- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

DIS dipole picture cross section at finite energy

Magnus Bertilsson, Tuomas Lappi, Heikki Mäntysaari, and Xuan-Bo Tong

PDF

Phys. Rev. D 114, 054030 – Published 15 September, 2026

DOI: https://doi.org/10.1103/b9d2-h4w7

Abstract

We implement a finite-energy constraint in the dipole picture of deep inelastic scattering, by restricting the invariant mass of the produced partonic system by the virtual photon-target center of mass energy. We show that, for Q2=1  GeV2, the effect of this constraint can reach up to ∼35% for charm quarks and ∼7% for light quarks at x=0.01 but then rapidly decreases at smaller x or larger Q2.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (59)

  1. E. D. Bloom et al., High-energy inelastic e p scattering at 6-degrees and 10-degrees, Phys. Rev. Lett. 23, 930 (1969).
  2. L. V. Gribov, E. M. Levin, and M. G. Ryskin, Semihard processes in QCD, Phys. Rep. 100, 1 (1983).
  3. L. D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei at small transverse momentum, Phys. Rev. D 49, 3352 (1994).
  4. L. D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phys. Rev. D 49, 2233 (1994).
  5. E. Iancu and R. Venugopalan, The Color Glass Condensate and High-Energy Scattering in QCD (World Scientific, Singapore, 2003), pp. 249–3363, 10.1142/9789812795533_0005.
  6. O. Garcia-Montero and S. Schlichting, Effective theories for nuclei at high energies, Eur. Phys. J. A 61, 54 (2025).
  7. A. Morreale and F. Salazar, Mining for gluon saturation at colliders, Universe 7, 312 (2021).
  8. R. Abdul Khalek et al., Science requirements and detector concepts for the electron-ion collider: EIC yellow report, Nucl. Phys. A1026, 122447 (2022).
  9. V. Andreev et al. (H1 Collaboration), Measurement of inclusive ep cross sections at high Q2 at s=225 and 252 GeV and of the longitudinal proton structure function FL at HERA, Eur. Phys. J. C 74, 2814 (2014).
  10. H. Abramowicz et al. (ZEUS Collaboration), Deep inelastic cross-section measurements at large y with the ZEUS detector at HERA, Phys. Rev. D 90, 072002 (2014).
  11. H. Abramowicz et al. (H1 and ZEUS Collaborations), Combination of measurements of inclusive deep inelastic e±p scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75, 580 (2015).
  12. H. Abramowicz et al. (H1 and ZEUS Collaborations), Combination and QCD analysis of charm and beauty production cross-section measurements in deep inelastic ep scattering at HERA, Eur. Phys. J. C 78, 473 (2018).
  13. J. L. Albacete, N. Armesto, J. G. Milhano, and C. A. Salgado, Non-linear QCD meets data: A Global analysis of lepton-proton scattering with running coupling BK evolution, Phys. Rev. D 80, 034031 (2009).
  14. J. L. Albacete, N. Armesto, J. G. Milhano, P. Quiroga-Arias, and C. A. Salgado, AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks, Eur. Phys. J. C 71, 1705 (2011).
  15. C. Casuga, M. Karhunen, and H. Mäntysaari, Inferring the initial condition for the Balitsky-Kovchegov equation, Phys. Rev. D 109, 054018 (2024).
  16. M. Gao, Z.-B. Kang, J. Penttala, and D. Y. Shao, Determination of the initial condition for the Balitsky-Kovchegov equation with transformers, J. High Energy Phys. 03 (2026) 034.
  17. H. Mäntysaari and B. Schenke, Confronting impact parameter dependent JIMWLK evolution with HERA data, Phys. Rev. D 98, 034013 (2018).
  18. G. Beuf, H. Hänninen, T. Lappi, and H. Mäntysaari, Color glass condensate at next-to-leading order meets HERA data, Phys. Rev. D 102, 074028 (2020).
  19. H. Hänninen, A. Kykkänen, and H. Schlüter, Reconstruction of the dipole amplitude in the dipole picture as a mathematical inverse problem, Phys. Rev. D 112, 094026 (2025).
  20. H. Hänninen, H. Mäntysaari, R. Paatelainen, and J. Penttala, Proton structure functions at next-to-leading order in the dipole picture with massive quarks, Phys. Rev. Lett. 130, 192301 (2023).
  21. C. Casuga, H. Hänninen, and H. Mäntysaari, Initial condition for the Balitsky-Kovchegov equation at next-to-leading order, Phys. Rev. D 112, 034003 (2025).
  22. H. Mäntysaari and P. Zurita, In depth analysis of the combined HERA data in the dipole models with and without saturation, Phys. Rev. D 98, 036002 (2018).
  23. N. N. Nikolaev and B. G. Zakharov, Color transparency and scaling properties of nuclear shadowing in deep inelastic scattering, Z. Phys. C 49, 607 (1991).
  24. N. Nikolaev and B. G. Zakharov, Pomeron structure function and diffraction dissociation of virtual photons in perturbative QCD, Z. Phys. C 53, 331 (1992).
  25. A. H. Mueller and B. Patel, Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes, Nucl. Phys. B425, 471 (1994).
  26. G. Beuf, Dipole factorization for DIS at NLO: Combining the qq¯ and qq¯g contributions, Phys. Rev. D 96, 074033 (2017).
  27. H. Hänninen, T. Lappi, and R. Paatelainen, One-loop corrections to light cone wave functions: The dipole picture DIS cross section, Ann. Phys. (Amsterdam) 393, 358 (2018).
  28. G. Beuf, T. Lappi, and R. Paatelainen, Massive quarks at one loop in the dipole picture of deep inelastic scattering, Phys. Rev. Lett. 129, 072001 (2022).
  29. G. Beuf, T. Lappi, and R. Paatelainen, Massive quarks in NLO dipole factorization for DIS: Transverse photon, Phys. Rev. D 106, 034013 (2022).
  30. G. Beuf, T. Lappi, and R. Paatelainen, Massive quarks in NLO dipole factorization for DIS: Longitudinal photon, Phys. Rev. D 104, 056032 (2021).
  31. J. Kuokkanen, K. Rummukainen, and H. Weigert, HERA-data in the light of small x evolution with state of the art NLO input, Nucl. Phys. A875, 29 (2012).
  32. G. Beuf, Improving the kinematics for low-x QCD evolution equations in coordinate space, Phys. Rev. D 89, 074039 (2014).
  33. K. Watanabe, B.-W. Xiao, F. Yuan, and D. Zaslavsky, Implementing the exact kinematical constraint in the saturation formalism, Phys. Rev. D 92, 034026 (2015).
  34. G. A. Chirilli, Sub-eikonal corrections to scattering amplitudes at high energy, J. High Energy Phys. 01 (2019) 118.
  35. G. A. Chirilli, High-energy operator product expansion at sub-eikonal level, J. High Energy Phys. 06 (2021) 096.
  36. P. Agostini, T. Altinoluk, and N. Armesto, Next-to-eikonal corrections to dijet production in deep inelastic scattering in the dilute limit of the color glass condensate, J. High Energy Phys. 07 (2024) 137.
  37. 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. D 111, 014010 (2025).
  38. T. Altinoluk, G. Beuf, J. Favrel, and M. Fucilla, Next-to-leading order corrections to the next-to-eikonal DIS structure functions, J. High Energy Phys. 06 (2026) 124.
  39. P. Agostini, T. Altinoluk, N. Armesto, G. Beuf, F. Cougoulic, and S. Mulani, Dijet production in DIS off a large nucleus at next-to-eikonal accuracy in a Gaussian model within the CGC framework, Phys. Rev. D 113, 054035 (2026).
  40. G. A. Chirilli, B.-W. Xiao, and F. Yuan, One-loop factorization for inclusive hadron production in pA collisions in the saturation formalism, Phys. Rev. Lett. 108, 122301 (2012).
  41. H. Mäntysaari and J. Penttala, Exclusive heavy vector meson production at next-to-leading order in the dipole picture, Phys. Lett. B 823, 136723 (2021).
  42. G. Beuf, T. Lappi, H. Mäntysaari, R. Paatelainen, and J. Penttala, Diffractive deep inelastic scattering at NLO in the dipole picture, J. High Energy Phys. 05 (2024) 024.
  43. F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan, Universality of unintegrated gluon distributions at small x, Phys. Rev. D 83, 105005 (2011).
  44. Y. V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy (Oxford University Press, Cambridge, 2013), Vol. 33.
  45. H. Mäntysaari, J. Penttala, F. Salazar, and B. Schenke, Finite-size effects on small-x evolution and saturation in proton and nuclear targets, Phys. Rev. D 111, 054033 (2025).
  46. M. Bertilsson, Precision_probes_EIC, https://github.com/Magsnush/Precision_probes_EIC (2025).
  47. T. Lappi and H. Mäntysaari, Forward dihadron correlations in deuteron-gold collisions with the Gaussian approximation of JIMWLK, Nucl. Phys. A908, 51 (2013).
  48. Y. V. Kovchegov, Small x F2 structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60, 034008 (1999).
  49. I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B463, 99 (1996).
  50. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B504, 415 (1997).
  51. J. Jalilian-Marian, A. Kovner, and H. Weigert, The Wilson renormalization group for low x physics: Gluon evolution at finite parton density, Phys. Rev. D 59, 014015 (1998).
  52. A. Kovner, J. G. Milhano, and H. Weigert, Relating different approaches to nonlinear QCD evolution at finite gluon density, Phys. Rev. D 62, 114005 (2000).
  53. E. Iancu, A. Leonidov, and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1, Nucl. Phys. A692, 583 (2001).
  54. A. H. Mueller, A simple derivation of the JIMWLK equation, Phys. Lett. B 523, 243 (2001).
  55. A. Dumitru, J. Jalilian-Marian, T. Lappi, B. Schenke, and R. Venugopalan, Renormalization group evolution of multi-gluon correlators in high energy QCD, Phys. Lett. B 706, 219 (2011).
  56. Y. V. Kovchegov, J. Kuokkanen, K. Rummukainen, and H. Weigert, Subleading-Nc corrections in non-linear small-x evolution, Nucl. Phys. A823, 47 (2009).
  57. T. Lappi and H. Mäntysaari, Single inclusive particle production at high energy from HERA data to proton-nucleus collisions, Phys. Rev. D 88, 114020 (2013).
  58. See Supplemental Material at http://link.aps.org/supplemental/10.1103/b9d2-h4w7 for relative difference of structure functions for both light and charm quarks.
  59. R. Boussarie and Y. Mehtar-Tani, A novel formulation of the unintegrated gluon distribution for DIS, Phys. Lett. B 831, 137125 (2022).

Outline

Information

Phys. Rev. D 114, 054030– Published 15 September, 2026
  • Received 3 February 2026
  • Accepted 13 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation