Theoretical analysis of the leptonic decays $B\to \mathrm{\ell }\mathrm{\ell }{\mathrm{\ell }}^{\mathrm{\prime }}{\bar{\nu }}_{{\mathrm{\ell }}^{\mathrm{\prime }}}$

We discuss the amplitude of the $B\to l^{+}{l}^{-}{l}^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}$ decays and the differential decay rate $d^2\mathrm{\Gamma }\mathrm{/}d{q}^{2}d{q}^{\mathrm{\prime }2}$, $q$ the momentum of the $l^{+}{l}^{-}$ pair emitted from the electromagnetic vertex and $q^{\mathrm{\prime }}$ the momentum of the $l^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}$ pair emitted from the weak vertex. For the relevant form factors, we construct dispersion representations in $q^2$ which consistently take into account the Ward identity constraints at $q^2=0$ and the contributions of light vector resonances. This allows a reliable description of the form factors in the range $0<q^2\le 1$ GeV${}^2$ that saturates around 99% of the decay rate. The differential decay rate behaves at small $q^2$ as $d\Gamma(B\to l^+l^-l’ \nu’)/dq^2\propto 1/q^2$inthelimit$m’l=0$, but contains also more singlular contribution of order $m_l^{\mathrm{\prime }2}\mathrm{/}{q}^4$, which we take into account. For the case $m_l^{\mathrm{\prime }}\le m_l$, the latter may be neglected and one obtains a mild logarithmic dependence of $\mathrm{\Gamma }(B\to l^{+}{l}^{-}{l}^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }})$ on $m_l$. For the case $m_l\ll m_l^{\mathrm{\prime }}$, however, the $m_l^{\mathrm{\prime }2}\mathrm{/}{q}^4$ terms dominate the decay rate leading to $\mathrm{\Gamma }(B\to l^{+}{l}^{-}{l}^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }})\sim m_l^{\mathrm{\prime }2}\mathrm{/}{m}_l^2$. We find the following features of the four-lepton $B$-decays: (i) The decay rates $\mathrm{\Gamma }(B\to {\mu }^{+}{\mu }^{-}(\mu {\nu }_{\mu },e{\nu }_e))$ are fully dominated by the region of light vector resonances $q^2\simeq M_{\rho }^2,M_{\omega }^2$; (ii) The decay rate $\mathrm{\Gamma }(B\to e^{+}{e}^{-}e{\nu }_e)$ receives comparable contributions from the region near $q^2\sim 4m_e^2$ and from the resonance region; (iii) One finds a strong enhancement of the decay rate $\Gamma(B\to e^+e^- \mu \nu\mu)\sim m_\mu^2/m_e^2$whichisdominatedbytheregion$q^2\sim 4m_e^2$duetotheterms$O(m_\mu^2/q^4)$ in the differential distribution.