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Factorization of Hard Processes in QCD

John C. Collins, Davison E. Soper, George Sterman

TL;DR

The paper formalizes factorization theorems for hard QCD processes by separating short-distance, perturbatively calculable hard scattering from long-distance, nonperturbative parton distributions and fragmentation functions. It develops operator-based definitions of PDFs, introduces eikonal (Wilson line) techniques, and analyzes the renormalization-group structure via Altarelli–Parisi kernels. The work presents both a scalar- theory and a gauge-theory perspective, using a φ^3 toy model to illuminate leading regions, subtraction methods, and the role of final-state interactions and Ward identities in establishing gauge-invariant factorization. Together, these elements provide a comprehensive framework for predicting high-energy hadronic cross sections through universal PDFs and calculable hard parts, enabling precise QCD phenomenology across DIS, DY, and jet-related processes.

Abstract

We summarize the standard factorization theorems for hard processes in QCD, and describe their proofs.

Factorization of Hard Processes in QCD

TL;DR

The paper formalizes factorization theorems for hard QCD processes by separating short-distance, perturbatively calculable hard scattering from long-distance, nonperturbative parton distributions and fragmentation functions. It develops operator-based definitions of PDFs, introduces eikonal (Wilson line) techniques, and analyzes the renormalization-group structure via Altarelli–Parisi kernels. The work presents both a scalar- theory and a gauge-theory perspective, using a φ^3 toy model to illuminate leading regions, subtraction methods, and the role of final-state interactions and Ward identities in establishing gauge-invariant factorization. Together, these elements provide a comprehensive framework for predicting high-energy hadronic cross sections through universal PDFs and calculable hard parts, enabling precise QCD phenomenology across DIS, DY, and jet-related processes.

Abstract

We summarize the standard factorization theorems for hard processes in QCD, and describe their proofs.

Paper Structure

This paper contains 36 sections, 133 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Deeply Inelastic Scattering
  3. Single Particle Inclusive Annihilation
  4. Drell-Yan
  5. Factorization in the Parton Model
  6. Calculation of the Hard Scattering Cross Section
  7. Relation to the renormalization group
  8. The parton distribution functions
  9. Operator Definitions
  10. Feynman rules and eikonal lines
  11. Renormalization
  12. Relation to Structure Functions
  13. Other Parton Distributions
  14. Factorization for $\phi^3$ Theory
  15. Deeply inelastic scattering
  16. ...and 21 more sections

Figures (19)

  • Figure 1: Born amplitude for the Drell-Yan process.
  • Figure 2: Order $\alpha_{s}$ contributions to the Drell-Yan cross section.
  • Figure 3: (a) Feynman rules for eikonal lines in the amplitude and its complex conjugate. (b) A general contribution to a parton distribution.
  • Figure 4: One loop corrections to quark distribution, Eq. (\ref{['eq:43']}).
  • Figure 5: Deeply inelastic scattering.
  • ...and 14 more figures