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.
Categories: hep-ph hep-th