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Quark Imaging in the Proton Via Quantum Phase-Space Distributions

A. V. Belitsky, Xiangdong Ji, Feng Yuan

TL;DR

This work extends the concept of phase-space distributions to quantum chromodynamics by formulating quark and gluon Wigner distributions inside the proton. It shows how these master distributions encompass both elastic form factors and parton densities, and how reductions yield transverse-momentum dependent distributions (TMDs) and generalized parton distributions (GPDs), enabling 3D imaging in the proton rest frame. The paper provides a GPD-parametrization framework and illustrates 3D quark images with the MIT bag model, highlighting the role of x-dependent localization and orbital angular momentum. Overall, the approach offers a unified, physically rich picture of parton dynamics with potential for lattice calculations and experimental access via GPDs and related observables.

Abstract

We develop the concept of quantum phase-space (Wigner) distributions for quarks and gluons in the proton. To appreciate their physical content, we analyze the contraints from special relativity on the interpretation of elastic form factors, and examine the physics of the Feynman parton distributions in the proton's rest frame. We relate the quark Wigner functions to the transverse-momentum dependent parton distributions and generalized parton distributions, emphasizing the physical role of the skewness parameter. We show that the Wigner functions allow to visualize quantum quarks and gluons using the language of the classical phase space. We present two examples of the quark Wigner distributions and point out some model-independent features.

Quark Imaging in the Proton Via Quantum Phase-Space Distributions

TL;DR

This work extends the concept of phase-space distributions to quantum chromodynamics by formulating quark and gluon Wigner distributions inside the proton. It shows how these master distributions encompass both elastic form factors and parton densities, and how reductions yield transverse-momentum dependent distributions (TMDs) and generalized parton distributions (GPDs), enabling 3D imaging in the proton rest frame. The paper provides a GPD-parametrization framework and illustrates 3D quark images with the MIT bag model, highlighting the role of x-dependent localization and orbital angular momentum. Overall, the approach offers a unified, physically rich picture of parton dynamics with potential for lattice calculations and experimental access via GPDs and related observables.

Abstract

We develop the concept of quantum phase-space (Wigner) distributions for quarks and gluons in the proton. To appreciate their physical content, we analyze the contraints from special relativity on the interpretation of elastic form factors, and examine the physics of the Feynman parton distributions in the proton's rest frame. We relate the quark Wigner functions to the transverse-momentum dependent parton distributions and generalized parton distributions, emphasizing the physical role of the skewness parameter. We show that the Wigner functions allow to visualize quantum quarks and gluons using the language of the classical phase space. We present two examples of the quark Wigner distributions and point out some model-independent features.

Paper Structure

This paper contains 11 sections, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relativity Constraint On Interpretation of Form Factors and Parton Distributions
  3. The Proton Form Factors and Scheme-Dependent Charge Distributions
  4. Parton Distributions As Seen in the Rest Frame of the Proton
  5. Quantum Phase-Space (Wigner) Distributions
  6. General Aspects of Wigner Distributions
  7. Quantum Phase-Space Quark Distributions in the Proton
  8. Three-Dimensional Images of the Quarks in the Proton
  9. A GPD Parametrization
  10. The MIT Bag Model
  11. Summary and Conclusions

Figures (3)

  • Figure 1: The $u$-quark phase-space charge distribution at different values of the Feynman momentum for non-factorizable ansatz of generalized parton distributions (\ref{['NonfactorGPD']}). The vertical and horizontal axis corresponds to $z$ and $|\vec{r}_\perp|$, respectively, measured in femtometers. The [dashed] contours separate regions of positive [darker areas] and negative [lighter areas] densities. Below each contour plot we presented the shape of three-dimensional isodensity contours [$\rho = {\rm const}$].
  • Figure 2: The phase-space charge distribution for the $u$-quark at negative Feynman momentum $x = -0.05$ and $x = - 0.4$ [two left panels] and $d$-quark for positive $x = 0.4$ and $x = 0.6$ [two right panels].
  • Figure 3: The phase-space charge density $\rho_+ (\vec{r}, x)$ calculated in the bag model for values of Feynman momentum $x = 0.1, 0.33, 0.5, 0.9$.