Freeze-out radii extracted from three-pion cumulants in pp, p-Pb and Pb-Pb collisions at the LHC

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Physics Letters B

www.elsevier.com/locate/physletb

Freeze-out radii extracted from three-pion cumulants in pp, p–Pb and Pb–Pb collisions at the LHC

.ALICE Collaboration

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

Articlehistory:

Received17April2014

Receivedinrevisedform6October2014 Accepted12October2014

Availableonline18October2014 Editor: L.Rolandi

In high-energy collisions,the spatio-temporalsizeof theparticleproduction regioncanbe measured using the Bose–Einstein correlations of identical bosons at low relative momentum. The source radii are typically extracted using two-pion correlations, and characterize the system at the last stageofinteraction, calledkinetic freeze-out.Inlow-multiplicitycollisions, unlikeinhigh-multiplicity collisions, two-pion correlations are substantially altered by background correlations, e.g. mini-jets.

Such correlations can be suppressed using three-pion cumulant correlations. We present the first measurementsofthesizeofthesystematfreeze-outextractedfromthree-pioncumulantcorrelationsin pp,p–PbandPb–PbcollisionsattheLHCwithALICE.Atsimilarmultiplicity,theinvariantradiiextracted in p–Pbcollisions are found to be5–15% largerthanthose inpp, while those inPb–Pb are 35–55%

largerthanthoseinp–Pb.Ourmeasurements disfavormodelswhichincorporatesubstantiallystronger collectiveexpansioninp–Pbascomparedtoppcollisionsatsimilarmultiplicity.

©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP³.

1. Introduction

Therole of initial andfinal-stateeffects ininterpreting differ- encesbetweenPb–Pbandppcollisionsisexpectedtobeclarified with p–Pb collisions [1]. However, the results obtained from p–

Pbcollisions at√

s_NN=⁵.02 TeV[2–10]havenotbeenconclusive sincetheycanbeexplainedassumingeitherahydrodynamicphase duringthe evolutionofthesystem[11–13] ortheformationofa ColorGlassCondensate (CGC)intheinitialstate[14,15].

Asin Pb–Pb collisions,the presence ofa hydrodynamic phase in high-multiplicity p–Pb collisions is expected to lead to a fac- torof1.5–2 largerfreeze-outradiithaninppcollisionsatsimilar multiplicity[16].Incontrast,aCGCinitialstatemodel(IP-Glasma), withoutahydrodynamicphase,predictssimilarfreeze-outradiiin p–Pbandppcollisions[17].Ameasurementofthefreeze-outradii inthetwosystemswillthusleadtoadditionalexperimentalcon- straintsontheinterpretationofthep–Pbdata.

Theextractionoffreeze-outradiicanbeachievedusingidenti- calbosoncorrelationsatlowrelativemomentum,whicharedom- inated by quantum statistics (QS) and final-state Coulomb and strong interactions (FSIs). Both FSIs and QS correlations encode information about the femtoscopic space–time structure of the particleemitting source at kinetic freeze-out [18–20]. The calcu- lationofFSIcorrelationsallowsfortheisolationofQScorrelations.

E-mailaddress:[email protected].

Typically, two-pion QS correlationsare used to extract the char- acteristic radius ofthe source [21–27].However, higher-order QS correlations can be used as well [28–32]. The novel features of higher-orderQScorrelationsareextractedusingthecumulantfor whichalllowerordercorrelationsareremoved[33,34].Themaxi- mumofthethree-pioncumulantQScorrelationisafactoroftwo largerthanfortwo-pionQScorrelations[33–36].Inadditiontothe increasedsignal, three-pion cumulants alsoremove contributions from two-particle background correlations unrelated to QS (e.g.

frommini-jets [24,26]). The combinedeffect ofan increasedsig- nalanddecreasedbackgroundisadvantageousinlowmultiplicity systemswhereasubstantialbackgroundexists.

In this Letter, we present measurements of freeze-out radii extracted using three-pion cumulant QS correlations. The invari- antradii areextractedinintervalsofmultiplicity andtriplet mo- mentum in pp (√

s=^{7 TeV), p–Pb (}√

s_NN=⁵.02 TeV) and Pb–

Pb (√

sNN=².76 TeV) which allows fora comparisonofthe var- ious systems. The radii extracted fromthree-pion cumulants are alsocomparedtothosefromtwo-pioncorrelations.

The Letter is organized into 5 remaining sections. Section 2 explainstheexperimentalsetupandeventselection.Section3de- scribes the identification of pions, as well as the measurement oftheeventmultiplicity.Section4explainsthethree-pioncumu- lant analysistechniqueused toextractthe sourceradii.Section 5 presentsthemeasuredsourceradii.Finally,Section 6summarizes theresultsreportedintheLetter.

http://dx.doi.org/10.1016/j.physletb.2014.10.034

0370-2693/©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP³.

2. Experimentalsetupandeventselection

Datafrompp, p–Pb,andPb–Pb collisions attheLHCrecorded withALICE[37]areanalyzed.Thedataforppcollisionsweretaken duringthe 2010pp runat√

s=^{7 TeV,forp–Pbcollisions during} the2013run at√

sNN=⁵.02 TeV,andforPb–Pbduring the2010 and 2011 runs at √

sNN=².76 TeV. For p–Pb, the proton beam energy was 4 TeV while for the lead beam it was 1.58 TeV per nucleon.Thus,thenucleon–nucleoncenter-of-masssystemmoved with respect to the ALICE laboratory system with a rapidity of

−⁰.465, i.e. in the direction of the proton beam. The pseudora- pidityinthelaboratorysystemisdenotedwith

η

throughoutthis Letter, which for the pp and Pb–Pb systems coincides with the pseudorapidityinthecenter-of-masssystem.

Thetriggerconditionsareslightlydifferentforeachofthethree collision systems. Forpp collisions, the VZERO detectors [38] lo- catedintheforwardandbackwardregionsofthedetector,aswell astheSiliconPixelDetector(SPD)atmid-rapidityareusedtoform a minimum-bias trigger by requiringat least one hit in the SPD oreither ofthe VZEROdetectors [39]. ForPb–Pb andp–Pb colli- sions,thetriggerisformedbyrequiringsimultaneoushitsinboth VZEROdetectors. Inaddition, high-multiplicitytriggers inpp and p–PbcollisionsbasedontheSPDareused.Twoadditionaltriggers inPb–PbareusedbasedontheVZEROsignalamplitudewhichen- hanced the statistics forcentral and semi-central collisions [38].

Approximately 164,115, and 52 million events are used for pp, p–Pb, and Pb–Pb collisions, respectively. For pp and p–Pb, the highmultiplicitytriggers accountforlessthan3% ofthecollected events.ForPb–Pb,thecentralandsemi-centraltriggersaccountfor about40% and52% ofthecollectedevents,respectively.

TheInner TrackingSystem(ITS) andTimeProjection Chamber (TPC) located atmid-rapidity are used for particle tracking [40].

TheITSconsistsof6layers ofsilicondetectors:siliconpixel(lay- ers1, 2), silicon drift (layers 3,4), and siliconstrip (layers5, 6) detectors.The ITSprovideshighspatial resolutionof theprimary vertex. The TPC alone is used formomentum and charge deter- minationofparticles via their curvatureinthe0.5 Tlongitudinal magneticfield,sinceclustersharingwithintheITScausesa small momentumbiasforparticlepairsatlowrelativemomentum.

TheTPCadditionallyprovidesparticleidentificationcapabilities through the specific ionization energyloss (dE/dx). The Time Of Flight(TOF)detectorisalsousedtoselectparticlesathighermo- menta. To ensure uniformtracking, the z-coordinate(beam-axis) oftheprimaryvertexisrequiredtobewithinadistanceof10 cm fromthedetectorcenter.Eventswithlessthanthreereconstructed chargedpionsare rejected,whichremovesabout25%and10% of thelow-multiplicityeventsinppandp–Pb,respectively.

3. Trackselectionandmultiplicityintervals

Trackswith total momentumless than1.0 GeV/c are usedto ensuregoodparticleidentification.Wealsorequiretransversemo- mentum p_T>0.16 GeV/c,andpseudorapidity|

η

|<0.8.Toensure good momentum resolutiona minimum of 70tracking points in theTPCarerequired.Chargedpionsareselectediftheyarewithin 2standarddeviations (

σ

) oftheexpectedpiondE/dxvalue [41].

For momenta greater than 0.6 GeV/c, high purity is maintained with TOF by selecting particles within 2

σ

of the expected pion time-of-flight.Additionally,trackswhichare within2

σ

oftheex- pectedkaonorprotondE/dxortime-of-flightvaluesarerejected.

Theeffectsoftrackmergingandsplittingareminimizedbasedon the spatial separation of tracks in the TPC as described in [42].

Forthree-pioncorrelationsthepaircutsareappliedtoeachofthe threepairsinthetriplet.

Similar as in [10], the analysis is performed in intervals of multiplicity which are defined by the reconstructed number of charged pions, N^rec_pions, in the above-mentioned kinematic range.

Foreachmultiplicity interval,thecorrespondingmeanacceptance andefficiencycorrected valueofthetotalcharged-pionmultiplic- ity, ^Npions^,andthetotalcharged-particle multiplicity,^Nch^,are determined using detector simulations with PYTHIA [43], DPM- JET [44],andHIJING [45] eventgenerators. Thesystematicuncer- tainty of^Nch ^{and N}pions^{is determined by comparing PYTHIA} to PHOJET (pp) [46], DPMJET to HIJING (p–Pb), and HIJING to AMPT (Pb–Pb)[47],andamountstoabout5%.Themultiplicityin- tervals,^Npions^,Nch^{,aswell astheaveragecentralityin Pb–Pb} andfractionalcrosssectionsinpp andp–PbaregiveninTable 1.

ThecollisioncentralityinPb–Pbisdeterminedusingthecharged- particle multiplicity in the VZERO detectors [38]. As mentioned above,thecenter-of-massreferenceframeforp–Pbcollisionsdoes not coincidewiththelaboratory frame,where^Nch^ismeasured.

However, fromstudiesusingDPMJETandHIJING atthegenerator level,thedifferenceto^Nch^{measuredinthe}center-of-massisex- pectedtobesmallerthan3%.

4. Analysistechnique

To extract the source radii,one can measuretwo- and three- particlecorrelation functionsasinRef. [42].Thetwo-particlecor- relationfunction

C2

(

p1

,

p2

) = α

2

N₂

(

p₁

,

p₂

)

N1

(

p1

)

N1

(

p2

)

⁽¹⁾

is constructedusing themomenta p_i,andis definedastheratio oftheinclusivetwo-particlespectrum overtheproduct ofthein- clusivesingle-particlespectra.BothareprojectedontotheLorentz invariant relative momentum q=

−(^p1−^p2)^μ(p₁−^p2)μ and the average piontransverse momentum kT= |^pT,1+ ^pT,2|/2. The numerator of the correlation function is formed by all pairs of particlesfromthesameevent.Thedenominatorisformedbytak- ing one particle from one event and the second particle from another eventwithin the samemultiplicity interval. The normal- izationfactor,

α

2,isdeterminedsuchthatthecorrelationfunction equals unity in a certain interval of relative momentum q. The location of the interval is sufficientlyabove the dominant region ofQS+^FSIcorrelationsandsufficiently narrowtoavoidthe influ- enceofnon-femtoscopiccorrelationsatlargerelativemomentum.

As thewidthofQS+^FSIcorrelationsisdifferentinallthree colli- sionsystems,ourchoiceforthenormalizationintervaldependson themultiplicityinterval.ForPb–Pb,thenormalizationintervalsare 0.15<q<0.175 GeV/cforN^rec_pions≥^{400 and0}.3<q<0.35 GeV/c for N_pions^rec <400. For pp and p–Pb the normalization interval is 1.0<q<1.2 GeV/c.

Following [48,49],the two-particle QS distributions, N^QS₂ , and correlations, C₂^QS,areextractedfromthemeasureddistributionsin intervalsofkTassuming

C2

(

q

) =

N

1

−

f_c²

+

f_c²K2

(

q

)

C₂^QS

(

q

)

B

(

q

).

(2)

The parameter f_c² characterizes the combined dilution effect of weak decays and long-lived resonance decays in the “core/halo”

picture [50,51].In Pb–Pb,it was estimatedtobe 0.7±⁰.05 with mixed-chargetwo-pioncorrelations[42].Thesameprocedureper- formed in pp and p–Pb data results in compatible values. The FSI correlation is given by K2(q), which includes Coulomb and strong interactions. For low multiplicities (N^rec_pions<150), K2(q) is calculated iteratively using the Fourier transform of the FSI corrected correlation functions.Forhighermultiplicities (N^rec_pions≥ 150), K₂(q)iscalculatedasinRef.[42]usingtheTHERMINATOR2

Table 1

Multiplicityintervalsasdeterminedbythereconstructednumberofchargedpions,N^rec_pions,withallofthetrackselectioncuts (p<1.0 GeV/c, pT>0.16 GeV/c,|η_|<0.8).

Npionsstandsfortheacceptancecorrectedaveragenumberofchargedpions,andNchforcorrespondingacceptancecorrectednumberofchargedparticlesinthesame kinematicrange.TheuncertaintiesonNchareabout5%.TheRMSwidthoftheNchdistributionineachintervalrangesfrom10%to35%forthehighestandlowest multiplicityintervals,respectively.TheaveragecentralityforPb–Pbinpercentiles,aswellasthefractionalcross-sectionsofthemultiplicityintervalsforp–Pbandppare alsogiven.TheRMSwidthsforthecentralitiesrangefromabout2 to4 percentilesforcentralandperipheralcollisions,respectively.

N_pions^rec Pb–Pb data p–Pb data pp data

Cent Npions Nch Fraction Npions Nch Fraction Npions Nch

[3,5) – – – 0.10 – – 0.23 4.0 4.6

[⁵,10) – – – 0.20 8.5 9.8 0.31 7.7 8.6

[¹⁰,15) – – – 0.18 15 17 0.12 13 15

[¹⁵,20) – – – 0.14 20 23 0.05 18 20

[²⁰,30) 77% 26 36 0.17 29 33 0.03 24 27

[³⁰,40) 73% 37 50 0.07 40 45 0.003 34 37

[⁴⁰,50) 70% 49 64 0.03 51 57 1×^{10−4 44 47}

[⁵⁰,70) 66% 66 84 0.01 63 71 – – –

[70,100) 60% 95 118 – – – – – –

[100,150) 53% 142 172 – – – – – –

[¹⁵⁰,200) 48% 213 253 – – – – – –

[²⁰⁰,260) 43% 276 326 – – – – – –

[²⁶⁰,320) 38% 343 403 – – – – – –

[³²⁰,400) 33% 426 498 – – – – – –

[400,500) 28% 534 622 – – – – – –

[500,600) 22% 654 760 – – – – – –

[600,700) 18% 777 901 – – – – – –

[⁷⁰⁰,850) 13% 931 1076 – – – – – –

[⁸⁵⁰,1050) 7.4% 1225 1413 – – – – – –

[¹⁰⁵⁰,2000) 2.6% 1590 1830 – – – – – –

model [52,53]. B(q) represents the non-femtoscopic background correlation,andis takenfromPYTHIAandDPMJETforppandp–

Pb, respectively[24,26]. Itis set equalto unityfor Pb–Pb,where nosignificantbackgroundisexpected.InEq.(2),N ^{istheresidual} normalizationofthefitwhichtypicallydiffersfromunityby0.01.

Thesame-chargetwo-pionQScorrelationcan beparametrized byanexponential

C^QS₂

(

q

) =

¹

+ λ

e^−Rinvq

,

(3) aswellasbyaGaussianorEdgeworthexpansion

C^QS₂

(

q

) =

¹

+ λ

E_w²

(

R_invq

)

e^−R2invq2 (4) E_w

(

R_invq

) =

¹

+

∞

n=3

κ

n

n

! ( √

2

)

^nHn

(

R_invq

),

(5) whereEw(Rinvq)characterizes deviationsfromGaussian behavior, H_n are theHermitepolynomials, and

κ

n aretheEdgeworthcoef- ficients[54].Thefirsttwo relevantEdgeworthcoefficients (

κ

3,

κ

4) arefoundtobesufficienttodescribethenon-Gaussian featuresat lowrelativemomentum.TheGaussianfunctionalformisobtained withEw=^{1 (}

κ

n=^{0)inEq.(4).Theparameter}λcharacterizes an apparentsuppressionfromanincorrectlyassumedfunctionalform ofC₂^QS andthe suppressiondue to possiblepion coherence [55].

The parameter Rinv is the characteristic radius fromtwo-particle correlationsevaluatedinthepair-restframe.Theeffectiveintercept parameterfortheEdgeworthfitisgivenbyλe=λE²_w(0)[54].The effectiveinterceptcanbebelowthechaoticlimitof 1.0forpartially coherentemission[36,42,55].Theextractedeffectiveinterceptpa- rameter is found to strongly depend on the assumed functional formofC₂^QS.

Thethree-particlecorrelationfunction C3

(

p1

,

p2

,

p3

) = α

3

N₃

(

p₁

,

p₂

,

p₃

)

N1

(

p1

)

N1

(

p2

)

N1

(

p3

)

⁽⁶⁾

is defined as the ratio of the inclusive three-particle spectrum over theproduct of the inclusivesingle-particle spectra.In anal- ogytothe two-pioncase, it isprojected ontothe Lorentz invari- ant Q₃=

q²₁₂+^q2₃₁+^q2₂₃ ^{and the average pion transverse mo-}

mentum KT,3= ^|pT,1+pT,2₃ ^+pT,3|.The numeratorof C3 isformedby taking three particles from the same event. The denominator is formedbytakingeachofthethreeparticlesfromdifferentevents.

Thenormalizationfactor,

α

3,isdetermined suchthatthecorrela- tionfunctionequalsunityintheintervalofQ3whereeachpairqi j liesinthesameintervalgivenbeforefortwo-pioncorrelations.

The extraction of the full three-pion QS distribution, N^QS₃ , in intervalsofK_T,3 isdoneasinRef.[42]bymeasuring

N₃

(

p₁

,

p₂

,

p₃

) =

^f1N₁

(

p₁

)

N₁

(

p₂

)

N₁

(

p₃

) +

f2

N2

(

p1

,

p2

)

N1

(

p3

) +

N2

(

p3

,

p1

)

N1

(

p2

)

+

N2

(

p2

,

p3

)

N1

(

p1

)

+

^f3K3

(

q12

,

q31

,

q23

)

N^QS₃

(

p1

,

p2

,

p3

),

(7) wherethefractions f1=(1−^fc)³+^3fc(1− ^fc)²−³(1− ^fc)(1− f_c²)= −⁰.08, f2 =¹− ^fc = ⁰.16, and f3 = ^fc³ = ⁰.59 using f_c²=⁰.7 as in the two-pion case. The term N2(pi,pj)N1(pk) is formed by taking two particles from the same event and the third particle from a mixed event. All three-particle distribu- tions are normalized to each other in the same way as for

α

3. K3(q12,q31,q23) denotes the three-pion FSI correlation, which in the generalized Riverside (GRS) approach [42,56,57] is approx- imated by K₂(q₁₂)K₂(q₃₁)K₂(q₂₃). It was found to describe the

π

^±

π

^±

π

^∓three-bodyFSIcorrelationtothefewpercentlevel[42].

FromEq.(7)onecanextract N^QS₃ andconstructthethree-pionQS cumulantcorrelation

c₃

(

p₁

,

p₂

,

p₃

) =

N3

1

+

2N₁

(

p₁

)

N₁

(

p₂

)

N₁

(

p₃

)

−

^N₂^QS

(

p₁

,

p₂

)

N₁

(

p₃

)

−

^N₂^QS

(

p₃

,

p₁

)

N₁

(

p₂

) −

^NQS₂

(

p₂

,

p₃

)

N₁

(

p₁

) +

^N₃^QS

(

p₁

,

p₂

,

p₃

)

/

N₁

(

p₁

)

N₁

(

p₂

)

N₁

(

p₃

) ,

(8)

where N^QS₂ (pi,pj)N1(p_k)= [^N2(pi,pj)N1(p_k)− ^N1(pi)N1(pj)× N1(pk)(1− ^fc²)]/(f_c²K2). In Eq. (8), all two-pion QS correlations are explicitly subtracted [34]. The QS cumulant inthis form has

Fig. 1.DemonstrationoftheremovaloftheK⁰_s decayfromthree-pioncumulants.

Mixed-chargethree-pioncorrelationsareprojectedagainsttherelativemomentum ofamixed-chargepair(q^±∓₃₁ ).TheK_s⁰decayintoaπ⁺₊π⁻pairisvisibleasex- pectedaround0.4GeV/c.TheFSIenhancementofthemixed-chargepair“31”is alsovisibleatlowq^±∓₃₁ .FSIcorrectionsarenotapplied.Systematicuncertaintiesare shownbyshadedboxes.

FSIsremovedbeforeitsconstruction.N3 istheresidualnormaliza- tionofthefitwhichtypicallydiffersfromunityby0.02.

The three-pion same-charge cumulant correlations are then projected onto 3D pair relative momenta and fit with an expo- nential

c3

(

q12

,

q31

,

q23

) =

1

+ λ

3e^{−Rinv,3(q12+q31+q23)/2}

,

(9) aswellasaGaussianandanEdgeworthexpansion[54]

c3

(

q12

,

q31

,

q23

) =

¹

+ λ

3Ew

(

R_inv,3q12

)

Ew

(

R_inv,3q31

)

×

^Ew

(

R_inv,3q₂₃

)

e^{−R2inv,3Q32/2}

.

(10) R_inv,3andλ3aretheinvariantradiusandinterceptparametersex- tractedfromthree-pioncumulantcorrelations,respectively.Theef- fectiveinterceptparameterfortheEdgeworthfitisλe,3=λ3E³_w(0). Foranexactfunctionalformofc₃,λe,3reachesamaximumof 2.0 for fully chaotic pion emission. Deviations below and above 2.0 canfurtherbecausedbyincorrectrepresentationsofc3,e.g.Gaus- sian.Eq.(10)neglectstheeffectofthethree-pionphase[33]which was foundtobe consistentwithzeroforPb–Pbcentral andmid- centralcollisions[42].Wenotethattheextractedradiifromtwo- andthree-pioncorrelationsneednotexactlyagree,e.g.inthecase ofcoherentemission[58].

Themeasuredcorrelationfunctionsneedtobecorrectedforfi- nitetrackmomentumresolutionoftheTPCwhichcausesa slight broadeningof the correlation functionsandleads toa slight de- crease of the extracted radii. PYTHIA (pp), DPMJET (p–Pb) and HIJING (Pb–Pb)simulationsareusedtoestimate theeffectonthe fit parameters. After the correction, both fit parameters increase by about 2% (5%) for the lowest (highest) multiplicity interval.

Therelative systematicuncertaintyofthiscorrectionis conserva- tively taken to be 1%. The pion purity is estimated to be about 96%. Muonsare found to be the dominantsource of contamina- tion, forwhich we apply corrections to the correlation functions as described in Ref. [42]. The correction typically increases the radius (intercept) fitparameters by less than1% (5%). The corre- spondingsystematicuncertaintyisincludedinthe comparisonof themixed-chargedcorrelationwithunity (seebelow).

5. Results

The absenceoftwo-particle correlationsin thethree-pion cu- mulantcanbedemonstratedvia theremovalofknowntwo-body effects such as the decay of K_s⁰ into a

π

⁺+

π

⁻ pair (Fig. 1).

The mixed-charge three-pion correlation function (C^±±∓₃ ) pro- jectedontotheinvariantrelativemomentumofoneofthemixed- chargepairsinthetripletexhibitsthe K_s⁰peakasexpectedaround q^±∓=⁰.4 GeV/c,whileitisremovedinthecumulant.

InFig. 2wepresentthree-pioncorrelationfunctionsforsame- charge (top panels) andmixed-charge (bottom panels) tripletsin pp, p–Pb,andPb–Pb collision systemsinthree sample multiplic- ity intervals. For same-charge triplets, the three-pion cumulant QS correlation (c^±±±₃ ) isclearly visible. Formixed-chargetriplets the three-pion cumulant correlation function (c^±±∓₃ ) is consis- tent with unity, as expected when FSIs are removed. Gaussian, Edgeworth, and exponential fits are performed in three dimen- sions (q12,q31,q23).ConcerningEdgeworthfits,differentvaluesof the

κ

coefficients correspond to different spatial freeze-out pro- files. In order to make a meaningful comparison of the charac- teristicradii acrossallmultiplicity intervalsandcollisionsystems, wefix

κ

3=⁰.1 and

κ

4=⁰.5.Thevaluesaredeterminedfromthe average offree fitsto c^±±±₃ forall multiplicity intervals, KT,3 in- tervals andsystems. The RMS of both

κ

3 and

κ

4 distributions is 0.1. Thechosen

κ

coefficientsproduceasharpercorrelation func- tion which corresponds to larger tails in the source distribution.

AlsoshowninFig. 2are modelcalculationsofc₃ inPYTHIA(pp), DPMJET(p–Pb)andHIJING(Pb–Pb),whichdonotcontainQS+^FSI correlations and demonstrate that three-pion cumulants, in con- trasttotwo-pioncorrelations[24,26],donotcontainasignificant non-femtoscopicbackground,evenforlowmultiplicities.

ThesystematicuncertaintiesonC₃areconservativelyestimated to be 1% by comparing

π

⁺ to

π

⁻ correlation functions and by tightening the track merging and splitting cuts. The systematic uncertainty onc^±±±₃ isestimatedby the residualcorrelation ob- served withc^±±∓₃ relativeto unity.The residualcorrelationleads to a 4% uncertainty on λe,3 while having a negligible effect on R_inv,3.Theuncertaintyon fcleadstoanadditional10% uncertainty onc3−^{1 and}λe,3.Wealsoinvestigatedtheeffectofsetting fc=¹ andthus f₁=⁰,f₂=⁰,f₃=¹.0 inEq.(7)andfoundanegligible effectonR_inv,3,whilesignificantlyreducingλe,3asexpectedwhen thedilutionisnotadequatelytakenintoaccount.

Figs. 3(a) and3(b) show the three-pion Gaussian fit parame- ters forlow andhigh KT,3 intervals, respectively. The ^kT^values forlow (high)kT are0.25 (0.43)GeV/c.TheresultingpairkT dis- tributions in the triplet K_T,3 intervals have RMS widths for the low (high) KT,3 of 0.12 (0.14) in pp and p–Pb and 0.04 (0.09) GeV/c inPb–Pbcollisions. The^kT^{valuesforlow (high) K}T,3 are 0.24 (0.39)GeV/c.Wealsoshowthefitparametersextractedfrom two-pioncorrelationsinordertocomparetothoseextractedfrom three-pioncumulants.ForPb–Pb,theGaussianradiiextractedfrom three-pioncorrelationsareabout10%smallerthanthoseextracted fromtwo-pioncorrelations,whichmaybeduetothenon-Gaussian featuresofthecorrelationfunction.Aclearsuppressionbelowthe chaotic limit isobserved fortheeffectiveinterceptparameters in all multiplicity intervals.The suppressionmaybe causedby non- Gaussian features of thecorrelation function andalso by a finite coherentcomponentofpionemission[36,42,55].

The systematic uncertainties on the fit parameters are domi- nated by fit-range variations, especially in the case of Gaussian fits to non-Gaussian correlation functions. The chosen fit range forc₃ variessmoothly betweenQ₃=⁰.5 and0.1 GeV/cfromthe lowest multiplicity ppto thehighestmultiplicity Pb–Pbintervals.

For C2, the fit ranges are chosen to be √

2 times narrower. The characteristic widthof Gaussian three-pion cumulantQS correla- tions projected against Q3 is a factorof √

2 times that of Gaus- sian two-pion QS correlations projected against q [35,36]. As a variation we change the upper bound of the fit range by ±^30%

forthree-pioncorrelationsandtwo-pioncorrelationsinPb–Pb for N^rec_pions>50. For N^rec_pions<50,in Pb–Pb, the upperlimit ofthe fit

Fig. 2.Three-pioncorrelationfunctionsversusQ3for0.16<KT,3<0.3 GeV/cinpp,p–PbandPb–PbcollisiondatacomparedtoPYTHIA,DPMJETandHIJINGgenerator-level calculations.Toppanelsareforsame-chargetriplets,whilebottompanelsareformixed-chargetriplets.Twopointsatlow Q3 withlargestatisticaluncertaintiesarenot shownforthepp same-chargecorrelationfunction.

Fig. 3.Two- andthree-pionGaussianfitparametersversusNchinpp,p–PbandPb–PbcollisionsystemsforlowandhighkTand KT,3 intervals.Toppanelsshowthe Gaussianradii R^G_inv andR^G_inv,3 andbottompanelsshowtheeffectiveGaussianinterceptparametersλ^Ge andλ^G_e,3.Thesystematicuncertaintiesaredominatedbyfitrange variationsandareshownbybounding/dashedlinesandshadedboxesfortwo- andthree-particleparameters,respectively.Thedashedanddash-dottedlinesrepresentthe chaoticlimitsforλ^Ge andλ^G_e,3,respectively.

rangeisincreased tomatchthatinp–Pb(i.e.0.13to0.27 GeV/c).

Forppandp–Pb,owingtothelargerbackgroundpresentfortwo- pioncorrelations,weextendthefitrangetoq=¹.2 GeV/cforthe upper variation. The non-femtoscopic background in Eq. (2) has a non-negligible effecton the extractedradii in the extended fit

range. The resulting systematic uncertainties are fully correlated for three-pionfit parameters for each collision system, since the fit-rangevariationshavethesameeffectineachmultiplicityinter- val. The systematicuncertainties forthe two-pion fit parameters are largely correlated and are asymmetric due to the different

Fig. 4.Two- andthree-pionEdgeworthfitparametersversusNchinpp,p–PbandPb–PbcollisionsystemsforlowandhighkTand KT,3intervals.Toppanelsshowthe Edgeworthradii R_inv^Ew and R_inv^Ew_,3 andbottompanelsshowtheeffectiveinterceptparametersλ^Ee^w andλ^E_e,^w₃.Asdescribedinthetext,κ3and κ4 arefixedto0.1and0.5, respectively.SamedetailsasforFig. 3.

Fig. 5.Two- andthree-pionexponential fitparametersversus^Nch^{inpp,p–PbandPb–Pbcollisionsystemsforlowandhighk}Tand KT,3intervals.Toppanelsshowthe exponential radiiR^Exp_inv andR^Exp_inv,3scaleddownby√

πandbottompanelsshowtheeffectiveinterceptparametersλ^Expe andλ^Exp_e,3.SamedetailsasforFig. 3.

fit-range variations. We note that the radii in pp collisions at

√s=^{7 TeV from our previous two-pion} measurement [26] are about25% smallerthanthecentralvaluesextractedinthisanaly- sisalthoughcompatiblewithin systematicuncertainties.Thelarge difference is attributed to the narrower fit range in this analy- sis. In [24,26] the chosen Gaussian fit range was q<1.4 GeV/c, whilehereitisq<0.35 GeV/cforthelowestmultiplicityinterval.

Thenarrowerfitrangeischosenbasedonobservationsmadewith three-pioncumulantsforwhichtwo-pionbackgroundcorrelations are removed.It isobserved inFig. 2 thateven forlow multiplic-

ities, thedominantQS correlation iswell below Q3=⁰.5 GeV/c.

The presence of the non-femtoscopic backgrounds can also bias theradiifromtwo-pioncorrelationsinwidefitrangesandissup- pressedwiththree-pioncumulantcorrelations.

To further address the non-Gaussian features of the correla- tion functions, we also extract the fit parameters from an Edge- worth andexponentialparametrizationasshowninFigs. 4 and5.

We observethat theEdgeworth andexponential radii aresignifi- cantlylargerthantheGaussianradii.However,theyshouldnotbe directly compared astheycorrespond todifferentsource profiles.

GaussianradiicorrespondtothestandarddeviationofaGaussian sourceprofilewhereasexponentialradiicorrespondtotheFWHM ofa Cauchy source. The Edgeworth radii are model independent andaredefinedasthe2ndcumulantofthemeasured correlation function.Note that the exponential radii have been scaled down by √

π

as is often done to compare Gaussian and exponential radii [23]. Compared to the Gaussian radii, the two- and three- pionradii are in much better agreement for the Edgeworth and exponentialfits.Thissuggeststhat thediscrepancybetweentwo- andthree-pionGaussian radiiareindeedcausedby non-Gaussian features of the correlation function. Concerning the effective in- tercepts, we observe a substantial increase as compared to the Gaussiancase.

Thequalities ofthe Gaussian, Edgeworth,andexponential fits forthree-pioncumulantcorrelationsvarydependingonthemulti- plicityinterval.The

χ

²/NDF forthe3Dthree-pionGaussian,Edge- worth, andexponential fits in the highest multiplicity Pb–Pb in- tervalis8600/1436,4450/1436,and4030/1436,respectively.The

χ

²/NDF decreasessignificantly forlower multiplicity intervalsto about4170/7785 forperipheral Pb–Pb and 12400/17305 for pp and p–Pb multiplicity intervals, for all fit types. The Edgeworth

χ

²/NDF is a few percent smaller than for Gaussian fits in low multiplicity intervals. The exponential

χ

²/NDF is a few percent smallerthanforEdgeworthfitsinlowmultiplicityintervals.

Due to the asymmetry of the p–Pb colliding system, the ex- tractedfitparametersin−⁰.8<

η

<−⁰.4 and0.4<

η

<0.8 pseu- dorapidityintervalsarecompared.Theradiiandtheeffectiveinter- cept parameters inboth intervalsare consistent withinstatistical uncertainties.

The extracted radii in each multiplicity interval and system correspond to different^Nch ^{values. Tocompare the radii in pp} and p–Pb at the same ^Nch ^{value, we perform a linear fit to} thepp three-pionEdgeworth radii asa function of ^Nch^{1/3. We} then compare the extracted p–Pb three-pion Edgeworth radii to thevalueofthe ppfitevaluated atthesame^Nch^{.We findthat} theEdgeworth radii in p–Pb are on average 10±^{5% largerthan} forppintheregionofoverlappingmultiplicity.Thecomparisonof Pb–Pbto p–Pbradii is done similarlywhere the fitis performed to p–Pb data and compared to the two-pion Pb–Pb Edgeworth radii. The Edgeworth radii in Pb–Pb are found to be on average 45±^{10% larger thanforp–Pbintheregionof}overlappingmulti- plicity.Theratiocomparisonasitisdoneexploitsthecancellation ofthelargelycorrelatedsystematicuncertainties.

Tobeindependent oftheassumedfunctional formforc3,the same-chargethree-pioncumulantcorrelationfunctionsaredirectly compared between two collision systems at similar multiplicity.

Fig. 6(a)showsthat whilethe three-pioncorrelationfunctionsin ppandp–Pb collisionsaredifferent,theircharacteristicwidthsare similar. It is therefore the λe,3 valueswhich differ the most be- tweenthetwosystems.Fig. 6(b)showsthat thecorrelationfunc- tionsinp–PbandPb–Pb collisionsaregenerallyquitedifferent.

6. Summary

Three-pioncorrelationsofsame- andmixed-chargepionshave beenpresentedforpp (√

s=^{7 TeV),p–Pb (}√

sNN=⁵.02 TeV)and Pb–Pb (√

s_NN=².76 TeV) collisions at the LHC, measured with ALICE. Freeze-outradii using Gaussian, Edgeworth,and exponen- tial fits have been extracted from the three-pion cumulant QS correlation and presented in intervals of multiplicity and triplet momentum.Comparedtotheradiifromtwo-pioncorrelations,the radiifromthree-pioncumulantcorrelationsarelesssusceptibleto non-femtoscopicbackgroundcorrelationsduetotheincreasedQS signalandtheremovaloftwo-pionbackgrounds.

ThedeviationofGaussian fitsbelowthe observedcorrelations at low Q3 clearly demonstrates the importance of non-Gaussian features of the correlation functions. The effective intercept pa- rameters from Gaussian (exponential) fits are significantly below (above)thechaotic limits,whilethecorresponding Edgeworthef- fectiveinterceptsaremuchclosertothechaoticlimit.

Atsimilarmultiplicity,theinvariantradiiextractedfromEdge- worth fits in p–Pb collisions are found to be 5–15% larger than those in pp, while those in Pb–Pb are 35–55% larger than those in p–Pb. Hence, models which incorporate substantially stronger collective expansion in p–Pbthan pp collisions atsimilar multi- plicityare disfavored. The comparability of theextracted radii in pp and p–Pb collisions at similar multiplicity is consistent with expectationsfromCGCinitialconditions(IP-Glasma)withoutahy- drodynamicphase[17].Thesmallerradiiinp–Pbascompared to Pb–Pbcollisionsmaydemonstratetheimportanceofdifferentini- tial conditionson thefinal-state,or indicatesignificant collective expansionalreadyinperipheralPb–Pbcollisions.

Acknowledgements

We would like to thank Richard Lednický, Máté Csanád, and TamásCsörg ˝ofornumeroushelpfuldiscussions.

The ALICE Collaboration would like to thank all its engineers andtechniciansfortheir invaluablecontributions totheconstruc- tion of the experiment and the CERN accelerator teams for the outstandingperformanceoftheLHCcomplex.

The ALICECollaborationgratefullyacknowledges theresources andsupportprovidedby allGridcentresandtheWorldwide LHC ComputingGrid(WLCG)Collaboration.

The ALICE Collaboration acknowledges the following funding agencies fortheir supportin buildingandrunning theALICEde- tector: StateCommitteeofScience,World FederationofScientists (WFS) and Swiss Fonds Kidagan, Armenia, Conselho Nacional de DesenvolvimentoCientífico eTecnológico (CNPq),Financiadorade EstudoseProjetos(FINEP),FundaçãodeAmparoàPesquisadoEs- tado de São Paulo (FAPESP);National NaturalScience Foundation of China (NSFC), the Chinese Ministry of Education (CMOE) and the Ministry of Science andTechnology of the People’s Republic of China (MSTC); Ministry of Education and Youth of the Czech Republic; Danish Natural Science Research Council, the Carlsberg FoundationandtheDanishNationalResearchFoundation;TheEu- ropean Research Council under the European Community’s Sev- enthFrameworkProgramme; HelsinkiInstituteofPhysicsandthe AcademyofFinland;FrenchCNRS-IN2P3,the‘RegionPaysdeLoire’,

‘RegionAlsace’,‘RegionAuvergne’andCEA,France;GermanBMBF and the Helmholtz Association; General Secretariat for Research andTechnology,MinistryofDevelopment,Greece;HungarianOTKA andNationalOfficeforResearchandTechnology(NKTH); Depart- ment ofAtomic Energy, Government of India andDepartment of Science and Technology, Ministry of Science andTechnology, In- dia;IstitutoNazionalediFisicaNucleare(INFN)andCentroFermi– MuseoStoricodellaFisicaeCentroStudieRicerche“EnricoFermi”, Italy; MEXT Grant-in-Aid forSpecially PromotedResearch, Japan;

Joint Institute for Nuclear Research, Dubna; National Research Foundation of Korea (NRF); CONACYT, DGAPA, México; ALFA-EC andtheEPLANETProgram(EuropeanParticlePhysicsLatinAmeri- canNetwork);StichtingvoorFundamenteelOnderzoekderMaterie (FOM)andtheNederlandseOrganisatievoorWetenschappelijkOn- derzoek (NWO), Netherlands; Research Council ofNorway (NFR);

NationalScienceCentre,Poland;MinistryofNationalEducation/In- stitute forAtomic PhysicsandCNCS-UEFISCDI, Romania; Ministry ofEducationandScience ofRussianFederation, RussianAcademy ofSciences,RussianFederalAgencyofAtomicEnergy,RussianFed- eralAgencyforScienceandInnovationsandTheRussianFounda-