[Paper Review] Light-Ray Operators and their Application in QCD

B. Geyer, Müller, D.+1 more|ArXiv.org|Jun 8, 1994

Quantum Chromodynamics and Particle Interactions2 references3 citations

TL;DR

This paper introduces light-ray operators as a generalization of local operators in the operator product expansion, enabling a unified framework for nonperturbative parton distributions and wave functions in QCD. It derives evolution equations that include the Altarelli-Parisi and Brodsky-Lepage equations as special cases, showing the Altarelli-Parisi kernel emerges as a limiting case of a more general kernel, and applies the formalism to virtual Compton scattering near forward kinematics.

ABSTRACT

The nonperturbative parton distribution and wave functions are directly related to matrix elements of light-ray (nonlocal) operators. These operators are generalizations of the standard local operators known from the operator product expansion. The renormalization group equation for these operators leads to evolution equations for more general distribution amplitudes which include the Altarelli-Parisi and the Brodsky-Lepage equations as special cases. It is possible to derive the Altarelli-Parisi kernel as a limiting case of the extended Brodsky-Lepage kernel. As new application of the operator product expansion the virtual Compton scattering near forward direction is considered.

Motivation & Objective

• To extend the operator product expansion to nonlocal light-ray operators for describing nonperturbative parton distributions and wave functions.
• To derive evolution equations for generalized distribution amplitudes that encompass known results like the Altarelli-Parisi and Brodsky-Lepage equations.
• To demonstrate how the Altarelli-Parisi kernel arises as a limiting case of a broader Brodsky-Lepage-type kernel.
• To apply the formalism to the virtual Compton scattering process near forward kinematics, extending the utility of the operator product expansion.

Proposed method

• Formalism based on nonlocal light-ray operators, defined as path-ordered exponentials along light-like directions.
• Use of renormalization group equations for light-ray operators to derive evolution equations for distribution amplitudes.
• Derivation of the generalized evolution kernel that reduces to the Brodsky-Lepage kernel in specific limits.
• Application of the formalism to the virtual Compton scattering amplitude in the forward limit using the operator product expansion.
• Incorporation of renormalization effects through the use of the anomalous dimension matrix in the light-ray operator framework.
• Analysis of the operator structure in momentum space, focusing on the behavior near the forward region.

Experimental results

Research questions

• RQ1 How can nonlocal light-ray operators be used to generalize the standard operator product expansion in QCD?
• RQ2 What is the structure of the evolution kernel for generalized distribution amplitudes, and how does it relate to known evolution equations?
• RQ3 Can the Altarelli-Parisi evolution kernel be derived as a limiting case of a more general kernel within this framework?
• RQ4 How does the formalism apply to physical processes such as virtual Compton scattering near forward kinematics?
• RQ5 What are the renormalization group properties of light-ray operators and how do they affect distribution amplitudes?

Key findings

• The light-ray operator formalism successfully generalizes the operator product expansion to include nonlocal, nonperturbative parton distributions and wave functions.
• The evolution equations derived from light-ray operators reproduce the Brodsky-Lepage and Altarelli-Parisi equations as special cases.
• The Altarelli-Parisi kernel is shown to emerge as a limiting case of a more general evolution kernel derived from the extended formalism.
• The framework provides a consistent description of virtual Compton scattering near forward kinematics through the operator product expansion.
• The renormalization group analysis of light-ray operators leads to a unified evolution equation for generalized distribution amplitudes.
• The method establishes a direct connection between the structure of nonlocal operators and the dynamics of parton distributions in QCD.