1. QCD 相転移の臨界点近傍における 非平衡ダイナミクスについて 北沢正清（京大）, 国広悌二（京大基研 ), 根本幸雄 (RIKEN-BNL) 0 T  の１コメ ント Chiral symmetry breaking Color superconductivity (CSC) critical endpoint CONTENTS 1, Introduction 2, Collective Mode in CSC 3, Effective Equation for Collective Mode 4, Numerical Simulation 5, Summary and Outlooks

2. Main New Results given in This Talk derive equation for the collective mode of the pair field above CSC, position of pole: 0 T  (determine A, B microscopically)  (x) x x =2  /k in linear response theory.

3. Phase Transitions and Fluctuations in QCD fluctuation diverges.  ()() At the critical point of the second order phase transitions, fluctuation of order parameter field might be responsible for various observables. chiral transition critical end point (CEP) CSC transition 0 2SC T  CFL RHIC AGS SPS GSI,J-PARC another CEP? 1, 1, Introduction ? ?? light sigma meson susceptibilities baryon number, chiral, etc… transport coefficient ??? Stephanov, Rajagopal, Shuryak / Berdnikov, Rajagopal / Hatta, Ikeda / Fukushima / Fujii / Hatta, Stephanov

4. Spectral Function of Pair Field ε→ ０ （Ｔ → Ｔ Ｃ ） ＋ ＋・・・ As T is lowered toward T C, The peak of  becomes sharp. M.K., T.Koide, T.Kunihiro, Y.Nemoto, PRD 65, 091504 (2002) The peak survives up to e ~ 0.2 electric SC ： e ~ 0.005  (k=0,  =0) diverges at T=T C. Thouless criterion provided the second order transition, Fluctuation of pair field in CSC

5.  Dependence of Pseudogap Depth of the pseudogap hardly changes with . Pseudogap in quark DOS!

6. Model 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) 2, 2, Collective Mode in CSC

7. Response Function of Pair Field expectation value of induced pair field: external field: Linear Response Fourier transformation with Matsubara formalism Retarded Green function RPA approx.: where,

8. Analytic Properties of Q(k,  )

9. 1st term: 3,4th term: Im Q(k, w ) pair creation scattering  ~0 k~0 near T c Collective mode 2nd term: H.Fujii, PRD67 (2003) 094018 cf.) in the chiral phase transition

10. Since ImQ is free from UV divergence, we calculate it without cutoff. Then, we obtain a simple form, Re Q is calculated from the dispersion relation, Notice: which ensures (with 3-momentum cutoff) Cutoff Scheme

11. Spectral Function of Pair Field ε→ ０ （Ｔ → Ｔ Ｃ ） ＋ ＋・・・ As T is lowered toward T C, The peak of  becomes sharp. M.K., T.Koide, T.Kunihiro, Y.Nemoto, PRD 65, 091504 (2002) The peak survives up to e ~ 0.2 electric SC ： e ~ 0.005  (k=0,  =0) diverges at T=T C. Thouless criterion provided the second order transition, Collective Mode in CSC

12. Collective Mode pole of the response function pair field  ind (k,  (k)) can be created with an infinitesimal  ex Notice: pole locates in the lower half plane Pole of Collective Mode k z k z first sheet second sheet

13. Numerical Results  =0.4  =0.2  =0 k=200MeV k=100MeV k=0MeV k=300MeV  =400MeV Our calculation shows, linearquadratic for k=0  =0,0.2,…,0.8 k=0,50,100,… Poles locate in one direction in the complex plane. It is not pure imaginary. damped oscillation

14. 3, 3, Effective Equation for Collective Mode Near the crtical temperature,  -1 =g -1 +Q expands, Thouless criterion: Notice The solution of collective mode (  -1 =0) reads, here, : real : complex

15. : real : complex TDGL equation pure imaginary second time derivative term can appear when particle-hole symmetry is broken Notice: pure imaginary in sigma mode of  SB H. Fujii Particle-hole asymmetry in CSC caused the real part of . It decreases as  increases.

16. Numerical Check for  =400MeV (T c =40.04MeV) Lowest expansion reproduces the full calculations well. up to covers the region where valid collective mode appear. Im  (k) k :Lowest expansion :Full calculation

17. Time Evolution of Pair Field  =0.01  =0.05  =0.1  =0.5 k =0 MeV k =50 MeV k =100 MeV k =150 MeV k =200 MeV As T is lowered toward T c, lifetime of the collective mode becomes longer. large momentum mode is not affected at all near T c. Damped oscillation, but heavy damping 4, 4, Numerical Simulation

18. Fluctuations in Coordinate Space in infinite matter initial condition:  =0.5  =0.1  =0.01 200fm  t  200fm Long wave length (low momentum) fluctuations survives. t Time scale of CSC is longer than the one of  SB.

19. We calculated the collective mode of pair field in CSC. We derived effective equation which describes non-equilibrium dynamics of the pair-field near T c and low momentum, and confirmed that nature of the collective mode is damped oscillation. Summary The collective mode with pole near the origin might affect various observables (,k)(,k) collective mode: cf, in  SB: