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, (2004), M.K., T.Koide, T.Kunihiro and Y.Nemoto, hep-ph/ , M.K., T.Kunihiro and Y.Nemoto, in preparation× ２. 第６回 関西ＱＮＰセミナー 於：京大基研

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

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

The pseudogap survives up to  =0.05~0.1 ( 5~10% above T C ). Numerical Result : Density of State

Spectral Function of Quarks  [MeV] k [MeV]  = 0 MeV  = 0.05 quark part  - ( ,k) sharp peak with negative dispersion k [MeV]  [MeV] quasiparticle peak  ~ k

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

2, 2, Quarks above CSC phase transition T 

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

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

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

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

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

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

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

Rondom Phase Approx. (RPA)  ()() Thouless Criterion D R (0,0) diverges at T C - for second order phase transitions D.J. Thouless, AoP 10,553(1960) r.h.s. is equal to zero at T c due to the critical conditon. The fluctuation diverges at T c. Thermodynamic Potential

Softening of Pair Fluctuations Dynamical Structure Factor T =1.05T c The peak grows from e ~ 0.2 electric SC ： e ~  = 400 MeV

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

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

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

The pseudogap survives up to  =0.05~0.1 ( 5~10% above T C ). Numerical Result : Density of State

 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

Im   ,k) quasi-particle peak w = m –k peak of Im S w =k– m : collective mode : on-shell  |Im   | has peaks around  =  k, which is found to be the hole energy. |Im  - | k coincide at fermi surface. Re   ,k) w = m –k  -- 0  k Peak of |Im   | kFkF

Dispersion Relation of Quarks  =   (p) rapid increase around  =0  [MeV] k [MeV]  k kFkF 0  k kFkF Normal Super cf.)  = 400 MeV  =0.01

Dispersion Relation of Quarks  =   (p) rapid increase around  =0  [MeV] k [MeV] Re   ,k=k F )  [MeV]  = 400 MeV  =0.01 w.f. renormalization  still Fermi-liquid-like However,

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

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

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

Level Repulsion    p pFpF

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

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

3, 3, Quarks above chiral phase transition T 

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) scalar and pseudoscalar parts

Sigma Mode above T c Hatsuda, Kunihiro ( ’ 85) sharp peak in time-like region  -mode Spectral Function soft mode of CSC k  k  sharp peak around  = k =0

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

Spectral Function of Quarks  [MeV] k [MeV]  = 0 MeV  = 0.05 quark part  - ( ,k) sharp peak with negative dispersion k [MeV]  [MeV] quasiparticle peak  ~ k

Self Energy k [MeV]  [MeV] Two peaks in Im  produces five solutions of the dispersion relation.

Spectral Function of Quarks  [MeV] k [MeV]  = 0 MeV  = 0.05 positive energy part  - ( ,k) k [MeV]  [MeV]  - ( ,k)  + ( ,k) k [MeV]

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

Level Repulsion   m>m> m=m= dispersion relation m,-m for the CSC

Self Energy

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

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

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

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

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

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

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

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

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

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