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

Article

Reference

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

ATLAS Collaboration AKILLI, Ece (Collab.), et al.

Abstract

Correlations of two flow harmonics vn and vm via three- and four-particle cumulants are measured in 13 TeV pp , 5.02 TeV p +Pb, and 2.76 TeV peripheral Pb+Pb collisions with the ATLAS detector at the LHC. The goal is to understand the multi-particle nature of the long-range collective phenomenon in these collision systems. The large non-flow background from dijet production present in the standard cumulant method is suppressed using a method of subevent cumulants involving two, three and four subevents separated in pseudorapidity.

The results show a negative correlation between v2 and v3 and a positive correlation between v2 and v4 for all collision systems and over the full multiplicity range. However, the magnitudes of the correlations are found to depend on the event multiplicity, the choice of transverse momentum range and collision system. The relative correlation strength, obtained by normalisation of the cumulants with the 〈vn2〉 from a two-particle correlation analysis, is similar in the three collision systems and depends weakly on the event multiplicity and transverse momentum. These results based on the [...]

ATLAS Collaboration, AKILLI, Ece (Collab.), et al . Correlated long-range mixed-harmonic fluctuations measured in pp , p +Pb and low-multiplicity Pb+Pb collisions with the ATLAS detector. Physics Letters. B , 2019, vol. 789, p. 444-471

DOI : 10.1016/j.physletb.2018.11.065

Available at:

http://archive-ouverte.unige.ch/unige:113946

Disclaimer: layout of this document may differ from the published version.

1 / 1

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_ncos n(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.

whilepreservingthegenuinelong-rangemulti-particlecorrelations associated with the ridge. Here each subevent is a collection of particles in a given

η

range. This so-called “subevent method”

hasbeendemonstratedtomeasurereliablycn{⁴}^andscn,m{⁴}^[13, 16]. Incontrast,c_n{⁴} ^andscn,m{⁴}^{basedon thestandard cumu-} lant method are contaminated by non-flow correlations over the full multiplicity range in pp collisions and the low multiplicity regionin p+A collisions [16].Insmallcollisionsystems,measure- mentshavebeenperformedforcn{⁴}^withboththestandard [15, 17] and subevent methods [18], andfor sc_n,m{⁴} ^{with the stan-} dardmethod [19].Thesubeventmethodhasnotyetbeenusedto measurescn,m{⁴}^,andnomeasurements ofacn{³}^haveeverbeen attemptedinsmallcollisionsystems.

This Letter presents measurements of sc2,3{⁴}^{, sc}2,4{⁴} ^and ac₂{³}ⁱⁿ pp collisionsat√

s=^{13 TeV,p+Pbcollisionsat}√ s_NN= 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+p collisions,the−z directionisalwaysdefinedasthedirectionofthe Pbbeam.

Table 1

Thelistofdatasetsusedinthisanalysis.

Pb+Pb p+Pb pp

Integratedluminosity (year)

7 μb⁻¹(2010) 28 nb⁻¹(2013) 0.07 pb⁻¹(2015) 0.3 nb⁻¹(2016) 0.84 pb⁻¹(2016)

usingmethodssimilartothoseappliedintheofflineanalysis.The HLT enables the high-multiplicity track triggers (HMT) to select events according to the number of tracks having pT>0.4 GeV matched to theprimary vertex, N^HLT_ch . The differentHMTtriggers applyadditionalrequirementsoneitherthetotaltransverseenergy (E_T)inthecalorimetersorthenumberofhitsintheMBTSfound bytheL1trigger,aswellasonN^HLT_ch bytheHLTtrigger.Thepp and p+Pbdatawerecollectedusingcombinationsoftheminimum-bias andHMTtriggers.Theminimum-biastriggerrequiredeitherahit 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 √

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 data were 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+Pbcollisions 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+Pbeventswas found to be consistentwiththe efficiencyfrom thePb+PbMCsimulation [17].

4. Eventandtrackselection

The offline eventselection forthe pp and p+Pbdata 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+Pbdata, and0.001–0.006 forthe 2016 p+Pbdata. 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+Pbsystems 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 pp and2016 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

pairs,tripletsorquadrupletsinoneeventtoobtain{²}n^,{²}m^, {³}n^and

{⁴}n,m

.Then theobtainedvaluesareaveragedoveran eventensemblewithsimilar N^rec_ch toobtainsc_n,m{⁴}^andacn{³}^.In theabsenceofnon-flowcorrelations, scn,m{⁴}^andacn{³}^measure thecorrelationbetweenvnandvmorbetweenvn andv2n:

ac_n

{

³

} =

V²_nV^∗_2n

=

v²_nv_2ncos 2n

(

n

−

2n

)

,

(4)

sc_n,m

{

⁴

} =

v²_nv²_m

−

v_n² v²_m

,

(5)

wheretheaverages aretakenoverthe events.Thisanalysismea- suresthreetypesofcumulantsdefinedinEq. (3):sc2,3{⁴}^,sc2,4{⁴} andac2{³}^.

5.2.Cumulantsinthesubeventmethod

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.

The four-subevent cumulant method is only relevant for the symmetric cumulants scn,m{⁴}^{. Tracks in each event are divided} into four subevents a, b, c, and d, each covering one quarter of the

η

range:−

η

max<

η

a<−

η

max/2,−

η

max/2≤

η

b<0,0≤

η

c<

η

max/2,and

η

max/2≤

η

d<

η

max.Themulti-particleazimuthalcor- relationsandcumulantsarethenevaluatedas:

{

⁴

}

n,m

a,b|c,d

=

e^{in(φa1−φc2)+im(φ3b−φ4d)}

,

(9)

sc^a_n^,_,^b_m^|c,d

{

⁴

} = {

⁴

}

n,m

a,b|c,d

− {

²

}

n

a|c

{

²

}

m

b|d

.

(10) Thefour-subevent methodbasedonEqs. (9) and(10) shouldfur- ther suppress the residual non-flow contributions, for instance wheneachofthetwojetsfromthedijetfallsacrosstheboundary betweentwo neighbouringsubevents. Tomaximisethe statistical 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

}

^{2cos n}

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+PbandPb+Pbcollisions, respectively.Theuncertaintiesforac₂{³}^{varyintherangeof0–2%}

in pp collisions, and 0–1% in p+PbandPb+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 pp collisions,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 pp collisionsandlessthan2% 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.Thechangesforpp collisionsareinthe rangeof5–15%forsc_2,3{⁴}^,2–8%forsc2,4{⁴}^{and1–5%forac}2{³}^. Theranges forp+Pbcollisionsaremuchsmallerduetothemuch 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 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 pp dataset,thedifferenceswith

andwithoutpile-uprejectionareintherangeof0–7%forsc2,3{⁴}^, 2–15%for sc2,4{⁴} ^{and2–3% forac}2{³}^{.As a} cross-check,the pp dataaredividedintotwosampleswithapproximatelyequalnum- ber of events based on the

μ

value:

μ

>0.4 and

μ

<0.4, and theresults arecompared. No systematicdifferencesare observed betweenthetwoindependentdatasets.

The systematicuncertainties fromdifferent sources are added in quadrature to determine the total systematic uncertainty. In p+PbandPb+Pbcollisions,thetotaluncertaintiesareintherange of3–8% forsc_2,3{⁴}^{,1–5% forsc}2,4{⁴}^{and1–4%forac}2{³}^{.In pp} collisions, the total uncertainties are larger, mainly due to larger non-flowcontribution,largerpile-upandthelesssharpturn-onof theHMT triggers. Theyare in the ranges of10–20% for sc2,3{⁴}^, 10–20%forsc2,4{⁴}^{and2–5%forac}2{³}^{.Thetotalsystematicun-} certaintiesaregenerallysmallerthanthestatisticaluncertainties.

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 pp collisionsin0.3<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.

ThebottomrowofFig.1showssc2,3{⁴}^fromPb+Pbcollisions.

The results are consistent among all four methods across most of the Nch range. Inthe low Nch region, where thenon-flow contributionisexpectedtobesignificant,theuncertainties ofthe resultsaretoolargetodistinguishbetweendifferentmethods.

Theresultsforthesymmetric cumulantsc2,4{⁴}^{are presented} inFig.2.Thetoprowshowsthe sc_2,4{⁴}^{obtainedfromthestan-} dard, two-subevent, three-subevent and four-subevent methods from pp collisions in 0.3< pT<3 GeV (left panel) and 0.5<

p_T<5 GeV (right panel). The values of sc_2,4{⁴} ^{are positive for} all fourmethods.However, theresultsfromthestandard method are much largerthan those fromthesubevent methodsand also exhibit amuch strongerincrease towardsthe lower^Nch ^region.

Thisbehaviourisconsistentwiththeexpectationthatthestandard methodismore affectedby dijets.Significantdifferencesare also 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- tionsfromdijetsinpp collisions,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