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

TL;DR

This paper provides a theoretical analysis of the leptonic decay amplitudes of B mesons into four leptons, constructing dispersion representations for form factors and exploring decay rate behaviors across different lepton mass regimes.

Contribution

It introduces a dispersion-based method for form factor calculation in B decays, accounting for Ward identities and light vector resonances, and analyzes decay rate dependencies on lepton masses.

Findings

Decay rates are dominated by light vector resonance regions.

Significant enhancement in decay rate for certain lepton flavor combinations.

Differential decay rates exhibit singular behavior at small q^2 depending on lepton masses.

Abstract

We discuss the amplitude of the $B\to l^+l^-l'\nu'$ decays and the differential decay rate $d^2\Gamma/dq^2dq'^2$, $q$ the momentum of the $l^+l^-$ pair emitted from the electromagnetic vertex and $q'$ the momentum of the $l'\nu'$ 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 $d\Gamma(B\to l^+l^-l' \nu')/dq^2\propto 1/q^2$ 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 $\Gamma(B\to l^{+}l^{-}l'\nu')$ 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 $\Gamma(B\to l^{+}l^{-}l'\nu')\sim m'^2_l/m^2_l$. We find the following features of the four-lepton $B$-decays: (i) The decay rates $\Gamma\left(B\to \mu^+\mu^-(\mu \nu_\mu,e\nu_e)\right)$ are fully dominated by the region of light vector resonances $q^2\simeq M_\rho^2, M_\omega^2$; (ii) The decay rate $\Gamma(B\to e^{+}e^{-}e \nu_e)$ 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 $\Gamma(B\to e^+e^- \mu \nu_\mu)\sim m_\mu^2/m_e^2$ which is dominated by the region $q^2\sim 4m_e^2$ due to the terms $O(m_\mu^2/q^4)$ in the differential distribution.