$\mathcal{C}\mathcal{P}$ violation with an unbroken $\mathcal{C}\mathcal{P}$ transformation

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