Electroweak Effects in the Double Dalitz Decay $B_s\to l^{+}{l}^{-}{l}^{\mathrm{\prime }+}{l}^{\mathrm{\prime }-}$

We investigate the double Dalitz decays $B_s\to l^{+}{l}^{-}{l}^{\mathrm{\prime }+}{l}^{\mathrm{\prime }-}$ on the basis of the effective Hamiltonian for the transition $b\bar{s}\to l^{+}{l}^{-}$, and universal form factors suggested by QCD. The correlated mass spectrum of the two lepton pairs in the decay $B_s\to e^{+}{e}^{-}{\mu }^{+}{\mu }^{-}$ is derived in an efficient way, using a QED result for meson decays mediated by two virtual photons: $B_s\to {\gamma }^{\ast }{\gamma }^{\ast }\to e^{+}{e}^{-}{\mu }^{+}{\mu }^{-}$. A comment is made on the correlation between the planes of the two lepton pairs. The conversion ratios $\rho_{lll’l’}= \frac{\Gamma(B_s \to l^+ l^- l’^+ l’^-)}{\Gamma(B_s \to \gamma \gamma)}$areestimatedtobe$\rho_{eeee}=3 \times 10^{-4}, \rho_{ee\mu\mu}=9 \times 10^{-5} $\text{and}$ \rho_{\mu\mu\mu\mu}=3 \times 10^{-5}$,andareenhancedrelativetopureQEDby$10-30 %$.