Gluon generalized transverse momentum dependent parton distributions and Wigner functions of the Proton

TL;DR

This work computes leading-twist gluon GTMDs for unpolarized and longitudinally polarized gluons at nonzero skewness $ξ$ within a light-front gluon spectator model inspired by soft-wall AdS/QCD, using overlaps of two-particle LFWFs. Analytic expressions for the four $F^g_{1,1 ext{--}1,4}$ and four $G^g_{1,1 ext{--}1,4}$ GTMDs are derived, highlighting $ξ$-dependent azimuthal structures and connections to GPDs and TMDs in the forward and zero-skewness limits; the model also yields gluon OAM, spin–orbit correlations, and their relation to GTMDs such as $F_{1,4}^g$ and $G_{1,1}^g$. From the GTMDs, the authors compute Wigner distributions for various polarization configurations, reveal that Wigner functions are not positive definite, and analyze their behavior in impact-parameter, transverse-momentum, and mixed spaces to expose spin-orbit correlations and angular-momentum patterns. The numerical results show gluon GTMDs peaking at small $x$, nonzero $ξ$ generating azimuthal dependencies, and OAM and spin–orbit correlations that are negative, with forward-limit projections recovered as gluon GPDs and TMDs, in agreement with lattice and other model findings. Overall, the study provides a comprehensive gluon phase-space picture and elucidates how spin, OAM, and spin–orbit effects manifest in GTMDs and Wigner distributions, offering insights relevant to nucleon spin decomposition and to experimental access via GTMD-sensitive processes.

Abstract

We present a calculation of the leading-twist (twist-2) generalized transverse momentum distributions (GTMDs) for unpolarized and longitudinally polarized gluons in the proton, using a recently developed light-front gluon spectator model inspired by the soft-wall AdS/QCD prediction. We calculate the gluon GTMDs for non-zero skewness, using overlaps in terms of light-front wave functions. We also compute the gluon Wigner distributions for various polarization configurations of the proton target, including unpolarized, longitudinally polarized, and transversely polarized states. We present these Wigner distributions and the associated spin densities in both the transverse momentum plane and the transverse impact parameter plane. Furthermore, we evaluate the gluon orbital angular momentum and spin-orbit correlations, and extract the generalized parton distribution and transverse momentum-dependent distribution limits of the GTMDs.

Figures (11)

• Figure 1: Upper panel: Three-dimensional distributions of the four $F$-type gluon GTMDs: $F_{1,1}^{g},~F_{1,2}^{g},~F_{1,3}^{g}$ and $F_{1,4}^{g}$ as functions of the gluon longitudinal momentum fraction $x$ and the skewness parameter $\xi$, evaluated at fixed transverse momentum $k_{\perp}^{2}=0.3~\mathrm{GeV}^2$, transverse momentum transfer $\Delta_{\perp}^{2}=0.2~\mathrm{GeV}^2$, and $k_{\perp}\cdot\Delta_{\perp}=0$. Lower panel: Same as the upper panel but plotted as functions of $x$ and $\Delta_{\perp}^{2}$, with fixed $k_{\perp}^{2}=0.3~\mathrm{GeV}^2$ and $\xi=0.1$.
• Figure 2: Upper panel: Three-dimensional distributions of the four $G$-type gluon GTMDs: $G_{1,1}^{g},~G_{1,2}^{g},~G_{1,3}^{g}$ and $G_{1,4}^{g}$ as functions of the gluon longitudinal momentum fraction $x$ and the skewness parameter $\xi$, evaluated at fixed transverse momentum $k_{\perp}^{2}=0.3~\mathrm{GeV}^2$, transverse momentum transfer $\Delta_{\perp}^{2}=0.2~\mathrm{GeV}^2$, and $k_{\perp}\cdot\Delta_{\perp}=0$. Lower panel: Same distributions plotted as functions of $x$ and $\Delta_{\perp}^{2}$, with fixed $k_{\perp}^{2}=0.3~\mathrm{GeV}^2$ and $\xi=0.1$.
• Figure 3: Our model predictions for the gluon kinetic OAM $L_{z}^{g}$ (left panel), canonical OAM $\ell_{z}^{g}$ (middle panel) and the gluon spin-orbit correlation factor $\mathcal{C}_{z}^{g}$ (right panel) along with those reported in Refs. Tan:2023kblTan:2023vvi with an another spectator model by Lu et al. as a functions of gluon longitudinal momentum fraction $x$.
• Figure 4: Upper panel: The transverse Wigner distributions of unpolarized gluons in an unpolarized proton are shown in three configurations: in the transverse impact parameter plane at fixed transverse momentum $\mathbf{k}_\perp = 0.4\hat{e}_x~\text{GeV}$ (left), in the transverse momentum plane at fixed impact parameter $\mathbf{b}_{\perp} = 0.5\hat{e}_x~\text{fm}$ (middle), and in the mixed transverse plane after integrating over $k_x$ and $b_y$ (right). Lower panel: Same as the upper panel, but for longitudinally polarized gluons in an unpolarized proton.
• Figure 5: Upper panel: The transverse Wigner distributions of right-handed ($R$) linearly polarized gluons in an unpolarized proton are shown in three configurations: in the transverse impact parameter plane at fixed transverse momentum $\mathbf{k}_\perp = 0.4\hat{e}_x~\text{GeV}$ (left), in the transverse momentum plane at fixed impact parameter $\mathbf{b}_{\perp} = 0.5\hat{e}_x~\text{fm}$ (middle), and in the mixed transverse plane after integrating over $k_x$ and $b_y$ (right). Lower panel: Same as the upper panel, but for left-handed ($L$) linearly polarized gluons in an unpolarized proton.