Critical behavior of gauge theories and Coulomb gases in three and four dimensions

TL;DR

This paper investigates the critical phenomena in gauge theories with matter in three and four dimensions, revealing new insights into their phase structures, including the existence of gapless points and regions, and implications for related models like Coulomb gases and QED.

Contribution

It provides a controlled semiclassical analysis of critical regions in $SU(N)$ gauge theories with a $ heta$ term and fundamental fermions, uncovering novel critical behaviors and their relation to anomalies and topological structures.

Findings

Critical points in 4D theories occur at specific parameter values for all $N_F \\ge 1$.

3D Coulomb gases can exhibit gapless critical points, contrary to previous expectations.

Critical behavior in 3D QCD-like theories involves an interval in parameter space, not a point.

Abstract

Gauge theories with matter often have critical regions in their parameter space where gapless degrees of freedom emerge. Using controlled semiclassical calculations, we explore such critical regions in $SU(N)$ gauge theories with a topological $\theta$ term and $N_F$ fundamental fermions in four dimensions, as well as related field theories in three dimensions. In four-dimensional theories, we find that for all $N_F \ge 1$ the critical behavior always occurs at a point in parameter space. For $N_F>1$ this is consistent with the standard QCD expectations, while for $N_F=1$ our results are consistent with recent observations concerning 't Hooft anomalies. We also show how the $N$-branched structure of observables transmutes into the $N_F$-branched structure seen in chiral Lagrangians as the mass parameter is dialed. As a side benefit, our analysis of these 4D theories implies the unexpected result that 3D Coulomb gases can have gapless critical points. We also consider QCD-like parity-invariant theories in three dimensions, and find that their critical behavior is quite different. In particular, we show that their gapless region is an interval in parameter space, rather than a point. Our results have non-trivial implications for the infrared behavior of three-dimensional compact QED.