On the Standard Model prediction for $\mathcal{B}(B_{s,d} \to \mu^{+} \mu^{-})$

The decay Bs→μ+μ− is one of the milestones of the flavor program at the LHC. We reappraise its Standard Model prediction. First, by analyzing the theoretical rate in the light of its main parametric dependence, we highlight the importance of a complete evaluation of higher-order electroweak corrections, at present known only in the large-mt limit, and leaving sizable dependence on the definition of electroweak parameters. Using insights from a complete calculation of such corrections for $K\to\pi\nu\bar{\nu}$ decays, we find a scheme in which NLO electroweak corrections are likely to be negligible. Second, we address the issue of the correspondence between the initial and the final state detected by the experiments, and those used in the theoretical prediction. Particular attention is devoted to the effect of the soft radiation, which has not been discussed for this mode in the previous literature, and that can lead to O(10 %) corrections to the decay rate. The “non-radiative” branching ratio (which is equivalent to the branching ratio fully inclusive of bremsstrahlung radiation) is estimated to be (3.23±0.27)×10−9 for the flavor eigenstate, with the main uncertainty resulting from the value of $f_{B_{s}}$, followed by the uncertainty due to higher order electroweak corrections. Applying the same strategy to Bd→μ+μ−, we find for its non-radiative branching ratio (1.07±0.10)×10−10.