Physics Letters B 789 (2019) 444–471
Contents lists available atScienceDirect
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
r
a
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 v2
nfrom atwo-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=1vncos n(φ
− n
)
, where vn andn repre-sent the magnitude and the event-plane angle of the nth-order flowharmonic.Theyarealsoconvenientlyrepresentedbytheflow vector: Vn
=
vneinn.The vn are known to depend onthe colli-sionsystem,buthaveweakdependenceoncollisionenergies [6,7]. Theridgereflectsmulti-partondynamicsearlyinthecollisionand hasgeneratedsignificantinterestinthehigh-energyphysics com-munity. A key question is whetherthe long-range multi-particle collectivityreflects initial momentumcorrelation fromgluonsat- 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 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}
=
v4n−
2vn22 constrain the width of p(
vn)
[10], while thefour-particlesymmetriccumulantsscn,m{4}
=
v2nv2m
−
v2nv2m quantifythelowest-ordercorrelationbetweenvn andvm[12].The three-particle asymmetric cumulants such asacn{3
}
=
V2nV∗2n=
v2nv2ncos 2n
(
n−
2n)
[5,13] aresensitivetocorrelations involv-ingboththeflowmagnitudevn andflowphase
n.
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 suppressedbyrequiring corre-lation betweenparticles fromdifferentsubevents separatedinη
,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
v2n ob-tainedfromatwo-particlecorrelationanalysis [7] toquantifytheir relative correlation strength. The measurements suggest that the resultsobtainedwiththe standard methodare strongly contami-natedbycorrelationsfromnon-flowsources. Theresultsobtained withthethree-subeventmethodorthefour-subeventmethod pro-videnewevidenceoflong-rangethree- orfour-particleazimuthal correlations.TheLetterisorganisedasfollows.DetailsoftheATLASdetector, thetriggersystem, datasets,aswellaseventandtrackselections areprovided inSections 2to 4.Section 5describesthe standard andsubeventcumulantmethodsusedinthisanalysis.Theanalysis procedureandsystematicuncertaintiesaredescribedinSections6
and7,respectively.Themeasuredcumulantsarepresentedin Sec-tion8.AsummaryisgiveninSection9.
2. Detectorandtrigger
TheATLASdetector [20] providesnearlyfull solid-angle cover-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 the 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 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.
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, and7 μb−1 of Pb+Pb data at
√
sNN=
2.
76 TeV. The 2.
76 TeV Pb+Pbdatawerecollectedin2010.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 of0
.
465 of the nucleon–nucleon centre-of-mass frame towards the protonbeamdirectionrelativetotheATLASrestframe.The direc-tionofthePbbeamisalwaysdefinedtohavenegative pseudora-pidity. The 13 TeV pp data were collected during several special runsoftheLHCwithlowpile-upin2015and2016.Asummaryof thedatasetsusedinthisanalysisisshowninTable1.Thetrackreconstructionefficiencywasdeterminedusing simu-latedMonteCarlo(MC)eventsamples (Section4).The pp events were simulated withthe Pythia8 MC event generator [27] using theA2set oftunedparameters withMSTW2008LO parton distri-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.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+Pbwastakenfromthelargerp+Pbsample recon-structedwiththesamereconstructionalgorithm.Theefficiencyin p+Pbeventswas found to be consistentwiththe efficiencyfrom thePb+PbMCsimulation [17].
4. Eventandtrackselection
The offline eventselection forthe pp and p+Pbdata requires atleastonereconstructedvertexwithitslongitudinalposition sat-isfying
|
zvtx|
<
100 mmrelative tothe nominalinteractionpoint.Thevertexisrequiredtohaveatleasttwo associatedtrackswith pT
>
0.
4 GeV. The mean number of collisions per bunch446 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, Nn,byusingthelocationofthepeak corre-spondingtoasingleneutron.ThedistributionofNn ineventswith pile-upisbroaderthanthatfortheeventswithoutpile-up.Hence asimplerequirementontheZDCsignaldistributionisusedto fur-thersuppresseventswithpile-up,whileretainingmorethan98% of eventswithout pile-up. The impact ofresidual pile-up, atthe levelof
10−3,isstudiedbycomparingtheresultsobtainedfrom datawithdifferentμ
values.TheofflineeventselectionforthePb+Pbdatarequires
|
zvtx|
<
100 mm.Theselection alsorequiresatimedifference
|
t|
<
3 ns betweensignalsintheMBTStriggercountersoneithersideofthe interactionpoint tosuppressnon-collisionbackgrounds. A coinci-dence between the ZDC signals at forward and backward pseu-dorapidityisrequiredtoreject avarietyofbackgroundprocesses, whilemaintaininghighefficiencyforinelasticprocesses.The frac-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+Pb analy-ses,tracksarerequiredtohavea pT-dependentminimumnumber
ofhitsin theSCT. The transverse (d0) andlongitudinal(z0 sin
θ
)impactparametersofthetrackrelativeto thevertexare required tobelessthan1.5 mm.Additionalrequirements
|
d0|/
σ
d0<
3 and|
z0sinθ
|/
σ
z0<
3 are imposed, whereσ
d0 andσ
z0 are theun-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,thetrack selec-tioncriteriamakeuseoftheIBL,asdescribedinRefs. [14,25].For the pp and2016 p+Pbanalyses,thetracks arerequiredtosatisfy
|
dBL0|
<
1.
5 mm and|
z0sinθ
|
<
1.
5 mm, where dBL0 is thetrans-verseimpactparameterofthetrackrelativetothebeamline (BL). Thecumulantsarecalculatedusingtrackspassingtheabove se-lectionrequirements,andhaving
|
η
|
<
2.
5 and 0.
3<
pT<
3 GeVor 0
.
5<
pT<
5 GeV. These two pT ranges are chosen becausethey 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 used forcom-patibility with the requirements in the HLT selections described above. Due to different trigger requirements, most of the p+Pb events with Nrecch
>
150 are provided by the 2013 dataset,while the2016datasetprovidesmostoftheeventsatlowerNrecch.
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 oftrackη
,pTandthenumberofreconstructedcharged-particletracks,butaveragedoverthefullrangeinazimuth.Theefficiencies are
simi-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 Nrecch>
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 Nrecch to obtain the efficiency-corrected aver-age number of charged particles per event,
Nch=
b Nrecch
. The valueofthecorrectionfactorisobtainedfromtheMCsamples de-scribed above, and is found to be nearly independent of Nrecch in therangeusedinthisanalysis, Nrecch
<
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}
arethenstudiedasafunctionof
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].Itwasextended re-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=
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.The averagesfromEq. (1) canbe ex-pressed intermsofper-particle normalisedflowvectorsq
n;l with l=
1,
2...
ineachevent [11]: qn;l≡
j wljeinφj j wlj,
(2)wherethesumrunsoveralltracksintheeventandwjisaweight assigned to the jth track. This weight is constructed to correct for both detectornon-uniformity andtracking inefficiency 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“”representsa weighted averageof{
k}
over anevent ensemble with similar NrecInthestandardcumulantmethoddescribedabove,allk-particle multipletsinvolvedin
{
k}n
and{
k}n
,mareselectedusingtracks inthe entire ID acceptanceof
|
η
|
<
η
max=
2.
5. To suppressfur-therthenon-flowcorrelationsthattypicallyinvolveafewparticles withina localisedregion in
η
,the tracksare dividedintoseveral subevents, each covering a uniqueη
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-particleazimuthalcor-relationsareevaluatedas:
{
2}
na|b=
ein(φa1−φb2),
{
3}
n2a|b=
ein(φa1+φa2−2φ3b),
{
4}
n,m 2a|2b=
ein(φa1−φ2b)+im(φa3−φ4b),
(6)where the superscript or subscript a (b) indicates tracks chosen fromthe subeventa (b).Herethe three- andfour-particle cumu-lantsaredefinedas:
ac2an|b
{
3} = {
3}
n2a|b,
sc2an,m|2b{
4} =
{
4}
n,m 2a|2b− {
2}
na|b{
2}
ma|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 andη
max/
3<
η
c<
η
max.Themulti-particleazimuthalcorrelationsandcumulantsarethenevaluatedas:
{
3}
na,b|c=
ein(φa1+φb2−2φ3c),
{
4}
n,m a,b|2c=
ein(φ1a−φ2c)+im(φb3−φc4),
(7) and acan,b|c{
3} = {
3}
na,b|c,
scan,,bm|2c{
4} = {
4}
n,ma,b|2c− {
2}
na|c{
2}
mb|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, theη
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. NormalisedcumulantsAlthoughthecumulantsreflectthenatureofthecorrelation be-tween vn and vm,their magnitudesalsodependonthesquare of singleflow harmonicsv2
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 v22v23
−
1,
(11) nsc2,4{
4} =
sc2,4{
4}
v2{
2}
2v4{
2}
2=
v2 2v24 v2 2v2 4
−
1,
(12) nac2{
3} =
ac2{
3}
2v2
{
2}
4+
c2{
4}
c4{
2}
=
v22v4cos 4(
2−
4)
v42
v24
,
(13) wherethe vn{
2}
2=
v2n
areflowharmonicsobtainedusinga two-particlecorrelationmethodbasedonaperipheralsubtraction tech-nique [7,14], andc2{
4}
=
v4 2−
2v2 2 2are four-particlecumulant resultsfromRefs. [17,18]. Thisdefinitionfornac2
{
3}
ismotivatedbyRef. [36].
6. Analysisprocedure
Themeasurement ofthescn,m{4
}
andac2{
3}
followsthesameanalysis 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
,mfrom Eqs. (1), (6), (7) and (9) are calculated for each event from particles in one of two different pT ranges,
0
.
3<
pT<
3 GeVand 0.
5<
pT<
5 GeV. The numbersofrecon-structedchargedparticlesinthesepT rangesaredenotedby Nsel1ch
andNsel2
ch ,respectively.
In thesecond step, thecorrelators
{
k}
for 0.
3<
pT<
3 GeV(0
.
5<
pT<
5 GeV) areaveraged overeventswiththe same Nsel1ch(Nchsel2)toobtain
{
k}
,andthensc2,3{
4}
,sc2,4{
4}
andac2{
3}
.Thesc2,3
{
4}
,sc2,4{
4}
andac2{
3}
values are then averaged in broadermultiplicityrangesoftheeventensemble,weightedbynumberof events,toobtainstatisticallysignificantresults.
In the third step, the sc2,3
{
4}
, sc2,4{
4}
and ac2{
3}
valuesob-tained fora given Nsel1ch or Nsel2ch are mapped to
Nrecch,the aver-agenumberofreconstructedchargedparticleswith pT>
0.
4 GeV.448 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471
Themappingprocedureisnecessarysothat sc2,3
{
4}
,sc2,4{
4}
andac2
{
3}
obtainedforthetwodifferent pT rangescan becomparedusingacommonx-axisdefinedby
Nrecch.TheNrecch valueisthen converted to Nch, the efficiency-corrected average number ofchargedparticleswithpT
>
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 ofη
. Allreconstructedchargedparticleswith pT>
0.
3 GeVareenteredintoa two-dimensional histogram N
(φ,
η
)
,andthe weightfactor isthenobtainedasd(φ,
η
)
≡
N(
η
)
/
N(φ,
η
)
,whereN(
η
)
isthe trackdensityaveraged overφ
in thegivenη
bin.Thisprocedure removesmostoftheφ
-dependentnon-uniformity inthedetector acceptance [17].In order to calculate the normalised cumulants from Eqs. (11)–(13),theflowharmonicsvn{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 Nrecch.The de-tailsofthecorrectionprocedurearegivenintheAppendixAand arediscussedbrieflybelow.ThestandardprocedureofRefs. [7,14] firstconstructsa
φ
dis-tributionforpairsoftrackswith|
η
|
>
2:theper-trigger-particle yield Y(φ)
foragiven Nrecch range.Thedominatingnon-flow jet
peak at
φ
∼
π
is estimatedusing low-multiplicity events with Nrecch<
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 nφ
,
where superscripts “peri” and “tmp” indicate quantities for the Nrecch
<
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 independentofNrecch,andrequiresasmallcorrectionifvn{2
}
doeschangewithNchrec(AppendixA).In p+PbandPb+Pbcollisions,this correctionin the Nrecch
>
100 regionamounts toa 2–6%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 isnearlyindependentofNchrec[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,trackreconstructionefficiency,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-particlecu-mulantsac2
{
3}
,sincethe|
scn,m{4}|
valuesaremuchsmallerthanthoseforac2
{
3}
.Thesystematicuncertaintiesaregenerallysimilaramong 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 therange 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, the re-quirementon thesignificanceofimpactparameters,
|
d0|/
σ
d0 and|
z0sinθ
|/
σ
z0 are also varied fromlessthan 3 to lessthan 2.Foreachvariation,thetrackingefficiencyisre-evaluatedandthe anal-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<
pT<
5 GeVthanthoseobtainedfor0.
3<
pT<
3 GeV.Previousmeasurementsindicatethattheazimuthalcorrelations (both the flow andnon-flow components) have a strong depen-denceon pT,buta relativelyweakdependenceon
η
[5,7].There-fore, pT-dependent systematic effects in the trackreconstruction
efficiencycouldaffectthecumulantvalues.Theuncertaintyinthe trackreconstruction efficiencyismainlydueto differencesinthe detectorconditionsandmaterial descriptionbetweenthe simula-tion and the data. The efficiency uncertainty varies between 1% and4%,depending ontrack
η
andpT [7,17].Its impactonmulti-particlecumulantsisevaluatedbyrepeatingtheanalysiswiththe trackingefficiencyvariedupanddownbyitscorresponding uncer-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,theevaluateduncertaintyistypicallylessthan3%formostofthe
Nchranges.Most eventsin pp and p+Pb collisions are collected with the HMT triggerswith severalonline Nrecch thresholds. Inorder to es-timate the possible bias due to trigger inefficiency asa function of
Nch,theoffline Nrecch requirementsarechanged suchthat theHMT triggerefficiencyisatleast 50%or80%. Theresults are ob-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
Nchregion.Thenominalanalysisisperformedusingthe50%efficiency selection and the differences betweenthe nominal re-sults andthose fromthe80% efficiency selection areincluded in thesystematicuncertainty.Thechangesforpp collisionsareinthe rangeof5–15%forsc2,3
{
4}
,2–8%forsc2,4{
4}
and1–5%forac2{
3}
.Theranges forp+Pbcollisionsaremuchsmallerduetothemuch sharperturn-onofthetriggerefficiencyandlargersignal:theyare estimatedtobe 1–3%forsc2,3
{
4}
,2–4%forsc2,4{
4}
and1–2%forac2
{
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,thedifferenceswithThe vn{2
}
values used to obtain normalised cumulants from Eqs. (11)–(13) aremeasuredfollowingtheprescriptionofthe pre-viousATLAS publications [7,14],resulting invery similar system-aticuncertainties. The correction for the biasof the template fit procedure,asdescribedinSection6,reducesthesensitivitytothe choiceoftheperipheral Nrecch bin.The uncertaintiesofnormalised
cumulantsare obtainedby propagationof theuncertainties from theoriginalcumulantsandvn{2
}
,takingintoaccountthatthe cor-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 obtainedfromthe standard,two-, three- and four-subevent methodsfrom pp collisionsin0
.
3<
pT<
3 GeV(leftpanel)and0.
5<
pT<
5 GeV(right panel). The values from the standard method are positive overthefull
Nch range,andare larger atlower Nch orinthehigher pT range.Thisbehaviour suggeststhat the sc2,3
{
4}
valuesfromthe standard methodin pp collisions, including those from Ref. [19],are strongly influenced by non-flow effects inall
Nchand pT ranges [16]. In contrast, the values from the subevent
methods are negative over the full
Nch range, and they areslightlymorenegative atlowest
Nch andalsomorenegative athigher pT.Theresultsare consistentamongthe varioussubevent
methods for 0
.
3<
pT<
3 GeV.For the high pT region of 0.
5<
pT
<
5 GeV,resultsfromthetwo-subeventmethodaresystemati-callylowerthanthosefromthethree- andfour-subeventmethods, suggestingthatthetwo-subeventmethodmaybeaffectedby neg-ative non-flow contributions. Such negative non-flow correlation hasbeenobservedinaPythia8 calculation [16].
Themiddlerowof Fig.1 showssc2,3
{
4}
from p+Pbcollisions.At
Nch>
140,the valuesare negative andconsistentamong allfourmethods,reflectinggenuinelong-rangecollectivecorrelations. At
Nch<
140, the values are different between the standardmethod and the subevent methods. The sc2,3
{
4}
from thestan-dardmethodchangessignaround
Nch∼
80 andremainspositiveatlower
Nch,reflecting thecontributionfromnon-flowcorrela-tions.Incontrast,thesc2,3
{
4}
fromvarioussubeventmethodsarenegativeandconsistentwitheachotherat
Nch<
140,suggestingthattheymainlyreflectthegenuinelong-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 themeasurement,nodifferencesareobservedbetweenthethree- and four-subevent methods. This comparison suggests that the two-subeventmethodmaynotbesufficienttorejectnon-flow correla-tionsfromdijetsinpp 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 may not vanish even atlarge
Nchval-ues.All subeventmethods suggest anincrease of sc2,4
{
4}
towardlower
NchforNch<
40,whichmayreflectsomeresidualnon-flowcorrelationsinthisregion.
ThebottomrowofFig.2showssc2,4
{
4}
fromPb+Pbcollisions.Thesc2,4
{
4}
valuesincreasegraduallywithNchforallfourmeth-ods. This increase reflects the known fact that the v2 increases
with
NchinPb+Pbcollisions [37].The valuesfromthestandardmethod are systematically larger than those from the subevent methods, andthis difference varies slowly with
Nch,similar tothe behaviour observed in p+Pb collisions in the high
Nchre-gion.
The resultsfortheasymmetric cumulantac2
{
3}
are presentedinFig. 3.The top rowshowstheresults obtainedfromthe 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 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 differencesdecrease and disappear at
Nch>
100. The ac2{
3}
values fromthe three-subeventmethodshow a slightincrease for
Nch<
40but are nearly constant for
Nch>
40. This behaviour suggeststhatinthethree-subeventmethod,thenon-flowcontributionmay play some role at
Nch<
40, but is negligible for Nch>
40.Therefore, the ac2
{
3}
from the three-subevent method supportstheexistence ofathree-particlelong-rangecollective flowthat is nearly independent of
Nch in pp collisions, consistentwith the Nch-independentbehaviourofv2 and v4 observedpreviouslyinthetwo-particlecorrelationanalysis [7].
ThemiddleandbottomrowsofFig.3showac2
{
3}
from p+Pband Pb+Pb collisions, respectively. The ac2
{
3}
values from the450 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471
Fig. 1. The symmetriccumulantsc2,3{4}asafunctionofNchfor0.3<pT<3 GeV(leftpanels)and0.5<pT<5 GeV (rightpanels)obtainedfor pp 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+PbcollisionsandNch∼
80 inPb+Pbcollisions.Inthesubevent methods,theinfluenceofnon-flow contributions isvery smallfor
Nch>
60 inboth collisionsystems,andthere-forethe
Nchdependenceofac2{
3}
reflectstheNchdependenceofthe v2 and v4. The ac2
{
3}
valuesfrom thesubevent methodsincrease with
Nch, and the increase is stronger in Pb+Pbcolli-sions. This is consistent with previous observations that v2 and
v4increasewith
NchmorestronglyinPb+Pbthaninp+Pbcolli-sions [17].
The valuesofsc2,4
{
4}
andac2{
3}
,which areboth measuresofcorrelations between v2 and v4, show significant differences
be-tween thestandardmethodandthesubevent methods,asshown in Figs. 2 and 3. The
Nch dependence of these differencesde-creasesgraduallywith
Nch,andisconsistentwithaninfluenceofnon-flowthatisexpectedtoscaleas1
/
Nch.However,thesedif-ferencesseemtopersistfor
Nch>
200 inp+Pbcollisionsandfor Nch>
150 inPb+Pbcollisions,whichisnotcompatiblewiththeFig. 2. The symmetriccumulantsc2,4{4}asafunctionofNchfor0.3<pT<3 GeV(leftpanels)and0.5<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-lationeffectsarefoundto belargefor v4 andstronglycorrelated
with v2, andtherefore they are expected to reduce the sc2,4
{
4}
andac2
{
3}
in the subeventmethod. Therefore, the observeddif-ferencesbetweenthe standardmethod andsubevent method re-flectthecombinedcontributionfromnon-flowcorrelations,which dominates in the low
Nch region, and decorrelation, which ismoreimportant atlarge
Nch (see furtherdiscussion intheAp-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. The asymmetriccumulantac2{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 patterns havepreviously been observed inlarge collisionsystems [42–44],butarenowconfirmedalsointhesmall collisionsystems,oncenon-floweffectsareadequatelysuppressed. Inthemultiplicityrangecoveredbythe pp collisions,
Nch<
150,theresultsforsymmetriccumulantssc2,3
{
4}
andsc2,4{
4}
aresim-ilaramongthethree systems.Intherange
Nch>
150,|
sc2,3{
4}|
andsc2,4
{
4}
arelargerinPb+Pbthaninp+Pbcollisions.Theresultsfor ac2
{
3}
are similar among the three systems at Nch<
100,buttheydeviatefromeachother athigher
Nch.The pp dataareapproximatelyconstantordecreaseslightlywith
Nch,whilethep+PbandPb+Pb data show significant increasesas a functionof
Nch.The bottomrowshowstheresultsforthehigher pT rangeof0
.
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. Thenormalised cumulants generally show a much weaker
Nchde-pendence at
Nch>
100, where the statistical uncertainties aresmall.Thisbehaviourimpliesthatthestrong
Nchdependenceofthescn,m{4
}
andac2{
3}
valuesreflectstheNchdependenceofthevn values,andthesedependencesare removedinthe normalised cumulants.Thenormalised cumulantsarealsosimilaramong dif-ferent collisionsystems atlarge
Nch,althoughsome differencesFig. 4. The Nchdependenceofsc2,3{4}(leftpanels),sc2,4{4}(middlepanels)andac2{3}(rightpanels)in0.3<pT<3 GeV(toprow)and0.5<pT<5 GeV(bottomrow)
obtainedfor pp collisions(solidcircles),p+Pbcollisions(opencircles)andlow-multiplicityPb+Pbcollisions(opensquares).Theerrorbarsandshadedboxesrepresentthe statisticalandsystematicuncertainties,respectively.
attherelativelevelof20–30%areobservedforsmaller
Nch.Theonlyexception isnsc2,3
{
4}
,whose valuesinthe pp collisions areverydifferentfromthoseinp+PbandPb+Pbcollisions.Incontrast, the sc2,3
{
4}
values in Fig. 4 are close among different systems.Thissuggeststhatthe
v23valuesfromthetemplatefitmethod [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 tounderestimatetheoddflow harmonics,duetothepresenceof alargeaway-side peak atφ
∼
π
inthetwo-particle correlation function.Thecomparisonofsc2,3{
4}
andnsc2,3{
4}
amongdifferentcollision systems provides indirect evidenceof this underestima-tionof
v23.Fig.5 showsthat thenormalisedcumulantsareconsistent be-tween0
.
3<
pT<
3 GeVand0.
5<
pT<
5 GeV.Ontheotherhand,themagnitudesofthecumulantsinFig.4differby alargefactor betweenthetwopTranges:aboutafactorofthreeforsc2,3
{
4}
andsc2,4
{
4}
,anda factoroftwoforac2{
3}
.Theseresultssuggest thatthe pT dependenceofsc2,3
{
4}
,sc2,4{
4}
andac2{
3}
largelyreflectsthepT dependenceofthevn atthesingle-particlelevel.
9. Discussion
Three- and four-particle cumulants involving correlations be-tweentwo harmonicsofdifferentorder vn andvm are measured in
√
s=
13 TeV pp,√
sNN=
5.
02 TeV p+Pb,andlow-multiplicity√
sNN
=
2.
76 TeV Pb+Pb collisions with the ATLAS detector attheLHC,with totalintegratedluminosities 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}
andsc2,4
{
4}
,and the three-particleasymmetric cumulant ac2{
3}
.Thesymmetriccumulants scn,m{4
}
=
vn2v2m
−
v2nvm2 probe the cor-relation ofthe flow magnitudes,while the asymmetric cumulant ac2
{
3} =
v22v4cos 4(
2−
4)
issensitivetocorrelationsinvolvingboth the flowmagnitude vn andflow phase
n.They are calcu-lated using the standard cumulant method,as well 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 inpp collisions, as well as over the low
Nch range in p+Pb andPb+Pbcollisions. The differencesarelarger forparticles athigher pT oratsmaller
Nch.When analysedwiththestandard methodin pp collisions,thisbehaviouriscompatiblewiththedominance ofthe non-flowcorrelationsratherthan thelong-rangecollective flow correlations. Systematic, but much smaller, differences are alsoobserved inthelow
Nch regionbetweenthetwo-subeventmethodandthree- orfour-subeventmethods,whichindicatethat the two-subevent method may still be affected by correlations arising fromjets. Onthe other handno differencesare observed between the three-subevent and four-subevent methods, within experimentaluncertainties,suggestingthatmethodswiththreeor moresubeventsaresufficienttorejectnon-flow correlationsfrom jets. Therefore,thethree-subevent methodisusedto presentthe mainresultsinthisanalysis.
The three-subevent methodprovides a measurement of nega-tive sc2,3
{
4}
and positive sc2,4{
4}
andac2{
3}
over nearly thefull Nch range and in all three collision systems. These results454 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471
Fig. 5. The Nchdependenceofnsc2,3{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-viously beenobserved inlarge collision systems [42–44], butare now confirmed in small collision systems, once non-flow effects areadequatelysuppressed.Thevaluesofsc2,3
{
4}
andsc2,4{
4}
areconsistent in pp and p+Pb collisions over the same
Nch range,buttheir magnitudesatlarge
Nchare muchsmaller thanthoseforPb+Pb collisions.The valuesof ac2
{
3}
are similar atvery low Nch among the three systems, but are very different at large Nch.Ontheotherhand,afterscalingbythev2n
estimatedfrom atwo-particle analysis [7,14], theresultingnormalisedcumulants nsc2,3{
4}
,nsc2,4{
4}
andnac2{
3}
showamuchweakerdependenceon
Nch,and their valuesare much closerto each other amongthe three systems. The magnitudes of the normalised cumulants are also similar to each other for 0
.
5<
pT<
5 GeV as well as0
.
3<
pT<
3 GeV. This suggests that the Nch, pT and systemdependenceofthesc2,3
{
4}
,sc2,4{
4}
andac2{
3}
reflectmostlythe Nch,pTandsystemdependenceof v2n,buttherelativestrengths ofthecorrelationsaresimilarforthethreecollisionsystems.
The new results obtained with the subevent cumulant tech-nique provide further evidence that the ridge is indeed a long-rangecollectivephenomenoninvolvingmanyparticlesdistributed across a broad rapidity interval. The similarity between differ-entcollisionsystemsfornsc2,3
{
4}
,nsc2,4{
4}
andnac2{
3}
,andtheweakdependenceoftheseobservablesonthepTrangeand
Nch,largelyfree fromnon-flow effects,provideanimportantinput 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 successful operation ofthe LHC, as well asthe supportstaff fromour institutions withoutwhom ATLAScouldnotbeoperatedefficiently.
WeacknowledgethesupportofANPCyT,Argentina;YerPhI, Ar-menia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azer-baijan; SSTC, Belarus; CNPq and FAPESP,Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT,Chile; CAS,MOSTand NSFC,China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic;DNRFandDNSRC,Denmark;IN2P3-CNRS,CEA-DRF/IRFU, France; SRNSFG, Georgia; BMBF, HGF, andMPG, 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 and Wallenberg Foundation, Sweden;SERI,SNSFandCantonsofBernandGeneva, 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, andthe OntarioInnovation Trust,Canada; EPLANET,ERC, ERDF,FP7, Hori-zon 2020and Marie Skłodowska-Curie Actions, European Union; Investissements d’Avenir Labex and Idex, ANR, Région Auvergne andFondationPartagerleSavoir,France;DFGandAvHFoundation, Germany;Herakleitos,ThalesandAristeiaprogrammesco-financed by EU-ESFandtheGreek NSRF;BSF,GIFandMinerva,Israel;BRF, Norway; CERCA Programme Generalitat de Catalunya,Generalitat
Fig. 6. The valuesofvn{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 crucialcomputing support 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.Majorcontributorsofcomputingresources arelistedin Ref. [45].
Appendix A. Improvementtothetemplatefitprocedure
In order to separate the long-range ridge from other non-flowsources, especiallydijets,the ATLAS Collaborationdeveloped atemplatefittingproceduredescribedinRefs. [7,14].Thefirststep istoconstructa
φ
distributionofparticlepairswithlarge pseu-dorapidityseparation|
η
|
>
2,theso-called“per-trigger”particle yield, Y(φ)
, for a given Nchrec 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-flowcomponentispeakedatφ
∼
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 Nrecch
<
20,anda constantpedestal modulatedbycos
(
nφ)
forn≥
2 [7,14]: Y(φ)
=
F Y(φ)
peri+
Gtmp 1+
2 ∞ n=2 vn{
2,
tmp}
2cos nφ
.
(14)The scale factor F and pedestal Gtmp are fixed by the fit, and vn{2
,
tmp}
arecalculated froma Fouriertransform. Onthe other hand,both Y(φ)
andY(φ)
peri contain a dijetcomponent and flowcomponent:Y
(φ)
=
Y(φ)
centjet+
Gcent 1+
2 ∞ n=2 vn{
2}
2cos nφ
,
(15)Y
(φ)
peri=
Y(φ)
perijet+
Gperi 1+
2 ∞ n=2 vn{
2,
peri}
2cos nφ
.
(16) Withtheassumptionthat theshapeofthedijetcomponentis in-dependent of Nrecch, and the magnitudes of the dijet components arerelatedbythescalefactorF :Y(φ)
jetcent=
F Y(φ)
perijet ,Eq. (14) canbewrittenas:Y
(φ)
=
Y(φ)
centjet+ (
Gtmp+
F Gperi)
+
2 ∞ n=2 Gtmpvn{
2,
tmp}
2+
F Gperivn{
2,
peri}
2×
cos nφ.
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 ifand only if vn{2}
is independent of Nrecch. 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 Nrecch interval
(40
≤
Nrecch<
60) inthis analysis, by using vn{2,
tmp}
of the sec-ond Nrecch interval (20≤
Nchrec<
40) as an estimate of the true flow harmonics.Sincethe non-flow contributionprimarily affects the oddharmonics, the v3{
2,
tmp}
2 maybecome negative in thefirstfew Nrecch intervalsin pp collisions.Insuchcases,the correc-tionstartsfromthesecond Nrecch intervalwithpositive v3
{
2,
tmp}
2(60
≤
Nrecch
<
80)byusingv3{
2,
tmp}
fromtheprevious Nrecchinter-val(40
≤
Nrecch<
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 thisassump-tion is studied by changing the default peripheral interval from Nrecch
<
20 to Nrecch<
10 and10≤
Nrecch<
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}
2valuesobtainedviaEq. (14)dif-fersubstantiallyforthedifferentNrecch ranges.
Inadditiontothetemplatefitwithandwithouttheabove men-tioned correction procedure, the ATLAS and CMS collaborations
456 The ATLAS Collaboration / Physics Letters B 789 (2019) 444–471
Fig. 7. The v2(leftcolumn),v3 (middlecolumn)and v4 (rightcolumn)obtainedfromtwo-particlecorrelationsin0.3<pT<3 GeVinpp (toprow),p+Pb(middlerow)
andPb+Pb(bottomrow)collisions.Ineachpanel,theyarecomparedbetweenthreemethods:directFouriertransformation(solidcircles),templatefit(opencircles)andthe improvedtemplatefit(opensquares).Theerrorbarsandshadedboxesrepresentthestatisticalandsystematicuncertainties,respectively.
alsocalculateddirectlythe vn{2
}
valuesviaaFouriertransformof theY(φ)
distributionwithoutdijetsubtraction [7,19].The differ-encesbetweenthedirectFouriertransformandtemplatefitreflect mainlythe away-sidejet contributionsubtracted by thetemplate fitprocedure,andthereforegivea senseofthemagnitudeof un-known systematic uncertainties associated with the template fit procedure. If these differences are too large, the vn{
2,
tmp}
val-uesmaybesensitivetothesystematiceffectsassociatedwiththe assumptionthattheshapeofY(φ)
jetisindependentofNrecch.Fig.7 comparesthe vn{2
}
in 0.
3<
pT<
3 GeV obtainedfromY
(φ)
usingthreemethods:a directFouriertransform(solid cir-cles), a template fit (open circles) and a template fit corrected for the bias (open squares), as described above. The systematic uncertainties forthe template fit results are nearly the same as those fromRef. [7]. Fig. 7showsthat the changes introduced bythe correction procedure described above are small in all cases andforallharmonics.Thevaluesoftheeven-orderharmonics, v2
and v4, are also quite similar to those obtained from the direct
Fourier transformation, reflecting the fact that the dijet correla-tions havevery little influence on the even-orderharmonics. On the other hand,significant differences are observed between the direct Fouriertransformandtemplate fitfor v3,especially inthe
pp collisions, dueto the influence ofY
(φ)
jet,a trend observedanddiscussedpreviouslyinRefs. [7,15].Thetemplatefitprocedure isabletosubtractthedijetcorrelationsandchangethesignofv3,
butalsointroducesalargeuncertaintyassociatedwiththe proce-dure. AsdiscussedinSection8.2,thebehaviourofthesymmetric cumulantssc2,3
{
4}
inFig.4andnormalisedcumulantsnsc2,3{
4}
inFig.5inpp collisions,suggestthatthev3valuesfromthetemplate