1. Zhao Zhang ( Kyoto University ) Vector-vector interaction, Charge neutrality and the number of QCD critical points contents  Introduction to QCD phase diagram  Competition between chiral condensate and diquark condensate Effect from vector-vector four quark interaction Effect from electrical chemical potential μ e under charge-neutrality & β-equilibrium  Multiple critical-point structures for chiral restoration  Summary & Outlook Collaborators: K. Fukushima, T. Kunihiro

2. 1. 1 Schematic Phase Diagram of QCD ?  What’ s the possible phase of non-CFL ?  What ‘s the effect of non-CFL on chiral restoration ? New critical and in dense QCD ?  New critical points and crossover in dense QCD ? Low T region : Two critical points (end of first order transition) Liquid-gas transition confirmed by experiment Chiral critical point existence and location ? Recent review on CSC: Alford, Rajagopal, Schaefer and Schmitt, RMP, (’08) 2 nd order phase transition, massless mode, critical fluctuations

3. Ruster, Werth, Buballa, Shovkovy and Rischke, PRD (’05) See also, Abuki and Kunihiro, NPA (’06) 1.2 Phase diagram from 3-flavor NJL model Strong diquark couplingWeak diquark coupling The possible candidate for non-CFL: 2SC or g2SC The influence of 2SC (g2SC) on chiral restoration ?

4. 1.3 The number of CP’s in QCD phase diagram ? The results from NJL: 1,2,3,4,0 are all possible Mechanisms: Vector interaction and charge-neutrality T μ T T T μ μ μ

5. The key for the appearance of miltiple critical-point structures The competition between chiral condensate and diquark condensate: I.The emergence of coexisting phase for chiral symmetry breaking and color superconducting phase II.Abnormal T dependence of the gap of the mismatching diquark condensate in coexisting region

6.  Order parameterChiral Symmetry Breaking 2. Competition between  SB and CSC 2.1 Chiral symmetry breaking phase (  SB) In the chiral limit (m q =0): m q M q current quark mass dynamical quark mass

7.  2.2 Color superconducting phase ( CSC)  Cooper instability: BCS: In sufficiently cold fermionic matter, any attractive interaction leads to the instability to form Cooper pairs.  QCD at asymptotic density: weak interaction: asymptotic freedom  quark Fermi surface Attractive channel: one-gluon exchange interaction  QCD at moderate density: Instanton-induced four-quark interaction models support CSC, for example NJL model Many internal degrees of freedom of quark: spin, color, flavor, patterns of quark-quark cooper paring are more complicated than that of normal SC u CFL dSCuSC2SC u ds u ds u dssd u

8.  Chiral restoration with increasing μ E 0 E 0 Large μ and small M means large Fermi sphere and strong Cooper instability  Baryon density suppresses the quark-antiquark pairing. 2.3 Competition between at moderate density  Formation of quark Cooper pair with increasing  M M

9. 2.4.1 Repulsive vector interaction  However, vector interaction channel also naturally appears in the effective theories.  Instanton-anti-instanton molecule model, Schaefer,Shuryak (‘98)  Renormalization-group analysis, N.Evans et al. (‘99), Schaefer, Wilczek (‘99)  Dyson-Schwinger equation model of QCD, Cahill, Roberts (‘85) G V /G S =0.25 G V /G S =0.5 Hadron spectroscopy: Klimt,Luts,&Weise (’90), Roberts, Williams (’94)  In the 4-quark interaction models, scalar and pseudoscalar interaction channels have been extensively used to study  SB and CSC 2.4 Two mechanisms for enhancing the competition between  SB and CSC

10. Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96 Chiral restoration is weakened and delayed towards higher μ With increasing G V: density-density correlation Quark number density: 1. Negative dynamical chemical potential: 2. Negative contribution to the free energy, vector-type quark-antiquark condensate In NJL: dynamical quark mass

11. Enhancing the competition between  SB and CSC Kitazawa, Koide, Kunihiro & Nemoto (’02) 2 critical points  COE becomes broader with increasing Gv due to the delay of  SB restoration  the competition of two condensates in COE leads to new CP Coexistence (COE) region is the key for the appearance of two CP’s: 1 critical pointno critical point

12. μ u =μ - 2μ e /3 μ d =μ + μ e /3 μ μ e  100MeV for μ  400MeV 2.4.2 Electrical chemical potential (μ e ) under charge-neutrality & β-equilibrium vs Role 1: μ e > 0, an effective vector interaction Role 2: μ e > 0, the mismatch between μ u and μ d.

13. Effect from mismatch quark-quark paring under charge-neutrality & β-equilibrium ρ d > ρ u >ρ s Mismatch paring or pair breaking Standard BCS paring For 2-flavor asymmetric homogenous CSC: (1)Abnormal thermal behavior of diquark gap pFpF p Double effects of T : The competition between two condensates may be enhanced with increasing T Fermi sphere is smeared by T (2)Chromomagnetic instability: imaginary Meissner mass

14. Schematic critical-point structure with charge-neutrality and CSC Possible critical-point structures induced by μ e weakstrong “survivor” “remnant”

15. β-equilibrium condition Zhang, Fukushima and Kunihiro, PRD, (’09), arXiv: 0808.3371 Local charge-neutrality Two constraints for bulk matter: Results from a simple 2-flavor NJL: 3. Multi-critical-point structures of QCD ? 3.1 Multi-critical-point structures induced by μ e under charge-neutrality & β-equilibrium

16. Result : Impact on chiral restoration without CSC  Increasing Q μ corresponds to increasing G V  Chiral restoration is delayed and weakened with Increasing Q μ Same effect as vector interaction (Q d = -1/3 fixed). In nature, electric charges : Q u = 2/3, Q d = -1/3

17. Without charge-neutrality With charge-neutrality First report on 3 critical-point structure Intermediate coupling: G D / G S = 0.75  COE is significantly broadened by nonzero μe  Line E-F becomes crossover due to the abnormal thermal behavior of diquark condensate  D and E are free from chromomagnetic instaibility, the fate of F is unknown 1 CP 3 CP’s ! Result : Impact on chiral restoration with and without charge- neutrality including CSC

18. A general phenomenon for asymmetric homogenoues superconductivity or superfluidity system Leading to 3 critical points structure Large Small In COE region In CSC region

19. For stronger diquark coupling cases 2 CP’s no CP For weaker diquark coupling cases 1 CP

20.  The multiple critical point structures are not observed in traditional NJL model if the first order chiral phase transition is relatively strong. Ruster, Werth, Buballa, Shovkovy and Rischke, PRD (’05)

21. 3.2 Combined effects of vector-interaction and charge- neutrality&β-equilibrium on chiral phase transition  For more real case, both vector interaction and neutral charge-constraint should be taken into account simultaneously  Combined effect on chiral transition should be more significant. Role 1:helping ue to fulfill the multi-critical-point structure Role 2: effect on chromomagnetic instability Suppressing the chromomagnetic instability ? suppressing

22. Results from a nonlocal two flavor NJL with different sets of model parameters We used 3 sets of the model Parameters to check the model- parameter sensitivity of the main results. Model parameters fixing (Gs and Λ): taking Gv as free parameter with G D /Gs =0.75 Two flavor NJL model with vector interaction

23. Parameters set 1 : M(p=0)= 400 MeV, the stronger first order chiral transition case

25. Increasing Gv/Gs Parameters set 2 : M(p=0)= 367.5 MeV, the relatively strong chiral first order trannsition case

26. Parameters set 3 : M(p=0)= 330 MeV, the weak chiral transition case In contrast to the case with parameters set 1, the remnant first vanishes with increasing Gv

27. Effect of Vector interaction on chromomagnetic instability With increasing G v /G s :  The ratio  μ/Δ is effectively suppressed  Unstable region shrinks to larger μ and lower T Unstable region

28. 3.3 Extension to 3 flavor case  Effect from μ e with charge-neutrality & β-equilibrium For 3 flavor limit, μ e=0. For 2+1 flavor case, μ e still plays important role on chiral restoration due to the relatively heavy s quark.  Effect from vector interaction and UA(1) breaking interaction? Mean field thermal potential : Results from 2+1 flavor NJL under charge-neutrality  vector-vector interaction  ’t Hooft six-quark interaction  Only considering 2CSC  Other possible cubic coupling terms are ignored

29. Order of critical-point numbers with increasing Gv/Gs: 1,2,4,2,0. For example: 2 CP’s4 CP’s2 CP’s All of the critical-point structures appearing for 2 flavor case are confirmed in 2+1 flavor case with some range of model parameters.

30.  Vector interaction can effectively suppress the unstable asymmetric homogenous CSC region related to chromomagnetic instability. Summary :  Vector interaction can delay and weaken chiral restoration  Vector interaction can delay and weaken chiral restoration, enhance the the competition between chiral condensate and diquark condensate.  Under charge-neutrality & β-equilibrium, electric chemical potential can be regarded as an effective vector interaction.  The abnormal thermal behavior of mismatch paring can enhance the competition in COE with increasing T. 5 types of critical-point structure are found in NJL study, Multi-critical-point structures are possible to happen in QCD Outlook:  Effect of Vector interaction on asymmetric inhomogeneous CSC, especially the role of space-like vector-type quark-antiquark condensate. Z.Zhang and T. Kunihiro, in preparation