Elliptic Flow Analysis at RHIC with the Lee-Yang Zeroes Method in a Relativistic Transport Approach

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by  Xianglei Zhu

Elliptic Flow Analysis at RHIC with the Lee-Yang Zeroes Method in a Relativistic
Transport Approach
Xianglei Zhu,1,2,3 Marcus Bleicher,2 and Horst St¨ocker1,2
1Frankfurt Institute for Advanced Studies (FIAS),
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
2Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
3Physics Department, Tsinghua University, Beijing 100084, China
The Lee-Yang zeroes method is applied to study elliptic flow (v2) in Au+Au collisions at √s =
200A GeV, with the UrQMD model. In this transport approach, the true event plane is known
andboththenonfloweffectsandevent-by-eventv2 fluctuationsexist. Althoughthelowresolutions
prohibit the application of the method for most central and peripheral collisions, the integral and
6 differential elliptic flow from theLee-Yangzeroes method agrees with theexact v2 values verywell
for semi-central collisions.
0
0
PACSnumbers: 25.75.Ld,25.75.Dw,25.75.Gz
2

n
a Anisotropic flow [1, 2, 3, 4], or more specifically, ellip- roesmethodwasdevelopedtoanalyzethe collectiveflow
J tic flow (v ), which is the secondFourier harmonic [5] in [29, 30, 31]. This new method enables the extraction of
2
7 thetransversedistributionoftheemittedparticles,isex- the “true” collective flow from the genuine correlation
1 pectedtobesensitivetotheearlypressuregradientsand between a large number of particles. It is expected to

therefore to the equation of state (EOS) of the formed give the cleanest values of the genuine collective flow. It
1
fireball in heavy-ion collisions [5, 6, 7, 8, 9, 10]. In addi- is,infact,alsoeasiertoapplytodatathanthecumulant
v
9 tion, the elliptic flowofhighpT particlesis relatedto jet method. This method has been applied in the flow anal-
4 fragmentation and energy loss of the primordially pro- ysis at SIS/GSI [26]. Here it was found that the v1 and
0 ducedhardantiquark-quarkpairwhentravelingthrough v values extracted from this new method do agree with
2
1 thehotQCDmedium[11]. Therefore,understandingthe those from the higher order cumulant method.
0
microscopic nature of the elliptic flow is of great impor- Inthe presentpaper,the Lee-Yangzeroesmethod will
6
tance for the study of the key questions of experimental be tested in the integral and differential v analysis of
0 2
andtheoreticalrelativisticheavyioncollisions,thetrans- UrQMD events at RHIC energy. The UrQMD model
/
h port coefficients and the pressure, namely the equation [32, 33] is well suited for the test: it has been shown in
t
- of state. [27]thatboththe nonfloweffects andthev2 fluctuations
cl To measure flow, experiments usually use the reaction [36], which can affect the flow data, do exist naturally
u plane method [12, 13] or the equivalent two-particle cor- in the model. Most importantly, the true reaction plane
n relationmethod[12,14,15]. However,thesetwo-particle angleΦRisknownbyconstructionintheUrQMDmodel.
v: correlation-based methods suffer from correlations un- This allowsfor the direct calculationofthe exactelliptic
i related to the reaction plane. These additional contri- flow from its basic definition, i.e. v2 = cos2(φ ΦR) .
X h − i
butions are usually dubbed nonflow effects [16], such as Through the present analysis, it will be demonstrated
ar theoveralltransversemomentumconservation,smallan- that the Lee-Yang zeroes method works well for the v2
gle azimuthal correlationsdue to final state interactions, measurements at RHIC energy.
resonance decays, jet production [17] and quantum cor- In order to determine the “integrated” elliptic flow,
relations due to the HBT effect [18]. In order to de- which is defined as
crease the contribution of the nonflow effects to the flow
M
measurements,many-particlecumulantmethodwaspro-
V = ω cos(n(φ Φ )) ,
posed [19]. In this method, usually 4- and 6-particle cu- 2 * j j − R +
j=1
mulants are used to estimate the collective flow. This X
method has been applied to the flow analysis at RHIC,
the generating function of the Lee-Yang zeroes method
SPS and SIS/GSI [14, 15, 20, 21, 22, 23, 24, 25, 26].
is introduced and defined as [29, 30, 31]
AndithasalsobeentestedintheUrQMDmodelin[27].
It was found that the two-particle correlation method is M
stronglyaffectedbynonfloweffectsintheUrQMDmodel, Gθ(ir)= [1+irω cos(φ θ)] .
j j
such as jet production and resonance decays, while the * − +
j=1
Y
4- and 6-particle cumulants give the true v values with
2
betterthan<5%accuracyforthesemi-centralcollisions In the above definitions, M is the number of particles
atRHIC,eventhoughthismethodisaffectedseverelyby used to estimate the integral flow in each event, φ is
j
the rather large v fluctuations for the central and very the particle azimuthalangle,r is a positive realvariable,
2
peripheralcollisions[28]. Morerecently,theLee-Yangze- θ is an arbitrary reference angle and ω is the particle
j
2
weight. Itisworthstressingthatinouranalysis,the1/M
6
weightis usedto decreasethe effect ofevent-by-eventM UrQMD(2.2) Au+Au s1/2=200AGeV
fluctuations. Therefore, V2 = v2 in our analysis. Fig. 1 2.5) 5
showsthe amplitudes ofthe generatingfunction Gθ(ir) <
as a function of r for θ = 0 at different centralit|ies. For| ηs (|| 4
the semi-central bins (e.g. 20-30%), with the increase e
cl 3
rof=r,0,|Gthθe(nirr)e|adchecerseaassehsarrappmidilnyimfruommwthhiechvaisluveeroyfc1loaset parti
to 0. This appearence of the sharp minimum indicates of all 2 v {∞} 1/M w.
that there exists a zero ofthe generatingfunction. After %) 1 v2{∞} unit w.
ifinntdeginrgaltehlleipptoicsifltioownirs0θdoeftetrhmeinfierdstazsero of |Gθ(ir)|, the v(2 0 2 v2
vθ =j /rθ, 0 10 20 30 40 50 60 70
2 01 0
% Most Central
wherej =2.40483isthefirstrootoftheBesselfunction
01
J0(x). The final value of the integral v2 is the average FIG.2: (Coloronline)TheUrQMDmodelresultsforintegral
of v2θ over 5 equally spaced values of θ from 0 to 4π/5, v2 values for Au+Au collisions at √s = 200A GeV. Results
and the statistical uncertainty of the final integral v2 is from the Lee-Yang zeroes method (v2{∞}) with unit weight
about a factor of 2 smaller [30]. and 1/M weight are compared to the exact v2 in different
centrality bins. (see text for details.)
UrQMD(2.2) Au+Au s1/2=200 AGeV 0-5%
1 105--2100%% 12 UrQMD(2.2) v2{∞}
20-30% Au+Au s1/2=200AGeV (20-30%) v2
0.8 30-40% 10
40-50%
θ(ir)| 0.6 50-60% <2.5) 8
|G 0.4 η%) (|| 6
(2 4
0.2 v
2
0
0 50 100 150 200 250 300 0
r 0 2 4 6 8 10
P (GeV/c)
T
θ
FIG.1: (Coloronline) G (ir) asafunctionofr forθ=0at
| |
different centralities. FIG.3: TheUrQMDmodelresultsforv2(pT)insemi-central
(20-30%)Au+Aucollisions at√s=200AGeV.Resultsfrom
Fig. 2 shows the Lee-Yang zeroes results on the cen- the Lee-Yang zeroes method (v2 ) are compared to the
{∞}
trality dependence of the integral v2. For the integral exact v2.
v analysis, all particles in the pseudorapidity region
2
η <2.5areusedandthenumberofparticlesfromevent
| |
toeventfluctuatesineachcentralitybin. Thecentralities whetherthe methodcanbeappliedornot[29,30,31]. If
inouranalysisareselectedaccordingtothesamegeomet- this parameter is too small, i.e., χ 0.5, the statistical
≤
rical fractions of the total cross section (0-5%,5-10%,10- errorwillbetoolarge,andthemethodisinapplicable. χ
20%,20-30%,30-40%,40-50%,50-60%,60-70%) as used by is roughly given by v2√M. Therefore, it will be smaller
the STAR experiment [14]. Here, however, we use im- for the most central bin where v2 is small, and for pe-
pact parameter cuts instead of multiplicity cuts. More ripheral bins where M is small. For the feasibility test
than 1.3 106 minimum bias events are used in the inte- withtheUrQMDmodel,χisabout1inthe semi-central
·
gral v analysis. bin (20-30%), while it is close to 0.5 in the most central
2
The v values extracted from the Lee-Yang zeroes (0-5%) and peripheral bins (>40%). In fact, as is shown
2
method with 1/M weight (solid circles) agrees with the in Fig. 1, the first minimum of Gθ(ir) is not compat-
| |
exact v (open triangles) very well for the semi-central ible with a zero for most central (0-5%) and peripheral
2
bins (about 5-40%), as has been seen in Fig. 2. How- (>40%) bins. Therefore, the Lee-Yang zeroes method
ever, the Lee-Yang zeroes method does not work well can not be used in most central (0-5%) and peripheral
for the most central (0-5%) and peripheral (>40%) bins (>40%) bins from the UrQMD model.
in the present model study: the resolution parameter χ, We have also tried the Lee-Yang zeroes method with
whichisrelatedtotheevent-planeresolution[35],decides unit weight. These results are shown as open circles in
3
Fig. 2. For the semi-central bins (10-40%), it slightly trality bin with better than 2% accuracy, However it is
undershoots the exact v , which indicates that the ze- still necessary to see whether it reproduces the differen-
2
roesmethodwithunitweightisnotcompletelyfreefrom tial v correctly. Especially at large transverse momenta
2
the effect of M fluctuations. However, the effect of M (p ),nonflowcontributionsareexpectedtobelarge. Fig.
T
fluctuations is small. Experimentally, the centrality is 3 and Fig. 4 shows the comparisons of the Lee-Yang ze-
usually selected according to the multiplicity M of the roesresultsforv (p )andv (η)totheexactv values. It
2 T 2 2
events. Inthiscase,theM fluctuationswillbenarrower, isclearthattheresultsfromtheLee-Yangzeroesmethod
and the effect of M fluctuations on the zeroes method agree with the exact v values very well for the whole
2
will be weaker. The present Lee-Yang zeroes analysis p - and η-range. This shows that the Lee-Yang zeroes
T
based on the M selected centralities (not shown in this method indeed reproduced exactly the genuine elliptic
Letter) indeed tells that the integral v values from the flow values for these centrality bins very well.
2
1/M weight and unit weight agree well with each other To summarize, we have applied the Lee-Yang zeroes
and the M fluctuations have almost negligible effects. method in the v analysis of the UrQMD model at
2
200A GeV RHIC Au+Au collisions. Only for the semi-
5 central bins (5-40%), is this method actually applicable.
v {∞}
UrQMD(2.2) 2 Both the integral and differential v from the Lee-Yang
Au+Au s1/2=200AGeV (20-30%) v2 zeroes method agree well with the e2xact values in these
4
bins. The UrQMD model includes many of the nonflow
correlations which are also seen in the real data. Hence,
3
these good agreements show that the Lee-Yang zeroes
%)
(2 method is not sensitive to the nonflow correlations and
v 2 v fluctuationsforthe semi-centralbinsandindeedgives
2
thegenuineellipticflowthere. Itshouldbenotedthatthe
1 current UrQMD model underpredict the RHIC v data
2
by about 40%. Therefore, in the experiments at RHIC
0 or LHC, the zeroes method will work over a somewhat
broader range of centrality bins.
-4 -2 0 2 4
η
FIG. 4: The UrQMD model results for v2(η) in semi-central
Acknowledgments
(20-30%)Au+Aucollisions at√s=200AGeV.Resultsfrom
the Lee-Yang zeroes method (v2 ) are compared to the
{∞}
exact v2. This work was triggered by a discussion with Dr. A.
PoskanzeratLBL.TheauthorsthankDrs. A.Poskanzer,
After determining rθ, the differential v can be esti- J.-Y. Ollitrault and R. Snellings for stimulating discus-
0 2
matedfromEq.(9)in[31]. Forthetestofthedifferential sions. We are grateful to the Center for the Scientific
flow, we use more than 6 105 semi-central events (with Computing (CSC) at Frankfurt for the computing re-
·
impact parameters from 6.7 to 8.3 fm, corresponding to sources. This work was supported by GSI and BMBF.
about 20% to 30% of the total cross section). From the X.Z.thankstheFrankfurtInternationalGraduateSchool
above results on the integral v , we know that the Lee- for Science (FIGSS) at the J. W. Goethe-Universit¨at for
2
Yang zeroes method reproduces the exact v in this cen- financial support.
2
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