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Eikonal spin-dependent odderon and gluon Sivers function of a proton and its small-x evolution

Sanjin Benić1, Adrian Dumitru2,3, Florian Hechenberger4, and Tomasz Stebel5

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Phys. Rev. D 114, 034006 – Published 4 August, 2026

DOI: https://doi.org/10.1103/wwmj-yxtg

Abstract

The matrix element in the proton of the eikonal Odderon operator, with a helicity flip, has been shown to correspond to the dipole gluon Sivers function. We employ a three quark light-front model of the proton to determine the Sivers function at moderately small x0∼0.1 and transverse momentum k⊥≲1  GeV. The model light-cone (LC) wave function predicts the properties of xf1T⊥g(x,k⊥) such as its overall magnitude, the position of its peak in k⊥, and its behavior at small k⊥. We then compute numerically the Balitsky-Fadin-Kuraev-Lipatov anomalous dimension characterizing the power-law tail at k⊥≳1.5  GeV of the gluon Sivers function at small (but preasymptotic) LC momentum fractions x∼x0e−1/αs: xf1T⊥g(x,k⊥)∼k⊥−3.3.

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References (64)

  1. D. W. Sivers, Single spin production asymmetries from the hard scattering of point-like constituents, Phys. Rev. D 41, 83 (1990).
  2. R. Boussarie et al., TMD handbook, arXiv:2304.03302.
  3. D. Boer, C. Lorcé, C. Pisano, and J. Zhou, The gluon sivers distribution: Status and future prospects, Adv. High Energy Phys. 2015, 371396 (2015).
  4. A. Accardi et al., Electron Ion Collider: The next QCD frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52, 268 (2016).
  5. R. Abdul Khalek et al., Science requirements and detector concepts for the electron-ion collider: EIC Yellow Report, Nucl. Phys. A1026, 122447 (2022).
  6. D. Boer, M. G. Echevarria, P. Mulders, and J. Zhou, Single spin asymmetries from a single Wilson loop, Phys. Rev. Lett. 116, 122001 (2016).
  7. J. Zhou, Transverse single spin asymmetries at small x and the anomalous magnetic moment, Phys. Rev. D 89, 074050 (2014).
  8. X. Yao, Y. Hagiwara, and Y. Hatta, Computing the gluon Sivers function at small-x, Phys. Lett. B 790, 361 (2019).
  9. R. Boussarie, Y. Hatta, L. Szymanowski, and S. Wallon, Probing the gluon Sivers function with an unpolarized target: GTMD distributions and the odderons, Phys. Rev. Lett. 124, 172501 (2020).
  10. H. Dong, D.-X. Zheng, and J. Zhou, Sea quark sivers distribution, Phys. Lett. B 788, 401 (2019).
  11. Y. V. Kovchegov and M. G. Santiago, Quark sivers function at small x: Spin-dependent odderon and the sub-eikonal evolution, J. High Energy Phys. 11 (2021) 200; 09 (2022) 186(E).
  12. Y. V. Kovchegov and M. G. Santiago, T-odd leading-twist quark TMDs at small x, J. High Energy Phys. 11 (2022) 098.
  13. F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan, Universality of unintegrated gluon distributions at small x, Phys. Rev. D 83, 105005 (2011).
  14. B. Pasquini, S. Cazzaniga, and S. Boffi, Transverse momentum dependent parton distributions in a light-cone quark model, Phys. Rev. D 78, 034025 (2008).
  15. M. G. Ryskin, Odderon and polarization phenomena in QCD, Sov. J. Nucl. Phys. 46, 337 (1987).
  16. B. G. Zakharov, Quark—diquark model of nucleon and properties of Pomeron and Odderon NN scattering amplitudes in QCD Born approximation. (In Russian), Sov. J. Nucl. Phys. 49, 860 (1989).
  17. A. H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL pomeron, Nucl. Phys. B415, 373 (1994).
  18. S. Jeon and R. Venugopalan, A classical odderon in QCD at high energies, Phys. Rev. D 71, 125003 (2005).
  19. L. Zheng, E. C. Aschenauer, J. H. Lee, B.-W. Xiao, and Z.-B. Yin, Accessing the gluon sivers function at a future electron-ion collider, Phys. Rev. D 98, 034011 (2018).
  20. S. Benic, A. Dumitru, L. Motyka, and T. Stebel, Gluon Sivers function from forward exclusive χc1 photoproduction on unpolarized protons, Phys. Rev. D 111, 054008 (2025).
  21. J. P. Ma, Diffractive photoproduction of eta(c), Nucl. Phys. A727, 333 (2003).
  22. H. Mäntysaari, Y. Tawabutr, and X.-B. Tong, Nucleon energy correlators for the odderon, Phys. Rev. D 112, 114027 (2025).
  23. S. Bhattacharya, Z.-B. Kang, D. Padilla, and J. Penttala, Probing the Sivers asymmetry with transverse energy-energy correlators in the small-x regime, arXiv:2504.10475.
  24. Y. Hatta and O. V. Teryaev, Single spin asymmetry in e+p→e′+B↑+X, arXiv:2602.07682.
  25. C. Alexandrou et al., Precision QCD with the electron-ion collider, arXiv:2604.04765.
  26. B. Pasquini and F. Yuan, Sivers and Boer-Mulders functions in light-cone quark models, Phys. Rev. D 81, 114013 (2010).
  27. E. Iancu and R. Venugopalan, The color glass condensate and high-energy scattering in QCD, in Quark-Gluon Plasma 4, edited by R. C. Hwa and X.-N. Wang (World Scientific, Singapore, 2004), pp. 249–363.
  28. Y. V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy (Oxford University Press, New York, 2013), Vol. 33.
  29. S. Benić and A. Dumitru, Off forward non-s-channel helicity conserving contributions to exclusive vector quarkonium production from the spin dependent BFKL pomeron, Phys. Rev. D 112, 034025 (2025).
  30. A. Dumitru, G. A. Miller, and R. Venugopalan, Extracting many-body color charge correlators in the proton from exclusive DIS at large bjorken x, Phys. Rev. D 98, 094004 (2018).
  31. M. Diehl and P. Hagler, Spin densities in the transverse plane and generalized transversity distributions, Eur. Phys. J. C 44, 87 (2005).
  32. Z.-B. Kang and J.-W. Qiu, Single transverse-spin asymmetry for D-meson production in semi-inclusive deep inelastic scattering, Phys. Rev. D 78, 034005 (2008).
  33. A. Dumitru and T. Stebel, Multiquark matrix elements in the proton and three gluon exchange for exclusive ηc production in photon-proton diffractive scattering, Phys. Rev. D 99, 094038 (2019).
  34. A. Dumitru, V. Skokov, and T. Stebel, Subfemtometer scale color charge correlations in the proton, Phys. Rev. D 101, 054004 (2020).
  35. C. Lorce, B. Pasquini, X. Xiong, and F. Yuan, The quark orbital angular momentum from Wigner distributions and light-cone wave functions, Phys. Rev. D 85, 114006 (2012).
  36. A. Dumitru and R. Paatelainen, Sub-femtometer scale color charge fluctuations in a proton made of three quarks and a gluon, Phys. Rev. D 103, 034026 (2021); 108, 059901(E) (2023).
  37. A. Dumitru, H. Mäntysaari, and R. Paatelainen, Cubic color charge correlator in a proton made of three quarks and a gluon, Phys. Rev. D 105, 036007 (2022); 109, 119901(E) (2024).
  38. Y. V. Kovchegov, L. Szymanowski, and S. Wallon, Perturbative odderon in the dipole model, Phys. Lett. B 586, 267 (2004).
  39. Y. Hatta, E. Iancu, K. Itakura, and L. McLerran, Odderon in the color glass condensate, Nucl. Phys. A760, 172 (2005).
  40. L. N. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Sov. J. Nucl. Phys. 23, 338 (1976).
  41. E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, The Pomeranchuk singularity in non-Abelian gauge theories, Sov. Phys. JETP 45, 199 (1977).
  42. I. I. Balitsky and L. N. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28, 822 (1978).
  43. L. Motyka, Nonlinear evolution of Pomeron and odderon in momentum space, Phys. Lett. B 637, 185 (2006).
  44. S. Benić, A. Dumitru, A. Kaushik, L. Motyka, and T. Stebel, Photon-Odderon interference in exclusive χc charmonium production at the electron-ion collider, Phys. Rev. D 110, 014025 (2024).
  45. A. Dumitru, H. Mäntysaari, and R. Paatelainen, High-energy dipole scattering amplitude from evolution of low-energy proton light-cone wave functions, Phys. Rev. D 107, 114024 (2023).
  46. J. Bartels, High-energy behavior in a non-Abelian gauge theory (II): First corrections to Tn→m beyond the leading lns approximation, Nucl. Phys. B175, 365 (1980).
  47. J. Kwiecinski and M. Praszalowicz, Three gluon integral equation and odd c singlet regge singularities in QCD, Phys. Lett. 94B, 413 (1980).
  48. U. D’Alesio, F. Murgia, and C. Pisano, Towards a first estimate of the gluon Sivers function from AN data in pp collisions at RHIC, J. High Energy Phys. 09 (2015) 119.
  49. A. Adare et al. (PHENIX Collaboration), Measurement of transverse-single-spin asymmetries for midrapidity and forward-rapidity production of hadrons in polarized p+p collisions at s=200 and 62.4 GeV, Phys. Rev. D 90, 012006 (2014).
  50. M. Burkardt, Sivers mechanism for gluons, Phys. Rev. D 69, 091501 (2004).
  51. E. Iancu, A. H. Mueller, D. N. Triantafyllopoulos, and S. Y. Wei, Gluon dipole factorisation for diffractive dijets, J. High Energy Phys. 10 (2022) 103.
  52. S. Hauksson, E. Iancu, A. H. Mueller, D. N. Triantafyllopoulos, and S. Y. Wei, TMD factorisation for diffractive jets in photon-nucleus interactions, J. High Energy Phys. 06 (2024) 180.
  53. L. Zheng, E. C. Aschenauer, J. H. Lee, and B.-W. Xiao, Probing gluon saturation through dihadron correlations at an electron-ion collider, Phys. Rev. D 89, 074037 (2014).
  54. D. Boer, Y. Hagiwara, J. Zhou, and Y.-j. Zhou, Scale evolution of T-odd gluon TMDs at small x, Phys. Rev. D 105, 096017 (2022).
  55. K. Cassar, Z. Wang, X. Chu, and E.-C. Aschenauer, Investigating the broadening phenomenon in two-particle correlations induced by gluon saturation, Phys. Rev. D 112, 034034 (2025).
  56. P. Caucal, Z.-B. Kang, P. Korcyl, F. Salazar, B. Schenke, T. Stebel, R. Venugopalan, and W. Zhao, Probing gluon saturation with forward di-hadron correlations in proton-nucleus collisions, Phys. Lett. B 879, 140599 (2026).
  57. G. Beuf, Improving the kinematics for low-x QCD evolution equations in coordinate space, Phys. Rev. D 89, 074039 (2014).
  58. P. Caucal, F. Salazar, B. Schenke, and R. Venugopalan, Back-to-back inclusive dijets in DIS at small x: Sudakov suppression and gluon saturation at NLO, J. High Energy Phys. 11 (2022) 169.
  59. A. Dumitru, H. Mäntysaari, and R. Paatelainen, Stronger C-odd color charge correlations in the proton at higher energy, Phys. Rev. D 107, L011501 (2023); 111, 059902(E) (2025).
  60. F. Hechenberger, The eikonal spin-dependent odderon and gluon sivers function of a proton, and its small-x evolution, Zenodo, (2026) 10.5281/zenodo.18703045.
  61. F. Hechenberger, Nucleon-lccqm, GitHub, https://github.com/HechenMountain/nucleon-lccqm (2026).
  62. F. Schlumpf, Relativistic constituent quark model of electroweak properties of baryons, Phys. Rev. D 47, 4114 (1993); 49, 6246(E) (1994).
  63. S. J. Brodsky and F. Schlumpf, Wave function independent relations between the nucleon axial coupling g(A) and the nucleon magnetic moments, Phys. Lett. B 329, 111 (1994).
  64. Z. Ye, J. Arrington, R. J. Hill, and G. Lee, Proton and neutron electromagnetic form factors and uncertainties, Phys. Lett. B 777, 8 (2018).

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Phys. Rev. D 114, 034006– Published 4 August, 2026
  • Received 30 March 2026
  • Accepted 8 July 2026
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