Quark Wigner distributions in a light-cone spectator model

TL;DR

The paper computes all $16$ twist-two quark Wigner distributions for $u$ and $d$ in a light-cone spectator model that includes scalar and axial-vector spectators and Melosh–Wigner rotations, providing a comprehensive phase-space picture of the proton’s quark structure. By defining the Wigner operator with various Dirac structures and evaluating overlaps of light-cone wave functions, the authors connect GTMDs to TMDs and IPDs, while truncating the Wilson line to a unit operator to focus on T-even parts. The results reveal flavor-dependent central concentration in coordinate space, dipole and quadrupole spin-orbit patterns, and notable differences between longitudinal and transverse polarizations, illustrating how Wigner distributions encode spin-orbit correlations beyond what TMDs and GPDs alone offer. The work emphasizes the potential of Wigner distributions to bridge TMDs and GPDs, while acknowledging the need for nontrivial Wilson lines and more realistic dynamics in future studies.

Abstract

We investigate the quark Wigner distributions in a light-cone spectator model. The Wigner distribution, as a quasi-distribution function, provides the most general one-parton information in a hadron. Combining the polarization configurations, unpolarized, longitudinal polarized or transversal polarized, of the quark and the proton, we can define 16 independent Wigner distributions at leading twist. We calculate all these Wigner distributions for the $u$ quark and the $d$ quark respectively. In our calculation, both the scalar and the axial-vector spectators are included, and the Melosh-Wigner rotation effects for both the quark and the axial-vector spectator are taken into account. The results provide us a very rich picture of the quark structure in the proton.

Figures (11)

• Figure 1: (Color online). Unpolarized Wigner distributions $\rho_{_\mathrm{UU}}$ and mixing distributions $\tilde{\rho}_{_\mathrm{UU}}$ for the $u$ quark (upper) and the $d$ quark (lower). The first column the Wigner distributions in transverse coordinate space with definite transverse momentum $\bm{k}_\perp=0.3\,\textrm{GeV}\,\hat{\bm{e}}_y$. The second column are the Wigner distributions in transverse momentum space with definite transverse coordinate $\bm{b}_\perp=0.4\,\textrm{fm}\,\hat{\bm{e}}_y$. The third column are the mixing distributions $\tilde{\rho}_{_\mathrm{UU}}$.
• Figure 2: (Color online). Unpol-longitudinal Wigner distributions $\rho_{_\mathrm{UL}}$ and mixing distributions $\tilde{\rho}_{_\mathrm{UL}}$ for the $u$ quark (upper) and the $d$ quark (lower). The first column the Wigner distributions in transverse coordinate space with definite transverse momentum $\bm{k}_\perp=0.3\,\textrm{GeV}\,\hat{\bm{e}}_y$. The second column are the Wigner distributions in transverse momentum space with definite transverse coordinate $\bm{b}_\perp=0.4\,\textrm{fm}\,\hat{\bm{e}}_y$. The third column are the mixing distributions $\tilde{\rho}_{_\mathrm{UL}}$.
• Figure 3: (Color online). Unpol-transverse Wigner distributions $\rho_{_\mathrm{UT}}$ and mixing distributions $\bar{\rho}_{_\mathrm{UT}}$ for $u$ quark (upper) and $d$ quark (lower). The first column are the distributions in transverse coordinate space with fixed transverse momentum $\bm{k}_\perp=0.3\,\textrm{GeV}\,\hat{\bm{e}}_x$ parallel to the quark polarization, and the second column are those with fixed transverse momentum $\bm{k}_\perp=0.3\,\textrm{GeV}\,\hat{\bm{e}}_y$ perpendicular to the quark polarization. The third column are the distributions in transverse momentrum space with fixed transverse coordinate $\bm{b}_\perp=0.4\,\textrm{fm}\,\hat{\bm{e}}_y$ perpendicular to the quark polarization. The fourth column are the mixing distributions.
• Figure 4: (Color online). Longi-unpolarized Wigner distributions $\rho_{_\mathrm{LU}}$ and mixing distributions $\tilde{\rho}_{_\mathrm{LU}}$ for $u$ quark (upper) and $d$ quark (lower). The first column the Wigner distributions in transverse coordinate space with definite transverse momentum $\bm{k}_\perp=0.3\,\textrm{GeV}\,\hat{\bm{e}}_y$. The second column are the Wigner distributions in transverse momentum space with definite transverse coordinate $\bm{b}_\perp=0.4\,\textrm{fm}\,\hat{\bm{e}}_y$. The third column are the mixing distributions $\tilde{\rho}_{_\mathrm{LU}}$.
• Figure 5: (Color online). Longitudinal Wigner distributions $\rho_{_\mathrm{LL}}$ and mixing distributions $\tilde{\rho}_{_\mathrm{LL}}$ for the $u$ quark (upper) and the $d$ quark (lower). The first column the Wigner distributions in transverse coordinate space with definite transverse momentum $\bm{k}_\perp=0.3\,\textrm{GeV}\,\hat{\bm{e}}_y$. The second column are the Wigner distributions in transverse momentum space with definite transverse coordinate $\bm{b}_\perp=0.4\,\textrm{fm}\,\hat{\bm{e}}_y$. The third column are the mixing distributions $\tilde{\rho}_{_\mathrm{LL}}$.