1. Thermal phase transition of color superconductivity with Ginzburg-Landau effective action on the lattice M. Ohtani ( RIKEN ) with S. Digal (Univ. of Tokyo) T. Hatsuda (Univ. of Tokyo) XQCD, Aug 2 @ Swansea Introduction GL effective action Phase diagram in weak gauge coupling Phase transition on the lattice Summary

2. Δ ~ 100MeV T c ~ 60MeV Introduction Non-perturbative analysis of colorsuper transition Ｔ μ Hadrons Quark-Gluon Plasma Color Superconductivity RHIC 170MeV ~400MeV N ☆ Cores    qq   0

3. ¶ no sign problem bosonic T-  dependence: m, i, ,  g (( Ginzburg-Landau effective action  GL action in terms of the quark pair field  fc (x) & gauge field  Iida & Baym PRD 65 (2002) 014022 { discretize & rescale SU f (3)  SU c (3) Higgs on Lattice 2 couplings for quartic terms ○○○

4. mean field without gluon  Iida & Baym PRD 63 (2001) 074018 mean field (ungauged) normal  CFL normal  2SC   unbound 2 nd order transition            as T @ T c (MF) 1 = 2 in weak coupling

5. weak gauge coupling limit mean field (ungauged) perturbative analysis  Matsuura,Hatsuda,Iida,Baym PRD 69 (2004) 074012 Normal  CFL normal  2SC   unbound 2 nd order transition gluonic fluctuation |  | 3 term 1 st order transition normal  2SC   normal  2SC  CFL unbound normal  CFL

6. normal unbound Phase diagram in weak gauge coupling CFL 2SC  T/T c (MF)    

7. Phase diagram in weak gauge coupling

8. Analytic results for large  mean field (ungauged) perturbative analysis Normal  CFL normal  2SC   unbound 2 nd order transition gluonic fluctuation 1 st order transition normal  2SC   normal  2SC  CFL unbound normal  CFL

9. parameters Setup for Monte-Carlo simulation Lattice size L t = 2, L s = 12, 16, 24, 32, 40 @ RIKEN Super Combined Cluster pseudo heat-bath method for gauge field generalized update-algorithm of SU(2) Higgs-field  = 5.1  0.7   c in pure YM  take several pairs of ( 1, 2 ), scanning  { with 3,000 － 60,000 configurations  Bunk, NP(Proc.Suppl) 42 (‘95), 556 update

10. broken phase  Phase identification (Tr  †  ) 1/2 large order param. ⇔ broken phase plateau jump @  c update step phase transition to ‘color super’ Tr  x †  x ¶ Tr  x †  x  0 even in sym. phase thermal fluctuation 

11. identifying the phases by eigenvalues of  y  diagonalization matrix elements of  †   †    ・・・ CFL a  †   b ・・・ 2SC b ¶  †  :  gauge invariant

12. Hadron (Quark-Gluon Plasma) Color Superconducting state  5.1 4.8 5.6 0.08 0.16 Phase diagram with i fixed 3.6 CFL 2SC normal ● Similar trends with SU(2) Higgs ● no clear signal of end points as i 1 = 2 =.0005 

13. 2SC CFL 1 st order transition: Hysteresis & boundary shift initial config. = a thermalized config. with slightly different   Hysteresis : different configs. with same   Put 3 configs in spatial sub-domain  Thermalize it with fixed  Polyakov loop normal  CFL 2SC

14. Phase diagram with  fixed 1 2 CFL 2SC 1 st order transition CFL w/ metastable 2SC 2SC  CFL lattice simulation metastable 2SC: 2SC observed in hysteresis & disappeared in boundary shift test perturbative analysis 2SC   2SC  CFL unbound CFL

15. Free energy by perturbation    =   normal CFL    †        2SC  Iida,Matsuura,Tachibana,Hatsuda PRD 71 (2005) 054003 ● largest barrier btw normal &CFL ●   metastable 2SC

16. Summary and outlook  GL approach with quark pair field  & gauge on lattice  SU(3) Higgs model  eigenvalues of  †  to identify the phases  1 st order trans. to CFL & 2SC phases in coupling space  We observed hysteresis. transition points  boundary shift with mixed domain config.   metastable 2SC state in transition from normal to CFL, which is consistent with perturbative analysis  charge neutrality, quark mass effects, correction to scaling, phase diagram in T -  …