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Transverse momentum dependent parton distributions at small $x$

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Paul Caucal, Farid Salazar

High Energy Physics - Phenomenology · Submitted 19 Oct 2026 · 300 references

Pith's verdict: A thorough, well-structured review that accurately surveys the small-x TMD field and honestly flags its open problems.

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arXiv ID2610.26004 (version 1)
Pith ID5JYVIG2D, Pith's permanent record of this paper and its review
CitationFarid Salazar, Paul Caucal. "Transverse momentum dependent parton distributions at small $x$." arXiv:2610.26004 (2026). https://pith.science/paper/5JYVIG2D
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The paper

The paper claims

This review establishes that the Color Glass Condensate effective theory provides a unified, predictive framework for transverse-momentum-dependent parton distributions at small x, connecting saturation physics to standard TMD…

The reading

This review synthesizes our current understanding of transverse-momentum-dependent parton distributions in the small-x regime. It demonstrates that the Color Glass Condensate effective theory provides a unified description, mapping the operator definitions of TMDs onto Wilson-line correlators. These correlators satisfy non-linear evolution equations (BK, JIMWLK) that resum large logarithms of 1/x, while Sudakov logarithms of the hard scale are resummed via Collins-Soper-Sterman evolution. The framework extends to polarized and diffractive distributions, and makes testable predictions for experiments at the EIC, RHIC, and LHC. A key achievement is the joint resummation of small-x and Sudakov logarithms, which is essential for precision phenomenology.

What carries the argument

The central objects are the Wilson-line correlators of the Color Glass Condensate, specifically the dipole S-matrix S(r) = (1/N_c) <Tr[U(0)U†(r)]>_x and the Weizsäcker-Williams correlator < Tr[∂_i U(x) U†(y) ∂_i U(y) U†(x)] >_x. These encode multiple scattering and saturation effects. The argument is carried by the JIMWLK renormalization-group equation, which governs the x-evolution of these correlators, and the Collins-Soper-Sterman equation, which governs the $Q^{2}$-evolution of the TMDs. The key mechanism is the identification of the TMD operator definitions with these Wilson-line correlators in the small-x limit, and the demonstration that the perturbative corrections factorize into a universal high-energy evolution kernel and process-dependent coefficient functions.

Extended reading notes

Core claim

The central claim of this review is that the Color Glass Condensate effective theory offers a systematic and predictive framework for transverse-momentum-dependent parton distributions at small Bjorken x. At leading power in x, all gluon and sea-quark TMDs can be expressed in terms of a small number of Wilson-line correlators: the dipole, the Weizsäcker-Williams, and the adjoint dipole. These correlators are the building blocks for inclusive, diffractive, and target-fragmentation observables. Their x-dependence is governed by the JIMWLK/Balitsky-Kovchegov equations, which resum the large logarithms of 1/x, while their $Q^{2}$-dependence is controlled by an evolution equation analogous to Collins-Soper-Sterman, which resums Sudakov logarithms. Remarkably, the two resummations can be consistently separated through a lifetime-ordering prescription. Beyond the eikonal approximation, sub-eikonal corrections provide access to helicity and other spin-dependent TMDs, with their small-x asymptotics now known at double-logarithmic accuracy. The review thus establishes that the CGC framework can serve as the foundation for a unified description of TMDs from moderate to small x, with direct phenomenological applications for current and future colliders.

Load-bearing premise

The entire small-x TMD framework assumes the kinematic hierarchy $Λ_QCD^{2}$ << $Q_s^{2}$, $k_T^{2}$ << $Q^{2}$ << s, which justifies the simultaneous resummation of small-x and Sudakov logarithms but fails when these hierarchies are not satisfied.

Editorial extensions

If this is right

  • The joint resummation of small-x and Sudakov logarithms, demonstrated for back-to-back dijet production in DIS, can be extended to other processes, providing precise predictions for the EIC.
  • Diffractive TMDs, such as the gluon diffractive distribution, are expressed in terms of the dipole correlator and exhibit a 1/k_T^4 tail, making diffractive dijets a clean probe of saturation.
  • The small-x asymptotics of all eight leading-twist quark TMDs and several gluon TMDs in the helicity sector are now known at large N_c in the double-logarithmic approximation, enabling global fits to polarized data.
  • For forward dihadron production in pA collisions, the combination of BK evolution, CSS resummation, and improved TMD (ITMD) hard factors describes STAR data on the nuclear suppression of the away-side peak.
  • Sub-eikonal corrections to unpolarized TMDs provide a systematic pathway toward the moderate-x regime, while sub-eikonal spin-dependent TMDs reveal the proton's spin structure at small x.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The Wilson-line building blocks of small-x TMDs also control generalized TMDs (GTMDs) and Wigner distributions, so the same framework may be used to image the proton's phase-space structure at small x.
  • The observed 'wash out' of the Cronin peak by small-x evolution suggests that nuclear modification ratios in TMD observables will follow a characteristic pattern: suppression at moderate transverse momenta, with a turnover scale that grows as x decreases.
  • The fact that most small-x TMDs can be expressed in terms of a single dipole correlator in the Gaussian approximation implies that future measurements of different TMD observables (e.g., dijets, photon-jet, heavy quark pairs) will be correlated, providing a consistency check of the saturation picture.
Original abstract

Transverse-momentum-dependent (TMD) parton distributions provide a three-dimensional description of the momentum structure of hadrons. They are probed in proton-proton and proton-nucleus collisions at the LHC and RHIC, and are a central part of the physics program of the future Electron-Ion Collider (EIC). While TMD factorization is well established at moderate Bjorken $x$, the small-$x$ regime requires a different treatment, since large logarithms of $1/x$ must be resummed and non-linear effects due to high gluon densities become important. The Color Glass Condensate (CGC) effective theory provides a natural framework to describe TMDs in this saturation regime and to connect them with standard TMD factorization. This chapter reviews quark and gluon TMDs at small $x$, covering operator definitions, gauge-link structures, and Wilson-line correlators in the CGC for unpolarized and polarized distributions. We discuss factorization for inclusive and diffractive observables, highlighting the role of process dependence and the emergence of a limited set of Wilson-line operators. We then summarize the evolution of these distributions, namely the resummation of logarithms of $1/x$ by the Balitsky-Kovchegov and JIMWLK equations, and of Sudakov logarithms of the hard scale through Collins-Soper-Sterman evolution. We also review TMDs beyond the eikonal approximation, including the sub-eikonal Wilson-line operators that give access to helicity and other spin-dependent distributions, their double-logarithmic small-$x$ evolution and the resulting asymptotics, and the sub-eikonal corrections to the eikonal distributions, which provide a first step toward the moderate-$x$ regime. Finally, we discuss phenomenological implications for present and future experiments, with emphasis on the EIC, and outline the theoretical challenges toward a unified description of TMDs.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 7,313 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.30 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.16 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Transverse-momentum-dependent (TMD) parton distributions provide a three-dimensional description of the momentum structure of hadrons. They are probed in proton-proton and proton-nucleus collisions at the LHC and RHIC, and are a central part of the physics program of the future Electron-Ion Collider (EIC). While TMD factorization is well established at moderate Bjorken $x$, the small-$x$ regime requires a different treatment, since large logarithms of $1/x$ must be resummed and non-linear effects due to high gluon densities become important. The Color Glass Condensate (CGC) effective theory provides a natural framework to describe TMDs in this saturation regime and to connect them with standard TMD factorization. This chapter reviews quark and gluon TMDs at small $x$, covering operator definitions, gauge-link structures, and Wilson-line correlators in the CGC for unpolarized and polarized distributions. We discuss factorization for inclusive and diffractive observables, highlighting the role of process dependence and the emergence of a limited set of Wilson-line operators. We then summarize the evolution of these distributions, namely the resummation of logarithms of $1/x$ by the Balitsky-Kovchegov and JIMWLK equations, and of Sudakov logarithms of the hard scale through Collins-Soper-Sterman evolution. We also review TMDs beyond the eikonal approximation, including the sub-eikonal Wilson-line operators that give access to helicity and other spin-dependent distributions, their double-logarithmic small-$x$ evolution and the resulting asymptotics, and the sub-eikonal corrections to the eikonal distributions, which provide a first step toward the moderate-$x$ regime. Finally, we discuss phenomenological implications for present and future experiments, with emphasis on the EIC, and outline the theoretical challenges toward a unified description of TMDs.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Transverse-momentum-dependent (TMD) parton distributions provide a three-dimensional description of the momentum structure of hadrons. background 1.00 0.04 0.04 asserted 1.00
s1 They are probed in proton-proton and proton-nucleus collisions at the LHC and RHIC, and are a central part of the physics program of the future Electron-Ion Collider (EIC). background 1.00 0.05 0.03 asserted 1.00
s2 While TMD factorization is well established at moderate Bjorken $x$, the small-$x$ regime requires a different treatment, since large logarithms of $1/x$ must be resummed and non-linear effects due to high gluon densities become important. background 0.75 0.05 0.08 asserted 1.00
s3 The Color Glass Condensate (CGC) effective theory provides a natural framework to describe TMDs in this saturation regime and to connect them with standard TMD factorization. background 0.70 0.08 0.15 asserted 0.99
s4 This chapter reviews quark and gluon TMDs at small $x$, covering operator definitions, gauge-link structures, and Wilson-line correlators in the CGC for unpolarized and polarized distributions. contribution 0.96 0.27 0.07 asserted 0.92
s5 We discuss factorization for inclusive and diffractive observables, highlighting the role of process dependence and the emergence of a limited set of Wilson-line operators. contribution 0.97 0.56 0.09 asserted 0.89
s6 We then summarize the evolution of these distributions, namely the resummation of logarithms of $1/x$ by the Balitsky-Kovchegov and JIMWLK equations, and of Sudakov logarithms of the hard scale through Collins-Soper-Sterman evolution. contribution 0.72 0.17 0.07 asserted 0.97
s7 We also review TMDs beyond the eikonal approximation, including the sub-eikonal Wilson-line operators that give access to helicity and other spin-dependent distributions, their double-logarithmic small-$x$ evolution and the resulting asymptotics, and the sub-eikonal corrections to the eikonal distributions, which provide a first step toward the moderate-$x$ regime. contribution 0.93 0.31 0.11 asserted 0.98
s8 Finally, we discuss phenomenological implications for present and future experiments, with emphasis on the EIC, and outline the theoretical challenges toward a unified description of TMDs. future work 0.58 0.47 0.17 asserted 0.64

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Gauge-link structures of the two gluon TMDs that dominate small-𝑥 physics, in the (𝑥 + , 𝝃⊥) plane, with the target shockwave at 𝑥 + = 0. Each correlator contains two staple-shaped Wilson links in the fundamental representation: 𝑊 (𝜉, 0) (blue), running from 0 to 𝜉, and 𝑊† (𝜉, 0) (red), running back from 𝜉 to 0, each through light-cone infinity. (a) The WW distribution: both staples extend to 𝑥 + = +∞, so that the active gluon undergoes only final-state interactions. (b) The dipole distribution: one staple extends to +∞ and the other to −∞, so that both initial- and final-state…
Figure 2. Small-𝑥 gluon TMDs, normalized by the transverse area 𝑆⊥, for a lead nucleus (solid) and a proton (dashed) at 𝑥 = 10−2 , 10−4 and 10−6 . F (1) 𝑞𝑔 and FWW are the dipole and WW distributions of Eqs. (5) and (3), FAdj is the adjoint dipole distribution, and F (1) 𝑔𝑔 , F (2) 𝑞𝑔 and F (3) 𝑔𝑔 are the additional distributions that enter two-particle production in forward 𝑝𝐴 collisions [51]. All distributions are expressed through the dipole in the Gaussian approximation at large 𝑁𝑐, and the dipole is evolved by the running-coupling BK equation [141] from an MV initial condition at 𝑥0 =…
Figure 3. Small-𝑥 sea-quark TMDs 𝑥𝑞(𝑛) , normalized by the transverse area 𝑆⊥, for a lead nucleus (solid) and a proton (dashed) at 𝑥 = 10−2 , 10−4 and 10−6 , obtained from the same evolved dipole as in
Figure 4. Phase space of the NLO gluon in back-to-back dijet production in DIS, in terms of its transverse momentum 𝑘𝑔⊥ and momentum fraction 𝑧𝑔 = 𝑘 + 𝑔 /𝑞 + . Red: small-𝑥 evolution of the target, below the line 𝜏𝑔 = 𝜏𝛾 ∼ 1/𝑞 − , that is 𝑧𝑔 = 𝑘 2 𝑔⊥/𝑄 2 (𝑘 − 𝑔 = 𝑥𝑃− ), and above the dotted line 𝑘 − 𝑔 = 𝑃 − . Blue: Sudakov factor of the TMD PDF. Hatched: real and virtual emissions cancel. The horizontal dashed line is a cut at fixed 𝑘 + 𝑔 , which, for naive high-energy factorization along the projectile light-cone momentum, marks the separation between evolution and NLO coeff
Figure 6. Away-side coincidence probability of forward di-𝜋 0 production (2.6 < 𝜂 < 4.0) in 𝑝 𝑝 (blue) and 𝑝Au (red) collisions at √ 𝑠𝑁 𝑁 = 200 GeV, for several bins of trigger and associated transverse momenta, compared with STAR data [266]. The calculation combines running￾coupling BK evolution, perturbative and non-perturbative Sudakov resummation and ITMD hard factors, and the theory curves are multiplied by an overall factor of 1.5. The bands correspond to a variation of 𝑄 2 𝑠0,Au by a factor of 1.2. Figure from [273]. although self-normalizing the correlation discarded the information…

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.26004 by the authors.

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Discussion

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