Scaling of Elliptic Flow for a fluid at Finite Shear Viscosity

V. Greco V. Greco M. Colonna M. Colonna M. Di Toro M. Di Toro G. FeriniG. Ferini

From the Coulomb Barrier to the Quark-Gluon Plasma, Erice (Sicily) 22 Sept. 2008From the Coulomb Barrier to the Quark-Gluon Plasma, Erice (Sicily) 22 Sept. 2008

University of CataniaINFN-LNS

Reminder of v2 at RHIC

Evidences for non-ideal hydrodynamics

Transport approach with only 22 scatterings

time dependent cross section renormalized to fix /s/s

Relation between vv22(p(pTT)/<v)/<v22>> and vv22(p(pTT)/)/

scaling ?

effects of freeze-out

Relation of /s/s with coalescence (QNS)

v2(pT) not only <v2> hints at /s < 3/4/s < 3/4

+ coalescence at intermediate pT

Ou

tlin

eMomentum anisotropy measure of Momentum anisotropy measure of /s/s

Ferini et al., 0805. 4814 [nucl-th]

A measure of the Interaction: Elliptic FlowA measure of the Interaction: Elliptic Flow

x

yz

px

py

v2 is the 2nd harmonic Fourier coeff.of the distribution of particles.

nn

TT

ndp

dN

ddp

dN)cos(v21

Perform a Fourier expansion of the momentum space particle distributions

22

22

2 2cosyx

yx

pp

ppv

22

22

xy

xyx

Free streaming

v2=0

The analysis can be extended !

Good probe of early pressure

c2s=dP/d

CASCADE =10 mb

Similar trend in hydro

0)(

0)(

xj

xT

B

HydrodynamicsHydrodynamicsNo microscopic details

(mean free path -> 0, =0)

+ EoS

Parton cascadeParton cascade

v2 saturation pattern reproduced

Large=10-15mb (coalesc.includ.)

• Good description of hadron spectra and v2(pT)

• Mass ordering of v2

versus pT

D. Molnar & M. Gyulassy, NPA 697 (02)

First stage of RHICFirst stage of RHIC

22 Ifp

Parton elastic 22 interactions

It’s not that perfect …It’s not that perfect …

Is it really zero ideal hydro

(zero shear viscosity) ?

B. I. Abelev et al., (STAR), PRC77 (08)

STAR, J. Phys. G34 (2007)

Not too peripheral

Not too high pT

Not too high harmonics

,...),,,(v 22 sizesytembcK s

Relation betweenRelation betweenxxand vand v22

Ideal HydrodynamicsIdeal Hydrodynamics:

Independent of - impact parameter - system size

Bhalerao et al., PLB627(2005)

2v

time

Effect of finite Effect of finite ss ?! ?!

Data show evidence for deviation from hydro scaling v2/

Hydrodynamics

22

22

xy

xyx

Au+Au@200 GeV

STAR, PRC77(08)

y

v

A

F xx

Small viscosity Large cross sections Strong couplings beyond pQCD

Shear ViscosityShear Viscosity

1)Quantum mechanism s > 1/15 :R. Lacey et al., PRL99(2006)

2) 4 SYM + Gauge theory g ∞:

4

1

.

Mins

Smaller than anyother known fluid!

λ15

1 p

s

13

1

T

tE

Can we constrain Can we constrain /s with /s with vv22??

Kinetic Theory

Study of dissipative effects on <vStudy of dissipative effects on <v22>>

How sensitive is elliptic flow to finite /s?

Z. Xu & C. Greiner, PRL 101(08)

Agreement for s=0.3 – 0.6 /s=0.15 – 0.08

Dependence on freeze-out

Viscous Hydro Cascade (2<->2,2<->3)

P. Romatschke, PRL99 (07)

Dependence on relaxation time

II0 order expansion with green terms (D. Rischke)

Transport approachTransport approach

...22 Ifp

Solved discretizing the space in x, ycells

Collision integral not solved with the geometrical interpretation, but with a local stochastic sampling

Z. Xhu, C. Greiner, PRC71(04)

In the limit

t0 and 3x0

exact solutions of the Boltzmann equation

Convergency of v2 results tested against variable

t and 3x discretization and test particle number.

3x

Cross section for fixed Cross section for fixed /s/s

We simulate a constant shear viscosity during the HIC

cost.)4(15

4

Tn

p

s tr

sTn

p

/

1

415

4

Relativistic Kinetic theory Cascade code

We have used pQCD-likecross section with screening mass

The viscosity is kept constant varying s

=cell index in the r-space

s

m

mtdt

d D

D

s

22

19 2

22

2

2sind

ddtr

Evolution of cross section with Evolution of cross section with TemperatureTemperatureA rough estimate of (T) can be done using ss

p

/

1

15

4

Neglecting and inserting in (*)

4

1

s3

2

45

2T

g

T

Ps

2

1

Ttr

At T=200 MeV tr10 mb

In our code it is evaluated locally (different from D. Molnar arXiV:0806.0026)

(*)

Elliptic flow sensitive to the Shear Viscosity

Au+Au @ 200 AGeV

b=9 fm

b=7 fm

b=5 fm

b=3 fm

Sensitivity increasing at larger pT

Intermediate pT can say more about /s

50% increase

“strong evidence for hydrodynamic scaling of v2 over a broad

selection of the elliptic flow data”

Does vDoes v22/<v/<v22> scaling validate ideal hydro?> scaling validate ideal hydro?

PHENIX PRL 98, 162301 (2007)

Scaling with Centrality and System Size

Such scalings holds alsoSuch scalings holds alsoat finite viscosity?at finite viscosity?

Scaling outside the hydro region

vv22// and v and v22/<v/<v22> as a function of p> as a function of pTT

Scaling for bothboth v2/<v2> and v2/ for bothboth Au+Au and Cu+Cu

Small /s does not break v2/ the scaling

Agreement with PHENIX data for v2/<v2>

Violation of the scaling at higher /s

/s1/4 closer to data, but…

Au+Au & Cu+Cu@200 AGeV

/s=1/2

/s=1/

PHENIX, PRL98 (2007)

Of course it is more Of course it is more complex…complex… STAR, PRC77 (2008)

v2/ does not scale!

Can a cascade approach account for this?

Freeze-out is crucial!

v2/<v2> scales!

Elliptic flow sensitive to freeze outElliptic flow sensitive to freeze out

Effect of freeze-out increasing with b

For <c=0.7 GeV/fm3 collisions are switched off

b=3 fm

/s=1/4Effect of freeze-out

vv22// and vand v22/<v/<v22> with freeze-out> with freeze-out

VV22// broken in a way similar to STAR data

Agreement with PHENIX and STAR scaling of v2/<v2>

The freeze-out lowers the V2(pT) at higher pT …

(about 40% in b=3-9 (about 40% in b=3-9 fm)fm)

No freeze-out No freeze-out

/s=1/4

v2/ scaling broken v2/<v2> scaling kept!

Cascade can get both features:

Short Reminder …Short Reminder …

/3)(p3v)(pv

/2)(p2v)(pv

Tq2,TB2,

Tq2,TM2,

Enhancement of vEnhancement of v22Quark Number ScalingQuark Number Scaling

n

p

nT

2V1

Molnar and Voloshin, PRL91 (03)Fries-Nonaka-Muller-Bass, PRC68(03)

2

22)2()(

T

T

q

T

T

M ppd

dNαp

pd

dN

3

22)3()(

T

T

q

T

T

B ppd

dNp

pd

dN

)2cos(v21φ 2q

TT

q

TT

q

dpp

dN

ddpp

dN

Considering only momentum spacex - p correlation neglected

narrow wave function

v2 for baryon is larger and saturates at higher pT

v2q fitted from v2

GKL, PRC68(03)

vv2q2q(p(pTT) fitted. Is it reasonable) fitted. Is it reasonablethe vthe v22 needed by coalescence? needed by coalescence?

vv22(p(pTT) as a measure of ) as a measure of /s/sv2/<v2> scaling reproduced, what about v2 absolute value?

/s >3/4 too low v2(pT) at pT1.5 GeV/c

for quantitative estimate an EOS with phase transition ( ≠ 3p) needed!

lower the estimate the /s

PHENIX/s 0.1-0.2 + freeze-outOpen the room to need coalescence in the region of QNS

Finite /s shape for v2(pT)

consistent with that needed by coalescence

Freeze-out with Freeze-out with /s from QGP /s from QGP HG HG

Smooth transition of /s from minimal value (1/4) to the value typical of a pion-kaon gas (7/4)

Previous results Previous results with sudden with sudden freeze-out freeze-out confirmedconfirmed

GeV/fm3]

/s

1/4

7/4

1.70.3

HG

QGP

=0.7GeV/fm3

Preliminary

v2/<v2> () scaling holds at finite /s up to 0.15Freeze-out at =0.7 GeV/fm3 or

/s change from 1/4 7/4 (in cross-over

region):

Transport at finite Transport at finite /s pave the way for /s pave the way for

consistency:consistency:

breaking of v2(pT)/ scaling

persistence of v2(pT)/<v2> scaling

v2(pT) need presence of coalescence

at pT> 1.5 GeV with 1/4</s<3/4

SummarySummary

Au+Au @ 200 AGeV

Scaling of time evolution with the system size

Hydrodynamics Cascade

As in hydro in the early evolution v2/ scales with system size

At the end a significant breaking is observed

221 11 yxR

If Elliptic Flow is very large

To balance the minimum a v4 > (10v2-1)/34 is required

v4 > 4.4% if v2=25%

222

4224

4 )(

6)4cos(

yx

yyxx

pp

ppppv

STAR, J. Phys. G34 (2007)

v2 and v4 contain rich information on /s

Study of dissipative effects at RHICTransport approach

Z. Xu et al., 0711.0961 [nucl-th]

With only 22 collisions, RHIC v2 is obtained by using a growing cross section 2/3 which yields /s1/4 on average (D. Molnar, 0806.0026 [nucl-th])

Agreement with data for s=0.3 – 0.6 /s=0.15 – 0.08

GeV/fm3]

/s

1/4

7/4

1.70.3

HG

QGP