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Scaling Characteristics ofAzimuthal Anisotropy at RHIC

Michael IssahSUNY Stony Brook

for the PHENIX Collaboration

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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

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Elliptic Flow

y

x

py

px

coordinate-space-anisotropy momentum-space-anisotropy

Initial/final conditions, dof, EOS

Elliptic flow strength determined principally by EOS and initial eccentricity

v2 px

2 py2

px2 py

2

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• High energy densities are achieved, higher than required for phase transition to occur (~ 1 GeV/fm3)

Energy density

PRL87, 052301 (2001)

Central collisionsperipheral collisions

thermalization time (0 ~ 0.2 – 1 fm/c)

Bj~ 5 – 15 GeV/fm3

dy

dE

RT

Bj0

2

11

Extrapolation From EExtrapolation From ETT

DistributionsDistributions

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Hydrodynamic description of v2

Elliptic flow well described by hydrodynamic models up to pT ~ 1.5 GeV/c

Perfect fluid

Hydro

by H

uovin

en e

t al.

hydro

tuned t

o fi

t ce

ntr

al

spect

ra d

ata

.

PRC 72 (05) 014904

200 GeV Au+Aumin-bias

F. Wang, QM2005

PRL 91, 2003 (PHENIX)

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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

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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 ?

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Is thermalization achieved ?

Large v2 indicative of high degree of thermalization of produced matter

Are there other observables showing that the matter is thermalized ? Eccentricity scaled v2

Ideal hydrodynamics is scale invariant. If the matter behaves hydrodynamically and is thermalized, v2 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

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Determination of eccentricity

Eccentricity usually obtained from a Glauber Model

One can also use experimental quantity sensitive to initial eccentricity, like the integrated v2

“Integrated v2 reflects momentum anisotropy of bulk matter and saturates within the first 3-4 fm/c just after collision” (Gyulassy,Hirano nucl-th/050604)

Integrated v2 is proportional to the eccentricity

2 2

2 2

y x

y x

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Eccentricity scaling

Eccentricity scaling observed in hydrodynamic model over a broad range of centralities

Bhalerao, Blaizot, Borghini, Ollitrault , nucl-th/0508009

R: measure ofsize of system

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Eccentricity scaling and system size

v2 scales with eccentricityand across system size

PHENIX Preliminary

PHENIX Preliminary

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Can we make an estimate of cs ?

Energy dependence at RHIC energies seem to indicate a soft equation of state. How soft ?

We can make an estimate of cs from elliptic flow measurements

Bhalerao, Blaizot, Borghini, Ollitrault , nucl-th/0508009

Definition of v2 in model typically 2 times larger than with usual definition

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Estimation of cs

Equation of state: relation between pressure and

energy density

cs ~ 0.35 ± 0.5(cs

2 ~ 0.12), soft EOSF. Karsch, hep-lat/0601013

v2/ecc for <pT> ~ 0.5 GeV/c

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Energy dependence of elliptic flow

Saturation of azimuthal anisotropy observed at RHIC energies

Kolb, Heinz, nucl-th/0305084

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Transverse kinetic energyof 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 = KEcoll + KEth

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0 1 2 3 4 5

v 2

0.00

0.05

0.10

0.15

0.20

0.25

0.30 s 200 GeVNNAu Au

0SK

p

fsTy

5 < Centrality < 30 %

K

(STAR)

(PHENIX)

(STAR)

(PHENIX)

(PHENIX)

22 0 3 01 2

20 1 1

~ 1 ..T

T k Tk kv y m

T k m k m

22 0 3 01 2

20 1 1

~ 1 ..T

T k Tk kv y m

T k m k m

2fsT m Ty k y m 2fsT m Ty k y m

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0

( )

( )

I wv

I w

Buda-Lund Modelnucl-th/0310040 R.Lacey, QM2005

•Equivalent to a transverse kinetic energy•Non-relativistic expression

Approximate scaling variable

Relativistic effects are importantUse relativistic formula

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Scaling v2 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

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PHENIX preliminary data

Transverse kinetic energy scaling works for a large selection of particles

Transverse kinetic energy scaling

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Usual test for quark degrees of freedom

STAR preliminary200 GeV Au+Au

Constituent quark scaling works above pT/n ~ 1 GeV/c

M. Oldenburg, QM2005

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Quark mass matters !

Scaling works

Scaling holds over the whole range of KET

PHENIX preliminary data

Test for partonic degrees of freedom

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Universal scaling across centralities

Scaling observed across centrality and particle species

PHENIX preliminary data

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Scaling works for other particles too!

Universal scaling : do phi mesons and d scale ?

PHENIX preliminary data

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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 KET

(<=1GeV) and separately at higher KET , showing the relevance of the quark degrees of freedom

Scaling with KET/n leads to universal scaling of elliptic flow over a broad range of centralties and particle species