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Magnetic susceptibility of a hot hadronic medium and quark degrees of freedom near the QCD crossover point

Rupam Samanta* and Wojciech Broniowski†

  • *Contact author: rupam.samanta@ifj.edu.pl
  • †Contact author: wojciech.broniowski@ifj.edu.pl
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Phys. Rev. C 113, 045201 – Published 6 April, 2026

DOI: https://doi.org/10.1103/d2gg-mgjj

Abstract

The lattice QCD results for the temperature-dependent magnetic susceptibility of the medium below the crossover temperature cannot be reconciled with the widely used Hadron Resonance Gas model, also amended with the physical magnetic moments of hadrons or the pion–vector-meson loops. As noticed earlier, one observes a substantially too strong diamagnetism at temperatures in the range above ≈120MeV compared to the lattice. This hints at a presence of quarks significantly below the QCD crossover temperature, which are needed as a source of paramagnetism. However, the pions must be retained to describe the diamagnetism data at low temperatures. Therefore, we consider here a quark-meson approach, where the temperature-dependent quark masses are fixed in a model-free way using the baryon-baryon and baryon-strangeness susceptibilities from the lattice at zero magnetic field. The constituent quarks possess anomalous magnetic moments estimated from the octet baryon magnetic moments. The vacuum quark-loop and meson-loop contributions are duly incorporated. We show that in such a framework, one can describe the magnetic susceptibility up to the crossover point. The qualitative conclusion is that the QCD degrees of freedom must extend far below the crossover temperature, down to ≈120MeV.

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References (83)

  1. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer, and K. K. Szabo, The QCD phase diagram for external magnetic fields, J. High Energy Phys. 02 (2012) 044.
  2. G. S. Bali, F. Bruckmann, M. Constantinou, M. Costa, G. Endrodi, S. D. Katz, H. Panagopoulos, and A. Schafer, Magnetic susceptibility of QCD at zero and at finite temperature from the lattice, Phys. Rev. D 86, 094512 (2012).
  3. M. D'Elia, Lattice QCD simulations in external background fields, Lect. Notes Phys. 871, 181 (2013).
  4. G. Endrödi, QCD equation of state at nonzero magnetic fields in the Hadron Resonance Gas model, J. High Energy Phys. 04 (2013) 023.
  5. C. Bonati, M. D'Elia, M. Mariti, F. Negro, and F. Sanfilippo, Magnetic susceptibility of strongly interacting matter across the deconfinement transition, Phys. Rev. Lett. 111, 182001 (2013).
  6. C. Bonati, M. D'Elia, M. Mariti, F. Negro, and F. Sanfilippo, Magnetic susceptibility and equation of state of Nf=2+1 QCD with physical quark masses, Phys. Rev. D 89, 054506 (2014).
  7. G. S. Bali, F. Bruckmann, G. Endrödi, S. D. Katz, and A. Schäfer, The QCD equation of state in background magnetic fields, J. High Energy Phys. 08 (2014) 177.
  8. G. S. Bali, G. Endrődi, and S. Piemonte, Magnetic susceptibility of QCD matter and its decomposition from the lattice, J. High Energy Phys. 07 (2020) 183.
  9. G. Endrődi and G. Markó, On electric fields in hot QCD: Perturbation theory, J. High Energy Phys. 12 (2022) 015.
  10. H.-T. Ding, J.-B. Gu, A. Kumar, S.-T. Li, and J.-H. Liu, Baryon electric charge correlation as a magnetometer of QCD, Phys. Rev. Lett. 132, 201903 (2024).
  11. B. B. Brandt, G. Endrődi, G. Markó, and A. D. M. Valois, Steady electric currents in magnetized QCD and their use for the equation of state, J. High Energy Phys. 07 (2024) 027.
  12. G. Endrődi, QCD with background electromagnetic fields on the lattice: A review, Prog. Part. Nucl. Phys. 141, 104153 (2025).
  13. N. Astrakhantsev, V. V. Braguta, A. Y. Kotov, and A. A. Roenko, QCD equation of state at nonzero baryon density in an external magnetic field, Phys. Rev. D 109, 094511 (2024).
  14. H.-T. Ding, J.-B. Gu, A. Kumar, and S.-T. Li, Second order fluctuations of conserved charges in external magnetic fields, Phys. Rev. D 111, 114522 (2025).
  15. J. O. Andersen, W. R. Naylor, and A. Tranberg, Phase diagram of QCD in a magnetic field: A review, Rev. Mod. Phys. 88, 025001 (2016).
  16. P. Adhikari, et al., Strongly interacting matter in extreme magnetic fields, Prog. Part. Nucl. Phys. 146, 104199 (2026).
  17. D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, The Effects of topological charge change in heavy ion collisions: “Event by event P and CP violation”, Nucl. Phys. A 803, 227 (2008).
  18. K. Fukushima, D. E. Kharzeev, and H. J. Warringa, The chiral magnetic effect, Phys. Rev. D 78, 074033 (2008).
  19. D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions—A status report, Prog. Part. Nucl. Phys. 88, 1 (2016).
  20. J. O. Andersen, QCD phase diagram in a constant magnetic background: Inverse magnetic catalysis: Where models meet the lattice, Eur. Phys. J. A 57, 189 (2021).
  21. F. Karsch and E. Laermann, Susceptibilities, the specific heat and a cumulant in two-flavor QCD, Phys. Rev. D 50, 6954 (1994).
  22. C. Sasaki, B. Friman, and K. Redlich, Susceptibilities and the phase structure of a chiral model with Polyakov loops, Phys. Rev. D 75, 074013 (2007).
  23. A. Bazavov, et al., The chiral and deconfinement aspects of the QCD transition, Phys. Rev. D 85, 054503 (2012).
  24. P. Adhikari and J. O. Andersen, Quark condensates and magnetization in chiral perturbation theory in a uniform magnetic field, arXiv:2102.01080.
  25. M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Signatures of the tricritical point in QCD, Phys. Rev. Lett. 81, 4816 (1998).
  26. Z. Fodor and S. D. Katz, Critical point of QCD at finite T and μ, lattice results for physical quark masses, J. High Energy Phys. 04 (2004) 050.
  27. Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, The order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature (London) 443, 675 (2006).
  28. K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rep. Prog. Phys. 74, 014001 (2011).
  29. E. Fermi, High-energy nuclear events, Prog. Theor. Phys. 5, 570 (1950).
  30. I. Y. Pomeranchuk, On the theory of multiple particle production in a single collision, Dokl. Akad. Nauk SSSR 78, 889 (1951).
  31. R. Hagedorn, Statistical thermodynamics of strong interactions at high-energies, Nuovo Cim. Suppl. 3, 147 (1965).
  32. P. Koch, B. Muller, and J. Rafelski, Strangeness in relativistic heavy ion collisions, Phys. Rep. 142, 167 (1986).
  33. R. Dashen, S.-K. Ma, and H. J. Bernstein, S-matrix formulation of statistical mechanics, Phys. Rev. 187, 345 (1969).
  34. R. Venugopalan and M. Prakash, Thermal properties of interacting hadrons, Nucl. Phys. A 546, 718 (1992).
  35. F. Karsch, K. Redlich, and A. Tawfik, Thermodynamics at nonzero baryon number density: A Comparison of lattice and hadron resonance gas model calculations, Phys. Lett. B 571, 67 (2003).
  36. M. Gyulassy and L. McLerran, New forms of QCD matter discovered at RHIC, Nucl. Phys. A 750, 30 (2005).
  37. P. Huovinen and P. Petreczky, QCD equation of state and hadron resonance gas, Nucl. Phys. A 837, 26 (2010).
  38. A. Bazavov, et al. (HotQCD Collaboration), Equation of state in(2+1)-flavor QCD, Phys. Rev. D 90, 094503 (2014).
  39. H.-T. Ding, F. Karsch, and S. Mukherjee, Thermodynamics of strong-interaction matter from Lattice QCD, Int. J. Mod. Phys. E 24, 1530007 (2015).
  40. S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. Szabo, Fluctuations of conserved charges at finite temperature from lattice QCD, J. High Energy Phys. 01 (2012) 138.
  41. A. Bazavov, et al. (HotQCD Collaboration), Fluctuations and correlations of net baryon number, electric charge, and strangeness: A comparison of lattice QCD results with the hadron resonance gas model, Phys. Rev. D 86, 034509 (2012).
  42. R. Bellwied, S. Borsanyi, Z. Fodor, S. D. Katz, A. Pasztor, C. Ratti, and K. K. Szabo, Fluctuations and correlations in high temperature QCD, Phys. Rev. D 92, 114505 (2015).
  43. S. Borsanyi, Z. Fodor, J. N. Guenther, S. K. Katz, K. K. Szabo, A. Pasztor, I. Portillo, and C. Ratti, Higher order fluctuations and correlations of conserved charges from lattice QCD, J. High Energy Phys. 10 (2018) 205.
  44. D. Bollweg, J. Goswami, O. Kaczmarek, F. Karsch, S. Mukherjee, P. Petreczky, C. Schmidt, and P. Scior(HotQCD Collaboration), Second order cumulants of conserved charge fluctuations revisited: Vanishing chemical potentials, Phys. Rev. D 104, 074512 (2021).
  45. H. T. Ding, S. T. Li, Q. Shi, and X. D. Wang, Fluctuations and correlations of net baryon number, electric charge and strangeness in a background magnetic field, Eur. Phys. J. A 57, 202 (2021).
  46. R. Samanta and W. Broniowski, Magnetic properties of the hadron resonance gas with physical magnetic moments, Phys. Rev. C 112, 045202 (2025).
  47. A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Decoding the phase structure of QCD via particle production at high energy, Nature (London) 561, 321 (2018).
  48. C. P. Hofmann, Diamagnetic and paramagnetic phases in low-energy quantum chromodynamics, Phys. Lett. B 818, 136384 (2021).
  49. L. McLerran, K. Redlich, and C. Sasaki, Quarkyonic matter and chiral symmetry breaking, Nucl. Phys. A 824, 86 (2009).
  50. L. McLerran and S. Reddy, Quarkyonic matter and neutron stars, Phys. Rev. Lett. 122, 122701 (2019).
  51. M. Marczenko, L. McLerran, K. Redlich, and C. Sasaki, Reaching percolation and conformal limits in neutron stars, Phys. Rev. C 107, 025802 (2023).
  52. B.-J. Schaefer and J. Wambach, The phase diagram of the quark meson model, Nucl. Phys. A 757, 479 (2005).
  53. K. Kamikado and T. Kanazawa, Magnetic susceptibility of a strongly interacting thermal medium with 2+1 quark flavors, J. High Energy Phys. 01 (2015) 129.
  54. E. Ruiz Arriola, Pion structure at high-energies and low-energies in chiral quark models, Acta Phys. Pol. B 33, 4443 (2002).
  55. W. Broniowski, E. Ruiz Arriola, and P. Sanchez-Puertas, Baryonic form factors of the pion and kaon in a chiral quark model, Phys. Rev. D 106, 036001 (2022).
  56. S. Pal, Strongly interacting matter under extreme conditions, arXiv:2501.13123.
  57. M. Marczenko, M. Szymański, P. M. Lo, B. Karmakar, P. Huovinen, C. Sasaki, and K. Redlich, Magnetic effects in the hadron resonance gas, Phys. Rev. C 110, 065203 (2024).
  58. V. Vovchenko, Magnetic field effect on hadron yield ratios and fluctuations in a hadron resonance gas, Phys. Rev. C 110, 034914 (2024).
  59. P. Huovinen, M. Marczenko, M. Szymański, B. Karmakar, P. M. Lo, C. Sasaki, and K. Redlich, Hadron resonance gas is not a good model for hadronic matter in a strong magnetic field, in 31st International Conference on Ultra-relativistic Nucleus-Nucleus Collisions (unpublished), arXiv:2510.01431.
  60. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, Course of Theoretical Physics (Butterworth-Heinemann, Oxford, 1980), Vol. 5.
  61. R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed. (Elsevier, Amsterdam, 2011).
  62. W. Broniowski and W. Florkowski, Explanation of the RHIC p⊥ spectra in a thermal model with expansion, Phys. Rev. Lett. 87, 272302 (2001).
  63. W.-Y. Tsai and A. Yildiz, Motion of charged particles in a homogeneous magnetic field, Phys. Rev. D 4, 3643 (1971).
  64. M. G. de Paoli, D. P. Menezes, L. B. Castro, and C. C. Barros Jr., The Rarita-Schwinger particles under the influence of strong magnetic fields, J. Phys. G: Nucl. Part. Phys. 40, 055007 (2013).
  65. W. Broniowski, W. Florkowski, and L. Y. Glozman, Update of the Hagedorn mass spectrum, Phys. Rev. D 70, 117503 (2004).
  66. M. Moinester, Pion polarizability 2022 status report, arXiv:2205.09954.
  67. F. He, D. B. Leinweber, A. W. Thomas, and P. Wang, Chiral extrapolation of the magnetic polarizability of the neutral pion, Phys. Rev. D 102, 114509 (2020).
  68. S. Navas, et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  69. H. A. Weldon, Covariant Calculations at Finite Temperature: The Relativistic Plasma, Phys. Rev. D 26, 1394 (1982).
  70. Y. Fujimoto and H. Yamada, A supplementary remark on finite temperature perturbation, Z. Phys. C 37, 265 (1988).
  71. Y.-S. Oh, A. I. Titov, and T. S. H. Lee, Electromagnetic production of vector mesons at low-energies, in NSTAR2000: The Physics of Excited Nucleons, pp. 255–262, arXiv:nucl-th/0004055.
  72. A. Gökalp and O. Yilmaz, The coupling constants gρπγandgωπγ as derived from QCD sum rules, Eur. Phys. J. C 24, 117 (2002).
  73. S. Mao, Correlations and fluctuations in a magnetized PNJL model with and without the inverse magnetic catalysis effect, Chin. Phys. C 49, 063106 (2025).
  74. J. Mei, R. Wen, S. Mao, M. Huang, and K. Xu, Magnetic catalysis and diamagnetism from pion fluctuations, Phys. Rev. D 110, 034024 (2024).
  75. V. Mykhaylova, M. Bluhm, K. Redlich, and C. Sasaki, Quark-flavor dependence of the shear viscosity in a quasiparticle model, Phys. Rev. D 100, 034002 (2019).
  76. P. Chakraborty and J. I. Kapusta, Quasiparticle theory of shear and bulk viscosities of hadronic matter, Phys. Rev. C 83, 014906 (2011).
  77. P. J. A. Bicudo, J. E. F. T. Ribeiro, and R. Fernandes, The anomalous magnetic moment of quarks, Phys. Rev. C 59, 1107 (1999).
  78. K. Xu, J. Chao, and M. Huang, Effect of the anomalous magnetic moment of quarks on magnetized QCD matter and meson spectra, Phys. Rev. D 103, 076015 (2021).
  79. S. Ghosh, N. Chaudhuri, P. Roy, and S. Sarkar, Thermomagnetic modification of the anomalous magnetic moment of quarks using the NJL model, Phys. Rev. D 103, 116008 (2021).
  80. W. Thirring, Electromagnetic properties of hadrons in the static su6 model, in Quantum Electrodynamics, edited by P. Urban (Springer, Vienna, 1965), pp. 205–211.
  81. G. Karl, Baryon magnetic moments and the spin of the proton, Phys. Rev. D 45, 247 (1992).
  82. S. Capstick and B. D. Keister, Baryon magnetic moments in a relativistic quark model, arXiv:nucl-th/9611055.
  83. G. Passarino and M. J. G. Veltman, One loop corrections for e+e− annihilation into μ+μ− in the Weinberg model, Nucl. Phys. B 160, 151 (1979).

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Phys. Rev. C 113, 045201– Published 6 April, 2026
  • Received 27 November 2025
  • Revised 11 February 2026
  • Accepted 13 March 2026
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