A precision relation between Γ(K → μ + μ − )(t) and B K L → μ + μ − / B K L → γγ $$ \mathcal{B}\left({K}_L\to {\mu}^{+}{\mu}^{-}\right)/\mathcal{B}\left({K}_L\to \gamma \gamma \right) $$

Abstract We find that the phase appearing in the unitarity relation between B K L → μ + μ − $$ \mathcal{B}\left({K}_L\to {\mu}^{+}{\mu}^{-}\right) $$ and B K L → γγ $$ \mathcal{B}\left({K}_L\to \gamma \gamma \right) $$ is equal to the phase shift in the interference term of the time- dependent K → μ + μ − 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 → μ + μ − decay rate to extract the CKM matrix element combination ∣ V ts V td sin β + β s ∣ ≈ A 2 λ 5 η ¯ $$ \mid {V}_{ts}{V}_{td}\sin \left(\beta +{\beta}_s\right)\mid \approx {A}^2{\lambda}^5\overline{\eta} $$ . We find a model-independent theoretically clean prediction, cos2 φ 0 = 0.96 ± 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 φ 0, reducing a four-fold ambiguity in the angle φ 0 to a two-fold one.