  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.
