Dispersion relations for B−→ℓ−ν¯ℓℓ′−ℓ′+ form factors

Abstract

Using dispersive methods, we study the $B\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}$ form factors underlying the decay ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\ell}}{\ensuremath{\ell}}^{\ensuremath{'}\ensuremath{-}}{\ensuremath{\ell}}^{\ensuremath{'}+}$. We discuss the ambiguity that arises from a separation of the full ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\ell}}{\ensuremath{\ell}}^{\ensuremath{'}\ensuremath{-}}{\ensuremath{\ell}}^{\ensuremath{'}+}$ 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 $B\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}$ form factors to the well-known $B\ensuremath{\rightarrow}V$, $V=\ensuremath{\omega}(782),\ensuremath{\rho}(770)$, analogs. Using the combination of a series expansion in a conformal variable and a vector-meson-dominance ansatz to parametrize the $B\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}$ form factors, we infer the values of the associated unknown parameters from the available input on $B\ensuremath{\rightarrow}V$. The phenomenological application of our formalism includes the determination of the branching ratios and forward-backward asymmetries of the process ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\ell}}{\ensuremath{\ell}}^{\ensuremath{'}\ensuremath{-}}{\ensuremath{\ell}}^{\ensuremath{'}+}$.