Fermionic bound states in Minkowski-space: Light-cone singularities and structure

TL;DR

This paper develops a Minkowski-space approach to solve the Bethe-Salpeter equation for fermionic bound states, effectively handling singularities and providing numerical results for various interaction kernels, including applications to quark-antiquark systems.

Contribution

It introduces a novel Minkowski-space method using Nakanishi representation and null-plane projection to analyze fermionic bound states, extending to quark-antiquark systems with lattice-informed parameters.

Findings

Stable numerical solutions for fermionic bound states in relativistic regimes.

Dependence of binding energies on coupling constants and light-front amplitudes.

Signatures of spin degrees of freedom in light-front amplitudes of a mock pion.

Abstract

The Bethe-Salpeter equation for two-body bound system with spin $1/2$ constituent is addressed directly in the Minkowski space. In order to accomplish this aim we use the Nakanishi integral representation of the Bethe-Salpeter amplitude and exploit the formal tool represented by the exact projection onto the null-plane. This formal step allows one i) to deal with end-point singularities one meets and ii) to find stable results, up to strongly relativistic regimes, that settles in strongly bound systems. We apply this technique to obtain the numerical dependence of the binding energies upon the coupling constants and the light-front amplitudes for a fermion-fermion $0^+$ state with interaction kernels, in ladder approximation, corresponding to scalar-, pseudoscalar- and vector boson exchanges, respectively. After completing the numerical survey of the previous cases, we extend our approach to a quark-antiquark system in $0^-$ state, taking both constituent-fermion and exchanged boson masses, from lattice calculations. Interestingly, the calculated light-front amplitudes for such a mock pion show peculiar signatures of the spin degrees of freedom.