Solving Bethe-Salpeter equation for two fermions in Minkowski space

TL;DR

This paper extends a Minkowski space solution method for the Bethe-Salpeter equation to two-fermion systems, providing numerical solutions and analyzing bound states with various exchange kernels.

Contribution

It introduces a novel approach to solve the Bethe-Salpeter equation for two fermions in Minkowski space using Nakanishi representation and light-front projection, including regularization of singularities.

Findings

Numerical solutions for J=0 bound states with different exchange kernels.

Binding energies closely match Euclidean space results.

Amplitudes in Minkowski space are successfully obtained.

Abstract

The method of solving the Bethe-Salpeter equation in Minkowski space, which we developed previously for spinless particles, is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original Bethe-Salpeter amplitudes. The numerical solutions for the J=0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. The stability of the scalar and positronium states without vertex form factor is discussed. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained.