$B_s^0\to {\ell }^{+}{\ell }^{-}\gamma$ as a Test of Lepton Flavor Universality

Abstract

We discuss a number of strategies to reduce the $\mathcal{B}(B_s^0\to {\ell }^{+}{\ell }^{-}\gamma )$ theoretical error, and make such a measurement a new probe of the interactions that are interesting in the light of present-day flavor discrepancies. In particular, for low di-lepton invariant mass we propose to exploit the close parenthood between $\mathcal{B}(B_s^0\to {\ell }^{+}{\ell }^{-}\gamma )$ and the measured $\mathcal{B}(B_s^0\to \varphi (\to K^{+}{K}^{-})\gamma )$. For high $q^2$, conversely, we exploit the fact that the decay is dominated by two form-factor combinations, plus contributions from broad charmonium that we model accordingly. We construct the ratio $R_{\gamma }$, akin to $R_K$ and likewise sensitive to lepton-universality violation. Provided the two rates in this ratio are integrated in a suitable region that minimises bremsstrahlung contributions while maximising statistics, the ratio is very close to unity and the form-factor dependence cancels to an extent that makes it a new valuable probe of lepton-universality violating contributions in the effective Hamiltonian. We finally speculate on additional ideas to extract short-distance information from resonance regions, which are theoretically interesting but statistically limited at present.

Comments: 21 pages, 4 figures. v4: in appendix removed equation already present in main text