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

TL;DR

This paper establishes a precise relation between decay phases of kaons and proposes a Standard Model test with about 2% accuracy, enhancing understanding of CP violation and CKM matrix elements.

Contribution

It provides a model-independent prediction for the phase relation in kaon decays and discusses how future measurements can test the Standard Model with high precision.

Findings

The phase in the unitarity relation equals the phase shift in decay interference.

A 2% precision Standard Model test is feasible at future kaon facilities.

Theoretical analysis favors negative cosine solutions, reducing ambiguity.

Abstract

We find that the phase appearing in the unitarity relation between $\mathcal{B}(K_L\rightarrow \mu^+\mu^-)$ and $\mathcal{B}(K_L\rightarrow \gamma\gamma)$ is equal to the phase shift in the interference term of the time-dependent $K\rightarrow \mu^+\mu^-$ 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 $K\rightarrow \mu^+\mu^-$ decay rate to extract the CKM matrix element combination $\vert V_{ts} V_{td} \sin(\beta+\beta_s)\vert\approx A^2\lambda^5\bar\eta$. We find a model-independent theoretically clean prediction, $\cos^2\varphi_0 = 0.96 \pm 0.03$. 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 $\cos\varphi_0$, reducing a four-fold ambiguity in the angle $\varphi_0$ to a two-fold one.