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Pion-kaon femtoscopy and the lifetime of the hadronic
phase in Pb
−Pb collisions at
√
s_NN = 2.76 TeV
Shreyasi Acharya, Dagmar Adamova, Alexander Adler, Jonatan Adolfsson,
Madan Mohan Aggarwal, Shahrukh Agha, Gianluca Aglieri Rinella,
Michelangelo Agnello, Neelima Agrawal, Zubayer Ahammed, et al.
To cite this version:
Shreyasi Acharya, Dagmar Adamova, Alexander Adler, Jonatan Adolfsson, Madan Mohan Aggarwal,
et al.. Pion-kaon femtoscopy and the lifetime of the hadronic phase in Pb
−Pb collisions at
√
s_NN
Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
Pion–kaon
femtoscopy
and
the
lifetime
of
the
hadronic
phase
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
r
a
c
t
Articlehistory:
Received28August2020
Receivedinrevisedform8December2020 Accepted11December2020
Availableonline17December2020 Editor:M.Doser
Inthispaper,thefirstfemtoscopicanalysisofpion–kaoncorrelationsattheLHCisreported.Theanalysis wasperformedonthePb–Pbcollisiondataat√sNN=2.76 TeVrecordedwiththeALICEdetector.The
non-identicalparticle correlations probe the spatio-temporalseparation between sources ofdifferent particlespeciesaswellastheaveragesourcesizeoftheemittingsystem.Thesizesofthepionandkaon sourcesincreasewithcentrality,and pionsareemittedcloser tothecentreofthesystemand/orlater thankaons.Thisisnaturallyexpectedinasystemwithstrongradialflowandisqualitativelyreproduced by hydrodynamic models. ALICE data on pion–kaon emission asymmetry are consistent with (3+1)-dimensionalviscoushydrodynamicscoupledtoastatisticalhadronisationmodel,resonancepropagation, anddecaycodeTHERMINATOR2 calculation,withanadditionaltimedelaybetween1and 2 fm/c for
kaons.Thedelaycanbeinterpretedasevidenceforasignificanthadronicrescatteringphaseinheavy-ion collisionsattheLHC.
©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Themaingoaloftheheavy-ionprogrammeattheLargeHadron Collider (LHC) is to study the deconfined state of strongly inter-acting matter. This state, where the relevant degrees offreedom are quarks and gluons, is called the quark-gluon plasma (QGP). Experimental resultsfrom RHICsuggest that the QGPbehaves as a fluid with small specific viscosity [1–4]. The characteristics in momentum space can be accessed from radial and elliptic flow, transversemomentumspectraorfromevent-by-eventfluctuations. The space-time structure, relevant forthe size andpressure gra-dients ofthe system, can be accessedusing two-particle correla-tions.
Non-identical particle correlationsare sensitive to the relative space-time emissionshifts ofdifferentparticle species [5–7]. The difference betweenmeanemission space-timecoordinatesoftwo particlespeciesatfreeze-outiscalledemissionasymmetry.It oc-curs asa consequence ofthe collective expansion of thesystem, the presence of short-lived resonances decaying into the consid-eredparticles,theradialflowoftheseresonances,andthe possibil-ityofhavingadditionalrescatteringbetweenthechemicaland ki-neticboundariesoftheevolutionofthesystem [7].Measurements of correlationsofnon-identical particles inlow-energy heavy-ion collisions allowed one to establish an emission time ordering of
E-mailaddress:alice-publications@cern.ch.
thenuclearfragments [8,9].Inrelativisticheavy-ioncollisionsthey providedindependent evidenceofcollective transverseexpansion inAu–Au collisions at
√
sNN=
130 GeV atthe Relativistic Heavy IonCollider(RHIC) [10].TheHanbury BrownandTwiss(HBT) [11–16] pion correlation radii are a measure of the source size of pions of a given mo-mentum. Together with measurements of the elliptic flow and the transverse momentum spectra of identified particles they havebeenfundamentalinidentifyingtherelevantstagesof ultra-relativisticheavy-ioncollisions andtheir properties [17]. Further-more, a recent measurement of the kaon femtoscopic radii in Pb–Pb collisions [18] showedthat (when compared forthesame eventcentrality andpair mT) they are systematically larger than the ones frompions and those predicted by models based on a hydrodynamic evolutioncoupledto statisticalhadronisation.Only after including the hadronic rescatteringphase could the model [19] reproduce thedata forpions andkaons simultaneously. The meanemissiontimeofkaons(11.6 fm
/
c) andofpions(9.5 fm/
c)werereported [18].Thedifferenceisattributedtotherescattering throughtheK∗resonance.
Particleyieldsandspectraaddfurthersupporttomodelswhich include the formation of a dense hadronic phase in the final stages of the evolution of the fireball created in heavy-ion col-lisions. The suppression or the enhancement of the yield (with respecttoppcollisions)ofshort-livedresonancesdueto rescatter-ing (suppression) orregeneration (enhancement) in the hadronic phase has beenproposed asan observable for theestimation of
https://doi.org/10.1016/j.physletb.2020.136030
0370-2693/©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
the lifetime and properties of the hadronic phase [20–22]. The measurements of severalresonances, fromthevery short-lived
ρ
meson(
τ
=
1.
4 fm/
c),K∗ (τ
=
4 fm/
c),(
1520)
(τ
=
10 fm/
c)to longer-livedφ
(τ
=
46 fm/
c), demonstratestrong suppression of short-lived resonances incentral collisions[23–25].The observed suppression can result from a long-lasting hadronic rescattering phase.Recently, pion–kaon correlations were studied theoretically with a (3+1) viscous hydrodynamic model [26], coupled to the statistical hadronisation, resonance decay, and propagation code THERMINATOR 2 [28]. The model uses a parameterisation of the equation ofstate interpolatingbetweenthelatticeresults[27] for high temperatures and the hadron gas equation of state at low temperatures. The hadronisation occurs via the Cooper-Frye for-malism withoutdistinction betweenchemical andkinetic freeze-out. No further interactions betweenthe hadronsare considered, however, the emission time of each species can be delayed by hand,mimickingtheeffectofrescattering.The femtoscopic emis-sion asymmetry was shown to be highly sensitive to this delay. Moreover, it can be decoupled from other mechanisms like flow orresonancecontributionspresentatfreeze-out,includingtheK∗ resonance [28]. This approach has been explored for pion–kaon pairs.Detailedpredictions fordifferentemission scenariosforthe pion–kaon radii and their emission asymmetry as a function of thesourcevolumehavebeenmadeforPb–Pbcollisionsat
√
sNN= 2.76TeVin [28].Inthiswork
π
+K+,π
−K+,π
+K−,andπ
−K−momentum corre-lations are analysed usingthe femtoscopy technique. Two meth-ods are used to evaluate the emission asymmetry in order to strengthen theresults.The first methoddecomposes the correla-tionsintotermsofonedimensionalsphericalharmonic(SH) coef-ficients [29] whilethesecondoneisbasedontheCartesian repre-sentationofthecorrelationfunction [5].ThesourcesizeparameterRoutandtheemissionasymmetry
μ
outaremeasuredasafunction ofthe cuberootoftheaverage charged-particlemultiplicity den-sitydNch/
dη
1/3.Finally,theobtainedresultsarecomparedwith detailedmodelcalculations [28] assumingthepreviouslyfound de-layedkaonemission[18].2. Dataselection
Inthispaper, pion–kaoncorrelation resultsobtainedwithPb– Pbcollisionsat
√
sNN =2.76TeVarepresented.Thismeasurement used40million eventscollectedby ALICEin2011. Adetailed de-scriptionoftheALICEdetectoranditsperformanceintheLHCRun 1(2009–2013)isgivenin[30,31].Eventswere triggeredandclassifiedaccordingtotheir central-itywas determined usingthe measured signal amplitudesin the V0 detectors [32]. Three trigger configurations were used: mini-mum bias, semi-central (10–50% collision centrality), andcentral (0–10% collision centrality) [32]. The analyses were performed in six centrality classes: (0–5%), (5–10%), (10–20%), (20–30%), (30–40%),and(40–50%),separatelyforpositiveandnegative mag-neticfieldpolarity.Thereconstructedprimaryvertexisrequiredto liewithin
±
7 cmofthenominalinteractionpointalongthebeam axis inorder to haveuniformtracking andparticle identification performance.Charged particle trackingis performedusingthe Time Projec-tion Chamber (TPC) [30,33] and the Inner Tracking System (ITS) [30].The ITSallowsforhighspatialresolutionindeterminingthe primary collisionvertex.Inthisanalysis,thedeterminationofthe track momenta was performed using tracks reconstructed only from TPC signals and constrained to the primary vertex. A TPC tracksegmentisreconstructedfromatleast70spacepoints (clus-ters) out of a maximum of 159. The
χ
2 of the track fit, nor-malised to the number ofdegrees of freedom, is required to beTable 1
Single particle selection criteria, together with particle identification variationsusedforuncertaintyestimation.
Track selection
pT 0.19<pT<1.5 GeV/c
|η| <0.8
DCAtransversetoprimaryvertex <2.4 cm DCAlongitudinaltoprimaryvertex <3.0 cm Kaon selection
Default Loose Strict
Nσ ,TPC(forp<0.4 GeV/c) <2 <2.5 <2
Nσ ,TPC(for0.4<p<0.45 GeV/c) <1 <2 <1
Nσ ,TPC(forp>0.45 GeV/c) <3 <3 <2
Nσ ,TOF(for0.5<p<0.8 GeV/c) <2 <3 <2
Nσ ,TOF(for0.8<p<1.0 GeV/c) <1.5 <2.5 <1.5
Nσ ,TOF(for1.0<p<1.5 GeV/c) <1 <2 <1 Pion selection
Default Loose Strict
Nσ ,TPC(for p<0.5 GeV/c) <3 <3 <2.5
N2
σ ,TPC+N2σ ,TOF(for p>0.5 GeV/c) <3 <3 <2.5
χ
2/
ndf<
2. The distances of closest approach (DCA) of a track to theprimary vertex inthe transverse (DCAxy) andlongitudinal (DCAz) directionsare requiredtobe lessthan2.4cmand3.2cm, respectively.These selectionsare imposed toreduce the contam-ination fromsecondary tracks originatingfrom weak decays and from interaction with the detector material. The transverse mo-menta and pseudorapidities of pions and kaons were restricted to 0.
19<
pT<
1.
5 GeV/
c and|
η
|
<
0.
8. All selectionsare sum-marisedinTable1.The charged-particle tracks are identified as pions and kaons usingthecombinedinformationoftheirspecificionisationenergy loss(dE
/
dx)intheTPCandthetime-of-flightinformationfromthe Time-Of-Flight (TOF)detectors [34].For eachreconstructed parti-cle, thesignalsfrom boththe TPCandthe TOF(dE/
dx andtime offlight,respectively)arecomparedwiththeonespredictedfora pionorkaon. A value Nσ isassigned toeach track denotingthe numberofstandarddeviationsbetweenthemeasuredtrackdE/
dx ortimeofflightandtheexpectedone.Forpions,thesignal(dE/
dx forpT<
500 MeV/
c,combineddE/
dx andtimeofflightabovethis value)isallowed todiffer fromthecalculationby 3σ
.Forkaons, five selections were used, as detailed in Table 1, together with variations used for uncertainty estimation. The selection criteria are optimised to obtain a high-purity sample while maximising efficiency,especially in the regions where separating kaons from other particle species are challenging. The purity was estimated fromMonteCarlosimulationsusingtheHIJING [35] event genera-torcoupledtotheGEANT3 [36] transportpackageandwasfound tobeabove98%forboththepionandkaonsamples.Theidentifiedtracksfromeacheventarecombinedintopairs. Two-particle detectoracceptanceeffects,includingtracksplitting, track merging, as well as effects coming from
γ
→
e+e− con-version, contribute to the measured distributions. The following selectionsare applied to reduce these effects.For pairs oftracks within|
η
|
<
0.
1 an exclusionon the fractionof merged points is introduced. The merged fractionis definedas the ratioof the numberof steps of1 cm considered inthe TPCradius range for whichthedistancebetweenthetracksislessthan3 cmtothe to-talnumberof steps.Pairswitha mergedfractionabove 3% were removed.Thee+e−pairsoriginatingfromphotonconversionscan be misidentified asa real pion–kaon pair andit is necessary to removespurious correlations arisingfromsuch pairs. Thesepairs are removedif their invariant mass, assuming the electron mass forbothparticles,islessthan0.002GeV/
c2,andtherelativepolar angle,θ
,betweenthetwotracksislessthan0.008rad.3. Correlationfunctions
ThefemtoscopiccorrelationfunctionC
(
k∗)
,asafunctionofthe pionandkaonrelativethree-momentak∗=
12(
p∗π−
p∗K)
inthepair restframe(PRF)indicatedwiththeasterisk,isconstructedasC
(
k∗)
=
N
A(
k∗
)
B
(
k∗)
,
(1)where A
(
k∗)
is thedistributionconstructed fromthesameevent and B(
k∗)
is the reference distribution from particles belonging to differenteventsusingtheeventmixingmethod [37].The nor-malisation constantN
is used to ensure that the ratio reaches unityoutsidethemomentumrangewherethecorrelationfunction is affectedby finalstate interactions, i.e.0.
15<
k∗<
0.
20 GeV/
c,wherek∗
=
k∗.The averagetransversemomentum ofpionsand kaons belongingto pairs withk∗<
40 MeV/c is 0.
27 GeV/c (std. dev. 0.
07 GeV/c) and 0.
93 GeV/c (std. dev. 0.
23 GeV/c), respec-tively,independentofcentrality.Thefirstandsecondmomentsofthedistributionofthe spatio-temporalseparationofemissionpointsinthePRFcanbeobtained from correlation functionseither inthe three-dimensional Carte-sian representation [5] or using its decomposition into spherical harmonics (SH) [29,38]. The three-momentum and position dif-ferencescanbe projectedontothe out-side-longorthogonal axes, wherethelongaxisisthebeamaxis,theout axisisinthe direc-tionofthetransversepairvelocityinthelaboratorysystem,while the side axis is perpendicular to the long andout axes [39,40]. At midrapidity, theemission asymmetry – displacementbetween pionandkaonsources–canexistonlyintheoutdirection [28].In thiswork,theemissionasymmetryintheoutdirectionisobtained withtwodifferentmethodsandtheyareexplainedhereafter.
The SH decompositionallows oneto projectthe three-dimen-sionalinformationcontainedinthecorrelationfunctionintoaset of one-dimensional distributions. The method applied here uses thedirectdecompositionof A
(
k∗)
and B(
k∗)
during thefillingof thediscretedistributions [29].ThenumeratorcanbewrittenasA
(
k∗)
=
√
4π
∞ l=0 l m=0 Alm(
k∗)
Ylm(θ
∗,
ϕ
∗),
(2)where Ylm
(θ
∗,
ϕ
∗)
are the spherical harmonics and Aml(
k∗)
=
1 4π
4π A
(
k∗)
Ym
l ∗
(θ
∗,
ϕ
∗)
d∗.Asimilar definitionisvalidalsofor
thedenominator.Thel
<
3 termsfromtheinfinitesetof numera-toranddenominatordistributionsarefilledforeachreconstructed pairusingthecorrespondingYlm(θ
∗,
ϕ
∗)
weightforitsθ
∗ andϕ
∗angles.Fromtheseone-dimensionaldistributions,thecomponents ofthecorrelationfunctioncanbecalculatedfollowingthemethod introducedin [29].
The femtoscopic information relevant for the emission asym-metry measurementiscontainedintwoone-dimensional correla-tion functions, C00 and the real part of C11, where Cij is defined as Aij
/
Bij.The C00 andC11 functions are mostly sensitive to the source size andthe emissionasymmetry, respectively [29]. Addi-tionally, the values of C01 (asymmetry in the long direction) andC11 are checked for zero emission asymmetry. Their deviations from zeromayindicate track reconstruction problemsin the de-tector. Higher order componentsare smallandirrelevant forthis analysis.
The C0
0,
C11,andC11 componentsofthecorrelation function intheSHrepresentationareshowninFig.1forthedifferentpairs. Forlike-signpairs,the C00 correlationgoesbelowunityatlowk∗, reflecting therepulsive character ofthe mutualCoulomb interac-tion. Forunlike-signpairs,the effectisopposite (see alsoFig. 2). For theC1
1 correlation function,the deviationfrom unityis di-rectlyrelatedtotheemissionasymmetrybetweenthetwoparticle
Fig. 1. TheC0
0(toppanel),C 1
1 (middlepanel),and C11(bottompanel)SH com-ponentsofthechargedpion–kaonfemtoscopiccorrelationfunctionsforPb–Pb col-lisionsat√sNN=2.76TeVinthe5–10%centralityclass,positivefieldpolarity.The differentchargecombinationsofpionsandkaonsareshownwithdifferentcolours andmarkers.Thestatisticalandsystematicuncertaintiesareshownasverticalbars andboxes,respectively.
species. The
C1
1 should beflat by symmetryandthus is agood checkfordetectorandanalysisbiases.
For the Cartesian representation analysis, the reconstructed pairsweredividedintotwodifferentcorrelationfunctions,namely
C+
(
k∗)
andC−(
k∗)
,wherethesignreflectsthesignofk∗out.These correlation functionsrepresenttwo differentscenarios wherethe firstparticle(byconstructionthepion)isfasterorslowerthanthe second one (the kaon). Thedifference between themreflects the space-timeemissionasymmetry.Itcan be observedfromFig.2 that thecorrelation functionis notexactlyequal tounityatlargevaluesofk∗,buthassome in-trinsicslopemainlyduetothepresenceofellipticflow,resonance decays, and due to global conservation of energy and momen-tum.Thesebackground correlationshaveto be subtracted before fittingthecorrelationfunctionsinboththeSHandCartesian rep-resentations.Theproceduretoestimatethenon-femtoscopic back-groundis described in detail in [41], whereit is shownthat for
π
±K±pairsthenon-femtoscopicbaselinecanbeparameterisedby acommon6th orderpolynomialfunctionforallpaircombinations. Thesameapproachisusedtocorrecttheeffectofnon-femtoscopic backgroundin thepresent analysisandthe resultingbackground estimationisshowninFig.2asasolid blacklineforthe C00 and
C11 componentsofpion–kaonpairsofdifferentchargesign com-binations.4. Fittingofthecorrelationfunctions
Theexperimental correlationfunctionsinbothrepresentations arecomparedtotheoreticalfunctionscalculatedwiththesoftware packageCorrFit [42].Thesefunctionsarecalculatedas
Fig. 2. TheC0
0 (toppanel)andC 1
1 (bottom panel)componentsofthepion–kaon correlation functionsinthe 5–10%centrality classshowingthe non-femtoscopic backgroundinthe spherical-harmonicrepresentation,positive fieldpolarity. The backgroundfit corresponds toa6th order polynomial functioncommonfor all chargecombinations.Thetwostructuresvisibleinthecorrelationfunctionat0.11 GeV/c andat0.29GeV/c correspondtotheremainingeffectfromtrackmerging andtheK∗ resonance,respectively.Thestatisticalandsystematicuncertaintiesare shownasverticalbarsandboxes,respectively.
C
(
k∗)
=
S
(
r∗)
|
πK(
r∗,
k∗)
|
2d4r∗S
(
r∗)
d4r∗,
(3)wherethefour-vectorr∗
=
x∗π−
x∗K isthespace-timeposition dif-ference of a pionanda kaon, S(
r∗)
is thesource emission func-tion which is the probability of emitting a pair of particles at a given position difference. The possible dependenceof the source on k∗ has been neglected. This approximation has been proven forradiilargerthan1–2 fm [15].πKisthepion–kaonpairwave function.ItaccountsfortheCoulombandstrongfinal-state inter-actions (FSI), the former being dominant for the correlation ef-fect [28].
In order to be able to compare the resulting radii to those obtained fromidentical-particlefemtoscopy, we parameterise the source in the longitudinallycomoving coordinate system(LCMS), defined foreach pair such that the longitudinal pair momentum vanishes.Therelativetwo-particlesourcecanbeexpressedas
S
(
r)
∝
exp−
[
rout−
μ
out]
2 2R2out−
r2 side 2R2side−
r2long 2R2long,
(4)where Rout,Rside,and Rlongarethefemtoscopicradiiinthethree directionsand
μ
out istheemission asymmetry.Inorder toavoid a large set of fitting parameters, the relations Rside=
Rout andRlong
=
1.
3Rout areused,whicharebasedonmeasuredradiifrom identical pionfemtoscopyfromthe sameexperimental data [16]. Inthisapproach onlytwoindependent parametersare neededto characterise the correlation function for the whole system:μ
out andRout.Inorderto(numerically)computethefitfunction corre-spondingtoEq. (3),therelativepositionsbetweenpionsandkaonsaresampledfromEq. (4),whiletheir momentaaresampledfrom therespectiveexperimental distributions fromthesamedataset. The positions and momenta are then boosted fromthe LCMS to the PRF. The fitvalue isthe mean wave function squared inthe PRF.
Thefittingprocedure alsoaccountsforthe purityofthe sam-ple,defined asthe percentageofthe properly identified primary particlepairsoriginatingfromthe3DGaussianprofile,referredto asthe“Gaussiancore”.Productsofdecaysoflonglivedresonances areconsidered asnot correlated.Following themethod proposed in [7],thevaluesforthepurityparameterdependonthe misiden-tification,onthesecondarycontaminationfromweakdecays,and onthepercentageofpionsandkaonsthatcomefromstrongly de-cayingresonances constituting the long-rangetails in the source distribution,outsidetheGaussian core.These threepurityfactors aredenotedas p, f ,and g,respectively.The pairpurity(also re-ferred to asthe primary fraction) is evaluated independently for each centrality class and magnetic field polarity and is defined as:
Pπ±K±
=
pπ±·
pK±·
fπ±·
fK±·
g.
(5)All parameters except g are obtainedfrom a detailedsimulation ofthedetectorresponsecalculated using theHIJING MonteCarlo modelwith particle transport performed by GEANT3. The g
val-uesaretakenfromacalculationin [7] followingthemethodology usedin [28].Thetotalvalueofthepairpurityis0.73forthe0–5% centrality class and decreases smoothly to 0.61 for the 40–50% centralityclass.
The experimental finite momentum resolution has been in-corporated in the fitting procedure. The ideal three-momenta of 20 000randomlyselectedpairsfromanalyseddataperk∗ binused inthefittingroutineweresmearedbythemomentum-dependent experimentalmomentumandangularresolutions.Thesewere ob-tainedfromMonteCarlosimulations usinga detaileddescription oftheexperimentalset-up.
Each of the correlation functions obtained for the six event centralities, four charge combinations, and two polarities of the electric field have been fitted independently. The values of the radii and emission asymmetry are obtained using a
χ
2 minimi-sationin the Rout−
μ
out plane. The fitting isdone in the range 0<
k∗<
0.
1 GeV/
c usingtheCorrFitpackage [42].Afitexampleof theC00(
k∗)
andC11(
k∗)
partsofthecorrelationfunctionforπ
−K− andπ
−K+ isshowninFig.3.Notethatthepoorχ
2 valuesreflect theresidualdeviationsfromaGaussiandistribution,ratherthanan improperlyperformedfit.Thenon-Gaussianitycomesmainlyfrom combiningdifferent pair transverse momenta, representing three spatialdimensionsinaone-dimensional correlationfunction,and thepresence ofdaughtersofshort-lived (uptoω
) resonance de-cays.Thesystematicuncertaintiesareestimatedbyvaryingthe parti-cleidentificationandselectioncriteria, thenormalisationrangeof thecorrelationfunctions, thebackgroundfitrangeofthe polyno-mialthatisusedforestimationofnon-femtoscopiccontributions, thefit range,andthemomentum resolutionparameters usedfor smearing.Valuesofthesevariationsandtheirindividual contribu-tionstothesystematicuncertaintyaresummarisedinTable2.All thesystematicuncertaintiesare evaluatedindependentlyforeach centralityclass andthe maximum value isreported inthe table. Theprimarypairfractionsaretreatedseparately.Theyintroducea significantandcorrelatedsystematicerrorforallcentralities.
Thefinaluncertaintyisobtainedcombiningthesystematicand statisticaluncertainties using the covarianceellipses method.For eachoftheeightfitresults(paircombinationsandmagneticfield polarities)aswellasforeach systematicvariation,104 pointsare generated following a two-dimensional Gaussian distribution in
Fig. 3. TheC0
0(k∗)andC11(k∗)partsofthecorrelationfunctionfor(left)π−K−and(right)π−K+pairs,shownasmarkersforthe5–10%centrality,withthecorresponding fitscalculatedusingtheCorrFitpackageshownasdashedlines.Onlyhalfofthestatisticsisused,correspondingtoonemagneticfield(positivefieldpolarity).Thestatistical andsystematicuncertaintiesareshownasverticallinesandboxes,respectively.
Table 2
InputparameterstoCorrFitusedtofitthecorrelationfunctionsandvariationofrelevantparametersandrangesused fortheevaluationofthesystematicuncertaintiesofRoutandμout.Thefirstthreeuncertaintysourcesaffectthe cor-relationfunctionsandarevisualisedinFigs.1and2.Theuncertaintieswereestimatedforallthecentralityranges independentlyandmaximumvalueisreported.Thevariationofprimarypairfractionswasnotincludedinthe covari-anceellipsecalculationandisshownseparatelyasacorrelatedmodel-dependentsystematicuncertaintyindicatedwith a† symbol.UncertaintiesfromfitsusingonlyCoulombinteraction,indicatedwithsymbol‡,arenotincludedinthe fi-nalsystematicuncertainty.Therangesindicatedwith§symbolincludeexclusionof0.1–0.125GeV/c and0.265–0.315 GeV/c,toaccountforsplittingeffectsandK∗resonance.
Uncertaintysource Default value Variations max Rout(%) maxμout(%) PID Default in Table1 Loose and strict in Table1 3.0 12.0 Backgroundfitrange (k∗inGeV/c) 0.0–0.5§ 0.0–0.265§, 0.125–0.5§ 2.6 17.3 Normalisationrange (k∗inGeV/c) 0.15–0.2 0.1–0.12, 0.18–0.21 3.3 18.0
Fitrange(k∗inGeV/c) 0–0.1 0–0.08/0.12, 0.005–0.1 3.7 13.4 Momentumresolution Procedure from [30,31] +12% 3.6 10.3
Primaryfraction† In Sec.4 ±10% 15.0† 20.0†
Analysistype SH Cartesian coordinates 1.6 3.1
πK‡ Strong and Coulomb Coulomb only 33.0‡ 8.7‡
the Rout–
μ
out space, where the mean and covariance are taken fromthefit.Thecovarianceellipsesarecalculatedfromthesample of generatedpoints ineach centrality bin.The systematic uncer-taintiesusedforthefinal resultareobtainedusing1σ
covariance ellipses. Negligible correlation between Rout–μ
out parameters is observed.Additionally, the analysiswas done inthe Cartesian represen-tation [5] using the projected C+ and C− correlation functions shown inFig. 4. The results ofthis analysisare fully compatible with those from SH within uncertainties. However, these results are not incorporated asanother source ofsystematic uncertainty sincetheCartesianmethodyieldsthreetimeslargerstatistical un-certaintiesof
μ
out.Fits tocorrelationfunctionsconsideringonlyCoulomb interac-tionshowasystematicandcentrality-dependentdecreaseforRout of the order of 33% with a significantly increased
χ
2 of the fit. For this reason these are not included in the evaluation of the uncertainties. However, the effect on the asymmetry parameter,supportingthe predictionmadein [28],is about9%,in linewith othervariationsanddemonstratingtheprevalenceoftheCoulomb interactionfortheemissionasymmetrymeasurement.
5. Results
The final extractedradii, Rout, andemission asymmetry,
μ
out, arecalculatedasaweightedaveragesbetweenthevaluesobtained from the analysis of correlation functions corresponding to two magnetic field polarities and four possible charge combinations ofchargedpion–kaonpairs,usingthe SHrepresentation.The ob-tainedvalues are shown asa function of dNch/
dη
1/3 in Fig. 5. Theradiusincreasessmoothlyfrom4 fmto9 fmwhengoingfrom the40–50%centralityintervalto0–5%.Atthesametime,the emis-sionasymmetry evolvesfroma starting value ofμ
out= −
2.
5 fm andreachesμ
out= −
4 fmforthemostcentralevents.Inthesame figure,thepredictionspublishedin [28] areshownaslinesfor dif-ferenthypotheses of theextra delay for kaons, starting fromtheFig. 4. Pion–kaoncorrelationfunctionsintheCartesianrepresentationforallchargecombinations.TheC−isonthenegativesideofthek∗axeswhileC+isonthepositive. ThefemtoscopicfitsareshownasasolidblacklineandwerecomputedusingtheCorrFitpackage.Thestatisticalandsystematicuncertaintiesaresmallerthanthemarkers.
Fig. 5. Pion–kaonsourcesize(upperpanel)andemissionasymmetry(lowerpanel) forPb–Pbcollisionsat√sNN=2.76TeVasafunctionofdNch/dη1/3.Thesolid linesshow predictionsfrom calculationofsourcesize andemission asymmetry usingtheTHERMINATOR2modelwithdefaultandselectedvaluesofadditional delaywithameantimeof τandwidthσtforkaons [28].Thestatisticaland systematicuncertaintiesarecombinedandshownassquarebrackets.The uncer-taintyrelatedtothefractionofprimarypairsisreportedseparatelyasacorrelated model-dependentsystematicuncertaintyof±15%(20%).
default setting with no delay to a maximum of 3.2 fm
/
c extraemission time.Thisdelay reducestheasymmetryproduced natu-rallywhichoriginatesfromthecollectivebehaviourofthe expand-ingsystemcreatedinthecollisionsmodelledwithTHERMINATOR 2 [43].Theagreementbetweenthemeasuredandpredictedradiiis goodforperipheraleventsbutmeasurementsarelargerby1.5 fm forthemostcentralevents.Ontheotherhand,theemission asym-metry measurement follows thepredicted trends forall centrali-ties.Thedatapointsliebetweenthecurvescorrespondingtotime delaysof1.0and2.1fm
/
c.Themodel-dependentsystematicerrorsof15%and20%forthe radii and asymmetry, respectively, are present also in the
theo-retical prediction,asthesame valuesforthefraction ofparticles withintheGaussiancoreareusedtoobtaintheradiiandemission asymmetry [7]. Therefore, this additional systematic uncertainty wouldsynchronouslymovetheresultsupanddownandthe pre-dictionlineswithoutchangingtheirinterpretation.
6. Discussion
In thiswork the first femtoscopy analysis ofpion–kaon pairs at the LHC is presented. The collective behaviour of the matter createdinPb–Pb collisions generatesa naturalasymmetry inthe emissionofpionsandkaonsduetotheir differentmasses.Thisis relatedtothekaon emissiondistribution,which ismorestrongly influencedby flowthan pions [7]. Theanalysiswas implemented usingthesphericalharmonicsandtheCartesian representationof the femtoscopic correlation function. The non-femtoscopic back-groundpresentintherawratioswassubtractedusingacombined fitto thefourpossible charge combinations.The final resultsare compared to state-of-the-art hydrodynamical calculations where an additional delay for kaons was introduced to mimic the be-haviourduringthehadronrescatteringphase.
The radii values predicted by the theoretical calculation [28] have several assumptions included in the particle distributions whicharedifferentfromtheexperiment.Oneofthemisthatthe presence ofthe strong interaction does not modify the emission asymmetry visiblein the correlation functions. Ouranalysis con-firms this statement; removal of strong interaction from the fit hassignificantinfluenceontheradii(33%)butmoderateinfluence on the emission asymmetry (9%). Even though pions and kaons havebeenselectedaccordingtoALICEacceptanceandmomentum ranges,theoptimisationofthepurityofthedatasamplemodified the transverse momentum distribution. This experimental effect biases the distributions towards lower momentum values,hence itincreasesthesourceradii.
Theobtainedwidthoftherelativepion–kaonsource,Rout,can be compared to the pion and kaon source radii extracted from identical-particle correlation analyses added in quadrature. The pion–kaon pairs used in the current analysis are predominantly composed of soft pions (0
.
2≤
mT≤
0.
3 GeV/c) and hard kaons (1.
0≤
mT≤
1.
3 GeV/c).ThepionandkaonsourceradiimeasuredTable 3
Centrality-averaged difference between the μout predicted using THERMINA-TOR with different values of the added kaon delay τ [28] and the one mea-suredinthisanalysis,dividedbythetotal uncertaintyofthemeasurementσexp.
τ (μTHERM out −μ exp out)/σexp no delay −3.62 1.0 fm/c −1.02 2.1 fm/c 2.15 3.2 fm/c 5.26
fortheserangesoftransversemass(mT)in0–10%centralcollisions were7–8.5 fmand4–5 fm,respectively [18].Addedinquadrature, thisyields8–10 fm,wellinagreementwiththemostcentralpion– kaonpointinFig.5.Similarly,for30–50%centralityclass,thepion andkaonsourcesare4–4.5 fmand2–3 fm,respectively,andtheir combinationyields4.5–5.5 fm,againinreasonableagreementwith theaverageoftwomostperipheralintervalsinFig.5.
The emission asymmetry presented here coincides with the predictions calculated includinga delay of the kaon emission of 1.0–2.1fm
/
c.Thedifference betweentheμ
out valuespredictedin Ref. [28] andthemeasuredvalue,averagedovercentralityand nor-malisedto thetotaluncertaintyofourmeasurement,isshownin Table3.The values obtained for the emission asymmetry are in line with those predicted by the hydrokinetic model [19], the bro-ken mT scaling of the radii of kaons with respect to pions ob-served in [18],andfromthe short-lived resonances measuredby ALICE [23–25]. This measurement is anotherconfirmation of the hadronrescatteringphase.
In order to better understand the relevant effects influencing theemissionasymmetry,itwouldbenaturaltocontinuethe stud-ies measuringothersystems.It wouldbe especiallyinterestingto measurethe
π
pandKpsystemsandprobethevalidityofthe re-lationμ
πoutp=
μ
πoutK+
μ
Kp
out [7]. Final-stateinteractions such asthe onestakingplaceinalong-lastingrescatteringphasemight mod-ifyordistortthispicture.
In summary, the firstmeasurement ofthe emission asymme-try of pions and kaons for different centralities at the LHC has been performed. Rout was measured to be 9 fm forcentral col-lisionsanddecreasesasafunctionofcentralityto4.5 fmformore peripheralcollisions.Atthesametime,themagnitudeofthe emis-sion asymmetry changed from
μ
out= −
4.
5 fmtoμ
out= −
2 fm. This confirms the importance of the collective expansion of the system with the pions emitted closer to the centreof the colli-sionand/orlaterthankaons.However,thecollectivemotionisnot enough toreproducethetrendoftheemissionasymmetry which accordingto state-of-the-artmodels basedon3+1 viscous hydro-dynamicsdemandsanadditionaltimedelayof1–2fm/
c forkaons in orderto reproduce themeasured trend. Thisobservationis in agreement witha hydrodynamic evolution of theexpanding sys-temandfavorsastrongerradialflowincentralcollisionstogether witha dense andlong-lasting hadronicrescatteringphase at the endoftheevolutionofthefireballatLHCenergies.Declarationofcompetinginterest
Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.
Acknowledgements
The ALICE Collaboration would like to thank all its engineers andtechniciansfortheirinvaluablecontributionstothe
construc-tion of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collab-oration gratefully acknowledges the resources and support pro-videdbyallGridcentresandtheWorldwideLHCComputingGrid (WLCG) collaboration. The ALICE Collaboration acknowledges the followingfundingagencies fortheir support inbuildingand run-ningthe ALICE detector: A. I. AlikhanyanNational Science Labo-ratory (Yerevan Physics Institute) Foundation (ANSL), State Com-mitteeofScienceandWorldFederationofScientists(WFS), Arme-nia; Austrian Academy ofSciences, Austrian Science Fund (FWF): [M 2467-N36]andNationalstiftungfürForschung,Technologieund Entwicklung,Austria;MinistryofCommunicationsandHigh Tech-nologies, National Nuclear Research Center, Azerbaijan; Conselho Nacionalde Desenvolvimento Científico e Tecnológico (CNPq), Fi-nanciadorade Estudos e Projetos(Finep), Fundaçãode Amparoà Pesquisa doEstado de São Paulo (FAPESP)andUniversidade Fed-eraldoRioGrandedoSul(UFRGS),Brazil;MinistryofEducationof China (MOEC),MinistryofScience & Technologyof China(MSTC) andNational NaturalScience Foundation ofChina (NSFC),China; Ministry of Science and Education and Croatian Science Foun-dation, Croatia; Centro de Aplicaciones Tecnológicas yDesarrollo Nuclear (CEADEN),Cubaenergía,Cuba;The Ministryof Education, YouthandSports oftheCzechRepublic,CzechRepublic;The Dan-ishCouncilforIndependentResearchNaturalSciences,theVillum Fonden and Danish National Research Foundation (DNRF), Den-mark;Helsinki InstituteofPhysics (HIP),Finland; Commissariatà l’Énergie Atomique (CEA) and Institut National de Physique Nu-cléaire etdePhysique desParticules (IN2P3)andCentre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum fürSchwerionenforschungGmbH,Germany;GeneralSecretariatfor ResearchandTechnology,MinistryofEducation,Researchand Re-ligions,Greece;NationalResearchDevelopmentandInnovation Of-fice,Hungary;DepartmentofAtomicEnergy,GovernmentofIndia (DAE),DepartmentofScienceandTechnology,GovernmentofIndia (DST),University Grants Commission, Governmentof India(UGC) andCouncil ofScientificandIndustrial Research(CSIR),India; In-donesian Institute of Sciences, Indonesia; Centro Fermi - Museo StoricodellaFisicaeCentroStudieRicercheEnricoFermiand Isti-tutoNazionalediFisicaNucleare(INFN),Italy;Institutefor Innova-tiveScienceandTechnology,NagasakiInstitute ofAppliedScience (IIST),JapaneseMinistryofEducation,Culture,Sports,Scienceand Technology(MEXT)andJapanSocietyforthePromotionofScience (JSPS)KAKENHI, Japan; Consejo Nacional de Cienciay Tecnología (CONACYT),throughFondode CooperaciónInternacionalen Cien-cia y Tecnología (FONCICYT) and Dirección General de Asuntos delPersonalAcademico(DGAPA),Mexico;NederlandseOrganisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Re-search Council of Norway, Norway; Commission on Science and TechnologyforSustainableDevelopmentintheSouth(COMSATS), Pakistan;PontificiaUniversidadCatólicadelPerú,Peru;Ministryof Science andHigher Education,National Science Centre and WUT ID-UB,Poland;KoreaInstituteofScienceandTechnology Informa-tion and National Research Foundation of Korea (NRF), Republic of Korea;Ministry of Education and Scientific Research, Institute of Atomic Physics and Ministry of Research and Innovation and Institute of Atomic Physics, Romania; Joint Institute for Nuclear Research(JINR),MinistryofEducation andScienceoftheRussian Federation, National Research Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Basic Research, Russia;Ministry ofEducation, Science,Research andSportofthe Slovak Republic, Slovakia; National ResearchFoundation ofSouth Africa,SouthAfrica;SwedishResearchCouncil(VR)andKnut& Al-iceWallenbergFoundation(KAW),Sweden;EuropeanOrganization forNuclearResearch,Switzerland;SuranareeUniversityof Technol-ogy(SUT), NationalScience andTechnology DevelopmentAgency
(NSDTA) and Office of the Higher Education Commission under NRU projectofThailand,Thailand;TurkishAtomicEnergy Agency (TAEK),Turkey;NationalAcademyofSciencesofUkraine,Ukraine; ScienceandTechnologyFacilitiesCouncil(STFC),UnitedKingdom; NationalScienceFoundationoftheUnitedStatesofAmerica(NSF) andUnitedStatesDepartmentofEnergy,OfficeofNuclear Physics (DOENP),UnitedStatesofAmerica.
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