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Pomeron evolution, entanglement entropy, and Abramovskii-Gribov-Kancheli cutting rules

Mustapha Ouchen and Alex Prygarin

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Phys. Rev. D 112, 094027 – Published 17 November, 2025

DOI: https://doi.org/10.1103/cpvq-f5gx

Abstract

We use Pomeron evolution equation in zero transverse dimension based on the Abramovskii-Gribov-Kancheli (AGK) cutting rules to calculate the von Neumann entropy and q moments that are used as an experimental test for the Koba-Nielsen-Olesen scaling. In order to avoid the negative probabilities that emerge from the negative AGK weights in the Minkowski space, we reformulate the Pomeron evolution in the Euclidean space. The resulting positive definite probabilities for cut and uncut Pomerons are used in calculating the q moments. The comparison to the experimental data shows that our AGK-based model successfully describes q-moment dependence on the mean multiplicity without any adjustable parameter in the experimental data of the p−p collisions by the ALICE Collaboration.

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References (59)

  1. E. Levin and A. Prygarin, Inclusive gluon production in the dipole approach: AGK cutting rules, Phys. Rev. C 78, 065202 (2008).
  2. V. A. Abramovsky, V. N. Gribov, and O. V. Kancheli, Character of inclusive spectra and fluctuations produced in inelastic processes by multi—pomeron exchange, Yad. Fiz. 18, 595 (1973) [Sov. J. Nucl. Phys. 18, 308 (1973)].
  3. D. E. Kharzeev and E. M. Levin, Deep inelastic scattering as a probe of entanglement, Phys. Rev. D 95, 114008 (2017).
  4. D. E. Kharzeev and E. Levin, Deep inelastic scattering as a probe of entanglement: Confronting experimental data, Phys. Rev. D 104, L031503 (2021).
  5. K. Zhang, K. Hao, D. Kharzeev, and V. Korepin, Entanglement entropy production in deep inelastic scattering, Phys. Rev. D 105, 014002 (2022).
  6. A. Florio, D. Frenklakh, K. Ikeda, D. Kharzeev, V. Korepin, S. Shi, and K. Yu, Real-time nonperturbative dynamics of jet production in Schwinger model: Quantum entanglement and vacuum modification, Phys. Rev. Lett. 131, 021902 (2023).
  7. M. Hentschinski, D. E. Kharzeev, K. Kutak, and Z. Tu, Probing the onset of maximal entanglement inside the proton in diffractive deep inelastic scattering, Phys. Rev. Lett. 131, 241901 (2023).
  8. M. Hentschinski, Entangelement entropy in high energy collisions of electrons and protons, Rev. Mex. Fis. Suppl. 4, 021110 (2023).
  9. K. Kutak, Entanglement entropy of proton and its relation to thermodynamics entropy, arXiv:2310.18510.
  10. K. Kutak, Entanglement entropy and proton’s structure, Acta Phys. Pol. B Proc. Suppl. 17, 2-A3 (2024).
  11. K. Kutak, Entanglement entropy, krylov complexity, and DIS data, Acta Phys. Pol. B Proc. Suppl. 18, 5-A4 (2025).
  12. L. S. Moriggi, G. S. Ramos, and M. V. T. Machado, Multiplicity dependence of the pT-spectra for identified particles and its relationship with partonic entropy, Phys. Rev. D 110, 034005 (2024).
  13. P. Caputa and K. Kutak, Krylov complexity and gluon cascades in the high energy limit, Phys. Rev. D 110, 085011 (2024).
  14. S. Bhattacharya, R. Boussarie, and Y. Hatta, Spin-orbit entanglement in the Color Glass Condensate, Phys. Lett. B 859, 139134 (2024).
  15. Y. Liu, M. A. Nowak, and I. Zahed, Mueller’s dipole wave function in QCD: Emergent Koba-Nielsen-Olesen scaling in the double logarithm limit, Phys. Rev. D 108, 034017 (2023).
  16. Y. Hatta and J. Montgomery, Maximally entangled gluons for any x, Phys. Rev. D 111, 014024 (2025).
  17. E. Levin, Particle production in a toy model: Multiplicity distribution and entropy, Phys. Rev. D 111, 016019 (2025).
  18. A. Kovner, M. Lublinsky, and M. Serino, Entanglement entropy, entropy production and time evolution in high energy QCD, Phys. Lett. B 792, 4 (2019).
  19. N. Armesto, F. Dominguez, A. Kovner, M. Lublinsky, and V. Skokov, The Color Glass Condensate density matrix: Lindblad evolution, entanglement entropy and Wigner functional, J. High Energy Phys. 05 (2019) 025.
  20. R. Peschanski and S. Seki, Evaluation of entanglement entropy in high energy elastic scattering, Phys. Rev. D 100, 076012 (2019).
  21. G. S. Ramos and M. V. T. Machado, Investigating entanglement entropy at small-x in DIS off protons and nuclei, Phys. Rev. D 101, 074040 (2020).
  22. Y. Afik and J. R. M. de Nova, Quantum information with top quarks in QCD, Quantum 6, 820 (2022).
  23. G. S. Ramos and M. V. T. Machado, Investigating the QCD dynamical entropy in high-energy hadronic collisions, Phys. Rev. D 105, 094009 (2022).
  24. H. Duan, A. Kovner, and V. V. Skokov, Classical entanglement and entropy, Int. J. Mod. Phys. A 39, 2450104 (2024).
  25. A. Dumitru, A. Kovner, and V. V. Skokov, Entanglement entropy of the proton in coordinate space, Phys. Rev. D 108, 014014 (2023).
  26. H. G. Dosch, G. F. de Teramond, and S. J. Brodsky, Entropy from entangled parton states and high-energy scattering behavior, Phys. Lett. B 850, 138521 (2024).
  27. A. Hutson and R. Bellwied, Exploring particle production and thermal-like behavior through quantum entanglement, arXiv:2410.17429.
  28. L. S. Moriggi and M. V. T. Machado, Precise determination of the Pomeron intercept via a scaling entropy analysis, Phys. Rev. D 111, 014017 (2025).
  29. G. S. Ramos, Investigating notions of entropy and its production in high energy particle collisions, arXiv:2502.01038.
  30. W. Qi, Z. Guo, and B. W. Xiao, Studying maximal entanglement and Bell nonlocality at an electron-ion collider, arXiv:2506.12889.
  31. G. S. Ramos, L. S. Moriggi, and M. V. T. Machado, Investigating QCD dynamical entropy in high-energy nuclear collisions, Phys. Lett. B 868, 139737 (2025).
  32. K. A. Mamo, Entanglement, trace anomaly and confinement in QCD, arXiv:2507.00176.
  33. H. Duan, C. Akkaya, A. Kovner, and V. V. Skokov, Entanglement, partial set of measurements, and diagonality of the density matrix in the parton model, Phys. Rev. D 101, 036017 (2020).
  34. A. Baty, P. Gardner, and W. Li, Novel observables for exploring QCD collective evolution and quantum entanglement within individual jets, Phys. Rev. C 107, 064908 (2023).
  35. A. Florio and D. E. Kharzeev, Gibbs entropy from entanglement in electric quenches, Phys. Rev. D 104, 056021 (2021).
  36. S. Grieninger, D. E. Kharzeev, and I. Zahed, Entanglement in a holographic Schwinger pair with confinement, Phys. Rev. D 108, 086030 (2023).
  37. G. A. Miller, Entanglement of elastic and inelastic scattering, Phys. Rev. C 108, L041601 (2023).
  38. U. Gürsoy, D. E. Kharzeev, and J. F. Pedraza, Universal rapidity scaling of entanglement entropy inside hadrons from conformal invariance, Phys. Rev. D 110, 074008 (2024).
  39. J. Barata, W. Gong, and R. Venugopalan, Realtime dynamics of hyperon spin correlations from string fragmentation in a deformed four-flavor Schwinger model, Phys. Rev. D 109, 116003 (2024).
  40. S. Grieninger, K. Ikeda, D. E. Kharzeev, and I. Zahed, Entanglement in massive Schwinger model at finite temperature and density, Phys. Rev. D 109, 016023 (2024).
  41. B. J. J. Khor, D. M. Kurkcuoglu, T. J. Hobbs, G. N. Perdue, and I. Klich, Confinement and kink entanglement asymmetry on a quantum Ising chain, Quantum 8, 1462 (2024).
  42. A. Florio, Two-fermion negativity and confinement in the Schwinger model, Phys. Rev. D 109, L071501 (2024).
  43. A. J. Barr, M. Fabbrichesi, R. Floreanini, E. Gabrielli, and L. Marzola, Quantum entanglement and Bell inequality violation at colliders, Prog. Part. Nucl. Phys. 139, 104134 (2024).
  44. Y. Meurice, Lower bounds on entanglement entropy without twin copy, Phys. Rev. Res. 7, L022023 (2025).
  45. A. M. Polyakov, A similarity hypothesis in the strong interactions. I. Multiple hadron production in e+e− annihilation, Zh. Eksp. Teor. Fiz. 59, 542 (1970) [JETP 32, 296 (1971)].
  46. Z. Koba, H. B. Nielsen, and P. Olesen, Scaling of multiplicity distributions in high-energy hadron collisions, Nucl. Phys. B40, 317 (1972).
  47. E. Levin and A. Prygarin, The BFKL Pomeron Calculus in zero transverse dimension: Summation of the Pomeron loops and the generating functional for the multiparticle production processes, Eur. Phys. J. C 53, 385 (2008).
  48. M. Kozlov, E. Levin, and A. Prygarin, The BFKL Pomeron calculus in the dipole approach, Nucl. Phys. A792, 122 (2007).
  49. A. Kormilitzin, E. Levin, and A. Prygarin, Multiparticle production in the mean field approximation of high density QCD, Nucl. Phys. A813, 1 (2008).
  50. J. Adam et al. (ALICE Collaboration), Charged-particle multiplicities in proton–proton collisions at s=0.9 to 8 TeV, Eur. Phys. J. C 77, 33 (2017).
  51. N. Johnson, A. Kemp, and S. Kotz, Univariate Discrete Distributions, 1st ed., Wiley Series in Probability and Statistics (Wiley, New York, 2005), 10.1002/0471715816.
  52. A. R. Wilcox, Indices of qualitative variation (1967), https://www.osti.gov/servlets/purl/4167340.
  53. J. Bartels, L. N. Lipatov, and A. Sabio Vera, N=4 supersymmetric Yang Mills scattering amplitudes at high energies: The Regge cut contribution, Eur. Phys. J. C 65, 587 (2010).
  54. J. Bartels, L. N. Lipatov, and A. Sabio Vera, BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80, 045002 (2009).
  55. L. N. Lipatov and A. Prygarin, Mandelstam cuts and light-like Wilson loops in N=4 SUSY, Phys. Rev. D 83, 045020 (2011).
  56. L. N. Lipatov and A. Prygarin, BFKL approach and six-particle MHV amplitude in N=4 super Yang-Mills, Phys. Rev. D 83, 125001 (2011).
  57. J. Bartels, L. N. Lipatov, and A. Prygarin, MHV amplitude for 3→3 gluon scattering in Regge limit, Phys. Lett. B 705, 507 (2011).
  58. J. Bartels, A. Kormilitzin, L. N. Lipatov, and A. Prygarin, BFKL approach and 2→5 maximally helicity violating amplitude in N=4 super-Yang-Mills theory, Phys. Rev. D 86, 065026 (2012).
  59. See Supplemental Material at http://link.aps.org/supplemental/10.1103/cpvq-f5gx for a mathematica file with the main results.

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Phys. Rev. D 112, 094027– Published 17 November, 2025
  • Received 19 August 2025
  • Accepted 10 October 2025
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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.

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