CP violation in K → μ+μ− with and without time dependence through a tagged analysis

We point out that using current knowledge of \( \mathcal{B}\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right) \) and \( \mathcal{B}\left({K}_L^0\to \gamma \gamma \right) \), one can extract short-distance information from the combined measurement of the time-integrated CP asymmetry, A_CP(K⁰ → μ⁺μ⁻), and of \( \mathcal{B}\left({K}_S^0\to {\mu}^{+}{\mu}^{-}\right) \). We discuss the interplay between this set of observables, and demonstrate that determining sign[A_CP(K⁰ → μ⁺μ⁻)] would eliminate the discrete ambiguity in the Standard Model prediction for \( \mathcal{B}\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right) \). We then move on to feasibility studies within an LHCb-like setup, using both time-integrated and time-dependent information, employing K⁰ and \( {\overline{K}}^0 \) tagging methods. We find that, within an optimistic scenario, the short-distance amplitude, proportional to the CKM parameter combination \( \mid {A}^2{\lambda}^5\overline{\eta}\mid \), could be constrained by LHCb at the level of about 35% of its Standard Model value, and the discrete ambiguity in \( \mathcal{B}{\left({K}_L^0\to {\mu}^{+}{\mu}^{-}\right)}_{\textrm{SM}} \) could be resolved at more than 3σ by the end of the high luminosity LHC.