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Scaling of Elliptic Flow for a fluid at Finite Shear Viscosity. V. Greco M. Colonna M. Di Toro G. Ferini. University of Catania INFN-LNS. From the Coulomb Barrier to the Quark-Gluon Plasma, Erice (Sicily) 22 Sept. 2008. Momentum anisotropy measure of /s. Reminder of v 2 at RHIC - PowerPoint PPT Presentation
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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