Dispersion relations for $B^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\ell }^{\prime -}{\ell }^{\prime +}$ form factors

Abstract

Using dispersive methods, we study the $B\to {\gamma }^{\ast }$ form factors underlying the decay $B^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\ell }^{\prime -}{\ell }^{\prime +}$. We discuss the ambiguity that arises from a separation of the full $B^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\ell }^{\prime -}{\ell }^{\prime +}$ amplitude into a hadronic tensor and a final-state-radiation piece, including effects from nonvanishing lepton masses. For the eligibility of a dispersive treatment, we propose a decomposition of the hadronic part that leads to four form factors that are free of kinematic singularities. By establishing a set of dispersion relations, we then relate the $B\to {\gamma }^{\ast }$ form factors to the well-known $B\to V$, $V=\omega (782),\rho (770)$, analogues. Using the combination of a series expansion in a conformal variable and a vector-meson-dominance ansatz to parameterize the $B\to {\gamma }^{\ast }$ form factors, we infer the values of the associated unknown parameters from the available input on $B\to V$. The phenomenological application of our formalism includes the determination of the branching ratios and forward-backward asymmetries of the process $B^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\ell }^{\prime -}{\ell }^{\prime +}$ for the two lightest leptons.

Comments: 23 pages, 6 figures, 5 tables; v2: branching ratios with tau leptons included, version published in Phys. Rev. D