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

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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 ra c t

Articlehistory:

Received22May2017

Receivedinrevisedform10July2017 Accepted27July2017

Availableonline4August2017 Editor:L.Rolandi

Thesecondandthethirdorderanisotropicflow,V2andV3,aremostlydeterminedbythecorresponding initial spatialanisotropy coefficients,ε2and ε3,inthe initialdensitydistribution.Inaddition totheir dependence onthe sameorderinitialanisotropycoefficient, higherorderanisotropicflow, Vn (n>3), can alsohave asignificant contribution from lower order initial anisotropy coefficients, whichleads 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 atthe Large HadronCollider. The} measurementsaredoneforparticlesinthepseudorapidityrange|η_|<0.8 andthetransversemomentum range0.2<pT<5.0 GeV/casafunctionofcollisioncentrality.Theresultsarecomparedwiththeoretical calculationsandprovideimportantconstraintsontheinitialconditions,includinginitialspatialgeometry anditsfluctuations,aswellastheratiooftheshearviscositytoentropydensityoftheproducedsystem.

©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

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 existathightem- 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 1/4

π

[8].AnisotropicflowischaracterisedusingaFourierdecom- positionofthe particleazimuthal distributioninthe planetrans- versetothebeamdirection[9,10]:

dN d

ϕ ^∝

¹

⁺

²

∞ n=¹

v_ncos

[

ⁿ

( ϕ −

n

)],

⁽¹⁾

whereN isthenumberofproducedparticles,

ϕ

istheazimuthal angleoftheparticleandnisthenth 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

≡ −

rⁿe^inφ

^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 v_n/

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

/sderivedindirectlyfrom 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

/sthan V2 and V3 dueto morepro- 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 correlationsbetweendifferentor- der anisotropic flow angles and amplitudes shed light on both theearlystage dynamicsandthetransport propertiesofthecre- 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 thantheindivid- ual vn measurements [24–28,31,32]. It is a challenge for current theoreticalmodelstoprovidequantitativedescriptions ofthecor- relationsbetweendifferentorderflowcoefficients.

Asdiscussedabove,itisknownthatthelowerorderanisotropic flowVn(n=^{2,3)islargelydeterminedbyalinearresponseofthe} systemto the corresponding

ε

n (except in peripheral collisions).

Higher order anisotropic flow V_n with n> 3 have contributions notonlyfromthelinearresponseofthesystemto

ε

n,butalsocon- tributionsproportionaltotheproductof

ε

2 and/or

ε

3.Thesecon- tributionsareusually callednon-linearresponse[25,34] inhigher orderanisotropic flow. Fora single event, Vn withn= ^{4, 5 and} 6can bedecomposedintotheso-calledlinearandthenon-linear contributions,accordingto

V4

=

^V4^NL

+

^V4^L

= χ

4,22

(

V2

)

²

+

^V4^L

,

(3) V5

=

V₅^NL

+

V₅^L

= χ

5,32V2V3

+

V₅^L

,

(4) V6

=

V₆^NL

+

V₆^L

= χ

6,222

(

V2

)

³

+ χ

6,33

(

V3

)

²

+ χ

6,42V2V₄^L

+

V₆^L

.

(5) Here

χ

n,mk is a new observable called the non-linear mode co- efficient[34] andV_n^NL representsthenon-linearmode whichhas contributionsfrommodeswithlowerorderanisotropycoefficients.

The V_n^L termrepresents the linear mode, which was naïvelyex- pected from the linear response of the system to the same or- der

ε

n. However, a recenthydrodynamic calculation showed that V_n^Lisnotdrivenbythelinearresponsetothestandardlymoment- defined

ε

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

ε

_n [30,35]. For example, V₄^L is ex- pectedtobedrivenbythe4th-ordercumulant-definedanisotropy coefficientanditscorrespondinginitialsymmetryplanewhichcan becalculatedas

ε

₄eⁱ⁴⁴

≡ −

z⁴

−

³

z²

2

r⁴

= ε

4eⁱ⁴⁴

+

³

r²

2

r⁴

ε

²₂eⁱ⁴²

,

(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_n^NL and V_n^L, are uncorrelated(e.g. V_n^L is perpendicularto V_n^NL), theycan beisolated. Oneofthe proposed approachestovalidatetheassumptionthat V_n^NL andV_n^Lareuncor- 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

/swithbetterprecision[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 particleswithrespecttothesym- metryplanen asv_n= ^cosn(

ϕ

−n)^.Sincen isunknownex- perimentally,thesimplestapproachtoobtainvnisusing2-particle correlations:

vn

{

²

} =

^cosn

( ϕ

1

− ϕ

2

)

^1/2

=

v_n²

1/2

.

(9)

Here ^{denotes theaverage overall particles ina single event} andthenanaverageofoverallevents,indicatestheeventaver- ageofoverallevents,and

ϕ

irepresentstheazimuthalangleofthe i-thparticle.Theanalysedeventsaredividedintotwosub-events AandB,separatedbyapseudorapiditygap, tosuppressnon-flow effects.Thelatteraretheazimuthal correlationsnot associatedto thecommonsymmetry plane n,such asjetsandresonancede- cays.Thus,wemodifyEq.(9)to

vn

{

²

} =

^cos

(

n

ϕ

₁^A

₋

n

ϕ

₂^B

)

^1/2

=

v²_n

1/2

.

(10)

Here

ϕ

₁^Aand

ϕ

₂^B areselectedfromsubeventAandB,respectively.

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

V4

(

V₂^∗

)

²v₂²

A

V₄

(

V₂^∗

)

²

A

v₂²

=

^cos

(

4

ϕ

₁^A

+

²

ϕ

₂^A

−

²

ϕ

₃^B

−

²

ϕ

₄^B

−

²

ϕ

₅^B

)

^cos

(

4

ϕ

₁^A

₋

2

ϕ

₂^B

₋

2

ϕ

₃^B

)

cos

(

2

ϕ

₁^A

₋

2

ϕ

₂^B

) ,

(11)

v₂⁶

v₂⁴ v₂²

=

^cos

(

2

ϕ

₁^A

₊

2

ϕ

₂^A

₊

2

ϕ

₃^A

₋

2

ϕ

₄^B

₋

2

ϕ

₅^B

₋

2

ϕ

₆^B

)

cos

(

2

ϕ

₁^A

+

2

ϕ

₂^A

−

2

ϕ

₃^B

−

2

ϕ

₄^B

)

cos

(

2

ϕ

₁^A

−

2

ϕ

₂^B

) .

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

V5V₃^∗V₂^∗v₂²

A

V₅V₃^∗V₂^∗

A

v₂²

=

^cos

(

5

ϕ

₁^A

+

²

ϕ

₂^A

−

³

ϕ

₃^B

−

²

ϕ

₄^B

−

²

ϕ

₅^B

)

^cos

(

5

ϕ

₁^A

₋

3

ϕ

₂^B

₋

2

ϕ

₃^B

)

cos

(

2

ϕ

₁^A

₋

2

ϕ

₂^B

) ,

(13)

v₂⁴v₃²

v₂²v₃² v₂²

=

^cos

(

3

ϕ

₁^A

+

²

ϕ

₂^A

+

²

ϕ

₃^A

−

³

ϕ

₄^B

−

²

ϕ

₅^B

−

²

ϕ

₆^B

)

^cos

(

3

ϕ

₁^A

₊

2

ϕ

₂^A

₋

3

ϕ

₃^B

₋

2

ϕ

₄^B

)

cos

(

2

ϕ

₁^A

₋

2

ϕ

₂^B

) .

(14) ThemagnitudeofV_n^NL wasdenotedasvn{m}^(heremisthe lower orderflow symmetry plane andm=²,3) in [34]. Theno- tation v_n,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 V_n with 2 or/and 3 [34].

Forsub-eventAwecandefine:

v₄^A_,22

=

^cos

(

4

ϕ

₁^A

₋

2

ϕ

₂^B

₋

2

ϕ

₃^B

)

^cos

(

2

ϕ

₁^A

₊

2

ϕ

₂^A

₋

2

ϕ

₃^B

₋

2

ϕ

₄^B

) ,

(15)

v₅^A_,32

=

^cos

(

5

ϕ

₁^A

−

³

ϕ

₂^B

−

²

ϕ

₃^B

)

^cos

(

3

ϕ

₁^A

+

²

ϕ

₂^A

−

³

ϕ

₃^B

−

²

ϕ

₄^B

) ,

(16) v₆^A_,222

=

^cos

(

6

ϕ

₁^A

−

²

ϕ

₂^B

−

²

ϕ

₃^B

−

²

ϕ

₄^B

)

cos

(

2

ϕ

₁^A

+

2

ϕ

₂^A

+

2

ϕ

₃^A

−

2

ϕ

₄^B

−

2

ϕ

₅^B

−

2

ϕ

₆^B

) ,

(17) v₆^A_,33

=

^cos

(

6

ϕ

₁^A

₋

3

ϕ

₂^B

₋

3

ϕ

₃^B

)

^cos

(

3

ϕ

₁^A

₊

3

ϕ

₂^A

₋

3

ϕ

₃^B

₋

3

ϕ

₄^B

) .

(18)

Similarly, one can obtain v_n^B_,mk for sub-event B. The average of v_n^A_,mk andv_n^B_,mk,definedasv_n,mk,quantifies themagnitudeofthe non-linear mode in higher order anisotropic flow, which can be writtenas[36]:

v_4,22

=

^v4v²₂cos

(

4

₄

−

⁴

₂

)

^v4₂

≈

^v4cos

(

4

4

−

⁴

2

),

⁽¹⁹⁾ v_5,32

=

^v5v₃v₂cos

(

5

5

−

³

3

−

²

2

)

^v2₃^v2₂

≈

v5cos

(

5

5

−

3

3

−

2

2

),

(20) v6,222

=

^v6v³₂cos

(

6

6

−

⁶

2

)

^v6₂

≈

^v6cos

(

6

6

−

⁶

2

) ,

(21) v_6,33

=

^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 v^L₆ is currentlynotavailable. Therefore,weonlyfocuson thetwonon- linearmodesofV₆ withoutdiscussingv^L₆.AccordingtoEqs.(3)to (4),themagnitudesofthelinearmodeinhigherorderanisotropic flowcanbecalculatedas:

v^L₄

=

v₄²

{

²

} −

^v₄²_,22

,

(23) v^L₅

=

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 correlationsbetweendifferentor- der flow symmetry planes if the correlations between different orderflow coefficientsareweak. Theyarevery similarto theso- called weighted event-plane correlationsmeasured by theATLAS Collaboration[33].Thedifferencesareasfollows:^v22v²₃^isusedin Eq.(20)and(26),while^v2₂^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

^v4₂

⁽²⁹⁾

χ

5,32

=

^v5,32

^v2₂^v2₃

(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-massen- ergy of √

sNN=².76 TeVin 2010. Minimumbias Pb–Pb collision eventswere triggeredby thecoincidenceofsignalsintheV0de- tector [37,38], with an efficiency of 98.4% of the hadronic cross section [39].The V0 detectoris composed oftwo arrays ofscin- 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 andbycorrelatingreconstructedclustersandtrackletswiththeSil- icon Pixel Detectors (SPD). The fraction of pile-up events in the datasampleisfoundtobenegligibleafterapplyingdedicatedpile- 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 amplitudeandcentralityintervalsweredefinedfollowingthepro- 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 ofthereconstructedvertexalongthebeamdirectionfromthecen- treoftheALICEdetectorto9,8and7cmdonotshowadifference ofmorethan2%comparedtothedefaultselectioncriteriaforvar- iousmeasurements. Thecontributions frompile-upevents tothe finalsystematicuncertaintyarefoundto benegligible.Thesensi- 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 v_n,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.Severaldifferentap- 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.Itwasfoundthatthepossiblere- mainingnon-flow effects arelessthan 10.5%,11% and7%for vn,

ρ

n,mkand

χ

n,mk,respectively.Theyaretakenintoaccountinthefi- nalsystematicuncertainty.The systematicuncertainties evaluated foreachsourcementionedabovewereaddedinquadraturetoob- 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,independentoftheinitialconditionsandtheidealorvis- 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-linearmodesinhigheror- deranisotropicflowinPb–Pbcollisionsat√_s

NN=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 v^L₄ dependsweaklyoncentralityandisthelarger contributionto v4{²}^{forthecentralityrange0–30%. Thenon-linearmode, v}4,22, increases monotonicallyas thecentrality decreases andsaturates around centrality percentile 50%, becoming the dominantsource forcentralityintervalsabove 40%.Similar trendsofcentralityde- 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 v_6,33 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 thismethodonecanutiliselargefluctuationsintheinitialgeome- 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-floweffectsandprovidesaro- 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,whileitslightlyoverestimatestheresultsforV5.

Thecentralitydependenceof

ρ

n,mk,whichquantifiestheangu- larcorrelationsbetweendifferentorderflow symmetry planes,is

Fig. 2.Centralitydependenceofv4(left),v5(middle)andv6(right)inPb–Pbcollisionsat√_s

NN=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 fromcen- tral to peripheral collisions, which suggeststhat the correlations between 2 and 4 are stronger in peripheral than in central collisions. It implies that V^NL₄ 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

-dependentfluctuationsoftheflowsymmetryplaneandtheflow 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 asin- troduced inSec. 2: the term

v²₂v²₃1/2

isused in

ρ

5,32, whereas v²₂1/2

v²₃1/2

is usedin the“event-plane correlations”[33]. Con- sidering theknown anti-correlations between v2 and v3 [26,27], v²₂v²₃1/2

couldbeupto10%lowerthan v²₂1/2

v²₃1/2

depending on thecentralityclass[27],leading toa slightlylarger

ρ

5,32 than ^cos(55−³3−²2)w fromATLAS.

Ithasbeenobservedinhydrodynamicandtransportmodelcal- 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.Itisthereforeexpectedthatthesehydrodynamiccalcu- 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

/softheQGPinhydrodynamiccalcula- tions.

Fig. 4presentsthemeasurementsofthenon-linearmodecoef- ficientsasafunctionofcollisioncentrality.Itisobservedthat

χ

4,22 and

χ

6,222 decrease modestly from central to mid-central colli- sions,andstayalmost constantfrommid-centraltomoreperiph- eralcollisions.For

χ

5,32and

χ

6,33strongcentralitydependenceis not observed either. Thus, the dramatic increase of v_n,mk shown in Fig. 2 appears to be mainly dueto the increase of v2 and/or v₃ fromcentraltoperipheralcollisionsandnottheincreaseofthe non-linearmodecoefficient.Itisalsonoteworthythattherelation- ship of

χ

4,22∼

χ

6,33≈^χ5,32₂ ^is approximatelyvalid, aspredicted by hydrodynamic calculations[34].Thecomparisons toevent-by- eventviscoushydrodynamiccalculationsfromVISH2+1[44]and from IP-Glasma+MUSIC+UrQMD [48] are also presented in Fig. 4.VISH2+1showsthat

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√_s

NN=².76 TeV.Hydro- dynamiccalculationsfromVISH2+1[44]areshowninshadedareasandtheone fromIP-Glasma+MUSIC+UrQMD[48]areshownwithopenbands.(Forinter- pretationofthereferencestocolourinthisfigurelegend,thereaderisreferredto thewebversionofthisarticle.)

thecurveswithdifferent

η

/svaluesforVISH2+1areverysim- 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- drodynamiccalculationusingsmoothinitialGaussiandensitypro- files [34], whereas an opposite prediction was obtained in the ideal hydrodynamic calculation evolving genuinely bumpy initial conditionsobtainedfromaMonteCarlosamplingoftheinitialnu- 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-Glasmainitialcon- 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

/sinthehydrodynamiccalculations 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 initialcon- 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

/satfreeze-out.Themeasurements of

χ

5,32 and

χ

6,33 do not further constrain the

η

/s during sys- temevolution,however,theyprovideuniqueinformationon

η

/sat 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

/sisusedinthehydrodynamiccalculations.Suchaweak

centrality dependence of

χ

5,32 and

χ

6,33 is observed in data as well.Themeasurementspresentedheresuggestasmall

η

/svalue atfreeze-out,whichcanbeusefultoconstrainthetemperaturede- 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 weakcentralitydepen- 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 toperiph- eralcollisions. Itisshownthat thisis mainlyduetotheincrease of lower order flow coefficients v2 and v3. The correlationsbe- tween different flow symmetry planes are measured. The results are compatible withthe previous “event-plane correlation” mea- surements,andcanbequantitativelydescribedbycalculationsus- ing the IP-Glasma+MUSIC+UrQMD framework. Furthermore, non-linearmode coefficients, whichhave differentsensitivitiesto the shearviscosity over entropydensity ratio

η

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

η

/s.Inaddition,theMC- 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 totheconstruc- tionoftheexperimentandtheCERNacceleratorteamsfortheout- 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 NacionaldeDesenvolvimentoCientí- ficoe Tecnológico(CNPq),UniversidadeFederal doRioGrandedo Sul (UFRGS),Financiadorade Estudose Projetos(Finep)andFun- dação de Amparo à Pesquisa do Estado de São Paulo (FAPESP),