[Paper Review] Quark orbital motions from Wigner distributions

Tianbo Liu|arXiv (Cornell University)|Jun 30, 2014

High-Energy Particle Collisions Research1 references5 citations

TL;DR

This paper investigates quark orbital motions in a polarized proton using Wigner distributions within a light-cone spectator model, incorporating both scalar and axial-vector spectators and gauge link effects via one-gluon exchange. It finds that both u and d quarks exhibit positive orbital angular momentum at small x, but the d quark shows a sign reversal at large x, revealing nontrivial spin-orbit correlations in the nucleon structure.

ABSTRACT

We investigate quark Wigner distributions in a light-cone spectator model. Both the scalar and the axial-vector spectators are included. The light-cone wave functions are derived from effective quark-spectator-nucleon vertex and then generalized by adjusting the power of energy denominators. The gauge link is taken into account by introducing relative phases to the light-cone amplitudes, and the phases are estimated from one gluon exchange interactions. The mixing distributions, which describe the correlation between transverse coordinate and transverse momentum and represent quark orbital motions, are calculated from the Wigner distributions. We find both $u$ quark and $d$ quark have positive orbital angular momentum in a polarized proton at small $x$ region, but a sign change is observed at large $x$ region for the $d$ quark. Besides, some model relations between Wigner distributions with different polarization configurations are found.

Motivation & Objective

• To investigate quark orbital motions in the nucleon using Wigner distributions as a phase-space representation of partonic structure.
• To model the nucleon as a quark-spectator system with both scalar and axial-vector spectators to study spin and momentum correlations.
• To include gauge link effects via relative phases from one-gluon exchange to ensure gauge invariance in Wigner distributions.
• To calculate mixing distributions that provide probability interpretations of quark transverse coordinate–momentum correlations, revealing orbital motion.
• To analyze the orbital angular momentum and spin-orbit correlators for u and d quarks across different momentum fractions x.

Proposed method

• Derive light-cone wave functions from effective quark-spectator-nucleon vertex interactions in the spectator model.
• Generalize perturbative amplitudes by adjusting the power of energy denominators to preserve Lorentz invariance and frame independence.
• Introduce relative phases to light-cone amplitudes to model the gauge link (Wilson line) in the Wigner operator, estimated from one-gluon exchange.
• Construct ten independent quark Wigner distributions by combining quark and nucleon polarization states (unpolarized, longitudinal, transverse).
• Compute mixing distributions by integrating over one transverse coordinate and one transverse momentum, enabling probability interpretation of orbital motion.
• Calculate orbital angular momentum and spin-orbit correlators as integrals over mixing distributions to quantify spin-orbit alignment.

Experimental results

Research questions

• RQ1 How do quark orbital motions, as encoded in Wigner distributions, depend on the longitudinal momentum fraction x in a polarized proton?
• RQ2 What is the role of the gauge link (Wilson line) in shaping the structure of quark Wigner distributions, and how is it implemented in the light-cone formalism?
• RQ3 Do u and d quarks exhibit distinct orbital angular momentum behavior across the x-range, particularly in the transition from small to large x?
• RQ4 How do spin-orbit correlations manifest in the mixing distributions, and what is their sign and magnitude for different quark flavors?
• RQ5 Are there model-independent relations between Wigner distributions with different polarization configurations, and how do they relate to TMDs and IPDs?

Key findings

• Both u and d quarks exhibit positive orbital angular momentum in a longitudinally polarized proton at small x, indicating a preference for anticlockwise motion.
• The d quark shows a sign change in orbital angular momentum at large x, transitioning from positive to negative, indicating a reversal in preferred orbital motion direction.
• The unpolarized mixing distributions exhibit left-right and top-bottom symmetry, confirming no preferred direction in coordinate-momentum space for unpolarized quarks.
• The unpol-longitudinal mixing distributions show that both u and d quarks prefer clockwise motion in a polarized proton, implying negative spin-orbit correlation.
• The spin-orbit correlator can be negative even when both quark spin and orbital angular momentum are positive, due to dominant contributions from specific Fock state components.
• Model relations between Wigner distributions with different polarization configurations are found, which reduce to known relations between TMDs and IPDs upon integration.