CP violation in $K\to {\mu }^{+}{\mu }^{-}$ with and without time dependence through a tagged analysis

Abstract

We point out that using current knowledge of $\mathcal{B}(K_L^0\to {\mu }^{+}{\mu }^{-})$ and $\mathcal{B}(K_L^0\to \gamma \gamma )$, one can extract short-distance information from the combined measurement of the time-integrated CP asymmetry, $A_{\mathrm{CP}}(K^0\to {\mu }^{+}{\mu }^{-})$, and of $\mathcal{B}(K_S^0\to {\mu }^{+}{\mu }^{-})$. We discuss the interplay between this set of observables, and demonstrate that determining $\mathrm{sign}[A_{\mathrm{CP}}(K^0\to {\mu }^{+}{\mu }^{-})]$ would eliminate the discrete ambiguity in the Standard Model prediction for $\mathcal{B}(K_L^0\to {\mu }^{+}{\mu }^{-})$. We then move on to feasibility studies within an LHCb-like setup, using both time-integrated and time-dependent information, employing $K^0$ and $\overline{K}{}^0$ tagging methods. We find that, within an optimistic scenario, the short-distance amplitude, proportional to the CKM parameter combination $|A^2{\lambda }^5\bar{\eta }|$, could be constrained by LHCb at the level of about $35\%$ of its Standard Model value, and the discrete ambiguity in $\mathcal{B}(K_L^0\to {\mu }^{+}{\mu }^{-}{)}_{\mathrm{SM}}$ could be resolved at more than $3\sigma$ by the end of the high luminosity LHC.