Sudakov Resummation for Subleading SCET Currents and Heavy-to-Light Form Factors

TL;DR

The paper develops a comprehensive RG-improved description of Sudakov resummation for subleading SCET currents in heavy-to-light form factors at large recoil. By performing a two-stage QCD→SCET_I→SCET_II matching and exploiting conformal symmetry in the hard-collinear sector, it derives the anomalous dimensions, solves the evolution equations, and establishes universality in the hard-scattering coefficients through the H_M factors. It demonstrates that soft-overlap and hard-scattering contributions can be treated within a unified factorization framework, with explicit one-loop jet-function results and RG-improved resummation for the hard-scattering terms, and provides numerical results illustrating the relative size of matching and running effects. The work also clarifies the conditions under which spin-symmetry relations are preserved and discusses the asymptotic Sudakov behavior, highlighting the practical impact of λ_B and other hadronic inputs on phenomenology of B→P,V transitions.

Abstract

The hard-scattering contributions to heavy-to-light form factors at large recoil are studied systematically in soft-collinear effective theory (SCET). Large logarithms arising from multiple energy scales are resummed by matching QCD onto SCET in two stages via an intermediate effective theory. Anomalous dimensions in the intermediate theory are computed, and their form is shown to be constrained by conformal symmetry. Renormalization-group evolution equations are solved to give a complete leading-order analysis of the hard-scattering contributions, in which all single and double logarithms are resummed. In two cases, spin-symmetry relations for the soft-overlap contributions to form factors are shown not to be broken at any order in perturbation theory by hard-scattering corrections. One-loop matching calculations in the two effective theories are performed in sample cases, for which the relative importance of renormalization-group evolution and matching corrections is investigated. The asymptotic behavior of Sudakov logarithms appearing in the coefficient functions of the soft-overlap and hard-scattering contributions to form factors is analyzed.

Figures (10)

• Figure 1: Two-step matching procedure for a hard-scattering contribution. The dashed lines in the SCET${}_{\rm I}$ diagram represent hard-collinear fields, those in the SCET${}_{\rm II}$ four-quark operator denote collinear fields. In all cases, the heavy quark is depicted as a double line, and the wavy line represents the flavor-changing current.
• Figure 2: One-loop QCD diagrams contributing to the matching calculation for the subleading scalar current. The external gluon can be attached at any of the places marked by a cross.
• Figure 3: $\hbox{SCET}_{\rm I}$ graphs contributing to the anomalous dimensions of the subleading heavy-collinear currents $J_j^B$, represented by a crossed circle. Full lines denote soft fields, dashed lines hard-collinear fields.
• Figure 4: Functions $U_\Gamma(u,\mu_h,\mu)$ in (\ref{['ULO']}) for $\mu_h=4.8$ GeV and $\mu=1.55$ GeV. The curves correspond to RG evolution with $\gamma_1+2\gamma_2$ (dotted) and $\gamma_1$ (dashed). The left and right plots show results derived by using 20 and 40 basis functions $\psi_n$, respectively. The solid lines were obtained by numerical integration of the evolution equation.
• Figure 5: Results for the Wilson coefficient $C_S^B(E,u,\mu_i)$ at $E=m_b/2$. The dashed lines represent the tree-level (gray) and one-loop (black) coefficients at the high scale $\mu_h=4.8$ GeV. The solid lines are obtained after evolving both coefficients with the leading-order anomalous dimension down to the intermediate scale $\mu_i=1.55$ GeV.