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Effects of equilibrium coexisting phases in the first-order chiral transition within the linear σ model with quarks

R. M. Aguirre

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Phys. Rev. C 113, 045204 – Published 7 April, 2026

DOI: https://doi.org/10.1103/13s1-hm5v

Abstract

The first order chiral phase transition for quark matter with flavor imbalance is studied using the Linear σ model with quarks, also known as quark-meson model. Special attention is paid to the role of the scalar isovector meson. The general consensus presently is that the chiral transition changes from a smooth crossover to first-order at low temperatures. This transition is assumed to be discontinuous with unstable or metastable intermediate states. However, if multiple charges are simultaneously conserved the system could undergo a continuous change through a coexistence of equilibrium states. Under such assumption the bulk properties are analyzed and several remarkable effects for the speed of sound and the susceptibilities are stressed.

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References (75)

  1. M. G. Alford, A. Kapustin, and F. Wilczek, Imaginary chemical potential and finite fermion density on the lattice,Phys. Rev. D 59, 054502 (1999).
  2. D. T. Son and M. A. Stephanov, QCD at finite isospin density, Phys. Rev. Lett. 86, 592 (2001).
  3. Y. Nishida, Phase structures of strong coupling lattice QCD with finite baryon and isospin density, Phys. Rev. D 69, 094501 (2004).
  4. S. Ejiri, Y. Maezawa, N. Ukita, S. Aoki, T. Hatsuda, N. Ishii, K. Kanaya, and T. Umeda (WHOT-QCD Collaboration), Equation of state and heavy-quark free energy at finite temperature and density in two flavor lattice QCD with Wilson quark action, Phys. Rev. D 82, 014508 (2010).
  5. K. Aryal, C. Constantinou, R. L. S. Farias, and V. Dexheimer, High-energy phase diagrams with charge and isospin axes under heavy-ion collision and stellar conditions, Phys. Rev. D 102, 076016 (2020).
  6. A. Prakash, I. Gupta, M. Breschi, R. Kashyap, D. Radice, S. Bernuzzi, D. Logoteta, and B. S. Sathyaprakash, Detectability of QCD phase transitions in binary neutron star mergers: Bayesian inference with the next generation gravitational wave detectors, Phys. Rev. D 109, 103008 (2024).
  7. D. Toublan and J. B. Kogut, Isospin chemical potential and the QCD phase diagram at nonzero temperature and baryon chemical potential, Phys. Lett. B 564, 212 (2003).
  8. M. Frank, M. Buballa, and M. Oertel, Flavor mixing effects on the QCD phase diagram at nonvanishing isospin chemical potential: One or two phase transitions? Phys. Lett. B 562, 221 (2003).
  9. A. Barducci, R. Casalbuoni, G. Pettini, and L. Ravagli, A calculation of the QCD phase diagram at finite temperature, and baryon and isospin chemical potentials, Phys. Rev. D 69, 096004 (2004).
  10. A. Barducci, G. Pettini, L. Ravagli, and R. Casalbuoni, Ladder QCD at finite isospin chemical potential, Phys. Lett. B 564, 217 (2003).
  11. M. Loewe and C. Villavicencio, Two-flavor condensates in chiral dynamics: Temperature and isospin density effects, Phys. Rev. D 71, 094001 (2005).
  12. L.-y. He, M. Jin, and P.-f. Zhuang, Pion superfluidity and meson properties at finite isospin density, Phys. Rev. D 71, 116001 (2005).
  13. L. He and P. Zhuang, Phase structure of Nambu–Jona-Lasinio model at finite isospin density, Phys. Lett. B 615, 93 (2005).
  14. Z. Zhang and Y.-x. Liu, Two-flavor QCD phases and condensates at finite isospin chemical potential, Phys. Rev. C 75, 035201 (2007).
  15. S. Mukherjee, M. G. Mustafa, and R. Ray, Thermodynamics of the Polyakov-Nambu-Jona-Lasinio model with nonzero baryon and isospin chemical potentials, Phys. Rev. D 75, 094015 (2007).
  16. P. Kovacs and Z. Szep, Influence of the isospin and hypercharge chemical potentials on the location of the critical end point in the µB-T phase diagram of the SU(3)L×SU(3)R chiral quark model, Phys. Rev. D 77, 065016 (2008).
  17. C.-f. Mu, L.-y. He, and Y.-x. Liu, Evaluating the phase diagram at finite isospin and baryon chemical potentials in the Nambu–Jona-Lasinio model, Phys. Rev. D 82, 056006 (2010).
  18. H. Abuki, Fate of chiral critical point under the strong isospin asymmetry, Phys. Rev. D 87, 094006 (2013).
  19. T. Xia, L. He, and P. Zhuang, Three-flavor Nambu–Jona-Lasinio model at finite isospin chemical potential, Phys. Rev. D 88, 056013 (2013).
  20. R. Stiele, E. S. Fraga, and J. Schaffner-Bielich, Thermodynamics of (2+1)-flavor strongly interacting matter at nonzero isospin, Phys. Lett. B 729, 72 (2014).
  21. H. Liu, J. Xu, L.-W. Chen, and K.-J. Sun, Isospin properties of quark matter from a 3-flavor NJL model, Phys. Rev. D 94, 065032 (2016).
  22. S. S. Avancini, A. Bandyopadhyay, D. C. Duarte, and R. L. S. Farias, Cold QCD at finite isospin density: Confronting effective models with recent lattice data, Phys. Rev. D 100, 116002 (2019).
  23. B. S. Lopes, S. S. Avancini, A. Bandyopadhyay, D. C. Duarte, and R. L. S. Farias, Hot QCD at finite isospin density: Confronting the SU(3) Nambu–Jona-Lasinio model with recent lattice data, Phys. Rev. D 103, 076023 (2021).
  24. P. Adhikari and J. O. Andersen, QCD at finite isospin density: Chiral perturbation theory confronts lattice data, Phys. Lett. B 804, 135352 (2020).
  25. P. Costa, R. Câmara Pereira, and C. Providência, Role of the conserved charges in the chiral symmetry restoration phase transition, Phys. Rev. D 102, 054010 (2020).
  26. P.-C. Chu, X.-H. Li, H. Liu, M. Ju, and Y. Zhou, Properties of isospin asymmetric quark matter in quark stars, Phys. Rev. C 108, 025808 (2023).
  27. A. Ayala, A. Bandyopadhyay, R. L. S. Farias, L. A. Hernández, and J. L. Hernández, QCD equation of state at finite isospin density from the linear sigma model with quarks: The cold case, Phys. Rev. D 107, 074027 (2023).
  28. A. Ayala, B. S. Lopes, R. L. S. Farias, and L. C. Parra, On the origin of the peak of the sound velocity for isospin imbalanced strongly interacting matter, Phys. Lett. B 864, 139396 (2025).
  29. S. K. Ghosh, A. Lahiri, S. Majumder, M. G. Mustafa, S. Raha, and R. Ray, Quark number susceptibility: Revisited with fluctuation-dissipation theorem in mean field theories, Phys. Rev. D 90, 054030 (2014).
  30. D. Parganlija, F. Giacosa, and D. H. Rischke, Vacuum properties of mesons in a linear sigma model with vector mesons and global chiral invariance, Phys. Rev. D 82, 054024 (2010).
  31. D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, and D. H. Rischke, Meson vacuum phenomenology in a three-flavor linear sigma model with (axial-)vector mesons, Phys. Rev. D 87, 014011 (2013).
  32. P. Kovács, Z. Szép, and G. Wolf, Existence of the critical endpoint in the vector meson extended linear sigma model, Phys. Rev. D 93, 114014 (2016).
  33. S. Gasiorowicz and D. A. Geffen, Effective Lagrangians and field algebras with chiral symmetry, Rev. Mod. Phys. 41, 531 (1969).
  34. D. Metzger, H. Meyer-Ortmanns, and H. J. Pirner, Chiral symmetry restoration at finite temperature in the linear sigma model, Phys. Lett. B 321, 66 (1994); 328, 547(E) (1994).
  35. J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich, Chiral symmetry restoration at nonzero temperature in the SU(3)r×SU(3)l linear sigma model, Phys. Rev. D 62, 085008 (2000).
  36. B.-J. Schaefer and M. Wagner, The three-flavor chiral phase structure in hot and dense QCD matter, Phys. Rev. D 79, 014018 (2009).
  37. A. Ayala, L. A. Hernández, M. Loewe, J. C. Rojas, and R. Zamora, On the critical end point in a two-flavor linear sigma model coupled to quarks, Eur. Phys. J. A 56, 71 (2020).
  38. L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys. Rev. D 9, 3320 (1974).
  39. N. Petropoulos, Linear sigma model and chiral symmetry at finite temperature, J. Phys. G: Nucl. Part. Phys. 25, 2225 (1999).
  40. J. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10, 2428 (1974).
  41. C. Sasaki, B. Friman, and K. Redlich, Quark number fluctuations in a chiral model at finite baryon chemical potential, Phys. Rev. D 75, 054026 (2007).
  42. C. Sasaki, B. Friman, and K. Redlich, Density fluctuations in the presence of spinodal instabilities, Phys. Rev. Lett. 99, 232301 (2007).
  43. C. Sasaki, B. Friman, and K. Redlich, Chiral phase transition in the presence of spinodal decomposition, Phys. Rev. D 77, 034024 (2008).
  44. B. Stokic, B. Friman, and K. Redlich, Kurtosis and compressibility near the chiral phase transition, Phys. Lett. B 673, 192 (2009).
  45. N. K. Glendenning, First order phase transitions with more than one conserved charge: Consequences for neutron stars, Phys. Rev. D 46, 1274 (1992).
  46. R. V. Gavai and S. Gupta, Quark number susceptibilities, strangeness and dynamical confinement, Phys. Rev. D 64, 074506 (2001).
  47. L. Ferroni and V. Koch, Mean field approach to flavor susceptibilities with a vector interaction, Phys. Rev. C 83, 045205 (2011).
  48. C. Ratti, S. Roessner, and W. Weise, Quark number susceptibilities: Lattice QCD versus PNJL model, Phys. Lett. B 649, 57 (2007).
  49. G. A. Almasi, B. Friman, and K. Redlich, Baryon number fluctuations in chiral effective models and their phenomenological implications, Phys. Rev. D 96, 014027 (2017).
  50. V. Skokov, B. Friman, E. Nakano, K. Redlich, and B. J. Schaefer, Vacuum fluctuations and the thermodynamics of chiral models, Phys. Rev. D 82, 034029 (2010).
  51. V. Skokov, B. Stokic, B. Friman, and K. Redlich, Meson fluctuations and thermodynamics of the Polyakov loop extended quark-meson model, Phys. Rev. C 82, 015206 (2010).
  52. V. Skokov, B. Friman, and K. Redlich, Quark number fluctuations in the Polyakov loop-extended quark-meson model at finite baryon density, Phys. Rev. C 83, 054904 (2011).
  53. M. Asakawa, U. W. Heinz, and B. Muller, Fluctuation probes of quark deconfinement, Phys. Rev. Lett. 85, 2072 (2000).
  54. R. V. Gavai and S. Gupta, Fluctuations, strangeness and quasi-quarks in heavy-ion collisions from lattice QCD, Phys. Rev. D 73, 014004 (2006).
  55. S. Datta, R. V. Gavai, and S. Gupta, Quark number susceptibilities and equation of state at finite chemical potential in staggered QCD with Nt=8, Phys. Rev. D 95, 054512 (2017).
  56. R. M. Aguirre, Coexisting phases in the chiral transition within the Linear sigma model with quarks, arXiv:2511.11822.
  57. H. Goldberg, Chiral symmetry restoration at finite temperature in the SU(3) × SU(3) Linear sigma model, Phys. Lett. B 131, 133 (1983).
  58. G. Amelino-Camelia and S.-Y. Pi, Selfconsistent improvement of the finite temperature effective potential, Phys. Rev. D 47, 2356 (1993).
  59. D. Röder, J. Ruppert, and D. H. Rischke, Chiral symmetry restoration in linear sigma models with different numbers of quark flavors, Phys. Rev. D 68, 016003 (2003).
  60. S. K. Ghosh, T. K. Mukherjee, M. G. Mustafa, and R. Ray, Susceptibilities and speed of sound from Polyakov-Nambu-Jona-Lasinio model, Phys. Rev. D 73, 114007 (2006).
  61. J. M. Pawlowski and F. Rennecke, Higher order quark-mesonic scattering processes and the phase structure of QCD, Phys. Rev. D 90, 076002 (2014).
  62. S. K. Rai and V. K. Tiwari, On-shell versus curvature mass parameter fixing schemes in the quark-meson model and its phase diagrams, Phys. Rev. D 105, 094010 (2022).
  63. N. Bilic and H. Nikolic, Chiral symmetry restoration in the linear sigma model at nonzero temperature and baryon density, Eur. Phys. J. C 6, 515 (1999).
  64. B.-J. Schaefer and M. Wagner, QCD critical region and higher moments for three-flavor models, Phys. Rev. D 85, 034027 (2012).
  65. A. Biguet, H. Hansen, T. Brugière, P. Costa, and P. Borgnat, Sensitivity of predictions in an effective model – application to the chiral critical end point position in the Nambu–Jona-Lasinio model, Eur. Phys. J. A 51, 121 (2015).
  66. H. Heiselberg and M. Hjorth-Jensen, Phases of dense matter in neutron stars, Phys. Rep. 328, 237 (2000).
  67. P. Costa, M. C. Ruivo, and C. A. de Sousa, Thermodynamics and critical behavior in the Nambu–Jona-Lasinio model of QCD, Phys. Rev. D 77, 096001 (2008).
  68. L.-M. Liu, W.-H. Zhou, J. Xu, and G.-X. Peng, Isospin effect on quark matter instabilities, Phys. Lett. B 822, 136694 (2021).
  69. A. Zacchi and J. Schaffner-Bielich, Implications of the fermion vacuum term in the extended SU(3) quark meson model on compact star properties, Phys. Rev. D 100, 123024 (2019).
  70. F. G. Gardim, G. Giacalone, M. Luzum, and J.-Y. Ollitrault, Thermodynamics of hot strong-interaction matter from ultrarelativistic nuclear collisions, Nat. Phys. 16, 615 (2020).
  71. Y.-S. Mu, J.-A. Sun, L. Yan, and X.-G. Huang, Extracting the speed of sound in heavy-ion collisions: A study of quantum-initiated fluctuations and thermalization, Phys. Rev. Lett. 135, 162301 (2025).
  72. S. Hernández-Ortiz, R. M. von Dossow, and A. Raya, Constraining the position of the CEP through the speed of sound in the LSMq, Eur. Phys. J. A 60, 154 (2024).
  73. M. D'Agostino, et al., Negative heat capacity in the critical region of nuclear fragmentation: An Experimental evidence of the liquid gas phase transition, Phys. Lett. B 473, 219 (2000).
  74. B. Borderie, et al., Negative heat capacity for hot nuclei using formulation from the microcanonical ensemble INDRA Collaboration, Eur. Phys. J. A 56, 101 (2020).
  75. J. N. De, S. K. Samaddar, S. Shlomo, and J. B. Natowitz, Continuous phase transition and negative specific heat in finite nuclei, Phys. Rev. C 73, 034602 (2006).

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Phys. Rev. C 113, 045204– Published 7 April, 2026
  • Received 23 November 2025
  • Revised 27 January 2026
  • Accepted 13 March 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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