1. WWND, San Diego1 Scaling Characteristics of Azimuthal Anisotropy at RHIC Michael Issah SUNY Stony Brook for the PHENIX Collaboration

2. WWND, San Diego2 Outline  Introduction  What can we learn from scaling characteristics of azimuthal anisotropy  Eccentricity scaling and thermalization  Speed of sound estimation  Scaling with transverse kinetic energy and implications  Summary

3. WWND, San Diego3 Elliptic Flow y x pypy pxpx coordinate-space-anisotropy  momentum-space-anisotropy Initial/final conditions, dof, EOS Elliptic flow strength determined principally by EOS and initial eccentricity

4. WWND, San Diego4 High energy densities are achieved, higher than required for phase transition to occur (~ 1 GeV/fm 3 ) Energy density PRL87, 052301 (2001) Central collisions peripheral collisions thermalization time (  0 ~ 0.2 – 1 fm/c)  Bj  ~ 5 – 15 GeV/fm 3 Extrapolation From E T Distributions

5. WWND, San Diego5 Hydrodynamic description of v 2 Elliptic flow well described by hydrodynamic models up to p T ~ 1.5 GeV/c Perfect fluid Hydro by Huovinen et al. hydro tuned to fit central spectra data. PRC 72 (05) 014904 200 GeV Au+Au min-bias F. Wang, QM2005 PRL 91, 2003 (PHENIX)

6. WWND, San Diego6 Important issues  Some important issues have been raised about:  The range of validity of perfect fluid hydrodynamics  The importance of viscosity effects and where they become important  Estimates of properties of the fluid : speed of sound, latent heat  Whether we can gain access to quark degrees of freedom

7. WWND, San Diego7 Exploring scaling properties  Scaling properties in science relate macroscopic observables to underlying system properties  In heavy-ion collisions, they can serve to find simple laws relating measured anisotropy to system properties and/or degrees of freedom  Eccentricity scaling  System size scaling  Mass scaling and constituent quark scaling  What can be learnt from these scaling properties ?

8. WWND, San Diego8 Is thermalization achieved ?  Large v 2 indicative of high degree of thermalization of produced matter  Are there other observables showing that the matter is thermalized ? Eccentricity scaled v 2  Ideal hydrodynamics is scale invariant. If the matter behaves hydrodynamically and is thermalized, v 2 should be independent of system size  Do we observe such independence in the data?  Data for different colliding systems (Au+Au, Cu+Cu) available to test this

9. WWND, San Diego9 Determination of eccentricity  Eccentricity usually obtained from a Glauber Model  One can also use experimental quantity sensitive to initial eccentricity, like the integrated v 2  “Integrated v 2 reflects momentum anisotropy of bulk matter and saturates within the first 3-4 fm/c just after collision” (Gyulassy,Hirano nucl- th/050604)  Integrated v 2 is proportional to the eccentricity

10. WWND, San Diego10 Eccentricity scaling Eccentricity scaling observed in hydrodynamic model over a broad range of centralities Bhalerao, Blaizot, Borghini, Ollitrault, nucl-th/0508009 R: measure of size of system

11. WWND, San Diego11 Eccentricity scaling and system size v 2 scales with eccentricity and across system size PHENIX Preliminary

12. WWND, San Diego12 Can we make an estimate of c s ?  Energy dependence at RHIC energies seem to indicate a soft equation of state. How soft ?  We can make an estimate of c s from elliptic flow measurements Bhalerao, Blaizot, Borghini, Ollitrault, nucl-th/0508009 Definition of v 2 in model typically 2 times larger than with usual definition

13. WWND, San Diego13 Estimation of c s Equation of state: relation between pressure and energy density c s ~ 0.35 ± 0.5 (c s 2 ~ 0.12), so ft EOS F. Karsch, hep-lat/0601013 v 2 /ecc for ~ 0.5 GeV/c

14. WWND, San Diego14 Energy dependence of elliptic flow Saturation of azimuthal anisotropy observed at RHIC energies Kolb, Heinz, nucl-th/0305084

15. WWND, San Diego15 Transverse kinetic energy of a particle in a relativistic fluid PID scaling Velocity of a particle in a non-relativistic perfect fluid Ollitrault, NPA638 Pressure is a measure of average kinetic energy: Elliptic flow, being driven by pressure gradients, should be sensitive to the collective transverse kinetic energy Average kinetic energy of a particle: KE = KE coll + KE th

16. WWND, San Diego16 Buda-Lund Model nucl-th/0310040 R.Lacey, QM2005 Equivalent to a transverse kinetic energy Non-relativistic expression Approximate scaling variable Relativistic effects are important Use relativistic formula

17. WWND, San Diego17 Scaling v 2 with transverse kinetic energy Scaling holds up to 1 GeV Scaling breaks Mesons scale together Baryons scale together Possible hint of quark degrees of freedom PHENIX preliminary data

18. WWND, San Diego18 PHENIX preliminary data Transverse kinetic energy scaling works for a large selection of particles Transverse kinetic energy scaling

19. WWND, San Diego19 Usual test for quark degrees of freedom STAR preliminary 200 GeV Au+Au Constituent quark scaling works above p T /n ~ 1 GeV/c M. Oldenburg, QM2005

20. WWND, San Diego20 Quark mass matters ! Scaling works Scaling holds over the whole range of KE T PHENIX preliminary data Test for partonic degrees of freedom

21. WWND, San Diego21 Universal scaling across centralities Scaling observed across centrality and particle species PHENIX preliminary data

22. WWND, San Diego22 Scaling works for other particles too ! Universal scaling : do phi mesons and d scale ? PHENIX preliminary data

23. WWND, San Diego23 Summary  Eccentricity scaling holds over a broad range of centralities and is indicative of thermalization of matter produced at RHIC  Hydrodynamic model comparison leads to an estimate of the speed of sound. Data compatible with soft EOS  Transverse kinetic energy is an appropriate variable to scale elliptic flow; related to pressure gradients  Baryons and mesons scale together at low KE T (<=1GeV) and separately at higher KE T, showing the relevance of the quark degrees of freedom  Scaling with KE T /n leads to universal scaling of elliptic flow over a broad range of centralties and particle species