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

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.