A precision relation between Γ(K → μ + μ − )(t) and $$ \mathcal{B}\left({K}_L\to {\mu}^{+}{\mu}^{-}\right)/\mathcal{B}\left({K}_L\to \gamma \gamma \right) $$
Avital Dery (Department of Physics, LEPP, Cornell University, Ithaca, NY, 14853, USA); Mitrajyoti Ghosh (Department of Physics, LEPP, Cornell University, Ithaca, NY, 14853, USA); Yuval Grossman (Department of Physics, LEPP, Cornell University, Ithaca, NY, 14853, USA); Teppei Kitahara (Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya, 464-8602, Japan, CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China, Institute for Advanced Research, Nagoya University, Nagoya, 464-8601, Japan, KEK Theory Center, IPNS, KEK, Tsukuba, 305-0801, Japan); Stefan Schacht (Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, United Kingdom)
We find that the phase appearing in the unitarity relation between $$ \mathcal{B}\left({K}_L\to {\mu}^{+}{\mu}^{-}\right) $$ and $$ \mathcal{B}\left({K}_L\to \gamma \gamma \right) $$ 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 $$ \mid {V}_{ts}{V}_{td}\sin \left(\beta +{\beta}_s\right)\mid \approx {A}^2{\lambda}^5\overline{\eta} $$ . 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.
