QCD factorization of the four-lepton decay $B^{-}\to \ell {\bar{\nu }}_{\ell }{\ell }^{(\prime )}{\bar{\ell }}^{(\prime )}$

Abstract

Motivated by the first search for the rare charged-current $B$ decay to four leptons, $\ell {\bar{\nu }}_{\ell }{\ell }^{(\prime )}{\bar{\ell }}^{(\prime )}$, we calculate the decay amplitude with factorization methods. We obtain the $B\to {\gamma }^{\ast }$ form factors, which depend on the invariant masses of the two lepton pairs, at leading power in an expansion in ${\Lambda }_{\mathrm{QCD}}/m_b$ to next-to-leading order in ${\alpha }_s$, and at $\mathcal{O}({\alpha }_s^0)$ at next-to-leading power. Our calculations predict branching fractions of a few times $10^{-8}$ in the ${\ell }^{(\prime )}{\bar{\ell }}^{(\prime )}$ mass-squared bin up to $q^2=1{\text{GeV}}^2$ with $n_{+}q>3$GeV. The branching fraction rapidly drops with increasing $q^2$. An important further motivation for this investigation has been to explore the sensitivity of the decay rate to the inverse moment ${\lambda }_B$ of the leading-twist $B$ meson light-cone distribution amplitude. We find that in the small-$q^2$ bin, the sensitivity to ${\lambda }_B$ is almost comparable to $B^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }\gamma$ when ${\lambda }_B$ is small, but with an added uncertainty from the light-meson intermediate resonance contribution. The sensitivity degrades with larger $q^2$.

Comments: 22 pages, 7 figures; v2: study of the sensitivity to the shape of the B-LCDA added (Sec. 4.4). Matches published version