1. Hydrodynamic Approaches to Relativistic Heavy Ion Collisions Tetsufumi Hirano RIKEN BNL Research Center

2. Contents Introduction: dynamics of heavy ion collisions Hydrodynamic Models –Equation of State –Initial Condition –Freezeout Success and Failure of Hydrodynamic approaches at RHIC –Elliptic Flow –HBT puzzle Summary

3. Introduction 1: Space-Time Evolution of Heavy Ion Collision z (collision axis) t QGP phase Cross over? 0 hadrons photons leptons Hadron phase jets z x Reaction plane Time scale ~10 fm/c

4. Introduction 2: Static to Dynamic Lattice QCD simulations F.Karsch et al. (’00) STATIC QCD matter Matter produce in heavy ion collisions is DYNAMIC. Full 3D simulation by T.H. and Y.Nara (’04) Powerful and reliable 1 st principle calculations Currently, small size and no time evolution Space-time evolution Expansion Cool down Phase transition … One possible description is HYDRODYNAMICS.

5. Basics of Hydrodynamics Hydrodynamic Equations Energy-momentum conservation Charge conservations (baryon, strangeness, etc…) For perfect fluids (neglecting viscosity), Energy densityPressure4-velocity Within ideal hydrodynamics, pressure gradient dP/dx is the driving force of collective flow.  Collective flow is believed to reflect information about EoS!  Phenomenon which connects 1 st principle with experiment Need equation of state (EoS) P(e,n B ) to close the system of eqs.  Hydro can be connected directly with lattice QCD Caveat: Thermalization, << ( typical system size)

6. Inputs for Hydrodynamic Simulations Final stage: Free streaming particles  Need decoupling prescription Intermediate stage: Hydrodynamics can be valid if thermalization is achieved.  Need EoS Initial stage: Particle production and pre-thermalization beyond hydrodynamics  Instead, initial conditions for hydro simulations t z Need modeling (1) EoS, (2) Initial cond., and (3) Decoupling

7. Main Ingredient: Equation of State Latent heat One can test many kinds of EoS in hydrodynamics. Lattice QCD predicts cross over phase transition. Nevertheless, energy density explosively increases in the vicinity of T c.  Looks like 1 st order. Lattice QCD simulations Typical EoS in hydro model H: resonance gas(RG) p=e/3 Q: QGP+RG F.Karsch et al. (’00) From P.Kolb and U.Heinz(’03)

8. Interface 1: Initial Condition Need initial conditions (energy density, flow velocity,…) Parametrize initial hydrodynamic field Take initial distribution from other calculations Initial time  0 ~ thermalization time ex.) In transverse plane, energy density or entropy density prop. to # of participants, # of binary collisions, or etc. Energy density from NeXus. (Left) Average over 30 events (Right) Event-by-event basis (Talk by Hama) T.H.(’02) x xx yy

9. Interface 2: Freezeout Need translation from thermodynamic variables to particle spectra to be observed. Sudden freezeout (Cooper-Frye formula) Continuous particle emission (Talk by Hama) Hadronic afterburner via Boltzmann eq. Hadronic Cascade (RQMD, UrQMD) QGP Fluid Teaney, Lauret, Shuryak Bass, Dumitru … QGP Fluid Hadron Fluid =0 =infinity T f.o. Escaping probability P f free (x,p)=Pf(x,p) 

10. Hydrodynamic Models @ RHIC Initial conditions Parametrization Taken from other model With/without fluctuation EoS Lattice inspired model With/without phase transition With/without chemical freeze out Decoupling Sudden freezeout Continuous emission Hadronic cascade There are many options: In addition, Dimension Boost inv. (Bjorken, ’83) 1D(r) + boost inv. + cylindrical sym. 2D(x,y) + boost inv. Full 3D Cartesian (t,x,y,z)  coordinate Each option reflects what one wants to study.

11. Success of Hydrodynamics --Elliptic Flow-- How the system respond to initial spatial anisotropy? Ollitrault (’92) Hydrodynamic expansion Initial spatial anisotropy Final momentum anisotropy INPUT OUTPUT Rescattering dN/d   Free streaming 0 22 dN/d   0 22 2v22v2 x y  Talk by Voloshin

12. Boltzmann to Hydro !? Molnar and Huovinen (’04) elastic cross section 47mb ~ inelastic cross section of pp at RHIC energy!? Still ~30% smaller than hydro result! Hydro ( ~0) is expected to gain maximum v2 among transport theories.  “hydrodynamic (maximum) limit”

13. Hydrodynamic Results of v 2 /  Hydrodynamic response is const. v 2 /  ~ 0.2 @ RHIC Exp. data reach hydrodynamic limit at RHIC for the first time. Exp. line is expected to bend at higher collision energy. (response)=(output)/(input) Number density per unit transverse area Dimension 2D+boost inv. Initial condition Parametrization EoS QGP + RG (chem. eq.) Decoupling Sudden freezeout STAR(’02) LHC? Kolb, Sollfrank, Heinz (’00)

14. Hydrodynamic Results of v 2 (p T,m) Dimension 2D+boost inv. Initial condition Parametrization EoS QGP + RG (chem. eq.) Decoupling Sudden freezeout PHENIX(’03) Correct p T dependence up to p T =1-1.5 GeV/c Mass ordering Deviation in intermediate ~ high p T regions  Other physics Jet quenching (Talk by Vitev) Recombination (Talk by Hwa) Not compatible with particle ratio  Need chem. freezeout mechanism Huovinen et al.(’01)

15. Hydrodynamic Results of v 2 (  ) Dimension Full 3D (  coordinate) Initial condition Parametrization EoS 1.QGP + RG (chem. eq.) 2.QGP + RG (chem. frozen) Decoupling Sudden freezeout Hydrodynamics works only at midrapidity? Forward rapidity at RHIC ~ Midrapidity at SPS? Heinz and Kolb (’04) T.H. and K.Tsuda(’02)

16. Hydrodynamic Results of v 2 (again) Dimension 2D+boost inv. Initial condition Parametrization EoS Parametrized by latent heat (LH8, LH16, LH-infinity) RG QGP+RG (chem. eq.) Decoupling Hadronic cascade (RQMD) Teaney, Lauret, Shuryak(’01) Large gap (~50% reduction) at SPS comes from finite or “viscosity”. Latent heat ~0.8 GeV/fm 3 is favored. Hadronic afterburner explains forward rapidity? (T.H. and Y.Nara, in progress)

17. Summary for Success of Hydrodynamics Description of elliptic flow parameter v 2 v 2 (p T,m) Up to 1-1.5 GeV/c v 2 (  ) Near midrapidity Multiplicity dependence Need cascade/viscosity for hadrons Phase transition with latent heat ~ 0.8 GeV/fm 3 is favored Future study: Forward rapidity by hydro+hadronic cascade Viscosity in QGP A lot of work should be done…

18. Failure of Hydrodynamics --HBT puzzle-- Talks by Magestro, Csorgo and Hama p1p1 p2p2 reaction plane z R long KTKT R out R side x y Two particle corr. fn. Bird’s eye viewView from beam axis q 1 2 C2C2 q 1/R

19. Source Function and Flow Long wave length Short wave length Source fn. from hydro x-y x-t Midrapidity & cylindrical symmetry From P.Kolb and U.Heinz(’03) K T : “Wave length” to extract radii Source fn.

20. Sensitivity to Chemical Composition Dimension Full 3D (  coordinate) Initial condition Parametrization EoS 1.QGP + RG (chem. eq.) 2.QGP + RG (chem. frozen) Decoupling Sudden freezeout T.H. and K.Tsuda (’02) SOLID LINE DASHED LINE R side R out R long R out / R side Note that exp. data of R out /R side slightly increase by considering core-halo picture R out / R side (hydro) > R out / R side (data)~1  HBT puzzle!!! HBT radii reflects last interaction points.  Problem of sudden freezeout?

21. Sensitivity to Freezeout (contd.) HBT radii from continuous particle emission model  Talk by Hama Dimension 1D+boost inv. + cylindrical sym. Initial condition Parametrization EoS QGP + RG (chem. eq.) Decoupling Hadronic afterburner by UrQMD Better in low p T region for T c =160 MeV case by smearing through cascade.  Still something is missing to interpret the data. (Absolute value?) STAR PHENIX Taken from D. Magestro, talk @ QM04 Hydro 200 Hydro 160 Hydro+cascade 200 Hydro+cascade 160 Soff, Bass, Dumitru (’01)

22. x-t Correlation of Source Function Why hydro doesn’t work? positive! x t Negative x-t correlation Positive? Negative? Typical source fn. from hydro x t Positive x-t correlation Hubble like flow? Csorgo et al. R out ~R side may require positive x-t corr.

23. Summary and Outlook From elliptic flow point of view, a hydro + cascade (RQMD) model with latent heat 0.8 GeV/fm 3 gives a good description at both SPS and RHIC (in low p T and near midrapidity).  Need full 3D hydro + hadronic cascade (a possible model to describe all rapidity region at RHIC) However, a similar model (hydro + UrQMD) fails to reproduce HBT radii.  Need a thorough search for initial conditions  Need more sophisticated description of the late stage (HBT is a quantum effects!)