M. Ohtani ( RIKEN ) with S. Digal ( Univ. of Tokyo )

1. Lattice study of color superconductivity with Ginzburg-Landau action M. Ohtani (RIKEN) with S. Digal (Univ. of Tokyo) T. Hatsuda (Univ. of Tokyo)  hep-lat/0511018 • Introduction • GL effective action • Phase diagram in weak gauge coupling • Phase transition on the lattice • Summary Feb 28 @ KEK-hadron06

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

3. { ¶ no sign problem no det D T-m dependence: m, li ,k,g ( discretize & rescale SUf (3) SUc(3) Higgs on Lattice ○ ○ ○ 2 couplings for quartic terms Ginzburg-Landau effective action Iida & Baym PRD 65 (2002)014022 GL action in terms of the quark pair field Fcf (x) & gauge field

4. l1 = l2 in weak coupling mean field without gluon Iida & Baym PRD 63 (2001)074018 mean field (ungauged) l2 normal  CFL as T D F~D D l1 F = 0 normal  2SC unbound @ Tc(MF) 0 F~0 D 2nd order transition

5. perturbative analysis |F |3 term , -F 2 ln F 2 term.. gluonic fluctuation Matsuura,Hatsuda,Iida,Baym PRD 69 (2004) 074012 T

6. perturbative analysis l2 normal  CFL gluonic fluctuation normal2SCCFL |F |3 term l1 normal 2SC unbound 1st order transition weak gauge coupling limit mean field (ungauged) l2 Normal  CFL l1 normal2SC unbound 2nd order transition

7. Setup for Monte-Carlo simulation parameters { b = 5.1 0.7 bc in pure YM take several pairs of (l1, l2 ), scanning k Lattice size Lt = 2 , Ls = 12, 16, 24, 32, 40 with 3,000－60,000 configurations update pseudo heat-bath method for gauge field generalized update-algorithm of SU(2) Higgs-field Bunk, NP(Proc.Suppl) 42 (‘95), 556 @ RIKEN Super Combined Cluster

8. phase transition to ‘color super’ ¶ Tr Fx†Fx¹ 0 even in sym. phase thermal fluctuation  broken phase plateau jump @ kc Phase identification Tr Fx†Fx (Tr F†F )1/2 update step large order param. ⇔ broken phase

9. F†F I・・・ CFL a F†F ~b ・・・ 2SC b ¶ F†F  : gauge invariant diagonalization identifying the phases by eigenvalues of F†F matrix elements of F†F 

10. Phase id. : consistent with / without gauge fix. identifying the phases by Tr(F†F )n Maximize  Re Tr Ux,μ  u/ Landau gauge fixing x,μ 1/(n-1) 3 ・・・ CFL = 2 ・・・ dSC 1 ・・・ 2SC [Tr F† F ]n Tr [(F†F )n ] # of quark pairing:

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

12. perturbative analysis l2 CFL 2SCCFL l1 2SC unbound Phase diagram with b fixed lattice simulation l2 CFL CFL w/metastable 2SC 2SCCFL metastable 2SC: 2SC observed in hysteresis & disappeared in boundary shift test 2SC l1 1st order transition

13. ●largest barrier btw normal &CFL ●$metastable 2SC Free energy by perturbation D1 = D2 2 D1 F†F~D1 D3 CFL normal 2SC D3

14. Summary and outlook • GL approach with quark pair field F & gauge on lattice •  SU(3) Higgs model • eigenvalues of F†F to identify the phases 1st order trans. toCFL & 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-m, …