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One-loop renormalization of quark TMD in the light-cone gauge: CSS evolution

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that the one-loop renormalized quark TMD in the target light-cone gauge satisfies exactly the standard Collins-Soper-Sterman evolution equations.

desk verdict Solid one-loop calculation that recovers the known CSS equations and attributes the Sudakov double log to the ML zero-mode, but the derivation leans on an unevaluated soft-factor cancellation that should be fixed before acceptance. read the letter →

arxiv 2505.20467 v1 pith:4C3NWZFC submitted 2025-05-26 hep-ph

Tolga Altinoluk , Guillaume Beuf , Jamal Jalilian-Marian This is my paper
classification hep-ph
keywords quarkTMDlight-conegaugeMandelstam-LeibbrandtprescriptionCSSevolutionSudakovdoublelogarithmbackgroundfieldmethodrapiditydivergencetransversemomentumdependence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper calculates the one-loop quantum corrections to the unpolarized quark transverse-momentum-dependent distribution (TMD) in the target light-cone gauge, using the background field formalism and the Mandelstam-Leibbrandt (ML) prescription for the extra singularity in the gluon propagator. It claims that after rapidity and ultraviolet renormalization, the quark TMD satisfies exactly the standard Collins-Soper-Sterman (CSS) evolution equations at one loop. If correct, this means the light-cone gauge, despite its extra propagator singularity, yields the same evolution as covariant gauges, and it identifies the ghost-like zero-mode term of the ML prescription as the source of the double-log Sudakov contribution from the transverse part of the gauge link at infinity.

What carries the argument

The load-bearing object is the ML-prescribed light-cone gauge gluon propagator, whose second form splits off a ghost-like zero-mode term with poles at $n\cdot k=0$. In the radiation-to-infinity diagrams, the $k^-$ integration picks this zero-mode pole and produces the constant $-1$ that turns a would-be single rapidity pole into the double-log structure. Combined with the pure rapidity regulator $(k^-/k^+)^{\eta/2}$ and the background-field non-renormalization relation $g_0 A_i^{(0)} = \mu^\epsilon g A_i$, which lets the transverse Wilson line be evaluated in bare perturbation theory, this yields the scale dependence encoded in the CSS equations.

What would settle it

Evaluate the soft factor that is supposed to cancel the Wilson-line self-energy diagram in the target light-cone gauge and check it against the Fig. 1e result $\alpha_s C_F/(2\pi)\,\Gamma(1-\epsilon)/(\epsilon(1-2\epsilon))\,(\pi\mu^2 b^2)^\epsilon\, q_{\rm Bckgd}(x,b;\mu^2)$. If the cancellation is not exact, Eq. (55) acquires an extra term, and the $\zeta$- and $\mu$-derivatives in Eqs. (83)--(84) shift away from the standard CSS coefficients.

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Extended reading notes

Core claim

The paper's central claim is that the renormalized quark TMD in the target light-cone gauge obeys the one-loop CSS equations $$\$mu^{2}$ \frac{d}{d\$mu^{2}$} q(x,b;\$mu^{2}$,\zeta) = \left[\frac{\alpha_s C_F}{2\pi}\left(\log\frac{\$mu^{2}$}{\zeta}+\frac{3}{2}\right)+O(\$alpha_s^{2}$)\right] q(x,b;\$mu^{2}$,\zeta)$$ and $$\zeta \frac{d}{d\zeta} q(x,b;\$mu^{2}$,\zeta) = \left[-\frac{\alpha_s C_F}{2\pi}\log\frac{\$mu^{2}$ $b^{2}$}{$c_0^{2}$}+O(\$alpha_s^{2}$)\right] q(x,b;\$mu^{2}$,\zeta).$$ The rapidity-divergent piece comes from the diagrams in which a gluon is emitted from the quark or antiquark to the transverse part of the gauge link at infinity, and the double-log Sudakov structure is produced by the zero-mode term, the constant $-1$ in the $k^-$ integral, that arises from the ML prescription. The ladder diagrams contribute only finite NLO corrections, and the Wilson-line self-energy diagram is discarded on the assumption that an uncalculated soft factor cancels it.

Load-bearing premise

The load-bearing premise is that the discarded Wilson-line self-energy diagram is exactly cancelled by a soft factor; the paper does not define or compute that soft factor, so if the cancellation fails the final CSS equations change.

Editorial extensions

If this is right

  • The one-loop anomalous dimensions of the quark TMD in the target light-cone gauge are the standard CSS coefficients, so CSS evolution is gauge-consistent for this operator.
  • The double-log Sudakov term is localized: it arises from the ghost-like zero mode of the ML prescription, not from the usual poles, so any light-cone gauge TMD calculation must keep that term to reproduce CSS resummation.
  • The background-field framework with the non-renormalization relation $g_0 A_i^{(0)} = \mu^\epsilon g A_i$ gives a streamlined route to renormalize TMDs, extendable to gluon TMDs and to the projectile light-cone gauge.
  • After rapidity subtraction, the remaining $\zeta$ dependence of the quark TMD is $\log(\mu^2 b^2/c_0^2)$, which is what drives the CSS evolution in impact-parameter space.

Reading between the lines

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

  • The paper leaves implicit that a direct computation of the soft factor is the decisive next test; until then, the cancellation assumption is what carries the final result.
  • The same zero-mode mechanism may explain why rapidity divergences and Sudakov logs are tied to residual gauge freedom, suggesting a link between light-cone gauge prescriptions and the structure of factorization proofs.
  • One could test the generality by computing the gluon TMD in the same setup: the CSS coefficients for gluons would be the analogous check that the mechanism is flavour-independent.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper computes the one-loop corrections to the unpolarized quark TMD in the target light-cone gauge (A^- = 0) using the background field formalism, dimensional regularization, the Mandelstam-Leibbrandt prescription for the light-cone gauge singularity, and the pure rapidity regulator. The authors obtain the total NLO correction from the radiation-to-infinity diagrams, argue that the ladder diagrams are finite, and combine this with light-cone gauge field renormalization constants taken from prior literature to extract the one-loop CSS evolution equations, Eqs. (83)-(84). They also identify the ghost-like zero-mode piece of the ML-prescribed gluon propagator as the origin of the double-log Sudakov contribution.

Significance. If the missing soft-factor step is properly supplied, the calculation provides a nontrivial consistency check: the quark TMD defined with the ML-prescribed light-cone gauge and the pure rapidity regulator is shown to reproduce the standard one-loop CSS evolution equations, and the zero-mode part of the ML propagator is identified as the source of the double-log Sudakov term. The calculation is explicit and parameter-free, and it connects the background-field/CGC-style framework with standard TMD renormalization. The paper does not provide machine-checkable code or new falsifiable predictions; its value is analytic and interpretive, and it would be a useful reference for future small-x/TMD studies if the soft-factor gap is closed.

major comments (2)
  1. [Sec. III D, App. B, Eq. (55)] The total NLO expression (55) is obtained by simply discarding the Wilson-line self-energy diagram 1e; the text explicitly states that no soft factor is defined or evaluated. Appendix B shows that this diagram gives the nonzero, UV-divergent contribution (B7), proportional to qBckgd. If the cancellation by the square-root soft factor invoked in Sec. III D is not exact, Eq. (55) acquires the extra term (B7), and the UV renormalization factor Z_UV in Eq. (80) must be modified. The assertion that, with the pure rapidity regulator, the rapidity divergences of the soft factor at +infinity and -infinity cancel exactly, leaving only self-energy removal, is not demonstrated. Because the added term is zeta-independent, Eq. (84) is likely unaffected, but the impact on Eq. (83) depends on the mu-dependence of the new counterterm, which is not assessed. This is a load-bearing premise for the derivation of the CSS equations.
  2. [Sec. III C, after Eq. (54)] The conclusion that diagram 1d contributes only a finite NLO correction is stated without any calculation; the paper only says it 'can be calculated in a similar way.' Since the extraction of the CSS equations from Eq. (55) relies on the absence of zeta-dependent or UV poles from all diagrams other than 1a and 1b, the pole structure of diagram 1d should be demonstrated explicitly, or at least the relevant integrals should be relegated to an appendix.
minor comments (5)
  1. [Sec. I] The text contains 'CCS equations', which should be 'CSS equations', and the phrase 'scales of the of the processes' contains a duplicated word.
  2. [Sec. II, Eq. (1)] The rapidity-regulator factor is not typeset unambiguously; it would be clearer to display it as ((k^- nu^+)/(k^+ nu^-))^{eta/2} or an equivalent explicit form.
  3. [Appendix B, after Eq. (B3)] The integration variable zeta introduced in the change of variables conflicts with the rapidity scale zeta defined in Eq. (3); using a different symbol, such as sigma' or u, would avoid confusion.
  4. [Abstract and Sec. III] The abstract and introduction state that the one-loop corrections are calculated, but the 'finite NLO' terms in Eqs. (31), (55) and (75) are never evaluated; please state explicitly that only the divergent parts needed for the renormalization and evolution are retained.
  5. [Sec. III C, Eq. (42)] The parameter epsilon_s is introduced for the reduced spacetime dimensionality; it would help to remind the reader that epsilon_s = epsilon in conventional dimensional regularization, since the case epsilon_s = 0 is used later.

Circularity Check

0 steps flagged · score 2.0 of 10

No meaningful circularity: the CSS equations are an independently known benchmark, not an input; the only self-citation is minor and not load-bearing, and the discarded Wilson-line self-energy diagram is an unverified cancellation assumption rather than a circular step.

full rationale

I walked the derivation chain from the operator definition (Eq. (2)/(6)) through the NLO diagrams (Figs. 1a-1d), the rapidity renormalization (Eqs. (71)-(74)), the field renormalization imported from Refs. [68-70] (Eqs. (56)-(69)), and the extraction of the CSS equations (Eqs. (83)-(84)). No step reduces to its own input by construction: the renormalization constants Z2 and \tilde Z2 are taken from independent prior literature, not from the target CSS result; the rapidity renormalization factor Zrap. is derived from the computed pole in Eq. (55), not assumed; and the final anomalous dimensions are benchmarked against the known CSS equations rather than fitted to them. The self-citation to the authors' previous work [59] is used for technical ML-prescription contour manipulations (e.g., the simplification around Eq. (47)), but those manipulations are re-derived in the text, so the citation is not load-bearing. The one flagged weakness is in Sec. III D and App. B: the paper states 'we do not explicitly define and evaluate any soft factor, but simply discard the contribution from the diagram Fig. 1e', and if the discarded contribution Eq. (B7) were not cancelled exactly it would modify Eq. (55) and hence ZUV in Eq. (80). This is an unverified assumption and a completeness gap, but it is not circularity: the soft-factor cancellation is not defined in terms of the CSS equations, and the statement is an omitted proof rather than an input-output equivalence. Thus the paper is essentially self-contained against an external benchmark, with only minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the calculation depends on standard perturbative QCD input (renormalization constants, propagators, prescriptions) and on one paper-specific assumption: the unverified cancellation of the Wilson line self-energy diagram by a soft factor.

assumptions (5)
  • ad hoc to paper A soft factor exists that exactly cancels the Wilson line self-energy diagram 1e; this factor is not defined or evaluated.
    The paper explicitly says 'we do not explicitly define and evaluate any soft factor, but simply discard the contribution from the diagram Fig. 1e' (Sec. III D). The final result depends on this cancellation.
  • domain assumption Background fields obey their equations of motion including quantum corrections, so one-point functions of fluctuation fields vanish.
    Invoked in Sec. III A to justify keeping only two-point correlators of fluctuations.
  • domain assumption The target is dilute, so expansion to two insertions of the background field is sufficient at order g^2.
    Sec. III A restricts the calculation to the dilute regime.
  • domain assumption The pure rapidity regulator with the ordering eta to 0 at fixed nonzero epsilon regulates only rapidity divergences.
    This ordering is assumed throughout (Sec. II) and is necessary for the pole structure used in Sec. V.
  • domain assumption The Mandelstam-Leibbrandt prescription is the correct resolution of the light-cone gauge propagator singularity; the propagator (8) with (9) is taken as input.
    Used throughout, quoted from Refs. [62,63,66].

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Cite this review

Pith. "Pith review of One-loop renormalization of quark TMD in the light-cone gauge: CSS evolution." pith.science (2026). https://pith.science/paper/4C3NWZFC

@misc{pith2026250520467,
  author       = {Pith},
  title        = {Pith review of: One-loop renormalization of quark TMD in the light-cone gauge: CSS evolution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4C3NWZFC}},
  note         = {Machine review of arXiv:2505.20467}
}
read the original abstract

We calculate the one-loop corrections to the quark TMD in the light-cone gauge using the background field formalism, with the Mandelstam-Leibbrandt (ML) prescription for the extra singularity present in the light-cone gauge propagator. We use the pure rapidity regulator for rapidity divergences. The Collins-Soper-Sterman (CSS) evolution equations are indeed obtained from the one loop renormalization of the quark TMD. In this setup, the double log contribution to the CSS resummation is found to come from the ghost-like zero-mode from the ML prescription, in the diagrams with a gluon propagator ending on the transverse part of the gauge link at infinity.

Figures

Figures reproduced from arXiv: 2505.20467 by the authors.

Figure 1
Figure 1. FIG. 1: NLO diagrams in the expansion of the quark TMD around its background contribution. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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Reference graph

Works this paper leans on

71 extracted references · 11 canonical work pages · cited by 1 Pith paper

  1. [1]

    C. J. Bomhof, P. J. Mulders, and F. Pijlman, The Construction of gauge-links in arbitrary hard processes, Eur. Phys. J. C 47, 147 (2006), arXiv:hep-ph/0601171

    arXiv 2006

  2. [2]

    projectile

    The transverse indices are indicated by latin lettersi, j, . . .. Moreover, transverse vectors are written in bold caracters, and with a euclidian scalar product, so thatx·y ≡ x+y− + x−y+ − x·y. 3 where P indicates ordering along the path, and there is as well a space-like Wilson line in the far future U (Y +, x + b, 0−; Y +, x, 0−) = P exp −iµϵg Z 1 0 dτ...

    work page 2023

  3. [3]

    Bacchetta, M

    A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P. J. Mulders, and M. Schlegel, Semi-inclusive deep inelastic scattering at small transverse momentum, JHEP02, 093, arXiv:hep-ph/0611265

    arXiv

  4. [4]

    Collins,Foundations of Perturbative QCD , Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Vol

    J. Collins,Foundations of Perturbative QCD , Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Vol. 32 (Cambridge University Press, 2023)

    work page 2023

  5. [5]

    Angeles-Martinezet al., Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects, Acta Phys

    R. Angeles-Martinezet al., Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects, Acta Phys. Polon. B46, 2501 (2015), arXiv:1507.05267 [hep-ph]

    arXiv 2015

  6. [6]

    Boussarie et al., TMD Handbook, (2023), arXiv:2304.03302 [hep-ph]

    R. Boussarie et al., TMD Handbook, (2023), arXiv:2304.03302 [hep-ph]

    arXiv 2023

  7. [7]

    J. C. Collins and D. E. Soper, Back-To-Back Jets in QCD, Nucl. Phys. B193, 381 (1981), [Erratum: Nucl.Phys.B 213, 545 (1983)]

    1981

  8. [8]

    J. C. Collins and D. E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B194, 445 (1982)

    1982

Show all 71 references
  1. [9]

    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. B250, 199 (1985)

    1985

  2. [10]

    Gelis, E

    F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, The Color Glass Condensate, Ann. Rev. Nucl. Part. Sci.60, 463 (2010), arXiv:1002.0333 [hep-ph]

    2010 arXiv

  3. [11]

    J. L. Albacete and C. Marquet, Gluon saturation and initial conditions for relativistic heavy ion collisions, Prog. Part. Nucl. Phys.76, 1 (2014), arXiv:1401.4866 [hep-ph]

    2014 arXiv

  4. [12]

    Blaizot, High gluon densities in heavy ion collisions, Rept

    J.-P. Blaizot, High gluon densities in heavy ion collisions, Rept. Prog. Phys.80, 032301 (2017), arXiv:1607.04448 [hep-ph]

    2017 arXiv

  5. [13]

    Balitsky, Operator expansion for high-energy scattering, Nucl

    I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B463, 99 (1996), arXiv:hep-ph/9509348

    1996 arXiv

  6. [14]

    Y. V. Kovchegov, Small x F(2) structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D60, 034008 (1999), arXiv:hep-ph/9901281

    1999 arXiv

  7. [15]

    Y.V.Kovchegov,UnitarizationoftheBFKLpomerononanucleus,Phys.Rev.D 61,074018(2000),arXiv:hep-ph/9905214

    2000 arXiv

  8. [16]

    Jalilian-Marian, A

    J. Jalilian-Marian, A. Kovner, L. D. McLerran, and H. Weigert, The Intrinsic glue distribution at very small x, Phys. Rev. D 55, 5414 (1997), arXiv:hep-ph/9606337

    1997 arXiv

  9. [17]

    Jalilian-Marian, A

    J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B504, 415 (1997), arXiv:hep-ph/9701284

    1997 arXiv

  10. [18]

    Jalilian-Marian, A

    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. D59, 014014 (1998), arXiv:hep-ph/9706377

    1998 arXiv

  11. [19]

    Jalilian-Marian, A

    J. Jalilian-Marian, A. Kovner, and H. Weigert, The Wilson renormalization group for low x physics: Gluon evolution at finite parton density, Phys. Rev. D59, 014015 (1998), arXiv:hep-ph/9709432. 18

    1998 arXiv

  12. [20]

    Kovner, J

    A. Kovner, J. G. Milhano, and H. Weigert, Relating different approaches to nonlinear QCD evolution at finite gluon density, Phys. Rev. D62, 114005 (2000), arXiv:hep-ph/0004014

    2000 arXiv

  13. [21]

    Weigert, Unitarity at small Bjorken x, Nucl

    H. Weigert, Unitarity at small Bjorken x, Nucl. Phys. A703, 823 (2002), arXiv:hep-ph/0004044

    2002 arXiv

  14. [22]

    Iancu, A

    E. Iancu, A. Leonidov, and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1., Nucl. Phys. A 692, 583 (2001), arXiv:hep-ph/0011241

    2001 arXiv

  15. [23]

    Iancu, A

    E. Iancu, A. Leonidov, and L. D. McLerran, The Renormalization group equation for the color glass condensate, Phys. Lett. B510, 133 (2001), arXiv:hep-ph/0102009

    2001 arXiv

  16. [24]

    Ferreiro, E

    E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nonlinear gluon evolution in the color glass condensate. 2., Nucl. Phys. A703, 489 (2002), arXiv:hep-ph/0109115

    2002 arXiv

  17. [25]

    Dumitru and J

    A. Dumitru and J. Jalilian-Marian, Forward quark jets from protons shattering the colored glass, Phys. Rev. Lett.89, 022301 (2002), arXiv:hep-ph/0204028

    2002 arXiv

  18. [26]

    Dumitru, A

    A. Dumitru, A. Hayashigaki, and J. Jalilian-Marian, The Color glass condensate and hadron production in the forward region, Nucl. Phys. A765, 464 (2006), arXiv:hep-ph/0506308

    2006 arXiv

  19. [27]

    Altinoluk and A

    T. Altinoluk and A. Kovner, Particle Production at High Energy and Large Transverse Momentum - ’The Hybrid Formal- ism’ Revisited, Phys. Rev. D83, 105004 (2011), arXiv:1102.5327 [hep-ph]

    2011 arXiv

  20. [28]

    G. A. Chirilli, B.-W. Xiao, and F. Yuan, One-loop Factorization for Inclusive Hadron Production inpA Collisions in the Saturation Formalism, Phys. Rev. Lett.108, 122301 (2012), arXiv:1112.1061 [hep-ph]

    2012 arXiv

  21. [29]

    Dominguez, C

    F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan, Universality of Unintegrated Gluon Distributions at small x, Phys. Rev. D83, 105005 (2011), arXiv:1101.0715 [hep-ph]

    2011 arXiv

  22. [30]

    Petreska, TMD gluon distributions at small x in the CGC theory, Int

    E. Petreska, TMD gluon distributions at small x in the CGC theory, Int. J. Mod. Phys. E 27, 1830003 (2018), arXiv:1804.04981 [hep-ph]

    2018 arXiv

  23. [31]

    Marquet, C

    C. Marquet, C. Roiesnel, and P. Taels, Linearly polarized small-x gluons in forward heavy-quark pair production, Phys. Rev. D97, 014004 (2018), arXiv:1710.05698 [hep-ph]

    2018 arXiv

  24. [32]

    Taels, T

    P. Taels, T. Altinoluk, G. Beuf, and C. Marquet, Dijet photoproduction at low x at next-to-leading order and its back-to- back limit, JHEP10, 184, arXiv:2204.11650 [hep-ph]

    arXiv

  25. [33]

    P.Caucal, F.Salazar, B.Schenke, T.Stebel,andR.Venugopalan,Back-to-BackInclusiveDijetsinDeepInelasticScattering at Small x: Complete NLO Results and Predictions, Phys. Rev. Lett.132, 081902 (2024), arXiv:2308.00022 [hep-ph]

    2024 arXiv

  26. [34]

    Caucal and F

    P. Caucal and F. Salazar, Dihadron correlations in small-x DIS at NLO: transverse momentum dependent fragmentation, JHEP 12, 130, arXiv:2405.19404 [hep-ph]

    arXiv

  27. [35]

    Taels, Forward production of a Drell-Yan pair and a jet at small x at next-to-leading order, JHEP 01, 005, arXiv:2308.02449 [hep-ph]

    P. Taels, Forward production of a Drell-Yan pair and a jet at small x at next-to-leading order, JHEP 01, 005, arXiv:2308.02449 [hep-ph]

    arXiv

  28. [36]

    Altinoluk, N

    T. Altinoluk, N. Armesto, A. Kovner, M. Lublinsky, and E. Petreska, Soft photon and two hard jets forward production in proton-nucleus collisions, JHEP04, 063, arXiv:1802.01398 [hep-ph]

    arXiv

  29. [37]

    Altinoluk, R

    T. Altinoluk, R. Boussarie, C. Marquet, and P. Taels, TMD factorization for dijets + photon production from the dilute- dense CGC framework, JHEP07, 079, arXiv:1810.11273 [hep-ph]

    arXiv

  30. [38]

    Altinoluk, R

    T. Altinoluk, R. Boussarie, C. Marquet, and P. Taels, Photoproduction of three jets in the CGC: gluon TMDs and dilute limit, JHEP07, 143, arXiv:2001.00765 [hep-ph]

    2001 arXiv

  31. [39]

    Kotko, K

    P. Kotko, K. Kutak, C. Marquet, E. Petreska, S. Sapeta, and A. van Hameren, Improved TMD factorization for forward dijet production in dilute-dense hadronic collisions, JHEP09, 106, arXiv:1503.03421 [hep-ph]

    arXiv

  32. [40]

    van Hameren, P

    A. van Hameren, P. Kotko, K. Kutak, C. Marquet, E. Petreska, and S. Sapeta, Forward di-jet production in p+Pb collisions in the small-x improved TMD factorization framework, JHEP12, 034, [Erratum: JHEP 02, 158 (2019)], arXiv:1607.03121 [hep-ph]

    2019 arXiv

  33. [41]

    M. Bury, A. van Hameren, P. Kotko, and K. Kutak, Forward trijet production in p-p and p-Pb collisions at LHC, JHEP 09, 175, arXiv:2006.13175 [hep-ph]

    2006 arXiv

  34. [42]

    Fujii, C

    H. Fujii, C. Marquet, and K. Watanabe, Comparison of improved TMD and CGC frameworks in forward quark dijet production, JHEP12, 181, arXiv:2006.16279 [hep-ph]

    2006 arXiv

  35. [43]

    Altinoluk, C

    T. Altinoluk, C. Marquet, and P. Taels, Low-x improved TMD approach to the lepto- and hadroproduction of a heavy-quark pair, JHEP06, 085, arXiv:2103.14495 [hep-ph]

    arXiv

  36. [44]

    Altinoluk and R

    T. Altinoluk and R. Boussarie, Lowx physics as an infinite twist (G)TMD framework: unravelling the origins of saturation, JHEP 10, 208, arXiv:1902.07930 [hep-ph]

    1902 arXiv

  37. [45]

    Altinoluk, R

    T. Altinoluk, R. Boussarie, and P. Kotko, Interplay of the CGC and TMD frameworks to all orders in kinematic twist, JHEP 05, 156, arXiv:1901.01175 [hep-ph]

    1901 arXiv

  38. [46]

    Boussarie and Y

    R. Boussarie and Y. Mehtar-Tani, Gauge invariance of transverse momentum dependent distributions at smallx, Phys. Rev. D103, 094012 (2021), arXiv:2001.06449 [hep-ph]

    2021 arXiv

  39. [47]

    Boussarie, H

    R. Boussarie, H. Mäntysaari, F. Salazar, and B. Schenke, The importance of kinematic twists and genuine saturation effects in dijet production at the Electron-Ion Collider, JHEP09, 178, arXiv:2106.11301 [hep-ph]

    arXiv

  40. [48]

    Caucal, M

    P. Caucal, M. G. Morales, E. Iancu, F. Salazar, and F. Yuan, Unveiling the sea: universality of the transverse momentum dependent quark distributions at smallx, (2025), arXiv:2503.16162 [hep-ph]

    2025

  41. [49]

    Altinoluk, N

    T. Altinoluk, N. Armesto, and G. Beuf, Probing quark transverse momentum distributions in the color glass conden- sate: Quark-gluon dijets in deep inelastic scattering at next-to-eikonal accuracy, Phys. Rev. D 108, 074023 (2023), arXiv:2303.12691 [hep-ph]

    2023 arXiv

  42. [50]

    Altinoluk, G

    T. Altinoluk, G. Beuf, A. Czajka, and C. Marquet, Back-to-back dijet production in DIS at next-to-eikonal accuracy and twist-3 gluon TMDs, Phys. Rev. D111, 014010 (2025), arXiv:2410.00612 [hep-ph]

    2025 arXiv

  43. [51]

    Altinoluk, G

    T. Altinoluk, G. Beuf, E. Blanco, and S. Mulani, Quark TMDs from back-to-back dijet production at forward rapidities 19 in pA collisions beyond eikonal accuracy in the CGC, (2024), arXiv:2412.08485 [hep-ph]

    2024

  44. [52]

    Marquet, B.-W

    C. Marquet, B.-W. Xiao, and F. Yuan, Semi-inclusive Deep Inelastic Scattering at small x, Phys. Lett. B682, 207 (2009), arXiv:0906.1454 [hep-ph]

    2009 arXiv

  45. [53]

    Altinoluk, J

    T. Altinoluk, J. Jalilian-Marian, and C. Marquet, Sudakov double logs in single-inclusive hadron production in DIS at small x from the color glass condensate formalism, Phys. Rev. D110, 094056 (2024), arXiv:2406.08277 [hep-ph]

    2024 arXiv

  46. [54]

    Caucal, E

    P. Caucal, E. Iancu, A. H. Mueller, and F. Yuan, Jet Definition and Transverse-Momentum-Dependent Factorization in Semi-Inclusive Deep-Inelastic Scattering, Phys. Rev. Lett.134, 061903 (2025), arXiv:2408.03129 [hep-ph]

    2025 arXiv

  47. [55]

    Altinoluk, N

    T. Altinoluk, N. Armesto, A. Kovner, and M. Lublinsky, Single inclusive particle production at next-to-leading order in proton-nucleus collisions at forward rapidities: Hybrid approach meets TMD factorization, Phys. Rev. D108, 074003 (2023), arXiv:2307.14922 [hep-ph]

    2023 arXiv

  48. [56]

    H. Duan, A. Kovner, and M. Lublinsky, Collins-Soper-Sterman Hamiltonian: High energy evolution of rapidity dependent observables, Phys. Rev. D111, 054022 (2025), arXiv:2407.15960 [hep-ph]

    2025 arXiv

  49. [57]

    H. Duan, A. Kovner, and M. Lublinsky, Born-Oppenheimer Renormalization group for High Energy Scattering: the Setup and the Wave Function, (2024), arXiv:2412.05085 [hep-ph]

    2024 arXiv

  50. [58]

    H. Duan, A. Kovner, and M. Lublinsky, Born-Oppenheimer Renormalization group for High Energy Scattering: CSS, DGLAP and all that, (2024), arXiv:2412.05097 [hep-ph]

    2024 arXiv

  51. [59]

    Caucal and E

    P. Caucal and E. Iancu, Evolution of the transverse-momentum dependent gluon distribution at small x, Phys. Rev. D 111, 074008 (2025), arXiv:2406.04238 [hep-ph]

    2025 arXiv

  52. [60]

    Altinoluk, G

    T. Altinoluk, G. Beuf, and J. Jalilian-Marian, Renormalization of the gluon distribution function in the background field formalism, (2023), arXiv:2305.11079 [hep-ph]

    2023 arXiv

  53. [61]

    Mukherjee, V

    S. Mukherjee, V. V. Skokov, A. Tarasov, and S. Tiwari, Unified description of DGLAP, CSS, and BFKL evolution: TMD factorization bridging large and small x, Phys. Rev. D109, 034035 (2024), arXiv:2311.16402 [hep-ph]

    2024 arXiv

  54. [62]

    Mukherjee, V

    S. Mukherjee, V. V. Skokov, A. Tarasov, and S. Tiwari, Perturbative Corrections to Quark TMDPDFs in the Background- Field Method: Gauge Invariance, Equations of Motion, and Multiple Interactions, (2025), arXiv:2502.15889 [hep-ph]

    2025 arXiv

  55. [63]

    Mandelstam, Light Cone Superspace and the Ultraviolet Finiteness of the N=4 Model, Nucl

    S. Mandelstam, Light Cone Superspace and the Ultraviolet Finiteness of the N=4 Model, Nucl. Phys. B213, 149 (1983)

    1983

  56. [64]

    Leibbrandt, The Light Cone Gauge in Yang-Mills Theory, Phys

    G. Leibbrandt, The Light Cone Gauge in Yang-Mills Theory, Phys. Rev. D29, 1699 (1984)

    1984

  57. [65]

    M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita, and H. X. Zhu, Subleading power rapidity divergences and power corrections for qT, JHEP04, 123, arXiv:1812.08189 [hep-ph]

    arXiv

  58. [66]

    A. V. Belitsky, X. Ji, and F. Yuan, Final state interactions and gauge invariant parton distributions, Nucl. Phys. B656, 165 (2003), arXiv:hep-ph/0208038

    2003 arXiv

  59. [67]

    Bassetto, M

    A. Bassetto, M. Dalbosco, I. Lazzizzera, and R. Soldati, Yang-Mills Theories in the Light Cone Gauge, Phys. Rev. D31, 2012 (1985)

    1985

  60. [68]

    Collins, Rapidity divergences and valid definitions of parton densities, PoSLC2008, 028 (2008), arXiv:0808.2665 [hep- ph]

    J. Collins, Rapidity divergences and valid definitions of parton densities, PoSLC2008, 028 (2008), arXiv:0808.2665 [hep- ph]

    2008 arXiv

  61. [69]

    Bassetto, M

    A. Bassetto, M. Dalbosco, and R. Soldati, Renormalization of the Yang-Mills Theories in the Light Cone Gauge, Phys. Rev. D36, 3138 (1987)

    1987

  62. [70]

    Bassetto, M

    A. Bassetto, M. Dalbosco, and R. Soldati, One Loop Renormalization of the Yang-Mills Theory With Dirac Fermions in the Light Cone Gauge, Phys. Rev. D33, 617 (1986)

    1986

  63. [71]

    Leibbrandt and S.-L

    G. Leibbrandt and S.-L. Nyeo, Generalized Ward Identity for the Quark Selfenergy and the Quark Quark - Gluon Vertex in the Light Cone Gauge, Phys. Lett. B140, 417 (1984)

    1984

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