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Two-flavor chirally imbalanced quark matter beyond large Nc

André G. da Silva1,*, Dyana C. Duarte1,†, Ricardo L. S. Farias1,‡, Marcus Benghi Pinto2,3,§, Rudnei O. Ramos4,∥, and William R. Tavares4,¶

  • *Contact author: andre-silva.as@acad.ufsm.br
  • †Contact author: dyana.duarte@ufsm.br
  • ‡Contact author: ricardo.farias@ufsm.br
  • §Contact author: marcus.benghi@ufsc.br
  • ∥Contact author: rudnei@uerj.br
  • Contact author: tavares.william@ce.uerj.br
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Phys. Rev. D 112, 076021 – Published 17 October, 2025

DOI: https://doi.org/10.1103/cks2-8m69

Abstract

We investigate a chirally imbalanced medium in the context of the two-flavor Nambu-Jona-Lasinio model using both the large-Nc (LN) and beyond large-Nc (BLN) approximations. To incorporate BLN effects, we consider the optimized perturbation theory (OPT) to the first nontrivial order, which includes two-loop (exchange) contributions. This procedure allows us to explicitly explore how finite Nc corrections affect the thermodynamics as well as the phase diagram of chirally imbalanced quark matter. We then compare the results obtained with a sharp three-dimensional cutoff—generically referred to as the traditional regularization scheme—and with an alternative procedure called the medium separation scheme (MSS). In the first case, we observe that the pseudocritical temperature decreases as the chiral chemical potential increases, an effect dubbed inverse chiral catalysis. On the other hand, when considering the MSS regularization, which properly isolates the medium contributions from the vacuum, we find the opposite result. We show that the results obtained with MSS are consistent with well-established lattice quantum chromodynamics data in both the LN and BLN approximations. Finally, we suggest that to cope with the high-density limit, the standard OPT interpolation prescription must be modified with the inclusion of an extra variational parameter.

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References (88)

  1. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. 1., Phys. Rev. 122, 345 (1961).
  2. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. II., Phys. Rev. 124, 246 (1961).
  3. P. de Forcrand, Simulating QCD at finite density, Proc. Sci., LAT2009 (2009) 010 [arXiv:1005.0539].
  4. A. Okopinska, Nonstandard expansion techniques for the effective potential in lambda phi**4 quantum field theory, Phys. Rev. D 35, 1835 (1987).
  5. A. Duncan and M. Moshe, Nonperturbative physics from interpolating actions, Phys. Lett. B 215, 352 (1988).
  6. V. I. Yukalov, Interplay between approximation theory and renormalization group, Phys. Part. Nucl. 50, 141 (2019).
  7. K. G. Klimenko, Gross-Neveu model and optimized expansion method, Z. Phys. C 50, 477 (1991).
  8. K. G. Klimenko, Nonlinear optimized expansion and the Gross-Neveu model, Z. Phys. C 60, 677 (1993).
  9. J. L. Kneur, M. B. Pinto, and R. O. Ramos, Thermodynamics and phase structure of the two-flavor Nambu-Jona-Lasinio model beyond large-Nc, Phys. Rev. C 81, 065205 (2010).
  10. R. L. S. Farias, G. Dallabona, G. Krein, and O. A. Battistel, Cutoff-independent regularization of four-fermion interactions for color superconductivity, Phys. Rev. C 73, 018201 (2006).
  11. R. L. S. Farias, D. C. Duarte, G. Krein, and R. O. Ramos, Thermodynamics of quark matter with a chiral imbalance, Phys. Rev. D 94, 074011 (2016).
  12. D. C. Duarte, R. L. S. Farias, and R. O. Ramos, Regularization issues for a cold and dense quark matter model in β-equilibrium, Phys. Rev. D 99, 016005 (2019).
  13. A. Das, D. Kumar, and H. Mishra, Chiral susceptibility in the Nambu-Jona-Lasinio model: A Wigner function approach, Phys. Rev. D 100, 094030 (2019).
  14. 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).
  15. B. S. Lopes, D. C. Duarte, R. L. S. Farias, and R. O. Ramos, Speed of sound peak in isospin QCD: A natural prediction of the Medium Separation Scheme, arXiv:2507.14343.
  16. K. Fukushima, D. E. Kharzeev, and H. J. Warringa, The chiral magnetic effect, Phys. Rev. D 78, 074033 (2008).
  17. A. Vilenkin, Equilibrium parity violating current in a magnetic field, Phys. Rev. D 22, 3080 (1980).
  18. D. T. Son and A. R. Zhitnitsky, Quantum anomalies in dense matter, Phys. Rev. D 70, 074018 (2004).
  19. M. A. Metlitski and A. R. Zhitnitsky, Anomalous axion interactions and topological currents in dense matter, Phys. Rev. D 72, 045011 (2005).
  20. A. Vilenkin, Macroscopic parity violating effects: Neutrino fluxes from rotating black holes and in rotating thermal radiation, Phys. Rev. D 20, 1807 (1979).
  21. N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam, and P. Surowka, Hydrodynamics from charged black branes, J. High Energy Phys. 01 (2011) 094.
  22. D. T. Son and P. Surowka, Hydrodynamics with triangle anomalies, Phys. Rev. Lett. 103, 191601 (2009).
  23. K. Landsteiner, E. Megias, and F. Pena-Benitez, Gravitational anomaly and transport, Phys. Rev. Lett. 107, 021601 (2011).
  24. M. Ruggieri, Z. Y. Lu, and G. X. Peng, Influence of chiral chemical potential, parallel electric, and magnetic fields on the critical temperature of QCD, Phys. Rev. D 94, 116003 (2016).
  25. T. G. Khunjua, K. G. Klimenko, and R. N. Zhokhov, Dualities in dense quark matter with isospin, chiral, and chiral isospin imbalance in the framework of the large−Nc limit of the NJL4 model, Phys. Rev. D 98, 054030 (2018).
  26. T. G. Khunjua, K. G. Klimenko, and R. N. Zhokhov, Chiral imbalanced hot and dense quark matter: NJL analysis at the physical point and comparison with lattice QCD, Eur. Phys. J. C 79, 151 (2019).
  27. T. G. Khunjua, K. G. Klimenko, and R. N. Zhokhov, Dense baryonic matter and applications of QCD phase diagram dualities, Particles 3, 62 (2020).
  28. T. G. Khunjua, K. G. Klimenko, and R. N. Zhokhov, Influence of chiral chemical potential μ5 on phase structure of the two-color quark matter, Phys. Rev. D 106, 045008 (2022).
  29. D. Suenaga, K. Suzuki, Y. Araki, and S. Yasui, Kondo effect driven by chirality imbalance, Phys. Rev. Res. 2, 023312 (2020).
  30. D. E. Kharzeev and H. U. Yee, Chiral Magnetic Wave, Phys. Rev. D 83, 085007 (2011).
  31. K. Rajagopal and A. V. Sadofyev, Chiral drag force, J. High Energy Phys. 10 (2015) 018.
  32. N. Yamamoto, Chiral Alfvén wave in anomalous hydrodynamics, Phys. Rev. Lett. 115, 141601 (2015).
  33. M. N. Chernodub, Chiral heat wave and mixing of magnetic, vortical and heat waves in chiral media, J. High Energy Phys. 01 (2016) 100.
  34. A. V. Sadofyev and Y. Yin, The charmonium dissociation in an “anomalous wind”, J. High Energy Phys. 01 (2016) 052.
  35. L. K. Yang, X. F. Luo, J. Segovia, and H. S. Zong, A brief review of chiral chemical potential and its physical effects, Symmetry 12, 2095 (2020).
  36. D. E. Kharzeev, J. Liao, and P. Tribedy, Chiral magnetic effect in heavy ion collisions: The present and future, Int. J. Mod. Phys. E 33, 2430007 (2024).
  37. Z. Song, J. Zhao, Z. Fang, and X. Dai, Detecting the chiral magnetic effect by lattice dynamics in Weyl semimetals, Phys. Rev. B 94, 214306 (2016).
  38. V. V. Braguta, M. I. Katsnelson, A. Y. Kotov, and A. M. Trunin, Catalysis of dynamical chiral symmetry breaking by chiral chemical potential in Dirac semimetals, Phys. Rev. B 100, 085117 (2019).
  39. K. Kamada, N. Yamamoto, and D. L. Yang, Chiral effects in astrophysics and cosmology, Prog. Part. Nucl. Phys. 129, 104016 (2023).
  40. J. Charbonneau and A. Zhitnitsky, Topological currents in neutron stars: Kicks, precession, toroidal fields, and magnetic helicity, J. Cosmol. Astropart. Phys. 08 (2010) 010.
  41. A. Ohnishi and N. Yamamoto, Magnetars and the chiral plasma instabilities, arXiv:1402.4760.
  42. V. V. Braguta, V. A. Goy, E. M. Ilgenfritz, A. Y. Kotov, A. V. Molochkov, M. Muller-Preussker, and B. Petersson, Two-color QCD with non-zero chiral chemical potential, J. High Energy Phys. 06 (2015) 094.
  43. V. V. Braguta, E. M. Ilgenfritz, A. Y. Kotov, B. Petersson, and S. A. Skinderev, Study of QCD phase diagram with non-zero chiral chemical potential, Phys. Rev. D 93, 034509 (2016).
  44. M. Ruggieri, The critical end point of quantum chromodynamics detected by chirally imbalanced quark matter, Phys. Rev. D 84, 014011 (2011).
  45. K. Fukushima, M. Ruggieri, and R. Gatto, Chiral magnetic effect in the PNJL model, Phys. Rev. D 81, 114031 (2010).
  46. J. Chao, P. Chu, and M. Huang, Inverse magnetic catalysis induced by sphalerons, Phys. Rev. D 88, 054009 (2013).
  47. L. Yu, H. Liu, and M. Huang, Spontaneous generation of local CP violation and inverse magnetic catalysis, Phys. Rev. D 90, 074009 (2014).
  48. L. Yu, H. Liu, and M. Huang, Effect of the chiral chemical potential on the chiral phase transition in the NJL model with different regularization schemes, Phys. Rev. D 94, 014026 (2016).
  49. M. N. Chernodub and A. S. Nedelin, Phase diagram of chirally imbalanced QCD matter, Phys. Rev. D 83, 105008 (2011).
  50. M. Hanada and N. Yamamoto, Universality of phase diagrams in QCD and QCD-like theories, Proc. Sci., LATTICE2011 (2011) 221 [arXiv:1111.3391].
  51. B. Wang, Y. L. Wang, Z. F. Cui, and H. S. Zong, Effect of the chiral chemical potential on the position of the critical endpoint, Phys. Rev. D 91, 034017 (2015).
  52. S. S. Xu, Z. F. Cui, B. Wang, Y. M. Shi, Y. C. Yang, and H. S. Zong, Chiral phase transition with a chiral chemical potential in the framework of Dyson-Schwinger equations, Phys. Rev. D 91, 056003 (2015).
  53. C. Shi, X. T. He, W. B. Jia, Q. W. Wang, S. S. Xu, and H. S. Zong, Chiral transition and the chiral charge density of the hot and dense QCD matter, J. High Energy Phys. 06 (2020) 122.
  54. Y. L. Tian, Z. F. Cui, B. Wang, Y. M. Shi, Y. C. Yang, and H. S. Zong, Dyson–Schwinger equations of chiral chemical potential, Chin. Phys. Lett. 32, 081101 (2015).
  55. Z. F. Cui, I. C. Cloet, Y. Lu, C. D. Roberts, S. M. Schmidt, S. S. Xu, and H. S. Zong, Critical endpoint in the presence of a chiral chemical potential, Phys. Rev. D 94, 071503 (2016).
  56. M. Frasca, Nonlocal Nambu-Jona-Lasinio model and chiral chemical potential, Eur. Phys. J. C 78, 790 (2018).
  57. M. Ruggieri and G. X. Peng, Critical temperature of chiral symmetry restoration for quark matter with a chiral chemical potential, J. Phys. G 43, 125101 (2016).
  58. M. Ruggieri and G. X. Peng, Critical temperature of chiral symmetry restoration for quark matter with a chiral chemical potential, arXiv:1602.03651.
  59. M. Ruggieri, M. N. Chernodub, and Z. Y. Lu, Topological susceptibility, divergent chiral density and phase diagram of chirally imbalanced QCD medium at finite temperature, Phys. Rev. D 102, 014031 (2020).
  60. R. L. Liu, M. Y. Lai, C. Shi, and H. S. Zong, Finite volume effects on QCD susceptibilities with a chiral chemical potential, Phys. Rev. D 102, 014014 (2020).
  61. R. L. Liu and H. S. Zong, QCD susceptibilities in the presence of the chiral chemical potential, Mod. Phys. Lett. A 35, 2050137 (2020).
  62. L. K. Yang, X. Luo, and H. S. Zong, QCD phase diagram in chiral imbalance with self-consistent mean field approximation, Phys. Rev. D 100, 094012 (2019).
  63. Z. Pan, Z. F. Cui, C. H. Chang, and H. S. Zong, Finite-volume effects on phase transition in the Polyakov-loop extended Nambu-Jona-Lasinio model with a chiral chemical potential, Int. J. Mod. Phys. A 32, 1750067 (2017).
  64. Y. Lu, Z. F. Cui, Z. Pan, C. H. Chang, and H. S. Zong, QCD phase diagram with a chiral chemical potential, Phys. Rev. D 93, 074037 (2016).
  65. D. Espriu, A. Gómez Nicola, and A. Vioque-Rodríguez, Chiral perturbation theory for nonzero chiral imbalance, J. High Energy Phys. 06 (2020) 122.
  66. A. Gómez Nicola, P. Roa-Bravo, and A. Vioque-Rodríguez, Pion scattering, light resonances, and chiral symmetry restoration at nonzero chiral imbalance and temperature, Phys. Rev. D 109, 034011 (2024).
  67. A. A. Andrianov, V. A. Andrianov, and D. Espriu, Chiral perturbation theory vs. Linear sigma model in a chiral imbalance medium, Particles 3, 15 (2020).
  68. V. V. Braguta and A. Y. Kotov, Catalysis of dynamical chiral symmetry breaking by chiral chemical potential, Phys. Rev. D 93, 105025 (2016).
  69. F. X. Azeredo, D. C. Duarte, R. L. S. Farias, G. Krein, and R. O. Ramos, Deconfinement and chiral phase transitions in quark matter with chiral imbalance, Phys. Rev. D 110, 076007 (2024).
  70. S. Ghosh, N. Chaudhuri, S. Sarkar, and P. Roy, Mass and spectral function of scalar and pseudoscalar mesons in a hot and chirally imbalanced medium using the two-flavor NJL model, Phys. Rev. D 109, 016021 (2024).
  71. S. Ghosh, N. Chaudhuri, P. Roy, and S. Sarkar, Shear and bulk viscous coefficients of a hot and chirally imbalanced quark matter using NJL model, Phys. Rev. D 111, 076012 (2025).
  72. J. L. Kneur, M. B. Pinto, and R. O. Ramos, Critical and tricritical points for the massless 2D Gross-Neveu model beyond large N, Phys. Rev. D 74, 125020 (2006).
  73. J. L. Kneur, M. B. Pinto, R. O. Ramos, and E. Staudt, Emergence of tricritical point and liquid-gas phase in the massless 2+1 dimensional Gross-Neveu model, Phys. Rev. D 76, 045020 (2007).
  74. J. L. Kneur, M. B. Pinto, R. O. Ramos, and E. Staudt, Updating the phase diagram of the Gross-Neveu model in 2+1 dimensions, Phys. Lett. B 657, 136 (2007).
  75. S. K. Gandhi and M. B. Pinto, delta expansion of models with chiral symmetry breaking, Phys. Rev. D 46, 2570 (1992).
  76. L. Ferroni, V. Koch, and M. B. Pinto, Multiple critical points in effective quark models, Phys. Rev. C 82, 055205 (2010).
  77. T. E. Restrepo, J. C. Macias, M. B. Pinto, and G. N. Ferrari, Dynamical generation of a repulsive vector contribution to the quark pressure, Phys. Rev. D 91, 065017 (2015).
  78. S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64, 649 (1992).
  79. K. Fukushima, Phase diagrams in the three-flavor Nambu-Jona-Lasinio model with the Polyakov loop, Phys. Rev. D 77, 114028 (2008); 78, 039902(E) (2008).
  80. K. Fukushima, Critical surface in hot and dense QCD with the vector interaction, Phys. Rev. D 78, 114019 (2008).
  81. M. Buballa, NJL-model analysis of dense quark matter, Phys. Rep. 407, 205 (2005).
  82. P. M. Stevenson, Optimized perturbation theory, Phys. Rev. D 23, 2916 (1981).
  83. P. Stevenson, Sense and nonsense in the renormalization scheme dependence problem, Nucl. Phys. B203, 472 (1982).
  84. 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).
  85. N. Y. Astrakhantsev, V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev, and A. A. Nikolaev, Lattice study of QCD at finite chiral density: Topology and confinement, Eur. Phys. J. A 57, 15 (2021).
  86. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW170817: Measurements of neutron star radii and equation of state, Phys. Rev. Lett. 121, 161101 (2018).
  87. M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, A. K. Harding, W. C. G. Ho, J. M. Lattimer et al., PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter, Astrophys. J. Lett. 887, L24 (2019).
  88. M. C. Miller, F. K. Lamb, A. J. Dittmann, S. Bogdanov, Z. Arzoumanian, K. C. Gendreau, S. Guillot, W. C. G. Ho, J. M. Lattimer, M. Loewenstein et al., The radius of PSR J0740+6620 from NICER and XMM-Newton data, Astrophys. J. Lett. 918, L28 (2021).

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Phys. Rev. D 112, 076021– Published 17 October, 2025
  • Received 9 September 2025
  • Accepted 30 September 2025
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