A precision relation between Γ(K → μ ⁺ μ ⁻)(t) and B(KL→^μ+μ−)/B(KL→γγ)

We find that the phase appearing in the unitarity relation between B(K_L→ μ⁺μ⁻) and B(K_L→ γγ) 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_tsV_tdsin (β+ β_s) ∣ ≈ A²λ⁵η¯. We find a model-independent theoretically clean prediction, cos²φ₀ = 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 φ₀, reducing a four-fold ambiguity in the angle φ₀ to a two-fold one.