Theoretical analysis of the leptonic decays $Bto ell ell ellbar u_{ell}$

We discuss the general properties of the amplitude of the $Bto l^+l^-l u$ decays and calculate the related kinematical distributions $d^2Gamma/dq^2dq^2$, $q$ the momentum of the $l^+l^-$ pair emitted from the electromagnetic vertex and $q$ the momentum of the $l u$ pair emitted from the weak vertex. We emphasize that electromagnetic gauge invariance imposes essential constraints on the $Bto gamma^*l u$ amplitude at small $q^2$ which in the end yield the behaviour of the differential branching fraction as $dGamma(Bto l^+l^-l u)/dq^2propto 1/q^2$ and a mild logarithmic dependence of $Gamma(Bto l^{+}l^{-}l u)$ on the lepton mass $m_l$. Consequently, (i) the main contribution to the decay rate $Gamma(Bto mu^+mu^-e u_e )$ comes from the region of light vector resonances $rho^0$ and $omega$, $q^2simeq M_rho^2, M_omega^2$ and (ii) the decay rate $Gamma(Bto e^{+}e^{-}mu u_mu)$ receives comparable contributions from the region of small $q^2$ and from the resonance region. As the result, the decay rate $Gamma(Bto e^+e^-mu u_mu)$ is only a factor $sim 2$ larger than $Gamma(Bto mu^+mu^-e u_e)$. We perform a detailed analysis of the uncertainties in the theoretical predictions for the decays $Bto l^+l^-l u$ in the Standard Model. We found that the theoretical expectations for such decays in the Standard Model are only marginally compatible with the recent upper limits of the LHCb collaboration.