The dual properties of chiral and isospin asymmetric dense quark matter formed of two-color quarks


In this paper the phase structure of dense baryon matter composed of u and d quarks with two colors has been investigated in the presence of baryon μB , isospin μI and chiral isospin μI5 chemical potentials in the framework of Nambu-Jona-Lasinio model with quark-antiquark and quark-quark interaction channels. In the chiral limit, it has been shown in the mean-field approximation that the duality between phases with spontaneous chiral symmetry breaking and condensation of charged pions, found in the three color case, remains valid in the two color case. In addition, it has been shown that there are two more dualities in the phase diagram in two color case, namely (as in the case with μI5 = 0), at μI5 ≠ 0 the general (μ, μI, μI5)-phase portrait of the model has dual symmetry between the phase with condensation of charged pions and the phase with diquark condensation. This duality stays exact even in the physical point, m0 ≠ 0. And at m0 = 0 the (μ, μI, μI5)- phase portrait becomes even more symmetrical, since dual symmetry between phases with spontaneous chiral symmetry breaking and diquark condensation appears. It is shown that due to the dualities the phase diagram is extremely symmetric and has interlacing structure. One can show that the phase portrait of two-color NJL model can be obtained just by duality properties from the results of investigations of three-color NJL model (it was noticed only after the numerical calculations have been performed). Three-color case shares only one duality of the two color one, and one can only see a facet of this enormously symmetric picture in the case of three colors. Using dualities only, it is possible to show that there are no mixed phases (phases with two non-zero condensates). This prediction of dualities is of great use, because for sure it can be shown by the direct calculations but it would be enormously more complicated and time-consuming numerically.

A preprint version of the article is available at ArXiv.


  1. [1]

    Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. 1., Phys. Rev.122 (1961) 345 [INSPIRE].

  2. [2]

    S.P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys.64 (1992) 649 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  3. [3]

    D. Ebert, H. Reinhardt and M.K. Volkov, Effective hadron theory of QCD, Prog. Part. Nucl. Phys.33 (1994) 1 [INSPIRE].

    ADS  Google Scholar 

  4. [4]

    T. Inagaki, T. Muta and S.D. Odintsov, Dynamical symmetry breaking in curved space-time: four fermion interactions, Prog. Theor. Phys. Suppl.127 (1997) 93 [hep-th/9711084] [INSPIRE].

    ADS  Google Scholar 

  5. [5]

    M. Buballa, NJLS model analysis of quark matter at large density, Phys. Rept.407 (2005) 205 [hep-ph/0402234] [INSPIRE].

  6. [6]

    A.A. Garibli, R.G. Jafarov and V.E. Rochev, Mean-field expansion, regularization issue, and multi-quark functions in Nambu–Jona-Lasinio model, Symmetry11 (2019) 668.

    MATH  Google Scholar 

  7. [7]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Charged pion condensation in dense quark matter: Nambu–Jona-Lasinio model study, Symmetry11 (2019) 778.

    Google Scholar 

  8. [8]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Dense baryonic matter and applications of QCD phase diagram dualities, Particles3 (2020) 62.

    Google Scholar 

  9. [9]

    J.B. Kogut, M.A. Stephanov and D. Toublan, On two color QCD with baryon chemical potential, Phys. Lett.B 464 (1999) 183 [hep-ph/9906346] [INSPIRE].

  10. [10]

    J.B. Kogut et al., QCD-like theories at finite baryon density, Nucl. Phys.B 582 (2000) 477 [hep-ph/0001171] [INSPIRE].

  11. [11]

    K. Splittorff, D.T. Son and M.A. Stephanov, QCD-like theories at finite baryon and isospin density, Phys. Rev.D 64 (2001) 016003 [hep-ph/0012274] [INSPIRE].

  12. [12]

    C. Ratti and W. Weise, Thermodynamics of two-colour QCD and the Nambu Jona-Lasinio model, Phys. Rev.D 70 (2004) 054013 [hep-ph/0406159] [INSPIRE].

  13. [13]

    D.C. Duarte et al., BEC-BCS crossover in a cold and magnetized two color NJLS model, Phys. Rev.D 93 (2016) 025017 [arXiv:1510.02756] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    J.O. Andersen and A.A. Cruz, Two-color QCD in a strong magnetic field: the role of the Polyakov loop, Phys. Rev.D 88 (2013) 025016 [arXiv:1211.7293] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    T. Brauner, K. Fukushima and Y. Hidaka, Two-color quark matter: U(1)Arestoration, superfluidity and quarkyonic phase, Phys. Rev.D 80 (2009) 074035 [Erratum ibid.D 81 (2010) 119904] [arXiv:0907.4905] [INSPIRE].

  16. [16]

    J.O. Andersen and T. Brauner, Phase diagram of two-color quark matter at nonzero baryon and isospin density, Phys. Rev.D 81 (2010) 096004 [arXiv:1001.5168] [INSPIRE].

    ADS  Google Scholar 

  17. [17]

    S. Imai, H. Toki and W. Weise, Quark-hadron matter at finite temperature and density in a two-color PNJL model, Nucl. Phys.A 913 (2013) 71 [arXiv:1210.1307] [INSPIRE].

    ADS  Google Scholar 

  18. [18]

    P. Adhikari, S.B. Beleznay and M. Mannarelli, Finite density two color chiral perturbation theory revisited, Eur. Phys. J.C 78 (2018) 441 [arXiv:1803.00490] [INSPIRE].

    ADS  Google Scholar 

  19. [19]

    P. Adhikari and H. Nguyen, Multicomponent superfluidity in two-color QCD at finite density at next-to-leading order, arXiv:2002.04052 [INSPIRE].

  20. [20]

    J. Chao, Phase diagram of two-color QCD matter at finite baryon and axial isospin densities, Chin. Phys.C 44 (2020) 034108 [arXiv:1808.01928] [INSPIRE].

    ADS  Google Scholar 

  21. [21]

    V.G. Bornyakov, V.V. Braguta, A.A. Nikolaev and R.N. Rogalyov, Effects of dense quark matter on gluon propagators in lattice QC2 D, arXiv:2003.00232 [INSPIRE].

  22. [22]

    D.T. Son and M.A. Stephanov, QCD at finite isospin density: From pion to quark-anti-quark condensation, Phys. Atom. Nucl.64 (2001) 834 [hep-ph/0011365] [INSPIRE].

  23. [23]

    M. Loewe and C. Villavicencio, Thermal pions at finite isospin chemical potential, Phys. Rev.D 67 (2003) 074034 [hep-ph/0212275] [INSPIRE].

  24. [24]

    D.C. Duarte, R.L.S. Farias and R.O. Ramos, Optimized perturbation theory for charged scalar fields at finite temperature and in an external magnetic field, Phys. Rev.D 84 (2011) 083525 [arXiv:1108.4428] [INSPIRE].

    ADS  Google Scholar 

  25. [25]

    D. Ebert, K.G. Klimenko, A.V. Tyukov and V.C. Zhukovsky, Pion condensation of quark matter in the static Einstein universe, Eur. Phys. J.C 58 (2008) 57 [arXiv:0804.0765] [INSPIRE].

    ADS  Google Scholar 

  26. [26]

    L.-y. He, M. Jin and P.-f. Zhuang, Pion superfluidity and meson properties at finite isospin density, Phys. Rev.D 71 (2005) 116001 [hep-ph/0503272] [INSPIRE].

  27. [27]

    D. Ebert and K.G. Klimenko, Gapless pion condensation in quark matter with finite baryon density, J. Phys.G 32 (2006) 599 [hep-ph/0507007] [INSPIRE].

  28. [28]

    D. Ebert and K.G. Klimenko, Pion condensation in electrically neutral cold matter with finite baryon density, Eur. Phys. J.C 46 (2006) 771 [hep-ph/0510222] [INSPIRE].

  29. [29]

    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 (2010) 056006 [INSPIRE].

    ADS  Google Scholar 

  30. [30]

    H. Abuki, R. Anglani, R. Gatto, G. Nardulli and M. Ruggieri, Chiral crossover, deconfinement and quarkyonic matter within a Nambu-Jona Lasinio model with the Polyakov loop, Phys. Rev.D 78 (2008) 034034 [arXiv:0805.1509] [INSPIRE].

    ADS  Google Scholar 

  31. [31]

    H. Abuki, R. Anglani, R. Gatto, M. Pellicoro and M. Ruggieri, The Fate of pion condensation in quark matter: From the chiral to the real world, Phys. Rev.D 79 (2009) 034032 [arXiv:0809.2658] [INSPIRE].

    ADS  Google Scholar 

  32. [32]

    R. Anglani, A microscopic study of pion condensation within Nambu-Jona-Lasinio model, Acta Phys. Polon. Supp.3 (2010) 735 [arXiv:1001.0471] [INSPIRE].

    Google Scholar 

  33. [33]

    J.O. Andersen and T. Brauner, Linear σ-model at finite density in the 1/N expansion to next-to-leading order, Phys. Rev.D 78 (2008) 014030 [arXiv:0804.4604] [INSPIRE].

    ADS  Google Scholar 

  34. [34]

    J.O. Andersen and L. Kyllingstad, Pion condensation in a two-flavor NJLS model: the role of charge neutrality, J. Phys.G 37 (2009) 015003 [hep-ph/0701033] [INSPIRE].

  35. [35]

    Y. Jiang, K. Ren, T. Xia and P. Zhuang, Meson screening mass in a strongly coupled pion superfluid, Eur. Phys. J.C 71 (2011) 1822 [arXiv:1104.0094] [INSPIRE].

    ADS  Google Scholar 

  36. [36]

    A. Folkestad and J.O. Andersen, Thermodynamics and phase diagrams of Polyakov-loop extended chiral models, Phys. Rev.D 99 (2019) 054006 [arXiv:1810.10573] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. [37]

    P. Adhikari, J.O. Andersen and P. Kneschke, Pion condensation and phase diagram in the Polyakov-loop quark-meson model, Phys. Rev.D 98 (2018) 074016 [arXiv:1805.08599] [INSPIRE].

    ADS  Google Scholar 

  38. [38]

    P. Adhikari, J.O. Andersen and P. Kneschke, Two-flavor chiral perturbation theory at nonzero isospin: Pion condensation at zero temperature, Eur. Phys. J.C 79 (2019) 874 [arXiv:1904.03887] [INSPIRE].

    ADS  Google Scholar 

  39. [39]

    J.O. Andersen, P. Adhikari and P. Kneschke, Pion condensation and QCD phase diagram at finite isospin density, PoS(Confinement2018)197 (2019) [arXiv:1810.00419] [INSPIRE].

  40. [40]

    D. Ebert, T.G. Khunjua, K.G. Klimenko and V.C. Zhukovsky, Charged pion condensation phenomenon of dense baryonic matter induced by finite volume: The NJL(2) model consideration, Int. J. Mod. Phys.A 27 (2012) 1250162 [arXiv:1106.2928] [INSPIRE].

    ADS  MATH  Google Scholar 

  41. [41]

    N.V. Gubina, K.G. Klimenko, S.G. Kurbanov and V.C. Zhukovsky, Inhomogeneous charged pion condensation phenomenon in the NJL2model with quark number and isospin chemical potentials, Phys. Rev.D 86 (2012) 085011 [arXiv:1206.2519] [INSPIRE].

    ADS  Google Scholar 

  42. [42]

    A. Mammarella and M. Mannarelli, Intriguing aspects of meson condensation, Phys. Rev.D 92 (2015) 085025 [arXiv:1507.02934] [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    S. Carignano et al., Scrutinizing the pion condensed phase, Eur. Phys. J.A 53 (2017) 35 [arXiv:1610.06097] [INSPIRE].

    ADS  Google Scholar 

  44. [44]

    M. Mannarelli, Meson condensation, Particles2 (2019) 411 [arXiv:1908.02042] [INSPIRE].

    Google Scholar 

  45. [45]

    J.O. Andersen and P. Kneschke, Bose-Einstein condensation and pion stars, arXiv:1807.08951 [INSPIRE].

  46. [46]

    B.B. Brandt et al., New class of compact stars: Pion stars, Phys. Rev.D 98 (2018) 094510 [arXiv:1802.06685] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  47. [47]

    K. Fukushima, D.E. Kharzeev and H.J. Warringa, The chiral magnetic effect, Phys. Rev.D 78 (2008) 074033 [arXiv:0808.3382] [INSPIRE].

    ADS  Google Scholar 

  48. [48]

    M.A. Metlitski and A.R. Zhitnitsky, Anomalous axion interactions and topological currents in dense matter, Phys. Rev.D 72 (2005) 045011 [hep-ph/0505072] [INSPIRE].

  49. [49]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Chiral imbalanced hot and dense quark matter: NJLS analysis at the physical point and comparison with lattice QCD, Eur. Phys. J.C 79 (2019) 151 [arXiv:1812.00772] [INSPIRE].

    ADS  Google Scholar 

  50. [50]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Dualities and inhomogeneous phases in dense quark matter with chiral and isospin imbalances in the framework of effective model, JHEP06 (2019) 006 [arXiv:1901.02855] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  51. [51]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, QCD phase diagram with chiral imbalance in NJLS model: duality and lattice QCD results, J. Phys. Conf. Ser.1390 (2019) 012015 [arXiv:1812.01392] [INSPIRE].

    Google Scholar 

  52. [52]

    M. Ruggieri, G.X. Peng and M. Chernodub, Chiral medium produced by parallel electric and magnetic fields, EPJ Web Conf.129 (2016) 00037 [arXiv:1609.04537] [INSPIRE].

    Google Scholar 

  53. [53]

    M. Ruggieri and G.X. Peng, Quark matter in a parallel electric and magnetic field background: chiral phase transition and equilibration of chiral density, Phys. Rev.D 93 (2016) 094021 [arXiv:1602.08994] [INSPIRE].

    ADS  Google Scholar 

  54. [54]

    R. Gatto and M. Ruggieri, Hot quark matter with an axial chemical potential, Phys. Rev.D 85 (2012) 054013 [arXiv:1110.4904] [INSPIRE].

    ADS  Google Scholar 

  55. [55]

    L. Yu, H. Liu and M. Huang, Spontaneous generation of local CP-violation and inverse magnetic catalysis, Phys. Rev.D 90 (2014) 074009 [arXiv:1404.6969] [INSPIRE].

    ADS  Google Scholar 

  56. [56]

    L. Yu, H. Liu and M. Huang, Effect of the chiral chemical potential on the chiral phase transition in the NJLS model with different regularization schemes, Phys. Rev.D 94 (2016) 014026 [arXiv:1511.03073] [INSPIRE].

    ADS  Google Scholar 

  57. [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 (2016) 125101 [arXiv:1602.05250] [INSPIRE].

    ADS  Google Scholar 

  58. [58]

    A.A. Andrianov, V.A. Andrianov and D. Espriu, Chiral perturbation theory vs. linear σ-model in a chiral imbalance medium, Particles3 (2020) 15 [arXiv:1908.09118] [INSPIRE].

  59. [59]

    D. Espriu, A.G. Nicola and A. Vioque-Rodríguez, Chiral perturbation theory for nonzero chiral imbalance, arXiv:2002.11696 [INSPIRE].

  60. [60]

    G. Cao and P. Zhuang, Effects of chiral imbalance and magnetic field on pion superfluidity and color superconductivity, Phys. Rev.D 92 (2015) 105030 [arXiv:1505.05307] [INSPIRE].

    ADS  Google Scholar 

  61. [61]

    V.V. Braguta and A.Yu. Kotov, Catalysis of dynamical chiral symmetry breaking by chiral chemical potential, Phys. Rev.D 93 (2016) 105025 [arXiv:1601.04957] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  62. [62]

    V.V. Braguta et al., Study of QCD phase diagram with non-zero chiral chemical potential, Phys. Rev.D 93 (2016) 034509 [arXiv:1512.05873] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  63. [63]

    N.Yu. Astrakhantsev et al., Lattice study of QCD at finite chiral density: topology and confinement, arXiv:1902.09325 [INSPIRE].

  64. [64]

    V.V. Braguta et al., Two-color QCD with non-zero chiral chemical potential, JHEP06 (2015) 094 [arXiv:1503.06670] [INSPIRE].

    ADS  Google Scholar 

  65. [65]

    V.V. Braguta, M.I. Katsnelson, A.Yu. Kotov and A.M. Trunin, Catalysis of dynamical chiral symmetry breaking by chiral chemical potential in Dirac semimetals, Phys. Rev.B 100 (2019) 085117 [arXiv:1904.07003] [INSPIRE].

    ADS  Google Scholar 

  66. [66]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Dense baryon matter with isospin and chiral imbalance in the framework of NJL4model at large Nc: duality between chiral symmetry breaking and charged pion condensation, Phys. Rev.D 97 (2018) 054036 [arXiv:1710.09706] [INSPIRE].

    ADS  Google Scholar 

  67. [67]

    T. Khunjua, K. Klimenko and R. Zhokhov, Dense quark matter with chiral and isospin imbalance: NJL-model consideration, EPJ Web Conf.191 (2018) 05015 [arXiv:1901.03049] [INSPIRE].

    Google Scholar 

  68. [68]

    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-Nclimit of the NJL4model, Phys. Rev.D 98 (2018) 054030 [arXiv:1804.01014] [INSPIRE].

    ADS  Google Scholar 

  69. [69]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Charged pion condensation and duality in dense and hot chirally and isospin asymmetric quark matter in the framework of the NJL2model, Phys. Rev.D 100 (2019) 034009 [arXiv:1907.04151] [INSPIRE].

    ADS  Google Scholar 

  70. [70]

    R.N. Zhokhov, K.G. Klimenko and T.G. Khunjua, Pion condensation in hot dense quark matter with isospin and chiral-isospin asymmetries within the Nambu—Jona-Lasinio model, Moscow Univ. Phys. Bull.74 (2019) 473 [INSPIRE].

    ADS  Google Scholar 

  71. [71]

    I.A. Shovkovy, Two lectures on color superconductivity, Found. Phys.35 (2005) 1309 [nucl-th/0410091] [INSPIRE].

  72. [72]

    M. Huang, Color superconductivity at moderate baryon density, Int. J. Mod. Phys.E 14 (2005) 675 [hep-ph/0409167] [INSPIRE].

  73. [73]

    K.G. Klimenko and D. Ebert, Mesons and diquarks in a dense quark medium with color superconductivity, Theor. Math. Phys.150 (2007) 82 [INSPIRE].

    MATH  Google Scholar 

  74. [74]

    M.G. Alford, A. Schmitt, K. Rajagopal and T. Schäfer, Color superconductivity in dense quark matter, Rev. Mod. Phys.80 (2008) 1455 [arXiv:0709.4635] [INSPIRE].

    ADS  Google Scholar 

  75. [75]

    E.J. Ferrer and V. de la Incera, Magnetism in dense quark matter, Lect. Notes Phys.871 (2013) 399 [arXiv: 1208.5179] [INSPIRE].

  76. [76]

    T.G. Khunjua, K.G. Klimenko and R.N. Zhokhov, Electrical neutrality and /3-equilibrium conditions in dense quark matter: generation of charged pion condensation by chiral imbalance, arXiv: 2005.05488 [INSPIRE].

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Khunjua, T., Klimenko, K. & Zhokhov, R. The dual properties of chiral and isospin asymmetric dense quark matter formed of two-color quarks. J. High Energ. Phys. 2020, 148 (2020).

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