- 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

Phase structure of quark matter and in-medium properties of mesons from Callan-Symanzik flows

Sebastian Töpfel1,2, Jan M. Pawlowski3,4, and Jens Braun1,2,4

  • 1Technische Universität Darmstadt, Department of Physics, Institut für Kernphysik, Theoriezentrum, Schlossgartenstraße 2, D-64289 Darmstadt, Germany
  • 2Helmholtz Research Academy Hesse for FAIR, Campus Darmstadt, D-64289 Darmstadt, Germany
  • 3Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
  • 4ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany

PDF

Phys. Rev. D 113, 076015 – Published 16 April, 2026

DOI: https://doi.org/10.1103/s55g-qgpk

Abstract

We compute meson spectral functions at finite temperature and density in the quark-meson model, supplemented with a computation of the phase diagram. In particular, we provide a detailed analysis of the nonanalytic structure of the meson two-point functions which is of great relevance for phenomenological applications, such as moat regimes and inhomogeneous phases. Furthermore, it is also relevant from a field-theoretical standpoint as it provides an insight into the applicability of derivative expansions of the effective action to studies of general fermion-boson models, both at zero and finite chemical potential. Our computation is based on a functional renormalization group setup that preserves causality, all spacetime symmetries, and the Silver-Blaze property. The combination of these properties can only be achieved by a Callan-Symanzik regulator. Instead of momentum shell integrations, renormalization group flows generated by such a regulator describe the change of the theory induced by a change of the masses of the mesons and quarks. A particular focus of our work lies on the construction of controlled Callan-Symanzik flows in the presence of spontaneous and explicit chiral symmetry breaking by means of chiral Ward-Takahashi identities.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (81)

  1. J. Braun et al., Renormalised spectral flows, SciPost Phys. Core 6, 061 (2023).
  2. J. Fehre, D. F. Litim, J. M. Pawlowski, and M. Reichert, Lorentzian quantum gravity and the graviton spectral function, Phys. Rev. Lett. 130, 081501 (2023).
  3. T. Gasenzer and J. M. Pawlowski, Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach, Phys. Lett. B 670, 135 (2008).
  4. T. Gasenzer, S. Kessler, and J. M. Pawlowski, Far-from-equilibrium quantum many-body dynamics, Eur. Phys. J. C 70, 423 (2010).
  5. S. Floerchinger, Analytic continuation of functional renormalization group equations, J. High Energy Phys. 05 (2012) 021.
  6. N. Strodthoff, B.-J. Schaefer, and L. von Smekal, Quark-meson-diquark model for two-color QCD, Phys. Rev. D 85, 074007 (2012).
  7. K. Kamikado, N. Strodthoff, L. von Smekal, and J. Wambach, Real-time correlation functions in the O(N) model from the functional renormalization group, Eur. Phys. J. C 74, 2806 (2014).
  8. R.-A. Tripolt, N. Strodthoff, L. von Smekal, and J. Wambach, Spectral functions for the quark-meson model phase diagram from the functional renormalization group, Phys. Rev. D 89, 034010 (2014).
  9. J. M. Pawlowski and N. Strodthoff, Real time correlation functions and the functional renormalization group, Phys. Rev. D 92, 094009 (2015).
  10. T. Yokota, T. Kunihiro, and K. Morita, Functional renormalization group analysis of the soft mode at the QCD critical point, Prog. Theor. Exp. Phys. 2016, 073D01 (2016).
  11. K. Kamikado, T. Kanazawa, and S. Uchino, Mobile impurity in a Fermi sea from the functional renormalization group analytically continued to real time, Phys. Rev. A 95, 013612 (2017).
  12. C. Jung, F. Rennecke, R.-A. Tripolt, L. von Smekal, and J. Wambach, In-medium spectral functions of vector- and axial-vector mesons from the functional renormalization group, Phys. Rev. D 95, 036020 (2017).
  13. J. M. Pawlowski, N. Strodthoff, and N. Wink, Finite temperature spectral functions in the O(N)-model, Phys. Rev. D 98, 074008 (2018).
  14. T. Yokota, T. Kunihiro, and K. Morita, Tachyonic instability of the scalar mode prior to the QCD critical point based on the functional renormalization-group method in the two-flavor case, Phys. Rev. D 96, 074028 (2017).
  15. Z. Wang and P. Zhuang, Meson spectral functions at finite temperature and isospin density with the functional renormalization group, Phys. Rev. D 96, 014006 (2017).
  16. R.-A. Tripolt, C. Jung, N. Tanji, L. von Smekal, and J. Wambach, In-medium spectral functions and dilepton rates with the functional renormalization group, Nucl. Phys. A982, 775 (2019).
  17. R.-A. Tripolt, J. Weyrich, L. von Smekal, and J. Wambach, Fermionic spectral functions with the functional renormalization group, Phys. Rev. D 98, 094002 (2018).
  18. L. Corell, A. K. Cyrol, M. Heller, and J. M. Pawlowski, Flowing with the temporal renormalization group, Phys. Rev. D 104, 025005 (2021).
  19. S. Huelsmann, S. Schlichting, and P. Scior, Spectral functions from the real-time functional renormalization group, Phys. Rev. D 102, 096004 (2020).
  20. C. Jung, J.-H. Otto, R.-A. Tripolt, and L. von Smekal, Self-consistent O(4) model spectral functions from analytically continued functional renormalization group flows, Phys. Rev. D 104, 094011 (2021).
  21. Y.-y. Tan, Y.-r. Chen, and W.-j. Fu, Real-time dynamics of the O(4) scalar theory within the fRG approach, SciPost Phys. 12, 026 (2022).
  22. M. Heller and J. M. Pawlowski, Causal temporal renormalisation group flow of the energy-momentum tensor, arXiv:2112.12652.
  23. J. V. Roth, D. Schweitzer, L. J. Sieke, and L. von Smekal, Real-time methods for spectral functions, Phys. Rev. D 105, 116017 (2022).
  24. J. V. Roth and L. von Smekal, Critical dynamics in a real-time formulation of the functional renormalization group, J. High Energy Phys. 10 (2023) 065.
  25. J. Horak, F. Ihssen, J. M. Pawlowski, J. Wessely, and N. Wink, Scalar spectral functions from the spectral functional renormalization group, Phys. Rev. D 110, 056009 (2024).
  26. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910, 1 (2021).
  27. G. Markó, U. Reinosa, and Z. Szép, Bose-Einstein condensation and Silver Blaze property from the two-loop Φ-derivable approximation, Phys. Rev. D 90, 125021 (2014).
  28. N. Khan, J. M. Pawlowski, F. Rennecke, and M. M. Scherer, The phase diagram of QC2D from functional methods, arXiv:1512.03673.
  29. J. Braun, T. Dörnfeld, B. Schallmo, and S. Töpfel, Renormalization group studies of dense relativistic systems, Phys. Rev. D 104, 096002 (2021).
  30. 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].
  31. F. Rennecke, R. D. Pisarski, and D. H. Rischke, Particle interferometry in a moat regime, Phys. Rev. D 107, 116011 (2023).
  32. V. Schon and M. Thies, 2-D model field theories at finite temperature and density, arXiv:hep-th/0008175.
  33. M. Buballa and S. Carignano, Inhomogeneous chiral condensates, Prog. Part. Nucl. Phys. 81, 39 (2015).
  34. D. F. Litim and J. M. Pawlowski, Perturbation theory and renormalization group equations, Phys. Rev. D 65, 081701 (2002).
  35. D. F. Litim and J. M. Pawlowski, Completeness and consistency of renormalisation group flows, Phys. Rev. D 66, 025030 (2002).
  36. A. Geißel, T. Gorda, and J. Braun, Pressure and speed of sound in two-flavor color-superconducting quark matter at next-to-leading order, Phys. Rev. D 110, 014034 (2024).
  37. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  38. J. M. Pawlowski, Aspects of the functional renormalisation group, Ann. Phys. (Amsterdam) 322, 2831 (2007).
  39. H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852, 287 (2012).
  40. F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, Towards quantitative precision for QCD at large densities, Phys. Rev. D 111, 036030 (2025).
  41. K. Otto, C. Busch, and B.-J. Schaefer, Regulator scheme dependence of the chiral phase transition at high densities, Phys. Rev. D 106, 094018 (2022).
  42. H. Gies and C. Wetterich, Universality of spontaneous chiral symmetry breaking in gauge theories, Phys. Rev. D 69, 025001 (2004).
  43. J. Braun, The QCD phase boundary from quark-gluon dynamics, Eur. Phys. J. C 64, 459 (2009).
  44. M. Mitter, J. M. Pawlowski, and N. Strodthoff, Chiral symmetry breaking in continuum QCD, Phys. Rev. D 91, 054035 (2015).
  45. J. Braun, L. Fister, J. M. Pawlowski, and F. Rennecke, From quarks and gluons to hadrons: Chiral symmetry breaking in dynamical QCD, Phys. Rev. D 94, 034016 (2016).
  46. A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Nonperturbative quark, gluon, and meson correlators of unquenched QCD, Phys. Rev. D 97, 054006 (2018).
  47. W.-j. Fu, J. M. Pawlowski, and F. Rennecke, QCD phase structure at finite temperature and density, Phys. Rev. D 101, 054032 (2020).
  48. F. Ihssen, J. M. Pawlowski, F. R. Sattler, and N. Wink, Towards quantitative precision in functional QCD I, arXiv:2408.08413.
  49. F. Gao and J. M. Pawlowski, Phase structure of (2+1)-flavor QCD and the magnetic equation of state, Phys. Rev. D 105, 094020 (2022).
  50. J. Braun et al., Soft modes in hot QCD matter, Phys. Rev. D 111, 094010 (2025).
  51. P. Isserstedt, C. S. Fischer, and T. Steinert, Thermodynamics from the quark condensate, Phys. Rev. D 103, 054012 (2021).
  52. J. Braun, M. Leonhardt, and J. M. Pawlowski, Renormalization group consistency and low-energy effective theories, SciPost Phys. 6, 056 (2019).
  53. A. J. Helmboldt, J. M. Pawlowski, and N. Strodthoff, Towards quantitative precision in the chiral crossover: Masses and fluctuation scales, Phys. Rev. D 91, 054010 (2015).
  54. T. D. Cohen, Functional integrals for QCD at nonzero chemical potential and zero density, Phys. Rev. Lett. 91, 222001 (2003).
  55. J. Braun, M. Leonhardt, and M. Pospiech, Fierz-complete NJL model study: Fixed points and phase structure at finite temperature and density, Phys. Rev. D 96, 076003 (2017).
  56. J. Braun, M. Leonhardt, and M. Pospiech, Fierz-complete NJL model study III: Emergence from quark-gluon dynamics, Phys. Rev. D 101, 036004 (2020).
  57. M. Leonhardt, M. Pospiech, B. Schallmo, J. Braun, C. Drischler, K. Hebeler, and A. Schwenk, Symmetric nuclear matter from the strong interaction, Phys. Rev. Lett. 125, 142502 (2020).
  58. S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992).
  59. J. Braun, S. Finkbeiner, F. Karbstein, and D. Roscher, Search for inhomogeneous phases in fermionic models, Phys. Rev. D 91, 116006 (2015).
  60. D. Roscher, J. Braun, and J. E. Drut, Phase structure of mass- and spin-imbalanced unitary Fermi gases, Phys. Rev. A 91, 053611 (2015).
  61. J. Braun, F. Karbstein, S. Rechenberger, and D. Roscher, Crystalline ground states in Polyakov-loop extended Nambu–Jona-Lasinio models, Phys. Rev. D 93, 014032 (2016).
  62. R.-A. Tripolt, B.-J. Schaefer, L. von Smekal, and J. Wambach, Low-temperature behavior of the quark-meson model, Phys. Rev. D 97, 034022 (2018).
  63. T. F. Motta, J. Bernhardt, M. Buballa, and C. S. Fischer, Toward a stability analysis of inhomogeneous phases in QCD, Phys. Rev. D 108, 114019 (2023).
  64. R. D. Pisarski and F. Rennecke, Signatures of moat regimes in heavy-ion collisions, Phys. Rev. Lett. 127, 152302 (2021).
  65. M. Haensch, F. Rennecke, and L. von Smekal, Medium induced mixing, spatial modulations, and critical modes in QCD, Phys. Rev. D 110, 036018 (2024).
  66. L. Pannullo, M. Wagner, and M. Winstel, Regularization effects in the Nambu–Jona-Lasinio model: Strong scheme dependence of inhomogeneous phases and persistence of the moat regime, Phys. Rev. D 110, 076006 (2024).
  67. W.-j. Fu, J. M. Pawlowski, R. D. Pisarski, F. Rennecke, R. Wen, and S. Yin, The QCD moat regime and its real-time properties, Phys. Rev. D 111, 094026 (2025).
  68. J. Friedel, XIV. The distribution of electrons round impurities in monovalent metals, London, Edinburgh, Dublin Philos. Mag. J. Sci. 43, 153 (1952).
  69. J. Friedel, Electronic structure of primary solid solutions in metals, Adv. Phys. 3, 446 (1954).
  70. J. Friedel, Metallic alloys, Nuovo Cimento 7, 287 (1958).
  71. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, Boston, 1971).
  72. F. Rennecke and S. Yin, Dissecting the moat regime at low energies I: Renormalization and the phase structure, arXiv:2510.06712.
  73. A. Koenigstein, L. Pannullo, S. Rechenberger, M. J. Steil, and M. Winstel, Detecting inhomogeneous chiral condensation from the bosonic two-point function in the (1+1)-dimensional Gross–Neveu model in the mean-field approximation, J. Phys. A 55, 375402 (2022).
  74. A. Koenigstein and L. Pannullo, Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions 1≤d<3. II. Nonzero temperature and chemical potential, Phys. Rev. D 109, 056015 (2024).
  75. A. Koenigstein and M. Winstel, Revisiting the spatially inhomogeneous condensates in the (1+1)-dimensional chiral Gross–Neveu model via the bosonic two-point function in the infinite-N limit, J. Phys. A 57, 335401 (2024).
  76. J. Braun, Thermodynamics of QCD low-energy models and the derivative expansion of the effective action, Phys. Rev. D 81, 016008 (2010).
  77. J. Braun and B. Schallmo, From quarks and gluons to color superconductivity at supranuclear densities, Phys. Rev. D 105, 036003 (2022).
  78. B.-J. Schaefer and M. Wagner, The three-flavor chiral phase structure in hot and dense QCD matter, Phys. Rev. D 79, 014018 (2009).
  79. J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113, 1 (2002).
  80. J. Braun et al. (fQCD Collaboration) (2026), https://fqcd-collaboration.github.io.
  81. S. Töpfel, A. Geißel, and J. Braun, Subtleties in the calculation of correlation functions for hot and dense systems, Phys. Rev. D 111, 016023 (2025).

Outline

Information

Phys. Rev. D 113, 076015– Published 16 April, 2026
  • Received 6 February 2025
  • Accepted 23 February 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