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  • Open Access

Assessing the reconstruction of the critical line in the QCD phase diagram from imaginary to real chemical potential

Fabrizio Murgana* and Marco Ruggieri†

  • *Contact author: fabrizio.murgana@dfa.unict.it
  • †Contact author: marco.ruggieri@dfa.unict.it
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Phys. Rev. D 112, 056001 – Published 8 September, 2025

DOI: https://doi.org/10.1103/vl7y-j84r

Abstract

We test a technique adopted in the lattice simulations framework, to reconstruct the chiral-phase boundary at real chemical potential, μ, via extrapolation from imaginary μ. We use a low-energy effective model, the quark-meson model, both in the mean-field approximation and within the functional renormalization group, the latter in the local potential approximation. The model provides results both for real and imaginary values of μ; thus a direct comparison can be performed between the prediction of the model for real values of μ and the ones obtained via extrapolation from the results at imaginary μ. We compute an effective convergence radius for the reconstruction technique, μconv, and find μconv≈263  MeV. This value sustains the validity of the reconstruction technique also for finite and moderate values of the chemical potential. On the other hand, within our model, μconv is smaller then the value of μ where we find the actual critical endpoint. Near this point of the phase diagram, we find a discrepancy between the actual phase boundary and the one obtained via extrapolation of ≈100%. Therefore, our results show that the location of the critical endpoint obtained via reconstruction from imaginary μ might be considered with due caution.

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References (69)

  1. K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rep. Prog. Phys. 74, 014001 (2010).
  2. A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto, and G. Pettini, Chiral symmetry breaking in QCD at finite temperature and density, Phys. Lett. B 231, 463 (1989).
  3. H. Meyer-Ortmanns, Phase transitions in quantum chromodynamics, Rev. Mod. Phys. 68, 473 (1996).
  4. T. K. Herbst, J. M. Pawlowski, and B.-J. Schaefer, Phase structure and thermodynamics of QCD, Phys. Rev. D 88, 014007 (2013).
  5. F. Murgana, V. Greco, M. Ruggieri, and D. Zappalà, Functional renormalization group study of thermodynamic geometry around the phase transition of quantum chromodynamics, Phys. Rev. D 109, 096017 (2024).
  6. M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Signatures of the tricritical point in QCD, Phys. Rev. Lett. 81, 4816 (1998).
  7. F. Wilczek, Application of the renormalization group to a second order QCD phase transition, Int. J. Mod. Phys. A 07, 3911 (1992); 07, 6951(E) (1992).
  8. M. Ruggieri, M. Tachibana, and V. Greco, Renormalized vs nonrenormalized chiral transition in a magnetic background, J. High Energy Phys. 07 (2013) 165.
  9. Y. Zhang, D. Zhang, and X. Luo, Experimental study of the QCD phase diagram in relativistic heavy-ion collisions, Nucl. Tech. 46, 040001 (2023).
  10. G. Roland, K. Safarik, and P. Steinberg, Heavy-ion collisions at the LHC, Prog. Part. Nucl. Phys. 77, 70 (2014).
  11. N. Armesto, N. Borghini, S. Jeon, and U. A. Wiedemann, Proceedings, Workshop on Heavy Ion Collisions at the LHC: Last Call for Predictions: Geneva, Switzerland, May 14—June 8, 2007 (IOP Publishing, Bristol, UK, 2008), Vol. 35.
  12. N. Xu (STAR Collaboration), An overview of STAR experimental results, Nucl. Phys. A931, 1 (2014).
  13. A. Bazavov et al. (HotQCD Collaboration), Chiral and deconfinement aspects of the QCD transition, Phys. Rev. D 85, 054503 (2012).
  14. C. Ratti and W. Weise, Thermodynamics of two-colour QCD and the Nambu Jona-Lasinio model, Phys. Rev. D 70, 054013 (2004).
  15. C. Ratti, M. A. Thaler, and W. Weise, Phases of QCD: Lattice thermodynamics and a field theoretical model, Phys. Rev. D 73, 014019 (2006).
  16. G. Pan and Z. Y. Meng, The sign problem in quantum monte carlo simulations, in Encyclopedia of Condensed Matter Physics (Second Edition), edited by T. Chakraborty (Academic Press, Oxford, 2024), 2nd ed., pp. 879–893.
  17. R. V. Gavai and S. Gupta, Pressure and nonlinear susceptibilities in QCD at finite chemical potentials, Phys. Rev. D 68, 034506 (2003).
  18. A. Bazavov et al., The QCD equation of state to O(μB6) from Lattice QCD, Phys. Rev. D 95, 054504 (2017).
  19. M. G. Alford, A. Kapustin, and F. Wilczek, Imaginary chemical potential and finite fermion density on the lattice, Phys. Rev. D 59, 054502 (1999).
  20. R. Bellwied, S. Borsanyi, Z. Fodor, J. Günther, S. D. Katz, C. Ratti, and K. K. Szabo, The QCD phase diagram from analytic continuation, Phys. Lett. B 751, 559 (2015).
  21. G. Parisi, On complex probabilities, Phys. Lett. 131B, 393 (1983).
  22. F. Attanasio, B. Jäger, and F. P. G. Ziegler, Complex Langevin simulations and the QCD phase diagram: Recent developments, Eur. Phys. J. A 56, 251 (2020).
  23. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. 1, Phys. Rev. 122, 345 (1961).
  24. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124, 246 (1961).
  25. V. Koch, Aspects of chiral symmetry, Int. J. Mod. Phys. E 06, 203 (1997).
  26. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, England, 1996), Vol. 2.
  27. K. Fukushima and V. Skokov, Polyakov loop modeling for hot QCD, Prog. Part. Nucl. Phys. 96, 154 (2017).
  28. J. Bernhardt and C. S. Fischer, From imaginary to real chemical potential QCD with functional methods, Eur. Phys. J. A 59, 181 (2023).
  29. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Addison-Wesley, Reading, MA, 1995), p. 842.
  30. M. Gell-Mann and M. Levy, The axial vector current in beta decay, Nuovo Cimento 16, 705 (1960).
  31. F. Murgana, A. Koenigstein, and D. H. Rischke, Reanalysis of critical exponents for the O(N) model via a hydrodynamic approach to the functional renormalization group, Phys. Rev. D 108, 116016 (2023).
  32. K. S. Jeong, F. Murgana, A. Dash, and D. H. Rischke, Functional Renormalization Group analysis of the quark-condensation pattern on the Fermi surface: A simple effective-model approach, arXiv:2407.13589.
  33. A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, J. Braun, M. Buballa, and D. H. Rischke, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases—Part I: The o(n) model, arXiv:2108.02504.
  34. J. Braun, M. Leonhardt, and J. M. Pawlowski, Renormalization group consistency and low-energy effective theories, SciPost Phys. 6, 56 (2019).
  35. J. Stoll, N. Zorbach, A. Koenigstein, M. J. Steil, and S. Rechenberger, Bosonic fluctuations in the (1+1)-dimensional Gross-Neveu(-Yukawa) model at varying μ and T and finite N, arXiv:2108.10616.
  36. U. Ellwanger, Flow equations for N point functions and bound states, Z. Phys. C 62, 503 (1994).
  37. C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993).
  38. T. R. Morris, On truncations of the exact renormalization group, Phys. Lett. B 334, 355 (1994).
  39. M. Reuter and C. Wetterich, Effective average action for gauge theories and exact evolution equations, Nucl. Phys. B417, 181 (1994).
  40. J. M. Pawlowski, Aspects of the functional renormalisation group, Ann. Phys. (N.Y.) 322, 2831 (2007).
  41. P. Kopietz, L. Bartosch, and F. Schütz, Introduction to the Functional Renormalization Group (Springer, Berlin, Heidelberg, 2010), Vol. 798.
  42. 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).
  43. J. Berges, N. Tetradis, and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rep. 363, 223 (2002).
  44. C. Wetterich, Average action and the renormalization group equations, Nucl. Phys. B352, 529 (1991).
  45. J. M. Pawlowski, Aspects of the functional renormalisation group, Ann. Phys. (N.Y.) 322, 2831 (2007).
  46. D. F. Litim, Critical exponents from optimized renormalization group flows, Nucl. Phys. B631, 128 (2002).
  47. D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001).
  48. L. Canet, B. Delamotte, D. Mouhanna, and J. Vidal, Optimization of the derivative expansion in the nonperturbative renormalization group, Phys. Rev. D 67, 065004 (2003).
  49. T. R. Morris, The exact renormalization group and approximate solutions, Int. J. Mod. Phys. A 09, 2411 (1994).
  50. B. Bergerhoff and C. Wetterich, Effective quark interactions and QCD propagators, Phys. Rev. D 57, 1591 (1998).
  51. J. A. Adams, J. Berges, S. Bornholdt, F. Freire, N. Tetradis, and C. Wetterich, Solving nonperturbative flow equations, Mod. Phys. Lett. A 10, 2367 (1995).
  52. E. Grossi and N. Wink, Resolving phase transitions with discontinuous Galerkin methods, SciPost Phys. Core 6, 071 (2023).
  53. E. Grossi, F. J. Ihssen, J. M. Pawlowski, and N. Wink, Shocks and quark-meson scatterings at large density, Phys. Rev. D 104, 016028 (2021).
  54. B. Delamotte, An introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852, 49 (2012).
  55. D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001).
  56. D. F. Litim, Critical exponents from optimised renormalisation group flows, Nucl. Phys. B631, 128 (2002).
  57. N. Zorbach, A. Koenigstein, and J. Braun, Functional renormalization group meets computational fluid dynamics: RG flows in a multi-dimensional field space, arXiv:2412.16053.
  58. 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–Nneveu model in the mean-field approximation, J. Phys. A 55, 375402 (2022).
  59. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160, 241 (2000).
  60. J. W. Thomas, Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations (Springer, New York, 1999).
  61. E. Bladé, M. Gómez-Valentín, J. Dolz, J. Aragón-Hernández, G. Corestein, and M. Sánchez-Juny, Integration of 1d and 2d finite volume schemes for computations of water flow in natural channels, Adv. Water Resour. 42, 17 (2012).
  62. G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 19, 1892 (1998).
  63. A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, J. Braun, M. Buballa, and D. H. Rischke, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. I. The O(N) model, Phys. Rev. D 106, 065012 (2022).
  64. A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, and J. Braun, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows, Phys. Rev. D 106, 065013 (2022).
  65. F. Ihssen, F. R. Sattler, and N. Wink, Numerical RG-time integration of the effective potential: Analysis and benchmark, Phys. Rev. D 107, 114009 (2023).
  66. K. Morita, V. Skokov, B. Friman, and K. Redlich, Role of mesonic fluctuations in the Polyakov loop extended quark-meson model at imaginary chemical potential, Phys. Rev. D 84, 074020 (2011).
  67. S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabo, QCD crossover at finite chemical potential from lattice simulations, Phys. Rev. Lett. 125, 052001 (2020).
  68. 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).
  69. F. Murgana, Supplementary data for “Assessing the reconstruction of the critical line in the QCD phase diagram from imaginary to real chemical potential”, https://drive.google.com/drive/folders/1Puc88P5GDnEzwTJhbuKrnOpaB2hqtdX9.

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Phys. Rev. D 112, 056001– Published 8 September, 2025
  • Received 12 May 2025
  • Accepted 19 July 2025
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