Instantaneous Bethe-Salpeter Approach to Pseudoscalar Mesons

TL;DR

This paper develops an instantaneous Bethe-Salpeter approach to describe light pseudoscalar mesons, capturing their dual nature as bound states and Goldstone bosons within quantum chromodynamics.

Contribution

It derives an exact relationship between Bethe-Salpeter solutions and the interquark potential, revealing its behavior at different quark masses and distances.

Findings

The potential has a logarithmically softened Coulomb singularity at the origin.

At large distances, the potential rises beyond bounds for massless quarks.

For heavy quarks, the potential approaches a finite limit.

Abstract

Light pseudoscalar mesons are Janus-type particles: Within quantum chromodynamics, they must be described as bound states of its fundamental degrees of freedom and as the (pseudo-) Goldstone bosons of its spontaneously broken chiral symmetry. This janiform nature of pions and kaons may be easily accommodated by the Bethe-Salpeter formalism in its instantaneous limit: Starting from the general behaviour of the Bethe-Salpeter solutions for light pseudoscalar mesons at large Euclidean momenta, we provide the exact relationship between the solutions of our bound-state equation and the underlying interactions, boiled down to a configuration-space potential depending on the interquark distance: the potential exhibits a logarithmically softened Coulomb singularity at the origin and, at large interquark distances, rises beyond bounds for massless quarks but approaches a non-positive finite limit for too heavy quarks.