Direct $CP$ Violation in $K to mu^+ mu^-$

A rare decay $K_L to mu^+ mu^- $ has been measured precisely, while a rare decay $K_S to mu^+ mu^- $ will be observed by an upgrade of the LHCb experiment. Although both processes are almost CP-conserving decays, we point out that an interference contribution between $K_L$ and $K_S$ in the kaon beam emerges from a genuine direct CP violation. It is found that the interference contribution can change $K_S to mu^+ mu^-$ standard-model predictions at $mathcal{O}(60%)$. We also stress that an unknown sign of $mathcal{A}(K_L to gamma gamma)$ can be determined by a measurement of the interference, which can much reduce a theoretical uncertainty of $mathcal{B}(K_L to mu^+ mu^-)$. We also investigate the interference in a new physics model, where the $epsilon_K / epsilon_K$ tension is explained by an additional $Z$-penguin contribution.