Linear and non-linear flow mode in Pb–Pb collisions at sNN=2.76 TeV

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Linear

and

non-linear

flow

mode

in

Pb–Pb

collisions

at

√

s

_NN

=

2

.

76 TeV

.

ALICE

Collaboration



a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received22May2017

Receivedinrevisedform10July2017 Accepted27July2017

Availableonline4August2017 Editor:L.Rolandi

The second and the third order anisotropic flow, V2and V3, are mostly determined by the corresponding

initial spatial anisotropy coefficients,

ε

2and

ε

3, in the initial density distribution. In addition to their

dependence on the same order initial anisotropy coefficient, higher order anisotropic flow, Vn (n >3),

can also have a significant contribution from lower order initial anisotropy coefficients, which leads to mode-coupling effects. In this Letter we investigate the linear and non-linear modes in higher order anisotropic flow Vn for n=4, 5, 6 with the ALICE detector at the Large Hadron Collider. The

measurements are done for particles in the pseudorapidity range |

η

|<0.8 and the transverse momentum range 0.2 <pT<5.0 GeV/c as

a function of collision centrality. The results are compared with theoretical

calculations and provide important constraints on the initial conditions, including initial spatial geometry and its fluctuations, as well as the ratio of the shear viscosity to entropy density of the produced system.

©2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3_.

1. Introduction

The primary goal of the ultra-relativistic heavy-ion collision programme at the Large Hadron Collider (LHC) is to study the properties of the Quark–Gluon Plasma (QGP), a novel state of stronglyinteractingmatter thatis proposedto existathigh tem-peratures and energy densities [1,2]. Studies ofazimuthal corre-lationsofproduced particleshavecontributedsignificantly tothe characterisation of the matter created in heavy-ion collisions [3, 4]. Anisotropic flow, which quantifies the anisotropy of the mo-mentum distribution of final state particles, is sensitive to the event-by-event fluctuatinginitial geometry ofthe overlapregion, togetherwiththetransportpropertiesandequationofstateofthe system[4–7].Thesuccessfuldescriptionofanisotropicflowresults by hydrodynamic calculations suggests that the created medium behaves asa nearly perfect fluid [4,5] with a shear viscosity to entropy density ratio,

η

/

s, close to a conjectured lower bound 1/4π [8].AnisotropicflowischaracterisedusingaFourier decom-positionofthe particleazimuthal distributioninthe plane trans-versetothebeamdirection[9,10]:

dN d

ϕ

∝

1

+

2 ∞



n=1 vncos

[

n

(

ϕ

− 

n

)

],

(1)

whereN isthenumberofproducedparticles,

ϕ

istheazimuthal angleoftheparticleand



nisthenth orderflowsymmetryplane.

 _{E-mailaddress:alice-publications@cern.ch.}

The nth order (complex)anisotropic flow Vn isdefined as: Vn

≡

vneinn,wherevn

= |

Vn|istheflowcoefficient,and



nrepresents theazimuthofVn inmomentumspace.Fornon-centralheavy-ion collisions,thedominantflowcoefficientisv2,referredtoaselliptic

flow.Non-vanishingvaluesofhigherflowcoefficientsv3–v6atthe

LHC are ascribed primarily to theresponse of theproduced QGP tofluctuationsoftheinitialenergydensityprofileofthecolliding nucleons[11–15].

The standard (moment-defined) initial anisotropy coefficients

ε

ntogetherwiththeircorrespondinginitialsymmetryplanes(also calledparticipantplanes)



n canbecalculatedfromthetransverse positions

(

r

,

φ)

oftheparticipatingnucleons

ε

neinn

≡ −



rneinφ





rn

_

(

for n

>

1

),

(2)

where

 

denotes the averageover the transverse position of all participating nucleons,

φ

is azimuthal angle, and n is the order of the coefficient [11,16]. It has been shown in [17,18] that V2

and V3 are mostly determined with the same order initial

spa-tialanisotropycoefficients

ε

2and

ε

3,respectively.Consideringthat

η/

s reducesthe hydrodynamic response of vn to

ε

n,it was pro-posedin[18–21]that vn

/ε

n (forn

=

2,3) couldbeadirectprobe to quantitatively constrain the

η

/

s of the QGP in hydrodynamic calculations.However,

ε

n cannotbedeterminedexperimentally. In-stead,theyareobtainedfromvarioustheoreticalmodels,resulting inlargeuncertaintiesintheestimated

η

/

s derivedindirectlyfrom

v2 andv3measurements[17,19].Ontheotherhand,higherorder

anisotropic flow Vn with n

>

3 probe smaller spatial scales and http://dx.doi.org/10.1016/j.physletb.2017.07.060

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

thusaremoresensitiveto

η

/

s than V2 and V3 dueto more

pro-nouncedviscouscorrections[16,22].Thus,thestudyofthefullset offlow coefficientsis expected to constrain both

ε

n and

η

/

s si-multaneously.However,itwasrealisedlaterthat Vn withn

>

3 is notlinearlycorrelatedwiththecorresponding

ε

n [16,22,23],which makes the extractionof

η

/

s from measurements of higherorder flow coefficientsless straightforward.In addition to the studyof flowcoefficients, the resultsof correlationsbetweendifferent or-der anisotropic flow angles and amplitudes shed light on both theearlystage dynamicsandthetransport propertiesofthe cre-atedQGP[24–32]. Inparticular,the characteristicpatternofflow symmetry plane correlations(also knownas angularcorrelations offlow-vectors) observedin experiments isreproduced quantita-tivelybytheoreticalcalculations[29–33].However,thecorrelations betweenflow coefficients (also known as amplitude correlations offlow-vectors),investigatedusingsymmetriccumulants,provide stricterconstraintsoninitialconditionsand

η

/

s thanthe individ-ual vn measurements [24–28,31,32]. It is a challenge for current theoreticalmodelstoprovidequantitativedescriptions ofthe cor-relationsbetweendifferentorderflowcoefficients.

Asdiscussedabove,itisknownthatthelowerorderanisotropic flowVn(n

=

2,3)islargelydeterminedbyalinearresponseofthe systemto the corresponding

ε

n (except in peripheral collisions). Higher order anisotropic flow Vn with n

>

3 have contributions notonlyfromthelinearresponseofthesystemto

ε

n,butalso con-tributionsproportionaltotheproductof

ε

2 and/or

ε

3.These

con-tributionsareusually callednon-linearresponse[25,34] inhigher orderanisotropic flow. Fora single event, Vn withn

=

4, 5 and 6can bedecomposedintotheso-calledlinearandthenon-linear contributions,accordingto

V4

=

V4NL

+

V4L

=

χ

4,22

(

V2

)

2

+

V4L

,

(3)

V5

=

V5NL

+

V5L

=

χ

5,32V2V3

+

V5L

,

(4)

V6

=

V6NL

+

V6L

=

χ

6,222

(

V2

)

3

+

χ

6,33

(

V3

)

2

+

χ

6,42V2V4L

+

V6L

.

(5)

Here

χ

n,mk is a new observable called the non-linear mode co-efficient[34] andV_nNL representsthenon-linearmode whichhas contributionsfrommodeswithlowerorderanisotropycoefficients. The V_nL termrepresents the linear mode, which was naïvely ex-pected from the linear response of the system to the same or-der

ε

n. However, a recenthydrodynamic calculation showed that

V_nLisnotdrivenbythelinearresponsetothestandardly moment-defined

ε

n introducedinEq.(2),butthecorresponding cumulant-definedanisotropy coefficient

ε

n [30,35]. For example, V4L is

ex-pectedtobedrivenbythe4th-ordercumulant-definedanisotropy coefficientanditscorrespondinginitialsymmetryplanewhichcan becalculatedas

ε

₄ei44

≡ −



z4



₋

₃



_z2



2



r4



=

ε

4e i44

+

3



r2



2



r4



ε

2 2ei42

,

(6)

where z

=

reiφ_{. The calculations forother order anisotropy}

coef-ficients and their corresponding initial symmetry planes can be found in [30,35]. If the non-linear and linear modes of higher order anisotropic flow, V_nNL and V_nL, are uncorrelated(e.g. V_nL is perpendicularto VnNL), theycan beisolated. Oneofthe proposed approachestovalidatetheassumptionthat VNL

n andVnLare uncor-relatedistestingthefollowingrelations[25]:



V4

(

V₂∗

)

2v₂2





V4

(

V₂∗

)

2

 

v₂2

 =



v₂6





v₂4

 

v₂2

,

(7)



V5V₃∗V₂∗v₂2





V5V₃∗V₂∗

 

v₂2

 =



v₂4v₃2





v₂2v₃2

 

v₂2

.

(8)

Iftheaboverelationsarevalid,onecouldcombinetheanalyses ofhigherorderanisotropic flowwithrespecttotheir correspond-ingsymmetryplanesandtotheplanesoflower orderanisotropic flow V2 or V3 to eliminate theuncertainty frominitial state

as-sumptionsandextract

η

/

s withbetterprecision[34].

In this Letter, the linear and non-linear modes in higher or-der anisotropicflow generationare studiedin Pb–Pbcollisions at

√

sNN

=

2.76 TeV with the ALICEdetector. The main observables

are introduced in Section 2 and the experimental setup is de-scribedinSection3.Section4presentsthestudyofthesystematic uncertaintiesoftheabovementionedobservables.Theresultsand their discussionare provided inSection 5.Section6 contains the summaryandconclusions.

2. Observablesandanalysismethods

Ideally, the flow coefficient vn can be measured via the az-imuthalcorrelationsofemitted particleswithrespecttothe sym-metryplane



n asvn

= 

cos n(ϕ

− n

)



.Since



n isunknown ex-perimentally,thesimplestapproachtoobtainvnisusing2-particle correlations: vn

{

2

} = 

cos n

(

ϕ

1

−

ϕ

2

)



1/2

=



v_n2



1/2

.

(9)

Here





denotes theaverage overall particles ina single event andthenanaverageofoverallevents,



indicatestheevent aver-ageofoverallevents,and

ϕ

irepresentstheazimuthalangleofthe

i-thparticle.Theanalysedeventsaredividedintotwosub-events AandB,separatedbyapseudorapiditygap, tosuppressnon-flow effects.Thelatteraretheazimuthal correlationsnot associatedto thecommonsymmetry plane



n,such asjetsandresonance de-cays.Thus,wemodifyEq.(9)to

vn

{

2

} = 

cos

(

n

ϕ

1A

−

n

ϕ

2B

)



1/2

=



v2_n



1/2

.

(10) Here

ϕ

A 1 and

ϕ

B

2 areselectedfromsubeventAandB,respectively.

Before introducing observables related to the linear and non-linearmodesinhigherorderanisotropicflow,itiscrucialtoverify whetherEqs.(7)–(8)areapplicable.The leftandrighthandsides ofEq.(7)areobtainedbyconstructingsuitablemulti-particle cor-relations[34]:



V4

(

V₂∗

)

2v₂2



A



V4

(

V2∗

)

2



A



v₂2



=



cos

(

4

ϕ

1A

+

2

ϕ

2A

−

2

ϕ

3B

−

2

ϕ

4B

−

2

ϕ

5B

)





cos

(

4

ϕ

A 1

−

2

ϕ

2B

−

2

ϕ

3B

)

 

cos

(

2

ϕ

1A

−

2

ϕ

2B

)



,

(11)



v₂6





v₂4

 

v₂2



=



cos

(

2

ϕ

1A

+

2

ϕ

2A

+

2

ϕ

3A

−

2

ϕ

4B

−

2

ϕ

5B

−

2

ϕ

6B

)





cos

(

2

ϕ

A 1

+

2

ϕ

2A

−

2

ϕ

3B

−

2

ϕ

4B

)

 

cos

(

2

ϕ

1A

−

2

ϕ

2B

)



.

(12)

Similarly, we can validate Eq. (8) by calculating both sides with[34]:



V5V₃∗V₂∗v₂2



A



V5V3∗V2∗



A



v₂2



=



cos

(

5

ϕ

1A

+

2

ϕ

2A

−

3

ϕ

3B

−

2

ϕ

4B

−

2

ϕ

5B

)





cos

(

5

ϕ

A 1

−

3

ϕ

2B

−

2

ϕ

3B

)

 

cos

(

2

ϕ

1A

−

2

ϕ

2B

)



,

(13)



v4 2 v32





v₂2v₃2

 

v₂2



=



cos

(

3

ϕ

A 1

+

2

ϕ

2A

+

2

ϕ

3A

−

3

ϕ

4B

−

2

ϕ

5B

−

2

ϕ

6B

)





cos

(

3

ϕ

A 1

+

2

ϕ

2A

−

3

ϕ

3B

−

2

ϕ

4B

)

 

cos

(

2

ϕ

1A

−

2

ϕ

2B

)



.

(14)

ThemagnitudeofVnNL wasdenotedasvn{m}(here



misthe lower orderflow symmetry plane andm

=

2,3) in [34]. The no-tation vn,mk, wheren specifies the order ofthe flow termwhile

m andk etc. denotethe contributinglower order flowsymmetry planes,isusedinthisLetter.Ifthelinearandnon-linearmodesare independent,thenthenon-linearmodeinhigherorderanisotropic flow can be analysed by correlating Vn with



2 or/and



3 [34].

Forsub-eventAwecandefine:

v₄A_,22

=



cos

(

4

ϕ

A 1

−

2

ϕ

2B

−

2

ϕ

3B

)







cos

(

2

ϕ

A 1

+

2

ϕ

2A

−

2

ϕ

3B

−

2

ϕ

4B

)



,

(15) v₅A_,32

=



cos

(

5

ϕ

A 1

−

3

ϕ

2B

−

2

ϕ

3B

)







cos

(

3

ϕ

A 1

+

2

ϕ

2A

−

3

ϕ

3B

−

2

ϕ

4B

)



,

(16) v₆A_,222

=



cos

(

6

ϕ

A 1

−

2

ϕ

2B

−

2

ϕ

3B

−

2

ϕ

4B

)







cos

(

2

ϕ

A 1

+

2

ϕ

2A

+

2

ϕ

3A

−

2

ϕ

4B

−

2

ϕ

5B

−

2

ϕ

6B

)



,

(17) v₆A_,33

=



cos

(

6

ϕ

A 1

−

3

ϕ

2B

−

3

ϕ

3B

)







cos

(

3

ϕ

A 1

+

3

ϕ

2A

−

3

ϕ

3B

−

3

ϕ

4B

)



.

(18)

Similarly, one can obtain v_nB_,mk for sub-event B. The average of

vA

n,mk andvnB,mk,definedasvn,mk,quantifies themagnitudeofthe non-linear mode in higher order anisotropic flow, which can be writtenas[36]: v4,22

=



v4v22cos

(

4



4

−

4



2

)







v4₂



≈ 

v4cos

(

4



4

−

4



2

)

,

(19) v5,32

=



v5v3v2cos

(

5



5

−

3



3

−

2



2

)







v2₃v2₂



≈ 

v5cos

(

5



5

−

3



3

−

2



2

)

,

(20) v6,222

=



v6v32cos

(

6



6

−

6



2

)







v6 2



≈ 

v6cos

(

6



6

−

6



2

)

,

(21) v6,33

=



v6v2₃cos

(

6



6

−

6



3

)







v4₃



≈ 

v6cos

(

6



6

−

6



3

)

.

(22)

The approximationis validifthe correlationbetweenlower (n

=

2,3)andhigher(n

>

3)flowcoefficientsisweak.

As canbe seen inEqs. (3)–(5),the calculation for V6 is more

complicatedthan V4 and V5,and theexact expression for vL₆ is

currentlynotavailable. Therefore,weonlyfocuson thetwo non-linearmodesofV6 withoutdiscussingvL6.AccordingtoEqs.(3)to (4),themagnitudesofthelinearmodeinhigherorderanisotropic flowcanbecalculatedas:

vL₄

=



v₄2

{

2

} −

v₄2_,22

,

(23) vL₅

=



v₅2

{

2

} −

v₅2_,32

.

(24)

Theratioofvn,mk to vn{2

}

,denotedas

ρ

n,mk,canbecalculated as:

ρ

4,22

=

v4,22 v4

{

2

}

= 

cos

(

4



4

−

4



2

)

,

(25)

ρ

5,32

=

v5,32 v5

{

2

}

= 

cos

(

5



5

−

3



3

−

2



2

)

,

(26)

ρ

6,222

=

v6,222 v6

{

2

}

= 

cos

(

6



6

−

6



2

)

,

(27)

ρ

6,33

=

v6,33 v6

{

2

}

= 

cos

(

6



6

−

6



3

)

.

(28)

These observablesmeasurethe correlationsbetweendifferent or-der flow symmetry planes if the correlations between different orderflow coefficientsareweak. Theyarevery similarto the so-called weighted event-plane correlationsmeasured by theATLAS Collaboration[33].Thedifferencesareasfollows:



v2₂v2₃



isusedin Eq.(20)and(26),while



v2

2



v23



wasusedin[33],whichdidnot

considertheanti-correlationsbetweenv2 andv3foundin[27].In

addition,multi-particlecorrelationsareusedfor v2 andv3 inthe

denominatoroftheobservables,whiletwo-particlecorrelationsare used intheevent-planecorrelationswhichmight bebiasedfrom fluctuationsofv2 andv3.

The non-linear mode coefficients

χ

n,mk in Eqs. (3) to (5) are definedas:

χ

4,22

=

v4,22





v4 2



(29)

χ

5,32

=

v5,32





v2₂v2₃



(30)

χ

6,222

=

v6,222





v6₂



(31)

χ

6,33

=

v6,33





v4₃



.

(32)

These quantifythe contributions ofthe non-linearmode andare expectedtobeindependentofv2 orv3.

Alloftheobservablesabovearebasedon2- andmulti-particle correlations, which can beobtained usingthegeneric framework foranisotropicflowanalysesintroducedinRef.[24].

3. Experimentalsetupanddataanalysis

The data samples analysed in this Letter were recorded by ALICE during thePb–Pb runs ofthe LHCat a centre-of-mass en-ergy of

√

sNN

=

2.76 TeVin 2010. Minimumbias Pb–Pb collision

eventswere triggeredby thecoincidenceofsignalsintheV0 de-tector [37,38], with an efficiency of 98.4% of the hadronic cross section [39].The V0 detectoris composed oftwo arrays of scin-tillator counters, V0-AandV0-C, whichcover the pseudorapidity ranges 2.8

<

η

<

5.1 and

−

3.7

<

η

<

−

1.7, respectively. Beam background events were rejected using the timing information fromtheV0andtheZeroDegreeCalorimeter(ZDC)[37]detectors andbycorrelatingreconstructedclustersandtrackletswiththe Sil-icon Pixel Detectors (SPD). The fraction of pile-up events in the datasampleisfoundtobenegligibleafterapplyingdedicated pile-upremovalcriteria[40].Onlyeventswithareconstructedprimary vertex within

±

10 cm fromthe nominal interaction point along the beam direction were used in this analysis. The primary ver-texwasestimatedusingtracksreconstructedbytheInnerTracking System (ITS) [37,41] and TimeProjection Chamber (TPC) [37,42]. The collision centrality was determined from the measured V0 amplitudeandcentralityintervalsweredefinedfollowingthe pro-ceduredescribedin[39].About13millionPb–Pbeventspassedall oftheeventselectioncriteria.

Tracksreconstructedusingthecombinedinformationfromthe TPCand ITS are used in this analysis. This combination ensures a high detection efficiency, optimum momentum resolution, and aminimum contributionfromphoton conversions andsecondary charged particles produced either in the detector material or fromweakdecays. Toreduce the contributionsfrom secondaries, charged tracks were required to have a distance of closest ap-proachtotheprimary vertexinthelongitudinal(z)directionand transverse (xy) plane smaller than 3.2 cm and 2.4 cm, respec-tively.Additionally, tracks were required to haveat least 70TPC spacepointsoutofthemaximum159.Theaverage

χ

2 _perdegree

offreedom ofthe trackfit to theTPC spacepoints was required to be below 2. In this study, tracks were selected in the pseu-dorapidity range

|

η

|

<

0.8 and the transverse momentum range 0.2

<

pT

<

5.0 GeV/c.

4. Systematicuncertainties

Numeroussources ofsystematicuncertaintywere investigated byvaryingtheeventandtrackselectionaswellastheuncertainty associated with the possible remaining non-flow effects in the analysis. The variation of theresults withthe collision centrality iscalculatedbyalternativelyusingtheTPCorSPDtoestimatethe eventmultiplicityandis foundto be lessthan3% forall observ-ables.Resultswithoppositepolaritiesofthemagneticfieldwithin theALICEdetectorandwithnarrowingthenominal

±

10cmrange ofthereconstructedvertexalongthebeamdirectionfromthe cen-treoftheALICEdetectorto9,8and7cmdonotshowadifference ofmorethan2%comparedtothedefaultselectioncriteriafor var-iousmeasurements. Thecontributions frompile-upevents tothe finalsystematicuncertaintyarefoundto benegligible.The sensi-tivityto the trackselection criteria was explored by varying the numberofTPCspacepoints andby usingtracksreconstructedin theTPC alone.Varying the numberof TPCspace points from70 to80,90and100out ofa possible158,results ina1–3% varia-tionof theresultsfor vn,within 1.5%for

ρ

n,mk and

χ

n,mk.Using TPC-onlytracksleadstoadifferenceoflessthan14%,17%and8% forvn,

ρ

n,mk and

χ

n,mk,respectively.Botheffectswereincludedin theevaluationofthesystematicuncertainty.Severaldifferent ap-proaches have been applied to estimate the effects of non-flow. Theseincludethe investigationofmulti-particlecorrelationswith various

|

η

|

gaps,theapplicationofthelike-signtechniquewhich correlatestwoparticleswitheitherallpositiveornegativecharges andsuppresssuchnon-flowasduetoresonancedecays,aswellas thecalculationsusingHIJINGMonteCarlosimulations[43],which donotincludeanisotropicflow.Itwasfoundthatthepossible re-mainingnon-flow effects arelessthan 10.5%,11% and7%for vn,

ρ

n,mkand

χ

n,mk,respectively.Theyaretakenintoaccountinthe fi-nalsystematicuncertainty.The systematicuncertainties evaluated foreachsourcementionedabovewereaddedinquadratureto ob-tainthetotalsystematicuncertaintyofthemeasurements.

5. Resultsanddiscussion

As discussed in Sec. 2, one can validate the assumption that linearandnon-linear modesinhigher orderanisotropic flow are uncorrelated via Eqs. (7) and (8). These have been tested in A Multi-Phase Transport (AMPT) model [25] as well as in the hy-drodynamiccalculations [44]. Good agreementbetween left- and right-hand sides of Eqs. (7) and (8) is found for all centrality classes,independentoftheinitialconditionsandtheidealor vis-cous fluid dynamics used in the calculations. Thus, it is crucial tocheck theseequalities indata,to further confirmthe assump-tion that the two components are uncorrelated and can be iso-latedindependently.Fig. 1 confirms thatthe agreementobserved

Fig. 1. Studyofrelationshipbetweenlinear and non-linearmodesinhigher or-deranisotropicflowinPb–Pbcollisionsat√sNN=2.76 TeV,accordingtoEqs.(7) and(8).

intheoreticalcalculationsisalsopresentinthedatadespitesmall deviationsfound incentral collisions whentestingEqs. (8).Their centralitydependencyare similarastheprevious theoretical pre-dictions [25,44]. The measurements support the assumption that higher order anisotropic flow Vn (n

>

3) can be modeled as the sumofindependentlinearandnon-linearmodes.

Themagnitudesoflinearandnon-linearmodesinhigherorder anisotropic flow are reported as a function of collision central-ity in Fig. 2. In this Letter, sub-events A and B are built in the pseudorapidity ranges

−

0.8

<

η

<

−

0.4 and 0.4

<

η

<

0.8, re-spectively,whichresultsinapseudorapiditygapof

|

η

|

>

0.8 for all presentedmeasurements. It canbe seen thatthe linearmode

vL

4 dependsweaklyoncentralityandisthelarger contributionto

v4

{

2

}

forthecentralityrange0–30%. Thenon-linearmode, v4,22,

increases monotonicallyas thecentrality decreases andsaturates around centrality percentile 50%, becoming the dominantsource forcentralityintervalsabove 40%.Similar trendsofcentrality de-pendencehavebeenobservedforV5,althoughv5,32becomes the

dominantcontributionincentralitypercentileabove30%.Onlytwo non-linearcomponentsv6,222and v6,33arediscussedforV6.Itis

showninFig. 2 (right)that v6,222 increasesmonotonically asthe

centrality decreases to centrality 50%, while v6,33 has a weaker

centralitydependencecomparedtov6,222.

The linear and non-linear modes in higher order anisotropic flow were investigated by the ATLAS Collaboration [26] using a differentapproachbasedon“EventShapeEngineering”[45].With thismethodonecanutiliselargefluctuationsintheinitial geome-tryofthesystemtoselecteventscorrespondingtoaspecificinitial shape. Theconclusion is qualitativelyconsistent withwhat is re-portedhere, although a direct comparisonis not possibledueto the different kinematic cuts (especially the integrated pT range)

usedinthetwomeasurements.Thehigherorderanisotropicflow induced by lower order anisotropic flow were alsomeasured us-ing the event-plane method at the LHC [14,46]. However, the measurements of the non-linear mode presented in this Letter are based on themulti-particle correlationsmethod witha

|

η

|

gap. This method makes it easier to measure an observable like

v5,32,whichislessstraightforwardtodefineusingtheeventplane

method [14,46]. In addition, as pointed out in [25,34,47], this new multi-particle correlations method should strongly suppress short-range(inpseudorapidity)non-floweffectsandprovidesa ro-bust measurement withoutany dependenceon the experimental acceptance. The measurements are compared to recent hydrody-namic calculationsfroma hybrid

IP

-

Glasma

+ MUSIC + UrQMD

model[48],inwhichrealisticevent-by-eventinitialconditionsare used and the hydrodynamic evolution takes into account both shear andbulkviscosity. It isshownthat thishydrodynamic cal-culationcould describe quantitatively thetotal magnitudes of V4

and V6,aswell asthe magnitudesoftheir linear andnon-linear

modes,whileitslightlyoverestimatestheresultsfor V5.

Thecentralitydependenceof

ρ

n,mk,whichquantifiesthe angu-larcorrelationsbetweendifferentorderflow symmetry planes,is

Fig. 2. Centralitydependenceofv4(left),v5(middle)andv6(right)inPb–Pbcollisionsat√sNN=2.76 TeV.Contributionsfromlinearandnon-linearmodesarepresented

withopenandsolidmarkers,respectively.ThehydrodynamiccalculationsfromIP-Glasma+ MUSIC + UrQMD[48]areshownforcomparison.

Fig. 3. Centralitydependenceofρn,mkinPb–Pbcollisionsat√sNN=2.76 TeV.ATLAS

measurementsbasedontheevent-planecorrelation[33]arepresentedwithopen markers.ThehydrodynamiccalculationsfromIP-Glasma+ MUSIC + UrQMD[48] areshownwithopenbands.(Forinterpretationofthereferencestocolourinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)

presentedin Fig. 3.It is observedthat

ρ

4,22 increases from

cen-tral to peripheral collisions, which suggeststhat the correlations between



2 and



4 are stronger in peripheral than in central

collisions. It implies that VNL₄ tends to align with V4 in more

peripheral collisions. The results of

ρ

6,33, which measures the

correlation of



3 and



6,do not exhibit a strong centrality

de-pendencewithinthestatisticaluncertainties.Asmentionedabove,

ρ

4,22 and

ρ

6,33 are similar to the previous “event-plane

correla-tion” measurements



cos(44

−

42

)

w

and



cos(66

−

63

)

w

in [33]. The comparisons between measurements of these ob-servablesare also presentedin Fig. 3. Theresults are compatible with each other, despite the different kinematic ranges used by ATLAS and this analysis. It should be also noted that the mea-surements of

ρ

n,mk presented in this Letter show the symmetry plane correlations at mid-pseudorapidity

|

η

|

<

0.8 while ATLAS measuredthesymmetryplanecorrelationsusing

−

4.8

<

η

<

−

0.5 and 0.5

<

η

<

4.8 for two-plane correlations, and using

−

2.7

<

η

<

−

0.5, 0.5

<

η

<

2.7 and 3.3

<

|

η

|

<

4.8 for 3-plane corre-lations [33]. Previous investigations suggest that there might be

η-dependent

fluctuationsoftheflowsymmetryplaneandtheflow magnitude[49,50]. As a consequence,one might expect a differ-ence when measuring the correlations of flow symmetry planes fromdifferentpseudorapidityregions.However,Fig. 3showsgood agreement between the ALICE and ATLAS measurements. There-fore, no obvious indication that the flow symmetry plane varies with

η

can be deduced fromthis comparison. It is noticeablein

Fig. 3that the

ρ

5,32measurement seemsslightlyhigherthanthe



cos(55

−

33

−

22

)

w

measurement. This is mainly dueto a

small difference betweenthe definitionsof the observable as in-troduced inSec. 2: the term



v2

2v23



1/2

isused in

ρ

5,32, whereas



v2₂



1/2



v2₃



1/2 is usedin the“event-plane correlations”[33]. Con-sidering theknown anti-correlations between v2 and v3 [26,27],



v2 2v23



1/2

couldbeupto10%lowerthan



v2 2



1/2



v2 3



1/2 depending on thecentralityclass[27],leading toa slightlylarger

ρ

5,32 than



cos(55

−

33

−

22

)

w

fromATLAS.

Ithasbeenobservedinhydrodynamicandtransportmodel cal-culations that the symmetry plane correlations, e.g. correlations of second and fourth order symmetry planes, change sign dur-ingthesystemevolution[29,30,32].Themeasuredflowsymmetry plane correlations could be nicely explained by the combination of contributions fromlinear and non-linear modes in higher or-der anisotropic flow [30]. This indicates that the flow symmetry plane correlation

ρ

n,mk carries important information about the dynamic evolution ofthe created system. In addition, the model calculations suggest that stronger initial symmetry plane corre-lations are reflected in stronger correlations between the flow symmetry planes in the final state [29,32]. And a larger value of

η

/

s of the QGP leads to weaker flow symmetry plane corre-lations in the final state. As pointed out in [29], the hydrody-namic calculations from

VISH2

+ 1

using Monte Carlo Glauber (MC-Glb) or Monte Carlo Kharzeev–Levin–Nardi (MC-KLN) initial conditions can only describe qualitatively the trends ofthe cen-trality dependence of the event-plane correlation measurements by ATLAS.Itisthereforeexpectedthatthesehydrodynamic calcu-lationscannotdescribethepresentedALICEmeasurements,which are compatiblewith theATLAS event-plane correlation measure-ments. Fig. 3 shows that the hydrodynamic calculations from

IP

-

Glasma

+ MUSIC + UrQMD

[48] reproduce nicely the mea-surements of symmetry plane correlations

ρ

n,mk. The measure-ments of

ρ

n,mk presented in this Letter, together with the com-parison tohydrodynamiccalculations, shouldplaceconstraintson theinitialconditionsand

η

/

s oftheQGPinhydrodynamic calcula-tions.

Fig. 4presentsthemeasurementsofthenon-linearmode coef-ficientsasafunctionofcollisioncentrality.Itisobservedthat

χ

4,22

and

χ

6,222 decrease modestly from central to mid-central

colli-sions,andstayalmost constantfrommid-centraltomore periph-eralcollisions.For

χ

5,32and

χ

6,33strongcentralitydependenceis

not observed either. Thus, the dramatic increase of vn,mk shown in Fig. 2 appears to be mainly dueto the increase of v2 and/or

v3 fromcentraltoperipheralcollisionsandnottheincreaseofthe

non-linearmodecoefficient.Itisalsonoteworthythatthe relation-ship of

χ

4,22

∼

χ

6,33

≈

χ5,32₂ is approximatelyvalid, aspredicted

by hydrodynamic calculations[34].Thecomparisons to event-by-eventviscoushydrodynamiccalculationsfrom

VISH2

+ 1

[44]and from

IP

-

Glasma

+ MUSIC + UrQMD

[48] are also presented in

Fig. 4.

VISH2

+ 1

showsthat

χ

4,22 calculationswithMC-Glb

ini-tial conditions are larger than those with MC-KLN initial condi-tions,i.e.

χ

4,22dependsontheinitialconditions.Atthesametime,

Fig. 4. CentralitydependenceofχinPb–Pbcollisionsat√sNN=2.76 TeV.

Hydro-dynamiccalculationsfromVISH2+ 1[44]areshowninshadedareasandtheone fromIP-Glasma+ MUSIC + UrQMD[48]areshownwithopenbands.(For inter-pretationofthereferencestocolourinthisfigurelegend,thereaderisreferredto thewebversionofthisarticle.)

thecurveswithdifferent

η

/

s valuesfor

VISH2

+ 1

arevery sim-ilar,indicating that

χ

4,22 isinsensitiveto

η

/

s.The measurements

favour IP-GlasmaandMC-KLN over MC-Glb initial conditions re-gardlessof

η

/

s.Thissuggeststhatthe

χ

4,22measurement canbe

usedto constrain the initial conditions, withless concernof the settingof

η

/

s

(

T

)

inhydrodynamiccalculationsthanpreviousflow observables.

It was predicted that

χ

6,222

<

χ

6,33 based on the ideal

hy-drodynamiccalculationusingsmoothinitialGaussiandensity pro-files [34], whereas an opposite prediction was obtained in the ideal hydrodynamic calculation evolving genuinely bumpy initial conditionsobtainedfromaMonteCarlosamplingoftheinitial nu-cleonpositionsinthecollidingnuclei[44].ItisseeninFig. 4that

χ

6,222

∼

χ

6,33 within the current uncertainties. The data are not

abletodiscriminatethedifferentpredictionsin[34]and[44]. Hy-drodynamiccalculationsusingMC-KLNandIP-Glasmainitial con-ditions give better descriptions of

χ

6,222, compared to the ones

using MC-Glb initial conditions. For

χ

5,32 noneof the

combina-tionsofinitialconditionsand

η

/

s inthehydrodynamiccalculations agreequantitatively withdata. Thismight be due tothe current difficulty of describing the anti-correlations between v2 and v3

in hydrodynamic calculations [27,51], which are involved in the calculation of

χ

5,32. Furthermore,

VISH2

+ 1

calculations show

that

χ

5,32 and

χ

6,33 are very weakly sensitiveto the initial

con-ditions,butdecreaseas

η

/

s increases. Theinvestigation withthe

VISH2

+ 1

hydrodynamic framework shows that the sensitivity of

χ

5,32 and

χ

6,33 to

η

/

s is not dueto sensitivityto shear

vis-couseffectsduring thebuildupofhydrodynamic flow.Instead,as foundin[44],itisduetothe

η

/

s atfreeze-out.Themeasurements of

χ

5,32 and

χ

6,33 do not further constrain the

η

/

s during

sys-temevolution,however,theyprovideuniqueinformationon

η

/

s at

freeze-outwhichwas poorlyknownandcannot beobtainedfrom otheranisotropicflowrelatedobservables.Furtherimprovementof model calculations on correlations between different order flow coefficients are necessary to better understand the comparison of

χ

5,32 obtained from data and hydrodynamic calculations. The

χ

6,33 results are consistent withhydrodynamic calculationsfrom

VISH2

+ 1

withMC-KLN initialconditions using

η

/

s

=

0.08 and

IP

-

Glasma

+ MUSIC + UrQMD

with a

η

/

s

=

0.095. It is shown that

χ

5,32 and

χ

6,33 have a weak centrality dependence if a

smaller

η

/

s isusedinthehydrodynamiccalculations.Suchaweak

centrality dependence of

χ

5,32 and

χ

6,33 is observed in data as

well.Themeasurementspresentedheresuggestasmall

η

/

s value

atfreeze-out,whichcanbeusefultoconstrainthetemperature de-pendenceoftheshearviscosityoverentropydensityratio,

η

/

s

(T),

in the development of hydrodynamic frameworks. These results suggest that future tuning of the parameterisations of

η

/

s

(T)

in hydrodynamic frameworks using the presented measurements is necessary.

6. Summary

The linear and non-linear modes in higher order anisotropic flow generation were studied with 2- and multi-particle corre-lations in Pb–Pb collisions at

√

sNN

=

2.76 TeV. The results

pre-sentedinthisLetter show that higherorderanisotropic flowcan be isolated into two independent contributions: the component thatarisesfromanon-linearresponseofthesystemtothelower orderinitialanisotropycoefficients

ε

2and/or

ε

3,andalinearmode

whichisdrivenbylinearresponseofthesystemtothesameorder cumulant-definedanisotropycoefficient. A weakcentrality depen-denceisobservedforthecontributionsfromlinearmodewhereas the contributions from non-linear mode increase dramatically as the collision centrality decreases, and it becomes the dominant source inhigher orderanisotropic flow inmid-central to periph-eralcollisions. Itisshownthat thisis mainlyduetotheincrease of lower order flow coefficients v2 and v3. The correlations

be-tween different flow symmetry planes are measured. The results are compatible withthe previous “event-plane correlation” mea-surements,andcanbequantitativelydescribedbycalculations us-ing the

IP

-

Glasma

+ MUSIC + UrQMD

framework. Furthermore, non-linearmode coefficients, whichhave differentsensitivitiesto the shearviscosity over entropydensity ratio

η

/

s and theinitial conditions,are presented inthisLetter. Comparisons to hydrody-namic calculations suggest that the data is described better by hydrodynamic calculationswithsmaller

η

/

s.Inaddition,the MC-Glbinitialconditionisdisfavouredbythepresentedresults.

Measurementsoflinearandnon-linearmodesinhigher order anisotropic flow and their comparison to hydrodynamic calcula-tionsprovidemorepreciseconstraintsontheinitialconditionsand temperaturedependenceof

η

/

s.Theseresultscouldalsooffernew insightsintothegeometryofthefluctuatinginitialstate andinto the dynamical evolution ofthe stronglyinteracting medium pro-ducedinrelativisticheavy-ioncollisionsattheLHC.

Acknowledgements

The ALICE Collaboration would like to thank all its engineers andtechniciansfortheir invaluablecontributions tothe construc-tionoftheexperimentandtheCERNacceleratorteamsforthe out-standingperformanceoftheLHCcomplex.TheALICECollaboration gratefully acknowledges the resources and support provided by all Gridcentres andtheWorldwide LHC ComputingGrid (WLCG) collaboration. The ALICE Collaboration acknowledges the follow-ing funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (YerevanPhysicsInstitute)Foundation (ANSL),State Committeeof Science andWorld Federation ofScientists (WFS), Armenia; Aus-trian Academy of Sciences and Österreichische Nationalstiftung für Forschung, Technologie undEntwicklung, Austria; Ministry of CommunicationsandHighTechnologies,NationalNuclearResearch Center, Azerbaijan;Conselho NacionaldeDesenvolvimento Cientí-ficoe Tecnológico(CNPq),UniversidadeFederal doRioGrandedo Sul (UFRGS),Financiadorade Estudose Projetos(Finep)and Fun-dação de Amparo à Pesquisa do Estado de São Paulo (FAPESP),

Brazil; Ministry of Science & Technology of China (MSTC), Na-tional Natural Science Foundation of China (NSFC) and Ministry of Education of China (MOEC), China; Ministry of Science, Edu-cationandSportsandCroatian Science Foundation, Croatia; Min-istryofEducation,YouthandSportsoftheCzech Republic,Czech Republic; The Danish Council for Independent Research — Nat-ural Sciences, the Carlsberg Foundation and Danish National Re-search Foundation (DNRF), Denmark;Helsinki Institute ofPhysics (HIP),Finland;Commissariatàl’EnergieAtomique(CEA)and Insti-tut Nationalde Physique Nucléaire etde Physique desParticules (IN2P3)and Centre Nationalde laRecherche Scientifique(CNRS), France; Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF) and GSI Helmholtzzentrum für Schwe-rionenforschungGmbH,Germany;GeneralSecretariatforResearch and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy, Government of India (DAE) and Council of Scientific and Industrial Research (CSIR), NewDelhi,India; IndonesianInstituteofScience, Indonesia; Cen-tro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche EnricoFermiandIstitutoNazionalediFisicaNucleare(INFN),Italy; InstituteforInnovativeScience andTechnology,NagasakiInstitute ofApplied Science (IIST),Japan Society forthe Promotionof Sci-ence(JSPS)KAKENHIandJapanese MinistryofEducation,Culture, Sports, Science and Technology (MEXT), Japan; Consejo Nacional deCienciayTecnología (CONACYT),throughFondodeCooperatión Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Académico (DGAPA), Mexico; NederlandseOrganisatievoorWetenschappelijkOnderzoek(NWO), Netherlands; The Research Council of Norway, Norway; Commis-sion on Science and Technology for Sustainable Development in theSouth(COMSATS),Pakistan;PontificiaUniversidadCatólicadel Perú,Peru;MinistryofScienceandHigherEducationandNational Science Centre, Poland; Korea Institute of Science and Technol-ogyInformationandNationalResearchFoundationofKorea(NRF), Republicof Korea; Ministryof Education andScientific Research, InstituteofAtomicPhysicsandRomanianNationalAgencyfor Sci-ence,Technology andInnovation,Romania; JointInstitute for Nu-clearResearch(JINR),MinistryofEducationandScienceofthe Rus-sian FederationandNationalResearchCentre KurchatovInstitute, Russia;Ministry ofEducation, Science,Research andSportofthe Slovak Republic, Slovakia; NationalResearch Foundation ofSouth Africa,SouthAfrica; Centrode AplicacionesTecnológicasy Desar-rolloNuclear(CEADEN),Cubaenergía,Cuba;MinisteriodeCienciae Innovación andCentrodeInvestigacionesEnergéticas, Medioambi-entalesyTecnológicas(CIEMAT),Spain;SwedishResearchCouncil (VR)andKnut&AliceWallenbergFoundation(KAW),Sweden; Eu-ropean Organization for Nuclear Research, Switzerland; National Science andTechnology DevelopmentAgency (NSDTA), Suranaree University of Technology (SUT) and Office of the Higher Educa-tionCommissionunderNRUprojectofThailand,Thailand;Turkish Atomic Energy Agency (TAEK), Turkey; National Academy of Sci-encesofUkraine,Ukraine;ScienceandTechnologyFacilities Coun-cil (STFC), United Kingdom; National Science Foundation of the United Statesof America(NSF) and United StatesDepartment of Energy,OfficeofNuclearPhysics(DOENP),UnitedStatesof Amer-ica.

References

[1]E.V.Shuryak,Quark–gluonplasmaandhadronicproductionofleptons,photons andpions,Phys.Lett.B78(1978)150,Yad.Fiz.28(1978)796.

[2]E.V.Shuryak,Quantumchromodynamicsandthetheoryofsuperdensematter, Phys.Rep.61(1980)71–158.

[3]J.-Y.Ollitrault,Anisotropyasasignatureoftransversecollectiveflow,Phys.Rev. D46(1992)229–245.

[4]S.A. Voloshin, A.M. Poskanzer, R. Snellings, Collective phenomena in non-centralnuclearcollisions,arXiv:0809.2949[nucl-ex].

[5]U.Heinz,R.Snellings,Collectiveflowandviscosityinrelativisticheavy-ion col-lisions,Annu.Rev.Nucl.Part.Sci.63(2013)123–151,arXiv:1301.2826[nucl-th]. [6]S.Pratt,E.Sangaline,P.Sorensen,H.Wang,Constrainingtheequation ofstate ofsuper-hadronicmatterfromheavy-ioncollisions,Phys.Rev.Lett.114(2015) 202301,arXiv:1501.04042[nucl-th].

[7]H.Song,Y.Zhou,K.Gajdosova,Collectiveflowandhydrodynamicsinlargeand smallsystemsattheLHC,Nucl.Sci.Tech.28 (7)(2017)99,arXiv:1703.00670 [nucl-th].

[8]P.Kovtun,D.T.Son,A.O.Starinets,Viscosityinstronglyinteractingquantum fieldtheories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601, arXiv:hep-th/0405231.

[9]S.Voloshin,Y.Zhang,FlowstudyinrelativisticnuclearcollisionsbyFourier expansionofAzimuthalparticledistributions,Z.Phys.C70(1996)665–672, arXiv:hep-ph/9407282.

[10]A.M.Poskanzer,S.A.Voloshin,Methodsforanalyzinganisotropicflowin rel-ativistic nuclear collisions, Phys. Rev. C 58 (1998) 1671–1678, arXiv:nucl-ex/9805001.

[11]B. Alver,G. Roland, Collision geometry fluctuations and triangularflow in heavy-ioncollisions,Phys.Rev.C81(2010)054905,arXiv:1003.0194[nucl-th], Phys.Rev.C82(2010)039903(Erratum).

[12]ALICECollaboration,K.Aamodt,etal.,Higherharmonicanisotropicflow mea-surementsofchargedparticlesinPb–Pbcollisionsat√sN N=2.76 TeV,Phys.

Rev.Lett.107(2011)032301,arXiv:1105.3865[nucl-ex].

[13]ATLASCollaboration,G.Aad,etal.,Measurementoftheazimuthalanisotropy forchargedparticleproductionin√sN N=2.76 TeVlead–leadcollisionswith

theATLASdetector,Phys.Rev.C86(2012)014907,arXiv:1203.3087[hep-ex]. [14]CMSCollaboration, S. Chatrchyan,et al., Measurement ofhigher-order

har-monicazimuthalanisotropyinPbPbcollisionsat√sN N=2.76 TeV,Phys.Rev.

C89 (4)(2014)044906,arXiv:1310.8651[nucl-ex].

[15]ALICECollaboration,J.Adam,et al.,Anisotropicflowofchargedparticlesin Pb–Pbcollisionsat√sNN=5.02 TeV,Phys.Rev.Lett.116 (13)(2016)132302,

arXiv:1602.01119[nucl-ex].

[16]B.H.Alver,C.Gombeaud,M.Luzum,J.-Y.Ollitrault,Triangularflowin hydrody-namicsandtransporttheory,Phys.Rev.C82(2010)034913,arXiv:1007.5469 [nucl-th].

[17]Z. Qiu, C. Shen, U. Heinz, Hydrodynamic elliptic and triangular flow in Pb–Pb collisions at √s=2.76 A TeV, Phys. Lett. B 707 (2012) 151–155, arXiv:1110.3033[nucl-th].

[18]H.Niemi,G.S.Denicol,H.Holopainen,P.Huovinen,Event-by-event distribu-tionsofazimuthalasymmetriesinultrarelativisticheavy-ioncollisions,Phys. Rev.C87 (5)(2013)054901,arXiv:1212.1008[nucl-th].

[19]H.Song,S.A.Bass,U.Heinz,T.Hirano,C.Shen,200 A GeVAu+Aucollisions serveanearlyperfectquark–gluonliquid,Phys.Rev.Lett.106(2011)192301, arXiv:1011.2783,Phys.Rev.Lett.109(2012)139904(Erratum).

[20]F.G.Gardim,J.Noronha-Hostler,M.Luzum,F.Grassi,Effectsofviscosityonthe mappingofinitialtofinalstate inheavyioncollisions, Phys.Rev.C91 (3) (2015)034902,arXiv:1411.2574[nucl-th].

[21]J.Fu,Centralitydependenceofmappingthehydrodynamicresponsetothe ini-tialgeometryinheavy-ioncollisions,Phys.Rev.C92 (2)(2015)024904. [22]D.Teaney,L.Yan,Nonlinearitiesintheharmonicspectrumofheavyion

col-lisionswithidealandviscoushydrodynamics,Phys.Rev.C86(2012)044908, arXiv:1206.1905[nucl-th].

[23]F.G.Gardim,F.Grassi,M.Luzum,J.-Y.Ollitrault,Mappingthe hydrodynamic responsetotheinitialgeometryinheavy-ioncollisions,Phys.Rev.C85(2012) 024908,arXiv:1111.6538[nucl-th].

[24]A.Bilandzic, C.H. Christensen,K. Gulbrandsen, A.Hansen, Y. Zhou,Generic frameworkforanisotropicflow analyseswithmultiparticleazimuthal corre-lations,Phys.Rev.C89 (6)(2014)064904,arXiv:1312.3572[nucl-ex]. [25]R.S.Bhalerao,J.-Y.Ollitrault,S.Pal,Characterizingflowfluctuationswith

mo-ments,Phys.Lett.B742(2015)94–98,arXiv:1411.5160[nucl-th].

[26]ATLASCollaboration,G.Aad,et al.,Measurementofthecorrelationbetween flowharmonicsofdifferentorderinlead–leadcollisionsat√sN N=2.76 TeV

withtheATLASdetector,Phys.Rev.C92 (3)(2015)034903,arXiv:1504.01289 [hep-ex].

[27]ALICECollaboration,J.Adam,etal.,Correlatedevent-by-eventfluctuationsof flowharmonicsinPb–Pbcollisionsat√sNN=2.76 TeV,Phys.Rev.Lett.117

(2016)182301,arXiv:1604.07663[nucl-ex].

[28]Y.Zhou,Reviewofanisotropicflowcorrelationsinultrarelativisticheavy-ion collisions, Adv. High Energy Phys. 2016 (2016) 9365637, arXiv:1607.05613 [nucl-ex].

[29]Z.Qiu,U.Heinz,Hydrodynamicevent-planecorrelationsinPb+Pbcollisions at√s=2.76 A TeV,Phys.Lett.B717(2012)261–265,arXiv:1208.1200 [nucl-th].

[30]D.Teaney,L.Yan,Event-planecorrelationsandhydrodynamicsimulations of heavyioncollisions,Phys.Rev.C90 (2)(2014)024902,arXiv:1312.3689 [nucl-th].

[31]H.Niemi,K.J.Eskola,R.Paatelainen,Event-by-eventfluctuationsina perturba-tiveQCD+saturation+hydrodynamicsmodel:determiningQCDmattershear viscosity inultrarelativisticheavy-ioncollisions, Phys. Rev.C 93 (2)(2016) 024907,arXiv:1505.02677[hep-ph].

[32]Y.Zhou,K.Xiao,Z.Feng,F.Liu,R.Snellings,Anisotropicdistributionsina mul-tiphasetransportmodel,Phys.Rev.C93 (3)(2016)034909,arXiv:1508.03306 [nucl-ex].

[33]ATLASCollaboration,G.Aad,etal.,Measurementofevent-planecorrelations in√sN N=2.76 TeVlead–leadcollisionswiththeATLASdetector,Phys.Rev.C

90 (2)(2014)024905,arXiv:1403.0489[hep-ex].

[34]L.Yan,J.-Y.Ollitrault,ν4,ν5,ν6,ν7:nonlinearhydrodynamicresponseversus

LHCdata,Phys.Lett.B744(2015)82–87,arXiv:1502.02502[nucl-th]. [35]J. Qian, U.Heinz, R. He, L. Huo, Differentialflow correlations in

relativis-ticheavy-ioncollisions,Phys.Rev.C95 (5)(2017)054908,arXiv:1703.04077 [nucl-th].

[36]R.S. Bhalerao,J.-Y.Ollitrault,S. Pal,Event-plane correlators,Phys.Rev.C88 (2013)024909,arXiv:1307.0980[nucl-th].

[37]ALICECollaboration,K.Aamodt,etal.,TheALICEexperimentattheCERNLHC, J. Instrum.3(2008)S08002.

[38]ALICECollaboration,E.Abbas,etal.,PerformanceoftheALICEVZEROsystem, J. Instrum.8(2013)P10016,arXiv:1306.3130[nucl-ex].

[39]ALICECollaboration,B.Abelev,et al.,CentralitydeterminationofPb–Pb col-lisionsat √sN N=2.76 TeV withALICE,Phys. Rev.C88 (4)(2013)044909,

arXiv:1301.4361[nucl-ex].

[40]ALICECollaboration,B.B.Abelev,etal.,PerformanceoftheALICEexperimentat theCERNLHC,Int.J.Mod.Phys. A29(2014)1430044,arXiv:1402.4476 [nucl-ex].

[41]ALICECollaboration,K.Aamodt,etal.,AlignmentoftheALICEinnertracking systemwith cosmic-raytracks,J. Instrum.5(2010)P03003,arXiv:1001.0502 [physics.ins-det].

[42]J. Alme, et al.,The ALICE TPC, a large3-dimensional trackingdevice with fastreadoutfor ultra-highmultiplicity events,Nucl. Instrum.MethodsPhys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 622 (2010) 316–367, arXiv:1001.1950[physics.ins-det].

[43]M.Gyulassy,X.-N.Wang,HIJING 1.0:a MonteCarloprogramfor partonand particle production inhigh-energyhadronicand nuclearcollisions, Comput. Phys.Commun.83(1994)307,arXiv:nucl-th/9502021.

[44]J.Qian,U.W.Heinz,J.Liu,Mode-couplingeffectsinanisotropicflowin heavy-ioncollisions,Phys.Rev.C93 (6)(2016)064901,arXiv:1602.02813[nucl-th]. [45]J. Schukraft, A. Timmins, S.A. Voloshin, Ultra-relativistic nuclear collisions:

event shapeengineering,Phys.Lett.B719(2013)394–398,arXiv:1208.4563 [nucl-ex].

[46]ALICECollaboration,B.Abelev,etal.,Anisotropicflowofchargedhadrons, pi-onsand(anti-)protonsmeasuredathightransversemomentuminPb–Pb col-lisionsat√sN N=2.76 TeV,Phys.Lett.B719(2013)18–28,arXiv:1205.5761

[nucl-ex].

[47]M. Luzum, J.-Y. Ollitrault,Eliminating experimentalbiasin anisotropic-flow measurements ofhigh-energynuclearcollisions, Phys. Rev.C87 (4) (2013) 044907,arXiv:1209.2323[nucl-ex].

[48]S.McDonald,C.Shen,F.Fillion-Gourdeau,S.Jeon,C.Gale,Hydrodynamic pre-dictionsforPb+Pbcollisionsat5.02 A TeV,arXiv:1609.02958[hep-ph]. [49]CMSCollaboration,V.Khachatryan,etal.,Evidencefortransversemomentum

andpseudorapiditydependenteventplanefluctuationsinPbPbandpPb colli-sions,Phys.Rev.C92 (3)(2015)034911,arXiv:1503.01692[nucl-ex]. [50]L.-G.Pang,G.-Y.Qin,V.Roy,X.-N.Wang,G.-L.Ma,Longitudinaldecorrelation

ofanisotropicflowsinheavy-ioncollisionsattheCERNLargeHadronCollider, Phys.Rev.C91 (4)(2015)044904,arXiv:1410.8690[nucl-th].

[51]X.Zhu,Y.Zhou,H.Xu,H.Song,Correlationsofflowharmonicsin2.76 A TeV Pb–Pbcollisions,Phys.Rev. C95 (4)(2017)044902,arXiv:1608.05305[nucl-th].

ALICECollaboration

S. Acharya

139

,

D. Adamová

96

,

J. Adolfsson

34

,

M.M. Aggarwal

101

,

G. Aglieri Rinella

35

,

M. Agnello

31

,

N. Agrawal

48

,

Z. Ahammed

139

,

N. Ahmad

17

,

S.U. Ahn

80

,

S. Aiola

143

,

A. Akindinov

65

,

S.N. Alam

139

,

J.L.B. Alba

114

,

D.S.D. Albuquerque

125

, D. Aleksandrov

92

,

B. Alessandro

59

,

R. Alfaro Molina

75

,

A. Alici

54

,

12

,

27

,

A. Alkin

3

,

J. Alme

22

, T. Alt

71

,

L. Altenkamper

22

,

I. Altsybeev

138

,

C. Alves Garcia Prado

124

,

M. An

7

, C. Andrei

89

, D. Andreou

35

,

H.A. Andrews

113

,

A. Andronic

109

,

V. Anguelov

106

,

C. Anson

99

,

T. Antiˇci ´c

110

,

F. Antinori

57

,

P. Antonioli

54

,

R. Anwar

127

, L. Aphecetche

117

,

H. Appelshäuser

71

,

S. Arcelli

27

,

R. Arnaldi

59

,

O.W. Arnold

107

,

36

,

I.C. Arsene

21

,

M. Arslandok

106

,

B. Audurier

117

, A. Augustinus

35

,

R. Averbeck

109

,

M.D. Azmi

17

, A. Badalà

56

,

Y.W. Baek

61

,

79

,

S. Bagnasco

59

, R. Bailhache

71

,

R. Bala

103

,

A. Baldisseri

76

,

M. Ball

45

, R.C. Baral

68

,

A.M. Barbano

26

,

R. Barbera

28

, F. Barile

33

,

53

, L. Barioglio

26

,

G.G. Barnaföldi

142

,

L.S. Barnby

95

,

113

,

V. Barret

82

,

P. Bartalini

7

,

K. Barth

35

,

E. Bartsch

71

,

M. Basile

27

,

N. Bastid

82

,

S. Basu

141

,

139

,

B. Bathen

72

,

G. Batigne

117

,

A. Batista Camejo

82

,

B. Batyunya

78

, P.C. Batzing

21

,

I.G. Bearden

93

,

H. Beck

106

,

C. Bedda

64

,

N.K. Behera

61

,

I. Belikov

135

,

F. Bellini

27

,

H. Bello Martinez

2

,

R. Bellwied

127

,

L.G.E. Beltran

123

,

V. Belyaev

85

, G. Bencedi

142

,

S. Beole

26

,

A. Bercuci

89

,

Y. Berdnikov

98

,

D. Berenyi

142

,

R.A. Bertens

130

,

D. Berzano

35

,

L. Betev

35

, A. Bhasin

103

, I.R. Bhat

103

,

A.K. Bhati

101

,

B. Bhattacharjee

44

,

J. Bhom

121

,

L. Bianchi

127

,

N. Bianchi

51

,

C. Bianchin

141

,

J. Bielˇcík

39

,

J. Bielˇcíková

96

,

A. Bilandzic

36

,

107

, G. Biro

142

,

R. Biswas

4

,

S. Biswas

4

,

J.T. Blair

122

, D. Blau

92

,

C. Blume

71

,

G. Boca

136

,

F. Bock

106

,

84

,

35

,

A. Bogdanov

85

,

L. Boldizsár

142

,

M. Bombara

40

,

G. Bonomi

137

,

M. Bonora

35

,

J. Book

71

,

H. Borel

76

,

A. Borissov

19

,

M. Borri

129

, E. Botta

26

,

C. Bourjau

93

,

P. Braun-Munzinger

109

,

M. Bregant

124

,

T.A. Broker

71

,

T.A. Browning

108

,

M. Broz

39

,

E.J. Brucken

46

,

E. Bruna

59

,

G.E. Bruno

33

, D. Budnikov

111

,

H. Buesching

71

,

S. Bufalino

31

,

P. Buhler

116

,

P. Buncic

35

,

O. Busch

133

,

Z. Buthelezi

77

,

J.B. Butt

15

,

J.T. Buxton

18

,

J. Cabala

119

,

D. Caffarri

35

,

94

,

H. Caines

143

,

A. Caliva

64

, E. Calvo Villar

114

,

P. Camerini

25

,

A.A. Capon

116

,

F. Carena

35

, W. Carena

35

,

F. Carnesecchi

27

,

12

,

J. Castillo Castellanos

76

,

A.J. Castro

130

, E.A.R. Casula

24

,

55

,

C. Ceballos Sanchez

9

,

P. Cerello

59

,

S. Chandra

139

, B. Chang

128

,

S. Chapeland

35

,

M. Chartier

129

,

J.L. Charvet

76

,

S. Chattopadhyay

139

,

S. Chattopadhyay

112

,

A. Chauvin

107

,

36

,

M. Cherney

99

,

C. Cheshkov

134

,

B. Cheynis

134

,

V. Chibante Barroso

35

,

D.D. Chinellato

125

,

S. Cho

61

,

P. Chochula

35

,

K. Choi

19

, M. Chojnacki

93

, S. Choudhury

139

,

T. Chowdhury

82

, P. Christakoglou

94

,

C.H. Christensen

93

,

P. Christiansen

34

, T. Chujo

133

, S.U. Chung

19

, C. Cicalo

55

, L. Cifarelli

12

,

27

,

F. Cindolo

54

, J. Cleymans

102

,

F. Colamaria

33

,

D. Colella

66

,

35

,

A. Collu

84

, M. Colocci

27

,

M. Concas

59

,

ii

,

G. Conesa Balbastre

83

,