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

Figure 1

The time distributions of $K\to {\mu }^{+}{\mu }^{-}$ [$I(t)$] are shown within the SM with several choices of $D$, which are normalized by the decay intensity from $0.1{\tau }_S$ to $1.45{\tau }_S$ (solid lines) and from $0.1{\tau }_S$ to $3{\tau }_S$ (dashed lines) with $D=0$. The left and right panels correspond to the positive and negative signs of $A_{L\gamma \gamma }^{\mu }$ in Eq. (11), respectively.

Figure 2

The effective branching ratio into ${\mu }^{+}{\mu }^{-}$ in Eq. (12) as a function of the dilution factor. The left and right panels correspond to the positive and negative signs of $A_{L\gamma \gamma }^{\mu }$ in Eq. (11), respectively. The SM predictions are represented by blue and red lines, where the darker bands stand for uncertainty from the interference in Eq. (10) and the lighter bands denote uncertainty from $A_{S\gamma \gamma }^{\mu }$ in Eq. (13). Gray bands represent $\mathcal{B}(K_S\to {\mu }^{+}{\mu }^{-}{)}_{\mathrm{SM}}$ in Eq. (2). The ${\epsilon }_K^{\prime }/{\epsilon }_K$ anomaly can be explained at $1\sigma$ in the green regions within the modified $Z$-coupling model.