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

Physics Letters B 789 (2019) 444–471

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Physics Letters B

www.elsevier.com/locate/physletb

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 ra c t

Articlehistory:

Received6July2018

Receivedinrevisedform7October2018 Accepted13November2018

Availableonline2January2019 Editor: D.F.Geesaman

Correlations oftwoflowharmonics vnand vm viathree- andfour-particlecumulantsaremeasuredin 13TeVpp,5.02TeVp+Pb,and2.76TeVperipheralPb+PbcollisionswiththeATLASdetectorattheLHC.

The goalistounderstandthe multi-particlenature ofthelong-rangecollectivephenomenoninthese collisionsystems.Thelargenon-flowbackgroundfromdijetproductionpresentinthestandardcumulant methodissuppressedusingamethodofsubeventcumulantsinvolvingtwo,threeand foursubevents separatedinpseudorapidity.Theresultsshowanegativecorrelationbetweenv2and v3 andapositive correlationbetweenv2andv4forallcollisionsystemsandoverthefullmultiplicityrange.However,the magnitudes ofthecorrelationsare foundtodependontheeventmultiplicity,thechoiceoftransverse momentumrangeand collisionsystem. Therelativecorrelation strength,obtainedbynormalisationof the cumulants withthe ^v2n^{from a}two-particle correlationanalysis, issimilar inthe threecollision systemsanddependsweaklyontheevent multiplicityand transversemomentum.Theseresultsbased onthesubevent methodsprovidestrongevidence ofasimilar long-rangemulti-particle collectivityin 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=¹v_ncosn(φ−n), where v_n and n repre- sent the magnitude and the event-plane angle of the nth-order flowharmonic.Theyarealsoconvenientlyrepresentedbytheflow vector: V_n=^vneⁱⁿⁿ.The v_n are known to depend onthe colli- sionsystem,buthaveweakdependenceoncollisionenergies [6,7].

Theridgereflectsmulti-partondynamicsearlyinthecollisionand hasgeneratedsignificantinterestinthehigh-energyphysicscom- munity. A key question is whetherthe long-range multi-particle collectivityreflects initial momentumcorrelation 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-particle correlation 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 v_n, as well as the correlated fluctuations between two flowharmonics, v_n andv_m.Theseevent-by-eventfluctuationsare often represented by probability density distributions p(v_n) and p(v_n,v_m), respectively. For instance, the four-particle cumulants cn{⁴}=

v⁴_n

−² v²_n2

constrain the width of p(vn) [10], while thefour-particlesymmetriccumulantsscn,m{⁴}=

v²_nv²_m

− v²_n v²_m quantifythelowest-ordercorrelationbetweenvn andvm[12].The three-particle asymmetric cumulants such asacn{³}=

V²_nV^∗_2n =

v_n²v_2ncos 2n(n−2n)

[5,13] aresensitivetocorrelationsinvolv- ingboththeflowmagnitudev_n andflowphasen.

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

η

gapbetweenthe two particles ineach pair anda peripheral subtractionprocedure [3–5,7,14,15]. Formulti-particlecumulants, the non-flowcontributions canbe suppressedbyrequiringcorre- lation betweenparticles fromdifferentsubevents 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.

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 normalisedby the

v²_n ob- tainedfromatwo-particlecorrelationanalysis [7] toquantifytheir relative correlation strength. The measurements suggest that the resultsobtainedwiththe standard methodare stronglycontami- natedbycorrelationsfromnon-flowsources. Theresultsobtained withthethree-subeventmethodorthefour-subeventmethodpro- videnewevidenceoflong-rangethree- orfour-particleazimuthal correlations.

TheLetterisorganisedasfollows.DetailsoftheATLASdetector, thetriggersystem, datasets,aswellaseventandtrackselections areprovided inSections 2to 4.Section 5describesthe standard andsubeventcumulantmethodsusedinthisanalysis.Theanalysis procedureandsystematicuncertaintiesaredescribedinSections6 and7,respectively.ThemeasuredcumulantsarepresentedinSec- tion8.AsummaryisgiveninSection9.

2. Detectorandtrigger

TheATLASdetector [20] providesnearlyfull solid-anglecover- agearound the collision point withtracking detectors, calorime- ters,andmuonchambers, andiswell suited formeasurementof multi-particlecorrelationsoveralarge pseudorapidityrange.¹ The measurements were performed using primarily the inner detec- tor(ID), minimum-biastrigger scintillators(MBTS) and thezero- 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 T axial magnetic field [21]. An ad- ditionalpixellayer, the “insertable B-layer”(IBL) [22] is installed betweenthe Run-1 (2010–2013)andRun-2(2015–2018) periods.

The MBTS detects charged particles within 2.1|

η

_|3.9 using two hodoscopes ofcounters positioned 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] consistsof a first-level (L1) trigger implemented using a combination of dedicated electron- icsandprogrammablelogic,andahigh-level trigger(HLT)imple- mentedinprocessors.TheHLTreconstructscharged-particletracks

1 ATLAStypicallyusesaright-handedcoordinatesystemwithitsoriginat the nominalinteractionpoint(IP)inthe centreofthe detectorandthe z-axisalong thebeampipe.Thex-axispointsfromtheIPtothecentreoftheLHCring,andthe y-axispointsupward.Cylindricalcoordinates(r,φ)areusedinthetransverseplane, φbeingtheazimuthalanglearoundthebeampipe.Bydefault,thepseudorapidityis definedintermsofthepolarangleθasη= −ln tan(θ/2).However,forasymmetric p+PborPb+pcollisions,the−^{zdirectionisalwaysdefinedasthedirectionofthe} Pbbeam.

andHMTtriggers.Theminimum-biastriggerrequiredeitherahit inatleastoneMBTScounter,orahitinatleastoneMBTScounter oneachside,oratleastonereconstructedtrackattheHLTseeded by a random trigger at L1. More detailed information about the triggersusedfortheppandp+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+Pb datarecordedat√

sNN=⁵.02 TeV, and 7 μb⁻¹ of Pb+Pb data at √

s_NN=².76 TeV. The 2.76 TeV Pb+Pb datawerecollectedin2010.The p+Pbdataweremainlycollected in2013,butalsoinclude0.3 nb⁻¹ ofdatacollectedin2016,which increase thenumber ofeventsatmoderatemultiplicity (see Sec- tion4).Duringboth p+Pbruns,theLHCwasconfiguredtoprovide a4 TeVprotonbeamanda1.57 TeVper-nucleonPbbeam, which produced collisions at √

sNN=⁵.02 TeV, with a rapidity shift of 0.465 of the nucleon–nucleon centre-of-mass frame towards the protonbeamdirectionrelativetotheATLASrestframe.Thedirec- tionofthePbbeamisalwaysdefinedtohavenegative pseudora- pidity. The 13 TeV pp datawere collected during several special runsoftheLHCwithlowpile-upin2015and2016.Asummaryof thedatasetsusedinthisanalysisisshowninTable1.

Thetrackreconstructionefficiencywasdeterminedusingsimu- latedMonteCarlo(MC)eventsamples (Section4).The pp events were simulated withthe Pythia8 MC event generator [27] using theA2set oftunedparameters withMSTW2008LO partondistri- bution functions [28]. The HIJING eventgenerator [29] was used to producePb+Pb and p+Pb collisions withthesame energyand the sameboost ofthe centre-of-mass systemasinthe data.The detector response was simulated using Geant4 [30,31] with de- tectorconditionsmatchingthoseduring thedata-taking.Thesim- 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+Pbwastakenfromthelargerp+Pbsamplerecon- structedwiththesamereconstructionalgorithm.Theefficiencyin p+Pb eventswas found to be consistentwiththe efficiencyfrom thePb+PbMCsimulation [17].

4. Eventandtrackselection

The offline eventselection forthe pp and p+Pb data requires atleastonereconstructedvertexwithitslongitudinalpositionsat- isfying|^zvtx|<100 mmrelative tothe nominalinteractionpoint.

Thevertexisrequiredtohaveatleasttwo associatedtrackswith pT>0.4 GeV. The mean number of collisions per bunch cross- ing,

μ

, was 0.002–0.8for the13 TeV pp data,0.03 forthe 2013

446 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471

p+Pb data, and0.001–0.006 forthe 2016 p+Pb data. In orderto suppressadditional interactions in the same bunch crossing (re- ferredto aspile-up)in pp collisions,eventscontaining additional verticeswithatleast fourassociatedtracks arerejected. In p+Pb collisions,eventswithmorethanone goodvertex, definedasany vertexforwhichthescalarsumofthepToftheassociatedtracks isgreater than5 GeV, arerejected. The remainingpile-up events are further suppressedby using the signal inthe ZDCin the di- rection of the Pb beam. This signal is calibrated to the number ofdetectedneutrons, N_n,byusingthelocationofthepeakcorre- spondingtoasingleneutron.ThedistributionofNn ineventswith pile-upisbroaderthanthatfortheeventswithoutpile-up.Hence asimplerequirementontheZDCsignaldistributionisusedtofur- thersuppresseventswithpile-up,whileretainingmorethan98%

of eventswithout pile-up. The impact ofresidual pile-up, atthe levelof10⁻³,isstudiedbycomparingtheresultsobtainedfrom datawithdifferent

μ

values.

TheofflineeventselectionforthePb+Pbdatarequires|^zvtx|<

100 mm.Theselection alsorequiresatimedifference |t|<3 ns betweensignalsintheMBTStriggercountersoneithersideofthe interactionpoint tosuppressnon-collisionbackgrounds. Acoinci- dence between the ZDC signals at forward and backward pseu- dorapidityisrequiredtoreject avarietyofbackgroundprocesses, whilemaintaininghighefficiencyforinelasticprocesses.Thefrac- tionofeventswithmorethanoneinteractionafterapplyingthese selectioncriteriaislessthan10⁻⁴.

Charged-particletracksandcollisionverticesarereconstructed using algorithms optimised forimproved performance forRun-2.

In order to comparedirectly with the pp and p+Pb systems us- ing event selections based on the multiplicity of the collisions, a subset of datafrom low-multiplicityPb+Pb collisions, collected during the 2010 LHC heavy-ion run with a minimum-bias trig- ger,wasanalysedusingthesametrackreconstructionalgorithmas thatusedforp+Pbcollisions.ForthePb+Pband2013p+Pbanaly- ses,tracksarerequiredtohavea pT-dependentminimumnumber ofhitsin theSCT. The transverse (d₀) andlongitudinal(z₀ sinθ) impactparametersofthetrackrelativeto thevertexare required tobelessthan1.5 mm.Additionalrequirements|^d0|/

σ

d₀<3 and

|^z0sinθ|/

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 detaileddescription of the track se- lectionforthe2010Pb+Pbdataand2013p+Pbdatacanbefound inRefs. [5,17].

Forall thedatatakensincethestartofRun-2,thetrackselec- tioncriteriamakeuseoftheIBL,asdescribedinRefs. [14,25].For the ppand2016 p+Pbanalyses,thetracks arerequiredtosatisfy

|^dBL₀ |<1.5 mm and |^z0sinθ|<1.5 mm, where d^BL₀ is the trans- verseimpactparameterofthetrackrelativetothebeamline (BL).

Thecumulantsarecalculatedusingtrackspassingtheabovese- 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 N_ch^rec), tracks with p_T>0.4 GeV and |

η

|<2.5 are used forcom- patibility with the requirements in the HLT selections described above. Due to different trigger requirements, most of the p+Pb events with N^rec_ch >150 are provided by the 2013 dataset,while the2016datasetprovidesmostoftheeventsatlowerN_ch^rec.

The efficiencyof thecombined trackreconstruction and track selection requirements is estimated using MC samples recon- structed withthesame algorithmsandselection requirements as indata.Efficiencies,

(

η

,pT), areevaluated asafunction oftrack

η

,pTandthenumberofreconstructedcharged-particletracks,but averagedoverthefullrangeinazimuth.Theefficiencies aresimi-

larforeventswiththesamemultiplicity.Forallcollisionsystems, the efficiency increases by about 4% as track pT increases from 0.3 GeVto 0.6GeV.Above 0.6GeV,the efficiencyisindependent 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 independent ofthe event multiplicity for N^rec_ch >40.

Forlower-multiplicityeventstheefficiencyissmallerbyupto3%

duetobroaderd0 andz0sinθ distributions [17].

The fraction of falsely 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^rec_ch to obtain the efficiency-corrected aver- age number of charged particles per event, ^Nch =^b

N^rec_ch . The valueofthecorrectionfactorisobtainedfromtheMCsamplesde- scribed above, and is found to be nearly independent of N^rec_ch in therangeusedinthisanalysis, N_ch^rec<400.Itsvalueandtheasso- ciated uncertainties are b=¹.29 ± ^{0.05 forthe Pb+Pband2013} p+Pbcollisions andb=¹.18± ^{0.05 forRun-2 p+Pb and pp colli-} sions [32].Bothscn,m{⁴}^andac2{³}^{arethenstudiedasafunction} of^Nch^.

5. Cumulantmethod

The multi-particlecumulantmethod [10] hastheadvantageof directly reducing non-flow correlations from jets and dijets. The mathematical framework for the standard cumulant is based on theQ-cumulantsdiscussedinRefs. [11,12,33].Itwasextendedre- cently to the caseof subevent cumulants in Refs. [13,16]. These methodsarebrieflysummarisedbelow.

5.1. Cumulantsinthestandardmethod

The standardcumulantmethodcalculates k-particleazimuthal correlations, {k}^{, in one event using a complex number nota-} tion [11,12]:

{

²

}

n

=

e^{in(φ1−φ2)}

, {

³

}

n

=

e^{in(φ1+φ2−2φ3)}

, {

4

}

n,m

=

e^{in(φ1−φ2)+im(φ3−φ4)}

,

(1)

where “^{” denotes a} single-event average over all pairs, triplets orquadruplets,respectively.The averagesfromEq. (1) canbeex- pressed intermsofper-particle normalisedflowvectorsq_n;l with l=¹,2...ineachevent [11]:

q_n;l

≡

j

w^l_je^inφj

j

w^l_j

,

(2)

wherethesumrunsoveralltracksintheeventandw_jisaweight assigned to the jth track. This weight is constructed to correct for both detectornon-uniformity andtracking inefficiency asex- 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“^{”representsa weighted averageof}{^k}^{over anevent} ensemble with similar N^rec_ch. One averages first over all distinct

Inthestandardcumulantmethoddescribedabove,allk-particle multipletsinvolvedin{^k}n^and

{^k}n,m

areselectedusingtracks inthe entire ID acceptanceof |

η

|<

η

max=².5. To suppressfur- therthenon-flowcorrelationsthattypicallyinvolveafewparticles withina localisedregion in

η

,the tracksare dividedintoseveral subevents, each covering a unique

η

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

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

η

max<

η

a<0 and0≤

η

b<

η

max.Theper-eventk-particleazimuthal cor- relationsareevaluatedas:

{

²

}

n

a|^b

=

e^{in(φa1−φb2)}

, {

³

}

n

2a|^b

=

e^{in(φa1+φa2−2φ3b)}

, {

4

}

n,m

2a|2b

=

e^{in(φa1−φ2b)+im(φa3−φ4b)}

,

(6)

where the superscript or subscript a (b) indicates tracks chosen fromthe subeventa (b). Herethe three- andfour-particlecumu- lantsaredefinedas:

ac^2a_n^|b

{

3

} = {

3

}

n

2a|^b

,

sc^2a_n,m^|2b

{

⁴

} =

{

⁴

}

n,m

2a|2b

− {

²

}

n

a|b

{

²

}

m

a|b

.

Thetwo-subeventmethodsuppresses correlationswithina single jet(intra-jetcorrelations),sinceparticles fromone jetusually fall inonesubevent.

Inthe three-subevent cumulant method,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

=

e^{in(φa1+φb2−2φ3c)}

, {

⁴

}

n,m

a,b|2c

=

e^{in(φ1a−φ2c)+im(φb3−φc4)}

,

(7)

and

ac^a_n^,b|c

{

³

} = {

³

}

n

a,b|^c

,

sc^a_n^,_,^b_m^|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

rangesfor thefour subevents are swapped with eachother,andtheresultsareaveragedtoobtainthefinalvalues.

5.3. Normalisedcumulants

Althoughthecumulantsreflectthenatureofthecorrelationbe- tween vn and vm,their magnitudesalsodependonthesquare of singleflow harmonicsv²_n andv²_m,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)

nsc_2,4

{

⁴

} =

^sc2,4

{

⁴

}

v₂

{

²

}

^2v4

{

²

}

²

=

v²₂v²₄

v²₂ v²₄

−

¹

,

(12)

nac2

{

3

} =

^ac2

{

³

}

2v2

{

²

}

⁴

+

^c2

{

⁴

}

c4

{

²

}

=

v²₂v4cos 4

(

2

−

4

)

v⁴₂ v²₄

,

(13)

wherethe vn{²}²= v²_n

areflowharmonicsobtainedusingatwo- particlecorrelationmethodbasedonaperipheralsubtractiontech- nique [7,14], andc₂{⁴}=

v⁴₂

−² v²₂2

are four-particlecumulant resultsfromRefs. [17,18]. Thisdefinitionfornac2{³}^ismotivated byRef. [36].

6. Analysisprocedure

Themeasurement ofthescn,m{⁴}^andac2{³} ^{followsthesame} analysis procedure as for the four-particle cumulants c_n{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 event from particles in one of two different p_T ranges, 0.3<pT<3 GeVand 0.5<pT<5 GeV. The numbersof recon- structedchargedparticlesinthesepT rangesaredenotedby N^sel1_ch andN^sel2_ch ,respectively.

In thesecond step, thecorrelators {^k}^{for 0}.3<pT<3 GeV (0.5<pT<5 GeV) areaveraged overeventswiththe same N^sel1_ch (N_ch^sel2)toobtain{^k}^,andthensc2,3{⁴}^,sc2,4{⁴}^andac2{³}^.The sc2,3{⁴}^,sc2,4{⁴} ^andac2{³} ^{values are then averaged in broader} multiplicityrangesoftheeventensemble,weightedbynumberof events,toobtainstatisticallysignificantresults.

In the third step, the sc2,3{⁴}^{, sc}2,4{⁴} ^{and ac}2{³} ^{values ob-} tained fora given N^sel1_ch or N^sel2_ch are mapped to

N^rec_ch

,the aver- agenumberofreconstructedchargedparticleswith p_T>0.4 GeV.

448 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471

Themappingprocedureisnecessarysothat sc2,3{⁴}^,sc2,4{⁴}^and ac2{³}^{obtainedforthetwodifferent p}T rangescan becompared usingacommonx-axisdefinedby

N^rec_ch .The

N^rec_ch

valueisthen converted to ^Nch^{, the} efficiency-corrected average number of chargedparticleswithp_T>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

. Allreconstructedchargedparticleswith pT>0.3 GeVareentered intoa two-dimensional histogram N(φ,

η

),andthe weightfactor isthenobtainedasd(φ,

η

)≡ ^N(

η

)/N(φ,

η

),whereN(

η

)isthe trackdensityaveraged overφ in thegiven

η

bin.Thisprocedure removesmostoftheφ-dependentnon-uniformity inthedetector acceptance [17].

In order to calculate the normalised cumulants from Eqs.

(11)–(13),theflowharmonicsv_n{²}^{areobtainedfroma“template} fit”oftwo-particleφ correlationasdescribedinRefs. [7,14].The vn{²} ^{values are calculated} identically to the procedure used in the previous ATLAS publications [7,14], butare furthercorrected fora bias,which exists onlyif vn{²} ^{changes with Nrec}_ch^{.The de-} tailsofthecorrectionprocedurearegivenintheAppendixAand arediscussedbrieflybelow.

ThestandardprocedureofRefs. [7,14] firstconstructsaφdis- tributionforpairsoftrackswith|

η

|>2:theper-trigger-particle yield Y(φ) foragiven N_ch^rec range.Thedominatingnon-flow jet peak at φ∼

π

is estimatedusing low-multiplicity events with N^rec_ch <20 andseparatedviaatemplatefitprocedure,andthehar- 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

v_n

{

²

,

tmp

}

^2cosn

φ

,

where superscripts “peri” and “tmp” indicate quantities for the N^rec_ch <20 eventclassandquantitiesafterthetemplatefitforthe eventclassofinterest,respectively.ThescalefactorF andpedestal G^tmp are fixed by the fit, and vn{²,tmp} ^{are calculated from a} Fouriertransform. Thisprocedureimplicitlyassumesthat v_n{²}^is independentofN^rec_ch,andrequiresasmallcorrectionifv_n{²}^does changewithN_ch^rec(AppendixA).In p+PbandPb+Pbcollisions,this correctionin the N^rec_ch >100 regionamounts toa 2–6%reduction forv2{²,tmp}^{anda4–9%reductionforv}3{²,tmp}^andv4{²,tmp}^. The correction is smaller for v₂{²,tmp} ^{in pp collisions as it is} nearlyindependentofN_ch^rec[7].

7. Systematicuncertainties

The evaluation of the systematic uncertainties follows closely theprocedureestablishedforthefour-particlecumulantscn{⁴}^and describedinRef. [18].Themainsourcesofsystematicuncertainties arerelatedto thedetectorazimuthal non-uniformity,trackselec- tion,trackreconstructionefficiency,triggerefficiencyandpile-up.

Duetotherelativelypoorstatisticsandlargernon-floweffects,the systematicuncertainties are typically larger in pp collisions. The systematic uncertainties are also generally larger, in percentage, forfour-particlecumulantssc_n,m{⁴}^thanforthethree-particlecu- mulantsac2{³}^,sincethe|^scn,m{⁴}|^{valuesaremuchsmallerthan} thoseforac2{³}^{.Thesystematic}uncertaintiesaregenerallysimilar among the two- and three- and four-subevent methods, but are different from those 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 factor d(φ,

η

). The impact of theweighting procedure is studiedby fixing the weight to unityand repeating the analysis. The results are mostly consistent with the nominal results. The corresponding uncertainties for scn,m{⁴} ^{vary in the} range of0–4%,0–2% and1–2% in pp, p+Pb andPb+Pbcollisions, respectively.Theuncertaintiesforac₂{³}^{varyintherangeof0–2%}

in pp collisions, and 0–1% in p+Pb andPb+Pb collisions, respec- tively.

The systematicuncertaintyassociated withthetrack selection is estimated by tightening the |^d0| ^and |^z0sinθ| requirements.

They are each varied from the default requirement of less than 1.5 mm tolessthan1 mm.In p+PbandPb+Pbcollisions, there- quirementon thesignificanceofimpactparameters, |^d0|/

σ

d0 and

|^z0sinθ|/

σ

z0 are also varied fromlessthan 3 to lessthan 2.For eachvariation,thetrackingefficiencyisre-evaluatedandtheanal- ysisisrepeated. Forac₂{³}^{,whichhasalargeflowsignal,thedif-} ferencesfromthenominalresultsareobservedtobelessthan2%

forallcollisionsystems.Forscn,m{⁴}^{,forwhichthesignalissmall,} the differencesfrom the nominal results are found to be in the rangeof2–10%in ppcollisions,2–7%in p+Pbcollisionsand2–4%

inPb+Pbcollisions.Thedifferencesaresmallerforresultsobtained for0.5<p_T<5 GeVthanthoseobtainedfor0.3<p_T<3 GeV.

Previousmeasurementsindicatethattheazimuthalcorrelations (both the flow andnon-flow components) have a strong depen- denceon p_T,buta relativelyweakdependenceon

η

[5,7].There- fore, pT-dependent systematic effects in the trackreconstruction efficiencycouldaffectthecumulantvalues.Theuncertaintyinthe trackreconstruction efficiencyismainlydueto differencesinthe detectorconditionsandmaterial descriptionbetweenthesimula- tion and the data. The efficiency uncertainty varies between 1%

and4%,depending ontrack

η

andpT [7,17].Its impactonmulti- particlecumulantsisevaluatedbyrepeatingtheanalysiswiththe trackingefficiencyvariedupanddownbyitscorrespondinguncer- taintyasafunctionoftrackpT.Forthestandardcumulantmethod, which is more sensitive to jets and dijets, the evaluated uncer- tainty amountsto2–6%in ppcollisionsandlessthan2% inp+Pb collisionsfor^Nch>100.Forthesubeventmethods,theevaluated uncertaintyistypicallylessthan3%formostofthe^Nch^ranges.

Most eventsin pp and p+Pb collisions are collected with the HMT triggerswith severalonline N^rec_ch thresholds. Inorder toes- timate the possible bias due to trigger inefficiency asa function of ^Nch^{,theoffline Nrec}_ch requirementsarechanged suchthat the HMT triggerefficiencyisatleast 50%or80%. Theresults areob- tained independently for each variation. These results are found to be consistent with each other forthe subevent methods, and show some differencesforthe standard cumulantmethodin the low^Nch^{region.Thenominalanalysisisperformedusingthe50%}

efficiency selection and the differences betweenthe nominal re- sults andthose fromthe80% efficiency selection areincluded in thesystematicuncertainty.Thechangesforppcollisionsareinthe rangeof5–15%forsc_2,3{⁴}^,2–8%forsc2,4{⁴}^{and1–5%forac}2{³}^. Theranges forp+Pb collisionsaremuchsmallerduetothemuch sharperturn-onofthetriggerefficiencyandlargersignal:theyare estimatedtobe 1–3%forsc2,3{⁴}^,2–4%forsc2,4{⁴}^and1–2%for ac₂{³}^.

Inthisanalysis,apile-uprejectioncriterionisappliedtoreject eventscontainingadditionalverticesin ppandp+Pbcollisions.In order to check the impact of residualpile-up, the analysisis re- peated without the pile-uprejection criterion.No differences are observed in p+Pbcollisions, asisexpected sincethe

μ

valuesin p+Pbaremodest.Forthe13 TeV ppdataset,thedifferenceswith

The vn{²} ^{values used to obtain normalised cumulants from} Eqs. (11)–(13) aremeasuredfollowingtheprescriptionofthepre- viousATLAS publications [7,14],resulting invery similar system- aticuncertainties. The correction for the biasof the template fit procedure,asdescribedinSection6,reducesthesensitivitytothe choiceoftheperipheral N^rec_ch bin.The uncertaintiesofnormalised cumulantsare obtainedby propagationof theuncertainties from theoriginalcumulantsandvn{²}^{,takingintoaccountthatthecor-} relatedsystematicuncertaintiespartiallycancelout.

8. Results

Theresults are presented intwo parts. Section 8.1presents a detailedcomparisonbetweenthe standard method andsubevent methods to demonstrate the ability of the subevent methods to suppress non-flow correlations. Section 8.2 compares the cumu- lantsamongpp,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 methodsfrom ppcollisionsin0.3<p_T<3 GeV(leftpanel)and0.5<p_T<5 GeV (right panel). The values from the standard method are positive overthefull ^Nch ^{range,andare larger atlower N}ch ^orinthe higher pT range.Thisbehaviour suggeststhat the sc2,3{⁴}^values fromthe standard methodin pp collisions, including those from Ref. [19],are strongly influenced 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 atlowest ^Nch ^{andalsomorenegative at} higher pT.Theresultsare consistentamongthe varioussubevent methods for 0.3<p_T<3 GeV.For the high p_T region of 0.5<

pT<5 GeV,resultsfromthetwo-subeventmethodaresystemati- callylowerthanthosefromthethree- andfour-subeventmethods, suggestingthatthetwo-subeventmethodmaybeaffectedbyneg- ative non-flow contributions. Such negative non-flow correlation hasbeenobservedinaPythia8 calculation [16].

Themiddlerowof Fig.1 showssc2,3{⁴} ^{from p+Pb} collisions.

At^Nch>140,the valuesare negative andconsistentamong all fourmethods,reflectinggenuinelong-rangecollectivecorrelations.

At Nch<140, the values are different between the standard method and the subevent methods. The sc_2,3{⁴} ^{from the stan-} dardmethodchangessignaround^Nch ∼^{80 andremainspositive} atlower^Nch^{,reflecting the}contributionfromnon-flow correla- tions.Incontrast,thesc2,3{⁴}^{fromvarioussubeventmethodsare} negativeandconsistentwitheachotheratNch<140,suggesting thattheymainlyreflectthegenuinelong-rangecorrelations.

observed between thetwo-subevent andthree- or four-subevent methods at low ^Nch^{, butthese} differencesdecrease anddisap- pear for ^Nch>100. Within the statistical uncertainties of the measurement,nodifferencesareobservedbetweenthethree- and four-subevent methods. This comparison suggests that the two- subeventmethodmaynotbesufficienttorejectnon-flowcorrela- tionsfromdijetsinppcollisions,andmethodswiththreeormore subeventsarerequiredtosuppressthenon-flowcontributionover themeasured^Nch^range.

The middlerowofFig. 2showssc2,4{⁴}^{from p+Pb}collisions.

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 may not vanish even atlarge ^Nch ^val- ues.All subeventmethods suggest anincrease of sc2,4{⁴} ^toward lower ^Nch^forNch<40,whichmayreflectsomeresidualnon- flowcorrelationsinthisregion.

ThebottomrowofFig.2showssc2,4{⁴}^fromPb+Pbcollisions.

Thesc_2,4{⁴}^{valuesincreasegraduallywithN}ch^{forallfourmeth-} ods. This increase reflects the known fact that the v2 increases with^Nch^inPb+Pbcollisions [37].The valuesfromthestandard method are systematically larger than those from the subevent methods, andthis difference 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} inFig. 3.The top rowshowstheresults obtainedfromthe stan- dard, two-subevent, and three-subevent methods from pp colli- sions in 0.3< p_T<3 GeV (left panel) and 0.5<p_T <5 GeV (right panel).Theresultsare positiveforall methods.The results from the standard method are much larger than those from the subeventmethods, consistentwiththe expectationthat the 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-subeventmethodshow a 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 theexistence ofathree-particlelong-rangecollective flowthat is nearly independent of^Nch ^{in pp} collisions, consistentwith the Nch-independentbehaviourofv2 and v4 observedpreviouslyin thetwo-particlecorrelationanalysis [7].

ThemiddleandbottomrowsofFig.3showac2{³}^{from p+Pb} and Pb+Pb collisions, respectively. The ac2{³} ^{values from the} standard method have a significant non-flow contribution up to

450 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471

Fig. 1.Thesymmetriccumulantsc2,3{⁴}^{asafunctionofN}ch^for0.3<pT<3 GeV(leftpanels)and0.5<pT<5 GeV (rightpanels)obtainedfor ppcollisions(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+PbcollisionsandN}ch ∼^{80 inPb+Pb}collisions.

Inthesubevent methods,theinfluenceofnon-flow contributions isvery smallfor^Nch>60 inboth collisionsystems,andthere- forethe^Nch^{dependenceofac}2{³}^{reflectstheN}ch^dependence ofthe v₂ and v₄. The ac₂{³} ^{valuesfrom thesubevent methods} increase with ^Nch^{, and the increase is stronger in Pb+Pb colli-} sions. This is consistent with previous observations that v2 and v4increasewithNchmorestronglyinPb+Pbthaninp+Pbcolli- sions [17].

The valuesofsc2,4{⁴} ^andac2{³}^{,which areboth measuresof} correlations between v2 and v4, show significant differences be- tween thestandardmethodandthesubevent methods,asshown in Figs. 2 and 3. The ^Nch ^{dependence of these} differences de- creasesgraduallywith^Nch^{,andisconsistentwithaninfluenceof} non-flowthatisexpectedtoscaleas1/^Nch^{.However,thesedif-} ferencesseemtopersistfor^Nch>200 inp+Pbcollisionsandfor Nch>150 inPb+Pbcollisions,whichisnotcompatiblewiththe predicted behaviour of non-flow correlations. The differences at

Fig. 2.Thesymmetriccumulantsc2,4{⁴}^{asafunctionofN}ch^for0.3<pT<3 GeV(leftpanels)and0.5<pT<5 GeV (rightpanels)obtainedfor ppcollisions(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^Nch^mayarisefromlongitudinalflowdecorrelations [38,39], whichhavebeenmeasuredbyCMS [40] andATLAS [41].Decorre- lationeffectsarefoundto belargefor v4 andstronglycorrelated with v2, andtherefore they are expected to reduce the sc2,4{⁴} andac₂{³} ^{in the subeventmethod. Therefore, the observed dif-} ferencesbetweenthe standardmethod andsubevent methodre- flectthecombinedcontributionfromnon-flowcorrelations,which dominates in the low ^Nch ^{region, and} decorrelation, which is moreimportant atlarge ^Nch ^{(see furtherdiscussion inthe 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 and a positive correlation between v₂ and v₄.

452 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471

Fig. 3.Theasymmetriccumulantac2{³}^{asafunctionofN}ch^for0.3<pT<3 GeV(leftpanels)and0.5<pT<5 GeV(rightpanels)obtainedforppcollisions(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 patterns havepreviously been observed inlarge collisionsystems [42–44],butarenowconfirmedalsointhesmall collisionsystems,oncenon-floweffectsareadequatelysuppressed.

Inthemultiplicityrangecoveredbythe ppcollisions,^Nch<150, theresultsforsymmetriccumulantssc2,3{⁴}^andsc2,4{⁴}^aresim- ilaramongthethree systems.Intherange^Nch>150,|^sc2,3{⁴}|

andsc2,4{⁴}^{arelargerinPb+Pbthaninp+Pb}collisions.Theresults for ac2{³} ^{are similar among the three systems at N}ch<100, buttheydeviatefromeachother athigher^Nch^{.The ppdataare} approximatelyconstantordecreaseslightlywithNch,whilethe p+Pb andPb+Pb data show significant increasesas a functionof

^Nch^{.The bottomrowshowstheresultsforthehigher p}T range of0.5<p_T<5 GeV,wheresimilartrendsareobserved.

Fig. 5 shows the results for normalised cumulants, nsc2,3{⁴}^, nsc2,4{⁴} ^{and nac}2{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^Nch^dependenceof thescn,m{⁴}^andac2{³}^{valuesreflectstheN}ch^{dependenceofthe} vn values,andthesedependencesare removedinthe normalised cumulants.Thenormalised cumulantsarealsosimilaramongdif- ferent collisionsystems atlarge ^Nch^{,althoughsome} differences