On the dynamical origin of the η ′ potential and the axion mass

We investigate the dynamics responsible for generating the potential of the η ′, 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 θ angle, and that this same potential largely determines the properties of the η ′ 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 η ′ 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 η ′ potential is also responsible for the axion potential. We present a simple derivation of the axion mass formula for an arbitrary number of flavors.