Bound States of a Minimally Interacting Spin-0 and Spin-1/2 Constituent in the Instantaneous Approximation

TL;DR

This paper numerically solves bound states of a spin-0 and spin-1/2 particle interacting via minimal electrodynamics using an improved method to handle singularities, revealing detailed bound-state properties.

Contribution

Develops a novel numerical method for solving integral equations with logarithmic singularities in the context of bound states in electrodynamics.

Findings

Accurate bound-state solutions with fewer basis functions.

Method effectively handles logarithmic singularities in kernels.

Identifies additional electrostatic potential terms in the nonrelativistic limit.

Abstract

Bound-state solutions are obtained numerically in the instantaneous approximation for a spin-0 and spin-1/2 constituent that interact via minimal electrodynamics. To solve the integral equations in momentum space, a method is developed for integrating over the logarithmic singularity in kernels, making it possible to use basis functions that essentially automatically satisfy the boundary conditions. For bound-state solutions that decrease rapidly at small and large values of momentum, accurate solutions are obtained with significantly fewer basis functions when the solution is expanded in terms of these more general basis functions. The presence of a derivative coupling in single-photon exchange complicates the construction of the Bethe-Salpeter equation in the instantaneous approximation and, in the nonrelativistic limit, gives rise to an additional electrostatic potential term that is second order in the coupling constant and decreases as the square of the distance between constituents.