The Symmetries and the General Harmonic Solution to Equations of Maxwell Electrodynamics with an Axion
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RETICAL AND MATHEMATICAL PHYSICS
The Symmetries and the General Harmonic Solution to Equations of Maxwell Electrodynamics with an Axion O. V. Kechkin1* and P. A. Mosharev2, 3** 1
Department of Physics, Moscow State University, Moscow, 119991 Russia Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991 Russia 3 Department of Higher Mathematics, National Research University Moscow Power Engineering Institute, Moscow, 111250 Russia 2
Received March 1, 2020; revised March 12, 2020; accepted March 16, 2020
Abstract—The effective Lagrangian is derived and a group of hidden symmetries is determined for Maxwell’s theory with an axion in the stationary case. The general harmonic solution is constructed, and it is shown that a central source can have only a Coulomb-type magnetic field. The electric field of such a source has a complex potential, which can be finite everywhere with a special choice of parameters. The differential scattering cross section of the test particle on a dione with an axionic charge is calculated within the first order of the perturbation theory. Keywords: electrodynamics with axion, monopoles, dions, exact solutions, symmetries. DOI: 10.3103/S0027134920030133
INTRODUCTION: THE MAXWELL ELECTRODYNAMICS WITH AN AXION Axions, as neutral bosons with zero spin, were first introduced in the context of a solution to the strong CP problem [1, 2]. Later, it was established that the field of an axion enters, in addition to the dilaton, gauge, and gravitational fields, the spectrum of massless excitations of a string (for instance, of a heterotic one). Axions are of profound interest as candidates for the role of the new physics [3, 4]. We refer readers who are interested in alternative ways of constructing nonlinear electrodynamics to the classical papers [5, 6] and to the modern works [7–17]. The dynamics of an axion and photon (that is, Maxwell) fields are described by the Lagrangian of the following form (here, we consider massless axions, in contrast to, e.g., [18]): 1 1 L = − (F μν Fμν + γκ F μν Fμν ) + ∂ μ κ∂μ κ. (1) 4 2 Here, Fμν = Aν,μ − Aμ,ν is the tensor of the electromagnetic field, Aμ is the 4-potential of the electromagnetic field (μ, ν = 0, 1, 2, 3), F μν = 12 αβμν Fαβ , where αβμν is the Levi–Chivita symbol, gμν = diag(1, −1, −1, −1) is the Minkowski metrics, κ is * **
the field of an axion, and γ is an arbitrary constant of an axion-Maxwell bond. The Euler–Lagrange equations corresponding to Eq. (1) are written as 1 (2) ∂ μ ∂μ κ + γ Fμν Fμν = 0, 4 (3) ∂ν F μν + γκ Fμν = 0. It follows from the second equation that the introduction of an axion into electrodynamics is equivalent to the appearance of an efficient current of electric charges in the Maxwell equations ∂ν F μν = J μ , namely, of the current J μ = −∂ν γκ F μν , and, due to antisymmetry of Fμν , the law of charge conservation (∂μ J μ ≡ 0) is identically satisfied for it. In contrast to [19], we will consider only stationary fields and seek for the symmetries of the stationary system and of the general harmon
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