Generalized transverse momentum distributions at small-$x$

Abstract: We compute the complete set of the leading-twist gluon and sea-quark generalized transverse momentum distributions (GTMDs) in the small-$x$, or eikonal, approximation at vanishing skewness $ξ= 0$. All the gluon GTMDs become expressed in terms of the basic gluon dipole operator featuring also proton helicity-flips. Consequently, we establish universal relations between otherwise distinct GTMDs that hold in the small-$x$ limit. The obtained results are systematically projected onto the transverse momentum dependent distributions (TMD) and the generalized parton distribution (GPD) cases, recovering known results where available. In case of sea-quarks all the GTMDs are given in terms of the gluon dipole convoluted with a hard kernel. We generalize the unpolarized sea-quark GTMD to non-zero momentum transfers and find new results for the proton helicity-flip distributions. We pay special attention to their perturbative, high transverse momentum, tails to confirm that it becomes governed by the small-$x$ gluon GPDs. The obtained results provide guidance to the phenomenological modeling of the GTMDs but also allow for their explicit computations.

• The paper derives the complete leading-twist gluon and sea-quark GTMDs at small-x and xi=0, reducing 16 gluon functions to three independent complex dipole amplitudes and identifying 12 nonzero functions as universal combinations of them.
• The analysis shows that gluon S- and D-wave GTMDs arise from the spin-independent dipole, P- and F-wave functions from spin-dependent dipoles, while all gluon helicity GTMDs and eight sea-quark helicity or transversity GTMDs vanish in the strict eikonal limit.
• The results reproduce established small-x TMD relations, connect GPDs and Odderon contributions to tri-gluon structures, and provide high-transverse-momentum sea-quark tails useful for phenomenology, while leaving skewness, Weizs01ckerWilliams links, and sub-eikonal effects for future work.

Overview and motivation

Generalized transverse momentum distributions (GTMDs) constitute the most general leading-twist parametrization of hadron structure, with 16 independent quark and 16 independent gluon functions at leading twist (Lorcé et al., 2013). Their experimental extraction is complicated by the sheer number of complex-valued functions and their multi-dimensional kinematic dependence. This paper by Benić, Hagiwara, Šarić, and Vivoda addresses this complexity in the high-energy regime: it computes the complete set of leading-twist gluon and sea-quark GTMDs at small-$x$ (the eikonal approximation) and vanishing skewness $\xi = 0$, expressing everything in terms of the gluon dipole operator $S_{\Lambda'\Lambda}(k,\Delta)$ built from longitudinal Wilson lines. The central structural result is that the dipole GTMD contains only three complex scalar functions — the helicity-non-flip $S$, and the helicity-flip $S^\perp_{1T}$ and $S_T$ — whose real and imaginary parts realize the Pomerons and Odderons, respectively. This reduction from 16 to 3 functions forces universal relations among otherwise distinct GTMDs.

The analysis uses the dipole-type gauge-link configuration; the authors note that the alternative Weizsäcker-Williams configuration, related to the gluon quadrupole operator, is left for future work. The small-$x$ matching starts from the factorized form

$$W^{ij}_{\Lambda'\Lambda} = \frac{2N_c}{x\alpha_S}\left(k^i + \tfrac{1}{2}\Delta^i\right)\left(k^j - \tfrac{1}{2}\Delta^j\right) S_{\Lambda'\Lambda}(k,\Delta),$$

which completely factorizes the gluon helicity indices and thereby underlies all subsequent relations.

Gluon GTMDs at small-$x$

Matching the helicity matrix elements onto the standard multipole classification ($S$, $\xi = 0$0, $\xi = 0$1, $\xi = 0$2 waves labeled by orbital angular momentum transfer $\xi = 0$3), the paper establishes that $\xi = 0$4-wave and $\xi = 0$5-wave gluon GTMDs are controlled by the spin-independent dipole, while $\xi = 0$6-wave and $\xi = 0$7-wave GTMDs are controlled by the spin-dependent dipoles $\xi = 0$8 and $\xi = 0$9. Concretely:

• The unpolarized GTMD satisfies $S_{\Lambda'\Lambda}(k,\Delta)$0, while the spin-orbit GTMD obeys $S_{\Lambda'\Lambda}(k,\Delta)$1, yielding the near-forward relation $S_{\Lambda'\Lambda}(k,\Delta)$2, which recovers the unpolarized–spin-orbit connection found previously (Bhattacharya et al., 2024).
• All four helicity-type GTMDs vanish identically in the strict eikonal limit.
• The $S_{\Lambda'\Lambda}(k,\Delta)$3-wave GTMDs satisfy $S_{\Lambda'\Lambda}(k,\Delta)$4, a notable result connecting distributions with different angular momentum transfer.
• The proton-helicity-flip $S_{\Lambda'\Lambda}(k,\Delta)$5-waves with $S_{\Lambda'\Lambda}(k,\Delta)$6 are expressible through those with $S_{\Lambda'\Lambda}(k,\Delta)$7, and the $S_{\Lambda'\Lambda}(k,\Delta)$8-wave relates to the $S_{\Lambda'\Lambda}(k,\Delta)$9-wave via $S$0.

In total, of the 16 gluon GTMDs only three are independent at small-$S$1: the remaining non-vanishing twelve are fully determined by the three Pomerons/Odderons. These relations hold separately for real and imaginary parts.

TMD and GPD limits

Projecting to $S$2 reproduces the known small-$S$3 TMD relations: $S$4 and $S$5, recovering the equality between unpolarized and linearly polarized gluon TMDs (Metz et al., 2011) and the transversely polarized-proton relations (Boer et al., 2015, Boer et al., 2016).

In the GPD limit, the paper clarifies the roles of the isotropic and elliptic Fourier components of the Pomeron. The unpolarized GPD $S$6 is governed by the isotropic Pomeron $S$7, while $S$8 receives contributions from both isotropic and elliptic pieces of the spin-dependent Pomerons. A new result is the combination

$S$9

showing that this linear combination of transversity GPDs is controlled entirely by the elliptic part of the spin-independent Pomeron. All gluon helicity GPDs vanish at small-$S^\perp_{1T}$0. On the Odderon side, the imaginary parts of the dipole are matched onto dynamical twist-3 tri-gluon GPDs $S^\perp_{1T}$1 and $S^\perp_{1T}$2 constructed from $S^\perp_{1T}$3 color structures, generalizing the known forward-limit connection between the spin-dependent Odderon and the tri-gluon PDF relevant for single-spin asymmetries (Zhou, 2013).

Sea-quark GTMDs

The sea-quark sector is computed via the background propagator method in covariant gauge, where the quark correlator reduces to hard kernels convoluted with the same gluon dipole $S^\perp_{1T}$4. Two structural facts drive the results: the hard kernels $S^\perp_{1T}$5 and $S^\perp_{1T}$6 are real and parton-helicity independent. Consequently:

• All helicity-type and transversity-type sea-quark GTMDs vanish in the strict eikonal limit (eight scalar functions are zero); obtaining them requires sub-eikonal corrections.
• Real parts of all surviving GTMDs are governed by Pomerons; imaginary parts by Odderons.
• Of the 16 sea-quark GTMDs, only six are non-zero: the unpolarized GTMD, the spin-orbit GTMD, and the four associated with transversely polarized protons.

New results include the off-forward generalization of the unpolarized sea-quark GTMD including the elliptic-Pomeron contribution, and the finding that its imaginary part is controlled by the spin-independent Odderon. For the proton-helicity-flip $S^\perp_{1T}$7-wave GTMDs, closed-form expressions involve the isotropic and elliptic components of $S^\perp_{1T}$8 and $S^\perp_{1T}$9, plus the spin-dependent Odderon $S_T$0 in the imaginary parts. Apart from the known unpolarized–spin-orbit relation $S_T$1 (which, unlike the gluon case, holds only for the real parts), no new relations among sea-quark GTMDs emerge at general kinematics.

Perturbative tails and consistency checks

Expanding the hard kernels at $S_T$2 allows complete factorization of the high-$S_T$3 tails in terms of small-$S_T$4 gluon GPDs. The forward limit of the unpolarized tail reproduces the classic result $S_T$5 [hep-ph/9809427], while the off-forward correction involves the new GPD combination $S_T$6. The imaginary parts of all three non-zero $S_T$7-wave GTMDs share the simple high-$S_T$8 relation $S_T$9, since all are sourced by the spin-dependent tri-gluon GPD $x$0. In the GPD limit, the sea-quark $x$1 and $x$2 exhibit a Pomeron decomposition identical in structure to their gluon counterparts, differing only by the replacement of $x$3 by the hard kernel $x$4.

A point of tension deserves note: for the spin-orbit GTMD tail, the authors find agreement with Kanazawa et al. (Kanazawa et al., 2014) in the forward limit but report a discrepancy with Bertone et al. (Bertone et al., 11 Feb 2025), whose result lacks the $x$5 piece and carries a linear rather than quadratic $x$6-dependence. This disagreement is stated plainly and not resolved within the paper.

Limitations and open questions

The analysis is restricted to vanishing skewness $x$7; skewness corrections, addressed recently elsewhere (Kovchegov et al., 10 Dec 2025, Bhattacharya et al., 2 Oct 2025), are outside its scope. The strict eikonal limit sets all helicity and transversity quark GTMDs and the gluon helicity GTMDs to zero, so sub-eikonal dynamics is required for these sectors. The Weizsäcker-Williams gauge-link configuration and diffractive GTMDs remain uncomputed here. Finally, the disagreement over the spin-orbit GTMD's perturbative tail with (Bertone et al., 11 Feb 2025) constitutes an open question requiring reconciliation.

Conclusion

This work provides a systematic and complete mapping of leading-twist gluon and sea-quark GTMDs onto the small-$x$8 dipole framework at $x$9, reducing the gluon sector to three independent complex functions and the sea-quark sector to six non-vanishing GTMDs expressed as dipole–hard-kernel convolutions. The resulting universal relations constrain phenomenological parametrizations relevant to EIC physics, and the analytic formulas serve directly as initial conditions for small-$$W^{ij}_{\Lambda'\Lambda} = \frac{2N_c}{x\alpha_S}\left(k^i + \tfrac{1}{2}\Delta^i\right)\left(k^j - \tfrac{1}{2}\Delta^j\right) S_{\Lambda'\Lambda}(k,\Delta),$$0 or Sudakov evolution in numerical computations based on Pomeron and Odderon models.