1. 高密度クォーク物質における カイラル凝縮とカラー超伝導の競 合 M. Kitazawa,T. Koide,Y. Nemoto and T.K. Prog. of Theor. Phys., 108, 929(2002) 国広 悌二 ( 京大基研） 東大特別講義 2005 年 12 月 5-7 日 Ref.

2. 1 1 Introduction Color Superconductivity(CSC) asymptotic freedom Fermi surface attractive channel in one-gluon exchange interaction Cooper instability:In sufficiently cold fermionic matter, any attractive interaction leads to the instability to form infinite Cooper pairs. QCD at high density: [ ３ ] C ×[ ３ ] C ＝ [ ３ ] C ＋ [ ６ ] C Attractive! Cold, dense quark matter is color superconducting D.Bailin and A.Love, Phys.Rep.107,325(’84) Ｔ  or  0 CSC Hadrons

3. Recent Progress in CSC (’98~) The di-quark gap  can become ~100MeV. The possibility to observe the CSC in neutron stars or heavy ion collisions Another symmetry breaking pattern Color-flavor locked (CFL) phase at high density(  q >>m s ) M.Alford et al., PLB422(’98)247 / R.Rapp et al., PRL81(’98)53 / D.T.Son,PRD59(’99)094019 ｕ ｄ ｄ ｄｕ ｕ (  q < m s ) ~ M.Alford,K.Rajagopal, F.Wilczek, Nucl.Phys.B537(’98)443 ｕ ｄ ｄｓ ｓ ｕ 2SC: CFL: R.Pisarski,D.Rischke(’99) T.Schaefer,F.Wilczek(’99) K.Rajagopal,F.Wilczek(’00)

4. Ｔ μ 0 Chiral Symmetry Broken 2SC Phase Diagram of QCD NJL-type 4-Fermi model Random matrix model Schwinger-Dyson eq. with OGE CFL End point of the 1st order transition M.Asakawa, K.Yazaki (’89) K. Rajagopal and F. Wilczek (’02), ”At the Frontier of Particle Physics / Handbook of QCD” Chap.35 Various models lead to qualitatively the same results. J.Berges, K.Rajagopal(’98) / T.Schwarz et al.(’00) B.Vanderheyden,A.Jackson(’01) M.Harada,S.Takagi(’02) / S.Takagi(’02) However, almost all previous works have considered only the scalar and pseudoscalar interaction in qq and qq channel.

5. Instanton-anti-instanton molecule model Shaefer,Shuryak (‘98) Renormalization-group analysis N.Evans et al. (‘99) The importance of the vector interaction is well known : Vector interaction naturally appears in the effective theories. Hadron spectroscopy Klimt,Luts,&Weise (’90) Chiral restoration Asakawa,Yazaki (’89) / Buballa,Oertel(’96) Vector Interaction density-density correlation m

6. Effects of G V on Chiral Restoration Chiral restoration is shifted to higher densities. The phase transition is weakened. As G V is increased, First Order Cross Over G V →Large Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96

7. Chiral Restoration at Finite   :Small  :Large Chiral condensate ( q-q condensate ) CSC ( q-q condensate ) E 0 E 0 Small Fermi sphere Large Fermi sphere leads to strong Cooper instability  Baryon density suppresses the formation of q-q pairing.

8. 2 2 Formulation Parameters: To reproduce the pion decay constant the chiral condensate : is varied in the moderate range. current quark mass Hatsuda,Kunihiro(’94) Nambu-Jona-Lasinio(NJL) model (2-flavors,3-colors) :

9. Thermodynamic Potential in mean field approximation :chiral condensate :di-quark condensate Quasi-particle energies: cf.)  -  model

10. Gap Equations The absolute minimum of  gives the equilibrium state. If there are several solutions, one must choose the absolute minimum for the equilibrium state. T=0 MeV,  =314 MeV G V =0 Gap equations ( the stationary condition):

11. Effect of Vector Interaction on  Vector interaction delays the chiral restoration toward larger . large M small  small m large  = Contour map of  in M D -  plane = T=0 MeV  =314 MeV m

12. 3 3 Numerical Results Phase Diagram Order Parameters M D :Chiral Condensate  :  Diquark Gap The existence of the coex. phase Berges,Rajagopal(’98):× Rapp et al.(’00) : ○ First Order Second Order Cross Over

13. As G V is increased… (1) The critical temperatures of the  SB and CSC hardly changes. It does not change at all in the T-  plane.

14. Another end point appears from lower temperature, and hence there can exist two end points in some range of G V ! (3) (4) The region of the coexisting phase becomes broader. (2) Appearance of the coexisting phase becomes robust. The first order transition between  SB and CSC phases is weakened and eventually disappears.

15. Order Parameters at T=0 (in the case of chiral limit)  [MeV] 300 400 Chiral restoration is delayed toward larger .  SB survives with larger Fermi surface. Stronger Cooper instability is stimulated with  SB. The region of the coexisting phase becomes broader.

16. Contour of  with G V /G S =0.35 5 MeV 12 MeV 15 MeV T= 22 MeV  Large fluctuation owing to the interplay between  SB and CSC is enhanced by G V.

17. End Point at Lower Temperature pFpF p pFpF p  SB CSC This effect plays a role similar to the temperature, and new end point appears from lower T. As G V is increased, Coexisting phase becomes broader.  becomes larger at the phase boundary between CSC and  SB. The Fermi surface becomes obscure.

18. Phase Diagram in 2-color Lattice Simulation J.B.Kogut et al. hep-lat/0205019

19. Summary The vector interaction enhances the interplay between  SB and CSC. More deep understanding about the appearance of the 2 endpoints Future Problems The phase structure is largely affected by the vector interaction especially near the border between  SB and CSC phases. Coexistence of  SB and CSC, 2 endpoints phase structure, Large fluctuation near the border between  SB and CSC The calculation including the electric and color charge neutrality.

20. Phase Diagram in the T  plane