Quark Wigner distributions and orbital angular momentum in light-front dressed quark model

We calculate the Wigner functions for a quark target dressed with a gluon at one loop in perturbation theory. The Wigner distributions give the combined position and momentum space information of the quark distributions and are related to both generalized parton distributions and transverse momentum dependent parton distributions. We calculate and compare the different definitions of quark orbital angular momentum and the spin-orbit correlations in this perturbative model. We compare our results with other model calculations.

Figure 1

Plots of the Wigner distributions vs $m$ (mass in GeV) for fixed values of $b_{\perp }$ and $k_{\perp }$ at ${\mathrm{\Delta }}_{\max }=1.0\mathrm{GeV}$. All the plots on the left [(a), (c), (e)] are for three fixed values of $b_{\perp }$ (0.1, 0.5, 1.0) in ${\mathrm{GeV}}^{-1}$ where $k_{\perp }=0.4\mathrm{GeV}$. 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 $b_{\perp }=0.4{\mathrm{GeV}}^{-1}$. For all plots we took $\overrightarrow{k_{\perp }}=k\widehat{j}$ and $\overrightarrow{b_{\perp }}=b\widehat{j}$.

Figure 2

3D plots of the Wigner distributions ${\rho }_{UU}$. Plots (a) and (b) are in $b$ space with $k_{\perp }=0.4\mathrm{GeV}$. Plots (c) and (d) are in $k$ space with $b_{\perp }=0.4{\mathrm{GeV}}^{-1}$. Plots (e) and (f) are in mixed space where $k_x$ and $b_y$ are integrated. All the plots on the left [(a), (c), (e)] are for ${\mathrm{\Delta }}_{\max }=1.0\mathrm{GeV}$. Plots on the right [(b), (d), (f)] are for ${\mathrm{\Delta }}_{\max }=5.0\mathrm{GeV}$. For all the plots we kept $m=0.33\mathrm{GeV}$, integrated out the $x$ variable, and took $\overrightarrow{k_{\perp }}=k\widehat{j}$ and $\overrightarrow{b_{\perp }}=b\widehat{j}$.

Figure 3

3D plots of the Wigner distributions ${\rho }_{LU}$. Plots (a) and (b) are in $b$ space with $k_{\perp }=0.4\mathrm{GeV}$. Plots (c) and (d) are in $k$ space with $b_{\perp }=0.4{\mathrm{GeV}}^{-1}$. Plots (e) and (f) are in mixed space where $k_x$ and $b_y$ are integrated. All the plots on the left [(a), (c), (e)] are for ${\mathrm{\Delta }}_{\max }=1.0\mathrm{GeV}$. Plots on the right [(b), (d), (f)] are for ${\mathrm{\Delta }}_{\max }=5.0\mathrm{GeV}$. For all the plots we kept $m=0.33\mathrm{GeV}$, integrated out the $x$ variable, and took $\overrightarrow{k_{\perp }}=k\widehat{j}$ and $\overrightarrow{b_{\perp }}=b\widehat{j}$.

Figure 4

3D plots of the Wigner distributions ${\rho }_{LL}$. Plots (a) and (b) are in $b$ space with $k_{\perp }=0.4\mathrm{GeV}$. Plots (c) and (d) are in $k$ space with $b_{\perp }=0.4{\mathrm{GeV}}^{-1}$. Plots (e) and (f) are in mixed space where $k_x$ and $b_y$ are integrated. All the plots on the left [(a), (c), (e)] are for ${\mathrm{\Delta }}_{\max }=1.0\mathrm{GeV}$. Plots on the right [(b), (d), (f)] are for ${\mathrm{\Delta }}_{\max }=5.0\mathrm{GeV}$. For all the plots we kept $m=0.33\mathrm{GeV}$, integrated out the $x$ variable, and took $\overrightarrow{k_{\perp }}=k\widehat{j}$ and $\overrightarrow{b_{\perp }}=b\widehat{j}$.

Figure 5

Plots of the Wigner distributions vs $b_{\perp }$ for different ${\mathrm{\Delta }}_{\max }$ (GeV) for a fixed value $k_{\perp }=0.4\mathrm{GeV}$ and $m=0.33\mathrm{GeV}$. $b_{\perp }$ is in ${\mathrm{GeV}}^{-1}$.

Figure 6

Plots of OAM (a) $L_z^q$ and (b) $l_z^q$ vs $m$ (GeV) for different values of $Q$ (GeV).