Vacuum Alignment with more Flavors
Abstract
We study the alignment of the vacuum in gauge theories with Nf 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 Nf=2,3 and 4 case. For Nf=2 and Nf=3 we reproduce earlier known results including the Dashen phase with spontaneous violation of the combined charge conjugation and parity symmetry, CP. For Nf=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 Nf-1 flavors are degenerate with positive mass m>0 and the mass of the Nf'th flavor becomes negative and equal to -m CP breaks spontaneously.