A Precision Relation between $\Gamma (K\to {\mu }^{+}{\mu }^{-})(t)$ and $\mathcal{B}(K_L\to {\mu }^{+}{\mu }^{-})/\mathcal{B}(K_L\to \gamma \gamma )$

Abstract

We find that the phase appearing in the unitarity relation between $\mathcal{B}(K_L\to {\mu }^{+}{\mu }^{-})$ and $\mathcal{B}(K_L\to \gamma \gamma )$ is equal to the phase shift in the interference term of the time-dependent $K\to {\mu }^{+}{\mu }^{-}$ decay. A probe of this relation at future kaon facilities constitutes a Standard Model test with a theory precision of about $2\%$. The phase has further importance for sensitivity studies regarding the measurement of the time-dependent $K\to {\mu }^{+}{\mu }^{-}$ decay rate to extract the CKM matrix element combination $|V_{ts}{V}_{td}\sin (\beta +{\beta }_s)|\approx A^2{\lambda }^5\bar{\eta }$. We find a model-independent theoretically clean prediction, ${\cos }^2{\phi }_0=0.96\pm 0.03$. The quoted error is a combination of the theoretical and experimental errors, and both of them are expected to shrink in the future. Using input from the large-$N_C$ limit within chiral perturbation theory, we find a theory preference towards solutions with negative $\cos {\phi }_0$, reducing a four-fold ambiguity in the angle ${\phi }_0$ to a two-fold one.