Measuring K0SK± interactions using pp collisions at √s=7 TeV

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Physics Letters B

www.elsevier.com/locate/physletb

Measuring K ⁰ _S K ^± interactions using pp collisions at √

s = ^{7 TeV}

.ALICE Collaboration

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

Articlehistory:

Received27September2018

Receivedinrevisedform28November2018 Accepted13December2018

Availableonline18December2018 Editor:L.Rolandi

We presentthefirstmeasurements offemtoscopiccorrelationsbetweentheK⁰_S and K^±particlesinpp collisions at √_s

=^{7 TeV measured by the ALICE} experiment. The observed femtoscopic correlations are consistentwith final-stateinteractionsproceeding solelyviathe a0(980) resonance.Theextracted kaon source radius and correlation strength parameters for K⁰_SK⁻ are found to be equal within the experimental uncertainties to those forK⁰_SK⁺.Results ofthe present study are comparedwith those fromidentical-kaonfemtoscopicstudiesalsoperformedwithppcollisionsat√_s

=^{7 TeVbyALICEand} withaK⁰_SK^±measurementinPb–Pbcollisionsat√_s

NN=².76 TeV.CombinedwiththePb–Pbresults,our ppanalysisisfoundtobecompatiblewiththeinterpretationofthea0(980)havingatetraquarkstructure insteadofthatofadiquark.

©^{2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Recently, by using Pb–Pb collisions at √

sNN=².76 TeV, the ALICEexperiment [1] has published the first-everstudy ofK⁰_SK^± femtoscopy [2].K⁰_SK^±femtoscopydiffersfromidentical-kaonfem- toscopy,forwhichanumberofstudiesexistintheliterature [3–6], in that the only pair interaction expected is a final-state inter- action (FSI)through the a0(980) resonance. It was found in that Pb–Pb study that the FSI in K⁰_SK^± proceeds solely through the a₀(980)resonance,i.e.withnocompetingnon-resonant channels, andtheextractedkaonsourceparametersagreewithpublishedre- sultsfromidentical-kaonstudiesinPb–Pbcollisions.Theseresults werefoundtobecompatiblewiththeinterpretationofthea0res- onance asatetraquark state ratherthana diquark¹ state [2,7–9].

A recenttheoretical calculation has shown that the ALICEPb–Pb results can indeed be described by a model based on the four- quarkmodel [10].

TheargumentgiveninRef. [2] fora tetraquarka₀ beingcom- patiblewiththePb–PbK⁰_SK^± resultstatedabove isbasedontwo factors: 1) the kaon source geometry, and2) an empirical selec- tionrule(for thesake ofsimplicityofnotation, “a₀”willbeused fortheremainder ofthispapertorepresent“a0(980)”). Forfactor 1),theproductioncrosssection ofthea0 resonanceinareaction channel such as K⁰K⁻→^a−0 should depend on whether the a⁻₀ is composed of du or dssu quarks, the former requiring the an- nihilation of the ss pairand the latterbeing a direct transfer of

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

1 Notethattheterm“diquark”willbeusedinthispapertoindicateaqiqjquark pair.

the valence quarks from the kaons to the a⁻₀. Since the femto- scopicsizeofthe0–10%mostcentralPb–Pbcollisionismeasured tobe5–6fm,thelargegeometryinthesecollisionsisfavorablefor the directtransfer ofquarksto thea₀, whereasnot favorable for theannihilationofthestrangequarksduetotheshort-rangedna- tureof thestronginteraction. Forfactor2), thedirect transferof thevalencequarksfromthekaons tothea⁻₀ isfavoredsincethis is an “OZIsuperallowed” reaction [9].The OZIrule canbe stated as “an inhibition associated with the creation or annihilation of quark lines” [9]. Thus, the annihilation of the strange quarks is suppressedbytheOZIrule.Both ofthesefactorsfavortheforma- tionofatetraquarka₀ andsuppresstheformationofadiquarka₀. As a resultof this, if the a0 were a diquark, one would expect competingnon-resonantchannelspresentand/or noFSIatall,i.e.

free-streaming, ofthe kaonpairthus diluting thestrength ofthe a0 resonant FSI. The fact that this isnot seen to be the casein Pb–Pbcollisionsfavorsthetetraquarka₀ interpretation.

Thegeometryofthekaonsourceisseentobeanimportantfac- torintheargumentgivenabove,i.e.thelargekaonsourceseenin Pb–Pb collisionssuppressestheannihilationofthestrange quarks in the kaon pair and enhances the direct transfer of quarks to the a₀. It is interesting to speculate on the dependence of the strength of the a0 resonant FSI on the size of the kaon source, particularlyfora verysmallsourceofsize ∼^{1 fm that wouldbe} obtainedin pp collisions[4,5]. Fora kaonsource ofsize ∼^{1 fm,} the kaons in a produced kaon pair would be overlapping with each otheratthesource,thusgivingageometricenhancementof thestrange-quarkannihilationchannelthatcouldcompetewith,or evendominateover,theOZIrulesuppressionofquarkannihilation.

Thus we might expect that the tetraquark a0 resonant FSI could be diluted or completely suppressed by competingnon-resonant https://doi.org/10.1016/j.physletb.2018.12.033

0370-2693/©^{2018PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

annihilation channels that could open up, whereas a diquark a0 resonantFSI,whichwasnotseentobesuppressedbyeithergeom- etryortheOZIruleinPb–Pb,wouldnotbediluted.Afemtoscopic measurementofK⁰_SK^±correlationsinppcollisionsshouldbeable totestthisbydeterminingthestrengthofthea₀FSIbymeasuring thefemtoscopicλparameter. Inmoreconcreteterms,ifwe were tocomparetheλparametersextractedinK⁰_SK^±femtoscopicmea- surementsinppcollisionsandPb–Pbcollisions,foratetraquarka0 wewouldexpectλ_K0

SK^±(PbPb)> λ_K0

SK^±(pp)whereasforadiquarka0 we would expect λ_K0

SK^±(PbPb)∼λ_K0

SK^±(pp). An independent check couldalsobemadeby comparingλfromK⁰_SK^± femtoscopyinpp collisionswithλfromidentical-kaonfemtoscopyinppcollisionsin asimilarwayaswasdoneforPb–Pbcollisions [2].Sinceweexpect identical-kaoncorrelationstogosolelythroughquantumstatistics (and FSI for neutral kaons), our expectation for a tetraquark a0 wouldbeλK K(pp)> λ_K0

SK^±(pp)whereasforadiquarka0 wewould expectλK K(pp)∼λ_K0

SK^±(pp).

In this Letter, femtoscopic correlations with the particle pair combinationsK⁰_SK^± are studiedforthe first timein pp collisions at√

s=^{7 TeVbythe ALICE}experiment.The physics goalsofthe presentK⁰_SK^±femtoscopystudyarethefollowing:1)showtowhat extenttheFSIthrough thea0 resonancedescribesthecorrelation functions,2) studytheK⁰ andK⁰ sources tosee ifthereare dif- ferencesin the sourceparameters, 3) comparethe results ofthe extractedkaonsourceparametersfromthepresentstudywiththe publishedresultsfromPb–Pbcollisions andidenticalkaonresults from pp collisions, and 4) see if the results from this pp study arecompatiblewitha tetraquarka0 assuggestedfromthePb–Pb study.

2. Descriptionofexperimentanddataselection

The ALICEexperiment andits performance in the LHC Run1 (2009–2013) are described in Ref. [1] and Refs. [11,12], respec- tively.About 370×¹⁰⁶ minimum-bias 7 TeV pp collision events takenin 2010 were used in this analysis. Events were classified using the measured amplitudes in the V0 detectors, which con- sistoftwo arraysof scintillatorslocated along thebeamline and covering the full azimuth [13,14]. Charged particles were recon- structed and identified with the central barrel detectors located within a solenoid magnet with a field strength of B= ±⁰.5 T.

ChargedparticletrackingwasperformedusingtheTimeProjection Chamber(TPC) [15] and the Inner Tracking System(ITS) [1]. The ITSallowedforhighspatialresolutionindeterminingtheprimary (collision)vertex. Amomentum resolutionof lessthan10MeV/c was typically obtained for the charged tracks of interest in this analysis [16]. The primary vertexwas obtainedfromthe ITS, the positionofthe primary vertexbeingconstrainedalong the beam direction(the “z-position”) tobe within ±^{10 cmofthecenterof} theALICEdetector.Inadditiontothestandardtrackqualityselec- tions [16], theselectionsbasedon thequalityoftrackfittingand thenumber ofdetected tracking points in the TPCwere used to ensurethatonlywell-reconstructedtracksweretakenintheanal- ysis [11,15,16].

Particleidentification(PID)forreconstructedtrackswascarried outusing boththe TPCandtheTime-of-Flight(TOF) detectorsin thepseudorapidityrange|

η

|<0.8 [11,12].ForthePIDsignalfrom both detectors, a value was assigned to each track denoting the number of standard deviations between the measured track in- formationandcalculatedvalues (Nσ) [6,11,12,16]. ForTPCPID,a parametrizedBethe–Blochformulawas usedtocalculatethespe- cific energy loss dE/dx in the detector expected for a particle withagivenmassandmomentum.ForPIDwithTOF,theparticle

masswas usedtocalculatetheexpectedtime-of-flightasafunc- tionoftracklengthandmomentum.Thisprocedurewasrepeated forfour“particlespecieshypotheses”,i.e.electron,pion,kaonand proton,and,foreachhypothesis,adifferentNσ valuewasobtained perdetector.

2.1. Kaonselection

Themethods usedtoselectandidentifyindividual K⁰_S andK^± particles are the sameasthose used forthe ALICEK⁰_SK⁰_S [4] and K^±K^± [5] analyses from√

s=7 TeVppcollisions. Theseare now describedbelow.

2.1.1. K⁰_Sselection

The K⁰_S particles were reconstructed from the decay K⁰_S →

π

⁺

π

⁻,withthedaughter

π

⁺and

π

⁻tracksdetectedintheTPC, ITSandTOF detectors.The secondaryvertexfinderusedtolocate theneutralkaondecaysemployed the“on-the-fly”reconstruction method [16],whichrecalculatesthedaughtertrackmomentadur- ing the original tracking process under the assumption that the tracks came from a decay vertex instead of the primary vertex.

Pions with pT>0.15 GeV/c were accepted (since for lower pT track finding efficiency drops rapidly) and the distance of clos- est approach to the primary vertex (DCA) of the reconstructed K⁰_S was requiredto be lessthan 0.3cmin alldirections. The re- quired Nσ valuesforthe pions were Nσ^TPC<3 (forall momenta) and N^TOFσ <3 for p>0.8 GeV/c. An invariant mass distribution for the

π

⁺

π

⁻ pairs was produced and the K⁰_S was defined to be resulting from a pair that fell into the invariant mass range 0.480<mπ⁺π⁻<0.515 GeV/c², corresponding to ±⁴.7

σ

, where

σ

=³.7 MeV/c²isthewidthofaGaussianfittotheinvariantmass distribution.

2.1.2. K^±selection

Chargedkaontracksweredetected usingtheTPCandTOFde- tectors, andwere accepted ifthey were within the range0.14<

pT<1.2 GeV/c in order to obtain good PID. The determination ofthe momentaofthe trackswas performedusingtracks recon- structedwiththeTPConlyandconstrainedtotheprimaryvertex.

In order to reduce the number of secondary tracks (for instance the chargedparticles produced inthe detectormaterial,particles fromweakdecays,etc.),theprimarychargedkaontrackswerese- lectedbasedontheDCA,suchthattheDCAtransversetothebeam directionwaslessthan2.4cmandtheDCAalongthebeamdirec- tion was less than 3.2 cm. If the TOF signal were not available, the required Nσ valuesforthe chargedkaons were N^TPCσ <2 for p_T<0.5 GeV/c,andthe trackwasrejectedfor p_T>0.5 GeV/c.If the TOFsignal were also available and pT>0.5 GeV/c: N^TPCσ <2 andN^TOFσ <2 (0.5<pT<1.2 GeV/c).

TheK⁰_SK^±experimentalpairpuritywasestimatedfromaMonte Carlo(MC) studybasedon PYTHIA [17] simulationswiththe Pe- rugia2011 tune [18], and using GEANT3 [19] to model particle transportthroughtheALICEdetectors.Thepuritywasdetermined from the fraction of the reconstructed MC simulated pairs that were identified asknown K⁰_SK^± pairs fromPYTHIA.The pairpu- ritywas estimatedto be ∼^{83% forall kinematicregions studied} in thisanalysis. The single-particle puritiesfor K⁰_S andK^± parti- clesusedinthisanalysiswereestimatedtobe∼^92%and∼^91%, respectively.The uncertaintyincalculatingthepairpurityisesti- matedtobe±^1%.

3. Analysismethods

3.1. Experimentalcorrelationfunctions

ThisanalysisstudiesthemomentumcorrelationsofK⁰_SK^±pairs usingthetwo-particlecorrelationfunction,definedas

C

(

k^∗

) =

^A

(

k^∗

)

B

(

k^∗

) ,

(1)

where A(k^∗) isthemeasured distributionofpairsfromthesame event, B(k^∗) is the reference distribution of pairs from mixed events,andk^∗ isthemagnitudeofthemomentumofeachofthe particlesinthepairrestframe(PRF),

k^∗

=

(

s

−

^m2_K0

−

^m2_K±

)

²

−

^4m2_K0m²_K±

4s (2)

where

s

=

m²_K0

+

m²_K±

+

2E_K0E_K±

−

2

p_K0

·

p_K± (3) andm_K0 (E_K0)andm_K± (E_K±)aretherestmasses(total energies) oftheK⁰_S andK^±,respectively.

Thedenominator B(k^∗)wasformedby mixingK⁰_S andK^± par- ticlesfromeacheventwithK^± andK⁰_S particles,respectively,from tenother events,whereeach eventhasatleastboth aK^± anda K⁰_S [2]. The vertices of the mixedevents were constrained to be within2cmofeachotherinthez-direction.

Two-trackeffects, suchasthe mergingof tworeal tracksinto onereconstructedtrackandthesplittingofonerealtrackintotwo reconstructedtracks,isanimportantissueforfemtoscopicstudies.

Thisanalysisdealtwiththeseeffectsusingthefollowingmethod.

For each kaon pair, the distance between the K⁰_S pion daughter trackandthesame-chargedK^±trackwascalculatedatuptonine points throughouttheTPC(every 20cmfrom85cm to245cm) andthenaveraged.Comparingpairsfromthesameeventtothose frommixed events,one observes a splittingpeak foran average separationof<11 cm. Tocorrectforthis,thisanalysisdemanded thatthesame-charge particlesfromeachkaonpairmusthavean averageTPCseparationofatleast13cm.Mixed-eventtrackswere normalizedby subtracting theprimary vertexposition fromeach usedtrackpoint.

Correlationfunctions were createdseparately forthe two dif- ferent charge combinations,K⁰_SK⁺ and K⁰_SK⁻, andforthree over- lapping/non-exclusive pair transverse momentum kT = |^pT,1 +

p_T,2|/^{2 ranges: all k}T, k_T<0.85 and k_T>0.85 GeV/c, where kT=⁰.85 GeV/c is the location of the peak of the kT distribu- tion.ThemeankTvaluesforthesethreebinswere0.66,0.49and 1.17 GeV/c, respectively. The raw K⁰_SK⁺ correlation functions for thesethreebinscomparedwiththosegeneratedfromPYTHIAsim- ulations withthe Perugia2011 tune and usingGEANT3 to model particletransportthroughtheALICEdetectorsareshowninFig.1.

The PYTHIA correlation functions are normalized to the data in thevicinityofk^∗=⁰.8 GeV/c.TherawK⁰_SK⁻correlationfunctions lookverysimilartothese.ItisseenthatalthoughPYTHIAqualita- tivelydescribesthetrendsofthebaselineofthedata,itdoesnot describeitquantitatively suchthat itcouldbe usedtomodelthe baseline directly.Instead,forthepresentanalysisthestrategy for dealingwiththebaselinewastodescribeitwithseveralfunctional formstobefittedtotheexperimentalcorrelationfunctionsandto usePYTHIA to test the appropriatenessof the proposed baseline functionalforms.

Three functional forms for the baseline were tested with PYTHIA:quadratic,Gaussianandexponential,givenby

Fig. 1.RawK⁰_SK⁺ correlationfunctionsforthethreekT binscomparedwiththose from PYTHIA.Theerror barsarestatistical. Thescaleof C(k^∗)is arbitrary.The PYTHIA correlationfunctionsarenormalized tothedata inthe vicinityofk^∗= 0.8 GeV/c.

C_quadratic

(

k^∗

) =

^a

(

1

−

^bk∗

+

^ck∗2

)

(4)

CGaussian

(

k^∗

) =

a

(

1

+

bexp

(−

ck^∗2

))

(5)

Cexponential

(

k^∗

) =

^a

(

1

+

^bexp

( −

^ck∗

))

(6) where a,b andc are fit parameters. Fig.2 showsfits of Eq. (4), Eq. (5) and Eq. (6) to the PYTHIA correlation functions shown in Fig. 1 for the three kT ranges used in this analysis. As seen, all three functionalforms do reasonablywell inrepresenting the PYTHIA correlation functions. Thus, all three forms were used in fittingthe experimentalcorrelation function andthe differentre- sultsobtainedwillbeusedtoestimatethesystematicuncertainty due to the baseline estimation. Of course there are an infinite number offunctionsone could tryto representthe baseline,but atleast thethree thatwere chosen forthiswork are simpleand representativeofthreebasicfunctionalforms.

Correlationfunctionswere correctedformomentumresolution effectsusingPYTHIAcalculations.Theparticlemomentumresolu- tioninALICEfortherelativelylow-momentumtracksusedinthe present analysis was <10 MeV/c [1]. Two correlation functions were generatedwithPYTHIA:one intermsofthe generator-level k^∗ and one in terms of the simulated detector-level k^∗. Because

Fig. 2.Comparisons offitsofthree possiblebaseline functionalformswith the PYTHIAcorrelationfunctionsthatwereshowninFig.1.Fitsweremadeinthek^∗ range0–0.8 GeV/c.ThescaleofC(k^∗)isarbitrary.

PYTHIA does not incorporate final-state interactions, simulated femtoscopicweights were determined usinga 9th-order polyno- mial fitin k^∗ to theexperimental correlation function forthe kT range considered. When filling the same-event distributions, i.e.

A(k^∗) in Eq. (1), kaon pairs were individually weighted by this 9th-order fit according to their generator-level k^∗. Then, the ra- tioofthe “ideal”correlation function to the“measured”one (for eachk^∗ bin) was multipliedtothe datacorrelation functionsbe- forethe fitprocedure. Thiscorrection mostly affectedthe lowest k^∗bins,increasingtheextractedsourceparametersby∼^2%.

3.2.Final-stateinteractionmodel

Thefinal-stateinteraction modelused inthepresentpp colli- sionanalysisfollowsthesameprinciplesastheonesusedforthe ALICEPb–Pbcollisionanalysis [2].ThemeasuredK⁰_SK^± correlation functionswere fit withformulas that include a parameterization whichincorporatesstrongFSI. Itwas assumedthat theFSIarises intheK⁰_SK^±channelsduetothenear-thresholdresonance,a0.This parameterizationwas introduced by R. Lednickyand isbased on the model by R. Lednicky and V.L. Lyuboshitz [20,21] (see also Ref. [3] formoredetailsonthisparameterization).

Using an equal emission time approximation in the PRF [20], the elastic K⁰_SK^± transition is written as a stationary solution _−k∗(r^∗)ofthescatteringprobleminthePRF.Thequantityr^∗rep- resentstheemissionseparationofthepairinthePRF,andthe−^k∗ subscriptreferstoareversaloftimefromtheemissionprocess.At largedistancesthishastheasymptoticformofasuperpositionof anincomingplanewaveandanoutgoingsphericalwave,

_−k∗

(

r^∗

) =

^{e−ik∗·r∗}

+

^f

(

k^∗

)

^e

ik^∗r^∗

r^∗

,

(7)

where f(k^∗) is the s-wave K⁰K⁻ or K⁰K⁺ scattering amplitude whose contribution is the s-wave isovector a0 resonance (see Eq. (11)inRef. [3])and

f

(

k^∗

) = γ

_a

0→KK

m²_a0

−

^s

−

ⁱ

( γ

_a

0→KKk^∗

+ γ

_{a0→π η}k_{π η}

) .

(8)

In Eq. (8), ma₀ is the mass of the a0 resonance, and

γ

_a

0→^KK and

γ

a0→π η are the couplings of the a0 resonance to the K⁰K⁻ (or K⁰K⁺) and

π η

channels, respectively. Also, s=⁴(m²

K⁰+^k∗2)

Table 1

Thea0massandcouplingparameters,allinGeV/c²,usedinthepresentstudy.

Reference ma₀ γa₀→KK γa₀→π η

Achasov2 [7] 1.003 0.8365 0.4580

andkπ ηdenotesthemomentumintheseconddecaychannel(

π η

) (seeTable1).

The correlation function due to the FSI is then calculated by integrating_−k∗(r^∗)intheKoonin–Prattequation[22,23],

CFSI

(

_k^∗

) =

d³

r^∗S

(

r^∗

)

_−k∗

(

r^∗

)

²

,

(9)

where S(r^∗)isaone-dimensionalGaussiansourcefunctionofthe PRFrelativedistance^r∗withaGaussianwidthR oftheform

S

(

r^∗

) ∼

^{e−r∗2/(4R2)}

.

(10) Equation (9) can be integrated analytically for K⁰_SK^± correla- tionswithFSIfortheone-dimensionalcase,withtheresult CFSI

(

k^∗

) =

1

+ λ α

1 2

^f

(

k^∗

)

R

²

+

^2Rf

(

k^∗

)

√ π

R F1

(

2k^∗R

)

−

^If

(

k^∗

)

R F2

(

2k^∗R

) +

C

,

(11)

where F1

(

z

) ≡

√ π

e^−z2erfi

(

z

)

2z

;

^F2

(

z

) ≡

¹

−

^e−z2

z

.

(12)

Intheabove equations

α

isthefractionofK⁰_SK^± pairsthat come from the K⁰K⁻ or K⁰K⁺ system, set to 0.5 assuming symmetry in K⁰ andK⁰ production[3], R is theradius parameter fromthe spherical Gaussian sourcedistribution givenin Eq. (10), and λ is the correlation strength. The correlation strength is unity in the idealcaseofpure a0-resonant FSI,perfectPID,aperfectGaussian kaon sourceand theabsence oflong-lived resonances whichde- cayintokaons.ThetermC isa calculatedcorrection factorthat takesintoaccountthedeviationofthesphericalwave assumption usedinEq. (7) intheinnerregionoftheshort-rangepotential(see theAppendix inRef. [3]).Its effectontheextracted R andλpa- rametersistoincrease themby∼^{14%.Notethattheformofthe} FSIterminEq. (11) differsfromtheformoftheFSItermforK⁰_SK⁰_S correlations(Eq. (9)ofRef. [3])byafactorof1/2 duetothenon- identicalparticlesinK⁰_SK^±correlationsandthustheabsenceofthe requirementtosymmetrizethewavefunctiongiveninEq. (7).

AsseeninEq. (8),theK⁰K⁻orK⁰K⁺ s-wavescatteringampli- tudedependsonthea₀massanddecaycouplings.FromtheALICE Pb–Pb collision K⁰_SK^± study [2], it was found that sourceparam- etersextractedwiththe“Achasov2”parameters ofRef. [7] agreed bestwiththeidenticalkaonmeasurements,thusinthepresentpp collisionstudyonlythe Achasov2parameters areused.Thesepa- rameters areshown inTable1. Sincethe correction factorC is foundtomainlydependon

γ

_a0KK¯ [3],itisjudgedthatthesystem- aticuncertaintyonthecalculationofC isnegligible.

The experimental K⁰_SK^± correlation functions, calculated using Eq. (1),werefitwiththeexpression

C

(

k^∗

) =

^CFSI

(

k^∗

)

C_baseline

(

k^∗

),

(13)

whereCbaseline(k^∗)isEq. (4),Eq. (5) orEq. (6).

The fitting strategy used was to carry out a 5-parameter fit ofEq. (13) tothe K⁰_SK^± experimental correlation functionstoex- tractR,λ,a,b andc foreachofthesix(k_T range)–(chargestate)

Fig. 3.CorrelationfunctionsdividedbyoneofthebaselinefunctionswithfitsfromEq. (13) forK⁰_SK⁺andK⁰_SK⁻andk^∗fitrange(0.0–0.6GeV/c)forthethreekTbinsand thequadraticbaselinefunctionassumption.Statistical(lines)andthequadraticsumofthestatisticalandsystematic(boxes)uncertaintiesareshown.Fork^∗>0.05 GeV/c, thesystematicuncertaintiesbecomenegligibleandtheboxesarenolongershown.

Fig. 4.Sample raw correlation functions for K⁰_SK⁺showing the fitted quadratic baseline function, Eq. (4). Statistical uncertainties are shown. The scale ofC(k^∗)is arbitrary.

combinations.Foreach ofthesesixcombinations,thethreebase- linefunctional forms,and two k^∗ fit ranges,(0.0–0.6 GeV/c) and (0.0–0.8 GeV/c), werefit,givingsixR andsixλ parametervalues foreachcombination.Thesesixvalueswerethenaveragedandthe variance calculated to obtain the final values for the parameters andanestimateofthecombinedsystematicuncertaintiesfromthe baselineassumptionsandfitrange,respectively.

4. Resultsanddiscussion

4.1. Fitstotheexperimentalcorrelationfunctions

Fig. 3 shows sample correlation functions divided by the quadraticbaselinefunctionwithfitsofEq. (13) forK⁰_SK^± andthe k^∗fitrange(0.0–0.6 GeV/c)forthethreekT bins.Thefitsusingthe otherbaselineassumptionsandtothewiderrange(0.0–0.8 GeV/c) aresimilar inquality. Comparingwiththequadraticbaseline, us- ing theGaussian baseline tends togive ∼^{10–20% smallersource} parameters whereas usingthe exponential baseline tends to give

∼^{10–20% larger source} parameters. The average

χ

²/ndf and p-

value over all ofthe fitsare 1.554 and0.172, respectively. Statis- tical (lines)and the quadraticsum ofthe statistical andsystem- atic(boxes) uncertaintiesare shown.The systematicuncertainties were determined by varying cuts on the data (see the discus- sion of the “cut systematicuncertainty” in the section below on

“Systematic Uncertainties”formore details). Fig. 4showssample raw correlationfunctionsfor K⁰_SK⁺ forthe threek_T bins andthe quadratic baselinefunction, Eq. (4), that was fitcorresponding to the 5-parameter fits of Eq. (13) to the K⁰_SK⁺ data presented in Fig.3.Statisticaluncertaintiesonthefitparameterswereobtained byconstructingthe1

σ

λvs. Rcontourandtakingtheerrorstobe attheextremeextentsofthecontour.Atypicalvalueofthecorre- lationcoefficientis0.642.Thismethodgivesthemostconservative estimatesofthestatisticaluncertainties.

The Achasov2 a₀ FSI parameterization coupled with the vari- ousbaselineassumptionsgivesagoodrepresentationofthesignal regionofthedata,i.e.reproducingtheenhancementinthek^∗ re- gion0.0–0.1 GeV/c andthesmalldipintheregion0.1–0.3 GeV/c.

A good representation of the signal region was also seen to be the case for the Pb–Pb analysis with the Achasov2 parameteri-

Table 2

FitresultsforaverageRandλshowingstatisticalandsystematicuncertaintiesfromK⁰_SK^±femtoscopywithppcollisionsat√

s=^{7 TeV.The“}[+/−]^{”inthefirstcolumn} referstoK⁰_SK⁺orK⁰_SK⁻.Seethetextforthedefinitionsofthevariousuncertainties.

Rorλ[+/−] kTcut (GeV/c)

Fit value

Statistical uncert.

Fit systematic uncert.

Cut systematic uncert.

Total systematic uncert.

Total quadratic uncert.

R[+]^{(fm) k}T<0.85 0.905 0.063 0.243 0.033 0.245 0.253

kT>0.85 0.788 0.077 0.168 0.031 0.171 0.188

AllkT 0.922 0.048 0.188 0.038 0.192 0.198

λ[+] kT<0.85 0.189 0.046 0.070 0.012 0.071 0.085

kT>0.85 0.222 0.080 0.066 0.015 0.068 0.105

AllkT 0.242 0.046 0.066 0.020 0.069 0.083

R[−]^{(fm) kT}<0.85 1.039 0.060 0.244 0.039 0.247 0.254

kT>0.85 0.786 0.082 0.145 0.032 0.148 0.169

AllkT 0.995 0.046 0.185 0.041 0.190 0.195

λ[−] ^kT<0.85 0.253 0.044 0.096 0.016 0.097 0.107

kT>0.85 0.208 0.084 0.038 0.016 0.042 0.094

AllkT 0.277 0.038 0.074 0.023 0.078 0.087

zation, which has a qualitatively different k^∗ dependence of the correlationfunctionthatisdominatedbyadipatlowk^∗ (compare presentFig.3withFig. 2fromRef. [2]).Theenhancementseenfor thesmall-R systematlowk^∗ isexpectedfromEq. (11) asacon- sequenceofthefirstterminthebracketsthat goesas1/R².This demonstratestheabilityofEq. (11) todescribetheFSIinboththe smallandlargesizeregimesasgoingthroughthea0resonance.

4.2.ExtractedR andλparameters

Theresultsforthe extractedaverage R andλ parameters and thestatisticalandsystematicuncertaintiesontheseforthepresent analysisofK⁰_SK^± femtoscopyfrom7TeV pp collisions areshown inTable2.Thestatisticaluncertaintiesgivenaretheaveragesover the6fits foreachcase. As canbe seen, R andλ forK⁰_SK⁺ agree withinthestatisticaluncertaintieswiththoseforK⁰_SK⁻inallcases.

4.3.Systematicuncertainties

Table 2 shows the total systematic uncertainties on the ex- tracted R and λ parameters. As is seen, formost casesthe total systematic uncertainty is larger than the statistical uncertainty.

The total systematic uncertainty is broken down in Table 2 into two main contributions, the “fit systematicuncertainty” and the

“cut systematic uncertainty”, andis the quadratic sum of these.

The fitsystematic uncertaintyis the combinedsystematic uncer- tainty due to the various baseline assumptions and varying the k^∗ fitrange,asdescribedearlier.Thecutsystematicuncertaintyis thesystematicuncertaintyrelatedtothevariouscutsmadeinthe dataanalysis.Todeterminethis,singleparticlecutswerevariedby

∼^{10%,andthevaluechosenfortheminimumseparationdistance} ofsame-sign tracks was varied by ∼^{20%. Taking the} upper-limit valuesof thevariations tobe conservative,this ledto additional errorsof4% for R and8% for λ.Asseeninthetable, thefitsys- tematicuncertaintydominatesoverthecutsystematicuncertainty inall cases,demonstratingthelarge uncertainties indetermining thenon-femtoscopicbaselineinppcollisions.The“totalquadratic uncertainty” is the quadratic sum of the “statistical uncertainty”

columnandthe“totalsystematicuncertainty”column.

4.4. ComparisonswithK⁰_SK^±resultsfromPb–Pbcollisionsat

√sNN=².76TeVandidentical-kaonresultsfromppcollisionsat

√s=^7TeV

Inthissection comparisonsofthepresentresultsfor R andλ withK⁰_SK^±measurementsfromALICE2.76 TeVPb–Pbcollisionsfor 0–10%centrality [2],andwithidentical-kaonmeasurements from ALICE7 TeV pp collisions [4,5] are presented.Since it is seen in Table 2that the extractedparameters forK⁰_SK⁺ agree within the statisticaluncertainties withthoseforK⁰_SK⁻inallcases,theseare averaged overweighted bythestatisticaluncertainties inthefol- lowingfigures.

Fig. 5 shows the comparison with the ALICE Pb–Pb collision K⁰_SK^± measurements.The λparameters havebeendividedby the pairpurityforeachcase,i.e.83%forthepresentppcollisionsand 88% for the Pb–Pb collisions [2], so that they can be compared onthesamebasis.Itisseenthat R for0–10%centralityPb–Pb is

∼^{5 fm,andis}significantlylargerthantheR∼^{1 fmmeasuredfor} pp collisions. Thisisexpectedsince R reflectsthe geometric size oftheinteractionregionofthecollision.Itissomewhatsurprising thatλforppcollisionsisseentobesignificantlylessthanthatfor Pb–Pbcollisions.Therearetwomainfactorseffectingthevalueof theλparameter:1)thedegreetowhichaGaussianfitsthecorre- lationfunction and2) theeffectoflong-livedresonancesdiluting thekaonsample. For1),itisseeninFig.3forppandinFig. 2of Ref. [2] forPb–PbthattheGaussianfunctionusedintheLedincky equation,Eqs. (10) and(11),fitsbothcollidingsystemswell,mini- mizingtheeffectof1).For2),theK^∗ decay(∼^{50 MeV)hasthe} largest influence on diluting the kaon sample, and it is unlikely that the multiplicity ratioof K/K^∗ changes dramaticallyin going from2.76 TeV to7 TeV. From theseargumentswe mightnaively expectλtobesimilarintheppandPb–Pbcases.

In order to properly compare the present results with the ALICE pp collision identical-kaon measurements, we must take the weighted average (weightedby their statistical uncertainties) over the multiplicity bins used in Refs. [4,5] since our present resultsaresummedoverallmultiplicity.Fig.6showsthecompar- ison betweenthepresentresultsfor R andλ andmeasurements fromtheidentical-kaon femtoscopyin7 TeVpp collisions.The R values are seen to agree between the present analysis and the identical kaon analyses within the uncertainties. The λ param- eters shown in Fig. 6 are each divided by their respective pair purities. Going fromthe lowest to the highestkT points, for the neutral-kaon pairs thepuritiesare 0.88and0.84 [4], andforthe

Fig. 5.RandλparametersextractedinthepresentanalysisfromK⁰_SK^±femtoscopy averagedoverK⁰_SK⁺andK⁰_SK⁻,alongwithacomparisonwithK⁰_SK^±resultsfrom ALICE2.76TeVPb–Pbcollisionsfor0–10%centrality [2].Thequadraticsumofthe statisticalandsystematicuncertaintiesisplottedfor allresultsasboxesandthe statisticaluncertaintiesaregivenaslines.Theλparametershavebeendividedby theirrespectivepairpuritiestofacilitatetheircomparison.

Fig. 6.Randpurity-normalizedλparametersextractedinthepresentanalysisfrom K⁰_SK^± femtoscopyaveraged overK⁰_SK⁺ and K⁰_SK⁻,along with comparisonswith identicalkaonresultsfromALICE7TeVppcollisionsaveragedovereventmultiplic- ity.Thequadraticsumofthestatisticalandsystematicuncertaintiesisplottedfor allresultsasboxesandthestatisticaluncertaintiesaregivenaslines.Alsoplotted asabluedashedlineisthesimpleaverageoftheidentical-kaonpurity-normalized λparameters.

charge-kaon pairs the purities are 0.84, 0.61, 0.79 and 1.0 [5], respectively. The purity-normalized λ parameters for the identi- cal kaons are seen to scatter in a wide range betweenvalues of

0.3–0.7,whereas the K⁰_SK^± valuesare seento liein thenarrower rangeof0.25–0.30.

Inordertohelpto clarifythecomparisonbetweenthepurity- normalizedλvaluesfromK⁰_SK^±andtheidentical-kaonresults,the simpleaverageovertheidenticalkaonpurity-normalizedλparam- etersisplottedasa bluedashedlineinFig.6.As seen,theK⁰_SK^± valuestendtobesmallerthantheaverageoftheidenticalkaons, as was more significantly the case for the comparison with the purity-normalizedλvaluesfromPb–PbseeninFig.5,howeverthe large scatterofthe identicalkaonsmakes itdifficult todrawany strongconclusionsfromthiscomparison.

4.5. Implicationsfromthepresentresultsforthea0tobeatetraquark state

The K⁰_SK^± FSI is described well by assuming it is due to the a0 resonancefor both pp and Pb–Pb collisions, asseen in Fig. 3 of the present work and in Fig. 2 of Ref. [2]. The R parameters extracted fromthismethod are also seen toagree within uncer- tainties withthe identical-kaon measurements for each of these collisionsystems.ForPb–Pbcollisions,itwasfoundthattheλpa- rameters extractedfromK⁰_SK^± alsoagree withthe corresponding identical-kaon measurements for Pb–Pb collisions indicating that the FSIbetweenthe kaonsgoes solelythrough thea₀ resonance.

Thepresentppcollisionresultsforλ,whicharesignificantlylower than the K⁰_SK^± values from Pb–Pb collisions seen in Fig. 5 and whichtendtobelowerthanthecorrespondingidentical-kaonval- ues in pp collisions seen in Fig. 6, imply that the FSI for these collisions does not go solely through the a0 resonance, i.e. non- resonant elastic channels and/or free-streaming are also present.

From the arguments given in the Introduction, this is the geo- metric effectthat wouldbe expectedin the caseof atetraquark a₀ since competing annihilation channels could open up in the smallersystemandcompetewiththeFSIthroughthea0,whereas foradiquarka0 theFSIshouldstill gosolelythroughthea0.The pp collisionresultsare thuscompatiblewiththeconclusionfrom thePb–Pbcollisionmeasurement [2] thatfavorstheinterpretation ofthea0 resonancetobeatetraquarkstate.

5. Summary

In summary, femtoscopic correlations with the particle pair combinationsK⁰_SK^± arestudiedinppcollisionsat√

s=^{7 TeVfor} the first time by the LHC ALICE experiment. Correlations in the K⁰_SK^±pairsareproducedbyfinal-stateinteractionswhichproceed through the a0 resonance. It is found that the a0 final-state in- teraction describes the shape of the measured K⁰_SK^± correlation functionswell.TheextractedradiusandλparametersforK⁰_SK⁻are found tobe equalwithin the experimentaluncertainties to those for K⁰_SK⁺. Results of the presentstudy are compared with those from identical-kaon femtoscopic studies also performed withpp collisions at √

s=^{7 TeVby ALICEandwitharecent ALICEK0}_S^K± measurementinPb–Pbcollisionsat√

sNN=².76 TeV.Thesecom- parisonssuggest thatnon-resonant elasticscatteringchannelsare presentinppcollisions,unlikeinPb–Pbcollisions.Itisourconclu- sionthatthepresentresults,incombinationwiththeALICEPb–Pb collision measurements,favortheinterpretationofthea₀ tobe a tetraquarkstate.

Acknowledgements

The ALICECollaboration would like to thank all its engineers andtechniciansfortheirinvaluablecontributionstotheconstruc- tionoftheexperimentandtheCERNacceleratorteamsfortheout- standingperformanceoftheLHCcomplex.TheALICECollaboration