QCD factorization of the four-lepton decay $$B^-\rightarrow \ell \bar{\nu }_\ell \ell ^{(\prime )} \bar{\ell }^{(\prime )}$$

Motivated by the first search for rare charged-current $B$ decay to four leptons, $\ell \bar{\nu}_\ell \ell^{(\prime)} \bar{\ell}^{(\prime)}$, we calculate amplitude with factorization methods. We obtain $B\to \gamma^*$ form factors, which depend on invariant masses of two lepton pairs, at leading power in an expansion $\Lambda_{\rm QCD}/m_b$ next-to-leading order $\alpha_s$, and $\mathcal{O}(\alpha_s^0)$ power. Our calculations predict branching fractions a few times $10^{-8}$ $\ell^{(\prime)} \bar{\ell}^{(\prime)}$ mass-squared bin up $q^2=1~\text{GeV}^2$ $n_+q>3~$GeV. The fraction rapidly drops increasing $q^2$. An important further motivation this investigation has been explore sensitivity rate inverse moment $\lambda_B$ leading-twist meson light-cone distribution amplitude. find that small-$q^2$ bin, is almost comparable $B^- \rightarrow \ell^- \bar{\nu}_\ell\gamma$ when small, but added uncertainty from light-meson intermediate resonance contribution. degrades larger