Correlation measurements between flow harmonics in Au+Au collisions at RHIC

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Physics

Letters

B

www.elsevier.com/locate/physletb

Correlation

measurements

between

flow

harmonics

in

Au

+

Au

collisions

at

RHIC

STAR

Collaboration

J. Adam

i

,

L. Adamczyk

a

,

J.R. Adams

ad

,

J.K. Adkins

t

,

G. Agakishiev

r

,

M.M. Aggarwal

af

,

Z. Ahammed

bc

,

N.N. Ajitanand

aq

,

I. Alekseev

p

,

aa

,

D.M. Anderson

as

,

R. Aoyama

aw

,

A. Aparin

r

,

D. Arkhipkin

c

,

E.C. Aschenauer

c

,

M.U. Ashraf

av

,

F. Atetalla

s

,

A. Attri

af

,

G.S. Averichev

r

,

X. Bai

g

,

V. Bairathi

ab

,

K. Barish

ay

,

A.J. Bassill

ay

,

A. Behera

aq

,

R. Bellwied

au

,

A. Bhasin

q

,

A.K. Bhati

af

,

P. Bhattarai

at

,

J. Bielcik

j

,

J. Bielcikova

k

,

L.C. Bland

c

,

I.G. Bordyuzhin

p

,

J. Bouchet

s

,

J.D. Brandenburg

ak

,

A.V. Brandin

aa

,

D. Brown

x

,

J. Bryslawskyj

ay

,

I. Bunzarov

r

,

J. Butterworth

ak

,

H. Caines

bf

,

M. Calderón de la Barca Sánchez

e

,

J.M. Campbell

ad

,

D. Cebra

e

,

I. Chakaberia

c

,

s

,

ao

,

P. Chaloupka

j

,

F.-H. Chang

ac

,

Z. Chang

c

,

N. Chankova-Bunzarova

r

,

A. Chatterjee

bc

,

S. Chattopadhyay

bc

,

J.H. Chen

ap

,

X. Chen

v

,

X. Chen

an

,

J. Cheng

av

,

M. Cherney

i

,

W. Christie

c

,

G. Contin

w

,

H.J. Crawford

d

,

S. Das

g

,

T.G. Dedovich

r

,

I.M. Deppner

az

,

A.A. Derevschikov

ah

,

L. Didenko

c

,

C. Dilks

ag

,

X. Dong

w

,

J.L. Drachenberg

u

,

J.C. Dunlop

c

,

L.G. Efimov

r

,

N. Elsey

be

,

J. Engelage

d

,

G. Eppley

ak

,

R. Esha

f

,

S. Esumi

aw

,

O. Evdokimov

h

,

J. Ewigleben

x

,

O. Eyser

c

,

R. Fatemi

t

,

S. Fazio

c

,

P. Federic

k

,

P. Federicova

j

,

J. Fedorisin

r

,

Z. Feng

g

,

P. Filip

r

,

E. Finch

ax

,

Y. Fisyak

c

,

C.E. Flores

e

,

L. Fulek

a

,

C.A. Gagliardi

as

,

F. Geurts

ak

,

A. Gibson

bb

,

D. Grosnick

bb

,

D.S. Gunarathne

ar

,

Y. Guo

s

,

A. Gupta

q

,

W. Guryn

c

,

A.I. Hamad

s

,

A. Hamed

as

,

A. Harlenderova

j

,

J.W. Harris

bf

,

L. He

ai

,

S. Heppelmann

ag

,

S. Heppelmann

e

,

N. Herrmann

az

,

A. Hirsch

ai

,

L. Holub

j

,

S. Horvat

bf

,

X. Huang

av

,

B. Huang

h

,

S.L. Huang

aq

,

T. Huang

ac

,

H.Z. Huang

f

,

T.J. Humanic

ad

,

P. Huo

aq

,

G. Igo

f

,

W.W. Jacobs

o

,

A. Jentsch

at

,

J. Jia

c

,

aq

,

K. Jiang

an

,

S. Jowzaee

be

,

E.G. Judd

d

,

S. Kabana

s

,

D. Kalinkin

o

,

K. Kang

av

,

D. Kapukchyan

ay

,

K. Kauder

be

,

H.W. Ke

c

,

D. Keane

s

,

A. Kechechyan

r

,

D.P. Kikoła

bd

,

C. Kim

ay

,

T.A. Kinghorn

e

,

I. Kisel

l

,

A. Kisiel

bd

,

L. Kochenda

aa

,

L.K. Kosarzewski

bd

,

A.F. Kraishan

ar

,

L. Kramarik

j

,

L. Krauth

ay

,

P. Kravtsov

aa

,

K. Krueger

b

,

N. Kulathunga

au

,

S. Kumar

af

,

L. Kumar

af

,

J. Kvapil

j

,

J.H. Kwasizur

o

,

R. Lacey

aq

,

J.M. Landgraf

c

,

K.D. Landry

f

,

J. Lauret

c

,

A. Lebedev

c

,

R. Lednicky

r

,

J.H. Lee

c

,

Y. Li

av

,

W. Li

ap

,

X. Li

an

,

C. Li

an

,

J. Lidrych

j

,

T. Lin

as

,

M.A. Lisa

ad

,

Y. Liu

as

,

H. Liu

o

,

F. Liu

g

,

P. Liu

aq

,

T. Ljubicic

c

,

W.J. Llope

be

,

M. Lomnitz

w

,

R.S. Longacre

c

,

S. Luo

h

,

X. Luo

g

,

R. Ma

c

,

Y.G. Ma

ap

,

G.L. Ma

ap

,

L. Ma

m

,

N. Magdy

aq

,

R. Majka

bf

,

D. Mallick

ab

,

S. Margetis

s

,

C. Markert

at

,

H.S. Matis

w

,

O. Matonoha

j

,

D. Mayes

ay

,

J.A. Mazer

al

,

K. Meehan

e

,

J.C. Mei

ao

,

N.G. Minaev

ah

,

S. Mioduszewski

as

,

D. Mishra

ab

,

S. Mizuno

w

,

B. Mohanty

ab

,

M.M. Mondal

n

,

I. Mooney

be

,

D.A. Morozov

ah

,

M.K. Mustafa

w

,

Md. Nasim

f

,

T.K. Nayak

bc

,

J.D. Negrete

ay

,

J.M. Nelson

d

,

D.B. Nemes

bf

,

M. Nie

ap

,

G. Nigmatkulov

aa

,

T. Niida

be

,

L.V. Nogach

ah

,

T. Nonaka

aw

,

S.B. Nurushev

ah

,

G. Odyniec

w

,

A. Ogawa

c

,

K. Oh

aj

,

V.A. Okorokov

aa

,

D. Olvitt Jr.

ar

,

B.S. Page

c

,

R. Pak

c

,

Y. Panebratsev

r

,

B. Pawlik

ae

,

H. Pei

g

,

E-mailaddress:mnasim2008@gmail.com(Md. Nasim).

https://doi.org/10.1016/j.physletb.2018.05.076

C. Perkins

d

,

J. Pluta

bd

,

K. Poniatowska

bd

,

J. Porter

w

,

M. Posik

ar

,

N.K. Pruthi

af

,

M. Przybycien

a

,

J. Putschke

be

,

A. Quintero

ar

,

S.K. Radhakrishnan

w

,

S. Ramachandran

t

,

R.L. Ray

at

,

R. Reed

x

,

H.G. Ritter

w

,

J.B. Roberts

ak

,

O.V. Rogachevskiy

r

,

J.L. Romero

e

,

L. Ruan

c

,

J. Rusnak

k

,

O. Rusnakova

j

,

N.R. Sahoo

as

,

P.K. Sahu

n

,

S. Salur

al

,

J. Sandweiss

bf

,

J. Schambach

at

,

A.M. Schmah

w

,

W.B. Schmidke

c

,

N. Schmitz

y

,

B.R. Schweid

aq

,

J. Seger

i

,

M. Sergeeva

f

,

R. Seto

ay

,

P. Seyboth

y

,

N. Shah

ap

,

E. Shahaliev

r

,

P.V. Shanmuganathan

x

,

M. Shao

an

,

W.Q. Shen

ap

,

F. Shen

ao

,

Z. Shi

w

,

S.S. Shi

g

,

Q.Y. Shou

ap

,

E.P. Sichtermann

w

,

R. Sikora

a

,

M. Simko

k

,

S. Singha

s

,

D. Smirnov

c

,

N. Smirnov

bf

,

W. Solyst

o

,

P. Sorensen

c

,

H.M. Spinka

b

,

B. Srivastava

ai

,

T.D.S. Stanislaus

bb

,

D.J. Stewart

bf

,

M. Strikhanov

aa

,

B. Stringfellow

ai

,

A.A.P. Suaide

am

,

T. Sugiura

aw

,

M. Sumbera

k

,

B. Summa

ag

,

X.M. Sun

g

,

X. Sun

g

,

Y. Sun

an

,

B. Surrow

ar

,

D.N. Svirida

p

,

A.H. Tang

c

,

Z. Tang

an

,

A. Taranenko

aa

,

T. Tarnowsky

z

,

J. Thäder

w

,

J.H. Thomas

w

,

A.R. Timmins

au

,

D. Tlusty

ak

,

T. Todoroki

c

,

M. Tokarev

r

,

C.A. TomKiel

x

,

S. Trentalange

f

,

R.E. Tribble

as

,

P. Tribedy

c

,

S.K. Tripathy

n

,

B.A. Trzeciak

j

,

O.D. Tsai

f

,

B. Tu

g

,

T. Ullrich

c

,

D.G. Underwood

b

,

I. Upsal

ad

,

G. Van Buren

c

,

J. Vanek

k

,

A.N. Vasiliev

ah

,

I. Vassiliev

l

,

F. Videbæk

c

,

S. Vokal

r

,

S.A. Voloshin

be

,

A. Vossen

o

,

F. Wang

ai

,

G. Wang

f

,

Y. Wang

av

,

Y. Wang

g

,

J.C. Webb

c

,

G. Webb

c

,

L. Wen

f

,

G.D. Westfall

z

,

H. Wieman

w

,

S.W. Wissink

o

,

R. Witt

ba

,

Y. Wu

s

,

Z.G. Xiao

av

,

W. Xie

ai

,

G. Xie

h

,

Z. Xu

c

,

J. Xu

g

,

Q.H. Xu

ao

,

Y.F. Xu

ap

,

N. Xu

w

,

C. Yang

ao

,

S. Yang

c

,

Q. Yang

ao

,

Y. Yang

ac

,

Z. Ye

h

,

Z. Ye

h

,

L. Yi

bf

,

K. Yip

c

,

I.-K. Yoo

aj

,

N. Yu

g

,

H. Zbroszczyk

bd

,

W. Zha

an

,

J.B. Zhang

g

,

X.P. Zhang

av

,

S. Zhang

ap

,

Z. Zhang

ap

,

L. Zhang

g

,

J. Zhang

v

,

J. Zhang

w

,

Y. Zhang

an

,

S. Zhang

an

,

J. Zhao

ai

,

C. Zhong

ap

,

C. Zhou

ap

,

L. Zhou

an

,

Z. Zhu

ao

,

X. Zhu

av

,

M. Zyzak

l a_{AGHUniversityofScienceandTechnology,FPACS,Cracow30-059,Poland}

b_{ArgonneNationalLaboratory,Argonne,IL 60439} c_{BrookhavenNationalLaboratory,Upton,NY 11973} d_{UniversityofCalifornia,Berkeley,CA 94720} e_{UniversityofCalifornia,Davis,CA 95616} f_{UniversityofCalifornia,LosAngeles,CA 90095} g_{CentralChinaNormalUniversity,Wuhan,Hubei430079} h_{UniversityofIllinoisatChicago,Chicago,IL 60607} i_{CreightonUniversity,Omaha,NE 68178}

j_{CzechTechnicalUniversityinPrague,FNSPE,Prague,11519,CzechRepublic} k_{NuclearPhysicsInstituteASCR,Prague25068,CzechRepublic}

l_{FrankfurtInstituteforAdvancedStudiesFIAS,Frankfurt60438,Germany} m_{FudanUniversity,Shanghai,200433}

n_{InstituteofPhysics,Bhubaneswar751005,India} o_{IndianaUniversity,Bloomington,IN 47408}

p_{AlikhanovInstituteforTheoreticalandExperimentalPhysics,Moscow117218,Russia} q_{UniversityofJammu,Jammu180001,India}

r_{JointInstituteforNuclearResearch,Dubna,141980,Russia} s_{KentStateUniversity,Kent,OH 44242}

t_{UniversityofKentucky,Lexington,KY 40506-0055} u_{LamarUniversity,PhysicsDepartment,Beaumont,TX 77710}

v_{InstituteofModernPhysics,ChineseAcademyofSciences,Lanzhou,Gansu730000} w_{LawrenceBerkeleyNationalLaboratory,Berkeley,CA 94720}

x_{LehighUniversity,Bethlehem,PA 18015}

y_{Max-Planck-InstitutfurPhysik,Munich80805,Germany} z_{MichiganStateUniversity,EastLansing,MI 48824}

aa_{NationalResearchNuclearUniversityMEPhI,Moscow115409,Russia} ab_{NationalInstituteofScienceEducationandResearch,HBNI,Jatni752050,India} ac_{NationalChengKungUniversity,Tainan70101}

ad_{OhioStateUniversity,Columbus,OH 43210}

ae_{InstituteofNuclearPhysicsPAN,Cracow31-342,Poland} af_{PanjabUniversity,Chandigarh160014,India}

ag_{PennsylvaniaStateUniversity,UniversityPark,PA 16802} ah_{InstituteofHighEnergyPhysics,Protvino142281,Russia} ai_{PurdueUniversity,WestLafayette,IN 47907}

aj_{PusanNationalUniversity,Pusan46241,RepublicofKorea} ak_{RiceUniversity,Houston,TX 77251}

al_{RutgersUniversity,Piscataway,NJ 08854}

am_{UniversidadedeSaoPaulo,SaoPaulo,05314-970,Brazil} an_{UniversityofScienceandTechnologyofChina,Hefei,Anhui 230026} ao_{ShandongUniversity,Jinan,Shandong250100}

ap_{ShanghaiInstituteofAppliedPhysics,ChineseAcademyofSciences,Shanghai201800} aq_{StateUniversityofNewYork,StonyBrook,NY 11794}

au_{UniversityofHouston,Houston,TX 77204} av_{TsinghuaUniversity,Beijing100084}

aw_{UniversityofTsukuba,Tsukuba,Ibaraki305-8571,Japan} ax_{SouthernConnecticutStateUniversity,NewHaven,CT 06515} ay_{UniversityofCalifornia,Riverside,CA 92521}

az_{UniversityofHeidelberg,Heidelberg,69120,Germany} ba_{UnitedStatesNavalAcademy,Annapolis,MD 21402} bb_{ValparaisoUniversity,Valparaiso,IN 46383}

bc_{VariableEnergyCyclotronCentre,Kolkata700064,India} bd_{WarsawUniversityofTechnology,Warsaw00-661,Poland} be_{WayneStateUniversity,Detroit,MI 48201}

bf_{YaleUniversity,NewHaven,CT 06520}

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Articlehistory: Received11March2018

Receivedinrevisedform30April2018 Accepted25May2018

Availableonline31May2018 Editor:M.Doser

Keywords: Collectivity Correlation Shearviscosity

Flow harmonics (vn) in the Fourier expansion of the azimuthal distribution of particlesare widely usedtoquantifytheanisotropyinparticleemissioninhigh-energyheavy-ioncollisions.Thesymmetric cumulants,SC(m,n),areusedtomeasurethecorrelationsbetweendifferentordersofflowharmonics. These correlations are used to constrain the initial conditions and the transport properties of the medium intheoretical models.In thisLetter, wepresent the firstmeasurements of the four-particle symmetriccumulantsinAu+Aucollisionsat√sN N=39 and200 GeVfromdatacollectedbytheSTAR experimentatRHIC.Weobservethatv2andv3areanti-correlatedinallcentralityintervalswithsimilar correlationstrengthsfrom39 GeVAu+Auto2.76 TeVPb+Pb(measuredbytheALICEexperiment).The

v2–v4 correlation seemsto be stronger at39 GeV thanathigher collisionenergies. The initial-stage anti-correlationsbetweensecondand thirdordereccentricitiesaresufficienttodescribethemeasured correlations between v2 and v3. The best description of v2–v4 correlations at √sN N =200 GeV is obtainedwithinclusionofthesystem’snonlinearresponsetoinitialeccentricitiesaccompaniedbythe viscous effect with

η

/s>0.08. Theoretical calculations usingdifferent initial conditions,equations of state and viscous coefficients need to be further exploredto extract

η

/s of the medium created at RHIC.

©2018TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

High-energyheavy-ion collisions at the Relativistic Heavy Ion Collider(RHIC) andthe LargeHadron Collider(LHC) are believed tohavecreatedaQCDmediumatextremelyhighenergydensities. Thepropertiesofthismediumareunderpersistentinvestigations. Anextensivelystudiedsubjectis“flow”,the collectiveanisotropic expansionofthemedium. Flow originatesfromtheinitial spatial anisotropy(and/orfluctuationsintheposition)ofthecolliding nu-cleons,anddevelopsduetothestronginteractionamongparticles produced in the collision [1–6]. The anisotropy in the momen-tumspaceisquantifiedbytheFouriercoefficientsoftheazimuthal emissiondistributionofproducedparticleswithrespecttothenth ordereventplane [7,8]:

dN d

ϕ

∝

1

+

∞



n=1 2vncos [n

(

ϕ

− 

n

)

]

,

(1) where

ϕ

is the azimuthal particle emission angle and



n isnth orderevent plane angle. These vn are the single-particle Fourier coefficientsthat can bederived withoutthe determinationofthe eventplane whichis discussedlater on.The flow harmonics, vn, quantifythe nth orderanisotropy of particles of interest,and its magnitudeimprints the initial anisotropy [1], the expansion dy-namics [9,10] and the equation of state of the medium [2,11]. Variouseffortshavebeenmadetounderstandthemeasured flow harmonics and extract the transport properties of the medium, butdifferentmodels needa differentvalueof

η

/

s (viscosityover

entropy density) to describe the same experimental

measure-ments [12,13].Forexample,the NeXSPheRIOmodelwith

η

/

s

=

0 (describedinRef. [12]) explainsall vn in all centralities whereas

the MUSIC model (described in Ref. [13]) indicates that a vis-cousmediumwith

η

/

s

∼

0

.

08 isneededtoexplainthevn datain Au

+

Aucollisions at

√

sN N

=

200 GeV. Thissuggests that besides the choice for

η

/

s, other transport properties, equation of state, andtheinitialstate alsoaffectsthesecalculationsandusingonly vn datait isdifficulttohavecontrol ontheseparameters. There-fore,moreinputsfromexperimentalobservablesarewarrantedto furtherconstraintheoreticalmodels.

Toprobe theinitial conditions,itisimportant tomeasure the distributionsof vn andevent-by-eventcorrelationsamong vn val-uesinaneventsample [14,15] astheevent-by-eventcorrelations between different orders of flow harmonics are theorized to be sensitive to the transport properties of the medium [16,17]. Re-cently, the four-particlesymmetric cumulants [18–22] have been proposed to unravel the initial-stage phenomena and the later-stagemediumproperties.Thesefour-particlesymmetriccumulants SC

(

m

,

n

)

aredefinedas [18] SC

(

m

,

n

)

≡ 

cos

(

m

ϕ

1

+

n

ϕ

2

−

m

ϕ

3

−

n

ϕ

4

)



c

= 

cos

(

m

ϕ

1

+

n

ϕ

2

−

m

ϕ

3

−

n

ϕ

4

)



− 

cos

[

m

(

ϕ

1

−

ϕ

2

)

] 

cos

[

n

(

ϕ

1

−

ϕ

2

)

]

=



v2_mv2_n



−



v2_m

 

v_n2



.

(2)

betweenv2

n and v2m;inotherwords, v2n beinglargerthan



vn2



in aneventenhances(suppresses)theprobability ofv2_m beinglarger than



v2

m



inthatsame event.The SC

(

m

,

n

)

observablesfocuson the correlationsbetweendifferent orders offlow harmonics, and facilitatethe quantitative comparison betweenexperimental data and model calculations. A normalized symmetric cumulant will facilitateaquantitativecomparisonbetweendifferentcollision en-ergiesor betweendataandmodel calculationsasthe magnitude ofsymmetriccumulantdependsonmagnitudeofflowharmonics. Thenormalizedsymmetriccumulant, N SC

(

m

,

n

)

,isdefinedas N SC

(

m

,

n

)

=



v

2

nv2m

 − 

v2n



vm2





vn2



vm2



.

(3)

The ALICE collaboration has recently measured SC

(

2

,

3

)

and SC

(

2

,

4

)

inPb

+

Pbcollisions at

√

sN N

=

2

.

76 TeV, andfoundthat thecentrality dependenceof SC

(

2

,

4

)

cannot be capturedby hy-drodynamics modelcalculationswitha constant

η

/

s [19].In this Letter,wepresentthefirstmeasurementsofsymmetriccumulants in Au

+

Au collisions at

√

sN N

=

39 and200 GeV. Due to limited statistics, resultsfromAu

+

Aucollisions at

√

sN N

=

62

.

4 GeV are notpresentedinthisletter.Section2discussesexperimentdetails andthe analysis method,and Section 3 describesthe results for SC

(

2

,

3

)

andSC

(

2

,

4

)

asfunctions ofcentralityandcomparesthem withavailablemodelpredictions.ThesummaryisinSection4.

2. Experiment and analysis

ThedatausedinthismeasurementwerecollectedfromAu

+

Au collisions bythe STAR [23] intheyears 2010(39 GeV)and2011 (200 GeV). We analyzed 1

.

1

×

108 and 4

×

108 minimum-bias Au

+

Au eventsat

√

sN N

=

39 and 200 GeV,respectively. The col-lisioncentralitydeterminationwas validatedby comparingMonte CarloGlaubercalculationstothecharged-hadronmultiplicity mea-suredwiththetimeprojection chamber(TPC)within a pseudora-piditywindowof

|

η

|

<

0

.

5.Thedetailedprocedurestoobtainthe simulated multiplicity are similar to that described in Ref. [24]. Events were required to have collision vertex positions (in the radialdirection)within2 cmofthebeamaxistoreduce contribu-tionsfrombeam-pipe(ataradiusof4 cm)interactions,andwithin alimiteddistancefromthecenterofthedetectoralongthebeam direction(

±

40 cm forthe 39 GeV data set and

±

30 cm forthe 200 GeVdataset).Chargedparticlesusedinthisanalysiswere re-constructedbytheSTARTPCwith

|

η

|

<

1

.

0.Thedistanceofclosest approach(DCA)ofatracktotheprimaryvertexwasrequiredtobe lessthan 3 cm.Wealsorequiredthenumberoffitpoints (nhits) used to reconstruct a track to be greater than 15 and the ratio ofthe numberof fitpoints tomaximum possiblehits (nhits/hit-max)tobe greaterthan0.52.Inaddition weappliedatransverse momentumcut(0

.

2

<

pT

<

2 GeV

/

c)tothechargedtracksto min-imizenonfloweffectse.g.low andhighpT cutsusedtominimize resonanceandjetcontribution,respectively.Thesedefaultcut set-tingswerelatervariedforasystematicanalysis.

Thetwo- andfour-particlecorrelationsinEq. (2) canbe evalu-atedintermsofflowvectors [25].Theflowvector(orQvector)for nth harmonicisdefinedas Qn,p

≡



kM=1w

p ke

inϕk_{,where M isthe}

multiplicityofanevent.Theweights(wk

=

weffwϕ )wereapplied tocorrectforthe pT-dependentefficiency(weff

=

_eff(1_pT)) andfor imperfectionsinthedetectoracceptance(wϕ ).Theevent-by-event v_n2 and v2_nv_m2 in Eq. (2) were calculated with the multi-particle Q -cumulantmethod [18,25]: v2_n

= 

cos

[

n

(

ϕ

1

−

ϕ

2

)

] =

N



2



n,−n D



2



n,−n (4) and v_n2v2_m

= 

cos

(

m

ϕ

1

+

n

ϕ

2

−

m

ϕ

3

−

n

ϕ

4

)



=

N



4



n,m,−n,−m D



4



n,m,−n,−m

,

(5) where N



2



n,−n

=

Qn,1Q−n,1

−

Q0,2

,

(6) D



2



n1,n2

=

N



2



0,0

=

Q 2 0,1

−

Q0,2

,

(7) N



4



n,m,−n,−m

=

Qn,1Qm,1Q−n,1Q−m,1

−

Qn+m,2Q−n,1Q−m,1

−

Qm,1Q0,2Q−m,1

−

Qn,1Qm−n,2Q−m,1

+

2Qm,3Q−m,1

−

Qm,1Q−n,1Qn−m,2

+

Qm−n,2Qn−m,2

−

Qn,1Q−n,1Q0,2

+

Q0,2Q0,2

+

2Q₋n,1Qn,3

−

Qn,1Qm,1Q−n−m,2

+

Qn₊m,2Q−n−m,2

+

2Qm,1Q−m,3

+

2Qn,1Q−n,3

−

6Q0,4

,

(8) D



4



n,m,−n,−m

=

N



4



0,0,0,0

=

Q₀4_,1

−

6Q₀2_,1Q0,2

+

3Q02,2

+

8Q0,1Q0,3

−

6Q0,4 (9) and Q₋n,p

=

Qn∗,p

.

(10)

TheweightsofM

(

M

−

1

)

andM

(

M

−

1

)(

M

−

2

)(

M

−

3

)

wereused to average the 2-particle and 4-particle correlations over events (secondaverageinEq. (2)).

The values of v2

n in the denominator of Eq. (3) are obtained with the 2-particle correlations with a pseudorapidity gap of

|

η

|

>

1

.

0 betweenthetwoparticlestosuppressfew-particle non-flowcorrelations [20].Theexpressionofv2

nintermsofflowvector withapseudorapiditygapcanbewrittenas

v_n2

=

Q A n,1

.

QnB,∗1 MA

.

MB

.

(11)

Here Q_nA_,1 andQ_nB_,1 aretheflowvectorsfromsub-events A andB, with MA and MB the corresponding multiplicities.Although, the eta-gapsuppressesfew-particlenonflowcontributions(mainlydue toshortrangecorrelation),nonflowduetolongrangecorrelations mightaffectthemagnitudeofmeasurednormalized SC

(

m

,

n

)

.

Reference [26] showsthat ifN SC

(

m

,

n

)

ismeasuredina wide centralityrange,wherethemultiplicitysignificantlyfluctuates,the measurementsofthesymmetriccumulantswillbebiasedbysuch fluctuations. This isknown asthe centrality-bin-width(CBW) ef-fect. Accordingly, in thisanalysis, the symmetric cumulantswere measured in small multiplicity windows (bin size equal to one) and then combined into 10% centrality bins to reduce statistical uncertainties. Notethatwehavechecked,usingtheAMPTmodel, themagnitudeof N SC

(

m

,

n

)

remainsunchangedifweuseimpact parameterbinsinsteadofmultiplicity.

The main systematic uncertainties came from 1) event and track selection cuts, and 2) corrections for the non-uniform az-imuthal acceptanceandefficiency.Twomethodswere adoptedto correct for the azimuthal dependence of the tracking efficiency:

Fig. 1. (Coloronline.) SC(2,4)× NpartandSC(2,3)× NpartasfunctionsofNpart inAu+Aucollisionsat√sN N=39 and200 GeV.HIJINGmodelresultsareshown

onlyfor200 GeV.Verticalslinesarestatisticaluncertaintiesandsystematicerrors areshownwithcapsymbols.

the re-centering correction was applied as a function of multi-plicity.Both the

ϕ

-weighting andthere-centering methods were appliedseparatelyforeachrunperiodofdatatakingand central-ityinterval. The difference between the two correction methods wasincludedinthesystematicuncertainty.Toestimatesystematic uncertaintyduetovariationoftracksandeventselectioncuts,we havevaried DCA,nhits, nhits/hitmax andVz values from default cutvalue.Thetotalsystematicuncertaintywasobtainedbyadding uncertaintiesfromdifferentsourcesinquadrature.Maximum con-tribution(

∼

8–12%)tothesystematicuncertaintycomesfromthe correctionfornon-uniformazimuthalacceptanceandefficiency.

3. Results

Fig.1presentsthesymmetriccumulantsSC

(

2

,

3

)

andSC

(

2

,

4

)

multiplied by the average number of participating nucleons,



Npart, asfunctionsofcentrality (represented by



Npart [24]) at midrapidity (

|

η

|

<

1

.

0) for charged hadrons in Au

+

Au collisions at

√

sN N

=

39 and 200 GeV. Systematic errors are shown with cap symbols. Due to limitedstatistics at 39 GeV, the SC

(

2

,

4

)

is measured in two wide centrality bins, 0–20% and 20–40%. Pos-itive values of SC

(

2

,

4

)

are observed for all centrality intervals atbothcollision energies, suggestiveof acorrelation between v2 and v4. On the other hand, the negative values of SC

(

2

,

3

)

re-vealtheanti-correlationbetweenv2andv3.Themagnitudeofthe SC

(

2

,

3

)

× 

Npartand SC

(

2

,

4

)

× 

Npartincreasesfromcentralto mid-peripheraleventsandthenagaindecreasesforveryperipheral events.ThisisalsotrueforSC

(

2

,

3

)

andSC

(

2

,

4

)

(notshown).An inherentfeatureofthesymmetriccumulantisthesuppressionof nonflow effects thanks to the use of the four-particle cumulant, wherenonflowrefers toazimuthal correlationsnotrelatedtothe reactionplane orientation,arising fromresonances, jets,quantum statistics,andso on. Fig.1 alsoshowsHIJING model calculations of SC

(

2

,

3

)

and SC

(

2

,

4

)

in Au

+

Aucollisions at

√

sN N

=

200 GeV [27,28].ComparisontotheHIJINGmodel,whichincludesonly non-flowphysics,suggeststhatnonfloweffectscannotexplainthedata ofnon-zerosymmetriccumulants.

Anisotropicflowisgeneratedbytheinitialgeometricanisotropy

coupled with a collective expansion of the produced medium.

There is an intense interest in understanding the origin of the initial-stage fluctuations and how these fluctuations manifest

Fig. 2. (Coloronline.) N SC(2,4)and N SC(2,3)asfunctionsofNpartinAu+Au collisions at √sN N=200 GeV. The normalized symmetriccumulants of spatial

eccentricities,N SC(m,n)ε_{,usingthe Glaubermodel,arealsoshown. N SC(m,n)}v

representnormalizedsymmetriccumulantsofflowcoefficientsusingequation (3).

themselves in correlations betweenmeasured particles. The nor-malized symmetric cumulantsevaluated inthe coordinate space, N SC

(

m

,

n

)

ε

= (



ε

2 n

ε

m2



−



ε

2 n

 

ε

2 m



)/(



ε

2 n

 

ε

2 m



)

,inAu

+

Aucollisionsat

√

sN N

=

200 GeV (usingtheMonteCarloGlauber model [26])are showninFig.2 andcompared withN SC

(

m

,

n

)

v measured inthe momentumspace [7].Ifonlyeccentricitydrivesvn,thenweexpect N SC

(

m

,

n

)

v

=

N SC

(

m

,

n

)

ε . Fig. 2 demonstrates that the initial-stage anti-correlation between the 2nd (

ε

2) and 3rd (

ε

3) order eccentricityismainlyresponsiblefortheobservedanti-correlation between v2 and v3. However, the correlation between

ε

2 and

ε

4 under-predicts the observed correlation between v2 and v4. ThedifferencebetweenN SC

(

2

,

4

)

vandN SC

(

2

,

4

)

ε increasesfrom centraltoperipheralcollisions.Theanisotropicflow v4 hasa con-tributionnotonlyfromthelinearresponseofthesystemto

ε

4,but alsohasacontributionproportionalto

ε

2

2.Therefore,theincreased difference between N SC

(

2

,

4

)

v and N SC

(

2

,

4

)

ε from central to peripheral collisions is presumably because

ε

2 has an increased contributioninv4 inmore-peripheralcollisions.Thisisconsistent withthe observation reportedby the ATLAS [29] and ALICE [19] experimentsinPb

+

Pbcollisionsat

√

sN N

=

2

.

76 TeV.Therelative contributionof

ε

2 inv4,ascomparedtothatof

ε

4 wassuggested todependontheviscouspropertiesofthemedium [30].Therefore, N SC

(

2

,

4

)

providesaprobeintothemediumproperties.

Fig. 3. (Coloronline.) N SC(2,4)and N SC(2,3)asfunctionsofaverageNpartin Au+Aucollisionsat√sN N=39 and200 GeV.ALICEresultsfor2.76TeVPb+Pb [19]

arealsoshownforcomparison.Verticallinesarestatisticaluncertainties.Systematic errorsareshownwithcapsymbols.TheCBWcorrectedresultsarelabeledby “Nar-rowMult.Bins”andCBWuncorrectedresultsarelabeledby“WideMult.Bins”.

Fig. 4. (Color online.) N SC(2,4) and N SC(2,3) as functions ofaverage Npart inAu+Au collisions at √sN N=200 GeV, with hydrodynamics [26] and AMPT

model [21] predictionsforcomparison.AllresultsareCBWcorrected.

faircomparison. ThereisaslightdifferenceinN SC

(

2

,

4

)

between results with and without the CBW correction; however, this ef-fectislargerinN SC

(

2

,

3

)

.TheuncorrectedvaluesofN SC

(

2

,

3

)

or N SC

(

2

,

4

)

at200 GeVand 2.76TeVare very closeto each other forallcentralityintervals.

WecompareinFig.4our measurementswithavailable model predictions [21,26] for N SC

(

2

,

3

)

and N SC

(

2

,

4

)

in Au

+

Au colli-sions at 200 GeV. The AMPT calculations [21] are shown for a partonicmediumwith

η

/

s

=

0

.

18 (i.e.,3 mbparton–parton inter-actioncross-section). In the AMPT model,

η

/

s in partonicmatter is estimated with the assumption that the partonic matter only consists of massless u and d quarks [31]. AMPT model calcula-tions with

η

/

s

=

0

.

18 are in agreementwith the vn magnitudes in peripheral collisions, but it over-predicts the vn data for the most-central collisions [21]. The N SC

(

2

,

3

)

and N SC

(

2

,

4

)

from dataare reasonablywell described by the AMPT model,however

an increasing deviation between data and AMPT model calcula-tionisobservedfor N SC

(

2

,

4

)

atperipheralcollisions.Predictions from an ideal hydrodynamics model (NexSPheRIO) [26] are also showninFig.4.TheidealhydrodynamicsmodelwithNexSPheRIO initial conditions is able to explain the anti-correlation between v2 andv3 withintheoreticaluncertainties,butunder-predictsthe correlation between v2 and v4. TheNexSPheRIOmodeldescribes the magnitudes of all flow harmonics (v2, v3 and v4) up to pT

∼

2.0 GeV/c within 10%, measured in all centrality intervals at the top RHIC energy [26]. The failure of the ideal hydrody-namics model for the v2–v4 correlation supports the idea that the symmetric cumulants provide additional constraints to theo-retical models.Liketheidealhydrodynamicsmodel,aviscous hy-drodynamicsmodel(MCKLNinitialconditionandwith

η

/

s

=

0

.

08) roughlyexplainstheN SC

(

2

,

3

)

dataandunder-predictsN SC

(

2

,

4

)

forperipheralcollisions.However,thepredictionfromtheviscous hydrodynamicsmodelforN SC

(

2

,

4

)

isclosertodatathantheideal hydrodynamicsmodel.Notethat,allpresentedmodels(hydroand transport)under-predict N SC

(

2

,

4

)

. Thismaybe improvedby re-visitingmodel ingredientssuch astheaverage initialstate, initial statefluctuations, energydeposition“smearing”,equationofstate, the appearance of other forms of transport, etc. Hence, a sound conclusionrequiresfurtherinvestigationalongthatline.

4. Conclusions

We have presented the first measurements of the

charge-inclusive four-particle symmetric cumulants as functionsof cen-trality at midrapidity in Au

+

Au collisions at

√

sN N

=

39 and 200 GeV.Thenewdataprovideadditionalconstraintsontheinitial conditionsandthetransportpropertiesintheoreticalmodels. Anti-correlationhasbeenobservedbetweenevent-by-eventfluctuations of v2 and v3,whiletheevent-by-event fluctuationsof v2 and v4 are found to be correlated. The initial-stage anti-correlation be-tween

ε

2 and

ε

3 appearstodescribetheobservedanti-correlation between v2 and v3,which seems tosupport theidea ofthe lin-earitybetween

ε

n andvn[7].However,theinitial-stagecorrelation alone is not sufficient to describe the measured correlation be-tween v2 and v4: the nonlinear hydrodynamic response of the mediumhastobeincludedtoreproducethedata.Thev2–v4 cor-relationseemstobedifferentbetween200 GeVand39 GeV,which could be attributedto the corresponding difference in theinitial conditionsand/orthetransportpropertiesofthemedium.Wehave comparedtheSTARmeasurementswithanumberofavailable the-oretical modelcalculations.All themodelsexplain thesymmetric cumulant between v2 and v3; however, none of them are able to describe the v2–v4 correlation forall centralities (though the hydrodynamics model(

η

/

s

=

0) could successfullyreproduce the individual flow harmonics). A viscoushydrodynamicsmodelwith

η

/

s

=

0

.

08 outperformstheidealhydrodynamicsmodelin explain-ing thedata, butstill seems tounder-predict N SC

(

2

,

4

)

,whereas theAMPTcalculationswith

η

/

s

=

0

.

18 areevenclosertothe mea-surement. A detailedcomparisonofthepresenteddatato models withdifferent equation ofstates,initial conditions,and transport coefficients is needed to determine these coefficients quantita-tively.

Acknowledgements

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

NERSC Center at LBNL, and the Open Science Grid consortium

ScienceFoundationofChina,ChineseAcademyofScience,the Min-istryofScienceandTechnologyofChinaandtheChineseMinistry ofEducation,the NationalResearch Foundation ofKorea, GAand

MSMT of the Czech Republic, Departmentof Atomic Energy and

DepartmentofScienceandTechnologyoftheGovernmentofIndia; theNationalScienceCentre ofPoland, NationalResearch Founda-tion,theMinistryofScience,EducationandSportsoftheRepublic ofCroatia,ROSATOM ofRussiaandGermanBundesministeriumfur Bildung,Wissenschaft,ForschungandTechnologie(BMBF)andthe HelmholtzAssociation.

References

[1]J.-Y.Ollitrault,Phys.Rev.D46(1992)229.

[2]D.Teaney,J.Lauret,E.V.Shuryak,Phys.Rev.Lett.86(2001)4783. [3]R.S.Bhalerao,J.-P.Blaizot,N.Borghini,etal.,Phys.Lett.B627(2005)49. [4]R.S.Bhalerao,J.-Y.Ollitrault,Phys.Lett.B641(2006)260.

[5]C.Adler,etal.,STARCollaboration,Phys.Rev.Lett.87(2001)182301. [6]R.Snellings,NewJ.Phys.13(2011)055008.

[7]B.Alver,G.Roland,Phys.Rev.C81(2010)054905,Erratum:Phys.Rev.C82 (2010)039903.

[8]A.M.Poskanzer,S.A.Voloshin,Phys.Rev.C58(1998)1671. [9]H.Sorge,Phys.Rev.Lett.78(1997)2309.

[10]P.F.Kolb,P.Huovinen,U.Heinz,etal.,Phys.Lett.B500(2001)232.

[11]U.Heinz,J.Phys.Conf.Ser.50(2006)230.

[12]F.G.Gardim,F.Grassi,M.Luzum,etal.,Phys.Rev.Lett.109(2012)202302. [13]B.Schenke,S.Jeon,C.Gale,Phys.Rev.C85(2012)024901.

[14]M.Miller,R.Snellings,arXiv:nucl-ex/0312008.

[15]B.Alver,etal.,PHOBOSCollaboration,Phys.Rev.Lett.98(2007)242302. [16]G.Aad,etal.,ATLASCollaboration,Phys.Rev.C90(2014)024905. [17]X.Zhu,Y.Zhou,H.Xu,etal.,Phys.Rev.C95(2017)044902.

[18]A.Bilandzic,C.H.Christensen,K. Gulbrandsen,etal.,Phys. Rev.C89(2014) 064904.

[19]J.Adam,etal.,ALICECollaboration,Phys.Rev.Lett.117(2016)182301. [20]Y.Zhou,K.Xiao,Z.Feng,etal.,Phys.Rev.C93(2016)034909. [21]M.Nasim,Phys.Rev.C95(2017)034905.

[22]A.M.Sirunyan,etal.,CMSCollaboration,Phys.Rev.Lett.120(2018)092301. [23]K.H. Ackermann, et al., STAR Collaboration, Nucl. Instrum. Methods A 499

(2003)624.

[24]B.I.Abelev,etal.,STARCollaboration,Phys.Rev.C81(2010)024911. [25]A.Bilandzic,R.Snellings,S.Voloshin,Phys.Rev.C83(2011)044913. [26]F.G. Gardim,F.Grassi,M.Luzum,etal.,Phys.Rev.C95(2017)034901,and

privatecommunication.

[27]M.Gyulassy,X.N.Wang,Comput.Phys.Commun.83(1994)307. [28]X.N.Wang,M.Gyulassy,Phys.Rev.D44(1991)3501.

[29]G.Aad,etal.,ATLASCollaboration,Phys.Rev.C92(2015)034903. [30]D.Teaney,L.Yan,Phys.Rev.C86(2012)044908.