Hydrodynamical Evolution near the QCD Critical End Point

1. Hydrodynamical Evolution near the QCD Critical End Point Duke University Chiho NONAKA in Collaboration with Masayuki Asakawa (Kyoto University) November, 2003@Collective Flow and QGP properties, BNL

2. RHIC T Critical end point CFL 2SC m GSI Critical End Point in QCD ? Suggestions • NJL model (Nf = 2) K. Yazaki and M.Asakawa., NPA 1989 • Lattice QCD • Reweighting • Z. Fodor and S. D. Katz • (JHEP 0203 (2002) 014) • Imaginary chemical potential • Forcrand and Philipsen hep-lat/0307020

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

4. T QGP Hadronic h r m EOS with CEP • 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 3d Ising Model QCD Same Universality Class

5. h : external magnetic field QCD Mapping T h r m EOS of 3-d Ising Model • Parametric Representation of EOS Guida and Zinn-Justin NPB486(97)626

6. T QGP Hadronic h r m 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

7. Equation of State CEP Baryon number density Entropy Density

8. Focusing and CEP

9. Comparison with Bag + Excluded Volume EOS • n /s trajectories in T- m plane B Bag Model + Excluded Volume Approximation (No End Point) With End Point = Usual Hydro Calculation Not Focused Focused

10. Sound velocity along n /s B /L /L TOTAL TOTAL • Clear difference between n /s=0.01 and 0.03 B Sound Velocity • Effect on Time Evolution • Collective flow EOS

11. r h faster (shorter) expansion Focusing Slowing out of Equilibrium • Berdnikov and Rajagopal’s Schematic Argument B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017 slower (longer) expansion Correlation length longer than xeq xeq x along r = const. line • Effect of Focusing on x? E h Time evolution : Bjorken’s solution along nB/s t0 = 1 fm, T0 = 200 MeV

12. x • Max. depends on n /s. eq B h • Trajectories pass through the region where • x is large. (focusing) r eq Correlation Length (I) • x eq Widom’s scaling low

13. t • x is larger than x at Tf. • Differences among xs on n /s are small. • In 3-d, the difference between x and x becomes • large due to transverse expansion. eq B eq Correlation Length (II) • x : time evolution (1-d) Model C (Halperin RMP49(77)435)

14. Consequences in Experiment (I) CERES:Nucl.Phys.A727(2003)97 • Fluctuations • CERES • 40,80,158 AGeV Pb+Au • collisions Mean PT Fluctuation No unusually large fluctuation CEP exists in near RHIC energy region ?

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

16. Its Consequences • Slowing out of equilibrium • Large fluctuation • Freeze out temperature at RHIC • Fluctuation Focusing CEP and Its Consequences Future task • Realistic hydro calculation with CEP