What do the scaling characteristics of elliptic flow reveal about the properties of the matter at RHIC ? Michael Issah Stony Brook University for the PHENIX Collaboration CIPANP 2006, Puerto Rico

2 Outline  Motivation  Eccentricity scaling and thermalization  Equation of state and speed of sound estimation  Dependence of elliptic flow on kinetic energy and degrees of freedom  Summary

CIPANP 2006, Puerto Rico3 Elliptic Flow y x pypy pxpx coordinate-space-anisotropy  momentum-space-anisotropy Initial/final conditions, dof, EOS Elliptic flow strength determined principally by initial conditions, initial eccentricity and EoS

CIPANP 2006, Puerto Rico4 High energy densities are achieved, higher than required for phase transition to occur (~ 1 GeV/fm 3 ) Are energy densities high enough? PRL87, (2001) Central collisions peripheral collisions thermalization time (  0 ~ 0.2 – 1 fm/c)  Bj  ~ 5 – 15 GeV/fm 3 Extrapolation From E T Distributions

CIPANP 2006, Puerto Rico5 Elliptic flow at RHIC Elliptic flow at RHIC  Large v 2 created at RHIC  Indicates early thermalization  Elliptic flow data agrees well with hydrodynamic predictions  Can one exploit scaling properties of v 2 to get insight into the degree of thermalization and test for hydrodynamic behavior ? Hydro by Huovinen et al. hydro tuned to fit central spectra data. PRC 72 (05) GeV Au+Au min-bias PRL 91, 2003 (PHENIX)

CIPANP 2006, Puerto Rico6 Eccentricity scaling as a test for the degree of thermalization  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

CIPANP 2006, Puerto Rico7 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 is proportional to the eccentricity STAR Collaboration

CIPANP 2006, Puerto Rico8 Eccentricity scaling Eccentricity scaling observed in hydrodynamic model over a broad range of centralities Bhalerao, Blaizot, Borghini, Ollitrault, nucl-th/ R: measure of size of system

CIPANP 2006, Puerto Rico9 Eccentricity scaling and system size v 2 scales with eccentricity and across system size PHENIX Preliminary k ~ 3.1 (obtained from data )

CIPANP 2006, Puerto Rico10 Energy dependence of elliptic flow  Saturation of azimuthal anisotropy observed at RHIC energies  Suggests the matter has a soft EoS

CIPANP 2006, Puerto Rico11 Estimate of the speed of sound (I)  Energy dependence at RHIC energies seem to indicate a soft equation of state. How soft ?  Estimate of c s from elliptic flow measurements can be made from eccentricity scaled v 2 Bhalerao, Blaizot, Borghini, Ollitrault, nucl-th/

CIPANP 2006, Puerto Rico12 Estimation of the speed of sound (II) Assume simple EoS: relation between pressure and energy density c s ~ 0.35 ± 0.05 (c s 2 ~ 0.12), so ft EOS F. Karsch, hep-lat/ v 2 /ε for ~ 0.45 GeV/c

CIPANP 2006, Puerto Rico13 Scaling breaks  Elliptic flow scales with KE T up to KE T ~1 GeV  Indicates hydrodynamic behavior  Possible hint of quark degrees of freedom become apparent at higher KE T Baryons scale together Mesons scale together PHENIX preliminary See H. Masui’s talk in this session Kinetic energy of a particle in a relativistic fluid: = m T – m Scaling v 2 with kinetic energy (I)

CIPANP 2006, Puerto Rico14 PHENIX preliminary Transverse kinetic energy scaling works for a large selection of particles Scaling v 2 with kinetic energy (II)

CIPANP 2006, Puerto Rico15  Constituent quark scaling works better with KE T as compared to p T  Hadron mass (hydro) scaling at low KE T (KE T < 1 GeV) is not perturbed by constituent quark scaling at higher KE T ! Test for partonic degrees of freedom PHENIX preliminary STAR preliminary 200 GeV Au+Au

CIPANP 2006, Puerto Rico16 KE T /n scaling and hydrodynamics  Scaling works for a broad set of particles  Supports the observation that all particles emerge from a common flow field  Hydrodynamic calculation not incompatible with KET/n scaling PHENIX preliminary Chiho Nonaka: 3-D Hydro R. Fries, SQM2006

CIPANP 2006, Puerto Rico17 KE T /n scaling across collision centralities KE T /n scaling observed across centralities

CIPANP 2006, Puerto Rico18 Summary  Scaling properties of elliptic flow provide a way to infer some properties of the matter formed in heavy- ion collisions at RHIC  Eccentricity scaling shows that the medium is highly thermalized, with a soft EoS  Transverse kinetic energy scaling indicates perfect fluid hydrodynamic behavior  In addition, scaling with KE T /n leads to comprehensive scaling of elliptic flow data, which prevails over a broad range of centralities  Taken together, these observations show that the matter is highly thermalized, has a soft equation of state and is made up of flowing constituent quarks