$B^*_{s,d} \to \mu^+ \mu^-$ and its impact on $B_{s,d} \to \mu^+ \mu^-$

The decay of $B^*_{s,d} \to \mu^+ \mu^-$~ and its impact on $B_{s,d} \to \mu^+ \mu^-$ is studied here. The $ \mu^+ \mu^-$ decay widths of vector mesons $B^*_{s,d}$ are about a factor of 700 larger than the corresponding scalar mesons $B_{s,d}$. The obtained ratio of the branching fractions $Br({B_{s,d}^*\to \mu^+\mu^-})/{Br({B_{s,d}\to\mu^+\mu^-})}$ is about $\frac{0.3 \times {\rm eV}}{{\Gamma(B^*_{s,d} \to B_{s,d} \gamma)}}$. At the same time, the hadronic contribution $B_{s,d} \to B^*_{s,d} \gamma \to \mu^+ \mu^-$ is estimated too. The relative increase of the amplitude of $B_{s,d}\to \mu^+\mu^-$ is about $(0.01\pm 0.006) \sqrt{\frac{{\Gamma(B^*_{s,d} \to B_{s,d} \gamma)}}{{100~ {\rm eV}}}}$. If we choose $\Gamma(B^*_{s,d} \to B_{s,d} \gamma)=2~$eV, the branching fractions of the vector mesons to lepton pair are $(6.2 \pm 0.6) \times 10^{-10}$ and $(1.7 \pm 0.2) \times 10^{-11}$ for $B^*_{s}$ and $B^*_{d}$ respectively. If we choose $\Gamma(B^*_{s,d} \to B_{s,d} \gamma)=200~$eV, the updated branching fractions of the scalar mesons to muon pair are $(3.78 \pm 0.25)\times 10^{-9}$ and $(1.09 \pm 0.09)\times 10^{-10}$ for $B_{s}$ and $B_{d}$ respectively. Further studies on $B^*_{s,d}$ are usefully here, including dielectron decay, two-body decay with $J/\psi$, and so on.