Insights on the current semi-leptonic $B$-decay discrepancies -- and how $B_s\to {\mu }^{+}{\mu }^{-}\gamma$ can help

Abstract

$B_s\to {\mu }^{+}{\mu }^{-}\gamma$, measured at high $q^2$ as a partially reconstructed decay, can probe the origin of the existing discrepancies in semi-leptonic $b\to s$ and $b\to c$ decays. We perform a complete study of this possibility. We start by reassessing the alleged discrepancies, with a focus on a unified EFT description. Using the SMEFT, we find that the tauonic Wilson coefficient required by $R(D^{(\ast )})$ implies a universal muonic Wilson coefficient of precisely the size required by semi-muonic BR data and, separately, by semi-muonic angular analyses. We thus identify reference scenarios. Importantly, $B_s\to {\mu }^{+}{\mu }^{-}\gamma$ offers a strategy to access them without being affected by the long-distance issues that hamper the prediction of semi-leptonic $B$ decays at low $q^2$. After quantifying to the best of our knowledge the $B_s\to {\mu }^{+}{\mu }^{-}\gamma$ experimental over the long haul, we infer the $B_s\to {\mu }^{+}{\mu }^{-}\gamma$ sensitivity to the couplings relevant to the anomalies. In the example of the real-$\delta C_{9,10}$ scenario, we find significances below 3$\sigma$. Such figure is to be compared with other single-observable sensitivities that one can expect from e.g. BR and angular data, whether at low or high $q^2$, and not affected by long-distance issues such as narrow resonances or intermediate charmed di-meson rescattering.

arXiv comment29 pages, 8 figures, 2 tables

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• Search for the $B_s^0\to {\mu }^{+}{\mu }^{-}\gamma$ decay
• Charming-loop contribution to $B_s\to \gamma \gamma$ decay
• Maximising the physics potential of $B^{\pm }\to {\pi }^{\pm }{\mu }^{+}{\mu }^{-}$ decays
• Efficient lattice QCD computation of radiative-leptonic-decay form factors at multiple positive and negative photon virtualities