1. Duke University Chiho NONAKA in Collaboration with Masayuki Asakawa (Kyoto University) Hydrodynamical Evolution near the QCD Critical End Point June 26, 2003@HIC03, McGill University, Montreal

2. C.NONAKA 06/26/2003 2 Critical End Point in QCD ? Critical End Point in QCD ? Z. Fodor and S. D. Katz ( JHEP 0203 (2002) 014 ) NJL model (Nf = 2) Lattice (with Reweighting) K. Yazaki and M.Asakawa., NPA 1989 Suggestions 2SC CFL T  RHIC GSI Critical end point

3. C.NONAKA 06/26/2003 3 Phenomenological Consequence ? Phenomenological Consequence ? Divergence of Fluctuation Correlation Length critical end point M. Stephanov, K. Rajagopal, and E.Shuryak, PRL81 (1998) 4816 Still we need to study EOS Focusing Dynamics (Time Evolution) Hadronic Observables : NOT directly reflect properties at E Fluctuation, Collective Flow If expansion is adiabatic.

4. C.NONAKA 06/26/2003 4 How to Construct EOS with CEP? Assumption Critical behavior dominates in a large region near end point Near QCD end point singular part of EOS Mapping Matching with known QGP and Hadronic entropy density Thermodynamical quantities EOS with CEP  r h T QGP Hadronic 3d Ising Model Same Universality Class QCD

5. C.NONAKA 06/26/2003 5 EOS of 3-d Ising Model Parametric Representation of EOS Guida and Zinn-Justin NPB486(97)626 h : external magnetic field QCD Mapping  T r h

6. C.NONAKA 06/26/2003 6 Thermodynamical Quantities Singular Part of EOS near Critical Point Gibbs free energy Entropy density Matching Entropy density Thermodynamical quantities Baryon number density, pressure, energy density  r h T QGP Hadronic

7. C.NONAKA 06/26/2003 7 Equation of State CEP Entropy Density Baryon number density

8. C.NONAKA 06/26/2003 8 Comparison with Bag + Excluded Volume EOS Comparison with Bag + Excluded Volume EOS With End Point Bag Model + Excluded Volume Approximation (No End Point) Focused Not Focused = Usual Hydro Calculation n /s trajectories in T-  plane B

9. C.NONAKA 06/26/2003 9 Slowing out of Equilibrium Slowing out of Equilibrium B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017 Berdnikov and Rajagopal’s Schematic Argument along r = const. line Correlation length longer than  eq h  faster (shorter) expansion r h slower (longer) expansion Effect of Focusing on  ? Focusing Time evolution : Bjorken’s solution along n B /s    fm, T 0 = 200 MeV   eq

10. C.NONAKA 06/26/2003 10 Correlation Length (I) Widom’s scaling low  eq depends on n /s.  Max. Trajectories pass through the region where  is large. (focusing) eq B r h

11. C.NONAKA 06/26/2003 11 Correlation Length (II)  time evolution (1-d) Model C (Halperin RMP49(77)435)  is larger than  at Tf. Differences among  s on n /s are small. In 3-d, the difference between  and  becomes large due to transverse expansion. eq B 

12. C.NONAKA 06/26/2003 12 Consequences in Experiment (I) CERES: nucl-ex/0305002 Fluctuations CERES 40,80,158 AGeV Pb+Au collisions No unusually large fluctuation CEP exists in near RHIC energy region ? Mean P T Fluctuation

13. C.NONAKA 06/26/2003 13 Consequences in Experiment (II) Xu and Kaneta, nucl-ex/0104021(QM2001) Kinetic Freeze-out Temperature J. Cleymans and K. Redlich, PRC, 1999 ? Low T comes from large flow. f ? Entropy density EOS with CEP EOS with CEP gives the natural explanation to the behavior of T. f

14. C.NONAKA 06/26/2003 14 CEP and Its Consequences Realistic hydro calculation with CEP Future task Slowing out of equilibrium Large fluctuation Freeze out temperature at RHIC Fluctuation Its Consequences Focusing

15. Back UP

16. C.NONAKA 06/26/2003 16 Hadronic Observables Fluctuations Mean transverse momentum fluctuation Charge fluctuations D-measure Dynamical charge fluctuation Balance function Collective Flow Effect of EOS Jeon and Koch PRL85(00)2076 Pruneau et al, Phys.Rev. C66 (02) 044904 Gazdzicki and Mrowczynski ZPC54(92)127 Korus and Mrowczynski, PRC64(01)054906 Asakawa, Heinz and Muller PRL85(00)2072 Bass, Danielewicz, Pratt, PRL85(2000)2689 Rischke et al. nucl-th/9504021

17. C.NONAKA 06/26/2003 17 Baryon Number Density Critical end point 1st order crossover Crossover :  st order :

18. C.NONAKA 06/26/2003 18 n B /S contours S nBnB n B /S Focusing !

19. C.NONAKA 06/26/2003 19 Focusing and CEP

20. C.NONAKA 06/26/2003 20 FocusingFocusing What is the focusing criterion ?  r h T h r CEP contours From our model Dominant terms Critical behavior etc.

21. C.NONAKA 06/26/2003 21 FocusingFocusing Scavenius et al. PRC64(2001)045202 Analyses from Linear sigma model & NJL model They found the Critical point in T-  plane. NJL model Sigma model The critical point does not serve as a “focusing” point ! Sigma model NJL model n B /S lines in  plane

22. C.NONAKA 06/26/2003 22 Hydrodynamical evolution Au+Au 150AGeV b=3 fm

23. C.NONAKA 06/26/2003 23 Relativistic Hydrodynamical Model Relativistic Hydrodynamical Equation Baryon Number Density Conservation Equation Lagrangian hydrodynamics Space-time evolution of volume element Effect of EoS Flux of fluid

24. C.NONAKA 06/26/2003 24 Sound Velocity Clear difference between n /s=0.01 and 0.03 B Effect on Time Evolution Collective flow EOS Sound velocity along n /s B /L TOTAL /L TOTAL