Azimuthal anisotropy of charged jet production in √sNN=2.76 TeV Pb-Pb collisions

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Physics Letters B

www.elsevier.com/locate/physletb

Azimuthal anisotropy of charged jet production in √

s _NN = ² . 76 TeV Pb–Pb collisions

.ALICE Collaboration

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received5October2015

Receivedinrevisedform30November2015 Accepted16December2015

Availableonline18December2015 Editor:M.Doser

Wepresentmeasurementsoftheazimuthaldependenceofchargedjetproductionincentralandsemi- central √_s

NN=².76 TeV Pb–Pbcollisions withrespecttothesecondharmonic eventplane,quantified as v^{ch jet}₂ .Jetfindingisperformedemployingtheanti-kTalgorithmwitharesolutionparameterR=⁰.2 usingcharged tracksfromtheALICE tracking system.The contributionoftheazimuthalanisotropy of the underlyingeventis takenintoaccountevent-by-event. Theremaining(statistical) region-to-region fluctuations are removedonanensemblebasis byunfoldingthe jetspectra fordifferentevent plane orientations independently. Significant non-zero v^{ch jet}₂ is observed insemi-central collisions (30–50%

centrality)for20<p^{ch jet}_T <90 GeV/c.Theazimuthaldependenceofthechargedjetproductionissimilar tothedependenceobservedforjetscomprisingbothchargedandneutralfragments,andcompatiblewith measurementsofthev2ofsinglechargedparticlesathigh pT.Goodagreementbetweenthedataand predictions fromJEWEL,an event generatorsimulatingparton shower evolutioninthe presence ofa denseQCDmedium,isfoundinsemi-centralcollisions.

©^{2015CERNforthebenefitoftheALICE}Collaboration.PublishedbyElsevierB.V.Thisisanopen accessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The aim of the heavy-ion program at the LHC is to study strongly interacting matter in ultra-relativistic nuclear collisions where the formation of a quark–gluon plasma (QGP), a decon- fined state of quarks and gluons, is expected [1]. Hard partons thatpropagatethroughthecollisionmediumloseenergyvia(mul- tiple) scattering and gluon radiation [2,3]. Jet measurements are usedtoexperimentally explorepartonenergylossinthehot and densemedium. StudiesattheLHCandRHIChaveshownthat jet andhigh-p_T single particle productioninheavy-ioncollisions are suppressed with respect to the expected production in a super- position of independent pp collisions [4–13]. This observation is consistent withenergy loss,which is further supported by mea- surementsofdijetenergy asymmetry anddi-hadronangularcor- relations[14–16].

Innon-centralPb–Pbcollisions,theinitialoverlapregionofthe collidingnucleiprojectedintotheplaneperpendiculartothebeam directionhas an approximately elliptic shape. Jetsemitted along theminoraxisoftheellipse(definedasthein-planedirection)on averagetraverselessmedium–andarethereforeexpectedtolose lessenergy–thanjetsthatareemittedalongthemajoraxisofthe

E-mailaddress:[email protected].

ellipse(theout-of-planedirection).Thedependenceofjetproduc- tionontheanglerelativetothesecond-harmonicsymmetryplane 2 (thesymmetry planeangles n define theorientationsofthe symmetry axesoftheinitialnucleon distributionofthecollision) can be used to probe the path-length dependence of jet energy loss. This dependence is quantified by the parameter v^{ch jet}₂ , the coefficient of the second term ina Fourierexpansion ofthe az- imuthaldistributionofjetsrelativetosymmetryplanesn,

dN d

ϕ

_jet

₋

n

∝

¹

+

∞

n=¹

2v_n^jetcos

n

ϕ

_jet

₋

_n

,

(1)

where

ϕ

jetdenotestheazimuthalangleofthejet.

Incentralcollisions, theaveragedistancethata jetpropagates through the medium isapproximately equal in the in-plane and out-of-plane directions, therefore a small v^{ch jet}₂ is expected. In semi-central collisionsthe averagein-mediumdistanceis shorter, while the relative difference between the average distances in- plane and out-of-plane is larger, hence a non-zero v^{ch jet}₂ is ex- pected. Fluctuationsin the initial distributionof nucleons within the overlap region can lead to additionalcontributions to v^{ch jet}₂ andhigherharmoniccoefficientsintheFourierdecomposition.

The path-length dependence of parton energy loss is of par- ticular interest because it is sensitive to the underlying energy- loss mechanism. For collisional (elastic) energyloss, the amount http://dx.doi.org/10.1016/j.physletb.2015.12.047

0370-2693/©^{2015CERNforthebenefitoftheALICE}Collaboration.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

of lost energy depends linearly on path length, while forradia- tive (inelastic) energy loss, the dependence is quadratic due to interferenceeffects[17,18].Some strong-interactionmodelsbased on the AdS/CFT correspondence suggest an even stronger path- length dependence [19,20]. Earlier studies of the v2 of high-pT single particles have already tested the path-length dependence of energy loss [21–25]. Comparisons of these results to theoret- ical calculations have shown that the v2 is sensitive to several aspects of the medium evolution, including the effects of longi- tudinal andtransverseexpansion andthelife time ofthe system until freeze-out[26].It is thereforeimportant to measuremulti- ple observablesthat are sensitive to the path-length dependence ofenergy loss,such asrecoil yields ofcharged particlesand jets [11,27,28].Jetsareexpectedtobetterrepresenttheoriginalparton kinematicsandprovidemoredetailedinformationonenergyloss.

TheoreticalpredictionsfromJEWEL,whichcouplespartonshower evolutiontothepresenceofaQCDmediumwithadensityderived from Glauber simulations [29,30], have shown that a finite v^jet₂ is expected for non-central collisions at the LHC. Similar results havebeen foundin v^jet₂ studies inheavy-ioncollisions generated by the AMPT model [31,32]. A first measurement of v^{calo jet}₂ of jetscomprisingboth chargedandneutralfragmentshasbeenre- portedby the ATLAS Collaboration [33].The results presented in this paper extend the v^{ch jet}₂ measurement to a lower p_T range (pT>30 GeV/cforcentralcollisions andpT>20 GeV/c forsemi- centralcollisions).

Inthisarticle,measurements of v^{ch jet}₂ of R=⁰.2 chargedjets reconstructedwiththeanti-k_T jetfinderalgorithminPb–Pbcolli- sionswith0–5%and30–50%collisioncentralityarepresented.The largest experimental challenge in jet analyses in heavy-ion colli- sions is the separation of the jet signal fromthe background of mostlylow-pT particlesfromtheunderlyingeventandfromunre- latedscatteringsthat takeplaceinthecollision.Thejet energyis correctedonajet-by-jetbasisusinganestimateofthebackground transversemomentumdensitywhichtakesintoaccountthedomi- nantflowharmonicsv2andv3 ofthebackgroundevent-by-event, aswillbedescribed inSections2.1and2.2.Thecoefficient v^{ch jet}₂ isobtainedfromp_T-differentialjetyieldsmeasuredwithrespectto theexperimentally accessibleeventplane EP,2,whichis recon- structedatforwardrapidities(2.8<

η

<5.1 and−³.7<

η

<−¹.7, Sec. 2.1). The reported v^{ch jet}₂ hasbeen correctedback to theaz- imuthalanisotropywithrespecttotheunderlyingsymmetryplane 2 by applying an event plane resolution correction (Sec. 2.4).

Jetsare reconstructedat mid-rapidity(|

η

jet|<0.7) using charged constituenttrackswithmomenta0.15<p_T<100 GeV/c,andare required to contain a charged hadron with pT≥^{3 GeV}/c. The in-plane andout-of-plane jet spectraare unfolded independently to take into account detectoreffects and remaining azimuthally- dependent fluctuations in the underlying event transverse mo- mentum density (Sec.2.3). The jet spectra are corrected back to particle-leveljetsconsistingofonlyprimarychargedparticlesfrom thecollision.

2. Experimentalsetupanddataanalysis

ALICEisadedicatedheavy-ionexperimentattheLHCatCERN.

A full overview of the detector layout and performance can be foundin [34,35].Thecentral barreldetectorsystem, coveringfull azimuth,ispositionedinasolenoidalmagnetwithafieldstrength of0.5 T.ItcomprisestheInnerTrackingSystem(ITS)builtfromsix layersofsilicondetectors(the SiliconPixel,Drift,andStripDetec- tors:SPD,SDDandSSD)andaTimeProjectionChamber(TPC).The twoinnerlayersoftheITS,whichcomprisetheSPD,arelocatedat 3.9and7.2 cmradialdistancefromthebeamaxis.

The data presented in this paper were recorded in the Pb–

Pb data taking periods in 2010 and 2011 at √

sNN=².76 TeV, using aminimum-biastrigger (2010)oran onlinecentralitytrig- ger for hadronic interactions (2011), which requires a minimum multiplicity in both the V0A and V0C detectors (discs of seg- mented scintillators coveringfull azimuth and2.8<

η

<5.1 and

−³.7<

η

<−¹.7,respectively). The V0detectors areused tode- termine event centrality based on the energy deposition in the scintillator tiles[36]andtheeventplane orientation,seeSec.2.1.

Centrality, determined fromthe sumoftheV0amplitudes,isex- pressed as percentiles of the total hadronic cross section, with 0–5% referring to the most central (largest multiplicity) events [36].Thetriggerisfullyefficientinazimuthinthepresentedcen- tralityranges.CentralityestimationusingtheV0systemdoesnot biastheEP,n determination[37].TimeinformationfromtheV0 detectors is used to reject beam-gas interactions fromthe event sampleandtheremainingcontributionofsuchinteractionsisneg- ligible.Onlyeventswithaprimaryvertexpositionwithin±^{10 cm} alongthebeamdirectionfromthenominalinteractionpointwere usedintheanalysis.Atotalof6.8×^{106 eventswith0–5%central-} ityand8.6×^{106 eventswith30–50%}centrality,correspondingto integratedluminosities of18and5.6 μb⁻¹,respectively,are used inthisanalysis.

ChargedparticletracksinthisanalysisaremeasuredbytheITS andTPCandareselectedinapseudorapidityrange|

η

_|<0.9 with transversemomenta0.15<pT<100 GeV/c.Toensureagoodmo- mentumresolution,trackswererequiredtohaveatleastthreehits per track inthe ITS. Since theSPD acceptanceis non-uniformin azimuthfor thedatasample used inthisanalysis, two classesof tracksareused.Thefirstclassrequiresatleastthreehitspertrack inthe ITS,withatleastonehit pertrackintheSPD. Thesecond class contains tracks without hits in the SPD, in which case the primary interaction vertexis used asan additionalconstraintfor themomentum determination.Foreachtrack, theexpectednum- berofTPCspacepointsiscalculatedbasedonitstrajectory;tracks are acceptediftheyhaveatleast80%oftheexpectedTPCspace- points, witha minimum of70 TPCpoints. Tracksproduced from interactions between particlesand thedetector,as well astracks originatingfromweakdecays(‘secondarytracks’)arerejected.The contribution ofsecondary tracks to thetrack sample isless than 10% fortracks with pT<1 GeV/c andnegligible fortracks with highertransversemomentum.

2.1. Eventplanedetermination

The coefficient v^{ch jet}₂ quantifies azimuthalanisotropy withre- spect to 2. The azimuthal anisotropy of the underlying event (‘backgroundflow’) isalsodescribed byaFourierserieswithhar- monics v_n= ^cos(n[

ϕ

−n]) ^{[38,39] where}

ϕ

denotes thetrack azimuthalangle.However,sincetheinitialdistributionofnucleons is not accessible experimentally,the eventplaneangles EP,n,i.e.

the axes ofsymmetry of thedensity ofoutgoing particlesin the transverse plane,are usedinplace ofn when measuring v^{ch jet}₂ andvn.

The event plane angles EP,2 and EP,3 in this study, cor- responding to the two dominant Fourier harmonics, are recon- structed using the V0 detectors. Each V0 array consists of four ringsintheradial direction,witheach ringcomprisingeightcells withthesameazimuthalsize.Thecalibratedamplitudeofthesig- nal in each cell, proportional to the multiplicity incident on the cell,isusedasaweight wcell intheconstructionoftheflowvec- tors Qn [40]

Q_n

=

cells

w_cellexp

(

i n

ϕ

cell

) .

(2)

Inordertoaccountforanon-uniformdetectorresponsewhichcan generatea bias in the EP,n azimuthal distribution, the compo- nents ofthe Qn-vectorsare adjusted using a re-centeringproce- dure[41,42].TheV0AandV0Cdetectorscoverdifferent

η

regions inwhich multiplicity N andbackground flow v_n maydiffer. The totalV0 Q-vector istherefore constructedusingweights

χ

n [40]

thatare approximatelyproportionaltothe eventplane resolution ineachdetector,

Q_n,V0

= χ

_n²_,V0AQ_n,V0A

+ χ

_n²_,V0CQ_n,V0C

,

(3) toachievetheoptimalcombinedeventplaneresolution.Theevent planesare reconstructedfromtherealandimaginarypartsof Q_n as

EP,n

=

^arctan

^[Qn] [Q_n]

/

n

.

(4)

The v^{ch jet}₂ itself is measured with respect tothe second har- monic event plane angle. It is corrected for the finite precision withwhichthetruesymmetryplaneismeasuredintheV0system byapplyinganeventplaneresolutioncorrection,seeSec.2.4.

2.2.Jetreconstructioninthepresenceofbackgroundflow

Jetfinding isperformedusingthe FastJet[43,44]implementa- tion of the infrared and collinear safe k_T and anti-k_T sequential recombinationalgorithmsusingthepTrecombinationschemeand takingmasslessjetconstituents.TheresolutionparameterR=⁰.2 determines the characteristic maximum distance of constituent trackstothejetaxisinthe

η

–

ϕ

plane.

In heavy-ion collisions, a large combinatorial background is presentfromparticles that arenot relatedto the hard scattering thatproducedagivenjet.Thisbackgroundissubtractedfromeach jet on an event-by-event basis. The anti-kT algorithm is used to findsignaljets.Afiducialcut of|

η

jet|<0.7 is appliedon thesig- naljets toensure that all jets are fullycontainedwithin theITS andTPCacceptances andedge effects are avoided.The contribu- tion of combinatorial (or ‘fake’) jets (clustered underlying event energy) to the measured jet spectrum is reduced by requiring thatreconstructed jetscontain atleastone chargedparticlewith pT>3 GeV/c andhaveanarea ofatleast0.56

π

R².Theseselec- tioncriterialeavethehardpartofthejetspectrumunalteredwhile significantlyreducing thenumberofcombinatorialjetswhichsta- bilizestheunfoldingprocedure[4,5,45].

The k_T-algorithm is used to estimate the average transverse momentumdensityoftheunderlyingevent,

ρ

ch^{,onanevent-by-} eventbasis.Thequantity

ρ

ch^{isthemedianofthe}distributionof p^raw_T,chjet/A(theratiooftransversemomentumtojetarea)ofrecon- structed R=⁰.2 k_T-jets, excluding the leading two jetsfromthe sampleasproposed in[46] andimplemented inearlierALICEjet studies[4,5,45].The kT jetsarerequired tolie within |

η

jet|<0.7 andhave anarea A>0.01.Thejet area A is determinedby em- bedding a fixed number of near zero-momentum ghostparticles per event prior to jet finding; the number of ghost particles in eachreconstructedjetthengivesadirectmeasureofthejetarea.

A ghostdensityof200particles perunitarea isused,sothat ap- proximately25ghostparticlesareclusteredintoajetwitharadius of 0.2.

Ineach event, theanisotropy ofthe underlyingeventismod- eledusingthedominant[47]flowharmonicsv2andv3,

ρ

ch

( ϕ ) = ρ

0

1

+

²

{

^v2cos

2

ϕ −

EP,2

+

^v3cos

3

ϕ ₋

EP,3

}

.

(5)

Fig. 1.Transversemomentumdensityofchargedtracksasafunctionofazimuthal angleforasingleeventfromthemostcentral0–5%eventclass.Datapoints(blue) aregivenwithstatisticaluncertaintiesonly.TheredcurveisthefitofEq.(5)to thedistribution,thegreenandgraycurves,obtainedfromthefitofEq.(5)aswell, showtheindependentcontributionsofv2and v3toρch(ϕ).Thedashedmagenta lineisthenormalizationconstantρ0.(Forinterpretationofthereferencestocolor inthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Here,

ρ

ch(

ϕ

)istheazimuthaldistributionofsummedtrackpT for tracks with 0.15<p_T<5 GeV/c and |

η

track|<0.9. The parame- ters

ρ

0 and vn are determined event-by-event from a fit of the right side of Eq. (5) to the data. The event plane angles EP,n are not fitted, but fixed to the V0 event plane angles. A single eventexampleofthisprocedureisillustrated inFig. 1,wherethe datapoints representthetransversemomentum densitydistribu- tioninasingle event,theredcurve representsthefull functional descriptionof

ρ

ch(

ϕ

)(Eq.(5)),thegreenandgraycurvesgive the contributionsoftheseparateharmonicsv2andv3,andthedashed magentalineisthenormalizationconstant

ρ

0.Toreduce thebias ofhard jets inthe estimatesof vn in Eq.(5) whileretaining az- imuthal uniformity, the leading jet in each eventis removed by rejecting all tracksfor which|

η

jet−

η

track|<R.The

η

separation betweenthetracksandtheV0detectorsalsoremovesshortrange correlationsbetweentheeventplanesandtracks.

ThenumberofbinstowhichEq.(5)isfittedissetonanevent- by-eventbasistothesquare rootofthenumberoftracks.Thefit maximizestheestimatedlikelihood[48],whichisbasedonaPois- son distribution for the bin content. Since the bin contents are not pure counts, butweighted by pT,the statisticaluncertainties on each bin

σ

i are estimated as the sum of the squares of the pT of the individual particles:

σ

i=

σ

( pT)= √ ^p2_T^{. A scaled} Poisson distribution P(xi/wi|^mui/wi) is used as the probability distributionforthedatapoints inthelikelihood calculation,with a scale factor wi=

σ

_i²/yi where yi is the bin content and

μ

i is theexpected signal fromthe fit function.The compatibility of each fit withthe data is tested by calculating the

χ

² and eval- uating the probability of finding a test statistic at leastas large asthe observed one inthe

χ

² distribution. When this probabil- ityis less than0.01, the average eventbackground density

ρ

ch is usedinstead of

ρ

ch(

ϕ

); thisoccursin 3% (mostcentral) to7%

(semi-central) ofevents.Theacceptance criterionis variedin the systematicstudies;thesensitivitytoitissmall.

Thecorrectedtransversemomentum p^chjet_T ofajetofarea A is calculatedfromthemeasuredrawjetmomentum,p^raw_T,chjet,as

p^chjet_T

=

p^raw_T,chjet

− ρ

ch localA (6)

where

ρ

ch local is obtained from integration of

ρ

ch(

ϕ

) around

ϕ

jet±^R

Fig. 2.TheδpTdistribution(Eq.(8))fromtherandomcone(RC)procedureasfunctionofconeazimuthalangleϕRCrelativetotheeventplane.Inpanel(a)theazimuthally- averagedbackgroundρch^hasbeensubtracted;inpanel(b)theazimuthallydependentρch(ϕ)fromanevent-by-eventfitofthe pT-densitywithEq.(5).Thesolidblack linerepresentsthemeanoftheδpTdistribution.

ρ

_{ch local}

₌ ρ

_ch

2R

ρ

0 ϕ

+^R

ϕ−^R

ρ

_ch

( ϕ )

d

ϕ .

(7)

Thepre-factoroftheintegral, ρch

2Rρ0,ischosensuchthatintegration over the full azimuth yields the average transverse momentum density

ρ

ch^{.ThevalidityofEq.(5)asa}descriptionofthecontri- butionofbackgroundflowtotheunderlyingeventenergyistested byplacingconesofradius R=⁰.2 atrandompositions(excluding thelocation ofthe leading jet)inthe

η

–φ plane andsubtracting theexpected summed transversemomentum in acone fromthe measuredtransversemomentuminthecone,

δ

p_T

=

p^tracks_T

− ρπ

R²

.

(8)

Here,

ρ

is theexpected transversemomentum density.This pro- cedure is repeatedmultiple timesper event, until the full phase spaceis covered, to obtain a distribution ofδp_T values.The δp_T distributionasafunctionoftheconeazimuthalangle

ϕ

RCrelative totheeventplaneEP,2 isshowninFig. 2.Inpanel(a)

ρ

ch^has beenusedfortheestimationoftheunderlyingeventsummed pT andinpanel(b)

ρ

ch(

ϕ

).Incorporatingazimuthal dependenceinto theunderlyingeventdescriptionleadstoasizablereductioninthe cosinemodulationoftheδpTdistribution.

Theeffectivenessofthesubtractionofbackgroundflowisquan- tified by comparing the expected and measured widths of the δpTdistributionintheabsenceofbackgroundflow,

σ

(δp_T^vn=0)(see Fig. 2(b)) to the expected and measured widths of the δp_T dis- tribution in thepresence of backgroundflow,

σ

(δp^v_Tⁿ) (Fig. 2(a)).

Assumingindependentparticleemission andPoissonianstatistics, theexpectedwidthoftheδpTdistributionintheabsenceofback- groundflow(v_n=^{0)isgivenby[45]}

σ (δ

p_T^vn=0

) =

N_A

σ

²

(

p_T

) +

^NA

^pT

^{2 (9)}

whereNAistheaverageexpectednumberoftrackswithinacone, ^pT ^{is the mean p}T of a single particle spectrum and

σ

(pT) is the standard deviationof thisspectrum. This expectationcan be extendedtoincludecontributionsfrombackgroundflowby intro- ducing non-Poissonian density fluctuations (the background flow harmonicsvn)[45],as

σ (δ

p_T^vn

) =

N_A

σ

²

(

p_T

) + (

N_A

+

^2N2_A

(

v²₂

+

^v2₃

))

^pT

²

.

(10)

Fig. 3.Centralitydependenceofthemeasuredandexpectedrelativechangeinthe δpTdistributionwidthfromusingtheazimuthallydependentρch localinsteadofthe medianρch.Thebluepointsgivetheexpectedreductionfromsimpleassumptions aboutthebehaviorofchargedparticlespectraandflowharmonics vn (following Eqs.(9)and(10)).TheredpointsusethemeasuredwidthsfromδpTdistributions directly.Statisticaluncertaintiesaresmallerthanthemarkersize.(Forinterpretation ofthereferencestocolorinthisfigurelegend,thereaderisreferredtotheweb versionofthisarticle.)

The measured widths areobtained fromthe δp_T distributions directly; thedistributions are constructedusingasthe transverse momentumdensity

ρ

inEq.(8)either

ρ

ch^toobtain

σ

(δp^v_Tⁿ)or

ρ

ch local for

σ

(δp^v_Tⁿ⁼⁰).

Fig. 3 shows the expected and measured relative change in the width of the δpT distribution, quantified as (

σ

(δp_T^vn) −

σ

(δp_T^vn=0))/

σ

(δp^v_Tⁿ), as function of collision centrality. The blue points give the expected reduction from Eqs. (9) and (10). The red points use the measured widths from δpT distributions. The expectedchangeisingoodquantitativeagreementwiththemea- sured changeover theentire centralityrange,indicating that the width of the δpT distributions can be understood in terms of a simpleindependentparticleemissionmodelwithbackgroundflow contributions.

The background subtraction, unfolding,and correction for the reactionplaneresolutionasdescribedinSections2.3and2.4were also validated using events consisting of PYTHIA jets embedded inheavy-ionbackgroundeventsandtoymodelevents.Inthefirst

study, full PYTHIA pp events were combined with reconstructed Pb–Pbcollisionstocreateeventswithacontrolledsignalandback- ground.ThesignaljetsfromPYTHIAhavenopreferredorientation, v^{ch jet}₂ =^{0,whiletheheavy-ioneventshavea non-zero v}2 ofthe softparticles. Jetsfound inthe events were matchedto the em- beddedPYTHIAjetsandtheanalysiswascarriedoutwithmatched jetsonly.Afterunfolding,thev^{ch jet}₂ wascompatiblewith0,asex- pected. The other study was based on events generated using a simplethermalmodelforsoftparticleproductionandadistribu- tionofhigh-pT particlesthat resemblesthejetspectrum,assug- gestedin[49].Anon-zerov2=⁰.07 wasintroducedformomenta p_T<5 GeV/ctomodelthebackgroundflowandtwovariationsat largepT>30 GeV/c: v2=^{0 or v}2=⁰.05.Inbothcases,theinput flowvalueswerecorrectlyreconstructedbytheanalysis.

2.3.Unfolding

Afterthesubtractionprocedurepresentedintheprevious sec- tion,themeasured jetspectrumisunfolded[50,51] tocorrectfor detectoreffectsandfluctuationsintheunderlyingeventtransverse momentumdensity.Mathematically,theunfoldedjetspectrumcan bederivedfromthemeasuredspectrumbysolving

M

(

p^rec_T,chjet

)

=

G

(

p^rec_T,chjet

,

p^gen_T,chjet

)

T

(

p^gen_T,chjet

) ε (

p^gen_T,chjet

)

dp^gen_T,chjet (11)

forT(p^gen_T,chjet),theunfoldedtruejetspectrum,whereM(p^rec_T,chjet)is themeasured jet spectrum, G(p^rec_T,chjet,p^gen_T,chjet) is afunctional de- scription(responsefunction) ofdistortions duetobackgroundfluc- tuations and detector response, and

ε

(p^gen_T,chjet) is the jet finding efficiency.The coefficient v^{ch jet}₂ is not affected by the efficiency, hence

ε

(p^gen_T,chjet)willbeomittedfromhereon.Sincethemeasured jetspectrumisbinned, Eq.(11)isdiscretizedby replacingthein- tegralbyamatrixmultiplication

Mm

=

Gm,t

·

T_t (12)

where T_t is the solution of the discretized equation (the prime indicates that T_t is not corrected for jet-finding efficiency). The combined response matrix G_m,t is the product of the response matricesfromdetectoreffects andtransversemomentumdensity fluctuations,thelatterofwhichareconstructedindependentlyfor thein-planeandout-of-planespectrabyembeddingrandomcones atspecificrelativeazimuthwithrespecttotheeventplane(seethe textbelowEq.(13)forthedefinitionoftheintervals).

Thedetectorresponse matrix isobtainedby matching pp jets generatedbyPYTHIA[52](‘particle-level’jets)tothesamejetsaf- tertransportthroughthedetector(‘detector-level’jets)byGEANT3 [53], where the detector conditions are tuned to those of the Pb–Pbdata-takingperiods.Particle-leveljetscontainonlyprimary chargedparticlesproducedbytheeventgenerator,whichcomprise allprompt chargedparticles producedin thecollision,aswell as productsof strongandelectromagneticdecays,while productsof weakdecaysofstrangehadronsarerejected.Matchingisbasedon theshortestdistanceinthe

η

–

ϕ

planebetweendetectorleveland particleleveljetsandisbijective,meaningthat thereisaone-to- onecorrespondence betweendetectorandparticlelevel jets. The response matrix for background fluctuations is constructed from theδpTdistributions,which,whennormalized,areprobabilitydis- tributionsforthechange ofthejet energycausedbybackground fluctuations.

SolvingEq.(12)requiresinversion ofGm,t andgenerallyleads to non-physical results which oscillate wildly due to the statis-

tical fluctuations of the measured jet yield. The unfolded solu- tion therefore needs to be regularized. In general this is done by introducing a penalty term for large local curvatures associ- atedwithoscillations.Variousalgorithmsforregularizedunfolding exist;the unfoldingmethodbased onthe SingularValue Decom- position(SVD unfolding)[54] isused inthisstudy.Acomparison totheunfoldedsolutionfrom

χ

²minimization[55]isusedinthe systematicstudies.

The measured jet spectrum is taken as input forthe unfold- ingroutineintherange30<p^chjet_T <105 GeV/cfor0–5%collision centralityand15<p^chjet_T <90 GeV/c for30–50%collisioncentral- ity. The lower bound corresponds to five times thewidth of the δp_T distribution,theupperboundistheedgeofthelastmeasured binwhichcontainsatleast10counts.Thisconfigurationwasfound toleadtoreliableunfoldedsolutionsinMonte Carlostudies[4,49].

The unfolded jet spectrum starts at0 GeV/c to allow forfeed-in oftrue jetswithlow p^chjet_T . Inaddition,combinatorial jetswhich are not rejected by the jet area and leading charged particle re- quirements are migrated to momenta lower than the minimum measured p^chjet_T . The unfolded solution ranges up to 200 GeV/c (0–5%) and170 GeV/c (30–50%)to allowformigration ofjetsto a p^chjet_T higher than themaximum measured momentum. As the data points of the unfolded solution are strongly correlated for p^chjet_T outsidetheexperimentallymeasuredinterval, v^{ch jet}₂ willbe reportedonlywithinthelimitsofthemeasuredjetspectra.

2.4. Evaluationofv^{ch jet}₂

Thecoefficientv^{ch jet}₂ iscalculatedfromthedifferencebetween the unfolded pT-differential jet yields in-plane (Nin) and out-of- plane (N_out) with respect to the second harmonic event plane, correctedforeventplaneresolution,

v^{ch jet}₂

(

p^chjet_T

) = π

4 1 R2

N_in

(

p^chjet_T

) −

^Nout

(

p^chjet_T

)

N_in

(

p^chjet_T

) +

^Nout

(

p^chjet_T

) .

(13) Eq. (13) is derived by integrating Eq. (1) for n=^{2, over inter-} vals

−^π₄,^π₄ and

3π 4 ,⁵₄^π

for N_in and π

4,³₄^π

and

5π 4 ,⁷₄^π

for Nout,substituting EP,2 for2.Eq.(13)is sensitivetocorre- lationsbetweeneven-orderharmonicsv2n andEP,2.Asaresult oftheintegrationlimitshowever,thefirstharmonicoftheFourier expansionthatcancontributetotheobservedv^{ch jet}₂ isv^{ch jet}₆ .The V0eventplane resolutionR2 isintroducedto accountforthefi- niteprecisionwithwhichthetruesymmetryplane2ismeasured intheV0systemandisdefinedas

R2

=

cos

2

_EP^V0_,2

−

2

.

(14)

Measuring eventplanes inmultiple

η

regions (sub-events) allows fortheevaluationoftheresolutiondirectlyfromdata[56,57].Us- ing the fullV0 detectorandnegative andpositive

η

sidesof the TPCassub-events,theresolutioninEq.(13)isevaluatedas R2

=

⎛

⎝

cos 2

_EP^V0_,2−_EP^TPC_,^,₂^η>0 cos 2

_EP^V0_,2−_EP^TPC_,^,₂^η<0

cos 2

_EP^TPC_,^,₂^η>0−_EP^TPC_,^,₂^η<0

⎞

⎠

1/2

.

(15) The eventplane resolutionR2 is found tobe 0.47 in0–5% cen- tralityand0.75in30–50%centralitywithnegligibleuncertainties.

TheEP,2 anglesintheTPCareobtainedfollowingtheprocedure

of Eq. (4) on tracks with 0.15<pT<4 GeV/c, using unit track weightsintheconstructionoftheflowvectorsQ2 (seeEq.(2)).

UsingtheV0detectorsforthereconstructionoftheeventplane guaranteesthatthejetaxisandeventplaneinformationaresepa- ratedinpseudorapidityby |

η

|>1 andthus removesautocorre- lationbiasesbetweenthesignal jetsandeventplane orientation.

The possiblenon-flow correlation betweenthe eventplane angle andjetsduetodi-jetswithonejetatmid-rapidityandonejetin theV0acceptancewas studiedusingthePYTHIAeventgenerator.

The rateof such di-jetconfigurations was found to be negligible (lessthan 1 per milleof thetotal di-jetrateat mid-rapidity) for p^chjet_T >20 GeV.Possibleeffectsfromback-to-backjetpairswitha jetineachoftheV0detectorsareevensmaller.

2.5. Systematicuncertainties

Themeasuredv^{ch jet}₂ iscorrectedforexperimentaleffects,such asthefiniteeventplaneresolutionanddetectoreffectsonthejet energyscaleaswellastheeffectsoftheuncorrelatedbackground and its fluctuations using the corrections outlined in the Sec- tions2.1–2.4.Hydrodynamicflow ofthe backgroundistakeninto accountevent-by-eventintheunderlyingeventdescription,resid- ualeffects areremoved by azimuthally dependent unfolding.The remaininguncertaintiesinthesecorrectionproceduresaretreated assystematicuncertainties.Systematicuncertaintieson v^{ch jet}₂ are groupedinto twocategories, shapeand correlated,basedon their point-to-point correlation. Shape uncertainties are anti-correlated betweenpartsoftheunfoldedspectrum:whentheyieldinpartof thespectrumincreases,itdecreaseselsewhereandviceversa.Cor- related uncertainties are correlated point-to-point. Both types of uncertaintieshoweverhavecontributionswhichleadtocorrelated changesofN_in andNout.

Correlateduncertaintiesareestimatedforthein-planeandout- of-planejet spectraindependently. Twosourcesofcorrelated un- certainties are considered: tracking efficiency and the inclusion of combinatorial jets in the measured jet spectrum. The domi- nant correlated uncertainty (^{10%) arises from tracking and is} estimatedbyconstructingadetectorresponsematrixwithatrack- ingefficiencyreducedby 4%(motivatedby studies[4]comparing reconstructed tracks to simulations of HIJING [58] events). The observed difference betweenthe nominaland modified unfolded solutionistakenasasymmetricuncertaintytoallowforanover- andunderestimation of the tracking efficiency. The sensitivityof theunfoldedresulttocombinatorialjetsistestedbychangingthe lower range of the unfolded solution from 0 to 5 GeV/c, which leads toan overall(correlated) increase oftheunfolded jetyield.

Both correlated uncertainties are added in quadrature andprop- agatedto v^{ch jet}₂ assuming that variations are strongly correlated betweenthe in-plane andout-of-plane jet spectra, whilestill al- lowingforeffects fromazimuthally-dependentvariations in track occupancyandreconstructionefficiency,bysettingthesamplecor- relationcoefficient

ρ

≡

σ

i,j/(

σ

i

σ

j)to0.75.

Shape uncertainties fall into three categories: assumptions in the unfolding procedure, feed-in of combinatorial jets, and the sensitivityof the unfolded solution to the shape of the underly- ing event energy distribution. The dominant contribution to the unfolding uncertainty is related to the regularization of the un- foldedsolution.The SVDalgorithm[54] regularizestheunfolding byomittingcomponentsofthemeasured spectrumforwhichthe singular value is smallandwhich amplify statisticalnoise inthe result.Toexplorethesensitivityoftheresulttotheregularization strength, theeffective rankof thematrix equation that is solved isvariedbychanginganintegerregularizationparameterkby±^1.

The SVDunfolding algorithm uses aprior spectrumas thestart- ingpointoftheunfolding;theresultoftheunfoldingistheratio

between the full spectrum and thisprior. The unfolded solution fromthe

χ

² algorithm[55] isusedasprior (default)aswellasa PYTHIAspectrum.Thebiasfromthechoiceofunfoldingalgorithm itselfistestedbycomparingtheresultsoftheSVDunfoldingand the

χ

² algorithm.

Thesamenominalunfoldingapproachisusedforthein-plane and out-of-plane jet spectra and the δpT distributions for the in-plane andout-of-plane background fluctuations are similar in width; the unfolding uncertainty is therefore strongly correlated between the in-plane and out-of-plane jet spectra. These corre- lations are taken into account by applying the variations in the unfolding procedure to the in-plane andout-of-plane jet spectra atthesametimeandcalculatingtheresultingvariationsofv^{ch jet}₂ . Thetotaluncertaintyfromunfoldingisdeterminedbyconstructing a distributionofall unfoldedsolutions ineach p^chjet_T intervaland assigningthewidthofthisdistributionasasystematicuncertainty.

The other two components of the shape uncertainty are the sensitivityof theunfolded solutionto combinatorialjetsandun- certainties arising from the description of the underlying event;

both are estimated on the in-plane andout-of-plane jet spectra independentlyandpropagatedtov^{ch jet}₂ asuncorrelated.Asystem- atic uncertainty is only assigned when the observed variation is found to bestatisticallyincompatible withthe nominalmeasure- ment.Theeffectofcombinatorialjetsistestedbyvaryingthemin- imump^chjet_T ofthemeasuredjetspectrumby±^{5 GeV}/c,effectively increasingordecreasingthepossiblecontributionofcombinatorial jet yield atlow jetmomentum. Totest theassumptions madein the fittingofEq.(5) themaximum p_T ofaccepted tracksis low- eredto 4 GeV/c. Additionally, theminimum p-valuethat is used as a goodness offit criterion is changed from0.01 (the nominal value)to 0.1.Theminimumrequireddistanceoftrackstothelead- ingjetaxisinpseudorapidityisenlargedto 0.3.

Table 1 gives an overview of the systematic uncertainties in terms of absolute uncertainties on v^{ch jet}₂ for all sources (where thetotaluncertaintyisthequadraticsumoftheseparatecompo- nents).HighstatisticsMonte Carlotestinghasbeenusedtoverify thatuncertainties labeled‘^{stat’areindeednegligiblecompared} tootheruncertainties.

3. Resultsanddiscussion

The coefficients v^{ch jet}₂ as function of p^chjet_T for 0–5% and 30–50%collisioncentralityarepresentedinFig. 4.Significantpos- itive v^{ch jet}₂ is observed insemi-central collisions andno (signifi- cant) pTdependenceisvisible.Theobservedbehaviorisindicative ofpath-length-dependent in-medium partonenergy loss.Theob- served v^{ch jet}₂ in central collisions is of similar magnitude. The systematic uncertainties on themeasurement however are larger than thoseonthe semi-central v^{ch jet}₂ data,inparticular atlower p^chjet_T ,asaresultofthelargerrelativebackgroundcontributionto themeasuredjetenergy.

The significance of the results is assessed by calculating a p-valueforthehypothesisthat v^{ch jet}₂ =^{0 overthepresentedmo-} mentum range.The p-valueisevaluated startingfromamodified

χ

² calculationthat takesintoaccount bothstatisticaland(corre- lated)systematicuncertainties,assuggestedin[59].Themodified

χ

² forthehypothesisv^{ch jet}₂ =

μ

i iscalculatedbyminimizing

˜

χ

²

( ε

corr

, ε

_shape

) =

_n

i=1

(

v_2,i

+ ε

corr

σ

corr,i

+ ε

shape

− μ

i

)

²

σ

_i²

+ ε

_corr²

+

¹ n

n

i=¹

ε

²_shape

σ

_shape² _,i

(16)

Table 1

Systematicuncertaintiesonv^{ch jet}₂ forvarioustransversemomentaandcentralities.Uncertaintiesincentralandsemi-centralcollisionsaregiveninthesamepTranges.The definitionsofshapeuncertaintyandcorrelateduncertaintyareexplainedinSec.2.5.Fieldswiththevalue‘stat’indicatethatnosystematiceffectcanberesolvedwithin thestatisticallimitsoftheanalysis.

p^chjet_T (GeV/c) Uncertainty onv^{ch jet}₂

30–40 60–70 80–90 30–40 60–70 80–90

Centrality (%) 0–5 30–50

Shape Unfolding 0.017 0.012 0.016 0.016 0.011 0.015

p^chjet_T -measured 0.013 ^{stat stat 0.024 stat stat}

ρch(ϕ)fit 0.015 ^{stat 0.016 stat stat stat}

Total 0.027 0.012 0.023 0.029 0.011 0.015

Correlated Tracking 0.009 0.009 0.009 0.007 0.007 0.007

p^chjet_T -unfolded ^{stat stat stat stat stat stat}

Total 0.009 0.009 0.009 0.007 0.007 0.007

Fig. 4.Second-orderharmoniccoefficientv^{ch jet}₂ asafunction of p^chjet_T for0–5%(a)and30–50%(b)collisioncentrality.Theerrorbarsonthepointsrepresentstatistical uncertainties,theopenandshadedboxesindicatetheshapeandcorrelateduncertainties(asexplainedinSec.2.5).

withrespect tothesystematicshifts

ε

shape,

ε

corr,where v2,i rep- resentthemeasureddata(npoints),

σ

iarestatisticaluncertainties and

σ

shape,i,

σ

corr,idenotethetwospecifictypesofsystematicun- certainties.Theparameter

ε

corrisameasureofthefullycorrelated shifts;ashiftofalldatapointsbythecorrelateduncertainty

σ

corr,i givesa totalcontributionto

χ

˜² ofone unit.Thesystematicshifts forthe shape uncertainty are taken to be of equal size for each point, since this gives the best agreement with the v^{ch jet}₂ =⁰ hypothesis andthus provides a conservative estimate of the sig- nificance;the penalty factor is constructed such that an average shiftofalldatapointsby

σ

shape addsoneunitto

χ

˜².

The p-value itself iscalculated usingthe

χ

² distribution with n−^{2 degrees of freedom. For} semi-central collisions a p-value of0.0009isfound,indicating significantpositive v^{ch jet}₂ .Itshould be noted that the most significant data points are at p^chjet_T <

60 GeV/c; the results in the range 60<p^chjet_T <100 GeV/c are compatible with v^{ch jet}₂ =^{0 (p-value 0.02). In central} collisions, a p-value withrespect to thehypothesis of v^{ch jet}₂ =^{0 of0.12is} foundwhichindicatesthat v^{ch jet}₂ iscompatiblewith0withintwo standarddeviations. Followingthe sameapproach an upperlimit ofv^{ch jet}₂ =⁰.088 isfoundwithinthesameconfidenceinterval.

3.1.Comparisontopreviousmeasurementsandmodelpredictions

Togeta betterqualitativeunderstandingoftheresults,the v₂ of single charged particles v^part₂ [21,22] and the ATLAS v^{calo jet}₂

measurement [33] are shown together with the v^{ch jet}₂ measure- mentinFig. 5.TheATLASresultisforjetswithresolutionparam- eter R=⁰.2 within |

η

_|<2.1 comprising both chargedand neu- tralfragments.The eventplane angleismeasured bytheforward calorimetersystemat3.2<|

η

|<4.9.Jetsarereconstructedbyap- plyingtheanti-kT algorithmtocalorimetertowers,afterwhich,in an iterativeprocedure,a flow-modulatedunderlying eventenergy issubtracted.Eachjetisrequiredtoliewithin

η

²+

ϕ

²<0.2 ofeitheracalorimeterclusterofpT>9 GeV/cora pT>10 GeV/c trackjet withresolutionparameterR=⁰.4 builtfromconstituent tracks of pT>4 GeV/c (the full reconstruction procedurecan be foundin[33,60]).

It is importantto realize that the energyscales ofthe ATLAS v^{calo jet}₂ andALICEv^{ch jet}₂ measurementsaredifferent(astheALICE jetsdonotincludeneutralfragments)whichcomplicatesa direct comparisonbetweenthetwomeasurements.ThecentralATLASre- sultsarealsoreportedin5–10%collisioncentrality.TheALICEand ATLASmeasurementsareinqualitativeagreement,bothindicating path-length-dependentpartonenergyloss.Giventheuncertainties, thedifferenceinthecentralvaluesofthemeasurementisnotsig- nificant.

Fig. 5alsoshowsthe v2 ofsinglechargedparticlesv^part₂ (from [21,22]),whichisexpectedtobemostlycausedbyin-mediumen- ergylossatintermediateandhighmomenta(p_T^{5 GeV}/c).Even thougha directquantitativecomparisonbetweenv^{ch jet}₂ andv^part₂ cannot be madeastheenergyscales forjetsandsingle particles aredifferent,themeasurementscanbecompared qualitatively,and

Fig. 5.Ellipticflowcoefficientv2ofchargedparticles[21,22](red,green)andR=0.2 fulljets(comprisingbothchargedandneutralfragments)measuredwithin|η|<2.1 [33](blue)superimposedontheresultsfromthecurrentanalysisofR=0.2 chargedjetsv^{ch jet}₂ .Inallmeasurements,statisticalerrorsarerepresentedbybarsandsystematic uncertaintiesbyshadedoropenboxes.NotethatthesamepartonpTcorrespondstodifferentsingleparticle,fulljetandchargedjetpT.ATLASv^{calo jet}₂ andCMSv2from [22,33]in30–50%centralityaretheweightedarithmeticmeansofmeasurementsin10%centralityintervalsusingtheinversesquareofstatisticaluncertaintiesasweights.

(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6.v^{ch jet}₂ ofR=⁰.2 chargedjetsobtainedfromtheJEWELMonte Carlo(redline)forcentral(a)andsemi-central(b)collisions comparedtodata.JEWELdatapointsare presentedwithonlystatisticaluncertainties.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

itcanbeseenthat forcentralevents,thesingleparticle v^part₂ and v^{ch jet}₂ areofsimilarmagnitudeandonlyweaklydependentonpT over a large range of pT (≈^{20–50 GeV}/c). For non-central colli- sions (30–50%), the measurements of v₂ for single particles and jetsare also inqualitative agreement in the pT range wherethe uncertaintiesallowforacomparison.

Fig. 6showsthev^{ch jet}₂ ofR=⁰.2 chargedjetsfromtheJEWEL Monte Carlo[29,30]comparedtothemeasured v^{ch jet}₂ .JEWELsim- ulates parton shower evolution in the presence of a dense QCD medium by generating hard scatterings according to a collision geometryfromaGlauber[61]densityprofile.A1DBjorkenexpan- sionisused tosimulatethetime evolution ofthemedium. After radiativeandcollisionalenergyloss,PYTHIAis usedtohadronize thefragmentstofinalstateparticles.

TheanalysisontheJEWELeventsisperformedwiththe same jet definitionandacceptancecriteriathat are usedforthe v^{ch jet}₂ analysisindata,usingthesymmetryplane2 fromthesimulated initial geometry as EP,2. The JEWEL Monte Carlo shows finite significant v^{ch jet}₂ in semi-central collisions; in central collisions

v^{ch jet}₂ is compatiblewithzero.The JEWELresultforsemi-central 30–50%collisionsiscompatiblewiththemeasuredvalues(p-value 0.4 using Eq. (16) with the JEWEL results as hypothesis

μ

i and the quadraticsumofthestatisticaluncertainties ofboth datasets as

σ

i inthe denominator ofthe firstsumof Eq.(16)). In central JEWEL collisions v^{ch jet}₂ is consistent with zero, while the mea- sured values are compatible with the JEWEL v^{ch jet}₂ within two standard deviations.Itshould alsobe notedthat JEWELcurrently uses an optical Glauber modelfor the initial state andtherefore doesnotincludefluctuationsintheparticipantdistributiondueto the spatial configuration ofnuclei inthe nucleus. This simplified treatment ofthe overlapgeometrymayunderestimate the v^{ch jet}₂ [38,62]. Thiscomparisonofv^{ch jet}₂ inJEWELtoexperimental data complements earlierstudiesofthepath-length-dependent parton energylossandmodelpredictionsforthejet RAA [5].

4. Conclusion

The azimuthal anisotropy of R=⁰.2 charged jet production, quantified as v^{ch jet}₂ , has been presented in central and semi-