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

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