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We consider the $B^0_s \to \mu^{+} \mu^{-} \gamma$ effective lifetime, andthe related CP-phase sensitive quantity $A_{\Delta\Gamma_s}^{\mu \mu \gamma}$,as a way to obtain qualitatively new insights on the current $B$-decaydiscrepancies. Through a fit comparing pre- to post-Moriond-2021 data weidentify a few theory benchmark scenarios addressing these discrepancies, andfeaturing large CP violation in addition. We then explore the possibility oftelling apart these scenarios with $A_{\Delta\Gamma_s}^{\mu \mu \gamma}$, onceresonance-modeling and form-factor uncertainties are taken into account. We doso in both regions of low and high invariant di-lepton mass-squared $q^2$. Forlow $q^2$, we show how to shape the integration range in order to reduce theimpact of the $\phi$-resonance modelling on the $A_{\Delta\Gamma_s}^{\mu \mu\gamma}$ prediction. For high $q^2$, we find that the corresponding pollutionfrom broad-charmonium resonances has a surprisingly small effect on$A_{\Delta\Gamma_s}^{\mu \mu \gamma}$. This is due to a number ofcancellations, that can be traced back to the complete dominance ofsemi-leptonic operator contributions for high $q^2$ -- at variance with low$q^2$ -- and to $A_{\Delta\Gamma_s}^{\mu \mu \gamma}$ behaving like aratio-of-amplitudes observable. Our study suggests that$A_{\Delta\Gamma_s}^{\mu \mu \gamma}$ is -- especially at high $q^2$ -- apotentially valuable probe of short-distance CP-violating effects in the verysame Wilson coefficients that are associated to current $b \to s$discrepancies. Its discriminating power, however, relies on progress inform-factor uncertainties. Interestingly, high $q^2$ is the region where $B^0_s\to \mu^{+} \mu^{-} \gamma$ is already being accessed experimentally, and theregion where form factors are more accessible through non-perturbative QCDmethods.

On the effective lifetime of $B_s \to \mu \mu \gamma$