Correlated long-range mixed-harmonic fluctuations measured in $pp$, $p$+Pb and low-multiplicity Pb+Pb collisions with the ATLAS detector

JID:PLB AID:34344 /SCO Doctopic: Experiments [m5Gv1.3; v1.250; Prn:4/01/2019; 8:57] P.1 (1-28)

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Correlated

long-range

mixed-harmonic

fluctuations

measured

in

pp,

p+Pb

and

low-multiplicity

Pb+Pb

collisions

with

the

ATLAS

detector

.The

ATLAS

Collaboration



a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received6July2018

Receivedinrevisedform7October2018 Accepted13November2018

Availableonlinexxxx Editor: D.F.Geesaman

Correlationsoftwoflowharmonicsvnand vm viathree- andfour-particlecumulantsaremeasuredin 13TeVpp,5.02TeVp+Pb,and2.76TeVperipheralPb+PbcollisionswiththeATLASdetectorattheLHC. The goalis tounderstandthemulti-particle natureofthe long-rangecollectivephenomenoninthese collisionsystems.Thelargenon-flowbackgroundfromdijetproductionpresentinthestandardcumulant methodissuppressedusingamethodofsubeventcumulantsinvolvingtwo,threeandfoursubevents separatedinpseudorapidity.Theresultsshowanegativecorrelationbetweenv2and v3andapositive correlationbetweenv2andv4forallcollisionsystemsandoverthefullmultiplicityrange.However,the magnitudes ofthecorrelationsare foundtodependontheeventmultiplicity,thechoiceoftransverse momentumrangeandcollisionsystem. Therelativecorrelationstrength, obtainedbynormalisationof the cumulantswith the v2

n from atwo-particlecorrelation analysis,is similar inthe threecollision systemsanddependsweaklyontheeventmultiplicityandtransverse momentum.Theseresultsbased onthesubeventmethods providestrongevidenceofasimilarlong-rangemulti-particlecollectivityin

pp,p+PbandperipheralPb+Pbcollisions.

©2019TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

One of the goals in the studies of azimuthal correlations in high-energy nuclear collisions at the Relativistic Heavy Ion Col-lider(RHIC)andtheLargeHadronCollider(LHC)istounderstand the multi-parton dynamics of QCD in the strongly coupled non-perturbative regime [1]. Measurements of azimuthal correlations in small collision systems, such as pp, p+A or d+A collisions, have revealed the ridge phenomenon [2–6]: enhanced produc-tion of particle pairs at small azimuthal angle separation,

φ,

extended over a wide range of pseudorapidity separation,



η

. The azimuthal structure has been related to harmonic modula-tion of particle densities, characterised by a Fourier expansion, dN/dφ

∝

1

+

2



∞_n=1vncos n(φ

− n

),

where vn and



n repre-sent the magnitude and the event-plane angle of the nth-order

flowharmonic.Theyarealsoconvenientlyrepresentedbytheflow vector: Vn

=

vneinn. The vn are knownto depend on the colli-sionsystem,buthaveweakdependenceoncollisionenergies [6,7]. Theridgereflectsmulti-partondynamicsearlyinthecollisionand hasgeneratedsignificantinterestinthehigh-energyphysics com-munity. A key question is whether the long-range multi-particle collectivityreflects initialmomentum correlation fromgluon

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

uration effects [8], ora final-statehydrodynamic response to the initialtransversecollisiongeometry [9].

Further insight into the ridge phenomenon is obtained via a multi-particlecorrelation technique, known ascumulants, involv-ing three or more particles [10–12]. The multi-particle cumu-lants probe the event-by-event fluctuation of a single flow har-monic vn, as well as the correlated fluctuations between two flowharmonics, vn andvm.Theseevent-by-eventfluctuationsare often represented by probability density distributions p

(

vn

)

and

p

(

vn

,

vm

),

respectively. For instance, the four-particle cumulants

cn{4

}

=



v4_n



−

2



v_n2

2

constrain the width of p

(

vn

)

[10], while thefour-particlesymmetriccumulantsscn,m{4

}

=



v_n2v2_m



−



v2_n

 

v_m2



quantifythelowest-ordercorrelationbetweenvn andvm[12].The three-particle asymmetric cumulantssuch as acn{3

}

=



V2_nV∗_2n



=



v2

nv2ncos 2n(n

− 2n

)



[5,13] aresensitivetocorrelations involv-ingboththeflowmagnitudevn andflowphase



n.

Oneofthechallengesinthestudyofazimuthalcorrelationsin smallcollisionsystemsishowtodistinguishthelong-rangeridge from“non-flow”correlationsinvolvingonlyafewparticles,suchas resonancedecays,jets, ordijets.Fortwo-particlecorrelations, the non-flowcontributioniscommonlysuppressedbyrequiringalarge



η

gap betweenthe two particlesin each pairand a peripheral subtractionprocedure [3–5,7,14,15]. Formulti-particlecumulants, thenon-flow contributionscan besuppressed byrequiring corre-lationbetween particlesfrom differentsubevents separatedin

η

,

https://doi.org/10.1016/j.physletb.2018.11.065

0370-2693/©2019TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 ac2{3

}

inpp collisions at s

=

13 TeV, p+Pbcollisionsat sNN

=

5.02 TeV andlow-multiplicityPb+Pbcollisionsat

√

sNN

=

2.76 TeV. They are obtained using two-, three- and four-subevent cumu-lant methods and are compared with results from the standard cumulantmethod.The cumulantsare normalised by the



vn2



ob-tainedfromatwo-particlecorrelationanalysis [7] toquantifytheir relative correlation strength. The measurements suggest that the resultsobtainedwiththe standard methodare strongly contami-natedbycorrelationsfromnon-flow sources.Theresultsobtained withthethree-subeventmethodorthefour-subeventmethod pro-videnewevidenceoflong-rangethree- orfour-particleazimuthal correlations.

TheLetterisorganisedasfollows.DetailsoftheATLASdetector, thetriggersystem, datasets,aswell aseventandtrackselections are providedin Sections2to 4.Section 5 describesthestandard andsubeventcumulantmethodsusedinthisanalysis.Theanalysis procedureandsystematicuncertaintiesaredescribedinSections6

and7,respectively.Themeasuredcumulantsarepresentedin Sec-tion8.AsummaryisgiveninSection9.

2. Detectorandtrigger

TheATLAS detector [20] providesnearly fullsolid-angle cover-age around the collision point with tracking detectors, calorime-ters,andmuon chambers,andis well suitedformeasurement of multi-particlecorrelationsoveralargepseudorapidityrange.1 The measurements were performed using primarily the inner detec-tor (ID),minimum-bias triggerscintillators (MBTS) andthe zero-degreecalorimeters(ZDC).TheIDdetectschargedparticleswithin

|

η

|

<

2.5 using a combination of a silicon pixel detector, a sili-conmicrostripdetector (SCT),andastraw-tubetransitionradiation tracker, all immersed in a 2 Taxial magnetic field [21]. An ad-ditionalpixel layer, the“insertable B-layer” (IBL) [22] is installed betweentheRun-1 (2010–2013)andRun-2 (2015–2018)periods. The MBTS detects charged particles within 2.1

 |

η

|



3.9 using two hodoscopes of counterspositioned at z

= ±

3.6 m. The ZDC, usedonlyinp+PbandPb+Pbcollisions,arepositionedat

±

140 m from the collision point, and detect neutral particles, primarily neutronsandphotons,with

|

η

|

>

8.3.

The ATLAS trigger system [23,24] consists of a first-level (L1) trigger implemented using a combination of dedicated electron-icsandprogrammablelogic, andahigh-level trigger(HLT) imple-mentedinprocessors.TheHLTreconstructscharged-particletracks

1 _{ATLAStypicallyusesaright-handedcoordinatesystemwithitsoriginatthe}

nominalinteractionpoint(IP)inthecentreofthedetectorandthe z-axisalong thebeampipe.Thex-axispointsfromtheIPtothecentreoftheLHCring,andthe y-axispointsupward.Cylindricalcoordinates(r,φ)areusedinthetransverseplane, φbeingtheazimuthalanglearoundthebeampipe.Bydefault,thepseudorapidityis definedintermsofthepolarangleθasη= −ln tan(θ/2).However,forasymmetric p+PborPb+p collisions,the−z directionisalwaysdefinedasthedirectionofthe Pbbeam.

andHMTtriggers.The minimum-biastriggerrequiredeithera hit inatleastoneMBTScounter,orahitinatleastoneMBTScounter oneachside,oratleastonereconstructedtrackattheHLTseeded by a random trigger at L1. More detailed information about the triggersusedforthepp and p+Pbdataandtheirperformancecan befoundinRefs. [7,25] andRefs. [5,26],respectively.

3. DatasetsandMonteCarlosimulations

This analysisis basedon ATLAS datasets corresponding to in-tegrated luminosities of 0.9 pb−1 of pp data recorded at

√

s

=

13 TeV, 28 nb−1 of p+Pbdatarecordedat

√

sNN

=

5.02 TeV,and 7 μb−1 of Pb+Pb data at

√

sNN

=

2.76 TeV. The 2.76 TeV Pb+Pb datawerecollectedin2010.The p+Pbdatawere mainlycollected in2013,butalsoinclude0.3 nb−1 ofdatacollectedin2016,which increase the numberofeventsatmoderate multiplicity (see Sec-tion4).Duringboth p+Pbruns,theLHCwasconfiguredtoprovide a4 TeVprotonbeamanda1.57 TeVper-nucleonPbbeam,which produced collisions at

√

sNN

=

5.02 TeV, witha rapidity shift of 0.465 of the nucleon–nucleon centre-of-mass frame towards the protonbeamdirectionrelativetotheATLAS restframe.The direc-tionofthePbbeamisalwaysdefinedtohavenegative pseudora-pidity. The 13 TeV pp data were collected during severalspecial runsoftheLHCwithlowpile-upin2015and2016.Asummaryof thedatasetsusedinthisanalysisisshowninTable1.

Thetrackreconstructionefficiencywasdeterminedusing simu-latedMonteCarlo(MC)eventsamples (Section4).The pp events

were simulatedwiththe Pythia8 MC eventgenerator [27] using the A2setoftunedparameters withMSTW2008LOparton distri-bution functions [28]. The HIJING eventgenerator [29] was used to producePb+Pb and p+Pbcollisions withthe sameenergyand the sameboost of thecentre-of-mass systemasin thedata. The detector response was simulated using Geant_{4 [30,31] with} de-tectorconditionsmatchingthoseduringthedata-taking.The sim-ulated events and data events are reconstructed with the same algorithms. The MC sample for Pb+Pb events in the multiplicity region of interest is very small, and so the track reconstruction efficiencyforPb+Pbwastakenfromthelarger p+Pbsample recon-structedwiththesamereconstructionalgorithm.Theefficiencyin

p+Pbevents was foundto be consistentwith theefficiencyfrom thePb+PbMCsimulation [17].

4. Eventandtrackselection

The offline eventselection forthe pp and p+Pbdata requires atleastonereconstructedvertexwithitslongitudinalposition sat-isfying

|

zvtx|

<

100 mmrelative to thenominalinteraction point. The vertexisrequiredtohaveatleasttwoassociatedtrackswith

pT

>

0.4 GeV. The mean number of collisions per bunch cross-ing,

μ

, was 0.002–0.8 forthe 13TeV pp data,0.03 for the2013

JID:PLB AID:34344 /SCO Doctopic: Experiments [m5Gv1.3; v1.250; Prn:4/01/2019; 8:57] P.3 (1-28) The ATLAS Collaboration / Physics Letters B•••(••••)•••–••• 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

p+Pb data,and 0.001–0.006for the 2016 p+Pb data.In order to suppressadditional interactions in the same bunch crossing (re-ferredtoaspile-up)in pp collisions,eventscontainingadditional verticeswithatleastfour associatedtracksare rejected. In p+Pb

collisions,eventswithmorethanonegoodvertex, definedasany vertexforwhichthescalarsumofthe pT oftheassociatedtracks isgreater than5GeV, are rejected. Theremaining pile-up events are furthersuppressed by using thesignal inthe ZDCin the di-rection of the Pb beam. This signal is calibrated to the number ofdetected neutrons,Nn,byusingthelocationofthepeak corre-spondingtoasingleneutron.ThedistributionofNn ineventswith pile-upisbroaderthanthatfortheeventswithoutpile-up.Hence asimplerequirementontheZDCsignaldistributionisusedto fur-thersuppresseventswithpile-up,whileretainingmorethan98% ofevents without pile-up.The impact of residual pile-up, atthe levelof



10−3,isstudiedbycomparingtheresultsobtainedfrom datawithdifferent

μ

values.

TheofflineeventselectionforthePb+Pbdatarequires

|

zvtx|

<

100 mm.Theselectionalsorequiresatimedifference

|

t

|

<

3 ns betweensignalsintheMBTStriggercountersoneithersideofthe interactionpointto suppressnon-collisionbackgrounds.A coinci-dence between the ZDC signals at forward and backward pseu-dorapidityisrequiredtorejecta varietyofbackgroundprocesses, whilemaintaininghighefficiencyforinelasticprocesses.The frac-tionofeventswithmorethanoneinteractionafterapplyingthese selectioncriteriaislessthan10−4_.

Charged-particletracksandcollision verticesarereconstructed usingalgorithms optimised forimproved performance for Run-2. Inorder to compare directly with the pp and p+Pb systems us-ing event selections based on the multiplicity of the collisions, a subset ofdata fromlow-multiplicity Pb+Pbcollisions, collected during the 2010 LHC heavy-ion run with a minimum-bias trig-ger,wasanalysedusingthesametrackreconstructionalgorithmas thatusedforp+Pbcollisions.ForthePb+Pband2013 p+Pb analy-ses,tracksarerequiredtohaveapT-dependentminimumnumber ofhitsinthe SCT. The transverse (d0) andlongitudinal (z0 sin

θ

) impactparametersofthetrackrelativetothevertexarerequired tobelessthan1.5 mm.Additionalrequirements

|

d0

|/

σ

d0

<

3 and

|

z0sin

θ

|/

σ

z0

<

3 are imposed, where

σ

d0 and

σ

z0 are the

un-certainties of the transverse and longitudinal impact parameter values,respectively. A more detailed description of the track se-lectionforthe2010Pb+Pbdataand2013 p+Pbdatacanbefound inRefs. [5,17].

Forallthedatatakensincethestart ofRun-2,thetrack selec-tioncriteriamakeuseoftheIBL,asdescribedinRefs. [14,25].For the pp and2016p+Pbanalyses,thetracksare requiredto satisfy

|

dBL₀

|

<

1.5 mm and

|

z0sin

θ

|

<

1.5 mm, wheredBL₀ is the trans-verseimpactparameterofthetrackrelativetothebeamline (BL). Thecumulantsarecalculatedusingtrackspassingtheabove se-lectionrequirements,andhaving

|

η

|

<

2.5 and 0.3

<

pT

<

3 GeV or 0.5

<

pT

<

5 GeV. These two pT ranges are chosen because they were often used in the previous ridge measurements at the LHC [6,7,14,15,17]. However, to count the number of recon-structed charged particles for event-class definition (denoted by

Nrec

ch), tracks with pT

>

0.4 GeV and

|

η

|

<

2.5 are usedfor com-patibility with the requirements in the HLT selections described above. Due to different trigger requirements, most of the p+Pb

eventswith Nrec_ch

>

150 are provided by the 2013 dataset,while the2016datasetprovidesmostoftheeventsatlower Nrec

ch. Theefficiency ofthe combined trackreconstruction andtrack selection requirements is estimated using MC samples recon-structedwiththe samealgorithms andselection requirementsas indata.Efficiencies,

(

η

,

pT

),

are evaluated asa functionof track

η

,pTandthenumberofreconstructedcharged-particletracks,but averagedoverthefullrangeinazimuth.Theefficienciesare

simi-larforeventswiththesamemultiplicity.Forallcollisionsystems, the efficiency increases by about 4% as track pT increases from 0.3 GeVto0.6GeV. Above0.6GeV, theefficiencyis independent of pT and reaches 86% (72%) for Run-1 pp and p+Pb, and 83% (70%)forPb+PbandRun-2 p+Pbcollisions,at

η

≈

0 (

|

η

|

>

2).The efficiency is independentof the eventmultiplicity for Nrec_ch

>

40. Forlower-multiplicityeventstheefficiencyissmallerbyupto3% duetobroaderd0andz0sin

θ

distributions [17].

The fraction offalsely reconstructed charged-particle tracks is alsoestimatedandfoundtobenegligiblysmallinalldatasets.This fractiondecreaseswithincreasingtrackpT,andevenatthelowest transversemomentaof0.3 GeVitisbelow1%ofthetotalnumber oftracks.Therefore,thereisnocorrectionforthepresenceofsuch tracksintheanalysis.

In the simulated events, the reconstruction efficiency reduces the measured charged-particle multiplicity relative to the gener-atedmultiplicityforprimarychargedparticles.Acorrectionfactor

b is used to correct N_chrec to obtain the efficiency-corrected aver-age number of charged particles per event,



Nch =b



Nrec

ch



. The valueofthecorrectionfactorisobtainedfromtheMCsamples de-scribed above, and isfound to be nearly independent of Nrec_ch in therangeusedinthisanalysis, Nrec

ch

<

400.Itsvalueandthe asso-ciated uncertainties are b

=

1.29

±

0.05 forthePb+Pb and2013

p+Pbcollisionsandb

=

1.18

±

0.05 forRun-2 p+Pb and pp

colli-sions [32].Bothscn,m{4

}

andac2{3

}

arethenstudiedasafunction

of



Nch.

5. Cumulantmethod

Themulti-particlecumulantmethod [10] hastheadvantageof directly reducing non-flow correlations from jetsand dijets. The mathematical framework for the standard cumulant is based on theQ-cumulantsdiscussedinRefs. [11,12,33].Itwasextended re-cently to the case of subevent cumulants in Refs. [13,16]. These methodsarebrieflysummarisedbelow.

5.1. Cumulantsinthestandardmethod

The standardcumulantmethodcalculatesk-particle azimuthal correlations,

{

k

}

, in one event using a complex number nota-tion [11,12]:

{

2

}

n

 =



ein(φ1−φ2)



_,

{

₃

}

n

 =



ein(φ1+φ2−2φ3)



_,



{

4

}

n,m



=



ein(φ1−φ2)+im(φ3−φ4)



,

(1)

where “



” denotes a single-event average over all pairs, triplets orquadruplets,respectively.Theaverages fromEq. (1) canbe ex-pressedintermsofper-particle normalisedflowvectorsq_n;l with

l

=

1,2...ineachevent [11]: q_n;l

≡



j wl_jeinφj





j wl_j

,

(2)

wherethesumrunsoveralltracksintheeventandwjisaweight assigned to the jth track. This weight is constructed to correct for both detectornon-uniformity and trackinginefficiency as ex-plainedinSection6.

The multi-particle asymmetric and symmetric cumulants are obtainedfrom

{

k

}

as:

acn

{

3

} = {

3

}

n

 ,

scn,m

{

4

} =



{

4

}

n,m



 − {

2

}

n

 {

2

}

m

 ,

(3) where“



” representsaweighted average of

{

k

}

overan event ensemble with similar Nrec

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

5.2. Cumulantsinthesubeventmethod

Inthestandardcumulantmethoddescribedabove,allk-particle

multipletsinvolvedin

{

k

}n

and



{

k

}n

,m



are selectedusingtracks in theentire ID acceptance of

|

η

|

<

η

max

=

2.5.To suppress fur-therthenon-flowcorrelationsthattypicallyinvolveafewparticles within alocalised regionin

η

, thetracksare dividedinto several subevents, each covering a unique

η

interval. The multi-particle correlations are then constructed by only correlating tracks be-tweendifferentsubevents.

Inthe two-subevent cumulantmethod, thetracks are divided into two subevents, labelled by a and b, according to

−

η

max

<

η

a

<

0 and 0

≤

η

b

<

η

max.The per-eventk-particle azimuthal cor-relationsareevaluatedas:

{

2

}

n



a|b

=



ein(φa1−φb2)



,

{

3

}

n



2a|b

=



ein(φ1a+φa2−2φ3b)



,



{

4

}

n,m



2a|2b

=



ein(φa1−φb2)+im(φa3−φb4)



,

(6)

where the superscript or subscript a (b) indicates tracks chosen fromthesubeventa (b).Here thethree- andfour-particle cumu-lantsaredefinedas:

ac2an|b

{

3

} = {

3

}

n



2a|b

,

scn2a,m|2b

{

4

} =



{

4

}

n,m





2a|2b

− {

2

}

n



a|b

{

2

}

m



a|b

.

Thetwo-subevent methodsuppressescorrelationswithina single jet(intra-jetcorrelations),since particlesfromonejet usuallyfall inonesubevent.

In the three-subevent cumulantmethod, tracks in each event are divided into three subevents a, b and c, each covering one third of the

η

range,

−

η

max

<

η

a

<

−

η

max

/3,

|

η

b|

≤

η

max

/3 and

η

max

/3

<

η

c

<

η

max.Themulti-particleazimuthalcorrelationsand cumulantsarethenevaluatedas:

{

3

}

n



a,b|c

=



ein(φ1a+φ2b−2φc3)



,



{

4

}

n,m



a,b|2c

=



ein(φa1−φ2c)+im(φ3b−φ4c)



,

(7) and aca_n,b|c

{

3

} = {

3

}

n



a,b|c

,

scan,,bm|2c

{

4

} = {

4

}

n,m



a,b|2c

− {

2

}

n



a|c

{

2

}

m



b|c

.

(8) Since a dijet event usually produces particles in at most two subevents, the three-subevent method efficiently suppresses the non-flow contribution from inter-jet correlations associated with dijets. To maximise the statistical precision, the

η

range for subevent a is swapped with that for subevent b or c, and the resultsareaveragedtoobtainthefinalvalues.

precision, the

η

ranges forthe foursubevents are swapped with eachother,andtheresultsareaveragedtoobtainthefinalvalues.

5.3. Normalisedcumulants

Althoughthecumulantsreflectthenatureofthecorrelation be-tween vn andvm,their magnitudesalsodepend onthesquareof single flowharmonics v2

n andv2m,seeEq. (4).Thedependenceon the single flow harmonics can be scaled out via the normalised cumulants [34,35]: nsc2,3

{

4

} =

sc2,3

{

4

}

v2

{

2

}

2v3

{

2

}

2

=



v2₂v2₃





v2₂

 

v2₃

 −

1

,

(11) nsc2,4

{

4

} =

sc2,4

{

4

}

v2

{

2

}

2v4

{

2

}

2

=



v2 2v24





v2 2

 

v2 4

 −

1

,

(12) nac2

{

3

} =

ac2

{

3

}

2v2

{

2

}

4

₊

_c 2

{

4

}



c4

{

2

}

=



v2₂v4cos 4

(

2

− 

4

)





v4₂

 

v2₄



,

(13)

wherethevn{2

}

2

=



v_n2



areflowharmonicsobtainedusinga two-particlecorrelationmethodbasedonaperipheralsubtraction tech-nique [7,14],andc2{4

}

=



v4 2



−

2



v2 2

2

are four-particlecumulant resultsfromRefs. [17,18].Thisdefinitionfornac2{3

}

ismotivated byRef. [36].

6. Analysisprocedure

The measurementofthescn,m{4

}

andac2{3

}

followsthesame

analysis procedure as for the four-particle cumulants cn{4

}

in Ref. [18].Themulti-particlecumulantsarecalculatedinthreesteps using charged particles with

|

η

|

<

2.5. In the first step,

{

2

}n

,

{

3

}n

and



{

4

}n

,m



from Eqs. (1), (6), (7) and (9) are calculated for each eventfrom particles in one oftwo different pT ranges, 0.3

<

pT

<

3 GeVand 0.5

<

pT

<

5 GeV. The numbers of recon-structedchargedparticlesinthesepT rangesaredenotedby Nsel1_ch andNsel2

ch ,respectively.

In the second step,the correlators

{

k

}

for0.3

<

pT

<

3 GeV (0.5

<

pT

<

5 GeV)areaveraged overeventswiththesame Nsel1_ch (Nsel2_ch )toobtain

{

k

}

,andthensc2,3{4

}

,sc2,4{4

}

andac2{3

}

.The

sc2,3{4

}

, sc2,4{4

}

and ac2{3

}

values are then averaged in broader

multiplicityrangesoftheeventensemble,weightedbynumberof events,toobtainstatisticallysignificantresults.

In the third step, the sc2,3{4

}

, sc2,4{4

}

andac2{3

}

values

ob-tained fora given Nsel1_ch or N_chsel2 are mapped to



N_chrec



, the aver-agenumberofreconstructedchargedparticleswithpT

>

0.4 GeV.

JID:PLB AID:34344 /SCO Doctopic: Experiments [m5Gv1.3; v1.250; Prn:4/01/2019; 8:57] P.5 (1-28) The ATLAS Collaboration / Physics Letters B•••(••••)•••–••• 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 Themappingprocedureisnecessarysothatsc2,3{4

}

,sc2,4{4

}

and

ac2

{

3

}

obtainedforthe twodifferent pT rangescanbe compared usinga commonx-axisdefinedby



Nrec_ch



.The



Nrec_ch



value isthen converted to



Nch, the efficiency-corrected average number of chargedparticleswith pT

>

0.4 GeV,asdiscussedinSection4.

In order to account for detector inefficiencies and non-uni-formity,particleweightsusedinEq. (2) aredefinedas:

w

(φ,

η

,

pT

)

=

d

(φ,

η

)/

(

η

,

pT

) .

Theadditionalweightfactord

(φ,

η

)

accountsfornon-uniformities in the azimuthal acceptance of the detector as a function of

η

. AllreconstructedchargedparticleswithpT

>

0.3 GeVareentered intoa two-dimensionalhistogram N

(φ,

η

),

andthe weightfactor isthenobtainedasd

(φ,

η

)

≡ 

N

(

η

)

 /

N

(φ,

η

),

where



N

(

η

)



isthe trackdensityaveragedover

φ

inthe given

η

bin.Thisprocedure removesmostofthe

φ-dependent

non-uniformityinthedetector acceptance [17].

In order to calculate the normalised cumulants from Eqs. (11)–(13),theflowharmonicsvn{2

}

areobtainedfroma“template fit”oftwo-particle

φ

correlationasdescribedinRefs. [7,14].The

vn{2

}

values are calculated identically to the procedure used in theprevious ATLAS publications [7,14], butare further corrected fora bias,which existsonly if vn{2

}

changeswith N_chrec.The de-tailsofthecorrection procedurearegivenintheAppendixAand arediscussedbrieflybelow.

ThestandardprocedureofRefs. [7,14] firstconstructsa

φ

dis-tributionforpairsoftrackswith

|

η

|

>

2:theper-trigger-particle yield Y

(φ)

foragiven Nrec

ch range.The dominatingnon-flow jet peak at

φ

∼

π

is estimatedusing low-multiplicity eventswith

N_chrec

<

20 andseparatedviaatemplatefitprocedure,andthe har-monic modulation of the remaining component is taken as the

vn{2

}

2 _[7]: Y

(φ)

=

F Y

(φ)

peri

+

Gtmp



1

+

2 ∞



n=2 vn

{

2

,

tmp

}

2cos n

φ

,

where superscripts “peri” and “tmp” indicate quantities for the

N_chrec

<

20 eventclassandquantitiesafterthetemplatefitforthe eventclassofinterest,respectively.ThescalefactorF andpedestal

Gtmp are fixed by the fit, and vn{2,tmp

}

are calculated from a Fouriertransform.Thisprocedureimplicitlyassumesthat vn{2

}

is independentof Nrec

ch,andrequiresasmallcorrectionifvn{2

}

does changewithN_chrec(AppendixA).In p+PbandPb+Pbcollisions,this correctioninthe Nrec_ch

>

100 region amountsto a2–6% reduction forv2{2,tmp

}

anda4–9%reductionforv3{2,tmp

}

andv4{2,tmp

}

. The correction is smaller for v2{2,tmp

}

in pp collisions as it is nearlyindependentofNrec_ch [7].

7.Systematicuncertainties

The evaluation of the systematic uncertainties follows closely theprocedureestablishedforthefour-particlecumulantscn{4

}

and describedinRef. [18].Themainsourcesofsystematicuncertainties arerelatedto thedetectorazimuthal non-uniformity,track selec-tion,trackreconstruction efficiency,triggerefficiencyandpile-up. Duetotherelativelypoorstatisticsandlargernon-floweffects,the systematicuncertainties are typically larger in pp collisions. The systematic uncertainties are also generally larger, in percentage, forfour-particlecumulantsscn,m{4

}

thanforthethree-particle

cu-mulantsac2{3

}

,sincethe

|

scn,m{4

}|

valuesare muchsmallerthan

thoseforac2{3

}

.Thesystematicuncertaintiesaregenerallysimilar among the two- and three- and four-subevent methods, but are different fromthose for the standard method, which is strongly

influenced by non-flow correlations. The following discussion fo-cusesonthethree-subeventmethod,whichisthedefaultmethod usedtopresentthefinalresults.

The effect of detector azimuthal non-uniformity is accounted for usingthe weight factord

(φ,

η

).

The impact ofthe weighting procedure isstudied by fixing the weight tounity andrepeating the analysis. The results are mostly consistent with the nominal results. The corresponding uncertainties for scn,m{4

}

vary in the

rangeof 0–4%,0–2% and1–2% in pp, p+PbandPb+Pbcollisions, respectively.Theuncertaintiesforac2{3

}

varyintherangeof0–2% in pp collisions, and0–1% in p+Pb andPb+Pb collisions, respec-tively.

The systematicuncertaintyassociated withthetrackselection is estimated by tightening the

|

d0| and

|

z0sin

θ

|

requirements. They are each varied from the default requirement of less than 1.5 mmto lessthan 1 mm.In p+PbandPb+Pbcollisions,the re-quirementonthe significanceofimpactparameters,

|

d0|/

σ

d0 and

|

z0sin

θ

|/

σ

z0 are alsovaried from lessthan 3 toless than2. For

eachvariation,thetrackingefficiencyisre-evaluatedandthe anal-ysisisrepeated.Forac2{3

}

,whichhasalargeflowsignal,the dif-ferencesfromthenominalresultsareobservedtobelessthan2% forallcollisionsystems.Forscn,m{4

}

,forwhichthesignalissmall,

the differences from the nominal results are found to be in the rangeof2–10%inpp collisions,2–7%in p+Pbcollisionsand2–4% inPb+Pbcollisions.Thedifferencesaresmallerforresultsobtained for0.5

<

pT

<

5 GeVthanthoseobtainedfor0.3

<

pT

<

3 GeV.

Previousmeasurementsindicatethattheazimuthalcorrelations (both the flow and non-flow components) have a strong depen-denceon pT,butarelativelyweak dependenceon

η

[5,7]. There-fore, pT-dependent systematic effects inthe track reconstruction efficiencycouldaffectthecumulantvalues.Theuncertaintyinthe trackreconstruction efficiencyismainlyduetodifferencesinthe detectorconditionsandmaterialdescription betweenthe simula-tion and the data. The efficiency uncertainty varies between 1% and4%,dependingontrack

η

and pT [7,17].Its impacton multi-particlecumulantsisevaluatedby repeatingtheanalysiswiththe trackingefficiencyvariedupanddownbyitscorresponding uncer-taintyasafunctionoftrackpT.Forthestandardcumulantmethod, which is more sensitive to jets and dijets, the evaluated uncer-taintyamounts to2–6%inpp collisionsandlessthan2%in p+Pb

collisionsfor



Nch >100.Forthesubeventmethods,theevaluated uncertaintyistypicallylessthan3%formostofthe



Nchranges.

Mostevents in pp and p+Pb collisions are collected with the HMTtriggers withseveralonline Nrec_ch thresholds. Inorderto es-timate the possible bias dueto trigger inefficiency asa function of



Nch,the offline Nrec

ch requirementsare changedsuch that the HMTtrigger efficiencyisatleast50% or80%. Theresultsare ob-tained independently foreach variation. These results are found to be consistent witheach other for the subevent methods, and show some differencesfor thestandard cumulant methodin the low



Nchregion.Thenominalanalysisisperformedusingthe50% efficiency selection and the differencesbetween the nominal re-sults andthosefrom the80% efficiencyselection are included in thesystematicuncertainty.Thechangesforpp collisionsareinthe rangeof5–15%forsc2,3{4

}

,2–8%forsc2,4{4

}

and1–5%forac2{3

}

.

Therangesfor p+Pbcollisions aremuchsmallerdueto themuch sharperturn-onofthetriggerefficiencyandlargersignal:theyare estimatedtobe 1–3%forsc2,3{4

}

,2–4%forsc2,4{4

}

and1–2%for

ac2{3

}

.

Inthisanalysis,apile-uprejectioncriterionisappliedtoreject eventscontainingadditionalverticesin pp andp+Pbcollisions.In order to check the impact ofresidual pile-up, theanalysis is re-peated without thepile-up rejectioncriterion. No differencesare observed in p+Pbcollisions, asisexpectedsince the

μ

valuesin

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 certaintiesaregenerallysmallerthanthestatisticaluncertainties.

The vn

{

2

}

values used to obtain normalised cumulants from Eqs. (11)–(13) aremeasuredfollowingtheprescriptionofthe pre-vious ATLAS publications [7,14], resultinginvery similar system-atic uncertainties. The correction for the bias ofthe template fit procedure,asdescribedinSection6,reducesthesensitivitytothe choiceoftheperipheral Nrec

ch bin.Theuncertaintiesofnormalised cumulantsare obtainedby propagation ofthe uncertainties from theoriginalcumulantsandvn{2

}

,takingintoaccountthatthe cor-relatedsystematicuncertaintiespartiallycancelout.

8. Results

The resultsare presented intwo parts.Section 8.1 presentsa detailedcomparisonbetween thestandard method andsubevent methods to demonstrate the ability of the subevent methods to suppress non-flow correlations. Section 8.2 compares the cumu-lantsamong pp,p+PbandPb+Pbcollisionstoprovideinsightinto thecommonnatureofcollectivityinthesesystems.

8.1. Comparisonbetweenstandardandsubeventmethods

The top row of Fig. 1 compares the sc2,3{4

}

values obtained

fromthe standard, two-,three- and four-subevent methods from

pp collisionsin0.3

<

pT

<

3 GeV(leftpanel)and0.5

<

pT

<

5 GeV (right panel). The values from the standard method are positive overthe full



Nchrange,andarelarger atlower



Nchorinthe higher pT range.This behaviour suggeststhat thesc2,3{4

}

values

fromthe standard method in pp collisions, including those from Ref. [19], are stronglyinfluenced by non-flow effects inall



Nch

and pT ranges [16]. In contrast, the values from the subevent methods are negative over the full



Nch range, and they are slightlymorenegative at lowest



Nch andalsomore negative at higher pT.The resultsareconsistent amongthevarioussubevent methods for 0.3

<

pT

<

3 GeV. Forthe high pT region of 0.5

<

pT

<

5 GeV,resultsfromthetwo-subeventmethodare systemati-callylowerthanthosefromthethree- andfour-subeventmethods, suggestingthatthetwo-subeventmethodmaybeaffectedby neg-ative non-flow contributions. Such negative non-flow correlation hasbeenobservedinaPythia8 calculation [16].

Themiddle rowofFig.1 showssc2,3{4

}

from p+Pbcollisions.

At



Nch >140, thevaluesare negative andconsistent amongall fourmethods,reflectinggenuinelong-rangecollectivecorrelations. At



Nch <140, the values are different between the standard method and the subevent methods. The sc2,3{4

}

from the

stan-dardmethodchangessignaround



Nch ∼80 andremainspositive atlower



Nch,reflectingthe contributionfromnon-flow correla-tions.Incontrast,thesc2,3{4

}

fromvarioussubeventmethodsare

negativeandconsistentwitheachotherat



Nch <140,suggesting thattheymainlyreflectthegenuinelong-rangecorrelations.

methodis moreaffectedby dijets.Significantdifferencesare also observed between the two-subevent andthree- or four-subevent methods at low



Nch, butthese differencesdecrease and disap-pear for



Nch >100. Within the statistical uncertainties of the measurement,nodifferencesareobservedbetweenthethree- and four-subevent methods. This comparison suggests that the two-subeventmethodmaynotbesufficienttorejectnon-flow correla-tionsfromdijetsin pp collisions,andmethodswiththreeormore subeventsarerequiredtosuppressthenon-flowcontributionover themeasured



Nchrange.

The middlerowofFig. 2showssc2,4{4

}

from p+Pbcollisions.

Significantdifferencesareobservedbetweenthestandardmethod and the subevent methods over the full



Nch range. However, nodifferencesareobservedamongthevarioussubeventmethods. These results suggest that the standard method is contaminated by large contributions from non-flow correlations at low



Nch, and thesecontributions maynot vanish even at large



Nch val-ues.All subeventmethods suggest an increase ofsc2,4{4

}

toward

lower



Nchfor



Nch <40,whichmayreflectsome residual non-flowcorrelationsinthisregion.

ThebottomrowofFig.2showssc2,4{4

}

fromPb+Pbcollisions.

Thesc2,4{4

}

valuesincreasegraduallywith



Nchforallfour

meth-ods. This increase reflects the known fact that the v2 increases with



NchinPb+Pbcollisions [37].The valuesfromthe standard method are systematically larger than those from the subevent methods, and thisdifference varies slowly with



Nch, similar to the behaviour observed in p+Pb collisions in the high



Nch re-gion.

The resultsfortheasymmetric cumulantac2{3

}

are presented in Fig.3.The top rowshowsthe resultsobtainedfromthe stan-dard, two-subevent, and three-subevent methods from pp

colli-sions in 0.3

<

pT

<

3 GeV (left panel) and 0.5

<

pT

<

5 GeV (right panel).Theresultsare positive forallmethods. Theresults from the standard method are much larger than those from the subevent methods,consistent withtheexpectationthatthe stan-dardmethodismoreaffectedbynon-flowcorrelationsfromdijets. Significantdifferencesarealsoobservedbetweenthetwo-subevent and three-subevent methods at low



Nch, but these differences decrease and disappear at



Nch >100. The ac2{3

}

values from the three-subeventmethod showa slightincrease for



Nch <40 but are nearly constant for



Nch >40. This behaviour suggests thatinthethree-subeventmethod,thenon-flowcontributionmay play some role at



Nch <40, but is negligible for



Nch >40. Therefore, the ac2{3

}

from the three-subevent method supports the existenceofa three-particlelong-rangecollectiveflowthat is nearly independent of



Nchin pp collisions, consistent withthe



Nch-independentbehaviourof v2 andv4 observedpreviouslyin thetwo-particlecorrelationanalysis [7].

The middleandbottomrowsofFig.3show ac2{3

}

fromp+Pb

and Pb+Pb collisions, respectively. The ac2{3

}

values from the standard method have a significant non-flow contribution up to

JID:PLB AID:34344 /SCO Doctopic: Experiments [m5Gv1.3; v1.250; Prn:4/01/2019; 8:57] P.7 (1-28) The ATLAS Collaboration / Physics Letters B•••(••••)•••–••• 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Fig. 1. Thesymmetriccumulantsc2,3{4}asafunctionofNchfor0.3<pT<3 GeV(leftpanels)and0.5<pT<5 GeV (rightpanels)obtainedforpp collisions(toprow),

p+Pbcollisions(middlerow)andlow-multiplicityPb+Pbcollisions(bottomrow).Ineachpanel,thesc2,3{4}isobtainedfromthestandardmethod(filledsymbol),the

two-subeventmethod(opencircles),three-subeventmethod(opensquares)andfour-subeventmethod(opendiamonds).Theerrorbarsandshadedboxesrepresentthe statisticalandsystematicuncertainties,respectively.



Nch ∼200 inp+Pbcollisionsand



Nch ∼80 inPb+Pbcollisions. Inthesubeventmethods, theinfluenceofnon-flow contributions isverysmall for



Nch >60 in both collisionsystems,and there-forethe



Nchdependenceofac2{3

}

reflectsthe



Nchdependence ofthe v2 and v4. The ac2{3

}

values fromthe subevent methods increase with



Nch, andthe increase is stronger in Pb+Pb colli-sions. This is consistent with previous observations that v2 and

v4 increasewith



NchmorestronglyinPb+Pbthanin p+Pb colli-sions [17].

Thevaluesof sc2,4{4

}

andac2{3

}

,whichare bothmeasures of

correlations between v2 and v4, show significant differences be-tweenthestandard methodandthesubeventmethods,asshown in Figs. 2 and3. The



Nch dependence of thesedifferences de-creasesgraduallywith



Nch,andisconsistentwithaninfluenceof non-flowthatisexpectedtoscaleas1/



Nch.However,these dif-ferencesseemtopersistfor



Nch >200 inp+Pbcollisionsandfor



Nch >150 inPb+Pbcollisions,whichisnot compatiblewiththe predicted behaviour of non-flow correlations. The differences at

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Fig. 2. Thesymmetriccumulantsc2,4{4}asafunctionofNchfor 0.3<pT<3 GeV(leftpanels)and 0.5<pT<5 GeV (rightpanels)obtainedfor pp collisions(top

row),p+Pbcollisions(middlerow)andlow-multiplicityPb+Pbcollisions(bottomrow).Ineachpanel,thesc2,4{4}isobtainedfromthestandardmethod(filledsymbol),

two-subeventmethod(opencircles),three-subeventmethod(opensquares)andfour-subeventmethod(opendiamonds).Theerrorbarsandshadedboxesrepresentthe statisticalandsystematicuncertainties,respectively.

large



Nchmayarisefromlongitudinalflowdecorrelations [38,39], whichhavebeenmeasuredbyCMS [40] andATLAS [41]. Decorre-lationeffectsarefound tobelarge forv4 andstronglycorrelated with v2, and therefore they are expected to reduce the sc2,4{4

}

andac2{3

}

in the subevent method.Therefore, the observed dif-ferencesbetweenthe standard methodandsubevent method re-flectthecombinedcontributionfromnon-flowcorrelations,which dominates in the low



Nch region, and decorrelation, which is moreimportant atlarge



Nch (see furtherdiscussion in the Ap-pendixB).

The results presented above suggest that the three-subevent method is sufficient to suppress mostof the non-flow effects. It isthereforeusedasthedefaultmethodforthediscussionbelow.

8.2. Comparisonbetweencollisionsystems

Fig. 4 shows a direct comparison of cumulants for the three collisionsystems.Thethreepanelsinthetoprowshowtheresults forsc2,3{4

}

,sc2,4{4

}

andac2{3

}

,respectively,for0.3

<

pT

<

3 GeV. These results support the existence of a negative correlation be-tween v2 and v3 anda positive correlation between v2 and v4.

JID:PLB AID:34344 /SCO Doctopic: Experiments [m5Gv1.3; v1.250; Prn:4/01/2019; 8:57] P.9 (1-28) The ATLAS Collaboration / Physics Letters B•••(••••)•••–••• 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Fig. 3. Theasymmetriccumulantac2{3}asafunctionofNchfor0.3<pT<3 GeV(leftpanels)and0.5<pT<5 GeV(rightpanels)obtainedforpp collisions(toprow),p+Pb

collisions(middlerow)andlow-multiplicityPb+Pbcollisions(bottomrow).Ineachpanel,theac2{3}isobtainedfromthestandardmethod(filledsymbol),two-subevent

method(opencircles),andthree-subeventmethod(opensquares).Theerrorbarsandshadedboxesrepresentthestatisticalandsystematicuncertainties,respectively.

Such correlation patternshave previously been observedin large collisionsystems [42–44],butarenowconfirmedalsointhesmall collisionsystems,oncenon-floweffectsareadequatelysuppressed. Inthemultiplicityrangecoveredbythepp collisions,



Nch <150, theresultsforsymmetriccumulantssc2,3{4

}

andsc2,4{4

}

are

sim-ilaramongthethreesystems.Inthe range



Nch >150,

|

sc2,3{4

}|

andsc2,4{4

}

arelargerinPb+Pbthaninp+Pbcollisions.Theresults

for ac2{3

}

are similar among the three systems at



Nch <100, buttheydeviatefromeachotherathigher



Nch.The pp dataare approximatelyconstantordecreaseslightlywith



Nch,whilethe

p+Pb andPb+Pb datashow significant increasesasa function of



Nch.The bottomrowshowstheresultsforthehigher pT range of0.5

<

pT

<

5 GeV,wheresimilartrendsareobserved.

Fig. 5 shows the results for normalised cumulants, nsc2,3{4

}

,

nsc2,4{4

}

and nac2{3

}

, compared among the three systems. The

normalised cumulants generally show a much weaker



Nch de-pendence at



Nch >100, where the statistical uncertainties are small.Thisbehaviourimpliesthatthestrong



Nchdependenceof thescn,m{4

}

andac2{3

}

valuesreflectsthe



Nchdependenceofthe

vn values,andthesedependencesare removedinthenormalised cumulants.Thenormalisedcumulantsare alsosimilaramong dif-ferentcollision systemsat large



Nch, althoughsome differences

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Fig. 4. TheNchdependenceofsc2,3{4}(leftpanels),sc2,4{4}(middlepanels)andac2{3}(rightpanels)in0.3<pT<3 GeV(toprow)and0.5<pT<5 GeV(bottomrow)

obtainedforpp collisions(solidcircles),p+Pbcollisions(opencircles)andlow-multiplicityPb+Pbcollisions(opensquares).Theerrorbarsandshadedboxesrepresentthe statisticalandsystematicuncertainties,respectively.

attherelativelevelof20–30%areobservedforsmaller



Nch.The onlyexception isnsc2,3{4

}

, whosevaluesinthe pp collisionsare

verydifferentfromthoseinp+PbandPb+Pbcollisions.Incontrast, the sc2,3{4

}

values in Fig. 4 are close among different systems.

Thissuggeststhatthe



v2₃



valuesfromthetemplatefitmethod [7] maybesignificantlyunderestimated.AspointedoutinRef. [7] and emphasised in Appendix A, the template fit method, and other methods based on peripheral subtraction in general [5,15], tend tounderestimatetheodd flowharmonics,duetothepresenceof alarge away-sidepeak at

φ

∼

π

inthe two-particlecorrelation function.Thecomparisonofsc2,3{4

}

andnsc2,3{4

}

amongdifferent

collision systems provides indirect evidence ofthis underestima-tionof



v2₃



.

Fig.5showsthat thenormalisedcumulantsare consistent be-tween0.3

<

pT

<

3 GeVand0.5

<

pT

<

5 GeV.Ontheotherhand, themagnitudesofthecumulantsinFig.4differby alarge factor betweenthetwopTranges:aboutafactorofthreeforsc2,3{4

}

and

sc2,4{4

}

,andafactoroftwo forac2{3

}

.Theseresultssuggestthat

the pT dependenceofsc2,3{4

}

,sc2,4{4

}

andac2{3

}

largelyreflects

thepTdependenceofthe vnatthesingle-particlelevel.

9. Discussion

Three- and four-particle cumulants involving correlations be-tweentwo harmonicsofdifferentorder vn and vm aremeasured in

√

s

=

13 TeV pp,

√

sNN

=

5.02 TeV p+Pb, andlow-multiplicity

√

sNN

=

2.76 TeV Pb+Pb collisions with the ATLAS detector at theLHC, withtotalintegratedluminosities of0.9 pb−1,28 nb−1, and 7 μb−1, respectively. The correlation between vn and vm is studied using four-particle symmetric cumulants, sc2,3{4

}

and

sc2,4{4

}

, andthe three-particle asymmetric cumulantac2{3

}

. The

symmetric cumulantsscn,m{4

}

=



v2_nv_m2



−



v2_n

 

v2_m



probethe cor-relation ofthe flow magnitudes, whilethe asymmetric cumulant ac2{3

} =



v2

2v4cos 4(2

− 4

)



issensitivetocorrelationsinvolving both the flow magnitude vn and flow phase



n. Theyare calcu-lated using the standard cumulant method, aswell asthe two-, three- and four-subevent methods to suppress non-flow effects. Thefinalresultsarepresentedasafunctionoftheaveragenumber ofchargedparticleswithpT

>

0.4 GeV,



Nch.

Significant differences are observed between the standard method and the subevent methods over the full



Nch range in

pp collisions, as well as over the low



Nch range in p+Pb and Pb+Pb collisions.The differencesare largerforparticles athigher

pT oratsmaller



Nch.Whenanalysedwiththestandardmethod in pp collisions, thisbehaviouriscompatiblewiththedominance ofthe non-flow correlationsratherthanthe long-rangecollective flow correlations. Systematic, but much smaller, differences are also observedinthe low



Nch regionbetweenthe two-subevent methodandthree- orfour-subeventmethods,whichindicatethat the two-subevent method may still be affected by correlations arising from jets. Onthe other hand nodifferences are observed between the three-subevent and four-subevent methods, within experimentaluncertainties,suggestingthatmethodswiththreeor moresubeventsaresufficienttorejectnon-flow correlationsfrom jets. Therefore,thethree-subevent methodisused topresentthe mainresultsinthisanalysis.

The three-subevent method provides ameasurement of nega-tive sc2,3{4

}

andpositive sc2,4{4

}

and ac2{3

}

over nearly the full



Nch range and in all three collision systems. These results in-dicate a negative correlation between v2 and v3 and a positive

JID:PLB AID:34344 /SCO Doctopic: Experiments [m5Gv1.3; v1.250; Prn:4/01/2019; 8:57] P.11 (1-28) The ATLAS Collaboration / Physics Letters B•••(••••)•••–••• 11 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Fig. 5. TheNchdependenceofnsc2,3{4}(leftpanels),nsc2,4{4}(middlepanels)andnac2{3}(rightpanels)in0.3<pT<3 GeV(toprow)and0.5<pT<5 GeV(bottom

row)obtainedforpp collisions(solidcircles),p+Pbcollisions(opencircles)andlow-multiplicityPb+Pbcollisions(opensquares).Theerrorbarsandshadedboxesrepresent thestatisticalandsystematicuncertainties,respectively.

correlationbetweenv2andv4.Suchcorrelationpatternshave pre-viouslybeen observed inlarge collisionsystems [42–44], butare now confirmed in small collision systems, once non-flow effects areadequatelysuppressed.Thevaluesofsc2,3{4

}

andsc2,4{4

}

are

consistent in pp and p+Pbcollisions over the same



Nch range, buttheirmagnitudes atlarge



Nchare muchsmallerthan those forPb+Pbcollisions. The valuesof ac2{3

}

are similar atvery low



Nch among the three systems, but are very different at large



Nch.Ontheother hand,afterscalingbythe



v2_n



estimatedfrom atwo-particleanalysis [7,14],the resultingnormalisedcumulants nsc2,3{4

}

,nsc2,4{4

}

andnac2{3

}

showamuchweakerdependence

on



Nch, andtheir valuesare much closerto each other among the three systems. The magnitudes of the normalised cumulants are also similar to each other for 0.5

<

pT

<

5 GeV as well as 0.3

<

pT

<

3 GeV. This suggests that the



Nch, pT and system dependenceofthe sc2,3{4

}

, sc2,4{4

}

andac2{3

}

reflectmostly the



Nch,pTandsystemdependenceof



vn2



,buttherelativestrengths ofthecorrelationsaresimilarforthethreecollisionsystems.

The new results obtained with the subevent cumulant tech-nique provide further evidence that the ridge is indeed a long-rangecollective phenomenoninvolvingmanyparticlesdistributed across a broad rapidity interval. The similarity between differ-entcollisionsystemsfornsc2,3{4

}

,nsc2,4{4

}

andnac2{3

}

,andthe

weakdependenceoftheseobservablesonthepT rangeand



Nch, largelyfreefromnon-floweffects,providean importantinput to-wardsunderstandingthespace–timedynamicsandtheproperties of the medium created in small collision systems. These results provideinputstodistinguishbetweenmodelsbasedoninitial-state momentumcorrelationsandmodelsbasedonfinal-state hydrody-namics.

Acknowledgements

We thank CERN forthe very successfuloperation of the LHC, aswell as thesupport staff fromour institutionswithout whom ATLAScouldnotbeoperatedefficiently.

WeacknowledgethesupportofANPCyT,Argentina;YerPhI, Ar-menia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azer-baijan; SSTC, Belarus; CNPq andFAPESP, Brazil; NSERC, NRC and CFI,Canada; CERN; CONICYT, Chile;CAS, MOSTandNSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic;DNRFandDNSRC,Denmark;IN2P3-CNRS,CEA-DRF/IRFU, France; SRNSFG, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece;RGC,Hong KongSAR, China;ISFandBenoziyoCenter, Is-rael; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; NWO, Netherlands; RCN,Norway;MNiSW andNCN, Poland;FCT, Portu-gal;MNE/IFA,Romania;MESofRussiaandNRCKI,Russian Feder-ation;JINR;MESTD,Serbia;MSSR,Slovakia;ARRSandMIZŠ, Slove-nia; DST/NRF, South Africa; MINECO, Spain; SRC andWallenberg Foundation,Sweden;SERI,SNSFandCantons ofBernandGeneva, Switzerland;MOST,Taiwan;TAEK, Turkey;STFC,UnitedKingdom; DOE and NSF, United States of America. In addition, individ-ual groupsandmembers havereceived support fromBCKDF, the Canada Council, CANARIE,CRC, Compute Canada,FQRNT,and the OntarioInnovation Trust,Canada; EPLANET,ERC,ERDF, FP7, Hori-zon 2020 and Marie Skłodowska-Curie Actions, European Union; Investissements d’Avenir Labex and Idex, ANR, Région Auvergne andFondationPartagerleSavoir,France;DFGandAvHFoundation, Germany;Herakleitos,ThalesandAristeiaprogrammesco-financed byEU-ESF andtheGreekNSRF;BSF,GIFandMinerva, Israel;BRF, Norway; CERCA Programme Generalitat de Catalunya, Generalitat

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Fig. 6. Thevaluesofvn{2,tmp}2obtainedfollowingthetemplatefitproceduregiveninEq. (14) [7] inpp collisionsforn=2 (leftpanel),n=3 (middlepanel)andn=4 (rightpanel).Ineachpanel,thevaluesarecalculatedforthreeperipheralNrecch intervals:N

rec ch <20,N

rec

ch <10 and10≤N rec

ch <20.Onlystatisticaluncertaintiesareshown.

Valenciana,Spain;theRoyalSocietyandLeverhulmeTrust,United Kingdom.

The crucial computingsupport fromall WLCG partners is ac-knowledged gratefully, in particular from CERN, the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Swe-den),CC-IN2P3(France),KIT/GridKA(Germany),INFN-CNAF(Italy), NL-T1(Netherlands),PIC(Spain),ASGC(Taiwan),RAL(UK)andBNL (USA),theTier-2facilitiesworldwideandlargenon-WLCGresource providers.Majorcontributorsofcomputingresourcesarelisted in Ref. [45].

Appendix A. Improvementtothetemplatefitprocedure

In order to separate the long-range ridge from other non-flow sources,especially dijets,theATLAS Collaborationdeveloped atemplatefittingproceduredescribedinRefs. [7,14].Thefirststep istoconstructa

φ

distributionofparticlepairswithlarge pseu-dorapidityseparation

|

η

|

>

2,theso-called“per-trigger”particle yield, Y

(φ),

for a given Nrec_ch range. The

|

η

|

>

2 requirement suppresses the intra-jet and other short-range correlations, and in small collision systems the resulting Y

(φ)

distributions are knowntobedominatedbyaway-sidejetcorrelations [4,5,14].This away-sidenon-flowcomponentispeakedat

φ

∼

π

,andleadsto asignificantbiasintheflowcoefficientsvn,especiallyfortheodd harmonics.

Tosubtracttheaway-sidejetcorrelations,themeasuredY

(φ)

distribution in a given Nrec

ch interval is assumed to be a sum ofa scaled “peripheral” distribution Y

(φ)

peri, obtainedfor low-multiplicity events Nrec

ch

<

20,anda constant pedestalmodulated bycos(n

φ)

forn

≥

2 [7,14]: Y

(φ)

=

F Y

(φ)

peri

+

Gtmp



1

+

2 ∞



n=2 vn

{

2

,

tmp

}

2cos n

φ

.

(14)

The scale factor F and pedestal Gtmp are fixed by the fit, and

vn{2,tmp

}

are calculatedfrom aFourier transform. Onthe other hand,both Y

(φ)

and Y

(φ)

peri containa dijet componentand flowcomponent:

Y

(φ)

=

Y

(φ)

cent_jet

+

Gcent



1

+

2 ∞



n=2 vn

{

2

}

2cos n

φ

,

(15)

Y

(φ)

peri

=

Y

(φ)

peri_jet

+

Gperi



1

+

2 ∞



n=2 vn

{

2

,

peri

}

2cos n

φ

.

(16) Withtheassumptionthattheshapeofthedijetcomponentis in-dependent of N_chrec,and the magnitudes of the dijet components arerelatedbythescalefactorF :Y

(φ)

_jetcent

=

F Y

(φ)

peri_jet ,Eq. (14) canbewrittenas:

Y

(φ)

=

Y

(φ)

cent_jet

+ (

Gtmp

+

F Gperi

)

+

2 ∞



n=2



Gtmpvn

{

2

,

tmp

}

2

+

F Gperivn

{

2

,

peri

}

2



×

cos n

φ.

Comparing with Eqs. (15) and (16), one obtains Gcent

₌

_Gtmp

₊

F Gperiandthefollowingrelation:

vn

{

2

}

2

=

vn

{

2

,

tmp

}

2

−

F Gperi Gcent



vn

{

2

,

tmp

}

2

−

vn

{

2

,

peri

}

2



,

which shows that vn

{

2,tmp

}

from the template fit differs from the true vn{2

}

by a correction term that vanishes if andonly if

vn{2

}

is independent of Nrec_ch. Since the true flow harmonics in the peripheral interval vn{2,peri

}

are unknown in principle, the correction is applied starting from the third-lowest Nrec

ch interval (40

≤

N_chrec

<

60) in thisanalysis, by using vn{2,tmp

}

ofthe sec-ond Nrec_ch interval (20

≤

N_chrec

<

40) as an estimate of the true flow harmonics.Since thenon-flow contributionprimarily affects the odd harmonics,the v3{2,tmp

}

2 _{maybecome negative in the} first few Nrec_ch intervalsin pp collisions.Insuch cases,the correc-tionstartsfromthesecond Nrec_ch intervalwithpositivev3{2,tmp

}

2 (60

≤

Nrec

ch

<

80)byusingv3

{

2,tmp

}

fromtheprevious Nrecch inter-val(40

≤

N_chrec

<

60).

One important feature of the template fit analysis is the as-sumption that the dijet component Y

(φ)

jet is independent of



Nch. In Ref. [7], the uncertainty associated with this assump-tion is studied by changing the default peripheral interval from

Nrec_ch

<

20 to Nrec_ch

<

10 and 10

≤

N_chrec

<

20.Itwas foundthat the

vn{2,tmp

}

valuesarerelativelyinsensitivetothechoiceof periph-eralintervalforn

=

2 andn

=

4,butthesensitivityismuchlarger for n

=

3. This finding is reproduced in Fig. 6 for pp collisions,

whichshowsthatthev3

{

2,tmp

}

2 valuesobtainedviaEq. (14) dif-fersubstantiallyforthedifferentNrec_ch ranges.

Inadditiontothetemplatefitwithandwithouttheabove men-tioned correction procedure, the ATLAS and CMS collaborations