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

We point out that using current knowledge of ${\cal B}(K^0_L\to\mu^+\mu^-)$ and ${\cal B}(K^0_L\to \gamma\gamma)$, one can extract short-distance information from the combined measurement of the time-integrated CP asymmetry, $A_{\rm CP}(K^0\to\mu^+\mu^-)$, and of ${\cal B}(K^0_S\to\mu^+\mu^-)$. We discuss the interplay between this set of observables, and demonstrate that determining ${\rm sign}[A_{\rm CP}(K^0\to\mu^+\mu^-)]$ would eliminate the discrete ambiguity in the Standard Model prediction for ${\cal B}(K^0_L\to\mu^+\mu^-)$. 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 ${\cal B}(K^0_L\to\mu^+\mu^-)_{\rm SM}$ could be resolved at more than $3\sigma$ by the end of the high luminosity LHC.