Direct photon elliptic flow in Pb–Pb collisions at √sNN=2.76 TeV

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Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Direct photon elliptic ﬂow 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:

Received23May2018

Receivedinrevisedform16October2018 Accepted19November2018

Availableonline23November2018 Editor:L.Rolandi

Theellipticﬂowofinclusiveanddirectphotonswasmeasuredatmid-rapidityintwocentralityclasses 0–20% and 20–40% in Pb–Pb collisions at √_s

NN=².76 TeV by ALICE. Photons were detected with the highly segmentedelectromagneticcalorimeter PHOS and viaconversions inthe detectormaterial with the e⁺e⁻ pairs reconstructed in the central tracking system. The results of the two methods were combinedand the direct-photon elliptic ﬂowwas extracted inthetransverse momentumrange 0.9<pT<6.2 GeV/c.Acomparison toRHICdata showsasimilarmagnitudeofthemeasureddirect- photon ellipticﬂow.Hydrodynamic andtransportmodelcalculationsare systematicallylowerthanthe data,butarefoundtobecompatible.

©²⁰¹⁸^TheÂuthor.^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Thetheoryofthestronginteraction,QuantumChromoDynam- ics (QCD), predicts a transition from ordinary hadronic matter to a new state where quarks and gluons are no longer con- ﬁned to hadrons [1,2]. Lattice calculations predict a chiral and deconﬁnement crossover transitions over the temperature range 145–163 MeV[1,2], which is accessiblein collisions of ultrarela- tivisticheavyions.Thecreationandstudyofthepropertiesofthis hotstronglyinteractingmatter–Quark–GluonPlasma(QGP)–are themainobjectivesoftheALICEexperiment.

Thehotstronglyinteractingmatter,createdinnucleus–nucleus collisions,expands,coolsandfinallytransformstoordinaryhadro- nicmatter. Toexperimentally studythe quark matter properties, severalobservableswereproposed.Here,weconcentrateonstudy- ing the development of collectiveflow usingdirectphotons. Direct photonsarethephotonsnotoriginatingfromhadronicdecaysbut produced in electromagnetic interactions. Unlike hadrons, direct photonsare produced atall stagesofthe collision. Incomingnu- cleipassingthrougheachotherproducedirectphotonsinscatter- ingsoftheir partonicconstituents.Inaddition, (thermal)photons are emitted in the deconfined quark–gluon plasma and hadronic matter, characterized bythe thermaldistributions ofpartons and hadrons,respectively.Sincethemeanfreepathofaphotoninhot matterismuchlargerthanthetypicalsizesofthecreatedfireball [3],directphotonsescapethecollisionzoneunaffected,delivering direct informationon the conditions at the productiontime and onthedevelopmentofcollectiveflow.

E-mailaddress:alice-publications@cern.ch.

The observations ofa strong azimuthal asymmetry of particle production over a wide rapidity range in nucleus–nucleus collisions was oneofthekey resultsobtainedatRHIC[4–7] and LHC [8–12] energies. It was interpreted as a consequence of collec- tive expansion – collectiveflow –of thematter having aninitial spatial asymmetry, which is more prominent in collisions with non-zeroimpactparameter.Toquantifythecollectiveflow,theaz- imuthal distributions of final state particles are expanded in the series1+²

vncos[ⁿ(

ϕ

−R P)]^[13],^depending^on^the^difference betweentheparticleazimuthalangle

ϕ

andthereactionplaneori- entationR P,definedbytheimpactparameterandbeamaxis.At mid-rapiditythesecondharmonicv2(ellipticflow)reflectstheex- pansionofthealmond-likeshapeofthehotmattercreatedbythe mutual penetration of the collidingnuclei. Higher harmonics v₃, v4,etc.aresensitivetofluctuationsoftheinitialshapeofthecre- ated hot matter and are typically much smaller than v2, except forcentralcollisions,where v2 decreases duetoamoresymmet- ric geometry. Collectiveflow is sensitive to the equation of state ofhotmatterandtheamountofshearviscosity.Theinitialspatial asymmetryoftheexpandingfireballdiminisheswithtime,forany equation of state. Forstrongly interactingmatter thisasymmetry translatesintoanazimuthalanisotropyinmomentumspace,while forfreestreamingweaklyinteractingmatterthereisnofinalpar- ticleazimuthalanisotropy.

Hadrons providethepossibilitytotestwithhighprecision the flowpatternofthelateststageofthecollision,whenthehotmat- ter decouples intofinal particles. Complementary tothem, direct photonsprovide thepossibilityto investigatethedevelopment of flow during the evolution of hot matter. First calculations pre- dicted that the photon emission rate from the hot quark–gluon orhadronmatterincreaseswithtemperatureas∝^T²êxp(−Êγ/T) [14], whereEγ isthephotonenergyandT isthetemperatureof https://doi.org/10.1016/j.physletb.2018.11.039

0370-2693/©²⁰¹⁸^TheÂuthor.^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

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ALICE Collaboration / Direct photon elliptic ﬂow in Pb–Pb 309

thematter. Then, thelow transverse momentum (pT^{4 GeV}/c) partis controlledby thecoolerlateststage,andthehigh pT part (pT^{5 GeV}/c)ofthe spectrumbythehot initialstate. However, detailedcalculationswhich include the full hydrodynamic evolution(seee.g. [15])show thatcontributions ofallstagesarecom- parable for all pT regions asa higher temperature of the initial stageiscompensatedbyalargerspace–timevolumeandstronger radial flow of the later stages. Since the observed direct-photon flowistheconvolutionofallstagesofthecollision,includingthe contributionfromtheinitial stagewhen theflowpatternhasnot yet developed, the calculations predict much smaller azimuthal anisotropyforthermalphotonsthanforhadrons[16,17].

Thefirstmeasurementofadirect-photonspectruminrelativis- ticnucleus–nucleuscollisionswaspresentedbytheWA98collabo- ration [18], and later also by the PHENIX Collaboration [19–22], and by the ALICE Collaboration [23]. The first measurement of elliptic flow of direct photons in Au–Au collisions at √

sNN = 200 GeV was performedby the PHENIX Collaboration [24]. Sur- prisingly, it was found to be close to the ﬂow of hadrons [25].

RecentPHENIXresults, presentingmoreprecise measurementsof ellipticandtriangularflow extendedto lower pT [26],confirmed thisearlyresult.Thediscrepancybetweenexperimentalresultsand theorypredictionstriggeredasetoftheoreticalstudies,whichcan besplitintotwo classes.Themain ideainthefirstclassofmod- els[27–44] istoincreasetheemissionofdirectphotonsfromthe laterstagesofthecollisionand/orsuppressemissionoftheinitial stage. In the second class of models [45–47], a new azimuthally asymmetricsourceofdirect photonsisconsideredlike jet-matter interactionsorsynchrotronradiationinthefieldofcollidingnuclei.

Thesetheoreticalefforts considerablyreduce thediscrepancy,but consistentreproductionofboththedirect-photonspectraandflow isstill missing.The measurementofdirect-photon flowathigher collision energy is important asan independent confirmation of the results at lower energy, and could also allow to disentangle betweendifferentcontributions.

Inthispaper,we presentthefirstmeasurement ofthedirect- photonflowinPb–PbcollisionsattheLHCandcompareourfind- ingstoRHICresultsandtopredictionsofhydrodynamicaswellas transportmodels.

2. Detectorsetup

Thedirectphotonflowisbasedonthemeasurementoftheel- lipticflowofinclusivephotonsandtheestimationofthecontribu- tionofdecayphotonsusingtheavailablehadronflowresults.Pho- tonsarereconstructed viatwo independentmethods: thePhoton ConversionMethod(PCM)andwiththeelectromagneticcalorime- terPHOS.

Intheconversionmethod,theelectronandpositrontracksfrom photonconversionsare measuredwiththeInnerTrackingSystem (ITS)and/ortheTimeProjectionChamber(TPC).TheITS[48] consists of two layers of Silicon Pixel Detectors (SPD) positioned at radial distancesof 3.9 cmand7.6 cm, two layers ofSiliconDrift Detectors(SDD)at15.0 cmand23.9 cm,andtwolayersofSilicon StripDetectors(SSD)at38.0 cm and43.0 cm.Thetwoinnermost layers cover a pseudorapidity range of|

η

|<2 and |

η

|<1.4, respectively.TheTPC[49] isalarge(85 m³)cylindricaldriftdetector ﬁlledwithaNe–CO2–N2 (85.7/9.5/4.8%)gasmixture.Itcoversthe pseudorapidityrange|

η

|<0.9 overthefull azimuthalanglewith amaximumtracklengthof159reconstructed spacepoints.With thesolenoidal magnetic field of B=⁰.5 T, electron andpositron trackscanbereconstructeddownto p_T≈^{50 MeV}/c.TheTPCpro- vides particle identification via the measurement of the specific energyloss(dE/dx) witha resolutionof5.2%inppcollisions and 6.5% in central Pb–Pb collisions [50]. The ITS and the TPC were

alignedwithrespecttoeachothertotheleveloflessthan100 μm usingcosmic-rayandpp collisiondata[51].Particle identiﬁcation isalsoprovided bythe Time-of-Flight(TOF) detector[52] located ataradialdistanceof370<r<399 cm.Thisdetectorconsistsof MultigapResistivePlateChambers(MRPC)andprovidestimingin- formationwithanintrinsicresolutionof50 ps.

PHOS[53] isan electromagneticcalorimeterwhichconsistsof three modules installed at a radial distance of 4.6 m from the interaction point. It subtends 260^◦<

ϕ

<320^◦ in azimuth and

|

η

_|<0.12 in pseudorapidity. Each module consists of 3584 detector cells arranged ina matrix of 64×^{56 lead} ^tungstate^crys- tals each of size 2.2×².2×^{18 cm}³^. ^The ^signal ^from êach ^cell ismeasured by an avalanchephotodiode (APD)associatedwitha low-noisecharge-sensitivepreamplifier.Toincreasethelightyield, reduce electronic noise,andimproveenergy resolution,theAPDs andpreamplifiersare cooledto atemperatureof −²⁵^◦^C.^The ^re- sultingenergyresolutionis

σ

E/E=(1.8%/E)⊕(3.3%/√

E)⊕¹.1%, where E isinunits ofGeV. The energydeposition ineach PHOS celliscalibratedinpp collisionsbyaligningthe

π

⁰ peakposition inthetwo-photoninvariantmassdistribution.

FortheminimumbiastriggerinthePb–Pb runandeventplane orientation calculation,two scintillatorarray detectors (V0–Aand V0–C) [54] are used, which subtend 2.8<

η

<5.1 and −³.7<

η

<−¹.7,respectively. EachoftheV0 arraysconsistsof32chan- nels and is segmented in four rings in the radial direction, and each ringis dividedintoeight sectorsintheazimuthal direction.

ThesumofthesignalamplitudesoftheV0–AandV0–Cdetectors servesasameasureofcentralityinthePb–Pb collisions.

3. Dataanalysis

ThisanalysisisbasedondatarecordedbytheALICEexperiment in the ﬁrst LHC heavy-ion run in the fall of 2010. The detector readout was triggered by the minimum bias interaction trigger based on signals from the V0–A, V0–C, and SPD detectors. The eﬃciencyfortriggeringonaPb–Pb hadronicinteractionrangedbe- tween98.4%and99.7%.Theeventsaredividedintothecentraland semi-centralcentralityclasses0–20%and20–40%,respectively,ac- cordingtotheV0–AandV0–Csummedamplitudes[55].Toensure auniformtrackacceptanceinpseudorapidity

η

,onlyeventswitha primaryvertexwithin±^{10 cm from}^the^nominalinteractionpoint along thebeamline(z-direction)areused.Afterofflineeventse- lection, 13.6×¹⁰⁶ êventsâre âvailable ^for^the ^PCM ânalysisând 18.8×¹⁰⁶ êvents^for^the^PHOSânalysis.

The direct-photonellipticflow isextractedon astatisticalba- sisbysubtractingtheellipticflowofphotonsfromhadron decays from the inclusive photon elliptic flow. We assume that in each bin ofthe photon transverse momentum the measured inclusive photonflowcanbedecomposedas

v^γ₂^,^inc

=

^N^γ^,^dir

N_γ_,_incv^γ₂^,^dir

+

^N^γ^,^dec

N_γ_,_incv^γ₂^,^dec

,

(1) whereNγ,inc=^Nγ,dir+^Nγ,decistheinclusivephotonyieldwhich can be decomposed into the contributions of direct (Nγ,dir) and decay(Nγ,dec) photons. The vγ,inc

2 , vγ,dir

2 and vγ,dec

2 are thecor- respondingphotonflows.Itisconvenienttoexpressdirect-photon flowintermsoftheratioRγ=^Nγ,inc/Nγ,dec,theinclusivephoton flow vγ,inc

2 ,andthedecayphotonﬂowvγ,dec

2 :

v^γ₂^,^dir

=

^v

γ,inc

2 R_γ

−

^v^γ₂^,^dec

R_γ

−

1

.

(2)

The ratioRγ was measuredinthesamedataset in[23],whereas vγ,dec

2 is calculated with a simulation of photons from decays

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which is also known as cocktail simulation. The PCM and PHOS measurements of inclusive photon ﬂow are performed indepen- dently.Theyarethencombinedandusedwiththecombinedratio Rγ aswellasthecalculateddecayphotonﬂow.

Thephoton ellipticﬂow v₂ iscalculatedwiththeScalarProd- uct(SP)method,whichisa two-particlecorrelationmethod[56], usingapseudorapiditygapof|

η

_|>0.9 betweenthephotonand thereferenceflow particles.The appliedgapreduces correlations not relatedtotheeventplane n,such asthe onesduetoreso- nancedecaysandjets,knownasnon-flow effects.TheSPmethod usesthe Q-vector,computedfromasetofreferenceflowparticles (RFP)definedas:

Q

n

=

i∈^RFP

w_ieⁱⁿ^ϕⁱ

,

(3)

where

ϕ

i istheazimuthal angleofthe i-thRFP,n istheorderof theharmonicandw_i isa weightappliedforevery RFP.TheRFPs aretakenfromtheV0–AandV0–Cdetectors.Sincethesedetectors donotprovidetrackinginformation,wesumovertheV0–A/V0–C cells,whiletheamplitudesofthesignalfromeachcell,whichare proportionaltothenumberofparticlesthat causea hit,areused asaweight wi.Thenon-uniformityofthedetectorazimuthaleffi- ciencyistakenintoaccountbyapplyingtheinverseoftheevent- averagedsignalasaweightforeachoftheV0 segments,together witha recenteringprocedure [50,57].More specifically, theellip- ticflow v2 iscalculatedusingtheunitflowvectoru2=êⁱ²^ϕ ^built fromreconstructedphotons

v2

=

^u

2

·

^Q_M²^A_A^∗

u₂

·

^Q_M²^C_C^∗

Q₂^A

M_A

·

^Q_M²^C_C^∗

,

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wherethetwopairs ofbracketsinthenumeratorindicateanav- erage over all photonsand over all events; MA and MC are the estimates of multiplicity from the V0–A andV0–C detectors, respectively;and Q₂^A^∗, Q₂^C^∗ are thecomplexconjugatesoftheﬂow vectorcalculatedinsub-eventAandC,respectively.

In the PCM analysis, photons converting into e⁺e⁻ pairs are reconstructedwithanalgorithmwhichsearchesfordisplacedver- tices with two oppositely charged daughter tracks. Only good quality TPCtrackswitha transversemomentum above50 MeV/c and a pseudorapidity of |

η

_|<0.9 are considered. The vertex ﬁnding algorithm uses the Kalman ﬁlter technique for the decay/conversion point and four-momentum determination of the neutral parent particle (V⁰) [58]. Further selection is performed on the level of the reconstructed V⁰. Only V⁰s with a conversion points at radii between 5< R<180 cm are accepted such that the

π

⁰ and

η

-meson Dalitz decays are rejected and to ensure a good coverage by the tracking detectors of the conversion daughters. To identify an e⁺e⁻ pair, the speciﬁc energy loss (dE/dx)intheTPC[50] ofbothdaughtersisused.The transverse momentum componentqT ofthe electron momentum, pe, withrespecttotheV⁰ momentum-vectorisrestrictedtobeq_T<

0.05 1−(

α

/0.95)² GeV/c,where

α

isthe energy asymmetry of the conversion daughters. Random associations of electrons and positronsarefurtherreducedbyselectingV⁰swithcos(θ )>0.85, where θ is the pointing angle, which is the angle between the momentum-vector ofthee⁺e⁻ pairandthevector that connects theprimaryvertexandtheconversionpoint.Basedontheinvari- ant mass of the e⁺e⁻ pair andthe pointing angle of the V⁰ to theprimary vertex,thevertexﬁndercalculatesa

χ

² valuewhich reﬂects the levelof consistency withthe hypothesis that the V⁰

Table 1

Summaryoftherelativesystematicuncertainties(in%)oftheinclusivephotonel- lipticﬂowinthePCMandPHOSanalysis,andofthedecayphotonsimulation.All contributionsareexpectedtobecorrelatedinpTwiththemagnitudeoftherelative uncertaintyvaryingpoint-by-point.

Centrality 0–20% 20–40%

pT(GeV/c) 2.0 5.0 2.0 5.0

PCM

Photon selection 2.4 4.2 2.1 4.0

Energy resolution 1.0 1.0 1.0 1.0

Eﬃciency 3 3 1.9 1.9

Total 4.0 5.3 3.0 4.5

PHOS

Eﬃciency & contamination 3.0 3.0 0.7 0.7

Event plane ﬂatness 2.0 2.0 1.4 1.4

Total 3.5 3.5 1.6 1.6

Decay photon calculation

Parameterization ofv^π₂ 1.3 3.6 0.8 2.2

η/π⁰normalization 1.7 3.2 1.7 2.4

Total 2.2 4.8 1.9 3.3

comesfromaphotonoriginatingfromtheprimaryvertex.Aselec- tion basedonthis

χ

² value isusedtofurtherreduce contamination in thephoton sample. The main sources ofbackgroundthat remain after these selection criteria are V0s reconstructed from

π

^±e^∓,

π

^±

π

^∓,

π

^±K^∓ ande^±K^∓pairs,whichisimportanttotake intoaccountasshownin[59].Theellipticflowofthisbackground issubtractedusingaside-bandmethodapproach.Inthismethod, the dE/dx informationof bothconversion daughters iscombined into a1-dimensionalquantity.Thesignal isa peakeddistribution andtheside-bandsaredominatedbybackgroundsources. The v2 oftheside-bandsismeasuredandsubtractedfromthemainsignal region using the purityof the photon sample, which isobtained byfittingMonteCarlotemplatestothedata.Thecorrectiontothe measured inclusive photon flow is ofthe order of5% for central and2.5%forsemi-centralcollisions,respectively.

Thesystematicuncertaintiesoftheinclusivephotonﬂowmea- sured with PCM are summarized in Table 1. The uncertainties related to the photon selection (|

η

_|, R, min pT, qT,

χ

²/ndf and cos(θ ))areobtainedbyvaryingtheselectioncriteria,andthesys- tematic uncertainties relatedto the contamination of thephoton sample are quantiﬁedby theuncertaintyonthe backgroundﬂow subtraction. Theenergyresolutionuncertainties,whicharedueto detector resolution effects and bremsstrahlung of electrons, are estimated by comparing vγ,inc

2 distributions as a function of the reconstructedandtruepTusingMCsimulations.Theuncertainties related to the variation of reconstruction eﬃciency in- and out- of-plane are calculated from studying the photon reconstruction eﬃciencyasafunctionofthetrackmultiplicity.Formostofthese sources onlyasmalldependenceon pT andcollisioncentralityis observed.

InthePHOSanalysis,thesamephotonselectioncriteriaareap- plied as in the direct-photon spectra analysis [23]. Cells with a common edge with another cell that are both above the energy thresholdof25MeVarecombinedintoclusterswhichareusedas photoncandidates.Toestimatethephoton energy,theenergiesof allcells oronlythosewithcenterswithinaradius R_core=³.5 cm from thecluster center ofgravity are summed. Comparedto the fullclusterenergy,thecoreenergyislesssensitivetooverlapswith low-energyclustersinahighmultiplicityenvironment,andiswell reproducedby GEANT3MonteCarlosimulations[23]. The fullen- ergyisusedforthesystematicuncertaintyestimate.Thecontribu- tionofhadronicclustersisreducedbyrequiringEcluster>0.3 GeV, N_cells>2 and by accepting only clusters above a minimum lat-

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Fig. 1.Comparisonofthemeasuredinclusivephotonflow(v^{γ ,}₂înc)totheindividualPCMandPHOSmeasurements(v^{γ ,}₂înd)inthe0–20%(left)and20–40%(right)centrality classes.Theindividualresultsaredividedbythecombined v^{γ ,}₂înc.Theverticalbarsoneachdatapointindicatethestatisticaluncertaintiesandtheboxesindicatethe systematicuncertainties.

eralclusterdispersion[60].Thelatterselectionrejectsrareevents when hadronspunch through the crystal andhadronically inter- act with APD, producing a large signal in one cell of a cluster, not proportional to the energy deposition. In addition to these cuts, we also apply a p_T-dependent dispersion cut and perform achargedparticleveto(CPV).TheCPVremovesclustersbasedon theminimal distancebetweenthe PHOScluster positionandthe positionofextrapolatedchargedtracks onthe PHOSsurface,and isusedtosuppresshadroncontribution[60]. Bothdispersionand CPVcutsaretuned usingpp collisiondatatoprovide thephoton reconstructionandidentiﬁcationeﬃciencyatthelevelof96–99%.

MeasurementswithdifferentcombinationsofdispersionandCPV cutsare used forthe estimate of systematicuncertainties. Possi- blepileupcontributionfromother bunchcrossingsisremoved by aloosecutontheclusterarrival time|^t|<150 ns,whichissmall comparedtoaminimumtimebetweenbunchcrossingsof525 ns.

To estimate the reconstruction and identiﬁcation eﬃciencies and correction for energy smearing with their possible depen- denceontheanglewithrespecttotheeventplane,weembedded simulated photon clusters into real data events and applied the standard reconstruction procedure. PHOS properties (energy and position resolutions, residual de-calibration, absolute calibration, non-linear energy response) are tuned in the simulation to re- produce the pT-dependence of the

π

⁰ peak position and width [60].Thecorrectionfortheeventplanedependenceoftherecon- structionand identification efficiencies, which comes as additive totheobservedphotonflow,islessthan10⁻³ bothincentraland mid-central collisions and is comparableto the statisticaluncer- tainties of the embedding procedure. The correction due to the energy smearing, is estimated to be 4% and 1% for central and semi-centralcollisions,respectively.Thecontaminationofthepho- tonsample measured withPHOSoriginatesmainly from

π

^± and

¯

p,ⁿ¯annihilation,withothercontributionsbeingmuchsmaller.The applicationofthedispersionandCPVcutsreducestheoverallcon- taminationatpT≈¹.5 GeV/cfromabout15%to2–3%anddownto 1–2%atpT∼^{3–4 GeV}/c.Toestimateandsubtractthehadroncon- tribution,thePHOSresponsematricesareconstructedfor

π

^±,K,p and^p¯ ûsing^real^dataôr^Monte^Carlosimulationsandconvoluted withthemeasuredspectra,flowandrelativeyieldsofhadrons.

Systematicuncertaintiesoftheinclusivephotonflowmeasured with PHOS are summarized in Table 1. They can be split into two groups: contributions related to the contamination and dependence ofreconstruction,identificationandsmearingefficiency ontheanglewithrespecttotheeventplane,anduncertaintiesre- latedtotheflatnessoftheeventplanecalculation,theeventplane resolution andthe contributionofnon-flow effects.Uncertainties ofthe firstgroup are estimatedby comparingthefully corrected photonflow measuredwithdifferentsetsofidentificationcriteria andwithfull andcoreenergy.Uncertaintiesof thesecond group areestimatedbycomparinginclusivephotonflowsmeasuredsep- arately withthe V0–A andV0–C detectors. Note that because of thelimitedazimuthalacceptance,PHOSismuchmoresensitiveto thenon-flatnessoftheeventplanedistributioncomparedtoPCM.

Inthecombinationoftheinclusivephotonv2resultsfromPCM andPHOS,both measurements aretreatedasindependent.Possi- ble correlations dueto the use of the same V0Aand V0C event plane vectors are found to be negligible. To take into account correlationsofthe individual measurementsin binsoftransverse momentum, we describethe measured inclusivephoton flows as vectors^v^γ₂^,înc^,^PCM^,^v^γ₂^,înc^,^PHOS^,^where^the^vector^components^corre- spondto themeasured pT bins, andthecorrelations ofthetotal uncertainties are described by covariance matrices Vv2,PCM and Vv₂,PHOS,respectively. The elements of thecovariance matrixare calculatedassuminguncorrelatedstatisticaluncertaintiesandfully correlated(

ρ

=¹)systematicuncertainties; V_{i j}=^Vstat,i j+^Vsyst,i j, where Vsyst,i j=

ρσ

syst,i

σ

syst,j,for pT bini and j.Then, thecom- binedinclusivephotonﬂowisthevector

v^γ₂^,^inc

= (

V_v⁻¹

2,PCM

+

^V_v⁻₂¹_,_PHOS

)

⁻¹

× (

V⁻_v¹

2,PCM

v^γ₂^,^inc^,^PCM

+

^V_v⁻₂¹_,_PHOS

v^γ₂^,^inc^,^PHOS

).

(5) The inclusivephoton v₂ measured withPCMandPHOSarecom- paredinFig.1,whichshowsthe ratiooftheindividualvalues to the combinedﬂow. The PCMand PHOSmeasurements arefound to be consistent witheach other with p-values of0.93 and0.43 forthecentralityclasses0–20%and20–40%,respectively.

The decay photon ﬂow is estimated using a cocktail simulation.Decaysthatcontribute morethan1% ofthetotaldecaypho-

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Fig. 2.Ellipticﬂowofdecayphotonsfromπ⁰,η,ω,andthetotalcocktailsimulationasafunctionoftransversemomentuminthe0–20%(left)and20–40%(right)centrality classes.Thebandrepresentsthetotaluncertaintyofthetotalcocktailsimulation.

ton yield are taken intoaccount:

π

⁰→²

γ

,

η

→²

γ

,

ω

→

γ π

⁰, K_s⁰→²

π

⁰→⁴

γ

.Other contributions arenegligible comparedto thesystematicuncertaintiesofthecocktail.Indecaysof

η

and

ω

mesons only photonsproduced directly in decays are accounted, while those coming from daughter

π

⁰ decays are already ac- countedin

π

⁰ contribution.TheK⁰_s decaydoesnotcontributesig- niﬁcantlytothephotonsamplemeasuredwiththePCMapproach.

Therefore, we correct the PHOS measurement for this contribu- tionbeforecombiningthePHOSandPCMmeasurements.Herewe usethesameapproach asin thedirect-photonspectrum analysis [23],butthistime thesimulationoftheellipticflowisadded.To estimate the ellipticflow of neutralpions, a parametrization has been made of the charged pion flow measured under the same conditions,i.e., chargedpions measured intheTPCandreference particlesintheV0–AandV0–Cdetectors[61,62] areused.Toes- timatethecontributionof

η

and

ω

mesons,themeasured elliptic flowofchargedandneutralkaons[61] isscaled,assumingscaling with the transverse kinetic energy K ET=^mT−^m. ^The ^compar- ison of different contributions and overall decay photon flow is shownin Fig. 2. The v2 contributions were added withweights, proportionaltotherelativedecayphotonyieldofamesonintotal decayyield[23].Thewidthofthecoloredbandrepresentsthesys- tematicuncertainties ofthedecayphotonellipticflow vγ,dec

2 .The

decay photon ﬂow is mainly determined by the

π

⁰ ﬂow, while other contributions make relatively small corrections: the

η

and

ω

contributions slightlyreduce the decay photon elliptic ﬂow at p_T<2 GeV/c andincrease itcompared tothe

π

⁰ contributionat higher pT.The systematicuncertainties ofthedecayphoton ﬂow are summarized in Table 1. The largest uncertainties come from theparametrizationofthechargedpionellipticﬂowandfromthe relativeyield

η

/

π

⁰.

4. Results The vγ,inc

2 measured in two centrality classes are shown in Fig.3.Theellipticﬂowcoeﬃcientsofinclusivephotonsanddecay photonsareverysimilar overthefullrange0.9<pT<6.2 GeV/c.

Asthefractionofdirectphotonovertheinclusivephotonyieldis relativelysmall,∼^10%ⁱⁿôur^pT range[23],thecollectiveflowof inclusivephotonsisdominatedbythedecayphotonflow.Inmod- els based on relativistic hydrodynamics the medium is assumed

to be in or close to local thermal equilibrium. An equation of stateisusedtorelatethermodynamicquantitiesliketemperature, energy density, and pressure. Photon production is modeled by foldingthe space–timeevolutionofa collisionwithtemperature- dependent photon production rates in the QGP and the hadron gas.Anotherapproachistaken, e.g.,inthePHSDtransportmodel in which the QGP degrees of freedom are modeled as massive strongly-interactingquasi-particles[63].Forbothclassesofmodels the development of a strong early elliptic flow, necessary to re- producetheobserveddirect-photonflow,givesrisetoalargepion elliptic flow atfreeze-out andthereforeto a large inclusivepho- tonellipticflow.Itisthereforeanimportanttesttocheckwhether amodelcandescribeboththeinclusiveandthedirect-photonelliptic flow. The prediction of the hydrodynamic model described in[64] fortheinclusivephoton v2 intherange1<pT<3 GeV/c isabout40%abovethedata,thoughthemagnitudeoftheelliptic flow of unidentifiedhadrons is reproduced within 10–20% accu- racy inthis p_T range [65].The PHSDmodel [63] alsopredictsan

∼^40%^higherînclusive^photon^flow,êven^thoughît^reproduces^the unidentifiedhadronflowwell.

Thedirect-photonv2iscalculatedfromthecombinedPCMand PHOSphoton excess Rγ [23], thecombinedinclusive v2,andthe calculated decay photon v2. In the propagation of uncertainties, therelativelysmallsigniﬁcanceofthephotonexcessofabout1–3 standard deviations(dependingonthecentralityclass andp_T interval) requiresspecialattention. This isillustrated fora selected p_T interval in the left panel ofFig. 4 which showsthe obtained vγ,dir

2 andits uncertaintyasa function ofthe photonexcess Rγ. TheGaussianfunctioninthispanelrepresentsthemeasuredvalue ofRγ inthispT interval(dashedline)andits1

σ

totaluncertainty (darkblueshadedarea).ForRγ¹.05 onelosesthesensitivityto vγ,dir

2 astheuncertainties, indicated bythe redshaded band, in- creasedrastically.WiththecurrentuncertaintiesonRγ wecannot ruleoutcompletelythat Rγ¹.05.

Weaddressthelimitedsigniﬁcanceofthedirect-photonexcess by employing aBayesian approach. The parameters Rγ,t, vγ,inc,t

2 ,

vγ,dec,t

2 denotingthetruevaluescarry theindex“t”andthemea- suredquantities Rγ,m,vγ,dec,m

2 , vγ,dec,m

2 theindex“m”.Notethat Rγ,t is restrictedtoits physicallyallowed range(Rγ,t≥^1), ^while

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Fig. 3.Ellipticﬂowofinclusivephotonsanddecayphotons,comparedtohydrodynamic[31] andtransportPHSD[30] modelpredictionsinthe0–20%(left)and20–40%

(right)centralityclasses.Theverticalbarsoneachdatapointindicatethestatisticaluncertaintiesandtheboxesindicatethesizesofthetotaluncertainties.

Fig. 4.Left:Centralvalue(solidredline)anduncertaintyofthedirect-photonv2foraselectedpTinterval.Theupperandloweredgesoftheredshadedareacorrespondto thetotaluncertaintyofv^{γ ,}₂^dirasobtainedfromlinearGaussianpropagationoftheuncertaintiesσ(v^{γ ,}₂^inc)andσ(v^{γ ,}₂^dec).TheGaussian(witharbitrarynormalization)reﬂects themeasuredvalueofRγ inthispTinterval(bluedashedline)andits±¹σ uncertainty(dark-blueshadedinterval).Right:Posteriordistributionofthetruevalueofv^{γ ,}₂^dir forthesameintervalintheBayesianapproach.Notethatthedistributionhasanon-Gaussianshape,implyingthatthe±2σintervaltypicallycorrespondstoaprobabilityof lessthan95.45%aswouldbethecaseforaGaussian.

themeasuredvalue Rγ,m canﬂuctuatebelowunity.Theposterior distributionofthetrueparameterscanbewrittenas

P

( ϑ |

^m

) ∝

^P

(

m

| ϑ) π ( ϑ ),

π ( ϑ ) ≡ π (

R

_γ_,_t

) = (

R_γ_,_t_,₁

−

¹

, ...,

R_γ_,_t_,_n

−

¹

),

⁽⁶⁾

whereinm=(Rγ,m,vγ,inc,m

2 ,vγ,dec,m

2 ),ϑ=(Rγ,t,vγ,inc,t 2 ,vγ,dec,t

2 ).

Hereweusethenotation introducedinEq. (5):vectorsrepresent sets of measurements in different pT bins andn is the number ofthesebins.The function

π

(Rγ,t)encodes theprior knowledge aboutRγ.ThemultivariateHeaviside functioncorrespondstoa constant(improper) priorfor Rγ,t≥^1.^Theprobability toobserve acertainsetofmeasured valuesgiventhetruevaluesismodeled withmultivariate GaussiansG(^x;

μ

,V)(where

μ

is thevectorof meanvaluesandV isthecovariancematrix):

P

(

m

| ϑ) =

x=^Rγ,vγ,inc 2 ,vγ,dec

2

G

(

xm

;

xt

,

Vx

).

(7)

Bysamplingthe posteriordistribution P(ϑ| ^m),we obtaintriplets (Rγ,vγ,inc

2 ,vγ,dec

2 )foreach pT binfromwhichwecalculatevγ,dir 2 accordingto Eq. (2).An example ofthe resultingdistribution for vγ,dir

2 isshown inFig. 4(right panel).The mediansof the vγ,dir 2 distributions are taken as central values. The lower and upper edges oftheerrorbars correspondtovalues ofv^dir₂ atwhich the integralofthev2 distributionis15.87%and84.13%ofthetotalin- tegral. In case of a Gaussian distribution thiscorresponds to 1

σ

uncertainties.

Theresultsforthedirect-photonellipticﬂowforthetwocen- tralityclasses, 0–20% and20–40%,are shown inFig. 5. The total uncertainties, reﬂecting the Bayesian posterior distributions, are shown as boxes, and the error bars represent statistical uncer-

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Fig. 5.EllipticﬂowofdirectphotonscomparedwithPHENIXresults[26] forthe0–20%(left)and20–40%(right)centralityclasses.Theverticalbarsoneachdatapoint indicatethestatisticaluncertaintiesandtheboxesthetotaluncertainty.

Fig. 6.Ellipticﬂowofdirectphotonscomparedtomodelcalculationsinthe0–20%(left)and20–40%(right)centralityclasses.Theverticalbarsoneachdatapointindicate thestatisticaluncertaintiesandtheboxesthetotaluncertainty.

tainties. The correlation of v₂_,_dir points for different p_T bins as quantifiedbythecorrelationmatrixisstrongatlow pT^{2 GeV}/c (correlation coefficients typically in the range 0.6–0.75) whereas the uncertainties at high pT are dominated by statistical uncertainties.We compareour resultsto measurements madeatRHIC energiesbythePHENIXcollaboration[26].Theinclusivephotonv₂ wasmeasuredbyPHENIXthroughthereconstructionofe⁺e⁻pairs from photon conversions and withan electromagnetic calorimeter.Thedirect-photonellipticflowinAu–AucollisionsatRHICand inPb–Pbcollisions attheLHCare foundtobe compatiblewithin uncertainties.A simpleexplanationofthelargeandsimilardirect- photonellipticflowforpT^{2 GeV}/c atRHICandtheLHCisthat thebulkofthethermaldirectphotonsisproducedlateattemper- aturesclosetothetransitiontemperatureTc.Thisisinterestingas naïvelyone wouldexpectthe T² temperaturedependenceofthe

photonemissionratetomaketheearly hotQGPphaseafterther- malizationalsothebrightestphase.

Fig.6comparesthemeasureddirect-photonellipticﬂowvγ,dir 2 totheestimateddecayphotonellipticﬂowvγ,dec

2 ,markedascock- tail,andtothepredictionsofseveraltheoreticalmodels.Similarly tomeasurementsatRHICenergies[24],weﬁndthatthedirectand decayphoton ellipticﬂow are similar. Wecompare ourmeasure- mentstostate-of-the-arthydrodynamicmodelcalculations[31,66]

and thePHSDtransport model [63]. The measured direct-photon elliptic ﬂow is systematically higherthan theoretical predictions, butisstillcompatible.

Inordertoquantifythedeviationofthedirect-photon v₂mea- surementfroma certainhypothesiswithafrequentist p-valueor, equivalently,thecorrespondingnumberofstandarddeviations,we use aBayesian-inspired method[67].In thisapproach,the likeli- hoodL(^v^γ₂^,^inc^,^m|^v^γ₂^,^dir^,^t)servesasateststatisticandisobtainedby

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integratingover the nuisance parameters ^v^γ₂^,^dec^,^t ^and Rγ,t using theirBayesianposteriordistributionsasweights.Wefocusonthe interval 0.9<pT<2.1 GeV/c inwhich the contributionof ther- malphotonsis expectedtobe important.The signiﬁcance ofthe deviationfrom the hypothesis vγ,dir,t

2 =^{0 for} ^individual ^pT bins isintherange1.8–2.1

σ

forthe0–20%classand0.9–1.5

σ

forthe 20–40%class.Wealsogoastepfurtherandestimatethecombined signiﬁcanceofthedeviationfromthehypothesisvγ,dir

2 ≡^{0 for}^this pT interval. Thistestsinaddition howwell the shapeof vγ,inc,m

2 asa functionof pT agrees withvγ,dec,m

2 /Rγ,i.e., withtheexpec- tationforvγ,dir

2 ≡^0.^Weêstimate^the^covariance^matrix^describing thecorrelation by characterizing the differentsources of system- aticuncertaintiesof Rγ,theinclusive,andthedecayphotonflow aseitherfullyuncorrelatedor fullycorrelated in pT. Varyingthe assumptions about the correlation of the data points we obtain significancesof typically lessthan 1

σ

forboth centrality classes.

Whiletheappliedmethodisessentialforameaningfulcomparison ofthevγ,dir

2 datawithdifferentmodelpredictions,themethodsto estimatethecovariancematrixcanbeimprovedinfutureanalyses.

5. Conclusions

Insummary,wereporttheﬁrstmeasurementofellipticﬂowof inclusiveanddirect photonsasa function of transverse momen- tumintherange0.9<pT<6.2 GeV/cforcentralandsemi-central Pb–Pbcollisionsat√

sNN=².76 TeV.Theellipticﬂowofinclusive photonswas measured withthescalarproductmethod,indepen- dentlyintheelectromagneticcalorimeterPHOSandwiththepho- tonconversionmethodwherethereferenceparticlesinbothcases were measured by the V0–A and V0–C detectors. The combined inclusivephotonvγ,inc

2 ,togetherwiththecalculateddecayphoton vγ,dec

2 andthepreviously measured Rγ are usedto calculatethe elliptic ﬂow of direct photons. The measured direct-photon ﬂow vγ,dir

2 appearstobe closeto thedecayphoton ﬂowforbothcen- tralityclasses,similar to observations atlower collision energies.

Moreover,themeasured vγ,dir

2 issimilar tothemeasurements by the PHENIX collaboration at RHIC. The considered hydrodynamic andtransportmodelspredictalargerinclusivephotonellipticflow (byapproximately 40%) anda smaller direct-photon elliptic flow than observed. With current uncertainties, however, these models are consistent with the presented direct-photon elliptic flow data. Future measurements using a larger statistics dataset will greatlyincreasetheprecisionofthismeasurementandallowusto extendthe measurementtohigher pT,since thestatisticaluncer- taintyisdominatingthe totaluncertaintyfor p_T>2.0 GeV/c and pT>3.0 GeV/cforthePHOSandPCMinclusivephotonflowmea- surement, respectively. Inaddition, a larger statisticsdataset will alsohelptoconstrainthesystematicuncertaintiesontheinclusive anddecayphotonflow,aswellasthemeasurementofRγ overthe wholep_Trange.Afurtherreductionofthesystematicuncertainties is expected from improved detector knowledge. For instance, in caseofPCMthelargestsystematicuncertaintyinthemeasurement of Rγ isrelated to modeling the material inwhich the photons convert.Calibrating regions ofthe detectorwithlesswell known materialbudget basedonregions withvery well knownmaterial mightsignificantlyreducetheoverallmaterialbudgetuncertainty.

TheRγ measurementcanbeimprovedfurtherbymeasuringneu- tralpion and etameson spectra in a combinedPCM-calorimeter approachinwhichonedecayphotonismeasuredthroughconver- sionandtheotherwithacalorimeter.

Acknowledgements

We wouldlike to thank ElenaBratkovskaya andJean-François Paquetforprovidingcalculationsshowninthispaperandforuse- fuldiscussions.

The ALICE Collaboration would like to thank all its engineers andtechniciansfortheir invaluablecontributions totheconstruc- tionoftheexperimentandtheCERNacceleratorteamsfortheout- standingperformanceoftheLHCcomplex.TheALICECollaboration gratefully acknowledges the resources and support provided by all Gridcentres andtheWorldwide LHC ComputingGrid (WLCG) collaboration. The ALICE Collaboration acknowledges the follow- ing funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (YerevanPhysicsInstitute)Foundation (ANSL),State Committeeof Science andWorld Federation ofScientists (WFS), Armenia; Aus- trian Academy of Sciences and Österreichische Nationalstiftung für Forschung, Technologie undEntwicklung, Austria; Ministry of CommunicationsandHighTechnologies,NationalNuclearResearch Center, Azerbaijan;Conselho NacionaldeDesenvolvimentoCientí- ficoe Tecnológico(CNPq),UniversidadeFederal doRioGrandedo Sul (UFRGS),Financiadorade Estudose Projetos(Finep)andFun- dação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil; Ministry of Science & Technology of China (MSTC), Na- tionalNaturalScienceFoundationofChina(NSFC)andMinistryof EducationofChina(MOEC),China;MinistryofScienceandEduca- tion,Croatia;MinistryofEducation,YouthandSportsoftheCzech Republic,CzechRepublic;TheDanishCouncilforIndependentRe- search – Natural Sciences, the Carlsberg Foundation and Danish NationalResearchFoundation (DNRF),Denmark;HelsinkiInstitute ofPhysics(HIP),Finland;Commissariatàl’EnergieAtomique(CEA) andInstitutNationaldePhysiqueNucléaireetdePhysiquedesPar- ticules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF) and GSI Helmholtzzentrum fürSchwerionenforschungGmbH,Germany;GeneralSecretariatfor ResearchandTechnology,MinistryofEducation,ResearchandRe- ligions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy, Government of India (DAE), Department of Science andTechnology, Government of India(DST), University Grants Commission, Governmentof In- dia (UGC)andCouncil of Scientific andIndustrialResearch, India (CSIR), India; Indonesian Institute of Sciences, Indonesia; Centro Fermi –Museo Storicodella Fisica e CentroStudi e Ricerche En- ricoFermiandInstitutoNazionale diFisica Nucleare (INFN),Italy;

Institute forInnovativeScienceandTechnology,NagasakiInstitute of AppliedScience (IIST),Japan Societyforthe Promotion ofSci- ence(JSPS)KAKENHIandJapanese MinistryofEducation,Culture, Sports, Science and Technology (MEXT), Japan; Consejo Nacional deCiencia(CONACYT)yTecnología,throughFondodeCooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico;

NederlandseOrganisatievoorWetenschappelijkOnderzoek(NWO), Netherlands; The ResearchCouncil ofNorway, Norway; Commis- sion on Science and Technology for Sustainable Development in theSouth(COMSATS),Pakistan;PontiﬁciaUniversidadCatólicadel Perú,Peru;MinistryofScienceandHigherEducationandNational Science Centre, Poland; Korea Institute of Science and Technol- ogyInformationandNationalResearchFoundationofKorea(NRF), Republic ofKorea; Ministry ofEducation and Scientiﬁc Research, Institute of Atomic Physics and Romanian National Agency for Science, Technology and Innovation, Romania; Joint Institute for Nuclear Research(JINR),MinistryofEducation andScience ofthe Russian FederationandNationalResearchCentre KurchatovInsti- tute, Russia; Ministry of Education, Science, Research and Sport