第十届 QCD 相变与相对论重离子物理研讨会, August Z. Zhang,

1. 1 QCD critical point Z. Zhang,  Chiral restoration (m=0) well defined Order parameter: Chiral condensate Two-flavor with the U(1)A anomaly 2nd-order phase transition with O(4) universality class Critical point for m>0 ? It’s Location? Only one ? It’s number ? How to detect it in HIC Expt. ? Pisarski and welczek, PRD 29, 338(1984)

1.2. Conjectured phase diagram of QCD Z. Zhang, Common feature: At least one QCD critical point included. Its existence and location is dynamical problem, determined by QCD itself Problem: Nonperturbative The first principle at finite density, sign problem QCD models and QCD-like theory K. Fukushima & T. Hatsuda

1.3. Critical point structure from NJL model: one or zero Z. Zhang, N. Bratovic, T. Hstsuda, and W. Weise, PLB 719 (2003) flavor PNJL C. Sasaki, B. Friman, and K.Redlich, PRD75, (2007) 2-flavor NJL

1.4. Multiple critical points in NJL model with CSC Z. Zhang, M. Kitazawa, T. Koide, Y. Nemoto and T.Kunihiro, PTP (’022) Abuki, Baym, Hatsuda and Yamamoto, PRD 81 (2010) Axial anomaly Z. Z., K. Fukushima, T.Kunihiro, PRD79 (2009) Charge neutrality Vector interaction + Charge neutrality + Axial anomaly Vector interaction Z. Z., and T. Kunihiro, PRD80(2009)014015; PRD83 (2011) Denied by H. Basler and M. Buballa in PRD 82 (2010)

1.5Two critical points at finite isospin chemical potential Z. Zhang, B. Klein, D. Toulblan and J.J.M. Verbaarschot, PRD 68 (2003) D. Toulblan and J.B. Kogut, PLB 564 (2003) 212 Random Matrix Model NJL Model ( without flavor-mixing ) ( without flavor-mixing )

1.6 The effect of Axial anomaly-mixing Z. Zhang, M. Frank, M. Buballa and M. Oertel, PLB 562 (2003) 221 The fate of the two critical points depends on the degree of the U(1)A asymmetry ! Effect of the flavor-mixing induced by axial anomaly Two-flavor NJL model ‘t Hooft interaction Rough estimate: Mass flavor-mixing Two first-order lines for

1.6 U(1)A restoration at finite T in lattice simulations Z. Zhang, Recent lattice formulations on the U(1)A symmetry restoration at finite T (1) A.Bazavov, et al. (HotQCD Collaboration) PRD 86, (2012) Lattice: Domain wall fermions Chiral symmetry restoration U(1)A symmetry restoration Dirac spectral density The axial anomaly effect may be suppressed significantly near the chiral boundary ? ! Banks-Casher relation

Z. Zhang, G. Cossu, S. Aoki et al. PRD 87, (2013) Lattice: Overlap fermions Dirac Spectral density Recent lattice formulations on the U(1)A symmetry restoration at finite T (2) 1.7 U(1)A restoration at finite T in lattice simulations

2. Non-anomaly flavor-mixing induced by vector interactions Z. Zhang, Both Lattice results suggest the axial U(1)A symmetry is effectively restored in the Chiral symmetric phase (for T>=Tc). If it is true, we can expect that the same thing may happen at finite density. The question: Can the two critical points at finite isospin chemical potential survive if the anomaly flavor-mixing is suppressed significantly near the chiral boundary ? Other possible flavor-mixing ? One possibility : Mismatched vector interactions in the isoscalar and Isovector channels may lead to a non-anomaly flavor mixing at finite Isospin chemical potential.

Z. Zhang, Non-anomaly flavor-mixing induced by vector interactions General four-fermion interactions with the flavor symmetry Vector-isoscalar interactionVector-isovector interaction Chiral symmetry does not require the two couplings are identical in the four-fermion interaction model Two types of vector interactions in NJL type model:

Z. Zhang, C. Sasaki, B. Friman, and K.Redlich, PRD75, (2007) The argument for the mismatched vector interactions by C. Sasaki et al. L. Ferroni and V. Koch, PRC 83,045205(2011) 2. Non-anomaly flavor-mixing induced by vector interactions Very important for the determination of the off-diagonal susceptibility

2. Non-anomaly flavor-mixing induced by vector interactions Z. Zhang, The most popular version of NJL model : At the mean filed level, usually only Hartree contribution is considered Including the Fock term: >0 One gluon exchange

2. Non-anomaly flavor-mixing induced by vector interactions Z. Zhang, Curvature: Recent lattice calculation: Paolo Cea et al. PRD 85, (2012) N. Bratovic, T. Hstsuda, and W. Weise, PLB 719 (2003)131 < 0 ! Near Tc for zero

Z. Zhang, Non-anomaly flavor-mixing induced by vector interactions Our lagrangian : Baryon and Isospin chemical potentials: Effective quark chemical potentials are shifted by vector interactions Non-anomaly Flavor-mixing Isospin asymmetry is weakened

2. Non-anomaly flavor-mixing induced by vector interactions Z. Zhang, Mass flavor-mixing due to the ‘t Hooft interaction Thermal potential at the mean field level Two types of flavor-mixing !

Z. Zhang, Phase diagrams at small isospin chemical potentials under the influence of the non-anomaly flavor-mixing (without the anomaly flavor-mixing ) must be much stronger than To change the two 1 st order lines into one,

Z. Zhang, Phase diagrams at finite isospin chemical potential under the influence of vector interactions and the weak anomaly flavor-mixing.

Z. Zhang, Phase diagram at finite isospin chemical potential under the influence of vector interactions and the weak anomaly flavor-mixing.

Z. Zhang, M. Frank, M. Buballa and M. Oertel, PLB 562 (2003) 221 Our results vs

Z. Zhang,

For Nc=3 ? The mean field calculations based on QCD models, such as (P)NJL, (P)QM and RMM etc., support such a conclusion. based on the QCD inequalities and the large- Nc orbifold equivalence

Z. Zhang, M. Hanada, Y. Matruo, and N. Yamamoto, PRD 86, (2012) In our case, ≠ ( if the mismatched vector interactions are included ) Mean field level

5. Summary and outlook Z. Zhang,