Direct $CP$ Violation in $K\to {\mu }^{+}{\mu }^{-}$

Abstract

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 ${ϵ}_K^{\prime }/{ϵ}_K$ tension is explained by an additional $Z$-penguin contribution.

Comments: 6 pages, 2 figures; references added, version published in PRL