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Directed flow from parton spin-orbit coupling in $pp$ and $pA$ collisions

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Pith's one-line read A new source of directed flow in forward proton-proton and proton-nucleus collisions arises from parton spin-orbit coupling and directly probes the proton's double helicity PDFs.

desk verdict A plausible new mechanism for forward directed flow from gluon spin-orbit coupling and double helicity PDFs; the quantitative claim leans on one imported relation that the authors should be asked to back up. read the letter →

arxiv 2505.05172 v2 pith:6AR3ID3U submitted 2025-05-08 hep-ph hep-exnucl-exnucl-th

Sanjin Benić , Yoshitaka Hatta This is my paper · ORCID
classification hep-phhep-exnucl-exnucl-th PACS 12.38.-t13.85.-t
keywords directedflowdoublepartonscatteringhelicityPDFgluonspin-orbitcouplingsmall-xQCDcolorglasscondensateforwardrapidityazimuthalcorrelations
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

The paper claims that forward-rapidity proton-proton and proton-nucleus collisions contain a previously overlooked source of the two-particle azimuthal correlation called directed flow ($v_1$). The source is the square of the helicity-dependent term in the elementary quark-gluon scattering amplitude, which becomes sizable because gluons at small $x$ have a strong spin-orbit correlation. If the derivation is right, measuring this correlation gives a direct handle on the proton's double helicity parton distribution $F_{\Delta q\Delta q}$ and on the small-$x$ gluon spin-orbit coupling, quantities that are otherwise hard to isolate. The predicted signal is a $v_1$ that falls roughly as $1/|P|$, which distinguishes it from hydrodynamic and odderon-based explanations.

What carries the argument

The machinery is the double parton scattering cross section built from the squared helicity-dependent quark-gluon amplitude, together with a single imported small-$x$ identity. The identity, Eq. (17), says that the gluon spin-orbit correlation $C_g(x,k,\Delta)$, scaled by the nucleon mass squared, is proportional to the dipole $S$-matrix $S(x,k,\Delta)$ with a known coefficient. The double helicity PDF $F_{\Delta q\Delta q}(x_1,x_2,\Delta)$ plays the role of the projectile-side weight; after a Gaussian model for its $\Delta$ dependence and a factorized ansatz for the target's $S$-matrix, the $\Delta$ integral collapses and leaves a compact expression for $v_1$ that depends mainly on the ratio $|F_{\Delta q\Delta q}|/F_{qq}$ and not on the detailed form of $S$.

What would settle it

A decisive check is to measure forward rapidity $v_1$ in $pp$ collisions after standard nonflow subtraction and compare its $|P|$ dependence with the predicted $1/|P|$ behavior: observing $v_1\propto|P|$ in the low-$|P|$ region, or a sharp peak near $|P|\sim0.2$ GeV of the odderon type, would rule out this mechanism as the dominant source.

Watch

Extended reading notes

Core claim

The central discovery is that a $\cos(\phi_1-\phi_2)$ two-particle correlation, interpreted as directed flow, appears in unpolarized $pp$ and $pA$ collisions at forward rapidity from the square of the helicity-flip term of the quark-gluon amplitude (Eq. 1). Combined with the small-$x$ relation $x C_g(x,k,\Delta)/M^2 \approx -N_c/(8\pi^4 \alpha_s)\,S(x,k,\Delta)$ (Eq. 17), the corresponding double parton scattering cross section (Eq. 18) becomes proportional to $F_{\Delta q\Delta q}(x_1,x_2,\Delta) S(x'_1,k_1,-\Delta) S(x'_2,k_2,\Delta)$. The sign of the gluon spin-orbit coupling is lost because the distribution is squared, but its magnitude survives in the size of the correlation. The ratio of this contribution to the unpolarized double parton cross section yields $v_1 \simeq (1/|P|)\sqrt{|F_{\Delta q\Delta q}|/F_{qq}}$ in the no-fragmentation limit, so forward $v_1$ is a direct probe of the double helicity PDF.

Load-bearing premise

The load-bearing premise is that the small-$x$ relation (17) imported from earlier work is correct in sign, magnitude, and kinematic range at $x'\sim10^{-2}$ to $10^{-4}$ and $|\Delta|\sim0.1$ to $1$ GeV, together with the factorization of the target's double-gluon distribution into single distributions (Eq. 9).

Editorial extensions

If this is right

  • Forward two-particle correlation data at high-energy hadron colliders already exist, so the predicted $v_1$ can be searched for in reanalyzed samples without new instrumentation.
  • If confirmed, the sign of $v_1$ in a chosen channel such as $uu\to\pi^+\pi^+$ fixes the sign of $F_{\Delta u\Delta u}$ and helps settle conflicting model predictions.
  • The predicted $1/|P|$ falloff and the $A^{-1/3}$ nuclear-size dependence separate this mechanism from hydrodynamic ($v_1\propto|P|$) and odderon (low-$|P|$ peak) interpretations.
  • At the partonic level, $v_1$ is nearly independent of the dipole $S$-matrix; the only sizable model dependence enters through fragmentation functions.
  • In the maximally negative scenario $F_{\Delta u\Delta u}=-F_{uu}$, the numerical estimates put $|v_1|$ at the percent level for $|P|\sim1$ GeV at both collider energies considered, which is in the measurable range.

Reading between the lines

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

  • If the same mechanism is evaluated with a quasi-real photon as the projectile, the correlation should appear in photoproduction-like final states, offering an independent test at future lepton-hadron colliders.
  • Because the sign of the gluon spin-orbit coupling is lost when the amplitude is squared, measuring this observable alone cannot determine whether gluon helicity and orbital angular momentum are aligned or anti-aligned; a polarized-beam version would be needed to recover that sign.
  • The neglected color-octet double PDF could contaminate the $S^2$ form with a color-quadrupole term; a model or lattice estimate of that octet distribution would show how safe the large-$N_c$ truncation is.
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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 proposes a new mechanism for directed flow (a cos(phi1-phi2) two-particle azimuthal correlation) in forward proton-proton and proton-nucleus collisions. The mechanism is double parton scattering in which the square of the helicity-dependent quark-gluon amplitude (1) is combined with the small-x gluon spin-orbit relation (17). The resulting cross section (18) is proportional to the proton's double helicity parton distribution F_Delta q Delta q times the square of the dipole S-matrix, and the ratio of (18) to the unpolarized DPS cross section gives v1 ~ (1/|P|) sqrt(|F_Delta q Delta q|/F_qq) with an A^{-1/3} target dependence. Numerical estimates are presented for LHC and RHIC kinematics using the GBW and MVe dipole models, with the maximally negative double-helicity PDF ansatz (26).

Significance. If the central relation (17) is valid in the kinematics used, the paper identifies a genuinely new, experimentally testable source of directed flow in cold QCD collisions, with distinctive 1/|P| and A^{-1/3} scalings that are falsifiable with LHCb forward data. The derivation is transparent about its main approximations: the color-octet DPDF is neglected in (10), the double-gluon distribution is factorized in (9), and the Gaussian and factorized-Delta models are introduced in (22) and (23). The paper also correctly emphasizes that the observable can probe the poorly known double helicity DPDF, whose sign can in principle be read off from the sign of v1. The central weakness is the complete dependence of the numerical predictions on the imported small-x spin-orbit relation (17), whose quantitative validity at the k and Delta values of Figures 1 and 2 is not established in the manuscript.

major comments (2)
  1. [II, Eqs. (17)-(18)] The v1 signal is built on the imported relation x C_g(x,k,Delta)/M^2 ~ -N_c/(8 pi^4 alpha_s) S(x,k,Delta). The paper states only 'valid when x << 1', but the numerical applications involve x' ~ 10^-2 to 10^-4, k ~ 0.5-3 GeV and |Delta| ~ 0.1-1 GeV. Since C_g enters squared in Eq. (18), a moderate error in the coefficient shifts v1 by the same relative amount, and a breakdown of the relation away from k=Delta=0 would remove the predicted cos(phi1-phi2) correlation entirely. The authors should provide an independent derivation of (17), or at least a quantitative estimate of the O(k^2/M^2), O(Delta^2/M^2) and subleading-x corrections, with an explicit statement of the range of (k, Delta, x') over which the relation is accurate for the kinematics of Figures 1 and 2.
  2. [III, Eqs. (24)-(25)] The Gaussian integration as written does not reproduce the quoted coefficient. With F_Delta q Delta q proportional to exp(-Delta^2/(2 sigma^2)) and S(x',k,Delta) proportional to exp(-R_A^2 Delta^2/4), the Delta-integrals in Eq. (20) yield the inverse of beta = 1/(2 sigma^2) + R_A^2/2, not 2/(sigma^2 + R_A^2). The correct analytic expression for v1 is v1 ~ 1/(2|P|) sqrt(2 sigma^2/(1 + sigma^2 R_A^2)), which for sigma = 0.26 GeV and R_A = 1.2 fm (A = 1) gives approximately 0.098/|P|; the formula displayed in Eq. (25) gives a different value. The equations should be corrected, and the unit convention relating sigma (GeV) and R_A (fm) should be stated explicitly, because all numerical results in Figures 1 and 2 are generated from Eq. (24).
minor comments (5)
  1. [III, Eq. (26)] The plotted magnitudes in Figures 1 and 2 are upper limits obtained by saturating the positivity bound with r = -1. The text does mention this, but the abstract and introduction should more prominently state that the central quantitative claim is conditional on this model input, given that the literature quoted in the same paragraph reports both negative and positive values of r.
  2. [II, Eq. (15)] The ellipsis in Eq. (15) hides the subleading terms in the expansion of the helicity GTMD; a brief statement about the expected size of these terms would help the reader judge the validity of the subsequent substitution.
  3. [II, Eq. (6) and Eq. (18)] The factor of 1/2 for identical partons is stated after Eq. (6) but is not explicitly carried through Eq. (18) or the ratio (20); the authors should state whether both the numerator and denominator of the v1 ratio include the identical-particle symmetry factor, since the cancellation in the ratio is not automatic.
  4. [III, Figures 1 and 2] The comparison of v1 magnitudes between LHC and RHIC is made with different yref and Pref values; the text should note that these are different kinematic reference choices and that an apples-to-apples comparison across energies would require identical reference selections.
  5. [III, text after Eq. (24)] The sentence 'Fig. 2 show the results' should read 'Fig. 2 shows the results'; there is also a missing comma after 'from [23]' in the sentence introducing Eq. (26).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a self-contained DPS calculation that uses a parameter-free prior small-x spin-orbit relation as an external input.

full rationale

The paper's central chain is not circular. The new cross-section term (12)-(18) is obtained by squaring the helicity-dependent piece of the amplitude (1) and following standard double-parton-scattering factorization; the key step, Eq. (17), is labeled 'a crucial use of the formula [8, 21]' and is a parameter-free small-x relation imported from prior work whose stated assumption (x << 1) does not include the target observable v1. The numerical estimates depend on explicitly acknowledged model inputs: the Gaussian width sigma = 0.26 GeV, the maximally negative scenario F_Delta u Delta u = -F_uu (Eq. 26), and the GBW/MVe dipole models; none of these is fitted to the predicted v1, and the paper presents the result as an upper limit ('max scenario'). The v1 ~ 1/|P| shape and A^{-1/3} scaling follow analytically from (24)-(25), not from a fitted ansatz. The claim that v1 probes F_Delta q Delta q is a factorization statement, not a definitional identity, since F_Delta q Delta q is an independent matrix element. The footnotes flag neglected color-octet and <SS>-correlation contributions as model uncertainties, which are correctness risks rather than circularity. Therefore no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central v1 formula (20)/(24) rests on the small-x relation (17), the factorization approximations (9), (22), (23), and the ad hoc maximal DPDF input (26). None of these are new entities; they are QCD approximations and model choices. The free parameters sigma, R_A, and the ratio r set the numerical scale of the prediction. Without independent determination of F_Delta q Delta q, the magnitude of v1 is an upper-bound estimate.

free parameters (3)
  • sigma (Gaussian width of double PDFs) = 0.26 GeV
    Introduced in Eq. (22) as the common transverse width of Fqq and FDeltaqDeltaq, taken from Ref. [25]. It sets the overall prefactor in the v1 estimate (25): 0.098 = 1/(4 sqrt(1/sigma^2 + R_A^2)) for A=1 with no fragmentation.
  • R_A (target radius) = 1.2 A^{1/3} fm
    Standard nuclear radius used in the Delta-dependence of the dipole S-matrix (Eq. 23). It controls the A^{-1/3} scaling of v1 and the numerical prefactor in (24)-(25).
  • r = F_Delta u Delta u / F_uu = -1 (maximal negative scenario)
    The double helicity DPDF is essentially unconstrained; the paper assumes saturation of the positivity bound with negative sign in Eq. (26) to obtain an upper limit. All plotted v1 values scale as sqrt(|r|); changing r to -0.1 reduces v1 by roughly one third.
assumptions (6)
  • domain assumption Small-x spin-orbit relation: x C_g(x,k,Delta)/M^2 approximately -N_c/(8 pi^4 alpha_s) S(x,k,Delta)
    Imported from Refs. [8,21] and used at Eq. (17) to convert the helicity gluon GTMD into the dipole S-matrix. This is the key step that produces the S^2 structure of Eq. (18). Its validity at the relevant x' and Delta is assumed.
  • domain assumption Saturation of the target double-gluon distribution by single-nucleon intermediate states
    Used in Eq. (9) to factorize the double gluon distribution into a product of single gluon GTMDs, giving the factorized S(x1')S(x2') form. This is equivalent to a large-Nc-like approximation and is not controlled beyond the stated assumption.
  • ad hoc to paper Gaussian model for the Delta-dependence of double PDFs (Eq. 22)
    Assumes Fqq and FDeltaqDeltaq have the same Gaussian width sigma in impact parameter space; no derivation is given, and the width could differ between the two distributions.
  • ad hoc to paper Factorized Delta-dependence of the dipole S-matrix (Eq. 23)
    Assumes S(x',k,Delta) = S(x',k) exp(-R_A^2 Delta^2/4), which allows the Delta integral to be performed analytically in (24). The factorization is a modeling choice.
  • domain assumption Neglect of the color-octet double parton distribution (Eq. 10)
    The color-singlet Fqq term is kept and the octet F^8_qq is dropped based on large-Nc counting and the inequality Fqq >= |F^8_qq| [36]. The paper notes this but does not quantify the correction.
  • ad hoc to paper Maximal negative double helicity PDF: F_Delta u Delta u = -F_uu (Eq. 26)
    The sign and magnitude of FDeltaqDeltaq are unknown; the paper adopts saturation of the positivity bound with negative sign to estimate the maximum size of v1. This assumption fixes the normalization of all numerical predictions.

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Pith. "Pith review of Directed flow from parton spin-orbit coupling in $pp$ and $pA$ collisions." pith.science (2026). https://pith.science/paper/6AR3ID3U

@misc{pith2026250505172,
  author       = {Pith},
  title        = {Pith review of: Directed flow from parton spin-orbit coupling in $pp$ and $pA$ collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AR3ID3U}},
  note         = {Machine review of arXiv:2505.05172}
}
abstract

We point out a novel mechanism to generate $\cos \phi$ two-particle azimuthal correlation (`directed flow') in unpolarized proton-proton and proton-nucleus collisions in the forward rapidity region of the projectile proton. This is a direct consequence of the recently discovered strong spin-orbit coupling in gluons at small-$x$. The observable simultaneously serves as a unique probe into the double helicity parton distribution functions of the proton.

Figures

Figures reproduced from arXiv: 2505.05172 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Reference graph

Works this paper leans on

59 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [21]

    Directed flow from parton spin-orbit coupling in $pp$ and $pA$ collisions

    (see also [22]). We will contrast our finding with this reference. As is already implied in the above argument, our observable simultaneously probes the double helicity parton distribution function (PDF) F∆q∆q∼⟨Sq zSq z⟩∼⟨ (¯qγ+γ5q)2⟩ of the projectile proton which is nonvanishing even if the proton is unpolarized. In contrast to the double unpolarized PD...

    work page Pith review arXiv 2025

  2. [1]

    Mayer and H

    M. Mayer and H. Jensen, Elementary theory of nuclear shell structure (John Wiley & Sons, Inc., New York, 1955). 8

    work page 1955

  3. [2]

    Fermi, Nuovo Cim

    E. Fermi, Nuovo Cim. 10, 407 (1954)

    work page 1954

  4. [3]

    R. L. Varner, W. J. Thompson, T. L. McAbee, E. J. Ludwig, and T. B. Clegg, Phys. Rept. 201, 57 (1991)

    work page 1991

  5. [4]

    Lorce and B

    C. Lorce and B. Pasquini, Phys. Rev. D 84, 014015 (2011), 1106.0139

    arXiv 2011

  6. [5]

    Kanazawa, C

    K. Kanazawa, C. Lorc´ e, A. Metz, B. Pasquini, and M. Schlegel, Phys. Rev. D 90, 014028 (2014), 1403.5226

    arXiv 2014

  7. [6]

    Mukherjee, S

    A. Mukherjee, S. Nair, and V. K. Ojha, Phys. Rev. D 90, 014024 (2014), 1403.6233

    arXiv 2014

  8. [7]

    Quark spin-orbit correlations in the proton

    M. Engelhardt et al., PoS LA TTICE2021, 413 (2022), 2112.13464

    work page Pith review arXiv 2022

Show all 59 references
  1. [8]

    Bhattacharya, R

    S. Bhattacharya, R. Boussarie, and Y. Hatta, Phys. Lett. B 859, 139134 (2024), 2404.04208

    2024 arXiv

  2. [9]

    Hatta and J

    Y. Hatta and J. Schoenleber, JHEP 09, 154 (2024), 2404.18872

    2024 arXiv

  3. [10]

    Bhattacharya, K

    S. Bhattacharya, K. Cichy, M. Constantinou, X. Gao, A. Metz, J. Miller, S. Mukherjee, P. Petreczky, F. Steffens, and Y. Zhao, JHEP 01, 146 (2025), 2410.03539

    2025 arXiv

  4. [11]

    Hatta and J

    Y. Hatta and J. Montgomery, Phys. Rev. D 111, 014024 (2025), 2410.16082

    2025 arXiv

  5. [12]

    Agrawal and R

    S. Agrawal and R. Abir (2025), 2505.21048

    2025 arXiv

  6. [13]

    Bhattacharya, A

    S. Bhattacharya, A. Metz, and J. Zhou, Phys. Lett. B 771, 396 (2017), [Erratum: Phys.Lett.B 810, 135866 (2020)], 1702.04387

    2017 arXiv

  7. [14]

    Boussarie, Y

    R. Boussarie, Y. Hatta, B.-W. Xiao, and F. Yuan, Phys. Rev. D 98, 074015 (2018), 1807.08697

    2018 arXiv

  8. [15]

    M. G. Echevarria, P. A. Gutierrez Garcia, and I. Scimemi, Phys. Lett. B 840, 137881 (2023), 2208.00021

    2023 arXiv

  9. [16]

    Bhattacharya, D

    S. Bhattacharya, D. Zheng, and J. Zhou, Phys. Rev. Lett. 133, 051901 (2024), 2312.01309

    2024 arXiv

  10. [17]

    Bhattacharya, R

    S. Bhattacharya, R. Boussarie, and Y. Hatta, Phys. Rev. D 111, 034019 (2025), 2404.04209

    2025 arXiv

  11. [18]

    Teaney and L

    D. Teaney and L. Yan, Phys. Rev. C 83, 064904 (2011), 1010.1876

    2011 arXiv

  12. [19]

    Aad et al

    G. Aad et al. (ATLAS), Phys. Rev. C 90, 044906 (2014), 1409.1792

    2014 arXiv

  13. [20]

    Adamczyk et al

    L. Adamczyk et al. (STAR), Phys. Lett. B 747, 265 (2015), 1502.07652

    2015 arXiv

  14. [22]

    Soudi and A

    I. Soudi and A. Majumder, Phys. Rev. C 111, 024901 (2025), 2404.05287

    2025 arXiv

  15. [23]

    D. Boer, T. Van Daal, P. J. Mulders, and E. Petreska, JHEP 07, 140 (2018), 1805.05219

    2018 arXiv

  16. [24]

    Diehl, D

    M. Diehl, D. Ostermeier, and A. Schafer, JHEP 03, 089 (2012), [Erratum: JHEP 03, 001 (2016)], 1111.0910

    2012 arXiv

  17. [25]

    J. R. Gaunt and W. J. Stirling, JHEP 03, 005 (2010), 0910.4347

    2010 arXiv

  18. [26]

    Rinaldi, S

    M. Rinaldi, S. Scopetta, M. Traini, and V. Vento, JHEP 12, 028 (2014), 1409.1500

    2014 arXiv

  19. [27]

    Chang, A

    H.-M. Chang, A. V. Manohar, and W. J. Waalewijn, Phys. Rev. D 87, 034009 (2013), 1211.3132

    2013 arXiv

  20. [28]

    M. G. Echevarria, T. Kasemets, P. J. Mulders, and C. Pisano, JHEP 04, 034 (2015), 1501.07291

    2015 arXiv

  21. [29]

    Kasemets and M

    T. Kasemets and M. Diehl, JHEP 01, 121 (2013), 1210.5434

    2013 arXiv

  22. [30]

    Strikman and W

    M. Strikman and W. Vogelsang, Phys. Rev. D 83, 034029 (2011), 1009.6123

    2011 arXiv

  23. [31]

    Dumitru, A

    A. Dumitru, A. Hayashigaki, and J. Jalilian-Marian, Nucl. Phys. A 765, 464 (2006), hep-ph/0506308

    2006 arXiv

  24. [32]

    Lappi and H

    T. Lappi and H. Mantysaari, Nucl. Phys. A 908, 51 (2013), 1209.2853

    2013 arXiv

  25. [33]

    Stasto, B.-W

    A. Stasto, B.-W. Xiao, and F. Yuan, Phys. Lett. B 716, 430 (2012), 1109.1817

    2012 arXiv

  26. [34]

    leads to the cos 2( ϕP1−ϕP2) correlation [33, 41, 42]. More relevant to the present work is the cos( ϕP1−ϕP2) correlation [21] dσ∼ cos(ϕP1−ϕP2)Fqq⊗O(k1)⊗O(k2), (11) 1 It should be mentioned that correlations neglected in the approximation ⟨SS⟩≈ SS in (6) can give rise to nontr...

    2019

  27. [35]

    Hagiwara, Y

    Y. Hagiwara, Y. Hatta, B.-W. Xiao, and F. Yuan, Phys. Lett. B 771, 374 (2017), 1701.04254

    2017 arXiv

  28. [36]

    Hatta, B.-W

    Y. Hatta, B.-W. Xiao, and F. Yuan, Phys. Rev. Lett. 116, 202301 (2016), 1601.01585

    2016 arXiv

  29. [37]

    Mekhfi, Phys

    M. Mekhfi, Phys. Rev. D 32, 2380 (1985)

    1985

  30. [38]

    Kasemets and P

    T. Kasemets and P. J. Mulders, Phys. Rev. D 91, 014015 (2015), 1411.0726

    2015 arXiv

  31. [39]

    Hatta and A

    Y. Hatta and A. H. Mueller, Nucl. Phys. A 789, 285 (2007), hep-ph/0702023

    2007 arXiv

  32. [40]

    Dumitru, F

    A. Dumitru, F. Gelis, L. McLerran, and R. Venugopalan, Nucl. Phys. A 810, 91 (2008), 0804.3858

    2008 arXiv

  33. [41]

    Kovner and M

    A. Kovner and M. Lublinsky, Phys. Rev. D 83, 034017 (2011), 1012.3398

    2011 arXiv

  34. [42]

    Y. V. Kovchegov and D. E. Wertepny, Nucl. Phys. A 906, 50 (2013), 1212.1195

    2013 arXiv

  35. [43]

    Levin and A

    E. Levin and A. H. Rezaeian, Phys. Rev. D 84, 034031 (2011), 1105.3275

    2011 arXiv

  36. [44]

    Iancu and A

    E. Iancu and A. H. Rezaeian, Phys. Rev. D 95, 094003 (2017), 1702.03943

    2017 arXiv

  37. [45]

    Hagiwara, Y

    Y. Hagiwara, Y. Hatta, and T. Ueda, Phys. Rev. D 94, 094036 (2016), 1609.05773

    2016 arXiv

  38. [46]

    Dumitru, H

    A. Dumitru, H. M¨ antysaari, and R. Paatelainen, Phys. Rev. D 107, L011501 (2023), [Erratum: Phys.Rev.D 111, 059902 (2025)], 2210.05390

    2023 arXiv

  39. [47]

    Borghini, P

    N. Borghini, P. M. Dinh, and J.-Y. Ollitrault, Phys. Rev. C 62, 034902 (2000), nucl-th/0004026

    2000 arXiv

  40. [48]

    T.-C. Peng, Z. Hu, S. Nair, S. Xu, X. Liu, C. Mondal, X. Zhao, and J. P. Vary (2024), 2410.11574

    2024 arXiv

  41. [49]

    Diehl, T

    M. Diehl, T. Kasemets, and S. Keane, JHEP 05, 118 (2014), 1401.1233

    2014 arXiv

  42. [50]

    Golec-Biernat and S

    K. Golec-Biernat and S. Sapeta, JHEP 03, 102 (2018), 1711.11360

    2018 arXiv

  43. [51]

    Lappi and H

    T. Lappi and H. M¨ antysaari, Phys. Rev. D88, 114020 (2013), 1309.6963

    2013 arXiv

  44. [52]

    Balitsky, Nucl

    I. Balitsky, Nucl. Phys. B 463, 99 (1996), hep-ph/9509348

    1996 arXiv

  45. [53]

    Y. V. Kovchegov, Phys. Rev. D 60, 034008 (1999), hep-ph/9901281

    1999 arXiv

  46. [54]

    Salazar, B

    F. Salazar, B. Schenke, and A. Soto-Ontoso, Phys. Lett. B 827, 136952 (2022), 2112.04611

    2022 arXiv

  47. [55]

    Dulat, T.-J

    S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, Phys. Rev. D 93, 033006 (2016), 1506.07443

    2016 arXiv

  48. [56]

    J. J. Ethier, N. Sato, and W. Melnitchouk, Phys. Rev. Lett. 119, 132001 (2017), 1705.05889

    2017 arXiv

  49. [57]

    Aaij et al

    R. Aaij et al. (LHCb), Phys. Lett. B 762, 473 (2016), 1512.00439

    2016 arXiv

  50. [58]

    Aaij et al

    R. Aaij et al. (LHCb), Phys. Rev. C 109, 054908 (2024), 2311.09985

    2024

  51. [59]

    Y. Shi, L. Wang, S.-Y. Wei, B.-W. Xiao, and L. Zheng, Phys. Rev. D 103, 054017 (2021), 2008.03569

    2021 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.