Gluon GTMDs: Probing Nucleon Tomography

• Gluon GTMDs are generalized two-parton correlation functions that encode longitudinal momentum, transverse momentum, and spatial positions of gluons in nucleons.
• They provide a unified framework bridging GPDs, TMDs, and Wigner distributions, enabling detailed phase-space imaging and insights into gluon orbital angular momentum.
• Experimental access via exclusive processes like double quarkonium and J/ψ production offers practical means to measure spin–orbit correlations and other key QCD observables.

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 (Bhattacharya et al., 2018). These correlators can be projected onto unpolarized and helicity distributions, e.g. via $\delta_\perp^{ij}$ and $-i\varepsilon_\perp^{ij}$ (Chakrabarti et al., 17 Sep 2025, Tan et al., 2024).

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

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

(Chakrabarti et al., 17 Sep 2025, Tan et al., 2024).

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)

(Tan et al., 2024, Chakrabarti et al., 17 Sep 2025, Bhattacharya et al., 2018)

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 $E^g, \widetilde{H}^g, \widetilde{E}^g$ (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025).

• Wigner distributions: Fourier transforming GTMDs in $\Delta_\perp\to b_\perp$ at $\xi=0$ yields gluon Wigner distributions in $(x,k_\perp, b_\perp)$ (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025).

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 (Hagiwara et al., 2016).

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 $T_{UU}+T_{LL}-T_{XX}-T_{YY}$ (Bhattacharya et al., 2018).

• 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 (Bhattacharya et al., 24 Jan 2026). 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 (Bhattacharya et al., 24 Jan 2026).

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: $$x F_0(k,\Delta) = \frac{N_c}{2\pi^2\alpha_s}(\Delta^2/4 - k^2) \int dr\,db\,d\phi_{br}~ J_0(kr) J_0(b\Delta) T_Y(r,b,\phi_{br})$$

$$x F_1(k,\Delta) = -\frac{N_c}{2\pi^2\alpha_s}(\Delta^2/4 - k^2) \int dr\,db\,d\phi_{br}~ J_2(kr) J_2(b\Delta) \cos2\phi_{br} T_Y(r,b,\phi_{br})$$

(Hagiwara et al., 2016). 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 (Echevarria et al., 2016). 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 (Echevarria et al., 2016). 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) (Chakrabarti et al., 17 Sep 2025, Tan et al., 2024).

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 $\mathcal C_z^g \sim -15.6$ (Chakrabarti et al., 17 Sep 2025).

• Exclusive double quarkonium production and gluon GTMDs: (Bhattacharya et al., 2018)
• Wigner/GTMDs in the CGC and small-$x$: (Hagiwara et al., 2016)
• Light-front model calculations and IPDs: (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025)
• Experimental signatures at the EIC: (Bhattacharya et al., 24 Jan 2026)
• Evolution and soft-factor subtraction: (Echevarria et al., 2016)