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Soft-collinear effective theory and heavy-to-light currents beyond leading power

M. Beneke, A. P. Chapovsky, M. Diehl, Th. Feldmann

TL;DR

The paper extends soft-collinear effective theory (SCET) to second order in the expansion parameter λ, constructing a gauge-invariant effective Lagrangian that couples collinear and ultrasoft modes and deriving heavy-to-light currents up to λ^2. It provides a detailed λ-expansion framework, including Wilson-line structures and multipole expansions for ultrasoft fields, and demonstrates a non-renormalization property for the tree-level SCET Lagrangian against hard loops. The authors apply this framework to heavy-to-light meson transitions at large recoil, clarifying leading-power factorization into soft-overlap and hard-scattering contributions and outlining power-suppressed corrections and their impact on form-factor relations. These results lay groundwork for a systematic, OPE-inapplicable treatment of power corrections in hard QCD processes and inform predictions for exclusive B decays into light mesons.

Abstract

An important unresolved question in strong interaction physics concerns the parameterization of power-suppressed long-distance effects to hard processes that do not admit an operator product expansion (OPE). Recently Bauer et al.\ have developed an effective field theory framework that allows one to formulate the problem of soft-collinear factorization in terms of fields and operators. We extend the formulation of soft-collinear effective theory, previously worked out to leading order, to second order in a power series in the inverse of the hard scale. We give the effective Lagrangian and the expansion of ``currents'' that produce collinear particles in heavy quark decay. This is the first step towards a theory of power corrections to hard processes where the OPE cannot be used. We apply this framework to heavy-to-light meson transition form factors at large recoil energy.

Soft-collinear effective theory and heavy-to-light currents beyond leading power

TL;DR

The paper extends soft-collinear effective theory (SCET) to second order in the expansion parameter λ, constructing a gauge-invariant effective Lagrangian that couples collinear and ultrasoft modes and deriving heavy-to-light currents up to λ^2. It provides a detailed λ-expansion framework, including Wilson-line structures and multipole expansions for ultrasoft fields, and demonstrates a non-renormalization property for the tree-level SCET Lagrangian against hard loops. The authors apply this framework to heavy-to-light meson transitions at large recoil, clarifying leading-power factorization into soft-overlap and hard-scattering contributions and outlining power-suppressed corrections and their impact on form-factor relations. These results lay groundwork for a systematic, OPE-inapplicable treatment of power corrections in hard QCD processes and inform predictions for exclusive B decays into light mesons.

Abstract

An important unresolved question in strong interaction physics concerns the parameterization of power-suppressed long-distance effects to hard processes that do not admit an operator product expansion (OPE). Recently Bauer et al.\ have developed an effective field theory framework that allows one to formulate the problem of soft-collinear factorization in terms of fields and operators. We extend the formulation of soft-collinear effective theory, previously worked out to leading order, to second order in a power series in the inverse of the hard scale. We give the effective Lagrangian and the expansion of ``currents'' that produce collinear particles in heavy quark decay. This is the first step towards a theory of power corrections to hard processes where the OPE cannot be used. We apply this framework to heavy-to-light meson transition form factors at large recoil energy.

Paper Structure

This paper contains 20 sections, 142 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Power counting and fields
  3. Kinematical definitions
  4. Fields
  5. Gauge transformations
  6. Wilson lines
  7. Effective Lagrangian
  8. Collinear Lagrangian
  9. Including ultrasoft quarks
  10. Expansion in $\lambda$
  11. No renormalization
  12. Heavy quark Lagrangian
  13. Heavy-to-light current to order $\lambda^2$
  14. Derivation of the effective current
  15. Reparameterization constraints
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Kinematics of heavy quark decay into a single cluster of collinear and ultrasoft particles.
  • Figure 2: Collinear interaction of the heavy quark.
  • Figure 3: Tree-level matching of the heavy quark current.
  • Figure :