[Paper Review] Parton correlation functions and factorization in deep inelastic scattering

T. C. Rogers|ArXiv.org|Dec 7, 2007

Particle physics theoretical and experimental studies6 references3 citations

TL;DR

This paper proposes a perturbative QCD factorization formalism that preserves exact four-momentum conservation in both initial and final states by introducing parton correlation functions (PCFs), which depend on all components of parton four-momentum. The approach uses rigorous operator definitions for non-perturbative factors to ensure universality and consistency across processes, enabling higher-precision calculations in deep inelastic scattering beyond standard collinear factorization.

ABSTRACT

We outline the basic properties of a pertubative QCD factorization formalism that maintains exact over-all kinematics in both the initial and final states. Such a treatment requires the use of non-perturbative factors that depend on all components of parton four-momentum. These objects are referred to as parton correlation functions. We describe the complications faced in defining parton correlation functions and discuss recent progress. Emphasis is placed on the need for precise operator definitions in a complete derivation of factorization.

Motivation & Objective

• To develop a factorization framework in perturbative QCD that maintains exact over-all kinematics, including transverse momentum and invariant energy conservation.
• To address the limitations of standard collinear factorization, which approximates parton momenta and risks large errors in high-precision or final-state-sensitive processes.
• To establish a consistent, operator-based definition of non-perturbative factors—parton correlation functions (PCFs)—to ensure universality across different processes.
• To overcome the shortcomings of $k_T$-unintegrated PDFs, which lack reliable operator definitions and suffer from rapidity divergences.
• To lay the foundation for a practical, higher-order-computable formalism by deriving a factorization formula with PCFs that can be fitted to data and evolved in multiple rapidity variables.

Proposed method

• Formulate a factorization framework based on graphs with full initial and final state structure, avoiding the standard handbag diagram approximation.
• Use topological factorization via Ward identities to disentangle soft, jet, and target contributions into separate parton correlation functions (S, J, F).
• Define PCFs using gauge-invariant operator insertions, including Wilson lines at light-cone infinity to ensure exact gauge invariance.
• Apply double-counting subtractions to handle overlapping divergences and maintain consistency in higher-order calculations.
• Derive a factorization formula of the form $\sigma = C \otimes F \otimes J \otimes S + \mathcal{O}((\Lambda/Q)^a |\sigma|)$, where $C$ is a hard scattering coefficient and $F, J, S$ are PCFs.
• Ensure universality by grounding all non-perturbative factors in precise operator definitions that can be compared across processes.

Experimental results

Research questions

• RQ1 Can a factorization formalism be constructed that preserves exact four-momentum conservation in both initial and final states in deep inelastic scattering?
• RQ2 What are the necessary operator definitions for non-perturbative factors like parton correlation functions to ensure universality across different processes?
• RQ3 Why do standard $k_T$-unintegrated PDF definitions fail in a rigorous factorization framework, and how can they be improved?
• RQ4 How can rapidity divergences in $k_T$-PDFs be resolved through proper gauge-invariant definitions and Wilson lines?
• RQ5 Can the resulting PCF-based formalism be extended to higher orders and applied to multiple processes with consistent evolution equations?

Key findings

• The paper establishes a fully unintegrated factorization formalism using parton correlation functions (PCFs) that depend on all components of parton four-momentum, ensuring exact kinematic conservation.
• The proposed formalism avoids the approximations of standard collinear factorization, particularly the assumption that struck partons carry momentum only in the plus direction.
• Operator definitions for PCFs are essential for universality, as they allow consistent comparison and cross-process application of non-perturbative factors.
• The standard $k_T$-unintegrated PDF definition is shown to be inconsistent due to rapidity divergences and lack of proper gauge invariance, invalidating its use in rigorous factorization.
• The factorization formula $\sigma = C \otimes F \otimes J \otimes S + \mathcal{O}((\Lambda/Q)^a |\sigma|)$ is derived at lowest order, with $C$ as a smooth function rather than a generalized function like a $\delta$-function.
• The framework enables higher-order corrections via double-counting subtractions and provides a path toward evolution equations in multiple rapidity variables, potentially linking to equations like CCFM.