[Paper Review] Twist-2 Light-Ray Operators: Anomalous Dimensions and Evolution Equations

J. Blümlein, B. Geyer+1 more|ArXiv.org|Nov 20, 1997

Crystallography and Radiation Phenomena3 citations

TL;DR

This paper computes the anomalous dimensions of twist-2 light-ray operators in quantum chromodynamics (QCD) at one-loop order (O(αs)) for both non-singlet and singlet cases in unpolarized and polarized deep inelastic scattering. It derives evolution equations for structure functions, partition functions, and wave functions via Fourier transforms of matrix elements, extending Radyushkin's non-singlet solution to the singlet case, with connections to Altarelli-Parisi and Brodsky-Lepage kernels as special cases.

ABSTRACT

The non-singlet and singlet anomalous dimensions of the twist--2 light-ray operators for unpolarized and polarized deep inelastic scattering are calculated in $O(α_s)$. We apply these results for the derivation of evolution equations for partition functions, structure functions, and wave functions which are defined as Fourier transforms of the matrix elements of the light-ray operators. Special cases are the Altarelli-Parisi and Brodsky-Lepage kernels. Finally we extend Radyushkin's solution from the non-singlet to the singlet case.

Motivation & Objective

• To compute the non-singlet and singlet anomalous dimensions of twist-2 light-ray operators in QCD at O(αs) for both unpolarized and polarized deep inelastic scattering.
• To derive evolution equations for partition functions, structure functions, and wave functions defined as Fourier transforms of matrix elements of light-ray operators.
• To generalize Radyushkin's non-singlet solution to the singlet case, enabling a unified treatment of flavor-singlet evolution in QCD.
• To establish connections between the derived evolution kernels and known results such as the Altarelli-Parisi and Brodsky-Lepage kernels.
• To provide a systematic framework for analyzing the evolution of parton distributions and light-ray operators in high-energy scattering processes.

Proposed method

• Utilizes perturbative QCD to compute one-loop matrix elements of twist-2 light-ray operators in the context of deep inelastic scattering.
• Applies Fourier transformation to matrix elements of light-ray operators to define structure functions, wave functions, and partition functions.
• Derives evolution equations for these functions using the computed anomalous dimensions in both non-singlet and singlet channels.
• Extends Radyushkin’s solution, originally valid for non-singlet operators, to the more complex singlet case involving flavor-singlet currents.
• Relates the resulting evolution kernels to the well-known Altarelli-Parisi and Brodsky-Lepage kernels as limiting cases.
• Employs standard renormalization and factorization techniques in the light-cone operator product expansion framework to ensure consistency with QCD evolution.

Experimental results

Research questions

• RQ1 What are the one-loop anomalous dimensions of twist-2 light-ray operators in the non-singlet and singlet channels for unpolarized and polarized deep inelastic scattering?
• RQ2 How can evolution equations for structure functions, partition functions, and wave functions be derived from matrix elements of light-ray operators via Fourier transforms?
• RQ3 To what extent can Radyushkin’s non-singlet solution for light-ray operator evolution be generalized to the singlet case?
• RQ4 How do the derived evolution kernels relate to the established Altarelli-Parisi and Brodsky-Lepage kernels in QCD?
• RQ5 What is the role of flavor-singlet contributions in the evolution of light-ray operators in deep inelastic scattering?

Key findings

• The paper computes the O(αs) anomalous dimensions for both non-singlet and singlet twist-2 light-ray operators in unpolarized and polarized deep inelastic scattering.
• Evolution equations for structure functions, partition functions, and wave functions are derived as Fourier transforms of matrix elements of twist-2 light-ray operators.
• The singlet case of the evolution equations is systematically derived, extending Radyushkin’s non-singlet solution to include flavor-singlet contributions.
• The resulting evolution kernels reduce to the known Altarelli-Parisi and Brodsky-Lepage kernels in appropriate limits.
• The formalism provides a consistent framework for studying the evolution of parton distributions and light-ray operators in QCD at next-to-leading order.
• The results are valid for both polarized and unpolarized scattering processes, enabling a unified treatment of spin-dependent and spin-independent structure functions.