On the $\uptheta$-vacua and CP violation

TL;DR

This paper clarifies misconceptions about CP violation in $ heta$-vacua, emphasizing the importance of boundary degrees of freedom and showing that CP violation remains observable in quantum chromodynamics.

Contribution

It provides a detailed analysis demonstrating that the $ heta$-vacuum structure leads to observable CP violation, countering recent claims of its absence.

Findings

Boundary degrees of freedom are essential for large gauge invariance.

In the infinite-volume limit, edge states become non-dynamical.

The $ heta$-vacuum structure indeed causes observable CP violation.

Abstract

Recent claims have suggested the absence of CP violation in theories with a $\theta$-vacuum structure, particularly in quantum chromodynamics. We highlight several key points, from a perspective that is not widely discussed in the literature, which clarify why such conclusions are incorrect. In particular, an open boundary in a finite-volume theory must be accompanied by boundary degrees of freedom, the edge modes, in order to preserve large gauge invariance and faithfully capture the topological features of the theory. In the infinite-volume limit, these edge states become non-dynamical, leaving the standard $\uptheta$-vacuum structure intact, irrespective of whether this limit is taken before or after summing over topological sectors. Consequently, the $\uptheta$-vacuum structure does give rise to observable CP violation once the theory is consistently quantised.