San Diego

1. Beijing San Diego

2. Symmetry Restoration and Breaking • Is there a first order phase transition in QCD? • Under what conditions will the phase transition and, hence, the chiral symmetry restoration and breaking of center Z(3) symmetry occur?

3. Finite Density QCD Phase Transition from Lattice Calculation • Finite Density Algorithm with Canonical Approach • Winding Number Expansion Method • Results on NF = 2, and 4 with Wilson Fermion • NF = 3 with Clover Fermion χQCDCollaboration: Anyi Li, Andrei Alexandru, KFL, and XiangfeiMeng PKU, June 29, 2009

4. Quark Gluon Plasma (QGP) Phase Hardonic Phase

5. RHIC : Au-Au collisions with 200 GeV/A PHOBOS BRAHMS PHENIX STAR Relativitic Heavy Ion Collider

6. RHIC Relativistic (ideal) hydrodynamics + initial cond. + freezeout + … Soft physics from lattice QCD -- EoS, Tc etc Hard physics from pQCD -- jet energy loss etc L. Landau (1953) Current Methodology to study QGP T. Hatsuda Lattice ‘06

7. Phase DiagramUnder Study T Crossover μ We want to study here

8. Finite density lattice QCD calculations: Multi-parameter Reweighting Taylor Expansion Imaginary Chemical Potential Canonical Ensemble with Reweighting …

11. Overlap problem

12. Overlap problem

16. Canonical partition function

17. Canonical Ensemble Approach: is real

18. T T S S A0 A0 T T S S A1 A2

19. Overlap Problem

20. Avoid the Overlap Problem KFL (Int. Jou. Mod. Phys. B16, 2017 (2002)) The earlier procedure is like projection after variation (Peierls and Yoccoz) Need variation after projection (Zeh-Rouhaninejad-Yoccoz) Accept/reject based on detB. Unfortunately, this introduces fluctuation problem! Because

21. Canonical ensembles Discrete Fourier transform Canonical approach K. F. Liu, QCD and Numerical Analysis Vol. III (Springer,New York, 2005),p. 101. Andrei Alexandru, Manfried Faber, Ivan Horva´th,Keh-Fei Liu, PRD 72, 114513 (2005) Standard HMC Accept/Reject Phase Continues Fourier transform Useful for large k WNEM 21 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

22. Instability of discrete Fourier transform It’s difficult to pick up the high frequency modes with discrete Fourier transform Winding number expansion in canonical approach to finite densityXiangfeiMeng - Lattice 2008 Williamsburg

23. Winding number expansion (I) In QCD Tr log loop loop expansion In particle number space Where Winding number expansion in canonical approach to finite densityXiangfeiMeng - Lattice 2008 Williamsburg

24. T T S S A0 A0 T T S S A1 A2

25. Winding number expansion (II) For So The first order of winding number expansion Here the important is that the FT integration of the first order term has analytic solution is Bessel function of the first kind . Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg

26. Winding number expansion test (I) K=12 Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg

27. Winding number expansion test (III) Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg

29. Observables Polyakov loop Baryon chemical potential Phase 29 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

30. Phase Diagrams T T First order ? ρq ρh Mixed phase ρ μ

31. T plasma hadrons coexistent ρ Phase diagram Two flavors Four flavors ? T plasma hadrons coexistent ρ 31 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

32. Baryon Chemical Potential for Nf = 4 (mπ ~ 1 GeV) 32 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

33. Phase Boundaries from Maxwell Construction Nf = 4 Wilson gauge + fermion action Maxwell construction : determine phase boundary 33 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

35. T plasma hadrons coexistent ρ Phase diagram Two flavors Four flavors ? T plasma hadrons coexistent ρ 35 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

36. Baryon Chemical Potential (mπ ~ 1 GeV) 36 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg

37. Three Flavors (Preliminary) mπ ~0.8 GeV

39. Critical End Point

40. PolyakovLoop

41. Quark Condensate

42. We are here

43. Summary • Canonical Ensemble Approach overcomes the overlap problem and alleviates the fluctuation problem. No sign problem for T > 0.8 Tc. • Exact algorithm which can discern a first order phase transition from finite volumes. • Wilson fermion on 63 x 4 lattice with mπ ~ 1 GeV shows NF = 2 is 2nd order (?) down to 0.83 Tc, NF = 4 is first order. • Iwasaki + Clover fermion on 63 x 4 lattice with mπ ~ 0.8 GeV results for NF = 3 shows an unambiguous first order phase transition at finite density.

44. Future • Smaller masses • Larger volume  HNMC (A. Alexandru, et al. arXiv:0711.2678) • Chiral fermion action • Lower temperature (sign problem)

45. Phase Boundaries Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62 4 flavor (taste) staggered fermion

46. Polyakov Loop

47. Winding number expansion (IV) The parameters of Winding number expansion---Fourier series The recursion of Bessel function Winding number expansion in canonical approach to finite density Xiangfei Meng - Lattice 2008 Williamsburg

48. Phase Boundary (Preliminary) Nf = 4 This work Ph. Forcrand,S.Kratochvila, Nucl. Phys. B (Proc. Suppl.) 153 (2006) 62 48 Finite density simulation with the canonical ensemble Anyi Li - Lattice 2008 Williamsburg