Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space

TL;DR

This paper introduces a novel analytical technique for solving the two-fermion Bethe-Salpeter equation directly in Minkowski space, overcoming singularity issues and enabling realistic relativistic bound state calculations.

Contribution

It provides an exact analytical treatment of singularities in the Bethe-Salpeter equation, expanding the applicability of Minkowski space solutions to complex, realistic systems with various spins.

Findings

Successful analytical handling of singularities in Minkowski space

Validation through comparison with existing Euclidean and Minkowski results

Demonstration of potential for non-perturbative relativistic calculations

Abstract

Actual solutions of the Bethe-Salpeter equation for a two-fermion bound system are becoming available directly in Minkowski space, by virtue of a novel technique, based on the so-called Nakanishi integral representation of the Bethe-Salpeter amplitude and improved by expressing the relevant momenta through light-front components, i.e. $k^\pm=k^0 \pm k^3$. We solve a crucial problem that widens the applicability of the method to real situations by providing an analytically exact treatment of the singularities plaguing the two-fermion problem in Minkowski space, irrespective of the complexity of the irreducible Bethe-Salpeter kernel. This paves the way for feasible numerical investigations of relativistic composite systems, with any spin degrees of freedom. We present a thorough comparison with existing numerical results, evaluated in both Minkowski and Euclidean space, fully corroborating our analytical treatment, as well as fresh light-front amplitudes illustrating the potentiality of non perturbative calculations performed directly in Minkowski space.