On cusps in the η′ potential

A bstract The large N analysis of QCD states that the potential for the η′ meson develops cusps at η′ = π / N f , 3 π / N f , ⋯ , with N f 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 N f 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 N f = N − 1, N , and N + 1. We argue that the analysis of the N f = N case should be subject to the above anomaly argument, and thus there should be a cusp; while the N f = N ± 1 cases are consistent, as N f and N are coprime. We discuss how this cuspy/smooth transition can be understood. For N f < N , we find that the number of branches of the η′ potential is gcd( N , N f ), 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 N f and N are coprime. The s-confinement in supersymmetric QCD at N f = N + 1 is a famous example, and the argument generalizes for any number of fermions in the adjoint representation.