Factorization, the light-cone distribution amplitude of the B-meson and the radiative decay B -> gamma l nu_l

TL;DR

This paper verifies QCD factorization for the radiative decay B → γℓνℓ in the heavy-quark limit at one loop, showing the amplitude factorizes into a perturbative hard-scattering kernel convolved with the B-meson LCDA Φ_B. It computes the one-loop hard kernel, demonstrates the cancellation of infrared mass singularities, and uses SCET to resum Sudakov logarithms, with the resummed coefficient C(μ_F) and a mostly nonperturbative LCDA governing the decay. The study provides explicit results for the one-loop kernel, analyzes the scale dependence, and discusses how the three-scale structure (m_b, √(m_b Λ_QCD), Λ_QCD) enters the factorization and resummation. A phenomenological exploration for B^+ → e^+νγ highlights large sensitivity to λ_B and its inverse moments, underscoring the potential to extract nonperturbative input from data and informing factorization in two-body non-leptonic B decays. Overall, the work supports the applicability of QCD factorization to hard-collinear B decays and clarifies the role of the B-meson LCDA in hard spectator-like contributions.

Abstract

We study the radiative decay B -> gamma l nu_l in the framework of QCD factorization. We demonstrate explicitly that, in the heavy-quark limit and at one-loop order in perturbation theory, the amplitude does factorize, i.e. that it can be written as a convolution of a perturbatively calculable hard-scattering amplitude with the (non-perturbative) light-cone distribution amplitude of the B-meson. We evaluate the hard-scattering amplitude at one-loop order and verify that the large logarithms are those expected from a study of the b->u transition in the Soft-Collinear Effective Theory. Assuming that this is also the case at higher orders, we resum the large logarithms and perform an exploratory phenomenological analysis. The questions addressed in this study are also relevant for the applications of the QCD factorization formalism to two-body non-leptonic B-decays, in particular to the component of the amplitude arising from hard spectator interactions.

Figures (12)

• Figure 1: Graphical representation of the factorization formula. Only one of the two form-factor terms in eq. (\ref{['fff']}) is shown for simplicity.
• Figure 2: Schematic representation of the ${B\to\gamma\ell\nu_\ell}$ decay. The momentum of the $B$-meson is denoted by $p$ and that of the photon ($\gamma$) by $q$.
• Figure 3: Tree-level diagrams for the ${B\to\gamma\ell\nu_\ell}$ decay with the two-parton external state $|b\bar{u}\rangle$.
• Figure 4: Tree-level diagram for the ${B\to\gamma\ell\nu_\ell}$ decay with the three-parton external state $|b\bar{u}g\rangle$. Only the leading twist contribution is shown.
• Figure 5: Electromagnetic vertex: contributions to the form factors (left) and $\Phi^{(1)\,\textrm{em}}\otimes T^{(0)}$ (right). The dashed line in the right-hand diagram represents the path-ordered exponential.