Peeking into the $\theta$ vacuum

TL;DR

This paper introduces a subvolume method to analyze the $ heta$ dependence of the vacuum energy in 4D SU(N) Yang-Mills theory, revealing spontaneous CP violation at $ heta=\pi$ and challenging previous theoretical assumptions.

Contribution

The paper presents a novel subvolume approach for lattice studies of $ heta$ dependence in SU(N) Yang-Mills theory, with results indicating spontaneous CP violation at $ heta=\pi$ for N=2.

Findings

Evidence of spontaneous CP violation at $ heta=\pi$

Qualitative difference in $ heta$ dependence from high temperature case

Method successfully applied to SU(2) Yang-Mills at $T=1.2T_c$

Abstract

We propose a subvolume method to study the $\theta$ dependence of the free energy density of the four-dimensional SU($N$) Yang-Mills theory on the lattice. As an attempt, the method is first applied to SU(2) Yang-Mills theory at $T=1.2\,T_c$ to understand the systematics of the method. We then proceed to the calculation of the vacuum energy density and obtain the $\theta$ dependence qualitatively different from the high temperature case. The numerical results combined with the theoretical requirements provide the evidence for the spontaneous CP violation at $\theta = \pi$, which is in accordance with the large $N$ prediction and indicates that the similarity between 4d SU($N$) and 2d CP$^{N-1}$ theories does not hold for $N$=2.