Anisotropic flow in event-by-event ideal hydrodynamic simulations of sqrt(s_{NN})=200 GeV Au+Au collisions

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by  Fernando G. Gardim

Anisotropic flow in event-by-event ideal hydrodynamic simulations of √sNN=200 GeV
Au+Au collisions
Fernando G. Gardim,1 Fr´ed´erique Grassi,1 Matthew Luzum,2 and Jean-Yves Ollitrault2
1Instituto de F´ısica, Universidade de S˜ao Paulo, C.P. 66318, 05315-970, S˜ao Paulo-SP, Brazil
2CNRS, URA2306, IPhT, Institut de physique th´eorique de Saclay, F-91191 Gif-sur-Yvette, France
(Dated: January 18, 2013)
We simulate top-energy Au+Au collisions using ideal hydrodynamics in order to make the first
comparison to the complete set of mid-rapidity flow measurements made by the PHENIX Col-
laboration. A simultaneous calculation of v2, v3, v4, and the first event-by-event calculation of
quadrangular flow defined with respect to the v2 event plane (v4{Ψ2}) gives good agreement with
measuredvalues,includingthedependenceonbothtransversemomentumandcentrality. Thispro-
vides confirmation that the collision system is indeed well described as a quark-gluon plasma with
3
anextremelysmallviscosity,andthatcorrelations aredominantlygeneratedfrom collectiveeffects.
1
0 In addition we present a prediction for v5.
2
PACSnumbers: 25.75.Ld,24.10.Nz

n
a
J I. INTRODUCTION the extraction of a precise finite value is hampered by

7 poor knowledge of the earliest stages of the collision, as
1 Evidence suggests that in a collision between ultra- well as other uncertainties [10].

relativistic heavy nuclei, a strongly-interacting, low- An important recent development was the realization
]
h viscosity fluid — the quark-gluon plasma (QGP) — is oftheimportanceofquantumfluctuations,whichinpar-
-t created. The clearest indication of this behavior is seen ticular implies an event-by-event breaking of the sym-
l in the azimuthal anisotropy [1] among the bulk of emit- metry naivelyimplied by the collisionofidenticalnuclei.
c
u tedparticles. Intheory,onecharacterizesthisanisotropy Specifically,thecoefficientsvnaregenerallynon-zeroalso
n in terms of a single-particle probability distribution for for odd n [11], the event plane angles do not necessar-
[ each collision event. By writing this distribution as a ily point in the same direction as the impact parame-

Fourierserieswithrespectto theazimuthalangleofout- ter [12, 13], and these quantities fluctuate significantly
2
v going particles φ, one can define flow coefficients vn and from one event to another, even at a fixed impact pa-
2 event plane angles Ψ : rameter [14].
n
8 These insights led to the possibility that all of the
∞
8 2πdN measuredlong-rangecorrelationsmaybegeneratedsolely
2 N dφ =1+2Xvncosn(φ−Ψn), (1) from collective behavior [11, 15].
3. n=1 Several new flow observables — specifically ones im-
0 vneinΨn einφ , (2) plied by the presence of event-by-event fluctuations —
≡h i
2 were recently measured for the first time [16–20]. Stud-
where the brackets indicate an average over the single
1 ies of these new observables indicate that, individually,
v: particle probability, and the event plane angles Ψn are they indeed appear to have properties that are consis-
chosen such that v are the (positive) magnitudes of the
i n tent with a hydrodynamic origin [15, 21–23]. However,
X complex Fourier coefficients.
they have not yet all been reproduced in a single calcu-
Experimentally, one measures the azimuthal depen-
r lationwithin one model using a single set of parameters.
a dence of event-averaged correlations between detected
This has left some lingering doubt about whether the
particles. Thesemeasurementsindicatethepresenceofa
interpretation in terms of collective behavior is indeed
very large “elliptic flow” coefficient v2 [2, 3], which typ-
correct [24]. In addition, each measurement provides an
ically can only be reproduced in calculations where the
independent constraint on theory, so identifying models
systemis modeledas a strongly-interactingfluid. In this
and sets of parameters that can simultaneously satisfy
picture, the large momentum anisotropy is generated as
all the constraints is a necessary first step in reducing
ahydrodynamicresponsetothespatialanisotropyofthe
various theoretical uncertainties.
nuclear overlap region in collisions of non-zero impact
In this Letter we perform state-of-the-art ideal hydro-
parameter. Itevenappearsthatthe createdquark-gluon
dynamiccalculationsandcomparetheresultstothefirst
plasma must be an almost perfect (zero viscosity) fluid,
measurements[16] ofthese new observablesat Relativis-
witharatioofshearviscositytoentropydensityη/sthat
tic Heavy-Ion Collider (RHIC) as well as previous mea-
is at mosta few times 1/(4π), a value that wasfamously
conjecturedtobeauniversallowerbound[4]1. However, surements by the same collaboration [25]. Other groups
anditmayevenbepossibletohaveanarbitrarilysmallvalue[8],
1 The bound is now known to be violated in some theories [5–7], though theeffective viscositymaystillhaveafinitebound[9].
2
have presented calculations from some of these observ- in order to show the size of the effect of fluctuations on
ables using event-by-event ideal [26–29] or viscous [30] event-plane analyses.
hydrodynamics, or transport models [31]. The present Likewise, the measured value v4 Ψ2 depends on the
study encompasses simultaneously, for the first time, all resolution [37], and is usually clos{e to} v4v22cos(4Ψ4
h −
the measured flow observables at midrapidity. 4Ψ2) /p v24 , but with increasing resolution approaches
i h i
v4cos(4Ψ4 4Ψ2) .
h − i
II. OBSERVABLES
III. RESULTS
All the experimental results considered here were ob-
tained using the event-plane method [32]. With this
Using the hydrodynamic code NeXSPheRIO [38], we
method, one first identifies an event plane Ψ in each
n simulatetop-energyAu-AucollisionsatRHIC.Thiscode
event using a specific detector at forward rapidity, and
solves the equations of ideal relativistic hydrodynamics
then calculates the correlation of particles near midra-
using fluctuating initial conditions from the eventgener-
pidity with this event plane, e.g.,
ator NeXus [39].
NeXus aims at a realistic and consistent approach of
v Ψ cosn(φ Ψ ) , (3)
n{ n}≡h − n i the initial stage of nuclear collisions [39]. It is a Monte-
Carlo generator which takes into account not only the
where the brackets indicate an average over particles in
fluctuations of nucleon positions within nuclei [30], but
a largenumber ofevents. Arapidity gapwith the event-
alsofluctuations atthe partoniclevel: the momentum of
plane detector suppresses nonflow correlations [15, 33].
each nucleon is shared between one or several “partici-
At RHIC, “triangular flow” v3 Ψ3 and “quadrangu-
{ } pants” and a “remnant”, which implies non-trivial dy-
lar flow” v4 Ψ4 were measured for the first time, as a
{ } namical fluctuations in each nucleon-nucleon collisions.
function of the particle transversemomentum p in vari-
t
The resulting full energy-momentum tensor is matched
ous centrality classes by the PHENIX collaboration [16]
to a hydrodynamic form, resulting in a fluctuating flow
(preliminary data from STAR have now also been pre-
fieldinadditiontoafluctuatinginitialenergydensity,in
sented [20]).
all three spatial dimensions, with the transverse length
Previously, a different quadrangular flow observable
scale of the fluctuations set mostly by the size of the in-
has been measured, defined with respect to Ψ2 [25, 34].
coming nucleons.
We use a different notation for this quantity to avoid
At the end of the hydrodynamic evolution, discrete
confusion:
particles are emitted using a Monte-Carlo generator2.
NeXSPheRIOprovidesagooddescriptionofrapidityand
v4 Ψ2 cos4(φ Ψ2) . (4)
{ }≡h − i transverse momentum spectra [43], elliptic flow v2 [44],
vn is analyzed using a large sample of events, and its and the rapidity-even v1 observable, directed flow at
value fluctuates from one event to the other. These fluc- midrapidity [45]. In addition, it is known to reproduce
tuations (which were not appreciated when the method the long-range structures observed in two-particle corre-
wasdeveloped),combinedwiththeuseofafinite sample lations [46]. All parameters were fixed from these earlier
of particles in the analysis, cause the measured value to investigations,before anyofthe new observables(v3, v4)
deviate from the event average of the theoretical coeffi- were measured — nothing has been tuned here.
cientsdefinedinEq.(1). Generally,v Ψ liesbetween For this work, we generated 110 NeXus events each in
n n
{ }
the mean value and the root-mean-square (rms) value 5%centralityclassesupto60%centrality,solvingthehy-
of v . One can parameterize the resulting measurement drodynamic equations independently for each event. As
n
as [35]: in Ref. [47], at the end of each hydro event, we run the
Monte-Carlo generator many times, so that we can do
v Ψ vα 1/α, (5) the flow analysis using approximately 6 105 particles
n{ n}≃h ni ×
perevent. Thissignificantlyreducesstatisticalnoiseand
where here the brackets indicate an average over events. allowsfor anaccuratedeterminationofv andΨ inev-
n n
The value of α depends on the event plane resolution ery event. It also suppresses non-flow correlations from,
Res Ψ v √N [36]: If the resolution is poor, α 2, e.g.,particledecays. Thesequantitiesarethencalculated
n n
{ } ∼ ≃
andthemeasuredv isarmsvalue,whileiftheresolution by Eq. (2), with the average taken over all particles in
n
is large, α 1, and the result gets closer to the mean the pseudorapidity interval 1 < η < 1. The procedure
≃ −
value. used to measure v in hydrodynamics thus mimics the
n
The most recent data from PHENIX has a maximum
event plane resolution of 0.74 (for v2 around 30% cen-
trality [25]) and much smaller for v3 and v4 [16], which
implies α > 1.81 [36]. So in general the results are very 2 Freeze-out occurs at a constant temperature. Hadrons do not
close to a rms value of v . Nevertheless, in the follow- rescatter after freeze-out [40–42], but resonance decays are im-
n
ing we compute both limiting cases α = 2 and α = 1 plemented.
3
00-10% 10-20% 20-30% 30-40% 40-50% 50-60%
0.24
NeXSPheRIO+
2 0.16 NeXSPheRIO-
v PHENIX
0.08
0
NeXSPheRIO+
0.12 NeXSPheRIO-
3 PHENIX
v
0.06
0
NeXSPheRIO+
0.08 NeXSPheRIO-
4 PHENIX
v 0.04
0
0.06
NeXSPheRIO+
NeXSPheRIO-
0.04
5
v
0.02
0
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
p @GeV(cid:144)cD
T
FIG. 1. (Color online) Results for vn{Ψn} for n = 2–5, compared to published data from the PHENIX collaboration [16].
Closed and open symbols correspond to two different ways of averaging overevents(mean and rms value, respectively). Error
bars represent statistical uncertainty from the finite number of events. The left column (0–10%) represents the 10% most
central collisions, which each column to theright increasingly peripheral.
experimentalprocedure,withtwodifferences: (i)thereis v4. The pt dependence is a generic feature of ideal hy-
no need for a rapidity gap, because nonflow correlations drodynamics [21]. The magnitude and centrality depen-
are negligible; (ii) there is no need for a resolution cor- dence of v , on the other hand, depend on the initial
n
rection, because the huge multiplicity per event ensures conditions: v2 is mostly driven by the almond shape of
that the resolution is close to 1 for all events [47]. the overlaparea,which depends on the particular model
Fig. 1 displays v as a function of the particle trans- used [49], while higher harmonics are mostly driven by
n
verse momentum p , averagedover events in a centrality initial fluctuations [11], which explains why they have a
t
class. The averageoverevents is estimated in two differ- mild centrality dependence [50].
entwaysinordertoillustratetheeffectofevent-by-event
flow fluctuations on the experimental analysis. The first
estimate, labeled NeXSPheRIO-, is a plain mean value
Originally, quadrangular flow v4 had been measured
(corresponding to α = 1 in Eq. (5)). The second esti-
with respect to the event-plane of elliptic flow. Recent
mate, labeled NeXSPheRIO+ is a weigthed average
resultsshowthatv4 Ψ2 [25]issmallerthanv4 Ψ4 [16],
{ } { }
v cosn(φ Ψ ) typically by a factor 2 for peripheral collisions, and by
v+ Ψ h n − n i. (6)
n{ n}≡ phvn2i samfaalcletor,r i5t ifsormceeansutrraeldcwoliltihsioansb.ettAerlthreoluagtihvev4a{cΨcu2r}aciys
The average of vn+{Ψn} over pt is the rms average of vn than v4{Ψ4}, because of the better resolution on Ψ2.
(α = 2 in Eq. (5)). For Gaussian flow fluctuations [48], This makes v4 Ψ2 a useful quantity for detailed model
{ }
the ratio of the rms to the mean is p4/π 1.13 for v3 comparisons [51]. As in the case of vn Ψn , we per-
≃ { }
and v5, and closer to 1 for v2 and v4. form the average over events in two different ways in
Fig. 1 shows that our event-by-event ideal hydrody- order to illustrate the effect of event-by-event flow fluc-
namic calculation reproduces well the observed central- tuations. The first estimate, labeled NeXSPheRIO-, is
ity and transverse momentum dependence of v2, v3 and a plain mean value, as in Eq. (4). The second estimate,
4
3
00-05% 05-10% 10-15% 15-20%
2
1
3
20-25% 25-30% 30-35% 35-40%
22
v
(cid:144) 2
<
2
Y
1
8
4
v
3
40-45% 45-50% 50-60%
2 NeXSPheRIO+
NeXSPheRIO-
1
PHENIX
0
0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2
p @GeV(cid:144)cD
T
FIG.2. (Coloronline)Resultsforthefirstevent-by-eventhydrodynamiccalculationofv4{Ψ2}/v2{Ψ2}2,comparedtopublished
datafromthePHENIXcollaboration [25]. AsinFig.1,closedandopensymbolscorrespondtotwodifferentwaysofaveraging
overevents(seetext),errorbarsrepresentstatisticaluncertaintyfromthefinitenumberofevents,andsmallerpercentilerefers
to more central collisions
labeled NeXSPheRIO+ is a weigthed average: tion using ideal hydrodynamics, i.e., with negligible vis-
cosity. These results prove that no non-zero QGP vis-
v4+{Ψ2}≡ hv22cosp4(vφ24−Ψ2)i. (7) ccoaslcituylaitsiorneqrueqirueidretsoarenpergolidguibceletvhiessceosditayta—. Ikneefapcint,gtehvis-
h i erythingelsefixed,aviscositythesizeoftheconjectured
The actual event-plane value is expected to lie between bound η/s = 1/4π would suppress v and destroy the
n
these two limits, depending on the resolution [37]. remarkable fit to data. In addition, the ratio v4/v22(pT)
Since v4 Ψ2 can be generated by elliptic flow as a depends stronglyonη/s, andany non-zerovalue usually
second ord{er eff}ect [52], we scale it by v2 Ψ2 2 for each tends to destroy the flat curve that ideal hydrodynamics
{ }
pt. Hereafter, we denote v4 Ψ2 and v2 Ψ2 simply by predicts [51, 53].
{ } { }
v4 and v2. Fig. 2 displays this first event-by-event hy-
However, this requirement of negligible viscosity de-
drodynamic calculation of v4/v22 as a function of pt for pends crucially on aspects of the model which are not
different centralities. The measured ratio is remarkably
entirely constrained. In particular, although the NeXus
constantasafunctionofp ,andincreasesmildlyforcen-
t model provides an honest effort at a reasonable descrip-
tralcollisions. Idealhydrodynamicspredictsv4/v22 ≃1/2 tion of the physics, with many realistic elements, there
at high p for a single event [52]. For all centralities, the
t is considerable uncertainty about the early stages of a
measuredvalueofv4/v22 isgreaterthan1/2,evenathigh heavy-ioncollisionandtheresultinginitialconditionsfor
pt. This can be explained [37] by v2 fluctuations, ex- hydrodynamic evolution. In principle, another model,
cept for the two most central bins, where one expects
coupled to viscous hydrodynamics might well be able to
v4/v22 ≃ 1 [37], smaller than the measured value, which fit these data. For example Ref. [30] presents event-by-
is between 1.5 and 2 for the most central bin. For these
event viscous hydrodynamic calculations with Glauber
two central bins, our results from event-by-event hydro-
initialconditions thatrequireavalue closeto η/s=0.08
dynamics are in good agreement with experiment (first
togivereasonableagreementwiththequantitiesinFig.1
two panels in Fig. 2). This shows that other sources of
fltroiwbuflteucttouva4ti/ovn22s.,Aothsiemritlahranfinvd2inflguchtausatbieoennsraelponoret,ecdonin- saceetnnssteirtvaielvrecaoltlolciestinhotenrsai.nliittSiieeascl,ocntohdnoldyu,igthaioltnthhso,euytghhuenvedff4e/ervcpt22reoidsficnntoontv-3vzeefroryor
a transportcalculationwith v4 andv2 both defined with
viscosity depends significantly on the way it is imple-
respect to the direction of the impact parameter [31].
mentedatfreezeout[53],andthecorrectimplementation
Our calculated v4/v22 is slightly higher than data for the is an open issue.
next two bins (10 20%). Above 20% centrality, data
−
are within the range spanned by our calculations. Thus, this workis only a firststep in identifying mod-
Thecalculationsshownheresimulatethesystemevolu- elsthatarecompatiblewithdata,andstrongconclusions
5
cannotyetbedrawnabout,e.g.,theprecisevalueofη/s. in a single calculation, this provides a complete, unified
Although the success of these calculations are an impor- picture of the bulk evolution of a heavy-ion collision as
tantmilestone,provingthatatthispointnolowerbound an extremely low-viscosity fluid. Indeed, for our model
can yet be placed on η/s, we can not yet make a precise of initial conditions, a negligible viscosity is required for
statement about an upper bound – only that it still ap- agoodfitto allmid-rapidityflowobservables. Therefore
pearsunlikely thata value significantly largerthan 1/4π nolowerbound cancurrentlybe placedonthe shearvis-
will be possible. cosity of the quark-gluon plasma. Further study will be
needed to determine a reliable upper bound, but finding
models (such as this one) that are compatible with all
IV. CONCLUSIONS measured data is a significant first step.
Using an ideal hydrodynamic model with fluctuating
initialconditions,wehaveperformedthe firstsimultane-
ACKNOWLEDGMENTS
ouscalculationofv2 Ψ2 ,v3 Ψ3 ,v4 Ψ4 andv4 Ψ2 as
{ } { } { } { }
a function of transverse momentum and centrality. Our
results are in good agreement with the most recent ex- We thank Yogiro Hama for useful discussion. This
perimental results for all the observablesat RHIC, at all work is funded by Cofecub under project Uc Ph
centralities and in a wide range of transverse momen- 113/08;2007.1.875.43.9, by FAPESP under projects
tum. This provides convincing confirmation of the cur- 09/50180-0and09/16860-3,andby CNPqunder project
rentparadigmthatcollectiveeffectsalonecanexplainall 301141/2010-0. ML is supported by the European Re-
long-range correlations in the soft sector. Further, since search Council under the Advanced Investigator Grant
allsuchmeasuredcorrelationsaregeneratedconsistently ERC-AD-267258.
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