Factorization and Sudakov Resummation in Leptonic Radiative B Decay

TL;DR

This work uses soft-collinear effective theory (SCET) to prove a QCD factorization theorem for the radiative leptonic decay B→γ l ν at leading power in Λ/m_b, expressing the amplitude as a convolution of a perturbatively calculable hard kernel with the B-meson LCDA φ_+^B. The authors perform a two-step perturbative matching to separate hard and hard-collinear scales into a hard function H and a jet function J, and they derive renormalization-group equations that enable Sudakov resummation down to low scales. They demonstrate the absence of endpoint divergences and non-valence Fock-state contributions at leading power, and provide the NLO hard-scattering kernel T with scale-dependent logs that cancel with LCDA evolution. The analysis clarifies the relation between heavy-collinear current anomalous dimensions and cusp anomalous dimensions, and yields RG-improved predictions with moderate radiative corrections, applicable to related decays such as B→γγ and B→γ l^+ l^-.

Abstract

Soft-collinear effective theory is used to prove factorization of the B->gamma+l+nu decay amplitude at leading power in Lambda/m_b, including a demonstration of the absence of non-valence Fock states and of the finiteness of the convolution integral in the factorization formula. Large logarithms entering the hard-scattering kernel are resummed by performing a two-step perturbative matching onto the low-energy effective theory, and by solving evolution equations derived from the renormalization properties of the leading-order B-meson light-cone distribution amplitude. As a byproduct, the evolution equation for heavy-collinear current operators in soft-collinear effective theory is derived.

Figures (5)

• Figure 1: Tree-level matching calculation for the Wilson coefficients $C_1$ and $C_2$, without (left) and with (right) an external soft gluon. The dashed line denotes the flavor-changing weak current. The resulting non-local operators in SCET are denoted by a crossed circle. Double lines represent effective heavy-quark fields in HQET.
• Figure 2: One-loop diagrams in the full theory contributing at leading power to the $B\to\gamma l\nu$ decay amplitude.
• Figure 3: One-loop diagrams in the effective theory whose contribution to the amplitude needs to be subtracted in the calculation of the Wilson coefficients.
• Figure 4: RG-improved predictions for the hard-scattering kernel at maximum photon energy. Left: Results at $\mu=1$ GeV. The bands refer to different values of the intermediate matching scale: $\mu_i^2=\Lambda_h m_b$ (center), $2\Lambda_h m_b$ (top), $0.5\Lambda_h m_b$ (bottom). Their width reflects the sensitivity to the high-energy matching scale $\mu_h^2$, varied between $2m_b^2$ and $0.5m_b^2$. The dashed line shows the result obtained at one-loop order. Right: Dependence of the kernel on the renormalization scale $\mu$, varied between 0.75 GeV and 2.0 GeV as indicated on the curves.
• Figure 5: Energy dependence of the convolution integral $I(E_\gamma)$ normalized to its tree-level value, assuming the model (\ref{['model']}) for the LCDA at a low scale $\mu_0$ such that $\alpha_s(\mu_0)=0.5$ (top), 0.75 (center), and 1.0 (bottom). The matching scales are set to their default values. The solid curves correspond to the resummed kernel, while the dashed ones are obtained at one-loop order.