Gluon bound state and asymptotic freedom derived from the Bethe--Salpeter equation

TL;DR

This paper derives and solves the Bethe-Salpeter equations for gluon and ghost bound states in a massive Yang-Mills theory, demonstrating the impact of ghosts on gluon bound states and confirming consistency with asymptotic freedom.

Contribution

It provides a systematic derivation of coupled Bethe-Salpeter equations for gluons and ghosts and presents numerical solutions showing their effects and consistency with asymptotic freedom.

Findings

Ghosts influence two-gluon bound states.

Solutions align with asymptotic freedom.

Bound states depend on gauge coupling.

Abstract

In this paper we study the two-body bound states for gluons and ghosts in a massive Yang-Mills theory which is obtained by generalizing the ordinary massless Yang-Mills theory in a manifestly Lorentz covariant gauge. First, we give a systematic derivation of the coupled Bethe-Salpeter equations for gluons and ghosts by using the Cornwall-Jackiw-Tomboulis effective action of the composite operators within the framework of the path integral quantization. Then, we obtain the numerical solutions for the Bethe-Salpeter amplitude representing the simultaneous bound states of gluons and ghosts by solving the homogeneous Bethe-Salpeter equation in the ladder approximation. We study how the inclusion of ghosts affects the two-gluon bound states in the cases of the standing and running gauge coupling constant. Moreover, we show explicitly that the approximate solutions obtained for the gluon-gluon amplitude are consistent with the ultraviolet asymptotic freedom signaled by the negative $\beta$ function.