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$B_s \to \mu^+ \mu^- \gamma$, measured at high $q^2$ as a partiallyreconstructed decay, can probe the origin of the existing discrepancies insemi-leptonic $b \to s$ and $b \to c$ decays. We perform a complete study ofthis possibility. We start by reassessing the alleged discrepancies, with afocus on a unified EFT description. Using the SMEFT, we find that the tauonicWilson coefficient required by $R(D^{(*)})$ implies a universal muonic Wilsoncoefficient of precisely the size required by semi-muonic BR data and,separately, by semi-muonic angular analyses. We thus identify referencescenarios. Importantly, $B_s \to \mu^+ \mu^- \gamma$ offers a strategy toaccess them without being affected by the long-distance issues that hamper theprediction of semi-leptonic $B$ decays at low $q^2$. After quantifying to thebest of our knowledge the $B_s \to \mu^+ \mu^- \gamma$ experimental over thelong haul, we infer the $B_s \to \mu^+ \mu^- \gamma$ sensitivity to thecouplings relevant to the anomalies. In the example of the real-$\deltaC_{9,10}$ scenario, we find significances below 3$\sigma$. Such figure is to becompared with other single-observable sensitivities that one can expect frome.g. BR and angular data, whether at low or high $q^2$, and not affected bylong-distance issues such as narrow resonances or intermediate charmed di-mesonrescattering.

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