- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Thermodynamics in symmetry-improved Cornwall-Jackiw-Tomboulis formalism: Application to the low-energy effective theory of QCD

Yuepeng Guan1,*, Mamiya Kawaguchi2,†, Shinya Matsuzaki1,‡, and Akio Tomiya3,4,§

  • *Contact author: guanyp22@mails.jlu.edu.cn
  • †Contact author: mamiya@aust.edu.cn
  • ‡Contact author: synya@jlu.edu.cn
  • §Contact author: akio@yukawa.kyoto-u.ac.jp
PDF

Phys. Rev. D 114, 054015 – Published 8 September, 2026

DOI: https://doi.org/10.1103/gqzf-qvl6

Abstract

We study the thermodynamics of the symmetry-improved Cornwall-Jackiw-Tomboulis formalism and apply it to a low-energy effective theory of QCD. In the symmetry-improved formulation, Ward-Takahashi identities are restored by auxiliary sources whose values are fixed self-consistently by the equilibrium state. While this construction improves the symmetry properties of the loopwise truncated two-particle-irreducible (2PI) theory, it also makes the thermodynamic interpretation of the pressure nontrivial. We formulate several pressure prescriptions, including the conventional vacuum-subtracted pressure, a source-matched subtraction, and a pulled-back pressure in which the explicit source-induced energy shift is removed. Using the three-flavor linear sigma model with quarks, we analyze the equation of state, isentropic trajectories, adiabatic sound velocity, and trace anomaly across the chiral transition. We find that the global thermodynamic structure is stable under the different prescriptions, while quantitative differences are concentrated near the crossover and first-order transition region. These results establish a practical framework for constructing thermodynamically consistent observables in symmetry-improved 2PI approaches.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. D. Boyanovsky, H. J. de Vega, and D. J. Schwarz, Phase transitions in the early and the present universe, Annu. Rev. Nucl. Part. Sci. 56, 441 (2006).
  2. A. Addazi, T. Lundberg, A. Marcianò, R. Pasechnik, and M. Šumbera, Cosmology from strong interactions, Universe 8, 451 (2022).
  3. E. S. Fraga, A. Kurkela, and A. Vuorinen, Neutron star structure from QCD, Eur. Phys. J. A 52, 49 (2016).
  4. K. Fukushima, QCD phase diagram and astrophysical implications, J. Subatomic Part. Cosmol. 3, 100066 (2025).
  5. P. Foka and M. A. Janik, An overview of experimental results from ultra-relativistic heavy-ion collisions at the CERN LHC: Bulk properties and dynamical evolution, Rev. Phys. 1, 154 (2016).
  6. K. Fukushima, B. Mohanty, and N. Xu, Little-Bang and femto-nova in nucleus-nucleus collisions, AAPPS Bull. 31, 1 (2021).
  7. M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Signatures of the tricritical point in QCD, Phys. Rev. Lett. 81, 4816 (1998).
  8. M. A. Stephanov, Non-Gaussian fluctuations near the QCD critical point, Phys. Rev. Lett. 102, 032301 (2009).
  9. M. Stephanov, QCD critical point: Recent developments, EPJ Web Conf. 314, 00042 (2024).
  10. S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabó, QCD equation of state at nonzero chemical potential: Continuum results with physical quark masses at order mu2, J. High Energy Phys. 08 (2012) 053.
  11. S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and K. K. Szabo, Full result for the QCD equation of state with 2+1 flavors, Phys. Lett. B 730, 99 (2014).
  12. A. Bazavov et al. (HotQCD Collaboration), Equation of state in (2+1)-flavor QCD, Phys. Rev. D 90, 094503 (2014).
  13. A. Bazavov et al. (HotQCD Collaboration), Chiral crossover in QCD at zero and non-zero chemical potentials, Phys. Lett. B 795, 15 (2019).
  14. K. Nagata, Finite-density lattice QCD and sign problem: Current status and open problems, Prog. Part. Nucl. Phys. 127, 103991 (2022).
  15. K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rep. Prog. Phys. 74, 014001 (2011).
  16. J. Braun, M. Leonhardt, and J. M. Pawlowski, Renormalization group consistency and low-energy effective theories, SciPost Phys. 6, 056 (2019).
  17. H. Zhang, D. Hou, T. Kojo, and B. Qin, Functional renormalization group study of the quark-meson model with ω meson, Phys. Rev. D 96, 114029 (2017).
  18. M. Osman, D. Hou, W. Wang, and H. Zhang, Functional renormalization group study of the quark-meson model with ω and ρ vector mesons, Eur. Phys. J. C 85, 804 (2025).
  19. M. Drews, T. Hell, B. Klein, and W. Weise, Thermodynamic phases and mesonic fluctuations in a chiral nucleon-meson model, Phys. Rev. D 88, 096011 (2013).
  20. W.-j. Fu, J. M. Pawlowski, and F. Rennecke, Strangeness neutrality and QCD thermodynamics, SciPost Phys. Core 2, 002 (2020).
  21. H. Hansen, R. Stiele, and P. Costa, Quark and Polyakov-loop correlations in effective models at zero and nonvanishing density, Phys. Rev. D 101, 094001 (2020).
  22. J. O. Andersen and M. Strickland, Resummation in hot field theories, Ann. Phys. (Amsterdam) 317, 281 (2005).
  23. D. Curtin, P. Meade, and H. Ramani, Thermal resummation and phase transitions, Eur. Phys. J. C 78, 787 (2018).
  24. L. Niemi, P. Schicho, and T. V. I. Tenkanen, Singlet-assisted electroweak phase transition at two loops, Phys. Rev. D 103, 115035 (2021).
  25. J. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10, 2428 (1974).
  26. G. Baym, Selfconsistent approximation in many body systems, Phys. Rev. 127, 1391 (1962).
  27. N. Petropoulos, Linear sigma model and chiral symmetry at finite temperature, J. Phys. G 25, 2225 (1999).
  28. J. T. Lenaghan and D. H. Rischke, The O(N) model at finite temperature: Renormalization of the gap equations in Hartree and large N approximation, J. Phys. G 26, 431 (2000).
  29. J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich, Chiral symmetry restoration at nonzero temperature in the SU(3)(r) x SU(3)(l) linear sigma model, Phys. Rev. D 62, 085008 (2000).
  30. D. Roder, 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).
  31. A. Pilaftsis and D. Teresi, Symmetry improved CJT effective action, Nucl. Phys. B874, 594 (2013).
  32. H. Mao, On the symmetry improved CJT formalism in the O(4) linear sigma model, Nucl. Phys. A925, 185 (2014).
  33. A. Pilaftsis and D. Teresi, Symmetry-improved 2PI approach to the Goldstone-boson IR problem of the SM effective potential, Nucl. Phys. B906, 381 (2016).
  34. B. Garbrecht and P. Millington, Constraining the effective action by a method of external sources, Nucl. Phys. B906, 105 (2016).
  35. M. Kobayashi and T. Maskawa, Chiral symmetry and eta-x mixing, Prog. Theor. Phys. 44, 1422 (1970).
  36. M. Kobayashi, H. Kondo, and T. Maskawa, Symmetry breaking of the chiral u(3) x u(3) and the quark model, Prog. Theor. Phys. 45, 1955 (1971).
  37. G. ’t Hooft, Symmetry breaking through Bell-Jackiw anomalies, Phys. Rev. Lett. 37, 8 (1976).
  38. G. ’ Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14, 3432 (1976).
  39. A. A. Osipov and B. Hiller, Path integral bosonization of the ’t Hooft determinant: Fluctuations and multiple vacua, Phys. Lett. B 539, 76 (2002).
  40. A. A. Osipov and B. Hiller, Path integral bosonization of the ’t Hooft determinant: Quasiclassical corrections, Eur. Phys. J. C 35, 223 (2004).
  41. A. A. Osipov, B. Hiller, J. Moreira, and A. H. Blin, Stationary phase corrections in the process of bosonization of multi-quark interactions, Eur. Phys. J. C 46, 225 (2006).
  42. J.-P. Blaizot, J. M. Pawlowski, and U. Reinosa, Functional renormalization group and 2PI effective action formalism, Ann. Phys. (Amsterdam) 431, 168549 (2021).
  43. J. I. Kapusta and C. Gale, Finite-temperature Field Theory: Principles and Applications, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2011).
  44. B.-J. Schaefer and M. Wagner, The three-flavor chiral phase structure in hot and dense QCD matter, Phys. Rev. D 79, 014018 (2009).
  45. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  46. Y. Guan, M. Kawaguchi, S. Matsuzaki, and A. Tomiya, Columbia plot based on symmetry-improved CJT formalism in linear sigma model, Phys. Rev. D 113, 014013 (2026).
  47. 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).
  48. M. Kawaguchi, S. Matsuzaki, and A. Tomiya, Analysis of nonperturbative flavor violation at chiral crossover criticality in QCD, Phys. Rev. D 103, 054034 (2021).
  49. T. D. Cohen, Functional integrals for QCD at nonzero chemical potential and zero density, Phys. Rev. Lett. 91, 222001 (2003).
  50. W.-j. Fu, J. M. Pawlowski, F. Rennecke, and B.-J. Schaefer, Baryon number fluctuations at finite temperature and density, Phys. Rev. D 94, 116020 (2016).
  51. W.-j. Fu, J. M. Pawlowski, and F. Rennecke, QCD phase structure at finite temperature and density, Phys. Rev. D 101, 054032 (2020).
  52. D. Blaschke and D. Ebert, Variational path-integral approach to back-reactions of composite mesons in the Nambu-Jona-Lasinio model, Nucl. Phys. B921, 753 (2017).
  53. F. Rennecke and R. D. Pisarski, Moat regimes in QCD and their signatures in heavy-ion collisions, Proc. Sci. CPOD2021 (2022) 016. [arXiv:2110.02625].
  54. W.-j. Fu, J. M. Pawlowski, R. D. Pisarski, F. Rennecke, R. Wen, and S. Yin, QCD moat regime and its real-time properties, Phys. Rev. D 111, 094026 (2025).
  55. B.-J. Schaefer, M. Wagner, and J. Wambach, Thermodynamics of (2+1)-flavor QCD: Confronting models with lattice studies, Phys. Rev. D 81, 074013 (2010).
  56. T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phase structure and thermodynamics of QCD, Phys. Rev. D 88, 014007 (2013).
  57. S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabó (Wuppertal-Budapest Collaboration), Is there still any Tc mystery in lattice QCD? results with physical masses in the continuum limit III, J. High Energy Phys. 09 (2010) 073.
  58. M. Mitter and B.-J. Schaefer, Fluctuations and the axial anomaly with three quark flavors, Phys. Rev. D 89, 054027 (2014).
  59. A. Bazavov et al., The chiral and deconfinement aspects of the QCD transition, Phys. Rev. D 85, 054503 (2012).
  60. M. Motta, R. Stiele, W. M. Alberico, and A. Beraudo, Isentropic evolution of the matter in heavy-ion collisions and the search for the critical endpoint, Eur. Phys. J. C 80, 770 (2020).

Outline

Information

Phys. Rev. D 114, 054015– Published 8 September, 2026
  • Received 4 June 2026
  • Accepted 20 August 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation