---
title: Gluon Wigner Distributions in QCD
url: https://www.emergentmind.com/topics/gluon-wigner-distributions
type: topic
---

# Gluon Wigner Distributions in QCD

Searching arXiv for recent and foundational papers on gluon Wigner distributions.
Gluon Wigner distributions are the most differential phase-space objects used to describe gluons in QCD, encoding the joint dependence of gluons on longitudinal momentum fraction, transverse momentum, and transverse position, and, in generalized formulations, on a boost-invariant longitudinal coordinate as well [1711.04602]. They are quasi-probability distributions rather than ordinary probabilities, because they can become negative, but they unify the information content of generalized transverse-momentum dependent distributions (GTMDs), generalized parton distributions (GPDs), and transverse-momentum dependent distributions (TMDs) within a single framework [1511.00922]. In practice, gluon Wigner distributions are defined from off-forward correlators of gluon field strengths \(F^{+i}\) with polarization projectors \(\Gamma^{ij}\), and they have been investigated in perturbative dressed-quark models, light-cone spectator models, AdS/QCD-inspired spectator constructions, and small-\(x\) Color Glass Condensate formulations [1709.00943].

## 1. Definition and kinematic content

At leading twist, the gluon Wigner distribution is defined on the light front at \(z^+=0\) as a Fourier transform in the transverse momentum transfer \(\boldsymbol{\Delta}_\perp\) of a bilocal gluon correlator built from field strengths \(F^{+i}\) [1711.04602]. In the dressed-quark formulation, a standard operator definition is
\[
\begin{aligned}
x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}}
\int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \\
&\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0},
\end{aligned}
\]
with \(x=k^+/p^+\), transverse momentum \(\boldsymbol{k}_\perp\), and impact parameter \(\boldsymbol{b}_\perp\) conjugate to \(\boldsymbol{\Delta}_\perp\) [1711.04602]. The field-strength component is
\[
F^{+i}=\partial^+A^i-\partial^iA^+ + g f^{abc} A^{+a}A^{ib},
\]
with suppressed color indices [1711.04602].

This definition makes explicit that the natural reduced phase space for gluons is five-dimensional, \((x,\boldsymbol{k}_\perp,\boldsymbol{b}_\perp)\), after fixing light-front time and integrating out the light-cone energy variable [1511.00922]. In more general, nonzero-skewness formulations, the Wigner distribution can also depend on a boost-invariant longitudinal coordinate \(\sigma\), producing a mixed longitudinal-position representation in addition to transverse phase space [2309.03917]. In that setting the gluon phase-space density is described as
\[
\rho(x,\mathbf{b}_\perp,\sigma;\mathbf{p}_\perp;\text{spins}),
\]
with \(\sigma=\tfrac12 b^-P^+\) or equivalently the corresponding boost-invariant longitudinal variable used in the Fourier transform over skewness \(\xi\) [2603.24694].

The polarization structure is selected by \(\Gamma^{ij}\). At twist two, the standard choices are
\[
\Gamma^{ij}=\delta_\perp^{ij},\qquad
\Gamma^{ij}=-i\epsilon_\perp^{ij},\qquad
\Gamma^{ij}=\Gamma^{RR},\qquad
\Gamma^{ij}=\Gamma^{LL},
\]
corresponding to unpolarized, longitudinally polarized, and circular or linearly polarized gluon sectors [1711.04602]. Different target polarizations then produce the usual \(UU\), \(UL\), \(LU\), \(LL\), and transverse-polarization analogues [1709.00943].

A central structural fact is that Wigner distributions are Fourier transforms of GTMDs, GTMDs reduce to GPDs upon integration over \(\boldsymbol{k}_\perp\), and GTMDs reduce to TMDs in the forward limit \(\Delta_\perp\to 0\) [1711.04602]. This is why gluon Wigner distributions are repeatedly described as “mother distributions” [1511.00922].

## 2. Gauge links, gauge choices, and operator variants

For gluons, gauge invariance is more intricate than for quarks because the correlator contains two color-charged field-strength operators. The dressed-quark analyses explicitly note that a gluon Wigner distribution “need two gauge links for color gauge invariance” [1711.04602]. In the practical calculations of the perturbative light-front model, the choice is light-cone gauge \(A^+=0\) and the gauge links are set to unity [1711.04602]. The same simplification is adopted in related model studies of gluon GTMDs and Wigner distributions [1706.10183].

This simplification suppresses process dependence and removes T-odd effects in those model calculations [1706.10183]. It also means that distinctions between different Wilson-line topologies are not resolved there, even though they are conceptually essential. In particular, gluon Wigner distributions can be defined with \(++\) or \(+-\) gauge-link combinations, corresponding respectively to Weizsäcker–Williams-type and dipole-type gluon distributions [1711.04602]. Hatta and collaborators showed that both of these give the same orbital angular momentum distribution of the gluon [1711.04602].

At small \(x\), the operator structure is commonly recast in terms of Wilson lines and dipole \(S\)-matrices. In that regime, the gluon Wigner distribution is related to the dipole amplitude \(T_Y(\boldsymbol{r},\boldsymbol{b})=1-S_Y(\boldsymbol{r},\boldsymbol{b})\) through
\[
x\,W(x,\boldsymbol{b},\boldsymbol{k}) = -\frac{2 N_c}{\alpha_s}\int \frac{d^2\boldsymbol{r}}{(2\pi)^2}\, e^{i \boldsymbol{k}\cdot\boldsymbol{r}}\, \left(\frac{1}{4}\nabla_{\boldsymbol{b}}^2 + \boldsymbol{k}^2\right)\, T_Y(\boldsymbol{r},\boldsymbol{b}),
\]
which provides the CGC realization of the gluon Wigner distribution [1609.05773]. In this formulation, the Wilson lines are the primary dynamical objects and the Wigner distribution inherits its structure from the impact-parameter-dependent dipole amplitude [1609.05773].

A further distinction appears between Wigner and Husimi distributions at small \(x\). The Husimi distribution is obtained by Gaussian smearing of the Wigner distribution in both \(\boldsymbol{b}\) and \(\boldsymbol{k}\), and in the CGC calculation it comes out positive everywhere within numerical accuracy, whereas the Wigner distribution is not positive definite [1609.05773]. This does not replace the Wigner distribution, but it changes interpretational emphasis from exact phase-space quasi-density to coarse-grained semiclassical density [1609.05773].

## 3. Model realizations and overlap representations

A large fraction of the explicit literature computes gluon Wigner distributions from overlaps of light-front wave functions. The simplest field-theoretic setting is the dressed-quark model, in which the target is a quark dressed by a gluon at one loop and the Fock space is truncated to \(|q\rangle\) and \(|qg\rangle\) sectors [1711.04602]. The dressed state is expanded as
\[
\begin{aligned}
\big|p^{+}, \boldsymbol{p}_{\perp}, s \big\rangle &= \Phi^{s}(p)\, b^{\dagger}_{s}(p) | 0 \rangle \\
&\quad + \sum_{s_1, s_2} \int \cdots \, \Phi^{s}_{s_1 s_2}(p; p_1,p_2)\, b^{\dagger}_{s_1}(p_1) a^{\dagger}_{s_2}(p_2)\, | 0 \rangle,
\end{aligned}
\]
with \(b^\dagger\) and \(a^\dagger\) creating quark and gluon states, respectively [1711.04602]. The two-particle quark–gluon LFWF is obtained in light-front Hamiltonian perturbation theory and contains the full spinor and polarization dependence required to evaluate the gluon correlator [1711.04602].

Within this model, only the two-particle Fock sector contributes to gluon Wigner distributions because the single-particle state contains no gluon [1711.04602]. The correlator is therefore represented as an overlap of two-body LFWFs evaluated at shifted transverse arguments determined by \(\boldsymbol{\Delta}_\perp\). Symbolically,
\[
W_{\sigma,\sigma'}^{[\Gamma]}(x,\boldsymbol{k}_\perp,\boldsymbol{b}_\perp)
\sim \int \frac{d^2\boldsymbol{\Delta}_\perp}{(2\pi)^2}
e^{-i\boldsymbol{\Delta}_\perp\cdot \boldsymbol{b}_\perp}
\sum \Psi^{*\,\sigma'}\, \mathcal{O}^{[\Gamma]}\,\Psi^\sigma,
\]
where \(\mathcal{O}^{[\Gamma]}\) contracts gluon polarization indices according to the chosen \(\Gamma^{ij}\) [1711.04602].

The light-cone spectator model introduces a different proton-level realization. In that construction, the proton is treated as an active gluon plus a spin-\(\tfrac12\) spectator with effective mass \(M_X\), and the LFWFs are modeled with a QED-like helicity structure together with an AdS/QCD-inspired soft-wall radial function [2603.24694]. The proton state is written as a two-body Fock expansion in active-gluon and spectator helicities, and the parameters are fixed from gluon PDFs at \(Q_0=2\) GeV [2603.24694]. This setup yields closed-form twist-2 gluon GTMDs \(F_{1,i}^g\) and \(G_{1,i}^g\), which are then Fourier transformed to Wigner distributions [2603.24694].

A related proton-level spectator construction was used to calculate transverse and mixed gluon Wigner distributions at zero skewness, together with canonical gluon OAM and spin–orbit correlations [2312.07997]. That model again treats the gluon as the active constituent but differs from the dressed-quark calculation in that the target is proton-like rather than a perturbative dressed quark [2312.07997]. By contrast, the CGC approach does not rely on LFWF overlap language; it instead computes the Wigner distribution from Wilson-line correlators evolved with BK or JIMWLK dynamics [1609.05773], [1902.05087].

## 4. Polarization structure and characteristic phase-space patterns

The leading-twist polarization classification for gluon Wigner distributions is extensive. In the dressed-quark literature, explicit distributions are defined for unpolarized, longitudinally polarized, and linearly polarized gluons inside unpolarized, longitudinally polarized, and transversely polarized targets [1709.00943]. The minimal and most frequently discussed set comprises \(W_{UU}\), \(W_{UL}\), \(W_{LU}\), and \(W_{LL}\) [1511.00922].

Several recurring geometric patterns emerge. For the unpolarized gluon in an unpolarized target, \(W_{UU}\) in impact-parameter space displays a central positive peak in the dressed-quark model [1711.04602]. In the later proton spectator model, the corresponding impact-parameter distribution is circularly symmetric around \(\boldsymbol{b}_\perp=0\), while the transverse-momentum-space distribution is circularly symmetric with a central negative maximum at \(\boldsymbol{p}_\perp=0\) for the first Mellin moment studied there [2603.24694]. These differences reflect model dependence rather than a contradiction of definitions.

Spin-dependent distributions typically show dipole or quadrupole structures. In the dressed-quark analyses, \(W_{UL}\), describing a longitudinally polarized gluon in an unpolarized target, “shows dipole-like structure” in impact-parameter space [1711.04602]. In the more detailed 2017 gluon study, \(W^{LU}\) and \(W^{UL}\) both display dipole patterns in both impact-parameter and transverse-momentum space, while quadrupole structures emerge in mixed spaces [1709.00943]. In the 2018 three-dimensional imaging study, \(W_{TU}^g\) exhibits a dipole-like structure in \(\boldsymbol{b}_\perp\) space and a quadrupole-like structure in \(\boldsymbol{k}_\perp\) space [1802.07249].

These patterns are not merely visual motifs. The dipole structures are tied to factors proportional to \((\boldsymbol{\Delta}_\perp\times\boldsymbol{k}_\perp)_z\), which become \((\boldsymbol{b}_\perp\times\boldsymbol{k}_\perp)_z\) after Fourier transform and therefore signal spin–orbit coupling [1802.07249]. The quadrupole patterns for linearly polarized or mixed-spin configurations indicate higher multipole correlations in phase space [1709.00943].

At nonzero skewness, the boost-invariant longitudinal coordinate \(\sigma\) introduces an additional pattern: diffraction-like oscillations. In the dressed-quark nonzero-skewness calculations, the \(\sigma\)-space gluon Wigner distributions exhibit oscillatory structures “reminiscent of the single-slit interference phenomenon in optics” [2507.13691]. In the proton spectator model inspired by AdS/QCD, all leading-twist gluon Wigner distributions in \(\sigma\)-space show oscillatory behavior analogous to diffraction, with the pattern more sensitive to \(x\) than to \(-t\) [2603.24694].

## 5. Orbital angular momentum, spin–orbit correlations, and spin decomposition

One of the main reasons gluon Wigner distributions are studied is that specific GTMDs and phase-space moments encode gluon orbital angular momentum and spin–orbit correlations. In the canonical phase-space form, the gluon OAM is written schematically as
\[
L_g^z \sim \int dx\, d^2\boldsymbol{k}_\perp\, d^2\boldsymbol{b}_\perp\,
(\boldsymbol{b}_\perp\times\boldsymbol{k}_\perp)_z\,
W_{\text{appropriate gluon polarizations}}(x,\boldsymbol{k}_\perp,\boldsymbol{b}_\perp),
\]
with the precise polarization channel determined by the GTMD decomposition [1711.04602].

In GTMD language, the canonical gluon OAM is associated with the gluon analogue of \(F_{1,4}\), while the gluon spin–orbit correlation is controlled by \(G_{1,1}\) [1501.03728]. The perturbative dressed-quark calculation found that canonical and kinetic gluon OAM are both nonzero and distinct, and that both decrease in magnitude as the quark mass increases [1511.00922]. The same work emphasized that, unlike in the quark sector of that model, canonical gluon OAM and gluon spin–orbit correlation are numerically different [1511.00922].

The 2015 detailed dressed-quark study derived explicit expressions for canonical gluon OAM \(l_z^g\), kinetic gluon OAM \(L_z^g\), and the gluon spin–orbit correlation \(C_z^g\) in terms of integrals over GTMDs [1501.03728]. There, canonical and kinetic gluon OAM differ because of the distinct GTMD and GPD structures entering their definitions [1501.03728]. The explicit expressions show that \(C_z^g\) is controlled by \(G_{1,1}^g\), and the sign structure suggests anti-alignment of gluon spin and gluon orbital motion in parts of the dressed-quark phase space [1501.03728].

In the proton spectator model with nonzero skewness and \(\sigma\)-space analysis, the canonical gluon OAM is related to \(\rho_{LU}\) and the spin–orbit correlation to \(\rho_{UL}\) [2603.24694]. That model gives
\[
l_z^g \approx -0.375,\qquad
C_z^g \approx -15.6,\qquad
s^g \simeq 0.215
\]
for the full \(x\)-range quoted there [2603.24694]. The negative sign of \(l_z^g\) means that the canonical gluon OAM tends to be anti-aligned with the proton spin, and the negative \(C_z^g\) indicates strong anti-alignment between gluon spin and gluon OAM [2603.24694].

A different proton spectator model obtained a negative canonical gluon OAM \(l_z^g\approx -0.333\) and negative spin–orbit correlation, both dominated by small \(x\), again indicating anti-alignment of gluon spin and OAM in that framework [2312.07997]. This suggests a degree of qualitative stability across spectator-type models, although the magnitudes remain model-dependent. A plausible implication is that gluon phase-space distortions linked to OAM are robust structural features, but their integrated values are sensitive to the assumed wave function and gauge-link realization.

## 6. Small-\(x\) formulations, elliptic gluon Wigner distributions, and phenomenology

At small \(x\), gluon Wigner distributions acquire a distinct geometric and phenomenological role through their relation to impact-parameter-dependent dipole amplitudes. In the CGC framework, the Wigner distribution is expressed through the dipole \(S\)-matrix, whose evolution is governed by the BK or JIMWLK equations [1609.05773], [1902.05087]. This makes saturation physics, transverse geometry, and angular correlations part of the same object.

A particularly important component is the elliptic gluon Wigner distribution, the \(\cos 2(\phi_b-\phi_k)\) harmonic in the angular decomposition
\[
G(\boldsymbol b_\perp,\boldsymbol k_\perp)
=
G^0(b_\perp,k_\perp)
+
2\cos 2(\phi_b-\phi_k)\,\widetilde G(b_\perp,k_\perp)
+\cdots,
\]
where \(\widetilde G\) is the elliptic component [1701.04254]. Physically, this term encodes a quadrupole deformation of the gluon phase space and a correlation between the direction of transverse momentum and the direction of impact parameter [1701.04254].

This elliptic component has direct phenomenological consequences. In double parton scattering in \(pp\) or \(pA\) collisions, it generates a \(\cos 2(\phi_{k_1}-\phi_{k_2})\) modulation and hence a nonzero \(v_2\) in two-particle correlations [1701.04254]. In that formulation, the elliptic flow parameter arises entirely from the product of elliptic gluon Wigner distributions of the target [1701.04254]. This provides a purely initial-state mechanism for azimuthal anisotropy in small systems.

The same small-\(x\) Wigner structure can be accessed in exclusive diffractive dijet production. In ultraperipheral \(pA\) collisions, the dipole gluon Wigner distribution enters the exclusive diffractive dijet cross section, and both its isotropic and elliptic components can be reconstructed from the measured angular dependence [1706.01765]. The \(\cos 2(\phi_P-\phi_\Delta)\) modulation directly probes the elliptic component \(\tilde S\) or \(\widetilde G\) [1706.01765]. Similarly, coherent diffractive dijet production at an EIC was shown in a full CGC+JIMWLK treatment to be sensitive to the leading anisotropy of the gluon Wigner distribution, with the predicted elliptic modulation depending strongly on the growth of the proton with decreasing \(x\) [1902.05087].

In the CGC calculations, the angular anisotropy of the Wigner distribution decreases with decreasing \(x\) because the proton grows in impact parameter and spatial gradients become smoother [1902.05087]. This produces a phenomenologically testable energy dependence of the dijet elliptic coefficient. The same studies also show that the Husimi and Wigner anisotropies differ at low transverse momentum because smearing suppresses the geometric correlation, but they converge at larger \(P\) where both distributions are positive and less sensitive to coarse graining [1609.05773], [1902.05087].

## 7. Open issues, limitations, and broader significance

Several limitations are repeatedly emphasized across the literature. In the perturbative dressed-quark and spectator calculations, the target is not a full nucleon but a simplified quark–gluon or gluon–spectator system, often with a truncated Fock space and no explicit higher-order evolution [1711.04602], [2312.07997]. Gauge links are commonly set to unity in light-cone gauge, so process dependence and T-odd structures are not addressed [1711.04602]. In the small-\(x\) CGC calculations, impact-parameter evolution requires regulators or model assumptions to control long-range tails, and the Husimi distribution depends on a smearing scale not fixed by first principles [1609.05773], [1902.05087].

Experimental access also remains indirect. Several theoretical proposals exist for probing gluon GTMDs and Wigner distributions, including diffractive dijet production in DIS and ultraperipheral collisions, virtual photon–nucleus quasielastic scattering, and future EIC measurements [1711.04602], [2603.24694]. Yet no direct experimental extraction of a gluon Wigner distribution currently exists [1711.04602].

Despite these limitations, the significance of gluon Wigner distributions is clear. They provide the most complete one-parton description of gluons in a hadron, unify transverse imaging and momentum tomography, encode spin–orbit and orbital-angular-momentum information, and connect proton structure studies to small-\(x\) saturation physics [1511.00922], [1609.05773]. The extension to boost-invariant longitudinal position space adds an additional layer of tomography, effectively producing a “6D” imaging language with \((x,\mathbf{p}_\perp)\) in momentum space and \((\mathbf{b}_\perp,\sigma)\) in position space [2603.24694].

A plausible implication is that future progress will require combining several presently separate lines of work: gauge-link-complete operator definitions, realistic proton wave functions or lattice-computable correlators, and phenomenological channels sensitive to both isotropic and elliptic components. The existing body of work already shows that gluon Wigner distributions are not only formal extensions of GTMDs, but operationally useful structures that organize orbital motion, polarization correlations, and geometric features of the gluon content of hadrons across both moderate-\(x\) and small-\(x\) regimes [1709.00943], [1902.05087].

Source: https://www.emergentmind.com/topics/gluon-wigner-distributions