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External Operators and Anomalous Dimensions in Soft-Collinear Effective Theory

Thomas Becher, Richard J. Hill, Bjoern O. Lange, Matthias Neubert

TL;DR

Becher, Hill, Lange, and Neubert show that SCET must include soft-collinear messenger modes that couple soft and collinear sectors. They extend SCET to external currents and four-quark operators, compute their anomalous dimensions, and demonstrate that soft-collinear exchanges are essential for the correct UV behavior and RG evolution. The decoupling analysis reveals that color-singlet four-quark operators factorize into the sum of current anomalous dimensions, while color-octet operators receive nontrivial soft-collinear contributions, impacting factorization proofs for exclusive B decays. Overall, the work strengthens the SCET framework for rigorous factorization in B physics and clarifies how external kinematics feed into operator running at high scales.

Abstract

It has recently been argued that soft-collinear effective theory for processes involving both soft and collinear partons contains a new soft-collinear mode, which can communicate between the soft and collinear sectors of the theory. The formalism incorporating the corresponding fields into the effective Lagrangian is extended to include external current and four-quark operators relevant to weak interactions. An explicit calculation of the anomalous dimensions of these operators reveals that soft-collinear modes are needed for correctly describing the ultraviolet behavior of the effective theory.

External Operators and Anomalous Dimensions in Soft-Collinear Effective Theory

TL;DR

Becher, Hill, Lange, and Neubert show that SCET must include soft-collinear messenger modes that couple soft and collinear sectors. They extend SCET to external currents and four-quark operators, compute their anomalous dimensions, and demonstrate that soft-collinear exchanges are essential for the correct UV behavior and RG evolution. The decoupling analysis reveals that color-singlet four-quark operators factorize into the sum of current anomalous dimensions, while color-octet operators receive nontrivial soft-collinear contributions, impacting factorization proofs for exclusive B decays. Overall, the work strengthens the SCET framework for rigorous factorization in B physics and clarifies how external kinematics feed into operator running at high scales.

Abstract

It has recently been argued that soft-collinear effective theory for processes involving both soft and collinear partons contains a new soft-collinear mode, which can communicate between the soft and collinear sectors of the theory. The formalism incorporating the corresponding fields into the effective Lagrangian is extended to include external current and four-quark operators relevant to weak interactions. An explicit calculation of the anomalous dimensions of these operators reveals that soft-collinear modes are needed for correctly describing the ultraviolet behavior of the effective theory.

Paper Structure

This paper contains 6 sections, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. SCET fields and interactions at leading order
  3. Soft-collinear current operators
  4. Four-quark operators
  5. Decoupling transformation
  6. Conclusions

Figures (2)

  • Figure 1: SCET graphs contributing to the anomalous dimension of a soft-collinear current. Full lines denote soft fields, dashed lines collinear fields, and dotted lines soft-collinear fields.
  • Figure 2: SCET graphs contributing to the anomalous dimension of the four-quark operators $Q_{(C)}(\omega,\sigma)$. Full lines denote soft fields, dashed lines collinear fields, and dotted lines soft-collinear fields.