illustrate the behaviour of the potentials (\[pot2\]) and (\[pot3\]) for $\epsilon_1 \approx 10^{11}$ and $\epsilon_2 \approx 10^{20}$. The effect of decreasing the value of the parameter $\epsilon_1$ is to increase the height of the plateau and the relative maximum of the potentials in the quadratic and cubic cases respectively. This reflects the fact that decreasing this parameter is equivalent to increasing the energy scales involved. In this sense there exists no continuous transformation from an $R^{2}$–theory to the ordinary Einstein–Hilbert action as this parameter approaches zero. In the neighbourhood of the origin of $\bar{\phi}$ corresponding to smaller values of $R$ the quadratic term in the action dominates and the potentials in this region are equivalent. This can be seen by expanding the last of the three terms in the square brackets of Eq. (\[pot3\]). The first-order contribution cancels the remaining terms in $U_{f_3}$ and the second-order term reduces the form of $U_{f_3}$ to that of $U_{f_2}$. Hence the two potentials are effectively identical if the third- and higher-order terms in the expansion can be neglected. It is straightforward to show that this is a consistent approximation if ł[consistent]{} | |\_[limit]{} ( ) . For polynomial Lagrangians with $f(R)=\left(\,\sum_{k=1}^{n}\,\epsilon_{k-1}\, R^k\,\right)\,/\,2\kappa^{2}$, the detailed form of the corresponding potential $U(\bar{\phi} )$ is extremely complicated and generally not expressible in an analytically closed form. Nevertheless, one can determine the qualitative behaviour of the potential at small and large $\bar{\phi}$. Close to the origin the quadratic term in the action again dominates and the potential in this region is therefore similar to Eq. (\[pot2\]). The asymptotic behaviour at infinity, however, depends critically upon the combination of the highest degree $n$ of the polynomial and the dimensionality $D$ of the space-time [@m]. More precisely, for $D>2n$ the potential is unbounded from above, for $D=2n$ it flatens into a plateau and for $D<2n$ the potential has an exponentially decaying tail [@b]. In particular, if $D<2n$ the effective scalar field potential $U(\bar{\phi})$ is qualitatively equivalent to the cubic potential (\[pot3\]). As a result, when $D=4$ the qualitative behaviour of $U(\bar{\phi})$ does not change relative to the cubic case as terms with $n>3$ are considered, although the relative position of the maximum of $U(\bar{\phi})$ will be $n$-dependent. This implies that the $n=2$ contribution is rather special in four dimensions, whereas the $R^3$-term is in fact a more generic perturbation. Thus, it is instructive to consider this case further. Behaviour of the Wave Function {#Psi} ============================== Within the context of the spatially closed FLRW minisuperspace, the Wheeler–DeWitt equation derived from theory (\[conformal\]) has been solved for an arbitrary potential, subject to the condition that the momentum operator for the scalar field can be neglected [@v1; @v2]. This is self-consistent if $|dV/d\phi | \ll {\rm max} \{ |V|, a^{-2} \}$, where $a$ represents the cosmological scale factor and ł[rescale]{} V  U  | . The WKB approximations of the wave functions satisfying the quantum tunneling boundary condition ($\Psi_{V}$) and the Hartle–Hawking no-boundary proposal ($\Psi_{HH}$) then take the forms [@v2] \_[V]{} &=& (1-a\^2V)\^[-1/4]{} \ \_[HH]{} &=& (1-a\^2V)\^[-1/4]{} in the classically forbidden (Euclidian signature) region defined by $a^2\,V<1$, and \_[V]{} &=& e\^[i/4]{}(a\^2V-1)\^[-1/4]{} \ \_[HH]{} &=& 2(a\^2V-1)\^[-1/4]{} in the classically allowed (Lorentzian signature) region $a^2\,V>1$. Substituting for $V(\phi)$ from the potentials of the quadratic and cubic Lagrangians of Section \[Lagrange\], it can readily be seen that the wave functions corresponding to the quadratic and cubic theories have very different types of behaviour, at least for large $\phi$. In the quadratic case both $\Psi_{V}$ and $\Psi_{HH}$ remain bounded. However, for the cubic case $\Psi_{HH}$ becomes divergent in the classically allowed region whilst $\Psi_{V}$ remains regular. In this sense then the qualitative behaviour of the wave function satisfying the no-boundary proposal is fragile with respect to cubic perturbations to the action. This is significant because often the quadratic corrections to the action are taken as representative of higher-order perturbations. To proceed it is important to ensure that for the regimes under consideration the conformal transformation (\[conf\]) remains non-singular. This is the case if the condition $df(R)/dR\neq 0$ is valid for all values of $R$. The conformal transformation is singular at the point R = - , in the $R^2$–theory and at the point R = -for the $R^{3}$–theory. Since $\epsilon_{1}$ and $\epsilon_{2}$ are taken to be positive, these conditions imply that in both cases the problematic values of $R$ lie in the region $R<0$. However, for a classical, spatially closed FLRW model, the Ricci curvature scalar is given by R = 6(1-q)()\^[2]{} +  , where $q \equiv -\ddot{a}\,a / \dot{a}^{2}$ defines the deceleration parameter and a dot denotes differentiation with respect to cosmic proper time. Now if, as is generally assumed, the Universe tunnels into the Lorentzian region in an inflationary phase ($q<0$), it follows that $R$ will be positive-definite. Thus, the conformal transformation is self-consistent in these theories. Interpretation of the Wave Function =================================== In the previous section we saw that the wave functions corresponding to the tunneling and the Hartle–Hawking boundary conditions have qualitatively different modes of behaviour for the quadratic and cubic theories. To see what predictive effects such changes might have, we employ the notion of a probability density $\rho$ as is usually done. For the cases of the tunneling and the Hartle–Hawking boundary conditions respectively, $\rho$ takes the form [@v2] \_[V]{}(a,) &=& C\_[V]{}\[rhot\]\ \_[HH]{}(a,) &=& C\_[HH]{}\[rhohh\] on surfaces of constant scale factor in the classically allowed region of minisuperspace, where the normalisation constants $C_{V}$ and $C_{HH}$ are given by C\_[V]{}\^[-1]{} &=& \_[V()&gt;0]{} d \ C\_[HH]{}\^[-1]{} &=& \_[V()&gt;0]{} d  . Since $\rho (\phi)$ is usually not normalisable, the common practice is to employ the notion of a conditional probability [@hall91]. One argues that the initial values of the scalar field must lie in the range $\phi_{min}<\phi_{i}<\phi_{P}$. The lower limit $\phi_{min}$ follows from the requirement that the Universe expands at least until the formation of large-scale structure and the upper bound follows from the condition that $V(\phi_P )\approx 1$, since the minisuperspace approximation is unlikely to be valid when the potential energy of the matter sector exceeds the Planck density. However, in a chaotic inflationary scenario there is a critical value of the scalar field, $\phi_{suf}$, and sufficient inflation occurs if $\phi_i > \phi_{suf}$ but not for $\phi_i<\phi_{suf}$. We must therefore calculate the conditional probability that sufficient inflation occurs given that $\phi_i$ is bounded by $\phi_{min}$ and $\phi_P$. This quantity takes the form [@hall91] \[condprob\] P(\_[i]{}&gt;\_[suf]{}|\_[min]{}&lt;\_[i]{}&lt;\_[P]{}) =  , and allows us to determine which of the two boundary conditions considered here “naturally” predicts a phase of sufficiently long inflationary expansion. Sufficient inflation is a prediction of a theory if $P\approx 1$, whereas it is not if $P \ll 1$. For standard reheating the minimum amount of inflation that solves the horizon problem is determined by the condition $N\equiv \ln (a_f / a_i ) \approx 65$, where subscripts $i$ and $f$ denote the values of the scale factor at the onset and end of inflation respectively [@guth]. It is then straightforward to deduce from the classical field equations that \[efold\] N  65  6 \_[\_[f]{}]{}\^[\_[suf]{}]{} V() ( )\^[-1]{}d , where the value of the scalar field at the end of inflation, $\phi_f$, is computed from the relation ł[end]{} \_[= \_f]{}\^[2]{} = 1 . This condition corresponds to the breakdown of the slow-roll approximation [@st1984].[^1] Once $\phi_f$ is known, the value of $\phi_{suf}$ can be determined numerically by evaluating the integral in Eq. (\[efold\]). To understand how the probability densities (\[rhot\]) and (\[rhohh\]) change in the quadratic and cubic cases, we shall consider them in turn. Since (\[rhot\]) and (\[rhohh\]) are usually [*not*]{} normalisable (unless the range of values that $\phi$ can take is bounded), we set the “normalisation constants” equal to one as is the common practice. The Quadratic case {#quadratic} ------------------ To begin with, we note that the shape of $V(\phi)$ does not qualitatively change with changes in the coupling constant $\epsilon_{1}$. This parameter only fixes the height of the plateau and as a result leaves the shapes of the two probability densities unchanged. Consequently the qualitative behaviours of the probability densities are robust with respect to changes in $\epsilon_{1}$. Figure 2a gives a plot of $\rho_{V}$ showing that it starts at zero when $\phi=0$ and asymptotically approaches a constant value. On the other hand, as can be seen from Figure 2b, $\rho_{HH}$ decreases from infinity and asymptotically approaches a constant value. We should emphasise here that since the probability distribution functions () and () typically take values of the order $\exp(\pm 10^{14})$, we, for the sake of graphical representation, applied non-linear scalings of the kinds $\tilde{\rho}_{V}={\rho_V}^{1/C}$ and $\tilde{\rho}_{HH} =\ln\left({\rho_{HH}}^{1/C}\right)$ respectively (where $C$ is a constant) to the two probability distribution functions. Note, however, that the values of the argument $\phi$ remain uneffected by this scaling. Contrary to the claim of Biswas and Guha [@bg93], the two probability distribution functions reveal [*no*]{} qualitative changes as compared to the case of “chaotic” type potentials (e.g. $V(\phi)=m^{2}\,\phi^{2}/2$) as discussed by Vilenkin [@v2] and Halliwell [@hall91]. This means that the tunneling wave function has its maximum nucleation probability for the Universe coming into existence somewhere on the plateau of the potential $V(\phi)$, whereas the Hartle–Hawking wave function peaks near the true minimum of the potential at $\phi=0$. Translated into initial values of the Ricci curvature scalar, this means that the tunneling wave function prefers values of $ R_{i}$ near the Planck scale, whereas the no-boundary wave function favours a Universe of large initial size, i.e. small $R_{i}$ [@mijetal89]. **Figures 2a & 2b** We now consider the conditional probability (\[condprob\]). The range of values of $\phi_{i}$ is specified by the range of initial values $R_i$ . In Planckian units, where $R_{P}=1$, we deduce that $\phi_{P}=13.0$. The value of $\phi_f$ is calculated from (\[end\]) to be $\phi_f =0.38$ and condition (\[efold\]) is therefore satisfied for $\phi_{suf} = 2.27$. Since the conditional probability measure (\[condprob\]) essentially amounts to a comparison of areas between the $\rho(\phi)$ curve and the positive $\phi$-axis in Figures 2a and 2b, it seems obvious that the tunneling wave function leads to sufficient inflation whereas the no-boundary wave function does not. This is in line with the conclusions of Vilenkin [@v2] and Mijić et al [@mijetal89] and in contrast to what is claimed by Biswas and Guha [@bg93]. The Cubic Case {#cubic} -------------- We now consider the effects of adding a cubic term to the action. In general $\rho_{V}$ is peaked around the maximum of $V(\phi)$ at $\phi_{max}$ and falls off to zero on both sides. In contrast $\rho_{HH}$ decreases from infinity near $\phi=0$ to a minimum at $\phi_{max}$ and diverges again as $\phi\rightarrow\infty$. In this sense the presence of the cubic term drastically alters the shapes of the two probability distributions. This qualitative behaviour is illustrated in Figures 3a and 3b for $\epsilon_1 =10^{11}$ and $\epsilon_2 =10^{20}$. **Figures 3a & 3b** Now, regarding the location of the maximum nucleation probability, the tunneling case is unambiguous since there is only a single peak in the probability distribution function. Note, however, that in the cubic case this wave function favours [*smaller*]{} values of the initial curvature $R_i$ (viz. $\phi_i$) as compared to those in the quadratic case, where they are of Planckian order. On the other hand, the case of the Hartle–Hawking boundary condition is ambiguous because of the presence of two peaks in the probability distribution function, corresponding respectively to low and high values of $R_{i}$. From a practical point of view, the question arises as to whether the Vilenkin wave function still predicts a phase of sufficiently long inflationary expansion immediately after tunneling into the Lorentzian signature region. To investigate this, we confined ourselves to the region on the left of the maximum in the potential (\[pot3\]), i.e. $\phi \le \phi_{max}$. Although inflation occurs on both sides of the turning point, there is no end to the superluminal expansion if the field rolls down the right-hand side and consequently there is no reasonable mechanism of reheating [@b]. On the basis of these physical considerations it is therefore more appropriate to [*identify*]{} the upper limit $\phi_P$ of the integrals in Eq. (\[condprob\]) with $\phi_{max}$ rather than with the Planck limit. The specific value of the conditional probability depends on the magnitude of $\epsilon_2$ and it is therefore necessary to determine the relevant range of values for this parameter. We noted in Section that $\epsilon_2$ is bounded from above by the condition $\epsilon_2 \ll {\epsilon_1}^2$. As $\epsilon_2$ is decreased relative to a [*fixed*]{} $\epsilon_1$, the location of the maximum is shifted to larger values of $\phi$ and eventually beyond the Planck limit $\phi_{P}$. This follows since the model reduces to the $R^2$–theory for which the potential exhibits a plateau, i.e. the maximum is effectively located at infinity in this case. However, according to condition (\[consistent\]) the region over which the cubic and quadratic potentials are equivalent also increases as $\epsilon_2$ decreases. The question then is whether $\phi_{limit}$ grows faster or slower than $\phi_{max}$. By explicitly calculating the values of $\phi_{max}$ and $\phi_{limit}$ it is found that $\phi_{limit}$ exceeds $\phi_{max}$ for all parameter values $\epsilon_2 \le 10^{20}$. This implies that the $R^2$– and $R^3$–theories are equivalent for $\phi < \phi_{max} $ in this range. Hence the results in Section for $R^2$–theory may be carried over directly to the cubic case in this region of the variable $\phi$, although there is the important difference that the upper bound on $\phi_i$ is now identified with $\phi_{max}$ and not $\phi_P$. For any given $\epsilon_2$ the end of inflation occurs at $\phi_f =0.38$ as in the $R^2$-case, since the $R^3$-contribution is negligible at very small $\phi$. Unfortunately a direct numerical integration of Eq. (\[condprob\]) can not be performed, because the integrands are typically of the orders of of $\exp(\pm 10^{14})$. However, since the probability density $\rho$ is a single valued, positive-definite function of $\phi$, it follows that a handle on the qualitative behaviour of the conditional probability can be obtained by investigating how the area under the $\rho (\phi)$ curve changes as $\epsilon_2$ changes. The problem then reduces to determining how the limits of the integrals in the numerator and denominator vary as the parameters of the theory are altered. The dependences of the parameters of interest on $\epsilon_2$ are summarised in Table . We find that $\phi_{suf}$ for the potential (\[pot3\]) settles at the same value as in the quadratic case when $\epsilon_{2}$ is of order $10^{18}$ or smaller. We also find that $\phi_{max}$ rapidly approaches $\phi_{suf}$ in the region $10^{18}\le\epsilon_{2}\le10^{20}$. This implies that the integral in the numerator of the conditional probability (\[condprob\]) becomes [*much smaller*]{} than the term in the denominator for $\epsilon_2 \ge 10^{18}$. Consequently the Vilenkin scheme does not predict a phase of sufficiently long inflation in this region, contrary to the results for the $R^{2}$–model. We further note that for the same range of initial values of $\phi$, the Hartle–Hawking wave function shows no qualitative change from the quadratic case. Consequently, it appears that neither boundary condition predicts inflation for this choice of the parameters $\epsilon_{1}$ and $\epsilon_{2}$. This behaviour occurs because the presence of the cubic perturbation severely restricts the range of initial field values $\phi_i$ for which a phase of sufficiently long inflationary expansion is likely. Including the full range of values of $\phi_{i}$ up to the Planck limit $\phi_{P}$ would not significantly improve this result in the Vilenkin scheme. In the Hartle–Hawking case, however, the integral in the numerator of (\[condprob\]) would have a large contribution from the second peak in $\rho_{HH}$. However, this range of $\phi_{i}$ was excluded, as discussed above, in order to avoid the problem of exiting the inflationary expansion. ---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- -- $\epsilon_{2}$ $10^{20}$ $10^{18}$ $10^{16}$ $10^{14}$ $10^{12}$ $10^{10}$ $10^{8}$ $10^{6}$ $\phi_{P}$ $23.6$ $21.3$ $19.0$ $16.7$ $14.4$ $13.1$ $13.0$ $13.0$ $\phi_{limit}$ $2.65$ $4.95$ $7.25$ $9.56$ $11.9$ $14.2$ $16.5$ $18.8$ $\phi_{max}$ $1.59$ $2.68$ $3.78$ $4.94$ $7.06$ $9.34$ $11.7$ $13.0$ $\phi_{suf}$ $1.59$ $2.24$ $2.27$ $2.27$ $2.27$ $2.27$ $2.27$ $2.27$ ---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- -- : Summarising, for different values of $\epsilon_2$, the values of the scalar field corresponding to $R_P=1$ $(\phi_P)$, the limit of $\phi$ below which the $R^2$– and $R^3$–potentials are equivalent $(\phi_{limit})$, the location of the maximum in the potential $(\phi_{max})$ and the values of the field that just lead to sufficient inflation $(\phi_{suf})$. We specify $\epsilon_1 =10^{11}$ throughout due to microwave background considerations. As $\epsilon_2$ increases to order of $10^{18}$, the magnitudes of the quantities $\phi_{max}$ and $\phi_{suf}$ become comparable to one another and this implies that the numerator in the conditional probability approaches zero. This suggests that the conditional probability will become significantly smaller than unity for values of $\epsilon_2 \ge 10^{18}$. ł[tab1]{} Even though the conditional probability $P$ of Eq. () cannot be estimated numerically in this case, nevertheless, we present a set of values of “scaled conditional probabilities” in Appendix A which are obtained by applying a non-linear scaling to the probability distribution functions as discussed in Section . These values, which may be treated as qualitative indicators of $P$, also support the conclusions given in this section. Discussion and Conclusions ========================== In this paper we have investigated how the probability of realising sufficient inflation from quantum cosmology is altered when higher-order corrections to the Einstein–Hilbert action are introduced. Our results confirm that the addition of quadratic terms to the action does not reverse the conclusions of Vilenkin [@v1; @v2] regarding the effects of boundary conditions on the likelihood of sufficient inflation, in contrast to some recent claims [@bg93]. On the other hand, cubic perturbations can produce qualitative changes to the nature of the probability distribution function $\rho(\phi)$. From a physical point of view one is confined to consider initial values of the scalar field that allow an exit from the inflationary expansion. As a result the important physical (as opposed to purely mathematical) consequences of cubic perturbations are that they restrict the measure of allowed initial field values $\phi_{i}$ that lead to sufficient inflation. This is in agreement with the classical arguments [@b]. By considering the conditional probability (\[condprob\]) (see also Appendix A) we have argued that if the coupling constant $\epsilon_{2}$, which determines the strength of the $R^{3}$-contribution to the Lagrangian, exceeds a critical value, neither the tunneling nor the no-boundary boundary conditions predict an epoch of sufficient inflation, in the sense that the conditional probability is significantly less than unity in both cases. Our results appear to exhibit some generality in four-dimensions. As discussed in Section , the qualitative shape of the self-interaction potential $V(\phi)$ remains unaltered if general polynomial perturbations with a highest order term $\epsilon_{n-1}\,R^n$ are considered. In general this result is true when $D<2n$. This immediately implies that neither of the two probability distributions $\rho_V$ and $\rho_{HH}$ for the $n=3$ case will be qualitatively affected under $n>3$ perturbations. The qualitative conclusions drawn for the case of cubic perturbations in Section therefore remain robust under higher-order perturbations to the action, although of course the details of what happens will depend on how the precise location of the maximum in the potential $V(\phi)$ is related to the highest-order term. However, the consequences of the quadratic and the cubic perturbations (as well as those of general polynomial types) depend crucially on the values of the free parameters of the system, namely $\epsilon_k~(k=1, \ldots , n-1)~,~D,~n$, as well as on the initial field values $\phi_i$. In particular, the dimensionality $D$ of the space-time is crucial in deciding the maximum degree $n$ of perturbations allowed ($D<2n$ say) above which the perturbations would be qualitatively inconsequential, i.e. the system would be robust. Finally we remark that inflation is possible, at least at the classical level, if the field is initially placed to the right of the maximum in Eq. (\[pot3\]) and given sufficient kinetic energy to travel over the hill towards $\phi=0$. Unfortunately, our analysis can not consider this possibility since the scalar field momentum operator in the Wheeler–DeWitt equation then becomes important and the solutions (\[rhot\]) and (\[rhohh\]) are no longer valid. Furthermore, if one is prepared to include the effects of the $R^3$-contribution in the action, the cubic term $R {\Box} R$ should also be considered. In this case the effective theory resembles Einstein gravity minimally coupled to two scalar fields after a suitable conformal transformation on the metric [@b] and in principle a similar analysis to the one presented here can be followed for this more general case. We shall return to some of these questions in future. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Jonathan J Halliwell for helpful remarks. HvE is supported by a Grant from the Drapers’ Society at QMW. JEL is supported by the Science and Engineering Research Council (SERC), UK, and is supported at Fermilab by the DOE and NASA under Grant No. NAGW-2381. RT is supported by the SERC, UK, under Grant No. H09454. Appendix A {#appendix-a .unnumbered} ========== As was pointed out in Section , the integrands involved in the definition of conditional probability typically have magnitudes of order $\exp(\pm 10^{14})$, which makes the numerical calculation of the integrals not possible in practice. Now due to the nature of these numbers no linear scaling of the probability function $\rho$ can bypass this difficulty. The question then arises as to whether appropriate non-linear scalings exist which keep the conditional probability $P$ invariant. To see that there do [*not*]{}, recall that the only scalings that leave the Wheeler–DeWitt equation, $H\,\Psi=0$, of the $D$-dimensional minisuperspace models of Quantum Cosmology invariant are given by $\tilde{H}=\Omega^{-2}\,H,\ \tilde{\Psi}=\Omega^{\gamma}\,\Psi \hspace{0.5mm} \rightarrow \hspace{0.5mm}\tilde{H}\,\tilde{\Psi}= \Omega^{\gamma-2}\,H\,\Psi=0$, ($\Omega(q)$ is an arbitrary function of the minisuperspace co-ordinates $q$) provided $\gamma$ and $\xi$ (a free parameter in the Wheeler–DeWitt equation) are given by $\gamma=(2-D)/2$ and $\xi=-(D-2)/8(D-1)$ respectively [@hall88]. Effectively this amounts to a redefinition of the potential $U(q)$ and the DeWitt metric of minisuperspace $f^{\alpha\beta}(q)$, which occur in the Hamilton operator $H$. More importantly, under such scale transformations the conserved probability current density $j^{\alpha}$ defined from $\Psi$ remains unchanged. This freedom, however, is not of much use in bypassing the numerical difficulty mentioned above in order to obtain quantitative values for $P$. Nevertheless, if we confine ourselves to qualitative information, we may choose non-linear (but monotonic) scalings of $\rho$, which, while violating the invariance properties of the model, would nevertheless supply us with a qualitative indicator of $P$. This is not dissimilar to the way non-linear scalings of functions are employed for the purpose of graphical representation. To calculate a qualitative indicator of $P$ we define the [*non-linearly scaled conditional probability*]{} $\tilde{P}$ as \[scalconpr\] P(\_[i]{}&gt;\_[suf]{}|\_[min]{}&lt;\_[i]{}&lt;\_[P]{})  , where $C$ is the index of non-linear scaling. Clearly such a scaling will not change the qualitative behaviour of $\rho_{V}$ and the values of its argument $\phi$, and therefore the values of the boundaries of the integrals occurring in Eq. () (as listed in Table ) remain the same. Furthermore, such scalings leave $P$ invariant in the limiting cases where $P=0$ and $P=1$. Here we chose $C=10^{14}$. Table gives the values of $\tilde{P}$ as a function of $\epsilon_{2}$ for the Vilenkin wave function in the case of the $R^3$–model, calculated for $\epsilon_{1}=10^{11}$ and the boundary values of $\phi$ given in Table . We approximated $\phi_{min}$ by $\phi_{f}=0.38$. For the corresponding value of $\tilde P$ for the Vilenkin model in the $R^{2}$-case of Section we found $\tilde{P}=0.85$. ---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- -- $\epsilon_{2}$ $10^{20}$ $10^{18}$ $10^{16}$ $10^{14}$ $10^{12}$ $10^{10}$ $10^{8}$ $10^{6}$ $\tilde P$ $0.00$ $0.20$ $0.45$ $0.59$ $0.72$ $0.79$ $0.84$ $0.85$ ---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- -- : Behaviour of the non-linearly scaled conditional probability distribution $\tilde{P}$ for the Vilenkin wave function $\Psi_V$ in the $R^3$–model of Section . We specify $\epsilon_1=10^{11}$ throughout. ł[tab2]{} As can be seen from Table , the behaviour of $\tilde P$ supports the conclusions drawn in Section on the basis of qualitative analysis. [99]{} DeWitt B S 1967 [*Phys. 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D*]{} [**38**]{} 2468 Figure Captions {#figure-captions .unnumbered} =============== [*Figure 1:*]{} (a) The effective self-interaction potential (\[pot2\]) corresponding to the $R^2$–theory with $\epsilon_{1}=10^{11}$. The scalar field and magnitude of the potential have been rescaled via Eq. (\[rescale\]) to enable easy comparison with the results of Section 4; (b) The rescaled effective interaction potential (\[pot3\]) corresponding to the $R^3$–theory with $\epsilon_{1}=10^{11}$ and $\epsilon_{2}=10^{20}$. [*Figure 2:*]{} (a) The Vilenkin probability distribution $\rho_V(\phi)$ for the $R^{2}$–theory with a rescaling $\rho_V(\phi) = \left[\ \exp (- 2/3V )\ \right]^{10^{-14}}$; (b) The Hartle–Hawking probability distribution $\rho_{HH} (\phi)$ for the $R^{2}$–theory with a rescaling $\rho_{HH}(\phi) = \ln \left[\ \exp ( 2/3V)\ \right]^{10^{-14}}$. We choose these particular rescaled values of $\rho(\phi)$ in order to obtain easily interpretable plots from our numerical programme. [*Figure 3:*]{} (a) The Vilenkin probability distribution $\rho_V(\phi)$ for the $R^{3}$–theory with the same rescaling as for Figure 2a; (b) The Hartle–Hawking probability distribution $\rho_{HH} (\phi)$ for the $R^{3}$–theory with the same rescaling as for Figure 2b. [^1]: Strictly speaking, conditions (\[efold\]) and (\[end\]) are only valid in spatially flat FLRW models, but we are considering spatially closed cases in this work. However, during inflation the curvature term in the Friedmann equation is redshifted to zero within one Hubble expansion time and the Universe effectively becomes spatially flat at an exponentially fast rate. For our purposes, therefore, these expressions remain valid.

--- abstract: 'We present results from a Partial-Wave Analysis (PWA) of diffractive dissociation of 190 [ $\text{GeV}\! / c$]{} [$\pi^-$]{} into [${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} final states on nuclear targets. A PWA of the data sample taken during a COMPASS pilot run in 2004 on a [$\text{Pb}$]{} target showed a significant spin-exotic ${\ensuremath{J^{PC}}}= 1^{-+}$ resonance consistent with the controversial ${\ensuremath{\pi_1}}(1600)$, which is considered to be a candidate for a non-[$q$]{}[${ \overline{q}}$]{} mesonic state. In 2008 COMPASS collected a large diffractive [${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} data sample using a hydrogen target. A first comparison with the 2004 data shows a strong target dependence of the production strength of states with spin projections $M = 0$ and $1$.' address: | Excellence Cluster Universe, Technische Universität München,\ Boltzmannstr. 2, 85748 Garching, Germany.\ bgrube@ph.tum.de author: - BORIS GRUBE for the COMPASS Collaboration title: | DIFFRACTIVE DISSOCIATION OF 190 [ $\text{GeV}\! / c$]{} [$\pi^-$]{} \ INTO [${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} FINAL STATES AT COMPASS --- The COmmon Muon and Proton Apparatus for Structure and Spectroscopy (COMPASS)[@compass] is a fixed-target experiment at the CERN Super Proton Synchrotron. It is a two-stage spectrometer that covers a wide range of scattering angles and particle momenta with high angular resolution. The target is surrounded by a Recoil Proton Detector (RPD) that measures the time of flight of the recoil protons. COMPASS uses the M2 beamline which can deliver secondary hadron beams with a momentum of up to 300[ $\text{GeV}\! / c$]{} and a maximum intensity of [ ]{} $\text{sec}^{-1}$. The negative hadron beam consists of 96.0 % [$\pi^-$]{}and 3.5 % [$K^-$]{}. Two ChErenkov Differential counters with Achromatic Ring focus (CEDAR) upstream of the target are used to identify the incoming beam particles. During a pilot run in 2004 and subsequent data taking periods in 2008 and 2009 COMPASS has acquired large data sets of diffractive dissociation of 190[ $\text{GeV}\! / c$]{} [$\pi^-$]{} on H$_2$, Ni, W, and [$\text{Pb}$]{}targets. In these events the beam pion is excited to some resonance $X^-$ via $t$-channel Reggeon exchange with the target. At 190[ $\text{GeV}\! / c$]{}the process is dominated by Pomeron exchange. Diffractive reactions are known to exhibit a rich spectrum of produced states and are characterized by two kinematic variables: the square of the total center-of-mass energy and the squared four-momentum transfer from the incoming beam particle to the target, $t = (p_\text{beam} - p_X)^2$. It is customary to use the variable $t' = {{|{t}|}} - {{|{t}|}}_\text{min}$ instead of $t$, where ${{|{t}|}}_\text{min}$ is the minimum value of ${{|{t}|}}$ allowed by kinematics. In 2004 the trigger selected one incoming and at least two outgoing charged particles, whereas in 2008 a signal from the recoil proton was required in the RPD. In the offline event selection diffractive events were enriched by an exclusivity cut of $\pm 4$ GeV around the nominal beam energy. The $t'$ region between 0.1 and 1.0[ $(\text{GeV}\! / c)^2$]{} was selected for the analysis (see [[Fig. \[fig:tPrime\]]{}]{}). ![[${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} invariant mass distribution of the selected data sample for $t' \in [0.1, 1.0]{~\ensuremath{(\text{GeV}\! / c)^2}}$.[]{data-label="fig:threePiMass"}](tprime_zoom){width="\textwidth"} ![[${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} invariant mass distribution of the selected data sample for $t' \in [0.1, 1.0]{~\ensuremath{(\text{GeV}\! / c)^2}}$.[]{data-label="fig:threePiMass"}](Invariant_Mass_of_2008_data){width="\textwidth"} [[Figure \[fig:threePiMass\]]{}]{} shows the [${\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{} invariant mass distribution of the selected 2008 data sample. It exhibits clear structures in the mass regions of the well-known resonances ${\ensuremath{a_1}}(1260)$, ${\ensuremath{a_2}}(1320)$, and ${\ensuremath{\pi_2}}(1670)$. In order to find and disentangle the various resonances in the data, a PWA was performed, in which the total cross section was assumed to factorize into a resonance and a recoil vertex. The isobar model[@isobar] is used to decompose the decay $X^- \to {\ensuremath{{\ensuremath{\pi^-}}{\ensuremath{\pi^+}}{\ensuremath{\pi^-}}}}$ into a chain of successive two-body decays: The $X^-$ with quantum numbers [$J^{PC}$]{} and spin projection $M^\epsilon$ decays into a di-pion resonance, the so-called isobar, and a bachelor pion. The isobar has spin $S$ and a relative orbital angular momentum $L$ with respect to ${\ensuremath{\pi^-}}_\text{bachelor}$. A partial wave is thus defined by ${\ensuremath{J^{PC}}}M^\epsilon[\text{isobar}]L$, where $\epsilon = \pm 1$ is the reflectivity[@reflectivity]. The production amplitudes are determined by extended maximum likelihood fits performed in 40[ $\text{MeV}\! / c^2$]{} wide bins of the three-pion invariant mass $m_X$. In these fits no assumption is made on the produced resonances $X^-$ other then that their production strengths are constant within a $m_X$ bin. The PWA model includes five [${\ensuremath{\pi^+}}{\ensuremath{\pi^-}}$]{}isobars[@compassExotic]: ${\ensuremath{\pi}}{\ensuremath{\pi}}$ $s$-wave, ${\ensuremath{\rho}}(770)$, ${\ensuremath{f_0}}(980)$, ${\ensuremath{f_2}}(1270)$, and ${\ensuremath{\rho_3}}(1690)$. It consists of 41 partial waves with $J \leq 4$ and $M \leq 1$ plus one incoherent isotropic background wave. In order to describe the data, mostly positive reflectivity waves are needed. This corresponds to production with natural parity exchange. The three most dominant waves [$1^{++}$ $0^{+}$ $[{\ensuremath{\rho}}{\ensuremath{\pi}}] S$]{}, [$2^{++}$ $1^{+}$ $[{\ensuremath{\rho}}{\ensuremath{\pi}}] D$]{}, and [$2^{-+}$ $0^{+}$ $[{\ensuremath{f_2}}{\ensuremath{\pi}}] S$]{} contain resonant structures that correspond to the ${\ensuremath{a_1}}(1260)$, ${\ensuremath{a_2}}(1320)$, and ${\ensuremath{\pi_2}}(1670)$, respectively. The resonance parameters extracted from the 2004 data are in good agreement with the PDG values[@compassExotic]. In addition the 2004 data exhibit a resonant peak around 1660[ $\text{MeV}\! / c^2$]{} in the spin-exotic [$1^{-+}$ $1^{+}$ $[{\ensuremath{\rho}}{\ensuremath{\pi}}] P$]{} wave consistent with the disputed ${\ensuremath{\pi_1}}(1600)$[@compassExotic]. A first comparison of the 2008 H$_2$ data with the 2004 Pb data without acceptance corrections shows a surprisingly large dependence on the target material. The data — normalized to the narrow $a_2(1320)$ resonance in the [$2^{++}$ $1^{+}$ $[{\ensuremath{\rho}}{\ensuremath{\pi}}] D$]{} wave — exhibit a strong suppresssion of $M = 1$ waves on the H$_2$ target, whereas the corresponding $M = 0$ waves are enhanced such that the intensity sum over $M$ remains about the same. As an example [[Fig. \[fig:MDep\]]{}]{} shows this effect for the $a_1(1260)$ peak in the ${\ensuremath{J^{PC}}}= 1^{++}$ waves. ![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M1_2004 "fig:"){width="\textwidth"}\ ![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M1_2008 "fig:"){width="\textwidth"} ![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M0_2004 "fig:"){width="\textwidth"}\ ![Normalized intensity sums of the ${\ensuremath{J^{PC}}}= 1^{++}$ partial waves for different spin projection quantum numbers $M = 1$ on the left and $M = 0$ on the right hand side. The top row shows data from the Pb, the bottom row data from the H$_2$ target. The wave intensities are dominated by a broad structure around 1.2[ $\text{GeV}\! / c^2$]{} which is the ${\ensuremath{a_1}}(1260)$.[]{data-label="fig:MDep"}](a1_M0_2008 "fig:"){width="\textwidth"} Acknowledgments {#acknowledgments .unnumbered} =============== This work is supported by the German BMBF, the Maier-Leibnitz-Labor der LMU und TU München, the DFG Cluster of Excellence *Origin and Structure of the Universe*, and CERN-RFBR grant 08-02-91009. [0]{} P. Abbon [*et al.*]{}, *Nucl. Instrum. Meth.* **A577**, 455 (2007). S. U. Chung and T. L. Trueman, *Phys. Rev.* **D11** 633, (1975). J. D. Hansen [*et al.*]{}, *Nucl. Phys.* **B81**, 403 (1974). M. G. Alekseev [*et al.*]{} \[COMPASS Collaboration\], *Phys. Rev. Lett.* **104**, 241803 (2010).

--- abstract: 'For cosmologies including scale dependence of both the cosmological and the gravitational constant, an additional consistency condition dictated by the Bianchi identities emerges, even if the energy-momentum tensor of ordinary matter stays individually conserved. For renormalization-group (RG) approaches it is shown that such a consistency relation ineluctably fixes the RG scale (which may have an explicit as well as an implicit time dependence), provided that the solutions of the RG equation for both quantities are known. Hence, contrary to the procedures employed in the recent literature, we argue that there is no more freedom in identification of the RG scale in terms of the cosmic time in such cosmologies. We carefully set the RG scale for the RG evolution phrased in a quantum gravity framework based on the hypothetical existence of an infrared (IR) fixed point, for the perturbative regime within the same framework, as well as for an evolution within quantum field theory (QFT) in a curved background. In the latter case, the implications of the scale setting for the particle spectrum are also briefly discussed.' author: - 'A. Babić' - 'B.Guberina' - 'R. Horvat' - 'H. Štefančić [^1]' title: | Renormalization-group running cosmologies\ - a scale-setting procedure --- Recently, indisputable evidence has been mounting to suggest that the expansion of our universe is accelerating owing to the nonvanishing value of unclustered dark energy with negative pressure, see [@1]. The crucial evidence for the existence of dark energy \[or the cosmological constant (CC)\] relies on the CMB observations [@2] which strongly support a spatially flat universe as predicted by inflationary models. By combinations of all data a current picture of the universe emerges, in which about 2/3 of the critical energy density of the present universe is made up by a background dark energy with the parameter of the equation of state $-1.38 \leq w \leq -0.82$ at $95\% $ confidence level [@3]. Pressed by these data, theorists now need explain not only why the CC is small, but also why dark-energy domination over ordinary matter density has occurred for redshifts $z { \setlength{\unitlength}{12pt} \begin{picture}(1.4,1.) \put(.7,-0.3){\makebox(0.0,1.)[t]{$<$}} \put(.7,-0.3){\makebox(0.0,1.)[b]{$\sim$}} \end{picture}1}$ (the coincidence problem). Although models with a truly static CC fit these data well, they have two additional drawbacks (besides the coincidence problem): (1) they cannot theoretically explain why the CC is today small but nonvanishing; (2) they have a problem to ensure a phase of inflation, an epoch in the early universe when the CC dominated other forms of energy density. Assessing the possibility to have dynamical dark energy, rolling scalar field models (quintessence fields) [@35] with generic attractor properties [@4] that make the present dark energy density insensitive to the broad range of unknown initial conditions, have been aimed at dealing with the coincidence problem. Still, a quintessence potential has to be fine-tuned to yield the present ratio of ordinary matter to quintessence energy density, at the same time allowing a phase dominated by matter so that structure can form; therefore these models cannot address the coincidence problem. In addition, such models may have difficulties in achieving the current quintessence equation of state with its parameter below -0.8 [@5]. Although extended models with spatially homogeneous light scalar fields based on a nonlinear kinetic energy (k-essence) [@6] have been put forward, it seems that today a trustworthy solution to the coincidence problem, beyond invoking an anthropic principle [@7], is still lacking. Recently, motivated by the observational data, models of dark energy with the supernegative equation of state ($w < -1$) have been introduced [@Cald]. This form of dark energy, named phantom energy, leads to many interesting phenomena, the most striking being the possibility of divergence of the scale factor in finite time, the so-called “Big Rip" effect [@Cald2]. Another class of variable dark-energy models which could successfully mimic quintessence models and may also shed some light on the coincidence problem, have been put forward recently. They are based on the observation [@8; @9] that even a “true" CC (with the equation of state being precisely -1) cannot be fixed to any definite constant (including zero) owing to the renormalization-group (RG) running effects. In [@9], the variation of the CC arises solely from particle field fluctuations, without introducing any quintessence-like scalar fields. Particle contributions to the RG running of $\Lambda $ due to vacuum fluctuations of massive fields have been properly derived in [@10], with a somewhat unexpected outcome that more massive fields do play a dominant role in the running at any scale. In the model [@11], the RG running is due to non-perturbative quantum gravity effects and the hypothesis of the existence of an IR attractive RG fixed point. Both models [@9; @11] also promote the gravitational constant to a RG running quantity [^2], with a prominent scaling behavior found in [@11]. In both models the presence of quintessence-like scalar fields is redundant and not required for consistency with observational data. It should be noted that in the above RG running models the amount of running will depend not only on the parameters of an underlying physical theory, but also on the characteristic RG scale, which must be correctly identified. Several different scenarios for RG-scale adoption have been contemplated in the literature. In [@12] the RG scale was identified with the Hubble parameter, which for the present time is $H_0 \sim 10^{-33} \;\mbox{\rm eV}$. This leads to extremely slow running of the CC and the gravitational constant. Another choice, given by the fourth root of the total energy density [@9; @10], produces much faster running of the CC. The advantage of such a choice is that it entails a direct association with particle momenta (and therefore with the temperature of interacting particle species) in the past radiation epoch. In the models [@8; @85; @11] the RG scale (cutoff) was identified with the inverse of the cosmological time, which is essentially equivalent to the previous case of the Hubble parameter for cosmologies with $a \propto t^n $. Another choice for the relevant cutoff with the implicit time dependence in the form of the inverse scale factor was also analyzed [@85]. With the above choices, it was found that no consistent solution existed in the former case for curved universes (aside for the radiation-dominated era) and even for flat universes in the latter case. In the present paper, we argue that the consistency relation as dictated by the Bianchi identity, which relates the time dependencies of the CC, the gravitational constant and the ordinary matter density, does unambiguously set the RG scale once the RG solutions are known. Hence, any conclusion about the consistency of cosmological solutions with an arbitrary choice of the RG scale may be quite misleading. Our scale-setting procedure works well for both flat and curved universes as well as for the matter background with an arbitrary equation of state. We note that if the ordinary energy-momentum tensor is not separately conserved (for instance, owing to the interaction between matter and the CC, which causes a transfer of energy from matter to the CC or vice versa), then the consistency relation becomes more complicated and the scale-setting procedure is no longer straightforward. In the following, we explain the scale-setting procedure in detail and apply it to several relevant RG solutions obtained in quantum gravity and in QFT in a curved background. [**Scale-setting procedure: importance and limitations**]{} {#sec2} =========================================================== In many cosmological approaches based on the renormalization-group running of the fundamental quantities such as $\rho_{\Lambda}$ and $G$, the fundamental procedure of the underlying quantum (field) theory specifies the running of the aforementioned quantities in terms of the appropriate running scale. From the viewpoint of quantum field theory, the running scale has a natural interpretation in the form of the scaling factor of external momenta or the energy cutoff. The formulation of the consistent cosmological model comprising running quantities requires translation of the running in terms of the running scale to the evolution of these quantities in terms of cosmological variables. This identification of the running scale with a specific function of cosmological variables is a crucial step in all approaches relying on the RGE approach for the dynamics of quantities such as $\rho_{\Lambda}$ and $G$. All efforts that have been undertaken in this direction so far have concentrated on the argumentation in favor of some seemingly “natural” choices for the RGE scale, without a proper procedure of their derivation. What is meant by “natural” choices is to a great extent a matter of taste and depends strongly on the outcome one wishes to achieve. Thus, one should not be surprised that the literature contains choices for the RG scale which, at present, differ up to 30 orders of magnitude, see, e.g., Refs.\[13, 14, 16\]. Needless to say, with different choices for the RG scale one automatically selects different cosmologies, thus making RG approaches devoid of any firm and generic prediction. In addition to noting that even a phenomenological setting of the RG scale is not an obvious matter in cosmological setups, one also finds for both RG cosmologies discussed in this paper (for detailed discussions, see the subsequent chapters below) that the situation with the scale setting is intrinsically not free from ambiguities. For instance, for the RG evolution in a conventional field-theoretical model in the classical curved background (see chapter 6), one derives the beta functions from the loop expansion. The main ambiguity comes from the fact that the beta functions are strongly dependent on the renormalization scheme. For example, in the MS-scheme the RG shows how the various parameters depend on the RG scale $\mu $. On the other hand, in the mass-dependent schemes there is no $\mu $, and one directly gains the dependence on the external signal (like the typical energy of interaction). Aside from the UV limit where both schemes are proved to be equivalent, this identification fails, especially for massive fields. As we deal here with vacuum graphs with no external legs, not only the physical sense of $\mu $ becomes more troublesome, but also decoupling of heavy-particle species remains ambiguous in both schemes. In the RG approach in quantum gravity (see chapters 3 and 4), which is essentially nonperturbative, one is shown how a given observable is related to the IR cutoff in the theory. Hence, the role of $\mu $ is undertaken by the IR cutoff $k$. The cosmological interpretation of $k$ is less ambiguous now because it should approximately match the size of the system; one can envisage several candidates like the Hubble distance, the particle horizon, or the event horizon. Still, the intrinsic interpretation of $k$ is still ambiguous as any link to the external signal is missing. Despite the phenomenologically interesting results and more or less strong physical motivation behind some choices for the RGE scale, the fact remains that the RGE scale-setting procedure is largely arbitrary. A naturally arising question is whether some physical argument might remove this apparent arbitrariness and introduce a procedure that would set the RGE scale automatically in terms of cosmological variables. This paper is centered around such a procedure for a broad class of models based on the RGE approach. The class of RGE-based cosmological models is characterized by the following properties. The only running quantities taken into consideration are the cosmological term $\Lambda$ (or equivalently the cosmological term energy density $\rho_{\Lambda}= \Lambda/(8 \pi G)$) and the gravitational coupling constant $G$. These running quantities depend on a single running scale, generally denoted by $\mu$. Other components of the content of the universe (such as nonrelativistic matter or radiation) evolve in a standard way, i.e. there is no exchange of energy-momentum between these components and the dynamical cosmological term. For the sake of simplicity and clarity of exposition, in all considerations put forward in this paper we assume that there exists only one additional matter component described by the equation of state $p_{m}= w \rho_{m}$, where $w$ is constant. The inclusion of additional matter components and/or more general equations of state is straightforward. The physical principle behind the scale-setting procedure is conceptually simple. The RGE improved Einstein equation for these cosmologies acquires the form $$\label{eq:Ein} R^{\alpha\beta} - \frac{1}{2} g^{\alpha\beta} R = -8 \pi G(\mu)( T_{m}^{\alpha\beta} + T_{\Lambda}^{\alpha\beta}(\mu)) \, ,$$ where $R^{\alpha\beta}$ and $R$ denote the Ricci tensor and scalar respectively, while $T_{m}^{\alpha\beta}$ and $T_{\Lambda}^{\alpha\beta}(\mu) = \rho_{\Lambda} (\mu) g^{\alpha\beta}$ stand for matter and cosmological term energy-momentum tensors, respectively. The only physical requirement is that equation (\[eq:Ein\]) must maintain its general covariance with the quantities $G$ and $T_{\Lambda}^{\alpha\beta}$ having the $\mu$ (and implicitly $t$) dependence. This requirement translates into the following equation relating $G$, $\rho_{ \Lambda}$ and $\rho_{m}$: $$\label{eq:evol} \dot{G}(\mu)(\rho_{m}+\rho_{\Lambda}(\mu)) + G(\mu)\dot{\rho_{\Lambda}}(\mu) =0\, .$$ Here dots denote time derivatives. Assuming the nonvanishing $\dot{\mu}$ throughout the evolution of the universe, we arrive at the equation $$\label{eq:muset} \rho_{m} = -\rho_{\Lambda}(\mu) - G(\mu) \left( \frac{d \rho_{\Lambda}(\mu)}{d \mu} \right) \left( \frac{d G(\mu)}{d \mu} \right)^{-1} \, .$$ On the left-hand side of the above equation we have a function of the scale factor $a$, while on the right-hand side we have a function of $\mu$, i.e. we have an expression of the form $\rho_{m}=f(\mu)$. The inversion of this expression in principle gives the scale $\mu$ in terms of $\rho_{m}$ in the form $\mu=f^{-1}(\rho_{m})$. In the cases in which the inversion of the function $f$ gives multiple possibilities for the choice of $\mu$, we assume that one of them can be selected on some physical grounds (such as the positivity of the scale in the case of $f(x)=x^2$). Once the procedure of the scale setting is completed, we formally have the quantities $\rho_{\Lambda}$ and $G$ as functions of $\rho_{m}$ [^3]. The matter energy density has the usual scaling behavior with the scale factor $$\label{eq:scalmatt} \rho_{m}=\rho_{m,0}\left( \frac{a}{a_{0}} \right)^{-3(1+w)} \, .$$ The set of equations necessary for the description of the evolution of the universe is completed by the Friedmann equation $$\label{eq:Friedmann} \left(\frac{\dot{a}}{a} \right)^{2} + \frac{K}{a^{2}} = \frac{8 \pi}{3} G (\rho_{m}+\rho_{\Lambda})\, .$$ In the following sections we present several examples of the functioning of the procedure and point out some physically interesting effects in the evolution of the cosmological models based on RGE and our scale-setting procedure. Before doing this, we make some comments on the character of our scale-setting procedure as well as on cases where it is not applicable. First of all, it is clearly seen that our procedure lacks the first-principle connection to quantum gravity considerations, and therefore should be considered as purely phenomenological. Concerning our basic Eq. (2), we have assumed that $\rho_m $ is intrinsically independent of $\mu $. [^4] As seen from (2), as long as $\rho_m $ retains its canonical form, the scale is univocally fixed and does not even implicitly depend on $\mu $ in (2). The situation changes when, for instance, an interaction between matter and the CC and/or $G$ is turned on. In this case, Eq. (2) generalizes to $$\dot{G}(\rho_{\Lambda } + \rho_m ) + G \dot{\rho }_{\Lambda } + G (\dot{\rho }_{m} + 3H\rho_m (1 + w )) = 0 \;.$$ We see from the above equation for the interacting matter that any deviation from the canonical case (4) depends decisively on $\mu $. Hence, one is not able to fix the scale before specifying such interactions [*a priori*]{}. Also, the same conclusion is reached for other two extra cases derived from (6): (i) for the constant $G$ and the running $\Lambda $, and (ii) for the constant (or zero) $\Lambda $ and the running $G$. Thus, one is restricted to the cosmological setup defined by (2) for the full applicability of our scale-setting procedure. As a first step in the next section, we shortly review the nonperturbative quantum gravity as derived by Reuter et al. [@8; @85] and apply the scale setting procedure for the running $\Lambda (k)$ and $G(k)$ in the perturbative regime, which interpolates the behavior of $\Lambda (k)$ and $G(k)$ between the UV fixed point and the infrared limit. Exact renormalization-group approach in quantum gravity {#sec2.5} ======================================================= In the quantum-gravity approach of [@QG1; @QG2] the metric is the fundamental dynamical variable. It is possible to construct a scale-dependent gravitational action $\Gamma_k [g_{\mu\nu}]$ and derive the appropriate evolution equation. The properties of the constructed “effective average action", especially a built-in infrared cutoff, are welcome in the case of quantum gravity since it is possible to study low-momentum behavior in a nonperturbative way. In such a way, quantum gravity can be used to describe gravitational phenomena at very large distances. A systematic study of the application of the exact renormalization-group approach to quantum gravity[^5] was done by Reuter [@8]. There appears the effective average action $\Gamma_k [g_{\mu\nu}]$, which is basically a Wilsonian coarse-grained free energy [@QG2; @reuwett]. This action depends on metrics and the momentum scale $k$ interpreted as an infrared cutoff in the following way. If one has a physical system with a size $L$, then the mass parameter $k \sim 1/L$ defines an infrared cutoff. This is the fundamental step in deriving the average effective action in the Wilson-Kadanoff formulation of the effective action. The main difference between the effective average action $\Gamma _k [g_{\mu\nu}]$ and the ordinary effective action $\Gamma [g_{\mu\nu}]$ is in the fact that the path integral which defines $\Gamma _k$ integrates only the quantum fluctuations with the momenta $p^2 > k^2$, thus describing the dynamics of the metrics averaged over the volume with size $(k^{-1})^3$. The derived effective field theory is valid near the scale $k$. For any scale $k$, there is an $\Gamma_k$ which is an effective field theory at that scale. This means that all gravitational phenomena are correctly described at tree level by $\Gamma_k$, including the loop effects with $p^2\simeq k^2$. Quantum fluctuations for $p^2 > k^2$ are integrated out, and large-distance metric fluctuations, $p^2 < k^2$, are not included. Of course, in the limit $k\rightarrow 0$, the infrared cutoff “disappears" and the original effective action $\Gamma$ is recovered. On the other hand, the Einstein-Hilbert action can be interpreted as fundamental theory in the limit $k\rightarrow \infty$. The determination of the infrared cutoff (i. e., the scale $k$) is by no means trivial in reality. It is rather simple in a massless theory such as massless QED, where the inverse of a distance is the only mass scale present in the theory. However, a real situation one encounters is the variety of mass scales and the proper identification, if any, is not trivial at all. The correct way to proceed would be to study the RG flow of the effective action $\Gamma _k [g_{\mu\nu}]$ and make the identification of the infrared cutoff by inspecting the RG evolution. Once the infrared cutoff is identified, one should solve the Bianchi identities and the conservation laws for matter [@Martin]. On the other hand, one can start vice versa, i. e., use the Bianchi identities, etc., to fix a scale and look whether it is possible to give a physical interpretation to the scale. It is clear that there should exist a certain physical mechanism which effectively stops the running in the infrared. Obviously, it is a function of all possible scales that appear in a certain case and/or characterize a physical system under consideration — it may include particle momenta, field strengths, the curvature of spacetime, etc. However, looking at the global system described by Einstein equations, the conditions of homogeneity and isotropy lead to a possible natural choice of $k$ to be proportional to the cosmological time $t$, $k=k(t)$. If the dependence $k(t)$ is known, the $G(k)$ and $\Lambda (k)$ become the functions $G(t)$ and $\Lambda (t)$. The choice $k\propto \frac{1}{t}$ seems to be a plausible one. The cosmological time $t$ describes the temporal distance between some event-point $P$ and the big-bang. For the effective field theory $\Gamma_k$ to be valid at $t$, one need not integrate quantum fluctuations with momenta smaller than $\frac{1}{t}$, $p^2\ll \frac{1}{t^2}$. Namely, at the time $t$, the fluctuations with frequencies smaller than $1/t$ should not play any role yet, and the meaning of the infrared cutoff becomes obvious. Nonperturbative solutions of the RG equations are obtained by truncation of the original infinite dimensional space of all action functionals to some specific finite dimensional space which appears relevant to a given physical problem. The usual truncation is the reduction to the Einstein-Hilbert action. This Ansatz leads to the coupled system [@8; @85] of equations for $G(k)$ and $\Lambda (k)$. For the case $\Lambda < k^2$, it is simple and easy to solve. It leads to two attractive fixed points $g_*$, $\beta (g_*)=0$. The ultraviolet (non-gaussian) fixed point is $$g_*^{\mathrm{UV}}=\frac{1}{\omega '},$$ where $\omega '$ is the number which is calculated, and an infrared (gaussian) fixed point at $g_*^{\mathrm{IR}}=0$. All trajectories which are attracted towards $g_*^{\mathrm{IR}}=0$ for $k=0$ and towards $g_*^{\mathrm{UV}}$ for $k\rightarrow \infty$ lead to the following form of $G (k)$: $$G(k)=\frac{G_0}{1+\omega G_0 k^2} \, ,$$ which for small $k$ reads $$G(k)=G_0-\omega G_0^2 k^2 + {\mathcal{ O}}(k^4) \, . \label{eq:g2}$$ Since for the Einstein-Hilbert truncation $G(k)$ does not run between the scale where it is experimentally determined to be the Newton constant $G_{\mathrm{N} }$, and a cosmological scale for which $k\sim 0$, $G_0$ can be safely identified as $G_0\equiv G(k=0)=G_{\mathrm{N}}$. For $k^2\gg G_0^{-1}$, the fixed-point behavior sets in and $G(k)$ becomes $$G(k)\sim \frac{1}{\omega k^2} \, ,$$ in accordance with the asymptotic running predicted by Polyakov [@polyakov]. The cosmological constant $\Lambda (k)$ then reads $$\Lambda (k)=\Lambda_0+\nu G_0 k^4\left[ 1+{\mathcal{O}}(G_0 k^2)\right] \, . \label{eq:g4}$$ Equations (\[eq:g2\]) and (\[eq:g4\]) define the so-called perturbative regime of the infrared cutoff $k$. The solutions (\[eq:g2\]) and (\[eq:g4\]) are basically expansions in the dimensionless ratio $(k/M_{Pl })^2$. By inspection one sees that renormalization effects are important only for the infrared cutoff $k$ approaching $M_{Pl}$. Equation (\[eq:g2\]) shows that increasing $k$ leads to decreasing $G (k)$ — a first sign of asymptotic freedom of pure quantum gravity [@8]. If we apply our scale-setting procedure to the quantum gravity we reviewed, i.e. using (\[eq:g2\]) and (\[eq:g4\]) for $G(k)$ and $\Lambda (k)$, and the expression (\[eq:muset\]) for $k$, we obtain $$k=\sqrt{\frac{4\pi\omega}{\nu}\rho_{m}G_0},$$ and $$k_0\sim\sqrt{\rho_{m, 0}G_0}\sim H_0$$ for the “present" value of the infrared cutoff $k_0$. Quantum-gravity model with an IR fixed point {#sec3} ============================================ In the formalism of quantum gravity introduced in [@8; @85], it is possible to set up renormalization-group equations for the cosmologically relevant quantities $\Lambda$ and $G$. From the phenomenological point of view, the assumption of the existence of the IR fixed point in the RGE flow [@11] is of special interest (for related work see also [@Litim]).[^6] In this setting, the infrared cutoff $k$ plays the role of the general RGE scale $\mu$. In the infrared limit, the running of $\Lambda$ and $G$ can be expressed as $$\label{eq:LamIR} \Lambda(k) = \lambda_{*} k^2 \, ,$$ $$\label{eq:GIR} G(k) = \frac{g_{*}}{ k^2} \, ,$$ where $\lambda_{*}$ and $g_{*}$ are constants and $k$ is the infrared cutoff. The application of the procedure explained in section \[sec2\] to the running parameters (\[eq:LamIR\]) and (\[eq:GIR\]) results in expressing the scale $k$ in terms of cosmological quantities (and implicitly of time). Insertion of (\[eq:LamIR\]) and (\[eq:GIR\]) into (\[eq:muset\]) finally gives $$\label{eq:IRset} k = \left( \frac{8 \pi g_{*}}{\lambda_{*}} \rho_{m} \right)^{1/4} \, .$$ It is important to note that the result of this scale-setting procedure is not explicitly dependent on the topology $K$ of the universe. One may, however, argue that there exists a certain level of implicit dependence since the expansion of the universe (and therefore the dependence of $\rho_{m}$ on time) depends on $K$. Moreover, the scale-setting procedure of this paper unambiguously identifies the scale $k$ in the model of quantum gravity with the IR fixed point. The result (\[eq:IRset\]) leads to the following general result: $$\label{eq:LamrelG} \Lambda = 8 \pi G \rho_{m} = (8 \pi \lambda_{*} g_{*} \rho_{m})^{1/2} \, .$$ Combining the expressions (\[eq:scalmatt\]), (\[eq:Friedmann\]) and (\[eq:LamrelG\]) allows the dependence of the scale factor on time. It is convenient to introduce the parameters $\rho_{c} = 3 H^{2}/(8 \pi G)$, $\Omega_{\Lambda} = \Lambda/(8 \pi G \rho_{c})$, $\Omega_{m}=\rho_{m}/\rho_{c}$ and $\Omega_{K}=-K/(H^2 a^2)$. In terms of these parameters equation (\[eq:LamrelG\]) can be written as $$\label{eq:Omegas} \Omega_{\Lambda} = \Omega_{m} \, .$$ Using the relation $\Omega_{\Lambda} + \Omega_{m} + \Omega_{K} = 1$ we obtain an implicit expression for the scale factor of the universe: $$\label{eq:aodt} H_{0} (t-t') = \int_{a'/a_{0}}^{a/a_{0}} [\Omega_{K}^{0} + (1-\Omega_{K}^{0}) u^{(1-3w)/2} ]^{-1/2} \, d u \, .$$ Throughout this paper the subscript or the superscript $0$ refers to the present epoch of the evolution of the universe. From the expression (\[eq:aodt\]) it is evident that the expansion of the universe depends on two parameters: $w$ and $\Omega_{K}^{0}$. The general solution of (\[eq:aodt\]) can be given in terms of the confluent hypergeometric functions. However, it is far more instructive to consider the form that the solutions acquire for some special choices of these parameters. We concentrate on three interesting cases in the limit $t' \rightarrow 0$ ($a' \rightarrow 0$) to make a comparison with the results of [@11]: [**i) $\mathbf{\Omega_{K}^{0}=0}$, $\mathbf{w}$ arbitrary.**]{} For the flat universe one obtains $$\label{eq:flat} \frac{a}{a_{0}}=\left( \frac{3}{4} (1+w) H_{0} t \right)^{\frac{4}{3(1+w)}} \, ,$$ which leads to the following expression for the scale $k$: $$\label{eq:kflat} k = \frac{4}{3(1+w)H_{0}} \left( \frac{8 \pi g_{*}}{\lambda_{*}} \rho_{m,0} \right)^{1/4} \frac{1}{t} = \frac{1}{1+w} \left( \frac{8}{3 \lambda_{*}} \right)^{1/2} \frac{1}{t} \, .$$ The results obtained above are in complete agreement with the results of [@11] for the flat case. However, in the case of [@11], the choice $k=\xi/t$ was introduced on phenomenological grounds and it was found that consistency was achieved in an otherwise overdetermined set of equations. In our case, the setting of the scale $k$ is a result of the systematic procedure and confirms the choice of [@11]. [**ii) $\mathbf{w=1/3}$, $\mathbf{\Omega_{K}^{0}}$ arbitrary.**]{} In this cosmological model the universe of arbitrary curvature has a simple evolution law for the scale factor $$\label{eq:wrad} \frac{a}{a_{0}}= H_{0} t \, ,$$ which allows us to express $k$ as $$\label{eq:krad} k = \frac{1}{H_{0}} \left( \frac{8 \pi g_{*}}{\lambda_{*}} \rho_{m,0} \right)^{1/4} \frac{1}{t} \, .$$ The result displayed above further explains why the Ansatz $k=\xi/t$ functions well also for cosmologies of any curvature including the radiation component only. We have found agreement with [@11] in this case as well. [**iii) $\mathbf{w=0}$, $\mathbf{\Omega_{K}^{0} \ge 0}$.**]{} The universe containing nonrelativistic matter only and having arbitrary curvature has the law of evolution of the scale factor given by the following expression: $$\begin{aligned} \label{eq:nonrel} H_{0} t &=& \frac{4}{3(1-\Omega_{K}^{0})^{2}} \left[ \left( \Omega_{K}^{0}+(1-\Omega_{K}^{0})\left( \frac{a}{a_{0}} \right)^{1/2} \right)^{1/2} \left( (1-\Omega_{K}^{0}) \left( \frac{a}{a_{0}}\right)^{1/2} - 2 \Omega_{K}^{0} \right) \right. \nonumber \\ &+& 2 ( \left. \Omega_{K}^{0})^{3/2} \right] \, .\end{aligned}$$ From the expression given above it is possible to obtain the scale factor $a$ as a function of time. In this case, the scale $k$ can no longer be expressed in the form of Ansatz $k = \xi/t$. However, it is still possible to uniquely define the scale $k$ by choosing the only real solution of the cubic equation (\[eq:nonrel\]). The product $t (a/a_{0})^{-3/4}$ (since $k \sim (a/a_{0})^{-3/4}$) as a function of time is displayed in Fig. \[fig:1\]. It is clear from the figure that the choice $k \sim 1/t$ is adequate for the flat universe, while for the curved one the scale may differ from the $\xi/t$. It is important to stress here that persisting with the $k \sim 1/t $ identification (as done in [@11]) may lead to misleading conclusions about consistency of the cosmological solutions to the system of the equations as given by (\[eq:evol\]), (\[eq:Friedmann\]), (\[eq:LamIR\]) and (\[eq:GIR\]). The observation in [@11] was that consistent solutions to the above system (with $K = +1$ or $K = -1$) exist only for a radiation dominated universe ($w = +1/3 $). The final conclusion in [@11] actuated by this observation was that our universe could fall into the basin of attraction induced by the IR fixed point only if it is spatially flat ($K=0 $). On the contrary, our scale-setting procedure shows that consistent solutions to the above system can be obtained for a universe having arbitrary curvature and for the matter equation of state with arbitrary $w$. The preceding paragraphs demonstrate how our scale-setting procedure leads to the mathematically consistent choice for the scale $k$ for any geometry or the matter content of the universe. Once the specification of the scale $k$ is available, one can consider its physical interpretation. The formulation of the underlying formalism, which leads to the RGE equations for $\Lambda$ and $G$, may generally put some constraints on the physically acceptable choices for the scale $k$. One such constraint is that this scale should have a geometrical interpretation [@11]. This requirement is of qualitative nature and still allows for many geometrically motivated choices for $k$ [@85]. In finding the solution of the dynamics of the universe, one, therefore, has to obtain a solution for $k$ with an acceptable geometrical interpretation which at the same time satisfies Bianchi identity. One possible approach is first to choose an [*Ansatz*]{} for $k$ in such a way that it has a proper geometrical interpretation. Then, in the second step, one checks whether the set of equations describing the cosmology has a mathematically consistent solution. In the case of the mathematical inconsistency one should either discard the universe with a given geometry and the matter content as unphysical (should one decide to insist on the Ansatz) or try with some other suitable choice for $k$. It is conceivable that in some cases of physical interest one might exhaust the list of obvious Ansätze without finding the one which would lead to a mathematically consistent set of cosmological equations. Such an outcome certainly would not imply that the appropriate choice for $k$ does not exist for a given geometry or the matter content, but that some nontrivial choice might exist which still has a satisfactory geometrical interpretation. A different approach is based on the scale-setting procedure introduced in this paper. The scale-setting procedure always yields a mathematically consistent solution for $k$. Given this solution, it can be further examined to test whether it meets the physical requirements. This procedure is [*systematic*]{} and allows treatment of any geometry or the matter content of the universe. For example, from Fig. \[fig:1\] one can see that for the universe with a small curvature the scale $k$ obtained by our scale-setting procedure is reasonably close (at times when the IR fixed point is supposed to dominate) to the scale obtained for the flat case and it still has a satisfactory geometrical interpretation. In this way the models with small, but nonvanishing curvatures are physically acceptable as a possible description of the universe within the RGE framework discussed in this section. This conclusion is expected since the consistency of the model for the flat space and inconsistency for the arbitrarily small $|\Omega_{K}^{0}|$ would represent a certain discontinuity and would be very surprising. Cosmological implications {#sec4} ========================= An interesting extension of the example given in the preceding section is the study of a more general running of $\Lambda$ and $G$ in the infrared region. Namely, it is interesting to consider the running in the form $$\label{eq:Lamgen} \Lambda(k) = A k^{\alpha} \, ,$$ $$\label{eq:Ggen} G(k) = \frac{B}{ k^{\beta}} \, ,$$ where $A$, $B$, $\alpha$ and $\beta$ are positive constants. This general example clearly comprises the case of the IR fixed point considered in the preceding section, but also allows for a much more general cosmological evolution. The application of our scale-setting procedure to the RGE-based cosmology yields the following expression for the scale $k$: $$\label{eq:scalgen} k=\left( \frac{8 \pi \beta B}{\alpha A} \rho_{m} \right)^{1/(\alpha + \beta)} \, .$$ The scale identified in such a manner results in the following laws of evolution for the quantities $\Lambda$ and $G$: $$\label{eq:Lamlaw} \Lambda = \left( \frac{8 \pi \beta}{\alpha} \right)^{\alpha/(\alpha+\beta)} B^{\alpha/(\alpha+\beta)} A^{\beta/(\alpha+\beta)} \rho_{m}^{\alpha/(\alpha+\beta)} \, ,$$ $$\label{eq:Glaw} G = \left( \frac{8 \pi \beta}{\alpha} \right)^{-\beta/(\alpha+\beta)} B^{\alpha/(\alpha+\beta)} A^{\beta/(\alpha+\beta)} \rho_{m}^{-\beta/(\alpha+\beta)} \, .$$ These two evolution laws reveal an interesting feature of the cosmology determined by (\[eq:Lamgen\]) and (\[eq:Ggen\]). Namely, $$\label{eq:ratiogen} \frac{\rho_{\Lambda}}{\rho_{m}} = \frac{\Lambda}{8 \pi G \rho_{m}} = \frac{\beta}{\alpha} \, .$$ The ratio of the shares in the total energy content of the universe of the two components is determined by two exponents $\alpha$ and $\beta$. If we consider a scenario of the cosmological evolution in which the running of the type given by (\[eq:Lamgen\]) and (\[eq:Ggen\]) sets in relatively recently (on a cosmological time scale) and is preceded by a long period of a very mild running of $\Lambda$ and $G$, then it might be possible to explain the results of various cosmological observations [@1; @2] for a combination of exponents $\beta/\alpha \approx 2$. To verify this hypothesis, it would be necessary to perform an analysis similar to that made in [@bentivegna]. If we combine the results (\[eq:Lamlaw\]) and (\[eq:Glaw\]) with (\[eq:scalmatt\]), we obtain the scaling laws $$\label{eq:asymlaw} \Lambda \sim G \rho_{m} \sim a^{-3(1+w) \alpha/(\alpha+\beta)} \, .$$ The most interesting consequence of such a scaling law is the possibility that in the distant future $\Lambda$ and $G \rho_{m}$ decrease faster than the curvature term, i.e. faster than $a^{-2}$. For this to happen, the following condition must be satisfied: $$\label{eq:cond} \frac{\beta}{\alpha} < \frac{1+3w}{2} \, .$$ Therefore, for the parameters $\alpha$, $\beta$ and $w$ in such a relation, in the distant future the universe becomes curvature dominated and may, in the case of the closed universe, cease expanding and revert to contraction. From (\[eq:ratiogen\]) it is clear that presently the universe cannot be described by the RGE-based cosmological model given by (\[eq:Lamgen\]) and (\[eq:Ggen\]), with $\alpha$ and $\beta$ such that the universe becomes curvature dominated in the future. However, it is possible that this sort of running sets in at some future instant and may lead to the curvature-dominated universe. RGE cosmological models from quantum field theory on curved space-time {#sec5} ====================================================================== As a last example, let us consider a generic case when both $\rho_{\Lambda}$ and $G$ can be expressed as series in the square of the RGE scale $\mu$. This case naturally appears in the considerations of RGE for $\rho_{\Lambda}$ and $G$ in the formalism of quantum field theory in curved spacetime [@9; @10; @12] when the influence of mass thresholds on the RGE flow is properly taken into account [@10]. Quantum field theory on curved space-time has its motivation in effective field theories, especially in the very successfull chiral perturbation theory which is a low-energy nonlinear realization of QCD. The framework of the effective field theories enables one to renormalize otherwise nonrenormalizable theories. Gravity, with quantum corrections being small up to the very high Planck scale, appears to be, [*prima facie*]{}, an even better effective field theory, which is expected to govern the effects of quantum gravity at low-energy scale without the knowledge of the true quantum gravity. However, the reality is not so simple, since the possible existence of the singularity in the future, corresponding to the wavelength probed of the order of the size of the universe - deeply in the infrared region - may break the validity of the effective theory in the infrared limit. As a matter of fact, it was shown [@Tsamis] that quantum gravity may lead to the very strong renormalization effects at large distances induced by infrared divergences. In the following discussions we assume the validity of the effective theory at present time and present scales persists in the infrared region. However, we are aware that possible infrared effects in quantum gravity may influence our conclusions in an unexpected way. It is, for example, well known [@Martin] that, in nonperturbative quantum gravity, the G runs to the infinity in the infrared limit, which is an artifact of the Einstein-Hilbert truncation, i. e., the absence of high derivative terms. The next important question is the determination and the meaning of the RGE scales. In the $\bar{MS}$ scheme, the $\mu$ dependence in the effective action is compensated by the running of the parameter $\Lambda$ as in QED where the $\mu$ dependence is compensated by the running charge $e(\mu)$. The overall effective action which contains a running $\Lambda(\mu)$ is therefore scale independent, as required. The physical interpretation of the RG scale in the high derivative sector can be achieved calculating [@Gorbar] the polarization operator of gravitons arising from the particle loops in the linearized gravity. In the RG equation which contains the derivative with respect to the arbitray renormalization parameter $\mu$, one usually eliminates the $\mu$ parameter trading it for a Euclidean momentum $p$. In the physical mass-dependent renormalization scheme (e. g. momentum subtraction scheme) $\mu$ is traded for a certain Euclidean momentum $p$. In QCD, for example, one writes an RG equation with respect to the momentum scaling parameter $\lambda$, $p\rightarrow \lambda p$, and eliminates the derivatives with respect to $\mu$. The trade is performed by identifying the new scale with the typical average energy (momentum) of the physical process. On the other hand, the use of the physical mass-dependent renormalization scheme allows us to control the decoupling of the massive particles in an unambigous way. In the UV limit, the physical mass-dependent renormalization scheme will coincide with the results obtained using the minimal subtraction scheme, but the latter fails to describe the low momentum behavior of the beta functions. The problem that appears is the following: as regards the running of $\Lambda$ and $(1/G)$ on the curved background there is no known method of calculating that would be compatible with some physical scheme (e. g. the momentum subtraction scheme). One is inevitably forced to use phenomenology and an intuition if one wants to determine the meaning of the RG scale. Intuitively, one expects the scale to coincide with the typical momentum of gravitinos which appear in the vertices of the Green functions. However, this “determination" relies heavily on our feeling of the underlying phenomenology. In the following we restrict our considerations to the field theory defined at the classical background, the approach developed in [@9; @10]. Such a model was based on the observation [@9] that even a “true” CC in such theories cannot be fixed to any definite constant (including zero) owing to the renormalization-group (RG) running effects. The variation of the CC arises solely from particle field fluctuations, without introducing any quintessence-like scalar fields. Particle contributions to the RG running of the CC which are due to vacuum fluctuations of massive fields have been properly derived in [@10], with a somewhat peculiar outcome that more massive fields do play a dominant role in the running at any scale. When the RGE scale $\mu$ is less than all masses in the theory, we can write [@10; @12] $$\label{eq:Lamexp} \rho_{\Lambda} = \sum_{n=0}^{\infty} C_{n} \mu^{2n} \, ,$$ $$\label{eq:Gexp} G^{-1} = \sum_{n=0}^{\infty} D_{n} \mu^{2n} \, .$$ We assume that both series converge well and can be well approximated by retaining just a first few terms. The application of the scale-setting procedure yields the expression $$\label{eq:rhommu} \rho_{m} = -C_{0} + \frac{C_{1} D_{0}}{D_{1}} + 2 \frac{D_{0}}{D_{1}}\left( C_{2} - \frac{C_{1} D_{2}}{D_{1}} \right) \mu^{2} + \dots \, ,$$ which finally leads to the identification of the scale $\mu$ $$\label{eq:murhom} \mu^{2} \simeq \frac{1}{2} \frac{1 - \frac{D_{1}}{C_{1} D_{0}} (\rho_{m} + C_{0})}{\frac{D_{2}}{D_{1}}-\frac{C_{2}}{C_{1}}} \, .$$ We can cast more light on this general expression by more precisely specifying the values of the coefficients $C_{i}$ and $D_{i}$. From the studies of the cosmologies with the running $\rho_{\Lambda}$ in the formalism of quantum field theory in curved spacetime [@9; @10; @12], we know that generally $C_{1} \sim m_{max}^{2}$, $C_{2} \sim N_{b}-N_{f} \sim 1$, $C_{3} \sim 1/m_{min}^{2}$, etc. Here $m_{max}$ and $m_{min}$ denote the largest and the smallest masses of (massive) fields in the theory, respectively, and $N_{b}$ and $N_{f}$ stand for the number of bosonic and fermionic massive degrees of freedom in the theory, respectively. The value of $C_{0}$ does not follow from the argumentation of the same type, but its value could hardly surpass $\rho_{\Lambda,0}$. Adopting the same line of reasoning for $G$, on dimensional grounds, we expect $D_{1} \sim 1$, $D_{2} \sim 1/m_{min}^{2}$, etc. For the value of $D_{0}$ it is natural to take $M_{Pl}^{2}$. Any other choice would imply a very strong running for $G$ which would not be consistent with the observational data on $G$ variation [@G]. Taking these values of the coefficients $C_{i}$ and $D_{i}$, we obtain the following expression for the scale $\mu$: $$\label{eq:muconcrete} \mu^{2} \simeq \frac{1}{2} \gamma_{1} m_{min}^{2} \left[ 1 - \frac{\gamma_{2}}{M_{Pl}^{2} m_{max}^{2}} (\rho_{m} + C_{0}) \right] \, .$$ Here $\gamma_{1}$ and $\gamma_{2}$ represent constants of order 1. This result for the RGE scale implies a very slow running of the scale $\mu$ in the vicinity of $\mu \simeq m_{min}$. This value of the scale is also only marginally acceptable as far as the convergence of the expansions (\[eq:Lamexp\]) and (\[eq:Gexp\]) is concerned. Furthermore, when the solution (\[eq:muconcrete\]) is inserted into (\[eq:Lamexp\]), one arrives at $\rho_{\Lambda } \sim m_{max}^2 m_{min}^2$. If for $m_{min}$ one takes the mass scale revealed in recent neutrino oscillation experiments, of order $10^{-3} \; \mbox{\rm eV}$, one obtains $\rho_{\Lambda }^0 $, which is obviously far too large for any $m_{max}$. In the extreme case, we can set $m_{max} \sim M_{Pl}$ and saturate $\rho_{\Lambda }^0 $ with $m_{min} \sim m_{quintessence} \sim 10^{-33} \;\mbox{\rm eV}$. Let us, however, recall that from the beginning of our presentation we have proclaimed quintessence degrees of freedom as redundant ones in the present approach. Although some intermediate scale for $m_{max}$ and $m_{min}$ are also possible, we do not consider this case a plausible one. In the model put forward in [@10], it is pointed out that some amount of fine-tuning needs to be introduced into the expansion (\[eq:Lamexp\]) in order to maintain the consistency with the results of primordial nucleosynthesis. This fine-tuning is realized by the requirement $C_{1} \simeq 0$. If we adopt this constraint, the solution for $\mu^{2}$ becomes $$\label{eq:muC10} \mu^{2} \simeq \frac{1}{2} \frac{\rho_{m} + C_{0}}{M_{Pl}^{2}} \, ,$$ which at the present epoch of the evolution of the universe is quite well approximated by $\mu \simeq H$, a choice advocated in [@12]. However, this value of the scale leads to an extremely slow variation of $\rho_{\Lambda}$ with the leading term in the scale $\mu$ of the type $\mu^{4} \simeq H^{4}$. The value of the coefficient $C_{1} \sim m_{max}^{2}$ leads to the too strong running, while the fine-tuned value $C_{1} \simeq 0$ leads to the negligible running. It is natural to ask whether some intermediate value for $C_{1}$ might lead to a satisfactory running scheme. Assuming that $C_{1}$ has the value around $m_{min}$, the natural value of the scale becomes $$\label{eq:C1mmin} \mu^{2} = \gamma_{3} \left( m_{min}^{2} - \gamma_{4}\frac{\rho_{m} + C_{0}}{M_{Pl}^{2}} \right) \, .$$ Here $\gamma_{3}$ and $\gamma_{4}$ refer to dimensionless constants of the order 1. This value of the scale $\mu$ can approximately reproduce the present value of the cosmological term energy density $\rho_{\Lambda, 0}$ even with the assumption $C_{0}=0$. From (\[eq:C1mmin\]) it is also evident that the scale changes very slowly. Another problem is the issue of convergence of the series (\[eq:Lamexp\]) and (\[eq:Gexp\]). However, as the ratio $\frac{\rho_{m} + C_{0}}{M_{Pl}^{2}}$ is at present time (and also during the matter-dominated era) proportional to $H^{2}$, with this choice of $C_{1}$, we effectively reproduce the evolution of the models elaborated in [@12] where the variability of the $\rho_{\Lambda}$ is described with the $H^{2}$ term. It is interesting to note that if one allows the violation of the decoupling theorem for $1/G$, it is possible to obtain a RGE based cosmology, consistent with our scale-setting procedure, in which $\mu \sim H$ and $\rho_{\Lambda}$ varies with $H^{2}$ [@H2]. The general discussion given in this section indicates that the application of the scale-setting procedure to the RGE cosmology determined by (\[eq:Lamexp\]) and (\[eq:Gexp\]) yields results at variance with observational bounds for the values of the coefficients $C_{i}$ and $D_{i}$ determined on dimesional grounds only. To achieve the consistency of the model with the observational results, it is necessary to impose restrictions on some of these coefficients. The mechanism behind these restrictions, however, still remains to be clarified. Conclusions =========== The procedure of the scale setting in RGE-based cosmologies presented in this paper translates well-founded models of quantum theories to unambiguously defined cosmological models. The underlying argument behind the entire procedure, the maintaining of general covariance at the level of the Einstein equation, is conceptually simple and theoretically convincing. The details and the origin of the RGE for $G$ and $\rho_{\Lambda}$ are immaterial for the functioning of the procedure, which is, in this respect, quite robust. In several examples given in this paper, the applicability of the procedure was demonstrated in two cases with RGE of quite different backgrounds: quantum gravity and quantum field theory in curved spacetime. In both cases, the scale-setting procedure provides insight beyond the phenomenologically motivated choices for the RGE running scale. With the availability of this scale-setting procedure, emphasis is now put on the more precise formulation of RGE for the cosmologically relevant quantities in the underlying fundamental theories. [**Acknowledgements.**]{} The authors would like to thank J. Solà for useful comments on the manuscript. This work was supported by the Ministry of Science, Education and Sport of the Republic of Croatia under the contract No. 0098002 and 0098011. [88]{} S. Pearlmutter et al., Astrophys. J. 517 (1999) 565; A. G. Reiss et al., Astronom. J. 116 (1998) 1009. C. L. Bennett et al., astro-ph/0302207; D. N. Spregel et al., astro-ph/0302209; H. V. P. Peiris et al., astro-ph/0302225. A. Melchiorri, L. Mersini, C. Odman and M. Trodden, Phys. Rev. D 68 (2003) 043509. P. J. E. Peebles, B. Ratra, Astrophys. J. Lett., 325 L 17 (1988); C. Wetterich, Nucl. Phys. B 302 (1988) 668; R. R. Caldwell, R. Dave, P. J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582. P. J. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. 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Lett. B 311 (1993) 27; O. Bertolami, J. Garcia-Bellido, Int. J. Mod. Phys. D 5 (1996) 363. I. Shapiro and J. Sola, JHEP 0202 (2002) 006; I. Shapiro and J. Sola, C. Espana-Bonet and P. Ruiz-Lapuente, Phys. Lett. B 574 (2003) 149; C. Espana-Bonet, P. Ruiz-Lapuente, I. Shapiro and J. Sola, JCAP 0402 (2004) 006. I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective Action in Quantum Gravity, IOP, Bristol, 1992. J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. 363 (2002) 223. S. Falkenberg and S. D. Odintsov, Int. J. Mod. Phys. A 13 (1998) 607; L. N. Granda and S. D. Odintsov, Grav. Cosmol.  4 (1998) 85; L. N. Granda and S. D. Odintsov, Phys. Lett. B 409 (1997) 206. D.F. Litim, Phys. Rev. Lett. 92 (2004) 201301; D.F. Litim, Phys. Rev. D 64 (2001) 105007; F. Freire, D.F. Litim and J.M. Pawlowski, Phys. Lett. B 495 (2000) 256. M. Reuter and C. Wetterich, Nucl. Phys. B 417 (1994) 181; M. Reuter and C. Wetterich, Nucl. Phys. B 427 (1994) 291; M. Reuter and C. Wetterich, Nucl. Phys. 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On leave of absence from the Theoretical Physics Division, Rudjer Bošković Institute, Zagreb, Croatia. [^2]: For interesting work on the scale dependent gravitational coupling see [@bertolami]. [^3]: However, it is necessary to stress that these dependences are not fundamental, but derived in a scale-setting procedure. [^4]: We can safely ignore weak logarithmic running of masses contained in $\rho_m $. [^5]: Similar problems were also treated in [@Odintsov; @Litim]. [^6]: In the effective framework of Ref. [@8; @85] based on the Einstein-Hilbert approximation, an analysis of IR effects leading to the existence of the IR fixed point in the RGE flow is not possible. It would require truncations beyond the Einstein-Hilbert approximation, containing nonlocal invariants, for instance. Hence, the existence of the IR fixed point was hypothesized in [@85].

--- abstract: 'The complex transmission of $\rm Y_{1-x}Pr_xBa_2Cu_3O_7$ single crystal thin films has been measured in the range 0.2-1.0 THz using time domain spectroscopy. The complex conductivity is calculated without using a Kramers-Kronig analysis. All of the superconducting samples show a peak in $\sigma_1(T)$ below $T_c$. The underdoped samples show a deviation from $1/(\alpha+\beta T)$ behavior above $T_c$ that may be linked with the onset of a spin gap.' address: 'Max-Planck-Institut für Festkörperforschung, Postfach 800665, 70506 Stuttgart, Germany' author: - 'J.O. White[^1], R. Buhleier, S.D. Brorson[^2], I.E. Trofimov[^3], H.-U. Habermeier, and J. Kuhl' title: 'Complex conductivity of $\rm Y_{1-x}Pr_xBa_2Cu_3O_7$ thin films measured by coherent terahertz spectroscopy' --- \#1 \#2[$\rm Y_{#1}Pr_{#2}BCO$]{} \#1 \#2[$\rm Y_{#1}Pr_{#2}Ba_2Cu_3O_7$]{} INTRODUCTION ============ The spectroscopic data on the high-$T_c$ superconducting compound $\rm Y_1Ba_2Cu_3O_7$ measured in the far infrared (FIR) as well as the GHz-region has revealed many interesting features. The new technique of THz-time-domain spectroscopy bridges the gap between the two regimes. Here we report results of a THz investigation into the xYPrBCO) system. We investigated five YPrBCO samples having Pr composition $x$ = 0.0, 0.2, 0.3, 0.4, and 1.0. The films are grown to a thickness of 150 nm by pulsed laser deposition [@habermeier91] onto NdGaO$_3$ substrates. Our spectroscopic technique involves a time-domain measurement of the electric field of a microwave pulse transmitted through the sample [@brorson94]. A Fourier transform yields the complex transmission spectrum. To calculate the conductivity, we make use of a multiple reflection formula for the field transmitted through the YBCO layer. RESULTS ======= The conductivity spectra at 50 K are shown in Fig. \[sigma\_nu\]. The addition of Pr to YBCO should have at least two interrelated effects: a) The suppression of $T_c$ changes the partitioning between normal and superconducting carriers. b) The total number of carriers $N$ (or their mobility) may be reduced. Both factors a) and b) cause $\sigma_2$ to decrease with \[Pr\]. Only normal carriers contribute to $\sigma_1$ for $\omega \neq 0$, but now factors a) and b) compete. At 50 K, $\sigma_1$ decreases with \[Pr\], therefore the effect of a reduction in $N$ dominates the effect of the shift in $T_c$. For 30% and 40% Pr, we observe a frequency dependent $\sigma_1$ and can directly measure $\tau(T)$, the quasi-particle scattering time. Pure PrBCO is a dielectric at 50 K, as seen by a conductivity proportional to frequency. Examining $\sigma_2(T)$ at a fixed frequency (Fig. \[sigma\_T\]a), we see that it is close to zero at high temperature, but rises sharply at the onset of superconductivity, thus providing an ac measurement of $T_c$ (Table 1). In all of the superconducting alloys, $\sigma_1$ displays a peak below $T_c$ which has been seen previously only in pure YBCO. It has been attributed to a sharp rise in the scattering time dominating the effect of a decrease in the number of normal carriers [@nuss91]. The normal state behavior of our samples is particularly interesting because other [*underdoped*]{} materials such as oxygen deprived (123)YBCO undergo a phase transition associated with the opening of a spin gap at $T_D>T_c$ [@ito93]. If the normal carriers couple strongly to spin fluctuations, the opening of a spin gap should be accompanied by an [*increase*]{} in the scattering time $\tau$, giving rise to an [*enhancement*]{} in $\sigma_1$ below $T_D$ for $\omega<1/\tau$. For pure (optimally doped) YBCO, at 480 GHz, $\sigma_1$ shows only a single transition at $T_c$. For the (underdoped) alloys, $\sigma_1$ has two transitions, one at $T_c$, the other at a higher temperature which increases with \[Pr\]. To accentuate the two transitions, Fig. \[sigma\_T\]b is shaded in the region bounded by $T_c$, the experimental curve, and a dashed line representing $1 / (\alpha+\beta T)$ behavior. The higher transition temperature seen at 480 GHz matches that of a transition also observed in the dc resistivity. We evaluate the penetration depth $\lambda_L$ and the plasma frequency $\omega_p$ by fitting the data to a two fluid model of the form: $$\sigma(\omega) = {\epsilon_0 \omega_p^2 \tau \over {1-i\omega\tau}}x_n + {1 \over {\mu_0\lambda_L^2}}\left(- \pi\delta(\omega) + {i \over \omega} \right) x_s, \label{2fluid}$$ where $x_n$ and $x_s$ are the fractions of normal and superconducting carriers. The results are shown in Table 1. The decrease in $\omega_p$ with \[Pr\] supports the theory that Pr suppresses superconductivity by reducing the population of mobile holes in the $\rm CuO_2$ planes. $x$ $d$ (nm) $\lambda_L$ (nm) $\omega_p$ (cm$^{-1}$) $T_c^{\rm ac}$ (K) ----- ---------- ------------------ ------------------------ -------------------- 0.0 155 170 9500 92 0.2 134 350 4600 72 0.3 170 380 4200 59 0.4 170 590 2700 41 [9]{} H. U. Habermeier [*et al.*]{}, Physica C [**180**]{}, 17 (1991). S. D. Brorson [*et al.*]{}, Phys. Rev. B [**49**]{}, 6185 (1994). M. C. Nuss [*et al.*]{}, Phys. Rev. Lett [**66**]{}, 3305 (1991). T. Ito, K. Takenaka, and S. Uchida, Phys. Rev. Lett. [**70**]{}, 3995 (1993). [^1]: on leave from: Hughes Research Laboratories, Malibu. [^2]: present addr.: Teledanmark Laboratories, Copenhagen. [^3]: present address: Rutgers University, Piscataway, NJ.

--- abstract: 'We have used an extension of our slow light technique to provide a method for inducing small density defects in a Bose-Einstein condensate. These sub-resolution, micron-sized defects evolve into large amplitude sound waves. We present an experimental observation and theoretical investigation of the resulting breakdown of superfluidity. We observe directly the decay of the narrow density defects into solitons, the onset of the ‘snake’ instability, and the subsequent nucleation of vortices.' author: - | \ \ \ \ \ \ \ \ \ \ \ \ \ title: 'Observation of Quantum Shock Waves Created with Ultra Compressed Slow Light Pulses in a Bose-Einstein Condensate' --- Superfluidity in Bose condensed systems represents conditions where frictionless flow occurs because it is energetically impossible to create excitations. When these conditions are not satisfied, various excitations develop, and experiments on superfluid helium, for example, have provided evidence that the nucleation of vortex rings occurs when ions move through the fluid faster than a critical speed ([*1,2*]{}). Under similar conditions, shock waves would occur in a normal fluid ([*3*]{}). Such discontinuities are not allowed in a superfluid and instead topological defects, such as quantized vortices and solitons, are nucleated when the spatial scale of density variations becomes on the order of the healing length. This is the length scale at which the kinetic energy, associated with spatial variations of the macroscopic condensate wave function, becomes on the order of the atom-atom interaction energy ([*2,4*]{}). It is therefore also the minimum length over which the density of a condensate can adjust. Bose-Einstein condensates (BECs) of alkali atoms ([*5*]{}) have provided a system for the study of superfluidity, which is theoretically more tractable than liquid helium and allows greater experimental control. An exciting recent development is the production of solitons and vortices. Experiments have employed techniques that manipulated the phase of the BEC ([*6-9*]{}), or provided the system with a high angular momentum which makes vortex formation energetically favorable ([*10,11*]{}). However, a direct observation of the formation of vortices via the breakdown of superfluidity has been lacking. Rather, rapid heating from ’stirring’ with a focused laser beam above a critical velocity was observed as indirect evidence of this process ([*12,13*]{}). In this context, it is natural to ask what would happen if one were to impose a sharp density feature in a BEC with a spatial scale on the order of the healing length. Optical resolution limits have prevented direct creation of this kind of excitation. In this paper we present an experimental demonstration of creation of such defects in sodium Bose-Einstein condensates, using a novel extension of the method of ultra slow light pulse propagation ([*14*]{}) via electromagnetically induced transparency (EIT) ([*15,16*]{}). Our slow-light setup is described in ([*14*]{}). By illuminating a BEC with a ‘coupling’ laser, we create transparency and low group velocity for a ‘probe’ laser pulse subsequently sent into the cloud. In a geometry where the coupling and probe laser beams propagate at right angles, we can control the propagation of the probe pulse from the side. By introducing a spatial variation of the coupling field, along the pulse propagation direction, we vary the group velocity of the probe pulse across the cloud. Here we accomplish this by blocking half of the coupling beam so it illuminates only the left hand side ($z<0$) of the condensate, setting up a light ‘roadblock’. In this way, we compress the probe pulse to a small spatial region at the center of the BEC, while bypassing the usual bandwidth requirements for slow light ([*17*]{}). The technique produces a short wavelength excitation by suddenly removing a narrow disk of the condensate, with the width of the disk determined by the width of the compressed probe pulse. We find that this excitation results in short wavelength, large amplitude sound waves that shed off ‘gray’ solitons ([*18-20*]{}), and we make the connection to the formation of shock waves in classical fluids. The ‘snake’ (Kadomtsev-Petviashvili) instability is predicted to cause solitons to decay into vortices ([*21-24*]{}). This has been observed with optical solitons ([*25*]{}) and recently the JILA group predicted and observed that solitons in a BEC decay into vortex rings ([*9*]{}). Here we present a direct observation of the dynamics of the snake instability in a BEC and the subsequent nucleation of vortices. The images of the evolution are compared to numerical propagation of the Gross-Pitaevskii equation in two dimensions. Details of our Bose-Einstein condensation apparatus can be found in ([*26*]{}). We use condensates with about 1.5 million sodium atoms in the state $|1 \rangle \equiv |3S_{1/2},F=1,M_F=-1 \rangle$ and trapped in a 4-Dee magnet. For the experiments presented here, the magnetic trap has an oscillator frequency $\omega_z=(2\pi) 21$ Hz along its symmetry direction and a frequency $\omega_x = \omega_y = 3.8 \omega_z$ in the transverse directions. The peak density of the condensates is then $9.1 \times 10^{13}~\mathrm{cm}^{-3}$. The temperature is $T \sim 0.5\,T_c$, where $T_c=300$ nK is the critical temperature for condensation, and so the vast majority ($\sim 90 \%$) of the atoms occupy the ground state. To create slow and spatially localized light pulses, the coupling beam propagates along the $x$ axis (Fig. 2B), is resonant with the $D_1$ transition from the unoccupied hyperfine state $|2 \rangle \equiv |3S_{1/2},F=2,M_F=-2 \rangle$ to the excited level $ |3 \rangle \equiv |3P_{1/2}, F=2, M_F=-2 \rangle$, and has a Rabi frequency $\Omega_c = (2\pi) 15$ MHz ([*27*]{}). We inject probe pulses along the $z$ axis, resonant with the transition and with peak Rabi frequency $\Omega_p = (2\pi) 2.5$ MHz. The pulses are Gaussian shaped with a $1/e$ half-width of $\tau = 1.3\;\mu$s. With the entire BEC illuminated by the coupling beam, we observe probe pulse delays of $4\;\mu$s for propagation through the condensates, corresponding to group velocities of 18 m/s at the center of the clouds. A pulse with a temporal half-width $\tau$ is spatially compressed from a length $2 \, c \tau$ in vacuum to ([*14,17,28,29*]{}) $$L=2 \tau V_g = 2 \tau \frac{|\Omega_c|^2}{\Gamma f_{13} \sigma_0 n_c}$$ inside the cloud, where $\Gamma = (2 \pi) 10$ MHz is the decay rate of state $|3 \rangle$, $n_c$ is the cloud density, $\sigma_0 = 1.65 \times 10^{-9}\;\mathrm{cm}^2$ is the absorption cross-section for light resonant with a two-level atom, and $f_{13}=1/2$ is the oscillator strength of the $|1 \rangle \rightarrow |3 \rangle$ transition. The atoms are constantly being driven by the light fields into a dark state, a coherent superposition of the two hyperfine states $|1 \rangle$ and $ |2 \rangle$ ([*15*]{}). In the dark state, the ratio of the two population amplitudes is varying in space and time with the electric field amplitude of the probe pulse as $$\psi_2 = - \frac{\Omega_p}{\Omega_c}\,\psi_1,$$ where $\psi_1,\;\psi_2$ are the macroscopic condensate wave functions associated with the two states $|1 \rangle$ and $ |2 \rangle$. For the parameters listed above, the probe pulse is spatially compressed from 0.8 km in free space to only 50 $\mu$m at the center of the cloud, at which point it is completely contained within the atomic medium. The corresponding peak density of atoms in $|2 \rangle$, proportional to $|\psi_2|^2$, is $1/34$ of the total atom density. The $|1 \rangle$ atoms have a corresponding density depression. From Eqs. 1 and 2, it is clear that to minimize the spatial scale of the density defect, we need to use short pulse widths and low coupling intensities. However, for all the frequency components of the probe pulse to be contained within the transmission window for propagation through the BEC ([*17*]{}), we need a pulse with a temporal width $\tau$ of at least $2 \sqrt{D} \Gamma / {\Omega_c}^2 \approx 0.3~\mu$s (here $D \approx 520$ is the optical density of a condensate for on-resonance two level transitions) to avoid severe attenuation and distortion. Furthermore, we see from Eq. 2 that to maximize the amplitude of the density depression would favor use of a peak Rabi frequency for the probe of $\Omega_p \sim \Omega_c$. This also severely distorts the pulse. Both of these distortion effects accumulate as the pulse propagates through large optical densities. This motivated us to introduce a roadblock in the condensate for a light pulse approaching from the left hand ($z<0$) side. By imaging a razor blade onto the right half of the condensate, we ramp the coupling beam from full to zero intensity over the course of a $12\;\mu$m region in the middle of the condensate, determined by the optical resolution of the imaging system. In the illuminated region ($z<0$), our bandwidth and weak-probe requirements are well satisfied and we get undistorted, unattenuated propagation through the first half of the cloud to the high-density, central region of the condensate. As the pulse enters the roadblock region of low coupling intensity, it is slowed down and spatially compressed. The exact shape and size of the defects which are created with this method are dependent on when absorption effects become important. To accurately model the pulse compression and defect formation, we account for the dynamics of the slow light pulses, the coupling field, and the atoms self-consistently. At sufficiently low temperatures, the dynamics of the two-component condensate can be modelled with coupled Gross-Pitaevskii (GP) equations ([*4,5*]{}). Here we include terms to account for the resonant two-photon light coupling between the two components: $$\begin{aligned} i \hbar \frac{\partial}{\partial t}\psi_1 & = & \left(-\frac{\hbar^2 \nabla^2}{2m} + V_1(\mathbf{r}) + U_{11} |\psi_1|^2 + U_{12} |\psi_2|^2 \right) \psi_1 \nonumber \\ & & \hspace{2 cm} - i \frac{\mid{\Omega_p}\mid^2}{2 \Gamma} \psi_1 - i \frac{{\Omega_p}^\ast \Omega_c}{2 \Gamma} \psi_2 \nonumber - i N_c \sigma_e \hbar \frac{k_{2 \gamma}}{2 m} |\psi_2|^2 \psi_1, \\ i \hbar \frac{\partial}{\partial t}\psi_2 & = & \left(-\frac{\hbar^2 (\nabla^2 + i \mathbf{k_{2 \gamma}} \cdot \nabla )}{2m} + V_2(\mathbf{r}) + U_{22} |\psi_2|^2 + U_{12} |\psi_1|^2 \right) \psi_2 \nonumber \\ & & \hspace{2 cm}- i \frac{\mid{\Omega_c}\mid^2}{2 \Gamma} \psi_2 - i \frac{\Omega_p {\Omega_c}^\ast}{2 \Gamma} \psi_1 - i N_c \sigma_e \hbar \frac{k_{2 \gamma}}{2 m} |\psi_1|^2 \psi_2.\end{aligned}$$ Here $V_1(\mathbf{r}) = \frac{1}{2} m {\omega_z}^2( \lambda^2 (x^2+y^2) + z^2)$, where $m$ is the mass of the sodium atoms, and $\lambda=3.8$. Due to the magnetic moment of atoms in state $|2 \rangle$, $V_2(\mathbf{r})= - 2 V_1(\mathbf{r})$, and atoms in this state are repelled from the trap. The EIT process involves absorption of a probe photon and stimulated emission of a coupling photon, leading to a $4.1$ cm/s recoil velocity. This is described by a term in the second equation, containing $\mathbf{k_{2 \gamma}} = \mathbf{k_p - k_c}$, the difference in wave vectors between the two laser beams. (Here we use a gauge where the recoil momentum is transformed away.) Atom-atom interactions are characterized by the scattering lengths, $a_{ij}$, via $U_{ij} = 4 \pi N_c \hbar^2 a_{ij}/m$, where $a_{11}=2.75\;\mathrm{nm}$, $a_{12}=a_{22} = 1.20\,a_{11}$ ([*30*]{}), and $N_c$ is the total number of condensate atoms. To obtain the light coupling terms in Eq. 3, we have adiabatically eliminated the excited state amplitude $\psi_3$ ([*31*]{}), as the relaxation from spontaneous emission occurs much faster than the light coupling and external atomic dynamics driving $\psi_3$. In our model, atoms in $|3 \rangle$ that spontaneously emit are assumed to be lost from the condensate, which is why the light coupling terms are non-Hermitian. Finally, the last term in each equation accounts for losses due to elastic collisions between high momentum $|2\rangle$ atoms and the nearly stationary $|1\rangle$ atoms ($\sigma_e = 8 \pi {a_{12}}^2$) ([*32*]{}). The spatial dynamics of the light fields are described classically with Maxwell’s equations in a slowly varying envelope approximation, again using adiabatic elimination of $\psi_3$ : $$\begin{aligned} \frac{\partial}{\partial z}\Omega_p & = & - \frac{1}{2}f_{13} \sigma_0 N_c(\Omega_p |\psi_1|^2 + \Omega_c {\psi_1}^\ast \psi_2), \nonumber \\ \frac{\partial}{\partial x}\Omega_c & = & - \frac{1}{2}f_{23} \sigma_0 N_c(\Omega_c |\psi_2 \mid^2 + \Omega_p \psi_1 {\psi_2}^\ast).\end{aligned}$$ In the region where the coupling beam is illuminating the BEC ($z<0$), the light coupling terms dominate the atomic dynamics and solving Eqs. 3 and 4 reduces to Eqs. 1 and 2 above. We have performed numerical simulations in two dimensions ($x$ and $z$) to track the behavior of the light fields and the atoms. The probe and coupling fields were propagated according to Eq. 4 with a second order Runge-Kutta algorithm ([*33*]{}) and the atomic mean fields were propagated according to Eq. 3, with an Alternating-Direction Implicit variation of the Crank-Nicolson algorithm ([*33,34*]{}). In this way, Eqs. 3 and 4 were solved self-consistently ([*35*]{}). Profiles of the probe pulse intensity along $z$, through $x=0$, are shown in Fig. 1A. As the pulse runs into the roadblock, a dramatic compression of the probe pulse’s spatial length occurs. When the probe pulse enters the low coupling region, the Rabi frequency $|\Omega_p|$ becomes on the order of $|\Omega_c|$. So the density of state $|2 \rangle$ atoms, $N_c |\psi_2|^2$, increases in a narrow region, which is accompanied by a decrease in $N_c |\psi_1|^2$ (Fig. 1B). The half-width of the defect is $2\;\mu$m. As the compression develops, absorption/spontaneous emission events eventually start to remove atoms from the condensate and reduce the probe intensity. Experimental results are shown in Fig. 2. Fig. 2A is an in-trap image of the original condensate of $|1 \rangle$ atoms, Fig. 2B diagrams the beam geometry, and Fig. 2C shows a series of images of state $|2 \rangle$ atoms as the pulse propagates into the roadblock. The corresponding optical density (OD) profiles along $z$ through $x=0$ are also shown. The OD is defined to be $- \mathrm{ln}(I/I_0)$, where $(I/I_0)$ is the transmission coefficient. All imaging is done with near resonant laser beams propagating along the $y$-axis, and with a duration of 10 $\mu$s. There is clearly a build-up of a dense, narrow sample of $|2 \rangle$ atoms at the center of the BEC as the pulse propagates to the right. Note that the pulse reaches the roadblock at the top and bottom edges of the cloud before the roadblock is reached at the center, which is a consequence of the transverse variation in the density of the BEC, with the largest density along the center line. After the pulse compression is achieved, we shut off the coupling beam to avoid heating and phase shifts of the atom cloud due to extended exposure to the coupling laser, and the subsequent dynamics of the condensate are observed. (We observed that exposure to the coupling laser alone, for the exposure times used to create defects, causes no excitations of the condensates). In considering the dynamics resulting from this excitation of a condensate, it should be noted that the roadblock ‘instantaneously’ removes a spatially selected part of $\psi_1$. The entire light compression happens in approximately $15\;\mu\mathrm{s}$. After the pulse is stopped and the coupling laser turned off, the $|2 \rangle$ atoms remaining in the condensate $\psi_2$ have a $4.1$ cm/s recoil and atoms which have undergone absorption and spontaneous emission events have a similarly sized but randomly directed recoil. So the $\psi_2$ component and the other recoiling atoms interact with $\psi_1$ for less than $0.5$ ms before leaving the region. Both of these time scales are short compared with the several millisecond timescale over which most of the subsequent dynamics of $\psi_1$ occur, as discussed in the following. We first considered the one-dimensional dynamics along the $z$ axis. Snapshots of both condensate components, obtained from numerical propagation in 1D according to Eqs. 3 and 4, are shown at various times after the pulse is stopped at the roadblock (and the coupling laser turned off) (Fig. 3A). In the linear hydrodynamic regime, where the density defect has a relative amplitude $A \ll 1$ and a half-width $\delta \gg \xi$ (here $\xi= 1/\sqrt{8 \pi N_c |\psi_{1}|^2 a_{11}}$ is the local healing length which is $0.4\;\mu\mathrm{m}$ at the center of the ground state condensate in our experiment), one expects to see two density waves propagating in opposite directions at the local sound speed, $c_s = \sqrt{U_{11} |\psi_1 \mid^2/m}$, as seen experimentally in ([*36*]{}). However, for the parameters used in our experiment, the sound waves are seen to shed off sharp features propagating at lower velocities. Examination of the width, speed, and the phase jump across these features shows that they are gray solitons. Describing the slowly varying background wave function of the condensate with $\psi^{(0)}_{1}\!,$ the wave function in the vicinity of a gray soliton centered at $z_0$ is ([*18-20*]{}): $$\psi_1(z,t) = \psi^{(0)}_{1}(z,t) \left( i \sqrt{1-\beta^2} + \beta \; \mathrm{tanh}\left(\frac{\beta}{\sqrt{2}~\xi} (z-z_0)\right) \right).$$ The dimensionless constant $\beta$ characterizes the ‘grayness’, with $\beta = 1$ corresponding to a stationary soliton with a $100\%$ density depletion. With $\beta \ne 1$, the solitons travel at a fraction of the local sound velocity, $c_s \sqrt{1-\beta^2}$. As seen in the figure, after a shedding event, the remaining part of the sound wave continues to propagate at a reduced amplitude. Our numerical simulations show that the solitons eventually reach a point where their central density is zero and then oscillate back to the other side, in agreement with the discussions in ([*18,20*]{}). In Fig. 3A, each of the two sound waves shed off two solitons. By considering the available free energy created by a defect, one finds that, when the defect size is somewhat larger than the healing length and the defect amplitude, $A$, is on the order of unity, the number of solitons that can be created is approximately $\sqrt{A} \delta /(2 \xi)$. One obtains a simple physical estimate of the conditions necessary for soliton shedding by calculating the difference in sound speed associated with the difference in atom density between the center and back edge of the sound wave. As confirmed by our numerical simulations, this difference leads to development of a steeper back edge and an increasingly sharp jump in the phase of the wave function. This is the analog of shock wave formation from large amplitude sound waves in a classical fluid ([*3*]{}). When the spatial width of the back edge has decreased to the width of a soliton with amplitude $\beta = \sqrt{A/2}$ (according to Eq. 5), such a soliton is shed off the back. It’s subsonic speed causes it to separate from the remaining sound wave. Furthermore, by creating defects with sizes on the order of the healing length, we excite collective modes of the condensate, with wave vectors on the order of the inverse healing length. In this regime, the Bogoliubov dispersion relation is not linear ([*4,5*]{}), and accordingly some of the sound wave will disperse into smaller ripples, as seen in Fig. 3A. Considering the evolution of a defect of relative density amplitude $A$ and half-width $\delta$ in an otherwise homogenous medium, we estimate that solitons of amplitude $\beta = \sqrt{A/2}$ will be created after the two resulting sound waves have propagated a distance $$z_{sol} = \frac{2 \delta}{A}\left( \frac{1 - \frac{\xi}{2 \delta}}{1 - \frac{{\pi}^2 \xi^2}{\delta^2 A}}\right).$$ This is in agreement with our numerical calculations. We conclude that the minimum soliton formation length is obtained for large amplitude defects with a width just a few times the healing length. This dictates the defect width picked in the experiments presented here. Narrower defects disperse, whereas larger defects, comparable to the cloud size, couple to collective, nonlocalized excitations of the condensate. We explored the soliton formation experimentally by creating defects in a BEC with the light roadblock. We controlled the size of the defects by varying the intensity of the probe pulses, which had a width $\tau = 1.3~\mu$s. OD images of state $|1 \rangle$ condensates are shown (Fig. 4) in one particular case. Immediately after the defect is created, the trap is turned off, and the cloud evolves and expands for 1 ms and 10 ms, respectively. As seen from the 1 ms picture, a single deep defect is formed initially, which results in creation of 5 solitons after 10 ms of condensate dynamics and expansion. The initial defect created in the trap could not be resolved with our imaging system, which has a resolution of 5 $\mu$m. By varying the probe intensity, we find that the number of solitons formed scales linearly with the probe pulse energy, as expected. To study the stability of the solitons, we first performed 2D numerical simulations of Eqs. 3 and 4. Fig. 3B shows density profiles, $N_c |\psi_1|^2$, obtained for the same parameters as used in Fig. 1B. Again, the profiles are shown at various times after the pulse is compressed and stopped. The deepest soliton (the one closest to the center) is observed to quickly curl and eventually collapse into a vortex pair. The wave function develops a $2 \pi$ phase shift in a small circle around the vortex cores, which shows that they are singly quantized vortices. Also, the core radius is approximately the healing length. (Upon collapse, a small sound wave between the two vortices carries away some of the remaining soliton energy.) This decay can be understood as resulting from variation in propagation speed along the transverse soliton front. As discussed in ([*22-25*]{}), a small deviation will be enhanced by the nonlinearity in the Gross-Pitaevskii equation, and thus, the soliton collapses about the deepest (and therefore slowest) point. Fig. 5 shows experimental images of state $|1 \rangle$ condensates. After the defect is created, the condensate of $|1 \rangle$ atoms is left in the trap for a varying amount of time (as indicated on the figures). The trap was then abruptly turned off and, $15$ ms after release, we imaged a selected slice of the expanded condensate, with a thickness of 30 $\mu$m along $y$ ([*37*]{}). The release time of $15$ ms is picked large enough that the condensate structures are resolvable with our imaging system ([*38*]{}). The slice was optically pumped from state $|1 \rangle$ to the $|3S_{1/2},\,F=2 \rangle$ manifold for $10\;\mu\mathrm{s}$ before it was imaged with absorption imaging by a laser beam nearly resonant with the transition from $|3S_{1/2},\,F~=~2 \rangle$ to $|3P_{3/2},\,F~=~3 \rangle$. The total pump and imaging time was small enough that no significant motion due to photon recoils occurred during the exposure. The slice was selected at the center of the condensate by placing a slit in the path of the pump beam. For the data in Fig. 5A, it is seen that the deepest soliton curls as it leaves certain sections behind, and at 1.2 ms it has nucleated vortices. This is a direct observation of the snake instability. In Fig. 5B, at 0.5 ms, the snake instability has caused a complicated curving structure in one of the solitons, and vortices are observed after 2.5 ms. The vortices are seen to persist for many milliseconds and slowly drift towards the condensate edge. We observed them even after $30$ ms of trap dynamics, long enough to study the interaction of vortices with sound waves reflected off the condensate boundaries. Preliminary results, obtained by varying the $y$ position of the imaged condensate slices, indicate a complicated 3D structure of the vortices. In addition, the defect has induced a collective motion of the condensate whereby atoms, originally in the sides of the condensate, attempt to fill in the defect. This leads to a narrow and dense central region, which then slowly relaxes (Fig. 5B). We performed the experiment with a variety of Rabi frequencies for the probe pulses, and saw nucleation of vortices only for the peak $\Omega_p > (2 \pi) 1.4$ MHz. The free energy of a vortex is substantially smaller near the border of the condensate where the density is smaller, so smaller (and thus lower energy) defects will form vortices very near the condensate edges, seen as ‘notches’ in Figs. 3B and 5. In conclusion, we have studied and observed how small wavelength excitations cause a breakdown of superfluidity in a BEC. Our results show how localized defects in a superfluid will quite generally either disperse into high frequency ripples or end up in the form of topological defects such as solitons and vortices, and we have obtained an analytic expression for the transition between the two regimes. By varying our experimental parameters, we can create differently sized and shaped defects, and also control the number of defects created, allowing studies of a myriad of effects. Among them are soliton-soliton collisions, more extensive studies of soliton stability, soliton-sound wave collisions, vortex-soliton interactions, vortex dynamics, interaction between vortices, and the interaction between the BEC collective motion and vortices. References {#references .unnumbered} ========== 1. R.J. Donnelly, [*Quantized vortices in Helium II*]{} (Cambridge Univ. Press, Cambridge, 1991). 2. 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Julienne, Phys. Rev. Lett. [**84**]{}, 5462 (2000). 33. W. H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, [*Numerical Recipes in C, Second Edition*]{} (Cambridge University Press, Cambridge, 1992). 34. S.E. Koonin and D.C. Merideth, [*Computational Physics*]{}, (Addison-Wesley, Reading, MA, 1990). 35. For propagation of the GP equation in 1D, we typically used a spatial grid with 4000 points and $dz=0.040\;\mu$m. In 2D simulations, we typically used a $750 \times 750$ grid with $dz=0.21\;\mu$m and $dx=0.057\;\mu$m. To solve the equations self-consistently, we kept track of the wave functions at previous time points and projected forward to second order. Smaller time steps and grid spacing were also used to assure convergence of the results. To mimic the nonlinear interaction strength at the center of a [*three*]{} dimensional cloud we put in an effective condensate radius (calculated with the Thomas-Fermi approximation ([*39*]{})) in the dimensions which were not treated dynamically. In all calculations, the initial condition was the ground state condensate wave function with all atoms in $|1 \rangle$, obtained by propagating a Thomas-Fermi wave function in imaginary time. 36. M.R. Andrews, [*et al.*]{}, Phys. Rev. Lett. [**79**]{}, 553 (1997). 37. M.R. Andrews [*et al.*]{}, Science [**275**]{}, 637 (1997). 38. F. Dalfavo and M. Modugno, Phys. Rev. A [**61**]{}, 023605 (2000). 39. G. Baym and C. Pethick, Phys. Rev. Lett. [**76**]{}, 6 (1996). 40. Supplementary material is available at [*Science*]{} Online at www.sciencemag.org. 41. This work was supported by the Rowland Institute for Science, the Defense Advanced Research Projects Agency, the U.S. Airforce Office of Scientific Research, the U.S. Army Research Office OSD Multidisciplinary University Research Initiative Program, the Harvard Materials Research Science and Engineering Center sponsored by the National Science Foundation, and by the Carlsberg Foundation, Denmark. C.S. is supported by a National Defense Science and Engineering Grant sponsored by the Department of Defense. Figure Captions {#figure-captions .unnumbered} =============== **Fig. 1.** **(A)** Compression of a probe pulse at the light ’roadblock’, according to 2D numerical simulations of Eqs. 3 and 4. The solid curves indicate probe intensity profiles along $z$ (at $x=0$), normalized to the peak input intensity. The snapshots are taken at a sequence of times, indicated in the figure, where $t=0$ is defined as the time when the center of the probe pulse enters the BEC from the left. For reference, the atomic density profile of the original condensate is plotted (in arbitrary units) as a dashed curve. The gray shading indicates the relative coupling input intensity as a function of $z$, with white corresponding to full intensity, and the darkest shade of gray corresponding to zero. The spatial turn off of the coupling field is centered at $z=0$ and occurs over $12\;\mu$m, as in the experiment. The number of condensate atoms is $1.2 \times 10^6$ atoms, the peak density is $6.9 \times 10^{13}\; \mathrm{cm}^{-3}$, and the coupling Rabi frequency is $\Omega_c=(2 \pi) 8.0$ MHz. The probe pulse has a peak Rabi frequency of $\Omega_p=(2 \pi)2.5$ MHz and a $1/e$ half-width of $\tau = 1.3 \;\mu$s. **(B)** Creation of a narrow density defect in a BEC. Density profiles of the two condensate components, $N_c|\psi_1|^2$ (dashed) and $N_c|\psi_2|^2$ (solid), along $z$ at $x=0$ for a sequence of times. Note that the $z$ range of the plot is restricted to a narrow region around the roadblock at the cloud center. The densities are normalized relative to the peak density of the original condensate indicated by the red dashed curve. The other curves correspond to times $1 \;\mu$s (green), $4 \;\mu$s (blue), and $14\; \mu$s (black). The width of the probe pulse is $\tau = 4 \;\mu$s and the other parameters are the same as in (A). An animated version is provided in the supplemental material ([*40*]{}) as Animation 1.\ \ **Fig. 2.** **(A)** Experimental in-trap OD image of a typical BEC before illumination by the probe pulse and coupling field. The condensate contains $1.5 \times 10^6$ atoms. The imaging beam was -30 MHz detuned from the $|1 \rangle \rightarrow |3P_{1/2}, F=2, M_F = -1 \rangle$ transition. **(B)** Top view of the beam configuration used to create and study localized defects in a BEC as discussed in the text. **(C)** Build up of state $|2 \rangle$ atoms at the road block. In-trap OD images (left) show the transfer of atoms from $|1\rangle$ to $|2 \rangle$ as the probe pulse propagates through the condensate and runs into the roadblock. The atoms in $|2\rangle$ were imaged with a laser beam -13 MHz detuned from the $| 2\rangle \rightarrow |3P_{3/2}, F=3, M_F = -2\rangle$ transition. To allow imaging, the probe pulse propagation was stopped at various times, indicated in the figure, by switching the coupling beam off ([*29*]{}). The figures on the right show the corresponding line cuts along the probe propagation direction through the center of the BEC. The probe pulse had a Rabi frequency $\Omega_p=(2 \pi) 2.4$ MHz and the coupling Rabi frequency was $\Omega_c= (2 \pi) 14.6$ MHz.\ \ **Fig. 3.** **(A)** Formation of solitons from a density defect. The plots show results of a 1D numerical simulation of Eqs. 3 and 4. The light roadblock forms a defect and the subsequent formation of solitons is seen. The defect is set up with the same parameters for the light fields as in Fig. 1B. The number of condensate atoms is $N_c=1.2 \times 10^6$ and the peak density is $7.5 \times 10^{13}\;\mathrm{cm}^{-3}$. The times in the plots indicate the evolution time relative to the time when the probe pulse stops at the roadblock and the coupling beam is switched off (at $t=8\;\mu\mathrm{s}$ with $t=0$ as defined in Fig. 1A). The solid and dashed curves show the densities of $|1\rangle$ atoms ($N_c | \psi_1|^2$) and $|2\rangle$ atoms ($N_c |\psi_2|^2$). The phase of $\psi_1$ is shown in each case with a dotted curve (with an arbitrary constant added for graphical clarity). In the first two frames, the $|2\rangle$ atoms quickly leave due to the momentum recoil, leaving a large-amplitude, narrow defect in $\psi_1$ (Because this is a 1D simulation, the momentum kick in the $x$ direction is ignored). **(B)** The snake instability and the nucleation of vortices. The plots show the density $N_c |\psi_1|^2$ from a numerical simulation in 2D, with white corresponding to zero density and black to the peak density ($6.9 \times 10^{13}\;\mathrm{cm}^{-3}$). The parameters are the same as in Fig. 1B and the times indicated are relative to the coupling beam turn-off at $t=21\;\mu\mathrm{s}$. The solitons curl about their deepest point, eventually breaking and forming vortex pairs of opposite circulation (seen first at 3.5 ms). Several vortices are formed and the last frame shows the vortices slowly moving towards the edge of the condensate. At later times, they interact with sound waves which have reflected off the condensate boundaries. Animated versions (Animations 2 and 3) are provided in the supplemental material ([*40*]{}).\ \ **Fig. 4.** Experimental OD images and line cuts (at $x=0$) of a localized defect (top) and the resulting formation of solitons (bottom) in a condensate of $|1 \rangle$ atoms. The imaging beam was detuned $-30$ MHz and $-20$ MHz, respectively, from the $|\,3S_{1/2},\;F=2\rangle \rightarrow |\,3P_{3/2}, F=3\rangle$ transition. Prior to imaging, the atoms were optically pumped to $|\,3S_{1/2}, F=2\rangle$ for $10\;\mu\mathrm{s}$. The probe pulse had a peak Rabi frequency $\Omega_p = (2 \pi) 2.4$ MHz. The coupling laser had a Rabi frequency of $\Omega_c = (2 \pi) 14.9$ MHz, was turned on $6\;\mu\mathrm{s}$ before the probe pulse maximum, and had a duration of $18\;\mu\mathrm{s}$.\ \ **Fig. 5.** Experimental OD images of a $|1\rangle$ condensate, showing development of the snake instability and the nucleation of vortices. In each case, the BEC was allowed to evolve in the trap for a variable amount of time after defect creation. **(A)** The deepest soliton (nearest the condensate center) is observed to curl due to the snake instability and eventually break, nucleating vortices at 1.2 ms. Defects were produced in BECs with $1.9\times 10^6$ atoms by probe pulses with a peak $\Omega_p = (2 \pi) 2.4$ MHz, and a coupling laser with $\Omega_c = (2 \pi)14.6$ MHz. The imaging beam was $-5$ MHz detuned from the $|\,3S_{1/2},\,F=2\rangle \rightarrow |\,3P_{3/2},\,F=3\rangle$ transition. **(B)** The snake instability and behavior of vortices at later times. The parameters in this series are the same as in (A), except that the peak $\Omega_p = (2\pi) 2.0$ MHz, the number of atoms in the BECs was $1.4 \times 10^6\!,$ and the pictures were taken with the imaging beam on resonance. Animation Captions {#animation-captions .unnumbered} ================== **Animation 1.** An animation, based on 2D numerical calculations, showing creation of a narrow density defect in a BEC by the light roadblock. The parameters and conventions are the same as in Fig. 1B. Successive frames are spaced by $1~\mu$s. The solid curve shows the build-up of $|2 \rangle$ atoms as the probe pulse runs into the roadblock, and the dashed curve shows the corresponding depletion of the density of the condensate of $|1 \rangle$ atoms. **Animation 2.** Animation of 40 ms of BEC dynamics based on 1D numerical simulations. Parameters and conventions are the same as in Fig. 3A, but the phase is not plotted here. Successive frames are spaced by 0.25 ms. The animation shows a narrow density defect in the $|1 \rangle$ condensate decaying into four solitons due to the steepening of the back edge of the sound waves. The high frequency ripples are due to the nonlinear part of the Bogoliubov dispersion curve. When the solitons reach a point where their amplitude, $\beta$, equals unity they turn around. Upon reaching the center of the condensate, they pass through each other unaffected. **Animation 3.** Animation showing 30 ms of dynamics of the state $|1 \rangle$ condensate, based on 2D numerical simulations. Parameters and conventions are the same as in Fig. 3B. (The plot range of each frame is $96.8~ \mu$m$~\times~ 31.2~\mu$m). Successive frames are spaced by 0.4 ms. The narrow density defect in the condensate decays into several solitons and the deepest solitons decay, via the snake instability, into vortices and release their remaining energy as sound waves. The vortices drift slowly, while some of the sound waves reflect off the condensate boundaries and subsequently interact with the vortices.

--- abstract: 'Nanopore based sequencing has demonstrated significant potential for the development of fast, accurate, and cost-efficient fingerprinting techniques for next generation molecular detection and sequencing. We propose a specific multi-layered graphene-based nanopore device architecture for the recognition of single DNA bases. Molecular detection and analysis can be accomplished through the detection of transverse currents as the molecule or DNA base translocates through the nanopore. To increase the overall signal-to-noise ratio and the accuracy, we implement a new ”multi-point cross-correlation” technique for identification of DNA bases or other molecules on the molecular level. we demonstrate that the cross-correlations between each nanopore will greatly enhance the transverse current signal for each molecule. We implement first-principles transport calculations for DNA bases surveyed across a multi-layered graphene nanopore system to illustrate the advantages of proposed geometry. A time-series analysis of the cross-correlation functions illustrates the potential of this method for enhancing the signal-to-noise ratio. This work constitutes a significant step forward in facilitating fingerprinting of single biomolecules using solid state technology.' author: - Towfiq Ahmed - 'Jason T. Haraldsen' - 'John J. Rehr' - Massimiliano Di Ventra - Ivan Schuller - 'Alexander V. Balatsky' title: Correlation dynamics and enhanced signals for serial DNA sequencing --- Introduction ============ ![ Illustration of the three layer graphene based nanopore as a possible multilayered sequencing device. [**[b]{}**]{} Schematic of transmission currents through two graphene layers where isolated DNA bases pass through the nanopores. The current vs. time spectra are recorded for each layer independently. A cross correlation between the current data from multipores reveals useful information by increasing signal to noise ratio as described in the text; [**[c]{}**]{} Hydrogen capped graphene nanoribbons and the DNA bases inside the pore. Here, only the flat orientation of the DNA bases are shown. The other angular orientations are shown in the supplementary section.are shown. The other angular orientations are shown in the supplementary section.[]{data-label="f1"}](fig_1.eps){width="3.5in"} With applications ranging from explosives and drug detection to DNA sequencing and biomolecular identification, the ability to detect specific molecules and/or molecular series presents many challenges for scientists. With a specific need for timely and accurate measurements and evaluation, it is essential that researchers investigate both the manner of detection as well as explore new and improved computational methods for analysis to keep up with the growing pace of the individual fields. The field of DNA sequencing is rapidly evolving due to increasing support and technology. As this occurs, sequencing techniques are challenged by the need for a rapid increase of accuracy, speed, and resolution for smaller amounts of material [@xprize]. Nanopore-based sequencing [@Zwolak2008; @Branton2008] and serial methods [@towfiq_dna; @Kilina2007] provide promising alternatives to the well established Sanger method [@sanger], particularly for identifying single DNA bases using transverse conductance [@ventra_2005; @ventra_2006]. Such an approach relies on the ability to resolve the electronic fingerprints of DNA one relevant unit at a time (‘serial’) as DNA translocates through a nanochannel. It has been established that experimental methods are capable of achieving single-base resolution, which has prompted investigations into the local electrical properties of single DNA bases [@tanaka; @Yarotski2009]. Concurrently, the theoretical underpinnings of this approach have been continuously developing [@towfiq_dna; @Kilina2011; @Kilina2007; @ventra_2005; @ventra_2006]. The single-molecule sensitivity of nanopore sequencing has been recently demonstrated by Kawai [*[et al.]{}*]{} [@kawai] and Lindsay[*[et al.]{}*]{} [@Chang2010]. The sequence of DNA/RNA oligomers and microRNA by tunneling has also been demonstrated [@kawai2012]. Despite such high-quality experimental methods, the most pressing challenge in serial sequencing lies in overcoming effects of noise that lead to a small signal to noise (S/N) ratio in the measured current $I$. The signal fluctuations generally originate from thermal agitation and bond formation between base and nanopore/electrode walls or interactions with a substrate. In an effort to avoid these limitations, we propose the sequential measurement of transverse current cross-correlations, as obtained from multiple pairs of electrodes. The experimental set up for such a nanopore arrangement is schematically shown in \[f1\]. To be specific, we focus on graphene as the porous material, because it is atomically thick and exhibits extraordinary thermal and electronic properties. Besides these geometric advantages and good conductivity, graphene also possesses high tensile strength and can endure a high transmembrane pressure environment [@graphene_mechanical]. Consequently, graphene has been proposed as an effective substrate and conducting medium for nanopore sequencing by numerous groups [@branton; @merchant; @schneider; @prezhdo; @tanaka; @scheicher]. We emphasize, however, that the method for nanopore sequencing may be useful in any other method in which serial measurements (e.g., time series) are made to ascertain individual properties (resistivity here) of the bases. Although this challenge is much more severe for protein based or solid state nanopores, the nature of an atomically thick graphene nanopore wall cannot completely rule out the $\pi-\pi$ stacking between carbon and DNA bases. In addition, vibration and other electronic fluctuations present in the graphene membrane can significantly mask the conductance signals, making it difficult to differentiate the individual DNA bases. Previous theoretical [@Kilina2007; @Kilina2011] and experimental [@tanaka] studies of the interactions between DNA bases and graphene derivatives have revealed the local electronic structure of single bases. The experimental realization of a single layer graphene-based nanopore device is made possible by combining several state of the art techniques e.g., mechanical exfoliation from graphite on SiO$_2$ substrate. Transverse tunneling current(conductance) measurements, as the single strand (ss)DNA translocates through a monolayer graphene nanopore, were previously reported by Schneider [*et al.*]{} [@schneider]. AFM studies [@Yarotski2009] and theoretical simulations of scanning tunneling spectroscopy (STS) [@towfiq_dna] support the identification of electronic features with varying spatial extent and intensity near the HOMO-LUMO band. To make nanopore sequencing and detection a viable method for determining translocating molecules, one must overcome this the noise to signal problem. Therefore, we propose a multilayered graphene device in which the transverse conductance is measured through each nanopore independently, as a series of DNA bases or other molecules translocates through them (see  \[f1\]). As molecules translocate, they create a time dependent sequence of translocation currents through each of the layers. One then monitors the translocation currents at different pores and acquires a record of sequential current of the same base as it arrives and moves through the individual pores (shown in  \[f2\]). The time series of the cross correlation currents can then be used to reduce the uncorrelated, independent noise source, and hence enhance the signal to noise ratio and improve the differentiation between bases. While our device is being discussed under the idea of DNA sequencing, the general method and device setup can be used for any molecule small enough to fit through a nanopore. While we are focusing on the area of DNA sequencing and biomolecules, this cross-correlation method for data analysis of the transverse currents can be utilized for the analysis of any molecular series given the proper understanding of the molecules electronic properties. Results and Discussion ====================== We first discuss our first-principles calculations of transmittance for individual DNA bases inside the graphene nanopore, as presented in  \[f3\]. Then in  \[f4\], we show the partial signal recovery using our time-simulation model with three layer graphene nanopores and the cross-correlation between the corresponding signals. In our first-principles approach, for each DNA base, we have taken three random angular orientation with the graphene membrane, while calculating the transmittance between the two electrodes with 0.7 V bias voltage. The configuration averaged transmittance for A, C, G, and T are shown in the solid blue curve in  \[f3\](a)-(d). The conductance of a pure graphene nanoribbon with hydrogenated nanopore is shown in solid red curve in  \[f3\] for comparison. The transmittance curve is analogous to the non-equilibrium density of states in the presence of the bias voltage where the zero of energy is the Fermi energy of the central graphene region. The vertical dashed lines are at -0.35 eV and +0.35 eV, which are the chemical potentials of the left and right electrodes respectively. For each base ( \[f3\](a)-(d)), the transmittance curve (solid blue line) in between the left and right electrode chemical potentials is significantly enhanced compared to the pure graphene membrane with a nanopore (solid red line). The features in this region are characteristic of the four bases. For example, a comparison of the Guanine transmittance ( \[f3\](c)) with that of Thymine ( \[f3\](d)), shows the presence of a characteristic broad peak. For a systematically study of the difference between the transmittance among the four bases, we also plotted the difference curves (the top three) in  \[f3\](a)-(d). If the signatures of one or more of the DNA bases are known prior to the detection, the difference curve may provide the signature of an unknown base. For example, if one knows the transmittance of Thymine, a comparison of the characteristic features of difference-squared transmittance $(A-T)^2$, $(G-T)^2$, $(C-T)^2$, helps identify the unknown base.  \[f3\](a),(c), and (d) show the difference-curves contain several (up to three) dominant peaks in between the vertical dashed lines. In principle, it is possible to calculate a large number of configurations and maintain a complete data-base of such characteristic difference curves for the sequencing purpose. Such methods are challenged by two major limitations. The first one is prior knowledge of the exact location of one or more kinds of DNA base, either from the transmittance curve or form other technique. The second one is the presence of significant noise in the data, which makes it difficult for the detection of any single base. Some bases exhibit characteristic features in the transmittance curve, which make them easily detectable. For example, the Thymine ( \[f3\](d) solid blue line) has a very low conductance compared to the others which (in agreement with previous calculations [@ventra_2005; @ventra_2006]) shown by the low peak amplitude near 0 eV. However, even the detection of Thymine can be difficult in the presence of noisy data. To illustrate the specifics of the approach, we present the simulation of a time-series for three graphene nanopore layers with the test sequence A$_0$C$_0$A$_2$G$_2$T$_1$C$_2$G$_1$T$_2$ in Fig.4. In nanopore based DNA sequencing, the current ($I(t)$) is the measured quantity rather than the transmittance ($T(E)$). Thus, we calculated the current from the transmittance. Using the parameters described previously we simulated time-dependent current spectra $I_{L-1}$, $I_{L-2}$, and $I_{L-3}$ for our test sequence, as shown in  \[f4\](a). The low current amplitude for Thymine in the case of T$_1$ and T$_2$ is expected from the transmittance curve in  \[f3\](d), but the natural noise present in the data makes it difficult to confirm the presence of T$_1$ at the expected location. In  \[f4\](b), we present the cross-correlation between the current spectra from different pairs of graphene layers. For each pair, the cross-correlation is plotted as a function of time-delay within the -10 $\mu$s to +10 $\mu$s range. The cross-correlation spectrum is approximately symmetric around mid point of the total range due to the overlaps between similar pairs of peaks from opposite ends of the original data. Therefore, we only focus on the positive time-delay. The correlation spectrum inside the highlighted dashed box in  \[f4\](b) is enhanced in  \[f4\](c). By comparing peaks between  \[f4\](a) and (c), we confirm the presence of Thymine with T$_1$ configuration. Although the amplitudes of the current spectrum do not translate directly into the amplitudes of the cross-correlation spectrum, they confirm the existence of T$_1$. Thus, a time-series analysis using current cross-correlations $\langle I_{i}(t) \otimes I_{j}(t) \rangle$ recovers all eight peaks in our test sequence ( \[f4\](b)). The suppression of white noise is substantial and the peaks at time-delay=0 in the correlation function ( \[f4\](b)) are enhanced. We can easily extend this approach to three-point or higher $N$-point correlations, which we demonstrate here, to exponentially reduce the noise-to-signal ratio. The two-point cross-correlation is generally expressed with a single parameter as in $$R^{(2)}(\tau)=\int_0^T I_1(t) I_2(t-\tau) dt,$$ where the time interval is between $0$ to $T$. The three-point correlation is a function of two independent variables $$R^{(3)}(\tau, \tau')=\int_0^T I_1(t) I_2(t-\tau) I_3(t-\tau') dt.$$ We can simplify the description of triple correlation function in the complete two dimensional parametric space by constraining it to the line $\tau' = 2 \tau$ as in  \[f5\](b). Thus the constrained triple correlation function becomes, $$R^{(3)}(\tau)=\int_0^T I_1(t) I_2(t-\tau) I_3(t-2 \tau) dt.$$ Following this procedure we can measure currents from $N$ independent graphene layers and calculate constrained $N$-point correlation as $$\begin{aligned} R^{(N)}(\tau)=\int_0^T I_1(t)&I_2(t-\tau) I_3(t-2 \tau) .... \nonumber \\ &.... I_N(t-(N-1)\tau) dt.\end{aligned}$$ The three panels in  \[f5\](a) show our calculated current signal from a single layer as well as the two and three point cross-correlation functions from the corresponding two and three independent graphene nanopores. The test sequence used here is A$_0$C$_0$A$_2$G$_2$T$_1$C$_2$G$_1$T$_2$C$_1$. Using two, three and four point cross-correlation functions, we estimated the ratios between the average signal and average noise in each case, as shown in Table.1 in the supplementary section. We confirm the exponential drop in the noise to signal ratio as shown in  \[f5\](c). The computational details and the table containing the results are also given in the supplementary section. Computational Method ==================== In this work, we ignore the background contribution from the large phosphate backbone typically present in a single stranded DNA (ssDNA). This simplification is based on the assumption that by identifying and subtracting the background noise coming from the heavy and rigid backbone structure. one can isolate the relevant signal from the individual bases. More specifically we have built on earlier work [@towfiq_dna; @Kilina2011; @ventra_2005; @scheicher] to model the pore conductance containing a molecule in two steps: 1) First, we carried out [*[ab initio]{}*]{} calculations of transmission ($T(E)$) and current ($I$) as a single DNA base translocates through the nanopore of a graphene mono-layer. 2) Then, we simulate the time-dependence of the current data by adopting a simple model with multi-layered graphene nanopores with added statistical noise and broadening. Calculations of transmission were performed taking each DNA base inside the nanopore with three different angular orientations, and using the Landauer-Buttikker [@land] formalism implemented in the [*ab initio*]{} software ATK [@mads_2]. We emphasize that out approach does not rely nor requires a geometry optimization of molecules in the pores. The translocation is a dynamical process with significant variations of configurations found for molecules inside a pore. Thus, the same molecule can arrive in different orientations at each pore, a process which contributes to the configuration noise sources that we address here. Therefore, we do not optimize the configurations and instead use the set of various configurations as the set, from which the random sampling is taken. In these calculations, we have taken a graphene nanoribbon with 208 carbon atoms in the conduction region, where the nanopore is constructed by removing center carbon atoms and capping the inner wall with hydrogen atoms, since hydrogenated edges were found [@scheicher] to enhance the average experimental conductivity. The bias voltage between the left and right electrodes is fixed as +0.35 and -0.35 eV. In this work, the nanopore dimension is much smaller than that modeled by other groups [@postma; @prezhdo]. The details and various parameters of our first-principles calculations can be found in the supplementary section. To demonstrate the recoverability of current ($I(t)$) signals from noise, we show the relation between noise coming from different layers. For simplicity, we consider the dominant noise primarily from two sources. As the bases translocate through the [*[i]{}*]{}-th graphene nanopore layer, the vibration in the DNA backbone may influence individual base plane to land with random angular orientation with the graphene plane, causing a configuration-noise $S_i^C(t)$. The additional noise, such as thermal vibration of the graphene membrane at the $i$-th nanopore, is defined as $S_i^A(t)$. Thus the total noise of $i$-th nanopore can be expressed as $$S_i(t)=S_i^C(t)+S_i^A(t).$$ The correlation between the two layers is therefore given by $$\begin{aligned} <S_i(t)&\cdot S_j(t')>\,=\,<S_i^C(t) \cdot S_j^C(t')> \nonumber \\ &+<S_i^C(t) \cdot S_j^A(t')> +<S_i^A(t) \cdot S_j^C(t')> \nonumber \\ &+<S_i^A(t) \cdot S_j^A(t')>.\end{aligned}$$ Here $t'=t+\Delta t$. For $i \neq j$, the contribution from the last three terms on the right side of Eq. 2 are negligible due the weakly or uncorrelated signals in separate nanopores. Since the DNA bases are strongly attached to the ssDNA backbone, the configuration-noise between two membranes mainly contributes to the first term in Eq. 2. Therefore, the noise can be approximated as $$<S_i \cdot S_j>\,\approx\,<S_i^C \cdot S_j^C> ,$$ where, for $i = j$, all the terms on right side of Eq. 2 survive and contribute significantly to the total noise. Since the noise between $i$ and $j$ is uncorrelated, a comparison of their signals will enhance the individual base signals by reducing the noise to signal ratio. There are two extreme limits in which we can take advantage of the above observation. These limits relate to the rate of base translocation compared to the typical vibrational frequency of the bases facing the electrodes. When this occurs, the above cross correlations allow us to reduce the [*intrinsic*]{} noise due to random orientations. On the other hand, when the translocation rate is slower than the vibrational frequency, the uncorrelated noise is eliminated and the only one that survives is the correlated one. We focus here on the second case since experimentally the latter situation is more likely [@Zwolak2008; @Branton2008]. As an example, we show the low current amplitude for Thymine in  \[f4\](a), and in  \[f4\](c) the enhancement of the signal to noise ratio. We have taken a test sequence A$_0$C$_0$A$_2$G$_2$T$_1$C$_2$G$_1$T$_2$, where the subscripts imply different angular orientations of the bases inside the pore. The time dependence of this sequence is modeled by taking the time interval between two consecutive bases $ \tau = 1.0 \;\mu\text{s}$, including a random Gaussian uncertainly between the interval with $\sigma_{\tau}= \pm 0.2 \;\mu\text{s}$. Each current signal is also broadened using a random Gaussian broadening with $\sigma_{broad}=0.2\,\mu A$. To simulate a realistic experiment with background noise, we have also included additive white Gaussian noise. We assume that with the applied field in the vertical direction, the average elapsed time between two translocating bases is $\tau \approx 1.0 \;\mu\text{s}$. The time-distance between two consecutive graphene layers is set to $\Delta t \approx 0.2 \;\mu\text{s}$. Conclusions =========== We implement first-principles calculation of transmittance for a systematic study of the identification of single DNA bases or other biomolecules translocating through graphene nanopores. To eliminate the high background noise, we propose a multilayered graphene-based nanopore device combined with a multi-point cross-correlation method to substantially improve the signal to noise ratio of the electronic readout of biomolecules. To illustrate this approach, we adopted a statistical method for simulating the time-dependent current spectrum. The enhanced resolution is produced by the multiple translocation readouts of the same bases of the same molecule through the pores. The cross-correlated signals from each pair of electrodes will suppress the uncorrelated noise produced by each single translocation event. In this way, thymine can serve as a “reference molecule” for identifying other molecules from the difference transmittance curves. We also demonstrate the recovery of signals associated with different configurations by taking cross-correlations between different pairs of graphene layers. This study provides a promising method for an enhanced signal to noise ratio in the multipore graphene based devices (or any other serial sequencing device), and their potential applicability as a next generation biomolecular detection technique. While we focus on the correlations in DNA bases, this cross-correlation method can be used for any molecule or molecular series for detection or identification purposes. We are grateful to K.T. Wikfeld, K. Zakharchenko and Svetlana Kilina for useful discussions. This work is supported by the Center for Integrated Nanotechnologies at Los Alamos, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396. Work at NORDITA was supported by VR 621-2012-2983 and ERC 321031-DM. MD acknowledges partial support from the National Institutes of Health. IKS is supported by AFOSR FA9550-10-1-0409. @ifundefined [24]{} FOUNDATION, X. P. Archon Genomics X PRIZE. 2013; <http://www.genomics.xprize.org> Zwolak, M.; [Di Ventra]{}, M. *Rev. Mod. 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**THE CONSTITUTIVE RELATIONS AND THE** **MAGNETOELECTRIC EFFECT FOR  MOVING MEDIA** Tomislav Ivezić *Ruer Bošković Institute, P.O.B. 180, 10002 Zagreb, Croatia* E-mail: ivezic@irb.hr**** In this paper the constitutive relations for moving media with homogeneous and isotropic electric and magnetic properties are presented as the connections between the generalized magnetization-polarization bivector $\mathcal{M}$ and the electromagnetic field $F$. Using the decompositions of $F$ and $\mathcal{M}$, it is shown how the polarization vector $P(x)$ and the magnetization vector $M(x)$ depend on $E$, $B$ and two different velocity vectors, $u$ - the bulk velocity vector of the medium, and $v$ - the velocity vector of the observers who measure $E$ and $B$ fields. These constitutive relations with four-dimensional geometric quantities, which correctly transform under the Lorentz transformations (LT), are compared with Minkowski’s constitutive relations with the 3-vectors and several essential differences are pointed out. They are caused by the fact that, contrary to the general opinion, the usual transformations of the 3-vectors $\mathbf{E}$, $\mathbf{B}$, $\mathbf{P}$, $\mathbf{M}$, etc. are not the LT. The physical explanation is presented for the existence of the magnetoelectric effect in moving media that essentially differs from the traditional one. *Keywords:* Constitutive relations; moving media; magnetoelectric effect. **1. Introduction** In this paper the constitutive relations for moving media with homogeneous and isotropic electric and magnetic properties are presented using the abstract four-dimensional (4D) geometric quantities and their representations in the standard basis. These results are compared with Minkowski’s constitutive relations$^{1}$ with the 3-vectors and with Minkowski’s constitutive relations that are obtained by means of exterior forms.$^{2}$ The paper is organized as follows. First, in Sec. 2, we present a review of some previous results that are important for the theory presented here. In the recent paper,$^{3}$ a formulation of the field equations for moving media is developed by the generalization of an axiomatic geometric formulation of the electromagnetism in vacuum.$^{4}$ As mentioned in Ref. 3 almost the entire physics literature deals with the electromagnetic excitation tensor $\mathcal{H}$, i.e. with the electric $D$ and magnetic $H$ excitations (see Eq. (\[h1\])) and the constitutive relations refer to the connections between them and $F$, i.e. $E $ and $B$, respectively. But, as shown in Ref. 3, it is physically better founded to formulate the field equations for moving media in terms of $F$ and the generalized magnetization-polarization bivector $\mathcal{M}$ instead of, as usual, $F$ and $\mathcal{H}$. In Sec. 2, the decompositions of $F$ (\[E2\]) and $\mathcal{M}$ (\[M1\]) are presented. $F$ is decomposed into vectors $E$, $B$ and $v$ - the velocity vector of the observers who measure $E$ and $B$ fields. $\mathcal{M}$ is decomposed into the polarization vector $P$, the magnetization vector $M$ and $u$ - the bulk velocity vector of the medium. In Ref. 3, the field equations are written in terms of $F$ and $\mathcal{M}$ and also in terms of vectors $E$, $B$, $P$, $M $ and the velocity vectors $u$ and $v$. These field equations are also quoted in Sec. 2. Furthermore, in Sec. 2, the usual transformations (UT) of the 3-vectors $\mathbf{E}$ and $\mathbf{B}$, (\[JCB\]), and of $\mathbf{P}$ and $\mathbf{M}$, (\[ps\]), are written and their difference relative to the Lorentz transformations (LT) of vectors, as 4D geometric quantities, e.g., the electric field vector $E$, (\[T1\]), is pointed out. In Sec. 3, we formulate the constitutive relations as the relations between $\mathcal{M}$ and $F$, Eqs. (\[cr1\]) and (\[cr2\]). Then, using the decompositions of $F$ (\[E2\]) and $\mathcal{M}$ (\[M1\]) we get from (\[cr1\]) and (\[cr2\]) how $P(x)$, (\[P\]), and $M(x)$, (\[M\]), depend on $E$, $B$ and two different velocity vectors, $u$ and $v$. The constitutive relations (\[P\]) and (\[M\]) are the basic relations that are obtained in this paper and they are not reported in previous approaches. The last term in (\[P\]) and (\[M\]) describes the magnetoelectric effect in a moving dielectric in a new way. In Sec. 4, we have represented all 4D geometric quantities from (\[P\]) and (\[M\]) in the standard basis in order to compare them with some usual formulations. This procedure yields Eqs. (\[po\]) and (\[ma\]). In Sec. 5, Minkowski’s constitutive relations with the 3-vectors (\[de\]) are quoted. The equations (\[de\]) are considered to be the fundamental constitutive equations for moving media in the whole physics community. In Sec. 5.1, the relations (\[de\]) are written in equivalent forms asconstitutive equations which explicitly express the 3-vectors $\mathbf{P}$ and $\mathbf{M}$ as the functions of the 3-vectors $\mathbf{E}$ and $\mathbf{B}$, (\[pl\]) and (\[mg\]), or (\[mp\]). These forms of Minkowski’s constitutive relations are compared with our relations (\[po\]) and ([ma]{}), i.e., with Eqs. (\[pc\]) and (\[ma1\]), and several essential differences are pointed out. It is argued that these differences appear since Minkowski’s constitutive relations are with the 3-vectors and they are derived using the UT and not the LT. In Sec. 5.2, it is shown that the same differences remain in the low velocity limit. In Sec. 5.3, it is presented the comparison with Minkowski’s constitutive relations that are obtained by means of exterior forms.$^{2}$ It is shown that the constitutive relations with exterior forms from Ref. 2 are completely equivalent to Minkowski’s constitutive relations with the 3-vectors and, accordingly, they also differ from the relations obtained in this paper. In Sec. 6, Eq. (\[i\]) represents the interaction term in the Lagrangian for the interaction between the electromagnetic field $F$ and the dipole moment bivector $D$, whereas Eq. (\[1\]) is its low velocity limit. The last two terms in (\[i\]), or (\[1\]), contain the direct interaction of $E$ with the magnetic dipole moment vector$\ m$ and $B$ with the electric dipole moment vector $d$. These terms give the physical explanation for the existence of the magnetoelectric effect in moving media. That explanation markedly differs from the traditional one. In Sec. 7, some remarks are given, which refer to the general constitutive relations. In Sec. 8, the conclusions are presented. **2. A Short Review of Some Previous Results** We shall deal with 4D geometric quantities, i.e. in the geometric algebra formalism. For the exposition of the geometric algebra see Ref. 5. The generators of the spacetime algebra are four basis vectors $\left\{ \gamma _{\mu }\right\} ,\mu =0...3,$ satisfying $\gamma _{\mu }\cdot \gamma _{\nu }=\eta _{\mu \nu }=diag(+---)$. This basis, the standard basis, is a right-handed orthonormal frame of vectors in the Minkowski spacetime $M^{4}$ with $\gamma _{0}$ in the forward light cone, $\gamma _{0}^{2}=1$ and $\gamma _{k}^{2}=-1$ ($k=1,2,3$). The standard basis $\left\{ \gamma _{\mu }\right\} $ corresponds to Einstein’s system of coordinates in which the Einstein synchronization of distant clocks$^{6}$ and Cartesian space coordinates $x^{i}$ are used in the chosen inertial frame of reference. First, we briefly expose the formulation of the field equations from Ref. 3. As shown in Ref. 3, the field equations (5), $\partial (\varepsilon _{0}F+\mathcal{M})=j^{(C)}/c$; $\partial \cdot (\varepsilon _{0}F+\mathcal{M})=j^{(C)}/c$, $\partial \wedge F=0$, can be taken as the primary equations for the electromagnetism in moving media. The bivector $F=F(x)$ represents the electromagnetic field and, as shown in Ref. 4, it can be taken as the primary quantity for the whole electromagnetism. $j^{(C)}$ is the conduction current density of the *free* charges. $\mathcal{M}$ is the generalized magnetization-polarization bivector $\mathcal{M=M}(x)$, which is connected with the magnetization-polarization current density of the *bound* charges $j^{(\mathcal{M})}=-c\partial \mathcal{M}=-c\partial \cdot \mathcal{M}$. (According to Eq. (3),$^{3}$ the total current density vector $j$ can be decomposed as $j=j^{(C)}+j^{(\mathcal{M})}$.) The field equation with sources is written in the ‘source representation’ in Eq. (7)$^{3}$ $\partial \cdot \varepsilon _{0}F=j^{(C)}/c-\partial \cdot \mathcal{M}$; the sources of $F$ are the true currents $j^{(C)}$ and the magnetization-polarization current density $\partial \cdot \mathcal{M}$. In most materials $\mathcal{M}$ is a function of the field $F$ and this dependence is determined by the constitutive relations. In all previous formulations of the electromagnetism in media (at rest, or moving), the electromagnetic excitation tensor is introduced as $\mathcal{H}=\varepsilon _{0}F+\mathcal{M}$. Then, the constitutive relations refer to the connections between $\mathcal{H}$ and $F$. However, as discussed in Ref. 3, physically different kind of entities are mixed in that definition for $\mathcal{H}$; an electromagnetic field $F$ and a matter field $\mathcal{M}$. Furthermore, in general, two different velocity vectors, $v$ - the velocity of the observers and $u$ - the velocity of the moving medium, enter into the decompositions of $F$ and $\mathcal{M}$, respectively. For $F$, that decomposition is given by Eq. (10)$^{3}$ $$F=E\wedge v/c+(IcB)\cdot v/c, \label{E2}$$ where the electric and magnetic fields are represented by vectors $E(x)$ and $B(x)$ and $I$ is the unit pseudoscalar. There is no rest frame for the field $F$, that is, for $E$ and $B$, and therefore the vector $v$ in the decomposition (\[E2\]) is interpreted as the velocity vector of the observers who measure $E$ and $B$ fields. Then $E(x)$ and $B(x)$ are defined with respect to $v$, i.e., with respect to the observer, as $$E=F\cdot v/c,\quad B=-(1/c)I(F\wedge v/c). \label{E1}$$It also holds that $E\cdot v=B\cdot v=0$; both $E$ and $B$ are space-like vectors and they depend not only on $F$ but on $v$ as well. Similarly, in Eq. (12),$^{3}$ the bivector $\mathcal{M(}x\mathcal{)}$ is decomposed into two vectors, the polarization vector $P(x)$ and the magnetization vector $M(x)$ and the unit time-like vector $u/c$ $$\mathcal{M}=P\wedge u/c+(MI)\cdot u/c^{2}. \label{M1}$$There is the rest frame for a medium, i.e., for $\mathcal{M}$, or $P$ and $M$, and therefore the vector $u$ in the decomposition (\[M1\]) is identified with bulk velocity vector of the medium in spacetime. Then, $P(x)$ and $M(x)$ are defined with respect to $u$ as$$P=\mathcal{M}\cdot u/c,\quad M=cI(\mathcal{M}\wedge u/c) \label{M2}$$and it holds that $P\cdot u=M\cdot u=0$. As in the case with $F$, it is visible from (\[M2\]) that $P$ and $M$ depend not only on $\mathcal{M}$ but on $u$ as well. Usually, only the velocity vector $u$ of the moving medium is taken into account, or the case $u=v$ is considered,$^{7}$ i.e., it is supposed that the observer frame is comoving with medium, or both decompositions (\[E2\]) and (\[M1\]) are made with the same velocity vector, either $u$ or $v$.$^{8}$ Such assumptions enable the introduction of the electromagnetic excitation bivector $\mathcal{H}$, and, by using (\[E2\]) and (\[M1\]), one finds the decomposition of $\mathcal{H}$ into the electric and magnetic excitations (other names of which are ‘electric displacement’ and ‘magnetic field intensity’)$$\mathcal{H=}D\wedge u/c+(IH)\cdot u/c^{2} \label{h1}$$(Eq. (14)$^{3}$), where, as usual, the electric displacement vector $D=\varepsilon _{0}E+P$ and the magnetic field intensity vector $H=(1/\mu _{0})B-M$ are introduced. In Ref. 3, inserting the decompositions of $F$ and $\mathcal{M}$ (Eqs. ([E2]{}) and (\[M1\]) here) into the field equation, the general form of the field equation for a magnetized and polarized moving medium with $E(x)$, $B(x)$, $P(x)$ and $M(x)$ (the Ampèrian form) is obtained in the form of Eq. (15), i.e., (16) (the vector part, with sources), or (18) in the ‘source representation’ $\partial \cdot \{\varepsilon _{0}[E\wedge v/c+(IB)\cdot v]\}=j^{(C)}/c-\partial \cdot \lbrack P\wedge u/c+(1/c^{2})(MI)\cdot u]$, and (17) (the trivector part, without sources) $\partial \wedge \lbrack E\wedge v/c+(IB)\cdot v]=0$. In contrast to all previous results, in the vector part, i.e., the part with sources, of the field equation there are two different velocities $u$ and $v$. From these field equations it is concluded$^{3}$ that in the field equation with sources, (16) or (18) in Ref. 3, *the usual Ampèr-Maxwell law and Gauss’s law are inseparably connected in one law.* Similarly, *Faraday’s law and the law that expresses the absence of magnetic charge are also inseparably connected in one law,* the field equation without sources, Eq. (17).$^{3}$ As shown in Secs. 6 and 7 in Ref. 3, this inseparability is an essential difference relative to the usual Maxwell equations with the 3-vectors. Next, we mention an important result regarding the usual formulation of electromagnetism, as in Ref. 9, which is presented in Ref. 10 and discussed in Refs. 11, 12 and 3. It is argued$^{10}$ that an individual vector has no dimension; the dimension is associated with *the dimension of its domain*. Hence, the time-dependent $\mathbf{E(r,}t\mathbf{)}$, $\mathbf{B(r,}t\mathbf{)}$, $\mathbf{D}(\mathbf{r},t)$ etc. cannot be the 3-vectors, since they are defined on the spacetime. Therefore, we use the term ‘vector’ for a geometric quantity, which is defined on the spacetime and which always has in some basis of that spacetime, e.g., the standard basis $\{\gamma _{\mu }\} $, four components (some of them can be zero). Note that vectors are usually called the 4-vectors. However, an incorrect expression, the 3-vector, will still remain for the usual $\mathbf{E(r,}t\mathbf{)}$, $\mathbf{B(r,}t\mathbf{)}$, $\mathbf{D}(\mathbf{r},t)$ etc.. Moreover, recently,$^{13-17,11} $ it is proved that, contrary to the general belief, the UT of the 3-vectors of the electric and magnetic fields, $\mathbf{E(r,}t\mathbf{)}$ and $\mathbf{B(r,}t\mathbf{)}$ respectively, see, e.g., Eqs. (11.148) and (11.149) in Ref. 9, differ from the LT (boosts) of the corresponding 4D quantities that represent the electric and magnetic fields. As explained,$^{13-17,11}$ the fundamental difference between the UT and the LT of the electric and magnetic fields is that in the UT, e.g., the components of the transformed $\mathbf{E}^{\prime }$ are expressed by the mixture of components of $\mathbf{E}$ and $\mathbf{B}$, and similarly for $\mathbf{B}^{\prime }$, Eq. (11.148).$^{9}$ The UT of the 3-vectors $\mathbf{E}$ and $\mathbf{B}$ are given, e.g., by Eqs. (11.149)$^{9}$ and they are $$\begin{aligned} \mathbf{E}^{\prime } &=&\gamma (\mathbf{E}+\mathbf{\beta \times }c\mathbf{B)-}(\gamma ^{2}/(1+\gamma ))\mathbf{\beta (\beta \cdot E),} \notag \\ \mathbf{B}^{\prime } &=&\gamma (\mathbf{B}-(1/c)\mathbf{\beta \times E)-}(\gamma ^{2}/(1+\gamma ))\mathbf{\beta (\beta \cdot B),} \label{JCB}\end{aligned}$$ where $\mathbf{E}^{\prime }$, $\mathbf{E}$, $\mathbf{\beta }$ and $\mathbf{B}^{\prime }$, $\mathbf{B}$ are all 3-vectors. All what is stated for the 3-vectors $\mathbf{E}$ and $\mathbf{B}$ and their UT holds in the same measure for the couple of the 3-vectors $\mathbf{P}$ and $\mathbf{M}$ and their UT $$\begin{aligned} \mathbf{P} &=&\gamma (\mathbf{P}^{\prime }+\mathbf{\beta \times M}^{\prime }/c\mathbf{)-}(\gamma ^{2}/(1+\gamma ))\mathbf{\beta (\beta \cdot P}^{\prime }\mathbf{),} \notag \\ \mathbf{M} &=&\gamma (\mathbf{M}^{\prime }-\mathbf{\beta \times }c\mathbf{P}^{\prime }\mathbf{)-}(\gamma ^{2}/(1+\gamma ))\mathbf{\beta (\beta \cdot M}^{\prime }\mathbf{),} \label{ps}\end{aligned}$$ see the equations, e.g., Eq. (4.2),$^{18}$ or Eqs. (18-68) - (18-71),$^{19}$ or (6.78a) and (6.81a),$^{20}$ etc. However, *the correct LT always transform the 4D algebraic object (vector, bivector) representing the electric field only to the electric field, and similarly for the magnetic field.* In order to explain this fundamental difference between the LT and the UT let us introduce the frame of ‘fiducial’ observers as the frame in which the observers who measure fields $E$ and $B$ are at rest. That frame with the standard basis $\{\gamma _{\mu }\}$ in it is called the $\gamma _{0}$-frame. In the $\gamma _{0}$-frame $v=c\gamma _{0}$ and therefore $E$ from (\[E1\]) becomes $E=F\cdot \gamma _{0}$ and it transforms under the active LT (Eqs. (10) and (11) in Ref. 11) in such a manner that both $F$ and the velocity of the observer $v=c\gamma _{0}$ are transformed by the LT, see Eq. (12)$^{11}$ ($E=F\cdot \gamma _{0}\longrightarrow E^{\prime }=R(F\cdot \gamma _{0})\widetilde{R}=(RF\widetilde{R})\cdot (R\gamma _{0}\widetilde{R})$). As explained in Ref. 11, *Minkowski, in Sec.* 11.6,$^{1}$*showed that both factors of the vector* $E$, *as the product of one bivector and one vector, has to be transformed by the LT.* However, it is worth mentioning that Minkowski in all other parts of Ref. 1 dealt with the usual 3-vectors $\mathbf{E}$, $\mathbf{B}$, $\mathbf{D}$, etc.. These correct LT give that $$E^{\prime }=E+\gamma (E\cdot \beta )\{\gamma _{0}-(\gamma /(1+\gamma ))\beta \}, \label{T1}$$Eq. (13).$^{11}$ *In the same way vector* $B$ *transforms and vectors* $P$, $M$ *as well*, *but for* $P$ *and* $M$ *the LT,* *like* (\[T1\]), *are the transformations from the rest frame of the medium* ($u=c\gamma _{0}$). For boosts in the direction $\gamma _{1}$ one has to take in that equation that $\beta =\beta \gamma _{1}$ (on the l.h.s. is vector $\beta $ and on the r.h.s. $\beta $ is a scalar). Hence, in the standard basis and when $\beta =\beta \gamma _{1}$ that equation becomes $$E^{\prime \nu }\gamma _{\nu }=-\beta \gamma E^{1}\gamma _{0}+\gamma E^{1}\gamma _{1}+E^{2}\gamma _{2}+E^{3}\gamma _{3}, \label{T2}$$what is Eq. (14).$^{11}$ The most important result is that *the electric field vector* $E$ *transforms by the LT again to the electric field vector* $E^{\prime }$*; there is no mixing with the magnetic field* $B$. The same happens with vectors $P$ and $M$. On the other hand, if in the transformation of $E=F\cdot \gamma _{0}$ only $F $ is transformed by the LT $R$, but not the velocity of the observer $v=c\gamma _{0}$ ($E=F\cdot \gamma _{0}\longrightarrow E_{F}^{\prime }=(RF\widetilde{R})\cdot \gamma _{0}$, Eq. (15)$^{11}$), then, in the standard basis and when $\beta =\beta \gamma _{1}$, one finds $$E_{F}^{\prime \nu }\gamma _{\nu }=E^{1}\gamma _{1}+\gamma (E^{2}-c\beta B^{3})\gamma _{2}+\gamma (E^{3}+c\beta B^{2})\gamma _{3}. \label{J2}$$ what is Eq. (17).$^{11}$ It is visible from the comparison of Eq. (\[J2\]) with Eq. (11.148)$^{9}$ that the transformations of components (taken in the standard basis) of $E_{F}^{\prime }$ are exactly the same as the transformations of $E_{x,y,z}$ from Eq. (11.148),$^{9}$ i.e., the components from (\[JCB\]). **3. Constitutive Relations for Moving Media in Geometric Terms** Let us consider the case of a simple medium with homogenous and isotropic electric and magnetic properties. In accordance with Sec. 2 we formulate the constitutive relations for moving media using the generalized magnetization-polarization bivector $\mathcal{M}$ and the electromagnetic field bivector $F$$$\mathcal{M}\cdot u=\varepsilon _{0}\chi _{E}F\cdot u, \label{cr1}$$$$(I\mathcal{M})\cdot u=(\chi _{B}/\mu _{0}c^{2})u\cdot (IF) \label{cr2}$$and the electric and magnetic susceptibility ($\chi _{E}$, $\chi _{B}$). Using the decompositions of $F$ (\[E2\]) and $\mathcal{M}$ (\[M1\]) we get from (\[cr1\]) how the polarization vector $P(x)$ depends on $E$, $B$ and $u$, $v$,$$P=(\varepsilon _{0}\chi _{E}/c)\{(1/c)[(u\cdot v)E-(u\cdot E)v]+(u\wedge v\wedge B)I\} \label{P}$$and from (\[cr2\]) how the magnetization vector $M(x)$ depends on $E$, $B$ and $u$, $v$,$$M=\varepsilon _{0}\chi _{B}\{[(u\cdot v)B-(u\cdot B)v]-(1/c)(u\wedge v\wedge E)I\}. \label{M}$$Observe that $P$ in (\[P\]) and $M$ in (\[M\]) contain both velocities $u $, the bulk velocity vector of the medium, and $v$, the velocity vector of the observers. The relations (\[P\]) and (\[M\]) are the basic relations that are obtained in this paper. In our approach, the relations (\[P\]) and (\[M\]) replace the constitutive relations with the 3-vectors (\[pl\]) and (\[mg\]), which are equivalent to Minkowski’s equations (\[de\]). The equations (\[pl\]), (\[mg\]) and (\[de\]) are given below in Sec. 5. In the special case when $v=u$, an observer frame comoving with medium, ([P]{}) and (\[M\]) yield $$P=\varepsilon _{0}\chi _{E}E,\qquad M=(\chi _{B}/\mu _{0})B, \label{pm}$$which are the familiar linear constitutive laws for the case of an electrically neutral, isotropic, non-dispersive, polarizable medium at rest. Introducing the relative permittivity $\varepsilon _{r}$, $\varepsilon _{r}=1+\chi _{E}$, and the relative permeability $\mu _{r}$, $\mu _{r}=1/(1-\chi _{B})$, the constitutive laws (\[pm\]) can be written as $P=\varepsilon _{0}(\varepsilon _{r}-1)E$, $M=(1/\mu _{0})(1-1/\mu _{r})B$. In this case ($v=u$) the excitations $D$, $D=\varepsilon _{0}E+P$ and $H$, $H=(1/\mu _{0})B-M$ can be introduced, as in Sec. 2, and the rest frame constitutive relations take the usual forms $$D=\varepsilon E,\qquad H=(1/\mu )B, \label{dh1}$$where $\varepsilon =\varepsilon _{0}\varepsilon _{r}$ and $\mu =\mu _{0}\mu _{r}$. The last term in (\[P\]) and (\[M\]) describes the magnetoelectric effects in a moving dielectric. According to the last term in (\[P\]) a moving dielectric becomes electrically polarized when placed in a magnetic field, the Wilsons’ experiment.$^{21}$ Similarly, according to the last term in (\[M\]) a moving dielectric becomes magnetized when it is placed in an electric field, Röntgen’s experiment.$^{22}$ **4. Constitutive Relations for Moving Media in the Standard Basis** The equations (\[cr1\]) - (\[dh1\]) are all coordinate-free relations. In order to compare them with some usual formulations we have to represent all 4D geometric quantities from them in the standard basis $\left\{ \gamma _{\mu }\right\} $. Then, in the $\left\{ \gamma _{\mu }\right\} $ basis, Eq. (\[P\]) becomes$$P^{\mu }\gamma _{\mu }=(\varepsilon _{0}\chi _{E}/c)[(1/c)(E^{\mu }v^{\nu }-E^{\nu }v^{\mu })+\varepsilon ^{\mu \nu \alpha \beta }v_{\alpha }B_{\beta }]u_{\nu }\gamma _{\mu }, \label{po}$$whereas (\[M\]) takes the form $$M^{\mu }\gamma _{\mu }=\varepsilon _{0}\chi _{B}[(B^{\mu }v^{\nu }-B^{\nu }v^{\mu })+(1/c)\varepsilon ^{\mu \nu \alpha \beta }E_{\alpha }v_{\beta }]u_{\nu }\gamma _{\mu }. \label{ma}$$ We shall examine Eqs. (\[po\]) and (\[ma\]) taking that the laboratory frame, the $S$ frame, is the $\gamma _{0}$-frame ($v=c\gamma _{0}$, $v^{\mu }=(c,0,0,0)$, $E^{0}=B^{0}=0$) in which the material medium, the $S^{\prime } $ frame, is moving with velocity $u$, $u^{\nu }=(\gamma _{u}c,\gamma _{u}U^{1},\gamma _{u}U^{2},\gamma _{u}U^{3})$, $U^{k}$ are the components of the 3-velocity $\mathbf{U}$ and $\beta _{u}=\left\vert \mathbf{U}\right\vert /c$. Then, in the laboratory frame, Eq. (\[po\]) becomes$$P=\varepsilon _{0}\chi _{E}\gamma _{u}[E^{i}(U^{i}/c)\gamma _{0}+(E^{i}+\varepsilon ^{0ijk}U_{j}B_{k})\gamma _{i}]. \label{pc}$$It can be seen from (\[pc\]) that, e.g., if $E^{\mu }=(0,0,0,0)$, $B^{\mu }=(0,0,0,-B^{3})$, $u^{\mu }=(\gamma _{u}c,\gamma _{u}U^{1},0,0)$, the components of the polarization are $$P^{\mu }=(0,0,P^{2}=\varepsilon _{0}\chi _{E}\gamma _{u}U^{1}B^{3},0); \label{pwi}$$these components correspond to the ‘translational’ version of Wilsons’ experiment. As stated on page 13,$^{23}$ the magnetoelectric effect ‘in a moving dielectric is of course anisotropic in the sense that the polarization depends upon an applied field which is perpendicular to the direction of motion and this polarization is then perpendicular to both the applied field and the direction of motion.’ Comparing $P^{2}$ from (\[pwi\]) with $P_{y}$ from Eq. (1.14)$^{23}$ (see also the term $\chi _{(em)}^{\alpha \beta }$ in Fig. 3.3 and Eq. (2.9)$^{23}$), we see that for the above conditions (the electric field is absent) these two expressions differ only in the $\gamma _{u}$ factor; $P^{2}$ contains $\gamma _{u}$ whereas $P_{y}$ contains $\gamma _{u}^{2}$. However, it is worth mentioning that the complete $P_{y}$ (both, the 3-vectors of the electric and magnetic fields exist) from Eq. (1.14),$^{23}$ or the components of the 3-vector $\mathbf{P}$ from Eq. (2.9) and Fig. 3.3,$^{23}$ can be compared only with the spatial components of (\[pc\]), but $P $ in (\[pc\]) contains the temporal component as well. Again, the *spatial components* of (\[pc\]) and the complete $P_{y}$ differ only in the $\gamma _{u}$ factor. Similarly, from (\[ma\]), we find $$M=(\chi _{B}/\mu _{0})\gamma _{u}[B^{i}(U^{i}/c)\gamma _{0}+(B^{i}-\varepsilon ^{0ijk}U_{j}E_{k}/c^{2})\gamma _{i}]. \label{ma1}$$ In the case of low velocities of the medium, $\beta _{u}\ll 1$, the equations (\[pc\]) and (\[ma1\]) are almost unchanged to first order of $\beta _{u}$; only it is taken that $\gamma _{u}\simeq 1$. Observe that in (\[pc\]) and (\[ma1\]) the terms with $\gamma _{0}$ are comparable to the terms containing $U_{j}B_{k}\gamma _{i}$ and $(1/c^{2})U_{j}E_{k}\gamma _{i}$, respectively. Thus, to first order of $\beta _{u}$, the terms $P^{0}$ and $M^{0}$ *cannot be neglected* relative to the complete $P^{i}$ and $M^{i} $, respectively. The constitutive relations (\[pc\]) and (\[ma1\]) significantly differ from all previous constitutive equations for such simple moving media. They will be compared with the usual formulations with the 3-vectors. **5. Comparison with Minkowski’s Constitutive Relations** **and with Their Derivations** Let us examine some of the usual derivations of the constitutive relations for moving media. As already mentioned, almost the entire physics literature deals with the electromagnetic excitation tensor $\mathcal{H}$, i.e. with the electric $D$ and magnetic $H$ excitations and the constitutive relations refer to the connections between them and $F$, i.e. $E$ and $B$, respectively. Minkowski$^{1}$ was the first who derived the constitutive relations for the 3-vectors $$\begin{aligned} \mathbf{D}+(1/c^{2})\mathbf{U}\times \mathbf{H} &=&\varepsilon (\mathbf{E}+\mathbf{U}\times \mathbf{B}) \notag \\ \mathbf{B}-(1/c^{2})\mathbf{U}\times \mathbf{E} &=&\mu (\mathbf{H}-\mathbf{U}\times \mathbf{D}), \label{de}\end{aligned}$$Eqs. (C) and (D) in Sec. 8.$^{1}$ From that time, the equations (\[de\]) are considered to be the fundamental constitutive equations for moving media in the whole physics community. They are subsequently derived in different ways and used in numerous papers and textbooks on the electromagnetism. **5.1 Comparison with Minkowski’s constitutive relations that are** **obtained by means of the 3-vectors**$^{24}$ **and** **in the covariant approach**$^{25}$**** One way of the derivation of Eqs. (\[de\]) is presented in Ref. 24. There, these equations, i.e. Eqs. (7),$^{24}$ are derived using Minkowski’s crucial hypothesis that the relations with the 3-vectors $\mathbf{D}=\varepsilon \mathbf{E}$, $\mathbf{H}=(1/\mu )\mathbf{B}$, which correspond to (\[dh1\]), retain their form in the moving frame $S^{\prime } $ with the same $\varepsilon $ and $\mu $, i. e., $\mathbf{D}^{\prime }=\varepsilon \mathbf{E}^{\prime }$, $\mathbf{H}^{\prime }=(1/\mu )\mathbf{B}^{\prime }$. Then, in Ref. 24, *the UT* (\[JCB\]) *and the similar UT for* $\mathbf{D}^{\prime }$ *and* $\mathbf{H}^{\prime }$ (Eqs. (3)$^{24}$) *are used to obtain Minkowski’s constitutive equations* (\[de\]). Furthermore, in Ref. 24, the constitutive relations, which explicitly express $\mathbf{D}$ and $\mathbf{H}$ in terms of $\mathbf{E}$ and $\mathbf{B}$, are derived (Eqs. (10)$^{24}$) from (\[de\]). Introducing $\mathbf{D}=\varepsilon _{0}\mathbf{E}+\mathbf{P}$ and $\mathbf{H}=(1/\mu _{0})\mathbf{B}-\mathbf{M}$ into Eqs. (10)$^{24}$ we find the constitutive equations which explicitly express $\mathbf{P}$ as a function of $\mathbf{E}$ and $\mathbf{B}$ $$\begin{aligned} \mathbf{P} &=&\gamma ^{2}\varepsilon _{0}\{\chi _{E}[\mathbf{E}-c^{-2}\mathbf{U}(\mathbf{U\cdot E})+\mathbf{U}\times \mathbf{B}] \notag \\ &&+\chi _{B}[\mathbf{U\times (B}-c^{-2}\mathbf{U}\times \mathbf{E)]}\} \label{pl}\end{aligned}$$and also $\mathbf{M}$ in terms of $\mathbf{E}$ and $\mathbf{B}$ $$\begin{aligned} \mathbf{M} &=&(\gamma ^{2}/\mu _{0})\{\chi _{B}[\mathbf{B}-c^{-2}\mathbf{U}(\mathbf{U\cdot B})-c^{-2}\mathbf{U}\times \mathbf{E}] \notag \\ &&-c^{-2}\chi _{E}[\mathbf{U\times (E}+\mathbf{U}\times \mathbf{B)}]\}. \label{mg}\end{aligned}$$The equations (\[pl\]) and (\[mg\]) are equivalent to Minkowski’s equations (\[de\]). Introducing the relative permittivity $\varepsilon _{r}$ and the relative permeability $\mu _{r}$ instead of the susceptibilities $\chi _{E}$ and $\chi _{B}$ the equations (\[pl\]) and (\[mg\]) become$$\begin{aligned} \mathbf{P} &=&\varepsilon _{0}(\varepsilon _{r}-1)\mathbf{E}+\gamma ^{2}\varepsilon _{0}(\varepsilon _{r}-1/\mu _{r})\mathbf{U\times (B}-c^{-2}\mathbf{U}\times \mathbf{E),} \notag \\ \mathbf{M} &=&\mu _{0}^{-1}(1-1/\mu _{r})\mathbf{B}-\gamma ^{2}\varepsilon _{0}(\varepsilon _{r}-1/\mu _{r})\mathbf{U\times (E}+\mathbf{U}\times \mathbf{B).} \label{mp}\end{aligned}$$and they are, in the same way as Eqs. (\[pl\]) and (\[mg\]), equivalent to Minkowski’s equations (\[de\]). Similarly, in Ref. 25, the relations (\[de\]) (Eqs. (76.9)$^{25}$) are obtained by the covariant generalization (only components, implicitly taken in the standard basis) of the relations with the 3-vectors $\mathbf{D}=\varepsilon \mathbf{E}$, $\mathbf{H}=(1/\mu )\mathbf{B}$, which correspond to our Eq. (\[dh1\]). The covariant generalizations from Ref. 25 are $\mathcal{H}^{\lambda \mu }u_{\mu }=\varepsilon F^{\lambda \mu }u_{\mu }$ ((76.7)$^{25}$) and $^{\ast }F^{\lambda \mu }u_{\mu }=\mu ^{\ast }\mathcal{H}^{\lambda \mu }u_{\mu }$ ((76.8)$^{25}$), where $^{\ast }F^{\lambda \mu }$ and $^{\ast }\mathcal{H}^{\lambda \mu }$ are the dual tensors, i.e., only the components ($^{\ast }F^{\alpha \beta }=(1/2)\varepsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta }$). Then, Minkowski’s constitutive equations (\[de\]) with the 3-vectors are derived by *Minkowski’s identifications,*$^{1}$* of components of* $F^{\alpha \beta }$ *with components of the 3-vectors* $\mathbf{E}$ *and* $\mathbf{B}$ ($E^{i}=F^{i0}$, $B^{i}=(1/2c)\varepsilon ^{ijk0}F_{jk}$, see, e.g., Eq. (11.137)$^{9}$) *and by the similar identifications* *for* $\mathcal{H}^{\alpha \beta }$ *and the 3-vectors* $\mathbf{D}$ *and* $\mathbf{H}$ ($D^{i}=\mathcal{H}^{i0}$, $H^{i}=(c/2)\varepsilon ^{ijk0}\mathcal{H}_{jk}$). However, as discussed, e.g. in Sec. 2,$^{11}$ the components are not the whole physical quantity. The mentioned identifications of components are synchronization dependent and the UT (\[JCB\]) (and the similar ones for $\mathbf{D}^{\prime }$ and $\mathbf{H}^{\prime }$) significantly differ from the LT (\[T1\]). On the other hand, our equations (\[cr1\]) and (\[cr2\]) deal with coordinate-free, 4D geometric quantities. Instead of Minkowski’s identifications of components, and the same ones in Ref. 25, the mathematically correct decompositions of $F$ (\[E2\]) and $\mathcal{M}$ (\[M1\]) are used for the derivation of the constitutive equations (\[P\]) and (\[M\]) (and, in the standard basis, (\[po\]) and (\[ma\])) from (\[cr1\]) and (\[cr2\]), respectively. As mentioned above, the equations (\[pl\]) and (\[mg\]), or (\[mp\]), which are equivalent to (\[de\]), can be compared with Eqs. (\[pc\]) and (\[ma1\]). There are important differences between them. 1\) The equations (\[pc\]) and (\[ma1\]) are with correctly defined 4D quantities that properly transform under the LT, whereas it is not the case with Eqs. (\[pl\]) and (\[mg\]), or (\[mp\]), with the 3-vectors that transform according to the UT. 2\) Furthermore, (\[pc\]) and (\[ma1\]) contain the term with $\gamma _{0} $, which cannot exist in the approach with the 3-vectors, i.e. in ([pl]{}) and (\[mg\]), or (\[mp\]). 3\) Also, in (\[pc\]), the polarization vector $P$ contains only the electric susceptibility $\chi _{E}$ and not $\chi _{B}$, whereas, in ([pl]{}), the polarization 3-vector $\mathbf{P}$ contains both susceptibilities $\chi _{E}$ and $\chi _{B}$, (or, in (\[mp\]), both, $\varepsilon _{r}$ and $\mu _{r}$). Similarly, in (\[ma1\]), there is only $\chi _{B}$ and not $\chi _{E}$, whereas, in (\[mg\]), there are both susceptibilities $\chi _{B}$ and $\chi _{E}$, (or, in (\[mp\]), both, $\mu _{r}$ and $\varepsilon _{r}$ ). It can be seen from the derivation of (\[pl\]) and (\[mg\]), or (\[mp\]), that both susceptibilities appear in the expressions for $\mathbf{P}$ (\[pl\]) and $\mathbf{M}$ (\[mg\]) because the UT (\[JCB\]) and the similar ones for $\mathbf{D}^{\prime }$ and $\mathbf{H}^{\prime }$ (Eqs. (3)$^{24}$), or, equivalently, the UT (\[ps\]), are used and in them, e.g., in (\[JCB\]), $\mathbf{E}^{\prime }$ *is expressed by the mixture of* $\mathbf{E}$ *and* $\mathbf{B}$, and similarly, in (\[ps\]), $\mathbf{P}^{\prime }$ *is expressed by the mixture of* $\mathbf{P}$ *and* $\mathbf{M}$. **5.2. Comparison of the low velocity limits** These differences remain in the low velocity limit, $\beta _{u}\ll 1$, as well. That limit is already examined for vectors $P$ and $M$ from (\[pc\]) and (\[ma1\]), respectively. Only $\gamma _{u}\simeq 1$ is taken in (\[pc\]) and (\[ma1\]). In Ref. 24, the low velocity limit, i.e., the quasi-static approximation, is obtained in the same way by putting $\gamma _{u}\simeq 1$ in Eqs. (10) for $\mathbf{D}$ and $\mathbf{H}$ expressed in terms of $\mathbf{E}$ and $\mathbf{B}$, which led to Eqs. (11).$^{24}$ They, Eqs. (11),$^{24}$ are commonly used in literature. In the formulation with the 3-vectors $\mathbf{P}$ and $\mathbf{M}$ the equations (\[mp\]) are derived from Eqs. (10).$^{24}$ Therefore, in that approximation, the usual 3-vectors $\mathbf{P}$ and $\mathbf{M}$ are given by (\[mp\]) but with $\gamma _{u}\simeq 1$. This means that again both $\varepsilon _{r}$ and $\mu _{r}$, i.e., both susceptibilities $\chi _{E}$ and $\chi _{B}$, appear in the quasi-static approximation in the expressions for $\mathbf{P}$ and $\mathbf{M}$. In Ref. 24, it is argued that such a procedure with $\gamma _{u}\simeq 1$ does not give a correct quasi-static approximation, because the obtained equations, (11),$^{24}$ ‘do not obey the group additivity property.’ Hence, according to Ref. 24, the equations (11)$^{24}$ have to be replaced by two well-defined Galilean limits of Minkowski’s constitutive equations ((\[de\]) here), the magnetic and electric limits, i.e. with two sets of low-velocity formulae. These two sets are Eqs. (12) and (13) in Ref.24 and it is stated there: ‘Both sets (12) and (13) do form a group.’ However, the UT (\[JCB\]) and (\[ps\]) and the similar UT for $\mathbf{D}^{\prime }$ and $\mathbf{H}^{\prime }$ (Eqs. (3)$^{24}$) are not the LT and therefore constitutive equations with the 3-vectors, (\[de\]), or Eqs. (10),$^{24}$, and also (\[pl\]) and (\[mg\]), or (\[mp\]), here, are not the *relativistic* constitutive equations and also Eqs. (12) and (13)$^{24}$ are not a quasi-static approximation of the relativistically correct constitutive relations. Thus, as stated above, all differences from Sec. 5.1 remain in the low velocity limit as well. In the special case of moving ($\beta _{u}\ll 1$) nonmagnetic ($\mu _{r}=1$, i.e. $\chi _{B}=0$) media the 3D polarization $\mathbf{P}$ from (\[mp\]), to first order of $\beta _{u}$, becomes $$\mathbf{P=}\varepsilon _{0}\chi _{E}(\mathbf{E}+\mathbf{U}\times \mathbf{B}). \label{pc1}$$ The equation (\[pc1\]) is Eq. (9-19) (2),$^{19}$ or the last equation in Problem 6.8.$^{20}$ But, even for $\beta _{u}\ll 1$, $P$ from (\[pc\]) will have the same form for both cases $\chi _{B}\neq 0$ and $\chi _{B}=0$, because it does not depend on $\chi _{B}$ ($P\simeq \varepsilon _{0}\chi _{E}[E^{i}(U^{i}/c)\gamma _{0}+(E^{i}+\varepsilon ^{0ijk}U_{j}B_{k})\gamma _{i}]$). As already stated, to first order of $\beta _{u}$, $P^{0}$ cannot be neglected relative to $P^{i}$, which means that we have not an equation that would correspond to (\[pc1\]). **5.3. Comparison with constitutive equations that are obtained** **by means of exterior forms**$^{2}$**** It is worth noting that Minkowski’s constitutive equations ([de]{}) are also obtained in Ref. 2 using exterior calculus, i.e., with the use of abstract 4D geometric quantities, Eqs. (E.4.28) and (E.4.29).$^{2}$ At that place (p. 353) Hehl and Obukhov stated: ‘Originally, the constitutive relations (E.4.28), (E.4.29) were derived by Minkowski with the help of Lorentz transformations for the case of a flat space time and a uniformly moving medium. We stress, however, that the Lorentz group never entered the scene in our derivation above.’ It can be concluded from these statements that the authors$^{2}$ believe, as all others, that the UT ([JCB]{}) are the relativistically correct LT. However, as discussed in Sec. 2, the LT are given by Eq. (\[T1\]) and not by (\[JCB\]). Furthermore, in Ref. 2, the expressions for the polarization $P$ and the magnetization $M$, the equations (E.4.30) and (E.4.31), respectively, are derived from the constitutive relations (E.4.28), (E.4.29). The expressions for $P$ and $M,$ (E.4.30) and (E.4.31), respectively, from Ref. 2, are very similar to the equations with the 3-vectors, (\[pl\]) and (\[mg\]), respectively, which are obtained from (\[de\]). Indeed, all quantities in the constitutive relations (E.4.28), (E.4.29) and in the expressions for $P$ and $M$, (E.4.30) and (E.4.31), respectively, from Ref. 2, are in the 3D subspace, i.e., like the usual 3-vectors. (Observe that $v$ that enters into these relations is the 3-velocity 1-form, (E.4.27), or the 3-velocity vector. Also, the Hodge star operators in these expressions are defined by the 3-space metrics of the laboratory and material foliations.) This means that these results$^{2}$ also significantly differ from the constitutive relations (\[P\]) and (\[M\]) in which all quantities are the 4D geometric quantities and there is no the space-time split. Let us describe in more detail the calculation from Ref. 2, which led to the mentioned results.$^{2}$ In Ref. 2, Minkowski’s constitutive relations (E.4.28), (E.4.29) and the expressions for $P$ and $M$, (E.4.30) and (E.4.31), respectively, are derived using (1+3) - splitting of spacetime both in the laboratory frame and in the frame of moving macroscopic matter, the material frame. The starting relations in that derivation are the decompositions ((1+3) - splitting) of the electromagnetic excitation tensor $\mathcal{H}$ and of $F$ in the laboratory frame (unprimed quantities), ($\mathcal{H=}-H\wedge dt+D$, $F=E\wedge dt+B$) Eq. (E.4.14), and in the material frame (primed quantities), ($\mathcal{H=}-H^{\prime }\wedge dt^{\prime }+D^{\prime }$, $F=E^{\prime }\wedge dt^{\prime }+B^{\prime }$) Eq. (E.4.15). (In both equations our notations is used.) There is an assumption in connection with the decompositions (E.4.14) and (E.4.15) in Ref. 2. It is: ‘Clearly, we preserve the same symbols $\mathcal{H}$ (our notation) and $F$ on the left-hand sides of (E.4.14) and (E.4.15) because these are just the same physical objects. In contrast, the right-hand sides are of course different, hence we use primes.’ From the mathematical point of view there are no reasons for such argumentations. If $H$, $dt$, $D$, $E$, $B$ are transformed then $\mathcal{H} $ and $F$ have to be transformed as well, as can be concluded going to a uniformly moving medium, i.e., when the LT are applicable. Let us explain these assertions in more detail. *There is no mathematical procedure by which in one equation some parts of it are transformed and the other parts are not. It is not possible that there is one law for the transformations of some 4D geometric quantities and another law for the transformations of the other quantities.* As mentioned, e.g., in Ref. 11, *any multivector* $M$ *transforms by the active LT in the same way*, Eq. (11),$^{11}$ $M\rightarrow M^{\prime }=RM\widetilde{R}$, where the Lorentz boost $R$ is given by Eq. (10).$^{11}$ Hence, if $R$ is applied to (\[E2\]), $R(F=E\wedge v/c+(IcB)\cdot v/c)\widetilde{R}$, then $F$ *is also transformed*, together with $E$, $B$ and $v$. Thus, the assertion that $F$ is unchanged under the active LT is without any mathematical justification. In addition, let us compare the relations (E.4.14) and (E.4.15) with the consideration from Ref. 26. In Sec. 4 ‘Premetric electrodynamics in exterior calculus’ in Ref. 26, it is argued that the decomposition of $\mathcal{H}$, Eq. (31),$^{26}$ if put in matrix form, yields the identification (6)$_{1}$ in Ref. 26. Furthermore, the field strength $F$ is decomposed in the same way as $\mathcal{H}$, i.e. ‘into two pieces: one along the 1-dimensional time $t$ and another one embedded in 3-dimensional space ($a,b=1,2,3.$).’ Thus, in the notation,$^{26}$ $F=E\wedge dt+B=E_{a}dx^{a}\wedge dt+(1/2)B_{ab}dx^{a}\wedge dx^{b}$, Eq. (34).$^{26}$ If put in matrix form, it yields the usual identification (6)$_{2}$ in Ref. 26, i.e., in our notation, $E^{i}=F^{i0}$, $B^{i}=(1/2c)\varepsilon ^{ijk0}F_{jk}$. The decompositions (31) and (34)$^{26}$ are the same as the decompositions (E.4.14).$^{2}$ As already stated, in the material frame, the same decompositions of $\mathcal{H}$ and $F$ ($F=E^{\prime }\wedge dt^{\prime }+B^{\prime }$) are performed, (E.4.15),$^{2}$ and they yield the usual identifications (6)$_{1}$ and (6)$_{2}$ in Ref. 26, but now with primes, $E^{\prime i}=F^{\prime i0}$, $B^{\prime i}=(1/2c)\varepsilon ^{ijk0}F_{jk}^{\prime }$. This discussion reveals that the (1+3) - splitting of spacetime, i.e., *the usual identifications* (6)$_{1}$, (6)$_{2}$ *in both frames*, are the real cause that, although the authors$^{2}$ worked with exterior forms in the 4D spacetime, they obtained the same results for Minkowski’s constitutive relations and for $D$ and $H$ ((E.4.25) and (E.4.26)$^{2}$), i.e., for $P$ and $M$, as those that were found by means of the 3-vectors and their UT, e.g., in Ref. 24. However, as already discussed, e.g., in Refs. 27 and 11, and also in Ref. 12, *the (1+3) - splitting of spacetime and the usual identifications of components of, e.g.,* $F^{\alpha \beta }$ *(implicitly taken in the standard basis) with components of the 3D* $\mathbf{E}$ *and* $\mathbf{B}$ *in both frames are synchronization dependent and they are meaningless in the ‘r’ synchronization.* We note that *different synchronizations are nothing else than different conventions and physics must not depend on conventions.* (In Ref. 26, Hehl (Eq. (6)$_{2}$) quotes Minkowski’s identifications from Sec. 3$^{1}$ and states: ‘We stress that the identifications in (6) are premetric, they are valid on any (well-behaved) differential manifold that can be split locally into time and space.’ This means that Hehl considers that 1+3 decomposition, i.e., the space-time split, is also - premetric. But, if a nonstandard basis, the $\{r_{\mu }\}$ basis with the ‘r’ synchronization is used, i.e., when the appropriate metric is used, then it is not possible to make the usual identifications, that is, 1+3 decomposition, i.e., the space-time split. That metric is discussed in, e.g., the text below Eq. (14)$^{12}$: ‘.. the components $g_{\mu \nu ,r}$ of the metric tensor $g_{ab}$ are $g_{ii,r}=0$, and all other components are $=1$.’ Hence, both, the usual identifications (Eqs. (6)$^{26}$), and 1+3 decomposition, i.e., the space-time split, are meaningful *only* when the Minkowski metric, e.g., $diag(1,-1,-1,-1)$, is used. Thus, these identifications depend on the chosen metric and therefore they are not premetric. (For the $\{r_{\mu }\}$ basis see, for example, Refs. 27, 12.) In the $\{r_{\mu }\}$ basis, it holds that $F_{r}^{10}=E^{1}+cB^{3}-cB^{2}$. The relation for $F_{r}^{10}$ shows that *the components have no definite physical meaning* since they are dependent on the chosen synchronization. Only in the $\{\gamma _{\mu }\}$ basis does it hold that $E^{i}=F^{i0}$, $P^{i}=\mathcal{M}^{i0}$, etc. According to that, the usual 3-vectors $\mathbf{E}$, $\mathbf{B}$, $\mathbf{D}$, $\mathbf{H}$, $\mathbf{P}$, $\mathbf{M}$, etc., where, e.g., $\mathbf{P=}\mathcal{M}^{10}\mathbf{i}+\mathcal{M}^{20}\mathbf{j}+\mathcal{M}^{30}\mathbf{k}$ ($\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are the unit 3-vectors) *have no definite physical meaning,* since the components $\mathcal{M}^{i0}$ are dependent on the chosen synchronization.) Comparing, for example, the decompositions of $F$ from Ref. 2 and 26 (the above quoted Eq. (34)$^{26}$) with our decomposition $F=E\wedge v/c+(IcB)\cdot v/c$, Eq. (\[E2\]), i.e., in some arbitrary basis $\left\{ e_{\mu }\right\} $ (it can be the standard basis $\left\{ \gamma _{\mu }\right\} $, or the $\left\{ r_{\mu }\right\} $ basis with the ‘r’ synchronization, etc.) $F=(\delta _{\quad \mu \nu }^{\alpha \beta }E_{e}^{\mu }v_{e}^{\nu }+c\varepsilon ^{\alpha \beta \mu \nu }v_{e,\mu }B_{e,\nu })e_{\alpha }\wedge e_{\beta }$ ($\alpha ,\beta ,\mu ,\nu =0,1,2,3$ and $E_{e}^{\mu }$, $B_{e}^{\mu }$, $v_{e}^{\mu }$ are the components in the $\left\{ e_{\mu }\right\} $ basis), we see that there are important differences between them. All quantities, including $v$, in our relations are the 4D quantities and there is no (1+3) - splitting of spacetime, which means that they are more general than the decompositions from Ref. 2 and 26. If, for example, the magnetization 1-form $M$, Eq. (E.4.31),$^{2}$ would be written in the $\left\{ \gamma _{\mu }\right\} $ basis then it would contain in the material frame *and in the laboratory frame* only spatial components and not the time component in contrast to the relation (\[ma1\]) (and (\[pc\])). Observe also that relations (\[P\]) and (\[M\]), or in the $\left\{ \gamma _{\mu }\right\} $ basis (\[po\]) and (\[ma\]), contain both the velocity of the observer $v$ and the velocity of the considered medium $u$, whereas there is only one velocity, the velocity of the considered medium, in the equations (E.4.25)-(E.4.31).$^{2}$ Furthermore, the obvious similarity of the results with exterior forms$^{2}$ and the usual results with the 3-vectors and their UT, e.g. from Ref. 18 (or 24), can be nicely seen comparing Sec. E.4.5 under the title ‘The experiments of Röntgen and Wilson$\And $Wilson’ in Ref. 2 and Secs. 5.2 and 5.3 in Ref. 18. It is visible from Eq. (E.4.49) and Fig. E.4.2$^{2}$ that in the matter-free region, region (2), there is only an electric field $E_{(2)}$, whereas there are *both* the electric field $E_{(1)}$ and *the induced magnetic field* $B_{(1)}$ in the slab moving with *constant* velocity. Completely the same result for that case is obtained in the third equation in (5.18) in Sec. 5.2,$^{18}$ where it is exclusively dealt with the 3-vectors and their UT. The same conclusion holds for the comparison of the treatments of the Wilsons’ experiment, Sec. E.4.5$^{2}$ and Sec. 5.3.$^{18}$ From the above discussion it is visible that either by using the 3-vectors and their UT, or by using the modern mathematical language, the exterior forms, always, in all previous treatments, it is obtained that, e.g. the electric field in one frame is ‘seen’ as slightly changed electric field *and* *the induced magnetic field* in the frame relatively moving with *constant* velocity. The same holds for the polarization and the magnetization and also for the electric and magnetic excitations. **6. The Physical Explanation of the Magnetoelectric Effect** **in Moving Media** The consideration in the previous sections shows that there are important differences in the form of Minkowski’s constitutive relations ([de]{})**,** or equivalently the constitutive relations for $\mathbf{P}$ (\[pl\]) and $\mathbf{M}$ (\[mg\]), i.e., (\[mp\]), and the constitutive relations for vectors $P$ (\[P\]) and $M$ (\[M\]). However, there is also a significant difference in physical interpretation of these constitutive relations. Particularly, this refers to the interpretation of the magnetoelectric effect for moving media. In previous approaches it is considered that the term which describes how the magnetic field influences the polarization ($\gamma ^{2}\varepsilon _{0}(\varepsilon _{r}-1/\mu _{r})\mathbf{U\times B}$ in Eq. (\[mp\])) and the term which describes how the electric field influences the magnetization ($-\gamma ^{2}\varepsilon _{0}(\varepsilon _{r}-1/\mu _{r})\mathbf{U\times E} $ in Eq. (\[mp\])) are determined by the UT (\[JCB\]) for $\mathbf{E}$ and $\mathbf{B}$. Namely, according to the UT (\[JCB\]), in the rest frame of a moving medium the magnetic field 3-vector from the laboratory frame is ‘seen’ as a slightly changed magnetic field 3-vector and an *induced electric field 3-vector*. That induced electric field 3-vector interacts with the electric dipole moments 3-vectors by the interaction term $\mathbf{E}\cdot \mathbf{d}$ giving the polarization $\mathbf{P}$ as a 3-vector. Similarly, in the rest frame of a moving medium the electric field from the laboratory frame is ‘seen’ as a slightly changed electric field and an *induced magnetic field*. That induced magnetic field interacts with the magnetic dipole moments by the interaction term $\mathbf{B}\cdot \mathbf{m}$ giving the magnetization $\mathbf{M}$ (all are 3-vectors). In my opinion, it is very strange that, e.g., the magnetic field from a permanent magnet is ‘seen’ as an *induced electric field* in a relatively moving frame. What is the physical reason which causes that one field becomes another field in a relatively moving frame? How can it be that a mere uniform moving transforms one field to another one? It cannot be that the transformations describe the physics, but, on the contrary, the physics has to describe how to get the correct transformations of the fields. If the electric and magnetic fields are interpreted as vectors defined on the 4D spacetime then it is very natural to have that a magnetic field vector remains the magnetic field vector in a relatively moving frame. There is no physical cause that can change one field to another one in a relatively moving frame. According to the LT (\[T1\]), an electric field vector transforms again to the electric field vector and similarly for a magnetic field vector. Hence, the explanation for the magnetoelectric effect in moving media is completely different in our approach. Instead of dealing with the electric and magnetic dipole moments 3-vectors $\mathbf{d}$ and $\mathbf{m}$ we deal with the dipole moment bivector $D$, as a primary quantity for dipole moments, which does have a definite physical reality. The decomposition of $D$ into the electric and magnetic dipole moments vectors $d$ and $m$, respectively, and the unit time-like vector $u/c $, and the expressions for $d$ and $m$, which are obtained from $D$ and determined relative to $u$ are$$\begin{aligned} D &=&d\wedge u/c+(mI)\cdot u/c^{2}, \notag \\ d &=&D\cdot u/c,\quad m=cI(D\wedge u/c); \label{di}\end{aligned}$$compare with (\[M1\]) and (\[M2\]). In this case the vector $u$ is the bulk velocity vector of the medium as in (\[M1\]) and (\[M2\]). The interaction term in the Lagrangian for the interaction between the electromagnetic field $F$ and the dipole moment bivector $D$ can be written as a sum of two terms$$\begin{aligned} L_{int} &=&F\cdot D=(1/c^{2})[-(E\cdot d+B\cdot m)(v\cdot u)+(E\cdot u)(v\cdot d) \label{i} \\ &&+(B\cdot u)(v\cdot m)]+(1/c^{3})[(E\wedge m-c^{2}B\wedge d)\wedge v\wedge u]I. \notag\end{aligned}$$In the tensor formulation, the relations (\[di\]) and (\[i\]) are given by Eqs. (2) and (3), respectively, in Ref. 12 (they are first reported in Ref. 28). Observe that every term on the r.h.s. of (\[i\]) contains both velocities $u$ and $v$. As seen from the last two terms they contain the direct interaction of $E$ with $m$, and $B$ with $d$. *These terms give the physical explanation for the existence of the magnetoelectric effect in moving media.* Moreover, there is no need for any transformation. We only need to represent $E$, $d$, $B$, $m$, $u$ and $v$ from (\[i\]) in the standard basis and then to choose the laboratory frame as our $\gamma _{0}$-frame. It can be seen from the discussion of Eq. (25)$^{12}$ that in the laboratory frame, as the $\gamma _{0}$-frame, and in the low velocity limit, we can neglect the contributions to $L_{int}$ from the terms with $d^{0}$ and $m^{0}$; they are $u^{2}/c^{2}$ of the usual terms $E\cdot d$ or $B\cdot m$. Then, what remains from (\[i\]) is $$L_{int}=-((E_{i}d^{i})+(B_{i}m^{i}))-(1/c^{2})\varepsilon ^{0ijk}(E_{i}m_{k}-c^{2}B_{i}d_{k})u_{j}. \label{1}$$This is, to order $0(u^{2}/c^{2})$, relativistically correct expression with vectors for $L_{int}$. The last two terms that contain the direct interactions between $E$ and $m$ and between $B$ and $d$ are not taken into account in any of the previous investigations of the magnetoelectric effect in moving media. They are $u/c$ of the usual terms with the direct interaction of $E$ with $d$ and $B$ with $m$. (The expression (\[1\]) is first reported in Ref. 29 in connection with the EDM searches.) **7. A Brief Discussion of the General Constitutive Relations** In this paper, the consideration is restricted to the constitutive relations and the magnetoelectric effect in moving media with homogeneous and isotropic electric and magnetic properties. For them, the rest frame constitutive relations are given by Eqs. (\[dh1\]). The general constitutive relations which cover dielectric, magnetic and magnetoelectric behavior are considered, e.g., in Ref. 26 and in more detail in Refs. 30, 2, 31 and 23. Mainly, e.g., Refs. 2, 26, 30, 31, these more general constitutive relations link the electromagnetic excitation $\mathcal{H}$ and $F$, $\mathcal{H}_{\alpha \beta }=\kappa _{\alpha \beta }{}^{{}\gamma \delta }F_{\gamma \delta }$ (our notation), Eq. (17),$^{30}$ where ‘$\kappa _{\alpha \beta }{}^{{}\gamma \delta }(x)$ is the twisted constitutive tensor of type $\left[ \begin{array}{c} 2 \\ 2\end{array}\right] $’ or, in another form, e.g., $^{\ast }\mathcal{H}^{\alpha \beta }=\chi ^{\alpha \beta \gamma \delta }F_{\gamma \delta }$ (our notation), Eq. (21),$^{26}$ where $^{\ast }\mathcal{H}^{\alpha \beta }=(1/2)\varepsilon ^{\alpha \beta \gamma \delta }\mathcal{H}_{\gamma \delta }$, $\chi ^{\alpha \beta \gamma \delta }$ is ‘a *constitutive tensor density* of rank 4 and weight +1, with the dimension \[$\chi $\]=1/resistance.’ In Refs. 23, the general constitutive relations are established by expressing $\mathcal{M}$ as a linear function of the electromagnetic field $F$, $\mu _{0}c\mathcal{M}_{\alpha \beta }=(1/2)\xi _{\alpha \beta }^{\gamma \delta }F_{\gamma \delta } $ (our notation), Eq. (2.45),$^{23}$ where ‘$\xi _{\alpha \beta }^{\gamma \delta }$ is the *general susceptibility tensor*, which is dimensionless because of the choice of the constant $\mu _{0}c$.’ However, in all these treatments, all quantities in the equations are not geometric quantities but components in the standard basis, which means that the components of $F$, $\mathcal{H}$, $\mathcal{M}$ are obtained by the usual identifications, i.e., they are considered to be the components of the 3-vectors $\mathbf{E}$ and $\mathbf{B}$, $\mathbf{D}$ and $\mathbf{H}$, $\mathbf{P}$ and $\mathbf{M}$, respectively. This can be nicely seen from Eqs. (2.47) and (2.48)$^{23}$ and from their comparison with Eq. (2.9).$^{23} $ Hence, these relations are not so general relations and they are not premetric, as stated, e.g., in Ref. 26, because, as already mentioned in Sec. 5.3, the usual identifications and the space-time split are meaningless, e.g., in the $\{r_{\mu }\}$ basis with the ‘r’ synchronization. In contrast with the usual covariant approach with coordinate-dependent quantities, all relations (\[E2\]) - (\[h1\]), (\[T1\]), (\[cr1\]) - (\[dh1\]) and (\[di\]) - (\[i\]) are coordinate-free relations, which are written in terms of the abstract 4D geometric quantities. **8. Conclusions** The constitutive relations for $P$ (\[P\]) and $M$ (\[M\]) contain both velocity vectors $u$ and $v$ and thus they differ from all previous expressions. They are formulated in terms of coordinate-free quantities that correctly transform under the LT (\[T1\]), whereas, as explained in Secs. 5 - 5.2, it is not the case with Minkowski’s constitutive relations (\[pl\]) and (\[mg\]), or (\[mp\]), with the 3-vectors that transform according to the UT (\[JCB\]) and (\[ps\]), which are not the LT. As discussed in Sec. 5.3, Minkowski’s constitutive relations for $P$ and $M$, (E.4.30) and (E.4.31), respectively, from Ref. 2, are obtained by the (1+3) - splitting of spacetime and consequently they are equivalent to the usual expressions with the 3-vectors (\[pl\]) and (\[mg\]). Hence, these relations,$^{2}$ which are obtained using exterior forms, also differ from our results (\[P\]) and (\[M\]). The differences that are quoted at the end of Sec. 5.1, points 1) - 3), could be used for the experimental examination and comparison of the results presented here and the constitutive relations which are obtained in the usual formulations either with the 3-vectors or with exterior forms. Furthermore, a completely new physical explanation of the magnetoelectric effect in moving media that is presented in Sec. 6, Eq. (\[i\]), or Eq. (\[1\]), offers the possibility for the experimental investigations of the magnetoelectric effect from the relativistically correct point of view. 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--- abstract: 'An approximation model based on convolutional neural networks (CNNs) is proposed for flow field predictions. The CNN is used to predict the velocity and pressure field in unseen flow conditions and geometries given the pixelated shape of the object. In particular, we consider Reynolds Averaged Navier-Stokes (RANS) flow solutions over airfoil shapes. The CNN can automatically detect essential features with minimal human supervision and shown to effectively estimate the velocity and pressure field orders of magnitude faster than the RANS solver, making it possible to study the impact of the airfoil shape and operating conditions on the aerodynamic forces and the flow field in near-real time. The use of specific convolution operations, parameter sharing, and robustness to noise are shown to enhance the predictive capabilities of CNN. We explore the network architecture and its effectiveness in predicting the flow field for different airfoil shapes, angles of attack, and Reynolds numbers.' author: - Saakaar Bhatnagar - Yaser Afshar - Shaowu Pan - Karthik Duraisamy - Shailendra Kaushik date: 'Received: date / Revised version: date' title: Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks --- Introduction {#intro} ============ With advances in computing power and computational algorithms, simulation-based design and optimization has matured to a level that it plays a significant role in an industrial setting. In many practical engineering applications, however, the analysis of the flow field tends to be the most computationally intensive and time-consuming part of the process. These drawbacks make the design process tedious, time consuming, and costly, requiring a significant amount of user intervention in design explorations, thus proving to be a barrier between designers from the engineering process. Data-driven methods have the potential to augment [@Duraisamy:2019] or replace [@Guo:2016] these expensive high-fidelity analyses with less expensive approximations. Learning representations from the data, especially in the presence of spatial and temporal dependencies, have traditionally been limited to hand-crafting of features by domain experts. Over the past few years, deep learning approaches [@Bengio:2009; @Schmidhuber:2015] have shown significant successes in learning from data, and have been successfully used in the development of novel computational approaches [@Raissi:2018; @Raissi:2018b; @Raissi:2019]. Deep learning presents a fast alternative solution as an efficient function approximation technique in high-dimensional spaces. Deep learning architectures such as deep neural networks (DNNs), routinely used in data mining, are well-suited for application on big, high-dimensional data sets, to extract multi-scale features. Deep convolutional neural networks (CNN) belong to a class of DNNs, most commonly applied to the analysis of visual imagery. Previous works [@Lecun:1998; @Taylor:2010; @Zuo:2015] have illustrated the promise of CNNs to learn high-level features even when the data has strong spatial and temporal correlations. Increasing attention being received by CNNs in fluid mechanics partly originates from their potential benefit of flexibility in the shape representation and scalability for 3D and transient problems. Figure \[fig:fig1\] illustrates the simplified layout of a typical CNN, LeNet-5 [@Lecun:1998] applied to the handwritten digit recognition task. The main advantage of a CNN is that it exploits the low dimensional high-level abstraction by convolution. The key idea of CNN is to learn the representation and then to use a fully connected standard layer to fit the relationship between the high-level representation and output. State of the art in application of CNNs in fluid dynamics {#State_of_the_art} --------------------------------------------------------- The use of deep neural networks in computational fluid dynamics recently has been explored in some rudimentary contexts. [@Guo:2016] reported the analysis and prediction of non-uniform steady laminar flow fields around bluff body objects by employing a convolutional neural network (CNN). The authors reported a computational cost lower than that required for numerical simulations by GPU-accelerated CFD solver. Though this work was pioneering in the sense that it demonstrated generalization capabilities, and that CNNs can enable a rapid estimation of the flow field, emphasis was on qualitative estimates of the velocity field, rather than on precise aerodynamic characteristics. [@Miyanawala:2017] used a CNN to predict aerodynamic force coefficients of bluff bodies at a low Reynolds number for different bluff body shapes. They presented a data-driven method using CNN and the stochastic gradient-descent for the model reduction of the Navier-Stokes equations in unsteady flow problems. [@Lee:2017; @Lee:2018] used a generative adversarial network (GAN) to predict unsteady laminar vortex shedding over a circular cylinder. They presented the capability of successfully learning and predicting both spatial and temporal characteristics of the laminar vortex shedding phenomenon. [@hennigh:2017b] presented an approach to use a DNN to compress both the computation time and memory usage of the Lattice Boltzmann flow simulations. The author employed convolutional autoencoders and residual connections in an entirely differentiable scheme to shorten the state size of simulation and learn the dynamics of this compressed form. [@Tompson:2016] proposed a data-driven approach for calculating numerical solutions to the inviscid Euler equations for fluid flow. In this approach, an approximate inference of the sparse linear system used to enforce the Navier-Stokes incompressibility condition, the “pressure projection” step. This approach cannot guarantee an exact solution pressure projection step, but they showed that it empirically produces very stable divergence-free velocity fields whose runtime and accuracy is better than the Jacobi method while being orders of magnitude faster. [@Zhang:2017] employed a CNN as feature extractor for a low dimensional surrogate modeling. They presented the potential of learning and predicting lift coefficients using the geometric information of airfoil and operating parameters like Reynolds number, Mach number, and angle of attack. However, the output is not the flow field around the airfoil but the pressure coefficients at several locations. It is unclear whether this model would have good performance in predicting the drag and pressure coefficient when producing the flow field at the same time. The primary contribution of the present work is a framework that can be used to predict the flow field around different geometries under variable flow conditions. Towards this goal and following [@Guo:2016], we propose a framework with a general and flexible approximation model for near real-time prediction of non-uniform steady RANS flow in a domain based on convolutional neural networks. In this framework, the flow field can be extracted from simulation data by learning the relationship between an input feature extracted from geometry and the ground truth from a RANS simulation. Then without standard convergence requirements of the RANS solver, and its number of iterations and runtime, which are irrelevant to the prediction process, we can directly predict the flow behavior in a fraction of the time. In contrast to previous studies, the present work is focused on a more rigorous characterization of aerodynamic characteristics. The present study also improves on computational aspects. For instance, [@Guo:2016] use an separated decoder, whereas the present work employs shared-encoding and decoding layers, which are computationally efficient compared to the separated alternatives. Methodology {#Methodology} ============ CFD Simulation {#CFD} -------------- In this work, flow computations and analyses are performed using the OVERTURNS CFD code [@Duraisamy:2005; @Lakshminarayan:2010]. This code solves the compressible RANS equations using a preconditioned dual-time scheme [@Pandya:2003]. Iterative solutions are pursued using the implicit approximate factorization method [@Pulliam:1981]. Low Mach preconditioning [@Turkel:1999] is used to improve both convergence properties and the accuracy of the spatial discretization. A third order Monotonic Upwind Scheme for Conservation Laws (MUSCL) [@VanLeer:1979] with Koren’s limiter [@Koren:1993] and Roe’s flux difference splitting [@Roe:1986] is used to compute the inviscid terms. Second order accurate central differencing is used for the viscous terms. The RANS closure is the SA [@Spalart:1992] turbulence model and $\gamma - \overline{Re_{\theta t}}$ model [@Medida:2011] is used to capture the effect of the flow transition. No-slip boundary conditions imposed on the airfoil surface. The governing equations are provided in the Appendix. Simulations are performed over the S805 [@Somers:1997a], S809 [@Somers:1997b], and S814 [@Somers:2004] airfoils. S809 and S814 are among a family of airfoils which contain a region of pressure recovery along the upper surface which induces a smooth transition from laminar to turbulent flow (so-called “transition-ramp”). These airfoils are utilized in wind turbines [@Aranake:2012]. Computations are performed using structured C-meshes with dimensions $394 \times 124$ in the wrap-around and normal directions respectively. Figure \[fig:fig2\] shows the airfoils and their near-body meshes. [0.45]{} [0.45]{} [0.45]{} Simulations are performed at Reynolds numbers $0.5,~1,~2,~\text{and}~3 \times 10^6$, respectively, and a low Mach number of $0.2$ is selected to be representative of wind turbine conditions. At each Reynolds number, the simulation is performed for different airfoils with a sweep of angles of attack from $\alpha=0^{\circ}$ to $\alpha=20^{\circ}$. The OVERTURNS CFD code has been validated for relevant wind turbine applications in [@Aranake:2012]. Convolutional Neural Networks {#CNN} ----------------------------- In this study, we consider the convolutional neural network to extract relevant features from fluid dynamics data and to predict the entire flow field in near real-time. The objective is a properly trained CNN which can construct the flow field around an airfoil in a non-uniform turbulence field, using only the shape of the airfoil and fluid flow characteristics of the free stream in the form of the angle of attack and Reynolds number. In this section, we describe the structure and components of the proposed CNN. Network Structure {#Structure} ----------------- To develop suitable CNN architectures for variable flow conditions and airfoil shapes, we build our model based on an encoder-decoder CNN, similar to the model proposed by [@Guo:2016]. Encoder-decoder CNNs are most widely used for machine translation from a source language to a target language [@Chollampatt:2018]. The encoder-decoder CNN has three main components: a stack of convolution layers, followed by a dense layer and subsequently another stack of convolution layers. Figure \[fig:fig3\] illustrates the proposed CNN architecture designed in this work. [@Guo:2016] used a shared-encoder but separated decoder. We conjecture that the separated decoder may be a limiting performance factor. To address this issue, we designed shared-encoding and decoding layers in our configuration, which save computations compared to the separated alternatives. Explicitly, the weights of the layers of the decoder are shared where they are responsible for extracting high-level representations of pressure and different velocity components. This design provides the same accuracy of the separated decoders but, it is almost utilized fifty percent fewer parameters compared to the separated alternatives. Also, in the work of [@Guo:2016], the authors used only one low Reynolds number for all the experiments, but here, the architecture is trained with four high Reynolds numbers, three airfoils with different shapes and 21 different angles of attacks. In this architecture, we use three convolution layers both in the shared-encoding and decoding parts. The inputs to the network are the airfoil shape and the free stream conditions of the fluid flow. We use the convolution layers to extract the geometry representation from the inputs. The decoding layers use this representation in convolution layers and generate the mapping from the extracted geometry representation to the pressure field and different components of the velocity. The network uses the Reynolds number, the angle of attack, and the shape of the airfoil in the form of $150 \times 150$ 2D array created for each data entry. The geometry representation has to be extracted from the RANS mesh and fed to the network with images. Using images in CNNs allows encoding specific properties into the architecture, and reducing the number of parameters in the network. Geometry Representation {#Geometry} ------------------------ A wide range of approaches are employed to capture shape details and to classify points into a learnable format. Among popular examples are methods like implicit functions in image reconstruction [@Hoppe:1992; @Carr:2001; @Kazhdan:2006; @Fuhrmann:2014], or shape representation and classification [@Zhang:2004a; @Ling:2007; @Xu:2015; @Fernando:2015]. In applications such as rendering and segmentation and in extracting structural information of different shapes, signed distance functions (SDF) are widely used. SDF provides a universal representation of different geometry shapes and represents a grid sampling of the minimum distance to the surface of an object. It also works efficiently with neural networks for shape learning. In this study, to capture shape details in different object representations, and following [@Guo:2016; @Prantl:2017], we use the SDF sampled on a Cartesian grid. [@Guo:2016] reported the effectiveness of SDF in representing the geometry shapes for CNNs. The authors empirically showed that the values of SDF on the Cartesian grid provide not only local geometry details but also contain additional information on the global geometry structure. Signed Distance Function {#SDF} ------------------------ A mathematical definition of the signed distance function of a set of points $\textbf{X}$ determines the minimum distance of each given point $\textbf{x} \in \textbf{X}$ from the boundary of an object $\partial\Omega$. $$\begin{aligned} \textrm{SDF}(\textbf{x}) = \left\{\begin{matrix} d(\textbf{x}, \partial \Omega) & ~\textbf{x}\notin\Omega \\ 0 & ~~~\textbf{x}\in\partial\Omega\\ -d(\textbf{x}, \partial \Omega) & ~\textbf{x}\in\Omega \end{matrix}\right., \label{eq:eq1}\end{aligned}$$ where, $\Omega$ denotes the object, and $d(\textbf{x}, \partial \Omega) = \min_{\textbf{x}_I \in \partial \Omega}{\left( |\textbf{x} - \textbf{x}_I |\right )}$ measures the shortest distance of each given point $\textbf{x}$ from the object boundary points. The distance sign determines whether the given point is inside or outside of the object. Figure \[fig:fig4\] illustrates the signed distance function contour plot for a S814 airfoil. Here, the SDF has positive values at points which are outside of the airfoil, and it decreases as the point approaches the boundary of the airfoil where the SDF is zero, and it takes negative values inside the airfoil. Fast marching method [@Sethian:1996] and fast sweeping method [@Zhao:2005] are among the popular algorithms for calculating the signed distance function. To generate a signed distance function, we use the CFD input structured C-mesh information and define the points around the object (airfoil). Figure \[fig:fig5\] shows the C-mesh representation of an airfoil (S814) and its boundary points on a Cartesian grid. We find the distance of Cartesian grid points from the object boundary points, using the fast marching method [@Sethian:1996]. To find out whether a given point is inside, outside, or just on the surface of the object, we search the boundary points and compute the scalar product between the normal vector at the nearest boundary points and the vector from the given point to the nearest one and judge the function sign from the scalar product value. For other non-convex objects, one can also use different approaches of crossing number or winding number method which are common in ray casting [@Foley:1995]. After pre-processing the CFD mesh files, we use the SDF as an input to feed the encoder-decoder architecture with multiple layers of convolutions. Convolution layers in the encoding-decoding part extract all the geometry features from the SDF. Convolutional Encoder-Decoder Approach {#Encoder} -------------------------------------- To learn all the geometry features from an input SDF, we compose the encoder and decoder with convolution layers and convolutional filters. Every convolutional layer is composed of 300 convolutional filters. Therefore, a convolution produces a set of 300 activation maps. Every convolution in our design is wrapped by a non-linear Swish activation function [@Ramachandran:2017]. Swish is defined as $x.\sigma(\beta x)$ where $\sigma(z)=(1+exp(-z))^{-1}$ is the sigmoid function and $\beta$ is either a constant or a trainable parameter. The resulting activation maps are the encoding of the input in a low dimensional space of parameters to learn. The decoding operation is a convolution as well, where the encoding architecture fixes the hyper-parameters of the decoding convolution. Compared to the encoding convolution layer, here a convolution layer has reversed forward and backward passes. This inverse operation is sometimes referred to “deconvolution”. The decoding operation unravels the high-level features encoded and transformed by the encoding layers and generates the mapping to the pressure field and different components of the velocity. When we use the CNN, neurons in the same feature map plane have identical weights so that the network can study concurrently, and it learns implicitly from the training data. The training phase of the CNN comprises the input function, the feed-forward process, and the back-propagation process. Data Preparation {#Data} ---------------- In total, a set of 252 RANS simulations were performed. This data includes our CFD predictions for three different S805, S809, and S814 airfoils. The training data-set consists of $85$ percent of the full set, and the remaining data sets are used for testing, as shown in Fig. \[fig:fig6\]. The test points are chosen uniformly at random on the feature space, providing an unbiased evaluation of a model fit on the training data-set while tuning the model’s hyper-parameters. Figure \[fig:fig7\] shows the x-component of the velocity field ($U$) around the S814 airfoil on the structured C-mesh. The simulation is performed at an angle of attack of $\alpha = 9^\circ$ and with the Reynolds number of $3\times10^6$. The CFD data has to be interpolated onto a $150\times 150$ Cartesian grid which contains the SDF. A triangulation-based scattered data interpolation method [@Amidror:2002] is used. After the interpolation of the data to the Cartesian grid, the interior points masked and the velocity is set to zero. The comparison of the reconstructed data in Fig. \[fig:fig8\] and the CFD data in Fig. \[fig:fig7\] shows evidence of interpolation errors. The interpolated data is normalized using the standard score normalization by subtracting the mean from the data and dividing the difference by the standard deviation of the data. Scaling the data causes each feature to contribute approximately proportionately to the training, and also results in a faster convergence of the network [@Aksoy:2000]. Network Training and Hyper-parameter Study {#Training} ------------------------------------------ The network learns different weights during the training phase to predict the flow fields. In each iteration, a batch of data undergoes the feed-forward process followed by a back-propagation (see Sec. \[Encoder\]). For a given set of input and ground truth data, the model minimizes a total loss function which is a combination of two specific loss functions and an L2 regularization as follows: $$\begin{aligned} \label{eq:MSE} &\text{MSE}_\text{shared} = \frac{1}{m (n_x-2) (n_y-2)}~\sum_{l=1}^{m} \sum_{j=2}^{n_y-1} \sum_{i=2}^{n_x-1}\\ &\nonumber \quad \big[(U^l_{{ij}_\text{truth}}-U^l_{{ij}_\text{pred}})^2~+~(V^l_{{ij}_\text{truth}}-V^l_{{ij}_\text{pred}})^2~+~\\ &\nonumber \quad ~(P^l_{{ij}_\text{truth}}-P^l_{{ij}_\text{pred}})^2 \big], \\ \label{eq:GSshared} &\text{GS}_\text{shared} = \frac{1}{6m(n_x-2)(n_y-2)}~\sum_{l=1}^{m} \sum_{j=2}^{n_y-1} \sum_{i=2}^{n_x-1} \\ &\nonumber \quad [(\frac{\partial P^l}{\partial x}_{{ij}_\text{truth}} - \frac{\partial P^l}{\partial x}_{{ij}_\text{pred}})^2 ~+~(\frac{\partial P^l}{\partial y}_{{ij}_\text{truth}} - \frac{\partial P^l}{\partial y}_{{ij}_\text{pred}})^2 ~+~ \\ &\nonumber \quad ~ (\frac{\partial U^l}{\partial x}_{{ij}_\text{truth}} - \frac{\partial U^l}{\partial x}_{{ij}_\text{pred}})^2 ~+~(\frac{\partial U^l}{\partial y}_{{ij}_\text{truth}} -\frac{\partial U^l}{\partial y}_{{ij}_\text{pred}})^2 ~+~ \\ &\nonumber \quad ~ (\frac{\partial V^l}{\partial x}_{{ij}_\text{truth}} - \frac{\partial V^l}{\partial x}_{{ij}_\text{pred}})^2 ~+~(\frac{\partial V^l}{\partial y}_{{ij}_\text{truth}} - \frac{\partial V^l}{\partial y}_{{ij}_\text{pred}})^2],\\ \label{eq:L2} &\text{L2}_\text{regularization} = \frac{1}{2m}\sum_{l=1}^{L}\sum_{i=1}^{n_l}(\theta^l_{i})^2,\end{aligned}$$ where $U$, and $V$ are the x-component and y-component of the velocity field respectively, and $P$ is the scalar pressure field. $m$ is the batch size, $n_x$ is the number of grid points along the x-direction, $n_y$ is the number of grid points along the y-direction, and $L$ is the number of layers with trainable weights, and $n_l$ represents number of trainable weights in layer $l$. MSE is the mean squared error, and GS is gradient sharpening or gradient difference loss (GDL) [@Mathieu:2015; @Lee:2018]. In this paper, we use gradient sharpening based on a central difference operator. The network was trained for $30,000$ epochs with a batch size of $214$ data points, which took 33 GPU hours. For the separated decoder, the following loss functions are used: $$\begin{aligned} \label{eq:MSEseparate} &\text{MSE}_\text{separated} = \frac{1}{m (n_x-2) (n_y-2)}~\sum_{l=1}^{m} \sum_{j=2}^{n_y-1} \sum_{i=2}^{n_x-1}\\ &\nonumber \quad \big[(X^l_{{ij}_\text{truth}}-X^l_{{ij}_\text{pred}})^2],\\ \label{eq:GSseparate} &\text{GS}_\text{separated} = \frac{1}{2m(n_x-2)(n_y-2)}~\sum_{l=1}^{m} \sum_{j=2}^{n_y-1} \sum_{i=2}^{n_x-1} \\ &\nonumber \quad [(\frac{\partial X^l}{\partial x}_{{ij}_\text{truth}} - \frac{\partial X^l}{\partial x}_{{ij}_\text{pred}})^2 ~+~(\frac{\partial X^l}{\partial y}_{{ij}_\text{truth}} - \frac{\partial X^l}{\partial y}_{{ij}_\text{pred}})^2],\end{aligned}$$ where $X$ stands for $U,~V$ or $P$. Finding the optimal set of hyper-parameters for the network is an empirical task and is done by performing a grid search consisting of an interval of values of each hyper-parameter, and training many networks with several different combinations of these hyper-parameters. The resulting networks are compared based on generalization tendency and the difference between the truth and prediction. Results and Discussion {#Results} ====================== We first show the capability of the designed network architecture to accurately estimate the velocity and pressure field around different airfoils given only the airfoil shape. Then, we quantitatively assess the error measurement followed by a sequence of results which demonstrate usability, accuracy and effectiveness of the network. Figure \[fig:fig9\] illustrates the training and validation results from the network. It shows the working concept of the proposed structure, by incorporating the fluid flow characteristics and airfoil geometry. Results are presented at the epoch number with the lowest validation error. Model validation {#Model} ---------------- The Absolute percent error (APE) or the unsigned percentage error is used as a metric for comparison: $$\begin{aligned} \text{APE}=\frac{|\text{Prediction} - \text{Truth} |} {|\text{Truth}|} \times 100. \label{eq:APE}\end{aligned}$$ The mean value of the absolute percent error (MAPE) is standard as a Loss function for regression problems. Here, model evaluation is done using MAPE due to the very intuitive interpretation regarding the relative error and its ease of use. In this paper, the MAPE between the prediction and the truth is calculated in the wake region of an airfoil and the entire flow field around the airfoil. Here, the wake region of the airfoil is an area defined as $\{(x,y)|x\in\left[1.1, 1.5\right],y\in \left[-0.5, 0.5\right]\}$, and $\{(x,y)|x\in\left[-0.5, 1.5\right],y\in \left[-0.5, 0.5\right]\}$ is the entire flow field area around the airfoil. The predictions contain $2-3\%$ of points with an error value greater than $100\%$, which are treated as outliers and not included in the reported errors. Numerical simulations {#Experiments} --------------------- ### Angle of attack variation {#Performance1} At a fixed Reynolds number ($Re=1\times10^6$) and fixed airfoil shape (S805), we consider simulations with angles of attack of one-degree increments from $\alpha=0^{\circ}$ to $\alpha=20^{\circ}$. By using this small set of data (21 data points), we train the network with 50 filters instead of the aforementioned 300 filters in each layer (see Sec. \[Encoder\] for more details). The total loss function comprises only an MSE and with no regularization during training. Thus the cost function over the training set is presented as, $$\begin{aligned} \text{Cost} = \lambda_\text{MSE} \times \text{MSE}, \label{eq:cost1}\end{aligned}$$ where $\lambda_\text{MSE}$ is a user defined parameter (here it is $\lambda_\text{MSE}=1$). After the network training is complete, testing is performed on four unseen angles of attacks, $\alpha=2.5^\circ,~7.5^\circ,~12.5^\circ,~\text{and}~19.5^\circ$ respectively. Figure \[fig:fig10\] shows the comparison between the network prediction and the actual observation from the CFD simulation for the x-component of the velocity field around the S805 airfoil at an angle of attack of $\alpha = 12.5^\circ$. A visual comparison shows that the prediction is in agreement with the truth. [0.45]{} [0.45]{} [0.45]{} [0.45]{} Table. \[tab:tab1separate\] and \[tab:tab1shared\] present the MAPE calculated in the wake region and the entire flow field around the S805 airfoil (see Fig.\[fig:fig10\]), where the fluid flow characteristics are the angle of attack of $\alpha = 12.5^\circ$ and the Reynolds number of $1\times10^6$. [![width 1pt]{}l![width 1pt]{}c|c|c![width 1pt]{}]{}   & $\mathbf{U}$ & $\mathbf{V}$ & $\mathbf{P}$\ **Error in the wake region** & 24.9% & 10.15% & 24.97%\ **Error in the entire flow** & 13.51% & 11.92% & 13.50%\ [![width 1pt]{}l![width 1pt]{}c|c|c![width 1pt]{}]{}   & $\mathbf{U}$ & $\mathbf{V}$ & $\mathbf{P}$\ **Error in the wake region** & 15.08% & 7.98% & 14.82%\ **Error in the entire flow** & 9.62% & 8.65% & 7.31%\ The results in table. \[tab:tab1separate\] and \[tab:tab1shared\], illustrate that the errors in the wake region are generally similar to the errors in the entire flow field. This trend is true not only for this case but also in subsequent experiments. Figure \[fig:fig11\] shows the comparison between the CFD result and the network prediction of the x-component velocity profile of the airfoil wake at $x=1.1$ (downstream location from the leading edge). [0.5]{} [0.5]{} ### Shape, angle of attack, and Reynolds number variation {#Shape} We train the network using $85$ percent of the 252 RANS simulation data-sets, with the variation of the airfoil shape, angle of attack and Reynolds number. Every convolutional layer is composed of 300 convolutional filters (see Sec.\[Encoder\] for more details). The total loss function during training comprises an MSE loss function with the L2 regularization. Thus, the cost function over the training set is presented as, $$\begin{aligned} \text{Cost} = \lambda_\text{MSE} \times \text{MSE} + \lambda_\text{L2} \times \text{L2}_\text{regularization}, \label{eq:cost2}\end{aligned}$$ where $\lambda_\text{MSE}=1$ and $\lambda_\text{L2}=10^{-5}$ are user defined parameters. Figures. \[fig:fig102\] and \[fig:fig103\] present the comparisons between the network predictions and observations for the x-component of the velocity field around the S809 and S814 airfoils at $(\alpha = 1^\circ,~Re = 1\times10^6)$ and $(\alpha = 19^\circ,~Re = 3\times10^6)$. [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} Quantitative results are presented in Tables \[tab:tab2separate\] and \[tab:tab2shared\]. [![width 1pt]{}c|c|c|c|c|c![width 1pt]{}]{} **Airfoil** & **AOA** & **Re** & **Variable** & **Error in the** & **Error in the**\   &   & $\times 10^6$ &   & **wake region** & **entire flow**\ **S809** & $1^\circ$ & 1 & U & 12.25% & 10.35%\ **S809** & $1^\circ$ & 1 & V & 24.27% & 11.53%\ **S809** & $1^\circ$ & 1 & P & 5.14% & 8.40%\ **S814** & $19^\circ$ & 3 & U & 30.80% & 13.13%\ **S814** & $19^\circ$ & 3 & V & 10.43% & 5.49%\ **S814** & $19^\circ$ & 3 & P & 13.84% & 5.70%\ [![width 1pt]{}c|c|c|c|c|c![width 1pt]{}]{} **Airfoil** & **AOA** & **Re** & **Variable** & **Error in the** & **Error in the**\   &   & $\times 10^6$ &   & **wake region** & **entire flow**\ **S809** & $1^\circ$ & 1 & U & 11.43% & 7.79%\ **S809** & $1^\circ$ & 1 & V & 15.53% & 8.74%\ **S809** & $1^\circ$ & 1 & P & 5.76% & 7.36%\ **S814** & $19^\circ$ & 3 & U & 27.23% & 13.20%\ **S814** & $19^\circ$ & 3 & V & 5.57% & 4.69%\ **S814** & $19^\circ$ & 3 & P & 12.93% & 5.71%\ ### Shape, angle of attack, and Reynolds number variation with gradient sharpening {#Gradient} To penalize the difference of the gradient in the loss function, and to address the lack of sharpness in predictions, we use gradient sharpening (GS) [@Mathieu:2015; @Lee:2018] in the loss functions combination and present the cost function over the training set as, $$\begin{aligned} \text{Cost} = \lambda_\text{MSE} \times \text{MSE} + \lambda_\text{GS} \times \text{GS} +\lambda_\text{L2} \times \text{L2}_\text{regularization}, \label{eq:cost3}\end{aligned}$$ where $\lambda_\text{MSE},~\lambda_\text{GS}~\text{and}~\lambda_\text{L2}$ are the user defined parameters and their values are set via systematic experimentation, as $0.9,~0.1~\text{and}~10^{-5}$ respectively. Figures. \[fig:fig104\] and \[fig:fig105\] present the comparisons between the network predictions with and without GS loss for the x-component of the velocity field around S809 and S814 airfoils respectively. [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} Visual comparisons of the predictions and the absolute difference with and without GS as illustrated in Figs. \[fig:fig104\] and \[fig:fig105\] are proofs of further gains and sharpness in the network predictions. The “absolute difference”\[absolute\] between the prediction and ground truth, for example, is defined as the absolute difference in the subtraction of each element in prediction from the corresponding element in ground truth. The MAPE for the components of the velocity field and pressure of the airfoils (S809 and S814 discussed above) are presented in table. \[tab:tab3separate\] and \[tab:tab3shared\]. The errors are reported in the wake region and the entire flow field around the airfoils with and without GS. The predictions with GS in the loss function compared to not having it show significantly reduced errors in the wake region of the airfoil (twenty percent or more in the x-component of the velocity and pressure predictions) and obvious gains and sharpness in the entire flow field around the airfoil. To further compare the accuracy of the network predictions, we use three probes around different airfoils in different flow conditions. These probes are leading edge probe (LE), trailing-edge probe (TE), and the probe at the wake region of an airfoil. Figure \[fig:fig200\] illustrates these three probes around different airfoils, S805, S809, and S814, respectively. [0.45]{} [0.45]{} [0.45]{} Table. \[tab:tab4\] presents the APE (Eq. \[eq:APE\]) at the probe locations (LE, TE, and wake region probe). Figures. \[fig:fig17\], \[fig:fig18\] and \[fig:fig19\] illustrate the flow-field predictions with gradient sharpening in the loss function and in comparison with the reference results from the OVERTURNS CFD code. [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} [0.32]{} Figures. \[fig:fig500\] and \[fig:fig501\] illustrate the x-component velocity profile of the airfoil wake at $x=1.1$ (downstream location from the leading edge). These predictions include GS in the loss function. As a further comparison of the network prediction accuracy, we consider the pressure distribution on the upper and lower boundaries. Figures. \[fig:fig22\], \[fig:fig23\], and \[fig:fig24\] depicts the Ground truth vs. Predictions of the normalized pressure using the standard score normalization along the surface of the S805, S809, and S814 airfoils respectively. It is noteworthy that the surface with a one-pixel gap adjacent to the airfoil surface is used to obtain the pressure values. This change is due to the masking of the airfoil as an input during the training. Overall, results are in good agreement with the ground truth simulation results in the entire range of angles of attacks and Reynolds numbers for the three different airfoils. ### Prediction for unseen airfoil shapes {#Unseen} To further explore the predictive ability and accuracy of the trained network, three unseen geometries are considered as shown in Figure \[fig:fig25\]). The first one, denoted by “new airfoil” is an averaged shape of S809 and S814 airfoils. In addition, the S807 and S819 airfoils are also considered. Figures \[fig:fig26\], \[fig:fig27\], \[fig:fig28\] illustrate the prediction of the network on the unseen airfoils in comparison to CFD simulations. [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [0.45]{} [![width 1pt]{}c|c|c|c![width 1pt]{}]{} **Airfoil** & **Variable** & **Error in the** & **Error in the**\   &   & **wake region** & **entire flow**\ **New** & U & 10.55% & 6.41%\ **New** & V & 5.71% & 8.59%\ **New** & P & 4.41% & 5.27%\ **S807** & U & 11.35% & 8.4%\ **S807** & V & 2.2% & 8.8%\ **S807** & P & 6.0% & 7.6%\ **S819** & U & 13.3% & 10.3%\ **S819** & V & 2.4% & 8.6%\ **S819** & P & 5.5% & 7.5%\ Table. \[tab:tab6\] provides a quantification of the results, and suggests good generalization properties of the network to an unseen shape. Conclusions and Future Work {#Conclusion} =========================== A flexible approximation model based on convolutional neural networks was developed for efficient prediction of aerodynamic flow fields. Shared-encoding and decoding was used and found to be computationally more efficient compared to separated alternatives. The use of convolution operations, parameter sharing and robustness to noise using the gradient sharpening were shown to enhance predictive capabilities. The Reynolds number, angle of attack, and the shape of the airfoil in the form of a signed distance function are used as inputs to the network and the outputs are the velocity and pressure fields. The framework was utilized to predict the Reynolds Averaged Navier–Stokes flow field around different airfoil geometries under variable flow conditions. The network predictions on a single GPU were four orders of magnitude faster compared to the RANS solver, at mean square error levels of less than 10% over the entire flow field. Predictions were possible with a small number of training simulations, and accuracy improvements were demonstrated by employing gradient sharpening. Furthermore, the capability of the network was evaluated for unseen airfoil shapes. The results illustrate that the CNNs can enable near real-time simulation-based design and optimization, opening avenues for an efficient design process. It is noteworthy that using three airfoil shapes in training, is a data limitation and reduces the general prediction behavior for unseen airfoil geometries from other families. Future work will seek to use a rich data set including multiple airfoil families in training and to augment the training data-sets to convert a set of input data into a broader set of slightly altered data [@Shijie:2017] using operations such as translation and rotation. This augmentation would effectively help the network from learning irrelevant patterns, and substantially boost the performance. Furthermore, exploring physical loss functions can be helpful in explicitly imposing physical constraints such as the conservation of mass and momentum to the networks. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported by General Motors Corporation under a contract titled “Deep Learning and Reduced Order Modeling for Automotive Aerodynamics.” Computing resources were provided by the NSF via grant 1531752 MRI: Acquisition of Conflux, A Novel Platform for Data-Driven Computational Physics (Tech. Monitor: Stefan Robila). Appendix: Governing equations {#appendix-governing-equations .unnumbered} ============================= The RANS equations are derived by ensemble-averaging the conservation equations of mass, momentum and energy. These equations, for compressible flow are given by: $$\begin{aligned} &\frac{\partial\bar\rho}{\partial t}+\frac{\partial\left(\bar\rho\hat u_i\right)}{\partial x_i} =0 \\ &\frac{\partial\left(\bar\rho\hat u_i\right)}{\partial t}+\frac{\partial\left(\bar\rho\hat u_i\hat u_j\right)}{\partial x_j} =-\frac{\partial\bar p}{\partial x_i}+\frac{\partial\bar\sigma_{ij}}{\partial x_j}+\frac{\partial\tau_{ij}}{\partial x_j} \\ &\frac{\partial\left(\bar\rho\hat E\right)}{\partial t}+\frac{\partial\left(\bar\rho\hat H\hat u_j\right)}{\partial x_j} = \frac{\partial}{\partial x_j}\left(\bar\sigma_{ij}\hat u_i+\overline{\sigma_{ij}u_i''}\right) - \\ &\nonumber \quad\quad \frac{\partial}{\partial x_j}\left(-\hat\kappa\frac{\partial\hat T}{\partial x_j}+c_P\overline{\rho u_j'' T''}-\hat u_i\tau_{ij}+\frac{1}{2}\overline{\rho u_i'' u_i'' u_j''}\right),\end{aligned}$$ where the overbar indicates conventional time-average mean, $u_i$ is the fluid velocity, $\rho$ is the density, $p$ is the pressure, $\tau_{ij}$ is the Reynolds stress term, $c_P$ is the heat capacity at constant pressure, and $\kappa$ is the kinetic energy of the fluctuating field (local turbulent kinetic energy). The density weighted time averaging (Favre averaging) of any quantity $\xi$, denoted by $\hat\xi$ is given as $\hat\xi=\overline{\rho\xi}/\bar\rho$, where, $$\begin{aligned} &\hat H=\hat E+\frac{\bar p}{\bar\rho},\\ &\bar\sigma_{ij}=\mu_t\left(\frac{\partial\hat u_i}{\partial x_j}+\frac{\partial\hat u_j}{\partial x_i}-\frac{2}{3}\frac{\partial\hat u_k}{\partial x_k}\delta_{ij}\right),\\ &\tau_{ij}=-\overline{\rho u_i'' u_j''},\\ &k=\frac{\widehat{u_i''^2}+\widehat{v_i''^2}+\widehat{w_i''^2}}{2},\\ &\bar p = (\gamma-1)\bar\rho\left[\hat E - \frac{\hat u^2+\hat v^2+\hat w^2}{2} - k\right].\end{aligned}$$ To provide closure to the above equations, we use the model proposed by [@Spalart:1992]. In this closure, the Boussinesq hypothesis relates the Reynolds stress and the effect of turbulence as an eddy viscosity $\mu_t$. Employing the Boussinesq approach, and Reynolds Analogy a transport equation for a working variable $\tilde\nu$ is solved to estimate the eddy viscosity field at every iteration. $$\begin{aligned} \label{eq:eq10} &\frac{\partial \tilde{\nu}}{\partial t}+u_j\frac{\partial \tilde{\nu}}{\partial x_j} = C_{b1}\left[1-f_{t2}\right]\tilde{S}\tilde{\nu} + \\ \nonumber &~\frac{1}{\sigma}\left\lbrace\nabla\cdot\left[\left(\nu+\tilde{\nu}\right)\nabla\tilde{\nu}\right]+C_{b2}\left|\nabla\tilde{\nu}\right|^2\right\rbrace -\left[C_{w1}f_w-\frac{C_{b1}}{\kappa^2}f_{t2}\right]\left(\frac{\tilde{\nu}}{d}\right)^2.\end{aligned}$$ The turbulent eddy viscosity is computed as $\mu_t=\bar\rho\tilde\nu f_{v1}$, where, $$\begin{aligned} &\nonumber f_{v1}=\frac{\chi^3}{\chi^3+C_{v1}^3},~\chi=\frac{\tilde{\nu}}{\nu},~\nu=\frac{\mu}{\bar\rho},\\ &\nonumber f_{t2}=C_{t3} \exp \left(-C_{t4}\chi^2\right),\\ &\nonumber \tilde{S}=S+\frac{\tilde{\nu}}{\kappa^2 d^2}f_{v2},\\ &\nonumber S=\sqrt{2\Omega_{ij}\Omega{ij}},~f_{v2}=1-\frac{\chi}{1+\chi f_{v1}},\\ &\nonumber f_w=g\left[\frac{1+C_{w3}^6}{g^6+C_{w3}^6}\right]^{1/6}, \\ &\nonumber g=r+C_{w2}(r^6-r),~ r=\frac{\tilde{\nu}}{\tilde{S}\kappa^2d^2}, \\ &\nonumber C_{w1}=\frac{C_{b1}}{\kappa^2}+\frac{1+C_{b2}}{\sigma},~ C_{b1}=0.1355,~\sigma=2/3,~C_{b2}=0.622,\\ &\nonumber \kappa=0.41,~C_{w2}=0.3,~ C_{w3}=2.0,~C_{v1}=7.1,~C_{t3}=1.2,~C_{t4}=0.5.\end{aligned}$$ The first term on the right hand side of this Eq. \[eq:eq10\] is the production term for $\tilde\nu$ while the second term represents dissipation. 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--- abstract: 'We analyse the proposal of sliding phases (SP) in layers hosting global U(1) symmetric variables with finite inter-layer Josephson coupling. Based on the Kosterlitz-Thouless renormalization group (RG) approach, such phases were predicted to exist in various layered (or 1D quantum coupled) systems. The key in the RG argument is treating the coupling as though the variables are non-compact. Large scale Monte Carlo simulations of a layered model, where the SP is supposed to exist, finds no indication of such a phase. Instead, 3D behavior is observed. This result is consistent with the asymptotically exact analytical solution. A generic argument against SP in translationally invariant systems with short range interactions is provided. We have also suggested an alternative model for the SP – adding long-range interactions to the inter-layer Josephson term.' author: - 'S. Vayl' - 'A.B. Kuklov' - 'V. Oganesyan' title: 'Sliding phases in U(1) symmetric systems – mirage of the renormalization group' --- Introduction ============ The idea behind the SP was put forward by Efetov in 1979 in the context of layered superconductor with parallel magnetic field [@Efetov]. It was suggested that the field can suppress the inter-layer Josephson coupling so that the low energy properties of this 3D system can be described as being essentially of 2D character. Later, in Ref. [@Korshunov] it was shown that the frustration due to the magnetic field is not sufficient to fully suppress the coupling. In the context of quantum 1D chains (equivalent to 2D classical layers by the virtue of the quantum to classical mapping) the possibility of the decoupling between chains has also been explored [@PWA; @Wen]. The main argument for such a decoupling is based on the scaling dimensions of the Josephson coupling determined with respect to the Luttinger liquid parameter in each chain: if it is larger than 2, the coupling should become irrelevant [@Wen]. These proposals have been criticized in Refs.[@Castellani; @Fabrizio] where it was shown that the inter-chain tunneling is always relevant. Further argumentation in favor of the SP and, actually, the name [*Sliding Phases*]{} have been proposed in Refs.[@Lubensky; @Toner] where the inter-layer gradient couplings between classical XY variables in each layer have been considered in addition to the Josephson one. Such gradient terms can independently control the scaling dimensions of the Josephson coupling and of the vortex fugacity in each layer so that the first one can become irrelevant above some temperature $T_d$ (of the [*dimensional reduction*]{} [@Sondhi]) which is below the temperature of the Berizinskii-Kosterlitz-Thouless (BKT) transition in the layers. Thus, there is a range of temperatures where the SP are supposed to exist. This approach was also developed for the case of quantum 1D Luttinger liquids coupled by both the Josephson and the gradient terms [@Kane_2001; @Ashwin_2001] which are the analog of the Andreev-Bashkin drag effect [@AB_effect]. It is important to note that the proposal of SP is based on applying the RG logic to compact variables characterized by global U(1) symmetry. While these early suggestions were more of a purely academic interest, expanding capabilities of ultra-cold-atoms techniques in recent years emphasize the importance of these suggestions especially in the context of possible new phases in composite lattices [@Cazalila] and in the presence of disorder [@Demler]. In more general terms, the question is if it is possible to realize a phase transition from a low- to higher- dimensional behavior. Here we will analyze a simplest classical XY system characterized by the gradient interactions and the Josephson coupling $u$. The gradient terms are chosen in such a way that the SP is supposed to exist in some range where the renormalized value $u_r$ of $u$ scales to zero as layers size $L$ grows. We will present the results of the large scale Monte Carlo simulations of this system. Our analysis is based on the dual formulation of the model – in terms of the closed loops. The main result is that no SP state exists in such a system. Instead, the value of $u_r$ is always finite. This behavior will be compared with the standard asymmetric XY layered model where no SP are expected to occur. We will also derive the analytical result for $u_r$ in the asymptotic limit when the intra-layer stiffness is much larger than $u$. The numerical results have been found to be consistent with the analytical solutions for both models. Our paper is organized as follows. In Sec.\[Sec:I\] we introduce the bilayer model and provide the RG solution for SP. Then, we construct the dual representation in Sec. \[sec:dual\]. Using the duality we have found the asymptotic analytical solution for the renormalized Josephson coupling $u_r$ in Sec. \[sec:AS\]. The Monte Carlo simulations of the bilayer model are presented in Sec.\[sec:num\]. Then, in Sec.\[Sec:Nz\] we present the results on a stack of bilayers along the same lines as for the bilayer. Finally, in Sec. \[sec:dis\] we discuss the implications of our analytical and numerical results and also provide an alternative model for the SP. Bilayer model of SP {#Sec:I} =================== Here we introduce a model of two asymmetric parallel layers, each being a square lattice of linear size $L=1,2,3,...$ (in terms of the inter-site shortest distance) characterized by two fields $\psi_1=\exp(\phi_{1})$ and $\psi_2=\exp(\phi_{2})$ on the layers $z=1,2$, respectively.The action can be written as H&=& - \_[ij]{} \[t\_1 (\_[ij]{} \_1 - A\_[ij]{}) +t\_2 (\_[ij]{} \_2 -g\_2 A\_[ij]{})\ &+& A\^2\_[ij]{}\] - \_i u(\_2(i)-\_1(i)) \[2N\] where $t_1>,t_2>0, g>0$ and $g_2$ are parameters; $\langle ij\rangle$ denotes summation over nearest neighbor sites within each layer; $\nabla_{ij} \phi_a \equiv \phi_a(i) - \phi_a(j)$; $A_{ij}$ is a bond vector field (that is, $A_{ij}=-A_{ji}$) oriented along the bond $\langle ij\rangle$. It is introduced in order to generate the “current-current” interaction (cf. [@Lubensky; @Toner; @Sondhi; @Kane_2001; @Ashwin_2001]) consistent with the compact nature of the fields $\phi_{1,2}$. This action is to be used in the partition function Z=DA D\_[1]{} D\_[2]{} (- H) \[ZZ2\] where the temperature is absorbed into the the parameters $t_1,t_2,u,g$. Our focus is on verifying the applicability of the RG analysis to the renormalization of the Josephson coupling $u$. Hence, we will not discuss physical origins of the variables and the parameters. The RG solution for bilayer {#sec:RG} --------------------------- In the approximation ignoring compact nature of the variables, the terms $-\cos(\nabla_{ij} \phi_1 - A_{ij})$ and $-\cos(\nabla_{ij} \phi_2 -g_2 A_{ij})$ are replaced by $(\nabla_{ij} \phi_1 - A_{ij})^2/2$ and $(\nabla_{ij} \phi_2 -g_2 A_{ij})^2/2$, respectively. Then, the gaussian integration over $A_{ij}$ can be carried out explicitly in Eq.(\[ZZ2\]), so that (\[2N\]) in terms of the remaining variables becomes H\_0= \_[ij]{} K\_[ab]{} \_[ij]{} \_a \_[ij]{} \_b - \_i u(\_2-\_1), \[2N0\] where the matrix $K_{ab}, \, a,b =1,2$ is related to the original parameters as K\_[11]{}&=& , K\_[22]{}= ,\ K\_[12]{} &=& - . \[Ktg\] . The stability of $H_0$ is guaranteed by K\_[11]{}K\_[22]{} - K\_[12]{}\^2= &gt;0. \[stab\] The condition for SP can be obtained along the lines of the logic [@Wen; @Lubensky; @Toner; @Sondhi; @Kane_2001; @Ashwin_2001] which ignores the compactness of $\phi_{1,2}$. Specifically, introducing the variables $\varphi =\phi_1 + \phi_2$ and $\theta= \phi_2 -\phi_1$ and, then, integrating out $\varphi$, the resulting partition function becomes Z\_0=D\^[-H\_]{}, H\_ = d\^2 x , \[Z\_0\] where the notation K= \[K\] is introduced and the long wave limit is considered – so that the summation along the layers is replaced by the integration $\int d^2 x ...$. As long as the compactness of $\theta$ is ignored, Eqs. (\[Z\_0\]) represent the standard Sine-Gordon model in 2D. The RG analysis predicts (see in, e.g., [@Lubensky_book]) that at $K<K_d= 1/(8\pi)$ the renormalization renders the Josephson coupling $u$ irrelevant in the thermo-limit $L\to \infty$. More specifically, the renormalized $u$ should flow to zero as $u_r \sim u L^{b}\to 0,\, b= 2(1- K_d/K)<0$. Such a behavior is supposed to occur together with the persistence of the algebraic order along the planes. This requirement imposes further restrictions on the values of $K_{ab}$. Without loss of generality let’s assume $K_{11} <K_{22}$ and introduce the notations: $T=1/K_{11}$ as a measure of temperature, and $Y=K_{22}/K_{11} >1,\, X=K_{12}/K_{11} $. Then, the condition $K<1/(8\pi)$ for SP becomes T&gt;T\_d=. \[SSP\] In order to guarantee the algebraic order in each layer no BKT transition should occur in the layers. In order to determine possible types of vortices responsible for the transition, we examine the form (\[2N0\]) by “reinstating” the compactness of the variables in the limit $u=0$ (which is supposed to renormalize to zero). In the presence of the gradient coupling a vortex may have a composite structure [@Babaev; @Kaurov]: $q_1$ circulations in the component 1 may be bound to $q_2$ circulations in the component 2. Finding the condition for the proliferation of such composite vortices can be achieved along the line of Kosterlitz-Thouless argument developed for simple XY-model in 2D. The free energy of such a complex is F\_v= L -2 L. \[vort\] Then, $F_v\propto \pi [(q_1+ Xq_2)^2 + (Y-X^2)q^2_2] - 2T $, and the stability against the BKT transition is guaranteed by the positivity of $F_v$ or T &lt; T\_[(q\_1,q\_2)]{}= \[(q\_1+ Xq\_2)\^2 + (Y-X\^2)q\^2\_2\], \[TBKT\] where the minimization with respect to $q_1, q_2$ must be performed. Proliferation of simple vortices $q_1=\pm 1,\,\, q_2=0$ corresponds to $ T_{(1,0)}=\pi/2$. It is, however, not always a minimal value as long as $X\neq 0$. Let’s also note that, since $Y>1$ by definition, there are no solutions for $T_d < T_{(q_1,q_2)}$ if $X=0$. Let’s look for a solution when $X$ is integer, that is, $X=1,2,3,...$. Then, keeping in mind the stability requirement (\[stab\]), that is, $Y-X^2=\delta >0$, the minimal vortex corresponding to $q_1 =- Xq_2, q_2=\pm 1$ has free energy lower than that of the simple vortex if $\delta <1$. In this case, the solution for $T_d < T_{(X,-1)}$ exists if $X\geq 3$. Proliferation of the composite vortex pairs corresponds to disordering of the original fields $\exp(i\phi_{1,2})$, while the composite field $\Psi= \exp(i (\phi_1 + X \phi_2))$ remains (algebraically) ordered. This mechanism constitutes the formation of thermally induced bound phases (or using the language of superfluidity – [*Thermally Paired Superfluid*]{}, TPS, [@TPS]). In other words, the system behaves in a such a way that the algebraic order persists only in the composite field $\Psi=\psi_1 (\psi_2)^X$.\[Since $X>1$ we call such a composite phase as thermally bound superfluid (TBS) by analogy with the TPS\]. This effect does not require that $X$ is necessarily integer. If $X$ is non-integer, its closest integer part will determine the power of $\psi_2$. If $\delta >1$, the lowest energy vortex is the simple one, and the solution for the inequality $T_d < T_{(q_1,q_2)}$ also requires $X\geq 3$. For $X>>1$, $T_d \to 0$ while $ T_{(q_1,q_2)} \to (\pi/2) {\rm min}(1,\delta)$ as long as $\delta $ is kept constant. Such a limit corresponds to the largest range of $T$ where SP are to be anticipated for the two-layer model. However, for practical purposes of simulations using too large $X$ leads to slower convergence. Thus, we choose $X=5,\, Y=25.5$ corresponding to a reasonably wide range where SP is anticipated to exist. Taking into account Eq.(\[K\]), the condition for the SP can be written as &lt; T &lt; . \[SPT\] Keeping in mind the chosen values $X=5, \delta=1/2$, this becomes $8\pi/73 < T < \pi /4$ or $ 0.344< T<0.785$. The results of the simulations will be conducted at the “temperature” $T$ in the middle of the interval $(T_d,T_{BKT})$, that is, $T\approx 0.565$. More specifically, $K_{11}=1/T,\, K_{22}= 25.5 K_{11},\, K_{12}=5K_{11}$. In terms of the original parameters $t_1, t_2, g_2, g$, the relations $gt_2|g_2|(1-5|g_2|)=5$, $gt_1(5.1|g_2|-1)=1$ , $t_1\approx 0.177 |g_2|/[(1-4|g_2|)(5.1|g_2|-1)$ and $10/51 < |g_2| <1/5$ must be satisfied. \[Such a fine tuning is of no concern, since the focus is on the consistency of the paradigm of SP rather than on physical origin of the values\]. Concluding this section, the presented analysis based on the RG finds that it is possible to find a range of temperatures where the sequence of phases is as presented in Fig. \[PD\] in the panel (a): at $T<T_d$ the Josephson coupling is relevant. We call this range as Josephson phase in Fig. \[PD\]. At $ T_d <T< T_{(q_1,q_2)}$ there is the SP where the symmetry $U(1)$ is promoted to U(1)$\times$ U(1). In the range $ T_{(q_1,q_2)}<T<T_n$ the TBS phase is characterized by the composite field $\Psi$. Thus, the broken symmetry is partially restored through the subgroup $Z_N$, where $N=1+[X]$. At higher temperatures $T>T_n$ the composite field $\Psi$ becomes disordered. -8mm -5mm Dual representation {#sec:dual} ------------------- The partition function $Z$ can be evaluated by the high-temperature expansion method (see e.g. in [@Parisi]) in terms of $t_1,t_2,u$ and the explicit integration over the variables. This approach allows obtaining $Z$ in terms of the integer bond variables – powers of the corresponding Taylor series. Since the resulting configurational space consists of closed loops of the bond currents, further simulations can be effectively performed by the Worm Algorithm [@WA]. As will be also shown, the language of loops also allows obtaining analytic expressions for the renormalized Josephson coupling $u_r$ which are exact in the asymptotic limit. We will be utilizing the Villain approximation [@Villain] for the cosines to obtain the so called J-current version [@Jcurr] of Eqs.(\[ZZ2\]),(\[2N\]): Z=\_[{m\_[a,ij]{}, m\_i}]{} DDA [e]{}\^[-H\_V]{}, \[Vill\] H\_V&=& \_[ij]{} \[ (\_[ij]{} \_1 - A\_[ij]{} +2m\_[1,ij]{})\^2\ &+& (\_[ij]{} \_2 -g\_2 A\_[ij]{} +2m\_[1,ij]{} )\^2 + A\^2\_[ij]{}\]\ &+& \_i (\_2(i)-\_1(i) +2m\_i)\^2, \[2Nv\] where $m_{a,ij}= - m_{a,ji}=0,\pm 1, \pm 2,...$ ($a=1,2$) are integer numbers defined along bonds between two nearest sites $i$ and $j$ along the planes and $m_i=0,\pm 1, \pm2, ...$ is an integer assigned to a site $i$ and oriented from the layer 1 to the layer 2. The Villain approximation proves to be very accurate for establishing the transition points as well as in general if the effective constants $\tilde{t}_1,\tilde{t}_2, u_V$ are properly expressed in terms of the corresponding bare values $t_1,t_2,u$ (see in Ref.[@Kleinert]). The “renormalization” can be essentially ignored for $t_1,t_2 \geq 1$, so that in what follows we will be using $\tilde{t}_1=t_1,\, \tilde{t}_2=t_2$. Similarly, for the Josephson coupling $u \sim 1$ one should take $u_V=u$ and, if $u<<1$, the corresponding relation is $u_V=1/(2 \ln (2/u))$ [@Villain; @Kleinert]. After using the Poisson identity for each integer and performing the integrations over $\phi_i$ and $A$, the resulting expression becomes Z=\_[{J\_[1,ij]{}},{J\_[2,ij]{}}, {J\_[z,i]{}}]{} [e]{}\^[-H\_J]{}, \[Z2\] H\_J= \_[ij ; a,b]{} (K\^[-1]{})\_[ab]{} J\_[a,ij]{} J\_[b,ij]{} + \_i J\_i\^2, \[H\_J\] where $a,b=1,2$ labels layers and $(K^{-1})_{ab}$ is the matrix inverse to $K_{ab}$ introduced in Eqs.(\[2N0\]),(\[Ktg\]); The summation runs over the integer bond currents $ J_{1,ij}, \, J_{2,ij}$ (oriented from site $i$ to site $j$ so that $ J_{a,ij}=- J_{a,ji},\, a=1,2$) within each corresponding layer 1,2 as well as over the integer currents $J_{i}$ oriented along the bond connecting the site $i$ in the layer 1 to the site $i$ in the layer 2. All the configurations are restricted by the Kirchhoff’s current conservation rule – the total of all J-currents incoming to any site must be equal to the total of all outcoming currents from the same site. It is useful to note that Eqs.(\[Z2\],\[H\_J\]) can also be obtained directly from Eq.(\[2N0\]) by reinstating the compact nature of the variables: $ \nabla_{ij} \phi_1 \to \nabla_{ij} \phi_1 +2\pi m_{1, ij}$ and $ \nabla_{ij} \phi_2 \to \nabla_{ij} \phi_2 +2\pi m_{2, ij}$ so that the matrix $K_{ab} $ is viewed as being independent from the parameters in the action (\[2N\]). The system (\[Z2\],\[H\_J\]) features statistics of closed loops. If $u=0$, there are two sorts of loops – one in each layer. Thus, each configuration is characterized by definite values of the windings $W_{a, \alpha}$ in the $a$th layer along the $\alpha=\hat{x},\hat{y}$ directions of the planes. It is straightforward to show that statistics of these windings determine the renormalized values $\tilde{K}_{ab}$ of the matrix $K_{ab}$ along the line of the approach [@Ceperley]. More specifically \_[ab]{} = \_[=,]{} W\_[a,]{} W\_[b,]{}. \[KR\] This expressions are valid for periodic boundary conditions (PBC). It is important to note that $\tilde{K}_{ab}$ represents an exact linear response (at zero momentum) with respect to Thouless phase twists. In other words, if there are externally imposed infinitesimal constant gradients $\nabla_\alpha \phi_{1,2} \to 0$ (violating the PBC) of the phases $\phi_{1,2}$, the free energy acquires the contribution $\delta F=\frac{1}{2}L^2 \sum_{a,b, \alpha} \tilde{K}_{ab}\nabla_\alpha \phi_{a}\nabla_\alpha \phi_{b} $. On the other hand, in the presence of the gradient the integrand of the partition function gets the factor $\exp(i L\sum_{a,\alpha} W_{a,\alpha} \nabla_\alpha \phi_a)$. Comparing both expressions leads to the relation (\[KR\]). As a test of consistency, we have checked numerically that in the regime where the SP state is supposed to exist (that is, $X=5,Y=25.5, T\approx 0.565$), the deviations of $\tilde{K}_{ab}$ from the bare values $K_{ab}$ are within the statistical error less than 1% for all tested sizes of the layers $10\leq L \leq 1000$. Significant deviations are observed only as the system approaches fully disordered state – that is, where the fields $\psi_{1,2}$ as well as the composite one $\Psi$ become disordered. In this case, $\tilde{K}_{ab}$ flow to zero as $L$ increases. The deviations remain small (about 2-3%) even in the regime where $\Psi$ is the only ordered field. The emergence of the TBS is detected by observing that windings $W_{a,\alpha}$ in the layers 1 and 2 are changing exactly by the increment $\Delta W_1= 1$, $\Delta W_2= X$ (plus or minus), respectively. At finite values of $u$ the loops belong to both layers so that no separate windings can be introduced. However, the sums $W_\alpha = W_{1,\alpha} + W_{2,\alpha}$ remain well defined and can be used to evaluate the rigidity $\rho_{\alpha}$ of the fields along the layers. In a general case of $N_z$ symmetric (with respect to the $x,y$ directions) layers $\rho=\rho_x =\rho_y$ : &=&\_W\^2\_ \[stif22\]\ W\_ &=& \_[ij , a=1,2,...N\_z]{} J\_[a,ij]{}, \[WW\] where for a given $\alpha=\hat{x},\hat{y}$ in (\[WW\]) the bond $\langle ij \rangle$ (as well as $J_{a,ij}$) is oriented along the direction $\alpha$. Our focus here on the renormalized value $u_r$ of the Josephson coupling $u$ in the SP regime. If the periodic boundary conditions are also imposed perpendicular to the layers (along $z$-direction), the inter-layer response $u_r$ is given by windings $W_z$ along $z$-direction: u\_r=W\^2\_z, W\_z = \_[i]{} J\_i, \[WU\] where the summation $\sum_i$ of the currents $J_i$ (oriented along $z$-direction) is performed over all sites of all layers. Similarly to the cases (\[KR\]) and (\[stif22\]), Eq.(\[WU\]) represents the full linear response at zero momentum – that is, the renormalized value $u_r$ of the Josephson coupling $u$. At this point, we should comment on how to interpret the PBC for two layers, $N_z=2$. While in the case $N_z\geq 3$ it is a natural procedure to link the $z=N_z$th layer to the first one, $z=1$, by the Josephson term, the case $N_z=2$ needs an auxiliary construction because the layers 1 and 2 are coupled already directly. The formal procedure, then, consists of adding a third layer, $z=3$, with no rigidity along $x,y$ directions and coupled by the Josephson term to both layers, $z=1,2$. If the coupling $u_{13}$ between the layers 1 and 3 and the coupling $u_{23}$ between the layers 2 and 3 add up as $1/u_{13} + 1/u_{23}=1/u_V$, in the dual action (\[H\_J\]) the sum in the last term can be extended to the layers $z=1,2,3$, while the first term is still confined to the layers $z=1,2$. The key to this procedure is the Kirchhoff’s rule: the J-current from a site $(x,y)$ along $z$-direction from the layer 2 to the layer 3 must be exactly the same as the current from the site $(x,y)$ in the layer 3 to the layer 1. Then, in the form (\[H\_J\]) the same value $u_V$ can be used for the currents from the layer 1 to the layer 2 directly or through the layer 3. Asymptotic expression for $u_r$ {#sec:AS} ------------------------------- As mentioned above, the dual representation allows obtaining analytically the asymptotic values for $u_r$ . Let’s begin with the trivial case of zero stiffnesses $K_{ab}$ and arbitrary number of layers, $N_z=2,3,4,$. In this case, the action (\[H\_J\]) trivially becomes H\_A= \_[i]{} J\^2\_i, J\_i=0,1, 2,... \[A\] where the summation runs over all sites $i$ of only [*one*]{} layer, say, $z=1$. In this expression the Kirchhoff rule dictates that the current $J_i$ at a given site along $z$-direction must be the same for all values of $z$. Thus, such a current with $J_i=\pm 1$ constitutes one closed loop characterized by the winding $W=J_i$. This allows constructing the partition function exactly as Z\_A=\^[L\^2]{} \[ZA\] where $L^2$ is the number of sites in one layer. The stiffness (\[WU\]) can be found by taking into account that the total winding along z-direction is $W_z=\sum_i J_i$, where the summation runs over $L^2$ sites of only one layer. Then, using statistical independence of different sites we find u\_r=. \[UR1\] This expression shows that, as long as $N_z$ is finite, the Josephson coupling remains relevant even if there is no in-plane order. In the limit $u_V <<1$ only the term $W=1$ is important, so that Eq.(\[UR1\]) becomes u\_r= . \[URR\] Eq.(\[URR\]) can be used even in the case when there is finite stiffness along the layers, with $N_z$ substituted by some effective value $M$, that is, u\_r= . \[GenM\] This is easily justified in the “ideal gas” approximation of rare fluctuations of the J-currents in z-direction. In the case of $K_{12}=0$ the asymptotic asymmetric limit corresponds to $K_{11}=K_{22} >> 1$ (so that the system is well above the disordering transition) and $u_V<<1$. The loop proliferation can be viewed from the perspective of the Worm Algorithm [@WA] where one open end of a string of J-currents walks randomly until it meets another open end so that the loop is formed. Then, the most of the path is residing in a layer with only occasional jumps between neighboring layers. These jumps are controlled by the exponential smallness of $\exp(-1/2u_V)$. Thus, the full action can be well approximated by $H_A$, Eq.(\[A\]), with $N_z=1$. This leads to Eq.(\[GenM\]) with $M=1$: u\_r= . \[urXY\] Later we will present the numerical evidence that the stiffness perpendicular to the layers of a simple XY layered model in the asymptotic limit can well be described by the above equation. Below we will show that for the case of the two asymmetric layers, the effective value of $M$ is $M=2$ in Eq.(\[GenM\]). Numerical results for $N_z=2$ {#sec:num} ----------------------------- Here we present the results of Monte Carlo simulations of the bilayer in the regime of SP in the limit $u_V<<1$. The action (\[H\_J\]) can be represented in the notations $T,X,Y,\delta$ as H\_J= \_[ij]{}+ \_i , \[H12\] where $J_{1,ij}$ and $J_{2,ij}$ refer to the inplane bond currents in the layers 1 and 2, respectively; the actual values of the parameters used in the simulations have been discussed at the end of Sec.\[sec:RG\]: $X=5, \delta=1/2, T=(T_d + T_{(X,-1)})/2 \approx 0.565$. The structure of the loops is determined by the energy of creating a J-current element along a given direction. A typical energy to create an additional J-current element in the plane 2 can be estimated as $ \delta E_2 \approx T/(2\delta) \approx 0.5$. Thus, large loops with a typical values $|\vec{J}_2|=1$ can exist in the plane 2. In contrast, the energy to create an isolated element in the plane 1 (with no $J_2$ currents along the same bond in the layer 2) costs much larger energy: $\delta E_1 \approx T(1+ X^2/\delta)/2 \approx 15 $ . Thus, the probability to create such an element is exponentially suppressed as $\sim \exp(-15)$, and no entropy contribution (due to 4 optional directions along the plane) can compensate for such a low value. This implies that no large isolated loops can exist in the layer 1. The only option to create a large loop in the layer 1 is if each element $J_{1,ij}$ is coupled to currents $J_{2,ij}=XJ_{1,ij}$ along the same bonds in the layer 2. A typical energy of this combined element is $\delta E_{12} \approx T/2 \approx 0.25$. This strong asymmetry between the layers has immediate implication for the windings along $z$-direction – the minimal length $M$ of the element $J_i$ must be $M=2$ in Eq.(\[GenM\]). Thus, the stiffness $u_r$ in the limit $u<<1$ becomes u\_r = 4[e]{}\^[-1/u\_V]{}= u\^2, \[UR3\] where the asymptotic expression $ u_V=\frac{1}{2 \ln (2/u)}$ [@Villain; @Kleinert] has been used. The results of the simulations is shown in Fig.\[figN2\]. -8mm -5mm The first striking feature to notice is that $u_r$ does not depend on the layers size $L$ over 2 orders of magnitude of $L$ and over 7 orders of $u_V$ (which is actually $\sim \ln(1/u)$ of the bare coupling). Second, the numerically found value $u_r$ follows the analytical result (\[UR3\]) with high accuracy – even for values $u_V \sim 1$. Both features are in the striking conflict with the RG prediction stating that $u_r$ should decay as $\propto L^{2(1-T/T_d)}\approx L^{-1.28} \to 0$ in the SP regime ($T>T_d$). It should be also noted that the stiffness along the layers (\[stif22\]) remains finite and much larger than $u_r$, that is, $\rho=32.3 \pm 0.1$ for all simulated sizes from $L=8$ to $L=960$. This justifies the validity of Eq.(\[UR3\]) even in the case $u_V \sim 1$. Extending the two-layer model to arbitrary number $N_z/2$ of pairs of layers {#Sec:Nz} ============================================================================ As it became evident from the previous analysis, no SP can occur in the double layer. Referring to the sketch of the possibilities, Fig. \[PD\], the option (b) is realized. Here we will address a possibility of SP in a $N_z$-layers setup. In other words, we will be looking for a behavior where the renormalized stiffness (the inter-layer Josephson coupling) $u_r$ decays as a function of $N_z$ in the limit $L \to \infty$, while the stiffness along planes remains finite.\[This would be a “weaker” version of the SP\]. RG solution {#sec:RGNz} ----------- We consider the PBC setup: the odd $z=1,3,5,7,...$ and the even $z=2,4,6,..$ layers are characterized by the inplane stiffnesses $K_{11}$ and $K_{22} > K_{11}$, respectively, with the nearest layers coupled by the current-current term $\propto K_{12}$ (the same for all pairs of layers) as well as by the Josephson coupling $- u \sum_{x,y,z}\cos(\phi_{z+1} -\phi_z)$ , where $\psi_z(x,y)=\exp(i\phi_z(x,y))$ is the XY variable defined on a site $(x,y)$ belonging to the layer $z$. In the linearized with respect to $\nabla_{ij} \phi_z$ approximation analogous to Eq.(\[2N0\]) the model becomes H\_0&=& \_[z=1,3,5,...]{}{ \_[ij]{} H\_[z;ij]{}\ &-& u \_[x,y]{} \[ (\_[z+1]{}-\_z) + (\_[z-1]{}-\_z))\]} \[NNz\] where the summation runs over odd values of $z$ and the notation H\_[z;ij]{} = (\_[ij]{} \_z)\^2 + (\_[ij]{} \_[z+1]{})\^2\ + K\_[12]{}\_[ij]{} \_[z]{}(\_[ij]{} \_[z+1]{}+\_[ij]{} \_[z-1]{}) \[HNNz\] is used. If the compact nature of $\phi_z$ is ignored, the one-loop RG flow equation for $u_r$ reads =(2- (\_[z+1]{} - \_z)\^2 \_S ) u\_r \[RGN\] where $\langle ... \rangle_S$ refers to the RG shell integration over the inplane momenta $\Lambda/(1+S) < |\vec{q}| <\Lambda$, with $\Lambda \sim 1/L$ being the cutoff and $S \to 0$. We note that, due to the PBC along $z$-direction, the mean $\langle (\phi_{z+1} - \phi_z)^2\rangle$ does not depend on $z$. Using discrete Fourier representation along $z$ direction with doubled unit cell containing two layers (the odd and the even) with two sorts of phases $\phi_z=\phi^{(1)}(z)$ and $\phi_z=\phi^{(2)}(z)$ along odd and even layers, respectively, the part $H_{z;ij}$ can be diagonalized and the correlator in Eq.(\[RGN\]) found. This gives Eq.(\[RGN\]) rewritten as =2(1- )u\_r, \[RGN2\] where T\^[-1]{}\_d=\^[(Nz/2)-1]{}\_[m=0]{}, \[TdNz\] and the wavevectors along $z$ take values dictated by the periodic boundary conditions $q_m=4\pi m/N_z,\, m=0,1,2,..., (N_z/2) -1$. Here we use the same notations $T=1/K_{11}, X=K_{12}/K_{11}, Y=K_{22}/K_{11}$ introduced in Sec.\[sec:RG\]. Thus, at $T>T_d$ RG predicts irrelevance of $u_r$. The upper limit on $T$ can be obtained from the requirement of no free 2D vortices in the limit $u=0$. There are, actually, two options: i) looking for a composite vortex characterized by phase windings $q_1$ and $q_2$ in odd and even layers, respectively, forming a string of length $N_z$ perpendicular to the layers; ii) considering independent vortices $q_1=\pm 1$ only in odd layers (characterized by smallest stiffness $K_{11}$) and characterized by the temperature $T_{(1,0)}= \pi/2$. As the analysis shows, the option when composite vortices form finite strings (say, $q_2=1$ in layers $z=1,3$ and $q_1=-2X$ in the layer $z=2$ by analogy with the $N_z=2$ case) costs larger energy than in the case ii). The option i) is characterized by the vortex energy $E_v= \pi K_{11} (N_z/2)[(q_1 +2Xq_2)^2 + (Y-4X^2)q_2^2]\propto N_z$. The minimum for a $2X$ integer is achieved at $q_1=-2Xq_2,\, q_2=\pm 1$. Thus, in order to compensate for the factor $N_z >>1$, the system must be very close to the instability $0<Y-4X^2 <2/N_z$. Simulations in this region turn out to be problematic. Thus, we will conduct simulations in the range $T_d<T<T_{(1,0)}$, provided the condition $T_d <T_{(1,0)}$ holds in the limit where $\delta=Y-4X^2$ remains finite for $N_z>>1$. Specifically, the condition $T_d <\pi/2$ reads \^[Nz/2 -1]{}\_[m=0]{} &gt; 8. \[TdT\] It can surely be achieved for large enough $X$ in the limit $ N_z >>1$. Replacing the summation by integration in this limit and considering $\delta <<1$, Eq.(\[TdT\]) gives $ \delta < (X /4\sqrt{2})^2$. For the simulations we have chosen $\delta =0.3$ and $X=6$, which gives $T_d \approx 0.983$ with $T= 1.28$ chosen in the middle of the interval between $T_{(1,0)}=\pi/2\approx 1.57$ and $T_d$. The chosen value of $T_d$ corresponds to the limit $N_z \to \infty$, and for any finite $N_z$, the actual $T_d$ from Eq.(\[TdNz\]) is below this value. At this point we note that for any finite $u$ the system is 3D and, strictly, speaking the notion of the BKT transition becomes inadequate for large enough $N_z$: even at $T>\pi/2$ the odd layers would still have XY order due to the proximity to even layers. Here, however, we presume that SP scenario holds and $u_r$ vanishes at large $L$. Practically, simulations are performed at finite $u$ and we always control that the XY stiffness (helicity modulus) along the layers remains finite and independent of $L$. Dual formulation ---------------- The dual formulation in terms of the closed loops of integer J-currents (along bonds in and between the layers) can be achieved similarly to the case $N_z=2$ by reinstating the compactness of $\phi_z$ in Eqs.(\[NNz\],\[HNNz\]) through the Villain approach: $\nabla_{ij} \phi_z \to (\nabla_{ij} \phi_z + 2\pi m_{z,ij})$ along the planes and $-u\cos(\phi_{z+1} - \phi_z) \to (u_V/2)(\phi_{z+1} - \phi_z + 2\pi m_{i,z})$ for Josephson coupling, where $m_{z,ij}$ refers to an arbitrary (oriented) integer defined on the bond $ij$ belonging to the plane $z$ and $m_{z,i}$ stands for an integer on a bond connecting site $i$ in the plane $z$ to the same site in the plane $z+1$. The partition function is obtained as a result of integration over all $\phi_z(i)$ and summations over all bond integers. The J-currents enter through the Poisson identity $\sum_{m=0,\pm 1, \pm 2,..} f(m) \equiv \sum_{J=0,\pm 1, \pm 2,..} \int dx \exp(2\pi \i J x)f(x)$ applied to each bond integer. This allows explicit integration over all phases $\phi_z$ as well as over the bond integers $m_{z,ij}, m_{i,z}$. There are two types of J-currents: inplane $J^{(a)}_{z,ij},\, a=1,2$ within each “elementary cell” (along $z$) and between the planes $J_{i,z}$. The label $a=1$ refers to J-current defined on the bond $ij$ belonging to a plane with odd $z$. Accordingly, $J^{(2)}_{z,ij}$ stands for the inplane current on the layer with even $z$. Then, $J_{i,z}$ denotes the current from the site $i$ from the plane $z$ to the plane $z+1$. The integration over phases $\phi$ generates the Kirchhoff constraint — similarly to the bilayer case. Finally the ensemble can be represented as Z&=& \_[{}]{} (- H\_J ),\ H\_J&=& \_[ij; z,z’]{}V\_[ab]{}(z-z’) J\^[(a)]{}\_[z,ij]{}J\^[(b)]{}\_[z’,ij’]{}\ &+& \_[i,z]{} J\^2\_[z,i]{}, \[Hdual\] where the matrix $V_{ab}(z-z')$ is defined in terms of the matrix $K_{ab}$. It reflects the asymmetry between odd and even layers. Explicitly, $V_{11} (z)=YV_{22}(z)$, for $z=z-z'$ being even, describes the interaction between odd layers, and $V_{22}(z)$ is defined between even layers; $V_{12}(z)= - X [V_{22}(z+1) +V_{22}(z-1)]$ refers to the interaction between odd and even layers (that is, $z$ is odd), and V\_[22]{}(z)= \_[q\_m]{} , \[Vzz22\] with $ z=0,\pm 2, \pm 4, ...$ and the summation running over $q_m=4\pi m/N, m=0,1,..., N/2 -1$. The dual formulation for $N_z$ layers allows obtaining the asymptotic expression for $u_r$ within the same logic used for deriving Eq.(\[UR3\]). We will repeat it here. The loop formation can be viewed as a process of random walks of two ends of a broken loop – exactly along the line of the Worm Algorithm [@WA]. Such a walk of each end is controlled by energetics of creating one bond element $|J|=1$ in a randomly chosen direction – either along a given plane or perpendicular to it. Very similar to the case of the two layers, the energy to create such an element alone along the odd layer costs energy $>>T \sim 1$, while the same element along an even layer costs energy $\sim 1$. The only option for creating a loop in an odd layer is if its energy is compensated by parallel elements in the even plane. This feature is caused by the strong current-current interaction $\sim X$. Thus, if the walk occurs along $z$-direction from some even layer $z$ toward the neighboring odd layer $z+1$, the subsequent move along the odd layer will be too energetically costly so that the walker would either move further toward $z+2$ layer or will go back to the original layer $z$. Thus, the inter-layer elements are characterized by either $J_{i,z}=J_{i,z+1}=\pm 1$ or $J_{i,z}=J_{i,z+1}=0$. The weight of such a process is either $\exp(-1/u_V)$ or $1$, respectively. Even if the walker makes a step or two along the layer $z+1$ (which is a highly improbable event) and then chooses to go toward the layer $z+2$, the contribution to the partition function will be further reduced exponentially by the energy of the element $J$ along the odd plane. Thus, such processes can be ignored, and we arrive at the conclusion that $u_r$ given by Eq.(\[UR3\]) must be valid for arbitrary $N_z$ in the asymptotic limit. Numerical results ----------------- The model (\[Hdual\]) has been simulated by the Worm Algorithm [@WA]. The renormalized inter-layer stiffness $u_r$ was found for a wide range of layer sizes, $6\leq L \leq 640$ and $10 \leq N<\leq 40$. The resulting data is presented in Figs. \[fig3\],\[fig4\]. As can be seen in Fig. \[fig3\], the solution (\[UR3\]) plays the role of the envelop for the family of the curves $u_r$ vs $1/u_V$ for various $L$ and $N$. We note that the stiffness $\rho$ along the layers (as determined by Eq.(\[stif22\])) remains independent of the sizes and much larger ($\rho=22.6 \pm 0.5$) than $u_r$. This justifies the applicability of the asymptotic limit for Eq.(\[UR3\]). We have also controlled that the system is far enough from any possible composite phases [@Kaurov] state by measuring the lowest order correlator $\langle \exp(i\phi_z(x,y))\exp(-i\phi_{z'}(x',y')) \rangle$ and observing that it exhibits long-range order for values $u\sim 1$ in the limit $N_z\sim L$.\[In the composite phase state such a correlator is short ranged\]. Thus, the system is well in the putative SP state. Its behavior, however, is drastically different from the RG prediction. -8mm -5mm -8mm -5mm -8mm -5mm -8mm -5mm At this point we should discuss the deviations of the numerical curves from the analytical result seen in Fig.\[fig3\]. To some extent this behavior could have been interpreted as the evidence of SP. There is, however, one important observation: the value of $u_V=u^*$ below which such suppression begins [*decreases*]{} as (u\^\*)\^[-1]{} = (L\^2/N\_z), =1.000.02 \[UgM2\] for $L^2/N_z >>1$ in the main logarithmic approximation. This behavior is demonstrated in Fig.\[fig4\], where the value $u^*$ corresponds the offset for $u_r$ being taken at 1/10 of the value given by the analytical expression (\[UR3\]). The suppression of $u^*\to 0$ in the limit $L\to \infty$ indicates that it is a finite size effect. In other words, starting from small $L$ at some $u_V <1$, the renormalized stiffness $u_r$ can be well below the asymptotic limit (\[UR3\]). As $L$ increases while $u_V$ is kept fixed, $u_r$ eventually approaches the limit (\[UR3\]). This feature is clearly in a stark contrast with the RG prediction of SP where $u_r$ is supposed to flow to zero for fixed $u_V$ as a power of $L \to \infty$. The deviation from the limit (\[UR3\]) has a very simple explanation: it is essentially a consequence of the generic exponential suppression of any order in a quasi-1D limit $N_z \to \infty$. \[There is no such suppression in the case of $N_z=2$ because the loops along z-direction are independent from the inplane ones\]. In the context of our system it can be interpreted as the effect of zero modes (in each layer) fluctuations. Indeed, excitations along the planes renormalize $u_V$ to $u_r$ at short distances. Then, as long as $u_r L^2 << K_{11}$ the only remaining lowest energy degrees of freedoms are zero modes, that is, excitations with $\phi_z=\phi^{(0)}_{z}$ being independent from the $x,y$ positions along the planes (and dependent on $z$). Then, the effective action becomes S\_0= -u\_rL\^2 \_z\[(\^[(0)]{}\_[z+1]{} - \^[(0)]{}\_[z]{})\]. \[zero\] It should be used in the partition function $Z=\int D\phi^{(0)}_z \exp(-S_0)$. Its analytical evaluation gives u\^[(0)]{}\_r \~( -N\_z/(u\_rL\^2)) \[zero\] for the value of the stiffness along $z$-direction – that is, the renormalised $u_r$ on large scale. Keeping in mind that at short scales $u_r \sim \exp(-1/u_V)$ for $u_V<<1$, one arrives at the relation (\[UgM2\]) with $\gamma=1$. It corresponds to the requirement $N_z/(u_rL^2) \approx 1$ for some $u_V=u^*$ so that Eq.(\[zero\]) describes the exponential suppression at $u_V <u^*$. Clearly, such a quasi-1D suppression is also present in the standard XY model (where no SP are anticipated to exist). In order to demonstrate this explicitly we have also simulated a simple XY model given by the system Z\_[XY]{}&=&D \_z (-H\_[XY]{}), \[Hxy\]\ H\_[XY]{}&=&- \_[ij ,z]{} \[ (\_[ij]{} \_z) + u (\_[z]{} \_z)\], with some $\tilde{K}>>1$ (guaranteeing that no BKT transition occurs in each layer for $u=0$), and $0<u<<\tilde{K}$. In the dual representation this system is described by H\_[XY]{} \_[XY]{} = \_[ij,z]{} J\_[z,ij]{}\^2 + \_[i,z]{} J\_[i,z]{}\^2, \[Hxy2\] where $J_{ij,z}$ and $J_{i,z}$ are the same J-currents introduced above for the model (\[Hdual\]). The results of the simulations of this model are presented in Fig.\[fig1\],\[fig2\]. In the asymptotic limit the inter-layer stiffness is described by Eq.(\[urXY\]). Then, according to the above discussion the value $u^*$ determining the start of the deviations is given by $(u^*)^{-1} = 2\ln(L^2/N_z)$, that is, with the slope $\gamma=2$ which should be compared with the numerical value $\gamma=1.95 \pm 0.05$ in Fig.\[fig2\]. Thus, both models demonstrate essentially the same 3D behavior. The only difference is the slope of the renormalized Josephson coupling $\ln u_r$ vs its bare value $u_V$. It is determined by the minimal length of the elementary J-current in the direction perpendicular to the layers. Discussion {#sec:dis} ========== The RG approach to 2D systems proves to be every effective in many cases including 2D XY model when it can be mapped on the Sine-Gordon (SG) one [@Polyakov]. A successful implementation of the RG analysis to the Josephson coupling was demonstrated in Ref.[@Kane_Fisher] where a single weak link can make one channel Luttinger liquid insulating. The merit of RG, however, should be taken with caution when applied to the dimensional reduction situations in layered systems. In this case there is no exact mapping between XY and SG representations at finite inter-layer Josephson coupling, and the approximation ignoring the compact nature of the variables becomes uncontrolled. As our analysis of one particular layered system shows, no SP exists in such a system despite the RG prediction: the system shows essentially the 3D behavior of the asymmetric XY model. While it is not obvious to us what is wrong with treating Josephson coupling between 2D layers by RG in the original representation of fields, the dual formulation in terms of the closed loops gives a very important insight. Specifically, the SP means that as layer size $L\to \infty$, a suppression of the Josephson coupling between layers would require that the number of times elements of closed loops fluctuate between layers must scale slower than $L^2$ so that the density of such events is zero in the limit $L=\infty$. The loops statistics, however, is controlled by local energies of creating finite elements and the entropy due to 6 directions in 3D vs 4 along layers. Thus, as long as there is a finite energy to cross between neighboring layers, the entropy will lead to a finite density of crossings for large enough $L$. Similar argument can be applied to quantum wires in terms of the quantum to classical mapping where imaginary time is treated as an extra dimension. The dual approach and the argumentation along the line of the numerical algorithm [@WA], treating closed loops formation as a process of the worm head wondering around and eventually finding its tail, allowed us to expose the actual stages of the renormalization of the Josephson coupling: i) At small scales Josephson coupling is controlled by exponentially suppressed random and independent (in the asymptotic limit) events of crossings between layers. It can be viewed as an ideal gas of J-currents between the layers. This stage leads to the renormalized coupling, in general, represented by Eq.(\[GenM\]) with $M=1,2,3,...$. ii) If the number of layers $N_z$ increases, with $L$ being fixed, quasi 1D fluctuations further suppress the coupling exponentially as demonstrated in Eq.(\[zero\]). Here we have discussed a local model characterized by short range interactions between the inter-layer J-current elements. This feature in combination with the low density of such elements justifies the “ideal gas” approximation for them, which in its turn leads to finite values of the renormalized inter-layer Josephson coupling. The question may be raised if a presence of long-range forces between the inter-plane J-currents $J_i$ can change the situation and lead to the SP or its weaker version – where $u_r \to 0$ with the growing number of layers $N_z$ in the limit $L=\infty$. In this respect we note that in order to realize this, fluctuations of the difference of the J-currents with positive and negative orientations must be macroscopically suppressed. In this case the fluctuation of the winding numbers in $z$-direction $\langle W_z^2 \rangle $ will scale slower than $L^2$ so that $ u_r \sim \langle W^2\rangle/L^2 \to 0$. This may be caused by interactions between the inter-layer J-currents decaying not faster than the second power of their separation. More specifically, the following additional repulsive term H\_[SP]{}=\_[i,j]{} U(\_i -\_j) J\_i J\_j \[SPX\] in the simple XY J-current model (\[Hxy2\]) with $ U(\vec{x})$ having the long range tail $\sim 1/|\vec{x}|^\sigma$ with $\sigma <2$ will generate the energy contribution $\sim W^2_z L^{-\sigma}$ in terms of the windings in z-direction. Consequently, the renormalized Josephson coupling (\[WU\]) would scale as $u_r \sim L^{\sigma -2} \to 0$. In one particular example long-range forces are introduced into the inter-layer Josephson in the standard XY model (\[Hxy\]) by some effective gauge-type term $- u\cos(\nabla_z \phi - g_z A_z) + (\vec{\nabla} A_z)^2$, where $\vec{\nabla} A_z$ refers to the derivatives along the layers of some soft mode $A_z$, with $g_z$ being a parameter. 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--- abstract: | This paper focuses on reinforcement learning (RL) with limited prior knowledge. In the domain of swarm robotics for instance, the expert can hardly design a reward function or demonstrate the target behavior, forbidding the use of both standard RL and inverse reinforcement learning. Although with a limited expertise, the human expert is still often able to emit preferences and rank the agent demonstrations. Earlier work has presented an iterative preference-based RL framework: expert preferences are exploited to learn an approximate policy return, thus enabling the agent to achieve direct policy search. Iteratively, the agent selects a new candidate policy and demonstrates it; the expert ranks the new demonstration comparatively to the previous best one; the expert’s ranking feedback enables the agent to refine the approximate policy return, and the process is iterated.\ In this paper, preference-based reinforcement learning is combined with active ranking in order to decrease the number of ranking queries to the expert needed to yield a satisfactory policy. Experiments on the mountain car and the cancer treatment testbeds witness that a couple of dozen rankings enable to learn a competent policy. author: - Riad Akrour - Marc Schoenauer - Michèle Sebag bibliography: - 'ourbib.bib' title: 'APRIL: Active Preference-learning based Reinforcement Learning' --- \ **Acknowledgments**. The first author is funded by FP7 European Project [*Symbrion*]{}, FET IP 216342, <http://symbrion.eu/>. This work is also partly funded by ANR Franco-Japanese project Sydinmalas ANR-08-BLAN-0178-01.

--- abstract: 'We review the status of non-perturbative analyses of multi-jet event shape distributions and mean values, highlighting the physical insight on QCD dynamics they can provide.' author: - 'A. Banfi' title: 'Why multi-jet studies?' --- INTRODUCTION ============ In every QCD observable, perturbative (PT) and non-perturbative (NP) dynamics are inseparable, the reason being that the observed degrees of freedom are not quarks and gluons, the elementary particles entering the QCD bare Lagrangian, but hadrons, whose description in terms of partons is well beyond the domain of PT theory. Fortunately, at least for sufficiently inclusive observables, the difference between parton and hadron level predictions is suppressed by inverse powers of the hard scale of the process. As far as the latter is short-distance dominated, one can safely compute QCD observables using the PT parton language, and interpret the discrepancy with experimental data, in case any is seen, as the need for NP hadronisation corrections. One can then adopt two complementary approaches. One is to consider observables whose hadronisation corrections are almost negligible, for instance total cross sections or inclusive non-QCD particle distributions. These observables can be computed in PT QCD and exploited to determine the value of the coupling ${\alpha_s}$ [@alpha-exp]. The other approach is to consider observables which are very sensitive to NP physics, in order to have an insight on the hadronisation mechanism. The best known example is event shape variable distributions and mean values. These variables are constructed by combining final state momenta to obtain a number that gives an idea of the geometrical properties of hadron energy-momentum flow. The value of an event shape $V$ is related to the scale at which hadrons are probed, so that measuring event shape distributions makes it possible to study physics at very different scales, which range from the domain of PT QCD ($V\sim 1$) down to the confinement region ($V \sim \Lambda_\mathrm{QCD}$) where the quark/gluon language is scarcely applicable. For this reason, although shape variables have been used to measure ${\alpha_s}$, they are ideal for investigating properties of QCD dynamics. So far, both experimental and theoretical investigations have been restricted only to two-jet event shapes (see [@DSreview] for a review). Here we will discuss how the extension of such studies to multi-jet event shapes can shed further light on the interplay between PT and NP physics in QCD observables. TWO-JET STUDIES =============== NP correction to event shapes {#sec:NPcorr} ----------------------------- Experimental data clearly indicates that PT QCD alone is not enough to predict event-shape distributions and mean values [@evshape-exp]. Let us consider the mother of all event shapes, the thrust [@thrustdef] $$\label{eq:thrust} T = \max_{\vec n_T}\frac{ \sum_h|\vec p_h \cdot \vec n_T|}{\sum_h |\vec p_h|}\>, \qquad \tau \equiv 1-T\>.$$ The thrust is a measure of particle alignment, and is a typical two-jet variable, since it vanishes in the limit of two narrow jets. As one can see from fig. \[fig:meanT\], the dependence of the mean value of $1-T$ on the $e^+e^-$ centre-of-mass energy $Q$ is correctly described only after adding to the QCD fixed order prediction a NP $1/Q$-suppressed correction $$\label{eq:tau-mean} {\left\langle \tau \right\rangle}={\left\langle \tau \right\rangle}_{\mathrm{PT}}+ {\left\langle \tau \right\rangle}_{\mathrm{NP}}\>,\qquad {\left\langle \tau \right\rangle}_{\mathrm{PT}}= {\alpha_s}(Q) \>\tau_1+{\alpha_s}^2(Q)\> \tau_2+\ldots \>,\qquad {\left\langle \tau \right\rangle}_{\mathrm{NP}}\simeq \frac{C_\tau}{Q}\>,$$ where $C_\tau\simeq 1\mathrm{GeV}$ when ${\left\langle \tau \right\rangle}_{\mathrm{PT}}$ is evaluated at next-to-leading order (NLO). {width=".5\textwidth"} One might think that such a discrepancy could be removed by including higher orders in the PT expansion. Actually Sterman observed that a term $18 \>\alpha_s^3$ can already mimic a $1/Q$ behaviour [@Sterman]. However, theoretical analyses show that the PT series is an intrinsically ill-defined object, since it is doomed to diverge factorially (for a review, see [@Beneke]). Attempts to regularise such a divergence give rise to a power-suppressed ambiguity, known as infrared (IR) renormalon. Looking more closely at the origin of the divergence, one can see that it arises when resumming [*renormalon chain*]{} graphs containing an arbitrary number of linked quark or gluon bubbles [@renormalon]. A renormalon resummation for the thrust mean value yields $$\label{eq:tau-PT} {\left\langle \tau \right\rangle}_{{\mathrm{PT}}} \simeq \frac{4C_F{\alpha_s}}{\pi}\sum_{n=0}^\infty n!\left(\frac{2\beta_0{\alpha_s}}{4\pi}\right)^n n^{\beta_1} \simeq 2 C_F \int\frac{dk_t}{k_t}d\eta \> \frac{{\alpha_s}(k_t)}{\pi} \>\frac{k_t}{Q}e^{-|\eta|}\>,$$ where $\beta_0$ and $\beta_1$ are the first two coefficients of the QCD beta function. After a renormalon analysis, the series in eq. (\[eq:tau-PT\]) gives a $1/Q$ ambiguity. The last equality in eq. (\[eq:tau-PT\]) results from the fact that the series can be seen as the PT expansion of the integral of the running coupling ${\alpha_s}(k_t)$ down to the infrared. Since the PT coupling develops a Landau singularity at low momenta, one may be tempted to ascribe the divergence to the presence of the Landau pole in the $k_t$ integration contour. However, eq. (\[eq:tau-PT\]) shows that the divergence of ${\left\langle \tau \right\rangle}_{\mathrm{PT}}$ is determined only by $\beta_0$ and $\beta_1$, i.e. it is independent of the particular coupling adopted. This naive observation is confirmed by more refined theoretical analyses, which show that IR renormalons are always present whatever is the behaviour of the coupling at low scales [@DU]. The main message of this discussion is that, in order to obtain a satisfactory [*theoretical*]{} understanding of event shape observables, PT theory must always be supplemented with information on QCD dynamics at low scales. In particular, a size of the power correction of around $1\mathrm{GeV}$ suggests that NP effects arise from partons which have started the blanching process that leads to the formation of hadrons. Two-jet shapes and the Feynman tube model {#sec:Feynman} ----------------------------------------- Consider now the specific case of an event-shape $V$ in $e^+ e^-$ annihilation that vanishes in the two-jet limit (thrust, $C$-parameter, heavy-jet mass, etc.). The mean value ${\left\langle V \right\rangle}$ will receive its main contribution from hard particles whose transverse momenta (with respect to the thrust axis) are of the order of the hard scale $Q$. There are however soft hadrons, with transverse momenta up to about $1\mathrm{GeV}$, whose contribution to ${\left\langle V \right\rangle}$ cannot be fully computed with PT techniques. The Feynman tube model [@tube] gives a phenomenological description of such hadrons by assuming that their distribution is uniform in rapidity: $$\label{eq:dnh} \frac{dn_h}{d\ln k_t d\eta} = \Phi_h(k_t)\>.$$ Observing also that their contribution $\delta V$ to $V$ is additive $$\label{eq:deltaV} \delta V \simeq \sum_i \frac{k_{ti}}{Q} f_V(\eta_i)\>,$$ one obtains a $1/Q$-suppressed NP correction ${\left\langle V \right\rangle}_{\mathrm{NP}}$ to ${\left\langle V \right\rangle}$, given by $$\label{eq:VNP} {\left\langle V \right\rangle}_{\mathrm{NP}}= \sum_h \int \frac{dk_t}{k_t}d\eta \>\Phi_h(k_t) \frac{k_t}{Q} f_V(\eta) = c_V\frac{{\left\langle k_t \right\rangle}_{\mathrm{NP}}}{Q}\>,\qquad c_V = \int_{-\!\infty}^\infty\!\!\! d\eta \>f_V(\eta)\>,$$ where $c_V$ is a calculable coefficient and $$\label{eq:ktnp} {\left\langle k_t \right\rangle}_{\mathrm{NP}}= \sum_h \int \frac{dk_t}{k_t} \Phi_h(k_t)\> k_t\>,$$ is the average transverse momentum of the produced hadrons, and represents a genuine NP quantity. The assumption of eq. (\[eq:dnh\]) makes it possible to completely factorise rapidity and transverse momentum dependence. Furthermore, eq. (\[eq:VNP\]) implies that power corrections to two-jet event shapes are [*universal*]{}, in the sense that, after computing the variable dependent coefficient $c_V$, they depend only on the NP parameter ${\left\langle k_t \right\rangle}_{\mathrm{NP}}$, which is the same for all variables. A comment is in order concerning the validity of eq. (\[eq:dnh\]). Since in a hard collision most particles are aligned along the thrust axis, one expects that the distribution of hadrons away from the jets should be invariant over small boosts along the thrust axis direction, but that, as soon as one moves forward in rapidity, eq. (\[eq:dnh\]) gets modified. Fortunately, for most event shapes $f_V(\eta)$ strongly suppresses the contribution of hadrons at large rapidities, so that leading power corrections are determined only by soft hadrons in a central rapidity region, which are well described by eq. (\[eq:dnh\]).[^1] This very same approach can be used to compute power correction to event-shape distributions. Since $V=V_{\mathrm{PT}}+\delta V$ with $\delta V \sim \epsilon/Q$, the distribution $d\sigma/dV$ can be seen as a convolution of a PT distribution $d\sigma_{\mathrm{PT}}/dV_{\mathrm{PT}}$ and a NP shape function $f_{\mathrm{NP}}(\epsilon)$ [@shape-fun] $$\label{eq:dist-PT-NP} \frac{1}{\sigma}\frac{d\sigma}{dV} = \int dV_{\mathrm{PT}}\> d\epsilon \>\frac{1}{\sigma}\frac{d\sigma_{\mathrm{PT}}}{dV_{\mathrm{PT}}}(V_{\mathrm{PT}}) \>f_{\mathrm{NP}}(\epsilon)\> \delta\left(V-V_{\mathrm{PT}}-\frac{\epsilon}{Q}\right)\>.$$ In the region $VQ \gg {\left\langle \epsilon \right\rangle}$ one can expand $f_{\mathrm{NP}}(\epsilon)$ around ${\left\langle \epsilon \right\rangle}$ and obtain $$\frac{1}{\sigma}\frac{d\sigma}{dV} \simeq \frac{1}{\sigma}\frac{d\sigma_{\mathrm{PT}}}{dV} \left(V-\frac{{\left\langle \epsilon \right\rangle}}{Q}\right)\>.$$ Recalling now that ${\left\langle \epsilon \right\rangle}\!/Q = {\left\langle V \right\rangle}_{\mathrm{NP}}$, one observes that leading power corrections to $d\sigma/dV$ result simply in a shift of the corresponding PT distribution, whose magnitude is the same as the power correction to ${\left\langle V \right\rangle}$ [@DW]. To conclude, the basic assumption of the Feynman tube model, that the distribution of soft (central) hadrons is uniform in rapidity, implies that leading $1/Q$ NP corrections both to mean values and distributions can be expressed in terms of a single universal parameter ${\left\langle k_t \right\rangle}_{\mathrm{NP}}$. This universality hypothesis has been tested against experimental data and is found to hold within $20\%$ accuracy [@DSreview]. EXTENSION TO MULTI-JET EVENT SHAPES =================================== Intra-jet hadron distribution {#sec:intra-jet} ----------------------------- When moving from two-jet to multi-jet events one encounters difficulties in extending eq. (\[eq:dnh\]). One can reasonably think that particles inside one of the jets (inter-jet hadrons) are produced uniformly in rapidity and azimuth with respect to the jet axis. However, for particles at large angles with respect to all jets (intra-jet hadrons), since there is no natural way to define $\eta$ and $\phi$, one does not expect $dn_h$ to have a simple expression. To find a reasonable assumption for $dn_h$, we start from two-jet events and consider the PT probability $dw(k)$ for the emission of a soft dressed gluon $k$ off a back-to-back quark-antiquark system (whose momenta are $p$ and $\bar p$) in a colour singlet [@thrust-np]: $$\label{eq:dw-2jet} dw(k) = C_F \frac{dk_t^2}{k_t^2} d\eta \frac{d\phi}{2\pi} \frac{{\alpha_s}(k_t)}{\pi}\>,\qquad \eta = \frac 12\ln\frac{\bar p k}{pk}\>,\qquad k_t^2 = \frac{(2pk)(2k \bar p)}{2p\bar p}\>,$$ where $\eta$ and $k_t$, the gluon rapidity and transverse momentum, have been written in a Lorentz invariant form and the coupling is taken in the physical CMW scheme [@CMW]. The inter-jet hadron distribution in eq. (\[eq:dnh\]) can be obtained from eq. (\[eq:dw-2jet\]) by replacing $2\>C_F {\alpha_s}(k_t)/\pi$ with $\sum_h \Phi_h(k_t)$. In this way $dn_h$ might be interpreted as an effective measure of QCD interaction strength in the infrared. When the number of hard emitters is larger than two, the distribution $dw(k)$ can be written as a sum of contributions from all dipoles formed by the hard partons [@CatGraz]: $$\label{eq:dw-multi} dw(k) = \sum_{i<j}(-\vec T_i\cdot \vec T_j) \frac{d\kappa^2_{ij}}{\kappa^2_{ij}} d\eta_{ij} \frac{d\phi_{ij}}{2\pi} \frac{{\alpha_s}(\kappa_{ij})}{\pi}\>,\qquad \eta_{ij} = \frac 12 \ln\frac{p_j k}{p_i k}\>, \qquad \kappa_{ij}^2 = \frac{(2p_i k)(2k p_j)}{2p_i p_j}\>,$$ where now $\kappa_{ij}$, $\eta_{ij}$ and $\phi_{ij}$ are the transverse momentum, rapidity and azimuth in the $(ij)$-dipole centre-of-mass frame, and the coupling is again in the CMW scheme. The vector $\vec T_i$ represents the colour charge of hard parton $p_i$. From colour conservation one has that for less than four emitters $\vec T_i\cdot \vec T_j$ are numbers, while starting from four partons they are actual matrices in colour space. We now postulate that, after a suitable extension of the CMW coupling at low scales has been introduced, eq. (\[eq:dw-multi\]) gives the distribution of intra-jet hadrons in a multi-jet event. This corresponds to the so-called local parton-hadron duality (LPHD) hypothesis (see [@LPHD] for a recent review), which states that hadron flow is determined by parton flow. This assumption has solid phenomenological bases, since it describes very well hadron multiplicity flows [@multi] and string/drag effects in three-jet events [@drag]. Before constructing such an extended coupling, we first discuss to what extent the soft large-angle approximation is sufficient to compute $1/Q$ power corrections to event shapes. In fact, $\kappa_{ij}$ can reach the NP domain not only for soft large-angle, but also for hard collinear emissions. Considering emissions collinear to a given leg, we can classify event shapes according to whether they are damped in rapidity or not. If $V$ is damped in rapidity, soft collinear particles with transverse momentum $k_{ti}$ and rapidity $\eta_i$ with respect to the given leg contribute to $V$ with a correction $$\label{eq:deltaV-damped} \delta V \simeq \sum_i \frac{k_{ti}}{Q} f_V(\eta_i)\>,\qquad f_V(\eta) \sim e^{-\eta} \>, \quad\>\eta\to\infty\>.$$ This implies that only large angle emissions contribute to $\delta V$, up to higher power corrections. If $V$ is uniform in rapidity, we have $\delta V \sim \sum_i k_{ti}$, so that one might think that all rapidities contribute equally to $\delta V$. However, due to PT radiation, each hard parton takes a recoil, and acquires a transverse momentum $p_t \sim VQ$. But one measures transverse momenta with respect to a fixed axis (e.g. the thrust axis), and not with respect to the hard emitter’s direction (the jet broadenings [@broadNP] are a relevant example). Since, as we have seen in the previous section, leading power corrections come from the region $\delta V \ll V$, we must have then $k_{ti}/Q \sim \delta V \ll p_t/Q \sim V$. This means that particles giving the leading contribution to $\delta V$ must be soft, and displaced from the hard emitting parton, i.e. soft at large angles. To conclude, eq. (\[eq:dw-multi\]) gives the physically correct starting point to compute $1/Q$ power corrections to event-shape variables. Dokshitzer-Marchesini-Webber (DMW) extension of CMW coupling {#sec:DMW} ------------------------------------------------------------ The CMW coupling can be extended at low scales via the dispersion relation [@DMW] $$\label{eq:DMW} {\alpha_s}(\kappa) = -\int_0^\infty\!\!\frac{dm^2}{(m^2+\kappa^2)}\>\rho_s(m^2)= \kappa^2 \int_0^\infty\!\!\!\frac{dm^2}{(m^2+\kappa^2)^2}\>{\alpha_\mathrm{eff}}(m^2)\>,$$ where ${\alpha_\mathrm{eff}}(m^2)$ is the logarithmic derivative of the spectral density $\rho_s(m^2)$, and is itself a QCD coupling. This approach automatically incorporates the LPHD hypothesis by assuming that hadronisation corrections are due to [*gluers*]{}, extra-soft gluons with transverse momenta $\kappa \sim \Lambda_\mathrm{QCD}$, whose emission probability is ruled by the NP part of the dispersive coupling. Within the dispersive DMW approach, power corrections to event shapes can be computed by considering the emission of a single gluer, and following a standard procedure [@np-ee]: 1. real and virtual corrections are combined to obtain the dispersive representation of the QCD coupling. This means that in eq. (\[eq:dw-multi\]), for each dipole, one can make the substitution $$\label{eq:dw-DMW} \frac{{\alpha_s}(\kappa)}{\kappa^2} = \int_0^\infty\!\!\!\frac{dm^2}{(m^2+\kappa^2)^2}{\alpha_\mathrm{eff}}(m^2)\>;$$ 2. the gluer is given a mass $m^2$, i.e. it is allowed to decay inclusively. In practice this results in replacing everywhere in the event-shape definition $\kappa^2$ with $\kappa^2+m^2$, so that the contribution to a variable $\delta V^{(ij)}_\mathrm{gluer}$ of a gluer emitted off the $(ij)$-dipole is $$\label{eq:dV-gluer} \delta V_\mathrm{gluer}^{(ij)} = \frac{\sqrt{\kappa^2+m^2}}{Q} f_V^{(ij)}(\eta,\phi)\>,$$ where $\kappa$, $\eta$ and $\phi$ are, as usual, considered in the emitting dipole centre-of-mass frame. Then the leading power correction to ${\left\langle V \right\rangle}$ becomes $$\label{eq:dV-naive} {\left\langle V \right\rangle}^\mathrm{naive}_{\mathrm{NP}}= \frac{{\left\langle \kappa \right\rangle}_\mathrm{naive}}{Q} \sum_{i<j} (-\vec T_i \cdot \vec T_j)\> c_V^{(ij)}\>, \qquad c_V^{(ij)} = \int d\eta \frac{d\phi}{2\pi} f_V^{(ij)}(\eta,\phi)\>,$$ where $$\label{eq:kappa-naive} {\left\langle \kappa \right\rangle}_\mathrm{naive} = \int_0^\infty \frac{d\kappa^2 dm^2}{(m^2+\kappa^2)^2} \sqrt{\kappa^2+m^2} \>\frac{{\alpha_\mathrm{eff}}(m^2)}{\pi} \simeq \frac{2}{\pi} \int_0^\infty \frac{dm^2}{m^2} m \>\delta {\alpha_\mathrm{eff}}(m^2)\>,$$ turns out to be related to the $1/2$-moment of $\delta {\alpha_\mathrm{eff}}(m^2)$, the NP part of the dispersive coupling; 3. non-inclusiveness of event shapes with respect to secondary gluon decay is accounted for by replacing ${\left\langle \kappa \right\rangle}_\mathrm{naive}$ with ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}\equiv {\cal M}{\left\langle \kappa \right\rangle}_\mathrm{naive}$, where ${\cal M}$ is the so-called [*Milan*]{} factor [@thrust-np; @Milan]. It is also customary to rewrite ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}$ in terms of $\alpha_0(\mu_I)$, the average of the dispersive coupling below the merging scale $\mu_I$, in such a way that the sum of PT and NP contributions to ${\left\langle V \right\rangle}$ is free of IR renormalons: $$\label{eq:alpha0} {\left\langle \kappa \right\rangle}_{\mathrm{NP}}= \frac{4 \mu_I}{\pi^2}{\cal M} \left(\alpha_0(\mu_I)-{\alpha_s}(Q)+{\cal O}({\alpha_s}^2)\right)\>, \qquad \alpha_0(\mu_I) = \int_0^{\mu_I}\frac{dk}{\mu_I}{\alpha_s}(k)\>.$$ We are now able to discuss what kind of information on QCD dynamics we obtain from power corrections to two- or multi-jet event shapes. From eq. (\[eq:dV-naive\]) the power correction to a two-jet variable can be written in the form $$\label{eq:dV-2jet} {\left\langle V \right\rangle}_{\mathrm{NP}}= C_F \frac{{\left\langle \kappa \right\rangle}_{\mathrm{NP}}}{Q} c_V = \frac{{\left\langle k_t \right\rangle}_{\mathrm{NP}}}{Q} c_V\>,$$ where ${\left\langle k_t \right\rangle}_{\mathrm{NP}}$ is the NP parameter appearing in eq. (\[eq:ktnp\]), and the last equality has been obtained by comparing eq. (\[eq:VNP\]) and eq. (\[eq:dV-naive\]). We see then that both the dispersive approach and the Feynman tube model give the same result for ${\left\langle V \right\rangle}_{\mathrm{NP}}$. In fact universality of leading NP corrections to two-jet event shapes implies simply that central soft hadrons are produced uniformly in rapidity. When increasing the number of jets, the NP correction to a multi-jet variable is given by $$\label{eq:dV-multi} {\left\langle V \right\rangle}_{\mathrm{NP}}=\frac{{\left\langle \kappa \right\rangle}_{\mathrm{NP}}}{Q} \>\tilde c_V\>, \qquad \tilde c_V\equiv\sum_{i<j} (-\vec T_i \cdot \vec T_j) \> c_V^{(ij)}\>.$$ The main features of this PT QCD inspired correction are: 1. the same parameter ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}$ determines the magnitude of power corrections for [*all*]{} event shapes and for [*any number*]{} of hard jets; 2. ${\left\langle V \right\rangle}_{\mathrm{NP}}$ depends on the [*colour*]{} charges of emitting partons through the factors $\vec T_i \cdot \vec T_j$; 3. the power correction is sensitive to the [*geometry*]{} of the underlying hard event (the angles between the jets) through the coefficients $c_V^{(ij)}$. This colour and geometry dependence is a highly non-trivial property that, once established, would imply that hadronisation preserves the main characteristics of partonic energy-momentum flow. Three-jet event shapes in $e^+e^-$ annihilation {#sec:3jet-ee} ----------------------------------------------- We now consider the specific case of three-jet events in $e^+e^-$ annihilation. At Born level a three-jet event is made up of a quark $p_q$, an antiquark $p_{\bar q}$ and a gluon $p_g$. Transverse momenta are, as usual, defined with respect to the thrust axis $\vec n_T$, and one defines an event-plane as the one containing $\vec n_T$ and thrust major axis $\vec n_M$, defined as the one maximising the projection of transverse momenta. Every transverse momentum $\vec k_t$ can be decomposed into an in-plane component $k^\mathrm{in}$ and an out-of-plane component $k^{\mathrm{out}}$ as follows $$\label{eq:in-out} \vec k_t = k^\mathrm{in} \>\vec n_\mathrm{in} + k^{\mathrm{out}}\>\vec n_{\mathrm{out}}\>,\qquad \vec n_\mathrm{in}=\vec n_M\>, \quad \vec n_{\mathrm{out}}\equiv \vec n_T\times\vec n_M\>.$$ The two variables for which there exists a prediction for leading power corrections are the $D$-parameter [@dpar] and the thrust-minor $T_m$ (a.k.a. ${K_\mathrm{out}}$) [@kout-ee]. The $D$-parameter is defined as the determinant of the tensor $$\label{eq:Dpar} \theta^{\alpha\beta} Q \equiv \sum_h \frac{p_h^\alpha p_h^\beta}{|\vec p_h|}\>, \qquad D \equiv 27 \det \theta = 27 \lambda_1 \lambda_2 \lambda_3 \>.$$ The soft particle contribution $\delta D$ is given by $$\label{eq:deltaD} \delta D \simeq 27 \lambda_1 \lambda_2 \sum_i \frac{\kappa_i^2 \sin^2 \phi_i}{\omega_i Q} \>,$$ where $\kappa_i$ is the transverse momentum in the emitting dipole centre-of-mass frame and $\phi_i=0$ for an emission inside the event plane. Due to the presence of the energy $\omega_i$ in each denominator of eq. (\[eq:deltaD\]), this variable is damped in rapidity, and the coefficient $\tilde c_D$ is given by $$\label{eq:cD} \tilde c_D = C_F c_D^q(T,T_M) + C_F c_D^{\bar q}(T,T_M) + C_A c_D^g(T,T_M)\>,$$ where $c_D^q$, $c_D^{\bar q}$ and $c_D^g$ depend on the event geometry through $T$ and $T_M$. The thrust-minor $T_m$ measures the out-of-event-plane momentum flow: $$\label{eq:Tm} T_m Q = {K_\mathrm{out}}\equiv \sum_h |p_h^{\mathrm{out}}| \>,\qquad \delta{K_\mathrm{out}}\simeq \sum_i \kappa_i |\sin \phi_i| \>.$$ This variable is uniform in rapidity, so that one needs to consider the fact that each hard emitting parton recoils against PT radiation. This results in $$\label{eq:ckout} \tilde c_{{K_\mathrm{out}}} \simeq {\left\langle |\sin\phi| \right\rangle}\left(C_F\ln\frac{Q_{q\bar q}}{|p_q^{\mathrm{out}}|}+ C_F\ln\frac{Q_{q\bar q}}{|p_{\bar q}^{\mathrm{out}}|}+ C_A\ln\frac{Q_{qg} Q_{g\bar q}}{Q_{q\bar q} |p_g^{\mathrm{out}}| }\right)\>,\qquad {\left\langle |\sin\phi| \right\rangle}\equiv\int_0^{2\pi}\frac{d\phi}{2\pi}|\sin \phi|=\frac 2\pi \>,$$ where $Q^2_{ij}=2 p_j \cdot p_j$. We notice that for each hard emitter the power correction is proportional to its colour charge and to the rapidity interval available to the emitted gluon. The fact that the NP correction depends on hard parton recoil implies that the final answer will be obtained after averaging $\tilde c_{{K_\mathrm{out}}}$ over the PT recoil distribution, which results in a complicated but very interesting interplay between PT and NP physics. Fig. \[fig:Dpar\] shows the comparison between theoretical predictions [@dpar] and experimental data [@aleph-data] for the $D$-parameter differential distribution and mean values. Three-jet events are selected by imposing a lower limit $y_\mathrm{cut}\!=\!0.1$ on the (Durham) three-jet resolution $y_3$. The distribution in fig. \[fig:Dpar\] represents the first theoretical prediction for a three-jet variable at the state-of-the-art accuracy, which includes all-order PT resummation of logarithmic enhanced contribution to next-to-leading logarithmic (NLL) accuracy, matching to the corresponding NLO distribution obtained with <span style="font-variant:small-caps;">nlojet</span>++ [@nlojet], and $1/Q$ NP power corrections. For each Born configuration, the leading NP correction results in a shift of the PT distribution. The presented distribution is also slightly squeezed since the shift depends on the event geometry and one has to integrate over different Born configurations. ![Theoretical predictions [@dpar] for distribution (left) and mean values (right) of the $D$-parameter compared to ALEPH data [@aleph-data], for $Q=91.2\mathrm{GeV}$. Mean values for the data are computed from the corresponding distributions, with a $2\%$ estimated experimental error and an extra $1\%$ systematic error to account for the extraction of mean values from distributions [@hasko].[]{data-label="fig:Dpar"}](dpar_ee-th+dat.eps "fig:"){width=".45\textwidth" height=".3\textheight"} ![Theoretical predictions [@dpar] for distribution (left) and mean values (right) of the $D$-parameter compared to ALEPH data [@aleph-data], for $Q=91.2\mathrm{GeV}$. Mean values for the data are computed from the corresponding distributions, with a $2\%$ estimated experimental error and an extra $1\%$ systematic error to account for the extraction of mean values from distributions [@hasko].[]{data-label="fig:Dpar"}](dpar_ee-means.eps "fig:"){width=".45\textwidth" height=".3\textheight"} The comparison to data is overall quite good. One can observe that the shift is quite large, about twice as much as in two-jet event shapes. This is mainly due to NP radiation off a gluon, which has a large colour charge. Although a full analysis of theoretical uncertainties remains to be performed [@AG-hardwork], the magnitude of the shift suggests that leading power corrections might not describe with sufficient accuracy the distribution around the peak, but that higher powers, or even a shape function, might be needed. Three-jet event shapes in hadron-hadron collisions {#sec:3jet-hh} -------------------------------------------------- One can construct three-jet event shapes also in hadron-hadron collisions. The simplest case is to consider a process with an electro-weak vector boson $Z^0$ or $W^\pm$ recoiling against a high-$p_t$ jet. The process involves three hard emitters, two incoming ($p_1$ and $p_2$) and one outgoing ($p_3$). The event plane is the one containing the beam and the momentum $\vec q$ of the vector boson, so that we have $\vec n_\mathrm{out} = (\vec q \times \vec p_1)/(|\vec q||\vec p_1|)$. Since one does not measure inside the beam pipe, any measured hadron $p_h$ will have $|\eta_h| < \eta_0$, where $\eta_0$ is the maximum available rapidity. Therefore the out-of-plane momentum flow ${K_\mathrm{out}}$ has to be defined as [@kout-hh] $$\label{eq:kout-hh} {K_\mathrm{out}}\equiv \sum_h |p_h^{\mathrm{out}}| \times \Theta(\eta_0-|\eta_h|)\>,$$ In the soft limit $\delta{K_\mathrm{out}}$ gets two contributions, one from particles emitted directly from the three hard partons, the other from hadrons produced in beam remnant interactions, the so-called soft underlying event [@pedestal]: $$\label{eq:deltakout-hh} \delta{K_\mathrm{out}}= \delta{K_\mathrm{out}}^\mathrm{gluer}+\delta{K_\mathrm{out}}^\mathrm{remnant}\>,\qquad \delta{K_\mathrm{out}}^\mathrm{gluer} \simeq {\sum_i}^\prime \kappa_i |\sin\phi_i| \>, \qquad \delta{K_\mathrm{out}}^\mathrm{remnant} \simeq{\sum_i}^\prime k_{ti} |\sin\phi_i|\>,$$ where each primed sum is restricted to particles outside the beam pipe. In the term $\delta{K_\mathrm{out}}^\mathrm{gluer}$ the transverse momentum $\kappa_i$ is considered as usual in the emitting dipole centre-of-mass frame, while in $\delta{K_\mathrm{out}}^\mathrm{remnant}$ each $k_{ti}$ is in the laboratory frame. As before, the gluer contribution depends on the same parameter ${\left\langle \kappa \right\rangle}_{\mathrm{NP}}$ as for $e^+e^-$ event shapes, and is given by: $$\label{eq:deltakout-gluer} \delta{K_\mathrm{out}}^\mathrm{gluer} = {\left\langle \kappa \right\rangle}_{\mathrm{NP}}\> \tilde c_{{K_\mathrm{out}}} \>, \qquad \tilde c_{{K_\mathrm{out}}} \simeq {\left\langle |\sin\phi| \right\rangle} \left(C_1(\eta_0-\eta_3)+C_2(\eta_0+\eta_3)+ C_3 \ln\frac{Q_t}{|p_3^{\mathrm{out}}|}\right)\>,$$ where $Q_t$ is the transverse momentum of the vector boson, which represents also the hard scale of the process, and $C_i$ is the colour charge of hard parton $p_i$. As in the $e^+e^-$ case, for each emitting hard parton, the power correction is proportional to the rapidity interval available to the emitted gluer. The beam remnant contribution can be computed by assuming that the distribution of hadrons produced in soft beam-remnant interactions is uniform in rapidity, which gives $$\label{eq:deltakout-remnant} \delta{K_\mathrm{out}}^\mathrm{remnant} = {\left\langle |\sin\phi| \right\rangle}\times{\left\langle k_t \right\rangle}_\mathrm{remnant} \times 2\eta_0\>.$$ We have then access to the new NP parameter ${\left\langle k_t \right\rangle}_\mathrm{remnant}$, the average transverse momentum of hadrons produced in beam remnant interactions. This very same parameter will appear in power corrections to event shapes in hadronic dijet production [@hh-shapes]. A RICH PHENOMENOLOGICAL PROGRAM {#sec:pheno} =============================== The considerations we have presented so far can be seen as the starting point for a rich phenomenological program. First of all one has to keep in mind that multi-jet event-shape observables are not ideal to perform pure PT studies aimed at precise determinations of the strong coupling ${\alpha_s}$. Large theoretical uncertainties are associated to PT higher orders and to NP power corrections, and jet selection cuts reduce the number of available events. However, from the point of view of understanding QCD dynamics, they are an incredibly valuable tool. A research program has already started with the study of power corrections to three-jet event shapes in $e^+e^-$ annihilation and DIS [@AG-hardwork], aimed at determining whether the NP parameter $\alpha_0$ is the same as for two-jet event shapes. Once the universality of $\alpha_0$ is established, one could move to considering hadron-hadron collisions. There one could test our understanding of beam-remnant interactions in a complementary way with respect to what has been done so far. At the moment the only observable used to test models of the underlying event is the away-from-jet particle/energy flow [@Field], for which we have a less clear theoretical understanding than we have for event shapes. Provided our modelling of the underlying event proves to be correct, we have access to the NP parameter ${\left\langle k_t \right\rangle}_\mathrm{remnant}$, and are able to use this parameter to study event-shapes in hadronic dijet production [@hh-shapes], where a lot of progress has been done in recent years, especially since when the automated resummation program <span style="font-variant:small-caps;">caesar</span> has become available [@caesar]. It was Yuri Dokshitzer and Pino Marchesini who initiated me to multi-jet studies. I therefore thank them and Giulia Zanderighi for a many-year collaboration on this subject. I am also grateful to the organisers for the possibility to participate to this very interesting and stimulating workshop. 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--- abstract: 'Recent evidence for pentaquark baryons is critically reviewed in the light of new high statistics data. The search of the WA89 experiment for the $\Xi^{--}(1860)$ is presented in detail and consequences of its non-observations are discussed.' author: - | Josef Pochodzalla\ [*Institut für Kernphysik, Universität Mainz*]{}\ title: | PENTAQUARKS - FACTS AND MYSTERIES\ or SISYPHUS AT WORK --- =11.6pt The Myth of Sisyphus ==================== Giving these days a talk on pentaquarks or - even worse - writing afterwards a report for the proceedings reminds very much on Sisyphus, a man eternally condemned to roll a rock to the top of a mountain, whence the stone would fall back to its own weight. Having just finished the transparencies for the talk, the next paper with a new – positive or negative – result appears. In that sense, the present manuscript written during june 2004 represents an updated version of the talk given at the PANDA workshop in march 2004. But may be there is even a deeper link between the pentaquark search and the destiny of Sisyphus. Since its advent in 1964 the quark model[@QUARK] is very much appreciated for describing the vast amount of strongly interacting particles, the so called hadron-zoo. Experimentally there is no doubt of the existence of baryons, made up of three quarks, and mesons, consisting of a quark anti-quark pair. A priori the quark model imposes no upper limit on the number of quarks/anti-quarks a hadron can be built of. However, it is widely agreed upon, that the colour quantum numbers of the constituents should add up to the colour neutral state. As a consequence physicist desperately seek for exotic quark and gluon structures which differ from the well known meson and baryon structure. Narrow resonances with exotic quark content would be of course particularly welcome because the theoretical interpretation would be very much simplified. In the past many new particles have been spotted like the tetra quark $U$(3100)[@WA62_U; @BIS2_U], the $f_J$(2230) seen first by the MARK III collaboration[@fj2230_1] and the $S$(1936)[@S1936]. Unfortunately none of these narrow resonances survived detailed studies with high statistics. So here we go again... The Experimental Situation of the $\Theta^+(1530)$ ================================================== At present twelve experimental groups have reported evidence for a narrow baryonic resonance in the KN channel at a mass of about 1530[2]{} (see Refs. 6-17) (for an updated list of references see[@Theta:Lit]). Based on previous predictions[@Theta:Diakonov] (for some earlier references see also[@Theta:Walliser]) this resonance was - because of its exotic quark content - interpreted as a pentaquark state. As a consequence already the first observations triggered a flood of theoretical papers which is still increasing with an increment of about one paper each second day (top part of Fig. \[fig:WA8901\]). Figure \[fig:WA8902\] shows the first nine published results which gave evidence for the existence of the so called $\Theta^+(1530)$. Unlike in the original publications I prefer to show here the data points including the statistical error bars. Obviously a common drawback of the individual observations is the limited statistics and hence limited confidence[@Bit00] of the peaks. A little bit disturbing is also the fact that the magnitude of the effect is nearly independent of the experimental situation. Because of the low statistics it is important to note that any cuts applied during the search process can modify the statistical significance of an a priori unknown peak unless the cuts are determined with an independent data sample or Monte Carlo data (see e.g.[@Neeb]). The low statistics of the experiments shown in Fig. \[fig:WA8902\] did usually not allow to separate the data in two distinct data samples. It is furthermore interesting that the position of the various peaks are not fully consistent. Indeed already quite early doubts have been raised because of possible experimental artifacts[@Theta:Dzierba; @Theta:Zavertyaev]. A recent analysis of the HYPER-CP collaboration also underlines the necessity to remove so called ghost tracks, i.e near-duplicate tracks, during the analysis[@Theta:HYPERCP]. Using the positive track from a $\Lambda$ decay twice as a $\pi^+$ and a proton produces a peak near 1.54[2]{} (cf. also the discussion on Fig. \[fig:WA8907\] below). Finally, even if the observed peaks were real, more conventional processes can not be excluded completely at the moment[@Theta:Nussinov; @Theta:Kahana; @Theta:Kishimoto; @Theta:Bicudo] (see however Ref.[@Theta:Llanes]). Since the beginning of this year also quite a number of negative results became available (see lower part of Fig. \[fig:WA8901\]). No signals of the $\Theta^+(1530)$ could be found by BES[@Theta:BES], HERA-B[@Xi:HERAB], OPAL[@Theta:OPAL], PHENIX[@Theta:PHENIX], DELPHI[@Theta:DELPHI], ALEPH[@Theta:ALEPH], HYPER-CP[@Theta:HYPERCP], E690[@Theta:E690], CDF[@Theta:CDF] and BABAR[@Theta:BABAR]. Although a direct comparison of the positive and negative results is quite difficult, the discovery potential of the various experiment can be judged by the observed yield of known resonances. Whereas the experiments with a positive result have – if mentioned in the publications at all – typical $\Lambda(1520)$ yields of at most a few hundred, the experiments with negative outcome report in several cases a few thousand identified $\Lambda(1520)$ events. So while counting naively just the number of reported results, the situation is presently at near-balance (see Fig. \[fig:WA8901\]), it seems that the critics have gained already an advantage. It is therefore indisputable that further high-statistics experiments are needed to establish the observed resonance beyond any doubt. Once this has been achieved – preliminary high statistics data of the LEPS collaboration seem to confirm their first observation[@Theta:LEPSD] – the observation and non-observation of these resonance in different reactions may help to shed some light on the production mechanism and possibly also on the internal structure of these exotic states. ![*$x_F$ distribution of positive (open symbols) and negative (closed symbols) $\Sigma$ resonances studied by WA89.*[]{data-label="fig:WA8904"}](WA89_all0.eps){width="5.7cm"} ![*$x_F$ distribution of positive (open symbols) and negative (closed symbols) $\Sigma$ resonances studied by WA89.*[]{data-label="fig:WA8904"}](WA89_sigma.eps){width="5.2cm"} The $\Xi(1860)$ - Another Stone for Sisyphus? ============================================= The interpretation of the observed peaks in terms of a five-quark state was significantly strengthened by the subsequent observation of another member of the anticipated antidecuplet of pentaquarks. Based on 1640 $\Xi^-$ candidates produced in p+p interactions at 160[1]{} beam momentum, both in the $\Xi^-\pi^+$ and the $\Xi^-\pi^-$ channels narrow peak structures at an invariant mass of 1.860[2]{} were observed by the NA49 collaboration[@Xi:NA49]. Possible signals of a $\Xi^*$ resonance at 1.860[2]{} decaying into ${\Xi^-}\pi^+$ and $Y\overline{K}$ were reported already 1977 for K$^-$p interactions at 2.87[1]{}[@Bri77]. However, no corresponding signals have been seen in other K$^-$ induced reactions (for a compilation and a discussion of these data see Ref.[@Xi:Fischer]). A preliminary analysis of proton-nucleus interactions at 9201 by the HERA-B collaboration using a total of 19000 reconstructed $\Xi^-$ and $\overline{\Xi}^+$ events, shows no indication for the $\Xi^{--}$ nor the $\Theta^+$ resonances[@Xi:HERAB]. Searches for the $\Xi(1860)$ resonances are also being performed by the ZEUS, CDF, ALEPH, E690 and the BABAR collaboration. The ZEUS data comprise 1361 $\Xi^-$ and 1303 $\overline{\Xi}^+$ events, the CDF sample contains 19150 $\Xi^-$ and 16736 $\overline{\Xi}^+$ and the ALEPH collaboration collected about 1800 $\Xi^-$ . Negative – though still preliminary – results have been reported by all three collaborations at the DIS04 conference[@Xi:CDFZEUS]. The E690[@Theta:E690] and BABAR[@Xi:BABAR] experiments could not find a significant signal despite a large data sample of 512000 and 258000 observed $\Xi^-$, respectively. First preliminary results of the WA89 collaboration were presented at the HYP03 conference already in october 2003[@HYP03]. The final result presented in the following section are available in Ref. [@Xi:WA89] ![*Upper histogram: $x_F$ distribution of the observed $\Xi^-$ events within a $\pm$2$\sigma$ mass window. Lower histogram: $x_F$ distribution of the observed $\Xi^-\pi^-$ pairs within the mass range between 1.82 and 1.90 [2]{}. In both cases the background has been subtracted by means of sideband events.*[]{data-label="fig:WA8906"}](WA89_penta01.eps){width="5.7cm"} ![*Upper histogram: $x_F$ distribution of the observed $\Xi^-$ events within a $\pm$2$\sigma$ mass window. Lower histogram: $x_F$ distribution of the observed $\Xi^-\pi^-$ pairs within the mass range between 1.82 and 1.90 [2]{}. In both cases the background has been subtracted by means of sideband events.*[]{data-label="fig:WA8906"}](WA89_penta02.eps){width="5.7cm"} The Hyperon Beam Experiment WA89 ================================ The hyperon beam experiment WA89 had the primary goal to study charmed particles and their decays. At the same time it collected a high statistics data sample of hyperons and hyperon resonances. The hyperon beamline[@WA89:beam] selected $\Sigma^-$ hyperons with a mean momentum of 340 1 and a momentum spread of $\sigma (p)/p=9\%$. In addition the beam contained small admixtures of $K^-$ (2.1%) and $\Xi^-$ (1.3%)[@WA89:xi]. The trajectories of incoming and outgoing particles were measured in silicon microstrip detectors upstream and downstream of the target. The experimental target itself consisted of one copper slab with a thickness of 0.025 $\lambda_I$ in beam direction, followed by three carbon (diamond powder) slabs of 0.008 $\lambda_I$ each, where $\lambda_I$ is the interaction length. The momenta of the decay particles were measured in a magnetic spectrometer equipped with MWPCs and drift chambers. In order to allow hyperons and $K^0_S$ emerging from the target to decay in front of the magnet the target was placed 13.6m upstream of the center of the spectrometer magnet. The symbols in Fig. \[fig:WA8903\] mark the cross sections per nucleon for strange and charmed hadrons produced in $\Sigma^-$ induced reactions at 345[1]{}. In cases where the branching ratio of the observed decay channel is not known only lower limits are indicated by the vertical arrows. Typical for most hadronic interactions in this energy regime the cross sections follow roughly a mass dependence $\propto exp(-\Delta m/150MeV)$ as indicated by the straight line. The importance of the projectile for the hyperon production is illustrated by the $x_F$ distributions of positive (open symbols) and negative (closed symbols) $\Sigma$ resonances shown in Fig. \[fig:WA8904\]. Whereas at large $x_F$ a significant enhancement of negative hyperons of nearly a factor of 10 is observed for the ground state, the decuplet resonance at 1385 2 shows an enhancement of less than 3. Considering the fact that for the $\Sigma^{\pm}_{1660}$ only values for $\sigma \cdot BR$ are given, the large cross section for $\Sigma^-_{1660}$ seems particularly striking (see closed triangles in Fig. \[fig:WA8904\]). Furthermore, the $\Sigma^-_{1660}$ shows again an enhancement over the $\Sigma^+_{1660}$ beyond a factor of 10. This is significantly larger than for $\Sigma_{1385}$ but comparable to that of the ground state hyperons. Assuming that the observed $\Sigma_{1660}$ is a $J^P=1/2^+$ octet state, the strong leading effect for the $\Sigma_{1660}$ as compared to the rather weak effect of the $\Sigma_{1385}$ decuplet may be related to the $[ds]$ diquark structure. In the $J^P=3/2^+$ decuplet hyperon the $[ds]$ diquarks have spin 1, while in the $\Sigma_{1660}$ the $[ds]$ diquarks have predominantly spin 0. Search for the exotic $\Xi^{--}(1860)$ Resonance ================================================ Since statistics is the key point when looking for new particles, we also included interactions in the tracking detectors (silicon detectors and plastic scintillator) located close to these targets in our search for the S=-2 resonance in $\Sigma^-$ induced reactions. $\Xi^-$ were reconstructed in the decay chain $\Xi^- \rightarrow \Lambda\pi^- \rightarrow p\pi^-\pi^-$. The invariant mass distributions of the $\Xi^-$ candidates are shown Fig. \[fig:WA8905\] for two regions of the total momentum of the $\Lambda\pi$ pair. The cut at 80[1]{} corresponds to an $x_F$ value of about 0.25 (see below). The WA89 analysis is based on a total of 676k $\Xi^-$ candidates observed over a background of 170k $p\pi^-\pi^-$ combinations. Out of these candidates 240k, 281k and 155k can be attributed to the C, Cu and “Si+C+H” target, respectively. Because of the strangeness content of the $\Sigma^-$ beam also the cross sections for $\Xi$ resonances are shifted towards large $x_F$ with respect to the $\Sigma^-$-nucleon cm-system[@WA89:xistar]. Since in the WA89 setup the efficiency drops significantly at $x_F<$0.1 the yield of $\Xi^-$ peaks at $x_F \approx$ 0.2 (upper histogram in Fig. \[fig:WA8906\]). $\Xi^-\pi^-$ pairs within the mass range of 1.82 to 1.90 [2]{} are shifted to even larger $x_F$ (lower histogram in Fig. \[fig:WA8906\]). For comparison, the $\Xi^-$ events observed by NA49 are distributed over an $x_F$ range between -0.25 and +0.25[@NA49:Barna]. ![*Effective mass distribution of $\Xi^-\pi^-$ combinations with $x_F(\Xi^-\pi^-)\leq$0.15 (part a), $x_F(\Xi^-\pi^-)\leq$0.3 (part b) and $x_F(\Xi^-\pi^-) > $0.3 (part c). In each plot the lower and upper histogram correspond to the carbon and copper target, respectively.*[]{data-label="fig:WA8908"}](WA89_penta03.eps){width="5.5cm"} ![*Effective mass distribution of $\Xi^-\pi^-$ combinations with $x_F(\Xi^-\pi^-)\leq$0.15 (part a), $x_F(\Xi^-\pi^-)\leq$0.3 (part b) and $x_F(\Xi^-\pi^-) > $0.3 (part c). In each plot the lower and upper histogram correspond to the carbon and copper target, respectively.*[]{data-label="fig:WA8908"}](WA89_penta04.eps){width="5.5cm"} Fig. \[fig:WA8907\] shows the invariant mass spectrum of all observed $\Xi^-\pi^-$ pairs. Fig. \[fig:WA8907\]b shows an extended view of the region around a mass of 1.8622 marked by the arrows. All reactions, including also interactions in the tracking detectors close to the C and Cu targets, contribute to this figure. The structure observed at around 1.52 in the upper histogram of Fig. \[fig:WA8907\]a is caused by events where the negative pion from the decay of the $\Xi^-$ was wrongly reconstructed as a double track. As can be seen from the lower histogram in Fig. \[fig:WA8907\]a, these fake pairs are reduced substantially by subtracting background from $\Xi^-$ sideband events. The NA49 collaboration has observed a ratio of $\Xi^{--}$ to $\Xi^-$ candidates of about 1/40. If we assume the same [*relative*]{} production cross sections over the full kinematic range for the reaction in question and similar [*relative*]{} detection efficiencies $[{\varepsilon}(\Xi^{--})/{\varepsilon}(\Xi^-)]_{WA89}\approx [{\varepsilon}(\Xi^{--})/{\varepsilon}(\Xi^-)]_{NA49}$ we would expect of the order of 17000 $\Xi^{--} \rightarrow \Xi^-+\pi^-$ events in our full data sample. The FWHM of the peaks observed by NA49 is 172 and is limited by the experimental resolution. Since in our experiment the resolution is expected to be slightly smaller $\approx$ 10[2]{} (FWHM), this excess should be concentrated in less than 6 channels in Fig. \[fig:WA8907\]b. Obviously, no such enhancement can be seen in the spectra. The $\Xi(1860)$ events observed by NA49 are concentrated at small $x_F$. For a better comparison with the NA49 experiment we therefore scanned our data for different ranges of $x_F$. Fig. \[fig:WA8908\] shows the effective mass distributions of $\Xi^-\pi^-$ combinations with $x_F(\Xi^-\pi^-)\leq$0.15, $\leq$0.3 and $> $0.3 in the region around 1.862[2]{}. In each panel, the upper and lower histograms correspond to reactions with the carbon and copper target, respectively. No background subtraction was applied to these spectra. Assuming again a $\Xi^{--}$ to $\Xi^-$ ratio of 1/40 as observed by NA49 and considering now only the $x_F$ range between 0 and 0.15, we estimate that approximately 700 and 900 $\Xi^{--} \rightarrow \Xi^-\pi^-$ events should be seen in Fig. \[fig:WA8908\]a for the C and Cu target, respectively. None of these spectra shows evidence for a statistically significant signal around 1.862[2]{}, nor does such a signal appear in any other sub-sample. Upper limits on the production cross sections were estimated separately for the copper and carbon targets, in five bins of $x_F$ between $x_F=0.15$ and $x_F=0.9$. Assuming a dependence of the cross section on the mass number as $\sigma_{nucl}\propto \sigma_0\cdot A^{2/3}$, where $\sigma_0$ is the cross section [*per nucleon*]{}, we obtained the limits on $BR\cdot d\sigma_0 /dx_F$. Limits on the integrated production cross sections $\sigma$ were then calculated by summing quadratically the contributions $ d\sigma /dx_F \cdot \Delta x_F$ in the five individual $x_F$ bins. The results are $BR\cdot \sigma_{max}(0.15<x_F<0.9)$= 16 and 55 $\mu b$ per nucleus in case of the carbon and copper target, respectively. An extrapolation to the cross sections per nucleon yields the two values $BR\cdot \sigma_{0,max} = 3.1\, \mu b$ for the carbon and $3.5\, \mu b$ for the copper target, in excellent agreement with each other. As can be seen from Fig. \[fig:WA8903\], these limits do not exceed the production cross sections of all other observed $\Xi^*$ resonances. At large $x_F$ a significant fraction of the $\Xi^-$ are produced by interactions induced by the $\Xi^-$ beam contamination[@WA89:xi; @WA89:xipol]. Even if we were to assume that the $\Xi^{--}(1860)$ production can be attributed exclusively to the 1.3% $\Xi^-$ admixture in the beam, we obtain e.g. for the carbon target and $x_F\geq$0.5 a limit for the $\Xi^{--}$ production by $\Xi^-$ of 740$\mu$b. For comparison, even this large 3$\sigma$ limit corresponds to only 4% of the $\Xi^-$ production cross section in $\Xi^-$+Be interactions at 116[1]{} in the same kinematic range[@Bia87]. Finally we note that the $\Xi^- \pi^+$ mass distribution observed in the 1993 data set by WA89 has already been published some years ago[@WA89:ximpip]. This combination is dominated by the peak from $\Xi^0(1530)$ decays (see Fig. \[fig:WA8909\]). The observed central mass was in good agreement with the known value of M = 1531.8 $\pm$ 0.3[2]{}[@pdg]. Unfolding the observed width with the width of the $\Xi_{1530}$ of $\Gamma = 9.1$[2]{}[@pdg] gave an experimental resolution of $\sigma _{\Xi^{0}(1530)}$ = 3.7[2]{}. Furthermore, a weak resonance signal with a width of $\Gamma = 10 \pm 6 \Mevc2\ $ is visible at $M = 1686 \pm 4 \Mevc2\ $ above a large background. In the mass region of the $\Xi^0(1860)$ no enhancement over the uncorrelated background can be seen in the WA89 data. Quintessence ============ After an euphoric stage with many favorable reports within a short time we have now reached a phase which is much more unclear. Counting just the number of reported results the situation of the $\Theta^{+}(1530)$ is presently at most near-balance between sightings and non-sightings. It seems, however, that – because of the higher statistics – the non-sighting experiments gain the preponderance. Considering the seven non-observations of the $\Xi^{--}(1860)$ resonance compared to the single claim in favor of it by the NA49 collaboration, this pentaquarks seems to stand of very shaky ground at present. [*If*]{}, nonetheless, the $\Xi^{--}$ signal observed by the NA49 collaboration is real, then the non-observation in the WA89 experiment – as well as the other experiments – is not easily understood. Generally particle ratios do not vary significantly for the beam momentum range in question (1601 vs. 3401) [@Let03; @Liu04]. The fact that the $\Theta^+(1530)$ has been seen in reactions on complex nuclei [@Theta:NEUTRINO; @Theta:SVD] makes also the different targets (hydrogen vs. C, Si, Cu) an unlikely cause for the discrepancy. The internal structure of the $\Sigma^-$ projectile or of the $\Xi^{--}(1860)$ could be a more plausible reason for the rather low limit of the $\Xi^{--}(1860)$/$\Xi^-$ ratio. It is well known, that a transfer of a strange quark from the beam projectile to the produced hadron enhances the production cross sections in particular at large $x_F$ (see, for instance, Fig. \[fig:WA8904\]). The different leading effects for octet and decuplet $\Sigma$ states[@WA89:sigma] even hint at an $[sd]$ diquark transfer from the $\Sigma^-$ projectile[@WA89:poc01]. The production of a pentaquark containing correlated quark-quark pairs (see e.g. Ref.[@Jaf03]) would probably benefit from such a diquark transfer. However, for example in case of an extended $\overline{K}-N-\overline{K}$ molecular structure of the $\Xi(1860)$[@Bic04] an $[sd]$ diquark transfer may not necessarily enhance the $\Xi^{--}$ production leading also to a narrower $x_F$ distribution. As a consequence the cross section in $\Sigma^-$ induced reactions might not exceed the one for production in pp interactions. The latter cross section is predicted to be $\sim$ 4$\mu$b[@Liu04] which is then close to our limit. Thus, if future high statistics experiments will confirm the production of the $\Xi^{--}(1860)$ resonance in proton-proton interaction, the non-observation with the $\Sigma^-$ beam would point to a very exceptional production mechanism possibly related to an exotic structure of the $\Xi^{--}(1860)$. However, the possible non-observation by the E690 collaboration [@Theta:E690] in 800[1]{} p-p interactions may even ruin this argument in favor of the $\Xi^{--}(1860)$ resonance. Keeping in mind the past searches for exotic quark structures and looking at the present contradictory data, we can therefore not exclude that the stone of Sisyphus is just about to roll back downhill and that the quest for exotic pentaquark states may end where it began. May be QCD is indeed sticking to two and three valence quarks only and may be we are just lacking the right argument for this beautiful simplicity. In this situation it might be helpful to recall what Albert Camus said about the poor Sisyphus. 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--- abstract: | Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c=eH_ce$. Then $U_c$ is filtered by order of differential operators, with associated graded ring $\operatorname{gr}U_c={\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]^{{W}}$ where ${{W}}$ is the $n$-th symmetric group. We construct a filtered ${\mathbb{Z}}$-algebra $B$ such that, under mild conditions on $c$: - the category $B{\text{-}{\textsf}{qgr}}$ of graded noetherian $B$-modules modulo torsion is equivalent to $U_c{\text{-}{\textsf}{mod}}$; - the associated graded ${\mathbb{Z}}$-algebra $\operatorname{gr}B$ has $\operatorname{gr}B{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }\operatorname{Hilb(n)}$, the category of coherent sheaves on the Hilbert scheme of points in the plane. This can be regarded as saying that $U_c$ simultaneously gives a noncommutative deformation of ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ and of its resolution of singularities $\operatorname{Hilb(n)}\to {\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$. As the companion paper [@GS2] shows, this result is a powerful tool for studying the representation theory of $H_c$ and its relationship to $\operatorname{Hilb(n)}$. address: - 'Department of Mathematics, Glasgow University, Glasgow G12 8QW, Scotland' - 'Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA.' author: - 'I. Gordon' - 'J. T. Stafford' title: Rational Cherednik algebras and Hilbert schemes --- [^1] Introduction ============ {#sec101} This is the first of two closely related papers on rational Cherednik algebras. In their short history, Cherednik algebras have been influential in a surprising range of subjects: for example they have been used to answer questions in integrable systems, combinatorics, and symplectic quotient singularities (see [@BEGqi; @gordc; @BFG; @GK]). In this paper we strengthen the connections between Cherednik algebras and geometry by showing that they can be regarded as noncommutative deformations of Hilbert schemes of points in the plane. In the sequel [@GS2] this will be used to show the close relationship between modules over the Cherednik algebra and sheaves on the Hilbert scheme as well as to answer various open problems about these modules. {#intro-1.2} Fix $c\in {\mathbb{C}}$. We assume throughout the paper that $c\notin \frac{1}{2} + {\mathbb{Z}}$ and, for simplicity, we will also assume that $c\not\in \mathbb{R}_{\leq 0}$ in this introduction, see and for the more general case. Let $H_c= H_{1,c}$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c = eH_ce$. The formal definition of $H_c$ is given in but one may regard it as a deformation of the twisted group ring $D({\mathfrak{h}})\ast {{W}}$, where $D({\mathfrak{h}})$ is the ring of differential operators on ${\mathfrak{h}}\cong {\mathbb{C}}^{n-1}$ with the natural action of the symmetric group ${{W}}= \mathfrak{S}_n$. The algebra $U_c$ is then the corresponding deformation of the fixed ring $D({\mathfrak{h}})^{{W}}$. The algebras $U_c$ and $H_c$ have a natural filtration by order of differential operators with associated graded rings $\operatorname{gr}{U_c} \cong {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^\ast]^{{{W}}}$ and $\operatorname{gr}{H_c} \cong {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^\ast]\ast {{{W}}}$. Thus we may also regard $U_c$ as a deformation of ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^\ast]^{{{W}}}$. In this introduction we will mostly be concerned with $U_c$, but since $U_c$ and $H_c$ are Morita equivalent (Corollary \[morrat-cor\]) the results we prove for $U_c$ also apply to $H_c$. It is well-known that ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ has a crepant resolution $\operatorname{Hilb(n)}\to {\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$, where $\operatorname{Hilb(n)}$ is a variant on the Hilbert scheme of $n$ points in the plane (see for the formal definition). The ring $U_c$ has finite global homological dimension (see Corollary \[gldim\]) and so one should expect that it has the properties of a smooth deformation of ${{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}^{{W}}$; in other words its properties should be more closely related to those of $\operatorname{Hilb(n)}$ than to ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$. Hints of this have been reported in [@gordc] and [@BEGfd]: finite dimensional $H_c$-modules deform the sections of some remarkable sheaves on $\operatorname{Hilb(n)}$. The main aim of this paper is to formalise this idea by showing that there is a second way of passing to associated graded objects that maps $U_c{\text{-}{\textsf}{mod}}$ precisely to $\operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$. {#intro-1.3} We take our cue from the theory of semisimple Lie algebras. When $n=2$, $U_c$ is isomorphic to a factor of $U(\mathfrak{sl}_2)$ [@EG Section 8] and, for all $n$, the properties of $U_c$ are similar to those of $U(\mathfrak{g})/P$, where $P$ is a minimal primitive ideal in the enveloping algebra of a complex semisimple Lie algebra $\mathfrak{g}$ (see, for example [@ginz; @GGOR; @guay]). The intuition from the last paragraph not only holds for enveloping algebras but can also be formalised through the Beilinson-Bernstein equivalences of categories. This gives a diagram $$\begin{CD} D_{\mathcal{B}} @< \sim << U(\mathfrak{g})/P \\ @V \operatorname{gr}VV @VV \operatorname{gr}V \\ {\mathcal{O}}_{T^*\mathcal{B}} @<\tau << {\mathcal{O}}(\mathcal{N}) \end{CD}$$ where $\mathcal{B}=G/B$ is the flag variety, the primitive ideal $P$ has trivial central character and $\tau: T^*\mathcal{B}\to \mathcal{N}$ is the Springer resolution of the nullcone $\mathcal{N}$. The Morita equivalence from the sheaf of differential operators $D_{\mathcal{B}}$ to $ U(\mathfrak{g})/P$ is obtained by taking global sections under the identification $U(\mathfrak{g})/P\cong D(\mathcal{B})$. Ginzburg has raised the question of whether a similar phenomenon holds for Cherednik algebras (see [@GK Conjecture 1.6] for a variant on this conjecture). In other words, can one complete the following diagram? $$\begin{CD} ? @< \sim << U_c \\ @V \operatorname{gr}VV @VV \operatorname{gr}V \\ {\mathcal{O}}_{\operatorname{Hilb(n)}} @< \tau<<{\mathcal{O}}({\mathfrak{h}}\oplus {\mathfrak{h}}^*/{{W}}) \end{CD}$$ The main result of the paper gives a positive answer to this question. Given a graded ring $R$, we write $R{\text{-}{\textsf}{qgr}}$ for the quotient category of noetherian graded $R$-modules modulo those of finite length. Main Theorem {#mainthm-intro} ------------ *There exists a graded ring $B$, filtered by order of differential operators, such that* 1. there is an equivalence of categories $U_c {\text{-}{\textsf}{mod}}\simeq B{\text{-}{\textsf}{qgr}}$; 2. there is an equivalence of categories $\operatorname{gr}B{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$. {#intro-1.4} The construction of $B$ needs some explanation. For $n>2$, it can be shown that the Hilbert scheme $\operatorname{Hilb(n)}$ is not a cotangent bundle, so we cannot use sheaves of differential operators as a non-commutative model. Instead we take as our starting point Haiman’s description of $\operatorname{Hilb(n)}$ as a blow-up of ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ and deform this to a non-commutative setting. Set $A^0 = {\mathcal{O}}({\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}})$ with ideal $I=A^1\delta$, where $\delta $ is the discriminant and $A^1 = {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^\epsilon$ the module of anti-invariants. Then [@haidis Proposition 2.6] proves that $\operatorname{Hilb(n)}={\textsf}{Proj}\, A$ where $A=A^0[tI]$ is the Rees ring of $I$ (see Section \[sect-haiman\] for the details). Unfortunately one cannot construct $B$ as an analogous Rees ring over $U_c$, since $U_c$ is a simple ring for generic values of $c$. We circumvent this problem by using [*${\mathbb{Z}}$-algebras*]{} (see Section \[zalg\]). Specifically, the ring $B$ from Theorem \[mainthm-intro\] is an algebra $B=\bigoplus_{i\geq j\geq 0}B_{ij}$ whose multiplication is defined in matrix fashion: $B_{ij}B_{jk}\subseteq B_{ik}$ but $B_{ij}B_{\ell k}=0$ when $j\not= \ell$. The diagonal terms are just $B_{ii}=U_{c+i}$ while the off-diagonal terms $B_{ij}$ are given as the appropriate tensor products of the $(U_{d+1}, U_{d})$-bimodules $Q_{d}^{d+1} = eH_{d+1}\delta e$. The shift functors $S_d : U_d {\text{-}{\textsf}{mod}}\rightarrow U_{d+1}{\text{-}{\textsf}{mod}}$ given by tensoring with $Q_{d}^{d+1}$ are important operators in the theory of Cherednik algebras and have already played a crucial role in combinatorics and representation theory; see, for example, [@BEGfd; @BEGqi; @gordc]. A good way to think of the functor $S_d$ is as the analogue of the translation functor [@BG] from Lie theory. In order to have control over $B$ we need to know that the $Q_{d}^{d+1}$ are progenerators for all $d\in c+{\mathbb{N}}$; equivalently that the $S_d$ are Morita equivalences. This is a conjecture from [@GGOR Remark 5.17] which we answer with: Theorem {#morrat-intro} ------- \[Corollary \[morrat-cor\]\] [*The shift functor $S_d$ is a Morita equivalence for all $d\in c+{\mathbb{N}}$.* ]{} The significance of this result is that $B$ now has rather pleasant properties; in particular Theorem \[mainthm-intro\](1) is an easy consequence. For the second assertion of Theorem \[mainthm-intro\], we note that it is easy to obtain a ${\mathbb{Z}}$-algebra $\widehat{A} =\bigoplus_{i\geq j\geq 0} A_{ij}$ from the graded algebra $A=\bigoplus_{k\geq 0}I^k$ for which $A{\text{-}{\textsf}{qgr}}\simeq \widehat{A}{\text{-}{\textsf}{qgr}}$. One simply takes $A_{ij}=I^{i-j}$ for each $i,j$. Thus the main step in the proof of Theorem \[mainthm-intro\] is given by: Proposition {#pre-cohh-intro} ----------- \[Theorem \[main\]\] [*Under the filtration induced from the order filtration of differential operators, $\operatorname{gr}B_{ij} \cong A_{i-j} = I^{i-j}$ and so $\operatorname{gr}B\cong \widehat{A}$ as ${\mathbb{Z}}$-algebras.* ]{} In this result the inclusion $ I^{i-j}\subseteq \operatorname{gr}B_{ij}$ is straightforward. The opposite inclusion is much more subtle as it is difficult to keep close control of the filtration on $B_{ij}$. Our proof leans heavily on the work of Haiman in [@hai3] and [@hai1] surrounding the $n!$ and polygraph theorems and the strategy is outlined in more detail in за. Applications {#intro-1.10} ------------ Theorem \[mainthm-intro\] gives a powerful technique for relating $H_c$- or $U_c$-modules to sheaves on $\operatorname{Hilb(n)}$: given a $U_c$-module $M$ with a good filtration $\Lambda$ we obtain a filtered object $(\widetilde{M},\Lambda)\in B{\text{-}{\textsf}{qgr}}$ by tensoring with $B$ and then a coherent sheaf $\Phi_\Lambda (M)\in \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$ by taking the associated graded module. This process is studied in [@GS2] where we show there that the subtle combinatorics and geometry of $\operatorname{Hilb(n)}$ is reflected in the representation theory of $U_c$ and $H_c$. Let $\Delta_c(\mu)$ be the [*standard $H_c$-module*]{} corresponding to $\mu\in{{\textsf}{Irrep}({{W}})}$ (this is the analogue of a Verma module) with unique simple factor $L_c(\mu)$. These modules have a natural good filtration $\Lambda$ and we mention a couple of illustrative results from [@GS2]. - Suppose that $c=1/n+k$ for $k\in {\mathbb{N}}$, so that $L_c(\operatorname{{\textsf}{triv}})$ is the unique finite dimensional simple $H_c$-module. Then $\Phi_\Lambda (eL_c(\operatorname{{\textsf}{triv}})) \cong {\mathcal{O}}_{Z_n}\otimes\mathcal{L}^{k}$, where $Z_n=\tau^{-1}(0)$ is the [*punctual Hilbert scheme*]{} and $\mathcal{L}={\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$ is the Serre twisting sheaf . - For any $c$, the characteristic cycle of $\Phi_\Lambda (e\Delta_c(\mu))$ equals $\sum_{\lambda}K_{\mu\lambda}[Z_\lambda]$, where $K_{\mu\lambda}$ are Kostka numbers and the $Z_{\lambda }$ are particular irreducible components of $\tau^{-1}({\mathfrak{h}}\oplus \{{\bf 0} \}/{{W}})$. The first of these results is used in [@GS2] to show that the natural bigraded structure on $\operatorname{gr}_\Lambda (eL_{1/n+k})$ coincides with that on $\mathrm{H}^0(Z_n, {\mathcal{L}}^k)$, thus confirming a conjecture of [@BEGfd]. The second of these results illustrates the subtlety of $\Phi$: if one passes directly from $U_c$ to $\operatorname{gr}U_c\cong {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]^{{W}}$ then $\operatorname{gr}_\Lambda (e\Delta_c(\mu))\cong {\mathbb{C}}[{\mathfrak{h}}]\otimes \mu$ for any $\mu$ and $c$. Thus the support variety of $\operatorname{gr}_\Lambda e\Delta_c(\mu)$ is independent of $\mu$. We prove one such correspondence in this paper. Let ${\mathcal{P}}$ denote the Procesi bundle on $\operatorname{Hilb(n)}$, the vector bundle of rank $n!$ coming from Haiman’s $n!$ theorem, see . Then Corollary \[cohh-subsect\] proves: Corollary {#cohh-intro} --------- If $eH_c$ is given the order filtration $\Lambda$, then $\Phi_\Lambda (eH_c)={\mathcal{P}}$. {#sect-1.10} One reason why Theorem \[mainthm-intro\] provides a strong bridge between Hilbert schemes and Cherednik algebras is that the construction of $B$ carries within it key elements of both theories. For instance, we have already mentioned that the shift functor $S_c$ is an analogue of the translation functor from Lie theory. It is also the analogue of the shift functor in $\operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$ given by tensoring with ${\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$. Indeed, given a $U_c$-module $M$ with a good filtration $\Lambda$, it is easy to show that $\Phi_{\Gamma}(Q^{c+1}_{c} \otimes M)= {\mathcal{O}}_{\operatorname{Hilb(n)}}(1)\otimes\Phi_\Lambda(M)$, for the appropriate filtration $\Gamma$ (see [@GS2]). Similarly, Corollary \[cohh-intro\] can be interpreted as saying that $H_c$ is a noncommutative analogue of the isospectral scheme $X_n$, as defined in (see for further details). {#intro-1.18} The ${\mathbb{Z}}$-algebra has the virtue that it exists whenever one has an analogue of the translation principle; that is, one has algebras $R_i$ and progenerative $(R_{i+1},R_{i})$-bimodules $Q_{i,i+1}$ (these algebras can also be indexed by more general lattices than ${\mathbb{Z}}$). One can then construct a ${\mathbb{Z}}$-algebra as we have done and Theorem \[mainthm-intro\](1) will still hold. It is not clear when Theorem \[mainthm-intro\](2) will hold and, even when it is true, it will undoubtedly be rather subtle. Hilbert schemes realise crepant resolutions for the symplectic quotient singularity $({\mathbb{C}}^2)^n/G$ whenever $G$ is the wreath product of a finite subgroup of $SL_2({\mathbb{C}})$ with the symmetric group ${{W}}$, see [@wang Theorem 4.2]. We believe that our methods will generalise to the symplectic reflection algebras $H_{c}(G)=H_{1,c}(G)$ associated with $(({\mathbb{C}}^2)^n, G)$ to give non-commutative deformations of those Hilbert schemes. Even when there is no crepant resolution of such a singularity (by [@GK] this happens for Weyl groups $G$ of types other than A and B) the ${\mathbb{Z}}$-algebra associated to $H_c(G)$ will still contain interesting information, as [@gordc] demonstrates. For a Weyl group, the analogue of Theorem \[morrat-intro\] is at least known for sufficiently large values of the defining parameter $c$ [@BEGfd Proposition 4.3], but little is known for small values of $c$. The translation principle obviously holds for factors of enveloping algebras of semisimple Lie algebras and we can prove an analogue of Theorem \[mainthm-intro\] in this case. However, the proof uses nontrivial Lie theoretic results, notably the Beilinson-Bernstein equivalence of categories, and it is unclear whether this approach carries information that cannot be obtained from that equivalence. It would be interesting to see if the recent work [@BK; @tan] on the Beilinson-Bernstein equivalence for quantised enveloping algebras can be understood in a ${\mathbb{Z}}$-algebra framework. {#intro-1.19} The paper is organised as follows. In Section \[sect-rationalchered\] we recall the needed facts about rational Cherednik algebras, while in Section \[shift\] we prove Theorem \[morrat-intro\]. In Section \[sect-haiman\] we describe some of Haiman’s work on Hilbert schemes, adapted to the variety $\operatorname{Hilb(n)}$, and use it to describe various Poincaré series that will be needed in the proof of Theorem \[mainthm-intro\]. Section \[zalg\] proves the results about ${\mathbb{Z}}$-algebras that were mentioned earlier in this introduction. Section \[sect-filt\] is the heart of the paper: in it we prove Theorem \[mainthm-intro\](2). This is derived from an analogous result about the associated graded module of $B_{k0}\otimes_{U_c}eH_c$ that also implies Corollary \[cohh-intro\]. Section \[sect7\] then gives a reinterpretation of Theorem \[mainthm-intro\] in terms of a tensor product filtration of $B_{ij}$. In Appendix \[app-a\] we prove the following result that may be of independent interest: [*Suppose that $R=\bigoplus_{i\geq 0}R_i$ is an ${\mathbb{N}}$-graded algebra over a field $k$, with $R_0=k$. If $P$ is a right $R$-module that is both graded and projective, then $P$ is graded-free in the sense that $P$ has a free basis of homogeneous elements.*]{} This is a graded analogue of a classic result from [@Kap] for which we do not know a reference. Acknowledgement --------------- We would like to thank Victor Ginzburg for bringing his conjecture to our attention, since it really formed the starting point for this work. We would also like to thank Tom Nevins and Catharina Stroppel for suggesting many improvements to us. Rational Cherednik algebras {#sect-rationalchered} =========================== {#rcadef} In this section we define the rational Cherednik algebras (which will always be of type $A$ in this paper) and give some of the basic properties that will be needed in the body of the paper. Let ${{W}}=\mathfrak{S}_n$ \[symmetric-defn\] be the [*symmetric group*]{} on $n$ letters, regarded as the Weyl group of type $A_{n-1}$ acting on its $(n-1)$-dimensional representation ${\mathfrak{h}}\subset {\mathbb{C}}^n$ by permutations. Let $\mathcal{S}=\{s=(i,j)\ \text{with}\ i<j\}\subset {{W}}$ \[involution-defn\] denote the reflections, with reflecting hyperplanes $\alpha_s=0$. We make similar definitions for ${\mathfrak{h}}^*$ and normalise $\alpha_s^{\vee} \in {\mathfrak{h}}$ so that $\alpha_s(\alpha_s^{\vee}) = 2$. Given $c\in {\mathbb{C}}$, [*the rational Cherednik algebra of type $A_{n-1}$*]{} is the ${\mathbb{C}}$-algebra $H_c$\[hc-defn\] generated by the vector spaces ${\mathfrak{h}}$ and ${\mathfrak{h}}^*$ and the group ${{W}}$ with defining relations $$\begin{aligned} wxw^{-1} = w(x), \quad wyw^{-1} = w(y), & &\text{for all }y\in {\mathfrak{h}}, x\in {\mathfrak{h}}^*, w\in {{W}}\\ x_1x_2 = x_2x_1, \quad y_1y_2 = y_2y_1, & &\text{for all }y_i\in {\mathfrak{h}}, x_j\in {\mathfrak{h}}^*\\ yx - xy = x(y) - \sum_{s\in \mathcal{S}} c\alpha_s (y) x(\alpha_s^{\vee})s, & &\text{for all }y\in {\mathfrak{h}}, x\in {\mathfrak{h}}^*.\end{aligned}$$ We should note that the definition of the Cherednik algebra is not uniform throughout the literature. The definition we are using agrees with that in [@BEGqi; @BEGfd; @EG; @guay] but [*not*]{} that from [@GGOR] where our constant $c$ equals $-k_1$ for their constant $k_1$ (see [@GGOR Remark 3.1]). {#subsec-3.2} We write the coordinate ring of an affine variety $V$ as ${\mathbb{C}}[V]$. By [@EG Theorem 1.3], the subalgebra of $ H_c$ generated by $ {\mathfrak{h}}^*$ can and will be identified with ${\mathbb{C}}[{\mathfrak{h}}]$, while $ {\mathfrak{h}}$ generates a copy of ${\mathbb{C}}[{\mathfrak{h}}^*]$ inside $H_c$ and the elements $w\in {{W}}$ span a copy of the group algebra ${\mathbb{C}}{{W}}$ in $H_c$. Fix once and for all dual bases $\{ x_i \}$ and $\{ y_i \}$ of ${\mathfrak{h}}^*$ and ${\mathfrak{h}}$ respectively; thus ${\mathbb{C}}[{\mathfrak{h}}]={\mathbb{C}}[x_1,\dots,x_{n-1}]$ and ${\mathbb{C}}[{\mathfrak{h}}^*]={\mathbb{C}}[y_1,\dots,y_{n-1}]$. By [@EG Theorem 1.3] there is a Poincaré-Birkhoff-Witt isomorphism of ${\mathbb{C}}$-vector spaces $$\label{PBW} {\mathbb{C}}[{\mathfrak{h}}]\otimes_{{\mathbb{C}}} {\mathbb{C}}{{W}}\otimes_{{\mathbb{C}}} {\mathbb{C}}[{\mathfrak{h}}^*] \xrightarrow{\sim} \ H_c.$$ Filter $H_c$ by\[filt-defn\] $\operatorname{{\textsf}{ord}}^0 H_c = {\mathbb{C}}[{\mathfrak{h}}]\ast W$, $\operatorname{{\textsf}{ord}}^1H_c = {\mathfrak{h}}+ \operatorname{{\textsf}{ord}}^0H_c$ and $\operatorname{{\textsf}{ord}}^iH_c = (\operatorname{{\textsf}{ord}}^1H_c)^i$ for $i>1$, and define the [*associated graded ring*]{} to be $\operatorname{{\textsf}{ogr}}H_c=\bigoplus \operatorname{{\textsf}{ogr}}^n H_c$, where $\operatorname{{\textsf}{ogr}}^n H_c = \operatorname{{\textsf}{ord}}^n H_c/\operatorname{{\textsf}{ord}}^{n-1}H_c$. Then is equivalent to the assertion that $\operatorname{{\textsf}{ogr}}H_c$ is isomorphic to the skew group ring ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\ast {{W}}$ defined by $\sigma f = \sigma(f) \sigma$, for $\sigma\in {{W}}$ and $f\in {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]$. The Dunkl-Cherednik representation {#dunch} ---------------------------------- Let $\delta \in {\mathbb{C}}[{\mathfrak{h}}]$\[delta-defn\] denote the discriminant polynomial $\delta = \prod_{s\in \mathcal{S}} \alpha_s$. Thus $\delta$ transforms under ${{W}}$ by the sign representation and ${\mathfrak{h}^{\text{reg}}}={\mathfrak{h}}\setminus \{\delta=0\}$ is the subset of ${\mathfrak{h}}$ on which the action of ${{W}}$ is free. By [@EG Proposition 4.5] there is an injective algebra morphism $\theta_c : H_c \to D ({\mathfrak{h}^{\text{reg}}}) \ast {{W}}, $\[theta-defn\] where $D(Z)$ denotes the [*ring of differential operators*]{} on an affine variety $Z$. Under $\theta_c$ the elements of ${\mathbb{C}}[{\mathfrak{h}}]$ are identified with the multiplication operators while, by [@EG p.280] and in the notation of , $y_i\in{\mathfrak{h}}$ is sent to the [*Dunkl operator* ]{} $$\label{dunkop}\theta_c (y_i) = \partial_{i} - \sum_{s\in S} c\alpha_s(y_i)\alpha_s^{-1}(1-s), \qquad \text{where}\ \partial_{i}= \partial/\partial x_i.$$ Since $\delta$ acts ad-nilpotently on $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$, the set $\{\delta^n\}$ forms an Ore set in that ring. As observed in [@BEGqi p.288]), $\theta_c$ becomes an isomorphism on inverting $\delta$; that is, $$\label{locdunk} {H_c^{\text{reg}}} = H_c[\delta^{-1}] \cong D({\mathfrak{h}^{\text{reg}}})\ast {{W}}.$$ For any variety $Z$, there is a natural filtration on $D(Z)$ by order of operators and this induces a filtration on $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ and its subalgebras by defining elements of ${{W}}$ to have order zero. If $R$ is a subalgebra (or submodule) of $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$, we write the operators of order $\leq n$ as $\operatorname{{\textsf}{ord}}^n(R)$. \[order-filt-defn\] When $R=H_c$, $\operatorname{{\textsf}{ord}}$ is clearly the same filtration as that defined in . The associated graded ring of $R$ will be written $\operatorname{{\textsf}{ogr}}(R)=\bigoplus \operatorname{{\textsf}{ogr}}^n(R)$, where $\operatorname{{\textsf}{ogr}}^n(R)=\operatorname{{\textsf}{ord}}^n(R)/\operatorname{{\textsf}{ord}}^{n-1}(R)$, and the resulting graded structure of $\operatorname{{\textsf}{ogr}}(R)$ will be called the [*order*]{} or [*$\operatorname{{\textsf}{ogr}}$ gradation*]{}.\[ogr-defn\] (This will be only one of several filtrations used in this paper.) {#gradingsec} The rings of differential operators $D({\mathfrak{h}})$ and $D({\mathfrak{h}^{\text{reg}}})$ also have a graded structure, given by the adjoint action $[{\mathbf{E}},-]$ of the [*Euler operator* ]{} ${\mathbf{E}}=\sum x_i\partial_{i}\in D({\mathfrak{h}})$. \[Euler-defn\] We will call this the [*Euler grading*]{} and write $\operatorname{{\mathbf{E}}\text{-deg}}$\[Euler-deg-defn\] for the corresponding degree function; thus $\operatorname{{\mathbf{E}}\text{-deg}}x_i=1$ and $\operatorname{{\mathbf{E}}\text{-deg}}\partial_{i}=-1$. Since ${\mathbf{E}}\in D({\mathfrak{h}})^{{W}}$, ${\mathbf{E}}$ commutes with ${{W}}$ in $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ and so this grading extends to that ring with $\operatorname{{\mathbf{E}}\text{-deg}}{{W}}=0$. By inspection, implies that the $y_i$ also have degree $-1$ and so each $H_c$ is also graded under $[{\mathbf{E}},-]$ and we continue to call this the Euler grading. It is well-known and easy to check that the ${\mathbf{E}}$-grading is compatible with the order filtration on $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$, in the sense that $[{\mathbf{E}}, \operatorname{{\textsf}{ord}}^n D({\mathfrak{h}^{\text{reg}}})\ast {{W}}]\subseteq \operatorname{{\textsf}{ord}}^n D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ for all $n\geq 0$. We therefore obtain an induced grading, again called the ${\mathbf{E}}$-grading, on the associated graded ring $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}^{\text{reg}}})\ast {{W}}\cong {\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^*]\ast W$. Clearly this is again given by $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}^*=1$ (which we define to mean that $\operatorname{{\mathbf{E}}\text{-deg}}(x)=1$ for every $0\not=x\in {\mathfrak{h}}^*$) while $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}=-1$ and $\operatorname{{\mathbf{E}}\text{-deg}}{{W}}=0$. One should note that, in general, ${\mathbf{E}}\notin H_c$. However, there is a natural element in $H_c$ that has the same adjoint action. Indeed, let $$\label{hdefn} {\mathbf{h}}= {\mathbf{h}}_c = \frac{1}{2} \sum_{i=1}^{n-1} (x_iy_i + y_ix_i) \in H_c.$$ \[boldh-defn\] This is independent of the choice of basis. By [@BEGqi (2.6)] we have $$\label{grading} [{\mathbf{h}},x] = x, \quad [{\mathbf{h}}, y] = - y, \quad \text{and} \quad [{\mathbf{h}}, w ] = 0 \qquad\mathrm{for \ all}\ x\in{\mathfrak{h}}^*,y\in {\mathfrak{h}}\ \mathrm{and}\ w\in {{W}}.$$ Thus commutation with ${\mathbf{h}}$ also induces the Euler grading on $H_c$. The spherical subalgebra {#shiftfunct} ------------------------ Let $e\in {\mathbb{C}}{{W}}$ be the trivial idempotent and $e_-\in {\mathbb{C}}{{W}}$\[e-defn\] be the sign idempotent; thus $e = |{{W}}|^{-1} \sum_{w\in {{W}}} w$ and $e_- = |{{W}}|^{-1} \sum_{w\in{{W}}} \text{sign}(w)w.$ The main algebra of study in this paper is not the Cherednik algebra itself, but its [*spherical subalgebra*]{} ${U}_c=eH_ce$\[spherical-defn\] and the related algebra ${U}^-_c=e_-H_ce_-$. We will use frequently and without comment that $\delta$ is a ${{W}}$-anti-invariant and so $e_-\delta =\delta e$. Also, as $\operatorname{{\mathbf{E}}\text{-deg}}{{W}}=0$, both $U_c$ and $U^-_c$ have an induced ${\mathbf{E}}$-graded structure. Partitions {#dominance} ---------- The rest of this section is devoted to the definition and basic properties of category $\mathcal{O}_c$. Since its structure depends upon the combinatorics of ${{W}}$-representations, we begin with the relevant notions from that theory. We write a partition of $n$ as $\mu = (\mu_1\geq \mu_2 \geq \cdots \geq \mu_l > 0)$, with the understanding that $\mu_i =0$ for $i>l$. The *Ferrers diagram* of $\mu$ is the set of lattice points\[d-mu-defn\] $$d(\mu) = \{ (i,j)\in {\mathbb{N}}\times {\mathbb{N}}: j < \mu_{i+1}\}.$$ Following the French style, the diagram is drawn with the $i$-axis vertical and the $j$-axis horizontal, so the parts of $\mu$ are the lengths of the rows, and $(0,0)$ is the lower left corner. The *arm* $a(x)$ and the *leg* $l(x)$ of a point $x\in d(\mu)$ denote the number of points strictly to the right of $x$ and above $x$, respectively. The [*hook length*]{} $h(x)$ is $1+a(x)+l(x)$. For example: $$\label{e:arm-leg-pix} \mu =(5,5,4,3,1)\qquad \begin{array}[c]{cccccc} \bullet & \hbox to 0pt{\hss $\scriptstyle l(x)$\hss }\\ \cline{2-2} \bullet & \multicolumn{1}{|c|}{\bullet }& \bullet \\ \bullet & \multicolumn{1}{|c|}{\bullet }& \bullet & \bullet \\ \cline{2-5} \bullet & \multicolumn{1}{|c|}{\llap{${}_{x}$}\bullet } & \bullet & \bullet& \multicolumn{1}{c|}{\bullet }& {\scriptstyle a(x)} \\ \cline{2-5} \llap{${}_{(0,0)}$} \bullet & \bullet & \bullet & \bullet & \bullet \end{array} \qquad a(x) = 3,\quad l(x) = 2, \quad h(x)=6.$$ The [*transpose partition $\mu^t$*]{} is obtained from $\mu$ by exchanging the rows and columns of $\mu$. We will always use the [*dominance ordering*]{}\[dominance-defn\] of partitions as in [@MacD p.7]; thus if $\lambda$ and $\mu$ are partitions of $n$ then $\lambda\geq \mu$ if and only if $\sum_{i=1}^k \lambda_i \geq \sum_{i=1}^k\mu_i$ for all $k\geq 1$. Let ${{\textsf}{Irrep}({{W}})}$\[irred-defn\] denote the set of simple ${{W}}$-modules, up to isomorphism. As usual, irreducible representations of ${{W}}$ will be parametrised by partitions of $n$. We use the ordering on ${{\textsf}{Irrep}({{W}})}$ arising from the dominance ordering; thus, as in [@MacD Example 1, p.116], the [trivial representation]{} $\operatorname{{\textsf}{triv}}$\[triv-defn\] is labelled by $(n)$ while the [sign representation]{} $\operatorname{{\textsf}{sign}}$\[sign-defn\] is parametrised by $(1^n)$ and so $\operatorname{{\textsf}{triv}}>\operatorname{{\textsf}{sign}}$. Note that the operation on ${{\textsf}{Irrep}({{W}})}$ given by tensoring by $\operatorname{{\textsf}{sign}}$ corresponds to the transposition of partitions. Category $\mathcal{O}_c$ {#subsec-3.7} ------------------------ (See [@GGOR] and [@BEGqi Definition 2.4].) Let $\mathcal{O}_c$\[cat-O-defn\] be the abelian category of finitely-generated $H_c$-modules $M$ which are locally nilpotent for the subalgebra $\mathbb{C}[{\mathfrak{h}}^*]\subset H_c$. By [@guay Theorem 3] $\mathcal{O}_c$ is a highest weight category. Given $\mu \in {{\textsf}{Irrep}({{W}})}$, we define $\Delta_c(\mu)$,\[standard-defn\] an object of $\mathcal{O}_c$ called the *standard module*, to be the induced module $\Delta_c(\mu) = H_c\otimes_{\mathbb{C}[{\mathfrak{h}}^*]\ast {{W}}} \mu,$ where $\mathbb{C}[{\mathfrak{h}}^*]\ast {{W}}$ acts on $\mu$ by $pw\cdot m = p(0) (w\cdot m)$ for $p\in \mathbb{C}[{\mathfrak{h}}^*]$, $w\in {{W}}$ and $m\in \mu$. It is shown in [@BEGqi Section 2] that each $\Delta_c(\mu)$ has a unique simple quotient $L_c(\mu)$,\[L-defn\] that the set $\{ L_c(\mu) : \mu \in {{\textsf}{Irrep}(W)}\}$ provides a complete list of non-isomorphic simple objects in $\mathcal{O}_c$ and that every object in $\mathcal{O}_c$ has finite length. Note that it follows from the PBW Theorem \[PBW\] that the standard module $\Delta_c(\mu)$ is a free left $\mathbb{C}[{\mathfrak{h}}]$-module of rank $\dim(\mu)$. The ${{\textsf}{KZ}}$ functor {#subsec-3.11} ----------------------------- Let $M\in \mathcal{O}_c$. Then its localisation ${M^{\text{reg}}}=M[\delta^{-1}]$ at the powers of $\delta$ is a $W$-equivariant $D$-module on ${\mathfrak{h}^{\text{reg}}}$ in the sense that ${M^{\text{reg}}}$ is a ${{W}}$-equivariant vector bundle on ${\mathfrak{h}^{\text{reg}}}$ with a flat $W$-equivariant connection. On taking the germs of horizontal sections on ${\mathfrak{h}^{\text{reg}}}/{{W}}$ we get a representation of the braid group $B_n = \pi_1 ({\mathfrak{h}^{\text{reg}}}/{{W}})$. This representation factors through the [*Hecke algebra*]{} ${\mathcal{H}_{q}}$\[hecke-defn\] of ${{W}}$ with parameter $q= \exp (2\pi i c)$ [@GGOR Theorem 5.13]. In this way we have the [*Knizhnik-Zamolodchikov functor*]{} \[KZ-defn\] ${{\textsf}{KZ}}:\mathcal{O}_c \rightarrow {\mathcal{H}_{q}}{\text{-}{\textsf}{mod}}.$ There is an anti-involution $\iota$ on ${\mathcal{H}_{q}}$ induced by $\iota(T_w) = T_{w^{-1}}$. Given a module $V$ for ${\mathcal{H}_{q}}$, the space $V^{\ast} = \operatorname{Hom}_{{\mathcal{H}_{q}}}(V, {\mathbb{C}})$ becomes an ${\mathcal{H}_{q}}$-module via the rule $h\cdot f (v) = f(\iota(h)v)$. The images of the standard modules under ${{\textsf}{KZ}}$ are known, [@GGOR Remark 6.9 and Corollary 6.10]. For $c\in {\mathbb{R}}_{\geq 0}$ and $\mu\in {{\textsf}{Irrep}({{W}})}$ $$\label{kzsp} {{\textsf}{KZ}}(\Delta_c(\mu)) \cong Sp_q(\mu)^{\ast}$$ where $Sp_q(\mu)$\[specht-defn\] is the so-called [*Specht module*]{} associated to $\mu$. (The dual module appears since the defining relations for the rational Cherednik algebra given in [@GGOR] are normalised differently to ; as remarked in , our parameter $c$ corresponds to $-k_1$ in [@GGOR].) Now suppose that $M\in \mathcal{O}_c$ has a filtration $$0 = M_0\subset M_1\subset \cdots \subset M_{t-1} \subset M_t =M$$ such that $M_i/M_{i-1}$ is a standard module for all $1\leq i \leq t$. If $N\in \mathcal{O}_c$ and $c\notin \frac{1}{2}+{\mathbb{Z}}$ then [@GGOR Proposition 5.9] implies that $$\label{kzhom} \operatorname{Hom}_{H_c}(N,M) = \operatorname{Hom}_{{\mathcal{H}_{q}}}({{\textsf}{KZ}}(N), {{\textsf}{KZ}}(M)).$$ Morita equivalence of Cherednik algebras {#shift} ======================================== {#sec301} A powerful technique in the theory of semisimple Lie algebras is the translation principle, given by tensoring with a finite dimensional module, in part because it gives an equivalence of categories between the ${\mathcal{O}}$ categories (and the Harish-Chandra categories) corresponding to distinct central characters [@BG]. One interpretation of this is that tensoring with a module of $\mathfrak{k}$-finite vectors gives a Morita equivalence between the corresponding factors of the enveloping algebra [@JS Corollary 4.13]. Although it does not involve finite dimensional modules, there is a natural analogue of this procedure for Cherednik algebras, given by the Heckman-Opdam shift functors defined in . These functors have proved useful in a number of papers (see, for example, [@BEGqi; @BEGfd; @gordc]) and for particular values of $c$ these functors are known to give equivalences of categories between $H_c$, $U_{c}$ and $U_{c+1}$ (see, for example, [@BEGqi Theorem 8.1] and [@BEGfd Proposition 4.3]). It is an open problem to determine precisely when these equivalences exist [@GGOR Remark 5.17] and this question is crucial to our ${\mathbb{Z}}$-algebra approach to Cherednik algebras. We give an essentially complete answer to this question in Corollary \[morrat-cor\] and Remark \[morrat-cor-remark\]. We also prove that the equivalence $H_c\to H_{c+1}$ maps category $\mathcal{O}_c$ to $\mathcal{O}_{c+1}$ and sends the standard module $\Delta_c(\mu)$ to $\Delta_{c+1}(\mu)$, see Proposition \[shiftonO\]. {#subsec-4.0} Fix $c\in {\mathbb{C}}$ and keep the notation of . If we identify $H_{c}$ with its image in $D ({\mathfrak{h}^{\text{reg}}})\ast W$ via the Dunkl operator then, by [@BEGfd Proposition 4.1], there is an identity $$\label{conj} {U}_c \ =\ \delta^{-1} {U}^-_{c+1}\delta \ = \ e\delta^{-1} H_{c+1}\delta e.$$ In particular, \[Q-defn\] $Q_c^{c+1}= eH_{c+1}e_-\delta = eH_{c+1}\delta e$ is a $({U}_{c+1}, {U}_c)$-bi-submodule of $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$. The shift functors mentioned above are given by \[shift-defn\] $$S_c: {U}_c {\text{-}{\textsf}{mod}}\to {U}_{c+1}{\text{-}{\textsf}{mod}}: \qquad N\mapsto Q_c^{c+1} \otimes_{{U}_c} N$$ and $$\widetilde{S}_c: H_{c} {\text{-}{\textsf}{mod}}\to H_{c+1}{\text{-}{\textsf}{mod}}: \qquad M\mapsto H_{c+1}e_-\delta \otimes_{{U}_{c}} eM.$$ {#subsec-4.1} When $c$ is a positive real number, the Morita equivalence between $U_c$ and $ U_{c+1}$ is given by $S_c$ and we begin with that case. The general case, proved in Corollary \[morrat-cor\], will be an easy consequence. \[morrat\] Assume that $c\in {\mathbb{R}}_{\geq 0}$ with $c\notin \frac{1}{2} + \mathbb{Z}$. Then both shift functors $\widetilde{S}_c: H_{c}{\text{-}{\textsf}{mod}}\to H_{c+1}{\text{-}{\textsf}{mod}}$ and $S_c: {U}_c{\text{-}{\textsf}{mod}}\to {U}_{c+1}{\text{-}{\textsf}{mod}}$ are Morita equivalences. Moreover, the idempotent functor $E_c: H_{c} {\text{-}{\textsf}{mod}}\to {U}_c{\text{-}{\textsf}{mod}}$ given by $M\mapsto eM$ is a Morita equivalence. In order to prove that $S_c$ is an equivalence, we need to show that $Q=Q_c^{c+1}$ is a projective generator for ${U}_{c+1}{\text{-}{\textsf}{mod}}$, with endomorphism ring $\mathrm{End}_{U_{c+1}}(Q) =U_{c}$. Arguing as in [@EG Theorem 1.5(iv)] the dual $Q^\ast=\operatorname{Hom}_{{U}_{c+1}}(Q,{U}_{c+1})$ is $P = \delta^{-1} e_-H_{c+1}e.$ By the dual basis lemma, $Q$ is a projective ${U}_{c+1}$-module with $\mathrm{End}_{U_{c+1}}(Q) = U_{c}$ if and only if $PQ={U}_c$ while $Q$ is a generator if and only if $QP={U}_{c+1}$. Substituting in the given formulæ for $Q$ and $P$ shows that we need to prove that $$\label{morrat1} H_{c+1}e_-H_{c+1} =H_{c+1}\qquad\text{and} \qquad H_{c+1}eH_{c+1} = H_{c+1}\quad\text{for}\ c\geq 0.$$ Similarly, as $H_{c}e$ is a projective left $H_{c}$-module, $E_c$ will be a Morita equivalence if we prove that $$\label{morrat11} H_ceH_c = H_c \quad \text{for}\ c\geq 0.$$ Since $\widetilde{S}_{c}=E^{-1}_{c+1}\circ S_{c}\circ E_{c}$, Equations \[morrat1\] and \[morrat11\] will suffice to prove the theorem. The proof of Theorem \[morrat\] will be through a series of lemmas and we begin with the first equality in . Set $d=c+1$; thus $d\in {\mathbb{R}}_{\geq 1}$, with $d\notin \frac{1}{2}+{\mathbb{Z}}$. Reduction to Category ${\mathcal{O}}$ {#subsec-4.2} ------------------------------------- If $H_de_-H_d$ is a *proper* two-sided ideal of $H_d$ it must be contained in a primitive ideal, and hence, by [@ginz Generalized Duflo Theorem], annihilate an object from category $\mathcal{O}_d$. Thus it is enough to show that $e_-$ does not annihilate any simple module belonging to $\mathcal{O}_d$. To do this we first show in Corollary \[poono\] that the composition factors of $\Delta_d(\mu)$ are of the form $L_d(\lambda)$ for $\lambda \leq \mu$. Under the ${\mathbb{Z}}$-strings ordering such a result is proved in [@guay] but as we work with the dominance ordering of partitions and representations, as defined in , this definitely requires work, see also . We then show that the lowest weight copy of the sign module for ${{W}}$ in $\Delta_d(\mu)$ does not occur in any standard module $\Delta_d(\lambda)$ for $\lambda < \mu$. Since $L_d(\mu)$ is the head (that is, the unique simple factor module) of $\Delta_d(\mu)$ it will follow that $e_-L_d(\mu)\neq 0$. Lemma. {#basiccom} ------ [*Let $c\in {\mathbb{R}}_{\geq 0}$ with $c\notin \frac{1}{2}+\mathbb{Z}$. If $\mathrm{Hom}_{H_c}(\Delta_c(\lambda),\,\Delta_c(\mu))\not=0$ for $\lambda,\mu\in {{\textsf}{Irrep}({{W}})}$, then $\lambda \leq \mu$ in the dominance ordering.* ]{} Let $S_q = S_q(n,n)$\[schur-defn\] be the $q$-Schur algebra defined in [@DJ Section 1], where $q = \exp(2\pi i c)$. It is conjectured in [@GGOR Remark 5.17] that $S_q{\text{-}{\textsf}{mod}}$ is equivalent to ${\mathcal{O}}_c$. We cannot prove this, but we will show that there is a relationship which implies the lemma. For each $\mu\in {{\textsf}{Irrep}({{W}})}$ there is an $S_q$-module $W_q(\mu)$, \[q-weyl-defn\] called the [*$q$-Weyl module*]{}. By [@DJ Corollary 8.6], there is an isomorphism $$\label{qsch}\operatorname{Hom}_{\mathcal{H}_q}(Sp_q(\mu), Sp_q(\lambda)) \cong \operatorname{Hom}_{S_q}(W_q(\lambda),W_q(\mu)).$$ On the other hand, by and we have $$\label{tf} \operatorname{Hom}_{H_c}(\Delta_c(\lambda), \Delta_c(\mu)) \cong \operatorname{Hom}_{\mathcal{H}_q}(Sp_q(\lambda)^{\ast}, Sp_q(\mu)^{\ast}) \cong \operatorname{Hom}_{{\mathcal{H}_{q}}}(Sp_q(\mu), Sp_q(\lambda)).$$ Each $W_q(\nu)$ has a simple head $F_q(\nu)$,\[F-defn\] [@DJ Theorem 4.6] and $\{ F_q(\nu): \nu\in {{\textsf}{Irrep}({{W}})}\}$ is a complete, repetition-free list of the simple $S_q$-modules up to isomorphism, [@DJ Theorem 8.8]. Furthermore, $F_q(\lambda)$ is a composition factor of $W_q(\mu)$ only if $\lambda\leq \mu$, [@DJ Corollary 8.9]. By and a non-zero homomorphism $\phi: \Delta_c(\lambda)\to \Delta_c(\mu)$ implies the existence of a non-zero homomorphism $\phi ':W_q(\lambda)\to W_q(\mu)$. Thus $F_q(\lambda)$ must be a composition factor of $W_q(\mu)$ and so $\lambda\leq \mu$. Corollary {#poono} --------- [*Assume that $c\in {\mathbb{R}}_{\geq 0}$, with $c\notin \frac{1}{2}+\mathbb{Z}$. If $ [\Delta_c(\mu) : L_c(\lambda) ] \neq 0$ for $\lambda,\mu\in {{\textsf}{Irrep}({{W}})}$, then $ \lambda \leq \mu$ in the dominance ordering.* ]{} \(1) For arbitrary $c$ and $\mu$, the unique occurrence of $L_c(\mu)$ as a composition factor of $\Delta_c(\mu)$ is as its head—see, for example, the discussion after Lemma 7 in [@guay Section 2]. \(2) Since $\operatorname{{\textsf}{sign}}$ is minimal in the dominance ordering, the lemma and the above remark imply that $\Delta_c(\operatorname{{\textsf}{sign}})$ is irreducible for all $c\in {\mathbb{R}}_{\geq 0}$. This can also be deduced from [@guay]. We argue by induction on $\mu$. More precisely, suppose that $[\Delta_c(\mu): L_c(\lambda)] \neq 0$ for some $\mu \neq \lambda$ and that the lemma holds for any $\nu < \mu$. (The induction starts since there are only finitely many $\sigma$ with $\sigma<\mu$.) Let $P_c(\lambda)$ \[P-defn\] denote the projective cover of $\Delta_c(\lambda)$, as in [@GGOR Section 3.5], and write $K$ for the kernel of the associated homomorphism $\phi: P_c(\lambda) \to \Delta_c(\mu)$. By [@guay Proposition 13] there is a $\Delta$-filtration of $P_c(\lambda)$ $$P_c(\nu) = M_0 \supset M_1 \supset \cdots \supset M_t = 0$$ with each factor $M_j/M_{j+1}$ of the form $\Delta_c(\lambda_j)$ for some $\lambda_j\in {{\textsf}{Irrep}({{W}})}$. Thus there exists $i$ such that $M_i+K/K \neq 0$ but $M_{i+1}+K/K = 0$. This gives a non-zero composition $$\psi: \Delta_c(\lambda_i) \cong M_i / M_{i+1} \longrightarrow (M_i + K)/K \longrightarrow P_c(\lambda)/K \longrightarrow \Delta_c(\mu).$$ By Lemma \[basiccom\], $\lambda_i \leq\mu$. If $\lambda_i=\mu$ then the first remark after the statement of the lemma would imply that $\psi$ and hence $\phi$ are surjective, contradicting the fact that $\lambda\not=\mu$. Thus $\lambda_i<\mu$. By BGG reciprocity [@guay Theorem 19], $[P_c(\lambda): \Delta_c(\lambda_i)] = [\Delta_c(\lambda_i): L_c(\lambda)]\neq 0$ and so, by induction, $\lambda \leq \lambda_i$. Thus $\lambda < \mu$. {#order-remark} A result analogous to Corollary \[poono\] is proved as part of the proof of [@guay Proposition 13]. However the ${\mathbb{Z}}$-strings ordering used in [@guay] is different from the dominance ordering. An explicit example where the orderings differ can be found when $n=8$, by taking $\lambda = (6,1,1)$ and $\mu = (4,4)$. In this case $\lambda$ and $\mu$ are incomparable in the dominance ordering, but comparable in the ${\mathbb{Z}}$-strings ordering. The canonical grading on $\mathcal{O}_c$. {#cangrad} ----------------------------------------- The final ingredient we need for the proof of Theorem \[morrat\] is a canonical grading on ${\mathcal{O}}_c$. Let ${\mathbf{h}}_c\in H_c$ be defined as in . Then, for $M\in {\mathcal{O}}_c$ and $\alpha\in{\mathbb{C}}$, set $$W_{\alpha}(M) = \{ m\in M : ({\mathbf{h}}_c - \alpha)^km = 0 \text{ for $k\gg 0$}\}.$$\[cangrad-defn\] By [@GGOR (2.4.1)] this gives the *canonical grading* $M = \sum_{\alpha\in {\mathbb{C}}} W_{\alpha}(M).$ This observation has two useful consequences. First, if $\theta:M_1\rightarrow M_2$ is an $H_c$-module homomorphism with $M_i\in {\mathcal{O}}_c$, then $\theta (W_{\alpha}(M_1)) \subseteq W_{\alpha}(M_2)$ for each $\alpha\in{\mathbb{C}}$. Secondly, if $p\in H_c$ has $\operatorname{{\mathbf{E}}\text{-deg}}p = t$, then , implies that $p\cdot W_{\alpha}(M) \subseteq W_{\alpha+t}(M)$. Note that the standard module $\Delta_c(\mu)$ is therefore a lowest weight module since it is generated as a ${\mathbb{C}}[{\mathfrak{h}}]$-module by the space $1\otimes \mu$. {#fkdeg} To describe the graded structure of the standard modules we need a little notation. Recall that the space of coinvariants ${\mathbb{C}}[{\mathfrak{h}}]^{\text{co} {{W}}} = {{\mathbb{C}}[{\mathfrak{h}}]}\big/{{\mathbb{C}}[{\mathfrak{h}}]_+^{{{W}}}{\mathbb{C}}[{\mathfrak{h}}]}$ is a finite dimensional graded algebra isomorphic as a ${{W}}$-module to the regular representation. As in [@op], the polynomials $$\label{fakedegrees} f_{\mu}(v) = \sum_{i\geq 0} [{\mathbb{C}}[{\mathfrak{h}}]^{\text{co} {{W}}}_i : \mu] v^i$$ are called the *fake degrees*\[fake-defn\] of $\mu\in{{\textsf}{Irrep}({{W}})}$. We define $n(\mu)$ to be the lowest power of $v$ appearing in $f_{\mu}(v)$; thus, $f_{\mu}(v) = a v^{n(\mu) }+ \text{higher order terms.}$ In the notation of [@hai3], $n(\mu)$ is equal to the [*partition statistic*]{} $ \sum_{i}\mu_i (i-1)$ (see the proof of [@babyv Theorem 6.4]). Finally, implies that $$\label{fakedegrees2} f_{\mu^t}(1)=\dim \mu^t = \dim \mu = f_\mu(1) \qquad\rm{for}\ \mu\in {{\textsf}{Irrep}({{W}})}.$$ {#subsec-3.10} Given a graded ${{W}}$-module $M=\sum_{\alpha\in{\mathbb{C}}} W_\alpha(M)$ we define its [*graded Poincaré series*]{} to be $$p(M,v,{{W}}) = \sum_{\alpha\in {\mathbb{C}}} v^\alpha \sum_{\lambda\in {{\textsf}{Irrep}({{W}})}} [W_\alpha(M) : \lambda][\lambda].$$\[W-poincare\] This is easily determined for standard modules. [(1)]{} Under the canonical grading, the subspace $1\otimes \mu$ of $\Delta_c(\mu)$ has weight $m+c(n(\mu) - n(\mu^{t}))$, where $m=(n-1)/2$. [(2)]{} The Poincaré series of $\Delta_c(\mu)$ as a graded ${{W}}$-module is $$\label{polystand} p(\Delta_c(\mu), v, {{W}}) = v^{m+c(n(\mu)-n(\mu^t))} \frac{\sum_{\lambda}f_{\lambda}(v) [\lambda\otimes \mu]}{\prod_{i=2}^n (1-v^i)}.$$ \(1) We need to compute the action of ${\mathbf{h}}=\frac{1}{2}\sum_{i=1}^{n-1}(x_iy_i+y_ix_i)$ on the space $1\otimes \mu$. By the defining relations of $H_c$ from , and the fact that the $\{x_i\}$ and $\{y_i\}$ are dual bases, we obtain $$\begin{array}{rl} {\mathbf{h}}= \sum_i x_iy_i + (n-1)/2 -\frac{1}{2}\sum_{s\in \mathcal{S}} \sum_{i}c\alpha_s(y_i)x_i(\alpha_s^\vee)s =& \sum x_iy_i + (n-1)/2 -\frac{c}{2}\sum_{s\in \mathcal{S}} \alpha_s(\alpha_s^\vee)s\\ \noalign{\vskip 5pt} =&\sum x_iy_i + m -c\sum_{s\in \mathcal{S}}s. \end{array}$$ The action of $\sum(1-s)$ on $\lambda\in {{\textsf}{Irrep}({{W}})}$ can be derived from [@BM; @Lu]. More precisely, $\lambda$ is special by [@Lu (4.2.2)] and so $n(\lambda)= b_\lambda=a_\lambda$ in the notation of [@Lu]. Therefore, by [@BM Section 4.21] and [@Lu Section 4.1 and (5.11.5)], $\sum_s(1-s)$ acts on $\lambda\in {{\textsf}{Irrep}({{W}})}$ with weight $N+n(\lambda)-n(\lambda^t)$, where $N=n(n-1)/2$ is the cardinality of $\mathcal{S}$. Thus $\sum_ss$ acts on $1\otimes \mu$ with weight $-(n(\mu) - n(\mu^t))$ and hence ${\mathbf{h}}$ acts with weight $m+c(n(\mu) - n(\mu^t))$. \(2) As graded ${{W}}$-modules, $\Delta(\mu)\cong ({\mathbb{C}}[{\mathfrak{h}}]\otimes \mu)[k]$ for $k=m+c(n(\mu) - n(\mu^t))$. The shift arises from the fact that, by (1), the generator $1\otimes \mu$ of $\Delta_c(\mu)$ lives in degree $k$. The Chevalley-Shephard-Todd Theorem implies that, as graded ${{W}}$-modules, ${\mathbb{C}}[{\mathfrak{h}}]\cong {\mathbb{C}}[{\mathfrak{h}}]^{{W}}\otimes {\mathbb{C}}[{\mathfrak{h}}]^{\text{co} {{W}}}$. Now ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ is a polynomial ring with generators in degrees $2\leq i\leq n$ and so its Poincaré polynomial is $ \prod_{i=2}^n (1-v^{i})^{-1}$. On the other hand, the coinvariant ring $ {\mathbb{C}}[{\mathfrak{h}}]^{\text{co}{{W}}}$ has graded Poincaré polynomial $\sum_{\lambda} \sum_i [{\mathbb{C}}[{\mathfrak{h}}]_i^{\text{co} {{W}}} : \lambda] [\lambda]v^i$. By definition, this is just $\sum_{\lambda} f_{\lambda}(v)[\lambda]$. Combining these observations gives . Completion of the proof of Theorem \[morrat\] {#subsec-4.5} --------------------------------------------- We first prove that $H_de_-H_d=H_d$ (where $d=c+1$, as before). Since $\mu\cong \mu^*$ for symmetric groups, the sign representation is a direct summand of $\mu\otimes \nu$ if and only if $\nu =\mu^t$. Thus implies that $\operatorname{{\textsf}{sign}}$ first appears in $\Delta_d(\mu)$ in the weight space $$m+d(n(\mu)-n(\mu^t)) + n(\mu^t) = m+ dn({\mu}) - (d-1)n({\mu^t}) \qquad\text{where}\quad m=(n-1)/2.$$ If $\lambda \leq \mu$ then $n({\lambda}) \geq n(\mu)$ by [@Shi Theorem B and Proposition 1.6]. Moreover, as $\lambda^t\geq \mu^t$, we have $n(\lambda^t)\leq n(\mu^t)$. Since $d\in \mathbb{R}_{\geq 1}$, $$m+dn({\lambda}) -(d-1)n({\lambda^t}) \geq m+dn({\mu}) -(d-1)n({\mu^t})$$ with equality if and only if $ \lambda=\mu$. It follows that the copy of $\operatorname{{\textsf}{sign}}$ appearing in the lowest possible weight space of $\Delta_d(\mu)$ is never a weight of $\Delta_d(\lambda)$ for $\lambda < \mu$. By Corollary \[poono\], this means that this copy of $\operatorname{{\textsf}{sign}}$ is a weight for $L_d(\mu)$ and hence that $e_-L_d(\mu)\not=0$. By this implies that $H_de_-H_d =H_d$, and so the first equality of is proven. It remains to show that $H_ceH_c=H_c$ for $c\in {\mathbb{R}}_{\geq 0}$. The argument of shows that we need to prove that $e$ does not annihilate any simple module from $\mathcal{O}_c$. In this case $\operatorname{{\textsf}{triv}}$ appears in $\mu\otimes \nu$ if and only if $\nu=\mu$. Therefore, now implies that $\operatorname{{\textsf}{triv}}$ first appears in $\Delta_c(\mu)$ in degree $m+c(n(\mu) -n(\mu^t))+n(\mu).$ Let $\lambda\leq \mu$. Then $$m +c(n(\lambda)-n(\lambda^t))+n(\lambda)\ =\ m+(c+1)n(\lambda) -cn(\lambda^t) \ \geq\ m+c(n(\mu)-n(\mu^t))+n(\mu),$$ with equality if and only if $\lambda=\mu$. This means that $\operatorname{{\textsf}{triv}}$ appears in $\Delta_c(\lambda)$ in a higher degree than its first appearance in $\Delta_c(\mu)$. In particular, the simple quotient $L_c(\mu)$ of $\Delta_c(\mu)$ contains a copy of $\operatorname{{\textsf}{triv}}$ and so it cannot be annihilated by $e$. This therefore completes the proof of and and hence proves the theorem. General equivalences {#subsec-4.55} -------------------- We now give the promised extension of Theorem \[morrat\] to more general values of $c$. Since it requires no extra work, and it is put to crucial use in [@BFG], we will also prove the result over more general base fields. Thus if $k$ is a subfield of ${\mathbb{C}}$, with $c\in k$, let $H(k)_c$ denote the $k$-algebra defined by the generators and relations from . We write $U(k)_c$, $Q(k)_c^{c+1}$, etc, for the corresponding objects defined over $k$. \[morrat-hyp\] Set $\mathcal{C} = \{z: z=\frac{m}{d}\ \mathrm{where}\ m, d \in {\mathbb{Z}}\text{ with } 2\leq d\leq n \text{ and } z\notin {\mathbb{Z}}\}.$ Assume that $c\in {\mathbb{C}}$ is such that $c\notin \frac{1}{2} + \mathbb{Z}$. If $c$ is a rational number with $-1<c<0$ assume further that $c\not\in \mathcal{C}$. Corollary {#morrat-cor} --------- *Let $k\subseteq {\mathbb{C}}$ be a field and assume that $c\in k$ satisfies Hypothesis \[morrat-hyp\].* [(1)]{} $U(k)_c$ and $H(k)_{c}$ are Morita equivalent. If $c \notin (-2,-1)_{\mathcal{C}} =\{z\in \mathcal{C} : -2<z<0\}$, then $U(k)_c$ is Morita equivalent to $U(k)_{c+1}$. [(2)]{} Let $a=-c$. Then $H(k)_{a}$ is Morita equivalent to $U(k)^-_{a} = e_-H(k)_{a}e_-$. If $a \notin (1,2)_{\mathcal{C}}$, then $U(k)^-_{a}$ is Morita equivalent to $U(k)^-_{a-1}$. \(1) We start with the case $U_c=U({\mathbb{C}})_c$. If $c\not\in \mathcal{C}$ then it follows from [@BEGqi Theorem 8.1] and [@DJ2 Theorem 4.3] that $H_c$, $U_c$ and $U_c^-$ are simple, Morita equivalent rings (see the introduction to [@BEG3] for the details). Since this also applies to $H_{c+1}$ the conditions are trivially satisfied and the result follows. Thus we may assume that $c\in \mathcal{C}$. If $c\geq -1$, then necessarily $c\geq 0$ and so the result follows from Proposition \[morrat\]. Otherwise $c\leq -1$. In this case the discussion before [@De Remark 2.2] shows that there is an isomorphism $\chi: H_c\to H_{-c}$ satisfying $\chi(e_-)=e$. Thus, for any $c$, implies that $U_c\cong U^-_{-c} \cong eH_{-c-1}e=U_{-c-1}$. The result for $c\leq -1$ therefore follows from the cases already discussed. Finally, let $k$ be an arbitrary subfield of ${\mathbb{C}}$ and consider $U(k)_c$. In order to prove, for example, that $U(k)_c$ is Morita equivalent to $U(k)_{c+1}$ we need to prove that $Q(k)P(k)=U(k)_{c+1}$ and $P(k)Q(k)=U(k)_c$. By construction, $Q({\mathbb{C}})=Q(k)\otimes_k{\mathbb{C}}$, and similarly for $P({\mathbb{C}})$. By the earlier part of the proof, $U({\mathbb{C}})_c/P({\mathbb{C}})Q({\mathbb{C}})=0$. The faithful flatness of $U({\mathbb{C}})_c=U(k)_c\otimes_k{\mathbb{C}}$ as a $U(k)_c$-module then ensures that $U(k)_c/P(k)Q(k)=0$, whence $PQ=0$. All the other steps in the proof follow in exactly the same way. \(2) Using the identity $U_c\cong U^-_{-c}$, this follows from part (1). Remarks {#morrat-cor-remark} ------- \(1) The condition that $c\notin \frac{1}{2} + \mathbb{Z}$ is needed in Theorem \[morrat\] and Corollary \[morrat-cor\] in order to apply and may be unnecessary. This is the case when $n=2$ as $U_c$ is Morita equivalent to $U_{c+1} $ if and only if $c\not= -\frac{3}{2}, -\frac{1}{2}$ (see, for example, [@EG Proposition 8.2]). The point about the excluded cases is that $U_{-\frac{1}{2}}$ is simple but the two neighbouring algebras, $U_{\frac{1}{2}}$, $U_{-\frac{3}{2}}$ are not. Combining [@EG Proposition 8.2] with [@St Theorem B] shows that $U_{-\frac{1}{2}}$ has infinite global dimension, and so the next Corollary \[gldim\] also fails for this value of $c$. \(2) This also shows that the hypothesis $c\notin (-2,0)_{\mathcal{C}}$ is serious. Indeed, for any $n\geq 2$, let $c=-m/n\in (-1,0)_{\mathcal C}$. Then one can prove that the factor module $V_c =\Delta_c(\operatorname{{\textsf}{sign}})/I_c$ considered in [@CE Theorem 3.2] does not contain a copy of the ${{W}}$-module $\operatorname{{\textsf}{triv}}$ (we thank Pavel Etingof for this fact). In particular $eV_c=0$ and so [*the functor $E_c$ is not an equivalence*]{}. If we further assume that $(m,n)=1$, then $V_c$ is the unique irreducible finite dimensional $H_c$-module by [@CE Corollary 3.3] and [@BEGfd Theorem 1.2(ii)]. Since $U_c=\mathrm{End}_{H_c}(eH_c)$, this implies that $U_c$ has no finite dimensional modules. However, by Corollary \[morrat-cor\](1) and [@BEGfd Theorem 1.2] $U_{c\pm 1}{\text{-}{\textsf}{mod}}$ does have such modules and so [*there is no equivalence between $U_c $ and $U_{c\pm 1} $*]{}. Corollary {#gldim} --------- [*Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\]. Then $H_c$ and $U_c$ have finite homological global dimension and satisfy the Auslander-Gorenstein conditions and Cohen-Macaulay conditions of [@lev].*]{} Since this result takes us a little far afield, the details of the proof are left to the interested reader. Standard techniques show that $\operatorname{{\textsf}{ogr}}H_c\cong {{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast{{W}}$ and hence $H_c$ have the given properties (see, for example, [@Br Theorem 4.4]). By Corollary \[morrat-cor\], $U_c$ is Morita equivalent to $H_c$ and it follows that $U_c$ also has these properties. The shift functor on $\mathcal{O}_c$ {#subsec-4.6} ------------------------------------ Many computations for ${U}_c$ reduce to computations in category $\mathcal{O}$ and so it is important to know that, under the hypotheses of Theorem \[morrat\], $S_c$ does provide an equivalence between the corresponding categories. This is the point of the next result. \[shiftonO\] Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\] and that $c\notin {\mathbb{Q}}_{\leq -1}$. Then the shift functor $\widetilde{S}_c$ restricts to an equivalence between $\mathcal{O}_{c}$ and $\mathcal{O}_{c+1}$ such that $\widetilde{S}_c(\Delta_{c}(\lambda)) \cong \Delta_{c+1}(\lambda)$ for all partitions $\lambda$ of $n$. Thus $S_c(e\Delta_{c}(\lambda)) =e\Delta_{c+1}(\lambda)$. By Corollary \[morrat-cor\](2), an analogue of the proposition also holds when $c\in \mathbb{Q}_{\leq -1}$, provided that one shifts in a negative direction. The final assertion of the proposition is immediate from the previous one combined with Corollary \[morrat-cor\](1). We begin by showing that $\widetilde{S}_c$ restricts to an equivalence between $\mathcal{O}_{c}$ and $\mathcal{O}_{c+1}$. Fix $M\in \mathcal{O}_{c}$. Let $\mathcal{I}_t={\mathbb{C}}[{\mathfrak{h}}^*]^{{{W}}}_{\geq t}$ denote the ${{W}}$-invariant elements of ${\mathbb{C}}[{\mathfrak{h}}^*]$ of degree at least $t$ and set $I_t=\mathcal{I}_t{\mathbb{C}}[{\mathfrak{h}}^\ast ]$, Then ${\mathbb{C}}[{\mathfrak{h}}^\ast]/I_t$ is a finite dimensional algebra and so all homogeneous elements of ${\mathbb{C}}[{\mathfrak{h}}^*]$ of large degree belong to $I_t$. Thus it is enough to show that, if $\widetilde{m}=he_- \delta\otimes em \in \widetilde{S}_c(M)= H_{c+1}e_-\delta \otimes_{{U}_c} eM$, for some $h\in H_{c+1}$ and $m\in M$, then $\widetilde{m}$ is annihilated by $\mathcal{I}_t$ for $t\gg 0$. Recall the ${\mathbf{E}}$-grading on $H_c$ from . Since ${\mathbb{C}}[{\mathfrak{h}}^*]$ acts locally nilpotently on $M$, the PBW isomorphism shows that any homogeneous element of $H_c$ of sufficiently large negative ${\mathbf{E}}$-degree annihilates $m\in M$. Thus, assume that $qm=0$ for all $q\in H_c$ with $\operatorname{{\mathbf{E}}\text{-deg}}(q)\leq -t$ and let $p\in {\mathbb{C}}[{\mathfrak{h}}^*]^{{{W}}}_{\geq t}$. Then $$phe_-\delta \otimes em \ =\ [p,h]e_-\delta \otimes em + h\delta \delta^{-1} pe_- \delta \otimes em \ =\ [p,h]e_-\delta \otimes em + he_- \delta \otimes \delta^{-1} p \delta em.$$ Since $\operatorname{{\mathbf{E}}\text{-deg}}\delta^{-1}p \delta=\operatorname{{\mathbf{E}}\text{-deg}}p \leq -t$, we have $\delta^{-1} p\delta em = 0$ by the hypothesis on $t$. Therefore $p(he_- \delta \otimes em) = [p,h]e_- \delta \otimes em$ for any such $p$. Since the choice of $t$ was independent of $h$, this implies that $p^r (he_- \delta \otimes em) = \operatorname{ad}(p)^r(h)(e_- \delta \otimes em)$, for any $r>0$. Now, $p$ commutes with both ${\mathbb{C}}[{{W}}]$ and ${\mathbb{C}}[{\mathfrak{h}}^*]$, and so the defining relations of $H_{c+1}$ from ensure that the adjoint action of $p\in {\mathbb{C}}[{\mathfrak{h}}^*]^{{{W}}}$ on $H_{c+1}$ is locally nilpotent (see also [@BEGqi Lemma 3.3(v)]). Therefore a sufficiently large power of $p$ annihilates $he_-\delta \otimes em$. Thus $\widetilde{S}_c(M)\in \mathcal{O}_{c+1}$ and $\widetilde{S}_c$ does restrict to the desired equivalence. It remains to compute $\widetilde{S}_c(\Delta_{c}(\lambda))$ and we begin with the analogous problem on ${H_{c+1}^{\text{reg}}}$. In the notation of , $${\widetilde{S}_c(\Delta_{c}(\lambda))^{\text{reg}}} = {H_{c+1}^{\text{reg}}}e_-\delta \otimes_{\delta^{-1}{U}^-_{c+1}\delta } e\Delta_c(\lambda).$$ By , ${H_{c+1}^{\text{reg}}} \cong A=D({\mathfrak{h}^{\text{reg}}})\ast W$ and so ${\widetilde{S}_c(\Delta_{c}(\lambda))^{\text{reg}}} \cong Ae_-\delta \otimes_{B} {e\Delta_{c} (\lambda)^{\text{reg}}},$ where $B=\delta^{-1} e_-Ae_-\delta $. On the other hand, induces an isomorphism $$\theta: Ae_-\delta \otimes_{B} e{\Delta_c(\lambda)^{\text{reg}}} \longrightarrow Ae \otimes_{eAe} e{\Delta_c(\lambda)^{\text{reg}}}; \qquad ae_-\delta \otimes em\mapsto a\delta e\otimes em.$$ Combined with the identity $H_ceH_c=H_c$ from Corollary \[morrat-cor\](1), this implies that $$\label{nonzeromap1} {\widetilde{S}_c(\Delta_{c}(\lambda))^{\text{reg}}} \cong A e \otimes_{eA e} {e\Delta_{c}(\lambda)^{\text{reg}}} \cong {\left(H_{c}e \otimes_{{U}_c} e\Delta_{c}(\lambda)\right)^{\text{reg}}} \cong{\Delta_{c}(\lambda)^{\text{reg}}}\not=0.$$ If $c\not\in \mathcal{C}$, we are done. Indeed, in this case [@BEGqi Corollary 2.11] implies that $\Delta_{c+1}(\lambda)$, $\Delta_{c}(\lambda)$ and hence $\widetilde{S}_c(\Delta_{c}(\lambda))$ are all simple modules. The isomorphism implies that $\widetilde{S}_c(\Delta_{c}(\lambda))\hookrightarrow {\Delta_{c+1}(\lambda)^{\text{reg}}}$. Under this embedding, $\widetilde{S}_c(\Delta_{c}(\lambda))\cap \Delta_{c+1}(\lambda)\not=0$ and hence $\widetilde{S}_c(\Delta_{c}(\lambda))= \Delta_{c+1}(\lambda)$. We may therefore assume that $c\in \mathcal{C}$, in which case Hypothesis \[morrat-hyp\] implies that $c\geq 0$ and we can use the KZ-functor from . By and , $ {{\textsf}{KZ}}(\widetilde{S}_c(\Delta_{c}(\lambda))) \cong {{\textsf}{KZ}}(\Delta_{c}(\lambda)) \cong Sp_q(\lambda)^{\ast}.$ By and we therefore have $$\label{nonzeromap} \operatorname{Hom}_{H_{c+1}}(\widetilde{S}_c(\Delta_{c}(\lambda)), \Delta_{c+1}(\lambda)) \cong \operatorname{Hom}_{S_q}(W_q(\lambda), W_q(\lambda)) = {\mathbb{C}}.$$ It follows from Corollary \[poono\] that the composition factors of $\Delta_{c+1}(\lambda)$ are of the form $L_{c+1}(\nu)$ with $\nu\leq \lambda$ in the dominance ordering. We will show by an ascending induction on this ordering that $\widetilde{S}_c(\Delta_{c}(\lambda)) \cong \Delta_{c+1}(\lambda)$. If $\lambda$ is minimal in the dominance ordering then $\lambda=\operatorname{{\textsf}{sign}}$ and so both $\Delta_{c+1}(\lambda)$ and $\widetilde{S}_c(\Delta_{c}(\lambda))$ are simple by Remark \[poono\]. By there is a non-zero map from $\widetilde{S}_c(\Delta_{c}(\lambda))$ to $\Delta_{c+1}(\lambda)$ which therefore must be an isomorphism. This begins the induction. Let $\lambda$ be arbitrary and suppose that, for all $\nu < \lambda$ in the dominance ordering, we have $\widetilde{S}_c(\Delta_{c}(\nu))\cong\Delta_{c+1}(\nu)$, and hence that $\widetilde{S}_c(L_{c}(\nu)) \cong L_{c+1}(\nu)$. Since $\widetilde{S}_c$ is an equivalence, $\widetilde{S}_c(\Delta_{c}(\mu))$ has simple head $\widetilde{S}_c(L_{c}(\mu))$ for each $\mu$. By $\widetilde{S}_c(L_{c}(\lambda))$ is therefore isomorphic to a composition factor of $\Delta_{c+1}(\lambda)$. But, by Corollary \[poono\] and the remark thereafter, the composition factors of $\Delta_{c+1}(\lambda)$, except for the head, are of the form $L_{c+1}(\nu)$ for $\nu < \lambda$. Thus the non-zero map must send the head of $\widetilde{S}_c(\Delta_{c}(\lambda))$ to the head of $\Delta_{c+1}(\lambda)$ and so induce an isomorphism $\widetilde{S}_c(\Delta_{c}(\lambda))\xrightarrow{\sim} \Delta_{c+1}(\lambda)$. This completes the inductive step, and hence the proof of the proposition. The Hilbert scheme {#sect-haiman} ================== {#sect401} Haiman’s work on Hilbert schemes gives detailed information about their structure, in particular as “Proj” of appropriate Rees rings. The resulting formulæ for the Poincaré series of these rings will be crucial to the proof of the main theorem in Section \[sect-filt\]. In this section, we briefly describe the relevant results from the literature and use this to derive the appropriate Poincaré series. {#subsec-5.1} Let $\operatorname{Hilb^n{\mathbb{C}}^2}$\[hin-defn\] be [*the Hilbert scheme of $n$ points on the plane*]{}, which we realise as the set of ideals of colength $n$ in the polynomial ring ${\mathbb{C}}[{\mathbb{C}}^2]$. Similarly, identify the variety $S^n{\mathbb{C}}^2$ of *unordered* $n$-tuples of points in ${\mathbb{C}}^2$ with the categorical quotient $ {\mathbb{C}}^{2n}/{{W}}$ under the diagonal action of ${{W}}$ on ${\mathbb{C}}[{\mathbb{C}}^{2n}]$. Then the map\[tau-defn\] $\tau : \operatorname{Hilb^n{\mathbb{C}}^2}\to S^n{\mathbb{C}}^2={\mathbb{C}}^{2n}/{{W}}$ which sends an ideal to its support (counted with multiplicity) is a resolution of singularities (see, for example, [@Nak Theorem 1.15]). We will actually be interested in a resolution of singularities for ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^{{W}}$ rather than ${\mathbb{C}}[{\mathbb{C}}^{2n}]^{{W}}$, simply because the associated graded ring of $U_c$ is ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^{{W}}$. The results we need follow easily from the corresponding results on $\operatorname{Hilb^n{\mathbb{C}}^2}$ and so we begin with the latter. The (isospectral) Hilbert scheme {#isospecsec} -------------------------------- Following [@hai3 Definition 3.2.4] [*the isospectral Hilbert scheme*]{} ${\mathbb{X}}_n$\[isospec-defn\] is the reduced fibre product $$\begin{CD} {\mathbb{X}}_n @> f_1 >> {\mathbb{C}}^{2n} \\ @V \rho_1 VV @VVV \\ \operatorname{Hilb^n{\mathbb{C}}^2}@> \tau >> {\mathbb{C}}^{2n}/{{W}}. \end{CD}$$ It is a highly non-trivial fact (see [@hai3 Theorem 3.1 and the proof of Proposition 3.7.4]) that $\rho_1$ is a flat map of degree $n!$. Haiman has given a description of both $\operatorname{Hilb^n{\mathbb{C}}^2}$ and ${\mathbb{X}}_n$ as Proj of appropriate graded rings and we recall this description since it will be extremely important to us. Let $ {\mathbb{A}}^1={\mathbb{C}}[{\mathbb{C}}^{2n}]^{{\epsilon}}$ \[AAA-1-defn\] be the space of ${{W}}$-alternating polynomials in ${\mathbb{C}}[{\mathbb{C}}^{2n}]$ and write $ {\mathbb{J}}^1= {\mathbb{C}}[{\mathbb{C}}^{2n}]{\mathbb{A}}^1$ for the ideal generated by ${\mathbb{A}}^1$. For $d\geq 1$ define ${\mathbb{A}}^d$ and ${\mathbb{J}}^d$ to be the respective $d^{\text{th}}$ powers of ${\mathbb{A}}^1$ and ${\mathbb{J}}^1$ using multiplication in ${\mathbb{C}}[{\mathbb{C}}^{2n}]$; thus \[JJJ-defn\] ${\mathbb{J}}^d={\mathbb{C}}[{\mathbb{C}}^{2n}] {\mathbb{A}}^d$. Finally, set ${\mathbb{J}}^0={\mathbb{C}}[{\mathbb{C}}^{2n}]$, ${\mathbb{A}}^0={\mathbb{C}}[{\mathbb{C}}^{2n}]^{{W}}$ and ${\mathbb{A}}=\bigoplus_{d\geq 0} {\mathbb{A}}^d\cong {\mathbb{A}}^0[t{\mathbb{A}}^1]$. Then [@haidis Proposition 2.6] proves that $$\label{AAA-alg-defn} \operatorname{Hilb^n{\mathbb{C}}^2}\cong \operatorname{Proj}{\mathbb{A}}\mathrm{ \ as\ a\ scheme\ over\ } \operatorname{Spec}{\mathbb{A}}^0 = {\mathbb{C}}^{2n}/{{W}}$$ Similarly, ${\mathbb{X}}_n \cong \operatorname{Proj}{\mathbb{S}}$, where ${\mathbb{S}}={\mathbb{C}}[{\mathbb{C}}^{2n}][t{\mathbb{J}}^1],$ is the blowup of ${\mathbb{C}}^{2n}$ at ${\mathbb{J}}^1$ [@hai3 Proposition 3.4.2]. {#subsec-5.5} Observe that ${\mathbb{J}}^d$ is generated by its ${{W}}$-alternating or ${{W}}$-invariant elements, respectively, depending on whether $d$ is odd or even. Following Haiman we refer to these elements as having *correct parity*. \[corpar\] [(1)]{} For any $d\geq 0$, ${\mathbb{A}}^d$ consists of the elements of ${\mathbb{J}}^d$ with the correct parity. [(2)]{} If ${\mathbb{C}}^n$ denotes the first copy of that space in ${\mathbb{C}}^{2n}$, then ${\mathbb{J}}^d$ is a free module over both ${\mathbb{C}}[{\mathbb{C}}^n]$ and ${\mathbb{C}}[{\mathbb{C}}^n]^{{{W}}}$. \(1) The statement is clearly true for $d=0,1$. Assume, by induction, that it is true for $d-1$. We will suppose that $d$ is even, the argument in the odd case being similar. Since ${\mathbb{A}}^1$ generates the ideal ${\mathbb{J}}^1$, any element $x\in {\mathbb{J}}^d$ can be decomposed as $x= \sum_i p_i q_i$ where $p_i \in {\mathbb{J}}^{d-1}$ and $q_i \in {\mathbb{A}}^1$. Since $q_ie = e_-q_i$ we have $(p_iq_i)e = (p_i e_-)q_i$ for all $i$. If $x$ has the correct parity then $x = xe = \sum_i (p_iq_i)e = \sum_i (p_ie_-q_i).$ But ${\mathbb{J}}^{d-1}e_{-}$ is the subset of ${{W}}$-alternating elements of ${\mathbb{J}}^{d-1}$ and so ${\mathbb{J}}^{d-1}e_{-}={\mathbb{A}}^{d-1}$ by induction. Thus $x\in {\mathbb{A}}^{d-1} {\mathbb{A}}^1 ={\mathbb{A}}^d$. \(2) By [@hai3 Proposition 4.11.1] ${\mathbb{J}}^d$ is a projective module over ${\mathbb{C}}[{\mathbb{C}}^n]$ and hence over ${\mathbb{C}}[{\mathbb{C}}^n]^{{W}}$. Since ${\mathbb{C}}[{\mathbb{C}}^n]$ and ${\mathbb{C}}[{\mathbb{C}}^n]^{{W}}$ are polynomial rings, any such projective module is free by the Quillen-Suslin Theorem. Geometric interpretation {#geomint} ------------------------ There is a geometric description of both ${\mathbb{A}}^d$ and ${\mathbb{J}}^d$. Let ${\mathcal{B}}_1$\[tauto-defn\] be the [*tautological rank $n$ vector bundle*]{} on $\operatorname{Hilb^n{\mathbb{C}}^2}$ and let ${\mathcal{P}}_1 = (\rho_1)_{\ast}{\mathcal{O}}_{{\mathbb{X}}_n}$ \[PP1-defn\] denote the [*Procesi bundle*]{} of rank $n!$ arising from the map $\rho_1 : {\mathbb{X}}_n \to \operatorname{Hilb^n{\mathbb{C}}^2}$. Write ${\mathcal{L}}_1 = \bigwedge^n {\mathcal{B}}_1$\[LLL-defn\] for the determinant bundle of ${\mathcal{B}}_1$. By [@haidis Proposition 2.12] ${\mathcal{L}}_1$ is also the canonical ample line bundle ${\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}(1)$ associated to the presentation $\operatorname{Hilb^n{\mathbb{C}}^2}\cong \operatorname{Proj}{\mathbb{A}}$. {#subsec-5.7} Set $l=dn$ for some $d\geq 1$ and write \[RR-defn\] ${\mathbb{R}}(n, l) = H^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1\otimes {\mathcal{B}}_1^{l}).$ One should note that ${\mathbb{R}}(n,l)$ is defined in [@hai1] to be the coordinate ring of the polygraph $Z(n,l)$ but, by [@hai1 Theorem 2.1], it is also isomorphic to the given ring of global sections. There is an action of ${{W}}\times {{W}}^d$ on ${\mathcal{P}}_1\otimes {\mathcal{B}}_1^{l}$, with ${{W}}$ acting fibrewise on ${\mathcal{P}}_1$ and ${{W}}^d\subset \mathfrak{S}_{l}$ acting on ${\mathcal{B}}_1^{l}$ by permutations. By construction, $({\mathcal{P}}_1)^{{{W}}} ={\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}$ and $({\mathcal{B}}_1^l)^{{\epsilon}_d} = {\mathcal{L}}_1^d$, where ${\epsilon}_d$\[epsilon-defn\] denotes the sign representation of ${{W}}^d$. The proof of [@hai3 Proposition 4.11.1] shows that ${\mathbb{J}}^d \cong {\mathbb{R}}(n, l)^{{\epsilon}_d}$. On the other hand, the action of ${{W}}^d$ is trivial on $\operatorname{Hilb^n{\mathbb{C}}^2}$, so ${\mathcal{P}}_1\otimes {\mathcal{B}}_1^{l}$ is a direct sum of its isotypic components. Hence $$\label{cohj} {\mathbb{J}}^d \cong {\mathbb{R}}(n, l)^{{\epsilon}_d} = H^0(\operatorname{Hilb^n{\mathbb{C}}^2}, ({\mathcal{P}}_1\otimes {\mathcal{L}}_1)^{{\epsilon}_d}) =H^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1\otimes {\mathcal{L}}_1^d).$$ It is not true, however, that the natural ${{W}}$-action on the two sides agrees. Indeed, thanks to the proof of [@hai2 Proposition 4.2] the isomorphism written ${{W}}$-equivariantly is $$\label{cohj1} {\mathbb{J}}^d \otimes {\epsilon}^{\otimes d} \cong H^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1\otimes {\mathcal{L}}_1^d).$$ The reason for this is that the isomorphism in is given by the ${\mathbb{C}}[{\mathbb{C}}^{2n}]$-module homomorphism sending (in the notation of [@hai2]) the generators $\Delta_{t_1}(\mathbf{a},\mathbf{b})\cdots \Delta_{t_d}(\mathbf{a},\mathbf{b})$ on the right hand side to their evaluations on the left hand side: $\mathbf{a}\mapsto \mathbf{x}$, $\mathbf{b}\mapsto \mathbf{y}$. The element $\Delta_{t_j}(\mathbf{a},\mathbf{b})$ has a trivial ${{W}}$-action as no $\mathbf{x}$’s or $\mathbf{y}$’s are involved, whereas its specialisation has a ${{W}}$-action of ${\epsilon}^{\otimes d}$ since that specialisation is the product of $d$ determinants. As a result, and Lemma \[corpar\] combine to prove: \[coha\] There is an isomorphism of ${\mathbb{A}}^0$-modules ${\mathbb{A}}^d \cong {\mathbb{R}}(n,l)^{{{W}}\times {\epsilon}_d} = H^0 (\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{L}}_1^d) .$ (Bi)graded characters {#bigr1} --------------------- There is a bigrading\[bigrad-defn\] on ${\mathbb{C}}[{\mathbb{C}}^{2n}]={\mathbb{C}}[\mathbf{x},\mathbf{y}]$ with $\deg x_i = (1,0)$ and $\deg y_j = (0,1)$ which, as in [@hai1 (12)], arises from the action of ${\mathbb{T}}^2= ({\mathbb{C}}^*)^2$ on the plane ${\mathbb{C}}^2$ given by $(\alpha, \beta )\cdot (u,v) = (\alpha^{-1} u, \beta^{-1} v)$ for $(u,v)\in {\mathbb{C}}^2$. This action extends to $\operatorname{Hilb^n{\mathbb{C}}^2}$, and the bundles ${\mathcal{P}}_1$, ${\mathcal{B}}_1$, ${\mathcal{L}}_1$ are naturally ${\mathbb{T}}^2$-equivariant. The isomorphisms from and Lemma \[coha\] respect the induced bigradings. Of course, the sections $M$ of any one of these modules obtains an induced action of ${\mathbb{T}}^2$ and this is equivalent to a ${\mathbb{Z}}^2$-grading $M=\bigoplus M_{i,j}$; explicitly, an element $f\in M$ is homogeneous of weight $(i,j)$ if $(\alpha,\beta)f=\alpha^i\beta^jf$. The ${\mathbb{T}}^2$–fixed points of $\operatorname{Hilb^n{\mathbb{C}}^2}$ are precisely the ideals $I_\mu$ that are associated to partitions $\mu$ of $n$ by \[I-mu-defn\] $$I_{\mu} = {\mathbb{C}}\cdot \{ x^ry^s : (r,s)\notin d(\mu)\} \subseteq {\mathbb{C}}[x,y],$$ see [@hai1 Proposition 3.1]. The set of monomials $\mathcal{B}_{\mu} = \{ x^ry^s : (r,s)\in d(\mu) \}$ that are not in $I_{\mu}$ form a natural ${\mathbb{C}}$-basis of ${\mathbb{C}}[x,y]/I_{\mu}$. {#subsec-5.10} For a bigraded space $V = \sum_{i,j}V_{i,j}$ with finite dimensional weight spaces we define the [*bigraded Poincaré series*]{} \[Poincare-defn\] of $V$ to be $$p(V,s,t) = \sum_{i,j} \dim(V_{i,j})s^it^j.$$ Haiman has calculated the bigraded Poincaré series of ${\mathbb{R}}(n,l)$ and a similar calculation will allow us to find the bigraded Poincaré series of ${\mathbb{J}}^d$. For a pair of partitions $\lambda, \mu$ let $K_{\lambda \mu}(t,s)$ \[kostka-defn\] be the [*Kostka–Macdonald coefficients*]{} defined in [@MacD VI, (8.11)]. Set $$\Omega(\mu)= \prod_{x\in d(\mu)}(1-s^{1+l(x)}t^{-a(x)})(1-s^{-l(x)}t^{1+a(x)}) \qquad\mathrm{and}\qquad P_{\mu}(s,t) = \sum_{\lambda} s^{n(\mu)}K_{\lambda \mu}(t,s^{-1}) f_{\lambda}(1).$$ We remark that many of the formulæ we cite from Haiman’s papers are given in terms of Frobenius series $\mathcal{F}_M(z;s,t)$ but, as in [@hai2 (6.5)], we can always specialise these to Hilbert series $p(M,s,t)$ by specialising $s_\lambda(z)$ to $f_\lambda(1)=\dim \lambda$. \[bigr-hain\] Under the ${\mathbb{T}}^2$-bigraded structure, the bigraded Poincaré series of ${\mathbb{J}}^d$ is $$p({\mathbb{J}}^d, s, t) = \sum_{\mu} P_{\mu}(s,t) \, \Omega(\mu)^{-1} s^{dn(\mu)}t^{dn(\mu^t)}.$$ By [@hai1 Theorem 2.1] $H^i(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1 \otimes {\mathcal{L}}_1^d) = 0$ for $i>0$, while $H^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1 \otimes {\mathcal{L}}_1^d) = {\mathbb{J}}^d$ by . Thus, in the notation of [@hai1 Section 3], $p({\mathbb{J}}^d, s, t)=\chi_{{\mathcal{P}}_1 \otimes {\mathcal{L}}_1^d}(s,t)$ and so, by [@hai1 Proposition 3.2], $$\label{bigr11} p({\mathbb{J}}^d, s, t)=\sum_\mu p({\mathcal{P}}_1\otimes {\mathcal{L}}_1^d(I_\mu),s,t)\,\, \Omega(\mu)^{-1} =\sum_\mu p({\mathcal{P}}_1(I_{\mu}),s,t) p({\mathcal{L}}_1(I_\mu),s,t)^d\,\, \Omega(\mu)^{-1}.$$ Here we have used the fact that, as $I_\mu$ defines a finite dimensional scheme, we can identify the sheaf ${\mathcal{P}}_1\otimes {\mathcal{L}}_1^d(I_\mu)$ with its global sections, and so $p({\mathcal{P}}_1\otimes {\mathcal{L}}_1^d(I_\mu),s,t)$ is naturally defined. We now evaluate the right hand side of . It is proved in [@haidis (3.9)], using the notation of [@haidis (1.9)], that $p({\mathcal{L}}_1(I_{\mu}),s, t) = \prod_{x\in d(\mu)} s^{l'(x)}t^{a'(x)}= s^{n(\mu)}t^{n(\mu^t)}$. On the other hand, by [@hai1 Proposition 3.4] (which is proved in [@hai3 Section 3.9] and uses the notation of [@hai1 (46)]), $p({\mathcal{P}}_1(I_{\mu}),s, t) = P_{\mu}(s,t)$. Substituting these observations into shows that $$p({\mathbb{J}}^d, s, t)=\sum_\mu P_{\mu}(s,t)\, \Omega(\mu)^{-1} s^{dn(\mu)}t^{dn(\mu^t)},$$ as required. Blowing up $({\mathfrak{h}}\oplus {\mathfrak{h}}^*)/{{W}}$ {#hi-defn-sec} ---------------------------------------------------------- All the results described so far have natural analogues for the subvariety ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$ of ${\mathbb{C}}^{2n}$. Geometrically, this follows from the observation that the natural additive action of ${\mathbb{C}}^2$ by translation on $\operatorname{Hilb^n{\mathbb{C}}^2}$ gives a decomposition $\operatorname{Hilb^n{\mathbb{C}}^2}= {\mathbb{C}}^2\times \left( \operatorname{Hilb^n{\mathbb{C}}^2}\right)/{\mathbb{C}}^2$ into a product of varieties [@Nak p.10]. Unravelling the actions shows that $ \operatorname{Hilb^n{\mathbb{C}}^2}/{\mathbb{C}}^2$ provides a resolution of singularities for ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$. However, since we need the algebraic consequences of Haiman’s results, we will take a more algebraic approach. We emphasise that the embedding ${\mathfrak{h}}\oplus {\mathfrak{h}}^* \hookrightarrow {\mathbb{C}}^{2n}$ is always given by embedding ${\mathfrak{h}}$ into the first copy of ${\mathbb{C}}^n$ and ${\mathfrak{h}}^*$ into the second copy. To fix notation, let ${\mathfrak{h}}$ be the hypersurface $\mathbf{z}=0$ in ${\mathbb{C}}^{n}$ and similarly let ${\mathfrak{h}}^*$ be the hypersurface $\mathbf{z}^*=0$ in the second copy of ${\mathbb{C}}^{n}$; thus ${\mathbb{C}}[{\mathbb{C}}^{2n}]={\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*][\mathbf{z},\mathbf{z}^*]$. Since $\mathbf{z},\mathbf{z}^*\in {\mathbb{C}}[{\mathbb{C}}^{2n}]^{{W}}$, this induces the decomposition ${\mathbb{C}}[{\mathbb{C}}^{2n}]^{{W}}={\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]^{{W}}[\mathbf{z},\mathbf{z}^*]$. Following the lead of , we set $$A^1 = {{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}^{{\epsilon}}\ \subset \ {\mathbb{A}}^1={\mathbb{C}}[{\mathbb{C}}^{2n}]^{{\epsilon}}\qquad\text{and}\qquad J^1 = {{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}A^1\ \subset \ {\mathbb{J}}^1= {\mathbb{C}}[{\mathbb{C}}^{2n}]{\mathbb{A}}^1.$$ \[A-1-defn\] We then define $A^0={{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}^{{W}}$, $J^0={{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}$ and, for $d>1$, take $A^d=(A^1)^d$ and $J^d = (J^1)^d$ for the respective $d^{\text{th}}$ powers using the multiplication in ${{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}$. Finally, we write $$A=\bigoplus_{i\geq 0}A^i \cong A^0[A^1t]\qquad\text{and}\qquad S = \bigoplus_{i\geq 0}J^i \cong {{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}[J^1t]$$ for the corresponding Rees rings. The next result is basic observation about these objects. \[hi-basic-lem\] [(1)]{} For $d\geq 0$, ${\mathbb{A}}^d = A^d[\mathbf{z},\mathbf{z}^*]$ is the set of polynomials with coefficients from $A^d$. Similarly, ${\mathbb{J}}^d = J^d[\mathbf{z},\mathbf{z}^*]$. [(2)]{} Each $J^d$ is a free module over ${\mathbb{C}}[{\mathfrak{h}}]$ and ${{\mathbb{C}}[{\mathfrak{h}}]}^{{{W}}}$. \(1) By definition, ${\mathbb{A}}^1=\left({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}[\mathbf{z},\mathbf{z}^*]\right)^{\epsilon}= {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]^{{\epsilon}} [\mathbf{z},\mathbf{z}^*] =A^1[\mathbf{z},\mathbf{z}^*]$ as polynomial extensions. Thus ${\mathbb{A}}^d = (A^1[\mathbf{z},\mathbf{z}^*])^d = (A^1)^d[\mathbf{z},\mathbf{z}^*] = A^d[\mathbf{z},\mathbf{z}^*]$ and ${\mathbb{J}}^d = A^d[\mathbf{z},\mathbf{z}^*]{{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}= J^d[\mathbf{z},\mathbf{z}^*].$ \(2) By part (1) and Lemma \[corpar\], ${\mathbb{J}}^d = J^d[\mathbf{z},\mathbf{z}^*]$ is a free module over ${{\mathbb{C}}[{\mathfrak{h}}]}[\mathbf{z}]$ and hence over ${{\mathbb{C}}[{\mathfrak{h}}]}$. Therefore, so is its ${{\mathbb{C}}[{\mathfrak{h}}]}$-module summand $J^d$. {#hi-defn-sec2} Recall the resolution of singularities $\tau: \operatorname{Hilb^n{\mathbb{C}}^2}\to {\mathbb{C}}^{2n}/{{W}}$ defined in and define $\operatorname{Hilb(n)}=\tau^{-1} ({\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}})$ \[hi-defn\], with the resulting morphism $\tau: \operatorname{Hilb(n)}\to {\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$. Using the identifications of , the basic properties of $\operatorname{Hilb(n)}$ are easy to determine. \[hi-basic-lem2\] [(1)]{} $\operatorname{Hilb(n)}=\operatorname{Proj}(A) $ and $\tau :\operatorname{Hilb(n)}\to {\mathfrak{h}}\oplus {\mathfrak{h}}^*/{{W}}$ is a resolution of singularities. 1. Moreover $\tau$ is a crepant resolution: that is $\omega_{\operatorname{Hilb(n)}} \cong {\mathcal{O}}_{\operatorname{Hilb(n)}}$. 2. Set $X_n=\operatorname{Proj}(S)$. Then $X_n$ is the reduced fibre product $$\begin{CD} X_n @> >> {\mathfrak{h}}\oplus {\mathfrak{h}}^* \\ @V \rho VV @VVV \\ \operatorname{Hilb(n)}@> \tau >> {\mathfrak{h}}\oplus {\mathfrak{h}}^*/{{W}}. \end{CD}$$ and the map $\rho$ is flat of degree $n!$. \(1) Recall from that $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Proj}({\mathbb{A}})$. By Lemma \[hi-basic-lem\], ${\mathbb{A}}= A[\mathbf{z},\mathbf{z}^*]$. The maps $A\hookrightarrow {\mathbb{A}}$ and ${\mathbb{C}}[\mathbf{z},\mathbf{z}^*] \hookrightarrow {\mathbb{A}}$ give maps $\operatorname{Hilb^n{\mathbb{C}}^2}\to \operatorname{Proj}(A)$ and $\operatorname{Hilb^n{\mathbb{C}}^2}\to\mathrm{Spec}({\mathbb{C}}[\mathbf{z},\mathbf{z}^*]) \cong {\mathbb{C}}^2$ and hence, by universality, a map $\operatorname{Hilb^n{\mathbb{C}}^2}\to \operatorname{Proj}(A)\times {\mathbb{C}}^2$. It is easy to check that this is an isomorphism locally and hence globally. The identification of ${\mathfrak{h}}\oplus{\mathfrak{h}}^*$ with the subvariety $\mathbf{z}=0=\mathbf{z}^*$ of ${\mathbb{C}}^{2n}$ easily yields $\operatorname{Hilb(n)}=\operatorname{Proj}(A)$ and so $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Hilb(n)}\times {\mathbb{C}}^2$. Since $\operatorname{Hilb^n{\mathbb{C}}^2}$ is a resolution of singularities of ${\mathbb{C}}^2/{{W}}$, the result follows. \(2) By [@har Exercise II.8.3(b)] $\omega_{\operatorname{Hilb^n{\mathbb{C}}^2}} \cong \omega_{\operatorname{Hilb(n)}} \boxtimes \omega_{{\mathbb{C}}^2}$, the external tensor product on $\operatorname{Hilb^n{\mathbb{C}}^2}= \operatorname{Hilb(n)}\times {\mathbb{C}}^2$. Now (2) follows since $\omega_{\operatorname{Hilb^n{\mathbb{C}}^2}} \cong {\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}$ by [@hai3 Proposition 3.6.3]. \(3) As in part (1), $\mathbb{S}= \bigoplus {\mathbb{J}}^d =S[\mathbf{z},\mathbf{z}^*]$ and $\operatorname{Proj}(\mathbb{S}) \cong \operatorname{Proj}(S)\times {\mathbb{C}}^2$. The assertions of the corollary now follow from the corresponding results for ${\mathbb{X}}=\operatorname{Proj}(\mathbb{S})$ that were stated in . We also have analogues of ${\mathcal{P}}_1$ and ${\mathcal{L}}_1$ for $\operatorname{Hilb(n)}$. These are defined in the same way: ${\mathcal{P}}= \rho_{\ast}{\mathcal{O}}_{X_n}$\[PP-defn\] is the [*Procesi bundle*]{} on $\operatorname{Hilb(n)}$ of rank $n!$ arising from the map $\rho : X_n \to \operatorname{Hilb(n)}$ while ${\mathcal{L}}$\[LL-A-defn\] is the canonical ample line bundle ${\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$ associated to the presentation $\operatorname{Hilb(n)}\cong \operatorname{Proj}A$. {#section} Since ${\bf z}$ and ${\bf z}^*$ are bihomogeneous, the bigradings of to pass $\operatorname{Hilb(n)}$. Therefore, Lemma \[hi-basic-lem\](1) implies that $p(J^m, s, t)=(1-s)(1-t) p({\mathbb{J}}^m, s, t)$. Substituting this formula into Proposition \[bigr-hain\] gives: \[bigr\] The bigraded Poincaré series of $J^d$ is $$\qquad\qquad \qquad\qquad\qquad\qquad p(J^d, s, t) = \sum_{\mu} P_{\mu}(s,t)(1-s)(1-t)\, \Omega(\mu)^{-1} s^{dn(\mu)}t^{dn(\mu^t)}. \hfill \qquad\qquad\qquad \qquad\hfill\qed$$ {#babyverma} In Corollary \[gr\] we will give a singly graded analogue of Corollary \[bigr\] that will be needed in the proof of the Theorem \[mainthm-intro\]. In the proof we will need the following combinatorial formulæfor the fake degrees $f_\mu(v)$, as defined in . Let $\mu\in{{\textsf}{Irrep}({{W}})}$. Then [(1)]{} $ f_{\mu}(v) = v^N f_{\mu^t}(v^{-1})$, where $N=n(n-1)/2$, [(2)]{} $f_{\mu}(v)\prod_{x\in d(\mu)} (1 - v^{h(x)}) = v^{n(\mu)}\prod_{i=1}^n (1-v^i),$ where $h(x)=1+a(x)+l(x)$ as in , [(3)]{} $\sum_{\lambda} v^{n(\mu)}K_{\lambda \mu}(v^{-1},v^{-1})f_{\mu}(v^{-1}) f_{\lambda}(1) = \sum_{\lambda} f_{\lambda}(v^{-1}) f_{\mu}(1)f_{\lambda}(1).$ \(1) This is a well-known formula (see, for example, [@op p.453]). (2,3) Up to a change of notation, these are both proved within the proof of [@babyv Theorem 6.4]—see the displayed equations immediately after, respectively immediately before [@babyv (18)]. {#gr} The ${\mathbf{E}}$-grading from descends naturally to $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}})\cong {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]$ and we will use the same notation there; thus $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}^*=1$ and $\operatorname{{\mathbf{E}}\text{-deg}}{\mathfrak{h}}= -1$. For an ${\mathbf{E}}$-graded module (or, indeed, any ${\mathbb{Z}}$-graded module) $M=\bigoplus_{i\in{\mathbb{Z}}}M_i$, we write the corresponding Poincaré series as $p(M,v)=\sum v^i\dim_{\mathbb{C}}M_i$. Set $$\label{factorial-defn} [n]_v! = \frac{\prod_{i=1}^n (1-v^i)}{(1-v)^n} .$$ Under the ${\mathbf{E}}$-grading, the module $\overline{J^d} = {J^d}/{{{\mathbb{C}}[{\mathfrak{h}}]}^{{{W}}}_+J^d}$  has Poincaré series $$\label{Z-grading0} p(\overline{J^d}, v) = \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-d(n(\mu) - n(\mu^t))}[n]_v!}{\prod_{i=2}^n (1-v^{-i})} .$$ Since ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}_+$ is ${\mathbf{E}}$-graded, so is $\overline{J^d}$, and so the result does make sense. By Lemma \[corpar\](2), the fundamental invariants of ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ form an r-sequence in $J^d$ for any $d\geq 0$. Since these elements have degrees $2\leq r\leq n$, Corollary \[bigr\] implies that $\overline{J^d}$ has Poincaré series $$\label{Z-grading15} p(\overline{J^d}, v) = \left( (1-t)\prod_{i=1}^n(1- s^i)\sum_{\mu} P_{\mu}(s,t)\, \Omega(\mu)^{-1} s^{dn(\mu)}t^{dn(\mu^t)} \right)_{s=v, t=v^{-1}}$$ where $P_\mu$ and $\Omega(\mu)$ are defined in . Lemma \[babyverma\](2) implies that $$\label{Z-grading16} \Big(\Omega(\mu)\Big)_{s=v,t=v^{-1}} =f_{\mu}(v)^{-1}f_{\mu}(v^{-1})^{-1} \prod_{i=1}^n(1-v^i)(1-v^{-i}).$$ This gives $$\label{Z-grading2} p(\overline{J^d}, v) = \frac{ \sum_{\mu} P_{\mu}(v,v^{-1})f_{\mu}(v)f_{\mu}(v^{-1}) v^{dn(\mu)}v^{-dn(\mu^t)} } {\prod_{i=2}^n (1-v^{-i})}.$$ By Lemma \[babyverma\](3) the numerator of this expression can be described as $$\label{numerator} \sum_{\mu} \left(\sum_{\lambda} f_{\lambda}(v^{-1}) f_{\lambda}(1)\right) f_{\mu}(1) f_{\mu}(v) v^{d(n(\mu)-n(\mu^t))}.$$ Applying Lemma \[babyverma\](1) and using the equality $f_{\mu}(1) = f_{\mu^t}(1)$ from we find that equals $$\label{rearrange} \sum_{\mu} \left(\sum_{\lambda}f_{\lambda}(v^{-1}) f_{\lambda}(1)\right) f_{\mu^t}(1) f_{\mu^t}(v^{-1}) v^N v^{-d(n(\mu^t) - n(\mu))}.$$ The standard formula $\sum \dim {\mathbb{C}}[{\mathfrak{h}}]^{\text{co}{{W}}}_iv^{-i}=[n]_{v^{-1}}!$ shows that the fake degrees satisfy the identity $$\sum_{\lambda} f_{\lambda}(v^{-1}) f_{\lambda}(1)= \frac{\prod_{i=1}^n (1-v^{-i})}{(1-v^{-1})^n} = [n]_{v^{-1}}!.$$ Applying this and to we find that $$\label{weredone} p(\overline{J^d}, v) = \frac{\sum_{\mu} f_{\mu^t}(1) f_{\mu^t}(v^{-1}) v^{-d(n(\mu^t) - n(\mu))} v^N[n]_{v^{-1}}!}{\prod_{i=2}^n (1-v^{-i})}.$$ After changing the order of summation from $\mu$ to $\mu^t$ and using the equality $$v^N [n]_{v^{-1}} = v^N\frac{\prod_{i=1}^n (1-v^{-i})}{(1-v^{-1})^n} = \frac{\prod_{i=1}^n (1-v^i)}{(1-v)^n} = [n]_v!,$$ becomes the required equality , and so the corollary is proved. ${\mathbb{Z}}$–algebras {#zalg} ======================= {#Z-alg-defn} Typically in noncommutative algebra—and certainly in our case—one cannot apply the Rees ring construction since one is working with just right modules or homomorphism groups rather than bimodules. One way round this is to use ${\mathbb{Z}}$-algebras and in this section we describe the basic properties that we need from this theory. The reader is referred to [@BP] or [@SV Section 11] for the more general theory and to [@bgs Section 3] for applications of ${\mathbb{Z}}$-algebras to Koszul duality. Throughout this paper a [*${\mathbb{Z}}$-algebra*]{} will mean a [*lower triangular ${\mathbb{Z}}$-algebra*]{}. By definition, this is a (non-unital) algebra $B=\bigoplus_{i\geq j\geq 0} B_{ij}$, where multiplication is defined in matrix fashion: $B_{ij}B_{jk}\subseteq B_{ik}$ for $i\geq j\geq k\geq 0$ but $B_{ij}B_{\ell k}=0$ if $j\not=\ell$. Although $B$ cannot have a unit element, we do require that each subalgebra $B_{ii}$ has a unit element $1_i$ such that $1_ib_{ij}=b_{ij}=b_{ij}1_j$, for all $b_{ij}\in B_{ij}$. {#Z-alg-defn2} Let $B$ be a ${\mathbb{Z}}$-algebra. We define the category $B{{\textsf}{\text{-}Grmod}}$ to be the category of ${\mathbb{N}}$-graded left $B$-modules $M=\bigoplus_{i\in {\mathbb{N}}} M_i$ such that $B_{ij}M_j\subseteq M_i$ for all $i\geq j$ and $B_{ij}M_k=0$ if $k\not=j$. Homomorphisms are defined to be graded homomorphisms of degree zero. The subcategory of noetherian graded left $B$-modules will be denoted $B{{\textsf}{\text{-}grmod}}$. In all examples considered in this paper $B{{\textsf}{\text{-}grmod}}$ will consist precisely of the finitely generated graded left $B$-modules. A module $M\in B{{\textsf}{\text{-}Grmod}}$ is [*bounded*]{} if $M_n = 0$ for all but finitely many $n\in {\mathbb{Z}}$ and [*torsion*]{} if it is a direct limit of bounded modules. We let $B{\text{-}{\textsf}{Tors}}$ denote the full subcategory of torsion modules in $B{{\textsf}{\text{-}Grmod}}$ and write $B{\text{-}{\textsf}{tors}}$ for the analogous subcategory of $B{\text{-}{\textsf}{qgr}}$. The corresponding quotient categories are written $B{\text{-}{\textsf}{Qgr}}= B{{\textsf}{\text{-}Grmod}}/B{\text{-}{\textsf}{Tors}}$ and $B{\text{-}{\textsf}{qgr}}= B{{\textsf}{\text{-}grmod}}/B{\text{-}{\textsf}{tors}}$.\[qgr-defn\] Write $\pi(M)$ for the image in $B{\text{-}{\textsf}{Qgr}}$ of $M\in B{{\textsf}{\text{-}Grmod}}$. {#zalgex1} There are two basic examples of ${\mathbb{Z}}$-algebras that will interest us. For the first, suppose that $S= \bigoplus_{n\geq 0} S_n$ is an ${\mathbb{N}}$-graded algebra. As in [@bgs Example 3.1.3] we can canonically associate a ${\mathbb{Z}}$-algebra $\widehat{S}=\bigoplus_{i\geq j\geq 0}\widehat{S}_{ij}$ to $S$ by setting $\widehat{S}_{ij} = S_{i-j}$ with multiplication induced from that in $S$. Define categories $S{{\textsf}{\text{-}Grmod}},\dots,S{\text{-}{\textsf}{qgr}}$ in the usual manner. In particular, $S{{\textsf}{\text{-}Grmod}}$ denotes the category of ${\mathbb{Z}}$-graded $S$-modules, from which the other definitions follow as in the last paragraph. We then let $S{{\textsf}{\text{-}Grmod}}_{\geq 0}$ denote the full subcategory of $S{{\textsf}{\text{-}Grmod}}$ consisting of ${\mathbb{N}}$-graded $S$-modules $M=\bigoplus_{i\in{\mathbb{N}}}M_i$. It is immediate from the definitions that the identity map $\iota: M=\bigoplus_{i\in{\mathbb{N}}}M_i\mapsto M=\bigoplus_{i\in{\mathbb{N}}}M_i$ gives equivalences of categories $S{{\textsf}{\text{-}Grmod}}_{\geq 0}\simeq \widehat{S}{{\textsf}{\text{-}Grmod}}$ and $S{{\textsf}{\text{-}grmod}}_{\geq 0}\simeq\widehat{S}{{\textsf}{\text{-}grmod}}$. For any module $M\in S{{\textsf}{\text{-}Grmod}}$, one has $\pi(M)=\pi(M_{\geq 0})$ in $S{\text{-}{\textsf}{Qgr}}$ and so $\iota$ induces category equivalences $$\label{zalgex11} S{\text{-}{\textsf}{Qgr}}\simeq \widehat{S}{\text{-}{\textsf}{Qgr}}\qquad \text{and} \qquad S{\text{-}{\textsf}{qgr}}\simeq \widehat{S}{\text{-}{\textsf}{qgr}}.$$ {#zalgex2} For the second class of examples, suppose that we are given noetherian algebras $R_n$ for $n\in {\mathbb{N}}$ with $(R_i, R_j)$-bimodules $R_{ij}$, for $i> j\geq 0$. Assume, moreover, that there are morphisms $\theta_{ij}^{jk} : R_{ij}\otimes_{R_j}R_{jk}\to R_{ik}$ satisfying the the obvious associativity conditions. Then we can define a ${\mathbb{Z}}$-algebra $R_{\mathbb{Z}}$ by $R_{\mathbb{Z}}=\bigoplus_{i\geq j\geq 0}R_{ij}$, where $R_{ii}=R_i$ for all $i$. A particular example of this construction is the one that interests us. Suppose that $\{R_n : n\in {\mathbb{N}}\}$ are Morita equivalent algebras, with the equivalence induced from the progenerative $(R_{n+1},R_{n})$-bimodules $P_n$. Define $R_{ij} = P_{i-1}\otimes_{R_{i-1}}\otimes\cdots \otimes_{R_{j+2}} P_{j+1}\otimes_{R_{j+1}}P_j$ and $R_{jj}=R_j$, for $i>j\geq 0$. Tensor products provide the isomorphisms $\theta_\bullet^\bullet$ and associativity is automatic. The corresponding ${\mathbb{Z}}$-algebra $R_{\mathbb{Z}}=\bigoplus_{i\geq j\geq 0} R_{ij}$ will be called the [*Morita ${\mathbb{Z}}$-algebra associated to the data $\{R_n,P_n : n\in {\mathbb{N}}\}$*]{}. {#Zalgequiv} Write $R{\text{-}{\textsf}{mod}}$ for the category of finitely generated left modules over a noetherian ring $R$. Although easy, the next result provides the foundation for our approach to $U_c$: in order to study $U_c{\text{-}{\textsf}{mod}}$ it suffices to study $R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$, for any Morita ${\mathbb{Z}}$-algebra $R_{\mathbb{Z}}$ with $R_0\cong U_c$. Suppose that $R_{\mathbb{Z}}$ is the Morita ${\mathbb{Z}}$-algebra associated to the data $\{R_n,P_n : n\in {\mathbb{N}}\}$, where $R_0$ is noetherian. 1. Each finitely generated graded left $R_{\mathbb{Z}}$-module is noetherian. 2. The association $\phi: M \mapsto \bigoplus_{n\in {\mathbb{N}}}R_{n0}\otimes_{R_0}M$ induces an equivalence of categories between $R_0{\text{-}{\textsf}{mod}}$ and $R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$. \(1) Any finitely generated graded left $R_{\mathbb{Z}}$-module $M$ is a graded image of $\bigoplus_{a_i} \big(\bigoplus_{j\geq a_i} R_{ja_i}\big) \otimes_{R_{a_i}} R_{a_i},$ for some $a_i\in {\mathbb{N}}$ and so we may assume that $M= \bigoplus_{j\geq a} R_{ja},$ for some $a\geq 0$. Let $L\subseteq M$ be a graded submodule and write $R^*_{ij}$ for the dual of the progenerator $R_{ij}$. Then $$X(j)=R^*_{ja}\otimes_{R_j}L_j \subseteq R^*_{ja}\otimes_{R_j}M_j = R^*_{ja}\otimes R_{ja} \xrightarrow{\sim} R_a,\qquad \text{for}\ j\geq a.$$ As $R_a$ is Morita equivalent to $R_0$, it is noetherian and so $\sum_{j\geq a}X(j)=\sum_{i=a}^b X(i)$, for some $b\geq a$. Now, $$L_k = R_{ka}X(k) \subseteq \sum_{i=a}^b R_{ka}X(i) = \sum_{i=a}^b R_{ki}R_{ia}X(i) =\sum_{i=a}^b R_{ki}L_i \qquad\text{for}\ k\geq a.$$ Thus $L$ is generated by $L_j$ for $b\geq j\geq a$. Finally, as each $L_i$ is a submodule of the noetherian left $R_i$-module $R_{ia}$, it is finitely generated and hence so is $L$. \(2) Certainly $\phi(M)\in R_{\mathbb{Z}}{{\textsf}{\text{-}Grmod}}$ and, as $\phi(M)$ is finitely generated by the generators of ${}_{R_0}M$, one has $\phi(M)\in R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$. Thus $\Phi (M)=\pi\phi(M)\in R_{{\mathbb{Z}}}{\text{-}{\textsf}{qgr}}$. Since $\Phi$ sends $R_0$-module homomorphisms to graded $R_{\mathbb{Z}}$-module homomorphisms, $\Phi$ is a functor. Conversely, suppose that $\widetilde{N}\in R_{\mathbb{Z}}{\text{-}{\textsf}{qgr}}$ and pick a preimage $N\in R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$. Then $N$ is generated by $\bigoplus_{i=0}^a N_i$, for some $a$, and so $N_j=R_{ja}N_a$, for all $j\geq a$. For $j\geq i\geq a$ we have natural maps of $R_a$-modules $$\theta_{ji}: R^*_{ia}\otimes N_i \cong R^*_{ia}\otimes R^*_{ji}\otimes R_{ji}\otimes N_i \cong R^*_{ja}\otimes (R_{ji}\otimes N_i ) \twoheadrightarrow R^*_{ja}\otimes N_j,$$ where the tensor products are over the appropriate $R_k$. By the associativity of tensor products, $\theta_{ki}=\theta_{kj}\theta_{ji}$, for all $k\geq j\geq i\geq a$. Since each $N_i$ is a noetherian $R_i$-module, each $R^*_{ia}\otimes N_i$ is a noetherian $R_a$-module and so $\theta_{ji}$ is an isomorphism for all $j\geq i\gg 0$. Equivalently, $N_j\cong R_{ji}\otimes N_i$ for all such $j\geq i$. Set $\Theta(\widetilde{N}) =R^*_{j0}\otimes N_j\in R_0{\text{-}{\textsf}{mod}}$ for some $j\gg 0$. Since any two preimages of $\widetilde{N}$ in $R_{\mathbb{Z}}{{\textsf}{\text{-}grmod}}$ agree in high degree, $\Theta(\widetilde{N})$ is independent of the choice of $N$. Moreover, as $R^*_{j0}=R^*_{k0}R_{kj}$, $$\phi(\Theta(\widetilde{N}))_{\geq j} \cong \bigoplus_{k\geq j} R_{k0}\otimes R^*_{j0}\otimes N_j \cong \bigoplus_{k\geq j} R_{kj}\otimes N_j = \bigoplus_{k\geq j}N_k,$$ and so $\Phi\Theta(\widetilde{N}) = \widetilde{N}.$ Checking that $\Theta$ and $\Phi$ are inverse equivalences is now routine. {#section-1} We remark that many of the standard techniques and results concerned with associated graded modules for unital algebras extend routinely to ${\mathbb{Z}}$-algebras. These only appear in peripheral ways in this paper and so we refer the reader to [@GS2] for a discussion of these results. The main theorem {#sect-filt} ================ {#sect601} In this section we prove the main theorem of the paper by proving Theorem \[mainthm-intro\] from the introduction. Indeed we will prove more generally that a version of that theorem holds for all values of $c\in {\mathbb{C}}$ that satisfy Hypothesis \[morrat-hyp\]. As was true with Corollary \[morrat-cor\] and Proposition \[shiftonO\], the theorem will take slightly different forms depending on whether $c\in \mathbb{Q}_{\leq -1}$ or not, so it is convenient to separate the cases with Hypothesis {#main-hyp} ---------- [*The element $c\in{\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\] but $c\not\in \mathbb{Q}_{\leq -1}$.* ]{} {#subsec-6.1} Assume that Hypothesis \[main-hyp\] holds. By Corollary \[morrat-cor\] there is a Morita equivalence $S_c: {U}_c{\text{-}{\textsf}{mod}}\to {U}_{c+1}{\text{-}{\textsf}{mod}}$ given by $S_c(M) = Q_c^{c+1} \otimes_{{U}_c} M$, where $Q_c^{c+1}=eH_{c+1}e_-\delta\subset D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$ is considered as a right ${U}_c$-module via . Following we can therefore define a Morita ${\mathbb{Z}}$-algebra $B(c)= B=U_{\mathbb{Z}}$\[B-ring-defn\] associated to the data $\{U_{c+i},\, Q_{c+i}^{c+i+1}; i\in {\mathbb{N}}\}$; thus $B=\bigoplus_{i\geq j\geq 0}B_{ij}$ where, for integers $i>j\geq 0$, $$\label{Mij-defn} B_{jj}={U}_{c+j} \qquad\text{and}\qquad B_{ij} = \ Q_{c+i-1}^{c+i}Q_{c+i-2}^{c+i-1}\cdots Q_{c+j}^{c+j+1},$$ where the multiplication in taken in $D({\mathfrak{h}^{\text{reg}}})\ast W$. Note that, by Corollary \[morrat-cor\], we have a natural isomorphism $$\label{tpdef} B_{ij}\ \cong\ Q_{c+i-1}^{c+i}\otimes_{{U}_{c+i-1}}Q_{c+i-2}^{c+i-1} \otimes_{{U}_{c+i-2}}\cdots\otimes_{{U}_{c+j+1}} Q_{c+j}^{c+j+1},$$ and so this does accord with the definition in . The Main Theorem {#subsec-6.6} ---------------- Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\]. The differential operator filtration $\operatorname{{\textsf}{ord}}$ on $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$, as defined in , induces filtrations on the subspaces $B_{ij}$ and hence on $B$, which we will again write as $\operatorname{{\textsf}{ord}}$. The fact that these filtrations are induced from that of $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$ ensures that the associated graded object $$\operatorname{{\textsf}{ogr}}B = \bigoplus_{i\geq j\geq 0}\operatorname{{\textsf}{ogr}}B_{ij}$$ is also a ${\mathbb{Z}}$-algebra. Similarly, recall from the ${\mathbb{N}}$-graded algebra $A=\bigoplus_{i\geq 0} A^{i}$ associated to $\operatorname{Hilb(n)}$. In this section it is more convenient to use the isomorphic algebra $A=\bigoplus_{i\geq 0} A^{i}\delta^i$ to which we canonically associate the ${\mathbb{Z}}$-algebra \[Aij-defn\] $\widehat{A} = \bigoplus_{i\geq j \geq 0} A^{i-j}\delta^{i-j},$ in the notation of . \[main\] Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\] and define $B$ and $\widehat{A}$ as above. Then: 1. There is an equivalence of categories ${U}_c{\text{-}{\textsf}{mod}}\ \xrightarrow{\sim}\ B{\text{-}{\textsf}{qgr}}$. 2. There is an equality $\operatorname{{\textsf}{ogr}}B = e \widehat{A}e$ and hence a graded ${\mathbb{Z}}$-algebra isomorphism $\operatorname{{\textsf}{ogr}}B \cong \widehat{A}$. 3. $\operatorname{{\textsf}{ogr}}B{\text{-}{\textsf}{qgr}}\simeq\operatorname{{\textsf}{Coh} }\operatorname{Hilb(n)}$. Combining Theorem \[main\] with Corollary \[morrat-cor\] and the isomorphism $U_c\cong U_{-c-1}$ from the proof of that result gives: \[main-cor\] [(1)]{} Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[morrat-hyp\]. Then there exists a ${\mathbb{Z}}$-algebra $B'$ such that $U_c{\text{-}{\textsf}{mod}}\simeq B'{\text{-}{\textsf}{qgr}}$ and $\operatorname{{\textsf}{ogr}}B\cong \widehat{A}$. Thus $\operatorname{{\textsf}{ogr}}B'{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$. [(2)]{} If $c\in {\mathbb{C}}$ with $c\not\in \frac{1}{2}+{\mathbb{Z}}$, then $H_c{\text{-}{\textsf}{mod}}\simeq B''{\text{-}{\textsf}{qgr}}$ and $\operatorname{{\textsf}{ogr}}B''{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$ for some ${\mathbb{Z}}$-algebra $B''$. {#app-to-main} Analogues of Theorem \[main\] also hold for certain important $U_{c+k}$-modules and we will derive the theorem from one of these. The module in question is the $(U_{c+k},H_c)$-bimodule $N(k)=B_{k0}eH_c$ with the induced $\operatorname{{\textsf}{ord}}$ filtration coming from the inclusion $N(k)\subset D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}$. Recall the definition of $J^d$ from . \[pre-cohh\] Assume that $c\in{\mathbb{C}}$ satisfies Hypothesis \[main-hyp\] and let $k\in {\mathbb{N}}$. Then $\operatorname{{\textsf}{ogr}}N(k)=e J^k\delta^k$ as submodules of $\operatorname{{\textsf}{ogr}}D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}={\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\ast {{W}}$. Outline of the proof of the theorem and proposition {#surjstrat} --------------------------------------------------- [*For the rest of the section, we will assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\].*]{} Thus the notation from and is available and, by Corollary \[morrat-cor\], $N(k)\cong B_{k0}\otimes_{U_c} eH_c$ is a progenerative $(U_{c+k},\,H_c)$-bimodule. As will be shown in , Theorem \[main\] follows easily from Proposition \[pre-cohh\], so we need only discuss the proof of the latter result. This is nontrivial and will take most of the section but, in outline, is as follows. It is easy to see that $eJ^k\delta^k\subseteq \operatorname{{\textsf}{ogr}}N(k)$ (see Lemma \[thetainjA\]). The other inclusion is considerably harder. The philosophy behind the proof is to note that we can grade both $J^{k}\delta^{k} $ and $ N(k)$ by the ${\mathbf{E}}$-gradation. This is not immediately useful since the graded pieces of the two sides are infinite dimensional but both sides have factor modules for which the graded pieces are finite dimensional. For $eJ^{k}\delta^{k}\cong J^k\delta^k$ the factor is the module $\overline{J^{k}}\delta^{k}$ described by Corollary \[gr\], while the analogous factor $\overline{N(k)}$ of $\operatorname{{\textsf}{ogr}}N(k)$ is described in and Corollary \[poincare-S2A\] and is related to the standard modules $\Delta_{c+k}(\mu)$. The key observation is that these factors have the same Poincaré series and so they are naturally isomorphic as graded vector spaces. The proof of the theorem then amounts to lifting this isomorphism to give the desired equality $eJ^k\delta^k = \operatorname{{\textsf}{ogr}}N(k)$. This also shows that the result has to be non-trivial. Indeed, an alternative proof of the proposition (or the theorem) would also provide an alternative proof to a number of the results from [@hai3]. {#abstract-products} We start with two elementary observations that will be used frequently. If $R=\bigcup F^iR$ is a filtered ring and $r\in F^mR\smallsetminus F^{m-1}R$, we write $\sigma(r)=[r+F^{m-1}R]\in \operatorname{gr}_F^mR$ for the [*principal symbol*]{}\[princ-symbol-defn\] of $r$. Let $R=\bigcup F^iR$ be a filtered $k$-algebra, for a field $k$. [(1)]{} Let $A$, $B$ be subspaces of $R $ and give $A$, $B$ and $AB$ the induced filtration $F$. Then $(\operatorname{gr}_FA)(\operatorname{gr}_FB)\subseteq \operatorname{gr}_FAB$, as subspaces of $\operatorname{gr}_FR$. Indeed, if $a\in A$ and $b\in B$ satisfy $\sigma(a)\sigma(b)\not=0$, then $\sigma(a)\sigma(b)= \sigma(ab)$. [(2)]{} Suppose that $A=\bigcup F^iA$ is a filtered right $R$-module and that $B=\bigcup F^iB$ is a filtered left $R$-module and give the vector space $A\otimes_RB$ the *tensor product filtration*: $F^n(A\otimes B)=\sum_j F^jA\otimes F^{n-j}B$. Then there is a natural surjection $\operatorname{gr}_FA\otimes_{\mathrm{gr}\,R}\operatorname{gr}_FB\twoheadrightarrow \operatorname{gr}_F(A\otimes_RB)$. \(1) Identify $\operatorname{gr}_FA=\bigoplus (F^nA+F^{n-1}R)/F^{n-1}R\subseteq \operatorname{gr}_FR$ so that the result makes sense. Suppose that $\bar{a}\in \operatorname{gr}_F^nA$ and $\bar{b}\in \operatorname{gr}_F^mB$ are such that $\bar{a}\bar{b}\not=0$ in $\operatorname{gr}_FR$. Lift $\bar{a}$ to $a\in F^nA$ and $\bar{b}$ to $b\in F^mB$. Then, as elements of $\operatorname{gr}_FR$, one has $\bar{a}\bar{b} = [a+F^{n-1}R][b+F^{m-1}R] \subseteq [ab+F^{n+m-1}R]$. Since $\bar{a}\bar{b}\not=0$, $ab\in F^{n+m}R\smallsetminus F^{n+m-1}R$, whence $\bar{a}\bar{b}=\sigma(ab)$ is the image of $ab$ in $\operatorname{gr}_F(AB)$. \(2) Define a map $\rho: \operatorname{gr}_FA\times \operatorname{gr}_FB \to \operatorname{gr}_F(A\otimes_RB)$ by $\rho(\bar{a}, \bar{b} ) = [a\otimes b + F^{n+m-1}(A\otimes B)]$, for $\bar{a}\in \operatorname{gr}_F^nA$, $\bar{b}\in \operatorname{gr}_F^mB$ and where the rest of the notation is the same as for part (1). This clearly defines a ${\mathbb{C}}$-bilinear map that is $\operatorname{gr}_F R$-balanced in the sense that $\rho(\bar{a}\bar{r}, \bar{b}) = \rho(\bar{a}, \bar{r} \bar{b})$ for $\bar{r}\in \operatorname{gr}^s_FR$. By universality, $\rho$ therefore induces a map $ \operatorname{gr}_FA\otimes_{\operatorname{gr}R}\operatorname{gr}_FB\to \operatorname{gr}_F(A\otimes_RB)$. It is surjective since $F^{n+m}(A\otimes B)/ F^{n+m-1}(A\otimes B)$ is spanned by elements of the given form $[a\otimes b + F^{n+m-1}(A\otimes B)].$ Lemma {#grade-elements} ----- Let $R=\bigcup_{i\geq 0}F^iR$ be a filtered ring, pick $r\in R$ and let $ I$ be a subset of $R$. Under the induced filtrations, $\operatorname{gr}_F(rI) = \sigma(r)\operatorname{gr}_F(I)$ in the following cases: 1. $\sigma(r)$ is regular in $\operatorname{gr}_FR$; 2. $ r=r^2 \in F^0(R)$ and $rI\subseteq I$. Assume that $r\in F^sR\smallsetminus F^{s-1}R$. We claim that, in both cases, it suffices to prove that $F^n(rI) = rF^{n-s}I$ for all $n\geq s$. Indeed, if this is true then the identity $F^m(rI) = rI\cap F^mR$ implies that the $n^{\mathrm{th}}$ summand of $\operatorname{gr}(rI)$ equals $$\frac{F^n(rI)}{F^{n-1}(rI)} \ = \ \frac{F^n(rI)}{F^n(rI)\cap F^{n-1}R} \ \cong\ \frac{ F^n(rI) + F^{n-1}R}{F^{n-1}R} \ = \ \frac{rF^{n-s}I + F^{n-1}R}{F^{n-1}R},$$ which is the $n^{\mathrm{th}}$ summand of $\sigma(r)gr(I)$. \(1) In this case, $rt\in F^n(rI) = rI\cap F^n(R) \Leftrightarrow t\in I\ \mathrm{and}\ t\in F^{n-s}R,$ as required. \(2) Here, $rF^{n}I \subseteq F^n(rI)$ whence $rF^{n}I = r^2F^{n}I \subseteq rF^n(rI) \subseteq rF^nI$. Since $rF^n(rI) = F^n(rI) $ this implies that $rF^n(I)=F^n(rI)$. [**Example**]{}. It is easy to check that some hypotheses are required for the lemma to hold. For example, filter the polynomial ring $R={\mathbb{C}}[x,y]$ by $x,xy\in F^0R$ but $y\in F^1R$. Then $x,xy\in F^0(xR)$, yet $xy\not\in \sigma(x)\operatorname{gr}_FR$. {#section-2} We now turn to the proof of Proposition \[pre-cohh\]. As was mentioned in the inclusion $J^k\delta^k e \subseteq \operatorname{{\textsf}{ogr}}N(k)$ is easy. \[thetainjA\] [(1)]{} For $i\geq j\geq 0$ we have $e(A^{i-j}\delta^{i-j})e \subseteq \operatorname{{\textsf}{ogr}}B_{ij}$. [(2)]{} The inclusion of part (1) is an equality for $i=j$ and for $i=j+1$. [(3)]{} For $k\geq 0$ there is an inclusion $eJ^{k}\delta^{k} \subseteq \operatorname{{\textsf}{ogr}}N(k)$ of left $eA^0e$-modules. This is an equality for $k=0$. \(2) By the PBW Theorem \[PBW\], $\operatorname{{\textsf}{ogr}}B_{ii} = e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast {{W}})e$ and so the claim holds for $i=j$. Similarly, since $ e,\delta\in\operatorname{{\textsf}{ord}}^0(D({{\mathfrak{h}}^{\text{reg}}})\ast{{W}})$ and $\delta$ is regular in $\operatorname{{\textsf}{ogr}}(D({{\mathfrak{h}}^{\text{reg}}})\ast{{W}})$, Lemma \[grade-elements\] implies that $$\operatorname{{\textsf}{ogr}}B_{j+1,j} = \operatorname{{\textsf}{ogr}}( eH_{c+j+1}e_- \delta) = \operatorname{{\textsf}{ogr}}(eH_{c+j+1}e_-)\delta= e(\operatorname{{\textsf}{ogr}}H_{c+j+1})e_-\delta = e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast{{W}})e_-\delta = eA^1\delta e.$$ \(1) Combining part (2) with Lemma \[abstract-products\](1) and induction shows that $$(eA^1\delta^1e)^{i-j} \ = \ \operatorname{{\textsf}{ogr}}B_{i,i-1}\operatorname{{\textsf}{ogr}}B_{i-1,i-2}\cdots \operatorname{{\textsf}{ogr}}B_{j+1,j} \ \subseteq\ \operatorname{{\textsf}{ogr}}\left( B_{i,i-1} \cdots B_{j+1,j} \right) \ = \ \operatorname{{\textsf}{ogr}}B_{ij}.$$ \(3) When $k=0$, the assertion $eJ^k\delta^{k} = \operatorname{{\textsf}{ogr}}N(k)$ is just the statement that $e{{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}= e({{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\ast {{W}}) $. When $k>0$, part (i) and Lemma \[abstract-products\] give $eJ^k\delta^k = eA^k\delta^ke{\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]\ast {{W}}\subseteq \operatorname{{\textsf}{ogr}}B_{k0}\operatorname{{\textsf}{ogr}}(eH_c)\subseteq \operatorname{{\textsf}{ogr}}N(k).$ {#h-defn} The next several results will be aimed at getting a more detailed understanding of the bimodule structure of $N(k)$ and its factors. For the most part we are interested in their graded structure for which the actions of the elements ${\mathbf{h}}_{c+t}\in H_{c+t}$ from are particularly useful. Given an $({U}_{c+s},{U}_{c+t})$-bimodule $M$, define $${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}m = {\mathbf{h}}_{c+s} m - m{\mathbf{h}}_{c+t} \text{ for any $m\in M$}.$$ When $s=t$ this is just the adjoint action of ${\mathbf{h}}_{c+s}$ on $M$. \[diaggrad\] [(1)]{} $ e{\mathbf{h}}_{c+t-1} e = \delta^{-1}e_-{\mathbf{h}}_{c+t} e_-\delta$. [(2)]{} The action of ${\mathbf{h}}$ is diagonalisable on the modules $N(i)$, $B_{ij}$ and $M(i)=H_{c+i}eB_{i0}$, for any $i\geq j\geq 0$. \(1) Use the first paragraph of the proof of [@gordc Theorem 4.10]. \(2) We start with the $B_{ij}$. If $b_1\in B_{i\ell}$ and $b_2\in B_{\ell j}$, then ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}(b_1b_2) = ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_1)b_2 + b_1({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_2)$. Thus, by induction, it suffices to prove the result for each $B_{t,t-1} = e H_{c+t}\delta e$. Clearly $e{\mathbf{h}}={\mathbf{h}}e$. Thus, by part (1), for any $m\in H_{c+t}$ we have $$\label{diaggrad1} {\mathbf{h}}{ \,{}_{^{^{\bullet}}}}e m \delta e \ =\ {\mathbf{h}}_{c+t}e m \delta e - e m \delta e {\mathbf{h}}_{c+t-1} \ = \ e {\mathbf{h}}_{c+t} m \delta e - e m \delta (\delta^{-1}e_-{\mathbf{h}}_{c+t}e_-\delta) \ =\ e ([{\mathbf{h}}_{c+t}, m]) \delta e.$$ By $H_{c+t}$ is diagonalisable under the adjoint ${\mathbf{h}}_{c+t}$-action and so the result for $B_{ij}$ follows. The same argument works for the modules $N(i)$ and $M(i)$ if one uses the decompositions $N(i)=(B_{i0})(eH_c)$ and $M(i)=(H_{c+i}e)(B_{i0})$. {#B-freeA} The factors of $N(k)$ that most interest us are defined as follows. Since $N(k)$ is a $({U}_{c+k},H_c)$-bimodule, the embeddings ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}\hookrightarrow {U}_{c+k}$ and ${\mathbb{C}}[{\mathfrak{h}}^*]\hookrightarrow H_c$ make $N(k)$ into a $({\mathbb{C}}[{\mathfrak{h}}]^{{{W}}},\, {\mathbb{C}}[{\mathfrak{h}}^*])$-bimodule. Let ${\mathbb{C}}$ be the trivial module over either ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$ or ${\mathbb{C}}[{\mathfrak{h}}^*]$ and set $ \overline{N(k)} = {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}} N(k)$ and $ \underline{N(k)} = N(k) \otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]} {\mathbb{C}}.$ As ${\mathbb{C}}$ is a graded ${\mathbf{h}}$-module, the adjoint action of ${\mathbf{h}}$ on $N(k)$ from Lemma \[diaggrad\] induces a ${\mathbb{Z}}$-grading, again called the ${\mathbf{h}}$-grading, on both $\overline{N(k)}$ and $\underline{N(k)}$. If an element $b$ from any of these three modules has degree $n$ in this grading we write $\operatorname{{\mathbf{h}}\text{-deg}}(b)=n$.\[hdeg-defn\] The reader should note that, as will be explained in , this is [*not*]{} the same as the ${\mathbf{E}}$-gradation on these modules. The next result gives the elementary properties of these modules. \[Bbar-freeA\] [(1)]{} For any $i\geq j\geq 0$, $B_{ij}\subseteq U_{c+i}\cap U_{c+j}$. 1. For $k\geq 0$, both $N(k)$ and $U_{c+k}$ are free left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-modules, while $N(k)$ is a free right ${\mathbb{C}}[{\mathfrak{h}}^*]$-module. 2. $\underline{N(k)}$ is a finitely generated, free left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module. 3. Similarly, $\overline{N(k)}$ is a finitely generated, free right ${\mathbb{C}}[{\mathfrak{h}}^*]$-module. We will use frequently and without comment the fact that ${\mathbb{C}}[{\mathfrak{h}}^*]$ is a free ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-module. Moreover, as ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$ is a polynomial ring, any projective ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$ is free by the Quillen-Suslin Theorem. \(1) By induction, we may assume that $i=j+1$. The inclusion $B_{ij}=eH_{c+i}\delta e\subseteq U_{c+i}$ is immediate. If $p\in H_{c+i}$ then, by , $$epe_-\delta\ =\ e\delta^{-1}\delta p e_- \delta\ =\ \delta^{-1} e_- \delta p e_-\delta\ \in\ \delta^{-1} e_-H_{c+i}e_-\delta\ =\ U_{c+j}.$$ \(2) By the PBW Theorem \[PBW\], each $H_d$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]$-module and as a right ${\mathbb{C}}[{\mathfrak{h}}^*]$-module. Therefore, $H_d$ is a free left ${\mathbb{C}}[{\mathfrak{h}}]^W$-module as is its summand $H_de$. Under the left action of ${{W}}$, $(H_de)^{{W}}=eH_de$ since, if $fe\in (H_{d}e)^W$, then $fe =|{{W}}|^{-1}\sum_{w\in {{W}}} wfe = efe$. But $ (H_{d}e)^W$ is a ${{W}}$-module summand of $H_{d}e$, while the actions of ${{W}}$ and ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ commute. Thus $U_{d}=(H_{d}e)^{{W}}$ is a ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module summand of $H_{d}e$ and hence is free. By Corollary \[morrat-cor\], $N(k)\cong B_{k0}\otimes_{U_c}eH_c$ is a projective left $U_{c+k}$-module and hence a free left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module. On the other hand, $N(k)$ is a projective right $H_c$-module and hence a projective right ${\mathbb{C}}[{\mathfrak{h}}^*]$-module. \(3) Set $X=H_{c}\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]} {\mathbb{C}}$. Clearly $X\in \mathcal{O}_{c}$ in the sense of and, by , $X \cong {\mathbb{C}}[{\mathfrak{h}}]\otimes_{{\mathbb{C}}} {\mathbb{C}}{{W}}$ as left ${\mathbb{C}}[{\mathfrak{h}}]\ast {{W}}$-modules. Thus $X$ is a finitely generated free left ${\mathbb{C}}[{\mathfrak{h}}]$-module and so, by [@GGOR Proposition 2.21], $X$ has a filtration whose factors are standard modules. By definition, $\underline{N(k)} = eM$ where $M = \widetilde{S}_{c+k-1}\circ \cdots \circ \widetilde{S}_{c}(X)$, in the notation of . By Proposition \[shiftonO\] $M$ also has a finite filtration by standard modules and so [@GGOR Proposition 2.21] shows that $M$ is a finitely generated free module over ${\mathbb{C}}[{\mathfrak{h}}]$ and hence over ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$. Thus, so is its summand $eM$. \(4) We first show that $N(k)$ is a finitely generated right module over $R=({\mathbb{C}}[{\mathfrak{h}}]^{{W}})^{\mathrm{op}} \otimes_{\mathbb{C}}{\mathbb{C}}[{\mathfrak{h}}^*]$. By part (1), $B_{k0}\subseteq U_{c}$ and so $N(k)\subseteq eH_c$. Thus $\operatorname{{\textsf}{ogr}}N(k) \subseteq \operatorname{{\textsf}{ogr}}H_{c} = {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\ast {{W}}$, which is certainly a noetherian ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}\otimes {\mathbb{C}}[{\mathfrak{h}}^*]$-module. Since the $\operatorname{{\textsf}{ord}}$ filtration on $N(k)$ is the one induced from $D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}$, the actions of ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ and ${\mathbb{C}}[{\mathfrak{h}}^*]$ on $\operatorname{{\textsf}{ogr}}N(k)$ are the natural ones induced from the actions of those rings on $N(k)\subset D({{\mathfrak{h}}^{\text{reg}}})\ast{{W}}$. In other words, the given $R$-module structure of $\operatorname{{\textsf}{ogr}}N(k) $ is the one induced from the $R$-module structure of $N(k)$. Since the former module is finitely generated, so is the latter. Let $y_1,\ldots ,y_{n-1}$ be the generators of ${\mathbb{C}}[{\mathfrak{h}}^*]$ and let $q_1, \ldots ,q_{n-1}$ be the fundamental invariants of ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$. By (2), the $\{y_j\}$ form an r-sequence in $N(k)$, while (3) implies that the $\{q_j\}$ form an r-sequence in the factor $\underline{N(k)}=N(k)/\sum N(k)y_j$ as a module over ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}=R/\sum y_jR$. Thus $\Sigma = \{y_\ell ,q_{m} : 1\leq \ell, m\leq n-1\}$ is a regular sequence for the right $R$-module $N(k)$. In particular, if $\mathfrak{n}=\sum y_iR+q_jR$, then $\Sigma$ is an r-sequence for the $R_{\mathfrak{n}}$-module $N(k)_\mathfrak{n}$. By the Auslander-Buchsbaum formula [@matsumura Ex. 4, p.114], $N(k)_\mathfrak{n}$ is therefore free as a $R_{\mathfrak{n}}$-module. Finally, consider $\overline{N(k)}=N(k)/\sum q_jN(k)$. Under the induced ${\mathbf{h}}$-grading, $\overline{N(k)}$ is a finitely generated, graded ${\mathbb{C}}[{\mathfrak{h}}^*]$-module and so corresponds to a ${\mathbb{C}}^*$-equivariant coherent sheaf on ${\mathfrak{h}}^*$. As a result the locus where $\overline{N(k)}$ is not free is a ${\mathbb{C}}^*$-stable closed subvariety of ${\mathfrak{h}}^*$. If this locus is non-empty it must contain the unique ${\mathbb{C}}^*$-fixed point $\mathfrak{p}=(y_1,\dots,y_{n-1})$ for this expanding ${\mathbb{C}}^*$-action. But then $(\overline{N(k)})_{\mathfrak{p}}$ would not be free, contradicting the conclusion of the last paragraph. {#poincare-start} We next need to understand the graded structure of the modules $\overline{N(k)}$ and $\underline{N(k)}$ under the ${\mathbf{h}}$-grading. To do this, we express $\underline{N(0)}$ as a weighted sum of standard modules in the Grothendieck group $G_0(U_c)$ and then to use Proposition \[shiftonO\] to write $\underline{N(k)}=B_{k0}\otimes \underline{N(0)}$ in a similar manner. This is quite delicate since there are some subtle shifts involved and we first want to understand these shifts for $B_{ij}\otimes \Delta_c(\mu)$. We will need to work with the following graded version $\widetilde{{\mathcal{O}}}_d$\[cat-O-gr-defn\] of ${\mathcal{O}}_d$ constructed in [@GGOR Section 2.4]. The objects $M$ in $\widetilde{{\mathcal{O}}}_d$ are finitely generated $H_d$-modules on which ${\mathbb{C}}[{\mathfrak{h}}^*]$ acts locally nilpotently and which come equipped with a $\mathbb{Z}$-grading $M = \bigoplus_{r\in {\mathbb{Z}}} M_{r}$ such that $p M_{r} \subseteq M_{r+\ell}$ for each $p\in H_d$ with $\operatorname{{\mathbf{E}}\text{-deg}}(p)=\ell$. The morphisms are homogeneous $H_d$-module homomorphisms of degree zero. A [*graded standard module*]{} \[graded-standard-defn\] $\widetilde{\Delta}_d(\mu)$, isomorphic to $\Delta_d(\mu)$ as an ungraded module, is given by setting $\widetilde{\Delta}_d(\mu)_r = {\mathbb{C}}[{\mathfrak{h}}]_r\otimes \mu$. By local nilpotence and finite generation, each weight space of a module $M\in \widetilde{{\mathcal{O}}}_d$ is finite dimensional and so $M$ has a well-defined Poincaré series. There is a degree shift functor\[module-shift-defn\] $[1]$ in $\widetilde{{\mathcal{O}}}_d$ defined by $M[1]_r = M_{r-1}$. By abuse of notation, $\widetilde{{\mathcal{O}}}_d$ will also denote the corresponding category of graded $U_d$-modules. \[standAAA\] Fix $i\geq j\geq 0$ and $\mu\in{{\textsf}{Irrep}({{W}})}$. Give $B_{ij}$ the adjoint ${\mathbf{h}}$-grading and let $B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)$ have the grading this induces. Then $B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)\in \widetilde{{\mathcal{O}}}_{c+i}$ and, as elements of that category, $$B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu) \cong e\widetilde{\Delta}_{c+i}[(i-j)(n(\mu)-n(\mu^t))].$$ Write $\nabla = B_{ij}\otimes_{U_{c+j}}e\widetilde{\Delta}_{c+j}(\mu)$ and let $\deg_{c+u} $ denote the degree function in $\widetilde{{\mathcal{O}}}_{c+u}$. By hypothesis, the graded structure of an element $b\otimes v\in \nabla$ is given by $\deg (b\otimes v) = \operatorname{{\mathbf{h}}\text{-deg}}(b)+\deg_{c+j}(v)$. Proposition \[shiftonO\] implies that (as ungraded modules) $$\label{grot3} \nabla =S_{c+i}\circ \cdots \circ S_{c+j+1}(e\Delta_{c+j}(\mu)) \cong e\Delta_{c+i}(\mu).$$ Thus, under its given grading, $\nabla \in \widetilde{{\mathcal{O}}}_{c+i}$. Unfortunately, it is not easy to write the generator $e\otimes \mu$ of $e\Delta_{c+i}(\mu)$ as an element of $\nabla$ and for this reason the shift in the grading in is subtle. In order to understand this we will use the canonical grading from and we write the corresponding degree function as $\deg_{\mathrm{can}}$. The advantage of this grading is that it is simply given by the left multiplication of ${\mathbf{h}}_{c+i}$. Thus, as is an isomorphism of left $U_{c+i}$-modules and hence of left ${\mathbb{C}}[{\mathbf{h}}_{c+i}]$-modules, it is automatically a graded isomorphism under the canonical grading. Since ${\mathfrak{h}}^*$ has ${\mathbf{E}}$-degree $1$, the canonical grading on $\Delta_d(\mu)$, for any $d\in {\mathbb{C}}$, is a shift of the grading on $\widetilde{\Delta}_d(\mu)$. The shift is easy to compute. By definition, the generator $1\otimes \mu$ of $\widetilde{\Delta}_d(\mu)$ has $\deg_d(1\otimes \mu)=0$ whereas, by Proposition \[subsec-3.10\], the generator $1\otimes\mu$ of $\Delta_d(\mu)$ has $$\deg_{\mathrm{can}}(1\otimes \mu)= D(d,\mu) =(n-1)/2+ d(n(\mu)-n({\mu^t})).$$ We may therefore regard $\Delta_d(\mu)$ as being in $\widetilde{{\mathcal{O}}}_d$, in which case $$\label{shft} \Delta_d(\mu)= \widetilde{\Delta}_d(\mu)[D(d,\mu)].$$ Let $b\in B_{ij}$ with $\operatorname{{\mathbf{h}}\text{-deg}}(b)=r$ and suppose that $v\in e\Delta_{c+j}(\mu)$ has $\deg_{\mathrm{can}}(v)=s$. Then $${\mathbf{h}}_{c+i}\cdot b\otimes v = ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b)\otimes v +b\,{\mathbf{h}}_{c+j}\otimes v \ =\ ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b)\otimes v +b\otimes {\mathbf{h}}_{c+j}v =(r+s)b\otimes v.$$ Thus $\deg_{\mathrm{can}}(b\otimes v) =\operatorname{{\mathbf{h}}\text{-deg}}(b)+ \deg_{\mathrm{can}}(v)$. Finally, implies that $$\begin{aligned} \deg_{c+i}(b\otimes v) &=& \deg_{\mathrm{can}}(b\otimes v) - D(c+i, \mu) \ = \ \operatorname{{\mathbf{h}}\text{-deg}}(b) + \deg_{\mathrm{can}}(v) - D(c+i,\mu) \\ &=& \operatorname{{\mathbf{h}}\text{-deg}}(b) + \deg_{c+j}(v) + D(c+j,\mu) - D(c+i,\mu) \\ &=& \deg (b\otimes v) + (j-i)(n(\mu) - n(\mu^t)),\end{aligned}$$ as required. {#poincare-sa-sect} Given a ${\mathbb{Z}}$-graded complex vector space $M = \bigoplus_{r\in{\mathbb{Z}}}M_r$ such that $\dim_{{\mathbb{C}}} M_r$ is finite for all $r$ then, as in , we define the Poincaré series\[formal-Poincare-defn\] of $M$ to be $p(M,v) = \sum v^r \dim_{{\mathbb{C}}} M_r.$ Each $N(k)$ is graded via the adjoint ${\mathbf{h}}$ action from , although of course the summands are infinite dimensional. Thus in order to understand the more detailed structure of $N(k)$ and $\operatorname{{\textsf}{ogr}}N(k)$ we will consider the Poincaré series of the factor modules $\overline{N(k)}$ and $\underline{N(k)}$. \[poincare-SA\] If $\overline{N(k)}$ as graded via the adjoint ${\mathbf{h}}$ action on $N(k)$, then its Poincaré series is $$\label{poincare-SA00}p(\overline{N(k)}, v) = \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-k(n(\mu) - n(\mu^t))}[n]_v!}{\prod_{i=2}^n (1-v^{-i})}.$$ We first calculate the Poincaré series for $\underline{N(k)}$, and we begin with $\underline{N(0)}$. As in the proof of Lemma \[Bbar-freeA\](3), $X = H_{c} \otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]} {\mathbb{C}}$ is an object of $\widetilde{{\mathcal{O}}}_{c}$, where the grading is the natural one defined by $\deg(1\otimes 1)=0$. By construction, $eX \cong \underline{N(0)}$ and this is a [*graded*]{} isomorphism since the adjoint ${\mathbf{h}}$-graded structure of $\underline{N(0)}=U_c/I$ is simply defined by $\operatorname{{\mathbf{h}}\text{-deg}}(e)=0$. Thus, as elements of the Grothendieck group $G_0(\widetilde{{\mathcal{O}}}_{c})$, we can write $ [X] = \sum_{\mu} p_{\mu} [\widetilde{\Delta}_{c}(\mu)]$ for some $p_{\mu} \in {\mathbb{Z}}[v,v^{-1}]$. By we have a graded isomorphism $X\cong {\mathbb{C}}[{\mathfrak{h}}]\otimes {\mathbb{C}}{{W}}$. Applying $({\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]}-)$ to the formula $ [X] = \sum_\mu p_\mu [\widetilde{\Delta}_{c}(\mu)]$ therefore yields ${\mathbb{C}}{{W}}= \sum_{\mu} p_{\mu} [\mu].$ It follows from that $p_{\mu} = f_{\mu}(1)$ and so $ [\,\underline{N(0)}\,] = \sum_{\mu} f_{\mu}(1) [e\widetilde{\Delta}_{c}(\mu)] . $ Combining this formula with Lemma \[standAAA\] shows that $$\label{verygrot} [\,\underline{N(k)}\,] = \sum_{\mu} f_{\mu}(1)v^{k(n(\mu)-n(\mu^t))} [e\widetilde{\Delta}_{c+k}(\mu)] .$$ The Poincaré series of $N(k)$ is now easy to compute. First, in the [*canonical grading*]{}, shows that $$p(\Delta_d(\mu), v, W) = v^{D(d,\mu)}\frac{\sum_{\lambda} f_{\lambda}(v) [\lambda\otimes \mu]}{\prod_{i=2}^n(1-v^i)} \qquad\mathrm{and\ so}\qquad p(e\Delta_d(\mu), v)= v^{D(d,\mu)} \frac{f_{\mu}(v)}{\prod_{i=2}^n (1-v^i)}$$ for any $d\in {\mathbb{C}}$. Therefore, implies that $p(e\widetilde{\Delta}_d(\mu), v)= f_{\mu}(v) \prod_{i=2}^n (1-v^i)^{-1} $ in the graded category $\widetilde{{\mathcal{O}}}_{d}$. Combined with this shows that $$\label{wrongsideformulaA} p(\underline{N(k)},v) \ = \ \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) - n(\mu^t))}}{\prod_{i=2}^n (1-v^{i})}.$$ Finally, we calculate the Poincaré series of $\overline{N(k)}$. By Lemma \[Bbar-freeA\](2,3), an ${\mathbf{h}}$-homogeneous basis for this module is given by lifting a homogeneous ${\mathbb{C}}$-basis from $\overline{N(k)}\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]} {\mathbb{C}}= {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}} {\underline{N(k)}}.$ Thus, combining with the formulæ $p({\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}, v) = \prod_{i=2}^n (1-v^i)^{-1}$ and $ p({\mathbb{C}}[{\mathfrak{h}}^*], v) = (1-v^{-1})^{n-1}$ gives $$\label{wrongsideformula2A} p(\overline{N(k)}, v) = \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) - n(\mu^t))}}{(1-v^{-1})^{n-1}}.$$ This needs to be adjusted to yield . Set $N=n(n-1)/2$. Then Lemma \[babyverma\](1) and combine to show that $$\begin{aligned} \sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) - n(\mu^t))} &=& \sum_{\mu} f_{\mu^t}(1)f_{\mu^t}(v^{-1}) v^{k(n(\mu) - n(\mu^t))}\\ &=& v^N\sum_{\lambda} f_{\lambda}(1)f_{\lambda}(v^{-1}) v^{k(n(\lambda^t) - n(\lambda))}.\end{aligned}$$ Moreover, rearranging gives $$[n]_v! \ = \ \frac{\prod_{i=1}^n (1-v^i)}{(1-v)^n} \ =\ v^N\frac{\prod_{i=1}^n (1-v^{-i})}{(1-v^{-1})^n}.$$ Combining these formulæ with gives . {#poincare-S2A} Recall the Euler gradation $\operatorname{{\mathbf{E}}\text{-deg}}$ on $D({{\mathfrak{h}}^{\text{reg}}})\ast {{W}}$ and its subrings from . Since $e$, $e_-$ and $\delta$ are homogeneous under this action, each $Q_{c+\ell}^{c+\ell+1}$ and hence each $B_{ij}$ and $N(k)$ is also graded under this action. As in , this induces a graded structure, again called $\operatorname{{\mathbf{E}}\text{-deg}}$, on $\operatorname{{\textsf}{ogr}}B_{ij}$ and $\operatorname{{\textsf}{ogr}}N(k)$. Since the fundamental invariants of ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ are ${\mathbf{E}}$-homogeneous, the ${\mathbf{E}}$-grading on $N(k)$ descends to gradings on $\overline{N(k)}$ and $\underline{N(k)}$. Similarly, each $A^{u}\delta^{u}$ and $J^u\delta^u$ has an ${\mathbf{E}}$-grading induced from that on ${\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]$ and hence so does $A=\bigoplus_{u\geq 0}A^u\delta^u.$ However, the ${\mathbf{E}}$-grading on $B_{k0}$ and hence on $N(k)$ is [*not*]{} equal to the adjoint ${\mathbf{h}}$-grading. The problem is that, in , the adjoint ${\mathbf{h}}$ action does not “see” the element $\delta$. Thus if we wish to relate the Poincaré series of $N(k)$ to that of $J^{k}\delta^{k}$ we need the following slight modification of Proposition \[poincare-SA\]. Let $k\geq 0$, set $N=n(n-1)/2$ and write $K=kN$. 1. If $b\in B_{ij}$ for $i\geq j\geq 0$ is homogeneous under the ${\mathbf{h}}$-grading then it is homogeneous in the ${\mathbf{E}}$-grading and $\operatorname{{\mathbf{E}}\text{-deg}}b = (i-j)N + \operatorname{{\mathbf{h}}\text{-deg}}b.$ 2. Under the ${\mathbf{E}}$-grading, $\overline{N(k)}$ has Poincaré series $$p(\overline{N(k)}, v) = v^{K} \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-k(n(\mu) - n(\mu^t))}[n]_v!}{\prod_{i=2}^n (1-v^{-i})}.$$ while $\underline{N(k)}$ has Poincare series $p(\underline{N(k)},v) = v^{K}\displaystyle \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v) v^{k(n(\mu) - n(\mu^t))}}{\prod_{i=2}^n (1-v^{i})}.$ \(1) If $b_1\in B_{ik}$ and $b_2\in B_{kj}$ then ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}(b_1b_2) = ({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_1)b_2 + b_1({\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b_2)$ and $[{\mathbf{E}}, b_1b_2] = [{\mathbf{E}},b_1]b_2 + b_1[{\mathbf{E}},b_2]$. By induction, it therefore suffices to prove the result when $b=em\delta e\in B_{k,k-1}= eH_{c+k} \delta e$, for some $k>0$. By we see that ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b = e[{\mathbf{h}}_{c+k}, m]\delta e$ whereas $[{\mathbf{E}}, b ] = e[{\mathbf{E}}, m] \delta e + em[{\mathbf{E}}, \delta] e$. By , $ [{\mathbf{h}}_{c+k}, m]=[{\mathbf{E}}, m]$ and so the two gradings differ by $\operatorname{{\mathbf{E}}\text{-deg}}\delta = N$. \(2) This follows from part (1) combined with Proposition \[poincare-SA\], respectively . {#filter-injA} Fix $k\geq 0$ and for notational simplicity write $\mathcal{J}=e J^k\delta^k$ and $\mathcal{N}=N(k)$. The final step in the proof of Proposition \[pre-cohh\] is to show that the inclusion $\Theta: \mathcal{J} \hookrightarrow \operatorname{{\textsf}{ogr}}\mathcal{N}$ from Lemma \[thetainjA\](3) is surjective. In order to effectively use Corollary \[poincare-S2A\], we do this by lifting $\Theta$ to a ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$-module map $\theta: \mathcal{J}\to \mathcal{N}$. The order filtration on $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ induces a graded structure on $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}^{\text{reg}}})\ast {{W}}\cong {\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus{\mathfrak{h}}^*]\ast{{W}}$ and hence on $\operatorname{{\textsf}{ogr}}{\mathcal{N}}$, which we call the *order gradation*; thus $\deg_{\operatorname{{\textsf}{ord}}} ({\mathbb{C}}[{\mathfrak{h}}]\ast {{W}})=0$, while $\deg_{\operatorname{{\textsf}{ord}}} {\mathfrak{h}}=1$. We will use the same terminology for the induced grading on the rings $A^0={\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^W$ and $A$ and the module ${\mathcal{J}}$. Let ${\mathcal{N}}^m=\operatorname{{\textsf}{ord}}^m{\mathcal{N}}$ denote the elements in ${\mathcal{N}}$ of order $\leq m$. Similarly, write ${\mathcal{J}}= \bigoplus_{m\geq 0} \operatorname{{\textsf}{ogr}}^m {\mathcal{J}}$ for the graded structure of ${\mathcal{J}}$ under the $\operatorname{{\textsf}{ord}}$ gradation and write the induced order filtration as ${\mathcal{J}}= \bigcup {\mathcal{J}}^m$, for ${\mathcal{J}}^m = \operatorname{{\textsf}{ord}}^m {\mathcal{J}}= \bigoplus_{0\leq i\leq m} \operatorname{{\textsf}{ogr}}^i{\mathcal{J}}$. There exists an injective map $\theta : {\mathcal{J}}\hookrightarrow {\mathcal{N}}$ of left $\mathbb C[{\mathfrak{h}}]^W$-modules such that: 1. $\theta$ is a graded homomorphism under the ${\mathbf{E}}$-gradation and is a filtered homomorphism under the order filtration. 2. The associated graded map $\operatorname{{\textsf}{ogr}}\, \theta: {\mathcal{J}}\to \operatorname{{\textsf}{ogr}}{\mathcal{N}}$ induced by $\theta$ is precisely $\operatorname{{\textsf}{ogr}}\theta = \Theta$. Trivially, $\Theta$ is an ${\mathbf{E}}$-graded map (by which we always mean a graded map of degree zero), as well as being graded under the $\operatorname{{\textsf}{ord}}$ gradation. For any $m$, $\operatorname{{\textsf}{ogr}}^m{\mathcal{J}}$ is an ${\mathbf{E}}$-graded $\mathbb C[{\mathfrak{h}}]^W$-module. By Corollary \[hi-basic-lem\](2) ${\mathcal{J}}$ is a free left $\mathbb C[{\mathfrak{h}}]^W$-module, and hence so is each summand $\operatorname{{\textsf}{ogr}}^m{\mathcal{J}}$. Thus we may pick an ${\mathbf{E}}$-homogeneous free basis $\{a_{jm}\}$ for $\operatorname{{\textsf}{ogr}}^m{\mathcal{J}}$. Now $a_{jm}=\Theta(a_{jm})\in \operatorname{{\textsf}{ogr}}^m {\mathcal{N}}={{\mathcal{N}}}^m/{\mathcal{N}}^{m-1}$ and the surjection $\pi_m: {\mathcal{N}}^m\to {{\mathcal{N}}}^m/{\mathcal{N}}^{m-1}$ is an ${\mathbf{E}}$-graded surjection. Thus, for each ${j,m}$ we can pick an ${\mathbf{E}}$-homogeneous preimage $\theta(a_{jm})\in {\mathcal{N}}^m$ of $\Theta(a_{jm})$. Define $\theta$ to be the $\mathbb C[{\mathfrak{h}}]^W$-module map induced by the map $ a_{jm}\mapsto \theta(a_{jm})$ on basis elements. Since $\pi_m$ is a left ${\mathbb{C}}[{\mathfrak{h}}]^W$-module map, a straightforward induction on orders of elements ensures that the $\theta(a_{jm})\in {\mathcal{N}}^m$ are a free basis for the module they generate. The other conclusions of the lemma follow automatically from the construction of $\theta$. {#step-1} As happens with many questions about ${{W}}$-invariants, it is easy to prove that $\Theta$ is surjective on ${\mathfrak{h}^{\text{reg}}}$. Given a left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module $M$, we will write $M[\delta^{-2}]$ for the localisation ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}[\delta^{-2}]\otimes_{{\mathbb{C}}[{\mathfrak{h}}]^{{W}}}M$. Clearly, when $M$ is a left ${\mathbb{C}}[{\mathfrak{h}}]$-module, $M[\delta^{-2}]$ is naturally isomorphic to ${\mathbb{C}}[{\mathfrak{h}}][\delta^{-1}]\otimes_{{\mathbb{C}}[{\mathfrak{h}}]}M$. \(1) The inclusion $ \Theta[\delta^{-2}] : {\mathcal{J}}[\theta^{-2}] \hookrightarrow (\operatorname{{\textsf}{ogr}}{\mathcal{N}})[\delta^{-2}]$ is an equality. \(2) The induced map $ \theta[\delta^{-2}] : {\mathcal{J}}[\theta^{-2}] \to {\mathcal{N}}[\delta^{-2}]$ is an isomorphism. This map is graded under the ${\mathbf{E}}$-grading and is a filtered isomorphism under the order filtration, in the sense that $\theta[\delta^{-2}]$ maps $ \operatorname{{\textsf}{ord}}^n{\mathcal{J}}[\delta^{-2}] $ isomorphically to $\operatorname{{\textsf}{ord}}^n{\mathcal{N}}[\delta^{-2}]$ for each $n$. \(1) By $B_{k,k-1}[\delta^{-2}] = eH_{c+k}\delta[\delta^{-2}]e= e (D({\mathfrak{h}^{\text{reg}}})\ast {{W}})e,$ for any $k\in{\mathbb{C}}$. Repeated application of this shows that $B_{ij}[\delta^{-2}] = e (D({\mathfrak{h}^{\text{reg}}})\ast {{W}})e$ and hence, by Corollary \[morrat-cor\], that ${\mathcal{N}}[\delta^{-2}] = e (D({\mathfrak{h}^{\text{reg}}})\ast {{W}})eH_c=e(D({\mathfrak{h}^{\text{reg}}})\ast {{W}}) .$ Since $\operatorname{{\textsf}{ord}}(\delta^2)=0$, we deduce that $(\operatorname{{\textsf}{ogr}}{\mathcal{N}})[\delta^{-2}] = e({\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^{\ast}]\ast {{W}}).$ On the other hand, since $\delta^{2k}\in J^k\delta^k\subseteq {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]$, certainly ${\mathcal{J}}[\delta^{-2}] = e{\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^{\ast}]= e( {\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus {\mathfrak{h}}^{\ast}]\ast {{W}})$. Since $\Theta$ is given by inclusion, $ \Theta[\delta^{-2}]$ is therefore an isomorphism. \(2) By Lemma \[filter-injA\], $\theta $ and hence $ \theta[\delta^{-2}] $ are graded maps under the ${\mathbf{E}}$-gradation and filtered under the order filtration. Since $\operatorname{gr}( \theta[\delta^{-2}] )= \Theta[\delta^{-2}] $ is an isomorphism, necessarily $\theta [\delta^{-2}] $ is a filtered isomorphism. Notation {#eqpoi-sect} -------- As in , set ${\mathcal{J}}=eJ^k\delta^k$, ${\mathcal{N}}=N(k)$ and write $\theta({\mathcal{J}})^m=\operatorname{{\textsf}{ord}}^m \theta({\mathcal{J}})=\theta({\mathcal{J}})\cap {\mathcal{N}}^m$ for all $m\geq 0$. We rewrite the ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-basis of $\theta({\mathcal{J}})$ constructed in the proof of Lemma \[filter-injA\] as $\{a_{g\ell m}\}$, where each $a_{g\ell m}$ is $g$-homogeneous under the ${\mathbf{E}}$-gradation and has order exactly $\ell$. Since these were induced from the bases $\{ a_{c\ell }\}$ of $\operatorname{{\textsf}{ogr}}^\ell {\mathcal{J}}$, the set $\{a_{g\ell m} : \ell\leq t\}$ does give a basis of $\theta({\mathcal{J}})^t$. By Lemma \[Bbar-freeA\](2), ${\mathcal{N}}$ is a free left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module and it is certainly graded. Thus, by Theorem \[graded-proj-thm\], it is graded-free. We may therefore pick a $ {\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-basis $\{ b_{g u}\}$ of ${\mathcal{N}}$ where, again, each $b_{g u} $ is ${\mathbf{E}}$-homogeneous of degree $g$ but of unspecified order. This basis is far from unique and one cannot expect that $\{b_{gu} : b_{gu} \in {\mathcal{N}}^m\}$ forms a basis of ${\mathcal{N}}^m$; indeed at this stage we do not even know that ${\mathcal{N}}^m$ is a free $\mathbb C[{\mathfrak{h}}]^W$-module. {#subsec-step42} We are now ready to put these observations together to prove the hard part of Proposition \[pre-cohh\]. \[grsameA\] Fix $k\geq 0$ and set ${\mathcal{J}}= eJ^k\delta^k$ and ${\mathcal{N}}=N(k)$. Then the map $\theta:{\mathcal{J}}\to{\mathcal{N}}$ is an isomorphism. Set $\mathfrak{m}= {\mathbb{C}}[{\mathfrak{h}}]^{W}_+$ and note that ${\mathcal{N}}/\mathfrak{m}{\mathcal{N}}=\overline{N(k)}$. On the other hand, in the notation of Corollary \[gr\], ${\mathcal{J}}/\mathfrak{m}{\mathcal{J}}\cong \overline{J^{k}}[K]$ is the shift of $\overline{J^k}$ by $\deg \delta^{k} = K=kn(n-1)/2$. By Corollaries \[gr\] and \[poincare-S2A\], we therefore have an equality of Poincaré series under the ${\mathbf{E}}$-gradation: $$\label{eqpoiA} p( {\mathcal{J}}/\mathfrak{m}{\mathcal{J}}, v) = v^{K}\frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1})v^{-k(n(\mu) - n(\mu^t))}[n]_v!} {\prod_{i=2}^n (1-v^{-i})} = p( {{\mathcal{N}}/\mathfrak{m}{\mathcal{N}}}, v).$$ Keep the ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-bases of $\theta({\mathcal{J}})\cong {\mathcal{J}}$ and ${\mathcal{N}}$ described in Notation \[eqpoi-sect\]. We write ${a(g\ell m)} = g$ whenever $a_{g\ell m}$ exists for that choice of $g,\ell,m$; thus $\sum_{g\ell m} v^{a(g\ell m)}$ denotes the sum $\sum v^g$, where one has one copy of $v^g$ for each $\ell,m$ for which $a_{g\ell m}$ exists. Define ${b(gu)}$ analogously. Since the bases $\{a_{g\ell m}\}$ and $\{b_{gu}\}$ induce ${\mathbb{C}}$-bases of ${\mathcal{J}}/\mathfrak{m}{\mathcal{J}}$, respectively $\overline{N(k)}$, can be reinterpreted as $$\begin{aligned} \label{eqpoi2} \sum_{g,\ell,m} v^{a(g\ell m)} &= & v^{K} \frac{\sum_{\mu} f_{\mu}(v^{-1})f_{\mu}(v)v^{-k(n(\mu)- n(\mu^t))}[n]_v!} {\prod_{i=2}^n (1-v^{-i})} = \sum_ {g,u} v^{b(gu)}.\end{aligned}$$ We note that has several consequences for the $a(g\ell m)$ and $b(gu)$. 1. For fixed $g$, there exist only finitely many elements $a_{g\ell m}$ and $b_{gu}$. This is because the middle expression in is a well-defined series. 2. There exists a universal upper bound $a(g\ell m)\leq T$. This is because the numerator in the middle expression in (\[eqpoi2\]) is a finite sum of polynomials. However, there is no universal lower bound. 3. For any $g_0$, the number of $a_{g\ell m}$ with $g=g_0$ equals the number of $b_{gu}$ with $g=g_0$. This is simply because $\sum v^{a(g\ell m)} = \sum v^{b(gu)}$ and the numbers are finite by ($\dagger$1). We aim to adjust the basis $\{b_{gu}\}$ to be equal to the basis $\{a_{g\ell m}\}$, and we achieve this by a downwards induction on $g$. The induction starts since, by ($\dagger$3), there are no basis elements $b_{gu}$ with $g>T$. Let $-\infty < G\leq T$ and, by induction, suppose that $\{b_{gu} : u\in {\mathbb{Z}}\}= \{a_{g\ell m} : \ell,m \in {\mathbb{Z}}\}$ for all $g>G$. Suppose that there exists a basis element $b_{Gw} \not\in \{a_{G\ell m}\}$. By Lemma \[step-1\](2), $\theta({\mathcal{J}})[\delta^{-2}]={\mathcal{N}}[\delta^{-2}]$ and so there exists a homogeneous element $\mathbf{x}^m\in {\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ of ${\mathbf{E}}$-degree $m$ such that $\mathbf{x}^mb_{Gw}\in \theta({\mathcal{J}})$. Thus we have the ${\mathbf{E}}$-homogeneous equation $$\label{bigger-kk} \mathbf{x}^mb_{Gw} = \sum_{g<G} c_{gfh}a_{gfh} +\sum c_{Gfh}a_{Gfh}+ \sum_{g>G} c'_{gz}b_{gz},$$ where $c_{gfh}, c'_{gz}\in \mathbb C[{\mathfrak{h}}]^W$ and summation over $f,h,z$ is suppressed. Since $\theta({\mathcal{J}})\subseteq {\mathcal{N}}$, we may write each $a_{gfh}$ as an ${\mathbf{E}}$-homogeneous sum $a_{gfh} =\sum d_{\bullet}b_{uz}$ for some $d_\bullet = d_{fghuz} \in {\mathbb{C}}[{\mathfrak{h}}]^W$ and obtain $$\label{bigger-k} \mathbf{x}^mb_{Gw} = \sum_{g<G} c_{gfh}d_{\bullet}b_{uz} +\sum c_{Gfh}d_{\bullet}b_{uz} + \sum_{g>G} c'_{gz}b_{gz}.$$ Both the last two displayed equations are ${\mathbf{E}}$-homogeneous of ${\mathbf{E}}$-degree $G+m$ and so, by , each element $c_{gfh}$ must have ${\mathbf{E}}$-degree $\geq m$. Thus the $b_{uz} $ appearing in the first two terms on the right hand side of must have ${\mathbf{E}}$-degree $\leq G$. Thus the only appearance of $b_{gz}$ with $g>G$ is in the third sum. Since the $b_{uz}$ are a ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-basis of ${\mathcal{N}}$, that third term $\sum_{g>G} c'_{gz}b_{gz}$ is actually zero. Now consider where the specific term $b_{Gw}$ appears on the right hand side of . For $g<G$, implies that $\operatorname{{\mathbf{E}}\text{-deg}}c_{gfh}>m$ for each $f, h$ and so $b_{Gw}$ cannot appear in the first sum. Thus it must appear nontrivially in some term $c_{Gf'h'}d'b_{Gw}$ in the second sum. In this case, implies that $\operatorname{{\mathbf{E}}\text{-deg}}c_{Gf'h'} =m$. Hence $d'\in\mathbb C\smallsetminus\{0\}$ and $$a_{Gf'h'} = d'b_{Gw} +\sum_{(uz)\not= (Gw)} d_{uz}'' b_{uz}.$$ Thus we can replace $b_{Gw}$ by $a_{Gf'h'}$ in our basis for ${\mathcal{N}}$. By ($\dagger$3), the sets $\{a_{G\ell m} : \ell,m\in {\mathbb{Z}}\}$ and $\{b_{Gu} : u\in {\mathbb{Z}}\}$ have equal finite cardinality. After a finite number of steps we therefore have $\{b_{Gu}\} \subseteq \{a_{G\ell m}\}$ and hence $\{b_{Gu}\}= \{a_{G\ell m}\}$. This completes the inductive step and hence the proof of the lemma. We can now pull everything together and prove both Theorem \[main\] and Proposition \[pre-cohh\]. Proof of Proposition \[pre-cohh\] {#subsec-6.21A} --------------------------------- Recall from Lemma \[thetainjA\] that $\Theta : eJ^k\delta^{k}\to \operatorname{{\textsf}{ogr}}N(k)$ is the natural inclusion. On the other hand, for any $k \geq 0$, Proposition \[grsameA\] implies that the map $\theta: eJ^k\delta^{k}\to N(k)$ is an isomorphism. Lemma \[filter-injA\](2) therefore implies that $\operatorname{gr}_{\Lambda} N(k) = \operatorname{{\textsf}{ogr}}\theta(eJ^k\delta^{k}) = \Theta(eJ^k\delta^k) = eJ^k\delta^k$. Proof of Theorem \[main\] {#subsec-6.21} ------------------------- \(1) This is immediate from Corollary \[morrat-cor\](1) and Lemma \[Zalgequiv\]. \(2) Fix $i\geq j\geq 0$. Since $c+j$ still satisfies Hypothesis \[main-hyp\], Proposition \[pre-cohh\] implies that $\operatorname{{\textsf}{ogr}}B_{ij}eH_{c+j} =eJ^{i-j}\delta^{i-j}$. Multiplying this identity on the right by $e$ and applying Lemma \[grade-elements\] and Corollary \[morrat-cor\](1) gives $$eJ^{i-j}\delta^{i-j} e = \operatorname{{\textsf}{ogr}}(B _{ij}eH_{c+j})e =\operatorname{{\textsf}{ogr}}(B_{ij} eH_{c+j}e) =\operatorname{{\textsf}{ogr}}B_{ij} .$$ Since $\delta$ transforms under ${{W}}$ by the sign representation, Lemma \[corpar\](1) shows that $eJ^{i-j}\delta^{i-j} e= eA^{i-j}\delta^{i-j}e$. Combining these observations gives $\operatorname{{\textsf}{ogr}}B_{ij} = eA^{i-j}\delta^{i-j} e$. Therefore, $\operatorname{{\textsf}{ogr}}B= \bigoplus \operatorname{{\textsf}{ogr}}B_{ij} = e\widehat{A}e\cong \widehat{A}$, as graded vector spaces. In order to ensure that this is an isomorphism of graded ${\mathbb{Z}}$-algebras we need to check that the multiplication in $\operatorname{{\textsf}{ogr}}B$ coming from the tensor product multiplication in $B$ is the same as the natural multiplication in $\widehat{A}$. This follows from Lemma \[abstract-products\](1). \(3) The equivalences $\operatorname{{\textsf}{ogr}}(B){\text{-}{\textsf}{qgr}}\simeq A{\text{-}{\textsf}{qgr}}\simeq \operatorname{{\textsf}{Coh} }(\operatorname{Hilb(n)})$ follow from (2) combined with , respectively Corollary \[hi-basic-lem2\](1). Corollary {#order-free} --------- [*Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\] and pick $i\geq j\geq 0$. Then, for $m\geq 0$, each of the modules $\operatorname{{\textsf}{ord}}^mN(i)$, $\operatorname{{\textsf}{ogr}}^mN(i)$, $\operatorname{{\textsf}{ord}}^m B_{ij}$ and $\operatorname{{\textsf}{ogr}}^m B_{ij}$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module.*]{} By construction and Proposition \[pre-cohh\], the map $\Theta: \operatorname{{\textsf}{ogr}}N(i)\to eJ^{i}\delta^i$ is an isomorphism of $\operatorname{{\textsf}{ord}}$-graded modules. Thus $\operatorname{{\textsf}{ogr}}^mN(i)\cong \operatorname{{\textsf}{ogr}}^m eJ^{i}\delta^i$ is a free ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module by Lemma \[hi-basic-lem\]. By induction on $m$, it follows that $\operatorname{{\textsf}{ord}}^mN(i)$ is also free. The analogous results for $B_{ij}$ follow by multiplying everything on the right by $e$. {#cohh-subsect} We end the section by noting that Proposition \[pre-cohh\] provides an interesting connection between $H_c$-modules and the isospectral scheme $X_n$ defined in . Adjusting to the conventions of this section, we identify $\operatorname{Hilb(n)}= \operatorname{Proj}\widetilde{A}$, for $\widetilde{A}=\bigoplus A^k\delta^k$. By construction, the Procesi bundle ${\mathcal{P}}=\rho_*{\mathcal{O}}_{X_n}$ from is then just the image in $\operatorname{{\textsf}{Coh} }\, \operatorname{Hilb(n)}$ of the $\widetilde{A}$-module $\bigoplus J^k\delta^k$. Thus the next result is an immediate consequence of Proposition \[pre-cohh\]. Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\]. Let $e\widetilde{H}_c= \bigoplus_{k\geq 0}B_{k0}\otimes_{U_c} e\widetilde{H}_c$ be the $B$-module associated to the $U_c$-module $eH_c$ and filter each $B_{k0}\otimes_{U_c}eH_c \cong B_{k0}eH_c$ by the $\operatorname{{\textsf}{ord}}$ filtration. Set $\operatorname{{\textsf}{ogr}}e\widetilde{H}_c = \bigoplus \operatorname{{\textsf}{ogr}}B_{k0}eH_c$. Then the sheaf associated to $\operatorname{{\textsf}{ogr}}e\widetilde{H}_c$ in $\operatorname{{\textsf}{Coh} }\operatorname{Hilb(n)}$ is the Procesi bundle ${\mathcal{P}}$. {#cohh-subsect-chat} Just as Theorem \[main\] can be interpreted as saying that $U_c$ provides a noncommutative model for $\operatorname{Hilb(n)}$, so Corollary \[cohh-subsect\] can be interpreted as saying that the algebra $H_c$ provides a noncommutative model for $X_n$. Here is one aspect of this analogy. It follows from [@BKR] and [@hai1] that there is an equivalence $\xi$ of derived categories between ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$ and $\operatorname{Hilb(n)}$ that is induced by a Fourier-Mukai transform over ${\mathcal{P}}$. Now pass to the noncommutative situation, replacing ${\mathfrak{h}}\oplus{\mathfrak{h}}^*/ {{W}}$, $\operatorname{Hilb(n)}$ and ${\mathcal{P}}$ by $H_c{\text{-}{\textsf}{mod}}$, $B{\text{-}{\textsf}{qgr}}$ and $eH_c$, respectively. Then Corollary \[morrat-cor\] shows that $eH_c$ still induces a derived equivalence between the two categories. Indeed, it is even a equivalence of categories. The fact that derived equivalences in the commutative case can become full equivalences in the noncommutative case happens elsewhere and is in accord with the philosophy behind [@GK Conjecture 1.6] (see [@GK Remark 1.7]). As will be justified in [@GS2], Corollary \[cohh-subsect\] therefore “sees” the equivalence $\xi$ and this provides some intriguing connections between sheaves on $\operatorname{Hilb(n)}$ and modules over $H_c$. {#section-3} If one considers Cherednik algebras in characteristic $p>0$, where $H_c$ is a finite module over its centre, then the relationship between $H_c$ and $\operatorname{Hilb(n)}$ becomes closer still. For example, [@BFG] shows that there is even a derived equivalence between $H_c$ and an Azumaya algebra over a Frobenius twist of $\operatorname{Hilb(n)}$. Similarly in characteristic zero, symplectic reflection algebras with parameter $t=0$ are finite modules over their centre, and [@GSm Theorem 1.2] shows that there are often derived equivalences between these algebras and varieties that deform Hilbert schemes. Tensor product filtrations {#sect7} ========================== {#sect701} The tensor product decomposition of the $B_{ij}$ can be used to give a second filtration on that module by inducing a filtration on $B_{ij}$ from the $\operatorname{{\textsf}{ord}}$ filtration on the tensorands. It turns out that the main theorem is essentially equivalence to the assertion that the two filtrations are equal. In this short section we give the details behind this assertion. Analogues of this result also hold for the module $N(k)$ defined in and the module $M(k)=H_{c+k}eB_{k0} =H_{c+k}\delta e B_{k-1,0}$ defined in and so we begin by giving a general context for all three results. {#tens-defn-sect} For fixed $i\geq j\geq 0$ we are interested in the following tensor product decompositions $$\label{tensor-1} B_{ij}\cong Q_{c+i-1}^{c+i}\otimes Q_{c+i-2}^{c+i-1}\otimes\cdots \otimes Q_{c+j}^{c+j+1},$$ $$\label{tensor-101} N(i)\cong Q_{c+i-1}^{c+i}\otimes \cdots \otimes Q_{c}^{c+1}\otimes eH_c\qquad\mathrm{or}\qquad N(i) \cong B_{i0}\otimes eH_c$$ and $$\label{app-c-cor} M(i)\cong H_{c+i}\delta e\otimes_{U_{c+i-1}} B_{i-1,i-2}\otimes \cdots\otimes_{U_{c+1}} B_{10} \qquad\mathrm{or}\qquad M(i) \cong H_{c+i}\delta e\otimes_{U_{c+i-1}}B_{i-1,0}$$ where the tensor products are over the appropriate rings $U_k$. Corresponding to these decompositions we have the [*tensor product filtration*]{} $\operatorname{{\textsf}{ten}}$ defined by the following convention: Given a module $C=C_1\otimes\cdots \otimes C_r$, where each $C_j$ is filtered by the $\operatorname{{\textsf}{ord}}$ filtration, define $$\label{tensor-2} \operatorname{{\textsf}{ten}}^n(C) = \Big\{\sum c_{1}\otimes \cdots\otimes c_r, \ \mathrm{where}\ c_m\in \operatorname{{\textsf}{ord}}^{\ell(m)}(C_m) \ \mathrm{with}\ \sum_{m=1}^{r} \ell(m)\leq n\Big\}.$$ As usual, we will write the associated graded module as $\operatorname{{\textsf}{tgr}}C = \bigoplus \operatorname{{\textsf}{ten}}^n C/\operatorname{{\textsf}{ten}}^{n-1} C$. \[ord-tens\] Assume that $c\in {\mathbb{C}}$ satisfies Hypothesis \[main-hyp\]. Let $C$ denote one of the objects $B_{ij}$, $N(i)$ or $M(i)$ and consider the tensor product filtrations induced from one of the tensor product decompositions (\[tensor-1\]–\[app-c-cor\]). Then $\operatorname{{\textsf}{ord}}^mC=\operatorname{{\textsf}{ten}}^mC$, for all $m\geq 0$. We will prove the result for the decomposition and the first decomposition in each of and . The proof in the remaining cases is left to the reader as it uses essentially the same argument, although one needs to use the conclusion of the lemma for . In each of the three cases we are given a decomposition $C=C_1\otimes\cdots \otimes C_r$, say with $\operatorname{{\textsf}{ogr}}C_j=D_j$ and $\operatorname{{\textsf}{ogr}}C=D$. Moreover, by Theorem \[main\], respectively Proposition \[pre-cohh\] combined with Lemma \[thetainjA\], respectively Proposition \[app-c-prop\] combined with Lemma \[thetainjC\], there is an equality $D_1\cdots D_r=D$ given by multiplication in $D({\mathfrak{h}^{\text{reg}}})\ast{{W}}$. Equivalently, the natural multiplication map $\chi: D_1\otimes \cdots \otimes D_r\to D$ is surjective. Consider the graded map $\chi$ in more detail. Given elements $\bar{\alpha}_j\in \operatorname{{\textsf}{ogr}}^{m(j)} D_j$, with $m = \sum m(j)$, lift the $\bar{\alpha}_j$ to elements $\alpha_j\in ord^{m(j)}C_j$. Then $\chi$ is defined by $$\chi(\bar{\alpha}_1\otimes\cdots\otimes \bar{\alpha}_r) =\left(\alpha_1\cdots\alpha_r + \operatorname{{\textsf}{ord}}^{m - 1}C\right)/\operatorname{{\textsf}{ord}}^{m - 1}C.$$ By the definition of the $\operatorname{{\textsf}{ten}}$ filtration, this says that image of $\chi$ is contained in (and indeed equal to) $\bigoplus_{m} \bigl(\operatorname{{\textsf}{ten}}^m C + \operatorname{{\textsf}{ord}}^{m-1} C\bigr)/\operatorname{{\textsf}{ord}}^{m-1} C$. But $\chi$ is surjective. By induction on $m$ we therefore have $\operatorname{{\textsf}{ord}}^mC= \operatorname{{\textsf}{ten}}^m C + \operatorname{{\textsf}{ord}}^{m-1} C = \operatorname{{\textsf}{ten}}^m C$. {#ord-tens-chat} The equality of filtrations given by Lemma \[ord-tens\] is not merely a formality; indeed the result for $B_{ij}$ is essentially the same result as Theorem \[main\]. To see this, suppose that $\operatorname{{\textsf}{ogr}}B_{ij}=\operatorname{{\textsf}{tgr}}B_{ij}$ for all $i\geq j\geq0$. As Lemma \[thetainjA\](2) shows, $\operatorname{{\textsf}{ogr}}B_{\ell+1,\ell} = A^1\delta$ for each $\ell$ and so, by Lemma \[abstract-products\](2), we get a surjection $\chi$ from $E=(A^1\delta)^{\otimes(i-j)}$ onto $\operatorname{{\textsf}{tgr}}B_{ij}=\operatorname{{\textsf}{ogr}}B_{ij}$. The multiplication map $\phi: E\to (A^1\delta)^{i-j}$ is surjective and its kernel is the largest torsion $A^0$-submodule of $(A^1\delta)^{i-j}$. On the other hand $\operatorname{{\textsf}{ogr}}B_{ij}\subseteq e{\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]^{{W}}$ is a torsion-free $A^0$-module and so $\mathrm{ker}(\phi) \subseteq \mathrm{ker}(\chi)$. Thus $\operatorname{{\textsf}{ogr}}B_{ij} = E/\mathrm{ker}(\chi) $ is a homomorphic image of $ (A^1\delta)^{i-j} $. Since $ (A^1\delta)^{i-j} $ is a right ideal of the domain $A^0$, any proper factor of $ (A^1\delta)^{i-j} $ will be torsion. Thus $\mathrm{ker}(\phi)= \mathrm{ker}(\chi)$ and $\operatorname{{\textsf}{ogr}}B_{ij} \cong (A^1\delta)^{i-j}$. {#order-counter} The observation in suggests that Lemma \[ord-tens\] will only hold for very special decompositions and this is indeed the case. In essence, Theorem \[main\] says that the identity $B_{ij}\cong B_{i,i-1}\otimes\cdots\otimes B_{j+1,j}$ is a filtered isomorphism. On the other hand, an identity like $H_c\cong H_ce\otimes_{U_c} eH_c$ from Theorem \[morrat\] is clearly not filtered; in writing the element $1$ as an element of $H_ce \otimes eH_c$ an easy computation shows that one needs to use commutators of elements from ${\mathbb{C}}[{\mathfrak{h}}]$ and ${\mathbb{C}}[{\mathfrak{h}}^*]$ and so $1\notin \operatorname{{\textsf}{ten}}^0(H_c)$. However, $ge=ge\cdot 1 \in \operatorname{{\textsf}{ten}}^0(H_c)$ for any $0\not=g\in {\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ and so $\sigma(ge)\sigma(1)=0$ in $\operatorname{{\textsf}{tgr}}H_c$. On the other hand, as $1$ is a regular element of $\operatorname{{\textsf}{ogr}}H_c$, no such equation is possible $\operatorname{{\textsf}{ogr}}H_c$. Thus $\operatorname{{\textsf}{ten}}H_c\not\cong \operatorname{{\textsf}{ogr}}H_c$. As a second example, it is easy to check that Lemma \[ord-tens\] will fail for $M(i)$ if one introduces one more tensor product, $M(i) \cong H_{c+i}e \otimes_{U_{c+i}} B_{i0}$. Indeed, Lemma \[thetainjC\] implies that $ \operatorname{{\textsf}{ogr}}M(1)={\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]\delta e$. On the other hand, for the given decomposition Lemmas \[thetainjA\] and \[abstract-products\] imply that $\operatorname{{\textsf}{tgr}}H_c$ is a homomorphic image of $ T=\operatorname{{\textsf}{ogr}}H_{c+1}e\otimes_{U_{c+1}}\operatorname{{\textsf}{ogr}}Q_c^{c+1}\cong {\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]e\otimes_{A^0} A^1\delta e.$ Clearly the image of $T$ in $\operatorname{{\textsf}{ogr}}M(1)$ is just ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]e A^1\delta e=J^1\delta e$. By the argument of the second paragraph of , this is also the image of $\operatorname{{\textsf}{tgr}}M(1)$ in $\operatorname{{\textsf}{ogr}}M(1)$. Graded projective modules {#app-a} ========================= {#section-4} The aim of this appendix is to prove the following graded analogue of a well-know result of Kaplansky [@Kap Theorem 2], for which we do not know a reference. \[graded-proj-thm\] Let $A=\bigoplus_{i\geq 0} A_i$ be a connected ${\mathbb{N}}$-graded $k$-algebra (thus $A_0=k$). Let $P$ be a right $A$-module that is both graded and projective. Then $P$ is a *graded-free* $A$-module in the sense that $P$ has a free basis of homogeneous elements. Throughout this proof all graded maps are graded maps of degree zero. We will write the degree of a homogeneous element $x\in P$ as $|x|$. An observation of Eilenberg [@Eil Section 1] shows that $P$ is graded projective in the sense that there is a graded isomorphism $F\cong P\oplus Q$, for some $A$-module $Q$ and graded-free $A$-module $F$. We need a minor variant on this result, so we give the proof. Take a graded surjection $\phi: F=\bigoplus f_iA\twoheadrightarrow P$ and an ungraded splitting $\theta: P\to F$. If $p_i=\phi(f_i)$, then write $\theta(p_i)=g_i+h_i$, where $g_i$ is the homogeneous component of $\theta(p_i)$ with $|g_i|=|p_i|$. Then check that the map $p_i\mapsto g_i$ also splits $\phi$. This proof also shows that, if $P$ is countably generated, then we can take $F$ to be a countably generated graded-free module. The heart of the proof of the theorem is contained in the next two sublemmas. Sublemma {#graded-proj-sublemma1} -------- [*Under the hypotheses of the theorem, $P$ is a graded direct sum of countably generated $A$-modules.*]{} The proof of [@Kap Theorem 1] also works in the category of graded modules. Sublemma {#graded-proj-sublemma} -------- [*Keep the hypotheses of the theorem and assume that $P$ is countably generated. If $x\in P$ then there exists a graded-free direct summand $G$ of $P$ such that $x\in G$.* ]{} By the result of Eilenberg described above, we may pick a graded isomorphism $F\cong P\oplus Q$, for some $A$-module $Q$ and countably generated graded-free $A$-module $F$. Select a homogeneous basis $\{u_i : i\in \mathbb N\}$ for $F$ such that there is a graded expression $x=\sum_{i=1}^n u_ia_i$, with $a_i\in A$ and $n$ as small as possible. We first claim that no $a_j$ can be written as a [*left*]{} linear combination of the other $a_\ell$. Indeed, suppose that $a_n=\sum_{i=1}^{n-1} r_ia_i$, for some $r_i\in A$. By taking the appropriate component we may assume that each $r_i$ is homogeneous with $|r_i|=|a_n|-|a_i|$. It follows that $|u_nr_i|=|u_i|$ and hence that $u_i'=u_i+u_nr_i$ is homogeneous. However $$\sum_{i=1}^{n-1} u_i'a_i =\sum_{i=1}^{n-1} u_ia_i + u_n(\sum_{i=1}^{n-1} r_ia_i) =x.$$ This contradicts the minimality of $n$ and proves the claim. Reorder the basis $\{u_\ell\}$ so that $|u_i|\leq |u_{i+1}|$ for $1\leq i\leq n$ and write $u_i=p_i+q_i$, for $p_i\in P$, $q_i\in Q$, all of the same degree. Notice that $P\ni x=\sum u_ia_i=\sum p_ia_i + \sum q_ia_i$ and so $\sum q_ia_i\in P\cap Q=0$. Hence $$\label{kap-1} x= \sum_{i=1}^n u_ia_i=\sum_{i=1}^n p_ia_i$$ Next write each $p_i$ as a homogeneous sum $p_i=\sum_{j=1}^nu_jc_{ji} + t_i$, where $t_i\in \sum_{i>n}u_iA$. Then $$x = \sum_{i=1}^n u_ia_i=\sum p_ia_i = \sum_{i,j=1}^n u_jc_{ji}a_i +\sum_{i=1}^n t_i a_i.$$ Since $\{u_i\}$ is a basis, $$\label{kap2} a_j=\sum_{i=1}^n c_{ji}a_i\qquad\text{for}\quad 1\leq j\leq n.$$ We claim that $c_{ji}=0$ for $i<j$ and that $|c_{ji}|>0$ whenever $i>j$ (and $c_{ji}\not=0$). Since $|u_i|\leq |u_{i+1}|$, we have $|a_i|\geq |a_{i+1}|$ for each $i$. Also $|c_{ji}|=|u_i|-|u_j|$ for all $i,j$ and so $c_{ji}=0$ if $|u_i|<|u_j|$. Thus both parts of the claim are clear when $|u_i|\not= |u_{j}|$; equivalently, when $|a_i|\not= |a_{j}|$. So, suppose that $|a_i|=|a_j|$, for some $i\not =j$ and that $c_{ji}\not=0$. Then $c_{ji}\in k^*$ and so expresses $a_i$ as a left linear combination of the other $a_\ell$. This contradicts the initial minimality assumption on $n$ and proves the claim. Note that $c_{jj}=1$ for all $j$, since otherwise would express $a_j$ as a left linear combination of the other $a_\ell$. The last paragraph implies that $C=(c_{ji})$ is an upper triangular matrix, with units on the diagonal and so it is invertible. In particular, $\{p_1,\dots,p_n\}\cup \{u_{n+\ell} : \ell >0\}$ is a basis for $F$. Thus $G=\sum_{i=1}^n p_iA$ is a graded-free direct summand of $F$ contained in $P$. Thus $G$ is also a graded-free direct summand of $P$ which, by , contains $x$. {#section-5} The proof of the theorem follows from the sublemmas by an easy induction. By Sublemma \[graded-proj-sublemma1\] we may assume that $P$ is countably generated, say by homogeneous elements $z_i $ for $i\in \mathbb N$. By induction, suppose that there is a graded decomposition $P=Q_1\oplus\cdots \oplus Q_n\oplus R_n$, where each $Q_i$ is graded-free and $z_i\in Q_1\oplus\cdots \oplus Q_i$, for $1\leq i\leq n$. By Sublemma \[graded-proj-sublemma\] this does hold when $n=1$. Write $z_{n+1} = q+r$ as a homogeneous sum, where $ q\in \sum Q_j$ and $r\in R_n$. Since $R_n$ also satisfies the hypotheses of Sublemma \[graded-proj-sublemma\], $R_n$ has a graded-free summand $Q_{n+1}$ containing $r$, completing the inductive step. Finally, $$\widetilde{P}\ = \ \lim_{n\to \infty} \big( Q_1\oplus\cdots \oplus Q_n\big) \ \cong \ \bigoplus_{i= 1}^\infty Q_i$$ is a graded-free submodule of $P$ that contains each $z_i$. Therefore $P=\widetilde{P}$. Another module {#C} ============== {#app-c-1} Fix $c\in {\mathbb{C}}$ that satisfies Hypothesis \[main-hyp\] and an integer $k\geq 0$. For applications in [@GS2] we will need an analogue of Proposition \[pre-cohh\] for the left $H_{c+k}$-module $M(k) = H_{c+k}eB_{k0}\subseteq D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$. As before, we filter $M(k)$ by the induced order filtration $\operatorname{{\textsf}{ord}}$, so that $\operatorname{{\textsf}{ogr}}M(k)\subseteq \operatorname{{\textsf}{ogr}}D({\mathfrak{h}^{\text{reg}}})\ast{{W}}= {\mathbb{C}}[{{\mathfrak{h}}^{\text{reg}}}\oplus {\mathfrak{h}}^*]\ast {{W}}.$ The aim of this appendix is then to prove: \[app-c-prop\] The left $H_{c+k}$-module $M(k) = H_{c+k}eB_{k0}$ satisfies $\operatorname{{\textsf}{ogr}}M(k) = J^{k-1}\delta^ke.$ Recall that Proposition \[pre-cohh\] showed that the module $N(k)=B_{k0}\otimes eH_c$ had associated graded ring $eJ^k\delta^k$. In a sense, Proposition \[app-c-prop\] is just a left-right analogue of that result and so much of the present proof is formally very similar to that of Proposition \[pre-cohh\]. We should first explain why the two results involve different powers of $J^1$. The reason is that one can write $M(k) = H_{c+k}eH_{c+k}\delta eB_{k-1,0}$. By Corollary \[morrat-cor\] and the left hand end of this expression collapses to give $ M(k) = H_{c+k}\delta eB_{k-1,0}.$ In particular, $M(1) = H_{c+1}\delta e$. A routine computation using Lemmas \[abstract-products\] and \[grade-elements\] then gives: Lemma {#thetainjC} ----- [*$\operatorname{{\textsf}{ogr}}M(1) = {{\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]}\delta e$ while $J^{k-1}\delta^{k}e \subseteq \operatorname{{\textsf}{ogr}}M(k)$ for all $k\geq 1$.*]{} It takes considerably more work to show that $J^{k-1}\delta^{k}e $ actually equals $ \operatorname{{\textsf}{ogr}}M(k)$ for $k>1$. The proofs of the first few steps in this argument are very similar to those of Lemmas \[Bbar-freeA\], \[filter-injA\] and \[step-1\] in the proof of Proposition \[pre-cohh\] and so we will just indicate how to modify the earlier proofs to work here. {#B-freeC} Since $M(k)$ is a $(H_{c+k},{U}_c)$-bimodule, the embeddings ${\mathbb{C}}[{\mathfrak{h}}]\hookrightarrow H_{c+k}$ and ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}\hookrightarrow {U}_c$ make $M(k)$ into a $({\mathbb{C}}[{\mathfrak{h}}],\, {\mathbb{C}}[{\mathfrak{h}}^*]^{{W}})$-bimodule. Let ${\mathbb{C}}$ be the trivial module over either ${\mathbb{C}}[{\mathfrak{h}}]$ or ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$ and set $ \overline{M(k)} = {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]} M(k)$ and $ \underline{M(k)} = M(k) \otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}} {\mathbb{C}}.$ \[Bbar-freeC\] [(1) ]{} $M(k)$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]$-module and a right ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-module. 1. $\underline{M(k)}$ is a finitely generated, free left ${\mathbb{C}}[{\mathfrak{h}}]$-module. 2. Analogously, $\overline{M(k)}$ is a finitely generated, free right ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-module. \(1) By Corollary \[morrat-cor\], $M(k)$ is projective as a left $H_{c+k}$-module and as a right ${U}_c$-module. By , $H_{c+k}$ and hence $M(k)$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]$-module. Similarly, the argument of Lemma \[Bbar-freeA\](2) shows that $U_c$ and hence $M(k)$ are free right ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-modules. \(2) This is contained in the proof of Lemma \[Bbar-freeA\](3). \(3) Mimic the proof of Lemma \[Bbar-freeA\](4). {#filter-injC} Using the conventions from , each $M(k)$ and $J^{k-1}\delta^{k}e$ is ${\mathbf{E}}$-graded. Since ${\mathbb{C}}[{\mathfrak{h}}]_+$ is ${\mathbf{E}}$-graded, the ${\mathbf{E}}$-grading on $M(k)$ descends to one on $\overline{M(k)}$. Similarly, $J^{k-1}\delta^{k}e$ has the order grading $\operatorname{{\textsf}{ogr}}$ from . Write $\Theta: J^{k-1}\delta^ke\hookrightarrow \operatorname{{\textsf}{ogr}}M(k)$ for the inclusion from Lemma \[thetainjC\]. There exists an injective map $\theta : J^{k-1}\delta^ke\hookrightarrow M(k)$ of left $\mathbb C[{\mathfrak{h}}]$-modules such that: 1. $\theta$ is an ${\mathbf{E}}$-graded homomorphism and is a filtered homomorphism under the order filtration. 2. The associated graded map $\operatorname{{\textsf}{ogr}}\theta: J^{k-1}\delta^ke \to \operatorname{{\textsf}{ogr}}M(k)$ induced by $\theta$ is precisely $\operatorname{{\textsf}{ogr}}\theta = \Theta$. 3. In the notation of , the inclusion $ \theta[\delta^{-2}] : (J^{k-1}\delta^k e)[\delta^{-2}] \to M(k)[\delta^{-2}] $ is an isomorphism. This map is ${\mathbf{E}}$-graded and is a filtered isomorphism under the order filtration. (1,2) As in the proof of Lemma \[filter-injA\], one constructs $\theta$ by lifting a ${\mathbf{E}}$-homogeneous basis of the free ${\mathbb{C}}[{\mathfrak{h}}]$-module $\operatorname{{\textsf}{ogr}}^n(J^{k-1}\delta^k)e$ to a set of ${\mathbf{E}}$-homogeneous elements in $\operatorname{{\textsf}{ord}}^n M(k)$. \(3) This is essentially the same as the proof of Lemma \[step-1\]. {#section-6} By Lemma \[diaggrad\], $M(k)$ is graded under the adjoint ${\mathbf{h}}$-action and, as both copies of ${\mathbb{C}}$ are ${\mathbf{h}}$-graded modules, this grading restricts to one on $\overline{M(k)} $ and $ \underline{M(k)}$. In each case, we call this [*the ${\mathbf{h}}$-grading*]{}. For the reasons given in , this does not equal the ${\mathbf{E}}$-grading. \[poincare-SC\] If $\overline{M(k)}$ is graded via the adjoint ${\mathbf{h}}$ action, then it has Poincaré series $$p(\overline{M(k)}, v) = \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-(k-1)(n(\mu) - n(\mu^t))}}{\prod_{i=2}^n (1-v^{-i})}.$$ This is similar to the proof of Proposition \[poincare-SA\] except that we use the module $Y=H_ce\otimes_{R}{\mathbb{C}}$, where $R=e{\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}e$, in place of $X=H_c\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]}{\mathbb{C}}$. As in that proposition, $Y$ is an object in $\widetilde{{\mathcal{O}}}_c$ and so we can write $ [Y] = \sum_{\mu} p_{\mu} [\widetilde{\Delta}_{c}(\mu)]$ for some $p_{\mu} \in {\mathbb{Z}}[v,v^{-1}]$. To calculate the $p_{\mu}$ note that, by , $Y\cong {\mathbb{C}}[{\mathfrak{h}}]\otimes {\mathbb{C}}[{\mathfrak{h}}^*]^{\text{co}{{W}}}$. Applying $({\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]}-)$ to the equation $ [Y] =\sum p_{\mu} [\widetilde{\Delta}_{c}(\mu)]$ therefore yields $[{\mathbb{C}}[{\mathfrak{h}}^*]^{\text{co}{{W}}}] = \sum_{\mu} p_{\mu} [\mu].$ Thus implies that $p_{\mu} = f_{\mu}(v^{-1})$ (this is a polynomial in $v^{-1}$ rather than $v$ since ${\mathbb{C}}[{\mathfrak{h}}^*]$ is negatively ${\mathbf{E}}$-graded) and so, as an element of $G_0(\widetilde{{\mathcal{O}}}_{c})$, $$\label{grot22} [Y] = \sum_{\mu} f_{\mu}(v^{-1}) [\widetilde{\Delta}_{c}(\mu)] .$$ Now consider $\underline{M(k)}$, which we can write as $H_{c+k}e\otimes_{U_{c+k}} B_{k0}\otimes_{U_c} eY$. By and Corollary \[morrat-cor\], $H_{c+k}e\otimes_{U_{c+k}} e\widetilde{\Delta}_{c+k}(\lambda) \cong \widetilde{\Delta}_{c+k}(\lambda)$. Thus and Lemma \[standAAA\] combine to show that $$[\underline{M(k)} ] = \sum_{\mu} f_{\mu}(v^{-1})v^{k(n(\mu)-n(\mu^t)} [\widetilde{\Delta}_{c+k}(\mu)] .$$ As graded vector spaces, $\widetilde{\Delta}_{c+k}(\mu) \cong {\mathbb{C}}[{\mathfrak{h}}]\otimes \mu$ and so $p(\widetilde{\Delta}_{c+k}(\mu), v) = f_\mu(1) (1-v)^{-(n-1)}$ by . Therefore, $$\label{wrongsideformulaC} p(\underline{M(k)}, v) = \frac{\sum_{\mu} f_\mu(1)f_{\mu}(v^{-1})v^{k(n(\mu)-n(\mu^t)}} {(1-v)^{(n-1)}}.$$ By parts (2) and (3) of Lemma \[Bbar-freeC\], a homogeneous basis for $\overline{M(k)}$ is given by lifting a homogeneous ${\mathbb{C}}$-basis for $\overline{M(k)}\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}} {\mathbb{C}}= {\mathbb{C}}\otimes_{{\mathbb{C}}[{\mathfrak{h}}]} {\underline{M(k)}}.$ Thus, combining with the formulæ $p({\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}, v) = \prod_{i=2}^n (1-v^{-i})^{-1}$ and $ p({\mathbb{C}}[{\mathfrak{h}}], v) = (1-v)^{n-1}$ gives $$\label{wrongsideformula2C} p(\overline{M(k)}, v) = \frac{\sum_{\mu} f_{\mu}(v^{-1})f_{\mu}(1) v^{k(n(\mu) - n(\mu^t))}}{\prod_{i=2}^n (1-v^{-i})}.$$ By [@op Theorem 8] the fake degrees satisfy $f_{\mu}(v^{-1}) = f_{\mu^t}(v^{-1})v^{n(\mu^t)-n(\mu)}.$ Combined with this implies that $$f_{\mu}(v^{-1}) f_{\mu}(1) v^{k(n(\mu)-n(\mu^t))} = f_{\mu^t}(v^{-1})f_{\mu^t}(1) v^{-(k-1)(n(\mu^t)-n(\mu))}.$$ Substituting this into gives the stated formula for $p(\overline{M(k)},\, v)$. {#poincare-S2C} As was true for Corollary \[poincare-S2A\], we need to slightly modify Proposition \[poincare-SC\] in order to compute the Poincaré series for $\overline{M(k)}$ under the ${\mathbf{E}}$-grading. Set $K=kn(n-1)/2$ and $\mathfrak{n}={\mathbb{C}}[{\mathfrak{h}}]_+$. Under the ${\mathbf{E}}$-grading there is an equality of Poincaré series $$\label{poincare-sss} \phantom{\frac{\displaystyle \int}{\displaystyle \int}} p(\overline{M(k)}, v) = v^{K} \frac{\sum_{\mu} f_{\mu}(1)f_{\mu}(v^{-1}) v^{-(k-1)(n(\mu) - n(\mu^t))}}{\prod_{i=2}^n (1-v^{-i})}= p(J^{k-1}\delta^k/\mathfrak{n}J^{k-1}\delta^k,\,v).$$ Equation \[diaggrad1\] continues to hold if we replace $em\delta e$ by $m\delta e$. Thus the argument of Corollary \[poincare-S2A\](1) combined with Proposition \[poincare-SC\] and the formula $M(k)=H_{c+k}\delta e B_{k-1,0}$ gives the first equality of . In order to obtain the second equality in , note that $p(J^{k-1}\delta^k/\mathfrak{n}J^{k-1}\delta^k,\,v) = v^Kp(J^{k-1} /\mathfrak{n}J^{k-1},\,v)$. Set $p(v)=p(J^{k-1} /\mathfrak{n}J^{k-1},\,v)$ and $q(v)=p(J^{k-1}/\mathfrak{m}J^{k-1},\,v)$, where $\mathfrak{m}={\mathbb{C}}[{\mathfrak{h}}]^{{W}}_+$ The Poincaré series $q(v)$ has been computed in Corollary \[gr\]. Since that series was obtained by specialising the bigraded Poincaré series $p(J^d, s, t)$ from Corollary \[bigr\], it follows immediately that $$p( v)\ = \ \frac{p({\mathbb{C}}[{\mathfrak{h}}],v)}{p({\mathbb{C}}[{\mathfrak{h}}]^{{W}},v)} \, \,q(v) \ = \ \frac{(1-v)^{n-1}}{\prod_{i=2}^{n}(1-v^i)} \, \, q(v) \ = \ \frac{q(v)}{[n]_v!}$$ where the final equality uses . Substituting these observations into Corollary \[gr\] gives the second equality in . Proof of proposition \[app-c-prop\] {#subsec-6.21C} ----------------------------------- We first show that the map $\theta:J^{k-1}\delta^ke \to M(k)$ is an isomorphism for all $k\geq 1$. This is analogue of Proposition \[grsameA\]. In that case, a purely formal argument showed that Proposition \[grsameA\] followed from . The same argument can be used, essentially without change, to show that the bijectivity of $\theta$ follows from . Combined with Lemma \[filter-injC\](ii) this says that $\operatorname{{\textsf}{ogr}}M(k) = \operatorname{{\textsf}{ogr}}\theta(J^{k-1}\delta^{k}e) = J^{k-1}\delta^ke$, as required. Index of Notation {#index .unnumbered} ================= [2]{} ${\mathbb{A}}^1$, $A^1$, alternating polynomials, ${\mathbb{A}}=\bigoplus {\mathbb{A}}^i$, $A=\bigoplus A^i$, , $\widehat{A} = \bigoplus_{i\geq j \geq 0} A^{i-j}$, ${\mathcal{B}}_1$, the tautological rank $n$ bundle, $B=\bigoplus B_{ij}$ for $B_{ij}= \prod_{a=u}^{v-1} Q_{a}^{a+1}$, canonical grading $W_\alpha $, $d(\mu)= \{ (i,j)\in {\mathbb{N}}\times {\mathbb{N}}: j < \mu_{i+1}\}$, $\delta=\prod_{s\in \mathcal{S}} \alpha_s$, $\Delta_c(\mu) $, the standard module, $\widehat{\Delta}_c(\mu) $, the graded standard module, dominance ordering on ${{\textsf}{Irrep}({{W}})}$, Dunkl-Cherednik representation $\theta_c$, ${\mathbf{E}}=\sum x_i\delta_i$, the Euler operator, $\operatorname{{\mathbf{E}}\text{-deg}}$, the Euler grading, $e, e_-$, trivial and sign idempotents, fake degrees $f_\mu$, $H_c$, the rational Cherednik algebra, ${\mathfrak{h}},{\mathfrak{h}}^*$, ${\mathfrak{h}^{\text{reg}}}$, $\mathbf{h} = \mathbf{h}_c= \frac{1}{2} \sum_{i=1}^{n-1} x_iy_i + y_ix_i \in H_c$, $\operatorname{{\mathbf{h}}\text{-deg}}$, the ${\mathbf{h}}$-grading, Hecke algebra ${\mathcal{H}_{q}}$, Hilbert schemes $\operatorname{Hilb^n{\mathbb{C}}^2}$,  $\operatorname{Hilb(n)}$, , $I_\mu$, monomial ideal for a partition $\mu$, ${\mathbb{J}}^1= {\mathbb{C}}[{\mathbb{C}}^{2n}]{\mathbb{A}}^1$, $J^1= {\mathbb{C}}[{\mathfrak{h}}\oplus{\mathfrak{h}}^*]A^1$, , $L_c(\mu)$, simple factor of $\Delta_c(\mu)$, ${\mathcal{L}}_1 ={\mathcal{O}}_{\operatorname{Hilb^n{\mathbb{C}}^2}}(1)$,  ${\mathcal{L}}={\mathcal{O}}_{\operatorname{Hilb(n)}}(1)$, , $\mathcal{O}_c$, category $\mathcal{O}$ for $H_c$, $\widetilde{{\mathcal{O}}}_c$, graded category $\mathcal{O}$ for $H_c$, $[n]_v! = (1-v)^{-n}\prod_{i=1}^n (1-v^i)$, $N(k)=B_{k0}eH_c$, $\overline{N(k)} = {\mathbb{C}}\otimes N(k)$, $\underline{N(k)} = N(k)\otimes {\mathbb{C}}$, $\operatorname{{\textsf}{ord}},\operatorname{{\textsf}{ogr}}$, order filtration and order gradation, ${\mathcal{P}}_1$, ${\mathcal{P}}$, the rank $n!$ Procesi bundles, , $p(M,v)$, Poincaré series, $p(V,s,t)$, bigraded Poincaré series, $p(M,v,{{W}})$, ${{W}}$-graded Poincaré series, $\operatorname{{\textsf}{qgr}}$, $\operatorname{{\textsf}{Qgr}}$, quotient categories, $Q^{c+1}_{c} = eH_{c+1}e_-\delta =e H_{c+1}\delta e$, $\mathbb{R}(n,l)= {\rm H}^0(\operatorname{Hilb^n{\mathbb{C}}^2}, {\mathcal{P}}_1\otimes {\mathcal{B}}_1^{l})$, $\rho_1: {\mathbb{X}}_n\to \operatorname{Hilb^n{\mathbb{C}}^2}$, $\rho: X_n\to \operatorname{Hilb(n)}$, , $\mathbb{S} = \oplus \mathbb{J}^i$, $S=\oplus {J}^i$, , $S_q = S_q(n,n)$, $q$-Schur algebra, $\mathcal{S}$, the reflections in ${{W}}$, $\sigma(r)$, the principal symbol of $r$, $\operatorname{{\textsf}{sign}}$, the sign representation of ${{W}}$, Specht module $Sp_q(\mu)$, $ \tau: \operatorname{Hilb^n{\mathbb{C}}^2}\rightarrow {\mathbb{C}}^{2n}/{{W}}$, $ \tau: \operatorname{Hilb(n)}\rightarrow {\mathfrak{h}}\oplus{\mathfrak{h}}^*/{{W}}$, $\operatorname{{\textsf}{triv}}$, the trivial representation of ${{W}}$, $U_c=eH_{c}e$, the spherical subalgebra, $U_c^-=e_-H_ce_-$, the anti-spherical subalgebra, ${{W}}=\mathfrak{S}_n$, the symmetric group, $\mathbb{X}_n$, $X_n$, isospectral Hilbert schemes, , [GGOR]{} E. Backelin and K. Kremnitzer, Quantum flag varieties, equivariant quantum $\mathcal{D}$-modules, and localization of quantum groups, math.QA/0401108. 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--- author: - 'Akash Kumar, Anand Sengupta, and Shashikiran Ganesh' title: Autonomous Dome for Robotic Telescope --- Introduction {#sec:intro} ============ Physical Research Laboratory operates a robotic 50cm telescope, Autonomous Telescope for Variability Studies(ATVS), at its observatory at Mount Abu (latitude : $24^\circ 39^m 9^s$ North, longitude: $72^\circ 46^m 47^s$ East, altitude : $1680 m$), India. The operating conditions are quite tough in terms of range in humidity, temperature and wind. The telescope is protected from the environment by a fibre glass dome manufactured by Sirius Observatories.\ Dome controller for autonomous operations ========================================= The telescope operates in the robotic mode[@Ganesh2013] and efforts are on way to fully automate it using Remote Telescope System (RTS2, <http://rts2.org/>). The observatory has over 200 clear nights in a year but needs to be completely closed and sealed during the Indian monsoon season (generally mid June to September) every year. Apart from this there are occasions when the skies become cloudy and/or wind conditions also increase drastically. For autonomous operations to succeed, therefore, we need to have the dome controller be very reliable with independent inputs/monitoring of the weather conditions so that the shutters could be closed automatically in case of bad weather.\ Humidity being a major cause of repeated failures of electronic boards, it has not been found feasible to use commercially available solutions for controlling the dome. Hence we decided to go for a locally engineered solution using cheap, general purpose, electronic boards available locally. The objective of this work was to make an autonomous dome controller for a robotic telescope using easily replaceable electronics. The controller is in charge of the dome functions like closing and opening of dome’s shutter, clockwise and counter-clockwise motion of the dome to track the telescope. It also keeps the record of the position of the dome and can be controlled by a Windows or Linux based computer using appropriate drivers.\ We built the dome controller using the ubiquitous Arduino boards. An Arduino is an open source physical computing platform based on a simple micro-controller board. It consists of a physical programmable circuit board. It can be programmed via the USB port of the computer using the Arduino IDE(integrated Development Environment) built with the Processing platform. The Arduino IDE includes support for various electronic components such as encoders and other sensors, relay boards etc. We used one Arduino board to control the dome’s shutter movement (opening and closing) and another one to control the dome rotation (clockwise or counter-clockwise). Both the boards communicate with each other using RF transceivers. This is necessitated because the dome rotation controller is connected to the PC and the shutter controller is connected via the dome controller. The shutter controller, powered by a battery, is mounted in a box on the dome and rotates with the dome. Thus wireless communication is a must. For monitoring the orientation of the dome we use an incremental rotary encoder which converts the angular motion of the dome into a series of digital pulses which encode the movement of the dome. To control the motion of motors of the shutter and dome we have used semiconductor relay boards with physical limit switches . These semiconductor relay boards are operated using digital pulses from the Arduino microcontroller board.\ Drive Development Phases {#sec:errors} ======================== ![View of Sirius Dome housing the PRL 50cm telescope at Mount Abu (left panel). Block diagram of dome and shutter control logic (right panel).[]{data-label="fig:simple"}](Dome_pic.PNG){width="\columnwidth"} The work was divided in different phases and work of all the phases was accomplished one by one and finally integrated to make the drive. The different phases were\ (i) Making a programmable controller of the dome’s shutter\ (ii) Making a programmable controller of the dome’s rotation\ (iii) Conversion of the dome’s motion into digital code\ (iv) Setting up a wireless connection between the dome’s rotation controller and the dome’s shutter\ (v) Making it compatible with RTS2.\ In the first phase, a mechanical controller circuit for the dome’s shutter was made using limit switches and was successfully tested. After that a programmable control circuit for the dome’s shutter was made using an arduino board and semiconductor relays to control the motors and was coded for opening, intermediate stop and closing operations of the shutter. Work on the position encoder was also started to record the dome’s position.\ Simultaneously, work on the rotation part of the dome was started and a programmable control circuit was made using another Arduino board and was coded for clockwise, counter-clockwise motion and for sensing home position. Then the position encoder was integrated with this controller and the code was modified accordingly. Work on the wireless communication between rotation and shutter controllers was started using radio frequency transceivers and a communication protocol was established and encoded for transferring the command to the controller boards.\ The electronic circuits have been tested at the observatory and software coding has been developed for synchronizing the dome orientation with the azimuth being pointed by the telescope. We plan to integrate the shutter control board with independent cloud, rain and wind sensors to allow quick closure of the shutters independent of any computer control / manual intervention.\ [*Acknowledgement*]{} [**The first author would like to acknowledge support received from the organizers towards local hospitality which enabled his participation in the workshop. His travel to Malaga was funded by IIT Gandhinagar. Work at PRL is funded by the Dept of Space, Govt. of India. We thank colleagues at PRL for their support of this work.**]{} Ganesh, S., Baliyan, K. S., Chandra, S., Joshi, U. C., Kalyaan, A., Mathur, S. N, Automated telescope for variability studies, 31st ASI Meeting, ASI Conference Series, 2013, Vol. 9, pp 99 Edited by Pushpa Khare & C. H. Ishwara-Chandra

--- abstract: 'We consider the pair production of color triplet spin–$\frac{3}{2}$ quarks and their subsequent decays at the LHC. This particle, if produced, will most likely decay into top quark and gluon, bottom quark and gluon, or a light quark jet and gluon, depending on the quantum number of the spin–$\frac{3}{2}$ particle. This would lead to signals with $t\bar{t}jj$, $b\bar{b}jj$, or $4j$ in the final states. We present a detailed analysis of the signals and backgrounds at $\sqrt{s}= 7$, $8$ and $14$ TeV and show the reach for such particles by solving for observable mass values for the spin–$\frac{3}{2}$ quarks through its decay products.' author: - 'Duane A. Dicus$^{1,}$[^1], Durmus Karabacak$^{2,}$[^2], S. Nandi$^{2,}$[^3], and Santosh Kumar Rai$^{2,}$[^4]' title: --- \[sec:intro\]Introduction ========================= The Standard Model (SM) of particle physics has been extensively tested by many independent experiments and the results are in agreement with the predictions of the SM. The Large Hadron Collider (LHC) at CERN is designed to explore the energy and intensity frontier which could show physics beyond the SM. The initial results released by the ATLAS and CMS experiments not only confirm the predictions of the SM, including the discovery of the Higgs boson [@cms:2012gu; @atlas:2012gk], but have also started pushing the energy scale required by new physics models including exotic fermions and gauge bosons which are not present in the SM. Among exotic fermions one possible new particle is a spin–$\frac{3}{2}$ excitation of quarks. We will assume this spin–$\frac{3}{2}$ particle to be a color triplet like an ordinary quark and consider the pair production and the decay of such an exotic particle at the LHC. It is not outside the realm of possibility that a spin–$\frac{3}{2}$ quark could exist as a fundamental particle. We could also have spin–$\frac{3}{2}$ bound states of ordinary quarks with gluons or the Higgs boson. There are also theoretical models in which spin–$\frac{3}{2}$ quarks arise as bound states of three heavy quarks for sufficiently strong Yukawa couplings [@tay]. The masses of these bound states are typically expected to be a few TeV. A heavy spin–$\frac{3}{2}$ quark could also exist as the lightest Regge recurrences of light spin–$\frac{1}{2}$ quarks or as Kaluza-Klein modes in string theory if one or more of the compactification radii is of the order of the weak scale rather than the Planck scale and such weak compactification in the framework of both string theory and field theory has been popular [@sure]. In this work we restrict ourselves to the collider production of point-like spin–$\frac{3}{2}$ color triplet quarks. The production of spin–$\frac{3}{2}$ quarks by hadronic collisions has been previously considered by Moussallam and Soni [@mous] and by Dicus, Gibbons, and Nandi [@Dicus:1998yc]. There are several studies on production of spin–$\sp$ fermions at lepton colliders [@Walsh:1999pb; @Almeida:1995yp; @Cakir:2007wn] and also the virtual effects of such particles on $t\bar{t}$ production [@Stirling:2011ya]. Our paper is organized as follows. In Section \[sec:frules\], we give the Feynman rules relevant for the production of spin–$\frac{3}{2}$ quarks. In Section \[sec:cross\], we give the explicit analytic formulae for the squares of the amplitude, various subprocess cross sections and total production cross sections. In Section \[sec:signals\], we present the analysis of the signal of spin–$\frac{3}{2}$ particle decaying into light jets or into heavy flavor modes. Here we make the physics analysis of relevant background and signal for three different decay scenarios. Section \[sec:summary\] contains a summary. \[sec:frules\]Feynman Rules for Spin–$\frac{3}{2}$ Particles ============================================================ The Lagrangian and the equations of motion for a free spin–$\sp$ particle of mass $M$ can be written as [@rari; @mold] $$\label{eq:lag} \mathcal{L} = \bar{\psi}_{\alpha} \Lambda^{\alpha\beta}\psi_\beta$$ $$\Lambda^{\alpha\beta}\psi_\beta=0$$ where $$\Lambda_{\alpha\beta} =(i\slashed \partial- M)g_{\alpha\beta} + iA(\gamma_\alpha \partial_\beta+\gamma_\beta \partial_\alpha) +\frac{iB}{2}\gamma_\alpha \slashed \partial \gamma_\beta + CM\gamma_\alpha \gamma_\beta $$ with $B \equiv 3A^2+2A+1$ and $C \equiv 3A^2+3A+1$. The parameter $A$ is arbitrary except that $A \not=-\frac{1}{2}$. The field $\psi_\alpha$ satisfies the subsidiary conditions $$\begin{aligned} \gamma^\alpha \psi_\alpha &= 0 \label{R1}\\ \partial^\alpha \psi_\alpha & =0. \label{R2}\end{aligned}$$ The propagator $S_{\alpha\beta}$ is given by $$\begin{split} S_{\alpha\beta}(p)=&\frac{1}{\slashed p -M} \bigg[g_{\alpha\beta} - \frac{1}{3}\gamma_\alpha \gamma_\beta - \frac{2}{3 M^2}p_\alpha p_\beta + \frac{1}{3M}(p_\alpha \gamma_\beta - p_\beta \gamma_\alpha)\bigg]\\ & + \Bigg\{\frac{a^2}{6M^2} \slashed p \gamma_\alpha \gamma_\beta - \frac{ab}{3M}\gamma_\alpha \gamma_\beta + \frac{a}{3M^2}\gamma_\alpha p_\beta + \frac{ab}{3M^2}\gamma_\beta p_\alpha \Bigg\} \end{split}$$ where $$\begin{aligned} a =\frac{A+1}{2A+1} \quad \mbox{and} \quad b =\frac{A}{2A+1} ~~ .\end{aligned}$$ From Eq.(\[R1\]) and Eq.(\[R2\]) the terms depending on the parameter $A$ in the propagator vanish on the mass shell. A redefinition of the spin–$\frac{3}{2}$ field $\psi_\alpha$ allows one to remove the $A$ dependent terms in the propagator [@pasc]. However, in our analysis we have kept the $A$ dependence in the propagator and in the interaction vertices and used the disappearance of A as a check on our calculations. The minimal substitution in Eq.(\[eq:lag\]) gives the interaction of spin–$\frac{3}{2}$ quarks with gluon and photon fields, $$\mathcal{L}_I = g \bar{\psi}_\alpha \bigg( \frac{B}{2} \gamma^\alpha \gamma^\mu \gamma^\beta + A g^{\alpha\mu}\gamma^\beta + A \gamma^\alpha g^{\mu\beta} + g^{\beta\alpha}\gamma^\mu \bigg) T_a \psi_\beta A_\mu^a\,\,,$$ where $g$ is the coupling constant, $T_a$’s are the group generators and $A_\mu^a$ are the gauge fields. For on-shell particles only the last term is nonzero. \[sec:cross\]Calculation of Cross Sections ========================================== In this section we provide the expressions necessary for the process, $$\label{eq:ppQQ} p p \rightarrow Q_{3/2} \bar{Q}_{3/2} + X\,\,$$ where $Q_{3/2}$ is the spin–$\sp$ quark. There are two subprocesses which contribute, $q\bar{q}$ annihilation and gluon fusion. The Feynman diagrams are shown in Fig.\[feyndiag\] where *(a)* represents the $q\bar{q}$ annihilation while *(b)–(d)* represent the $t$,$u$ and $s$-channel contributions of the gluon fusion subprocess respectively. Just as for top quark production the largest contribution to the production of spin–$\sp$ at LHC energies is through gluon fusion. ![*The leading order (LO) Feynman diagrams for the pair production of spin–$\sp$ quarks through (a) $q\bar{q}$ and $gg$ initial states in (b) $t$-channel, (c) $u$-channel and (d) $s$-channel.*[]{data-label="feyndiag"}](Fig1.eps){width="6.6in" height="1.1in"} The t-channel amplitude shown in Fig.\[feyndiag\] is given by $$\label{eq:mt} \begin{split} \mathcal{M}_t =~ & g_s^2\bar{u}_\rho (p)\big(g^{\rho\alpha} \gamma^\mu + A g^{\mu \rho} \gamma^\alpha \big) T_a \epsilon^{a}_\mu(k) \\ & \Bigg\{\frac{1}{\slashed Q -M}\Bigg[g_{\alpha\beta} - \frac{1}{3}\gamma_\alpha \gamma_\beta - \frac{2}{3 M^2}Q_\alpha Q_\beta + \frac{1}{3M}\big(Q_\alpha \gamma_\beta - Q_\beta \gamma_\alpha \big)\Bigg]\\ & + \frac{a^2}{6 M^2}\slashed Q \gamma_\alpha \gamma_\beta - \frac{ab}{3 M}\gamma_\alpha \gamma_\beta + \frac{a}{3M^2}\gamma_\alpha Q_\beta + \frac{ab}{3M^2}\gamma_\beta Q_\alpha \Bigg\}\\ & \big(g^{\sigma\beta} \gamma^\nu + A g^{\nu\sigma} \gamma^\beta \big) T_b \epsilon_\nu^b(k') v_\sigma(p')\quad , \end{split}$$ while the $u$ channel amplitude has a similar form due to crossing symmetry, $$\begin{split} \mathcal{M}_u =~ & g_s^2\bar{u}_\rho (p)\big(g^{\rho\beta} \gamma^\nu + A g^{\rho \nu} \gamma^\beta \big) T_b \epsilon^{b}_\nu(k') \\ & \Bigg\{\frac{1}{\slashed Q^{'} -M}\Bigg[g_{\beta\alpha} - \frac{1}{3}\gamma_\beta \gamma_\alpha - \frac{2}{3 M^2}Q^{'}_\beta Q^{'}_\alpha + \frac{1}{3M}\big(Q^{'}_\beta \gamma_\alpha - Q^{'}_\alpha \gamma_\beta \big)\Bigg]\\ & + \frac{a^2}{6 M^2}\slashed Q^{'} \gamma_\beta \gamma_\alpha - \frac{ab}{3 M}\gamma_\beta \gamma_\alpha + \frac{a}{3M^2}\gamma_\beta Q^{'}_\alpha + \frac{ab}{3M^2}\gamma_\alpha Q^{'}_\beta \Bigg\}\\ & \big(g^{\sigma\alpha} \gamma^\mu + A g^{\sigma\mu} \gamma^\alpha \big) T_a \epsilon_\mu^a(k) v_\sigma(p')\,, \end{split}$$ where $\slashed{Q}=\slashed{p}-\slashed{k}$, $\slashed{Q}'=\slashed{k}-\slashed{p}'$. The amplitude for the $s$-channel contribution has a much simpler form because the $A$ dependence goes away for the spin–$\sp$ particles produced on-shell, $$\label{eq:ms} \begin{split} \mathcal{M}_s =& -ig_s^2 f_{abc} \bar{u}^\rho(p)\gamma^\alpha T^c v_\rho(p')\frac{1}{\hat{s}} \epsilon_{\mu}^a(k) \epsilon_{\nu}^b (k') \\ & \bigg[g^{\mu\alpha}(2k+k^{'})^{\nu} - g^{\alpha\nu}(2k^{'}+k)^{\mu} + g^{\nu\mu}(k^{'}-k)^\alpha\bigg]~. \end{split}$$ The $\epsilon^a$’s represent the gluon fields while the spin–$\sp$ particles are denoted by the $u$ and $v$ spinors carrying Lorentz indices. From the expressions $\mathcal{M}_t$ and $\mathcal{M}_u$ we see that off-shell spin–$\sp$ particle exchange leads to an explicit dependence on the contact parameter $A$. Although we have this dependence in the amplitudes, the final results should be independent of $A$. Indeed, we find that this dependence goes away not only from the final total result but also from each individual contribution such as $\Sigma |\mathcal{M}_t|^2$ or $\Sigma |\mathcal{M}_u|^2$ or the cross terms. This was verified by calculating the amplitude squares and all interference terms in both axial-gauge and Feynman-gauge. Using Eqs.(\[eq:mt\]-\[eq:ms\]), the full spin and color averaged matrix amplitude square for the gluon-gluon subprocess is $$\begin{aligned} \sum |\mathcal{M}|^2_{GG} &=& \frac{g_s^4}{1944}\left[-2106 - \frac{5832 M^2}{\sh}+\frac{112 \sh}{M^2}-\frac{272 \sh^2}{M^4} + \frac{39 \sh^3}{M^6}-\frac{2592 M^4 \sh^2}{u'^2 t'^2} - \frac{48 \sh^4}{u'^2t'^2} \right. \nonumber \\ &+& \left. \frac{5832 M^4}{u' t'}+\frac{2592 M^2 \sh}{u' t'} + \frac{539 \sh^2}{u' t'}+\frac{4 \sh^3}{M^2 u' t'} + \frac{33 \sh^4}{M^4 u' t'}+\frac{521 u' t'}{M^4} + \frac{2916 u' t'}{\sh^2} \right. \nonumber \\ &-& \left. \frac{121 \sh u' t'}{M^6}+\frac{4 \sh^2 u' t'}{M^8} - \frac{8 u'^2 t'^2}{M^8}\right] \label{eq:matggsq}\end{aligned}$$ where $t'$ and $u'$ are related to the usual definitions of the Mandelstam variables $t$ and $u$ in the parton center-of-mass frame as $t'=t-M^2$ and $u'=u-M^2$. The total cross section for the gluon-gluon subprocess is then $$\label{eq:gg2QQbar} \begin{split} \hat{\sigma}(gg\rightarrow Q_{3/2} \bar{Q}_{3/2}) =&\frac{\pi\alpha_s^2}{116640~ \hat{s}} \Bigg\{ 60 ~\ln\bigg(\frac{1+\beta}{1-\beta}\bigg) \bigg[66 y^2+8y+886 + 5184\frac{1}{y} +1296\frac{1}{y^2}\bigg] \\ &+\beta\bigg[24 y^4+1178y^3-13626 y^2+11380 y -97200 -602640\frac{1}{y}\bigg]\Bigg\} \end{split}$$ where $\alpha_s \equiv g^2_s/4\pi$ , $y\equiv \hat{s}/M^2$ and $\beta \equiv \sqrt{1-4/y}$. This expression for the total subprocess cross section agrees with Ref.[@Dicus:1998yc], but disagrees with Ref.[@mous]. However Ref.[@mous] has an algebraic error which, when corrected, gives agreement with Eq.(\[eq:gg2QQbar\]) [@mous2]. The pair production of the spin–$\sp$ quarks will also have contributions coming from the amplitude for the quark-antiquark annihilation subprocess which is given by $$\label{eq:mqq} \mathcal{M}_{q\bar{q}}=-i g^2\frac{1}{\hat{s}}\bar{u}^{\rho}(p)T_a\gamma^\mu v_\rho(p^{'}) \bar{u}(k)\gamma_\mu v(k')~.$$ The spin and color averaged matrix amplitude square for the quark-antiquark process is $$\begin{aligned} \sum |\mathcal{M}_{q\bar{q}}|^2 &=& \frac{4g_s^4}{81 M^4 \sh^2} \left[36 \sh M^6 - 2 \sh M^2 (\sh+2t')^2 + \sh^2 (\sh^2+2 \sh t'+2 t'^2) \right. \nonumber \\ &&\left. + 2 M^4 (\sh^2+18 \sh t'+18 t'^2)\right] \label{eq:matqqsq}\end{aligned}$$ and the total cross section for this subprocess is $$\label{eq:qq2QQbar} \hat{\sigma}(q\bar{q}\rightarrow Q_{3/2} \bar{Q}_{3/2})= \frac{\pi \alpha_s^2}{81 \hat{s}}\beta \Bigg[\frac{8}{3}y^2-\frac{16}{3}y-\frac{16}{3}+96\frac{1}{y}\Bigg]~.$$ To obtain the production cross section we convolute Eq.(\[eq:gg2QQbar\]) and Eq.(\[eq:qq2QQbar\]) with the parton distribution functions (PDF). $$\label{prodcros} \begin{split} \sigma({pp \rightarrow Q_{3/2} \bar{Q}_{3/2}+X})=& \left\{ \sum_{i=1}^{5} \int dx_1 \int dx_2 ~\mathcal{F}_{q_i}(x_1,Q^2)\times \mathcal{F}_{\bar{q}_i}(x_2,Q^2) \times \hat{\sigma}(q_i\bar{q}_i\rightarrow Q \bar{Q}) \right\} \\ & + \int dx_1 \int dx_2 ~\mathcal{F}_g (x_1,Q^2) \times \mathcal{F}_g (x_2,Q^2) \times \hat{\sigma}(gg\rightarrow Q \bar{Q}), \end{split}$$ where $\mathcal{F}_{q_i}$, $\mathcal{F}_{\bar{q}_i}$ and $\mathcal{F}_{g}$ represent the respective PDF’s for partons (quark, antiquark and gluons) in the colliding protons, while $Q$ is the factorization scale. In Fig.\[prodfig\] we plot the leading-order production cross section for the process $p p \rightarrow Q_{3/2} \bar{Q}_{3/2}+X$ at center of mass energies of 7, 8 and 14 TeV as a function of the spin–$\sp$ quark mass $M$. ![*The production cross sections for $p p \rightarrow Q_{3/2} \bar{Q}_{3/2}+X$ at the LHC as a function of spin–$\sp$ quark mass $M$ at center-of-mass energies, $E_{CM} = 7, 8$ and $14$ TeV. We have chosen the scale as $Q=M$, the mass of the spin–$\sp$ quark.*[]{data-label="prodfig"}](cs-all.eps){width="3.4in"} We set the factorization scale $Q$ equal to $M$, and used the [CTEQ6$\ell$1]{} parton distribution functions [@Pumplin:2002vw]. This production cross section is larger than any spin-$\frac{1}{2}$ colored fermion of same mass such as a fourth-generation quark or an excited quark. This is not unexpected, as the cross section given in Eq.(\[eq:gg2QQbar\]) grows with energy as $\hat{s}^3$ which violates unitarity at high energies. We assume that the interactions given in Sec.II represent an effective interaction such that, at higher energies, higher order contributions will be important and the cross section will be damped by some form factors dependent on the scale of the new physics. Some explicit ways to address this have been discussed in Refs.[@Hassanain:2009at; @Stirling:2011ya]. There is some natural enhancement, however, because the particles carry additional spin degree of freedom when compared to spin-$\frac{1}{2}$ fermions. We find that for the 7 TeV run of the LHC, the pair production of a colored spin–$\sp$ exotic fermion has cross sections in excess of a few hundred femtobarns (fb) for masses as high as 600 GeV. At the current run of the LHC, with a center of mass energy of 8 TeV, cross sections in excess of 100 fb are obtained for masses up to 750 GeV. Therefore a strong case can be made to search for such exotics in the current and upcoming LHC data, just as that being done for coloron like particles. Any search for these exotics would crucially depend on how the particle decays and what is produced in the final state so let us now discuss how these particles will decay. Higher dimension-five operators would lead to interactions between the massive spin–$\sp$ states and the spin–$\frac{1}{2}$ states such as [@Stirling:2011ya] $$\label{dim5} \mathcal{L}_{dim-5} = i\frac{g_s}{\Lambda}\bar{\psi}_\alpha\left( g^{\alpha\beta} + A \gamma^\alpha \gamma^\beta \right) \gamma^\nu T^a \frac{(1\pm \gamma_5)}{2} \xi F^a_{\beta\nu} + H.C.$$ where $F^a_{\beta\nu}$ represents the field tensor of the gauge field and $\xi$ is the spin–$\frac{1}{2}$ fermion. $\Lambda$ determines the scale of some new physics which, for example, could be the scale which remedies the unitarity violation seen in the cross section. Note that large values of scale $\Lambda$ would imply that the interaction strength weakens. We will assume that the colored spin–$\sp$ will decay promptly to a gluon and a spin–$\frac{1}{2}$ fermion (which in our case is a SM quark) with 100% branching probability. So there is no need for us to calculate a branching ratio and thus no need to use Eq.(\[dim5\]). The only thing we need is for $\Lambda$ to be large enough such that Eq.(\[dim5\]) does not change the production cross section significantly. Thus if the quantum numbers dictate a decay to a particular family of quarks we can have three different scenarios corresponding to the decay of the spin–$\sp$ particle to one of the three SM quark families, a light SM quark and a gluon ($Q_{3/2}\to q g$), or a heavy quark and a gluon ($Q_{3/2}\to b g$ or $Q_{3/2}\to t g$). We will now analyze each of these signals and the corresponding SM background representative of the type of decay. \[sec:signals\] Signals at the LHC ================================== \[sec:4j\]Four jet final state ------------------------------ As mentioned above we assume that the spin–$\sp$ colored fermion can decay to a SM quark and a gluon. If the quark happens to belong to the first two families of the SM quarks, then these quarks will hadronize and form jets, as will the gluons, leading to four jets in the final state. All the jets will carry large transverse momenta ($p_T$) as they are byproducts of a heavy particle decay. However, with final states only comprised of jets the signal will be overwhelmed by the huge QCD background which would also be characterized by high $p_T$ jets. Therefore to extract the signal from the huge -- -------- ------- ------- ------- ------- ------ -- 500 600 700 800 900 1000 326. 124. 48.6 18.8 7.2 2.8 134. 51.9 24.9 11.5 5.1 2.1 65.2 21.0 10.1 5.7 3.0 1.5 194. 61.2 27.6 15.1 8.1 4.1 106. 32.2 12.6 6.6 4.1 2.4 58.1 17.6 6.5 3.0 1.8 1.2 4842. 1549. 569.4 242.2 120.8 69.7 3271. 1074. 399.7 167.6 79.5 43.3 2184.3 746.9 280.8 117.6 54.9 28.4 -- -------- ------- ------- ------- ------- ------ -- : *The signal cross section for the $4j$ final state coming from the pair production of spin–$\sp$ quarks of mass M with $\sqrt{s}=7, 8~\text{and}~14$ TeV as the cut on the transverse momenta of the jets is varied. Also shown is the QCD background which has been estimated using [`M`adgraph 5]{} [@mad5].*[]{data-label="tab:4j"} background, one needs to devise some specific conditions on the kinematics of the final state particles and also put the focus on to the uniqueness of the signal coming from the new particles. The most obvious feature that the signal will exhibit is a peak in the invariant mass distribution of a pair of jets coming from the decay of the spin–$\sp$ quark. In comparison, the QCD background would trail off for high invariant mass values of the dijet. This signal could be mimicked by other new physics scenarios where new colored particles produced in pairs decay hadronically to dijets. In fact the CMS Collaboration has made an initial analysis on such particles at LHC with $\sqrt{s}=7$ TeV using 2.2 fb$^{-1}$ of integrated luminosity and put a lower limit on the mass of coloron-type particles to be 580 GeV [@cmsnote]. We have used the CMS analysis to obtain an effective lower limit of $M \sim 490$ GeV on the mass of a spin–$\sp$ quark which decays into a light quark and gluon. Note that the bound is lower than the coloron mass bound because for similar masses the pair production cross section for spin–$\sp$ quarks is smaller than the pair production of colorons. The search strategies at CMS did not include stronger cuts on the $p_T$ of the jets, which should further suppress the large QCD background for the $4j$ final state. In Table \[tab:4j\] we summarize the signal cross section with different set of $p_T$ cuts on the jets in the final state and also highlight how the cuts affect the QCD background. In addition to the $p_T$ cut, the jets must lie within the rapidity gap of $|\eta_j|<2.5$ and the jets are isolated in the $(\eta,\phi)$ plane satisfying $\Delta R_{jj}> 0.5$, where $\Delta R$ is defined as $\Delta R=\sqrt{(\Delta\eta)^2+(\Delta\phi)^2}$. A minimum cut on the invariant mass of each dijet pair has been also implemented for both signal and background, given by $M_{jj}>10$ GeV. As one would expect, for stronger requirements on the jet $p_T$, the QCD background begins to fall off rapidly. The signal is affected more by the $p_T$ cuts for smaller values of the spin–$\sp$ quark mass because the jets have higher $p_T$ if they come from the decay of heavier spin–$\sp$ quark. The numbers in Table \[tab:4j\] demonstrate this, as stronger cuts are shown to effect the background more by suppressing it at times by more than 90% which improves the signal to background ratio significantly. Thus with 2.2 fb$^{-1}$ integrated luminosity(L) at $\sqrt{s}=7$ TeV, and a $p_T$ cut of 200 GeV, we find that the ratio of signal to square root of background, $S/\sqrt{B}\equiv\,L\sigma_s/\sqrt{L\sigma_b}$ is about $4.4$ for a spin–$\sp$ quark with mass $M=500$ GeV which suggests a significant improvement in the mass reach for such exotic particles. At $\sqrt{s}=7$ TeV, the stronger cuts are not helpful as they also suppress the signal by a large amount. It is worth pointing out here that our analysis, done at the leading-order parton level, does not correspond to the exact numbers seen at the experiments as no detector level effects have been included. However, after accounting for the suppression in events due to hadronization and fragmentation effects, detector efficiencies and acceptance, the strong cuts would still help in improving the mass reach for colored particles which are pair produced and decay hadronically to a pair of jets. \[sec:bbjj\]Final state with two $b$–jets and two light jets ------------------------------------------------------------ In this section we consider the scenario where the spin–$\sp$ quark quantum numbers dictate its decay to a bottom quark and a gluon so the pair produced spin–$\sp$ quarks lead to a final state with two $b$-jets and two light jets ($2b2j$) all carrying large transverse momenta. This final state is already included in the $4j$ analysis when no heavy flavor tagging is applied on the events. However, recent analysis at both ATLAS and CMS have shown that a very high efficiency for $b$-tagging may be obtained [@atlasbtag; @cmsbtag]. Dependent on the transverse momenta of the b-jets, the efficiencies could be as high as 70% for jets with $p_T> 100$ GeV. So even though we lose part of the events due to limited efficiencies, the QCD background is reduced significantly as the $b$-jet production forms a small subset of the full $4j$ background. On the other hand, the pair production cross section for the spin–$\sp$ quark remains unaffected even if its quantum numbers correspond to a bottom quark. Therefore, the signal events will benefit from such flavor tagging and improve the signal to background ratio. ![*The invariant mass distribution of the leading jet and leading $b$-jet for the SM background and the superposed signal coming from the production of spin–$\sp$ quarks with the SM background. Distributions are shown for three different values of mass of the spin–$\sp$ quark and at different center of mass energies, viz. (a) $M=500$ GeV; $\sqrt{s}=7$ TeV, (b) $M=600$ GeV; $\sqrt{s}=8$ TeV and (c) $M=1$ TeV; $\sqrt{s}=14$ TeV.* []{data-label="fig:MinvJ1B1"}](Mj1b1_500.eps "fig:"){width="2.1in"} ![*The invariant mass distribution of the leading jet and leading $b$-jet for the SM background and the superposed signal coming from the production of spin–$\sp$ quarks with the SM background. Distributions are shown for three different values of mass of the spin–$\sp$ quark and at different center of mass energies, viz. (a) $M=500$ GeV; $\sqrt{s}=7$ TeV, (b) $M=600$ GeV; $\sqrt{s}=8$ TeV and (c) $M=1$ TeV; $\sqrt{s}=14$ TeV.* []{data-label="fig:MinvJ1B1"}](Mj1b1_600.eps "fig:"){width="2.1in"} ![*The invariant mass distribution of the leading jet and leading $b$-jet for the SM background and the superposed signal coming from the production of spin–$\sp$ quarks with the SM background. Distributions are shown for three different values of mass of the spin–$\sp$ quark and at different center of mass energies, viz. (a) $M=500$ GeV; $\sqrt{s}=7$ TeV, (b) $M=600$ GeV; $\sqrt{s}=8$ TeV and (c) $M=1$ TeV; $\sqrt{s}=14$ TeV.* []{data-label="fig:MinvJ1B1"}](Mj1b1_1000.eps "fig:"){width="2.1in"} In Fig.\[fig:MinvJ1B1\] and Fig.\[fig:MinvJ2B1\] we plot the invariant mass distribution of the two final state jets with the leading $b$-jet. Note that in the analysis for a resonant particle, the resonance is not seen in the two $b$-jet invariant mass but is seen in the light quark jet and $b$-jet. This will reduce the QCD background significantly. We plot the invariant mass distribution for three different values of the spin–$\sp$ quark mass and at three different center of mass energies. Both the light quark jets and the $b$-jets are ordered according to their $p_T$ and we call the leading $b$-jet as $b_1$ and the subleading $b$-jet as $b_2$ with similar notation for the light quark jets. The events used in the plots presented in Fig.\[fig:MinvJ1B1\] and Fig.\[fig:MinvJ2B1\] for both the signal and the background satisfy the following kinematic selection cuts: - Both the light quark jets and $b$-jets have a minimum transverse momenta $p_T>150$ GeV and lie within the rapidity gap of $|\eta|<2.5$. - To resolve the final states in the detector they should be well separated. To achieve this we require that they satisfy $\Delta R_{ij} > 0.7$ with $i,j$ representing the $b$-jets and the light quark jets. As above the variable $\Delta R_{ij}$ defines the separation of two particles in the ($\eta,\phi$) plane of the detector with $\Delta R_{ij}=\sqrt{(\eta_i-\eta_j)^2+(\phi_i-\phi_j)^2}$, where $\eta$ and $\phi$ represent the pseudo-rapidity and azimuthal angle of the particles respectively. - To suppress large contributions of gluon splitting into two ($b$) jets we demand that the minimum invariant mass of two ($b$)-jets satisfy $M^{inv}_{ij} > 10$ GeV. - We also demand that there are no additional jets with $p_T > 150$ GeV. ![*The invariant mass distribution of the sub-leading jet and leading $b$-jet for the SM background and the superposed signal coming from the production of spin–$\sp$ quarks with the SM background. The choices of $M$ and $\sqrt{s}$ are the same as in Fig.\[fig:MinvJ1B1\].* []{data-label="fig:MinvJ2B1"}](Mj2b1_500.eps "fig:"){width="2.1in"} ![*The invariant mass distribution of the sub-leading jet and leading $b$-jet for the SM background and the superposed signal coming from the production of spin–$\sp$ quarks with the SM background. The choices of $M$ and $\sqrt{s}$ are the same as in Fig.\[fig:MinvJ1B1\].* []{data-label="fig:MinvJ2B1"}](Mj2b1_600.eps "fig:"){width="2.1in"} ![*The invariant mass distribution of the sub-leading jet and leading $b$-jet for the SM background and the superposed signal coming from the production of spin–$\sp$ quarks with the SM background. The choices of $M$ and $\sqrt{s}$ are the same as in Fig.\[fig:MinvJ1B1\].* []{data-label="fig:MinvJ2B1"}](Mj2b1_1000.eps "fig:"){width="2.1in"} A clear resonance is observed in both the $M_{j_1b_1}$ and $M_{j_2b_1}$ distributions in the bin corresponding to the spin–$\sp$ quark mass. It is interesting to observe that both the leading and subleading jet forms a resonance in the invariant mass with the leading $b$-jet. As we have ordered the jets according to their $p_T$, their respective points of origin become immaterial and therefore both the combinations show an invariant mass peak. However, the subleading jet gives the more pronounced peak with the leading $b$-jet which seems to make it the favorable combination. We have used three different values for the spin–$\sp$ quark mass, $M=500$ GeV, $600$ GeV, and $1$ TeV at $\sqrt{s}=7,8$ and $14$ TeV respectively. As the larger center of mass energy gives a bigger pair production cross section (Fig.\[prodfig\]), we choose larger values for the spin–$\sp$ quark mass for higher $\sqrt{s}$ to show that the signal will be significantly greater even for the larger values of mass which are inaccessible with lower center of mass energies. We use the same set of kinematic cuts for the analysis done at $\sqrt{s}=7$ and $8$ TeV. However, as in the case of $4j$ final states, stronger cuts on the transverse momenta of both the $b$-jet and the light quark jet would be useful in improving the signal to background ratio. We therefore modify the cut on transverse momenta and demand that $p_T>400$ GeV for the jets at $\sqrt{s}=14$ TeV. For our analysis of both the signal and background, we have considered a $b$-tagging efficiency of 50% while the mistag rate for light quark jets tagged as $b$-jets is taken as 1%. Both the $b$-tag efficiency and the mistag rates are dependent on the transverse momenta ($p_T$) and rapidity ($\eta$) and our choices do not include these effects. To do such detailed analysis one would also need to include various other systematics including showering and hadronization effects at the LHC and detector-level simulations which is beyond the scope of this work. So we assume that our choice for the efficiencies and the mistag rate is a good approximation when averaged over the entire range of transverse momenta for the quarks within the allowed rapidity gap. With the above set of cuts the signal cross section for different values of the spin–$\sp$ mass along with the SM background are shown in Table \[tab:2b2j\]. When compared with $4j$ analysis, the reach for spin–$\sp$ quarks in the $2b2j$ channel is found to be improved significantly. For example, for $M=1$ TeV with an integrated luminosity of 10 fb$^{-1}$ for $\sqrt{s}=14$ TeV and a $p_T> 400$ GeV cut on the jets, the $S/\sqrt{B}\simeq 4$ in the $4j$ final state while it becomes $S/\sqrt{B}\simeq 15$ in the $2b2j$ final state. ------------------- ------- -------------------- ------- ------ ------ ------ ------- $pp\to 2b2j$ SM background (fb) 500 600 700 800 900 1000 $\sqrt{s}=7~TeV$ 182.5 55.0 17.6 5.9 2.1 0.7 351.3 $\sqrt{s}=8~TeV$ 403.0 124.8 41.6 14.7 5.5 2.1 608.9 $\sqrt{s}=14~TeV$ 584.8 275.4 123.4 57.6 29.7 17.1 12.9 ------------------- ------- -------------------- ------- ------ ------ ------ ------- : *The signal cross section for the $2b2j$ final state at LHC with $\sqrt{s}=7,8~\text{and}~14$ TeV for different choices of the mass $M$. Note that the $p_T$ cut on the jets is 150 GeV for $\sqrt{s}=7~\text{and}~8$ TeV while it is 400 GeV for $\sqrt{s}=14$ TeV. We have included a $b$-tag efficiency $\epsilon_b=0.5$ in cross sections.*[]{data-label="tab:2b2j"} \[sec:ttjj\]Final state with $t\bar{t}$ and two light jets ---------------------------------------------------------- Finally we specialize to the case where the spin–$\sp$ quark carries quantum numbers similar to the top quark and therefore decays to a top quark and a gluon. This would lead to a $t\bar{t}$ final state with two additional jets with large transverse momenta through the process chain given by $pp\,\longrightarrow\,Q_{3/2}\bar{Q}_{3/2}\,\longrightarrow\,t\bar{t}gg$. This would be a very nice signal which would not only provide a strong hint for physics beyond the SM but would also effect the inclusive top quark pair production if the additional jets are not triggered upon. However, as the production cross section of the heavier $Q_{3/2}$ particles are small compared to the pair production of $t\bar{t}$ (about 10% of $\sigma_{t\bar{t}}$ for $M=400$ GeV) the new physics signal is more --------------------- --------- --------------- ---------- --------- $pp\to t\bar{t}jj$ SM background ($p_T^j > 100$ GeV) 500 800 1000 $\sqrt{s}=7~TeV$ 1.11 pb 21.7 fb 2.4 fb 2.12 pb $\sqrt{s}=8~TeV$ 2.38 pb 53.4 fb 6.8 fb 3.55 pb $\sqrt{s}=14~TeV$ 49.4 pb 1.46 pb 249\. fb 24.7 pb --------------------- --------- --------------- ---------- --------- : *The signal cross section for the $t\bar{t}jj$ final state coming from the pair production of spin–$\sp$ quarks with $\sqrt{s}=7, 8~\text{and}~14$ TeV for different choices of the mass $M$ for a fixed cut of $100$ GeV on the transverse momenta of the jets. Also shown is the dominant QCD background in SM which has been estimated using [`M`adgraph 5]{}.*[]{data-label="tab:ttjj"} pronounced when the additional jets with high $p_T$ are triggered on. We look at the $t\bar{t}jj$ signal and SM background and consider a 100 GeV cut on the transverse momenta of the additional (nontop) jets. Note that by demanding two jets with $p_T>100$ GeV along with a $t\bar{t}$ pair would completely eliminate the large background coming from the pair production of $pp\to t\bar{t}$. We generate the SM background using `MadGraph 5` for $pp\to t\bar{t}jj$ and $pp\to t\bar{t}jj(+j)$ with some additional basic acceptance cuts of $|\eta_j|<2.5$ and $\Delta R_{jj}>0.5$. We list the cross section for the signal and background for different values of the spin–$\sp$ quark mass in Table \[tab:ttjj\]. A quick comparison of the signal with the background shows that the although the background is quite large when compared to the signal for $M=1$ TeV, our experience from the previous analysis of $4j$ and $2b2j$ signal implies that stronger cuts on the transverse momenta of the jets will suppress the background further. As before the signal will not change much for large values of $M$. [**Variable**]{} Cut ${\mathcal C}_1$ Cut ${\mathcal C}_2$ ------------------------------------------ ---------------------- ---------------------- $p_T^{\ell,b}$ $>10,20$ GeV $>10,20$ GeV $\bm{p_T^j}$ $>50$ GeV $>200$ GeV $|\eta|$ $<2.5$ $<2.5$ $\bm{\Delta R_{jj}}$ $>0.4$ $>0.7$ $\Delta R_{\ell\ell,\ell j, \ell b, bj}$ $>0.2$ $>0.2$ : *Two different set of cuts ${\mathcal C}_1$ and ${\mathcal C}_2$, imposed on the final state $\ell^+\ell^-bbjj\slashed{E}_T$ where the cuts are different only on the kinematic variables shown in bold. Not listed is a b-tagging efficiency of $\epsilon_b\,=\,0.5$ for both sets.*[]{data-label="tab:cuts"} To put this in perspective let us now consider the full decay of the top quarks in the final state and look more closely at the signal and SM background for two different set of cuts on the transverse momenta of the jets. To analyze the signal we focus on the semileptonic decay mode of the produced top quark leading to the following final state: $$\begin{aligned} p p \longrightarrow & (Q_{3/2} \to t g) \longrightarrow (t \to b W^+) g \longrightarrow (W^+ \to \ell^+ \nu_\ell) b g \nonumber \\ \hookrightarrow & (\bar{Q}_{3/2} \to \bar{t} g) \longrightarrow (\bar{t} \to \bar{b} W^-) g \longrightarrow (W^- \to \ell^- \bar{\nu}_\ell) \bar{b} g \nonumber \\ \hookrightarrow \ell^+ & \ell^- b \bar{b} j j \slashed{E}_T \end{aligned}$$ where we restrict ourselves to the choice of $\ell = e, \mu$ for the charged lepton. As it is very difficult to differentiate between $b$ and $\bar{b}$ even with heavy flavor tagging of the jets, we are looking at a final state with a pair of charged leptons ($\ell^+_i\ell^-_j$), two hard $b$-jets, two hard light quark jets and missing transverse momenta. We define two set of cuts which we list in Table \[tab:cuts\]. The results in Table \[tab:2l2b2j\] show that going from the cuts $\mathcal{C}_1$ to the cuts $\mathcal{C}_2$ drastically reduces the background without much change in the signal. The conclusions to be drawn from Table \[tab:2l2b2j\] are the following: 1) at $\sqrt{s}\,=\,8$ TeV the $p_T$ cuts extend the reach above $M\,=\,500$ GeV but well below $M\,=\,800$ TeV the signal cross-section becomes too small to be observed (independent of the cuts). 2) At $\sqrt{s}\,=\,14$ TeV the stronger $p_T$ cuts seem unnecessary for $M$ near $500$ GeV but become essential for $M$ equal $800$ GeV. At $M\,=\,1000$ GeV the cross-section is small but could be seen when the integrated luminosity exceeds about $200$ fb$^{-1}$. --------------------------------------- --------------- -------------------- --------------- -------------- $pp\to \ell^+\ell^-bbjj\slashed{E_T}$ SM background (fb) 500 800 1000 $\sqrt{s}=8~TeV$ 20.1 (7.8) 0.4 (0.3) 0.055 (0.045) 93.2 (2.9) $\sqrt{s}=14~TeV$ 385.9 (186.1) 11.2 (8.2) 1.9 (1.6) 522.8 (26.7) --------------------------------------- --------------- -------------------- --------------- -------------- : The signal cross section for the $ \ell^+\ell^-bbjj\slashed{E}_T$ final state with $\sqrt{s}=8~\text{and}~14$ TeV for different choices of the mass $M$ for the cuts ${\mathcal C}_1 ({\mathcal C}_2)$ shown in Table \[tab:cuts\]. []{data-label="tab:2l2b2j"} \[sec:summary\] Discussion and Summary ====================================== In this work we have focused on the signals for colored spin–$\sp$ fermions at the LHC. These particles will have large production cross sections and can be discovered through resonances in different channels depending on their decay properties. We have presented complete analytic expressions for the parton-level matrix amplitudes and cross sections used in our calculations. We considered three different scenarios for the higher spin fermion mixing with SM quarks which dictates the decay modes. We find that such an exotic fermion can decay hadronically to two light jets or into a gluon and heavy quark flavors. This leads to three different final state topologies $4j$, $2b2j$ and $t\bar{t}jj$. We did a detailed analysis of the three different cases and show that a strong cut on the transverse momenta of the final state jets is very useful in suppressing the otherwise large QCD background for hadronic final states at LHC. We have compared our results with a CMS study on $4j$ final states and extracted a lower bound of 490 GeV on the spin–$\sp$ quark mass. We further showed that this reach can be improved by using stronger cuts on the $p_T$ of the jets; the details are given in Table \[tab:4j\]. Given that only a limited amount of luminosity will be collected at $\sqrt{s}\,=\,8$ TeV the reach in $M$ for this final state is between $600$ GeV and $700$ GeV. At $\sqrt{s}\,=\,14$ TeV the reach easily exceeds $M\,=\,1$ TeV. We then considered the case where the spin–$\sp$ quark decays to a gluon and a bottom quark and showed, in Figs. \[fig:MinvJ1B1\] and \[fig:MinvJ2B1\], that the event characteristics of such a final state leads to a clear invariant mass peak when the $b$-jet is paired with the gluon jets. We also showed, in Table \[tab:2b2j\], that the SM background is suppressed in this final state which would lead to a better reach for spin–$\sp$ quark mass. Finally we focused on the signal where the spin–$\sp$ quark decays to a top quark and gluon where the signal and background are shown, for nominal cuts, in Table \[tab:ttjj\]. 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[^1]: Electronic address: dicus@physics.utexas.edu [^2]: Electronic address: durmas@ostatemail.okstate.edu [^3]: Electronic address: s.nandi@okstate.edu [^4]: Electronic address: santosh.rai@okstate.edu

--- abstract: 'The mid infrared emission of early type galaxies traces the presence of intermediate age stellar populations as well as even tiny amounts of ongoing star formation. Here we discuss high S/N [*Spitzer*]{} IRS spectra of a sample of Virgo early type galaxies, with particular reference to NGC 4435. We show that, by combining mid infrared spectroscopic observations with existing broad band fluxes, it is possible to obtain a very clean picture of the nuclear activity in this galaxy.' date: '?? and in revised form ??' --- Introduction ============ With the advent of the [*Spitzer Space Telescope*]{} new frontiers have been opened in the study of stellar population content of early-type galaxies (ETGs) and, in particular, the ability to quantify the occurrence and strength of the [*rejuvenation*]{} episodes. By means of mid infrared (MIR) observations it is possible to detect the presence of intermediate age stellar populations in passively evolving galaxies, and measure even tiny amounts of ongoing star formation activity. Bressan, Granato & Silva (1998) suggested that the MIR spectral region of old and intermediate age stellar populations should be affected by the presence of mass-losing oxigen-rich AGB giants. Their integrated emission around 10$\mu$m should be clearly seen in passively evolving galaxies; its analysis, in combination with UV, optical and NIR observations, should provide accurate age-metallicity ranking, unbiased by the age-metallicity degeneracy. Moreover, ongoing star formation can be easily detected in the MIR, from the presence of prominent emission features such as PAHs and atomic or molecular emission lines (e.g. Kaneda et al. 2005, Bressan et al. 2006a,b, Panuzzo et al. 2007). Last but not least, MIR nebular lines constitute a strong diagnostic to disentangle star formation and AGN activity and they also allow a [*direct and perhaps unique*]{} determination of the chemical abundance of the surrounding gas (Panuzzo et al. 2007). For the above reasons we begun a systematic study of the properties of ETGs in the mid infrared spectral region with the [*Spitzer Space Telescope*]{}. Here we report on the results obtained with [*Spitzer*]{} IRS (Houck et al. 2004) MIR spectroscopic observations of a sample of ETGs in the Virgo cluster (Bressan et al 2006a,b; Bressan et al 2007). Early-type galaxies in the mid infrared. ======================================== Eighteen ETGs among those that define the colour-magnitude relation of the Virgo cluster (Bower, Lucy & Ellis 1992) were observed in standard staring mode with the low resolution IRS modules between 5 and 20$\mu$m, in January and July 2005. The calibration and spectra extraction procedures are discussed in detail in Bressan et al. (2006a). The spectra of these galaxies are shown in Bressan et al. (2006a) and Bressan et al. (2007). For thirteen galaxies (76%) of our sample, the MIR spectrum is characterized by the presence of a broad emission features above 10$\mu$m, [*without any other narrow emission feature*]{}. The analysis of the IRS spectra indicates that the [*10$\mu$m feature*]{} has an extended spatial distribution; moreover its spatial distribution is consistent with that obtained below 8$\mu$m, where the spectra are dominated by stellar photospheres. This result has been confirmed by the analysis of [*Spitzer*]{} IRS Peak-Up imaging observations in the blue (16$\mu$m) filter of selected galaxies (Annibali et al. in preparation). It is also in agreement with previous ISOCAM observations that indicated spatially resolved emission at both 6.7 and 15 $\mu$m (Athey et al. 2002, Ferrari et al 2002, Xilouris et al. 2004). In view of these considerations and based on preliminary fits with our models of passively evolving old simple stellar populations, we argued that we have detected the 10$\mu$m features, due to silicate emission from the circumstellar envelopes of mass losing AGB stars, as predicted by Bressan et al. (1998). Bressan et al. (2007) have recently shown that the 10$\mu$m feature observed in early type galaxies is similar in shape but about a factor four larger than the [*semi empirical*]{} one obtained for the globular cluster 47 Tuc, consistent with a metallicity variation of the same order. We are now computing new isochrones and SSP models that account for a more realistic description of the AGB phase and of their dusty envelopes. ![Comparison between the observed SED of the central region of NGC 4435 and the GRASIL model. The thick solid line represents the model for the total SED, i.e. the starburst component plus the old stellar component; the three dots-dashed dark green line represents the contribution from the old stellar population, and the dashed blue line represents the total contribution from the burst of star formation, the dotted red line represents the emission from molecular clouds, the dot-dashed green line represents the diffuse medium emission and the thin solid cyan line denotes the emission from stars of the starburst component without applying the extinction from dust. The filled red circles are the broad band data. *Left*: Comparison from 0.1$\mu$m to 100 MHz. *Right*: Comparison for the MIR wavelengths. The thickest solid blue line represents the IRS *Spitzer* spectrum. []{data-label="active"}](bressan_fig1.eps "fig:"){width="50.00000%"} ![Comparison between the observed SED of the central region of NGC 4435 and the GRASIL model. The thick solid line represents the model for the total SED, i.e. the starburst component plus the old stellar component; the three dots-dashed dark green line represents the contribution from the old stellar population, and the dashed blue line represents the total contribution from the burst of star formation, the dotted red line represents the emission from molecular clouds, the dot-dashed green line represents the diffuse medium emission and the thin solid cyan line denotes the emission from stars of the starburst component without applying the extinction from dust. The filled red circles are the broad band data. *Left*: Comparison from 0.1$\mu$m to 100 MHz. *Right*: Comparison for the MIR wavelengths. The thickest solid blue line represents the IRS *Spitzer* spectrum. []{data-label="active"}](bressan_fig2.eps "fig:"){width="50.00000%"} Among bright Virgo cluster ETGs observed by our team, four galaxies (24%) show various levels of activity. NGC 4636 (optically classified as a LINER) shows low ionization emission lines (\[ArII\]7$\mu$m, \[NeII\]12.8$\mu$m, \[NeIII\]15.5$\mu$m and \[SIII\]18.7$\mu$m) on a continuum similar to other passive galaxies. NGC 4486 (M87) shows the same emission lines on a continuum dominated by the AGN emission above 8$\mu$m. The broad continuum feature above 10$\mu$m in M87 could be caused by silicate emission from the dusty torus (Siebenmorgen et al. 2005, Hao et al. 2005). NGC 4550 shows some PAH emission features while the MIR SED of NGC 4435 is characteristic of a star forming object. The panchromatic SED of NGC 4435 ================================ NGC 4435 is an S0 galaxy interacting with NGC 4438 and it hosts a circumnuclear disk. Panuzzo et al. (2007) combined the *Spitzer* IRS spectra of NGC 4435 with IRAC and MIPS archival data and existing broad band measurements from X-ray to radio wavelengths to obtain an accurate panchromatic spectral energy distribution (SED) of this galaxy. The SED was analysed with GRASIL (Silva et al. 1998) and well reproduced at all wavelengths. The analysis shows that the circumnuclear disk experienced a burst of star formation activity which is now fading. The IRS data themselves provide precise answers on important questions such as the nature of the nuclear activity suspected from optical (Ho et al. 1997) and X-ray (Machacek et al. 2004) observations, and the metallicity of the gas in the circumnuclear disk. We fail to detect any high excitation nebular emission lines in the IRS spectrum; the \[NeIII\]15.5/\[NeII\]12.8 ratio constrains the contribution of a possible AGN to the ionizing flux to be less than 2%. The upper limit on the temperature derived from H$_2$ S(1) and S(2) rotational lines is lower than expected for AGN excitation and PAH features are well reproduced by star formation models. Moreover, the X-ray emission is within the range expected from X-ray binaries in an advanced phase of the starburst. As for the metallicity of the nuclear disk, the comparison of observed MIR nebular lines with those predicted by the GRASIL model (Panuzzo et al 2003) indicates that it is almost solar. This is one of the first accurate [*direct*]{} estimates of the gas metallicity in ETGs. The age of the starburst, $\sim$180 Myr, corresponds to the epoch of the onset of the interaction with NGC 4438 derived from dynamical simulations (Combes et al. 1988). The mass of stars born during the starburst ($\sim 1.22\times10^8~M_\odot$) amounts to about 1.5% of the stellar mass sampled by the central 5 arcsec aperture. Conclusions =========== We have obtained with *Spitzer* IRS mid infrared spectra of ETGs selected along the colour-magnitude relation of the Virgo cluster. The mid infrared SED of most of our ETGs shows a clear broad emission around 10$\mu$m and longward as predicted in Bressan et al. (1998) which is likely due to dusty mass losing AGB stars. In the remaining fraction of galaxies (24%) we detect signatures of activity at various levels. The analysis of the IRS spectrum of NGC 4435 testifies to the superb capability of Spitzer to probe the nature of this type of activity and supports the notion that ETGs with relatively strong hydrogen absorption features are due to recent small rejuvenation episodes, rather than being the result of delayed galaxy formation (Bressan et al. 1996). A. B., G.L. G. and L. S. thank INAOE for warm hospitality. Athey, A., Bregman, J., Bregman, J., Temi, P., & Sauvage, M. 2002, ApJ, 571, 272 Bower, R. G., Lucey, J. R., Ellis, R. S. 1992, MNRAS, 254, 601 Bressan, A., et al. 2006, ArXiv Astrophysics e-prints, arXiv:astro-ph/0701620 Bressan, A., et al. 2006, ApJl, 639, L55 Bressan, A., et al. 2006, ArXiv Astrophysics e-prints, arXiv:astro-ph/0604068 Bressan, A., Granato, G.L., Silva, L. 1998, AA, 332, 135 Bressan, A., Chiosi, C., & Tantalo, R. 1996, A[&]{}A, 311, 425 Combes, F., Dupraz, C., Casoli, F., & Pagani, L. 1988, A&A, 203, 9 Ferrari, F., Pastoriza, M. G., Macchetto, F. D. et al.  2002, A[&]{}A, 389, 355 Hao, L., et al. 2005, ApJl, 625, L75 Ho, L. C., Filippenko, A. V., & Sargent, W. L. 1997, ApJS, 112, 315 Houck, J.R. 2004, ApJS, 154, 18 Kaneda, H., Onaka, T., & Sakon, I. 2005, ApJl, 632, L83 Machacek, M. E., Jones, C., & Forman, W. R. 2004, ApJ, 610, 183 Panuzzo, P., et al. 2007, ApJ in press (astro-ph/0610316) Panuzzo, P., Bressan, A., Granato, G. L., Silva, L., & Danese, L. 2003, A[&]{}A, 409, 99 Siebenmorgen, R., Haas, M., Kr[ü]{}gel, E., & Schulz, B. 2005, A[&]{}A, 436, L5 Silva, L., Granato, G. L., Bressan, A., & Danese, L. 1998, ApJ, 509, 103 Xilouris, E. M. et al.  2004, A[&]{}A, 416, 41

--- abstract: 'In this paper, we present a method of applying integral action to enhance the robustness of energy shaping controllers for underactuated mechanical systems with matched disturbances. Previous works on this problem have required a number of technical assumptions to be satisfied, restricting the class of systems for which the proposed solution applies. The design proposed in this paper relaxes some of these technical assumptions.' author: - 'Joel Ferguson$^{1}$, Alejandro Donaire$^{2}$, Romeo Ortega$^{3}$ and Richard H. Middleton$^{1}$[^1][^2][^3]' bibliography: - 'libraryURLRemoved.bib' title: '**Matched disturbance rejection for energy-shaping controlled underactuated mechanical systems** ' --- at (current page.south) ; Introduction ============ Interconnection and damping assignment passivity-based control (IDA-PBC) is a nonlinear control method whereby the closed-loop system is a passive port-Hamiltonian (pH) system with desired characteristics to comply with the control objectives [@Ortega2004]. Many systematic solutions have been proposed for the stabilization of nonlinear systems using IDA-PBC, but the general procedure is still limited by the designers ability to solve the so called *matching equations*. Although the matching equation are difficult to solve in some cases, IDA-PBC has been successful applied to a variety of nonlinear systems such as electrical machines [@Petrovic2001; @Gonzalez2008], power converters [@Rodriguez2000; @Rodriguez2001] and underactuated mechanical systems [@Acosta2005]-[@Donaire2016a]. In general, the equilibrium of a mechanical system stabilised with IDA-PBC will be shifted when an external disturbance acts on the system. In this paper we are interested in robustifying IDA-PBC [*vis-á-vis*]{} constant external disturbances. A general design for the addition of integral action to pH systems with the objective of rejecting disturbances was first presented in [@Donaire2009] and further discussed in [@Ortega2012]. The approach relies on a (possibly implicit) change of coordinates to satisfy the matching equations. The integral action scheme was tailored to fully actuated mechanical systems in [@Romero2013a] and underactuated mechanical systems in [@Donaire2016]. While in both cases the required change of coordinates to satisfy the matching equations were given explicitly, a number of technical assumption were imposed to do so. In both cases, the proposed integral action controllers were shown to preserve the desired equilibrium of the system, rejecting the effects of an unknown matched disturbance. More recently, an alternative method for the addition of integral action to pH systems was presented in [@Ferguson2015], [@Ferguson]. In these works, the controller is constructed from the open-loop dynamics of the plant. The energy function of the controller is chosen such that it couples the plant and controller states, which allows the matching equations to be satisfied by construction. In addition, the control system studied in [@Ferguson] has a physical interpretation and is shown to be equivalent to a control by interconnection (CbI) scheme, another PBC technique [@Ortega2007]. The method in [@Ferguson2015] was shown to be applicable to mechanical systems with constant mass matrix. In this paper, we extend the integral action design proposed in [@Ferguson2015] to underactuated mechanical systems subject to matched disturbances. The assumption of a constant mass matrix is relaxed, and general mechanical systems are considered. The method proposed in this paper is constructed to directly satisfy the matching equations without the need of the technical assumptions previously used in [@Donaire2016]. Specifically, the presented scheme allows the open-loop mass matrix, shaped mass matrix and input mapping matrix to be state dependant. [**Notation.**]{} In this paper we use the following notation: Let $x \in\mathbb{R}^n$, $x_1\in\mathbb{R}^m$, $x_2\in\mathbb{R}^s$. For real valued function $\mathcal{H}(x)$, $\nabla\mathcal{H}\triangleq \left(\frac{\partial \mathcal{H}}{\partial x}\right)^\top$. For functions $\mathcal{G}(x_1,x_2)\in\mathbb{R}$, $\nabla_{x_i}\mathcal{G}\triangleq \left(\frac{\partial \mathcal{G}}{\partial x_i}\right)^\top$ where $i \in\{1,2\}$. For fixed elements $x^\star\in \mathbb{R}^n$, we denote $\nabla \mathcal{H}^\star\triangleq \nabla \mathcal{H}(x)|_{x=x^\star}$. For vector valued functions $\mathcal{C}(x)\in\mathbb{R}^m$, $\nabla_x \mathcal{C}$ denotes the transposed Jacobian matrix $\left(\frac{\partial \mathcal{C}}{\partial x}\right)^\top$. Problem Formulation {#ProbForm} =================== In this paper, we consider mechanical systems that have been stabilised using IDA-PBC. This class of systems can be expressed as[^4]: $$\label{mecdist} \begin{split} \begin{bmatrix} \dot{q} \\ \dot{\bp} \end{bmatrix} &= \underbrace{ \begin{bmatrix} 0_{n\times n} & M^{-1}(q)\mathbf{M}_d(q) \\ -\mathbf{M}_d(q)M^{-1}(q) & \mathbf{J}_2(q,\bp)-R_d(q) \end{bmatrix}}_{F_m(q,\mathbf{p})} \begin{bmatrix} \nabla_q \mathbf{H}_d \\ \nabla_\mathbf{p} \mathbf{H}_d \end{bmatrix} \\ &\phantom{---}+ \underbrace{ \begin{bmatrix} 0_{m\times n} & G^\top(q) \end{bmatrix}^\top}_{G_m(q)} (u-d) \\ \mathbf{y} &= G^\top(q)\nabla_\mathbf{p} \mathbf{H}_d, \end{split}$$ with Hamiltonian $$\label{mechOLHam} \mathbf{H}_d(q,\mathbf{p})= \frac 12 \bp^\top \mathbf{M}_d^{-1}(q) \bp + V_d(q),$$ where $q,\mathbf{p} \in \mathbb{R}^n$ are the generalised configuration and momentum vectors respectively, $n$ is the number of degrees of freedom of the system, $u\in\mathbb{R}^m$ is the input, $y\in\mathbb{R}^m$ is the output, $d\in\mathbb{R}^m$ is a constant disturbance, $M(q) > 0$ and $\mathbf{M}_d(q) >0$ are the open-loop and shaped mass matrices of the system respectively, $V_d(q)$ is the shaped potential energy, $G(q)$ is the full-rank input matrix, $R_d (q)= G(q)K_p(q)G^\top(q)$ for some $K_p(q) \geq 0$ is the damping matrix and $\mathbf{J}_2(q,\mathbf{p}) = -\mathbf{J}_2^\top(q,\mathbf{p})$ is a skew-symmetric matrix. We assume that has a strict minimum at the desired operating point $(q,\mathbf{p}) = (q^\star,0_{n\times 1})$. For the remainder of the paper, the explicit state dependency of terms and various mapping are assumed and omitted. The control objective is to develop a dynamic controller $u=\beta(q,\bp,\zeta)$, where $\zeta \in \mathbb{R}^m$ is the state of the controller, that ensures asymptotic stability of the desired equilibrium $(q,\bp,\zeta) = (q^{\star},0,\zeta^\star)$, for some $\zeta^\star \in \mathbb{R}^m$, even under the action of constant disturbances $d$. Previous Work {#PrevSol} ============= A nonlinear PID controller was proposed in [@Donaire2016] as a solution to the matched disturbance rejection problem. Under the assumptions: - $G$ and $\mathbf{M}_d$ are constant - $G^\perp\nabla_q(\bp^\top M^{-1}\bp) = 0_{(n-m)\times 1}$, the control law was proposed to be $$\begin{split} u = &-\left[ K_pG^\top \mathbf{M}_d^{-1}GK_1G^\top M^{-1} + K_1G^\top \dot{M}^{-1} + K_2K_I \right. \\ &\left. \times(K_2^\top + K_3^\top G^\top \mathbf{M}_d^{-1}GK_1)G^\top M^{-1} \right]\nabla V_d \\ & -\left[K_1G^\top M^{-1}\nabla^2V_dM^{-1} + (G^\top G)^{-1}G^\top J_2\mathbf{M}_d^{-1} \right. \\ &\left. + K_2K_IK_3^\top G^\top \mathbf{M}_d^{-1} \right]\bp \\ & -(K_P G^\top \mathbf{M}_d^{-1} GK_2 + K_3)K_I\zeta \\ \dot{\zeta} = &(K_2^\top G^\top M^{-1} + K_3^\top G^\top \mathbf{M}_d^{-1}GK_1G^\top M^{-1})\nabla V_d \\ &+ K_3^\top G^\top \mathbf{M}_d^{-1}\bp, \end{split}$$ where $K_1 > 0$, $K_P > 0$, $K_I > 0$, $K_3 > 0$ and $$K_2 = (G^\top \mathbf{M}_d^{-1}G)^{-1}.$$ The resulting closed-loop can be expressed as $$\label{DonaireCL} \begin{split} \begin{bmatrix} \dot{q} \\ \dot{z}_2 \\ \dot{\zeta} \end{bmatrix} &= \begin{bmatrix} -\Gamma_1 & M^{-1}M_d & -\Gamma_2 \\ -M_d M^{-1} & -GK_pG^\top & -GK_3 \\ \Gamma_2^\top & K_3^\top G^\top & -K_3^\top \end{bmatrix} \nabla H_z \\ H_z &= \frac12 z_2^\top\mathbf{M}_d^{-1}z_2 + V_d(q) + \frac{1}{2}(\zeta-\alpha)^\top K_I(\zeta-\alpha) \end{split}$$ where, $$\label{Donairez2} \begin{split} z_2 &= \bp+GK_1G^\top M^{-1}\nabla V_d+GK_2K_I(\zeta-\alpha) \\ \Gamma_1 &= M^{-1}GK_1G^\top M^{-1} \\ \Gamma_2 &= M^{-1}GK_2 \\ \alpha &= K_I^{-1}(K_p + K_3)^{-1}d. \end{split}$$ The closed-loop system was shown to have a stable equilibrium at $(q,\bp,\zeta) = (q^\star,0_{n\times 1},\alpha)$. Furthermore, if the output signal $$y_{d3} = \begin{bmatrix} G^\top M^{-1}\nabla V_d \\ G^\top \mathbf{M}_d^{-1}z_2 \\ K_I(z_3-\alpha) \end{bmatrix}$$ is detectable, then the equilibrium is asymptotically stable. The assumptions P.1 and P.2 are necessary to ensure that the dynamics of $z_2$ in match the dynamics of $\mathbf{p}$ in , using the transformation . Integral Action for Underactuated Mechanical Systems ==================================================== In this section we propose an alternative method to add integral action to mechanical systems. This is achieved by first performing a momentum transformation such that the disturbance is pre-multiplied by the identity, rather than $G$. The integral action control law is then defined in the transformed coordinates. The resulting closed-loop is shown to be unique and preserves the desired operating point $q^\star$ of the original system. Momentum transformation {#momentumTransform} ----------------------- To solve the integral action problem, we transform the dynamics such that the disturbance is pre-multiplied by the identity, rather than $G$. Such a transformation is always possible utilising the following matrix: $$\label{Ttransform} T(q) = \begin{bmatrix} \{G^\top G\}^{-1}G^\top \\ G^\perp \end{bmatrix},$$ where $G^\perp \in \mathbb{R}^{m\times n}$ is a full-rank, left annihilator of $G$. \[momLemma\] Consider the system under the change of momentum coordinates $p= T\bp$. The dynamics can be equivalently expressed as $$\label{mecp} \begin{split} \begin{bmatrix} \dot{q} \\ \dot{p}_1 \\ \dot{p}_2 \end{bmatrix} &= \begin{bmatrix} 0_{n \times n} & S_1 & S_2 \\ -S_1^\top & S_{31}-K_p & S_{32} \\ -S_2^\top & -S_{32}^\top & S_{34} \end{bmatrix} \begin{bmatrix} \nabla_{q}\calH_d \\ \nabla_{p_1}\calH_d \\ \nabla_{p_2}\calH_d \end{bmatrix} \\ &\phantom{---} + \begin{bmatrix} 0_{m\times n} & I_{m\times m} & 0_{m\times s} \end{bmatrix}^\top (u-d) \\ y &= \nabla_{p_1}\calH_d \\ \calH_d&=\frac12 p^\top M_d^{-1}(q) p + V_d(q), \end{split}$$ where $p = \operatorname{col}(p_1,p_2)$, $p_1 \in \mathbb R^m$, $p_2 \in \mathbb R^s$, $s=n-m$, $$\label{s3} \begin{split} M_d &= T \mathbf{M}_d T^{\top} \\ S_1 &= M^{-1}\mathbf{M}_dG\{G^\top G\}^{-1} \\ S_2 &= M^{-1}\mathbf{M}_dG^{\perp\top} \\ S_{31} &= \{G^\top G\}^{-1}G^\top J_p G\{G^\top G\}^{-1} \\ S_{32} &= \{G^\top G\}^{-1}G^\top J_p G^{\perp\top} \\ S_{34} &= G^{\perp}J_p G^{\perp\top} \\ \end{split}$$ and $J_p$ is defined by $$\label{Jp} \begin{split} J_p &= \mathbf{M}_d M^{-1}\nabla_q^\top(T^{-1}p) -\nabla_q(T^{-1}p) M^{-1}\mathbf{M}_d \\ &\phantom{---}+\mathbf J_2(q,T^{-1}p). \end{split}$$ As $J_p = -J_p^\top$, both $S_{31}$ and $S_{34}$ are skew-symmetric. The proof of this lemma follows along the lines of the proof of [@Fujimoto2001a Lemma 2], [@Venkatraman2010a Proposition 1] and [@Duindam208 Theorem 1], therefore the full proof is omitted. An outline of the proof, however, can be found in the Appendix. Importantly, the output of the system under the change of momentum, $y$, remains unchanged. Indeed, $$\label{outputEquv} \begin{split} \mathbf{y} &= G^\top\nabla_\mathbf{p} \mathbf{H}_d \\ &= G^\top T^\top\nabla_{p}\calH_d \\ &= G^\top \begin{bmatrix} G\{G^\top G\}^{-1} & (G^\perp)^\top \end{bmatrix} \nabla_{p}\calH_d \\ &= \begin{bmatrix} I_{m} & 0_{m\times s} \end{bmatrix} \nabla_{p}\calH_d \\ &= y. \end{split}$$ Integral action control law --------------------------- The integral action control law is now proposed for the underactuated mechanical system described in $(q,p)$ coordinates by . \[propiacl\] Consider the system in closed-loop with the controller \[controlLaw\] $$\begin{aligned} u &= (-S_{31}+K_p+J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1} \calH_d \nonumber \\ &\phantom{---}+ (J_{c_1}-R_{c_1}) \nabla_{p_1}\calH_c \label{iacontroller} \\ \dot{\zeta}&=-R_{c_2}\nabla_{p_1}\mathcal{H}_d -S_1^\top\nabla_{q}\mathcal{H}_d + S_{32}\nabla_{p_2}\mathcal{H}_d, \label{iazeta} \end{aligned}$$ where $$\mathcal{H}_c = \frac12(p_1-\zeta)^\top K_I(p_1-\zeta),$$ $\zeta \in \mathbb{R}^m$, $K_I > 0$ and $J_{c_1}=-J_{c_1}^\top$, $R_{c_1} > 0$ $R_{c_2} > 0$ are constant matrices free to be chosen. Then, the closed-loop dynamics can be written in the pH form, $$\label{iacl} \begin{bmatrix} \dot{q} \\ \dot{p}_1 \\ \dot{p}_2 \\ \dot{\zeta} \end{bmatrix} = F(x) \begin{bmatrix} \nabla_{q}\mathcal{H}_{cl} \\ \nabla_{p_1}\mathcal{H}_{cl} \\ \nabla_{p_2}\mathcal{H}_{cl} \\ \nabla_{ \zeta}\mathcal{H}_{cl} \end{bmatrix} - \begin{bmatrix} 0_{n\times 1} \\ d \\ 0_{s\times 1} \\ 0_{m\times 1} \end{bmatrix},$$ where $$\label{Fx} F(x)= \begin{bmatrix} 0_{n \times n} & S_1 & S_2 & S_1 \\ -S_1^\top & J_{c_1}-R_{c_1}-R_{c_2} & S_{32} & -R_{c_2} \\ -S_2^\top & -S_{32}^\top & S_{34} & -S_{32}^\top \\ -S_1^\top & -R_{c_2} & S_{32} & -R_{c_2} \\ \end{bmatrix}$$ and $\mathcal{H}_{cl}:\mathbb{R}^{2n+m}\to\mathbb{R}$ is the closed-loop Hamiltonian defined as $$\mathcal{H}_{cl}(q,p_1,p_2,\zeta)= \mathcal{H}_d(q,p_1,p_2) + \mathcal{H}_c(p_1,\zeta).$$ First notice that $\nabla_{p_1}\mathcal{H}_c = -\nabla_{\zeta}\mathcal{H}_c$. Due to this relationship, the dynamics of $q$ and $p_2$ in are equivalent to the dynamics of $q$ and $p_2$ in . Considering the dynamics of $\zeta$ in and using $\nabla_{p_1}\mathcal{H}_c = -\nabla_{\zeta}\mathcal{H}_c$ yields $$\begin{split} \dot{\zeta} &= -R_{c_2}\nabla_{p_1}\mathcal{H}_d -S_1^\top\nabla_{q}\mathcal{H}_d + S_{32}\nabla_{p_2}\mathcal{H}_d \\ &= -R_{c_2}(\nabla_{p_1}\mathcal{H}_d + \nabla_{p_1}\mathcal{H}_c - \nabla_{p_1}\mathcal{H}_c) -S_1^\top\nabla_{q}\mathcal{H}_d \\ &\phantom{---} + S_{32}\nabla_{p_2}\mathcal{H}_d \\ &= -R_{c_2}\nabla_{p_1}\mathcal{H}_{cl} -R_{c_2}\nabla_{\zeta}\mathcal{H}_{cl} -S_1^\top\nabla_{q}\mathcal{H}_{cl} + S_{32}\nabla_{p_2}\mathcal{H}_{cl} \\ \end{split}$$ which matches the dynamics of $\zeta$ in . Finally, considering the dynamics of $p_1$ in , $$\begin{split} \dot{p}_1 &= -S_1^\top\nabla_{q}\calH_d + (S_{31}-K_p)\nabla_{p_1}\calH_d + S_{32}\nabla_{p_2}\calH_d \\ &\phantom{---}+ u - d \\ &= -S_1^\top\nabla_{q}\calH_d + (J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1}\calH_d + S_{32}\nabla_{p_2}\calH_d \\ &\phantom{---} + (J_{c_1}-R_{c_1}) \nabla_{p_1}\calH_c - d \\ &= -S_1^\top\nabla_{q}\calH_d + (J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1}\calH_d + S_{32}\nabla_{p_2}\calH_d \\ &\phantom{---} + (J_{c_1}-R_{c_1}-R_{c_2}) \nabla_{p_1}\calH_c + R_{c_2} \nabla_{p_1}\calH_c - d \\ &= -S_1^\top\nabla_{q}\calH_{cl} + (J_{c_1}-R_{c_1}-R_{c_2})\nabla_{p_1}\calH_{cl} \\ &\phantom{---} + S_{32}\nabla_{p_2}\calH_{cl}- R_{c_2} \nabla_{\zeta}\calH_{cl} - d, \end{split}$$ which is equivalent to the dynamics of $p_1$ in . In the case that $S_{31}$ and $K_v$ are constant, The choice $J_{c_1} = S_{31}$, $R_{c_1} = K_v$ can be made to simplify the control law . Stability --------- For the remainder of this section, the stability properties of the closed-loop system are considered. It is shown that the integral action control preserves the desired operating point $q^\star$ of the open-loop system. Further, if the original system is detectable, then the closed-loop system is asymptotically stable. The closed-loop system has an isolated equilibrium point $$\label{equilibrium} (q,p,\zeta) = (q^\star,0_{n\times 1},-K_I^{-1}(J_{c_1}-R_{c_1})^{-1}d).$$ The dynamics of $q$ in can be simplified to $$\begin{split} \dot{q} &= M^{-1}\mathbf{M}_d T^\top\nabla_p\mathcal{H}_d \\ &=M^{-1}\mathbf{M}_d T^\top M_d^{-1}p. \end{split}$$ As $M, \mathbf{M}_d, M_d, T$ are full-rank, $p=0_{n\times 1}$ and $\nabla_p \mathcal{H}_{d} = 0_{n\times 1}$ at any equilibrium. As $\nabla_{p_2} \mathcal{H}_{cl} = \nabla_{p_2} \mathcal{H}_{d}$, $$\label{equli3} \nabla_{p_2} \mathcal{H}_{cl} = 0_{s\times 1}.$$ The difference between the dynamics of $p_1$ and $\zeta$ are given by $\dot{p}_1 - \dot{\zeta} = (J_{c_1} - R_{c_1})\nabla_{p_1}\mathcal{H}_{cl} - d$. As $\nabla_p\mathcal{H}_d = 0_{n\times 1}$, $$\label{equli1} \nabla_{p_1}\mathcal{H}_{cl} = -\nabla_{\zeta}\mathcal{H}_{cl} = (J_{c_1} - R_{c_1})^{-1}d.$$ Recalling that $-\nabla_{\zeta}\mathcal{H}_{cl} = -\nabla_{\zeta}\mathcal{H}_{c} = K_I(p_1-\zeta)$ and $p_1 = 0$, can be rearranged to find $\zeta = -K_I^{-1}(J_{c_1}-R_{c_1})^{-1}d$. Substituting the equilibrium gradients and into and considering the dynamics of $p$, it results in $$\dot{p} = -\begin{bmatrix} S_1 & S_2 \end{bmatrix}^\top\nabla_q\mathcal{H}_{cl},$$ which implies that $\nabla_q\mathcal{H}_{cl} = \nabla_q\mathcal{H}_{d} = 0_{n\times 1}$ at any equilibrium as $\begin{bmatrix} S_1 & S_2 \end{bmatrix}$ is full-rank. The equilibrium gradient $\nabla_q\mathcal{H}_{d} = 0_{n\times 1}$ is satisfied by $q = q^\star$. \[propmatched\] Consider system subject to unknown matched disturbance in closed-loop with the controller . The following properties hold: (i) The equilibrium of the closed-loop system is stable. \[stability\]\ (ii) If the output $$\label{detectOutput} y_{p_1}=\begin{bmatrix} \nabla_{p_1}\mathcal{H}_d \\ \nabla_{p_1} \calH_{c} - (J_{c_1} - R_{c_1})^{-1}d \end{bmatrix}$$ is detectable, the equilibrium is asymptotically stable. \[asympStability\]\ (iii) If the shaped potential energy $V_d$ is radially unbounded, then the stability properties are global. \[globStable\] To verify , consider the function $$\label{iamatchlyap} \mathcal{W} = \mathcal{H}_d(q,p_1,p_2) + \frac12(z - z^\star)^\top K_I(z - z^\star),$$ where $z = p_1 - \zeta$ and $z^\star = p_1^\star - \zeta^\star = K_I^{-1}(J_{c_1}-R_{c_1})^{-1}d$, as a Lyapunov candidate for the system. $\mathcal{W}$ has a strict minimum at as $\mathcal{H}_d$ is strictly minimised by $(q,p)=(q^\star,0_{n\times 1})$ and $K_I > 0$. Defining $w = \operatorname{col}(q,p_1,p_2,\zeta)$, the closed-loop dynamics can be equivalently expressed as $$\label{iaclMod} \dot w = F(x) \underbrace{ \begin{bmatrix} \nabla_{q}\mathcal{H}_{cl} \\ \nabla_{p_1}\mathcal{H}_{cl} - (J_{c_1}-R_{c_1})^{-1}d \\ \nabla_{p_2}\mathcal{H}_{cl} \\ \nabla_{ \zeta}\mathcal{H}_{cl} + (J_{c_1}-R_{c_1})^{-1}d \end{bmatrix}}_{\nabla_w\mathcal{W}}.$$ The equilibrium is stable since $F+F^\top \leq 0$, which implies that $\dot{\mathcal{W}}\leq 0$ along the trajectories of the closed-loop system. The claim follows by considering the structure of $F$ and invoking LaSalle’s invariance principle. Finally, to verify , first note that the component of $\mathcal{W}$ associated with the controller and disturbance, $\frac12(z - z^\star)^\top K_I(z - z^\star)$, is radially unbounded in $z$. Then, recalling that $\mathcal{H}_d$ is of the form and $M_d^{-1} > 0$, it is clear that $\mathcal{H}_d$ is radially unbounded in $p$. Finally, if $V_d$ is radially unbounded in $q$, then $\mathcal{W}$ is radially unbounded. This implies that the closed-loop system is globally stable. \[CorrDetect\] If the output of the system is detectable when $d=0_{m\times 1}$ and $u=0_{m\times 1}$, then the closed-loop system is asymptotically stable. By Proposition \[propmatched\], the equilibrium of the closed-loop is asymptotically stable if $y_{p_1}$ is detectable. The control action , evaluated at $y_{p_1} = 0_{2m\times 1}$ is $u = d$. Further, using , the output of resolves to be $\mathbf{y} = y = \nabla_{p_1}\mathcal{H}_d = 0_{m\times 1}$. Substituting $u = d$ and $\mathbf{y} = 0_{m\times 1}$ into recovers the zero dynamics of the original, undisturbed system. Thus, if is detectable when $d=0_{m\times 1}$ and $u=0_{m\times 1}$, then the closed-loop system is asymptotically stable. Cart Pendulum Example ===================== In this section, we apply the presented integral action scheme to the cart pendulum system. For the existing IDA-PBC laws, the shaped mass matrix $\mathbf{M}_d$ is not constant so the integral action scheme of [@Donaire2016] cannot be used. Stabilisation control of the cart pendulum using IDA-PBC was solved in [@Acosta2005]. After partial feedback linearisation, the cart pendulum can be modelled as a pH system of the form $$\label{PendubotOL} \begin{split} \begin{bmatrix} \dot q \\ \dot{\mathbf{p}} \end{bmatrix} &= \begin{bmatrix} 0_{2\times 2} & I_{2\times 2} \\ -I_{2\times 2} & 0_{2\times 2} \end{bmatrix} \nabla \mathcal{H} \\ &\phantom{---} + \begin{bmatrix} 0_{2\times 1} \\ \mathbf{G} \end{bmatrix} \left(u-d\frac{1}{m_c+m_p\sin^2\theta}\right) \\ \mathcal{H} &= \frac12 \mathbf{p}^\top M^{-1}\mathbf{p} + \mathcal{V}, \end{split}$$ where $q = \begin{bmatrix} q_1 & q_2 \end{bmatrix}^\top$ is the configuration vector containing the angle of the pendulum from vertical and the horizontal position of the car respectively, $\mathbf{p} = \begin{bmatrix} \mathbf{p}_1 & \mathbf{p}_2 \end{bmatrix}^\top$ is the generalised momenta, $$\begin{split} M &= I_{2\times 2} \\ \mathbf{G} &= \begin{bmatrix} -b\cos(q_1) \\ 1 \end{bmatrix} \\ \mathcal{V} &= a\cos(q_1), \end{split}$$ $m_c$ and $m_p$ are the masses of the cart and pendulum respectively, $a = \frac{g}{l}$, $b = \frac{1}{l}$, $g$ is the acceleration due to gravity and $l$ is the length of the pendulum. The disturbance $d$ is an unknown constant force collocated with the input $u$. Note that the system is not in the form as the disturbance is not constant. In the remainder of this section, the undisturbed system will be stabilised using IDA-PBC and the resulting closed-loop will be converted into the form by defining a new input mapping matrix and input. Energy shaping -------------- In the case that $d = 0_{m\times 1}$, the cart pendulum can be stabilised around a desired equilibrium $(q_1,q_2,\mathbf{p}) = (0,q_{2}^\star,0_{2\times 1})$ using the IDA-PBC law $$\label{PendubotESCtrl} \begin{split} u &= \{\mathbf{G}^\top \mathbf{G}\}^{-1}\mathbf{G}^\top\{ \nabla_q\mathcal{H} - \mathbf{M}_dM^{-1}\nabla_q\mathbf{H}_d + \mathbf{J}_2\mathbf{M}_d^{-1}\mathbf{p} \} \\ &\phantom{---}- \frac{1}{(m_c+m_p\sin^2\theta)^2}K_p\mathbf{G}^\top \mathbf{M}_d^{-1}\mathbf{p} + u', \end{split}$$ where $$\begin{split} \mathbf{M}_d &= \begin{bmatrix} \frac{kb^2}{3}\cos^3q_1 & -\frac{kb}{2}\cos^2q_1 \\ -\frac{kb}{2}\cos^2q_1 & k\cos q_1 + m_{22}^0 \end{bmatrix} \\ V_d &= \frac{3a}{kb^2cos^2 q_1} + \frac{P}{2}\left[q_2 - q_{2}^\star + \frac{3}{b}\log\left(\sec q_1 + \tan q_1\right) \right. \\ &\left.\phantom{---------} + \frac{6m_{22}^0}{kb}\tan^2 q_1\right] \\ \mathbf{J}_2 &= \mathbf{p}^\top \mathbf{M}_d^{-1}\alpha \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \\ \alpha &= \frac{k\gamma_1}{2}\sin q_1 \begin{bmatrix} -b\cos q_1 \\ 1 \end{bmatrix} \\ \gamma_1 &= -\frac{kb^2}{6}\cos^3q_1, \end{split}$$ $P > 0$, $k>0$, $m_{22}^0>0$ are tuning parameters, $K_p > 0$ is a constant used for damping injection and $u'$ is an additional input for further control design. The cart pendulum , together with the control law , results in the closed-loop $$\label{ESCarkDist} \begin{split} \begin{bmatrix} \dot q \\ \dot{\mathbf{p}} \end{bmatrix} &= \begin{bmatrix} 0_{2\times 2} & M^{-1}\mathbf{M}_d \\ -\mathbf{M}_dM^{-1} & \mathbf{J}_2 - GK_p G^\top \end{bmatrix} \nabla \mathbf{H}_d \\ &\phantom{---}+ \begin{bmatrix} 0_{1\times 2} & G^\top \end{bmatrix}^\top (\tilde u - d) \\ \mathbf{H}_d &= \frac12 \mathbf{p}^\top M_d^{-1}(q)\mathbf{p} + V_d(q), \end{split}$$ where $\tilde u = (M+m\sin^2\theta)u'$, and $G = \frac{1}{m_c+m_p\sin^2\theta}\mathbf{G}$ Clearly, the closed-loop system is of the form . Integral action --------------- Before the integral action control law can be applied, the momentum must be transformed as per Section \[momentumTransform\]. Taking $G^\perp = (m_c+m_p\sin^2\theta)\begin{bmatrix} 1 & b\cos q_1 \end{bmatrix}$, the necessary momentum transformation is $p = T\mathbf{p}$ with $$T(q) = (m_c+m_p\sin^2\theta) \begin{bmatrix} \frac{-b\cos q_1}{b^2\cos^2 q_1 + 1} & \frac{1}{b^2\cos^2 q_1 + 1} \\ 1 & b\cos q_1 \end{bmatrix},$$ and results in the transformed Hamiltonian $$\calH_d=\frac12 p^\top \underbrace{T^{-\top}\mathbf M_d^{-1}(q) T^{-1}}_{M_d^{-1}(q)}p + V_d(q).$$ In the new momentum coordinates, the $S$ matrices can be resolved as per : $$\begin{split} S_1 &= \frac{m_c+m_p\sin^2\theta}{b^2\cos^2 q_1 + 1} \begin{bmatrix} -\frac{kb^3}{3}\cos^4q_1 -\frac{kb}{2}\cos^2q_1 \\ \frac{kb^2}{2}\cos^3q_1 + k\cos q_1 + m_{22}^0 \end{bmatrix} \\ S_{31} &= 0 \\ S_{32} &= \frac{1}{b^2\cos^2 q_1 + 1} \begin{bmatrix} -b\cos(q_1) \\ 1 \end{bmatrix}^\top\\ &\phantom{--}\left[\mathbf{M}_d M^{-1}\nabla_q^\top(T^{-1}(q)p)-\nabla_q(T^{-1}(q)p) M^{-1}\mathbf{M}_d\right.\\ &\left.\phantom{---} + J_2(q,p)\right] \begin{bmatrix} 1 \\ b\cos(q_1) \end{bmatrix}. \end{split}$$ where $$\begin{split} J_2 &= p^\top T^{-\top} \mathbf{M}_d^{-1}\alpha \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}. \\ \end{split}$$ As $S_{31} = 0$ and $K_p$ is constant, the integral control law is simplified by making the selection $R_{c_1} = K_p, J_{c_1} = 0$ which results in \[CartPendIntLaw\] $$\begin{aligned} \tilde u &= -R_{c_2}\nabla_{p_1} \calH_d -K_p \nabla_{p_1}\calH_c \\ \dot{\zeta}&=-R_{c_2}\nabla_{p_1}\mathcal{H}_d -S_1^\top\nabla_{q}\mathcal{H}_d + S_{32}\nabla_{p_2}\mathcal{H}_d. \end{aligned}$$ As discussed in [@Acosta2005], $V_d$ is radially unbounded on the domain $Q = \left\lbrace(-\frac{\pi}{2},\frac{\pi}{2})\times\mathbb{R}\right\rbrace$ and the system with $\tilde u = d = 0$ is detectable. Thus, by Proposition \[propmatched\] and Corollary \[CorrDetect\], the closed-loop system is asymptotically stable with region of attraction given by the set $\{Q\times \mathbb{R}^2\times \mathbb{R}\}$. Numerical simulation -------------------- The cart pendulum was simulated using the following plant parameters: $g = 9.8, M = I_{2\times 2}, l = 1, m_c = 1, m_p = 1$. The desired cart position was selected to be $q_2^\star = 0$ and the energy shaping control law was implemented with the controller parameters $k=1, m_{22}^0 = 1, P = 1, K_p = 10$. To reject the effects the disturbance $d$, the control law was applied with the controller storage function $\mathcal{H}_c(p_1,\zeta) = \frac12 K_I(p_1-\zeta)^2$ and $K_I = 0.05$. The system was simulated for 60 seconds with state of the plant initialised at $(q_1(0),q_2(0),p(0)) = (0,1,0_{2\times 1})$ and the controller initialised at $\zeta(0) = 0$. For the time interval $t\in[0,30)$ the disturbance was set to $d=0$. At $t=30s$, a disturbance of $d=2$ was applied for the remainder of the simulation. Figure \[Cartfig2\] shows that the cart pendulum, together with the integral action control law, tends towards the desired equilibrium on the time interval $t\in[0,30)$. At $t=30$, the disturbance $d=2$ is applied and the states move away from the desired equilibrium. On the time interval $t\in[30,60]$, the integral control compensates for the disturbance and the system again approaches the desired equilibrium. ![The cart pendulum in closed-loop with an energy shaping controller and integral action subject to a constant disturbance. The system tends toward the desired final position $(q_1,q_2) = (0,0)$ on the interval $t\in[0,30)$. At $t=30$, a disturbance is applied to the system. The integral control compensates for the disturbance and the system tends toward the equilibrium.[]{data-label="Cartfig2"}](DistRejc){width="49.00000%"} Conclusions =========== In this paper, a method to robustify IDA-PBC via the addition of integral action to underactuated mechanical systems was presented. The method relaxes technical assumptions required by previous solutions. The control scheme preserves the desired equilibrium of the open-loop system, rejecting the effects of an unknown matched disturbance. Further, the closed-loop system was shown to be asymptotically stable provided that the passive output of the open-loop system is detectable. \[app1\] *Proof of Lemma \[momLemma\]:* Let $x_m = \operatorname{col}(q,p)$, $\mathbf{x}_m = \operatorname{col}(q,\mathbf{p})$ and $x_m = g_t(\mathbf{x}_m) = (q,T\mathbf{p})$. The transformed Hamiltonian is defined as $$\begin{split} \mathcal{H}_d(q,p) &= \mathbf{H}_d(q,T^{-1}(q)p) \\ &= \frac 12 p^\top \underbrace{T^{-\top}(q) \mathbf{M}_d^{-1}(q) T^{-1}(q)}_{M_d^{-1}(q)}p + V_d(q). \end{split}$$ Utilising the differential of $g_t$ (see [@lee2012introduction]) can be equivalently expressed in $x_m$ as $$\label{dynXm} \begin{split} \dot{x}_m &= \left\lbrace \nabla_{\mathbf{x}_m}^\top g_t F_m \nabla_{\mathbf{x}_m}g_t \right\rbrace \big|_{\mathbf{x}_m = g_t^{-1}(q,p)} \nabla_{x_m}\mathcal{H}_d \\ &\phantom{---}+ \left\lbrace \nabla_{\mathbf{x}_m}^\top g_tG_m \right\rbrace \big|_{\mathbf{x}_m = g_t^{-1}(q,p)} (u-d_m) \\ &= \left\lbrace \begin{bmatrix} I_{l\times l} & 0_{l\times l} \\ \nabla_q^\top\left(T\mathbf{p}\right) & T \end{bmatrix} \begin{bmatrix} 0_{l\times l} & M^{-1}\mathbf{M}_d \\ -\mathbf{M}_dM^{-1} & \mathbf{J}_2-R_d \end{bmatrix} \right. \\ &\phantom{} \left. \times \begin{bmatrix} I_{l\times l} & \nabla_q\left(T\mathbf{p}\right) \\ 0_{l\times l} & T^\top \end{bmatrix} \right\rbrace \bigg|_{\mathbf{x}_m = g_t^{-1}(q,p)} \begin{bmatrix} \nabla_q \mathcal{H}_d \\ \nabla_p \mathcal{H}_d \end{bmatrix} \\ & + \left\lbrace \begin{bmatrix} I_{l\times l} & 0_{l\times l} \\ \nabla_q^\top\left(T\mathbf{p}\right) & T \end{bmatrix} \begin{bmatrix} 0_{l\times m} \\ G \end{bmatrix} \right\rbrace \bigg|_{\mathbf{x}_m = g_t^{-1}(q,p)} (u-d_m) \\ &= \begin{bmatrix} 0_{n\times n} & M^{-1}\mathbf{M}_dT^\top \\ -T\mathbf{M}_dM^{-1} & T(J_p-R_d)T^\top \end{bmatrix} \begin{bmatrix} \nabla_q \mathcal{H}_d \\ \nabla_p \mathcal{H}_d \end{bmatrix} \\ &\phantom{---}+ \begin{bmatrix} 0_{n\times m} \\ TG \end{bmatrix} (u-d), \end{split}$$ where $J_p$ is defined in . Recalling that $R_d (q)= G(q)K_p(q)G^\top(q)$, the term $TR_dT^\top$ can be simplified to $$\begin{split} TR_dT^\top &= \begin{bmatrix} \{G^\top G\}^{-1}G^\top \\ G^\perp \end{bmatrix} GK_pG^\top \begin{bmatrix} \{G^\top G\}^{-1}G^\top \\ G^\perp \end{bmatrix}^\top \\ &= \begin{bmatrix} K_p & 0_{m\times s} \\ 0_{s\times m} & 0_{s\times s} \end{bmatrix}. \end{split}$$ Finally, subdividing the momentum variable of into $p = \operatorname{col}(p_1,p_2)$ and substituting $T$ by its definition recovers the dynamics .   [^1]: $^{1}$Joel Ferguson and Richard H. Middleton are with School of Electrical Engineering and Computing and PRC CDSC, The University of Newcastle, Callaghan, NSW 2308, Australia. [Email: Joel.Ferguson@uon.edu.au, Richard.Middleton@newcastle.edu.au]{} [^2]: $^{2}$Alejandro Donaire is with the Department of Electrical Engineering and Information Theory and PRISMA Lab, University of Naples Federico II, Napoli 80125, Italy, and with the School of Electrical Eng. and Comp. Sc. of the Queensland University of Technology, Brisbane, QLD, Australia. [Email: Alejandro.Donaire@unina.it]{} [^3]: $^{3}$Romeo Ortega is with Laboratoire des Signaux et Systèmes, CNRS-SUPELEC, 91192, Gif-sur-Yvette, France [Email: romeo.ortega@lss.supelec.fr]{} [^4]: See [@Donaire2016] for the detailed explanation and motivation of the problem formulation.

--- abstract: 'Stationary whirling of slender and homogeneous (continuous) elastic shafts rotating around their axis, with pin-pin boundary condition at the ends, is revisited by considering the complete deformations in the cross section of the shaft. The stability against a synchronous sinusoidal disturbance of any wave length is investigated and the analytic expression of the buckling amplitude is derived in the weakly non-linear regime by considering both geometric and material (hyper-elastic) non-linearities. The bifurcation is super-critical in the long wave length domain for any elastic constitutive law, and sub-critical in the short wave length limit for a limited range of non-linear material parameters.' author: - Serge Mora title: | Synchronous whirling of spinning homogeneous elastic cylinders:\ linear and weakly non-linear analyses --- Introduction ============ A homogeneous and balanced elastic cylinder rotating around its axis is unstable beyond a critical angular velocity, leading to transverse deformations and whirling if the ends of the cylinder are constraint for instance with bearings. This instability results from the competition between the destabilizing effect of the centrifugal force that tends to drive the cylinder away from the axis of rotation, and the elastic forces opposed to the deformation. The whirling of rotating cylinders, as well as the propagation of vibrations in the neighborhood of the critical angular velocity, have been extensively investigated in the context of rotor-dynamics [@Kramer1993; @Genta2005] because of their damaging effects on the smooth running of rotating machinery such as compressors, pumps, turbines, turbochargers, jet engines [@Chen2005]. Understanding the stability of spinning shafts and their post-buckling behavior is crucial for the success in the design of this kind of rotating systems. While most of the studies have dealt with small deformations linearized at leading order [@Ehrich1964], few studies have considered non-linear effects [@Noah1995; @Yamamoto2012]. The non-linear dynamic behaviour of a uniform, slender rotating shaft made of a viscoelastic material with external damping mechanism has been studied by considering geometric non-linearities resulting from large transverse displacements [@Shaw1989; @Kurnik1994; @Hosseini2013]. Using the center manifold technique [@Henry1981] and the normal form method, the effects of external and internal damping on the whirling of rotating shafts have been investigated in terms of Hopf or double eigenvalues bifurcations. By pushing expansions up to order 2 in terms of the characteristic magnitude of the infinitesimal strain, $\varepsilon$, but Hookean elasticity for the strain-stress relation, the whirling amplitude in steady state configurations have been computed as the radius of a limit cycle in phase portraits [@Hosseini2013]. However, the intrinsic non-linear features of material constitutive law have been neglected in these studies. Indeed, order $\varepsilon^2$ in the expansion of the governing equations originates both from geometrical non-linearities (arising from the expression of the local curvature of the center line of the cylinder) together with non-linearities in the constitutive law of the elastic material. These last non-linearities are essential in order to fulfilled the requirement of material objectivity [@Ogden1984]. An expansion of the bending energy based on a scalar non-linear constitutive law [@Haslach1985] has been proposed in order to calculate non-synchronous whirling of rotating shafts [@Cveticanin1998]. Because of the scalar features of the constitutive law used by the author, this approach is limited to deformation with large wave length (compared with the radius of the shaft) and the issues related to Poisson effect are ignored. In addition, the rotating shaft was supposed to be not extensible which is not relevant for pin-pin ends since the extension of the center-line with pin-pin ends is of order $\varepsilon^2$ and cannot be neglected. A linear analysis of the whirling bifurcation of infinite rotating cylinders under axial tension has been developed in [@Ogden1980a], based on non-linear constitutive equations in three dimensions so that this analysis is relevant for any wave length of the deformation, but the non-linear analysis is still missing. In previous papers [@Richard2018; @Richard2019], the bifurcations of spinning undeformable shafts, surrounded by a compliant elastic layer, have been investigated both in the linear and the non-linear regimes, under plane strain assumption.\ In this paper, a non-linear analysis of the [*stationary*]{} whirling of [*homogeneous*]{} rotating cylinders is developed, based on the hypothesis of negligible external damping [@Ehrich1964] so that the system is conservative. The steady states are reached once transient vibrations are damped thanks to dissipative processes (internal damping) occurring inside the elastic material. The cylinders are supposed to be slender, their length $L$ being far larger than the radius $r_0$. The elastic material is assumed to be isotropic and incompressible. The buckling amplitude of synchronous and steady sinusoidal perturbations of any wave length is calculated without any further assumption for the constitutive law of the elastic material. The analysis relies on the complete three dimensional equations so that the results are relevant for any wave length of the whirling, including wave length of the same order of magnitude as the radius of the shaft. The complete (non-linear) equations governing the equilibrium steady states are derived in Section \[sec : base equations\]. A Lagrange multiplier accounts for the incompressibility constraint and the equations for the three components of displacement field are established in strong form. Section \[sec : linear\] is devoted to the linear stability analysis. The critical angular velocity is found to depend on the shear modulus of the elastic material, its mass density, the radius of the rotating cylinder, and in a non trivial manner on the ratio of the wave length of the deformation to the radius of the cylinder. The weakly non-linear analysis of the bifurcation is carried out in Section \[sec : non linear\]. The bifurcation is found to be super-critical for neo-Hookean materials, and can be sub-critical at small wave length for particular constitutive laws. Predictions of sections \[sec : linear\]-\[sec : non linear\] are checked in Section \[sec : FEM\] by means of numerical simulations based on the Finite Element Method. The last part (Section \[sec : conclusion\]) of the paper is devoted to a conclusion. Equilibrium equations based on a finite strain theory {#sec : base equations} ===================================================== In this section the non-linear equations governing the equilibrium (steady) configurations of a rotating elastic cylinder are derived, considering an arbitrary hyper-elastic incompressible isotropic material. Let $r_{0}$ denote the radius of the undeformed cylinder, $\rho$ its mass density and $\mu$ its initial shear modulus, [*i.e.*]{} the shear modulus for infinitesimal strain. The cylinder is spun with an angular velocity $\omega$ about its axis, as sketched in Figure \[fig : scheme\]. ![Sketches of an elastic cylinder of length $L$ and radius $r_0$ rotating around its axis with the angular velocity $\omega$. The two ends of the cylinder are pinned at the axis. (a) Three dimensional view of the reference (unbuckled) configuration. (b) Side view of this reference configuration. (c) Side view of a perturbation of characteristic wave vector $k$ parallel to the axis.[]{data-label="fig : scheme"}](scheme.pdf){width="55.00000%"} In the co-rotating frame, both the elastic force and the centrifugal force are conservative. The equilibrium can therefore be derived from the condition that the total potential energy is stationary. The position $\mathbf{R}$ of a material point in the deformed configuration is given as a map ${\mathbf{R}}(\mathbf{r})$ in terms of the position $\mathbf{r}$ in the undeformed configuration. For an isotropic and incompressible elastic material, the strain energy density is a function of the two first invariants, $I_1$ and $I_2$, of Green’s deformation tensor $\mathbf{C}=\mathbf{F}^T\cdot\mathbf{F}$, where $\mathbf{F}=\partial \mathbf{R}/\partial \mathbf{r}$ is the deformation gradient: $$\begin{array}{lll} I_1 & = & \mathrm{tr}~ \mathbf{C} - 3,\\ I_2 & = & \frac{1}{2}\left(\left(\mathrm{tr}~ \mathbf{C}\right)^2 - \mathrm{tr}~\left(\mathbf{C}^2\right) \right) - 3. \end{array} \label{eq:invariants}$$ The strain energy density is then written as $\mu\, W(I_1,I_2)$ where $W$ is the dimensionless strain energy density. For the strain energy $\mu\, W(I_1,I_2)$ to be consistent with the initial shear modulus $\mu$, the following normalization condition must be enforced: $$\frac{\partial W}{\partial I_1}(0,0) + \frac{\partial W}{\partial I_2}(0,0)=\frac{1}{2}. \label{eqn : tangent modulus}$$ For an incompressible neo-Hookean solid [@Ogden1984; @Macosko94] and for an incompressible Mooney-Rivlin solid [@Mooney1940; @Rivlin1948], the dimensionless strain energy density are respectively $W=\frac{1}{2}(I_1-3)$ and $W=\frac{1}{2}\left(\beta(I_1-3)+(1-\beta)(I_2-3)\right)$, with $\beta$ a material constant in the range $[0;1]$. Incompressibility of the elastic material imposes the condition $\mathcal{D}(\mathbf{r}) = 1$, where $\mathcal{D}=\det \mathbf{F}$ is the Jacobian of the transformation. To characterize equilibrium configurations, we seek stationary points of the augmented energy $${\cal E}=\int_{ 0<r<r_0;~0<z<L} \mathrm{d}\mathbf{r} \,\left( \mu\,{W}(I_1,I_2) -\frac{1}{2}\rho\,\omega^2\, \left(\mathbf{R} \cdot \mathbf{R}-(\mathbf{R}\cdot \mathbf{e}_z)^2\right) +\mu\,q\,({\cal D}-1) \right). \label{eqn : energy general}$$ The terms in the integrand are the strain energy, the potential of the centrifugal force, and the Lagrange term taking care of the incompressibility constraint ${\cal D}= 1$ by means of a Lagrange multiplier $q(\mathbf{r})$. From Eq. \[eqn : energy general\], the equilibrium of the system is governed by the two dimensionless parameters in the problem, namely $\alpha=\rho \, r_0^2\, \omega^2/ \mu$ and the ratio $L/r_0$. We use cylindrical coordinates, with $r$ the distance to the axis, $\theta$ the angle and $z$ the height in the unperturbed state (Figure \[fig : scheme\]). Let $\mathbf{e_r}$, $\mathbf{e_\theta}$,$\mathbf{e_z}$ be the orthonormal basis vectors associated with coordinates $r$, $\theta$ and $z$ respectively. In the deformed configuration, the position $\mathbf{R}$ of a material point is $\mathbf{R}=\mathbf{r}+u(r,\theta,z)\mathbf{e_r}+v(r,\theta,z)\mathbf{e_\theta}+w(r,\theta,z)\mathbf{e_z}$ and the deformation gradient $\mathbf{F} =\nabla\mathbf{R} ({\mathbf{r}})$ is: $$\mathbf{F}=\left( \begin{array}{ccc} 1+u_{,r}&(u_{,\theta}-v)/r&u_{,z}\\ \\ v_{,r}&(v_{,\theta}+u)/r+1&v_{,z}\\ \\ w_{,r}&w_{,\theta}/r&w_{,z}+1 \end{array} \right), \label{eqn : 3d F}$$ where a comma in subscript denotes a partial derivative. Expressions of ${\cal D}$, $I_1$ and $I_2$ are directly deduced from Eq. \[eqn : 3d F\]. The equilibrium equations are derived from the condition that the first variation of Eq. \[eqn : energy general\] with respect to the unknowns $u(r,\theta,z)$, $v(r,\theta,z)$, $w(r,\theta,z)$ and $q(r,\theta,z)$ is zero. Let $\mathbf{t}=(u,v,w,q)$ denote the collection of unknowns, and $\delta \mathbf{t}=(\delta u,\delta v, \delta w, \delta q)$ a virtual displacement that is kinematically admissible (abbreviated as ‘k.a.’), as imposed by the boundary conditions. The field $\mathbf{t}(r,\theta,z)$ is a solution of the problem if $$\forall \delta \mathbf{t}\;\mathrm{k.a.}, \quad D{\cal E}(\alpha,\mathbf{t})\left[\delta \mathbf{t} \right]=0. \label{eqn : general1}$$ $D{\cal E}(\alpha,\mathbf{t})\left[\delta \mathbf{t} \right]$ denotes the first variation of the energy evaluated in the configuration $\mathbf{t}$ with an increment $\delta \mathbf{t}$, also known as the first Gâteaux derivative of the functional ${\cal E}$ [@gateaux]. Note that the dependence of ${\cal E}$ with $L/r_0$ is not explicitly written in Eq. \[eqn : general1\] because it is a fixed parameter in the system, contrary to $\alpha$. Defining $${\cal G}=r\left\{W(I_1,I_2)+q({\cal D}-1) -\frac{1}{2}\frac{\alpha}{r_0^2}\left((r+u)^2+v^2\right)\right\}$$ and integrating by parts Eq. \[eqn : general1\], we obtain the equations in the interior of the body as $$\begin{aligned} {\cal D}-1&=&0, \label{eqn : 3d CP0 brute}\\ \frac{\partial {\cal G}}{\partial u}-\frac{\partial}{\partial r}\left(\frac{\partial {\cal G}}{\partial u_{,r}} \right)-\frac{\partial}{\partial \theta}\left(\frac{\partial{\cal G}}{\partial u_{,\theta}} \right)-\frac{\partial}{\partial z}\left(\frac{\partial{\cal G}}{\partial u_{,z}} \right)&=&0, \label{eqn : 3d CP1 brute}\\ \frac{\partial {\cal G}}{\partial v}-\frac{\partial}{\partial r}\left(\frac{\partial{\cal G}}{\partial v_{,r}} \right)-\frac{\partial}{\partial \theta}\left(\frac{\partial{\cal G}}{\partial v_{,\theta}} \right)-\frac{\partial}{\partial z}\left(\frac{\partial{\cal G}}{\partial v_{,z}} \right)&=&0, \label{eqn : 3d CP2 brute}\\ \frac{\partial {\cal G}}{\partial w}-\frac{\partial}{\partial r}\left(\frac{\partial{\cal G}}{\partial w_{,r}} \right)-\frac{\partial}{\partial \theta}\left(\frac{\partial{\cal G}}{\partial w_{,\theta}} \right)-\frac{\partial}{\partial z}\left(\frac{\partial{\cal G}}{\partial w_{,z}} \right)&=&0. \label{eqn : 3d CP3 brute}\end{aligned}$$ The first equation (Eq. \[eqn : 3d CP0 brute\]) is the incompressibility constraint and the three other equations (Eqs. \[eqn : 3d CP1 brute\]-\[eqn : 3d CP3 brute\]) are the equilibrium in the radial, circumferential and longitudinal directions, respectively. These equations are complemented by the condition of zero traction at the lateral boundary $r=r_0$, $$\left. \frac{\partial{\cal G}}{\partial u_{,r}}\right|_{r=r_0}=\left. \frac{\partial{\cal G}}{\partial v_{,r}}\right|_{r=r_0}=\left. \frac{\partial{\cal G}}{\partial w_{,r}}\right|_{r=r_0}=0. \label{eqn : 3d BC3 brute}$$ In addition, the pin-pin condition at the ends imposes: $$u(0,\theta,z)=v(0,\theta,z)=w(0,\theta,z)=0 \mbox{ for } z=0 \mbox{ and } z=L \label{eqn : boundary ends 1}$$ and $$\frac{\partial{\cal G}}{\partial u_{,z}}=\frac{\partial{\cal G}}{\partial v_{,z}}=\frac{\partial{\cal G}}{\partial w_{,z}}=0 \mbox{ for } z=0 \mbox{ and } z=L. \label{eqn : boundary ends 2}$$ The three last boundary conditions (Eqs. \[eqn : boundary ends 2\]) originate from the variation of the augmented energy at the vicinity of the ends. Since end effects are expected to spread in a domain of characteristic size $r_0$, their relative contribution to the total augmented energy is of order $r_0/L$. Hence, within the hypothesis of a slender shaft ($r_0 \gg L$), boundary conditions Eqs. \[eqn : boundary ends 2\] is negligible. This simplification makes possible the harmonic decomposition of the deformation (with unique wave length and unique circumferential wave number, see Section \[sec : linearization\]).\ The equilibrium configurations are the solutions of the system formed by Eqs. \[eqn : 3d CP0 brute\]-\[eqn : boundary ends 1\]. Because of non-linearities in the equations, the analytic resolution is out of reach. In Section \[sec : unbuckled\] the system is resolved in the reference (undeformed) configuration. Then the magnitude of the displacement is assumed to scale as a small parameter, $\varepsilon$, so that it is infinitely smaller than the other length scales ($r_0$ and $L$). Eqs. \[eqn : 3d CP0 brute\]-\[eqn : boundary ends 1\] is resolved at linear order (order $\varepsilon$) in Section \[sec : linearization\], and at order $\varepsilon^2$ in Section \[sec : non linear\]. Finally, they are solved numerically by means of finite elements in Section \[sec : FEM\]. linear bifurcation analysis {#sec : linear} =========================== Unbuckled solution {#sec : unbuckled} ------------------ We start by analyzing the unbuckled configuration (base state), and label all quantities relevant to it using a subscript ‘$0$’. In this configuration, $u_{0}(r,\theta,z)=0$, $v_{0}(r,\theta,z)=0$ and $w_{0}(r,\theta,z)=0$. The Lagrange multiplier $q$ is found from the radial equilibrium Eq. \[eqn : 3d CP1 brute\] and Eq. \[eqn : 3d BC3 brute\] as $$q_0=\frac{\alpha}{2}\left(1-(r/r_0)^2\right)+\beta-2. \label{eqn : base lagrange multiplier}$$ Altogether, the unbuckled solution of Eqs. \[eqn : 3d CP0 brute\]-\[eqn : 3d BC3 brute\] is written as $\mathbf{t}_0=(u_0,v_0,,w_0,q_0)$. Linearization of the equations {#sec : linearization} ------------------------------ A small perturbation is added to the unbuckled solution, and the equations of Section \[sec : base equations\] are linearized with respect to the amplitude of the perturbation, $$\mathbf{t}=\mathbf{t}_0+\varepsilon \mathbf{t}_1 =\left(\varepsilon u_1(r,\theta,z),\varepsilon v_1(r,\theta,z),\varepsilon w_1(r,\theta,z),q_0(r,\theta,z)+\varepsilon q_1(r,\theta,z)\right). \label{eq:order1expansion}$$ We first assume a harmonic $\theta$ and $z$ dependence of any variation of the perturbation of $u$, $v$, $w$ and $q$: $$\left\{ \begin{array}{l} u_1=u^+_1(r,\theta,z)={\cal R}e \left(f_u(r)e^{\mathrm{i}\theta+\mathrm{i}kz} \right) \\ v_1=v^+_1(r,\theta,z)={\cal R}e \left(-\mathrm{i}f_v(r)e^{\mathrm{i}\theta+\mathrm{i}kz} \right) \\ w_1=w_1^+(r,\theta,z)={\cal R}e \left(-\mathrm{i}f_w(r)e^{\mathrm{i}\theta+\mathrm{i}kz} \right) \\ q_1=q^+_1(r,\theta,z)={\cal R}e \left(f_q(r)e^{\mathrm{i}\theta+\mathrm{i}kz} \right) \end{array} \right. \label{eqn : helicoidal +}$$ where $k$ is the axial wave number. ${\cal R}e$ denotes the real part. The conventional complex factor $(-\mathrm{i})$ has been included for convenience, anticipating on the fact that the phase of $v_{1}$ and $w_1$ are shifted by $\pi/2$ compared to the phase of the two other unknowns. At linear order in $\varepsilon$, Eqs. \[eqn : 3d CP0 brute\]-\[eqn : 3d BC3 brute\] yield respectively: $$r{{d f_u}\over{dr}}+f_u+kr f_w+f_v=0 \label{eqn : 3d CP0 lin}$$ $$-{{d^2 f_u}\over{d r^2}}-\frac{1}{r}{{d f_u}\over{dr}}+(k^2+\frac{2}{r^2})f_u+\left({{\alpha }\over{r_{0}^2}}+\frac{2}{r^2}\right)f_v+{{\alpha krf_w}\over{r_{0}^2}}-{{d f_q}\over{dr}} =0 \label{eqn : 3d CP1 lin}$$ $$\left(\frac{2}{r}+{{\alpha r}\over{r_{0}^2}}\right)f_u-r{{ d^2 f_v}\over{dr^2}}-{{d f_v}\over{dr}}+(\frac{2}{r}+k^2r)f_v +f_q=0$$ $${{\alpha rf_u}\over{r_{0}^2}}-\frac{1}{k}{{d^2 f_w}\over{dr^2}}-\frac{1}{rk}{{d f_w}\over{dr}}+\left(\frac{1}{kr^2}+k\right)f_w+f_q=0. \label{eqn : 3d CP3 lin}$$ The boundary conditions Eqs. \[eqn : 3d CP0 brute\]-\[eqn : 3d CP3 brute\] at order $\varepsilon$ are respectively: $$r_0{{d f_u}\over{dr}}-kr_0 f_w-f_v-f_u +r_0f_q=0 ~~~~~~ \mbox{ at } r=r_0, \label{eqn : 3d BC1 lin}$$ $$r_0{{d f_v}\over{dr}}-f_v -f_u=0 ~~~~~~ \mbox{ at } r=r_0, \label{eqn : 3d BC2 lin}$$ $$r_0 {{d f_w}\over{dr}}-kr_0 f_u=0 ~~~~~~ \mbox{ at } r=r_0. \label{eqn : 3d BC3 lin}$$ After the elimination of $f_v$, $f_w$ and $f_q$ in Eqs. \[eqn : 3d CP0 lin\]-\[eqn : 3d CP3 lin\], one obtains an order 6 differential equation for $f_u$: $$\begin{split} r^5 {{d^6 f_u}\over{d r^6}} +&9 r ^4 {{d^5 f_u}\over{d r^5}} f_u+\left( 9 r^3-3 k^2 r^5\right) {{d^4 f_u}\over{d r^4}}+\left(-18 k^2 r^4-12 r^2\right){{d^3 f_u}\over{d r^3}}+\left(3 k^4 r^5-9 k^2 r^3+9 r\right) {{d^2 f_u}\over{d r^2}}\\ &+\left(9 k^4 r^4+9 k^2 r^2-9\right) {{d f_u}\over{d r}} -k^6 r^5 f_u=0 \end{split} \label{eqn : 3d equa diff}$$ and, after substitutions in Eqs. \[eqn : 3d BC1 lin\]-\[eqn : 3d BC3 lin\], one obtains the boundary conditions at $r=r_0$ in term of $f_u$: $$\begin{split} -r_0^3 {{d^4 f_u}\over{d r^4}} f_u-6 r_0^2 {{d^3 f_u}\over{d r^3}}+ \left(2 k^2 r_0^3-3 r_0\right) {{d^2 f_u}\over{d r^2}}+\left(14 k^2 r_0^2+3\right) {{d f_u}\over{d r}}+\left(-k^4 r_0^3-4 \alpha k^2 r_0\right) f_u=0 \label{eqn : 3d BC1} \end{split}$$ $$\begin{split} -r_0^5 {{d^6 f_u}\over{d r^6}}&-9 r_0^4 {{d^5 f_u}\over{d r^5}} + \left(2 k^2 r_0^5-10 r_0^3\right) {{d^4 f_u}\over{d r^4}} +\left(16 k^2 r_0^4+6 r_0^2\right) {{d^3 f_u}\over{d r^3}}+\left(-k^4 r_0^5+16 k^2 r_0^3-12 r_0\right) {{d^2 f_u}\over{d r^2}}\\ &+\left(-7 k^4 r_0^4-8 k^2 r_0^2+12 \right) {{d f_u}\over{d r}}-5 k^ 4 r_0^3 f_u=0 \label{eqn : 3d BC2} \end{split}$$ $$\begin{split} r_0^5 {{d^6 f_u}\over{d r^6}} &+9 r_0 ^4 {{d^5 f_u}\over{d r^5}} +\left( 10 r_0^3-2 k^2 r_0^5\right) {{d^4 f_u}\over{d r^4}} +\left(-16 k^2 r_0^4-6 r_0^2\right){{d^3 f_u }\over{d r^3}} +\left(k^4 r_0^5-24 k^ 2 r_0^3+12 r_0\right) {{d^2 f_u}\over{d r^2}} \\ &+\left(7 k^4 r_0^4-12\right) {{d f_u}\over{d r}} -3 k^4 r_0^3 f_u=0. \label{eqn : 3d BC3} \end{split}$$ General solution ---------------- Let $s_1(kr)$, $s_2(kr)$ and $s_3(kr)$ be three independent solutions of Eqs. \[eqn : 3d equa diff\]-\[eqn : 3d BC3\] that do not diverge, as well as their first derivative, at $r=0$. These solutions are sought as series expansions in the form: $$s_i(kr)=\sum_{m=0}^\infty a_m(kr)^m.$$ The condition for $s_i(kr)$ to be a solution of Eq. \[eqn : 3d equa diff\] is, for $m\ge 6$: $$a_{m-6}-3a_{m-4}(m-4)(m-2)+3a_{m-2}m(m-2)^2(m-4) -a_m(m-4)(m-2)^2m^2(2+m)=0,$$ where $a_0$, $a_2$ and $a_4$ are constants that are not fixed up to now. Coefficients $a_m$ with an odd index have to be 0. In order to build three independent solutions of Eq. \[eqn : 3d equa diff\], we choose $a_0=1$, $a_2=a_4=0$ for $s_1(kr)$ ; $a_0=a_4=0$ and $a_2=1$ for $s_2(kr)$ ; and $a_0=a_2=0$ and $a_4=1$ for $s_3(kr)$. Writing now the general solution $f_u(r)$ of Eq. \[eqn : 3d equa diff\] as: $$f_u(r)=As_1(kr)+Bs_2(kr)+Cs_3(kr),$$ and substituting this expression in the boundary conditions Eqs. \[eqn : 3d BC1\]-\[eqn : 3d BC3\], one gets a linear system of 3 homogeneous equations with three unknowns $A$, $B$ and $C$. The condition for a non-zero deformation, [*i.e.*]{} $(A,B,C)\ne(0,0,0)$, is obtained by imposing the determinant of the linear system to be zero, leading to the condition for $\alpha$, $\alpha=\alpha_c$ with $\alpha_c$: $$\alpha_c=\frac{3}{4}(kr_0)^4-\frac{5}{24}(kr_0)^6 -\frac{19}{1536}(kr_0)^8 + \cdots \label{eqn : alphac}$$ Higher orders in the expansion can be calculated as well. For $\alpha=\alpha_c$, the system is neutrally stable against a perturbation of wave number $k$. $\alpha_c$ is plotted as a function of $kr_0$ in Figure \[fig : linear threshold\]. ![Solid line: Critical value of $\alpha$ at the instability onset, as a function of $k$, calculated from Eq. \[eqn : alphac\] at order 30 in $kr_0$ (higher orders in the expansion lead to indistinguishable curves). Dashed line: First term in the expansion of $\alpha_c$ with respect to $kr_0$. Filled circles: Threshold $\alpha^*$ obtained from FEM simulations (see Eq. \[eqn : star\] of Section \[sec : FEM\]).[]{data-label="fig : linear threshold"}](alpha_log.pdf){width="45.00000%"} Taking the first term in the expansion Eq. \[eqn : alphac\], one recovers the well known expression of the linear threshold calculated in the long wave length limit in the framework of Hookean elasticity, $\alpha_c=\frac{3}{4}(kr_0)^4$ (see the dashed line in Figure \[fig : linear threshold\]). The expressions of functions $f_u(r)$, $f_v(r)$, $f_w(r)$ and $f_q(r)$, with the condition $f_u(r_0)=\xi$ ($\varepsilon \xi$ will be referred as the buckling amplitude) are: $$\begin{aligned} \frac{f_u(r)}{\xi}&=&1+\left(1-\frac{r^2}{r_0^2}\right)\frac{(kr_0)^2}{4}-\left(3-\frac{4r^2}{r_0^2}+\frac{r^4}{r_0^4}\right)\frac{3(kr_0)^4}{64}+\left(23-\frac{39r^2}{r_0^2}+\frac{21r^4}{r_0^4}-\frac{5r^6}{r_0^6}\right)\frac{(kr_0)^6}{2304}+\cdots\\ \frac{f_v(r)}{\xi}&=&-1-\left(1+\frac{r^2}{r_0^2}\right)\frac{(kr_0)^2}{4}+\left(9+\frac{12r^2}{r_0^2}-\frac{r^4}{r_0^4}\right)\frac{(kr_0)^4}{64}-\left(23+\frac{135r^2}{r_0^2}-\frac{39r^4}{r_0^4}+\frac{r^6}{r_0^6}\right)\frac{(kr_0)^6}{2304}+\cdots\\ \frac{f_w(r)}{\xi}&=&\frac{r}{r_0}(kr_0)-\left(3\frac{r}{r_0}-\frac{r^3}{r_0^3}\right)\frac{(kr_0)^3}{4}+\left(\frac{7r}{r_0}-\frac{4r^3}{r_0^3}+\frac{r^5}{r_0^5}\right)\frac{(kr_0)^5}{64}+\cdots\\ f_q(r)&=&\frac{\xi}{r_0}\left\{ \frac{r}{r_0}(kr_0)^2-\left(4\frac{r}{r_0}-\frac{r^3}{r_0^3}\right)\frac{(kr_0)^4}{8}-\left(\frac{39r}{r_0}-\frac{42r^3}{r_0^3}-\frac{r^5}{r^5_0}\right)\frac{(kr_0)^6}{192}+\cdots \right\}\end{aligned}$$ Indeed, the form of Eq. \[eqn : helicoidal +\] corresponds to a right helical deformation along the z axis. The left helical deformation can be deduced from the previous one (with the transformation $k \leftrightarrow -k$) : $$\left\{ \begin{array}{l} u_1=u^-_1(r,\theta,z)={\cal R}e \left(f_u(r)e^{\mathrm{i}\theta-\mathrm{i}kz} \right) \\ v_1=v^-_1(r,\theta,z)={\cal R}e \left(-\mathrm{i}f_v(r)e^{\mathrm{i}\theta-\mathrm{i}kz} \right) \\ w_1=w_1^-(r,\theta,z)={\cal R}e \left(\mathrm{i}f_w(r)e^{\mathrm{i}\theta-\mathrm{i}kz} \right) \\ q_1=q^-_1(r,\theta,z)={\cal R}e \left(f_q(r)e^{\mathrm{i}\theta-\mathrm{i}kz} \right) \end{array} \right. \label{eqn : helicoidal -}$$ Up to now, the boundary conditions at the ends of the cylinder, Eq. \[eqn : boundary ends 1\], have not been taken into account. In that case, the general solution at linear order of the problem consists of any linear combinations of the solutions Eqs. \[eqn : helicoidal +\] and \[eqn : helicoidal -\]. Imposing now the boundary condition Eq. \[eqn : boundary ends 1\] yields the unique (up to the buckling amplitude) solution of the complete problem at linear order: $$\begin{aligned} u_1&=&f_u(r)\sin \theta \sin kz \nonumber\\ v_1&=&-f_v(r) \cos \theta \sin kz \nonumber\\ w_1&=&-f_w(r) \sin \theta \cos kz \label{eqn : t linear}\\ q_1&=&f_q(r) \sin \theta \sin kz \nonumber\end{aligned}$$ with $k=n\pi/L$ and $n$ an integer. Hence, at linear order, only discrete values of wave number $k$ are admissible for the system to be neutral against sinusoidal perturbations, which corresponds to discrete values of the control parameter $\alpha$. In the following, we start from a value of $\alpha$ at which the system is neutrally stable ($\alpha=\alpha_c$), and we consider a quasi-static increase of $\alpha$. The deformation is not harmonic anymore, and we calculated the expression of the corresponding mode, including the buckling amplitude. Weakly non-linear analysis {#sec : non linear} ========================== Introduction ------------ In this section, we carry out a Koiter expansion [@Koiter-On-the-stability-of-an-elastic-equilibrium-1945; @Hutchinson-Imperfection-sensitivity-of-externally-1967; @Hutchinson-Koiter-Postbuckling-theory-1970; @Budiansky-Theory-of-buckling-and-post-buckling-1974; @Peek-Triantafyllidis-Worst-shapes-of-imperfections-1992; @Peek-Kheyrkhahan-Postbuckling-behavior-and-imperfection-1993; @Heijden2009koiter] of the bifurcated solution in the vicinity of a bifurcation point. The displacement field and the Lagrange multiplier are expanded to order 3 in terms of an arc-length parameter $\varepsilon$ defined as [@Chakrabarti2018]: $$\begin{aligned} \alpha & =&\alpha_{\mathrm{c}} +\alpha_2 \varepsilon^2 \label{eqn : definition of alpha2}\\ \mathbf{t}(\alpha) & = &\mathbf{t}_{0}(\alpha) + \varepsilon\mathbf{t}_{1} + \varepsilon^2\mathbf{t}_{2} +\varepsilon^3\mathbf{t}_{3}+ \cdots \label{eqn : definition of xi}\end{aligned}$$ where $\alpha_c$ is the critical dimensionless angular velocity determined from the linear bifurcation analysis, see Eq. \[eqn : alphac\]. The base solution $\mathbf{t}_0$ depends on the load $\alpha$ through $q_0$. The first-order correction $\mathbf{t}_1$ is the linear mode calculated in Section \[sec : linear\] and normalized so that the buckling amplitude is $\varepsilon \xi$. Second-order correction to the displacement {#sec : second order} ------------------------------------------- The second order displacements $\mathbf{t}_2=(u_{2},t_{2},z_2,q_{2})$ results from the non-linear interaction of the linear mode $\mathbf{t}_{1}$ with itself. As a result, it involves a superposition of Fourier modes having wave numbers $\pm k$ with respect to the variable $z$, and circumferential wave numbers $\pm 1$. Hence, we seek the second-order correction $\mathbf{t}_{2}$ to the displacement as: $$\begin{split} u_2(r,\theta,z)=&g_{u1}(r)+g_{u2}(r)\sin(2kz)+g_{u3}(r)\cos(2kz)+g_{u4}(r)\sin(2\theta)+g_{u5}(r)\cos(2\theta)+g_{u6}\sin(2\theta)\sin(2kz)\\& +g_{u7}\sin(2\theta)\cos(2kz)+g_{u8}(r)\cos(2\theta)\sin(2kz)+g_{u9}(r)\cos(2\theta)\cos(2kz) \label{eqn : gu} \end{split}$$ $$\begin{split} v_2(r,\theta,z)=&g_{v1}(r)+g_{v2}(r)\sin(2kz)+g_{v3}(r)\cos(2kz)+g_{v4}(r)\cos(2\theta)+g_{v5}(r)\sin(2\theta)+g_{v6}(r)\cos(2\theta)\sin(2kz)\\&+g_{v7}(r)\cos(2\theta)\cos(2kz)+g_{v8}\sin(2\theta)\sin(2kz)+g_{v9}\sin(2\theta)\cos(2kz) \label{eqn : gv} \end{split}$$ $$\begin{split} w_2(r,\theta,z)=&g_{w1}(r)+g_{w2}(r)\cos(2kz)+g_{w3}(r)\sin(2kz)+g_{w4}(r)\sin(2\theta)+g_{w5}(r)\cos(2\theta)+g_{w6}\sin(2\theta)\cos(2kz)\\&+g_{w7}\sin(2\theta)\sin(2kz)+g_{w8}(r)\cos(2\theta)\cos(2kz) +g_{w9}(r)\cos(2\theta)\sin(2kz) \label{eqn : gw} \end{split}$$ $$\begin{split} q_2(r,\theta,z)=&g_{q1}(r)+g_{q2}(r)\sin(2kz)+g_{q3}(r)\cos(2kz)+g_{q4}(r)\sin(2\theta)+g_{q5}(r)\cos(2\theta)+g_{q6}\sin(2\theta)\sin(2kz)\\& +g_{q7}\sin(2\theta)\cos(2kz)+g_{q8}(r)\cos(2\theta)\sin(2kz)+g_{q9}(r)\cos(2\theta)\cos(2kz). \label{eqn : gq} \end{split}$$ The calculation of the deformation at order $\varepsilon^2$ requires to take into account a series expansion of the dimensionless strain energy $W$ at order 2 in terms of $I_1-3$ and $I_2-3$. Here, we consider the most general form for this expansion, without any restriction to a specific kind of constitutive equation: $$W=\frac{1}{2}\beta(I_1-3)+\frac{1}{2}\left(1-\beta\right)(I_2-3)+\gamma_{11}(I_1-3)^2+\gamma_{12}(I_1-3)(I_2-3)+\gamma_{22}(I_2-3)^2\cdots, \label{eqn : 3rd order W}$$ where $\beta$, $\gamma_{11}$, $\gamma_{12}$ and $\gamma_{22}$ are constant parameters that depend on the material properties. For instance, $\beta=1$ and $\gamma_{11}=\gamma_{12}=\gamma_{22}=0$ for an incompressible neo-Hookean solid [@Ogden1984; @Macosko94], and $\gamma_{11}=\gamma_{12}=\gamma_{22}=0$ for an incompressible Mooney-Rivlin solid [@Mooney1940; @Rivlin1948]. The unknown functions in Eq. \[eqn : gu\]-\[eqn : gq\] are found by solving at order $\varepsilon^2$ the differential equations Eqs. \[eqn : 3d CP0 brute\]-\[eqn : 3d CP3 brute\] with the boundary conditions Eqs. \[eqn : 3d BC3 brute\]. Inserting Eqs. \[eqn : definition of alpha2\]-\[eqn : definition of xi\] into the Cauchy-Poisson Eqs. \[eqn : 3d CP0 brute\]-\[eqn : 3d CP3 brute\] and in the boundary condition Eqs. \[eqn : 3d BC3 brute\] at order $\varepsilon^2$ yields: $$\begin{aligned} g_{u1}(r)&=&\frac{\xi^2}{r_0}\left\{-\frac{r}{4r_0}(r_0 k)^2+\left(4 \frac{r}{r}-\frac{r^3}{r^3_0}\right) \frac{(kr_0)^4}{32}+\left(7 \frac{r}{r_0}-6 \frac{r^3}{r_0^3}+2 \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{128} +\cdots \right\} \nonumber \\ g_{u3}(r)&=&\frac{\xi^2}{r_0}\left\{-\frac{r}{8r_0}( kr_0)^2-\left((2 \beta-9) \frac{r}{r_0}+3 \frac{r^3}{r^3_0}\right) \frac{(kr_0)^4}{32}+\left((116 \beta-230) \frac{r}{r_0} +54 \frac{r^3}{r_0^3}+(12 \beta-33) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1152} +\cdots \right\}\nonumber\\ g_{u5}(r)&=&\frac{\xi^2}{r_0}\left\{\frac{r}{8r_0} (kr_0)^2-\frac{r^3}{16r_0^3}(kr_0)^4+\left((150 \beta-269) \frac{r}{r_0}+(20 \beta+198) \frac{r^3}{r_0^3}+(-30 \beta-19) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1536} +\cdots \right\}\nonumber\\ g_{u9}(r)&=&\frac{\xi^2}{r_0}\left\{\frac{r}{8r_0} (kr_0)^2-\left(6 \frac{r}{r_0}-3 \frac{r^3}{r_0^3}\right) \frac{(kr_0)^4}{16}+\left((2 \beta+561) \frac{r}{r_0}+(60 \beta-422) \frac{r^3}{r_0^3}+(-26 \beta+111) \frac{r^5}{r^5_0}\right)\frac{(kr_0)^6}{1536} +\cdots \right\}\nonumber\end{aligned}$$ $$\begin{aligned} g_{v5}(r)&=&\frac{\xi^2}{r_0}\left\{-(\frac{r}{8r_0} (kr_0)^2-\left((150 \beta-269) \frac{r}{r_0}+(40 \beta-36) \frac{r^3}{r_0^3}+(-90 \beta+75) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1536} +\cdots \right\}\nonumber\\ g_{v9}(r)&=&\frac{\xi^2}{r_0}\left\{-\frac{r}{8r_0} (kr_0)^2+\left(3 \frac{r}{r_0}-\frac{r^3}{r^3_0}\right) \frac{(kr_0)^4}{8}-\left((2 \beta+561) \frac{r}{r_0}+(-8 \beta-348) \frac{r^3}{r_0^3}+(-14 \beta+49)\frac{r^5}{r_0^5}\right) \frac{(kr_0)^6}{1536}+\cdots \right\}\nonumber\end{aligned}$$ $$\begin{aligned} g_{w3}(r)&=&\frac{\xi^2}{r_0}\left\{-\frac{kr_0}{8}+(2 \beta-5)\frac{(kr_0)^3}{32}-\left((116 \beta-293)+(36 \beta-63) \frac{r^4}{r^4_0}\right) \frac{(kr_0)^5}{1152} +\cdots \right\}\nonumber\\ g_{w9}(r)&=&\frac{\xi^2}{r_0}\left\{-\frac{r^2}{8r_0^2} (kr_0)^3-\left((16 \beta-44) \frac{r^2}{r_0^2}+(-8 \beta+31) \frac{r^4}{r^4_0}\right) \frac{(kr_0)^5}{192} +\cdots \right\}\nonumber\end{aligned}$$ $$\begin{aligned} g_{q1}(r)&=&\xi^2k^2\left\{\frac{1}{4}-\left(3-8(\beta-1 -6 \gamma) \frac{r^2}{r_0^2}\right) \frac{(kr_0)^2}{16}\right. \nonumber\\ &&\left. - \left((-12 \beta+29+256\gamma)+(64 \beta-120-832\gamma) \frac{r^2}{r_0^2}+(-12 \beta+33+320\gamma) \frac{r^4}{r_0^4}\right)\frac{(kr_0)^4}{128} +\cdots \right\}\nonumber\\ g_{q3}(r)&=&\xi^2k^2\left\{\left(\beta+1+4(6 \gamma-\beta+1) \frac{r^2}{r_0^2}\right) \frac{(kr_0)^2}{8}\right. \nonumber \\ && \left.-\left((-44 \beta+515+2304\gamma)+(-720 \beta+252+2880\gamma)\frac{r^2}{r_0^2}+(180 \beta-153-576\gamma) \frac{r^4}{r_0^4}\right) \frac{(kr_0)^4}{1152} +\cdots \right\}\nonumber\\ g_{q5}(r)&=&\xi^2 k^2\left\{(24 \gamma-4 \beta+7) \frac{r^2}{r_0^2} \frac{(kr_0)^2}{8}-\left((-258 \beta+489+2496\gamma) \frac{r^2}{r_0^2}+(-24 \beta-74-960\gamma) \frac{r^4}{r^4_0}\right) \frac{(kr_0)^4}{384} +\cdots \right\}\nonumber\\ g_{q9}(r)&=&\xi^2 k^2\left\{-(24 \gamma-4 \beta+7) \frac{r^2}{r_0^2}\frac{(kr_0)^2}{8}+\left((-218 \beta+289+960\gamma) \frac{r^2}{r_0^2}+(72 \beta-106-192\gamma) \frac{r^4}{r_0^4}\right)\frac{(kr_0)^4}{384} +\cdots \right\}\nonumber\end{aligned}$$ with $\gamma=\gamma_{11}+\gamma_{12}+\gamma_{22}$. The other functions $g$ defined in Eqs. \[eqn : gu\]-\[eqn : gq\] are equal to zero. Note that the boundary conditions Eqs. \[eqn : boundary ends 1\] at $Z=0$ and $Z=L$ are fulfilled at order $\varepsilon^2$. Amplitude equation {#sec : amplitude} ------------------ The Koiter method proceeds by inserting the expansion in Eqs. \[eqn : definition of alpha2\]–\[eqn : definition of xi\] into the non-linear equilibrium written earlier in Eq. \[eqn : general1\] as $$\forall \delta{\mathbf{t}},\quad D{\cal E}\left(\alpha_{\mathrm{c}} +\alpha_2 \varepsilon^2, \mathbf{t}_{0}(\alpha) + \varepsilon\mathbf{t}_{1} + \varepsilon^2\mathbf{t}_{2} +\varepsilon^3\mathbf{t}_{3}+ \cdots\right)[\delta \mathbf{t}]=0, \label{eqn : equilibrium condition total}$$ where $\delta \mathbf{t}(r,\theta,z)$ is the set of virtual functions $\bigl(\delta{u},\delta{v},\delta{w},\delta{q}\bigl)$ that represent infinitesimal increments of the displacements (including the Lagrange multiplier) satisfying the kinematic boundary conditions. Eq. \[eqn : equilibrium condition total\] is then expanded order by order in $\varepsilon$ [@Heijden2009koiter; @Triantafyllidis-Stability-of-solids:-from-2011; @Chakrabarti2018]. Order $\varepsilon$ of Eq. \[eqn : equilibrium condition total\] yields the linear bifurcation problem : $$\forall \delta{\mathbf{t}},\quad D^2{\cal E}\left(\alpha_{\mathrm{c}}\right) \left[\mathbf{t}_{1},\delta \mathbf{t}\right]=0. \label{eq:KoiterOrder2}$$ Order $\varepsilon^{2}$ yields the equations for the second-order correction $\mathbf{t}_{2}$. One obtains at $\varepsilon^{3}$ the equation: $$\begin{gathered} \forall \delta{\mathbf{t}},\quad D^2\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot \left[\mathbf{t}_{3},\delta \mathbf{t}\right]+ D^3\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot \left[\mathbf{t}_{2},\mathbf{t}_{1},\delta \mathbf{t}\right] \\ {}+\alpha_2 \left.\frac{\mathrm{d} D^2\mathcal{E}(\alpha,\mathbf{t}_{0}(\alpha))}{\mathrm{d}\alpha} \right|_{\alpha=\alpha_{\mathrm{c}}} \cdot \left[\mathbf{t}_{1},\delta \mathbf{t}\right] +\frac{1}{6} D^4\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot \left[\mathbf{t}_{1},\mathbf{t}_{1},\mathbf{t}_{1},\delta \mathbf{t}\right] = 0 \textrm{.} \nonumber\end{gathered}$$ Upon insertion of the particular virtual displacement $\delta \mathbf{t} = \mathbf{t}_{1}$, the first term cancels out by Eq. \[eq:KoiterOrder2\] and we are left with $$D^3\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot \left[\mathbf{t}_{2},\mathbf{t}_{1},\mathbf{t}_1\right] +\alpha_2\left.\frac{\mathrm{d} D^2\mathcal{E}(\alpha,\mathbf{t}_{0}(\alpha))}{\mathrm{d}\alpha} \right|_{\alpha=\alpha_{\mathrm{c}}} \cdot \left[\mathbf{t}_{1},\mathbf{t}_1\right] +\frac{1}{6} D^4\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot \left[\mathbf{t}_{1},\mathbf{t}_{1},\mathbf{t}_{1},\mathbf{t}_1\right] = 0 \textrm{.} \label{eqn : gateau4}$$ $D^2{\cal E}\left[\delta \mathbf{t};\mathbf{t}_1\right]$ denotes the second Gâteaux derivative of ${\cal E}$, which is a bi-linear symmetric form on the increment $\mathbf{t}_1$ and on the virtual increment $\delta \mathbf{t}$. Similarly, $D^3{\cal E}\left[\delta \mathbf{t};\mathbf{t}_1;\mathbf{t}_2 \right]$ is the third Gâteaux derivative (a tri-linear symmetric form). From Eqs. \[eqn : definition of alpha2\]–\[eqn : definition of xi\], the value of $\alpha_{2}$ finally allows to express the buckling amplitude $\varepsilon \xi$ as a function of the increment of the load $\alpha-\alpha_{c} = \alpha_{2}\varepsilon^{2}$. The quantities appearing Eq. \[eqn : gateau4\] are calculated with the help of a symbolic calculation language from the explicit expression of the functions $f$ (linear order) and the functions $g$ (second order): $$\begin{aligned} D^3\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot [\mathbf{t}_{2},\mathbf{t}_{1},\mathbf{t}_1] &=&\frac{\mu \pi^2 r_0^2}{k} \left(\frac{\xi}{r_0} \right)^4\left\{ -\frac{3(kr_0)^4}{8}(16 \gamma-2 \beta+1 )+(kr_0)^6(2 \gamma-\frac{5}{16})+\cdots \right\},\\ D^4\mathcal{E}(\alpha_{\mathrm{c}},\mathbf{t}_{0}(\alpha_{\mathrm{c}}))\cdot [\mathbf{t}_{1},\mathbf{t}_{1},\mathbf{t}_{1},\mathbf{t}_1 &=&\frac{\mu \pi^2 r_0^2}{k} \left(\frac{\xi}{r_0} \right)^4\left\{\frac{9(kr_0)^4}{2}(8 \gamma- \beta+1)-\frac{3 (kr_0)^6}{8}(32\gamma-6 \beta+7) +\cdots \right\},\\ \left. \frac{\mathrm{d} D^2\mathcal{E}(\alpha,\mathbf{t}_{0}(\alpha))}{\mathrm{d}\alpha}\right|_{\alpha=\alpha_{\mathrm{c}}}&=&-\frac{\mu \pi^2 r_0^2}{k}\frac{1}{2} \left(\frac{\xi}{r_0} \right)^2. \end{aligned}$$ Eq. \[eqn : gateau4\] then yields the sought relation between the scaled load increment $\alpha_2$ and amplitude $\xi$: $$\alpha_2=\kappa \frac{\xi^2}{r_0^2}, \label{eqn : kappa2}$$ with $$\kappa=\frac{3}{4} (kr_0)^4+\frac{3}{4}(\beta-2) (kr_0)^6+\left(\frac{27}{8} \gamma-\frac{3}{32} \beta^2-\frac{61}{96} \beta+\frac{105}{128}\right) (kr_0)^8+\cdots \label{eqn : kappa}$$ Multiplying both sides of Eq. \[eqn : kappa2\] by $\varepsilon^{2}$ and identifying ([*i*]{}) the load increment $\alpha-\alpha_{c}$ from Eq. \[eqn : definition of alpha2\] and ([*ii*]{}) the true buckling amplitude $\zeta = \varepsilon\,\xi$, we find the amplitude equation as $$\left( \frac{\zeta}{r_0}\right)^2=\frac{1}{\kappa}\left( \alpha-\alpha_c\right). \label{eqn : zeta order 2}$$ ![$\kappa$ defined by Eq. \[eqn : kappa\] calculated from series expansions at order 30 in $kr_0$ for a neo-Hookean constitutive law $(\beta;\gamma)=(1;0)$; two Mooney-Rivlin constitutive laws $(\beta;\gamma)=(0.5;0)$ and $(\beta;\gamma)=(1;0)$ ; and a constitutive law with $(\beta;\gamma)=(0;-0.5)$.[]{data-label="fig : kappa"}](kappa.pdf){width="55.00000%"} $\kappa$ is plotted as a function of $kr_0$ for different constitutive laws in Figure \[fig : kappa\]. The order in the expansion in $kr_0$ in Eq. \[eqn : kappa\] is high enough to have no visible effect of it in this plot. Interestingly, $\kappa$ can be positive or negative, depending on the constitutive law and on the values of $kr_0$. Based on the amplitude equation in Eq. \[eqn : zeta order 2\], the bifurcation is super critical (continuous) if $\kappa>0$ ($\zeta=r_0\sqrt{(\alpha-\alpha_c)/\kappa}$) and sub-critical (discontinuous) otherwise ($\zeta=r_0\sqrt{(\alpha_c-\alpha)/\kappa}$). The bifurcated branch is found above the critical load $\alpha_c$ in the super critical case, and below $\alpha_c$ in the sub critical case. ![Sketches of the buckling amplitude close to the bifurcation point. (a) for $\kappa<0$ the bifurcation is sub-critical. (b) for $\kappa>0$ the bifurcation is super-critical.[]{data-label="fig : bifurcations"}](bifurcations.pdf){width="55.00000%"} Comparison with finite element simulations {#sec : FEM} ========================================== In this section, the complete non-linear problem defined by Eq. \[eqn : general1\] is implemented by using the open source tool for solving partial differential equations FEniCS [@Fenics2012]. The goal is to check whether the numerical simulations well capture the results of both the linear and the non-linear analysis of sections \[sec : linear\]-\[sec : non linear\]. We consider a semicircular solid cylinder $\Omega$ of radius $r_0$ and height $2 \pi /k$. A Cartesian coordinates system ($x,y,z$) with the base vectors ($\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z$) is chosen such that $(x,y,z) \in \Omega \Leftrightarrow \sqrt{x^2+y^2}\leq r_0$, $x \geq 0$ and $0 \leq z \leq 2 \pi / k$. An incompressible and isotropic elastic solid (mass density $\rho$ ; shear modulus $\mu$) occupying the domain $\Omega$ in its reference configuration is subjected to the action of the centrifugal volume force $\rho \omega^2 (x\mathbf{e}_x+y\mathbf{e}_y)$. The lateral surface ($\sqrt{x^2+y^2}= r_0$) of the cylinder is traction free, the displacements in the direction of $\mathbf{e}_x$ are set to zero for $x=0$ (so that $x=0$ is a plane of symmetry), and periodic boundary conditions along axis $z$ with wave number $k$ are implemented.\ The displacement vector $\mathbf{u}$ is discretized using Lagrange finite elements with a quadratic interpolation, and the Lagrange multiplier $q$ with a linear interpolation. The non-linear problem in $\mathbf{u}-q$ is solved using a Newton algorithm based on a direct parallel solver (`MUMPS`). Quasi-static simulations are performed by setting $\mu=1$, $r_0=1$, and slowly varying $\alpha\equiv \rho \omega^2$ up to the desired value. For each $\alpha$ the displacement field and the Lagrange multiplier are computed. Simulations are carried out for different values of the wave number and different elastic constitutive laws. ![Snapshots from FEM simulations of a neo-Hookean spinning cylinder with $kr_0=\frac{\pi}{4}$, for $\frac{\alpha-\alpha_c}{\alpha_c}=0$ (a), $\num{8.4e-4}$ (b), $\num{4.2e-2}$ (c) and $\num{0.1}$ (d). The buckling amplitudes are respectively equal to $0$ (a), $0.037$ (b), $0.29$ (c) and $0.45$ (d). Colors indicate the normalized displacement along $x$-direction, $u_x/r_0$.[]{data-label="fig : snap"}](snap.pdf){width="45.00000%"} ![Gray symbols: Normalized buckling amplitude $\zeta/r_0$ as a function of $\alpha$ for the neo-Hookean constitutive law, obtained by FEM simulations for $kr_0=\pi/15$. Solid line is obtained by fitting $\zeta/r_0=\sqrt{(\alpha-\alpha^*)/\kappa^*}$ via $\alpha^*$ and $\kappa^*$: $\alpha^*=0.00144$ and $\kappa^*=0.00137$. Inset : same data plotted in log-scales.[]{data-label="fig : p030"}](p030.pdf){width="45.00000%"} Starting from the undisturbed base system ($\mathbf{u}=0$), $\alpha$ is gradually increased with increments $\delta \alpha=1/100000$. The deformation is almost zero until a critical value of $\alpha$ for which the deformation begins to increase (as a function of $\alpha$) abruptly. Due to the boundary conditions imposed in the simulations, these deformations are consistent with those investigated in Secs. \[sec : linear\]-\[sec : non linear\] (see Figure \[fig : snap\]). Accordingly with the definition in Section \[sec : amplitude\] of buckling amplitude $\zeta$, the buckling amplitude in the simulations is computed as the maximum displacement of the material points located at the lateral boundary of the cylinder. The normalized buckling amplitude $\zeta/r_0$ computed from the simulations for a neo-Hookean constitutive law and the wave number $kr_0=\pi/15$ is plotted as a function of $\alpha$ in Figure \[fig : p030\]. Fitting $\zeta/r_0$ with the function $f(\alpha)$ defined by : $$\begin{aligned} f(\alpha)=0 &\mbox{ for }& \alpha<\alpha^* \nonumber \\ f(\alpha)=\sqrt{\left(\alpha-\alpha_c\right)/\kappa^*} &\mbox{ for }& \alpha>\alpha^*, \label{eqn : star}\end{aligned}$$ for $\zeta/r_0<0.2$ gives values of the threshold $\alpha^*$ computed from the simulations, as well as the coefficient $\kappa^*$. ![$\kappa$ as a function of $kr_0$ in log-log scale for different constitutive laws. Solid lines result from of Eq. \[eqn : kappa\] (calculated at order 30 in $kr_0$; the plot would not change by increasing the order). Filled circles results from fits of the buckling amplitude calculated by FEM simulation, as presented in Figure \[fig : p030\]. []{data-label="fig : kappa log"}](kappa_log.pdf){width="50.00000%"} In Figure \[fig : linear threshold\], $\alpha^*$ is plotted together with the theoretical prediction of the linear threshold, Eq. \[eqn : alphac\]. A comparison of $\kappa^*$ with the theoretical prediction based on the weakly non-linear analysis, Eq. \[eqn : kappa\], is shown in Figure \[fig : kappa log\] for different wave numbers and different constitutive laws. The good agreement between theory and simulations clearly validates the results of Sections \[sec : linear\] and \[sec : non linear\]. In addition, the simulations show that the prediction of Eqs. \[eqn : kappa\]-\[eqn : zeta order 2\] remains good for finite values of $\zeta/r_0$ (Figure \[fig : p030\]). Discrepancies with the square root expression of the weakly non-linear analysis are barely observable in log-log scales (inset of Figure \[fig : p030\]). For instance, for a neo-Hookean constitutive law and $kr_0=15\pi$, differences are smaller than 1% for $\zeta/r_0<90\%$.\ The sub-critical nature of the bifurcation, unveiled in Figure \[fig : kappa\] for certain values of $\beta$ and $\gamma$ and certain values of the wave number, is also captured by the FEM simulations. In those cases, the load $\alpha$ has to be gradually decreased from a value larger than the instability threshold, and the buckling amplitude is found to grow as $\alpha$ continues to decrease below the critical load (Figure \[fig : q0065\]). For these sub-critical bifurcations, the range of the load in which the buckling amplitude follows a square root law is more reduced compared to the super-critical case. ![Filled circles: Normalized buckling amplitude $\zeta/r_0$ as a function of $\alpha$ for a Mooney-Rivlin constitutive law with $\beta=0$, obtained by FEM simulations for $kr_0=\pi/3.25$. Solid line is the prediction of the weakly non-linear theory of Section \[sec : non linear\].[]{data-label="fig : q0065"}](q0065.pdf){width="50.00000%"} Concluding remarks {#sec : conclusion} ================== The non-linearities driving the buckling amplitude in the bifurcation of an initially straight spinning cylinder arise both from the geometry and the elastic response of the material. They simultaneously appear at order 2 in expansions with respect to the amplitude of the deformations. The buckling amplitude has been calculated in the weakly non-linear regime for different wave numbers and for any isotropic and isochoric constitutive law of the elastic material. Since the calculation relies on a Koiter expansion of the deformation calculated from a base undeformed configuration, the obtained analytic expression is limited to infinitesimal deformations. It has been complemented with numerical simulations, showing that the analytic expression is indeed relevant beyond the limit of the small infinitesimal deformations. In the long wave length limit ($kr_0\ll 1$), $\alpha_c \sim \kappa \sim \frac{3}{4}(kr_0)^4$ (from Eqs. \[eqn : alphac\] and \[eqn : kappa\]). Hence, $\zeta=r_0\sqrt{\alpha/\alpha_c-1}$. This formula differs from the expression proposed in [@Hosseini2013] and established in the long wave length limit through a one-dimensional model and by ignoring material non-linearities. This discrepancy shows that an approach based on Hookean elasticity for calculating the buckling amplitude is not relevant even in the long wave length limit. Indeed, calculating the buckling amplitude using reduction to one dimensional model would require a reduction consistent with the non-linear material constitutive law of the elastic material [@Audoly2019]. Non-linearities in the elastic material properties are a key ingredient for the study of the whirling instability: the buckling amplitude at the instability onset, and also the nature of the bifurcation (sub-critical or super-critical) depend on the coefficients appearing in the second order expansion of the strain energy density. Indeed, non-linear elastic properties are important in many systems or devices in which elastic bodies are subjected to finite deformations, as in soft robotics and surgery. As in the whirling instability investigated here, these deformations can be associated to instabilities [@Mora_softmatter2011; @Mora_prl2014; @Mora_softmatter2019] that lead to dramatic change in the system behaviour. The development of rigorous frameworks and methodologies for predicting, understanding and analysing these instabilities is then an important task. The deformations considered in this paper being stationary, the equilibrium configurations have been analyzed by minimizing the total energy of the system, since energy dissipation processes are not relevant. A study of the issue of the transient regimes, [*i.e.*]{} the way the previously investigated steady states are reached, would required more complex formulations in which the dissipative processes have to be accounted for together with the material and geometric non-linearities in dynamical equations.\ [**Acknowledgments:**]{} Corrado Maurini is thanked for his help with FEniCS.\ 0.05[**Conflict of Interest:**]{} The author declares that he has no conflict of interest. [10]{} E. Krämer, editor. . Springer-Verlag, 1993. G. Genta, editor. . Springer, 2005. W.J. Chen and E.J. Gunter, editors. . Trafford, 2005. F.F. Ehrich. Shaft whirl induced by rotor internal damping. , 31:279–282, 1964. S. Noah and P. Sundarajan. 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Worst shapes of imperfections for space trusses with many simultaneously buckling members. , 29:2385–2402, 1992. R. Peek and M. Kheyrkhahan. Postbuckling behavior and imperfection sensitivity of elastic structures by the [Lyapunov-Schmidt-Koiter]{} approach. , 108(3):261–279, 1993. A. van der Heijden. . Cambridge University Press Cambridge, 2009. A. Chakrabarti, S. Mora, F. Richard, T. Phou, J.M. Fromental, Y. Pomeau, and B. Audoly. Selection of hexagonal buckling patterns by the elastic rayleigh-taylor instability. , 121:234–257, 2018. Nicolas Triantafyllidis. . , 2011. A. Logg, K.A. Mardal, and G. Wells. . Springer, 2012. C. Lestringant and B. Audoly. Asymptotically exact strain-gradient models for nonlinear slender elastic structures: A systematic derivation method. , page 103730, 2019. S. Mora, M. Abkarian, H. Tabuteau, and Y. Pomeau. Surface instability of soft solids under strain. , 7:10612–10619, 2011. S. Mora, T. Phou, J. M. Fromental, and Y. Pomeau. 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--- abstract: 'We calculate the ratio $R_{\ell\ell}$ of same sign (SS) to opposite sign (OS) dileptons in type I and generalized inverse seesaw models and show that it can be anywhere between 0 and 1 depending on the detailed texture of the right-handed neutrino mass matrix. Measurement of $R_{\ell\ell}$ in hadron colliders can therefore provide a way to probe the nature of seesaw mechanism and also to distinguish between the two types of seesaw mechanisms. We work within the framework of left-right symmetric model as an example. We emphasize that coherence of the final states in the $W_R$ decay is crucial for this discussion and it requires the right-handed neutrinos to be highly degenerate. We isolate the range of parameters in the model where this effect is observable at the LHC and future colliders.' author: - Arindam Das - 'P. S. Bhupal Dev' - 'Rabindra N. Mohapatra' title: Same Sign versus Opposite Sign Dileptons as a Probe of Low Scale Seesaw Mechanisms --- Introduction ============ Different kinds of seesaw mechanism have been proposed as ultraviolet (UV)-complete theories that lead to the dimension-5 Weinberg operator [@weinberg] for understanding small neutrino masses. Two of them are the so called type-I [@seesaw1; @seesaw2; @seesaw3; @seesaw4; @seesaw5] and inverse seesaw [@ISS1; @ISS2], which have been widely discussed in the literature. The type-I seesaw involves adding SM-singlet heavy fermions $N$ with Majorana masses that violate lepton number maximally, whereas in the inverse seesaw, one adds two SM-singlet heavy neutrinos $N$ and $S$ and a small $L$-violating mass for one set of the new singlet fermions. A simple UV-complete extension of the Standard Model (SM) that incorporates all the key ingredients of both type I seesaw and inverse seesaw and leads to them naturally is the left-right symmetric model [@LR1; @LR2; @LR3]. No extra symmetries need to be added to generate the right texture for getting tiny neutrino masses. The right-handed neutrino (RHN), predicted by anomaly considerations in this theory, couples to the right-handed (RH) gauge boson $W_R$ and is the source of the lepton number violating (LNV) signal [@KS] we will discuss. In this paper we will work within the framework of the minimal left-right model and assume that $W_R$ is kinematically accessible to the colliders. In other words, for $\sqrt s=14$ TeV LHC, we assume the mass of the $W_R$ boson to be less than 5 TeV or so [@Ferrari:2000sp; @Deppisch:2015qwa]. A key predictions for the TeV-scale left-right type-I seesaw model is that it leads to a spectacular LNV signal in hadron colliders in the form of two same-sign leptons and two jets with no missing energy [@KS]. This arises from the production and decay of heavy RHNs, both mediated by the $W_R$ gauge boson in the $s$-channel. The Majorana nature of the RHN dictates that the final states with same-sign (SS) dileptons ($\ell^\pm\ell^\pm$) appear in equal number with opposite-sign (OS) dilepton states ($\ell^\pm\ell^\mp$). In other words, the minimal left-right type-I seesaw prediction is that the ratio of the number of events in the two final states, $R_{\ell\ell}\equiv N_{\rm SS}/N_{\rm OS}=1$. This in fact is considered a ‘smoking gun’ signal for TeV-scale type-I seesaw in general[^1] and, more specifically, for the left-right seesaw model and has been extensively studied in the literature, both for the LHC [@KS; @Ferrari:2000sp; @Deppisch:2015qwa; @Nemevsek:2011hz; @Das:2012ii; @AguilarSaavedra:2012gf; @Han:2012vk; @Chen:2013fna; @Khachatryan:2014dka; @Ng:2015hba; @Dev:2015kca; @Gluza:2016qqv; @Mitra:2016kov; @Ruiz:2017nip; @Aad:2015xaa; @Dev:2016dja; @Roitgrund:2017byx], as well as other future colliders [@Lindner:2016lxq; @Mondal:2015zba; @Biswal:2017nfl; @Golling:2016gvc]. On the other hand, in the inverse seesaw mechanism, lepton number breaking is very small, because the heavy singlet neutrino ($N$) is paired with another singlet fermion ($S$) to form a pseudo-Dirac pair and the Majorana nature of the neutrino emerges from a keV-scale Majorana mass $\mu_S$ of $S$ fermion (for TeV-scale seesaw). This model when embedded into the TeV-scale left-right framework exhibits some interesting features. There are two versions of this model: the minimal version where there is no majorana mass for the $N$ [@Dev:2009aw; @An:2011uq; @Chen:2011hc] and a second more general one where there is a Majorana mass $\mu_R$ for $N$ [@Dev:2012sg; @Awasthi:2013ff; @Dev:2015pga]. In the minimal version, the leading order prediction for collider signal is that final states will approximately conserve lepton number, implying that $R_{\ell\ell}\simeq 0$ [@Chen:2011hc]. In the more general inverse seesaw, which can also arise from left-right seesaw models [@Dev:2015pga], the neutrino mass formula remains unaffected at the tree-level, although there is an unavoidable one-loop contribution from electroweak radiative corrections [@Dev:2012sg]; however the $N$ fermion has a potentially large Majorana mass that breaks lepton number by two units. The question remains as to how do the dilepton final states look like in this general case i.e. is $R_{\ell\ell}=1$ or different? This question has been recently studied in some special cases [@Dev:2015pga; @Anamiati:2016uxp; @Antusch:2017ebe] and was shown that due to interference between two heavy Majorana neutrino mass eigenstates, one could in principle realize a scenario with $R_{\ell\ell}$ anywhere between 0 and 1. The goal of this study is to do a more general analysis and discuss whether analyzing dilepton states in a hadron collider via production of a $W_R$ boson, one can probe the details of the RHN mass matrix and distinguish between the type-I and general inverse seesaw mechanisms. The rest of the paper is organized as follows. In Section \[sec:coh\] we discuss the coherence condition for interference between two heavy Majorana neutrino mass eigenstates, which plays a crucial role in our discussion. In Section \[sec:typeI\], we apply the coherence conditions to discuss the nature of dilepton final states in type-I seesaw. In Section \[sec:inv\] we explain the general inverse seesaw model. In Section \[sec:inv2\] we apply the coherence conditions for the inverse seesaw case to get the $R_{\ell\ell}$ as a function of parameters of inverse seesaw model. We give our conclusions in Section \[sec:con\]. Some useful three-body decay widths for the RHN are listed in Appendix \[sec:app\]. Coherence Conditions for Interference {#sec:coh} ===================================== When a $W_R$ gauge boson is produced in proton-proton collisions, it decays into flavor eigenstates of the RHNs $N_{\ell}$ along with the corresponding charged lepton $\ell_R$ (where $\ell=e,\mu,\tau$). For simplicity, let us consider two RHNs, say $N_e$ and $N_\mu$. When these flavor eigenstates evolve, they do so as linear combination of mass eigenstates $N_{1,2}$. The $N_{1,2}$ are linear combinations of $N_e$ and $N_\mu$ in the type-I seesaw case and of $N$ and $S$ in the inverse seesaw case. The $N_{1,2}$ are Majorana fermions and they will evolve and interfere as they produce the charged leptons (along with two jets) in their final state. Only if the coherence condition (discussed below) is satisfied, they will interfere; otherwise they will simply give equal number of SS and OS dilepton final states. The coherence conditions for light neutrinos have been discussed inRefs. [@Kayser:1981ye; @Akhmedov:2007fk]. There are two conditions that must be satisfied for interference between the two states to take place: (i) coherence in emission and (ii) the coherence must be maintained till the RHNs decay i.e. for their full decay length. The results imply that the first condition is satisfied when the uncertainty in their mass square exceeds their actual mass difference. We now transplant their argument to the case of two RHNs at hand. Denoting by $\sigma_{m^2}$ the mass uncertainty, we get for the coherence condition $\sigma_{m^2}$ $\geq \Delta M^2\equiv |M^2_1-M^2_2|$. The $\sigma_{m^2}$ in this case is estimated to be $2\sqrt{2} E\Gamma_{W_R}$ where $E$ is the average energy of the RHN eigenstates and $\Gamma_{W_R}$ is the width of the $W_R$ which causes the uncertainly in the energy of the produced heavy neutrino state. Thus, in our case, coherence in emission occurs when $$\begin{aligned} \Delta M^2 \ \leq \ 2\sqrt{2}E\Gamma_{W_R} \, . \label{eq:coh1}\end{aligned}$$ For TeV-scale $W_R$ and RHNs, this is satisfied when the mass difference between the states is less than few hundred GeV, where we have estimated $\Gamma_{W_R}\simeq (g^2/12\pi) M_{W_R}$, setting the $SU(2)_L$ and $SU(2)_R$ couplings to be equal, i.e. $g_L=g_R\equiv g$. Turning to the second condition, we take the decay length $L$ as $L=1/\Gamma_N$ and using the results of Ref. [@Akhmedov:2007fk], require that $L\leq \sigma_x/\delta v_g$, where $\sigma_x$ is the size of the RHN wave packets and $\delta v_g$ is the difference between the group velocities of the individual RHNs. We have $\sigma_x \sim ( \sigma_E )^{-1}\sim (\Delta M^2/2\sqrt{2}E)^{-1}$ and $\delta v_g\sim (\Delta M^2/2E^2)$. Putting theses together, we get $$\begin{aligned} L \ \equiv \ \frac{1}{\Gamma_N} \ < \ \frac{4\sqrt{2}E^3}{(\Delta M^2)^2} \, . \label{eq:coh2}\end{aligned}$$ This implies a stringent condition on the mass difference between the two interfering RHNs. For instance, for $M_{W_R}=5$ TeV, $M_{N}\simeq 1$ TeV, and $E\sim 2$ TeV, we get the coherence condition $\Delta M\equiv |M_1-M_2|\leq $ GeV. Note that this condition is more stringent than what condition (i) alone would have implied \[cf. Eq. \] and requires a degeneracy of one part in $10^3$ between the two RHN masses for interference to take place. In deriving this, we have used the decay width formula for the RHNs given in Appendix \[sec:app\]. From this discussion, we conclude that if interference effect is observed, it will imply constraints on the mass matrix of both the type I and inverse seesaw, helping to further elucidate the nature of the seesaw. It will for example imply that there are at least two nearly degenerate RHN states, consistent with the general expectation from many TeV-scale seesaw models [@Pilaftsis:1991ug; @Gluza:2002vs; @Kersten:2007vk; @Xing:2009in; @He:2009ua; @Adhikari:2010yt; @Ibarra:2010xw; @Mitra:2011qr; @Dev:2013oxa; @Chattopadhyay:2017zvs], which require the quasi-degeneracy to satisfy the neutrino oscillation data. This is also the requirement for successful resonant leptogenesis via the out-of-equilibrium decay of TeV scale RHNs [@Pilaftsis:2003gt; @Dev:2017wwc]. Same Sign vs Opposite Sign Dilepton Events in Type-I seesaw {#sec:typeI} =========================================================== Let us first briefly recapitulate the well known field theoretic argument of why for Majorana RHNs the final states in its decay have equal number of both sign leptons. For concreteness, we illustrate this in the context of left-right model but the argument is general. In the left-right model, the decay of $N$ can be assumed to occur via the emission of a virtual $W_R$ boson and it comes from the RH gauge interaction $$\begin{aligned} {\cal L}_I \ = \ \frac{g}{\sqrt{2}}\bar{\ell}_R\gamma_\mu N W^{-,\mu}_R+\frac{g}{\sqrt{2}} N^TC^{-1}\gamma_\mu\ell_RW^{+,\mu}_R \label{eq:1}\end{aligned}$$ The second term in the above equation is nothing but the hermitian conjugate of the first one after we use the Majorana condition for $N$ i.e. $N=C\bar{N}^T$ (where $C$ is the charge conjugation operator). Now note that in both terms the $N$ field is annihilated but the final state from the first term is an $\ell^-$ whereas that from the second term is an $\ell^+$, while both the amplitudes are the same i.e. $g/\sqrt{2}$. This is the basic reason for equal number of SS and OS dileptons in the final states which for a $pp$ collision leads to their ratio $R_{\ell\ell}=1$. To see how interference between two RHN states affects the ratio $R_{\ell\ell}$, let us consider the simple case of type-I seesaw with only two heavy neutrino flavors $(N_e,N_\mu)$. This case has been discussed in some details in Refs. [@Bray:2007ru; @Gluza:2015goa; @Carmona:2016oxx; @Gluza:2016qqv]. Here we emphasize the importance of the coherence condition and present new analytic results on the effect on different flavor combinations of the final states. One can easily generalize this to more flavors, but the main conclusion of this section remains unchanged. Including the effect of $CP$ violation, we can write the flavor eigenstates as the following combinations of the mass eigenstates: $$\begin{aligned} N_e \ & = & \ c_\theta N_1+s_\theta e^{i\delta}N_2\, , \nonumber \\ N_\mu\ & = & \ -s_\theta N_1+c_\theta e^{i \delta}N_2 \, , \label{eq:3.2}\end{aligned}$$ where $\delta$ is the $CP$ phase in the RHN mixing, $\theta$ is the mixing angle in this sector, and $c_\theta\equiv \cos\theta,\, s_\theta \equiv \sin\theta$. For the general $2\times 2$ RHN mass matrix $$\begin{aligned} {\cal M}_N \ = \ \begin{pmatrix} M_1 & Me^{i\phi} \\ Me^{i\phi} & M_2 \end{pmatrix}\, , \label{mass}\end{aligned}$$ the mixing angle is given by $$\begin{aligned} \theta \ = \ \frac{1}{2}\tan^{-1}\left|\frac{2M}{M_1-M_2}\right| \, . \label{eq:theta}\end{aligned}$$ Substituting Eqs.  in the interaction Lagrangian for RHNs in Eq. , we get $$\begin{aligned} \mathcal{L}_I &\ = \ &\frac{g}{\sqrt{2}}\Big[\bar{e}_R\gamma_\mu (c_\theta N_1+s_\theta e^{i\delta}N_2)W^{-,\mu}_R+(c_\theta N_1+s_\theta e^{-i\delta}N_2)^TC^{-1}\gamma_\mu e_RW^{+,\mu}_R \nonumber \\ &+&\bar{\mu}_R\gamma_\mu (-s_\theta N_1+c_\theta e^{i\delta}N_2)W^{-,\mu}_R+ (-s_\theta N_1+c_\theta e^{-i\delta}N_2)^TC^{-1}\gamma_\mu\mu_R W^{+,\mu}_R \Big] \end{aligned}$$ where we have assumed that RH charged leptons are the mass eigenstate. Using the coherence conditions, we can write the time evolution of the amplitudes for SS and OS final states as follows: $$\begin{aligned} A_{{\rm OS},ee}(t)&\ = \ &c^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+s^2_\theta e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, , \label{eq:OSee} \\ A_{{\rm SS},ee}(t)&\ = \ &c^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+s^2_\theta e^{-2i\delta} e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, ,\label{eq:SSee} \\ A_{{\rm OS},\mu\mu}(t)&\ = \ &s^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+c^2_\theta e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, ,\label{eq:OSmumu} \\ A_{{\rm SS},\mu\mu}(t)&\ = \ &s^2_\theta e^{-iE_1 t -\frac{\Gamma_1t}{2}}+c^2_\theta e^{-2i\delta}e^{-iE_2 t -\frac{\Gamma_2t}{2}} \, , \label{eq:SSmumu}\\ A_{{\rm OS},e\mu}(t)&=&-c_\theta s_\theta \Big[ e^{-iE_1 t -\frac{\Gamma_1t}{2}}- e^{-iE_2 t -\frac{\Gamma_2t}{2}}\Big] \ =\ A_{{\rm OS},\mu e}(t)\, , \label{eq:OSemu} \\ A_{{\rm SS},e\mu}(t)&\ = \ &-c_\theta s_\theta \Big[ e^{-iE_1 t -\frac{\Gamma_1t}{2}}- e^{-2i\delta}e^{-iE_2 t -\frac{\Gamma_2t}{2}}\Big]\ = \ A_{{\rm SS},\mu e} (t) \, ,\label{eq:SSemu}\end{aligned}$$ where $\Gamma_{1,2}$ are the total decay widths of the two mass eigenstates $N_{1,2}$. We adopt the following procedure to get the ratio of SS and OS final states [@Anamiati:2016uxp]: $$\begin{aligned} R_{\ell\ell} \ = \ \frac{\int^\infty_0 dt\left |A_{{\rm SS},\ell\ell}(t)\right|^2}{\int^\infty_0 dt\left |A_{{\rm OS},\ell\ell}(t)\right|^2} \ \equiv \ \frac{N_{{\rm SS},\ell\ell}}{N_{{\rm OS},\ell\ell}} \, . \label{Rll}\end{aligned}$$ In order to illustrate the effect of the interference between the two states, we make the simplifying assumption that the two RHNs are non-relativistic (which is a good approximation when the $W_R$ mass is slightly larger than two times the RHN mass) and approximate $E_{1,2}\simeq M_{N_{1,2}}\simeq M_N\pm \Delta M/2$, where $\Delta M\equiv M_{N_1}-M_{N_2}$ is the mass splitting between the two RHN mass eigenstates. The eigenvalues $M_{N_{1,2}}$ can be obtained by calculating the eigenvalues of Eq. . Then from Eqs.  and , the number of SS and OS dielectron events are respectively given by $$\begin{aligned} N_{{\rm OS}, ee} & \ = \ & \Gamma_{\rm avg}\left[\frac{c^4_\theta}{\Gamma_1}+\frac{s^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\frac{\Gamma_1+\Gamma_2}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right] \, ,\\ N_{{\rm SS},ee} & \ = \ & \Gamma_{\rm avg}\left[\frac{c^4_\theta}{\Gamma_1}+\frac{s^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\left\{\frac{(\Gamma_1+\Gamma_2)\cos{2\delta}}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}-\frac{2\Delta M \sin{2\delta} }{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right\}\right] \, ,\end{aligned}$$ where $\Gamma_{\rm avg}\equiv (\Gamma_1+\Gamma_2)/2$. Similarly, for dimuon events, we have from Eqs.  and respectively, $$\begin{aligned} N_{{\rm OS},\mu\mu} & \ = \ & \Gamma_{\rm avg}\left[\frac{s^4_\theta}{\Gamma_1}+\frac{c^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\frac{\Gamma_1+\Gamma_2}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right] \, , \\ N_{{\rm SS},\mu\mu} & \ = \ & \Gamma_{\rm avg}\left[\frac{s^4_\theta}{\Gamma_1}+\frac{c^4_\theta}{\Gamma_2}+c^2_\theta s^2_\theta\left\{\frac{(\Gamma_1+\Gamma_2)\cos{2\delta}}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}-\frac{2\Delta M\sin{2\delta} }{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right\}\right] \, .\end{aligned}$$ Finally, for the $e\mu$ events, we have from Eqs.  and respectively $$\begin{aligned} N_{{\rm OS},e\mu} & \ = \ & N_{{\rm OS}, \mu e} \ = \ \Gamma_{\rm avg} \: c^2_\theta s^2_{\theta}\left[\frac{1}{\Gamma_1}+\frac{1}{\Gamma_2}-\frac{\Gamma_1+\Gamma_2}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right] \, ,\\ \ N_{{\rm SS},e\mu}& \ = \ & N_{{\rm SS}, \mu e} \ = \ \Gamma_{\rm avg} \: c^2_\theta s^2_{\theta}\left[\frac{1}{\Gamma_1}+\frac{1}{\Gamma_2}-\left\{\frac{(\Gamma_1+\Gamma_2)\cos{2\delta}}{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}-\frac{2\Delta M\sin{2\delta} }{\left(\frac{\Gamma_1+\Gamma_2}{2}\right)^2+(\Delta M)^2}\right\}\right] \, .\nonumber\\\end{aligned}$$ Expanding these equations out, we find that if there is no $CP$ phase i.e. $\delta=0$, we get $$\begin{aligned} N_{{\rm OS},ee} \ = \ N_{{\rm SS},ee}\, ; \quad N_{{\rm OS},\mu\mu} \ = \ N_{{\rm SS},\mu\mu} \, ; \quad N_{{\rm OS}, e\mu} \ = \ N_{{\rm SS},e\mu} \, , \end{aligned}$$ as expected for purely Majorana RHNs. However, in the presence of a non-zero $CP$ phase, we find $$\begin{aligned} N_{{\rm OS},\ell\ell} \ \neq \ N_{{\rm SS},\ell\ell} \, , \quad {\rm or} \quad R_{\ell\ell}\neq 1 \, , \end{aligned}$$ as illustrated in Figure \[fig:Rll-typeI\]. We emphasize again that these arguments are true only if the two RHN states satisfy the coherence conditions  and . Let us apply our findings to the special case where the RHN mass matrix is of the form $M_N=M\tau_1$ where $\tau_1$ is the first Pauli matrix. In this case $\theta=\pi/4$ and $\delta=\pi/2$. Also in this case, $\Gamma_1=\Gamma_2$ and $\Delta M=0$. Substituting this in Eq. , we get $N_{{\rm SS},ee}=N_{{\rm SS},\mu\mu}=0$ and only $N_{{\rm SS},e\mu}\neq 0$ as we would expect from the structure of the RHN mass matrix.[^2] In Figure \[fig:Rll-typeI\], we show the variation of $R_{\ell\ell}$ (for $\ell\ell=ee,\mu\mu$) as a function of the $CP$ phase $\delta$ for different values of $\Delta M/\Gamma_{\rm avg}$. As for the RHN decay widths, we have used the three-body decay widths of $N_\ell\to W_R^*\ell \to q\bar{q}'\ell$ (see Appendix \[sec:app\]). For numerical purposes, we have chosen a fixed value of $M_{W_R}=5$ TeV and $M_{N_1}=500$ GeV, but our main results are independent of the choice of the exact mass values, as long as $M_{N_{1,2}}< M_{W_R}$, which is anyway required from vacuum stability arguments [@Mohapatra:1986pj; @Maiezza:2016bzp]. We find that for $\delta=0$ and $\pi$ (i.e. no $CP$ violation), $R_{\ell\ell}=1$, as discussed above. But for $\delta=\pi/2$ and $\Delta M=0$, $R_{\ell\ell}=0$, i.e. there is a completely destructive interference between the two RHN mass eigenstates in the SS channel. The degree of interference decreases rapidly as we increase $\Delta M$ and as $\Delta M$ becomes much larger than $\Gamma_{\rm avg}$, there is virtually no interference, leading to the limit $R_{\ell\ell}\to 1$, as expected for purely Majorana RHNs. In the intermediate range of $\Delta M/\Gamma_{\rm avg}$, we have $R_{\ell\ell}>1$, i.e. enhanced SS signal even compared to the purely Majorana case, for certain choices of the $CP$ phase $\delta$, when the constructive interference is maximum in Eqs.  and . Note here that both $ee$ and $\mu\mu$ channels lead to almost identical predictions for the ratio $R_{\ell\ell}$ because for $\Delta M\ll M$ in Eq. , $\theta\approx \pi/4$, so $c_\theta\simeq s_\theta\simeq 1/\sqrt{2}$. General Inverse Seesaw Case {#sec:inv} =========================== We start this section by briefly reviewing the inverse seesaw extension of the left-right symmetric model. The model is based on the gauge group $SU(3)_c\times SU(2)_L\times SU(2)_R \times U(1)_{B-L}$ gauge group [@LR1; @LR2; @LR3] with scalar sector consisting of two $SU(2)$ doublets $\chi_R^{}(1,1,2,+1)$, together with a bidoublet $\phi(2,2,0)$ and a $B-L=2$ triplet $\Delta_R(1,3,+2)$, while the fermion sector contains not only the usual $SU(2)$ doublets of the left-right model i.e. $Q_L^{}(3,2,1,+\frac{1}{3})$, $Q_R^{}(3,1,2,+\frac{1}{3})$, $L^{}(1,2,1,-1)$ and $R^{}(1,1,2,-1)$, but also additional $SU(2)$ singlets $S_a$ (with $a=1,2,3$). Note that we are working with a model where parity symmetry breaking scale $M_P$ and the $SU(2)_R$ symmetry breaking scale $v_R$ are different with $M_P\gg v_R$ [@Chang:1983fu]. To discuss the inverse seesaw in this model, we need the leptonic Yukawa couplings: $$\begin{aligned} {\cal L}_Y \ = \ h_l\bar{L}\phi R+h_\nu \bar{R}\chi_R S +f\bar{R}^C \Delta_R R+\mu_s \bar{S}^CS+{\rm H.c.} \end{aligned}$$ After symmetry breaking by the vacumm expectation values of the Higgs fields i.e. ${\rm Diag} \langle\phi \rangle=(\kappa, \kappa')$; $\langle \chi^0_R\rangle = \sigma_R$ and $\langle\Delta^0_R\rangle = v_R$, we get the neutral fermion mass matrix of the form: $$\begin{aligned} {\cal M} \ &=& \ \left(\begin{array}{ccc} 0 & M_D & 0\\ M^T_D & \mu_R & M_N\\ 0 & M^T_N & \mu_S \end{array}\right) \label{eq:4.2}\end{aligned}$$ where $M_D=h_l\sqrt{\kappa^2+\kappa'^2}$, $\mu_R=fv_R$ and $M_N=h_\nu \sigma_R$.[^3] It leads to the formula for light neutrino mass matrix at tree-level:[^4] $$\begin{aligned} M_\nu \ = \ (M_DM_N^{-1}) \mu_S(M_DM_N^{-1})^T \, . \label{eq:4.3}\end{aligned}$$ This is the inverse seesaw mechanism at work for the most general case where each entry in Eq.  is a $3\times 3$ matrix corresponding to three flavors. For simplicity below we consider a single family version of this matrix to illustrate our discussion of SS and OS dilepton plus two jets in $pp$ collision. We note that this analysis can be applicable to realistic situation with flavor in the following way: Consider the case when $M_D, M_N, \mu_R$ are all diagonal $3\times 3$ matrices and let all neutrino flavor mixings reside in the $\mu_S$ matrix. This is the so-called flavor-diagonal scenario. Proper choice of the $\mu_S$ matrix can explain the observed neutrino oscillation results but since each element of the $\mu_S$ matrix is very small compared to other matrices in the problem i.e. $\mu_R, M_N$, they will not affect our conclusions about the ratio $R_{\ell\ell}$ for each flavor. Of course one could also consider flavor structures in $\mu_R$ and/or $M_N$. The analysis is then more complicated and we do not consider it here.[^5] $R_{\ell\ell}$ in the Inverse Seesaw Case {#sec:inv2} ========================================= In order to study the final states with SS or OS dileptons, we consider a simplified yet realistic case where $\mu_R$ and $M_N$ are $2\times 2$ diagonal matrices so that all neutrino mixings arise from the matrix $\mu_S$, which does not have any effect on $R_{\ell\ell}$. We consider the eigenstates of the mass matrix . We do this in stages and for the parameter domain where $M_N\gg \mu_R \gg M_D \gg \mu_S$, we can first diagonalize the lower $2\times 2$ matrix and get the following eigenstates with real eigenvalues $$\begin{aligned} {\cal N}_{1} \ &=& \ c_\alpha N+s_\alpha S \, ; \\ {\cal N}_{2} \ &=& \ i(-s_\alpha N+c_\alpha S) \, .\end{aligned}$$ Using these we can rewrite the $W_R$-induced charged-current interactions as $$\begin{aligned} {\cal L}_I \ = \ \frac{g}{\sqrt{2}}\bar{\ell}_R\gamma_\mu (c_\alpha {\cal N}_1+is_\alpha {\cal N}_2)W^{-,\mu}_R+\frac{g}{\sqrt{2}} (c_\alpha {\cal N}_1-is_\alpha {\cal N}_2)^TC\gamma_\mu\ell_RW^{+,\mu}_R \label{eq:laginv}\end{aligned}$$ To calculate the OS and SS event numbers, we need to discuss whether there is coherence between the decays of $\mathcal{N}_{1,2}$ when produced in $W_R$ decay as well as the requirement for maintaining coherence over the decay length of the RHNs. For the inverse seesaw case, the discussions in Ref. [@Akhmedov:2007fk] lead us to the same coherence condition in the emission as stated in Sec. \[sec:coh\] except in this case, $\Delta M$ denotes the mass difference between the $N$ and $S$ fermions. The condition on parameters from coherence length considerations are different here since the Dirac Yukawa couplings which dominate the decay of $N,S$ states are expected to be much larger for inverse seesaw than the type I case. The decay length in this case is therefore much shorter than the type I case i.e. $L\sim \frac{12\pi}{h^2 M_N}$. Choosing $h\sim 0.1$ and $M_N\sim $ TeV, we get for $\Delta M\sim 100$ GeV, as compared to 1 GeV or so in the type I case. When the coherence condition is satisfied, recalling that the first term in the Lagrangian is responsible for the production of OS events and second one for SS events, we can write the amplitudes for OS and SS events as follows: $$\begin{aligned} A_{\rm OS}(t) \ &=& \ c^2_\alpha e^{-iE_1t-\frac{\Gamma_1}{2}t}+s^2_\alpha e^{-iE_2t-\frac{\Gamma_2}{2}t} \, , \nonumber \\ A_{\rm SS}(t) \ &=& \ c^2_\alpha e^{-iE_1t-\frac{\Gamma_1}{2}t}-s^2_\alpha e^{-iE_2t-\frac{\Gamma_2}{2}t} \, .\end{aligned}$$ We approximate $E_{1,2}\simeq M_{1,2}\pm \Delta M/2$ as before and use the expression in Eq.  to obtain for the OS and SS events respectively $$\begin{aligned} N_{\rm OS} \ & = \ \Gamma_{\rm avg} \left[ \frac{c_\alpha^4}{\Gamma_1}+\frac{s_\alpha^4}{\Gamma_2}-\frac{c_\alpha^2 s_\alpha^2(\Gamma_1+\Gamma_2)}{\frac{(\Gamma_1+\Gamma_2)^2}{4}+(\Delta M)^2}\right], \label{eq:OS} \\ N_{\rm SS} \ & = \ \Gamma_{\rm avg} \left[ \frac{c_\alpha^4}{\Gamma_1}+\frac{s_\alpha^4}{\Gamma_2}+\frac{c_\alpha^2s_\alpha^2(\Gamma_1+\Gamma_2)}{\frac{(\Gamma_1+\Gamma_2)^2}{4}+(\Delta M)^2} \right] \, . \label{eq:SS}\end{aligned}$$ Using the RHN decay widths given in Appendix \[sec:app\], we have plotted in Fig. \[fig:Rll-ISS\] the ratio $R_{\ell\ell}=N_{\rm SS}/N_{\rm OS}$ as a function of $\mu_R$ for different RHN masses. Here we have chosen a fixed value of $M_{W_R}=5$ TeV for illustration. We find that smaller values of $\mu_R$ favors the OS signal whereas higher values of $\mu_R$ favor the SS signal. For lower values of $M_N$, the range of $\mu_R$ increases where $R_{\ell\ell}\to 1$. Now we can look at three special cases: [**Case (i): $\mu_R=0$**]{}: This is the case which has been considered in Refs. [@Anamiati:2016uxp; @Antusch:2017ebe]. In this case for a TeV $M_N$, fitting neutrino mass scale requires that $\mu_S\leq $ keV. This means that $\Delta M \sim \mu_S\sim $ keV and the coherence condition is very well satisfied. Furthermore, in this case $c_\alpha=s_\alpha = \frac{1}{\sqrt{2}}$. Using the fact that we have also $\Gamma_1=\Gamma_2$, we get from Eqs.  and , that $$\begin{aligned} R_{\ell\ell} \ = \ \frac{(\Delta M)^2}{2\Gamma^2+(\Delta M)^2} \end{aligned}$$ in agreement with the result in Ref. [@Anamiati:2016uxp]. Note that for TeV-scale $M_{N}$, typically $\Gamma\sim 10-100$ keV and $\Delta M\sim 1$ keV, leading to $R_{\ell\ell}\lesssim 1\%$. Thus to get large $R_{\ell\ell}$ in inverse seesaw models, one must include the effect of $\mu_R$. [**Case (ii) $\mu_R\ll M_N$**]{}: In this case, in general $\alpha$ is different from $\pi/4$ and we do not expect $\Gamma_1$ and $\Gamma_2$ to be equal. If we assume that $\Gamma_1\sim \Gamma_0 c^2_\alpha$ and $\Gamma_2\sim \Gamma_0 s^2_\alpha$, we get $$\begin{aligned} R_{\ell\ell} \ = \ \frac{\cos^2 2\alpha+\frac{4(\Delta M)^2}{\Gamma^2_0}}{1+\sin^22\alpha+\frac{4(\Delta M)^2}{\Gamma^2_0}} \, . \end{aligned}$$ For the case when $\frac{4(\Delta M)^2}{\Gamma^2_0}\ll 1$, it reduces to the formula in Ref. [@Dev:2015kca]. In this case, $R_{\ell\ell}$ can be significant; see Figure \[fig:Rll-ISS\]. [**Case (iii): Hierarchical masses i.e. $\mu_R \gg M_N$:**]{} In this case, the two eigenstates ${\cal N}_{1,2}$ have a large mass difference i.e. $(\Delta M)^2\gg \Gamma^2_{1,2}$. In this case, there is no coherence and we have therefore $R_{\ell\ell}=1$ as in the type-I seesaw case since the two Majorana eigenstates both lead to equal number of SS and OS dilepton states. Conclusion {#sec:con} ========== We show that in generic TeV scale $W_R$ models for type I and general inverse seesaw models, the ratio $R_{\ell\ell}$, of the number of same sign ($N_{\rm SS}$) and opposite sign ($N_{\rm OS}$) dilepton states need not be the same when summed over different flavors, contrary to general expectations. This can happen when there is a high degree of degeneracy between the RHNs produced in $W_R$ decay. The degree of degeneracy depends on whether it is type I or inverse seesaw case, and is determined by the coherence condition which in turn depends on the magnitude of the Dirac Yukawa couplings in the theory. For generic choice of parameters, in the first case, the degeneracy has to be at the level of one part in a thousand for TeV scale RHNs whereas in the case of inverse seesaw, it can be a factor of ten or less. Thus observation of the ratio $R_{\ell\ell}$ can in principle, allow us to probe the deeper structure of the RHN mass matrix in the type I seesaw case and the $(N,S)$ sector mass matrix in the inverse seesaw case. We find that in the case of type I seesaw, one needs $CP$ violation to get $R_{\ell\ell}$ different from one, whereas for the inverse seesaw, it is the parameter $\mu_R$ which governs $R_{\ell\ell}$. We believe that the connection between $R_{\ell\ell}\neq 1$ and near degeneracy of RHN states is already an important conclusion, since it is known that low scale leptogenesis in TeV scale seesaw models already requires near degeneracy of RHN states. Our main goal in this work was to derive the analytic results for $R_{\ell\ell}$ in the singlet seesaw scenario, and to show as a proof of principle that it can be different from 1 in the parameter space relevant for the LHC. This result is valid irrespective of the details of the collider simulation of the OS and SS events, with their respective signal and background efficiencies, which can be done in a straightforward manner for any given benchmark point following the existing experimental analyses; see e.g. Ref. [@Khachatryan:2014dka]. Also in the case of inverse seesaw, we have ignored detailed flavor effects, since our goal has been merely to illustrate an interesting phenomenon involving lepton [*number*]{} violation. A detailed collider analysis (including flavor effects) is a bit premature at this stage and might be more appropriate in scrutinizing the different seesaw models, only if there is a statistically significant observation of dilepton plus two jet signal (either SS or OS) in the future.[^6] Acknowledgement {#acknowledgement .unnumbered} =============== We gratefully acknowledge the local hospitality provided at the ACFI workshop on ‘Neutrinos at the High-Energy Frontier’ at UMass, Amherst, where part of this work was done. R.N.M. was supported by the US National Science Foundation under Grant No. PHY1620074. Partial Decay Widths of $N$ {#sec:app} =========================== In the left-right model, the RHN has three-body decays through an off-shell $W_R$ (for $M_N<M_{W_R}$): $N_\ell\to W_R^*\ell \to \ell q\bar{q}^{\prime}$. This is in addition to the usual two-body decay modes of the RHN: $N\to W\ell, \, Z\nu, \, h\nu$, induced by its mixing with the light neutrinos. In this analysis, we choose the region of parameter space where the light-heavy neutrino mixing is small enough to ensure that the three-body decay is dominant over the two-body one [@Chen:2013fna]. For light-quark final states, the corresponding three-body decay width is given by [@Gluza:2015goa; @Das:2016akd] $$\Gamma(N \to q\bar{q}^\prime \ell ) \ = \ \frac{g_R^4}{2048 \pi^3} M_{N} \frac{12}{x}\left[1-\frac{x}{2}-\frac{x^2}{6}+\frac{1-x}{x}\ln (1-x)\right] ~,$$ with $x = M_{N}^2/M_{W_R}^2$. Here we neglect the SM quark and lepton masses. For the $N \to t\bar{b} \ell$ decay channel, we have [@Dobrescu:2015jvn; @Das:2016akd] $$\begin{aligned} \Gamma (N \to \overline{b} t \ell) \ = \ \frac{g_R^4}{2048 \pi^3} M_{N} F_t(x, y) ~,\end{aligned}$$ where $$\begin{aligned} F_t(x, y) &\ = \ \frac{12}{x} \biggl[ (1-y) -\frac{x}{2}(1-y^2) -\frac{x^2}{6}\left( 1-\frac{3}{2}y + \frac{3}{2}y^2 -y^3 \right) \nonumber \\[3pt] &-\frac{5x^3y}{8}(1-y^2) +\frac{x^4y^2(1-y)}{4} -\frac{x^3y^2}{4}(4+x^2y)\ln y \nonumber \\ &+ \frac{1-x}{x} \ln \left(\frac{1-x}{1-xy}\right) \left\{1-\frac{xy}{4}\left[4+x+x^2-x^3 y^2 (1+x)\right]\right\} \biggr] ~,\end{aligned}$$ with $y=m_t^2/M_N^2$ and $m_t$ is the top mass. These decay widths have been used in our numerical analysis for $R_{\ell\ell}$ (see Figures \[fig:Rll-typeI\] and \[fig:Rll-ISS\]) with a benchmark value of $M_{W_R}=5$ TeV and $M_N=500$ GeV. As long as $M_N\ll M_{W_R}$, the actual values of these masses do not affect our final results. 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--- abstract: 'The effective evolution of an inhomogeneous cosmological model may be described in terms of spatially averaged variables. We point out that in this context, quite naturally, a measure arises which is identical to a fluid model of the ‘Kullback–Leibler Relative Information Entropy’, expressing the distinguishability of the local inhomogeneous mass density field from its spatial average on arbitrary compact domains. We discuss the time–evolution of ‘effective information’ and explore some implications. We conjecture that the information content of the Universe – measured by Relative Information Entropy of a cosmological model containing dust matter – is increasing.' author: - 'Akio Hosoya[^1]' - 'Thomas Buchert[^2]' - 'Masaaki Morita[^3]' title: 'Information Entropy in Cosmology[^4]' --- A Measure of Inhomogeneity in the Universe ========================================== Cosmology is based on the hypothesis of simplicity called the cosmological principle, i.e. homogeneity and isotropy. The departure of the actual mass distribution from the homogeneous universe model is quantified in terms of density contrast or a statistical quantity like the two–point correlation function, which both have been studied either by perturbation theory or numerical simulations. Behind these investigations there is a belief that the Universe is homogeneous on some large enough scale. This belief has to be quantitatively confronted with observation, explicitly introducing a measure of inhomogeneity for a domain of the Universe. In this [*Letter*]{} we propose a measure which quantifies the distinguishability of the actual mass distribution from its spatial average, borrowing a well–known concept in standard information theory. Suppose we are told that the probability distribution is $\{q_{i}\}$ and would like to examine how close this distribution is to the actual one $\{p_{i}\}$ by carrying out observations or coin tossing; the relevant quantity in information theory is the [*relative entropy*]{}, $${\cal S} \lbrace p || q\rbrace = \sum_{i} p_{i}\ln \frac{p_{i}}{q_{i}}\;\;,$$ which is positive for ${q_{i}}\neq {p_{i}}$, and zero if the actual distribution $\{p_{i}\}$ agrees with the presumed one $\{q_{i}\}$. Note that this relative entropy is not symmetric for the two distributions $\{p_{i}\}$ and $\{q_{i}\}$. It is known that this measure always decreases or stays the same under Markovian stochastic processes (i.e., a [*linear*]{} positive map). Namely, the actual distribution becomes less and less distinguishable from the priorly informed distribution due to the random process. In cosmology we are interested in how the real matter distribution is different from its spatial average. For a continuum the relevant quantity would be $$\label{entropy} \frac{{\cal S} \lbrace \varrho || \langle\varrho\rangle_{\cal D}\rbrace}{V_{\cal D}} \;=\; \Bigl\langle \varrho \ln \frac{\varrho}{\langle\varrho\rangle_{\cal D}}\Bigr\rangle_{\cal D}\;\;,$$ where $\varrho$ is the actual distribution and $\langle \cdots \rangle_{\cal D}$ its spatial average in the volume $V_{\cal D}$ on the compact domain $\cal D$ of the manifold $\Sigma$. We shall conjecture that the measure ${\cal S} \lbrace \varrho || \langle\varrho\rangle_{\Sigma}\rbrace$ continues to grow indefinitely, if $\Sigma$ is compact. The resolution of the apparent discrepancy between the gravitational system and the ordinary stochastic system will be, (i) we are considering in cosmology a non–isolated system defined by a comoving region $\cal D$ in contrast to an isolated system for an ordinary stochastic process, and (ii) the time evolution dictated by Einstein’s equations induces a negative feed–back due to the attractive nature of the gravitational force, which tends to make the matter distribution more and more inhomogeneous. Deduction of the Measure ======================== To begin with let us emphasize that the functional (\[entropy\]), known as the ‘Kullback–Leibler Relative Information Entropy’ ([*cf*]{} [@kullback], [@kullback_leibler], [@cover:entropy]) is not assumed as a measure [*a priori*]{}, rather it can be [*deduced*]{} from a fundamental kinematical relation that refers to the [*non–commutativity*]{} of two operations: spatially averaging and evolving the material mass density field. The specific form of the information measure is, thus, inherently determined by the physical problem at hand, and does not need to be justified empirically or axiomatically as is the common status of information measures in the literature. We define the averaging operation in terms of Riemannian volume integration, restricting attention to scalar functions $\Psi (t,X^i)$, $$\label{average} \langle \Psi (t, X^i)\rangle_{\cal D}: = \frac{1}{V_{\cal D}}\int_{\cal D} \sqrt{g} d^3 X \;\;\Psi (t, X^i) \;\;\;,$$ with the Riemannian volume element $d\mu_g := \sqrt{g} d^3 X$, $g:=\det(g_{ij})$, and the volume of an arbitrary compact domain, $V_{\cal D}(t) : = \int_{\cal D} \sqrt{g} d^3 X$; $X^i$ are coordinates in a $t=const.$ hypersurface (with $3-$metric $g_{ij}$) that are comoving with fluid elements of dust: $$ds^2 = -dt^2 + g_{ij}dX^idX^j \;\;.$$ It is evident from the above setting that we predefine a simple time–orthogonal foliation (which restricts the matter to an irrotational dust continuum) in order to simplify the framework in which we discuss our measure as a concept of a [*spatial*]{} average. We wish to emphasize that the formalism below could be carried over to more general settings (e.g. to perfect fluids or scalar fields ([*cf.*]{} [@buchert:grgfluid]) with possibly further interesting implications. The above–mentioned ‘non–commutativity’ has been fruitfully exploited in previous work on the averaging problem of inhomogeneous cosmologies [@buchert:average; @buchert:grgdust; @buchert:grgfluid; @buchert:onaverage; @buchertcarforaPRL], and can be compactly written in terms of a [*commutation rule*]{} for the averaging of a scalar field $\Psi$: $$\begin{aligned} \label{commutationrule} \langle \Psi{\dot \rangle}_{\cal D} - \langle{\dot \Psi}\rangle_{\cal D} = \langle \Psi\theta\rangle_{\cal D} - \langle \Psi\rangle_{\cal D}\langle\theta\rangle_{\cal D} \nonumber \\ =\langle\Psi\delta\theta\rangle_{\cal D} =\langle\theta\delta\Psi\rangle_{\cal D} =\langle\delta\Psi\delta\theta\rangle_{\cal D}\;\;\;,\end{aligned}$$ where $\theta$ denotes the local expansion rate (as minus the trace of the extrinsic curvature of the hypersurfaces $t=const.$). We have rewritten the r.h.s. of the first equality in terms of the deviations of the local fields from their spatial averages, $\delta\Psi := \Psi - \langle\Psi\rangle_{\cal D}$ and $\delta\theta := \theta - \langle\theta\rangle_{\cal D}$. The key–statement of the [*commutation rule*]{} (\[commutationrule\]) is that the operations [*spatial averaging*]{} and [*time evolution*]{} do not commute. In cosmology we may think of initial conditions at the epoch of last scattering, when the fluctuations imprinted on the Cosmic Microwave Background are considered to be averaged–out on a restframe of a standard Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology. In this picture the evolution of the Universe is described by first averaging–out (or ignoring) inhomogeneities and then evolving the average distribution by a homogeneous (in the above case homogeneous–isotropic) universe model. A realistic model would first evolve the inhomogeneous fields and, at the present epoch, the resulting fields would have to be evaluated by spatial averaging to obtain the final values of, e.g., the averaged density field. In particular, this comment applies to all cosmological parameters (see, e.g., [@buchert:grgdust] and [@buchertcarforaPRL]). Let us illustrate this statement for the mass density field. Setting $\Psi = \varrho$, Eq. (\[commutationrule\]) reads: $$\label{commutationdensity1} \langle \varrho{\dot\rangle}_{\cal D} + \langle\theta\rangle_{\cal D}\langle \varrho\rangle_{\cal D} = \langle{\dot \varrho} + \theta\varrho\rangle_{\cal D} \;\;\;.$$ Since the r.h.s. vanishes due to the continuity equation, we also have a continuity equation for the averages: $$\label{continuity2} \langle \varrho{\dot\rangle}_{\cal D} + \langle\theta\rangle_{\cal D}\langle \varrho\rangle_{\cal D} = 0\;\;\;,$$ which simply expresses the conservation of the total material mass, $M_{\cal D} = \int_{\cal D} \sqrt{g} d^3 X\;\varrho$, in our comoving and synchronous gauge setting. A fairly general insight that, in principle, will not depend on some specialized setting, can be obtained by rewriting Eq. (\[commutationdensity1\]): the notion of ‘non–commutativity’ mentioned above comes into the fore by observing that the time–evolution of the average density does not coincide with the average of the locally evolved density: $$\label{commutationdensity2} \langle \varrho{\dot\rangle}_{\cal D} - \langle{\dot \varrho}\rangle_{\cal D} = \langle \varrho\theta\rangle_{\cal D} - \langle \varrho\rangle_{\cal D}\langle\theta\rangle_{\cal D} =\langle\delta\varrho\delta\theta\rangle_{\cal D}\;\;\;.$$ For the fluctuation terms on the r.h.s., which would vanish in the FLRW model without any perturbation, we can give a deeper interpretation. For this end let us ask, which functional will reproduce these terms upon performing the time–derivative. First, note that for the averaged expansion rate $\langle\theta\rangle_{\cal D}$ the corresponding functional is the volume according to $$\label{averageexpansion} \langle\theta\rangle_{\cal D} = \frac{{\dot V}_{\cal D}}{V_{\cal D}} =:3H_{\cal D} \;\;\;.$$ The latter equality demonstrates that this quantity may be regarded as an [*effective Hubble function*]{}, which will show up in our discussion later. Interestingly, the answer is provided, for $\varrho > 0$, by the functional ${\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace$, Eq. (\[entropy\]), so that the source of non–commutativity in Eq. (\[commutationdensity2\]) is given (up to the sign) by the production of [*Relative Information Entropy*]{}, defined as to measure the deviations from the average mass density due to the development of inhomogeneities: $$\label{relativeentropy} \langle \varrho{\dot\rangle}_{\cal D} - \langle {\dot\varrho}\rangle_{\cal D} = -\frac{{\dot {\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace}}{V_{\cal D}}\;\;.$$ This measure can actually be inferred from its definition in phase space in terms of the one–particle distribution function for dust matter, i.e. the matter density multiplied by a delta–function distribution in velocity space [@hosoya:infoentropy]. It is here, where generalizations of the matter model, e.g. supported by pressure, vorticity and/or velocity dispersion could be implemented, resulting in more general entropies after taking velocity moments in phase space. The reader may ask, whether this measure is superior to the density fluctuation measure, which also provides a generally growing and positive–definite valuation of the density distribution. Let us give some answers to this question before we proceed. A standard index of inhomogeneity in cosmology is the density contrast $\delta:=\frac{\delta\varrho}{\langle\varrho\rangle_{\cal D}}$ and the derived positive measure $(\Delta\varrho)^2 : = \langle \varrho^2 \rangle_{\cal D} - \langle\varrho\rangle_{\cal D}^2$. The Relative Information Entropy or the distinguishability, Eq. (\[entropy\]), may have further implications by exploiting results from information theory. At the present stage we do not claim that this measure is superior to the density fluctuation, but rather it is [*complementary*]{}. This can be illustrated by pointing out that both measures are “cousins” in a $1$–parameter family of inhomogeneity measures defined by $${\cal F}_{\alpha} \lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace :=\frac{\langle\varrho\rangle_{\cal D}}{\alpha}\left[\Bigl\langle \left(\frac{\varrho}{\langle\varrho\rangle_{\cal D}} \right)^{\alpha+1}\Bigr\rangle_{\cal D}-1\right]\;\;,\nonumber$$ with $\alpha$ being a real parameter. In the limit $\alpha\rightarrow 0$ the formula reproduces the relative entropy, ${\cal F}_{\alpha \rightarrow 0}\rightarrow {\cal S}/V_{\cal D}$, whereas $\alpha=1$ reproduces the density fluctuation, ${\cal F}_{\alpha =1} = (\Delta\varrho)^2 / \langle\varrho\rangle_{\cal D}$. This interpolating formula is known as the [*Tsallis relative entropy*]{}. It should be emphasized that the limit $\alpha \rightarrow 0$ is singled out as the only measure that exactly provides the source of non–commutativity with regard to the density evolution. Properties of the Measure ========================= The measure ${\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace$ forms one of the central concepts in information theory [@cover:entropy]; ${\cal S}=0$ (“zero structure”) is attained by the homogeneous mass distribution, $\varrho = \langle\varrho\rangle_{\cal D}$. First, for strictly positive mass density, $\varrho > 0$, ${\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace$ is positive definite, which can be readily confirmed, i.e. it is indeed a [*measure*]{}. Let us have a closer look at the total time–derivative of our measure. Following from what has been said above, we may write the total [*Relative Information Entropy production*]{} as follows: $$\label{entropyproduction} \frac{{\dot {\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace}}{V_{\cal D}} = -\langle\delta\varrho\theta\rangle_{\cal D} = -\langle\varrho\delta\theta\rangle_{\cal D} = -\langle\delta\varrho\delta\theta\rangle_{\cal D} \;\;.$$ The last quantity is bounded according to Schwarz’ inequality, so that we obtain: $$\label{bound1} \Big\vert\frac{{\dot {\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace}}{V_{\cal D}} \Big\vert=|\langle\delta\varrho\delta\theta\rangle_{\cal D}| \;\le\; \Delta\varrho \Delta\theta \;\;,$$ with the positive–definite fluctuation amplitudes $$\Delta\varrho := \sqrt{\langle(\delta\varrho)^2\rangle}\;\;;\;\; \Delta\theta := \sqrt{\langle(\delta\theta)^2\rangle}\;\;.$$ This inequality states that the temporal change of the ratio between the [*distinguishability*]{} of the density distribution from the homogeneous distribution and the volume is bounded by the density and expansion fluctuation amplitudes. We may say that the production of information in the Universe and its volume expansion are competing. We may look more closely at bounds as well as kinematical and dynamical conditions for the total second time–derivative of ${\cal S}\lbrace\varrho || \langle\varrho\rangle_{\cal D}\rbrace$. In [@hosoya:infoentropy] we give sufficient conditions for the [*time–convexity*]{} of our measure. Let us put one of them into perspective. We consider the question under which condition the time–derivative of the Relative Information Entropy production is positive. A straightforward calculation provides: $$\label{Pdot2} \frac{{\ddot{\cal S}}}{V_{\cal D}} = -\langle\delta\varrho\delta{\dot\theta}\rangle_{\cal D} + \langle \varrho\rangle_{\cal D} (\Delta\theta)^2 \;\;.$$ Raychaudhuri’s equation, $$\label{raychaudhuri} \dot\theta = \Lambda - 4\pi G\varrho - \frac{1}{3}\theta^2 - 2\sigma^2\;\;,$$ with the rate of shear $\sigma := \sqrt{\frac{1}{2}\sigma^i_{\;j} \sigma^j_{\;i}}$, the shear tensor $\sigma_{ij}$ being minus the trace–free part of the extrinsic curvature), together with the [*commutation rule*]{} (\[commutationrule\]) yields: $$\begin{aligned} \frac{{\ddot{\cal S}}}{V_{\cal D}}& = 4\pi G(\Delta \varrho)^2+\langle\varrho\rangle_{\cal D}(\Delta\theta)^2 + \frac{1} {3}\langle\delta\varrho\delta\theta^2\rangle_{\cal D}+2\langle\delta \varrho\delta\sigma^2\rangle_{\cal D}\\ \geq & 4\pi G (\Delta \varrho)^2 - \Delta \varrho[\frac{1}{3} \Delta\theta^2 + 2\Delta\sigma^2] + \langle\varrho\rangle_{\cal D} (\Delta\theta)^2\;\;.\end{aligned}$$ The r.h.s. is positive, if $$\label{sufficient} \frac{1}{2} \frac{\Delta(\frac{1}{3}\theta^2)+\Delta(2\sigma^2)} {\Delta\theta} \leq \sqrt{4\pi G\langle\varrho\rangle_{\cal D}} = \frac{1}{t_{F_{\cal D}}}\;\;,$$ where $t_{F_{\cal D}}$ denotes the [*effective free–fall time*]{} on $\cal D$. Eq. (\[sufficient\]) provides a sufficient condition for the time–convexity of the Relative Information Entropy, which can be met, if gravity dominates over expansion and shear fluctuations. Time–convexity implies that entropy production eventually becomes positive, i.e. the structure eventually surfaces and its rate of formation increases. Discussion and Conjecture ========================= Looking at Eq. (\[entropyproduction\]) we appreciate that the source, i.e., the averaged Relative Information Entropy production density, can be positive or negative. In cosmology, the processes of a relative accumulation of matter (cluster formation) and a relative dilution of matter (void formation) create [*structure*]{} compared with the average distribution. Following from Eq. (\[entropyproduction\]), information entropy is produced if, on average, there are overdense fluid elements ($\delta\varrho > 0$) which are contracting ($\theta < 0$), or underdense elements ($\delta\varrho < 0$) which are expanding ($\theta > 0$), respectively. With regard to cosmological structure formation these two states are generically encountered in a self–gravitating system, i.e., for large enough times and looking at some regional scale, an asymmetry of states is created due to the coupling of the expansion rate to the rate of change of the density through the continuity equation. We know from a calculation of the measure in linear perturbation theory that the growing–mode solution supports states with $\lbrace \delta\varrho > 0\;,\;\theta < 0 \rbrace$ (contracting clusters) and $\lbrace \delta\varrho < 0\;,\;\theta > 0 \rbrace$ (expanding voids). Thus, [*for sufficiently large times*]{}, i.e. when the decaying mode disappears, our measure will increase. Looking at Eq. (\[sufficient\]) we conclude that also in the case of the second time–derivative we have the possibility of time–concavity of the Relative Information Entropy. However, we have evidence that, at least for large enough times and on sufficiently large scales, time–convexity [*always*]{} holds for a self–gravitating continuum of dust. In particular, in the linear perturbation theory, our measure is [*always*]{} time–convex [@hosoya:infoentropy]. We can illustrate roughly the physical content of the sufficient condition (\[sufficient\]) as follows. Concentrating on the linear regime by considering the case in which fluctuations of a quantity are small compared with their average values, we may expand the quadratic expressions and keep only the leading terms: $$\Delta(\theta^2) \;\approx \; 2|\langle \theta \rangle_{\cal D}|\Delta\theta \;\;\;;\;\;\; \Delta(\sigma^2) \;\approx \; 2|\langle \sigma \rangle_{\cal D}|\Delta\sigma \;\;\;.$$ In this limit, if we additionally think of a large domain featuring approximately vanishing average shear, $\langle \sigma \rangle_{\cal D}\approx 0$, the sufficient condition (\[sufficient\]) reduces to the inequality $$\label{times} t_{F_{\cal D}} \; \le \; t_{H_{\cal D}} =: | H_{\cal D}^{-1} | \;\;,$$ i.e., if the [*effective free–fall time*]{} on $\cal D$ is smaller than the [*effective Hubble time*]{} $t_{H_{\cal D}}$, with $H_{\cal D}$ defined in Eq. (\[averageexpansion\]), then time–convexity of our measure is ensured under the given assumptions. The expectation that both positivity of the Relative Information Entropy production and time–convexity, which are supported by the linear perturbation results, will hold in the dust continuum [*generically*]{}, at least for large enough times and on sufficiently large scales of averaging, establishes the following. [*Conjecture:*]{} The Relative Information Entropy of a dust matter model ${\cal S} \lbrace \varrho || \langle\varrho\rangle_{\Sigma}\rbrace$ is, for sufficiently large times, globally (i.e. averaged over the whole compact manifold $\Sigma$) an increasing function of time. We are currently investigating nonlinear exact solutions for spherically–symmetric domains [@hosoya:infoentropy], which may provide further support for our conjecture. A note is in order as for the relation to observational constraints. In our context a generalized form of Friedmann’s differential equation governs the averaged expansion (\[averageexpansion\]), and a set of four effective cosmological parameters can be defined [@buchert:grgdust], [@buchertcarforaPRL]. Assuming that, on sufficiently large scales of averaging, kinematical fluctuations and the averaged $3-$Ricci curvature have negligible contributions, then the sum of the cosmological parameters for the matter content and the cosmological term have to add up to $1$; the former is indeed given by the fraction of the two competing times: $$\Omega^m_{\cal D}:= \frac{8\pi G \langle \varrho\rangle_{\cal D}}{3 H_{\cal D}^2} = \frac{2}{3}\frac{t_{H_{\cal D}}^2}{t_{F_{\cal D}}^2} \;\;.$$ Refering to observational results, e.g. by WMAP ([*Wilkinson Microwave Anisotropy Probe*]{})[@WMAP], its contribution is $\Omega^m_{\cal D}\approx 0.3$ and, thus, $t_{F_{\cal D}}$ is slightly larger than $t_{H_{\cal D}}$. Note that this does not immediately imply that our measure is not time–convex, because the condition (\[times\]) derives from the sufficient condition (\[sufficient\]), which only provides a rough estimation and is not very stringent. On cosmological scales both times are indeed very similar, so that we should make the estimation tighter to see whether or not time–convexity holds; this we postpone to the future work [@hosoya:infoentropy]. We contemplate that the measure that we propose in the present [*Letter*]{} not only incorporates an assessment of structure, but may turn out to be a fundamental quantity in many other respects, e.g. for the study of Black Holes and the Early Universe. We would like to thank Alvaro Domínguez, José Gaite and Atsushi Taruya for constant interest and discussions. TB acknowledges hospitality during a visit in 2001 at the Hosoya Laboratory of the Tokyo Institute of Technology, where the main work on this subject was done with support by The Japanese Society for the Promotion of Science (JSPS). He also acknowledges hospitality at the University of Tokyo with support by the Research Center for the Early Universe (RESCEU, Tokyo), COE Monkasho Grant, where this Letter was written. This work is also partially supported by the Sonderforschungsbereich SFB 375 ‘Astroparticle physics’ by the German science foundation DFG. [2004]{} T. Buchert, G. R. G. [**32**]{}, 105 (2000). T. Buchert, G. R. G. [**33**]{}, 1381 (2001). T. Buchert, in: [*9th JGRG Meeting, Hiroshima 1999*]{}, Y. Eriguchi et al. (eds.), pp. 306–321 (2000). T. Buchert and M. Carfora, Phys. Rev. Lett. [**90**]{}, 31101-1-4 (2003). T. Buchert and J. Ehlers, Astron. Astrophys. [**320**]{}, 1 (1997). T. M. Cover and J. A. Thomas, [*Elements of Information Theory*]{} Wiley, N.Y. (1991). A. Hosoya, T. Buchert, and M. Morita, in prep. (2004). S. Kullback, [*Information Theory and Statistics*]{}, Wiley, N.Y. (1959). S. Kullback, R. A. Leibler, [*On information and sufficiency*]{}, Ann. Math. Statistics [**22**]{} (1951), pp 79–86. D. N. Spergel et al. Ap.J. Suppl. [**148**]{}, 175 (2003). [^1]: ahosoya@th.phys.titech.ac.jp [^2]: buchert@theorie.physik.uni-muenchen.de [^3]: masaaki@cosmos.phys.ocha.ac.jp [^4]: a short version of this [*Letter*]{} received honorable mention in the GRG essay competition 2003

--- abstract: 'The influence of Rashba spin-orbit coupling on zero conductance resonances appearing in one-dimensional conducting rings asymmetrically coupled to two leads is investigated. For this purpose, the transmission function of the corresponding one-electron scattering problem is derived analytically and analyzed in the complex energy plane with focus on the zero-pole structure characteristic of transmission (anti)resonances. The lifting of real conductance zeros due to spin-orbit coupling in the asymmetric Aharonov-Casher ring is related to the breaking of spin reversal symmetry in analogy to the time-reversal symmetry breaking in the asymmetric Aharonov-Bohm ring.' author: - Urs Aeberhard - Katsunori Wakabayashi - Manfred Sigrist bibliography: - 'paper.bib' title: 'Effect of spin-orbit coupling on zero-conductance resonances in asymmetrically coupled one-dimensional rings' --- Introduction\[sec:1\] ===================== An important feature of one-dimensional ring shaped conductors or electronic devices is the appearance of quantum interference effects under the influence of electromagnetic potentials, known as Aharonov-Bohm[@ab:59] (AB) and Aharonov-Casher[@ac:84] (AC) effect. In numerous investigations, the transmission properties of mesoscopic AB and AC-rings coupled to current leads were studied under various aspects such as AB-flux and coupling dependence of resonances[@buettiker:84], geometric (Berry) phases[@loss:90; @ady:92; @alg:93; @quian:94; @hentschel:03] and spin flip, precession and interference effects[@yi:97; @nitta:99; @hentschel2:03; @frustaglia:04; @molnar:04]. Most of the investigated models use symmetrically coupled rings. There are however mesoscopic systems like nanographite ribbons showing conductance properties that are based on asymmetric configurations [@wakabayashi:01], giving rise to a specific dip structure of anti-resonances (zero-conductance resonances) in the model transmission. The effects of asymmetry on the transmission were considered mainly in quantum network models[@wu:91; @yi:03; @bercioux:04]. In quasi 1d systems, real conductance zeros appear under the condition of conserved time reversal symmetry[@lee:99; @lee:01] (TRS). The (anti)resonances in the transmission due to local quasi-bound states correspond to a specific zero-pole structure in the complex energy plane[@porod:93; @porod:94; @deo:94; @price:93]. The application of an external magnetic field modifies this zero-pole structure, shifting the transmission zeros away from the real axis, with the shift as a function of the AB-phase[@kim:02]. Thus, the lifting of conductance zeros is related to the breaking of TRS. In this paper, the influence of spin-orbit coupling (SOC) on zero-conductance resonances in asymmetrically coupled rings is investigated by means of an AC-ring where an effective in-plane magnetic field results from the *Rashba* effect[@rashba:60] of moving electrons in the presence of an electric field perpendicular to the ring plane, as considered in Ref. and . This means that the role of time reversal symmetry is now transfered to inversion symmetry (parity). We will show that parity connected with the Rashba spin orbit coupling can be viewed in an analogous way as the case of time reversal symmetry for spinless particles. This paper is organized as follows. In Sec. \[sec:2\], a single-particle description of the one-dimensional ring subject to Rashba-SOC in terms of Hamiltonian, eigenstates and eigenenergies is given, following Ref. . The section concludes with the results for the transmission of the asymmetric AC-ring in the one-electron scattering picture which is derived in the appendix. The analytic expression for the transmission function is analyzed in Sec. \[sec:3\] with focus on geometry and SOC dependence of the transmission zeros. Sec. \[sec:4\] contains a symmetry argument which establishes an analogy between formation and lifting of the zeros due to Rashba-SOC in the AC-ring and the corresponding effects on spinless electrons due to the magnetic field in the AB-ring. The main results are summarized in the conclusions of Sec. \[sec:5\]. AC-ring in single particle picture \[sec:2\] ============================================ The coupling of electron spin and orbital degrees of freedom is due to the magnetic field generated in the reference frame of a moving electron by an electric field in the reference frame of the laboratory. In two dimensional systems (e.g. due to the presence of a confinement potential along a specific direction), an important contribution of electric fields is the Rashba effect, a consequence of lack of inversion symmetry, that causes a spin band-splitting proportional to the momentum. In the ring system under consideration, the Rashba field results from the asymmetric confinement along the direction perpendicular to the ring plane. Hamiltonian ----------- In the following investigation of one-dimensional rings, $z$ is chosen as the direction of confinement, perpendicular to the plane of motion. The various SO-coupling mechanisms are accounted for using the following model Hamiltonian: $$\hat{H}_{SO}=\frac{\alpha}{\hbar}(\hat{\vec{\sigma}}\times\hat{\vec{p}})_{z}=i\alpha\left(\hat{\sigma}_{y}{\frac{\partial}{\partial x}}-\hat{\sigma}_{x}{\frac{\partial}{\partial y}}\right),$$ where $\frac{\hbar}{2}\hat{\vec{\sigma}}$ is the spin operator in terms of the Pauli spin matrices, $\hat{\vec{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ and $\alpha$ is the Rashba parameter characterizing the strength of the SOC corresponding to an electric field $\vec E_{R}=\left(0,0,E_{z}\right)$ in $z$-direction, arising from a potential $V(z)$ due to structural or confinement asymmetry. In polar coordinates $x=r\cos\varphi$ and $y=r\sin\varphi$ the total Hamiltonian in effective mass approximation reads[@meijer:02] $$\begin{aligned} \hat{H}(r,\varphi)=&-\frac{\hbar^{2}}{2m^{*}}\left[{\partial^{2}_{ r}}+\frac{1}{r}{\frac{\partial}{\partial r}}+\frac{1}{r^{2}}{\partial^{2}_{ \varphi}}\right]-\frac{i\alpha}{r}(\cos\varphi\sigma_{x}\nonumber\\&+\sin\varphi\sigma_{y}){\frac{\partial}{\partial \varphi}}+i\alpha(\cos\varphi\sigma_{y}-\sin\varphi\sigma_{x}){\frac{\partial}{\partial r}}, \end{aligned}$$ with the effective mass $m^{*}$. In the case of a one-dimensional ring, a confining potential $V(r)$ needs to be added in order to force the electron wave functions to be localized on the ring in the radial direction. It is shown in Ref. that the exact form of the confining potential is not essential. A simple possibility is the harmonic potential centered around $r=\r$, $V(r)=\frac{1}{2}K(r-\r)^{2}$ where $ \r $ is the radius of the ring. ![(a) Momentum dependent in-plane Rashba field $\vec{B}_{R}$, (b) Up and down spin eigenstates do not generally align with the Rashba field $\vec{B}_{R}$, but make a tilt angle $\theta$ with the electric field $\vec{E}_{R}$ perpendicular to the ring plane ($\vec{E}_{R}$, $\vec{B}_{R}$ and $\vec{v}_{g}$ form an orthogonal coordinate system). []{data-label="fig:tiltangle1"}](fig1a.eps "fig:"){width="2.5in"}![(a) Momentum dependent in-plane Rashba field $\vec{B}_{R}$, (b) Up and down spin eigenstates do not generally align with the Rashba field $\vec{B}_{R}$, but make a tilt angle $\theta$ with the electric field $\vec{E}_{R}$ perpendicular to the ring plane ($\vec{E}_{R}$, $\vec{B}_{R}$ and $\vec{v}_{g}$ form an orthogonal coordinate system). []{data-label="fig:tiltangle1"}](fig1b.eps "fig:"){width="2.6in"} Considering only the lowest radial mode, the resulting one-dimensional Hamiltonian for fixed radius $\r$ is (see Ref. for a complete derivation) $$\begin{aligned} \hat{H}_{1D}(\varphi)=&\langle R_{0}(r)|\hat{H}(r,\varphi)|R_{0}(r)\rangle\nonumber\\ =&-\frac{\hbar^2}{2m^{*}\r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}-\frac{i\alpha}{\r}(\cos\varphi\sigma_{x}\nonumber\\&+\sin\varphi\sigma_{y})\frac{\partial}{\partial\varphi}-\frac{i\alpha}{2\r}(\cos\varphi\sigma_{y}-\sin\varphi\sigma_{x}). \label{eq:1dham} \end{aligned}$$ The last term in the above expression for the 1D-Hamiltonian encodes the correction due to the radial confinement. The Hamiltonian in Eq.(\[eq:1dham\]) can be written in a dimensionless form[@molnar:04], $$H = \frac{2m^{*}\r^{2}}{\hbar^{2}}\hat H_{1D}=\left(-i{\frac{\partial}{\partial \varphi}}+\frac{\beta}{2}\sigma_{r}\right)^{2}\label{eq:Hamiltonian}$$ where $\beta=2\alpha m^{*}\r/\hbar^{2}$ is the dimensionless SOC-constant, $\sigma_{r}=\cos\varphi\sigma_{x}+\sin\varphi\sigma_{y}$ and the additive constant $-\beta^{2}/4$ was neglected[^1]. Eigenstates and energy spectrum ------------------------------- The eigenstates of Hamiltonian follow as the solution of the time-independent Schrödinger equation and have the general form [@frustaglia:04; @molnar:04] $$\Psi_{n}^{\sigma}(\varphi)=e^{i n\varphi}\chi^{\sigma}(\varphi), \label{eq:ringeigstate}$$ where $n$ is the orbital quantum number and $\sigma=\uparrow,\downarrow\cong\pm1$ labels the spin. For the isolated ring, $n\in\mathbbm{Z}$, but when coupled to leads, $n$ can adopt any real number allowed by energy, depending on spin and direction of motion. The spinors $\chi^{\sigma}(\varphi)$ are generally not aligned with the momentum dependent and spatially varying Rashba-field $\vec{B}_{R}(r)=2\beta(\hat z\times \vec p)/\varrho$, but make a tilt angle $\tilde\theta=\pi/2-\theta$ given by $\tan\theta=-\beta$ relative to the direction of the electric field $\vec{E}_{R}$ (see Fig. \[fig:tiltangle1\]). The energy eigenvalues of the states in Eq.(\[eq:ringeigstate\]) are[@molnar:04] $$E_{n}^{\sigma}=\left(n-\Phi_{AC}^{\sigma}/2\pi\right)^{2}.\label{eq:eigenenergy}$$ with the Aharonov-Casher phase[@frustaglia:04] $$\Phi_{AC}^{\sigma}=-\pi\left(1-\sigma\sqrt{\beta^{2}+1}\right).$$ At fixed energy $E$, the dispersion relation yields the quantum numbers $n_{\lambda}^{\sigma}(E)$ through $$n_{\lambda}^{\sigma}(E)=\lambda \sqrt{E} +\Phi_{AC}^{\sigma}/2\pi,\quad \lambda=\pm.$$ For a plane wave arriving from lead I with wave vector $k$ we get $$n_{\lambda}^{\sigma}(k)=\lambda k\r+\Phi_{AC}^{\sigma}/2\pi. \label{eq:n}$$ The sense of propagation is determined by the sign of the group velocity, which in the latter case is given by $$v_{g,\lambda}^{\sigma}=\frac{\hbar}{2m^{*}\r}\frac{dE_{n_{\lambda}^{\sigma}}^{\sigma}}{dn_{\lambda}^{\sigma}}=\frac{\hbar}{2m^{*}\r}(n_{\lambda}^{\sigma}-\Phi_{AC}^{\sigma}/2\pi)=\lambda k\r,$$ $\lambda$ thus encoding the traveling direction. The quantum numbers for different spin and sense of propagation are related by $$n_{\lambda}^{\sigma}=-\left(n_{-\lambda}^{-\sigma}+1\right).\label{eq:qn}$$The corresponding eigenstates of the closed ring are $$\begin{aligned} \Psi_{\lambda}^{\sigma}(\varphi)&=e^{in_{\lambda}^{\sigma}\varphi} \left(\begin{array}{c} \sin\Big(\frac{\theta}{2}+\frac{\pi}{4}(1+\sigma)\Big)\\ -\cos\Big(\frac{\theta}{2}+\frac{\pi}{4}(1+\sigma)\Big)e^{i\varphi}\\ \end{array}\right) \frac{1}{\sqrt{2\pi}},\label{eq:es}\\ \sigma&=\pm1,\quad\lambda=\pm.\nonumber\end{aligned}$$ These eigenstates differ from the solutions of the free system by the phase factors in the spin part. Current ------- In order to investigate transport in our quantum mechanical system, an expression for the probability current density is requested. The probability current density $j$ is determined by inserting the Schrödinger equation $$i\hbar\frac{\partial\Psi}{\partial t}=H\Psi,$$ with $H$ from Eq.(\[eq:Hamiltonian\]), and its adjoint into the continuity equation imposed by probability conservation, $$\frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial \varphi}=0,$$ where $\rho=|\Psi|^2$ denotes the probability density. The probability current density can be expressed in terms of velocity operators: $$j=\frac{1}{2}\left(\Psi^{\dagger}(\hat{v}\Psi)+\Psi(\hat{v}\Psi)^{\dagger}\right). \label{eq:current}$$ The velocity operators are derived from the Hamiltonian by[@molenkamp:01], $$\hat{v}=\frac{\partial \hat{H}}{\partial\hat{p}}$$ where $\hat{p}$ is the momentum operator, whose explicit form depends on the coordinate system[^2]: $$\hat{p}_{\varphi}=-i{\frac{\partial}{\partial \varphi}}~(\mathrm{ring})\quad \mathrm{and}\quad\hat{p}_{x}=-i\r{\frac{\partial}{\partial x}}~(\mathrm{leads}).$$ In absence of SOC, only the kinetic energy term of the Hamiltonian contributes to the velocity operators, which in this case are $$\hat{v}_{0}(\varphi)=-2i{\frac{\partial}{\partial \varphi}}\quad \mathrm{and}\quad \hat{v}_{0}(x)=-2i\r{\frac{\partial}{\partial x}}.\label{eq:v0}$$ For finite SOC, $H_{SO}$ yields an additional term for the ring (assuming zero SOC in the leads) $$\hat{v}_{SO}(\varphi)=\beta\sigma_{r}(\varphi)~\mathrm{and}~\hat{v'}_{SO}(\varphi')=\beta\sigma_{r}'(\varphi'),\label{eq:vso}$$ where $\sigma_{r}'(\varphi')\equiv\sigma_{r}(-\varphi')=\cos\varphi'\sigma_{x}-\sin\varphi'\sigma_{y}$. The total velocity operator to consider in the expression of the probability current density given by Eq. is $$\hat v=\hat{v}_{0}+\hat{v}_{SO}.$$ The above results will be used when investigating the lead and ring-currents in the appendix. Transmission amplitude from the one-electron scattering formalism ----------------------------------------------------------------- Conductance in mesoscopic structures can be expressed by means of the Landauer conductance formula, which in our case reads $$G=\frac{e^{2}}{h}\sum_{\sigma=\uparrow,\downarrow}|T_{\sigma}|^{2},\label{eq:land}$$ where $T_{\sigma}$ is the (spin dependent) transmission amplitude[^3]. The previously obtained expressions for wavefunction and current are now used to calculate the transmission amplitude for the ring system from the proper requirements on wave function continuity and probability current conservation[@griffiths:53]. The calculation is performed in the appendix and follows Ref. and . It yields the transmission amplitude $$T_{\sigma}(\phi,\beta,\gamma)=\frac{4i\Big[e^{i\frac{\Phi_{AC}^{\sigma}}{2}(1-\gamma)}\sin\big(\frac{\phi}{2}(1+\gamma)\big)+e^{-i\frac{\Phi_{AC}^{\sigma}}{2}(1+\gamma)}\sin\big(\frac{\phi}{2}(1-\gamma)\big)\Big]}{\cos\phi\gamma-5\cos\phi+4\cos\Phi_{AC}^{\sigma}+4i\sin\phi} \label{eq:trm}$$ as a function of energy ($\phi=2\pi k\r$), SOC ($\Phi_{AC}^{\sigma}(\beta)$) and asymmetry ($\gamma=(1-R)/(1+R)$ ), where $R$ stands for the ratio of lower and upper ring arm lengths (see Fig.\[fig:tiltangle1\]). In the following discussion of transmission and conductance, spin index $\sigma$ refers to the spinors in the ring eigenstates in Eq. , whereas the standard spinor basis (eigenvectors of $\sigma_{z})$ are labeled by $s$. Geometry and SOC dependence of transmission zeros \[sec:3\] =========================================================== Free system $(\beta=0)$ ----------------------- The transmission function in Eq. displays a peculiar resonant behavior characterized by a set of zeros and poles. The transmission zeros are obtained from Eq. as the solution of $$\sin\Big(\frac{\phi}{2}(\gamma-1)\Big)=e^{-i\Phi_{AC}^{\sigma}}\sin\Big(\frac{\phi}{2}(\gamma+1)\Big). \label{eq:z}$$ For $\beta=0$, the phase factor equals unity, and Eq. simplifies to $$\cos\Big(\frac{\phi}{2}\gamma\Big)\sin\Big(\frac{\phi}{2}\Big)=0,$$ what yields zeros at $$\phi_{0,1}=2m\pi \quad \textrm{and}\quad \phi_{0,2}=(2m+1)\pi/\gamma,\quad m\in\mathbbm{Z}.$$ Obviously, there are two types of zeros. The zeros of the first kind at $\phi_{0,1}$ correspond to the eigenstates of the closed ring, whereas the zeros of second type at $\phi_{0,2}$ are given by the geometry dependent interference condition for nodes at the right junction[@yi:03] and appear only in an asymmetric configuration ($\gamma\neq0$). The poles related to transmission resonances are determined by $$\cos\phi\gamma-5\cos\phi+4\cos\Phi_{AC}^{\sigma}+4i\sin\phi=0.$$ Fig. \[fig:nosoc\] shows the conductance in absence of SOC ($\beta=0$) for symmetry ($R=1$) and asymmetry parameters $R=1/2$ and $R=(2\sqrt{3}-1)/(2\sqrt{3}+1)\approx0.55$. The oscillation in the conductance for the symmetric configuration is due to the coupling of lead and ring, which does not correspond to perfect transmission and therefore leads to resonances as a consequence of backscattering effects[@buettiker:84]. These resonances however do not give rise to conductance zeros: from Eq. follows that zeros and poles of the conductance compensate each other and yield a finite value. In the asymmetric ring ($R=1/2$, $R\approx0.55$), both types of zeros appear. \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$$]{} \[t\]\[c\][$\phi$]{} \[\]\[r\]\[0.8\][$R=1$]{} \[\]\[r\]\[0.8\][$R=1/2$]{} \[\]\[r\]\[0.8\][$R\approx0.55$]{} ![Conductance for $\beta=0$ in symmetric ($R=1$) and asymmetric system ($R=1/2$ and $R=(2\sqrt{3}-1)/(2\sqrt{3}+1)\approx0.55$. For the lead-ring coupling assumed in the present model ($\epsilon=4/9$), transmission is not perfect even in case of equal branch length. In the asymmetric system, periodical transmission zeros appear.[]{data-label="fig:nosoc"}](fig3.eps "fig:"){width="3.2in"} By examination of the transmission amplitude in the complex energy plane we find a certain connection between the conductance zeros and transmission resonances. Zeros on the real axis are accompanied by nearby poles in complex plane (Fig. \[fig:polesnosoc\] ) and \[fig:zeropole\]). Fig. \[fig:zeropole\] shows zeros (a) and poles (b) at $R=1/2$ separately . A similar feature is known from the quantum waveguide systems with an attached resonator [@porod:94]. In the present case a pair of poles is associated with each zero of the first kind at $\phi_{0,1}$. The real part of the energies of the zeros and poles are not exactly identical, which results in an asymmetric shape of the resonance (Fano type) [@fano:61]. These characteristic features can be clearly observed in Fig. \[fig:nosoc\], e.g. at $\phi_{0,1}=2\pi$. Note that at $\phi=0$ and $\phi=6\pi$ both numerator and denominator of the transmission amplitude vanish simultaneously for $R=1/2$, such that they annihilate at these places, as can be easily observed in Fig. \[fig:nosoc\]. (0,0)(5,6.0) \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[\]\[\][$$]{} \[\]\[\][$Im~\phi$]{} (2.2,4.6)[![Zero-pole structure of $G$ in the complex plane for $\beta=0$ and (a) $R=1$, (b) $R=1/2$, (c) $R\approx0.55$. The zeros lie on the real axis, whereas the poles have a finite imaginary part.[]{data-label="fig:polesnosoc"}](fig4a.eps "fig:"){width="3.5in"}]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[\]\[\][$$]{} \[\]\[\][$Im~\phi$]{} (2.2,2.3)[![Zero-pole structure of $G$ in the complex plane for $\beta=0$ and (a) $R=1$, (b) $R=1/2$, (c) $R\approx0.55$. The zeros lie on the real axis, whereas the poles have a finite imaginary part.[]{data-label="fig:polesnosoc"}](fig4b.eps "fig:"){width="3.5in"}]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$12\pi$]{} \[\]\[\][$Re~\phi$]{} \[\]\[\][$Im~\phi$]{} \[\]\[\]\[0.6\][$0$]{} (2.2,.0)[![Zero-pole structure of $G$ in the complex plane for $\beta=0$ and (a) $R=1$, (b) $R=1/2$, (c) $R\approx0.55$. The zeros lie on the real axis, whereas the poles have a finite imaginary part.[]{data-label="fig:polesnosoc"}](fig4c.eps "fig:"){width="3.5in"}]{} (-1.8,5.7)[$(a)$]{} (-1.8,3.4)[$(b)$]{} (-1.8,1.2)[$(c)$]{} (0,0)(5,3.6) \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[b\][$(a)$]{} \[b\][$(b)$]{} \[c\]\[r\][$$]{} \[c\]\[c\][$Im~\phi$]{} (2.2,2.45)[![(a) zeros and (b) poles of $G$ in the asymmetric case ($R=1/2$) for $\beta=0$.[]{data-label="fig:zeropole"}](fig5a.eps "fig:"){width="3.5in"}]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$12\pi$]{} \[c\]\[r\][$Re~\phi$]{} \[c\]\[c\][$Im~\phi$]{} \[\]\[\]\[0.6\][$0$]{} (2.15,.0)[![(a) zeros and (b) poles of $G$ in the asymmetric case ($R=1/2$) for $\beta=0$.[]{data-label="fig:zeropole"}](fig5b.eps "fig:"){width="3.8in"}]{} (-1.8,3.6)[$(a)$]{} (-1.8,1.2)[$(b)$]{} Finite Rashba-SOC $(\beta\neq0)$ -------------------------------- There are two remarkable features in the transmission characteristics arising as effects of SOC. The first is the finite transmission probability in the spin channel opposite to the incident spin orientation. This is the result of spin precession along the ring branches due to SOC as considered in Ref. . The conductance zeros in the opposite channel correspond to a frequency of precession which reproduces the incident spin orientation at the right junction. The second aspect, and the one on which we will concentrate in the following, is the lifting of certain conductance zeros in the incident channel. These features can be observed in Fig. \[fig:soc\] where the transmission for finite SOC is displayed. (0,0)(5,13) \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$$]{} \[b\][$(a)$]{} \[b\][$(b)$]{} \[t\]\[c\][$\phi$]{} \[\]\[r\]\[0.8\][$s=\uparrow$]{} \[\]\[r\]\[0.8\][$s=\downarrow$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} (2.2,11.0)[![Transmission probability for nonzero SOC $\beta=0.6$ from $s=\uparrow$ into $s=\uparrow$ and $s=\downarrow$ spin channels for symmetric ((a) $R=1$) and asymmetric system ((b) $R=1/2$, (c) $R\approx0.55$). In the symmetric AC-ring, spin-orbit interaction causes the appearance of transmission zeros, in the asymmetric configuration however, the latter are partially lifted. For particular asymmetry ratios $R$, the geometry dependent zeros persist.[]{data-label="fig:soc"}](fig6a.eps "fig:"){width="3.2in"}]{} (2.2,6.75)[![Transmission probability for nonzero SOC $\beta=0.6$ from $s=\uparrow$ into $s=\uparrow$ and $s=\downarrow$ spin channels for symmetric ((a) $R=1$) and asymmetric system ((b) $R=1/2$, (c) $R\approx0.55$). In the symmetric AC-ring, spin-orbit interaction causes the appearance of transmission zeros, in the asymmetric configuration however, the latter are partially lifted. For particular asymmetry ratios $R$, the geometry dependent zeros persist.[]{data-label="fig:soc"}](fig6b.eps "fig:"){width="3.2in"}]{} (2.2,2.50)[![Transmission probability for nonzero SOC $\beta=0.6$ from $s=\uparrow$ into $s=\uparrow$ and $s=\downarrow$ spin channels for symmetric ((a) $R=1$) and asymmetric system ((b) $R=1/2$, (c) $R\approx0.55$). In the symmetric AC-ring, spin-orbit interaction causes the appearance of transmission zeros, in the asymmetric configuration however, the latter are partially lifted. For particular asymmetry ratios $R$, the geometry dependent zeros persist.[]{data-label="fig:soc"}](fig6c.eps "fig:"){width="3.2in"}]{} (-2.0,13.)[$(a)$]{} (-2.0,8.8)[$(b)$]{} (-2.0,4.5)[$(c)$]{} It is instructive to analyze the modification of the transmission amplitude in the complex energy plane. Fig.\[fig:polessoc\] shows the lifting of the zeros of the first kind as well as the emergence of zeros that were canceled by poles in the free system. The shifting of zeros and poles away from the real axis is displayed in Fig. \[fig:zeropolesoc\]. In the conductance, the zeros of the first kind appear no longer. They are still present in the up- and down transmission amplitudes, but different spin components are shifted in opposite directions, as it is shown in Fig. \[fig:zeroshift\]. (0,0)(5,6.0) \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[\]\[\][$$]{} \[\]\[\][$Im~\phi$]{} (2.2,4.6)[![Zero-pole structure of $G$ in the complex plane for $\beta=0.6$ and (a) $R=1$, (b) $R=1/2$, (c) $R\approx0.55$.[]{data-label="fig:polessoc"}](fig7a.eps "fig:"){width="3.5in"}]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[\]\[\][$$]{} \[\]\[\][$Im~\phi$]{} (2.2,2.3)[![Zero-pole structure of $G$ in the complex plane for $\beta=0.6$ and (a) $R=1$, (b) $R=1/2$, (c) $R\approx0.55$.[]{data-label="fig:polessoc"}](fig7b.eps "fig:"){width="3.5in"}]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$12\pi$]{} \[\]\[\][$Re~\phi$]{} \[\]\[\][$Im~\phi$]{} \[\]\[\]\[0.6\][$0$]{} (2.2,.0)[![Zero-pole structure of $G$ in the complex plane for $\beta=0.6$ and (a) $R=1$, (b) $R=1/2$, (c) $R\approx0.55$.[]{data-label="fig:polessoc"}](fig7c.eps "fig:"){width="3.5in"}]{} (-1.8,5.7)[$(a)$]{} (-1.8,3.4)[$(b)$]{} (-1.8,1.2)[$(c)$]{} (0,0)(5,3.7) \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[b\][$(a)$]{} \[b\][$(b)$]{} \[c\]\[r\][$$]{} \[c\]\[c\][$Im~\phi$]{} (2.2,2.3)[![(a) zeros and (b) poles of $G$ in the asymmetric case ($R=1/2$) for $\beta=0.6$.[]{data-label="fig:zeropolesoc"}](fig8a.eps "fig:"){width="3.6in"}]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$12\pi$]{} \[c\]\[r\][$Re~\phi$]{} \[c\]\[c\][$Im~\phi$]{} \[\]\[\]\[0.6\][$0$]{} (2.2,.0)[![(a) zeros and (b) poles of $G$ in the asymmetric case ($R=1/2$) for $\beta=0.6$.[]{data-label="fig:zeropolesoc"}](fig8b.eps "fig:"){width="3.6in"}]{} (-1.8,3.4)[$(a)$]{} (-1.8,1.2)[$(b)$]{} To study the behavior of the transmission zeros under the influence of SOC, an expansion around the zeros in the AC-phase $\Phi_{AC}^{\sigma}~(mod~2\pi)$ of the transmission probability $\mathscr{T}_{\sigma}=|T_{\sigma}|^{2}$ is performed: $$\begin{aligned} \mathscr{T}_{\sigma}(\Phi_{AC}^{\sigma})& =8\csc(\pi m\gamma)\left(\Phi_{AC}^{\sigma}\right)^{2}+ O\big[\left(\Phi_{AC}^{\sigma}\right)^{3}\big]\nonumber \\ &\quad\mathrm{at}\quad\phi_{0,1}=2m\pi,\label{eq:expan1}\\ \nonumber\\ \mathscr{T}_{\sigma}(\Phi_{AC}^{\sigma})& =\frac{16\left[1+\cos\big((2m+1)\pi/\gamma\big)\right]}{\left[5-3 \cos\big((2m+1)\pi/\gamma\big)\right]^{2}}\left(\Phi_{AC}^{\sigma}\right)^{2}\nonumber\\ &+O\big[\left(\Phi_{AC}^{\sigma}\right)^{3}\big]\nonumber\\ &\quad\mathrm{at}\quad\phi_{0,2}=(2m+1)\pi/\gamma.\label{eq:expan2}\end{aligned}$$ Eq. shows that the zeros of the first type are removed by the action of SOC for all values of $\gamma$. For the zeros of the second type however there are geometries where the zeros persist even in presence of the interaction. From Eq. follows the geometry condition for persistent zeros: $$\begin{aligned} \gamma_{per}=\frac{2m+1}{2n+1}\quad\Leftrightarrow\quad R_{per}=\frac{m+n+1}{n-m},\\ \nonumber\\ n,m\in\mathbb{Z},~n>m.\nonumber\end{aligned}$$ (0,0)(2,2.2) \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$-\pi$]{} \[\]\[\]\[0.6\][$0$]{} \[\]\[\]\[0.6\][$\pi$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[\]\[\]\[0.6\][$$]{} \[c\]\[c\][$$]{} \[c\]\[r\][$$]{} \[\]\[\][$a)$]{} \[\]\[\][$b)$]{} \[\]\[\][$c)$]{} \[\]\[\][$d)$]{} \[\]\[\]\[0.6\][$2\pi$]{} \[\]\[\]\[0.6\][$4\pi$]{} \[\]\[\]\[0.6\][$6\pi$]{} \[\]\[\]\[0.6\][$8\pi$]{} \[\]\[\]\[0.6\][$10\pi$]{} \[\]\[\]\[0.6\][$12\pi$]{} \[c\]\[c\][$Re~\phi$]{} \[c\]\[c\][$Im~\phi$]{} ![Zeros of a) $T_{\uparrow}$ and b) $T_{\downarrow}$ for $\beta=0.6$ and $R=1/2$. SOC shifts the zeros away from the real axis, the direction of the shift depending on the spin.[]{data-label="fig:zeroshift"}](fig9.eps "fig:"){width="2.3in"} (-3.7,2.2)[$a)$]{} (0,2.2)[$b)$]{} Analogy to AB-ring and symmetry argument\[sec:4\] ================================================= It was shown [@wakabayashi:01; @wakabayashi:00b] for the AB-ring that zero conductance energies belong to states of vanishing vorticity, i.e. the circular currents in the loop system change sign at these energies. The zero conductance resonances can therefore be regarded as the signatures of destructive interference resulting from the superposition of circular currents of opposite chirality corresponding to degenerate resonant states of the loop system. The possibility of superposition is due to the degeneracy of the two chiral states as a consequence of time reversal symmetry in absence of external fields. A magnetic field, respectively the resulting flux $\Phi$ through the loop, destroys this degeneracy as a consequence of broken TRS (Fig. \[fig:symmetry\]). ![Resonant states and broken symmetry for AB-ring (E=energy).[]{data-label="fig:symmetry"}](fig10.eps){width="2.0in"} In the present case of a one-dimensional ring subject to Rashba SOC, the role of the magnetic flux $\Phi$ is played by the Rashba term depending on the coupling $\beta$. In fact, the transmission function in Eq. equals the expression obtained in Ref. for the asymmetric AB-ring, except that the AB-phase $\Phi_{AB}=2\pi\Phi/\Phi_{0}$ is replaced by the (spin dependent) AC-phase $\Phi_{AC}^{\sigma}$. In analogy to the AB-ring, there are conductance zeros due to resonant states of different chirality for the free system at $\beta=0$. At finite SOC, configurations of opposite spin *and* chirality are still degenerate as a consequence of time reversal symmetry: with the time reversal operator given by $\hat{T}=-i\sigma_{y} \hat K$, where $\hat K$ is the operator for complex conjugation, and using the relations in Eq., we find $$\hat T \Psi_{n,\lambda}^{\sigma}=-\sigma\Psi_{n,-\lambda}^{-\sigma}.$$ For a fixed spin orientation however, states of opposite chirality are no longer degenerate as parity is broken for $\beta\neq0$. The situation with SOC is illustrated in Fig. \[fig:symmetrysoc\]. ![Resonant states and broken parity for ring subject to Rashba SOC. []{data-label="fig:symmetrysoc"}](fig11.eps){width="2.0in"} The vanishing of circular currents corresponding to time reversed degenerate states is easily derived: from Eqs. , follow the currents $$\begin{aligned} j_{\lambda}^{\sigma}& =2\left(n_{\lambda}^{\sigma}+\sin^{2}\Big(\frac{\theta}{2}+\frac{\pi}{4}(1-\sigma)\Big)\right)+\sigma\beta\sin\theta\\ &\sigma=\pm1,\quad\lambda=\pm. \nonumber\end{aligned}$$ The total circular current of time reversed states has to vanish such that $$j_{tot}\left(\Psi_{\lambda}^{\sigma},\Psi_{-\lambda}^{-\sigma}\right) =2\left(n_{\lambda}^{\sigma}+1+n_{-\lambda}^{-\sigma}\right)\equiv0\quad\forall\beta,$$ whereas the total circular current for states of a equal spin disappear only for $\beta \to 0$, $$\begin{aligned} j_{tot}\left(\Psi_{+}^{\uparrow},\Psi_{-}^{\uparrow}\right)& =2\left(n_{+}^{\uparrow}+n_{-}^{\uparrow}+2\sin^{2}\frac{\theta}{2}+\beta\sin\theta\right),\\ j_{tot}\left(\Psi_{+}^{\downarrow},\Psi_{-}^{\downarrow}\right)& =2\left(n_{+}^{\downarrow}+n_{-}^{\downarrow}+2\cos^{2}\frac{\theta}{2}-\beta\sin\theta\right). \end{aligned}$$ The symmetry breaking analogy between AB-rings and rings subject to Rashba-SOC appears already in the corresponding Hamiltonians and their symmetries. For Rashba SOC, the normalized magnetic flux $\Phi/\Phi_{0}$ breaking time reversal symmetry in the AB-ring is replaced by the spin dependent vector potential $A(\varphi)$ which respects the TRS of $\hat{H}$, i.e. $$\big[\hat H,\hat T\big]_{\Psi_{n,\lambda}^{\sigma}}=\left(n+1+ \Phi_{AC}^{-\sigma}\right)^{2}-\left(n-\Phi_{AC}^{\sigma}\right)^{2}\equiv0\quad\forall \beta,$$ but changes under spin reversal, and which is related to the Aharonov-Casher phase by Eq. for the eigenenergies. The main results of this analysis are summarized in Tab. \[tab:analogy\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ AB-ring ring with Rashba-SOC ----------------------------------------------------------------- -------------------------------------------------------------------------- --------------------------------------------------------------------------------- $\begin{array}{l}\textrm{ext.}\\ \textrm{field}\\ $\vec{B}=(0,0,B_{z})$ $\vec{E}=(0,0,E_{z})$ \end{array}$ $\begin{array}{l}\textrm{Hamil-}\\\textrm{tonian}\\\end{array}$ $\hat{H}=\frac{1}{2m\r^{2}}\left(\frac{\hbar}{i}\frac{\partial}{\partial $\hat{H}=\frac{1}{2m\r^{2}} \varphi}+\frac{\Phi}{\Phi_{0}}\right)^{2}$ \left(\frac{\hbar}{i}{\frac{\partial}{\partial \varphi}}+A(\varphi)\right)^{2}$ $\Phi_{0}=\frac{hc}{e}$ $A(\varphi)=\frac{\beta\hbar^{2}}{2}\sigma_{r}(\varphi)$ $\begin{array}{l} time reversal $\hat{T}$ spin parity $\hat{P}_{s}$ \textrm{broken} \\ \textrm{symm.}\\ \end{array}$ $\begin{array}{lcl}[\hat{H},\hat{T}]&= $\begin{array}{lcl}[\hat{H},\hat{P}_{s}]&=&-i\beta &\frac{2\hbar k}{m}\frac{\Phi}{\Phi_{0}}\\&=&0\Leftrightarrow\Phi=0 \sin\varphi\sigma_{z}\\&=&0\Leftrightarrow\beta=0\end{array}$ \end{array}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ : Symmetry breaking analogy between AB- and Rashba-rings.[]{data-label="tab:analogy"} In Ref. , a relation was established between the breaking of TRS by a magnetic field in an AB-ring and the location of the transmission zeros in the complex plane. It was shown that real zeros appear if the flux is an integer or half integer multiple of the flux quantum $\Phi_{0}$, and are shifted off the real axis for other flux values. Due to the analogy to the AB-ring, the behavior of the transmission zeros of the ring subject to Rashba-SOC follows the same rules, now depending on the value of the AC-phase. This implies the periodical dependence of transmission properties on the value of the SOC-constant $\beta$: real transmission zeros demand a (half) integer AC-Phase, $\Phi_{AC}^{\sigma}/2\pi=(2m+1)/2$, $m\in\mathbbm{Z}$, which is satisfied by[@frustaglia:04] $\beta=\sqrt{4(m+1)^{2}-1}$. It was shown by Lee and co-workers that conductance zeros occur generically in quasi-1D systems if time reversal is a symmetry[@lee:99; @lee:01]. In the proof, they used the constraints laid upon the elements of the scattering matrix describing the system of spinless particles by the symmetry and unitarity requirement. In the case of particles with spin, these conditions are reproduced only in the presence of time reversal symmetry [*and*]{} parity, apart from special situations (geometries). As in Ref. , it is possible to combine the AB- and AC-effects by the inclusion of a finite magnetic flux in the Hamiltonian , $$H=\left(-i{\frac{\partial}{\partial \varphi}}+\frac{\beta}{2}\sigma_{r}-\frac{\Phi}{\Phi_{0}}\right)^{2}.\label{eq:HamiltonianFlux}$$ Eq. becomes[@molnar:04] $$n_{\lambda}^{\sigma}(k)=\lambda k\r+\big(\Phi_{AC}^{\sigma}+\Phi_{AB}\big)/2\pi. \label{eq:nflux}$$ Thus, the AB-effect contributes just a spin independent additive phase, i.e. in eq. , the AC-phase has to be replaced by the sum of AB- and AC-phases. Conclusions\[sec:5\] ==================== In summary, we have shown that zero conductance resonances appearing as a signature of interfering resonant states of the loop system, and as a consequence of its asymmetry, behave in a similar way under the influence of a magnetic flux through the loop as in presence of a perpendicular electric field generating Rashba spin-orbit coupling. Real conductance zeros are lifted by the influence of these external fields, being shifted into the complex plane depending on the value of the AB(AC)-phase. In the case of the magnetic flux, it is the breaking of time reversal symmetry which destroys the energetic degeneracy of states with opposite chirality, preventing the destructive interference leading to the zeros. For Rashba SOC, time reversal symmetry is respected, but not spin reversal symmetry, which again leads to a chiral dependence in the energy of the loop wave function and eventually to the lifting of the conductance zeros. 0.4 cm We gratefully acknowledge the financial support of this study by a Grant of the Swiss Nationalfonds. Derivation of the transmission amplitude in the single-electron scattering picture ================================================================================== This derivation follows Ref. and . System geometry and coordinates are shown in Fig. \[fig:tiltangle1\]. Asymmetry is introduced by choosing different lengths $l_{up}$ and $l_{low}$ for upper and lower branches of the ring in the figure, and is expressed by means of the asymmetry factor $R=l_{low}/l_{up}$. This leads to different phases at the left junction ($A$): $$\varphi(A)=\frac{2\pi}{R+1}\equiv\varphi_{A},\qquad \varphi'(A)=\frac{2\pi R}{R+1}\equiv\varphi'_{A}$$ The connection of leads and ring is described by the application of spin-dependent Griffith’s boundary conditions [@griffiths:53], which demand a) continuity of the wave function and b) probability current conservation at the junctions (A) and (B)[^4]. To be able to apply the boundary conditions, we need the wave functions of leads and branches. The wavefunctions $\Psi_{I}$ and $\Psi_{II}$ for incoming and outgoing leads respectively, can be expanded in terms of the spinors $\chi^{\sigma}$ at the junctions, $$\begin{aligned} \Psi_{I}(x)&=\sum_{\sigma=\uparrow,\downarrow}\Psi_{I}^{\sigma}(x)\chi^{\sigma}(\varphi_{A}),\quad x\in[-\infty,0],\\ \Psi_{II}(x')&=\sum_{\sigma=\uparrow,\downarrow}\Psi_{II}^{\sigma}(x')\chi^{\sigma}(0),\quad x'\in[0,\infty], \end{aligned}$$ The expansion coefficients are the orbital wave functions $$\begin{aligned} \Psi_{I}^{\sigma}(x)&=i_{\sigma}e^{ikx}+r_{\sigma}e^{-ikx},\\ \Psi_{II}^{\sigma}(x')&=t_{\sigma}e^{ikx'}\end{aligned}$$ where we assume an incident plane wave from the left with wave number $k$. The coefficients $i_{\sigma}$ of the incoming wave are chosen such that $\sum_{\sigma}|i_{\sigma}|^{2}=1$. $r_{\sigma}$ and $t_{\sigma}$ are the spin dependent reflection and transmission coefficients, respectively. A similar expansion in terms of the ring eigenstates in (\[eq:es\]) yields the wave functions $\Psi_{up}$ and $\Psi_{low}$ of upper and lower branches, respectively: $$\begin{aligned} \Psi_{up}(\varphi)&=\sum_{\sigma=\uparrow,\downarrow}\Psi_{up}^{\sigma}(\varphi)\chi^{\sigma}(\varphi),\quad\varphi\in[0,\varphi_{A}],\\ \Psi_{low}(\varphi')&=\sum_{\sigma=\uparrow,\downarrow}\Psi_{low}^{\sigma}(\varphi')\chi^{\sigma}(-\varphi'),\quad\varphi'\in[0,\varphi'_{A}],\end{aligned}$$ with the corresponding orbital components $$\begin{aligned} \Psi_{up}^{\sigma}(\varphi)&=\sum_{\lambda=+,-}a_{\lambda}^{\sigma}e^{in_{\lambda}^{\sigma}\varphi},\\ \Psi_{low}^{\sigma}(\varphi')&=\sum_{\lambda=+,-}b_{\lambda}^{\sigma}e^{-in_{\lambda}^{\sigma}\varphi'}, \end{aligned}$$ where $n_{\lambda}^{\sigma}$ is given by Eq.(\[eq:n\]). Imposing the boundary conditions mentioned above, it is now possible to relate the transmission and reflection coefficients $r_{\sigma}$ and $t_{\sigma}$ to the input parameters $i_{\sigma}$. The continuity conditions for the wave function demand $\Psi_{II}^{\sigma}(0)=\Psi_{up}^{\sigma}(0)=\Psi_{low}^{\sigma}(0)$ and $\Psi_{I}^{\sigma}(0)=\Psi_{up}^{\sigma}(\varphi_{A})=\Psi_{low}^{\sigma}(\varphi'_{A})$, yielding the equations $$\begin{aligned} \sum_{\lambda=+,-}a_{\lambda}^{\sigma}&=\sum_{\lambda=+,-}b_{\lambda}^{\sigma}=t_{\sigma},\label{eq:cont1}\\ \sum_{\lambda=+,-}a_{\lambda}^{\sigma}e^{in_{\lambda}^{\sigma}\varphi_{A}}&=\sum_{\lambda=+,-}b_{\lambda}^{\sigma}e^{-in_{\lambda}^{\sigma}\varphi'_{A}}=r_{\sigma}+i_{\sigma}. \label{eq:cont2}\end{aligned}$$ Probability current density conservation requires $j_{up}^{\sigma}+j_{low}^{\sigma}+j_{I(II)}^{\sigma}=0$ at the junctions. The current densities follow evaluating the expressions derived in Sec.\[sec:2\] for the wave functions above. The (dimensionless) ring currents read $$\begin{aligned} j_{up}^{\sigma}(\varphi)=&\frac{1}{2}\Big(\big(\Psi_{up}^{\sigma}\chi^{\sigma}\big)^{\dagger}\big(\hat{v}\Psi_{up}^{\sigma}\chi^{\sigma}\big)\nonumber\\ &+\Psi_{up}^{\sigma}\chi^{\sigma}\big(\hat{v}\Psi_{up}^{\sigma}\chi^{\sigma}\big)^{\dagger}\Big)(\varphi),\\ j_{low}^{\sigma}(\varphi')=&\frac{1}{2}\Big(\big(\Psi_{low}^{\sigma}\chi_{-}^{\sigma}\big)^{\dagger}\big(\hat{v}'\Psi_{low}^{\sigma}\chi_{-}^{\sigma}\big)\nonumber\\ &+\Psi_{low}^{\sigma}\chi_{-}^{\sigma}\big(\hat{v}'\Psi_{low}^{\sigma}\chi_{-}^{\sigma}\big)^{\dagger}\Big)(\varphi'),\end{aligned}$$ where $\hat{v}(\varphi)=\hat{v}_{0}(\varphi)+\hat{v}_{SO}(\varphi)$, $\hat{v}'(\varphi)=\hat{v}_{0}(\varphi)-\hat{v}_{SO}(\varphi)$ and $\chi_{-}^{\sigma}(\varphi')=\chi^{\sigma}(-\varphi')$. The currents in the leads are given by $$\begin{aligned} j_{I}^{\sigma}(x)=&\frac{1}{2}\Big(\big(\Psi_{I}^{\sigma}\chi_{A}^{\sigma}\big)^{\dagger}\big(\hat{v}_{0}\Psi_{I}^{\sigma}\chi_{A}^{\sigma}\big)\nonumber\\ &+\Psi_{I}^{\sigma}\chi_{A}^{\sigma}\big(\hat{v}_{0}\Psi_{I}^{\sigma}\chi_{A}^{\sigma}\big)^{\dagger}\Big)(x),\\ j_{II}^{\sigma}(x')=&\frac{1}{2}\Big(\big(\Psi_{II}^{\sigma}\chi_{B}^{\sigma}\big)^{\dagger}\big(\hat{v}_{0}\Psi_{II}^{\sigma}\chi_{B}^{\sigma}\big)\nonumber\\ &+\Psi_{II}^{\sigma}\chi_{B}^{\sigma}\big(\hat{v}_{0}\Psi_{II}^{\sigma}\chi_{B}^{\sigma}\big)^{\dagger}\Big)(x'), \end{aligned}$$ where $\chi_{A(B)}^{\sigma}=\chi_{\sigma}(\varphi(A(B))$. Using the equality of the wave function at the junctions and noting that $\hat{v}_{SO}(\varphi)\chi^{\sigma}(\varphi)=-\hat{v}_{SO}(\varphi)\chi_{-}^{\sigma}(\varphi')$, the probability current density conservation condition simplifies to $$\begin{aligned} \hat{v}_{0}\Psi_{up}^{\sigma}\big|_{\varphi=0\left(\varphi_{A}\right)}&+\hat{v}_{0}\Psi_{low}^{\sigma}\big|_{\varphi'=0\left(\varphi'_{A}\right)} +\hat{v}_{0}\Psi_{I(II)}^{\sigma}\big|_{x(x')=0}=0.\nonumber\\\end{aligned}$$ From that follows an additional pair of equations for the coefficients: $$\begin{aligned} &\sum_{\lambda=+,-}a_{\lambda}^{\sigma}\frac{n_{\lambda}^{\sigma}}{k\r}-\sum_{\lambda=+,-}b_{\lambda}^{\sigma}\frac{n_{\lambda}^{\sigma}}{k\r}+t_{\sigma}=0,\\ &\sum_{\lambda=+,-}a_{\lambda}^{\sigma}e^{in_{\lambda}^{\sigma}\varphi_{A}}\frac{n_{\lambda}^{\sigma}}{k\r}-\sum_{\lambda=+,-}b_{\lambda}^{\sigma}e^{-in_{\lambda}^{\sigma}\varphi'_{A}}\frac{n_{\lambda}^{\sigma}}{k\r}+i_{\sigma}-r_{\sigma}=0. \end{aligned}$$ Together with Eqs. and , we now have enough equations to determine the coefficient set $\{r_{\sigma}, ~t_{\sigma},~a_{\lambda}^{\sigma},~b_{\lambda}^{\sigma}\}$, $\lambda=\pm$, for both spin polarizations $\sigma=\uparrow,\downarrow$ as a function of the input coefficients $i_{\sigma}$, the incident wave number $k$, ring radius $\r$ and SOC-constant $\beta$. For an incident current from the right, an analogous calculation is performed with $\{i_{\sigma},~r_{\sigma}\}$ (left lead) and $\{t_{\sigma},~0\}$ (right lead) replaced by $\{0,~t_{\sigma}'\}$ and $\{r_{\sigma}',~i_{\sigma}'\}$, respectively. This enables us to formulate the scattering matrix of the ring system: $\vec o=\underline{S}\vec i$, where $\vec o$ stands for outgoing and $\vec i$ for incoming wave coefficients. The relations can be written as $t_{\sigma}^{(')}=\sum_{\sigma'}T_{\sigma\sigma'}^{(')}i_{\sigma'}^{(')}$, $r_{\sigma}^{(')}=\sum_{\sigma'}R_{\sigma\sigma'}^{(')}i_{\sigma'}^{(')}$. A careful examination shows that no spin flip amplitudes for transmission or reflection in this spinor basis are present, and a possible modification of the spinor is only due to a difference between propagating channels. Thus, the scattering matrix reads $$\underline{S}=\left(\begin{array}{rrrr} R_{\uparrow}&0&T_{\uparrow}'&0\\ 0&R_{\downarrow}&0&T_{\downarrow}'\\ T_{\uparrow}&0&R_{\uparrow}'&0\\ 0&T_{\downarrow}&0&R_{\downarrow}'\\ \end{array}\right).\label{eq:smat}$$ The overall conductance then follows from the entries of the scattering matrix by means of the Landauer conductance formula shown in Eq., with the spin dependent transmission amplitude given by Eq.. The corresponding expression for the reflection amplitude is $$\begin{aligned} R_{\sigma}(\phi,\beta,\gamma)&=&\frac{\cos\phi\gamma+3 \cos\phi-4\cos\Phi_{AC}^{\sigma}}{\cos\phi\gamma-5\cos\phi+4\cos\Phi_{AC}^{\sigma}+4i\sin\phi}.\nonumber\\ \end{aligned}$$ The (time reversed) functions for incident wave in the right lead are related to those above by $T'_{\sigma}(\Phi_{AC}^{\sigma})=T_{\sigma}(-\Phi_{AC}^{\sigma})$ and $R'_{\sigma}=R_{\sigma}$. The transmission and reflection coefficients with respect to the standard $\sigma_{z}$-basis $\{|s\rangle\}$ are obtained by the corresponding spin rotation $\Lambda (\{|s\rangle\}\rightarrow\{|\sigma\rangle\})$ of the diagonal transmission and reflection blocks in the scattering matrix , e.g. for the transmission $$T_{ss'}=\langle s'|\Lambda^{-1}(0)\circ \big[T_{\sigma\sigma'}\big]\circ\Lambda(\varphi_{A})|s\rangle,$$ where $$\Lambda(\varphi)=\left(\begin{array}{cc} \cos\frac{\theta}{2}&e^{-i\varphi}\sin\frac{\theta}{2}\\ \sin\frac{\theta}{2}&-e^{-i\varphi}\cos\frac{\theta}{2} \end{array}\right),$$ and $$\big[T_{\sigma\sigma'}\big]=\left(\begin{array}{cc} T_{\uparrow}&0\\ 0&T_{\downarrow} \end{array}\right),$$ and analogously for the reflection coefficients. [^1]: The neglected term in produces a spin independent modification of the spectrum, which should not influence the interference effects. [^2]: Note that using $H$, we are still working with dimensionless quantities. [^3]: Generally, $ G=\frac{e^{2}}{h}\sum_{\sigma,\sigma'=\uparrow,\downarrow}|T_{\sigma\sigma'}|^{2}$, where $T_{\sigma\sigma'},~\sigma\neq\sigma'$ is the spin flip amplitude. The conductance does not depend on the choice of spinor basis, i.e. it is invariant under spin rotation, we can therefore make use of the ring-spinor basis where the spin flip amplitudes vanish, and define $T_{\sigma\sigma}\equiv T_{\sigma}$. [^4]: In the scattering matrix approach of \[\], these conditions correspond to a coupling parameter $\epsilon=4/9$, which is below the value of $\epsilon=1/2$ for perfect transmission.

--- abstract: | We introduce a new class of non-standard variable-length codes, called *adaptive codes*. This class of codes associates a variable-length codeword to the symbol being encoded depending on the previous symbols in the input data string. An efficient algorithm for constructing adaptive codes of order one is presented. Then, we introduce a natural generalization of adaptive codes, called *GA codes*. [**Keywords:**]{} adaptive mechanisms, compression rate, data compression, entropy, prefix codes, variable-length codes. title: '****' --- Introduction ============ The theory of variable-length codes [@bp1] originated in concrete problems of information transmission. Especially by its language theoretic branch, the field has produced a great number of results, most of them with multiple applications in engineering and computer science. Intuitively, a *variable-length code* is a set of strings such that any concatenation of these strings can be uniquely decoded. We introduce a new class of non-standard variable-length codes, called *adaptive codes*, which associate a variable-length codeword to the symbol being encoded depending on the previous symbols in the input data string. The paper is organized into six sections. After this introductory section, the definition of adaptive codes and several theoretical remarks are given in Section 2, as well as some characterization results for adaptive codes. The main results of this paper are presented in Section 3, where we focus on designing an algorithm for constructing adaptive codes of order one. In Section 4, we compute the entropy bounds for this algorithm. A natural generalization of adaptive codes is presented in Section 5. Finally, the last section contains a few concluding remarks. Before ending this introductory section, let us present some useful notation used throughout the paper [@rs1; @as1], and then review some basic concepts. We denote by $|S|$ the *cardinality* of a set $S$; if $x$ is a string of finite length, then $|x|$ denotes the length of $x$. The *empty string* is denoted by $\lambda$. For an alphabet $\Sigma$, we denote by $\Sigma^{*}$ the set $\bigcup_{n=0}^{\infty}\Sigma^{n}$, and by $\Sigma^{+}$ the set $\bigcup_{n=1}^{\infty}\Sigma^{n}$, where $\Sigma^{0}$ is defined by $\{\lambda\}$. Let us denote by $\Sigma^{\leq n}$ the set $\bigcup_{i=0}^{n}\Sigma^{i}$ and by $\Sigma^{\geq n}$ the set $\bigcup_{i=n}^{\infty}\Sigma^{i}$. Let $X$ be a finite and nonempty subset of $\Sigma^{+}$, and $w\in\Sigma^{+}$. A *decomposition of w* over $X$ is any sequence of words $u_{1}, u_{2}, \ldots, u_{h}$ with $u_{i}\in X$, $1\leq i\leq h$, such that $w=u_{1}u_{2}\ldots u_{h}$. A *code* over $\Sigma$ is any nonempty set $C\subseteq\Sigma^{+}$ such that each word $w\in\Sigma^{+}$ has at most one decomposition over $C$. A *prefix code* over $\Sigma$ is any code $C$ over $\Sigma$ such that no word in $C$ is proper prefix of another word in $C$. Adaptive Codes ============== In this section we introduce a new class of non-standard variable-length codes, called adaptive codes. These codes are based on adaptive mechanisms, that is, the variable-length codeword associated to the symbol being encoded depends on the previous symbols in the input data string. Let $\Sigma$ and $\Delta$ be two alphabets. A function ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$, with $n\geq{1}$, is called an if its unique homomorphic extension ${\overline{c}:\Sigma^{*}\rightarrow\Delta^{*}}$ given by: - $\overline{c}(\lambda)=\lambda$, - $\overline{c}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=$ $c(\sigma_{1},\lambda)$ $c(\sigma_{2},\sigma_{1})$ $\ldots$ $c(\sigma_{n-1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-2}})$ $c(\sigma_{n},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-1}})$ $c(\sigma_{n+1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n}})$ $c(\sigma_{n+2},\sigma_{2}\sigma_{3}\ldots\sigma_{n+1})$ $c(\sigma_{n+3},\sigma_{3}\sigma_{4}\ldots\sigma_{n+2})\ldots$ $c(\sigma_{m},\sigma_{m-n}\sigma_{m-n+1}\ldots\sigma_{m-1})$, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$, is injective. Let us take an example in order to better understand the adaptive mechanisms presented in the definition above. Let $\Sigma=\{{\texttt{\textup{a}}},{\texttt{\textup{b}}}\}$, $\Delta=\{0,1\}$ be alphabets, and ${c:\Sigma\times\Sigma^{\leq{2}}\rightarrow\Delta^{+}}$ a function given by the table below. One can easily verify that the function $\overline{c}$ is injective, and according to Definition 2.1, $c$ is an adaptive code of order two. Let $x={\texttt{\textup{abaa}}}\in\Sigma^{+}$. Using the definition above, we encode $x$ by $\overline{c}(x)=c({\texttt{\textup{a}}},\lambda)c({\texttt{\textup{b}}},{\texttt{\textup{a}}})c({\texttt{\textup{a}}},{\texttt{\textup{ab}}})c({\texttt{\textup{a}}},{\texttt{\textup{ba}}})=0101$. Let ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$ be an adaptive code of order $n$, $n\geq{1}$. We denote by $C_{c, \sigma_{1}\sigma_{2}\ldots\sigma_{h}}$ the set $\{c(\sigma,\sigma_{1}\sigma_{2}\ldots\sigma_{h}) \mid \sigma\in\Sigma\}$, for all $\sigma_{1}\sigma_{2}\ldots\sigma_{h}\in\Sigma^{\leq{n}}-\{\lambda\}$, and by $C_{c, \lambda}$ the set $\{c(\sigma,\lambda) \mid \sigma\in\Sigma\}$. We write $C_{\sigma_{1}\sigma_{2}\ldots\sigma_{h}}$ instead of $C_{c, \sigma_{1}\sigma_{2}\ldots\sigma_{h}}$, and $C_{\lambda}$ instead of $C_{c, \lambda}$ whenever there is no confusion. If $w\in\Sigma^{+}$ then we denote by $w(i)$ the $i$-th symbol of $w$. In the rest of this paper we denote by ${\it AC}(\Sigma,\Delta,n)$ the set $\{{c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}} \mid$ $c$ is an adaptive code of order $n\}$. Let $\Sigma$ and $\Delta$ be two alphabets, and let ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$ be a function. If $C_{{\sigma_{1}\sigma_{2}\ldots\sigma_{h}}}$ is a prefix code, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{h}}\in\Sigma^{\leq{n}}$, then $c\in{{\it AC}(\Sigma,\Delta,n)}$. **Proof** Let us assume that $C_{{\sigma_{1}\sigma_{2}\ldots\sigma_{h}}}$ is prefix code, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{h}}\in\Sigma^{\leq{n}}$, but $c\notin{{\it AC}(\Sigma,\Delta,n)}$. By Definition 2.1, the unique homomorphic extension of c, denoted by $\overline{c}$, is not injective. This implies that $\exists$ $u\sigma u', u\sigma'u''\in\Sigma^{+}$, with $\sigma,\sigma '\in\Sigma$ and $u,u',u''\in\Sigma^{*}$, such that $\sigma\neq\sigma'$ and $(*)$ $ $ $\overline{c}(u\sigma u')=\overline{c}(u\sigma'u'')$. We can rewrite $(*)$ by $(**)$ $ $ $\overline{c}(u)c(\sigma,P_{n}(u))\overline{c}(u')=$ $\overline{c}(u)c(\sigma',P_{n}(u))\overline{c}(u'')$, where $P_{n}(u)$ is given by $$P_{n}(u)= \left\{ \begin{array}{ll} \lambda & \textrm{if $u=\lambda$,} \\ u_{1}\ldots u_{q} & \textrm{if $u=u_{1}u_{2}\ldots u_{q}$ and $u_{1},u_{2},\ldots,u_{q}\in\Sigma$ and $q\leq{n}$,} \\ u_{q-n+1}\ldots u_{q} & \textrm{if $u=u_{1}u_{2}\ldots u_{q}$ and $u_{1},u_{2},\ldots,u_{q}\in\Sigma$ and $q>n$.} \end{array} \right.$$ By hypothesis, $C_{P_{n}(u)}$ is a prefix code and $c(\sigma,P_{n}(u)),c(\sigma',P_{n}(u))\in{C_{P_{n}(u)}}$. Therefore, the set $\{c(\sigma,P_{n}(u)),c(\sigma',P_{n}(u))\}$ is a prefix code. But the equality $(**)$ can hold if and only if $\{c(\sigma,P_{n}(u)),c(\sigma',P_{n}(u))\}$ is not a prefix set. Hence, our assumption leads to a contradiction. $\diamondsuit$ The converse of Theorem 2.1 does not hold. We can prove this by taking a counter-example. Let us consider $\Sigma=\{{\texttt{\textup{a}}},{\texttt{\textup{b}}}\}$ and $\Delta=\{0,1\}$ two alphabets, and ${c:\Sigma\times\Sigma^{\leq{2}}\rightarrow\Delta^{+}}$ a function given by the table below. One can verify that the unique homomorphic extension of $c$, denoted by $\overline{c}$, is injective. Therefore, we conclude that the function $c$ is an adaptive code of order two. Let $\Sigma$, $\Delta$, and ${\it Bool}=\{{\it True}, {\it False}\}$ be alphabets. We define the function ${\it Prefix}:{\it AC}(\Sigma,\Delta,n)\rightarrow {\it Bool}$ by: $${\it Prefix}(c)= \left\{ \begin{array}{ll} {\it True} & \textrm{if $C_{u}$ is a prefix code, for all $u\in\Sigma^{\leq{n}}$,} \\ {\it False} & \textrm{otherwise.} \end{array} \right.$$ The function *Prefix* can now be used to translate the hypothesis in Theorem 2.1: if ${c:\Sigma\times\Sigma^{\leq{n}}\rightarrow\Delta^{+}}$ is a function satisfying ${\it Prefix}(c)={\it True}$, then we conclude that $c\in{{\it AC}(\Sigma,\Delta,n)}$. Let $c\in{{\it AC}(\Sigma,\Delta,n)}$ be an adaptive code satisfying ${\it Prefix}(c)={\it True}$. Then, the algorithm **Decoder** described below requires a linear time. (340,240) (30,211)[**Decoder**$(c,u)$]{} (30,199)[input:$c\in{{\it AC}(\Sigma,\Delta,n)}$ *such that* ${\it Prefix}(c)={\it True}$ *and* $u\in\Delta^{+}$;]{} (30,187)[output:$w\in\Sigma^{+}$ *such that* $\overline{c}(w)=u$;]{} (30,175)[begin]{} (15,163)[1. $w:=\lambda$; $i:=1$; $Last:=\lambda$; $length:=|u|;$]{} (15,151)[2.while $i\leq{length}$ do]{} (15,139)[begin]{} (15,127)[3. *Let* $\sigma\in\Sigma$ *be the unique symbol of $\Sigma$ with the property*]{} (15,115)[ *that* $c(\sigma,Last)$ *is prefix of* $u(i)\cdot u(i+1)\cdot\ldots\cdot u(length)$;]{} (15,103)[4.$w:=w\cdot\sigma$;]{} (15,91)[5.$i:=i+|c(\sigma,Last)|$;]{} (15,79)[6.if $|Last|<n$]{} (15,67)[7.then $Last:=Last\cdot\sigma$;]{} (15,55)[8.else $ $ $Last:=Last(|Last|-n+2)\cdot\ldots\cdot Last(|Last|)\cdot\sigma$;]{} (15,43)[end]{} (15,31)[9.return $w$;]{} (30,19)[end]{} (8,227)[(1,0)[327]{}]{} (8,9)[(1,0)[327]{}]{} (8,227)[(0,-1)[218]{}]{} (335,227)[(0,-1)[218]{}]{} In the third step of the algorithm given above, the symbol denoted by $\sigma$ is unique with that property due to the input restrictions. One can easily verify that the while loop in algorithm **** is iterated $$|u|-\sum_{i=1}^{h}(|c(w_{i},P_{n}(w_{1}w_{2}\ldots w_{i-1}))|-1)$$ times, where $w=w_{1}w_{2}\ldots w_{h}$, and $P_{n}$ is the function given in Theorem 2.1. In practice, we can use only adaptive codes satisfying the equality ${\it Prefix}(c)={\it True}$, since designing a decoding algorithm for the other case requires additional information and more complicated techniques. Data Compression using Adaptive Codes ===================================== The construction of adaptive codes requires different approaches, depending on the structure of the input data strings. In this section, we focus on data compression using adaptive codes of order one. Let $\Sigma$ be an alphabet and $w=w_{1}w_{2}\ldots w_{h}\in\Sigma^{\geq{2}}$, with $w_{i}\in\Sigma$, for all $i\in\{1,2,\ldots,h\}$. A subword $uu$ of $w$, with $u\in\Sigma$, is called a of w. Let $\Sigma$ be an alphabet and $w=w_{1}w_{2}\ldots w_{h}\in\Sigma^{\geq{2}}$. It is useful to consider the following notations: 1. ${\it Pairs}(w)=\{i \mid 1\le i\le |w|-1, w_i=w_{i+1}\}$, 2. ${\it NRpairs}(w)=|{\it Pairs}(w)|$, 3. ${\it Prate}(w)=\frac{{\it NRpairs}(w)}{|w|}$. The main goal of this section is to design an algorithm for constructing adaptive codes of order one, under the assumption that the input data strings have a large number of pairs. Let $\Sigma=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{h}\}$ and $\Delta=\{0,1\}$ be alphabets, $c\in{{\it AC}(\Sigma,\Delta,1)}$ an adaptive code of order one, and $w\in\Sigma^{+}$. We denote by $A_{c}$ the matrix given by: $ A_{c} = \left(\begin{array}{ccccc} c(\sigma_{1},\sigma_{1}) & c(\sigma_{1},\sigma_{2}) & \ldots & c(\sigma_{1},\sigma_{h}) & c(\sigma_{1},\lambda) \\ c(\sigma_{2},\sigma_{1}) & c(\sigma_{2},\sigma_{2}) & \ldots & c(\sigma_{2},\sigma_{h}) & c(\sigma_{2},\lambda) \\ & & \ldots & & \\ c(\sigma_{h},\sigma_{1}) & c(\sigma_{h},\sigma_{2}) & \ldots & c(\sigma_{h},\sigma_{h}) & c(\sigma_{h},\lambda) \end{array} \right) $. Let us denote by **Huffman**$({\it EF}(w),n)$ the well-known Huffman’s algorithm [@ds1], where $n\geq{1}$, and ${\it EF}(w)$ is the matrix given below. $ {\it EF}(w) = \left(\begin{array}{ccccc} \sigma_{1} & \sigma_{2} & \ldots & \sigma_{n} \\ f(\sigma_{1},w) & f(\sigma_{2},w) & \ldots & f(\sigma_{n},w) \end{array} \right) $. We assume that the first row of the matrix ${\it EF}(w)$ contains the symbols which are being encoded, while the second row contains their frequencies, that is, $f(\sigma_{i})$ is the frequency of the symbol $\sigma_{i}$ in $w$. Also, we assume that **Huffman**$({\it EF}(w),n)$ is the matrix given by $ \textbf{Huffman}({\it EF}(w),n) = \left(\begin{array}{ccccc} H(\sigma_{1},w) & H(\sigma_{2},w) & \ldots & H(\sigma_{n},w) \end{array} \right) $ where $H(\sigma_{i},w)$ is the codeword associated to the symbol $\sigma_{i}$ by Huffman’s algorithm. The algorithm **Builder** described further on takes linear time, and constructs an adaptive code of order one satisfying ${\it Prefix}(c)={\it True}$. Let $c:\Sigma\times\Sigma^{\leq{1}}\rightarrow\{0,1\}^{+}$ be a function given by the matrix **Builder**$(c)$. Then, $c\in{{\it AC}(\Sigma,\{0,1\},1)}$ and ${\it Prefix}(c)={\it True}$. **Proof** Applying the algorithm **Builder** to the function $c$, one can easily verify that ${\it Prefix}(c)={\it True}$. Therefore, according to **Theorem 2.1**, $c$ is an adaptive code of order one, that is, $c\in{{\it AC}(\Sigma,\{0,1\},1)}$. $\diamondsuit$ (340,282) (15,268)[**Builder**(c)]{} (15,256)[input:$c:\Sigma\times\Sigma^{\leq{1}}\rightarrow\{0,1\}^{+}$, $\Sigma=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{h}\}$;]{} (15,244)[output:$A_{c}$ *such that* $c\in{{\it AC}(\Sigma,\{0,1\},1)}$ *and* ${\it Prefix}(c)={\it True}$;]{} (15,232)[begin]{} (0,220)[$1.$for $i:=1$ to $h$ do $A_{c}(i,i):=0$;]{} (0,198)[2. $E:= \left( \begin{array}{cccc} \sigma_{2} & \sigma_{3} & \ldots & \sigma_{h} \\ 0 & 0 & \ldots & 0 \end{array} \right) $;]{} (0,176)[3.$X:=$**Huffman**$(E,h-1)$;]{} (0,164)[4.for $i:=2$ to $h$ do]{} (0,152)[ begin]{} (0,140)[5. $A_{c}(1,i):=1\cdot X(1,i-1)$;]{} (0,128)[6.$X(1,i-1):=1\cdot X(1,i-1)$;]{} (0,116)[7. $A_{c}(i,1):=X(1,i-1)$;]{} (0,104)[ end]{} (0,92)[8.for $j:=2$ to $h$ do]{} (0,80)[ begin]{} (0,68)[9.for $i:=2$ to $j-1$ do $A_{c}(i,j):=X(1,i-1)$;]{} (-5,56)[10.for $i:=j+1$ to $h$ do $A_{c}(i,j):=X(1,i-1)$;]{} (0,44)[ end]{} (-5,32)[11.for $i:=1$ to $h$ do $A_{c}(i,h+1):=A_{c}(1,i)$;]{} (-5,20)[12.return $A_{c}$;]{} (15,8)[end]{} (-8,282)[(1,0)[330]{}]{} (-8,2)[(1,0)[330]{}]{} (-8,282)[(0,-1)[280]{}]{} (322,282)[(0,-1)[280]{}]{} Let $c:\{{\texttt{\textup{a}}},{\texttt{\textup{b}}},{\texttt{\textup{c}}}\}\times\{{\texttt{\textup{a}}},{\texttt{\textup{b}}},{\texttt{\textup{c}}}\}^{\leq{1}}\rightarrow\{0,1\}^{+}$ be a function. One can verify that $A_{c}$ is the matrix given below. $A_{c}=\left( \begin{array}{cccc} 0 & 10 & 11 & 0 \\ 11 & 0 & 10 & 10 \\ 10 & 11 & 0 & 11 \end{array} \right) $. Let $w={\texttt{\textup{abbbcabccaabccabbcba}}}$ be an input data string. It is easy to verify that ${\it Pairs}(w)=\{2,3,8,10,13,16\}$, ${\it NRpairs}(w)=6$, and ${\it Prate}(w)=0.3$. Encoding the string $w$ by $c$ requires the computation of $\overline{c}(w)$. Using Definition 2.1, we get that $|\overline{c}(w)|=33$. Let us apply Huffman’s algorithm to the data string $w$ in order to make a comparison between the results. If we denote by ${\it Huffman}(w)$ the codeword associated to $w$ by Huffman’s algorithm, we get that $|Huffman(w)|=32$. An even better result can be obtained when the input data string has a larger number of pairs, as shown in the following example. Let $c:\{{\texttt{\textup{a}}},{\texttt{\textup{b}}},{\texttt{\textup{c}}}\}\times\{{\texttt{\textup{a}}},{\texttt{\textup{b}}},{\texttt{\textup{c}}}\}^{\leq{1}}\rightarrow\{0,1\}^{+}$ be an adaptive code of order one given as in the previous example, and $w={\texttt{\textup{abbbccbccaabccaaacba}}}$ an input data string. One can verify that ${\it Pairs}(w)=\{2,3,5,8,10,13,15,16\}$, ${\it NRpairs}(w)=8$, ${\it Prate}(w)=0.4$, and $|\overline{c}(w)|=31$. Encoding the string $w$ by Huffman’s algorithm, we get that $|{\it Huffman}(w)|=34$. The results obtained in the previous examples are summarized in the table below, which shows that we get substantial improvements for input data strings having a larger number of pairs. Builder: Entropy Bounds ======================= In this section, we focus on computing the entropy bounds for the algorithm described in section 3. Given that our algorithm is based on Huffman’s algorithm, let us first recall the entropy bounds for Huffman codes. Let $\Sigma$ be an alphabet, $x$ a data string of length $n$ over $\Sigma$ and $k$ the length of the encoder output, when the input is $x$. The , denoted by $R(x)$, is defined by $$R(x)=\frac{k}{n}.$$ Let $R(x)$ be the compression rate in codebits per datasample, computed after encoding the data string $x$ by the Huffman algorithm. One can obtain upper and lower bounds on $R(x)$ before encoding the data string $x$ by computing the entropy denoted by $H(x)$. Let $x$ be a data string of length $n$, $(F_{1},F_{2},\ldots,F_{h})$ the vector of frequencies of the symbols in $x$ and $k$ the length of the encoder output. The entropy $H(x)$ of $x$ is defined by $$H(x)=\frac{1}{n}\sum_{i=1}^{h}F_{i}\log_{2}(\frac{n}{F_{i}}).$$ Let $L_{i}$ be the length of the codeword associated to the symbol with the frequency $F_{i}$ by the Huffman algorithm, $1\leq{i}\leq{h}$. Then, the compression rate $R(x)$ can be re-written by $$R(x)=\frac{1}{n}\sum_{i=1}^{h}F_{i}L_{i}.$$ If we relate the entropy $H(x)$ to the compression rate $R(x)$, we obtain the following inequalities: $$H(x)\leq{R(x)}\leq{H(x)+1}.$$ Let $\Sigma=\{\sigma_{1},\sigma_{2},\ldots,\sigma_{t}\}$ be an alphabet and $c:\Sigma\times\Sigma^{\leq{1}}\rightarrow\{0,1\}^{+}$ an adaptive code of order one constructed as shown in section 3. Also, consider $w=w_{1}w_{2}\ldots{w_{s}}\in{\Sigma^{+}}$, $w_{i}\in\Sigma$, $1\leq{i}\leq{s}$, and $p$ the number of symbols occurring in $w$. We denote by $R_{A}(w)$ the compression rate obtained when encoding the string $w$ by $\overline{c}$ and by $H_{A}(w)$ the entropy of $w$. It is useful to consider the following notations: 1. ${\it EH}(w)=\{i$ $\mid$ $2\leq{i}\leq{s}$ and $w_{i}\neq{w_{i-1}}\}$, 2. ${\it LNotHuffman}(w)=|c(w_{1},\lambda)|+\sum_{i\in{{\it Pairs}(w)}}|c(w_{i+1},w_{i})|$, 3. ${\it LHuffman}(w)$ is the entropy of $w_{j_{1}}w_{j_{2}}\ldots{w_{j_{r}}}$, $j_{k}\in{{\it EH}(w)}$, $1\leq{k}\leq{r}$, 4. $H_{A}(w)={\it LNotHuffman}(w)+{\it LHuffman}(w)$. It is easy to verify that ${\it LNotHuffman}(w)={\it NRPairs}(w)+|c(w_{1},\lambda)|$. Using the notation above, we get that $${\it LHuffman}(w)=\sum_{i\in{{\it EH}(w)}}\{ \frac{1}{N(w_{i})}\sum_{q\in{{\it Prev}(w_{i})}}[F_{q}(w_{i})(1+\log_{2}\frac{N(w_{i})}{F_{q}(w_{i})})]\},$$ where - $N(w_{i})=|\{j$ $\mid$ $j\in{{\it EH}(w)}$ and $w_{j}={w_{i}}\}|$, - ${\it Prev}(w_{i})=\{j$ $\mid$ $j+1\in{{\it EH}(w)}$ and $w_{j+1}=w_{i}\}$, - $F_{q}(w_{i})=|\{j$ $\mid$ $j\in{{\it EH}(w)}$ and $w_{j}=w_{i}$ and $w_{j-1}=w_{q}\}|$, $q\in{{\it Prev}(w_{i})}$. Finally, we can relate the entropy $H_{A}(w)$ to the compression rate $R_{A}(w)$ by the following inequalities: $$H_{A}(w)\leq{R_{A}(w)}\leq{H_{A}(w)}+1,$$ where $R_{A}(w)$ is given by $$R_{A}(w)=\frac{|c(w_{1},\lambda)c(w_{2},w_{1})\ldots c(w_{s},w_{s-1})|}{s}.$$ GA Codes ======== In this section, we introduce a natural generalization of adaptive codes (of any order), called *GA codes* (**G**eneralized **A**daptive codes). Theorem 5.1 proves that adaptive codes are particular cases of GA codes. Let $\Sigma$ and $\Delta$ be two alphabets and $F:N^{*}\times\Sigma^{+}\rightarrow\Sigma^{*}$ a function, where $N$ is the set of natural numbers, and $N^{*}=N-\{0\}$. A function $c_{F}:\Sigma\times\Sigma^{*}\rightarrow\Delta^{+}$ is called a if its unique homomorphic extension $\overline{c_{F}}:\Sigma^{*}\rightarrow\Delta^{*}$ given by - $\overline{c_{F}}(\lambda)=\lambda$, - $\overline{c_{F}}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=c_{F}(\sigma_{1},F(1,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))\ldots c_{F}(\sigma_{m},F(m,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))$, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$, is injective. The function $F$ in Definition 5.1 is called the of the GA code $c_{F}$. Clearly, a GA code $c_{F}$ can be constructed if its adaptive function $F$ is already constructed. Let $\Sigma$ and $\Delta$ be two alphabets. We denote by $GAC(\Sigma,\Delta)$ the set $\{c_{F}:\Sigma\times\Sigma^{*}\rightarrow\Delta^{+}$ $\mid$ $c_{F}$ is a GA code$\}$. Let $\Sigma$ and $\Delta$ be alphabets. Then, $AC(\Sigma,\Delta,n)\subset{GAC(\Sigma,\Delta)}$, for all $n\geq{1}$. **Proof** Let $c_{F}\in{AC(\Sigma,\Delta,n)}$ be an adaptive code of order $n$, $n\geq{1}$, and $F:N^{*}\times\Sigma^{+}\rightarrow\Sigma^{*}$ a function given by: $$F(i,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}})= \left\{ \begin{array}{ll} \lambda & \textrm{if $i=1$ or $i>m$,} \\ {\sigma_{1}\sigma_{2}\ldots\sigma_{i-1}} & \textrm{if $2\leq{i}\leq{m}$ and $2\leq{i}\leq{n+1}$,} \\ \sigma_{i-n}\sigma_{i-n+1}\ldots\sigma_{i-1} & \textrm{if $2\leq{i}\leq{m}$ and $i>n+1$,} \end{array} \right.$$ for all $i\geq{1}$ and ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in{\Sigma^{+}}$. One can verify that $|F(i,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}})|\leq{n}$, for all $i\geq{1}$ and ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in{\Sigma^{+}}$. According to Definition 2.1, the function $\overline{c_{F}}$ is given by: - $\overline{c_{F}}(\lambda)=\lambda$, - $\overline{c_{F}}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=$ $c_{F}(\sigma_{1},\lambda)$ $c_{F}(\sigma_{2},\sigma_{1})$ $\ldots$ $c_{F}(\sigma_{n-1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-2}})$ $c_{F}(\sigma_{n},{\sigma_{1}\sigma_{2}\ldots\sigma_{n-1}})$ $c_{F}(\sigma_{n+1},{\sigma_{1}\sigma_{2}\ldots\sigma_{n}})$ $c_{F}(\sigma_{n+2},\sigma_{2}\sigma_{3}\ldots\sigma_{n+1})$ $c_{F}(\sigma_{n+3},\sigma_{3}\sigma_{4}\ldots\sigma_{n+2})\ldots$ $c_{F}(\sigma_{m},\sigma_{m-n}\sigma_{m-n+1}\ldots\sigma_{m-1})$, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$. It is easy to remark that - $\overline{c_{F}}({\sigma_{1}\sigma_{2}\ldots\sigma_{m}})=c_{F}(\sigma_{1},F(1,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))\ldots c_{F}(\sigma_{m},F(m,{\sigma_{1}\sigma_{2}\ldots\sigma_{m}}))$, for all ${\sigma_{1}\sigma_{2}\ldots\sigma_{m}}\in\Sigma^{+}$, which proves the theorem. $\diamondsuit$ Conclusions and Future Work =========================== We introduced a new class of non-standard variable-length codes, called adaptive codes, which associate a variable-length codeword to the symbol being encoded depending on the previous symbols in the input data string. The main results of this paper are presented in Section 3, where we have shown that if an input data string $x$ has a significant number of pairs, then a good compression rate is achieved when encoding $x$ by adaptive codes of order one. In a further paper devoted to adaptive codes, we intend to extend the algorithm **Builder** to adaptive codes of any order.

--- abstract: 'We study the convexity of the entropy functional along particular interpolating curves defined on the space of finitely supported probability measures on a graph.' author: - 'Erwan Hillion [^1]' title: 'Entropy along $W_{1,+}$-geodesics on graphs' --- Introduction ============ The Wasserstein distance $W_p(\mu_0,\mu_1)$ between two finitely supported probability measures on a metric space $(X,d)$ with its Borel $\sigma$-algebra is defined for $p \geq 1$ by $$\label{eq:WassersteinDef} W_p(\mu_0,\mu_1)^p := \inf_{\pi \in \Pi(\mu_0,\mu_1)} \int_{X \times X} d(x_0,x_1)^p d\pi(x_0,x_1),$$ where $\Pi(\mu_0,\mu_1)$ is the (non-empty) set of couplings between $\mu_0$ and $\mu_1$, i.e. the set of probability measures on $X \times X$ having $\mu_0$ and $\mu_1$ as marginals. The optimization problem defined by equation  is called the Monge-Kantorovitch problem and any minimizer for  is called optimal coupling between $\mu_0$ and $\mu_1$. For a comprehensive study of optimal transportation theory, the reader is referred to the textbooks [@VillaniBook1] and [@VillaniBook2] by Villani. Under mild conditions, it is possible to show that the set $\Pi_p(\mu_0,\mu_1)$ of optimal couplings between $\mu_0$ and $\mu_1$ is non-empty. Furthermore, under the additional assumptions that $p>1$, $(X,d)$ is the Euclidean space $(\mathbb{R}^d,|.|)$ and $\mu_0$ is absolutely continuous with respect to the Lebesgue measure, one can prove the existence of a measurable map $T : \mathbb{R}^d \rightarrow \mathbb{R}^d$ such that the coupling $\pi := (Id \times T)_*\mu_0$ is a minimizer for . In particular, $\mu_1$ is the pushforward of $\mu_0$ by the application $T$: $\mu_1 := T_*\mu_0$ and equation  can be rewritten $$W_p(\mu_0,\mu_1)^p = \int_{\mathbb{R}^d} |x-T(x)|^p d\mu_0(x).$$ It is possible to go further by considering, for $0 \leq t \leq 1$, the measure $\mu_t := (T_t)_*\mu_0$, where the application $T_t : \mathbb{R}^d \rightarrow \mathbb{R}^d$ is defined as the barycenter $T_t(x) := (1-t)x + t T(x)$. One can then show that the family $(\mu_t)_{t \in [0,1]}$ is a geodesic for the Wasserstein distance $W_p$, in the sense that $$W_p(\mu_0,\mu_1) = \sup_{0 = t_0 \leq t_1 \cdots \leq t_{n} = 1} \sum_{i=0}^{n-1} W_p(\mu_{t_i},\mu_{t_{i+1}}).$$ Moreover, a fundamental property of optimal couplings asserts that $T_t$ is injective, which allows us to define unambiguously a velocity field $(v_t)_{t \in [0,1]}$ by $$v_t(T_t(x)) := T(x)-x.$$ The terminology ’velocity field’ comes from the fact that, if we write $d\mu_t(x) = f_t(x) dx$, then the density $f_t(x)$ satisfy, at least formally, the transport equation $$\label{eq:velocitytransport} \frac{\partial}{\partial t} f_t(x) + \operatorname{div}(v_t(x) f_t(x)) = 0.$$ Moreover, the velocity field $v_t(x)$ satisfies the Hamilton-Jacobi-type equation $$\label{eq:velocityoptimalHJ} \frac{\partial}{\partial t} v_t(x) + \frac{1}{2} \operatorname{grad}|v_t(x)|^2 = 0,$$ which can be simplified into $$\label{eq:velocityoptimal} \frac{\partial}{\partial t} v_t(x) = - \operatorname{div}(v_t(x)) v_t(x).$$ In [@BenamouBrenier], Benamou and Brenier proved that both equations  and  can be used to give a characterization of $W_p$-geodesics, more precisely we have: \[th:BenamouBrenier\] Given two finitely supported probability measures $d\mu_0(x) := f_0(x) dx$ and $d\mu_1(x) := f_1(x) dx$, we have $$\label{eq:MKBenamou} W_p(\mu_0,\mu_1)^p = \inf \int_0^1 \int_{\mathbb{R}^d} |v_t(x)|^p d\mu_t(x),$$ where the infimum is taken over the set of curves $(\mu_t)_{t \in [0,1]} = (f_t(x) dx)_{t \in [0,1]}$ joining the prescribed measures $\mu_0$ and $\mu_1$, and where $(v_t(x))_{t \in [0,1]}$ is a velocity field such that equation  holds. Moreover, the formal optimality condition for the optimization problem  is given by equation . Theorem \[th:BenamouBrenier\] is also true for families of probability measures defined on a Riemannian manifold, having smooth enough densities with respect to the Riemannian volume measure. However, in this framework, equations and are no longer equivalent. The optimality condition  is the starting point of the article [@HillionGeodesic] by the author. The main idea is the following: given two distinct probability measures $f_0,f_1$ on a graph $G$, there is no interpolating curve $(f_t)_{t \in [0,1]}$ with a finite length for the Wasserstein $W_p$, for any $p>1$. However, in generic cases there are infinitely many geodesics $(f_t)_{t \in [0,1]}$ for the $W_1$ distance. The aim of [@HillionGeodesic] is to choose among this set a particular $W_1$-geodesic satisfying a discrete version of equation . These interpolating curves are called $W_{1,+}$-geodesics on $G$; we recall their basic properties in Section 2. The purpose of this article is to study the behaviour of the entropy functional along a $W_{1,+}$-geodesic $(f_t(x))_{t \in [0,1], x \in G}$ on a graph $G$. More precisely, we will study the convexity of the function $t \mapsto H(t)$ defined by $$H(t) := \sum_{x \in G} f_t(x) \log(f_t(x)),$$ where by convention $0 \log 0 = 0$. The methods used to prove such convexity properties are adapted from the previous article [@HillionContraction] by the author, and use the first-order-calculus formalism introduced in [@HillionGeodesic]. The motivation behind this research work comes from Sturm-Lott-Villani theory, developed in the articles [@SturmRicci01], [@SturmRicci02] and  [@LottVillani]. The main idea of this theory is the following: it is possible to obtain some information about the geometry of a measured length space $(X,d,\nu)$ by studying the behaviour of entropy functionals along $W_2$-geodesics on the space of probability measures over $(X,d)$. A major result asserts that a compact Riemannian manifold $(M,g)$ satisfies the Ricci curvature bound $\operatorname{Ric}\geq K g$ if and only if each pair of absolutely continuous probability measures $\mu_0, \mu_1$ can be joined by a Wasserstein $W_2$-geodesic $(\mu_t)_{t \in [0,1]}$ such that $$\label{eq:LVSconvexity} H(\mu_t) \leq (1-t) H(\mu_0) + t H(\mu_1) - K \frac{t(1-t)}{2} W_2(\mu_0,\mu_1)^2,$$ where the relative entropy $H(\mu)$ is defined by $H(\mu): = \int_{M} \rho \log(\rho) d\operatorname{vol}$ if $d\mu = \rho . d\operatorname{vol}$ and by $H(\mu)=\infty$ if $\mu$ is not absolutely continuous with respect to the Riemannian volume measure. It is then possible to define the curvature condition ’$\operatorname{Ric}\geq K$’ on a measured length space $(X,d,\nu)$ if Equation  is satisfied for any $W_2$-Wasserstein geodesic on $\mathcal{P}_2(X)$. Several geometric theorems and functional inequalities holding on Riemannian manifolds satisfying a Ricci curvature bound are still valid in the framework of measured length spaces with a curvature condition ’$\operatorname{Ric}\geq K$’. The generalization of Sturm-Lott-Villani theory to discrete setting has been the subject of many research works, each leading to its own definition of Ricci curvature bounds on graphs, among which we can cite papers by Ollivier [@OllivierRicci] and Erbar-Maas [@ErbarMaas]. The latter is based on the study of a discrete version of the minimization problem  for $p=2$, whereas our approach is based on a discrete version of equation  characterizing the solutions of . Another important work in discrete Sturm-Lott-Villani theory is [@GRST] which, like this present work, is based on the study of the behaviour of the entropy functional along mixtures of binomial measures. The results proven in our paper show that the convexity properties of the entropy along $W_{1,+}$-geodesics are linked with some intuitive notion of curvature bounds on graphs. However, it seems that our study of the convexity of the entropy does not lead to a definition of Ricci curvature bounds strong enough to imply important functional inequalities, such as the modified logarithmic Sobolev inequality introduced in [@BobkovLedoux]. Our article is outlined as follows: in Section \[sec:Rappel\], we recall the definition and basic properties of the $W_1$-orientation and $W_{1,+}$-geodesics, which are developed in the previous article [@HillionGeodesic]. We also introduce the notion of canonical $W_{1,+}$-geodesic, see Theorem \[th:CanonicalGeodesic\], which will be used in Section \[sec:ProductSpaces\]. In Section \[sec:GeneralFormula\], we begin the study of the entropy function $H(t)$ along a $W_{1,+}$-geodesic on a graph; we use the Benamou-Brenier equation , which is at the heart of definition of $W_{1,+}$-geodesics, to obtain bounds on the second derivative $H''(t)$. The calculations done in this section are inspired by those done in the previous article [@HillionContraction] by the author; the results obtained are also linked with the more general theory of entropic interpolations, developed by Léonard in a recent series of articles, including [@LeoConvex], [@LeoLazy] and [@LeonardSurvey]. In Section \[sec:ProductSpaces\], we refine the calculations done in Section \[sec:GeneralFormula\] to prove a tensorization property. This property allows us to give bounds on the second derivative $H''(t)$ when the underlying graph is a product graph. Interesting examples are given by $\mathbb{Z}^n$, the cube $\{0,1\}^n$, or more generally by the Cayley graph of a finitely generated abelian group. In the Appendix, we present two additional results on families of probability measures on $\mathbb{Z}$. We first prove that, along a $W_{1,+}$-geodesic on $\mathbb{Z}$, other types of functionals are convex, belonging to the family of Renyi entropies functionals. The second part of the Appendix is devoted to another type of interpolation of probability measures on $\mathbb{Z}$, defined as a mixture of binomial distributions with respect to a $W_2$-optimal coupling. $W_{1,+}$-geodesics on graphs {#sec:Rappel} ============================= In this section, we first recall the main definitions and properties of [@HillionGeodesic]. The reader is referred to this paper for detailed proofs and additional explanations. We then introduce the new notion of canonical $W_{1,+}$-geodesic, which will be used in the study of product spaces in Section \[sec:ProductSpaces\]. Definition and construction --------------------------- Let $G$ be a locally finite, connected graph. We denote by $d$ the usual graph distance on $G$ and by $x \sim y$ the adjacency relation on $G$, meaning that $(x,y)$ is an edge of $G$. A curve of length $n$ on $G$ is an application $\gamma : \{0,\ldots n\} \rightarrow G$ satisfying $\gamma(i) \sim \gamma(i+1)$. A geodesic between two vertices $x$ and $y$ is a curve of minimal length joining $x$ to $y$. The set of geodesics between $x$ and $y$ is denoted by $\Gamma_{x,y}$ and its cardinality by $|\Gamma_{x,y}|$. The set of all geodesic curves of $G$ is denoted by $\Gamma(G)$. Let $f_0,f_1$ be two finitely supported probability distributions on $G$. We denote by $\Pi_1(f_0,f_1)$ the set of $W_1$-optimal couplings between $f_0$ and $f_1$, i.e. the set of couplings between $f_0$ and $f_1$ which minimize the functional $$I_1(\pi) := \sum_{x,y \in G} d(x,y) \pi(x,y).$$ Using properties of supports of optimal couplings, one can prove that the following definition in unambiguous: Let $f_0,f_1$ be two finitely supported probability measures on $G$. - The $W_1$-orientation on $G$ with respect to $f_0,f_1$ is constructed in the following way: a couple $(x,y)$ of adjacent vertices is oriented by $x \rightarrow y$ if there exists an optimal coupling $\pi \in \Pi_1(f_0,f_1)$ and a geodesic $\gamma \in \Gamma(G)$ of length $n$ such that $(\gamma(0),\gamma(n)) \in \operatorname{Supp}(\pi)$ and such that there exists $i \in \{0,\ldots n-1\}$ with $\gamma(i)=x$ and $\gamma(i+1)=y$. - Let $x_1 \in G$. The set $\mathcal{E}(x_1)$, resp. $\mathcal{F}(x_1)$, is the (possibly empty) set of vertices $x_0 \in G$, resp. $x_2 \in G$, such that $x_0 \rightarrow x_1$, resp. $x_1 \rightarrow x_2$. - An oriented path on $G$ is a mapping $\gamma : \{0,\ldots n\}$ with $\gamma(i) \rightarrow \gamma(i+1)$. - The $W_1$-orientation w.r.t. $f_0,f_1$ induces a partial order on the vertices of $G$: we denote $x \leq y$ if there exists an oriented path $x = \gamma(0) \rightarrow \cdots \rightarrow \gamma(n)=y$. One important property of this orientation is the fact that every oriented path is a geodesic: \[prop:OrientedGeodesic\] If we have $\gamma(0) \rightarrow \cdots \rightarrow \gamma(n)$ then $d(\gamma(0),\gamma(n)) = n$. A particular subset of geodesics on the oriented $G$ is given by extremal geodesics: Let $\gamma : \gamma(0) \rightarrow \cdots \rightarrow \gamma(n)$ be a geodesic on the oriented $G$. We say that $\gamma$ is an extremal geodesic, and we write $\gamma \in \operatorname{E\Gamma}$ if it cannot be extended in a longer geodesic, i.e. if the sets $\mathcal{E}(\gamma(0))$ and $\mathcal{F}(\gamma(n))$ are empty. The introduction of an orientation makes possible the introduction of a first-order calculus on $G$. We first define: The oriented edge graph $(E(G),\rightarrow)$ associated to $(G,\rightarrow)$ is defined as follows: its vertices are denoted by $(x_0x_1)$, where $x_0 \rightarrow x_1 \in G$ and its oriented edges join each couple $(x_0x_1) \rightarrow (y_0y_1)$ such that $x_1=y_0$. The oriented graph of oriented triples $(T(G),\rightarrow)$ is the graph $(E(E(G)),\rightarrow)$: its vertices are the triples $(x_0x_1x_2)$ with $x_0 \rightarrow x_1 \rightarrow x_2$ and its edges are defined between each couple $(x_0x_1x_2) \rightarrow (x_1x_2x_3)$. When the choice of the orientation on $G$ is unambiguous, we will often write $E(G), T(G)$ instead of $(E(G),\rightarrow), (T(G),\rightarrow)$. The divergence of a function $g : E(G) \rightarrow \mathbb{R}$ defined on the oriented edges of $G$ is the function $\nabla \cdot g : G \rightarrow \mathbb{R}$ defined by: $$\nabla \cdot g (x_1)= \sum_{x_2 \in \mathcal{F}(x_1)} g(x_1x_2) - \sum_{x_0 \in \mathcal{E}(x_1)} g(x_0x_1).$$ We define similarly the divergence $\nabla \cdot h : E(G) \rightarrow \mathbb{R}$ of a function $h : T(G) \rightarrow \mathbb{R}$ defined on the oriented triples of $G$. We denote $\nabla_2 \cdot h := \nabla \cdot (\nabla \cdot h)$. This first-order differential operator on the oriented graph allows us to introduce a discrete version of the formal optimality condition , on which is based the definition of $W_{1,+}$-geodesics: Let $G$ be a graph, $W_1$-oriented with respect to a couple of probability measures $f_0,f_1$. A family $(f_t)=(f_t)_{t \in [0,1]}$ is said to be a $W_{1,+}$-geodesic if: 1. The curve $(f_t)$ is a $W_1$-geodesic. 2. There exist two families $(g_t)$ and $(h_t)$ defined respectively on $E(G)$ and $T(G)$, such that: $$\frac{\partial}{\partial t} f_t = - \nabla \cdot g_t \ , \ \frac{\partial}{\partial t} g_t = - \nabla \cdot h_t.$$ 3. For every $(xy) \in E(G)$ we have $g_t(xy)>0$. 4. The triple $(f_t,g_t,h_t)$ satisfies the Benamou-Brenier equation $$\label{eq:BBcondition} \forall (x_0x_1x_2) \in T(G) \ , \ f_t(x_1)h_t(x_0x_1x_2) = g_t(x_0x_1) g_t(x_1x_2).$$ Let us fix a couple $f_0$, $f_1$ of probability measures on $G$ and endow $G$ with the $W_1$-orientation with respect to $f_0,f_1$. The existence of a $W_{1,+}$-interpolation $(f_t)$ joining $f_0$ to $f_1$ is the main result of [@HillionGeodesic]. Moreover, any such curve $(f_t)$ can be seen as a mixture of binomial families of distributions with respect to a coupling which is solution of a certain minimization problem. Canonical $W_{1,+}$-geodesics ----------------------------- In this paper we are mostly interested in particular $W_{1,+}$-geodesics, called canonical $W_{1,+}$-geodesics, which correspond to the case where $\forall \gamma \in \operatorname{E\Gamma}\ , \ C(\gamma)=1$, with the notations of [@HillionGeodesic]. The existence, uniqueness, and construction of such curves can be summed up by the following: \[th:CanonicalGeodesic\] Let $x_0 \leq \cdots \leq x_n \in G$ be an oriented $n+1$-uples of vertices of $G$. We define: $$m(x_0,\ldots, x_n) := \frac{|\Gamma_{x_0,\cdots, x_n}|}{|\operatorname{E\Gamma}|},$$ where $\Gamma_{x_0,\cdots, x_n}$ is the set of extremal geodesics visiting $x_0,\ldots, x_n$: $$\Gamma_{x_0,\cdots x_n} := \{\gamma \in \operatorname{E\Gamma}\ : \ \exists i_0 \leq \cdots \leq i_n \ ,\ \gamma(i_k)= x_k\}.$$ There exists a unique couple of families of functions $P_t(x),Q_t(x)$, defined for $x \in G$ and $t \in [0,1]$, such that each $t \mapsto P_t(x)$ and $t \mapsto Q_t(x)$ is positive and polynomial in $t$, and satisfying the following property: let us consider the families of functions $(f_t)$, $(g_t)$, $(h_t)$ respectively defined on $G$, $E(G)$ and $T(G)$ by $$\begin{aligned} f_t(x_0) &:=& m(x_0)P_t(x_0)Q_t(x_0), \\ g_t(x_0x_1) &:=& m(x_0,x_1) P_t(x_0) Q_t(x_1), \\ h_t(x_0x_1x_2) &:=& m(x_0,x_1,x_2) P_t(x_0)Q_t(x_2). \end{aligned}$$ Then the triple $(f_t,g_t,h_t)$ satisfies all the items of the definition of a $W_{1,+}$-geodesic. Such a curve will be called canonical $W_{1,+}$-geodesic joining $f_0$ to $f_1$. The reason why we introduce these particular geodesics comes from the following property, which will be used in Section \[sec:ProductSpaces\]: \[prop:CanonicalTriple\] If the triple $(f_t,g_t,h_t)$ defines a canonical $W_{1,+}$-geodesic, then for any oriented triple $(x_0x_1x_2) \in T(G)$, the quantity $h(x_0x_1x_2)$ does not depend on $x_1$, and therefore can be written $h(x_0x_2)$. **Proof:** It suffices to show that the cardinality $|\Gamma_{x_0,x_1,x_2}|$ does not depend on $x_1$. This comes from the fact that every $\gamma \in \Gamma_{x_0,x_1,x_2}$ can be written $$\gamma \ : \ \gamma(0) \rightarrow \cdots \rightarrow \gamma(i)=x_0 \rightarrow x_1 \rightarrow x_2=\gamma(i+2) \rightarrow \cdots \rightarrow \gamma(n).$$ We thus have $|\Gamma_{x_0,x_1,x_2}| = A(x_0) B(x_2)$, where $A(x_0)$ is the number of oriented paths joining some $\gamma(0)$ such that $\mathcal{E}(\gamma(0))=\emptyset$ to $x_0$ and where $B(x_2)$ is defined similarly. $\square$ General bounds on $H''(t)$ {#sec:GeneralFormula} ========================== In this section, we adapt the method used in [@HillionContraction] to prove the convexity of the entropy along the contraction of a probability measure on $\mathbb{Z}$ to the more general framework of $W_{1,+}$-geodesics on a graph. We then apply this method in the cases where $G$ is the graph $\mathbb{Z}$ or a complete graph. We finally study the behaviour, along a $W_{1,+}$-geodesic, of the relative entropy with respect to a log-concave reference probability measure and discuss why the hypothesis of a uniform bound on the second derivative $H''(t)$ may not be by itself a sufficient condition for interseting functional inequalities to hold. Benamou-Brenier triples ----------------------- Let $G$ be a graph, endowed with the $W_1$-orientation with respect to a couple of probability distributions $f_0,f_1$ on $G$. A Benamou-Brenier triple, or BB-triple, on $(G,\rightarrow)$, is a triple of positive functions $f,g,h$ defined respectively on $G$, $E(G)$ and $T(G)$ such that $$\forall (x_0x_1x_2) \in T(G) \ , \ h(x_0x_1x_2)f(x_1) = g(x_0x_1)g(x_1x_2).$$ It is clear that, if a triple $(f_t,g_t,h_t)$ defines a $W_{1,+}$-geodesic, then for each $t \in [0,1]$, $(f_t,g_t,h_t)$ is a BB-triple. Other types of BB-triples will be considered in Section \[sec:ProductSpaces\]. The functional $\mathcal{I}$ is defined for every BB-triple on $(G,\rightarrow)$ by $$\label{eq:defI} \mathcal{I}(f,g,h) := \sum_{x \in G} \nabla_2 \cdot h(x) \log(f(x)) + \frac{(\nabla \cdot g(x))^2}{f(x)}.$$ Let us consider $(f_t,g_t,h_t)$ defining a $W_{1,+}$-geodesic on $G$. The entropy $H(t)$ of $f_t$ satisfies $$H''(t) = \mathcal{I}(f_t,g_t,h_t).$$ **Proof:** This simply comes from the definition of the families $(g_t)_{t \in [0,1]}$ and $(h_t)_{t \in [0,1]}$: $$\frac{\partial}{\partial t}f_t(x) = - \nabla \cdot g_t(x) \ , \ \frac{\partial^2}{\partial t^2} f_t(x) = \nabla_2 \cdot h_t(x). \ \square$$ Integration by parts on $G$ --------------------------- In order to obtain bounds on $H''(t)$, we first use integration by parts to transform the sum in : \[prop:Itelescopic\] For any BB-triple $(f,g,h)$ we have $$\begin{aligned} \mathcal{I}(f,g,h) &=& \sum_{x \in G} \left[ \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) \log\left(\frac{f(x) h_t(x x_1 x_2)}{g(x x_1)^2}\right) \right] \\ && + \sum_{x \in G} \left[ \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_{-2} \in \mathcal{E}(x_{-1})} h(x_{-2}x_{-1}x) \log\left(\frac{f(x) h_t(x_{-2}x_{-1}x)}{g(x_{-1}x)^2}\right) \right] \\ && + \sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)}.\end{aligned}$$ **Proof:** We add to the sum defining $\mathcal{I}(f,g,h)$ (see equation ) the following telescopic sums $$0 = \sum_{x \in G} \nabla_2 \cdot (h \log(h))(x) \ , \ 0= -2 \sum_{x \in G} \nabla \cdot \left(g \nabla \cdot h\right)(x).$$ The proposition is then proven by noticing that, $(f,g,h)$ being a BB-triple, we have $$\forall x \in G \ , \ -2 \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} h(x_{-1}xx_1) \log\left(\frac{h(x_{-1}xx_1) f(x)}{g(x_{-1}x)g(xx_1)} \right) =0. \ \square$$ Combining Proposition \[prop:Itelescopic\] with the elementary inequality $\log(x) \geq 1-1/x$ allows us to obtain bounds on $\mathcal{I}(f,g,h)$: \[prop:Iboundnaif\] For any triple $(f,g,h)$ we have $$\label{eq:Iboundnaif} \mathcal{I}(f,g,h) \geq \sum_{(x_0x_1) \in E(G)} \frac{g(x_0x_1)^2}{f(x_0)}\left(1-\left|\mathcal{F}(x_1)\right| \right) + \frac{g(x_0x_1)^2}{f(x_1)}\left(1-\left|\mathcal{E}(x_0)\right| \right).$$ **Proof:** The inequality $\log(x) \geq 1-1/x$ implies $$\begin{aligned} \mathcal{I}(f,g,h) &\geq& \sum_{x \in G} \left[ \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) - \frac{g(x x_1)^2}{f(x)} \right] \\ && + \sum_{x \in G} \left[ \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_{-2} \in \mathcal{E}(x_{-1})} h(x_{-2}x_{-1}x) - \frac{g(x_{-1}x)^2}{f(x)} \right] \\ && + \sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)}.\end{aligned}$$ The following are obvious: $$\sum_{x_2 \in \mathcal{F}(x_1)} \frac{g(x x_1)^2}{f(x)} = |\mathcal{F}(x_1)| \frac{g(x x_1)^2}{f(x)} \ , \ \sum_{x_{-2} \in \mathcal{E}(x_{-1})} \frac{g(x_{-1}x)^2}{f(x)} = |\mathcal{E}(x_{-1})| \frac{g(x_{-1}x)^2}{f(x)}.$$ Moreover, we have: $$\begin{aligned} \sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) &=& \sum_{(x_0x_1x_2) \in T(G)} h(x_0x_1x_2) \\ &=& \sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(x_{-1}x)g(xx_1)}{f(x)},\end{aligned}$$ and similarly: $$\sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \sum_{x_2 \in \mathcal{F}(x_1)} h(xx_1x_2) = \sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(x_{-1}x)g(xx_1)}{f(x)}.$$ Expanding $\sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)}$ allows us to find similar terms: $$\begin{aligned} \sum_{x \in G} \frac{(\nabla \cdot g(x))^2}{f(x)} &=& \sum_{x \in G} \frac{\left(\sum_{x_1 \in \mathcal{F}(x)} g(xx_1) - \sum_{x_{-1} \in \mathcal{E}(x)} g(x_{-1}x)\right)^2}{f(x)} \\ &\geq & \sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(xx_1)^2}{f(x)}+\sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \frac{g(x_{-1}x)^2}{f(x)} \\ && -2 \sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \sum_{x_1 \in \mathcal{F}(x)} \frac{g(x_{-1}x)g(xx_1)}{f(x)}.\end{aligned}$$ We used the fact that $g$ is non-negative to apply the inequality $$\left(\sum_{x_1 \in \mathcal{F}(x)} g(xx_1) \right)^2 \geq \sum_{x_1 \in \mathcal{F}(x)} g(xx_1)^2,$$ which is far from being optimal, unless $|\mathcal{F}(x)|=0$ or $1$. Combining these estimations leads to $$\begin{aligned} \mathcal{I}(f,g,h) &\geq& \sum_{x \in G} \sum_{x_1 \in \mathcal{F}(x)} \left(1-|\mathcal{F}(x_1)|\right) \frac{g(x x_1)^2}{f(x)}\\&&+\sum_{x \in G} \sum_{x_{-1} \in \mathcal{E}(x)} \left(1-|\mathcal{E}(x_{-1})|\right)\frac{g(x_{-1}x)^2}{f(x)},\end{aligned}$$ which, up to a change of indices, is exactly inequality . $\square$ The bound obtained in Proposition \[prop:Iboundnaif\] is interesting in two fundamental cases: \[cor:IboundCube\] Let $(f,g,h)$ be a BB-triple of functions on $(G,\rightarrow)$ where $G$ is the complete graph with $n$ points, $W_1$-oriented with respect to some couple $(f_0,f_1)$. We have $$\mathcal{I}(f,g,h) \geq \sum_{(x_0x_1) \in E(G)} g(x_0x_1)^2 \left(\frac{1}{f(x_0)}+\frac{1}{f(x_1)} \right).$$ **Proof:** We apply Proposition \[prop:Iboundnaif\], using the fact that, if $(x_0x_1) \in E(G)$, then the sets $\mathcal{E}(x_0)$ and $\mathcal{F}(x_1)$ are empty, or the equivalent fact that the set of oriented triple $T(G)$ is empty: indeed if there exists $(x_0x_1x_2) \in T(G)$ then, by Proposition \[prop:OrientedGeodesic\], we have $d(x_0,x_2)=2$, which is a contradiction. $\square$ \[cor:IboundZ\] Let $(f,g,h)$ be a (finitely supported) BB-triple of functions on $(G,\rightarrow)$ where $G$ is the graph $\mathbb{Z}$, $W_1$-oriented with respect to some couple $(f_0,f_1)$. T $\mathcal{I}(f,g,h) \geq 0.$ **Proof:** We use this time the fact that each vertex of $\mathbb{Z}$ has two neighbours, which implies that, for every $x \in \mathbb{Z}$, $|\mathcal{E}(x)|+|\mathcal{F}(x)| \leq 2$. In particular, if $(x_0x_1) \in E(G)$, then $\mathcal{E}(x_1)$ is non-empty (as it contains $x_0$), so $|\mathcal{F}(x_1)| \leq 1$. Similarly we have $|\mathcal{E}(x_0)| \leq 1$. Applying Proposition \[prop:Iboundnaif\] leads to the result. $\square$ **Remark.** Corollary \[cor:IboundZ\] can be extended to the framework of cyclic graphs $\mathbb{Z}_r$ for $r \geq 2$, because in this case every vertex has also two neighbours. About the convexity of the relative entropy ------------------------------------------- We have been so far interested in the behaviour of the Shannon entropy functional $H(f) := \sum_{x \in G} f(x) \log(f(x))$ along $W_{1,+}$ geodesics on $G$. However, the functional which is considered in Sturm-Lott-Villani theory are the relative entropy $H_\nu$ with respect to some reference probability measure $\nu$. In this paragraph, we present some results about the behaviour of $H_\nu$ along $W_{1,+}$-geodesics on graphs. Let $\nu$ be a probability measure fully supported on $G$. The relative entropy $H_\nu(f)$ of a probability measure $f$ on $G$ is defined by $$H_\nu(f) := \sum_{x \in G} f(x) \log\left(\frac{f(x)}{\nu(x)}\right).$$ **Remark.** Let $(f_t)_{t \in [0,1]}$ be a $W_{1,+}$-geodesic supported on a finite subset of vertices $A \subset G$. Let $\nu$ be the uniform probability distribution on $A$. Then the Shannon and relative entropies are linked by $$H_\nu(f_t) = H(f_t) + \log(|A|)$$ so the convexity of $t \mapsto H_\nu(f_t)$ is equivalent to the convexity of $t \mapsto H(f_t)$. As in the Riemannian case, it is interesting to consider log-concave reference measures: We endow $G$ with a reference measure $\nu(x) := \exp(-V(x))$. We suppose that there exists $K>0$ such that, for every geodesic path of length $2$ $\gamma_0,\gamma_1,\gamma_2$ we have $$V(\gamma_0)-2 V(\gamma_1) + V(\gamma_2) \geq K.$$ Let $(f_t)$ be a $W_{1,+}$-geodesic, $H(t)$ be the Shannon entropy of $f_t$ and $H_\nu(t)$ its relative entropy. Then $$\label{eq:EntroRelative} H_\nu''(f_t) \geq H''(t) + K W^2(f_0,f_1),$$ where $$W^2(f_0,f_1) := \sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2)$$ does not depend on $t$. **Proof:** We have $$H_\nu(t) - H(t) = - \sum_{x \in G} f_t(x) \log(\nu(x)) = \sum_{x \in G} f_t(x) V(x),$$ and by differentiating twice with respect to $t$ we have $$H_\nu''(t) = H''(t) + \sum_{x \in G} (\nabla_2 \cdot h_t)(x) V(x) = \sum_{(x_0x_1x_2) \in T(G)} h_t(x) (V(x_2)-2V(x_1)+V(x_0)),$$ which, by the convexity assumption made on $V$, proves equation . In order to prove that $\sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2)$ does not depend on $t$, we first use the Benamou-Brenier condition  to write $$\frac{\partial}{\partial t} h_t(x_0x_1x_2) = - \sum_{x_{-1} \in \mathcal{E}(x_0)} \frac{g_t(x_{-1}x_0) g_t(x_0x_1) g_t(x_1x_2)}{f_t(x_0)f_t(x_1)}+ \sum_{x_3 \in \mathcal{F}(x_2)} \frac{g_t(x_0x_1)g_t(x_1x_2)g_t(x_2x_3)}{f_t(x_1)f_t(x_2)}.$$ A simple change of indices then show that $$\frac{\partial}{\partial t} \sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2) = \sum_{x_0 \rightarrow \cdots \rightarrow x_3 \in G} \frac{g_t(x_0x_1)g_t(x_1x_2)g_t(x_2x_3)}{f_t(x_1)f_t(x_2)}-\frac{g_t(x_0x_1)g_t(x_1x_2)g_t(x_2x_3)}{f_t(x_1)f_t(x_2)}=0,$$ so $\sum_{(x_0x_1x_2) \in T(G)} h_t(x_0x_1x_2)$ does not depend on $t$. $\square$ **Remark.** One major difference with the continuous case is the fact that, although acting as the Wasserstein distance $W_2$, the quantity $W(f_0,f_1)$ does not define a distance on $\mathcal{P}(G)$. For instance, if $f_0$ and $f_1$ are two Dirac distributions at two adjacent vertices, we have $W(f_0,f_1)=0$. A different perspective on the same issue consists in writing $$W^2(f_0,f_1) = \sum_{x_1 \in G} f_t(x_1) V_{+,t}(x_1) V_{-,t}(x_1),$$ where $V_{+,t}(x_1) := \sum_{x_2 \in \mathcal{F}(x_1)} \frac{g_t(x_1x_2)}{f_t(x_1)}$ and $V_{-,t}(x_1) := \sum_{x_0 \in \mathcal{E}(x_1)} \frac{g_t(x_0x_1)}{f_t(x_1)}$ are the two velocity functions, which can be written $W^2 = \langle V_{+,t} , V_{-,t}\rangle$ for the scalar product with respect to $f_t$. This formula is the discrete analogue of the Benamou-Brenier formula  for $p=2$, but in the continuous setting we have $W_2^2 = <v_t,v_t>= ||v_t||^2$ for the scalar product with respect to $f_t$. The fact that $V_{+,t} \neq V_{-,t}$ is a major obstacle to a generalization of the HWI inequality which holds for instance in the measured length space $(\mathbb{R}^d,\exp(-V(x))dx)$ (see [@LottVillani] for a proof of this fact). Product of graphs {#sec:ProductSpaces} ================= Let $G_1$ and $G_2$ be two locally finite and connected graphs. In this section we study the behaviour of the entropy along $W_{1,+}$-geodesics defined on the product graph $G := G_1 \times G_2$ endowed with the usual product metric $$d_G((x_1,x_2),(y_1,y_2)) := d_{G_1}(x_1,y_1)+d_{G_2}(x_2,y_2).$$ The $W_1$-orientation on a product graph ---------------------------------------- The neighbours of a vertex $(x_1,x_2)$ in $G$ are the vertices $(x_1,y_2)$, where $d_{G_2}(x_2,y_2)=1$ and $(y_1,x_2)$ where $d_{G_1}(x_1,y_1)=1$. From this fact we easily deduce the following description of geodesic curves in $G$: \[prop:GeodProduct\] Let $\gamma \in \Gamma(x,y)$ be a geodesic on $G$, where $(x,y)=((x_1,x_2),(y_1,y_2))$. There exist two geodesics $\gamma_1 \in \Gamma(x_1,y_1)$, $\gamma_2 \in \Gamma(x_2,y_2)$ defined respectively on $G_1$ and $G_2$, and an application $$\phi : \{0,\ldots, d(x,y)\} \rightarrow \{0,\ldots, d_1(x_1,y_1)\}$$ with $\phi(0)=0$, $\phi(d(x,y))=d_1(x_1,y_1)$ and $\phi(k+1)-\phi(k) \in \{0,1\}$, such that $$\gamma(k) = (\gamma_1(\phi(k)),\gamma_2(k-\phi(k))).$$ In particular, the cardinality of $\Gamma(x,y)$ satisfies $$|\Gamma(x,y)| = \binom{d(x,y)}{d(x_1,y_1)}|\Gamma(x_1,y_1)||\Gamma(x_2,y_2)|.$$ If $f$ is a probability distribution on $G$, we denote by $f^{(1)}$, $f^{(2)}$ its marginals on $G_1$ and $G_2$. To a coupling $\pi$ between two distributions $f_0,f_1$, which can be seen as a probability measure on $$G \times G = (G_1 \times G_2) \times (G_1 \times G_2) = (G_1 \times G_1) \times (G_2 \times G_2),$$ we associate the marginal couplings $\pi^{(1)}$ on $G_1 \times G_1$ between $f_0^{(1)}$ and $f_1^{(1)}$ and $\pi_2$ on $G_2 \times G_2$ between $f_0^{(2)}$ and $f_1^{(2)}$. We then describe the $W_{1,+}$-orientation on $G$ with respect to a couple of measures $f_0,f_1$. \[prop:W1OrientProduct\] Let $f_0,f_1 \in \mathcal{P}(G)$. For $i=1,2$ we define $$\mathcal{E}_i(x^{(i)}) := \{y^{(i)} \in G_i \ : \ y^{(i)} \rightarrow x^{(i)} \} \ , \ \mathcal{F}_i(x^{(i)}) := \{z^{(i)} \in G_i \ : \ x^{(i)} \rightarrow z^{(i)} \}$$ for the $W_1$ orientation on $G_i$ between $f_0^{(i)}$ and $f_1^{(i)}$. The $W_1$-orientation between $f_0$ and $f_1$ is then described by $$\begin{aligned} \mathcal{E}(x) &=& \left( \bigcup_{y^{(2)} \in \mathcal{E}_2(x^{(2)})} (x^{(1)},y^{(2)}) \right) \bigcup \left( \bigcup_{y^{(1)} \in \mathcal{E}_2(x^{(1)})} (y^{(1)},x^{(2)}) \right)\\ &=:& \mathcal{E}_1(x) \cup \mathcal{E}_2(x), \\ \mathcal{F}(x) &=& \left( \bigcup_{y^{(2)} \in \mathcal{F}_2(x^{(2)})} (x^{(1)},y^{(2)}) \right) \bigcup \left( \bigcup_{y^{(1)} \in \mathcal{F}_2(x^{(1)})} (y^{(1)},x^{(2)}) \right)\\ &=:& \mathcal{F}_1(x) \cup \mathcal{F}_2(x).\end{aligned}$$ **Proof:** Let $\pi \in \Pi(f_0,f_1)$ be a coupling between $f_0$ and $f_1$. We have $$\begin{aligned} I_1(\pi) &=& \sum_{(x_1,x_2),(y_1,y_2) \in G \times G} d((x_1,x_2),(y_1,y_2)) \pi((x_1,x_2),(y_1,y_2)) \\ &=& \sum_{(x_1,y_1) \in G_1 \times G_1} \sum_{(x_2,y_2) \in G_2\times G_2} d_1(x_1,y_1)+d_2(x_2,y_2) \pi((x_1,x_2),(y_1,y_2)) \\ &=& \sum_{(x_1,y_1) \in G_1 \times G_1} d_1(x_1,y_1) \pi^{(1)}(x_1,y_1) + \sum_{(x_2,y_2) \in G_2\times G_2}d_2(x_2,y_2) \pi^ {(2)}(x_2,y_2) \\ &=& I_1(\pi^{(1)})+ I_1(\pi^{(2)}),\end{aligned}$$ which proves that $\pi$ is $W_1$-optimal between $f_0$ and $f_1$ (for the distance $d_G$) if and only if its marginals $\pi^{(1)}$, $\pi^{(2)}$ are $W_1$-optimal between $f_0^{(1)}$ and $f_1^{(1)}$, resp $f_0^{(2)}$ and $f_1^{(2)}$ for the distance $d_{G_1}$, resp. $d_{G_2}$. We now fix a $W_1$-optimal coupling $\pi \in \Pi_1(f_0,f_1)$. Let $x=(x_1,x_2)$ and $y=(y_1,y_2)$ be two vertices of $G$ such that $\pi(x,y)>0$. We then have $\pi^{(1)}(x_1,y_1)>0$ and $\pi^{(2)}(x_2,y_2)>0$ for the marginal couplings, which are also $W_1$-optimal. Let $\gamma \in \Gamma_G(x,y)$, and $\gamma_1 \in \Gamma_{G_1}(x_1,y_1)$, $\gamma_2 \in \Gamma_{G_2}(x_2,y_2)$, $\phi : \{0,\ldots, d(x,y)\} \rightarrow \{0,\ldots, d_1(x_1,y_1)\}$ be associated to $\gamma$ by Proposition \[prop:GeodProduct\]. For $k \in \{0,\ldots, d(x,y)-1\}$, we have $$\gamma(k) = (\gamma_1(\phi(k)),\gamma_2(k-\phi(k))) \ , \ \gamma(k+1) = (\gamma_1(\phi(k)),\gamma_2(k-\phi(k+1)+1)).$$ If $\phi(k+1)=\phi(k)+1$, resp. $\phi(k+1)=\phi(k)$, then $\gamma(k+1) \in \mathcal{E}_1(\gamma(k))$, resp. $\gamma(k+1) \in \mathcal{E}_2(\gamma(k))$. Conversely, let us consider a vertex $x=(x_1,x_2) \in G$. We suppose that $x=\gamma(k)$ for some geodesic $\gamma$ of length $n$ such that $\pi(\gamma(0),\gamma(n))>0$ for a $W_1$-optimal coupling $\pi \in \Pi_1(f_0,f_1)$. We denote by $\gamma^{(1)}, \gamma^{(2)}$ the projections of $\gamma$, as defined in Proposition \[prop:GeodProduct\], $n_1$ and $n_2$ their respective lengths, and $\pi^{(1)}$, $\pi^{(2)}$ the marginals of $\pi$. Let $y \in \mathcal{E}_1(x)$. We have $y=(y_1,x_2)$ with $y_1 \in \mathcal{E}_1(x_1)$. There exists a $W_1$-optimal coupling $\tilde{\pi}^{(1)} \in \Pi_1(f_0,f_1)$ and a geodesic $\tilde{\gamma}_1$ on $G_1$, of length $\tilde{n}_1$, such that $\tilde{\gamma}_1(k_1)=x_1$, $\tilde{\gamma}_1(k_1+1)=y_1$ and $\tilde{\pi}^{(1)}(\tilde{\gamma}_1(0),\tilde{\gamma}_1(\tilde{n}_1)) > 0$. Let $\tilde{\pi}$ be any coupling between $f_0$ and $f_1$ having $\tilde{\pi}^{(1)}$ and $\pi^{(2)}$ as marginals and $\gamma$ be a geodesic of $G$ having $\tilde{\gamma_1}$ and $\gamma_2$ as projections. Then there exists some $k$ for which $\tilde{\gamma}(k)=x$, $\tilde{\gamma}(k+1)=y$. Furthermore $\tilde{\pi}$ is $W_1$-optimal between $f_0$ and $f_1$ and $\tilde{\pi}(\gamma(0),\gamma(\tilde{n}_1+n_2)) > 0$, which proves that $y \in \mathcal{E}(x)$. We can prove similarly that, if $y \in \mathcal{E}_2(x)$ then $y \in \mathcal{E}(x)$, which finishes the proof. $\square$ An immediate consequence of Proposition \[prop:W1OrientProduct\] is a decomposition of the divergence operator: \[prop:divdecomp\] The divergence $\nabla \cdot g$ of a function $g : E(G) \rightarrow \mathbb{R}$ can be written $\nabla \cdot g= \nabla^{(1)} \cdot g+\nabla^{(2)} \cdot g$ where $$\nabla^{(i)} \cdot g(x_1) := \sum_{x_2 \in \mathcal{F}_i(x_1)} g(x_1 x_2) - \sum_{x_0 \in \mathcal{E}_i(x_1)} g(x_0x_1).$$ Similarly, the second order divergence operator of a function $h : T(G) \rightarrow \mathbb{R}$ can be written $$\nabla_2 \cdot h = \nabla_2^{(11)} \cdot h +\nabla_2^{(12)} \cdot h+\nabla_2^{(21)} \cdot h+\nabla_2^{(22)} \cdot h,$$ with $\nabla_2^{(ij)} := \nabla^{(i)} \circ \nabla^{(j)}$. The structure of the oriented graph $(G_1 \times G_2)$ is better understood by introducing oriented product squares: An oriented product square of $G$ is a 4-uple of vertices $(x_0,x_1,x_1',x_2) \in G^4$ such that $x_1 \in \mathcal{F}_1(x_0)$, $x_1' \in \mathcal{F}_2(x_0)$, $x_2 \in \mathcal{F}_2(x_1)$ and $x_2 \in \mathcal{F}_1(x_1')$. We denote by $S(G)$ the set of oriented product squares of $G$. \[prop:OrientedSquares\] Let $x_0 \in G$. The following sets are all in bijection: - $\mathcal{A}_1$:= $\mathcal{F}_1(x_0) \times \mathcal{F}_2(x_0)$. - $\mathcal{A}_2$:= $\{ x_2 \in G \ : \ \exists x_1,x_1' \in G\times G \ , \ (x_0,x_1,x_1',x_2) \in S(G) \}$. - $\mathcal{A}_3$:= $\{ (x_0x_1x_2) \in T(G) \ : x_1 \in \mathcal{F}_1(x_0) , x_2 \in \mathcal{F}_2(x_1) \ \}$. - $\mathcal{A}_4$:= $\{ (x_0x_1'x_2) \in T(G) \ : x_1' \in \mathcal{F}_2(x_0) , x_2 \in \mathcal{F}_1(x_1') \ \}$. **Proof:** Let us fix $x_1 \in \mathcal{F}_1(x_0)$ and $x_1' \in \mathcal{F}_2(x_0)$. We write $x_0=(x_0^{(1)},x_0^{(2)})$ in $G_1 \times G_2$. There exist a unique $x_1^{(1)} \in \mathcal{F}_{G_1}(x_0^{(1)})$ and a unique $x_1'^{(2)} \in \mathcal{F}_{G_{2}}(x_0^{(2)})$ such that $x_1 = (x_1^{(1)},x_0^{(2)})$ and $x_1'=(x_0^{(1)},x_1'^{(2)})$ in $G_1 \times G_2$. We then set $x_2 := (x_1^{(1)},x_1'^{(2)})$ and it is easy to see that $(x_0,x_1,x_1',x_2) \in S(G)$. $\square$ Proposition \[prop:OrientedSquares\] shows that an oriented square $(x_0x_1x_1'x_2)$ is uniquely determined by the couple $x_0,x_2$. We will use the notation $(x_0x_2) \in S(G)$ to denote such squares. We will also denote the two midpoints $x_1,x_1'$ respectively by $m_1(x_0x_2)$ and $m_2(x_0x_2)$. Let $(f_t)$ be a $W_{1,+}$-geodesic on $G$. There exist two families of functions $(g_t)$ and $(h_t)$, defined respectively on $E(G)$ and $T(G)$, such that $\frac{\partial}{\partial t}f = - \nabla \cdot g$, $\frac{\partial}{\partial t}g = - \nabla \cdot h$ and satisfying $$\forall (x_0x_1x_2) \in T(G) \ , \ f_t(x_1)h_t(x_0x_1x_2) = g_t(x_0x_1)g_t(x_1x_2).$$ Given a vertex $x^{(2)}$, we now define, for $(x_0^{(1)}x_1^{(1)}x_2^{(1)}) \in T(G_1)$, the functions $$\begin{aligned} \label{eq:BBprojection} f_{t,x^{(2)}}(x_0^{(1)}) &:=& f_t(x_0^{(1)},x^{(2)}), \\ g_{t,x^{(2)}}(x_0^{(1)}x_1^{(1)}) &:=& g_t((x_0^{(1)},x^{(2)})(x_1^{(1)},x^{(2)})), \\ h_{t,x^{(2)}}(x_0^{(1)}x_1^{(1)}x_2^{(1)}) &:= & h_t((x_0^{(1)},x^{(2)})(x_1^{(1)},x^{(2)})(x_2^{(1)},x^{(2)})).\end{aligned}$$ The triple of functions $(f_{t,x^{(2)}},g_{t,x^{(2)}},h_{t,x^{(2)}})$ is then a BB-triple on $G_1$. Given $x^{(1)} \in G_1$, we define similarly the BB-triples of functions $(f_{t,x^{(1)}},g_{t,x_{(1)}},h_{t,x_{(1)}})$ on $G_2$. The divergence of $g_{t,x^{(2)}} : E(G_1) \rightarrow \mathbb{R}$ satisfies the relation $$(\nabla \cdot g_{t,x^{(2)}})(x^{(1)}) = (\nabla^{(1)} \cdot g_t)(x^{(1)},x^{(2)}).$$ The second order divergence $h_{t,x^{(2)}} : T(G_1) \rightarrow \mathbb{R}$ satisfies $$(\nabla_2 \cdot h_{t,x^{(2)}})(x^{(1)}) = (\nabla^{(11)} \cdot h_t)(x^{(1)},x^{(2)}).$$ A tensorization result ---------------------- We are now able to state the tensorization theorem: \[th:EntroTensorization\] Let $(f_t,g_t,h_t)$ be a canonical $W_{1,+}$-geodesic on $G$ and $H(t)$ denote the entropy of $f_t$. Then: $$H''(t) \geq \sum_{x^{(2)} \in G_2} \mathcal{I}(f_{t,x^{(2)}},g_{t,x^{(2)}},h_{t,x^{(2)}}) + \sum_{x^{(1)} \in G_1} \mathcal{I}(f_{t,x^{(1)}},g_{t,x^{(1)}},h_{t,x^{(1)}}).$$ **Proof:** We apply Proposition \[prop:divdecomp\]: $$\begin{aligned} \sum_{x^{(2)} \in G_2} \mathcal{I}(f_{t,x^{(2)}},g_{t,x^{(2)}},h_{t,x^{(2)}}) &=& \sum_{x^{(2)} \in G_2} \left( \sum_{x^{(1)} \in G_1} \nabla_2 \cdot h_{t,x^{(2)}}(x^{(1)}) \log(f_{t,x^{(2)}}(x^{(1)})) \right) \\ && + \sum_{x^{(2)} \in G_2} \left( \sum_{x^{(1)} \in G_1} \frac{(\nabla \cdot g_{t,x^{(2)}}(x^{(1)}))^2}{f_{t,x^{(2)}}(x^{(1)})} \right) \\ &=& \sum_{x \in G} \nabla_2^{(11)} \cdot h_t(x) \log(f_t(x)) + \frac{(\nabla^{(1)} \cdot g_t(x))^2}{f_t(x)}.\end{aligned}$$ Similarly, $$\sum_{x^{(1)} \in G_1} \mathcal{I}(f_{t,x^{(1)}},g_{t,x^{(1)}},h_{t,x^{(1)}}) = \sum_{x \in G} \nabla_2^{(22)} \cdot h_t(x) \log(f_t(x)) + \frac{(\nabla^{(2)} \cdot g_t(x))^2}{f_t(x)}.$$ To prove Theorem \[th:EntroTensorization\], it thus suffices to show the inequality $$\label{eq:CrossTerms} \sum_{x \in G} (\nabla_2^{(12)}+\nabla_2^{(21)}) \cdot h_t(x) \log(f_t(x)) + 2 \frac{\nabla^{(1)} g_t(x) \nabla^{(2)} \cdot g_t(x)}{f_t(x)} \geq 0.$$ By considering the telescopic sums $$\sum_{x \in G} \nabla^{(12)} \cdot h_t \log(h_t)(x)=0 \ , \ -2\sum_{x\in G} \nabla^{(1)} \cdot (\nabla^{(2)} \cdot h_t \log(g_t))=0,$$ we prove, as in Proposition \[prop:Itelescopic\], that $$\begin{aligned} \sum_{x \in G} \nabla_2^{(12)} \cdot h_t(x) \log(f_t(x)) &=& \sum_{(x_0x_1x_2) \in T^{(12)}(G)} h(x_0x_1x_2) \log\left(\frac{f_t(x_0)h_t(x_0x_1x_2)}{g_t(x_0x_1)^2}\right)\\&&+\sum_{(x_0x_1x_2) \in T^{(12)}(G)} h(x_0x_1x_2)\log\left(\frac{f_t(x_2)h_t(x_0x_1x_2)}{g_t(x_1x_2)^2}\right),\end{aligned}$$ where $T^{(12)}(G)$ is the set of oriented triples $(x_0x_1x_2) \in T(G)$ such that $x_0 \in \mathcal{E}_1(x_1)$ and $x_1 \in \mathcal{E}_2(x_2)$. We now use the bijection between $T^{(12)}(G)$ and $S(G)$, proven in Proposition \[prop:OrientedSquares\], and the fact that $h(x_0x_1x_2)$ does not depend on $x_1$, which comes from the assumption that $(f_t)$ is canonical and from Proposition \[prop:CanonicalTriple\], to write: $$\sum_{x \in G} \nabla_2^{(12)} \cdot h_t(x) \log(f_t(x)) = \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \left( \log \left(\frac{f(x_0)h(x_0x_2)}{g(x_0m_1(x_0,x_2))^2 } \right)+\log \left(\frac{f(x_2)h(x_0x_2)}{g(m_1(x_0,x_2)x_2)^2 } \right)\right).$$ Similarly, we have: $$\sum_{x \in G} \nabla_2^{(21)} \cdot h_t(x) \log(f_t(x)) = \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \left( \log \left(\frac{f(x_0)h(x_0x_2)}{g(x_0m_2(x_0,x_2))^2 } \right)+\log \left(\frac{f(x_2)h(x_0x_2)}{g(m_2(x_0,x_2)x_2)^2 } \right)\right).$$ Adding both equations and using the inequality $\log(x) \geq 1-1/x$ gives: $$\begin{aligned} \sum_{x \in G} (\nabla_2^{(12)}+\nabla_2^{(21)}) \cdot h_t(x) \log(f_t(x)) &=& 2 \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \log \left(\frac{f(x_0)h(x_0x_2)}{g(x_0m_1(x_0,x_2))g(x_0m_2(x_0,x_2)) } \right) \\&&+ 2 \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \log \left(\frac{f(x_2)h(x_0x_2)}{g(m_1(x_0,x_2)x_2)g(m_2(x_0,x_2)x_2) } \right) \\ &\geq& 4 \sum_{(x_0x_2) \in S(G)} h(x_0x_2) \\ && - 2 \sum_{(x_0x_2) \in S(G)} \frac{g(x_0m_1(x_0,x_2))g(x_0m_2(x_0,x_2))}{f_t(x_0)} \\ && - 2 \sum_{(x_0x_2) \in S(G)} \frac{g(m_1(x_0,x_2)x_2)g(m_2(x_0,x_2)x_2)}{f_t(x_2)}.\end{aligned}$$ We use again the bijection in Proposition \[prop:OrientedSquares\] to write $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} \frac{g(x_0m_1(x_0,x_2))g(x_0m_2(x_0,x_2)}{f_t(x_0)} &=& \sum_{x_0 \in G} \left( \sum_{(x_1,x_1') \in \mathcal{F}_1(x_0) \times \mathcal{F}_2(x_0)} \frac{g(x_0x_1)g(x_0x_1')}{f(x_0)} \right)\\ &=& \sum_{x_0 \in G} \frac{\sum_{x_1 \in \mathcal{F}_1(x_0)} g(x_0x_1) \cdot \sum_{x_1' \in \mathcal{F}_2(x_0)} g(x_0x_1')}{f(x_0)} \end{aligned}$$ and $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} \frac{g(m_1(x_0,x_2)x_2)g(m_2(x_0,x_2)x_2)}{f_t(x_2)} = \sum_{x_0 \in G} \frac{\sum_{x_{-1} \in \mathcal{E}_1(x_0)} g(x_{-1}x_0) \cdot \sum_{x_{-1}' \in \mathcal{E}_2(x_0)} g(x_{-1}'x_0)}{f(x_0)}. \end{aligned}$$ We also have: $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} h(x_0x_2) &=& \sum_{(x_{-1}x_0x_1) \in T^{(12)}(G)} h(x_{-1}x_0x_1) \\ &=& \sum_{x_0 \in G} \sum_{x_{-1} \in \mathcal{E}_1(x_0) \ , \ x_1 \in \mathcal{F}_2(x_0)} \frac{g(x_{-1}x_0) g(x_0x_1)}{f(x_0)} \\ &=& \sum_{x_0 \in G} \frac{\sum_{x_{-1} \in \mathcal{E}_1(x_0)} g(x_{-1}x_0) \cdot \sum_{x_{1} \in \mathcal{F}_2(x_0)} g(x_{1}x_0)}{f(x_0)},\end{aligned}$$ and: $$\begin{aligned} \sum_{(x_0x_2) \in S(G)} h(x_0x_2) &=& \sum_{(x_{-1}x_0x_1) \in T^{(21)}(G)} h(x_{-1}x_0x_1) \\ &=& \sum_{x_0 \in G} \frac{\sum_{x_{-1} \in \mathcal{E}_2(x_0)} g(x_{-1}x_0) \cdot \sum_{x_{1} \in \mathcal{F}_1(x_0)} g(x_{1}x_0)}{f(x_0)}.\end{aligned}$$ Adding the last four identities gives: $$\begin{aligned} \sum_{x \in G} (\nabla_2^{(12)}+\nabla_2^{(21)}) \cdot h_t(x) \log(f_t(x)) & \geq & -\sum_{x_0 \in G} \frac{\nabla^{(1)} \cdot g(x_0) \nabla^{(2)} \cdot g(x_0)}{f(x_0)},\end{aligned}$$ which is exactly the inequality  we wanted to obtain. $\square$ Examples -------- The tensorization Theorem \[th:EntroTensorization\] is generalized to products of more than two graphs: let $G = G_1 \times \cdots \times G_p$. For $i = 1,\ldots,p$, we denote by $\hat{G}_i$ the product $G_1 \times \cdots G_p$, where $G_i$ is omitted. Given some vertex $\hat{x} \in \hat{G}_i$ and a BB-triple $(f_t,g_t,h_t)$ on $G$, we define a BB-triple $(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}})$ as in equation . We then have: \[cor:EntroTensoMultiple\] Let $(f_t,g_t,h_t)$ be a BB-triple on $G$. The entropy $H(t)$ of $f_t$ satisfies $$H''(t) \geq \sum_{i=1}^p \sum_{\hat{x} \in \hat{G}_i} \mathcal{I}(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}}).$$ Applying Corollary \[cor:EntroTensoMultiple\] to the examples studied in Section 3 allows us to obtain interesting bounds on the second derivative $H''(t)$ in other important cases: \[prop:EntroTensoExemples\] Theorem \[th:EntroTensorization\] can be applied in the following fundamental examples: - The entropy $H(t)$ along a $W_{1,+}$-geodesic $(f_t)_{t \in [0,1]}$ on $\mathbb{Z}^n$ is a convex function of $t$. - Let $(f_t,g_t,h_t)$ be a $W_{1,+}$-geodesic on the cube $\mathbb{Z}_2^n$. Then $$\mathcal{I}(f_t,g_t,h_t) = \sum_{(x_0x_1) \in E(G)} g_t(x_0x_1)^2 \left(\frac{1}{f_t(x_0)} + \frac{1}{f_t(x_1)} \right) \geq 0.$$ **Proof:** The first point follows directly from Corollary \[cor:IboundZ\]. To prove the second point, we notice that the cube is described by the product $G_1 \times \cdots \times G_n$ where each $G_i$ is the two-point graph $\mathbb{Z}_2$. Each $\hat{G}_i$ is isometric to the $n-1$-dimensional cube. To each $\hat{x} \in \hat{G}_i$, we associate two vertices $\hat{x}_0,\hat{x}_1 \in G$ by setting the $i$-th coordiante to $0$ or $1$. If $\hat{x}_0 \rightarrow \hat{x}_1$ in $G$, we define $g_t(\hat{x}) := g_t(\hat{x}_0 \hat{x}_1)$. If $\hat{x}_1 \rightarrow \hat{x}_0$ we define $g_t(\hat{x}) := g_t(\hat{x}_1 \hat{x}_0)$. Finally if the edge $(\hat{x}_0 \hat{x}_1)$ is not oriented in $G$ we set $g_t(\hat{x}):=0$. In any case, we have, by Corollary \[cor:IboundCube\], $$\mathcal{I}(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}}) = g_t(\hat{x})^2 \left(\frac{1}{f_t(\hat{x}_0)}+\frac{1}{f_t(\hat{x}_1)} \right).$$ A (non-ordered) edge $(x_0 x_1)$ of $G$ is described in the following way: $x_0$ and $x_1$ differ by exactly one coordinate. In other terms, there is a bijection between the set of edges of $G$ and the disjoint union $\bigcup_{i=1}^p \hat{G}_i$. We can then write $$\sum_{i=1}^n \sum_{\hat{x} \in \hat{G}_i} \mathcal{I}(f_{t,\hat{x}} ,g_{t,\hat{x}},h_{t,\hat{x}}) = \sum_{(x_0x_1) \in (E(G),\rightarrow)} g_t(x_0x_1)^2 \left(\frac{1}{f_t(x_0)}+\frac{1}{f(x_1)} \right),$$ which is what we wanted. $\square$ These two examples can be seen as particular cases of a more general theorem: \[th:GroupCurvature\] Let $G$ be the Cayley graph of a finitely generated abelian group, with a set of generators $T=(\tau_1,\ldots, \tau_q)$. Let $(f_t)$ be a $W_{1,+}$-interpolation on $G$ and $H(t)$ the entropy of $f_t$. Then : $$\label{eq:IboundGroup} H''(t) \geq \sum_{(x_0 x_1) \in \tilde{E}(G)} g_t(x_0x_1)^2 \left(\frac{1}{f_t(x_0)}+\frac{1}{f(x_1)} \right),$$ where $\tilde{E}(G)$ is the subset of oriented edges $(x_0 \rightarrow x_1) \in E(G)$ such that $x_1 = \tau_i x_0$ for some generator $\tau_i \in T$ such that $\tau_i^2 = id$. **Proof:** Theorem \[th:GroupCurvature\] can be proven with the help of Theorem \[th:EntroTensorization\]. Indeed, any finitely generated abelian group is isomorphic to the direct product $$\mathbb{Z}^n \times \mathbb{Z}_2^{n_2} \times \cdots \times \mathbb{Z}_p^{n_p} \times \cdots,$$ where all but a finite number of coefficients $n_p$ are equal to $0$. As we have proven that $\mathcal{I}(f,g,h) \geq 0$ for any BB-triple on $\mathbb{Z}_p$ or on $\mathbb{Z}$, a direct application of Theorem \[th:EntroTensorization\] gives that $H''(t) \geq 0$. The more precise bound given in equation  is proven as in the second point of Proposition \[prop:EntroTensoExemples\]. $\square$ Appendix: further results on $W_{1,+}$-geodesics on $\mathbb{Z}$. ================================================================= Renyi entropy along $W_{1,+}$-geodesics on $\mathbb{Z}$. -------------------------------------------------------- In this appendix we prove that, along a $W_{1,+}$-geodesic on $\mathbb{Z}$, not only the relative entropy is convex, but also a larger class of functionals belonging to the family of Renyi entropies: given a probability distribution $(f(k))_{k \in \mathbb{Z}}$ and a parameter $0<p<1$, we set $$H_p(f) := -\sum_{k \in \mathbb{Z}} f(k)^p .$$ The relative entropy $H(f) := \sum_k f(k) \log(f(k))$ can be seen as a limit case of Renyi entropy as the parameter $p \rightarrow 1$, in the sense that $$H_p(f) = -1 + (1-p) H(f) + o((1-p)^2).$$ We then have: \[th:RenyiZ\] Let $(f_t)_{t \in [0,1]}$ be a $W_{1,+}$ geodesic on $\mathbb{Z}$. Then $t \mapsto H_p(f_t)$ is convex. In order to have simpler notations, we are going to prove Theorem \[th:RenyiZ\] under the additional assumption that $f_0$ is stochastically dominated by $f_1$ (see also Theorem \[th:EntroBinoW2\]). Under this assumption, the $W_1$-orientation on $\mathbb{Z}$ is simply described by orienting the edge $(k,k+1)$ by $k \rightarrow k+1$. If $g : E(G) \rightarrow \mathbb{R}$ is a function defined on oriented edges, we can then simply write $g(k)$ instead of $g(k,k+1)$ and the divergence operator $(\nabla \cdot g)(k) = (g(k)-g(k-1)$ can be seen as the left derivative of $g$. We will denote $\nabla g(k) := (\nabla \cdot g)(k)$. Similarly, if $k \rightarrow k+1 \rightarrow k+2$ is an oriented triple, we will write $h(k)$ instead of $h(k,k+1,k+2)$ and $(\nabla_2 \cdot h)(k)$ will be the twice left derivative $\nabla_2 h(k) := h(k)-2h(k-1)+h(k-2)$. With these notations, the Benamou-Brenier condition  is written $$\label{eq:BBCondZ} h_t(k-1) f_t(k) = g_t(k-1) vg_t(k) .$$ The proof of Theorem \[th:RenyiZ\] is based on two technical lemmas: For every triple of non negative numbers $f,g,h$ we have $$\label{eq:HolderLike} h f^{p-1} \leq \frac{1}{2-p} h^{2-p}g^{2p-2}+ \frac{1-p}{2-p} g^2 f^{p-2}.$$ **Proof:** The convexity of the exponential function implies the inequality: $$\forall a,b >0 \ , \ \forall \alpha,\beta >0 \ s.t. \ \frac{1}{\alpha}+\frac{1}{\beta}=1 \ , \ ab \leq \frac{a^\alpha}{\alpha}+ \frac{b^\beta}{\beta},$$ Setting $$\alpha := 2-p \ , \ \beta := \frac{2-p}{1-p} \ , \ a:= h g^{-2\frac{1-p}{2-p}} \ , \ b:= g^{2\frac{1-p}{2-p}} f^{p-1},$$ we obtain as wanted. $\square$ For every $x \geq 0$ we have $$\psi(x) := (1-p)(x-1)^2 - \frac{x^p-1}{2-p}-\frac{1}{2-p}x^{2-p}-\frac{1-p}{2-p}x^2+2x-1 \geq 0.$$ **Proof:** Let us compute the first derivatives of $h$: $$\psi'(x) = 2(1-p)(x-1) -\frac{p}{2-p} x^{p-1} -x^{1-p}-2\frac{1-p}{2-p}x+2,$$ $$\psi''(x) = 2(1-p) + \frac{p(1-p)}{(2-p)} x^{p-2}-(1-p)x^{-p}-2 \frac{1-p}{2-p},$$ $$\psi^{(3)}(x) = p(1-p) (x^{-p-1}-x^{p-3}).$$ As $p<1$, we have $-p-1>p-3$, so $\psi^{(3)}(x)$ is negative for $0 \leq x \leq 1$ and positive for $x\geq 1$. This means that $\psi''(x) \geq \psi''(1)=0$, so $\psi$ is convex. As we have $\psi(1)=0$ and $\psi'(1)=0$, we deduce that $\psi(x) \geq 0$ for every $x \geq 0$. $\square$ **Proof of Theorem \[th:RenyiZ\]:** The second derivative of $H_p(f_t)$ satisfies: $$-H_p''(t) = p \sum_{k \in \mathbb{Z}} \nabla_2 h_t(k) f_t(k)^{p-1} + p(p-1) \sum_{k \in \mathbb{Z}} (\nabla g_t(k))^2 f_t(k)^{p-2}.$$ As $p \in (0,1)$, we have $p(1-p) < 0$. To prove the convexity of $t \mapsto H_p(t)$, we first apply twice the inequality with $(f,g,h)=(f_t(k),g_t(k),h_t(k))$ and $(f,g,h)=(f_t(k),g_t(k-1),h_t(k-2))$ and then change indices to write $$\begin{aligned} \sum_{k \geq 0} \nabla_2(h_t(k)) f_t(k)^{p-1} &\leq& \sum_{k \geq 0} \frac{1}{2-p} h_t(k)^{2-p} g_t(k)^{2p-2}+ \frac{1-p}{2-p} g_t(k)^2 f_t(k)^{p-2}\\ && -2h_t(k-1) f_t(k)^{p-1} \\ &&+ \frac{1}{2-p} h_t(k-2)^{2-p} g_t(k-1)^{2p-2} + \frac{1-p}{2-p} g_t(k-1)^2 f_t(k)^{p-2} \\ &=& \sum_{k \geq 0} \frac{1}{2-p} h_t(k-1)^{2-p} g_t(k-1)^{2p-2}+ \frac{1-p}{2-p} g_t(k)^2 f_t(k)^{p-2}\\ && -2h_t(k-1) f_t(k)^{p-1} \\ &&+ \frac{1}{2-p} h_t(k-1)^{2-p} g_t(k)^{2p-2} + \frac{1-p}{2-p} g_t(k-1)^2 f_t(k)^{p-2} .\end{aligned}$$ We denote $v_{+,t}(k):= \frac{g_t(k)}{f_t(k)}$ and $v_{-,t}(k) :=\frac{g_t(k-1)}{f_t(k)}$. The Benamou-Brenier equation  is then written $h_t(k-1)=v_{+,t}(k)v_{-,t}(k) f_t(k)$ and we have $$\begin{aligned} \sum_{k \geq 0} \nabla_2(h_t(k)) f_t(k)^{p-1} &\leq& \sum_{k \geq 0} \frac{1}{2-p} v_{+,t}(k)^{2-p} v_{-,t}(k)^p f_t(k)^p + \frac{1-p}{2-p} v_{+,t}(k)^2 f_t(k)^p\\ && - 2 v_{+,t}(k)v_{-,t}(k) f_t(k)^p + \frac{1}{2-p} v_{+,t}(k)^p v_{-,t}(k)^{2-p} f_t(k)^p + \frac{1-p}{2-p} v_{-,t}(k)^2 f_t(k)^p.\end{aligned}$$ With the same notations, we have $$(g_t(k)-g_t(k-1))^2 f_t(k)^{p-2} = (v_{+,t}(k)^2-2v_{+,t}(k)v_{-,t}(k)+v_{-,t}(k)^2) f_t(k)^{p},$$ and $$0 = \sum_{k \geq 0} g_t(k)^{p}-g_t(k-1)^{p} = \sum_{k \geq 0} (v_{+,t}(k)^p-v_{-,t}(k)^p)f_t(k)^p.$$ We use these estimations and the positivity of $\psi$ to write $$\begin{aligned} -\frac{1}{p} H_p''(t) &=& \sum_{k \geq 0} \nabla_2(h_t(k)) f_t(k)^{p-1}-(1-p) \sum_{k\geq 0} \nabla (g_t(k))^2 f_t(k)^p \\ && -\frac{1}{2-p} \sum_{k \geq 0} g_t(k)^{p}-g_t(k-1)^{p} \\ &\leq& \sum_{k \geq 0} v_{+,t}(k)^2 f_t(k)^p \psi\left(\frac{v_{-,t}(k)}{v_{+,t}(k)}\right) \\ &\leq& 0,\end{aligned}$$ which finishes the proof of the theorem. $\square$ Binomial mixtures and $W_2$-optimal couplings. ---------------------------------------------- In this appendix we consider two finitely supported probability measures $f_0$ and $f_1$ on $\mathbb{Z}$. Through this article and the previous one (see [@HillionGeodesic]), we have seen that, by considering a mixture of binomial measures with respect to a proper coupling between $f_0$ and $f_1$, it is possible to construct a $W_1$-geodesic $(f_t)_{t \in [0,1]}$ which satisfies a Benamou-Brenier condition  which is a discrete analogue of a characterization of $W_2$-geodesics on the real line. Another natural way to generalize the notion of $W_2$-geodesic from the continuous setting to the discrete setting is the following: Let $\pi$ be the unique $W_2$-optimal coupling between $f_0$ and $f_1$. The binomial/$W_2$ interpolation $(f_t)_{t \in [0,1]}$ is defined by: $$f_t(k) : =\sum_{(i,j)} \pi_{i,j} \operatorname{bin}_{(i,j),t}(k),$$ where $\operatorname{bin}_{(i,j),t}$ is the binomial family between $i$ and $j$. Basic theorems on optimal transportation give the existence and uniqueness of a $W_2$-optimal coupling $\pi$ between $f_0$ and $f_1$. Thus the binomial/$W_2$ interpolation $(f_t)_{t \in [0,1]}$ exists and is unique. The question of the convexity of the entropy along $(f_t)$ is still open. The particular case where $f_1$ is a translation of $f_0$ has been studied by the author in [@HillionTranslation]. In this appendix we prove the more general: \[th:EntroBinoW2\] We make the following assumptions: 1. The measure $f_0$ is stochastically dominated by $f_1$ : $f_0 << f_1$, which means that for each $l \in \mathbb{Z}$, $\sum_{l \leq k} f_0(l) \geq \sum_{l \leq k} f_1(l)$. 2. Each $f_t$ is log-concave, i.e. that the inequality $f_t(k+1)^2 \geq f_t(k)f_t(k+2)$ holds for any $t \in [0,1]$ and $k \in \mathbb{Z}$. Then the entropy $H(t)$ of $f_t$ is a convex function of $t$. The stochastic domination assumption is not necessary but allows us to give a simpler proof. We will use it through the following: \[lem:W2coupling\] We suppose that $f_0 << f_1$. Then the $W_2$-optimal coupling $\pi$ between $f_0$ and $f_1$ satisfies the following: - If $\pi(i,j)>0$ then $i \leq j$. - If $\pi(i_1,j_1)>0$ and $\pi(i_2,j_2)>0$ then $(i_2-i_1)(j_2-j_1) \geq 0$. **Remark.** In particular the stochastic domination assumption allows us to use the same notations $g(k):=g(k,k+1)$, $h(k) := h(k,k+1,k+2)$ as in the first part of the Appendix. **Proof of Theorem \[th:EntroBinoW2\]:** Using the first point of Lemma \[lem:W2coupling\], we can write $$f_t(k) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i).$$ We now define the families of functions $(g_t)_{t \in [0,1]}$ and $(h_t)_{t \in [0,1]}$ by: $$\begin{aligned} g_t(k) &:=& \sum_{i \leq j} \pi(i,j) (j-i) \operatorname{bin}_{(j-i),t}(k-i) \\ h_t(k) &:= & \sum_{i \leq j} (j-i)(j-i-1) \pi(i,j) \operatorname{bin}_{(j-i-2),t}(k-i),\end{aligned}$$ so we have $$\frac{\partial}{\partial t} f_t(k) = -\nabla g_t(k) \ , \ \frac{\partial}{\partial t} g_t(k) = -\nabla h_t(k).$$ The study of the entropy of $f_t$ is similar to case of $W_{1,+}$-interpolations. We have: $$\begin{aligned} H''(t) &=& \sum_{k \in \mathbb{Z}} \nabla_2h_t(k) \log(f_t(k)) + \sum_{k \in \mathbb{Z}} \frac{(\nabla g_t(k))^2}{f_t(k)}.\end{aligned}$$ The major difference with $W_{1,+}$-interpolations comes from the fact that $f_t(k)h_t(k-1)$ is a priori not equal to $g_t(k)g_t(k-1)$. Let us introduce the family of functions $\tilde{h_t}(k) := \frac{g_t(k)g_t(k+1)}{f_t(k+1)}$. The triple $(f_t,g_t,\tilde{h}_t)$ is a BB-triple on $\mathbb{Z}$, so we have: $$\begin{aligned} H''(t) &=& \sum_{k \in \mathbb{Z}} \nabla_2(h_t-\tilde{h}_t)(k) \log(f_t(k))+\sum_{k \in \mathbb{Z}} \nabla_2\tilde{h}_t(k) \log(f_t(k)) + \sum_{k \in \mathbb{Z}} \frac{(\nabla g_t(k))^2}{f_t(k)} \\ &\geq& \sum_{k \in \mathbb{Z}} \nabla_2(h_t-\tilde{h}_t)(k) \log(f_t(k)) \\ &=& \sum_{k \in \mathbb{Z}} (h_t-\tilde{h}_t)(k) \nabla_2 \log(f_t(k+2)).\end{aligned}$$ By the assumption on the log-concavity of $f_t$, it thus suffices to show that $h_t \leq \tilde{h}_t$. to prove this fact, we notice that we can write $g_t(k)$, $g_t(k-1)$ and $h_t(k-1)$ under the form: $$g_t(k) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i) \frac{j-k}{1-t} \ , \ g_t(k) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i) \frac{k-i}{t},$$ $$h_t(k-1) = \sum_{i \leq j} \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i) \frac{(j-k)(k-i)}{t(1-t)}.$$ Let us denote, for $i \leq j$, $a(i,j) := \pi(i,j) \operatorname{bin}_{(j-i),t}(k-i)$. Then $g_t(k)g_t(k-1)-f_t(k)h_t(k-1)$ can be seen as a quadratic form in the variables $(a(i,j))_{(i,j) \in \operatorname{Supp}(\pi)}$. The coefficient associated to $a(i,j)^2$ is $$\frac{j-k}{1-t} \frac{k-i}{t} - \frac{(j-k)(k-i)}{t(1-t)} =0 .$$ If $(i_1,j_1) \neq (i_2,j_2)$ are in $\operatorname{Supp}(\pi)$ then the coefficient associated to $a(i_1,j_1) a(i_2,j_2)$ is $$\begin{aligned} \frac{j_1-k}{1-t}\frac{k-i_2}{t}+\frac{j_2-k}{1-t}\frac{k-i_1}{t}-\frac{(j_1-k)(k-i_1)}{t(1-t)}-\frac{(j_2-k)(k-i_2)}{t(1-t)} &=& \frac{(j_2-j_1)(i_2-i_1)}{t(1-t)} \\&\geq& 0.\end{aligned}$$ This shows that $h_t \leq \tilde{h}_t$, and finishes the proof of Theorem \[th:EntroBinoW2\]. $\square$ [10]{} Jean-David Benamou and Yann Brenier. A numerical method for the optimal time-continuous mass transport problem and related problems. In [*Monge [A]{}mpère equation: applications to geometry and optimization ([D]{}eerfield [B]{}each, [FL]{}, 1997)*]{}, volume 226 of [*Contemp. Math.*]{}, pages 1–11. Amer. Math. Soc., Providence, RI, 1999. Serguei Bobkov and Michel Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. , 156(2), 347-365. Matthias Erbar and Jan Maas. Ricci curvature of finite [M]{}arkov chains via convexity of the entropy. , 206(3):997–1038, 2012. Nathael Gozlan, Cyril Roberto, Paul-Marie Samson and Prasad Tetali. Displacement convexity of entropy and related inequalities on graphs. , 1-48, 2012. Erwan Hillion. Concavity of entropy along binomial convolutions. , 17:1–9, 2012. Erwan Hillion. Contraction of measures on graphs. , 2014 DOI 10.1007/s11118-014-9388-7 Erwan Hillion. $W_{1,+}$-interpolation of probability measures on graphs. , 2014 DOI 10.1007/s11118-014-9388-7 Christian Léonard. On the convexity of the entropy along entropic interpolations. . Christian Léonard. Lazy random walks and optimal transport on graphs. , arXiv:1308.0226. Christian L[é]{}onard. A survey of the Schrödinger problem and some of its connections with optimal transport. . 34 (2014), no. 4, 1533–1574. John Lott and C[é]{}dric Villani. Ricci curvature for metric-measure spaces via optimal transport. , 169(3):903–991, 2009. Yann Ollivier. Ricci curvature of Markov chains on metric spaces. , 256(3), 810-864. Karl-Theodor Sturm. On the geometry of metric measure spaces. [I]{}. , 196(1):65–131, 2006. Karl-Theodor Sturm. On the geometry of metric measure spaces. [II]{}. , 196(1):133–177, 2006. Cédric Villani. Topics in optimal transportation. , vol. 58. Cédric Villani. Optimal transport: old and new. , vol. 338. [^1]: Department of Mathematics, University of Luxembourg, erwan.hillion@uni.lu

--- abstract: | Hyperfine structures of the triplet $n^3S-$states in the four-electron Be-atom(s) and Be-like ions are considered. It is shown that to determine the hyperfine structure splitting in such atomic systems one needs to know the triplet electron density at the central atomic nucleus $\rho_T(0)$. We have developed the procedure which allows allows one to determine such an electron density $\rho_T(0)$ for arbitrary four-electron atoms and ions. PACS number(s): 31.30.Gs, 31.15.vj and 32.15.Fn author: - 'Alexei M. Frolov' title: 'On the hyperfine structure of the triplet $n^{3}S-$states of the four-electron atoms and ions' --- Introduction ============ In this communication we develop the new ab-initio method which can be applied for accurate evaluation of the hyperfine structure splittings in the triplet ${}^3S-$states of the four-electron atoms and/or ions. As follows from experiments such triplet $S(L = 0)-$states have an interesting hyperfine structure. For simplicity, let us consider, the triplet $2^3S-$state of the four-electron beryllium atom(s) (${}^{7}$Be, ${}^{9}$Be and ${}^{\infty}$Be). In general, if $F$ is the total electron-nuclear spin of the $2^3S-$state of an atom and $I_N$ is the spin of atomic nucleus (Be) and $I_N \ge 1$, then in experiments one can observe splitting of this state into a triplet of states. The total spin of these states equals $F = I_N + 1, I_N$ and $I_N - 1$, respectively. If $I_N = \frac12$, then we can see only a doublet of states with $F = \frac12$ and $\frac32$. This method can also be used to determine the hyperfine structure splitting for an arbitrary bound (triplet) $n^3S-$states in four-electron atoms and ions. This includes the triplet $2^3S-$state of the four-electron Be atoms (different isotopes). Our analyisis of the hyperfine structure of the four-electron atoms and ions is based on the generalization of the method developed earlier by Fermi [@Fermi] for the doublet $n^{2}S-$states of three-electron Li-atom(s) and Li-like ions (see also [@Fro2016]). First, we need to introduce the triplet electron density in a few-electron atom/ion. Formally, the triplet electron density is the spatial two-electron density distribution of the two atomic electrons which form one triplet pair. If we have a number of such pairs in atom/ion, then we need take into account all triplet electron pairs. Singlet electron pairs do not contribute to the triplet electron density. In reality, for accurate evaluations of the hyperfine structure splitting in the triplet states of four-electron atoms/ions one needs to know the triplet electron density at the central atomic nucleus which has non-zero electric charge $Q e$. The general definition of the electron density at the central atomic nucleus is written in from $\rho(0) = \sum^{N_e}_{i=1} \langle \delta({\bf r}_{iA}) \rangle$, where $i$ is the electron’s index, while index $A$ means the central atomic nucleus. For instance, for the singlet ground $1^1S-$state in the two-electron helium atom one finds $\rho_{S}(0) \approx$ 1.8104293185013928, while for the triplet $2^3S-$state of the helium-3 atom we have $\rho_{T}(0) \approx$ 1.31963500836957 [@Fro2007]. The last numerical value leads to the following hyperfine structure splitting in the $2^3S-$state of the ${}^{3}$He atom: $\Delta E_{hss}$ = 6740.452154 $MHz$ [@Fro2007] (see also [@BS]). The corresponding experimental value is $\Delta E_{hss}$ = 6739.701171(16) $MHz$ [@PRA70]. However, such a definition of the electron density cannot be used for the triplet states in atoms/ions, if the total number of bound electrons exceeds two. The reason is obvious, since all atoms/ions with more than two bound electrons always have the shell electronic structure. This means that the internal electrons form a number of closed electron shells which have zero spin, i.e. singlet electron shells. The electrons from the outer-most shell(s) can interact with the nuclear spin $I_N$, its numerical value it differs from zero. In general, this leads to the appearance of the hyperfine structure splitting in atoms and ions, if the total spin of outer-most electrons exceeds zero. This leads to the appearance of the hyperfine structure splitting in $N_e-$electron atoms and ions, where $N_e \ge 3$. It is clear that in actual atoms and ions we have small interactions between electrons from internal and outermost electron shells, e.g., electronic correlations, spin-spin interactions, etc. As follows from this picture the analysis and numerical computations of the hyperfine structure splitting are significantly more complicated than for the two-electron helium atom. In the lowest-order approximation we need to define the triplet electron density at the atomic nucleus must be given in a different manner. An alternative definition of the triplet electron density at the atomic nucleus ($A$) can be written in the form (see, e.g., [@Fermi], [@Fro2016], [@Lars]) $$\rho_T(0) = \sum^{N_e}_{i=1} \langle \delta({\bf r}_{iA}) (\sigma_{z})_i \rangle = \langle \Psi \mid \sum^{N_e}_{i=1} \delta({\bf r}_{iA}) (\sigma_{z})_i \mid \Psi \rangle \; \; \; \label{eq1}$$ where $\delta({\bf r}_{iA})$ is the electron-nucleus delta-function (the symbol $A$ designates the atomic nucleus) and $(\sigma_{z})_i$ is the $\sigma_z$ matrix of the $i$-th atomic electron, i.e. $(\sigma_{z})_i \alpha(i) = \alpha(i)$ and $(\sigma_{z})_i \beta(i) = - \beta(i)$ (see, e.g, [@LLQ], [@Dirac]). In Eq.(\[eq1\]) and everywhere below we assume that the wave function of the bound $2^{3}S-$state of the four-electron atom/ion has unit norm. As follows from Eq.(\[eq1\]) the triplet electron density $\rho_T(0)$ equals zero identically for an arbitrary singlet state in a few-electron atom, including the two-electron He atom. For the triplet states in the helium-3 atom the new definition of the electron density, Eq.(\[eq1\]), leads to the same hyperfine structure splitting as mentioned above. For the ground $2^{2}S-$state in the three-electron Li atoms and analogous ions such a definition of the doublet electron density at the central atomic nucleus, Eq.(\[eq1\]), allows one to evaluate the correct numerical values of the hyperfine structure splittings (see, e.g., [@Fro2016]) which are in good agreement with the known experimental values [@Liatom]. By using this definition of the triplet electron density at the atomic nucleus, Eq.(\[eq1\]), we can write the following formula (Fermi-Segré formula (see, e.g., [@LLQ])) for the hyperfine structure splitting of the $2^3S-$states in the Be atom (see, e.g., [@LLQ]) $$\Delta E_{hf} = \frac{8 \pi \alpha^2}{3} \mu_B \mu_N g_e g_N \; \; \rho_T(0) \; \; \frac12 [F(F + 1) - I_N(I_N + 1) - S_e(S_e + 1)] \; \; \label{eq2}$$ where ${\bf S}_{e}$ is the total electron spin of the atom, ${\bf I}_{N}$ is the spin of the nucleus in those isotopes of the Be atom(s) for which $\mid {\bf I}_{N} \mid \ne 0$ and ${\bf F}$ is the total angular momentum operator ${\bf F} = {\bf L} + {\bf S} = {\bf S}_e + {\bf I}_N$ of the four-electron atom/ion. For the triplet $S-$states in the four-electron atoms/ions the vector-operator ${\bf F} = {\bf S}_e + {\bf I}_N$ can be considered as the total spin of the atom, i.e. the sum of the electron and nuclear spins. Also, in this formula $\alpha \approx 7.297352568 \cdot 10^{-3} (\approx \frac{1}{137})$ is the dimensionless fine structure constant, $\mu_B$ is the Bohr magneton ($\mu_B = \frac12$ in atomic units) and $\mu_N = \mu_B \frac{m_e}{m_p}$, where $\frac{m_p}{m_e}$ = 1836.15267261 is the ratio of the proton and electron masses. The notation $g_e$ in Eq.(\[eq2\]) means the electron gyromagnetic ratio $g_e$ = -2.00223193043718 [@CRC]. The factor $g_N$ for the ${}^9$Be nucleus is $g_{N} = \frac{f_N}{I_N} \approx$ -1.177432 $\Bigl(\frac23\Bigr)$ = -0.7849547, since $f_N = -1.177432$ and $I_N = \frac32$. Finally, the formula for the hyperfine structure splitting $\Delta E_{hf}$ in the $2^3S-$state of the four-electron ${}^{7}$Be and ${}^{9}$Be atoms takes the form $$\begin{aligned} \Delta E_{hf}(MHz) = 314.061338965 \; \rho_T(0) \; [F(F + 1) - I_N(I_N + 1) - S_e(S_e + 1)] \label{eq3}\end{aligned}$$ where the factor 6.579 683 920 61$\cdot 10^9$ ($MHz/a.u.$) has been used to re-calculate the $\Delta E_{hf}$ energy from atomic units to $MegaHertz$. The Fermi-Segré formula, Eq.(\[eq2\]) and Eq.(\[eq3\]), is correct for all $S-$bound triplet states in the four-electron Be-atom(s) and Be-like ions. The same formula is applied to other four-electron atoms and ions which can be found in the bound triplet $S-$states. As follows from Eq.(\[eq3\]) in order to determine the numerical value of the hyperfine structure splitting $\Delta E_{hf}(MHz)$ one needs to evaluate the electron triplet density at the atomic nucleus $\rho_T(0)$. Accurate numerical evaluation of the $\rho_T(0)$ value is the main goal of this study. To perform such an evaluation we need to construct the accurate wave functions of the triplet $S-$states of the four-electron atoms and ions. In general, these wave functions are obtained as the solutions of the corresponding Schrödinger equation for the bound atomic state(s), i.e. $H \Psi = E \Psi$, where $H$ is the Hamiltonian operator defined below (see Eq.(\[Hamil\])) and $\Psi$ is the wave function and $E (< 0)$ is the total energy of the bound atomic state. This problem is considered in detail in the next Section. Hamiltonian and bound state wave functions ========================================== In the lowest-order approximation upon the fine-structure constant $\alpha$ we can consider the non-relativistic Schrödinger equation. The non-relativistic Hamiltonian $H$ of an arbitrary four-electron atomic system (i.e. atom, or ion) is written in the form [@LLQ] $$\begin{aligned} H = -\frac{\hbar^2}{2 m_e} \Bigl[ \sum^{4}_{i=1} \nabla^2_i + \frac{1}{M_A} \nabla^{2}_{5} \Bigr] - \sum^4_{i=1} \frac{Q e^2}{r_{i5}} + \sum^{3}_{i=1} \sum^{4}_{j=2 (j>i)} \frac{e^2}{r_{ij}} \label{Hamil}\end{aligned}$$ where $\hbar$ is the reduced Planck constant, $m_e$ is the electron mass, $e$ is the absolute value of the electric charge of the electron. Also, in this equation $Q e$ and $M_A$ are the electric charge and mass of the nucleus ($M_n \gg 1$) expressed in $e$ and $m_e$, respectively. Below, we consider the beryllium atom with the infinitely heavy atomic nucleus, i.e. when $M_A = \infty$, or $\frac{1}{M_A} = 0$ in Eq.(\[Hamil\]). We also discuss a few isotopes of the Be-atom with the finite nuclear masses $M_A$. In Eq.(\[Hamil\]) and everywhere below in this study the subscript 5 denotes the atomic nucleus, while subscripts 1, 2, 3 and 4 stand for electrons. Note that the four-electron Be atom has two independent series of bound states: singlet states and triplet states. The multiplicities of these states equal $2 \cdot 0 + 1 = 1$ (singlet) and $2 \cdot 1 + 1 = 3$ (triplet). Below, we consider only the triplet bound states in the Be atom(s) and Be-like ions. To determine the bound state wave function of the Be atom in its $2^{3}S-$state we need to solve the corresponding Schrödinger equation for the bound state(s): $H \Psi = E \Psi$, where $H$ is the Hamiltonian operator from Eq.(\[Hamil\]), while $E (< 0)$ is the total energy of the $2^{3}S-$state in the Be-atom. It is clear that the numerical value of $E$ must be lower than the total energy of the ground $2^2S-$state of the three-electron Be$^{+}$ ion $E \approx$ -14.3247631764657 $a.u.$ (otherwise, the $2^{3}S-$state in the Be-atom will be unstable, i.e. unbound). Now, consider the explicit construction of the trial wave function $\Psi$. In general, the wave functions of the bound $n^3S$-states in the Be atom are represented as the sum of products of the radial and spin functions. Each of these radial and/or spin functions depends upon spatial and spin coordinates of all four electrons. For the triplet states we can use only spin functions with $S = 1$ and $S_z = 1$, where $S$ is the total electron spin, i.e. ${\bf S} = {\bf s}_1 + {\bf s}_2 + {\bf s}_3 + {\bf s}_4$, of four-electrons and $S_z$ is its $z-$projection. Therefore, our spin function $\chi_{11}(1,2,3,4)$ is defined by the following equalities: ${\bf S}^2 \chi_{11}(1,2,3,4) = 1 (1 + 1) \chi_{11}(1,2,3,4) = 2 \chi_{11}(1,2,3,4)$ and $S_z \chi_{11}(1,2,3,4) = \chi_{11}(1,2,3,4)$. In general, there are two spin functions for each four-electron atom/ion in the triplet state. Below, we chose such functions in the form $\chi^{(1)}_{11} = \alpha \beta \alpha \alpha - \beta \alpha \alpha \alpha$ and $\chi^{(2)}_{11} = 2 \alpha \alpha \beta \alpha - \beta \alpha \alpha \alpha - \alpha \beta \alpha \alpha$. Finally, the total four-electron wave function of the triplet states in four-electron atoms and ions is represented in the form $$\begin{aligned} \Psi = {\cal A}_e [\psi(A;\{r_{ij}\}) (\alpha \beta \alpha \alpha - \beta \alpha \alpha \alpha)] + {\cal A}_e [\phi(B;\{r_{ij}\}) (2 \alpha \alpha \beta \alpha - \beta \alpha \alpha \alpha - \alpha \beta \alpha \alpha)] \label{equat2}\end{aligned}$$ where the notation $\{r_{ij}\}$ designates the ten relative coordinates (electron-nuclear and electron-electron coordinates) in the four-electron Be atom, while the notation ${\cal A}_e$ means the complete four-electron antisymmetrizer. The explicit formula for the ${\cal A}_e$ operator is $$\begin{aligned} {\cal A}_e = \hat{e} - \hat{P}_{12} - \hat{P}_{13} - \hat{P}_{23} - \hat{P}_{14} - \hat{P}_{24} - \hat{P}_{34} + \hat{P}_{123} + \hat{P}_{132} + \hat{P}_{124} + \hat{P}_{142} + \hat{P}_{134} + \hat{P}_{143} \nonumber \\ + \hat{P}_{234} + \hat{P}_{243} - \hat{P}_{1234} - \hat{P}_{1243} - \hat{P}_{1324} - \hat{P}_{1342} - \hat{P}_{1423} - \hat{P}_{1432} + \hat{P}_{12} \hat{P}_{34} + \hat{P}_{13} \hat{P}_{24} + \hat{P}_{14} \hat{P}_{23} \label{equat3}\end{aligned}$$ Here $\hat{e}$ is the identity permutation, while $\hat{P}_{ij}$ is the permutation of the spin and spatial coordinates of the $i-$th and $j-$th identical particles. Analogously, the notations $\hat{P}_{ijk}$ and $\hat{P}_{ijkl}$ stand for the consequtive permutations of the spin and spatial coordinates of the three and four identical particles (electrons). In real calculations one needs to know the explicit expressions for the spatial projectors only. These spatial projectors can be obtained, e.g., by applying the ${\cal A}_e$ operator to each component of the wave function in Eq.(\[equat2\]). At the second step we need to determine the scalar product (or spin integral) of the result and incident spin function. After the integration over all spin variables one finds the corresponding spatial projector. For instance, in the case of the first term in Eq.(\[equat2\]) we obtain the following spatial projector for the $\psi-$components of the total wave function $$\begin{aligned} {\cal P}_{\psi\psi} = \frac{1}{2 \sqrt{6}} (2 \hat{e} + 2 \hat{P}_{12} - \hat{P}_{13} - \hat{P}_{23} - \hat{P}_{14} - \hat{P}_{24} - 2 \hat{P}_{34} - 2 \hat{P}_{12} \hat{P}_{34} - \hat{P}_{123} - \hat{P}_{124} - \hat{P}_{132} \nonumber \\ - \hat{P}_{142} + \hat{P}_{134} + \hat{P}_{143} + \hat{P}_{234} + \hat{P}_{243} + \hat{P}_{1234} + \hat{P}_{1243} + \hat{P}_{1342} + \hat{P}_{1432}) \label{equat4}\end{aligned}$$ Analogous formulas have been found [@FroWa2010] for two other spatial projectors ${\cal P}_{\psi\phi} = {\cal P}_{\phi\psi}$ and ${\cal P}_{\phi\phi}$. These formulas for the ${\cal P}_{\psi\phi} = {\cal P}_{\phi\psi}$ and ${\cal P}_{\phi\phi}$ spatial projectors are significantly more complicated and they are not presented here (they can be found, e.g., in [@FroWa2010]). In actual bound state calculations we can always restrict ourselves to one spin function $\chi^{(1)}_{11}$ (or one spin configuration) and use the formula, Eq.(\[equat4\]). The functions $\psi(A;\{r_{ij}\})$ and $\phi(B;\{r_{ij}\})$ in Eq.(\[eq2\]) are the radial parts (or components) of the total wave function $\Psi$. For the bound states in various five-body systems these functions are approximated with the use of the KT-variational expansion written in ten-dimensional gaussoids [@KT]. Each of the spatial basis function in the KT expansion depends upon ten relative coordinates $r_{ij}$ only [@KT]. Here and everywhere below the notation $r_{ij} = \mid {\bf r}_i - {\bf r}_j \mid = r_{ji}$ means the $(ij)-$relative coordinate, i.e. the scalar distance between the particles $i$ and $j$ (${\bf r}_i$ are the corresponding Cartesian coordinates of the $i$-th particle). For instance, for the $\psi(A;\{r_{ij}\})$ function we have $$\begin{aligned} \psi(A;\{r_{ij}\}) = {\cal P} \sum^{N_A}_{k=1} C_K \exp(-\sum_{ij} a_{ij} r^{2}_{ij}) \label{equat5}\end{aligned}$$ where $N_A$ is the total number of basis function used in radial expansion, $C_k$ are the linear variational coefficients and ${\cal P} = {\cal P}_{\psi\psi}$ is the spatial projector defined by Eq.(\[equat4\]). The notations $A$ (or notations $A$ and $B$ in Eq.(\[equat2\])) stands for the corresponding set of the non-linear parameters $\{ a^{(k)}_{ij} \}$ (and $\{ b^{(k)}_{ij} \}$) in the radial wave functions, Eq.(\[equat5\]) (or Eq.(\[eq2\])). In actual calculations these two sets of non-linear parameters are optimized independently of each other. In general, the KT-variational expansion was found to be very effective for various few-body systems known in atomic, molecular and nuclear physics. A large number of fast algorithms have been developed recently for optimization of the non-linear parameters in the trial wave functions, Eq.(\[equat5\]), allow one to approximate the total energies $E$ and variational wave functions $\Psi$ to high and very high accuracy. The knowledge of the highly accurate wave function can be used to determine a large number of bound state properties. For the $2^{3}S-$state of the four-electron beryllium atom (${}^{\infty}$Be) some of the computed bound state properties expressed in atomic units can be found in Table I. The overall accuracy obtained for these expectation values is high, but there is a general problem related to the shell electronic structure of all few-electron atoms/ions, where the total number of bound electrons exceeds two. Indeed, in all current procedures the sums of all electron-nuclear and electron-electron expectation values are devided by the factors $N_e$ and $N_e (N_e + 1)/2$, respectively, where $N_e$ is the total number of bound electrons. Finally, all traces of the shell electronic structure in few-electron atoms/ions are lost from the results of such calculations. It is clear that few-body logic does not work well for actual atoms/ions with multi-shell electronic structure. This situation must be corrected in the future. Hyperfine structure splitting ============================= Let us evaluate the hyperfine structure splitting for the triplet $2^{3}S-$state of the ${}^{9}$Be atom. In our calculations we shall apply the numerical value of the electron density $\rho_T(0)$ at the atomic nucleus determined with the use of the formula $$\begin{aligned} \rho_T(0) = {\cal C} \langle {\cal A}_s (\Psi \chi^{(1)}_{11}) \mid \Bigl[ \sum^{4}_{i=1} \delta({\bf r}_{iA}) (\sigma_{z})_i \Bigr] \Psi \chi^{(1)}_{11} \rangle \; \; \; \label{eq7}\end{aligned}$$ where ${\cal C}$ is a normalization constant. Numerical computations of the overlap integrals between spin-functions included in Eq.(\[eq7\]) is significantly more complicated than for three-electron atomic systems. Indeed, the total number of terms in the left-hand side wave function of the Eq.(\[eq7\]) equals 24 and each of these terms must be multiplied by four (number of the electron-nucleus delta-functions). This means that we have 96 terms which contribute to the numerical value of the triplet electron density at the central atomic nucleus $\rho_T(0)$. Formally, it is difficult to present here all details of analytical computations of the electron density $\rho_T(0)$. However, we can illustrate such computations by considering the two terms which can be found in Eq.(\[eq7\]). First, consider the term in Eq.(\[eq7\]) which contains the permutation operator $\hat{P}_{13}$ (see, Eq.(\[equat4\]). Action of this operator on the spin function $\chi^{(1)}_{11} = \alpha \beta \alpha \alpha - \beta \alpha \alpha \alpha$ produces the function $\alpha \beta \alpha \alpha - \alpha \alpha \beta \alpha$. On the other hand, the explicit expression in the right-hand side of Eq.(\[eq7\]) takes the form $$\begin{aligned} \Bigl[ \sum^{4}_{i=1} \delta({\bf r}_{iA}) (\sigma_{z})_i \Bigr] \Psi \chi^{(1)}_{11} &=& \frac12 \delta({\bf r}_{1A}) (\alpha \beta \alpha \alpha + \beta \alpha \alpha \alpha) - \frac12 \delta({\bf r}_{2A}) (\alpha \beta \alpha \alpha + \beta \alpha \alpha \alpha) \nonumber \\ &+& \frac12 \delta({\bf r}_{3A}) (\alpha \beta \alpha \alpha - \beta \alpha \alpha \alpha) + \frac12 \delta({\bf r}_{4A}) (\alpha \beta \alpha \alpha - \beta \alpha \alpha \alpha) \Psi \label{eq71}\end{aligned}$$ where $\Psi$ is the spatal part of the total wave function. The following integration over spin variables leads to the formula for the matrix elements $$\begin{aligned} \frac12 \Bigr[ \langle \hat{P}_{13} \Psi_1 \mid \delta({\bf r}_{1A}) \Psi_2 \rangle - \langle \hat{P}_{13} \Psi_1 \mid \delta({\bf r}_{2A}) \Psi_2 \rangle + \langle \hat{P}_{13} \Psi_1 \mid \delta({\bf r}_{3A}) \Psi_2 \rangle + \langle \hat{P}_{13} \Psi_1 \mid \delta({\bf r}_{4A}) \Psi_2 \rangle \Bigr] \label{eq73}\end{aligned}$$ which contains only integrals over spatial variables. Analogously, the term which include the $\hat{P}_{12} \hat{P}_{34}$ permutation operator produces the result $$\begin{aligned} \hat{P}_{12} \hat{P}_{34} \chi^{(1)}_{11} = - \alpha \beta \alpha \alpha + \beta \alpha \alpha \alpha \label{eq75}\end{aligned}$$ which leads to the following formula $$\begin{aligned} - \langle \hat{P}_{12} \hat{P}_{34} \Psi_1 \mid \delta({\bf r}_{3A}) \Psi_2 \rangle - \langle \hat{P}_{12} \hat{P}_{34} \Psi_1 \mid \delta({\bf r}_{4A}) \Psi_2 \rangle \label{eq77}\end{aligned}$$ The explicit integration over electron spin variables of all other terms in Eq.(\[eq7\]) can be performed analogously. Note that the expectation values of some of the terms in Eq.(\[eq7\]) equal zero identically, e.g., for the $\hat{P}_{1324}$ and $\hat{P}_{1342}$ permutation operators, but the final formula for the $\rho_T(0)$ value still contains many dozens of terms. It is clear that the same formulas for the arising spatial projectors can also be applied for various variational expansions used for accurate calculations of the four-electron Be atom, e.g., for the exponential variational expansion and/or for the Hylleraas expansion. In other words, the multi-dimensional gaussoids used in this study is only one possible choice of the radial basis wave functions. In our calculations of the $2^3S$-state in the four-electron Be atom we have found the following numerical value of the triplet electron density at the atomic nucleus $\rho_T(0) \approx 0.7404721$. With this value Eq.(\[eq3\]) can be written in the form $$\begin{aligned} \Delta E_{hf}(MHz) &\approx& 232.553659 \; [F(F + 1) - I_N(I_N + 1) - S_e(S_e + 1)] = 232.553659 \nonumber \\ & & \times [F(F + 1) - \frac{23}{4}] \label{eq95}\end{aligned}$$ for the $2^3S-$state in the ${}^{9}$Be atom, where $f_N = -1.177432$ and $I_N = \frac32$. From this equation one finds that for the $2^3S-$state in the ${}^{9}$Be atom the hyperfine structure levels are $\varepsilon(F = \frac12)$ = -1162.768295 $MHz$, $\varepsilon(F = \frac32)$ = -465.107318 $MHz$ and $\varepsilon(F = \frac52)$ = 697.660978 $MHz$, respectively. These three levels with different energies determine the hyperfine structure of the ${}^{9}$Be atom. The differences between them are the corresponding hyperfine structure splittings. Analogously, for the ${}^{7}$Be atom, where $f_N = -1.39928$ and $I_N = \frac32$ one finds the hyperfine structure levels are $\varepsilon(F = \frac12)$ = -1381.853407 $MHz$, $\varepsilon(F = \frac32)$ = -552.741363 $MHz$ and $\varepsilon(F = \frac52)$ = 829.112044 $MHz$, respectively. Conclusion ========== We have developed the new ab-initio method which can be used to determine the hyperfine structure splitting of the bound triplet bound $S-$states of four-electron atoms and ions. Our method is based on the explicit derivation of the analytical formula for the operator $\sum^{4}_{i=1} \delta({\bf r}_{iA}) (\sigma_{z})_i$ in the case of the four-electron spin function $\chi^{(1)}_{11} = \alpha \beta \alpha \alpha - \beta \alpha \alpha \alpha$. This allows us to evaluate the hyperfine structure splitting in the triplet $2^{3}S-$state of the ${}^{9}$Be and ${}^{7}$Be atoms. It is clear that our method used in calculations of the hyperfine structure splitting for the triplet $2^{3}S-$state of the Be atom can easily be generalized to other four-electron atoms and ions which have at least one stable ${}^{3}S$-state. Unfortunately, at this time the direct comparison with the experimental results for the triplet $2^{3}S-$state in the ${}^{9}$Be and ${}^{7}$Be atoms is not possible, since the hyperfine structure splittings for the $2^{3}S-$state in the ${}^{9}$Be and ${}^{7}$Be atoms have never been measured. On the other hand, we note that a large number of experiments have been performed to observe the hyperfine structure splitting of various $P-, D-$ and other rotationally excited states in the Be atoms and Be-like ions (see, e.g., [@PhysRevA1] - [@CPL] and references therein). Similar experiments for the triplet $2^{3}S-$state in the ${}^{9}$Be and ${}^{7}$Be atoms are urgently needed, since their results can help to correct an additional factor(s) used in the formula, Eq.(\[eq95\]). Such factors can arise, since there is an obvious difference between the doublet $2^{2}S-$state of the Li atom and the triplet $2^{3}S-$state of the Be atom. Briefly this means that in contrast with the Li atom, in the four-electron Be-atom there are two interacting outer-most electrons and each of these electrons contribute to the actual hyperfine structure splitting. Note also that there are a few steps in our procedure which must be improved in the future computations. First, we need to improve the current accuracy of our wave functions. This means the better numerical accuracy of the wave functions constructed from the multi-dimensional (or ten-dimensional) gaussoids. It would be nice to use other basis sets of spatial functions in such calculations, since this can drastically improve the overall accuracy of the whole procedure. Second, in our current computations the second spin function $\chi^{(2)}_{11} = 2 \alpha \alpha \beta \alpha - \beta \alpha \alpha \alpha - \alpha \beta \alpha \alpha$ is not used. Very likely, this also reduces our overall accuracy even further. Nevertheless, this study indicates clearly that direct computation of the hyperfine structure splitting of the bound triplet $n^{3}S-$states of the four-electron atoms and ions are possible, since the corresponding analytical expression for the triplet electron density at the atomic nucleus has been derived. In the future our procedure will be modified to include two (or more) spin functions. Acknowledgments =============== This work was supported in part by the NSF through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics (ITAMP) at Harvard University and the Smithsonian Astrophysical Observatory. Also, I wish to thank James Babb (ITAMP) and David M. Wardlaw for stimulating discussion. [01]{} E. Fermi, Zeits. für Physik [**60**]{}, 320 (1930). A.M. Frolov, JETP Lett. [**103**]{}, 739 (2016). A.M. Frolov, J. Chem. Phys. [**126**]{}, 104302 (2007). H.A. Bethe and E.E. Salpeter, *Quantum Mechanics of One- and Two-Electron Atoms* (Plenum Publ. Corp., New York, 1977), §40. S.D. Rosner and A.M. Pipkin, Phys. Rev. A [**1**]{}, 571 (1970). S. Larsson, Phys. Rev. **169**, 49 (1968). L.D. Landau and E.M. Lifshitz, [*Quantum Mechanics: Non-Relativistic Theory*]{}, (3rd. ed. Pergamon Press, New York (1976)), Chpt. VI. P.A.M. Dirac, *The Principles of Quantum Mechanics* (4th ed., Oxford at the Clarendon Press, Oxford (UK) (1958)). R.G. Schlecht and D.W. McColm, Phys. Rev. **142**, 11 (1966). *CRC Handbook of Chemistry and Physics*, 95th Edition, Ed. W.M. Haynes, (Taylor and Francis Group, Boca Raton, FL, (2014-2015)). A.M. Frolov and David M. Wardlaw, JETP [**138**]{}, 5 - 15 (2010). N.N. Kolesnikov and V.I. Tarasov, Yad. Phys. **35**, 609 (1982) \[Sov. Phys. Nucl. Phys. **35**, 354 (1982)\]. P. Jönson and C. Froese Fischer, Phys. Rev. A **48**, 4113 (1993). S.N. Ray, T. Lee, and T.P. Das, Phys. Rev. A, **7**, 1469 (1973). C. Chen, J. At., Mol. and Opt. Phys., article ID 569876, 6 pp (2012). D.R. Beck and C.A. Nicolaides, Int. J. Quant. Chem. **26**, suppl. 18, 467 (1984). D. Sundholm and J. Olsen, Chem. Phys. Lett. **177**, 91 (1991). atom/ion $\langle r^{-2}_{eN} \rangle$ $\langle r^{-1}_{eN} \rangle$ $\langle r_{eN} \rangle$ $\langle r^2_{eN} \rangle$ $\langle r^3_{eN} \rangle$ $\langle r^4_{eN} \rangle$ ----------------- ------------------------------- ----------------------------------- ----------------------------------- ------------------------------- ------------------------------- -------------------------------- Be ($N$ = 800) 14.274895 2.0359815 2.6332278 17.19102 148.569 1472.9 Be ($N$ = 1000) 14.274896 2.0359818 2.6332253 17.19098 148.569 1472.9 Be ($N$ = 1200) 14.274897 2.0359820 2.6332235 17.19095 148.568 1472.8 atom/ion $\langle r^{-2}_{ee} \rangle$ $\langle r^{-1}_{ee} \rangle$ $\langle r_{ee} \rangle$ $\langle r^2_{ee} \rangle$ $\langle r^3_{ee} \rangle$ $\langle r^4_{ee} \rangle$ Be ($N$ = 800) 1.50343930 0.6192542 4.7138462 35.22448 320.596 3287.4 Be ($N$ = 1000) 1.50343917 0.6192544 4.7138420 35.22442 320.595 3287.4 Be ($N$ = 1200) 1.50343901 0.6192548 4.7138398 35.22435 320.593 3287.4 atom/ion $E$ $\langle \frac12 p^2_{e} \rangle$ $\langle \frac12 p^2_{N} \rangle$ $\langle \delta_{eN} \rangle$ $\langle \delta_{ee} \rangle$ $\langle \delta_{eee} \rangle$ Be ($N$ = 800) -14.430029018 3.60753776 14.89301041 8.740425 0.265321 0.0 Be ($N$ = 1000) -14.430029235 3.60753825 14.89301164 8.740425 0.265321 0.0 Be ($N$ = 1200) -14.430029456 3.60753933 14.89301235 8.740427 0.265319 0.0 : The expectation values of a number of electron-nuclear ($en$) and electron-electron ($ee$) properties (in $a.u.$) of the $2^{3}S-$state of the neutral Be (${}^{\infty}$Be) atom.

--- abstract: 'Spectroscopic studies of electronic phenomena in graphene are reviewed. A variety of methods and techniques are surveyed, from quasiparticle spectroscopies (tunneling, photoemission) to methods probing density and current response (infrared optics, Raman) to scanning probe nanoscopy and ultrafast pump-probe experiments. Vast complimentary information derived from these investigations is shown to highlight unusual properties of Dirac quasiparticles and many-body interaction effects in the physics of graphene.' author: - 'D. N. Basov' - 'M. M. Fogler' - 'A. Lanzara' - Feng Wang - 'Yuanbo Zhang ()' title: '*Colloquium*: Graphene spectroscopy' --- [UTF8]{}[gbsn]{} Introduction {#sec:Introduction} ============ Scope of this review -------------------- {width="6.00in"} Graphene is a single atomic layer of $sp^2$-hybridized carbon atoms arranged in a honeycomb lattice. This two-dimensional (2D) allotrope of carbon is characterized by a number of superlative virtues [@Geim2009gsa], e.g., a record-high electronic mobility at ambient conditions [@Morozov2008gic], exceptional mechanical strength [@Lee2008mot], and thermal conductivity [@Balandin2008stc; @Ghosh2008eht] Remarkable properties of graphene have ignited tremendous interest that resulted in approximately 50,000 publications at the time of writing. A number of authoritative reviews[^1] have been written to survey this body of literature but no single review can any longer cover the entire topic. The purpose of this Colloquium is to overview specifically the spectroscopic experiments that have helped to shape the modern understanding of the physical properties of graphene. While selected topics in graphene spectroscopy have been discussed,[^2] here we aim to present a panoramic view of physical phenomena in graphene emerging from both spectroscopy and imaging (Fig. \[fig:1.2.1\]C). Spectroscopic observables can be formally categorized as either quasiparticle or current/density response functions. The former are fermionic, the latter are bosonic. The former is traditionally measured by photoemission and tunneling spectroscopy, while the latter can be investigated by, e.g., optical spectroscopy. Yet it may be possible to infer both quasiparticle and collective properties from the same type of measurements. For example, fine anomalies of the quasiparticle spectra seen in photoemission can give information about interactions between quasiparticles and collective modes (Sec. \[sec:e-ph-pl\]) Conversely, optical conductivity, which is a collective response, enables one to infer, with some approximation, the parameters of a quasiparticle band-structure (Secs. \[sec:direct\], \[sec:renormalization\], \[sec:Landau\], and \[sec:BLG\]). Finding such connections is facilitated by spectacular tunability of graphene. For example, with photoemission or tunneling techniques one can monitor the chemical potential $\mu$ of graphene as a function of the electron concentration $N$ and thereby extract the thermodynamic density of states. The same physical quantity can be measured by a very different technique, the scanning single-electron transistor microscopy. In our analysis of such complementary data we focus on what we believe are the most pressing topics in the physics of graphene, e.g., many-body effects. Additionally, our review covers information obtained by scanned probes and out-of-equilibrium methods that greatly expand available means to study graphene in space and time domains. Finally, we briefly address phenomena that arise when physical properties of graphene are altered via its environment and nanostructuring. Graphene morphology {#sec:forms} ------------------- Graphene can be isolated or fabricated in a number of different forms, which is an important consideration in spectroscopy. Effectiveness of a given spectroscopic tool depends on the accessibility of the sample surface to the incident radiation. The size of the accessible area must normally be larger than the wavelength of the incident beam unless near-field probes are employed (Sec. \[sec:plasmons\]) Mosaic structure and defects may affect momentum and energy resolution of the measurement. Graphene differs widely in terms of these parameters depending on preparation method. Mechanical exfoliation of graphite typically produces single, bi-, and multi-layer graphene (SLG, BLG, and MLG, respectively) of a few $\mu\mathrm{m}$ in size, although occasionally samples of dimensions of hundreds of $\mu\mathrm{m}$ can be obtained. Exfoliated samples can be transferred onto insulating substrates, after which they can be gated and subject to transport measurements. The sign and the magnitude of carrier concentration $N$ in gated samples can be precisely controlled over a wide range. The lower bound on $|N| \sim 10^{10}\,\mathrm{cm}^{-2}$ is set by inhomogeneities (Sec. \[sec:Inhomogeneities\]). The upper bound $|N| \sim 10^{13}\,\mathrm{cm}^{-2}$ is limited by the dielectric breakdown strength of the substrate, although still higher $|N|$ are achievable by electrolytic gating .[^3] The carrier concentration can also be controlled by doping [@Chen2008cis]. Morphologically, exfoliated samples are single crystals. They hold the record for transport mobility $\mu_{tr}$ although it varies much with the type of the substrate. Currently, high-quality hexagonal boron nitride (hBN) substrates enable one to achieve $\mu_{tr} \sim 10^5\,\mathrm{cm}^2/\mathrm{Vs}$, which is about an order of magnitude higher than what is typical for graphene on SiO$_2$ and corresponds to $\mu\mathrm{m}$-scale mean-free path [@Dean2010bns; @Mayorov2011msb]. The highest mobility $\sim 10^6\,\mathrm{cm}^2/\mathrm{Vs}$ is demonstrated by exfoliated graphene that is suspended off a substrate and subject to current annealing [@Du2008abt; @Bolotin2008uem; @Elias2011dcr]. Mechanical instabilities limit the size of suspended devices to $1$–$2\,\mu\mathrm{m}$ and restrict the maximum $|N|$ to a few times $10^{11}\,\mathrm{cm}^{-2}$. Large-area graphene can be made by another method: epitaxial growth on SiC by thermal desorption of Si [@vanBommel1975laa]. Epitaxial graphene may contain a single or many dozens of layers. The initial layer (layer number $L = 0$) has strong covalent bonds to the SiC substrate and is electronically different from the ideal SLG [@deHeer2007eg]. The morphology and electron properties of the subsequent layers, $L > 0$, depend on which SiC crystal face it is grown: the Si-terminated $(0001)$ face or the C-terminated $(000\bar{1})$ face.[^4] According to @deHeer2011laa, the Si-face grown graphene is orientationally ordered and has the Bernal stacking (as in graphite). The structure of the C-face epitaxial graphene is consistent with a stacking where every other layer is rotated by approximately $\pm 7^\circ$ with respect to a certain average orientation. The rotations inhibit interlayer tunneling so that the band structure of each layer is similar to SLG (see also Sec. \[sec:substrate\]). The morphology of the epitaxial graphene after annealing resembles a carpet draping over the staircase [@Emtsev2009tws]. It is characterized by domains a few $\mu\mathrm{m}$ wide and up to $50\,\mu\mathrm{m}$ long that mirror the underlying SiC terraces [@Emtsev2009tws; @deHeer2011laa]. The graphene/SiC interface is charged, inducing the $n$-type doping of about $10^{13}\,\mathrm{cm}^{-2}$ in the first ($L = 1$) graphene layer. Other layers have much smaller carrier concentration because of screening. The screening length of about one layer was measured by ultrafast infrared (IR) spectroscopy [@Sun2010smo]. The doping of the surface layers can be altered by depositing charged impurities [@Ohta2006ces; @Zhou2008mti]. Relatively low mobility $\mu_{tr} = 500$–$10,000\,\mathrm{cm}^2/\mathrm{Vs}$, the inhomogeneity of the doping profile, and the lack of its *in situ* control can be seen as drawbacks of (the first generation of) epitaxial compared to exfoliated graphene. On the other hand, the much larger surface area of the epitaxial graphene is advantageous for spectroscopic studies and applications [@deHeer2007eg]. An important recent breakthrough is epitaxial growth of graphene on high-quality hBN substrates [@Yang2013egs]. Graphene samples of strikingly large 30-in width [@Bae2010rtr] can be produced by the chemical vapor deposition (CVD) on metallic surfaces, e.g., Ru,Ni or Cu that act as catalysts. CVD graphene can be transferred to insulating substrates making it amenable to gating and transport experiments [@Kim2009lsp; @Bae2010rtr]. The microstructure of CVD graphene sensitively depends on the roughness of the metallic substrate and the growth conditions. Typical structural defects of CVD graphene are wrinkles and folds induced by transfer process and also by thermal expansion of graphene upon cooling. Grain boundaries are other common defects that have been directly imaged by micro-Raman [@Li2010gfw], transmission electron microscopy [@Huang2011gag], scanning tunneling microscopy [@Tapaszto2012mep; @Koepke2013ase], and near-field microscopy [@Fei2013epp]. The corresponding domain sizes range between $1$–$20\,\mu\mathrm{m}$. On the other hand, graphene single crystals with dimension $\sim 0.5\,\mathrm{mm}$ have been grown on Cu by CVD [@Li2011lag]. Transport mobilities of CVD-grown graphene and epitaxial graphene on SiC are roughly on par. At the opposite extreme of spatial scales are nanocrystals and nanoribbons. Graphene crystals of nm-size can be synthesized by reduction of graphene oxide[^5] or by ultrasonic cleavage of graphite in an organic solvent [@Hernandez2008hyp; @Nair2012shp]. Laminates of such crystals can be of macroscopic size amenable to X-ray and Raman spectroscopy. Nanocrystals can also be grown epitaxially on patterned SiC surface [@deHeer2011laa]. Graphene nanoribbons (GNRs) can be produced by lithography, nanoparticle etching, and unzipping of carbon nanotubes. There have been a number of spectroscopic studies of GNRs by scanned probes [@Tao2011sre], transport[^6], and photoemission [@Siegel2008sde; @Zhou2008oot] but because of space limitations they could not be covered in this review. Electronic structure of graphene neglecting interactions {#sec:single} -------------------------------------------------------- In this section we summarize basic facts about the SLG band-structure within the independent electron approximation [@CastroNeto2009tep]. The nearest-neighbor carbon atoms in SLG form $sp^2$ bonds, which give rise to the $\pi$ and $\sigma$ electron bands. The $\sigma$-bands are relevant mostly for electronic phenomena at energies $\gtrsim 3\,\text{eV}$. The unique low-energy properties of graphene derive from the $\pi$-bands whose structure can be understood within the tight-binding model [@Wallace1947tbt]. If only the nearest-neighbor transfer integral $\gamma_0=3.0\pm 0.3\,\text{eV}$ (Fig. \[fig:1.2.1\]B) is included, the amplitudes $\psi_j$ of the Bloch functions on the two triangular sublattices $j = \mathrm{A}$ or $\mathrm{B}$ of the full honeycomb lattice can be found by diagonalizing the $2 \times 2$ Hamiltonian $$H_{\text{SLG}} = \begin{pmatrix} E_D & -{\gamma }_0 S_{\mathbf{k} } \\ -{\gamma }_0 S^*_{\mathbf{k} } & E_D \end{pmatrix}\,, \label{eqn:H_SLG}$$ where $E_D$ is the constant on-site energy, $\mathbf{k} = \left(k_x, k_y\right)$ is the in-plane crystal momentum, $S_{\mathbf{k} } = \exp({i k_x a / \sqrt{3}}) + 2 \exp({-i k_x a / 2\sqrt{3}}) \cos (k_y a / 2)$ represents the sum of the hopping amplitudes between a given site and its nearest neighbors, and $a = 2.461\,$Å is the lattice constant. The spectrum of $H_{\text{SLG}}$ has the form $\varepsilon_\pm(\mathbf{k} ) = E_D \pm \gamma_0 \left|S_{\mathbf{k} }\right|$ or $$\label{eqn:E} \varepsilon_\pm = E_D \pm \gamma_0 \sqrt{3 + 2 \cos k_y a + 4 \cos\frac{\sqrt{3} k_x a}{2}\, \cos\frac{k_y a}{2}}\,.$$ At energies $|\varepsilon - E_D| \ll \gamma_0$, this dispersion has an approximately conical shape $\varepsilon_\pm(\mathbf{k} ) = E_D \pm \hbar v_0 |\mathbf{k} - \mathbf{K}|$ with velocity $ v_0 = \frac{\sqrt{3}}{2}\, \frac{\gamma _0}{\hbar}\, a = 0.9\text{--}1.0 \times 10^8\,\text{cm} / \text{s} $ near the corners of the hexagonal Brillouin zone (BZ), see Fig. \[fig:1.2.1\]A. Only two of such corners are inequivalent, e.g., $\mathbf{K},\: \mathbf{K}' = \bigl(\frac{2\pi}{\sqrt{3}\, a}, \pm \frac{2\pi}{3 a}\bigr)$; the other four are obtained via reciprocal lattice translations. Near the $\mathbf{K}$ point, $H_{\text{SLG}}$ can be expanded to the first order in $q_\parallel$ and $q_\perp$ — the components of vector $\mathbf{q} = \mathbf{k} - \mathbf{K}$ parallel and perpendicular to $\mathbf{K}$, respectively. This expansion yields the 2D Dirac Hamiltonian $$H = E_D + \hbar v_0 (q_\parallel \sigma_x + q_\perp \sigma_y)\,, \label{eqn:Dirac}$$ which prompts analogies between graphene and quantum electrodynamics [@Katsnelson2007gnb]. Here $\sigma_x$, $\sigma_y$ are the Pauli matrices. Expansion near $\mathbf{K}'$ points gives a similar expression except for the sign of the $q_\parallel$-term. The eigenvector $\Psi = (\psi_A, \psi_B)^{\mathrm{T}}$ of $H$ can be thought of as a spinor. The direction of the corresponding pseudospin is parallel (antiparallel) for energy $\varepsilon_+$ ($\varepsilon_-$). The definite relation between the pseudospin and momentum directions is referred to as the *chirality*. The conical dispersion yields the single-particle density of states (DOS) $\nu(E)$ linear in $|E - E_D|$. Accounting for the four-fold degeneracy due to spin and valley, one finds $$\nu(E) = \frac{2}{\pi \hbar^2 v_0^2}\, |E - E_D|\,. \label{eqn:nu_0}$$ The frequently needed relations between the zero-temperature chemical potential $\mu$ (referenced to the Dirac point energy $E_D$), Fermi momentum $k_F$, and the carrier density $N$ read: $$k_F = \sqrt{\pi |N|\,}\,, \quad \mu \equiv E_F - E_D = \mathrm{sign}(N)\, \hbar v_0 k_F\,. \label{eqn:k_F}$$ For $E - E_D$ not small compared to $\gamma_0$, deviations from the simplified Dirac model arise. The spectrum exhibits saddle points at energies $E_D \pm \gamma_0$, which are reached at the three inequivalent points of the BZ: $\mathbf{M} = \bigl(\frac{2\pi}{\sqrt{3}\, a}, 0\bigr)$ and $\mathbf{M}', \mathbf{M}'' = \bigl(-\frac{\pi}{\sqrt{3}\, a}, \pm\frac{\pi}{a}\bigr)$, see Fig. \[fig:1.2.1\]A. The DOS has logarithmic van Hove singularities at these saddle-points. In the noninteracting electron picture, direct ($q = 0$) transitions between the conduction and valence band states of a given saddle-point would yield resonances at the energy $\hbar\omega = 2\gamma_0 \approx 5.4\,\mathrm{eV}$. (Actually observed resonances are red-shifted due to interaction effects, see Sec. \[sec:Excitations\].) Many-body effects and observables {#sec:interaction} --------------------------------- While the single-electron picture is the basis for our understanding of electron properties of graphene, it is certainly incomplete. One of the goals of the present review is to summarize spectroscopic evidence for many-body effects in graphene. In this section we introduce the relevant theoretical concepts. For simplicity, we assume that the temperature is zero and neglect disorder. The strength of Coulomb interaction $U(r) = e^2 / \kappa r$ in graphene is controlled by the ratio $$\alpha = \frac{e^2}{\kappa \hbar v_0}\,, \label{eqn:alpha}$$ where $\kappa$ is the effective dielectric constant of the environment. Assuming $v_0 \approx 1.0 \times 10^8\,\mathrm{cm}/\mathrm{s}$, for suspended graphene ($\kappa = 1$) one finds $\alpha \approx 2.3$, so that the interaction is quite strong. Somewhat weaker interaction $\alpha \approx 0.9$ is realized for graphene on the common SiO$_2$ substrate, $\kappa = (1 + \epsilon_{\mathrm{SiO}_2}) / 2 = 2.45$. For graphene grown on metals the long-range part of the interaction is absent, with only residual short-range interaction remaining. In general, spectroscopic techniques measure either quasiparticle or current (density) response functions. Within the framework of the Fermi-liquid theory [@Nozieres1999toq] interactions renormalize the quasiparticle properties, meaning they change them quantitatively. The current/density response is altered qualitatively due to emergence of collective modes. A striking theoretical prediction made two decades ago, @Gonzalez1994nfl is that Coulomb interaction among electrons should cause a logarithmically divergent renormalization of the Fermi velocity in undoped SLG, $$\frac{v(q)}{v(k_c)} = 1 + \frac14 \alpha(k_c) \ln \frac{k_c}{q} \:\:\text{at}\:\: k_F = 0\,, \label{eqn:v_k_RG}$$ which implies the negative curvature of the “reshaped” Dirac cones [@Elias2011dcr]. Here, $k_c$ is the high momentum cutoff and $q = |\mathbf{k} - \mathbf{K}|$ is again the momentum counted from the nearest Dirac point $\mathbf{K}$. The physical reason for the divergence of $v(q)$ is the lack of metallic screening in undoped SLG because of vanishing thermodynamic density of states (TDOS) $\nu_T = {d N} / {d (\mu + E_D)}$. While Eq.  can be obtained from the first-order perturbation theory [@Barlas2007cac; @Hwang2007dde; @Polini2007gap], the renormalization group (RG) approach of @Gonzalez1994nfl indicates that validity of this equation extends beyond the weak-coupling case $\alpha \ll 1$. It remains valid even at $\alpha \sim 1$ albeit in the asymptotic low-$q$ limit where the *running* coupling constant $\alpha(q) \equiv {e^2} / {\kappa \hbar v(q)} \ll 1$ is small. The RG flow equation underlying Eq. , $$\beta(\alpha) \equiv \frac{d \ln\alpha}{d \ln q} \simeq \frac{\alpha}{4}\,, \quad \alpha \ll 1\,, \label{eqn:beta_1}$$ is free of nonuniversal quantities $\kappa$ and $k_c$, and so in principle it can be used to compare the renormalization effects in different graphene materials. The problem is that the asymptotic low-$q$ regime is hardly accessible in current experiments where one typically deals with the nonperturbative case $\alpha \sim 1$. Theoretical estimates [@Gonzalez1999mfl; @Son2007qcp; @Foster2008gvl] of the $\beta$-function in this latter regime yield $$\beta \approx 0.2\,, \quad \alpha \sim 1. \label{eqn:beta_N}$$ The corresponding renormalized velocity scales as $$v(q) \sim q^{-\beta}\,. \label{eqn:v_power-law}$$ Distinguishing this weak power law from the logarithmic one  would still require a wide range of $q$. The gapless Dirac spectrum should become unstable once $\alpha$ exceeds some critical value [@Drut2009igi; @Khveshchenko2001gei; @Sheehy2007qcs]. It is unclear whether this transition may occur in SLG as no experimental evidence for it has been reported. In doped SLG the RG flow  is terminated at the Fermi momentum scale. Therefore, velocity renormalization should be described by the same formulas as in undoped one at $q \gg k_F$ but may have extra features at $q \leq k_F$. This expectation is born out by calculations [@DasSarma2007mbi]. The result for the Fermi velocity, written in our notations, is $$\frac{v_F}{v(k_c)} = 1 + \frac{\alpha}{\pi } \left(\ln \frac{1}{\alpha} - \frac{5}{3}\right) + \frac{\alpha}{4} \ln \frac{k_{c}}{k_{F} }\,, \quad \alpha \ll 1\,, \label{eqn:v_F_doped}$$ where $\alpha$ should be understood as $\alpha(k_F)$. Comparing with Eq. , we see that $v_F$ is larger than $v(q)$ in undoped SLG at the same momentum $q = k_F$ by an extra logarithmic term $\sim \alpha |\ln \alpha|$. This logarithmic enhancement of the Fermi velocity is generic for an electron gas with long-range Coulomb interactions in any dimension [@Giuliani2005qto]. As a result, the renormalized dispersion has an inflection point near $k_F$ \[see, e.g., [@Principi2012ttd; @DasSarma2013vra]\] and a positive (negative) curvature at smaller (larger) $q$. Renormalization makes the relation between observables and quasiparticle properties such as $v(q)$ more complicated than in the noninteracting case. For illustration, consider three key spectroscopic observables: the single-particle DOS $\nu(E)$, the TDOS $\nu_T$, and the threshold energy $\hbar \omega_{th}$ of interband optical absorption. Since for curved spectrum phase and group velocities are not equal, we must first clarify that by $v(q)$ we mean the latter, i.e., the slope of the dispersion curve $E(q)$. In theoretical literature, $E(q)$ is usually defined by the equation $$E(q) = \varepsilon(q) + \Sigma_1\bigl(q, E(q)\bigr)\,, \label{eqn:E_Dirac}$$ where $\Sigma(q,\omega) = \Sigma_1(q , \omega) + i \Sigma_2(q , \omega)$ is the electron self-energy and the subscripts $\pm$ are suppressed to lighten the notations. In experimental practice (Sec. \[sec:Dirac\]), more directly accessible than $\Sigma(q,\omega)$ is the spectral function $$A(q, \omega) = \frac{-2 \Sigma_2(q, \omega)} {[\omega - \varepsilon(q) - \Sigma_1(q, \omega)]^2 + [\Sigma_2(q, \omega)]^2}\,, \label{eqn:A}$$ and the more convenient definition of $E(q)$ is the energy $\omega$ at which $A(q, \omega)$ has a maximum. As long as this maximum is sharp so that the quasiparticles are well-defined, the two definitions are equivalent. For the velocity, they entail $$\frac{v(q)}{v_0} \equiv \frac{1}{\hbar v_0} \frac{d E}{d q} = \left(1 + \frac{\partial_q \Sigma_1}{\hbar v_0}\right) Z(q)\,. \label{eqn:v_q}$$ The three quantities in question, $\nu$, $\nu_T$, and $\hbar\omega_{th}$, are related to $v(q)$ as follows: $$\begin{aligned} \nu(E) &\simeq \frac{2}{\pi} \frac{q}{\hbar v(q)} Z(q)\,, \quad Z(q) \equiv \frac{1}{1 - \partial_E \Sigma_1}\,, \label{eqn:nu_E}\\ \nu_T(N) &\equiv \frac{d N}{d (\mu + E_D)} = \frac{2}{\pi} \frac{k_F}{\hbar v_F + Z(k_F) \partial_{k_F} \Sigma_1}\,, \label{eqn:TDOS}\\ \hbar \omega_{th} &= E_+(k_F) - E_-(k_F) + \Delta_{e h}\,. \label{eqn:omega_th}\end{aligned}$$ These formulas contain many-body corrections to the relations given in Sec. \[sec:single\] that enter through the derivatives of the self-energy, while Eq.  also has a vertex correction $\Delta_{e h}$. For example, the DOS $\nu(E)$ \[Eq. \], measurable by, e.g., scanning tunneling spectroscopy (STS) is multiplied by the quasiparticle weight $Z$. Near the Fermi level one usually finds $Z < 1$ [@Giuliani2005qto], so that the interactions diminish the DOS. Inferring $v_F$ from $\nu(E_F)$ using the formula $v_F \propto k_F / \nu(E_F)$ of the noninteracting theory would cause *overestimation* of the Fermi velocity, e.g., by the factor of $Z^{-1} = 1 + (1 / 2 + 1 / \pi) \alpha$ at $\alpha \ll 1$ [@DasSarma2007mbi]. \[In practice, the low-bias STS data may be influenced by disorder and finite momentum resolution, see Sec. \[sec:Dirac\].\] Away from the Fermi level the interaction may enhance rather than suppress $\nu(E)$. An example is the Dirac point region in a doped SLG where the DOS is predicted to be nonzero (U-shaped) [@LeBlanc2011eoe; @Principi2012ttd] rather than vanishing (V-shaped). Consider next the TDOS $\nu_T(N)$ given by Eq. , which follows from Eqs. –. The TDOS can be found by measuring capacitance between graphene and metallic gates, either stationary [@Ponomarenko2010dos; @Yu2013ipi] or scanned [@Martin2008ooe]. In the absence of interactions, the TDOS coincides with the DOS at the Fermi level. However, for repulsive Coulomb interactions the second term in the denominator of Eq.  is negative [@Giuliani2005qto]. (This term can be written in terms of parameter $F_s^0 < 0$ of the Landau Fermi-liquid theory.) Hence, while $\nu(E_F)$ is suppressed, $\nu_T$ is enhanced compared to the bare DOS. Extracting $v_F$ from $\nu_T(N)$ [@Yu2013ipi] may lead to *underestimation*. ![\[fig:Excitations\] Schematic dispersion of electron density excitations in SLG (lines). The horizontal axis corresponds to the $\Gamma$–$\text{K}$ cut through the Brillouin zone. All excitations experience Landau damping inside the electron-hole pair continuum (shaded). ](Basov_Fig2){width="3.20in"} The third quantity $\hbar \omega_{th}$ \[Eq. \] stands for the threshold energy required to excite an electron-hole pair with zero total momentum in the process of optical absorption. Without interactions $\hbar \omega_{th} = 2\mu = 2 \hbar v_0 k_F$ (see Fig. \[fig:Excitations\]), and so the bare velocity is equal to $\omega_{th}\, /\, 2 k_F$. Using the same formula for interacting system [@Li2008dcd] may lead to *underestimation* of the renormalized $v_F$, for two reasons. First, $v_F$ is the *group* velocity at the Fermi momentum while the ratio $[E_+(k_F) - E_-(k_F)] / (2 \hbar k_F)$ gives the average *phase* velocity of the electron and hole at $q = k_F$. If the dispersion has the inflection point near $k_F$, as surmised above, the group velocity must be higher than the phase one. Second, the threshold energy of the electron-hole pair is reduced by the vertex (or excitonic) correction $\Delta_{e h} < 0$ due to their Coulomb attraction. Let us now turn to the collective response of SLG at arbitrary $\omega$ and $\mathbf{k}$. The simplest type of such a process is excitation of a single particle-hole pair by moving a quasiparticle from an occupied state of momentum $\mathbf{p}$ and energy $E(\mathbf{p}) \leq E_F$ to an empty state of momentum $\mathbf{p}+\mathbf{k} $ and energy $E(\mathbf{p}+\mathbf{k} ) \geq E_F$. (The subscripts $\pm$ of all $E$’s are again suppressed.) The particle-hole continuum that consists of all possible $\bigl(E(\mathbf{p}+\mathbf{k} ) - E(\mathbf{p}), \mathbf{k} \bigr)$ points is sketched in Fig. \[fig:Excitations\]. If the energy and the in-plane momentum of an electromagnetic excitation falls inside this continuum, it undergoes damping when passing through graphene. The conductivity $\sigma(\mathbf{k}, \omega) = \sigma' + i \sigma''$ has a finite real part $\sigma'$ in this region. Collective modes can be viewed as superpositions of many interacting particle-hole excitations. A number of such modes have been predicted for graphene. Weakly damped modes exist outside the particle-hole continuum, in the three unshaded regions of Fig. \[fig:Excitations\]. At low energy the boundaries of these triangular-shaped regions have the slope $\pm \hbar v_F$. Collective excitations near the $\Gamma$-point (the left unshaded triangle in Fig. \[fig:Excitations\]) are Dirac plasmons. These excitations, reviewed in Sec. \[sec:plasmons\], can be thought of as coherent superpositions of intraband electron-hole pairs from the same valley. The excitations near the $\mathrm{K}$ point (the right unshaded triangle) involve electrons and holes of different valleys. Such intervalley plasmons [@Tudorovskiy2010ipi] are yet to be seen experimentally. Also shown in Fig. \[fig:Excitations\] are the “M-point exciton” that originates from mixing of electron and hole states near the $\mathrm{M}$-points of the BZ (Sec. \[sec:single\]) and its finite-momentum extension, which is sometimes called by a potentially confusing term “$\pi$-plasmon.” Two other collective modes have been theoretically predicted but not yet observed and not shown in Fig. \[fig:Excitations\]. One is the excitonic plasmon [@Gangadharaiah2008crf] — a single interband electron-hole pair marginally bound by Coulomb attraction. Its dispersion curve is supposed to run near the bottom of the electron-hole continuum. The other mode [@Mikhailov2007nem] is predicted to appear in the range $1.66 |\mu| < \hbar\omega < 2 |\mu|$ where $\sigma'' < 0$. Unlike all the previously mentioned collective modes, which are TM-polarized, this one is TE-polarized. It is confined to graphene only weakly, which makes it hardly distinguishable from an electromagnetic wave traveling along graphene. Besides electron density, collective modes may involve electron spin. Further discussion of these and of many other interaction effects in graphene can be found in a recent topical review [@Kotov2012eei]. Quasiparticle properties {#sec:quasiparticles} ======================== Dirac spectrum and chirality {#sec:Dirac} ---------------------------- {width="6.50in"} The first experimental determination of the SLG quasiparticle spectrum was obtained by analyzing the Shubnikov-de Haas oscillations (SdHO) in magnetoresistance [@Novoselov2005tda; @Zhang2005efm]. This analysis yields the cyclotron mass $$m = \hbar k_F / v_F \label{eqn:m}$$ and therefore the Fermi velocity $v_F$. The lack of dependence of $v_F \approx 1.0 \times 10^8\, \mathrm{cm}/\mathrm{s}$ on the Fermi momentum $k_F$ in those early measurements was consistent with the linear Dirac spectrum at energies below $0.2\,\mathrm{eV}$. Direct mapping of the $\pi$-band dispersion over a range of several eV [@Zhou2006lee; @Bostwick2007qdi] was achieved soon thereafter by the angle-resolved photoemission (ARPES) experiments. This experimental technique, illustrated by Fig. \[fig:ARPES\](a), measures the electron spectral function \[Eq. \] weighted by the square of the matrix element $M(\mathbf{k}, \nu)$ of interaction between an incident photon of frequency $\nu$ and an ejected photoelectron of momentum $\mathbf{k}$, see Eq.  below. The representative dispersion curves measured for epitaxial graphene on SiC are shown in Fig. \[fig:ARPES\](b) and (c), where red (black) color corresponds to high (low) intensity. The “dark corridor” [@Gierz2010idc] $\Gamma$–$\mathrm{K}$ along which one of the two dispersion lines is conspicuously missing, Fig. \[fig:ARPES\](c), occurs due to the selection rules for the matrix element $M(\mathbf{k}, \nu)$ known from prior work on graphite [@Shirley1995bzs; @Daimon1995sfi]. The full angular dependence of the ARPES intensity is depicted in Fig. \[fig:ARPES\](d). The ARPES measurements have been carried out on epitaxial graphene grown on a variety of substrates, on free-standing samples [@Knox2011mar], and on multilayered samples with weak interlayer interactions [@Sprinkle2009fdo]. The tight-binding model (Sec. \[sec:single\]) accounts for the main features of all these spectra. However, there are also subtle deviations. For example, the slope of the dispersion near the Dirac point varies systematically with the background dielectric constant $\kappa$ \[Fig. \[fig:ARPES\](d)\], which is consistent with the theoretically predicted velocity renormalization, see Secs. \[sec:interaction\] and \[sec:renormalization\]. Certain additional features near the Dirac point (see Fig. \[fig:4.1.1\]) have been interpreted[^7] as evidence for substrate-induced energy gaps, Sec. \[sec:substrate\]. For graphene on SiC, an alternative explanation invokes electron-plasmon coupling @Bostwick2007qdi, see Fig. \[fig:3.6.2\] in Sec. \[sec:e-ph-pl\]. Complimentary evidence for the Dirac dispersion of quasiparticles comes the tunneling and thermodynamic DOS measurements by means of scanned probes. The Dirac point manifests itself as a local minimum marked by the arrows in the STS tunneling spectra of Fig. \[fig:3.1.1\]a. The U- rather than the V-shaped form of this minimum (Sec. \[sec:single\]) is due to disorder smearing. The STS data obtained by @Zhang2008gpi (Fig. \[fig:3.1.1\]a) also exhibit a prominent suppression at zero bias for all gate voltages. To explain it [@Zhang2008gpi] proposed that this feature arises because of a limitation on the possible momentum transfer in tunneling. This limitation is lifted via inelastic tunneling accompanied by the emission of a BZ-boundary acoustic phonon of energy $\hbar \omega_0 = 63\,\mathrm{meV}$. This energy must be subtracted from the tip-sample bias $e V$ to obtain the tunneling electron energy inside the sample. By tuning the electron density $N$ with a backgate [@Zhang2008gpi; @Deshpande2009srs; @Brar2010ooc], one can change the Fermi energy $E_F$ with respect to the Dirac point $E_D$. Taking the former as the reference point (i.e., assuming $E_F \equiv 0$ for now) one obtains the relation $|E_D| = |e V_D| - \hbar \omega_0$. As shown in Fig. \[fig:3.1.1\]c, thus defined $|E_D|$ is proportional to $|N|^{1/2}$, as expected for the linear dispersion, Eq. . The same zero-bias gap feature is observed in other graphene samples studied by the Berkeley group, e.g., SLG on hBN[@Decker2011lepa]. Yet it is not seen in STS experiments of other groups, see, e.g., Fig. \[fig:el-ph\](c), Sec. \[sec:renormalization\], and Sec. \[sec:Landau\] below.[^8] ![\[fig:3.1.1\] (Color online) Spectroscopic determination of the Dirac dispersion in SLG. Panel a: the STS tunneling spectra $dI/dV$ taken at the same spatial point and different gate voltages $V_g$. Curves are vertically displaced for clarity. The arrows indicate the positions of the $dI/dV$ minima $V_D$. Panel c: the distance $|E_D|$ between the Dirac point and the Fermi level as a function of $V_g$ obtained from the data in panel a. The line is the fit to $E_D \propto |V_g|^{1/2}$. The insets are cartoons showing the electron occupation of the Dirac cones. After @Zhang2008gpi. Panel b: optical conductivity of SLG at different gate voltages with respect to the neutrality point. Panel d: the gate voltage dependence of the interband absorption threshold “$2E_F$” obtained from the data in panel b. After @Li2008dcd. ](Basov_Fig4){width="3.55in"} The $\mu(N)$ dependence can be more directly inferred from the TDOS $\nu_T(N)$ measured by the scanning single-electron transistor microscopy (SSETM) [@Martin2008ooe]. Unlike the STS spectra in Fig. \[fig:3.1.1\]a, the SSETM data are not obscured by the zero-bias feature. They show a finite and position-dependent TDOS at the neutrality point $N = 0$, reflecting once again the presence of disorder in graphene on SiO$_2$ substrate, see also Sec. \[sec:Inhomogeneities\]. The most definitive observation of the Dirac TDOS has been made using exfoliated graphene on hBN [@Yankowitz2012esd; @Yu2013ipi]. Similar to SSETM, the TDOS was extracted from the capacitance measurements; however, it was the capacitance between the sample and the global backgate rather than between the sample and the local probe. Let us now turn to the chirality of graphene quasiparticles. Recall that chirality refers to the phase relation between the sublattice amplitudes $\psi_j = \psi_j(\mathbf{k} )$, $j = \mathrm{A}, \mathrm{B}$, of the quasiparticle wavefunctions (Sec. \[sec:single\]). The chirality has been independently verified by several techniques. First, it naturally explains the presence of the special half-filled Landau level at the Dirac point seen in magnetotransport [@Novoselov2005tda; @Zhang2005efm]. Next, in the STS experiments the quasiparticle chirality is revealed by the LDOS features observed near impurities and step edges, see @Rutter2007sai [@Mallet2007eso; @Zhang2009oos; @Deshpande2009srs] and Sec. \[sec:Inhomogeneities\]. The chirality influences the angular distribution of the quasiparticle scattering by these defects, suppressing the backscattering [@Brihuega2008qci; @Xue2012lwl], in agreement with theoretical predictions [@Ando2002dca; @Katsnelson2006cta]. Finally, in ARPES the chirality manifests itself via the selection rules for the matrix element $$M(\mathbf{k}, \nu) = \frac{e}{c} \int d \mathbf{r}\, \Psi_f^*(\mathbf{r}) ({\mathbf{A}} \hat{\mathbf{v}}) \Psi_i(\mathbf{r}) \label{eqn:M}$$ that describes coupling of electrons to the vector potential ${\mathbf{A}}$ of the photon. Here the Coulomb gauge $\bm{\nabla} {\mathbf{A}} = \varphi = 0$ is assumed and $\hat{\mathbf{v}} = -i \hbar \bm{\nabla} / m$ is the velocity operator. The matrix element $M(\mathbf{k}, \nu )$ depends on the relative phase of $\psi_\mathrm{A}$ and $\psi_\mathrm{B}$. Based on symmetry considerations, the general form of $M(\mathbf{k}, \nu)$ at small $\mathbf{q} = \mathbf{k} - \mathbf{K}$ must be $$M(\mathbf{k}, \nu) = (c_1 \mathbf{K} + c_2 \mathbf{q}) \cdot {\mathbf{A}} \sum_{j = \mathrm{A}, \mathrm{B}} e^{-i \mathbf{K} \bm{\tau}_j} \psi_j(\mathbf{k} ) \label{eqn:M1}$$ if spin-orbit (SO) interaction effects can be ignored. Here $\bm{\tau}_j$ are the positions of $j$th atom in the unit cell and $\mathbf{K}$ is the nearest Dirac point. The coefficients $c_1$ and $c_2$ cannot be obtained solely from symmetry; however, regardless of their values, when $\mathbf{q}$ is parallel (antiparallel) to $\mathbf{K}$ for the states in the conduction (valence) band, the sum over $j$ in Eq.  vanishes and so does $M(\mathbf{k}, \nu)$. This explains the low-intensity “dark corridor” in the observed ARPES signal, Fig. \[fig:ARPES\](c) and (d). The ARPES selection rules are also relevant for BLG. Experimentally, the orientation of the low intensity directions rotates by $\pm 180^{\circ }$ ($\pm 90^{\circ}$) in SLG (BLG) when the photon polarization vector ${\mathbf{A}}$ is switched between two orientations, parallel and perpendicular to $\mathbf{K}$ [@Gierz2010idc; @Hwang2011dmo; @Liu2011vec]. @Hwang2011dmo discussed how these rotation angles can be linked to the Berry phase — a quantity closely related to chirality — in SLG and BLG. However, their theoretical model for the matrix element $M(\mathbf{k}, \nu)$ has been a subject of controversy, which appears to be rooted in different assumption about the final state wavefunction $\Psi_f(\mathbf{r})$ in Eq. . At very high energies $h \nu$, the conventional approximation of $\Psi_f(\mathbf{r})$ by a plane wave should be adequate [@Shirley1995bzs; @Mucha-Kruczynski2008cog]. In this case one can replace the velocity operator $\hat{\mathbf{v}}$ by $\hbar \mathbf{k} / m$ leading to $c_1 = c_2$ in Eq. . On the other hand, @Hwang2011dmo replaced $\hat{\mathbf{v}}$ by the band velocity $v\, \mathbf{q} / |\mathbf{q}|$. This is perhaps appropriate at low energies $h\nu$ at which $\Psi_f \approx \Psi_i$ near the graphene plane. The corresponding $c_1$ is equal to zero, which is admissible. However, $c_2 \propto 1 / |\mathbf{q}|$ diverges at $\mathbf{q} \to 0$, in contradiction to the $\mathbf{k}\cdot \mathbf{p}$ perturbation theory [@Yu1996fos]. In view of this problem and because other ARPES experiments and calculations @Gierz2010idc indicate a nontrivial $\nu$-dependence of $M(\mathbf{k}, \nu)$, further study of this question is desirable. Renormalization of Dirac spectrum {#sec:renormalization} --------------------------------- {width="7.0in"} Experimental verification of the many-body renormalization of the Dirac spectrum in graphene and its Fermi velocity $v_F$ in particular has been sought after in many spectroscopic studies. Some of these studies may be subject to interpretation because $v_F$ usually enters the observables in combination with other quantities, see Sec. \[sec:interaction\]. In addition, when the change in $v_F$ is small, one cannot completely exclude single-particle effects. Probably the first experimental indication for $v_F$ renormalization in graphene came from infrared absorption/transmission spectroscopy [@Li2008dcd] of exfoliated SLG on amorphous SiO$_2$ (a-SiO$_2$). This study found that $v_F$ increases from $1.0 \times 10^8\, \mathrm{cm}/\mathrm{s}$ to a $15\%$ higher value as the carrier density $N$ decreases from $3.0$ to $0.7 \times 10^{12}\, \mathrm{cm}^{-2}$, see Fig. \[fig:v\_F\]d. Next came an STS study of Landau level spectra [@Luican2011qll], which found a $25\%$ enhancement of $v_F$ (fifth row Table \[tbl:v\_F\]) in the same range of $N$. Substrate $\kappa$ $v\,(10^8\,\mathrm{cm}/\mathrm{s})$ Method Source ------------ ---------- ------------------------------------- ------------- --------- SiC (0001) 7.26 1.15(2) ARPES Hwang 2.0 ARPES Siegel 1.20(5) Capacitance Yu SiO$_2$ 1.80 2.5(3) ARPES Hwang a-SiO$_2$ 2.45 1.47(5) STS Luican 3.0(1) SdH Elias 2.6(2) Transport Oksanen : The Fermi velocity of SLG in excess of nominal bare value of $0.85\times 10^8\,\mathrm{cm}/\mathrm{s}$. In the last column, Hwang, Siegel, Yu, Luican, Elias, and Oksanen stand for [@Hwang2012fve], [@Siegel2013ccs], [@Yu2013ipi], [@Luican2011qll], [@Elias2011dcr], and [@Oksanen2013smm], respectively. \[tbl:v\_F\] A much broader range of $N$ has been explored in suspended graphene where $N$ as small as a few times $10^{9}\, \mathrm{cm}^{-2}$ can be accessed. Working with such ultra-clean suspended samples, @Elias2011dcr were able to carry out the analysis of the SdHO of the magnetoresistance over a two-decade-wide span of the carrier densities. This analysis yields the cyclotron mass \[Eq. \] and thence $v_F$. The Fermi velocity was shown to reach $v_F \approx 3.0 \times 10^8\, \mathrm{cm}/\mathrm{s}$, the largest value reported to date, cf. Table \[tbl:v\_F\]. @Elias2011dcr fitted their data (Fig. \[fig:v\_F\]A) to Eq.  for undoped graphene by treating $\alpha$ as an adjustable parameter. Figure 3 of @Elias2011dcr suggests another possible fit, to Eq.  with the exponent $\beta \approx 0.25$, which is close to Eq. . It would be better to compare the measured $v_F$ with the theoretical predictions for *doped* graphene, i.e., with the extension (or extrapolation) of Eq.  to the case in hand, $\alpha \sim 1$. From the measurements of quantum capacitance (the quantity proportional to the TDOS) of SLG on hBN, @Yu2013ipi found that $v_F$ increases by $\sim 15\%$ as $N$ varies from $5 \times 10^{12}$ down to a few times $10^{10}\,\mathrm{cm}^{-2}$, see Fig. \[fig:v\_F\]B. The vertex corrections were not included when the conversion of the quantum capacitance to $v_F$ was done. Therefore, this number represents the lower bound on $v_F$, see Sec. \[sec:interaction\]. Using substrates of different dielectric constant $\epsilon_{\mathrm{sub}}$ is another approach to study $v_F$ renormalization. An advantage of this approach is that a broad range of $N$ is not necessary in this case. Instead, the renormalization of velocity is driven by the change in the interaction strength $\alpha \propto 1 / \kappa$ where $\kappa = (1 + \epsilon_{\mathrm{sub}}) / 2$, see Eq. . A crude estimate of this effect is as follows. The dielectric screening by the substrate is effective at distances larger than the separation $d$ between graphene and the substrate. Hence, the momentum cutoff in Eqs.  and should be chosen $k_c \sim 1 / d$. If $d \lesssim 1\,\mathrm{nm}$ and $k_F^{-1} \sim 6\,\mathrm{nm}$, then $\ln(k_c / k_F) \lesssim 2$ and Eq.  entails $$\delta v_F \lesssim (1.0 \times 10^8\, \mathrm{cm}/\mathrm{s}) \times \delta\left(\frac{2}{\epsilon_{\mathrm{sub}} + 1}\right)\,. \label{eqn:delta_v_F}$$ where we use “$\delta$” to denote a change in a quantity. In a recent ARPES study [@Hwang2012fve] the smallest $v_F = (0.85\pm 0.05) \times 10^{8}\, \text{cm/s}$ was observed on metallic substrates. This number represents presumably the bare quasiparticle velocity in the absence of long-range Coulomb interactions. Note that it is close to the Fermi velocity $v_F = 0.81 \times 10^8\,\text{cm}/\text{s}$ measured in carbon nanotubes [@Liang2001fpi]. The ARPES results for three other substrates are reproduced in Fig. \[fig:v\_F\]D. They clearly demonstrate a prominent velocity enhancement near the Fermi level. Thus, graphene on (the carbon face of) SiC has $v_F$ that is only slightly larger than what is observed for metallic substrates [@Siegel2011mbi; @Hwang2012fve], which can be explained by the high $\kappa$. Graphene on hBN has $v_F$ close to that for SLG on a-SiO$_2$, which is consistent with the effective dielectric constants of hBN and a-SiO$_2$ being roughly equal [@Wang2012mdq; @Yu2013ipi]. A surprisingly large $v_F$ is found for graphene on crystalline SiO$_2$ (quartz), see Table \[tbl:v\_F\] and Fig. \[fig:v\_F\]D. {width="6.50in"} As mentioned above, renormalization of the quasiparticle velocity in SLG can also arise from single-particle physics. One example is the modification of the electron band-structure by external periodic potentials [@Park2008ngo; @Park2008abo; @Brey2009ezm; @Guinea2010bsa; @Wallbank2013gms]. Such potentials are realized in moiré superlattices that form when graphene is deposited on lattice-matched substrates, which we will discuss in Sec. \[sec:moire\]. Similar effects appear in misoriented graphene bilayers and multilayers that grow on the carbon face of SiC [@Haas2008wmg] (Sec. \[sec:forms\]) and are also common in CVD graphene grown on Ni [@Luican2011slb]. Calculations predict a strong dependence of the velocity on the twist angle [@Bistritzer2011mbi; @TramblydeLaissardiere2010lod; @LopesdosSantos2007gbw; @Shallcross2010eso; @Lopes_dos_Santos2012cmo]. The experimental value of $v_{F}$ reported for twisted graphene layers on the carbon face of SiC is $v_{F } \approx 1.10 \times 10^{8}\,\text{cm/s}$ [@Crassee2011mmo; @Miller2009otq; @Siegel2011mbi; @Sprinkle2009fdo]. Changes of $v_{F}$ up to 10% among different layers for graphene on the carbon-face of SiC have been deduced from SdH oscillations [@deHeer2007eg] and magneto-optical measurements [@Crassee2011mmo; @Crassee2011gfr]. In the latter case these changes have been attributed to electron-hole asymmetry and also to variation of the carrier density and dielectric screening among the graphene layers. No variation of $v_{F}$ as a function of twist angle was observed by ARPES and STS [@Miller2009otq; @Sprinkle2009fdo; @Siegel2011mbi; @Sadowski2006lls]. However, a $14\%$ decrease of $v_{F}$ at small twist angles was found in the STS study of CVD graphene transferred to the grid of a transmission electron microscope [@Luican2011slb]. Landau quantization {#sec:Landau} ------------------- Spectroscopy of Landau level (LL) quantization in a magnetic field is yet another way to probe quasiparticle properties of graphene. The linear dispersion of SLG leads to unequally spaced LLs: $E_{n} = E_D + \text{sgn}(n) v_0 \sqrt{2 e \hbar B |n|} $ (Fig. \[fig:Landau\_levels\]a), where $n > 0$ or $n < 0$ represents electrons or holes, respectively[@McClure1957bso; @Gusynin2006tod; @Jiang2007iso]. Each of the LLs has four-fold degeneracy due to the spin and valley degrees of freedom. Additionally, the electron-hole symmetric $n = 0$ LL gives rise to the extraordinary “half-integer” quantum Hall effect [@Novoselov2005tdg; @Zhang2005eoo], the observation of which back in 2005 was the watershed event that ignited the widespread interest in graphene. The LL spectrum of graphene has been probed using scanning tunneling spectroscopy (STS), IR spectroscopy, and Raman scattering. The STS of graphene LLs was first carried out in graphene on graphite samples, where suspended graphene is isolated from the substrate at macroscopic ridge-like defect in graphite [@Li2009sts]. Figure \[fig:Landau\_levels\]b displays the differential conductance of graphene versus tip-sample bias at different magnetic fields $B$ normal to the graphene surface. Well defined LDOS peaks corresponding to discrete LL states appear in the tunneling spectra. These LL peaks become more prominent and shift to higher energies in higher magnetic fields consistent with the expected $\sqrt{B |n|}$ law. Similar LL spectrum was also observed in epitaxial grown graphene layers on SiC [@Miller2009otq]. To examine the fine structure within a LL, @Song2010hrt performed high-resolution STS studies at temperatures as low as $10\,\text{mK}$ on epitaxial graphene. Figure \[fig:Landau\_levels\]c shows their data for the $n = 1$ LL at the magnetic field range where the LL1 starts to cross the Fermi energy (yellow line). The LL1 level is composed of four separate peaks, indicating that the valley and spin degeneracy is lifted. The larger energy splitting ($\Delta E_{v}$) is attributed to the lifting of valley degeneracy. It increases monotonically with the applied magnetic field with the effective $g$-factor of $18.4$. The smaller splitting ($\Delta E_{s} $) has an average $g$-factor close to $2$, presumably due to the electron spin. Quantitatively, this spin splitting shows a highly unusual dependence on the filling factor. Comparing the spectra at filling factors of $4$, $5$, and $6$, a clear enhancement of the spin splitting is observed at $\nu =5$, which can be attributed to many-body effects (exchange enhancement). In addition, new stable half-filled Landau levels appear at half fillings such as $9/2$ and $11/2$. Their origin is not yet clear. Landau level spectroscopy of graphene on SiO$_2$ was presented in @Luican2011qll and a similar study for graphene on hBN was reported in @Chae2012rgd. In the latter system, which has lower disorder, observation of many LLs was possible over a wide energy range. Deviations of the LL energies by about $\sim 10\%$ from the predictions of the single-particle theory were interpreted in terms of the Fermi velocity renormalization, see Fig. \[fig:v\_F\]C. This is in line with the results of other measurements discussed above (Table \[tbl:v\_F\]). The infrared (IR) spectroscopy provides another way to study the LL spectra [@Sadowski2006lls; @Jiang2007iso; @Henriksen2010iis]. The IR transitions between LLs have to satisfy the selection rule $\Delta |n|=\pm 1$, due to angular momentum conservation. Selection rules also apply to the circular polarization of light. As a result, graphene exhibits strong circular dichroism and Faraday effect [@Crassee2011gfr]. Figure \[fig:Landau\_levels\]d displays the experimental data of normalized IR transmission spectra through SLG at several magnetic fields @Jiang2007iso. The electron density is controlled so that Fermi energy lies between the $n = -1$ and $0$ LL (inset in Fig. \[fig:Landau\_levels\]d). Two transmission minima $T_{1} $ and $T_{2}$ are readily observable. The $T_{1} $ resonance corresponds to the $n=-1$ to $n=0$ intraband LL transition, and the $T_{2} $ resonance arises from the degenerate interband $n=-1$ to $n=2$ and $n=-2$ to $n=1$ transitions. The LL transition energies scales linearly with $\sqrt{B}$, as expected from the LL structure described above. A careful examination of the IR transitions as a function of electron filling factor further reveals that at zero filling factor, the $n=-1$ to $n=0$ (or $n=0$ to $n=1$) transition is shifted to a higher energy compared to that at the filling factor of $2$ and $-2$ [@Henriksen2010iis]. This shift was again tentatively attributed to interaction effects. Current/density response and collective modes {#sec:Excitations} ============================================= Optical conductivity {#sec:direct} -------------------- Traditionally measured by optical spectroscopy, the “optical” conductivity $\sigma(\omega) = \sigma'(\omega) + i \sigma''(\omega) \equiv \sigma(q = 0, \omega)$ quantifies the response of current to an external electric field in the low momenta $q \ll \omega / v_F$ region of the $q$-$\omega$ parameter space, see Fig. \[fig:Excitations\]. Both intraband and interband transitions contribute to the optical conductivity; we will start with the interband ones. In a charge-neutral SLG, which is a zero-gap semiconductor with the Fermi energy at the Dirac point, the interband transitions have no threshold. Particularly interesting is the range of (IR) frequencies $\hbar \omega \ll \gamma_0$, where quasiparticles behave as massless Dirac fermions. Since the Dirac spectrum has no characteristic frequency scale and neither does the Coulomb interaction, at zero temperature and in the absence of disorder the conductivity must be of the form $\sigma(\omega) = (e^2 / h) f(\alpha)$, where $\alpha$ is defined by Eq. . \[However, $\omega = 0$ is, strictly speaking, a singular point [@Ziegler2007mco].\] For the noninteracting case, $\alpha = 0$, the theory predicts[^9] $f(0) = \pi / 2$, so that $\sigma(\omega)$ is real and has the universal value of $$\sigma_0 = \frac{\pi}{2}\, \frac{e^2}{h}\,. \label{eqn:sigma_0}$$ The corresponding transmission coefficient $T = 1 - 4\pi \sigma(\omega) / c$ for suspended graphene is expressed solely in terms of the fine structure constant: $T = 1 - \pi (e^2 / \hbar c) \approx 0.977$.[^10] This prediction matches experimental data surprisingly well, with possible deviations not exceeding $15\%$ throughout the IR and visible spectral region [@Mak2008mot; @Nair2008fsc; @Li2008dcd]. This implies that the interaction correction $f(\alpha) - f(0)$ is numerically small even at $\alpha = 2.3$. At the level of the first-order perturbation theory this remarkable fact is explained by a nearly complete cancellations between self-energy and vertex contributions [@Mishchenko2008mci; @Sheehy2009oto; @Sodemann2012ict] Doping of graphene creates an effective threshold $\hbar \omega_{th}$ for interband absorption by the same mechanism as in the Burstein–Moss effect: the blue shift of the lowest energy of interband transitions in a doped semiconductor [@Yu1996fos]. Due to Pauli blocking, no direct interband transitions exist at $\hbar\omega < 2 |\mu|$ in the noninteracting electron picture, see Fig. \[fig:Excitations\]. Experimentally, the existence of such a threshold has been confirmed by IR spectroscopy of gated SLG [@Wang2008gvo; @Li2008dcd; @Horng2011dco]. As shown in Fig. \[fig:3.1.1\], the frequency position of the broadened step in $\sigma'(\omega)$ scales as the square-root of the gate voltage $V_g$, and so is proportional to $k_F$. This is consistent with the linear dispersion $2 |\mu| = 2 \hbar v_F k_F$ of the Dirac quasiparticles. This behavior is seen in both exfoliated [@Li2008dcd] and CVD-grown graphene [@Horng2011dco]. At the smallest gate voltages, deviations from the square-root law are seen @Li2008dcd, which may be due to an interplay of many-body effects, the velocity renormalization and the vertex corrections, see Secs. \[sec:interaction\] and \[sec:renormalization\]. Vertex corrections (which are also referred to as the excitonic effects) play a prominent role also in the optical energy range $4$–$6\,\text{eV}$. The dominant spectroscopic feature in this region is the interband transition that connects electron and hole states near the $\mathrm{M}$-point of the Brillouin zone, where the DOS has van Hove singularities (Sec. \[sec:single\]). This resonance is seen both in SLG and MLG samples.[^11] This resonance has been detected by EELS and dubbed “$\pi$-plasmon” \[see, e.g., @Eberlein2008pso\]. We prefer the term “$\mathrm{M}$-point exciton” to avoid confusion with the Dirac plasmon. Electron-electron interactions significantly renormalize the properties of this resonance. The position of the $\mathrm{M}$-point exciton is red shifted from the noninteracting value of $2\gamma_0$ by as much as $600\,\text{meV}$ in SLG samples [@Yang2009eeo; @Mak2011smb; @Chae2011efr]. The absorption peak has a Fano lineshape indicative of interaction effects. Let us now discuss the intraband transitions. The commonly used Drude model assumes that the intraband response of a conductor is a simple fraction: $$\label{eqn:sigma_Drude} \sigma_{\mathrm{intra}}(\omega) = \frac{i}{\pi}\, \frac{D}{\omega + i\gamma}\,,$$ For noninteracting electrons with an isotropic Fermi surface one generally finds [@Ashcroft1976ssp] $$D = \pi e^2 |N| / m\,, \label{eqn:D}$$ where $m$ is defined by Eq. . For Dirac electrons with $v_F = \mathrm{const}$ and $k_F = \sqrt{\pi |N|}$ both $m$ and $D$ scale as $|N|^{1/2}$. Parameter $D$ is known as the Drude weight. In the Drude model, the relaxation rate $\gamma$ is frequency-independent and can be related to the transport mobility $\mu_{tr}$ by $\hbar \gamma = e v_F / (k_F \mu_{tr})$. In exfoliated samples of typical mobility $\mu_{tr} \sim 10,000\,\text{cm}^{2}/ \text{Vs}$ and carrier density $N \sim 3\times 10^{11}\,\mathrm{cm}^{-2}$ one estimates $\gamma \sim 10\,\mathrm{meV}$. This is below the low-frequency cutoff of the IR microscopy [@Li2008dcd]. One can extend measurements to lower frequency provided larger area samples are used, such as epitaxial [@Choi2009ber; @Hofmann2011hcc] and CVD-grown graphene [@Horng2011dco; @Rouhi2012tgo; @Ren2012tai]. In both cases the gross features of the measured frequency dependence of IR conductivity comply with the Drude model. Note that such samples have relatively low mobility (Sec. \[sec:forms\]) and so show wider Drude peaks in $\sigma'(\omega)$. The intraband response as a function of the carrier density has been studied using a gated CVD-grown graphene [@Horng2011dco]. The experimentally observed Drude weight was found to be 20–50% smaller than predicted by Eq. , see Fig. \[fig:3.2.1\]. The reduction was larger on the electron ($\mu > 0$) side where the transport mobility was also lower. At the same time, the optical sum rule $\int_0^\infty \sigma'(\omega) d \omega = \mathrm{const}$ was apparently obeyed [@Horng2011dco]. The conservation of the total optical weight was made possible by a residual conductivity in the interval $\gamma \ll \omega \ll 2 |\mu| - \gamma$, first observed by @Li2008dcd. In this region of frequencies both interband and intraband transitions should be suppressed yet the conductivity remains no smaller than $\sigma'(\omega) \approx 0.5 e^2 / h$, see Fig. \[fig:3.1.1\]b. Redistribution of the optical weight is common to correlated electron systems [@Millis2004oca; @Qazilbash2009eci; @Basov2011eoc], and so the residual conductivity of graphene is suggestive of interaction effects. Calculation of such effects is more difficult than for the undoped graphene but an extensive theoretical literature already exists on the subject. For example, the role of interaction in the conductivity sum rule was tackled in [@Sabio2008fsr], the renormalization of $D$ was discussed in [@Abedpour2007rou; @Levitov2013eei]. The residual conductivity remains the most challenging problem. So far, theoretical calculations that consider electron-phonon[^12] or electron-electron[^13] interactions predict relatively small corrections to $\sigma'(\omega)$ inside the interbad-intraband gap $0 < \hbar \omega < 2 |\mu|$. Such corrections can however be enhanced by disorder [@Kechedzhi2013pad; @Principi2013idd]. ![\[fig:3.2.1\] (Color online) Gating-induced change $\Delta \sigma'(\omega) = \sigma'(\omega) - \sigma'_{\mathrm{CNP}}(\omega)$ in the optical conductivity of SLG. Solid lines are the fits assuming Drude model for both $\sigma(\omega)$ and $\sigma_{\mathrm{CNP}}(\omega)$. The latter is the conductivity at the charge-neutrality point. Its Drude form is chosen to account for inhomogeneous local doping, cf. Sec. \[sec:Inhomogeneities\]. After [@Horng2011dco].](Basov_Fig7){width="2.2in"} Plasmons {#sec:plasmons} -------- {width="6.0in"} A plasmon is a collective mode of charge-density oscillation in a system with itinerant charge carriers. Plasmons have been extensively investigated both in classical and quantum plasmas. The dispersion relation of plasmons in a 2D conductor is given by the equation $$\label{eqn:q_p} q_p(\omega) = \frac{i}{2\pi}\, \frac{\kappa(\omega) \omega}{\sigma(q_p, \omega)}\,,$$ where $\kappa(\omega)$ is the average of the dielectric functions of the media on the two sides, see, e.g., [@Fei2012gto; @Grigorenko2012gp]. At $q \ll k_F$ the $q$-dependence of $\sigma(q, \omega)$ can be neglected, and so the plasmon dispersion is determined by the optical conductivity $\sigma(\omega)$ discussed above. This implies that $\sigma(\omega)$, which is usually measured by optical spectroscopy, can also be inferred by studying plasmons [@Fei2012gto; @Chen2012oni]. (Actually, optics probes transverse rather than longitudinal response but at $q \ll\omega / v_F$ the two coincide.) Note that $q_p = q_p' + i q_p''$ is a complex number. Its real part determines the plasmon wavelength $\lambda_p = 2\pi / q_p'$ and the imaginary part characterizes dissipation. The condition for the propagating plasmon mode to exist is $q_p'' \ll q_p'$ or $\sigma' \ll \sigma''$, assuming $\kappa$ is real. In SLG this condition is satisfied (both in theory and in experiment) at frequencies that are smaller or comparable to $|\mu| / \hbar$. In particular, at $\hbar \omega \ll |\mu|$, one can use Eqs.  and to express the plasmon dispersion in terms of the Drude weight $D$: $$\label{eqn:omega_I} \omega_p(q) = \sqrt{\frac{2}{\kappa}\, D q}\,.$$ This $\sqrt{q}$-behavior is a well-known property of 2D plasmons. Using for $D$ with Eq.  for $m$, one finds $$\label{eqn:omega_p_q} \omega_p(q) = \sqrt{\frac{2 \sqrt{\pi}\,e^2}{\kappa\hbar v_F}}\ v_F |N|^{1 /4} q^{1/2}\,, \quad q \ll k_F\,.$$ The $|N|^{1/4}$-scaling of the plasmon frequency at fixed $q$ should be contrasted with $\omega_p \propto |N|^{1/2}$ scaling well known for the 2D electron gas with a parabolic energy spectrum (2DEG). The difference is due to the $D \propto N$ dependence in the latter system. Another qualitative difference is the effect of electron interactions on $D$. In 2DEG, interactions do not change $D$, which is the statement of Kohn’s theorem [@Giuliani2005qto]. In graphene, interactions renormalize the Drude weight [@Abedpour2007rou; @Levitov2013eei], which causes quantitative deviations from Eq. . Qualitative deviations from this equation occur however only at $q \sim k_F$ where the plasmon dispersion curve enters the particle-hole continuum, see Fig. \[fig:Excitations\]. At such momenta the Drude model  breaks down and a microscopic approach such as the random-phase approximation (RPA) becomes necessary [@Wunsch2006dpo; @Hwang2007dfs; @Jablan2009pig]. The RPA predicts that inside the particle-hole continuum the plasmon survives as a broad resonance that disperses with velocity that approaches a constant value $v_F$ at large $q$. Experimental measurements of the plasmon dispersion over a broad range of $q$ have been obtained by means of electron energy loss spectroscopy (EELS). Such experiments[^14] have confirmed the $\omega_p \propto \sqrt{q}$ scaling at small momenta and a kink in the dispersion in the vicinity of the particle-hole continnum. EELS study carried out by @Pfnur2011mpe reported two distinct plasmon modes, a result yet to be verified through other observations. The IR spectroscopy of graphene ribbons [@Ju2011gpf; @Yan2013dpm] and disks [@Yan2012ist; @Yan2012pcg; @Fang2013gth] offered a complementary method to probe the plasmon dispersion. The experimental signature of the plasmon mode is the absorption resonance whose frequency $\omega_{\mathrm{res}}$ is observed to scale as the inverse square root of the ribbon width $W$ (or disk radius $R$). This scaling agrees with the theoretical results relating $\omega_{\mathrm{res}}$ to the plasmon dispersion in an unbounded graphene sheet \[Eq. \]. For the ribbon, it reads $\omega_{\mathrm{res}} \approx \omega_p(2.3 / W)$ [@Nikitin2011ewt]. The same relation can be deduced from the previous work [@Eliasson1986mms] on plasmons in 2DEG stripes. In fact, most of the results obtained in the context of plasmons in 2DEG in semiconductors [@Demel990ndr; @Demel1991odp; @Kukushkin2003ore] and also electrons on the surface of a liquid $^4$He [@Glattli1985dhe; @Mast1985obe] directly apply to graphene whenever the Drude model holds. As shown theoretically and experimentally in that earlier work, the spectrum of plasmons in ribbons/stripes is split into a set of discrete modes dispersing as $\omega_l(q_\parallel) \approx \omega_p\bigl(\sqrt{q_l^2 + q_\parallel^2}\,\bigr)$, as a function of the longitudinal momentum $q_\parallel$ and the mode number $l = 1, 2, \ldots$, with $q_l = (\pi l - \delta_l) / W$ having the meaning of the transverse momentum. Numerical results [@Eliasson1986mms; @Nikitin2011ewt] suggest that the phase shift parameter is equal to $\delta_l \approx \pi / 4$ at $q_\parallel = 0$. The resonance mode detected in graphene ribbons [@Ju2011gpf; @Yan2013dpm] is evidently the $l = 1$ mode. Probing $q_\parallel \neq 0$ modes in ribbons with conventional optics is challenging and has not been done is graphene \[It may be possible with a grating coupler [@Demel1991odp].\] On the other hand, working with graphene disks, one can effectively access the quantized values $q_\parallel = m / R$, where $m$ is the azimuthal quantum number. The observed mode [@Yan2012ist; @Yan2012pcg; @Fang2013gth] is evidently the dipolar one, $m = l = 1$, which has the highest optical weight. An additional mode that appears in both in ribbons and disks is the *edge* plasmon. We will talk about it at the end of this section where we discuss the effects of magnetic field. The correspondence between the ribbon and bulk plasmon dispersions enables one to also verify the $|N|^{1/4}$-scaling predicted by Eq. . This has been accomplished by electrostatic gating of graphene micro-ribbons immersed in ionic gel [@Ju2011gpf] and monitoring their resonance frequency. Plasmons in graphene are believed to strongly interact with electrons. Using the ARPES @Bostwick2007qdi [@Bostwick2007rog; @Bostwick2010oop] observed characteristic departure of the quasiparticle dispersion from linearity near the Dirac point energy accompanied by an additional dispersion branch. These features, discussed in more detail in Sec. \[sec:e-ph-pl\], were interpreted in terms of plasmarons: bound states of electrons and plasmons [@Lundqvist1967sps]. @Walter2011esa [@Walter2011eso] demonstrated that the details of the plasmaron spectrum are sensitive to dielectric environment of graphene. @Carbotte2012eop proposed that plasmaron features can be detected in near-field optical measurements, which allow one to probe the IR response at momenta $q \gg \omega / c$. Complementary insights on the interaction between plasmons and quasiparticles have been provided by the STS. Based on the gate dependence of the tunneling spectra, @Brar2010ooc distinguished phonon and plasmon effects on the quasiparticle self-energy. Plasmons in graphene strongly interact with surface phonons of polar substrates such as SiC, SiO$_2$, and BN. Dispersion of mixed plasmon-phonon modes in graphene on SiC was investigated experimentally using high-resolution EELS [@Liu2008pda; @Liu2010pps; @Koch2010spp] and modeled theoretically by @Hwang2010ppc. Theoretical dispersion curves [@Fei2011ino] for graphene on SiO$_2$ are shown in the inset of Fig. \[fig:3.3.1\]b. The dispersion characteristic of mixed plasmon-phonon modes in nanoribbons measured via far-field IR spectroscopy was reported in [@Yan2012pcg; @Yan2013dpm]. In the near-field IR nanoscopy study of graphene micro-crystals on SiO$_2$ [@Fei2011ino] the oscillator strength of the plasmon-phonon surface modes was shown to be significantly enhanced by the presence of graphene, Fig. \[fig:3.3.1\]a. The strength of this effect can be controlled by electrostatic doping, in agreement with theoretical calculations @Fei2011ino. Imaging of plasmon propagation in real-space [@Fei2012gto; @Chen2012oni] \[Fig. \[fig:3.3.1\](left)\] have led to the first direct determination of both real and imaginary parts of the plasmon momentum $q_p = q_p' + i q_p''$ as a function of doping. In terms of potential applications of these modes, an important characteristic is the confinement factor $\lambda_p / \lambda_0$, where $\lambda_p = 2 \pi / q_p'$ is the plasmon wavelength and $\lambda_0 = 2 \pi c / \omega$ to the wavelength of light in vacuum. Experimentally determined confinement factor in exfoliated graphene @Fei2012gto was $\sim 65$ in the mid-IR spectral range $\omega \approx 800\,\mathrm{cm}^{-1}$. According to Eqs , the scale for the confinement is set by the inverse fine-structure constant, $\lambda_0 / \lambda_p = (\kappa / 2)(\hbar c / e^2)(\hbar \omega / \,|\mu|\,)$, with stronger confinement achieved at higher frequencies. The propagation length of the plasmons $\sim 0.5\, \lambda_p = 100$–$150\,\mathrm{nm}$ is consistent with the residual conductivity $\sigma' \approx 0.5 e^2 / h$ measured by the conventional IR spectroscopy [@Li2008dcd]. Possible origins of this residual conductivity have already been discussed above, Sec. \[sec:direct\]. In confined structures one additional mechanism of plasmon damping is scattering by the edges [@Yan2013dpm]. Despite the observed losses, the plasmonic figures of merits demonstrated by @Fei2012gto [@Chen2012oni] compare well against the benchmarks set by noble metals. Even though surface plasmons in metals can be confined to scales of the order of tens of $\mathrm{nm}$, their propagation length in this regime is plagued by giant losses and does not exceed $0.1 \lambda_p \sim 5\, \text{nm}$ for Ag/Si interface [@Jablan2009pig]. This consideration has not been taken into account in a recent critique of graphene plasmonics [@Tassin2012aco]. Further improvements in the figures of merits are anticipated for graphene with higher electronic mobility. The key forte of graphene in the context of plasmonics is the control over the plasmon frequency and propagation direction [@Mishchenko2010gpi; @Vakil2011tou] by gating. The properties of graphene plasmons get modified in the presence of a transverse magnetic field $B$. The magnetoplasmon dispersion is obtained from Eq.  by replacing $\sigma$ with the longitudinal conductivity $\sigma_{x x}$. For instance, instead of the Drude model , one would use its finite-$B$ analog, the Drude-Lorentz model [@Ashcroft1976ssp], which yields another well-known dispersion relation [@Chiu1974pot] $$\label{eqn:omega_mp} \omega_{mp}(q) = \sqrt{\omega_p^2(q) + \omega_c^2}\,.$$ This magnetoplasmon spectrum is gapped at the cyclotron frequency $\omega_c = e B / m c$ defined through the effective mass $m$ \[Eq. \]. Equation  is valid at small enough $B$ where Landau quantization can be ignored. At large $B$, quantum treatment is necessary. In the absence of interactions, the magnetoplasmon gap at $q = 0$ is given by $E_{n + 1} - E_n$, the energy difference between the lowest unoccupied $n + 1$ and the highest occupied $n$ Landau levels. Unlike the case of a 2DEG, where the Kohn’s theorem holds, renormalization of the Fermi velocity by interactions directly affects the cyclotron gap. This many-body effect has been observed by magneto-optical spectroscopy, Sec. \[sec:Landau\]. Probing finite-$q$ magnetoplasmons optically is possible via the finite-size effects, such as the mode quantization in graphene disks. As known from previous experimental [@Glattli1985dhe; @Mast1985obe; @Demel990ndr; @Demel1991odp; @Kukushkin2003ore], numerical [@Eliasson1986mms], and analytical [@Volkov1988eml] studies of other 2D systems, the single plasmon resonance at $B = 0$ splits into two. The upper mode whose frequency increases with $B$ can be regarded the bulk magnetoplasmon with $q \approx 1 / R$, where $R$ is the disk radius. The lower mode whose frequency drops with $B$ can be interpreted as the edge magnetoplasmon, which propagates around the disk in the anti-cyclotron direction. Both the bulk-like and the edge-like modes have been detected by the IR spectroscopy of graphene disk arrays [@Yan2012ist; @Yan2012pcg]. Additionally, in epitaxial graphene with a random ribbon-like microstructure, the $B$-field induced splitting of the Drude peak into a high- and a low-frequency branch was observed and interpreted in similar terms [@Crassee2012itp]. The distinguishing property of the edge magnetoplasmon is chirality: its the propagation direction is linked to that of the magnetic field. This property has been verified in graphene systems by time-domain spectroscopy [@Petkovic2013cdv; @Kumada2013pti], which also allowed extraction of the edge magnetoplasmon velocity. Other interesting properties of magnetoplasmons, such as splitting of the classical magnetoplasmon dispersion  into multiple branches have been predicted theoretically [@Roldan2009cmd; @Goerbig2011epg] and their similarities and differences with the 2DEG case have been discussed. These effects still await their experimental confirmation. Phonons {#sec:phonons} ------- Raman spectroscopy is the most widely used tool for probing optical phonons in graphene and related materials [@Ferrari2006rso; @Ferrari2007rso; @Dresselhaus2010poc; @Dresselhaus2012rsc]. Quantitative studies of the Raman modes can provide rich information on graphene electron-phonon interaction, electronic structure, as well as on graphene layer thickness, edges, doping, and strain. Because graphene has the same $s p^{2}$ bonding and hexagonal carbon lattice, its phonon band-structure is almost identical to that in graphite. Figure \[fig:3.4.1\]a shows calculated dispersion of the optical phonon branches in graphene (lines) [@Piscanec2004kaa] as well as the experimental data of graphite (symbols) [@Maultzsch2004pdi]. One feature of these dispersions is the discontinuity in the frequency derivative at the $\Gamma$ and $\mathrm{K}$ points in the highest optical branches. This discontinuity known as the Kohn anomaly arises from the unusual electron-phonon coupling in graphitic materials [@Piscanec2004kaa]. ![\[fig:3.4.1\] (Color online) Phonon dispersion and Raman spectroscopy of graphene. (a) Calculated phonon dispersion of SLG [@Piscanec2004kaa] (symbols) compared with the experimental data for graphite [@Mohr2007pdo] (lines). (b) Raman spectra of graphene and graphite measured at $514\,\text{nm}$ laser excitation showing the $G$ and the $2D$ Raman peaks [@Ferrari2006rso]. (c) The evolution of the $2D$ Raman peak with the number of graphene layers [@Ferrari2006rso]. (d) Schematics of the $G$-mode and the $2D$-mode Raman scattering processes.](Basov_Fig9){width="3.2in"} Figure \[fig:3.4.1\]b displays typical Raman spectra of SLG and graphite. They show the same qualitative Raman modes, with the two most prominent features being the $G$-mode ($\approx 1580\,\text{cm}^{-1}$) and the $2D$-mode ($\approx 2700\,\text{cm}^{-1}$, also known as $G'$ mode). The other weak but very informative Raman feature is the $D$-mode ($\approx 1350\,\text{cm}^{-1}$). The lineshape of $2D$ mode is very different in SLG, MLG, and graphite (Fig. \[fig:3.4.1\]c) [@Ferrari2006rso]. As illustrated in Fig. \[fig:3.4.1\]d, the $G$-peak arises from Raman scattering of the $\Gamma$-point phonon. The $2D$-peak, on the other hand, is a two-phonon process involving emission of two $\mathrm{K}$-point optical phonons. The $D$-peak is a double resonance process like the $2D$-peak. It requires structural defects to relax the momentum conservation constraint. {width="7.0in"} A detailed theory of the $G$-mode Raman signal was presented in @Basko2008ioc [@Basko2008tor; @Basko2009eei; @Basko2009cot]. The capability of controlling the electron Fermi energy through electrical gating helped to elucidate electron-phonon interactions [@Yan2007efe; @Pisana2007bot; @Malard2008ood; @Das2008mdb] and the quantum interference between different intermediate excitation pathways [@Chen2011cil; @Kalbac2010tio]. The frequency and linewidth of the Raman $G$-mode reflect the energy and lifetime of the optical phonon at the $\Gamma$ point. The $\Gamma$-point phonon experiences Landau damping by particle-hole excitations if its energy exceeds $2 |\mu|$ (see Fig. \[fig:Excitations\]). As a result, the parameters of the Raman $G$-mode depend on the carrier concentration, as demonstrated experimentally [@Yan2007efe; @Pisana2007bot]. The $G$-mode Raman shows a reduced damping and a blue shift when the Fermi energy is larger than one half of the phonon energy, so that the phonon decay pathway into electron-hole pairs gets blocked. When the Fermi energy in graphene is increased further, some of the intermediate electronic transitions necessary for Raman scattering become blocked. This reduces destructive interference among different pathways and increases the $G$-mode signal [@Chen2011cil]. The Raman scattering that gives rise to the $2D$ mode involves emission of two BZ-boundary phonons close to the $\mathrm{K}$-point. Being a two-phonon process, it still has large intensity, which is explained by the triple-resonance mechanism (Fig. \[fig:3.4.1\]d), where every intermediate step involves a resonant electronic excitation [@Basko2008tor; @Basko2009eei]. Due to smallness of the phonon energy compared with the incident photon energy $\hbar\omega$, the momenta $\mathbf{k}$ of the intermediate electron states are restricted to $\hbar\omega \approx E(\mathbf{k})$, where $E(\mathbf{k})$ is the electron dispersion (Sec. \[sec:single\]). The phonon momentum (relative to a $\mathrm{K}$-point phonon) then equals $2 (\mathbf{k} - \mathbf{K})$. Consequently, phonons and intermediate electronic transitions with specific momentum can be excited by varying incident photon energy for $2D$ Raman modes. This allows one to map the dispersion of both the phonon and the electrons. Once the phonon dispersion is known, Raman scattering can be used to probe electronic band-structure changes with a fixed laser excitation. For example, it can distinguish SLG, BLG, and MLG due to their different electronic dispersions [@Ferrari2006rso]. In BLG and MLG there are several conduction and valence bands (Sec. \[sec:BLG\]). Hence, valence electrons at more than one momentum $\mathbf{k}$ can satisfy the $\hbar \omega = E(\mathbf{k})$ relation. This leads to an apparent broadening and asymmetry of the $2D$ Raman peaks for BLG and MLG, compared to those for SLG [@Ferrari2006rso]. The Raman $D$-mode (short for the defect-mode) requires the existence of atomically sharp defects to provide the required momentum matching to scatter a zone boundary phonon close to $\mathrm{K}$-point. The intensity of the $D$-peak is used to characterize the sample quality of graphene [@Malard2009rsi; @Dresselhaus2010poc; @Ferrari2007rso]. The $D$-mode is also useful for probing graphene edges, which can be considered as line defects. Experiments show that the $D$-peak is indeed the strongest at graphene edges [@Graf2007srr; @Gupta2009pge; @Casiraghi2009rso], and that the $D$-mode intensity is at maximum for light polarization parallel to the edge and at minimum for the perpendicular polarization [@Casiraghi2009rso]. For ideal edges, theory predicts that the $D$-mode Raman peak intensity is zero for zigzag edges but large for armchair ones [@Casiraghi2009rso]. In addition to the effects discussed above, the intensity and frequency of Raman peaks also depend on the substrate [@Wang2008rso; @Ni2008uso; @*Ni2009uso; @Lee2008rso; @Berciaud2008pti], temperature [@Calizo2007tdo], and strain [@Yu2008rmi; @Proctor2009hpr; @Mohiuddin2009usi; @Huang2009psa] through their effects on the phonon dispersion and electron Fermi energy. Electron-phonon and electron-plasmon interaction {#sec:e-ph-pl} ------------------------------------------------ The energies and lifetimes of charge carriers in graphene are significantly affected by interactions with plasmons and phonons. The electron-phonon (el-ph) interaction results in a variety of novel phenomena discussed in Sec. \[sec:phonons\]. The ARPES has been used to probe the signature of the el-ph interaction in the electronic spectra of graphene [@Bostwick2007qdi; @McChesney2007meo; @McChesney2010evh; @Zhou2008dft] via the measurement of the quasiparticle velocity $v$. The el-ph coupling constant is usually defined by $\lambda = v_0 / v - 1$ [@Ashcroft1976ssp]. However, electron-electron (el-el) interaction also contributes to velocity renormalization (Secs. \[sec:interaction\] and \[sec:renormalization\]). Hence, thus defined $\lambda$ gives a good estimate of el-ph coupling only if el-el interaction is screened, which is the case for graphene on a metallic substrate @Siegel2011mbi. The el-ph interaction in graphene strongly depends on the carrier concentration, as shown in Fig. \[fig:el-ph\]a,b. @Siegel2011mbi have reported a large reduction of $\lambda$ for quasi-free-standing graphene with $E_{F}$ close to the Dirac point $E_{D}$. The overall reduction of the el-ph interaction can be reproduced by theoretical calculations [@Park2007vra]. However, to account for the fine features of the quasiparticle dispersion, the el-el interaction has to be included [@Siegel2011mbi; @Zhou2008dft; @Lazzeri2008iot]. At high doping $\lambda$ appears to be enhanced, reaching values $\lambda \sim 2$, and strongly anisotropic, similar to what is observed in graphite [@Zhou2006lee; @Leem2008eol; @Park2008epi] and in the intercalated compound CaC$_{6}$ [@Valla1999mbe]. @Calandra2007eso argued these effects result from distortion of the graphene bands that hybridize with a new Ca-related band. On the other hand, @Park2008vhs suggested that the anisotropy of $\lambda$ comes from the nonlinear band dispersion of the graphene bands at high doping. From a high resolution ARPES study @Zhou2008kaa concluded that the electron-phonon coupling is dominated by the following phonon modes: $A_{1g}$ phonon at approximately $150\pm 15\,\text{meV}$ near the BZ corner, $E_{2g}$ phonon ($\sim 200\, \text{meV}$) at the zone center, and the out-of-plane phonon at $60\, \text{meV}$. Among these, the $A_{1g}$ phonon is the one that mostly contribute to $\lambda$ and mainly responsible for the kinks in the ARPES and in the tunneling spectra [@Li2009sts], see Fig. \[fig:3.6.2\]b,c. The contribution of a specific phonon mode to $\lambda$ can also be determined by studying how the Raman signal varies as a function of the applied magnetic field. These magneto-Raman studies focused on the $E_{2g}$ phonon [@Faugeras2009tep; @Faugeras2011mrs], as the $A_{1g}$ phonon is Raman inactive in high quality graphene samples. The origin of the large discrepancy (Fig. \[fig:el-ph\]d) between theoretically predicted and experimentally measured values of $\lambda$ is debated.[^15] @Siegel2012epc found a good agreement with the theory (Fig. \[fig:el-ph\]d) using the bare velocity $v_0$ measured for graphene grown on Cu where the el-el interaction is expected to be screened. Electron-plasmon interaction is also believed to play an important role in renormalizing the band structure of graphene. @Bostwick2007qdi [@Bostwick2007rog; @Bostwick2010oop] have argued that this interaction is responsible for the anomalous departure from the linear dispersion observed in epitaxial graphene grown on the Si face of SiC. @Bostwick2010oop have provided evidence (Fig. \[fig:3.6.2\]) for a well-resolved plasmaron band in the ARPES spectra of a “freestanding” graphene sample in which hydrogen has been intercalated between graphene and SiC to make negligible the interaction between the two. The plasmaron of momentum $\mathbf{k}$ is a bound state of a hole with momentum $\mathbf{k} + \mathbf{q}$ and a plasmons of momentum $-\mathbf{q}$ [@Lundqvist1967sps]. Theoretical calculations [@Polini2008pat] within the $GW$ approximation predict that the plasmaron band appears at finite charge densities. Its energy separation from the primary quasiparticle band is proportional to $\mu$ with a coefficient that depends on the Coulomb interaction strength $\alpha$, which in turn depends on the dielectric environment of graphene. Quantitative aspects of these calculations were disputed by @Lischner2013pos who included vertex corrections neglected in the $GW$ scheme. Compared to @Polini2008pat, for the same $\alpha$ @Lischner2013pos find a broader plasmaron peak at a smaller separation from the primary band, which appears to be in a better agreement with the experiments of @Bostwick2010oop. No evidence of the plasmaron band has been reported in samples where decoupling of graphene from the buffer later was achieved by either gold or fluorine intercalation [@Walter2011hpd; @Starodub2011ipo]. This has been attributed to a stronger dielectric screening by the buffer layer. An alternative interpretation of the apparent nonlinearity of the Dirac spectrum of graphene on SiC invokes a substrate-induced band gap [@Zhou2007sib; @Zhou2008kaa; @Zhou2008oot; @Benfatto2008sso; @Kim2008ooa], see Sec. \[sec:substrate\] below. ![\[fig:3.6.2\] (Color online) The ARPES dispersion of doped ($N = 8 \times 10^{10}\,\text{cm}^{-2}$) graphene perpendicular (a) and parallel (b) to the $\Gamma$-$\mathrm{K}$ direction [@Bostwick2010oop]. The dashed black lines are guides to the eye for the dispersion of the hole and plasmaron bands; the solid red line goes through the Dirac point. The inset shows a schematic of the renormalized spectrum in the presence of plasmarons.](Basov_Fig11){width="3.1in"} Induced effects {#sec:Induced} =============== Inhomogeneities and disorder {#sec:Inhomogeneities} ---------------------------- Intentional and unintentional doping by charged impurities plays a very important role in the electronic phenomena of graphene. It is unclear if there is a single dominant source of unintentional doping even for most studied type of samples: exfoliated graphene on SiO$_2$. In addition to adsorbates from the ambient atmosphere, doping could also result from charged defects in SiO$_2$ [@Adam2007sct; @Wehling2007mdo; @Schedin2007doi; @Zhou2008mti; @Coletti2010cna] lithographic residues [@Dan2009iro], and metal contacts [@Connolly2010sgm]. The dopants introduce not only a change in the average carrier concentration but also charge inhomogeneities and scattering. Near the graphene neutrality point inhomogeneities of either sign can arise, which are often referred to as the electron-hole puddles [@Geim2007rg]. Thus, even at the neutrality point the graphene is always locally doped. This is a qualitative explanation for nonvanishing conductivity [@Geim2007rg; @Tan2007mos; @Chen2008cis] and TDOS [@Martin2008ooe]. A more detailed model [@Adam2007sct; @Hwang2007ctt; @Shklovskii2007smo; @Rossi2008gsg] invokes a system of conducting electron-rich and hole-rich regions separated by $p$-$n$ junctions [@Cheianov2007tfo; @Zhang2008nsa]. The transport involves percolation through the $p$ and $n$ regions aided by tunneling across the junctions [@Cheianov2007rrn; @DasSarma2011eti]. Many elements of this semiclassical model hark back to the earlier studies of two-dimensional [@Efros1993dos; @Fogler2004nsa] and three-dimensional [@Efros1984epd] electron systems in semiconductors. However, the puddle model may not be quantitatively reliable for graphene. The correlation length of the density inhomogeneities is typically very short. For SLG on SiO$_2$ it was consistently estimated to be of the order of $20\,\mathrm{nm}$ using several complementary scanned probes microscopy techniques [@Deshpande2011icd; @Luican2011qll; @Berezovsky2010ict; @Martin2008ooe]. A typical electron-hole “puddle” is also too small to contain even a single charge [@Martin2008ooe]. Therefore, the inhomogeneities in question may be better described as quantum interference patterns of disorder-scattered electron waves rather than large semiclassical puddles. The situation may change if Coulomb interactions among electrons and impurities is screened. The crossover to the semiclassical regime is predicted to occur [@Fogler2009npo] for graphene on a substrate of high dielectric constant $\kappa \gg 1$. Suppression of density inhomogeneities in one graphene layer due to screening by a nearby second layer has been invoked to explain the observed localization transition in graphene-hBN-graphene structures [@Ponomarenko2011tmi]. The inhomogeneities may also be induced by elastic strain and ripples [@Brey2008eic; @Guinea2008msa; @Gibertini2010edd]. Electron density inside the highly strained graphene bubbles [@Bunch2008iam; @Levy2010sip; @Georgiou2011gbw] is undoubtedly inhomogeneous. However, the relation between strain and electron density is nonlocal. Indeed, no local correlations between the carrier density in graphene and the roughness of SiO$_2$ substrate is evident in scanned probe images [@Martin2008ooe; @Deshpande2011icd; @Zhang2009oos]. The hypothesis that unintentional doping is caused by impurities trapped under graphene is supported by some micro-Raman experiments showing that proximity to the SiO$_{2}$ substrate results in increase of carrier density [@Berciaud2008pti; @Ni2009pci; @Bukowska2011rso]. Yet other micro-Raman measurements [@Casiraghi2009rso] have not observed such correlations. {width="5.0in"} Charge inhomogeneities can be reduced by either removing the substrate [@Du2008abt; @Knox2011mar] or using a high-quality hBN substrate [@Dean2010bns]. The random charge fluctuations of exfoliated graphene on hBN are at least two orders of magnitude smaller than those on SiO$_2$ according to the STM studies [@Xue2011stm; @Decker2011lep]. (However, in such structures periodic charge oscillations may appear instead of random ones, see Sec. \[sec:moire\].) These random fluctuations are on par with the values estimated from transport data for free-standing graphene [@Du2008abt]. The electronic mobility of graphene on hBN approaches $\sim 10^5\,\mathrm{cm}^2/\mathrm{Vs}$ implying the mean-free path of several hundreds nm [@Du2008abt; @Dean2010bns]. Although detrimental for transport properties, impurities can play a role of elementary perturbations that help reveal useful physical information. We can give two examples. First, disorder-induced LDOS fluctuations seen in STS [@Rutter2007sai; @Zhang2009oos] reveal the dominant momenta for inter- and intra-valley scattering and therefore shed light on chirality and energy spectrum of the quasiparticles. Second, by utilizing ionized Co adatoms one can study screening properties of graphene. The screening cloud surrounding the adatoms was shown to have a qualitatively different profile depending on the total charge of the adatom cluster. In the sub-critical case this profile is governed essentially by the linear response dielectic constant of graphene. Theoretical modeling of the STS spectra [@Brar2011gci; @Wang2012mdq] suggests the enhanced value $\epsilon \approx 3.0$ of this constant, which is indicative of many-body interactions [@Sodemann2012ict]. In the super-critical case [@Wang2013oac] sharp resonances in the local DOS appear, which is the hallmark of a nonlinear screening with intriguing analogy to “atomic collapse” of super-heavy elements. Substrate-induced doping {#sec:substrate} ------------------------ Metallic substrates induce a strong doping of graphene, which is readily seen by the ARPES (Fig. \[fig:4.1.1\]a). The chemical potential $\mu = E_{F} - E_D$ measured with respect to the Dirac point ranges from approximately $0.5\,\text{eV}$ for Cu (111) [@Gao2010ego] and Cu films [@Siegel2012epc; @Walter2011eso] to $2\,\text{eV}$ for other transition metals, such as Ni (111) [@Nagashima1994eso; @Dedkov2008rei; @Varykhalov2008eam], Ru (0001) [@Himpsel1982abd; @Enderlein2010tfo; @Sutter2009], and Co (0001) [@Rader2009ita]. An exception to this is graphene on Ir (111) [@N'Diaye2006tdi; @Pletikosic2009dca], where the surface states of the substrate cause pinning of $\mu$ near zero. Naively, graphene is $n$-doped if $W_G > W_M$ and $p$-doped otherwise, where $W_G = 4.5\,\text{eV}$ is the work function of pristine graphene and $W_M$ is that of the metal. In fact, the charge transfer is affected by chemical interaction between graphene and the metal and by their equilibrium separation [@Giovannetti2008dgw]. The amount of charge transfer can be modified by intercalation. Fluorine intercalation yields a large $p$-type doping of graphene [@Walter2011hpd]. Hydrogen intercalation leads to decoupling of graphene from the substrate [@Riedl2009qfs], as evidenced by the ARPES dispersions, Fig. \[fig:4.1.1\](b) and (c), typical of suspended graphene, cf. Fig. \[fig:4.1.1\](d). Similar effects can be obtained by Au intercalation [@Gierz2008ahd]. When gold atoms are intercalated between graphene and a Ni (111) substrate [@Varykhalov2008eam], $\mu$ drops down to $25\,\text{meV}$, corresponding to the two orders of magnitude decrease in the carrier concentration. Moiré patterns and energy gaps {#sec:moire} ------------------------------ When the lattice constants of the graphene layer and the substrate differ by a small relative amount $\delta$ and/or misoriented by an angle $\phi$ a moiré supelattice arises [@Marchini2007stm; @N'Diaye2006tdi; @Wang2008coo; @Wintterlin2009gom]. The electron dispersion in the presence of the moiré superlattice gets modified as a result of hybridization of the original Dirac cones with their replicas folded into a superlattice Brillouin zone (sBZ). Such replicas have been seen in the ARPES spectra of graphene on Ir (111) [@Pletikosic2009dca] although they may also be due to the final-state diffraction [@Sutter2009]. The most striking experimental manifestations of the moiré superlattice effects have recently been observed in SGL on hBN. This system has $\delta = 1.8\%$, so that the moiré period can be as long as $14\,\mathrm{nm}$, which can be easily imaged by scanned probes (Fig. \[fig:Moire\], insets). Dependence of the moiré period on the misorientation angle $\phi$ is very sharp, Fig. \[fig:Moire\]A, so achieving large period requires precise alignment. It has been predicted theoretically that at the intersections of the replica and the main bands new Dirac points appear, Fig. \[fig:Moire\]C. For the practical case of a weak superlattice potential, these points have energy $$E_D^m \simeq E_D \pm \frac{2\pi}{\sqrt{3}}\, \frac{\hbar v}{\Lambda}\,, \label{eqn:moire_DP}$$ where $\Lambda$ is the moiré period. \[For the opposite limit of strong modulation, see [@Brey2009ezm].\] The extra Dirac points are characterized by a modified and generally, anisotropic quasiparticle velocity [@Park2008ngo; @Guinea2010bsa; @Wallbank2013gms]. The original Dirac point at the center of the sBZ remains gapless, at least, within the scope of generic theoretical models of the moiré superlattice which preserve sublattice symmetry. First attempts to identify extra Dirac points in SLG/hBN structures were unsuccessful because these points were outside the experimental energy window [@Xue2011stm]. In more recent experiments, which utilized precisely aligned ($\phi < 0.5^\circ$) structures, the new Dirac points are clearly evidenced by additional minima of the DOS measured by STS [@Yankowitz2012esd], Fig. \[fig:Moire\]B. The unmistakable signatures of the second-generation Dirac points in transport include peaks in longitudinal resistance and the sign change of the Hall resistance [@Ponomarenko2013cdf; @Dean2013hba; @Yankowitz2012esd; @Yang2013egs], Fig. \[fig:Moire\]C and D. Additional Landau level (LL) fans emerging from these extra Dirac points are seen in magnetotransport [@Ponomarenko2013cdf; @Dean2013hba; @Hunt2013mdf] and the gate capacitance measurements. The detailed structure of such LLs is predicted to be fractal, as spectacularly illustrated by the iconic image of the “Hofstadter butterfly” [@Hofstadter1976elw]. Experiments [@Hunt2013mdf] in strong fields $B > 20\,\mathrm{T}$ demonstrate additional quantum Hall plateaus and a large gap (tens of meV) at the neutrality point, the physical origin of which remains to be understood. {width="5.5in"} Opening a band gap at the Dirac point is indeed the most often cited effect that can enable wider applications of graphene. Inducing gap by confinement in various graphene superstructures such as quantum dots, ribbons, *etc.*, has proved to be problematic due to disorder effects. Inducing a gap through graphene/substrate interaction seems an attractive alternative. The most straightforward mechanism of the gap generation is breaking the sublattice symmetry of graphene [@Brey2006eso; @Giovannetti2007sib; @Nakada1996esi; @Nilsson2007tta; @Trauzettel2007sqi] For a hypothetical commensurate SLG/hBN structure, a band gap $\sim 50\,\mathrm{meV}$ was predicted [@Giovannetti2007sib; @Slawinska2010egt]. Theoretically, the gap can also be induced by hybridization of the two valleys [@Manes2007eat]. A small band gap in graphene can also be induced by a spin-orbit coupling of Rashba type [@Kane2005qsh]. The curvature of the graphene sheet is predicted to enhance the Rashba splitting [@Huertas-Hernando2006soc; @Kuemmeth2008cos]. Existence of substrate-induced gaps have been indicated by many ARPES experiments. A very wide spread of gap values has been reported, which remains unexplained. One of the earliest ARPES studies [@Oshima1997ute] claimed the largest gap so far, $1.3\,\text{eV}$, for a “soft” SLG on TaC (111). Large gaps have also been reported for graphene on certain metallic substrates. @Brugger2009coe found $\sim 1\,\mathrm{eV}$ gap for SLG on Ru (0001). @Nagashima1994eso observed $0.7$–$1.3\,\text{eV}$ gaps in SLG on Ni (111) intercalated with alkaline metals. For Ru (0001) covered a with monolayer of gold [@Enderlein2010tfo] and for Ir (111) substrates [@Starodub2011ipo] the gap is $0.2\,\text{eV}$. The effect of intercalants is counterintuitive because these are metals to which graphene interacts weakly. The gap for SLG on Cu is $0.3$–$0.4\,\text{eV}$. @Zhou2007sib made a case for the $0.26\,\text{eV}$ gap in epitaxial graphene on SiC. As discussed in Sec. \[sec:e-ph-pl\], a competing interpretation of these data is in terms of plasmarons [@Bostwick2007qdi; @Bostwick2007rog; @Bostwick2010oop]. Comparable gaps were found for other semiconducting substrates [@Siegel2011mbi; @Walter2011eso]. For graphene on graphite the gap of $20\,\text{meV}$ [@Li2009sts; @Siegel2011mbi] was reported. The STS @Kawasaki2002dal observed a $0.5\,\text{eV}$ for SLG/single-layer hBN/Ni (111) structure. The largest Rashba splitting $13\pm 3\,\text{meV}$ has been reported for graphene on magnetic substrate Ni (111) intercalated by Au [@Varykhalov2008eam]. The mechanism behind this enhancement is still unknown. No Rashba splitting has been observed on another magnetic substrate, Co (0001) intercalated by Au [@Rader2009ita]. Although intrinsic spin-orbit (SO) coupling is also responsible for the opening a band gap, $\Delta_{\mathrm{SO}}$, in pure SLG it is predicted to be extremely small, e.g., $10^{-3}$–$10^{-2}\,\mathrm{meV}$ [@Kane2005qsh]. A broken symmetry at the interface of two SLG can somewhat amplify this gap [@Schmidt2010ese]. Impurities in graphene resulting in $s p^3$ type deformation of the flat graphene are also predicted to enhance $\Delta_{SO}$ up to $7\,\mathrm{meV}$ [@CastroNeto2009iis]. In fact, a recent experimental study on hydrogenated graphene revealed a drastically enhanced $\Delta_{\mathrm{SO}}$ of $2.5\,\mathrm{meV}$ [@Balakrishnan2013ces]. Alternatively, the interactions of charge carriers in graphene with heavy atoms such as In and Tl adsorbed on graphene are predicted to enhance $\Delta_{\mathrm{SO}}$ up to $7\,\mathrm{meV}$ and $21\,\mathrm{meV}$, respectively [@Weeks2011erq]. Elastic strain {#sec:strain} -------------- ![\[fig:4.2.1\] (Color online) STM images and STS spectra taken at $7.5\,\text{K}$. (A) Graphene monolayer patch on Pt(111) with four nanobubbles at the graphene-Pt border and one in the patch interior. Residual ethylene molecules and a small hexagonal graphene patch can be seen in the lower right (3D $z$-scale enhanced $4.6\times$). (Inset) High resolution image of a graphene nanobubble showing distorted honeycomb lattice resulting from strain in the bubble ($\text{max}\, z = 1.6\,\text{nm}$, 3D $z$-scale enhanced $2\times$). (B) STS spectra of bare Pt(111), flat graphene on Pt(111) (shifted upward by $3 \times 10^{-11}\, \Omega^{-1}$), and the center of a graphene bubble (shifted upward by $9 \times 10^{-11}\, \Omega^{-1}$). The peaks in the graphene bubble spectrum indicate the formation of pseudo-Landau levels. (C) Normalized peak energy versus $\text{sgn}\,(n) \sqrt{|n|}\,$ for peaks observed on five different nanobubbles follow expected scaling behavior (see text). Adapted from @Levy2010sip.](Basov_Fig14){width="2.4in"} A controlled uniaxial strain can be readily introduced into graphene by stretching the flexible substrate. The strain modifies graphene phonon energy spectrum, which is effectively probed by Raman spectroscopy. Under uniaxial strain the $G$ and $2D$ phonon bands display significant red shift proportional to the applied strain: a result of the anharmonicity of the interatomic potentials in graphene [@Ni2008uso; @*Ni2009uso; @Huang2009psa; @Mohiuddin2009usi; @Tsoukleri2009sag]. Meanwhile the $sp^{2}$ bonds of graphene lengthen/shorten in the direction parallel/perpendicular to the strain axis. This reduces the $C_{6}$ symmetry of the honeycomb lattice to $C_{2}$, and splits the doubly degenerate $G$ band into two singlet bands, $G^{+}$ and $G^{-}$, with normal modes perpendicular and parallel to the strain axis, respectively. The polarization of Raman scattered light for the $G^{+}$ and $G^{-}$ modes is thus expected to depend on the direction of the strain axis relative to the crystal orientation: a conjecture verified by @Huang2009psa and @Mohiuddin2009usi. Strain also introduces profound modifications to graphene electronic structure. The defining topology feature of graphene electronic band, namely the degeneracy of conical electron and hole bands at Dirac points, is protected by the inversion symmetry of graphene lattice [@Hasegawa2006zmo; @Kishigi2008ego; @Wunsch2008dpe]. A small perturbation in form of mechanical strain does not lift the degeneracy, but deforms the energy bands and shifts the Dirac points in both the energy and the momentum space. The former is equivalent to a scalar potential also known as the deformation potential. A general nonuniform strain generates a spatially varying dilation of the graphene lattice and therefore local ion density. The deformation potential arises because the corresponding Coulomb potential is only partially screened by electrons [@Suzuura2002pae; @Kim2008gaa; @Guinea2008gfi]. Next, shifting the Dirac point in the $k$-space away from the $\mathrm{K}$ ($\mathrm{K}'$) points [@Pereira2009seo; @Farjam2009cob; @Ni2008uso; @*Ni2009uso; @Dietl2008nmf]. is analogous to the effect induced by an external magnetic field applied perpendicular to the graphene plane. One can parametrize the mechanical strain by a gauge field $\mathbf{A}$ [@Iordanskii1985dal; @Kane1997ssa; @Sasaki2005leg; @Morpurgo2006isl; @Katsnelson2007gnb; @Fogler2008pfa] and define the pseudomagnetic field $B_s = \bm{\nabla} \times \mathbf{A}$. The strain-induced $\mathbf{A}$ and $B_s$ have opposite signs at two valleys $\mathrm{K}$ and $\mathrm{K}'$, so that the time-reversal symmetry is preserved. It is possible to engineer a special nonuniform strain for which $B_s$ is approximately constant in a finite-size region. If such pseudomagnetic field is strong enough, it can lead to Landau quantization and quantum Hall-like states [@Guinea2010ega; @Guinea2010gqp]. In a recent experiment [@Levy2010sip], such an unusual Landau quantization has been observed in highly strained graphene nanobubbles. The strain arises upon cooling because of a mismatch in the thermal expansion coefficients of graphene and the Pt substrate (Fig. \[fig:4.2.1\]A). The pseudo-Landau levels, manifested as local density of states peaks, are probed by STS (Figs. \[fig:4.2.1\]B and C). Their energies follow the theoretically predicted $\text{sgn}(n) \sqrt{|n|}$ behavior with a gigantic $B_s \sim 300\,\text{T}$, see Fig. \[fig:4.2.1\]C. Photo-induced effects {#sec:photo-induced} --------------------- Optical spectroscopy with ultrafast laser excitation pulses provides a unique tool to probe the dynamic evolution of electrons and phonons in graphene, including the cooling of the non-equilibrium quasiparticle plasma through electron-electron and electron-phonon interactions and the relaxation of hot optical phonons. These processes are not only of fundamental interest due to the unusual electronic structure in graphene, but also important for technological applications of high-field electronics and nonlinear photonic devices [@Xia2009ugp; @Bao2009alg; @Zhang2010gml; @Sun2010gml; @Bonaccorso2010gpa]. Response of graphene to a pulsed laser excitation has been studied by several complementary ultrafast spectroscopy techniques. For example, pump-probe IR/visible spectroscopy [@Sun2008uro; @Dawlaty2008mou; @Newson2009uck; @Huang2010uta] and pump-probe THz spectroscopy [@George2008uop] has been employed to track the time evolution of optical absorption and transmission by graphene. Ultrafast photoluminescence [@Lui2010upf; @Liu2010nbp; @Stoehr2010fol] has been used to monitor light emission by non-equilibrium electron gas. Time-resolved Raman spectroscopy [@Yan2009trr; @Chatzakis2011tdo] has explored generation and decay of hot optical phonons. @Breusing2009ucd studied graphite using $7\,\mathrm{fs}$ $1.55\,\mathrm{eV}$ pump pulses and a broadband probe pulses with energy spectrum from $1.2$ to $2\,\mathrm{eV}$. Figure \[fig:4.4.1\]a shows the observed increase of transmission during the first $150\,\mathrm{eV}$ with sub-$10\,\mathrm{fs}$ time resolution. This phenomenon is attributed to partial Pauli blocking of the optical transition by photo-excited electron-hole pairs. The change in transmission scales linearly with the pump influence and decays with two time constants of $13\,\mathrm{fs}$ and $100\,\mathrm{fs}$. The former characterizes electron-electron interactions, which cause energy redistribution within the conduction and valence band, as well as relaxation of occupation factors by Auger processes. The second time constant describes interaction of quasiparticles with optical phonons. The emission of optical phonons with energy $\approx 0.2\,\mathrm{eV}$ cools down electron-hole plasma. Once its temperature drops below this number, emission of optical phonons becomes ineffective. Eventual equilibration of the electron and lattice temperatures is achieved by emission of acoustic phonons on a time scale of $\sim 2\,\mathrm{ps}$. ![\[fig:4.4.1\] Ultrafast dynamics of excited electrons in graphene. (a) Spectrally integrated transmission change as a function of pump-probe delay (open circles). Solid line is the numerical fit, and dash-dotted line is cross correlation of pump and probe pulses. The inset shows linear dependence of the maximum transmission change on the absorbed pump fluence. The transmission increase is due to photo-induced Pauli blocking of interband transitions. The ultrafast decay is due to thermalization between the electrons/holes and optical phonons. (b) Transient transmission changes at probe photon energies of $1.24\,\text{eV}$ (solid), $1.55\,\text{eV}$ (dash-dotted) and $1.77\,\text{eV}$ (dashed) for short and long delays. The slower decay at picosecond time scale is due to equilibration with acoustic phonons [@Breusing2009ucd].](Basov_Fig15){width="3.3in"} The decay dynamics of graphene in the $8\,\text{ps}$ temporal range is shown in Fig. \[fig:4.4.1\]b with three different probe photon energies. In addition to the fast decay processes described above, it shows a slower relaxation process with a time constant of $1.4\, \text{ps}$. This picosecond time scale reflects partial thermalization between the hot electron/holes and optical phonons with the acoustic phonons in graphene. We note that the photo-induced transmission change can become negative at certain probe photon energies at longer delay. This is because optical excitation not only leads to Pauli blocking of interband transitions, but also increased high frequency absorption from intraband transitions. At longer pump-probe delay, the intraband absorption can dominate over the Pauli blocking effects at some probe photon energies. Similar photo-induced transmission decreases have also been observed in optical pump-THz probe measurements, where photo-induced intraband transitions always dominate [@George2008uop; @Strait2011vsc; @Wright2009sno]. Ultrafast photoluminescence monitors the light emission from the highly non-equilibrium electrons after femtosecond pump excitations [@Lui2010upf; @Liu2010nbp; @Stoehr2010fol]. Broad light emission across the visible spectral range ($1.7$–$3.5\, \text{eV}$) was observed with femtosecond near-IR laser excitation, where the incident photon has an energy of $1.5\,\text{eV}$. This unusual blue-shifted photoluminescence exhibits a nonlinear dependence on the laser fluence, and it has a dominant relaxation time within $100\,\text{fs}$. This nonlinear blue-shifted luminescence was attributed to recombination of hot electron-hole plasma generated right after the femtosecond excitation. In addition to the electron dynamics, researchers were able to probe the phonon dynamics specifically using time-resolved Raman spectroscopy [@Chatzakis2011tdo; @Yan2009trr]. Such studies show a decay lifetime of $2.5\,\text{ps}$ for the BZ-center $G$-mode phonons. This time scale corresponds to the cooling of the optical phonons through anharmonic coupling to acoustic phonons, and the $2.5\,\text{ps}$ time constant is similar to that obtained in pump-probe transmission spectroscopy. Bilayer and multilayer graphene {#sec:BLG} =============================== There has been a rapidly increasing interest in graphene systems with more than one layer (for an early review, see @Nilsson2008epo). The electronic structure of BLG and MLG is distinctly different from that of SLG. These differences give rise to many new phenomena, ranging from a tunable bandgap [@McCann2006lld; @McCann2006agi; @Castro2007bbg; @Oostinga2008gii; @Zhang2009doo] to strongly correlated ground states [@Feldman2009bss; @Bao2010moa; @Weitz2010bss; @Mayorov2011ids; @Velasco2012tso]. Unfortunately, space limitations and the open debate on the nature of these low-energy states [@Nilsson2006eei; @Min2008pmi; @Barlas2010aec; @Nandkishore2010dsa; @Nandkishore2010qah; @Nandkishore2011pke; @Vafek2010mbi] do not permit us to describe them in any detail. In this short section we confine ourselves to discussing “higher” energy properties of these materials that have been measured by ARPES and by optical spectroscopy. We begin with discussing the quasiparticle dispersion of BLG. For the Bernal stacked BLG, which is the most energetically favorable structure, it is conventionally described by means of five parameters. They include four hopping integrals $\gamma_0 = 3.0\,\text{eV}$, $\gamma_1 = 0.41\,\text{eV}$, $\gamma_3 = 0.3\,\text{eV}$, $\gamma_{4} = 0.15\,\text{eV}$, and also the on-site energy shift $\Delta' = 0.018\,\text{eV}$ (Fig. \[fig:1.2.1\]B).[^16] When BLG is subject to an electric field due to external gates or charged impurities, the sixth parameter, a scalar potential $\pm \Delta / 2$ on the two layers must be included. The interlayer bias $\Delta = e E d$ is given by the product of the layer separation $d = 0.335\,\mathrm{nm}$ and the electric field $E$ between the layers. {width="7.20in"} The BLG has four atoms in the unit cell, and so the electron spectrum consists of four bands. The two outer (lowest and highest energy) bands are hyperbolic, with the extremal values at approximately $\pm \gamma_1$ reached at the BZ corners. The shape of the two inner bands is more intricate. At high energies they are nested with the outer bands. At low energies their dispersion depends on the relation between the gap $\Delta$ and the hopping integral $\gamma_3$, which causes the trigonal warping. At $\Delta \gg \gamma_3 / (\gamma_0 \gamma_1)$, the trigonal warping is a small effect. The bands are shaped as sombreros, e.g., the conduction band has a local maximum at energy $\Delta / 2$ at $q = 0$ and a local minimum — the bottom of the sombrero — at the ring $|\mathbf{q}| \approx \Delta / \sqrt{2}\, \hbar v_0$, see Fig. \[fig:5.1.1\]d (right) [@McCann2006lld]. As $\Delta$ is decreased, the trigonal warping of the bottom of the sombrero becomes more and more pronounced. In the absence of the interlayer bias, $\Delta = 0$, the parabolic extrema split into the four conical Dirac points. This reconstruction is an example of the Lifshitz transition. The linear rather than parabolic shape of the bands in the symmetric BLG is supported by the linear-$T$ dependence of low-temperature electric conductivity in extremely clean suspended BLG [@Mayorov2011ids]. However, in less perfect samples, the quadruple Dirac cones structure is smeared by disorder. It is therefore common to approximate the inner bands by hyperboloids touching at a point, see Fig. \[fig:5.1.1\]d (left). In order to vary $\Delta$, one has to apply an external electric field to BLG. This can be achieved experimentally through electrostatic gating or doping. If $D_{t}$ and $D_{b}$ are electric displacement fields on the two sides of BLG (Fig. \[fig:5.1.1\]c), then the interlayer electric field $E$ is determined by their mean $\bar{D} = (D_{b} + D_{t}) / 2$. Notably, $E$ is smaller than $\bar{D}$ due to screening effects. Calculations within the self-consistent Hartree approximation [@McCann2006lld] predict a factor of two or so reduction in typical experimental conditions. The difference $D = D_{b} - D_{t}$ of the displacement fields produces a net carrier doping, and so the Fermi energy shift (Fig. \[fig:5.1.1\]d). Unless $D_t$ and $D_b$ are precisely equal or precisely opposite, the modification of the band gap $\Delta$ and the shift of the Fermi energy $E_F$ occur simultaneously [@McCann2006lld; @McCann2006agi; @Castro2007bbg]. The control of electronic structure of BLG was first revealed in ARPES studies of potassium-doped epitaxial graphene on SiC [@Ohta2006ces]. Figures \[fig:5.1.1\]e-g display the evolution of the ARPES spectra with doping. As prepared, BLG is $n$-type doped. This corresponds to a finite $D_b$ and zero $D_{t}$, leading to a nonzero bandgap (Fig. \[fig:5.1.1\]e). Potassium adsorption generates a finite $D_{t}$. When its value is the same as $D_{b}$, one obtains an electron doped gapless BLG (Fig. \[fig:5.1.1\]f). With further increase in potassium doping, the bandgap reappears (Fig. \[fig:5.1.1\]g). Tuning of BLG electron structure can be also achieved via coupling to different substrates [@Siegel2010qmg]. Complimentary insights on the band structure of BLG has been provided through IR spectroscopy. There is a total of six inter-band optical transitions possible in this material. The near-perfect nesting of two conductions bands results in a strong absorption peak at mid-IR energy $\gamma_1$, when the transition between them is activated by $n$-type doping. A refined estimate of $\gamma_1 = 0.40\pm 0.01\,\mathrm{eV}$ has been obtained by monitoring the lineshape and position of this peak in a gated BLG structure as a function of the gate voltage and modeling these spectra theoretically [@Zhang2009doo]. Similarly, the $p$-type doping activates transition between the two valence bands. From slight differences of $p$- and $n$-type spectra, the electron-hole symmetry breaking parameters $\gamma_4$ and $\Delta'$ have been inferred. The parameters obtained from the IR experiments are corroborated by those derived from the Raman spectroscopy [@Malard2007pte] and the capacitance measurements of the TDOS [@Henriksen2010mot]. The bandgap tuning by electrical gating was demonstrated using IR spectroscopy through monitoring the gate-induced change in other three transitions shown in Fig. \[fig:5.1.1\]d by arrows [@Mak2009ooa; @Kuzmenko2009dot; @Zhang2009doo]. The dependence on the bandgap on the mean displacement field $\bar{D}$ (Fig. \[fig:5.1.1\]h) was found to be in agreement with the theory [@McCann2006lld]. On the other hand, observing the predicted bandgap value from electrical transport measurements has been challenging.[^17] Gated BLG typically exhibits an insulating behavior only at $T < 1\,\text{K}$, suggesting a very narrow gap [@Oostinga2008gii]. This is because electrical transport is extremely sensitive to defects and impurities, and a very high quality graphene is required to reach the intrinsic BLG behavior. Recent transport studies [@Xia2010gfe] however demonstrate transport gaps closer to those obtained through IR spectroscopy. Electron-phonon coupling in gated BLG shows up in tunable electron-phonon Fano resonances [@Tang2010atp; @Kuzmenko2009gti; @Cappelluti2010psa]. There is a host of other effects that originate from unique gate-tunable electronic structure in BLG that have been predicted theoretically and are amenable to spectroscopic studies, e.g., a rich Landau level spectrum structure [@Zhang2011mcb]. However, we must leave this topic now to at least briefly discuss MLG. The electronic structure of MLG has been investigated experimentally using optical spectroscopy. Figure \[fig:5.1.1\]a displays a set of IR absorption spectra from $L = 1$ to $8$ layer graphene over the photon-energy range of $0.2$–$0.9\,\text{eV}$ [@Mak2010teo]. At energies in the range $\gamma_1 \ll \hbar\omega \ll \gamma_0$ the MLG is expected to behave in the first approximation as a stack of uncoupled SLG, each possessing the universal optical conductivity $\sigma_0$, Eq. . Indeed, at energies above $0.8\,\text{eV}$, the measured optical absorption scales linearly with $L$. However, at lower energies, the absorption becomes highly structured and distinct for different $L$. The evolution of the absorption spectra as a function of $L$ can be visualized from the false color plot Fig. \[fig:5.1.1\]b in which the principle transition energies are marked by the solid curves. The shape of these curves can be understood through zone-folding of the graphite band-structure. In particular, a gapless band is present if $L$ is odd and absent if it is even [@Mak2010teo]. Outlook {#sec:outlook} ======= A wealth of spectroscopic data analyzed in this review has provided a panoramic picture of electronic phenomena in graphene. The concept of 2D Dirac quasiparticles offers a unifying description of the gross features revealed in all spectroscopic and transport probes. At the same time, pronounced and reproducible deviations from the predictions of noninteracting models of graphene have been documented. An outstanding challenge for future research is probing these many-body effects using specimens of record-high electron mobility where the role of disorder is further reduced. Due to space limitations, we have not been able to cover some topics at all, e.g., nanostructured graphene, spin phenomena, graphene at ultra-high doping, or fractional quantum Hall effect in graphene [@Goerbig2011epg]. We also covered some others, e.g., BLG and MLG in insufficient detail. These topics are being actively explored, and the consensus is still being reached (for review, see [@McCann2013epb]). In addition, a particularly interesting class of materials for future research are hybrid multilayer structures and superlattices assembled from graphene and other ultrathin atomic crystals, such as hBN, MoS$_2$, *etc.*[@Geim2013vdw]. Besides fundamental research, graphene and its spectroscopy have inspired a number of applications. For example, the spectroscopic studies motivated the development of novel experimental tools and methods compatible with the architecture of gatable devices. Novel scanning spectroscopies have advanced by exploiting the unique aspect of graphene that it is unobstructed by other interfaces. Controlled modification of graphene properties has been demonstrated through using elastic strain, interactions with the substrate, adatoms, and/or other graphene layers. These new experimental approaches are expected to find applications in other areas of condensed matter physics. Examples of viable device concepts spawned by photon-based spectroscopies include compact (passive) optical components, photodetectors and bolometers, and saturable absorbers. In addition, standard plasmonics figures of merit show competitiveness or even superiority of graphene as a plasmonic medium compared to more seasoned metal-based technologies. An unresolved question is whether graphene is suitable for achieving population inversion and lasing. Acknowledgments {#acknowledgments .unnumbered} =============== D. B., M. F., and F. W. acknowledge support from ONR under Grant No. N0014-13-0464. The work at UCSD is also supported by DOE-BES under Contract No. DE-FG02-00ER45799, by AFOSR Grant No. FA9550-09-1-0566, by NSF Grant No. DMR-1337356, by ARO Grant No. W911NF-13-1-0210, and also by UCOP and FENA. Additionally, F. W. is supported by DOE-BES under Contracts No. DE-SC0003949 and No. DE-AC02-05CH11231. A. L. acknowledges support from the Novel $sp^2$-bonded Materials Program at Lawrence Berkeley National Laboratory, funded by the DOE Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05CH11231. Y. Z. is supported by NSF of China through Grant No. 11034001 and and MOST of China through Grant No. 2011CB921802. [^1]: See @CastroNeto2009tep [@Peres2010ctt; @DasSarma2011eti; @Kotov2012eei; @Katsnelson2012gci; @McCann2013epb]. [^2]: See @Orlita2010des for optics, @Ni2008rsa [@Dresselhaus2012rsc] for Raman scattering, for scanning tunneling spectroscopy, and @Connolly2010nog for other scanned probes. [^3]: See @Mak2009ooa [@Xia2009moq; @Efetov2010cep; @Ju2011gpf; @Newaz2012pcs]. [^4]: See @Berger2004ueg [@Berger2006eca; @Forbeaux1998hgo; @Charrier2002ssd; @Nagashima1993eso; @Rollings2006sac; @Ohta2006ces; @Emtsev2009tws]. [^5]: This can be done chemically [@Boehm1962; @Dikin2007pac] or via IR irradiation [@El-Kady2012lso]. [^6]: See @Han2007ebg [@Liu2008eco; @Todd2008qdb; @Stampfer2009egi; @Oostinga2010mtg; @Gallagher2010dig; @Han2010eti]. [^7]: See @Zhou2007sib [@Gao2010ego; @Siegel2012epc; @Walter2011eso; @Nagashima1994eso; @Dedkov2008rei; @Varykhalov2008eam; @Himpsel1982abd; @Enderlein2010tfo; @Sutter2009; @Rader2009ita; @Papagno2012lbg]. [^8]: See also [@Deshpande2009srs; @Li2012stm; @Song2010hrt; @Xue2011stm; @Yankowitz2012esd; @Chae2012rgd]. [^9]: See @Ludwig1994iqh for Dirac fermions in general and @Ando2002dca [@Gusynin2006tod; @Peres2006epo; @Ziegler2007mco; @Falkovsky2007std; @Stauber2008oco] for SLG. [^10]: See @Abergel2007vog [@Blake2007mgv; @Roddaro2007tov; @Ni2007gtd]. [^11]: See @Fei2008heo [@Santoso2011oor; @Mak2011smb; @Kravets2010seo; @Chae2011efr]. [^12]: See @Stauber2008cos [@Peres2008tic; @Hwang2012ose; @Scharf2013eos]. [^13]: See @Grushin2009eoc [@Peres2010eei; @Hwang2012ose; @Carbotte2012eop; @Principi2013ild]. [^14]: See @Liu2008pda [@Liu2010pps; @Koch2010spp; @Shin2011cot; @Tegenkamp2011peh]. [^15]: See @Bianchi2010epc [@Bostwick2007rog; @Park2008vhs; @Park2007vra; @McChesney2007meo; @McChesney2010evh; @McChesney2008sca; @Zhou2008kaa; @Calandra2007epc; @Filleter2009fad; @Grueneis2009esa]. [^16]: For a discussion of these numerical values and comparison with graphite, see @Li2009bsa [@Zhang2008dot; @Kuzmenko2009iso; @Kuzmenko2009gti]. [^17]: See @Szafranek2010eoo [@Xia2010gfe; @Castro2007bbg; @Oostinga2008gii].

--- abstract: 'Stackelberg Games are gaining importance in the last years due to the raise of Adversarial Machine Learning (AML). Within this context, a new paradigm must be faced: in classical game theory, intervening agents were humans whose decisions are generally discrete and low dimensional. In AML, decisions are made by algorithms and are usually continuous and high dimensional, e.g. choosing the weights of a neural network. As closed form solutions for Stackelberg games generally do not exist, it is mandatory to have efficient algorithms to search for numerical solutions. We study two different procedures for solving this type of games using gradient methods. We study time and space scalability of both approaches and discuss in which situation it is more appropriate to use each of them. Finally, we illustrate their use in an adversarial prediction problem.' author: - Roi Naveiro - David Ríos Insua title: 'Gradient Methods for Solving Stackelberg Games[^1]' --- Introduction {#sec:intro} ============ Over the last decade, the introduction of machine learning applications in numerous fields has grown tremendously. In particular, applications in security settings have grown substantially, [@mcdaniel2016machine]. In this domain, it is frequently the case that the data distribution at application time is different of the training data distribution, thus violating one of the key assumptions in machine learning. This difference between training and test distributions generally comes from the presence of adaptive adversaries who deliberately manipulate data to avoid being detected. The field of Adversarial Machine Learning (AML) studies, among other things, how to guarantee the security of machine learning algorithms against adversarial perturbations [@biggio2018wild]. A possible approach consists of modelling the interaction between the learning algorithm and the adversary as a game in which one agent controls the predictive model parameters while the other manipulates input data. Several different game theoretic models of this problem have been proposed, as reviewed in [@voro2018]. In particular, [@bruckner2011stackelberg] view adversarial learning as a Stackelberg game in which, a *leader* (she), the defender in the security jargon, makes her decision about choosing the parameters in a learning model, and, then, the *follower* or attacker (he), after having observed the leader’s decision, chooses an optimal data transformation. Mathematically, finding Nash equilibria of such Stackelberg games requires solving a bilevel optimization problem, which, in general cannot be undertaken analytically, [@sinha2018review], and numerical approaches are required. However, standard techniques are not able to deal with continuous and high dimensional decision spaces, as those appearing in AML applications. In this paper, we propose two procedures to solve Stackelberg games in the new paradigm of AML and study their time and space scalability. In particular, one of the proposed solutions scales efficiently in time with the dimension of the decision space, at the cost of more memory requirements. The other scales well in space, but requires more time. The paper is organized as follows: in Section \[sec:stack\_games\] we define Stackelberg games. Section \[sec:solution\_method\] presents the proposed solution methods as well as a discussion of the scalability of both approaches. The proposed solutions are illustrated with an AML experiment in Section \[sec:experiments\]. Finally, we conclude and present some lines for future research. Stackelberg games {#sec:stack_games} ================= We consider a class of sequential games between two agents: the first one makes her decision, and then, after having observed the decision, the second one implements his response. These games have received various names in the literature including sequential Defend-Attack [@Brown:2006] or Stackelberg [@Gibbons:1992; @tambe2011security] games. As an example, consider adversarial prediction problems, [@bruckner2011stackelberg]. In them, the first agent chooses the parameters of a certain predictive model; the second agent, after having observed such parameters, chooses an optimal data transformation to fool the first agent as much as possible, so as to obtain some benefit. As we focus on applications of Stackelberg games to AML, we restrict ourselves to the case in which the Defender ($D$) chooses her defense $\alpha \in \mathbb{R}^n$ and, then, the Attacker ($A$) chooses his attack $\beta \in \mathbb{R}^m$, after having observed $\alpha$. The corresponding bi-agent influence diagram, [@BAIDS], is shown in Fig. \[fig:baid1\]. The dashed arc between nodes $D$ and $A$ reflects that the Defender choice is observed by the Attacker. The utility function of the Defender, $u_D(\alpha, \beta)$, depends on both, her decision, and the attacker’s decision. Similarly, the Attacker’s utility function has the form $u_A(\alpha, \beta)$. In this type of games, it is assumed that the Defender knows $u_A(\alpha, \beta)$. This assumption is known as the common knowledge hypothesis. \(A) [$D$]{}; (B) \[right of=A\] [$A$]{}; (C) \[below of=A\] [$U_D$]{}; (E) \[below of=B\] [$U_A$]{}; \(B) edge node (C) edge node (E) (A) edge node (C) edge node (E) edge\[out = 90, in = 90, dashed\] node (B); Mathematically, finding Nash equilibrium of Stackelberg games requires solving a bilevel optimization problem, [@bard1991some]. The defender’s utility is called *upper level* or *outer* objective function while the attacker’s one is referred to as *lower level* or *inner* objective function. Similarly, the upper and lower level optimization problems, correspond to the defender’s and the attacker’s problem, respectively. These problems are also referred to as outer and inner problems. It is generally assumed that the attacker will act rationally in the sense that he will choose an action that maximizes his utility, [@french2000statistical], given the disclosed defender’s decision $\alpha$. Assuming that there is a unique global maximum of the attacker’s utility for each $\alpha$, and calling it $\beta^*(\alpha)$, a Stackelberg equilibrium is identified using backward induction: the defender has to choose $\alpha^*$ that maximizes her utility subject to the attacker’s response $\beta^*(\alpha)$. Mathematically, the problem to be solved by the defender is $$\label{bilevel} \begin{aligned} & \operatorname*{arg\,max}_{\alpha} & & u_D[\alpha, \beta^*(\alpha)]\\ & \text{s.t.} & & \beta^*(\alpha) = \operatorname*{arg\,max}_{\beta} u_A(\alpha,\beta). \end{aligned}$$ The pair $\left( \alpha^{*}, \beta^* (\alpha^{*}) \right)$ is a Nash equilibrium and a sub-game perfect equilibrium [@Hargreaves:2004]. When the attacker problem has more than one global maximum, several types of equilibrium have been proposed. The two more important are the optimistic and the pessimistic solutions, [@sinha2018review]. In an optimistic position, the defender expects the attacker to choose the optimal solution which gives the higher upper level utility. On the other hand, the pessimistic approach suggests that the defender should optimize for the worst case attacker solution. In this paper, we just deal with the case in which the inner utility has a unique global maximum. Solution Method {#sec:solution_method} =============== Bilevel optimization problems can rarely be solved analytically. Indeed even extremely simple instances of bilevel problems have been shown to be NP-hard, [@jeroslow1985polynomial]. Thus, numerical techniques are required. Several classical and evolutionary approaches have been proposed to solve , as reviewed by [@sinha2018review]. When the inner problem adheres to certain regularity conditions, it is possible to reduce the bilevel optimization problem to a single level one replacing the inner problem with its Karush-Kuhn-Tucker (KKT) conditions. Then, evolutionary techniques could be used to solve this single-level problem, thus making possible to relax the upper level requirements. As, in general, this single-level reduction is not feasible, several other approaches have been proposed, such as nested evolutionary algorithms or metamodeling-based methods. However, most of these approaches lack scalability: increasing the number of upper level variables produces an exponential increase on the number of lower level tasks required to be solved being thus impossible to apply these techniques to solve large scale bilevel problems as the ones appearing in the context of AML. In [@bruckner2011stackelberg] the authors face the problem of solving Stackelberg games in the AML context. However, they focus on a very particular type of game which can be reformulated as a quadratic program. In this paper, we provide more general procedures to solve Stackelberg games that are useful in the AML paradigm in which decision spaces are continuous and high dimensional. To this end, we focus on gradient ascent techniques to solve bilevel optimization problems. Let us assume that for any $\alpha$ the solution of the inner problem is unique. This solution defines an implicit function $\beta^*(\alpha)$. Thus, problem may be viewed solely in terms of the defender decisions $\alpha$. The underlying idea behind gradient ascent techniques is the following: given a defender decision $\alpha \in \mathbb{R}^n$ a direction along which the defender’s utility increases while maintaining feasibility must be found, and then, we move $\alpha$ in that direction. Thus, the major issue of ascent methods is to find the gradient of $u_D(\alpha, \beta^*(\alpha))$. In [@kolstad1990derivative] the authors provide a method to approximate such gradient that work for relatively large classical optimization problems but it is clearly insufficient to deal with the typical bilevel problems appearing in AML. Recently, [@franceschi2017forward] proposed forward and reverse-based methods for computing the gradient of the validation error in certain hyperparamenter optimization problems that appear in Deep Learning. Structurally, hyperparameter optimization problems are similar to Stackelberg games. We adapt their methodology to this domain. In particular we propose two alternative approaches to compute the gradient of $u_D[\alpha, \beta^*(\alpha)]$ with different memory and running time requirements. We refer to these approaches as backward and forward solutions, respectively. ### Notation {#notation .unnumbered} For the sake of clarity, we use the following notation: the gradient will be denoted as ${\mathop{}\! \mathrm{d}}_x$; the partial derivative as $\partial_x$. Similarly, second partial derivatives will be denoted as $\partial^2_x$ and $\partial_x \partial_y$. We shall use this notation indistinctly for the unidimensional and multidimensional cases. For instance, if $f(x,y)$ is a scalar function, $x$ is a $p$-dimensional vector and $y$ is a $q$-dimensional vector, then $\partial^2_x f(x,y)$ is the $p \times p$ matrix whose $(i,j)$ entry is $\partial_{x_i} \partial_{x_j} f(x,y)$, where $x_i$ is the $i$-th component of the vector $x$. Similarly, $\partial_x \partial_y f(x,y)$ is a $p \times q$ matrix whose $i,j$ entry is $\partial_{x_i} \partial_{y_j} f(x,y)$. Backward solution {#sec:backward} ----------------- We propose here a new gradient ascent approach to solve the bilevel problem whose running time scales well with the defender’s decision space dimension. In particular, we propose to approximate problem by the following PDE-constrained optimization problem, [@hinze2008optimization] $$\label{alternative} \begin{aligned} & \operatorname*{arg\,max}_{\alpha} & & u_D\left[\alpha, \beta(\alpha, T)\right] \\ & \text{s.t.} & & \partial_t \beta(\alpha, t) = \partial_{\beta} u_A [\alpha, \beta(\alpha, t)]\\ & & & \beta(\alpha, 0) = 0. \end{aligned}$$ The idea is formalized in the next proposition, that can be proved using the results in [@bottou1998online]. Suppose that the following assumptions hold 1. The attacker problem, the inner problem in , has a unique solution $\beta^*(\alpha)$ for each defender decision $\alpha$. 2. For all $\epsilon>0$ and all $\alpha$, $$\begin{aligned} \inf_{\Vert \beta^*(\alpha) - \beta \Vert_2^2 > \epsilon} \left \langle \beta - \beta^*(\alpha), \partial_{\beta}u_A [\alpha, \beta] \right \rangle > 0. \end{aligned}$$ If $\beta(\alpha, t)$ satisfies the differential equation $$\label{diffeq} \partial_t \beta(\alpha, t) = \partial_{\beta} u_A [\alpha, \beta(\alpha, t)]$$ then $\beta(\alpha, t) \rightarrow \beta^*(\alpha)$ as $t \rightarrow \infty$, with rate $\mathcal{O}\left(\frac{1}{t}\right)$. The idea in $\eqref{alternative}$ is thus to constrain the trajectories $\beta(\alpha, t)$ to satisfy $\eqref{diffeq}$ and approximate the defender’s problem using $\beta(\alpha, T)$ with $T \gg 1$, instead of $\beta^*(\alpha)$. We propose solving problem using gradient ascent and the adjoint method, [@pontryagin2018mathematical], to compute the total derivative of the defender utility function with respect to her decision. The adjoint method defines an *adjoint function* $\lambda(t)$ satisfying the *adjoint equation* $$\label{adjointeq} {\mathop{}\! \mathrm{d}}_t \lambda (t) = - \lambda(t) ~ \partial_{\beta}^2 u_A[\alpha, \beta(\alpha, t)].$$ In terms of the adjoint function, the derivative of the defender utility with respect to her decision would be written as $$\label{derivative} {\mathop{}\! \mathrm{d}}_\alpha u_D[\alpha, \beta(\alpha, T)] = \partial_\alpha u_D[\alpha, \beta(\alpha, T)] - \int_0^T \lambda(t) \partial_{\alpha} \partial_{\beta} u_A [\alpha, \beta(\alpha, t)] {\mathop{}\! \mathrm{d}}t.$$ In Appendix \[prf\], we prove that if $\lambda(t)$ satisfies the adjoint equation , the derivative of the defender utility can be written as in . Algorithmically, we can proceed by discretizing via Euler method, and approximate the derivative discretizing the integral on the left hand side. This leads to Algorithm \[alg:aprox\_der\_bw\]. $\beta_0(\alpha) = 0$ $\beta_t(\alpha) = \beta_{t-1}(\alpha) + \eta \partial_{\beta}u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}}$ $\lambda_T = - \partial_{\beta} u_D(\alpha, \beta)\Big \vert_{\beta_{T}}$ ${\mathop{}\! \mathrm{d}}_{\alpha} u_D = \partial_{\alpha} u_D[\alpha, \beta_T(\alpha)]$ ${\mathop{}\! \mathrm{d}}_{\alpha} u_D = {\mathop{}\! \mathrm{d}}_{\alpha} u_D - \eta \lambda_{t+1} \partial_\alpha \partial_\beta u_A(\alpha, \beta)\Big \vert_{\beta_{t+1}}$ $\lambda_t = \lambda_{t+1} \left[I + \eta \partial^2_{\beta}u_A(\alpha, \beta)\Big \vert_{\beta_{t+1}} \right]$ **return** ${\mathop{}\! \mathrm{d}}_{\alpha} u_D$ Once we are able to compute this derivative, we can solve the defender’s problem using gradient ascent. Regarding its complexity, note that by basic facts of Automatic Differentiation (AD), [@griewank2008evaluating], if $\tau (n,m)$ is the time required to evaluate $u_D(\alpha, \beta)$ and $u_A(\alpha, \beta)$, then computing derivatives of these functions requires time $\mathcal{O}(\tau (n,m))$. Thus the first for loop in Algorithm \[alg:aprox\_der\_bw\] requires time $\mathcal{O}(T \tau (n,m))$. In the second loop, we need to compute second derivatives, which appear always multiplying the vector $\lambda_t$. By basic results of AD, Hessian vector products have the same time complexity as function evaluations. Thus in our case, we can compute second derivatives in time $\mathcal{O}(\tau (n,m))$ being the time complexity of the second for loop $\mathcal{O}(T\tau (n,m))$. Thus, overall, Algorithm \[alg:aprox\_der\_bw\] runs in time $\mathcal{O}(T\tau (n,m))$. Regarding space complexity, as it is necessary to store the values of $\beta_t(\alpha)$ produced in the first loop for later usage in the second one, if $\sigma(n,m)$ is the space requirement for storing each $\beta_t(\alpha)$, then $\mathcal{O}(T \sigma(n,m))$ is the space complexity of the backward algorithm. In certain applications where space complexity is critical, the backward solution could be infeasible as, within each iteration, it requires storing the whole trace $\beta_t(\alpha)$. In this particular cases, the forward solution proposed in the next section, solves this issue at a cost of loosing time scalability. Forward solution ---------------- In this case, we approximate by $$\label{alternative_fw} \begin{aligned} & \operatorname*{arg\,max}_{\alpha} & & u_D\left[\alpha, \beta_T(\alpha)\right] \\ & \text{s.t} & & \beta_t(\alpha) = \beta_{t-1}(\alpha) + \eta_t \partial_{\beta}u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}} && t = 1, \dots, T\\ & & & \beta_0(\alpha) = 0. \end{aligned}$$ The idea here is that, for each defense $\alpha$, we condition on a dynamical system that under certain conditions converges to $\beta^*(\alpha)$, the optimal solution for the attacker when the defender plays $\alpha$. Thus, we can approximate the defender’s utility by $u_D\left[\alpha, \beta_T(\alpha)\right]$, with $T \gg 1$. This idea is formalized in the next proposition that can be proved using the results of [@bottou1998online]. Suppose that the following assumptions hold 1. The attacker problem (the inner problem in ) has a unique solution $\beta^*(\alpha)$ for each defender decision $\alpha$. 2. For all $\epsilon>0$ and $\alpha$ $$\inf_{\Vert \beta - \beta^*(\alpha) \Vert_2^2 > \epsilon} \left \langle \beta - \beta^*(\alpha), \partial_{\beta}u_A [\alpha, \beta] \right \rangle > 0$$ 3. For some $A,B>0$ and all $\alpha$ $$\Vert \partial_{\beta}u_A [\alpha, \beta] \Vert_{2}^2 \leq A + B \Vert \beta - \beta^*(\alpha) \Vert_{2}^2$$ If for all $t$, $\beta_t(\alpha)$ satisfies $$\label{dynsys} \beta_t(\alpha) = \beta_{t-1}(\alpha) + \eta \partial_{\beta}u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}}$$ Then, $\beta_t(\alpha)$ converges to $\beta^*(\alpha)$, with rate $\mathcal{O}\left(\frac{1}{t}\right)$. We propose solving problem using gradient ascent. To that end, we need to compute ${\mathop{}\! \mathrm{d}}_\alpha u_D(\alpha, \beta_T(\alpha))$. Using the chain rule we have $$\begin{aligned} {\mathop{}\! \mathrm{d}}_\alpha u_D[\alpha, \beta_T(\alpha)] = \partial_\alpha u_D[\alpha, \beta_T(\alpha)] + \partial_{\beta_T} u_D[\alpha, \beta_T(\alpha)] {\mathop{}\! \mathrm{d}}_\alpha \beta_T(\alpha)\end{aligned}$$ To obtain ${\mathop{}\! \mathrm{d}}_\alpha \beta_T(\alpha)$, we can sequentially compute ${\mathop{}\! \mathrm{d}}_\alpha \beta_t(\alpha)$ taking derivatives in $$\begin{aligned} {\mathop{}\! \mathrm{d}}_\alpha \beta_t(\alpha) = {\mathop{}\! \mathrm{d}}_\alpha \beta_{t-1}(\alpha) + \eta_{t-1} \left[ \partial_\alpha \partial_{\beta} u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}} + \partial^2_\beta u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}} {\mathop{}\! \mathrm{d}}_\alpha \beta_{t-1}(\alpha) \right]\end{aligned}$$ This induces a dynamical system in ${\mathop{}\! \mathrm{d}}_\alpha \beta_t(\alpha)$ that can be iterated in parallel to the dynamical system in $\beta_t(\alpha)$. The whole procedure is described in Algorithm \[alg:aprox\_der\_fw\]. $\beta_0(\alpha) = 0$ ${\mathop{}\! \mathrm{d}}_\alpha \beta_0(\alpha) = 0$ $\beta_t(\alpha) = \beta_{t-1}(\alpha) + \eta \partial_{\beta}u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}}$ ${\mathop{}\! \mathrm{d}}_\alpha \beta_t(\alpha) = {\mathop{}\! \mathrm{d}}_\alpha \beta_{t-1}(\alpha) + \eta_{t-1} \left[ \partial_\alpha \partial_{\beta} u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}} + \partial^2_\beta u_A(\alpha, \beta)\Big \vert_{\beta_{t-1}} {\mathop{}\! \mathrm{d}}_\alpha \beta_{t-1}(\alpha) \right]$ ${\mathop{}\! \mathrm{d}}_\alpha u_D = \partial_\alpha u_D[\alpha, \beta_T(\alpha)] + \partial_{\beta_T} u_D[\alpha, \beta_T(\alpha)] {\mathop{}\! \mathrm{d}}_\alpha \beta_T(\alpha)$ **return** ${\mathop{}\! \mathrm{d}}_{\alpha} u_D$ Once we are able to compute this derivative, we can solve the defender’s problem using gradient ascent. Regarding time complexity, note that the bottleneck in Algorithm \[alg:aprox\_der\_fw\] is that we need to compute second derivatives of $u_A(\alpha,\beta)$. In particular, computing $\partial^2_\beta u_A(\alpha, \beta)$ requires time $\mathcal{O}(m \tau(m,n))$ as it requires computing $m$ Hessian vector products, one with each of the $m$ the unitary vectors. On the other hand, computing $\partial_\alpha \partial_{\beta} u_A(\alpha, \beta)$ requires computing $n$ Hessian vector products and thus time $\mathcal{O}(n \tau(m,n))$, while if we compute the derivative in the other way, first we derive with respect to $\beta$ and then with respect to $\alpha$, the time complexity is $\mathcal{O}(m \tau(m,n))$. Thus, we derive first with respect to the variable with the biggest dimension. Then, the time complexity of computing $\partial_\alpha \partial_{\beta} u_A(\alpha, \beta)$ is $\mathcal{O}(\min(n,m) \tau(m,n))$. Finally, as $\partial^2_\beta u_A(\alpha, \beta)$ and $\partial_\alpha \partial_{\beta} u_A(\alpha, \beta)$ could be computed in parallel, then the overall time complexity of the forward solution is $\mathcal{O}(\max[\min(n,m), m]T \tau(m,n)) = \mathcal{O}(mT \tau(m,n)) $. Regarding space, as in this case the values $\beta_t(\alpha)$ are overwritten at each iteration, we do not need to store all of them and the overall space complexity is $\mathcal{O}( \sigma(m,n))$. Experiments {#sec:experiments} =========== We illustrate now the proposed approaches. We start with a conceptual example in which we empirically test the scalability properties of both algorithms. Then, we apply the algorithms to solve a problem in the context of adversarial regression. All the code used for these examples has been written in python using the pytorch library for Automatic Differentiation, [@paszke2017automatic]; and is available at <https://github.com/roinaveiro/GM_SG>. Conceptual Example ------------------ We use a simple example to illustrate the scalability of the proposed approaches. Consider that the attacker’s and defender’s decisions are both vectors in $\mathbb{R}^n$. The attacker’s utility takes the form $u_A(\alpha, \beta) = -\sum_{i=1}^n 3(\beta_i - \alpha_j)^2$ and the defender’s one is $u_D(\alpha, \beta) = -\sum_{i=1}^n (7 \alpha_i + \beta_j^2)$. In this case, the equilibrium can be computed analytically using backward induction: for a given defense $\alpha \in \mathbb{R}^n$ we see that $\beta^*(\alpha) = \alpha$; substituting in the outer problem, the equilibrium is reached at $\alpha_j^* = -3.5, \beta_j^*(\alpha^*) = -3.5$ with $j=1, \dots, m$. We apply the proposed methods to this problem to test their scalability empirically. The parameters were chosen as follows: the learning rate $\eta$ of Algorithms \[alg:aprox\_der\_bw\] and \[alg:aprox\_der\_fw\] was set to 0.1; similarly, the learning rate of the gradient ascent used to solve the outer problem was also set to 0.1. Finally, all gradient ascents were run for $T=40$, enough to reach convergence. ![Backward and Forward running times versus the dimension of decision spaces.[]{data-label="fig:time_comp"}](time_comp.eps){width="0.8\linewidth"} Figure \[fig:time\_comp\] shows running times for increasing number of dimensions of the decision spaces (in this problem both the attacker’s and the defender’s decision space have the same dimension). As we discussed, the forward running time increases linearly with the number of dimensions while the backward solution remains approximately constant. This obviously comes at the cost of having more memory requirements, as in Algorithm \[alg:aprox\_der\_bw\] we need to store the whole trace $\beta_t(\alpha)$. Thus, in problems where the dimension of $\beta$ is very large the memory cost of the backward solution would become prohibitive and we would need to switch to the forward solution, as long as the dimension of $\alpha$ is small enough. In contrast, if the dimension of $\alpha$ is very big, the forward solution would become infeasible in time, thus being the backward optimal provided that the dimension of $\beta$ is such that it is possible to store the whole trace $\beta_t(\alpha)$. An application to adversarial regression {#exp:adv} ---------------------------------------- ### Problem statement We illustrate an application of the proposed methodology to adversarial regression problems, [@grosshans2013bayesian]. They are a specific class of prediction games, [@bruckner2011stackelberg], played between a *learner* of a regression model and a *data generator*, who tries to fool the learner modifying input data at application time, inducing a change between the data distribution at training and test time, with the aim of confusing the data generator and attain a benefit. Given a feature vector $x \in \mathbb{R}^p$ and its corresponding target value $y \in \mathbb{R}$, the learner’s decision is to choose the weight vector $w \in \mathbb{R}^p$ of a linear model $f_w(x) = x^\top w$, that minimizes the theoretical costs at application time, given by $$\begin{aligned} \theta_l (w, \bar{p}, c_l) = \int c_l(x,y) (f_w(x) - y)^2 {\mathop{}\! \mathrm{d}}\bar{p}(x,y),\end{aligned}$$ where $c_l(x,y) \in \mathbb{R}^+$ reflects instance-specific costs and $\bar{p}(x,y)$ is the data distribution at test time. To do so, the learner has a training matrix $X \in \mathbb{R}^{n\times p}$ and a vector of target values $y\in \mathbb{R}^n$, that is a sample from distribution $p(x,y)$ at training time. The data generator aims at changing features of test instances to induce a transformation in the data distribution from $p(x,y)$ to $\bar{p}(x,y)$. Let $z(x,y)$ be the data generator’s target value for instance $x$ with real value $y$, i.e. he aims at transforming $x$ to make the learner predict $z(x,y)$ instead of $y$. The data generator aims at choosing the transformation that minimizes the theoretical costs given by $$\begin{aligned} \theta_d (w, \bar{p}, c_d) = \int c_d(x,y) (f_w(x) - z(x,y))^2 {\mathop{}\! \mathrm{d}}\bar{p}(x,y) + \Omega_d(p,\bar{p})\end{aligned}$$ where $\Omega_d(p,\bar{p})$ is the incurred cost when transforming $p$ to $\bar{p}$ and $c_d(x,y)$ are instance specific costs. As the theoretical costs defined above depend on the unknown distributions $p$ and $\bar{p}$, we focus on their regularized empirical counterparts, given by $$\begin{aligned} \widehat{\theta}_l(w, \bar{X}, c_l) &=& \sum_{i=1}^n c_{l,i} (f_w(\bar{x}_i) - y_i)^2 + \Omega_l(f_w),\\ \widehat{\theta}_d(w, \bar{X}, c_d) &=& \sum_{i=1}^n c_{d,i} (f_w(\bar{x}_i) - z_i)^2 + \Omega_d(X, \bar{X}).\end{aligned}$$ In addition, we assume that the learner acts first, choosing a weight vector $w$. Then the data generator, after observing $w$, chooses his optimal data transformation. Thus, the problem to be solved by the learner is $$\label{advstack} \begin{aligned} & \operatorname*{arg\,min}_{w} & & \widehat{\theta}_l(w, T(X,w,c_d), c_l) \\ & \text{s.t.} & & T(X,w,c_d) = \operatorname*{arg\,min}_{X'} \widehat{\theta}_d(w, X', c_d), \end{aligned}$$ where $T(X,w,c_d)$ is the attacker’s optimal transformation for a given choice $w$ of weight vector. has the same form as , except that it is formulated in terms of costs rather than utilities. In addition, it is easy to see that if $\Omega_d(X, \bar{X})$ is equal to the squared Frobenius norm of the difference matrix $\Vert X - \bar{X} \Vert_F^2$, then the attacker’s problem has a unique solution. Thus, we can use the proposed solution techniques to look for Nash equilibria in this type of game, taking care of performing gradient descent instead of gradient ascent, as we are minimizing costs here. ### Experimental results We apply the results to the UCI white wine dataset, [@UCI]. This contains real information about 4898 wines, that consists of 11 quality indicators plus a wine quality score. $R_J$ and $R_D$ are two competing wine brands. $R_D$ has implemented a system to automatically measure wine quality using a regression over the available quality indicators: each wine is described by a vector of 11 entries, one per quality indicator. Wine quality ranges between 0 and 10. $R_J$, aware of the actual superiority of its competitor’s wines, decides to hack $R_D$’s system by manipulating the value of several quality indicators, to artificially decrease $R_D$’s quality rates. However, $R_D$ is aware of the possibility of being hacked, and decides to use adversarial methods to train its system. In particular, $R_D$ models the situation as a Stackelberg game. It is obvious that the target value of his enemy is $z(x,y) = 0$ for every possible wine. In addition, $R_D$ was able to filter some information about $R_J$’s wine-specific costs $c_{d,i}$. As basic model, a regular ridge regression, [@friedman2001elements], was trained using eleven principal components as features. The regularization strength was chosen using repeated hold-out validation, [@kim2009estimating], with ten repetitions. As performance metric we used the root mean squared error (RMSE), estimated via repeated hold-out. We compare the performance of two different learners against an adversary whose wine specific costs $c_{d,i}$ are fixed: The first one, referred to as Nash, assumes that the wine specific costs are common knowledge and plays Nash equilibria of the Stackelberg game defined in . The second learner, refer to as raw, is a non adversarial one and uses a ridge regression model. To this end, we split the data in two parts, $2/3$ for training purposes and the remaining $1/3$ for test. The training set is used to compute the weights $w$ of the regression problem. Those weights are observed by the adversary, and used to attack the test set. Then, the RMSE is computed using this attacked test set and the previously computed weights. In order to solve , we use the backward solution solution method of Section \[sec:backward\] due to its better time scalability. The hyperparameters were chosen as follows: number of epochs $T$ to compute the gradient in Algorithm \[alg:aprox\_der\_bw\], 100; the learning rate $\eta$ in this same Algorithm, was set to 0.01. Within the gradient descent optimization used to optimize the defender’s cost function, the number of epochs was set to 350 and the learning rate to $10^{-6}$. Finally, we assumed that the wine specific costs were the same for all instances and called the common value $c_d$. We studied how $c_d$ affects the RMSE for different solutions. Notice that, in this case, the dimension of the attacker’s decision space is huge. He has to modify the training data to minimize his costs. If there are $k$ instances in the training set, each of dimension $n$, the dimension of the attacker’s decision space is $n\times k$. In this case $k=3263$ ($2/3$ of 4898) and $n=11$. Thus the forward solution is impractical in this case, and we did not compute it. ![Convergence for several initial points.[]{data-label="convergence"}](rmse_vs_cw.eps){height="1.9in" width="1\linewidth"} ![Convergence for several initial points.[]{data-label="convergence"}](convergence.eps){height="1.82in" width="0.95\linewidth"} We show in Figure \[performance\], the RMSE for different values of the wine specific cost. We observe that Nash outperforms systematically the adversary unaware regression method. In the limit $c_d \rightarrow 0$, we see that $\widehat{\theta}_d(w, \bar{X}, c_d) \rightarrow \Omega_d(X, \bar{X})$. Thus, in this situation, the adversary will not manipulate the data. Consequently Nash and ridge regression solutions will coincide, as shown in Figure \[performance\]. However, as $c_d$ increases, data manipulation is bigger, and the RMSE of the adversary unaware method also increases. On the other hand, the Nash solution RMSE remains almost constant. We have also computed the average and standard deviation of training times. In an Intel Core i7-3630UM, 2.40GHz × 8 computer, the average training time is $131.6$ seconds with $2.7$ seconds standard deviation. This corresponds approximately to $2.66$ seconds per outer epoch. Each outer epoch involves running Algorithm \[alg:aprox\_der\_bw\] with 100 inner epochs. Finally, to illustrate convergence of the proposed approach, we solve using gradient descent with the backward method for 20 different random initializations of the defender’s decisions $\omega$. Results are depicted in Figure \[convergence\]. As can be seen, all paths converge with less than 150 epochs. Discussion {#sec:discussion} ========== The demand for scalable solutions of Stackelberg Games has increased in the last years due to the use of such games to model confrontations within Adversarial Machine Learning problems. In this paper, we have focused on gradient methods for solving Stackelberg Games, providing two different approaches to compute the gradient of the defender’s utility function: the forward and backward solutions. In particular, we have shown that the backward solution scales well in time with the defender’s decision space dimension, at a cost of more memory requirements. On the other hand, the forward solution scales poorly in time with this dimension, but well in space. We have provided empirical support of the scalability properties of both approaches using a simple example. In addition, we have solved an AML problem using the backward solution in a reasonable amount of time. In this problem, the defender’s decision space is continuous with dimension 11. The attacker’s decision space is also continuous with dimension $\mathcal{O}(10^4)$, as we showed in Section \[exp:adv\]. To the best of our knowledge, none of previous numerical techniques for solving Stackelberg games could deal, in reasonable time, with such high dimensional continuous decision spaces. Apart from scalability properties, a major advantage of the proposed framework is that it could be directly implemented in any Automatic Differentiation library such as PyTorch (the one used in this example) or TensorFlow, and thus benefit from the computational advantages of such implementations. We could extend the framework in several ways. First, as we discussed, the backward solution has poor space scalability. This is generally not an issue in most applications. Nevertheless, if space complexity is critical it is possible to reduce it at a cost of introducing a numerical error, as proposed in [@maclaurin2015gradient] in hyperparameter optimization problems. Instead of storing the whole trace $\beta_t(\alpha)$ in the first for loop of Algorithm \[alg:aprox\_der\_bw\] to use it in the second loop, we could sequentially undo its gradient update at each step of the second for loop. Obviously, this would introduce some numerical error. Another possible line of work would be to extend the framework to deal with Bayesian Stackelberg games, that are widely used to model situations in AML in which there is not common knowledge of the adversary’s parameters. In this line, the ultimate goal would be to apply the proposed algorithms to solve Adversarial Risk Analyisis (ARA, [@rios2009adversarial]) problems in AML, [@naveiro2019adversarial]. Throughout the paper, we have focused on exact gradient methods. However, it would be interesting to extend the proposed algorithms to work with stochastic gradient methods. In addition, in [@mokhtari2019unified] the authors propose several variants of Gradient Ascent to solve saddle point problems. It could be worth investigating how to extend such techniques to general Stackelberg Games. Finally, we highlight that one of the most important contributions of the paper is the derivation of the backward solution formulating the Stackelberg game as a PDE-constrained optimization problem and using the adjoint method. This provides a general and scalable framework that could be used to seek for Nash equilibria in other types of sequential games. Exploring this, is another possible line of future work. Proof of the adjoint method {#prf} =========================== The Lagrangian of problem is $$\mathcal{L} = u_D[\alpha, \beta(\alpha, T)] + \int_{0}^T \lambda(t) \left \lbrace {\mathop{}\! \mathrm{d}}_t \beta(\alpha, t) - \partial_{\beta} u_A [\alpha, \beta(\alpha, t)] \right \rbrace dt + \mu \beta(\alpha, 0).$$ As the constraints hold, by construction we have that $d_{\alpha}\mathcal{L} = d_{\alpha}u_D$ and $$\begin{aligned} {\mathop{}\! \mathrm{d}}_\alpha \mathcal{L} &=& \partial_\alpha u_D[\alpha, \beta(\alpha, T)] + \partial_\beta u_D[\alpha, \beta(\alpha, T)] {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, T) + \mu {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, 0) \nonumber \\ &+& \int_0^T \lambda(t) \left \lbrace {\mathop{}\! \mathrm{d}}_t {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, t) - \partial_\alpha \partial_{\beta} u_A [\alpha, \beta(\alpha, t)] - \partial^2_{\beta} u_A [\alpha, \beta(\alpha, t)] {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, t) \right \rbrace {\mathop{}\! \mathrm{d}}t. \nonumber \\ \label{first_adj}\end{aligned}$$ Integrating by parts, we have $$\begin{aligned} \int_0^T \lambda(t) {\mathop{}\! \mathrm{d}}_t {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, t) {\mathop{}\! \mathrm{d}}t = \left[\lambda(t){\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, t) \right]_0^T - \int_0^T {\mathop{}\! \mathrm{d}}_t\lambda(t) {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, t) {\mathop{}\! \mathrm{d}}t\end{aligned}$$ Inserting this in and grouping the terms conveniently we have $$\begin{aligned} {\mathop{}\! \mathrm{d}}_\alpha \mathcal{L} &=& \partial_\alpha u_D[\alpha, \beta(\alpha, T)] + \Big\lbrace \partial_\beta u_D[\alpha, \beta(\alpha, T)] + \lambda(T) \Big\rbrace {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, T) + \lbrace\mu- \lambda(0) \rbrace {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, 0) \nonumber \\ &+& \int_0^T \left \lbrace - {\mathop{}\! \mathrm{d}}_t\lambda(t) - \lambda(t)\partial^2_{\beta} u_A [\alpha, \beta(\alpha, t)]\right \rbrace {\mathop{}\! \mathrm{d}}_\alpha \beta(\alpha, t) -\lambda(t) \partial_\alpha \partial_{\beta} u_A [\alpha, \beta(\alpha, t)] {\mathop{}\! \mathrm{d}}t\end{aligned}$$ Since the constraints hold, we may choose freely the Lagrange multipliers. In particular, we may choose them so that we can avoid calculating the derivatives of $\beta(\alpha, t)$ with respect to $\alpha$ (as this is computationally expensive). Thus, we have that $\lambda$ satisfies the adjoint equation $$\begin{aligned} {\mathop{}\! \mathrm{d}}_t\lambda(t) = - \lambda(t)\partial^2_{\beta} u_A [\alpha, \beta(\alpha, t)]\end{aligned}$$ with $\lambda(T) = - \partial_\beta u_D[\alpha, \beta(\alpha, T)]$, and $\mu = \lambda(0)$. Using this, the derivative is computed as $$\begin{aligned} {\mathop{}\! \mathrm{d}}_\alpha \mathcal{L} &=& \partial_\alpha u_D[\alpha, \beta(\alpha, T)] - \int_0^T \lambda(t) \partial_\alpha \partial_{\beta} u_A [\alpha, \beta(\alpha, t)] {\mathop{}\! \mathrm{d}}t\end{aligned}$$ [^1]: R.N. acknowledges the Spanish Ministry for his grant FPU15-03636. The work of D.R.I. is supported by the Spanish Ministry program MTM2017-86875-C3-1-R and the AXA-ICMAT Chair on Adversarial Risk Analysis. D.R.I also acknowledges the support of the EU’s Horizon 2020 project 815003-2 TRUSTONOMY and the RTC-2017-6593-7 project.

--- abstract: 'Do experiments based on superconducting loops segmented with Josephson junctions (e.g., flux qubits) show macroscopic quantum behavior in the sense of Schrödinger’s cat example? Various arguments based on microscopic and phenomenological models were recently adduced in this debate. We approach this problem by adapting –to flux qubits– the framework of large-scale quantum coherence, which was already successfully applied to spin ensembles and photonic systems. We show that contemporary experiments might show quantum coherence more than one hundred times larger than experiments in the classical regime. However, we argue that the often used demonstration of an avoided crossing in the energy spectrum is not sufficient to conclude about the presence of large-scale quantum coherence. Alternative, rigorous witnesses are proposed.' author: - 'Florian Fröwis$^1$, Benjamin Yadin$^{2,3}$, Nicolas Gisin$^1$' bibliography: - 'SQUID.bib' title: 'Insufficiency of avoided crossings for witnessing large-scale quantum coherence in flux qubits' --- Introduction {#sec:introduction} ============ The experimental demonstration of a massive object in a superposition of two well separated positions is generally considered as a positive test of quantum mechanics on large scales. Typical examples are interference of large molecules or proposed experiments with levitating nanospheres [@Arndt_Testing_2014]. These situations are often compared to Schrödinger’s thought experiment of a cat in a superposition of dead and alive [@Schrodinger_gegenwartige_1935]. It was argued that superconducting quantum interference devices (SQUIDs), that is, superconducting loops segmented with Josephson junctions, can exhibit a similar characteristic [@Leggett_Macroscopic_1980; @Leggett_Testing_2002]. In certain parameter regimes, the magnetic flux through the loop can be seen as an appropriate analog to the position of a massive object, where the capacitance of the circuit plays the role of the mass (see Fig. \[fig:schematics\]). The nonlinearity of the Josephson junction can lead to an effective double-well potential, in which the minima of the wells correspond to well separated (i.e., “macroscopically distinct”) flux states. There has been a debate about the precise implications of successfully demonstrating a coherent superposition between the two wells [@Leggett_Testing_2002; @Korsbakken_Electronic_2009; @Korsbakken_size_2010; @Leggett_Note_2016; @Bjork_size_2004; @Marquardt_Measuring_2008; @Nimmrichter_Macroscopicity_2013]. On the one hand, it was argued that recent experiments [@Friedman_Quantum_2000; @Wal_Quantum_2000; @Hime_Solid-State_2006] operate in the 100 nA or $\mu$A regime implying the presence of up to $10^{9}$ electrons. This, together with the experimental evidence of “macroscopic coherence”, should be seen as a genuine “macroscopic quantum effect” [@Leggett_Testing_2002]. On the other hand, arguments based on microscopic modeling suggest that effectively at most a few thousand electrons make the difference between the two states localized in each well [@Korsbakken_Electronic_2009; @Korsbakken_size_2010] (see [@Leggett_Note_2016] for a critique of this argument). Further contributions also assign an “effective size”, that is, a number that should, for example, reflect the number of electrons that effectively participate in the observed quantum effect [@Bjork_size_2004; @Marquardt_Measuring_2008; @Nimmrichter_Macroscopicity_2013] (see also table \[tab:results\]). While the differences in the precise frameworks of [@Leggett_Testing_2002; @Korsbakken_Electronic_2009; @Korsbakken_size_2010; @Leggett_Note_2016; @Bjork_size_2004; @Marquardt_Measuring_2008; @Nimmrichter_Macroscopicity_2013] are expected to lead to some deviation in the obtained results, it is astonishing by how much they vary. We find effective sizes ranging from two [@Marquardt_Measuring_2008] to 10$^{10}$ [@Leggett_Testing_2002]. In addition, none of the theory papers present a conclusive, testable witness for their claim. ![\[fig:schematics\] (a) Schematics of a superposition in the flux coordinate $\phi$ in both wells (thick: probability distribution; thin: potential). Experiments witnessing coherence between the two wells are sometimes considered to resemble a Schrödinger-cat situation if the inter-well distance and the system size (e.g., number of participating electrons) are large. (b) Most basic schematics of a superconducting ring with a single Josephson junction (cross; see also Sec. \[sec:squid-experiment-\]). In a certain parameter regime the magnetic flux $\phi$ through the ring is an appropriate choice for the relevant degree of freedom [@Friedman_Quantum_2000; @Wal_Quantum_2000; @Leggett_Testing_2002; @Caldeira_Introduction_2014]. The SQUID is then called flux qubit. The external flux $\phi_x$ controls the effective potential for $\phi$. In practice, the single Josephson junction is replaced by several junctions for an *in situ* control of some experimental parameters [@Friedman_Quantum_2000; @Wal_Quantum_2000; @Hime_Solid-State_2006].](DoubleWell2.pdf){width="\columnwidth"} In this paper, we argue that one important aspect of the large-scale quantum nature of these experiments is the amount and the spread of quantum coherence in the flux coordinate since it is a direct test of the superposition principle. As already successfully done for spins and photons [@Frowis_Measures_2012; @Frowis_Linking_2015; @Oudot_Two-mode_2015], we rescale the coherence of the target state by the coherence of a classical reference state (i.e., the state confined in a single well). In this way, we introduce an “effective size” that quantifies the scale of the quantum effect. The advantage of this approach is the applicability to experimental data. After a short review of the motivation and the theory of the framework in Sec. \[sec:conc-large-quant\], the phenomenological model of the flux qubit experiments is presented in Sec. \[sec:squid-experiment-\]. Identifying the flux as a relevant observable, we show that indeed the ideally generated target states show large quantum coherence. With the parameters from experiment [@Friedman_Quantum_2000], we find that the quantum coherence of the target states in flux basis is almost two hundred times larger than of a classical reference state (i.e., the ground state of a single potential well). Ideal cat states could even reach numbers 1000 times larger than that of the reference state. The experimental proof for the generation of the eigenstates was argued to be the avoided crossing in the energy spectrum when tuning the imbalance of the two well minima. In Sec. \[sec:cert-large-quant\], we show that this evidence is not sufficient to conclude the presence of large-scale quantum coherence. We do so by presenting a simple dephasing model in the flux basis that leads to a drastic reduction of the quantum coherence (to the level of the classical reference state) while keeping the feature of the avoided crossing. The framework of large-scale quantum coherence provides testable witnesses, which often turn out to be feasible in practice [@Frowis_Lower_2017]. In Sec. \[sec:suff-exper-test\], we discuss the possibility of lower-bounding large quantum coherence by witnessing a strong response of the system to flux dephasing. The paper is summarized and discussed in Sec. \[sec:discussion\]. Framework of large quantum coherence {#sec:conc-large-quant} ==================================== One of the simplest and most fundamental questions in quantum mechanics is the validity of the superposition principle on all scales. There exist several alternative models that, for example, prevent a persistent superposition of two significantly distinct positions for massive particles [@Bassi_Models_2013]. Given an observable $X$ with a natural macroscopic limit (such as position or magnetization), quantum coherence between two far-distant parts of the spectrum arguably challenges our classical intuition more than quantum coherence constraint to a small (microscopic) regime [@Schrodinger_gegenwartige_1935]. For pure states, a superposition between far-distant spectral parts of $X$ implies that the wave function has a large spread. The simplest way to measure the spread of a state $| \psi \rangle $ is the variance $\mathrm{var}_{\psi}(X)$. For mixed states, the variance is no longer a faithful measure of coherence since it does not distinguish between coherent superposition and incoherent mixture. The convex roof construction is a well-known technique to overcome the shortcomings of the variance. Given a mixed state $\rho$, one considers the infinite set of all pure state decompositions (PSD) $\rho = \sum_i p_i \left| \psi_i \right\rangle\!\left\langle \psi_i\right| $ and minimizes the average variance $$\label{eq:1} \mathcal{I}(\rho,X) = \min_{\mathrm{PSD}}\sum_i p_i \mathrm{var}_{\psi_i}(X).$$ Since the incoherent part is eliminated with the convex roof construction, we call $\mathcal{I}(\rho,X)$ the quantum coherence of $\rho$ in the spectrum $X$. Measuring a certain value of $\mathcal{I}$ experimentally falsifies all collapse models that forbid the superposition principle to be valid on the order of $\sqrt{\mathcal{I}}$. Notably, $4 \mathcal{I}$ is the so-called quantum Fisher information [@Helstrom_Quantum_1976; @Braunstein_Statistical_1994; @Yu_Quantum_2013]. This implies that large quantum coherence is necessary to reach high sensitivity in parameter estimation protocols where $X$ is the generator of the parameter shift. The intuitive motivation to choose the convex roof of the variance is made more rigorous in a recent resource theory of “macroscopic coherence” [@Yadin_General_2016]. It arguably makes sense to compare $\mathcal{I}(\rho,X)$ to a reference state $| \psi_{\mathrm{ref}} \rangle $, which behaves “maximally” classically among all pure quantum states. Like the choice for the observable $X$, identifying $| \psi_{\mathrm{ref}} \rangle $ is physically motivated and hence situation-dependent. Nevertheless it brings some conceptual advantages, as we can avoid unwanted dependencies on scaling factors of $X$. Furthermore, it helps to compare systems with various numbers of modes or particles. One defines $$\label{eq:2} \mathcal{I}_{\mathrm{rel}}(\rho,X) = \frac{\mathcal{I}(\rho,X) }{\mathcal{I}(\psi_{\mathrm{ref}},X)},$$ which tells us how much larger the quantum coherence of $\rho$ is compared to that of the reference state. The observable $X$ and the reference state have to be physically motivated. Let us consider two examples [@Frowis_Measures_2012; @Frowis_Linking_2015]. Quadrature operators $X = e^{i\theta} a + e^{-i\theta} a^{\dagger}$ for phase space states (sums of quadrature operators for many modes) and collective spin operators $X = \sum_{i = 1}^N \sigma^{(i)}$ proved to be reasonable choices. The reference states are the coherent state in phase space and the spin-coherent state (i.e., parallel spins in a product state) in spin ensembles, respectively. It was shown that $\mathcal{I}_{\mathrm{rel}}(\rho,X)$ can then be interpreted as an effective size as it is connected to the number of microscopic entities (photons or spins) that effectively contribute to the large-scale quantum effect [@Frowis_Macroscopic_2017]. Generally, let us consider a classical reference state composed of uncorrelated microscopic constituents (“particles”) and an additive observable with a macroscopic limit. Then the variance is linear in the number of particles $N$, that is, $\mathrm{var}(X) = N \overline{\mathrm{var}}(x_0)$, where $x_0$ is the corresponding one-particle operator and $\overline{\mathrm{var}}$ is the average variance per particle. Then, the effective size $\mathcal{I}_{\mathrm{rel}}(\rho,X)$ gives the spread of the quantum coherence per particle and per microscopic unit, which justifies the use of the variance rather than, for example, the square root of it. Short summary of SQUID physics {#sec:squid-experiment-} ============================== ![\[fig:friedman\] (a) Center: two examples of the double-well potential $U$ (curved, blue line; Eq. (\[eq:5\])) for $\Phi_x = 0.499 \Phi_0$ and $\Phi_x = 0.5 \Phi_0$. The difference $\epsilon$ between the two minima for $\Phi_x = 0.499 \Phi_0$ is artificially increased for a clearer distinction between the two cases. Top: absolute square of the wave functions of the target states $| 0 \rangle $ and $| 1 \rangle $ at $\Phi_x = 0.499 \Phi_0$ in the experiment of Ref. [@Friedman_Quantum_2000], shifted by the respective eigenenergy. The confinement of each in a single well is clearly visible. Bottom: The same at $\Phi_x = 0.5 \Phi_0$. The wave functions are both spread over both wells. (b) Energies of the target states as a function of $\Phi_x$ (solid lines). The avoid crossing at $\Phi_x = 0.5 \Phi_0$ is clearly visible. The dashed lines are the energies of the target states after dephasing as described in Eq. (\[eq:6\]) with $\Gamma = \sqrt{\hbar/(C \omega)}$. The energies are shifted, but the avoided crossing is still present. The inset shows the energy of the ground state before (solid) and after (dashed) dephasing with the same $\Gamma$.\[fig:EffectDephaseFixCorrLength\]\[fig:WFdetail\]](WF_Detail_Collection_THz.pdf){width="\columnwidth"} In the discussion on whether experiments with superconducting devices show aspects of a Schrödinger-cat state, some approaches [@Marquardt_Measuring_2008; @Korsbakken_Electronic_2009; @Korsbakken_size_2010] work with a microscopic model. This leads to discussions about details like the proper choice of the “microscopic unit” or of characteristic quantities [@Leggett_Note_2016]. In the presented approach based on quantum coherence $\mathcal{I}$, it is more natural to work with a phenomenological model of a single degree of freedom, the total flux $\Phi$ and the charge $Q$ as the conjugate variable (i.e., $[\Phi,Q] = i \hbar$). To discuss a specific example, let us consider the SQUID experiment of Friedman *et al.* [@Friedman_Quantum_2000]. The effective potential depends on the the inductance $L$ and the critical current $I_c$. In addition, it is controlled by an external flux $\Phi_x$ (see Fig. \[fig:schematics\] (b)). This results in a quadratic part with inductive energy $E_L = \Phi_0^2/(2L)$ and in a nonlinear part from the junction with Josephson coupling energy $E_J = I_c \Phi_0/(2\pi)$ ($\Phi_0$ is the magnetic flux quantum), that is, $$\label{eq:5} U(\phi) = E_L \left( \frac{\Phi-\Phi_x }{\Phi_0} \right)^2 + E_J \cos \left( 2\pi\Phi/\Phi_0 \right).$$ The Hamiltonian is completed by the “kinetic” energy $\frac{1}{2C}Q^2$ where $C$ is the capacitance. For later, we introduce the charging energy $E_C = e^2/(2C)$ and the angular frequency of the hypothetical $LC$ circuit, $\omega = 1/\sqrt{L C}$. The experimental parameters given in Ref. [@Friedman_Quantum_2000] are $E_C = 188$ MHz, $E_L = 13$ THz and $E_J = 159$ GHz. In the following we use these numbers when it comes to numeric calculations, but we generally assume $E_J, E_L \gg E_C$. Depending on $\Phi_x$, the potential $U$ can exhibit an effective double-well structure (see Fig. \[fig:friedman\] (a)). At $\Phi_x = \frac{1}{2} \Phi_0$, there exist two global minima separated by roughly $0.655\Phi_0$. This implies that coherent tunneling is possible and the wave functions of the eigenstates have finite contributions in both wells. This leads to statements like “the wave function lives in both wells”. Given the large separation and the involvement of up to 10$^9$ Cooper pairs, some physicists have called this a Schrödinger-cat situation. The “classical” regime is at $\Phi_x = 0$, as the potential has a (deep) single well with a well-defined flux state as its ground state. While a small anharmonicity is always present, the ground state in this regime behaves like the ground state of an harmonic oscillator if $E_L, E_J \gg E_C$. For example, the variance of $\Phi$ and $Q$ are minimal in the sense that the Heisenberg uncertainty relation is basically tight [^1]. Hence, this state is chosen as the reference state as it behaves most classically. As argued in Refs. [@Friedman_Quantum_2000; @Leggett_Testing_2002], the experimental proof of the coherent superposition of “left and right” is the resolution of the predicted avoided crossing by tuning the imbalance between left and right well (see Fig. \[fig:friedman\] (a)). Since the energy gap is proportional to the tunneling probability, the splitting between the lowest two eigenstates is very tiny for the given energies. Hence the initial ground state is driven to highly excited states just below the barrier. There, the minimal energy gap is pronounced enough to be measurable (see Fig. \[fig:friedman\] (b)). While the discussion in this paper is adapted to the experiment of Ref. [@Friedman_Quantum_2000], other experimental setups lead to conceptually similar results. For example, the three-junction setup of van der Wal *et al.* [@Wal_Quantum_2000] can be modeled with two independent flux coordinates. A suitable coordinate transform leads to an effective double well potential similar to the one discussed before [@Orlando_Superconducting_1999]. Hence, we will find qualitative similar results, which are later given without discussing details. We only mention that a further parameter of the experiment is $\alpha$, which is the ratio of the Josephson coupling energy of one junction to the other two (identical) junctions. The parameters from Ref. [@Wal_Quantum_2000] are $E_J/E_C = 38$ and $\alpha = 0.8$. Large quantum coherence in flux qubits {#sec:appl-superc-devic} ====================================== In this section, we apply the framework of large quantum coherence to the flux qubit. Following on the discussion in Sec. \[sec:conc-large-quant\], we choose a target state (i.e., the state which is supposed to exhibit large quantum coherence), an observable $X$ for which we evaluate the spread of coherence, and a classical reference state to which we want to compare the target state. From Sec. \[sec:squid-experiment-\], it is evident that eigenstates (e.g., ground states) at $\Phi_x = 1/2 \Phi_0$ are good candidates for “cat states” if we choose $X = \Phi$. The ground state at $\Phi_x = 0$ serves as the classical reference state. In Sec. \[sec:effective-size-ideal\], we take the ground state at $\Phi_x = 1/2 \Phi_0$ as the target state, while in Sec. \[sec:effect-size-actu\] we numerically analyze the target states of the experiment in Ref. [@Friedman_Quantum_2000]. Since in both cases we treat the ideal situation, the quantum coherence $\mathcal{I}$ simplifies to the variance. Effective size of the ideal cat state {#sec:effective-size-ideal} ------------------------------------- We first calculate the variance of the classical reference state. For this, we notice that we are in the regime $E_L/E_C \approx 7.17 \times 10^4 \gg 1$, which implies a rather deep potential in the classical situation $\Phi_x = 0$. It can be easily shown that the anharmonic part of the potential does not significantly influence the ground state (despite a large $E_J$). Hence, to obtain an analytical result, we approximate $U$ from Eq. (\[eq:5\]) by an harmonic oscillator. From the second-order Taylor series of $U$ at $\Phi_x = 0$, we can extract the effective trapping frequency $\omega_{\mathrm{cl}} = \omega \sqrt{1+2\pi^2 E_J/E_L}$ and find $$\label{eq:8} \mathrm{var}_{\mathrm{ref}}(\Phi) \approx \frac{\hbar}{2 C \omega_{\mathrm{cl}}} = \frac{1}{2\pi} \sqrt{\frac{E_C}{E_L + 2\pi^2 E_J}} \Phi_0^2.$$ We now turn to the ground state at $\Phi_x = \Phi_0/2$, which is an equally weighted superposition of being in the left and right well. For simplicity, this state is called “cat state” in the following. Since the distance $d$ between the minima is much larger than the spread of the wave packet in one well, we approximate the variance of the cat state by $d^2$. In the present parameter regime, $d$ is in the order of $\Phi_0$ with a slight dependency on $E_C$ and $E_J$. We numerically find $\mathrm{var}_{\mathrm{cat}}(\Phi) \approx 0.655^2\Phi_0^2$. Together with Eq. (\[eq:8\]), we have $$\label{eq:10} \mathcal{I}_{\mathrm{rel}} (\Phi)\approx 0.86 \pi \sqrt{\frac{E_L + 2\pi^2 E_J}{E_C}} \approx 1315.$$ In the limit of a dominant Josephson energy $E_J/E_L \propto I_c L/\Phi_0 \gg 1$ (while keeping $E_L \gg E_C$), one has $\mathrm{var}_{\mathrm{cat}} (\Phi) \approx \Phi_0^2$ and $\mathcal{I}_{\mathrm{rel}} \approx 2 \sqrt{2}\pi^2 \sqrt{E_J/E_C} \approx 2565$. The calculation for the experiment of Ref. [@Wal_Quantum_2000] is analogous, but gives a fully analytic expression. We find $$\label{eq:11} \mathcal{I}_{\mathrm{rel}} = 4 \arccos^2 \left( \frac{1}{2\alpha} \right) \sqrt{4\alpha + 2} \sqrt{\frac{E_J}{E_C}} \approx 45.$$ For both experiments, we note the scaling of the effective size with $\sqrt{{E_J}/{E_C}}$. Effective size of the target states of Ref. [@Friedman_Quantum_2000] {#sec:effect-size-actu} -------------------------------------------------------------------- We now numerically calculate the effective size of $| 0 \rangle $ and $| 1 \rangle $ (see Fig. \[fig:friedman\] (a)) that were targeted in the experiment of Ref. [@Friedman_Quantum_2000]. For this, we first have to numerically diagonalize the Hamiltonian. As detailed in Ref. [@Everitt_Superconducting_2004], one analytically calculates the eigenbasis of the Hamiltonian in the case of $E_J = 0$ (since it reduces to the familiar $LC$ circuit) and then expresses the general $H$ in this basis. With the experimental parameters mentioned in Sec. \[sec:squid-experiment-\], we can diagonalize $H$ with high precision and reasonable cut-off dimensions. Here, we are interested in the regime around $\Phi_x = \frac{1}{2} \Phi_0$. As shown in Fig. \[fig:WFdetail\] (a), at $\Phi_x = \frac{1}{2}\Phi_0$, the wave functions are indeed spread over the entire classically allowed range. For the classical reference state, we again use Eq. (\[eq:8\]). The eigenstate $| 1 \rangle $ exhibits a spread of $\mathrm{var}_{\left| 1\right\rangle }(\Phi) \approx 6.32 \times 10^{-2} \Phi_0^2$. We find $\mathcal{I}_{\mathrm{rel}} \approx 194$. For the second state $| 0 \rangle $, the spread is slightly smaller. Reference SUNY [@Friedman_Quantum_2000] Delft [@Wal_Quantum_2000] --------------------------------------------------- ------------------------------- --------------------------- Effective size[^2] $\mathcal{I}_{\mathrm{rel}}$ 1315 / 194 45 Leggett[^3] [@Leggett_Testing_2002] $10^{10} / 10^{10}$ $10^{6} / 10^{10}$ Björk and Mana [@Bjork_size_2004] 33 - Marquardt *et al.* [@Marquardt_Measuring_2008] - $\lesssim 2$ Korsbakken *et al.* [@Korsbakken_Electronic_2009] 3800 - 5750 42 : \[tab:results\] Comparison of effective size $\mathcal{I}_{\mathrm{rel}}$ with results from literature for experiments [@Friedman_Quantum_2000; @Wal_Quantum_2000]. Our method is conceptually closest to the work of Björk and Mana [@Bjork_size_2004], whose result has to be squared to match ours. Interestingly, we obtain similar numbers as [@Korsbakken_Electronic_2009] (in the scenario of Sec. \[sec:effective-size-ideal\]) despite the different approach. See Sec. \[sec:discussion\] and Ref. [@Frowis_Macroscopic_2017] for further discussion. Certifying large quantum coherence in experiments {#sec:cert-large-quant} ================================================= We now turn to the question of how to certify the presence of large quantum coherence from the experimental data. We first show that the demonstration of an avoided crossing in the energy spectrum is not sufficient to witness wide-spread coherence. To this end we introduce a specific dephasing model that largely preserves the energy gap but significantly reduces the quantum coherence. Hence, we discuss alternative schemes at the end of the section. Robustness of avoided crossing, fragility of quantum coherence {#sec:robustn-avoid-cross} -------------------------------------------------------------- ![\[fig:EffectDephasingFixPhix\] Impact of dephasing as a function of the correlation length $\Gamma$ for the two eigenstates $| 0 \rangle $ and $| 1 \rangle $ (see Sec. \[sec:effect-size-actu\] and Fig. \[fig:WFdetail\] (a)) at fixed $\Phi_{x} = \frac{1}{2}\Phi_{0}$. (a) The relative quantum coherence $\mathcal{I}_{\mathrm{rel}}$ drops quickly when increasing the dephasing strength (i.e., decreasing $\Gamma$). For $\Gamma \approx 4 \sqrt{\mathrm{var}_{\mathrm{cl}}(\Phi)}$, one has $\mathcal{I}_{\mathrm{rel}} \approx 1$, that is, the spread of the quantum coherence is at the same order as the classical reference state. (b) Relative difference of the energy gap $\Delta E$ under dephasing ($\Delta E_0$ denotes the energy gap without dephasing). The energy gap $\Delta E$ is basically invariant for large $\Gamma$. Only for $\Gamma \lesssim 4 \sqrt{\mathrm{var}_{\mathrm{cl}}(\Phi)}$ the relative difference becomes experimentally relevant. The gray zone highlights the parameter range of $\Gamma$ where $\mathcal{I}_{\mathrm{rel}} \leq 1$ and $1-\Delta E/\Delta E_0 \leq 10^{-3}$. The yellow circles mark $\Gamma = \sqrt{\hbar/(C \omega)}$, which is also used in Fig. \[fig:friedman\] (b). The relative difference saturates at around 10$^{-10}$ for large $\Gamma$ because of numerical precision.](AvoidedCrossing_FixedPhix_New2.pdf){width="\columnwidth"} In this section, we consider a simple dephasing model in the flux basis and discuss its consequences. Suppose that the eigenstates are subject to dephasing $$\label{eq:6} \rho(\Phi,\Phi^{\prime}) \rightarrow e^{- (\Phi-\Phi^{\prime})^2/ (2\Gamma^2)}\rho(\Phi,\Phi^{\prime}),$$ meaning that coherence-elements $\rho(\Phi,\Phi^{\prime})$ are damped by a Gaussian function on a scale $\Gamma$ (called correlation length in the following). From the discussion in Sec. \[sec:conc-large-quant\] it is not surprising that this map reduces the quantum coherence $\mathcal{I}$, which is calculated using the spectral decomposition $\rho = \sum_i \lambda_i \left| \psi_i \right\rangle\!\left\langle \psi_i\right| $ and $$\label{eq:7} \mathcal{I}(\rho,\Phi) = \sum_{i < j} \frac{(\lambda_i - \lambda_j)^{2}}{\lambda_i + \lambda_j} \left| \left\langle \psi_i \right| \Phi \left| \psi_j \right\rangle \right|^2$$ (see, e.g., Ref. [@Braunstein_Statistical_1994]). The result for $\Phi_x = \frac{1}{2}\Phi_{0}$ and variable $\Gamma$ is presented in Fig. \[fig:EffectDephasingFixPhix\], which shows the fragility of the quantum coherence under flux dephasing. In contrast, the energy gap $\Delta E$ (i.e., the difference in the energy expectation value between the dephased eigenstates at $\Phi_x = \frac{1}{2}\Phi_0$) is rather stable (see Fig. \[fig:EffectDephasingFixPhix\] (b)). Only for a correlation length $\Gamma \lesssim \sqrt{\hbar/(C \omega)}$ (i.e., close to the width of the classical reference state) $\Delta E$ deviates significantly from the energy gap between the original eigenstates, denoted by $\Delta E_0$. Note that the energies of the states are increased by the map (\[eq:6\]). Since this affects all states equally strong (including the ground state), the differences in energies remain basically unchanged. From these findings, we conclude that the mere observation of an avoided crossing is not sufficient to prove large quantum coherence in the system. We emphasize that this conclusion cannot be found by reducing the flux coordinate to a two-dimensional space (left well/ right well), which is often done in qualitative discussions. Sufficient experimental test {#sec:suff-exper-test} ---------------------------- We now discuss the possibility for a sufficient experimental test to certify a minimal quantum coherence $\mathcal{I}(\rho,\Phi)$. As shown in Ref. [@Frowis_Detecting_2016], we can employ the following dynamical protocol. For this, we have to be able to implement $V_t = \exp(-i \Phi t)$. We first choose a suitable measurement with measurement operators $\{\Pi_k\}$ for the outcomes $\{x_k\}$. Then, we measure the probability distributions $p_k = \mathrm{Tr}\rho \Pi_k$ and $q_k = \mathrm{Tr}V_t\rho V_t^{\dagger} \Pi_k$. With this, we calculate the so-called Bhattacharyya coefficient $B = \sum_k \sqrt{p_k q_k}$. One finds [@Frowis_Detecting_2016] $$\label{eq:4} \mathcal{I}(\rho,\Phi) \geq \frac{1}{t^2} \arccos^2 B.$$ In essence, inequality Eq. (\[eq:4\]) witnesses the response of the system to a dynamical process generated by $\Phi$. High susceptibility implies the presence of large-scale quantum coherence. Using this bound, one finds certified quantum coherence which is (in the cold atom experiment of Ref. [@Hosten_Measurement_2016], for example) up to $71$ larger than that of classical reference states [@Frowis_Lower_2017]. As we show now, we can relax the requirements of this protocol. For instance, suppose we are not able to fully control $t$ (i.e., the duration of the unitary), but we only assume that $t$ follows a certain distribution $\mu(t)$ with $\int dt \mu(t) = 1$. Then, the quantum state after the non-unitary dynamics is described by $$\label{eq:9} \mathcal{E}(\rho) = \int_{-\infty}^{\infty} dt \mu(t) V_t \rho V_t^{\dagger}.$$ For example, $$\mu(t) = \frac{1}{\sqrt{2\pi} \Gamma} \exp(- \Gamma^2 t^2/2)\label{eq:15}$$ leads to an effective dephasing in the $\Phi$ basis exactly as modeled in Eq. (\[eq:6\]). We redefine $q_k = \mathrm{Tr} \mathcal{E}(\rho) \Pi_k$ for the calculation of $B$. Following Ref. [@Frowis_Detecting_2016] and using the quantum fidelity $F$, one can show that $$\begin{split} B &\geq F(\rho, \mathcal{E}(\rho)) \geq \int_{-\infty}^{\infty} dt \mu(t) F(\rho, V_t \rho V_t^{\dagger})\label{eq:14} \\ & \geq \int_{- \pi/(2\sqrt{\mathcal{I}})}^{ \pi/(2\sqrt{\mathcal{I}})} dt \mu(t) \cos \left( \sqrt{\mathcal{I}} t \right), \end{split}$$ where for the last step we used $F(\rho, V_t \rho V_t^{\dagger}) \geq \cos \left( \sqrt{\mathcal{I}} t \right)$ in the interval $t \in [- \pi/(2\sqrt{\mathcal{I}}), \pi/(2\sqrt{\mathcal{I}})]$ [@Uhlmann_gauge_1991; @Frowis_Kind_2012]. Finally, one has to invert inequality (\[eq:14\]) to obtain a lower bound on the quantum coherence $\mathcal{I}$. This is generally not analytically possible, but one can find either numerical solutions or approximations. The numerical solution is an optimization problem where one finds the maximal $\mathcal{I}$ such that the inequality in Eq.  is fulfilled given the data, $B$, and the model of the dynamics, $\mu(t)$. As an example for an analytical approximation, we consider Eq. (\[eq:15\]). In the weak dephasing limit (i.e., $\Gamma^2 \gg \mathcal{I}$), we can safely move the integration limits in the second line of Eq.  from $\pm \pi/(2 \sqrt{\mathcal{I}})$ to $\pm \infty$, which allows us to find $B \gtrsim \exp (-\Gamma^2 \mathcal{I}/2)$, from which $$\label{eq:12} \mathcal{I}(\rho,\Phi) \gtrsim 2\Gamma^2 \log 1/B$$ follows. Recent experiments with flux qubits (e.g., by Knee *et al.* [@Knee_strict_2016a]) are conceptually and technically very close to the presented framework. A standard way of implementing measurements is based on the Josephson Bifurcation Amplifier [@Siddiqi_RF-Driven_2004; @Lupascu_Quantum_2007; @Nakano_Quantum_2009], which can be modeled by a dephasing process similar to Eq. (\[eq:9\]). This opens the opportunity for witnessing large quantum coherence with flux qubits. Then, the protocol in Ref. [@Knee_strict_2016a], which was employed to falsify so-called macrorealistic theories, is of the same spirit as the procedure presented here, namely to witness the susceptibility of a system with respect to a specific influence. The difference is that here we do need to control the experiment to a certain degree. In particular, the dynamics of Eq. (\[eq:9\]) should be well characterized (e.g., $\Gamma$ in Eq. (\[eq:15\]) should be sufficiently known), while assumptions on the initial state $\rho$ and the measurement $\{\Pi_k\}$ are not necessary. Discussion and summary {#sec:discussion} ====================== In this paper, we proposed to consider the extent of quantum coherence in the flux coordinate as one relevant aspect of the “macroscopic quantum nature” of flux qubits. While the framework of large quantum coherence arguably has an operational meaning in the resource theory of asymmetry [@Marvian_How_2016; @Yadin_General_2016] and metrology [@Helstrom_Quantum_1976; @Braunstein_Statistical_1994], its interpretation as an “effective size” (i.e., how many Cooper pairs or electrons effectively contribute to the quantum phenomenon) is less clear. From a mathematical point of view, the framework of large quantum coherence is directly applicable to the SQUID experiments. The choice of observable (here, the flux $\Phi$) and the reference state (ground state of the “classical” regime $\Phi_x = 0$) are physically motivated. The scaling of the effective size in the ideal case (Sec. \[sec:effective-size-ideal\]) with $\sqrt{E_J/E_C}$ is interesting as the ratio $E_J/E_C$ is sometimes argued to be the relevant scale of the experiment [@Marquardt_Measuring_2008]. In this context, it is interesting to compare our results with the analysis by Korsbakken *et al.* [@Korsbakken_Electronic_2009; @Korsbakken_size_2010] [^4]. The authors work with a microscopic model to investigate the effective number of participating Cooper pairs in the cat state. They find numbers around 3800 - 5750 for Ref. [@Friedman_Quantum_2000] and 42 for Ref. [@Wal_Quantum_2000]. Since their results depend on experimentally “difficult” quantities like the Fermi velocity, this work was criticized by Leggett in Ref. [@Leggett_Note_2016] where he constructs an “obvious” example of a Schrödinger-cat state that still gives a very low effect size with Korsbakken *et al.*’s method. Nevertheless, we note that our results (which only depend on the phenomenological model) and the results of Korsbakken *et al.* are in the same order. It would be interesting to see whether this is pure coincidence or whether there is some underlying connection. We have shown that the experimental observation of avoided crossing in the energy spectrum is not a conclusive witness for the presence of large-scale quantum coherence. In contrast, the framework of large quantum coherence provides powerful lower bounds that are testable in practice. Here, we extended a recently developed protocol [@Frowis_Detecting_2016] to nonunitary dynamics, which is conceptually and technically close to existing experiments [@Knee_strict_2016a]. Note that Eq.  is a general way to lower bound the spread of quantum coherence for an observable $X$. For its successful application, it is necessary to faithfully implement the unitary $V_t = \exp(-i t X)$ in the experiment and to observe the impact by measuring the system before and after its application (on different runs). The control of $t$ can be relaxed to having knowledge about its distribution over many runs, $\mu(t)$. In contrast, no knowledge about the measurement is necessary to guarantee the correctness of Eq. . The presented work gives a first insight that wide-spread quantum coherence might be present in SQUID experiments. We see our results as a starting point for further investigations. As mentioned before, the microscopical interpretation of $\mathcal{I}_{\mathrm{rel}}$ is open. Furthermore, one should apply the framework of large quantum coherence to other experiments and parameter regimes, such as $E_J > E_C > E_L$ (i.e., fluxonium) and $E_J \approx E_L \gg E_C$ (c-shunt flux qubit). A well-justified choice of the relevant observable as well as the target and reference state will lead to further insight to meso- and macroscopic quantum phenomena. *Acknowledgments.—* We thank Daniel Estève and his research team, Amir Caldeira, Bill Munro, George Knee, Pavel Sekatski and Wolfgang Dür for inspiring discussions. This work was supported by the National Swiss Science Foundation (SNF), grant No. 172590, the European Research Council (ERC MEC), the EPSRC and Wolfson College, University of Oxford. [^1]: For the present energies, we find $\Delta \Phi \Delta Q \approx \hbar/2 (1+4\times 10^{-7}) \geq \hbar/2$. [^2]: The first number refers to the ideal scenario (Sec. \[sec:effective-size-ideal\]), the second to the implemented protocol (Sec. \[sec:effect-size-actu\]). [^3]: Leggett uses two measures: disconnectivity and extensive difference. [^4]: Further discussion regarding the theoretical estimation of the “effective size” of the SQUID done in other papers is provided in Ref. [@Frowis_Macroscopic_2017].

--- abstract: 'Modern face recognition systems leverage datasets containing images of hundreds of thousands of *specific* individuals’ faces to train deep convolutional neural networks to learn an embedding space that maps an *arbitrary* individual’s face to a vector representation of their identity. The performance of a face recognition system in face verification (1:1) and face identification (1:N) tasks is directly related to the ability of an embedding space to discriminate between identities. Recently, there has been significant public scrutiny into the source and privacy implications of large-scale face recognition training datasets such as MS-Celeb-1M and MegaFace, as many people are uncomfortable with their face being used to train dual-use technologies that can enable mass surveillance. However, the impact of an individual’s inclusion in training data on a derived system’s ability to recognize them has not previously been studied. In this work, we audit ArcFace, a state-of-the-art, open source face recognition system, in a large-scale face identification experiment with more than one million distractor images. We find a Rank-1 face identification accuracy of 79.71% for individuals present in the model’s training data and an accuracy of 75.73% for those not present. This modest difference in accuracy demonstrates that face recognition systems using deep learning work better for individuals they are trained on, which has serious privacy implications when one considers all major open source face recognition training datasets do not obtain informed consent from individuals during their collection.' author: - Chris Dulhanty - Alexander Wong bibliography: - 'AIES-bib.bib' title: Investigating the Impact of Inclusion in Face Recognition Training Data on Individual Face Identification --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10002978.10003029.10003032&lt;/concept\_id&gt; &lt;concept\_desc&gt;Security and privacy Social aspects of security and privacy&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010178.10010224.10010225.10010231&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Visual content-based indexing and retrieval&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010521.10010542.10010294&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Neural networks&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003456.10003462.10003487&lt;/concept\_id&gt; &lt;concept\_desc&gt;Social and professional topics Surveillance&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; Introduction ============ Face recognition systems using Deep Convolutional Neural Networks (DCNNs) depend on the collection of large image datasets containing thousands of sets of *specific* individuals’ faces for training. Using this data, DCNNs learn a set of parameters that can map an *arbitrary* individual’s face to a feature representation, or *faceprint*, that has small intra-class and large inter-class variability. The ability of a face recognition system to distinguish between identities within this embedding space depends on the size and diversity of its training data, along with its model capacity and underlying algorithms. Face recognition systems have benefited from the enabling power of Internet in the collection of large-scale image datasets and from hardware improvements in enabling efficient training of large models. Recently, increased attention to face recognition by academia, industry and government has brought new researchers, ideas and funding to the field, leading to performance improvements on benchmark tasks Labelled Faces in the Wild (LFW) [@LFWTech] and MegaFace [@nech2017level]. Consequently, face recognition systems are now being integrated into consumer and industrial electronic devices and offered as application programming interfaces (APIs) by providers such as Amazon, Microsoft, IBM, Megvii and Kairos. However, along with improved performance has come increased public discourse on the ethics of face recognition systems and their development. Algorithmic auditing of commercial face analysis applications has uncovered disparate performance for intersectional groups across several tasks. Poor performance for darker skinned females by commercial face analysis APIs has been reported by Buolamwini, Gebru and Raji [@buolamwini2018gender; @Raji2019], as has lower accuracy in face identification by commercial systems with respect to lower (darker) skin reflectance by researchers at the US Department of Homeland Security [@cook2019demographic]. As bias in training data begets bias in model performance, efforts to create more diverse datasets for these tasks have resulted. IBM’s Diversity in Faces dataset [@merler2019diversity], released in January 2019, is a direct response to this body of research. Using ten established coding schemes from scientific literature, researchers annotated one million face images in an effort to advance the study of fairness and accuracy in face recognition. However, this dataset has seen public scrutiny from a different, but equally notable perspective. A March 2019 investigation by NBC News into the origins of the dataset brought to the public conversation the issue of informed consent in large-scale academic image datasets, as IBM leveraged images from Flickr with a Creative Commons Licence without notifying content owners of their use [@solon_2019]. To rationalize the collection of large-scale image datasets without explicit consent of individuals, some computer vision researchers appeal to the non-commercial nature of their work. However, work by Harvey *et al.* at MegaPixels have found that authors’ stated limitations on dataset use do not translate to real-world restrictions [@megapixels]. In the case of Microsoft’s MS-Celeb-1M dataset, authors included an explicit “non-commercial research purpose only" clause with the dataset, which was the largest publicly-available face recognition dataset at the time. However, as the dataset has been cited in published works by the research arms of many commercial entities, findings cannot easily be isolated from improvements in product offerings. As a direct result of MegaPixel’s work on the ethics, origins, and privacy implications of face recognition datasets, MS-Celeb-1M [@guo2016ms], Stanford’s Brainwash dataset [@stewart2016end] and Duke’s Multi-Target, Multi-Camera dataset [@ristani2016MTMC] were removed from their authors’ websites in June 2019. However, in the case of MS-Celeb-1M, the data remains accessible via torrents, derived datasets and other hosts [@megapixels]. In addition to issues of bias and informed consent in data collection, the general use of face recognition systems by commercial and government agencies has been raised by civil rights groups and research centers, as there is no oversight for its deployment in civil society [@aclu; @whittaker2018ai]. For these and other reasons, multiple cities in the United States have banned the use of face recognition systems for law enforcement purposes [@conger_2019; @wu_2019; @ravani_2019]. Many people are concerned with their identify being used to train the dual-use technology that is face recognition. With reports of face recognition being used by law enforcement entities to identify protesters in London [@bowcott_2018] and Hong Kong [@mozur_2019], and measures enacted to ban face masks in the latter location [@yu_2019], there is merit in understanding the impact of one’s inclusion in the training data that fuels the development of these systems. In an effort to inform the conversation about informed consent and privacy in the domain of face recognition, we conduct experiments on a state-of-the-art system. The goal of this work is to determine the impact of an individual’s inclusion in face recognition training data on a derived system’s ability to recognize them. To the best of the authors’ knowledge, this is the first paper to investigate this relationship. The remainder of this paper is organized in the following manner; section two outlines ethical considerations for some decisions in the design and implementation of this work, section three provides background for the taxonomy, algorithms and data used in face recognition research, section four outlines the design of experiments used to address the research question, section five presents our results and adds discussion and the paper concludes in section six. Ethical Considerations ====================== Intent ------ The intent of this work is to investigate the performance of face recognition systems with respect to inclusion in training datasets. While one interpretation of this work may be to motivate efforts to mitigate demographic bias in the development of face recognition systems, it should be noted that increasing the performance of face recognition systems in any context can increase their ability to be used for oppressive purposes. In addition, due to historical societal injustices against marginalized populations and racially-biased police practices in the United States, a disproportionate number of African Americans and Hispanics are present in mugshot databases, often used by law enforcement agencies as data sources for face recognition systems [@naacp; @garvie2016perpetual]. These populations are therefore poised to receive a greater burden of the effects of improved face recognition systems. We therefore position this work as informing the discussion on data privacy and consent when it comes to face recognition systems and do not advocate for technical improvements without a larger discussion on the appropriate use and legality of the technology. Use of MS-Celeb-1M ------------------ As noted in the introduction, the MS-Celeb-1M dataset was removed from Microsoft’s website in June 2019. In a response to a Financial Times inquiry, Microsoft stated the website was retired “because the research challenge is over” [@murgia_2019]. However, a version of this dataset with detected and aligned faces from a “cleaned” subset of the original images is available from the Intelligent Behaviour and Understanding Group (iBUG) at Imperial College London. The dataset was offered as training data for the “Lightweight Face Recognition Challenge & Workshop”[^1] the group organized at ICCV 2019. The group has pre-trained face recognition models available as benchmarks for the challenge, trained on this data. As this work aims to conduct experiments in a realistic setting in order to better inform the conversation around data collection processes, the analysis of a state-of-the-art model, trained on a large dataset is necessary to gain insights that are applicable to commercial applications. We therefore have decided to use the MS-Celeb-1M dataset, through its derived version offered for the ICCV 2019 Workshop, for the limited scope of this work. **Dataset** **Year Released** **\# Identities** **\# Images** **Informed Consent Obtained?** **Source** ------------------------ ------------------- ------------------- --------------- -------------------------------- -------------------------- CASIA WebFace 2014 10,575 494K No [@yi2014learning] CelebA 2015 10,177 203K No [@liu2015faceattributes] VGGFace 2015 2,622 2.6M No [@BMVC2015_41] MS-Celeb-1M 2016 99,952 10.0M No [@guo2016ms] UMDFaces 2016 8,277 368K No [@bansal2017umdfaces] MegaFace (Challenge 2) 2016 672,057 4.7M No [@nech2017level] VGGFace2 2018 9,131 3.3M No [@cao2018vggface2] \[tab:data\] Background ========== Face Recognition Tasks ---------------------- Within the domain of face recognition lies two categories of tasks: *face verification* and *face identification* [@learned2016labeled]. In face verification, the goal is to assess if a presented image matches with the reference image of an individual, often to grant access to a physical device or location. Unlocking a smartphone with one’s face provides an example of face verification; a person presents their face to a phone and it is verified against a reference image of the known owner of the device. This task is referred to as 1:1 matching, as there is only one individual that the presented face image is compared against. In order to confirm a match, a threshold of similarity must be met, which can be set by the developer of a system to meet a specific level of security. Performance of a system on face verification tasks is reported in terms of accuracy; the number of correct verifications of all verification attempts. In face identification, a *gallery* of known identities is constructed from face images of individuals in advance of testing. Subsequently, a face image of unknown identity is presented to the system as the *probe*. The probe is then matched for similarity with all images in the gallery, constituting 1:N matching. If the system guarantees that the identity of the probe is within the gallery of identities, the problem is considered *closed-set face identification*, otherwise it is considered *open-set face identification*. Closed-set face identification tasks are common in academic benchmarks, as galleries are carefully constructed by their authors to contain all probes. In open-set face identification, a confidence threshold must be set to reject matches that do not meet a certain level of similarity. The selection of an appropriate threshold is especially relevant in high-risk applications such as law enforcement in which false positives have significant implications. Face identification performance is reported in terms of accuracy in returning the correct identity of a probe from the gallery, or in the open-set case, no identity if the probe does not exist in the galley. Common performance metrics include Rank-1 accuracy; of all identification attempts, the number of times the correct identity in the gallery is the most similar identity to the probe, and Rank-10 accuracy; the number of times the correct identity is in the ten most similar identities to the probe. Deep Face Recognition --------------------- Rapid improvements in image classification in the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) [@russakovsky2015imagenet] by AlexNet [@krizhevsky2012imagenet], ZFNet [@zeiler2014visualizing], GoogLeNet [@szegedy2015going] and ResNet [@he2016deep] from 2012 to 2015 cemented the DCNN as the standard method in computer vision research and applications. While early uses of convolutional neural networks in face verification showed preliminary success [@chopra2005learning; @huang2012learning], it was not until the introduction of the aforementioned network architectures that the modern era of deep face recognition was in full swing. Coupled with innovations in loss function design and access to larger image datasets, modern face recognition systems have improved state-of-the-art performance on benchmark face verification and identification tasks significantly in the past six years. For a complete survey of the development of deep face recognition systems, please refer to the review paper by Wang and Deng [@wang2018deep]; the following is a brief summary of major milestones. The first system to adapt findings from ILSVRC to face recognition was Facebook’s DeepFace [@taigman2014deepface], published in 2014 by Taigman *et al.*. The nine-layer AlexNet-based model was trained on a private dataset of 4.4M images of 4K identities and achieved state-of-the-art accuracy on face verification tasks LFW and YouTube Faces (YTF) [@wolf2011face], reducing the error rate by more than 50% on the latter task. Following this work, Google introduced FaceNet in 2015 with a major innovation in loss function design [@schroff2015facenet]. While the standard *softmax* loss function optimized inter-class differences, researchers found that intra-class differences remained high, problematic in the domain of face recognition. To rectify this problem, the *triplet loss* was introduced to jointly minimize the Euclidean distance between an anchor example and a positive example of the same identity and maximize the distance between an anchor and negative example. Using a ZFNet-based model and a private dataset of 200M images of 8M identities, they achieved state-of-the-art performance on LFW and YTF. Innovations in loss functions dominated the next wave of improvements in benchmark tasks, motivated by improving discrimination between classes by making features more separable. Wen *et al.* introduced the Center Loss in 2016 [@wen2016discriminative], followed by Liu *et al.* with the Angular Softmax in 2017 [@liu2017sphereface]. The Large Margin Cosine Loss was introduced in 2018 by Wang *et al.* [@wang2018cosface], and in 2019, Deng *et al.* incorporated the Additive Angular Margin Loss into the ArcFace model [@deng2019arcface], considered state-of-the-art on multiple face recognition benchmarks when published. {width="1.0\linewidth"} Face Recognition Training Datasets ---------------------------------- Access to large-scale face recognition training datasets has been essential to the development of modern solutions by the academic community. While early published resulted in the DCNN-era of face recognition came out of companies with access to massive private datasets, such as Facebook’s 500M images and 10M identities [@taigman2015web] and Google’s 200M images and 8M identities [@schroff2015facenet], the release of several open-source datasets in the ensuing years has allowed researchers to train models at scale. A summary of notable face recognition training datasets of the past six years is provided in Table \[tab:data\]. These datasets catalyzed the field of face recognition and lead to great advances in model performance on benchmark tasks. They largely consist of celebrity identities and copyrighted images scraped from the internet. One exception is MegaFace, which is derived from the YFCC100M dataset of 100M photos with a Creative Commons Licence, from 550K personal Flickr accounts [@thomee2015yfcc100m]. While the Creative Commons Licence permits the fair use of images, including in this context, Ryan Merkley, CEO of Creative Commons, noted the trouble of conflating copyright with privacy in a March 2019 statement: “... copyright is not a good tool to protect individual privacy, to address research ethics in AI development, or to regulate the use of surveillance tools employed online. Those issues rightly belong in the public policy space, and good solutions will consider both the law and the community norms of CC licenses and content shared online in general” [@cc_2019]. While MegaFace contains unknown, non-celebrity identities, an October 2019 investigation by the New York Times demonstrated that account metadata associated with images in the dataset allows for a trivial real-world identification of individuals [@hill_2019]. In all datasets, no informed consent was sought or obtained for individuals contained therein. Methodology =========== Face Recognition Model ---------------------- ### Training Data We employ a cleaned version of the MS-Celeb-1M dataset [@guo2016ms] as training data for a face recognition model in this work. This dataset was prepared for the ICCV 2019 Lightweight Face Recognition Challenge [@deng2019lightweight]. All face images were preprocessed by the RetinaFace model for face detection and alignment [@deng2019retinaface]. A similarity transformation was applied to each detected face using five predicted face landmarks to generate normalized face crops of 112 x 112 pixels. As the original version of this dataset has been shown to exhibit considerable inter-class noise, efforts have been made to automatically clean the dataset [@jin2018community]. In the case of this version, after face detection and alignment, cleaning was performed by a semi-automatic refinement strategy. First, a pre-trained ArcFace model [@deng2019arcface] was used to automatically remove outlier images of each identity. A manual removal of incorrectly labelled images by “ethnicity-specific annotators" followed to result in a dataset of 5,179,510 images of 93,431 identities. We refer to this dataset as *MS1M-RetinaFace*. ### Model We select the ArcFace model [@deng2019arcface] to study in this work. ArcFace employs the Additive Angular Margin Loss and a ResNet100 backbone to arrive at a 512-dimensional feature representation of an input image. The model achieves a verification accuracy of 99.83% on LFW and Rank-1 identification accuracy of 81.91% on the MegaFace Challenge 1 with one million distractors, considered state-of-the-art results. We select the model for study as is the top academic, open-source entrant on the National Institute of Standards and Technology (NIST) Face Recognition Vendor Test (FRVT) 1:1 Verification[^2], a benchmark used by many commercial entities to validate the performance of their face recognition systems. Pre-trained weights for this model were provided by iBUG. **Metric** **Probe Set** **All** **Males** **Females** --------------------------- --------------- --------- ----------- ------------- **Rank-1 Accuracy (%)** In-Domain 79.71 78.50 80.93 **** Out-of-Domain 75.73 77.30 74.17 **Rank-10 Accuracy (%)** In-Domain 90.82 90.92 90.73 **** Out-of-Domain 86.58 88.59 84.57 **Rank-100 Accuracy (%)** In-Domain 92.72 92.52 92.92 **** Out-of-Domain 89.22 90.59 87.84 Experiments ----------- To determine the effect of inclusion in the training data of a face recognition system on its ability to identify an individual, we frame the problem as a closed-set face identification task. We construct two probe datasets and perform face identification on a gallery of one million distractor images. We assess the performance of the model on the probe datasets in terms of Rank-1, Rank-10 and Rank-100 identification accuracies. A visual representation of the datasets used in this work is shown in Figure \[fig:experiments\]. ### Probe Data We construct two probe datasets from the VGGFace2 dataset [@cao2018vggface2]. Using regular expressions, we match identities in VGGFace2 by name with the identify list of MS1M-RetinaFace. We find 5,902 VGGFace2 identities present in MS1M-RetinaFace and 3,229 VGGFace2 identities not present in the training dataset. In each of these two groups, we randomly select 500 male identities and 500 female identities for evaluation, based on gender labels provided by VGGFace2 metadata. For each identity, we randomly select 50 images and perform face detection and alignment with the Multi-task Cascaded Convolutional Network (MTCNN) [@zhang2016joint] to generate normalized face crops of size 112 x 112 pixels. We refer to the set of 50,000 images of 1000 identities present in the training data as the *in-domain probe set* and the set of 50,000 images of 1000 identities not present in the training data at the *out-of-domain probe set*. We then generate 512-dimensional feature representations for all images in the in-domain and out-of-domain probe sets by running them through ArcFace. ### Gallery Data We leverage the MegaFace Challenge 1 “Distractor” dataset [@kemelmacher2016megaface] of 1,027,058 images of 690,572 identities to form the basis of the *gallery*. We again apply MTCNN to generate normalized face crops of 112 x 112 pixels for each image and run each image through ArcFace to generate 512D feature representations of all images in the gallery. ### Evaluation Protocol The experiments conducted in this work follow the protocol of MegaFace Challenge 1, with our probe sets in place of the standard FaceScrub test set [@ng2014data]. We employ the Linux development kit offered by MegaFace to perform evaluation. Each probe set is evaluated following Algorithm \[alg\]; a written description of this protocol follows. A probe set contains 1000 identities, each with 50 images represented as 512D features. For each identity, we iterate over their images, adding one image to the gallery at a time, which we will refer to as *the needle*. We then iterate over the remaining 49 images, using each one as a probe. We rank all images in the gallery by L2 distance in feature space to the probe, and record the position of the needle in the ranked list. We report results for each probe set in terms of Rank-1, Rank-10 and Rank-100 face identification accuracies. $r_1, r_{10}, r_{100} = 0$;\ gallery contains 1M distractor images;\ $\text{Rank-1\textsubscript{Acc.}} = r_1 / (1000\times50\times49)$;\ $\text{Rank-10\textsubscript{Acc.}} = r_{10} / (1000\times50\times49)$;\ $\text{Rank-100\textsubscript{Acc.}} = r_{100} / (1000\times50\times49)$; Results and Discussion ====================== We present results of the experiments in Table \[tab:res\] for Ranks 1, 10 and 100. We find there is a modest increase in face identification accuracy for identities present in the training data, compared to those who are not. In-domain identities have a 4.0% higher identification accuracy than out-of-domain identities at Rank-1, 4.2% higher at Rank-10, and 3.5% higher at Rank-100. Although not a significant margin, these results suggest that modern DCNN-based face recognition systems are biased towards individuals they are trained on. The disparate performance between probe sets suggests some amount of overfitting has occurred in the model. Although the model generalizes well to new identities, as evidenced by results on benchmarks LFW, MegaFace and on NIST’s FRVT, these results indicate that the 93k identities the system is trained on are more easily identifiable in a large-scale study. As the model’s Additive Angular Margin Loss sought to increase discrimination between classes by making features more separable, it appears the model has learned to map identities to the same feature representation more consistently for those it has seen before. We also investigated the role of gender in the performance of the face recognition model. We find small differences in performance between genders for in-domain identities, but a 3 - 4% decrease in performance for females compared to males who are out-of-domain, across all ranks. These results suggest that a gender bias exists in the face recognition model towards female identities. As the model has a smaller drop in face identification accuracy between domains for males, it has a greater ability to generalize to new male identities. While we do not have gender labels available for all identities in MS1M-RetinaFace, recent work has demonstrated that large-scale face recognition datasets are largely biased towards lighter-skinned males [@merler2019diversity]. A representational bias in MS1M-RetinaFace may account for this disparate performance across genders. Looking at these results in a different way, the consistent performance for in-domain identities across genders is perhaps more evidence that the model is overfitting to identities it has seen before. If the model only had a gender bias, we would have seen disparate performance for genders on both probe sets, however, these results suggest the model may also exhibit a “training inclusion bias”. Results of this study lead to the question; is the bias towards individuals in training data truly a consequence of overtraining, or is this a fundamental element of deep face recognition models? If we look to the manner by which the model was trained, overfitting in a traditional sense seems unlikely, as early stopping was employed, and results on held-out test identities demonstrate strong generalization. Perhaps there is a generalization gap in performance between in-domain and out-of-domain identities that is not apparent in current validation protocols, and increased regularization can mitigate this gap. Further testing on different training datasets and model architectures will be necessary to gather more evidence to answer this question. We did not analyze the effect of skin type on face recognition model performance in this study, as skin type annotations were not available to us at the time. However, two considerations were made to attempt to control for effects of skin type in these results. First, the selection of 1000 identities for each probe set is far larger than what is used in the standard protocol of MegaFace Challenge 1, where 80 identities are sampled from FaceScrub. Having a larger sample size helps to control for identities who may have either superior or poor performance due to possible model bias. In addition, the approach of random sampling in-domain and out-of-domain probe sets ensures both contain a similar distribution of identities with respect to skin type, with the assumption that the identities common to MS1M-RetinaFace and VGGFace2 and the identities distinct to VGGFace2 follow the same distribution of skin type. As both MS1M-RetinaFace and VGGFace2 use the popularity of celebrities online to construct identity lists, this assumption seems to be reasonable. Having said this, the role of skin type in the performance of the model is a very important relationship to study, and this is planned for future work. Fitzpatrick skin type [@fitzpatrick1988validity] annotations will need to be collected for all individuals in VGGFace2 such that sampling can be done to ensure even representation in probe sets across gender and skin type, and to determine intersectional accuracy. The results of this study are quite concerning from a privacy and informed consent perspective. As described in the background section on Face Recognition Training Datasets, there does not exist a major open-source dataset that gathers informed consent from the individuals it contains. Without these individuals’ knowledge or permission, the systems trained on their identities have a greater ability to identify them. As face recognition becomes more powerful and ubiquitous, the ability for misuse becomes greater. While MS-Celeb-1M contains only “celebrity" identities, this classification of an individual should not negate informed consent in the development of powerful surveillance technologies. Face recognition systems are unique among biometrics as the face can be easily captured at distance without one’s knowledge. The face uniquely identifies an individual, and it is difficult to opt-out of these systems without wearing a mask or other means of obfuscation, drawing undue attention to one’s self. From a legal perspective, the concept of informed consent in the analysis of images of individuals’ faces has traction in some jurisdictions. As reported by the New York Times with reference to potential financial liabilities of MegaFace [@hill_2019], the Illinois Biometric Information Privacy Act [@bipa2008] is a State law enacted in 2008 that gives Illinois residents the right to seek financial compensation from entities using their face scans without their informed consent. The experiments in this work aim to simulate a real-world testing environment of a state-of-the-art face recognition system, with a gallery of more than one million images. These findings, therefore, may hold for systems that are currently deployed in the real-world. Conclusion ========== In this work we present the first study to investigate the role of inclusion in face recognition training data on a derived system’s ability to identify an individual. Through the construction of two sets of probe data that overlap and are distinct from the training data of a state-of-the-art system, we conduct a large-scale face identification experiment. We find a modest 4% improvement in face identification accuracy for individuals who are present in training data, which is highly problematic given the norm in the field is to not gather informed consent in the collection of training datasets. Future work will apply this methodology to more models, training datasets and distance metrics (i.e. cosine distance) to see if results are consistent. Following prior work [@buolamwini2018gender; @Raji2019; @cook2019demographic], analysis of face recognition model bias with respect to gender, skin type and their intersections in large-scale face identification tasks is needed, as well as tying results to representational bias in training data. Additionally, the relationship between the *number of images* of an individual in training data and their ability to be identified is an interesting area of study. Finally, analysis of a face recognition model’s feature space directly provides an alternative to a task-based auditing approach, and may be fruitful for understating nuances of inter- and intra-class differences. We would like to thank the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs Program for their support. [^1]: <https://ibug.doc.ic.ac.uk/resources/lightweight-face-recognition-challenge-workshop/> [^2]: <https://www.nist.gov/programs-projects/frvt-11-verification>

--- abstract: 'Correlation effects are important for making predictions in the $\delta $ phase of Pu. Using a realistic treatment of the intra–atomic Coulomb correlations we address the long-standing problem of computing ground state properties. The equilibrium volume is obtained in good agreement with experiment when taking into account Hubbard $U$ of the order 4 eV. For this $U,$ the calculation predicts a 5f$^{5}$ atomic–like configuration with L=5, S=5/2, and J=5/2 and shows a nearly complete compensation between spin and orbital magnetic moments.' author: - 'S. Y. Savrasov$^{\ast }$ and G. Kotliar$^{+}$' - '$^{\ast }$Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1,  70569 Stuttgart, Germany.' - | $^{+}$Department of Physics and Astronomy and Center for Condensed Matter Theory\ Rutgers University, Piscataway, NJ 08854–8019 date: July 1999 title: 'Ground State Theory of $\delta $–Pu' --- Metallic plutonium is a key material in the energy industry and understanding its physical properties is of fundamental and technological interest [@Pu]. Despite intensive investigations [@PuBook], its extremely rich phase diagram with six crystal structures as well as its unique magnetic properties are not well understood. It is therefore of great interest to study the ground state of Pu by modern theoretical methods using first principles electronic structure calculations, which take into account the possible strong correlation among the f electrons. Density functional theory [@DFT] in its local density or generalized gradient approximations (LDA or GGA) is a well-established tool for dealing with such problems. This theory does an excellent job of predicting ground-state properties of an enormous class of materials. However, when applied to Pu [@Pucalc1; @Pucalc2], it runs into serious problems. Calculations of the high-temperature fcc $\delta$ phase have given an equilibrium atomic volume up to 35% lower than experiment [@Pucalc1]. This is the largest discrepancy ever known in density functional based calculations and points to a fundamental failure of existing approximations to the exchange-correlation energy functional. Many physical properties of this phase are puzzling: large values of the linear term in the specific heat coefficient and of the electrical resistivity are reminiscent of the physical properties of strongly-correlated heavy-fermion systems. On the other hand, the magnetic susceptibility is small and weakly temperature dependent [@fournier]. Moreover, early LDA calculations [@Pucalc2] predicted $\delta $–Pu to be magnetic with a total moment of 2.1 Bohr magnetons in disagreement with experiments. The reason for these difficulties has been understood for a long time: Pu is located on the border between light actinides with itinerant 5f–electrons and the heavy actinides with localized 5f electrons[@Johannson] . Near this localization-delocalization boundary, the large intra-atomic Coulomb interaction as well as the itineracy of the f electrons have to be considered on the same footing, and it is expected that correlations must be responsible for the anomalous properties. The parameter governing the importance of correlations in electronic structure calculations is the ratio between effective Hubbard interaction $U$ and the bandwidth $W.$ When the distance between atoms is small, the correlation effects may be not important, since the hybridization, and consequently the bandwidth become large. The low-temperature $\alpha $ phase of Pu has an atomic volume which is 25% smaller than the volume of $\delta $ phase. To the extent that the complicated monoclinic structure of the $\alpha $ phase can be modelled by the simplified fcc lattice, it becomes clear that the LDA or GGA calculations which ignore the large effective $U$ converge to the low volume $\alpha $ phase (for which $U/W < 1$). When volume is increased, this ratio is turned around, and LDA loses its predictive power. This results in the long-standing problem of accurate prediction of the volume of $\delta $–Pu. In the present work it will be shown that a proper treatment of Coulomb correlations allows us to compute the equilibrium atomic volume of $\delta $–Pu in good agreement with experiment. Moreover, our calculations suggest that there is a nearly complete compensation between the spin and the orbital contributions to the total magnetic moment which is consistent with experiment. Thus the strong correlation effects in $\delta $–Pu are not manifest in the static magnetic properties. To incorporate the effects of correlations we use the LDA + U approach of Anisimov and coworkers [@LDA+U]. This approach recognizes that the failure of LDA is related to the fact that it omits the Hubbard like interaction among electrons in the same shell, irrespectively of their spin orientation. A new orbital–dependent correction to the LDA functional was introduced to describe this effect. In its most recent, rotationally invariant representation, the correction to the LDA functional has the following form [@LDA+Urecent]: $$\Delta E[n]=\frac{1}{2}\sum_{\{\gamma \}}(U_{\gamma _{1}\gamma _{2}\gamma _{3}\gamma _{4}}-U_{\gamma _{1}\gamma _{2}\gamma _{4}\gamma _{3}})n_{\gamma _{1}\gamma _{2}}^{c}n_{\gamma _{3}\gamma _{4}}^{c}-E_{dc} \label{e2}$$ where $n_{\gamma _{1}\gamma _{2}}^{c}$ is the occupancy matrix for the correlated orbital (d or f), and $\gamma $ stands for the combined spin, ($s$), and azimuthal quantum number,($m$), indexes. The electron–electron correlation matrix $U_{\gamma _{1}\gamma _{2}\gamma _{3}\gamma _{4}}=\left\langle m_{1}m_{3}\left| v_{C}\right| m_{2}m_{4}\right\rangle \delta _{s_{1}s_{2}}\delta _{s_{3}s_{4}}$ can be expressed via Slater integrals $F^{(i)} $, $ i=0,2,4,6$ in the standard manner [@LDA+Urecent]. The term $E_{dc}$ accounts for the double counting effects. This scheme, known as the ”LDA+U method”, gives substantial improvements over the LDA in many cases[@LDA+Ureview]. The value of the $U$ matrix is an input which can be obtained from a constrained LDA calculations [@CLDA], or just taken from the experiment. The philosophy of this approach is that the delocalized s p d electrons are well described by the LDA while the energetics of the more localized f electrons require the explicit introduction of the Hubbard U. In the spirit of this method, in this work we will treat the s p d electrons by the generalized gradient approximation (GGA) [@GGA] which is believed to be more accurate that the LDA. Our implementation of the GGA+U functional is based on the localized–orbital representation provided by the linear–muffin–tin–orbital (LMTO) method for electronic structure calculations [@OKA]. It is important to include spin–orbit coupling effects which are not negligible for 5f electrons of Pu. Our calculations include non-spherical terms of the charge density and potential both within the atomic spheres and in the interstitial region [@Sav]. All low-lying semi-core states are treated together with the valence states in a common Hamiltonian matrix in order to avoid unnecessary uncertainties. These calculations are spin polarized and assume the existence of long–range magnetic order. For simplicity, the magnetic order is taken to be ferromagnetic [@Ferro]. We now report our results on the calculated equilibrium volume. To analyze the importance of the correlation effects, our calculations have been performed for several different values of $U$ varying from 0 to 4 eV. For $U$=4 eV we use standard choice of Slater integrals: $F^{(2)}$=10 eV, $F^{(4)}$=7 eV, and $F^{(6)}$=5 eV [@Pu]. For other U’s we have scaled these values proportionally. For each set of $F$’s a full self–consistent cycle minimizing the LDA/GGA+U functionals has been performed for a number of atomic volumes. We calculated the total energy $E$ as a function of both $V$ and $U$. For fixed $U,$ the theoretical equilibrium, $V_{calc},$ is given by the minimum of $E(V)$. Fig. 1 shows the dependence of the calculated–to–experimental equilibrium volume ratio $V_{calc}/V_{exp}$ as a function of the input $U.$ It is clearly seen that the $U$=0 result (LDA) predicts an equilibrium volume which is 38% off the experimental result and the use of GGA gives only slightly improved result ($V_{calc}/V_{exp}$=0.66). On the other hand, switching on a very large repulsion between 5f electrons obviously leads to an overestimate of the inter-atomic distances. An optimal $U$ deduced from this analysis is found to be close to 4 eV when using the GGA expressions for the exchange and correlation. This estimate of the intra-atomic correlation energy is in excellent agreement with the published conventional data [@PuU]: The value of $U$ deduced from the total energy differences was found to be 4.5 eV. Atomic spectral data give similar value close to 4 eV. Thus, it is demonstrated how significant it is to properly treat Coulomb correlations in predicting the equilibrium properties of this actinide. We now discuss the calculated GGA+U electronic structure of $\delta $–Pu for the optimal value of $U$=4 eV. Fig. 2 shows the energy bands in the vicinity of the Fermi level. They originate from the extremely wide 6s–band strongly mixed with the 5d–orbitals which are strongly hybridized among themselves. The resulting band complex has a bandwidth of the order of 20 eV. On top of this structure there exist a weakly hybridized set of levels originating from the 5f–orbitals. In order to understand the physics behind the formation of spin and orbital moment in the f–shell, it is instructive to visualize the orbital characters as ”fat bands”[@Fat]. The one–electron wave function has an expansion $\psi _{{\bf k}j}({\bf r})=\sum A _{lms}^{{\bf k}j} \phi _{lms}({\bf r})$ where $\phi _{lms}({\bf r})$ are the solutions of the radial Schrödinger equation normalized to unity within atomic sphere. The information about partial $lms$ characters of the state with given ${{\bf k}j}$ is contained in the coefficients $|A _{lms}^{{\bf k}j}|^2$. Sum over all $lms$ in the latter quantity gives unity (we neglect by a small contribution from the interstitial region) since one band carries one electron per cell. At the same time, sum over all $j$ in $|A _{lms}^{{\bf k}j}|^2$ is also equal to one since each $lms$ describes one state. Fixing a particular $lms$, we can visualize this partial character on top of the band structure by widening each band $E_{{\bf k}j}$ proportionally to $|A_{lms}^{{\bf k}j}|^2$. A maximum width $\Delta $ which corresponds to $\sum _j |A_{lms}^{{\bf k}j}|^2$=1 should be appropriately chosen. Now, at the absence of hybridization, each band originates from a particular $lms$ state, and therefore there exists only one ”fat band” for given $lms$ which has the maximum width $\Delta $. When hybridization is switched on, there can be many bands which have the particular $lms$ character, they will all be widened as $\Delta |A_{lms}^{{\bf k}j}|^2$, while the sum of individual widths for all bands is now equal to $\Delta $. The width of the band is then proportional to its $lms$ character. This technique [@Fat] gives us an important information on the distribution of atomic levels as well as their hybridization in a solid. For f–electrons of Pu, it is convenient to work in the spherical harmonics representation in which the f–f block of the Hamiltonian is found to be nearly diagonal. The result of such ”fat bands” analysis for 5f–orbitals is shown on Fig. 2. In order to distinguish the states with different $m$’s and spins we have used different colors. (-3 $\equiv $ red, -2 $\equiv $ green, -1 $\equiv $ blue, 0 $\equiv $ magenta, +1 $\equiv $ cyan, +2 $\equiv $ yellow, +3 $% \equiv $ gray). Two consequences are seen from this coloured spaghetti: First, spin–up and spin–down bands are all split by the values governed by the effective $U$ and the occupancies of the levels. These are just the well known lower and upper Hubbard subbands. Second, only spin–up states with $m$=-3,-2,-1,0, and +1 are occupied while all other states are empty. This simply implies 5f$^{5}$ like atomic configuration for $\delta $–Pu which is filled according to the Hund rule. Note that spin-orbit coupling is crucial for the existence of such an occupation scheme. In the absence of spin orbit coupling the occupancies of the levels with $\pm m$ are the same which automatically produces zero orbital moment. Besides providing the experimentally observed volume of the $\delta $–Pu, our calculation suggests a simple picture of the electronic structure of this material and sheds new light on its puzzling physical properties discussed in the introduction. The ”fat bands” shown in Fig. 2, suggest a physical picture in which the f electrons are in atomic states forming a multiplet of the 5f$^5$ configuration with L=5, S=5/2 spin orbit coupled to J=5/2. Crystal fields can split this multiplet into a doubly degenerate state transforming according to $\Gamma_7$ representation of the cubic group and a quartet transforming according to $\Gamma_8$ representation [@PuBook], but cannot remove the orbital degeneracy completely. In a dynamic picture, the f electrons will fluctuate between the degenerate configurations, until this degeneracy is removed by the Kondo effect with the delocalized electrons in the s-p-d band. Therefore the experimentally observed characteristic heavy fermion behavior in this system, namely, the large high–temperature resistivity and the large linear T coefficient of the specific heat arises naturally in this picture [@PuBook]. This heavy fermion behavior however should not appear in the magnetic susceptibility. The GGA + U calculation suggests that the magnetic moment of the low lying configurations of the f electrons is much smaller than the $5 \mu_B $ that one would obtain if we ignore the orbital angular momentum and assumed that the spin is fully polarized. The combination of strong Coulomb interactions and spin–orbit coupling reduce the crystal–field effects and give rise to a large magnetic moment which nearly cancels the spin moment. In an atomic picture, the $5f^5$ configuration with L=5, S=5/2 and J=5/2 has a total moment given by $M_{tot}$ $=\mu _{B}gJ=0.7\mu _{B}$, with Lande’s g-factor of 0.28. This simple relation breaks down in the presence of crystal fields, but in both the $\Gamma_7$ or the $\Gamma_8$ representation the g factor is further reduced from the atomic estimate. The GGA + U calculation gives a spin moment $ M_{S}$ 5.1 Bohr magnetons which is slightly increased relative to the 5 Bohr magnetons expected in a pure f$^{5}$ atomic configuration due to the polarization of the band electrons outside the muffin–tin shell. Evaluation of the orbital and total moments is in general a more difficult problem [@Cal]. We have estimated the average of $\langle {\bf k}j\left| l_{z}\right| {\bf k}j\rangle $ summed over all occupied states $|{\bf k}% j\rangle .$ This leads to a value $M_{L}$=-3.9 $\mu _{B.}$ for the orbital moment. The total calculated moment $M_{tot}=M_{S}+M_{L}$ is thus reduced to 1.2 $\mu _{B}$ . It worth noting that an atomic analogue of this estimate, $ M_{tot}$ $=\mu _{B}\left| L-2S\right| $ gives exactly zero for our 5f$^{5}$ ground state[@Another]. A remarkable outcome of the calculation is clearly seen: [*A nearly complete compensation of spin and orbital contributions occurs*]{} for metallic $\delta $–Pu. In this picture the weakly temperature independent susceptibility which is observed in $\delta $–Pu [@PuBook; @fournier] is the result of a a very large Van Vleck contribution and of a very small magnetic moment which results from the near cancellation of two large orbital and spin moments. In a recent paper [@Eriksson] Eriksson and coworkers introduced a different approach to the anomalous properties of $\delta$ plutonium. In their calculation a fraction of the f-electrons is treated as core electrons while the rest are treated as delocalized. Using a combination of the constrained LDA calculation with the atomic multiplets data they obtain the correct equilibrium volume when four f-electrons are part of the core, while one f electron is itinerant. The basic difference between the methods, is the different treatment of the f electrons. In this paper all the f electrons on equal footing, and their itineracy is reduced by the Hubbard U relative to the predictions of LDA or GGA calculations. Since our approach and that of Eriksson et. al. lead to different ground state configurations of the localized f electrons ($f^5$ vs $f^4$), further experimental spectroscopic studies of $\delta $–Pu would be of interest. In conclusion, using a realistic value of the Hubbard $U$=4 eV incorporated into the density functional GGA calculation, we have been able to describe ground state properties of $\delta $–Pu in good agreement with experimental data. This theory correctly predicts the equilibrium volume of the $\delta $ phase and suggest that nearly complete cancellation occurs between spin and orbital moments. The main shortcoming of the present calculation is the assumed long range spin and orbital order. This is the essential limitation of the LDA+U approach (or of any [*static*]{} mean field theory ) in order to capture the effects of correlations this approach it has to impose some form of long–range order. Static mean field theories are unable to capture subtle many–body effects such as the formation of local moments and their subsequent quenching via the Kondo effect. These deficiencies will be removed by ab initio [*dynamical*]{} mean field [@DMFT] calculations for which codes are currently being developed. We believe however, that our main conclusions, i.e. that correlations lead to the correct lattice constant and a reduction of the moment, relative to the LDA results, are robust consequences of the strong correlations presented in this material, and will be reproduced by more accurate treatments of the electron correlations. The authors are indebted to E. Abrahams, O. K. Andersen, O. Gunnarsson, A. I. Lichtenstein, and J. R. Schrieffer for many helpful discussions. The work was supported by the DOE division of basic energy sciences, grant No. DE-FG02-99ER45761. For a review, see, [*e.g*]{}., [*Plutonium - A General Survey*]{}, edited by K. H. Lieser (Verlag Chemie, 1974) For a review, see, [*e.g.*]{}, [*The Actinides: Electronic Strucuture and Related Properties*]{}, edited by A. J. Freeman and J. B. Darby, Vols. 1 and 2 (Academic Press, New York, 1974); For a review, see, [*e.g*]{}., [*Theory of the Inhomogeneous Electron Gas*]{}, edited by S. Lundqvist and S. H. March (Plenum, New York, 1983). For recent calculations, see, P. Söderlind, O. Eriksson, B. Johansson, and J. M. Wills, Phys. Rev. B [**50**]{}, 7291 (1994); J. C. Boettger, M. D. Jones, R. C. Albers, and D. J. Singh, cond-mat/9811199 I. V. Solovyev, A. I. Liechtenstein, V. A. Gubanov, V. P. Antropov, and O. K. Andersen, Phys. Rev. B [**43**]{}, 14414 (1991); S. Meot-Raymond and J. Fournier J. Alloys and Compounds [**232**]{} 119 (1996). B. Joahnsson J. Magn. Magn. Mater 47 (1985) 231; V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B [**44**]{}, 943 (1991); A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995); For a review, see, V. I. Anisimov, F. Aryasetiawan, and A. I. Liechtenstein, J. Phys.: Cond. Matter [**9**]{}, 767 (1997); P. H. Dederichs, S. Blügel, R. Zeller, and H. Akai, Phys. Rev .Lett. [**53**]{}, 2512 (1984); J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996). O. K. Andersen, Phys. Rev. B [**12**]{}, 3060 (1975). A short description of the method can be found in: S. Y. Savrasov, Phys. Rev. B [**54**]{}, 16470 (1996); In fact, our own and previous [@Pucalc2] total energy calculations suggest that the antiferromagnetic state has slightly lower energy. A realistic theory should not assume any long–range order at all. Unfortunately, it is very hard to treat short–range order with the existing techniques (through the use of supercells), while alternative methods based on dynamical mean–field theory and Green functions are still in developmental stage. See, [*e.g*]{}., J. P. Desclaux and A. J. Freeman, in[*Handbook on the Physics and Chemistry of the Actinides*]{}, edited by A. J.. Freeman and G. H. Lander (Elsevier, Amsterdam, 1984), Vol. 1; O. Jepsen and O. K. Andersen, Z. Phys. B [**97**]{}, 35 (1997); M. Singh, J. Callaway, and C. S. Wang, Phys. Rev. B[**14**]{}, 1214 (1976); Note that we have found another metastable solution of the functional \[e2\] corresponding to the occupation of the levels with $% m=-3,-2,-1,+1,+2.$ The orbital moment of this configuration is about 3 and carries a large magnetic moment. The energy of this configuration is however found to lie 0.5 Ry higher than the energy of the ground state configuration with high orbital moment. O. Eriksson, J. D. Becker, A. V. Balatsky, and J. M. Wills, J. Alloys and Comp. [**287**]{}, 1 (1999); A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. [**68**]{}, 13 (1996); FIGURE CAPTIONS Fig.1. Calculated theoretical volume (normalized to the experiment) of $\delta $–Pu as a function of the Hubbard $U$ within the LDA+U and GGA+U approaches. Fig.2. Calculated energy bands of $\delta $–Pu using GGA+U method with $U$=4 eV. Spin and orbital characters of the f-bands are shown with the color: ($m$=-3 $\equiv $ red, -2 $\equiv $ green, -1 $\equiv $ blue, 0 $\equiv $ magenta, +1 $\equiv $ cyan, +2 $\equiv $ yellow, +3 $% \equiv $ gray). Boxes from the left and from the right show approximate positions of the f levels.

--- abstract: 'Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links between two techniques, that of the symmetry reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraic-geometric point of view, which allows the introduction of specific first-order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems based on the introduction of specific rotation matrices enabled us to weaken the integrability condition. It allows us to establish the connection between the structure of the set of integral elements and the possibility of the construction of specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions have been obtained in explicit form including the general integral for the latter system of equations.' author: - | A.M. Grundland[^1],\ Centre de Recherche Mathématiques, Université du Montréal,\ C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C 3J7, Canada\ and Département de mathématiques et informatiques, Université du Québec,\ Trois-Rivières, (QC) G9A 5H7, Canada\ - | V. Lamothe[^2],\ Département de Mathématiques et Statisque, Université de Montréal,\ C.P. 6128, Succc. Centre-ville, Montréal, (QC) H3C 3J7, Canada\ title: 'Multimode solutions of first-order quasilinear systems obtained from Riemann invariants. Part I. ' --- [Running Title: Multimode solutions of quasilinear systems. Part I.\ PACS numbers: 02.30.Jr; Secondary 02.70.-c\ Keywords: symmetry reduction method, generalized method of characteristics, Riemann invariants, multimode solutions]{} Introduction {#intro} ============ Riemann waves represent a very important class of solution of nonlinear first-order systems of partial differential equations (PDEs). They are ubiquitous in the differential equations of mathematical physics, since they appear in all multidimensional hyperbolic systems and constitute their elementary solutions. Their characteristic feature is that, in most cases, they are expressible only in implicit form. For a homogeneous hyperbolic quasilinear system of first-order PDEs, \[eq:1\] \_\^[i]{}(u)=0,i=1,…,p,=1,…, q,=1,…, m (where $\mathcal{A}^1,\ldots,\mathcal{A}^p$ are $q\times m$ matrix functions of an unknown function $u$ and we adopt the convention that repeated indices are summed unless one of them is in a bracket), a Riemann wave solution is defined by the equation $u=f(r(x,u))$, where $f:{\mathbb{R}}{\rightarrow}{\mathbb{R}}^q$, and the function $r(x,u)={\lambda}_i(u)x^i$ is called the Riemann invariant associated with the vector $\lambda$ satisfying the equation $\ker{\left( {\lambda}_i\mathcal{A}^i(u) \right)}\neq 0$. These solutions have rank at most equal to one. They are building blocks for constructing more general types of solutions describing nonlinear superpositions of many waves ($k$-waves), which are very interesting from the physical point of view. Until now, the only way to approach this task was through the generalized method of characteristics (GMC) [@Burnat:1972; @CourantHilbert:1962; @Jeffrey:1976; @JohnKlainerman:1984; @Perad:1985; @Rozdestvenski:1983] and more recently through the conditional symmetry method (CSM) [@AblowitzClarkson; @ConteGrundHuard:2009; @Fushchych:1991; @GrundHuard:2006; @GrundHuard:2007; @OlverVorobev1995]. The GMC relies on treating Riemann invariants as new dependent variables (which remain constant along the appropriate characteristic curves of the initial system (\[eq:1\]) and constitute a set of invariants of the Abelian algebra of some vector field $X_a=\xi_a^i(u){\partial}_x^i$ with ${\lambda}_i^a\xi_a^i=0$ for $1\leq a\leq k<p$. This leads to the reduction of the dimension of the problem. The most important theoretical results obtained with the use of the GMC or CSM [@GrundHuard:2007] include the finding of necessary and sufficient conditions for the existence of Riemann $k$-waves in multidimensional systems. It was shown [@Perad:1985] that these solutions depend on $k$ arbitrary functions of one variable. Some criteria were also found [@Perad:1985] for determining the elastic or nonelastic character of the superposition of Riemann waves described by hyperbolic systems, which is particularly useful in physical applications. In applications to fluid dynamics and nonlinear field theory, many new and interesting results were obtained [@Boillat:1965; @Burnat:1972; @DubrovinNovikov:1983; @FerapontovKhus:2004:1; @FerapontovKhus:2004:2; @JohnKlainerman:1984; @Mises:1958; @Rozdestvenski:1983; @Whitham:1974; @Zakharov:1998]. Both the GMC and CSM methods, like all other techniques for solving PDEs, have their limitations. This fact has motivated the authors to search for the means of constructing larger classes of multiple wave solutions expressible in terms of Riemann invariants by allowing the introduction of complex integral elements in the place of real simple integral elements (with which the solutions of hyperbolic systems are built [@Perad:1985]). This idea originated from the work of S. Sobolev [@Sobolev:1934] in which he solved the wave equation by using the associated complex wave vectors. We are particularly interested in the construction of nonlinear superpositions of elementary simple mode solutions, and the proposed analysis indicates that the language of conditional symmetries is an effective tool for this purpose. This approach is applied to the nonstationary irrotational flow of a ideal plastic material in its elliptic region. A further extension of the proposed method to the case of inhomogeneous systems is proposed in order to be applicable either in elliptic or hyperbolic regions. This allows for a wider range of physical applications. The approach is based on the use of rotation matrices which obey certain algebraic conditions and allow us to write the reduced system in terms of Riemann invariants in the sense that each derivative of dependent variables is equal to an algebraic expression (see equation (\[eq:ms:12\])). We discuss in detail the sufficient conditions for the existence of multimode solutions. This approach is applied to a system describing a propagation of shock waves intensity in the nonlinear interaction of waves and particles. The general integral of this system has been constructed in an explicit form depending on two arbitrary functions of one variable. #### The organization of this paper is as follows. Section \[sec:2\] contains a detailed account of the generalized method of characteristics for first-order quasilinear systems of PDEs in many dimensions based on complex characteristic elements. In Section \[sec:3\] we formulate the problem of multimode solutions expressible in terms of Riemann invariants by means of a group theoretical approach. This allows us to formulate the necessary and sufficient conditions for constructing these types of solutions. In Section \[sec:4\], the usefulness of the method developed in Section \[sec:3\] is illustrated by an example of the ideal plasticity in $(2+1)$ dimensions, in which we find several bounded solutions. Moreover, we have drawn extrusion dies and the flow inside it (limiting ourselves to the region where the gradient catastrophe does not occur). Sections \[sec:5\] and \[sec:6\] comprise a new approach to solving inhomogeneous elliptic systems to obtain simple wave and simple mode solutions. In Section \[sec:7\], we have shown a example of a simple mode solution using the method presented in Section \[sec:6\]. The technique is applied on a system describing the propagation of shock waves for the nonlinear interaction of waves and particles. We obtain its general solution. Section \[sec:8\] summarizes the results obtained and contains some suggestions regarding further developments. The method of characteristics for complex integral elements. {#sec:2} ============================================================ The methodological approach assumed in this section is based on the generalized method of characteristics which has been extensively developed (*e.g. in* [@Burnat:1972; @DoyleGrundland:1996; @Grundland:1974; @GrundlandTafel:1996; @Perad:1985] and references therein) for multidimensional homogeneous and inhomogeneous systems of first-order PDEs. A specific feature of that approach is an algebraic and geometric point of view. An algebraization of systems of PDEs was made possible by representing the general integral elements as linear combinations of some special elements associated with those vector fields which generate characteristic curves in the spaces of independent variables $X$ and dependent variables $U$, respectively (see [@GrundlandZelazny:1983; @Perad:1985]). The introduction of these elements (called simple integral elements) proved to be very useful for constructing certain classes of rank-$k$ solutions in closed form. These integral elements proved to correspond to Riemann wave solutions in the case of nonelliptic systems and serve to construct multiple waves ($k$-waves) as a superposition of several single Riemann waves. #### The generalized method of characteristics for solving quasilinear hyperbolic first-order systems can be extended to the case of complex characteristic elements (see *e.g.* [@Perad:1985; @Sobolev:1934]). These elements were introduced not only for elliptic systems but also for hyperbolic systems by allowing the wave vectors to be the complex solutions of the dispersion relation associated with the initial system of equations. The starting point is to make an algebraization, according to [@Burnat:1972; @Grundland:1974; @Perad:1985], of a first-order system of PDEs (\[eq:1\]) in $p$ independent and $q$ dependent variables written in its matrix form \[eq:2.1\] &\^i(u)u\_i=0,i=1,…,p,\ &x=(x\^1,…,x\^p)X\^p, u=(u\^1,…,u\^q)U\^q, where $\mathcal{A}^i(u)={\left( \mathcal{A}^{\mu i}_\alpha(u) \right)}$ are $m\times q$ matrix functions of $u$ and we denote the partial derivatives by $u^\alpha_i={\partial}u^\alpha/{\partial}x^i$. The matrix $L_i^\alpha$ satisfying the conditions[@Burnat:1972] \[eq:intelem\] u\_i\^[{ L\_i\^:\_\^[i]{}L\_i\^=0, =1,…,q }]{} at some open given point $u_0\in U$ is called an integral element of the system (\[eq:2.1\]). This matrix $L={\left( {{\partial}u^\alpha}/{{\partial}x^i} \right)}$ is a matrix of the tangent mapping $du:X{\rightarrow}T_uU$ given by the formula $$X\ni (\delta x^i){\rightarrow}(\delta u^\alpha)\in T_uU,\quad \text{where }\delta u^\alpha=u_i^\alpha \delta x^i.$$The tangent mapping $du(x)$ determines an element of linear space $L(X,T_uU)$, which can be identified with the tensor product $T_uU\otimes X^\ast$, (where $X^\ast$ is the dual space of $X$, [*i.e.* ]{}the space of linear forms). It is well known [@Burnat:1972; @Grundland:1974; @Perad:1985] that each element of this tensor product can be represented as a finite sum of simple tensors of the form $$L=\gamma\otimes \lambda,$$where $\lambda\in X^\ast$ is a covector and $\gamma=\gamma^\alpha\frac{{\partial}}{{\partial}u^\alpha}\in T_uU$ is a tangent vector at the point $u\in U$. Hence, the integral element $L_i^\alpha$ is called a simple element if ${\operatorname{rank}}(L_i^\alpha)=1.$ To determine a simple integral element $L_i^\alpha$ we have to find a vector field $\gamma\in T_uU$ and a covector $\lambda\in X^\ast$ satisfying the so-called wave relation \[eq:2.2\] [( \_i\^[i]{}\_(u) )]{}\^=0,=1,…,m. The necessary and sufficient condition for the existence of a nonzero solution $\gamma$ for the equation (\[eq:2.2\]) is \[eq:2.3\] [( \_i\^i(u) )]{}&lt;(m,q). This relation is known as the dispersion relation. If the covector $\lambda=\lambda_i(u)dx^i$ satisfies the dispersion relation (\[eq:2.3\]) then there exists a polarization vector $\gamma\in T_uU$ satisfying the wave relation (\[eq:2.2\]). The algebraic approach which has been used in [@Burnat:1972; @Grundland:1974; @Perad:1985] for hyperbolic systems of equations (\[eq:2.1\]) allows the construction of certain classes of $k$-wave solutions admitting $k$ arbitrary functions of one variable. The replacement of the matrix of derivatives $u_i^\alpha$ in the system of equations (\[eq:2.1\]) by the simple real element $L_i^\alpha$ allows us to construct more general classes of solutions by replacing the real elements with complex ones. A specific form of solution $u(x)$ of an elliptic system (\[eq:2.1\]) is postulated for which the tangent mapping $du(x)$ is a sum of a complex element and its complex conjugate \[eq:2.4\] du\^(x)=(x)\^(u)\_i(u)dx\^i+|(x)|\^(u)|\_i(u)dx\^i,=1,…,q, where $\gamma=(\gamma^1,\ldots, \gamma^q)\in{\mathbb{C}}^q$ and $\lambda=(\lambda_1,\ldots,\lambda_p)\in {\mathbb{C}}^p$ satisfy \[eq:star\] \_i\_\^[i]{}(u)\^=0,|\_i\_\^[i]{}(u)|\^=0. Here the quantity $\xi(x)\neq 0$ is treated as a complex function of the real variables $x$. In what follows we assume that the vectors $\gamma$ and $\bar{\gamma}$ are linearly independent. The proposed form of solution (\[eq:2.4\]) is more general than the one proposed in [@Jeffrey:1976; @Rozdestvenski:1983] for which the derivatives $u^\alpha_i$ are represented by a real simple element leading to a simple Riemann wave solution. To distinguish this situation from the one proposed in (\[eq:2.4\]), we call the real-valued solution associated with a complex element and its complex conjugate a simple mode solution in accordance with [@Perad:1985]. This means that all first-order derivatives of $u^\alpha$ with respect to $x^i$ are decomposable in the following way \[eq:star2\] =(x)\^(u)\_i(u)+|(x)|\^(u)|\_i(u), or alternatively $$\frac{{\partial}u^\alpha}{{\partial}x^i}=i{\left( \xi(x) \gamma^\alpha(u)\lambda_i(u)-\bar{\xi}(x)\bar{\gamma}(u)^\alpha\bar{\lambda}_i(u) \right)},$$where a set of functions $(\lambda,\gamma)$ and their conjugates $(\bar{\lambda},\bar{\gamma})$ on $U$ satisfy the wave relation (\[eq:star\]). Similarly, as in the case of $k$-waves, for hyperbolic systems [@Perad:1985], we have to find the necessary and sufficient conditions for the existence of solutions of type (\[eq:2.4\]). These conditions are called involutivity conditions. #### First we derive a number of necessary conditions on the vector fields $(\gamma,\lambda)$ and their complex conjugate $(\bar{\gamma},\bar{\lambda})$ as a requirement for the existence of rank-$2$ (simple mode) solutions of the homogeneous system (\[eq:2.1\]). Namely, closing (\[eq:2.4\]) by exterior differentiation, we obtain the following 2-forms \[eq:2.5\] (d+d)+d+|(d||+|d|)+|d||=0, which have to satisfy (\[eq:2.4\]). Using (\[eq:2.4\]) we get \[eq:2.6\] d=||\_[,|]{}+|\_[,]{},d=|\_[,|]{}|+\_[,]{}d, where we have used the following notation $$\lambda_{,\gamma}=\gamma^\alpha\frac{{\partial}}{{\partial}u^\alpha}\lambda,\quad \gamma_{,\bar{\gamma}}=\bar{\gamma}^\alpha\frac{{\partial}}{{\partial}u^\alpha}\gamma.$$Substituting (\[eq:2.6\]) into the prolonged system (\[eq:2.5\]) we obtain \[eq:2.8\] +|+||=0, whenever the differential (\[eq:2.4\]) holds. The commutator of vector fields $\gamma$ and $\bar{\gamma}$, is denoted by $${\left[ \gamma,\bar{\gamma} \right]}={\left( \gamma,\bar{\gamma} \right)}_u+\gamma_{,\lambda_i}\lambda_{i,\bar{\gamma}}-\bar{\gamma}_{,\lambda_i}\lambda_{i,\gamma},$$while by ${\left( \gamma,\bar{\gamma} \right)}_u$ we denote a part of the commutator which contains the differentiation with respect to the variables $u^\alpha$, [*i.e.* ]{}$${\left( \gamma,\bar{\gamma} \right)}_u=\bar{\gamma}^\alpha\frac{{\partial}}{{\partial}u^\alpha}\left.\gamma(u,\lambda)\right|_{\lambda=const.} -\gamma^\alpha\frac{{\partial}}{{\partial}u^\alpha}\left.\bar{\gamma}(u,\bar{\lambda})\right|_{\bar{\lambda}=const.}.$$Let $\Phi$ be an annihilator of the vectors $\gamma$ and $\bar{\gamma}$, [*i.e.* ]{}$$<\omega \lrcorner\gamma>=0, \qquad <\omega \lrcorner \bar{\gamma}>=0, \quad \omega\in\Phi=\operatorname{An}{\left\{ \gamma,\bar{\gamma} \right\}}.$$Here, by the parenthesis $<\omega\lrcorner \gamma>$, we denote the contraction of the 1-form $\omega\in T_u^\ast U$ with the vector field $\gamma\in T_uU$ and similarly for the vector field $\bar{\gamma}\in T_uU$. Multiplying the equations (\[eq:2.8\]) by the 1-form $\omega\in\Phi$ we get \[eq:2.9\] |&lt;&gt;|=0. We look for the compatibility condition for which the system (\[eq:2.9\]) does not provide any algebraic constraints on the coefficients $\xi$ and $\bar{\xi}$. This postulate means that the profile of the simple modes associated with $\gamma\otimes \lambda$ and $\bar{\gamma}\otimes \bar{\lambda}$ can be chosen in an arbitrary way for the initial (or boundary) conditions. It follows from (\[eq:2.9\]) that the commutator of the vector fields $\gamma$ and $\bar{\gamma}$ is a linear combination of $\gamma$ and $\bar{\gamma}$, where the coefficients are not necessarily constant, [*i.e.* ]{}$${\left[ \gamma,\bar{\gamma} \right]}\in\operatorname{span}{\left\{ \gamma,\bar{\gamma} \right\}}.$$So, the Frobenius theorem is satisfied. That is, at every point $u_0$ of the space of dependent variables $U$, there exists a tangent surface $S$ spanned by the vector fields $\gamma$ and $\bar{\gamma}$ passing through the point $u_0\in U$. Moreover the above condition implies that there exists a complex-valued function $\alpha$ such that \[eq:2.10\] =-||, since $\overline{{\left[ \gamma,\bar{\gamma} \right]}}={\left[ \bar{\gamma},\gamma \right]}=-{\left[ \gamma,\bar{\gamma} \right]}$ holds. Next, from the vector fields $\gamma$ and $\bar{\gamma}$ defined on $U$-space, we can construct the coframe $\Psi$, that is the set of 1-forms $\sigma_1$ and $\sigma_2$ defined on $U$ satisfying the conditions \[eq:2.11\] &&lt;\_1&gt;=1,&&&lt;\_1|&gt;=0,\_1,\_2,\ &&lt;\_2&gt;=0, &&&lt;\_2|&gt;=1. Substituting (\[eq:2.6\]) and (\[eq:2.10\]) into the prolonged system (\[eq:2.8\]) and multiplying by $\sigma_1$ and $\sigma_2$ respectively, we get \[eq:2.11b\] &(i)&& d+[( ||\_[,|]{}+|\_[,]{} )]{}+||=0,\ &(ii) &&d||+|[( |\_[,]{}+||\_[,|]{} )]{}+|||=0. If $d\xi=d\bar{\xi}$ then equation (\[eq:2.11b\].i) is a complex conjugate of (\[eq:2.11b\].ii). So one can consider one of them, say (\[eq:2.11b\].i). We look for the condition of integrability such that the system (\[eq:2.11b\].i) does not impose any restriction on the form of the coefficients $\xi$ and $\bar{\xi}$. By exterior multiplication of (\[eq:2.11b\].i) by $\lambda$, we obtain \[eq:2.12i\] \_[,|]{}|=0, and its respective complex conjugate equation \[eq:2.12ii\] |\_[,]{}|=0. Note that if conditions (\[eq:2.10\]), (\[eq:2.12i\]) and (\[eq:2.12ii\]) are satisfied, then from the Cartan lemma[@Cartan:1953] the system (\[eq:2.11b\]) and consequently the set of 1-forms (\[eq:2.8\]) has nontrivial solutions for $d\xi$ and $d\bar{\xi}$. So under these circumstances the following result holds [@Perad:1985]. \[prop:1\] The necessary condition for the Pfaffian system (\[eq:2.4\]) to possess a simple mode solution is that the following two constraints on the complex-valued vector fields $\gamma$ and $\lambda$ be satisfied $$(i)\ \quad {\left[ \gamma,\bar{\gamma} \right]}=\alpha \gamma -\bar{\alpha} \bar{\gamma},$$for any complex-valued function $\alpha\in{\mathbb{C}}$, and $$(ii)\quad \lambda_{,\bar{\gamma}}, \bar{\gamma}_{,\gamma}\in\operatorname{span}{\left\{ \lambda,\bar{\lambda} \right\}}.$$ #### Note that the conditions (\[eq:2.10\]), (\[eq:2.12i\]) and (\[eq:2.12ii\]) are strong requirements on the functions $\gamma={\left( \gamma^1,\ldots,\gamma^q \right)}$ and $\lambda=(\lambda_1,\ldots,\lambda_p)$ and their complex conjugates which satisfy the wave relation (\[eq:2.2\]). Consequently, the postulated form of a one-mode solution $u(x)$ of the elliptic system (\[eq:2.1\]), required by the generalized method of characteristics (GMC), is such that all first-order derivatives of $u(x)$ with respect to $x^i$ are decomposable in the form (\[eq:2.4\]). This restriction is a strong limitation on the admissible class of solutions of (\[eq:2.1\]). So far, there have been few known examples of such solutions. Therefore, it is worth developing this idea by weakening the integrability conditions (\[eq:2.10\]), (\[eq:2.12i\]) and (\[eq:2.12ii\]) in order to construct multi-mode solutions (considered to be nonlinear superpositions of simple modes). In the next section we propose an alternative way of constructing multi-mode solutions (expressed in terms of Riemann invariants) which are obtained from a version of the conditional symmetry method (CSM) [@GrundHuard:2007] by adapting it to the elliptic systems. In Section \[sec:5\] and \[sec:6\] a further refinement of the above technique will be presented, which consist of generalising the idea of mode solutions for quasilinear systems (\[eq:2.1\]). This idea is based on the specific factorization of integral elements (\[eq:intelem\]) through the introduction of some rotation matrices by weakening the restrictions impose by the wave relations (\[eq:2.2\]). This approach allows us to go deeper into the geometrical aspects of solving systems (\[eq:2.1\]) and enables us to obtain new results. This constitutes the objective of this paper. Conditional symmetry method and multimode solutions in terms of Riemann Invariants. {#sec:3} =================================================================================== \[sec:condsym\] In this section, we examine certain aspects of the conditional symmetry method in the context of Riemann invariants for which the wave vectors $\lambda$ and $\bar{\lambda}$ are complex solutions of the dispersion relation (\[eq:2.3\]) associated with the original system (\[eq:2.1\]). Let us consider a nondegenerate first-order quasilinear elliptic system of PDEs (\[eq:2.1\]) in its matrix form \[eq:3.1\] \^1(u)u\_1+…+\^p(u)u\_p=0, where $\mathcal{A}^1,\ldots,\mathcal{A}^p$ are $m\times q$ real-valued matrix functions of $u$. Let us suppose that there exist $k$ linearly independent non-conjugate complex-valued wave vectors \[eq:wv\] \^(u)=[( \_1\^A(u),…,\_p\^A(u) )]{}\^p,\^A|\^BA,B[{ 1,…,k&lt;p }]{}, which satisfy the dispersion relation (\[eq:2.3\]). One should note that in (\[eq:wv\]) we do not require indices $A\neq B$, which means that real wave vectors are excluded from our consideration (we do not consider here the mixed case for wave vectors involving real and complex wave vectors). Under the above hypotheses, the $k$ wave vectors (\[eq:wv\]) and their complex conjugates $$\bar{{\lambda}}^A(u)={\left( \bar{{\lambda}}_1^A(u),\ldots,\bar{{\lambda}}_p^A(u) \right)}\in{\mathbb{C}}^p,\quad A=1,\ldots,k,$$satisfy the dispersion relation (\[eq:2.3\]). In what follows it is useful to introduce the notation $c.c.$ which means the complex conjugate of the previous term or equation. This notation is convenient for computational purposes allowing the presentation of some expressions in abbreviated form. #### Let us suppose that there exists a unique solution $u(x)$ of the system (\[eq:3.1\]) of the form \[eq:3.5\] u=f(r\^1(x,u),…, r\^k(x,u),|[r]{}\^1(x,u),…,|[r]{}\^k(x,u))+c.c., where the complex-valued functions $r^A,\bar{r}^A:{\mathbb{R}}^p\times {\mathbb{R}}^q{\rightarrow}{\mathbb{C}}$ are called the Riemann invariants associated respectively to wave vectors $\lambda^A$, $\bar{\lambda}^A$ and are defined by \[eq:3.2\] r\^A(x,u)=\_i\^A(u)x\^i,|[r]{}\^A(x,u)=|\_i\^A(u)x\^i,A=1,…,k, where $c.c.$ means complex conjugated previous term. Note that the functions $u(x)$ are defined implicitly in terms of $u^\alpha, x^i$, $r^A$ and $\bar{r}^A$. For any function $f:{\mathbb{C}}^k{\rightarrow}{\mathbb{C}}^q$ and its complex conjugate, the equation (\[eq:3.5\]) determines a unique real-valued function $u(x)$ on a neighborhood of the origin $x=0$. Note also that an analogue analysis as presented below can be performed if one replace the postulated form of the solution (\[eq:3.5\]) written in the Riemann invariants by the expression $$u=i{\left( f(r^1(x,u),\ldots,r^k(x,u),\bar{r}^1(x,u),\ldots,\bar{r}^k(x,u))-c.c. \right)}.$$Therefore we omit this case. #### The Jacobi matrix of derivatives of $u(x)$ is given by \[eq:3.6\] u=(u\^\_i)=[( I\_q- )]{}\^[-1]{} [( [( + )]{}+c.c. )]{}, or equivalently as \[eq:3.7\] u=[( + )]{}[( M\^1+M\^2| )]{}+c.c., where c.c. means the complex conjugate of the previous term. The $k\times k$ matrices $M^1$ and $M^2$ are defined by \[eq:3.7b\] M\^1=&\^[-1]{},\ M\^2&=-M\^1 [( + )]{}[( I\_k- )]{}\^[-1]{}, \[eq:3.8\] =[( )]{}\^[qk]{},=[( )]{}\^[qk]{},=(\_i\^A)\^[kp]{}, \[eq:3.9\] =[( )]{}=[( x\^i )]{}\^[kq]{},r=(r\^1,…,r\^k), and their respective conjugate equations. The matrices $I_q$ and $I_k$ are the $q\times q$ and $k\times k$ identity matrices respectively. We use the implicit function theorem to obtain the following conditions ensuring that functions $r^A,\bar{r}^A$ and $u^\alpha$ are expressible as graphs over some open subset $\mathcal{D}\subset {\mathbb{R}}^p$, \[eq:3.10\] [( I\_q- )]{}0 or \[eq:3.11\] &[( I\_k- )]{}0,0. So the inverse matrix in (\[eq:3.6\]) or in (\[eq:3.7\]) is well-defined in the vicinity of $x=0$, since \[eq:3.12\] =0,=0 x=0. Note that on the hypersurface defined by equation (\[eq:3.5\]) and one of the expressions (\[eq:3.10\]) or (\[eq:3.11\]) equal to zero, the gradient of the function $u(x)$ becomes infinite for some value of $x^i$. So the multimode solution, given by expression (\[eq:3.5\]) loses its sense on this hypersurface. Consequently, some types of discontinuities, [*i.e.* ]{}shock waves can occur. In what follows, we search for solutions defined on a neighborhood of $x=0$ under the assumption that the conditions (\[eq:3.10\]) or (\[eq:3.11\]) are different from zero. This means that if the initial data is sufficiently small, then there exists a time interval ${\left[ t_0,T \right]}$, $T>t_0$ (where we denote the independent variables by $t=x^0, \tilde{x}=x^1,\dots, x^n$ where $p=n+1$) in which the gradient catastrophe for the solution $u(t,\tilde{x})$ of the system (\[eq:2.1\]) does not take place [@GrundlandVassiliou:1991; @Rozdestvenski:1983]. #### Note that the Jacobian matrix of $u(x)$ has at most rank equal to $2k$. It follows that the proposed solution (\[eq:3.5\]) is also at most of rank-$2k$. Its image is a $2k$-dimensional submanifold ${\mathbb{S}}_{2k}$ in the first jet space $J^1=J^1(X\times U)$. #### Let us introduce a set of $p-2k$ linearly independent vectors $\xi_a:{\mathbb{R}}^q{\rightarrow}{\mathbb{C}}^p$ defined by \[eq:3.13\] \_a(u)=[( \_a\^1(u),…,\_a\^p(u) )]{},a=1,…,p-2k, satisfying the orthogonality conditions \[eq:3.14\] \_i\^A\_a\^i=0,|\_i\^A\_a\^i=0,A=1,…,k,a=1,…,p-2k, for a set of $2k$ linearly independent wave vectors ${\left\{ \lambda^1,\ldots,\lambda^k, \bar{\lambda}^1,\ldots, \bar{\lambda}^k \right\}}$. It should be noted that the set ${\left\{ \lambda^1,\ldots,\lambda^k,\bar{\lambda^1},\ldots,\bar{\lambda}^k,\xi^1,\ldots,\xi^{p-2k} \right\}}$ forms a basis for the space of independent variables $X$. Note also, that the vectors $\xi_a$ are not uniquely defined since they obey the homogeneous conditions (\[eq:3.14\]). As a consequence of equation (\[eq:3.6\]) or (\[eq:3.7\]), the graph $\Gamma={\left\{ x,u(x) \right\}}$ is invariant under the family of first-order differential operators \[eq:3.15i\] X\_a=\_a\^i(u),a=1,…,p-2 k, defined on $X\times U$ space. Since the vector fields $X_a$ do not include vectors tangent to the direction of $u$, they form an Abelian distribution on $X\times U$ space, [*i.e.* ]{} \[eq:3.16\] =0,a, b=1,…,p-2k. The set ${\left\{ r^1,\ldots, r^k,\bar{r}^1,\ldots,\bar{r}^k,u^1,\ldots u^q \right\}}$ constitutes a complete set of invariants of the Abelian algebra $\mathcal{L}$ generated by the vector fields (\[eq:3.15i\]). So geometrically, the characterization of the proposed solution (\[eq:3.5\]) of equations (\[eq:3.1\]) can be interpreted in the following way. If $u(x)$ is a $q$-component function defined on a neighborhood of the origin $x=0$ such that the graph of the solution $\Gamma={\left\{ (x,u(x)) \right\}}$ is invariant under a set of $p-2k$ vector fields $X_a$ with the orthogonality property (\[eq:3.14\]), then for some function $f$ the expression $u(x)$ is a solution of equation (\[eq:3.5\]). Hence the group-invariant solutions of the system (\[eq:3.1\]) consist of those functions $u=f(x)$ which satisfy the overdetermined system composed of the initial system (\[eq:3.1\]) together with the invariance conditions \[eq:3.17i\] \_a\^iu\_i\^=0,i=1,…,p,a=1,…,p-2k, ensuring that the characteristics of the vector fields $X_a$ are equal to zero. #### It should be noted that, in general, the conditions (\[eq:3.17i\]) are weaker than the differential constraints (\[eq:2.4\]) required by the generalized method of characteristics, since the latter method is subjected to the algebraic conditions (\[eq:2.2\]). In fact, equations (\[eq:3.17i\]) imply that there exist complex-valued matrix functions $\Phi_A^\alpha(x,u)$ and $\overline{\Phi}_A^\alpha(x,u)$ defined on the first jet space $J=J(X\times U)$ such that all first derivatives of $u$ with respect to $x^i$ are decomposable in the following way \[eq:3.18\] u\_i\^=\_A\^(x,u)\_i\^A+\_A\^(x,u)|\_i\^A, where \[eq:3.19\] \_A\^&=[( I\_q- )]{}\^[-1]{} [( + )]{},\ \_A\^&=[( I\_q- )]{}\^[-1]{} [( + )]{}, or \[eq:3.20\] \_A\^&=[( + )]{}M\^1+[( + )]{}\^2,\ \_A\^&=[( + )]{}\^1+[( + )]{}M\^2. The matrices $\Phi_A^\alpha\lambda_i^A$ and $\overline{\Phi}_A^\alpha\bar{\lambda}_i^A$ appearing in equation (\[eq:3.18\]) do not necessarily satisfy the wave relation (\[eq:2.2\]). As a result of this fact, the restrictions on the initial data at $t=0$ are eased, so we are able to consider more diverse types of modes in the superpositions than in the case of the generalized method of characteristics described in Section \[sec:2\]. #### Let us now proceed to solve the overdetermined system composed of equations (\[eq:3.1\]) and differential constraints (\[eq:3.17i\]) \[eq:3.21\] \^[i]{}\_(u)u\_i\^=0,\_a\^i(u)u\_i\^=0. Substituting (\[eq:3.6\]) or (\[eq:3.7\]) into (\[eq:3.1\]) yields the trace condition \[eq:3.22\] [( \^\^[-1]{} [( [( + )]{}+c.c. )]{} )]{}=0, or \[eq:3.23\] [( \^ )]{}=0, on the wave vectors $\lambda$ and $\bar{\lambda}$ and on the functions $f$ and $\bar{f}$, where $\mathcal{A}^1,\ldots,\mathcal{A}^q$ are $p\times q$ matrix functions of $u$ ([*i.e.* ]{}$\mathcal{A}^\mu=(\mathcal{A}^{\mu i}_\alpha(u))\in {\mathbb{R}}^{p\times q}, \mu=1,\ldots, m$). For the given initial system of equations (\[eq:3.1\]), the matrices $\mathcal{A}^\mu$ are known functions of $u$ and the trace conditions (\[eq:3.22\]) or (\[eq:3.23\]) are conditions on the functions $f$, $\bar{f}$, $\lambda$, $\bar{\lambda}$ (or on $\xi$ due to the orthogonality conditions (\[eq:3.14\])). From the computational point of view, it is useful to split $x^i$ into $x^{i_A}$ and $x^{i_a}$ and to choose a basis for the wave vector $\lambda^A$ and $\bar{\lambda}^A$ such that \[eq:3.24\] \^A=dx\^[i\_A]{}+\_[i\_a]{}\^Adx\^[i\_a]{},|\^A=dx\^[i\_A]{}+|\_[i\_a]{}\^Adx\^[i\_a]{}, A=1,…, k, where $(i_A,i_a)$ is a permutation of $(1,\ldots, p)$. So, the expressions (\[eq:3.9\]) become \[eq:3.25\] =x\^[i\_a]{},=x\^[i\_a]{}. Substituting (\[eq:3.25\]) into (\[eq:3.22\]) (or (\[eq:3.23\])) yields \[eq:3.26\] [( \^\^[-1]{}[( + )]{})]{}=0,=1…,m, or \[eq:3.27\] [( \^\^[-1]{})]{}=0,=1…,m, where $R=(r^1,\ldots, r^k,\bar{r}^1,\ldots,\bar{r}^k)^T$ and \[eq:3.28\] Q\_a=[( + )]{}+[( + )]{}=[( + )]{} \_a\^[qq]{},\ K\_a= ( [cc]{} &\ & ) =\_a[( + )]{}\^[2k2k]{}, and for simplicity of notation, we note \[eq:3.28b\] = ( &\ &| ) \^[2kp]{},\_a= ( &\ & ) \^[2kq]{},=[( , )]{}\^[q2k]{}, for $i_A$ fixed and $i_a=1,\ldots, p-1$. In (\[eq:3.28\]) the $2k\times 2k$ matrix $K_a$ is defined in terms of the $k\times k$ subblocks of the form ${{\partial}\lambda_{i_a}}/{{\partial}u}{\left( {{\partial}f}/{{\partial}r}+{{\partial}\bar{f}}/{{\partial}r} \right)}$, where $\eta_a$ is a matrix form of the block ${\partial}\lambda_{a}/{\partial}u$ over the block ${\partial}\bar{\lambda}_a/{\partial}u$. The notation ${\left( {{\partial}f}/{{\partial}r}, {{\partial}f}/{{\partial}\bar{r}} \right)}$ represents the matrix formed of the left block ${\partial}f/{\partial}r$ and the right block ${\partial}f/{\partial}\bar{r}$. Note that the functions $r^A$, $\bar{r}^A$ and $x^{i_a}$ are functionally independent in a neighborhood of $x=0$ and the matrix functions $\mathcal{A}^\mu$, ${\partial}f/{\partial}r$, ${\partial}f/{\partial}\bar{r}$, ${\partial}\bar{f}/{\partial}r$, ${\partial}\bar{f}/{\partial}\bar{r}$, $Q_a$ and $K_a$ depend on $r$ and $\bar{r}$ only. So, equations (\[eq:3.26\]) (or (\[eq:3.27\])) have to be satisfied for any value of coordinates $x^{i_a}$. As a consequence, we have some constraints on these matrix functions. From the Cayley-Hamilton theorem, we know that for any $n\times n$ invertible matrix $M$, the expression $(M^{-1}\det M)$ is a polynomial in $M$ of order $(n-1)$. Thus, using the tracelessness of the expression $\mathcal{A}^\mu{\left( I_q-Q_ax^{i_a} \right)}^{-1}({\partial}f/{\partial}R)\Lambda$, we can replace equations (\[eq:3.26\]) by the following condition \[eq:3.29\] [( \^Q[( + )]{})]{}=0,Q=(I\_q-Q\_ax\^[i\_a]{})\^[qq]{}, where $\operatorname{adj}M$ denotes the adjoint of the matrix $M$. As a consequence the matrix $Q$ is a polynomial of order $(q-1)$ in $x^{i_a}$. Taking (\[eq:3.29\]) and all its partial derivatives with respect to $x^{i_a}$ (with $r$, $\bar{r}$ fixed at $x=0$), we obtain the following conditions for the matrix functions $f(r,\bar{r})$ and $\lambda(f(r,\bar{r}))$ \[eq:3.30\] [( \^)]{}=0,=1,…,m, \[eq:3.31\] [( \^Q\_[(a\_1.]{}…Q\_[a\_s)]{}[( + )]{})]{}=0, where $s=1,\ldots,q-1$ and $(a_1,\ldots, a_s)$ denotes the symmetrization over all indices in the bracket. A similar procedure can be applied to system (\[eq:3.27\]) to yield (\[eq:3.30\]) and \[eq:3.32\] [( \^K\_[(a\_1]{}…K\_[a\_s)]{})]{}=0, where now $s=1,\ldots,2k-1$. Equations (\[eq:3.30\]) represent the initial value conditions on a surface in the space of independent variables $X$, given at $x^{i_a}=0$. Note that equations (\[eq:3.31\]) (or (\[eq:3.32\])) form the conditions required for the preservation of the property (\[eq:3.30\]) along the flows represented by the vector fields (\[eq:3.15i\]). Equation (\[eq:3.24\]) allows us to express $X_a$ in the form \[eq:3.33\] X\_a=\_[i\_a]{}-\_[i\_a]{}\^A\_[i\_A]{}-|\_[i\_a]{}\^A\_[i\_A]{},A=1,…,k. Substituting expressions (\[eq:3.28\]) into (\[eq:3.31\]) or (\[eq:3.32\]) and simplifying gives the unified form \[eq:3.34\] [( \^[( + )]{}\_[(a\_1.]{}[( + )]{}…\_[[a\_s]{})]{}[( + )]{})]{}=0. The index $s$ is either $\max(s)=q-1$ or $\max(s)=2k-1$, we choose the one which is more convenient from the computational point of view. In this case, for $k\geq 1$ the two approaches, CSM and GMC, become essentially different and, as we demonstrate in the following example, the CSM can provide rank-$2k$ solutions which are not Riemann $2k$-waves as defined by the GMC since we weaken the integrability conditions (\[eq:2.1\]) for the wave vectors $\lambda$ and $\bar{\lambda}$. #### A change of variable on $X\times U$ allows us to rectify the vector fields $X_a$ and considerably simplify the structure of the overdetermined system (\[eq:3.21\]) which classifies the preceding construction of multimodes solutions. For this system, in the new coordinates, we derive the necessary and sufficient conditions for the existence of rank-$2k$ solutions of the form (\[eq:3.5\]). Suppose that there exists an invertible $2k\times 2k$ subblock matrix \[eq:3.35\] H=(\^s\_t), 1s,t2k, of the larger matrix $\Lambda\in {\mathbb{C}}^{2k\times p}$, then the independent vector fields $X_a$ can be written as \[eq:3.36\] X\_a=\_[x\^[a+2k]{}]{}-(H\^[-1]{})\^s\_t\^t\_[a+2k]{}\_[x\^A]{}, which have the required form (\[eq:3.15i\]) and for which the orthogonality conditions (\[eq:3.14\]) are fulfilled. We introduce new coordinate functions \[eq:3.37\] z\^1=r\^1(x,u),…, z\^[k]{}=r\^[k]{}(x,u),z\^[k+1]{}=|[r]{}\^1(x,u),…, z\^[2k]{}=|[r]{}\^[k]{}(x,u),\ z\^[2k+1]{}=x\^[2k+1]{},…,z\^p=x\^p,v\^1=u\^1,…, v\^q=u\^q, on $X\times U$ space which allow us to rectify the vector fields (\[eq:3.36\]). As a result, we get \[eq:3.38\] X\_1=,…, X\_[p-2k]{}=. The $p$-dimensional submanifold invariant under $X_{1},\ldots,X_{p-2k}$, is defined by equations of the form \[eq:3.39\] v=f(z\^1,…, z\^k, |[z]{}\^1,…, |[z]{}\^k)+c.c. for an arbitrary function $f:X{\rightarrow}U$. The expression (\[eq:3.39\]) is the general solution of the invariance conditions \[eq:3.40\] v\_[z\^[2k+1]{}]{},…, v\_[z\^p]{}=0. In general, the initial system (\[eq:3.1\]) described in the new coordinates $(x,v)\in X\times U$ is a nonlinear system of first-order PDEs, of the form \[eq:3.42\] &\^[l]{}\_(v)=0,&&i=1,…, k,\ &\^[l]{}\_(v)=0,&&i=k+1,…, 2k,\ &\^[i]{}\_(v)=0, &&i=2k+1,…,p. We obtain the following Jacobi matrix in the coordinates $(z,\bar{z},v)$ \[eq:3.43\] =[( \^[-1]{} )]{}\^j\_l\^l\_i\^[pp]{},=\_s\^i-x\^l, whenever the invariance conditions (\[eq:3.40\]) are satisfied. Appending to the system (\[eq:3.42\]) the invariance condition (\[eq:3.40\]), we obtain the quasilinear reduced system of PDEs \[eq:3.44\] [( \^(v)[( I\_[q]{}- )]{}\^[-1]{})]{}=0,,…,=0,=1,…,m, or \[eq:3.45\] [( \^(v)\^[-1]{})]{}=0,,…,=0,=1,…,m. #### We now provide some basic definitions that we require for the use of the conditional symmetry method in order to encompass the use of Riemann invariants. #### A vector field $X_a$ is called a conditional symmetry of the original system (\[eq:3.1\]) if $X_a$ is tangent to the manifold ${\mathbb{S}}={\mathbb{S}}_\Delta\cap {\mathbb{S}}_Q$, [*i.e.* ]{} \[eq:3.46\] .\^[(1)]{}X\_a|\_T\_[(x,u\^[(1)]{})]{}, where the first prolongation of $X_a$ is given by \[eq:3.47\] \^[(1)]{}X\_a=X\_a-\^i\_[a,u\^]{}u\_j\^u\_i\^,a=1,…,p-2k and the submanifolds of the solution spaces are given by \[eq:3.48\] \_=[{ (x,u\^[(1)]{}):\_\^[i]{}u\_i\^=0,=1,…,m }]{}, and \[eq:3.49\] \_Q=[{ (x,u\^[(1)]{}):\_a\^i(u)u\_i\^=0,=1,…, q, a=1,…,p-2k }]{}. Consequently, an Abelian Lie algebra $\mathcal{L}$ generated by the vector fields $X_1,\ldots, X_{p-2k}$ is called a conditional symmetry algebra of the original system (\[eq:3.1\]) if the conditions \[eq:3.50\] .\^[(1)]{}X\_a[( \^iu\_i )]{}|\_=0,a=1,…, p-2k, are satisfied. #### Supposing that $\mathcal{L}$, spanned by the vector fields $X_1,\ldots, X_{p-2k}$, is a conditional symmetry algebra of the system (\[eq:3.1\]), a solution $u=f(x)$ is said to be a conditionally invariant solution of the system (\[eq:3.1\]) if the graph $\Gamma={\left\{ (x,f(x)) \right\}}$ is invariant under the vector fields $X_1,\ldots, X_{p-2k}$. #### **Proposition** 2. *A nondegenerate quasilinear hyperbolic system of first-order PDEs (\[eq:3.1\]) in $p$ independent variables and $q$ dependent variables admits a $p-2k$-dimensional symmetry algebra $\mathcal{L}$ if and only if ($p-2k$) linearly independent vector fields $X_1,\ldots, X_{p-2k}$ satisfy the conditions (\[eq:3.30\]) and (\[eq:3.34\]) on some neighborhood of $(x_0,u_0)$ of $S$. The solutions of (\[eq:3.1\]) which are invariant under the Lie algebra $\mathcal{L}$ are precisely $k$-mode solutions of the form (\[eq:3.5\]).* #### Proof. The proof of this proposition is essentially similar to that of the proposition in [@GrundHuard:2007]. For the sake of completeness we now give it. Let us express the vector fields $X_a$ in the new coordinates $(z,v)$ on $X\times U$. Equations (\[eq:3.38\]) and (\[eq:3.47\]) imply that \[eq:3.60\] \^[(1)]{}X\_a=X\_a,a=1,…, p-2k. The symmetry criterion that has to be satisfied for $G$ to be the symmetry group of the overdetermined system (\[eq:3.44\]) (or (\[eq:3.45\])) requires that the vector fields $X_a$ of $G$ satisfy $X_a(\Delta)=0$, whenever equation (\[eq:3.44\]) (or (\[eq:3.45\])) is satisfied. Thus the symmetry criterion applied to the invariance conditions (\[eq:3.40\]) vanishes identically. Applying this criterion to the system (\[eq:3.42\]) in the new coordinates, carrying out the differentiation and taking into account the conditions (\[eq:3.30\]) and (\[eq:3.34\]), we obtain the equations which are identically satisfied. #### The converse is also true. The assumption that the system (\[eq:3.1\]) is nondegenerate means, according to [@Olver:Application_of_Lie], that it is locally solvable and is of maximal rank at every point $(x_0,u_0)\in S$. Therefore, the infinitesimal symmetry condition is a necessary and sufficient condition for the existence of the symmetry group $G$ of the overdetermined system (\[eq:3.21\]). Since the vector fields $X_a$ form an Abelian distribution on $X\times U$, it follows that conditions (\[eq:3.30\]) and (\[eq:3.34\]) are satisfied. The solutions of the overdetermined system (\[eq:3.21\]) are invariant under the algebra $\mathcal{L}$ generated by the $p-2k$ vector fields $X_1,\ldots,X_{p-2k}$. The invariants of the group $G$ of such vector fields are provided by the functions ${\left\{ r^1,\ldots, r^k,\bar{r}^1,\ldots, \bar{r}^k,u^1,\ldots,u^q \right\}}$. So the general multimode solution of (\[eq:3.1\]) takes the required form (\[eq:3.5\]).  $\Box$ The ideal plastic flow. {#sec:4} ======================= In this section we would like to illustrate the proposed approach for constructing multimode solutions with the example of the ideal nonstationary irrotational planar ideal plastic flow subjected to an external force due to a work function $V$ (potential if $V_t=0$). Under the above assumptions the examined model is governed by a quasilinear elliptic homogenous system of five equations in (2+1) dimensions of the form [@Chak:2006; @Hill:1998; @Kat:1] \[eq:4.1\] &(a) &&\_x- [( \_x 2+\_y 2)]{}+[( V\_x-u\_t - u u\_x - v u\_y )]{}=0,\ &(b) &&\_y- [( \_x2- \_y 2)]{}+[( V\_y-v\_t - u v\_x - v v\_y )]{}=0,\ &(c) &&(u\_y+v\_x)2+ (u\_x-v\_y)2=0,\ &(d) && u\_x+v\_y=0,\ &(e) && u\_y-v\_x=0. Equations (\[eq:4.1\].a) and (\[eq:4.1\].b) involve the independent variables $\sigma, \theta, u, v$ and the potential $V$ is a given function of $(t,x,y)\in{\mathbb{R}}^3$. The stress tensor is defined by the mean pressure $\sigma$ and the angle $\theta$ relative to the $x$-axis minus $\pi/4$. Equation (\[eq:4.1\].c) represents the Saint-Venant-Von Mises plasticity equation. Equations (\[eq:4.1\].d) and (\[eq:4.1\].e) for the velocities $u$ and $v$ (along the $x$-axis and $y$-axis respectively) correspond to the incompressibility and the irrotationality of the flow of the plastic material. The independent variables are denoted by $(x^i)=(t,x,y)\in X\subset {\mathbb{R}}^3$ and the unknown functions by $(u^\alpha)=(\sigma,\theta, u,v)\in U\subset {\mathbb{R}}^4$. The first-order partial derivatives of $\sigma$ with respect to the independent variables $x$ and $y$ are denoted $\sigma_x$ and $\sigma_y$ respectively. The first-order partial derivatives of $u$, $v$ and $\theta$ are denoted similarly. #### Before using the approach described in Section \[sec:condsym\] to obtain solutions of the system (\[eq:4.1\]), we begin by expressing the system (\[eq:4.1\]) in a form which is more convenient for the purpose of computation. In the first place, we note that the equations (\[eq:4.1\].a) and (\[eq:4.1\].b) governing the average pressure $\sigma$ can be solved by quadrature when the angle $\theta$ satisfies the compatibility conditions for the mixed derivatives of $\sigma$ with respect to $x$ and $y$. This equation takes the form \[eq:4.2\] 2[( \_[x]{}\^2-\_[xy]{}-\_y\^2 )]{}(2)+[( \_[xx]{}-4\_x\_y-\_[yy]{} )]{}(2)=0. If $\theta$ satisfies (\[eq:4.2\]), then the pressure $\sigma$ can be expressed in terms of the velocities $u$, $v$, of the angle $\theta$ and of the potential $V$ with the form \[eq:4.3\] (t,x,y)&=-V(t,x,y)+[12]{}(2(t,x,y))++u\_t(t,x,y)dx\ & +\_y(t,x,y)(2(t,x,y))dx+c\_0(t). It should be noted that in order to obtain the compatibility condition (\[eq:4.2\]), one has to make use of equation (\[eq:4.1\].e), that is, we must suppose that the flow is irrotational. In order to find a solution of the original system (\[eq:4.1\]), it is still necessary to find a solution of the overdetermined system consisting of equations (\[eq:4.2\]), (\[eq:4.1\].c)-(\[eq:4.1\].e) for the three unknown functions $u,v$ and $\theta$. In order to reduce the order of the PDE (\[eq:4.2\]), we introduce the dependent variables $\phi$ and $\psi$, which are defined by the equations \[eq:4.4\] =\_x,=\_y. Therefore, we have to solve the following well-determined first-order system of 4 equations in the 4 unknowns $\phi$, $\psi$, $u$, $v$ \[eq:4.5\] (a)&&&[( 2\^2-(\_y-\_x)-2\^2 )]{}u\_y+[( \_x+4-\_y )]{}u\_x=0,\ (b)&&&\_y-\_x=0,\ (c)&&&u\_x+v\_y=0,\ (d)&&&u\_y-v\_x=0. Time appears in system (\[eq:4.5\]) only as a parameter since the system does not involve any derivatives with respect to time. Therefore, when solving the system (\[eq:4.5\]) we treat the unknowns $\phi, \psi, u, v$ in (\[eq:4.5\]) as functions of $x$ and $y$ only, and subsequently we replace the integration constant by arbitrary function of $t$. Since equation (\[eq:4.5\].a) is not quasilinear, the CSM as presented in Section \[sec:condsym\] cannot be applied to the entire system (\[eq:4.5\]). Nevertheless, the CSM can be used for the subsystem $\Delta$ of 3 equations in the 4 unknowns consisting of (\[eq:4.5\].b)-(\[eq:4.5\].d), which is then underdetermined. Next, we use the degrees of freedom of the obtained solution to satisfy equation (\[eq:4.5\].a). The subsystem $\Delta$ can be written in the following matrix form: \[eq:4.6\] ( [cccc]{} 0 & -1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & -1 ) ( [c]{}\ \ u\ v ) + ( [cccc]{} 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1\ 0 & 0 & 1 & 0 ) ( [c]{}\ \ u\ v ) = ( [c]{} 0\ 0\ 0\ 0 ) . The dispersion relation (\[eq:2.3\]) takes the form \[eq:4.7\] ( [cccc]{} \_2 & -\_1 & 0 & 0\ 0 & 0 & \_1 & \_2\ 0 & 0 & \_2 & -\_1 ) &lt;3. By solving the dispersion relation (\[eq:4.7\]), we obtain the wave vector $\lambda=(\lambda_1,\lambda_2)=(1,i)$ and its complex conjugate $\bar{\lambda}=(1,-i)$. Hence, the Riemann invariants associated with the wave vectors $\lambda$ and $\bar{\lambda}$ are \[eq:4.9\] r=x+iy,|[r]{}=x-iy. We look for a nontrivial real solution of (\[eq:4.5\].b)-(\[eq:4.5\].d) of the form \[eq:4.10\] =&f\_1(r,|[r]{})+\_1(r,|[r]{}),&&=f\_2(r,|[r]{})+\_2(r,|[r]{}),\ u=&f\_3(r,|[r]{})+\_3(r,|[r]{}), &&v=f\_4(r,|[r]{})+\_4(r,|[r]{}). Introducing the notation $\phi=u^1$, $\psi=u^2$, $u=u^3$, $v=u^4$ and $$\Lambda={\left( \begin{array}{cc} 1 & i \\ 1 & -i \end{array} \right)},$$we observe that the matrices $\eta_{a_j}$ defined by (\[eq:3.28b\]) all vanish. Consequently, all trace conditions (\[eq:3.34\]) used to obtain solutions of the form (\[eq:4.10\]) are identically satisfied. We still have to consider the trace conditions (\[eq:3.30\]) where the matrices $\mathcal{A}^\mu$, $\mu=1,2,3$, are given by $$\mathcal{A}^1={\left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array} \right)},\qquad \mathcal{A}^2={\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)},\qquad \mathcal{A}^3={\left( \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array} \right)}.$$Conditions (\[eq:3.30\]) take the form \[eq:4.11\] &(a)+--+i[( +++ )]{},\ &(b)++++i[( +-- )]{},\ &(c)+--+i[( +++ )]{}. Condition (\[eq:4.11\].a) implies that \[eq:4.12\] e[( )]{}=-m[( )]{}, while conditions (\[eq:4.11\].b) and (\[eq:4.11\].c) imply that the solutions for velocities $u$ and $v$ take the particular form \[eq:4.13\] u=h(r)+|[h]{}(|[r]{}),v=i[( h(r)-|[h]{}(|[r]{}) )]{}, where $r$ and $\bar{r}$ are given by (\[eq:4.9\]). Note that the most general solution for $u$ and $v$ of the trace conditions (\[eq:3.30\]-\[eq:3.32\]) is exactely the one given by the expression (\[eq:4.13\]). The function $h$ and its complex conjugate are also determined using condition (\[eq:4.12\]) and the PDE (\[eq:4.5\].a). This allows us to also determine the functions $\phi$ and $\psi$. For computational purposes, it is convenient to introduce velocities of the form (\[eq:4.13\]) in equation (\[eq:4.1\].c) in order to determine the angle $\theta$ in the form \[eq:4.14\] =[( i )]{}, where we denote $h^{(n)}=d^nh/dr^n$. Next, we substitute (\[eq:4.14\]) into equation (\[eq:4.2\]) in order to determine the function $h$ and its complex conjugate. Proceeding in this way, we find that $h$ and $\bar{h}$ satisfy the third-order ODE which is separable in $r$ and $\bar{r}$. It is therefore equivalent to the system \[eq:4.15\] (a)2-3=,(b)2 -3=, where $\Omega$ is a real separation constant. Equation (\[eq:4.15\].b) can be solved in a way similar way to (\[eq:4.15\].a). Defining \[eq:4.16\] g(r)=h\^[(1)]{}(r), equation (\[eq:4.15\].a) can be written as \[eq:4.17\] gg”-3/2 (g’)\^2+(/2) g\^3=0. According to [@Kamke:1979] equation (\[eq:4.17\]) admits a first integral which can be rewritten in the equivalent form \[eq:4.19\] =d. By the change of variable \[eq:4.20\] g=[( )]{}, equation (\[eq:4.19\]) can be transformed to $2\exp{\left( (2\Omega)^{-1}(\omega^2-c_1) \right)}d\omega=\pm \Omega d\xi,$ for which the solution, in term of the inverse error function $\operatorname{erf}^{-1}$, is \[eq:4.21\] =, where $c_2$ is a constant of integration. We substitute (\[eq:4.20\]) into (\[eq:4.16\]) with $\omega$ given by (\[eq:4.21\]) and integrate the result. We obtain the function $h(r)$ and its complex conjugate in terms of the error functions $\operatorname{erfi}$ and $\operatorname{erf}^{-1}$ as \[eq:4.22\] h(r)&=-+c\_3,\ |[h]{}(|[r]{})&=-+|[c]{}\_3, where the $c_i\in{\mathbb{C}}$ are integration constants and the real separation constant $\Omega$ is non zero. Since equations (\[eq:4.22\]) solve the system (\[eq:4.15\]), we have that $\theta$ given by (\[eq:4.14\]) satisfies the compatibility condition (\[eq:4.2\]) for $\sigma$. One should note that no derivatives with respect to time appear in the PDE (\[eq:4.2\]). Consequently, the equation (\[eq:4.2\]) is still satisfied if, in equations (\[eq:4.13\]) and (\[eq:4.14\]), we replace the function $h(r)$ and its complex conjugate respectively by \[eq:4.22bis\] h(t,r)&=-+c\_3(t),\ |[h]{}(t,|[r]{})&=-+|[c]{}\_3(t), which are obtained by substituting the arbitrary complex functions $c_i(t)$ in place of the integration constant $c_i$ and the arbitrary real function $\Omega(t)$ in place of the separation constant $\Omega$. Hence, the general solution of the system (\[eq:4.1\]) takes the form \[eq:4.23\] u(t,x,y)&=h(t,r)+|[h]{}(t,|[r]{}),\ v(t,x,y)&=i[( h(t,r)-|[h]{}(t,|[r]{}) )]{},\ (t,x,y)&=[( i )]{},\ (t,x,y)&=-V(t,x,y)+[12]{}(2(t,x,y))++u\_t(t,x,y)dx\ & +\_y(t,x,y)(2(t,x,y))dx+\_0(t), where the functions $h$ and $\bar{h}$ are defined by (\[eq:4.22bis\]). Note that the solution (\[eq:4.23\]) is a real-valued solution. #### Let us now consider the situation when $\Omega$ is identically equal to zero in the system of ODEs (\[eq:4.15\]). This leads to two possible independent solutions for $h(r)$ depending on whether its second order derivative $h^{(2)}(r)$ vanishes or not. These solutions are respectively \[eq:4.24\] (i) &h(r)=c\_1r+c\_2,&&h\^[(2)]{}(r)=0,\ (ii)&h(r)=+c\_3,&&h\^[(2)]{}(r)0, and their complex conjugates, where the $c_i$ are integration constant. Let us consider separately two cases: when the function $h$ is given by (\[eq:4.24\].i) and when $h$ is of the form (\[eq:4.24\].ii). #### Case i. In this case, the real solution of the original system (\[eq:4.1\]) takes the form \[eq:4.25\] u(t,x,y)=&4[( e(c\_1(t))x+m(c\_1(t))y+e(c\_2(t)) )]{},\ v(t,x,y)=&4[( m(c\_1(t))x-e(c\_1(t))y+m(c\_2(t)) )]{},\ (t,x,y)=& { &-[( )]{},&&m(c\_1(t))0,\ & && m(c\_1(t))=0, .\ (t,x,y)=&-V(t,x,y)+[( 2|c\_1(t)|\^2+e(\_1(t)) )]{}x\^2 -2m(\_1(t))xy\ &+[( 2|c\_1(t)|\^2-e(\_1(t)) )]{}y\^2+2[( 2e[( c\_1(t)|[c]{}\_2(t) )]{}+e(\_2(t)) )]{}x\ &-2[( 2m[( c\_1(t)|[c]{}\_2(t) )]{}+m(\_2(t)) )]{}y+\_0(t), where the real function $\sigma_0(t)$ and the complex functions $c_1(t)$, $c_2(t)$ are arbitrary functions of time and $\dot{c}_1(t)$, $\dot{c}_2(t)$ their respective derivatives with respect to $t$. It should be noted that if the potential $V$, the real function $\sigma_0(t)$ and the complex functions $c_1(t)$, $c_2(t)$, are all bounded and are their derivatives $\dot{c}_1(t)$, $\dot{c}_2(t)$ are also bounded, then the solution (\[eq:4.19\]) is bounded. Moreover, if we take these functions to be of the form \[eq:4.25\] \_0(t)=a\_0e\^[-s\_0t]{},c\_i(t)=a\_je\^[-s\_jt]{}+ib\_je\^[-q\_jt]{},j=1,2, where $0<q_j,s_j\in {\mathbb{R}}$, $a_j,b_j\in {\mathbb{R}}$, $j=0,1,2$, then we obtain a damped solution. Another physically interesting situation occurs when the arbitrary functions $c_i(t)$ are constant in solution (\[eq:4.19\]). In this case, we obtain a stationary solution for velocities $u$ and $v$. This allows us to draw the shape of those extrusion dies which are admissible by requiring that the tool walls coincide with the lines of flow generated by the velocities $u$ and $v$. This is necessary since we suppose that the flow is incompressible. From the practical point of view, it is convenient to press the plastic material through the die rectilinearly with constant speed. If we feed the die in this way, the plasticity region limits are curves defined by the ODE [@Lamothe:1; @Lamothe:2] \[eq:4.26\] =, where $U_0$ and $V_0$ are constants representing the feeding velocity (or extraction velocity) of the die along the $x$-axis and $y$-axis respectively. Equation (\[eq:4.26\]) is a consequence of the hypothesis that the flow is incompressible and the mass is conserved. This condition reduces to the boundary conditions described in [@Czyz:1974] when we require that the limits of the plasticity region correspond to the slip lines (which correspond to the characteristic curves of the original system (\[eq:4.1\])), that is, when we require that $dy/dx=\tan\theta$ or $dy/dx=-\cot\theta$. Here, we use the weakened condition (\[eq:4.26\]) because no constraints are imposed on the flow lines which are in contact with the tool walls. This is so because, for a given solution and given parameters, we choose the walls of the extrusion die to lie along the flow lines. For the purpose of illustrating the applicability of the method, we have drawn in figure \[fig:1\] the shape of a tool and the flow of matter inside the extrusion die for the following parameters : $\mathcal{R}e(c_1(t))=1$, $\mathcal{I}m(c_1(t))=0$, $\mathcal{R}e(c_2(t))=0$, $\mathcal{I}m(c_2(t))=0$. The feeding velocity of the tool is $U_0=5.95$, $V_0=0$ and the tool expels the matter at a velocity of $U_1=24.05$, $V_1=0$. The plasticity region at the opening is bounded by curve $C_1$ and at the exit by the curve $C_2$. This extrusion die can thin a plate or rod of ideal plastic material. ![Extrusion die corresponding to the solution (\[eq:4.19\]).[]{data-label="fig:1"}](outil_cas_i.jpg){width="3.5in"} #### Case ii. If $h$ is defined by (\[eq:4.17\].ii), then the corresponding nontrival solution of system (\[eq:4.1\]) takes the form \[eq:4.27\] u(t,x,y)=&e(c\_3(t))+2,\ v(t,x,y)=&e(c\_3(t))+2,\ (t,x,y)=&-(\^[-1]{}\ &), where the mean pressure $\sigma$ is given by (\[eq:4.3\]), where we substitute the values of functions $u,v,\theta$ given by (\[eq:4.21\]). The complex functions $c_i(t)$, $i=1,2,3$, and the real function $\sigma_0(t)$ which appear in this solution are arbitrary. For any time $t_0$, solution (\[eq:4.3\]), (\[eq:4.27\]) has a singularity at point $(x_0,y_0)$ which satisfies the equation $$x_0^2+y_0^2+2{\left( \mathcal{R}e(c_2(t_0))x_0+\mathcal{I}m(c_2(t_0))y_0 \right)}+|c_2(t_0)|^2=0.$$This singularity is stationary if the function $c_2(t)$ is constant. Otherwise, its position varies with time. If the functions $\sigma_0$ and $c_i$, $i=1,2,3$, are of the form (\[eq:4.25\]) defined in a region of the $xy$-plane on a time interval $[T_0,T)$ where the gradient catastrophe does not occur, then the solution is bounded and damped. In figure \[fig:2\], we have drawn the shape of an extrusion die corresponding to the solution (\[eq:4.27\]) for the following choice of parameters: $\mathcal{R}e{\left( c_1(t) \right)}=0$, $\mathcal{I}m{\left( c_1(t) \right)}=0$, $\mathcal{R}e{\left( c_2(t) \right)}=0$, $\mathcal{I}m{\left( c_2(t) \right)}=-0.5$, $\mathcal{R}e{\left( c_3(t) \right)}=-0.5$ and $\mathcal{I}m{\left( c_3(t) \right)}=0$. The feeding velocity has component $U_0=0.2$, $V_0=0.2$, and the extraction of material is performed at the velocity $U_1=0.2$, $V_1=-0.2$. This type of tool can be used to bend a rod by extrusion without having to fold it. Finally, we should emphasize that the flow changes considerably when the parameters are varied and we have the freedom to choose the walls of the tool among the flow lines for certain fixed parameters $c_i(t)$. Moreover, the velocity and the orientation of the feeding (extraction) can vary somewhat for a given shape of the tool. Consequently, many types of extrusion dies can be drawn. ![Extrusion die corresponding to the solution (\[eq:4.21\]).[]{data-label="fig:2"}](outil_cas_ii.jpg){width="3.5in"} Simple wave solutions of inhomogeneous quasilinear system. {#sec:ms} ========================================================== \[sec:5\] Consider an inhogeneous first-order system of $q$ quasilinear PDEs in $p$ independent variables and $q$ unknowns of the form \[eq:SW:1\] \^[i]{}\_(u) u\^\_i=b\^(u),,=1,…,q,i=1,…,p. Let us underline that the system can be either hyperbolic or elliptic. We are looking for real solutions describing the propagation of a simple wave which can be realized by the system (\[eq:SW:1\]). We postulate a form of the solution $u$ in terms of a Riemann invariant $r$, [*i.e.* ]{} \[eq:SW:2\] u=f(r),r=\_i(u) x\^i, i=1,…,p, where $\lambda(u)={\left( \lambda_1(u),\ldots,\lambda_p(u) \right)}$ is a real-valued wave vector. We evaluate the Jacobian matrix $u^\beta_i$ by applying the chain rule: $$u^\beta_i=\frac{{\partial}f^\beta}{{\partial}r}{\left( r_{x^i}+r_{u^\alpha}u^\alpha_i \right)}=\frac{{\partial}f^\beta}{{\partial}r}{\left( \lambda_i+\lambda_{i,u^\alpha}x^iu^\alpha_i \right)}.$$We assume that the matrix \[eq:SW:4\] =[( I\_q- )]{}\^[qq]{}, is invertible, where we have used the following notation ${\partial}f/{\partial}r={\left( {\partial}f^1/{\partial}r,\ldots, {\partial}f^q/{\partial}r \right)}^T$, ${\partial}r/{\partial}u ={\left( {\partial}r/{\partial}u^1,\ldots,{\partial}r/{\partial}u^q \right)}$. The Jacobian matrix ${\partial}u$ takes the form \[eq:SW:5\] u=\^[-1]{}\^[qq]{}. Replacing the Jacobian matrix (\[eq:SW:5\]) into the original system (\[eq:SW:1\]), we get \[eq:SW:7\] \^[i]{}\_[( \^[-1]{} )]{}\^\_\_i=b\^. Note that the expression $\Phi^{-1}\frac{{\partial}f}{{\partial}r}\in {\mathbb{R}}^q$ is a contravariant vector as well as $b\in{\mathbb{R}}^q$. Hence there exists a nonzero scalar function $\Omega=\Omega(x,u)$, a rotation matrix $L=L(x,u)\in SO(q)$ and a vector $\tau=\tau(x,u)\in{\mathbb{R}}^q$ such that \[eq:SW:8\] \^[-1]{}=L b+,\^i\_i=0. It should be noted that we assume appropriate levels of differentiability of the functions $\Omega$, $\lambda$, $L$, $\tau$ and $b$, as necessary in order to justify all the following steps. Using relation (\[eq:SW:8\]), we eliminate the vector $\Phi^{-1}\frac{{\partial}f}{{\partial}r}\in {\mathbb{R}}^q$ from equation (\[eq:SW:7\]), which allows us to factor out the vector $b$ on the right after regrouping all terms on the left of the resulting equation. Therefore, we obtain the condition \[eq:SW:13\] [( \^[i]{}\_i L-I\_q )]{}b=0, on the scalar function $\Omega$, the wave vector $\lambda$ and the rotation matrix $L$. This implies that we have the following dispersion relation \[eq:SW:14\] [( \^[i]{}\_i L-I\_q )]{}=0. Once a scalar function $\Omega$, a wave vector $\lambda$ and a matrix $L$ satisfying (\[eq:SW:13\]) have been obtained, equation (\[eq:SW:8\]) must be used in order to determine the function $f$. Replacing the expression (\[eq:SW:4\]) for the matrix $\Phi$ into equation (\[eq:SW:8\]) and solving for the vector $\frac{{\partial}f}{{\partial}r}$ and taking into account the relation ${\partial}r/{\partial}u=({\partial}\lambda_i/{\partial}u) x^i$, we find that \[eq:SW:15\] =, which cannot admit the gradient catastrophe. Indeed, if we suppose that $1+\Omega({\partial}\lambda_i/{\partial}u) x^i L b=0$ when we proposed from equation (\[eq:SW:8\]) to equation (\[eq:SW:15\]), we easily conclude that $\Omega L b=0$. Consequently, since the matrix $\Phi$ is invertible, we conclude from (\[eq:SW:8\]) that the solution for $f(r)$ is constant, so it cannot admit the gradient catastrophe if $\det \Phi\neq 0$. #### Up till now, we cannot be sure that system (\[eq:SW:15\]) for $f(r)$ is well-defined in the sense that it represents a system for $f(r)$ express in terms of $r$ only. To ensure this, we begin by introducing vector fields orthogonal to the wave vector $\lambda$, that is, vector fields of the form \[eq:SW:16\] X\_a=\^i\_a(u)\_[x\^i]{},a=1,…,p-1,i=1,…,p, where \[eq:SW:17\] \_a\^i\_i=0,a=1,…,p-1,(\_a\^i)=p-1. Consequently, the wave vector $\lambda$ together with the vectors $\xi_a$, $a=1,\ldots,p-1$ form a basis for ${\mathbb{R}}^p$. Moreover, let us note that the vector fields $X_a$ form an Abelian Lie algebra of dimension $p-1$. Application of the vector field (\[eq:SW:16\]) to equation (\[eq:SW:15\]) cancels out the left side since we suppose that $f=f(r)$, so it must be the same on the right side. Therefore, the conditions for the system of ODEs (\[eq:SW:15\]) to be well-defined in terms of $r$ are \[eq:SW:18\] X\_a=0,a=1,…,p-1. #### In summary, system (\[eq:SW:1\]) admits a simple wave solution if the following conditions are satisfied: - there exist a scalar function $\Omega(x,u)$, a wave vector $\lambda(u)$ and a rotation matrix $L(x,u)$ satisfying the wave equation (\[eq:SW:13\]); - there exist $p-1$ vector fields (\[eq:SW:16\]) which satisfy the orthogonality relation (\[eq:SW:17\]); - the right-hand side of equation (\[eq:SW:15\]) is annihilated by the vector fields (\[eq:SW:16\]), [*i.e.* ]{}conditions (\[eq:SW:18\]) are satisfied; - $\det \Phi\neq 0$, where $\Phi$ is given by (\[eq:SW:4\]); - $1+\Omega \frac{{\partial}\lambda_i}{{\partial}u}Lbx^i\neq 0$, $\lambda\in C^1$. These conditions are sufficient, but not necessary, since the vanishing of $\det\Phi$ does not imply that solutions of form (\[eq:SW:2\]) do not exist. We finish the present analysis of the simple wave solutions of system (\[eq:SW:1\]) by several remarks: - In general, there are more parameters (arbitrary functions) defining $\Omega$ and the rotation matrix $L$ than the minimum number required to satisfy condition (\[eq:SW:13\]). The remaining arbitrary quantities are arbitrary functions of the $x$’s and $u$’s, which have to be used to satisfy the conditions (\[eq:SW:18\]). However, in the particular case when the system (\[eq:SW:1\]) has two equations in two dependent variables, the two-dimensional matrix $L$ is defined by a single parameter and $\Omega$ is the only other parameter available to satisfy the condition (\[eq:SW:13\]). Thus, since the system is autonomous (it can be expressed solely in terms of the $u$’s and their derivatives), it is clear that, in that case, $\Omega$ and $L$ depend only on the dependent variables $u$’s. - Supposing that $b$ is continous and assuming that conditions (i-v) are satisfied, the right side of equation (\[eq:SW:15\]) is continuous, which ensures the existence and uniqueness of the solution for $f(r)$ (see for example [@Ince]). - A sufficient condition for the system (\[eq:SW:15\]) to be expressible in terms of $r$ only ([*i.e.* ]{}well-defined) is: $$\frac{{\partial}\lambda_i}{{\partial}u}L b=0,\qquad i=1,\ldots, p.$$This condition is trivially satisfied when the vector $\lambda$ is constant. More generally, it is sufficient to require that $\frac{{\partial}\lambda_i}{{\partial}u}Lbx^i$ be proportional to $r=\lambda_i x^i$, [*i.e.* ]{}that there exist $\beta(x,u)$ such that $$\frac{{\partial}\lambda_i}{{\partial}u}L b=\beta(x,u)\lambda_i,\qquad i=1,\ldots,p.$$ - If the matrix $A^i\lambda_i$ is invertible, then $$\frac{{\partial}\lambda_i}{{\partial}u}Lbx^i=\frac{{\partial}\lambda_i}{{\partial}u}(\mathrm{A}^j\lambda_j)^{-1}(\mathrm{A}^j\lambda_j) Lb x^i=\frac{1}{\Omega}\frac{{\partial}\lambda_i}{{\partial}u}(\mathcal{A}^j\lambda_j)^{-1}bx^i$$is satisfied due to condition (\[eq:SW:13\]). From the previous remark, we deduce $$\frac{1}{\Omega}\frac{{\partial}\lambda_i}{{\partial}u}(\mathcal{A}^ j\lambda_j)^{-1}b=\beta(u) \lambda_i,\quad i=1,\ldots,p,$$which is a sufficient condition where the rotation matrix $L$ is not involved when the matrix $A^j\lambda_j$ is invertible. We note that the approach presented above generalizes the results obtained in [@GrundlandZelazny:1983] where the simple states have been constructed with wave vectors $\lambda$ of constant direction for hyperbolic inhomogeneous systems (\[eq:SW:1\]). Simple mode solutions for inhomogeneous quasilinear system. {#sec:6} =========================================================== We now generalize the concept of simple wave solutions for inhomogeneous, quasilinear, systems of form (\[eq:SW:1\]). As in the case of the simple wave, the system can be either hyperbolic or elliptic. We look for a real solution, in terms of a Riemann invariant $r$ and its complex conjugate $\bar{r}$, of the form \[eq:ms:1\] u=f(r,|[r]{}),r(u,x)=\_i(u)x\^i,|[r]{}(u,x)=|\_i(u)x\^i,i=1,…, p, where $\lambda(u)={\left( \lambda_1(u),\ldots,\lambda_p(u) \right)}$ is a complex wave vector and $\bar{\lambda}(u)$ its complex conjugate. The Jacobian matrix takes the form \[eq:ms:2\] u=\^[-1]{}[( + | )]{}\^[qp]{}, where we assume that the matrix \[eq:ms:3\] =[( I\_q-- )]{}\^[qq]{}. is invertible. Replacing the Jacobian matrix (\[eq:ms:2\]) into the system (\[eq:SW:1\]), we obtain \[eq:ms:4\] \^[i]{}\^[-1]{}[( + | )]{}=b,\^i=[( A\^[i]{}\_)]{}\^[qq]{}. We introduce a rotation matrix $L=L(x,u)\in SO(q,{\mathbb{C}})$ and its complex conjuguate $\bar{L}=\bar{L}(x,u)\in SO(q,{\mathbb{C}})$ such that the relations \[eq:ms:5\] \^[-1]{}=L b + ,\^[-1]{}=| |[L]{} b+ |, hold, where $\Omega(x,u)$ and its conjugate $\bar{\Omega}$ are scalar complex functions, while $\tau(x,u)$ and its complex conjugate vector $\bar{\tau}(x,u)$ satisfy \[eq:ms:7\] \^i\_i +\^i|\_i |=0. The vectors $\tau$ and $\bar{\tau}$ can be seen as characteristic vectors of the homogeneous part of equation (\[eq:ms:4\]). We eliminate the vectors $\Phi^{-1}({\partial}f/{\partial}r)$ and $\Phi^{-1}({\partial}f/{\partial}\bar{r})$ from equation (\[eq:ms:4\]) using equations (\[eq:ms:5\]). Considering the equation (\[eq:ms:7\]), we obtain, as a condition on functions $\Omega$, $\lambda$, $L$ and their complex conjugates, the relation \[eq:ms:8\] [( \^i[( \_iL+|\_i||[L]{} )]{}-I\_q )]{}b=0. The vector $b$ can be identified with an eigenvector of equation (\[eq:ms:8\]). Since $b$ is known from the initial system (\[eq:SW:1\]), the equation has to be satisfied by an appropriate choice of functions $\Omega$, $\bar{\Omega}$, of wave vectors $\lambda$, $\bar{\lambda}$ and of rotation matrices $L$ and $\bar{L}$. In particular, this requires that the dispersion relation \[eq:ms:10\] [( \^i[( \_i L+|\_i|[L]{} )]{}-I\_q )]{}=0 holds. Equation (\[eq:ms:10\]) is an additional condition on $\lambda$ and $\bar{\lambda}$, $L$ and $\bar{L}$ for the equation (\[eq:ms:8\]) to have a solution. Multiplying equations (\[eq:ms:5\]) on the left by the matrix $\Phi$, writing the matrix $\Phi$ explicitly using the notations (\[eq:ms:3\]), and then solving for ${\partial}f/{\partial}r$ and ${\partial}f/{\partial}\bar{r}$, we obtain the system \[eq:ms:12\] =,=, where the scalar functions $\sigma_1$ and $\sigma_2$ and their complex conjugates are defined by the equations \[eq:ms:13\] \_1=(L b+),\_2=(L b+). In order to ensure that system (\[eq:ms:12\]) is well-defined in the sense that it can be expressed as a system for $f$ in terms of $r$ and $\bar{r}$, we introduce the vector fields \[eq:ms:14\] X\_a=\^i\_a(u)\_[x\^i]{},a=1,…, p-2, where the complex coefficients $\xi^i_a(u)$ satisfy the orthogonality relations \[eq:ms:15\] \^i\_a\_i=0,\^i\_a|\_i=0. Next, we apply them to equations (\[eq:ms:12\]), which gives us the following conditions &X\_a=0,\ &X\_a=0, $a=1,\ldots, p-2$, since the vector fields $X_a$ annihilate all the functions $f(r,\bar{r})$. #### Remark 1: The system (\[eq:ms:12\]) for $f(r,\bar{r})$ is not necessarily integrable. However, it is possible to use the arbitrary functions defining the function $\Omega(x,u)$, the wave vectors $\lambda(u)$, $\bar{\lambda}(u)$, the vectors $\tau(x,u)$, $\bar{\tau}(x,u)$ and the matrices of rotation $L(x,u)$, $\bar{L}(x,u)$, in order to satisfy the compatibility conditions of the system (\[eq:ms:12\]). Example. Nonlinear interaction of waves and particles. {#sec:7} ====================================================== Let us consider the inhomogeneous system describing the propagation of shock waves intensity in nonlinear interaction of waves and particles [@Luneburg:1964] \[eq:e1:1\] u\_x+\_y=2\^[1/2]{}a (u/2)(/2),u\_y-\_x=-2\^[1/2]{}a(u/2)(/2). It should be noted that the compatibility condition of the mixed derivatives of $\phi$ corresponds to the Liouville equation \[eq:e1:2\] u\_[xx]{}+u\_[yy]{}=a\^2. Therefore, each solution of the system (\[eq:e1:1\]) also gives us a solution of the Liouville equation (\[eq:e1:2\]). We use the methods presented in Section \[sec:6\] to obtain the general solution of the system (\[eq:e1:1\]). First, write the system (\[eq:e1:1\]) in the matrix form \[eq:e1:3\] ( [c c]{} 1 & 0\ 0 & -1 ) ( [c ]{} u\_x\ \_x ) + ( [c c]{} 0 & 1\ 1 & 0 ) ( [c]{} u\_y\ \_y ) = ( [c]{} b\_1\ b\_2 ) , where $$b_1=2^{1/2}a \exp(u/2)\sin(\phi/2),\qquad b_2=2^{1/2}a\exp{u/2}\cos(\phi/2).$$The matrices $\mathcal{A}^i$ introduced in equation (\[eq:ms:4\]) are given by $$\mathcal{A}^1={\left( \begin{array}{c c} 1 & 0\\ 0 & -1 \end{array} \right)},\qquad \mathcal{A}^2={\left( \begin{array}{c c} 0 & 1\\ 1 & 0 \end{array} \right)}.$$Condition (\[eq:ms:8\]) is then satisfied by the scalar function $\Omega$, the wave vector $\lambda$ and the rotation matrix $L$, defined by $$\Omega=12^{1/4}(1-\epsilon^i),\quad \lambda=(1,i),\quad L={\left( \begin{array}{c c} l_{11} & l_{12}\\ l_{21} & l_{22}\end{array} \right)},\quad \epsilon=\pm 1,$$together with their complex conjugates, where $$\begin{aligned} l_{11}&=l_{22}=-108^{1/4}i{\left( \frac{3^{1/2}\epsilon (b_1+ib_2)^2+i(b_1-ib_2)^2}{6(1-\epsilon i) (b_1^2+b_2^2)} \right)}\\ l_{12}&=-l_{21}=108^{1/4}i{\left( \frac{3^{1/2}\epsilon i (b_1+ib_2)^2+(b_1-ib_2)^2}{6(1-\epsilon i) (b_1^2+b_2^2)} \right)} \end{aligned}$$Since the wave vectors $\lambda$ and $\bar{\lambda}$ are constant, the quantities $\sigma_1$, $\sigma_2$ (and their conjugates) are zero. So, we obtain the system (\[eq:ms:12\]) written in terms of the Riemann invariants $r=x+ i y$ and $\bar{r}=x- i y$ \[eq:e1:5\] u\_r&=24\^[-1/2]{}a[( -i(3\^[1/2]{}+3 i )(/2)+(3\^[1/2]{}-3 i )(/2) )]{}(u/2)+\ \_r&=24\^[-1/2]{}a[( (3\^[1/2]{}-3 i )(/2)-i(3\^[1/2]{}+3 i )(/2) )]{}(u/2)+i\ u\_[|[r]{}]{}&=24\^[-1/2]{}a[( i(3\^[1/2]{}-3 i )(/2)+(3\^[1/2]{}+3 i )(/2) )]{}(u/2)+|\ \_[|[r]{}]{}&=24\^[-1/2]{}a[( i(3\^[1/2]{}-3 i )(/2)+(3\^[1/2]{}+3 i )(/2) )]{}(u/2)-i|\ where $\sigma(r,\bar{r})$ is an arbitrary scalar function defining the vector $\tau=(\sigma,i \sigma)^T$, which satisfies condition (\[eq:ms:7\]). The system (\[eq:e1:5\]) can be written in the more compact form \[eq:e1:6\] &a)&&[( u+i )]{}=2\^[-1/2]{}a i [( )]{},\ &b)&&[( u-i )]{}=-2\^[-1/2]{}a i [( )]{},\ &c)&&[( u-i )]{}=-(3/2)\^[1/2]{}a [( )]{}+2,\ &d)&&[( u+i )]{}=-(3/2)\^[1/2]{}a [( )]{}+2|.\ Since, the equations (\[eq:e1:6\].c) and (\[eq:e1:6\].d) are complex conjugates they can be satisfied by defining the quantity $\sigma$ annihilating (\[eq:e1:6\].c). Consequently, there are only the two equations (\[eq:e1:6\].a) and (\[eq:e1:6\].b) to solve. After the change of variable \[eq:e1:7\] f=u+i ,|[f]{}=u-i , equations (\[eq:e1:6\].a) and (\[eq:e1:6\].b) take the form \[eq:e1:8\] f\_r=2\^[-1/2]{} i a|f|\^2,f\_[|[r]{}]{}=-2\^[-1/2]{} i a|f|\^2, which have general solution \[eq:e1:8\] f(r,|[r]{})=, where $\psi$ and $\bar{\psi}$ are arbitrary functions of $r$ and $\bar{r}$, respectively. Replacing the solution (\[eq:e1:8\]) into equations (\[eq:e1:7\]), solving for $u$ and $\phi$, and using the Riemann invariants $r$ and $\bar{r}$, we obtain the general solution of the system (\[eq:ms:1\]). This solution depends on one arbitrary complex function $\psi$ of one complex variable $r$ and its complex conjugate function. u&=2[( )]{},\ &=-i[( - )]{}+2 n, where $n$ is odd and the term $2 n\pi$ is associated with the admissible branches of the logarithm. It is easily verified that the solution for $u$ satisfies the Liouville equation (\[eq:ms:2\]). Final remarks. {#sec:8} ============== The generalized method of characteristics was originally devised for solving first-order quasilinear hyperbolic systems. The proposed techniques described in Sections \[sec:3\], \[sec:5\] and \[sec:6\] allow us to extend the applicability of this approach not only to hyperbolic systems but also to encompass elliptic and mixed (parabolic) type systems, both homogeneous and inhomogeneous. A variant of the conditional symmetry method for obtaining multimode solutions has been proposed for these types of systems. We have demonstrated the usefulness of this approach through the examples of nonlinear interaction of waves and particles and of the ideal plasticity in $(2+1)$ dimensions in its elliptic region. New classes of real solutions have been constructed in closed form, some of them bounded. Some of the obtained solutions described a stationary flow for an appropriate choice of parameters. For these solutions, we have drawn extrusion dies and the vector fields which define the flow inside a region where the gradient catastrophe does not occur. #### The proposed approach for constructing multimode solutions can be used in several potential applications arising from systems describing nonlinear phenomena in physics. It should be noted that in the multidimensional case, for many physical models, there are few known examples of multimode solutions written in terms of Riemann invariants for elliptic systems. This is a motivating factor for the elaboration of the generalization of the methods presented in Sections \[sec:5\] and \[sec:6\] through the introduction of rotation matrices in the factorization of the Jacobian matrices (\[eq:ms:2\]). This fact weakens the integrability condition required in the expression (\[eq:ms:12\]). The approach proposed in this paper offers a new and promising way to construct and investigate such types of solutions. This makes our approach attractive since it can widen the potential range of applications leading to more diverse types of solutions. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported by a research grant from the Natural Sciences and Engineering Council of Canada (NSERC). [10]{} , , London Math.Soc. Lecture note ser 149, Cambrige Univ. Press. (1992). G. Boillat, , Gauthier-Villars, Paris, (1965). , The method of Riemann invariants and its applications to the theory of plasticity, ,part 1, 26, 6, 1974, 817-838 and part 2, 24, 1, (1972), 3-26. , , Chapt. 1: Les systèmes différentielles en involution, Gauthier-Villars, Paris, (1953). J. Chakrabarty, , Elsevier, (2006). R. Conte, A.M. Grundland, B. Huard, Elliptic solutions of isentropic ideal compressible fluid flow in $(3+1)$ dimensions, ,Math. Theor. 42, 135203, (2009) (14pp) R. Courant, D. Hilbert, , Interscience, New-York, (1962). J. Czyz, Construction of a flow of an ideal plastic material in a die, on the basis of the method of [R]{}iemann invariants, , 26(4), (1974), 589-616. , Simple waves and invariant solutions of quasiliear systems, ,37, 6, (1996), 2969-2979. , Hamiltonnian formalism of one-dimensional systems of hydrodynamic type, ,27, (1983), 665-669. , On the integrability of $(2+1)$ dimensional quasilinear systems, ,248, (2004), 187-206. , The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type, ,37, (2004), 2949-2963. , Conditional symmetry of equations of mathematical Physics, ,43, (1991), 1456-1470. A.M. Grundland, B. Huard, Riemann invariants and rank-$k$ solutions of hyperbolic systems, , 13,3 , (2006) 393-419. A.M. Grundland, B. Huard, Conditional symmetries and Riemann invariants for hyperbolic systems of PDEs, , Math. Theor. 40, (2007), 4093-4123. , Symmetry reduction and Riemann wave solutions, ,198, (1996), 879-892. A.M. Grundland, P. Vassiliou, On the solvability of the Cauchy problem for the Riemann double waves by the Monge-Darboux method. , (1991) 221-278. A.M. Grundland, R. Zelazny, Simple waves in quasilinear systems, Part I and Part II, , 24, 9, (1983), 2305-2329. , Riemann invariants for nonhomgeneous systems of quasilinear partial differential equations, , Ser Sci Techn 22, (1974), 273-282. R. Hill, , Oxford University press, 1998. E.L. Ince, , Dover Publ. New-York, 1956. A. Jeffrey, , Pitman Publ., 1976. , Almost global existence of nonlinear wave equations in three space dimensions, ,37, (1984), 443-455. E. Kamke, , , I B.G. Teubner Stuttgart, 1983. L. Katchanov, , Éditions Mir, Moscou, 1975. , . Publ. Brasil Math. Soc., Sao Paulo, 1967. V. Lamothe, Symmetry group analysis of an ideal plastic flow. , 53, 3, 033704, (2012). V. Lamothe, Group analysis of an ideal plasticity model, , 45, 285203, (2012). , , University of California Press, 1964. R. Mises, , Acad. Press, New-York, 1958. , , Springer-Verlag, New-York, 1986. , , Hankbook of Lie group analysis, Ed. N. Ibragimov, vol.3 chapt., 11 CRC Press, London, 1995 Z. Peradzynski, Geometry of interactions of Riemann waves, , Ed. Debnath, Research notes in mathematics no 111, Pitman Avanced Publ. Boston, 1985. B.L. Rozdestvenski, N.N. Janenko, Systems of Quasilinear Equations and their Applications to Gas Dynamics, , vol. 55, AMS Providence 1983. , Functionally invariant solutions of wave equations, ,5, (1934), 259-264 , , John-willey Publ, New-York, 1974. , , in serie Advances of Modern Mathematics, 1998. [^1]: grundlan@crm.umontreal.ca [^2]: lamothe@crm.umontreal.ca

--- abstract: 'We discuss some fundamental concerns regarding the recent proposal of Dimopoulos and Giudice for dynamically aligning the soft masses of the sfermions in the minimal supersymmetric standard model (MSSM) with the corresponding fermion masses to suppress flavor changing neutral currents. We show that the phenomenologically-favored presence of right-handed neutrinos in the theory, even if only at very high scales, generically disaligns the slepton mass matrices. Further suppression is then needed to meet the current upper bound on the rate for $\mu \to e\gamma$. Planned improvements in the search for $\mu \to e\gamma$ should easily detect this rare mode. (With improved sensitivity $\mu\to 3e$ may also be seen.) By measuring the helicity of the amplitude for $\mu \to e\gamma$ such experiments could distinguish between unified and non-unified models at very high energies; by inserting the various MSSM parameters as they become available, the mixing in the leptonic Yukawa couplings can be extracted; and by combining the results with those of various neutrino experiments some information about the right-handed neutrino Majorana matrices can also be gained.' --- -.3in 8.5in .1in .1in 6.6in \#1[.3ex]{} December 1995UND-HEP-95-US01\ RU-95-72\ hep-ph/9512354 .2in [**Disoriented Sleptons**]{} .3in Riccardo Rattazzi[^1]\ [*Department of Physics and Astronomy\ Rutgers University\ Piscataway, NJ 08855*]{} Uri Sarid[^2]\ [*Department of Physics\ University of Notre Dame\ Notre Dame, IN 46556*]{} .2in The most promising candidates for a fundamental theory underlying the standard model have been supersymmetric (SUSY) models. There are compelling theoretical and phenomenological reasons to believe that nature is supersymmetric on microscopic scales, and that the observed asymmetry at low energies between bosons and fermions is due to spontaneous SUSY breaking. Much attention has been focused on the highly-successful minimal supersymmetric extension of the standard model, the MSSM. In this paper we address some aspects of this model and of its extension to include neutrino masses. In particular, we analyze the implications of neutrino masses to a new mechanism recently introduced by Dimopoulos, Giudice and Tetradis (DGT) [@ref:dgt] to ameliorate the flavor problem of the MSSM. In this analysis we present their mechanism somewhat differently from their original work, and then focus on its implications to rare leptonic processes such as $\mu\to e\gamma$. If the DGT mechanism is operative at a very large momentum scale $\Lambda$, then data about such rare processes can be combined with results from direct SUSY searches and from various neutrino experiments to reveal important information about the leptonic couplings at the scale $\Lambda$. The long-standing problem which DGT have sought to solve is that of flavor-changing neutral currents (FCNC) in the MSSM. If the soft mass matrices of squarks and sleptons are—as expected—not too far above the electroweak scale, and if they are neither proportional to unit matrices nor aligned with the corresponding fermion mass matrices, then they induce unacceptably large contributions to various FCNC processes, in particular neutral kaon oscillations and $\mu \to e \gamma$. Various mechanisms have been suggested to overcome this difficulty: if the gauginos are somewhat heavier than expected, they would raise the squark masses and make them roughly proportional to the unit matrix (though the difficulties in the leptonic sector would be harder to overcome); if the soft masses start out universal—proportional to the unit matrix—at a very high scale, they typically remain so in their first- and second-generation entries, and so contribute little to the two sensitive FCNC processes mentioned above; and if some flavor symmetries align the squarks with the quarks and the sleptons with the leptons, then once again the FCNC contributions can be suppressed. But the first two solutions have serious shortcomings: gaugino dominance requires unnaturally heavy gauginos and moreover is not very effective for $\mu \to e \gamma$; and universal soft masses seem an unlikely outcome of various theories at the highest scales. The third approach postulates a set of approximate symmetries to explain both the observed fermion mass matrices and their alignment with the squark and slepton masses [@ref:hor]. It is somewhat similar in spirit to the approach of DGT, in that the same suppression mechanism which works in the quark and lepton sectors is applied to the SUSY-breaking sector, and generically yields similar FCNC suppression. The actual amount of suppression, though, varies considerably depending on the horizontal symmetries used, and can be stronger or weaker than in the DGT scenario. In any case, a thoroughly novel mechanism for suppressing FCNC’s is very welcome. The recent proposal of DGT introduces just such a mechanism: a model, or more correctly a paradigm, in which the squark and slepton mass matrices are dynamically aligned with those of the corresponding fermions. In this letter we first present the idea and the assumptions of DGT somewhat differently than in the original proposal, focusing on the intrinsic link between any such dynamics and various fundamental concerns about the vacuum energy. We then show that even if such a mechanism is viable, and can greatly improve the situation in the quark sector, we do not expect it to be nearly sufficient in the lepton sector. In the paper of DTG individual lepton numbers were conserved, and therefore not surprisingly the slepton and lepton mass matrices were exactly aligned and no FCNC processes such as $\mu \to e \gamma$ could occur. However, realistically we expect the lepton number symmetries to be violated in the neutrino sector for a variety of phenomenological reasons. As we show below, such violations will induce a misalignment between leptons and sleptons. Though we can not at present predict the degree of misalignment precisely, we expect the effect to be phenomenologically important, and to yield valuable information about the leptonic flavor violations at very high energies. In the standard model, flavor violation comes about exclusively through the Cabibbo-Kobayashi-Maskawa mixing matrix. We can choose a basis for the quark fields such that the gauge interactions are flavor-conserving, as are the Yukawa couplings of the leptons $Y_E = \widehat Y_E$ (the hat indicates that the matrix is diagonal) and of the down-type quarks $Y_D = \widehat Y_D$, but then there is no more freedom to diagonalize the Yukawa couplings of the up-type quarks $Y_U$: they are given by $Y_U = K^\dagger \widehat Y_U$, where $K$ is the CKM matrix. Since the Yukawa couplings of the first two quark generations are small and the mixing with the third generation is small, the standard model exhibits very feeble FCNCs. In the lepton sector, flavor is exactly conserved. The minimal extension of this standard model introduces eight more potentially flavor-violating matrices: the five scalar mass matrices $\tilde m_Q^2$, $\tilde m_U^2$, $\tilde m_D^2$, $\tilde m_L^2$, and $\tilde m_E^2$, and the three trilinear scalar coupling matrices $A_E$, $A_D$, and $A_U$. We will choose once again to keep the gauge and gaugino interactions flavor-diagonal, and to do so we always rotate superpartners together. If we then stay with the above choice of basis for quark and lepton fields, we have no more freedom to diagonalize the eight new soft-SUSY-breaking matrices. If their off-diagonal terms are not suppressed relative to the diagonal ones, unacceptably large FCNCs can arise, as discussed above. We will concentrate first on the scalar masses, and then return to a discussion of the $A$ terms. DGT have proposed that these scalar mass matrices be promoted to dynamical fields rather than be treated as mere parameters. The advantage is that there may then exist a dynamical relaxation mechanism which would align these matrices with the Yukawa matrices and thereby minimize the flavor-changing interactions. Such a situation may arise in string theory, where the low-energy field theory parameters are often dynamically determined by the vacuum expectation values of certain fields. The fundamental, microscopic theory and the low-energy effective field theory are matched at a scale $\Lambda$ which we take to be of order the string or Planck scales, but could also be some lower scale. The Yukawa couplings are assumed to be fixed by the fundamental theory, perhaps by expectation values of fields with very large masses, so they are simply parameters of the effective theory. As for the scalar mass matrices, it is conceivable that their eigenvalues and orientations are determined by different mechanisms. We will only consider the “disoriented” scenario in which the eigenvalues are first fixed by some dynamics responsible for supersymmetry breaking, and then the orientations are dynamically determined by a set of light moduli fields. (We call them “moduli”, with an abuse of language, because they would correspond to flat directions when either the Yukawa couplings vanish or supersymmetry is unbroken.) If we denote the scalar masses by $3\times 3$ matrices $\tilde m_I^2$ where $I$ runs over the five fields $Q$ (squark doublets), $U$ (up-type antisquark singlets), $D$ (down-type antisquark singlets), $L$ (slepton doublets) and $E$ (charged slepton singlets), and diagonalize them by means of unitary matrices $U_I$ as in Refs. [@ref:dgt], $$\tilde m_I^2 = U_I^{\dagger} \widehat m_I^2 U_I\,,\qquad I = Q,U,D,L,E\,, \label{eq:uadef}$$ where $\widehat m_I^2$ is a real diagonal matrix, then the disoriented assumption amounts to promoting $U_I$ to dynamical fields, whose expectation value is determined by minimizing an effective potential. One could further expand the set of dynamical fields to include the eigenvalues $\widehat m_I^2$, resulting in the “plastication” scenario of DGT, but we limit our discussion of this scenario to a brief remark towards the end of this work. To determine what is the physics which fixes the alignment of the $U_I$, we need to examine the exact effective potential of the theory, which includes the effects of quantum fluctuations from all scales. After allowing for possible new physics beyond the MSSM at some intermediate SUSY-invariant scale $M$ between the cutoff $\Lambda$ and the effective SUSY-breaking scale $\tilde m \sim m_Z$ in the observable sector, we may write the effective potential generically as $$V_{\rm eff} = c_4 {\cal O}(\Lambda^4) + c'_4 {\cal O}(\Lambda^2 M^2) + c''_4 {\cal O}(M^4) + c_2 {\cal O}(\Lambda^2 \tilde m^2) + c'_2 {\cal O}(M^2 \tilde m^2) + c_0 {\cal O}(\tilde m^4) + \ldots \label{eq:veff}$$ where we have omitted a constant and any terms smaller than $\sim \tilde m^4$. Since the first three terms do not involve any supersymmetry breaking, they must vanish inasmuch as the vacuum energy of a supersymmetric theory vanishes: $c_4 = c'_4 = c''_4 = 0$. Assuming a hierarchy $\Lambda \gg M \gg \tilde m$, the dominant term would then be the ${\cal O}(\Lambda^2 \tilde m^2)$ part, which appears as a quadratic divergence in the low-energy effective theory. We will discuss such divergences, and the possibility that they are absent, further below, and argue that not only are they expected, but that even if they vanish the main results of our work will not change. Therefore, we will assume that the ${\cal O}(\Lambda^2 \tilde m^2)$ terms dominate and determine the orientation of the scalar masses. We note in passing that whatever mechanism is ultimately responsible for cancelling the cosmological constant, namely the constant term in $V_{\rm eff}$, may have implications for the other terms in this potential, but we cannot yet speculate on what those implications may be. The direct consequence of this assumption, as noted already in Ref. [@ref:dgt], is that the physics which determines the alignment is the physics at the cutoff scale $\Lambda$, namely, just at the scale in which the effective low-energy theory breaks down. All higher-dimension “irrelevant” operators are in principle as relevant as the renormalizable “relevant” ones. (For instance, operators with four derivatives and a non-trivial flavor structure contribute to $c_2$ already at 1-loop order, potentially competing in an important way with the aligning “force” determined by the low energy Yukawa couplings.) Therefore it is really the entire fundamental theory at the scale $\Lambda$, rather than just its low-energy sector (the MSSM plus any additional new physics below $\Lambda$), which sets the dynamics of the scalar mass orientations. Since our experimental knowledge is limited to the low-energy sector while our theoretical understanding of the fundamental theory is not sufficiently advanced to calculated $V_{\rm eff}$, we cannot proceed further without some strong assumptions. This is a serious and apparently inherent weakness of this approach to solving the flavor problem. However, it can also be viewed favorably as affording us a window into the fundamental theory at the scale $\Lambda$: sensitivity to such scales means that our predictions are not independent of this unknown realm, and therefore that they may be used to experimentally probe it. What, then, can we say about the ${\cal O}(\Lambda^2 \tilde m^2)$ terms? First, consider the radiative contribution of the MSSM modes. The MSSM would possess a global $\rm U(3)^5$ flavor symmetry if not only the dynamical fields $\tilde m_I^2$ would transform under this symmetry but also the Yukawa couplings would transform appropriately. Since the Yukawa couplings are in fact fixed parameters, they are the spurions which carry the information about $\rm U(3)^5$ breaking. Hence, to lowest order in these Yukawa couplings, the MSSM modes contribute $$\begin{aligned} V_{\rm eff}^{\rm MSSM} &=& {\Lambda^2 \over \left(16\pi^2\right)^2} \left[ c_Q {\rm Tr}\,\, \tilde m_Q^2 \left( K^{\dagger} \widehat Y_U \widehat Y_U^{\dagger} K + k_Q \widehat Y_D \widehat Y_D^{\dagger} \right) + \right.\nonumber\\ & & \phantom{\Lambda^2 \over \left(16\pi^2\right)^2}\,\,\, c_U {\rm Tr}\,\, \tilde m_U^2 \widehat Y_U^{\dagger} \widehat Y_U + c_D {\rm Tr}\,\, \tilde m_D^2 \widehat Y_D^{\dagger} \widehat Y_D + \label{eq:veffmssm}\\ & & \phantom{[\Lambda^2 \over \left(16\pi^2\right)^2}\,\,\left. c_L {\rm Tr}\,\, \tilde m_L^2 \widehat Y_E \widehat Y_E^{\dagger} + c_E {\rm Tr}\,\, \tilde m_E^2 \widehat Y_E^{\dagger} \widehat Y_E\right] \nonumber\end{aligned}$$ to the full effective potential. The $c_I$ and also $k_Q$ are numerical (scalar) coefficients which can only be calculated once the matching conditions are specified at the cutoff scale. Fortunately, we only need to assume that they do not vanish. We also expect $k_Q$ to be of order one; indeed in the low energy MSSM, $k_Q=1$ to lowest order in the Yukawa couplings and up to small hypercharge effects. Then these MSSM contributions align $\tilde m_U^2$ with the [*diagonal*]{} mass-squared matrix $\sim Y_U^{\dagger} \widehat Y_U$ of the up-type quarks, $\tilde m_D^2$ with the [*diagonal*]{} mass-squared matrix $\sim Y_D^{\dagger} \widehat Y_D$ of the down-type quarks, and $\tilde m_Q^2$ with the [*non-diagonal*]{} linear combination $K^{\dagger} \widehat Y_U \widehat Y_U^{\dagger} K + k_Q \widehat Y_D \widehat Y_D^{\dagger}$. In the leptonic sector, there is only one spurion, the diagonal Yukawa matrix $Y_E$, so both $\tilde m_L^2$ and $\tilde m_E^2$ align with it and become diagonal: individual lepton numbers are conserved. Operators with higher powers of the Yukawa couplings will not significantly change the minimum configuration of the sfermion masses. This observation, based upon direct inspection of such operators, is independent of any additional suppressions from powers of $1/16\pi^2$. One underlying reason is that the minimum configuration corresponds to points of enhanced flavor symmetry, thus off diagonal entries in the configuration will always be proportional to the appropriate CKM mixings violating those symmetries. Additional suppressions arise because the Yukawa coupling eigenvalues are hierarchical and mostly $\ll 1$. For example, the $i,j$ entry of the matrix to which $\tilde m^2_U$ aligns will be of order $(m_u)_i (m_u)_j \theta_{ij}$ (in order to account for the chiral quantum numbers of the $U$ multiplet); this matrix is close to being diagonal because when $i < j$, $(m_u)_i (m_u)_i \ll (m_u)_i (m_u)_j \ll (m_u)_j (m_u)_j$, and also $\theta_{ij} \ll 1$. Of course, the MSSM modes are but a small part of the theory at the cutoff scale, and we have already seen that it is this full fundamental theory which determined the alignment of the scalar masses. To make progress, therefore, we must make some strong assumption about the remaining physics. If it is completely arbitrary, the partial alignment which would result from the low-energy modes is generically destroyed. On the other hand, if the remaining physics is related to the Yukawa couplings, in the sense that it preserves the same approximate symmetries, then the alignment can be preserved. This will therefore be our working assumption: - the [*explicit*]{} (rather than spontaneous) flavor violations in the theory at the cutoff are [*entirely*]{} parametrized by the Yukawa couplings, treated as spurions. It follows from this [*minimality assumption*]{} that the complete effective action has the same form as Eq. (\[eq:veffmssm\]) but with different coefficients, $c_I \to c'_I$ and $k_Q \to k'_Q$. The effective action will be minimized when the soft masses are as closely aligned as possible with the Yukawa couplings, that is, when the approximate flavor symmetries are maximized. (The ground state of many physical systems is the state of enhanced symmetry, so our results may have quite a wide range of applicability.) The minimality assumption itself is very restrictive, and hence quite predictive. Deviations from our predictions would indicate that there are new flavor violations in the cutoff theory which would normally be almost inaccessible to experimental probes. We will make this assumption throughout this work, and derive some quantitative phenomenological consequences. It should be noted, however, that even if there are other sources of flavor violation which misalign the scalar masses, they are unlikely to cancel the ones we can calculate from our effective potential, and hence our predictions serve very generally as rough lower bounds on the expected experimental signals. Under the minimality assumption, the mass matrix of the squark doublets aligns with the linear combination $K^{\dagger} \widehat Y_U \widehat Y_U^{\dagger} K + k'_Q \widehat Y_D \widehat Y_D^{\dagger}$. If we assume $k'_Q \sim {\cal O}(1)$ as suggested by $V_{\rm eff}^{\rm MSSM}$, then the second term in the linear combination may be ignored relative to the first, and $m_Q^2$ aligns approximately with the up-type Yukawa couplings, that is, it is misaligned by the CKM matrix $K$ relative to the diagonal down-type quark masses. Thus the strength of FCNCs in the quark-squark sector is suppressed by the small off-diagonal matrix elements of $K$, and this is the extent to which the disorientation mechanism alleviates the flavor problem in this sector. (Recall that, unlike the orientations, the eigenvalues of the squark mass matrices are fixed parameters in our current non-plasticated discussion.) Disorientation sets the generic values of such quantities as $\epsilon_k$, $\epsilon'/\epsilon_K$ and $B\bar B$ and $D \bar D$ mixings at (or below) their experimental values or bounds. On the other hand for the $K_L- K_S$ mass difference it is less effective[@ref:dgt], requiring a further suppression $${\left(\tilde m_{Q2}^2 - \tilde m_{Q1}^2\right) \over \tilde m_Q^2} < 0.1 \left({\tilde m_Q \over 300\,\rm GeV}\right)\,. \label{eq:klks}$$ We see that the disorientation mechanism suffices for satisfying most phenomenological bounds, but that some other solution—such as an accidental approximate degeneracy, an approximate universality, or perhaps plastication [@ref:dgt]—is still needed, at least for $\Delta m_K$. (The usual bounds on supersymmetric parameters from $K_L - K_S$ mass difference are not greatly alleviated by disorientation simply because disorientation can only suppress flavor-changing neutral currents by a factor of the appropriate CKM angle, but a much greater suppression is mandated by the experimental value of $\Delta m_K$, and is provided in the Standard Model by the lightness of the charm quark. Other quantities, such as $\epsilon_K$, which are sufficiently suppressed in the standard model by small CKM mixing angles, are indeed similarly suppressed in a disoriented scenario.) What of the leptonic sector? It is well-known that for generic soft masses the bounds on FCNCs from the radiative $\mu$ decay process $\mu \to e \gamma$ are considerably stronger than the $K_L - K_S$ bounds. Under the minimality assumption, the alignment is essentially determined by the Yukawa couplings of the low-energy modes, which in the MSSM conserve individual lepton numbers. Therefore [@ref:dgt] all FCNCs in the lepton sector vanish. However, there is considerable evidence that individual lepton numbers are not in fact conserved in nature, and therefore that there is indeed low-energy physics beyond the MSSM. The main indication comes from the solar neutrino flux, whose observed deficit can be explained by resonant (MSW) neutrino oscillations [@ref:msw] favoring neutrino masses in the $10^{-3}$ eV range [@ref:neutmass]. Other, perhaps less compelling, indications come from the atmospheric neutrino problem and from the density fluctuations at large scales measured by COBE [@ref:cobe]. This latter observation can be more easily explained provided a substantial fraction of the dark matter is hot [@ref:fluct; @ref:hot]. With the MSSM particle content the only way to obtain that is to have some neutrino mass (presumably the $\nu_\tau$) in the eV range. Once we allow for flavor violation in the leptonic sector, then it is bound to show up in the slepton mass matrices. Almost all models for neutrino masses involve a Majorana mass matrix $M_N$ for the singlet (“right-handed”) neutrino states, which also interact with the lepton doublets through a new set of Yukawa couplings $Y_N = K_L^\dagger \widehat Y_N$; here we use the basis defined above in which $Y_E = \widehat Y_E$ is diagonal, so $K_L$ is the leptonic analog of the CKM matrix $K$. In the quark sector there is a natural choice for which up-type quark to group with a given down-type quark in a single generation: one defines the three generations so as to make the CKM matrix close to the identity. In the leptonic sector we will find it convenient to define $K_L$, in the above basis, as the matrix which brings $Y_N$ into a diagonal form with [*increasing*]{} diagonal entries. As we will see below, the slepton masses will be aligned with $K_L$, so to avoid excessive flavor-changing processes $K_L$ will have to be close to the identity up to a permutation matrix. For simplicity we will assume that this permutation matrix is the identity. The eigenvalues of $M_N$ are much larger than the weak scale, making the observed neutrinos mostly left-handed and very light via the see-saw mechanism. Leptonic flavor violation is parametrized by the misalignment matrix $K_L$ and by $M_N$. We will assume, in the spirit of our previous minimality assumption and in agreement with phenomenological expectations, that $M_N \ll \Lambda$ (where again $\Lambda$ is of order the Planck or string scale). Therefore the effective potential for the slepton masses will be of the form (to lowest order in the Yukawa couplings) $$\begin{aligned} V_{\rm eff}^{\rm leptonic} &=& {\Lambda^2 \over \left(16\pi^2\right)^2} \left[ c'_L {\rm Tr}\,\, \tilde m_L^2 \left( K_L^{\dagger} \widehat Y_N \widehat Y_N^{\dagger} K_L + k'_L \widehat Y_E \widehat Y_E^{\dagger} \right) + \right.\nonumber\\ & & \phantom{\Lambda^2 \over \left(16\pi^2\right)^2}\,\,\left. c'_N {\rm Tr}\,\, \tilde m_N^2 \widehat Y_N^{\dagger} \widehat Y_N + c'_E {\rm Tr}\,\, \tilde m_E^2 \widehat Y_E^{\dagger} \widehat Y_E \right]\,. \label{eq:vefflept}\end{aligned}$$ The SU(2)-singlet charged slepton masses align to lowest order with the mass-squared matrix of the charged leptons, and hence are diagonal in the basis we have chosen: $$\tilde m_E^2 \simeq \widehat m_E^2\,. \label{eq:mealign}$$ But the presence of the right-handed neutrinos at the cutoff scale, and therefore of the Yukawa couplings $Y_N$ as spurions violating leptonic flavor symmetries, is enough to misalign the SU(2)-doublet slepton mass matrices. The misalignment is frozen in at the scale $\Lambda$, and so it remains even after the right-handed modes are integrated out of the low-energy theory. The sensitivity to the theory at the cutoff scale implies an insensitivity to the details of the decoupling of the right-handed neutrinos and to the flavor violation in $M_N$. Moreover, as in the quark sector, we will assume that $k'_L \sim {\cal O}(1)$ and that the Yukawa couplings of the neutrinos are larger than those of the charged leptons: then $m_L^2$ aligns to a good precision with the Yukawa couplings of the right-handed neutrinos and therefore is misaligned by the leptonic CKM matrix $K_L$ relative to the charged-lepton masses, $$\tilde m_L^2 \simeq K_L^{\dagger} \widehat m_L^2 K_L\,, \label{eq:mlalign}$$ much as $\tilde m_Q^2$ was misaligned by the CKM matrix $K$ relative to the down-type quark masses. Consequently, the strength of leptonic FCNCs is sensitive only to the matrix $K_L$ and not to the overall unknown size of the neutrino Yukawa couplings. By measuring processes such as $\mu \to e \gamma$ (and $\mu\to 3e$) and $\tau \to \mu \gamma$, we can directly probe these leptonic mixing angles. We stress that the above result does not necessarily require Yukawa couplings $\sim 1$ in the neutrino sector—having $Y_N \roughly > Y_E$ suffices. Another potential source for flavor violations is the set of $A$-term matrices. As discussed in Ref. [@ref:dgt], various assumptions could be made about the $A$ terms: they are at once similar to the Yukawa couplings, since they couple “left”- and “right”-handed modes, and to the scalar masses, since they break supersymmetry softly. If the orientation as well as the eigenvalues of the $A$ terms were fixed at the cut-off scale (like the Yukawa couplings), then to satisfy phenomenological bounds they would need to either be negligibly small or closely aligned with the Yukawa couplings; this would also follow from our minimality hypothesis (which would require $A_i \propto Y_i$ or $A_i = 0$), and is also one component of the commonly-made universality assumption. (Notice that fixed $A$-term orientations allow the $U_I$ soft-masses orientations to be regarded as pseudo-Goldstone bosons of the flavor symmetries.) Alternatively, the $A$ terms may have fixed eigenvalues but dynamically-determined orientations (like the soft masses): $$A_i = V_i^{\dagger} \widehat A_i W_i\,,\qquad i = U,D,N,E. \label{eq:widef}$$ Without loss of generality, we may arrange the (diagonal) entries of $\hat A_i$ in ascending order. We will assume in particular that the $V_i$ and $W_i$ fields are independent of the $U_I$ fields. (If they were all regarded strictly as pseudo-Goldstone bosons of flavor symmetries then there would be fewer independent fields—and consequently insufficient freedom to align both the sfermion masses and the $A$ terms.) What of the fixed eigenvalues $\widehat A_i$? If they are too large, undesirable minima develop [@ref:dersav] in the full MSSM scalar potential which spontaneously break the electromagnetic gauge symmetry. The limiting values for $\widehat A_i$ (unless the sfermions are extremely heavy) are roughly the corresponding [*fermion*]{} masses: $$\widehat A_i \,\roughly{<}\, m_i\,, \qquad i = u,d,n,e. \label{eq:aibound}$$ Thus we must suppose—as is always done—that the $A$ terms have sufficiently small eigenvalues, at most comparable to the hierarchical masses of the charged leptons. If in fact the eigenvalues are comparable to the hierarchical charged lepton masses, then they must also be at least roughly aligned, rather than antialigned, with the corresponding leptons in order to satisfy Eq. (\[eq:aibound\]). We now turn to the issue of alignment, namely the expectation values of $V_i$ and $W_i$. The alignment of the $A_i$ is determined under the minimality assumption by a spurionic analysis similar to that for the sfermion masses. To proceed, we further postulate that the only parameters breaking the $\rm U(1)_R$ symmetry of the MSSM are the soft ones, namely $A_i$, the bilinear Higgs coupling $B$, and the gaugino mass parametrized by $M_{1/2}$, all of which transform in the same way under R. This symmetry will allow only terms involving $AA^\dagger$ and $A^\dagger M_{1/2}$ but not any $AA$ terms. Then the operators determining the alignment of $A_E$ are given to lowest order by $$V_{\rm eff}^{\rm A} \sim c^{\prime A}_{N} M_{1/2}^\dagger {\rm Tr}\,A_N Y_N^\dagger + c^{\prime A}_{E} M_{1/2}^\dagger {\rm Tr}\, A_E Y_E^\dagger + c^{\prime A}_{EN} {\rm Tr}\, A_E Y_E^\dagger Y_N A_N^\dagger + c^{\prime A}_{EE} {\rm Tr}\, A_E A_E^\dagger Y_N Y_N^\dagger + {\rm h.c.} \label{eq:veffa}$$ What alignment do these terms induce? The first operator, ${\rm Tr}\,A_N Y_N^\dagger$, will likely dominate the alignment of $A_N$. It can be positive or negative, and has its largest magnitude when $W_N \simeq \unity$ (the unit matrix) and $V_N \simeq K_L$; hence for $c^{\prime A}_{N}$ of any sign, the first term in the potential is always minimized when $A_N \simeq K_L^\dagger \widehat A_N$ (up to an overall sign). Similarly, the second operator is minimized when $V_E \simeq W_E \simeq \unity$ so it favors $A_E \simeq \widehat A_E$. Next, to analyze the third term, we recall that $V_N \simeq K_L$ and $W_N \simeq \unity$, that $\widehat A_N$ and $\widehat Y_N$ are arranged in increasing order, and that $\widehat A_E$ and $\widehat Y_E$ are strongly hierarchical (unless the former is negligibly small); then, up to terms of order $K_{Lij} m_i/m_j$ for $i < j$, the third term is minimized when $A_E \simeq \widehat K_L^\dagger A_E$. Finally, the last operator in this potential is positive semidefinite, and attains its largest magnitude when $V_E \simeq K_L$. So if the coefficient $c^{\prime A}_{EE}$ is nonnegligible, it must be negative in order to lead to approximate alignment (rather than antialignment) of the charged sleptons with the charged leptons. Therefore the leptonic $A$ term relevant at low energies has the form $$A_E \simeq \tilde V_E^\dagger \widehat A_E \label{eq:aealign}$$ where the fixed matrix $\tilde V_E$ is, up to phases, $\simeq \unity$ if the second operator in $V_{\rm eff}^{\rm A}$ dominates over the second and third, and otherwise has entries comparable to $K_L$. We comment below on the possible CP-violating effects of these phases. We should add that the above form for $A_E$ is valid at the cut-off scale $\Lambda$. RG evolution to low energies will add two types of terms to the cut-off expressions: one from the gauge sector and the other from the Yukawa sector. The gauge contribution to $A_E$ is a diagonal matrix proportional to $M_{1/2} Y_E$ (we will use an approximate proportionality constant of $-0.3$). The gauge contribution to the soft masses adds universal terms $\propto M_{1/2}^2 \unity$, but these will not change the form of the mass matrices. Note that, in accordance with low-energy measurements of the gauge couplings and with the MSSM RG equations, we have assumed that the three gauge couplings approximately unify near the cut-off scale and that the gauginos have a common mass $M_{1/2}$ at that scale. The Yukawa sector contributions were first partially analyzed in Ref. [@ref:bm] and were recently analyzed in detail in Ref. [@ref:hmty]. We will comment on these contributions briefly below. Before discussing the phenomenology let us recollect our assumptions. First, we assumed that the slepton mass matrix orientations were dynamical degrees of freedom fixed by a cut-off scale effective potential in which the flavor violations are entirely due to the Yukawa couplings (the minimality assumption), and second, when right handed neutrinos are added to the MSSM to account for neutrino masses, we assume that $Y_N$ are at least as large (in a matrix sense) as $Y_E$. The result is Eqs. (\[eq:mealign\]) and (\[eq:mlalign\]) for slepton masses: namely, the “left-handed” \[that is, SU(2)-doublet\] soft slepton masses $\tilde m_L^2$ are misaligned relative to the charged lepton masses by the leptonic CKM matrix $K_L$, while the “right-handed” soft slepton masses $\tilde m_E^2$ are closely aligned with the charged leptons. If the $A$ terms are not negligible, then we further assumed that their orientations are dynamical according to Eq. (\[eq:widef\]), that their eigenvalues satisfy Eq. (\[eq:aibound\]), that alignment rather than antialignment results from the effective potential, and that $K_L$ is close to the identity. The last three requirements simply allow the stability of the electroweak vacuum. The resulting $A$ terms are misaligned on their left-hand side by at most $\sim K_L$ and are aligned with the charged leptons on their right-hand side. With the low-energy soft SUSY-breaking parameters at hand, we may study the expected phenomenology and compare the results to current experimental bounds and to future experimental potential. We will concentrate on the radiative flavor-changing muon decay $\mu \to e \gamma$ since it furnishes perhaps the most sensitive probe of these parameters, and since we anticipate its sensitivity to be greatly improved in the near future. With the above form for the soft terms, the radiative decay is completely dominated by one helicity amplitude ${1\over2} {\cal A}_{R \to L} \bar e_L \sigma_{\mu\nu} F^{\mu\nu} \mu_R$, while the other helicity amplitude vanishes to lowest order in $m_e/m_\mu$. We have independently and fully computed the dominant helicity amplitude in the MSSM, neglecting terms of order $m_e/m_\mu$, and compared our results with previous calculations[^3] We assume for brevity that mixing between the first two generations dominates; the generalization to full three-generation mixing is straightforward. Our result for the branching ratio is $\br(\mu \to e \gamma) = {\displaystyle{\tau_\mu m_\mu^3\over 16 \pi}} \left|{\cal A}_{R \to L}\right|^2$, where ${\cal A}_{R \to L} = {\displaystyle{e\over 16\pi^2}} m_\mu K_L^{1\mu*} K_L^{1e} \left[{\cal A}_A + {\cal A}_B +{\cal A}_C +{\cal A}_D +{\cal A}_E\right]$, and $$\begin{aligned} {\cal A}_A &=& \half \sum_{i=1}^4 \left(g^{\prime 2} \left|U_{i1}\right|^2 + g_2^2 \left|U_{i2}\right|^2 + 2 g' g_2 \,{\rm Re}\, U_{i1} U_{i2}^*\right) \left[{f(M_{0i}^2/\tilde m_{L1}^2) \over \tilde m_{L1}^2} - (L1 \leftrightarrow L2) \right] \label{eq:ampa} \\ {\cal A}_B &=& -g_2^2 \left(c_+^2 \left[ {g(M_{+1}^2/\tilde m_{L1}^2) \over \tilde m_{L1}^2} - (L1 \leftrightarrow L2) \right] + s_+^2 \left[ {g(M_{+2}^2/\tilde m_{L1}^2) \over \tilde m_{L1}^2} - (L1 \leftrightarrow L2) \right]\right) \phantom{A} \label{ampb} \\ {\cal A}_C &=& {g_2^2\over v_D} \left(C_1 \left[ {j(M_{+1}^2/\tilde m_{L1}^2) \over \tilde m_{L1}^2} - (L1 \leftrightarrow L2) \right] + C_2 \left[ {j(M_{+2}^2/\tilde m_{L1}^2) \over \tilde m_{L1}^2} - (L1 \leftrightarrow L2) \right]\right) \phantom{A} \label{ampc} \\ {\cal A}_D &=& -{g_2\over \sqrt{2}v_D} \sum_{i=1}^4 U_{i3}^* U_{i2} M_{0i} \left[{h(M_{0i}^2/\tilde m_{L1}^2) \over \tilde m_{L1}^2} - (L1 \leftrightarrow L2) \right] \label{ampd} \\ {\cal A}_E &=& -\sum_{i=1}^4 \left( U_{i1}^2 g^{\prime 2} + U_{i1} U_{i2} g' g_2\right)^* M_{0i} \times \label{ampe}\\ & & \qquad \left\{K_- \left[ {1\over \tilde m_{L1}^2-m_{E2}^2} \left({h(M_{0i}^2/\tilde m_{L1}^2)\over\tilde m_{L1}^2} - {h(M_{0i}^2/\tilde m_{E2}^2)\over\tilde m_{E2}^2} \right) - (L1 \leftrightarrow L2) \right] +\right. \nonumber\\ & & \qquad \left.\phantom{\{}K_+ \left[ {1\over \tilde m_{L1}^2-m_{E2}^2} \left({h(M_{0i}^2/\tilde m_{L1}^2)\over\tilde m_{L1}^2} - {h(M_{0i}^2/\tilde m_{E2}^2)\over\tilde m_{E2}^2} \right) + (L1 \leftrightarrow L2) \right] \right\} \nonumber\\\end{aligned}$$ The overall factor $K_L^{1\mu*} K_L^{1e}$ is the off-diagonal mixing in the leptonic CKM matrix; $U$ is the matrix which diagonalizes the neutralino mass matrix $M_0$ via $$\begin{aligned} U M_0 U^\dagger &=& U \left(\begin{array}{cccc} M_1 & 0 & -g_1 v_D/\sqrt{2} & g_1 v_U/\sqrt{2} \\ 0 & M_2 & g_2 v_D/\sqrt{2} & -g_2 v_U/\sqrt{2} \\ -g_1 v_D/\sqrt{2} & g_2 v_D/\sqrt{2} & 0 & \mu \\ g_1 v_U/\sqrt{2} & -g_2 v_U/\sqrt{2} & \mu & 0 \end{array}\right) U^\dagger \label{eq:neutmass}\\ &=& \phantom{U} \left(\begin{array}{cccc} M_{01} & 0 & 0 & 0 \\ 0 & M_{02} & 0 & 0 \\ 0 & 0 & M_{03} & 0 \\ 0 & 0 & 0 & M_{04} \end{array}\right)\,;\end{aligned}$$ $\tilde m_{Li}$ and $\tilde m_{Ei}$ are the mass eigenvalues of the left-handed sleptons; $v_U$ and $v_D$ are the up- and down-type Higgs boson mass parameters, satisfying $v_U^2 + v_D^2 = v^2 = (174\,\rm GeV)^2$ and $v_U/v_D = \tan\beta$ (so the standard model fermions have Dirac masses $m_i = Y_i v_{U,D}$); the chargino mass matrix is diagonalized via $$\left(\begin{array}{cc} M_2 & g_2 v_D \\ g_2 v_U & -\mu \end{array}\right) = \left(\begin{array}{cc} c_+ & -s_+ \\ s_+ & c_+ \end{array}\right) \left(\begin{array}{cc} M_{+1} & 0 \\ 0 & M_{+2} \end{array}\right) \left(\begin{array}{cc} c_- & s_- \\ -s_- & c_- \end{array}\right)$$ from which we obtain the useful parameter combinations $C_1 = M_{+1} s_- c_+/g_2$ and $C_2 = -M_{+2} s_+ c_-/g_2$; we also use $K_- = \mu \tan\beta - 0.3 M_{1/2} + (\tilde A_{E2,1} + \tilde A_{E2,2})/2$ and $K_+ = (\tilde A_{E2,1} - \tilde A_{E2,2})/2 \equiv \tilde A_{12}$ in which we expect $\tilde A_{E2,j} \equiv (\hat A_{E2}/Y_\mu) (K_L \tilde V_E^\dagger)^{j\mu*}/K_L^{j\mu*}$ to be between zero and the SUSY-breaking scale, as discussed above; and the four loop functions are defined via $f(x) = (2 x^3 + 3 x^2 - 6 x + 1 - 6 x^2 \ln x)/[12 (1-x)^4]$, $g(x) = (x^3 -6 x^2 +3 x + 2 +6 x \ln x)/[12 (1-x)^4]$, $h(x) = (- x^2 + 1 +2 x \ln x)/[2 (1-x)^3]$, and $j(x) = (x^2 -4 x+ 3 +2 \ln x)/[2 (1-x)^3]$. Various contour plots of the calculated branching ratio for $\mu \to e\gamma$ are shown in Fig. 1. In all the plots we have used a slepton degeneracy $\Delta \tilde m^2_L \equiv \tilde m_{L2}^2 - \tilde m_{L1}^2 = 0.1 \tilde m_L^2$ and a leptonic mixing $\left|K_L^{1\mu*} K_L^{1e}\right| = 0.04$. The horizontal axis spans values of the $\mu$ parameter between $-500\,\GeV$ and $500\,\GeV$, while the vertical axis spans the same range of the approximately-unified gaugino mass $M_{1/2}$ at the cut-off scale. For various values of $\tan\beta$ and $\tilde m_L$, the figures show contours of constant branching ratio normalized to the current experimental upper bound of $\br_{\rm exp}=4.9\times 10^{-11}$: the black, dark gray, light gray, and white regions indicate $\br/\br_{\rm exp} < 0.1$, $0.1 < \br/\br_{\rm exp} < 1$, $1< \br/\br_{\rm exp} < 10$, and $10 < \br/\br_{\rm exp}$, respectively. Also shown, as cross-hatched regions, are those parameter ranges excluded by LEP bounds on the lightest chargino and neutralino masses. In the top row of plots, in which $\tan\beta = 2$ while $\tilde m_L$ varies between 100 GeV and 500 GeV, we have used $\tilde A_{E2,1} = 0$ (which would result from $\tilde V_E = K_L$) and $\tilde A_{E2,2} = 100\,\GeV$. Making $\tilde A_{E2,1} \sim \tilde A_{E2,2}$ would not change the results significantly. For the remaining two rows of the figure, in which $\tan\beta = 2$ or $\tan\beta = 5$, we have set $\tilde A_{E2,2} = 0$. Making $\tilde A_{E2,2} = 100\,\GeV$ does not make much difference when $\tan\beta = 5$ so we omit the corresponding figure. To properly interpret these contours for any mixing, their scaling behavior is needed: $${\rm Br}(\mu \to e \gamma)= 4.9\times 10^{-11} \left [{{[\Delta \tilde m^2_L+f M_0 \tilde A_{12}]/\tilde m_L^2}\over 0.1}\right ]^2\left [{|K_L^{1\mu*} K_L^{1e}|\over 0.04}\right ]^2 \left [{300\,\GeV\over \tilde m_L}\right ]^4 F \label{eq:brratio}$$ where $F$ and $f$ arise from loop functions. Under different assumptions about the mixing angles and degree of degeneracy, the allowed regions will correspond to different contours in our plots. The value of $F$ is $\sim 1$ when $\mu \sim M_{1/2} \sim \tilde m_L$, but can be one or two orders of magnitude larger when $\mu$ or $M_{1/2}$ are hierarchically lower than the slepton mass. Thus, while light sleptons result as expected in very large branching ratios (unless $\mu$ is accurately tuned to produce a cancellation), simply raising the slepton masses without raising $\mu$ and $M_{1/2}$ does not immediately lower the branching ratio: to quickly suppress the branching ratio, the entire SUSY-breaking scale must be raised, which necessitates fine-tuning the electroweak scale. The amplitude responsible for this behavior is the oft-neglected ${\cal A}_C$, which is never negligible and which depends logarithmically on this hierarchy, dominating the amplitude by a factor of $\sim 10$ when the sfermions are $\sim 3$ times as heavy as the charginos. There is also a significant enhancement in the branching ratio when $\tan\beta$ is large, as in many other processes which then require suppression to agree with experiment [@ref:us]. Finally, the $A$ term contribution are often significant, especially when the “left-handed” soft-breaking masses are very nearly degenerate $(\Delta \tilde m^2_L \ll \tilde m_L^2)$ The size of leptonic mixings we have inserted is consistent with that suggested by the MSW solution of the solar neutrino problem. Actually, the $K_L$ mixings and the neutrino mixings observed at low energies (via neutrino oscillations of various sorts) are in general only indirectly related, via the Majorana mass matrix $M_N$. However, they are essentially equal when the eigenvalues of $M_N$ are all of the same order while those of $Y_N$ are strongly hierarchical. With such mixings, Fig. 1 and Eq. (\[eq:brratio\]) indicate that we need significant degeneracy in the slepton masses and a somewhat high SUSY-breaking scale to suppress $\mu\to e\gamma$ below its experimental bound—in fact, roughly the same degeneracy and SUSY-breaking scale as were needed to satisfy the neutral kaon mixing constraints. Thus the lepton sector fares no better (and no worse) than the quark sector in a disoriented scenario when neutrinos have sizeable Yukawa couplings at the cut-off scale. Admittedly, we have the freedom in the leptonic sector to assume that [*for some unknown reason*]{} the charged and neutral leptons are very closely aligned at the cut-off scale, in which case the mixing needed for the MSW scenario must be provided by the Majorana mass matrix of the right-handed neutrinos. Or we could assume that the Yukawa couplings of the neutrinos are much smaller than those of the charged leptons, though this seems unlikely. Otherwise, the SUSY-breaking scale must be at least several hundred GeV, implying the usual fine-tuning problems for the Z boson mass, or the left-handed slepton mass eigenvalues must be made highly degenerate. If the leptonic mixings $|K_L^{1\mu*} K_L^{1e}|$ are larger, say $\sim \sqrt{m_e/m_\mu} \simeq 0.07$ or even $\sim \theta_c \simeq 0.2$, then the SUSY-breaking scale must be raised further or the sleptons be made more degenerate to accommodate the experimental bounds. These arguments will be greatly strengthened by the planned improvement in the experimental searches for $\mu\to e\gamma$. Current proposals call for sensitivity to branching ratios as low as $10^{-14}$ [@ref:cooper]. If the disoriented scenario is correct, we certainly expect that $\mu\to e\gamma$ will be observed in this next generation of experiments. By measuring that rate we would gain some direct information about the leptonic CKM matrix. Of course, the other parameters affecting the branching ratio must be measured as well, but they will probably be determined within the coming decade. The above predictions of the disoriented scenario should be contrasted with the effects of RG evolution in the universal scenario [@ref:bm; @ref:hmty]. In the latter flavor violating slepton masses vanish by fiat at the cut-off scale, but are induced at lower scales by the neutrino Yukawa couplings via RG evolution: the diagonal slepton masses $\tilde m_0^2$ are augmented by $\delta \tilde m_L^2=(3/8\pi^2) \tilde m_0^2 Y_N Y_N^\dagger \ln(\Lambda/M_N)$, leading to a branching ratio given by Eq. (\[eq:brratio\]) but with a mass splitting $\Delta \tilde m_L^2/\tilde m_L^2=(3/8\pi^2) Y_{\nu_\mu}^2 \ln(\Lambda/M_N)$. Thus, unless the neutrino Yukawa coupling is of order one, the slepton masses are highly degenerate and hence this rare $\mu$ decay is greatly suppressed. The disoriented scenario in effect allows $\Delta \tilde m_L^2$ to be a free observable parameter while keeping the degree of misalignment between leptons and sleptons small, namely $\simeq K_L$. What does the disoriented scenario predict when $\tilde m_{L1}^2 = \tilde m_{L2}^2$ for some reason, such as plastication [@ref:dgt]? The only contributions to $\mu\to e\gamma$ are those proportional to off-diagonal $A$ terms \[namely the $K_+ = (\tilde A_{E2,1} - \tilde A_{E2,2})/2$ term in ${\cal A}_E$\] and those involving the third family left-handed sleptons. When $\tilde A_{E2,1} - \tilde A_{E2,2} \simeq 100\,\GeV$ and the slepton mass is $\simeq 250 \,\GeV$ the resulting rate is just below the present bound; for fixed gaugino mass, the rate decreases with the eighth power of the slepton mass, so heavy sleptons would only allow detection at the next generation of experiments. To account for mixing with the third family, the relevant mixing angle $K_L^{3\mu*} K_L^{3e}$ should be substituted for $K_L^{1\mu*} K_L^{1e}$ in the above calculation. Assuming these have the same size as their quark sector counterpart, this contribution alone yields a branching ratio roughly an order of magnitude below the current bound even if the slepton masses are not degenerate and are $\sim 100\,\GeV$. Our discussion so far has assumed that the orientations of squarks and sleptons are independent dynamical degrees of freedom. In the context of a grand-unified theory, the larger symmetry would typically reduce the number of independent orientations. As discussed in Ref. [@ref:dgt], the result in a unified disoriented model is a sleptonic mixing angle of order the Cabibbo angle $\theta_c \simeq K^{1s*} K^{1d}$, or perhaps of order $\sqrt{m_e/m_\mu}$ in a more detailed and realistic model. Since the mixing angle is larger than the $0.04$ we used above, the branching ratio of $\mu\to e\gamma$ is also larger. As a consequence we expect that in a disoriented GUT scenario the superpartners are quite heavy or the charged sleptons of the first two generations are highly degenerate. Third-generation and $A$ term effects are then important, and may dominate if $\tilde m_{L1}^2 - \tilde m_{L2}^2$ is sufficiently small. In very simple disoriented GUT models the misalignment may be only in the “right-handed” sector, but generically it is present in both sectors. In fact, a disoriented GUT scenario and a conventional GUT (see Ref. [@ref:bh] for a detailed analysis) have similar predictions. They both differ qualitatively, however, from the disoriented non-unified scenario in an important way: the helicity of the amplitudes. As we have shown, in a non-unified disoriented scenario with only the MSSM fields plus right-handed neutrinos at high scales, the process is completely dominated by the single helicity amplitude $\mu_R\to e_L \gamma$. On the other hand, in a realistic unified theory of flavor we expect flavor violations of comparable order in both the left- and right-handed sectors, while the minimal (and unrealistic) SU(5) model produces only right-handed mixing. Therefore in any unified theory we expect an amplitude for $\mu_L\to e_R \gamma$ at least as large as $\mu_R\to e_L \gamma$. Fortunately, in the planned experiments the decaying muon is polarized, so if sufficiently many $\mu\to e\gamma$ are observed, the angular distribution of the emitted electrons would reveal the helicity of the amplitude. A pure $\mu_R\to e_L \gamma$ result would be difficult to understand in a generic unified theory (disoriented or otherwise), but would be expected in a disoriented scenario if $\rm SU(3)\times SU(2)\times U(1)$ is the gauge group up to the cut-off scale. Another related rare $\mu$ decay is $\mu \to 3e$. It was recently observed [@ref:hmty] that, in contrast to previous statements in the literature, the amplitude which dominates this branching ratio is not the box diagram but rather the (photon) penguin diagram. Indeed, while the box contribution would be several orders of magnitude below the experimental bound, the penguin diagram yields a branching ratio [@ref:hmty] $${\br(\mu\to 3e)\over \br(\mu\to e\gamma)} \simeq {\alpha\over8\pi} \left({16\over3}\ln {m_\mu\over2m_e} - {14\over9}\right) \simeq 0.36 {\br_{\rm exp.\,bound}^{\rm present}(\mu\to 3e) \over \br_{\rm exp.\,bound}^{\rm present}(\mu\to e\gamma)} \label{eq:muee}$$ which is comparable to the experimental bound when $\mu\to e\gamma$ is close to its experimental bound. At present, $\mu\to 3e$ yields slightly weaker constraints than $\mu\to e\gamma$; only if the precision of $\mu\to 3e$ experiments keeps pace with the planned improvements in $\mu\to e\gamma$ searches will the former process remain competitive. As pointed out in Ref. [@ref:hmty], the penguin diagram is enhanced by $\ln(m_\mu/2m_e)$, which results from phase space integration as an electron and positron become collinear. (The coefficient of the log is just determined by the QED $\beta$ function by requiring the cancellation of the infrared divergences in the inclusive rate to order $\alpha$.) We should remark, however, that the experimental resolution may not allow highly collinear $e^+e^-$ pairs to be distinguished from other processes (including $\mu\to e\gamma$!), so the the denominator in the log should be replaced by the appropriate minimum resolvable energy. We have assumed throughout our discussion that the effective potential term $c_2 \tilde m^2 \Lambda^2$, which appears as a quadratic divergences in the low-energy theory, dominates and fixes the dynamics which aligns the soft masses. Such quadratic divergences are ubiquitous even in supersymmetric theories, when the supersymmetry is softly broken: while scalar masses are protected from quadratic divergences, the vacuum energy is not. Could $c_2$ vanish in a particular theory? Without a symmetry argument, assuming $c_2 = 0$ is akin to assuming the Higgs is light in a non-supersymmetric theory. Nevertheless, there have been studies where the vanishing of $c_2$ was invoked in order to proceed to a dynamical determination of the effective low energy parameters [@ref:kpz; @ref:kprz], and in particular of the gravitino mass itself. (Notice that not only $c_2$ but also any terms $c_2'$ arising from intermediate scales must vanish, presumably by the same mechanism.) While the implications of this idea are interesting, it is not clear yet how to implement it in an explicit field-theoretic model. As a matter of fact, to date the only available examples [@ref:fkz] satisfy $c_2 = 0$ only at 1-loop order (see [@ref:bpr] for an explicit example of its violation at 2-loop order). Moreover, if such an implementation were found, making $V_{\rm eff}={\cal O}(\tilde m^4)$, there would still be two contributions: one from the MSSM modes, and one which remains as a boundary term from matching the low- and high-energy theories. While the first contribution is determined to lowest order by the 1-loop RG evolution of the Higgs mass and of the cosmological constant, and is $\sim \tilde m^4 \log (\Lambda/\tilde m)/(4 \pi)^2$, the second contribution $\sim \xi_0 \tilde m^4$ is in principle unknown. The first contribution yields a phenomenology similar to the one we have studied throughout most of this paper. The second can only be controlled by making our minimality assumption (or some equivalently strong assumption)—and then, again, similar predictions would be made. We do not know the relative size of the two contributions. If we were to treat $\xi_0$ as a usual threshold correction arising from 1-loop field-theoretic diagrams, we would expect it to be $\xi_0 \sim 1/16\pi^2$ and hence subdominant in the limit $\log (\Lambda/\tilde m)\gg 1$, thus weakening the dependence on unknown cut-off physics. On the other hand, the MSSM contributions are suppressed by small Yukawa couplings. In the end, we must plead at least as much ignorance about the $c_2 = 0$ case as about the $c_2 \not = 0$ case, and so the minimality assumption is unavoidable. Finally, we comment briefly on CP violation. In Ref. [@ref:dt] the issue of dynamical CP phases was discussed in the context of the MSSM with universal soft terms. It was shown that, when the phases of $A$, $M_{1/2}$, $\mu$ and $B\mu$ are promoted to dynamical variables, the only CP-violating effects have a CKM origin, [*i.e.*]{} arise from the Jarlskog invariant $J$, and thus are suppressed. This was under the strong assumption that no new sources of explicit CP violation are present. The same conclusion can be reached in the non-universal case discussed here, under the parallel assumption that the coefficients in the effective potential are real. To understand this observation, consider the limit in which the quark and lepton Jarlskog invariants $J_{q,\ell}$ vanish, and choose a flavor basis in which the Yukawa matrices are real: in this basis also the soft terms relax to real matrices. In Ref. [@ref:dt], the invariant $J$ enters the effective potential at higher-loop order, so its effect on the CP-violating phases is further suppressed. In contrast, in a non-universal scenario, CP violation is already present in Eqs. (\[eq:mlalign\]) and (\[eq:aealign\]), and has no further loop suppressions. But as long as $J_\ell$ is not much larger than its quark counterpart $J_q$, no additional loop suppressions are needed to ensure that CP-violating quantities such as electric dipole moments are sufficiently small. We would like to acknowledge useful and stimulating discussions of various aspects of this work with M. Cooper, S. Dimopoulos, Y. Nir, S. Thomas and F. Zwirner. Figure Captions {#figure-captions .unnumbered} =============== Fig. 1: : Contours of constant branching ratio for $\mu\to e\gamma$ as functions of the $\mu$ parameter and the unified gaugino mass $M_{1/2}$ (approximately the wino mass) for various values of the slepton mass $\tilde m_L$, $\tan\beta$ and the $A$ parameter (as defined in the text). The black, dark gray, light gray, and white regions indicate $\br/\br_{\rm exp} < 0.1$, $0.1 < \br/\br_{\rm exp} < 1$, $1< \br/\br_{\rm exp} < 10$, and $10 < \br/\br_{\rm exp}$, respectively. 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Randall, EFI-95-21, May 1995, hep-ph/9505244. S. Dimopoulos and S. Thomas, SLAC-PUB-95-7010. [^1]: E-mail: rattazzi@physics.rutgers.edu [^2]: E-mail: sarid@particle.phys.nd.edu [^3]: Much of the existing literature omits parts of the amplitude, and in particular the important contribution ${\cal A}_C$ arising from chargino propagation with a mass insertion on the internal higgsino-wino line. In comparing our results with two of the complete calculations, we found one minor sign and normalization discrepancy with Ref. [@ref:sutter] and agreement with Ref. [@ref:hmty].

--- abstract: 'This paper investigates estimating the variance of a temporal-difference learning agent’s update target. Most reinforcement learning methods use an estimate of the value function, which captures how good it is for the agent to be in a particular state and is mathematically expressed as the expected sum of discounted future rewards (called the return). These values can be straightforwardly estimated by averaging batches of returns using Monte Carlo methods. However, if we wish to update the agent’s value estimates during learning–before terminal outcomes are observed–we must use a different estimation target called the $\lambda$-return, which truncates the return with the agent’s own estimate of the value function. Temporal difference learning methods estimate the expected $\lambda$-return for each state, allowing these methods to update online and incrementally, and in most cases achieve better generalization error and faster learning than Monte Carlo methods. Naturally one could attempt to estimate higher-order moments of the $\lambda$-return. This paper is about estimating the variance of the $\lambda$-return. Prior work has shown that given estimates of the variance of the $\lambda$-return, learning systems can be constructed to (1) mitigate risk in action selection, and (2) automatically adapt the parameters of the learning process itself to improve performance. Unfortunately, existing methods for estimating the variance of the $\lambda$-return are complex and not well understood empirically. We contribute a method for estimating the variance of the $\lambda$-return directly using policy evaluation methods from reinforcement learning. Our approach is significantly simpler than prior methods that independently estimate the second moment of the $\lambda$-return. Empirically our new approach behaves at least as well as existing approaches, but is generally more robust.' bibliography: - 'main.bib' title: 'Directly Estimating the Variance of the $\lambda$-Return Using Temporal-Difference Methods' --- Introduction ============ ![Each TD node takes as input a reward ${R}$, a discounting function $\gamma$, and features $\phi$. For the direct method (**top**) the squared TD error of the first-stage value estimator is used as the [meta-reward]{} for the second-stage [$V$]{} estimator. For VTD (**bottom**), a more complex computation is used for the [meta-reward]{} and an extra stage of computation is required. []{data-label="fig:networks"}](combined_graphic.pdf){width="\linewidth"} Conventionally in reinforcement learning, the agent estimates the expected value of the return—the discounted sum of future rewards, as an intermediate step to find an optimal policy. Given a trajectory of experience, the agent can average the returns observed from each state. To estimate the value function online—while the trajectory unfolds—we update the agent’s value estimates towards the expected $\lambda$-return. The $\lambda$-return has the same expected value as the return, but can be estimated online using a memory trace. Algorithms that estimate the expected value of the $\lambda$-return are called temporal-difference learning methods. The first moment, however, is not the only statistic that can be estimated. In addition to the expected value, we could estimate the variance of the $\lambda$-return. An estimate of the variance of the $\lambda$-return can be used in several ways to improve estimation and decision-making. @Sato2002 [@Ghavamzadeh; @Tamar2012; @Tamar2013b] use an estimate of the variance of the $\lambda$-return to design algorithms that account for risk in decision making. Specifically they formulate the agent’s objective as maximizing reward, while minimizing the variance of the $\lambda$-return. @White2016b estimated the variance of the $\lambda$-return, [$V$]{}, to automatically adapt the trace-decay parameter, $\lambda$, used in learning updates. This resulted in faster learning for the agent, but more importantly removed the need to tune $\lambda$ by hand. The variance [$V$]{} can be estimated directly or indirectly. Indirect estimation involves estimating the first moment (the value ${J}$) and second moment ($M$) of the return and taking their difference as: ${V}(s)=M(s)-{J}(s)^2$. @Sobel1982 were the first to formulate a Bellman operators for $M$. Later @Tamar2016 [@Tamar2013b; @Ghavamzadeh], extended @Sobel1982’s approach to estimating the variance for $\lambda = 0$ to $\lambda =1$. Finally, @White2016b introduced an estimation method called VTD, that supports off-policy learning [@Sutton2009; @Maei2011], state-dependent discounts and state-dependent trace-decay parameters. An alternative approach is to estimate the variance of the $\lambda$-return [$V$]{} directly. This has been considered by @Tamar2012, but they were unable to derive a Bellman operator—instead giving a Bellman-like operator—and considered only cost-to-go problems. In this paper, we show that one can use temporal-difference learning, a online method for estimating value functions [@Sutton1988], to estimate [$V$]{} directly. Our new method supports off-policy learning, state-dependent discounts, and state-dependent trace-decay parameters. We introduce a new Bellman operator for the variance of the $\lambda$-return, and further prove that even for a value function that does not satisfy the Bellman operator for the expected $\lambda$-return, the error in this recursive formulation is proportional to the error in the value function approximation. Interestingly, the Bellman operator for the second moment requires an unbiased estimate of the $\lambda$-return [@White2016b]; our Bellman operator for the variance avoids this term, and so has a simpler update. Both our direct method and VTD can be seen as a network of two TD estimators running sequentially (Figure \[fig:networks\]). Our goal is to understand the empirical properties of the direct and indirect approaches for estimating variance, as neither have yet been thoroughly studied. In general, we found that direct estimation is just as good as VTD, and in many cases better. Specifically, we observe that the direct approach is better behaved in the early stages of learning before the value function has converged. Further, we observe that the variance of the [$V$]{} estimates can be higher for VTD under several circumstances: (1) when there is a mismatch in step-size between the value estimator and the [$V$]{} estimator, (2) when traces are used with the value estimator, (3) when estimating [$V$]{} of the off-policy return, and (4) when there is error in the value estimate. Overall, we conclude that the direct approach to estimating [$V$]{} is both simpler and better behaved than VTD. The MDP Setting =============== We model the agent’s interaction with the environment as a finite Markov decision process (MDP) consisting of a finite set of states ${\mathcal{S}}$, a finite set of actions, ${\mathcal{A}}$, and a transition model $p: {\mathcal{S}}\times {\mathcal{S}}\times {\mathcal{A}}\rightarrow [0,1]$ defining the probability $p(s'|s, a)$ of transition from state $s$ to $s'$ when taking action $a$. In the policy evaluation setting considered in this paper, the agent follows a fixed policy $\pi(a|s)\in [0,1]$ that provides the probability of taking action $a$ in state $s$. At each timestep the agent receives a random reward $R_{t+1}$, dependent only on $S_t, A_t, S_{t+1}$. The return is the discounted sum of future rewards $$\begin{aligned} \label{eq:MCreturn} G_{t}&={R}_{t+1} + \gamma_{t+1}{R}_{t+2} + \gamma_{t+1}\gamma_{t+2}{R}_{t+3} + \ldots \\ &={R}_{t+1} + \gamma_{t+1}G_{t+1}. \end{aligned}$$ The discount function $\gamma: {\mathcal{S}}\rightarrow [0,1]$, with $\gamma_{t}\equiv\gamma(S_t)$, provides a variable level of discounting depending on the state [@Sutton2011]. The value of a state, ${j}(s)$, is defined as the expected return from state $s$ under a particular policy $\pi$ $$\begin{aligned} {j}(s)=&{\mathbb{E}}_{\pi}[G_t|S_t=s]. \label{eq:value}\end{aligned}$$ We use ${j}$ to indicate the true value function and ${J}$ the estimate. The TD-error is the difference between the one-step approximation and the current estimate: $$\begin{aligned} \delta_t={R}_{t+1}+\gamma_{t+1}{J}_t(S_{t+1})-{J}_t(S_t). \label{eq:TDerr}\end{aligned}$$ The $\lambda$-return $$G_t^{\lambda}={R}_{t+1}+\gamma_{t+1}(1-\lambda_{t+1}){J}_t(S_{t+1}) +\gamma_{t+1}\lambda_{t+1} G^{\lambda}_{t+1} $$ provides a bias-variance trade-off by incorporating ${J}$, which is a potentially lower-variance but biased estimate of the return. This trade-off is determined by a state-dependent trace-decay parameter, $\lambda_t\equiv\lambda(S_t) \in [0,1]$. When ${J}_t(S_{t+1})$ is equal to the expected return from $S_{t+1}=s$, then ${\mathbb{E}}_{\pi}[(1-\lambda_{t+1}){J}_t(S_{t+1}) +\gamma_{t+1}\lambda_{t+1} G^{\lambda}_{t+1}|S_{t+1}=s] = {\mathbb{E}}_{\pi}[G^{\lambda}_{t+1}|S_{t+1}=s]$, and so the $\lambda$-return is unbiased. Beneficially, however, the expected value ${J}_t(S_{t+1})$ is lower-variance than the sample $G^{\lambda}_{t+1}$. If ${J}_t$ is inaccurate, however, some bias is introduced. Therefore, when $\lambda=0$, the $\lambda$-return is lower-variance but can be biased. When $\lambda=1$, the $\lambda$-return equals the Monte Carlo return (Equation ); in this case, the update target exhibits more variance, but no bias. In the tabular setting evaluated in this paper, $\lambda$ does not affect the fixed point solution of the value estimate, only the rate at which learning occurs. It does, however, affect the observed variance of the return, which we estimate. The $\lambda$-return is implemented using traces as in the following [TD($\lambda$) ]{}algorithm, shown with accumulating traces: $$\begin{aligned} {E}_t(s)&\leftarrow\begin{cases} \gamma_t\lambda_t {E}_{t-1}(s) + 1 & s=S_t \\ \gamma_t\lambda_t {E}_{t-1}(s) & \forall s \in {\mathcal{S}}, s \ne S_t \end{cases}\\ {J}_{t+1}(S_t)&\leftarrow {J}_t(S_t) + \alpha\delta_t {E}_t(S_t) \end{aligned} \label{eq:td_replacing_traces}$$ Estimating the Variance of the Return {#sec:derivation} ===================================== When estimating [$V$]{}, we have both a value estimator and a variance estimator. The *value estimator* provides an estimate of the expected $\lambda$-return, known as the policy evaluation problem. The *variance estimator* provides an estimate of the variance of the $\lambda$-return. We show below how we can similarly use any TD method to learn the variance estimator, such as TD with accumulating traces (Equation \[eq:td\_replacing\_traces\]). Because we have two separate TD estimators—one each for [$J$]{} and [$V$]{} — they can select different trace-decay parameters for learning. In fact, as done by @White2016b, the value estimator can use a different trace-decay parameter than the $\lambda$-return for which we are estimating [$V$]{}. This is because the $\lambda$-return is defined for any given value function, regardless of how that value function is estimated. There are three possible trace-decay parameters: 1) the $\lambda$ of the $\lambda$-return for which variance is being estimated, 2) that used by the traces of the value estimator (${\kappa}$), 3) that used by the traces of the variance estimator (${\bar{\kappa}}$). We summarize the notation here for easy reference. Variables without the bar refer to the value estimator and variables with bars refer to the variance estimator. $$\begin{aligned} {j}-&\text{true value function of the target policy $\pi$.}\\ {J}-&\text{estimate of ${j}$.}\\ {R}-&\text{reward used in the value function estimate.}\\ \bar{{R}}-&\text{{meta-reward}{} used in the variance estimate.}\\ {\lambda}-&\text{bias-variance parameter of the target $\lambda$-return.}\\ {\kappa}-&\text{trace-decay parameter of the value estimator.}\\ {\bar{\kappa}}-&\text{trace-decay parameter of the secondary estimator.}\\ \gamma -&\text{discounting function used by the value estimator.}\\ \bar{\gamma} -&\text{discounting function used by the variance estimator.}\\ \delta_t -&\text{TD error of the value function at time $t$.}\\ \bar{\delta_t} -&\text{TD error of the variance estimator at time $t$.}\\ {M}-&\text{estimate of the second moment.}\\ {v}-&\text{true variance of the return.}\\ {V}-&\text{estimate of ${v}$.}\end{aligned}$$ Our direct algorithm, shown here, uses TD(0) to estimate variance. For an expanded implementation with traces in the off-policy setting see Appendix \[appendix:general\].\ \ **Direct Variance Algorithm** $$\begin{aligned} \bar{\gamma}_{t+1}& \leftarrow \gamma_{t+1}^2 {\lambda}_{t+1}^2\\ \bar{{R}}_{t+1}& \leftarrow \delta_t^2\\ \bar{\delta}_t& \leftarrow \bar{{R}}_{t+1} + \bar{\gamma}_{t+1}{V}_t(s') - {V}_t(s)\\ {V}_{t+1}(s)&\leftarrow {V}_t(s)+\bar{\alpha} \bar{\delta}_t \end{aligned} \label{alg:direct}$$ An alternative to this direct method is to instead estimate the second moment. The variant shown here is equivalent to on-policy VTD with no traces, ${\bar{\kappa}}=0$, and the step-size for the second set of weights set to 0. Further, the Tamar TD(0) algorithm ([@Tamar2016]) can be recovered from Equation \[alg:comparison\] by using ${\kappa}=0, {\lambda}=1, {\bar{\kappa}}=0$. This algorithm does not impose that the variance be non-negative.\ \ **Second Moment Algorithm (VTD)** $$\begin{aligned} \bar{\gamma}_{t+1}\leftarrow&\gamma_{t+1}^2 {\lambda}_{t+1}^2 \nonumber\\ \bar{{R}}_{t+1}\leftarrow&({R}_{t+1}+\gamma_{t+1}{J}_{t+1}(s'))^2- \bar{\gamma}_{t+1}{J}_{t+1}(s')^2 \nonumber\\ \bar{\delta}_t\leftarrow&\bar{{R}}_{t+1} + \bar{\gamma}_{t+1}{M}_t(s') - {M}_t(s) \label{alg:comparison} \\ {M}_{t+1}(s)\leftarrow&{M}_t(s)+\bar{\alpha} \bar{\delta}_t \nonumber\\ {V}_{t+1}(s) =& {M}_{t+1}(s)-{J}_{t+1}(s)^2 \nonumber\end{aligned}$$ Derivation of the Direct Method =============================== The derivation of the direct method follows from characterizing the Bellman operator for the variance of the $\lambda$-return. Theorem \[recurse\_var\] gives a Bellman equation for the variance ${v}$. It has precisely the form of a TD target with [meta-reward]{} $\bar{\delta}_t=\delta_t^2$ and discounting function $\bar{\gamma}_{t+1}=\gamma_{t+1}^2\lambda_{t+1}^2$. Therefore, we can conveniently estimate [$V$]{} using TD methods. Further, we show that even when the value function does not satisfy the Bellman equation, this results only in a proportional error in the variance estimator. We first show the result for the on-policy setting, for simplicity; the more general off-policy algorithm is provided in Appendix \[appendix:general\] This result provides the first general Bellman operator directly for the variance. The Bellman operators for the variance are general, in that they allow for either the episodic or continuing setting, by using variable $\gamma$. Interestingly, by directly estimating variance, we avoid a second term in the cumulant, that is present in approaches that estimate the second moment [@Tamar2013b; @Tamar2016; @White2016b]. While @Tamar2012 also developed an approach to directly estimate the variance, their method defined a non-linear Bellman operator and is restricted to cost-to-go problems. Follow-up work moved to estimating the second-moment instead [@Tamar2013b; @Tamar2016], but with simplifying assumptions that only considered expected reward from a state and assuming $\lambda = 1$. The work developing VTD generalizes to any $\lambda$, but does not characterize error when using an inaccurate value function. To have a well-defined solution to the fixed point, we need the discount to be less than one for some transition [@White2017; @Yu2015]. This corresponds to assuming that the policy is proper, for the cost-to-go setting [@Tamar2016]. The policy reaches a state $s$ where $\gamma(s) < 1$ in a finite number of steps. For any $s \in {\mathcal{S}}$, \[recurse\_var\] $$\begin{aligned} {j}(s)&={\mathbb{E}}\Big[ R_{t+1} + \gamma_{t+1} {J}(S_{t+1})\ | \ S_t=s\Big] \nonumber\\ {v}(s) &={\mathbb{E}}\Big[\delta_t^2+\gamma_{t+1}^2\lambda_{t+1}^2{v}(S_{t+1})\ | \ S_t=s\Big] \label{eq_bellman_var} $$ Further, for approximate value function ${J}$, if there is an $\epsilon: {\mathcal{S}}\rightarrow [0,\infty)$ bounding value estimates $({J}(s) - {j}(s))^2 \le \epsilon(s)$ and covariance terms $|{{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} ({j}(S_{t+1}) - {J}(S_{t+1})) |S_t = s \right]} | \le \epsilon(s)$, then $$\begin{aligned} \left|{V}(s) - {\mathbb{E}}\Big[\delta_t^2+\gamma_{t+1}^2\lambda_{t+1}^2{V}(S_{t+1})\ | \ S_t=s\Big] \right|\le 3 \epsilon(s)\end{aligned}$$ First we expand $G_{t}^{\lambda} - {j}(S_{t})$, from which we recover a series with the form of a return. $$\begin{aligned} G_{t}^{\lambda} - {j}(S_{t}) &= R_{t+1} + \gamma_{t+1} ( 1- \lambda_{t+1} ) {j}(S_{t+1}) - {j}(S_{t}) + \gamma_{t+1} \lambda_{t+1} (G_{t+1}^{\lambda} - {j}(S_{t+1})) \nonumber\\ &= R_{t+1} + \gamma_{t+1} {j}(S_{t+1}) - {j}(S_{t}) + \gamma_{t+1} \lambda_{t+1} (G_{t+1}^{\lambda} - {j}(S_{t+1})) \label{eq:delta-expansion} $$ The variance of $G_{t}^{\lambda}$ is therefore $$\begin{aligned} {v}(s) &= {{\mathbb{E}}\left[ \left(G_{t}^{\lambda}-{{\mathbb{E}}\left[ G_{t}^{\lambda}|S_t=s \right]}\right)^2|S_t=s \right]} \nonumber\\ &= {{\mathbb{E}}\left[ (G_{t}^{\lambda} - {j}(s))^2|S_t=s \right]} \label{eq_recursive_var}\\ &= {{\mathbb{E}}\left[ \Big(\delta_{t} + \gamma_{t+1}\lambda_{t+1}(G_{t+1}^{\lambda} - {j}(S_{t+1}))\Big)^2 |S_t=s \right]} \nonumber\\ &= {{\mathbb{E}}\left[ \delta_{t}^{2} | S_t = s \right]} + {{\mathbb{E}}\left[ \gamma_{t+1}^{2} \lambda_{t+1}^{2} (G_{t+1}^{\lambda} - {j}(S_{t+1}))^2 | S_t=s \right]} + 2 {{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} (G_{t+1}^{\lambda} - {j}(S_{t+1})) |S_t = s \right]}\nonumber\end{aligned}$$ Equation follows from Lemma \[delta\_var\] in the appendix, showing ${{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} (G_{t+1}^{\lambda} - {j}(S_{t+1})) |S_t = s \right]} = 0$.\ Now consider the case where we estimate the variance of the $\lambda$-return ${V}$ of an approximate value function ${J}$. $$\begin{aligned} {V}(s) &= {{\mathbb{E}}\left[ (G_{t}^{\lambda} - {j}(s) + {J}(s) - {J}(s) )^2|S_t=s \right]}\\ &= {{\mathbb{E}}\left[ (G_{t}^{\lambda} - {J}(s))^2 | S_t=s \right]} + ({J}(s) - {j}(s))^2 +2{{\mathbb{E}}\left[ G_{t}^{\lambda} - {J}(s)|S_t=s \right]} ({J}(s) - {j}(s)) . \end{aligned}$$ This last term simplifies to $$\begin{aligned} \!\!\!{{\mathbb{E}}\left[ G_{t}^{\lambda} \!-\! {J}(s)|S_t\!=\!s \right]} \!&= \! {{\mathbb{E}}\left[ G_{t}^{\lambda} \!-\! {j}(s)|S_t\!=\!s \right]} \!+\! {j}(s) - {J}(s)\\ &= {j}(s) - {J}(s)\end{aligned}$$ giving $ ({J}(s) - {j}(s))^2 + 2 ({j}(s) - {J}(s))({J}(s) - {j}(s)) = -({J}(s) - {j}(s))^2$. We can use the same recursive form, therefore, as , giving $$\begin{aligned} {V}(s) &= {{\mathbb{E}}\left[ \delta_t^2 + \gamma_{t+1}^{2} \lambda_{t+1}^{2}{V}(S_{t+1}) | S_t = s \right]} + 2 {{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} (G_{t+1}^{\lambda} - {J}(S_{t+1})) |S_t = s \right]} - ({J}(s) - {j}(s))^2\end{aligned}$$ For the second term, $$\begin{aligned} \bigg|{{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} (G_{t+1}^{\lambda} - {J}(S_{t+1})) |S_t = s \right]}\bigg| = &\bigg|{{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} (G_{t+1}^{\lambda} - {j}(S_{t+1})) |S_t = s \right]} \\ &+ {{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} ({j}(S_{t+1}) - {J}(S_{t+1})) |S_t = s \right]}\bigg|\\ = &\bigg|{{\mathbb{E}}\left[ \gamma_{t+1} \lambda_{t+1} \delta_{t} ({j}(S_{t+1}) - {J}(S_{t+1})) |S_t = s \right]} \bigg|\\ \le &\epsilon(s) .\end{aligned}$$ where the second equality follows from Lemma \[delta\_var\] and the last step from the assumption about bounded covariance terms. Therefore, $$\begin{aligned} \bigg|{V}(s) &- {{\mathbb{E}}\left[ \delta_t^2 + \gamma_{t+1}^{2} \lambda_{t+1}^{2}{V}(S_{t+1}) | S_t = s \right]}\bigg| \le 2 \epsilon(s) + ({J}(s) - {j}(s))^2 \le 3 \epsilon(s)\end{aligned}$$ Experiments =========== The primary purpose of these experiments is to demonstrate that both the [direct]{} method and VTD can approximate the true expected [$V$]{} under various conditions in the tabular setting. We consider two domains. The first is a deterministic chain, in Figure \[fig:chain\], which is useful for basic evaluation and gives results which are easy to interpret. The second is a more complex MDP, in Figure \[fig:complex\_mdp\], with different discount and trace-decay parameters in each state. For all experiments Algorithm \[eq:td\_replacing\_traces\] is used as the value estimator. Unless otherwise stated, traces are not used (${\kappa}={\bar{\kappa}}=0$). For each experimental setting 30 separate experiments were run and the estimates averaged, with standard deviation shown as shaded regions in the plots. The true values were determined by Monte Carlo estimation and are shown as dashed lines in the figures. Unless otherwise stated, the estimates are all initialized to zero. We look at the effects of relative step-size between the value estimator and the variance estimators in Section \[sec:exp:alpha\]. In Section \[sec:exp:state-dependent\] we use the complex domain to show that both algorithms can estimate the variance with state-dependent $\gamma$ and ${\lambda}$. In Section \[sec:exp:err\] we evaluate the two algorithms’ responses to errors in the value estimate. Section \[sec:exp:traces\] looks at the effect of using traces in the estimation method. Finally, in Section \[sec:exp:off-policy\] we examine the off-policy setting. ![Chain MDP with 4 non-terminal states and 1 terminal state. From each non-terminal state there is only a single action with a deterministic transition to the next state to the right. On each transition rewards are drawn from a normal distribution with mean and variance of 1.0. Evaluation was performed for ${\lambda}=0.9$, which was chosen because it is not at either extreme and because 0.9 is a commonly used value for many RL experimental domains.[]{data-label="fig:chain"}](experiments/chain_state_diagram.pdf){width="\linewidth"} ![Complex MDP, with state-based values for $\gamma$ and ${\lambda}$ and a stochastic policy. The state-dependent values of $\gamma$ and ${\lambda}$ are chosen to provide a range of values, with at least one state acting as a terminal state where $\gamma=0$. On-policy action probabilities are indicated by $\mu$ and off-policy ones by $\pi$.[]{data-label="fig:complex_mdp"}](complex_offpolicy_1/variance-complex-offpolicy.pdf){height="3in"} The Effect of Step-size {#sec:exp:alpha} ----------------------- We use the chain MDP to investigate the impact of step-size choice. In Figure \[fig:chain\_l1.0\_same\] all step-sizes are the same $(\alpha=\bar{\alpha}=0.001)$. Both algorithms behave similarly. For Figure \[fig:chain\_l1.0\_less\] the step-size of the value estimate, $(\alpha=0.01)$, is greater than that of the variance estimators, $(\bar{\alpha}=0.001)$. The [direct]{} algorithm smoothly approaches the correct value, while VTD first dips well below zero. This is to be expected as the estimates are initialized to zero and the variance is calculated as ${V}(s)={M}(s)-{J}(s)^2$. If the second moment lags behind the value estimate then the variance will be negative. In Figure \[fig:chain\_l1.0\_more\] the step-size for the variance estimators is larger than for the value estimator $(0.001=\alpha<\bar{\alpha}=0.01)$. While both methods overshoot the target, VTD has greater overshoot. For both cases of unequal step-size we see higher variance in the variance estimates for VTD. Figure \[fig:val\_init\_true\_alpha\_0\] explores this further. Here the value estimator is initialized to the true values and updates are turned off ($\alpha=0$). The variance estimators are initialized to zero and learn with $\bar{\alpha}=0.001$, chosen simply to match the step-sizes used in the previous experiments. Despite being given the true values the VTD algorithm produces higher variance in its estimates, suggesting that VTD is dependent on the value estimator tracking. This sensitivity to step-size is shown in Figure \[fig:alphas\]. All estimates are initialized to their true values. For each ratio we computed the average variance of the 30 runs of 2000 episodes. We can see that the [direct]{} method is largely insensitive to step-size ratio, but that VTD has higher mean squared error (MSE) except when the step-sizes are equal. This result holds for the other experimental settings of this paper, including the complex MDP, but further results are omitted for brevity. ![**Chain MDP** (${\lambda}=0.9$). Value estimate held fixed at the true values ($\alpha=0,\bar{\alpha}=0.001$). Notice the increased variance in the estimates for VTD, particularly in State 0.[]{data-label="fig:val_init_true_alpha_0"}](chain/val_init_true_alpha_0.png){width="\linewidth"} -0.5em ![**Chain MDP** (${\lambda}=0.9$). The MSE summed over all states as a function of ratios between the value step-size $\alpha$ (shown along the x-axis) and the variance step-size $\bar{\alpha}$ (shown as the 5 series). The [direct]{} algorithm is indicated by the solid lines and VTD is indicated by the dashed. The MSE of the VTD algorithm is higher than the [direct]{} algorithm, except when the step-size is the same for all estimators, $\alpha=\bar{\alpha}$ or for very small $\bar{\alpha}$.[]{data-label="fig:alphas"}](chain/alphas.png){height="2.0in"} -2em These results beg the question, would there ever be a situation where different step-sizes between value and variance estimators is justified? Methods which automatically set the step-sizes may produce different values which are specific to the performance of each estimator. One such algorithm is ADADELTA, which adapts the step-size based on the TD error of the estimator [@Zeiler2012]. Figure \[fig:adadelta\] shows that using a separate ADADELTA step-size calculation for each estimator results in higher variance for VTD as expected (ADADELTA: $\rho=0.99, \epsilon=1e-6$), given that the value estimator and VTD produce different TD errors. ![**Chain MDP** (${\lambda}=0.9$). Results using ADADELTA algorithm to automatically and independently set the step-sizes $\alpha$ and $\bar{\alpha}$. The step-sizes produced are given in Appendix \[sec:adadelta\_step\_size\].[]{data-label="fig:adadelta"}](chain/adadelta.png){width="\linewidth"} Estimating for State-dependent Gamma and Lambda. {#sec:exp:state-dependent} ------------------------------------------------ One of the contributions of VTD was the generalization to support state-based $\gamma$ and ${\lambda}$. Here we evaluate the complex MDP from Figure \[fig:complex\_mdp\] (in the on-policy setting, using $\mu$), which was designed for this scenario and which has a stochastic policy, is continuing, and has multiple possible actions from each state. Figure \[fig:complex\_mdp\_onpolicy\] shows that both algorithms estimate [$V$]{} with similar results. This experiment was run with all step-sizes equal ($\alpha=\bar{\alpha}=0.01$). ![**Complex MDP** evaluated on-policy with all step-sizes equal ($\alpha=\bar{\alpha}=0.01$).[]{data-label="fig:complex_mdp_onpolicy"}](complex_lambda/same_30_0_01_compare.png){width="\linewidth"} Variable Error in the Value Estimates {#sec:exp:err} ------------------------------------- The derivation of our [direct]{} algorithm assumes access to the true value function. The experiments of the previous sections demonstrate that both methods are robust under this assumption, in the sense that the value function was estimated from data and used to estimate [$V$]{}. It remains unclear, however, how well these methods perform when the value estimates converge to biased solutions. To examine this we again use the complex MDP shown by Figure \[fig:complex\_mdp\]. True values for the value functions and variance estimates are calculated from Monte Carlo simulation of 10,000,000 timesteps. For each run of the experiment each state of the value estimator was initialized to the true value plus an error (${J}(s)_0={j}(s)+\epsilon(s)$) drawn from a uniform distribution: $\epsilon(s)\in[-\zeta,\zeta]$, where $ \zeta=\max_s(|v(s)|)*\text{err ratio}$ (the maximum value in this domain is 1.55082409). The value estimate was held constant throughout the run $(\alpha=0.0)$. The experiment consisted of 120 runs of 80,000 timesteps. To look at the steady-state response of the algorithms we use only the last 10,000 timesteps in our calculations. Figure \[fig:complex\_error\_rand\_std\_dev\] plots the average variance estimate for each state. Additionally we show the average standard deviation of the estimates in the shaded regions. Sweeps over step-size were conducted, $\bar{\alpha}\in[0.05, 0.04, 0.03, 0.02, 0.01, 0.007, 0.005, 0.003, 0.001]$, and the MSE evaluated for each state. Each data point is for the step-size with the lowest MSE for that error ratio and state. While the average estimate is closer to the true values for VTD, the variance of the estimates is much larger. Further, the average estimates for VTD are either unchanged or move negative, while those of the [direct]{} algorithm tend toward positive bias. ![**Complex MDP**. For each run the value estimate of each state is offset by a random amount from uniform distribution whose size is a function of the Err Ratio and the maximum true value in the MDP. Standard deviation of the estimates is shown by shading.[]{data-label="fig:complex_error_rand_std_dev"}](complex_error/err_rand_std_dev.png){height="2in"} For Figure \[fig:best\_alpha\] the MSE is summed over all states. Again, for each error ratio the MSE was compared over the same step-sizes as before and for each point the smallest MSE is plotted. ![**Complex MDP**. The MSE computed for the last 10,000 timesteps of 120 runs summed over all states for the step-size with the lowest overall MSE at each error ratio. For each point the step-size used ($\alpha=\bar{\alpha}$) is displayed.[]{data-label="fig:best_alpha"}](complex_error/best_alpha.png){height="2in"} These results suggest the [direct]{} algorithm is less affected by error in ${J}$. Experiments with Traces {#sec:exp:traces} ----------------------- In this section we briefly look at the behavior of the complex domain when traces are used. For Figure \[fig:complex\_traces\_l0\_lb\_1\] traces are used for the variance estimators, but not for policy evaluation (${\kappa}=0.0,{\bar{\kappa}}=1.0$) and the step-sizes are all equal (0.01). Here we see no significant difference between VTD and the [direct]{} algorithm. For Figure \[fig:complex\_traces\_l1\_lb\_0\] we look at the opposite scenario, where traces are used for policy evaluation, but not in the variance estimators (${\kappa}=1.0, {\bar{\kappa}}=0.0$). Here we do see a difference, particularly the VTD method shows more variance in its estimates for State 0 and 3. Experiments in an Off-policy Setting {#sec:exp:off-policy} ------------------------------------ In the off-policy setting the agent follows a behavior policy $\mu$, but is estimating the value of a target policy $\pi$. The ratio between these two policies is called the importance sampling ratio, $\rho=\frac{\pi(s,a)}{\mu(s,a)}$, and is used to modify the value function update. We evaluate two different off-policy scenarios on the complex MDP. In the first scenario we estimate [$V$]{} under the target policy from off-policy samples. That is, we estimate [$V$]{} that would be observed if we were following the target policy. In this scenario ${\eta}=1,\bar{\rho}=\rho$. Figure \[fig:complex\_offpolicy\_same\] shows that both methods are able achieve the same results in this setting. ![**Complex MDP** estimating [$V$]{} from off-policy samples ($\alpha=\bar{\alpha}=0.01, {\eta}=1,\bar{\rho}=\rho$).[]{data-label="fig:complex_offpolicy_same"}](complex_offpolicy_1/same_30_0_01_0_01_0_01_compare_20000.png){width="\columnwidth"} In the second off-policy setting we estimate the variance of the off-policy return, which is the return being used to update the value estimator and is simply the multiplication of the $\lambda$-return by $\rho$. In this scenario $\bar{\rho}=1$ and ${\eta}=\rho$. Figure \[fig:complex\_offpolicy\_2\_same\] shows that both algorithms successfully estimate the return in this setting. However, despite having the same step-size as the value estimator, VTD produces higher variance in its estimates, as is most clearly seen in State 3. ![**Complex MDP** estimating the variance of the off-policy return ($\alpha=\bar{\alpha}=0.01,\bar{\rho}=1,{\eta}=\rho$).[]{data-label="fig:complex_offpolicy_2_same"}](complex_offpolicy_2/same_30_0_01_0_01_0_01_compare_20000.png){width="\linewidth"} Discussion ========== Both the [direct]{} method and VTD effectively estimate the variance across a range of settings, but the [direct]{} method is simpler and more robust. This simplicity alone makes the [direct]{} method preferable. The higher variance in estimates produced by VTD is likely due to the inherently larger target which VTD uses in its learning updates: ${\mathbb{E}}[X^2] \geq {\mathbb{E}}[(X - {\mathbb{E}}[X])^2]$; we show more explicitly how this affects the updates of VTD in Appendix \[sec:updates\]. One would expect the differences between the two approaches to be most pronounced for domains with larger returns than those demonstrated here. Our focus was simple MDPs. In such settings we can define clear experiments where the properties of these variance estimation algorithms can be carefully evaluated isolated from additional effects like state-aliasing due to function approximation. Consider the task of helicopter hovering formalized as a reinforcement learning task [@Ng2004]. In the most well-known variants of this problem the agent receives massive negative reward for crashing the helicopter (e.g., minus one million). In such problems the magnitude and variance of the return is large. In such cases, estimating the second moment may not be feasible from a statistical point of view, whereas the target of our direct variance estimate should be better behaved. We focused on the tabular case, where each state is represented uniquely. Future work will investigate extending our theoretical characterization and experiments to the function approximation case. Our algorithm extends naturally with little modification. To extend the theory, there have been some promising results characterizing fixed points under the projected Bellman operator for the second moment [@Tamar2016]. An extension to projected Bellman operators could also further help bound errors incurred from inaccuracies in the value function. Conclusion ========== In this paper we introduced a simple method for estimating the variance of the $\lambda$-return using temporal difference learning. Our approach is simpler than existing approaches, and appears to work better in practice. We performed an extensive empirical study. Our findings suggest that our new method outperforms VTD when: (1) there is a mismatch in step-size between the value estimator and the variance estimator, (2) traces are used with the value estimator, (3) estimating variances of the off-policy return, and (4) there is error in the value estimate. Acknowledgements {#acknowledgements .unnumbered} ================ Funding for this work was provided by the Natural Sciences and Engineering Research Council of Canada, Alberta Innovates, and Google DeepMind. ### References {#references .unnumbered} Variance Estimation in the Off-Policy Setting {#appendix:general} ============================================= Value estimates are made with respect to a target policy, $\pi$. If the behavior policy, $\mu$, is the same as the target policy then we say that samples are collected on-policy and when they are not the same, the samples are collected off-policy. A common approach for off-policy learning algorithms is to weight each update by the importance sampling ratio: $\rho_t=\frac{\pi(S_t,A_t)}{\mu(S_t,A_t)}$. Off-policy estimates are then implemented by multiplying the trace updates by $\rho_t$: $$\begin{aligned} {E}_t(s)&\leftarrow\begin{cases} \rho_t(\gamma_t\lambda_t {E}_{t-1}(s) + 1) & s=S_t\\ \rho_t\gamma_t\lambda_t {E}_{t-1}(s) & \forall s \in {\mathcal{S}}, s \ne S_t \end{cases}.\end{aligned}$$ There are two different scenarios to be considered in the off-policy setting. The first scenario is estimating the variance of the (on-policy) $\lambda$-return of the target policy, while following a different behavior policy. In the second setting, the goal is to estimate the variance of the off-policy $\lambda$-return. The off-policy $\lambda$-return is $$\!\!G_t^{\lambda} \!=\!\rho_{t} \Big( {R}_{t+1}+\gamma_{t+1}(1\!-\!\lambda_{t+1}){j}_t(S_{t+1}) +\gamma_{t+1}\lambda_{t+1} G^{\lambda}_{t+1} \Big). $$ where the multiplication by the potentially large importance sampling ratios can significantly increase variance. It is important to note that you would only ever estimate one or the other off-policy variance with a given estimator. Let ${\eta}$ be the weighting for the value estimator, and $\bar{\rho}$ the weighting for the variance estimator. If estimating the variance of the target return from off-policy samples, the first scenario, ${\eta}_t=1\ \forall t$ and $\bar{\rho}_t=\rho_t$. If estimating the variance of the off-policy return $\bar{\rho}_t=1\ \forall t$ and ${\eta}_t=\rho_t$. Here we present the resulting algorithms which use TD($\lambda$) estimators with accumulating traces. **Direct Variance Algorithm** $$\begin{aligned} \bar{{R}}_{t+1}& \leftarrow ({\eta}_t\delta_t + ({\eta}_t-1){J}_{t+1}(s))^2\\ \bar{\gamma}_{t+1}& \leftarrow \gamma_{t+1}^2 {\lambda}_{t+1}^2{\eta}_t^2\\ \bar{\delta}_t& \leftarrow \bar{{R}}_{t+1} + \bar{\gamma}_{t+1}{V}_t(s') - {V}_t(s)\\ \bar{{E}}_t(s)&\leftarrow\begin{cases} \bar{\rho}_t(\bar{\gamma}_t{\bar{\kappa}}_t \bar{{E}}_{t-1}(s) + 1) & s=S_t \\ \bar{\rho}_t(\bar{\gamma}_t{\bar{\kappa}}_t \bar{{E}}_{t-1}(s)) & \forall s \in {\mathcal{S}}, s \ne S_t \end{cases}\\ {V}_{t+1}(s)&\leftarrow {V}_t(s)+\bar{\alpha} \bar{\delta}_t \bar{{E}}_t(s) \end{aligned} \label{alg:direct_off_policy}$$ Variance is computed directly as ${V}_{t+1}(s)$. **Second Moment Algorithm** $$\begin{aligned} \bar{G}_t&\leftarrow {R}_{t+1} + \gamma_{t+1}(1-{\lambda}_{t+1}){J}_{t+1}(s')\\ \bar{{R}}_{t+1}& \leftarrow {\eta}_t^2 \bar{G}_t^2 + 2{\eta}_t^2 \gamma_{t+1} {\lambda}_{t+1} \bar{G}_t {J}_{t+1}(s')\\ \bar{\gamma}_{t+1}& \leftarrow {\eta}_t^2 \gamma_{t+1}^2 {\lambda}_{t+1}^2\\ \bar{\delta}_t& \leftarrow \bar{{R}}_{t+1} + \bar{\gamma}_{t+1}{M}_t(s') - {M}_t(s)\\ \bar{{E}}_t(s)&\leftarrow\begin{cases} \bar{\rho}_t(\bar{\gamma}_t{\bar{\kappa}}_t \bar{{E}}_{t-1}(s) + 1) & s=S_t \\ \bar{\rho}_t(\bar{\gamma}_t{\bar{\kappa}}_t \bar{{E}}_{t-1}(s)) & \forall s \in {\mathcal{S}}, s \ne S_t \end{cases}\\ {M}_{t+1}(s)&\leftarrow {M}_t(s)+\bar{\alpha} \bar{\delta}_t\bar{{E}}_t(s) \end{aligned} \label{alg:comparison_off_policy}$$ Variance is computed as ${V}_{t+1}(s)=M_{t+1}(s)-{J}_{t+1}(s)^2$. For convenience we summarize the variables used: $$\begin{aligned} {J}-&\text{estimated value function of the target policy $\pi$.}\\ {R}-&\text{reward used in the value function estimate.}\\ \bar{{R}}-&\text{{meta-reward}{} used in the variance estimate.}\\ {\lambda}-&\text{bias-variance parameter of the target $\lambda$-return.}\\ {\kappa}-&\text{trace-decay parameter of the value estimator.}\\ {\bar{\kappa}}-&\text{trace-decay parameter of the secondary estimator.}\\ \gamma -&\text{discounting function used by the {$J$}{} estimator.}\\ \bar{\gamma} -&\text{discounting function used by the {$V$}{} estimator.}\\ \delta_t -&\text{TD error of the value function at time $t$.}\\ \bar{\delta_t} -&\text{TD error of the variance estimator at time $t$.}\\ {M}-&\text{estimate of the second moment.}\\ {V}-&\text{estimate of the variance.}\\ \bar{\rho}-&\text{importance sampling ratio for estimating the }\\ &\text{variance of the target return from off-policy samples.}\\ {\eta}-&\text{importance sampling ratio used to estimate the} \\ &\text{variance of the off-policy return.}\end{aligned}$$ Bellman Operators for the Variance in the Off-Policy Setting ============================================================ \[delta\_var\] For ${j}(s) = {{\mathbb{E}}\left[ G_{t+1}^{\lambda}|S_t=s \right]}$, i.e., satisfying the Bellman equation, for any bounded function $b: {\mathcal{S}}\times {\mathcal{A}}\times \mathbb{R} \times {\mathcal{S}}\rightarrow \mathbb{R}$, $${\mathbb{E}}[b(S_t, A_t, R_{t+1}, S_{t+1})(G_{t+1}^\lambda-{j}(S_{t+1}))|S_t=s]=0$$ Let $b_t = b(S_t, A_t, R_{t+1}, S_{t+1})$. By the law of total expectation: $$\begin{aligned} {{\mathbb{E}}\left[ b_t (G_{t+1}^{\lambda}- {j}(S_{t+1})) |S_t=s \right]} = {\mathbb{E}}\left[ {\mathbb{E}}[b_{t} (G_{t+1}^{\lambda} \!- {j}(S_{t+1})) |S_t, A_{t}, S_{t+1}]|S_t=s\right]\end{aligned}$$ Given $S_{t}$, $A_t$, $R_{t+1}$ and $S_{t+1}$, $b_{t}$ is constant and can be moved outside of the expectation. Therefore, $$\begin{aligned} {\mathbb{E}}&[b_{t} (G_{t+1}^{\lambda} - {j}(S_{t+1})) \Big|S_{t}, A_{t}, R_{t+1}, S_{t+1}] ={{\mathbb{E}}\left[ b_{t} \big|S_{t}, A_{t}, R_{t+1}, S_{t+1} \right]} \times {{\mathbb{E}}\left[ G_{t+1}^{\lambda} - {j}(S_{t+1}) \big|S_{t}, A_{t}, R_{t+1}, S_{t+1} \right]}\end{aligned}$$ Because $${{\mathbb{E}}\left[ G_{t+1}^{\lambda} - {j}(S_{t+1}) \big|S_{t}, A_{t}, R_{t+1}, S_{t+1} \right]}=0$$ the result follows. \[recurse\_var\_off\_policy\] $$\begin{aligned} {v}(s) ={\mathbb{E}}[({\eta}_t\delta_t +({\eta}_t-1){j}(s))^2+\lambda_{t+1}^2\gamma_{t+1}^2{\eta}_t^2{v}(S_{t+1})|S_t=s]\end{aligned}$$ The proof is similar to the proof of Theorem \[recurse\_var\]. $$\begin{aligned} {v}(s) =&{\mathbb{E}}[\{G_t^\lambda - {j}(S_t)\}^2|S_t=s]\\ =& {\mathbb{E}}[\{{\eta}_t{R}_{t+1}+{\eta}_t\gamma_{t+1}(1-\lambda_{t+1}){j}(S_{t+1}) + {\eta}_t\gamma_{t+1}\lambda_{t+1} G_{t+1}^\lambda - v(s)\}^2|S_t=s]\\ =& {\mathbb{E}}[\{{\eta}_t{R}_{t+1}+{\eta}_t\gamma_{t+1}{j}(S_{t+1})-{\eta}_t {j}(s) + {\eta}_t {j}(s) - {\eta}_t\gamma_{t+1}\lambda_{t+1} {j}(S_{t+1})\\ \ &+ {\eta}_t\gamma_{t+1}\lambda_{t+1} G_{t+1}^\lambda-{j}(s)\}^2|S_t=s]\\ =& {\mathbb{E}}[\{({\eta}_t\delta_t\ + ({\eta}_t - 1){j}(s)) + {\eta}_t\gamma_{t+1}\lambda_{t+1}(G_{t+1}^\lambda-{j}(S_{t+1}))\}^2|S_t=s]\\ =& {\mathbb{E}}[({\eta}_t\delta_t + ({\eta}_t -1){j}(s))^2 +{\eta}_t^2\gamma_{t+1}^2\lambda_{t+1}^2(G_{t+1}^\lambda- {j}(S_{t+1}))^2\\ \ &+2{\eta}_t \gamma_{t+1}\lambda_{t+1} ({\eta}_t\delta_t + ({\eta}_t - 1){j}(s))(G_{t+1}^\lambda- {j}(S_{t+1}))|S_t=s]\\ =& {\mathbb{E}}[({\eta}_t\delta_t + ({\eta}_t - 1){j}(s))^2 +{\eta}_t^2\gamma_{t+1}^2\lambda_{t+1}^2(G_{t+1}^\lambda- {j}(S_{t+1}))^2\\ \ &+2{\eta}_t^2 \gamma_{t+1}\lambda_{t+1} \delta_t (G_{t+1}^\lambda- {j}(S_{t+1})) + 2{\eta}_t \gamma_{t+1}\lambda_{t+1}({\eta}_t-1){j}(s)(G_{t+1}^\lambda-{j}(S_{t+1}))|S_t=s]\end{aligned}$$ Using Lemma \[delta\_var\], with different fixed functions $b$, we can conclude that the last two terms are zero, giving $$\begin{aligned} {v}(s) =&{\mathbb{E}}[({\eta}_t\delta_t + ({\eta}_t-1){j}(s))^2 +{\eta}_t^2\gamma_{t+1}^2\lambda_{t+1}^2(G_{t+1}^\lambda-{j}(S_{t+1}))^2|S_t=s]\\ \intertext{By the law of total expectation} {v}(s) =&{\mathbb{E}}[({\eta}_t\delta_t + ({\eta}_t-1){j}(s))^2 +{\mathbb{E}}[{\eta}_t^2\gamma_{t+1}^2\lambda_{t+1}^2(G_{t+1}^\lambda-{j}(s'))^2|S_{t+1}=s']|S_t=s]\\ =&{\mathbb{E}}[({\eta}_t\delta_t+({\eta}_t-1){j}(s))^2 + {\eta}_t^2\gamma_{t+1}^2\lambda_{t+1}^2{v}(S_{t+1})|S_t=s].\end{aligned}$$ completing the proof. Theorem \[recurse\_var\_off\_policy\] gives a Bellman equation for ${V}(s)$ in the more general off-policy setting. The resulting TD algorithm uses [meta-reward]{} $({\eta}_t\delta_t +({\eta}_t-1){j}(s))^2$ and discounting function ${\eta}_t^2\gamma_{t+1}^2\lambda^2$. ADADELTA Step-Sizes {#sec:adadelta_step_size} =================== The step-sizes generated by the ADADELTA algorithm in Figure \[fig:adadelta\] are shown in Figure \[fig:adadelta\_stepsizes\]. As we evaluate in the tabular case at each timestep only the step-size for the current state has any impact. Thus, the values shown here are the average step-size used over each episode. ![**Chain MDP**. The average step-sizes computed for ADADELTA in Figure \[fig:adadelta\].[]{data-label="fig:adadelta_stepsizes"}](chain/adadelta_alphas.png){height="2in"} Variability in Updates {#sec:updates} ====================== In this section, we show the effective update to ${V}_t(s)$ on each timestep for each of the two algorithms in the on-policy setting. For notational clarity let ${r}={r}_{t+1},\alpha=\alpha_t, \gamma=\gamma_{t+1}, {\lambda}={\lambda}_{t+1},s=s_{t},s'=s_{t+1},\delta_t=\delta$. For the direct algorithm the change is just: $$\begin{aligned} \Delta {V}_t(s)=\bar{\alpha}(\delta^2 + \bar{\gamma}{V}_t(s')-{V}_t(s)). \label{eq:direct_update} \end{aligned}$$ The updates for the VTD algorithm are much more complicated to compute and we will make some assumptions about the domain in order to simplify the derivation. First we compute the change in the second moment and value estimators separately. We first expand the term $\delta^2$: $$\begin{aligned} \delta=&{r}+\gamma {J}_t(s')-{J}_t(s)\\ \delta^2=&({r}+\gamma {J}_t(s'))^2 - 2({r}+\gamma {J}_t(s')){J}_t(s)+ {J}_t(s)^2. $$ Now we expand the change in the second moment estimate, $M$. To simplify the expansion we make the assumption that at each transition the agent moves to a new state, i.e. $s_t\ne s_{t+1}\forall t$ (this is not required for our algorithm, but simplifies the expansions below). This assumption holds for both of the domains examined in this paper. This allows us to substitute ${J}_{t+1}(s')={J}_{t}(s)$, which greatly simplifies the updates. $$\begin{aligned} \Delta M(s)=&\bar{\alpha}[({r}+\gamma {J}_{t+1}(s'))^2 - \bar{\gamma}^2 {J}_{t+1}(s')^2 + \bar{\gamma}M_t(s')-M_t(s)]\\ =&\bar{\alpha}[({r}+\gamma {J}_{t}(s'))^2 - \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}M_t(s')-M_t(s)]\\ =&\bar{\alpha}[({r}+\gamma {J}_{t}(s'))^2 -2({r}+\gamma {J}_t(s')){J}_t(s) + {J}_t(s)^2 + 2({r}+\gamma {J}_t(s')){J}_t(s) - {J}_t(s)^2 \\ &- \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}M_t(s')-M_t(s)]\\ =&\bar{\alpha}[\delta^2 + 2({r}+\gamma {J}_t(s')){J}_t(s) - {J}_t(s)^2 - \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}M_t(s')-M_t(s)]\\ \intertext{Notice that from the definition of the TD error: $R+\gamma {J}_t(s')=\delta+{J}_t(s)$.} =&\bar{\alpha}[\delta^2 + 2(\delta+{J}_t(s)){J}_t(s) - {J}_t(s)^2 - \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}M_t(s')-M_t(s)]\\ =&\bar{\alpha}[\delta^2 + 2\delta {J}_t(s) + {J}_t(s)^2 - \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}M_t(s')-M_t(s)]\\ =&\bar{\alpha}[\delta^2 + (\bar{\gamma}M_t(s')-\bar{\gamma}{J}_t(s')^2) -(M_t(s)-{J}_{t}(s)^2) + 2\delta {J}_t(s) - \bar{\gamma}^2 {J}_{t}(s')^2 +\bar{\gamma}{J}_t(s')^2]\\ =&\bar{\alpha}[\delta^2 + \bar{\gamma}{V}_t(s') - {V}_t(s) + 2\delta {J}_t(s) - \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}{J}_t(s')^2]\\ =&\bar{\alpha}[\delta^2 + \bar{\gamma}{V}_t(s') - {V}_t(s)] + \bar{\alpha}[2\delta {J}_t(s) - \bar{\gamma}^2 {J}_{t}(s')^2 + \bar{\gamma}{J}_t(s')^2]\\\end{aligned}$$ The first half of this equation is the same as the update for the direct algorithm . Now we expand the change in the variance update for VTD: $$\begin{aligned} \Delta {V}_t(s) =&(M_{t+1}(s) - {J}_{t+1}(s)^2) - (M_{t}(s) - {J}_{t}(s)^2)\\ =&\Delta M(s) + {J}_{t}(s)^2 - {J}_{t+1}(s)^2\\ =&\Delta M(s) + {J}_{t}(s)^2 - (\alpha\delta + {J}_{t}(s))^2\\ =&\Delta M(s) + {J}_t(s)^2 - ((\alpha\delta)^2 + 2\alpha\delta {J}_t(s) + {J}_t(s)^2)\\ =&\Delta M(s) -(\alpha\delta)^2 - 2\alpha\delta {J}_t(s).\end{aligned}$$ Note that in the case that $\alpha=\bar{\alpha}$ this last term cancels out and we’re left with: $$\begin{aligned} \Delta {V}_t(s)=&\alpha[\delta^2 + \bar{\gamma}{V}_t(s') - {V}_t(s)] + \alpha {J}_t(s')^2(\bar{\gamma}- \bar{\gamma}^2) -(\alpha\delta)^2.\end{aligned}$$ This suggests that VTD will deviate from the direct method more when: $\alpha$ is larger, ${J}_t(s')$ is larger, $\bar{\gamma}=0.5$ and for large values of $\delta$. In general, we expect from this equation that the updates for the VTD will be larger than those of the direct method, suggesting a cause for the higher variance of variance estimates across runs as observed for VTD under a number of scenarios. We also empirically tested this hypothesis, with Table \[table:updates\] showing the updates for the two algorithms across the various experiments. For episodic tasks (chain MDP, Figures \[fig:chain\_l1.0\_same\]-\[fig:adadelta\]) the results show the average total absolute change over all states for a given episode averaged across runs and then averaged across all episodes. For the continuing case (complex MDP, Figures \[fig:complex\_mdp\_onpolicy\]-\[fig:complex\_offpolicy\_2\_same\]) the results are the average absolute change for a timestep averaged over all runs and then averaged across the entire run length. The experiments shaded in gray are those where the two algorithms behaved nearly identically. In this case we see that the average magnitude of updates is nearly identical. For the other experiments the VTD algorithm showed higher variance in its variance estimates across runs. For these experiments we see that the average magnitude of the VTD updates is much larger than for the direct algorithm. Fig. Value Snd Mmnt VTD Direct ------------------------------------- --------- ---------- --------- --------- \[fig:chain\_l1.0\_same\] 0.00332 0.0157 0.00415 0.00415 \[fig:chain\_l1.0\_less\] 0.0322 0.0165 0.143 0.00387 \[fig:chain\_l1.0\_more\] 0.00332 0.156 0.142 0.0419 \[fig:val\_init\_true\_alpha\_0\] 0.0 0.0166 0.0166 0.00381 \[fig:adadelta\] 0.0212 0.0306 0.0752 0.00884 \[fig:complex\_mdp\_onpolicy\] 0.00362 0.00675 0.00381 0.00385 \[fig:complex\_offpolicy\_same\] 0.00362 0.00461 0.00303 0.00307 \[fig:complex\_offpolicy\_2\_same\] 0.00362 0.0110 0.0116 0.00838 : Average updates for various experiments.[]{data-label="table:updates"}

[**Comment on “Phase Coexistence in Multifragmentation”**]{}\ In their letter Moretto et al. [@moretto96] propose the fragment charge distribution in nuclear multifragmentation to give a signal for the coexistence of nuclear liquid and vapor phase. To our opinion this signal is not usefull and misleading as fluctuations of different origin spoil it. Phase transitions in macro-physics are usually indicated by a peak in the specific heat e.g. $c_p(T)$ or by an anomaly of the caloric equation of state ($C\!E\!S$) $T(E)$ e.g. at constant pressure or volume. In closed finite systems, as e.g. highly excited nuclei, phase transitions are well indicated by the shape of the $C\!E\!S$ c.f.[@gross72; @gross95]. Inherent to phase transitions are large fluctuations at the transition which do not allow a clear phase separation in space or any other observable in small finite systems because of the nonvanishing coherence length of the phase fluctuations c.f. [@janke95; @gross150] and which differ at const.E and at const.T. E.g. even though the backbending of the $C\!E\!S$ is clearly seen for a 10-state Potts model at a lattice size of $100*100$ and the area under the oscillation of $T^{-1}(E)$ is close to the asymptotic value of the surface entropy no phase separation can be seen in the configurations. Therefore, the interpretation by ref.[@moretto96] is too naive and suffers from several further difficulties: Equations like formulas (1-3) of ref.[@moretto96] notice charge conservation only via the mean value but leave its fluctuation unrestricted. These fluctuations are usually substantial especially near to phase transitions. Moreover, in nuclear fragmentation one has to take care of the indistinguishability of identical fragments and the partition problem is not the Euler problem as is suggested in [@phair95]. The correct formula for the quantum partition of an integer $Z_0$ is given in [@gross110]. The experimental method used in ref.[@phair95; @moretto96] is of course not ideally suited to look for a phase transition in equilibrized nuclear systems. First of all this system is generated in a collision of two sizeable nuclei. The transverse energy $E_t$ does not give the total excitation energy $\varepsilon^*$ of the system nor is it neccessarily proportional to it. In fact the width in $\varepsilon^*$ at low $E_t$ can easily be of the order of a few MeV/nucleon[@moretto94a]. I.e. a fixed value of $E_t$ allows for considerable fluctuations of $\varepsilon^*$. It is therefore neccessary to investigate the signal of ref.[@moretto96] in a situation where we definitely [*have an equilibrized*]{} nuclear system with a [*sharply defined excitation energy*]{}. Lacking experimental data of this kind we investigated a model system from which we know by experience that it reproduces nuclear multifragmentation and which shows a nuclear phase-transition of first order towards fragmentation: The Berlin microcanonical Metropolis Monte Carlo $M\!M\!M\!C$-model [@gross95]. Here as also in other versions of statistical multifragmentation models like the Copenhagen model [@bondorf95] the phase transition towards fragmentation is clearly seen as anomaly in the $C\!E\!S$. In fig.1 we show the resultant parameter $cZ$ or better the quantity $cZ=ln\{P_n(Z)/P_{n+1}(Z)\}$ averaged over the IMF-multiplicities $n$ to get better statistics vs.$Z$. Here $P_n(Z)$ is the probability to find one fragment with charge $Z$ in events with n IMF’s. The two panels at $\varepsilon^*=5$ and $=6$ MeV/nucleon resemble the findings of Moretto. At lower excitation in contrast to ref.[@moretto96] $cZ$ is forced to rise with $Z$ as the emission of a second fragment with larger $Z$ is prohibited due to limited energy resources. We guess that at low [*transverse*]{} energy the experimental data of ref.[@moretto96] are overshadowed by deep inelastic collisions where some of the small fragments are likely from projectile break-up which as such have small transverse energies. Consequently, low total transverse energies do not really characterize the limitation to low excitation energies as indicated by the large width in $\varepsilon^*(E_t)$ [@moretto94a]. This is probably the reason for the vanishing quantity $c$ found in ref.[@moretto96] at low transverse energies. Conclusion: From all experience of microcanonical first order phase transitions in small systems one knows that it is normally rather difficult to see a clear phase separation even though the caloric equation of state gives an unambiguous signal, phase fluctuations are usually too large. Within the arguments of ref.[@phair95; @moretto96] there are at least [*two*]{} important conservation laws to be observed by the reaction: Conservation of charge [*and energy*]{}. The latter forces the “chemical potential” $c$ to rise again at low excitation energy. The observation of an anomaly in the caloric equation of state [@pochodzalla95] is still the best signal for a phase transition as was predicted in [@gross72; @bondorf95; @gross95]. Since long this is one of the classical signals for phase-transitions.\  \ D.H.E. Gross and A.S. Botvina Hahn-Meitner-Institut Berlin, Bereich Theoretische Physik,14109 Berlin, Germany\ L.G. Moretto et al. , 76:372–375, 1996. D.H.E. Gross et al. , 56:1544, 1986. D.H.E. Gross. , 53:605–658, 1990. D.H.E. Gross et al. , in print, 1996. W. Janke and S. Kappler. , 197:227, 1995. L. Phair et al. , 75:213, 1995. D.H.E. Gross et al. , 1:340–358, 1992. L.Moretto, private communication 1994 J.P. Bondorf et al. , 257:133–221, 1995. J. Pochodzalla et al. , 75:1040, 1995.

--- abstract: 'We review the physics potential at FAIR in the light of the existing data of the RHIC-BES program and the NA49/NA61 beam and system size scan. Special emphasize will be put on the potential of fluctuations, as well as dilepton observables.' address: - '$^{1}$ Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, and Institut für Theoretische Physik, Johann Wolfgang Goethe University, 60438 Frankfurt am Main, Germany' - '$^{2}$ SUBATECH, UMR 6457, Université de Nantes, Ecole des Mines de Nantes, IN2P3/CRNS. 4 rue Alfred Kastler, 44307 Nantes cedex 3, France' - '$^3$ CFTP, Departamento de Fisica, Instituto Superior Tecnico (Universidade Tecnica de Lisboa), Av. Rovisco Pais, 1049-001 Lisboa, Portugal' author: - 'M. Bleicher$^{1}$, M. Nahrgang$^{1,2}$, J. Steinheimer$^{1}$, Pedro Bicudo$^{3}$' title: 'Physics Prospects at FAIR [^1]' --- Introduction ============ The Facility for Antiproton and Ion Research, FAIR [@bib1a; @bib1b; @bib1c], will provide an extensive range of particle beams from protons and antiprotons to ion beams of all chemical elements up to the heaviest one, uranium, with in many respects world record intensities. As a joint effort of several countries the new facility builds, and substantially expands, on the present accelerator system at GSI, both in its research goals and its technical possibilities. Compared to the present GSI facility, an increase of a factor of 100 in primary beam intensities, and up to a factor of 10000 in secondary radioactive beam intensities, will be a technical property of the new facility. The main thrust of FAIR research focuses on the structure and evolution of matter on both a microscopic and on a cosmic scale. The approved FAIR research programme embraces 14 experiments, which form the four scientific pillars of FAIR and offers a large variety of unprecedented forefront research in hadron, nuclear, atomic and plasma physics as well as applied sciences. Already today, over 2500 scientists and engineers are involved in the design and preparation of the FAIR experiments. They are organized in the experimental collaborations APPA, CBM, NuSTAR, and PANDA. The CBM/HADES experiment is of particular interest for the understanding of highly compressed nuclear matter and its relevance for understanding fundamental aspects of the strong interaction. HADES [@bib3a; @bib3b; @bib3c] and CBM [@bib3d; @bib3e] at SIS100/300 will explore the QCD phase diagram in the region of very high baryon densities and moderate temperatures. This approach includes the study of the nuclear matter equation-of-state, the search for new forms of matter, the search for the predicted first order phase transition between hadronic and partonic matter, the QCD critical endpoint, and the chiral phase transition, which is related to the origin of hadron masses. It is intended to perform comprehensive measurements of hadrons, electrons, muons and photons created in collisions of heavy nuclei proton–nucleus, and proton–proton collisions at different beam energies. Most of the rare probes like lepton pairs, multi-strange hyperons and charm will be measured for the first time in the FAIR energy range. Dileptons ========= Dileptons represent a penetrating probe of the hot and dense nuclear matter created in heavy ion collisions at the CBM experiment at the FAIR facility. The analysis of the electromagnetic response of the dense and hot medium is tightly connected to the investigation of the in-medium modification of the vector meson properties. Vector mesons are ideally suited for this exploration, because they can directly decay into a lepton-antilepton pair. One therefore aims to infer information on the modifications induced by the medium on specific properties of the vector meson, such as its mass and/or its width, from the invariant mass dilepton spectra. In this work, we present a consistent calculation of the dilepton production at SPS energy within a model which attempts to take into account both the complexity of the dilepton rate in hot a dense medium as well as the complexity of the pre-, post-, and equilibrium heavy-ion dynamics. The latter is modelled with an integrated Boltzmann+hydrodynamics hybrid approach based on the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) transport model with an intermediate (3+1) dimensional ideal hydrodynamic stage [@Petersen:2008dd]. During the locally equilibrated hydrodynamical stage, dimuon emission is calculated locally in space-time according to the expression for the thermal equilibrium rate of dilepton emission per four-volume and four-momentum from a bath at temperature $T$ and baryon chemical potential $\mu_B$. During the local equilibrium phase, the radiation rate of the strongly interacting medium is standardly modelled using the vector meson dominance model and related to the spectral properties of the light vector mesons, with the $\rho$ meson having the dominant role [@Gale:1990pn; @Rapp:1999ej; @Ruppert:2007cr; @vanHees:2007th]. In-medium modifications of the $\rho$-meson spectral function due to scattering from hadrons in the heat bath are properly included in the model. Two additional sources of thermal radiation, namely emission from four-pion annihilation processes and from a thermalized partonic phase are included as well. As an input for the hydrodynamical part of the evolution we employ an equation of state in line with lattice data that follows from coupling the Polyakov loop to a chiral hadronic flavor-SU(3) model [@Steinheimer:2010ib]. The results we show for the dilepton spectra in In+In collisions at $E_{lab}=160 A$ GeV will be compared to fully acceptance corrected NA60 data [@Arnaldi:2008fw]. The data correspond to nearly minimum bias collisions, selecting events with a charged particle density $dN_{ch}/d\eta$$>$30. In Fig. \[fig1\] we show results for the invariant mass spectra of the excess dimuons in two slices in the transverse momentum of the dilepton pair $p_T$ when adopting a sudden freezeout approximation. The theoretical spectra are normalized to the corresponding average number of charged particles in an interval of one unit of rapidity around mid-rapidity. Results on more bins in transverse momentum as well as a detailed discussion on the applied model and results can be found in [@arXiv:1102.4574]. Of particular interest is that in the intermediate mass region, 1$<$$M$$<$1.5 GeV, we find that emission from the QGP accounts for about half of the total radiation. The remaining half is filled by the considered hadronic sources. The $4\pi$ annihilation alone is comparable to the QGP emission only for $M$$>$1.4 GeV. This offers the possibility to even quantitatively pin down the QGP contribution to the dilepton spectra and consequentely to the active degrees of freedom for the SIS beam energies. Fluctuations induced by a phase transition ========================================== At larger baryochemical potential, as achieved at FAIR, a first order phase transition is expected from model studies [@Bleicher:1998wu; @Scavenius:2000qd; @Ratti:2005jh; @arXiv:0704.3234]. Interesting observables could here be based on the growth of fluctuations due to the nonequilibrium effect of supercooling leading to nucleation and spinodal decomposition [@Csernai:1992tj; @Mishustin:1998eq; @Randrup:2010ax; @Chomaz:2003dz]. At zero baryochemical potential the nature of the phase transition of QCD is well understood from lattice QCD calculations, which show that it is an analytic crossover [@Aoki:2006we]. As a consequence there must be a critical point, which terminates the line of first order phase transitions. In equilibrium systems fluctuations and correlations of the order parameter diverge at the critical point. Coupling particles to the sigma field, the order parameter of chiral symmetry, leads to a nonmonotonic behaviour in fluctuations of net-charge or net-baryon number multiplicities [@Stephanov:1998dy; @Stephanov:1999zu]. The key ingredient is the correlation length which becomes infinite in a system at a critical point. In a realistic evolution of a heavy-ion collision, however, the growth of the correlation length is limited by the size of the system and by the finite time, which the dynamic systems spends at a critical point. Relaxation times also become infinite at the critical point, a phenomenon called critical slowing down. Even if the system is in equilibrium above the critical point it is necessarily driven out of equilibrium by passing trough the critical point. Assuming a phenomenological time evolution of the correlation length with parameters from the $3$d Ising universality class it was found that the correlation length does not grow beyond $2-3$ fm [@Berdnikov:1999ph]. The explicit propagation of fluctuations coupled to a dynamic model is a necessary step towards understanding the QCD phase diagram from heavy-ion collision experiments. Here we present results from a recently developped extention of chiral fluid dynamics [@Mishustin:1998yc; @Paech:2003fe], which self-consistently includes the nonequilibrium propagation of the fluctuation of the order parameter of chiral symmetry, the sigma field [@Nahrgang:2011mg; @Nahrgang:2011ll; @arXiv:1105.1962]. It is coupled to a fluid dynamic expansion, where the fluid is made out of quarks and antiquarks and acts as a locally equilibrated heat bath. Due to the interaction with the quark fluid the sigma fields is damped. This is taken into account by dissipative terms in the Langevin equation of motion of the chiral fields $$\partial_\mu\partial^\mu\sigma+\frac{\delta U}{\delta\sigma}+\frac{\delta \Omega_{\bar qq}}{\delta\sigma}+\eta\partial_t \sigma=\xi\, . \label{eq:equi_langevineq}$$ It contains a classical Mexican hat potential $U$, the quark contribution to the thermodynamic potential $\Omega_{\bar qq}$ to one-loop level, the damping coefficient $\eta$ and the stochastic noise field $\xi$. The dynamics of the quarks is reduced to the propagation of densities according to energy-momentum conservation, i.e. the equations of relativistic fluid dynamics $$\partial_\mu T^{\mu\nu}=S^\nu\, , \label{eq:fluidT}$$ where the source term $S^\nu$ accounts for the energy-momentum exchange between the fluid and the field. Here we apply a constant value of $\eta=2.2$/fm [@Biro:1997va]. The time evolution of the average value of the sigma field $\langle\sigma\rangle$ and the average temperature $\langle T\rangle$ is shown in figure \[fig:cf2\_hotregionsigmaeta22\]. The average is taken over an initially hot and dense sphere with radius $r=3$ fm. The phase transition temperature in a critical point scenario $T_c=139.88$ MeV is crossed at around $t=5$ fm, after which the slightly enhanced fluctuations fall off. The average sigma field smoothly relaxes towards its vacuum value. As the phase transition temperature of the first order phase transition, $T_c=123.27$ MeV, is lower than at a critical point the system relaxes later but the relaxational process itself is faster in a first order phase transition scenario. The vacuum value is reached around the same time, but the average sigma field shows strong oscillations and the fluctuations are enhanced in the first order phase transition scenario. ![The average of the sigma field (left) and of the temperature (right) for $\eta=2.2$/fm and both phase transition scenarios.[]{data-label="fig:cf2_hotregionsigmaeta22"}](hotregion_sigma_eta22.eps "fig:"){width=".45\textwidth"} ![The average of the sigma field (left) and of the temperature (right) for $\eta=2.2$/fm and both phase transition scenarios.[]{data-label="fig:cf2_hotregionsigmaeta22"}](hotregion_temp_eta22.eps "fig:"){width=".45\textwidth"} When the first order phase transition temperature is reached after $t\simeq 5$ fm large parts of the system are still in the chirally broken phase as the average value of the sigma field is still $\langle\sigma\rangle\lesssim10$ MeV. These large deviations of the sigma field from its equilibrium value is the nonequilibrium effect of supercooling. Due to the steep curvature in the effective potential experienced by the system once the barrier is overcome, the potential energy is transformed effectively in kinetic energy, which leads to the dissipation of energy via $\eta(\partial_t\sigma)^2$ in the source term. In figure \[fig:cf2\_hotregionsigmaeta22\] we can clearly observe the reheating effect at the first order phase transition. Between $t=7$ fm and $t=9$ fm the system is reheated from $T\simeq 118$ MeV below $T_c$ to $T\simeq 125$ MeV above $T_c$, followed by a subsequent cooling. Thus, the reheating causes the system to cross the phase transition two more times, once in the reverse direction from the low temperature phase to the high temperature phase around $t=8$ fm and again at around $t=9$ fm. This contributes to a slower relaxation of the average sigma field. The effective potential with a critical point is very flat at the transition temperature and reheating is not observed. Instead the cooling is slightly decelerated as seen in figure \[fig:cf2\_hotregionsigmaeta22\] between $t=5$ fm and $t=6$ fm. ![Time evolution of the deviation of the sigma field from the thermal equilibrium value for a scenario with a first order phase transition and with a critical point.[]{data-label="fig:flucgleich"}](flucgleichgewicht_cpfo22.eps){width=".75\textwidth"} The sigma field is locally initialized in equilibrium with the quark fluid. The subsequent time evolution of the averaged sigma fluctuations around the thermal equilibrium value plotted in figure \[fig:flucgleich\]. In a critical point scenario the fluctuations are slightly increased in the beginning of the expansion due to the dynamics of the system. As expected we do not observe an increase at the critical temperature. For the first order phase transition this is very different. Due to the effect of supercooling we find a large enhancement of nonequilibrium fluctuations after the phase transition temperature is reached. During relaxation parts of the system are reheated to temperatures above the phase transition which leads to a second increase in the fluctuations. Extentions of this model to include the deconfinement phase transition are presented in [@herold]. Summary ======= We presented results where for the first time dilepton emission is studied within a macro+micro hybrid approach. We found that three regions can be identified in the dilepton invariant spectra. The very low mass region of the spectrum is dominated by thermal radiation, the region around the $\rho$ meson peak is dominated by late stage cascade dilepton emission and the intermediate region receives both contributions from hadronic and QGP emission, with the QGP accounting for about half of the total emission. Such studies will be important to be able to connect dilepton measurements at the CBM experiment with the phase transition to the QGP. In the following we have shown results on the explicit propagation of fluctuations at the chiral phase transition within a dynamic model of heavy-ion collisions. During the expansion the system cools and crosses the phase transition where nonequilibrium effects occur. These are expecially dominant in a first order phase transition leading to supercooling and reheating of the entire system. Since nonequilibrium fluctuations are large at the first order phase transition and persist for some time below the phase transition there is a potential of observing the first order phase transition in heavy-ion collisions at the CBM experiment. This work was supported by the Hessian LOEWE initiative Helmholtz International Center for FAIR and used computational resources provided by the (L)CSC at Frankfurt. [99]{} H.H. Gutbrod et al. (Eds.) 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--- abstract: 'Monatomic nanowires of the nonmagnetic transition metals Ru, Rh, and Pd have been studied theoretically, using first-principles computational techniques, in order to investigate the possible onset of magnetism in these nanosystems. Our fully relativistic spin-polarized all-electron density functional calculations reveal the onset of Hund’s rule magnetism in nanowires of all three metals, with mean-field moments of 1.1, 0.3, and 0.7 $\mu_B$, respectively, at the equilibrium bond length. An analysis of the band structures indicates that the nanocontact superparamagnetic state suggested by our calculations should affect the ballistic conductance between tips made of Ru, Rh or Pd, leading to possible temperature and magnetic field dependent conductance.' address: - '$^1$Materialvetenskap, Brinellvägen 23, KTH, SE-10044 Stockholm, Sweden' - '$^2$Abdus Salam International Center for Theoretical Physics (ICTP), Strada Costiera 11, 34100 Trieste, Italy' - '$^3$International School for Advanced Studies (SISSA), via Beirut 2–4, 34014 Trieste, Italy' - '$^4$INFM DEMOCRITOS National Simulation Center, via Beirut 2–4, 34014 Trieste, Italy ' author: - 'A Delin$^{1,2}$ and E Tosatti$^{2,3,4}$' title: ' Electronic structure of 4d transition-metal monatomic wires ' --- Introduction ============ Reducing the dimensionality and size of a metallic object eventually leads to quantum confinement of the electrons in one or more dimensions. Examples of such systems are metallic nanowires, where the electrons are confined in two dimensions, but unconfined in the third dimension, along the wire. The ultimately smallest metallic wire consists of just a single metallic chain of atoms. Experimentally, long segments of such nanowires have been realized, in particular of Au[@kondo2000_helical; @rodrigues2000]. Production of shorter segments were recently reported for several other metals including the $4d$ transition metals Ru, Rh[@itakura2000], and Pd[@ugarte]. The quantum confinement of the electrons in the wires results in intriguing behaviour with respect to their mechanical, electrical and chemical properties and causes new physical phenomena to appear, for example quantized conductance[@wees1988] and helical geometries[@gulseren1998; @kondo2000_helical; @tosatti2001_tension]. Thus, the properties of these nanosystems may be dramatically different from the bulk properties of the same metals. In particular, it is interesting to explore whether and how nanowires of bulk nonmagnetic metals can become magnetic, and how other properties of these nanosystems in turn are affected by the presence of magnetism in the nanosystem, especially of a genuine Hund’s rule magnetic order parameter. We recently performed a similar study of the $5d$ metals Os, Ir, and Pt, which revealed intriguing magnetic properties of nanowire systems. In the present paper, we concentrate on the $4d$ transition metals Ru, Rh, and Pd, and contrast our results for these metals with our results for the corresponding $5d$ systems. We investigate the possibility of ferromagnetism[@note1] and its effect on other properties, notably quantized conductance for straight monostrand nanowires of these metals, using state-of-the-art all-electron computational methods based on density functional theory. We have also performed the corresponding calculations for the noble metal Ag, where no magnetism is expected, for comparison. We address here the physics of metallic nanowires suspended between two leads, where transmission electron microscope images on monostrand nanowires indicate straight wire geometries[@ugarte]. Nanowires can be stabilized in this way only temporarily, as the flow of atoms to the leads inevitably implies stretching and thinning, which eventually breaks the nanowire[@tosatti2001_tension]. A free, unsuspended chain of atoms would be totally unstable against an even larger set of deformations, for the final stable configuration will be a cluster, approximately spherical in shape, with a surface dominated by close-packed facets. In our calculations, we address strictly the straight wire geometry, with equidistant atoms. One could imagine more complicated monostrand wire geometries, for example zigzag geometries[@sanchezportal1999] or Peierls distortions, leading to di- tri- or multimerization[@peierls]. Such distorted configurations of an unsuspended monostrand wire may represent interesting local minima or saddle points in the total energy. When suspended between leads, however, local minima or saddle points of the string tension are to be considered instead of those of the energy, since they alone will correspond to long-lived, or “magic” nanowires[@tosatti2001_tension]. In Au, the zigzag deformations do not survive the string tension, and the same would happen, if they existed, in Ru, Rh, and Pd. Thus, we shall ignore zigzag distortions, since they are soft against tension, in the systems we address here. Peierls di-, tri- or multimerization distortions are critically dependent on a long wire as well as on a precise Fermi surface nesting, and would lead to insulating nanowires. In Ru, Rh, and Pd, the reported nanocontacts are three atoms long at most[@ugarte]. Moreover, there is no unique nesting since multiple bands cross the Fermi level, the precise Fermi crossings are tension-dependent, and the corresponding incommensurate order parameters are likely suppressed by size. The experimental evidence that nanocontacts of Ru, Rh, and Pd are consistently metallic further suggests neglecting Peierls distortions too until evidence to the contrary. In wires, the electrons are confined in two dimensions. Before investigating in more detail what effect that has on the magnetic properties of Ru, Rh, and Pd, let us summarize shortly what is known about the magnetic properties of these metals when the electrons are confined only in one dimension, or in all three. In a monolayer grown on, or sandwiched between, magnetically “inert” substrates such as Cu, Ag, Au, or graphite, the electrons are at least approximately confined in one dimension, opening up the possibility for two-dimensional magnetism. Search for two-dimensional magnetism in Ru, Rh or Pd in such systems has been conducted extensively both theoretically and experimentally. Starting with Ru, a monolayer of this metal has been observed to order ferromagnetically when grown on graphite[@pfandzelter1995], and when layered between graphene sheets[@suzuki2003]. No magnetism has been observed for Ru monolayers grown on Ag or Au surfaces. Theoretical calculations predict a Ru monolayer to be magnetic on graphite (under certain conditions)[@chen1997; @kruger1998], Ag[@eriksson1991; @blugel1992] and Au[@blugel1992], but nonmagnetic on Cu[@garcia1999]. What regards Rh metal, the only case in which two-dimensional magnetism has been observed is in a superlattice structure of Rh monolayers sandwiched between adjacent graphene sheets[@suzuki2003]. Monolayers of Rh grown on Ag, Au, or graphite have not shown any signs of magnetic order[@beckmann1997; @chado2001; @goldoni2001]. In great contrast to the experimental results, Rh monolayers have been predicted to order ferromagnetically on Cu[@garcia1999], Ag[@eriksson1991; @blugel1992], Au[@zhu1991; @blugel1992], and graphite[@chen1997; @kruger1998]. Finally, it has been predicted that a monolayer of Pd should be nonmagnetic on all substrates tested (Cu[@garcia1999], Ag[@eriksson1991; @redinger1995; @niklasson1997], and graphite[@chen1997; @kruger1998]). However, for Pd (and also Rh) films on Ag, calculations predict that the magnetic moment of the film is periodically suppressed and enhanced due to quantum well effects as a function of film thickness, giving rise to a finite ferromagnetic moment in certain films thicker than one monolayer[@niklasson1997]. All in all, the discrepancy between theory and experiment regarding magnetism in Ru and Rh monolayers appears to be rather large at present. One possible explanation for this discrepancy is diffusion of transition-metal atoms into the substrate, at least when the substrate is a noble metal. What regards Rh on graphite, the intricacies of this system have been discussed in detail in reference[@goldoni2001]. If we reduce the size of all three dimensions down to nanometer size, we end up with clusters. Small Ru, Rh, and Pd clusters have been predicted to have magnetic ground states[@galicia1993; @reddy1993; @vitos2000; @moseler2001]. Experimentally, magnetism has been observed in Rh and Pd clusters. Counter-intuitively, large Pd clusters appear to be magnetic whereas small Pd clusters are not[@cox1994; @sampedro2003; @taniyama1997]. Returning to magnetism in nanowires, it has been predicted that monatomic rows of Rh on Ag(001) are ferromagnetic, using a semi-empirical tight-binding method[@bazhanov2000]. Monoatomic rows of Ru, Rh, and Pd on vicinal surfaces of Ag have also been studied theoretically using a screened Korringa-Kohn-Rostocker Green function method[@bellini2001], predicting magnetism to appear in Ru and Rh chains, but not in Pd chains. Further, Spi[š]{}ák and Hafner[@spisak2003] predict ferromagnetism in Ru and Rh rows grown on the Ag (117) vicinal surface, using the projector augmented plane-wave computational method. Similarly, they also find a ferromagnetic ground state for Ru rows grown on Cu (117), and for freely hanging Ru and Rh mono-strand nanowires. Regarding the objects we study in this paper, i.e., monostrand wires hanging between leads, we can actually get some first clues about possible magnetism simply by starting from the atomic ground state of Ru, Rh, Pd, and Ag and analyzing the effect of a very weak hybridization of orbitals on adjacent atoms. In wire form, Ru and Rh could conceivably develop Hund’s rule magnetism, due to their partially empty narrow band-width $d$ shell. The Pd atom has in fact a filled $4d$ shell, but $sd$ hybridization in the wire enables $d\rightarrow s$ electron transfer, opening up the possibility to spin polarize Pd $d$-holes. Ag, on the other hand, is basically an $sp$ metal, with the $4d$ shell completely filled in all cases. Interestingly, it is in principle possible that wires of metals like Ag (a typical system that might be thought of as a jellium) in themselves magnetize under certain conditions, since even a jellium confined in a thin cylinder in principle magnetizes for certain radii of the cylinder[@zabala1998]. However, the moment formation in that case is weak and confined to very special radii or electron densities, and the associated energy gain is very small. That is of course so because exchange interactions, as described by Hund’s first rule, are not particularly strong in an $sp$ band metal or jellium. The situation is radically different for transition metals. Because of the partly occupied $d$ orbitals, their ability to magnetize is much stronger and of a fundamentally different nature compared to the jellium. In the discussion above we have completely neglected the issue of fluctuations. Thermal fluctuations in a nanowire are expected to be very large, and would destroy long range magnetic order in the absence of an external magnetic field. In earlier papers[@delin_5d_wires; @delin_pd_wire], we have argued in more detail how one might deal with fluctuations, and in what cases one can approximate some properties of the superparamagnetic state with those of a statically magnetized one (which is what we calculate), and we will not repeat those arguments here. Experimentally, evidence of one-dimensional superparamagnetism with fluctuations sufficiently slow on the time scale of the probe has been recently reported for Co atomic chains deposited on Pt surface steps[@gambardella2002]. ![ Sketch[@xcrysden] of the setup. Infinitely long wires extend along the $z$-direction and form a hexagonal mesh in the $xy$-plane. \[fig:figure\_1\] ](figure_1.eps) Method ====== The technical aspects of the present calculations are similar to those reported in our earlier papers on $5d$ metal nanowires[@delin_5d_wires]. For the present density-functional-based[@dft] electronic-structure calculations we used the all-electron full-potential linear muffin-tin orbital method (FP-LMTO)[@wills]. This method assumes no shape approximation of the potential or wave functions. The calculations were performed using the generalized gradient approximation (GGA)[@gga]. As a test, some calculations were also performed using the local density approximation (LDA)[@lda], giving results very similar to the GGA ones. Further, some calculations were double-checked using the WIEN code[@lapw; @wien97], again with very similar results. The calculations were performed with inherently three-dimensional codes, and thus the system simulated was an infinite two-dimensional array of infinitely long, straight wires. Figure \[fig:figure\_1\] shows a sketch of the setup. A one-dimensional Brillouin zone was used, i.e the k-points form a single line, stretching along the $z$-axis of the wire. The Bravais lattice in the $xy$-plane was chosen hexagonal. With this choice, the $d_{xy}$ and $d_{x^2 - y^2}$ orbitals become automatically degenerate, as they should for a single wire. Furthermore, we used non-overlapping muffin-tin spheres with a constant radius in the calculations of the equilibrium bond lengths $d$. The magnetic moments, bands structures, conductance-channel curves and band widths were calculated using muffin-tin spheres scaling with the bond length. Convergence of the magnetic moment was ensured with respect to k-point mesh density, Fourier mesh density, tail energies, and wire-wire vacuum distance. We performed both scalar relativistic (SR) calculations, and calculations including the spin-orbit coupling as well as the scalar-relativistic terms. The latter will be referred to as “fully relativistic” (FR) calculations in the following, although we are not strictly solving the full Dirac equation, or making use of current density functional theory. In the fully relativistic calculations, the spin axis was chosen to be aligned along the wire direction. [cccccl]{} & wire & bulk & bulk & moment & free\ & $d$ ([Å]{}) & $d$ ([Å]{}) & $d$ ([Å]{}) & ($\mu_B$) & atom\ metal & FR & FR & exp. & FR;SR & configuration\ & & & & &\ Ru & 2.27 & 2.70 & 2.71 & 1.1 & 4 ($^5 F_{5}$)\ Rh & 2.31 & 2.72 & 2.69 & 0.3 & 3 ($^4 F_{9/2}$)\ Pd & 2.56 & 2.78 & 2.75 & 0.7 & 0 ($^1 S_{0}$)\ Ag & 2.68 & 2.93 & 2.89 & - & 1 ($^2 S_{1/2}$)\ Results ======= Bond lengths and energetics --------------------------- In a monostrand nanowire, there are only two nearest neighbours, and therefore we expect the bond length minimizing the total energy to be smaller than in the bulk. This is indeed the case, as can be seen in Table \[tab:lattpar\], where calculated bond lengths for monowires and bulk are listed, together with the experimental bulk values. Our bulk GGA calculations for the equilibrium bond lengths are in very close agreement with the experimental values. Our calculated bond lengths for free-standing, monostrand wires are close to (within a few hundredths of an [Å]{}) the ones reported in reference[@bahn2001] (Pd and Ag), but significantly larger than (the differences are of the order 0.1 [Å]{}) the ones reported in reference[@spisak2003] (Ru, Rh, and Pd). We also wish to point out here that strictly speaking, a tip-suspended wire will not have a quasi-stable configuration at the bond length which minimizes the total energy, but at a slightly larger value since it is rather the string tension than the total energy which should attain a local minimum[@tosatti2001_tension]. Nevertheless, for simplicity, in the remainder of this paper, the bond length which minimizes the total energy will be called the equilibrium bond length. Table \[tab:lattpar\] also shows our calculated mean-field magnetic moments per atom at the equilibrium bond lengths. Note that the scalar relativistic and fully relativistic calculations predict the same magnetic moments within the precision given at the equilibrium bond length. Thus, the spin-orbit coupling appears to be unimportant what regards the existence and magnitude of the magnetic moments in the $4d$ metals Ru, Rh, and Pd. This makes a strong contrast to the situation in $5d$ nanowires of Os, Ir, and Pt, where relativistic effects were shown to be crucial for the correct description of the magnetic profiles[@delin_5d_wires]. Even for the $4d$ metal nanowires, however, it turns out that the spin-orbit coupling is by no means unimportant, as will be further elucidated in the analysis of the energetics, band structures, and conductance channels. Calculated magnetic moments for straight monostrand wires of Ru and Rh have been reported also by Spi[š]{}ák and Hafner[@spisak2003]. They found 0.98 $\mu_B$ for Ru, and 0.26 $\mu_B$ for Rh, which is in excellent agreement with our calculated spin moments. Our result for Pd differs from that of Bahn [*et al*]{}[@bahn2001] who found no magnetism in pseudopotential calculations for Pd monostrand nanowires. It seems possible that the disagreement could arise in this very borderline case due to the different methods used, in which case we would tend to trust our all-electron approach better. Experiments will have to be awaited in order to settle this question. The right-most column in Table \[tab:lattpar\] lists the experimental atomic ground state configuration, showing that the free Ru, Rh, Pd,and Ag atoms have spin moments of 4, 3, 0, and 1 $\mu_B$, respectively. Thus, we see that the predicted wire moments at the equilibrium bond length are much smaller than the magnetic moments of the free atom, except for Pd, where the atomic moment is zero, but the wire has a substantial magnetic moment of around 0.7 $\mu_B$. [cccccc]{} & $E_{\rm wire} - E_{\rm bulk}$ & $E_{\rm wire} - E_{\rm bulk}$ & $E_{\rm NM} - E_{\rm FM}$ & $E_{\rm NM} - E_{\rm FM}$\ & (eV) & (eV) & (meV) & (meV)\ metal & SR & FR & SR & FR\ & & & &\ Ru & 5.4 & 4.9 & 77 & 56\ Rh & 4.7 & 4.1 & 10 & 9\ Pd & 3.1 & 3.1 & 25 & 12\ Ag & 1.8 & 1.8 & - & -\ In order to analyze the relative stability of wire formation, we calculated the energy difference between wire and bulk. The results are displayed in Table\[tab:energies\]. The energy difference between wire and bulk is smallest for Ag (1.8eV), and increases as one goes left in the 4d series to Pd, Rh and Ru. Spin-orbit coupling somewhat reduces this energy difference with about 0.5eV for Ru and Rh, whereas it has no effect in Pd and Ag. Scalar-relativistic energy differences between monowire and bulk have been reported earlier for Pd and Ag, and our numbers agree well with that calculation[@bahn2001]. We also calculated the energy gain when the wire is allowed to spin polarize, $E_{\rm NM} - E_{\rm FM}$. This quantity is typically a few hundredths of an eV, and differs greatly from element to element. Spin-orbit coupling halves $E_{\rm NM} - E_{\rm FM}$ in the case of Pd, whereas the effect of spin-orbit coupling on magnetism is much smaller for Rh and Rh. In reference[@spisak2003], this energy difference was reported to be 39meV for Ru, and 6meV for Rh, i.e., smaller compared to the ones calculated in the present work (56 and 9meV, respectively). Magnetic moments and band structures ------------------------------------ The magnetic moments per atom as a function of nanowire bond length are shown in the left-most column of figure \[fig:figure\_2\]. The solid lines refer to the fully relativistic calculations, and the dotted lines to the scalar relativistic calculations. All the metals studied, except Ag, exhibit a magnetic moment for values of the bond lengths at equilibrium. Spin-orbit coupling apparently has a very limited effect on the magnetic profiles; for Ru and Rh, the difference is not even visible; for Pd however there is a small but visible quantitative difference. The magnetic profiles for Ru and Rh are quite similar to each other. For both metals, the magnetic moment increases with stretching, and reaches a plateau value for large bond lengths. The magnetic profile of Pd, on the other hand, is unique in that it has a maximum for bond lengths around the equilibrium value, and then decreases down to zero for stretched bond lengths (see also reference[@delin_pd_wire] for a more in-depth discussion of the Pd nanowire magnetic profile). The Ag wire is, unsurprisingly, firmly nonmagnetic, for the bond lengths studied. In order to shed some light onto the mechanisms behind the magnetic profiles displayed in figure \[fig:figure\_2\], we will now analyze the electronic structure of the wires with help of the band structures at different bond lengths. ![ Magnetic profiles and number of bands crossing the Fermi level for the four metals studied. The left-most column shows total magnetic moments per atom, as a function of bond length, both with spin-orbit coupling (FR, solid line) and without (SR, dotted line). The middle and right-most columns show the number of bands crossing the Fermi level as a function of bond length for the SR and FR calculation, respectively. The solid lines refer to ferromagnetic calculations, and the dotted lines are for nonmagnetic calculations. The dashed vertical lines point out the equilibrium bond lengths. \[fig:figure\_2\] ](figure_2.eps) ![ FR band structures, along the wire direction, at two different bond lengths (the equilibrium one, and a larger of 2.9 [Å]{}) for each element. The Fermi energy is at zero. Band doubling (present in panels a through f) indicates spin splitting due to magnetic order. \[fig:band\_structure\_fr\] ](figure_3.eps) ![ SR band structures, along the wire direction, at two different bond lengths (the equilibrium one, and a larger of 2.9 [Å]{}) for each element. The Fermi energy is at zero. Band doubling (present in panels a through f) indicates spin splitting due to magnetic order. \[fig:band\_structure\_sr\] ](figure_4.eps) ![ Character-resolved SR band structure for non-spin-polarized Pd, along the wire direction. The Fermi energy is at zero. The $d_{xz}$ and $d_{yz}$ orbitals are always degenerate, and the same is true for the orbital pair $d_{xy}$ and $d_{x^2-y^2}$. \[fig:fatbands\] ](figure_5.eps) Band structures for two different bond lengths, the equilibrium bond length, and a larger one of 2.9 [Å]{}, roughly representing two magnetic regimes, are shown in figure \[fig:band\_structure\_fr\] (fully relativistic calculation) and figure \[fig:band\_structure\_sr\] (scalar-relativistic calculation) for each of the four metals studied. The bands run from the zone center, $\Gamma$, to the zone edge, A, in the direction of the wire. The character of the bands close to the Fermi level is of critical importance for the moment formation, and therefore we also show character-resolved bands, see figure \[fig:fatbands\]. We found it useful to split up the $d$ character into three contributions: $d_z$, $(d_{xz},d_{yz})$ and $(d_{xy},d_{x^2-y^2})$. Thus, figure \[fig:fatbands\] has four panels, displaying separately the $s$, $d_z$, $(d_{xz},d_{yz})$ and $(d_{xy},d_{x^2-y^2})$ characters of the bands. The vertical error bars, or “thickness”, of the bands indicate the relative character weight. The data in figure \[fig:fatbands\] have been taken from a scalar-relativistic calculation for Pd. For the other metals, the relative weights of the orbitals for each band are qualitatively similar to the ones shown. From figure \[fig:fatbands\], we see that most bands in the vicinity of the Fermi level are of predominantly $d$ character. In fact, there are only two bands with some $s$ character crossing the Fermi level (see upper left panel in figure \[fig:fatbands\]). Of these, the highest lying band crosses the Fermi level closer to the zone center than the zone edge, at about one third of the distance between the zone center and zone edge. The second one of the two $s$-containing bands crosses the Fermi level very close to the zone edge (A). At that point, this band has some $s$ character, but is in fact dominated by $d_z$ character. At $\Gamma$ and A, (both critical points by symmetry), all band dispersions are horizontal. This gives rise to very sharp $1/\sqrt{E}$ band edge van Hove singularities, due to the one-dimensionality of the systems. If a band has mostly $d$ character at the edge, the exchange energy gain will be rather large if the band spin-splits so that one of the spin-channel band edges ends up above the Fermi level, and the other one below. We note in passing that, strictly speaking, in the fully relativistic calculations (figure \[fig:band\_structure\_fr\]) the spin-orbit coupling will mix the two spin channels so that, in general, an eigenvalue will have both majority and minority spin character. However, in the present calculations, this mixing is so small, typically just a few percent, that it is irrelevant for our qualitative discussion here. Thus, if a band edge ends up sufficiently near the Fermi level, we may expect a magnetic moment to develop. While apparently similar to the magnetization of the jellium wire[@zabala1998], magnetism here is much more substantial, since here the $d$ states involve a much stronger Hund’s rule exchange. We now go through all four metals, starting with Ru, analyzing how the band edges move as a function of bond length, and how this affects the magnetic state of the wires. [*Ru:*]{} The magnetic moment of Ru increases with the bond length. At the equilibrium bond length, the rather flat $(d_{xy},d_{x^2-y^2})$ bands are split around the Fermi level, creating a relatively large magnetic moment of 1.1 $\mu_B$, see panel a in figures \[fig:band\_structure\_fr\] and \[fig:band\_structure\_sr\]. The $(d_{xz},d_{yz})$ bands are still broad at this bond length and in principle unpolarized. As the bond length increases, the $(d_{xz},d_{yz})$ bands narrow down and eventually also spin-polarize. They split around the Fermi level, and causes a large magnetic moment of more than 3 $\mu_B$ for bond lengths larger than 2.7 [Å]{}. [*Rh:*]{} As in Ru, the magnetic moment of Rh increases with the bond length. In fact, the magnetic profile for Rh is very similar to the one of Ru, just shifted in magnitude. With one more electron than Ru, the bands of the Rh wire lie generally deeper. The result is that the flat $(d_{xy},d_{x^2-y^2})$ bands, just barely touch the Fermi level, instead of clearly crossing it, as in Ru. The result is a smaller splitting, and consequently a smaller moment. With increasing bond length, all the other bands also narrow down and eventually split around the Fermi level, just like for Ru. The result is a plateau value of the magnetic moment of about 2 $\mu_B$ for bond lengths above 2.6 [Å]{}. Thus, in both Ru and Rh, the flat $(d_{xy},d_{x^2-y^2})$ bands drive the formation of the magnetic moment, and the $(d_{xz},d_{yz})$ bands enhance it. [*Pd:*]{} In Pd, the very same flat $(d_{xy},d_{x^2-y^2})$ bands leading to Hund’s rule magnetism in Ru and Rh are entirely occupied at all bond lengths studied. The spin polarization is instead driven by the $s+d_{z^2}$ and $(d_{xz},d_{yz})$ bands, which have a high dispersion, and display one-dimensional band edges close to the Fermi level at $\Gamma$ and A. In the magnetic regime, it happens that these three band edges all are nearly degenerate in energy and close to the Fermi level. This accidental feature of the long Pd monostrand nanowire band structure dramatically increases the density of states, since divergent van Hove singularities are formed close to the Fermi level. For the stretched wire with bond length 2.9 [Å]{} (panel f in figures \[fig:band\_structure\_fr\] and \[fig:band\_structure\_sr\]), the $s$-dominated band with band edge at $\Gamma$ is split around the Fermi level. A more detailed discussion of the physics of Pd nanowires can be found in reference[@delin_pd_wire]. [*Ag:*]{} For Ag, basically an $sp$ metal, all the $d$ bands are filled and there is never any $d$-magnetism. The only band crossing the Fermi level is one of the $s$-dominated, high-dispersion, $s+d_{z^2}$ bands. Calculations have shown that monostrand wires of $sp$ metals, and even a jellium cylinder, can spin-split[@zabala1998; @ayuela2002]. For this to happen however, a band edge must be extremely close to the Fermi level. The $s+d_{z^2}$ band in the Ag wire has a band edge at $\Gamma$ approximately 1eV below the Fermi level, which is too far down to make a spin splitting possible. Ballistic conductance channels ------------------------------ As seen from the above discussion of the nanowire band structures, spin-splitting of bands does alter $n$, i.e., the number of bands (or channels) crossing the Fermi level. By virtue of the Landauer formula $$G = \frac{e^2}{h}\sum_i \tau_i,$$ where $\tau_i$ is the transmission through channel $i$, the maximum theoretical ballistic conductance has, in units of $\frac{1}{2}G_0 = e^2/h$, precisely the number of bands $n$ crossing the Fermi level as its upper limit. Thus, the conductance through the wires should be affected by the presence of magnetism. The middle column of figure \[fig:figure\_2\] (fully relativistic calculation) and right-most column of figure \[fig:figure\_2\] (scalar relativistic calculation) illustrate how $n$ is influenced by nanowire spin-polarization, bond length, and spin-orbit coupling. For Ru, Rh, and Pd in their magnetized state at the equilibrium bond length, $n$ is 9, 9, and 5, respectively, compared to 12, 10, and 8 in the nonmagnetic state when spin-orbit coupling is included in the calculation. In the scalar-relativistic calculations, the channel count at the equilibrium bond length for Ru and Rh is close but not identical to the channel count of the fully relativistic calculation. For Pd the two calculations give the same channel count at equilibrium bond length. For stretched wires, however, the scalar and fully relativistic calculations differ also for Pd in the number of conductance channels. In general, spin-polarization tends to decrease the number of channels. Should all the channels transmit fully, large ballistic conductances of $4.5 G_0$, $4.5 G_0$, and $2.5 G_0$ for Ru, Rh, and Pd, respectively would ensue, to be compared with nonmagnetic conductances of $6 G_0$, $5 G_0$, and $4 G_0$, respectively. In reality however, the situation will be quite different, due to the fact that most of the open channels have $d$ character. While the conductance of the $s$-dominated channels is generally close to one owing to nearly complete transmission, that of the $d$-channels is much smaller, with a high reflection at the lead-wire junction, generally dependent on the detailed junction geometry. Our calculations reveal that the occupation of each of the two $s$-channels is always finite for all bond lengths reported here, bringing an expected contribution close to $G_0$ to the total conductance. The $d$-channel contribution to the conductance is expected to be much smaller than $\frac{1}{2} G_0$ per channel. All in all, we may thus expect monatomic nanocontacts of Ru, Rh, and Pd to have a conductance above $G_0$ but well below $4.5 G_0$, $4.5 G_0$, and $2.5 G_0$, respectively. Since the scattering of the $d$ waves at the junctions depends highly on the geometry, whose details will change at every realization, we also expect the conductance histograms to exhibit peaks that could be both broad and poorly reproducible. Conductance histograms and traces for Ru and Rh nanocontacts have been measured by Itakura [*et al*]{}[@itakura2000]. In both these metals, they find a broad bump between $G_0$ and $2 G_0$, which is consistent with our results. Enomoto [*et al*]{}[@enomoto2002] have published conductance histograms and traces for Pd-Ag alloys. In their conductance histogram for pure Pd, there is no significant structure in the region $G_0$ - $4 G_0$, whereas for Ag, there is a sharp peak at $G_0$, as expected. Rodrigues [et al.]{}[@ugarte] have measured the conductance through pure Pd nanocontacts in ultra-high vacuum, and found a peak at very low conductance, around $0.5 G_0$. Discussion and conclusions ========================== Our all-electron calculations suggest that the Ru, Rh, and Pd monatomic nanowires exhibit spontaneous Hund’s rule magnetism for values of the bond length at and around equilibrium. The energy gain connected with the magnetic state is of the order of a few hundredths of an eV. This indicates that the magnetization could be stabilized against fluctuations at cryogenic temperatures, especially with the help of an external field. From a methodological point of view, the spin-orbit coupling is found to be important for a correct description of the energetics, and the number of $d$-dominated conductance channels. On the contrary, the spin moments themselves are rather insensitive to the spin-orbit coupling. How might this nanomagnetism be detected experimentally? Merely measuring the conduction through the wire at one single temperature and magnetic field strength will most probably not give conclusive information regarding the magnetic state of the atoms in the wire, since the transmission through $d$-channels is rather poor and vary greatly with geometry and can hardy be regarded as quantized. We speculate that the conductance may vary in the following qualitative way as a function of temperature and magnetic field strength. First, at low temperature and zero field a Kondo state between the conduction electrons and the nanomagnet could form, implying a high ballistic conductance[@costi]. Second, at high temperature and zero field Kondo-like effects could affect the ballistic conductance and cause it to drop[@costi]. Third, low temperature and a high magnetic field would take the nanowire to a magnetic, or in any case to a slowly fluctuating superparamagnetic regime. In this regime the number of conductance channels should diminish, and so should the conductance. At sufficiently low temperatures, the conductance should therefore be field sensitive, and that magnetoresistance would be a clear indication of a magnetic state. Therefore, a key experiment would be to measure ballistic conductance as a function of both temperature and external magnetic field. Fractional conductance peaks below $G_0$ have been observed experimentally, for example the $\frac{1}{2}G_0$ peak reported by Ono for Ni[@ono1999], and very recently by Rodrigues [*et al*]{} for Co, Pd and Pt[@ugarte], at room temperature and zero field. These results are intriguing, since we expect that the $s$-channels alone should yield a conductance larger than that. Impurities could be a possible explanation for these low-conductance peaks, as Untiedt [*et al*]{}[@untiedt2003] very recently demonstrated in the case of Pt. But several questions remain, such as why the $\frac{1}{2}G_0$ peak is much larger in Co and Pd than in Pt[@ugarte]. Therefore, it would be highly interesting to see if the half-conductance peak exists also in conductance histograms of Ru and Rh, and how the relative size of that peak, if it exists, varies with impurity concentration. We discussed in previous work[@smogunov2002], a possibility to obtain conductance $G_0$ from a magnetic transition metal nanowire with a magnetization reversal occurring inside the nanowire. This could in principle drop to $\frac{1}{2}G_0$ in an asymmetrical situation, with a net prevalence of majority spins over minority spins. We are however at the present time not able to explain how that kind of state could be sustained in Pd, at the experimental conditions of zero field and room temperature. Finally, we also note that the conductance histogram peaks in reference[@ugarte] for Co, Pd, and Pt, centered around $\frac{1}{2}G_0$, are rather broad, which suggests that they might not be caused by one single fully transmitting spin-polarized channel, but perhaps by several poorly conducting channels. In any case, more theory work will be needed to address the experimental data, explicitly including such elements as tip form, spin structures, impurities[@untiedt2003], strong correlations, and temperature as well as their effects on the system’s conductance. A.D. acknowledges financial support from the European Commission through contract no. HPMF-CT-2000-00827, STINT (Swedish Foundation for International Cooperation in Research and Higher Education), and VR (Swedish Research Council). Work at SISSA was also sponsored through TMR FULPROP, MIUR (COFIN and FIRB) and by INFM/F. 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--- abstract: 'Hybrid recommendation usually combines collaborative filtering with content-based filtering to exploit merits of both techniques. It is widely accepted that hybrid filtering outperforms the single algorithm, thus it has been the new trend in electronic commerce these years. In this paper, we propose a novel hybrid recommendation system based on weighted stochastic block model (WSBM). Our algorithm not only makes full use of content-based and collaborative filtering recommendation to solve the cold-start problem but also improves the accuracy of recommendation by selecting the nearest neighbor with WSBM. The experiment result shows that our proposed approach has better prediction and classification accuracy than traditional hybrid recommendation.' address: - 'Department of Mathematics, Jinan University, Guangzhou, China' - 'Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, China' author: - Yuchen Xiao - Ruzhe Zhong bibliography: - 'Recommendation.bib' title: A hybrid recommendation algorithm based on weighted stochastic block model --- Weighted stochastic block model,Hybrid recommendation. 90B15 ,91B74 ,93A30. Introduction ============ With the development of information technology and network technology, the scale of information from the Internet has rapidly increased in recent years. How to filter overloaded information effectively and recommend useful information to users has become a hot issue in recommendation systems. The current recommendation systems algorithms contain content-based, social-based [@wang2014online], context-aware, collaborative filtering [@candillier2007comparing; @bobadilla2012collaborative], knowledge-based [@shi2004intelligent; @bobadilla2013recommender], graph-based [@wei2013distinguishing] and hybrid recommendation [@porcel2012hybrid]. Among all the algorithms, content-based, collaborative filtering and hybrid recommendation are widely used. Content-based recommendation system [@van2000using; @salter2006cinemascreen] determined the preferences of users by their choices made before. Content-based recommendation extracts the description documents of items and determines whether the items suit users by comparing them with user’s preference. Collaborative filtering algorithm [@candillier2007comparing; @su2009survey] is one of the most widely used and mature recommendation methods [@burke2002hybrid]. Its main idea is to use the historical rating data of the user’s nearest neighbor to predict the rating of him or her, and then recommend the Top-N items for him or her. Breese divided the collaborative filtering algorithm into two parts: memory-based and model-based [@breese1998empirical]. Memory-based algorithm is simple but its accuracy will reduce with the increase of users, while model-based algorithm need to create new models frequently to adapt to the change of users or projects. It is evident that different recommendation methods have different advantages and disadvantages, so hybrid recommendation [@xu2005content; @borras2014intelligent] has become a commonly used algorithm in recommendation system. For example, Ariyoshi and Kamahara proposed a hybrid information recommendation method by applying singular value decomposition (SVD) on both collaborative filtering and content-based filtering respectively to reduce the cost of computation [@ariyoshi2010hybrid]. Lucas et al. divided the users into groups using personal demographic data (Demographic-based), content information of the items previously selected by the user (Content-based) and the information of other users (Collaborative filtering) [@lucas2013hybrid]. Rathachai et al. proposed a prediction model which is a combination of three scoring functions, and took collaborative filtering, community structure, and biological classification into account [@chawuthai2014link]. It can not only take full advantages of a variety of recommendation technologies but also avoid their disadvantages. The accuracy of the recommendation can be increased by using it. According to the users’ basic information and historical users’ rating data, we propose a hybrid recommendation based on WSBM. Our hybrid recommendation contains two parts: the content-based part and the collaborative filtering part. The content-based part predicts the users’ ratings through the similarities of item documentations, so as to recommend the new item. The collaborative filtering part combines the similarity of users’ basic information with the similarity of historical users’ rating data and constructed the overall similarity, which can achieve recommendation for new users. Moreover, in this part, we generate a user-user social network based on the same purchase records and adopt the WSBM to find the nearest neighbor, which improves the accuracy of the algorithm. Hybrid recommendation based on WSBM =================================== Content-based rating prediction ------------------------------- ### Building the feature document of items Vector space model (VSM) is used in this paper to build the feature document of items. We firstly extract the description documents of items from the web, and remove the stop words and function words. According to tf-idf [@kim2014noise], the feature weight of the description documents can be calculated, and the feature words above the given threshold $\sigma$ can be chosen to build a feature space. In the feature space, treat the feature words as key code and the weights as the values for the key code. Thus we get the document-word frequency matrix $X_{r\times s}$ (where $r$ is the number of feature words and $s$ is the number of the items contained in item set $N$). We use latent semantic analysis [@zhong2010novel] to map the document into a lower dimensional latent semantic space and we obtain an approximate matrix $X_M$ as follows $$\begin{aligned} \label{eq1} X_M={U_M}{S_M}{V_M}^T.\end{aligned}$$ ### Predicting rating According to the approximate document-word frequency matrix $X_M$, we calculate the similarity between item $I_j$ and item $I_k$ in (\[eq2\]) to obtain the similarity matrix for $j,k=1,2,\ldots,s$. To predict the rating of user $U_a$ for item $I_j$, we select the items which have been rated by user $U_a$ to form a reference set $I_c^*$. According to the similarity matrix and the reference set, we can predict the rating in (\[eq3\])[@ghauth2010learning] $$\begin{aligned} \label{eq2} sim(I_j,I_k)=\cos(I_j,I_k)=\frac{w_k\cdot w_j}{\lVert w_k\rVert\lVert w_j\rVert},\\ \label{eq3} p_{a,j} = \frac{\sum\limits_{k\ne j,I_k\in I_c^*}sim(I_j,I_k)p_{a,k}}{\sum\limits_{k\ne j,I_k\in I_c^*}sim(I_j,I_k)}.\end{aligned}$$ where $w_j$ is the $j^{th}$ column of $X_M$, $p_{a,j}$ is the rate of user $U_a$ for item $I_j$. Rating prediction of collaborative filtering -------------------------------------------- ### Selecting the nearest neighbor based on WSBM There are two commonly used methods to select the nearest neighbors: one is to select neighbors whose similarity to the target user is larger than a threshold [@kim2007effective], the other is to search for the neighbors who have the greatest $N$ similarities to the target user [@bobadilla2011framework]. Both of the two methods are not universal on different datasets for the reason that the best recommendation can be achieved on different threshold or $N$. As the community is considered to have a high correlation among users in social networks, we consider that users may have similar interests if they purchase same things. So we define the users as the vertexes of the network and if users have the same purchase, an edge is established between them, thus the numbers of same purchases are defined as the weights of edges. After constructing a weighted network, we adopt the WSBM to detect the community structure and to find the vertex’s nearest neighbors in the community. Without relying on the similarities, this method is more universal on different datasets. Based on Stochastic Block Model, WSBM [@aicher2013adapting] defines the distribution of edge as two parts: the existence distribution and the weight distribution. For vertexes in the same community, to avoid heavy-tailed degree distributions, the model revises the probability of edge’s connection between one vertex to another according to the vertex degree. In the WSBM, we define $A$ as the adjacency matrix of the weighted network $N$, and $A_{ij}$ as the weight of the edge between the vertexes $i$ and $j$. The integer $K$ denotes a fixed number of latent communities, and the vector $Z_{n\times 1}$ contains the community labels of each vertex. The WSBM defines a “bundle” of edges that run between each pair of communities ($kk'$) and assigns an edge existence parameter to each edge bundle $kk'$, which we represent collectively by the matrix $\theta_{K\times K}$. The existence probability of an edge $A_{ij}$ is given by the parameter $\theta_{Z_i}\theta_{Z_j}$ that depends only on the community memberships of vertexes $i$ and $j$. Therefore, the model is fully given by $\theta_{K\times K}$ and $Z_{n\times 1}$ [@aicher2014learning]. WSBM models an edge’s existence as a Bernoulli or binary random variable and an edge’s weight using an exponential family distribution. With the Bernoulli distribution to simulate the existence distribution, vertex degree information is added into the “edge-propensity” parameter $\phi_i$ to each vertex. As a result, the existence probability of an edge $A_{ij}$ is a Poisson random variable with mean $\phi_i\phi_j\theta_{Z_i}\theta_{Z_j}$. Because the maximum likelihood estimate of each propensity parameter $\phi_i$ is simply the vertex degree $d_i$, by fixing $\phi_i=d_i$, we can obtain the likelihood function for this model $$\begin{aligned} P(A|Z,\theta)\propto\prod_{ij}\exp(A_{ij}\cdot\log\theta_{Z_iZ_j}-d_id_j\cdot\theta_{Z_iZ_j}),\end{aligned}$$ which can be rewritten as $$\begin{aligned} P(A|Z,\theta)\propto\sum_{ij}T_{\epsilon}(A_{ij})\cdot\eta_{\epsilon}(\theta_{Z_iZ_j}).\end{aligned}$$ where $T_{\epsilon}=(A_{ij},-d_id_j)$ is the sufficient statistics and $\eta_{\epsilon}=(\log\theta_{Z_iZ_j},\theta_{Z_iZ_j})^\top$ is the natural parameters. With the exponential family distribution to simulate the existence of the distribution of the edges’ weight, for the given parameters $Z$ and $\theta$, we assume that $A_{ij}$ is conditionally independent, thus $A_{ij}$ is an exponential random variable parametrized by $\theta_{Z_iZ_j}$ over domain $\mathcal X$. Hence the likelihood has the form of an exponential family $$\begin{aligned} P(A|Z,\theta)\propto\exp(\sum_{ij}T_{\omega}(A_{ij})\cdot\eta_{\omega}(\theta_{Z_iZ_j})).\end{aligned}$$ Choosing an appropriate $(T_{\omega},\eta_{\omega})$, we can specify a stochastic block model, which weights are drawn from an exponential family distribution. Then we may combine their contributions in the likelihood function via a simple tuning parameter $\alpha\in[0,1]$ that determines their relative importance in inference $$\begin{aligned} \log P(A|Z,\theta)=\alpha\sum_{ij\in E}T_{\epsilon}(A_{ij})\cdot\eta_{\epsilon}(\theta_{Z_iZ_j})+(1-\alpha)\sum_{ij\in W}T_{\omega}(A_{ij})\cdot\eta_{\omega}(\theta_{Z_iZ_j}),\end{aligned}$$ where $E$ is the set of observed interactions (including non-edges) and $W$ is the set of weighted edges ($W\subset E$). In order to deeply understand the structure of the network, a reasonable number of latent block structure $K$ should be chosen. Moreover, we need to learn the parameters $Z$ and $\theta$ by exploiting maximum-likelihood function. Steps of the algorithm are as follows Determine the number of block structures $K$. Adopt the Bayesian model in search of the $K$ that maximizes the posterior probability $P(m|A)$, which is $$\begin{aligned} p(m|A)=\frac{p(A|m)p(m)}{\sum\limits_{m\in M}p(m,A)},\end{aligned}$$ where $m$ denotes a specific model, and $M$ denotes the model set with different $K$. Learn parameters $Z$ and $\theta$ - \(a) According to Variational Bayesian Model, the log-likelihood function is $$\begin{aligned} \log A=L(q)+KL(q\lVert p(\cdot|A)),\end{aligned}$$ where $\log A$ is a constant, and $$\begin{aligned} L(q)&=\int\sum\limits_Z q(Z,\theta)\log\frac{p(A,Z,\theta)}{q(Z,\theta)}d\theta,\\ KL(q\lVert p(\cdot|A))&=-\sum\limits_Z\int q(Z,\theta)\log\frac{p(Z,\theta|A)}{q(Z,\theta)}d\theta.\end{aligned}$$ - \(b) Maximize the lower bound $L(q)$, thus we get the minimum $KL$ distance, then $q(Z,\theta)$ is the nearest distribution to the posterior distribution $p(Z,\theta|A)$. - \(c) Based on total variation theory $$\begin{aligned} q(Z,\theta)=q_\theta(\theta)\prod_{i=1}^n q_Z(Z_i),\end{aligned}$$ and by using Expectation–maximization algorithm (EM algorithm) to optimize $q_Z(Z_i)$ within E-step and optimize $q_\theta(\theta)$ within M-step, both the optimal parameters $Z$ and $\theta$ can be figured out. Through the process above, all users in the network will be classified into $K$ communities. As for a certain user, the rest users in the same community are regarded as its nearest neighbors. We use $NN_a$ to represent the set of nearest neighbors of the user $U_a$. With the increase of web users, it is necessary to update the network. We can define a threshold of new users. When the number of new users reaches a specified limit, redetect the communities. Owing to the offline operation of community detection, the efficiency of online recommendation will not be affected. ### Similarity calculating 1. Similarity Based on Users’ Basic Information Albert et al. find that the user’s basic information such as gender, age, occupation, cultural background would have a relatively large impact on its interest preference [@albert2000error]. We calculate the similarity of basic information between target user $U_a$ and $U_i$ by constructing vector $\overrightarrow{use_i}$ from the basic information of user $U_i$ [@zhang2014collaborative],that is $$\begin{aligned} sim_1(U_i,U_a)=\cos(U_i,U_a)=\frac{\overrightarrow{use_i}\cdot\overrightarrow{use_a}}{\lVert\overrightarrow{use_i}\rVert\lVert\overrightarrow{use_a}\rVert}.\end{aligned}$$ 2. Similarity Based on Users’ Rating We collect the rating, evaluation of user’s and then clean, convert and entry them. Eventually we form a data matrix, containing each item evaluated by users $A=(R_{ij})_{m\times n}$, where $R_{ij}$ stands for the existing rating of user $U_a$ for item $I_j$, the formula presents as follows $$\begin{aligned} sim_2(U_i,U_a)=\cos({\vec{\vphantom{A}i}},{\vec{\vphantom{A}a}})=\frac{{\vec{\vphantom{A}i}}\cdot{\vec{\vphantom{A}a}}}{\lVert{\vec{\vphantom{A}i}}\rVert\lVert{\vec{\vphantom{A}a}}\rVert},\end{aligned}$$ where ${\vec{\vphantom{A}i}}$ and ${\vec{\vphantom{A}a}}$ are the rating vectors of user $U_i$ and $U_a$ respectively. Now we can define the integrated similarity $sim(U_i,U_a)$ as $$\begin{aligned} \label{eq4} sim(U_i,U_a)= \begin{cases} sim_1(U_i,U_a),&\text{$U_a$ is a new user,}\\ \alpha sim_1(U_i,U_a)+\beta sim_2(U_i,U_a),&\text{$U_a$ is an old user,} \end{cases}\end{aligned}$$ where $\alpha$ and $\beta$ are adjustable parameters which show the contribution of users’ basic information and users’ rating to the recommendation system. ### Predicting rating The reference set $U_a^*$ is formed by users included in $NN_a$ as well as having evaluated on $I_j$. We use all ratings of $U_i$ $(U_i\subset U_a^*)$ to predict $pp_{a,j}$ (the rating of user $U_a$ for item $I_j$) [@xie2013similarity], which is $$\begin{aligned} pp_{a,j}=\overline{R_a}+\frac{\sum\limits_{U_i\in NN_a}sim(U_i,U_a)(R_{i,j}-\overline{R_i})}{\sum\limits_{U_i\in NN_a}\lvert sim(U_i,U_a)\rvert},\end{aligned}$$ where $R_{i,j}$ is the rating of $U_i$ for item $I_j$, $\overline{R_i}$ and $\overline{R_a}$ are the average rating of $U_i$ and $U_a$ respectively. Producing recommendation ------------------------ Therefore, the final rating that target user $U_a$ given to item $I_j$ can be defined as follows $$\begin{aligned} \label{eq5} P_{a,j}=\gamma_1 p_{a,j}+\gamma_2 pp_{a,j},\end{aligned}$$ where $\gamma_1$ and $\gamma_2$ are adjustable parameters which shows the contribution of item-based and collaborative filtering to the recommendation system. In the initial stage of system operation, content-based is more accurate than collaborative filtering so $\gamma_1>\gamma_2$. And then collaborative filtering will be more accurate with the increasing number of users and rating, so $\gamma_1<\gamma_2$. Ultimately we recommend the items of which the ratings ranked top N to user $U_a$. Improving the rating of user ---------------------------- To some extent, the ratings of users can describe the preferences of them and the possible range is $0$ to $5$. The higher the ratings are the more users are interested in the items. If the recommendation system only considers users’ ratings, there might be some simulated trading to improve the ratings which will reduce the accuracy of recommendation. In addition to this, the ratings mainly depend on the first impression so most of the information on quality cannot be reflected. If we only consider the ratings, it is difficult to distinguish the qualities of items thus the accuracy of recommendation will be inaccurate. Additional comments can reflect the follow-up and long-term quality evaluation for items and they also synthesized feedbacks for service attitude and level. Therefore we can take the additional comments into account when calculate similarity, and the steps of additional comments processing are presented as follows Extract the keywords of original comments scored ($1$ to $5$) to form the keyword set ($K_1$ to $K_5$), and then take the union set of them to form the overall keywords set $K^*$. Calculate the weights of words in $K_l$ according to TF-IDF and construct the word frequency vector of rating $l$ which we denote as $V_l$ for $l=1,2,\ldots,5$. Extract the keywords of additional comment of user $U_a$ for item $I_j$, then calculate the weights of these words, thus we get the frequency vector of the additional comment which we denote as $AV_j$ for $j=1,2,\ldots,s$. Calculate $sim(AV_j,V_l)$ and denote the rating of the additional comment as the score $l^*$ when $sim(AV_j,V_l)$ is the highest, thus we get the appended rating of user $U_a$ for item $I_j(AR_{a,j})$. Therefore, we get the final rating $FR_{a,j}$ as follows $$\begin{aligned} FR_{a,j}=\eta_1R_{a,j}+\eta_2AR_{a,j},\end{aligned}$$ where $\eta_1$ and $\eta_2$ are adjustable parameters which show the contribution of users’ rating and users’ additional comments to the final ratings. For those without additional comments, they can be thought of having no particular like or dislike, so we define the appended rating of those people as a medium grade 3. Algorithm design ---------------- Our hybrid recommendation system based on WSBM is a combination of content-based filtering algorithm and collaborative filtering algorithm. The algorithm is expressed as follows Determine whether the user is a new user. If yes, go to \[step5\] Otherwise, go to \[step2\] Content-based rating prediction. Calculate the similarity between items after extracting feature documents of items. Then predict the rating $p_{a,j}$ of $U_a$ for item $I_j$ by using historical rating data. \[step2\] Selecting the nearest neighbor based on WSBM. Generate a user-user social network based on the same purchase records and structure the nearest neighbor set $NN_a$ of $U_a$ based on WSBM. Calculate the similarity $sim_2(U_i,U_a)$ by using the historical rating of $U_a$ and $U_i$ $(U_i\in NN_a)$. Calculate the similarity $sim_1(U_i,U_a)$ by using the basic information of $U_a$ and others. \[step5\] Calculate the integrated similarity $sim(U_i,U_a)$ which is defined by (\[eq4\]). Then predict the rating $pp_{a,j}$ of user $U_a$ for item $I_j$. Calculate the final predicted rating $P_{a,j}$ which is defined by (\[eq5\]). Producing Recommendation. Recommend the items of which the ratings rank top N to user $U_a$. The whole process is shown in Figure \[fig.flowchart\]. (start) \[startstop\] [Start]{}; (in1) \[io, below of = start\] [Initial $x_0=(x_{01},x_{02},\cdots)$]{}; (dec1) \[decision, below of = in1\] [New User?]{}; (pro2) \[process, below of = dec1, xshift = -8cm\] [Content-based rating prediction]{}; (pro3) \[process, below of = dec1\] [Selecting the nearest neighbor based on WSBM]{}; (pro4) \[process, below of = pro3\] [Similarity calculation (users’s rating)]{}; (pro5) \[process, below of = pro4\] [Similarity calculation (users’ basic information)]{}; (pro6) \[process, below of = pro5\] [Similarity and predicted rating calculation]{}; (pro7) \[process, below of = pro6\] [Final prediction rating]{}; (out1) \[io, below of=pro7\] [Output $x_0^*$]{}; (stop) \[startstop, below of=out1\] [stop]{}; (start) – (in1); (in1) – (dec1); (dec1) –node \[above\] [Y]{} (6,-4); (6,-4) – (6,-10); (6,-10) – (pro5); (0,-5) – (-8,-5); (-8,-5) – (pro2); (pro2) – (-8,-14); (-8,-14) – (pro7); (dec1) –node \[right\] [N]{} (pro3); (pro3) – (pro4); (pro4) – (pro5); (pro5) – (pro6); (pro6) – (pro7); (pro7) – (out1); (out1) – (stop); Experiments =========== An example of Improved Rating ----------------------------- In order to prove the improvements of the ratings, we give a small example. We extract the 50000 ratings with comments from a well-known Chinese online shopping website: Suning (<http://www.suning.com/>) through the web crawler technology. We prove the correctness of our algorithm with 25 additional comments and their initial ratings, improved ratings and additional comments are shown in Table \[table1\]. Initial ratings Improved ratings Appended Comments ----------------- ------------------ ----------------------------------------------------------------------------------- 5 3.5 Seems like a pregnant! 5 4.5 A pilling sweater! Poor quality! 4 4.5 Not bad! I like it. 5 4.5 It looks pretty good, but the color began to fading no longer than half a month! 5 4.5 Warm but rough! 4 4.5 An ideal piece! It is appropriate for autumn wear. 5 4.5 It has thread breakage. 5 3 Three stars. Look before you buy it! 5 4.5 I love it, but it is more expensive than that in Tmall. 5 5 Nice! I’ll be back! 5 4 Not so good. Little holes appeared at the sleeves. It might not be worth it. 5 4.5 I placed the order two weeks ago but I haven’t received it yet! What should I do? 5 4 It is worn-out though I only wear a few times! 5 5 Good quality! And it fits me very well! My friends think highly of it! 4 4.5 So far so good. Not bad. 1 1 They make a fool of the customers! Be careful of it! 5 4 Uncomfortable! It has an unmatched waist line. 4 4.5 A fair price! 2 3.5 It is worth buying. 4 4.5 Warm clothes! My parents are fond of them. We’ll be return customers! 4 4.5 It fits my mom very well! 2 3.5 Excellent quality and reasonable price! 5 4.5 I can’t really recommend it to you! 5 5 Cheap and comfortable! 5 5 Good! I will buy another one! : The evaluation of improving ratings[]{data-label="table1"} It can be seen from Table \[table1\] that differences do exist between initial rating and improved rating. If the additional comments are negative, the improved ratings will be lower than the initial one. Otherwise, the rating will be greater. In conclusion, the changes of the ratings are consistent with the additional comments and the improved ratings method has been proved. Experiment of recommendation accuracy ------------------------------------- ### Data - Dataset\ We use MovieLens dataset to evaluate our algorithm (<https://movielens.org/>). It is a public available dataset that consists of 100000 ratings from 943 users on 1682 movies. Each user has rated at least 20 objects and made a rating on a scale of 1 to 5. - Data Processing\ Users’ basic information contained in MovieLens is shown as follows User ID Age Gender Occupation --------- ----- -------- ------------ ... ... ... ... : The evaluation of improving ratings[]{data-label="table2"} For the sake of calculating conveniently, we turn information into digital form. Specifically, the age section is divided into seven groups: below 18, 18-25, 26-35, 36-45, 46-50, 51-56 and above 56, respectively with the representative integers from 1 to 7. Similarly, we represent 21 occupations with the integers 1 to 21, and genders with 0 and 1.Thus users’ basic information can be vectorized. Age Quantized Value Gender Quantized Value Occupation Quantized Value ---------- ----------------- ---------- ----------------- --------------- ----------------- $<18$ 1 male 1 administrator 1 $\cdots$ $\cdots$ female 2 $\cdots$ $\cdots$ $>56$ 7 $\cdots$ $\cdots$ doctor 21 : Process rule of u.user[]{data-label="table3"} According to the Bayesian model in WSBM, we figure out the optimal result: 4 block structures. After community detection, we get four communities the number of which is 273, 228, 236 and 206 respectively. ### The accuracy of recommendation At present, there are two commonly used types of evaluation metrics to evaluate the quality of recommendation system, prediction accuracy and classification accuracy. Prediction accuracy such as MAE and RMSE is to measure the compact degree between the predicted rating and actual rating. Classification accuracy such as Precision, recall and F-measure is to measure how accurately it can predict whether users like or dislike [@bobadilla2013recommender]. Without measure the accuracy of prediction directly, any deviation is allowed as long as it has no effect on classification. Mean absolute error (MAE), it measures the quality of recommendation by calculating the mean absolute error between the actual rating to predicted rating. We obtained the users’ predicted rating values set $\{p_1,\ldots,p_N\}$ and the actual rating values set $\{q_1,\ldots,q_N\}$. MAE is defined as follows $$\begin{aligned} MAE=\frac{\sum\limits_{i=1}^N\vert p_i-q_i\vert}{N}.\end{aligned}$$ To define the precision, recall and F-measure, we should firstly classify the items which have not been chosen or scored by users. There are four possibilities for a single item: The recommendations that users actually like are classified as True Positive (TP), and the others are classified as False Positive (FP). (For a five-grade marking system, the score of user-liked items is not less than 3).The items that are not recommended but that users actually like are classified as False Negative (FN), and the others are classified as True Negative (TN). Preference Recommended Not Recommended ------------ --------------------- --------------------- Like True Positive (TP) False Negative (FN) Dislike False Positive (FP) True Negative (TN) : Four possibilities for a single item[]{data-label="table4"} Precision is defined as the ratio of the number of items which users like as well as recommended by the system to the total number of the recommendation list. $$\begin{aligned} Precision=\frac{1}{N}\sum_u\frac{TP}{TP+FP}.\end{aligned}$$ Recall is defined as the ratio of the number of items which users like as well as recommended by the system to the total number of user-liked items. Therefore, the recall is given by $$\begin{aligned} Recall=\frac{1}{N}\sum_u\frac{TP}{TP+FN}.\end{aligned}$$ Therefore we get the F-measure $$\begin{aligned} \text{F-measure}=\frac{2\times Precision\times Recall}{Precision+Recall}.\end{aligned}$$ ### Performance of the method Considering the score prediction and classification accuracy of the algorithm, we use all of the evaluation metrics mentioned to evaluate the performance of our recommendation. As different divisions of training test have great impact on the algorithm accuracy, so we divided MovieLens into training set and test set according to different ratio and then compare MAE and RMSE of conventional and WSBM-based hybrid recommendation. According to the existing research [@del2008evaluation], precision, recall and F-measure have strong dependence on the length of recommendation. The evaluation metrics will have large variation if the length of recommendation list changes, so we fix the proportion of training set and test set (8:2) to compare precision, recall and F-measure of conventional and WSBM-based hybrid recommendation for different length of list. We define variable $\alpha, \beta, \gamma_1, \gamma_2$ as $0.2, 0.8, 0.4, 0.6$ and and obtain the following result. Table \[table5\] and Table \[table6\] show the MAE and RMSE of our hybrid recommendation based on WSBM and the traditional hybrid recommendation respectively. The tendency of evaluation metrics change with the different proportion of the training set can be shown in the line charts (Figure \[fig.mae\] and Figure \[fig.rmse\]). Define $P^*$ is the proportion of training set and test set. $P^*$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Ours 0.8299 0.8213 0.8202 0.8161 0.8153 0.7977 0.7790 0.7717 0.7683 Traditional 0.9869 0.9961 0.9593 0.9535 0.9515 0.9409 0.9141 0.8918 0.8884 Improvement ($\%$) 15.91 14.99 14.50 14.41 14.31 15.22 14.78 13.44 13.52 : MAE of our method and traditional method for MovieLens[]{data-label="table5"} ![The comparison of MAE between our method and traditional method[]{data-label="fig.mae"}](MAE.png){width="48.00000%"} Table \[table5\] and Figure \[fig.mae\] show that when the proportion of the training set varies from 0.1 to 0.9, MAE reduce in both methods and reach the lowest points (0.7683 in our hybrid recommendation and 0.8884 in traditional one). So, our hybrid recommendation performs better than the traditional one. For example, our method reduce MAE by $15.91\%$ compared with traditional when $P^*=0.1$. $P^*$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Ours 1.0217 1.0134 1.0118 1.0095 1.0071 0.9966 0.9781 0.9707 0.9678 Traditional 1.2288 1.2005 1.1940 1.1769 1.1699 1.1664 1.1072 1.0766 1.0712 Improvement ($\%$) 16.85 15.59 15.26 14.22 13.92 14.56 11.66 9.84 9.65 : RMSE of our method and traditional method for MovieLens[]{data-label="table6"} ![The comparison of RMSE between our method and traditional method[]{data-label="fig.rmse"}](RMSE.png){width="48.00000%"} Table \[table6\] and Figure \[fig.rmse\] indicate the increase of the proportion of training set from 0.1 to 0.9 results in decrease of RMSE in both methods. When the proportion of training set reaches 0.9, RMSE fall to 0.9678 in our method and 1.0712 in traditional method. It can be seen from the figure that our method achieves a lower RMSE than traditional method. When $P^*=0.1$, RMSE of our method is 16.85$\%$ lower than that of traditional method. To sum up, a larger size of the training set will lead to a better result in recommendation. Compared to the traditional hybrid recommendation, our method achieves better MAE and RMSE which means that selecting the nearest neighbors by WSBM improves the quality of recommendation. Table \[table7\], Table \[table8\] and Table \[table9\] show the precision, recall and F-measure of our hybrid recommendation based on WSBM and the traditional hybrid recommendation respectively. We also draw the line charts of them to learn the effects of different length of recommendation list (Figure \[fig.comparepre\], Figure \[fig.comparerecall\] and Figure \[fig.compareF\]). Define $L^*$ is the length of the recommendation list. $L^*$ 5 10 15 20 25 30 35 40 45 50 -------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Ours 0.7307 0.6823 0.6604 0.6460 0.6361 0.6290 0.6240 0.6210 0.6185 0.6166 Traditional 0.6466 0.6343 0.6270 0.6192 0.6176 0.6160 0.6142 0.6130 0.6116 0.6111 Improvement ($\%$) 13.02 7.57 5.33 4.33 3.00 2.12 1.58 1.30 1.12 0.91 $L^*$ 55 60 65 70 75 80 85 90 95 100 Ours 0.6148 0.6135 0.6126 0.6118 0.6110 0.6102 0.6097 0.6092 0.6088 0.6085 Traditional 0.6103 0.6098 0.6090 0.6089 0.6084 0.6082 0.6080 0.6076 0.6075 0.6074 Improvement ($\%$) 0.74 0.61 0.59 0.47 0.29 0.26 0.22 0.19 0.43 0.33 : Precision of our method and traditional method for MovieLens[]{data-label="table7"} ![The comparison of precision between our method and traditional method[]{data-label="fig.comparepre"}](Comparepre.png){width="48.00000%"} It can be seen from Table \[table7\] and Figure \[fig.comparepre\] that as $L^*$ increases, precision of the two methods decrease from 0.7307 in our method and 0.6466 in traditional method and finally converge to a fixed value. The precision of our method is always higher than that of traditional method, indicating a better accuracy of our method. When $L^*=5$, our method improves the precision by 13.02$\%$ compared with the traditional method. $L^*$ 5 10 15 20 25 30 35 40 45 50 -------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Ours 0.5668 0.7531 0.8418 0.8885 0.9187 0.9375 0.9506 0.9603 0.9676 0.9732 Traditional 0.5203 0.7221 0.8169 0.8662 0.9022 0.9255 0.9414 0.9527 0.9607 0.9674 Improvement ($\%$) 8.93 4.30 3.04 2.58 1.84 1.29 0.98 0.8 0.72 0.6 $L^*$ 55 60 65 70 75 80 85 90 95 100 Ours 0.9776 0.9812 0.9844 0.9870 0.9896 0.9913 0.9928 0.9938 0.9948 0.9956 Traditional 0.9727 0.9770 0.9803 0.9839 0.9865 0.9890 0.9907 0.9919 0.9932 0.9942 Improvement ($\%$) 0.51 0.43 0.42 0.32 0.32 0.24 0.21 0.19 0.16 0.14 : Recall of our method and traditional method for MovieLens[]{data-label="table8"} ![The comparison of recall between our method and traditional method[]{data-label="fig.comparerecall"}](Comparerecall.png){width="48.00000%"} Table \[table8\] and Figure \[fig.comparerecall\] show us that our hybrid recommendation get a better recall compared with the traditional hybrid recommendation. The increase of the length of the recommendation list from 5 to 30 leads to a dramatically raise of recall and finally the recall will converge to 1 in both methods. $L^*$ 5 10 15 20 25 30 35 40 45 50 -------------------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Ours 0.7402 0.7481 0.7518 0.7529 0.7534 0.7543 0.7546 0.7549 0.7402 0.7481 Traditional 0.7094 0.7222 0.7333 0.7397 0.7434 0.746 0.7474 0.749 0.7094 0.7222 Improvement ($\%$) 9.68 5.67 4.16 3.46 2.46 1.75 1.32 1.10 0.95 0.78 $L^*$ 55 60 65 70 75 80 85 90 95 100 Ours 0.7549 0.755 0.7552 0.7554 0.7555 0.7554 0.7555 0.7553 0.7554 0.7554 Traditional 0.75 0.7509 0.7513 0.7522 0.7526 0.7532 0.7535 0.7536 0.7539 0.7541 Improvement ($\%$) 0.64 0.54 0.51 0.42 0.38 0.29 0.26 0.22 0.19 0.17 : F-measureof our method and traditional method for MovieLens[]{data-label="table9"} ![The comparison of F-measure between our method and traditional method[]{data-label="fig.compareF"}](CompareF.png){width="48.00000%"} It can be seen from Table \[table9\] and Figure \[fig.compareF\] that there are rapid raises when $L^*$ changes from 5 to 25 and the values converge to a number around 0.75. Our method outperforms the traditional method. When $L^*=5$, our method improves the F-measure by 9.68$\%$ compared with the traditional method. In conclusion, curves of our method are always above those of the traditional method. When $L^*$ is small, obviously differences do exist between two methods which indicates that our method achieve a better classification accuracy when the recommended items are limited, that is, our method can distinguish the items which users like or dislike more accurately. Conclusions =========== In this paper, we have solved the cold-start problem of new users and new items by combining content-based and collaborative filtering algorithm. In the meantime, we apply WSBM algorithm to find communities with similar user preferences. It is more accurate and effective to predict ratings in communities instead of to an entire social network. As the score sparse will decrease the accuracy of recommendation, we improve the similarity by considering the users’ basic information to gain a more accurate recommendation. Experiments show that hybrid recommendation based on WSBM has better prediction and classification accuracy than the traditional method. However, our method mainly aimed at the users’ dominant behavior, such as buying and grading without taking into account the recessive behavior such as browsing and collection. Actually, considering the recessive behavior can achieve more accurate recommendation. In addition, considering the transfer of user’s interests could be the further research. Acknowledgements {#acknowledgements .unnumbered} ================ This work is supported by Guangdong Province Student’s Platform for Innovation and Entrepreneurship Training Program (Grant No.1055914152), and The Challenge Cup of the Ministry of Education of China (Grant No.15112005). Supplementary Material {#supplementary-material .unnumbered} ====================== Supplementary material (Algorithm codes and matrices files) will appear in the GitHub as soon as possible.

--- abstract: 'Without using conformal transformation, a simple type of five-dimensional $f(R)-$brane model is linearized directly in its higher-order frame. In this paper, the linearization is conducted in the equation of motion approach. We first derive all the linear perturbation equations without specifying a gauge condition. Then by taking the curvature gauge we derive the master equations of the linear perturbations. We show that these equations are equivalent to those obtained in the quadratical action approach \[Phys. Rev. D 95 (2017) 104060\], except the vector sector, in which a constraint equation can be obtained in the equation of motion approach but absent in the quadratical action approach. Our work sets an example on how to linearize higher-order theories without using conformal transformation, and might be useful for studying more complicated theories.' address: - 'School of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China' - 'School of Physical Science and Technology, Southwest University, Chongqing 400715, People’s Republic of China' - | Institute of Theoretical Physics, Lanzhou University,\ Lanzhou 730000, People’s Republic of China author: - Yuan Zhong - Ke Yang - 'Yu-Xiao Liu' title: 'Linearization of a warped $f(R)$ theory in the higher-order frame II: the equation of motion approach' --- $f(R)$ gravity ,linear perturbations,warped extra dimensions Introduction ============ In the last two decades, warped extra dimensions have been applied to explain the large hierarchy between the electroweak scale and the Planck scale [@RandallSundrum1999; @CabrerGersdorffQuiros2010; @CabrerGersdorffQuiros2011; @CabrerGersdorffQuiros2011; @RaychaudhuriSridhar2016], the splitting of fermion masses [@GherghettaPomarol2000], the reproduction of Newtonian gravity on a lower-dimensional hypersurface [@RandallSundrum1999a; @Gremm2000; @DeWolfeFreedmanGubserKarch2000; @CsakiErlichHollowoodShirman2000], and recently the LHC diphoton excess [@MegiasPujolasQuiros2016] and LHCb anomalies [@MegiasPanicoPujolasQuiros2016] (see [@Quiros2015; @Ponton2012; @Liu2017] for recent reviews on the theory and phenomenology of warped spaces). In a type of warped extra dimensional model, our world is described as a four-dimensional topological domain wall generated by a background scalar field in Einstein’s gravity [@RubakovShaposhnikov1983; @Gremm2000; @DeWolfeFreedmanGubserKarch2000; @CsakiErlichHollowoodShirman2000]. But it is also possible to generate pure geometric domain wall solutions in $f(R)$ theory [@ZhongLiu2016], where the gravitational Lagrangian is an arbitrary function of the scalar curvature (see [@Starobinsky1980; @BarrowOttewill1983; @NojiriOdintsov2003d; @CarrollDuvvuriTroddenTurner2004; @CapozzielloCarloniTroisi2003; @NojiriOdintsov2006] for early literatures and [@SotiriouFaraoni2010; @DeTsujikawa2010] for comprehensive reviews on $f(R)$ theory and its cosmological phenomenology). In this case, the domain wall is non-topological, because it connects two equivalent anti-de Sitter vacuum. More $f(R)$ domain wall solutions can be found in Refs.  [@ZhongLiu2016; @ParryPichlerDeeg2005; @AfonsoBazeiaMenezesPetrov2007; @HoffdaSilvaDias2011; @LiuZhongZhaoLi2011; @BazeiaMenezesPetrovSilva2013; @BazeiaLobaoMenezesPetrovSilva2014; @XuZhongYuLiu2015; @YuZhongGuLiu2015]. It is both important and interesting to study the linearization of domain wall solutions in a warped $f(R)$ gravity. Because the linearization not only tells us whether a solution is stable against small metric perturbation, but also offers the spectra of graviton and radion, which is important for phenomenological applications. As a higher-order curvature theory, $f(R)$ gravity might have some new features than Einstein’s theory. But a direct linearization of $f(R)$ domain wall is not easy, not only because one needs to carefully eliminate the residual gauge degrees of freedom, but also because the equation of motion in $f(R)$ gravity contains derivative up to fourth order. In literature, one usually rewrite the fourth-order $f(R)$ (the higher-order frame) as a second-order Einstein-scalar theory (the Einstein frame) by introducing a proper conformal transformation [@BarrowCotsakis1988; @Maeda1989; @Wands1994; @CapozzielloRitisMarino1997; @FaraoniGunzigNardone1999]. Therefore, to linearize a $f(R)$ domain wall, one can first do the conformal transformation and then conduct the linearization in the Einstein frame [@ZhongLiu2016]. To the authors’ knowledge, there is only a few works directly confront the linearization of $f(R)$ theory without using conformal transformation (see [@HwangNoh1996] for an example in $f(R)$ cosmology). If two frames are equivalent, the perturbation equations must be frame independent. But this conclusion is not obvious. Most importantly, when more general higher-order curvature theories are considered, the conformal transformation might not be convenient any more, then a direct analysis in the higher-order frame is inevitable. The aim of this work is to confront the linearization of $f(R)$ gravity with a warped geometry in the higher-order frame. In a previous work [@ZhongLiu2017], the linearization of $f(R)$ domain wall has been conducted in the quadratical action approach. In this paper, we redo the task in the equation of motion approach, and compare the results with those in Ref. [@ZhongLiu2017]. This paper is organized as follows. In next section, we briefly review the model and specify some conventions. The linearization of warped $f(R)$ domain walls is conducted in Sec. \[sec3\], where the metric perturbation is decomposed into scalar, tensor and vector parts. The gauge degrees of freedom will not be fixed until in Sec. \[sec4\], where the curvature gauge will be applied to simplify the scalar perturbation equation. The result is summarized in Sec. \[secSum\]. The model ========= In this paper, we consider a five-dimensional metric $f(R)$ gravity $$\begin{aligned} \label{action'} S=\frac{1}{2\kappa_5^2}\int d^5x\sqrt {-g}f(R).\end{aligned}$$ The corresponding Einstein field equations are $$\begin{aligned} \label{eqEE} R_{MN}f_R-\frac12g_{MN}f(R)+(g_{MN}\hat{\square}^{(5)} -\nabla_M\nabla_N)f_R=0,\end{aligned}$$ where $\hat{\square}^{(5)}=g^{MN}\nabla_{M}\nabla_{N}$ denotes the five-dimensional d’Alembertian operator defined by the metric $g_{MN}$ and the covariant derivative $\nabla_M$. The capital letters $M,N=0,1,2,3,5$ represent the bulk indices, and $f_R\equiv df(R)/dR$. A warped space is described by the following metric: $$\begin{aligned} \label{metric} ds^2=a^2(r)\eta_{MN}dx^M dx^N,\end{aligned}$$ where $\eta_{MN}=\textrm{diag}(-1,1,1,1,1)$ and $a(r)$ is the warp factor, which depends only on the extra dimension $r\equiv x^5$. Given the line element , it is easy to write the expressions of the connection, the Ricci tensor, the Ricci scalar and the last two terms in Eq. : $$\begin{aligned} \label{backquantities1} \Gamma^P_{MN}&=& 2\frac{{\delta _{(M}^P{\partial _{N)}}a}}{a} - {\eta _{MN}}\frac{{{\partial ^P}a}}{a},\\ R_{MN}&=&6\frac{\partial_{M}a\partial_{N}a}{a^2} -3\frac{\partial_{M}\partial_{N}a}{a} -2\eta_{MN}\left(\frac{a'}{a}\right)^2 -\eta_{MN}\frac{a''}{a},\\ R&=&-4 a^{-2}\left[\left(\frac{a'}{a}\right)^2+2\frac{a''}{a}\right],\\ \nabla_M\nabla_Nf_R &=& {\partial _M}{\partial _N}f_R - 2\frac{{\delta _{(M}^r{\partial _{N)}}a}}{a}f_R' + {\eta _{MN}}f_R'\frac{{a'}}{a},\\ \label{backquantities5} g_{MN}\hat{\square}^{(5)} f_R &=&{\eta _{MN}}({\square ^{(5)}}f_R + 3f_R'\frac{{a'}}{a})={\eta _{MN}}( f_R'' + 3f_R'\frac{{a'}}{a}).\end{aligned}$$ Here, we have used the following notations: 1. The primes represent derivatives with respect to the extra dimension $r$. 2. The bulk indices are always raised and lowered in terms of $\eta^{MN}$ and $\eta_{MN}$, respectively. For example, in the last term of $\Gamma^P_{MN}$ we used $\partial^{P}\equiv\eta^{PQ}\partial_{Q}$. 3. $\square^{(5)}\equiv\eta^{MN}\partial_M\partial_N\equiv\partial_M\partial^M$, and ${\square^{(4)}}\equiv\eta^{\mu\nu}\partial_\mu \partial_\nu\equiv\partial_\mu\partial^\mu$ are d’Alamber operators defined with Minkowski metrics and the ordinary partial derivatives. Obviously, $\square^{(5)}=\square^{(4)}+\partial_r\partial_r$. 4. The symmetrization bracket of tensor indices is defied as $$\begin{aligned} T^P_{~Q(M_1M_2\cdots M_n)}\equiv\frac{1}{n!}\left(T^P_{~QM_1M_2\cdots M_n}+\textrm{permutations of}~M_1,M_2,\cdots, M_n\right), \end{aligned}$$ for example, $$\begin{aligned} \frac{{\delta _{(M}^r{\partial _{N)}}a}}{a} \equiv\frac12\left(\frac{{\delta _{M}^r{\partial _{N}}a}}{a} +\frac{{\delta _{N}^r{\partial _{M}}a}}{a}\right). \end{aligned}$$ Substituting Eqs. - into the Einstein field equations , we get $$\begin{aligned} \label{backEinstein} &&{\eta _{MN}}\left( - \frac{1}{2}{a^2}f(R) + f_R'' - 2f_R{\left( {\frac{{a'}}{a}} \right)^2} + 2f_R'\frac{{a'}}{a} - f_R\frac{{a''}}{a} \right)\nonumber\\ &&+ 6f_R\frac{{{\partial _M}a{\partial _N}a}}{{{a^2}}} - 3f_R\frac{{{\partial _M}{\partial _N}a}}{a} - {\partial _M}{\partial _N}f_R + 2\frac{{\delta _{(M}^r{\partial _{N)}}a}}{a}f_R' = 0.\end{aligned}$$ Obviously, the non-trivial components are $$\begin{aligned} \label{Einstein1} a^2 f(R)+\left[4f_R\left(\frac{a'}{a}\right)^2+2f_R\frac{a''}{a}\right] -4f'_R\frac{a'}{a}-2f''_R=0,\end{aligned}$$ and $$\begin{aligned} 8f_R\left(\frac{a'}{a}\right)^2-8f_R\frac{a''}{a}+8f'_R\frac{a'}{a} -a^2f(R)=0.\end{aligned}$$ The summation of the above two equations leads to another useful identity: $$\begin{aligned} \label{EinsteinEq3} { - 2 f_R^\prime \frac{{a'}}{a} + 3{f_R}\frac{{a''}}{a} - 6{f_R}{{\left( {\frac{{a'}}{a}} \right)}^2} + f_R^{\prime \prime }} = 0,\end{aligned}$$ which is a second-order differential equation for $f_R(r)$, and can be analytically solved if the warped factor is simple enough, see Ref.  for some examples. The linearization of Einstein equations {#sec3} ======================================= Once the background solution is obtained, the next step is to consider the linear stability of the solution against small linear perturbation. Let us assume that the background solution is $\{a(r), f_R(r)\}$, and the metric perturbation is $$\begin{aligned} \delta g_{MN}=a(r)^2 h_{MN}(x^{\mu},~r),\end{aligned}$$ then the total metric reads $$\begin{aligned} g_{MN}=a(r)^2[\eta_{MN}+h_{MN}].\end{aligned}$$ By using the orthogonal relation of the total metric $$\begin{aligned} g^{MP}g_{PN}=\delta^M_{~N},\end{aligned}$$ one immediately concludes that, to the linear order $$\begin{aligned} \delta g^{MN}=-a(r)^{-2} h^{MN}, \quad h^{MN}\equiv \eta^{MP}\eta^{NQ}h_{PQ}.\end{aligned}$$ Similarly, one can derive the linear perturbations of the connection, the Ricci tensor and the scalar curvature: $$\begin{aligned} \label{pertGamma} \delta \Gamma^P_{MN}&=&\partial_{(M} h^P_{N)} -h_{MN}\frac{\partial^P a}{a} -\frac12\partial^P h_{MN} +{\eta}_{MN}\frac{a'}{a}h^{P}_r,\\ \label{pertRicci}\delta R_{MN}&=&\partial_P\partial_{(M}h^P_{N)} -\frac12\square^{(5)} h_{MN} -\frac32\frac{a'}{a}\partial_r h_{MN} -\frac{a''}{a}h_{MN} -2\left(\frac{a'}{a}\right)^2 h_{MN} \nonumber\\ &+&\eta_{MN}\frac{a'}{a}\partial_P h^{P}_r +\eta_{MN}\frac{a''}{a}h_{rr} +2\eta_{MN}\left(\frac{a'}{a}\right)^2h_{rr} -\frac12\partial_{M}\partial_{N}h \nonumber\\ &-&\frac12\eta_{MN}\frac{a'}{a}\partial_r h +3\frac{{a'}}{a}{\partial _{(M}}{h_{N)r}} - 2\frac{{a'}}{a}\frac{{{\partial _{(M}}a{h_{N)r}}}}{a} +2\left(\frac{a'}{a}\right)^2\eta_{r(M}h_{N)r},\\ \label{pertR} a^{2}\delta R&=&\partial_M\partial_N h^{MN} -\square^{(5)} h -4\frac{a'}{a} h' +8\frac{a'}{a}\partial_P h^P_r +8\frac{a''}{a}h_{rr} +4\left(\frac{a'}{a}\right)^2h_{rr}.\quad\quad\end{aligned}$$ Here $h$ is defined as $h\equiv\eta^{MN}h_{MN}$. The task of linearization is to derive the master equations for $h_{MN}$. In the equation of motion approach, the linear perturbation equations are obtained by perturbing all terms in the Einstein equations: $$\begin{aligned} &&\delta R_{MN}f_R+R_{MN}\delta f_R-\frac12\delta g_{MN}f(R) -\frac12g_{MN}f_R\delta R\nonumber\\ &+&\delta(g_{MN}\hat{\square}^{(5)} f_R)-\delta(\nabla_M\nabla_Nf_R)=0. \label{eqPertubedEE}\end{aligned}$$ The last two terms can be expanded as $$\begin{aligned} \delta(\nabla_M\nabla_N f_R)&=&(\partial_M\partial_N-\Gamma^P_{MN}\partial_P)\delta f_{R}-\delta \Gamma^P_{MN}\partial_P f_R,\\ \delta(g_{MN}\hat{\square}^{(5)} f_R)&=&\delta g_{MN}\hat{\square}^{(5)} f_R +g_{MN} \delta g^{PQ}(\nabla_P\nabla_Q f_R)\nonumber\\ &+&g_{MN}g^{PQ}\delta(\nabla_P\nabla_Q f_R),\end{aligned}$$ or more explicitly, $$\begin{aligned} \label{lastTerm1} \delta ({\nabla _M}{\nabla _N}{f_R}) &= &{\partial _M}{\partial _N}\delta {f_R} - 2\frac{{{\partial _{(M}}a{\partial _{N)}}\delta {f_R}}}{a} + {\eta _{MN}}\frac{{a'}}{a}{\partial _r}\delta {f_R}+ {f_R^{\prime}}\frac{{a'}}{a}{h_{MN}} \nonumber\\ & -& {f_R^{\prime}}{\partial _{(M}}{h_{N)r}} + \frac{1}{2}{f_R^{\prime}}{\partial _r}{h_{MN}} - {\eta _{MN}}{f_R^{\prime}}\frac{{a'}}{a}{h_{rr}},\\ \label{lastTerm2} \delta(g_{MN}\hat{\square}^{(5)} f_R) &=& {h_{MN}}(f_R'' + 3\frac{{a'}}{a}f_R') +{\eta _{MN}}\big(\square^{(5)} \delta f_R + 3\frac{{a'}}{a}{\partial _r}\delta f_R \nonumber\\ &- & f_R'{\partial ^P}{h_{Pr}} - 3f_R'\frac{{a'}}{a}{h_{rr}} + \frac{1}{2}f_R'{\partial _r}h - f_R''{h_{rr}}\big).\end{aligned}$$ Plugging Eqs. - and - into Eq. , after a long but straightforward calculation, we finally obtain the tensor form of the linearized Einstein equations: $$\begin{aligned} \label{generalEQ} &&- \frac{1}{2}f_R\square^{(5)} {h_{MN}} - \frac{3}{2}f_R\frac{{a'}}{a}{\partial _r}{h_{MN}} - \frac{1}{2}f_R'{\partial _r}{h_{MN}} - \frac{1}{2}f_R{\partial _M}{\partial _N}h - {\partial _M}{\partial _N}\delta f_R \nonumber\\ && + f_R{\partial _P}{\partial _{(M}}h_{N)}^P + 3f_R\frac{{a'}}{a}{\partial _{(M}}{h_{N)r}} - 2f_R\frac{{a'}}{a}\frac{{{\partial _{(M}}a{h_{N)r}}}}{a} + 2f_R{\left( {\frac{{a'}}{a}} \right)^2}{\eta _{r(M}}{h_{N)r}}\nonumber\\ && + 6\frac{{{\partial _M}a{\partial _N}a}}{{{a^2}}}\delta f_R - 3\frac{{{\partial _M}{\partial _N}a}}{a}\delta f_R + 2\frac{{{\partial _{(M}}a{\partial _{N)}}\delta f_R}}{a} + f_R'{\partial _{(M}}{h_{N)r}} \nonumber\\ && + {\eta _{MN}}\mathcal{I} = 0,\end{aligned}$$ where the scalar $\mathcal{I}$ is defined as $$\begin{aligned} \mathcal I&=& \square^{(5)} \delta f_R + \frac{1}{2}f_R\square^{(5)} h - \frac{1}{2}f_R{\partial _M}{\partial _N}{h^{MN}} - 3f_R\frac{{a'}}{a}{\partial _M}h_r^M - 3f_R\frac{{a''}}{a}{h_{rr}} \nonumber\\ &+& \frac{3}{2}f_R\frac{{a'}}{a}{\partial _r}h - 2f_R'\frac{{a'}}{a}{h_{rr}} - f_R'{\partial ^M}{h_{Mr}} + \frac{1}{2}f_R'{\partial _r}h - f_R''{h_{rr}} + 2\frac{{a'}}{a}{\partial _r}\delta f_R \nonumber\\ &-& 2{\left( {\frac{{a'}}{a}} \right)^2}\delta f_R - \frac{{a''}}{a}\delta f_R.\end{aligned}$$ Note that to derive Eq. , we have used the background Einstein equation to eliminate all the terms that proportional to $h_{MN}$. The nontrivial components of Eq. are $$\begin{aligned} \label{munucompo} (\mu,\nu)&:&f_R{\partial _M}{\partial _{(\mu }}h_{\nu )}^M - \frac{1}{2}f_R{\partial _\mu }{\partial _\nu }h - {\partial _\mu }{\partial _\nu }\delta f_R + 3f_R\frac{{a'}}{a}{\partial _{(\mu }}{h_{\nu )r}} + f_R^\prime {\partial _{(\mu }}{h_{\nu )r}}\nonumber\\ && - \frac{1}{2}f_R\square^{(5)}{h_{\mu \nu }} - \frac{3}{2}f_R\frac{{a'}}{a}{\partial _r}{h_{\mu \nu }} - \frac{1}{2}f_R^\prime {\partial _r}{h_{\mu \nu }} + {{\eta }_{\mu \nu }}{\cal I} = 0,\\ \label{pertmur} (\mu,r)&:& - \frac{1}{2}f_R\square^{(5)} {h_{\mu r}} + f_R{\partial _M}{\partial _{(\mu }}h_{r)}^M - \frac{1}{2}f_R{\partial _\mu }{\partial _r}h - {\partial _\mu }{\partial _r}\delta f_R + \frac{{a'}}{a}{\partial _\mu }\delta f_R\nonumber\\ && + \frac{3}{2}f_R\frac{{a'}}{a}{\partial _\mu }{h_{rr}} + \frac{1}{2}f_R^\prime {\partial _\mu }{h_{rr}} =0,\\ \label{pertrr} (r,r)&:&f_R{\partial _r}{\partial _M}h_r^M - \frac{1}{2}f_R\square^{(5)} {h_{rr}} + \frac{3}{2}f_R\frac{{a'}}{a}{\partial _r}{h_{rr}} + \frac{1}{2}f_R^\prime {\partial _r}{h_{rr}} - \frac{1}{2}f_R{\partial _r}{\partial _r}h\nonumber\\ &&- {\partial _r}{\partial _r}\delta f_R + 6{\left( {\frac{{a'}}{a}} \right)^2}\delta f_R - 3\frac{{a''}}{a}\delta f_R + 2\frac{{a'}}{a}{\partial _r}\delta f_R + {\cal I} = 0.\end{aligned}$$ It is well-known that the linear perturbations are invariant under the following gauge transformation: $$\begin{aligned} \label{geuge} \Delta h_{MN} &\equiv& \tilde{h}_{MN}-h_{MN}=-2\partial_{(M}\xi_{N)}-2\eta_{MN}\frac{a'}{a}\xi^r.\end{aligned}$$ Here, we use “$\Delta$" to indicate the change of perturbations, and $\xi^M\equiv \eta^{MN}\xi_N$ are parameters for an infinitesimal transformation of the coordinate $$\begin{aligned} x^{M}\to\tilde{x}^{M}=x^{M}+\xi^{M}(x^P).\end{aligned}$$ To proceed, we use the symmetry of the warped space and decompose the linear perturbation into scalar, tensor and vector parts [@Bardeen1980; @KodamaSasaki1984; @Weinberg2008]: \[decomposition\] $$\begin{aligned} {h_{\mu r}} &=& {\partial _\mu }F + {G_\mu },\\ {h_{\mu \nu }} &=& {\eta _{\mu \nu }}A + {\partial _\mu }{\partial _\nu }B + 2{\partial _{(\mu }}{C_{\nu )}} + {D_{\mu \nu }}. \end{aligned}$$ The advantage of this decomposition is, as we will see later, that different parts evolve independently. Note that $A, B, F, C_\mu, G_\mu, D_{\mu \nu }$ are all functions of $x^\mu$ and $r$. Among them, both $C_\mu$ and $G_\mu$ are transverse vectors, and $D_{\mu \nu }$ is a transverse and traceless (TT) tensor. In other words they satisfy the following equations: $$\begin{aligned} \partial^\mu C_\mu=&0&=\partial^\mu G_\mu,\\ \partial^\nu D_{\mu \nu }=&0&=D^\mu_\mu.\end{aligned}$$ The indices $\mu,\nu$ are raised by $\eta^{\mu\nu}$. Using these properties of the decomposed metric perturbations, we can rewrite the gauge transformation as follows $$\begin{aligned} \Delta A&=&-2\frac {a'}{a}\xi^r, \quad \Delta h_{rr}= -2\xi^{r\prime} -2\frac{a'}{a}\xi^r,\nonumber\\ \Delta B&=&-2\zeta,\quad \Delta F=-\xi^r-\zeta',\quad \Delta C_{\mu}=-\xi^{\perp}_{\mu},\\ \Delta G_\mu&=&-\xi_\mu^{\perp\prime},\quad \Delta D_{\mu\nu}=0.\nonumber\end{aligned}$$ Here, we applied the decomposition $\xi^\mu=\partial^\mu\zeta+\xi^{\perp\mu}$ such that $\partial_\mu\xi^{\perp\mu}=0$. Since only the gauge transformations of $B$ and $F$ depend on $\zeta$, they must appear together to ensure that the perturbation equations are independent of $\zeta$. Therefore, it is convenient to define $\psi = F - \frac{1}{2}B'$, whose gauge transformation only depends on $\xi^r$: $\Delta\psi=-\xi^r$. For the same reason, to make the perturbation equations gauge-invariant, $C_\mu$ and ${G_\mu }$ must appear together, and the only gauge-invariant combination of them is ${v_\mu } = {G_\mu } - C_\mu'$. Obviously, $v_\mu$ is also a transverse vector: $\partial^\mu v_\mu=0$. The tensor mode $D_{\mu\nu}$ is already gauge-invariant. Therefore, there is actually only one gauge degree of freedom $\xi^r$ to be fixed $$\begin{aligned} \Delta A&=&-2\frac {a'}{a}\xi^r, \quad \Delta h_{rr}= -2\xi^{r\prime} -2\frac{a'}{a}\xi^r, \quad \Delta\psi=-\xi^r.\end{aligned}$$ As we will see immediately, in $f(R)$ gravity the curvature perturbation also appears in the scalar perturbation equation, and its gauge transformation reads $$\Delta \delta^{(1)}R=-R'\xi^r.$$ Using the scalar-tensor-vector decomposition, we can expand the scalar $\mathcal{I}$ more explicitly as $$\begin{aligned} \label{scalarI} \mathcal{I} &= & \frac{1}{2}{f_R}{\square ^{(4)}}{h_{rr}} + \frac32{f_R}{\square ^{(4)}}A - {f_R}{\square ^{(4)}}\psi ' - 3{f_R}\frac{{a'}}{a}{\square ^{(4)}}\psi - {f_R^{\prime}}{\square ^{(4)}}\psi \nonumber \\ &+& 2{f_R}A'' - 3{f_R}\frac{{a''}}{a}{h_{rr}} - \frac{3}{2}{f_R}\frac{{a'}}{a}{h_{rr}^{\prime}} + 6{f_R}\frac{{a'}}{a}A' \nonumber \\ &-& 2{f_R^{\prime}}\frac{{a'}}{a}{h_{rr}} - \frac{1}{2}{f_R^{\prime}}{h_{rr}^{\prime}} + 2{f_R^{\prime}}A' - {f_R^{\prime\prime}}{h_{rr}} \nonumber \\ &+& {\square ^{(4)}}\delta {f_R} + \delta {f_R^{\prime\prime}} + 2\frac{{a'}}{a}{\partial _r}\delta {f_R} - 2{\left( {\frac{{a'}}{a}} \right)^2}\delta {f_R} - \frac{{a''}}{a}\delta {f_R}.\end{aligned}$$ Then, by inserting Eq.  into Eqs. - and using properties of $C_\mu, G_\mu$ and $D_{\mu\nu}$, we obtain the equation for the tensor perturbation: $$\begin{aligned} {\square^{(4)}}{D_{\mu \nu }} + 3\frac{{a'}}{a}D{'_{\mu \nu }} + \frac{{f_R^\prime }}{f_R}D{'_{\mu \nu }} + D''_{\mu \nu }=0.\end{aligned}$$ This result has been derived in Ref. [@ZhongLiuYang2011] in the equation of motion approach and in Ref. [@ZhongLiu2017] in quadratical action approach. We also obtain two equations for the vector modes $$\begin{aligned} \label{vector1} {\square^{(4)}}v_\mu &=&0,\\ \label{vector2} {\partial _{(\mu }}v{'_{\nu )}} + 3\frac{{a'}}{a}{\partial _{(\mu }}{v_{\nu )}} + \frac{f_R^\prime}{f_R} {\partial _{(\mu }}{v_{\nu )}}&=&0.\end{aligned}$$ Note that from the quadratic action approach, we did not get the constraint equation . Such a mismatch in the vector sector also appears in the Einstein-scalar theory [@Giovannini2001a; @ZhongLiu2013]. So far, we cannot tell the reason for this mismatch. One possibility is that the constraint equation comes from a boundary term in the quadratical action, which is neglected in Refs. [@Giovannini2001a; @ZhongLiu2013; @ZhongLiu2017]. For the scalar modes, we obtain four equations: $$\begin{aligned} \label{munuPP} 3\frac{{a'}}{a}\psi + \frac{{f_R^\prime }}{{{f_R}}}\psi + \psi ' - \frac{1}{2}{h_{rr}} - A - \frac{{\delta {f_R}}}{{{f_R}}} &=& 0,\\ \label{eqI} f_R\left({\square^{(4)}}A + \frac{{f_R^\prime }}{f_R}A' + 3\frac{{a'}}{a}A' + A''\right) - 2\mathcal{I} &=& 0,\\ \label{mu5S} - 2{\partial _r}\delta {f_R} + 2\frac{{a'}}{a}\delta {f_R} + 3{f_R}\frac{{a'}}{a}{h_{rr}} + f_R^\prime {h_{rr}} - 3{f_R}A' &=&0,\\ \label{5.16} 2{\square ^{(4)}}\psi ' - {\square ^{(4)}}{h_{rr}} + {\square ^{(4)}}A + 3\frac{{a'}}{a}{\partial _r}{h_{rr}} - 3A'' + 3\frac{{a'}}{a}A'+ \frac{{f_R^\prime }}{{{f_R}}}A' &+& \frac{{f_R^\prime }}{{{f_R}}}{\partial _r}{h_{rr}} \nonumber \\ - 2\frac{{{\partial _r}{\partial _r}\delta {f_R}}}{{{f_R}}} + 12{\left( {\frac{{a'}}{a}} \right)^2}\frac{{\delta {f_R}}}{{{f_R}}} - 6\frac{{a''}}{a}\frac{{\delta {f_R}}}{{{f_R}}} + 4\frac{{a'}}{a}\frac{{{\partial _r}\delta {f_R}}}{{{f_R}}} &=& 0.\end{aligned}$$ Using Eq. , the scalar $\mathcal{I}$ in Eq.  can be simplified further: $$\begin{aligned} \mathcal{I} &=& \frac{1}{2}{f_R}{\square ^{(4)}}A + 2{f_R}A'' - 3{f_R}\frac{{a''}}{a}{h_{rr}} - \frac{3}{2}{f_R}\frac{{a'}}{a}h_{rr}^\prime \nonumber \\ &+ &6{f_R}\frac{{a'}}{a}A' - 2f_R^\prime \frac{{a'}}{a}{h_{rr}} - \frac{1}{2}f_R^\prime h_{rr}^\prime + 2f_R^\prime A' - f_R^{\prime \prime }{h_{rr}} \nonumber \\ &+ & 2\frac{{a'}}{a}{\partial _r}\delta {f_R}+ \delta f_R^{\prime \prime } - 2{\left( {\frac{{a'}}{a}} \right)^2}\delta {f_R} - \frac{{a''}}{a}\delta {f_R}.\end{aligned}$$ Note that Eq.  can be derived from Eqs.  and . Besides, $\delta f_R=f_{RR}\delta R$ is not an independent perturbation mode, because the perturbation of the scalar curvature can be expanded in terms of other scalar perturbations: $$\begin{aligned} \label{delraR} {a^2}\delta R &= & - 16\frac{{a'}}{a}A' - 4A'' + 4\frac{{a'}}{a}h_{rr}' + 8\frac{{a''}}{a}{h_{rr}} + 4{\left( {\frac{{a'}}{a}} \right)^2}{h_{rr}} \nonumber \\ &+& 8\frac{{a'}}{a}{\square ^{(4)}}\psi + 2{\square ^{(4)}}\psi' - {\square ^{(4)}}{h_{rr}} - 3{\square ^{(4)}}A.\end{aligned}$$ This equation is derived by simply inserting Eqs.  into Eq. . The curvature gauge {#sec4} =================== To derive the final scalar perturbation equation, we need to eliminate the residual gauge degree of freedom $\xi^r$. One convenient way to fix the gauge is to choose the curvature gauge $\delta R=0$. In the quadratic action approach, we have derived the action of the scalar normal mode by using this gauge . We will check whether this gauge leads to the same scalar perturbation equation in the equation of motion approach. Under the curvature gauge, the independent scalar perturbation equations are $$\begin{aligned} \label{cuvgauge1} &&3\frac{{a'}}{a}\psi + \frac{{f_R^\prime }}{{{f_R}}}\psi + \psi ' - \frac{1}{2}{h_{rr}} - A = 0,\\ \label{cuvgauge3} &&3{f_R}\frac{{a'}}{a}{h_{rr}} + f_R^\prime {h_{rr}} - 3{f_R}A' =0,\\ \label{EqPsi1} &&3{\square ^{(4)}}A - 6\frac{{a'}}{a}{\square ^{(4)}}\psi - 2\frac{{f_R^\prime }}{{{f_R}}}{\square ^{(4)}}\psi + 3\frac{{a'}}{a}h_{rr}' \nonumber\\ &-& 3A'' + 3\frac{{a'}}{a}A' + \frac{{f_R^\prime }}{{{f_R}}}A' + \frac{{f_R^\prime }}{{{f_R}}}{h_{rr}'} = 0.\end{aligned}$$ The last equation is obtained from Eq.  after eliminating $\psi'$ by using Eq. . In addition, by eliminating $\psi'$ from Eq.  we get $$\begin{aligned} \label{EqPsi2} &&2\frac{{a'}}{a}{\square ^{(4)}}\psi - 2\frac{{f_R^\prime }}{{{f_R}}}{\square ^{(4)}}\psi - {\square ^{(4)}}A - 16\frac{{a'}}{a}A' \nonumber\\ &-& 4A'' + 4\frac{{a'}}{a}h_{rr}^\prime + 8\frac{{a''}}{a}{h_{rr}} + 4{\left( {\frac{{a'}}{a}} \right)^2}{h_{rr}} = 0.\end{aligned}$$ Using Eqs.  and to eliminate $\square^{(4)}\psi$ one would obtain an equation of $A$ and $h_{rr}$. But note that $h_{rr}$ can be eliminated by using the constraint equation . So, one finally would get a complicated equation of $A$, which will not be listed here. Our calculation shows that by defining a new variable $\mathcal{G} \equiv \theta A$ with $$\begin{aligned} \theta\equiv \frac{{{a^{3/2}}{f_R^{\prime}}}}{{\sqrt {3{f_R}} \left( {\frac{{a'}}{a} + \frac{{f_R^{\prime}}}{{3{f_R}}}} \right)}},\end{aligned}$$ the final equation can be written as $$\begin{aligned} \label{scalarPert} \square^{(4)}\mathcal{G}+\mathcal{G}''-\frac{\theta''}{\theta}\mathcal{G}=0.\end{aligned}$$ This equation is nothing but the same one obtained in the quadratical action apporach either in the Einstein frame  or in the higher-order frame [@ZhongLiu2017]. In sum, the curvature gauge leads to equivalent scalar perturbation equation, no matter from the quadratic action or the equation of motion approach. Using the theory of supersymmetric quantum mechanics, it is easy to show that any domain wall solution with $f_R>0$ is stable against linear perturbations (see Ref.  for detail). Summary {#secSum} ======= This work sets an example on how to directly linearize a higher-order gravitational theory in the equation of motion approach. For simplicity, we only consider a simple model, namely, a five-dimensional pure metric $f(R)$ gravity with warped geometry. In literature, one usually introduces a conformal transformation to rewrite the higher-order $f(R)$ theory into a second-order Einstein-scalar theory, and then conducts the linearization in the later frame. But the equivalence between these two different frames to the linear-order perturbations is seldom discussed. Besides, for more general higher-order curvature gravitational theories, conformal transformation might not be convenient any more. In that case, one needs to confront the direct linearization of the corresponding theory. In Ref. [@ZhongLiu2017] we discussed the first direct linearization of warped $f(R)$ domain wall in the quadratic action approach. The present work reconsiders the linearization of the same model of Ref. [@ZhongLiu2017] but follows a different approach, namely, the equation of motion approach. To compare with the results of Ref. [@ZhongLiu2017], we choose the curvature gauge to fix the gauge degrees of freedom. We find that the scalar and the tensor perturbation equations are consistent with those obtained in the quadratic action approach. For the vector mode, there are two equations, a wave equation and a constraint equation . In the quadratic action approach conducted in Ref. [@ZhongLiu2017], however, we only obtained the wave equation. This kind of mismatch in the vector sector also exists in Einstein-scalar theories [@Giovannini2001a; @ZhongLiu2013]. One possibility for the mismatch is that some boundary terms were neglected in the action approach. But to see if this hypothesis works, one needs a careful calculation, which will not be given here. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the National Natural Science Foundation of China (Grants No. 11605127, No. 11522541, No. 11375075 and No. 11647301). Yuan Zhong was also supported by China Postdoctoral Science Foundation (Grant No. 2016M592770). Y.-X. Liu was also supported by the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2016-k04). References {#references .unnumbered} ========== [45]{} natexlab\#1[\#1]{}\[2\][\#2]{} , , () . , , , () . , , , () . , , , , . , , () . , , () . , () . , , , , () . , , , , () . , , , () . , , , (). , () . , , . , , , arXiv:1707.08541. , , () . , , () . , () . , , () . , , () . , , , , () . , , , () . , , () . , , () . , , () . , , , () . , , , , () . , , () . , , , , () . , , , , () . , , , , , () . , , , , () . , , , , () . , , () . , () . , () . , , , () . , , , () . , , () . , , () . , () . , , () . , , , . , , , () . , () . , , () .

--- abstract: 'Given a state-of-the-art deep neural network classifier, we show the existence of a (image-agnostic) and very small perturbation vector that causes natural images to be misclassified with high probability. We propose a systematic algorithm for computing universal perturbations, and show that state-of-the-art deep neural networks are highly vulnerable to such perturbations, albeit being quasi-imperceptible to the human eye. We further empirically analyze these universal perturbations and show, in particular, that they generalize very well across neural networks. The surprising existence of universal perturbations reveals important geometric *correlations* among the high-dimensional decision boundary of classifiers. It further outlines potential security breaches with the existence of single directions in the input space that adversaries can possibly exploit to break a classifier on most natural images.[^1]' author: - | Seyed-Mohsen Moosavi-Dezfooli[^2][^3]\ [seyed.moosavi@epfl.ch]{} - | Alhussein Fawzi\ [alhussein.fawzi@epfl.ch]{} - '\' - | Omar Fawzi[^4]\ [omar.fawzi@ens-lyon.fr]{} - | Pascal Frossard\ [pascal.frossard@epfl.ch]{} bibliography: - 'bibliography.bib' title: Universal adversarial perturbations --- Acknowledgments {#acknowledgments .unnumbered} --------------- We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Tesla K40 GPU used for this research. [^1]: To encourage reproducible research, the code is available at [gitHub](https://github.com/LTS4/universal). Furthermore, a video demonstrating the effect of universal perturbations on a smartphone can be found [here](https://youtu.be/jhOu5yhe0rc). [^2]: The first two authors contributed equally to this work. [^3]: École Polytechnique Fédérale de Lausanne, Switzerland [^4]: ENS de Lyon, LIP, UMR 5668 ENS Lyon - CNRS - UCBL - INRIA, Université de Lyon, France

--- abstract: 'We derive expressions for the leading-order far-field flows generated by mobile colloids trapped at planar fluid-fluid interfaces. We consider both externally driven colloids and active colloids (swimmers) either adjacent to or adhered to the interface. In the latter case, we assume a pinned contact line. The Reynolds and capillary numbers are assumed small, in line with typical colloidal systems involving air- or alkane-aqueous interfaces. At clean (surfactant-free) interfaces, the hydrodynamic modes are essentially a restricted set of the usual Stokes multipoles in a bulk fluid. To leading order, driven colloids simply exert Stokelets parallel to the interface, while active colloids drive different kinds of fluid motion depending on their trapped configuration. We then consider how these modes are altered by the presence of an incompressible surfactant layer, which occurs at high Marangoni numbers. This limiting behavior is typical for colloidal-scale systems at small capillary numbers, even when scant surfactant is present. Compared to a clean interface, we find that incompressibility substantially weakens flow directed normal to the interface. Interestingly, for both driven and active colloids, we find that the leading-order flow normal to the interface is associated with colloid asymmetry with respect to the interfacial plane. Flow parallel to the interface, however, is not weakened. Moreover, surface-viscous stresses, if present, potentially generate very long-ranged flow on the interface itself and into the surrounding fluids. We examine the limiting forms of such flows. Our results have important implications for advective mass transport enhancement near fluid boundaries.' author: - 'Nicholas G. Chisholm' - 'Kathleen J. Stebe ,' bibliography: - 'main.bib' title: Driven and active colloids at fluid interfaces --- Intoduction {#sec:intro} =========== Fluid-fluid interfaces provide a rich setting for driven and active colloidal systems. Here, a “driven” colloid moves through a fluid due to external forces or torques, for example, a magnetic bead forced by a magnetic field. “Active” colloids, on the other hand, self propel by consuming a fuel source. For example, motile bacteria are active colloids that self-propel by the rotation of one or more flagella. Autophoretic nanorods or Janus particles are other examples of commonly studied active colloids. These catalytic swimmers self-propel via generation of chemical gradients that produce a propulsive layer of apparent fluid slip along the colloid surface. Past work on colloids adhered to interfaces has focused on their usefulness as Brownian rheological probes when embedded in biological lipid membranes or surfactant monolayers, where colloid motion is, in this case, “driven” by thermal fluctuations. For example, colloidal probes have been used to measure surface viscosity of a fluid interface as a function of surfactant concentration [@Sickert2007]. Such measurements require theoretical models of the mobility of the colloid. @Saffman1975 analytically computed the mobility of a flat disk embedded in a viscous, incompressible membrane separating two semi-infinite subphases in the limit of large Boussinesq number, a dimensionless number comparing the membrane viscosity to that of the surrounding fluid. This calculation was extended to moderate Boussinesq numbers by @Hughes1981 and to subphases of finite depth by @Stone1998. Later theoretical work quantified the response of a linearly viscoelastic membrane to an embedded point force [@Levine2002]. The effects of particle anisotropy have been quantified in the context of the mobility of a needle embedded in an incompressible Langmuir monolayer overlying a fluid of varying depth [@Fischer2004]. Finally, the impact of interfacial compressibility and surfactant solubility on the drag on a disk embedded in an interface above a thin film of fluid has also been quantified [@Elfring2016]. The dynamics of (three-dimensional) colloids that protrude into the surrounding fluid phases has also been characterized. Analytical and numerical analyses of the mobility of spheres [@Fischer2006; @Pozrikidis2007; @Stone2015; @Doerr2015; @Doerr2016] and thin filaments [@Fischer2006] can be found in the literature for clean and surfactant-laden interfaces in the limit of small capillary number, a dimensionless ratio of characteristic viscous stresses to interfacial tension. Active colloids are also strongly influenced by fluid interfaces. Motile bacteria have been the focus of much research in this context due to their relevance to human health and the environment. Seminal work by @Lauga2006 showed, via a resistive-force theory model, that circular trajectories of *E. coli* swimming near a solid boundary are caused by hydrodynamic interaction with the boundary. Similar results are found for free surfaces [@DiLeonardo2011], although the direction of circling is reversed. These theoretical models also predict that there is always an induced velocity toward the boundary, effectively trapping bacterium at the surface. More detailed boundary element simulations have shown the existence of stable trajectories of bacteria near solid boundaries, where the distance from the boundary and curvature of the trajectory reach a steady state [@Giacche2010]. In contrast, similar calculations show only unstable trajectories for swimmers near free surfaces; the swimmer inevitably crashes into the boundary unless it is initially angled steeply enough away to escape it altogether [@Pimponi2016]. Finally, @Shaik2017 analytically computed of the motion of a spherical “squirmer,” a common model for microorganism locomotion, near a weakly deformable interface. Others have investigated motion of autophoretic swimmers at fluid interfaces. Gold-platinum catalytic nanorods are highly motile at aqueous-alkane interfaces, and their rate of rotational diffusion can be used to measure interfacial shear viscosity [@Dhar2006]. Further experiments have shown that partially-wetted, self-propelled Janus particles at air-water interfaces move along circular trajectories with markedly decreased rotational diffusion as compared to their motion in a bulk fluid [@Wang2017]. Theoretical analysis has yielded analytical predictions of the linear and angular velocities of an autophoretic sphere straddling a surfactant-free interface with a freely-slipping, contact line [@Malgaretti2016]. This work has supplied valuable information about the influence of fluid interfaces on active colloid locomotion. Rather than developing detailed models for specific types of swimmers, an alternative approach is to use far-field models that capture universal features of colloid locomotion. For active colloids, this approach has been used to compute swimming trajectories near solid boundaries [@Spagnolie2012] and fluid interfaces [@Lopez2014]. Such methods are accurate when the colloid is separated from the boundary by a few body lengths [@Spagnolie2012]. Recent work has employed far-field models of active colloids to study trapping of microswimmers near surfactant-laden droplets [@Desai2018] and the density distribution of bacteria near fluid interfaces [@Ahmadzadegan2019]. While active and driven colloids near boundaries have been the subject of past theoretical analysis, the focus has largely been on computing drag (on driven colloids) or swimming trajectories (of active colloids) and how they are influenced by the boundary. The actual flows generated by such colloids at interfaces and the implications of these flows have received less attention. Aside from trapping due to hydrodynamic interactions, active colloids may be trapped at fluid interfaces by contact-line pinning, a phenomenon unique to fluid interfaces, in a variety of configurations, greatly affecting their motility and their induced fluid flows. For instance, recent work suggests that contact-line pinning traps *Pseusomonas aeruginosa* in a variety of different and persistent orientations at aqueous-hexadecane interfaces, leading to a distinct motility patterns [@Deng2020]. Bacteria may also become adhered to passive colloids already attached to the interface, towing them as cargo [@Vaccari2018]. The hydrodynamic implications of such trapped states have not been discussed. In this article, we use the multipole expansion method to examine the hydrodynamic modes generated by driven and active colloids at fluid interfaces in a variety of different trapped states. We focus on the modes that dominate in the far field, which may be observable in experiment. We focus on the case where the colloid is physically adhered to a fluid interface with a pinned contact line that constrains its motion. We also consider the case where the colloid is adjacent to the interface but not adhered, as might occur due to hydrodynamic trapping. This article is organized as follows. In \[sec:governing-eqs\], we develop the governing equations for the fluid motion due to colloids at two types of fluid interfaces: a clean, surfactant-free interface and an interface that is rendered incompressible by adsorbed surfactant. In \[sec:reciprocal-relations\], we develop a reciprocal relation that applies to two fluids in Stokes flow separated by either of these types of interface. In \[sec:clean-interfaces\], we develop a multipole expansion appropriate for colloids trapped at a clean interface, and we discuss the leading-order modes that are produced in the driven an active cases. We then compare these results to analogous results at an incompressible interface in \[sec:incompressible-interfaces\]. Finally, we conclude in \[sec:conclusion\] by discussing the implications of our results and opportunities for future research. Governing equations {#sec:governing-eqs} =================== Equations of motion ------------------- We consider a colloid adhered to a planar interface between two immiscible Newtonian fluids of viscosities $\mu_1$ and $\mu_2$, which are quiescent in the far field and together form an unbounded domain. We assume the resulting three-phase contact line is “pinned”, that is, it cannot move relative to the surface of the colloid. For simplicity, we further assume that the interface is flat and lies on the $z=0$ plane. The physical requirements for the assumption of a flat interface to hold are that (i) viscous stress due to flows generated by the particle are negligible compared to surface tension $\gamma$, which determines the equilibrium shape of the interface; (ii) the weight $mg$ of the colloid is also negligible compared to surface tension; and (iii) undulations in the contact line are negligibly small compared to the size of the colloid. Requirement (i) is formally satisfied when $\numCa = \mu U / \gamma \ll 1$, where $\numCa$ is the capillary number, $\mu$ is the fluid viscosity and $U$ is the characteristic velocity of the colloid. For typical colloidal systems at air-aqueous or alkane-aqueous interfaces, $\numCa = O(\num{e-7})$ to $O(\num{e-5})$. Requirement (ii) is satisfied when $\numBo = mg l^2 / \gamma \ll 1$, where $\numBo$ is the particle Bond number and $l$ is the characteristic length scale of the colloid. Requirement (iii) may not be generally satisfied. For isolated passive particles, nanometric contact line distortions alter the capillary energy that traps colloids on interfaces [@Stamou2000], and thermally activated fluctuations at the contact line are hypothesized to alter dissipation in the interface [@Boniello2015]. Neither effect is included here, but the results we present may form the basis for a perturbative method to treat the problem of undulated contact lines. At the colloidal scale, we may neglect the effects of fluid inertia and assume the flow on either side of the interface is governed by the Stokes equations, $$\label{eq:Stokes} \div{\ten\sigma} = - \grad p + \mu \laplace \vec u = \vec 0; \quad \div{\vec u} = 0,$$ subject to the appropriate boundary conditions on the colloid surface, where $\ten\sigma = -p \ten I + \mu [ \grad\vec u + {(\grad\vec u)}^\T ]$ is the stress tensor, $\ten I$ is the identity tensor, $\vec u = \vec u(\vec x)$ is the fluid velocity, $p = p(\vec x)$ is the hydrodynamic pressure, $\vec x = (x, y, z)$ is the position vector, and $\grad$ is the gradient operator with respect to $\vec x$. Let $V_1$, $V_2$, and $I$ denote the set of points in fluid 1, fluid 2, and on the interface, respectively. We assume that the viscosity changes abruptly across the interface as $\mu(z) = \mu_1 \Ind_{z>0}(z) + \mu_2 \Ind_{z<0}(z)$, where the indicator function $\Ind_P(z)$ is unity if condition $P$ is satisfied by its argument but otherwise vanishes. We further assume that fluid velocity is continuous across the fluid interface, that is, $\lr[]{\vec u}_I = \vec 0$, where $\lr[]{q_*}_I (\vec x) = (\lim_{z \to 0^+} - \lim_{z \to 0^-}) q_*(\vec x)$ denotes the “jump” in quantity $q_*$ across the interface going from fluid 2 to fluid 1. The general tangential stress balance on the fluid interface is $$\label{eq:iface-stress-bal-generic} \divS{\vec\varsigma} + \vec n \vdot \lr[]{\ten\sigma}_I \vdot \ten I_s = \vec 0 \quad $$ where $\vec\varsigma = \vec\varsigma(\vec x \in I)$ is the surface stress tensor, $\vec n$ is the unit normal to the interface pointing into fluid 1, $\ten I_s = \ten I - \vec{nn}$ is the surface projection tensor, and $\gradS = \ten I_s \vdot \grad$ is the surface gradient operator. The remaining normal component of the interfacial stress balance is replaced by the kinematic condition $\vec u \vdot \vec n = 0$ on $z=0$. Clean interface --------------- We call an interface “clean” if it is free of surfactant molecules. In the absence of temperature gradients, a clean interface is characterized by a uniform surface tension, $\vec\varsigma(\vec x) = \gamma_0 \ten I_s$. Then, $\divS{\vec\varsigma}$ vanishes and \[eq:iface-stress-bal-generic\] reduces to $$\label{eq:iface-stress-bal-clean} \vec n \vdot \lr[]{\ten\sigma}_I \vdot \ten I_s = \vec 0,$$ which states that the tangential stress on the fluid is continuous across the interface. Incompressible interface ------------------------ If surfactant is present, gradients in surfactant concentration due to flow exert Marangoni stresses on the surrounding fluids. At interfaces where $\numCa \gg 1$, these gradients need only be infinitesimal to balance viscous stresses due to colloid motion. As a result, the interface is constrained to surface-incompressible motion. To derive the most conservative estimate for the effects of these Marangoni stresses, consider trace surfactant concentrations, for which the surfactant can be approximated as a two dimensional ideal gas. In this case, the dependence of the surface pressure $\pi = \pi(\vec x \in I)$ on surfactant concentration $\Gamma = \Gamma(\vec x \in I)$ is given by $\pd\pi / \pd\Gamma = k_B T$, where $k_B$ is Boltzmann’s constant and $T$ is temperature. Scaling the surface pressure by viscous stresses $\scaled\pi = \pi / \bar\mu U$, where $\bar\mu = (\mu_1 + \mu_2)/2$ is the average surface viscosity, and letting $\scaled\Gamma = \Gamma / \bar\Gamma$, where $\bar\Gamma$ is the average surface concentration over the entire interface, we find $$\scaled\gradS \scaled\pi = \frac{k_B T \bar\Gamma}{\bar\mu U} \scaled\gradS \scaled\Gamma = \numMa\, \scaled\gradS \scaled\Gamma$$ where $\numMa$ is the dimensionless Marangoni number and $\scaled\gradS = l\gradS$. To evaluate $\numMa$, we consider typical parameter values for a colloid moving at $U = \SI{10}{\micro\meter/\second}$ at a hexadecane-water interface ($\gamma_0 \approx \SI{50}{\milli\newton/\meter}$) in the surface-gaseous state. The surfactant concentration required to produce a decrease in the surface tension is approximately $\bar\Gamma = \SI{2e3}{\text{molecules}/\micro\meter^2}$. Given $\bar\mu \approx \SI{1}{\milli\pascal\second}$, we estimate that $\numMa = O(10^3)$. Thus, very small perturbations in $\Gamma$ generate sufficient Marangoni stress to balance viscous stresses due to motion of the colloid. The large $\numMa$ limit has the important consequence that the fluid interface behaves as incompressible layer ($\divS{\vec u} = 0$ for $\vec x \in I$). Assuming bulk-insoluble surfactant, the non-dimensionalized surfactant mass balance on the interface is $$\label{eq:mass-transport-nd} \scaled\Gamma(\vec x) \scaled\gradS \vdot \scaled{\vec u} + \numMa^{-1} (\scaled{\vec u} \vdot \scaled\gradS) \scaled\pi = {(\numMa\,\numPe)}^{-1} \scaled\laplaceS \scaled\pi,$$ where $\scaled{\vec u} = \vec u / U$, $\numPe = U l / D_s$ is the Peclét number of the surfactant, and $D_s$ is the surface diffusivity of the adsorbed surfactant. Therefore, \[eq:mass-transport-nd\] implies that $\scaled\gradS \cdot \scaled{\vec u} \ll 1$ if $\numMa \gg 1$ and $\numPe_s \gtrsim \numMa^{-1}$. Assuming $l = \SI{10}{\micro\meter}$ and $D_s = \SI{e2}{\micro\meter^2/\second}$ (a typical value for small molecule surfactants), we have $\numPe_s = O(1)$, so surfactant diffusion does not typically restore compressibility of the interface. At larger surfactant concentrations, where departure from the surface-gaseous state is expected, the interface will generally remain incompressible because, except during phase transitions, $\pd\gamma / \pd\Gamma > k_B T$. Thus, we hereafter assume $\gradS \vdot \vec u = 0$ while discussing interfaces with surfactant. Dilute soluble surfactants also obey this constraint, as mass transport rates between the bulk and the interface are typically negligible. Here, we note that we may express the Marangoni number as $\numMa = E / \numCa$, where $E = \pd\scaled\gamma / \pd\scaled\Gamma$ is the Gibbs elasticity. Thus, interfacial incompressibility is the typical circumstance for any interfacial flow at low capillary number [@Blawzdziewicz1999]. A separate effect of increased surfactant concentration is the emergence of significant surface-viscous stresses due to interfacial shearing motion. If we assume Newtonian behavior of the surfactant, the interfacial stress tensor can be expressed as $$\label{eq:iface-stress-tensor-incompr} \vec\varsigma(\vec x) = -\pi(\vec x) \ten I_s + \mu_s \left[ \gradS\vec u + {(\gradS\vec u)}^\T \right]$$ for $\vec x \in I$, where $\mu_s$ is the surface viscosity. Then, inserting \[eq:iface-stress-tensor-incompr\] into \[eq:iface-stress-bal-generic\] yields the tangential stress balance for an incompressible, surfactant-laden interface $$\label{eq:iface-stress-bal-incompr} - \gradS\pi + \mu_s \laplaceS \vec u + \vec n \vdot \lr[]{\ten\sigma}_I \vdot \ten I_s = \vec 0,$$ which, in conjunction with the surface incompressibility condition, is analogous to the Stokes equations (if $\mu_s = 0$) for a two dimensional fluid that is forced by viscous stresses from the bulk phases. Reciprocal relation for two fluids separated by an interface {#sec:reciprocal-relations} ============================================================ Lorentz reciprocal theorem across an interface ---------------------------------------------- ![ A colloid, depicted in the center of the illustration, is surrounded by two arbitrary fluid regions $V^*_1 \subset V_1$ and $V^*_2 \subset V_2$, which meet at region $I^* \subset I$ on the interface. We assign the inward-facing normal vector $\hat{\vec n}$ to the boundaries of both of these regions. The unit normal $\vec n$ (sans hat) of the interface always points in the $+z$ direction. The boundary of $V_1^*$ consists of the colloid surface $S_1$, the interfacial region $I^*$, and the remaining outer surface $R^\text{o}_1$, with the boundaries of $V_2^*$ being similarly labeled. The boundary of $I^*$ (dashed line), denoted $\partial I^*$, has the counterclockwise oriented tangent vector $\hat{\vec t}$, and we define $\hat{\vec m} = \vec n \times \hat{\vec t}$, which points into $I^*$. The three-phase contact line is represented by the inner part of $\partial I^*$. []{data-label="fig:space"}](space.pdf) The Lorentz reciprocal theorem provides a relation between the velocity and stress fields of two arbitrary Stokes flows. We may extend this theorem to two fluid regions separated by a clean or incompressible interface, like those illustrated in \[fig:space\], as follows. Consider a region of fluid $V_\alpha^* \subset V_\alpha$ that is fully contained in fluid $\alpha$. Let $(\vec u, \ten\sigma)$ and $(\vec u', \ten\sigma')$ represent the velocity and stress fields of two independent solutions to the Stokes equations \[eq:Stokes\] in $V_\alpha^*$. Integration of $\div{(\ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u)}$ over $V_\alpha^*$ and application of the divergence theorem leads to the identity [@Kim1991] $$\label{eq:R-ident-bulk} \int_{V_\alpha^*} [(\div{\ten\sigma}) \vdot \vec u' - (\div\ten\sigma') \vdot \vec u] \dd V + \int_{\partial V_\alpha^*} (\ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u) \vdot \dd{\vec S} = 0$$ where $\partial V_\alpha^*$ denotes the boundary of $V_\alpha^*$, and $\dd{\vec S} = \hat{\vec n} \dd S$ points into $\partial V_\alpha^*$. If we extend \[eq:Stokes\] to the case where there is an external force density $\vec f(\vec x)$ on the fluid, then $\div{\ten\sigma} = -\vec f(\vec x)$ and $\div{\ten\sigma'} = -\vec f'(\vec x)$, which substituted into \[eq:R-ident-bulk\] gives the Lorentz reciprocal theorem, $$\label{eq:R-thm-bulk} \int_{V_\alpha^*} [\vec f(\vec x) \vdot \vec u' - \vec f'(\vec x) \vdot \vec u] \dd V = \int_{\partial V_\alpha^*} (\ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u) \vdot\dd{\vec S}.$$ If we add the pair of equations given by \[eq:R-thm-bulk\] for each of the two fluid phases $\alpha \in \{1, 2\}$, we obtain $$\begin{gathered} \label{eq:R-thm-sandwich} \int_{V^*} [\vec f(\vec x) \vdot \vec u' - \vec f'(\vec x) \vdot \vec u] \dd V \\ = \oint_R (\ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u) \vdot\dd{\vec S} + \int_{I^*} \left( \lr[]{\ten\sigma}_I \vdot \vec u'- \lr[]{\ten\sigma'}_I \vdot \vec u \right) \vdot \vec n \dd A,\end{gathered}$$ where $V^* := V_1^* \cup V_2^*$ is the union of the fluid volumes in each phase, $I^* := \partial V_1 \cap \partial V_2$ is the region (at the fluid interface) where $V_1^*$ and $V_2^*$ “touch”, and $R = \partial V^* \setminus I^*$ constitutes the remaining boundaries of $V_1^*$ and $V_2^*$ that are not adjacent to each other. For example, for the fluid region illustrated in \[fig:space\], $R = S_1 \cup S_2 \cup R^\text{o}_1 \cup R^\text{o}_2$, which includes both the surfaces of the colloid (the inner surfaces of $V^*$) and the outer surfaces of $V^*$. In the integral over $I^*$ in \[eq:R-thm-sandwich\], we have used the fact that the fluid velocity is continuous across the interface, $\lr[]{\vec u}_{I^*} = \lr[]{\vec u'}_{I^*} = \vec 0$. This term can be recast using the interfacial stress balance. Letting $\vec t^*$ be an arbitrary vector tangent to the interface, \[eq:iface-stress-bal-generic\] gives $$\label{eq:iface-tangential-stress-bal-forced} (\divS{\vec\varsigma}) \vdot \vec t^* + \vec n \vdot \lr[]{\ten\sigma}_I \vdot \vec t^* + \vec f_s \vdot \vec t^* = 0.$$ The final term of this equation accounts for an additional external surface force density $\vec f_s = \vec f_s(\vec x \in I)$ on the interface. Since there is no fluid flux through interface, both $\vec u$ and $\vec u'$ are tangent to the interface for $\vec x \in I$. Thus, using \[eq:iface-tangential-stress-bal-forced\], \[eq:R-thm-sandwich\] becomes $$\begin{gathered} \label{eq:R-thm-general-iface} \int_{V^*} \left[ \vec f \vdot \vec u' - \vec f' \vdot \vec u \right] \dd{V} + \int_{I^*} \left[ \vec f_s \vdot \vec u' - \vec f_s' \vdot \vec u \right] \dd A \\ = \oint_{R} \left( \ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u \right) \vdot \dd{\vec S} - \int_{I^*} \left[ (\divS{\vec\varsigma}) \vdot \vec u' - (\divS{\vec\varsigma'}) \vdot \vec u \right] \dd{A},\end{gathered}$$ where $\vec\varsigma$ and $\vec\varsigma'$ are the interfacial stress tensors associated with the unprimed and primed flows, respectively. Clean interface --------------- For a clean interface, $\vec\varsigma = -\ten I_s \gamma_0$ is constant, so the final integral in \[eq:R-thm-general-iface\] vanishes; $$\label{eq:R-thm-clean-iface} \int_{V^*} \left[ \vec f \vdot \vec u' - \vec f' \vdot \vec u \right] \dd{V} + \int_{I^*} \left[ \vec f_s \vdot \vec u' - \vec f_s' \vdot \vec u \right] \dd{A} = \oint_{R} \left( \ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u \right) \vdot \dd{\vec S}.$$ For completeness, we have included the possibility of an external surface force density exerted on the interface itself (even though there is physically no material adhered to it). If we assume $\vec f_s = \vec f_s' = \vec 0$, then the integral over $I^*$ in \[eq:R-thm-clean-iface\] also vanishes, which is the same as \[eq:R-thm-bulk\] with $\partial V_\alpha^*$ replaced by $R$. Incompressible interface ------------------------ Assuming an incompressible interface with Newtonian behavior, as described by \[eq:iface-stress-tensor-incompr\], there is a “surface” reciprocal identity for the interface that is analogous to \[eq:R-thm-bulk\] that is given by $$\label{eq:R-id-surface} \int_{I^*} \left[ (\divS{\vec\varsigma}) \vdot \vec u' - (\divS\vec\varsigma') \vdot \vec u \right] \dd A + \oint_{\partial I^*} { (\vec\varsigma \vdot \vec u' - \vec\varsigma' \vdot \vec u) } \vdot \hat{\vec m} \dd C = 0,$$ where the final term on the right-hand-side of this equation is a contour integral over the boundary of $I^*$, denoted $\partial I^*$. The unit vector $\hat{\vec m}$ points into $I^*$, meeting $\partial I^*$ at a right angle. It is defined as $\hat{\vec m} = \vec n \times \hat{\vec t}$, where $\hat{\vec t}$ is the counterclockwise-oriented unit tangent $\hat{\vec t}$ vector of $\partial I^*$ (see \[fig:space\]). in \[eq:R-thm-general-iface\] yields $$\begin{gathered} \label{eq:R-thm-incompressible-iface} \int_{V^*} \left[ \vec f \vdot \vec u' - \vec f' \vdot \vec u \right] \dd{V} + \int_{I^*} \left[ \vec f_s \vdot \vec u' - \vec f_s' \vdot \vec u \right] \dd A \\ = \oint_{R} \left( \ten\sigma \vdot \vec u' - \ten\sigma' \vdot \vec u \right) \vdot \dd{\vec S} + \oint_{\partial I^*} \left( \vec\varsigma \vdot \vec u' - \vec\varsigma' \vdot \vec u \right) \vdot \vec m \dd C.\end{gathered}$$ Comparing \[eq:R-thm-incompressible-iface\] to the analogous equation for a clean interface \[eq:R-thm-clean-iface\], we see that the final term is new. \[eq:R-thm-incompressible-iface\] involves a new contour integral over the boundary of $I^*$. This contour integral over the boundary of $I^*$ accounts for surface pressure gradients, or Marangoni stresses, that enforce the interfacial incompressibility constraint and, if $\mu_s > 0$, for surface-viscous dissipation. While we hereafter restrict ourselves to planar interfaces, we note that \[eq:R-thm-clean-iface,eq:R-thm-incompressible-iface\] hold even if the interface is curved, given that it has the same shape in both the primed and unprimed flow problems. Clean fluid interfaces {#sec:clean-interfaces} ====================== Green’s function {#ssec:clean-Green} ---------------- Due to the linearity of \[eq:Stokes\] and \[eq:iface-stress-bal-clean\], we may represent the velocity field due to a point force $\vec F$ located at $\vec y = (y_1, y_2, h)$ as $\ten G(\vec x, \vec y) \vdot \vec F$, where $\ten G$ is the Green’s function for two fluids separated by a clean interface, which satisfies \[eq:Stokes-G\] $$\begin{aligned} - \grad \vec P(\ten G; \vec x, \vec y) + \mu(z) \laplace \ten G(\vec x, \vec y) &= \begin{cases} \vec 0 & h = 0 \\ -\ten I \diracR3(\vec x - \vec y) & h \neq 0 \end{cases} \\ \label{eq:incompressible-G} \div{\ten G(\vec x, \vec y)} &= 0 \end{aligned}$$ for $\vec x \in V_1 \cup V_2$, subject to $|\vec u| \to 0$ as $|\vec x| \to \infty$ and \[eq:BCs-on-G\] $$\begin{aligned} \label{eq:iface-stress-bal-G} \ten I_s \vdot \lr[]{\vec n \vdot \ten T(\ten G; \vec x, \vec y)}_I &= \begin{cases} -\ten I_s \diracR2(\vec x - \vec y) & h = 0 \\ \vec 0 & h \neq 0 \end{cases} \\ \label{eq:iface-kinematics-G} \vec t^* \vdot \lr[]{\ten G(\vec x, \vec y)}_I &= \vec n \vdot \vec u = 0 \end{aligned}$$ for $\vec x \in I$. Here, $\vec P(\ten G;\!)$ is the (vectorial) pressure field associated with $\ten G$, $\ten T(\ten G;\!)$ is the stress tensor associated with $\ten G$, and $\diracR{n}(\vec x)$ is the Dirac delta in $\Reals^n$. Note that for $h = 0$, we take the force as being exerted on the interface itself rather than on one of the fluids. Solving \[eq:Stokes-G,eq:iface-stress-bal-G\] yields $$\label{eq:Gfun-clean} \ten G(\vec x, \vec y) = \begin{cases} [\ten J(\vec x - \vec y) + \ten U(\vec x, \vec y^*)] / \mu(h) & zh \geq 0 \\ \ten V(\vec x, \vec y) / \bar\mu & zh \leq 0 \\ \ten J(\vec x - \vec y) \vdot \ten I_s / \bar\mu & h = 0, \end{cases}$$ where $\vec y^* = (y_1, y_2, -h)$ is the “image” reflection of $\vec y$ through $z=0$ and $\ten J(\vec x) = (\ten I / r - \vec x \vec x / r^3) / 8\upi$ is the Oseen tensor. The tensors $\ten U$ and $\ten V$ represent hydrodynamic images that are necessary to satisfy continuity of tangential stress \[eq:iface-stress-bal-G\] and continuity of velocity at the interface. The image systems are given by [@Aderogba1978] $$\begin{aligned} \label{eq:upper-Blake-images} U_{ij}(\vec x, \vec\xi) &= (\kd^\para_{jk} - n_j n_k) \left[ J_{ik}(\vec x - \vec\xi) - \frac{\mu({\vec\xi \vdot \vec n})}{\bar\mu} V_{ik}(\vec x, \vec\xi) \right] \\ \label{eq:lower-Blake-images} V_{ij}(\vec x, \vec\xi) &= \left[ \kd^\para_{jk} + ({\vec\xi \vdot \vec n}) n_{k} \frac{\pd}{\pd\xi_j} + \frac12 ({\vec\xi \vdot \vec n})^2 \kd_{jk} \frac{\pd^2}{\pd\xi_l^2} \right] J_{ik}(\vec x - \vec\xi),\end{aligned}$$ where $\kd_{jk}$ is the Kronecker delta and $\kd^\para_{jk} = \kd_{jk} - n_j n_k$. The tensor indices $i,j,k,l \in \{1,2,3\}$ follow the Einstein summation convention. If, without loss of generality, we assume that the point force at $\vec y$ is located in the upper fluid ($h > 0$), then a Stokeslet, the Green’s function of the Stokes equations in an unbounded fluid, is induced at this point. The flow in the lower fluid ($z < 0$) comprises three image flows: a Stokeslet parallel to the interface, a Stokeslet dipole, and a degenerate Stokes quadrupole (a source doublet), all of which have their singular points at $\vec y$ (outside the physical domain of the lower fluid). These images correspond to each of the terms in \[eq:lower-Blake-images\], respectively. The image system for the upper fluid \[eq:upper-Blake-images\] is similar except that the image singularities are located at the image point $\vec y^*$. The image system $\ten U$ also includes an additional image Stokeslet that is the mirror reflection of the original forcing Stokeslet through $z=0$. Finally, we note that $\ten G$ is self-adjoint, $$\label{eq:self-adjoint-G} \ten G(\vec x, \vec y) = \ten G^\T(\vec y, \vec x),$$ which may be directly verified from \[eq:Gfun-clean\] or proven using \[eq:R-thm-clean-iface\] (see \[sec:app:self-adjointness\]). This property will prove useful in the following analysis. Multipole expansion ------------------- ### Expansion of the boundary integral equation {#sec:expansion-of-bie-clean} Using the Green’s function \[eq:Gfun-clean\] as the “primed” flow field in the reciprocal relation \[eq:R-thm-clean-iface\], we may generate a boundary integral equation for an object at the interface. Consider the interfacially-trapped colloid, illustrated in \[fig:space\], whose upper surface $S_1$ is in contact with fluid 1 and whose lower surface $S_2$ is in contact with fluid 2. An arbitrary volume of fluid $V^* = V^*_1 \cup V^*_2$ surrounds the colloid, which is bounded by $S_1$ and $S_2$ as well as the outer fluid surfaces $R^\text{o}_1$ and $R^\text{o}_2$. Using $\vec y$ as the variable of integration, we make the following substitutions into \[eq:R-thm-clean-iface\]: $\vec u'(\vec y) \to \ten G(\vec y, \vec x)$, $\ten\sigma'(\vec y) \to \ten T(\ten G; \vec y, \vec x)$, $\vec f' \to \ten I \diracR3(\vec y - \vec x)$, and $\vec f_s' \to \ten I \diracR2(\vec y - \vec x)$. We also assume that the external force densities vanish, $\vec f = \vec f_s = \vec 0$. Taking the limit of this equation as the outer surfaces $R^\text{o}_1$ and $R^\text{o}_2$ are made arbitrarily far from the colloid, we obtain the boundary integral representation of the velocity field, $$\label{eq:BIE-clean} \vec u(\vec x) = -\oint_{S_\text{c}} \ten G(\vec x, \vec y) \vdot [\ten\sigma \vdot \hat{\vec{n}} ](\vec y) \dd S(\vec y) + \oint_{S_\text{c}} {[\vec u \hat{\vec n}]}(\vec y) \odot \ten T (\ten G; \vec y, \vec x) \dd S(\vec y),$$ where $S_\text{c} = S_1 \cup S_2$ represents the surface of the colloid and the operator ‘$\odot$’ denotes complete contraction of its operands, e.g., $(\ten A \odot \ten B)_{j_1 \dots j_m} = A_{i_1 \dots i_n} B_{i_n \dots i_1 j_1 \dots j_m}$ if $\ten A$ is the tensor of lower rank and $(\ten A \odot \ten B)_{j_1 \cdots j_m} = A_{j_1 \cdots j_m i_1 \cdots i_n} B_{i_n \cdots i_1}$ if $\ten B$ is the tensor of lower rank. is clearly similar to the boundary integral equation for external Stokes flows in an unbounded bulk fluid [see, e.g., @Kim1991]. Indeed, it is derived in an analogous manner using the generalized reciprocal relation \[eq:R-thm-clean-iface\]. Integrals over $R^\text{o}_1$ and $R^\text{o}_2$ vanish because $\vec u \to \vec 0$ at points arbitrarily far from the colloid. Moreover, integrals over the interface itself do not appear in \[eq:BIE-clean\] because $\ten G$ and $\ten T$ implicitly account for transmission of hydrodynamic stresses through the interface. is valid as long as the volume of the colloid does not change and the colloid does not deform in a manner that would distort the flat shape of the pinned contact line. To generate a multipole expansion for $\vec u(\vec x)$, we replace $\ten G(\vec x, \vec y)$ and $\ten T(\ten G; \vec x, \vec y)$ in \[eq:BIE-clean\] with their Taylor series in $\vec y$ about an point on the interface as near as possible to the center of the colloid, which we designate as the origin $\vec 0$. This process is slightly complicated by the piecewise nature of $\ten G$ as $\vec y$ passes from one side of the interface to the other. In particular, certain components of $\grad_{\vec y} \ten G(\vec x, \vec y)$ contain a jump discontinuity over the interface at $z=0$. This difficulty is overcome by separating each integral in \[eq:BIE-clean\] into one over $S_1$ and another over $S_2$, so that the integrand is continuous over each surface of integration. Letting $\vec u^{(1)}$ and $\vec u^{(2)}$ denote the contributions from integration over $S_1$ and $S_2$, respectively, we may write the expansion as $\vec u(\vec x) = \vec u^{(1)} + \vec u^{(2)}$, where $$\label{eq:mpx-clean-upper} \begin{aligned} \vec u^{(1)}(\vec x) = -{\sum_{n=0}^\infty}\frac1{n!} {\left( \int_{S_1} [\uvec n \vdot \ten\sigma](\vec y)\,{\vec y^{\oprod n}}\dd S(\vec y) \right)} & \odot \left( {\lim_{\vec y \to \vec 0^+}} {\grad_{\vec y}^{\oprod n}}\ten G^\T(\vec x, \vec y) \right) \\ + {\sum_{n=0}^\infty}\frac1{n!} {\left( \int_{S_1} [\vec u \uvec n](\vec y)\,{\vec y^{\oprod n}}\dd S(\vec y) \right)} & \odot \left( {\lim_{\vec y \to \vec 0^+}} {\grad_{\vec y}^{\oprod n}}\ten T(\ten G; \vec y, \vec x) \right) \end{aligned}$$ and $$\label{eq:mpx-clean-lower} \begin{aligned} \vec u^{(2)}(\vec x) = -{\sum_{n=0}^\infty}\frac1{n!} {\left( \int_{S_2} [\uvec n \vdot \ten\sigma](\vec y)\,{\vec y^{\oprod n}}\dd S(\vec y) \right)} & \odot \left( {\lim_{\vec y \to \vec 0^-}} {\grad_{\vec y}^{\oprod n}}\ten G^\T(\vec x, \vec y) \right) \\ + {\sum_{n=0}^\infty}\frac1{n!} {\left( \int_{S_2} [\vec u \uvec n](\vec y)\,{\vec y^{\oprod n}}\dd S(\vec y) \right)} & \odot \left( {\lim_{\vec y \to \vec 0^-}} {\grad_{\vec y}^{\oprod n}}\ten T(\ten G; \vec y, \vec x) \right) \end{aligned}$$ Here, ${\vec y^{\oprod n}}= \vec y \vec y \cdots $ ($n$ times) denotes the $n$-fold tensor product and ${\grad_{\vec y}^{\oprod n}}$ similarly denotes the $n$-fold gradient operator. Writing $\ten T$ in terms of $\ten G$ as $$T_{ijk}(\ten G; \vec y, \vec x) = \kd_{ij} P_k(\ten G; \vec y, \vec x) + \mu(h) \left( \frac{\pd G_{kj}(\vec x, \vec y)}{\pd y_i} + \frac{\pd G_{ki}(\vec x, \vec y)}{\pd y_j} \right)$$ and collecting terms in $\ten G$, $\gradWRT{y} \ten G$, and so on for higher order gradients of $\ten G$, we arrive at the multipole expansion, \[eq:mpx-clean\] $$\vec u(\vec x) = \vec u^\text{m0}(\vec x) + \vec u^\text{m1}(\vec x) + \vec u^\text{m2}(\vec x) + \text{h.o.t}, \tag{\ref*{eq:mpx-clean}}$$ where $\vec u^\text{m0}$ is the force monopole (zeroth) moment, $\vec u^\text{m1}$ is the force dipole (first) moment, $\vec u^\text{m2}$ is the quadrupole (second) moment, and so on for higher order terms (h.o.t.). In particular, these first three moments are given by $$\begin{aligned} u^\text{m0}_i(\vec x) &= F^{(1)}_i G_{ij}({\vec x, \vec 0^{+}}) + F^{(2)}_i G_{ij}({\vec x, \vec 0^{-}}) \label{eq:mpx-clean-m0} \\ u^\text{m1}_i(\vec x) &= D^{(1)}_{jk} \frac{\pd G_{ij}}{\pd y_k}({\vec x, \vec 0^{+}}) + D^{(2)}_{jk} \frac{\pd G_{ij}}{\pd y_k}({\vec x, \vec 0^{-}}) \label{eq:mpx-clean-m1} \\ u^\text{m2}_i(\vec x) &= Q^{(1)}_{jkl} \frac{\pd G_{ij}}{\pd y_l \pd y_k}({\vec x, \vec 0^{+}}) + Q^{(2)}_{jkl} \frac{\pd G_{ij}}{\pd y_l \pd y_k}({\vec x, \vec 0^{-}}), \label{eq:mpx-clean-m2} \end{aligned}$$ where $\vec F^{(\nu)}$, $\ten D^{(\nu)}$, and $\ten Q^{(\nu)}$ are the monopole, dipole, and quadrupole coefficients for fluid $\nu \in \{1,2\}$, respectively. The shorthand notation $\vec 0^+$ indicates the limit as $\vec y$ approaches $\vec 0$ from above the interface (i.e., from fluid 1). Similarly, $\vec 0^-$ indicates the limit as $\vec y$ approaches $\vec 0$ from below. Note that if the colloid is wholly immersed in one fluid, then the multipole coefficients for the other fluid vanish. At distances, far enough from the colloid that points on the colloid surface are virtually indistinguishable from $\vec 0$, $|\vec x| \gg l $, the leading terms of \[eq:mpx-clean\] closely approximate $\vec u(\vec x)$. The reason is that, at points $|\vec x| \gg |\vec y|$, $\ten G(\vec x, \vec y)$ effectively appears as a Stokeslet and decays as $|\vec x - \vec y|^{-1}$; the image Stokes dipole and degenerate quadrupole terms contained in $\ten U$ and $\ten V$ in \[eq:Gfun-clean\] do not affect the far-field behavior of $\ten G$ because they decay more quickly than the Stokeslet terms. It follows that $\vec u^\text{m0}(\vec x) \sim r^{-1}$, where $r = |\vec x|$. Each successive multipole moment involves a higher-order gradient of $\ten G$. Thus, $\vec u^\text{m1}(\vec x) \sim r^{-2}$, $\vec u^\text{m2}(\vec x) \sim r^{-3}$ and so on for higher-order moments. The lowest order term with a nonzero coefficient dominates the far-field flow. This behavior is analogous to that of the multipole expansion for objects in a bulk fluid. ### Monopole moment The monopole moment corresponds to a point force exerted at the interface, which follows intuitively from the fact that at large distances $r \gg l$, the colloid is indistinguishable from a single point at the interface. The functional form of the flow is therefore just that of the Green’s function $\ten G$. The prefactors appearing in \[eq:mpx-clean-m0\] are given by $$\label{eq:monopole-coeff-clean} \vec F^{(\alpha)} = -\int_{S_\alpha} \ten\sigma \vdot \hat{\vec n} \dd S,$$ which is the force exerted on fluid $\alpha \in \{1,2\}$ due to motion of the colloid. There is no need to keep the separate limits on the right-hand side of \[eq:mpx-clean-m0\] because $\ten G(\vec x, \vec y)$ is continuous as $\vec y$ is moved across the interface for fixed $\vec x$. This property is not immediately obvious given the potential viscosity difference between the fluids. Recall, however, the boundary condition \[eq:iface-kinematics-G\] that demands continuity of $\ten G$ as $\vec x$ is brought across the interface for fixed $\vec y$. Since $\ten G$ is also self-adjoint \[eq:self-adjoint-G\], $\vec x$ implies continuity in $\vec y$. Indeed, one may verify directly that all three cases in \[eq:Gfun-clean\] are redundant; the first two cases of this equation reduce to the last as $h \to 0^\pm$. in \[eq:mpx-clean-m0\] yields the monopole moment as $$\label{eq:monopole-moment} u^\text{m0}_i(\vec x) = \frac{1}{\bar\mu} F_k \delta^\para_{kj} J_{ij}(\vec x),$$ where $\vec F = \vec F^{(1)} + \vec F^{(2)}$ is the total force exerted on both fluids. shows that $\vec u^\text{m0}$ is indistinguishable from a Stokeslet in an unbounded fluid of viscosity $\bar\mu$ associated with the effective force $\vec F \vdot \ten I_s$. The component of $\vec F$ normal to the interface does not contribute to the flow at leading order due to the presence of the interface. The “viscosity-averaged” Stokeslet represented by \[eq:monopole-moment\] possesses an axis of symmetry lying in the interfacial plane. The tangential shear stress therefore vanishes at $z = 0$, and \[eq:iface-stress-bal-clean\] is trivially satisfied. More generally, we will find that any mode with mirror symmetry of the velocity field about the interfacial plane has this property and is therefore a viscosity-averaged flow. ### Dipole moment The functional form of the dipole moment is given by $\gradWRT{y} \ten G(\vec x, \vec y)$ in the limit that $\vec y$ approaches the interface. Thus, this mode corresponds to the flow generated by a pair of opposite point forces at the interface that are displaced by an infinitesimal distance, or, more generally, a linear combination of such force doublets. Its prefactor for phase $\nu$ is given by $$\label{eq:dipole-coeff-clean} \ten D^{(\nu)} = \int_{S_\nu} \left[ -(\ten\sigma \vdot \hat{\vec n}) \vec y + \mu_\nu (\vec u \hat{\vec n} + \hat{\vec n} \vec u) \right] \dd{S(\vec y)},$$ which we decompose as $$\label{eq:dipole-coeff-clean-decomp} D^{(\nu)}_{jk} = S^{(\nu)}_{jk} + \frac12 \permut_{jkl} L^{(\nu)}_l + \frac13 D_{ii}^{(\nu)} \kd_{jk}$$ where $\ten\permut$ is the permutation tensor. Here, the irreducible tensor $S^{(\nu)}_{jk} = \frac12 ( D^{(\nu)}_{jk} + D^{(\nu)}_{kj} ) - \frac13 D^{(\nu)}_{ii} \kd_{jk}$ is associated with extensional (or contractile) stresses on the fluid, i.e., the stresslet at the interface, and $\vec L^{(\nu)}$ gives the torque exerted by the colloid on fluid $\nu$, $$\label{eq:torque-clean} \vec L^{(\nu)} = \ten\permut \vec{:} \ten D^{(\nu)} = -\int_{S_\nu} \vec y \times (\ten\sigma \vdot \hat{\vec n}) \dd{S(\vec y)},$$ and $\vec L = \vec L^{(1)} + \vec L^{(2)}$ is the total hydrodynamic torque on the system. The last term of \[eq:dipole-coeff-clean-decomp\] is associated with an isotropic stress, which does not produce flow due to fluid incompressibility \[eq:incompressible-G\]. Thus, it makes no contribution to $\vec u^\text{m1}$. We may rewrite \[eq:mpx-clean-m1\] as $$\begin{gathered} \label{eq:um1-clean-expanded} \newcommand\ab{{\alpha\beta}} u^{\text{m1}}_i(\vec x) = (D^{(1)}_\ab + D^{(2)}_\ab) \frac{\pd G_{i\alpha}}{\pd y_\beta}(\vec x, \vec 0) + D^{(1)}_{\alpha 3} \frac{\pd G_{i\alpha}}{\pd y_3}(\vec x, \vec 0^+) + D^{(2)}_{\alpha 3} \frac{\pd G_{i\alpha}}{\pd y_3}(\vec x, \vec 0^-) \\ + \left( D^{(1)}_{3\beta} + D^{(2)}_{3\beta} \right) \frac{\pd G_{i3}}{\pd y_\beta}(\vec x, \vec 0) + \left( D^{(1)}_{33} + D^{(2)}_{33} \right) \frac{\pd G_{i3}}{\pd y_3}(\vec x, \vec 0)\end{gathered}$$ where we introduce the convention that Greek tensor subscripts, here $\alpha \in \{1, 2\}$ and $\beta \in \{1, 2\}$, only run over the axes parallel to the interface. We have combined the separate limits in the first and penultimate terms of \[eq:um1-clean-expanded\] because gradients of $\ten G$ parallel to the interface are continuous. Furthermore, the same penultimate term vanishes; as we can see from \[eq:Gfun-clean\], $G_{i3}$ vanishes at all points on the interface for $\vec y = \vec 0$. We have also combined the limits in the final term of \[eq:um1-clean-expanded\] since $$\label{eq:G33-continuity-proof} 0 = \lr[]{\frac{\pd G_{\alpha i}(\vec y, \vec x)}{\pd y_\alpha}}_I = \lr[]{\frac{\pd G_{3i}(\vec y, \vec x)}{\pd y_\alpha}}_I = \lr[]{\frac{\pd G_{i3}(\vec x, \vec y)}{\pd y_3}}_I.$$ The first equality follows from continuity of parallel gradients, the second from \[eq:incompressible-G\] and the third from \[eq:self-adjoint-G\]. Note that the first two equalities swap the usual roles of $\vec x$ and $\vec y$. Finally, we must maintain the limits on the second and third terms of \[eq:um1-clean-expanded\] because $\lr[]{\pd G_{i\alpha} / \pd y_3}_I \neq 0$. The tangential stress balance on the interface \[eq:iface-stress-bal-G\] requires that $$\label{eq:iface-stress-jump-G} \newcommand*\xlim[1]{\lim_{\vec x \to \vec 0^{#1}}} \mu_1 \xlim+ \frac{\pd G_{\alpha k}(\vec x, \vec y)}{\pd x_3} - \mu_2 \xlim- \frac{\pd G_{\alpha k}(\vec x, \vec y)}{\pd x_3} = 0.$$ Therefore, applying \[eq:self-adjoint-G\] to \[eq:iface-stress-jump-G\], we find that the jump in $\pd G_{i\alpha}(\vec x, \vec y) / \pd y_3$ is by a factor of the viscosity ratio as $\vec y$ is moved across the interface for fixed $\vec x$. Putting \[eq:dipole-coeff-clean-decomp\] in \[eq:um1-clean-expanded\] and evaluating the necessary components of $\gradWRT{y} \ten G$, we may express the dipole moment explicitly in terms of the gradient of the Oseen tensor as $$\label{eq:dipole-moment} u^\text{m1}_i (\vec x) = -\frac{1}{\bar\mu} \left[ S^{\para}_{jk} + S^{\perp} n_j n_k + \frac12 \permut_{jk3} L_3 + \frac{\mu(-z)}{\mu_1} A^{(1)}_{jk} + \frac{\mu(-z)}{\mu_2} A^{(2)}_{jk} \right] \frac{\pd J_{ij}(\vec x)}{\pd x_k},$$ where $$\begin{aligned} S^{\para}_{jk} &= \left( \kd^\para_{j\alpha} \kd^\para_{k\beta} - \frac12 \kd^\para_{jk} \kd^\para_{\alpha\beta} \right) \left( S^{(1)}_{\alpha\beta} + S^{(2)}_{\alpha\beta} \right) \label{eq:dipole-coeff-para} \\ S^{\perp} &= S^{(1)}_{33} + S^{(2)}_{33} = -\kd^\para_{\alpha\beta} \left( S^{(1)}_{\alpha\beta} + S^{(2)}_{\alpha\beta} \right) \label{eq:dipole-coeff-perp}\\ A^{(\nu)}_{jk} &= \left( \kd^\para_{j\alpha} n_k + n_j \kd^\para_{k\alpha} \right) \left( S^{(\nu)}_{\alpha 3} - \frac12 \permut_{3\alpha\beta} L^{(\nu)}_\beta \right). \label{eq:dipole-coeff-asym}\end{aligned}$$ Each of the bracketed coefficients on the right-hand side of \[eq:dipole-moment\] make distinct contributions to the dipole moment at the interface. The first coefficient $\ten S^\para$, given by \[eq:dipole-coeff-para\], is a viscosity-averaged stresslet associated with extensional stresses produced by the colloid in the interfacial plane. Similarly, the second coefficient $S^{\perp}$, given by \[eq:dipole-coeff-perp\], is the viscosity-averaged stresslet perpendicular to the interface. Furthermore, $S^{(\nu)}_{33} = -S^{(\nu)}_{11} - S^{(\nu)}_{22}$ because $\ten S^{(\nu)}$ is traceless, so $\ten S^{\perp}$ accounts for extensional stress perpendicular to the interface and planar compression of the interface. The third coefficient is a viscosity-averaged rotlet, or point torque, about the $z$ axis of strength $L_3$. These viscosity-averaged flows exhibit mirror symmetry of the velocity field about $z=0$. Therefore, the tangential shear stress due to these modes vanishes on the interface, as is the case for the monopole moment. From the corresponding terms in \[eq:dipole-moment\], we see that the contributions to the dipole moment from $\ten A^{(1)}$ and $\ten A^{(2)}$ do not produce viscosity-averaged flows. As anticipated, the flow speed in one phase differs from that in the opposite phase by a factor of the viscosity ratio. (Intuitively, the flow is slower in the more viscous phase.) The difference in flow speed and the requirement that $\lr[]{\vec u}_I = \vec 0$ necessitates that $\vec u^\text{m1}(\vec x \in I) = \vec 0$. Interestingly, from \[eq:dipole-coeff-asym\], we see that the components of the stresslet $S^{(\nu)}_{i3}$ and torque $L^{(\nu)}_i$ for $i \in \{1,2\}$ contribute in a degenerate manner to $\ten A^{(\nu)}$. Although these modes are not viscosity-averaged, we see from \[eq:dipole-moment,eq:dipole-coeff-asym\] that the flow in the upper half-space ($z > 0$) of fluid 1 is equivalent to stresslet in an unbounded fluid (of viscosity $\bar\mu$), given by $\ten S^{\,\text{eff}}_\text{upper} = (\mu_2 / \mu_1) \ten A^{(1)} + \ten A^{(2)}$, with its singular point at $z=0$. For the lower fluid ($z < 0$), the effective stresslet is similarly $\ten S^{\,\text{eff}}_\text{lower} = \ten A^{(1)} + (\mu_1 / \mu_2) \ten A^{(2)}$. The quadrupolar and higher order moments of \[eq:mpx-clean\] can be similarly decomposed into two subsets of modes; one whose tangential stress vanishes at the interface and another whose velocity vanishes at the interface. Members of the former subset will be mirror-symmetric, viscosity-averaged flows and the latter will have velocities that differ by in magnitude by the viscosity ratio on either side of the interface. Here, we do not detail the higher-order modes further; the force monopole \[eq:monopole-moment\] and force dipole \[eq:dipole-moment\] describe the leading-order flows of driven and active colloids, respectively. Moreover, in many situations, we can infer the leading-order modes of different driven or active colloids in different configurations on or near the interface. Discussion ---------- ### Driven colloids For colloids driven by an external force $\vec F_\text{ext}$, it is clear that the monopole moment \[eq:monopole-moment\]—a viscosity averaged Stokeslet—is the leading-order far-field flow, except in the case that this force is exactly perpendicular to the interface, in which case $\vec u^\text{m0}$ vanishes. The force balance parallel to the interface is $\vec F \vdot \ten I_s = \vec F_\text{ext} \vdot \ten I_s$; this relationship holds whether or not the colloid is adhered or adjacent to the interface. For an adhered colloid, a purely perpendicular external force generates no motion because the pinned contact line is fixed on its surface. Of course, if the colloid is adjacent to the boundary, it may translate perpendicular to the interface, in which case a normal external force is balanced by the $z$-component of the hydrodynamic drag. For instance, consider the case where the external force $\vec F_\text{ext} = F_3 \bvec_z$ acts on a colloid immersed completely in fluid 1 that is distance $h$ from the interface. Then, from \[eq:monopole-coeff-clean\], $\vec F^{(1)} = \vec F = F_3 \bvec_z$ and $\vec F^{(2)} = \vec 0$. The monopole moment $\vec u^\text{m0}$ \[eq:monopole-moment\] vanishes because $\vec F \vdot \ten I_s = \vec 0$. The leading order flow therefore falls to the dipole moment. Recall that we “measure” $\ten D^{(1)}$ with respect to a point on the interface ($\vec x = \vec 0$), while the center of the colloid is located at $\vec x = h \bvec_z$. Substituting $\vec y = h \bvec_z + \vec y_\text{c}$ in \[eq:dipole-coeff-clean\], where $\vec y_\text{c}$ is the displacement from the center of the colloid, we find $$\label{eq:dipole-w-offset} \ten D^{(1)} = -\int_{S_1} (\ten\sigma \vdot \hat{\vec n}) (h \bvec_z + \vec y_\text{c}) \dd{S}(\vec y_\text{c}) = h F_3 \bvec_z \bvec_z + \ten D_\text{c},$$ where $\ten D_c$ is the dipole strength as measured from the colloid center. Thus, the external force on the colloid contributes a factor of $h F_3 \bvec_z \bvec_z$ to $\ten D^{(1)}$ (or a factor of $h F_3$ to $S^\perp$). When the particle is far from the interface, the contribution from the normal force dominates because $|\vec y_\text{c}| \sim l \ll h$. Otherwise, when $h \sim l$, contributions from $\ten D_\text{c}$ are generally significant and are sensitive to particle geometry, its distance to the interface, and the viscosity ratio. An external torque $\vec L_\text{ext}$ on the colloid also drives flow. We first consider a torque about the $z$-axis, $\vec L_\text{ext} = L_{\text{ext},z} \bvec_z$. This torque must be balanced hydrodynamically whether or not the colloid is adhered to the interface, $L_3 = L_{\text{ext},z}$. Note that pinned contact lines do not resist rotation about the $z$-axis. The torque induces a viscosity-averaged rotlet, given by the third bracketed term in \[eq:dipole-moment\]. For colloids that are axisymmetric about the $z$-axis, this is the only non-vanishing mode of \[eq:dipole-moment\]; it is readily shown that $S^\para_{jk} = S^\perp = A^{(\nu)}_{jk} = 0$ due to the azimuthal symmetry of the resulting flow. Of course, these coefficients are generally nonzero for general colloid geometries, so an external torque potentially produces all of the modes represented by \[eq:dipole-moment\]. ### Active colloids ![ Panels (a) and (b) illustrate a phoretic active colloid pinned to the interface. In (a), the active cap of the colloid generates a slip velocity that leads to in-plane swimming of a colloid pinned to the interface at velocity $\vec U$. In (b), the colloid active cap instead “pumps” fluid because contact line pinning prevents forward swimming. Panels (c) and (d) illustrate a bacterium also in swimming and pumping configurations. Thrust is generated by a rotating flagellum, which also produces a torque. In (c), this torque is balanced by capillary pinning, so there is a net hydrodynamic torque exerted on the fluid below the interface. For the vertically adhered bacterium (d), the hydrodynamic torque on the upper and lower fluid must vanish, since the body of the bacteria is free to counterrotate about the $z$ axis. The diagrams next to each illustration give appropriate minimal “point-force” models that correspond to the leading-order flows they are expected to generate. The arrows represent the orientation of these forces or torques (circular arrows) relative to the interface (dashed line). []{data-label="fig:active-colloid-configs"}](AC-configs.pdf){width="\linewidth"} ![ Dipolar hydrodynamic modes driven by an active colloid at a clean interface viewed level with the interface (top panels) and on the interfacial plane (bottom panels). The directed lines are streamlines in the laboratory frame and the gray lines are contours of constant flow speed. The vector $\vec e$ represents the alignment of the swimmer. (a) Force dipole (stresslet) mode expected for a swimmer moving parallel to the interface (pinned or unpinned), viewed along the interface. The configuration of the swimmer is like that in \[fig:active-colloid-configs\]a or c. (b) Stresslet due to an active colloid pinned at the interface, with a configuration as illustrated in \[fig:active-colloid-configs\]b or d. Modes (a-b) are the same as the force dipole in a bulk fluid with viscosity $\bar\mu$ and are axisymmetric about the swimmer alignment axis indicated by the vector $\vec e$. (c) Flow due to a point torque on the fluid where the torque vector $\vec L$ is parallel to the interface. Such a flow is expected for certain active colloids such as the bacterium illustrated in \[fig:active-colloid-configs\]c. A degenerate mode is generated by colloids adhered to the interface in an asymmetric manner, e.g., the colloid illustrated in \[fig:active-colloid-configs\]a. []{data-label="fig:clean-stresslets"}](active_clean.png){width="\linewidth"} Active colloids self-propel absent external forces or torques. For many kinds of active colloids, self-propulsion is generated by some active, thrust-producing part of the colloid that drives the remaining passive part, as illustrated in \[fig:active-colloid-configs\]; spatial separation of thrust and drag on the object generate a hydrodynamic dipole. Therefore, in a bulk fluid, an appropriate far-field model of an active colloid is that of a stresslet along the axis of swimming [@Lauga2009], which gives the velocity field $$\label{eq:bulk-axi-dipole} \vec u^\text{S}(\vec e; \vec x) = -\frac{D}{\mu_\text{b}} \vec e (\vec e \vdot \grad) \ten J(\vec x),$$ where $D$ is the strength of the force dipole, $\mu_\text{b}$ the viscosity of the bulk fluid, and $\vec e$ is a unit vector indicating the swimmer alignment. A similar model makes sense for an active colloid swimming parallel to the interface as illustrated in \[fig:active-colloid-configs\](a) and (c). Indeed, the same velocity field as \[eq:bulk-axi-dipole\] is produced by setting $\ten S^{\para} = D \vec e \vec e / 2$ and $S^\perp = -D/2$ in \[eq:dipole-moment\], with $\bar\mu$ replacing $\mu_\text{b}$. The resulting flow profile is illustrated in \[fig:clean-stresslets\]a. Interestingly, we find a similar mode if we set $S^\perp = D$ (with all other coefficients zero) and $\vec e = \vec n$ in \[eq:dipole-moment\]. This mode is expected of active colloids trapped perpendicular to the interface as depicted in \[fig:active-colloid-configs\](b) and (d). The colloid cannot self-propel in this configuration due to the pinned contact line, so the apparent stresslet \[eq:bulk-axi-dipole\] is not due to balancing hydrodynamic thrust and drag. Instead of swimming, the colloid becomes a fluid pump, resulting in a non-zero net hydrodynamic force on the colloid that is balanced by capillary forces. A minimal model for this pumping configuration is that of a point force exerted along the $z$-axis a small distance $\delta$ from the interface. While the monopole moment vanishes for a force in this direction, the dipole moment does not due to the small but finite separation of the force from the interface. The vertical point force gives $S^\perp = F \delta$ in \[eq:dipole-moment\], which is associated with the flow plotted in \[fig:clean-stresslets\]b. Viewed in the interfacial plane, this flow is sink-like for a pusher ($S^\perp > 0$) and source-like for a puller ($S^\perp < 0$). A pusher causes surface expansion ($\divS \vec u > 0$), as new interface must be created to replace the “sink.” Conversely, a puller causes surface compression. Another unique feature of active colloids pinned to interfaces is that they may exert an active hydrodynamic torque on the fluid about an axis parallel to the interface. This torque is balanced by surface tension at the pinned contact line. c illustrates this scenario for a motile bacterium pinned by its body and propelled by a rotating flagellum. The effect of this torque on the far-field flow enters through $\ten A^{(1)}$ for a torque on fluid 1 or $\ten A^{(2)}$ for a torque on fluid 2 \[eq:dipole-coeff-asym\]. The resulting flow profile is shown in \[fig:clean-stresslets\]c. The presence of this mode potentially discriminates the far-field flow of adhered versus unadhered swimmers; the net torque must vanish for detached active colloids that are adjacent to the interface. In the case of a bacterium, counterrotation of the body and flagellum instead produce a torque dipole in the far-field, a member of the higher-order quadrupole moment. A perpendicular configuration of the bacterium, as in \[fig:active-colloid-configs\](d) produces a torque dipole as well because the body may freely counterrotate in in the interface. ### Symmetry about the interfacial plane To conclude this discussion, we return to the motif of two major categories of modes: those which are weighted by the average viscosity, with vanishing tangential stress at the interface, and those whose velocity vanishes on the interface. In particular, the subset of dipolar modes corresponding to $\ten A^{(1)}$ and $\ten A^{(2)}$ in \[eq:dipole-moment\] are the only ones that fall into the latter category. The previous discussion associated $\ten A^{(1)}$ and $\ten A^{(2)}$ with a net hydrodynamic torque on the fluid adjacent to the interface about an axis parallel to the interface. Such torques might arise from active stresses or, for colloids adjacent to the interface, a driving external torque. However, this mode is not uniquely associated with these torques; from \[eq:dipole-coeff-asym\], we see that it also involves the components of the stresslet $S^{(\nu)}_{\alpha 3} $. To gain a better understanding of these modes, consider a spherical colloid of radius $a$ driven in rigid-body motion, which is adhered to the interface with a contact angle, such that half of the sphere is in each fluid. In this case, we may obtain the velocity field from that of a sphere moving in an unbounded fluid of uniform viscosity $\bar\mu$ [@Ranger1978; @Pozrikidis2007]. If the sphere translates at velocity $\vec U$ in the $z=0$ plane and rotates with angular velocity $\bvec_z \Omega_3$, the fluid velocity in the laboratory frame with its origin at the center of the sphere is $$\label{eq:sphere-flow} \vec u(\vec x) = \vec F \left(1 + \frac{a^2}{6} \nabla^2 \right) \vdot \ten J(\vec x) + \frac12 L_3 \bvec_z \vdot [\grad \times \ten J(\vec x)],$$ where $\vec F = 6 \upi \bar\mu \vec U a$ is the Stokes drag and $L_3 = 8 \upi \bar\mu \Omega_3$ is the torque. This velocity field is mirror-symmetric about the $z=0$ plane, so the tangential stress vanishes on $z=0$. It follows that \[eq:sphere-flow\] trivially satisfies \[eq:iface-stress-bal-clean\] and is therefore also the solution for two fluids of differing viscosities that average to $\bar\mu$; the flow is independent of the viscosity contrast. There is, of course, a normal stress jump across the interface in this case, but it is inconsequential at small $\numCa$—the interface remains flat. comprises a viscosity-averaged Stokeslet and degenerate quadrupole (or source doublet) at the center of the sphere. This solution implies that, for the sphere described above, the dipole moment completely vanishes unless there is an external torque about the $z$-axis, in which we obtain the viscosity-averaged rotlet described by \[eq:dipole-moment,eq:dipole-coeff-para\]. If there is no external torque on the sphere but it translates along, e.g., the $x$ axis, then we expect a torque about the $y$-axis for differing fluid viscosities. One might naively expect this hydrodynamic torque to produce flow, which clearly contributes to $\ten A^{(\nu)}$ \[eq:dipole-coeff-asym\]. However, for a sphere, it is readily shown that the final two bracketed terms of \[eq:dipole-moment\] cancel. More generally, we expect a viscosity-averaged flow to result for any driven or active colloid with mirror symmetry about $z=0$. If the boundary motion is symmetric about $z=0$, then the resulting fluid flow will reflect this symmetry. By the same arguments for a sphere presented above, identically vanishing tangential stress across the interface implies a viscosity averaged flow. Thus, $\ten A^{(\nu)}$ and the contribution they make to the dipole moment only contribute to the flow when there is some degree of asymmetry about the interfacial plane. For driven colloids, this asymmetry may come from an asymmetric colloid shape or an adhered configuration that places more of the colloid in one fluid (for a sphere, any contact angle other than will do). For active colloids, there will likely be asymmetry in activity or boundary motion, especially if the two fluid phases have differing viscosities or chemical properties. For example, the phoretic swimmer illustrated in \[fig:active-colloid-configs\]a is expected to produce a leading-order stresslet parallel to the interface due to hydrodynamic thrust and drag (\[fig:clean-stresslets\]a). However, we also expect a contribution from the asymmetric mode illustrated by \[fig:clean-stresslets\]c. In experiment, contact line pinning fixes colloids in random configurations at fluid interfaces, so such asymmetric adhered states are likely to be the norm. Incompressible interfaces and the role of surface viscosity {#sec:incompressible-interfaces} =========================================================== Green’s function {#ssec:dirty-Green} ---------------- We may define a Green’s function $\ten H$ for an incompressible interface that is analogous to that discussed in \[ssec:clean-Green\] for a clean interface. The major difference is that the interfacial stress balance \[eq:iface-stress-bal-G\] is replaced by \[eq:iface-stress-bal-H\] $$\begin{aligned} - \gradS \vec\Pi(\ten H;\!) + \mu_s \laplaceS \ten H + \ten I_s \vdot \lr[]{\vec n \vdot \ten T(\ten H;\!)}_I &= \begin{cases} -\ten I_s \diracR2(\vec x - \vec y) & h = 0 \\ \vec 0 & h \neq 0 \end{cases} \\ \label{eq:surf-div-H} \divS{\ten H} &= 0, \end{aligned}$$ where $\ten\Pi(\ten H;\!)$ is the (vectorial) surface pressure associated with $\ten H$, which enforces the surface incompressibility constraint \[eq:surf-div-H\]. Thus, $\ten H$ satisfies \[eq:Stokes-G\] subject to \[eq:iface-kinematics-G\] and \[eq:iface-stress-bal-H\], with $\ten G$ replaced by $\ten H$ in the former two equations. Like $\ten G$, $\ten H$ is self-adjoint (see \[sec:app:self-adjointness\]); $$\label{eq:self-adjoint-H} \ten H(\vec x, \vec y) = \ten H^\T(\vec y, \vec x).$$ The functional form of $\ten H$, given by [@Blawzdziewicz1999], is more complicated than that of $\ten G$ owing to the more complex interfacial mechanics. Interestingly, to determine the leading-order moments for colloids at interfaces, it suffices to know $\ten G$ and $\ten H(\vec x, \vec y)$ for $\vec y \in I$ only, that is, the flow due to a point force at the incompressible interface ($h=0$). The relevant derivations are outlined in \[sec:app:Gfuns\]. Letting $\ten H^0(\vec x) = \ten H(\vec x, \vec 0)$ and $\vec s = (x, y)$, we find that $$\begin{aligned} \label{eq:Hfun} { H^0_{\ga\gb}(\lenBq; \vec x) = \frac{1}{4\upi \bar\mu} R_0(\lenBq; s, z) \kd_{\ga\gb} + \frac{1}{2\upi \bar\mu} R_2(\lenBq; s, z) \Irr{\hat s_\ga \hat s_\gb} }\end{aligned}$$ and that $H^0_{3j} = H^0_{i3} = 0$, where $\hat{\vec s} = |\vec s| / s$ and $\Irr{\cdot}$ denotes the irreducible (traceless, symmetric) part of the enclosed tensor. Here, $\Irr{\hat s_\ga \hat s_\gb} = \hat s_\ga \hat s_\gb - \frac12 \kd_{\ga\gb}$. (Note that we regard this operation is being on a two-dimensional vector since $\ga, \gb = \{1,2\}$.) The functions $R_0$ and $R_2$, given by \[eq:rzFn\], depend on the Boussinesq length, $\lenBq = \mu_s / \bar\mu$, as well as position. The velocity field represented by $\ten H^0$ is everywhere parallel to the interface. As noted by @Stone2015, this result holds generally for Stokes flows driven by arbitrary incompressible surface motion. The $z$ component of the velocity vanishes on the interface as does its derivative in the $z$ direction; $[\pd u_3 / \pd x_3]_{z=0} = -\divS{\vec u} = 0$ because $\div{\vec u} = \vec 0$. The $z$ velocity and all its derivatives also vanish at infinity. As a Stokes flow, $\vec u$ is also biharmonic, $\nabla^4 u_3 = 0 $, but, due to its homogeneous behavior of $u_3$ at the boundaries, $u_3$ is just the trivial solution to this equation and $u_3 = 0$ everywhere. The vanishing behavior of $H_{3j}$ reflects this fact. At distances $r \gg \lenBq$, bulk viscous effects dominate over surface viscous effects. If $\lenBq$ is vanishingly small compared to the length scale of the colloid $l$, then surface-viscous effects are negligible, and the flow is only modified from that at a clean interface by Marangoni stresses. Thus, we define the dimensionless Boussinesq number as $\numBq = \lenBq/l$, which quantifies the relative importance of surface viscous to bulk viscous effects. In the limit $\numBq \to 0$, we obtain $R_n$ in closed form \[eq:rzFn|Bq=0\], and \[eq:Hfun\] reduces to $$\label{eq:Hfun0|Bq=0} \lr.|{H^0_{\ga\gb}}_{\numBq=0} = \frac{\kd_{\ga\gb}}{8\upi\bar\mu r} + \frac{(r - |z|)^2}{4\upi \bar\mu r s^2} \Irr{\hat s_\ga \hat s_\gb}.$$ Marangoni stresses do not affect the rate of decay of the flow from the origin, which remains as $r^{-1}$. The flow on the interface is purely radial (although not radially symmetric), and is given by $$\label{eq:Hfun0|Bq=0,z=0} \lr.|{H^0_{\ga\gb}(\vec x \in I)}_{\numBq=0} = \frac{\kd_{\ga\gb}}{8\upi\bar\mu r} \hat s_\ga \hat s_\gb.$$ In the opposite limit, $\numBq \gg 1$, surface viscosity has a dominant impact on the flow at distances $r \ll \lenBq$. Here, bulk viscous stresses from the surrounding fluid are very weak compared the interfacial stresses. Then, from \[eq:iface-stress-bal-H\], we recover the equations governing a two-dimensional Stokes flow [@Saffman1975]. Therefore, at distances $r \ll \lenBq$ from the colloid, $$\label{eq:Hfun0|Bq=oo} \lr.|{H^0_{\ga\gb}}_{\numBq\to\infty} \sim \frac{\hat s_\ga \hat s_\gb - \kd_{\ga\gb} \ln s}{4\pi\mu_s},$$ which is constant in $z$ (it is a two-dimensional flow field) and diverges logarithmically as $s$ is made large. Clearly, \[eq:Hfun0|Bq=oo\] cannot satisfy the homogenous boundary conditions at $r \to \infty$. Of course, this is Stokes’ paradox, and it is resolved by noting that \[eq:Hfun0|Bq=oo\] is not valid for $r \gtrsim \lenBq$, where bulk viscous effects inevitably become important. Despite the rather complicated form of $R_n$ for finite $\numBq$, its main “role” is simply to transition the flow field between the surface-viscosity-dominated, non-convergent behavior at distances $r \ll \lenBq$ from the colloid to the convergent, $1/r$ decay at distances $r \gg \lenBq$, where surface viscosity has a negligible effect. Interestingly, the surface pressure associated with $\ten H^0$ is independent of $\lenBq$ and is given quite simply by $\ten\Pi^0 = \vec s / 4\upi s^2$. Multipole expansion ------------------- ### Expansion of the boundary integral equation {#expansion-of-the-boundary-integral-equation} Using the reciprocal relation \[eq:R-thm-incompressible-iface\] for two fluids separated by an incompressible interface and following a procedure similar to that described in \[sec:expansion-of-bie-clean\], we obtain the boundary integral representation for $\vec u(\vec x)$ as $$\begin{gathered} \label{eq:BIE-incompr} u_k(\vec x) = - \oint_{S_\text{c}} { H_{kj}(\vec x, \vec y) (\hat n_i \sigma_{ij})(\vec y) }\dd S(\vec y) + \oint_{S_\text{c}} { (\hat n_i u_j)(\vec y) T_{ijk}(\ten H; \vec y, \vec x) }\dd S(\vec y) \\ - \oint_C { H_{k\beta}(\vec x, \vec y) (\hat m_\alpha \varsigma_{\alpha\beta})(\vec y) } \dd C(\vec y) + \oint_C { (\hat m_\alpha u_\beta) (\vec y) \Sigma_{\alpha\beta k}(\ten H; \vec y, \vec x) } \dd C(\vec y),\end{gathered}$$ for $\alpha,\beta,\gamma \in \{1,2\}$, where $\hat{\vec m} = \bvec_z \times \hat{\vec t}$, $C$ is the curve in the $z=0$ plane that runs along the three-phase contact line, and $\ten\Sigma(\ten H;\!)$ is the surface stress tensor associated with $\ten H$, which is given by $$\Sigma_{\ga\gb k}(\ten H; \vec y, \vec x) = -\kd_{\ga\gb} \Pi_k(\ten H; \vec y, \vec x) + \mu(h) \left( \frac{\pd H_{k\gb}(\vec x, \vec y)}{\pd y_\ga} + \frac{\pd H_{k\ga}(\vec x, \vec y)}{\pd y_\gb} \right).$$ Comparing \[eq:BIE-incompr\] to \[eq:BIE-clean\], there are two additional terms in \[eq:BIE-incompr\] that account for Marangoni forces and surface-viscous stresses at the contact line. Note that \[eq:BIE-incompr\] assumes that the hole in the interface created by an adhered colloid is of constant surface area. As before, we may generate a multipole expansion for $\vec u(\vec x)$ by replacing $\ten H$, $\ten T(\ten H;\!)$, and $\ten\Sigma(\ten H;\!)$ in \[eq:BIE-incompr\] with their respective Taylor series in $\vec y$ about the origin $\vec 0$, which we place at an appropriate point on the interface. We may write the expansion as $\vec u = \vec u^{(1)} + \vec u^{(2)} + \vec u^{(\text i)}$, where $$\begin{aligned} \label{eq:mpx-incompr-upper} \vec u^{(1)} &= \text{same as \cref{eq:mpx-clean-upper} with \(\ten G\) replaced by \(\ten H\),} \\ \label{eq:mpx-incompr-lower} \vec u^{(2)} &= \text{same as \cref{eq:mpx-clean-lower} with \(\ten G\) replaced by \(\ten H\),}\end{aligned}$$ and $$\newcommand*{\sum_{n=0}^\infty}{\sum_{n=0}^\infty} \newcommand*\intC[1]{\left( \int_C #1 \dd C(\vec y) \right)} \newcommand*{\vec y^{\oprod n}}{y_{\gc_1} \cdots y_{\gc_n}} \newcommand*\gradSyn[1]{\left. \frac{\pd^n #1}{\pd y_{\gc_1} \cdots \pd y_{\gc_n}} \right|_{\vec y = \vec 0}} \label{eq:mpx-incompr-iface} \begin{aligned} u^{(\text i)}_k (\vec x) = -&\sum_{n=0}^\infty { \frac1{n!} \intC{[\hat m_\ga \varsigma_{\ga\gb}](\vec y)\,{\vec y^{\oprod n}}} \gradSyn{H_{k\gb}(\vec x, \vec y \in I)} } \\ +&\sum_{n=0}^\infty { \frac1{n!} \intC{[u_\gb \hat m_\ga](\vec y)\,{\vec y^{\oprod n}}} \gradSyn{\Sigma_{\ga\gb k}(\ten H; \vec y \in I, \vec x)} }. \end{aligned}$$ Collecting terms from \[eq:mpx-incompr-upper,eq:mpx-incompr-lower,eq:mpx-incompr-iface\], we may write $\vec u$ as a multipole expansion analogous to that given by \[eq:mpx-clean-lower\], \[eq:mpx-incompr\] $$\vec u(\vec x) = \vec u^\text{m0}(\vec x) + \vec u^\text{m1}(\vec x) + \text{h.o.t}, \tag{\ref*{eq:mpx-incompr}}$$ where $$\begin{aligned} u^\text{m0}_i(\vec x) &= F^{(1)}_i H_{ij}({\vec x, \vec 0^{+}}) + F^{(2)}_i H_{ij}({\vec x, \vec 0^{-}}) + F^{(\text i)}_i H_{i\gb}({\vec x, \vec 0}) \label{eq:mpx-incompr-m0} \\ u^\text{m1}_i(\vec x) &= D^{(1)}_{jk} \frac{\pd H_{ij}}{\pd y_k}({\vec x, \vec 0^{+}}) + D^{(2)}_{jk} \frac{\pd H_{ij}}{\pd y_k}({\vec x, \vec 0^{-}}) + D^{(i)}_{\gb\gc} \frac{\pd H_{i\gb}}{\pd y_\gc}({\vec x, \vec 0}). \label{eq:mpx-incompr-m1} \end{aligned}$$ are analogous to \[eq:mpx-clean-m0,eq:mpx-clean-m1\], respectively, where \[eq:mpx-incompr-m0,eq:mpx-incompr-m1\] each contain an additional term to account for Marangoni and surface-viscous stresses exerted by the colloid on the interface at the contact line. While the particular functional form of the monopole and dipole moments are clearly modified by these interfacial stresses, their physical interpretation remains very similar to those found for a clean interface. ### Monopole Moment ![ Limiting forms of the surface-incompressible monopole moment for (a) $\numBq = 0$ and (b) $\numBq \gg 1$ viewed on the interfacial plane $z=0$. The direction of the point force exerted on the interfacial plane is indicated by the black arrow. The result for a clean interface (c) is also shown for comparison. Interestingly, a purely radial flow is recovered for $\numBq = 0$, while the angular dependence of the clean and the large-$\numBq$ incompressible interface are similar. []{data-label="fig:monopoles"}](monopoles.pdf){width="\linewidth"} Compared with that for a clean interface \[eq:monopole-moment\], the monopole moment also accounts for the force exerted on the interface by the colloid at the contact line due to Marangoni and surface-viscous stresses. This force is given by the prefactor of the final term in this equation, $$F^{(\text i)}_{\gb} = -\oint_C \hat m_\alpha \varsigma_{\alpha\beta} \dd C.$$ Again, $\ten H(\vec x, \vec y)$ is continuous as $\vec y$ is moved across the interface, so we may drop the separate limits in \[eq:mpx-incompr-m0\] to give $$\label{eq:monopole-moment-incompr} u^\text{m0}_i(\vec x) = \left( F^{(1)}_j + F^{(2)}_j + F^{(\text i)}_\beta \kd^\para_{\beta j} \right) H_{ij}(\vec x, \vec 0).$$ Like the clean-interface monopole, the incompressible monopole given by \[eq:monopole-moment\] does not depend on the viscosity contrast between the two fluids. Unlike the case for a clean interface, here, $\vec u^\text{m0}$ does not reduce to an effective, viscosity-averaged flow due to the nontrivial interfacial dynamics. shows the velocity field of the monopole moment in the limits $\numBq \to 0$ and $\numBq \to \infty$, which are given by \[eq:Hfun0|Bq=0,eq:Hfun0|Bq=oo\], respectively. ### Dipole Moment The dipole moment also has an additional contribution due to interfacial stresses given by the final term in \[eq:mpx-incompr-m1\], whose prefactor is $$D^{(\text i)}_{\beta\gamma} = \oint_C \left[ \hat m_\alpha \varsigma_{\alpha\beta} x_\gamma + \mu_s (\hat m_\beta u_\gamma + u_\gamma \hat m_\beta) \right] \dd C.$$ Noting that only the $z$-component of the gradient of $\ten H$ (with respect to $\vec x$ or $\vec y$) is discontinuous across the interface, we rewrite \[eq:mpx-incompr-m1\] as $$\begin{gathered} \label{eq:um1-incompr-expanded} \newcommand\ab{{\alpha\beta}} u^{\text{m1}}_i(\vec x) = \lr(){D^{(1)}_\ab + D^{(2)}_\ab + D^{(\text i)}_\ab} \frac{\pd H_{i\alpha}}{\pd y_\beta}(\vec x, \vec 0) + D^{(1)}_{\alpha 3} \frac{\pd H_{i\alpha}}{\pd y_3}(\vec x, \vec 0^+) + D^{(2)}_{\alpha 3} \frac{\pd H_{i\alpha}}{\pd y_3}(\vec x, \vec 0^-) \\ + \left( D^{(1)}_{3\beta} + D^{(2)}_{3\beta} \right) \frac{\pd H_{i3}}{\pd y_\beta}(\vec x, \vec 0) + \left( D^{(1)}_{33} + D^{(2)}_{33} \right) \frac{\pd H_{i3}}{\pd y_3}(\vec x, \vec 0).\end{gathered}$$ The fourth term of \[eq:mpx-incompr-m1\] vanishes because $H_{i3}(\vec x, \vec y)$ vanishes for $\vec y \in I$. Recall, the Green’s function for a clean interface had the same property due to the non-deformability of the interface. The final term of \[eq:um1-incompr-expanded\] also vanishes; the incompressibility of the interface and the surrounding fluid, $\div{\vec u} = \divS{\vec u} = \vec 0 $, implies that $$\label{eq:H-deriv-props} 0 = \lr.|{\frac{\pd H_{\alpha j}(\vec x, \vec y)}{\pd x_\alpha}}_{x_3 = 0} = \lr.|{\frac{\pd H_{3j}(\vec x, \vec y)}{\pd x_3}}_{x_3 = 0} = \lr.|{\frac{\pd H_{j3}(\vec x, \vec y)}{\pd y_3}}_{y_3 = 0},$$ where the final equality follows from \[eq:self-adjoint-H\]. We may decompose $\ten D^{(1)}$ and $\ten D^{(2)}$ into irreducible tensors as before \[eq:dipole-coeff-clean-decomp\]. A similar decomposition of $\ten D^{(\text i)}$ is $$\label{eq:iface-dipole-coeff-decomp} D^{(\text i)}_{\ga\gb} = S^{(\text i)}_{\ga\gb} + \frac12 \permut_{\ga\gb3} L^{(\text i)} + \frac12 \kd_{\ga\gb} D^{(\text i)}_{\gc\gc},$$ where the irreducible part of $\ten D^{(\text i)}$ is $$S^{(\text i)}_{\ga\gb} = \frac12 \lr(){D^{(\text i)}_{\ga\gb} + D^{(\text i)}_{\gb\ga}} - \frac12 \kd_{\ga\gb} D^{(\text i)}_{\gc\gc},$$ which represents the stresslet *on* the interface due to stresses at the contact line. Similarly, the pseudoscalar $L^{(\text i)}$, given by $$\label{eq:torque-on-iface} L^{(\text i)} = -\bvec_z \vdot \oint_C { \vec y \times [\hat{\vec m} \vdot \ten\varsigma](\vec y) } \dd C(\vec y),$$ is the torque (about the $z$ axis) exerted on the interface by the colloid. The total torque exerted on the surrounding system (both fluids and the interface) is therefore $\vec L = \vec L^{(1)} + \vec L^{(2)} + L^{(\text i)} \bvec_z$. Recalling the definition of $\ten\varsigma$ \[eq:iface-stress-tensor-incompr\], it is readily shown that surface pressure $\pi$ makes no contribution to $L^{(\text i)}$, and therefore $L^{(\text i)} = 0$ if $\mu_s = 0$. Finally, we note that applying the self-adjoint relation to the first equality in \[eq:H-deriv-props\] gives $ [{\pd H_{j \alpha}(\vec x, \vec y)} / {\pd y_\alpha}]_{y_3 = 0} = 0 $. Comparing this result with \[eq:um1-incompr-expanded\] reveals that the “surface” traces of $\ten D^{(1)}$, $\ten D^{(2)}$, and $\ten D^{(\text i)}$, e.g., $\kd_{\ga\gb} D^{(\nu)}_{\ga\gb}$, are of no dynamical significance. Incompressibility of the surrounding fluids further implies that $S^{(\text 1)}_{33}$ and $S^{(\text 2)}_{33}$ also have no affect on the flow. Recall that, for a clean interface, the modes associated with these components of the stresslet produced radially symmetric modes associated with local expansion or compression of the interface (see \[fig:clean-stresslets\]b). It is no surprise that these source/sink flows vanish for incompressible interfaces. One may easily verify that there exists no radially symmetric vector field on the interface that both satisfies $\divS{\vec u} = 0$ and vanishes at infinity. ![ Limiting forms of the surface-incompressible stresslet (dipole moment) for (a) $\numBq = 0$ and (b) $\numBq \gg 1$ viewed on the interfacial plane $z=0$. The result for a clean interface (c) is shown for comparison. The black arrows indicate the configuration of the force doublet. []{data-label="fig:dipoles"}](dipoles.pdf){width="\linewidth"} After dropping all vanishing terms, \[eq:dipole-coeff-clean-decomp\] and \[eq:iface-dipole-coeff-decomp\] in \[eq:um1-incompr-expanded\] gives $$\begin{gathered} \label{eq:um1-incompr-irred} u^{\text{m1}}_i(\vec x) = \lr(){S^\para_{\ga\gb} + \frac12 \permut_{{\ga\gb} 3} L_3} \frac{\pd H_{i\ga}}{\pd y_\gb}(\vec x, \vec 0) + \lr(){S^{(1)}_{l3} - \frac12 \permut_{3lm} L^{(1)}_m} \frac{\pd H_{i\ga}}{\pd y_3}(\vec x, \vec 0^+) \\ + \lr(){S^{(2)}_{l3} - \frac12 \permut_{3lm} L^{(2)}_m} \frac{\pd H_{i\ga}}{\pd y_3}(\vec x, \vec 0^-),\end{gathered}$$ where $$S^\para_{\ga\gb} = \left( \kd_{\ga\gc} \kd_{\gb\gd} - \frac12 \kd_{\ga\gb} \kd_{\gd\gc} \right) \left( S^{(1)}_{\gc\gd} + S^{(2)}_{\gc\gd} + S^{(\text i)}_{\gc\gd} \right).$$ The last two terms of \[eq:um1-incompr-irred\] are analogous to the asymmetric modes discussed for clean interfaces, given by the last two terms of \[eq:dipole-moment\]. Recall that, for a clean interface, these modes have vanishing velocity on the interface. Thus, these modes are also valid for incompressible interfaces. Indeed, for $\vec u(\vec x \in I) = \vec 0$, the interfacial stress balance \[eq:iface-stress-bal-incompr\] reduces to that for a clean interface \[eq:iface-stress-bal-clean\]. We also see from \[eq:um1-incompr-irred\] that the prefactors of these modes do not involve $\ten D^{(\text i)}$ and therefore have no explicit dependence on the interfacial stresses. Thus, we may simply “replace,” without modification, the last two terms in \[eq:um1-incompr-irred\] with the last two terms from \[eq:dipole-moment\], yielding the dipole moment as $$\label{eq:dipole-moment-incompr} u^\text{m1}_i (\vec x) = -\left( S^{\para}_{\alpha\beta} + \frac12 \permut_{\alpha\beta 3} L_3 \right) \frac{\pd H^0_{i\alpha}}{\pd y_\beta} + \left( \frac{\mu(-z)}{\bar\mu \mu_1} A^{(1)}_{jk} + \frac{\mu(-z)}{\bar\mu \mu_2} A^{(2)}_{jk} \right) \frac{\pd J_{ij}(\vec x)}{\pd x_k},$$ where $\ten A^{(\nu)}$ is given, as before, by \[eq:dipole-coeff-asym\]. In contrast, the first term of \[eq:dipole-moment-incompr\] is greatly affected by interfacial stresses. Evaluating the necessary gradient of $\ten H$, we find that (see \[sec:app:Gfuns\]) $$\label{eq:dipole-incompr-para} \lr.|{\frac{\pd H^0_{\ga\gb}}{\pd y_\gc}}_{\vec y = \vec 0} = -\lr.|{\frac{\pd H^0_{\ga\gb}}{\pd x_\gc}}_{\vec y = \vec 0} = \frac{R_1'}{8\upi \bar\mu} \left( 3\hat s_\gc \kd_{\ga\gb} - \hat s_\ga \kd_{\gb\gc} - \hat s_\gb \kd_{\gc\ga} \right) + \frac{R_3'}{2\upi \bar\mu} \Irr{\hat s_\ga \hat s_\gb \hat s_\gc},$$ where the functions $R'_n = R'_n(L_B; s, z)$ are given by \[eq:rzFn’\]. shows the velocity field expressed by \[eq:dipole-moment-incompr\] in the limits $\numBq \to 0$ and $\numBq \to \infty$, which are given by \[eq:Hfun0|Bq=0,eq:Hfun0|Bq=oo\], respectively. In the former limit, $R'_n$ reduces to the closed form expression given by \[eq:rzFn|Bq=0\] and decays spatially as $1/r^2$, which is the same as the far-field behavior of the dipole moment for a clean interface. We could also obtain the result for $\numBq = 0$ by evaluating the gradient of \[eq:Hfun0|Bq=0\] directly. We may similarly obtain the limiting behavior for $\numBq \to \infty$ as the two-dimensional Stokeslet dipole, i.e., the gradient of equation \[eq:Hfun0|Bq=oo\], which is given by $$\label{eq:Stokeslet-dipole-2d} \newcommand*\hs{\hat s} \lr.|{\frac{\pd H^0_{\ga\gb}}{\pd y_\gc}}_{\numBq\to\infty} \sim \frac{ 2 \hs_\ga \hs_\gb \hs_\gc - \hs_\ga \kd_{\gb\gc} - \hs_\gb \kd_{\gc\ga} + \hs_\gc \kd_{\ga\gb} }{4\upi\mu_s s}.$$ The velocity field represented by \[eq:Stokeslet-dipole-2d\] decays as $1/r$ and hence converges as $|\vec s| \to \infty$. This behavior contrasts with the monopole moment, which diverges logarithmically. However, as a three-dimensional flow, \[eq:Stokeslet-dipole-2d\] has no $z$ dependence and is constant in this direction, which is just another manifestation of Stokes’ paradox. For $\numBq$ large but finite, the required decay of the velocity along the $z$ direction occurs at distances $z \gtrsim \lenBq$. Far beyond the Boussinesq length, where $z/l \gg \numBq$, the $1/r^2$ decay of an inviscid interface is (eventually) recovered. Discussion ---------- In the context of trapped driven and active colloids, the interpretation of the leading-order monopole and dipole moments is largely the same as that discussed for clean interfaces. However, incompressibility dramatically restructures the behavior of these hydrodynamic modes. For instance, consider the dipolar mode associated with the $S_{33}$ component of the stresslet. At a clean interface, active colloids set up a source or sink flow on the interface (see \[fig:clean-stresslets\]b). These modes vanish for incompressible interfaces because the interface can no longer contract/expand to compensate for the source/sink. The remaining modes are significantly altered by surface incompressibility (see \[fig:monopoles,fig:dipoles\]) with the exception of the asymmetric modes. Recall that, at clean interfaces, the far-field fluid velocity both parallel and normal to the interface decays at the same rate: generally, $|\vec u| \sim r^{-1}$ for driven colloids and $|\vec u| \sim r^{-2}$ for active colloids (or colloids driven only by an external torque). If surface-viscous stresses are weak, $\numBq \ll 1$, then this far-field behavior also holds for the velocity components parallel to the interface. Namely, $|\vec u^\para| \sim r^{-1}$ for driven colloids and $|\vec u^\para| \sim r^{-2} $ for active colloids, where $\vec u^\para = \ten I_s \vdot \vec u$. However, the leading-order flow normal to the interface is significantly hindered. This hindrance is most severe for symmetric colloids, for which $\ten A^{(\nu)} = \vec 0$. In this case, the monopole and dipole moments only produce flow parallel to the interface, and $u^\text{m0}_3 = u^\text{m1}_3 = 0$. Hence, the fluid velocity normal to the interface is generally quadrupolar to leading order and decays at least as fast as $r^{-3}$. For driven and active colloids trapped in an asymmetric configuration, for which $\ten A^{(\alpha)} \neq \vec 0$, we recover the longer-ranged behavior $u_3 \sim r^{-2}$ associated with the dipole moment. Thus, this mode is of special importance. It may increase the rate at which colloids near the interface are transported toward or away from the interface via hydrodynamic interactions with driven or active colloids trapped at the interface. By the same mechanism, an “active sheet” of many trapped colloids at the interface may enhance mass transport in the $z$ direction. If the colloids comprising the active sheet move about randomly, this enhanced transport will likely lead to active diffusion. On the other hand, directed mass transport could be accomplished through organized motion of the active sheet. These possibilities are ripe opportunities for future research. While the flow normal to an incompressible interface is hindered in comparison to a clean interface, surface-viscous effects create very long-ranged flow parallel to the interface. Considering first the spatial behavior along the interfacial plane, we find that, for $\numBq \gg 1$ and distances $s \ll \lenBq$ from the colloid, $\vec u^\para \sim \ln s $ for the monopole moment and $\vec u^\para \sim s^{-1}$ for dipole moment. This behavior is simply that of a two-dimensional Stokes flow, which is recovered in the limit of highly viscous interfaces [@Saffman1975]. The divergent behavior of the velocity field is curtailed at distances $s \gtrsim \lenBq$, where bulk-viscous effects inevitably become important. To determine the spatial behavior along the $z$-axis, we observe from \[eq:rzFn,eq:rzFn’\] that, for $s \to 0$, $$\begin{aligned} R_n(\lenBq; 0, z) &\sim \frac{e^{2|z|/\lenBq}}{\lenBq} \ExpInt_1 \!\lr(){\frac{2|z|}{\lenBq}} \\ R'_n(\lenBq; 0, z) &\sim \frac{e^{2|z|/\lenBq}}{\lenBq\lr||z} \ExpInt_2 \!\lr(){\frac{2|z|}{\lenBq}},\end{aligned}$$ where $\ExpInt_p (x) = \int_1^\infty e^{-xt} / t^p \dd t$ is the generalized exponential integral. We note that [@Olver2010] $$\ExpInt_p(x) \sim \begin{cases} [{(-1)}^p / (p-1)!] x^{p-1} \ln x & \text{for}\ x \ll 1 \\ \ExpInt_p(x) \sim e^{-x}/x & \text{for}\ x \gg 1\ \text{and for all}\ p, \end{cases}$$ which implies that, for $|z| \ll \lenBq$, $R_n \sim \ln |z|$ and $R'_n \sim \ln |z|$. Recalling that $R_n$ and $R'_n$ govern the spatial behavior of the monopole and dipole moments, respectively, we see that both are logarithmically divergent in $z$ as $\numBq$ is made large. Therefore, the “lamellar” motion of the fluid strongly persists up to distances $z = O(\lenBq)$ into the surrounding fluid, regardless of whether the source of the motion is due to a driven or active colloid. At distances $z \gg \lenBq$, we find that $R_n \sim z^{-1}$ and $R'_n \sim z^{-2}$, so the far-field decay expected for $\numBq \ll 1$ is recovered. We expect this strong lateral fluid motion to significantly enhance spreading of a substance in directions parallel to the interface via Taylor dispersion. The shear flow driving Taylor dispersion is in this case generated by the motion of trapped colloids rather than motion of a bulk fluid relative to a no-slip boundary. The asymmetric modes that produce fluid motion normal to the interface are not modified by surface viscosity. Hence, $u_3 \sim 1/r^2$ for all $\numBq$. Future work will examine the implications of this fluid motion on transport and mixing rates. Conclusion {#sec:conclusion} ========== Summary ------- We have determined the leading order far-field flows generated by driven and active colloids trapped at planar fluid interfaces by a pinned contact line for $\numCa \gg 1$. Under these assumptions, the colloid is trapped in a fixed configuration and cannot move perpendicular to the interface. At clean interfaces devoid of surfactant, driven colloids produce “viscosity-averaged” Stokeslets when driven along the interface—the flow is no different than that expected for a colloid driven in an unbounded fluid of viscosity $\bar\mu$. Contact-line pinning at large $\numCa$ prevents such colloids from being driven normal to the interface. Similarly, active colloids produce viscosity-averaged force dipoles (stresslets) aligned in the swimming direction, similar to those generated by a swimmer in an unbounded fluid. This stresslet is associated with balanced hydrodynamic thrust and drag in the swimming direction. However, due to pinning of the contact line, such swimmers also generate additional “pumping” flows that are associated with net hydrodynamic forces and torques on the colloid that are supported by the interface. Some of these modes are associated with a net hydrodynamic force or torque on the colloid, which are supported by the pinned contact line, in contrast to swimmers in the bulk phase. These modes vanish if the colloid is adjacent, rather than adhered, to the interface. We consider surfactants, which render the interface incompressible even in the limit of scant surface concentrations. This constraint is generally applicable to driven and active colloids which move on interfaces for $\numCa \gg 1$. In this case, the flow modes associated with forced or self-propelled motion along the interface are altered significantly, even if the surfactant is dilute. An interesting feature of these modes is that they only induce “lamellar” fluid motion for which $u_z = 0$ at all distances $z$ from the interface. For active colloids in particular, the pumping mode associated “swimming” directly against the interface (i.e., in a perpendicular configuration) is eliminated for incompressible interfaces. We also find a set of force-dipole pumping modes that induce zero velocity at the interface and therefore persist independent of the interfacial mechanics. One such mode is produced when a pinned active colloid exerts a hydrodynamic torque on the interface. These modes may be of special importance to fluid mixing near boundaries—including solid ones—as they generate fluid motion normal to the interface. Future work and open issues --------------------------- A clear direction for future work is to probe experimental systems for signatures of the flow modes reported here. The differences we predict in the flow modes induced by active colloids in the adhered states versus adjacent states may be a useful in distinguishing these two cases in experiment. While we have determined the modes expected to dominate the far-field flow for driven and active colloids based on their trapped configuration the interface, comparison of our results to experimental datasets or computational results accounting for the near-field hydrodynamic details of particular colloids would be extremely valuable. Several open issues remain. We have not considered the effect of contact-line undulations on the flow. Interestingly, interfacial deformations due to such undulations are expected to decay at the same $1/r^2$ rate as the flow for an active colloid of negligible weight. Thus, these undulations may alter the flows we predict in interesting ways, especially because the contact line of an individual colloid may undulate randomly, being different for every colloid [@Stamou2000; @Kaz2012]. Another use for driven and active colloids at interfaces is to enhance mixing. Enhanced mixing in active colloidal suspensions has been studied extensively in bulk fluids [@Darnton2004; @Pushkin2013; @Lin2011; @Kasyap2014] and also in the vicinity of solid boundaries [@Mathijssen2015; @Mathijssen2018; @Kim2004]. Different behavior is expected for mobile fluid interfaces and will vary depending on the interfacial rheology and the adhered state of the active colloids that populate the interface. Using active or driven colloids to enhance transport processes presents an untapped dimension for interfacial engineering; interfaces are natural sites for many chemical reaction and separation processes. Our work emphasizes the importance of broken symmetry in the generation of mixing by active or passive colloids at interfaces. Such asymmetry naturally arises due to defects in colloid geometry, asymmetric trapped states, and, for active colloids, differences in activity in either fluid phase. Experimentalists seeking to enhance mixing using colloids at fluid interfaces should seek to design systems that maximize these sources of asymmetry. In addition, the effect of the interface on hydrodynamic interactions between swimmers at interfaces has yet to be investigated. Finally, while we have focused on far-field flows, detailed computations of the near-field hydrodynamics of specific types of active colloids in different adhered will also yield useful information such as the predicted trajectories of such colloids. The authors acknowledge useful discussions with Dr. Mehdi Molaei and Ms. Jiayi Deng as well as financial support from the National Science Foundation (NSF Grant No. DMR-1607878 and CBET-1943394) and the Gulf of Mexico Research Initiative. Declaration of Interests. The authors report no conflict of interest. Self-adjoint property of the Green’s functions {#sec:app:self-adjointness} ============================================== To show that the Green’s function $\ten G$ defined by \[eq:Gfun-clean\] is self-adjoint, i.e., $\ten G(\vec x, \vec y) = \ten G^\T(\vec y, \vec x)$, we make the following substitutions into \[eq:R-thm-clean-iface\]: $$\label{eq:sa-subs} \begin{aligned} \vec u(\vec x) &\to \ten G(\vec x, \vec y) \vdot \vec F, & \vec u'(\vec x) &\to \ten G(\vec x, \vec y') \vdot \vec F', \\ \ten\sigma(\vec x) &\to \ten T(\ten G; \vec x, \vec y) \vdot \vec F, & \ten\sigma'(\vec x) &\to \ten T(\ten G; \vec x, \vec y') \vdot \vec F', \\ \vec f(\vec x) &\to -\vec F \diracR3(\vec x - \vec y), & \vec f'(\vec x) &\to -\vec F \diracR3(\vec x - \vec y'), \\ \vec f_s(\vec x) &\to -\vec F \diracR2(\vec x - \vec y), & \vec f_s'(\vec x) &\to -\vec F \diracR2(\vec x - \vec y'). \end{aligned}$$ That is, we choose $\vec u$ as the flow field due to a point force $\vec F$ at $\vec y$ and $\vec u'$ the flow field due to another point force $\vec F'$ at point $\vec y'$. The point forces and their locations are arbitrary and may be exerted on either fluid or the interface. Each fluid domain is semi-infinite and bounded only by the interface. With the above substitutions, \[eq:R-thm-clean-iface\] becomes $$\begin{gathered} \label{eq:sa-intermediate} 0 = \int_{V^*} \left[ \diracR3(\vec x - \vec y) \vec F \vdot \ten G(\vec x, \vec y') \vdot \vec F' - \diracR3(\vec x - \vec y') \vec F' \vdot \ten G(\vec x, \vec y) \vdot \vec F \right] \dd V \\ + \int_{I^*} \left[ \diracR2(\vec x - \vec y) \vec F \vdot \ten G(\vec x, \vec y') \vdot \vec F' - \diracR2(\vec x - \vec y') \vec F' \vdot \ten G(\vec x, \vec y) \vdot \vec F \right] \dd A \\ + \oint_{R} \left\{ [\ten T(\vec x, \vec y) \vdot \vec F] \vdot [\ten G(\vec x, \vec y') \vdot \vec F'] - [\ten T(\vec x, \vec y') \vdot \vec F'] \vdot [\ten G(\vec x, \vec y) \vdot \vec F] \right\} \vdot \uvec n \dd S,\end{gathered}$$ where, for brevity, we omit $\ten G$ as an argument to $\ten T$. The integrations in \[eq:sa-intermediate\] are taken to be over an arbitrary volume that may contain points on the interface. If the boundaries of this volume in each fluid, represented by $R$, are made arbitrarily far from the points $\vec y$ and $\vec y'$, then the final integral in \[eq:sa-intermediate\] vanishes; $\ten G \sim r^{-1}$ and $\ten T(\ten G;\!) \sim 1/r^{-2}$, so this integral decays as $L_V^{-1}$ as $L_V \to \infty$, where $L_V$ is the characteristic size of the integration region. Then, using the definition of the Dirac delta, \[eq:sa-intermediate\] simplifies to $$\label{eq:G-is-self-adjoint} \vec F \vdot \ten G(\vec y, \vec y') \vdot \vec F' = \vec F' \vdot \ten G(\vec y', \vec y) \vdot \vec F.$$ Since $\vec F$, $\vec F'$, $\vec y$, and $\vec y'$ are all arbitrary, \[eq:G-is-self-adjoint\] implies that $\ten G(\vec y, \vec y') = \ten G^\T(\vec y', \vec y)$, that is, $\ten G$ is self-adjoint. Using the same procedure, it may be shown that $\ten H$ is also self-adjoint. Making a set of substitutions analogous to those appearing in \[eq:sa-subs\] along with the additional substitutions $\ten\varsigma(\vec x) \to \ten\Sigma(\ten H; \vec x, \vec y) \vdot \vec F$ and $\ten\varsigma(\vec x) \to \ten\Sigma(\ten H; \vec x, \vec y') \vdot \vec F'$ into \[eq:R-thm-incompressible-iface\], we find $$\begin{gathered} \label{eq:sa-intermediate-H} 0 = \vec F' \vdot \ten H(\vec y', \vec y) \vdot \vec F - \vec F \vdot \ten H(\vec y, \vec y') \vdot \vec F' \\ + \oint_{R} \left\{ [\ten T(\vec x, \vec y) \vdot \vec F] \vdot [\ten H(\vec x, \vec y') \vdot \vec F'] - [\ten T(\vec x, \vec y') \vdot \vec F'] \vdot [\ten H(\vec x, \vec y) \vdot \vec F] \right\} \vdot \uvec n \dd S \\ + \oint_{\partial I^*} \left\{ [\ten\Sigma(\vec x, \vec y) \vdot \vec F] \vdot [\ten H(\vec x, \vec y') \vdot \vec F'] - [\ten\Sigma(\vec x, \vec y') \vdot \vec F'] \vdot [\ten H(\vec x, \vec y) \vdot \vec F] \right\} \vdot \uvec m \dd C.\end{gathered}$$ In this case, an additional integral over $\partial I^*$ appears, which is the curve where our arbitrarily chosen fluid region intersects the interface. Both integrals in \[eq:sa-intermediate-H\] vanish as $L_V \to \infty$ provided that the Boussinesq length $\lenBq$ remains finite. For $r \gg \lenBq$, $\ten H$ and $\ten G$ share the same far-field decay behavior, i.e., $\ten H \sim r^{-1}$. From the remaining two terms in \[eq:sa-intermediate-H\], we find $\ten H(\vec y, \vec y') = \ten H^\T(\vec y', \vec y)$. Computation of the Green’s functions {#sec:app:Gfuns} ==================================== Here, we derive the Green’s functions $\ten G$ and $\ten H$ used in \[sec:clean-interfaces,sec:incompressible-interfaces\], respectively. Consider two immiscible fluids separated by a planar interface on the $z=0$ plane with a point force $\vec f$ applied at the point $(0, 0, h)$. Let $\FT f(\vec k) = \iint_{\Reals^2} f(\vec x^\para) \exp{[-i \vec k \vdot \vec s]} \dd^2 \vec s$ define the two-dimensional Fourier transform of $f(\vec s)$ in the $x$-$y$ plane, where $\vec s = (x, y)$ is the position vector on the interface. It is convenient to start with the boundary integral form of the Stokes equations for the upper and lower fluids, which, after applying the Fourier transform, are $$\begin{aligned} \label{eq:FT-upper-fluid} \Ind_{z > 0} \FT u^1_i(\vec k, z) &= - \FT T_{3 \ga i}(\ten J; \vec k, z) \FT v_\ga(\vec k) - \frac{1}{\mu_1} \FT J_{ij}(\vec k, z) \FT t^1_j(\vec k) + \frac{\Ind_{h>0}}{\mu_1} \FT J_{ij}(\vec k, z-h) f_j \\ \label{eq:FT-lower-fluid} \Ind_{z < 0} \FT u^2_i(\vec k, z) &= \FT T_{3 \ga i}(\ten J; \vec k, z) \FT v_\ga(\vec k) + \frac{1}{\mu_2} \FT J_{ij}(\vec k, z) \FT t^2_j(\vec k), + \frac{\Ind_{h<0}}{\mu_2} \FT J_{ij}(\vec k, z-h) f_j,\end{aligned}$$ respectively, where $J_{ij} = (\kd_{ij}/r + x_i x_j / r^3)/8\upi$ is the Oseen tensor, $T_{ijk}(\ten J;\!) = -\delta_{ij} P_k(\ten J;\!) + (\nabla_i J_{jk} - \nabla_j J_{ik})$, $\vec t^\nu = \bvec_z \vdot \vec\sigma^\nu$ is the surface traction at the interface, and $\vec v(\vec s) = \vec u(\vec s, z=0)$ is the surface velocity on the interface. For notational convenience, we hereafter omit $\ten J$ as an argument to $\ten T$. The Fourier transform of $\ten J$ is given by $$ \label{eq:FT-Stokeslet} \FT J_{ij}(\vec k, z) = \frac{\delta_{ij}}{2k} e^{-k|z|} + \frac{1}{4k^3} \FT{\nabla}_i \FT{\nabla}_j \left[ (1 + k|z|) e^{-k|z|} \right].$$ where $\FT{\nabla}_i = ik_i + \delta_{i3} (\partial/\partial z$). The interface, located at $z=0$, obeys the kinematic conditions $\bvec_z \vdot \vec v = 0$ (no flux) and $\vec u^1 = \vec u^2 = \vec v$ (continuity of velocity). These are accompanied by the dynamic condition given by the tangential stress balance at the interface, which in Fourier space is $$\label{eq:FT-interface-clean} \ten I_s \vdot ({[\FT{\vec t}]}_I + \Ind_{h = 0} {\vec f}) = \vec 0$$ for a clean interface and for an incompressible interface is $$\label{eq:FT-interface-incompr} \ten I_s \vdot ({[\FT{\vec t}]}_I + \Ind_{h=0} {\vec f}) = i \vec k \FT\pi + \mu_s k^2 \FT{\vec v}; \quad i\vec k \vdot \vec v = 0,$$ where $k = |\vec k|$. Multiplying \[eq:FT-upper-fluid,eq:FT-lower-fluid\] by $\mu_1$ and $\mu_2$, resp., taking the limit as $z \to 0^\pm$, resp., and adding the results gives $$\label{eq:Delta-force-balance-full} 2\bar\mu \kd_{i\gb} \FT v_\gb(\vec k) + \left( \mu_1 \lim_{z\to0^+} - \mu_2 \lim_{z\to0^-} \right) \FT T_{3 \ga i}(\vec k, z) \FT v_\ga (\vec k) + \FT J_{ij}(\vec k, 0) \lr[]{\FT t_j}_I (\vec k) = \FT J_{ij}(\vec k, -h) f_j,$$ where $\bar\mu = (\mu_1 + \mu_2) / 2$ is the average viscosity. Using \[eq:FT-Stokeslet\] and the definition of $\ten T$, we find that the second term on the left-hand side of \[eq:Delta-force-balance-full\] reduces to $$\label{eq:FT-double-layer-jump} \lim_{z\to0^\pm} \FT T_{3 \ga i} \FT v_\ga = - \left( \pm\frac{\delta_{i\ga}}{2} + \delta_{i3} \frac{ik_\ga}{2k} \right) \FT v_\ga,$$ which is the Stokes “double-layer” density for either side of the interface. For a clean interface, \[eq:FT-interface-clean,eq:FT-double-layer-jump\] in \[eq:Delta-force-balance-full\] yields, after a trivial Fourier inversion, $$\label{eq:FT-vel-surf-clean} \bar\mu \vec v(\vec s) = \ten I_s \vdot \ten J(\vec s - h \bvec_3) \vdot \vec f,$$ which shows that the fluid velocity at the interface is independent of the viscosity contrast and simply corresponds to the projection of the Oseen tensor, shifted to $z=h$, onto the interface at $z=0$. We may do the same for an incompressible interface by instead using \[eq:FT-interface-incompr\] in \[eq:Delta-force-balance-full\], from which we obtain $$\label{eq:Delta-force-balance} \left(\bar\mu + \frac12 \mu_s k\right) \FT v_\ga + \frac{ik_\ga}{4k} \pi = \FT J_{\ga j}|_{z=-h} f_j.$$ Taking the inner product of \[eq:Delta-force-balance\] with $i\vec k$ and solving for $\pi$ yields $$\label{eq:FT-surface-pressure} \begin{aligned} \FT\pi(\vec k) &= -\frac{4}{k} i\vec k \vdot \FT{\mathsfbi J}(\vec k, -h) \vdot \vec f \\ &= \frac{e^{-k |h|}}{k^2} \left[ (k |h| - 1) i\vec k + k^2 h \vec i_z \right] \vdot \vec f. \end{aligned}$$ The surface pressure is associated only with the Marangoni effect and depends neither on the bulk nor surface shear viscosities. Letting $\pi(\vec s, h) = \vec\Pi(\vec s, h) \vdot \vec f$ and carrying out the Fourier inversion to real space, we obtain $$\label{eq:surface-pressure} \newcommand*\rootrq{\sqrt{s^2 + h^2}} \vec\Pi(\vec s, h) = |h| \left( \gradS - \bvec_z \frac{\pd}{\pd h} \right) \frac{1}{4\upi\rootrq} + \frac{\vec s}{4\upi s^2} \left( 1 - \frac{|h|}{\rootrq} \right),$$ where $s = |\vec s|$. For $h = 0$, this result reduces to $\ten\Pi(\vec s, 0) = \vec s / 4\upi s^2$. Noting that $\vec v(\vec s) \equiv \ten H(\vec x \in I, \vec y) \vdot \vec f$ for $\vec y = (0,0,h)$, putting \[eq:FT-surface-pressure\] in \[eq:Delta-force-balance\] and solving for $\FT{\vec v}$ yields $$\label{eq:FT-vel-surf-incompr} \begin{aligned} \FT{\ten H} (\vec k, z=0, h) &= \frac{2}{2\bar\mu + \mu_s k} \left( \ten I_s - \frac{\vec{kk}}{k^2} \right) \vdot \FT{\ten J}(\vec k, -h) \\ &= \frac{e^{-k |h|}}{2\bar\mu + \mu_s k} \left( \frac{\ten I_s}{k} - \frac{\vec{kk}}{k^3} \right). \end{aligned}$$ Surface incompressibility of \[eq:FT-vel-surf-incompr\] is easily verified by contracting the right-hand side of this equation with $i\vec k$, thereby taking the divergence in Fourier space, which vanishes. We also see from \[eq:FT-vel-surf-incompr\] that a force perpendicular to the interface generates no interfacial flow; $H_{i3}(\vec x \in I, h) = 0$. We therefore conclude that the flow due to the $z$-component of the force is therefore the same as that for a rigid, no-slip wall, as is also noted by @Blawzdziewicz1999. Now, the self-adjoint property of $\ten H$ (see \[sec:app:self-adjointness\]) permits us to swap the roles of $h$ and $z$ in \[eq:FT-vel-surf-incompr\]; $$\label{eq:FT-Hfun} \FT{\ten H} (\vec k, z, h=0) = \frac{e^{-k |z|}}{2\bar\mu + \mu_s k} \left( \frac{\ten I_s}{k} - \frac{\vec{kk}}{k^3} \right).$$ Remarkably, from the interfacial flow profile due to a point force at $z = h$ \[eq:FT-vel-surf-incompr\], we automatically obtain the flow *at all points* $\vec x$ due to a point force at the interface ($h = 0$). Fourier inversion of \[eq:FT-Hfun\] to real space gives equation \[eq:Hfun\], $$\label{eq:app:Hfun} { H^0_{\ga\gb}(\lenBq; \vec x) = \frac{1}{4\upi \bar\mu} R_0(\lenBq; s, z) \kd_{\ga\gb} + \frac{1}{2\upi \bar\mu} R_2(\lenBq; s, z) \Irr{\hat s_\ga \hat s_\gb} }.$$ The functions $R_n$ are given by $$\label{eq:rzFn} R_n(\lenBq; s, z) = \int_0^\infty \frac{e^{-k|z|}}{2 + \lenBq k} J_n(ks) \dd k,$$ where $J_n$ is the Bessel function of the first kind of order $n$. In the special case that $\lenBq = 0$, we obtain $R_n$ in closed form as $$\label{eq:rzFn|Bq=0} R_n = \frac{(r - |z|)^n}{2r s^n}.$$ To obtain (surface) gradients of \[eq:app:Hfun\], we may take the tensor product of \[eq:FT-Hfun\] with $\vec k$ and repeat the Fourier inversion process to give $$\frac{\pd H^0_{\ga\gb}}{\pd x_\gc} = \frac{\pd H^0_{\ga\gb}}{\pd s_\gc} = \frac{R_1'}{8\upi \bar\mu} \left( \hat s_\ga \kd_{\gb\gc} + \hat s_\gb \kd_{\gc\ga} - 3\hat s_\gc \kd_{\ga\gb} \right) - \frac{R_3'}{2\upi \bar\mu} \Irr{\hat s_\ga \hat s_\gb \hat s_\gc},$$ where $$\label{eq:rzFn'} R_n(\lenBq; s, z) = \int_0^\infty \frac{k e^{-k|z|}}{2 + \lenBq k} J_n(ks) \dd k.$$ For $\lenBq = 0$, $R'_n$ reduces to $$\label{eq:rzFn'|Bq=0} R'_n = \frac{s^n (nr + |z|)}{2 r^3 (r + |z|)^n}.$$ To determine the full flow fields for $h \neq 0$, we can directly sum \[eq:FT-upper-fluid\] and \[eq:FT-lower-fluid\] to eliminate the Stokes double layer, which gives $$\label{eq:BIE-all-fluid} \FT u_i(\vec k, z; h) = - \frac1{\bar\mu} \FT J_{ij}(\vec k, z) \hat q_j(\vec k; h) + \frac1{\mu(h)} \FT J_{ij}(\vec k, z-h) f_j$$ where $$\hat{\vec q}(\vec k; h) := \bar\mu \left[ \frac{\FT{\vec t^1}(\vec k)}{\mu_1} - \frac{\FT{\vec t^2}(\vec k)}{\mu_2} \right] = \bar\mu \lr[]{\frac{\FT{\ten\sigma}(\vec k) \vdot \vec n}{\mu}}_I$$ is the Stokes single-layer density (in Fourier space). For a clean interface, setting $z=0$ in \[eq:BIE-all-fluid\] and putting \[eq:FT-vel-surf-clean\] into the result yields $$\label{eq:FT-single-layer-density-clean} \FT{\vec q}(k;h) = 4 k \lr(){\ten I - \frac{\mu(h)}{\bar\mu} \ten I_s} \vdot \FT{\ten J}(\vec k, -h) \vdot \vec f.$$ After inserting \[eq:FT-single-layer-density-clean\] back into \[eq:BIE-all-fluid\], lengthy algebraic manipulation, followed by inversion to real space, yields the velocity field in terms of the hydrodynamic image system described by $\ten G \vdot \vec f$ \[eq:Gfun-clean\]. A similar procedure could be used to determine $\ten H(\vec x, \vec y)$ for all $\vec y$, but we do not require that result in this paper. Others have performed such computations of $\ten H$ via other approaches [@Blawzdziewicz1999; @Fischer2006].

--- abstract: 'A consecutive formalism and analysis of *exactly solvable radial reflectionless potentials with barriers*, which in the spatial semiaxis of radial coordinate $r$ have one hole and one barrier, after which they fall down monotonously to zero with increasing of $r$, is presented. It has shown, that at their shape such potentials look qualitatively like radial scattering potentials in two-partial description of collision between particles and nuclei or radial decay potentials in the two-partial description of decay of compound spherical nuclear systems. An analysis shows, that the particle propagates without the smallest reflection and without change of an angle of motion (or tunneling) during its scattering inside the spherically symmetric field of the nucleus with such radial potential of interaction, i. e. the nuclear system with such interacting potential shows itself as *invisible* for the incident particle with any kinetic energy. An approach for construction of a hierarchy such reflectionless potentials is proposed, wave functions of the first potentials of this hierarchy are found.' author: - | Sergei P. Maydanyuk [^1]\ *Institute for Nuclear Research, National Academy of Sciences of Ukraine*\ *prosp. Nauki, 47, Kiev-28, 03680, Ukraine* bibliography: - 'Ref\_IMe.bib' title: Invisible nuclear system --- [**PACS numbers:**]{} 11.30.Pb 03.65.-w, 12.60.Jv 03.65.Xp, 03.65.Fd, [**Keywords:**]{} invisible nucleus, supersymmetry, exactly solvable model, reflectionless radial potentials, inverse power potentials, potentials of Gamov’s type, SUSY-hierarchy Introduction \[sec.1\] ====================== An interest to methods of supersymmetric quantum mechanics (SUSY QM) has been increasing every year. Initially constructed for a description of a symmetry between bosons and fermions in field theories, these methods during their development have formed completely independent section in quantum mechanics [@Cooper.1995.PRPLC]. Today, the methods of SUSY QM are a powerful tool for calculation and analysis of spectral characteristics of quantum systems, they have shown as extremely effective in obtaining of new types of *exactly solvable potentials* and in analysis of their properties, in an evident explanation of such unusual phenomena from the point of view of common sense as a *resonant tunneling*, a *reflectionless penetration* (or an *absolute transparency*) of the potentials (differed from the resonant tunneling by that it exists in a whole energy spectrum, where a coefficient of reflection is not only minimal but equals to zero also), *reinforcement of the barrier permeability* and *breaking of tunneling symmetry in opposite directions during the propagation of multiple of particles*, *absolute reflection for above-barrier energies*, *bound states in continuous energy spectra* of systems [@Zakhariev.1993.PHLTA; @Zakhariev.1994.PEPAN]. A number of papers has been increasing every year. Here, I should like to note a fine review [@Cooper.1995.PRPLC], to note intensively developed methods of *Nonlinear (also Polynomial, $N$-fold) supersymmetric quantum mechanics* in ), methods of *shape invariant potentials* with different types of parameters transformations (for example, see ), methods of a description of *self-similar potentials* studied by *Shabat* [@Shabat.1992.INPEE] and *Spiridonov* [@Spiridonov.1992.PRLTA; @Spiridonov.hep-th/0302046] and concerned with $q$-supersymmetry, methods of other types of potentials deformations and symmetries (for example, see [@Gomez-Ullate.quant-ph/0308062]), non-stationary approaches for a description of properties and behavior of quantum systems [@Samsonov.2002.Proc_IM]. One can note papers unified methods of supersymmetry with methods of inverse problem of quantum mechanics, and I should like to mention to nice monography [@Chadan.1977] and reviews [@Zakhariev.1994.PEPAN; @Zakhariev.1999.PEPAN] (with a literature list there). An essential progress has achieved in development of the methods of SUSY QM in spaces with different geometries [@Samsonov.1997.RusPhysJ], in non-commutative spaces [@Ghosh.2005.EPJC]. Having a powerful and universal apparatus, now the methods of SUSY QM find their application in a number of tasks of field theories, in QCD, in development of different models of quantum gravity, cosmology and other. However, in this paper I propose to pay attention into the reflectionless phenomenon in some types of spherical symmetric quantum systems (one note in development of SUSY QM formalism for different scattering problems). We find out a new type of radial exactly solvable reflectionless potential, which in its shape has one hole and one barrier, after which it falls down monotonously to zero with increasing of radial coordinate $r$ [@Maydanyuk.2005.APNYA]. Qualitatively, such potential looks like scattering potentials in two-partial description of collision between particle and spherically symmetric nucleus or decay potentials in the two-partial description of decay of compound spherical nuclear system. An analysis has shown that the particle propagates without the smallest reflection and without change of an angle of motion (or tunneling) in its scattering in the spherically symmetric field of the nucleus with such radial potential of interaction, i. e. the nuclear system with such potential shows itself as *invisible* for the incident particle with any kinetic energy. And this paper is devoted to an analysis of such radial reflection potentials. SUSY-interdependence between spectral characteristics of potentials partners in the radial problem \[sec.2\] ============================================================================================================ Darboux transformations \[sec.2.1\] ----------------------------------- Let’s consider a formalism of Darboux transformations in a problem about motion of a particle with mass $m$ in the spherically symmetric potential field (also see [@Andrianov.hep-th/9404061; @Bagrov.quant-ph/9804032]). The spherical symmetry of the potential allows to reduce this problem to the one-dimensional problem about the motion of this particle in the radial field $V(r)$, defined on the positive semiaxis of $r$, where wave function of such system looks like: $$\psi(r, \theta, \varphi) = \displaystyle\frac{\chi_{nl}(r)}{r} Y_{lm} (\theta, \varphi), \label{eq.2.1.1}$$ and the radial Schrödinger equation has a form: $$H \chi_{nl}(r) = -\displaystyle\frac{\hbar^{2}}{2m} \displaystyle\frac{d^{2} \chi_{nl}(r)}{dx^{2}} + \biggl(V_{n}(r) + \displaystyle\frac{l(l+1) \hbar^{2}}{2mr^{2}} \biggr) \chi_{nl}(r) = E_{n} \chi_{nl}(r) \label{eq.2.1.2}$$ and differs from the one-dimensional Schrödinger equation by a presence of a centrifugal term. One can reduce this equation to one-dimensional one by replacement: $$\bar{V}_{n}(r) = V_{n}(r) + \displaystyle\frac{l(l+1) \hbar^{2}}{2mr^{2}}. \label{eq.2.1.3}$$ As in the one-dimensional case, one can introduce operators $A_{1}$ and $A_{1}^{+}$: $$\begin{array}{ll} A_{1} = \displaystyle\frac{\hbar}{\sqrt{2m}} \displaystyle\frac{d}{dr} + W_{1}(r), & A_{1}^{+} = -\displaystyle\frac{\hbar}{\sqrt{2m}} \displaystyle\frac{d}{dr} + W_{1}(r), \end{array} \label{eq.2.1.4}$$ where $W_{1}(r)$ is a function, defined in the positive semiaxis $0 \le r < +\infty$ and continuous in it with an exception of some possible points of discontinuity. Then one can determine an interdependence between two hamiltonians of the propagation of the particle with mass $m$ in the fields $\bar{V}_{1}(r)$ and $\bar{V}_{2}(r)$: $$\begin{array}{l} H_{1} = A_{1}^{+} A_{1} + C_{1} = -\displaystyle\frac{\hbar^{2}}{2m} \displaystyle\frac{d^{2}}{dr^{2}} + \bar{V}_{1}(r), \\ H_{2} = A_{1} A_{1}^{+} + C_{1} = -\displaystyle\frac{\hbar^{2}}{2m} \displaystyle\frac{d^{2}}{dr^{2}} + \bar{V}_{2}(r), \end{array} \label{eq.2.1.5}$$ where each potential is expressed through one function $W_{1}(r)$: $$\begin{array}{ll} \bar{V}_{1}(r) = W_{1}^{2}(r) - \displaystyle\frac{\hbar}{\sqrt{2m}} \displaystyle\frac{d W_{1}(r)}{dr} + C_{1}, & \bar{V}_{2}(r) = W_{1}^{2}(x) + \displaystyle\frac{\hbar}{\sqrt{2m}} \displaystyle\frac{d W_{1}(r)}{dr} + C_{1}. \end{array} \label{eq.2.1.6}$$ One can find: $$\bar{V}_{2} (r) - \bar{V}_{1}(r) = V_{2} (r) - V_{1}(r) = 2\displaystyle\frac{\hbar}{\sqrt{2m}} \displaystyle\frac{d W_{1}(r)}{dr}. \label{eq.2.1.7}$$ The determination of the potentials $V_{1}(r)$ and $V_{2}(r)$ of two quantum systems on the basis of one function $W_{1}(r)$ establishes the interdependence between spectral characteristics (spectra of energy, wave functions, S-matrixes) of these systems. We shall consider this interdependence, as the interdependence given by Darboux transformations in the radial problem, and we shall name $W_{1}(r)$ as *superpotential*, potentials $V_{1}(r)$ and $V_{2}(r)$ as *supersymmetric potentials-partners*. Note, that there is a constant $C_{1}$ in the definition (\[eq.2.1.2\]) of the hamiltonians of two quantum systems. If to choose $C_{1}=E^{(1)}_{0}, E^{(2)}_{0} \ne E^{(1)}_{0}$ ($E^{(1)}_{0}$ and $E^{(2)}_{0}$ are the lowest levels of energy spectra of the first and second hamiltonians $H_{1}$ and $H_{2}$), then we obtain the most widely used construction two hamiltonians $H_{1}$ and $H_{2}$ in the one-dimensional case on the basis of the operators $A_{1}$ and $A_{1}^{+}$ (for example, see p. 287–289 in [@Cooper.1995.PRPLC]). However, this case corresponds to bound states in the discrete regions of the energy spectra of two studied quantum systems. For study of scattering, decay or synthesis processes in the radial consideration usually we deal with unbound states with the continuous region of the energy spectra (with the lowest energy levels $C_{1} = E^{(1)}_{0} = E^{(2)}_{0} = 0$) of quantum systems. Therefore, one need to use $C_{1}=0$ for obtaining the interdependence between the spectral characteristics of two systems on the basis of Darboux transformations (and we obtain a construction of hierarchy of potentials as in [@Maydanyuk.2005.APNYA], see p. 443–445): $$\begin{array}{l} H_{1} = A_{1}^{+} A_{1} = -\displaystyle\frac{\hbar^{2}}{2m} \displaystyle\frac{d^{2}}{dr^{2}} + \bar{V}_{1}(r), \\ H_{2} = A_{1} A_{1}^{+} = -\displaystyle\frac{\hbar^{2}}{2m} \displaystyle\frac{d^{2}}{dr^{2}} + \bar{V}_{2}(r). \end{array} \label{eq.2.1.8}$$ The interdependence between wave functions \[sec.2.3\] ------------------------------------------------------ We shall study two quantum systems, in each of which there is the scattering of the particle on the potential $V_{1}(r)$ or $V_{1}(r)$. Further, we shall not consider processes, concerned with loss of complete energy of systems (for example, dissipation, bremsstrahlung etc.). The energy spectra of these systems are *continuous*, and their lowest levels are *zero*. In accordence with (\[eq.2.1.8\]), we write: $$\begin{array}{l} H_{1} \chi^{(1)}_{k,l} = A_{1}^{+} A_{1} \chi^{(1)}_{k,l} = E^{(1)}_{k,l} \chi^{(1)}_{k,l}, \\ H_{2} \chi^{(2)}_{k^{\prime},l^{\prime}} = A_{1} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} = E^{(2)}_{k^{\prime},l^{\prime}} \chi^{(2)}_{k^{\prime},l^{\prime}}, \end{array} \label{eq.2.3.1}$$ where $E^{(1)}_{k, l}$ and $E^{(2)}_{k^{\prime}, l^{\prime}}$ are the energy levels of two systems with orbital quantum numbers $l$ and $l^{\prime}$, $\chi^{(1)}_{k,l}(x)$ and $\chi^{(2)}_{k^{\prime}, l^{\prime}}(x)$ are the radial components of wave functions, concerned with these levels, $k = \displaystyle\frac{1}{\hbar}\sqrt{2mE^{(1)}_{k,l}}$ and $k^{\prime} = \displaystyle\frac{1}{\hbar}\sqrt{2mE^{(2)}_{k^{\prime},l^{\prime}}}$ are wave vectors corresponding to the levels $E^{(1)}_{k,l}$ and $E^{(2)}_{k^{\prime}, l^{\prime}}$. From (\[eq.2.3.1\]) we obtain: $$H_{2} (A_{1} \chi^{(1)}_{k,l}) = A_{1} A_{1}^{+} (A_{1} \chi^{(1)}_{k,l}) = A_{1} (A_{1}^{+} A_{1} \chi^{(1)}_{k,l}) = A_{1} (E^{(1)}_{k,l} \chi^{(1)}_{k,l}) = E^{(1)}_{k,l} (A_{1} \chi^{(1)}_{k,l}). \label{eq.2.3.2}$$ We see, that the function $f(r)=A_{1} \chi^{(1)}_{k,l}(r)$ is the eigen-function of the operator $\hat{H}_{2}$ with quantum number $l$ to a constant factor, i. e. it represents the wave function $\chi^{(2)}_{k^{\prime},l}(r)$ of the hamiltonian $H_{2}$. The energy level $E^{(1)}_{k,l}$ must be the eigen-value of this operator exactly, i. e. it represents the energy level $E^{(2)}_{k^{\prime},l}$ of this hamiltonian. Here, new wave function and energy level have the same index ${k^{\prime}}$. One can write: $$\begin{array}{lcr} A_{1} \chi^{(1)}_{k,l} (r) = N_{2} \chi^{(2)}_{k^{\prime},l} (r), & E^{(1)}_{k,l} = E^{(2)}_{k^{\prime},l}, & N_{2} = const. \end{array} \label{eq.2.3.3}$$ Taking into account (\[eq.2.3.1\]), one can write: $$H_{1} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) = A_{1}^{+} A_{1} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) = A_{1}^{+} (A_{1} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) = A_{1}^{+} (E^{(2)}_{k^{\prime},l^{\prime}} \chi^{(2)}_{k^{\prime},l^{\prime}}) = E^{(2)}_{k^{\prime},l^{\prime}} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}). \label{eq.2.3.4}$$ and obtain: $$\begin{array}{lcr} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} (r) = N_{1} \chi^{(1)}_{k,l^{\prime}} (r), & E^{(2)}_{k^{\prime},l^{\prime}} = E^{(1)}_{k,l^{\prime}}, & N_{1} = const. \end{array} \label{eq.2.3.5}$$ Thus, we obtain the following interdependences between the wave functions and the levels of the continuous energy spectra of two systems SUSY-partners in the radial problem: $$\begin{array}{cccc} \chi^{(1)}_{k,l^{\prime}} (r) = \displaystyle\frac{1}{N_{1}} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} (r), & \chi^{(2)}_{k^{\prime},l} (r) = \displaystyle\frac{1}{N_{2}} A_{1} \chi^{(1)}_{k,l} (r), & E^{(1)}_{k,l} = E^{(2)}_{k^{\prime},l}, & E^{(1)}_{k,l^{\prime}} = E^{(2)}_{k^{\prime},l^{\prime}}. \end{array} \label{eq.2.3.6}$$ Darboux transformations establish the interdependence between the wave functions for the same energy levels of two systems. The coefficients $N_{1}$ and $N_{2}$ can be calculated from a normalization conditions for the wave functions (for the continuous energy spectra), and boundary condition are defined by scattering or decay process. The interdependence between amplitudes of transittion and reflection \[sec.2.4\] -------------------------------------------------------------------------------- For scattering the radial superpotential $W_{1}(r)$, the potentials $V_{1}(r)$ and $V_{2}(r)$ are finite in the whole spatial region of their definition and in asymptotic they tend to zero: $$\begin{array}{ll} W_{1} (r \to +\infty) = 0, & V_{1} (r \to +\infty) = V_{2} (r \to +\infty) = 0. \end{array} \label{eq.2.4.1}$$ Let’s find an interdependence between resonant and potential components of S-matrixes of these systems (for example, also see [@Andrianov.hep-th/9404061]). One can describe the particle motion in the direction to zero inside the fields $V_{1}(r)$ and $V_{2}(r)$ with use of plane waves $e^{-ikr}$ (we assume, that the plane waves of both systems have the same wave vectors $k$). In spatial asymptotic regions we obtain transmitted waves $T_{1}(k)e^{ikx}$ and $T_{2}(k)e^{ikx}$, which are formed in result of total propagation (with possible tunneling) through the potentials and describe the resonant scattering of the particle on the potentials, and reflected waves $R_{1}(k)e^{ikx}$ and $R_{2}(k)e^{ikx}$, which are formed in result of reflection from the potentials and describe the potential scattering of the particle on the potentials. For each process of scattering one can write components of wave functions, which are formed in result of the transmission through the potential and the reflection from it: $$\begin{array}{ll} \chi_{inc+ref}^{(1)}(k, r \to +\infty) = \bar{N}_{1} (e^{-ikr} + R_{1} e^{ikr}), & \chi_{tr}^{(1)}(k, r \to +\infty) \to \bar{N}_{1} T_{1} e^{ikr}, \\ \chi_{inc+ref}^{(2)}(k, r \to +\infty) = \bar{N}_{2} (e^{-ikr} + R_{2} e^{ikr}), & \chi_{tr}^{(2)}(k, r \to +\infty) \to \bar{N}_{2} T_{2} e^{ikr}, \end{array} \label{eq.2.4.2}$$ where the coefficients $\bar{N}_{1}$ and $\bar{N}_{2}$ can be found from the normalization conditions. Using (\[eq.2.4.2\]) for the wave functions in asymptotic region, taking into account the interdependence (\[eq.2.3.6\]) between them and definitions (\[eq.2.1.1\]) for the operators $A_{1}$ and $A_{1}^{+}$, we obtain: $$\begin{array}{l} \bar{N}_{1} \biggl( e^{-ikr} + R_{1} e^{ikr} \biggr) = \displaystyle\frac{\bar{N}_{2}}{N_{1}} \displaystyle\frac{ik\hbar}{\sqrt{2m}} \biggl(e^{-ikr} - R_{2} e^{ikr} \biggr), \\ \bar{N}_{1} T_{1} e^{ikr} = -\displaystyle\frac{\bar{N}_{2}}{N_{1}} \displaystyle\frac{ik\hbar}{\sqrt{2m}} T_{2} e^{ikr}. \end{array} \label{eq.2.4.3}$$ These expressions are carried out only, if items with the same exponents are equal between themselves. We find: $$\bar{N}_{1} = \displaystyle\frac{\bar{N}_{2}}{N_{1}} \displaystyle\frac{ik\hbar}{\sqrt{2m}} \label{eq.2.4.4}$$ and $$\begin{array}{cc} R_{1}(k) = - R_{2}(k), & T_{1}(k) = - T_{2}(k). \end{array} \label{eq.2.4.5}$$ Exp. (\[eq.2.4.5\]) establish the interdependence between the amplitudes of the transmission $T_{1}(k)$, $T_{2}(k)$ and the amplitudes of the reflection $R_{1}(k)$, $R_{2}(k)$ for the particle relatively two potentials. Squares of modules of the transmitted and reflected amplitudes represent the resonant and potential components of the S-matrixes for two systems. We see, that all these values do not depend on the normalized coefficients $N_{1}$, $N_{2}$, $\bar{N}_{1}$, $\bar{N}_{2}$. One can introduce the matrix of scattering $S_{l}(k)$ for $l$-partial wave: $$\chi_{nl}(r) \sim S_{l}(k) e^{ikr} - (-1)^{l} e^{-ikr} \label{eq.2.4.6}$$ and determine a phase shift $\delta_{l}(k)$: $$e^{i\delta_{l}(k)} = S_{l}(k). \label{eq.2.4.7}$$ Then with taking into account (\[eq.2.4.2\]), we find: $$S_{l}(k) = (-1)^{l+1} (R_{l}(k) + T_{l}(k)). \label{eq.2.4.8}$$ One can see, how these partial components of the S-matrixes and the phases for two systems are interdependent (also see [@Cooper.1995.PRPLC], p. 278–279): $$\begin{array}{cc} S_{l}^{(1)}(k) = - S_{l}^{(2)}(k), & \delta_{l}^{(1)}(k) = \delta_{l}^{(2)}(k) + \pi/2. \end{array} \label{eq.2.4.9}$$ Let’s consider a spherically symmetric quantum system with the radial potential, to which a zero amplitude of the reflection $R(k)$ of the wave function corresponds. The particle during its scattering in this field propagates into a center without the smallest reflection by the field. In particular, such is a nul radial potential. We shall name such quantum systems and their radial potentials as *reflectionless* or *absolutely transparent*. Then from (\[eq.2.4.5\]) one can see, that the potential-partner for the reflectionless potential is reflectionless also in that region, where it is finite. If such potential is finite on the whole region of its definition, then it is reflectionless completely (i. e. in standard definition of quantum mechanics). A series of the finite potentials of hierarchy, which contains the nul radial potential, should be reflectionless also. Using this simple idea and knowing a form of only one reflectionless potential, one can construct many new exactly solvable radial reflectionless potentials. Spherically symmetric systems with absolute transparency \[sec.3\] ================================================================== A radial reflectionless potentials with barriers \[sec.3.1\] ------------------------------------------------------------ In [@Maydanyuk.2005.APNYA] (see sec. 5.3.2, p. 459–462) an one-dimensional superpotential, defining a reflectionless potential which in semiaxis $0 < x < +\infty$ has one hole, one barrier and then with increasing of $x$ falls down monotonously to zero in asymptotic region, had found. As this superpotential is obtained on the basis of interdependence between two one-dimensional hamiltonians with continuous energy spectra, one can use it in the problem about scattering of a particle in the spherically symmetric field with a barrier and with orbital quantum number $l=0$. In such case, we have: $$\begin{array}{ll} W(r) = \displaystyle\frac{2\beta - \alpha}{f(\bar{r})} - \displaystyle\frac{\beta}{\bar{r}}, & \mbox{при } 2\beta \ne \alpha, \end{array} \label{eq.3.1.1}$$ where $$f(\bar{r}) = C(2\beta - \alpha) \bar{r}^{2\beta / \alpha} + \bar{r}. \label{eq.3.1.2}$$ Here $\bar{r} = r+r_{0}$, $\beta$ and $C$ are arbitrary real positive constants, $r_{0}$ is a positive number close to zero, and a designation $\alpha = \displaystyle\frac{\hbar}{\sqrt{2m}}$ is introduced. This superpotential is defined on the positive semiaxis of $r$ (at $r > r_{0}$). Let’s find potentials-partners for the superpotential (\[eq.3.1.1\]). In accordance with (\[eq.2.1.6\]), we obtain: $$\begin{array}{lcl} V_{1,2}(r) & = & \displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} - \displaystyle\frac{2\beta (2\beta - \alpha)} {\bar{r} f(\bar{r})} + \displaystyle\frac{\beta^{2}}{\bar{r}^{2}} \pm \\ & \pm & \Biggl( \displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} - \displaystyle\frac{2\beta (2\beta - \alpha)} {\bar{r} f(\bar{r})} + \displaystyle\frac{\alpha\beta}{\bar{r}^{2}} \Biggr) \end{array} \label{eq.3.1.3}$$ or $$\begin{array}{lcl} V_{1}(r) & = & \displaystyle\frac{\beta (\beta - \alpha)} {\bar{r}^{2}}, \\ V_{2}(r) & = & 2\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} - \displaystyle\frac{4\beta (2\beta-\alpha)} {\bar{r} f(\bar{r})} + \displaystyle\frac {\beta (\beta+\alpha)} {\bar{r}^{2}}. \end{array} \label{eq.3.1.4}$$ From (\[eq.3.1.4\]) one can see that at $\beta = \alpha$ the first potential $V_{1}(r)$ obtains zero value and, therefore, it becomes reflectionless. Then, according to (\[eq.2.4.5\]), if the second potential $V_{2}(r)$ is finite in a whole region of its definition, then it should be reflectionless also. At $\beta = \alpha$ we obtain: $$V_{2}(r) = \displaystyle\frac{2\alpha^{2}} {\biggl(r + r_{0} + \displaystyle\frac{1}{C\alpha}\biggr)^{2}}. \label{eq.3.1.5}$$ We see, that this potential is finite in the whole region of its definition at any values of the parameters $C>0$ and $r_{0} \ge 0$. Thus, we have obtained the reflectionless potential of the inverse power type with a shift to the left, which is defined on the whole positive semiaxis of $r$ (including $r=0$ and $r_{0}=0$). In accordance with [@Maydanyuk.2005.APNYA] (see p. 452–455, sec. 5.1.2), one can construct a hierarchy of the inverse power potentials, and a general solution of the potential with arbitrary number $n$ can be written down so: $$\begin{array}{ll} V_{n}(r) = \displaystyle\frac{\gamma_{n} \alpha^{2}}{\bar{r}^{2}}, & \gamma_{n \pm 1} = 1 + \gamma_{n} \pm \sqrt{4\gamma_{n}+1}. \end{array} \label{eq.3.1.6}$$ If to require, that the first potential $V_{1}(r)$ in this hierarchy must be constant (i. e. at $\gamma_{1}=0$ and $n=1$), then all hierarchy of the inverse power potentials (\[eq.3.1.6\]) becomes the *hierarchy of the reflectionless inverse power potentials*, and the solution (\[eq.3.1.5\]) becomes the general solution for the reflectionless inverse power potential. Note, that *when the hierarchy of the inverse power potentials becomes reflectionless, then the coefficients $\gamma_{n}$ become integer numbers*. We write its first values: $$\gamma_{n} = 0, 2, 6, 12, 20, 30, 42... \label{eq.3.1.7}$$ Now, if to calculate $\beta_{n}$ for given $\gamma_{n}$ with number $n$ from (\[eq.3.1.7\]) from the following condition: $$\beta_{n} (\beta_{n}-\alpha) = \gamma_{n} \alpha^{2}, \label{eq.3.1.8}$$ then the first potential $V_{1}(r)$ from (\[eq.3.1.4\]) becomes reflectionless inverse power potential (at $\beta =\beta_{n}$). The second potential $V_{2}(r)$ from (\[eq.3.1.4\]) is finite in the whole region of its definition (including $r=0$) and should be reflectionless also, however it is not inverse power potential. So, substituting the coefficients $\gamma_{n}$ with other numbers $n$ into the second expression (\[eq.3.1.4\]) for the potential $V_{2}(r)$, one can construct the whole hierarchy of the radial reflectionless potentials of this new type. In Fig. \[fig.1\] the potential $V_{2}(r)$ for the chosen values of the parameters $C$ and $\gamma_{n}$ is shown. From here one can see, that such potential has one hole and one barrier, after which it falls down monotonously to zero with increasing of the radial coordinate $r$. ![ A dependence of the radial potential $V_{2}(r)$ on $C$ and $\gamma_{n}$: (a) the barrier maximum and the hole minimum of this potential are changed along the axis $r$ at the change of $C$ (at $C = 0.01, 0.1, 0.3, 1.0, 2.5$, $\gamma_{n}=6$, $r_{0}=0.5$); (b) the barrier maximum of this potential practically does not changed along the axis $r$ at the change of $\gamma_{n}$ (at $C = 1$, $\gamma_{n}=2, 6, 12, 20 $, $r_{0}=0.5$). \[fig.1\]](f31a.eps "fig:"){width="57mm"} ![ A dependence of the radial potential $V_{2}(r)$ on $C$ and $\gamma_{n}$: (a) the barrier maximum and the hole minimum of this potential are changed along the axis $r$ at the change of $C$ (at $C = 0.01, 0.1, 0.3, 1.0, 2.5$, $\gamma_{n}=6$, $r_{0}=0.5$); (b) the barrier maximum of this potential practically does not changed along the axis $r$ at the change of $\gamma_{n}$ (at $C = 1$, $\gamma_{n}=2, 6, 12, 20 $, $r_{0}=0.5$). \[fig.1\]](f31b.eps "fig:"){width="57mm"} In its behavior such potential looks qualitatively like radial potentials with barriers used in theory of nuclear collisions for a description of scattering of particles on spherical nuclei, and for a description of decay and synthesis of nuclei of a spherical type also. This potential is reflectionless, if the parameter $\gamma_{n}$ has discrete values from the sequence (\[eq.3.1.7\]). For any reflectionless potential with given $\gamma_{n}$ one can displace continuously its barrier and hole along an axis $r$ by use of the parameter $C$. Such deformation of the shape of the reflectionless potential is shown in Fig. \[fig.2\]. ![The reflectionless radial exactly solvable potential $V_{2}(r)$ with the barrier. Continuous change of its shape at variation of $C$ ($\gamma_{n}=6$, $r_{0}=0.5$) \[fig.2\]](f32a.eps "fig:"){width="80mm"} ![The reflectionless radial exactly solvable potential $V_{2}(r)$ with the barrier. Continuous change of its shape at variation of $C$ ($\gamma_{n}=6$, $r_{0}=0.5$) \[fig.2\]](f32b.eps "fig:"){width="80mm"} An analysis of wave functions \[sec.3.2\] ----------------------------------------- ### Wave functions for the reflectionless inverse power potential \[sec.3.2.1\] Let’s find a radial wave function describing the scattering of the particle on the inverse power reflectionless potential (\[eq.3.1.5\]) at $l=0$. For the potential $V_{1}(r)$ with zero value of (\[eq.3.1.4\]) one can write its radial wave function (for arbitrary energy level concerned with wave vector $k$) at $l=0$ by such a way (at $\beta=\alpha$): $$\chi_{l=0}^{(1)}(k,r) = \bar{N}_{1} (e^{-ikr} - S_{l=0}^{(1)} e^{ikr}). \label{eq.3.2.1.1}$$ Then, one can find a radial wave function at $l=0$ for the reflectionless potential $V_{2}(r)$ of (\[eq.3.1.5\]) (for the energy level corresponding to the wave vector $k$) on the basis of the second expression of (\[eq.2.3.6\]). Taking into account (\[eq.2.1.4\]) and (\[eq.2.4.9\]), we obtain: $$\begin{array}{lcl} \chi_{l=0}^{(2)}(k,r) & = & \displaystyle\frac{1}{N_{2}} A_{1} \chi_{l=0}^{(1)}(k,r) = \displaystyle\frac{\bar{N}_{1}}{N_{2}} \biggl( \alpha\displaystyle\frac{d}{dr} + W(r) \biggr) \Bigl(e^{-ikr} - S_{l=0}^{(1)} e^{ikr}\Bigl) = \\ % & = & % i\displaystyle\frac{\bar{N}_{1}}{k\alpha N_{2}} % \biggl[ % \biggl( 1 + \displaystyle\frac{iW(r)}{k\alpha} \biggr) % e^{-ikr} - % S_{l=0}^{(2)} % \biggl( 1 - \displaystyle\frac{iW(r)}{k\alpha} \biggr) % e^{ikr} % \biggr] = \\ & = & \chi_{l=0}^{(-)}(k,r) - S_{l=0}^{(2)}\chi_{l=0}^{(+)}(k,r), \end{array} \label{eq.3.2.1.2}$$ where $$\chi_{l=0}^{(\pm)}(k,r) = \bar{N}_{2} \biggl( 1 \mp \displaystyle\frac{iW(r)}{k\alpha} \biggr) e^{\pm ikr} \label{eq.3.2.1.3}$$ and $$\bar{N}_{2} = i\displaystyle\frac{\bar{N}_{1}}{k\alpha N_{2}} \label{eq.3.2.1.4}$$ In accordance with main statements of quantum mechanics, for applying such form of the radial wave function to the description of scattering of the particle in the field of the potential $V_{2}(r)$, it needs to achieve a boundary requirement $\chi_{l=0}^{(2)}(k,r) \to 0$ at $r \to 0$, which gives a finiteness of the wave function (\[eq.2.1.1\]) at $r=0$ (and $S_{l=0}^{(2)}$ must have finite values and be not zero). One can see from (\[eq.3.2.1.2\]), that it is fulfilled only in case ($W(r)$ is real): $$\begin{array}{lcl} Re (S_{l=0}^{(2)}) = \displaystyle\frac{k^{2}\alpha^{2}-W^{2}(0)} {k^{2}\alpha^{2}+W^{2}(0)}, & Im (S_{l=0}^{(2)}) = \displaystyle\frac{2W(0)k\alpha}{k^{2}\alpha^{2}+W^{2}(0)}, \end{array} \label{eq.3.2.1.5}$$ where $$W(0) = -\displaystyle\frac{\alpha}{r_{0} + \displaystyle\frac{1}{C\alpha}}. \label{eq.3.2.1.6}$$ For the partial components of the S-matrix the following property $|S_{l=0}|^{2} = 1$ is fulfilled also. In limit $r \to 0$ we obtain the following expression for the radial wave function: $$\chi_{l=0}^{(2)}(k,r) = \bar{N}_{2} \biggl(1 + \displaystyle\frac{iW(0)}{k\alpha}\biggr) \biggl(e^{-ikr} - S_{l=0}^{(2)} \displaystyle\frac{k\alpha - iW(0)}{k\alpha + iW(0)} e^{ikr}\biggl), \label{eq.3.2.1.7}$$ which in its form coincides with Exp. (\[eq.3.2.1.1\]) for the wave function for the potential $V_{1}(r)$ from (\[eq.3.1.4\]) with zero value. In Fig. \[fig.3\] real and imaginary parts of the wave function near to the point $r=0$ are shown (here the starting formulas (\[eq.3.2.1.2\])–(\[eq.3.2.1.3\]) are taken). From Fig. \[fig.3\] (a, b) one can see a deformation of the imaginary part of this wave function with change of the wave vector $k$ and the parameter $C$. The real part of the wave function in its behavior looks like the imaginary part (see Fig. \[fig.3\] (c)). Here, one can see also that at such choice of the real and imaginary parts of the partial components of the S-matrix the wave function leaves from its zero value at $r=0$. ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f02.eps "fig:"){width="50mm"} ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f03.eps "fig:"){width="50mm"} ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f04.eps "fig:"){width="50mm"} In Fig. \[fig.4\] an evident picture of behavior of the imaginary part of the wave function close to point $r=0$ with continuous change of the wave vector $k$ and the parameter $C$ is shown. ![ Behavior of the imaginary part of the wave function (\[eq.3.2.1.2\]) depending on the wave vector $k$ and the parameter $C$ (the values $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$) are used): (a) dependence on the wave vector $k$ (at $C=1$); (b) dependence on the parameter $C$ (at $k=0.5$) \[fig.4\]](f06.eps "fig:"){width="80mm"} ![ Behavior of the imaginary part of the wave function (\[eq.3.2.1.2\]) depending on the wave vector $k$ and the parameter $C$ (the values $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$) are used): (a) dependence on the wave vector $k$ (at $C=1$); (b) dependence on the parameter $C$ (at $k=0.5$) \[fig.4\]](f07.eps "fig:"){width="80mm"} Note, that according to (\[eq.2.4.9\]), the condition (\[eq.3.2.1.5\]) can bring to not zero values of the radial wave function $\chi_{l=0}^{(1)}(k,r)$ at $r \to 0$ and can give discontinuity of the total wave function. However, a variation of the phase of $S_{l=0}^{(1)}$ does not change the form of the potential $V_{1}(r)$, which remains zero and reflectionless. In other words, the reflectionless potential $V_{1}(r)$ allows an arbitrariness in a choice of boundary conditions for the wave function at point $r=0$, and the chosen boundary conditions define the shape of the total wave function and a process proceeding in the field of the potential $V_{1}(r)$. There is a similar situation for the potential $V_{2}(r)$, which remains reflectionless with the variation of the S-matrix phase. Now let’s analyze the form of the wave function (\[eq.3.2.1.2\]) in asymptotic region. According to (\[eq.3.1.1\]), $W(r) \to 0$ at $r \to +\infty$ and we obtain: $$\chi_{l=0}^{(2)}(k,r) = \bar{N}_{2} \biggl(e^{-ikr} - S_{l=0}^{(2)} e^{ikr}\biggr). \label{eq.3.2.1.8}$$ One can see, that two components $\chi_{l=0}^{(\pm)}(k,r)$ in (\[eq.3.2.1.2\]) represent convergent and divergent waves, that can be useful for analysis of propagation of the particle in the field $V_{2}(r)$. Thus, we have found an *exact analytical division of the total radial wave function into its convergent and divergent components* (as for regular and singular Coulomb functions for the known Coulomb potential) in the description of scattering of the particle in the inverse power potential (\[eq.3.1.5\]). If for the convergent and divergent waves to define radial flows as: $$j^{\pm} (k,r) = \displaystyle\frac{i\hbar}{2m} \biggl( \chi_{l=0}^{(\pm)}(k,r) \displaystyle\frac{d \chi_{l=0}^{(\pm), *}(k,r)}{dr}- \chi_{l=0}^{(\pm), *}(k,r) \displaystyle\frac{d \chi_{l=0}^{(\pm)}(k,r)}{dr} \biggr), \label{eq.3.2.1.9}$$ then for both waves we obtain coincided absolute values of their flows: $$j^{\pm} (k,r) = \pm\displaystyle\frac{\hbar k}{m} |\bar{N}_{2}|^{2}. \label{eq.3.2.1.10}$$ We see, that the flows do not vary in dependence on $r$, and this gives a fulfillment of a conservation law for the flows from each wave and the total flow. Therefore, the convergent wave $\chi_{l=0}^{(-)}(k,r)$ propagates into the center without the smallest reflection by the field, because it is defined and is continuous on the whole region of the definition of the potential (\[eq.3.1.5\]) and it forms the constant radial flow $j^{-}(r)$. Now we can tell with confidence, that the *inverse power radial potential (\[eq.3.1.5\]), for which we have found the radial wave function (\[eq.3.2.1.2\])–(\[eq.3.2.1.4\]) for scattering, is reflectionless at $l=0$*. Further, one can find the radial wave functions at $l \ne 0$ on the basis of the same analysis, if for the radial wave function (\[eq.3.2.1.1\]) for the potential with zero value to use spherical Hankel functions instead of factors $\exp{(\pm ikr)}$. ### Wave functions for the reflectionless potential with the barrier \[sec.3.2.2\] One can use Exp. (\[eq.3.1.4\]) for calculation of a new reflectionless potential $V_{2}(r)$ with a barrier on the basis of the known reflectionless inverse power potential $V_{1}(r)$. Let’s assume, that these potentials are connected with one superpotential $W_{2}(r)$. Let’s consider the wave function for the reflectionless inverse power potential $V_{1}(r)$ at $l=0$ in the form: $$\chi_{l=0}^{(1)}(k,r) = \bar{N}_{1} \Bigl(f^{-}(r) e^{-ikr} - S_{l=0}^{(1)} f^{+}(r) e^{ikr} \Bigr). \label{eq.3.2.2.1}$$ Then the radial wave function at $l=0 $ for the reflectionless potential $V_{2}(r)$ with the barrier can be found on the basis of the second expression of (\[eq.2.3.6\]). Taking into account (\[eq.2.1.4\]) and (\[eq.2.4.9\]), we obtain: $$\begin{array}{lcl} \chi_{l=0}^{(2)}(k,r) & = & % \displaystyle\frac{1}{N_{2}} A_{2} \chi_{l=0}^{(1)}(k,r) = \displaystyle\frac{\bar{N}_{1}}{N_{2}} \biggl( \alpha\displaystyle\frac{d}{dr} + W_{2}(r) \biggr) \Bigl(f^{-}(r) e^{-ikr} - S_{l=0}^{(1)} f^{+}(r) e^{ikr} \Bigr) = \\ & = & \displaystyle\frac{\bar{N}_{1}}{N_{2}} \biggl[ \biggl(\alpha \displaystyle\frac{d f^{-}(r)}{dr} - ik\alpha f^{-}(r) + W_{2}(r) f^{-}(r) \biggr) e^{-ikr} - \\ & - & S_{l=0}^{(2)} \biggl(\alpha \displaystyle\frac{d f^{+}(r)}{dr} + ik\alpha f^{+}(r) + W_{2}(r) f^{+}(r) \biggr) e^{ikr} \biggr]. \end{array} \label{eq.3.2.2.2}$$ In this expression one can see the division of the total radial wave function into the convergent and divergent components, that can be interesting in analysis of scattering (with possible tunneling) of the particle in the field of the reflectionless potential $V_{2}(r)$ with the barrier. So, if to use the potential (\[eq.3.1.5\]) as the first reflectionless inverse power potential, then we find: $$\beta_{2} = 2 \alpha \label{eq.3.2.2.3}$$ and $$\begin{array}{lcl} f^{\pm}(r) = 1 \pm \displaystyle\frac{i} {k \biggl(\bar{r} + \displaystyle\frac{1}{C\alpha} \biggr)}, & \displaystyle\frac{d f^{\pm}(r)}{dr} = \mp \displaystyle\frac{i} {k \biggl(\bar{r} + \displaystyle\frac{1}{C\alpha}\biggr)^{2}}, & W_{2}(r) = \displaystyle\frac{\alpha}{\bar{r}} \displaystyle\frac{1 - 6C\alpha\bar{r}^{3}} {1 + 3C\alpha\bar{r}^{3}}. \end{array} \label{eq.3.2.2.4}$$ Substituting these expressions into (\[eq.3.2.2.2\]), one can find the total radial wave function for the reflectionless potential with the barrier. The value of the partial component of the S-matrix $S_{l=0}^{(2)}$ can be found from a boundary condition of this wave function at point $r=0$, as it was made in the previous paragraph for the inverse power reflectionless potential (\[eq.3.1.5\]). Conclusions \[sec.conclusions\] =============================== In finishing we note main conclusion and new results. - The new exactly solvable radial reflectionless potential with barrier, which in the spatial semiaxis of radial coordinate $r$ has one hole and one barrier, after which it falls down monotonously to zero with increasing of $r$, is proposed. It has shown, that at its shape such potential looks qualitatively like radial scattering potentials in two-partial description of collision between particles and nuclei or radial decay potentials in the two-partial description of decay of compound spherical nuclear systems. - The found reflectionless potential with the barrier depends on parameters $\gamma_{n}$ and $C$. One can deform the shape of this potential: by discrete values of $\gamma_{n}$ (from the sequence (\[eq.3.1.7\])) and by continuous values of $C$. The parameter $\gamma_{n}$ at its variation does not displace visibly a maximum of the barrier and a minimum of the hole along the semiaxis $r$, but it changes their absolute values. The parameter $C$ allows to displace continuously both the barrier maximum and the hole minimum. - A new approach for construction of a hierarchy of the radial reflectionless potentials with barriers is proposed. - An exact analytical form for the total radial wave function, its convergent and divergent components (as for regular and singular Coulomb functions for the known Coulomb potential) has found in the description of scattering of a particle in the field of the inverse power reflectionless potential and in the field of the reflectionless potential with the barrier (at $\beta=2\alpha$). - It has shown for the inverse power potential, that the radial flows for the convergent and divergent components of the radial wave function are constant on the whole semiaxis of $r$, have opposite directions and coincide by absolute values. This proves the reflectionless property of the inverse power potential (with a possible tunneling near the point $r=0$) on the whole semiaxis $r$. Such analysis is applicable for the found potential with the barrier also. The analysis has shown, that any selected region of the reflectionless potential with the barrier (with take into account both the barrier region, and the small vicinity near $r=0$) does not influence on the propagation of the particle. During scattering in the spherically symmetric field with such radial potential, the particle propagates through it without the smallest reflection and without any change of angle of direction of its motion (or tunneling). One can conclude, that the found radial potential with the barrier is reflectionless for the propagation of the particle with any kinetic energy. If to use it for the two-partial description of the scattering of the particle on the nucleus with the spherical shape, then one can conclude, that such nucleus shows itself as *invisible* for the incident particle. [^1]: E-mail: maidan@kinr.kiev.ua

--- abstract: 'We prove in this paper that the weighted volume – or generating function – of the set of integral transportation matrices between two integral histograms $r$ and $c$ of equal sum is a positive definite kernel of $r$ and $c$ when the set of considered weights forms a positive definite matrix. The computation of this quantity, despite being the subject of a significant research effort in algebraic statistics, remains an intractable challenge for histograms of even modest dimensions. We propose an alternative kernel which, rather than considering all matrices of the transportation polytope, only focuses on a sub-sample of its vertices known as its Northwestern corner solutions. The resulting kernel is positive definite and can be computed with a number of operations $O(R^2d)$ that grows linearly in the complexity of the dimension $d$, where $R^2$ – the total amount of sampled vertices – is a parameter that controls the complexity of the kernel.' address: 'Graduate School of Informatics, Kyoto University' author: - Marco Cuturi title: Positivity and Transportation --- Introduction ============ Suppose that among $30$ students in a classroom, $7$ and $23$ have light and dark colored eyes respectively. You are also told that $12$ of them have light hair while $18$ have dark hair. What are all the possible populations of the 4 subgroups of students with light/light, dark/dark, light/dark and dark/light eyes and hair color respectively? Such quantities can be arranged in a $2\times 2$ matrix whose row sum vector must be equal to $[7,23]^T$ and column sum vector must be equal to $[12,18]$, $\smallmat{3&4\\9&14}$ for instance, and more generally *any* integer values in the dots below that satisfy these constraints: $$\bordermatrix[{[]}]{& 12 & 18 \cr 7 & \bullet & \bullet\cr 23 & \bullet & \bullet \cr }$$ Alternatively, suppose that two bakeries in a small village produce daily $7$ and $23$ loafs of bread each, while two restaurants in the same area each need $12$ and $18$ loafs to serve their customers every day. What are all the possible morning delivery plans of bread loafs that the two bakeries and shops can agree upon? These seemingly trivial sets of matrices coincide, and are known in the statistics and optimization literature as the sets of *contingency tables* and *transportation plans* respectively. In statistics, the problem of enumerating all such tables arises naturally in hypothesis testing. Suppose that by entering the aforementioned classroom you observe that the actual repartition of these groups is $\smallmat{5&2\\7&16}$. Such an observation intuitively suggests that eye and hair color are related, but how confident should you be about this statement? In the $2\times 2$ case presented above, the Fisher exact test [@yates1934contingency] answers that question by computing the probabilities of *all* possible tables outcomes if one assumes that they have been generated as the product of independent Bernoulli variables with law $p_1=7/30$ and $p_2=12/30$. By comparing all these probabilities with that of the observed table, we can conclude how reliable an independence hypothesis would be. In optimization, given a $2\times 2$ cost matrix which describes the cost (in gas, calories or time) of bringing a loaf from each bakery to each shop, finding the delivery plan with minimal cost is known as a transportation problem. Transportation problems are an extremely general class of linear programs which are known to encompass all instances of network flows [@bertsimas1997introduction p.274]. Optimal transportation distances [@rachev1998mass; @villani09] are distances between probability densities which combine both perspectives outlined above, where the probabilistic view on contingency tables is matched with the goal of computing an optimal transportation plan between two marginal probabilities given a metric on the probability space of interest. Such distances have been widely used in computer vision following the impulsion of @rubner1997earth who used it to compare histograms of image features. When used in information retrieval tasks, transportation distances fare usually better in practice than other classical distances for histograms [@Pele-iccv2009]. Transportation distances have however two notable drawbacks. First, from a geometric point of view, transportation distances are deficient in the sense that they are not negative definite nor Hilbertian. Negative definiteness carries many favorable properties, among which the possibility to create Euclidean embeddings from which the metric can be accurately recovered, as well as the possibility to turn the distance into a positive definite kernel by simple exponentiation, as a radial basis function. Because of this deficiency, there is no known positive definite counterpart to transportation distances that can leverage the complexity of the set of contingency tables. Second, from a computational point of view, the computational cost of computing transportation distances grows in most cases of interest at least quadratically in the dimension $d$ of the histograms, which can be prohibitive for many applications. We try to address both issues in this work. The main contribution of this paper is theoretical: after providing some background material and motivation in Section \[sec:back\] we prove in Section \[sec:trans\] that the generating function of the set of all contingency tables between two integral histograms is a positive definite kernel. Our second contribution is practical: we propose in Section \[sec:nwc\] a positive definite kernel that leverages these ideas while still being computationally tractable. Background {#sec:back} ========== The Transportation Polytope and the Set of Contingency Tables ------------------------------------------------------------- We review in this section a few definitions, notations and results of interest to prove our result. In the following, we write $\dotprod{\,\cdot\,}{\cdot}$ for both the Frobenius dot-product and the usual dot-product of vectors. Given a dimension $d$ fixed throughout this paper, for two vectors $r,c\in \RR^d$, let $U(r,c)$ be the transportation polytope of $r$ and $c$, namely the subset of nonnegative matrices in $\RR^{d\times d}$ defined as: $$U(r,c)\defeq \{X\in\RR_+^{d\times d}\; |\; X\ones_d=r, X^T\ones_d=c\},$$ where $\ones_d$ is the $d$ dimensional vector of ones. $U(r,c)$ contains all nonnegative $d\times d$ matrices with row and column sums $r$ and $c$ respectively. It is easy to check that $U(r,c)$ is non-empty if and only if all coordinates of $r$ and $c$ are non-negative and if the total masses of $r$ and $c$ are the same, that is $r^T\ones_d=c^T\ones_d$. We will consider in most of this work *integral* vectors $r$ and $c$ taken in the set $\Sigma_N$ of $d$-dimensional integral histograms with total mass $N\in\NN$, $$\Sigma_d^N \defeq \{r \in\NN^{d} \;|\; r_1+\cdots+r_d = N\}.$$ We will also focus accordingly on the subset $\UU(r,c)$ of $U(r,c)$ that contains all integral transportation matrices, alternatively known as *contingency tables* [@lauritzen1982lectures; @everitt1992analysis]: $$\UU(r,c)\defeq U(r,c) \cap \NN^{d\times d}.$$ Weighted Volumes of Contingency Tables and Particular Cases of Positivity ------------------------------------------------------------------------- Ranging from early work by @yates1934contingency [@good1976] to @diaconisefron [@cryan2003polynomial; @chen2005sequential], the computation of elementary statistics about $\UU(r,c)$ has attracted considerable attention. Many of the ideas of this paper build upon recent work by @barvinok2008enumerating, most notably on his study of the generating function of $\UU(r,c)$, defined for $M\in \RR^{d\times d}$ as $$V(r,c\,;M)\defeq \sum_{X\in \UU(r,c)} e^{-\dotprod{X}{M}}.$$ The generating function can be related to the *weighted* volume [@barvinok2008enumerating p.2] of $\UU(r,c)$, defined for any nonnegative $d\times d$ matrix $K\in\RR_+^{d\times d}$ as: $$T(r,c\,;K) \defeq \sum_{X\in \UU(r,c)} \prod_{ij}^d k_{ij}^{x_{ij}}.$$ Both definitions are equivalent since if we agree that $k_{ij}=e^{-m_{ij}}$ then $T(r,c\,;K)=V(r,c\,;M)$. Because all of our results rely on $K$’s properties, we will mostly use the weighted volume formulation in this paper. Some sections in this paper, notably §\[subsec:rel\] below and §\[sec:nwc\], are better understood with the generating function formulation.  @cuturi07permanents [Prop.2] proved that the cardinal of the set $\UU(r,c)$ is a positive definite kernel of $r$ and $c$ using the Robinson-Schensted-Knuth bijection [@Knuth70] that maps each contigency table to a pair of Young tableaux with contents $r$ and $c$ and the same pattern. It is easy to see that the cardinal of $\UU(r,c)$ is equal to $T(r,c;\ones_{d\times d})$ or $V(r,c\,;\mathbf{0}_{d\times d})$.  @cuturi07permanents [Prop.1] also proved that $T(r,c\,;K)$ is a positive definite kernel of $r$ and $c$ if both are *binary* histograms and $K$ is a nonnegative $d\times d$ positive definite matrix. Since the computation of $T$ entails in that case the computation of the permanent of a Gram matrix, @cuturi07permanents called this kernel the permanent kernel. The main contribution of our paper is to prove in Theorem \[theo:genfpsd\] that the map $(r,c)\in\Sigma_d^N \mapsto T(r,c\,;K)$ is positive definite whenever $K$ is a $d\times d$ positive definite matrix. Relationships with the Optimal Transportation Distance {#subsec:rel} ------------------------------------------------------ Given a $d\times d$ cost matrix $M$, one can quantify the cost of mapping $r$ to $c$ using a transportation matrix $X$ as $\dotprod{X}{M}$. The minimum of this cost is called the optimal transportation cost, defined as: $$d_M(r,c) \defeq \min_{X\in U(r,c)} \dotprod{X}{M}.$$ A classical result of optimization in network flows [@bertsimas1997introduction Theo. 7.5] guarantees the existence of a contingency table $X^\star\in\UU(r,c)$ which achieves this minimum, as schematically represented in Figure \[fig:mainfig\]. Such an optimal table $X^\star$ can be obtained algorithmically in polynomial time [@ahuja1993network §9]. The minimal cost $d_M(r,c)$ turns out to be a distance [@villani09 §6.1] whenever the matrix $M$ is itself a metric. This distance is also known as the Wasserstein distance, Monge-Kantorovich’s, Mallow’s or Earth Mover’s [@rubner1997earth] in the computer vision literature. The transportation distance is not negative definite in the general case, as shown by counterexamples [@naor-2005] and embedding distortion results [@indyk2009]. Although some metrics $M$ can yield a negative definite distance[^1], characterizing the negative definiteness of $d_M$ remains an open question. Despite this fact, transportation distances have been used in practice to derive a *pseudo*-positive definite kernel: both @emd2004 [§4.C] or @emd2006 [§2.3] introduce the exponential of (minus) the minimum of $\dotprod{X}{M}$, $$\label{eq:km}k_M (r,c) = e^{-d_M(r,c)} = \exp\left(-\min_{X\in U(r,c)}\dotprod{X}{M}\right),$$ to form an undefinite kernel which can be used to compare histograms in practice. We prove that, although the value $\exp(- \langle X^\star, M\rangle)$ in itself is not a positive definite kernel, the sum of each term $\exp(- \langle X, M\rangle)$ over *all* possible contingency tables in $\UU(r,c)$ is positive definite when $M$ has suitable properties. The generating function $V_{rc}$ can be interpreted as the exponential of (minus) the soft-minimum of $\dotprod{X}{M}$ over all contingency tables, $$V(r,c\,;M) = \exp\left(-\,\underset{X\in \UU(r,c)}{\text{softmin}}\,\dotprod{X}{M}\right)= e^{\log\sum_{X\in \UU(r,c)} e^{-\dotprod{X}{M}}}= \sum_{X\in \UU(r,c)} e^{-\dotprod{X}{M}},$$ where the soft-minimum of a finite family of scalars $(u_i)$ is $$\,\underset{i}{\text{softmin}}\,u_i\,\defeq -\log\sum_{i} e^{-u_i}.$$ This expression relates our results in this work to previous applications of soft minimums to derive positive definite kernels from combinatorial distances for strings [@VerSaiAku04], time series [@cuturi07kernelSHORT] and trees [@shin2011mapping]. These ideas are summarized in Figure \[fig:mainfig\]. Generalized Permutations {#subsec:genperm} ------------------------ We close this section by providing some tools to prove the result. We write $S_N$ for the group of permutations over the set $\{1,\cdots,N\}$. For any vector $\alpha$ of size $N$ and permutation $\pi\in S_N$, we write $\alpha_\pi$ for the permuted vector with coordinates $\alpha_\pi=[\alpha_{\pi(1)}\,\alpha_{\pi(2)}\,\cdots\,\alpha_{\pi(N)}]$ and $\alpha_{p\cdot\cdot q}$ for the subvector $[\alpha_{p}\,\cdots\,\alpha_{q}]$ when $1\leq p \leq q \leq N$. For two vectors $\rho,\gamma$ of $\{1,\cdots,d\}^N$, the $2\times N$ array $$(\rho\,;\gamma) \defeq \begin{bmatrix} \rho_1 & \rho_2 &\cdots & \rho_N \\ \gamma_1 & \gamma_2 &\cdots & \gamma_N \\ \end{bmatrix},$$ is called a generalized permutation [@Knuth70]. To any generalized permutation $(\rho\,;\gamma)$ corresponds a $d\times d$ integral matrix $\chi(\rho\,;\gamma)$ defined as [@fulton1997young p.41]: $$\label{eq:fulton}[\chi(\rho\,;\gamma)]_{ij} \defeq \sum_{n=1}^N\ones_{\rho_t=i}\cdot\ones_{\gamma_t=j},\quad 1\leq i,j\leq d.$$ Consider the following example where $d=3,N=8$ and $$\rho=\begin{bmatrix}1\,2\,2\,2\,1\,3\,1\,3\;\end{bmatrix}, \gamma=\begin{bmatrix}1\,1\,2\,1\,3\,3\,3\,3\;\end{bmatrix}, (\rho\,;\gamma) = \begin{bmatrix}1\,2\,2\,2\,1\,3\,1\,3 \\1\,1\,2\,1\,3\,3\,3\,3\end{bmatrix}, \chi(\rho\,;\gamma) =\begin{bmatrix} 1 & 0 & 2\\ 2&1 &0 \\ 0 & 0 & 2 \end{bmatrix}.$$ If we consider now the permutation $\pi=[3\,6\,8\,5\,2\,1\,4\,7]$ we have that $$\rho=\begin{bmatrix}1\,2\,2\,2\,1\,3\,1\,3\;\end{bmatrix}, \gamma_\pi=\begin{bmatrix}2\,3\,3\,3\,1\,1\,1\,3\;\end{bmatrix}, (\rho\,;\gamma_\pi) = \begin{bmatrix}1\,2\,2\,2\,1\,3\,1\,3 \\2\,3\,3\,3\,1\,1\,1\,3\end{bmatrix}, \chi(\rho\,;\gamma_\pi) =\begin{bmatrix} 2 & 1 & 0\\ 0&0 &3 \\ 1 & 0 & 1 \end{bmatrix}.$$ Note that if $\rho$ and $\gamma$ have respectively $r_i$ and $c_i$ elements $i$ among their $N$ coefficients for all $1\leq i\leq d$, then $\chi(\rho\,;\gamma)\in \UU(r,c)$. One can see above that the corresponding histograms are $r=[3,3,2]$ and $c=[3,1,4]$ and that both $\chi(\rho\,;\gamma)$ and $\chi(\rho\,;\gamma_\pi)$ have row and column sums $r$ and $c$. The Weighted Volume as a Positive Definite Kernel {#sec:trans} ================================================= \[theo:genfpsd\] Let $K\in\RR_{+}^{d\times d}$. The map $(r,c)\mapsto T(r,c\,;K)$ is positive definite if $K$ is positive definite. The proof relies on the following observation: @barvinok2008enumerating showed that the weighted volume of $\bU(r,c)$ of two integral histograms $r$ and $c$ of total mass $N$ can be formulated as the expectation of the permanent of a random $N\times N$ matrix. To do so, @barvinok2008enumerating shows that the weighted volume – a sum indexed over all *contigency tables* $X\in\UU(r,c)$, can be rewritten as a sum indexed over all *permutations* $\pi$ in $S_N$, up to a correcting term known as the Fisher-Yates statistic (Equation  in the Appendix). The crux of @barvinok2008enumerating’s proof lies in a randomization scheme – using draws from the exponential law – to cancel out the Fisher-Yates statistic. We adopt a similar route to prove the positivity of $T$, by proving that the inverse of the Fisher-Yates statistic – defined as $\bk_2$ below – is itself positive definite to obtain the result. Suppose that $K\in\RR_+^{d\times d}$ is positive definite and consider two integral histograms $r,c$ in $\Sigma_d^N$. We represent $r$ as a $N$-dimensional vector $\rho\in\{1,\cdots,d\}^N$, $$\rho\defeq [\,\underbrace{1,\cdots,1}_{r_1 \text{ times }},\underbrace{2,\cdots,2}_{r_2 \text{ times }},\cdots,\underbrace{d,\cdots,d}_{r_d \text{ times }}\,],$$ and consider the analogous representation $\gamma$ for $c$. Let $\mathbf{k}_1$ and $\mathbf{k}_2$ be the following kernels on $(\rho,\gamma)$: $$\begin{aligned} \mathbf{k}_1(\rho,\gamma)&= \prod_{t=1}^N k(\rho_t,\gamma_t)\;, \text{ where } k(i,j) =k_{ij} \text{ for } 1\leq i,j\leq d,\\ \mathbf{k}_2(\rho,\gamma)&= \frac{1}{r_1!\cdots r_d!} \cdot \frac{1}{c_1!\cdots c_d!} \prod_{ij}^d x_{ij}!\;, \text{ where } X=\chi(\rho\,;\gamma). \quad (\text{see \S\ref{subsec:genperm}, Eq.~\eqref{eq:fulton}}) \end{aligned}$$ The kernel $\mathbf{k}_2$ is the inverse of the Fisher-Yates statistic (Equation  in the Appendix) associated to an integral transportation table $X$ and its marginals $r$ and $c$. $\mathbf{k}_1$ is trivially positive definite. The first group of terms of $\mathbf{k}_2$ is trivially positive definite as a product $f(r)f(c)$ where $f(r)=\frac{1}{r_1!\cdots r_d!}$. We prove that the other term, the product of factorials of $x_{ij}$, is positive definite in Lemma \[lem:fact\] using the proof strategy of a related result provided in Lemma \[lem:fac\]. Lemma \[lem:perm\] proves that when a kernel $\kappa$ on two vectors is symmetric (the definition is provided in the lemma), the sum $\sum_{\pi\in S_N}\kappa(\rho,\gamma_\pi)$ is itself positive definite. We use this result on the product $\kappa(\rho,\gamma)=\mathbf{k}_1(\rho,\gamma) \,\mathbf{k}_2(\rho,\gamma)$ which is trivially symmetric as the product of two symmetric kernels. We then prove in Lemma \[lem:decomp\] that $$\sum_{\pi\in S_N} \kappa(\rho,\gamma_\pi) = T(r,c\,;K).$$ Since the summation over all permutations in the left hand side is positive definite by Lemma \[lem:perm\], we conclude that $T(r,c\,;K)$ is itself a positive definite kernel as the product of two positive definite kernels. Northwestern Kernel {#sec:nwc} =================== The weighted volume $T(r,c\,;K)$ cannot be computed exactly even for small dimensions $d$, and approximations [@barvinok2008enumerating] are currently both too expensive and too loose to be of practical interest in a machine learning context. We adopt in this section an alternative approach, in which we propose to restrict the sum of elementary contributions $\exp(-\dotprod{X}{M})$ to a subset of extreme points of $U(r,c)$ and obtain a kernel whose computational complexity grows linearly in both the dimension $d$ and the size of the sample of extreme points. The main tool for this approach is provided by the Northwestern corner rule to generate a vertex of $U(r,c)$, which we recall in Section \[subsec:nwc\]. We define the Northwester kernel in Section \[subsec:sam\] and prove that it is positive definite. For any matrix $M\in\RR^{d\times d}$, we write $M_{\sigma\sigma'}$ for the row and column permuted matrix whose $i,j$ element is $m_{\sigma(i)\sigma'(j)}$. The Northwestern Corner Rule to Generate Vertices of $U(r,c)$ {#subsec:nwc} ------------------------------------------------------------- The Northwestern corner rule is a heuristic that produces a vertex of the polytope $U(r,c)$ in up to $2d$ operations. The rule starts by giving the highest possible value to $x_{11}$, and at each step when a highest possible value is given to entry $x_{ij}$ it moves on to $x_{ij+1}$ in case $x_{ij}$ filled column $j$, or $x_{i+1j}$ in case $x_{ij}$ filled row $i$. The rule proceeds until $x_{nn}$ has received a value. Here is an example of this sequence assuming $r=[2,5,3]$ and $c=[5,1,4]$: $$\begin{bmatrix} \bullet & 0 & 0 \\ 0 & 0 & 0 \\ 0& 0 & 0\end{bmatrix} \rightarrow \begin{bmatrix} 2 & 0 & 0 \\ \bullet & 0 & 0 \\ 0& 0 & 0\end{bmatrix} \rightarrow \begin{bmatrix} 2 & 0 & 0 \\ 3 & \bullet & 0 \\ 0& 0 & 0\end{bmatrix} \rightarrow \begin{bmatrix} 2 & 0 & 0 \\ 3 &1 &\bullet \\ 0& 0 & 0\end{bmatrix} \rightarrow \begin{bmatrix} 2 & 0 & 0 \\ 3 &1 &1 \\ 0& 0 & \bullet\end{bmatrix} \rightarrow \begin{bmatrix} 2 & 0 & 0 \\ 3 &1 &1 \\ 0& 0 & 3\end{bmatrix}$$ We write $\NW(r,c)$ for the unique Northwestern corner solution that can be obtained through this heuristic. There is, however, a much larger number of Northwestern corner solutions that can be obtained by permuting arbitrarily the order of $r$ and $c$ separately, computing the corresponding Northwestern corner table, and recovering a table of $\UU(r,c)$ by inverting again the order of columns and rows. Setting $\sigma=(3,1,2),\sigma'=(3,2,1)$ we have that $r_\sigma=[3,2,5], c_{\sigma'}=[4,1,5]$ and $\sigma^{-1}=(2,3,1),\sigma'=(3,2,1)$. Observe that: $$\NW(r_\sigma,c_\sigma') = \begin{bmatrix} 3 & 0 & 0 \\ 1 & 1 & 0 \\ 0& 0 & 5\end{bmatrix} \in \UU(r_\sigma,c_{\sigma'}),\;\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})= \begin{bmatrix} 0 & 1 & 1 \\ 5 & 0 & 0 \\ 0& 0 & 3\end{bmatrix}\in \UU(r,c).$$ Let $\Ncal(r,c)$ be the set of all Northwestern corner solutions that can be produced this way: $$\Ncal(r,c)\defeq\{ \NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'}), \sigma,\sigma'\in S_d\}.$$ Note that all Northwestern corner solutions only have by construction up to $2d-1$ nonzero elements. The Northwestern corner rule produces a table which is by construction unique for $r$ and $c$, but there is an exponential number of pairs or row/column permutations $(\sigma,\sigma')$ that may share the same table [@stougie2002polynomial p.2]. $\Ncal(r,c)$ is a subset of the set of extreme points of $U(r,c)$ [@brualdi2006combinatorial Corollary 8.1.4]. $\NW(r,c)$ is an optimal transportation between $r$ and $c$ if the cost matrix $M$ is a Monge matrix [@hoffman1961simple], that is a matrix $M$ that satisfies the inequalities $$\forall 1 \leq i,j,k,l\leq d, \quad m_{ij}+m_{kl}\leq m_{il}+m_{kj}.$$ Note however that a distance matrix cannot be a Monge matrix since the inequality above applied to $k=j$ and $l=i$ would imply that $0<2m_{ij}\leq m_{ii}+m_{jj}=0$. Random Sampling of Northwestern Corner Solutions {#subsec:sam} ------------------------------------------------ We propose in this section a kernel which uses arbitrary row/column permutations of $r$ and $c$ to recover extreme points of $\UU(r,c)$ and sum their individual contribution: Let $R$ be an arbitrary subset of permutations in $S_d$. The Northwestern kernel sampled on $R$ and parameterized by a matrix $M$, defined as $$N(r,c\,;K,R) \defeq \sum_{\sigma,\sigma'\in R} \exp\left(-\dotprod{M}{\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})}\right),$$ is a positive definite kernel if $K$, the element-wise exponential of $-M$, is positive definite. As in the proof of Theorem \[theo:genfpsd\], consider the representation of an integral histogram $r\in\Sigma_d^N$ as a $N$ dimensional vector $\rho$ that replicates $r_i$ times the index $i$ for all $i$ from $1$ to $d$. We also define, for any permutation $\sigma$ of $S_d$, the vector $\rho_\sigma$ as $$\rho_\sigma \defeq [\,\underbrace{\sigma(1),\cdots,\sigma(1)}_{r_{\sigma(1)} \text{ times }},\underbrace{\sigma(2),\cdots,\sigma(2)}_{r_{\sigma(2)} \text{ times }},\cdots,\underbrace{\sigma(d),\cdots,\sigma(d)}_{r_{\sigma(d)} \text{ times }}\,].$$ $\rho_\sigma$ for $\sigma\in S_d$ should not be confused with $\rho_\pi$ for $\pi\in S_N$ (§\[subsec:genperm\]): for any permutation $\sigma\in S_d$ there exists at least one permutation $\pi\in S_N$ such that $\rho_\sigma=\rho_\pi$ but the converse is not usually true. We show in Lemma \[lem:nwc\] that for $\sigma,\sigma'\in S_d$, $\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})=\chi(\rho_\sigma,\gamma_{\sigma'})$, and thus, $$N(r,c\,;K,R) = \sum_{\sigma,\sigma'\in R} e^{-\dotprod{M}{\chi(\rho_\sigma,\gamma_{\sigma'})}} = \sum_{\sigma,\sigma'\in R} \mathbf{k_1}(\rho_\sigma,\gamma_{\sigma'}),$$ where $\mathbf{k_1}$ is defined in Theorem \[theo:genfpsd\]. $N(r,c\,;K,R)$ is positive definite as a convolution kernel. \[lem:nwc\] Let $\sigma$ and $\sigma'$ be two permutations of $S_d$. Then $$\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})=\chi(\rho_\sigma,\gamma_{\sigma'}).$$ We write $E_{ij}$ for the $d\times d$ matrix of zeros except for the $(i,j)$ element set to $1$. We prove the result by induction on the total mass $N$. For $N=1$ the result is trivial since the only transportation matrix in $U(r,c)$ in that case is $E_{\sigma(i_1)\sigma(i_2)}$, where $i_1$ and $i_2$ are such that $r_{i_1}=c_{i_2}=1$. Suppose now that the result is true for all histograms of mass $N$ and consider the case where $r^T\ones_d=c^T\ones_d=N+1$. Let $i_1$ and $i_2$ be the smallest indices such that $r_{\sigma(i)}>0$ and $c_{\sigma'(i)}>0$ respectively. As a consequence, the first elements of $\rho_\sigma$ and $\gamma_{\sigma'}$ are $\sigma(i_1)$ and $\sigma(i_2)$ respectively. Consider the two vectors $\rho_*$ and $\gamma_*$ of length $N$ equal to $\rho_\sigma$ and $\gamma_{\sigma'}$ *without* these two first elements. Setting $\tilde{r}$ and $\tilde{c}$ to $r$ and $c$ except for the fact that $\tilde{r}_{\sigma(i_1)}=r_{\sigma(i_1)}-1$ and $\tilde{c}_{\sigma(i_2)}=r_{\sigma(i_2)}-1$, we have by induction that $\NW_{\sigma^{-1}\sigma'^{-1}}(\tilde{r}_\sigma,\tilde{c}_{\sigma'})=\chi(\rho_*,\gamma_*),$ since the two histograms have total mass $N$ and their representations are respectively $\rho_*$ and $\gamma_*$. By definition of the Northwestern corner rule, adding a unit of mass to the $i_1$’s and $i_2$’s components of $\tilde{r}_\sigma$ and $\tilde{c}_{\sigma'}$ only changes the very first iteration of the rule, since all coordinates of $\tilde{r}_\sigma$ and $\tilde{c}_{\sigma'}$ up to but not including $i_1$ and $i_2$ respectively are null by construction. Applying the rule yields a transportation table with an added unit in location $(i_1,i_2)$, providing thus the identity $$\NW(r_\sigma,c_{\sigma'}) = \NW(\tilde{r}_\sigma,\tilde{c}_{\sigma'}) + E_{i_1i_2},$$ which implies that $$\label{eq:nw}\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})= \NW_{\sigma^{-1}\sigma'^{-1}}(\tilde{r}_\sigma,\tilde{c}_{\sigma'}) + E_{\sigma(i_1)\sigma'(i_2)}.$$ By definition of $\chi$ we have that $$\label{eq:chi} \chi(\rho_\sigma\gamma_\sigma)= \chi(\rho_*,\gamma_*) + E_{\sigma(i_1)\sigma'(i_2)}$$ we get by combining Equations  and  above with the induction hypothesis that $\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})=\chi(\rho_\sigma,\gamma_{\sigma'})$. The evaluation of $N(r,c\,;K,R)$ requires $O(d\abs{R}^2)$ steps since computing each of the $\abs{R}^2$ contributions $\exp(-\dotprod{M}{\NW_{\sigma^{-1}\sigma'^{-1}}(r_\sigma,c_{\sigma'})})$ for a couple $\sigma,\sigma'$ requires up to $2d$ products. The size of $R\subset S_d$ can be controlled from a few permutations to an exhaustive enumeration, which would entail an overall complexity of the order of $O(dd!^2)$. Conclusion and Future Work ========================== We have proved in this paper that the fundamental ingredient of transportation distances, the polytope of contingency tables, can be used to define a positive definite kernel between two histograms. While the cost matrix of a transportation problem between two histograms $r$ and $c$ needs to be a distance matrix for the optimum to be itself a distance of $r$ and $c$, we have proved that the generating function of the same polytope is positive definite whenever the cost matrix is itself positive definite. This quantity is computationally intractable, and we have resorted to a summation that only considers a subset of extreme points of the polytope to define the north-western kernel. Future research includes the proposal of suitable subsets $R$ of permutations of $S_d$ tuned with data, as well as other approximation schemes. Appendix: Intermediate Results for the Proof of Theorem \[theo:genfpsd\] {#appendix-intermediate-results-for-the-proof-of-theoremtheogenfpsd .unnumbered} ======================================================================== \[lem:fac\]Let $a,b\in\{0,1\}^N$ be two binary vectors. The kernel $(a,b)\mapsto \dotprod{a}{b}!$ is positive definite. For $N=1$ the kernel is always equal to $1$ and is thus trivially positive definite. For $N>1$, the recursion $\dotprod{a}{b}!=\dotprod{a_1^{N-1}}{b_1^{N-1}}!\,(a_{N}b_{N}\dotprod{a_1^{N-1}}{b_1^{N-1}}+1)$ provides the expression $$\dotprod{a}{b}! = \prod_{t=1}^{N-1} \left(a_{t+1}b_{t+1}\dotprod{a_{1\cdot\cdot t}}{b_{1\cdot\cdot t}}+1\right),$$ which shows that $\dotprod{a}{b}!$ is the product of $N-1$ positive definite kernels on different features of $a$ and $b$. Rather than the lemma itself, we will use the identity above in the proof of Lemma \[lem:fact\]. We conjecture that this result can be extended to integral vectors. Numerical counterexamples show that this result cannot be generalized to vectors of $\RR^N$ through Euler’s or Hadamard’s $\Gamma$ function. \[lem:fact\]Let $\rho,\gamma\in\{1,\cdots,d\}^N$. The kernel $(\rho,\gamma)\mapsto \prod_{ij} x_{ij}!$, where $X=\chi(\rho;\gamma)$, is positive definite. An integral vector $\rho \in \{1,\cdots,d\}^N$ with $N$ components can be represented as a family of $d$ binary row vectors $\rho^1,\cdots,\rho^d$ of length $N$ where for $n\leq N$, $\rho^i_n\defeq \ones_{\rho_n=i}$. For instance, $$\text{if }\rho=\begin{bmatrix}1\,1\,2\,2\,2\,1\,3\,1\,3\,3\end{bmatrix},\text{ then } \begin{bmatrix}\rho^1\\\rho^2\\\rho^3\end{bmatrix}=\begin{bmatrix} 1&1&0&0&0&1&0&1&0&0\\ 0&0&1&1&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&1&1\\ \end{bmatrix}$$ These $d$ binary vector representations can be used to obtain the matrix $\chi(\rho\,;\gamma)$. Indeed, it is easy to check that if $X=\chi(\rho\,,\gamma)$ then $x_{ij}=\dotprod{\rho^i}{\gamma^j}$. As a consequence, we have that for all indices $i,j$ the coefficient $x_{ij}!=\dotprod{\rho^i}{\gamma^j}!$. We obtain that the product of factorials $$\prod_{ij}^d x_{ij}! = \prod_{i,j}^d\dotprod{\rho^i}{\gamma^j}!,$$ is thus a product of kernels evaluated on all possible pairs among the $d\times d$ representations for $\rho$ and $\gamma$. Although one might be tempted to interpret this product as a convolution kernel [@haussler99convolution] or a mapping kernel [@shin2008generalization], one should recall that such results only apply to *sums* of local kernels and not to *products*. Such products of kernels on parts are not, as simple counterexamples can show, positive definite in the general case. Using the decomposition which was used in the proof of Lemma \[lem:fac\], we have however that: $$\begin{aligned}\prod_{ij}^d x_{ij}! &= \prod_{i,j}^d\dotprod{\rho^i}{\gamma^j}! = \prod_{i,j}^d \prod_{t=1}^{N-1} \left(\rho^i_{t+1}\gamma^j_{t+1}\dotprod{\rho^i_{1\cdot\cdot t}}{\gamma^i_{1\cdot\cdot t}}+1\right),\\ &= \prod_{t=1}^{N-1} \prod_{i,j}^d \left(\rho^i_{t+1}\gamma^j_{t+1}\dotprod{\rho^i_{1\cdot\cdot t}}{\gamma^j_{1\cdot\cdot t}}+1\right) = \prod_{t=1}^{N-1} \left(1+\sum_{i,j}^d \rho^i_{t+1}\gamma^j_{t+1}\dotprod{\rho^i_{1\cdot\cdot t}}{\gamma^j_{1\cdot\cdot t}}\right), \end{aligned}$$ where we have used in the last operation the fact that only one of all $d^2$ products $(\rho^i_{t+1}\gamma^j_{t+1})_{ij}$ is nonzero, since $$\rho^i_{t+1}\gamma^j_{t+1}=\begin{cases} 1, \text{ if } \rho_{t+1}=i \text{ and } \gamma_{t+1}=j, \\ 0, \text {else.}\end{cases}$$ The product of factorials is thus a product of $N-1$ positive definite kernels indexed by $t$ and defined on $\rho$ and $\gamma$, where each of these $N-1$ kernel is $1$ plus a convolution kernel operating on the $d$ decompositions of $\rho_{1\cdot\cdot t}$ and $\gamma_{1\cdot\cdot t}$ as $d$ binary feature vectors, that is $$\prod_{ij}^d x_{ij}! = \prod_{t=1}^{N-1} \left(1+k_t(\rho,\gamma)\right);$$ where $$k_{t}(\rho,\gamma)=\sum_{i,j}^d h_t(\rho^i,\gamma^j) \text{ and } h_t(a,b) = a_{t+1}b_{t+1}\dotprod{a_{1\cdot\cdot t}}{b_{1\cdot\cdot t}}.$$ \[lem:perm\] Let $\alpha=(\alpha_1,\cdots,\alpha_N)$ and $\beta=(\beta_1,\cdots,\beta_N)$ be two lists of $N$ elements in a set $\Xcal$. Let $k$ be a symmetric kernel in $\Xcal^N$, that is a kernel invariant under a permutation of the order of both $\alpha$ and $\beta$: $\forall \pi\in S_N,\; k(\alpha,\beta)=k(\alpha_\pi,\beta_\pi).$ Then $(\alpha,\beta)\mapsto \sum_{\pi\in S_N} k(\alpha,\beta_{\pi})$ is positive definite. The function $g$ defined below is, by @haussler99convolution’s ([-@haussler99convolution]) convolution kernels framework, a positive definite kernel of $\alpha$ and $\beta$: $$g(\alpha,\beta)=\sum_{\pi'\in S_N} \sum_{\pi\in S_N} k(\alpha_{\pi'},\beta_{\pi}).$$ Using the symmetric property of $\kappa$, we have that $$g(\alpha,\beta)=\sum_{\pi'\in S_N} \sum_{\pi\in S_N} k(\alpha,\beta_{{\pi'}^{-1}\circ\pi}) = N!\sum_{\pi\in S_N} k(\alpha,\beta_{\pi}).$$ which proves the result. \[lem:decomp\] $\sum_{\pi\in S_N} \kappa(\rho,\gamma_\pi) = r_1!\cdots r_d! \cdot c_1!\cdots c_d! \,T(r,c\,;K)$ For any couple of vectors $\rho,\gamma$ we have that both $\mathbf{k}_1$ and $\mathbf{k}_2$ only depend on $X=\chi(\rho\;;\gamma)$. This is implicitly the case in the definition of $\mathbf{k}_2$ and one can check that $$\mathbf{k}_1(\rho,\gamma)= \prod_{t=1}^N k(\rho_t,\gamma_t) = \prod_{ij}^d k_{ij}^{x_{ij}}, \text{ where } X=\chi(\rho\;;\gamma).$$ With every permutation $\pi$ of we associate a transportation table $\chi(\rho\,;\gamma_\pi)$ which we call the pattern of $\pi$. Following [@barvinok2008enumerating §2,p.7], we know that the number of permutations $\pi$ that share the same pattern $X$ for $X\in \UU(r,c)$ only depends on $X$, $r$ and $c$ through a formula known as the Fisher-Yates statistic $n(X)$ of $X$, $$\label{eq:fy} n(X)\defeq \card\{\pi\in S_N | \,\chi(\rho\,;\gamma_\pi) = X\} = \frac{r_1!\cdots r_d! \cdot c_1!\cdots c_d!}{\prod_{ij}x_{ij}!}.$$ We thus have that $$\begin{aligned} \sum_{\pi\in S_N} \kappa(\rho,\gamma_\pi) &= \sum_{X\in \UU(r,c)} n(X)\, \mathbf{k}_1(\rho,\gamma_\pi) \mathbf{k}_2(\rho,\gamma_\pi) \\ & =\sum_{X\in \UU(r,c)} \frac{r_1!\cdots r_d! \cdot c_1!\cdots c_d!}{\prod_{ij}^d x_{ij}!} \prod_{ij}^d k_{ij}^{x_{ij}} \frac{\prod_{ij}^d x_{ij}!}{r_1!\cdots r_d! \cdot c_1!\cdots c_d!}= \,T(r,c\,;K).\end{aligned}$$ [^1]: Setting $M=\ones_{d\times d}-I_d$ yields the total variation distance between discrete probabilities, which is half the Manhattan or $l_1$ distance between $r$ and $c$. All these distances are known to be negative definite.

--- author: - Andrew Adamatzky title: 'Thirty eight things to do with live slime mould[^1]' --- Introduction {#introduction .unnumbered} ============ Acellular slime mould *P. polycephalum* has quite sophisticated life cycle [@stephenson1994myxomycetes], which includes fruit bodied, spores, single-cell amoebas, syncytium. At one phase of its cycle the slime mould becomes a plasmodium. The plasmodium is a coenocyte: nuclear divisions occur without cytokinesis. It is a single cell with thousands of nuclei. The plasmodium is a large cell. It grows up to tens centimetres when conditions are good. The plasmodium consumes microscopic particles and bacteria. During its foraging behaviour the plasmodium spans scattered sources of nutrients with a network of protoplasmic tubes. The plasmodium optimises it protoplasmic network to cover all sources of nutrients, stay away from repellents and minimise transportation of metabolites inside its body. The plasmodium’s ability to optimise its shape [@nakagaki2001path] attracted attention of biologists, then computer scientists [@adamatzky2010physarum] and engineers. Thus the field of slime mould computing was born. So far, the plasmodium is the only useful for computation stage of *P. polycephalum*’s life cycle. Therefore further we will use word ‘Physarum’ when referring to the plasmodium. Most computing and sensing devices made of the Physarum explore one or more key features of the Physarum’s physiology and behaviour: - the slime mould senses gradients of chemo attractants and repellents [@durham1976control; @ueda1976chemotaxis; @rakoczy2015application]; it responds to chemical or physical stimulation by changing patterns of electrical potential oscillations [@ridgway1976oscillations; @kishimoto1958rhythmicity] and protoplasmic tubes contractions [@wohlfarth1979oscillatory; @teplov1991continuum]; - it optimises its body to maximise its protoplasm streaming [@dietrich2015explaining]; and, - it is made of hundreds, if not thousands, of biochemical oscillators [@kauffman1975mitotic] with varied modes of coupling [@grebecki1978plasmodium]. Here we offer very short descriptions of actual working prototypes of Physarum based sensors, computers, actuators and controllers. Details can be found in pioneer book on Physarum machines [@adamatzky2010physarum] and the ‘bible’ of slime mould computing [@adamatzkyAdvancesPhysarum]. Optimisation and graphs ======================= Shortest path and maze {#path} ---------------------- Given a maze we want to find a shortest path between the central chamber and an exit. This was the first ever problem solved by Physarum. There are two Physarum processors which solve the maze. First prototype [@nakagaki2001path] works as follows. The slime mould is inoculated everywhere in a maze. The Physarym develops a network of protoplasmic tubes spanning all channels of the maze. This network represents all possible solutions. Then oat flakes are placed in a source and a destination site. Tube lying along the shortest (or near shortest) path between two sources of nutrients develop increased flow of cytoplasm. This tube becomes thicker. Tubes branching to sites without nutrients become smaller due to lack of cytoplasm flow. They eventually collapse. The sickest tube represents the shortest path between the sources of nutrients. The selection of the shortest protoplasmic tube is implemented via interaction of propagating bio-chemical, electric potential and contractile waves in the plasmodium’s body, see mathematical model in [@tero2006physarum]. The approach is not efficient because we must literally distribute the computing substrates everywhere in the physical representation of the problem. A number of computing elements would be proportional to a sum of lengths of the maze’s channels. Second prototype of the Physarum maze solver is based on Physarum’ chemo-attraction [@adamatzky2012slimemaze]. An oat flake is placed in the central chamber. The Physarum is inoculated somewhere in in a peripheral channel. The oat flake releases chemoattractants. The chemoattractants diffuse along the maze’s channels. The Physarum explores its vicinity by branching out protoplasmic tubes into opening of nearby channels. When a wave-front of diffusing attractants reaches Physarum, the Physarum halts lateral exploration. Instead it develops an active growing zone propagating along gradient of the attractants’ diffusion. The sickest tube represents the shortest path between the sources of nutrients. The approach is efficient because a number of computing elements would be proportional to a length of the shortest path. Towers of Hanoi --------------- Given $n$ discs, each of unique size, and three pegs, we want to move the entire stack to another peg by moving one top disk at a time and not placing a disk on top of smaller disc. The set of all possible configurations and moves of the puzzle forms a planar graph with $3n$ vertices. To solve the puzzle one must find shortest paths between configurations of pegs with discs on the graph [@hinz1989tower; @hinz1992shortest; @romik2006shortest]. Physarum solves shortest path (Sect. \[path\]) therefore it can solve Tower of Hanoi puzzle. This is experimentally demonstrated in [@reid2013solving]. Sometimes Physarum does not construct an optimal path initially. However, if its protoplasmic networks are damaged and then allowed to regrow the path closer to optimal then before develops [@reid2013solving]. Travelling salesman problem --------------------------- Given a graph with weighted edges, find a cyclic route on the graph, with a minimum sum of edge weights, spanning all nodes, where each node is visited just once. Commonly, weight of an edge is an Euclidean length of the edge. Physarum is used as component of an experimental device approximating the shortest cyclic route [@zhu2013amoeba]. A map of eight cities is considered. A set of solutions is represented by channels arranged in a star graph. The channels merge in a central chamber. There are eight channels for each city. Each channel encodes a city and the step when the city appears in the route. There are sixty four channels. Physarum is inoculated in the central chamber. The slime mould then propagates into the channels. A node of the data graph is assumed to be visited when its corresponding channel is colonised by Physarum. Growth of the Physarum in the star-shape is controlled by a recurrent neural network. The network takes ratios of colonizations of the channels as input and produces patterns of illumination, projected onto the channels, as output. The network is designed to prohibit revisiting of already visited nodes and simultaneous visits to multiple nodes. The Physarum propagates into channels and pulls back. Then propagates to other channels and pulls back from some of them. Eventually the system reaches a stable solution where no propagation occurs. The stable solution represents the minimal distance cyclic route on the data graph [@zhu2013amoeba]. A rather more natural approximate algorithm of solving the travelling salesman problem is based on aconstruction of an $\alpha$-shape (Sect. \[concavehull\]). This how humans solve the problem visually [@macgregor1996human]. An approximate solution of the travelling salesman problem by a shrinking blob of simulated Physarum is proposed in [@jones2014computation]. The Physarum is inoculated all over the convex hull of the data set. The Physarum’s blob shrinks. It adapts morphologically to the configuration of data nodes. The shrinkage halts when the Physarum no longer covers all data nodes. The algorithm is not implemented with real Physarum. Spanning tree ------------- A spanning tree of a finite planar set is a connected, undirected, acyclic planar graph, whose vertices are points of the planar set. The tree is a minimal spanning tree where sum of edge lengths is minimal [@nevsetvril2001otakar]. As algorithm for computing a spanning tree of a finite planar set based on morphogenesis of a neuron’s axonal tree was initially proposed [@adamatzky1991neural]: planar data points are marked by attractants (e.g. neurotrophins) and a neuroblast is placed at some site. Growth cones sprout new filopodia in the direction of maximal concentration of attractants. If two growth cones compete for the same site of attractants then a cone with highest energy (closest to previous site or branching point) wins. Fifteen years later we implemented the algorithm with Physarum [@adamatzky2008growing]. Degree of Physarum branching is inversely proportional to a quality of its substrate. Therefore to reduce a number of random branches we cultivate Physarum not on agar but just humid filter paper. Planar data set is represented by a configuration of oat flakes. Physarum is inoculated at one of the data sites. Physarum propagates to a virgin oat flake closest to the inoculation site. Physarum branches, if there are several virgin flakes nearby. It colonises next set of flakes. The propagation goes on until all data sites are spanned by a protoplasmic network. The protoplasmic network approximates the spanning tree. The resulted tree does not remain static though. Later cycles can be formed and the tree is transformed to one of proximity graphs, e.g. relative neighbourhood graph or Gabriel graph [@adamatzky2009developing]. Approximation of transport networks {#roadnetworks} ----------------------------------- Motorway networks are designed with an aim of efficient vehicular transportation of goods and passengers. Physarum protoplasmic networks evolved for efficient intra-cellular transportation of nutrients and metabolites. To uncover similarities between biological and human-made transport networks and to project behavioural traits of biological networks onto development of vehicular transport networks we conducted an evaluation and approximation of motorway networks by Physarum in fourteen geographical regions: Africa, Australia, Belgium, Brazil, Canada, China, Germany, Iberia, Italy, Malaysia, Mexico, The Netherlands, UK, and USA [@adamatzky2012bioevaluation]. We represented each region with an agar plate, imitated major urban areas with oat flakes, inoculated Physarum in a capital, and analysed structures of protoplasmic networks developed. We found that, the networks of protoplasmic tubes grown by Physarum match, at least partly, the networks of human-made transport arteries. The shape of a country and the exact spatial distribution of urban areas, represented by sources of nutrients, play a key role in determining the exact structure of the plasmodium network. In terms of absolute matching between Physarum networks and motorway networks the regions studied can be arranged in the following order of decreasing matching: Malaysia, Italy, Canada, Belgium, China, Africa, the Netherlands, Germany, UK, Australia, Iberia, Mexico, Brazil, USA. We compared the Physarum and the motorway graphs using such measures as average and longest shortest paths, average degrees, number of independent cycles, the Harary index, the $\Pi$-index and the Randić index. Using these measures we find that motorway networks in Belgium, Canada and China are most affine to Physarum networks. With regards to measures and topological indices we demonstrated that the Randić index could be considered as most bio-compatible measure of transport networks, because it matches very well the slime mould and man-made transport networks, yet efficiently discriminates between transport networks of different regions [@adamatzky2013motorways]. Many curious discoveries have been made. Just few of them are listed below. All segments of trans-African highways not represented by Physarum have components of non-paved roads [@adamatzky2013biological]. The east coast transport chain from the Melbourne urban area in the south to the Mackay area in the north, and the highways linking Alice Springs and Mount Isa and Cloncurry, are represented by the slime mould’s protoplasmic tubes in almost all experiments on approximation of Australian highways [@adamatzky2012slimeAustralia]. If the two parts of Belgium were separated with Brussels in Flanders, the Walloon region of the Belgian transport network would be represented by a single chain from Tournai in the north-west to the Liege area in the north-east and down to southernmost Arlon; motorway links connecting Brussels with Antwerp, Tournai, Mons, Charleroi and Namur, and links connecting Leuven with Liege and Antwerp with Genk and Turnhout, are redundant from the Physarum’s point of view [@adamatzky2012slimeBelgium]. The protoplasmic network forms a subnetwork of the man-made motorway network in the Netherlands; a flooding of large area around Amsterdam will lead to substantial increase in traffic at the boundary between flooded and non-flooded area, paralysis and abandonment of the transport network and migration of population from tshe Netherlands to Germany, France and Belgium [@adamatzky2013bioNetherlands]. Physarum imitates the 1947 year separation of Germany into East Germany and West Germany [@adamatzky2012schlauschleimer]. Mass migration {#migration} -------------- People migrate towards sources of safe life and higher income. Physarum migrates into environmentally conformable areas and towards source of nutrients. In [@adamatzky2013bio] we explored this analogy to imitate Mexican migration to USA. We have made a 3D Nylon terrain of USA and placed oat flakes, to act as sources of attractants and nutrients, to ten areas with highest concentration of migrants: New York, Jacksonville, Chicago, Dallas, Houston, Denver, Albuquerque, Phoenix, Los Angeles, San Jose. We inoculated Physarum in a locus between Ciudad Juárez and Nuevo Laredo, allowed it to colonise the template for five to ten days, and analysed routes of migration. From results of laboratory experiments we extracted topologies of migratory routes, and highlighted a role of elevations in shaping the human movement networks. Experimental archeology ----------------------- Experimental archeology uses analytical methods, imaginative experiments, or transformation of a matter [@ascher1961experimental; @ingersoll1977experimental; @coles1979experimental] in a context of human activity in the past. Knowing that Physarum is fruitful substrate for simulating transport routes (Sect. \[roadnetworks\]) and migration (Sect. \[migration\]) we explored foraging behaviour of the Physarum to imitate development of Roman roads in the Balkans [@evangelidis2015slime]. Agar plates were cut in a shape of Balkans area. We placed oat flakes in seventeen areas, corresponding to Roman provinces and major settlements and inoculated Physarum in Thessaloniki. We found that Physarum imitates growth of Roman roads to a larger extent. For example, the propagation of Physarum from Thessaloniki towards the area of Scopje and Stoboi matches the road aligned with the Valley of Axios, which is a key communication artery between the Balkan hinterland and Aegean area from Bronze age. A range of historical scenarios was uncovered [@evangelidis2015slime], including movement along via Diagonalis, the long diagonal axis that crossed central Balkan; propagation to the East towards Byzantium, towards the North along the coast of Euxeinus Pontus, and from Thessaloniki to Dyrrachion, along the western part of via Egnatia [@evangelidis2015slime]. Evacuation ---------- Evacuation is a rapid but temporary removal of people from the area of danger. Physarum moves away from sources of repellents or areas of uncomfortable environmental conditions. Would Physarum be able to find a shortest route of evacuation in geometrically constrained environment? To find out we undertook a series of experiments [@kalogeiton2015cellular]. We made a physical, scaled down, model of a whole floor of the real office building and inoculated Physarum in one of the rooms. In first scenario we placed a crystal of a salt in the room with Physarum in a hope that a gradient of sodium chloride diffusion would repel Physarum towards exist of the building along the shortest path. This did not happen. Physarum got lost in the template. By placing attractants near the exit we rectified the mishap and allowed Physarum to find a shortest route away from the ‘disaster’ [@kalogeiton2015cellular]. Evacuation is one of few problems where computer models of Physarum find better solution than the living Physarum [@kalogeiton2015biomimicry]. Space exploration ----------------- We employed the foraging behaviour of the Physarum to explore scenarios of future colonisation of the Moon and the Mars [@adamatzky2014slime]. We grown Physarum on three-dimensional templates of these planet and analysed formation of the exploration routes, dynamical reconfiguration of the transportation networks as a response to addition of hubs. The developed infrastructures were explored using proximity graphs and Physarum inspired algorithms of supply chain designs. Interesting insights about how various lunar missions will develop and how interactions between hubs and landing sites can be established are given in  [@adamatzky2014slime]. Geometry ======== Voronoi diagram {#voronoi} --------------- Let $\mathbf P$ be a non-empty finite set of planar points. A planar Voronoi diagram of the set $\mathbf P$ is a partition of the plane into such regions that, for any element of $\mathbf P$, a region corresponding to a unique point $p$ contains all those points of the plane which are closer to $p$ than to any other node of $\mathbf P$. A unique region $vor(p) = \{z \in {\mathbf R}^2: d(p,z) < d(p,m)\, \forall m \in {\mathbf R}^2, \, m \ne z \}$ assigned to the point $p$ is called a Voronoi cell of the point $p$. The boundary of the Voronoi cell of the point $p$ is built of segments of bisectors separating pairs of geographically closest points of the given planar set $\mathbf P$. A union of all boundaries of the Voronoi cells determines the planar Voronoi diagram: $VD({\mathbf P}) = \cup _{p \in {\mathbf P}} \partial vor(p)$ [@preparata1985computational]. The basic concept of constructing Voronoi diagrams with reaction–diffusion systems is based on an intuitive technique for detecting the bisector points separating two given points of the set $\mathbf P$. If we drop reagents at the two data points the diffusive waves, or phase waves if the computing substrate is active, travel outwards from the drops. The waves travel the same distance from the sites of origin before they meet one another. The points where the waves meet are the bisector points [@adamatzky2005reaction]. Plasmodium growing on a nutrient substrate from a single site of inoculation expands circularly as a typical diffusive or excitation wave. When two plasmodium waves encounter each other, they stop propagating. To approximate a Voronoi diagram with Physarum [@adamatzky2010physarum], we physically map a configuration of planar data points by inoculating plasmodia on a substrate. Plasmodium waves propagate circularly from each data point and stop when they collide with each other. Thus, the plasmodium waves approximate a Voronoi diagram, whose edges are the substrate’s loci not occupied by plasmodia. Time complexity of the Physarum computation is proportional to a maximal distance between two geographically neighbouring data points, which is capped by a diameter of the data planar set, and does not depend on a number of the data points. Delaunay triangulation ---------------------- Delaunay triangulation is a dual graph of Voronoi diagram. A Delaunay triangulation of a planar set is a triangulation of the set such that a circumcircle of any triangle does not contain a point of the set [@delaunay1934sphere]. There are two ways to approximate the Delaunay triangulation with Physarum. First is based on the setup of Voronoi diagram processor (Sect. \[voronoi\]). Previously, we wrote that when propagating fronts of Physarum meet they stop. This is true. But what happens after they stop is equally interesting. Physarum forms a bridge — a single protoplasmic tube —- connecting the stationary Physarum fronts. Such protoplasmic tubes typically span geographically neighbouring sites of inoculation and they cross sites of first contacts of growing wave fronts. These tubes represent edges of the Delaunay triangulation. This is why we proposed in [@shirakawa2009simultaneous] that the Voronoi diagram and the Delaunay triangulation are constructed simultaneously by Physarum growing on a nutrient agar. Second method of approximating the Delaunay triangulation is implemented on a non-nutrient agar. We represent planar data points by inoculation sites. The inoculants propagate and form a planar proximity graph spanning all inoculation sites. At the beginning of such development a Gabriel graph [@gabriel1969new] is formed. Then additional protoplasmic tubes emerge and the Gabriel graph is transformed to the Delaunay triangulation [@adamatzky2009developing]. Concave hull {#concavehull} ------------ The $\alpha$-shape of a planar set $\mathbf P$ is an intersection of the complement of all closed discs of radius $1/\alpha$ that includes no points of $\mathbf P$ [@edelsbrunner1983shape]. A concave hull is a connected $\alpha$-shape without holes. This is a non-convex polygon representing area occupied by $\mathbf P$. Given planar set $\mathbf P$ represented by physical objects Physarum must approximate concave hull of $\mathbf P$ by its thickest protoplasmic tube. We represent data points by somniferous pills placed directly on a non-nutrient agar. The pills emits attractants to ‘pull’ Physarum towards $\mathbf P$ but they also emit repellents preventing Physarum from growing inside $\mathbf P$ [@adamatzky2012slime]. The combination of long-distance attracting forces and short-distance (‘short’ is $O(D)$, where $D$ is a diameter of $\mathbf P$) repelling forces allows us to implement Jarvis wrapping algorithm [@jarvis1973identification]. We select a starting point which is extremal point of $\mathbf P$. We pull a rope to other extremal point. We continue until the set $\mathbf P$ is wrapped completely. We tested feasibility of the idea in laboratory experiments [@adamatzky2012slime]. In each experiment we arranged several half-pills in a random fashion near centre of a Petri dish and inoculated an oat flake colonised by Physarum few centimetres away from the set $\mathbf P$. Physarum propagates towards set $\mathbf P$ and starts enveloping the set with its body and the network of protoplasmic tubes. The plasmodium does not propagate inside configuration of pills. The plasmodium completes approximation of a shape by entirely enveloping $\mathbf P$ in a day or two. Computing circuits ================== Attraction-based logical gates ------------------------------ When two growing zones of separate Physarum cells meet they repel if there is a free space to deviate to. If there is no opportunity to deviate the cells merge. This feature is employed in the construction of Boolean logical gates — [not]{}, [or]{} and [and]{} —- in [@tsuda2004robust]. The gates are made of segments of agar gel along which the Physarum propagates. To implement input ‘1’ ([True]{}) in channel $x$ a piece of Physarum is inoculated in $x$ otherwise the input is considered to be ‘0’ ([False]{}). Attractants are placed in the end of the output channels to stimulate growth of the Physarum towards outputs. The Physarum propagates towards closest source of attractants along a shortest path. The gate [or]{} is a $ \begin{smallmatrix} \searrow & & \swarrow \\ & \downarrow & \end{smallmatrix} $ junction. Physarum placed in one of the inputs propagates towards the output. If each input contains the Physarum, the propagating cells merge and appear in the output as if they were a single cell. Even if both inputs are ‘1’ the Physarum cells have no space to avoid collision and therefore the merge and propagate into the output channel. The gate [and]{} looks like distorted ‘H’: $ \begin{smallmatrix} & & \, & \downarrow & & \downarrow\\ \hline \downarrow & & \, & & \downarrow& \\ \end{smallmatrix} $ When only one input is ‘1’ the Physarum propagates towards closes attractant and exits along right output channel. When both inputs are ‘1’ the Physarum from the right input channel propagates into the right output channel. The Physarum from the left input channel avoids merging with another Physarum and propagates towards left output channel. The left output channel realises [and]{}. Ballistic logical gates {#ballistic} ----------------------- In designs of ballistic gates [@adamatzky2010slimeballistic] we employ inertia of the Physarum growing zones. On a non-nutrient substrate the plasmodium propagates as a traveling localisation, as a compact wave-fragment of protoplasm. The plasmodium-localisation travels in its originally predetermined direction for a substantial period of time even when no gradient of chemo-attractants is present. We explore this feature of Physarum localisations to design a two-input two-output Boolean gates. The gate realising [and]{} on one output and [or]{} on another output look like horizontally flipped ‘K’: $ \begin{smallmatrix} \searrow & \downarrow \\ \swarrow & \downarrow \end{smallmatrix} . $ When left input is ‘1’ the Physarum propagates inertially along the vertical (on the right) output channel. The same happens when right input is ‘1’. If both inputs are ‘1’ then the Physarum from the right input propagates along vertical output channel but the Physarum from the left input repels from the right-input-Physarum and moves into the left output channel. The left output channel realises [and]{} and the right output channel realises [or]{}. The gate [not]{} is an asymmetric cross junction: $ \begin{smallmatrix} & | & \\ & \downarrow & \\ \rightarrow & & \rightarrow \\ & \downarrow \end{smallmatrix} . $ Vertical input channel is twice as long as horizontal input channel. Vertical input is constant [True]{}: Physarum is always inoculated their. Horizontal input is a variable. If variable input is ‘0’ then Physarum from the constant [True]{} vertical input propagates into the vertical output. If variable input is ‘1’ then Physarum from the input channel propagates into the horizontal output channel and blocks path of the Physarum representing constant [True]{}. Both ballistic gates work very well without attractants. However they work ever better when attractants are placed into output channels. Cascading of the gates into a binary adder is demonstrated in [@adamatzky2010slimeballistic]. Opto-electronics logical gates ------------------------------ In prototypes of repellent gates [@mayne2015slimegates], active growing zones of slime mould representing different inputs interact with each other by electronically switching light inputs and thus invoking photo avoidance. The gates [not]{} and [band]{} are constructed using this feature. The gate [not]{} is made of two electrodes. The electrodes are connected to a power supply. There is a green LED (with it is independent power supply) on one electrode. The Physarum is inoculated on another electrode. Input ‘1’ is represented by LED’s light on. Output is represented by the Physarum closing the circuit between two electrodes. When LED is illuminated (input ‘1’) the Physarum does not propagate between electrodes, thus output ’0’ is produced. When LED is off (input ‘0’) the Physarum closes the circuit by propagating between the electrodes. The gate [nand]{} is implemented with two LEDs. When both inputs are ‘0’ LEDs are off and the Physarum closes the circuit between its inoculation electrode and one of the LED electrodes, chosen at random. When both inputs are ‘1’, both LEDs are on, they repel Physarum. The Physarum does not propagate to any electrodes. When only one input is ‘1’, one LED is on and another LED is off, the Physarum propagates toward the electrode with non-illuminating LED and closes the circuit. Frequency based logical gates ----------------------------- The Physarum responds to stimulation with light, nutrients and heating by changing frequency of its electrical potential oscillations. We represent [True]{} and [False]{} values by different types of stimuli and apply threshold operations to frequencies of the Physarum oscillations. We represent Boolean inputs as intervals of oscillation frequency [@whiting2014slimefrequency]. Thus we experimentally implement [or]{}, [and]{}, [not]{}, [nor]{}, [nand]{}, [xor]{} and [xnor]{} gates, see details in [@whiting2014slimefrequency]. Micro-fluidic logical gates --------------------------- When a fragment of protoplasmic tube is mechanically stimulated, e..g. gently touched by a hair, a cytoplasmic flow in this fragment halts and the fragment’s resistivity to the flow dramatically increases. The cytoplasmic flow is then directed via adjacent protoplasmic tubes. A basic gate looks like a ‘Y’-junction of two protoplasmic tubes $x$ and $y$ with a horizontal bypass $z$ between them $$\includegraphics[scale=0.3]{xor_gate.pdf}.$$ Segments $x$ and $y$ are inputs. Segment $z$ is output. When both input segments are intact and there is a flow of cytoplasm between them there is no flow of cytoplasm via $z$. If one of the input tubes is mechanically stimulated the flow through this tube stops and the flow is diverted via the output tube $z$. If both input tubes are stimulated there is no flow via input or output tubes. Thus we implement [xor]{} gate [@adamatzky2014slimefluidic]. A mechanically stimulated fragment restores its flow of cytoplasm in one minute: the gate is reusable. By adding one more output (bypass) tube to [xor]{} gate we produce a gate with two inputs and two outputs: $z$ and $p$: $$\includegraphics[scale=0.3]{and_xor_gate.pdf}.$$ The output $z$ represents [xor]{}. The tube $p$ represents [nor]{} because a cytoplasmic flow is directed via $p$ if both tubes $x$ and $y$ are blocked. More complicated gates and memory devices can be found in [@adamatzky2014slimefluidic]. Intra-cellular collision based computing ---------------------------------------- The paradigm of a collision-based computing originates from the computational universality of the Game of Life, Fredkin-Toffoli conservative logic and the billiard-ball model with its cellular-automaton implementation [@adamatzky2002CBC]. A collision-based computer employs mobile localisations, e.g. gliders in Conway’s Game of Life cellular automata, to represent quanta of information in active non-linear media. Information values, e.g. truth values of logical variables, are given by either the absence or presence of the localizations or by other parameters such as direction or velocity. The localizations travel in space and collide with each other. The results of the collisions are interpreted as computation. Physarum ballistic gates (Sect. \[ballistic\]) are also based on collisions, or interactions, between active growing zones of Physarum. However, signals [*p*er se]{}, represented by growing body of Physarum, are not localised. Intra-cellular vesicles — up 100 nm droplets of liquid encapsulated by a lipid bilayer membrane — could be convenient representations of localised signals. In [@mayne2015computing] we outlined pathways towards collision-based computing with vesicles inside the Physarum cell. The vesicles travel along actin and tubulin network and collide with each other. The colliding vesicles may reflect, fuse or annihilate. The vesicles reflect in over half of the collisions observed. The vesicles fuse in one of seven collisions. The vesicles annihilate and unload their cargo in one of ten collisions. The vesicles becomes paired and travel as a single object in one of ten collisions. Based on the experimental observations, we derive soft spheres collision [@margolus2002universal] gates and also gates [not]{} and [fan-out]{}. Kolmogorov-Uspensky machine --------------------------- In 1950s Kolmogorov outlined a concept of an algorithmic process, an abstract machine, defined on a dynamically changing graph [@kolmogorov1953concept]. The structure later became known as Kolmogorov-Uspensky machine [@kolmogorov1958definition]. The machine is a computational process on a finite undirected connected graph with distinctly labelled nodes [@gurevich1988kolmogorov]. A computational process travels on the graph, activates nodes and removes or adds edges. A program for the machine specifies how to replace the neighbourhood of an active node with a new neighbourhood, depending on the labels of edges connected to the active node and the labels of the nodes in proximity to the active node [@blass2003algorithms]. The Kolmogorov-Uspensky machine is more flexible than a Turning machine because it recognises in real time some predicates not recognisable in real time by the Turing machine [@grigor1980kolmogoroff], it is stronger than any model of computation that requires $\Omega(n)$ time to access its memory [@cloteaux2006some; @shvachko1991different]. “Turing machines formalize computation as it is performed by a human. Kolmogorov-Uspensky machines formalize computation as it performed by a physical process.” [@blass2003algorithms]. We implement the Kolmogorov-Uspensky machine in the Physarum as follows [@adamatzky2007physarum]. Stationary nodes are represented by sources of nutrients. Dynamic nodes are assigned to branching site of the protoplasmic tubes. The stationary nodes are labelled by food colourings because Physarum exhibits an hierarchy of preferences to different colourings. An active zone in the storage graph is selected by inoculating the Physarum on one of the stationary nodes. An edge of the Kolmogorov-Uspensky machine is a protoplasmic tube connecting the nodes. Program and data are represented by the spatial configuration of stationary nodes. Results of the computation over a stationary data node are represented by the configuration of dynamic nodes and edges. The initial state of a Physarum machine (Physarum implementation of Kolmogorov-Uspensky machine) includes part of an input string (the part which represents the position of Physarum relative to stationary nodes), an empty output string, the current instruction in the program and the storage structure consisting of one isolated node. The physical graph structure developed by Physarum is the result of its computation. The Physarum machine halts when all data nodes are utilised. At every step of the computation there is an active node and an active zone (nodes neighbouring the active node). The active zone has a limited complexity: all elements of the zone are connected by some chain of edges to the initial node. The size of the active zone may vary depending on the computational task. In the Physarum machine, an active node is a trigger of contraction/excitation waves, which spread all over the plasmodium tree and cause pseudopodia to propagate, the shape to change and protoplasmic veins to annihilate. The active zone comprises stationary and/or dynamic nodes connected to an active node with tubes of protoplasm. Instructions of the Physarum machine are [input]{}, [output]{}, [go]{}, [halt]{}, [add node]{}, [remove node]{}, [add edge]{}, [remove edge]{}, [if]{}. The [input]{} is done via distribution of sources of nutrients. The [output]{} is recorded optically. The [set]{} instruction causes pointers to redirect. It is realised by placing a fresh nutrient source in the experimental container. When a new node is created, all pointers from the old node point to the new node [@adamatzky2007physarum]. Electronics =========== Wires ----- Protoplasmic tubes of the Physarum are conductive and therefore can be used as wires. A resistivity of Physarum protoplasmic tubes is of the same rank as resistivity of a cardiac and skeletal muscles of dogs and humans [@geddes1967specific]. By using 1-5 cm protoplasmic tube as a wire we can light up a LED, and keep it illuminated for days. We can power up a piezo audio transducer [@adamatzky2013physarumwire] using the Physarum wire. Due to high, comparing to conventional conductors, resistivity of the Physarum we must apply a high voltage to power loads. For example, to operate the LED array we should apply 10 V and 3.9 $\mu$A direct current. To produce a 30dB sound with buzzer we need to apply 8 V. The Physarum wires are self-growing. To connect two pins of a circuit with a wire we inoculate Physarum at one pin and place a source of attractants near another pin. The Physarum grows a protoplasmic tube connecting two pins. The Physarum propagates well, and relatively fast, 1-5 mm/h, on a bare surface of electronic boards. Growing Physarum circuits can be controlled by white and blue light, chemical and thermal gradients, and electrical fields. Physarum wires can be robustly routed with a wide range of organic volatiles [@de2014routing]. Physarum wires can self-repair after a substantial damage. After part of a protoplasmic tube is removed (1-2 mm segment) the tube restores its integrity in six to nine hours. Typically, a cytoplasm from cut-open ends spills out on a substrate. Each spilling of cytoplasm becomes covered by a cell wall and starts growing. In few hours growing parts of the tube meet with each other and merge. Restoration of tubes conductivity was confirmed by electrical measurements [@adamatzky2013physarumwire]. Physarum wire can transfer analogue signals below 19 KHz without distortion [@whiting2015transfer]. Physarum wires perform well in digital communication systems. For example, the protoplasmic tubes are used to establish communication between Arduino Mega and Digital 3-axis compass using I2C protocol. Valid magnometric data was confirmed by movement/rotation of the manometer and subsequent change in data receive alone the Physarum wire [@whiting2015transfer]. Physarum wires are not immortal. To make them last longer we cover them with conducting organic polymer polypyrrole [@de2015conducting]. A localised section of protoplasmic tube is treated with ferric chloride and exposed to vapour of pyrrole monomer. A 1 cm section of the treated tube has resistance 100 kOhm. The treatment is selective therefore we can produce live and functionalised Physarum wires in the same computing circuit. Low pass filter --------------- Physarum protoplasmic tubes are conductors. Do they modify analog or digital signals passing through them? Signal propagation in Physarum’s protoplasmic tubes was tested using a frequency response network analyser [@whiting2015transfer]. Sinusoidal voltage waveforms are sent via the protoplasmic tubes at frequencies from 10Hz to 4MHz. The transfer function of voltage waveform passing through the tube is frequency dependent. Signals of higher frequencies are dramatically reduced in strength while low frequency waveforms remain largely unaffected. The magnitude-frequency profile matches a low pass filter. In most cases there was some attenuation of the voltage through the wire at the pass-band frequency range with a mean attenuation of -6dB. The cut off frequency was defined as -3dB from the pass-band magnitude; the mean cut off frequency was 19KHz. The phase-frequency response showed the 45 degree phase shift to be aligned very closely to the cut off frequency in all protoplasmic tubes [@whiting2015transfer]. Oscillators ----------- An electronic oscillator is a device that produces periodic electronic signal. Electronic Physarum chips need oscillators to have a source of regularly spaced pulses. The experimental electronic oscillator, which converts direct current to alternating current signal, is made with Physarum as follows [@adamatzky2014slimeoscillator]. We made two electrodes setup — Physarum spans two electrodes with a single protoplasmic tube, apply direct current potential in a range 2 V to 15 V and measure electrical potential difference between the electrodes. When we apply an electrical potential to a protoplasmic tube we observe oscillations of the output electrical potential. The oscillations of the output potential are caused by periodic changes in resistance of the protoplasmic tube connecting the electrodes. Average resistance of a 10 mm protoplasmic tube is 3 MOhm. Resistance of the protoplasmic tube exhibits oscillatory behaviour with highly pronounced dominating frequency 0.014 Hz. The resistance oscillations have average amplitude 0.59 MOhm, minimum amplitude of resistance oscillations observed was 0.11 MOhm and maximum amplitude 1 MOhm. Oscillations in resistance observed are due to peristaltic contractions of the protoplasmic tube [@sun2009single]. Average output potential and average amplitude of output potential oscillations grow linearly with the increase of the input potential. Frequency of oscillations remains almost constant. Physarum oscillator produces the same frequency oscillations at 2 V and 15 V applied potential. A ratio of average amplitude of output potential oscillations to average output potential decreases by a power low with increase of input potential. Tactile sensor {#tactile} -------------- A tactile sensor is a device that responds to a physical contact between the device and an object. When a segment of a glass capillary is placed across protoplasmic tube, which spans reference and recording electrodes, Physarum demonstrates two types of responses to application of this load: an immediate response with a high-amplitude impulse and a prolonged response with changes in its oscillation pattern [@adamatzky2013slime]. The immediate response is a high-amplitude spike: its amplitude is 12.33 mV and its duration is 150 sec. The prolonged response is an envelop of increased amplitude oscillations. For example, an average amplitude of oscillations before stimulation is 2.3 mV and duration of each wave was 120 sec. The amplitude of waves in the prolonged response to stimulation is 5.29 mV with a period of a wave increased to 124 sec. Tactile sensor developed in  [@adamatzky2013slime] is non-reusable. While the load rests on the protoplasmic tube Physarum starts colonising the load. Removal of the load damages the protoplasmic tube. To rectify this deficiency we designed Physarum tactile bristle [@adamatzky2014tactile]. To make a tactile bristle with Physarum we stuck a bristle in the agar blob on the recording electrode [@adamatzky2014tactile]. In a couple of days after inoculation of Physarum to an agar blob on a references electrode the Physarum propagates to and colonises agar blob on a recording electrode. Physarum climbs up the bristle and occupies one third to a half of the bristle’s length. The sensor works by deflecting the bristle. A sensed object does not come into direct contact with Physarum but only with a tip of the bristle not-colonised by Physarum. A typical response of Physarum to deflection of the bristle is comprised of an immediate response —- a high-amplitude impulse, and a prolonged response. High-amplitude impulse is always well pronounced, prolonged response oscillations can sometimes be distorted by other factors, e.g. growing branches of a protoplasmic tube or additional strands of plasmodium propagating between the agar blobs. Responses are repeatable not only in different experiments but also during several rounds of stimulation in the same experiment [@adamatzky2014tactile]. Colour sensor {#colour} ------------- A colour sensor is a device that gives wavelength-dependent response when illuminated. Physarum is photo-sensitive. It changes pattern of electrical potential oscillatory activity when illuminated  [@block1981blue; @wohlfarth1981pathway]. Moreover the Physarum distinguishes the colour of illumination [@adamatzky2013towards]. We placed Physarum between two electrodes and illuminated it with red, green, blue or white light. We also illuminated Physarum with white light via transparent lens. Amount of light on the blob was 80-120 LUX for each colour. We say the Physarum recognises a colour of the light if it reacts to illumination with the colour by a unique changes in amplitude and periods of oscillatory activity. We found that Physarum recognises when red and blue light are switched on and when red light is switched off. Red and blue illuminations decrease frequency of oscillations. Red light increases amplitude of oscillations but blue light decreases the amplitude. Physarum does not differentiate between green and white lights. Switching off red light leads to increase of period and decrease of amplitude of oscillations [@adamatzky2013towards]. Chemical sensor {#chemical} --------------- A chemical sensor is a device that gives a selective response when exposed to a target chemical substance. Physarum senses and responds to volatile aromatic substances [@adamatzky2011attraction; @adamatzky2012simulating; @adamatzky2012physarum]. We studied Physarum’s binary preferences to various volatile chemicals [@delacycostello2013assessing] and derived an experimental mapping between subset of chemoattractants and chemorepellents: farnesene, tridecane, s(-)limonene, cis-3-hexenylacetate, geraniol, benzyl alcohol, linalool, nonanal and amplitude and frequency of electrical potential oscillation of Physarum [@whiting2014towards]. Physarum increases frequency of electrical potential oscillations when exposed to strongest attractants — farnesene, tridecane, s(-)limonene, cis-3-hexenylacetate. Exposure to repellents — linlool, benzyl alcohol, nonanal — leads to a decrease of oscillation frequency, and, for linlool and benzyl alcohol, increase of the oscillation amplitude. Physarum chemical sensor discriminates individual chemicals by changing amplitude and frequency of its electrical potential oscillations; it can detect chemical from a distance of several centimetres [@whiting2014towards]. Memristors ---------- A memristor is resistor with memory, which resistance depends on how much current had flown through the device [@chua1971memristor; @strukov2008missing]. Memristor is a material implication $\rightarrow$, a universal Boolean logical gate. In laboratory experiments [@gale2013slime] we demonstrated that protoplasmic tubes of Physarum show current versus voltage profiles consistent with memristive system. Experimental laboratory studies shown pronounced hysteresis and memristive effects exhibited by the slime mould [@gale2013slime]. Being a memristive element the slime mould’s protoplasmic tube can also act a a low level sequential logic element [@gale2013slime] operated with current spikes, or current transients. In such a device logical input bits are temporarily separated. Memristive properties of the slime mould’s protoplasmic tubes gives us a hope that a range of ‘classical’ memristor-based neuromorphic architectures can be implemented with Physarum. Memristor is an analog of a synaptic connection [@pershin2010experimental]. Being the living memristor each protoplasmic tube of Physarum may be seen as a synaptic element with memory, which state is modified depending on its pre-synaptic and post-synaptic activities. Therefore a network of Physarum’s protoplasmic tubes is an associate memory network. A memristor can be also made from Physarum bio-organic electrochemical transistor (Sect. \[transistor\]) by removing a drain electrode [@cifarelli2014non]. Schottky diodes --------------- A diode is a two-terminal passive non-linear device which conducts mainly in one direction. The diodes are used as current rectifiers to change alternating current to direct current. A forward voltage drop is a difference between electrical potentials of anode and cathode. A Schottky diode is a diode which has low forward voltage drop, comparing to other families of diodes. A device showing some resemblance to Schottky diode can be made of Physarum [@cifarelli2014non]. In this device, the Physarum spans two asymmetrical junction electrodes — gold and indium. Cyclic voltage-current characteristics are measured. The measurements reveal suppression of conductivity for low voltage values in the direction of the positive bias and rectification features, pronounced more for the low values of the bias voltage [@cifarelli2014non]. Physarum [*p*er se]{} is not a diode: the rectifying properties emerge due to combination of features of the asymmetric electrode junction with electrochemical activity of the Physarum [@cifarelli2014non]. Voltage divider --------------- Voltage divider is a circuit that produces a given fraction of an input voltage as an output voltage [@horowitz1989art]. A 10 mm protoplasmic tube acts as a simple divider: the output voltage is c. 0.9 of the input voltages, voltages tested up to 20 V [@adamatzky2013physarumwire]. A typical voltage divider has two resistances, represented by two protoplasmic tubes. The Physarum divider produces output with 12% accuracy. This error is linear and might be due to differences in the protoplasmic tubes’ resistances. Resistors in Physarum voltage divider can be made adjustable by applying illumination, heat, or loading the protoplasmic tubes with functional nano particles [@mayne2015toward; @whiting2015transfer]. For example, loading Physarum with magnetite lower resistance of protoplasmic tube to 10-20 KOhm, making it compatible by value with common resistors [@mayne2015toward]. One of the tube constituting the divider can be transformed to a potentiometer by making an output electrode a conductive micro needle [@whiting2015transfer]. Transistors {#transistor} ----------- A transistor is a three-terminal active device that power amplifies input signal. The additional power comes from an external source of power. An organic electrochemical transistor is a semiconducting polymer channel in contact with an electrolyte. Its functioning is based on the reversible doping of the polymer channel. A hybrid Physarum bio-organic electrochemical transistor is made by interfacing an organic semiconductor, poly-3,4-ethylenedioxythiophene doped with poly-styrene sulfonate, with the Physarum [@tarabella2015hybrid]. Physarum is used instead of electrolyte. Electrical measurements in three-terminal mode uncover characteristics similar to transistor operations. The device operates in a depletion mode similarly to standard electrolyte-gated transistors. The Physarum transistor works well with platinum, golden and silver electrodes. If the drain electrode is removed and the device becomes two-terminal, it exhibits cyclic voltage-current characteristics similar to memristors [@tarabella2015hybrid]. Thermic switch {#thermicswitch} -------------- A thermistor is resistor which resistance changes depending on its temperature. When we heat a Physarum protoplasmic tube up to 40^o^C it is resistance increases thousand times [@walter2015hybrid]. The temperature-induced increase of Physarum resistance is a threshold-wise. This is why the Physarum is not a thermistor but a thermic switch. The Physarum thermic switches are reusable. The full duty cycle, from the heat response to reforming, is tens of minutes. The Physarum thermic switches are successfully tested in hybrid circuits implementing analog summators, [and]{} and [nand]{} gates, and cascades of the gates into a Flip-Flop latch. The circuits performed well in multiple duty cycles with the same setups, producing reproducible results from one duty cycle to another [@walter2015hybrid]. Robotcs ======= Robot controllers ----------------- Physarum responds to stimuli by changing pattern of its electrical potential oscillations and cytoplasm shuttling. By interfacing the Physarum with actuators we can make the slime mould controller for robots. Two prototypes of such robotic controllers are made: controller for a hexapod robot [@tsuda2006robot] and controller for a robotic android head [@galeAndroid]. A Physarum controller for hexapod robot is made of a star shaped template [@tsuda2006robot]. It has six circular wells connected by the channels that meet at a single point. The Physarum grows inside the template. Physarum in each well acts as cytoplasm shuttle streaming oscillators. The Physarum oscillators in the wells coupled via Physarum body colonising the channels between the wells. Blue light used as a stimulus. The shuttle streaming of cytoplasm is measured via light absorbance. Oscillations of shuttle streaming in the wells, as a response to a stimulation with light, modulate phase and frequency of the robot legs’s movement and cause the robot to change its direction of movement. An electrical activity of Physarum in response to stimulations is converted to affective state in the design of Physarum emotional controller [@galeAndroid]. A Physarum is inoculated on a multi-electrode array and stimulated with nutrients (attractant) and light (repellent). Extracellular electrical potential is recorded. The recorded data is split into chunks. We employed a circumplex model of affect, where emotions are plotted in two-dimensions determined by the polarity and arousal level. The chunks are assigned polarity. Potential recorded during stimulation with attractant is given positive polarity. Data obtained during illumination of Physarum is assigned negative polarity. A level of arousal is proportional to amplitude of the electrical potential. Emotions are assigned to the data chunks, based on the polarity and the arousal of chunks and fed into an android robot. The data activate the motors placed in the positions matching sites of real muscles in a human face. Actuation of the motors causes movements of an artificial skin. The movements are expressed as affective facial expression of the android [@galeAndroid]. Actuators --------- Physarum contracts its body in a phase with oscillation of calcium waves. By placing a column of water on one side of the Physarum dumbbell shape we calculated that the Physarum weighting 5 mg, can lift up a load 36 times heavier than its own weight [@tsuda2012towards]. In [@adamatzky2010physarumboats] we studied how the Physarum can manipulate on a water surface. To make the Physarum propelling a ‘boat’ we take a small piece of a plastic foam, inoculate the Physarum on the foam and place this floater on a water surface. When Physarum is illuminated, it increases its peristaltic which transferred into a movement of the boat. If we allow Physarum to develop its protoplasmic tubes outside the float, on the water surface, then periodic contractions of tubes, stimulated by light, will make the Physarum boat propel away from the source of light. Another way to move the floater, is to place a stationary floater (anchor) with an attractants nearby the Physarum boat. Then Physarum develops protoplasmic tree towards the attractants and colonises the anchorr. Then Physarum straightens its tubes thus pulling the boat towards the anchor [@adamatzky2010physarumboats]. Nervous system -------------- The Physarum senses tactile, chemical and optical stimuli and converts the stimuli into characteristic patterns of its electrical potential oscillations. The electrical responses to stimuli may propagate along protoplasmic tubes for distances exceeding tens of centimetres, like impulses in neural pathways do. The Physarum makes decision about its propagation direction based on information fusion from thousands of spatially extended protoplasmic loci, similarly to a neurone collecting information from its dendritic tree. When growing on a non-nutrient substrate Physarum develops shapes resembling body of a single neuron. It looks like neuron — can it be behave as one? In [@adamatzky2015alife] we speculate on whether an alternative —- would-be — nervous systems can be developed and practically implemented from the slime mould. We uncover analogies between the slime mould and neurons, and demonstrate that the slime mould can play a role of primitive mechanoreceptors, photoreceptors, chemoreceptors; we also show how the Physarum neural pathways develop [@adamatzky2015alife]. Physarum neural networks do not have synapses represented as discrete structural elements. Synapse-like morphological contacts could not be formed. When two pieces of Physarum are inoculated at a distance form each other, they start exploring space around them and form branching networks of protoplasmic tubes. When two networks, grown from different sites of inoculation come into contact they usually fuse forming a single united network. However, there is a functional analog of synapses and an intrinsic feature of Physarum protoplasmic tubes which makes literally any loci of Physarum network a synapse. This is a memristive property (Sect. \[memristors\]). We explored the analogy between behaviour of neuron growth cones and Physarum active growing zones [@adamatzky2015alife]. To test if Physarum can develop information pathways we conducted several experiments on one to one scale models of human skull and brain. We used real scale models for the following reasons. First, to show that information pathways made of protoplasmic tubes can be tens of centimetres length and thus match lengths of neural pathways. Second, to demonstrate that — when propagating inside human skull — the plasmodium follows general anatomical trajectories of ocular and olfactory nerves. We found that morphology of information pathways developed by Physarum on a human skull matches well anatomy of the real nervous pathways. Impressive results we also obtained in imitation of sensorial innervation of the front scalp: we inoculated Physarum on the frontal bone above glabella and placed few oat flakes on the pariental bone. In two days Physarum developed an extensively branching tree of protoplasmic tubes. The tree spanned substantial part of the frontal lobe, even covering its lateral parts, crossed coronal sutura and developed actively branching growing zones moving towards the target site on the parietal bone [@adamatzky2015alife]. Illusions --------- A configuration of three Pac-Man shapes positioned at a vertices of an imaginary triangle and looking towards the centre of the triangle is a famous illusory contour [@kanizsa1976subjective]. When we look at this configuration of the shapes we are getting an impression of a white triangle defined by the Pac-Man shapes. The illusory contour disappears when the Pac-Man shapes are facing away from each other. Physarum shows tendency to ‘experience’ the same illusion as humans do [@tani2014kanizsa]. The Pac-Man shapes are made of a nutrient rich agar and placed on a non-nutrient agar. The Physarum is inoculated in the centre of each Pac-Man. The Physarum develops protoplasmic networks spanning the configuration of Pac-Man shapes. When spanning the configuration, when Pac-Man shapes are facing each other, the Physarum formed a network matching a contour of the illusory triangle in 4/5 of experiments [@tani2014kanizsa]. In the scenarios of away facing Pan-Man shapes the Physarum matched the illusory triangle only in the half of experiments and constructed spanning tree and other graphs in another half of experiments. Conclusion was that Physarum shows behaviour which mimic illusory impressions of humans [@tani2014kanizsa]. Exact mechanisms of such behaviour of Physarum are different from humans, however there may be some subtle analogies between how we visually scan pictures [@yarbus1967eye] and how the Physarum perceives its environment. Energy production ================= Modulation of energy generation ------------------------------- A microbial fuel cell is a biological electrochemical device which uses micro-organisms to convert energy of organic substates into an electrical energy. The micro-organisms colonise electrodes, metabolise organic material and donate electrons to the electrodes [@ieropoulos2005comparative]. A basic fuel cell is made of two electrodes and an ion selective membrane. Electrodes are made of folded carbon veil. Two scenarios are explored in [@taylor2015physarum] for anode and cathode sites of Physarum inoculation. When Physarum is housed on anode no significant difference between the test and the control fuel cells is observed. In experiments with Physarum inoculated on cathode a statistically significant power increase is observed. A peak open circuit voltage of Physarum fuel cell and control fuel cell was around 0.6V [@taylor2015physarum]. When 9.4kOhm external load is connected, the voltage drops to 0.2V in the control fuel cell and 0.25V in the Physarum fuel cell. During 7 days of experiments power generated by the Physarum fuel cell was 12.5–15 $\mu$W and power generated by the control fuel cell 10–11V [@taylor2015physarum]. Biodiesel production -------------------- This is not about computation, sensing, actuation or analog physical modelling. But we included this section because future unconventional devices will need energy. Biodiesel is a liquid fuel made of vegetable oil or animal fat and used in compression-ignition engines. It is produced by a conversion of a carboxylic acid ester into a different carboxylic acid ester by reacting lipids with an alcohol producing fatty acid esters [@ma1999biodiesel]. Biodiesel is advantageous to fossil fuels because it is biodegradable and has low toxicity. However, biodiesel is expensive to produce: cost of raw material is 2/3 of production costs. Slime moulds have relatively high concentration of lipids in their bodies. Thus they could be a good alternative to existing technologies of bio-mass production. *P. polycephalum* is found to be a champion, amongst *Myxomecetes*, in lipid production [@tran2012evaluating]. Physarum produces equivalent of 65 g of dry biomass and over 7 g of lipids per litre of culture medium in four days [@tran2012evaluating]. The biomass production rate exceeds that of algae [@chisti2007biodiesel]. Tran et al [@tran2015evaluation] show that cultivation of Physarum on 37.5 g per litre rice brans yields 7.5 g dry biomass and 0.9 g lipid in five days. Arts ==== Music generation ---------------- Physarum expresses its physiological states in patterns of its electrical potential oscillators [@adamatzky2011electrical]. In [@adamatzky2009music] 5-10 days recordings were processed and converted to sound track by mapping parameters of electrical potential oscillation to pitch, attack and duration of tones. First ever sound track was produced from Physarum’s electrical activity in  [@adamatzky2009music]. The music reflected a physiological transition of Physarum from a comfortable foraging state to a state of active search for disappearing nutrients to decision making state to transformation to sclerotium. When the track played to auditorium at various presentations, majority of people were getting a feeling of a dramatic development in the ‘life of Physarum’. Later the transformation of Physarum activity recording into sounds has been taken at a more professional, from music point of view, level in [@miranda2011sounds]. There electrical activity of Physarum was converted to parameters of sinusoidal oscillators; the rhythmic behaviour of Physarum was shown to produce different timbres. Memristive properties of Physarum (Sec. \[memristors\]) are used to generate musical responses in [@mirandamemristors]. A vocabulary notes are assigned voltage values. The Physarum current-voltage response to electrical stimulation is recorded. Discrete voltages are converted to notes. The notes are fed into a MIDI keyboard. During interactive music performance between human composer and Physarum, a feedback to the Physarum is implemented. Parts of well-know melody: Elgar’s Nimrod and Beethoven’s Für Elise were generated for live performances with Physarum in  [@mirandamemristors]. Modelling creativity -------------------- Creativity is manifested by divergent thinking and lack of lateral inhibition, making remote associations between ideas and concepts, switching between ideation, generating ideas of actualities, risk taking, nonconformity [@kuszewski2009genetics]. In [@adamatzky2013creativity] we show how to use live Physarum to analyse many scenarios of an individual creativity genesis. The divergent thinking is expressed by Physarum in its simultaneous reaction to several sources of attractants and repellents, and parallel implementation of sensorial fusion. Functional non-conformity of Physarum is manifested in its self-avoidance. Activity in both hemispheres and exchange of activities between hemispheres are considered to be attributes of human creativity [@kuszewski2009genetics]. In Physarum the hemispheres’ activity is represented by simultaneous oscillatory activity, with biochemical oscillators located to distant parts of Physarum body and oscillating with different frequencies and amplitudes. Interaction between ‘hemispheres’ is instantiated by waves of calcium waves and waves of contractile activity propagating along protoplasmic tubes. Cognitive control of divergent thinking is a requisite of creativity[@kuszewski2009genetics]. A person with extremely divergent thinking who is unable to control these associations would be potentially classified as mentally ill. However, those who can fit their high schizotypy (a range of personality characteristics ranging from normal to schizophrenia) traits into rigorous cognitive frameworks may be classified as gifted or even genius. Thus creativity could be positioned together with autism and schizophrenia in the same phase space. Physarum imitates cognitive control by tuning regularity of its protoplasmic network. A degree of branching of a Physarum network may be considered as a representation of a degree of schizotypy. Then ‘mathematical savant’ slime mould grows a low branching highly symmetrical protoplasmic networks and severely ‘autistic’ Physarum develops highly asymmetric low branching networks [@adamatzky2013creativity]. Things inspired by Physarum but never done with a real one ========================================================== 1. A non-quantum implementation of Shor’s factorisation algorithm [@blakey2014towards] is inspired by Physarum ability to retain the time-periods of stimulation, the anticipatory behaviour [@saigusa2008amoebae] 2. Single electron circuit solving maze problem [@shinde2014design] is based on a cellular automaton imitation of Physarum. 3. Particle based model of Physarum approximates moving average and low-pass filters in one-dimensional data sets and spatial computation of splines in two-dimensional data set [@jones2014material]. 4. Physarum logic is developed by interpreting basic features of the Physarum foraging behaviour in terms of process calculi and spatial logic without modal operators [@schumann2011physarum]. 5. Soft amoeboid robots [@piovanelli2012bio] —- models of coupled-oscillator-based robots [@umedachi2013fluid] are based on general principles of Physarum behaviour, especially coordination of distant parts of its cell. 6. Physarum concurrent games are proposed in [@schumann2014bio]. In these games rules can change, players update their strategies and actions, resistance points are reduced to payoffs. At the time these games were proposed, we were unaware about the lethal reaction followed a fusion between Physarum of two different strains [@carlile1972lethal]: a degree of lethality depends on the position and size of invasion between strains, which supports the ideas developed in [@schumann2014bio]. 7. Physarum is interfaced with a field-programmable array in [@mayne2015towardsFPGA]. The hybrid system performs predefined arithmetic operations derived by digital recognition of membrane potential oscillations. 8. A Physarum-inspired algorithm for solution of the Steiner tree problem is proposed in [@liu2015physarum] and applied to optimise the minimal exposure problem and worst-case coverage in wireless sensor networks [@liu2015physarum; @tsompanas2015cellular]. 9. An abstract implementation of reversible logical gates with Physarum is proposed in [@schumann2015conventional]. 10. Mechanisms of Physarum foraging behaviour are employed in robotics algorithm for simultaneous localisation and mapping [@kalogeiton2014hey]. 11. A range of algorithms for network optimisation is derived from a model of Physarum shortest path formation [@tero2006physarum; @bonifaci2012physarum]. Most of these algorithms are based on a feedback between traffic throughout a tube and the tube’s capacity. They include dynamical shortest path algorithms [@zhang2014improved], optimal communication paths in wireless sensor networks [@zhang2015physarumPPL; @dourvas2015hardware], supply chains design [@zhang2015physarumPPL], shortest path tree problem [@zhang2015physarum], design of fault tolerant graphs [@becker2015evaluating], and multi-cast routing [@liang2015new]. Behaviour of Physarum solver [@tero2006physarum] was also applied to deriving a shortest path on Riemann surface [@miyaji2008physarum]. 12. Physarum-inspired algorithm for learning Bayesian network structure from data is designed in [@schon2014physarum]. 13. Attraction based two-input two-output gate realising [and]{} and [or]{} and three-input two-output gate realising conjunction of three inputs and negation of one input with disjunction of two other inputs are constructed in particle-based model of Physarum [@jones2010towardsadder]; the gates are cascaded into a one-bit adder. 14. A nano-device aimed to solve Boolean satisfiability problem, inspired by optimisation of protoplasmic networks by Physarum, is designed in [@aono2012amoeba; @aono2015amoeba]. The device works on fluctuations generated from thermal energy in nanowires, electrical Brownian ratchets . 15. Solution of a the ‘exploration versus exploitation’ dilemma by Physarum making a choice between colonising nutrients and escaping illuminated areas is used in a tug of war model [@kim2010tug]: a parallel search of a space by collectives of locally correlated agents and decision making in situations of uncertainty. The ideas are developed further in a design of experimental device where a single photon solves the multi-armed bandit problem [@naruse2015single]. Post coitum omne animal triste est ================================== Excitement of being able to make computing, sensing and actuating devices with live Physarum eventually fades down. Let us wake up now. Is there any real use of the slime mould? Is it fast? No. Physarum is a very slow creature. It might take the Physarum several days to compute Voronoi diagram or a shortest path. Is it quick in responding to stimuli? Not really. Is it robust? No two experiments are the same. Physarum always behaves differently. Results of a computation performed by Physarum are only valid when at least dozen of experiments are done cause a single trial might mean nothing. Are the Physarum computing devices reliable? Failure rate is near 30%. The slime mould is not a miraculous computing substrate. It is just a user-friendly non-demanding living creature which changes its form and shape to stay comfortable in the fields of attractants and repellents. Why did we love Physarum then? Because by taking myriad of different shapes and exhibiting fantastically rich spectrum of electrical potential oscillations Physarum makes a unique fruitful material for interpretations of its behaviour in terms of purposeful transformation of data to results. 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--- abstract: 'We propose a notion of autoreducibility for infinite time computability and explore it and its connection with a notion of randomness for infinite time machines introduced in [@CaSc] and [@Ca3].' author: - Merlin Carl title: 'A note on Autoreducibility for Infinite Time Register Machines and parameter-free Ordinal Turing Machines' --- Autoreducibility for Infinite Time Register Machines ==================================================== The classical notion of autoreducibility can, for example, be found in [@DoHi]. We consider how this concept behaves in the context of infinitary machine models of computations. For the time being, we focus on Infinite Time Register Machines ($ITRM$s) (see [@ITRM] and [@ITRM2]) and ordinal Turing machines (see [@Ko]) - but the notion of course makes sense for other types like the Infinite Time Turing Machines ($ITTM$s, see [@HaLe]) as well. For $x\in ^{\omega}2$, we define $x_{\setminus n}$ as $x$ with its $n$th bit deleted (i.e. the bits up to $n$ are the same, the further bits are shifted one place to the left). We say that $x$ is $ITRM$-autoreducible iff there is an $ITRM$-program $P$ such that $P^{x_{\setminus n}}(n)\downarrow=x(n)$ for all $n\in\omega$. $x$ is called totally incompressible iff it is not $ITRM$-autoreducible, i.e. there is no $ITRM$-program $P$ such that $P^{x_{\setminus n}}(n)\downarrow=x(n)$ for all $n\in\omega$. If there is such a program, then we say that $P$ autoreduces $x$, $P$ is an autoreduction for $x$ or that $x$ is autoreducible via $P$. $x\in ^{\omega}2$ is $ITRM$-random in the measure sense iff there is no $ITRM$-decidable set $X$ of Lebesgue measure $0$ such that $x\in X$. $x\in ^{\omega}2$ is $ITRM$-random in the meager sense iff there is no $ITRM$-decidable meager set $X$ such that $x\in X$. We refer the reader to [@CaSc] and [@Ca3] for more information on $ITRM$-randomness, including that used in the course of this note. For the notion of $ITRM$-recognizability, we refer the reader to [@ITRM2], [@Ca] or [@Ca2]. No totally incompressible $x$ is $ITRM$-computable or even recognizable. $0^{\prime}_{ITRM}$, the real coding the halting problem for $ITRM$s, is $ITRM$-autoreducible. Clearly, if $P$ computes $x$, then $P$ is also an autoreduction for $x$. If $x$ is recognizable and $P$ recognizes $x$, we can easily retrieve a deleted bit by pluggin in $0$ and $1$ and letting $P$ run on both results to see for which one $P$ stops with output $1$. (The same idea works for finite subsets instead of single bits.) For $0^{\prime}_{ITRM}$, if a program index $j$ is given, it is easy to determine some index $i\neq j$ corresponding to a program that works in exactly the same way (by e.g. adding a meaningless line somewhere), so that the remaining bits allow us to reconstruct the $j$th bit. The autoreducibility of $0^{\prime}_{ITRM}$ also follows from the recognizability of $0^{\prime}_{ITRM}$ (see [@Ca2]). Let $x\in^{\omega}2$, $i\in\omega$. Then $\text{flip}(x,i)$ denotes the real obtained from $x$ by just changing the $i$th bit, i.e. $x\Delta\{i\}$. In the classical setting, no random real is autoreducible. This is still true for $ITRM$s: [\[randomnessimpliestotalincompressibility\]]{} If $x$ is $ITRM$-random, then $x$ is totally incompressible. (For the meager as well as for the measure $0$ interpretation of randomness.) Assume that $x$ is autoreducible via $P$. We show that $x$ is not $ITRM$-random. Let $X$ be the set of all $y$ which are autoreducible via $P$. Obviously, we have $x\in X$. $X$ is certainly decidable: Given $y$, use a halting problem solver for $P$ to see whether $P^{y_{\setminus n}}(n)\downarrow$ for all $n\in\omega$. If not, then $y\notin X$. Otherwise, carry out these $\omega$ many computations and check the results one after the other.\ Since $X$ is $ITRM$-decidable, it is provably $\Delta_{2}^{1}$, which implies that $X$ has the Baire property and thus is measurable. We show that $X$ must be of measure $0$. To see this, assume for a contradiction that $\mu(X)>0$. Note first, that, whenever $y$ is $P$-autoreducible and $z$ is a real that deviates from $y$ in exactly one digit (say, the $i$th bit), then $z$ is not $P$-autoreducible (since $P$ will compute the $i$th bit wrongly). By the Lebesgue density theorem, there is an open basic interval $I$ (i.e. consisting of all reals that start with a certain finite binary string $s$ length $k\in\omega$) such that the relative measure of $X$ in $I$ is $>\frac{1}{2}$. Let $X^{\prime}=X\cap I$, and let $X^{\prime}_0$ and $X^{\prime}_1$ be the subsets of $X^{\prime}$ consisting of those elements that have their $(k+1)$th digit equal to $0$ or $1$, respectively. Clearly, $X^{\prime}_{0}$ and $X^{\prime}_{1}$ are measurable, $X^{\prime}_{0}\cap X^{\prime}_{1}=\emptyset$ and $X^{\prime}=X^{\prime}_{0}\cup X^{\prime}_{1}$. Now define $\bar{X}^{\prime}_{0}$ and $\bar{X}^{\prime}_{1}$ by changing the $(k+1)$th bit of all elements of $X^{\prime}_{0}$ and $X^{\prime}_{1}$, respectively. Then all elements of $\bar{X}^{\prime}_{0}$ and $\bar{X}^{\prime}_{1}$ are elements of $I$ (as we have not changed the first $k$ bits), none of them is $P$-autoreducible (since they all deviate from $P$-autoreducible elements by exactly one bit, namely the $k$th), $\bar{X}^{\prime}_{0}\cap\bar{X}^{\prime}_{1}=\emptyset$ (elements of the former set have $1$ as their $(k+1)$th digit, for elements of $\bar{X}^{\prime}_{1}$ it is $0$) and $\mu(\bar{X}^{\prime}_{0})=\mu(X^{\prime}_{0})$, $\mu(\bar{X}^{\prime}_{1})=\mu(X^{\prime}_{1})$ (as the $\bar{X}^{\prime}_{i}$ are just translations of the $X^{\prime}_{i}$). As no element of the $\bar{X}^{\prime}_{i}$ is $P$-autoreducible, we have $(\bar{X}^{\prime}_{0}\cup\bar{X}^{\prime}_{1})\cap X^{\prime}=\emptyset$. Let $\bar{X}^{\prime}:=\bar{X}^{\prime}_{0}\cup\bar{X}^{\prime}_{1}$. Then we have\ $\mu_{I}(\bar{X}^{\prime})=\mu_{I}(\bar{X}^{\prime}_{0}\cup\bar{X}^{\prime}_{1})=\mu_{I}(\bar{X}^{\prime}_{0})+\mu_{I}(\bar{X}^{\prime}_{1})=\mu_{I}(X^{\prime}_{0})+\mu_{I}(X^{\prime}_{1})=\mu_{I}(X^{\prime})>\frac{1}{2}$ (where $\mu_{I}$ denotes the relative measure for $I$). So $X^{\prime}$ and $\bar{X}^{\prime}$ are two disjoint subsets of $I$ both with relative measure $>\frac{1}{2}$, a contradiction.\ For the meager version, we proceed similarly, taking $I$ to be an interval in which $X\cap I$ is comeager instead. That such an $I$ exists can be seen as follows: Suppose that $X$ is not meager. As above, $X$ is $ITRM$-decidable, hence provably $\Delta_{2}^{1}$ and therefore has the Baire property. Then, there is an open set $U$ such that $X\setminus U\cup U\setminus X$ is meager. In particular, $U$ is not empty. Hence $X$ is comeager in $U$. As $U$ is open, there is a nonempty open interval $I\subseteq U$. It is now obvious that $X\cap I$ is comeager in $I$, so $I$ is as desired. We then use the same argument as above, noting that two comeager subsets of $I$ cannot be disjoint. [\[Cohengenericsareincompressible\]]{} Let $x$ be Cohen-generic over $L_{\omega_{\omega}^{CK}+1}$. Then $x$ is totally incompressible (in the measure sense). Let $x$ be as in the assumption, and assume for a contradiction that $x$ is $P$-autoreducible. By the forcing theorem for provident sets (see [@Ma]), there must be a condition $p\subseteq x$ such that $p\Vdash\text{`}P\text{ autoreduces }\dot{x}\text{'}$, where $\dot{x}$ is a name for the generic real (i.e. for $\bigcup\dot{G}$, where $\dot{G}$ is the canonical name for the generic filter). Let $i\in\omega\setminus\text{dom}(p)$ and let $x^{\prime}:=\text{flip}(x,i)$. Then $x^{\prime}$ is still Cohen-generic over $L_{\omega_{\omega}^{CK}+1}$ and, as $p\subseteq x^{\prime}$, $p$ forces the $P$-autoreducibility of $x^{\prime}$; however, as $x^{\prime}$ differs from the $P$-autoreducible $x$ by only one bit, $x^{\prime}$ cannot be $P$-autoreducible, a contradiction. Thus $x$ is not $P$-autoreducible. [\[totallycompressiblesarerare\]]{} The set of $ITRM$-autoreducible reals has measure $0$. The proof of Theorem \[randomnessimpliestotalincompressibility\] shows that, for any $ITRM$-program $P$, the set of reals autoreducible via $P$ has measure $0$. As there are only countable many programs, the result follows. Denote by $IC_{ITRM}$ and $RA_{ITRM}$ the set of totally incompressible and $ITRM$-random reals, respectively (in the measure sense, for the time being). In this terminology, we showed above that $RA_{ITRM}\subseteq IC_{ITRM}$. However, the converse of Theorem \[randomnessimpliestotalincompressibility\] fails: [\[totalincompressibilitydoesnotimplyrandomness\]]{} $IC_{ITRM}\neq\subseteq RA_{ITRM}$, i.e. there is a real $x$ such that $x$ is totally incompressible, but not $ITRM$-random (in the measure sense). Let $X$ be an $ITRM$-decidable, comeager set of Lebesgue measure $0$. That such an $X$ exists is rather easy to see: A nice example for a comeager set of measure $0$ is the set of reals for which the zeros and ones in their binary representation are not equally distributed. It is straightforward to implement a decision procedure for this set on an $ITRM$.\ Now, the set of Cohen-generic reals over $L_{\omega_{\omega}^{CK}+1}$ is comeager and hence must intersect $X$. Let $x\in X$ be Cohen-generic over $L_{\omega_{\omega}^{CK}+1}$. By Corollary \[Cohengenericsareincompressible\], $x$ is totally incompressible. As $x\in X$ and $X$ is $ITRM$-decidable set of measure $0$, $x$ is not $ITRM$-random (in the measure sense). Thus $x\in RA_{ITRM}\setminus IC_{ITRM}$, as desired. In fact, the set of these reals is comeager, as the set $C$ of Cohen-generic reals over $L_{\omega_{\omega}^{CK}+1}$ is comeager, so that $C\cap X$ is also comeager and the proof shows that any element of $C\cap X$ is of this kind. Incompressibility and Randomness -------------------------------- We saw above that the following inclusions hold (where $C^{+}$ denotes the set of Cohen-generic reals over $L_{\omega_{\omega}^{CK}+1})$: $C\subsetneq RA_{ITRM}\subsetneq IC_{ITRM}$ (The first inclusion is proper because genericity for $\Pi_{1}$ and $\Sigma_1$-definable over $L_{\omega_{\omega}^{CK}}$ dense sets is sufficient, but not every such real is generic over $L_{\omega_{\omega}^{CK}+1}$, which requires intersection with every definable dense set, $Pi_{1}/\Sigma_1$ or not. We do not know whether $ITRM$-randomness can be characterized in terms of genericity in a natural way.)\ In this section, we consider the question how similar incompressibility is to randomness, i.e. which of the results obtained for random reals also hold for incompressibles.\ We start with an incompressible variant of the Kucera-Gacs theorem, which, as we recall, fails for $ITRM$-randomness, as no lost melody (an $ITRM$-recognizable real which is not $ITRM$-computable; this was shown in [@Ca3]) is reducible to a random real. [\[incompressiblekuceragacs\]]{} For every real $x$, there is a totally incompressible $y$ such that $x\leq_{ITRM} y$. Given $x$, let $y$ be Cohen-generic over $L_{\omega_{\omega}^{CK,x}+1}[x]$ and let $z:=x\oplus y$. Then certainly $x\leq_{ITRM}z$. Assume that $z$ is $P$-autoreducible for some program $P$. Hence, by the forcing theorem for provident sets [@Ma], there is a condition $p$ such that $p\Vdash$‘x$\oplus\bigcup\dot{G}$ is $P$-autoreducible’, where $\dot{G}$ is the canonical name for the generic filter. The same hence holds for every $y^{\prime}$ which is Cohen-generic over $L_{\omega_{\omega}^{CK,x}+1}[x]$ with $p\subseteq y^{\prime}$. Let $i\in\omega\setminus\text{dom}(p)$, $y^{\prime}:=\text{flip}(y,i)$, then $p$ forces the $P$-autoreducibility of $x\oplus y^{\prime}$. By absoluteness of computations, $x\oplus y^{\prime}$ is $P$-autoreducible. However, $x\oplus y^{\prime}$ differs from the $P$-autoreducible $x\oplus y$ in exactly one bit and hence cannot be $P$-autoreducible, a contradiction. Also, in contrast to the theorem that $ITRM$-computability from mutually $ITRM$-random reals implies plain $ITRM$-computability, mutually incompressibles can contain common non-trivial information ($COMP$ denotes the set of $ITRM$-computable reals): $x$ is totally incompressible relative to $y$ ($y$-incompressible, incompressible in $y$) iff there is no program $P$ such that $P^{x_{\setminus n}\oplus y}\downarrow=x(n)$ for all $n\in\omega$. If $x$ is $y$-incompressible and $y$ is $x$-incompressible, then $x$ and $y$ are mutually incompressible. [\[mutualincompressibility\]]{} There are mutually incompressible reals $y,z$ and a real $x\notin COMP$ such that $x\leq_{ITRM}y$ and $x\leq_{ITRM}z$. Let $y^{\prime}$, $z^{\prime}$ be mutually Cohen-generic over $L_{\omega_{\omega}^{CK,x}+1}[x]$, $y:=x\oplus y^{\prime}$, $z:=x\oplus z^{\prime}$ and apply the reasoning of the proof of Theorem \[incompressiblekuceragacs\]. Ordinal Turing Machines ======================= See [@Ko] for an introduction to ordinal Turing machines. For $OTM$s without parameters, define the notions of autoreducibility and total incompressibility as above for $ITRM$s. It turns out that there are no totally incompressible reals in $L$:\ [\[noOTMincompressibles\]]{} Assume $V=L$. Then there are no totally $OTM$-incompressible reals. Let $x\in L$. Our goal is to define a countable sequence $(P_{i}|i\in\omega)$ of programs deciding pairwise disjoint sets $(X_{i}|i\in\omega)$ with $\bigcup_{i\in\omega}X_{i}=\mathfrak{P}(\omega)$ such that if $y,z\in X$ differ only in finitely many bits, $y$ and $z$ are not in the same $X_{i}$. Once that is done, the proof is easy to finish: There is some $i\in\omega$ such that $x\in X_i$, without loss of generality let $i=0$. Then $X_{0}$ is decided by $P_{0}$. Now an autoreduction for $x$ works as follows: Given $n\in\omega$ and $x_{\setminus n}$, plug $0$ and $1$ in for the $i$th bit in $x_{\setminus n}$, getting reals $x_{0}$ and $x_{1}$, respectively, one of which is equal to $x$. Now use $P_{0}$ to decide whether $x_{0}\in X_{0}$ or $x_{1}\in X_{0}$. As $x_{0}$ and $x_{1}$ only differ in one bit and $X_{0}$ does not contain two (distinct) reals differing in only finitely many places, only one of $x_{0}$ and $x_{1}$ can be an element of $X_{0}$, and that is $x$, determining the $n$th digit of $x$.\ Now we construct $(P_{i}|i\in\omega)$ as follows: Let $(S_{i}|i\in\omega)$ be a natural enumeration of the finite sets of integers in order type $\omega$. Write $y\sim z$ iff $y$ and $z$ differ only in finitely many bits. For a real $a$, denote by $[a]_{0}$ the $<_{L}$-smallest real such that $[a]_{0}\sim a$. Then let $X_{i}:=\{[y]_{0}+_{b}S_{i}|y\in\mathfrak{P}^{L}(\omega\}$, where $+_{b}$ denotes the bitwise sum. Clearly, this is a countable partition of the constructible reals. Furthermore, there is a decision procedure for $X_{i}$ on an $OTM$ (which is in fact uniform in $i$) which works as follows: Given a real $a$ in the oracle, we can write $L$ on the tape until we arrive an $L$-level $L_{\alpha}\ni a$. Then, searching $L_{\alpha}$, we can identify $[a]_{0}$. Now compute the set $S$ of bits where $a$ and $[a]_{0}$ differ and compare it to our enumeration of finite subsets of $\omega$ fixed above: If $S=S_{i}$, then $a\in X_{i}$, otherwise $a\notin X_{i}$. Note that there are constructible reals which do not lie in any $OTM$-decidable null set, as the union $Y$ of all $OTM$-decidable null sets is an element in $L$ and, as a countable union of null sets, also a null set in $L$. Hence, at least in $L$, not every (parameter-free) $OTM$-random real is totally incompressible.\ Note that the situation will probably be quite different for Infinite Time Turing Machines ($ITTM$s), as they have neither the power to enumerate $L$ nor the ability to solve their own restricted halting program (like $ITRM$s).\ The $V=L$ hypothesis is probably unnecessarily strong here. However, even in rather mild extensions of $L$, $OTM$-incompressibles do exist: [\[OTMincomprinCohenextension\]]{} Let $x$ be Cohen-generic over $L$. Then $x$ is $OTM$-incompressible in $L[x]$ (and hence, by absoluteness of computations, in the real world). Assume for a contradiction that $x$ is $OTM$-autoreducible, say by the program $P$, where $x=\bigcup{G}$ and $G$ is a Cohen-generic filter over $L$. Then there is a finite $p\in G$ such that $p\Vdash\forall{n\in\omega}P^{x_{\setminus n}}(n)\downarrow=x(n)$. Let $i\in \omega\setminus\text{dom}(p)$, $x^{\prime}:=\text{flip}(x,i)$. Then $x^{\prime}\in L[x]$ is still Cohen-generic over $L$ and $p\subseteq x^{\prime}$ so that $p\Vdash \forall{n\in\omega}P^{x^{\prime}_{\setminus n}}(n)\downarrow=x^{\prime}(n)$. However, flipping a single bit cannot preserve $P$-autoreducibility, a contradiction. Hence $x$ is $OTM$-incompressible. Taking Theorem \[noOTMincompressibles\] and Theorem \[OTMincomprinCohenextension\] together, we get: [\[OTMincomprindependent\]]{} The existence of (parameter-free) $OTM$-incompressible reals is independent from $ZFC$. Consequently, the analogue of Theorem \[randomnessimpliestotalincompressibility\] for $OTM$s fails at least consistently: Every constructible $OTM$-random real provides a counterexample. For an $OTM$-program $P$, the set of $P$-autoreducibles is in general not decidable: [\[OTMPcompressibilityundecidable\]]{} Assume that $V=L$. Then there are $OTM$-programs $P$ such that $X_{P}:=\{x\mid \forall{n\in\omega}P^{x_{\setminus n}}(n)\downarrow=x(n)\}$ (i.e. the set of $P$-autoreducibles) is not $OTM$-decidable. Assume for a contradiction that $X_{P}$ is decidable for every $P$. By the same argument as in the proof of Theorem \[randomnessimpliestotalincompressibility\] then, $\mu(X_{P})=0$ for every $P$. Consequently, no $OTM$-autoreducible real is $OTM$-random, and hence, every $OTM$-random real is $OTM$-incompressible. However, the non-$OTM$-random reals are contained in a countable union of decidable null sets and hence form a null set themselves, so that the $OTM$-random reals have full measure, while, on the other hand, $OTM$-incompressibles do not exist in $L$, a contradiction. Note, however, that $P$-autoreducibility for $OTM$s is semidecidable by simply simultaneously running all $OTM$-programs on a real $x$ and checking whether one of them is an autoreduction. If such a program exists, it will eventually be found; otherwise, the search will not halt.\ We note further that such sets are in general not measurable: Assume $V=L$. Then there is an $OTM$-program $P$ such that the set of $P$-autoreducible reals is not measurable. In fact, there is a recursive set $I\subseteq\omega$ such that $\forall{x}P_{i}^{x}\downarrow=0\vee P_{i}^{x}\downarrow=1$, $S_{i}:=\{x\mid P_{i}^{x}\downarrow=1\}$ is not measurable, $S_{i}\cap S_{j}=\emptyset$ for $i\neq j$ and $\mathfrak{P}(\omega)=\bigcup_{i\in\omega}S_{i}$. Let $(s_{i}|i\in\omega)$ be an enumeration of $^{<\omega}\omega$ in order type $\omega$, denote by $x\sim y$ that $x$ and $y$ differ only at finitely many places, let $[x]_{\sim}$ be the $\sim$-equivalence class of $x$ and let $P_{i}$ be the program described in the proof of Theorem \[noOTMincompressibles\] that works as follows: Given $x$ in the oracle, determine the $<_{L}$-minimal representative $x_{0}$ of $[x]_{\sim}$, then output $x_{0}\Delta s_{i}$ (where $\Delta$ denotes the symmetric difference, that is we flip all the bits at places in $s_{i}$). Denoting, for $i\in\omega$, $E_{i}:=\{x\mid x=x_{0}\Delta s_{i}\}$, we have that $\mathfrak{P}(\omega)=\bigcup_{i\in\omega}E_{i}$, $E_{i}\cap E_{j}=\emptyset$ for $i\neq j$ and $P_{i}$ decides $E_{i}$ for all $i,j\in\omega$. Furthermore, it is well-known that none of the $E_i$ is measurable. Acknowledgements ================ We are indebted to Philipp Schlicht for several very helpful discussions on the subject and in particular for the proof idea for Theorem \[totalincompressibilitydoesnotimplyrandomness\]. M. Carl. The distribution of $ITRM$-recognizable reals. To appear in: Annals of Pure and Applied Logic, special issue from CiE 2012 M. Carl. Optimal Results on $ITRM$-recognizability. Preprint. arXiv:1306.5128v1 M. Carl. Algorithmic Randomness for Infinite Time Register Machines Preprint. arXiv:1401.1734v1 M. Carl, P. Schlicht. Infinite Computations with Random Oracles. Submitted. arXiv:1307.0160v3 M. Carl, P. Schlicht. Infinite Time Algorithmic Randomness. Work in progress. R.G. Downey, D. Hirschfeldt. Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer LLC $2010$ J. Hamkins, A. Lewis. Infinite Time Turing Machines. Journal of Symbolic Logic 65(2), 567-604 (2000) P. Koepke, R. Miller. An enhanced theory of infinite time register machines M. Carl, T. Fischbach, P. Koepke, R. Miller, M. Nasfi, G. Weckbecker. The basic theory of infinite time register machines P. Koepke. Turing computations on ordinals. Bulletin of Symbolic Logic 11 (2005), 377-397 A.R.D. Mathias. Provident sets and rudimentary set forcing. Preprint. Available at https://www.dpmms.cam.ac.uk/ ardm/fifofields3.pdf

--- author: - GIUSEPPE BOCCIGNONE bibliography: - 'levyeye.bib' title: A probabilistic tour of visual attention and gaze shift computational models --- This research was partially supported by the project “Interpreting emotions: a computational tool integrating facial expressions and biosignals based shape analysis and bayesian networks”, grant FIRB - *Future in Research* RBFR12VHR7\ Author’s address: G. Boccignone, Department of Computer Science, University of Milan, via Comelico 39/41, 20135 Milano, Italy; email: giuseppe.boccignone@unimi.it Introduction ============ As the french philosopher Merleau-Ponty put it, “vision is a gaze at grips with a visible world” [@maurice1945phenomenologie]. Goals and purposes, either internal or external, press the observer to maximise his information intake over time, by moment-to-moment sampling the most informative parts of the world. In natural vision this endless endeavour is accomplished through a sequence of eye movements such as saccades and smooth pursuit, followed by fixations. Gaze shifts require visual attention to precede them to their goal, which has been shown to enhance the perception of selected part of the visual field (in turn related to the foveal structure of the human eye, see for an extensive discussion of these aspects). The computational counterpart of using gaze shifts to enable a perceptual-motor analysis of the observed world can be traced back to pioneering work on active or animate vision [@aloimonos1988active; @Ballard; @bajcsy1992active]. The main concern at the time was to embody vision in the action-perception loop of an artificial agent that purposively acts upon the environment, an idea that grounds its roots in early cybernetics [@cordeschi2002discovery]. To such aim the sensory apparatus of the organism must be active and flexible, for instance, the vision system can manipulate the viewpoint of the camera(s) in order to investigate the environment and get better information from it. Surprisingly enough, the link between attention and active vision, notably when instantiated via gaze shifts (e.g, a moving camera), was overlooked in those early approaches, as lucidly remarked by . Indeed, active vision, as it has been proposed and used in computer vision, must include attention as a sub-problem [@rothenstein2008attention]. First and foremost when it must confront the computational load to achieve real-time processing (e.g., for autonomous robotics and videosurveillance). Nevertheless, the mainstream of computer vision has not dedicated to attentive processes and, more generally, to active perception much consideration. This is probably due to the original sin of conceiving vision as a pure information-processing task, a reconstruction process creating representations at increasingly levels of abstraction, a land where action had no place: the “from pixels to predicates” paradigm [@aloimonos1988active]. To make a long story short, the research field had a sudden burst when the paper by was published. Their work provided a sound and neat computational model (and the software simulation) to contend with the problem: in a nutshell, derive a saliency map and generate gaze shifts as the result of a Winner-Take-All (WTA) sequential selection of most salient locations. Since then, proposals and techniques have flourished. Under these circumstances, a deceptively simple question arises: Where are we now? A straight answer, which is the *leitmotiv* of this paper, is that whilst early active vision approaches overlooked attention, current approaches have betrayed purposive active perception. In this perspective, here we provide a critical discussion of a number of models and techniques. It will be by no means exhaustive, and yet, to some extent, idiosyncratic. Our purpose is not to offer a review (there are excellent ones the reader is urged to consult, e.g., [@BorItti2012; @borji2014salient; @bruce2015computational; @bylinskii2015towards]), but rather to spell in a probabilistic framework the variety of approaches, so to discuss in a principled way current limitations and to envisage intriguing directions of research, e.g., the hitherto neglected link between oculomotor behavior and emotion. In the following Section we highlight critical points of current approaches and open issues. In Section \[sec:prob\] we frame such problems in the language of probability. Section \[sec:action\] discusses possible routes to reconsider the problem of oculomotor behaviour within the action/perception loop. In Section \[sec:emo\] we explore the *terra incognita* of gaze shifts and emotion. A Mini review and open issues ============================= The aim of a computational model of attentive eye guidance is to answer the question *Where to Look Next?* by providing: 1. at the *computational theory level* (following ), an account of the mapping from visual data of a natural scene, say $\mathbf{I}$ (raw image data representing either a static picture or a stream of images), to a sequence of time-stamped gaze locations $(\mathbf{r}_{F_1}, t_1), (\mathbf{r}_{F_2}, t_2),\cdots$, namely $$\mathbf{I} \mapsto \{\mathbf{r}_{F_1}, t_1; \mathbf{r}_{F_2}, t_2;\cdots \}, \label{eq:mapping}$$ 2. at the *algorithmic level*, a procedure that simulates such mapping. A simple example of the problem we are facing is shown in Figure \[Fig:variab\]. ![Scan paths eye tracked from different human observers while viewing three pictures of different information content: outdoor (top row), indoor with meaningful objects (middle row), indoor with high semantic content (person and face, bottom row). The area of yellow disks marking fixations between saccades is proportional to fixation time (images freely available from the dataset).[]{data-label="Fig:variab"}](FigVariab.jpg) Under this conceptualization, when the input $\mathbf{I}$ is a static scene (a picture), the fixation duration time and saccade (lengths and directions) sequence are the only observables of the underlying guidance mechanism. When $\mathbf{I}$ stands for a time varying scene (e.g. a video), pursuit needs to be taken into account, too. We will adopt the generic terms of gaze shifts for either pursuit, saccades and fixational movements. In the following, for notational simplicity, we will write the time series $\{\mathbf{r}_{F_1}, t_1; \mathbf{r}_{F_2}, t_2;\cdots \}$ as the sequence $\{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$, unless the expanded form is needed. Also, we will generically refer to such sequence as scan path, though this term has a historically precise meaning in the eye movement literature [@privitera2006scanpath]. The common practice of computational approaches to derive the mapping (\[eq:mapping\]) is to conceive it as a two step procedure: 1. obtaining a suitable perceptual representation $\mathcal{W}$, i.e., $\mathbf{I} \mapsto \mathcal{W}$; 2. using $\mathcal{W}$ to generate the scan path, $\mathcal{W} \mapsto \{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$. It is important to remark that each gaze position $\mathbf{r}_{F}(t)$ sets a new field of view for perceiving the world, thus $\mathcal{W}=\{\mathcal{W}_{\mathbf{r}_{F}(1)}, \mathcal{W}_{\mathbf{r}_{F}(2)},\cdots \}$ should be a time-varying representation, even in the case of a static image input. This feedback effect of the moving gaze is hardly considered at the modelling stage [@zelinsky2008theory; @TatlerBallard2011eye]. By overviewing the field [@TatlerBallard2011eye; @BorItti2012; @bruce2015computational; @bylinskii2015towards], computational modelling has been mainly concerned with the first step: deriving a representation $\mathcal{W}$, typically in the form of a salience map. Yet, such step has recently evolved in a parallel research program, in which gaze shift prediction and simulation is not the focus, but salient object detection (for an in-depth review of this “second wave” of saliency-centered methods, see ). The second step, that is $\mathcal{W} \mapsto \{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$, which actually brings in the question of *how* we look rather than *where*, is seldom taken into account. Surprisingly, in spite of the fact that the most cited work in the field [@IttiKoch98] clearly addressed the *how* issue (gaze shifts as the result of a WTA sequential selection of most salient locations), most models simply overlook the problem. As a matter of fact, the representation $\mathcal{W}$, once computed, is usually validated in respect of its capacity for predicting the image regions that would be explored by the overt attentional shifts of human observers (in a task designed to minimize the role of top-down factors). Predictability is assessed according to some established evaluation measures (see , and , for a recent discussion). In other cases, if needed for practical purposes, e.g. for robotic applications, the problem of oculomotor action selection is solved by adopting some simple deterministic choice procedure that usually relies on selecting the gaze position $\mathbf{r}$ as the argument that maximizes a measure on the given representation $\mathcal{W}$. We will attack on the problem related to gaze shift generation in Sections \[sec:bias\] and \[sec:var\]. In the following Section we first discuss representational problems. Levels of representation and control {#sec:levels} ------------------------------------ The guidance of eye movements is likely to be influenced by a hierarchy of several interacting control loops, operating at different levels of processing. Each processing step exploits the most suitable representation of the viewed scene for its own level of abstraction. , in a plausible portrayal, have sorted out the following representational levels: 1) *salience*, 2) *objects*, 3) *values*, and 4) *plans*. Up to this date, the majority of computational models have retained a central place for low-level visual conspicuity [@TatlerBallard2011eye; @BorItti2012; @bruce2015computational]. The perceptual representation of the world $\mathcal{W}$ is usually epitomized in the form of a spatial saliency map, which is mostly derived bottom-up (early salience) following [@IttiKoch98]. The weakness of the bottom-up approach has been largely discussed (see, e.g. ). Indeed, the effect of early salience on attention is likely to be a correlational effect rather than an causal one [@foulsham2008], [@schutz2011eye]. Few examples are provided in Fig. \[Fig:itti\], where, as opposed to human scan paths (in free-viewing conditions), the scan path generated by using a salience-based representation [@IttiKoch98] does not spot semantically important objects (faces), the latter not being detected as regions of high contrast in colour, texture and luminance with respect to other regions of the picture. Under these circumstances, early saliency can be modulated to improve its fixation prediction. has considered prior knowledge on the typical spatial location of the search target, as well as contextual information (the gist of a scene, ). Further, object knowledge can be used to top-down tune early salience. In particular, when dealing with faces, a face detection step [@cerf2008predicting], [@postma2011], [@marat2013improving] or a prior for Bayesian integration with low level features [@bocc08tcsvt], can provide a reliable cue to complement early conspicuity maps. Indeed, faces drive attention in a direct fashion [@cerf2009faces] and the same holds for text regions [@cerf2008predicting; @BocCOGN2014]. It has been argued that salience has only an indirect effect on attention by acting through recognised objects: observers attend to interesting objects and salience contributes little extra information to fixation prediction [@EinhauserSpainPerona2008]. As a matter of fact, in the real world, most fixations are on task-relevant objects and this may or may not correlate with the saliency of regions of the visual array [@canosa2009real; @rothkopfBallard2007]. Notwithstanding, object-based information has been scarcely taken into account in computational models [@TatlerBallard2011eye]. There are of course exceptions to this state of affairs, most notable ones those provided by , , the Bayesian models discussed by and . The representational problem is just the light side of the eye guidance problem. When actual eye tracking data are considered, one has to confront with the dark side: regardless of the perceptual input, scan paths exhibit both systematic tendencies and notable inter- and intra-subject variability. As put it, where we choose to look next at any given moment in time is not completely deterministic, but neither is it completely random. Biases in oculomotor behaviour {#sec:bias} ------------------------------ Systematic tendencies or “biases” in oculomotor behaviour can be thought of as regularities that are common across all instances of, and manipulations to, behavioural tasks [@tatler2008systematic; @tatler2009prominence]. In that case case useful information about how the observers will move their eyes can be found. One remarkable example is the amplitude distribution of saccades and microsaccades that typically exhibit a positively skewed, long-tailed shape [@TatlerBallard2011eye; @dorr2010variability; @tatler2008systematic; @tatler2009prominence]. Other paradigmatic examples of systematic tendencies in scene viewing are: initiating saccades in the horizontal and vertical directions more frequently than in oblique directions; small amplitude saccades tending to be followed by long amplitude ones and vice versa [@tatler2008systematic; @tatler2009prominence]. Indeed, biases affecting the manner in which we explore scenes with our eyes are well known in the psychological literature (see for a thorough review), albeit underexploited in computational models. Such biases may arise from a number of sources. have suggested the following: biomechanical factors, saccade flight time and landing accuracy, uncertainty, distribution of objects of interest in the environment, task parameters. Understanding biases in eye guidance can provide powerful new insights into the decision about where to look in complex scenes. In a remarkable study, provided striking evidence that a model based solely on these biases and therefore blind to current visual information can outperform salience-based approaches. Further, the predictive performance of a salience-based model can be improved from $56\%$ to $80\%$ by including the probability of gaze shift directions and amplitudes. Failing to account properly for such characteristics results in scan patterns that are fairly different from those generated by human observers (which can be easily noticed in the example provided in Fig. \[Fig:itti\]) and eventually in distributions of saccade amplitudes and orientations that do not match those estimated from human eye behaviour. Variability {#sec:var} ----------- When looking at natural images or movies [@dorr2010variability] under a free-viewing or a general-purpose task, the relocation of gaze can be different among observers even though the same locations are taken into account. In practice, there is a small probability that two observers will fixate exactly the same location at exactly the same time. This effect is even more remarkable when free-viewing static images: consistency in fixation locations selected by observers decreases over the course of the first few fixations after stimulus onset [@TatlerBallard2011eye] and can become idiosyncratic. Such variations in individual scan paths (as regards chosen fixations, spatial scanning order, and fixation duration) still hold when the scene contains semantically rich “objects” (e.g., faces, see Figures \[Fig:variab\] and \[Fig:itti\]). Variability is also exhibited by the same subject along different trials on equal stimuli. ![A worst case analysis of scan path generation based on early salience (top left image, via ) vs. scan paths eye tracked from human subjects. Despite of inter-subject variability concerning the gaze relocation pattern and fixation duration time, human observers consistently fixate on the two faces. The simulated scan path fails to spot such semantically important objects that have not been highlighted as regions of high contrast in colour, texture and luminance. The area of yellow disks marking human fixations is proportional to fixation time ( dataset).[]{data-label="Fig:itti"}](FigItti.jpg) Randomness in motor responses is likely to be originated from endogenous stochastic variations that affect each stage between a sensory event and the motor response: sensing, information processing, movement planning and executing [@vanBeers2007sources]. It is worth noting that uncertainty comes into play since the earliest stage of visual processing: the human retina evolved such that high quality vision is restricted to the small part of the retina (about $2^{0}-5^{0}$ degrees of visual angle) aligned with the visual axis, the fovea at the centre of vision. Thus, for many visually-guided behaviours the coarse information from peripheral vision is insufficient [@strasburger2011peripheral]. In certain circumstances, uncertainty may promote almost “blind" visual exploration strategies [@tatler2009prominence; @over2007coarse], much like the behaviour of a foraging animal exploring the environment under incomplete information; indeed when animals have limited information about where targets (e.g., resource patches) are located, different random search strategies may provide different chances to find them . Indeed, few works have been trying to cope with the variability issue, after the early work by , . The glorious WTA scheme [@IttiKoch98], or variants such as the selection of the proto-object with the highest attentional weight [@anna] are deterministic procedures. Even when probabilistic frameworks are used to infer where to look next, the final decision is often taken via the maximum a posteriori (MAP) criterion which again is a deterministic procedure (technically, an $\arg\max$ operation, see ), or variants like the robust mean (arithmetic mean with maximum value) over candidate positions [@begum2010probabilistic]. As a result, for a chosen visual input $\mathbf{I}$ the mapping $\mathcal{W} \mapsto \{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$ will always generate the same scan path across different trials. As a last remark, the variability of visual scan paths has been considered a nuisance rather than an opportunity from a modelling standpoint. Nevertheless, beside theoretical relevance for modelling human behavior, the randomness of the process can be an advantage in computer vision and learning tasks. For instance, have reported that a stochastic attention selection mechanism (a refinement of the algorithm proposed in ) enables the i-Cub robot to explore its environment up to three times faster compared to the standard WTA mechanism [@IttiKoch98]. Indeed, stochasticity makes the robot sensitive to new signals and flexibly change its attention, which in turn enables efficient exploration of the environment as a basis for action learning [@nagai2009stability; @nagai2009bottom]. There are few notable exceptions to this current state of affairs, which will be discussed in Section \[sec:prior\]. Framing models in a probabilistic setting {#sec:prob} ========================================= We contend with the above issues by stating that observables such as fixation duration and gaze shift lengths and directions are random variables (RVs) that are generated by an underlying stochastic process. In other terms, the sequence $\{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$ is the realization of a stochastic process, and the ultimate goal of a computational theory is to develop a mathematical model that describes statistical properties of eye movements as closely as possible. The problem of answering the question *Where to Look Next?* in a formal way can be conveniently set in a probabilistic Bayesian framework. have re-phrased this question in terms of the posterior probability density function (pdf) $P(\mathbf{r} \mid \mathcal{W})$, which accounts for the plausibility of generating the gaze shift $\mathbf{r} = \mathbf{r}_{F}(t) - \mathbf{r}_{F}(t-1)$, after the perceptual evaluation $\mathcal{W}$. Formally, via Bayes’ rule: $$P(\mathbf{r} \mid \mathcal{W})= \frac{ P(\mathcal{W} \mid \mathbf{r} )}{P(\mathcal{W})} P(\mathbf{r}). \label{eq:BayesTatler}$$ In Eq. \[eq:BayesTatler\], the first term on the r.h.s. accounts for the likelihood $P(\mathcal{W} \mid \mathbf{r})$ of $\mathbf{r}$ when visual data (e.g., features, such as edges or colors) are observed under a gaze shift $\mathbf{r}_{F}(t) \rightarrow \mathbf{r}_{F}(t+1)$, normalized by $P(\mathcal{W})$, the evidence of the perceptual evaluation. As they put it, “The beauty of this approach is that the data could come from a variety of data sources such as simple feature cues, derivations such as Itti’s definition of salience, object-or other high-level sources”. The second term is the pdf $P(\mathbf{r})$ incorporating prior knowledge on gaze shift execution. The generative model behind Eq. \[eq:BayesTatler\] is shown in Fig. \[fig:tata\] shaped in the form of a Probabilistic Graphical Model (PGM, see for an introduction). A PGM is a graph where nodes (e.g., $\mathbf{r}$ and $\mathcal{W}$) denote RVs and directed arcs (arrows) encode conditional dependencies between RVs, e.g $P(\mathcal{W} \mid \mathbf{r})$. A node with no input arcs (for example $\mathbf{r}$) is associated with a prior probability, e.g., $P(\mathbf{r})$. Technically, as a whole, the PGM specifies at a glance a chosen factorization of the joint probability of all nodes. Thus, in Fig. \[fig:tata\] we can promptly read that $P(\mathcal{W} , \mathbf{r}) = P(\mathcal{W} \mid \mathbf{r})P(\mathbf{r})$. The PGM in Fig. \[fig:tatb\] represents the PGM in Fig. \[fig:tata\], but unrolled in time. Note that now the arc $\mathbf{r}_{F}(t) \rightarrow \mathbf{r}_{F}(t+1)$ makes explicit the dynamics of the gaze shift occurring with probability $P(\mathbf{r}_{F}(t+1) \mid \mathbf{r}_{F}(t))$. \[Fig:tat\] ![The dynamic PGM obtained by unrolling in time the PGM depicted in Fig. \[fig:tata\][]{data-label="fig:tatb"}](FigTatler.jpg) ![The dynamic PGM obtained by unrolling in time the PGM depicted in Fig. \[fig:tata\][]{data-label="fig:tatb"}](FigTatlerDyn.jpg) The probabilistic model represented in Fig. \[Fig:tat\] is generative in the sense that if all pdfs involved were fully specified, the attentive process could be simulated (via ancestral sampling, ) as: 1. Sampling the gaze shift from the prior: $$\mathbf{r}^{*} \sim P(\mathbf{r}); \label{eq:priorTatlersamp}$$ 2. Sampling the observation of the world under the gaze shift: $$\mathcal{W} ^{*} \sim P(\mathcal{W} \mid \mathbf{r}^{*}). \label{eq:likeTatlersamp}$$ Inferring the gaze shift $\mathbf{r}$ when $\mathcal{W}$ is known boils down to the inverse probability problem (inverting the arrows), which is solved via Bayes’ rule (Eq. \[eq:BayesTatler\]). In the remainder of this paper we will largely use PGMs to simplify the presentation and discussion of probabilistic models. We will see in brief (Section \[sec:like\]) that many current approaches previously mentioned can be accounted for by the likelihood term alone. But, crucial, and related to issues raised in Section \[sec:bias\], is the Bayesian prior $P(\mathbf{r})$. The prior, first {#sec:prior} ---------------- The prior $P(\mathbf{r})$ can be defined *prima facie* as the probability of shifting the gaze to a location *irrespective of the visual information* at that location, although the term “irrespective” should be used with some caution [@le2016introducing]. Indeed, the prior is apt to encapsulate any systematic tendency in the manner in which we explore scenes with our eyes. The striking result obtained by is that if we learn $P(\mathbf{r})$ from the actual observer’s behavior, then we can stochastically sample gaze shifts (Eq. \[eq:priorTatlersamp\]) so to obtain scan paths that, blind to visual information, out-perform feature-based accounts of eye guidance. Note that the apparent simplicity of the prior term $P(\mathbf{r})$ hides a number of subtleties. For instance, Tatler and Vincent expand the random vector $\mathbf{r}$ in terms of its components, amplitude $l$ and direction $\theta$. Thus, $P(\mathbf{r})= P(l, \theta)$. This simple statement paves the way to different options. First easy option: such RVs are marginally independent, thus, $P(l, \theta) = P(l) P(\theta)$. In this case, gaze guidance, solely relying on biases, could be simulated by expanding Eq. \[eq:priorTatlersamp\] via independent sampling of both components, i.e. at each time $t$, $l(t) \sim P(l(t)), \theta(t) \sim P(\theta(t))$. Alternative option: conjecture some kind of dependency, e.g. amplitude on direction so that $P(l, \theta) = P(l \mid \theta) P(\theta)$. In this case, the gaze shift sampling procedure would turn into the sequence $\widehat{\theta}(t) \sim P(\theta(t)), l(t) \sim P(l(t) \mid \widehat{\theta}(t) )$. Further: assume that there is some persistence in the direction of the shift, which give rise to a stochastic process in which subsequent directions are correlated, i.e., $\theta(t) \sim P(\theta(t) \mid \theta(t-1))$, and so on. To summarize, by simply taking into account the prior $P(\mathbf{r})$, a richness of possible behaviors and analyses are brought into the game. Unfortunately, most computational accounts of eye movements and visual attention have overlooked this opportunity, with some exceptions. For instance, propose a system model for saccade generation in a stochastic filtering framework. A prior on amplitude $P(l(t))$ is considered by learning a Gaussian mixture model from eye tracking data. This way one aspect of biases is indirectly taken into account. It is not clear if their model accounts for variability and whether and how oculomotor statistics compare to human data. In [@kimura2008dynamic], simple eye-movements patterns are straightforwardly incorporated as a prior of a dynamic Bayesian network to guide the sequence of eye focusing positions on videos. In a different vein, have recently addressed in-depth the bias problem and made the interesting point that viewing tendencies are not universal, but modulated by the semantic visual category of the stimulus. They learn the joint pdf $P(l, \theta)$ of saccade amplitudes and orientations via kernel density estimation; fixation duration is not taken into account. The model also brings in variability [@le2015saccadic] by generating a number $N_c$ of random locations according to conditional probability $P(\mathbf{r}_{F}(t) \mid \mathbf{r}_{F}(t-1))$ and the location with the highest saliency gain is chosen as the next fixation point. $N_c$ controls the degree of stochasticity. Others have tried to capture eye movements randomness [@keech2010eye1; @rutishauser2007probabilistic] but limiting to specific tasks such as conjunctive visual search. A few more exceptions can be found, but only in the very peculiar field of eye-movements in reading (see , for a discussion). The variability and bias issues have been explicitly addressed from first principles in the theoretical context of Lévy flights [@brockgeis; @bfpha04]. The perceptual component was limited to a minimal core (e.g., based on a bottom-up salience map) sufficient enough to support the eye guidance component. In particular in [@bfpha04] the long tail, positively skewed distribution of saccade amplitudes was shaped as a prior in the form of a Cauchy distribution, whilst randomness was addressed at the algorithmic level by prior sampling $\mathbf{r} \sim P(\mathbf{r}(t))$ followed by a Metropolis-like acceptance rule based on a deterministic saliency potential field. The degree of stochasticity was controlled via the “temperature” parameter of the Metropolis algorithm. The underlying eye guidance model was that of a random walker exploring the potential landscape (salience) according to a Langevin-like stochastic differential equation (SDE). The merit of such equation is the joint treatment of both the deterministic and the stochastic (variability) components behind eye guidance[^1]. This basic mechanism has been refined and generalized in [@BocFerAnnals2012] to composite $\alpha$-stable or Lévy random walks (the Cauchy law is but one instance of the class of $\alpha$-stable distributions), where, inspired by animal foraging behaviour, a twofold regime can be distinguished: local exploitation (fixational movements following Brownian motion) and large exploration/relocation (saccade following Lévy motion). What is interesting, with respect to the early model [@bfpha04], is that the choice between the “feed” or “fly” states is made by sampling from a Bernoulli distribution, $Bern(z \mid \pi)$, with the parameter $\pi$ sampled from the conjugate prior $Beta(\pi \mid \alpha, \beta)$. In turn, the behaviour of the Beta prior can be shaped via its hyperparameters $(\alpha, \beta)$, which, in an Empirical Bayes approximation, can be tuned as a function of the class of perceptual data at hand (in the vein of ) and of time spent in feeding (fixation duration). Most important, this approach paves the way to the possibility of treating visual exploration strategies in terms of *foraging* strategies [@wolfe2013time; @cain2012bayesian; @BocFerSMCB2013; @BocCOGN2014; @napboc_TIP2015]. We will further expand on this in Section \[sec:action\]. The unbearable lightness of the likelihood {#sec:like} ------------------------------------------ We noticed before, by inspecting Eq. \[eq:BayesTatler\] that the term $\frac{ P(\mathcal{W} \mid \mathbf{r} )} {P(\mathcal{W})} $ could be related to many models proposed in the literature. This is an optimistic view. Most of the approaches actually discard the dynamics of gaze shifts implicitly captured by the shift vector $\mathbf{r}(t)$. In practice, they are more likely to be described by a simplified version of Eq. \[eq:BayesTatler\]: $$P(\mathbf{r}_{F} \mid \mathcal{W}) = \frac{P(\mathcal{W} \mid \mathbf{r}_{F})} {P(\mathcal{W})} P(\mathbf{r}_{F}). \label{eq:BayesTatler2}$$ The difference between Eq. \[eq:BayesTatler\] and \[eq:BayesTatler2\] is subtle. The posterior $P(\mathbf{r}_{F} \mid \mathcal{W}) $ now answers the query “What is the probability of *fixating* at location $\mathbf{r}_{F}$ given visual data $\mathcal{W}$?” Further, the prior $P(\mathbf{r}_{F})$ simply accounts for the probability of spotting location $\mathbf{r}_{F}$. As a matter of fact, Eq. \[eq:BayesTatler2\] bears no dynamics. In probabilistic terms we may re-phrase this result as the outcome of an assumption of independence: $P(\mathbf{r}) = P(\mathbf{r}_{F}(t) - \mathbf{r}_{F}(t-1)) \nonumber \simeq P(\mathbf{r}_{F}(t) \mid \mathbf{r}_{F}(t-1)) = P(\mathbf{r}_{F}(t))$. To make things even clearer, let us explicitly substitute $\mathbf{r}_{F}$ with a RV $\mathbf{L}$ denoting locations in the scene, and $\mathcal{W}$ with RV $\mathbf{F}$ denoting features (whatever they may be); then, Eq. \[eq:BayesTatler2\] boils down to $$P(\mathbf{L} \mid \mathbf{F}) = \frac{P(\mathbf{F} \mid \mathbf{L})} {P(\mathbf{F})} P(\mathbf{L}). \label{eq:models}$$ The PGM underlying this inferential step is a very simple one and is represented in Figure \[fig:simple\]. A straightforward but principled use of Eq. \[eq:models\], which has been exploited by approaches that draw upon techniques borrowed from statistical machine learning [@murphy2012machine] is the following: consider $\mathbf{L}$ as a binary RV taking values in $\left[0,1\right]$ (or $\left[ -1, 1 \right]$), so that $P(\mathbf{L} = 1 \mid \mathbf{F})$ represents the probability for a pixel, a superpixel or a patch of being classified as salient. ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMsimple.jpg) ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMTorralba.jpg) ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMPoggio.jpg) ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMmy.jpg) \[Fig:models\] In the case the prior $P(\mathbf{L})$ is assumed to be uniform (no spatial bias, no preferred locations), then $ P(\mathbf{L} = 1 \mid \mathbf{F}) \simeq P( \mathbf{F} \mid \mathbf{L} = 1) $. The likelihood function $P( \mathbf{F} \mid \mathbf{L} = 1)$ can be determined in many ways; e.g., nonparametric kernel density estimation has been addressed by , who use center / surround local regression kernels for computing $\mathbf{F}$. More generally, taking into account the ratio $f(\mathbf{L})=\frac{P(\mathbf{L} = 1 \mid \mathbf{F})}{P(\mathbf{L} = 0 \mid \mathbf{F})}$ (or, commonly, the log-ratio) casts the saliency detection problem in a classification problem, in particular a discriminative one [@murphy2012machine], for which a variety of learning techniques are readily available. pioneered this approach by learning the saliency discriminant function $f(\mathbf{L})$ directly from human eye tracking data using a support vector machine (SVM). Their approach has paved the way to a relevant number of works from [@judd2009learning] – who trained a linear SVM from human fixation data using a set of low, middle and high-level features to define salient locations–, to most recent ones that wholeheartedly endorse machine learning trends. Henceforth, methods have been proposed relying on sparse representation of “feature words” (atoms) encoded in salient and non-salient dictionaries; these are either learned from local image patches [@yan2010visual; @lang2012saliency] or from eye tracking data of training images [@jiang2015image]. Graph-based learning is one other trend, from the seminal work of to (see the latter, for a brief review of this field). Crucially, for the research practice, data-driven learning methods allow to contend with large scale dynamic datasets. in the vein of and use SVM, but they remarkably exploit state-of-the art computer vision datasets (Hollywood-2 and UCF Sports) annotated with human eye movements collected under the ecological constraints of a visual action recognition task. As a general comment on (discriminative) machine learning-based methods, on the one hand it is embraceable the criticism by , who surmise that these techniques make “models data-dependent, thus influencing fair model comparison, slow, and to some extent, black-box.” But on the other hand, one important lesson of these approaches lies in that they provides a data-driven way of deriving the most relevant visual features as optimal predictors. The learned patterns can shape receptive fields (filters) that have equivalent or superior predictive power when compared against hand-crafted (and sometimes more complicated) models [@kienzle2009center]. Certainly, this lesson is at the base of the current exponentially growth of methods based on deep learning techniques [@lecun2015deep], in particular Convolutional Neural Networks (CNN, cfr. for a focused review), where the computed features seem to outperform, at least from an engineering perspective, most of, if not all, the state-of-the art features conceived in computer vision. Again, CNNs, as commonly exploited in the current practice, bring no significant conceptual novelty as to the use of Eq. \[eq:models\]: fixation prediction is formulated as a supervised binary classification problem (in some case, regression is addressed, ). For example, use a linear SVM for learning the saliency discriminant function $f(\mathbf{L})$ after a large-scale search for optimal features $\mathbf{F}$. Similarly, detect salient region via linear SVM fed with features computed from multi-layer sparse network model. [@lin2014saliency] use the simple normalization step [@IttiKoch98] to approximate $P(\mathbf{L} = 1 \mid \mathbf{F})$, where [@kruthiventi2015deepfix] use the last $1 \times 1$ convolutional layer of a fully convolutional net. Cogent here is the outstanding performance of CNN in learning and representing features that correlate well with eye fixations, like objects, faces, context. Clearly, one problem is the enormous amount of training data necessary to train these networks, and the engineering expertise required, which makes them difficult to apply for predicting saliency. However, by exploiting the well known network from [@AlexNetNIPS2012] as starting point, have given evidence that deep CNN trained on computer vision tasks like object detection boost saliency prediction. The network by has been optimized for object recognition using a massive dataset consisting of more than one million images, and results reported by on static pictures are impressive when compared to state-of-the-art methods, even to previous CNN-based proposals [@vig2014large]. Apart from the straightforward implementation via popular machine-learning algorithms, the “light” model described by Eq. \[eq:models\] is further amenable to a minimal model, which, surprisingly enough, is however capable of accounting for a large number of approaches. This can be easily appreciated by setting $P(\mathbf{F} \mid \mathbf{L}) = const., P(\mathbf{L})=const.$ so that Eq. \[eq:models\] reduces to $$P(\mathbf{L} \mid \mathbf{F}) \propto \frac{1}{P(\mathbf{F})}. \label{eq:modelItti}$$ Eq. \[eq:modelItti\] states that the probability of fixating a spatial location $\mathbf{L}= (x,y)$ is higher when “unlikely” features (unlikeliness $ \approx \frac{1} {P(\mathbf{F})}$) occur at that location. In a natural scene, it is typically the case of high contrast regions (with respect to either luminance, color, texture or motion). This is nothing but the salience-based component of the most prominent model in the literature [@IttiKoch98], which Eq. \[eq:modelItti\] re-phrases in probabilistic terms. A thorough reading of the review by is sufficient to gain the understanding that a great deal of computational models so far proposed (47 over 63 models) are much or less variations of this theme (albeit experimenting with different features, different weights for combining them, etc.) even when sophisticated probabilistic techniques are adopted to shape the distribution $P(\mathbf{F})$ (e.g., nonparametric Bayes techniques, ). Clearly, there are works that have tried to avoid weaknesses related to such a light-modelling of the perceptual input, and have tried to climb up the levels of the representation hierarchy [@schutz2011eye]. Some examples are summarized at a glance in Figure \[Fig:models\] (but see ). Nevertheless, in spite of its simplicity, Eq. \[eq:modelItti\] is apt to pave the way to interesting frameworks. For instance, by noting that $\log \frac{1} {P(\mathbf{F})}$ is nothing but Shannon’s Self- Information, information theoretic approaches become available at the algorithmic level. These approaches set computational constraints under the general assumption that saliency computation serves to maximize information sampled from the environment [@bruce2009saliency]. Keeping on with the information theory framework, and going back to Eq. \[eq:models\], a simple manipulation, $$\log P(\mathbf{L} \mid \mathbf{F}) - \log P(\mathbf{L}) = \log \frac{P(\mathbf{F} \mid \mathbf{L})} {P(\mathbf{F})} \label{eq:modelsKL}$$ sets the focus on the discrepancy, or dissimilarity, $\log P(\mathbf{L} \mid \mathbf{F}) - \log P(\mathbf{L}) = \log \frac{P(\mathbf{L} \mid \mathbf{F})}{P(\mathbf{L})}$ between the log-posterior and the log-prior. A (non-commutative) measure, formalizing this notion of dissimilarity is readily available in information theory, namely the Kullback-Leibler (K-L) divergence between two distributions $P(X)$ and $Q(X)$ [@Mackay]: $$D_{KL}(P(X) || Q(X))= \int_X \log \frac{P(x)} {Q(x)} P(x) dx \label{eq:KL}$$ Measuring differences between posterior and prior beliefs of the observers is however a general concept applicable across different levels of abstraction. For instance, one might consider the object-based model [@torralba2006contextual] in Fig. \[fig:torralba\], which can be used for inferring the joint posterior of gazing at certain kinds of objects $\mathbf{O}$ at location $\mathbf{L}$ of a viewed scene, namely, $P(\mathbf{O},\mathbf{L} \mid \mathbf{F}) \propto P(\mathbf{F} \mid \mathbf{O},\mathbf{L}) P(\mathbf{O},\mathbf{L})$. Then, $D_{KL}(P(\mathbf{O},\mathbf{L} \mid \mathbf{F}) || P(\mathbf{O},\mathbf{L}))$ is the average of the log-odd ratio, measuring the divergence between observer’s prior belief distribution on $(\mathbf{O},\mathbf{L})$ and his posterior belief distributions after perceptual data $\mathbf{F}$ have been gathered. Indeed, this is a statement that can be generalized to any model $\mathbf{M}$ in a model space $\mathcal{M}$ and new data observation $\mathbf{D}$ so to define the Bayesian surprise [@balditti2010] $\mathcal{S}(\mathbf{D},\mathbf{M})$: $\mathbf{D}$ is surprising if the posterior distribution resulting from observing $\mathbf{D}$ significantly differs from the prior distribution, i.e., $S(\mathbf{D},\mathbf{M}) = D_{KL}(P(\mathbf{M} \mid \mathbf{D} ) || P(\mathbf{M}))$. have shown that Bayesian surprise attracts human attention in dynamic natural scenes. To recap, Bayesian surprise is a measure of salience based on the K–L divergence. Eventually, note that the K-L divergence (\[eq:KL\]) is a flexible tool and can be used for different purposes. For instance, when dealing with models of perceptual evaluation such as those specified in Figs \[fig:torralba\], \[fig:poggio\], and \[fig:myPGM\], once the model has been detailed at the computational theory level via its PGM, then using the latter for learning inference and prediction brings in the algorithmic level. Indeed, for any Bayesian generative model other than trivial ones, such steps are usually performed in approximate form [@murphy2012machine]. Stochastic approximation resorting to algorithms such as Markov-chain Monte Carlo (MCMC) and Particle Filtering (PF) is one possible choice; the alternative choice is represented by deterministic optimization algorithms [@murphy2012machine] such as variational Bayes (VB) or belief propagation (BP, a message passing scheme exchanging beliefs between PGM nodes). For example, the model by , following the work of , relies upon BP message passing for inferential steps. Interestingly enough, has argued for a plausible neural implementation of BP. VB algorithms, on the other hand, are based on Eq. \[eq:KL\], where $P$ usually stands for a complete distribution and $Q$ is the approximating distribution; then, parameter (or model) learning is accomplished by minimizing the K-L divergence (as an example, the well known Expectation-Maximization algorithm, EM, can be considered a specific case of the VB algorithm, ). In Section \[sec:action\] we will also touch on a deeper interpretation of the K-L minimization / VB algorithm. But at this point a simple question arises: where have the eye movements gone? Making a step forward: back to the beginning of active vision {#sec:action} ============================================================= Visual perception coupled with gaze shifts should be considered the *Drosophila* of perception-action loops. Among the variety of active behaviors the organism can fluently engage to purposively act upon and perceive the world (e.g, moving the body, turning the head, manipulating objects), oculomotor behavior is the minimal, least energy, unit. To perform $3$-$4$ saccades per second, the organism roughly spends $300$ msecs to close the loop ($200$ msecs for motor preparation and execution, $100$ msecs left for perception). ![The perception-action loop unfolded in time as a dynamic PGM. $\mathcal{A}(t)$ denotes the ensemble of time-varying RVs defining the oculomotor action setting; $\mathcal{W}(t)$ stands for the ensemble of time-varying RVs characterising the scene as actively perceived by the observer; $\mathcal{G}$ summarizes the given goal(s), To simplify the graphics, conditional dependencies $\mathcal{G} \rightarrow \mathcal{A}(t+1)$ and $\mathcal{G} \rightarrow \mathcal{W}(t+1)$ have been omitted.[]{data-label="Fig:loop"}](FigLoop.jpg) One way to make justice of this forgotten link is going back to first principles by re-shaping the problem as the action-perception loop, which is presented in Figure \[Fig:loop\] in the form of a dynamic PGM. The model relies upon the following assumptions: - The scene that will be perceived at time $t+1$, namely $\mathcal{W}(t+1)$ is inferred from the raw data $\mathbf{I}$, gazed at $\mathbf{r}_{F}(t+1)$, under the goal $\mathcal{G}$ assigned to the observer, and is conditionally dependent on current perception $\mathcal{W}(t)$. Thus, the perceptual inference problem is summarised by the conditional distribution $P( \mathcal{W}(t+1)|\mathcal{W}(t), \mathbf{r}_{F}(t+1), \mathbf{I},\mathcal{G})$; - The external goal $\mathcal{G}$ being assigned, the oculomotor action setting at time $t+1$, $\mathcal{A}(t+1)$, is drawn conditionally on current action setting $\mathcal{A}(t)$ and the perceived scene $\mathcal{W}(t+1)$ under gaze position $\mathbf{r}_{F}(t+1)$; thus, its evolution in time is inferred according to the conditional distribution $P(\mathcal{A}(t+1) | \mathcal{A}(t), \mathcal{W}(t+1), \mathbf{r}_{F}(t+1), \mathcal{G})$. Note that the action setting dynamics $\mathcal{A}(t) \rightarrow \mathcal{A}(t+1)$ and the scene perception dynamics $\mathcal{W}(t) \rightarrow \mathcal{W}(t+1)$ are intertwined with one another by means of the gaze shift process $\mathbf{r}_{F}(t) \rightarrow \mathbf{r}_{F}(t+1)$: on the one hand next gaze position $\mathbf{r}_{F}(t+1)$ is used to define a distribution on $\mathcal{W}(t+1)$ and $\mathcal{A}(t+1)$; meanwhile, the probability distribution of $\mathbf{r}_{F}(t+1)$ is conditioned on current gaze position, $\mathcal{W}(t)$ and $\mathcal{A}(t)$, namely $P(\mathbf{r}_{F}(t+1)| \mathcal{A}(t), \mathcal{W}(t), \mathbf{r}_{F}(t))$. We have previously discussed the perceptual evaluation component $\mathcal{W}(t)$. A general way of defining the oculomotor executive control component $\mathcal{A}(t)$ is through the following ensemble of RVs: - $\{\mathbf{V}(t), \mathbf{R}(t)\}$: $\mathbf{V}(t)$ is a spatially defined RV used to provide a suitable probabilistic representation of value; $\mathbf{R}(t)$ is a binary RV defining whether or not a payoff (either positive or negative) is returned; - $\{\pi(t), z(t), \xi(t) \}$: an *oculomotor state representation* as defined via the multinomial RV $z(t)$, occurring with probability $\pi(t)$, and determining the choice of motor parameters $\xi(z,t)$ guiding the actual gaze relocation (e.g., lenght and direction of a saccade as opposed to those driving a smooth pursuit) ; - $\mathcal{D}(t)$: a set of state-dependent statistical decision rules to be applied on a set of candidate new gaze locations $\mathbf{r}_{new}(t+1)$ distributed according to the posterior pdf of $\mathbf{r}_{F}(t+1)$. In the end, the actual shift can be summarised as the statistical decision of selecting a particular gaze location $\mathbf{r}^{\star}_{F}(t+1)$ on the basis of $P(\mathbf{r}_{F}(t+1)| \mathcal{A}(t), \mathcal{W}(t), \mathbf{r}_{F}(t))$ so to maximize the expected payoff under the current goal $\mathcal{G}$, and the action/perception cycle boils down to the iteration of the following steps: 1. Sampling the gaze-dependent current perception: $$\mathcal{W}^{*}(t) \sim P( \mathcal{W}(t)|\mathbf{r}_{F}(t),\mathbf{F}(t), \mathbf{I}(t),\mathcal{G}); \label{eq:step1}$$ 2. Sampling the appropriate motor behavior (e.g., fixation or saccade): $$\mathcal{A}(t)^{*} \sim P(\mathcal{A}(t) | \mathcal{A}(t-1), \mathcal{W}^{*}(t),\mathcal{G}); \label{eq:step2}$$ 3. Sampling where to look next: $$\mathbf{r}_{F}(t+1) \sim P(\mathbf{r}_{F}(t+1)| \mathcal{A}(t)^{*} , \mathcal{W}^{*}(t), \mathbf{r}_{F}(t)). \label{eq:step3}$$ It is worth noticing that we have chosen to describe the observer’s action / perception cycle in terms of stochastic sampling based on the probabilistic model in Figure \[Fig:loop\]. However, one can recast the inferential problems in terms of deterministic optimization: in brief, optimising the probabilistic model $\mathcal{W}$ of how sensations are caused, so that the resulting predictions can select the optimal $\mathcal{A}$ to guide active sampling (gaze shift) of sensory data. One such approach, which is well known in theoretical neuroscience but, surprisingly, hitherto unconsidered in computer vision, relies on the free-energy principle [@friston2010free; @feldman2010attention; @friston2013anatomy]. Free-energy $\mathcal{F}$ is a quantity from statistical physics and information theory [@Mackay] that bounds the negative log-evidence of sensory data. Under simplifying assumptions, it boils down to the amount of prediction error of sensory data under a model. In such context, the action / perception cycle is the result of a dual minimization process: i) action reduces $\mathcal{F}$ by changing sensory input, namely by sampling (via gaze shifts) what one expects consistent with perceptual inferences; ii) perception reduces $\mathcal{F}$ by making inferences about the causes of sampled sensory signals and changing predictions. Friston defines this process “active inference”. To make a connection with Section \[sec:like\], by minimizing the free-energy, Bayesian surprise is maximised; indeed, in Bayesian learning, free energy minimization is a common rationale behind many optimisation techniques such as VB and BP [@Mackay; @murphy2012machine] The sampling scheme proposed is a general one and can be instantiated in different ways. For instance, in [@BocFerSMCB2013] the sampling step of Eq. \[eq:step3\] is performed through a generalization of the Langevin SDE equation used in [@BocFerAnnals2012]. Biases and variability are accounted for by the stochastic component of the equation. Since dealing with image sequences, the SDE provides operates in different dynamic modes: pursuit needs to be taken also into account in addition to saccades and fixational movements. Each mode is governed by a specific set of parameters of the $\alpha$-stable distribution estimated from eye tracking data. The choice among modes is accomplished by generalizing the method proposed in [@BocFerSMCB2013], using a Multinoulli distribution on $z(t)$ and with parameters $\pi(t)$ sampled from the conjugate prior, the Dirichlet distribution. In addition, the sampling step is used to accomplish an internal simulation step, where a number of candidates shifts is proposed and the most convenient is selected according to a decision rule. Also, the gaze dependent perception $\mathcal{W}$ can be modeled at any level of complexity (cfr. example in Figure \[Fig:levels\], Section \[sec:emo\]). In it is based on proto-objects sampled from time-varying saliency[^2]. Figure \[Fig:monica\] shows an excerpt of typical results of the model simulation, which compares human variability in gazing (top row) with that of two “simulated observers” (bottom rows) while viewing the `monica03` clip from the CRCNS eye-1 dataset. ![Top row: gaze positions (centers of the enhanced colored circles ) recorded from different observers on different frames of the `monica03` clip from the CRCNS eye-1 dataset, University of South California (freely available online). Middle and bottom rows show, on the same frames, the fovea position of two “observers” simulated by the method described in [@BocFerSMCB2013].[]{data-label="Fig:monica"}](Figmonica03.jpg) In and the perceptual evaluation component is extended to handle objects and task (external goal) levels by expanding on (cfr. Figure \[Fig:models\]), and the decision rule $\mathcal{D}(t)$ concerning the selection of the gaze is based on the expected reward according to the given goal $\mathcal{G}$ (see Section \[sec:emo\]). Meanwhile, nothing prevents to conceive more general perceptual evaluation and executive control components, by considering perceptual and action modalities other than the visual ones in the vein of , where eye movements and hand actions have been coupled with the goal of performing a drawing task. But most important, the action-perception cycle is by and large conceived in the foraging framework (see , for a thorough introduction), which at the most general level is summarized in Table \[metaphor\]. Visual foraging corresponds to the time-varying overt deployment of visual attention achieved through oculomotor actions, namely, gaze shifts. The forager feeds on patchily distributed preys or resources, spends its time traveling between patches or searching and handling food within patches. While searching, it gradually depletes the food, hence, the benefit of staying in the patch is likely to gradually diminish with time. Moment to moment, striving to maximize its foraging efficiency and energy intake, the forager should make decisions: Which is the best patch to search? Which prey, if any, should be chased within the patch? When to leave the current patch for a richer one? The spatial behavioral patterns exhibited by foraging animals (but also those detected in human mobility data) are remarkably close to those generated by gaze shifts [@viswanathan2011physics]. Figure \[Fig:spider\] presents an intriguing example in this respect. ![Monkey or human: can you tell the difference? The center image has been obtained by superimposing a typical trajectory of spider monkeys foraging in the forest of the Mexican Yucatan, as derived from [@ramos2004levy], on the “party picture” (left image) used in [@brockgeis]. The right image is an actual human scan path (modified after ) []{data-label="Fig:spider"}](FigParty.jpg) The fact that the physics underlying foraging overlaps with that of several other kinds of complex random searches and stochastic optimization problems [@viswanathan2011physics], and notably with that of visual exploration via gaze shifts [@viswanathan2011physics; @marlow2015temporal; @brockgeis], makes available a variety of analytical tools beyond classic metrics exploited in computer vision or psychology. For instance, in Figure \[Fig:ccdf\], it is shown how the gaze shift amplitude modes from human observers can be compared with those generated via simulation by using the complementary Cumulative Distribution Function (CCDF), which provides a precise description of the distribution of the gaze shift by considering its upper tail behavior. This can be defined as $\overline{F}(x)=P(| \mathbf{r}|>x)=1 - F(x)$, where $F$ is the cumulative distribution function (CDF) of amplitudes. Consideration of the upper tail, i.e. the CCDF of jump lengths is a standard convention in foraging, human mobility, and anomalous diffusion research [@viswanathan2011physics]. To sum up, foraging offers a novel perspective for formulating models and related evaluations of visual attention and oculomotor behavior [@wolfe2013time]. Unifying hypotheses such as the oculomotor continuum from exploration to fixation by can be reconsidered in the light of fundamental theorems of statistical mechanics [@weron2010generalization]. \[metaphor\] Interestingly enough, the reformulation of visual attention in terms of foraging theory is not simply an informing metaphor. It has been argued that what was once foraging for tangible resources in a physical space became, over evolutionary time, foraging in cognitive space for information related to those resources [@hills2006animal], and such adaptations play a fundamental role in goal-directed deployment of visual attention [@wolfe2013time]. Bringing value into the game: a doorway to affective modulation {#sec:emo} =============================================================== The introduction of a goal level, either exogenous (originating from outside the observer’s organism) or endogenous (internal) is not an innocent shift. From a classical cognitive perspective, the assignment of a task to the observer implicitly defines a value for every point of the space, in the sense that information in some points is more relevant than in others for the completion of the task; the shifting of the gaze on a particular point, in turn, determines the payoff that can be gained. There is a number of psychological and neurobiological studies showing the availability of value maps and loci of reward influencing the final gaze shift [@platt1999neural; @leon1999effect; @ikeda2003reward; @Hikosaka2006]. The payoff is nothing else that the value, with respect to the completion of the task, obtained by moving the fovea in a given position. Thus points associated with high values produce, when fixated, high payoffs since these fixations bring the observer closer to her/his goal. For instance, in [@BocCOGN2014] and in [@napboc_TIP2015], reward was introduced to make a choice among the candidate gaze shifts stochastically sampled according to Eq. \[eq:step3\], in terms of expected reward (e.g., tuned by the probability of finding the task-assigned object). Figure \[Fig:levels\] presents one example of the different scan paths obtained by progressively reducing the levels of representation in the perceptual evaluation component presented in Figure \[fig:myPGM\] [@BocCOGN2014; @napboc_TIP2015]. ![Different scan paths originated from the progressive reduction of representation levels in the perceptual evaluation component presented in Figure \[fig:myPGM\].[]{data-label="Fig:levels"}](FigLevels.jpg) Value set by the given goal can weight differently the objects within the scene, thus purposively biasing the scan path. Gaze is still uniformly deployed to relevant items within the scene (people, text) when object-based representation is exploited. When salience alone is used, the generated scan path fails in accounting for the relevant items and bears no relation with the semantics which can be attributed to the scene. The use of value and reward endow attentive models with the capability of handling complex task. For instance, considered the goal of text spotting in unconstrained urban environments, and its validation encompassed data gathered from a mobile eye tracking device [@clav2014]. extended the foraging framework to cope with the difficult problem of attentive monitoring of multiple video streams in a videosurveillance setting. Yet, developing eye guidance models based on reward is a difficult endeavour and computational models that use reward and uncertainty as central components are still in their infancy (but see the discussion by ). In this respect the remarkable work by Ballard and collegues counters the stream. Whilst salience, proto-objects and objects are representations that have been largely addressed in the context of human eye movements, albeit with different emphasis, in contrast, value has been neglected until recently [@schutz2011eye]. One reason is that in the real world there is seldom direct payoff (no orange juice for a primary reward) for making good eye movements or punishment for bad ones. However, the high attentional priority of ecologically pertinent stimuli can also be explained by mechanisms that do not implicate learning value through repeated pairings with reward. For example, a bias to attend to socially relevant stimuli is evident from infancy [@anderson2013value]. More generally, the selection of stimuli by attention has important implications for the survival and wellbeing of an organism, and attentional priority reflects the overall value of such selection (see for a discussion). Indeed, engaging attention with potentially harmful and beneficial stimuli guarantees that the relevant ones are selected early so to gauge the exact nature of the potential threat or opportunity and to readily initiate defensive or approach behavior. Under these circumstances, has proposed a broad definition of reward, which includes “not only the immediate primary rewards, but also other factors: the preference for a novel location or stimulus, the satisfaction of performing well or the desire to complete a given task." Such definition is consistent with the different psychological facets of reward [@berridge2003parsing]: i) learning (including explicit and implicit knowledge produced by associative conditioning and cognitive processes); ii) affect or emotion (implicit “liking” and conscious pleasure); iii) motivation (implicit incentive salience “wanting” and cognitive incentive goals). Thus, value representation level is central to both goal-driven affective and cognitive engagement with stimuli in the outside world. In this broader perspective, the effort to put value and reward into the game shows his inner worth in that, by accounting for the many aspects of “biological value" - salience, significance, unpredictability, affective content - , it paves the way to a wider dimension of information processing, as most recent results on the affective modulation of the visual processing stream advocate [@pessoa2008relationship; @pessoa2010emotion], and to the effective exploitation of computational attention models in the emerging domain of social signal processing [@vinciarelli2009social]. A number of important studies in the psychological literature (see, for a discussion, ) have addressed the relationship between overt attention behavior and emotional content of pictures. Many of them investigate specific issues related to individuals such as trait anxiety, social anxiety, spider phobia, and exploit restricted sets of stimuli such as emotional faces or spiders. In turn, the study by has exploited natural images and normal subjects, demonstrating an emotional bias both in attentional orienting and engagement among normal participants and using a wider range of emotional pictures. Results might be summarised as follows: i) emotionally pleasant and unpleasant pictures capture attention more readily than neutral pictures; ii) the emotional bias can be observed early in initial orienting and subsequent engagement of attention; iii) the early stimulus-driven attentional capture by emotional stimuli can be counteracted by goal-driven control in later stages of picture processing. Peculiarly relevant to our case, have shown that visual saliency does influence eye movements, but the effect is reliably reduced when an emotional object is present. Pictures containing negative objects were recognized more accurately and recalled in greater detail, and participants fixated more on negative objects than positive or neutral ones. Initial fixations were more likely to be on emotional objects than more visually salient neutral ones. Consistently with , the overall result suggest that the processing of emotional features occurs at a very early stage of perception. As a matter of fact, emotional factors are completely neglected in the realm of computational models of attention and gaze shifts. Some efforts have been spent in the field of social robotics, where motivational drives have an indirect influence on attention by influencing the behavioral context, which, in turn, is used to directly manipulate the gains of the attention system (e.g. by tuning the gains of different bottom-up saliency map, ). Yet, beyond these broadly related attempts, taking into account the specific influence of affect on eye-behaviour is not a central concern within this field (for a wide review, see ). Recent works in the image processing and pattern recognition community use eye tracking data for the inverse problem of the recognition of emotional content of images (e.g.,) or implicit tagging of videos [@soleymani2012multimodal]. Thus, they do not address the generative problem of how emotional factors contribute to the generation of gaze shifts in visual tasks. By contrast, neuroscience has shown that, crucially, cognitive and emotional contributions cannot be separated, as outlined in Figure \[Fig:neuro\]. ![Circuits for the processing of visual information and for the executive control [@pessoa2008relationship] show that action and perception components are inextricably linked through the mediation of the amygdala and the posterior orbitofrontal cortex (OFC). These structures are considered to be the neural substrate of emotional valence and arousal [@salzman2010emotion]. The ventral tegmental area (vTA) and the basal forebrain are responsible for diffuse neuromodulatory effects related to reward and attention. Anterior cingulate cortex (ACC) is likely to be involved in conflict detection and/or error monitoring but also in in computing the benefits and costs of acting by encoding the probability of reward. Nucleus accumbens is usually related to motivation.The lateral prefrontal cortex (LPFC) plays a central role at the cognitive level in maintaining and manipulating information, but also integrates this content with both affective and motivational information. Line thickness depicts approximate connection strength[]{data-label="Fig:neuro"}](FigNeuro.jpg) The scheme presented, which summarises an ongoing debate [@pessoa2008relationship], shows that responses from early and late visual cortex reflecting stimulus significance will be a result of simultaneous top-down modulation from fronto-parietal attentional regions (LPFC) and emotional modulation from the amygdala [@Mohanty2013]. On the one hand, stimulus’ affective value appears to drive attention and enhance the processing of emotionally modulated information. On the other hand, exogenously driven attention influences the outcome of affectively significant stimuli [@pessoa2008relationship]. As a prominent result, the cognitive or affective origin of the modulation is lost and stimulus’ effect on behaviour is both cognitive and emotional. At the same time, the cognitive control system (LPFC, ACC) guides behaviour while maintaining and manipulating goal-related information; however strategies for action dynamically incorporate value through the mediation of the nucleus accumbens, the amygdala, and the OFC. Eventually, basal forebrain cholinergic neurons provide regulation of arousal and attention [@goard2009basal], while dopamine neurons located in the vTA modulate the prediction and expectation of future rewards [@pessoa2008relationship]. It is to be noted in Figure \[Fig:neuro\] the central role of the amygdala and the OFC. It has been argued [@salzman2010emotion] that their tight interaction provides a suitable ground for representing, at the psychological level the core affect dimensions [@russell2003core] of valence (pleasure–displeasure conveyed by the visual stimuli) and arousal (activation–deactivation). From a computational standpoint, the observer’s core affect can in principle be modelled as a dynamic latent space [@vitale2014affective], which we surmise might be readily embedded within the loop as proposed in Fig. \[Fig:newloop\]. This way, gaze shifts would benefit from the crucial emotional mediation between the action control and perceptual evaluation components. ![The perception-action loop unfolded in time as a dynamic PGM as in Figure \[Fig:loop\], but integrated with the core affect latent dimension abstracted as the RV $\mathcal{E}(t)$. $\mathcal{C}(t)$ stands for a higher cognitive control RV. To simplify the graphics, conditional dependencies $\mathcal{G} \rightarrow \mathcal{A}(t+1)$ and $\mathcal{G} \rightarrow \mathcal{W}(t+1)$ have been omitted.[]{data-label="Fig:newloop"}](FigAffectLoop2.jpg) Conclusion ========== Time is ripe to abandon the marshland of mass production and evaluation of bottom-up saliency techniques. Such boundless effort is partially based on a fatally flawed assumption [@santini2008context]: that visual data have a meaning *per se*, which can be derived as a function of a certain representation of the data themselves. Meaning is an outcome of an interpretative process rather than a property of the viewed scene. It is the act of perceiving, contextual and situated, that gives a scene its meaning [@wittgenstein2010philosophical]. The course of modelling can be more fruitfully directed not only to climb the hierarchy of representation levels and to cope with overlooked aspects of eye guidance, but to eventually reappraise the observer within his natural setting: an active observer compelled to purposively exploit visual attention for accomplishing real-world tasks [@maurice1945phenomenologie]. [^1]: Matlab simulation is available for download at <http://www.mathworks.com/matlabcentral/fileexchange/38512-visual-scanpaths-via-constrained-levy-exploration-of-a-saliency-landscape> [^2]: Matlab simulation is available for download at <https://www.researchgate.net/publication/290816849_Ecological_sampling_of_gaze_shifts_Matlab_code>

--- abstract: 'We introduce a minimal model for a collection of self-propelled apolar active particles, also called as ‘active nematic’, on a two-dimensional substrate and study the order-disorder transition with the variation of density. The particles interact with their neighbours within the framework of the Lebwohl-Lasher model and move asymmetrically, along their orientation, to unoccupied nearest neighbour lattice sites. At a density lower than the equilibrium isotropic-nematic transition density, the active nematic shows a first order transition from the isotropic state to a banded state. The banded state extends over a range of density, and the scalar order parameter of the system shows a plateau like behaviour, similar to that of the magnetic systems. In the large density limit the active nematic shows a bistable behaviour between a homogeneous ordered state with global ordering and an inhomogeneous mixed state with local ordering. The study of the above phases with density variation is scant and gives significant insight of complex behaviours of many biological systems.' author: - Rakesh Das - Manoranjan Kumar - Shradha Mishra title: Density Induced Phases in Active Nematic --- [*Introduction*]{} :— Active systems are composed of [*self-propelled*]{} particles where each particle extracts energy from its surroundings and dissipates it through motion towards a direction determined by its orientation. These kind of systems are ubiquitous in nature, ranging from very small scale systems inside the cell to larger scales [@harada; @nedelec; @rauch; @benjacob; @animalgroups; @helbing; @feder; @kuusela31; @hubbard], vibrated granular media [@vnarayan; @kudrolli] etc., and have been studied extensively through experiments, theories and simulations [@sriramrmp; @tonertusr; @rev]. A collection of head-tail symmetric ‘apolar’ active particles with an average mutual parallel alignment is said to be in a ‘nematic’ state, whereas in an ‘isotropic’ state particles remain randomly oriented. An active system where fluid media do not play important role in emergence of ordered state, and thus the hydrodynamic interactions can be ignored, is called a ‘dry active system’ [@kemkemer; @vnarayan; @animalgroups; @serra; @schaller; @surrey]. Active nature of particles introduces a nonequilibrium coupling between density and orientation field, as represented in terms of curvature coupling current in literature [@sradititoner; @shradhanjop; @sriramrmp]. Such coupling in active nematic induces unusual properties like large density fluctuation [@sradititoner; @chateprl2006] and growth kinetics faster than $1/3$ as in usual conserved model [@shradhatrans2014]. Recent studies of the active nematic found a defect-ordered nematic state [@aparnaredner; @shimanatcomm; @yeomans] as opposed to the equilibrium nematic for high particle densities. Recent experiment on amolyiod flibrils [@ncommam] also found a phase with coexisting aligned and disordered fibril domains, similar to the defect-ordered state obtained in simulations. But few investigations have been done on the behaviours of the active nematic in various density limits, especially at low densities. Here we introduce a minimal model for two-dimensional active nematic and compare various ordering phases of active and equilibrium nematic in different density limits. The ordering in the system is characterised in terms of a scalar order parameter $S$ which is the positive eigen value of nematic order parameter $\Q$ [@pgdgenne] in two-dimensions. In the low density limit both active and equilibrium systems are in the isotropic (I) state with particles randomly oriented throughout the whole system (see Fig. \[fig:phase\_snap\](b) - I), resulting in a small $S$. The Phase diagram of the active nematic as a function of packing density $C$ (see Fig. \[fig:phase\_snap\](a)) shows a jump in $S$ close to $C=0.37$, whereas in the equilibrium case $S$ goes continuously to larger values and an isotropic to nematic (I-N) transition occurs close to $C=0.58$. In the equilibrium nematic (EN) state particles remain homogeneously oriented in the system (see Fig. \[fig:phase\_snap\](b) - EN). At $C=0.37$ the active system goes from the isotropic to a banded state (BS) where particles cluster and align in the perpendicular direction to the long axis of the band (see Fig. \[fig:phase\_snap\](b) - BS-1). With increasing density more number of particles participate in band formation (see Fig. \[fig:phase\_snap\](b) - BS-2) and $S$ follows a plateau over a range of density. In the large density limit active system shows bistability between a homogeneous ordered (HO) (see Fig. \[fig:phase\_snap\](b) - HO) and an inhomogeneous mixed (IM) or local ordered state (see Fig. \[fig:phase\_snap\](b) - IM). This IM state is very similar to defect-ordered nematic state in ref. [@aparnaredner; @shimanatcomm; @yeomans]. [*Model*]{} :— We consider a two dimensional square lattice. At each vertex ‘$i$’ we define an occupation variable $n_i$, which can take values $1$ (occupied) or $0$ (unoccupied), and an orientation variable $\theta_i$, which lies between $0$ and $\pi$ because of the apolar nature of the particles. Each particle interacts with its nearest neighbours through modified Lebwohl - Lasher Hamiltonian [@llasher] $$\mathcal{H} = -\epsilon \sum_{<ij>}n_i n_j \cos2(\theta_i-\theta_j) \label{eqll}$$ where $\epsilon$ is the interaction strength between two neighbouring particles. This model is analogous to the diluted XY-model with nonmagnetic impurities [@dilutedxymodel], where impurities and spins are analogous to vacancies and particles respectively in the present model. Orientation evolves through Monte - Carlo (MC) updates [@mcbinder] following the Hamiltonian in Eq. \[eqll\]. Unlike the diluted XY-model, particles also move on the lattice. Depending on the type of movement we define two kinds of models. (i) ‘Equilibrium model’ (EM) - a particle can diffuse to any unoccupied nearest-neighbouring site, and therefore satisfies the detailed balance condition. (ii) ‘Active model’ (AM) - a particle can move to only those unoccupied nearest-neighbouring sites which are in the direction that makes the least inclination with the particle orientation. Details of the model and particle movement are shown in Supplemental Material [@SM] section I. The asymmetric move of the active particles does not staisfy the detailed balance condition and arises in general because of the self-propelled nature of the particles in many biological [@kemkemer; @paxton] and granular systems [@vnarayan]. These moves produce an active curvature coupling current in coarse-grained hydrodynamic equations of motion [@shradhanjop; @sradititoner]. [*Numerical details*]{} :— We consider a collection of $N$ particles with random orientation $\theta_i \in [0,\pi]$, homogeneously distributed on a $L \times L$ square lattice ($L=150, 256, 512$) with periodic boundary condition. The packing density of the system is $C=N/(L \times L)$. We choose a particle randomly and move it to an unoccupied neighbouring site, followed by an orientation updation through MC. We use $10^6$ MC steps to evolve the system to its steady state and all the results have been obtained by averaging over next $2 \times 10^6$ MC steps. Twenty four realizations have been used for better statistics. We calculate the scalar order parameter $$S=\sqrt{(\frac{1}{N}\sum_i n_i \cos(2 \theta_i))^2+(\frac{1}{N}\sum_i n_i \sin(2 \theta_i))^2} \label{eqops}$$ which is small in the isotropic state and close to $1$ in the ordered state. First we calculate $S$ for EM as a function of inverse temperature $\beta= 1 / k_BT$ for different densities. As shown in Supplemental Material [@SM] section II, the critical temperature $T_c$ is approximated as $T_c(S=0.4)$. Critical temperature $T_c(C)$ decreases with the lowering of the packing density $C$ , similar trend is found in the study of diluted XY-model [@dilutedxymodel] for varying nonmagnetic site density. In rest of our calculations temperature is kept fixed at $\beta\epsilon = 2.0$ and packing density $C$ is varied from small values to complete filling $C=1.0$. [*Phase diagram*]{} :— At low densities, $C<0.37$, the active system is in the isotropic state where the particles with random orientation remain homogeneously distributed throughout the system, and therefore $S$ holds vanishingly small values. The jump occurs in $S$ at $C=0.37$. For $C \geq 0.37$ particles cluster in, and both ordered state with high local density and disordered state with low local density coexist (see Fig. \[fig:phase\_snap\](b) - BS-1). Mean alignment inside the band is perpendicular to the long axis of the band. In the moderate density limits, band formation in more favourable than lane formation (mean alignment parallel to the long axis of the structure) because the number of particles that can have translational motion is much larger in the banded state, and therefore entropy favours band formation. Similar mechanism create the bending instability close to order-disorder transition in the active polar systems [@chate; @shradhapre]. The above transition from I state to BS occurs at density lower than the corresponding equilibrium I-N transition density $C \simeq 0.58$ (see Fig. \[fig:phase\_snap\](a)). As we further increase $C$, unlike the equilibrium system where $S$ increases monotonically with $C$, the active system shows very small change in $S$ for a range of density. This plateau like appearance of $S$ with variation in $C$ is very similar to plateau phase in magnetization versus field curve of magnetic systems [@plateauphase]. If an energy gap exists between two consecutive magnetic states, a finite field is required for the magnetic system to go from the lower to the higher state. So until that finite field is applied, the increasing field keeps the system magnetization to be unchanged, and the system is called to be in the plateau phase. With increasing packing density in the plateau regime of the active nematic more particles participate in band formation (see Fig. \[fig:phase\_snap\](b) - BS-1 and BS-2). On further increment of density, close to equilibrium I-N transition $C \simeq 0.58$, transverse fluctuations lead the system to a mixed state [@shradhanjop; @shimaprl2011]. In the large $C$ limit active system shows a bistable behaviour with two distinct steady states; first, a state where $S$ is large and real space configuration is ‘homogeneous ordered’ (HO), and the second, an ‘inhomogeneous mixed’ (IM) state where $S$ realizes some moderate values. In the HO state though the particle orientation is homogeneous, large density inhomogeneity exists in the system (see Fig. \[fig:phase\_snap\](b) - HO). IM state is a local ordered state with many aligned clusters of high particle density. The system consists of many such aligned clusters of high density separated from low density disordered regions and mean alignment in each cluster is in different directions (see Fig. \[fig:phase\_snap\](b) - IM). IM state is similar to the defect-ordered state recently found in the study of ref. [@shimanatcomm; @aparnaredner; @yeomans], with large number of $\pm 1/2$ defects in high density active nematic. [*Renormalised mean field study for small $S$*]{}:— We also calculate the jump in the scalar order parameter $S$ and the shift in the transition density using the Renormalised mean field (RMF) method of the coupled coarse-grained hydrodynamic equations of motion for the number density $\rho({\bf r}, t) = \sum_{\alpha}\delta({\bf r}-{\bf R}_\alpha(t))$ and the order parameter $ {\bf w}_{i j}({\bf r}, t) = \rho({\bf r}, t) \Q({\bf r}, t) = \sum_{\alpha} ({\bf m}_{i \alpha} {\bf m}_{j \alpha} - \frac{1}{2}\delta_{i j}) \delta({\bf r}-{\bf R}_{\alpha}(t))$ for active nematic [@shradhanjop; @sradititoner]. $$\partial_{t}\rho=a_{0}\nabla_{i}\nabla_{j}{\bf w}_{ij}+D_{\rho}\nabla^{2}\rho \label{eqdensity}$$ and $$\begin{aligned} \partial_{t}{\bf w}_{ij} & =\left\{ \alpha_{1}\left(\rho\right)-\alpha_{2}\left(\mathbf{w}:\mathbf{w}\right)\right\} {\bf w}_{ij} \notag \\ & + \beta\left(\nabla_{i}\nabla_{j}-\frac{1}{2}\delta_{ij}\nabla^{2}\right)\rho+D_{\bf w}\nabla^{2}{\bf w}_{ij} \label{eqop}\end{aligned}$$ where, ${\bf m}_{\alpha} = (\cos(\theta_{\alpha}), \sin(\theta_{\alpha}))$ is the unit vector along the orientation $\theta_{\alpha}$ and ${\bf R}_{\alpha}(t)$ is the position of particle $\alpha$. We can obtain the number density $\rho$ by coarse-graining $C$ over small subvolume. Eqs. \[eqdensity\] and \[eqop\] can be obtained either from symmetry arguments as in ref. [@sradititoner] or from microscopic rule based model [@shradhanjop]. Density equation \[eqdensity\] is a continuity equation $\partial_{t}\rho = -\nabla \cdot {\bf J}$, because the total number of particles is conserved. Current $J_i = -a_0 \nabla_j {\bf w}_{ij} - D_{\rho} \nabla_i \rho$, where the first term consists of two parts, an active curvature coupling current ${\bf J}_a \propto a_0 {\rho \nabla_j \Q_{ij}}$ and anisotropic diffusion ${\bf J}_{p1} \propto \Q_{ij}\nabla_i \rho$, which can also be present in the equilibrium model. The second term in density equation is an isotropic diffusion ${\bf J}_{p2} \propto \nabla \rho$ term. First two terms in the order parameter equation ${\bf w}_{ij}$ is the alignment term. We choose $\alpha_1(\rho) = (\frac{\rho}{\rho_{IN}}-1)$ as a function of density which changes sign at some critical density $\rho_{IN}$. Third term is coupling to density and last term is diffusion in order parameter and written for equal elastic constant approximation for two-dimensional nematic. ![(Color online) $g_2(r)$ vs. $r$ on log-log scale at different densities. (a) Active nematic: ($\bigcirc$) and ($\square$) at low densities $g_2(r)$ decays exponentially, ($\diamond$) and ($+$) at intermediate density $g_2(r)$ decays algebraically, ($\bigtriangleup$) homogeneous ordered (algebraic decay), and inhomogeneous mixed (abrupt decay) at high density. (b) Equilibrium nematic ($\bigcirc$) and ($\square$) exponential decay of $g_2(r)$ at low densities and ($\diamond$) and ($+$) algebraic decay of $g_2(r)$ at high densities.[]{data-label="fig:correlation"}](g2_r_final.pdf){width="\linewidth"} A homogeneous steady state solution of Eqs. \[eqdensity\] and \[eqop\] gives a mean field transition from isotropic to nematic state at density $\rho_{IN}$ where $\alpha_1(\rho)$ changes sign. Using RMF method we calculate the effective free energy $f_{eff}(S)$ close to order-disorder transition when $S$ is small. We consider density fluctuations $\delta \rho$ and neglect order parameter fluctuations. The effective free energy $$f_{eff}\left(S\right)=-\frac{b_{2}}{2}S^{2}-\frac{b_{3}}{3}S^{3}+\frac{b_{4}}{4}S^{4} \label{eqfenergy}$$ where $b_{2}=\alpha_{1}(\rho)+\alpha_{1}^{\prime}(\rho)c$, where $c$ is a constant. $\alpha_1^{\prime}(\rho) = {\partial \alpha_1/\partial \rho|}_{\rho_0}$, $b_{3}=\frac{a_{0}\alpha_{1}^{\prime}(\rho)}{2D_{\rho}}$ and $b_{4}=\frac{1}{2}\alpha_{2}$. Both $b_3$ and $b_4$ are positive. A detail calculation for $f_{eff}$ is shown in Supplemental Material [@SM] section III. The density flcutuations introduce a new cubic order term protortional to the activity strength $a_0$, in the free energy $f_{eff} (S)$. The presence of such term produces a jump $\Delta S = S_{c}=\frac{2b_{3}}{3b_{4}} $ at a lower density $\rho_c = \rho_{IN}(1-\frac{2b_3^2}{9b_4} ) $. This type of jump and shift in transition because of flcutuations are also called as fluctuation dominated first order phase transition in statistical mechanics [@coleman] and widely studied in many systems [@fdfopt]. The jump in $S$ and the shift in $\rho_c$ is proportional to the activity parameter $a_0$ and for $a_0=0$ we recover the equilibrium transition. [*Two-point orientation correlation function*]{} :— To further characterise the system we also calculate the two-point orientation correlation $g_2(r) = <\sum_i n_i n_{i+r} \cos[2\left(\theta_i-\theta_{i+r}\right)] / \sum_i n_i n_i > $ at different packing densities, where $< . >$ signifies an average over many realisations. In Fig. \[fig:correlation\] we plot $g_2(r)$ vs. $r$ on log-log scale, for $C=0.30$, $0.36$, $0.38$, $0.52$ and $0.82$ for active model and $C=0.30$, $0.56$, $0.58$ and $0.78$ for equilibrium model. For very small packing density $C<0.37$, $g_2(r)$ decays exponentially in the active systems. Therefore the system is in short-range-ordered (SRO) isotropic state. At $C=0.38$, $g_2(r)$ decays following the power law $g_2(r) \simeq 1/r^{\eta(C)}$ and therefore the system is in a quasi-long-range-ordered (QLRO) state. At high packing densities correlation functions confirm the bistability in the active systems. For $C=0.82$ (see Fig. \[fig:correlation\](a)) $g_2(r)$ shows power law decay in HO state, whereas in IM state $g_2(r)$ decays abruptly after a distance $r$. The abrupt change in $g_2(r)$ at a certain distance indicates the presence of local ordered clusters. In contrast, the equilibrium systems show a transition from SRO isotropic state at low density $C\lsim0.56$ to QLRO nematic state at $C\gsim0.58$, similar to Berezinskii - Kosterlitz - Thouless (BKT) transition [@Berezinskii; @KT] in the diluted XY-model [@dilutedxymodel]. [*Orientation distribution*]{} :— We also compare the steady state properties of active and equilibrium models in the high density limit. First we calculate the steady state (static) orientation distribution $P(\theta)$ from one snapshot of particle orientation. Both active HO and equilibrium nematic show a Gaussian distribution of orientation (see Fig. \[fig:distribution\](a)). Hence in HO state orientation fluctuations of particles are of same kind as in equilibrium model and the system is in QLRO state. Distribution $P(\theta)$ in the IM state is very broad and spanning the whole range of orientation. Hence the system has many local ordered regions with different orientations. We also calculate the time averaged distribution of mean orientation of all the particles $P(\theta_m)$ in active HO and equilibrium nematic states. $P(\theta_m)$ is calculated from mean of all particle orientaions averaging over a long time (from $10^6$ to $3 \times 10^6$) in the steady state. $P(\theta_m)$ for HO is narrow in comparison to the broad distribution for equilibrium model (see Fig. \[fig:distribution\](b)). Narrow distribution of $P(\theta_m)$ implies that orientation autocorrelation in active system decay slowly as comapared to the corresponding equilibrium model which is in agreement with the slow decay of velocity autocorrelation in bacterial suspension [@wulib]. ![(Color online) (a) Steady state orientation distribution $P(\theta)$ of particles for HO, IM (active) and EN (equilibrium) states at high density. Lines are fit to Gaussian distribution for both HO and EN states. IM state shows very broad distribution of $P(\theta)$. (b) Plot of mean orientation distribution $P(\theta_m)$ averaged over a long time in the steady state for HO (active) and EN (equilibrium) states. $P(\theta_m)$ is very broad for EN in comparison to HO.[]{data-label="fig:distribution"}](distri_c85.pdf){width="\linewidth"} [*Summary*]{} :— In this letter we have introduced a minimal model for active nematic and found three distinct phases with the variation in density. At low densities the active nematic is in disordered isotropic state with very small correlation between the particles. With increasing density active nematic undergoes a fluctuation induced first order phase transition from the isotropic to the banded state where large number of particles participate in band formation. Large density fluctuations in the active systems change the nature of the transition and shift the transition density to smaller value as compared to the equilibrium isotropic nematic transition. At large densities equilibrium nematic is in QLRO nematic state, whereas active nematic goes from the banded state to either the homogeneous ordered (high $S$) or the inhomogeneous mixed (moderate $S$) state. This inhomogeneous mixed state is similar to the phase with coexisting aligned and disordered fibril domains found in recent experiment [@ncommam]. 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Renormalised mean field (RMF) study of active nematic for small scalar order parameter $S$ =============================================================================================== In this section we will write an effective renormalised mean field free energy for scalar order parameter $S$ for small $S$. We keep the density fluctuations and ignore the order parameter fluctuations in the coupled hydrodynamic equations of motion for active nematic. Density fluctuation produces a cubic order term in the effective free energy for scalar order parameter $S$ and such term produces a jump in $S$ at a new transition density $\rho_c$ lower than equilibrium I-N transition point $\rho_{IN}$. Shift in transition density and jump $\Delta S$ is directly proportional to the activity parameter $a_0$ and we recover equilibrium limit for zero $a_0$.\ We write coupled hydrodynamic equations of motion for density $\rho$ and order parameter ${\bf w} = \rho \Q$ where nematic order parameter [@pgdgenne] $$\begin{aligned} \Q({\bf r}, t)=S\left(\begin{array} {c c} \cos 2\theta ({\bf r}, t) & \sin 2\theta ({\bf r}, t) \\ \sin 2\theta ({\bf r}, t) & -\cos 2\theta ({\bf r}, t) \end{array} \right) \label{Q_tensor}\end{aligned}$$ $\theta$ being the coarse grained angle at position ${\bf r}$ and time $t$. Density equation $$\partial_{t}\rho=a_{0}\nabla_{i}\nabla_{j}{\bf w}_{ij}+D_{\rho}\nabla^{2}\rho \label{eqa1}$$ and order parameter equation ${\bf w}$ $$\partial_{t}{\bf w}_{ij}=\left\{ \alpha_{1}\left(\rho\right)-\alpha_{2}\left(\bf{w}:\bf{w}\right)\right\} {\bf w}_{ij}+\beta \left(\nabla_{i}\nabla_{j}-\frac{1}{2}\delta_{ij}\nabla^{2}\right)\rho+D_{{\bf w}}\nabla^{2}{\bf w}_{ij} \label{eqa2}$$ Density Eq. \[eqa1\] is a continuity equation $\partial \rho/\partial t = - \nabla \cdot {\bf J}$, where ${\bf J}$ has two parts, active and diffusive. Details of these two currents are given in the main text. $a_0$ is the activity parameter, present because of self-propelled nature of the particles, $\beta$ is the coupling of density in ${\bf w}$ equation. $D_{\rho}$ and $D_{\bf w}$ are the diffusion coefficients in density and order parameter equations respectively, $\alpha_1(\rho)$ and $\alpha_2$ are the alignment terms and ingeneral depends on the model parameters. For metric distance interacting models [@njopshradha] $\alpha_1(\rho)$ is a function of density and changes sign at some critical density. We choose $\alpha_1(\rho)= \frac{\rho}{\rho_{IN}}-1$ and $\alpha_2 = 1$. Steady state solution of homogeneous Eq. \[eqa1\] is $\rho=\rho_0$, we add small perturbation to mean density $\rho= \rho_0 + \delta \rho$. In the staedy state density fluctuation $\delta \rho$ can be obtained from Eq. \[eqa1\], $$\begin{aligned} & a_{0}\nabla_{i}\nabla_{j}{\bf w}_{ij}+D_{\rho}\nabla^{2}\delta \rho = 0 \notag \\ & \Rightarrow a_{0}\nabla_{j}{\bf w}_{ij}+D_{\rho}\nabla_{i}\delta \rho=constant=\mathbf{c_{1}} \label{eqa3}\end{aligned}$$ where ${\bf w}_{11} = -{\bf w}_{22} = \frac{S}{2} \cos(2 \theta)$ and ${\bf w}_{12} = {\bf w}_{21} = \frac{S}{2} \sin(2 \theta)$ and keep the lowest order terms in $S$ and $\theta$ $$\partial_x \delta \rho = -\frac{a_0}{D_{\rho}} \partial_x S \rightarrow \delta \rho(x)=-\frac{ a_0}{D_{\rho}} S + c \label{eqa4}$$ and $$\partial_y \delta \rho = \frac{a_0}{D_{\rho}} \partial_y S \rightarrow \delta \rho(y)=\frac{a_0}{D_{\rho}} S + c_1 \label{eqa5}$$ Here we assume nematic is aligned along one direction and there is variation only along the perpendicular direction. Hence we can choose either of equations \[eqa4\] or \[eqa5\]. Two constants $c$ and $c_1$ are the value of density where nematic order parameter is zero. We use Eq. \[eqa4\] and substitute the solution for density in equation for ${\bf w}_{ij}$ and obtain an effective equation for $S$. $$\partial_{t}S=\left\{ \alpha_{1}\left(\rho\right)-\frac{1}{2}\alpha_{2}S^{2}\right\} S + \mathcal{O}(\nabla^2 S) + \mathcal{O}(\nabla^2 \rho) \label{eqa6}$$ We neglect all the derivative terms and keep only polynomial in $S$, i.e. we neglect higher order fluctuations. We can do taylor expansion of $\alpha_1(\rho)$ about mean density $\rho_0$, $\alpha_1(\rho)=\alpha_1(\rho_{0}+\delta \rho)=\alpha_1(\rho_{0}) + \alpha_1^{\prime} \delta \rho$, where $\alpha_1^{\prime} = \frac{\partial \alpha_1}{\partial \rho}|_{\rho_{0}}$. Substitute the expression for $\delta \rho$ from Eq. \[eqa4\] hence $$\partial_{t}S=\left\{ \alpha_{1}\left(\rho_{0}\right)+\alpha_1^{\prime} \delta \rho-\frac{1}{2}\alpha_{2}S^{2}\right\} S \label{eqa7}$$ We can write an effective free energy for $S$ $$\partial_t {S} = -\frac{\delta f_{eff}(S)}{\delta S} \label{eqa8}$$ hence $$-\frac{\delta f_{eff}}{\delta S} =S\left\{{ \alpha_{1}\left(\rho_{0}\right)+\alpha} \left(\rho_{0}\right)\left(\frac{a_{0}}{2D_{\rho}}S+c_{1}\right)-\frac{1}{2}\alpha_{2}S^{2}\right\}$$ Therefore $$f_{eff}\left(S\right)=-\frac{b_{2}}{2}S^{2}-\frac{b_{3}}{3}S^{3}+\frac{b_{4}}{4}S^{4} \label{eqa9}$$ where $b_{2}=\alpha_{1}\left(\rho_{0}\right)+\alpha_1^{\prime} c$, $b_{3}=\frac{a_{0}\alpha_1^{\prime} }{2D_{\rho}}$ and $b_{4}=\frac{1}{2}\alpha_{2}$ and $c$ is a conatant. Hence fluctuation in density introduces a cubic order term in effective free energy $f_{eff}(S)$. Effective free energy in Eq. \[eqa9\] is similar to Landau free energy with cubic order term [@chaiklub]. We calculate jump $\Delta S$ and new critical density from coexistence condition for free energy. Steady state solutions of order parameter ($S=0$ for isotropic and $S \neq 0$ for ordered state) are given by $$\frac{\delta f_{eff}}{\delta S}=\left(-b_{2}-b_{3}S+b_{4}S^{2}\right)S=0$$ Non-zero $S$ is given by $-b_{2}-b_{3}S_c+b_{4}S_c^{2}=0$. Coexistence condition implies $$f_{eff}(S_c)=\left(-\frac{b_{2}}{2}-\frac{b_{3}}{3}S_c+\frac{b_{4}}{4}S_c^{2}\right)S_c^{2}=f_{eff}(S=0)=0$$ hence we get the solution $$S_{c}=-\frac{3b_{2}}{b_{3}}$$ and $$b_{2}^{c}=-\frac{2b_{3}^{2}}{9b_{4}}$$ Hence the jump at new critical point is $\Delta S = \frac{2 b_3}{3 b_4}$. Since $b_4 >0$ and hence $b_2^c <0$, the new critical density $$\rho_{c}=\rho_{IN}\left(1-\frac{2b_{3}^{2}}{9b_{4}}\right) < \rho_{IN} \label{eqrhoc}$$ is shifted to lower density in comparison to equilibrium $\rho_{IN}$. Eq. \[eqrhoc\] gives the expression for new transition density as given in main text. Hence using renormalised mean field theory we find a jump $\Delta S$ at a lower density as compared to equilibrium I-N transition density. [10]{} P. G. de Gennes and J. Prost, [*The Physics of Liquid Crystals*]{} (Clarendon Press, Oxford, 1995). E. Bertin, H. Chaté, F. Ginelli, S. Mishra, A. Peshkov and S. Ramaswamy, New J. of Phys. [**15**]{}, 085032 (2013). P. M. Chaikin and T. C. Lubensky, [*Principles of Condensed Matter Physics*]{} (Cambridge University Press, Cambridge, 2000).

--- abstract: 'The description of the proximity effect in superconducting/ferromagnetic heterostructures requires to use spin-dependent boundary conditions. Such boundary conditions must take into account the spin dependence of the phase shifts acquired by electrons upon scattering on the boundaries of ferromagnets. The present article shows that this property can strongly affect the critical temperature and the energy dependence of the density of states of diffusive heterostructures. These effects should allow a better caracterisation of diffusive superconductor/ferromagnet interfaces.' author: - Audrey Cottet title: 'Spectroscopy and critical temperature of diffusive superconducting/ferromagnetic hybrid structures with spin-active interfaces' --- Introduction ============ When a ferromagnetic metal ($F$) with uniform magnetization is connected to a BCS superconductor ($S$), the singlet electronic correlations characteristic of the $S$ phase can propagate into $F$ because electrons and holes with opposite spins and excitation energies are coupled coherently by Andreev reflections occurring at the $S/F$ interface. Remarkably, the ferromagnetic exchange field induces an energy shift between the coupled electrons and holes, which leads to spatial oscillations of the superconducting order parameter in $F$ [@Buzdin1982; @Golubov]. This effect has been observed experimentally through oscillations of the density of states (DOS) in $F$ with the thickness of $F$ [@TakisN], or oscillations of the critical current $I_{0}$ through $S/F/S$ structures [@Ryazanov; @TakisI; @SellierPRB; @Blum], with the thickness of $F$ or the temperature. The oscillations of $I_{0}$ have allowed to obtain $\pi$-junctions[@Guichard], i.e. Josephson junctions with $I_{0}<0$, which could be useful in the field of superconducting circuits [@Ioffe; @Taro]. A reentrant behavior of the superconducting critical temperature of $S/F$ bilayers with the thickness of $F$ has also been observed [@TcSF]. At last, some $F/S/F$ trilayers have shown a lower critical temperature for an antiparallel alignment of the magnetizations in the two $F$ layers as compared with the parallel alignment[@SSspinswitch], which should offer the possibility of realizing a superconducting spin-switch[@DeGennes; @TcFSFth]. ![a. Diffusive $F/S/F$[ ]{}trilayer consisting of a BCS superconductor $S$ with thickness $d_{S}$ placed between two ferromagnetic electrodes $F_{1}$ and $F_{2}$ with thickness $d_{F}$. In this picture, the directions of the magnetic polarizations in $F_{1}$ and $F_{2}$  are parallel \[antiparallel\], which corresponds to the configuration $\mathcal{C}=P$ $[AP]$. b. $S/F$[ ]{}bilayer consisting of a BCS superconductor $S$ with thickness $d_{S}/2$ contacted to a ferromagnetic electrode $F$ with thickness $d_{F}$.](structure.eps){width="0.7\linewidth"} For a theoretical understanding of the behavior of $S/F$ hybrid circuits, a proper description of the interfaces between the different materials is crucial. For a long time, the only boundary conditions available in the diffusive case were spin-independent boundary conditions derived for $S/$normal metal interfaces[@Kuprianov]. Recently, spin-dependent boundary conditions have been introduced for describing hybrid diffusive circuits combining BCS superconductors, normal metals and ferromagnetic insulators [@condmatHuertas]. These boundary conditions take into account the spin-polarization of the electronic transmission probabilities through the interface considered, but also the spin-dependence of the phase shifts acquired by electrons upon transmission or reflection by the interface. The first property generates widely known magnetoresistance effects[@magn]. The second property is less commonly taken into account. However, the Spin-Dependence of Interfacial Phase Shifts (SDIPS) can modify the behavior of many different types of mesoscopic circuits with ferromagnetic elements, like those including a diffusive normal metal island [@FNF], a resonant system[@CottetEurophys; @SST], a Coulomb blockade system[@CB; @Cottet06; @SST], or a Luttinger liquid[@Luttinger]. It has also been shown that the SDIPS has physical consequences in $S/F$ hybrid systems[@Tokuyasu; @otherBC; @mixing; @condmatHuertas]. One can note that, in some references, the SDIPS is called ”spin-mixing angle” or ”spin-rotation angle” (see e.g. Refs. ).** **In the diffusive $S/F$ case, the spin-dependent boundary conditions of Ref.  have been applied to different circuit geometries[@demoBC; @applications; @Morten; @cottet05; @Braude] but the only comparison to experimental data has been performed in Ref. . The authors of this Ref. have generalized the boundary conditions of Ref.  to the case of metallic $S/F$ interfaces with a superconducting proximity effect in $F$. They have showed that the SDIPS can induce a shift in the oscillations of the critical current of a $S/F/S$ Josephson junction or of the DOS of a $S/F$ bilayer with the thickness of $F$. Signatures of this effect have been identified in the hybrid structures of Refs. . Nevertheless, the problem of characterizing the SDIPS of diffusive $S/F$ interfaces has raised little attention so far, in spite of the numerous experiments performed. A good characterization of the properties of diffusive $S/F$ interfaces would be necessary for a better control of the superconducting proximity effect in diffusive heterostructures. The present article presents other consequences of the SDIPS than that studied in Ref. , which could be useful in this context. In particular, the SDIPS can generate an effective magnetic field in a diffusive $S$ in contact with a diffusive $F $, like found for a ballistic $S$ in contact with a ferromagnetic insulator [@Tokuyasu]. This effective field can be detected, in particular, through the DOS of the diffusive $F$ layer, with a visibility which depends on the thickness of $F$. A strong modification of the variations of the critical temperature of diffusive $S/F$ structures with the thickness of $F$ is also found. These effects should allow to characterize the SDIPS of diffusive $S/F$ interfaces through DOS and critical temperature measurements, by using the heterostructures currently fabricated. The calculations reported in this paper are also appropriate to the case of a diffusive $S$ layer contacted to a ferromagnetic insulator ($FI $). This paper is organized as follows: Section II presents the initial set of equations used to describe the heterostructures considered. The case of $F/S/F$ trilayers is mainly addressed, but the case of $S/F$ (or $S/FI$) bilayers follows straightforwardly. Section III specializes to the case of a weak proximity effect in $F$ and a superconducting layer with a relatively low thickness $d_{S}\leq\xi_{S}$, with $\xi_{S}$ the superconducting coherence length in $S$. The spatial evolution of the electronic correlations in the $S$ and $F$ layers is studied in Section III.A. The energy-dependent DOS of $S/F$ heterostructures is calculated in Section III.B. Section III.C considers briefly the limit of $S/FI$ bilayers. Section III.D discusses SDIPS-induced effective field effects in other types of systems. Section III.E compares the present work to other DOS calculations for data interpretation in $S/F$ heterostructures. Critical temperatures of $S/F$ circuits are calculated and discussed in Section III.F. Conclusions are presented in Section IV. Throughout the paper, I consider conventional BCS superconductors with a s-wave symmetry. Initial description of the problem ================================== This article mainly considers a diffusive $F/S/F$[ ]{}trilayer consisting of a BCS superconductor $S$ for $-d_{S}/2<x<d_{S}/2$, and ferromagnetic electrodes $F_{1}$ for $x\in\{-d_{S}/2-d_{F},-d_{S}/2\}$ and $F_{2}$ for $x\in\{d_{S}/2,d_{S}/2+d_{F}\}$ (see Figure 1.a). The magnetic polarization of the two ferromagnets can be parallel (configuration $\mathcal{C}=P$) or anti-parallel (configuration $\mathcal{C}=AP$), but the modulus $\left| E_{ex}\right| $ of the ferromagnetic exchange field is assumed to be the same in $F_{1}$ and $F_{2}$. Throughout the structure, the normal quasiparticle excitations and the superconducting condensate of pairs can be characterized with Usadel normal and anomalous Green’s functions $G_{n,\sigma}=\mathrm{sgn}(\omega_{n})\cos(\theta_{n,\sigma})$ and $F_{n,\sigma}=\sin(\theta_{n,\sigma})$, with $\theta_{n,\sigma}(x)$ the superconducting pairing angle, which depends on the spin direction $\sigma \in\{\uparrow,\downarrow\}$, the Matsubara frequency $\omega_{n}(T)=(2n+1)\pi k_{B}T$, and the coordinate $x$ (see e.g. Ref. ). The Usadel equation describing the spatial evolution of $\theta_{n,\sigma}$ writes $$\frac{\hbar D_{S}}{2}\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}% }=\left| \omega_{n}\right| \sin(\theta_{n,\sigma})-\Delta(x)\cos (\theta_{n,\sigma}) \label{UsadelS}%$$ in $S$ and $$\frac{\hbar D_{F}}{2}\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}% }=\left( \left| \omega_{n}\right| +iE_{ex}\sigma\mathrm{sgn}(\omega _{n})\right) \sin(\theta_{n,\sigma}) \label{UsadelF2}%$$ in $F_{1}$ and $F_{2}$, with $D_{F}$ the diffusion constant of the ferromagnets and $D_{S}$ the diffusion constant of $S$. The self-consistent superconducting gap $\Delta(x)$ occurring in (\[UsadelS\]) can be expressed as $$\Delta(x)=\frac{\pi k_{B}T\lambda}{2}\sum\limits_{\substack{\sigma \in\{\uparrow,\downarrow\}\\\omega_{n}(T)\in\{-\Omega_{D},\Omega_{D}\}}% }\sin(\theta_{n,\sigma}) \label{Delta}%$$ with $\Omega_{D}$ the Debye frequency of $S$, $\lambda^{-1}=2\pi k_{B}T_{_{c}% }^{BCS}\sum\nolimits_{\omega_{n}(T_{_{c}}^{BCS})\in\{0,\Omega_{D}\}}\omega _{n}^{-1}$ the BCS coupling constant and $T_{_{c}}^{BCS}$ the bulk transition temperature of $S$. I assume $\Delta=0$ in $F_{1}$ and $F_{2}$. The above equations must be supplemented with boundary conditions describing the interfaces between the different materials. First, one can use $$\left. \frac{\partial\theta_{n,\sigma}}{\partial x}\right| _{x=\pm (d_{S}/2+d_{F})}=0 \label{derZero}%$$ for the external sides of the structure. Secondly, the boundary conditions at the $S/F$ interfaces can be calculated by assuming that the interface potential locally dominates the Hamiltonian, i.e. at a short distance it causes only ordinary scattering (with no particle-hole mixing) (see e.g. Ref. ). This ordinary scattering can be described with transmission and reflection amplitudes $t_{n,\sigma}^{S(F)}$ and $r_{n,\sigma }^{S(F)}$ for electrons coming from the $S$($F$) side of the interface in channel $n$ with a spin direction $\sigma$. The phases of $t_{n,\sigma}% ^{S(F)}$ and $r_{n,\sigma}^{S(F)}$ can be spin-dependent due to the exchange field $E_{ex}$ in $F_{1(2)}$ and a possible spin-dependence of the barrier potential between $S$ and $F_{1(2)}$. Boundary conditions taking into account this so-called Spin-Dependence of Interfacial Phase Shifts (SDIPS) have been derived for $|t_{n,\uparrow}^{S}|^{2},|t_{n,\downarrow}^{S}|^{2}\ll1$ and a weakly polarized $F$[@condmatHuertas; @cottet05]. When there is no SDIPS, the boundary conditions involve the tunnel conductance $G_{T}=G_{Q}% \sum\nolimits_{n}T_{n}$ and the magnetoconductance $G_{MR}=G_{Q}% \sum\nolimits_{n}(|t_{n,\uparrow}^{S}|^{2}-|t_{n,\downarrow}^{S}|^{2})$, with $\uparrow(\downarrow)$ the majority(minority) spin direction in the $F$ electrode considered, $G_{Q}=e^{2}/h$, and $T_{n}=|t_{n,\uparrow}^{S}% |^{2}+|t_{n,\downarrow}^{S}|^{2}$. In the case of a finite SDIPS, one must also use the conductances $G_{\phi}^{F(S)}=2G_{Q}\sum\nolimits_{n}(\rho _{n}^{F(S)}-4[\tau_{n}^{S(F)}/T_{n}])$, $G_{\xi}^{F(S)}=-G_{Q}\sum \nolimits_{n}\tau_{n}^{S(F)}$ and $G_{\chi}^{F(S)}=G_{Q}\sum\nolimits_{n}% T_{n}(\rho_{n}^{F(S)}+\tau_{n}^{S(F)})/4$, with $\rho_{n}^{m}% =\operatorname{Im}[r_{n,\uparrow}^{m}r_{n,\downarrow}^{m~\ast}]$ and $\tau _{n}^{m}=\operatorname{Im}[t_{n,\uparrow}^{m}t_{n,\downarrow}^{m~\ast}]$ for $m\in\{S,F\}$. In the following, I will focus on the effects of $G_{\phi}^{F}$ and $G_{\phi}^{S}$, and I will assume $G_{MR}$, $G_{\xi}^{F(S)}$ and $G_{\chi }^{F(S)}$ to be negligible, like found with a simple barrier model in the limit $T_{n}\ll1$ and $E_{ex}\ll E_{F}$[@cottet05]. In this case, one finds that the boundary conditions for the $S/F$ interface located at $x=x_{j}=(-1)^{j}d_{s}/2$, with$\ j\in\{1,2\}$, write $$\xi_{F}\left. \frac{\partial\theta_{n,\sigma}^{F}}{\partial x}\right| _{x_{j}}=(-1)^{j}\gamma_{T}\sin[\theta_{n,\sigma}^{F}-\theta_{n,\sigma}% ^{S}]+i\gamma_{\phi}^{F}\epsilon_{n,\sigma}^{\mathcal{C},j}\sin[\theta _{n,\sigma}^{F}] \label{BCright}%$$ and $$\xi_{F}\left. \frac{\partial\theta_{n,\sigma}^{F}}{\partial x}\right| _{x_{j}}-\frac{\xi_{S}}{\gamma}\left. \frac{\partial\theta_{n,\sigma}^{S}% }{\partial x}\right| _{x_{j}}=\sum\limits_{m\in\{F,S\}}i\gamma_{\phi}% ^{m}\epsilon_{n,\sigma}^{\mathcal{C},j}\sin[\theta_{n,\sigma}^{m}] \label{BCleft}%$$ where the indices $S$ and $F$ indicate whether $\theta_{n,\sigma}$ and its derivative are taken at the $S$ or $F$ side of the interface. These equations involve the reduced conductances $\gamma_{T}=G_{T}\xi_{F}/A\sigma_{F}$ and $\gamma_{\phi}^{F(S)}=G_{\phi}^{F(S)}\xi_{F}/A\sigma_{F}$, the barrier asymmetry coefficient $\gamma=\xi_{S}\sigma_{F}/\xi_{F}\sigma_{S}$, the superconducting coherence lengthscale $\xi_{S}=(\hbar D_{S}/2\Delta _{BCS})^{1/2}$, the magnetic coherence lengthscale $\xi_{F}=(\hbar D_{F}/\left| E_{ex}\right| )^{1/2}$, the gap $\Delta_{BCS}$ for a bulk $S$, the normal state conductivity $\sigma_{F(S)}$ of the $F(S)$ material, and the junction area $A$. The coefficient $\epsilon_{n,j}^{\mathcal{C}}$ takes into account the direction of the ferromagnetic polarization of electrode $F_{j}$ in configuration $\mathcal{C}\in\{P,AP\}$. One can use the convention $\epsilon_{n,\sigma}^{P,j}=(-1)^{j}\sigma\mathrm{sgn}(\omega_{n})$ and $\epsilon_{n,\sigma}^{AP,j}=\sigma\mathrm{sgn}(\omega_{n})$, in which the factor $\mathrm{sgn}(\omega_{n})$ arising from the definition of $\theta_{n,\sigma}$ and the terms $(-1)^{j}$ and $\sigma$ arising from the boundary conditions have been included for compactness of the expressions. Note that in the presence of a finite SDIPS i.e. $\gamma_{\phi}^{F(S)}\neq0$, the right hand side of equation (\[BCleft\]) is not zero contrarily to what found in the spin-degenerate case [@Kuprianov]. In the general case, $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$ are different (see e.g. Appendix A). This implies that, with the present approximations, each interface is characterized by three parameters: $\gamma_{T}$, $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. For the sake of simplicity, symmetric $F/S/F$ trilayers are considered, so that $\gamma_{T}$, $\gamma_{\phi}^{F}$ and $\gamma_{\phi }^{S}$ are the same for the two $S/F$ interfaces. Before working out the above system of equations, it is interesting to note that the angle $\theta_{n,\sigma}$ calculated in the parallel configuration $\mathcal{C}=P$ for $x>0$ also corresponds to the angle $\theta_{n,\sigma}$ expected for a $S/F$ bilayer consisting of a superconductor $S$ for $0<x<d_{S}/2$, and a ferromagnetic electrode $F$ for $x\in\{d_{S}% /2,d_{S}/2+d_{F}\}$ (Figure 1.b). In practice, using a $F/S/F$ geometry can allow one to obtain more information on spin effects, as shown below. Case of a thin superconductor and a weak proximity effect in $F$ ================================================================ Spatial variations of the pairing angle --------------------------------------- I will assume that the amplitude of the superconducting correlations in $F_{1(2)}$ is weak, i.e. $\left| \theta_{n,\sigma}\right| \ll1$ for $x\in\{-d_{S}/2-d_{F},-d_{S}/2\}$ and $x\in\{d_{S}/2,d_{S}/2+d_{F}\}$ (*hypothesis 1)* so that one can develop the Usadel equation (\[UsadelF2\]) at first order in $\theta_{n,\sigma}$. This leads to $$\frac{\partial^{2}\theta_{n,\sigma}}{\partial x^{2}}-\left( \frac {k_{n,\sigma}^{\mathcal{C},j}}{\xi_{F}}\right) ^{2}\theta_{n,\sigma}=0 \label{UsadelF}%$$ in the ferromagnet $F_{j}$, with $j\in\{1,2\}$, $k_{n,\sigma}^{\mathcal{C}% ,j}=(2[i\eta_{n,\sigma}^{\mathcal{C},j}+\left| \omega_{n}/E_{ex}\right| )])^{1/2}$ and $\eta_{n,\sigma}^{\mathcal{C},j}=(-1)^{j}\epsilon_{n,\sigma }^{\mathcal{C},j}$. Combining Eqs. (\[derZero\]) and (\[UsadelF\]), one finds in $F_{j}$ $$\theta_{n,\sigma}(x)=\theta_{n,\sigma}^{F}(x_{j})\frac{\cosh\left( \left[ x-(-1)^{j}\left( d_{F}+\frac{d_{S}}{2}\right) \right] \frac{k_{n,\sigma }^{\mathcal{C},j}}{\xi_{F}}\right) }{\cosh\left( d_{F}\frac{k_{n,\sigma }^{\mathcal{C},j}}{\xi_{F}}\right) } \label{theta}%$$ This result together with boundary condition (\[BCright\]) leads to $$\theta_{n,\sigma}^{F}(x_{j})=\frac{\gamma_{T}\sin(\theta_{n,\sigma}^{S}% (x_{j}))}{\gamma_{T}\cos(\theta_{n,\sigma}^{S}(x_{j}))+i\gamma_{\phi}^{F}% \eta_{n,\sigma}^{\mathcal{C},j}+B_{n,\sigma}^{\mathcal{C},j}} \label{thetaF}%$$ with $B_{n,\sigma}^{\mathcal{C},j}=k_{n,\sigma}^{\mathcal{C},j}\tanh [d_{F}k_{n,\sigma}^{\mathcal{C},j}/\xi_{F}]$. This allows to rewrite the boundary condition (\[BCleft\]) in closed form with respect to $\theta_{n,\sigma}^{S}$, i.e. $$\xi_{S}\left. \frac{\partial\theta_{n,\sigma}^{S}}{\partial x}\right| _{x_{j}}=(-1)^{j+1}\mathcal{L}_{j,n,\sigma}^{\mathcal{C}}\sin[\theta _{n,\sigma}^{S}(x_{j})]$$ for the $S/F$ interface located at $x=x_{j}$, with $$\frac{\mathcal{L}_{j,n,\sigma}^{\mathcal{C}}}{\gamma}=\frac{\gamma _{T}(B_{n,\sigma}^{\mathcal{C},j}+i\gamma_{\phi}^{F}\eta_{n,\sigma }^{\mathcal{C},j})}{\gamma_{T}\cos(\theta_{n,\sigma}^{S})+B_{n,\sigma }^{\mathcal{C},j}+i\gamma_{\phi}^{F}\eta_{n,\sigma}^{\mathcal{C},j}}% +i\gamma_{\phi}^{S}\eta_{n,\sigma}^{\mathcal{C},j}%$$ In the following, I will assume $\left| E_{ex}\right| \gg\Delta_{BCS}$ like in most experiments, so that $k_{n,\sigma}^{\mathcal{C},j}=1+i\eta_{n,\sigma }^{\mathcal{C},j}$. I will also assume $d_{S}/\xi_{S}\leq1$, so that one can use, for the $\mathcal{C}$ configuration and $-d_{S}/2<x<d_{S}/2$, $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}% -\alpha_{n,\sigma}^{\mathcal{C}}(x/\xi_{S})-\beta_{n,\sigma}^{\mathcal{C}% }(x/\xi_{S})^{2} \label{thetaQuad}%$$ with $\left| \theta_{n,\sigma}(x)-\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% }\right| \ll1$ (*hypothesis 2*). Note that although experiments are often performed in the limit of thick superconducting layers $d_{S}>\xi_{S}$, assuming $d_{S}\leq\xi_{S}$ is not unrealistic since one can obtain diffusive superconducting layers with a thickness $d_{S}\sim\xi_{S}$ (see e.g. Ref. ). Furthermore, using relatively low values of $d_{S}$ is more favorable for obtaining efficient superconducting spin-switches[@Baladie]. Hypothesis 2 allows one to develop $\sin (\theta_{n,\sigma})$ and $\cos(\theta_{n,\sigma})$ at first order with respect to $\theta_{n,\sigma}-\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$ in $S$. Accordingly, I will neglect the space-dependence of $\Delta(x)$ and assign to it the value $\Delta^{\mathcal{C}}$ in configuration $\mathcal{C}$. Note that I do not make any assumption on the value of the angle $\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$, which is not necessarily close to the bulk BCS value. The coefficient $\mathcal{L}_{j,n,\sigma}% ^{\mathcal{C}}$ is transformed into its conjugate when the magnetic polarization of electrode $F_{j}$ is reversed. Therefore, I will note $\mathcal{L}_{1,n,\sigma}^{P}=\mathcal{L}_{2,n,\sigma}^{P(AP)}=\mathcal{L}% _{n}^{\sigma}$ and $\mathcal{L}_{1,n,\sigma}^{AP}=(\mathcal{L}_{n}^{\sigma })^{\ast}$. The above assumptions lead to $\alpha_{n,\sigma}^{P}=0$, $$\beta_{n,\sigma}^{P}=\frac{4\mathcal{L}_{n}^{\sigma}\sin(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})}{4\delta_{S}+\cos(\widetilde{\theta}_{n,\sigma }^{\mathcal{C}})\delta_{S}^{2}\mathcal{L}_{n}^{\sigma}} \label{bP}%$$$$\alpha_{n,\sigma}^{AP}=i\operatorname{Im}[\mathcal{L}_{n}^{\sigma}]\frac {4\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})-\cos(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})\delta_{S}^{2}\beta_{AP}}{4+2\operatorname{Re}% [\mathcal{L}_{n}^{\sigma}]\cos(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% })\delta_{S}}%$$ and $$\beta_{n,\sigma}^{AP}=\frac{\left( 8\operatorname{Re}[\mathcal{L}_{n}% ^{\sigma}]+4\left| \mathcal{L}_{n}^{\sigma}\right| ^{2}\cos(\widetilde {\theta}_{n,\sigma}^{\mathcal{C}})\delta_{S}\right) \sin(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})}{8\delta_{S}+6\operatorname{Re}[\mathcal{L}% _{n}^{\sigma}]\cos(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})\delta_{S}% ^{2}+\left| \mathcal{L}_{n}^{\sigma}\right| ^{2}\cos^{2}(\widetilde{\theta }_{n,\sigma}^{\mathcal{C}})\delta_{S}^{3}} \label{bAP}%$$ with $\delta_{S}=d_{S}/\xi_{S}$. On the other hand, from (\[UsadelS\]), one finds $$\beta_{n,\sigma}^{\mathcal{C}}=\frac{\Delta^{\mathcal{C}}\cos(\widetilde {\theta}_{n,\sigma}^{\mathcal{C}})-\left| \omega_{n}\right| \sin (\widetilde{\theta}_{n,\sigma}^{\mathcal{C}})}{2\Delta_{BCS}} \label{bc}%$$ The comparison between equations (\[bP\]), (\[bAP\]) and (\[bc\]) allows one to find $\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}$ as a function of $\Delta^{\mathcal{C}}$. Then, one has to calculate $\Delta^{\mathcal{C}}$ by using the self-consistency relation (\[Delta\]). I will study below the DOS and the critical temperature following from these Eqs., in a limit which leads to simple analytical expressions. Low-temperature density of states of $S/F$ heterostructures ----------------------------------------------------------- The DOS of the ferromagnets $F_{1}$ and $F_{2}$ of Figure 1.a can be probed at $x=\pm(d_{F}+d_{S}/2)$ by performing tunnelling spectroscopy through an insulating layer. So far, this quantity has been less measured[@TakisN; @Kontos04; @Courtois; @Reymond] than critical temperatures or supercurrents. However, this way of probing the superconducting proximity effect is very interesting because it allows one to obtain spectroscopic information. It has been shown that the zero-energy DOS of a $F$ layer in contact with a superconductor oscillates with the thickness of $F$. For certain thicknesses, this zero-energy DOS can even become higher than its normal state value $N_{0}$ [@theoDOS; @Zareyan; @Bergeret; @yokohama], as shown experimentally in Ref. . Remarkably, the SDIPS can shift these oscillations[@cottet05]. Although the energy dependence of the DOS of diffusive $S/F$ structures has raised some theoretical and experimental interest, the effect of the SDIPS on this energy dependence has not been investigated so far. For calculating analytically the low-temperature DOS of the structure of Fig. 1.a., one can assume $\gamma_{T}^{2}\cos(\theta_{n,\sigma}^{S}(x_{j}))/\left| (\right. \gamma_{T}\cos(\theta_{n,\sigma}^{S}(x_{j}))+i\gamma_{\phi}^{F}% \eta_{n,\sigma}^{P,2}+B_{n,\sigma}^{P,2})(\gamma_{T}+i\gamma_{\phi}^{S}% \eta_{n,\sigma}^{P,2}\left. )\right| \ll1$ (*hypothesis 3*), which leads to $\mathcal{L}_{n}^{\sigma}=\gamma\left( \gamma_{T}+i\gamma_{\phi}% ^{S}\sigma\mathrm{sgn}(\omega_{n})\right) $. This hypothesis is e.g. valid for $d_{F}\geq\xi_{F}$ and any value of $\gamma_{\phi}^{S(F)}$ if $\gamma_{T}$ is relatively small (see e.g. Fig. \[DOS1\]). I will also assume that the lowest order terms in $\delta_{S}$ prevail in the numerators and denominators of expressions (\[bP\]) and (\[bAP\]), i.e. $\beta_{n,\sigma}^{P}% \sim\mathcal{L}_{n}^{\sigma}\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% })\delta_{S}^{-1}$ and $\beta_{n,\sigma}^{AP}\sim\operatorname{Re}% [\mathcal{L}_{n}^{\sigma}]\sin(\widetilde{\theta}_{n,\sigma}^{\mathcal{C}% })\delta_{S}^{-1}$ (*hypothesis 4*). Taking into account hypothesis 3 and $\gamma\sim1$, hypothesis 4 is valid provided $\gamma_{T}$ and $\left| \gamma_{\phi}^{S}\right| $ are relatively small compared to $1$. Importantly, hypotheses 3 and 4 are less restrictive regarding the value of $\gamma_{\phi }^{F}$. Accordingly, I will often use $\gamma_{T},\left| \gamma_{\phi}% ^{S}\right| \ll\left| \gamma_{\phi}^{F}\right| $ in the Figs. of this paper. Hypotheses 3 and 4 lead to $$\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}=\mathrm{\arctan}\left( \frac{\Delta^{\mathcal{C}}}{\left| \omega_{n}\right| +\Omega_{n,\sigma }^{\mathcal{C}}}\right) \label{thetaS}%$$ with $$\Omega_{n,\sigma}^{P}=2\Delta_{BCS}\gamma(\gamma_{T}+i\gamma_{\phi}^{S}% \sigma\mathrm{sgn}(\omega_{n}))\delta_{S}^{-1} \label{omegaa}%$$ and $$\Omega_{n,\sigma}^{AP}=2\Delta_{BCS}\gamma\gamma_{T}\delta_{S}^{-1} \label{Om2}%$$ ![Zero energy density of states $N^{F}(\varepsilon=0)$ at $x=d_{F}+d_{S}/2$ as a function of $d_{F}$, for the $F/S/F$ structure of Figure 1.a in the $P$ configuration, with $\gamma=1$, $\gamma_{T}=0.12$, and $d_{S}/\xi_{S}=1$. The different curves correspond to different SDIPS parameters:$\ \gamma_{\phi}^{F}=\gamma_{\phi }^{S}=0$ (black dotted curve), $\{\gamma_{\phi}^{F}=-2.1$, $\gamma_{\phi}% ^{S}=0\}$ (black full curve), and $\{\gamma_{\phi}^{F}=-2.1$, $\gamma_{\phi }^{S}=0.12\}$ (red full curve). The low-temperature self-consistent gap $\Delta^{P}$ found for these three different cases is indicated at the bottom right of the Figure. Using a strong value for $\gamma_{\phi}^{F}$ allows one to change significantly the phase of the oscillations of $N_{\sigma}^{F}(0)$ with $d_{F}$ (effect similar to Ref. ). One can check that the hypotheses 1 to 4 are valid for the parameters used in this Figure. Note that for the different cases considered here, the critical temperature $T_{c}^{P}$ of the structure (see Sec. III.F) is such that $0.615T_{c}^{BCS}<T_{c}^{P}<0.635T_{c}^{BCS}$. The four red points correspond to the red full curves shown in Fig. \[DOS2\].[]{data-label="DOS1"}](DOS1.eps){width="0.8\linewidth"} Equations (\[thetaS\]) and (\[omegaa\]) show that an effective magnetic field $H_{eff}$ appears in the $S$ layer in the $P$ configuration, due to $\gamma_{\phi}^{S}\neq0$. From Eqs. (\[thetaS\]) and (\[omegaa\]), $H_{eff}$ can be expressed as $$g\mu_{B}H_{eff}=\frac{\hbar v_{F}^{S}}{d_{s}}\frac{2G_{\phi}^{S}\ell_{e}^{S}% }{3\sigma_{S}A}=2E_{TH}^{S}\frac{G_{\phi}^{S}}{G_{S}} \label{heff}%$$ Here, $v_{F}^{S}$ and $\ell_{e}^{S}$ denote the Fermi velocity and mean free path in $S$, and $G_{S}=\sigma_{S}A/d_{S}$ and $E_{TH}^{S}=\hbar D_{S}% /d_{S}^{2}$ denote the normal state conductance and the Thouless energy of the $S$ layer of Fig. 1.a. From Equations (\[thetaS\]) and (\[Om2\]), the effective field effect disappears in the $AP$ configuration because the two contacts are assumed to be symmetric, and therefore, their contributions to $H_{eff}$ compensate each other in the $AP$ case. Note that, in principle, the $\gamma_{\phi}^{F}$ term can induce an effective field analogue to $H_{eff}$ in the $F$ layer, but this effect is not relevant in the regime studied in this paper (see Appendix B). The effects of $H_{eff}$ on the DOS of the structure will be investigated in next paragraphs. In order to calculate $\Delta^{\mathcal{C}}$, one has to combine the self-consistency relation (\[Delta\]) with Eq. (\[thetaS\]), which gives, at low temperatures, $$\operatorname{Re}[\log(\frac{\Omega_{n,\sigma}^{\mathcal{C}}+\sqrt{\left( \Omega_{n,\sigma}^{\mathcal{C}}\right) ^{2}+\left( \Delta^{\mathcal{C}% }\right) ^{2}}}{\Delta_{BCS}})]=0 \label{DeltaEq}%$$ This equation can be solved numerically. The resulting $\Delta^{\mathcal{C}}$ is independent from the values of $n$ and $\sigma$ used in Eq. (\[DeltaEq\]). Then, the value of the pairing angle $\theta_{n,\sigma}$ in the ferromagnets can be found by using Eqs. (\[theta\]), (\[thetaF\]), (\[thetaQuad\]) and (\[thetaS\]). Note that for $\gamma\ll1$, the above Eqs. are in agreement with formula (5) of Ref. , obtained with rigid boundary conditions, i.e. $\theta_{n,\sigma}$ equal to its bulk value at the $S$ side. The energy dependence of $\theta_{n,\sigma}$ can be found by performing the analytic continuation $\omega_{n}=-i\varepsilon +\Gamma$ and $\mathrm{sgn}(\omega_{n})=1$ in the above equations. The rate $\Gamma=0.05$ is used to account for inelastic processes [@theseW]. At last, the density of states $N_{\sigma}(x,\varepsilon)$ at position $x$ for the spin direction $\sigma\in\{\uparrow,\downarrow\}$ can be calculated by using $N_{\sigma}(x,\varepsilon)=\left( N_{0}/2\right) \operatorname{Re}% [\cos[\theta_{n,\sigma}(x)]]$, where $N_{0}/2$ is the normal density of states per spin direction. ![Density of states $N^{F}(\varepsilon)$ at $x=d_{F}+d_{S}/2$ as a function of the energy $\varepsilon$, for the $F/S/F$ structure of Figure 1.a with $\gamma=1$, $\gamma_{T}=0.12$, $d_{S}/\xi_{S}=1$, $\gamma_{\phi}^{F}=-2.1$ and $\gamma_{\phi}^{S}=0.12$. The different panels correspond to different values of $d_{F}/\xi_{F}$. The red full curves correspond to the $P$ configuration and the blue dashed curves to the $AP$ configuration. For finite values of $\gamma_{\phi}^{S}$, some characteristic ”double-gap” structures appear in $N_{\sigma}^{F}(\varepsilon)$ in the $P$ case, due to a SDIPS-induced effective field appearing in $S$. The visibility of this effect strongly depends on the thickness $d_{F}$ of the $F$ layers. For the different cases considered here, one can check that hypotheses 1 to 4 are valid and that the critical temperature $T_{c}^{\mathcal{C}}$ of the structure (see Sec. III.F) is such that $0.62T_{c}^{BCS}<T_{c}^{\mathcal{C}}<0.67T_{c}^{BCS}$. Note that for the above parameters and $\gamma_{\phi}^{S}=0$, the curves obtained in the $P$ and $AP$ case would be identical and very close to the $AP$ curves shown here. []{data-label="DOS2"}](DOS2.eps){width="0.85\linewidth"} In the following, I will mainly focus on $N^{F}(\varepsilon)=\sum \nolimits_{\sigma\in\{\uparrow,\downarrow\}}N_{\sigma}(x=d_{F},\varepsilon)$. Figure \[DOS1\] shows the variations of the zero energy density of states $N^{F}(\varepsilon=0)$ as a function of $d_{F}$, for interface parameters $\gamma_{T}=\gamma_{\phi}^{S}=0.12$ and $\gamma_{\phi}^{F}=-2.1$. Importantly, the value $\gamma_{T}=0.12$ seems realistic, at least for the weakly polarized $_{0.9}$$_{0.1}$ bilayers used in Ref. , for which one finds $\gamma_{T}\sim0.15$ (see Ref. ). In addition, a simple barrier model suggests that the situation $\left| G_{\phi}^{F}\right| \gg G_{T}$, with $G_{\phi}^{F}<0$ can happen (see Appendix A). The value $\gamma_{\phi}^{F}=-2.1$ used in Fig. \[DOS1\] thus seems possible**.** One can see that $\gamma_{\phi}^{F}$ can change significantly the phase of the oscillations of $N^{F}(\varepsilon=0)$ with $d_{F}$ (this effect has already been studied in Ref.  in the case of rigid boundary conditions but I recall it here for the sake of completeness). In Fig. \[DOS1\], using for $\gamma_{\phi}^{F}$ a strong negative value allows to get $N^{F}(0)<N_{0}$ for the lowest values of $d_{F}% $, like often found in experiments. Note that from Fig. \[DOS1\], $\gamma_{\phi}^{S}$ can also shift the oscillations of $N^{F}(\varepsilon=0)$ with $d_{F}$. Here, this effect is much weaker than that of $\gamma_{\phi}% ^{F}$, but one has to keep in mind that the limit $\gamma_{\phi}^{S}\ll \gamma_{\phi}^{F}$ is considered. Equation (\[heff\]) shows that a measurement of $H_{eff}$ should allow to determine the conductance $G_{\phi}^{S}$ of a diffusive $S/F$ interface. In this context, studying the energy dependence of $N^{F}(\varepsilon)$ is very interesting, because it can allow to see clear signatures of $H_{eff}$, as shown below. Figure \[DOS2\] shows the energy dependence of $N^{F}% (\varepsilon)$ in the $P$ and $AP$ configurations, for a finite value of $\gamma_{\phi}^{S}$ and different values of $d_{F}$. For $\mathcal{C}=P$, $N^{F}(\varepsilon)$ shows some ”double-gap” structures which disappear if the device is switched to the $AP$ configuration. These double structures are an indirect manifestation of the effective magnetic field $H_{eff}$ occurring in $S$ in the $P$ configuration, due to $\gamma_{\phi}^{S}\neq0$. Although $H_{eff}$ is localized in the $S$ layer, the double-gap structure that this field produces in the DOS of $S$ is transmitted to the DOS of $F$ due to Andreev reflections occurring at the $S/F$ interfaces, as shown by Eq. (\[thetaF\]). Interestingly, Rowell and McMillan have already observed that an internal property of a $S$ layer can be seen through the superconducting proximity effect occurring in a nearby normal layer. More precisely, these authors have found that the DOS of an layer can reveal the phonon spectrum of an adjacent superconducting layer[@Rowell]. Remarkably, the visibility of $H_{eff}$ in $N^{F}(\varepsilon)$ is modulated by quantum interferences occurring in $F$. Indeed, $H_{eff}$ is more visible for certain values of $d_{F}$ (e.g. $d_{F}/\xi_{F}=1.0$ or $1.2$ in Fig. \[DOS2\]) than others (e.g. $d_{F}/\xi_{F}=2.1$ in Fig. \[DOS2\]), due to the $d_{F}$-dependence of Eq.(\[thetaF\]). It is useful to note that the SDIPS-induced effective field $H_{eff}$ should also occur in the $S/F$ bilayer of Figure 1.b. In this case, the Thouless energy and normal state conductance of the $S$ layer correspond to $\widetilde{E}_{TH}^{S}=4E_{TH}^{S}$ and $\widetilde{G}_{S}=2G_{S}$ respectively, so that one finds $g\mu_{B}H_{eff}=\widetilde{E}_{TH}^{S}% G_{\phi}^{S}/\widetilde{G}_{S}$. Double gap structures strikingly similar to those shown in Figure \[DOS2\] were indeed measured very recently by P. SanGiorgio et al., at the ferromagnetic side of diffusive bilayers, in the absence of any external magnetic field[@SanGiorgio]. Remarkably, the visibility of the observed double structures varies with $d_{F}$, as predicted above. Note that in $S/F$ bilayers, the field $H_{eff}$ should also be observable directly at the $S$ side by measuring $N^{S}% (\varepsilon)=\sum\nolimits_{\sigma\in\{\uparrow,\downarrow\}}N_{\sigma }(x=0,\varepsilon)$. However, for parameters comparable to those of Figure \[DOS2\], this should not enhance the resolution on $H_{eff}$ (see figure \[DOS3\], left). For certain values of $d_{F}$, $H_{eff}$ is even more visible in $N^{F}(\varepsilon)$ than in $N^{S}(\varepsilon)$(see Figure \[DOS2\]). Before concluding this section, I would like to emphasize that from Eqs. (\[thetaS\]) and (\[Om2\]), in the $AP$ configuration, the SDIPS-induced effective field $H_{eff}$ disappears for the $F/S/F$ structure considered in this paper because the two contacts are assumed to be symmetric and have thus opposite contributions to $H_{eff}$ in the $AP$ case. In the case of a dissymmetric structure, this should not be true anymore, but the SDIPS-induced effective field should nevertheless vary from the $P$ to the $AP$ case. This is one practical advantage of working with $F/S/F$ trilayers instead of $S/F$ bilayers. ![Energy dependent density of states $N^{S}(\varepsilon)$ measurable at the S side ($x=0$) of the $S/F$ structure of Figure 1.b. The left panel corresponds to the case of a metallic $F$ contact with parameters corresponding to that of Figs. \[DOS1\] and \[DOS2\]. In this case, the DOS measured at the $S$ side does not allow to resolve the SDIPS-induced effective field $H_{eff}$ better than a DOS measurement at the $F$ side of the structure, as can be seen from a comparison with Fig. \[DOS2\]. For certain values of $d_{F}$, $H_{eff}$ is even more visible in $N^{F}(\varepsilon)$ than $N^{S}(\varepsilon)$. The right panel corresponds to the case in which $F$ is not a metal but an insulating ferromagnet, i.e. $\gamma_{T}=0$ . In this case, one can use the reduced SDIPS parameter $\lambda_{\phi}^{S}=G_{\phi}^{S}\xi_{S}/A\sigma_{S}$. For $\lambda_{\phi}^{S}\neq0$, $N^{S}(\varepsilon)$ shows strong signatures of the effective field $H_{eff}$ $\ $induced by the $FI$ layer in $S$. One can check that the hypotheses 1 to 4 are valid for the parameters used in this Figure.[]{data-label="DOS3"}](DOS3.eps){width="1\linewidth"} Low-temperature density of states of $S/FI$ bilayers ---------------------------------------------------- Twenty years ago, internal Zeeman fields were observed in superconducting Al layers contacted to different types of ferromagnetic insulators ($FI$) (see Refs. ). Using a *ballistic* $S/FI$ bilayer model, Ref.  suggested that the observed internal fields could be induced by the SDIPS[@Tokuyasu]. However, the inadequacy of this ballistic approach for modeling the actual experiments was pointed out in Ref. . Most of the experiments on Al$/FI$ interfaces were interpreted by their authors in terms of a diffusive approach with no SDIPS, and an internal Zeeman field added arbitrarily in the Al layer (see Refs. ). The calculations of Section III.B. provide a microscopic justification for the use of such an internal field in the diffusive model. Indeed, using $\gamma_{T}=0$ in the above calculations allows one to address the case of diffusive $S/FI$ bilayers. One finds that the SDIPS-induced effective field $H_{eff}$ of Eq. (\[heff\]) can occur in a thin diffusive $S$ layer contacted to a $FI$ layer. This effective field effect can be seen e.g. in the density of states $N^{S}(\varepsilon)$ of the $S$ layer at $x=0$ (see Figure \[DOS3\], right). Remarkably, it was found experimentally[@Hao] that $H_{eff}$ scales with $d_{s}^{-1}$, in agreement with Eq. (\[heff\])[@NoteDG]. SDIPS-induced effective fields in other types of system ------------------------------------------------------- Interestingly, the SDIPS can induce effective field effects in other types of systems. First, the case of $S/N/FI$ trilayers with a thickness $d_{N}$ of normal metal $N$ has been studied theoretically[@demoBC; @applications]. In this case, a conductance $G_{\phi}^{N}$ similar to $G_{\phi}^{S}$ can be introduced to take into account the SDIPS for electrons reflected by the $FI$ layer. The $N$ layer is subject to an effective field $g\mu_{B}H_{eff}% ^{\prime}=E_{TH}G_{\phi}^{N}/G_{N}$ with $G_{N}$ the conductance of $N$. The expression of $H_{eff}^{\prime}$ is analogue to that of $H_{eff}$ (see Eq. \[heff\]), up to a factor 2 which accounts for the symmetry of the $F/S/F$ structure with respect to $x=0$ in the $P$ configuration. Secondly, an effective field $H_{eff}^{^{\prime\prime}}$ defined by $g\mu_{B}% H_{eff}^{^{\prime\prime}}=(\hbar v_{F}^{W}/2L)(\varphi_{L}^{\uparrow}% +\varphi_{R}^{\uparrow}-\varphi_{L}^{\downarrow}-\varphi_{R}^{\downarrow})$ has been predicted for a resonant single-channel ballistic wire with length $L$ placed between two ferromagnetic contacts[@CottetEurophys], with $\varphi_{L(R)}^{\sigma}$ the reflection phase of electrons with spin $\sigma$ incident from the wire onto the left(right) contact, and $v_{F}^{W}$ is the Fermi velocity in the wire. The expression of $H_{eff}^{^{\prime\prime}}$ also shows strong similarities with that of $H_{eff}$ (see Eq. \[heff\], middle term). In practice, signatures of the field $H_{eff}^{^{\prime\prime}}$ could be identified in a carbon nanotube contacted with two ferromagnetic contacts [@Cottet06; @Sahoo]. The fields $H_{eff}$, $H_{eff}^{^{\prime}}$ and $H_{eff}^{^{\prime\prime}}$ have the same physical origin: the energies of the states localized in the central conductor of the structure depend on spin due to the spin-dependent phase shifts acquired by electrons at the boundaries of this conductor. In all cases, the DOS of the central conductor reveals the existence of the SDIPS-induced effective field only if it already presents a strong energy dependence near the Fermi energy in the absence of a SDIPS. In the $F/S/F$ case, this energy dependence is provided by the existence of the superconducting gap in $S$[@cond]. In the $S/N/FI $ case, it is provided by the existence of a superconducting minigap in $N$. At last, in the case of the ballistic wire, it is provided by the existence of resonant states in the wire. Comparison between the present work and other models for data interpretation in $S/F$ heterostructures ------------------------------------------------------------------------------------------------------ For characterizing the properties of $S/F$ interfaces, one has to interpret the experimental data showing the oscillations of the density of states $N^{F}(\varepsilon)$ in $F$ with the thickness $d_{F}$ of $F$ (or the oscillations of the critical current $I_{0}$ of a $S/F/S$ Josephson with $d_{F}$). However, if one uses a simple description with spin-degenerate boundary conditions, the amplitude and the phase of these signals are not independent, which makes the agreement with experimental results impossible in most cases. The SDIPS concept can solve this problem since it produces a shift of the signals oscillations with respect to the $G_{\phi}^{S(F)}=0$ case. However, in many cases, the observed shifts were attributed to the existence of a magnetically dead layer (MDL) at the $F$ side of the $S/F$ interface (see e.g. Refs.  ). In other cases, the discrepancy between the theory and the data was solved by taking into account spin-scattering processes in the $F$ layer (see e.g. Ref. ). In order to have a better insight into superconducting proximity effect experiments, one must stress the importance of estimating experimentally the MDL thickness and the spin-scattering rate. In principle, spin-scattering rates can be estimated experimentally, as was done for instance for the alloy[@Pratt] which is frequently used in proximity effect measurements (see e.g. ). An experimental determination of the MDL thickness has also been performed in a few structures used to measure $T_{c}$ or $I_{0}% $[@Muhge; @Aarts; @Piano2; @Piano; @Bell], but, so far, this parameter has not been used for a real quantitative analysis of the data. In some situations, a model combining the SDIPS with spin-scattering and/or a MDL may be necessary. In any case, it is important to point out that descriptions based on spin-degenerate boundary conditions are, in principle, incomplete since they do not account for the effective field effect described in section III.B. Before concluding this section, it is interesting to note that the effective field effect produced by $G_{\phi}^{S}$ in $S$ or the phase shift of the spatial oscillations of the DOS provided by $G_{\phi}^{F}$ in $F$ will remain qualitatively similar when the signs of $G_{\phi}^{S}$ and $G_{\phi}^{F}$ are changed (not shown). The sign of $G_{\phi}^{F}$ in the experiments of Refs.  could be determined from a quantitative comparison between the theory and the data[@cottet05]. Below, we present a study of the critical temperature of $S/F$ structures which can give information on the signs of $G_{\phi}^{S}$ and $G_{\phi}^{F}$ through qualitative signatures. Critical temperature of $S/F$ heterostructures ---------------------------------------------- The critical temperature of $S/F$ hybrid structures has already been the topic of many theoretical (see e.g. Refs. ) and experimental (see e.g. Refs. ) studies, but the effects of the SDIPS on this quantity have raised little attention so far. I show below that the SDIPS can significantly modify the critical temperatures of $S/F$ diffusive structures. Calculating the critical temperature $T_{c}^{\mathcal{C}}$ of the structure of Figure 1.a in configuration $\mathcal{C}$ requires to consider the limit in which superconducting correlations are weak in $S$ as well as in $F$ (hypotheses 1 and 2 are then automatically satisfied). Equations (\[bP\]), (\[bAP\]) and (\[bc\]), then lead to ![Critical temperature $T_{c}^{P}$ for the $F/S/F$ structure of Figure 1.a in the $P$ configuration (top panels) and difference $\Delta T_{c}=T_{c}^{AP}-T_{c}^{P}$ between the critical temperatures in the $P$ and $AP$ configurations (bottom panels), as a function of the thickness $d_{F}$ of the $F$ layers. The left panels show the effect of a finite $\gamma_{\phi}^{S}$ and the right panels the effect of a finite $\gamma_{\phi}^{F}$. The four panels show the case $\gamma_{\phi}% ^{S}=\gamma_{\phi}^{F}=0$ with full red lines, for comparison. The other parameters used in the Figure are $\gamma=1$, $\gamma_{T}=0.12$ and $d_{S}% /\xi_{S}=1$. The SDIPS modifies $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ in a quantitative or qualitative way, depending on the case considered. The type of effects produced by the SDIPS on $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ strongly depend on the signs of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$.[]{data-label="FSFTc"}](FSFTc.eps){width="1\linewidth"} $$\widetilde{\theta}_{n,\sigma}^{\mathcal{C}}=\frac{\Delta^{\mathcal{C}}% }{\left| \omega_{n}\right| +2\Delta_{BCS}b_{n,\sigma}^{\mathcal{C}}% \delta_{S}^{-1}}%$$ with $$b_{n,\sigma}^{P}=\frac{4\mathcal{L}_{n,\sigma}^{0}}{4+\delta_{S}% \mathcal{L}_{n,\sigma}^{0}} \label{bp}%$$$$b_{n,\sigma}^{AP}=\frac{8\operatorname{Re}[\mathcal{L}_{n,\sigma}% ^{0}]+4\left| \mathcal{L}_{n,\sigma}^{0}\right| ^{2}\delta_{S}% }{8+6\operatorname{Re}[\mathcal{L}_{n,\sigma}^{0}]\delta_{S}+\left| \mathcal{L}_{n,\sigma}^{0}\right| ^{2}\delta_{S}^{2}} \label{bap}%$$ and $$\frac{\mathcal{L}_{n,\sigma}^{0}}{\gamma}=\frac{\gamma_{T}(B_{n,\sigma}% ^{P,2}+i\gamma_{\phi}^{F}\sigma\mathrm{sgn}(\omega_{n}))}{\gamma _{T}+B_{n,\sigma}^{P,2}+i\gamma_{\phi}^{F}\sigma\mathrm{sgn}(\omega_{n}% )}+i\gamma_{\phi}^{S}\sigma\mathrm{sgn}(\omega_{n})$$ These Eqs. together with (\[Delta\]) lead to $$\log\left( \frac{T_{c}^{BCS}}{T_{c}^{\mathcal{C}}}\right) =\operatorname{Re}% \left[ \Psi\left( \frac{1}{2}+\frac{b_{n,\sigma}^{\mathcal{C}}}{\delta _{S}\exp(\Gamma)}\frac{T_{c}^{BCS}}{T_{c}^{\mathcal{C}}}\right) \right] -\Psi\left( \frac{1}{2}\right) \label{Tc}%$$ where $\Gamma$ denotes Euler’s constant. The resulting $T_{c}^{\mathcal{C}}$ is independent from the values of $n$ and $\sigma$ used in Eq. (\[Tc\]). Note that in the case $\gamma_{\phi}^{S}=\gamma_{\phi}^{F}=0$ and $\delta _{S}\rightarrow0$, this equation is in agreement with Eqs. (18) and (19) of Ref. . Performing a numerical resolution of Eq. (\[Tc\]) together with (\[bp\]) and (\[bap\]), one obtains the results of Fig. \[FSFTc\], which shows the critical temperature $T_{c}^{P}$ of the structure in the $P$ configuration (top panels) and the difference $\Delta T_{c}=T_{c}^{AP}-T_{c}^{P}$ (bottom panels) as a function of $d_{F}$, for different interface parameters[@approx]. Here, for simplicity, I consider cases where $T_{c}^{\mathcal{C}}$ does not show a strongly reentrant behavior, i.e. a cancellation of $T_{c}^{\mathcal{C}}$ in a certain interval of $d_{F}$. Such a behavior can happen for instance for larger values of $\gamma_{T}$ (see e.g. Ref. ) or $\gamma_{\phi}^{(S)F}$, but this corresponds to a minority[@Tccoupe] of the experimental observations made so far. The four panels of Fig. \[FSFTc\] show the case $\gamma_{\phi}^{S}=\gamma_{\phi }^{F}=0$ with full red lines, for comparison with the other cases, where the SDIPS is finite. I first comment the results obtained for the $T_{c}^{P}% (d_{F})$ curves (top panels of Fig. \[FSFTc\]). In some cases, the SDIPS can modify quantitatively the $T_{c}^{P}(d_{F})$ curves, for instance by amplifying the dip expected in $T_{c}^{P}(d_{F})$ in the absence of a SDIPS (see e.g. dotted curve in the upper left panel, corresponding to $\gamma _{\phi}^{S}>0$). In some other cases the SDIPS can modify qualitatively the $T_{c}^{P}(d_{F})$ curves, for instance by transforming the minimum expected in $T_{c}^{P}(d_{F})$ into a maximum (see e.g. curve with circles in the upper right panel, corresponding to $\gamma_{\phi}^{F}<0$). I now comment the results obtained for the $\Delta T_{c}(d_{F})$ curves (bottom panels of Fig. \[FSFTc\]). In the range of parameters studied in this work, one always finds $\Delta T_{c}>0$ because the effects of the two $F$ layers on $S$ are partially compensated in the $AP$ case. In the absence of a SDIPS and for a low $\gamma_{T}$, the $\Delta T_{c}(d_{F})$ curve presents a maximum at a finite value of $d_{F}$ (see red full lines in bottom panels). In some cases, the SDIPS can modify quantitatively the $\Delta T_{c}(d_{F})$ curves, for instance by increasing the value of this maximum (see dotted curve in the bottom left panel, corresponding to $\gamma_{\phi}^{S}>0$). In other cases, the SDIPS can modify qualitatively the $\Delta T_{c}(d_{F})$ curves, for instance by turning the maximum expected in $\Delta T_{c}(d_{F})$ into a minimum (see dashed curve in the bottom left panel, corresponding to $\gamma_{\phi}^{S}<0$), or by increasing the complexity of the variations of $\Delta T_{c}$ with $d_{F}$ (see curve with circles in the bottom right panel, corresponding to $\gamma_{\phi}^{F}<0$), or by transforming $\Delta T_{c}(d_{F})$ into a monotonically decreasing curve with a reduced amplitude (see dot-dashed curve in bottom right panel, corresponding to $\gamma_{\phi }^{F}>0$). Remarkably, the type of effects produced by the SDIPS on the $T_{c}^{\mathcal{C}}(d_{F})$ and $\Delta T_{c}(d_{F})$ curves depend on the sign of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. Critical temperature measurements can thus be an interesting way to determine the values of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. For simplicity, I have shown here results for $\gamma_{\phi}^{F}\neq0$ and $\gamma_{\phi}^{S}\neq0$ separately. Nevertheless, the behavior predicted for $\gamma_{\phi}^{F}\neq0$ and $\gamma_{\phi}^{S}\neq0$ simultaneously remain highly informative on the values of $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$. Note that if $\gamma_{\phi}^{F}$ and $\gamma_{\phi}^{S}$ are increased compared to the values used in Fig. \[FSFTc\], the $T_{c}^{\mathcal{C}}(d_{F})$ curve gets some cancellation points like in Ref. , but the behaviors of $T_{c}^{\mathcal{C}}$ and $\Delta T_{c}$ remain, in many cases, qualitatively dependent on the signs of $\gamma_{\phi}^{F}$ and $\gamma_{\phi }^{S}$ (not shown). Before concluding, I note that with the approximations used in the previous section (III.B), the electronic correlations inside $S$ were affected by the presence of the $F$ electrodes through the parameter $\gamma_{\phi}^{S}$ only. Consequently, the self-consistent gap $\Delta^{\mathcal{C}}$ of the $S$ layer was independent from $d_{F}$, and it was furthermore identical for the $P$ and $AP$ configurations for $\gamma_{\phi}^{S}=0$. In the present section, I have not neglected the dependence of $T_{c}^{P}$ and $T_{c}^{AP}$ on $d_{F}$ and $\gamma_{\phi}^{F}$ because I have considered parameters for which hypothesis 3 is not acceptable anymore. In particular, I have used lower values for $d_{F}$[@regimes]. Conclusion ========== This article shows that the Spin-Dependence of Interfacial Phase Shifts (SDIPS) can have a large variety of signatures in diffusive superconducting/ferromagnetic ($S/F$) heterostructures. Ref. had already predicted that the SDIPS produces a phase shifting of the oscillations of the superconducting correlations with the thickness of $F $ layers. This article shows that this is not the only consequence of the SDIPS in $S/F$ circuits. In particular, the SDIPS can produce an effective magnetic field in a diffusive $S$ layer contacted to a diffusive $F$ layer. This effective field can be seen e.g. through the DOS of the diffusive $F$ layer, with a visibility which oscillates with the thickness of $F$. The SDIPS can also modify significantly the variations of the critical temperature of a $S/F$ bilayer or a $F/S/F$ trilayer with the thickness of $F $, either in a quantitative or in a qualitative way, depending on the regime of parameters considered and in particular the sign of the conductances $G_{\phi}^{S}$ and $G_{\phi}^{F}$ used to account for the SDIPS of the $S/F$ interfaces. In the case of a $F/S/F$ spin valve, this last result also holds for the thickness-dependence of the difference between the critical temperatures in the parallel and antiparallel lead configurations. These effects should help to determine the parameters $G_{\phi}^{S}$ and $G_{\phi}^{F}$ of diffusive $S/F$ interfaces. The calculations shown in this paper are also appropriate to the case of thin diffusive $S$ layers contacted to ferromagnetic insulators. *I thank P. SanGiorgio for showing me his experimental data prior to publication, which stimulated part III.B of this work. I acknowledge discussions with T. Kontos and W. Belzig. This work was supported by grants from the Swiss National Science Foundation and Région Ile-de-France.* Appendix A: Parameters of a $S/F$ interface from a Dirac barrier model ====================================================================== ![Conductances $G_{T}$ (full line), $\left| G_{\phi}^{S}\right| $ (dashed lines), and $\left| G_{\phi}^{F}\right| $ (dash-dotted lines) reduced by the number of channels $n$, as a function of the spin-averaged barrier strength $Z$ of a $S/F$ interface modeled with a Dirac barrier (see text). The curves were obtained with typical Fermi energies $E_{f}^{F}=0.8~\mathrm{eV}$ and $E_{f}% ^{S}=12~\mathrm{eV}$ in $F$ and $S$ respectively, a spin-polarization $P=0.06$ of the density of states in $F$, and a spin asymmetry $\alpha=0.06$ for the barrier. Note that for the set of parameters used this Figure, one finds $G_{\phi}^{F}<0$ and $G_{\phi}^{S}<0$. However, for other parameters (e.g. by using $\alpha<0$), one can reverse the signs of $G_{\phi}^{F}$ and $G_{\phi }^{S}$, or obtain opposite signs for $G_{\phi}^{F}$ and $G_{\phi}^{S}$ (not shown).[]{data-label="Dirac"}](Dirac.eps){width="0.85\linewidth"} The exact values of the conductances $G_{T}$, $G_{\phi}^{F}$ and $G_{\phi}% ^{S}$ of a $S/F$ interface depend on the details of this interface and on the microscopic structure of the contacted materials. Nevertheless, it is already interesting to study a simplified Dirac barrier model which shows that the parameters regime assumed in this paper is, in principle, possible. Neglecting the transverse part of the electrons motion, one finds $r_{n,\sigma}% ^{S(F)}=(k_{S(F)}^{\sigma}-k_{F(S)}^{\sigma}-iZ^{\sigma})/D^{\sigma}$ and $t_{n,\sigma}^{S(F)}=2(k_{S}^{\sigma}k_{F}^{\sigma})^{1/2}/D^{\sigma}$ with $D^{\sigma}=k_{S}^{\sigma}+k_{F}^{\sigma}+iZ^{\sigma}$. Here, $k_{S(F)}% ^{\sigma}$ is the Fermi electronic wavevector in $S(F)$ and $Z^{\sigma}$ is the strength of the Dirac barrier for electrons with spin $\sigma$. I use $\hbar k_{S}^{\sigma}=(2m_{e}E_{f}^{S})^{1/2}$, with $m_{e}$ the free electron mass and $E_{f}^{S}$ the Fermi level in $S$. For $F$, I use an s-band Stoner model, in which $\hbar k_{F}^{\sigma}=(2m_{e}E_{f}^{F}[1\pm2P/(1+P^{2}% )])^{1/2}$, with $P$ the spin-polarization of the density of states in $F$ and $E_{f}^{F}$ the Fermi level in $F$. I assume that the barrier can have a spin dependence $\alpha=(Z^{\downarrow}-Z^{\uparrow})/(Z^{\uparrow}+Z^{\downarrow })$ due to the ferromagnetic contact material used to form the interface. The results given by this approach are shown in Fig. \[Dirac\]. The conductances $G_{T}$, $\left| G_{\phi}^{F}\right| $ and $\left| G_{\phi}^{S}\right| $, reduced by the number of channels $n$, are shown as a function of the spin-averaged barrier strength $Z=(Z^{\uparrow}+Z^{\downarrow})/2$. The three types of conductances go to zero when $Z$ goes to infinity with $\alpha$ constant[@lim]. One can see that in the limit $T_{n}\ll1$ (i.e. here $G_{T}h/ne^{2}\ll1$) and $P\ll1$ in which the boundary conditions (\[BCright\],\[BCleft\]) have been derived, $\left| G_{\phi}^{F}\right| $ can be significantly stronger than $G_{T}$, as assumed in Figs. \[DOS1\] and \[DOS2\]. Note that for the set of parameters used in Fig. \[Dirac\], one finds $G_{\phi}^{F}<0$ and $G_{\phi}^{S}<0$. However, for other parameters (e.g. by using $\alpha<0$), one can reverse the signs of $G_{\phi}^{F}$ and $G_{\phi}^{S}$, or obtain opposite signs for $G_{\phi}^{F}$ and $G_{\phi}^{S}$ (not shown). This suggests that there is no fundamental constraint on the signs of $G_{\phi}^{F}$ and $G_{\phi}^{S}$ in the general case. Appendix B: Effective field effect in a $F$ layer ================================================= This appendix reconsiders the case of the $S/F$ bilayer of Fig.1.b, in the limit $d_{F}\ll\xi_{F}$ where $\theta_{n,\sigma}(x)$ can be approximated with a quadratic form in the $F$ layer. From Eqs. (\[UsadelF2\]) and (\[BCright\]), one finds, for $x\in\lbrack d_{S}/2,d_{S}/2+d_{F}]$, $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{F}+b_{n,\sigma}^{F}\left[ \frac{d_{S}-2x}{d_{F}}+\left( \frac{d_{S}-2x}{2d_{F}}\right) ^{2}\right]$$ with $$\widetilde{\theta}_{n,\sigma}^{F}=\arctan\left( \frac{\gamma_{T}% \sin(\widetilde{\theta}_{n,\sigma}^{S})}{\gamma_{T}\cos(\widetilde{\theta }_{n,\sigma}^{S})+\frac{2\xi_{F}}{d_{F}}\frac{\Omega_{n}}{E_{TH}^{F}}% +i\gamma_{\phi}^{F}\eta_{n,\sigma}^{P,2}}\right) \label{ThetaF}%$$$$b_{n,\sigma}^{F}=\frac{\Omega_{n}\sin(\widetilde{\theta}_{n,\sigma}^{F}% )}{E_{TH}^{F}}%$$ $\Omega_{n}=\left| \omega_{n}\right| +iE_{ex}\eta_{n,\sigma}^{P,2}$ and $\widetilde{\theta}_{n,\sigma}^{S}=\theta_{n,\sigma}^{S}(x=d_{S}/2)$. Here, $G_{F}=\sigma_{F}A/d_{F}$ and $E_{TH}^{F}=\hbar D_{F}/d_{F}^{2}$ denote the conductance and Thouless energy of the $F$ layer (note that $d_{F}\ll\xi _{F}\Longleftrightarrow E_{ex}\ll E_{TH}^{F}$). From completeness, I also give the analogous equations for $x\in\lbrack0,d_{S}/2]$. Assuming that $\theta_{n,\sigma}(x)$ can be approximated with a quadratic form in the $S$ layer, one finds, from Eqs. (\[UsadelS\]) and (\[BCleft\]), $$\theta_{n,\sigma}(x)=\widetilde{\theta}_{n,\sigma}^{S}+b_{n,\sigma}^{S}\left[ \frac{2(2x-d_{S})}{d_{S}}+\left( \frac{2x-d_{S}}{d_{S}}\right) ^{2}\right]$$ with $$\widetilde{\theta}_{n,\sigma}^{S}=\mathrm{\arctan}\left( \frac{\gamma \gamma_{T}\sin(\widetilde{\theta}_{n,\sigma}^{F})+\frac{4\xi_{S}}{d_{S}}% \frac{\Delta}{\widetilde{E}_{TH}^{S}}}{\gamma\gamma_{T}\cos(\widetilde{\theta }_{n,\sigma}^{F})+\frac{4\xi_{S}}{d_{S}}\frac{\left| \omega_{n}\right| }{\widetilde{E}_{TH}^{S}}+i\gamma\gamma_{\phi}^{S}\eta_{n,\sigma}^{P,2}% }\right) \label{ThetaS}%$$ and $$b_{n,\sigma}^{S}=\frac{\left| \omega_{n}\right| \sin(\widetilde{\theta }_{n,\sigma}^{S})-\Delta\cos(\widetilde{\theta}_{n,\sigma}^{S})}{\widetilde {E}_{TH}^{S}}%$$ The notations used in the above equations are the same as in Section III. In particular, the Thouless energy and the normal state conductance of the $S$ layer with thickness $d_{S}/2$ are denoted $\widetilde{G}_{S}=2\sigma _{S}A/d_{S}$ and $\widetilde{E}_{TH}^{S}=4\hbar D_{S}/d_{S}^{2}$, and one has $\eta_{n,\sigma}^{P,2}=\sigma\mathrm{sgn}(\omega_{n})$. In the limit $\widetilde{\theta}_{n,\sigma}^{F}\ll1$, Eq. (\[ThetaS\]) is in agreement with Eqs. (\[thetaS\]) and (\[omegaa\]). The analytic continuation of Eq. (\[ThetaS\]) shows that the $S$ layer is subject to the effective field $H_{eff}=\widetilde{E}_{TH}^{S}G_{\phi}^{S}/\widetilde{G}_{S}$, in agreement with Eq. (\[heff\]). The analytic continuation of Eq. (\[ThetaF\]) shows that the $F$ layer is subject to an analogue effective field $H_{eff}^{F}$, defined by $$g\mu_{B}H_{eff}^{F}=E_{TH}^{F}\frac{G_{\phi}^{F}}{G_{F}}%$$ It is interesting to compare Eqs. (\[ThetaF\]) and (\[thetaF\]). First, note that in the regime $\left| \widetilde{\theta}_{n,\sigma}^{F}\right| \ll1$, Eq. (\[ThetaF\]) can be recovered from the low $d_{F}$-limit of Eq. (\[thetaF\]). The interest of Eq. (\[ThetaF\]) is that it is valid beyond the regime of a weak superconducting proximity effect studied in part III. In particular, Eq. (\[ThetaF\]) indicates that, in principle, the field $E_{TH}^{F}$ can occur in a thin $F$ layer for any value of the tunneling conductance $G_{T}$ of the $S/F$ interface. Interestingly, in the case of a thick $F$ layer $d_{F}\geq\xi_{F}/2$, from Eq. (\[thetaF\]), the contribution of $G_{\phi}^{F}$ to the pairing angle $\theta_{n,\sigma}^{F}(x)$ cannot be put anymore under the form of an effective exchange field, due to the non-linearity of the $B_{n,\sigma}^{\mathcal{C},j}$ term. Thus, strictly speaking, the concept of a SDIPS-induced effective field is valid only in the limit of thin metallic layers. However, it might be possible, in principle, to observe reminiscences of the double-gap structure appearing in $N^{F}% (\varepsilon)$ in the regime of intermediate layer thicknesses $d_{F}\sim \xi_{F}$ at least, due to the continuity of the equations. Then, one can wonder why this effect does not appear in section III. This is due to the regime of parameters chosen: section III assumes $E_{ex}\gg\Delta_{BCS}$ like in most experiments, and it furthermore focuses on the typical energy range probed in superconducting proximity effect measurements, i.e. $\left| \varepsilon\right| \lesssim2\Delta_{BCS}$. In such a regime, one can neglect the term $\left| \omega_{n}\right| $ compared to $iE_{ex}\sigma \mathrm{sgn}(\omega_{n})$ in Eqs. (\[thetaF\]) and (\[ThetaF\]), so that $H_{eff}^{F}$ does not emerge in the model. Accordingly, signatures of the SDIPS-induced effective field $H_{eff}^{F}$ are not likely to be observed in practice. [99]{} A. I. Buzdin, L. N. Bulaevskii, and S. V. 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With the parameters of Fig. \[FSFTc\], the approximations $b_{n,\sigma}^{P}=\mathcal{L}_{n,\sigma}^{0}$ and $b_{n,\sigma}^{AP}% =\operatorname{Re}[\mathcal{L}_{n,\sigma}^{0}]$ used in Ref.  would be sufficient for calculating the $T_{c}^{P}$ and $T_{c}^{AP}$ curves, but not the difference $\Delta T_{c}$. Indeed, for the different cases studied in this Figure, the relative difference between the results obtained with these approximations and those given by Eqs. (\[bp\]) and (\[bap\]) is smaller than $\sim0.8\%$ for $T_{c}^{P}$ and $T_{c}^{AP}$, but it can reach $\sim18\%$ for $\Delta T_{c}$. This is why I have used the general equations (\[bp\]) and (\[bap\]) for plotting Figure \[FSFTc\]. L. R. Tagirov, I. A. Garifullin, N. N. Garifyanov, S. Ya. Khlebnikov, D. A. Tikhonov, K. Westerholt, and H. Zabel, J. Magn. Magn. Mater. **240**, 577 (2002). Note that both regimes $d_{F}\leq\xi_{F}$ (see e.g. Ref ) and $d_{F}\geq\xi_{F}$ (see e.g. Ref. ) seem to be accessible in practice. For $Z\rightarrow+\infty$, $G_{\phi}^{F(S)}$ vanishes because $\arg(r_{n,\sigma}^{F(S)})\rightarrow\pi$ and $\arg(t_{n,\sigma}% ^{F(S)})\rightarrow-\pi/2$ for $\sigma\in\{\uparrow,\downarrow\}$.