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

We consider the $ {B}_s^0 $→ μ$^{+}$μ$^{−}$γ 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 ϕ-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 → s discrepancies. Its discriminating power, however, relies on progress in form-factor uncertainties. Interestingly, high q$^{2}$ is the region where $ {B}_s^0 $→ μ$^{+}$μ$^{−}$γ is already being accessed experimentally, and the region where form factors are more accessible through non-perturbative QCD methods.