Vacuum Alignment with more Flavors

TL;DR

This paper investigates vacuum alignment in gauge theories with multiple flavors of fermions, revealing conditions for spontaneous CP violation across different flavor numbers and generalizing the results.

Contribution

It extends previous studies to theories with more flavors, identifying new regions of spontaneous CP violation and providing a general framework for vacuum alignment analysis.

Findings

Reproduces known results for Nf=2 and 3, including Dashen phase.

Identifies new CP violation regions for Nf=4.

Generalizes vacuum alignment conditions to arbitrary flavor number.

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 $N_f-1$ 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.