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Nonperturbative Evolution of Parton Quasi-Distributions

Anatoly Radyushkin

TL;DR

The paper develops a covariant framework based on parton virtuality distribution functions (VDFs) to connect quasi-distributions (PQDs) computed on the lattice with light-cone parton distributions (PDFs) and transverse momentum dependent distributions (TMDs). It shows that PQDs are fully determined by TMDs via a simple transform, and introduces soft-TMD models (Gaussian and simple non-Gaussian) to study nonperturbative evolution with the hadron momentum P. Numerical results demonstrate strong nonperturbative PQD evolution at moderate P, with Q(y,P) broad at small P and converging toward f(y) as P grows; the evolution patterns are robust across the models. The findings offer a practical approach to fit lattice PQDs with analytic soft-TMD models and extrapolate to the infinite-momentum limit, providing new insights into three-dimensional hadron structure from lattice QCD.

Abstract

Using our formalism of parton virtuality distribution functions (VDFs) we establish a connection between the transverse momentum dependent distributions (TMDs) ${\cal F} (x, k_\perp^2)$ and quasi-distributions $Q(y,P_z)$ introduced recently by X. Ji for lattice QCD extraction of parton distributions $f(x)$. We build models for PQDs from the VDF-based models for soft TMDs, and analyze the $P_z$ dependence of the resulting PQDs. We observe a strong nonperturbative evolution of PQDs for small and moderately large values of $P_z$ reflecting the transverse momentum dependence of TMDs. Thus, the study of PQDs on the lattice in the domain of strong nonperturbative effects opens a new perspective for investigation of the 3-dimensional hadron structure.

Nonperturbative Evolution of Parton Quasi-Distributions

TL;DR

The paper develops a covariant framework based on parton virtuality distribution functions (VDFs) to connect quasi-distributions (PQDs) computed on the lattice with light-cone parton distributions (PDFs) and transverse momentum dependent distributions (TMDs). It shows that PQDs are fully determined by TMDs via a simple transform, and introduces soft-TMD models (Gaussian and simple non-Gaussian) to study nonperturbative evolution with the hadron momentum P. Numerical results demonstrate strong nonperturbative PQD evolution at moderate P, with Q(y,P) broad at small P and converging toward f(y) as P grows; the evolution patterns are robust across the models. The findings offer a practical approach to fit lattice PQDs with analytic soft-TMD models and extrapolate to the infinite-momentum limit, providing new insights into three-dimensional hadron structure from lattice QCD.

Abstract

Using our formalism of parton virtuality distribution functions (VDFs) we establish a connection between the transverse momentum dependent distributions (TMDs) and quasi-distributions introduced recently by X. Ji for lattice QCD extraction of parton distributions . We build models for PQDs from the VDF-based models for soft TMDs, and analyze the dependence of the resulting PQDs. We observe a strong nonperturbative evolution of PQDs for small and moderately large values of reflecting the transverse momentum dependence of TMDs. Thus, the study of PQDs on the lattice in the domain of strong nonperturbative effects opens a new perspective for investigation of the 3-dimensional hadron structure.

Paper Structure

This paper contains 16 sections, 54 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Parton distributions
  3. Virtuality distribution functions
  4. Collinear PDFs and TMDs
  5. Quasi-Distributions
  6. QCD
  7. Spinor quarks
  8. Gauge fields
  9. Sum Rules
  10. Primordial TMDs
  11. Models for soft part
  12. Gaussian model
  13. Simple non-Gaussian models
  14. Numerical results
  15. Leading-order hard tail
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Structure of parton-hadron matrix element.
  • Figure 2: Evolution of $Q(y,P)$ in the Gaussian model for $P/\Lambda =3,5,10$ (from bottom to top at $y=0.2$) compared to the limiting PDF $f(y) = (1-y)^3 \theta (y)$.
  • Figure 3: Ratio $Q(y,P)/f(y)$ in the Gaussian model for $y =0.1, 0.3, 0.7$ (from bottom to top) and $f(y)=(1-y)^3$.
  • Figure 4: Evolution of $Q(y,P)$ in the $m=0$ model for $P/\Lambda =3,5,10$ (from bottom to top at $y=0.2$) compared to the limiting PDF $f(y) = (1-y)^3 \theta (y)$.
  • Figure 5: Ratio $Q(y,P)/f(y)$ in the $m=0$ model for $y =0.1, 0.3, 0.7$ (from bottom to top) and $f(y)=(1-y)^3$.