description of phobos data on v 2 ( h ) from s nn = 19.6 to 200 gev

1Quark Matter 2005, Aug. 4-9, Budapest A. Ster Hungary

Description of PHOBOS Description of PHOBOS data on vdata on v22(() from ) from ssNN NN = =

19.6 to 200 GeV19.6 to 200 GeVA. Ster1,2, M. Csanád3, T. Csörgő2

11 KFKI-RMKI, KFKI-RMKI, 22 KFKI-MFA, KFKI-MFA, 33 ELTE, Hungary ELTE, Hungary

• Buda-Lund hydrodynamical modelBuda-Lund hydrodynamical model• Buda-Lund ellipsoidal generalizationBuda-Lund ellipsoidal generalization• Buda-Lund fitting to PHOBOS recent data Buda-Lund fitting to PHOBOS recent data • ConclusionConclusion


Buda-Lund hydro modelBuda-Lund hydro model• 3D expansion with axial or ellipsoidal 3D expansion with axial or ellipsoidal

symmetrysymmetry

• Local thermal equilibriumLocal thermal equilibrium

• Analytic expressions for the observables Analytic expressions for the observables (no numerical simulations, but formulas)(no numerical simulations, but formulas)

• Reproduces known exact hydro solutions Reproduces known exact hydro solutions (nonrelativistic, Hubble, Bjorken limit)(nonrelativistic, Hubble, Bjorken limit)

• Core-halo picture (long lived Core-halo picture (long lived resonances)resonances)


An illustrative analogyAn illustrative analogy

• CoreCore SunSun• HaloHalo Solar windSolar wind• TT0,RHIC0,RHIC TT0,SUN 0,SUN 16 million K 16 million K • TTsurface,RHICsurface,RHIC TTsurface,SUN surface,SUN 6000 K 6000 K• RRGG Geometrical sizeGeometrical size• 00 Radiation lifetime Radiation lifetime • <<tt>> Radial flow of surface (~0)Radial flow of surface (~0)• Longitudinal expansion Longitudinal expansion

((~~0)0)

Fireball at RHICFireball at RHIC Fireball SunFireball Sun


Buda-Lund: spectra, HBT w/o Buda-Lund: spectra, HBT w/o puzzlespuzzles

J.Phys.G30: S1079-S1082, 2004; nucl-th/0403074


Ellipsoidal generalizationEllipsoidal generalization

• Axially symmetric case: RAxially symmetric case: RGG, u, utt

• Main axes of expanding ellipsoid: Main axes of expanding ellipsoid: X, Y, ZX, Y, Z• 3D expansion, 3 expansion rates: 3D expansion, 3 expansion rates: X, Y, ZX, Y, Z• Introducing velocity space eccentricityIntroducing velocity space eccentricity

• For Hubble type of expansions: For Hubble type of expansions: X(X())XX

• In this case: In this case: vv

• In addition: In addition: ZZ

22

22

YX

YXv


The ellipsoidal Buda-Lund The ellipsoidal Buda-Lund modelmodel

• The original model was developed for axial symmetry The original model was developed for axial symmetry central collisionscentral collisions

• In the most general hydrodynamical form (In the most general hydrodynamical form (‘Inspired by’ ‘Inspired by’ nonrelativistic solutions):nonrelativistic solutions):

• Assume special shapesAssume special shapes::• Generalized Cooper-Frye prefactor:Generalized Cooper-Frye prefactor:

• Four-velocity distribution:Four-velocity distribution:

• Temperature:Temperature:

• Fugacity:Fugacity:

M.Csanád, T.Csörgő, B. Lörstad: Nucl.Phys.A742:80-94,2004; nucl-th/0310040


Observables from Buda-Lund Observables from Buda-Lund hydrohydro

• Core-halo correction:Core-halo correction:

• One-particle spectrum with core-halo One-particle spectrum with core-halo correction:correction:

• Two-particle correlation functionTwo-particle correlation function::

• Flow coefficients:Flow coefficients:


The elliptic flowThe elliptic flow• One-particle spectrum:One-particle spectrum:

• The m-th Fourier component is the m-th flowThe m-th Fourier component is the m-th flow

• Depends on pseudorapidity and transverse Depends on pseudorapidity and transverse momentum momentum

• Pseudorapidity dependence not understood Pseudorapidity dependence not understood in other hydro models (but ref: plenary talk in other hydro models (but ref: plenary talk of Hama/SPHERIO)of Hama/SPHERIO)


Hydro predicts universal Hydro predicts universal scalingscaling

• Universal scaling variableUniversal scaling variable

• Universal scalig function:Universal scalig function:

• Elliptic flow depends on every physical Elliptic flow depends on every physical parameter (mass, colliding system, parameter (mass, colliding system, bombarding energy, pt, rapidity,centrality) bombarding energy, pt, rapidity,centrality) only through universal scaling variable only through universal scaling variable ww

• Do data collapse to the scaling curve of Do data collapse to the scaling curve of II1 1 / / II0 0 ??

x,y,

2

**

114 TTm

pw

t

t

)()(

0

12 wI

wIv


At large pseudorapiditiesAt large pseudorapidities• Under certain conditions, the even flows are:Under certain conditions, the even flows are:

, , where where

andand• Here Here ss is the space-time rapidity of the is the space-time rapidity of the

saddlepoint saddlepoint

• , and so, and so

• Rapidity grows Rapidity grows the asymmetry vanishes the asymmetry vanishes ((saddlepoint goes to the z axissaddlepoint goes to the z axis) ) elliptic flow elliptic flow vanishesvanishes

)()(

02 wI

wIv nn

)cosh( ymm stt

0 wys 0)0()0(

)(0

2 II

wv n

x,y,

2

**

114 TTmp

wt

t


Fits to PHOBOS dataFits to PHOBOS data

PHOBOS data from: Phys. Rev. Lett. 94, 122303 (2005)


Fit parametersFit parameters• Used (non-essential) model Used (non-essential) model

parameters:parameters:

• Fitted parameters:Fitted parameters:

TT00 RRss u’u’tt RRgg

175 175 MeVMeV 12.38 fm12.38 fm 1.51.5 13.5 13.5

fmfm

vv

200 200 GeVGeV

0.3940.394 0.0060.006 2.562.56 0.040.04

130 130 GeVGeV

0.3760.376 0.0050.005 2.462.46 0.040.04

62.4 62.4 GeVGeV

0.3490.349 0.0080.008 2.162.16 0.050.05

19.6 19.6 GeVGeV

0.2940.294 0.0290.029 1.701.70 0.250.25

Increasing parameter values with energy


Error contoursError contours


Universal scalingUniversal scaling• Data collapsing behavior to Data collapsing behavior to

theoretically predicted scaling theoretically predicted scaling functionfunction

The perfect fluid The perfect fluid extends from veryextends from verysmall to very largesmall to very largerapidities at RHICrapidities at RHIC

)()(

0

12 wI

wIv


ConclusionsConclusions

• Buda-Lund model describes vBuda-Lund model describes v22(() data @RHIC) data @RHIC

• The vanishing elliptic flow at large rapidities The vanishing elliptic flow at large rapidities is due to Hubble flow + finite longitudinal is due to Hubble flow + finite longitudinal sizesize

•vv22(()) data (2005) collapse to the data (2005) collapse to the

theoretically PREDICTED (2003) scaling theoretically PREDICTED (2003) scaling function of function of

• The perfect fluid is present in AuAu in the The perfect fluid is present in AuAu in the whole whole space space

)()(

0

12 wI

wIv


Sensitivity to the Equation of Sensitivity to the Equation of StateState

• Different initial conditions, different equation of stateDifferent initial conditions, different equation of state but exactly the same hadronic final state possible. (!!)but exactly the same hadronic final state possible. (!!)

• This is an exact, analytic result in hydro( !!).This is an exact, analytic result in hydro( !!).

cs2 = 2/3 c

s2 = 1/3


Time dependence Time dependence

• Blastwave or Cracow model type of cooling vs Blastwave or Cracow model type of cooling vs Buda-Lund typeBuda-Lund type of cooling, cs2= 2/3, half freeze-of cooling, cs2= 2/3, half freeze-out time (animated) out time (animated) http://csanad.web.elte.hu/phys/3danim/http://csanad.web.elte.hu/phys/3danim/

description of phobos data on v 2 ( h ) from s nn = 19.6 to 200 gev

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