  We investigate the dynamics responsible for generating the potential of the
$\eta'$, the (would-be) Goldstone boson associated with the anomalous axial
$U(1)$ symmetry of QCD. The standard lore posits that pure QCD dynamics
generates a confining potential with a branched structure as a function of the
$\theta$ angle, and that this same potential largely determines the properties
of the $\eta'$ once fermions are included. Here we test this picture by
examining a supersymmetric extension of QCD with a small amount of
supersymmetry breaking generated via anomaly mediation. For pure $SU(N)$ QCD
without flavors, we verify that there are $N$ branches generated by gaugino
condensation. Once quarks are introduced, the flavor effects qualitatively
change the strong dynamics of the pure theory. For $F$ flavors we find $|N-F|$
branches, whose dynamical origin is gaugino condensation in the unbroken
subgroup for $F<N-1$, and in the dual gauge group for $F >N+1$. For the special
cases of $F = N-1, N, N + 1$ we find no branches and the entire potential is
consistent with being a one-instanton effect. The number of branches is a
simple consequence of the selection rules of an anomalous $U(1)_R$ symmetry. We
find that the $\eta'$ mass does not vanish in the large $N$ limit for fixed
$F/N$, since the anomaly is non-vanishing. The same dynamics that is
responsible for the $\eta'$ potential is also responsible for the axion
potential. We present a simple derivation of the axion mass formula for an
arbitrary number of flavors.
