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Theoretical analysis of the leptonic decays $B\to \ell \ell \ell'\barν_{\ell'}$

We discuss the amplitude of the $B\to l^+l^-l'ν'$ decays and the differential decay rate $d^2Γ/dq^2dq'^2$, $q$ the momentum of the $l^+l^-$ pair emitted from the electromagnetic vertex and $q'$ the momentum of the $l'ν'$ 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 $dΓ(B\to l^+l^-l' ν')/dq^2\propto 1/q^2$ in the limit $m'_l=0$, but contains also more singlular contribution of order ${m'^2_l}/q^4$, which we take into account. For the case $m_l'\le m_l$, the latter may be neglected and one obtains a mild logarithmic dependence of $Γ(B\to l^{+}l^{-}l'ν')$ on $m_l$. For the case $m_l\ll m'_l$, however, the ${m'^2_l}/q^4$ terms dominate the decay rate leading to $Γ(B\to l^{+}l^{-}l'ν')\sim m'^2_l/m^2_l$. We find the following features of the four-lepton $B$-decays: (i) The decay rates $Γ\left(B\to μ^+μ^-(μν_μ,eν_e)\right)$ are fully dominated by the region of light vector resonances $q^2\simeq M_ρ^2, M_ω^2$; (ii) The decay rate $Γ(B\to e^{+}e^{-}e ν_e)$ 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 $Γ(B\to e^+e^- μν_μ)\sim m_μ^2/m_e^2$ which is dominated by the region $q^2\sim 4m_e^2$ due to the terms $O(m_μ^2/q^4)$ in the differential distribution.