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

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.