Chiral Vacuum Alignment and Spontaneous CP Violation by Four-Fermi Operators

TL;DR

This paper investigates how four-Fermi operators can induce spontaneous CP violation through vacuum alignment in models with chiral symmetry breaking, revealing critical transitions and potential light pseudo-Goldstone bosons.

Contribution

It provides an exact analysis for two-flavor models and a numerical study for three flavors, showing how vacuum solutions and CP breaking patterns emerge and transition.

Findings

Vacuum solutions fall into three classes with distinct CP breaking patterns.

Critical transitions occur where pseudo-Goldstone boson mass vanishes.

Imposing CP invariance on perturbations increases the probability of light pseudo-Goldstone bosons.

Abstract

In models where there is a global chiral symmetry which spontaneously breaks to its vectorial subgroup, the introduction of an explicit symmetry breaking perturbation will define the true vacuum of the theory. This true vacuum is found via the minimization of the expectation value of the perturbing hamiltonian between different vacua as prescribed by Dashen. The procedure of finding the correct vacuum of the theory may result in the spontaneous breaking of CP symmetry even if one initially demands CP invariance on the perturbation. In this work we study in detail models where the perturbation is provided by four-Fermi operators. We present exact treatment for models with two fermion flavors and study the three flavor case in depth numerically. We show that after Dashen procedure is applied the solutions for the true vacuum fall in three classes with different CP breaking patterns. Critical transitions is possible between these classes as one varies the parameters of the perturbation. We rigorously show that at these transitions a pseudo-Goldstone boson mass vanishes. We also advocate, and substantiate with numerical statistical analysis for various types of models, that if one imposes CP invariance on the perturbation before solving the vacuum alignment, the resulting vacuum structure will have a sizable probability for a light pseudo-Goldstone boson mass. That is a statistical variant of Peccei-Quinn mechanism can be speculated to operate.