1 Department of Physics, Chemistry and Pharmacy, Faculty of Science, SDU2 SDU eScience Centre, Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU3 CP3-Origins, Department of Physics, Chemistry and Pharmacy, Faculty of Science, SDU4 Department of Physics, Chemistry and Pharmacy, Faculty of Science, SDU
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 $N_f-1$ flavors are degenerate with positive mass $m>0$ and the mass of the $N_f$'th flavor becomes negative and equal to $-m$ CP breaks spontaneously.
Physical Review D (particles, Fields, Gravitation and Cosmology), 2014, Vol 89