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

受第一次搜索稀有带电电流B衰减到四个轻子的启发，\(\ell \bar{\nu }_\ell \ell ^{(\prime )} \bar{\ell }^{(\prime )}\)，我们用分解方法计算衰减幅度。我们获得了\(B\rightarrow \gamma ^*\)形状因子，它取决于两个轻子对的不变质量，在\(\Lambda _\mathrm{QCD}/m_b\)的扩展中处于领先地位到\(\alpha _s\) 中的次领先顺序，以及\(\mathcal {O}(\alpha _s^0)\)次领先的幂。我们的计算预测了\(\ell ^{(\prime )} \bar{\ell }^{(\prime )}\) 中几次\(10^{-8}\) 的分支分数质量平方 bin 高达\(q^2=1~\) GeV \(^2\)与\(n_+q>3~\) GeV。随着\(q^2\) 的增加，分支分数迅速下降。这项研究的另一个重要动机是探索衰减率对前导扭曲B介子光锥分布幅度的反矩\(\lambda_B\)的敏感性。我们发现在小\(q^2\) bin 中，对\(\lambda _B\)的敏感性几乎与\(B^- \rightarrow \mathrm {\ell }^- \bar{\nu }_{\mathrm {\ell }}\gamma \)当\(\lambda _B\)很小，但增加了光介子中间共振贡献的不确定性。灵敏度随着\(q^2\) 的增大而降低。