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Probing nuclear structure with the Balitsky-Kovchegov equation in full impact-parameter dependence

J. Cepila, M. Matas, and M. Vaculciak*

  • *Contact author: matej.vaculciak@fjfi.cvut.cz

Phys. Rev. C 113, 055204 – Published 7 May, 2026

DOI: https://doi.org/10.1103/mdrn-42dv

Abstract

Building on the newly available solution of the Balitsky-Kovchegov (BK) equation with the full impact-parameter dependence, we extend the study of parton evolution from proton to nuclear targets. Since a key part of the scientific program for future experimental facilities such as the EIC is to study gluon dynamics and shed new light on the phenomenon of parton saturation, we present predictions for key processes, such as deep-inelastic scattering or the diffractive production of vector mesons, on a variety of nuclear targets. Besides the standard BK equation, we employ its linearized version to identify a promising channel to search for gluon saturation in the nuclear collisions. Furthermore, we implement a tetrahedral model of oxygen to search for deviations from the standard, isotropic, Woods-Saxon approach. In addition to the future colliders, the presented results are also of interest for the current studies of nuclear vector-meson production at the LHC.

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References (72)

  1. M. G. Ryskin, Diffractive J/ψ electroproduction in LLA QCD, Z. Phys. C 57, 89 (1993).
  2. S. J. Brodsky, L. Frankfurt, J. F. Gunion, A. H. Mueller, and M. Strikman, Diffractive leptoproduction of vector mesons in QCD, Phys. Rev. D 50, 3134 (1994).
  3. I. P. Ivanov, N. N. Nikolaev, and A. A. Savin, Diffractive vector meson production at HERA: From soft to hard QCD, Phys. Part. Nuclei 37, 1 (2006).
  4. F. D. Aaron et al., Combined measurement and QCD analysis of the inclusive e± p scattering cross sections at HERA, J. High Energy Phys. 01 (2010) 109.
  5. P. Newman and M. Wing, The hadronic final state at HERA, Rev. Mod. Phys. 86, 1037 (2014).
  6. A. J. Baltz et al., The physics of ultraperipheral collisions at the LHC, Phys. Rep. 458, 1 (2008).
  7. J. G. Contreras and J. D. Tapia Takaki, Ultra-peripheral heavy-ion collisions at the LHC, Int. J. Mod. Phys. A 30, 1542012 (2015).
  8. S. R. Klein and H. Mäntysaari, Imaging the nucleus with high-energy photons, Nat. Rev. Phys. 1, 662 (2019).
  9. S. Acharya et al., Energy dependence of coherent photonuclear production of J/ψ mesons in ultra-peripheral Pb-Pb collisions at sNN=5.02TeV, J. High Energy Phys. 10 (2023) 119.
  10. 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).
  11. P. Achenbach et al., The present and future of QCD, Nucl. Phys. A 1047, 122874 (2024).
  12. J. L. Abelleira Fernandez, et al., A large hadron electron collider at CERN: Report on the physics and design concepts for machine and detector, J. Phys. G 39, 075001 (2012).
  13. D. Bendova, J. Cepila, J. G. Contreras, and M. Matas, Photonuclear J/ψ production at the LHC: Proton-based versus nuclear dipole scattering amplitudes, Phys. Lett. B 817, 136306 (2021).
  14. H. Mäntysaari, H. Roch, F. Salazar, B. Schenke, C. Shen, and W. Zhao, Global Bayesian analysis of J/ψ photoproduction on proton and lead targets, Phys. Rev. D 113, 014038 (2026).
  15. V. P. Goncalves, L. Santana, and W. Schäfer, Investigating the inclusive D0 photoproduction in ultraperipheral PbPb collisions at the Large Hadron Collider, Phys. Lett. B 869, 139859 (2025).
  16. R. Abdul Khalek et al., Science requirements and detector concepts for the Electron-Ion Collider: EIC Yellow Report, Nucl. Phys. A 1026, 122447 (2022).
  17. N. Armesto, Nuclear shadowing, J. Phys. G 32, R367 (2006).
  18. L. Frankfurt, V. Guzey, and M. Strikman, Leading twist nuclear shadowing phenomena in hard processes with nuclei, Phys. Rep. 512, 255 (2012).
  19. L. V. Gribov, E. M. Levin, and M. G. Ryskin, Semihard processes in QCD, Phys. Rept. 100, 1 (1983).
  20. A. H. Mueller and J.-w. Qiu, Gluon recombination and shadowing at small values of x, Nucl. Phys. B 268, 427 (1986).
  21. I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463, 99 (1996).
  22. Y. V. Kovchegov, Small x F2 structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60, 034008 (1999).
  23. 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).
  24. J. Cepila, J. G. Contreras, and M. Matas, Collinearly improved kernel suppresses Coulomb tails in the impact-parameter dependent Balitsky-Kovchegov evolution, Phys. Rev. D 99, 051502 (2019).
  25. D. Bendova, J. Cepila, J. G. Contreras, and M. Matas, Solution to the Balitsky-Kovchegov equation with the collinearly improved kernel including impact-parameter dependence, Phys. Rev. D 100, 054015 (2019).
  26. J. Berger and A. M. Stasto, Small-x nonlinear evolution with impact parameter and the structure function data, Phys. Rev. D 84, 094022 (2011).
  27. J. Cepila, J. G. Contreras, and M. Vaculciak, Solutions to the Balitsky-Kovchegov equation including the dipole orientation, Phys. Lett. B 848, 138360 (2024).
  28. J. Cepila, J. G. Contreras, and M. Vaculciak, Exclusive quarkonium photoproduction: Predictions with the Balitsky-Kovchegov equation including the full impact-parameter dependence, Phys. Rev. D 111, 056002 (2025).
  29. J. Cepila, J. G. Contreras, and M. Matas, Predictions for nuclear structure functions from the impact-parameter dependent Balitsky-Kovchegov equation, Phys. Rev. C 102, 044318 (2020).
  30. Y.-A. Li, S. Zhang, and Y.-G. Ma, Signatures of α-clustering in O16 by using a multiphase transport model, Phys. Rev. C 102, 054907 (2020).
  31. D. Behera, N. Mallick, S. Tripathy, S. Prasad, A. N. Mishra, and R. Sahoo, Predictions on global properties in O+O collisions at the Large Hadron Collider using a multi-phase transport model, Eur. Phys. J. A 58, 175 (2022).
  32. M. Matas, Nuclear helper (2024), https://github.com/MatasMarek/nuclear_helper.
  33. E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, The Pomeranchuk singularity in non-Abelian gauge theories, Sov. Phys. JETP 45, 199 (1977).
  34. I. I. Balitsky and L. N. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28, 822 (1978).
  35. L. N. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Sov. J. Nucl. Phys. 23, 338 (1976).
  36. A. H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL pomeron, Nucl. Phys. B 415, 373 (1994).
  37. Y. V. Kovchegov, Unitarization of the BFKL pomeron on a nucleus, Phys. Rev. D 61, 074018 (2000).
  38. I. Balitsky, Quark contribution to the small-x evolution of color dipole, Phys. Rev. D 75, 014001 (2007).
  39. Y. V. Kovchegov and H. Weigert, Triumvirate of running couplings in small-x evolution, Nucl. Phys. A 784, 188 (2007).
  40. B. Ducloué, E. Iancu, A. H. Mueller, G. Soyez, and D. N. Triantafyllopoulos, Non-linear evolution in QCD at high-energy beyond leading order, J. High Energy Phys. 04 (2019) 081.
  41. D. Bendova, J. Cepila, V. P. Gonçalves, and C. R. Sena, Deeply virtual Compton scattering at the EIC and LHeC: A comparison among saturation approaches, Eur. Phys. J. C 82, 99 (2022).
  42. F. D. Aaron et al., Diffractive electroproduction of ρ and Φ mesons at HERA, J. High Energy Phys. 05 (2010) 032.
  43. S. Chekanov et al., Exclusive electroproduction of Φ mesons at HERA, Nucl. Phys. B 718, 3 (2005).
  44. V. Andreev et al., Measurement of exclusive π+π− and ρ0 meson photoproduction at HERA, Eur. Phys. J. C 80, 1189 (2020).
  45. A. M. Sirunyan et al., Measurement of exclusive ρ(770)0 photoproduction in ultraperipheral pPb collisions at sNN= 5.02 TeV, Eur. Phys. J. C 79, 702 (2019).
  46. J. Breitweg et al., Elastic and proton dissociative ρ0 photoproduction at HERA, Eur. Phys. J. C 2, 247 (1998).
  47. S. Chekanov et al., Exclusive ρ0 production in deep inelastic scattering at HERA, PMC Phys. A 1, 6 (2007).
  48. M. Derrick et al., Measurement of elastic omega photoproduction at HERA, Z. Phys. C 73, 73 (1996).
  49. J. Breitweg et al., Measurement of exclusive omega electroproduction at HERA, Phys. Lett. B 487, 273 (2000).
  50. C. Alexa et al., Elastic and proton-dissociative photoproduction of J/ψ mesons at HERA, Eur. Phys. J. C 73, 2466 (2013).
  51. R. Aaij et al., Central exclusive production of J/ψ and ψ(2S) mesons in pp collisions at s=13TeV, J. High Energy Phys. 10 (2018) 167.
  52. B. B. Abelev et al., Exclusive J/ψ photoproduction off protons in ultra-peripheral p-Pb collisions at sNN=5.02 TeV, Phys. Rev. Lett. 113, 232504 (2014).
  53. A. Aktas et al., Elastic J/ψ production at HERA, Eur. Phys. J. C 46, 585 (2006).
  54. S. Chekanov et al., Exclusive photoproduction of J/ψ mesons at HERA, Eur. Phys. J. C 24, 345 (2002).
  55. R. Aaij et al., Updated measurements of exclusive J/ψ and ψ(2S) production cross-sections in pp collisions at s = 7 TeV, J. Phys. G 41, 055002 (2014).
  56. R. Aaij et al., Measurement of the exclusive Υ production cross-section in pp collisions at s=7 TeV and 8 TeV, J. High Energy Phys. 09 (2015) 084.
  57. C. M. S. collaboration, Measurement of exclusive Y photoproduction in pPb collisions at SNN=5.02TeV Eur. Phys. J. C 79, 277 (2019)
  58. J. Breitweg et al., Measurement of elastic Upsilon photoproduction at HERA, Phys. Lett. B 437, 432 (1998).
  59. S. Chekanov et al., Exclusive photoproduction of Upsilon mesons at HERA, Phys. Lett. B 680, 4 (2009).
  60. C. Adloff et al., Elastic photoproduction of J/ψ and Υ mesons at HERA, Phys. Lett. B 483, 23 (2000).
  61. F. D. Aaron et al., Deeply virtual Compton scattering and its beam charge asymmetry in e± collisions at HERA, Phys. Lett. B 681, 391 (2009).
  62. H. De Vries, C. De Jager, and C. De Vries, Nuclear charge and magnetization density distribution parameters from elastic electron scattering, Atom. Data Nucl. Data Tabl. 36, 495 (1987).
  63. A. H. Mueller, Small-x behavior and parton saturation: A QCD model, Nucl. Phys. B 335, 115 (1990).
  64. 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).
  65. H. Kowalski, L. Motyka, and G. Watt, Exclusive diffractive processes at HERA within the dipole picture, Phys. Rev. D 74, 074016 (2006).
  66. B. E. Cox, J. R. Forshaw, and R. Sandapen, Diffractive Υ production at the Tevatron and LHC, J. High Energy Phys. 06 (2009) 034.
  67. J. Nemchik, N. N. Nikolaev, E. Predazzi, and B. G. Zakharov, Color dipole phenomenology of diffractive electroproduction of light vector mesons at HERA, Z. Phys. C 75, 71 (1997).
  68. J. Nemchik, N. N. Nikolaev, and B. G. Zakharov, Scanning the BFKL pomeron in elastic production of vector mesons at HERA, Phys. Lett. B 341, 228 (1994).
  69. J. R. Forshaw, R. Sandapen, and G. Shaw, Color dipoles and ρ, ϕ electroproduction, Phys. Rev. D 69, 094013 (2004).
  70. N. Armesto and A. H. Rezaeian, Exclusive vector meson production at high energies and gluon saturation, Phys. Rev. D 90, 054003 (2014).
  71. S. Acharya et al., First measurement of the |t|-dependence of coherent J/ψ photonuclear production, Phys. Lett. B 817, 136280 (2021).
  72. A. Tumasyan et al., Probing small Bjorken-x nuclear gluonic structure via coherent J/ψ photoproduction in ultraperipheral Pb-Pb collisions at SNN=5.02TeV, Phys. Rev. Lett. 131, 262301 (2023).

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