$\mathcal{CP}$ violation with an unbroken $\mathcal{CP}$ transformation

TL;DR

This paper explores how a gauge theory can conserve $ ext{CP}$ symmetry at the fundamental level but still exhibit observable $ ext{CP}$ violation after spontaneous symmetry breaking, with implications for topological vacuum terms.

Contribution

It demonstrates that physical $ ext{CP}$ violation can occur in a theory where the original $ ext{CP}$ symmetry is unbroken, due to the automorphism structure of the residual symmetry group.

Findings

Explicit calculation of $ ext{CP}$ odd decay asymmetry shows violation.

Protection of topological vacuum terms like $ heta G ilde{G}$ is maintained.

The $ ext{CP}$ violation arises from the automorphism properties of the residual group.

Abstract

A $\mathcal{CP}$ conserving $\mathrm{SU}(3)$ gauge theory is spontaneously broken to $\mathrm{T}_7$ by the vacuum expectation value (VEV) of a $\boldsymbol{15}$-plet. Even though the $\mathrm{SU}(3)$-$\,\mathcal{CP}$ transformation is not broken by the VEV, the theory exhibits physical $\mathcal{CP}$ violation in the broken phase. This is because the $\mathrm{SU}(3)$-$\,\mathcal{CP}$ transformation corresponds to the unique order-two outer automorphism of $\mathrm{T}_7$, which is not a physical $\mathcal{CP}$ transformation for the $\mathrm{T}_7$ states, and there is no other possible $\mathcal{CP}$ transformation. We explicitly demonstrate that $\mathcal{CP}$ is violated by calculating a $\mathcal{CP}$ odd decay asymmetry in the broken phase. This scenario provides us with a natural protection for topological vacuum terms, ensuring that $\theta\,G_{\mu\nu}\widetilde{G}^{\mu\nu}$ is absent even though $\mathcal{CP}$ is violated for the physical states of the model.