Spontaneous CP Breaking in a QCD-like Theory

We examine the phase structure of a QCD-like theory at $\bar\theta =\pi$ obtained from supersymmetric $SU(N)$ QCD perturbed by a small amount of supersymmetry breaking via anomaly mediation (AMSB QCD). The spectrum of this theory matches that of QCD at the massless level, though the superpartners are not decoupled. In this theory it is possible to nail down the phase structure at $\bar\theta=\pi$ as a function of the quark masses and the number of flavors $F$. For one flavor we find that there is a critical quark mass, below which CP is unbroken, while above the critical mass CP is spontaneously broken. At the critical mass there is a second-order phase transition along with a massless $\eta'$. We are able to analytically solve for the minima and the critical mass for $N=2,3$ as well as for the large $N$ limit, while for other $N$ one can find numerical results. For two flavors, we find that CP is always broken as long as the quark masses are equal and non-zero, however there is a non-trivial phase boundary for unequal quark masses, which we find numerically. For $F\geq 3$ we obtain an intricate phase boundary which reproduces the various quark mass limits. All our results are in agreement with the predictions for ordinary QCD that were based on anomaly matching arguments for generalized symmetries and the effective chiral Lagrangian. We also briefly comment on the domain wall solutions first discussed by Draper, and are able to present analytic results for the simplest case of $SU(2)$ with one flavor.