---
title: 'Gluon GTMDs: Probing Nucleon Tomography'
url: https://www.emergentmind.com/topics/gluon-generalized-transverse-momentum-dependent-distributions-gtmds
type: topic
---

# Gluon GTMDs: Probing Nucleon Tomography

Generalized transverse momentum-dependent parton distributions (GTMDs) are the most general two-parton correlation functions within a hadron, encoding the joint dependence on the longitudinal momentum fraction, transverse momentum, and transverse spatial position of partons. Gluon GTMDs, defined for spin-½ hadronic targets, play a central role in mapping the multidimensional tomography of gluons in nucleons and nuclei. They bridge and generalize both generalized parton distributions (GPDs) and transverse momentum-dependent distributions (TMDs), serving as "mother distributions" for all two-parton correlation observables. Gluon GTMDs are key to understanding canonical orbital angular momentum (OAM) and spin–orbit correlations in QCD and are directly linked to gluon Wigner distributions, enabling phase-space imaging of the proton at small and moderate $x$.

## 1. Operator Definitions and Parametrization

The leading-twist, unintegrated, off-forward gluon GTMD correlator for a spin-½ hadronic target is defined as
\[
W^{ab\,ij}(x,{\vec k}_\perp,\xi,{\vec\Delta}_\perp)\,=\,\int\frac{dz^-\,d^2z_\perp}{(2\pi)^3\,P^+} e^{ixP^+z^--i{\vec k}_\perp\cdot {\vec z}_\perp} \langle p',\lambda'|F^{+i}_a(-\tfrac{z}{2})\,\mathcal W_{ab}\,F^{+j}_b(+\tfrac{z}{2})\,|p,\lambda\rangle \Big|_{z^+=0}\ .
\]
Here, $F^{\mu\nu}_a$ is the gluon field strength with color index $a$, $\mathcal W_{ab}$ is the adjoint gauge link (Wilson line), $P=(p+p')/2$, $\Delta=p'-p$, $x=k^+/P^+$, $\xi=-\Delta^+/(2P^+)$, and ${\vec k}_\perp$ is the average transverse gluon momentum [1802.10550]. These correlators can be projected onto unpolarized and helicity distributions, e.g. via $\delta_\perp^{ij}$ and $-i\varepsilon_\perp^{ij}$ [2509.14208, 2402.17162].

The leading-twist (twist-2) gluon GTMDs, parameterizing the correlator for a spin-½ target, are organized as follows:
- **F-type** (unpolarized target): $F_{1,1}$, $F_{1,2}$, $F_{1,3}$, $F_{1,4}$.
- **G-type** (longitudinally polarized gluons): $G_{1,1}$, $G_{1,2}$, $G_{1,3}$, $G_{1,4}$.
- **H-type** (linearly polarized gluons): $H_{1,n}^{\perp g}$, $n=1,\ldots,8$ (not detailed here).

In spinor language,
\[
W^g_{\lambda'\!,\lambda} = \frac{1}{2M} \bar u(p',\lambda') \bigg[ F_{1,1}^g + \frac{i\sigma^{i+}\,k_\perp^i}{P^+}F_{1,2}^g + \frac{i\sigma^{i+}\Delta_\perp^i}{P^+}F_{1,3}^g + \frac{i\sigma^{ij}k_\perp^i\Delta_\perp^j}{M^2}F_{1,4}^g \bigg] u(p,\lambda)~,
\]
and
\[
\widetilde W^g_{\lambda'\!,\lambda} = \frac{1}{2M}\bar u(p',\lambda')\bigg[-\frac{i\epsilon^{ij}_\perp k_\perp^i\Delta_\perp^j}{M^2}G_{1,1}^g +\frac{i\sigma^{i+}\gamma_5 k_\perp^i}{P^+}G_{1,2}^g + \frac{i\sigma^{i+}\gamma_5\Delta_\perp^i}{P^+}G_{1,3}^g + i\sigma^{+-}\gamma_5 G_{1,4}^g\bigg]u(p,\lambda)
\]
[2509.14208, 2402.17162].

## 2. Classification and Physical Interpretation

Gluon GTMDs encapsulate all leading-twist two-gluon correlations in the nucleon:
- $F_{1,1}^g$ represents the unpolarized gluon density (in forward limit yields $f_1^g$).
- $F_{1,4}^g$ encodes the canonical OAM of gluons: $\ell_z^g = -\int dx\,d^2k_\perp \frac{k_\perp^2}{M^2} F_{1,4}^g(x,0,k_\perp,0)$.
- $G_{1,1}^g$ is the gluon spin–orbit correlator: $\mathcal C_z^g = \int dx\,d^2k_\perp \frac{k_\perp^2}{M^2} G_{1,1}^g(x,0,k_\perp,0)$.
- $G_{1,4}^g$ is the longitudinal gluon helicity density (forward limit gives $g_{1L}^g$).
- $F_{1,2}^g$ and $G_{1,2}^g$ relate to gluon Sivers-type and "worm-gear" TMD effects, respectively.
- $F_{1,3}^g$ and $G_{1,3}^g$ encode skewness-induced distortions.

The sixteen GTMDs contain, via proper kinematic limits, all TMDs and GPDs at leading twist.

| Sector        | Example GTMD         | Forward limit             | Integrated ($k_\perp$) limit   |
|---------------|----------------------|---------------------------|-------------------------------|
| Unpolarized   | $F_{1,1}^g$          | $f_1^g$                   | $H^g(x,\xi,t)$                |
| OAM           | $F_{1,4}^g$          | $h_1^{\perp g}$ ($\sim$)  | No GPD analogue               |
| Helicity      | $G_{1,4}^g$          | $g_{1L}^g$                | $\widetilde H^g(x,\xi,t)$     |
| Spin–orbit    | $G_{1,1}^g$          | (none at TMD level)       | (none at GPD level)           |

[2402.17162, 2509.14208, 1802.10550]

## 3. Limits: Reduction to TMDs, GPDs, and Wigner Distributions

The GTMDs interpolate between TMDs, GPDs, and quark/gluon Wigner distributions:
- **TMD limit**: Forward ($\xi=0, \Delta_\perp=0$) yields TMDs; for example,
  \[
  F_{1,1}^g(x,0,0,\vec k_\perp^2,0) = f_1^g(x, k_\perp^2),\quad G_{1,4}^g(x, 0, 0, \vec k_\perp^2, 0) = g_{1L}^g(x, k_\perp^2)
  \]
- **GPD limit**: Integration over $k_\perp$ at fixed $\xi,\Delta_\perp$ gives GPDs:
  \[
  H^g(x, \xi, \Delta_\perp^2) = \int d^2k_\perp\,\left[ F_{1,1}^g + 2\xi^2\left(\frac{\Delta_\perp \cdot k_\perp}{\Delta_\perp^2} F_{1,2}^g + F_{1,3}^g \right)\right]
  \]
  and similarly for $E^g, \widetilde{H}^g, \widetilde{E}^g$ [2402.17162, 2509.14208].
- **Wigner distributions**: Fourier transforming GTMDs in $\Delta_\perp\to b_\perp$ at $\xi=0$ yields gluon Wigner distributions in $(x,k_\perp, b_\perp)$ [2402.17162, 2509.14208].

In the small-$x$ regime, gluon Wigner and GTMD distributions can be constructed from solutions to the impact-parameter-dependent Balitsky-Kovchegov equation, linking the GTMD to dipole $S$-matrices and saturation physics. The latter enables numerical evaluation of angular harmonics, including elliptic GTMDs ($\cos 2\phi$ component) relevant for diffractive dijet and exclusive vector-meson production [1609.05773].

## 4. Experimental Access and Phenomenology

Gluon GTMDs can be accessed via exclusive hard processes in hadronic and lepton-hadron collisions, with process-dependent selectivity for individual GTMDs:
- **Exclusive double quarkonium production**
  \[
  N N \to \eta_Q\,\eta_Q\,N N
  \]
  Here, the amplitude at leading order is a convolution of two gluon GTMD correlators. By forming polarization and azimuthal angle combinations in the final state, one projects out bilinear GTMD structures; for instance, $F_{1,4}$ is isolated by combinations such as $T_{UU}+T_{LL}-T_{XX}-T_{YY}$ [1802.10550].
- **Exclusive heavy meson production at the EIC**
  In exclusive $J/\psi$ electroproduction,
  \[
  e + p \to e' + J/\psi + p'
  \]
  twist-3 collinear factorization allows direct sensitivity to the $k_\perp$-moments of $F_{1,4}^g$ (OAM) and $G_{1,1}^g$ (spin–orbit), through the appearance of characteristic $\cos 2\phi$ and $\sin 2\phi$ azimuthal dependencies in the cross section [2601.17506]. The $\cos 2\phi$ modulation is polarization-independent and selects the canonical OAM, while $\sin 2\phi$ depends on target polarization and isolates the spin–orbit correlator.

Tables summarizing process sensitivity:

| Process                      | Measured Observable                   | Sensitive GTMD(s)        |
|------------------------------|----------------------------------------|--------------------------|
| $NN \to \eta_Q\eta_Q NN$     | Polarization, azimuthal weighting      | $F_{1,4}^g$, $G_{1,1}^g$ |
| $ep \to e' J/\psi p'$        | $\cos 2\phi$, $\sin 2\phi$ asymmetry   | $F_{1,4}^g$, $G_{1,1}^g$ |
| Diffractive dijet in DIS     | $\cos 2(\phi_{P_T}-\phi_\Delta)$       | Elliptic ($F_1$)         |

Theoretical and projected EIC studies indicate that few-percent level asymmetries in such processes are feasible for the extraction of gluon GTMDs [2601.17506].

## 5. Small-$x$ QCD and Saturation Domain

At small $x$, GTMDs are related to the quantum phase-space imaging of gluons in the color glass condensate (CGC) framework. The gluon GTMD is linked to the Fourier transform of the dipole forward $T$-matrix, enabling the calculation of both isotropic and elliptic ($\cos 2\phi$) GTMDs:
\[
x F_0(k,\Delta) = \frac{N_c}{2\pi^2\alpha_s}(\Delta^2/4 - k^2) \int dr\,db\,d\phi_{br}~ J_0(kr) J_0(b\Delta) T_Y(r,b,\phi_{br})
\]
\[
x F_1(k,\Delta) = -\frac{N_c}{2\pi^2\alpha_s}(\Delta^2/4 - k^2) \int dr\,db\,d\phi_{br}~ J_2(kr) J_2(b\Delta) \cos2\phi_{br} T_Y(r,b,\phi_{br})
\]
[1609.05773]. Both components peak at transverse momentum $k\sim Q_s(Y, b)$, the saturation scale, with the elliptic harmonic being numerically sub-leading but experimentally accessible in diffractive observables.

## 6. Evolution Properties and Renormalization

Gluon GTMDs, as nonlocal light-cone correlators with both rapidity and UV divergences, require soft factor subtraction for proper field-theoretic definition:
\[
\widetilde W^g(x,\xi,b_T; \mu, \zeta) = \Phi^g(x,\xi,b_T; \mu, \eta)\, \sqrt{S(b_T; \eta)}
\]
with $\Phi^g$ the unsubtracted correlator and $S(b_T)$ the gauge-invariant soft function [1602.06953]. Evolution proceeds via coupled renormalization-group (μ) and Collins-Soper (ζ) equations:
\[
\frac{d}{d\ln\mu} \widetilde W^g = [\Gamma_\text{cusp}^g \ln(\mu^2/\zeta) + \gamma^g]~\widetilde W^g~,\quad \frac{d}{d\ln\zeta} \widetilde W^g = -K_g(b_T;\mu)~\widetilde W^g
\]
where $\Gamma_\text{cusp}^g$ is the adjoint cusp anomalous dimension, $\gamma^g$ the non-cusp anomalous dimension, and $K_g$ the rapidity kernel. All gluon GTMDs—unpolarized, polarized, and helicity-flip—share an identical evolution kernel, as the evolution is spin-independent [1602.06953]. Solutions can be obtained for resummed evolution up to NNLL accuracy using known anomalous dimensions.

## 7. Model Implementations and Wigner Distributions

Light-front spectator and gluon-triquark models, particularly those anchored in soft-wall AdS/QCD, enable analytic and numerical evaluation of gluon GTMDs for nonzero skewness. These models represent GTMDs as overlaps of light-cone wave functions, providing closed-form results for all F- and G-type GTMDs across $(x, \xi, \vec k_\perp, \vec\Delta_\perp)$, and yield five-dimensional Wigner distributions for various polarization configurations ($UU$, $UL$, $LU$, $LL$, linearly polarized gluons) [2509.14208, 2402.17162].

The resulting Wigner distributions exhibit characteristic symmetry and multipole patterns in transverse momentum and impact parameter space, with $F_{1,4}^g$ and $G_{1,1}^g$ also providing direct access to the canonical gluon OAM and spin–orbit correlation, respectively. For example, model results at 2 GeV give total gluon angular momentum $J_z^g \sim 0.21$, kinetic OAM $L_z^g \sim -0.22$, canonical OAM $\ell_z^g \sim -0.38$, and spin–orbit correlation $\mathcal C_z^g \sim -15.6$ [2509.14208].

## References

- Exclusive double quarkonium production and gluon GTMDs: [1802.10550]
- Wigner/GTMDs in the CGC and small-$x$: [1609.05773]
- Light-front model calculations and IPDs: [2402.17162], [2509.14208]
- Experimental signatures at the EIC: [2601.17506]
- Evolution and soft-factor subtraction: [1602.06953]

Source: https://www.emergentmind.com/topics/gluon-generalized-transverse-momentum-dependent-distributions-gtmds