Schwinger - Dyson equation and NJL approximation in massive gauge theory with fermions

TL;DR

This paper studies chiral symmetry breaking in massive SU(N) gauge theories with fermions, using Dyson-Schwinger equations, and finds indications of a second order phase transition at a critical coupling.

Contribution

It demonstrates that chiral symmetry breaking is a strong coupling phenomenon and connects Dyson-Schwinger equations with an effective NJL model near the critical point.

Findings

Chiral symmetry breaking occurs at strong coupling.

Indications of a second order phase transition at a specific critical coupling.

Near the critical point, Dyson-Schwinger equations approximate an NJL gap equation.

Abstract

We consider massive $SU(N)$ gauge theory with fermions. Gauge bosons become massive due to the interaction with the scalar field, whose vacuum average provides the spontaneous breakdown of gauge symmetry. We investigate Dyson - Schwinger equation for the fermion propagator written in ladder approximation and in Landau gauge. Our analysis demonstrates that the chiral symmetry breaking in the considered theory is the strong coupling phenomenon. There are the indications that there appears the second order phase transition between chirally broken and symmetric phases of the theory at the value of coupling constant $\alpha_c = (1+\gamma)\times \frac{\pi}{3}\times \frac{1}{2 C_2(F)}$, where $0<\gamma<1$, and $\gamma$ depends on the scale, at which the fluctuations of the scalar field destroy the gauge boson mass. In the broken phase near the critical value of $\alpha$ the Dyson - Schwinger equation is approximated well by the gap equation of the effective Nambu - Joina - Lasinio model with the value of cutoff around gauge boson mass $M$ and the effective four - fermion coupling constant $\frac{4 \pi \alpha}{M^2}\times \frac{2C_2(F)}{N}$. The dynamical fermion mass $m$ may be essentially smaller than $M$.