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Gauge theories with matter often have critical regions in their parameterspace where gapless degrees of freedom emerge. Using controlled semiclassicalcalculations, we explore such critical regions in $SU(N)$ gauge theories with atopological $\theta$ term and $N_F$ fundamental fermions in four dimensions, aswell as related field theories in three dimensions. In four-dimensionaltheories, we find that for all $N_F \ge 1$ the critical behavior always occursat a point in parameter space. For $N_F>1$ this is consistent with the standardQCD expectations, while for $N_F=1$ our results are consistent with recentobservations concerning 't Hooft anomalies. We also show how the $N$-branchedstructure of observables transmutes into the $N_F$-branched structure seen inchiral Lagrangians as the mass parameter is dialed. As a side benefit, ouranalysis of these 4D theories implies the unexpected result that 3D Coulombgases can have gapless critical points. We also consider QCD-likeparity-invariant theories in three dimensions, and find that their criticalbehavior is quite different. In particular, we show that their gapless regionis an interval in parameter space, rather than a point. Our results havenon-trivial implications for the infrared behavior of three-dimensional compactQED.

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