# Solving the Bethe-Salpeter Equation in Minkowski Space for a Fermion-Scalar system

## TL;DR

This paper successfully solves the Bethe-Salpeter Equation for a fermion-scalar system in Minkowski space, providing new insights into bound state properties and interaction effects using advanced integral representation and light-front techniques.

## Contribution

It introduces the first Minkowski space solution for a fermion-scalar Bethe-Salpeter Equation using Nakanishi representation and light-front projection, extending previous methods to new systems.

## Key findings

- Correlation between binding energies and interaction coupling constants.
- Valence probabilities and light-front momentum distributions analyzed.
- Observation of repulsion effects and helicity conservation phenomena.

## Abstract

The ladder Bethe-Salpeter Equation of a bound (1/2)+ system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: i) a massive scalar exchange and ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully non-perturbative framework, as well as the onset of a scale-invariant regime.