1. Quark spectrum near chiral and color-superconducting phase transitions Masakiyo Kitazawa Kyoto Univ. M.K., T.Koide, T.Kunihiro and Y.Nemoto, PRD70,056003 (2004), M.K., T.Koide, T.Kunihiro and Y.Nemoto, hep-ph/0502035, M.K., T.Kunihiro and Y.Nemoto, in preparation. International Workshop on "Chiral Restoration in Nuclear Medium"

2. 1, 1, Introduction

3. 150~170MeV Phase Diagram of QCD Color Superconductivity(CSC) Hadrons T Chiral Symm. Broken 0  ~100MeV Hadronic excitations in QGP phase soft mode of chiral transition - Hatsuda, Kunihiro. qq bound state - Shuryak, Zahed; Brown, Lee, Rho, Shuryak. Lattice simulations – Asakawa, Hatsuda; etc. Pre-critical region of CSC large pair fluctuations  precursory phenomena of CSC M.K., et al., 2002,2004

4. Formation of Pseudogap in CSC M. K. et al. PRD70, 956003(2004) Let ’ s see the property of quarks near T c of (1) the CSC. (2) the  SB. the “pseudogap region” above the CSC phase.

5. Nambu-Jona-Lasinio model (2-flavor,chiral limit) ：  ： SU(2) F Pauli matrices ： SU(3) C Gell-Mann matrices C :charge conjugation operator so as to reproduce Parameters: Klevansky(1992), T.M.Schwarz et al.(1999) M.K. et al., (2002) 2SC is realized at low  and near T c. We neglect the gluon degree of freedom. Notice: NJL model

6. 2, 2, Quarks above CSC phase transition T 

7. Response Func. of Pair Field Response Fucntion D(k,  ) in linear response theory & RPA Spectral Function As T  T c, the peak grows. The soft mode of the CSC trans. ε→ ０ （Ｔ → Ｔ Ｃ ） for k=0 The peak grows from e ~ 0.2 electric SC ： e ~ 0.005  = 400 MeV M.K., et al., PRD 65, 091504 (2002)

8. T-matrix Approximation Quark Green function : :T-matrix Self-energy: Decomposition of G: positive energy part

9. Dispersion Relation of Quarks  =   (p) rapid increase around  =0  [MeV] k [MeV] 40 80 0 -40 -80 400 320 480 0  k kFkF 0  k kFkF Normal Super  = 400 MeV  =0.01 M. K. et al. 2002, 2004 w.f. renormalization  still Fermi-liquid-like However,

10. stronger diquark coupling G C Diquark Coupling Dependence GCGC ×1.3×1.5  = 400 MeV  =0.01

11. Resonant Scattering of Quarks G C =4.67GeV -2 Janko, Maly, Levin, PRB56,R11407 (1995)

12. Resonant Scattering of Quarks G C =4.67GeV -2 Mixing between quarks and holes  k  n f (  )

13. 3, 3, Quarks above chiral phase transition T 

14. Quarks at very high T 1-loop (g<<1) Hard Thermal Loop ( p, , m q <<T ) dispersion relations plasmino

15. Quarks at very high T 1-loop(g<<1) Hard Thermal Loop approximation( p, , m q <<T ) dispersion relations

16. Soft Mode of Chiral Transition Response Fucntion D(k,  )  fluctuations of the chiral order parameter Spectral Function ε→ ０ （Ｔ → Ｔ Ｃ ） for k=0 T  Hatsuda, Kunihiro ( ’ 85)

17. Quark Self-enrgy Quark Green function : :free quark progagator Self-energy: in the chiral limit

18. Spectral Function of Quarks  [MeV] k [MeV]  = 0 MeV T = 200MeV positive energy part  - ( ,k) k [MeV] sharp peak with negative dispersion  [MeV] quasiparticle peak  ~ k

19. Spectral Function of Quarks  [MeV] k [MeV]  = 0 MeV T = 200MeV positive energy part  - ( ,k) k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV]

20. Resonant Scatterings of Quarks These resonant scatterings affect the peaks of the spectral functions in a non-trivial way.

21. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.05

22. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.1

23. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.15

24. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.2

25. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.25

26. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.3

27. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.35

28. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.4

29. k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV] T dependence  = 0.5

30. SummarySummary The soft mode associated with the chiral and color-superconducting phase transitions strongly affects the property of quarks near T c. T 0  They can be understood through the resonant scattering of quarks. Future: finite quark mass, finite density, phenomenological applications

31.  0 ( ,k)  = 400 MeV  =0.01 Spectral Function of Quarks k  0  [MeV] quasi-particle peak,  =    k)~ k  Depression at Fermi surface Im   ,k=k F )  [MeV] The peak in Im  around  =0 owing to the decaying process: k [MeV] kFkF kFkF