Electron-Ion Collider: QCD Tomography

• Electron-Ion Collider is a high-luminosity accelerator facility that uses polarized electron-ion collisions to map the nucleon structure in momentum, spin, and spatial dimensions.
• It facilitates direct measurements of GTMDs, GPDs, and Wigner distributions, revealing detailed parton correlations and orbital angular momentum intricacies.
• The collider explores small-x dynamics and gluon saturation effects while investigating spin–orbit correlations and the nucleon spin puzzle in QCD.

The Electron-Ion Collider (EIC) is a proposed high-luminosity accelerator facility designed to probe the quark and gluon structure of nucleons and nuclei with unprecedented detail. By colliding high-energy polarized electrons with ions ranging from protons to heavy nuclei, the EIC enables direct access to novel observables sensitive to generalized transverse momentum dependent parton distributions (GTMDs), Wigner distributions, and the emergent properties of color interactions in QCD at small and moderate Bjorken-$x$. The facility's distinctive capability to resolve spatial, spin, and momentum correlations at the parton level underlies its centrality in the next phase of strong-interaction physics.

1. Scientific Motivation and Role in Hadronic Structure

The primary objective of the Electron-Ion Collider is to provide a multidimensional mapping of the nucleon and nuclear wave function in terms of both momentum and impact parameter degrees of freedom. This ambition arises from the recognition that traditional observables, such as inclusive deep inelastic structure functions or integrated parton densities, are insufficient to capture correlations relevant for orbital angular momentum (OAM), spin-orbit coupling, and spatial anisotropy of the color fields. The EIC specializes in:

• Accessing leading-twist gluon GTMDs $$F^g_{1,n}(x, \xi, \mathbf{k}_\perp, \mathbf{\Delta}_\perp)$$ and $$G^g_{1,n}(x, \xi, \mathbf{k}_\perp, \mathbf{\Delta}_\perp)$$, crucial for a full tomography of the nucleon spin decomposition and spatial structure (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025).
• Measuring the impact parameter dependent parton distributions (IPDs), which encode the transverse spatial profile of gluons and quarks as functions of $x$.
• Quantifying the dynamical mechanisms behind gluon OAM ($\ell^g_z$), spin-orbit correlations ($C^g_z$), and the transverse deformation of color fields induced by polarization and saturation effects.

These goals position the EIC as the central tool for exploring both the nonperturbative and perturbative regimes of QCD.

2. Formalism: GTMDs, GPDs, and Wigner Distributions

The EIC's exclusive and diffractive channels permit direct access to off-forward correlators of the type

$$W^{g[ij]}_{\lambda',\lambda}(x, P, \Delta, \mathbf{k}_\perp) = \int \frac{dz^-\, d^2 \mathbf{z}_\perp}{(2\pi)^3 P^+} e^{ik \cdot z} \langle p',\lambda'| F^{+i}_a(-\tfrac{z}{2})\, \mathcal{W}_{ab}\, F^{+j}_b(\tfrac{z}{2}) |p,\lambda \rangle_{z^+ = 0},$$

where $\mathbf{k}_\perp$ is the average transverse momentum, $\mathbf{\Delta}_\perp$ is the transverse momentum transfer, $\xi$ is the skewness, and $F^{\mu\nu}_a$ is the gluon field-strength tensor (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025, Bhattacharya et al., 24 Jan 2026, Bhattacharya et al., 2018).

At leading twist, these correlators are parametrized in terms of four "F-type" (unpolarized gluon) and four "G-type" (longitudinally polarized gluon) GTMDs, which can be projected onto generalized parton distributions (GPDs) by integrating over $\mathbf{k}_\perp$, and further into IPDs via Fourier transform over $\mathbf{\Delta}_\perp$. The resulting distributions,

$$\mathcal{X}(x, \mathbf{b}_\perp) = \int \frac{d^2 \mathbf{\Delta}_\perp}{(2\pi)^2} e^{-i \mathbf{\Delta}_\perp \cdot \mathbf{b}_\perp} X(x, 0, \mathbf{\Delta}_\perp^2),$$

probe the density of partons as a function of both longitudinal momentum and transverse position.

Key mechanisms in the EIC environment include the mapping of Wigner distributions to nontrivial multipole structures (monopole, dipole, quadrupole), exposing spin–OAM and spin–spin correlations in the nucleon (Chakrabarti et al., 17 Sep 2025). The gluon sector is of particular interest due to the unique sensitivity of $F^g_{1,4}$ (canonical OAM) and $G^g_{1,1}$ (spin–orbit correlation) GTMDs to genuinely quantum mechanical phase space correlations.

3. Exclusive Channels: Experimental Probes of Gluon Phase-Space Structure

The EIC's access to exclusive processes, most notably diffractive vector meson and heavy quarkonium production ($e\,p \to e'\, p'\, V$ where $V = J/\psi$, $\chi_{c1}$, and double-$\eta_Q$ production in $NN$ collisions), provides a direct window on GTMDs that encode orbital and spin correlations not accessible in the GPD or TMD frameworks (Bhattacharya et al., 24 Jan 2026, Bhattacharya et al., 2018). The twist-3 collinear factorization applied to exclusive electroproduction leads to distinct azimuthal modulations in the final-state cross sections, such as:

• The polarization-independent $\cos 2\phi$ modulation, attributed to interference between transverse virtual photon amplitudes,
• The polarization-dependent $\sin 2\phi$ modulation, sensitive to the proton helicity.

These modulations provide clean experimental signatures of $F^g_{1,4}$ and $G^g_{1,1}$ via the cross-section structure functions,

$$d\sigma_{TT}^0 / dt \sim | \Delta_\perp |^2 [ | \mathcal{F}_{1,4}^{TL} - \mathcal{G}_{1,4}^{TL} |^2 ],$$

$$d\sigma_{\sin 2\phi}^0 / dt \sim |\Delta_\perp|^2\, \mathrm{Re} \{ [\mathcal{F}_{1,1}^{TL} + \mathcal{G}_{1,1}^{TL}] [\mathcal{F}_{1,4}^{TL*} - \mathcal{G}_{1,4}^{TL*}] \},$$

and are directly related to the k$_\perp$-moments

$$L^g(x) = -\int d^2 k_\perp \frac{k_\perp^2}{M^2} F^g_{1,4}(x, 0, k_\perp, 0), \qquad C^g(x) = \int d^2 k_\perp \frac{k_\perp^2}{M^2} G^g_{1,1}(x, 0, k_\perp, 0).$$

The EIC's projected luminosity and ability to tag forward protons ensures sufficiently high rates for these channels to allow extraction of twist-3 asymmetries (Bhattacharya et al., 24 Jan 2026).

4. Small-$x$ Physics and Saturation Effects

The EIC is uniquely suited to exploring the gluon saturation regime at small-$x$, where nonlinear evolution and the emergence of the Color Glass Condensate (CGC) are expected to control partonic dynamics. At small-$x$, the phase space distributions of gluons are captured by Wigner and GTMD constructs derived from the impact-parameter dependent dipole $S$-matrix,

$$S_Y(r_\perp, b_\perp) = \frac{1}{N_c} \langle \mathrm{Tr}\, U(b_\perp + \tfrac{r_\perp}{2})\, U^\dagger(b_\perp - \tfrac{r_\perp}{2}) \rangle_Y,$$

with its evolution governed by the Balitsky-Kovchegov equation (Hagiwara et al., 2016). The resulting GTMDs exhibit:

• Pronounced peaks at the saturation momentum $Q_s(Y, \Delta)$, growing rapidly with increasing rapidity.
• A suppressed elliptic ($\cos 2\phi$) modulation $F_1(k, \Delta)$, which is a few percent of the monopole component $F_0(k, \Delta)$.

Measurement of diffractive dijet production, azimuthal angular correlations, and vector-meson channels in the EIC environment enables the mapping of GTMDs and Wigner distributions in the saturation regime, allowing the extraction of geometric scaling behavior and spatial anisotropy signatures.

5. Theoretical Foundations: Model Construction and Evolution

Representative calculations of leading-twist gluon GTMDs rely on overlap representations using light-front wave functions in spectator models inspired by soft-wall AdS/QCD dynamics (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025). The GTMDs are accessible in closed form, with explicit dependence on $x$, $\xi$, $k_\perp$, and $\Delta_\perp$. The connection to PDFs, GPDs, and TMDs is preserved in appropriate limits.

For practical phenomenology and scale dependence, it is imperative to include the proper rapidity and scale evolution of the GTMDs, ensuring cancellation of rapidity divergences via soft factors and providing evolution kernels associated with the cusp and noncusp anomalous dimensions. The evolution kernel for all (un)polarized gluon GTMDs is spin independent, with all large logarithms resumable to next-to-next-to-leading-logarithmic (NNLL) accuracy (Echevarria et al., 2016). Essential formulas:

• $\mu$-evolution:

$$\frac{d}{d\ln\mu} \ln \widetilde{W}^g(b_T; \mu, Q^2) = -\Gamma_{\mathrm{cusp}}^g(\alpha_s) \ln\frac{Q^2}{\mu^2} - \gamma^g(\alpha_s)$$

• $Q$-evolution:

$$\frac{d}{d\ln Q^2} \ln \widetilde{W}^g(b_T; \mu, Q^2) = - D^g(b_T; \mu)$$

This formalism underpins the readiness of EIC observables for global analyses and precise QCD factorization matching.

6. Phenomenological Impact and Prospects for Measurement

Numerical estimates in light-front spectator models show the spatial gluon profile of the nucleon is tightly localized in $b_\perp$, with distribution strength concentrated at low $x\, (\sim 0.01$-$0.2)$ (Tan et al., 2024, Chakrabarti et al., 17 Sep 2025). The canonical gluon OAM is found negative, indicating anti-alignment with nucleon spin; similarly, the gluon spin–orbit correlator is negative, signaling anti-alignment of gluon spin and OAM.

The extraction of GTMDs, particularly $F^g_{1,4}$ and $G^g_{1,1}$, is feasible via exclusive heavy quarkonium production and double quarkonium channels at the EIC. The projected five-fold differential cross section, azimuthal modulations, and polarization asymmetries provide distinct experimental handles for accessing the underlying phase-space structure of gluons (Bhattacharya et al., 24 Jan 2026, Bhattacharya et al., 2018). The EIC is expected to be able to separate twist-3 GTMD contributions from large unpolarized backgrounds through targeted polarization configurations and multi-dimensional analysis.

7. Outlook: The EIC's Role in QCD Tomography

The Electron-Ion Collider is poised to revolutionize the field of QCD tomography by enabling, for the first time, the direct and multidimensional mapping of nucleon and nuclear structure in momentum, position, and spin space. Its design synergizes high luminosity, polarization control, and kinematic reach, making it the definitive platform for precision studies of gluon and quark dynamics, the emergence of saturation, and the decomposition of proton spin in terms of OAM, spin-orbit, and multiparticle correlations.

Continued theoretical developments in factorization, soft-collinear resummation, and nonperturbative modeling will ensure accurate interpretation of EIC data. Phenomenological implications for hadronic structure—anti-alignment of gluon OAM, sizable spin-orbit correlations, and spatial anisotropy—will directly impact our understanding of the nucleon spin puzzle and the onset of saturation dynamics. The EIC is thereby essential for the next decade of hadron and nuclear physics, with its observables foundational for future progress in both experimental and theoretical QCD.