Theoretical analysis of the leptonic decays $B\to \ell \ell \ell {\bar{\nu }}_{\ell }$: Identical leptons in the final state

Abstract

We study the effects of the identical leptons in the final state of the $B^{+}\to {\ell }^{+}{\ell }^{-}{\ell }^{+}{\bar{\nu }}_{\ell }$ decay. The amplitude of the process is described by the same form factors as the amplitude of the $B\to \ell \ell {\ell }^{\prime }{\bar{\nu }}_{\ell }^{\prime }$ decay for non-identical leptons in the final state. However, the differential distributions are strongly different, as the $B^{+}\to {\ell }^{+}{\ell }^{-}{\ell }^{+}{\bar{\nu }}_{\ell }$ amplitude contains both the direct ($M_a$) and the exchange ($M_b$) diagrams. We calculate a number of the differential distirbutions. In particular, we propose an interesting observable that can be readily measured experimentally - the differential distribution over the invariant mass of the pair of leptons of the same charge, ${\ell }^{+}{\ell }^{+}$. The good news is that the interference between $M_a$ and $M_b$, $d{\mathcal{B}}_{ab}$, is found to be at the level of less than 1\% in all considered differental distributions and therefore can be neglected in the full kinematical region of this decay.

Comments: 9 pages. arXiv admin note: text overlap with arXiv:2107.07247