On cusps in the η' potential

The large N analysis of QCD states that the potential for the η' meson develops cusps at η' = π/ Nf, 3 π/Nf, , with Nf the number of flavors. Furthermore, the recent discussion of generalized anomalies tells us that even for finite N there should be cusps if N and Nf are not coprime, as one can show that the domain wall configuration of η' should support a Chern-Simons theory on it, i. e. , domains are not smoothly connected. On the other hand, there is a supporting argument for instanton-like, smooth potentials of η' from the analyses of softly-broken supersymmetric QCD for Nf= N-1, N, and N+1. We argue that the analysis of the Nf = N case should be subject to the above anomaly argument, and thus there should be a cusp; while the Nf = N 1 cases are consistent, as Nf and N are coprime. We discuss how this cuspy/smooth transition can be understood. For Nf< N, we find that the number of branches of the η' potential is gcd (N, Nf), which is the minimum number allowed by the anomaly. We also discuss the condition for s-confinement in QCD-like theories, and find that in general the anomaly matching of the θ periodicity indicates that s-confinement can only be possible when Nf and N are coprime. The s-confinement in supersymmetric QCD at Nf = N+1 is a famous example, and the argument generalizes for any number of fermions in the adjoint representation.