Quark Wigner Distributions and Orbital Angular Momentum in Light-front Dressed Quark Model

TL;DR

This work computes quark Wigner distributions in a light-front framework for a quark dressed with a gluon, revealing joint position–momentum information encoded in GTMDs and connecting to GPDs and TMDs. Using overlaps of two-particle light-front wave functions, the authors derive expressions for $\rho_{UU}$, $\rho_{LU}$, and $\rho_{LL}$ and evaluate twist-two GTMDs $F_{1i}$ and $G_{1i}$ to obtain both kinetic and canonical quark OAM. They show that canonical OAM, related to $F_{14}$, is not equal to kinetic OAM due to gluon contributions, with $F_{14}=-G_{11}$ and a negative spin–orbit correlation $C^{q}_{z}$. The results underscore the impact of gluonic degrees of freedom on quark OAM and spin correlations, and they highlight the qualitative differences from non-gluonic models. The analysis lays groundwork for future work on gluon Wigner distributions and transverse polarization effects.

Abstract

We calculate the Wigner functions for a quark target dressed with a gluon. These give a combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs). We calculate and compare the different definitions of quark orbital angular momentum in this model. We compare our results with other model calculations.

Figures (6)

• Figure 1: (Color online) Plots of the Wigner distributions vs m (mass in $GeV$) for fixed values of $b_\perp$ and $k_\perp$ at $\Delta_{max} = 1.0$ GeV. All the plots on the left (a,c,e) are for three fixed values of $b_\perp$ (0.1,0.5,1.0) in $GeV^{-1}$ where $k_\perp=0.4$ GeV. Plots on the right (b,d,f) are for three fixed values of $k_\perp$ (0.1,0.3,0.5) in $GeV$ where and $b_\perp=0.4$$GeV^{-1}$ . For all plots we took $\vec{k_\perp} = k \hat{j}$ and $\vec{b_\perp} = b \hat{j}$.
• Figure 2: (Color online) 3D plots of the Wigner distributions $\rho_{UU}$. Plots (a) and (b) are in $b$ space with $k_\perp = 0.4$ GeV. Plots (c) and (d) are in $k$ space with $b_\perp = 0.4$$GeV^{-1}$. Plots (e) and (f) are in mixed space where $k_x$ and $b_y$ are integrated. All the plots on the left panel (a,c,e) are for $\Delta_{max} = 1.0$ GeV. Plots on the right panel (b,d,f) are for $\Delta_{max} = 5.0$ GeV. For all the plots we kept $m = 0.33$ GeV, integrated out the $x$ variable and we took $\vec{k_\perp} = k \hat{j}$ and $\vec{b_\perp} = b \hat{j}$.
• Figure 3: (Color online) 3D plots of the Wigner distributions $\rho_{LU}$. Plots (a) and (b) are in $b$ space with $k_\perp = 0.4$ GeV. Plots (c) and (d) are in $k$ space with $b_\perp = 0.4$$GeV^{-1}$. Plots (e) and (f) are in mixed space where $k_x$ and $b_y$ are integrated. All the plots on the left panel (a,c,e) are for $\Delta_{max} = 1.0$ GeV. Plots on the right panel (b,d,f) are for $\Delta_{max} = 5.0$ GeV. For all the plots we kept $m = 0.33$ GeV, integrated out the $x$ variable and we took $\vec{k_\perp} = k \hat{j}$ and $\vec{b_\perp} = b \hat{j}$.
• Figure 4: (Color online) 3D plots of the Wigner distributions $\rho_{LL}$. Plots (a) and (b) are in $b$ space with $k_\perp = 0.4$ GeV. Plots (c) and (d) are in $k$ space with $b_\perp = 0.4$$GeV^{-1}$. Plots (e) and (f) are in mixed space where $k_x$ and $b_y$ are integrated. All the plots on the left panel (a,c,e) are for $\Delta_{max} = 1.0 GeV$. Plots on the right panel (b,d,f) are for $\Delta_{max} = 5.0$ GeV. For all the plots we kept $m = 0.33$ GeV, integrated out the $x$ variable and we took $\vec{k_\perp} = k \hat{j}$ and $\vec{b_\perp} = b \hat{j}$.
• Figure 5: (Color online) Plots of the Wigner distributions vs $b_\perp$ for different $\Delta_{max} (GeV)$ for a fixed value $k_\perp=0.4$ GeV and m = $0.33$ GeV. $b_\perp$ is in $GeV^{-1}$.