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Deep inelastic scattering as x -> 1 using soft-collinear effective theory

Aneesh V. Manohar

TL;DR

This work applies soft-collinear effective theory to the x→1 endpoint of deep inelastic scattering, enabling systematic resummation of Sudakov double logarithms by separating scales Q^2, Q^2(1−x), and Λ_QCD^2. By matching the QCD current onto SCET currents in both target and Breit frames, and then matching the bilocal operators onto parton distributions, the authors derive anomalous dimensions that are linear in ln μ (and ln N for moments) and show the resummed results are free of Landau pole singularities. The analysis yields explicit one-loop results for the SCET current and quark-distribution renormalization, reproduces known DIS moments, and provides an exponentiated form for F_N with controlled higher-order corrections. The work highlights frame-dependent infrared cancellations, connects to Collins-Soper bilocal operators, and clarifies differences between DIS and B decays, with implications for precision endpoint phenomenology.

Abstract

Soft-collinear effective theory (SCET) is used to sum Sudakov double-logarithms in the x ->1 endpoint region for the deep inelastic scattering structure function. The calculations are done in both the target rest frame and the Breit frame. The separation of scales in the effective theory implies that the anomalous dimension of the SCET current is linear in ln mu, and the anomalous dimension for the Nth moment of the structure function is linear in ln N, to all orders in perturbation theory. The SCET formulation is shown to be free of Landau pole singularities. Some important differences between the deep inelastic structure function and the shape function in B decay are discussed.

Deep inelastic scattering as x -> 1 using soft-collinear effective theory

TL;DR

This work applies soft-collinear effective theory to the x→1 endpoint of deep inelastic scattering, enabling systematic resummation of Sudakov double logarithms by separating scales Q^2, Q^2(1−x), and Λ_QCD^2. By matching the QCD current onto SCET currents in both target and Breit frames, and then matching the bilocal operators onto parton distributions, the authors derive anomalous dimensions that are linear in ln μ (and ln N for moments) and show the resummed results are free of Landau pole singularities. The analysis yields explicit one-loop results for the SCET current and quark-distribution renormalization, reproduces known DIS moments, and provides an exponentiated form for F_N with controlled higher-order corrections. The work highlights frame-dependent infrared cancellations, connects to Collins-Soper bilocal operators, and clarifies differences between DIS and B decays, with implications for precision endpoint phenomenology.

Abstract

Soft-collinear effective theory (SCET) is used to sum Sudakov double-logarithms in the x ->1 endpoint region for the deep inelastic scattering structure function. The calculations are done in both the target rest frame and the Breit frame. The separation of scales in the effective theory implies that the anomalous dimension of the SCET current is linear in ln mu, and the anomalous dimension for the Nth moment of the structure function is linear in ln N, to all orders in perturbation theory. The SCET formulation is shown to be free of Landau pole singularities. Some important differences between the deep inelastic structure function and the shape function in B decay are discussed.

Paper Structure

This paper contains 23 sections, 142 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Outline of Calculation
  3. Kinematics
  4. Target Rest Frame
  5. Breit Frame
  6. Power Counting
  7. Matching the current at $Q^2$ from QCD to SCET
  8. Anomalous dimension of the SCET current
  9. Breit Frame
  10. Target Rest Frame
  11. Matching at $Q^2(1-x)$ onto the parton distribution function
  12. Target Rest Frame
  13. Breit Frame
  14. Renormalization of Parton Distributions
  15. Target Rest Frame
  16. ...and 8 more sections

Figures (11)

  • Figure 1: The deep inelastic scattering process. The incoming lepton with energy $E$ scatters off a hadronic target with momentum $p$ to produce the final hadronic state $X$.
  • Figure 2: One-loop vertex correction to the electromagnetic current in QCD.
  • Figure 3: One loop correction to the electromagnetic vertex in the Breit frame from (a) ultrasoft, (b) $n$-collinear and (c) ${\bar{n}}$-collinear gluons.
  • Figure 4: One loop correction to the electromagnetic vertex in the target rest frame from (a) ultrasoft and (b) $n$-collinear gluons.
  • Figure 5: Tree graph for the product of two currents in the target rest frame.
  • ...and 6 more figures