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  • Open Access

Neural network extraction of chromoelectric and chromomagnetic gluon masses

Jie Mei1,*, Lingxiao Wang2,†, and Mei Huang1,‡

  • *Contact author: meijie@ucas.ac.cn
  • †Contact author: lingxiao.wang@riken.jp
  • ‡Contact author: huangmei@ucas.ac.cn
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Phys. Rev. D 114, 054044 – Published 23 September, 2026

DOI: https://doi.org/10.1103/l113-1z8w

Abstract

We present a neural network-based quasiparticle model to separate the contributions of chromoelectric and chromomagnetic gluons in SU(3) gauge theory. Using dual residual networks, we extract temperature-dependent thermal masses from lattice thermodynamic data of entropy density and trace anomaly. By incorporating physics-informed regularization that enforces hard thermal loop expectations at high temperatures, the trained models reproduce lattice results with high accuracy over T/Tc∈[1,10], capturing both the crossover behavior near Tc and the approach to perturbative scaling at high temperatures. The extracted masses exhibit physically reasonable behavior: they decrease sharply around Tc and increase linearly thereafter. Our analysis reveals that the effective suppression of the electric sector is thermodynamically necessary to reconcile the HTL-consistent magnetic behavior with the lattice QCD equation of state. Consequently, we find that the chromomagnetic sector contribution is compatible with chromoelectric part in the near-Tc region and dominates the system at high temperature region. Furthermore, we compute transport coefficients within the relaxation time approximation, finding that the temperature dependence of both shear and bulk viscosities and the relative contributions of the two sectors are consistent with recent lattice calculations, while their absolute scale is calibrated by the assumed relaxation time.

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References (78)

  1. A. Bazavov (HotQCD Collaboration), The QCD equation of state, Nucl. Phys. A931, 867 (2014).
  2. C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, C. Schmidt, and L. Scorzato, The QCD thermal phase transition in the presence of a small chemical potential, Phys. Rev. D 66, 074507 (2002).
  3. G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, and B. Petersson, Thermodynamics of SU(3) lattice gauge theory, Nucl. Phys. B469, 419 (1996).
  4. A. Peshier, B. Kampfer, O. P. Pavlenko, and G. Soff, A massive quasiparticle model of the SU(3) gluon plasma, Phys. Rev. D 54, 2399 (1996).
  5. M. Dirks, A. Niegawa, and K. Okano, A Slavnov-Taylor identity and equality of damping rates for static transverse and longitudinal gluons in hot QCD, Phys. Lett. B 461, 131 (1999).
  6. P. N. Meisinger, M. C. Ogilvie, and T. R. Miller, Gluon quasiparticles and the Polyakov loop, Phys. Lett. B 585, 149 (2004).
  7. T. Hatsuda and T. Kunihiro, QCD phenomenology based on a chiral effective Lagrangian, Phys. Rep. 247, 221 (1994).
  8. S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992).
  9. J. Mei and S. Mao, Inverse catalysis effect of the quark anomalous magnetic moment to chiral restoration and deconfinement phase transitions, Phys. Rev. D 102, 114035 (2020).
  10. J. Mei, T. Xia, and S. Mao, Mass spectra of neutral mesons K0, π0, η, η’ at finite magnetic field, temperature and quark chemical potential, Phys. Rev. D 107, 074018 (2023).
  11. J. Mei, R. Wen, S. Mao, M. Huang, and K. Xu, Magnetic catalysis and diamagnetism from pion fluctuations, Phys. Rev. D 110, 034024 (2024).
  12. K. Fukushima, Chiral effective model with the Polyakov loop, Phys. Lett. B 591, 277 (2004).
  13. C. Ratti, M. A. Thaler, and W. Weise, Phases of QCD: Lattice thermodynamics and a field theoretical model, Phys. Rev. D 73, 014019 (2006).
  14. S. Roessner, C. Ratti, and W. Weise, Polyakov loop, diquarks and the two-flavour phase diagram, Phys. Rev. D 75, 034007 (2007).
  15. B.-J. Schaefer, J. M. Pawlowski, and J. Wambach, The phase structure of the Polyakov–quark-meson model, Phys. Rev. D 76, 074023 (2007).
  16. A. Bhattacharyya, P. Deb, S. K. Ghosh, R. Ray, and S. Sur, Thermodynamic properties of strongly interacting matter in finite volume using Polyakov-Nambu-Jona-Lasinio model, Phys. Rev. D 87, 054009 (2013).
  17. R. Wen, C. Huang, and W.-J. Fu, Baryon number fluctuations in the 2+1 flavor low energy effective model, Phys. Rev. D 99, 094019 (2019).
  18. R. Wen and W.-j. Fu, Correlations of conserved charges and QCD phase structure, Chin. Phys. C 45, 044112 (2021).
  19. F. Xu and M. Huang, Electric and magnetic screenings of gluons in a model with dimension-2 gluon condensate, Chin. Phys. C 37, 014103 (2013).
  20. T. K. Herbst, M. Mitter, J. M. Pawlowski, B.-J. Schaefer, and R. Stiele, Thermodynamics of QCD at vanishing density, Phys. Lett. B 731, 248 (2014).
  21. P. J. Gunkel and C. S. Fischer, Locating the critical endpoint of QCD: Mesonic backcoupling effects, Phys. Rev. D 104, 054022 (2021).
  22. A. Nakamura, T. Saito, and S. Sakai, Lattice calculation of gluon screening masses, Phys. Rev. D 69, 014506 (2004).
  23. J.-P. Blaizot, E. Iancu, and A. Rebhan, Thermodynamics of the high temperature quark gluon plasma, in Quark–Gluon Plasma 3 (World Scientific, Singapore, 2003), pp. 60–122.
  24. U. Kraemmer and A. Rebhan, Advances in perturbative thermal field theory, Rep. Prog. Phys. 67, 351 (2004).
  25. J. O. Andersen and M. Strickland, Resummation in hot field theories, Ann. Phys. (Amsterdam) 317, 281 (2005).
  26. N. Su and K. Tywoniuk, Massless mode and positivity violation in hot QCD, Phys. Rev. Lett. 114, 161601 (2015).
  27. J. I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2011).
  28. R. A. Schneider and W. Weise, On the quasiparticle description of lattice QCD thermodynamics, Phys. Rev. C 64, 055201 (2001).
  29. A. Peshier, B. Kampfer, and G. Soff, From QCD lattice calculations to the equation of state of quark matter, Phys. Rev. D 66, 094003 (2002).
  30. M. Bluhm, B. Kampfer, R. Schulze, and D. Seipt, QCD equation of state: Physical quark masses and asymptotic temperatures, Prog. Part. Nucl. Phys. 62, 512 (2009).
  31. P. Alba, W. Alberico, M. Bluhm, V. Greco, C. Ratti, and M. Ruggieri, Polyakov loop and gluon quasiparticles: A self-consistent approach to Yang–Mills thermodynamics, Nucl. Phys. A934, 41 (2014).
  32. S. Koothottil and V. M. Bannur, Thermodynamic behavior of magnetized quark-gluon plasma within the self-consistent quasiparticle model, Phys. Rev. C 99, 035210 (2019).
  33. G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2, 303 (1989).
  34. K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators, Neural Netw. 2, 359 (1989).
  35. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  36. G. Aarts, K. Fukushima, T. Hatsuda, A. Ipp, S. Shi, L. Wang, and K. Zhou, Physics-driven learning for inverse problems in quantum chromodynamics, Nat. Rev. Phys. 7, 154 (2025).
  37. L.-G. Pang, K. Zhou, N. Su, H. Petersen, H. Stöcker, and X.-N. Wang, An equation-of-state-meter of quantum chromodynamics transition from deep learning, Nat. Commun. 9, 210 (2018).
  38. Z. Liu, W. Zhao, and H. Song, Principal component analysis of collective flow in relativistic heavy-ion collisions, Eur. Phys. J. C 79, 870 (2019).
  39. H. Huang, B. Xiao, Z. Liu, Z. Wu, Y. Mu, and H. Song, Applications of deep learning to relativistic hydrodynamics, Phys. Rev. Res. 3, 023256 (2021).
  40. S. Shi, K. Zhou, J. Zhao, S. Mukherjee, and P. Zhuang, Heavy quark potential in the quark-gluon plasma: Deep neural network meets lattice quantum chromodynamics, Phys. Rev. D 105, 014017 (2022).
  41. L. Wang, S. Shi, and K. Zhou, Reconstructing spectral functions via automatic differentiation, Phys. Rev. D 106, L051502 (2022).
  42. L. Wang, Deep learning for exploring hadron–hadron interactions, J. Subatomic Part. Cosmol. 3, 100024 (2025).
  43. L. Wang and J. Zhao, Learning hadron emitting sources with deep neural networks, Commun. Phys. 9, 90 (2026).
  44. L. Wang, T. Doi, T. Hatsuda, and Y. Lyu, Building hadron potentials from lattice QCD with deep neural networks, Proc. Sci. LATTICE2024 (2025) 076 [arXiv:2410.03082].
  45. K. Zhou, L. Wang, L.-G. Pang, and S. Shi, Exploring QCD matter in extreme conditions with machine learning, Prog. Part. Nucl. Phys. 135, 104084 (2024).
  46. X. Chen, Y. Chen, and K. Zhou, Data-driven Einstein-dilaton model for pure Yang-Mills thermodynamics, and glueball spectrum, Phys. Rev. D 112, 126025 (2025).
  47. W.-C. Dai, O.-Y. Luo, B. Chen, X. Chen, X.-Y. Zhu, and X.-H. Li, Extracting transport properties of quark-gluon plasma from the heavy-quark potential with neural networks in a holographic model, Phys. Lett. B 877, 140511 (2026).
  48. M. Mansouri, K. Bitaghsir Fadafan, and X. Chen, Holographic complex potential of a quarkonium from deep learning, Eur. Phys. J. C 86, 321 (2026).
  49. P. Baldi, P. Sadowski, and D. Whiteson, Searching for exotic particles in high-energy physics with deep learning, Nat. Commun. 5, 4308 (2014).
  50. P. Baldi, P. Sadowski, and D. Whiteson, Enhanced Higgs boson to τ+τ− search with deep learning, Phys. Rev. Lett. 114, 111801 (2015).
  51. J. Barnard, E. N. Dawe, M. J. Dolan, and N. Rajcic, Parton shower uncertainties in jet substructure analyses with deep neural networks, Phys. Rev. D 95, 014018 (2017).
  52. P. Broecker, J. Carrasquilla, R. G. Melko, and S. Trebst, Machine learning quantum phases of matter beyond the fermion sign problem, Sci. Rep. 7, 8823 (2017).
  53. A. Radovic, M. Williams, D. Rousseau, M. Kagan, D. Bonacorsi, A. Himmel, A. Aurisano, K. Terao, and T. Wongjirad, Machine learning at the energy and intensity frontiers of particle physics, Nature (London) 560, 41 (2018).
  54. L. Wang, G. Aarts, and K. Zhou, Diffusion models as stochastic quantization in lattice field theory, J. High Energy Phys. 05 (2024) 060.
  55. J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
  56. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019).
  57. F.-P. Li, H.-L. Lü, L.-G. Pang, and G.-Y. Qin, Deep-learning quasi-particle masses from QCD equation of state, Phys. Lett. B 844, 138088 (2023).
  58. F.-P. Li, L.-G. Pang, and G.-Y. Qin, QCD equation of state at finite μB using deep learning assisted quasi-parton model, Phys. Lett. B 868, 139692 (2025).
  59. F.-L. Liu, X.-Y. Wu, S. Cao, G.-Y. Qin, and X.-N. Wang, Constraining the equation of state with heavy quarks in the quasi-particle model of QCD matter, Phys. Lett. B 848, 138355 (2024).
  60. O. Soloveva, A. Palermo, and E. Bratkovskaya, Extraction of the microscopic properties of quasi-particles using deep neural networks, Phys. Rev. C 110, 034908 (2024).
  61. R. K. Tiwari, K. Goswami, S. Prasad, C. R. Singh, R. Sahoo, and M. Y. Jamal, Thermodynamic and transport properties of quark-gluon plasma at finite chemical potential with a DNN framework, arXiv:2604.05858.
  62. K. He, X. Zhang, S. Ren, and J. Sun, Deep residual learning for image recognition, in 2016 IEEE Conference on CVPR (IEEE, Piscataway, NJ, 2016), pp. 770–778.
  63. X. Yu, Z. Yu, and S. Ramalingam, Learning strict identity mappings in deep residual networks, in 2018 IEEE Conference on CVPR (IEEE, Piscataway, NJ, 2018), pp. 4432–4440.
  64. S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, Precision SU(3) lattice thermodynamics for a large temperature range, J. High Energy Phys. 07 (2012) 056.
  65. A. Rebhan, HTL quasiparticle picture of the thermodynamics of QCD, Nucl. Phys. A702, 111 (2002).
  66. M. Gockeler, R. Horsley, A. C. Irving, D. Pleiter, P. E. L. Rakow, G. Schierholz, and H. Stuben (QCDSF and UKQCD Collaborations), Determination of lambda in quenched and full QCD: An update, Nucl. Phys. B, Proc. Suppl. 140, 228 (2005).
  67. C. Ratti, S. Roessner, M. A. Thaler, and W. Weise, Thermodynamics of the PNJL model, Eur. Phys. J. C 49, 213 (2007).
  68. S. Roessner, T. Hell, C. Ratti, and W. Weise, The chiral and deconfinement crossover transitions: PNJL model beyond mean field, Nucl. Phys. A814, 118 (2008).
  69. V. Ozvenchuk, O. Linnyk, M. I. Gorenstein, E. L. Bratkovskaya, and W. Cassing, Shear, and bulk viscosities of strongly interacting, infinite parton-hadron matter within the parton-hadron-string dynamics transport approach, Phys. Rev. C 87, 064903 (2013).
  70. M. Bluhm, B. Kampfer, and K. Redlich, Bulk and shear viscosities of the gluon plasma in a quasiparticle description, Phys. Rev. C 84, 025201 (2011).
  71. P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94, 111601 (2005).
  72. H.-T. Ding, H.-T. Shu, and C. Zhang, Shear, and bulk viscosities of gluon plasma across the transition temperature from lattice QCD, Phys. Rev. D 113, 074503 (2026).
  73. N. Y. Astrakhantsev, V. V. Braguta, and A. Y. Kotov, Temperature dependence of the bulk viscosity within lattice simulation of SU(3) gluodynamics, Phys. Rev. D 98, 054515 (2018).
  74. H. B. Meyer, A calculation of the bulk viscosity in SU(3) gluodynamics, Phys. Rev. Lett. 100, 162001 (2008).
  75. M. Asakawa, T. Hatsuda, and Y. Nakahara, Maximum entropy analysis of the spectral functions in lattice QCD, Prog. Part. Nucl. Phys. 46, 459 (2001).
  76. Y. Burnier and A. Rothkopf, Bayesian approach to spectral function reconstruction for Euclidean quantum field theories, Phys. Rev. Lett. 111, 182003 (2013).
  77. N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, and A. Courville, On the spectral bias of neural networks, in Proceedings of the 36th International Conference on Machine Learning (ICML) (PMLR, 2019), Vol. 97, pp. 5301–5310.
  78. A. Jacot, F. Gabriel, and C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks, in Advances in Neural Information Processing Systems (NeurIPS) (Curran Associates, Red Hook, NY, 2018), Vol. 31, pp. 8571–8580.

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Phys. Rev. D 114, 054044– Published 23 September, 2026
  • Received 26 August 2025
  • Accepted 3 September 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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