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Entanglement entropy, Monte Carlo event generators, and soft gluon discovery

M. Hentschinski1, H. Jung2,3, and K. Kutak4,5

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Phys. Rev. D 113, 054024 – Published 16 March, 2026

DOI: https://doi.org/10.1103/q8s9-kd9s

Abstract

We study entropy production in deep inelastic scattering using Monte Carlo simulations. We show that the dominant contribution to entropy is due to soft gluons. This contribution is usually neglected in standard Monte Carlo approaches, since it does not affect hadronic spectra. However, it is relevant for entropy and multiplicity distributions, as we demonstrate with explicit calculations. We further show that as one includes soft gluons, making the Monte Carlo parton distributions closer to inclusive parton distribution functions (PDFs), the resulting entropy starts to grow with decreasing x. This provides further evidence that the bulk of the measured entropy originates from initial-state effects.

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References (68)

  1. F. Maltoni, C. Severi, S. Tentori, and E. Vryonidou, Quantum detection of new physics in top-quark pair production at the LHC, J. High Energy Phys. 03 (2024) 099.
  2. F. Maltoni, C. Severi, S. Tentori, and E. Vryonidou, Quantum tops at circular lepton colliders, J. High Energy Phys. 09 (2024) 001.
  3. Y. Afik et al., Quantum information meets high-energy physics: Input to the update of the European Strategy for Particle Physics, Eur. Phys. J. Plus 140, 855 (2025).
  4. Y. Hatta and J. Montgomery, Maximally entangled gluons for any x, Phys. Rev. D 111, 014024 (2025).
  5. S. Bhattacharya, R. Boussarie, and Y. Hatta, Spin-orbit entanglement in the color glass condensate, Phys. Lett. B 859, 139134 (2024).
  6. C. Altomonte, A. J. Barr, M. Eckstein, P. Horodecki, and K. Sakurai, Prospects for quantum process tomography at high energies, Quantum Sci. Technol. 10, 045060 (2025).
  7. Y. Zhang et al., Entanglement and Bell nonlocality in τ+τ− at the LHC using machine learning for neutrino reconstruction, arXiv:2504.01496.
  8. T. Han, M. Low, and Y. Su, Entanglement and Bell nonlocality in τ+τ− at the BEPC, J. High Energy Phys. 10 (2025) 217.
  9. P. Caputa and K. Kutak, Krylov complexity and gluon cascades in the high energy limit, Phys. Rev. D 110, 085011 (2024).
  10. W. Qi, Z. Guo, and B.-W. Xiao, Studying maximal entanglement and Bell nonlocality at an electron-ion collider, arXiv:2506.12889.
  11. K. Kutak, Gluon saturation and entropy production in proton–proton collisions, Phys. Lett. B 705, 217 (2011).
  12. K. Kutak, Entanglement entropy of proton and its relation to thermodynamics entropy, arXiv:2310.18510.
  13. R. Peschanski, Dynamical entropy of dense QCD states, Phys. Rev. D 87, 034042 (2013).
  14. A. Stoffers and I. Zahed, Holographic pomeron and entropy, Phys. Rev. D 88, 025038 (2013).
  15. A. Dumitru and E. Kolbusz, Quark pair angular correlations in the proton: Entropy versus entanglement negativity, Phys. Rev. D 108, 034011 (2023).
  16. A. Kovner and M. Lublinsky, Entanglement entropy and entropy production in the color glass condensate framework, Phys. Rev. D 92, 034016 (2015).
  17. J. Berges, S. Floerchinger, and R. Venugopalan, Dynamics of entanglement in expanding quantum fields, J. High Energy Phys. 04 (2018) 145.
  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. R. Peschanski and S. Seki, Evaluation of entanglement entropy in high energy elastic scattering, Phys. Rev. D 100, 076012 (2019).
  20. G. Dvali and R. Venugopalan, Classicalization and unitarization of wee partons in QCD and gravity: The CGC-black hole correspondence, Phys. Rev. D 105, 056026 (2022).
  21. K. Kutak and M. Praszałowicz, Entropy, purity and gluon cascades at high energies with recombinations and transitions to vacuum, Eur. Phys. J. C 85, 1215 (2025).
  22. D. E. Kharzeev and E. M. Levin, Deep inelastic scattering as a probe of entanglement, Phys. Rev. D 95, 114008 (2017).
  23. Z. Tu, D. E. Kharzeev, and T. Ullrich, Einstein-Podolsky-Rosen paradox and quantum entanglement at subnucleonic scales, Phys. Rev. Lett. 124, 062001 (2020).
  24. H1 Collaboration, Measurement of charged particle multiplicity distributions in DIS at HERA and its implication to entanglement entropy of partons, Eur. Phys. J. C 81, 212 (2021).
  25. T. Sjöstrand, S. Mrenna, and P. Skands, pythia 6.4 physics and manual, J. High Energy Phys. 05 (2006) 026.
  26. T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, An introduction to pythia 8.2, Comput. Phys. Commun. 191, 159 (2015).
  27. C. Bierlich et al., A comprehensive guide to the physics and usage of pythia 8.3, SciPost Phys. Codebases 2022, 8 (2022).
  28. G. A. Schuler and H. Spiesberger, DJANGO: The interface for the event generators HERACLES and LEPTO, in Workshop on Physics at HERA Hamburg, Germany, 1991 (DESY, Hamburg, 1991), pp. 1419–1432.
  29. H. Jung, The rapgap Monte Carlo version 3.3, http://projects.hepforge.org/rapgap/ (2021).
  30. H. Jung, Hard diffractive scattering in high-energy ep collisions and the Monte Carlo generator rapgap, Comput. Phys. Commun. 86, 147 (1995).
  31. M. Mendizabal, F. Guzman, H. Jung, and S. Taheri Monfared, On the role of soft gluons in collinear parton densities, Eur. Phys. J. C 84, 1299 (2024).
  32. I. Bubanja, H. Jung, A. Lelek, N. Raičević, and S. Taheri Monfared, Center-of-mass energy dependence of intrinsic-kT distributions obtained from Drell–Yan production, Eur. Phys. J. C 85, 278 (2025).
  33. I. Bubanja, H. Jung, N. Raicevic, and S. Taheri Monfared, Interplay of intrinsic motion of partons and soft gluon emissions in Drell–Yan production studied with pythia, Eur. Phys. J. C 85, 363 (2025).
  34. I. Bubanja et al., The small kT region in Drell–Yan production at next-to-leading order with the parton branching method, Eur. Phys. J. C 84, 154 (2024).
  35. F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Žlebčík, Collinear and TMD quark and gluon densities from parton branching solution of QCD evolution equations, J. High Energy Phys. 01 (2018) 070.
  36. F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Žlebčík, Soft-gluon resolution scale in QCD evolution equations, Phys. Lett. B 772, 446 (2017).
  37. D. E. Kharzeev, Quantum information approach to high energy interactions, Phil. Trans. A. Math. Phys. Eng. Sci. 380, 20210063 (2021).
  38. Y. Liu, M. A. Nowak, and I. Zahed, Rapidity evolution of the entanglement entropy in quarkonium: Parton and string duality, Phys. Rev. D 105, 114028 (2022).
  39. M. Hentschinski, D. E. Kharzeev, K. Kutak, and Z. Tu, QCD evolution of entanglement entropy, Rep. Prog. Phys. 87, 120501 (2024).
  40. D. E. Kharzeev, The maximal entanglement limit in statistical and high energy physics, arXiv:2601.00405.
  41. M. Hentschinski and K. Kutak, Evidence for the maximally entangled low x proton in deep inelastic scattering from H1 data, Eur. Phys. J. C 82, 111 (2022).
  42. M. Hentschinski, K. Kutak, and R. Straka, Maximally entangled proton and charged hadron multiplicity in deep inelastic scattering, Eur. Phys. J. C 82, 1147 (2022).
  43. 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).
  44. A. H. Mueller, Unitarity and the BFKL pomeron, Nucl. Phys. B437, 107 (1995).
  45. W. L. van Neerven and A. Vogt, Improved approximations for the three loop splitting functions in QCD, Phys. Lett. B 490, 111 (2000).
  46. S. Moch, J. A. M. Vermaseren, and A. Vogt, The three loop splitting functions in QCD: The nonsinglet case, Nucl. Phys. B688, 101 (2004).
  47. A. Vogt, S. Moch, and J. A. M. Vermaseren, The three-loop splitting functions in QCD: The singlet case, Nucl. Phys. B691, 129 (2004).
  48. J. Vermaseren, A. Vogt, and S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B724, 3 (2005).
  49. J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements, Nucl. Phys. B971, 115542 (2021).
  50. J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, The massless three-loop Wilson coefficients for the deep-inelastic structure functions F2, FL, xF3 and g1, J. High Energy Phys. 11 (2022) 156.
  51. J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock, The transition matrix element Agq(N) of the variable flavor number scheme at O(αs3), Nucl. Phys. B882, 263 (2014).
  52. J. Ablinger et al., The three-loop splitting functions Pqg(2) and Pgg(2,NF), Nucl. Phys. B922, 1 (2017).
  53. S. Moch, J. A. M. Vermaseren, and A. Vogt, The three-loop splitting functions in QCD: The Helicity-Dependent case, Nucl. Phys. B 889, 351 (2014).
  54. A. Behring, J. Blümlein, A. De Freitas, A. Goedicke, S. Klein, A. von Manteuffel, C. Schneider, and K. Schönwald, The polarized three-loop anomalous dimensions from on-shell massive operator matrix elements, Nucl. Phys. B948, 114753 (2019).
  55. J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements, J. High Energy Phys. 01 (2022) 193.
  56. R. K. Ellis, W. J. Stirling, and B. R. Webber, QCD and collider physics, Cambridge Monogr. Part. Phys., Nucl. Phys., Cosmol. 8, 1 (1996).
  57. A. Banfi, S. F. Ravasio, B. Jäger, A. Karlberg, F. Reichenbach, and G. Zanderighi, A POWHEG generator for deep inelastic scattering, J. High Energy Phys. 02 (2024) 023.
  58. C. Bierlich et al., Robust independent validation of experiment and theory: Rivet version 3, SciPost Phys. 8, 026 (2020).
  59. H. Jung, L. Lönnblad, M. Mendizabal, and S. Taheri Monfared, A parton shower consistent with parton densities at LO and NLO: PDF2ISR, Eur. Phys. J. C 85, 870 (2025).
  60. M. Ciafaloni, Coherence effects in initial jets at small Q2/s., Nucl. Phys. B296, 49 (1988).
  61. S. Catani, F. Fiorani, and G. Marchesini, QCD coherence in initial state radiation, Phys. Lett. B 234, 339 (1990).
  62. S. Catani, F. Fiorani, and G. Marchesini, Small x behavior of initial state radiation in perturbative QCD, Nucl. Phys. B336, 18 (1990).
  63. G. Marchesini, QCD coherence in the structure function and associated distributions at small x, Nucl. Phys. B445, 49 (1995).
  64. H. Jung, The CCFM Monte Carlo generator cascade, Comput. Phys. Commun. 143, 100 (2002).
  65. H. Jung, The cascade Monte Carlohttp://www.desy.de/~jung/cascade (2009).
  66. H. Jung et al., The CCFM Monte Carlo generator cascade version 2.2.03, Eur. Phys. J. C 70, 1237 (2010).
  67. S. Baranov et al., cascade3 A Monte Carlo event generator based on TMDs, Eur. Phys. J. C 81, 425 (2021).
  68. ALICE Collaboration, Physics of the ALICE forward calorimeter upgrade, Report No. ALICE-PUBLIC-2023-001 (2023).

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Phys. Rev. D 113, 054024– Published 16 March, 2026
  • Received 22 December 2025
  • Accepted 13 February 2026
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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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