A Precision Relation between $Γ(K\toμ^+μ^-)(t)$ and ${\cal B}(K_L\toμ^+μ^-)/{\cal B}(K_L\toγγ)$

Abstract

We find that the phase appearing in the unitarity relation between $\mathcal{B}(K_L\rightarrow μ^+μ^-)$ and $\mathcal{B}(K_L\rightarrow γγ)$ is equal to the phase shift in the interference term of the time-dependent $K\rightarrow μ^+μ^-$ 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 $K\rightarrow μ^+μ^-$ decay rate to extract the CKM matrix element combination $\vert V_{ts} V_{td} \sin(β+β_s)\vert\approx A^2λ^5\barη$. We find a model-independent theoretically clean prediction, $\cos^2φ_0 = 0.96 \pm 0.03$. 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 $\cosφ_0$, reducing a four-fold ambiguity in the angle $φ_0$ to a two-fold one.