Theta-vacuum: Phase Transitions and/or Symmetry Breaking at $\theta = \pi$

TL;DR

This paper discusses how quantum field theories with a theta-vacuum term exhibit either spontaneous CP symmetry breaking at  or a phase transition at some critical , based on their -dependence and probability distribution properties.

Contribution

It demonstrates that theories with a -vacuum term necessarily exhibit either CP violation at  or a phase transition at some , extending previous results to a broader class of models.

Findings

Either CP symmetry is spontaneously broken at  or a singularity occurs at some  between 0 and .

The result applies to models with a pure imaginary Euclidean action contribution, including QCD.

Illustrated with examples related to Witten's large N analysis of SU(N).

Abstract

Assuming that a quantum field theory with a $\theta$-vacuum term in the action shows non-trivial $\theta$-dependence and provided that some reasonable properties of the probability distribution function of the order parameter hold, we argue that the theory either breaks spontaneously CP at $\theta = \pi$ or shows a singular behavior at some critical $\theta_c$ between 0 and $\pi$. This result, which applies to any model with a pure imaginary contribution to the euclidean action consisting in a quantized charge coupled to a phase, as QCD, is illustrated with two simple examples; one of them intimately related to Witten's result on SU(N) in the large $N$ limit.