Elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in Pb-Pb collisions at √sNN = 2.76 TeV

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Physics Letters B

www.elsevier.com/locate/physletb

Elliptic flow of muons from heavy-flavour hadron decays at forward rapidity in Pb–Pb collisions at √

s _NN = ² . 76 TeV

.ALICE Collaboration

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

Articlehistory:

Received15July2015

Receivedinrevisedform19November2015 Accepted20November2015

Availableonline2December2015 Editor: L.Rolandi

Keywords:

LHC

ALICEexperiment Pb–Pbcollisions

Heavy-flavourdecaymuons Ellipticflow

The elliptic flow, v2, of muons from heavy-flavour hadron decays atforward rapidity (2.5< y<4) is measured inPb–Pb collisions at√_s

NN=².76 TeV with the ALICE detector atthe LHC. The scalar product,two- andfour-particleQ cumulantsandLee–Yangzerosmethodsareused.Thedependenceof the v2ofmuonsfromheavy-flavourhadrondecaysonthecollisioncentrality,intherange0–40%,and ontransversemomentum, pT,isstudiedintheinterval 3<pT<10 GeV/c.Apositive v2is observed withthe scalarproductandtwo-particle Q cumulantsinsemi-central collisions(10–20%and 20–40%

centrality classes) for the pT interval from 3 to about 5 GeV/c with a significance larger than 3σ, basedonthecombinationofstatisticalandsystematicuncertainties.Thev2magnitudetendstodecrease towardsmorecentralcollisionsandwithincreasingpT.Itbecomescompatiblewithzerointheinterval 6<pT<10 GeV/c.Theresultsarecomparedtomodelsdescribingtheinteractionofheavyquarksand openheavy-flavourhadronswiththehigh-densitymediumformedinhigh-energyheavy-ioncollisions.

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

1. Introduction

Experimentswith ultra-relativistic heavy-ion collisions aim at investigatingtheproperties ofstrongly-interacting matter atvery hightemperaturesandenergydensities.QuantumChromodynam- ics (QCD) calculations on the lattice predict, under these con- ditions, the formation of a Quark–Gluon Plasma (QGP), where color confinement vanishes and chiral symmetry is partially re- stored[1–5].Heavyquarks(charmandbeauty)arecreatedinini- tialhard-scatteringprocessesonatimescaleshorterthantheQGP formationtime.Subsequently,theyinteractwiththemediumcon- stituentsviainelastic[6,7]andelastic[8–10]processes.Therefore, heavyquarksare regardedaseffectiveprobes oftheQGPproper- ties.

Heavy-quark energy loss due to in-medium interactions can be studied by means ofthe nuclear modification factor R_AA, de- finedastheratiooftheyieldofheavy-flavourparticles measured in nucleus–nucleus (AA) collisions to that observed in proton–

proton (pp) collisions scaled by the number of binary nucleon–

nucleoncollisions.ThePHENIXandSTARCollaborationsmeasured, incentralAu–Aucollisions at√

sNN=^{200 GeV, astrongsuppres-} sion corresponding to a R_AA of about 0.2–0.3 for heavy-flavour decay electrons at mid-rapidity (y) and transverse momentum pT>5 GeV/c[11–17].Asimilarsuppressionwasalsomeasuredby

E-mailaddress:[email protected].

theSTARCollaborationformid-rapidityD⁰ mesons[18].A signif- icantsuppression wasalso observedbythePHENIX Collaboration at forwardrapidity formuons fromheavy-flavour hadron decays in central Cu–Cucollisions at √

sNN=^{200 GeV [19]. At the LHC,} theALICECollaborationreportedasimilar effectincentralPb–Pb collisions at √

sNN=².76 TeV for Dmesons atmid-rapidity [20]

and muons from heavy-flavour hadron decays at forward rapid- ity[21]intheinterval 2<p_T<16 GeV/cand4<p_T<10 GeV/c, respectively. The CMS Collaboration measured a significant sup- pression of non-prompt J/ψ from beauty-hadron decays in the interval 6.5<pT<30 GeV/c (3<pT<30 GeV/c) and|^y|<2.4 (1.6<|^y|<2.4)[22,23]. Afirstmeasurementofnon-prompt J/ψ by the ALICECollaborationat mid-rapidity(|^y|<0.8) andin the interval4.5<pT<10 GeV/c hasbeenrecentlypublished[24].

Further insights into the QGP evolution and the in-medium interactions can be gained from the study of the azimuthal anisotropy of particles carrying heavy quarks which, in contrast to light quarks, have experienced the full system evolution. The study of azimuthal anisotropy is a field of intense experimental andtheoreticalinvestigations(see [25]andreferencestherein).In non-central collisions,the initialspatial anisotropy ofthe overlap region, elongated in the direction perpendicular to the reaction plane, defined by the beam axis and the impact parameter of thecollision,isconvertedintoananisotropy inmomentumspace throughrescatterings[26].Experimentally,thestudyoftheparticle azimuthalanisotropyisbasedonaFourierexpansionofazimuthal distributionsgivenby:

http://dx.doi.org/10.1016/j.physletb.2015.11.059

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

d²N dp_Td

ϕ ⁼

1 2

π

dN dp_T

1

+

²

∞ n=¹

v_n

(

p_T

)

cos

[

ⁿ

( ϕ −

n

)]

,

(1)

where

ϕ

and pT are the particleazimuthal angle andtransverse momentum,respectively.The Fouriercoefficients, vn,characterize theanisotropyofproducedparticlesandnistheazimuthalangle of the initial-state symmetry plane for the nth harmonic, intro- ducedtoaccount fortheevent-by-eventfluctuationsoftheinitial nucleondensityprofile. Thesecond Fourier coefficient, v2, which canalsobe expressedas v₂= ^cos[²(

ϕ

−2)]^{,isnamed elliptic} flow.

The v2 of heavy-flavourhadronsis expectedto provide infor- mation on the collective expansion anddegree of thermalization ofheavy quarksinthemedium atlow pT (pT<2–3 GeV/c).The participation of heavy quarks in the collective expansion is ex- pected to give a positive v2 [26]. Moving towards intermediate p_T (3<p_T<6 GeV/c),the v₂ Fouriercoefficient isalsoexpected tobe sensitiveto thepresenceofrecombinationprocessesinthe hadronizationofheavyquarks[27,28].AthighpT (pT>6 GeV/c), thev₂ measurementcanconstrainthepath-lengthdependenceof thein-medium partonenergyloss, whichbecomes the dominant contributionto theazimuthal anisotropy andis alsopredictedto givea positive v2 [29,30],thuscomplementingthe RAA measure- ment.

The PHENIX Collaboration reported a positive v2 of heavy- flavour decay electrons at mid-rapidity in Au–Au collisions at

√s_NN =^{200 GeV, reaching a maximum value of about 0.15 at} pT=¹.5 GeV/c in semi-central collisions [14,15,31]. A similar behavior was also observed by the STAR Collaboration [32]. Re- cently,av2valuesignificantlylargerthanzerowasmeasuredforD mesonsatmid-rapidityinPb–Pbcollisionsat√

s_NN=².76 TeV[33, 34].Acomplementarymeasurementatthesameenergy,provided bytheheavy-flavourdecaymuonellipticflowatforwardrapidity (2.5<y<4), is of great interest in order to provide new con- straints for models that implement the heavy-quark interactions withthemedium. Finally,themeasurement isalsoimportantfor the interpretation of the J/ψ elliptic flow results at forward ra- pidity[35]intermsofaregenerationproductionfromdeconfined charmquarksinthemedium.

InthisLetter, wepresentthemeasurementoftheellipticflow of muons from heavy-flavour hadron decays at forward rapidity (2.5< y<4) in Pb–Pb collisions at √

sNN=².76 TeV recorded withthe ALICE detector.The elliptic flow is measured using dif- ferentmethods:scalarproduct[36],two- andfour-particle Q cu- mulants [37,38] andLee–Yang zeros [39–41]. These methods ex- hibitdifferentsensitivitiestoflowfluctuationsandcorrelationsnot relatedtotheazimuthal asymmetryintheinitial geometry(non- floweffects).The v₂ coefficientismeasuredasafunctionofp_T in theinterval 3<pT<10 GeV/c and inthree centralityclasses in therange0–40%.Thecentralitydependenceofv2 ispresentedin theinterval3<pT<5 GeV/c.

TheLetterisorganized asfollows.TheALICEdetector,withan emphasisonthemuonspectrometer,andthedatasamplearepre- sentedin Section 2.The analysisdetails, the methods forthe v2 measurement, the inclusive muon v₂ determination, the proce- dureforthesubtractionofthebackgroundofmuonsfromdecays oflight-flavourhadronsandthestudyofsystematicuncertainties, aredescribed inSection3.The v₂ resultsformuonsfromheavy- flavourdecaysarepresentedinSection4.The v₂ measurementin semi-centralcollisionsaswellasthepublished RAAincentralcol- lisions are compared to model calculations in Section 5. Finally, conclusionsaregiveninSection6.

2. ALICEexperimentanddatasample

The ALICE detector is described in detail in [42,43]. The ap- paratus is composedof aset ofcentral barreldetectors (pseudo- rapidity coverage|

η

|<0.9)located insideasolenoidmagnetthat generates a field of0.5 T parallelto the beamdirection, amuon spectrometer (−⁴<

η

<−².5¹) and a set of detectors forevent characterization andtriggering located in the forward andback- ward

η

regions.Themuonspectrometerconsistsofapassivefront absorber made of carbon, concrete and steel, a beam shield, a 3 T mdipole magnet,trackingchambers,a muonfilter(ironwall) andtrigger chambers. The muon trackingsystemis composed of five stations, each includingtwo planes ofcathod pad chambers, withthe thirdstationinside thedipole magnet. Themuon track- ing systemis completed by four trigger planes of resistiveplate chambers downstream of the iron wall, which absorbs hadrons thatpunchthroughthefrontabsorber,aswell assecondaryparti- clesproducedinsideitandlowmomentummuons(p<4 GeV/c).

Twoscintillatorarrays(V0)coveringthepseudo-rapidityinter- vals −³.7<

η

<−¹.7 and 2.8<

η

<5.1 are used for triggering, forcollision centralitydeterminationandforbeam-inducedback- ground rejection.The Zero DegreeCalorimeters (ZDC), located at 114 m fromthecentre ofthe detectoron both sides, can detect spectator protons and neutrons andare also used for the offline rejection ofbeam-inducedbackgroundandelectromagneticinter- actions. The Silicon PixelDetector (SPD), that composes the two innermost layers of the Inner Tracking System (ITS), is used for theinteractionvertexreconstruction.TheTimeProjectionChamber (TPC), whichmeasurescharged-particletrackswithfullazimuthal coveragein|

η

|<0.9,isusedinthisanalysisforthemeasurement ofthereferenceparticles(Section3.1).

TheresultspresentedinthisLetterareobtainedfromthedata sample recordedwithALICEduringthe2011Pb–Pbrun.Thedata were collected withaminimum-biastrigger requiringthecoinci- denceofsignalsinthetwoV0arraysinsynchronizationwiththe passage oftwo crossing bunches.In addition,the recordedevent sample was enriched with central and semi-central Pb–Pb colli- sionsbyapplyingthresholds,atthetriggerlevel,ontheV0signal amplitude.Thebeam-inducedbackground(beam–gasinteractions) was reduced by using the timing information from the V0 and ZDCdetectors.Furthermore,a minimalenergydepositintheZDC was requiredto rejectthe contributionfromelectromagneticPb–

Pb interactions. Onlyevents witha reconstructed primary vertex within ±^{10 cmfromthenominalpositionofthe} interactionver- texalongthebeamdirectionareanalyzed.ThePb–Pbcollisionsare classified accordingtotheir degree of centralityby means ofthe sum ofthe amplitudesof thesignals inthe V0detectorand the centrality classesare definedaspercentiles ofthe total hadronic Pb–Pb crosssection[44].The analysisiscarriedout inthreecen- trality classes: 0–10% (using the sample with trigger on central collisions),10–20%and20–40%(using thesamplewithtriggeron semi-central collisions).The analyzeddatasample corresponds to anintegratedluminosityof11.3 μb⁻¹ inthe0–10%centralityclass andof3.5 μb⁻¹ intheothertwocentralityclasses.

3. Dataanalysis

The elliptic flow of muonsfrom heavy-flavourhadron decays, vμ←^HF

2 ,isobtainedfromthemeasurement oftheinclusivemuon ellipticflow,vμ

2,bysubtractingtheellipticflowofmuonsfrompri-

1 IntheALICEreferenceframe,themuonspectrometercoversanegativeηrange andconsequentlyanegativeyrange.Inthefollowing,giventhatthecollidingsys- temissymmetric,theresultsarepresentedwithapositiveynotation.

marychargedpionandkaondecaysvμ←π,K

2 (Sections3.1and3.4), as:

v^μ₂^←HF

=

^v

μ

2

−

^fμ←π,K

·

^vμ₂^←π,K

1

−

^fμ←π,K

,

(2)

where fμ_←π,K is the muon background fraction, definedas the ratiooftheyield ofmuonsfromprimary chargedpionandkaon decaystothatofinclusivemuons.Themeasurementofthevμ←^HF

2 coefficientiscarriedoutintheinterval3<pT<10 GeV/cinorder tolimitthesystematicuncertaintyonthesubtractionofthemuon backgroundcontribution.

3.1.Trackselection

Theselectioncriteriaforparticles ofinterest,muontracks,are similarto those usedin theprevious analyses ofpp collisions at

√s=².76 TeV and7TeVandPb–Pbcollisionsat√

sNN=².76 TeV [21,45].The tracks are requiredto be within thegeometrical ac- ceptance of the muon spectrometer, with −⁴<

η

<−².5 and 170^◦< θabs<178^◦,whereθabs isthepolaranglemeasuredatthe endoftheabsorber.Inordertoimprovethemuonidentification,a reconstructedtrackinthetrackingchambersisrequiredtomatch atracksegment inthe triggerchambers. Thisleads to avery ef- ficientrejectionofthebackgroundproduced by chargedhadrons, which are absorbed in the iron wall. Furthermore, a cut on the product p·^{DCA ofthe track momentum p anddistance of clos-} estapproach(DCA)totheprimaryvertexisappliedtoremovethe beam-inducedbackgroundtracksandfaketrackscomingfromthe superpositionofseveralparticlescrossingthemuonspectrometer.

Due to multiple scatteringin the front absorber, the DCAdistri- bution of tracks coming form the interaction vertex is expected to be described by a Gaussian function, its width being depen- dentontheabsorbermaterialandproportionalto1/p.Background trackshaveaverybroaddistributioninp·^DCAandareeffectively rejected by a cut at 6

σ

, where

σ

is extracted from a Gaussian fit to the p·^DCA distribution measured intwo intervalsof θabs, correspondingtodifferentmaterialsinthefrontabsorber.Therel- ative momentum resolution of reconstructed tracks varies from about1%to4%fortrackswithmomentumbetween20 GeV/c and 100 GeV/c.Afterthecutsareapplied,intheregion p_T>3 GeV/c theresidualbackgroundtoheavy-flavourdecaymuonsconsistsof muonsfrom decaysof primary charged pions andkaons² andit amounts to 5–15%, depending on pT and on collision centrality (Section3.4).

Themid-rapiditycharged-particletracksusedtodeterminethe flowvector Qn orthegeneratingfunction(Section3.2)are called in the following reference particles. They are defined as tracks measured in the TPCin |

η

|<0.8. These are required to have at least 70 associated space points out of the maximum of 159, a

χ

² per degree of freedom (ndf) for the momentum fit in the range

χ

²/ndf<2 and a transverse momentum value in the in- terval0.2<p_T<5 GeV/c.Additionally,tracksarerejectediftheir distanceof closest approachto the primary vertex islarger than 3 cmintheplanetransversetothebeamdirectionorinthelon- gitudinaldirection.

3.2.Flowanalysismethods

The elliptic flow measurement is carried out using various methodsthat havedifferentsensitivities to flow fluctuationsand

2 Notethatthecontributionofmuonsfromsecondarylighthadrondecayspro- ducedinsidethefrontabsorberisnegligibleforpT>3 GeV/c[45].

non-flow effects [46]. Flow fluctuationsare mainlydue toevent- by-eventfluctuationsoftheinitial densityprofile,whilenon-flow effects correspond to correlations not related to the azimuthal anisotropy inthe initial state,such asresonance decays,jetsand Bose–Einsteincorrelationsbetweenidentical particles.It isworth mentioning that, inthe present analysis, mostof thesenon-flow effects arestrongly suppressedby introducing an

η

gapbetween reference particles andparticles of interest [47]. In thisanalysis, the scalarproduct [36],two- and four-particle Q cumulants [37, 38] andLee–Yang zeros [39–41] methods are employed. The de- scriptionofthesemethodswill belimitedtothefeaturesspecific to the present analysis. The following notations are introduced:

vμ(μ←^HF)

2 {^SP}^,referstothemeasurementusingthescalarproduct, vμ(μ←HF)

2 {²} ^{and vμ}₂^(μ←HF){⁴} ^{correspond to the ones using the} two-particle Q cumulants andfour-particle Q cumulants, while vμ(μ←^HF)

2 {^LYZ-Prod} ^{and vμ}₂^(μ←HF){^LYZ-Sum} ^{are obtained using} Lee–Yang zeros with product and sum generating functions. The superscripts

μ

and

μ

←^{HF refer to inclusivemuonsandmuons} from heavy-flavourhadron decays, respectively. It is worth men- tioning that thesemethods are more accurate than the standard eventplane method,which yields a measurement lying between the event-averaged mean value and the root-mean-square value inthe presenceofflow fluctuations[48,49].Moreover, themulti- particlecorrelation methods(four-particle Q cumulantsandLee–

Yang zeros)are less affected by non-flow correlationsthan two- particle correlation methods, but they cannot be used reliably when the muon flow magnitude is small and when the num- ber of muonsis small in the selected phase-spaceregion e.g. in centralandperipheralcollisions,respectively[37,39].Underthese conditions,thescalarproductandtwo-particlecumulantmethods providea v₂ measurementinawidercentralityrange.

The scalarproduct method[36,48], derived fromthestandard eventplane technique [48], is basedon the measurement of the flow vector Q_n [36] computed fromreference particles. In order to determine the ellipticflow, the Q2 vector in a givenevent is expressedas:

Q

₂

=

^N

j=¹ cos2

ϕ

j

,

N

j=¹ sin2

ϕ

j

,

(3)

where

ϕ

jistheparticleazimuthalangleandN isthemultiplicity ofreferenceparticles.

Withthismethodthe2ndharmoniccoefficientisgivenby:

v₂

{

^SP

} =

^Q2

·

^u2,i

( η ,

p_T

)

2

Q₂^A

·

^Q₂^B

,

(4)

where the brackets in the numerator indicate the average over muonsat forwardrapidity,in all events.The vector Q2 iscalcu- latedfromEq.(3)andthevectoru2,i=(cos 2

ϕ

i,sin 2

ϕ

i)istheunit vectoroftheithmuon.Inthedenominator,eachsampleofrefer- enceparticlesusedtocompute Q2isdividedintotwosub-samples ofsamemultiplicity insymmetrical

η

intervals, −⁰.8<

η

<−⁰.5 and 0.5<

η

<0.8, separated by a

η

gap of one unit of pseudo- rapidity, labeled withthe superscripts A and B and the brackets correspondtotheaverageoverevents.

Thecumulanttechnique[37,38]isbasedonacumulantexpan- sionofmulti-particleazimuthalcorrelations.Differentordercumu- lantshavedifferentsensitivitiestoflowfluctuations.Inthepresent analysis, two- andfour-particlecumulantsareusedtoextractthe muonellipticflow. Theresultspresentedinthe followingareob- tained from a direct calculation of multi-particle cumulants per- formed by usingthe Q-cumulanttechnique [38], whichis based on the moments of the magnitude of the flow vector Q₂. It is

worth mentioning that in this approach the cumulants are not biasedby the interferences between various harmonics. The ref- erenceellipticflowvaluesV2evaluatedfromthe2ndordercumu- lant c₂{²} ^{and4thorder cumulantc}2{⁴} ^{withreferenceparticles} are given by V2{²}=√

c2{²} ^{and V}2{⁴}=√⁴

−^c2{⁴}^,respectively.

Oncethereferenceellipticflowisestimated,themuonellipticflow withrespecttothereferenceellipticflowisobtainedfromthe2nd and4thordercumulantsaccordingto:

v2

{

2

} =

^d2

{

2

}

V₂

{

²

}

^andv2

{

4

} =

^d2

{

4

}

V2

{

4

}

³

,

(5)

where d2{²} ^andd2{⁴} ^{are the 2nd and 4thorder cumulants of} selectedmuons[38].

The Lee–Yangzeros method [39–41] relieson correlations in- volving all particles in the event. This is the limit of cumulants whentheorderofcumulantsgoestoinfinity.Themethodisbased on the location ofthe zeros in thecomplex plane, of a generat- ing functionof azimuthal correlations, whichrelates theposition ofthefirstminimumofthegeneratingfunctiontothemagnitude ofthereferenceellipticflowV₂ definedas:

V₂

≡

M

j=¹

cos

[

²

( ϕ

j

−

2

)]

events

,

(6)

whereM isthemultiplicityofreferenceparticlesandtheaverage istakenover allevents.Forthispurpose,thefollowing complex- valuedgeneratingfunctionisevaluatedasafunctionofapositive realvariablerandfew,typicallyfive,equallyspacedreferencean- glesϑ (LYZ-Prodmethod):

G^ϑ

(

ir

) ≡

M

j=1

(

1

+

ircos

[

2

( ϕ

j

− ϑ)])

events

.

(7)

Thefirstpositive minimumof|^Gϑ(ir)|^{,denotedasrϑ}₀^,allowsone to estimate V₂^ϑ, which can be written as V₂^ϑ = ^j01/r₀^ϑ, where j01².405 isthe firstrootofthe Besselfunction. Oncethefirst minimum r^ϑ₀ is determined, the differential muon elliptic flow is estimated with respect to the reference flow V₂^ϑ as detailed in[41].Finally,theresultis averagedover allϑ angles.An alter- nativeformofthegeneratingfunctionprovidedwiththeLYZ-Sum methodis:

G^ϑ

(

ir

) ≡

exp

ir

M

j=1

cos

[

2

( ϕ

j

− ϑ)]

events

.

(8)

Theversionofthemethodinvolvingaproductfortheconstruction of the generation function (Eq. (7)) was designedto disentangle interferences betweendifferentharmonics, whichis not the case withthegeneratingfunction usinga sumoftheindividual refer- enceparticlecontributions.Both generatingfunctionsareusedin thisanalysis.

Notethat, for all methods, autocorrelationeffects are avoided because the particles (muons) used in the determination of the flowarenotincludedintheestimationofthereferenceflow.

3.3. Inclusivemuonellipticflow

The ellipticflow ofinclusivemuons, vμ

2,isstudied withtwo- particle correlation methods (scalar product and two-particle Q cumulants)inthecentralityintervals0–10%,10–20%and20–40%.

In the 20–40% centrality interval, the multi-particle correlation methods(four-particle Q cumulantsandLee–Yangzeros)arealso used.

Severalsourcesofsystematicuncertaintyaffectingthemuonel- liptic flow measurement are considered. Thesetake into account the changes dueto thevariations ofthe referenceparticle selec- tion criteriaasin[33,34,50],to allowusto check therobustness ofthevμ

2 measurement.Sincethecollisionimpactparameterdis- tribution could slightly depend on the observable used for the centralitydetermination,a systematicuncertaintyis estimatedby repeating the analysisusingthe numberofclustersin theouter- most layer of the SPD and the number of tracks in the TPC as centralityestimators,insteadoftheV0signalamplitude. Thesys- tematicuncertaintyduetotheeffectofTPCtracksfromdifferent Pb–Pbcollisions piled-upinthesamerecordedeventisestimated byapplyingatightercuttoremoveoutliersinthemultiplicitydis- tribution ofreferenceparticles.Thisisdone byrequiringthat the centralityvaluesdeterminedusingtheV0signalamplitudeandthe numberofTPCtracksdonotdifferbymorethan5%.Anadditional systematic uncertaintyspecific to the scalar product isevaluated by varying the

η

gapbetweenthe two sub-events from1to 0.8

η

-units (see Eq. (4) and[36]). The various systematicuncertain- tiesareaddedinquadrature.Theytendtoincreasewithincreasing p_T (see Fig. 1).Asummaryofthesystematicuncertainties,inthe interval3<pT<4.5 GeV/c,ispresentedinTable 1.

Fig. 1showsthe pT-differentialmuonellipticflow(vμ 2) inthe 0–10%, 10–20% and 20–40% centrality classes as obtained using thevariousmethods.Thevaluesofvμ

2 slightlyincreasefromcen- tral tosemi-central collisions andthiseffect ismorepronounced in the pT interval 3<pT<4.5 GeV/c. The two-particle correla- tion methods(scalarproduct andtwo-particle Q cumulants)give consistent results over the whole pT range,indicating that these methodshaveasimilarsensitivitytonon-floweffects³andinpar- ticular to flow fluctuations. A similar agreement is found when comparingthemulti-particlecorrelation methods(four-particle Q cumulantsandLee–Yangzeros)toeachother.Nosignificantdiffer- encebetweenthe vμ

2 resultsextractedwithLee–Yangzerosusing either thesum orproduct generatingfunction is seen, hencein- dicating thatinterferencesbetweenharmonicsarenegligible [51].

Moreover, four-particle Q cumulants give comparable results as Lee–Yang zeros.The four-particle Q cumulantsandLee–Yang ze- rosareexpectedtobelessaffectedbynon-floweffectsthanscalar productortwo-particle Q cumulants[52].However,asmentioned non-flow effects are expected to be negligible, even with two- particle correlation techniques, due to the large

η

between ref- erenceparticlesandinclusivemuons.Finally,thecentralvaluesof vμ

2 obtainedwithfour-particleQ cumulantsorLee–Yangzerosare systematicallysmallerthanwithtwo-particlecorrelationmethods, althoughcompatiblewithinuncertainties.Suchdifferencesmayin- dicatethatinitialfluctuationsplayaroleinthedevelopmentofthe finalmomentum-spaceanisotropy.

3.4. Muonbackgroundsubtraction

The subtraction of the muon background contribution to the measured vμ

2 requires an estimate of the elliptic flow of muons from charged pion and kaon decays, vμ←π,K

2 , and of the back- ground fraction, fμ←π,K (see Eq. (2)). The determination of the vμ←π,K

2 coefficient requires two steps. First, the p_T- and

η

-differential v2 of charged particles measured in |

η

_|<2.5 by the ATLAS Collaboration inPb–Pb collisions [53] andthe pT dis- tributions of charged pions and kaons measured in |^y|<0.8 by

3 Notethat,inthisanalysis,mostnon-flowcorrelationsaresuppressed,evenwith two-particlecorrelationmethodssincereferenceparticlesandinclusivemuonsare separatedbyatleast1.7 η-units.However,itisworthmentioningthatthemain differencebetweenthetwomethodsistheηgapbetweenthetwosub-samples usedtocomputeQ2(Eq.(4))whichalsoallowstopartlyremovenon-floweffects.

Table 1

Systematicuncertaintysourcesaffectingtheinclusivemuonellipticflowmeasurementinthe0–10%,10–20%and 20–40%centralityclassesfortheinterval3<pT<4.5 GeV/c.Theyaregivenasapercentageofthev2value.

v^μ₂ analysis Source Systematic uncertainty (%)

0–10% 10–20% 20–40%

v^μ₂{^SP} Reference particles 3 1 3

Centrality selection 6 1 4

TPC pile-up 2 4 2

ηgap 13 1 1

v^μ₂{2} Reference particles 13 3 2

Centrality selection 14 3 6

TPC pile-up 8 1 4

v^μ₂{4} Reference particles 10

Centrality selection 1

TPC pile-up 1

v^μ₂{LYZ-Sum} Reference particles 4

Centrality selection 7

TPC pile-up 2

v^μ₂{LYZ-Prod} Reference particles 2

Centrality selection 8

TPC pile-up 2

Fig. 1.pT-differentialinclusivemuonv2in2.5<y<4 andvariouscentralityintervals,inPb–Pbcollisionsat√

sNN=².76 TeV.Thesymbolsareplacedatthecentreofthe pTintervaland,forvisibility,thepointsfromtwo-particleQcumulantsandLee–Yangzeroswithproductgeneratingfunctionareshiftedhorizontally.Theverticalerrorbars representthestatisticaluncertainty,thehorizontalerrorbarscorrespondtothewidthofthebin(notshownfortheshifteddatapoints)andtheopenboxesarethesystematic uncertainties.ThepTintervalsusedwiththeLee–Yangzerosmethodaredifferentwithrespecttotheothermethods.Upperpanels:resultsfromtwo-particlecorrelationflow methods(scalarproductandtwo-particle Q cumulants)inthe0–10%(left)and10–20%(right)centralityintervals.Lowerpanels:resultsinthe20–40%centralityinterval fromtwo-particlecorrelationflowmethods(scalarproductandtwo-particleQ cumulants)andfromfour-particleQ cumulants(left),andfromfour-particleQ cumulants andLee–Yangzeros(right).

the ALICE Collaboration in pp and Pb–Pb collisions [54,55] are extrapolated to forward rapidity. Then, the p_T distributions of muons from charged pion and kaon decays, needed to estimate

fμ←π,Kandvμ←π,K

2 ,aregeneratedaccordingtoasimulationtak- inginto accountthe decaykinematicsandtheeffectof thefront absorber.

The p_T- and

η

-differentialelliptic flowof chargedparticles in

|

η

|<2.5,v^ch₂ ,isextrapolatedtoforwardrapidityusing:

v^ch₂

(

pT

, η ) =

^F

( η ) ·

^vch2

(

pT

,

2

< | η _| <

2

.

5

),

(9) where v^ch₂ (pT,2<|

η

_|<2.5)is themeasured charged-particle el- liptic flow in 2<|

η

|<2.5 with the event plane method. Since

the v^ch₂ (pT) measured by the ATLAS Collaboration is affected by statistical fluctuations, it is assumed that in the interval 10<

pT<20 GeV/c,neededto simulatethedecaymuonsup to pT= 10 GeV/c, v^ch₂ remains constant with a value given by the one measured in the interval 10<pT<12 GeV/c. The extrapolation factor F(

η

) iscalculated by parameterizingthe

η

-differential v^ch₂ measuredbytheATLASCollaborationinvarious p_T intervalswith asecondorderpolynomial.Intheinterval7<pT<20 GeV/c,the ATLAS v^ch₂ doesnotshowa dependenceon

η

in|

η

_|<2.5.There- fore,for p_T>7 GeV/c, F(

η

)iscomputedastheaveragebetween aflatextrapolationfunctionandtheextrapolationfactorobtained withtheparabolicparameterizationin4<pT<7 GeV/c.

Themid-rapiditychargedpionandkaon pT distributionsmea- suredinPb–Pbcollisionsareextrapolatedtoforwardrapidityusing thesamestrategyasin[21]andsummarizedinthefollowing.As- sumingthatthenuclearmodificationfactorRπ,K

AA ofchargedpions andkaonsin Pb–Pbcollisions doesnot dependon rapidityup to y=^{4[21,56],the p}T distributionsofchargedpionsandkaons at forwardrapiditycanbeexpressedas:

dN_PbPb^π,K

dp_Tdy

=

^TAA

·

^d

σ

_pp^π,K

dp_Tdy

· [

^Rπ_AA^,K

(

p_T

)]

y=0

,

(10) where ^TAA ^{is the average nuclear overlap function in central-} ity classesunder study, estimated asdescribed in [57]. The sys- tematic uncertainty introduced by the assumption on Rπ,K

AA will be discussed later. The rapidity extrapolation ofthe mid-rapidity pionandkaon p_T-differentialcrosssectionsmeasuredinppcolli- sions[21,58]isdoneaccordingto:

d²

σ

_pp^π,K

dp_Tdy

=

d²

σ

_pp^π,K

dp_Tdy

y=⁰

·

^exp

−

^y2 2

σ

_y²

,

(11)

σ

y beingestimated fromMonte-Carlo eventgenerators (see [21]

fordetails).

The elliptic flow of muons from charged pion and kaon de- cays, vμ←π,K

2 , in 2.5< y<4 and in various centrality classes,⁴ is obtained by means of fast simulations using v^ch₂(

η

,p_T) given by Eq. (9) and charged pion and kaon pT distributions as ob- tained from Eqs. (10)–(11). The absorber effect is accounted for by rejectingthe pionsandkaonsthat donot decaywithin adis- tancecorrespondingtoone interactionlengthfromthebeginning of the absorber. The simulation was repeated twice, considering thatchargedparticlesareeitherallpionsorallkaons.

Thebackground fraction, fμ←π,K,is calculatedastheratio of thepT-differentialyieldofmuonsfromchargedpionandkaonde- cays in 2.5< y<4 obtained in the simulation to the measured p_T-differentialyieldofinclusivemuons.

The systematic uncertainties affecting the estimated vμ←π,K 2 aresummarizedinTable 2.Theyoriginatefromi)themethodused tomeasurethecharged-particle v^ch₂ inATLAS,ii)the

η

andpT ex- trapolation of v^ch₂ and iii) the treatment of the charged-particle v^ch₂ inthe fastsimulation procedure. As theeventplane method was used for the v^ch₂ measurement in ATLAS, the results range betweenthemean (^vch2^{)and R.M.S.(}

(v^ch₂ )^{2) ofthe true vch}2 valuesdue tofluctuations, depending onthe eventplane resolu- tionwhichvarieswiththecollisioncentrality[49].Accordingtoa Monte-CarloGlaubermodel[49],theratio

v²₂/v2isexpected

4 Thev^{μ←π ,}₂ ^Kofmuonsfromchargedpionandkaondecaysinthe20–40%cen- tralityclassisthenobtainedfromthemeanofthecharged-particlev2in20–30%

and30–40%centralityclasses,withanadditionalsystematicuncertaintyprovided bythedifferencewithrespecttotheresultsinthesetwocentralityclasses.

Table 2

Systematicuncertaintysourcesaffectingtheestimatedv^{μ←π ,}₂ ^Kfortheinterval3<

pT<10 GeV/c.Theyarestatedasapercentageofthev2 value.Thegivenrange reflectsthedependenceonthecollisioncentrality.

Source Systematic uncertainty (%)

Inputv^ch₂ bias 9

v^ch₂ ηextrapolation 9–12

v^ch₂ highpTextrapolation 13–15

πand K in fast simulations <1

to varyfromabout1.06 to1.15.Consequently,a conservativesys- tematicuncertaintyof15%isappliedtoaccount forthisbiasand ispropagatedto vμ←π,K

2 .Thesystematicuncertaintyduetothe

η

extrapolation of v^ch₂ is evaluated using severalfit functions(first andthirdorderpolynomials,andGaussian function)intheregion pT<7 GeV/c, andfor larger pT values an additional systematic uncertaintyduetotheextrapolationprocedureisconsidered. The latter is determined by comparing the results obtainedwith the twoextrapolationfunctionsusedintheinterval pT>7 GeV/c.The systematic uncertainty due to the assumption on v^ch₂ in the re- gion p_T>10 GeV/c is estimatedby varying v^ch₂ between0 and thevaluein10<p_T<12 GeV/c inthefastsimulations.Suchun- certaintyaffectsmainlythehighpT region(pT>7 GeV/c).Finally, the systematicuncertainty obtainedby treating charged particles separately aspionsandkaonsisfound tobenegligible.The vari- oussystematicuncertaintysourcesarepropagatedtotheestimated vμ←π,K

2 andaddedinquadrature.

The systematic uncertainty on fμ←π,K, detailed in [21], in- cludestheuncertaintyonthegenerated pTdistributionsofmuons from charged pionand kaon decays,and the uncertainty on the measured inclusivemuon pT distributions. The former originates from theinput chargedpion andkaon distributions, therapidity extrapolation andthe absorber effect.The systematicuncertainty on the measured inclusive muon yields contains the systematic uncertaintyondetectorresponse,residualmis-alignmentandcen- trality dependenceof the efficiency.This givesa total systematic uncertainty on fμ←π,K of about 21% in the interval 3<pT<

4.5 GeV/c withalmost nodependenceonthecollision centrality.

Finally, asdone forthe measurement of theheavy-flavour decay muon RAA [21], the systematic uncertainty dueto the unknown suppression of chargedparticles at forwardrapidity is calculated by varying fμ←π,K from 0 to two times the estimated value.

This corresponds to a variation of Rπ,K

AA (pT) at forward rapidity from0uptotwotimes[^Rπ_AA^,K(pT)]y=⁰.Thissystematicuncertainty amountsto10–30%intheinterval3<pT<4.5 GeV/c,depending onthecollisioncentralityandtheflowanalysismethod.

Fig. 2presentstheestimatedbackgroundellipticflow(vμ_←π,K

2 ,

left)andbackgroundfraction(fμ←π,K,right)asafunctionofpTin the0–10%,10–20%and20–40%centralityclasses.Theopenboxes representthesystematicuncertaintiespreviouslydiscussed,except forthesystematicuncertaintyduetotheunknownsuppressionof charged particles atforward rapidity which is treatedseparately.

The estimated vμ←π,K

2 and fμ←π,K decrease withincreasing p_T. Adecreasingtrendofthemagnitudeofvμ←π,K

2 fromsemi-central collisionstowardscentralcollisionsisalsoobserved.

Finally,thesystematicuncertaintyontheellipticflowofmuons fromheavy-flavourdecays,vμ←HF

2 ,containstwocontributions:the systematic uncertainties on vμ

2, vμ←π,K

2 and fμ←π,K propagated accordingtothedefinitionofvμ←^HF

2 giveninEq.(2),andthesys- tematic uncertainty due to the unknown suppression of charged particles atforward rapidity. The final systematic uncertainty on vμ←HF

2 isobtainedbyaddinginquadraturethetwocontributions.

Fig. 2.Estimatedbackgroundv2(v^{μ←π ,}₂ ^K,left)andbackgroundfraction(f^{μ←π ,K},right)asafunctionofpTin2.5<y<4 andvariouscentralityintervals,inPb–Pbcollisions at√

sNN=2.76 TeV.ThesymbolsareplacedatthecentreofthepTintervaland,forvisibility,thepointsforthecentralityclasses10–20%and20–40%areshiftedhorizontally.

Thehorizontalerrorbarscorrespondtothewidthofthebin(notshownfortheshiftedvalues)andtheopenboxesarethesystematicuncertainties.Seethetextfordetails.

Fig. 3.pT-differentialellipticflowofmuonsfromheavy-flavourdecays,v^μ←₂ ^HF,in2.5<y<4 andvariouscentralityintervals,inPb–Pbcollisionsat√_s

NN=².76 TeV.The symbolsareplacedatthecentreofthepTintervaland,forvisibility,thepointsfromtwo-particleQ cumulantsandLee–Yangzeroswithproductgeneratingfunctionare shiftedhorizontally.ThemeaningofthesymbolsisthesameasinFig. 1.Thehorizontalerrorbarsarenotplottedforshifteddatapoints.ThepTintervalsusedwiththe Lee–Yangzerosmethodaredifferentwithrespecttotheothermethods.Upperpanels:resultsfromtwo-particlecorrelationflowmethods(scalarproductandtwo-particle Q cumulants)inthe0–10%(left)and10–20%(right)centralityintervals.Lowerpanels:resultsinthe20–40%centralityintervalfromtwo-particlecorrelationflowmethods (scalarproductandtwo-particle Qcumulants)andfromfour-particleQ cumulants(left),andfromfour-particleQcumulantsandLee–Yangzeros(right).Seethetextfor details.

Itamountsto about12%–36%inthe interval3<pT<4.5 GeV/c, dependingonthecollisioncentralityandtheflowanalysismethod.

4. Results

Fig. 3 presentsthe p_T-differential ellipticflow ofmuons from heavy-flavourhadrondecays, vμ←^HF

2 , calculatedwithEq.(2).The resultsareshownforthe0–10%(upper,left),10–20%(upper,right) and20–40%(bottom)centralityclassesusingthesameflowmeth- ods as for the measurement of the inclusive muon elliptic flow

(Fig. 1). When comparing the results to those obtained for in- clusive muons (Fig. 1), one can notice that vμ←^HF

2 and vμ

2 are similarduetothesmallbackgroundfraction(5%to15%)inthe pT interval 3–10 GeV/c. The differences betweenthe various meth- ods are similar to those discussed for the measurement of the inclusive muon vμ

2 i.e. i) scalar product and two-particle Q cu- mulants give compatible results, ii) consistent results are also found with four-particle Q cumulants and Lee–Yang zeros, and iii) the vμ_←HF

2 values extracted from these multi-particle corre- lation methods are smaller, although still compatible within un-

Fig. 4.Ellipticflow ofmuonsfromheavy-flavourhadron decaysas afunctionofthe collisioncentralityin2.5<y<4 and3<pT<5 GeV/c,inPb–Pbcollisionsat

√sNN=2.76 TeV.Theresultsareobtainedwithscalarproductandtwo-particle Q cumulants.Verticalbars(openboxes)representthestatistical(systematic)uncertainty, thehorizontalerrorbarscorrespondtothewidthofthecentralitybin.Forvisibility,thepointsfromscalarproductareshiftedhorizontallyandthehorizontalerrorbarsare notplotted.

certainties, than the ones obtained with two-particle correlation methods. As mentioned in Section 3.3, such differences are ex- pectedif initial-statefluctuationsplay a role inthe development ofthefinalmomentum-spaceanisotropy.

Apositivevμ←^HF

2 isobservedatintermediatepTforthe20–40%

and10–20% centrality classeswith a significancelarger than 3

σ

when combining statistical and systematic uncertainties. In the 20–40% centrality class, the values of the significance in the in- terval 3<p_T<4 GeV/c (4<p_T<5.5 GeV/c) are4

σ

(3.2

σ

) and 4.3

σ

(3.8

σ

) with scalar product and two-particle Q cumulants, respectively. In the 10–20% centrality class and in the interval 3<pT<4.5 GeV/c, the valuesof the significancecorrespond to 4.4

σ

bothwithscalarproductandtwo-particle Q cumulants.This behavior resultsfromtheinterplay betweenthesignificant inter- actionofheavyquarkswiththeexpandingmedium andthepath- length dependenceofin-medium partonenergyloss [29,30].The vμ←^HF

2 ofmuonsfromheavy-flavourhadrondecaysdecreaseswith increasingp_Tandbecomescompatiblewithzerointhehighp_Tre- gion.

Fig. 4shows the centrality dependenceof the ellipticflow of muonsfromheavy-flavourhadrondecaysintheinterval3<pT<

5 GeV/c.Itisinvestigatedwithscalarproductandtwo-particle Q cumulants,whichcanbeappliedinawiderevent-multiplicity(i.e.

centrality)intervalcomparedtomulti-particlecorrelationmethods.

Asignificant decrease ofthe v₂ magnitude towardscentral colli- sionsisobserved.Thisisexpectedfromthedecreaseoftheinitial spatialanisotropyfromsemi-centraltocentralcollisions.

ALICEhas measured the ellipticflow of prompt Dmesons in

|^y|<0.8 in three centrality classes in the interval 0–50% with varioustwo-particlecorrelationmethods[33,34].Similartrendsas thosereportedhereformuonsfromheavy-flavourdecaysare ob- served,althoughindifferentp_Tandrapidityintervals.Inparticular, apositive v2wasobservedforDmesonsinsemi-centralcollisions in2<pT<6 GeV/c withasignificanceof5.7

σ

.

Thepositive ellipticflow ofmuonsfromheavy-flavourhadron decayshasbeenobservedina pTintervalfrom3toabout5 GeV/c where the charm contribution is expected to be dominant with respect to the beautycomponent according to perturbative QCD calculations[21].Thismeasurementsupportstheinterpretationof theJ/ψ positive v2 atforwardrapidity [35]interms ofa signifi- cantcontributiontoJ/ψproductionfromrecombinationofflowing charmquarksinthedeconfinedmedium.

5. Comparisonwithmodels

The resultspresented in this publication may constrain mod- els describing theinteractions ofheavy quarks withthemedium via elastic (collisional) and inelastic (radiative) processes, and in particular the partonenergyloss dependenceon the path-length withinthemedium.

The ellipticflow coefficient and the nuclear modification fac- tor of muons from heavy-flavour hadron decays [21] are com- paredtothefollowingthreemodels.TheMC@sHQ+ ^EPOStrans- port model [59] treats the propagation of heavy quarks in the medium including collisional and radiative energy loss, within a 3 + ¹ dimensional fluid dynamical expansion based on the EPOS model [60,61]. The hadronization of heavy quarks takes place at the transition temperature via recombination atlow pT and fragmentation at intermediate and high pT. The final-state hadronic interactions are not included in the model. TAMU [62]

is a transport model including only collisional processes via the Langevin equation. The hydrodynamical expansion is constrained by pT spectraandelliptic flow dataoflight-flavour hadrons. The hadronization is modeled including a component of recombina- tion of heavy quarks with light-flavourhadrons in the QGP. The diffusionofheavy-flavourmesonsinthehadronicphaseisalsoin- cluded.BAMPS[63–65]isapartonictransportmodelbasedonthe Boltzmann approachto multi-partonscatterings. Itincludes colli- sionalprocesseswitharunningstrongcouplingconstant.Thelack of radiative contributions is accounted for by scaling the binary cross section witha correction factor, tuned to describe the nu- clearmodificationfactorandellipticflowresultsatRHICenergies.

Vacuumfragmentationfunctionsareusedforthehadronization.

Fig. 5 shows a comparison of the three models with the measurement of the pT-differential elliptic flow of muons from heavy-flavour hadron decays in the 20–40% centrality class (up- per panel) and of the pT-differential nuclear modification factor of muons from heavy-flavour hadron decays in the 0–10% cen- tralityclass [21] (lower panel).Inthe interval 3<pT<5 GeV/c, theBAMPS modeldescribesthe vμ←^HF

2 datawithin uncertainties, while the TAMU andMC@sHQ+^{EPOS models give vμ}₂^{←HF values} lower than thedata. The three models describe the vμ←^HF

2 data

athigher pT,althoughthesizeable experimentaluncertainties af- fectthesignificanceofthecomparison.TheBAMPSmodeltendsto slightly underestimate the RAA of muons fromheavy-flavour de- cays inthe10% mostcentralcollisions,whiletheMC@sHQ+^EPOS