Effect of Dynamical SU(2) Gluons to the Gap Equation of Nambu--Jona-Lasinio Model in Constant Background Magnetic Field

TL;DR

This paper investigates how dynamical SU(2) gluons influence chiral symmetry breaking in the gauged Nambu--Jona-Lasinio model under constant magnetic fields, revealing enhanced symmetry breaking and complex dependencies on the background field.

Contribution

It provides a two-loop order analysis of gluon effects on the gap equation in 2+1 and 3+1 dimensions, including gauge-invariant formulation for electric and magnetic backgrounds.

Findings

Gluons enhance chiral symmetry breaking in 3+1 dimensions, especially with larger magnetic fields.

In 2+1 dimensions, gluon effects on symmetry breaking are non-monotonic with respect to background field strength.

Gluons induce symmetry breaking even without magnetic backgrounds in 3+1 dimensions.

Abstract

In order to estimate the effect of dynamical gluons to chiral condensate, the gap equation of SU(2) gauged Nambu--Jona-Lasinio model, under a constant background magnetic field, is investigated up to the two-loop order in 2+1 and 3+1 dimensions. We set up a general formulation allowing both cases of electric as well as magnetic background field. We rely on the proper time method to maintain gauge invariance. In 3+1 dimensions chiral symmetry breaking ($\chi$SB) is enhanced by gluons even in zero background magnetic field and becomes much striking as the background field grows larger. In 2+1 dimensions gluons also enhance $\chi$SB but whose dependence on the background field is not simple: dynamical mass is not a monotone function of background field for a fixed four-fermi coupling.