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Transport coefficients of chiral fluid dynamics using low-energy effective models

Pedro Nogarolli1,*, Gabriel S. Denicol2,†, and Eduardo S. Fraga1,‡

  • *Contact author: pedro.nogarolli@cern.ch
  • †Contact author: gsdenicol@id.uff.br
  • ‡Contact author: fraga@if.ufrj.br
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Phys. Rev. D 113, 034016 – Published 13 February, 2026

DOI: https://doi.org/10.1103/wdfm-gsgg

Abstract

We investigate the first-order transport coefficients of a fluid made of quasiparticles with a temperature-dependent mass extracted from chiral models. We describe this system using an effective kinetic theory, given by the relativistic Boltzmann equation coupled to a temperature-dependent background field determined from the thermal masses. We then simplify the collision term using the relaxation time approximation and implement a Chapman-Enskog expansion to calculate all first-order transport coefficients. In particular, we compute the bulk and shear viscosities using thermal masses extracted from the linear sigma model coupled with constituent quarks and the NJL model.

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References (64)

  1. J. Adams et al. (STAR Collaboration), Identified particle distributions in pp and Au+Au collisions at sNN=200  GeV, Phys. Rev. Lett. 92, 112301 (2004).
  2. Sean Gavin and Mohamed Abdel-Aziz, Measuring shear viscosity using correlations, Braz. J. Phys. 37, 1023 (2007).
  3. Charles Gale, Sangyong Jeon, and Bjoern Schenke, Hydrodynamic modeling of heavy-ion collisions, Int. J. Mod. Phys. A 28, 1340011 (2013).
  4. Peter F. Kolb and Ulrich Heinz, Hydrodynamic description of ultrarelativistic heavy-ion collisions, arXiv:nucl-th/0305084.
  5. Laszlo P. Csernai, Joseph I. Kapusta, and Larry D. McLerran, Strongly interacting low-viscosity matter created in relativistic nuclear collisions, Phys. Rev. Lett. 97, 152303 (2006).
  6. Peter Arnold, Guy David Moore, and Laurence G. Yaffe, Transport coefficients in high temperature gauge theories (I): Leading-log results, J. High Energy Phys. 11 (2000) 001.
  7. Govert Nijs, Wilke van der Schee, Umut Gürsoy, and Raimond Snellings, Bayesian analysis of heavy ion collisions with the heavy ion computational framework trajectum, Phys. Rev. C 103, 054909 (2021).
  8. J. E. Parkkila, A. Onnerstad, S. F. Taghavi, C. Mordasini, A. Bilandzic, M. Virta, and D. J. Kim, New constraints for QCD matter from improved bayesian parameter estimation in heavy-ion collisions at LHC, Phys. Lett. B 835, 137485 (2022).
  9. D. Everett et al. (JETSCAPE Collaboration), Multisystem bayesian constraints on the transport coefficients of QCD matter, Phys. Rev. C 103, 054904 (2021).
  10. Peter Arnold, Ça ğlar Doğan, and Guy D. Moore, Bulk viscosity of high-temperature QCD, Phys. Rev. D 74, 085021 (2006).
  11. Gabriel S. Denicol and Dirk H. Rischke, Microscopic Foundations of Relativistic Fluid Dynamics, 2nd ed., Lecture Notes in Physics (Springer International Publishing, New York, 2022).
  12. I. N. Mishustin and O. Scavenius, Chiral fluid dynamics and collapse of vacuum bubbles, Phys. Rev. Lett. 83, 3134 (1999).
  13. A. Dumitru, J. Brachmann, E. S. Fraga, W. Greiner, A. D. Jackson, J. T. Lenaghan, O. Scavenius, and Horst Stoecker, Hydrodynamic models for heavy ion collisions and beyond, Acta Phys. Hung. A 14, 121 (2001).
  14. K. Paech, Horst Stoecker, and A. Dumitru, Hydrodynamics near a chiral critical point, Phys. Rev. C 68, 044907 (2003).
  15. C. E. Aguiar, E. S. Fraga, and T. Kodama, Hydrodynamical instabilities beyond the chiral critical point, J. Phys. G 32, 179 (2006).
  16. Kerstin Paech and Adrian Dumitru, Density inhomogeneities in heavy ion collisions around the critical point, Phys. Lett. B 623, 200 (2005).
  17. Marlene Nahrgang, Christoph Herold, Stefan Leupold, Igor Mishustin, and Marcus Bleicher, The impact of dissipation and noise on fluctuations in chiral fluid dynamics, J. Phys. G 40, 055108 (2013).
  18. Marlene Nahrgang, Stefan Leupold, Christoph Herold, and Marcus Bleicher, Nonequilibrium chiral fluid dynamics including dissipation and noise, Phys. Rev. C 84, 024912 (2011).
  19. Christoph Herold, Marlene Nahrgang, Igor Mishustin, and Marcus Bleicher, Chiral fluid dynamics with explicit propagation of the Polyakov loop, Phys. Rev. C 87, 014907 (2013).
  20. Igor N. Mishustin, Tomoi Koide, Gabriel S. Denicol, and Giorgio Torrieri, Dynamics and stability of chiral fluid, Phys. At. Nucl. 77, 1130 (2014).
  21. Eduardo Grossi, Alexander Soloviev, Derek Teaney, and Fanglida Yan, Transport and hydrodynamics in the chiral limit, Phys. Rev. D 102, 014042 (2020).
  22. Marcus Bluhm et al., Dynamics of critical fluctuations: Theory—phenomenology—heavy-ion collisions, Nucl. Phys. A1003, 122016 (2020).
  23. Nora Weickgenannt and Jean-Paul Blaizot, Chiral hydrodynamics of expanding systems, Phys. Rev. D 109, 056012 (2024).
  24. Joseph I. Kapusta and Charles Gale, Finite-Temperature Field Theory: Principles and Applications, 2nd ed., Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2006).
  25. Gabriel S. Rocha, Maurício N. Ferreira, Gabriel S. Denicol, and Jorge Noronha, Transport coefficients of quasiparticle models within a new relaxation time approximation of the Boltzmann equation, Phys. Rev. D 106, 036022 (2022).
  26. P. Chakraborty and J. I. Kapusta, Departure from equilibrium of the quasiparticle distribution functions in high-energy nuclear collisions, Phys. Rev. C 95, 014907 (2017).
  27. Gabriel S. Rocha, Gabriel S. Denicol, and Jorge Noronha, Novel relaxation time approximation to the relativistic Boltzmann equation, Phys. Rev. Lett. 127, 042301 (2021).
  28. B. Lee, Chiral Dynamics (Gordon and Breach, New York, 1972).
  29. Robert D. Pisarski and Frank Wilczek, Remarks on the chiral phase transition in chromodynamics, Phys. Rev. D 29, 338 (1984).
  30. G. Fejos and T. Hatsuda, Order of the SU(Nf)×SU(Nf) chiral transition via the functional renormalization group, Phys. Rev. D 110, 016021 (2024).
  31. Benjamin W. Lee, Renormalization of the sigma model, Nucl. Phys. B9, 649 (1969).
  32. Murray Gell-Mann and M. Levy, The axial vector current in beta decay, Nuovo Cimento 16, 705 (1960).
  33. G. Baym and G. Grinstein, Phase transition in the sigma model at finite temperature, Phys. Rev. D 15, 2897 (1977).
  34. Alexander Bochkarev and Joseph I. Kapusta, Chiral symmetry at finite temperature: Linear versus nonlinear sigma models, Phys. Rev. D 54, 4066 (1996).
  35. Neven Bilic and Hrvoje Nikolic, Chiral symmetry restoration in the linear sigma model at nonzero temperature and baryon density, Eur. Phys. J. C 6, 515 (1999).
  36. Nicholas Petropoulos, Linear sigma model and chiral symmetry at finite temperature, J. Phys. G 25, 2225 (1999).
  37. O. Scavenius and A. Dumitru, A first order chiral phase transition may naturally lead to the ‘quench’ initial condition and strong soft pion fields, Phys. Rev. Lett. 83, 4697 (1999).
  38. Dirk Roder, Jorg Ruppert, and Dirk H. Rischke, Chiral symmetry restoration in linear sigma models with different numbers of quark flavors, Phys. Rev. D 68, 016003 (2003).
  39. O. Scavenius, A. Mocsy, I. N. Mishustin, and D. H. Rischke, Chiral phase transition within effective models with constituent quarks, Phys. Rev. C 64, 045202 (2001).
  40. O. Scavenius, A. Dumitru, E. S. Fraga, J. T. Lenaghan, and A. D. Jackson, First order chiral phase transition in high-energy collisions: Can nucleation prevent spinodal decomposition?, Phys. Rev. D 63, 116003 (2001).
  41. Eduardo S. Fraga and G. Krein, Can dissipation prevent explosive decomposition in high-energy heavy ion collisions?. Phys. Lett. B 614, 181 (2005).
  42. T. Koide and M. Maruyama, Enhancement of critical slowing down in chiral phase transition: Langevin dynamics approach, Nucl. Phys. A742, 95 (2004).
  43. C. Sasaki, B. Friman, and K. Redlich, Chiral phase transition in the presence of spinodal decomposition, Phys. Rev. D 77, 034024 (2008).
  44. G. Marko and Zs. Szep, Influence of the Polyakov loop on the chiral phase transition in the two flavor chiral quark model, Phys. Rev. D 82, 065021 (2010).
  45. Marlene Nahrgang, Stefan Leupold, and Marcus Bleicher, Equilibration and relaxation times at the chiral phase transition including reheating, Phys. Lett. B 711, 109 (2012).
  46. Leticia F. Palhares and Eduardo S. Fraga, Droplets in the cold and dense linear sigma model with quarks, Phys. Rev. D 82, 125018 (2010).
  47. Topi Kahara and Kimmo Tuominen, Effective models of two-flavor QCD: Finite μ and mq-dependence, Phys. Rev. D 82, 114026 (2010).
  48. D. Kroff and E. S. Fraga, Nucleating quark droplets in the core of magnetars, Phys. Rev. D 91, 025017 (2015).
  49. O. Scavenius, Á. Mócsy, I. N. Mishustin, and D. H. Rischke, Chiral phase transition within effective models with constituent quarks, Phys. Rev. C 64, 045202 (2001).
  50. U. Vogl and W. Weise, The Nambu and Jona Lasinio model: Its implications for hadrons and nuclei, Prog. Part. Nucl. Phys. 27, 195 (1991).
  51. S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992).
  52. Michael Buballa, NJL model analysis of quark matter at large density, Phys. Rep. 407, 205 (2005).
  53. Sangyong Jeon and Laurence G. Yaffe, From quantum field theory to hydrodynamics: Transport coefficients and effective kinetic theory, Phys. Rev. D 53, 5799 (1996).
  54. F. Debbasch and W. A. van Leeuwen, General relativistic Boltzmann equation, II: Manifestly covariant treatment, Physica (Amsterdam) 388A, 1818 (2009).
  55. Paul Romatschke, Relativistic (lattice) Boltzmann equation with nonideal equation of state, Phys. Rev. D 85, 065012 (2012).
  56. Mubarak Alqahtani, Mohammad Nopoush, and Michael Strickland, Quasiparticle equation of state for anisotropic hydrodynamics, Phys. Rev. C 92, 054910 (2015).
  57. Mark I. Gorenstein and Shin Nan Yang, Gluon plasma with a medium-dependent dispersion relation, Phys. Rev. D 52, 5206 (1995).
  58. J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica (Utrecht) 74, 466 (1974).
  59. Kevin Dusling, Guy D. Moore, and Derek Teaney, Radiative energy loss and v(2) spectra for viscous hydrodynamics, Phys. Rev. C 81, 034907 (2010).
  60. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases (Cambridge University Press, Cambridge, England, 1990).
  61. S. Chapman, On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. R. Soc. A 216, 279 (1916).
  62. David Enskog, Kinetische theorie der Vorgänge in Mässig verdünnten Gasen (Almqvist & Wiksells boktryckeri-a.-b., 1917).
  63. C. G. van Weert S. R. Groot, and W. A. van Leeuwen, Relativistic Kinetic Theory: Principles and Applications (North-Holland, Amsterdam, 1980).
  64. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Elsevier/Academic Press, Amsterdam, 2007), pp. xlviii+1171, translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).

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Phys. Rev. D 113, 034016– Published 13 February, 2026
  • Received 5 November 2024
  • Accepted 21 January 2026
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