Gluon Wigner Distributions in QCD

• Gluon Wigner distributions are five-dimensional quasi-probability phase-space functions that encode gluon kinematics in terms of momentum and spatial coordinates.
• They unify the information of GTMDs, GPDs, and TMDs through Fourier transforms, providing a comprehensive picture of gluon distributions in QCD.
• Model studies using light-front wave functions and CGC frameworks reveal detailed spin–orbit correlations and gluon orbital angular momentum insights.

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 (Mukherjee, 2017). 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 (Mukherjee, 2015). 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 (More et al., 2017).

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 $F^{+i}$ (Mukherjee, 2017). 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 $\Gamma^{ij}$0 (Mukherjee, 2017). The field-strength component is

$\Gamma^{ij}$1

with suppressed color indices (Mukherjee, 2017).

This definition makes explicit that the natural reduced phase space for gluons is five-dimensional, $\Gamma^{ij}$2, after fixing light-front time and integrating out the light-cone energy variable (Mukherjee, 2015). In more general, nonzero-skewness formulations, the Wigner distribution can also depend on a boost-invariant longitudinal coordinate $\Gamma^{ij}$3, producing a mixed longitudinal-position representation in addition to transverse phase space (Jana et al., 2023). In that setting the gluon phase-space density is described as

$\Gamma^{ij}$4

with $\Gamma^{ij}$5 or equivalently the corresponding boost-invariant longitudinal variable used in the Fourier transform over skewness $\Gamma^{ij}$6 (Sonia et al., 25 Mar 2026).

The polarization structure is selected by $\Gamma^{ij}$7. At twist two, the standard choices are

$\Gamma^{ij}$8

corresponding to unpolarized, longitudinally polarized, and circular or linearly polarized gluon sectors (Mukherjee, 2017). Different target polarizations then produce the usual $\Gamma^{ij}$9, $x$0, $x$1, $x$2, and transverse-polarization analogues (More et al., 2017).

A central structural fact is that Wigner distributions are Fourier transforms of GTMDs, GTMDs reduce to GPDs upon integration over $x$3, and GTMDs reduce to TMDs in the forward limit $x$4 (Mukherjee, 2017). This is why gluon Wigner distributions are repeatedly described as “mother distributions” (Mukherjee, 2015).

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” (Mukherjee, 2017). In the practical calculations of the perturbative light-front model, the choice is light-cone gauge $x$5 and the gauge links are set to unity (Mukherjee, 2017). The same simplification is adopted in related model studies of gluon GTMDs and Wigner distributions (More et al., 2017).

This simplification suppresses process dependence and removes T-odd effects in those model calculations (More et al., 2017). 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 $x$6 or $x$7 gauge-link combinations, corresponding respectively to Weizsäcker–Williams-type and dipole-type gluon distributions (Mukherjee, 2017). Hatta and collaborators showed that both of these give the same orbital angular momentum distribution of the gluon (Mukherjee, 2017).

At small $x$8, the operator structure is commonly recast in terms of Wilson lines and dipole $x$9-matrices. In that regime, the gluon Wigner distribution is related to the dipole amplitude $z^+=0$0 through

$z^+=0$1

which provides the CGC realization of the gluon Wigner distribution (Hagiwara et al., 2016). 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 (Hagiwara et al., 2016).

A further distinction appears between Wigner and Husimi distributions at small $z^+=0$2. The Husimi distribution is obtained by Gaussian smearing of the Wigner distribution in both $z^+=0$3 and $z^+=0$4, and in the CGC calculation it comes out positive everywhere within numerical accuracy, whereas the Wigner distribution is not positive definite (Hagiwara et al., 2016). This does not replace the Wigner distribution, but it changes interpretational emphasis from exact phase-space quasi-density to coarse-grained semiclassical density (Hagiwara et al., 2016).

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 $z^+=0$5 and $z^+=0$6 sectors (Mukherjee, 2017). The dressed state is expanded as

$z^+=0$7

with $z^+=0$8 and $z^+=0$9 creating quark and gluon states, respectively (Mukherjee, 2017). 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 (Mukherjee, 2017).

Within this model, only the two-particle Fock sector contributes to gluon Wigner distributions because the single-particle state contains no gluon (Mukherjee, 2017). The correlator is therefore represented as an overlap of two-body LFWFs evaluated at shifted transverse arguments determined by $\boldsymbol{\Delta}_\perp$0. Symbolically,

$\boldsymbol{\Delta}_\perp$1

where $\boldsymbol{\Delta}_\perp$2 contracts gluon polarization indices according to the chosen $\boldsymbol{\Delta}_\perp$3 (Mukherjee, 2017).

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-$\boldsymbol{\Delta}_\perp$4 spectator with effective mass $\boldsymbol{\Delta}_\perp$5, and the LFWFs are modeled with a QED-like helicity structure together with an AdS/QCD-inspired soft-wall radial function (Sonia et al., 25 Mar 2026). 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 $\boldsymbol{\Delta}_\perp$6 GeV (Sonia et al., 25 Mar 2026). This setup yields closed-form twist-2 gluon GTMDs $\boldsymbol{\Delta}_\perp$7 and $\boldsymbol{\Delta}_\perp$8, which are then Fourier transformed to Wigner distributions (Sonia et al., 25 Mar 2026).

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 (Tan et al., 2023). 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 (Tan et al., 2023). 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 (Hagiwara et al., 2016, Mäntysaari et al., 2019).

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 (More et al., 2017). The minimal and most frequently discussed set comprises $\boldsymbol{\Delta}_\perp$9, $F^{+i}$0, $F^{+i}$1, and $F^{+i}$2 (Mukherjee, 2015).

Several recurring geometric patterns emerge. For the unpolarized gluon in an unpolarized target, $F^{+i}$3 in impact-parameter space displays a central positive peak in the dressed-quark model (Mukherjee, 2017). In the later proton spectator model, the corresponding impact-parameter distribution is circularly symmetric around $F^{+i}$4, while the transverse-momentum-space distribution is circularly symmetric with a central negative maximum at $F^{+i}$5 for the first Mellin moment studied there (Sonia et al., 25 Mar 2026). 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, $F^{+i}$6, describing a longitudinally polarized gluon in an unpolarized target, “shows dipole-like structure” in impact-parameter space (Mukherjee, 2017). In the more detailed 2017 gluon study, $F^{+i}$7 and $F^{+i}$8 both display dipole patterns in both impact-parameter and transverse-momentum space, while quadrupole structures emerge in mixed spaces (More et al., 2017). In the 2018 three-dimensional imaging study, $F^{+i}$9 exhibits a dipole-like structure in $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$0 space and a quadrupole-like structure in $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$1 space (More et al., 2018).

These patterns are not merely visual motifs. The dipole structures are tied to factors proportional to $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$2, which become $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$3 after Fourier transform and therefore signal spin–orbit coupling (More et al., 2018). The quadrupole patterns for linearly polarized or mixed-spin configurations indicate higher multipole correlations in phase space (More et al., 2017).

At nonzero skewness, the boost-invariant longitudinal coordinate $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$4 introduces an additional pattern: diffraction-like oscillations. In the dressed-quark nonzero-skewness calculations, the $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$5-space gluon Wigner distributions exhibit oscillatory structures “reminiscent of the single-slit interference phenomenon in optics” (Jana et al., 18 Jul 2025). In the proton spectator model inspired by AdS/QCD, all leading-twist gluon Wigner distributions in $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$6-space show oscillatory behavior analogous to diffraction, with the pattern more sensitive to $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$7 than to $$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$8 (Sonia et al., 25 Mar 2026).

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

$$\begin{aligned} x\, W_{\sigma, \sigma'}(x,\boldsymbol{k}_{\perp},\boldsymbol{b}_{\perp}) &= \int \frac{d^2 \boldsymbol{\Delta}_{\perp}}{(2\pi)^2}\, e^{-i\boldsymbol{\Delta}_{\perp}\cdot\boldsymbol{b}_{\perp}} \int \frac{dz^{-} d^{2} \boldsymbol{z}_{\perp}}{2(2\pi)^3\, p^+}\, e^{i k\cdot z} \ &\quad \times \Big\langle p^{+}, -\tfrac{\boldsymbol{\Delta}_{\perp}}{2},\sigma' \Big| \Gamma^{ij}\, F^{+i}\Big( -\frac{z}{2}\Big)\, F^{+j}\Big( \frac{z}{2}\Big) \Big| p^{+}, \frac{\boldsymbol{\Delta}_{\perp}}{2},\sigma \Big\rangle \Big|_{z^{+}=0}, \end{aligned}$$9

with the precise polarization channel determined by the GTMD decomposition (Mukherjee, 2017).

In GTMD language, the canonical gluon OAM is associated with the gluon analogue of $x=k^+/p^+$0, while the gluon spin–orbit correlation is controlled by $x=k^+/p^+$1 (Mukherjee et al., 2015). 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 (Mukherjee, 2015). The same work emphasized that, unlike in the quark sector of that model, canonical gluon OAM and gluon spin–orbit correlation are numerically different (Mukherjee, 2015).

The 2015 detailed dressed-quark study derived explicit expressions for canonical gluon OAM $x=k^+/p^+$2, kinetic gluon OAM $x=k^+/p^+$3, and the gluon spin–orbit correlation $x=k^+/p^+$4 in terms of integrals over GTMDs (Mukherjee et al., 2015). There, canonical and kinetic gluon OAM differ because of the distinct GTMD and GPD structures entering their definitions (Mukherjee et al., 2015). The explicit expressions show that $x=k^+/p^+$5 is controlled by $x=k^+/p^+$6, and the sign structure suggests anti-alignment of gluon spin and gluon orbital motion in parts of the dressed-quark phase space (Mukherjee et al., 2015).

In the proton spectator model with nonzero skewness and $x=k^+/p^+$7-space analysis, the canonical gluon OAM is related to $x=k^+/p^+$8 and the spin–orbit correlation to $x=k^+/p^+$9 (Sonia et al., 25 Mar 2026). That model gives

$\boldsymbol{k}_\perp$0

for the full $\boldsymbol{k}_\perp$1-range quoted there (Sonia et al., 25 Mar 2026). The negative sign of $\boldsymbol{k}_\perp$2 means that the canonical gluon OAM tends to be anti-aligned with the proton spin, and the negative $\boldsymbol{k}_\perp$3 indicates strong anti-alignment between gluon spin and gluon OAM (Sonia et al., 25 Mar 2026).

A different proton spectator model obtained a negative canonical gluon OAM $\boldsymbol{k}_\perp$4 and negative spin–orbit correlation, both dominated by small $\boldsymbol{k}_\perp$5, again indicating anti-alignment of gluon spin and OAM in that framework (Tan et al., 2023). 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-$\boldsymbol{k}_\perp$6 formulations, elliptic gluon Wigner distributions, and phenomenology

At small $\boldsymbol{k}_\perp$7, 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 $\boldsymbol{k}_\perp$8-matrix, whose evolution is governed by the BK or JIMWLK equations (Hagiwara et al., 2016, Mäntysaari et al., 2019). 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 $\boldsymbol{k}_\perp$9 harmonic in the angular decomposition

$\boldsymbol{b}_\perp$0

where $\boldsymbol{b}_\perp$1 is the elliptic component (Hagiwara et al., 2017). 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 (Hagiwara et al., 2017).

This elliptic component has direct phenomenological consequences. In double parton scattering in $\boldsymbol{b}_\perp$2 or $\boldsymbol{b}_\perp$3 collisions, it generates a $\boldsymbol{b}_\perp$4 modulation and hence a nonzero $\boldsymbol{b}_\perp$5 in two-particle correlations (Hagiwara et al., 2017). In that formulation, the elliptic flow parameter arises entirely from the product of elliptic gluon Wigner distributions of the target (Hagiwara et al., 2017). This provides a purely initial-state mechanism for azimuthal anisotropy in small systems.

The same small-$\boldsymbol{b}_\perp$6 Wigner structure can be accessed in exclusive diffractive dijet production. In ultraperipheral $\boldsymbol{b}_\perp$7 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 (Hagiwara et al., 2017). The $\boldsymbol{b}_\perp$8 modulation directly probes the elliptic component $\boldsymbol{b}_\perp$9 or $\Gamma^{ij}$00 (Hagiwara et al., 2017). 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 $\Gamma^{ij}$01 (Mäntysaari et al., 2019).

In the CGC calculations, the angular anisotropy of the Wigner distribution decreases with decreasing $\Gamma^{ij}$02 because the proton grows in impact parameter and spatial gradients become smoother (Mäntysaari et al., 2019). 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 $\Gamma^{ij}$03 where both distributions are positive and less sensitive to coarse graining (Hagiwara et al., 2016, Mäntysaari et al., 2019).

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 (Mukherjee, 2017, Tan et al., 2023). Gauge links are commonly set to unity in light-cone gauge, so process dependence and T-odd structures are not addressed (Mukherjee, 2017). In the small-$\Gamma^{ij}$04 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 (Hagiwara et al., 2016, Mäntysaari et al., 2019).

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 (Mukherjee, 2017, Sonia et al., 25 Mar 2026). Yet no direct experimental extraction of a gluon Wigner distribution currently exists (Mukherjee, 2017).

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-$\Gamma^{ij}$05 saturation physics (Mukherjee, 2015, Hagiwara et al., 2016). The extension to boost-invariant longitudinal position space adds an additional layer of tomography, effectively producing a “6D” imaging language with $\Gamma^{ij}$06 in momentum space and $\Gamma^{ij}$07 in position space (Sonia et al., 25 Mar 2026).

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-$\Gamma^{ij}$08 and small-$\Gamma^{ij}$09 regimes (More et al., 2017, Mäntysaari et al., 2019).