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 Program Peterhof V.A. Fock Institute of Physics 24 – 27 September Please click on the title to view the abstract of the talk 24 September, Plenary session (Fock Hall) Lipatov Lev Nikolaevich (PNPI, Gatchina)  Double-logarithmic asymptotics of scattering amplitudes in gravity and supergravity Diehl Hans Werner (University Duisburg-Essen)   Recent advances in the study of the thermodynamic Casimir effect A survey of recent applications of the renormalization group approach to the study of thermodynamic Casimir forces near critical points is given. The successes and limitations of approaches based on the small $\epsilon=4-d$ expansion are explained. The large-$n$ limit of the $O(n)$ $\phi^4$ theory on a $d$-dimensional strip of thickness $L$ bounded by free surfaces is expressed exactly in terms of the eigenvalues and eigenfunctions of a self-consistent Schrödinger equation. Numerically exact results for the scaling functions of the $L$-dependent part of the excess free energy and the Casimir force are presented for $d=3$. Apart from brute-force Monte Carlo simulations of lattice model, the $n\to\infty$ limit provides so far the only single framework that is (i) capable to deal simultaneously with confined critical fluctuations, confined Goldstone modes, and dimensional crossover and (ii) exhibits qualitatively all features of the temperature-dependent Casimir force known from wetting experiments on He and Monte Carlo simulations of the XY model on a strip. Faddeev Ludwig Dmitrievich (PDMI RAS, SPb)  Volkov's Pentagon for the Modular Quantum Dilogarithm Arefieva Irina Yaroslavna (Steklov Mathematical Institute RAS, Moscow)   Scattering in quark gluon plazma A-24 Renormalization group methods and condensed matter (Fock Hall) Kompaniets Mikhail Vladimirovich (SPbSU, SPb)   Multiloop calculations in normalization point scheme Nalimov Mikhail Yurievich (SPbSU, SPb)   Temperature Green function formalism in Fermi system: superconducting phase transition. Temperature Green function formalism is applied to Fermi gas with density-density interaction. New field variables corresponding to critical modes are introduced. Usual mean-field theory can be applied in the new variables. A phase transition is investigated in the framework of the quantum-field renormalization group and $\epsilon$ -- expansion. As a result it is obtained that the character of the phase transition depends on fermion spin. Zalom Peter (Inst. Experim. Phys., Kosice)   Spatial parity violation and the turbulent magnetic Prandtl number Using the field theoretic renormalization group technique we have considered the influence of helicity (violation of spatial parity) on the diffusion processes of a passively advected vector field in the framework of the kinematic MHD turbulence. The influence of the helicity on the turbulent magnetic Prandtl number is analyzed. A nontrivial impact of the internal structure of the advected field on the diffusion rate is analytically described. Danco Michal (Inst. Experim. Phys., Kosice)   Effect Of Hydrodynamic Fluctuations On Phase Transition In Critical Models E and F The models E and F of critical dynamics are studied in the presence of random velocity fluctuations. Stochastic Navier-Stokes equation is used for generation of velocity fluctuations. Using Martin-Siggia-Rose procedure field theoretic model is obtained. The power counting and analysis of ultraviolet divergences are performed near the critical dimension of the models. Both of them are renormalized in the one-loop approximation using dimensional regularization in minimal subtraction scheme. Influence of hydrodynamic fluctuations on the stability near phase transition point is discussed. Komarova Marina (SPbSU, SPb)   Phase transitions types in critical dynamics B-24 Gravitation and cosmology (429 hall) Paston Sergey Alexandrovich (SPbSU, SPb)   Embeddings for solutions of Einstein equations The method of searching the embeddings of Riemannian spaces with a high enough symmetry in a flat ambient space is suggested. It is based on a procedure of construction of surfaces with a given symmetry. The method is used for the construction of the embeddings for the Schwarzschild solution in a six-dimensional ambient space and Friedmann-Robertson-Walker solutions in five-dimensional ambient space. All known variants of embeddings for these spaces are constructed using this method. Two new embeddings for Schwarzschild solution are constructed also. One of these new embeddings is asymptotically flat, while the other embeddings in a six-dimensional ambient space do not have this property. The asymptotically flat embedding can be used in the analysis of the many-body problem, as well as for the development of gravity description as a theory of a surface in a flat ambient space. /In collaboration with A.A. Sheykin/ Borowiec Andrzej (Univ. of Wroclaw, Wroclaw)   New Cosmological Models of Palatini Gravity Grib Andrey Anatolievich (Herzen University, SPb)   Active Galactic Nuclei as supercolliders of elementary particles It is shown that the energy of two colliding particles in the erhosphere of the rotating black hole close to the horizon can get values close to the Planckean scale.This occurs due to the possibility of very small radial velocity of one of the colliding particle called critical. The mechanism of getting critical particles by previous scattering and receiving critical momentum is analysed.The possibility to explain the observations of Ultra high energetic cosmic particles by the Auger group is discuused. Manida Sergey Nikolaevich (SPbSU, SPb)   Big Bang Distribution in R-space The Anti-de Sitter space in Beltrami coordinates is mostly suitable for modern cosmology because it has (quasi "Big Bang") singularity. It is possible to find the initial invariant distributions of the matter in the phase space, compute its dependance on time and compare the result with modern observations. C-24 Integrable models (Blue Hall) Bogoliubov Nikolay Mikhailovich (PDMI RAS, SPb)  Scalar products of the state vectors in the totally asymmetric exactly solvable models on a ring. The exactly solvable totally asymmetric models of the low dimensional non-equilibrium physics on a ring, namely the totally asymmetric simple exclusion process and the totally asymmetric simple zero range process, are considered. The Quantum Inverse Method allows to calculate the scalar products of the state vectors of the models and to represent the answers in the determinantal form. It is shown that the eigenvectors of the models form a complete orthogonal basis. The projections of the state vectors on a stationary states are studied. Isaev Alexey Petrovich (JINR, Dubna)   Discrete evolution operator for q-deformed top and Faddeev's modular double The structure of a cotangent bundle is investigated for quantum linear groups $GL_{q}(n)$ and $SL_{q}(n)$. Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on $SL_{q}(n)$ (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We calculate an evolution operator for the model of q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Connection between the two operators is given by a modular functional equation for Riemann theta function. Relations to the construction of the Faddeev's modular double will be discussed. Kapitonov Valery Stepanovich (SPbGTI(TU))  Grassmanian integral approach to the vertex models. The fermionization method is applied to the vertex models with boundary conditions. The six vertex model with the domain wall boundary conditions has been studied in detail. The two-point correlation function and the generating functional are given in an explicit form in the free fermion point. 25 September, Plenary session (Fock Hall) Kazakov Dmitry Igorevich (JINR, Dubna)   Gauge theories in spin-helicity formalism I would like to advocate the new approach to gauge theories put forward by Z.Bern, L.Dixon, D.Kosower et al which allows to perform calculations of gauge invariant quantities without fixing the gauge, without ghosts, etc and drastically simplifies the calculations. I will present some explicit examples and discuss the subtleties. Honkonen Juha (National Defense University, Helsinki)   Path-ordered Green functions in stochastic problems Andrianov Alexander Andreevich (SPbSU, SPb)   Flying of Vector Particles from a Parity Breaking (Chern-Simons) Medium to Vacuum and Back The problem of propagation of photons and massive vector mesons in the presence of Lorenz and CPT invariance violating medium (Chern-Simons action) will be dicussed when the parity-odd medium is separated from the vacuum along a hyperplane. The solutions in different half-spaces will be matched both in the case of space-like and time-like Chern-Simons action by means of the Bogolubov transformations. Accordingly the two different Fock vacua (zero particle states) happen to be mutually coherent states. The boundary mirror/transparency properties will be illustrated. They can help to register local parity violation in a finite volume of heavy ion fireball with nontrivial axial charge and/or of a star with cold axion condensate degrading to its boundary. A-25 Renormalization group methods and condensed matter (429 hall) Braun Mikhail Alexandrovich (SPbSU, SPb)   Quantized field energy in a medium with dispersion and absorption Remecky Richard (JINR, Dubna)   Passive Vector Advection in Developed Turbulence: Renormalization Group Results Lucivjansky Tomas (Pavol-Jozef Safarik University, Kosice)   Effect Of Compressibility On The Annihilation Process Annihilation processes, where the reacting particles are influenced by some external advective field, are one of the simplest examples of nonlinear statistical systems. This type of processes can be observed in miscellaneous chemical, biological or physical systems. In low space dimensions usual description by means of kinetic rate equation is not sufficient and the effect of density fluctuations must be taken into account. Using perturbative renormalization group we study the influence of random velocity field on the kinetics of single-species annihilation reaction at and below its critical dimension $d_c=2$. The advecting velocity field is modelled by the self-similar in space Gaussian variable finite correlated in time (Obukhov-Kraichnan model). Effect of the compressibility of velocity field is taken into account and the model is analyzed near its critical dimension by means of three-parameter expansion in $\epsilon,\Delta$ and $\eta$. Here $\epsilon$ is the deviation from the Kolmogorov scaling, $\Delta$ is the deviation from the (critical) space dimension $2$ and $\eta$ is the deviation from the parabolic dispersion law. Depending on the value of these exponents and the value of compressiblity parameter $\alpha$, the studied model can exhibit various asymptotic (long-time) regimes corresponding to the infrared fixed points of the renormalization group. The possible regimes are summarized and the decay rates for the mean particle number are calculated in the next-to-leading order of the pertubartion theory. Sokolov Aleksandr Ivanovich (SPbSU, SPb)   Phase transitions in two dimensions and pseudo-$\epsilon$ expansion The use of field-theoretical renormalization group for studying the critical behavior of two-dimensional models is discussed. RG functions of 2D $\lambda-\phi^4$ Euclidean n-vector theory are written down up to the five-loop terms. Numerical estimates obtained from these series by means of Pade-Borel-Leroy resummation are presented and compared with their exact counterparts known for $n = 1, 0, -1$. Then, pseudo-\epsilon expansions for the fixed-point location $g^*$, critical exponents and effective sextic coupling $g_6$ are derived from the five-loop series. The expansions obtained are shown to be more "friendly" than original RG series: the higher-order coefficients of pseudo-$\epsilon$ expansions for $g^*$, $g_6$ and $\gamma^{-1}$ turn out to be much smaller than their RG analogs. This enables one to resum the pseudo-$\epsilon$ expansions using simple Pade approximants, without addressing Borel-Leroy transformation. Moreover, numerical estimates extracted from the pseudo-$\epsilon$ expansions are found to be closer to the known exact values than those obtained from Pade-Borel-Leroy resummed five-loop RG series. Mizisin Lukas (Inst. Experim. Phys., Kosice)   Critical Behaviour Of Percolation Process Influenced by Random Velocity Field:Two-loop approximation Using perturbative renormalization group we study the influence of random velocity field on the critical behaviour of directed bond percolation process near its second-order phase transition between absorbing and active phase. Kraichnan model with finite correlation time for modelling advecting velocity field is considered. The field-theoretic renormalization group is applied to obtain information about possible large scale behavior. The model is analyzed near its critical dimension by means of three-parameter expansion in $\epsilon,\delta,\eta$, where $\epsilon$ is the deviation from the Kolmogorov scaling, $\delta$ the deviation from the critical space dimension $d_c$ and $\eta$ is the deviation from the parabolic dispersion law for the velocity correlator. Fixed points with corresponding regions of stability are evaluated to the second order in the perturbation scheme. D-25 Mathematical methods in QFT (Blue Hall) Derkachev Sergey Eduardovich (PDMI RAS, SPb)   Yang-Baxter equation and star-triangle relation The solution of the famous Yang-Baxter equation is usually called R-matrix. We consider the general scheme of the construction of R-matrix as integral operator. Everything is illustrated on various examples starting from the simplest one related to conformal group in two and four dimensions and finishing by elliptic Sklyanin algebra. Kotikov Anatoly Vasilievich (JINR, Dubna)   Calculation of complicated master integrals Several modern methods for calculation of complicated master integrals are considered. The results for two-loop two-point and three-point diagrams are presented. Lyakhovsky Vladimir Dmitrievich (SPbSU, SPb)  Expansion and approximation with multivariate Chebyshev polynomial bases Borzov Vadim (State University of Telecommunications, SPb)   Generalized Chebyshev oscillator The approach to construction of generalized oscillators, conтected with polynomials orthogonal on the real line, proposed by one of the authors early, is extended to the case of bivariate polynomials. As an interesting example we consider bivariate Chebyshev polynomials connected with representations of sl(3) algebra. The polynomials under consideration form a complete system of eigenfunctions for a quadratic hamiltonian of the constructed two-dimensional Chebyshev oscillator. We show that the concidered hamiltonian splits into the sum of two commuting operators. So it is possible to consider this two-dimensional oscillator as a system of two noninteracting oscillators. The generators of the first oscillator algebra leave invariant all N-particles subspaces while the ladder operators of the second oscillator transform neighboring N-particles subspaces one onto another. The representation of the ladder operators for two-dimensional Chebyshev oscillator by differential operators are given. /In collaboration with E.V.Damaskinski/ Malyshev Cyril Leonidovich (PDMI RAS, SPb)   Translational gauging, non-singular screw dislocations, and renormalization of the shear modulus E-25 Quantum field theory methods in elementary particles physics (Fock Hall) Bakulev Alexander Petrovich (JINR, Dubna)   Resummation in (F)APT with applications to hadron physics Gavrilov Sergey Petrovich (Herzen University, SPb)   Treatment of electron transport in graphene in the framework of Dirac model with unstable vacuum We treat electron transport in graphene monolayer close to the charge neutrality point in the framework of the Dirac model with unstable vacuum. It is shown that the electron-hole creation by a slowly varying electric field is crucial for the understanding the conductivity of graphene at low carrier density beyond linear response in dc. Self-consistent solutions of the Dirac-Maxwell equations are found in the strong-field approximation and it is shown that such solutions can explain the experimental super linear I-V of graphene devices that were recently reported. /In collaboration with D.M. Gitman, and N. Yokomizo from University of Sao Paulo, Brazil/ Kivel' Nikolay Alexandrovich (PINP, Gatchina)   Soft spectator scattering in the nucleon form factors at large $Q^2$ The proton form factors at large momentum transfer are dominated by two contributions which are associated with the hard and soft rescattering respectively. Motivated by a very active experimental form factor program at intermediate values of momentum transfers, $Q^{2}\sim5-15~\text{GeV}^{2}$, where an understanding in terms of only a hard scattering mechanism cannot yet be expected, we investigate the soft spectator scattering contribution using soft collinear effective theory (SCET). Within such description, the form factor is characterized, besides the hard scale $Q^2$, by a hard-collinear scale $Q \Lambda$, which arises due to presence of soft spectators, with virtuality $\Lambda^2$ ($\Lambda \sim 0.3-0.5$~GeV), such that $Q^{2}\gg Q\Lambda\gg \Lambda^{2}$. In case of nucleon FFs the soft spectator scattering mechanism contributes at leading power accuracy and therefore it must be considered in the systematic QCD factorization approach. We discuss how to generalize the well known Brodsky-Lepage collinear factorization in order to include the soft spectator contribution using the SCET technique. We carefully investigated the factorization of the soft and collinear modes in this case and demonstrate that even for the FF $F_{1}$ the pure collinear factorization could not be valid due to the mixing of the soft and collinear contributions. As a result this allows one to put specific constrains on the end-point behavior of the nucleon distribution amplitude. 26 September, Plenary session (Fock Hall) Lukierski Jerzy (Univ. of Wroclaw, Wroclaw)   From noncommutative space-timo to modified QFT We consider noncommutativity of space-time which is covariant under the action of quantum-deformed Poincare algebra. The consistency with quantum deformation of Poincare algebra leads to braided noncommutative field theory, with braid factors determined by the universal R-matrix. The technique will be applied to the kappa-deformation of free quantum fields. Wegner Franz (Ruprecht-Karls-Universitat Heidelberg)   Supersymmetry in Solid State Physics The notion of supersymmetry is used with at least three meanings. They are explained and several applications in solid state physics are given. Manashov Alexander (SPbSU & Regensburg University)   Kinematic power corrections in off-forward hard reactions. We develop a general approach to the calculation of kinematic corrections $\sim t/Q^2$, $\sim m^2/Q^2$ in hard processes which involve momentum transfer from the initial to the final hadron state. As the principal result, the complete expression is derived for the time-ordered product of two electromagnetic currents that includes all kinematic corrections to twist-four accuracy. The results are immediately applicable e.g. to the studies of deeply-virtual Compton scattering. 27 September, Plenary session (Fock Hall) Pajares Carlos (Santiago de Compostela Univ., Santiago de Compostela)   High density QCD and the new LHC data We compare the predictions of high density QCD with the LHC data on multiplicities,jet quenching,elliptic flow and long range rapidity correlations Please, write formulae using TeX notation, e.g: $\int \sin(x) dx$ Hnatic Michal (Inst. Experim. Phys., Kosice)   Princip of maximal randomness and parity violation in turbulence Physical quantity known as the helicity is a signature of parity violation in developed (magneto)hydrodynamic turbulence. The helicity similarly to the energy is time invariant in inviscid fluid. It occurs naturally in a wide variety of geophysical flows. In three-dimensional turbulence, contrary to the 2D turbulence, there is a joint direct cascade of both energy and helicity simultaneously from large to small scales. It can affect celebrated Kolmogorov scaling and change non-universal amplitudes like Kolmogorov constant in kinetic energy specrum and effective viscosity coefficient. A set of self-consistent equations in a statistical model of fully developed homogeneous isotropic turbulence, which is based on the principle of maximal randomness of the velocity field with a given energy spectral flux, is derived. The formal solution of these equations yields Kolmogorov exponents, but this solution leads to negative Kolmogorov constant ${\cal C}_k$ and negative viscosity, which thereafter induces the instability of turbulent flows. It has been demonstrated, that stable solution can be obtained by means of specific mechanism of spontaneous parity breaking. More precisely, the Kolmogorov solution becomes stable (both ${\cal C}_k$ and effective viscosity become positive) if one takes into account all allowable tensor structures in the statistical averages of velocity fluctuations including pseudotensor contributions generated by helicity. This solution predicts a large, closed to limit value ( equal to one), helical coefficient $\Theta$ in inertial range. The relationship obtained between ${\cal C}_k$ and $\Theta$ confirms this conclusion for the experimental value of ${\cal C}_k$. Antonov Nikolay Viktorovich (SPbSU, SPb)   Renormalization group in the problem of growing interface:  renormalizable model with infinitely many charges Within the framework of the field theoretic renormalization group (RG), a stochastic equation, proposed earlier for describing a randomly growing interface, is studied. Correct analysis of the model requires introducing an infinite number of counterterms and corresponding coupling constants (charges). The one-loop expression for the countertems can be obtained in a closed form, which allows to derive and analyze the corresponding RG equation. The analysis shows that the infinite-dimensional space of charges involves a two-dimensional surface of fixed points of the RG equations. It involves a region of infrared stability, and the model can exhibit a scale invariance with non-universal critical dimensions of the interface height and the time, satisfying a certain exact relation. Chetyrkin Konstantin (KIT, Karlsruhe)   Multiloop Renormalization Group Calculations: Current Status and Perspectives We outline the spectacular developments in the field of multiloop renormalization group from yearly 70-th till present time, which are heavily based on a proper use of versatile possibilities of the Bogouibov R-operation within dimensional regularization framework. Then we discuss future perspectives and open problems of multiloop RG calculations. In particular, we discuss the possibility of extending the 30-years old analytical calculation of the five-loop beta function in the $\phi^4$ model on the six-loop level. D-27 Mathematical methods in QFT (Fock Hall) Adzhemyan Loran Tsolakovich (SPbSU, SPb)   Fast algorithm for sector decomposition of diagrams at zero external momenta Sector decomposition method proved to be a great tool for the numerical computation of Feynman integrals. This approach significantly improves the convergence of the integrals at the cost of a substantial increase in the integrand (proportional to number of sectors). Therefore, one of the most important tasks is to find strategies, which give the minimal number of sectors. For diagrams at zero external momenta, the best of known strategies is a Speer sectors strategy. The proposed algorithm can reduce the number of sectors compared to Speer sectors for the most of Feynman diagrams of this type and hence reduce the time required for decomposition and integration. Marachevsky Valery (SPbSU,SPb)   Periodic geometries in the Casimir effect Casimir energy of two gratings separated by a vacuum slit is expressed in terms of Rayleigh coefficients. Comparison of theory and experiments is performed. Novozhilov Victor Yurievich (SPbSU, Spb)  Nazarov Anton Andreevich (SPbSU, SPb)   On singular elements in conformal field theory E-27 Quantum field theory methods in elementary particles physics (Blue Hall) Pis'mak Yury (SPbSU, SPb)   Chern-Simon Potential in Models of Nanophysics As a basic conception for the modeling of interaction of a macroscopic material body with quantum fields is considered the proposed by K.Symanzik approach. Its application in quantum electrodynamics enables one to establish the most general form of the action functional describing the interaction of 2-dimensional material surface with photon field. The models making it possible to calculate for thin films from non-ideal conducting material the Casimir energy, Casimir-Polder potential, characteristics of scattering processes and investigation of magneto- and electrostatic phenomena are presented. The specific of regularization and renormalization procedures used by calculations and the physical meaning of obtained results are discussed. Andrianov Vladimir Andreevich (SPbSU, SPb)   Local Parity violation in heavy ion collisions at extrime conditions Damaskinski Evgeny Victorovich (Mil. Eng.-Tech. University, SPb)   Generalized Chebyshev oscullator The approach to construction of generalized oscillators, conтected with polynomials orthogonal on the real line, proposed by one of the authors early, is extended to the case of bivariate polynomials. As an interesting example we consider bivariate Chebyshev polynomials connected with representations of sl(3) algebra. The polynomials under consideration form a complete system of eigenfunctions for a quadratic hamiltonian of the constructed two-dimensional Chebyshev oscillator. We show that the concidered hamiltonian splits into the sum of two commuting operators. So it is possible to consider this two-dimensional oscillator as a system of two noninteracting oscillators. The generators of the first oscillator algebra leave invariant all N-particles subspaces while the ladder operators of the second oscillator transform neighboring N-particles subspaces one onto another. The representation of the ladder operators for two-dimensional Chebyshev oscillator by differential operators are given. /In collaboration with V.V.Borzov/ Kudryavtsev V.A. (PNPI, Gatchina)   Born amplitudes for interaction of pi-, K-mesons and nucleons in composite superconformal string model. Composite superconformal string model is constructed to describe hadron amplitudes and hadron spectrum. It belongs to a new class of dual (string) models to be out of usual approaches. This model considers string scale alpha' as hadron scale is of order 1 GeV^(-2). There is supersymmetry on world two-dimensional surface only. Interacton amplitudes for pi-, K-mesons and nucleons are calculated within the framework of this model. The way of elimination of degeneracy in parity for baryons is found. (Joint report with A.N.Semenova.) Novikov Oleg Olegovich (SPbSU, SPb)   Scalar particles on self-gravitating thick brane Recent years, the models of beyond standard model physics based on the hypothesis that our universe is a four-dimensional space-time hypersurface (3-brane) embedded in a fundamental multi-dimensional space have become quite popular. The influence of gravity is especially interesting, playing an important role in a (de) localization of matter fields on the brane. The question arises, under what circumstances the localization of matter fields with spin zero on a brane is still possible when the minimal interaction with gravity is present? We consider a model of the domain wall formation with finite thickness (”thick” branes) by two scalar fields with quartic interaction characterized by mass $M$ and gravity in five-dimensional noncompact space-time, but with modified behavior of a metric factor in the bulk on both sides of the brane, i.e. with different anti-de Sitter geometries. Brane formation is possible when scalar field vacuum configurations have nontrivial topology. There are two types of these configurations corresponding to massless phase and massive phase controlled by mass parameter $\mu$. Due to mixing of scalar and gravitational degrees of freedom nonperturbative corrections of equations on second variation arise. This changes the spectrum in the scalar sector dramatically in comparison with model without gravity. Zero-mode exists in the case of asymmetric linear term with coefficient $|a|/>\frac{1}{3}\Bigl(2 \frac{\mu^2}{M^2}\Bigr)$ or in the case of symmetric linear term (defect of cosmological constant) with arbitrary positive coefficient $a$ or negative coefficient $a$ satisfying the same condition as in the case of asymmetric term. Also there is a light scalar state. Its leading order mass $\frac{m^2}{M^2}=2\frac{\mu^2}{M^2} O\Bigl(\frac{\mu^4}{M^4}\Bigr)$ is the same as in the similar model without gravity. /In collaboration with Alexander and Vladimir Andrianov/
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