Small-x TMD distributions: JIMWLK evolution and the Gaussian truncation

Florian Cougoulic; Piotr Korcyl

REVIEW

Small-x TMD distributions: JIMWLK evolution and the Gaussian truncation

Florian Cougoulic, Piotr Korcyl

High Energy Physics - Phenomenology · Submitted 31 Aug 2026 · 71 references

Pith's verdict: First systematic JIMWLK-vs-GT evolution benchmark for small-x TMDs; the representation-dependent slowdown is believable, though the finite-step extraction deserves a convergence check.

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CitationFlorian Cougoulic, Piotr Korcyl. "Small-x TMD distributions: JIMWLK evolution and the Gaussian truncation." arXiv:2608.30687 (2026). https://pith.science/paper/JV2TQKWW

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Editor's note

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Cougoulic and Korcyl give the first full-evolution benchmark of the Gaussian truncation (GT) against JIMWLK for the set of small-x TMD distributions relevant for dijet production. The central result is believable: GT works for the operators tied to the Laplacian of the dipole (Omega^3_qg, Omega^8a_gg and, less trivially, Omega^8s_gg) but systematically under-evolves the other irreps, with the mismatch growing with representation size; one TMD even misses the appearance of a node. They are candid about the singlet breaking the ordering, and they do not sell the rapidity-rescaling patch as more than a guess. Anyone building phenomenology on a Gaussian ansatz needs to know this.

The numerical side is standard and clean: MV initial condition, Langevin JIMWLK, IR-regulated kernel, fixed N_c=3, a large 6144^2 lattice, and fit tables in the appendix. The reconstruction of both fundamental and adjoint dipoles from the extracted gamma is a good consistency check. The corrected TMD-to-Omega matrix from their previous paper is a minor but useful service.

The soft spot is exactly the one flagged in the stress-test note. Equation (3.13) defines gamma^(evo) from a finite slice of size alpha_s deltaY = 0.05, so it is a slab average rather than an instantaneous two-point function, and it is then inserted into the ordered integrals in Eq. (4.5) as though it were the value at t = n deltaY. The three-parameter fit gamma = b r^2 + c r^delta with delta around 0.5 is purely phenomenological and, through the Laplacian, it controls the small-r region where TMD nodes live. This is the least secure link in the chain. Still, I do not think it undermines the central claim: the reported mismatch is a shift of several rapidity slices (GT at n=19 sitting near data at n=15-17), far larger than an O(deltaY) systematic accumulating over 19 steps. But the quantitative "speed mismatch" claim would be much stronger with a convergence check at smaller alpha_s deltaY or a central-difference extraction. A referee should ask for that, plus a more quantitative goodness-of-fit measure than eyeballing log-log plots.

This paper deserves a serious referee. It is a benchmark for a widely used approximation, clearly written, and honest about what it cannot explain. I would send it to review and expect it to be publishable after the systematic checks are either done or argued away.

Referee report

Significance

If the numerical comparison is quantitatively robust, the result is important for small-x phenomenology because it delimits where the computationally cheap Gaussian truncation can be trusted for TMD evolution and provides an operator-decomposition diagnostic for the failure. The paper is transparent about its main limitation: agreement for the Laplacian-of-dipole operators is not an independent test of the GT, and the authors explicitly flag the unresolved singlet behavior. Strengths include the detailed lattice/JIMWLK setup with an infrared regulator, the sector-specific use of fundamental and adjoint dipoles to mitigate Casimir-scaling violation, explicit fit tables, and a correction of a typo in the linear-combination matrix of the companion paper. The central quantitative claim, however, rests on an extracted intermediate function and on visual comparisons of figures, so the finite-rapidity-step and fit-ansatz systematics need to be controlled before the numerical conclusions can be taken at face value.

Major objections (3)

[Sec. 3.3, Eq. (3.13); used in Sec. 4.2, Eq. (4.5)] The extraction in Eq. (3.13) is the discrete version of the infinitesimal-step formula (3.8), but the actual step is α_s δY = 0.05. The quantity γ^{(evo),n}_r is therefore a slice average of γ^t_r, not the instantaneous value at t = n δY, and it is subsequently used in the ordered integrals of Eq. (4.5) as if it were the value at a single rapidity. No convergence check with smaller δY, and no alternative integration scheme for Eq. (4.5), is provided. Since γ^{(evo),n} grows with n and the exponentials in Eq. (4.5) contain differences of Casimirs, the resulting O(δY) systematic can accumulate over the 19 slices and is amplified in the higher-representation operators for which the paper claims the largest GT slowdown. A test at, e.g., α_s δY = 0.025, or a simultaneous fit of the dipole and of Ω^3_qg/Ω^{8a}_gg as suggested in footnote 6, is needed before the quantitative 'GT is two to four slices slower' statements in Sec. 4.3 are fully secure.
[Sec. 3.4, Eq. (3.14)] The three-parameter ansatz γ^{(evo),n}_r = b_n r^2 + c_n r^{δ_n}, with δ_n between roughly 0.49 and 0.88, is a purely phenomenological non-analytic form. Through the Laplacian in Eq. (4.5), the small-r behavior of this ansatz controls the region where the TMD nodes appear in Figs. 9 and 10. The paper does not report fit residuals, does not propagate the fit-parameter uncertainties into the GT curves, and does not check robustness against other functional forms (e.g., r^2 times a logarithm, or a spline). Because the ansatz is not derived from the BK/JIMWLK dynamics, a demonstration that the node positions and the reported ordering of evolution speeds are insensitive to the parametrization is required.
[Sec. 5, Figs. 9 and 10] The central claim that the GT fails for the majority of TMD distributions is supported by visual inspection only: the data are shown without statistical error bars and no goodness-of-fit measure is quoted for the individual TMD distributions. Given that some of the discrepancies are presented as quantitative (e.g., the GT curve at n=19 agreeing with data at n=15 or n=17), the paper should report either error bars, a chi-squared-like comparison, or the node positions with their statistical uncertainties for the representative distributions F^{(2)}_{qg}, F^{(3)}_{qg}, F^{(1)}_{gg}, and F^{(3)}_{gg}.

Minor objections (5)

[Sec. 5.2, Eq. (5.2)] The column vector on the right-hand side of Eq. (5.2) appears to contain six entries, the last being '8 Ω^1_gg', while seven TMD distributions are listed on the left. The duplication of the singlet operator and the factor 8 are not explained in the text; please clarify the normalization or correct the typo, since the linear combinations are the basis for the TMD conclusions.
[Sec. 4.3.2 and Sec. 4.4] The discussion states that the discrepancy grows with the dimension or charge of the irrep, but then reports that the singlet Ω^1_gg violates this ordering. The authors acknowledge this, but the conclusion would be clearer if the main claim were phrased as 'the GT underestimates the evolution speed in the majority of irreps' and the ordering were explicitly labeled as a partial, non-universal trend rather than a systematic pattern.
[Sec. 3.5 and Figs. 2-3] The statement that Eq. (3.16) recovers 'perfect agreement' with both the fundamental and the adjoint dipole is by construction for the dipole used to extract γ^{(evo)} in each sector; it is a useful self-consistency check but not an independent validation of the GT or of the extraction.
[Throughout] There are minor typos and formatting issues, including 'JIMWKL' in the Introduction, 'res. adjoint' in Sec. 3.5, 'TMDF' in Sec. 5.1, and the inconsistent rendering of 'Weizsäcker' in Sec. 5.2. These should be corrected in the final version.
[Sec. 2.2 and Appendix B] The numerical setup is described in detail, but no information is given about the number of realizations used for the ensemble averages or about the statistical errors of the data points. Adding this information, even in an appendix, would improve reproducibility and would allow the reader to assess the significance of the deviations seen in the figures.

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The paper

The paper claims

Gaussian truncation fails to reproduce full JIMWLK evolution for most small-x TMD distributions

The reading

This paper tests whether the Gaussian truncation of the Balitsky hierarchy—a computationally cheap approximation that expresses all operator expectations via a single two-point function—can correctly describe the rapidity evolution of small-x transverse-momentum-dependent (TMD) distributions. The authors numerically solve the JIMWLK equation on the lattice and compare the results for ten TMD distributions (three in the quark-gluon sector, seven in the gluon-gluon sector) with the Gaussian truncation predictions. They find that, although the approximation works well at the initial condition (McLerran-Venugopalan model), during evolution it systematically underestimates the evolution speed for the majority of distributions. The discrepancies are largest for operators in higher color representations, and in some cases the truncation predicts entirely wrong sign structures (e.g., missing nodes). The paper matters because many phenomenological studies rely on the Gaussian truncation for small-x observables; these results show that such an approximation can be unreliable once evolution is included.

What carries the argument

The central object is the two-point correlation function γ^t_r extracted from the dipole evolution, which under the Gaussian truncation determines all higher correlators via Wick contractions. The paper also uses the representation-theoretic decomposition of the relevant operators into irreducible components Ω^ω_ag (based on SU(3) birdtracks), which allows systematic comparison of the truncation's performance across different color channels. The full JIMWLK evolution is implemented numerically via the Langevin formulation on a 6144² lattice, with the initial condition given by the MV model.

Extended reading notes

Core claim

The Gaussian truncation of the Balitsky hierarchy, when applied to the JIMWLK evolution of small-x TMD distributions, fails to reproduce the full numerical evolution for all operators except those directly related to the Laplacian of the dipole ($Ω^{3}$_qg, $Ω^{{8a}}$_gg, $Ω^{{8s}}$_gg). For all other operators in the quark-gluon and gluon-gluon sectors, the truncation predicts a slower evolution than the full JIMWLK equation, with the mismatch growing for higher-dimensional representations. Moreover, for the TMD distributions that are linear combinations of the building-block operators, the failure can become qualitative: the truncation predicts extra nodes or misses nodes that appear in the full evolution. The ordering of the discrepancy by representation dimension suggests that the truncation misses multi-reggeon exchanges that accelerate the evolution in larger irreps.

Load-bearing premise

The extraction of the two-point function γ^(evo)_r from the dipole using Eq. (3.13) assumes an infinitesimal rapidity step, but the actual step is finite (α_s δY = 0.05), and the fit ansatz (3.14), b_n $r^{2}$ + c_n $r^{{δ_n}}$, is purely phenomenological with no theoretical justification for its functional form.

Editorial extensions

If this is right

Phenomenological studies that use the Gaussian truncation to compute small-x TMD distributions (e.g., for di-jet or heavy-boson production) likely underestimate the evolution speed, leading to incorrect predictions for momentum spectra and rapidity dependencies.
An effective rescaling of rapidity by an irrep-dependent factor (e.g., Y → Y_ω) may partially correct the truncation's speed, but a deeper theoretical understanding is needed before such a patch can be trusted.
The large-Nc limit (BK equation) cannot fully replace the JIMWLK evolution for TMD distributions, since the truncation's failure is linked to subleading-Nc effects that grow with representation dimension.
Future approximations must incorporate at least a three-point function or a more sophisticated treatment of multi-gluon correlations to capture the correct evolution of all TMD operators.

Reading between the lines

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

The systematic under-evolution in the Gaussian truncation likely arises from neglected multi-Reggeon exchanges (higher-order correlators) that become more important in higher color representations, consistent with the observed violation of Casimir scaling in the adjoint dipole evolution.
The fact that the singlet operator Ω^1_gg in the gluon-gluon sector also shows slower evolution, despite having the smallest Casimir, suggests that the failure is not solely governed by color factors but also by the operator's structure (derivatives of Wilson lines mixing with the dipole evolution).
A natural test of the inference that higher-point correlations accelerate the evolution would be to compare the GT and JIMWLK for the hexapole (six-point) operator, which should show even larger discrepancies than the quadrupole.
The method of extracting γ^t_r solely from the dipole is insufficient; a simultaneous fit including other simple operators (like Ω^15_qg) would improve the truncation's accuracy but would then lose the Gaussian truncation's main advantage.

Original abstract

We extend the systematic study of small-$x$ Transverse Momentum Dependent (TMD) distributions, initiated in [Cougoulic:2026drr], by including the energy evolution. We use the JIMWLK evolution equation to study the energy dependence of three distributions in the quark-gluon sector, $\mathcal{F}_{qg}$, and seven distributions in the gluon-gluon sector, $\mathcal{F}_{gg}$. The initial conditions are provided by the McLerran-Venugopalan model. We study these TMD distributions in position space and with $N_c = 3$ fixed. We then compare these results to the Gaussian truncation of the Balitsky hierarchy. Although the Gaussian truncation worked very well for the TMD distributions at the initial condition, now we identify some cases where the Gaussian truncation provides a reasonably good description, but in the majority of cases it fails to fully reproduce numerical data. Using the fixed representation building blocks, $\Omega_{ag}^\omega$, we are able to partially explain the outcomes. This work constitutes the next step with the goal of understanding the interplays between the large-$N_c$ limit and Gaussian truncation as well as the BK/JIMWLK evolution equations for phenomenologically relevant observables.

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, 2026-09-23: 1 request, 6,356 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.64 · human participants 0.01 · controlled comparison 0.47 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.98 · mentions code or data availability 0.02.

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

We extend the systematic study of small-$x$ Transverse Momentum Dependent (TMD) distributions, initiated in [Cougoulic:2026drr], by including the energy evolution. We use the JIMWLK evolution equation to study the energy dependence of three distributions in the quark-gluon sector, $\mathcal{F}_{qg}$, and seven distributions in the gluon-gluon sector, $\mathcal{F}_{gg}$. The initial conditions are provided by the McLerran-Venugopalan model. We study these TMD distributions in position space and with $N_c = 3$ fixed. We then compare these results to the Gaussian truncation of the Balitsky hierarchy. Although the Gaussian truncation worked very well for the TMD distributions at the initial condition, now we identify some cases where the Gaussian truncation provides a reasonably good description, but in the majority of cases it fails to fully reproduce numerical data. Using the fixed representation building blocks, $\Omega_{ag}^\omega$, we are able to partially explain the outcomes. This work constitutes the next step with the goal of understanding the interplays between the large-$N_c$ limit and Gaussian truncation as well as the BK/JIMWLK evolution equations for phenomenologically relevant observables.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We extend the systematic study of small-$x$ Transverse Momentum Dependent (TMD) distributions, initiated in [Cougoulic:2026drr], by including the energy evolution.	contribution 0.87	0.87	0.04	asserted 0.99
s1	We use the JIMWLK evolution equation to study the energy dependence of three distributions in the quark-gluon sector, $\mathcal{F}_{qg}$, and seven distributions in the gluon-gluon sector, $\mathcal{F}_{gg}$.	method 0.56	0.93	0.04	asserted 0.99
s2	The initial conditions are provided by the McLerran-Venugopalan model.	method 1.00	0.22	0.02	asserted 1.00
s3	We study these TMD distributions in position space and with $N_c = 3$ fixed.	method 0.97	0.91	0.02	asserted 0.98
s4	We then compare these results to the Gaussian truncation of the Balitsky hierarchy.	method 0.67	0.89	0.05	asserted 0.73
s5	Although the Gaussian truncation worked very well for the TMD distributions at the initial condition, now we identify some cases where the Gaussian truncation provides a reasonably good description, but in the majority of cases it fails to fully reproduce numerical data.	result 0.99	0.94	0.76	asserted 0.98
s6	Using the fixed representation building blocks, $\Omega_{ag}^\omega$, we are able to partially explain the outcomes.	interpretation 0.68	0.92	0.07	asserted 0.90
s7	This work constitutes the next step with the goal of understanding the interplays between the large-$N_c$ limit and Gaussian truncation as well as the BK/JIMWLK evolution equations for phenomenologically relevant observables.	future work 0.95	0.35	0.11	asserted 0.93

What kind of paper the classifier reports this is: simulation study 0.97 · evidence from simulation or computation 1.00.

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sources read

arXiv listing abstract v1

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full text
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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

References

Works cited (71)

[1]

F. Cougoulic, P. Korcyl, and T. Stebel, “Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations,”JHEP, vol. 08, p. 029, 2026

View on Pith 2026
[2]

R. Boussarieet al., “TMD Handbook,” 4 2023

View on Pith 2023
[3]

J. Collins,Foundations of Perturbative QCD, vol. 32. Cambridge University Press, 2011

View on Pith 2011
[4]

J. C. Collins, D. E. Soper, and G. F. Sterman, “Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production,”Nucl. Phys. B, vol. 250, pp. 199–224, 1985

View on Pith 1985
[5]

J.-w. Qiu and X.-f. Zhang, “QCD prediction for heavy boson transverse momentum distributions,”Phys. Rev. Lett., vol. 86, pp. 2724–2727, 2001

View on Pith 2001
[6]

X.-d. Ji, J.-P. Ma, and F. Yuan, “Qcd factorization for spin-dependent cross sections in dis and drell-yan processes at low transverse momentum,”Phys. Lett. B, vol. 597, pp. 299–308, 2004

View on Pith 2004
[7]

J. P. Ralston and D. E. Soper, “Production of Dimuons from High-Energy Polarized Proton Proton Collisions,”Nucl. Phys. B, vol. 152, p. 109, 1979

View on Pith 1979
[8]

D. W. Sivers, “Single Spin Production Asymmetries from the Hard Scattering of Point-Like Constituents,”Phys. Rev. D, vol. 41, p. 83, 1990

View on Pith 1990
[9]

D. W. Sivers, “Hard scattering scaling laws for single spin production asymmetries,”Phys. Rev. D, vol. 43, pp. 261–263, 1991

View on Pith 1991
[10]

J. C. Collins, “Fragmentation of transversely polarized quarks probed in transverse momentum distributions,”Nucl. Phys. B, vol. 396, pp. 161–182, 1993

View on Pith 1993
[11]

D. Boer, P. J. Mulders, and F. Pijlman, “Universality of T odd effects in single spin and azimuthal asymmetries,”Nucl. Phys. B, vol. 667, pp. 201–241, 2003

View on Pith 2003
[12]

L. N. Lipatov, “The Bare Pomeron in Quantum Chromodynamics,”Sov. Phys. JETP, vol. 63, pp. 904–912, 1986

View on Pith 1986
[13]

S. Catani, M. Ciafaloni, and F. Hautmann, “High-energy factorization and small x heavy flavor production,”Nucl. Phys. B, vol. 366, pp. 135–188, 1991

View on Pith 1991
[14]

F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, “The Color Glass Condensate,” Ann. Rev. Nucl. Part. Sci., vol. 60, pp. 463–489, 2010

View on Pith 2010
[15]

J. L. Albacete and C. Marquet, “Gluon saturation and initial conditions for relativistic heavy ion collisions,”Prog. Part. Nucl. Phys., vol. 76, pp. 1–42, 2014

View on Pith 2014
[16]

J.-P. Blaizot, “High gluon densities in heavy ion collisions,”Rept. Prog. Phys., vol. 80, no. 3, p. 032301, 2017

View on Pith 2017
[17]

A. Morreale and F. Salazar, “Mining for Gluon Saturation at Colliders,”Universe, vol. 7, no. 8, p. 312, 2021

View on Pith 2021
[18]

H. Raj and R. Venugopalan, “Qcd-gravity double copy in regge asymptotics: from 2→n amplitudes to radiation in shockwave collisions,”Acta Phys. Polon. B, vol. 56, no. 11, pp. 11–A1, 2025

View on Pith 2025
[19]

J. Jalilian-Marian, A. Kovner, and H. Weigert, “The Wilson renormalization group for low x physics: Gluon evolution at finite parton density,”Phys. Rev., vol. D59, p. 014015, 1998. – 28 –

View on Pith 1998
[20]

J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, “The Wilson renormalization group for low x physics: Towards the high density regime,”Phys. Rev., vol. D59, p. 014014, 1998

View on Pith 1998
[21]

H. Weigert, “Unitarity at small Bjorken x,”Nucl. Phys., vol. A703, pp. 823–860, 2002

View on Pith 2002
[22]

E. Iancu, A. Leonidov, and L. D. McLerran, “The renormalization group equation for the color glass condensate,”Phys. Lett., vol. B510, pp. 133–144, 2001

View on Pith 2001
[23]

E. Iancu, A. Leonidov, and L. D. McLerran, “Nonlinear gluon evolution in the color glass condensate. I,”Nucl. Phys., vol. A692, pp. 583–645, 2001

View on Pith 2001
[24]

E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, “Nonlinear gluon evolution in the color glass condensate. II,”Nucl. Phys., vol. A703, pp. 489–538, 2002

View on Pith 2002
[25]

I. Balitsky, “Operator expansion for high-energy scattering,”Nucl. Phys., vol. B463, pp. 99–160, 1996

View on Pith 1996
[26]

I. Balitsky, “Factorization for high-energy scattering,”Phys. Rev. Lett., vol. 81, pp. 2024–2027, 1998

View on Pith 2024
[27]

I. Balitsky, “Factorization and high-energy effective action,”Phys. Rev. D, vol. 60, p. 014020, 1999

View on Pith 1999
[28]

Y. V. Kovchegov, “Small-xF 2 structure function of a nucleus including multiple pomeron exchanges,”Phys. Rev., vol. D60, p. 034008, 1999

View on Pith 1999
[29]

Y. V. Kovchegov, “Unitarization of the BFKL pomeron on a nucleus,”Phys. Rev., vol. D61, p. 074018, 2000

View on Pith 2000
[30]

Y. V. Kovchegov and E. Levin,Quantum Chromodynamics at High Energy, vol. 33. Oxford University Press, 2013

View on Pith 2013
[31]

J.-P. Blaizot, E. Iancu, and H. Weigert, “Nonlinear gluon evolution in path integral form,” Nucl. Phys. A, vol. 713, pp. 441–469, 2003

View on Pith 2003
[32]

T. Lappi and H. M¨ antysaari, “On the running coupling in the JIMWLK equation,”Eur. Phys. J. C, vol. 73, no. 2, p. 2307, 2013

View on Pith 2013
[33]

I. Balitsky and G. A. Chirilli, “Next-to-leading order evolution of color dipoles,”Phys. Rev. D, vol. 77, p. 014019, 2008

View on Pith 2008
[34]

I. Balitsky and G. A. Chirilli, “Rapidity evolution of Wilson lines at the next-to-leading order,”Phys. Rev. D, vol. 88, p. 111501, 2013

View on Pith 2013
[35]

A. M. Stasto, B.-W. Xiao, and D. Zaslavsky, “Towards the Test of Saturation Physics Beyond Leading Logarithm,”Phys. Rev. Lett., vol. 112, no. 1, p. 012302, 2014

View on Pith 2014
[36]

T. Lappi and H. M¨ antysaari, “Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next to leading order,”Phys. Rev. D, vol. 91, no. 7, p. 074016, 2015

View on Pith 2015
[37]

B. Duclou´ e, T. Lappi, and Y. Zhu, “Single inclusive forward hadron production at next-to-leading order,”Phys. Rev. D, vol. 93, no. 11, p. 114016, 2016

View on Pith 2016
[38]

L. Motyka and A. M. Stasto, “Exact kinematics in the small x evolution of the color dipole and gluon cascade,”Phys. Rev. D, vol. 79, p. 085016, 2009

View on Pith 2009
[39]

G. Beuf, “Improving the kinematics for low-xQCD evolution equations in coordinate space,” Phys. Rev. D, vol. 89, no. 7, p. 074039, 2014. – 29 –

View on Pith 2014
[40]

E. Iancu, J. D. Madrigal, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, “Resumming double logarithms in the QCD evolution of color dipoles,”Phys. Lett. B, vol. 744, pp. 293–302, 2015

View on Pith 2015
[41]

B. Duclou´ e, E. Iancu, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, “Non-linear evolution in QCD at high-energy beyond leading order,”JHEP, vol. 04, p. 081, 2019

View on Pith 2019
[42]

T. Altinoluk, G. Beuf, and J. Penttala, “Exploring rapidity regularization schemes at low x with the DIS longitudinal structure function,”JHEP, vol. 04, p. 125, 2026

View on Pith 2026
[43]

R. Boussarie, P. Caucal, P. Korcyl, and Y. Mehtar-Tani, “Rotating the Color Glass Condensate,” 11 2025

View on Pith 2025
[44]

R. Boussarie, P. Caucal, and Y. Mehtar-Tani, “Collinear Structure of Nonlinear Small-x Evolution,” 9 2025

View on Pith 2025
[45]

G. Brunello, S. Caron-Huot, G. Crisanti, M. Giroux, and S. Smith, “High-energy evolution in planar QCD to three loops: the non-conformal contribution,”JHEP, vol. 11, p. 055, 2025

View on Pith 2025
[46]

A. Kovner and U. A. Wiedemann, “Eikonal evolution and gluon radiation,”Phys. Rev. D, vol. 64, p. 114002, 2001

View on Pith 2001
[47]

E. Iancu and L. D. McLerran, “Saturation and universality in QCD at small x,”Phys. Lett. B, vol. 510, pp. 145–154, 2001

View on Pith 2001
[48]

E. Iancu, A. Leonidov, and L. McLerran, “The Color glass condensate: An Introduction,” in Cargese Summer School on QCD Perspectives on Hot and Dense Matter, pp. 73–145, 2 2002

View on Pith 2002
[49]

E. Iancu, K. Itakura, and L. McLerran, “A Gaussian effective theory for gluon saturation,” Nucl. Phys. A, vol. 724, pp. 181–222, 2003

View on Pith 2003
[50]

H. Weigert, “Evolution at small x(bj): The Color glass condensate,”Prog. Part. Nucl. Phys., vol. 55, pp. 461–565, 2005

View on Pith 2005
[51]

Y. V. Kovchegov, J. Kuokkanen, K. Rummukainen, and H. Weigert, “Subleading-N(c) corrections in non-linear small-x evolution,”Nucl. Phys. A, vol. 823, pp. 47–82, 2009

View on Pith 2009
[52]

A. Dumitru and J. Jalilian-Marian, “Forward dijets in high-energy collisions: Evolution of QCD n-point functions beyond the dipole approximation,”Phys. Rev. D, vol. 82, p. 074023, 2010

View on Pith 2010
[53]

C. Marquet and H. Weigert, “New observables to test the Color Glass Condensate beyond the large-N c limit,”Nucl. Phys. A, vol. 843, pp. 68–97, 2010

View on Pith 2010
[54]

E. Iancu and D. N. Triantafyllopoulos, “JIMWLK evolution in the Gaussian approximation,” JHEP, vol. 04, p. 025, 2012

View on Pith 2012
[55]

T. Lappi and D. N. Triantafyllopoulos, “When JIMWLK evolution really matters: The example of incoherent diffraction,”Phys. Rev. D, vol. 114, no. 1, p. 014036, 2026

View on Pith 2026
[56]

C. Marquet, E. Petreska, and C. Roiesnel, “Transverse-momentum-dependent gluon distributions from JIMWLK evolution,”JHEP, vol. 10, p. 065, 2016

View on Pith 2016
[57]

M. Bury, P. Kotko, and K. Kutak, “TMD gluon distributions for multiparton processes,” Eur. Phys. J. C, vol. 79, no. 2, p. 152, 2019

View on Pith 2019
[58]

L. D. McLerran and R. Venugopalan, “Computing quark and gluon distribution functions for very large nuclei,”Phys. Rev., vol. D49, pp. 2233–2241, 1994. – 30 –

View on Pith 1994
[59]

L. D. McLerran and R. Venugopalan, “Gluon distribution functions for very large nuclei at small transverse momentum,”Phys. Rev., vol. D49, pp. 3352–3355, 1994

View on Pith 1994
[60]

L. D. McLerran and R. Venugopalan, “Green’s functions in the color field of a large nucleus,” Phys. Rev., vol. D50, pp. 2225–2233, 1994

View on Pith 1994
[61]

P. Korcyl, “Finite volume effects in the McLerran–Venugopalan initial condition for the JIMWLK equation,”Eur. Phys. J. C, vol. 82, no. 4, p. 369, 2022

View on Pith 2022
[62]

K. Rummukainen and H. Weigert, “Universal features of JIMWLK and BK evolution at small x,”Nucl. Phys. A, vol. 739, pp. 183–226, 2004

View on Pith 2004
[63]

P. Cvitanovic, “Group theory for Feynman diagrams in non-Abelian gauge theories,”Phys. Rev. D, vol. 14, pp. 1536–1553, 1976

View on Pith 1976
[64]

P. Cvitanovic,Group Theory: Birdtracks, Lie’s, and Exceptional Groups. Princeton University Press, 5 2020

View on Pith 2020
[65]

Y. L. Dokshitzer, “Perturbative QCD (and beyond),”Lect. Notes Phys., vol. 496, pp. 87–135, 1997

View on Pith 1997
[66]

S. Keppeler, “Birdtracks for SU(N),”SciPost Phys. Lect. Notes, vol. 3, p. 1, 2018

View on Pith 2018
[67]

S. Peign´ e,Color in QCD: An Introduction Featuring the Birdtrack Pictorial Technique. SpringerBriefs in Physics, Springer, 5 2024

View on Pith 2024
[68]

F. Cougoulic, P. Korcyl, and T. Stebel, “Improving the solver for the Balitsky-Kovchegov evolution equation with Automatic Differentiation,”Comput. Phys. Commun., vol. 313, p. 109616, 2025

View on Pith 2025
[69]

M. Li, “High Energy Evolution of Dipole Gluon Distribution Beyond Eikonal Approximation,” 7 2026

View on Pith 2026
[70]

S. Jeon and R. Venugopalan, “A Classical Odderon in QCD at high energies,”Phys. Rev. D, vol. 71, p. 125003, 2005

View on Pith 2005
[71]

S. Cali, K. Cichy, P. Korcyl, P. Kotko, K. Kutak, and C. Marquet, “On systematic effects in the numerical solutions of the JIMWLK equation,”Eur. Phys. J. C, vol. 81, no. 7, p. 663, 2021. – 31 –

View on Pith 2021

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Spin entanglement in $pp$ and $pA$ collisions: high-energy TMD effects and fragmentation probes
hep-ph 2026-11 accept novelty 7.0 of 10

The spin entanglement of quark-antiquark pairs in pp/pA collisions exhibits a cos(2phi) modulation that is maximally strong in dilute matter and suppressed by saturation effects, and the diffractive channel always pro...
Power corrections to heavy-quark spin correlations in small-$x$ deep-inelastic scattering
hep-ph 2026-11 conditional novelty 7.0 of 10

The azimuthally averaged heavy-quark spin density matrix in small-x DIS obtains target-dependent corrections at quartic order in dipole size, with mass-dependent asymptotics and a 3.5-3.7% increase in log-likelihood r...
The Colour Glass Condensate
hep-ph 2026-11 unverdicted

A pedagogical review of the Colour Glass Condensate effective theory and its applications to gluon saturation in high-energy QCD.
Transverse momentum dependent parton distributions at small $x$
hep-ph 2026-10 unverdicted

A comprehensive review of small-x TMDs in the CGC framework, covering their theoretical foundation, evolution, and phenomenology.

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 66 sentences without a confident label

Builds on 1 paper

In this study, we built upon previous research on small-$x$ TMD distributions by incorporating evolution into our framework in [ref] .

Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations · Discussion and prospects · 0.99

Uses the method or tool of 8 papers

The initial condition is generated according to the MV-model, and we refer to [ref] for further details.

Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations · Implementation of the JIMWLK evolution · 0.87
Gluon - Gluon: according to [ref] , we write TMD operators as:

TMD gluon distributions for multiparton processes · Definition of the operators · 0.93
Gluon - Gluon: according to [ref] , we write TMD operators as:

Transverse-momentum-dependent gluon distributions from JIMWLK evolution · Definition of the operators · 0.93

Compares with 1 paper

This situation is distinct from the cases explored in [ref] , where the dependence on the rapidity parametrization $y$ disappeared in favor of $Γ_$ being the potential integrated over the longitudinal profile.

When JIMWLK evolution really matters: the example of incoherent diffraction · Analytic results for the Gaussian truncation · 0.99

Cites as support 2 papers

While this program has obvious advantages, it has been shown, e.g. [ref] , that the GT does not automatically capture all relevant features of correlation functions for the quadrupole and the sextupole.

Forward dijets in high-energy collisions: evolution of QCD n-point functions beyond the dipole approximation · Introduction · 0.97
While this program has obvious advantages, it has been shown, e.g. [ref] , that the GT does not automatically capture all relevant features of correlation functions for the quadrupole and the sextupole.

When JIMWLK evolution really matters: the example of incoherent diffraction · Introduction · 0.97

Cites as background 26 papers

This approach has been widely used over the years, see e.g. [ref] and references therein.

Evolution at small x_bj: The Color Glass Condensate · Introduction · 0.99
This approach has been widely used over the years, see e.g. [ref] and references therein.

Forward dijets in high-energy collisions: evolution of QCD n-point functions beyond the dipole approximation · Introduction · 0.99
This approach has been widely used over the years, see e.g. [ref] and references therein.

JIMWLK evolution in the Gaussian approximation · Introduction · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

F. Cougoulic, P. Korcyl, and T. Stebel, “Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations,”JHEP, vol. 08, p. 029, 2026

View on Pith 2026

[2] [2]

R. Boussarieet al., “TMD Handbook,” 4 2023

View on Pith 2023

[3] [3]

J. Collins,Foundations of Perturbative QCD, vol. 32. Cambridge University Press, 2011

View on Pith 2011

[4] [4]

J. C. Collins, D. E. Soper, and G. F. Sterman, “Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production,”Nucl. Phys. B, vol. 250, pp. 199–224, 1985

View on Pith 1985

[5] [5]

J.-w. Qiu and X.-f. Zhang, “QCD prediction for heavy boson transverse momentum distributions,”Phys. Rev. Lett., vol. 86, pp. 2724–2727, 2001

View on Pith 2001

[6] [6]

X.-d. Ji, J.-P. Ma, and F. Yuan, “Qcd factorization for spin-dependent cross sections in dis and drell-yan processes at low transverse momentum,”Phys. Lett. B, vol. 597, pp. 299–308, 2004

View on Pith 2004

[7] [7]

J. P. Ralston and D. E. Soper, “Production of Dimuons from High-Energy Polarized Proton Proton Collisions,”Nucl. Phys. B, vol. 152, p. 109, 1979

View on Pith 1979

[8] [8]

D. W. Sivers, “Single Spin Production Asymmetries from the Hard Scattering of Point-Like Constituents,”Phys. Rev. D, vol. 41, p. 83, 1990

View on Pith 1990

[9] [9]

D. W. Sivers, “Hard scattering scaling laws for single spin production asymmetries,”Phys. Rev. D, vol. 43, pp. 261–263, 1991

View on Pith 1991

[10] [10]

J. C. Collins, “Fragmentation of transversely polarized quarks probed in transverse momentum distributions,”Nucl. Phys. B, vol. 396, pp. 161–182, 1993

View on Pith 1993

[11] [11]

D. Boer, P. J. Mulders, and F. Pijlman, “Universality of T odd effects in single spin and azimuthal asymmetries,”Nucl. Phys. B, vol. 667, pp. 201–241, 2003

View on Pith 2003

[12] [12]

L. N. Lipatov, “The Bare Pomeron in Quantum Chromodynamics,”Sov. Phys. JETP, vol. 63, pp. 904–912, 1986

View on Pith 1986

[13] [13]

S. Catani, M. Ciafaloni, and F. Hautmann, “High-energy factorization and small x heavy flavor production,”Nucl. Phys. B, vol. 366, pp. 135–188, 1991

View on Pith 1991

[14] [14]

F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, “The Color Glass Condensate,” Ann. Rev. Nucl. Part. Sci., vol. 60, pp. 463–489, 2010

View on Pith 2010

[15] [15]

J. L. Albacete and C. Marquet, “Gluon saturation and initial conditions for relativistic heavy ion collisions,”Prog. Part. Nucl. Phys., vol. 76, pp. 1–42, 2014

View on Pith 2014

[16] [16]

J.-P. Blaizot, “High gluon densities in heavy ion collisions,”Rept. Prog. Phys., vol. 80, no. 3, p. 032301, 2017

View on Pith 2017

[17] [17]

A. Morreale and F. Salazar, “Mining for Gluon Saturation at Colliders,”Universe, vol. 7, no. 8, p. 312, 2021

View on Pith 2021

[18] [18]

H. Raj and R. Venugopalan, “Qcd-gravity double copy in regge asymptotics: from 2→n amplitudes to radiation in shockwave collisions,”Acta Phys. Polon. B, vol. 56, no. 11, pp. 11–A1, 2025

View on Pith 2025

[19] [19]

J. Jalilian-Marian, A. Kovner, and H. Weigert, “The Wilson renormalization group for low x physics: Gluon evolution at finite parton density,”Phys. Rev., vol. D59, p. 014015, 1998. – 28 –

View on Pith 1998

[20] [20]

J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, “The Wilson renormalization group for low x physics: Towards the high density regime,”Phys. Rev., vol. D59, p. 014014, 1998

View on Pith 1998

[21] [21]

H. Weigert, “Unitarity at small Bjorken x,”Nucl. Phys., vol. A703, pp. 823–860, 2002

View on Pith 2002

[22] [22]

E. Iancu, A. Leonidov, and L. D. McLerran, “The renormalization group equation for the color glass condensate,”Phys. Lett., vol. B510, pp. 133–144, 2001

View on Pith 2001

[23] [23]

E. Iancu, A. Leonidov, and L. D. McLerran, “Nonlinear gluon evolution in the color glass condensate. I,”Nucl. Phys., vol. A692, pp. 583–645, 2001

View on Pith 2001

[24] [24]

E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, “Nonlinear gluon evolution in the color glass condensate. II,”Nucl. Phys., vol. A703, pp. 489–538, 2002

View on Pith 2002

[25] [25]

I. Balitsky, “Operator expansion for high-energy scattering,”Nucl. Phys., vol. B463, pp. 99–160, 1996

View on Pith 1996

[26] [26]

I. Balitsky, “Factorization for high-energy scattering,”Phys. Rev. Lett., vol. 81, pp. 2024–2027, 1998

View on Pith 2024

[27] [27]

I. Balitsky, “Factorization and high-energy effective action,”Phys. Rev. D, vol. 60, p. 014020, 1999

View on Pith 1999

[28] [28]

Y. V. Kovchegov, “Small-xF 2 structure function of a nucleus including multiple pomeron exchanges,”Phys. Rev., vol. D60, p. 034008, 1999

View on Pith 1999

[29] [29]

Y. V. Kovchegov, “Unitarization of the BFKL pomeron on a nucleus,”Phys. Rev., vol. D61, p. 074018, 2000

View on Pith 2000

[30] [30]

Y. V. Kovchegov and E. Levin,Quantum Chromodynamics at High Energy, vol. 33. Oxford University Press, 2013

View on Pith 2013

[31] [31]

J.-P. Blaizot, E. Iancu, and H. Weigert, “Nonlinear gluon evolution in path integral form,” Nucl. Phys. A, vol. 713, pp. 441–469, 2003

View on Pith 2003

[32] [32]

T. Lappi and H. M¨ antysaari, “On the running coupling in the JIMWLK equation,”Eur. Phys. J. C, vol. 73, no. 2, p. 2307, 2013

View on Pith 2013

[33] [33]

I. Balitsky and G. A. Chirilli, “Next-to-leading order evolution of color dipoles,”Phys. Rev. D, vol. 77, p. 014019, 2008

View on Pith 2008

[34] [34]

I. Balitsky and G. A. Chirilli, “Rapidity evolution of Wilson lines at the next-to-leading order,”Phys. Rev. D, vol. 88, p. 111501, 2013

View on Pith 2013

[35] [35]

A. M. Stasto, B.-W. Xiao, and D. Zaslavsky, “Towards the Test of Saturation Physics Beyond Leading Logarithm,”Phys. Rev. Lett., vol. 112, no. 1, p. 012302, 2014

View on Pith 2014

[36] [36]

T. Lappi and H. M¨ antysaari, “Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next to leading order,”Phys. Rev. D, vol. 91, no. 7, p. 074016, 2015

View on Pith 2015

[37] [37]

B. Duclou´ e, T. Lappi, and Y. Zhu, “Single inclusive forward hadron production at next-to-leading order,”Phys. Rev. D, vol. 93, no. 11, p. 114016, 2016

View on Pith 2016

[38] [38]

L. Motyka and A. M. Stasto, “Exact kinematics in the small x evolution of the color dipole and gluon cascade,”Phys. Rev. D, vol. 79, p. 085016, 2009

View on Pith 2009

[39] [39]

G. Beuf, “Improving the kinematics for low-xQCD evolution equations in coordinate space,” Phys. Rev. D, vol. 89, no. 7, p. 074039, 2014. – 29 –

View on Pith 2014

[40] [40]

E. Iancu, J. D. Madrigal, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, “Resumming double logarithms in the QCD evolution of color dipoles,”Phys. Lett. B, vol. 744, pp. 293–302, 2015

View on Pith 2015

[41] [41]

B. Duclou´ e, E. Iancu, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, “Non-linear evolution in QCD at high-energy beyond leading order,”JHEP, vol. 04, p. 081, 2019

View on Pith 2019

[42] [42]

T. Altinoluk, G. Beuf, and J. Penttala, “Exploring rapidity regularization schemes at low x with the DIS longitudinal structure function,”JHEP, vol. 04, p. 125, 2026

View on Pith 2026

[43] [43]

R. Boussarie, P. Caucal, P. Korcyl, and Y. Mehtar-Tani, “Rotating the Color Glass Condensate,” 11 2025

View on Pith 2025

[44] [44]

R. Boussarie, P. Caucal, and Y. Mehtar-Tani, “Collinear Structure of Nonlinear Small-x Evolution,” 9 2025

View on Pith 2025

[45] [45]

G. Brunello, S. Caron-Huot, G. Crisanti, M. Giroux, and S. Smith, “High-energy evolution in planar QCD to three loops: the non-conformal contribution,”JHEP, vol. 11, p. 055, 2025

View on Pith 2025

[46] [46]

A. Kovner and U. A. Wiedemann, “Eikonal evolution and gluon radiation,”Phys. Rev. D, vol. 64, p. 114002, 2001

View on Pith 2001

[47] [47]

E. Iancu and L. D. McLerran, “Saturation and universality in QCD at small x,”Phys. Lett. B, vol. 510, pp. 145–154, 2001

View on Pith 2001

[48] [48]

E. Iancu, A. Leonidov, and L. McLerran, “The Color glass condensate: An Introduction,” in Cargese Summer School on QCD Perspectives on Hot and Dense Matter, pp. 73–145, 2 2002

View on Pith 2002

[49] [49]

E. Iancu, K. Itakura, and L. McLerran, “A Gaussian effective theory for gluon saturation,” Nucl. Phys. A, vol. 724, pp. 181–222, 2003

View on Pith 2003

[50] [50]

H. Weigert, “Evolution at small x(bj): The Color glass condensate,”Prog. Part. Nucl. Phys., vol. 55, pp. 461–565, 2005

View on Pith 2005

[51] [51]

Y. V. Kovchegov, J. Kuokkanen, K. Rummukainen, and H. Weigert, “Subleading-N(c) corrections in non-linear small-x evolution,”Nucl. Phys. A, vol. 823, pp. 47–82, 2009

View on Pith 2009

[52] [52]

A. Dumitru and J. Jalilian-Marian, “Forward dijets in high-energy collisions: Evolution of QCD n-point functions beyond the dipole approximation,”Phys. Rev. D, vol. 82, p. 074023, 2010

View on Pith 2010

[53] [53]

C. Marquet and H. Weigert, “New observables to test the Color Glass Condensate beyond the large-N c limit,”Nucl. Phys. A, vol. 843, pp. 68–97, 2010

View on Pith 2010

[54] [54]

E. Iancu and D. N. Triantafyllopoulos, “JIMWLK evolution in the Gaussian approximation,” JHEP, vol. 04, p. 025, 2012

View on Pith 2012

[55] [55]

T. Lappi and D. N. Triantafyllopoulos, “When JIMWLK evolution really matters: The example of incoherent diffraction,”Phys. Rev. D, vol. 114, no. 1, p. 014036, 2026

View on Pith 2026

[56] [56]

C. Marquet, E. Petreska, and C. Roiesnel, “Transverse-momentum-dependent gluon distributions from JIMWLK evolution,”JHEP, vol. 10, p. 065, 2016

View on Pith 2016

[57] [57]

M. Bury, P. Kotko, and K. Kutak, “TMD gluon distributions for multiparton processes,” Eur. Phys. J. C, vol. 79, no. 2, p. 152, 2019

View on Pith 2019

[58] [58]

L. D. McLerran and R. Venugopalan, “Computing quark and gluon distribution functions for very large nuclei,”Phys. Rev., vol. D49, pp. 2233–2241, 1994. – 30 –

View on Pith 1994

[59] [59]

L. D. McLerran and R. Venugopalan, “Gluon distribution functions for very large nuclei at small transverse momentum,”Phys. Rev., vol. D49, pp. 3352–3355, 1994

View on Pith 1994

[60] [60]

L. D. McLerran and R. Venugopalan, “Green’s functions in the color field of a large nucleus,” Phys. Rev., vol. D50, pp. 2225–2233, 1994

View on Pith 1994

[61] [61]

P. Korcyl, “Finite volume effects in the McLerran–Venugopalan initial condition for the JIMWLK equation,”Eur. Phys. J. C, vol. 82, no. 4, p. 369, 2022

View on Pith 2022

[62] [62]

K. Rummukainen and H. Weigert, “Universal features of JIMWLK and BK evolution at small x,”Nucl. Phys. A, vol. 739, pp. 183–226, 2004

View on Pith 2004

[63] [63]

P. Cvitanovic, “Group theory for Feynman diagrams in non-Abelian gauge theories,”Phys. Rev. D, vol. 14, pp. 1536–1553, 1976

View on Pith 1976

[64] [64]

P. Cvitanovic,Group Theory: Birdtracks, Lie’s, and Exceptional Groups. Princeton University Press, 5 2020

View on Pith 2020

[65] [65]

Y. L. Dokshitzer, “Perturbative QCD (and beyond),”Lect. Notes Phys., vol. 496, pp. 87–135, 1997

View on Pith 1997

[66] [66]

S. Keppeler, “Birdtracks for SU(N),”SciPost Phys. Lect. Notes, vol. 3, p. 1, 2018

View on Pith 2018

[67] [67]

S. Peign´ e,Color in QCD: An Introduction Featuring the Birdtrack Pictorial Technique. SpringerBriefs in Physics, Springer, 5 2024

View on Pith 2024

[68] [68]

F. Cougoulic, P. Korcyl, and T. Stebel, “Improving the solver for the Balitsky-Kovchegov evolution equation with Automatic Differentiation,”Comput. Phys. Commun., vol. 313, p. 109616, 2025

View on Pith 2025

[69] [69]

M. Li, “High Energy Evolution of Dipole Gluon Distribution Beyond Eikonal Approximation,” 7 2026

View on Pith 2026

[70] [70]

S. Jeon and R. Venugopalan, “A Classical Odderon in QCD at high energies,”Phys. Rev. D, vol. 71, p. 125003, 2005

View on Pith 2005

[71] [71]

S. Cali, K. Cichy, P. Korcyl, P. Kotko, K. Kutak, and C. Marquet, “On systematic effects in the numerical solutions of the JIMWLK equation,”Eur. Phys. J. C, vol. 81, no. 7, p. 663, 2021. – 31 –

View on Pith 2021