Isolation of flow and nonflow correlations by two- and four-particle cumulant measurements of azimuthal harmonics in sNN=200GeV Au+Au collisions

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Isolation

of

flow

and

nonflow

correlations

by

two- and

four-particle

cumulant

measurements

of

azimuthal

harmonics

in

√

s

_NN

=

200 GeV

Au

+

Au

collisions

STAR

Collaboration

N.M. Abdelwahab

be

,

L. Adamczyk

a

,

J.K. Adkins

w

,

G. Agakishiev

u

,

M.M. Aggarwal

ai

,

Z. Ahammed

ba

,

I. Alekseev

s

,

J. Alford

v

,

C.D. Anson

af

,

A. Aparin

u

,

D. Arkhipkin

d

,

E.C. Aschenauer

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,

G.S. Averichev

u

,

A. Banerjee

ba

,

D.R. Beavis

d

,

R. Bellwied

aw

,

A. Bhasin

t

,

A.K. Bhati

ai

,

P. Bhattarai

av

,

J. Bielcik

m

,

J. Bielcikova

n

,

L.C. Bland

d

,

I.G. Bordyuzhin

s

,

W. Borowski

as

,

J. Bouchet

v

,

A.V. Brandin

ad

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S.G. Brovko

f

,

S. Bültmann

ag

,

I. Bunzarov

u

,

T.P. Burton

d

,

J. Butterworth

ao

,

H. Caines

bf

,

M. Calderón de la Barca Sánchez

f

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J.M. Campbell

af

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D. Cebra

f

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R. Cendejas

aj

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M.C. Cervantes

au

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P. Chaloupka

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D.L. Olvitt Jr.

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B.S. Page

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Y. Panebratsev

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*

Correspondingauthor.

E-mailaddress:l.yi@yale.edu(L. Yi). http://dx.doi.org/10.1016/j.physletb.2015.04.033

T. Pawlak

bb

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B. Pawlik

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C. Perkins

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A.M. Poskanzer

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Z.P. Zhang

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F. Zhao

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C. Zhong

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M. Zyzak

o

a_{AGHUniversityofScienceandTechnology,Cracow30-059,Poland} b_{ArgonneNationalLaboratory,Argonne,IL 60439,USA}

c_{UniversityofBirmingham,BirminghamB15 2TT,UnitedKingdom} d_{BrookhavenNationalLaboratory,Upton,NY 11973,USA} e_{UniversityofCalifornia,Berkeley,CA 94720,USA} f_{UniversityofCalifornia,Davis,CA 95616,USA} g_{UniversityofCalifornia,LosAngeles,CA 90095,USA} h_{UniversidadeEstadualdeCampinas,SaoPaulo13131,Brazil} i_{CentralChinaNormalUniversity(HZNU),Wuhan430079,China} j_{UniversityofIllinoisatChicago,Chicago,IL 60607,USA} k_{CracowUniversityofTechnology,Cracow31-155,Poland} l_{CreightonUniversity,Omaha,NE 68178,USA}

m_{CzechTechnicalUniversityinPrague,FNSPE,Prague, 115 19,Czech Republic} n_{NuclearPhysicsInstituteASCR,25068ˇRež/Prague,CzechRepublic} o_{FrankfurtInstituteforAdvancedStudiesFIAS,Frankfurt60438,Germany} p_{InstituteofPhysics,Bhubaneswar751005,India}

q_{IndianInstituteofTechnology,Mumbai400076,India} r_{IndianaUniversity,Bloomington,IN 47408,USA}

s_{AlikhanovInstituteforTheoreticalandExperimentalPhysics,Moscow 117218,Russia} t_{UniversityofJammu,Jammu180001,India}

u_{JointInstituteforNuclearResearch,Dubna,141 980,Russia} v_{KentStateUniversity,Kent,OH 44242,USA}

w_{UniversityofKentucky,Lexington,KY,40506-0055,USA} x

KoreaInstituteofScienceandTechnologyInformation,Daejeon 305-701,Republic of Korea y_{InstituteofModernPhysics,Lanzhou730000,China}

z_{LawrenceBerkeleyNationalLaboratory,Berkeley,CA 94720,USA} aa_{MassachusettsInstituteofTechnology,Cambridge,MA 02139-4307,USA} ab_{Max-Planck-InstitutfurPhysik,Munich 80805,Germany}

ac_{MichiganStateUniversity,EastLansing,MI 48824,USA} ad_{MoscowEngineeringPhysicsInstitute,Moscow115409,Russia}

ae_{NationalInstituteofScienceEducationandResearch,Bhubaneswar751005,India} af_{OhioStateUniversity,Columbus,OH 43210,USA}

ag_{OldDominionUniversity,Norfolk,VA 23529,USA} ah_{InstituteofNuclearPhysicsPAN,Cracow31-342,Poland} ai_{PanjabUniversity,Chandigarh160014,India}

aj_{PennsylvaniaStateUniversity,UniversityPark,PA 16802,USA} ak_{InstituteofHighEnergyPhysics,Protvino142281,Russia} al_{PurdueUniversity,WestLafayette,IN 47907,USA} am_{PusanNationalUniversity,Pusan609735,RepublicofKorea} an_{UniversityofRajasthan,Jaipur302004,India}

ao_{RiceUniversity,Houston,TX 77251,USA}

aq_{ShandongUniversity,Jinan,Shandong250100,China} ar_{ShanghaiInstituteofAppliedPhysics,Shanghai201800,China} as_{SUBATECH,Nantes44307,France}

at_{TempleUniversity,Philadelphia,PA 19122,USA} au_{TexasA&MUniversity,CollegeStation,TX 77843,USA} av_{UniversityofTexas,Austin,TX 78712,USA} aw_{UniversityofHouston,Houston,TX 77204,USA} ax_{TsinghuaUniversity,Beijing100084,China}

ay_{UnitedStatesNavalAcademy,Annapolis,MD,21402,USA} az_{ValparaisoUniversity,Valparaiso,IN 46383,USA} ba_{VariableEnergyCyclotronCentre,Kolkata700064,India} bb_{WarsawUniversityofTechnology,Warsaw00-661,Poland} bc_{UniversityofWashington,Seattle,WA 98195,USA} bd_{WayneStateUniversity,Detroit,MI 48201,USA}

be_{WorldLaboratoryforCosmologyandParticlePhysics(WLCAPP),Cairo 11571,Egypt} bf_{YaleUniversity,NewHaven,CT 06520,USA}

bg_{UniversityofZagreb,Zagreb,HR-10002,Croatia}

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Articlehistory:

Received6September2014

Receivedinrevisedform24February2015 Accepted19April2015

Availableonline21April2015 Editor: V.Metag

Keywords: Heavy-ion Flow Nonflow

Adata-drivenmethodwasappliedtoAu₊Aucollisionsat√sNN=200 GeV madewiththeSTAR

detec-toratRHICtoisolatepseudorapiditydistance

η

-dependentand

η

-independentcorrelationsbyusing two- andfour-particleazimuthalcumulantmeasurements.Weidentifieda

η

-independentcomponent ofthecorrelation,whichisdominatedbyanisotropicflowandflowfluctuations.Itwasalsofoundtobe independentof

η

withinthemeasuredrangeofpseudorapidity|

η

|<1.In20–30%centralAu+Au colli-sions,therelativeflowfluctuationwasfoundtobe34%±2%(stat.)±3%(sys.)forparticleswithtransverse momentum pT less than2 GeV/c.The

η

-dependentpart,attributedtononflowcorrelations,isfound

tobe5%±2%(sys.)relativetotheflowofthemeasuredsecondharmoniccumulantat|

η

|>0.7.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Heavy-ioncollisionsatultra-relativisticenergiesasproducedat theRelativisticHeavyIonCollider(RHIC)andattheLargeHadron Collider (LHC) provide means to study the Quark Gluon Plasma (QGP). In non-central collisions, the overlapregion of the collid-ingnucleiisanisotropic.Theenergydensitygradientconvertsthe initialcoordinate-spaceanisotropyintothefinalmomentum-space anisotropy, generally called anisotropic flow. As the system ex-pands,thecoordinate-spaceanisotropydiminishes.Hence, a mea-surement of flow is most sensitive to properties of the system at the early stage of the collision [1]. Through measurements of anisotropic flowandcomparisonwithhydrodynamic calculations, propertiesoftheearlystageofthecollisionmaybeextracted.One ofthe importantvariables,the ratioofthe shearviscosityto en-tropy densityof theQGP, was found to be not muchlarger than theconjecturedquantumlimitof1/4

π

[2].

Themomentum-spaceanisotropicflowcanbecharacterizedby Fouriercoefficients, vn,oftheoutgoingparticleazimuthal(φ) dis-tribution[3]: dN d

φ

∝

1

+

∞



n=1 2vncos n

(φ

− ψ

n

),

(1)

wheretheparticipantplaneischaracterizedbytheangle

ψ

n,given bytheinitialparticipantnucleon(orparton)configuration[4].The higherharmonicscanarisefrominitialfluctuationssuchthat

ψ

n is notnecessarilythesamefordifferentn.Because

ψ

n isnot experi-mentallyaccessible,theeventplane,constructedfromfinalparticle momenta,isusedasaproxyfortheinitial stateparticipantplane. Thedeterminationoftheanisotropicflowusesparticlecorrelations that are, however, contaminated by intrinsic particle correlations unrelatedtotheparticipantplane.Thosecorrelationsaregenerally called nonflow and are due to jet fragmentation and final state

interactions,suchasquantumstatistics,Coulombandstrong inter-actions,andresonancedecays[5].

Similarly, two- andmulti-particlecorrelationsarealsoused to measureanisotropy[6,5].Forexample,thetwo-particlecorrelation isgivenby: dN d

φ

∝

1

+

∞



n=1 2Vn

{

2

}

cos n

φ,

(2)

where

φ

is the azimuthal angle between the two particles. In theabsenceofnonflow,Eq.(2)followsfromEq.(1)withVn

{

2

}

=

vn,α vn,β (where

α

,

β

stand for the two particles). Otherwise, Vn

{

2

}

=

vn,α vn,β

+ δ

n,where

δ

n isthenonflowcontribution.Since evenasmalluncertaintyinflowcanintroducealargeerrorinthe extracted shear viscosity[7], it isimportant to separate nonflow contributionsfromflowmeasurements.

Thisarticledescribesamethodusedtoseparateflowand non-flow in adata-driven way,withminimal relianceon models. We measuretwo- andfour-particlecumulantswithdifferent pseudora-pidity(

η

)combinations.Byexploitingthesymmetryoftheaverage flowin

η

atmidrapidityinsymmetricheavy-ioncollisions,we sep-arate



η

-independentand



η

-dependentcontributions.We asso-ciatethe



η

-independentpartwithflow,whilethe



η

-dependent partisassociatedwithnonflow.Thisisbecauseflow isan event-wise many-particle azimuthal correlation,reflecting propertieson thesingle-particle level[1].Bycontrast,nonflow isafew-particle azimuthal correlation that depends on the



η

distancebetween theparticles.

2. Data analysis

2.1.Experimentdetailsanddataselection

This analysis principally relies on the STAR Time Projection Chamber (TPC) [8]. A total number of 25 million Au

+

Au colli-sionsat

√

sNN

=

200 GeV,collected withaminimum biastrigger in2004, were used.The eventsselected were requiredto havea primaryeventvertexwithin

|

zvtx

|

<

30 cm alongthebeamaxis (z) toensure nearly uniformdetector acceptance.The centrality def-initionwas basedon therawchargedparticlemultiplicity within

|

η

|

<

0.5 inTPC. The chargedparticle tracksused intheanalysis were required to satisfy the following conditions: the transverse momentum0.15

<

pT

<

2 GeV/c toremovehigh pT particles orig-inatingfromthejets;thedistanceofclosestapproachtotheevent vertex

|

dca

|

<

3 cm to ensurethat theparticlesare fromthe pri-mary collision vertex; the number of fit points along the track greater than 20out of45 maximum hit points, and the ratioof the numberof fit points along the track to the maximum num-berof possiblefit points largerthan 0.51forgoodprimary track reconstruction[9].Forthe particles usedinthispaper, the pseu-dorapidityregionwasrestrictedto

|

η

|

<

1.

2.2.Analysismethod

Inthisanalysis, theazimuthalanisotropyofthefinalstate par-ticles was calculated by the two- and four-particle Q-cumulant methodusingunit weightwithanon-uniformacceptance correc-tion [10]. By using the moment of the flow vector, thismethod makes multi-particle cumulant calculation faster without going overpairorahighermultiplet loop.Thenon-uniformacceptance correctionfor20–30%centralitywas0.7%forthesecondharmonic two-particlecumulant V2

{

2

}

,and0.5%forthe square rootofthe second harmonic four-particle cumulant V₂1/2

{

4

}

. The largest ac-ceptancecorrectionwas1.8%forV2

{

2

}

atthemostcentral,and1% forV₂1/2

{

4

}

atthemostperipheralcollisions.

Thetwo-particlecumulant,withoneparticleatpseudorapidity

η

α andanotherat

η

β,is[11]: V

{

2

} ≡ 

ei(φα−φβ)

 = 

_v

₍

_η

α

)

v

(

η

β)

 + δ(

η

)

≡ 

v

(

η

α

)



v

(

η

β)

 +

σ

(

η

α

)

σ

(

η

β

)

+

σ

(

η

)

+ δ(

η

,

η

α

,

η

β

),

(3)

where



η

= |

ηβ

−

η

α

|

.The doublebracketsrepresenttheaverage over particlepairs and the average over events,while the single bracketsarefortheaverageovereventsonly.Theharmonic num-ber n issuppressedtolightenthenotation.Theaverageflow,



v



, whichistheanisotropy parameterwithrespectto theparticipant plane,andtheflowfluctuation,

σ

,areonlyfunctionsof

η

,because flowreflectsthepropertyonthesingle-particlelevel.Both



v



and

σ

are



η

-independent quantities. However, because of the way the two-particle cumulant is measured, i.e., by two-particle cor-relation,therecould exista



η

-dependentflow fluctuation com-ponent. For example, the event planes determined by particles atdifferent

η

’s can be different [12]. In Eq. (3),

σ

denotes this



η

-dependentpartoftheflowfluctuation.The

δ

isthe contribu-tionfromnonflow,which isgenerallya functionof



η

,butmay alsodependon

η

.Forsimplicity,wewriteitintheformof

δ(

η

)

inthediscussionbelow.

Forthefour-particlecumulant,wetaketwoparticlesat

η

α and anothertwoat

η

β.Foreasierinterpretationoftheresults,wetake

thesquarerootofthefour-particlecumulant,whichhasthesame orderin



v



asthetwo-particlecumulant(itisjustthesame ob-servableasdiscussedinthereferences[16,17]).Itisgivenby:

V12

{

₄

} ≡





ei(φα+φα−φβ−φβ)



≈ 

v

(

η

α

)



v

(

η

β

)

 −

σ

(

η

α

)

σ

(

η

β

)

−

σ

(

η

),

(4)

wherethe approximationis thatthe flow fluctuationisrelatively smallcompared withthe average flow [13]. In V1/2

{

4

}

,the con-tributionfromthetwo-particlecorrelationsduetononfloweffects issuppressed,whilethecontributionfromthefour-particle corre-lations duetononflow effects

∝

1/M3 (M is multiplicity)andis, therefore,negligible[14,15].The fluctuationgivesnegative contri-butiontoV1/2

_{

₄

_}

_{,whilepositivetoV}

_{

₂

_}

_.

The two- andfour-particle cumulantswere measured for var-ious

(

η

α

,

η

β

)

pairs and quadruplets. Fig. 1 showsthe results for

20–30%central Au

+

Au collisions.Panels (a)and(b)are the two-particlesecond andthird harmoniccumulants,V2

{

2

}(

η

α

,

η

β

)

and V3

{

2

}(

η

α

,

η

β

),

respectively. Panel (c) is the square root of the

four-particle second harmonic cumulant, V₂1/2

{

4

}(

η

α

,

η

α

,

η

β

,

η

β

).

We observefromFig. 1that V2

{

2

}

decreases asthegapbetween

η

α and

η

β increases. Since the trackmerging (two particles

be-ing identified as one track) affects the region

|

η

|

<

0.05, the Vn

{

2

}

and Vn

{

4

}

points along the diagonal were excluded from furtheranalysis. V3

{

2

}

followsthesametrend,butthemagnitude issmaller.V3

{

2

}

decreasesmorerapidlywith



η

thanV2

{

2

}

does. V₂1/2

{

4

}

isroughlyconstantandthemagnitudeissmallerthanthat of V2

{

2

}

whichisconsistent withourexpectationthat V21/2

{

4

}

is lessaffectedbythenonflowandtheflowfluctuationisnegativein V₂1/2

{

4

}

.

In order to extract the values of the average flow,



v



, the



η

-dependent and



η

-independentflow fluctuations,

σ

and

σ

, andthenonflowcontribution,

δ,

wefollowananalysismethod de-scribedinRef.[16].

By takingthe difference between cumulants V

{

2

}

at

(

η

α

,

η

β

)

and

(

η

α

,

−

ηβ

),

wehave



V

{

2

} ≡

V

{

2

}(

η

α

,

η

β

)

−

V

{

2

}(

η

α

,

−

η

β

)

≡

V

{

2

}(

η

1

)

−

V

{

2

}(

η

2

)

= 

σ

+ δ,

(5) where

η

α

<

ηβ

<

0 or0

<

ηβ

<

η

α isrequired.Similarly,this dif-ferenceforV1/2

_{

₄

_}

_yields



V12

{

₄

} ≡

_V12

{

₄

}(

_η

_α

_,

_η

_β)

−

_V12

{

₄

}(

_η

_α

_,

−

_η

_β

₎

≡

V12

{

₄

}(

η

₁

₎

−

_V12

{

₄

}(

η

₂

₎

≈ −

σ

_.

₍₆₎

Here



η

1

≡

ηβ

−

η

α ,



η

2

≡ −

ηβ

−

η

α ,



σ

=

σ

(

η

1

)

−

σ

(

η

2

),

and

δ

= δ(

η

1

)

− δ(

η

2

).

Insymmetricheavy-ioncollisions,the difference of the two



η

-independent terms in Eqs. (3) and (4) iszero.Therefore thedifferencesinEqs. (5)and(6)depend only on the



η

-dependent terms: flow fluctuation



σ

and nonflow correlations

δ.

Ourgoalistoparameterizetheflowfluctuation



σ

and non-flow

δ.

The following part of this section is organized in this way:First,wediscusstheempiricalfunctionalformfor

D

(

η

)

=

σ

(

η

)

+ δ(

η

),

(7)

obtained from



V2

{

2

}

data.Second, we give the

σ

result from



V₂1/2

{

4

}

.UsingD and

σ

,

δ

canbedetermined.Third,wediscuss howtoobtain



v



and

σ

.

Fig. 1. Thesecond(a)andthird(b)harmonictwo-particlecumulantsfor(ηα,ηβ)pairsandthesecondharmonicfour-particlecumulantfor(ηα,ηα,ηβ,ηβ)quadrupletsfor 20–30%centralAu+Aucollisionsat√sNN=200 GeV.

Fig. 2. The(a)V2{2}and(b)V3{2}differencebetweenthepairsat(ηα,ηβ)and(ηα,−ηβ).Thedashedlinesarelinearfitsforeachdatasetofη1valueseparately.The solidcurvesareasinglefitofEq.(8)toalldatapointswithdifferentη1.(c)TheV21/2{4}differencebetweenquadrupletsat(ηα,ηα,ηβ,ηβ)and(ηα,ηα,−ηβ,−ηβ).The dashedlineisalinearfittothedatapoints.Thegraybandisthesystematicerror.Thedataarefrom20–30%centralAu+Aucollisionsat√sNN=200 GeV.



V

{

2

} =

D

(

η

1

)

−

D

(

η

2

)

=



a exp



−

η

1 b



+

A exp



−

η

2 1 2

σ

2



−



a exp



−

η

2 b



+

A exp



−

η

2 2 2

σ

2



,

(9)

followsfrom Eq.(5).Here is howthis functionalformis chosen. The measured two-particle second harmonic cumulantdifference



V2

{

2

}

is shown in Fig. 2(a). The data for each



η

1 value ap-pearstobelinearin



η

2

− 

η

1exceptnear



η

1

= 

η

2asshown by dashed lines inFig. 2(a) and(b).Moreover, the magnitude of



V2

{

2

}

decreases with increasing



η

1. Linear fits indicate that theinterceptdecreasesexponentiallywithincreasing



η

1,andthe slopesareallsimilar.Sowecandescribethisbehavior mathemat-icallyas a exp



−



η

1 b



+

k

(

η

2

− 

η

1

).

(10)

Inordertoexpressthemeasuredtwo-particlecumulantdifference intheformof

D

(

η

1

)

−

D

(

η

2

)

= [

σ

(

η

1

)

+ δ(

η

1

)

] − [

σ

(

η

2

)

+ δ(

η

2

)

]

= [

σ

(

η

1

)

−

σ

(

η

2

)

] + [δ(

η

1

)

− δ(

η

2

)

],

(11) we make two improvements to our initial guess of the D

(

η

)

function. First, we add a termaexp(

−

η2

b

)

that is small for all datawith



η

2 significantly largerthan



η

1.Second,becausethe lineartermisunboundedin



η

1 and



η

2,we choosetoreplace itwiththesubtractionoftwo wideGaussianterms.TheGaussian functionstendto zeroasthe exponentsbecome large,consistent withthe behavior of nonflow. The measured two-particle cumu-lant difference can then be described by Eq. (9). There are four

parameters inEq.(9), a,A

,

b,and

σ

,thatweredeterminedby fit-tingEq.(9)simultaneouslytoallmeasuredtwo-particlecumulant differencedatapointsofdifferent



η

1.Thefitresultsareshownin Fig. 2(a)asthe solid curveswith

χ

2

_/ndf

_≈

_1.The

_χ

2

_{/ndf values}

are about1 for all centrality classes exceptfor the most central it is about2.In the mostcentral collisions, thelargest contribu-tion to

χ

2

_{/ndf comes}

_{frompairs both atacceptance edgeof the} STAR TPC. The parameterizationis validwithin the fittingerrors. ThesameprocedurewasrepeatedforthethirdharmonicV3

{

2

}

as shownin Fig. 2(b).Thefitresults givethe



η

-dependentpartof the two-particlecumulantasEq.(8).Thus, theformofthe func-tion D isdata-driven.

Wethenfollowasimilarprocedureonthemeasureddifference of the square root of the four-particle cumulant, Eq. (4). We fit the



V₂1/2

{

4

}

=

σ

(

η

1

)

−

σ

(

η

2

)

byalinearfunctionk

(

η

2

−



η

1

),

asshown in Fig. 2(c). The slope k from the fit is

(1.1

±

0.8)

×

10−4.InFig. 2(c),eachdatapointistheaverageof



V₂1/2

{

4

}

forall



η

1 atsame



η

2

− 

η

1value.Withthe

σ

(

η

)

result,the contribution from nonflow,

δ,

can then also be determined from Eq.(7).

Subtracting theparameterized D ofEq.(8)fromthemeasured two-particlecumulants, V2

{

2

}

and V3

{

2

}

,yields,fromEq.(3),the



η

-independentterms



v2



≡ 

v



2

+

σ

2_{.Employingalso V}1/2

_{

₄

_}

from Eq.(4),the valuesof



v



and

σ

maybe individually deter-mined.

2.3. Systematicuncertainties

The systematicerrors for V

{

2

}

and V1/2

_{

₄

_}

_{are estimated by} varying eventandtrackquality cuts:theprimary eventvertexto

|

zvtx

|

<

25 cm; the number of fit points along the track greater than 15; the distance of closest approach to the event vertex

Fig. 3. Thedecomposedv2_{= v}2_+σ2_{forthesecond (a)andthird (b)harmonicsfor}

(ηα,ηβ)pairs.(c): Thetwo- andfour-particlecumulants,V2{2}(solidredsquares) andV1/22 {4}(solidbluetriangles),andthedecomposedv22(solidgreendots)asafunctionofηforoneparticlewhileaveragedoverηofthepartnerparticle.Thecyan bandontopofV21/2{4}pointspresentV

1/2

2 {4}+σ .(d): V3{2}(solidredsquares)andv23(solidgreendots)asafunctionofη.Thedashedlinesarethemeanvalue averagedoverηfor20–30%centralAu+Aucollisionsat√sNN=200 GeV.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtotheweb versionofthisarticle.)

The fitting error on the parameterized

σ

from



V1/2

_{

₄

_}

_is treatedasasystematicerror,whichis70%, since

σ

isconsistent withzeroinlessthan2-

σ

standarddeviation.Similarly,thefitting errors on the parameters used in the



η

-dependent correlation D aretreatedassystematicerrorsthatare propagatedthroughto thetotaluncertaintyonD.Inaddition,thereisasystematicerror on D thatisassociatedwiththechoice offittingfunctionshown asEq.(8),the magnitudeofwhichwas estimatedusingdifferent formsof thefitting function. Theforms tried included: an expo-nentialtermplus alinearterm, aGaussian functionplus alinear term,an exponentialfunction only,a Gaussianfunction only,and anexponentialfunctionplusatermoftheforme−12(

η σ )4. The total estimated uncertainty in the second harmonic of D

(

η

)

isan average of40% based onthe differentsources eval-uated.Thissystematicerroron D alsoappliestothedecomposed flowthrough



v2

_

₌

_V

_{

₂

_}

₋

_D.

3. Results and discussion

Fig. 3(a) and (b) shows thedecomposed flow with flow fluc-tuations



v

(

η

α

)

v

(

ηβ

)



(see Eq. (3)) for v2 and v3, respectively. The results are found to be independent of

η

for the measured pseudorapidity range

|

η

|

<

1. The observed decrease of V

{

2

}

in Fig. 1withincreasing



η

offdiagonalisduetocontributionsfrom nonflow and



η

-dependent fluctuations. Note that the analysis methoddoesnot make any assumptionabout the

η

dependence offlow; the flow can be



η

-independent but

η

-dependent. The observation that the decomposed flow and flow fluctuations are independentof

η

is,therefore,significant.

Fig. 3(c) and (d) shows the projections of



v

(

η

α

)

v

(

ηβ

)



in

Fig. 3(a) and (b) onto one

η

variable. The shaded band shows the systematic uncertainty, dominated by the systematic errors inthe subtracted D

(

η

)

term. For comparison, theprojection of the V2

{

2

}

is also shown, where the shaded band is the system-aticuncertainty.The projectionsare therespectivequantitiesasa functionof

η

ofoneparticleaveragedoverall

η

oftheother par-ticle.Theflowswith



η

-independentfluctuationaveragedover

η

are





v2₂

 =

6.27%

±

0.003%(stat

.)

₋+₀0._.08₀₇%(sys

.)

and





v2₃

 =

1.78%

±

0.008%(stat

.)

+₋0₀._.09₁₆%(sys

.)

forour pT range0.15

<

pT

<

2 GeV/c in the20–30%collisioncentralityrange.Thequoted statisticalerrors arefromthe V

{

2

}

measurements, whilethesystematicerrorsare dominatedby the parameterizationof D. Thedifference between V

{

2

}

and



v2



inFig. 3(c)isthe D

(

η

)

versus

η

ofoneparticle av-eragedoverall

η

oftheotherparticle.

Fig. 3(c) also shows the V₂1/2

{

4

}

projection as a function of

η

as the solid blue triangles. V₂1/2

{

4

}

is also independent of

η

. Thecyan bandshowsV₂1/2

{

4

}

+

σ

= 

v



2

−

σ

2_,withthe system-atic uncertainty that is dominated by the fitting uncertainty in

σ

.Thedifferencebetweenthedecomposed



v2

_

_{= }

_v

_

2

₊

_σ

2 _and V₂1/2

{

4

}

+

σ

= 

v



2

₋

_σ

2_{istheflowfluctuation,whichisalso} inde-pendentof

η

withinthemeasuredacceptance.Therelativeelliptic flowfluctuationisgivenby

σ

2



v2



=











v22

 − (

V 1 2 2

{

4

} +

σ

)



v2 2

 + (

V 1 2 2

{

4

} +

σ

)

=

34%

±

2%

(

stat

.)

±

3%

(

sys

.),

(12)

wherethesystematicerrorisdominatedbythoseinthe parame-terizationofD and

σ

.Themeasuredrelativefluctuationis consis-tentwiththatfromthePHOBOSexperiment[18]andtheprevious STARupperlimitmeasurement[19].

Often, a



η

-gap is applied to reduce nonflow contamination inflow measurements. Thenonflow D

¯

(

|

η

|)

withthe



η

-gapis calculatedas:

¯

D

(

|

η

|) =

2 |η|d



η

D

(

η

)

2

− |

η

|

.

(13)

Fig. 4. Theη-dependentcomponentofthetwo-particlecumulantwithη-gap,D in¯ Eq.(13),ofthesecond (a)andthird (b)harmonicsisshownasafunctionofη-gap

|η|>x.(x isthex-axisvalue.)Theshadedbandsaresystematicuncertainties.In(a)theestimatedσ isindicatedasthestraightline,withitsuncertaintyof±1 standard deviationasthecross-hatchedareafor20–30%centralAu+Aucollisionsat√sNN=200 GeV.

Fig. 5. Thenonflow,D¯2(soliddots),√δ2(openstars),

 ¯

D3(solidtriangles)andflow,

 v2

2/2 (opencircles),

 v2

3(opentriangles)resultsareshownasafunctionof centralitypercentileforthesecond (a)andthird (b)harmonics,respectively.Thestatisticalerrorsaresmallerthanthesymbolsizes.Thesystematicerrorsaredenotedby theverticalrectangles.

value)forthesecondandthirdharmonics,respectively.Thebands arethesystematicerrorsestimatedfromthefittingerrorsandthe differentfittingfunctionsasdescribedpreviously.Theseerrorsare correlatedbecause,forallthepoints shown,theerrors are calcu-latedfromthesameparametersinthefunction D.

Asnotedabove,D

¯

(

|

η

|)

iscomprisedoftwoparts:the contri-butionfromthe



η

-dependentflowfluctuation,

σ

,andtheterm representing the nonflow,

δ.

In Fig. 4(a), these two contributors areestimatedseparately.Thestraightlineisanestimateof

σ

.The cross-hatchedareaisitsuncertaintyof

±

1 standarddeviation.The differencebetweentheblacksolidpoints D

¯

(

|

η

|)

andthestraight line

σ

isthenonflow contribution.Forboththesecondharmonic andthe third harmonic shownin Fig. 4(a)and Fig. 4(b), respec-tively, D

¯

(

|

η

|)

decreases as the



η

-gap between two particles increases.When

|

η

|

>

0.6,D

¯

(

|

η

|)

isreducedtohalfofitsvalue when

|

η

|

>

0.

Fig. 5 shows





v2

_

_and

√

_{D for}

¯

_{all measured centralities for} the second harmonic (a) and the third harmonic (b).

|

η

|

>

0.7 [20] is used to present the D result.

¯

The errors on





v2

_

_and

√

¯

D areanti-correlated. Taking

|

η

|

>

0.7,therelativemagnitude

¯

D2

/



v2₂



=

5%

±

0.004%(stat

.)

±

2%(sys

.)

for20–30%centrality.Itis clearthat D

¯

2 increasesasthecollisionsbecomemoreperipheral.

The



η

-dependent nonflow contribution is mainly caused by near-side (small

φ)

correlations. These correlationsinclude jet-like correlations and resonance decays which decrease with in-creasing



η

. The



η

-independent correlation is dominated by anisotropicflow.However,thereshouldbea



η

-independent con-tribution fromnonflow,such asaway-side dijetcorrelations. This contributionshouldbesmallerthanthenear-sidenonflow contri-bution,because,inpart,someoftheaway-sidejetsareoutsidethe acceptanceand,therefore,undetected[21].

Fig. 6 shows

σ

2

/



v2



for all measured centralities. From the central to theperipheral collisions, the relative ellipticflow

fluc-Fig. 6. The relative elliptic flow fluctuation σ2/v2 centrality dependence in

√

sNN=200 GeV Au+Aucollisions. Thestatisticalerrors areshownbytheerror bars.Thesystematicerrorsaredenotedbytheverticalrectangles.

tuation slightly increases. The statistics are limited in the most peripheralcentralitybin.

4. Summary

We have analyzed two- and four-particle cumulantazimuthal anisotropies between pseudorapidity bins in Au

+

Au collisions

at

√

sNN

=

200 GeV from STAR. The



η

-dependent and the

Any comparisonwith flow data to extract the ratio ofthe shear viscositytoentropydensityandtodeterminetheinitialcondition shouldtake intoaccountnonflowcontaminationinflow measure-ment.

Acknowledgements

We thank the RHIC Operations Group and RCF at BNL, the

NERSC Center atLBNL, the KISTI Center in Korea, and the Open ScienceGridconsortiumforprovidingresourcesandsupport.This workwas supportedinpartby theOfficesofNPandHEP within theU.S. DOE Officeof Science,the U.S. NSF, CNRS/IN2P3,FAPESP CNPqofBrazil, theMinistryofEducation andScience ofthe Rus-sianFederation, NNSFC,CAS,MoSTandMoEofChina,the Korean ResearchFoundation,GAandMSMToftheCzechRepublic,FIASof Germany,DAE,DST,andCSIRofIndia,theNationalScienceCentre of Poland, National Research Foundation (NRF-2012004024), the MinistryofScience,EducationandSportsoftheRepublicof Croa-tia,andRosAtomofRussia.

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