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The evolution of parton distributions beyond leading order: the singlet case

R. K. Ellis, W. Vogelsang

TL;DR

The work addresses the calculation of two-loop (NLO) anomalous dimensions for space-like parton distributions in the singlet sector using the light-cone gauge with a PV regulator for $1/(n\cdot k)$.It presents a complete, well-documented factorization framework and derives the GLAP evolution equations from μ-dependence in $d=4-2\epsilon$ dimensions, connecting to the $x$-space kernels $P_{ij}(x,\alpha_S)$.The main results are explicit two-loop singlet and non-singlet splitting functions $P_{ij}^{(1)}$, expressed in terms of color structures and standard functions, and shown to be in agreement with Furmanski and Petronzio, with proper sum-rule consistency after endpoint treatments.By clarifying notation, integrals, and sub-diagram contributions, the paper strengthens the methodological basis for future extensions, such as polarized calculations or higher-loop analyses.

Abstract

A complete description of the calculation of anomalous dimensions (GLAP splitting functions) is given for parton distributions which appear in space-like processes. The calculation is performed in the light-cone gauge. The results are in agreement with the previous results of Furmanski and Petronzio.

The evolution of parton distributions beyond leading order: the singlet case

TL;DR

The work addresses the calculation of two-loop (NLO) anomalous dimensions for space-like parton distributions in the singlet sector using the light-cone gauge with a PV regulator for $1/(n\cdot k)$.It presents a complete, well-documented factorization framework and derives the GLAP evolution equations from μ-dependence in $d=4-2\epsilon$ dimensions, connecting to the $x$-space kernels $P_{ij}(x,\alpha_S)$.The main results are explicit two-loop singlet and non-singlet splitting functions $P_{ij}^{(1)}$, expressed in terms of color structures and standard functions, and shown to be in agreement with Furmanski and Petronzio, with proper sum-rule consistency after endpoint treatments.By clarifying notation, integrals, and sub-diagram contributions, the paper strengthens the methodological basis for future extensions, such as polarized calculations or higher-loop analyses.

Abstract

A complete description of the calculation of anomalous dimensions (GLAP splitting functions) is given for parton distributions which appear in space-like processes. The calculation is performed in the light-cone gauge. The results are in agreement with the previous results of Furmanski and Petronzio.

Paper Structure

This paper contains 14 sections, 103 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Calculation of anomalous dimensions
  3. Factorization
  4. Derivation of GLAP equation
  5. Non-singlet and singlet equations
  6. Renormalization constants
  7. Results
  8. Summary
  9. Virtual Integrals
  10. Two point function
  11. Three point function
  12. Real integrals
  13. Crossed ladder diagrams: topology b
  14. Real diagrams: topology cd

Figures (7)

  • Figure 1: Renormalization constants and the vertices which they renormalize
  • Figure 2: Basic topologies of the diagrams
  • Figure 3: Diagrams for the $qq$ part of the NLO non-singlet splitting functions
  • Figure 4: The $q\bar{q}$ diagrams
  • Figure 5: (a) Vertex correction graph (b) One parton emission (c) Two parton emission.
  • ...and 2 more figures