Vacuum alignment with more flavors

Abstract

We study the alignment of the vacuum in gauge theories with ${N}_{f}$ Dirac fermions transforming according to a complex representation of the gauge group. The alignment of the vacuum is produced by adding a small mass perturbation to the theory. We study in detail the ${N}_{f}=2,3$, and 4 case. For ${N}_{f}=2$ and ${N}_{f}=3$ we reproduce earlier known results including the Dashen phase with spontaneous violation of the combined charge conjugation and parity symmetry, $CP$. For ${N}_{f}=4$ we find regions with and without spontaneous $CP$ violation. We then generalize to an arbitrary number of flavors. Here it is shown that at the point where ${N}_{f}\ensuremath{-}1$ flavors are degenerate with positive mass $m>0$ and the mass of the ${N}_{f}$th flavor becomes negative and equal to $\ensuremath{-}m$, $CP$ breaks spontaneously.