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Non-universality of transverse momentum dependent parton distribution functions

C. J. Bomhof, P. J. Mulders

TL;DR

The paper analyzes why transverse momentum dependent parton distribution functions (TMDs) are generally nonuniversal in hadronic processes due to gauge-link (Wilson line) structures arising from initial- and final-state interactions. It introduces a decomposition of TMD correlators into universal, process-independent T-even and T-odd pieces plus universality-breaking terms governed by gluonic pole strengths, leading to the concept of gluonic pole cross sections that modify hard scattering parts. A central result is that universality-breaking components vanish upon $p_T$-integration and $p_T$-weighting, implying that integrated and certain weighted observables recover universal collinear behavior, while unintegrated observables retain nonuniversal contributions. The framework provides a gauge-invariant factorized form for unintegrated cross sections in terms of soft TMD correlators and hard parts (including gluonic-pole cross sections), and it catalogues the tree-level universality-breaking matrix elements, offering a structured path for future TMD-factorization studies in complex hadronic processes.

Abstract

In the field theoretical description of hadronic scattering processes, single transverse-spin asymmetries arise due to gluon initial and final state interactions. These interactions lead to process dependent Wilson lines in the operator definitions of transverse momentum dependent parton distribution functions. In particular for hadron-hadron scattering processes with hadronic final states this has important ramifications for possible factorization formulas in terms of (non)universal TMD parton distribution functions. In this paper we will systematically separate the universality-breaking parts of the TMD parton correlators from the universal T-even and T-odd parts. This might play an important role in future factorization studies for these processes. We also show that such factorization theorems will (amongst others) involve the gluonic pole cross sections, which have previously been shown to describe the hard partonic scattering in weighted spin asymmetries.

Non-universality of transverse momentum dependent parton distribution functions

TL;DR

The paper analyzes why transverse momentum dependent parton distribution functions (TMDs) are generally nonuniversal in hadronic processes due to gauge-link (Wilson line) structures arising from initial- and final-state interactions. It introduces a decomposition of TMD correlators into universal, process-independent T-even and T-odd pieces plus universality-breaking terms governed by gluonic pole strengths, leading to the concept of gluonic pole cross sections that modify hard scattering parts. A central result is that universality-breaking components vanish upon -integration and -weighting, implying that integrated and certain weighted observables recover universal collinear behavior, while unintegrated observables retain nonuniversal contributions. The framework provides a gauge-invariant factorized form for unintegrated cross sections in terms of soft TMD correlators and hard parts (including gluonic-pole cross sections), and it catalogues the tree-level universality-breaking matrix elements, offering a structured path for future TMD-factorization studies in complex hadronic processes.

Abstract

In the field theoretical description of hadronic scattering processes, single transverse-spin asymmetries arise due to gluon initial and final state interactions. These interactions lead to process dependent Wilson lines in the operator definitions of transverse momentum dependent parton distribution functions. In particular for hadron-hadron scattering processes with hadronic final states this has important ramifications for possible factorization formulas in terms of (non)universal TMD parton distribution functions. In this paper we will systematically separate the universality-breaking parts of the TMD parton correlators from the universal T-even and T-odd parts. This might play an important role in future factorization studies for these processes. We also show that such factorization theorems will (amongst others) involve the gluonic pole cross sections, which have previously been shown to describe the hard partonic scattering in weighted spin asymmetries.

Paper Structure

This paper contains 8 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Collinear Correlators
  3. Collinear functions in hadronic cross sections
  4. Transverse Momentum Dependent Correlators
  5. Parametrizations of parton correlators
  6. Transverse momentum dependent functions in hadronic cross sections
  7. Summary
  8. Universality-Breaking Matrix Elements

Figures (2)

  • Figure 1: Simplest structures (without loops) for gauge links and operators in quark correlators (a)-(b) and gluon correlators (c)-(f).
  • Figure 2: Possible behavior of the universal distribution function $f_1(x{,}p_T^2)$ (dashed line) and the process-dependent function $f_1^{(ab\rightarrow cd)}(x{,}p_T^2)$ (solid line) as a function of $|\boldsymbol p_T|^2$. Their difference-function, $\delta f_1^{(ab\rightarrow cd)}(x{,}p_T^2)$ (dash-dotted line), vanishes upon integration over $p_T$.