Table Of ContentHadronic dissipative effects on elliptic flow in ultrarelativistic heavy-ion collisions
Tetsufumi Hirano,1,∗ Ulrich Heinz,2 Dmitri Kharzeev,3 Roy Lacey,4 and Yasushi Nara5
1Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027, USA
2Department of Physics, Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA
3Nuclear Theory Group, Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA
4Department of Chemistry, SUNY Stony Brook, Stony Brook, NY 11794-3400, USA
5Institut fu¨r Theoretische Physik, J.W.Goethe-Universit¨at, Maxv.LaueStr.1, D-60438 Frankfurt, Germany
(Dated: February 7, 2008)
Westudytheellipticflowcoefficientv2(η,b)inAu+Aucollisionsat√s=200AGeVasafunction
of pseudorapidity η and impact parameter b. Using a hybrid approach which combines early ideal
fluiddynamicalevolution with late hadronicrescattering, we demonstratestrong dissipativeeffects
from the hadronic rescattering stage on the elliptic flow. With Glauber model initial conditions,
hadronic dissipation is shown to be sufficient to fully explain the differences between measured v2
6
valuesandidealhydrodynamicpredictions. InitialconditionsbasedontheColorGlass Condensate
0
modelgeneratelargerellipticflowandseemtorequireadditionaldissipationduringtheearlyquark-
0
gluon plasma stage in order toachieve agreement with experiment.
2
n PACSnumbers: 25.75.-q,25.75.Nq,12.38.Mh,12.38.Qk
a
J
7 OneoftheimportantnewdiscoveriesmadeattheRel- expansionphase. Inthepresentpaperweexplorethisis-
2 ativistic Heavy Ion Collider (RHIC) is the large ellip- suemorequantitatively,bytryingtoanswerthequestion
tic flow v2 in non-central Au+Au collisions [1]. At the how much of the observed deviation of v2 from the ideal
3
highest RHIC energy of √s = 200AGeV, the observed fluid prediction can be attributed to “late viscosity” in
v
v values near midrapidity (η <1), for not too large thedissipativehadronicphase,andwhetherornotsignif-
6 2
4 impact parameters (b<7fm)|a|n∼d transverse momenta icantadditional dissipative effects during the early QGP
0 (p <1.5GeV/c), agree∼with predictions from ideal fluid stagearerequiredforaquantitativeunderstandingofthe
T
1 dyna∼mics [2], including [3, 4] the predicted dependence data.
1
of v on the transverse momentum p and hadron rest
5 2 T Ourstudyisbasedonacomparisonofahybridmodel,
masses [5]. From these observations it has been con-
0 combininganidealfluiddynamicalQGPstagewithare-
/ cluded [6] that in these collisions a quark-gluon plasma
h alistic kinetic description of the hadronic stage (hadron
(QGP)iscreatedwhichthermalizesonaveryrapidtime
t cascade), with data on the centrality and rapidity de-
- scale τ < 1fm/c and subsequently evolves as an al-
l therm pendence of the p -integrated elliptic flow v (η,b) for
c most ideal fluid with exceptionally low viscosity. T 2
u charged hadrons [20]. We find that with Glauber model
n initial conditions [21], suitably generalized to account
On the other hand, the ideal fluid dynamical descrip-
: for the longitudinal structure of the initial fireball [22],
v tion gradually breaks down as one studies collisions at
i larger impact parameters and at lower energies [7] or hadronic dissipationis sufficient to explainthe data. On
X
the other hand, initial conditions based on the Color
moves away from midrapidity [8, 9, 10, 11]. This has
r Glass Condensate (CGC) model [23, 24] lead to larger
a been attributed alternatively to incomplete thermaliza-
elliptic flows which overpredict the data unless one ad-
tionofthe QGPduringthe earlystagesoftheexpansion
ditionally assumes that the early QGP stage possesses
[13] (“early viscosity”) and/or to dissipative effects dur-
significant shear viscosity, too, or that the QGP equa-
ing the late hadronic expansion stage [14, 15, 16] (“late
tion of state is significantly softer than usually assumed.
viscosity”). It has recently been argued [15, 17] that
Our analysis points to a need for a better understand-
quantum mechanics imposes a lower limit on the shear
ing of the initial conditions in heavy-ion collisions if one
viscosity of any medium, but that the shear viscosity
hopes to use experimental data to constrain the QGP
of the QGP can not exceed this lower limit by a large
viscosity and equation of state.
factor [18]. On the other hand, qualitative arguments
were presented in Ref. [16] which emphasize the impor- A (1+1)-dimensional hydro+cascade model was first
tance of hadronic dissipation and support a picture of a proposed in Ref. [25], putting emphasis on radial flow
“nearlyperfect fluid stronglycoupledQGP (sQGP)core in heavy-ion collisions. It was later extended to 2+1
andhighlydissipativehadroniccorona”inultrarelativis- dimensions for the study of elliptic flow near midrapid-
ticheavy-ioncollisions. Theimportanceofviscouseffects ity[14,26]. Bycombiningahydrodynamicdescriptionof
for a successful description of RHIC data on v and v theearlyexpansionstagewithahadrontransportmodel
2 4
was also emphasized in [19], although this work left it at the end we can implement a realistic treatment of the
open whether the corresponding lack of thermalization freeze-out process and of viscous effects during the fi-
occursmostlyatthe beginning ortowardsthe endofthe nalhadronicphase. Here we extendthe abovemodels to
2
full(3+1)-dimensionalhydrodynamics[27],inordertobe with ∆η=1.3 and σ =2.1, which are so chosen as to
η
abletostudytherapiditydependenceofellipticflow. Let reproduce the measured pseudorapidity distributions for
us briefly summarize our model. For the hydrodynamic chargedhadrons [35]. NA,B and N are the number of
part coll
stage,we solvethe conservationlaws∂ Tµν=0with the wounded nucleons in the two nuclei and the number of
µ
idealfluiddecompositionTµν=(e+p)uµuν pgµν (where binary nucleon-nucleon collisions, respectively, as calcu-
−
eandpareenergydensityandpressureanduµisthefluid latedfromthe Glauber modelnuclear thickness function
4-velocity) in Bjorken coordinates (τ,x ,η ) [9]. We T (x ) [21],
⊥ s A,B ⊥
neglect the finite (but at RHIC energy very small) net
baryon density. A massless ideal parton gas equation of dNA σin T (r ) B
part = T (r ) 1 1 NN B − , (3)
state(EOS)isemployedintheQGPphase(T >Tc=170 d2x⊥ A + " −(cid:18) − B (cid:19) #
MeV) while a hadronic resonance gas model is used at
T <Tc. Whenweusethehydrodynamiccodealltheway dNpBart = T (r ) 1 1 σNinNTA(r+) A , (4)
to final decoupling, we take into account [10] chemical d2x⊥ B − " −(cid:18) − A (cid:19) #
freezeout of the hadron abundances at T =170 MeV,
ch dN
separatedfromthermalfreezeoutofthemomentumspec- d2xcoll = σNinNTA(r+)TB(r−), (5)
tra at a lower decoupling temperature T , as required ⊥
dec
to reproduce the experimentally measured yields [28]. with the inelastic nucleon-nucleon cross section σin =
NN
For the hydro+cascade description, a hadronic trans- 42mb and r = x 1b 2 + y2 1/2 (where b is the im-
port model JAM [29] is employed for the late stage of ± ±2
pact parameter). The soft/hard fraction α=0.85 was
the expansion. JAM simulates nuclear collisions by in- (cid:2)(cid:0) (cid:1) (cid:3)
adjusted to reproduce the measured centrality depen-
dividual hadron-hadron collisions. Soft hadron produc-
dence [34] of the charged hadron multiplicity at midra-
tion in hadron-hadron scattering is modeled by exciting pidity. At η =0, Eq. (1) reduces to dS/(dη d2x )
s s ⊥
hadronic resonances and color strings. Color strings de- [α(NA +NB )+(1 α)N ]/(1+α) [36]; this param∝-
cay into hadrons after their formation time (τ 1fm/c) part part − coll
eterization is equivalent to the one used in Ref. [23],
∼
accordingtotheLundstringmodelPYTHIA[30]. Lead- 1−x(NA +NB ) + xN , with x=1−α. From
ing hadrons which contain original constituent quarks ∼ 2 part part coll 1+α
Eq. (1), we can compute the entropy density at the ini-
can scatter within their formation time with other
tialtimeτ =0.6fm/c[2]ofthehydrodynamicevolution,
0
hadrons assuming additive quark cross sections [31]. In s(τ ,x ,η )=dS/(τ dη d2x ), which provides the ini-
0 ⊥ s 0 s ⊥
the current study, it is initialized with output from the
tialenergydensityandpressuredistributionsthroughthe
above (3+1)-dimensional hydrodynamics by using the
tabulated EOS described above.
Cooper-Frye formalism [32] (rejecting backward going
The second type of initial conditions is based on the
particles)[14,26]. Weswitchfromhydrodynamicstothe
CGC model [37]. Specifically, we use the Kharzeev-
cascadeapproachattheswitchingtemperatureT =169
sw Levin-Nardi (KLN) approach [23] in the version previ-
MeV, i.e. just below the hadronization phase transition.
ously employedin [24]. In this approach,the energy dis-
We here study two types of initial conditions for the
tribution of produced gluons with rapidity y is given by
evolution. Thefirst,whichwecall“modifiedBGKinitial
the k -factorization formula [38]
T
condition” [22, 33], assumes an initial entropy distribu-
tion of massless partons according to dE 4π2N d2p pT d2k
T = c T Tα (Q2)
dS = C θ Y η fpp(η ) d2x⊥dy Nc2−1Z pT Z 4 s
dηsd2x⊥ 1+α b−| s| s × φA(x1,(pT+kT)2/4;x⊥)
α Yb(cid:0)−ηs dNpA(cid:1)art + Yb+ηs dNpBart × φB(x2,(pT−kT)2/4;x⊥), (6)
× " Yb d2x⊥ Yb d2x⊥ ! wherex =p exp( y)/√sandp isthetransversemo-
1,2 T T
±
mentum of the produced gluons. We choose an upper
dN
coll
+(1 α) , (1) limit of 3GeV/c for the p integration. For the uninte-
− d2x⊥ # T
grated gluon distribution function we use
where η =1ln[(t+z)/(t z)] is the space-time rapidity,
xC⊥==24(x.0,syi)s 2icshtohseenptoositrieo−pnrotdrauncsevetrhseemtoeatshuerebdeacmhaargxeisd, φA(x,kT2;x⊥)= 2π3κκαCCsFF(Q2s)Q2sQQ+2s2sΛ2,, kkT >≤ QQs,, (7)
hadron multiplicity in central collisions at midrapidity 2π3αs(Q2s) kT2+Λ2 T s
[34], Yb is the beam rapidity, and fpp is a suitable whereC =Nc2−1 andQ denotesthesaturationmomen-
parametrization of the shape of rapidity distribution in F 2Nc s
tum. We introduce a small regulator Λ = 0.2GeV/c in
pp collisions,
order to have a smooth distribution in the forward ra-
fpp(ηs)=exp −θ(|ηs|−∆η)(|ηs|−σ2∆η)2 , (2) pfeicdtietdy rbeygiionntr|oyd|u>c4in.5gatthiRsHsmICal(lortehgeurlaretogrio).nsTahree novoetraafl-l
(cid:20) η (cid:21)
3
normalization κ is determined by fitting the multiplicity 0.8
of charged hadron at midrapidity at √sNN = 200 GeV CGC
for the most central collisions. The saturation momen- 0.7 BGK (a =0.85)
n (a =1.0)
tum Qs of nucleus A in A+B collisions, needed in the 0.6 npart (a =0.0)
functionφA,isobtainedbysolvingthe followingimplicit binary
equation at fixed x and x : 0.5
⊥
2π2 dNA 0.4
Q2(x,x )= α (Q2)xG(x,Q2) part. (8) e
s ⊥ C s s s d2x 0.3
F ⊥
An analogous equation holds for the saturation momen- 0.2
tumofnucleusB inφ . Forthegluondistributionfunc-
B
0.1
tion G inside a nucleon we take the simple ansatz [23]
0
0 2 4 6 8 10 12 14
Q2+Λ2 b (fm)
xG(x,Q2)=Kln s x−λ(1 x)4 (9)
Λ2QCD ! −
FIG. 1: (Color online) Initial spatial eccentricity ε=hy2−x2i
withΛ=ΛQCD=0.2GeV.WechooseK=0.7andλ=0.2 hy2+x2i
at midrapidity as a function of impact parameter b, for
so that the average saturation momentum in the trans-
200AGeVAu+Aucollisions with CGC(solid red)andBGK
verseplane yields Q2(x=0.01) 2.0GeV2/c2 incentral
h s i∼ (dashedblue)initialconditions. Forcomparisonwealsoshow
200AGeV Au+Au collisions at RHIC. For the running
initial conditions where the initial parton density at midra-
coupling constantαs in Eq.(8) we use the standardper- pidityscaleswiththetransversedensityofwoundednucleons
turbative one–loop formula, but introducing a cut-off in (dotted green) and of binary collisions (dash-dotted black)
the infra-red region of small Q (i.e. near the surface of [21].
s
thenuclearoverlapregionwheretheproducedgluonden-
sityis low)by limiting the couplingconstanttoα 0.5.
s
≤
We canobtainthe energydensity distributionattime τ0 ing the hadronic stage, due to continued build-up of ra-
from Eq. (6), e(τ0,x⊥,ηs)=dET/(τ0dηsd2x⊥), where y dial flow, reproducing the proton data up to 2GeV/c
∼
is identified with ηs, and use this as the initial distribu- if one choosesTdec=100MeV asthe freeze-outtempera-
tion for the hydrodynamic evolution. ture. The hybrid hydro+cascademodel gives almost the
InFig.1 weshowthe initial eccentricityε=hhyy22+−xx22ii of same proton pT spectrum as ideal fluid dynamics with
the sourceatmidrapidity (ηs=0)forour twomodels for Tdec=100MeV, but further flattens the pion spectrum
the initial conditions. Here represents the average at larger pT, thereby improving the agreement with the
taken with respect to the iniht·ia·l·ienergy density distribu- pion data up to 2GeV/c. This can be understood as a
∼
tione(τ ,x ,η =0). WhiletheBGKmodelinterpolates consequenceofshearviscosityinthehadronicphase,with
0 ⊥ s
betweenthebinarycollisionandwoundednucleonscaling the fast longitudinal expansion generating positive addi-
curves(beingclosertothelatter),CGCinitialconditions tional viscous pressure components in the transverse di-
are seen to give much larger eccentricities. This can be rectionswhose effects onthe pT spectrumgrowquadrat-
tracedbacktoasteeperdropoftheenergydensityprofile ically with pT [18] and decrease with increasing hadron
near the edge in the CGC model. In ideal fluid dynam- mass. The agreement with the measured pT spectra is
ics, the largereccentricities ε resultin largerelliptic flow slightly better for CGC than for BGK initial conditions.
coefficients v [39]. As explained in [16], the chemically frozen hadronic
2
ThisisshownbythethinlinesinFig.2whichcompare EOSresultsinsmallerv valuesthanforahadronicEOS
2
ideal fluid dynamical calculations with RHIC data from whichassumeshadronicchemicalequilibriumalltheway
thePHOBOScollaboration[20]. Notethatthe hydrody- down to kinetic freeze-out at T . Correspondingly, our
dec
namic calculations shown here use an EOS in which the curveforBGKinitialconditionsliesbelowtheoneshown
hadronabundancesareheldfixedbelowT attheirchem- inFig.2 of[20]whichusesthe latterEOS.Itis notewor-
c
ical freeze-out values established during the hadroniza- thy that for central and semicentral collisions the data
tion process [10]. For this EOS it is known from earlier seem to lie consistently somewhat above the hydrody-
hydrodynamic studies [10, 40] that the slope of pion p namicpredictions withBGKinitialconditions. While at
T
spectrum stalls during the hadronic evolution. Without first sight this seems to argue against the validity of the
an additional transverse velocity kick already at the be- BGK model, it must be noted that event-by-event fluc-
ginning ofthe hydrodynamicstage[40]itreproducesthe tuations in the geometry of the nuclear overlap region,
pion data from central collisions at midrapidity only up which we don’t take into account [41], tend to signifi-
to 1GeV/c, falling off too steeply at largerp . In con- cantly increase the measured v in collisions with small
T 2
∼
trast, the proton p spectrum continues to flatten dur- impact parameters (large N values) [45].
T part
4
0.16
CGC, T =100MeV
dec
0.14 BGK, T =100MeV
dec
CGC, hydro+cascade 0.1 (a) 3-15% b=4.0fm hydro+cascade
0.12 BGK, hydro+cascade 0.08 T =100MeV
PHOBOS(hit) Tdec=169MeV
0.1 PHOBOS(track) 0.06 dec
2 PHOBOS
v
0.04
20.08
v
0.02
0.06
0
0.04 0.1 (b) 15-25% b=6.3fm
0.08
0.02
0.06
2
0 v 0.04
0 50 100 150 200 250 300 350 400
N 0.02
part
0
0.12 (c) 25-50% b=8.5fm
FIG. 2: (Color online) pT-integrated elliptic flow for charged
0.1
hadrons at midrapidity ( η < 1) from 200AGeV Au+Au
| | 0.08
collisions, as a function of the number Npart of participating 2
nucleons. The thin lines show the prediction from ideal fluid v 0.06
dynamics with a freeze-out temperature Tdec=100MeV, for 0.04
CGC (solid red) and BGK (dashed blue) initial conditions. 0.02
The thicklines (solid red for CGC and dashed bluefor BGK 0
-5 -4 -3 -2 -1 0 1 2 3 4 5
initialconditions)showthecorrespondingresultsfromthehy-
h
dro+cascadehybridmodel. Thedataarefrom thePHOBOS
Collaboration [20].
FIG. 3: (Color online) The pseudorapidity dependence of v2
for charged hadrons in (a) central (3-15%), (b) semicentral
The difference between the eccentricities given by the (15-25%), and (c) peripheral (25-50%) Au+Au collisions at
BGKandCGC initialconditions mayseemsurprisingin √s=200AGeV. The corresponding impact parameters are,
viewofthefactthatthecentralitydependenceofhadron respectively,b=4.0,6.3, and 8.5fm. Thehydrodynamicevo-
multiplicities in both approaches can be made numeri- lutionisinitializedwithmodifiedBGKinitialconditions. The
lines show the predictions from ideal fluid dynamics with
cally very similar by a proper choice of parameter α in
Eq. (1) [23]. When parameterized in terms of Eq. (1), Tdec=100 MeV (solid blue) and Tdec=169 MeV (dashed
green). The red circles show the corresponding results from
the main prediction of the CGC approach is the near
thehydro+cascadehybridmodel. Theblacksquaresaremea-
independence of α on the collision energy, which is con- surementsby thePHOBOS Collaboration [20].
firmed by the data. The reason for the big difference in
the eccentricities stems from the different entropy pro-
files predicted by the two approaches, especially in the p -integrated elliptic flow (shown in Fig. 2) and its p -
T T
regions where the density of produced particles is rela- slope, dv (p )/dp , at midrapidity, for both pions and
2 T T
tively small. While these differences contribute little to protons (to be published elsewhere). With CGC initial
the totalobservedmultiplicity, they appearquite impor- conditions even the QGP phase must exhibit significant
tant for the evaluation of eccentricity. The application dissipative effects if one want to reproduce the RHIC
of the CGC approach in the region of small parton den- data.
sity is of course questionable, and a better theoretical
InFigs.3and4weshowthemeasured[20]rapidityde-
understanding of the transitionfrom high to low density
pendence of v for three centrality classes, together with
2
regimes is clearly needed.
calculations at three representative impact parameters
The thick lines show the centrality dependence of b=4, 6.3 and8.5fmcorrespondingtothesecentralityse-
v from the hydro+cascade hybrid model. Whereas lections(i.e. adjustedtogivethecorrectaveragenumber
2
with BGK initial conditions the dissipative effects of the ofparticipantsN in eachcase asquotedin Ref. [20]).
part
hadronic phase in the cascade model reduce the purely Results fromBGK initialconditions areshowninFig. 3.
hydrodynamicv sufficientlytobringthetheoryinagree- Thelinesshowidealfluiddynamicalcalculationswithki-
2
ment with the data even for peripheral collisions, CGC netic decoupling assumed at T =100MeV (solid blue)
dec
initial conditions, with their larger eccentricities, cause and T =T =169MeV (dashed green). The dashed
dec sw
so much elliptic flow that even the hybrid model over- linesunderpredictthe dataatallimpactparametersand
predicts the data significantly. This is true for both the all but the most forward rapidities, indicating the need
5
line with experiment.
We conclude that the answer to the question, whether
all of the observed discrepancies between elliptic flow
0.1
(a) 3-15% b=4.0fm hydro+cascade dataandidealfluiddynamicalsimulationscanbeblamed
0.08 T =100MeV
dec on “late hadronic viscosity” and fully eliminated by em-
T =169MeV
0.06 dec ploying a hydro+cascade hybrid model such as the one
2 PHOBOS
v
0.04 studied here, depends on presently unknown details of
0.02 the initial state of the matter formed in the heavy-ion
collision. With BGK initial conditions hadronic dissipa-
0
0.1 tion seems to be able to reduce the elliptic flow enough
(b) 15-25% b=6.3fm
0.08 tobring the theoreticalpredictionsinline withthe data,
leavinglittleroomforadditionaldissipativeeffectsinthe
0.06
2 early QGP stage. CGC initial conditions yield signifi-
v
0.04
cantly more eccentric sources and produce larger than
0.02 observed elliptic flow even if dissipative effects in the
0 late hadronic stage are taken into account. In this case,
0.12 (c) 25-50% b=8.5fm the early QGP phase must either be significantly softer
0.1 than parametrized by our equation of state or exhibit
0.08 significant viscosity itself. With CGC initial conditions,
2
v 0.06 the excess over the v data from Au+Au collisions at
2
0.04
RHIC persists even at midrapidity in all but the most
0.02
central collisions; with such initial conditions, the stan-
0
dardclaim[2]thatatRHICenergiesthemeasuredelliptic
-5 -4 -3 -2 -1 0 1 2 3 4 5
h flow at midrapidity exhausts the theoretical upper limit
predicted by ideal fluid dynamics must be qualified.
We see that a data-based attempt to establish lim-
FIG. 4: (Color online) Sameas Fig. 3, exceptfor usingCGC its on the viscosity of the quark-gluon plasma requires,
instead of BGK initial conditions. among other things, a better understanding of the ini-
tial conditions of the fireball created in RHIC collisions.
Unfortunately, very few direct probes of the initial con-
forgeneratingadditionalellipticflowduringthehadronic ditions are available. In Ref. [22, 46] 3-dimensional jet
stage below Tc. The solid lines, on the other hand, tomographywas proposedto test the longitudinal struc-
stronglyoverpredicttheforwardrapiditydatainsemipe- ture of BGK and CGC initial conditions in noncentral
ripheraland peripheralcollisions, showing that ideal hy- collisions. A specific feature of CGC initial conditions is
drodynamics generates too much additional elliptic flow apredictedsignflipofthefirstFouriermomentv ofnu-
1
during the hadronic stage. The hydro+cascade hybrid clearmodificationfactorR (p ,y,φ)athighp asone
AA T T
model (red circles) gives a good description of the data movesawayfrommidrapidity[46]. Alternatively,onecan
over the entire rapidity range for all three centralities, tryto exploitthe factthat the large(evenif notperfect)
with the exceptionof the midrapidity regionin the most degree of thermalization observed in heavy-ioncollisions
central collision sample as already discussed above. The at RHIC limits the amount of entropy produced during
hadroniccascademodelprovidesjusttherightamountof theexpansion. Whentakingintoaccountthatfinalstate
dissipationtobringtheidealfluidpredictiondowntothe rescatterings in the medium produce only very small ef-
measured values, especially in very peripheral collisions fectsontheshapeoftherapiditydistribution,thefinally
and away from midrapidity. observedchargedhadronrapidity distributions therefore
The situation is different for the more eccentric CGC severely constrain the initial entropy and energy density
initial conditions, as shown in Fig. 4. Now v is over- profiles [23]. A better theoretical understanding of the
2
predictedatallcentralitiesandrapidities ifthe hadronic initial conditions, especially of the transition from the
phase isdescribedby idealfluiddynamics, andthe dissi- high density to small density regimes, is needed to ex-
pative effects of the hadronic cascade are no longer suf- tract the viscosity of quark–gluon plasma at the early
ficient to reduce v for the more peripheral bins enough stages. A systematic study of the charged hadron ra-
2
to obtain agreement with the data. In the peripheral pidity distributions for a variety of collision centralities,
sample (25-50%) the excess elliptic flow persists at al- center of mass energies and system sizes is needed to as-
most all rapidities and exists even if the fireball freezes sess which description of the initial state yields a more
out directly at hadronization (no hadronic evolution at consistentandefficientoveralldescriptionofallavailable
all). Significant dissipation in the early QGP phase is data.
needed in this case to bring the theoretical prediction in This work was supported by the U.S. DOE un-
6
der contracts DE-FG02-93ER40764 (T.H.), DE-FG02- (2001);D.KharzeevandE.Levin,ibid.B523,79(2001);
01ER41190 (U.H.), DE-AC02-98CH10886 (D.K.) and D. Kharzeev, E. Levin and M. Nardi, Phys. Rev. C 71,
DE-FG02-87ER40331.A008 (R.L.). Discussions with 054903 (2005); D. Kharzeev, E. Levin and M. Nardi,
Nucl. Phys. A 730, 448 (2004).
A. Adil, M. Gyulassy, and A.J. Kuhlman are gratefully
[24] T. Hirano and Y. Nara, Nucl. Phys.A 743, 305 (2004).
acknowledged.
[25] A. Dumitru, S.A. Bass, M. Bleicher, H. Stoecker and
W. Greiner, Phys. Lett. B 460, 411 (1999); S.A. Bass,
A. Dumitru, M. Bleicher, L. Bravina, E. Zabrodin,
H. Stoecker and W. Greiner, Phys. Rev. C 60, 021902
(1999); S.A. Bass and A. Dumitru, Phys. Rev. C 61,
∗ Correspond to [email protected] 064909 (2000).
[1] The experimental situation is summarized in B.B. Back [26] D.Teaney,J.LauretandE.V.Shuryak,Phys.Rev.Lett.
et al. [PHOBOS Collaboration], Nucl. Phys. A 757, 28 86, 4783 (2001).
(2005); J. Adams et al. [STAR Collaboration], Nucl. [27] C. Nonaka and S.A. Bass, nucl-th/0510038. [This paper
Phys.A 757,102(2005); K.Adcoxetal.[PHENIXCol- employs different hydrodynamic and hadronic cascade
laboration], Nucl.Phys. A 757, 184 (2005). codes from the present paper.]
[2] For a theoretical review see P.F. Kolb and U. Heinz, in [28] P. Braun-Munzinger, D. Magestro, K. Redlich and
Quark-Gluon Plasma 3, edited by R.C. Hwa and X.- J. Stachel, Phys.Lett. B 518, 41 (2001).
N. Wang (World Scientific, Singapore, 2004), p. 634 [29] Y. Nara, N. Otuka, A. Ohnishi, K. Niita and S. Chiba,
[nucl-th/0305084]. Phys. Rev.C 61, 024901 (2000).
[3] C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. [30] T.Sj¨ostrandetal.,Comp.Phys.Comm.135,238(2001).
87,182301(2001);J.Adamsetal.[STARCollaboration], [31] H.Sorge,Phys.Rev.C52,3291(1995);S.A.Bassetal.,
Phys.Rev.Lett. 92, 052302 (2001). Prog. Part. Nucl.Phys. 41, 255 (1998).
[4] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. [32] F. Cooper and G. Frye,Phys. Rev.D 10, 186 (1974).
Lett.89, 212301 (2002); S.S.Adleret al.[PHENIX Col- [33] S.J. Brodsky, J.F. Gunion and J.H. Kuhn, Phys. Rev.
laboration] Phys. Rev.Lett. 91, 182301 (2003). Lett. 39, 1120 (1977).
[5] P. Huovinen, P.F. Kolb, U. Heinz, P.V. Ruuskanen, and [34] B.B.Backetal.[PHOBOSCollaboration], Phys.Rev.C
S.A.Voloshin, Phys. Lett. B 503, 58 (2001). 65, 061901 (2002).
[6] U.HeinzandP.F.Kolb,Nucl.Phys.A702,269c(2002); [35] B. B. Back et al.,Phys. Rev.Lett. 91, 052303 (2003).
M.GyulassyandL.D.McLerran,Nucl.Phys.A 750,30 [36] A.J. Kuhlman and U. Heinz, Phys. Rev. C 72, 037901
(2005); E.V. Shuryak,Nucl. Phys.A 750, 64 (2005). (2005). [This paper uses a slightly smaller soft fraction
[7] C. Adler et al. [STAR Collaboration], Phys. Rev. C 66 α=0.75.]
034904 (2002); C.Altet al.[NA49Collaboration], Phys. [37] L.D. McLerran and R. Venugopalan, Phys. Rev. D 49,
Rev.C 68, 034903 (2003). 2233 (1994); 49, 3352 (1994); 50, 2225 (1994).
[8] B.B. Back et al. [PHOBOS Collaboration], Phys. Rev. [38] L.V.Gribov,E.M.Levin,andM.G. Ryskin,Phys.Rept.
Lett.89, 222301 (2002) 100, 1 (1983); E.Laenen and E.Levin,Ann.Rev.Nucl.
[9] T. Hirano, Phys. Rev.C 65, 011901 (2002). Part. Sci. 44, 199 (1994).
[10] T.HiranoandK.Tsuda,Phys.Rev.C66,054905(2002). [39] J. Y.Ollitrault, Phys. Rev.D 46, 229 (1992).
[11] A successful parametrization of the rapidity dependence [40] P.F.KolbandR.Rapp,Phys.Rev.C67,044903(2003).
of the elliptic flow coefficient, v2(η), using the “Buda- [41] The inclusion of event-by-eventfluctuations in both the
Lund model” has recently been presented in [12]. This geometricshapeanddensityprofileoftheinitialentropy
modelisbasedonarestrictedclassofselfsimilaranalyti- distribution is numerically expensive, and for our hy-
cal solutions tothenon-relativistic ideal fluid dynamical dro+cascadehybridmodelitexceedstheavailabletechni-
equations.The applicability ofthis modelto RHICdata cal resources. However,theirrole inaffectingtheextrac-
remains underdebate. tionofv2 fromtheoreticalcalculations andexperimental
[12] M. Csanad et al.,nucl-th/0512078. dataispresentlybeingintenselydiscussed[42,43,44,45].
[13] U.Heinz and P.F. Kolb, J. Phys.G 30, S1229 (2004). We point out in particular the work by the Sao Paulo –
[14] D. Teaney, J. Lauret, and E.V. Shuryak, Rio de Janeiro Collaboration [43, 44] who include such
nucl-th/0110037. fluctuationsintheiridealfluiddynamicalmodelandfind
[15] M. Gyulassy, nucl-th/0403032. thattheytendtoreducethehydrodynamicallypredicted
[16] T. Hirano and M. Gyulassy, nucl-th/0506049. elliptic flow, especially at forward rapidities. Whether
[17] P. Kovtun, D.T. Son and A.O. Starinets, Phys. Rev. thismighthelptoreducethediscrepancybetweentheory
Lett. 94, 111601 (2005); G. Policastro, D.T. Son and anddataforourhydro+cascadehybridwithCGCinitial
A.O.Starinets, Phys. Rev.Lett. 87, 081601 (2001). conditions remains to be studied.
[18] D.Teaney, Phys.Rev.C 68, 034913 (2003). [42] M. Gyulassy, D. H. Rischke and B. Zhang, Nucl. Phys.
[19] R.S. Bhalerao, J.-P. Blaizot, N. Borghini, and J.-Y. Ol- A 613, 397 (1997).
litrault, Phys. Lett. B 627, 49 (2005). [43] O. J. Socolowski, F. Grassi, Y. Hama and T. Kodama,
[20] B.B. Back et al. [PHOBOS Collaboration], Phys. Rev.Lett. 93, 182301 (2004).
nucl-ex/0407012 [44] Y. Hama et al., hep-ph/0510096; R. Andrade et al.,
[21] P.F. Kolb, U. Heinz, P. Huovinen, K.J. Eskola and nucl-th/0511021.
K.Tuominen, Nucl. Phys.A 696, 197 (2001). [45] M. Miller and R. Snellings, nucl-ex/0312008; X.l. Zhu,
[22] A. Adil and M. Gyulassy, Phys. Rev. C 72, 034907 M. Bleicher and H.Stocker, nucl-th/0509081.
(2005). [46] A. Adil, M. Gyulassy and T. Hirano, nucl-th/0509064.
[23] D. Kharzeev and M. Nardi, Phys. Lett. B 507, 121
