Pauli approximation for a vector particle with anomalous magnetic moment in an external Coulomb field

Abstract

Herein, a spin 1 particle with anomalous magnetic moment in an external Coulomb field is studied. We start with the relativistic tensor system of the Proca type in Cartesian coordinates. In these equations the Γ parameter is present related to an additional characteristic of the particle. In the case of an external magnetic field, it is interpreted as an anomalous magnetic moment. In the presence of an external electric field, additional interaction terms are presented as well; moreover, the terms of the first and second orders in parameter Γ appear. The case of an external Coulomb field is considered in detail. In the nonrelativistic approximation a Pauli type equation is obtained. In the nonrelativistic equation the separation of the variables with the use of spherical vectors is realized. One separate 2-nd order differential equation is found, in which additional interaction terms are missing. Besides, we derive systems of two coupled 2-nd order equations wherein linear and quadratic in parameter Γ interaction terms are presented. Previously, another approach was developed for analyzing the vector particle with anomalous magnetic moment. It was based on the use of tetrad formalism and separation of the variables in the Duffin – Kemmer equation with the help of the Wigner function. The nonrelativistic approximation was performed directly in the system of radial equations. Besides, previously formal Frobenius type solutions for an arising 4-th order differential equation were constructed; however, physically interpretable energy spectra were not found. We have proved that the radial equations derived by different methods are the same up to a simple liner transformation over two radial functions. In this paper, we have obtained a simpler 4-th order equation, the construction of Frobenius solutions becomes technically easier, but physical energy spectra are not found either.