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

TL;DR

This paper develops dispersive methods to analyze $B o ext{virtual photon}$ form factors relevant for the decay $B^- o ext{leptons}$, relating them to known vector-meson decays and enabling predictions of decay observables.

Contribution

It introduces a dispersive framework for $B o ext{virtual photon}$ form factors, relating them to $B o V$ decays, and provides a method to predict decay rates and asymmetries.

Findings

Derived dispersion relations for $B o ext{virtual photon}$ form factors.

Connected $B o ext{virtual photon}$ form factors to $B o V$ decays.

Predicted branching ratios and asymmetries for $B^- o ext{leptons}$ decays.

Abstract

Using dispersive methods, we study the $B \to \gamma^*$ 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 $B \to \gamma^*$ 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 $B \to \gamma^*$ 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.