カイラル相転移・カラー超伝導の 臨界温度近傍における クォークの準粒子描像

1. 第６回　関西ＱＮＰセミナー　於：京大基研 カイラル相転移・カラー超伝導の 臨界温度近傍における クォークの準粒子描像 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×２.

2. [３]c×[３]c＝[３]c＋[６]c Phase Diagram of QCD Color Superconductivity quark (fermion) system attractive channel in one-gluon exchange interaction. T RHIC 150~170MeV Cooper instability at sufficiently low T SU(3)c color-gauge symmetry is broken! Chiral Symm. Broken GSI,J-PARC Color Superconductivity Hadrons m Compact Stars 0 2SC pairing at low energy: d u

3. 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 M.K., et al., 2002,2004 • large pair fluctuations •  precursory phenomena of CSC Phase Diagram of QCD T 150~170MeV Chiral Symm. Broken ~100MeV Color Superconductivity(CSC) Hadrons m 0

4. Numerical Result : Density of State The pseudogap survives up to e =0.05~0.1 ( 5~10% above TC ).

5. quasiparticle peak w ~ k w [MeV] k[MeV] Spectral Function of Quarks m = 0 MeV e= 0.05 quark part sharp peak with negative dispersion r-(w,k) k[MeV] w [MeV]

6. TABLE OF CONTENTS 1, Introduction 2, Quarks above CSC phase transition 3, Quarks above chiral phase transition 4, Summary

7. 2,Quarks above CSC phase transition T m

8. Nature of CSC strong coupling! weak coupling D ~ 100MeV in electric SC D / EF ~ 0.0001 D / EF ~ 0.1 Mean field approx. works well. Short coherence length x. There exist large fluctuations of pair field. Large pair fluctuations can invalidate MFA. cause precursory phenomena of CSC. Matsuzaki, PRD62,017501 (2000) Abuki, Hatsuda, Itakura, PRD 65, 074014 (2002) cf.) Bosonization of Cooper pairs

9. F(D) D ペア場のゆらぎは、 集団モードを形成する。 極 T カラー超伝導 Tcで原点に到達 ソフトモード m ペア場のゆらぎ 二次相転移点では、秩序変数のゆらぎが発散。 ペア場D(x) for CSC クォーク対

10. Pair Fluctuations in Superconductors electric conductivity Precursory Phenomena in Alloys enhancement above Tc • Electric Conductivity • Specific Heat • etc… e ~10-3 Thouless, 1960 Aslamasov, Larkin, 1968 Maki, 1968, … e High-Tc Superconductor(HTSC) in quasi-two-dimensional cuprates 1986~ large fluctuations induced by strong coupling and quasi-two dimensionality pseudogap

11. Quarks in BCS Theory Quasi-particle energy: Density of State:

12. Pseudogap :Anomalous depression of the density of state near the Fermi surface in the normal phase. Conceptual phase diagram Renner et al.(‘96) The origin of the pseudogap in HTSC is still controversial.

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

14. Response Function of Pair Field Linear Response external field: expectation value of induced pair field: Retarded Green function Fourier Transformation T-matrix

15. Thouless Criterion D.J. Thouless, AoP 10,553(1960) DR(0,0) diverges at TC - for second order phase transitions W(D) Thermodynamic Potential D r.h.s. is equal to zero at Tc due to the critical conditon. The fluctuation diverges at Tc. Rondom Phase Approx. (RPA)

16. Softening of Pair Fluctuations Dynamical Structure Factor T =1.05Tc m= 400 MeV The peak grows frome ~ 0.2 electric SC：e ~ 0.005

17. Softening of Pair Fluctuations m= 400 MeV Dynamical Structure Factor Pole of Collective Mode pole: T =1.05Tc The pole approaches the origin as T is lowered toward Tc. The peak grows frome ~ 0.2 electric SC：e ~ 0.005 (the soft-mode of the CSC)

18. quark part :projection op. T-matrix Approximation Quark Green function : Decomposition of G:

19. Spectral Function of Quarks Spectral Function vanishes in the chiral limit from parity and rotational invariance spectral function of baryon density Density of State N(w)

20. Numerical Result : Density of State The pseudogap survives up to e =0.05~0.1 ( 5~10% above TC ).

21. Depression at Fermi surface Im S- (w,k=kF) The peak in ImS around w=0 owing to the decaying process: w [MeV] Spectral Function of Quarks m= 400 MeV e=0.01 r0(w,k) quasi-particle peak, w =w-(k)~ k-m w k 0 w [MeV] k[MeV] kF kF

22. Im S- (w,k) |Im S-| Peak of |Im S-| k w =m –k kF k -m 0 w m w |Im S-| has peaks around w =m-k, which is found to be the hole energy. : on-shell : collective mode w =m–k peak of ImS quasi-particle peak w =k–m Re S- (w,k) coincide at fermi surface.

23. rapid increase around w =0 Super Normal cf.) w w 0 k 0 kF k kF Dispersion Relation of Quarks m= 400 MeV e=0.01 80 w =w-(p) 40 w [MeV] 0 -40 -80 400 480 320 k [MeV]

24. However, Re S- (w,k=kF) w.f. renormalization still Fermi-liquid-like w [MeV] Dispersion Relation of Quarks m= 400 MeV e=0.01 80 w =w-(p) 40 w [MeV] 0 rapid increase around w =0 -40 -80 400 480 320 k [MeV]

25. stronger diquark coupling GC ×1.3 ×1.5 Diquark Coupling Dependence m= 400 MeV e=0.01 GC

26. Resonant Scattering of Quarks GC=4.67GeV-2 Janko,Maly, Levin, PRB56,R11407 (1995)

27. Resonant Scattering of Quarks GC=4.67GeV-2 Mixing between quarks and holes w kF k nf (w) w

28. w pF p Level Repulsion D+ D

29. Quarks at very high T • 1-loop (g<<1) • Hard Thermal Loop ( p, w, mq<<T ) dispersion relations plasmino plasmino

30. Quarks at very high T • 1-loop(g<<1) • Hard Thermal Loop approximation( p, w, mq<<T ) dispersion relations

31. 3,Quarks above chiral phase transition T m

32. Soft Mode of Chiral Transition Hatsuda, Kunihiro (’85) Response Fucntion D(k,w) scalar and pseudoscalar parts fluctuations of the chiral order parameter Spectral Function ε→０ （Ｔ→ＴＣ） fork=0 T m

33. Sigma Mode above Tc Hatsuda, Kunihiro (’85) Spectral Function sharp peak in time-like region w s -mode k soft mode of CSC w sharp peak around w = k =0 k

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

35. quasiparticle peak w ~ k w [MeV] k[MeV] Spectral Function of Quarks m = 0 MeV e= 0.05 quark part sharp peak with negative dispersion r-(w,k) k[MeV] w [MeV]

36. w [MeV] k[MeV] Self Energy Two peaks in ImS produces five solutions of the dispersion relation.

37. Spectral Function of Quarks m = 0 MeV e= 0.05 positive energy part w [MeV] r-(w,k) r+(w,k) r-(w,k) k[MeV] k[MeV] k[MeV] w [MeV]

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

39. dispersion relation m>D m=D Level Repulsion for the CSC D+ D m,-m m,-m

40. Self Energy

41. T dependence w [MeV] e = 0.05 r+(w,k) r-(w,k) k[MeV] k[MeV]

42. T dependence w [MeV] e = 0.1 r+(w,k) r-(w,k) k[MeV] k[MeV]

43. T dependence w [MeV] e = 0.15 r+(w,k) r-(w,k) k[MeV] k[MeV]

44. T dependence w [MeV] e = 0.2 r+(w,k) r-(w,k) k[MeV] k[MeV]

45. T dependence w [MeV] e = 0.25 r+(w,k) r-(w,k) k[MeV] k[MeV]

46. T dependence w [MeV] e = 0.3 r+(w,k) r-(w,k) k[MeV] k[MeV]

47. T dependence w [MeV] e = 0.35 r+(w,k) r-(w,k) k[MeV] k[MeV]

48. T dependence w [MeV] e = 0.4 r+(w,k) r-(w,k) k[MeV] k[MeV]

49. T dependence w [MeV] e = 0.5 r+(w,k) r-(w,k) k[MeV] k[MeV]

50. Summary The soft mode associated with the chiral and color-superconducting phase transitions drastically modifies the property of quarks near Tc. above CSC phase: Gap-like structure manifests itself! resonant scattering of quarks above chiral transition: Three peak structure appears! two resonant scatterings of quarks and anti-quarks Future: finite quark mass, finite density, phenomenological applications