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

TL;DR

This paper explores how the decay $B_s o \mu^+ \mu^- \gamma$ can help clarify current discrepancies in semi-leptonic B decays, offering a new strategy less affected by long-distance uncertainties.

Contribution

It provides a comprehensive analysis of $B_s o \mu^+ \mu^- \\gamma$ as a probe for B decay anomalies within the SMEFT framework, identifying reference scenarios and assessing experimental sensitivities.

Findings

$B_s o \\mu^+ \\mu^- \\gamma$ can access anomaly-related couplings.

Sensitivity to couplings is below 3σ in the real-$\\delta C_{9,10}$ scenario.

$B_s o \\mu^+ \\mu^- \\gamma$ offers a complementary probe unaffected by long-distance issues.

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 $R(D^{(*)})$ 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.