Study of the Λ–Λ interaction with femtoscopy correlations in pp and p–Pb collisions at the LHC

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Physics Letters B

www.elsevier.com/locate/physletb

Study of the – interaction with femtoscopy correlations in pp and p–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:

Received24May2019

Receivedinrevisedform26July2019 Accepted30July2019

Availableonline1August2019 Editor:L.Rolandi

Thiswork presentsnewconstraintsontheexistenceandthebindingenergyofapossible–bound state, the H-dibaryon, derived from –femtoscopic measurements by the ALICE collaboration. The resultsareobtainedfromanewmeasurementusingthefemtoscopytechniqueinpp collisionsat√

s= 13 TeVand p–Pb collisionsat√_s

NN=⁵.02 TeV, combined withpreviouslypublishedresults frompp collisionsat√_s=^{7 TeV.The}–scatteringparameterspace,spannedbytheinversescatteringlength f₀⁻¹ and the effectiveranged0,isconstrained by comparingthe measured–correlation function withcalculationsobtainedwithintheLednickýmodel.Thedataarecompatiblewithhypernucleiresults and latticecomputations,bothpredictingashallow attractiveinteraction,and permittotest different theoreticalapproaches describingthe–interaction.Theregioninthe(f₀⁻¹,d0) planewhichwould accommodatea–boundstateissubstantiallyrestrictedcomparedtopreviousstudies.Thebinding energyofthepossible–boundstateisestimatedwithinaneffective-rangeexpansionapproachand isfoundtobeB=³.2⁺₋¹₂^._.⁶₄(stat)⁺₋¹₁^._.⁸₀(syst) MeV.

©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introductionandphysicsmotivation

A detailed characterization of the – interaction is of fun- damentalinterestsinceitplays adecisiverole inthequantitative understanding of the hyperon(Y) appearance in dense neutron- richmatter,inproto-neutronandinneutronstars[1].Ifhyperons doappear atlarge densities andtheir fraction becomes sizeable, theY–Y interaction is expectedto play an importantrole in the equation ofstate of the system[2,3]. Evenif thehyperon densi- tiesin compact objects are negligible,the interplay betweenthe average separations and the – effective range determine the possible onset of phenomena such as fermion superfluidity, and henceinfluencethetransportpropertiesofthesystem[4–6].

Thecharacterizationofthe–interactionisstillanopenis- sue in experimental nuclear physics. The Nagara event, recently measured with the emulsion technique [7,8], reports a clear ev- idence fora double-hypernucleus ⁶He, witha small binding energy between the two s of B= ⁰.67±⁰.17 MeV. This value was obtainedby comparing thebinding energy ofthe two sinsidethedoublehypernucleus(B=⁶.91±⁰.16 MeV)with thebinding energy ofa single in a single-hypernucleus, how- ever, it might be influenced by three-body forces. Nevertheless, thisresultwas usedtoseta lowerlimit forthemassofthepre- dictedbutsofarnot observedH-dibaryon,apossibleboundstate composedofsixquarks(uuddss)[9].Severalexperimentalcollab- orations have been involved in the search for this state in the decaychannels H→p

π

andH→,in nuclear andelemen-

tary (e⁻e⁺) collisions, but no evidence has been found [10–12], even though an enhanced – production near threshold was measured byE224 andE522 atKEK-PS[13,14]. Theoreticalcalcu- lationsperformedwithinthechiralconstituentquarkmodelrelate theexistenceofaH-dibaryontoanoverbindingofthe⁶Hemea- suredintheNagaraevent [15,16].

Theoretical models constrained to the available nucleon–

nucleonandhyperon-nucleonexperimental data,assuming either asoft[15–17] orahard [18,19] repulsivecoreforthe–inter- action,predictdifferentscatteringlengths(f0)andeffectiveranges (d0). Throughoutthispaperthestandard signconvention infem- toscopy is used,accordingto which a positive f0 corresponds to an attractive interaction, while a negative scattering length cor- responds eitherto a repulsive potential (d0>|^f0|/2) or abound state (d0<|^f0|/2). It was reported that a smallvariation of the – repulsive core parametrization leads to inverse scattering lengthswithin −⁰.27 fm⁻¹< f₀⁻¹<4 fm⁻¹ andeffective ranges up to 16 fm [20]. Other calculations are directly constrained to the Nagara event and result in rather small scattering lengths andmoderateeffectiveranges,liketheFG(f₀⁻¹=¹.3 fm⁻¹;^d0= 6.59 fm) [21] and theHKMYY (f₀⁻¹= ¹.74 fm⁻¹;^d0= ⁶.45 fm) [22] models.Itisclearthatmoreexperimentaldataareneededto studytheprobleminamorequantitativeandmodel-independent way.

Analternativemethodtostudyhypernucleiistheinvestigation of momentum correlations of – pairs produced in hadron–

https://doi.org/10.1016/j.physletb.2019.134822

0370-2693/©^{2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

hadroncollisionsviathefemtoscopytechnique[23].TheSTARcol- laboration reported a – scatteringlength and effective range of f₀⁻¹= −⁰.91±⁰.31⁺₋⁰₀^._.⁰⁷₅₆ fm⁻¹ andd0= ⁸.52±².56⁺₋²₀^._.⁰⁹₇₄ fm, measuredinAu–Aucollisions at√

sNN=^{200 GeV [24].Theseval-} uescorrespondto arepulsive interaction; however,itwas shown thatthevaluesandthesignofthescatteringparametersstrongly depend on the treatment of feed-downcontributions fromweak decaystothemeasuredcorrelation.Are-analysisofthedataout- sidetheSTARcollaborationcametodifferentconclusions[20] and resultedinashallowattractiveinteraction.

In a pioneeringstudy [25], the – interaction was studied employing the femtoscopy technique in pp collisions at √

s=⁷ TeV.Thisstudydemonstratedthatthedataareconsistentwithei- ther a bound state or an attractive interaction, however, due to the small data sample no quantitative results were obtained. In thisletter,thesestudiesareextendedbyanalyzingfinal-statemo- mentum correlations in pp collisions at √

s=^{13 TeV and p–Pb} collisionsat√

sNN=⁵.02 TeV,recordedby ALICEduring LHCRun 2.Thesmallsystemsizeinpp and p–Pb givesrisetopronounced correlations fromstrong final-state interactions dueto the small relativedistanceatwhichparticlesareproduced.Hence,thelarge datasets enable a high-precision studyofthe –strong final- stateinteractionandprovidenewexperimentalconstraintsonthe scatteringparametersandtheexistenceofapossibleboundstate.

2. Dataanalysis

The analysis presented in this paper is based on the data samples collected by ALICE [26] during the Run 2 of the LHC (2015–2018) in pp collisions at √

s=^{13 TeV and} p–Pb collisions at√

s_NN=⁵.02 TeV,combinedwiththepreviouslyanalyzedRun1 datafrompp collisions at√

s=7 TeV [25].The eventandparticle candidateselectioncriteriafollowcloselytheprocedureappliedin theRun1analysis [25].

The events are triggered using two V0 detectors, which are small-angle plasticscintillator arraysplacedon eitherside ofthe collisionvertexatpseudorapidities2.8<

η

<5.1 and−³.7<

η

<

−¹.7 [27]. Minimum bias pp and p–Pb events are triggered by the requirement of coincidentsignals in both V0detectors, syn- chronouswiththe beamcrossingtime definedby the LHCclock.

The V0 detector is also used to reject background events stem- ming fromthe interaction ofbeamparticles withthe beampipe materialsorbeam-gasinteractions.Pile-upeventswithmorethan one collision per bunch crossing are rejected by evaluating the presenceofsecondaryeventvertices[27].Chargedparticlesarere- constructedbytheInner TrackingSystem(ITS) [26] andtheTime ProjectionChamber(TPC) [28],bothimmersedina0.5 Tsolenoidal magnetic field directed along the beam axis.A uniform detector coverage is assured by requiring the maximal deviation between thereconstructedprimaryvertex(PV)andthenominalinteraction pointtobesmallerthan10 cm.ThePVcanbereconstructedwith thecombinedinformationof theITSandTPC,andindependently withtheSiliconPixelDetector (SPD- oneof thethreesubdetec- torsoftheITS).Ifbothmethodsareavailable,thedifferenceofthe z-coordinatebetweenbothverticesisrequiredtobesmallerthan 5 mm.Afterapplyingtheseselection criteriatheremaining num- ber ofeventsis 1.0×^{109 forthe pp at} √

s=^{13 TeVsample and} 6.1×^{108 forp–Pb at}√

s_NN=⁵.02 TeV.Thiscorrespondstoabout 90%and84%ofallprocessedeventsinpp andp–Pb.

The–interactionisthemainfocusofthepresentstudy.As willbeexplainedinthenext section,thep–pcorrelationfunction isan essential inputforthe femtoscopicanalysisof–.There- fore, the reconstruction of both protons and particles will be described in the following paragraphs. To increase the statistical

significance ofthe result, the anti-particle pairs are measured as well.

The selection of the proton candidates follows the analysis strategy used forthe pp collisions at√

s=7 TeV [25]. The parti- cle identification(PID) is determined by thenumber of standard deviationsn

σ

betweenthehypothesisforaprotonandtheexper- imentalmeasurementofthespecificenergylossdE/dxintheTPC orthe timing informationfromtheTime-Of-Flight (TOF)detector [29]. Theanalyzedtracks areselectedwithin thekinematicrange 0.5<pT<4.05 GeV/c and |

η

_|<0.8. The PID is performedonly withtheTPCfortrackswithp<0.75 GeV/c,byrequiring|ⁿ

σ

|<3.

Tomaintain thepurityoftrackswith p>0.75 GeV/c,the|ⁿ

σ

|^is calculatedfromcombiningtheTPCandTOFinformation.Thecon- tributionofsecondaryparticles,whichstemfromelectromagnetic andweakdecaysorthedetectormaterial,areacontamination in thesignal.Thefractionsofprimary andsecondaryprotonsareex- tracted using Monte Carlo (MC) template fits to the distance of closest approachoftheparticlesto thePV [30].TheMC distribu- tions are generated usingPythia8.2 [31] forthe pp andDPMJET 3.0.5 [32] for the p–Pb case, filteredthrough the ALICEdetector andreconstructionalgorithm[26].Theprotonpurityinpp (p–Pb) isfoundtobe99(97)%withaprimaryfractionof85(86)%.

The particles are reconstructed via the decay →p

π

⁻, which hasabranching ratioof63.9%andc

τ

=⁷.89 cm [33].For the reconstruction of the the charge conjugate decay is em- ployed. The interaction rate of the LHC varied during different periods of the pp running. Tomaintain a constant purity that is independent of the interaction rate, in addition to the selection criteria usedfor the analysisof pp collisions at√

s=7 TeV [25], the charged decay tracks must either have a hit in one of the SPD or Silicon Strip Detector (SSD- ITS subdetector) layers or a matched TOF signal. After applyingall selection criteria thefinal andcandidatesareselectedina4 MeV/c²(∼³

σ

)masswin- dow aroundthenominalmass [33].Thefractions ofprimaryand secondaryparticlesareextractedsimilarlyastheprotons,while the observable for thetemplate fits is the cosine ofthe opening angle

α

betweenthemomentumandthevector pointingfrom thePVtothedecayvertex. Thepurityinpp (p–Pb)isfound tobe97(94)%withaprimaryfractionof59(50)%.Theexactcom- position ofsecondaries, aswell as the to ⁰ ratio,is fixed in the MC simulations, but is modeldependent. Therefore, the sys- tematicuncertaintiesincludea20% variationoftheratiosofthese contributions.

3. Analysisofthecorrelationfunction

The methodusedto investigatethe–interaction relieson particle pair correlations measured as a function of _k∗, defined asthe single-particlemomentum inthepairrestframe [23]. The observableofinterestC(^p1,^p2)isdefinedastheratiooftheprob- ability of measuring simultaneously two particles with momenta

p1 andp2,totheproductofthesingle-particleprobabilities:

C

(

^p

1

,

^p

2

) =

^P

(

^p

1

,

^p

2

)

P

(

^p

1

)

P

(

^p

2

) .

(1)

In the absence of correlations, the numerator factorizes and the correlationfunctionbecomesunity.Thefemtoscopyformalism [23]

relates the correlation function for a pair of particles, to their effective two-particleemitting source function S(r) andthe two- particlewavefunction(k^∗,^r):

C

(

k^∗

) =

S

(

r

) | (

_k

^∗

,

r

) |

²d³r

−−−−→

^k∗→∞ 1

,

(2)

Table 1

Theweightparameters(Eq. (4))λ^pp_i andλ^p_i^–Pboftheindividualcomponentsofthep–p,p–,p–⁻and–correlationfunctions.Thesub-indexesareusedtoindicate themotherparticleincaseoffeed-down.Onlythenon-flatfeed-down(residual)contributionsarelistedindividually,whileallothercontributionsarelistedas“flatresiduals (res.)”.Allmisidentified(fake)pairsareassumedtobeuncorrelated,thusresultinginaflatcorrelationsignal.

p–p p– p–⁻ –

Pair λ^pp_i

(%)

λ^p_i^–Pb (%)

Pair λ^pp_i

(%)

λ^p_i^–Pb (%)

Pair λ^pp_i

(%)

λ^p_i^–Pb (%)

Pair λ^pp_i

(%)

λ^p_i^–Pb (%)

pp 74.8 72.8 p 50.3 41.5 p⁻ 55.5 50.8 33.8 23.9

pp 15.1 16.1 p 0 16.8 13.8 p⁻₍₁₅₃₀₎− 8.8 8.1

p⁻ 8.3 12.1

flat res. 8.1 8.0 flat res. 20.4 24.9 flat res. 30.3 28.3 flat res. 59.8 64.0

fakes 2.0 3.1 fakes 4.2 7.7 fakes 5.4 12.8 fakes 6.4 12.1

whereristherelativedistancebetweenthepointsofemissionof thetwoparticles.ThisdefinitionofC(k^∗) assumesthat theemis- sion source is not dependent on k^∗, it is spherically symmetric andtheemissionofallparticles issimultaneous.TheEPOStrans- port model [34] predicts an emission source that does not fully satisfy the above assumptions. However, it was verified that the abovesimplificationsresultinverymilddeviationsinthecorrela- tionfunctions,whicharenegligibleforthepresentanalysis.

Forasphericalsymmetricpotentialtheangulardependenceof thewave-functionistriviallyintegratedout.Thus thedirectionof k^∗ becomes irrelevant on the left-hand side of Eq. (2). Particles withlarge relative momentaq^∗=^{2k∗ are not} correlated, leading toC(k^∗→ ∞)=^1.

The stronginteraction has a typical range ofa few femtome- tersandthusasignificantmodificationofthewavefunctionwith respecttoits asymptoticformisexpectedonlyforr^{2 fm.Con-} sequently,forsmallemissionsources thecorrelationfunctionwill be particularly sensitive to the strong interaction potential. Ex- perimentally, a small emission source can be formed in pp and p–Pb collisions[25,35].Inthecurrentanalysis, itisassumedthat theemission profile isGaussian andthat the p–p and–sys- temsarecharacterizedbyacommonsourcesizer₀=^rp–p=^r–, which is determined by fitting the p–p correlation function and thenusedfortheinvestigationofthe–interaction.Inpp colli- sionstheeffectofmini-jetsisonlypresentforbaryoncorrelations betweenparticleandanti-particle[25],hencetheinvestigateddata provideacleanenvironmenttoextractthefemtoscopicsignal.

Twodifferentframeworksareavailable forthe computationof C(k^∗).Thefirsttoolusedinthisanalysisisthe“CorrelationAnaly- sisToolusingtheSchrödingerequation”(CATS)[35].Here,alocal potentialV(r)isusedastheinputtoanumericalevaluationofthe wave function and the corresponding correlation function. CATS deliversan exactsolutionandthistool isusedto modelthep–p correlation using a Coulomb and an Argonne v18 potential [36]

forthe strong interaction.The known p–p interaction allows the sourcesizer0 tobeextractedfromthefittothemeasured corre- lationfunction.

Thesecond tool is theLednický model[37], which assumesa Gaussianemission sourceandevaluates thewave function inthe effective-rangeexpansion. In thisapproach,the interactionis pa- rameterizedintermsofthescatteringlength f0 andtheeffective ranged0. This approach produces a very accurate approximation forC(k^∗) in cased0^r0, while for smallervalues of r0 the ap- proximatesolutionmaybecomeunstable,inparticularfornegative valuesof f0 [25]. However, it isknown that the Lednickýmodel can be used to model the p– correlation function even for a sourcesizeofr₀=¹.2 fm,withadeviationfromtheexactsolution oflessthan 4% [35].It istherefore expectedthat thismodelcan successfullybeused tostudythe–interaction, eveninsmall collision systems.Nevertheless, thevalidity of the approximation willbefurtherverifiedinthenextsection.

Experimentally,thecorrelationfunctionisdefinedas C_exp

(

k^∗

) =

N ^Nsame

(

k^∗

)

N_mixed

(

k^∗

)

k^∗→∞

−−−−→

¹

,

(3) where Nsame(k^∗) and Nmixed(k^∗) are the same and mixed event distributions,while N ^isa normalizationconstant determinedby the condition that particlepairs withlarge relative momentaare notcorrelated.InsmallcollisionsystemsCexp(k^∗)oftenhasalong- rangetaildue tomomentum conservation,andarelatedapprox- imately linear non-femtoscopic background extending to low k^∗ [25]. The latter isincorporated by includinga linear termin the fitfunction.

Toincrease thestatisticalsignificance ofCexp(k^∗)the particle- particle (PP) and antiparticle-antiparticle (PP) correlations are combinedusingtheir weighted meanCexp(k^∗)=NPPCexp,PP(k^∗)⊕ NPPC_exp,PP(k^∗), with the normalization performed in the range 240<k^∗<340 MeV/c,whichisunaffectedbyfemtoscopiccorre- lations.ItwasverifiedthatNPPCexp,PP(k^∗)=NPPC_exp,PP(k^∗)within thestatisticaluncertainties.

The systematic uncertainties of the experimental correlation functionareevaluatedbyvaryingtheselectioncriteriaofthepro- ton and candidates within 20%, following the procedure used fortheanalysisofthepp collisionsat√

s=7 TeV [25].Neverthe- less,byperformingaBarlowtest [38],thesystematicuncertainties were found to be insignificant compared to thestatistical uncer- tainties.

Momentum resolution effects modify the correlation function byatmost10% andareaccountedforbycorrectingthetheoretical correlation function [25]. The measured experimental correlation functioncontainsnotonlythecorrelationsignalofinterest,butad- ditionallyaccumulatesresidualcontributionsfromfeed-downpar- ticles. These are considered in the theoretical description of the correlationby usingthelineardecompositionofthetotalcorrela- tionfunctioninto

C_tot

(

k^∗

) =

i

λ

iC_i

(

k^∗

),

(4)

wherethe sumrunsover all contributions, theλ parameters are theweight factorsforthedifferentcontributionstothe totalcor- relation and i =0 corresponds to the primary correlation. The λ coefficients are determined in a data-driven approach by per- forming Monte Carlo template fits to the data, using Pythia and DPMJET in pp and p–Pb collisions, respectively. The values ob- tainedaresummarizedinTable1.Thesystematicuncertaintiesare determined from the variation of the composition of secondary contributions, and the to ⁰ ratio. The individual contribu- tionsCi(k^∗)aremodeledeitherusingCATSortheLednickýmodel.

Thenon-genuine(i =⁰⁾contributionsincludeadditionalkinematic effects which lead to a smearing of their corresponding correla- tion functions [39]. As thecorrelation strength oftheseresiduals is strongly damped one can assume that C_{i =0}(k^∗)≈^{1 [40]. The}

Fig. 1.Resultsforthefitofthepp dataat√

s=^{13 TeV.Thep–p}correlationfunction(leftpanel)isfittedwithCATS(blueline)andthe–correlationfunction(right panel)isfittedwiththeLednickýmodel(yellowline).ThedashedlinerepresentsthelinearbaselinefromEq.(5),whilethedarkdashed-dottedlineontopofthe–data showstheexpectedcorrelationbasedonquantumstatisticsalone,incaseofastronginteractionpotentialcompatiblewithzero.

onlysignificant contribution is p–→^{p–p, where thep–}inter- actionismodeledusingthescatteringparametersfromanext-to- leading order(NLO)

χ

EFT calculation [41] and thecorresponding correlation function is computed using the Lednický model. The remaining residuals are considered flat,apart from p–⁻→^p–, p– ⁰→^p–andp–(1530)⁻→^p–−,wheretheinteractioncan bemodeled.Forthep–⁻ interactionarecentlatticeQCDpoten- tial,fromtheHALQCDcollaboration [42,43],isused.Thep– ⁰ is modeledasin [44],whilep–(1530)⁻isevaluatedbytakingonly theCoulombinteractionintoaccount.

Afterall correctionshavebeenappliedtoCtot(k^∗),thefinal fit functionis obtainedbymultiplying itwitha linearbaseline (a+ bk^∗)describingthenormalizationandnon-femtoscopybackground [25]

C_fit

(

k^∗

) = (

a

+

^bk∗

)

Ctot

(

k^∗

).

(5) Fig. 1 shows an example of the p–p and – correlation func- tions measured in pp collisions at √

s=^{13 TeV, together with} the fit functions. The p–p experimental data show a flat behav- ior intherange200<k^∗<400 MeV/c,thusbydefaulttheslope of the baseline is assumed to be zero (b=^{0) and the corre-} lation is fitted in the range k^∗<375 MeV/c. The resulting r0 values are 1.182±0.008(stat)⁺₋⁰₀^._.⁰⁰⁵₀₀₂(syst) fm in pp collisions at

√s=^{13 TeVand1}.427±⁰.007(stat)⁺₋⁰₀^._.⁰⁰¹₀₁₄(syst) fminp–Pb colli- sionsat√

sNN=⁵.02 TeV.Inpp collisionsat√

s=^{7 TeVthesource} sizeisr₀=¹.125±⁰.018(stat)⁺₋⁰₀^._.⁰⁵⁸₀₃₅(syst) fm [25].

Thesystematicuncertaintiesoftheradiusr0 areevaluatedfol- lowingtheprescriptionestablished during theanalysisofpp col- lisionsat√

s=7 TeV [25].Theupperlimitofthefitrangeforthe p–ppairsisvaried withink^∗∈ {³⁵⁰,375,400}^MeV/c andthein- puttotheλparametersismodified by20%,keepingprimary and secondaryfractionsconstant.

Two further systematic variations are performed for the p–p correlation. The first concerns the possible effect of non-femto- scopy contributions to the correlation functions, which can be modeled by a linear baseline (see Eq. (5)) with the inclusion of basafreefitparameter.Thefinalsystematicvariationistomodel thep–feed-downcontributionbyusingaleading-order(LO)[41, 45] computationtomodeltheinteraction.The effectofthelatter isnegligible,asthetransformationtothe p–psystemsmearsthe differencesobservedinthepurep–correlationfunctionout.

Toinvestigatethe–interactionthesourcesizesarefixedto the above results and the – correlations from all three data sets are fitted simultaneously in order to extract the scattering

parameters. The correlation functions show a slight non-flat be- havior atlarge k^∗,especiallyforthe pp collisionsat√

s=^{13 TeV} (rightpanelinFig.1).Thusthefitisperformedbyallowinganon- zeroslope parameterb (see Eq. (5)).The fitrangeisextended to k^∗<460 MeV/c in order to better constrain the linear baseline.

Duetothesmallprimaryλparameters(seeTable1)the–cor- relationsignalisquiteweak andthefitshowsa slightsystematic enhancement comparedto theexpected Ctot(k^∗)dueto quantum statisticsonly,suggestiveofanattractiveinteraction.However,the current statisticaluncertainties donot allow the –scattering parameters to be extractedfromthe fit.Therefore, an alternative approach to study the –interaction will be presented in the nextsection.Systematicuncertaintiesrelatedtothe–emission sourcemayarisefromseveraldifferenteffects,whicharediscussed intherestofthissection.

Previousstudieshaverevealedthattheemissionsourcecanbe elongated along some of the spatial directions andhave a mul- tiplicity or mT dependence [46,47]. In the present analysis it is assumed that the correlation function can be modeled by an ef- fective Gaussian source.The validity ofthis statement is verified by a simple toy MonteCarlo, inwhich a data-drivenmultiplicity dependenceisintroducedintothesourcefunctionandtheresult- ing theoreticalp–pcorrelation functioncomputedwithCATS. The deviationsbetweenthisresultandacorrelationfunctionobtained withaneffectiveGaussiansourceprofilearenegligible.

Possibledifferencesintheeffectiveemittingsourcesofp–pand –pairs duetothestrong decaysofbroadresonances andmT scaling are evaluated via simulations and estimated to have at mosta5% effectontheeffectivesourcesizer0.Thisistakeninto account by includingan additional systematicuncertaintyon the r– valueextractedfromthefittothep–pcorrelation.

4. Results

Inordertoextractthe–scatteringparameters,thecorrela- tionfunctionsmeasuredinpp collisionsat√

s=^{7,13 TeVaswell} asinp–Pb collisionsat√

sNN=⁵.02 TeVarefittedsimultaneously.

The rightpanel inFig.1showsthe–correlationfunctionob- tained in pp collisions at √

s=^{13 TeV together with the result} fromthefit.

Since the uncertainties of thescattering parameters are large, differentmodelpredictionsaretestedonthebasis oftheir agree- mentwiththemeasuredcorrelationfunctions.

One optionis to usea localpotential and obtain C(k^∗) based ontheexactsolutionfromCATS, withthesourcesizefixedtothe value obtainedfromthe fit to the p–p correlations. Manyof the

Fig. 2.–correlationsmeasuredinpp collisionsat√

s=^{13 TeV(leftpanel)and}p–Pb collisionsat√_s

NN=⁵.02 TeV(rightpanel)togetherwiththefunctionscomputed bythedifferentmodels [20].ThetestedpotentialsareconvertedtocorrelationfunctionsusingCATSandthebaselineisrefittedforeachmodel.Theeffectsofmomentum resolutionandresidualsareincludedinthetheorycurves.

existingmodelpredictionsaresummarizedin [20] andthecorre- spondingpotentials V(r)are parametrizedinalocalformusinga double-Gaussianfunction.Thecorrelationfunctiondependsonthe natureof the underlyinginteraction and Fig.2 showsthe exper- imental – correlations measured in pp collisions at √

s=¹³ TeV (left panel) and p–Pb collisions at √

sNN=⁵.02 TeV (right panel)togetherwiththecorrelation functionsobtainedfordiffer- entmeson-exchangeinteraction potentialsemployingCATS. Mod- els with a strongly attractive interaction (f₀^{−11 and positive),} liketheEhime[17] potential,resultinalargeenhancementofthe correlation function at low momenta which overshoots the data significantlybothinpp andp–Pb collisions.The sameisvalidfor potentialscorresponding to ashallow boundstate (f₀⁻¹→^{0 and} negative),e.g.NF44[19].

Theother testedpotentialscorrespond eithertoabound state orashallowattractive(f₀⁻¹¹⁾non-bindinginteraction.However, thosetwoverydifferentscenariosresultinsimilarcorrelationsand are difficult to separate. This isevident fromFig. 2 asall of the ESC08[48],HKMYY[22] andNijmegenND46[18] modelsproduce comparableresultsandarecompatiblewiththeexperimentaldata, eventhoughtheirscatteringparametersaredifferent.Inparticular, ND46predictsaboundstate,whiletheESC08andHKMYYmodels describeashallowattractivepotential andthelatterisconsistent withhypernucleidata[7,8].

TheLednickýmodelcanbeusedtocomputeC(k^∗)forany f₀⁻¹ andd0. Thus a scan over the scatteringparameters can be pre- formedandtheagreementtotheexperimentaldatacanbequan- tified.The Lednický model breaks down for source sizes smaller than the effective range, especially when dealing with repulsive interactions [25], as it produces unphysical negative correlation functions.As thereare norealistic models predicting such anin- teraction, thisstudy is not affected. Nevertheless,all models de- scribedin [20] are explicitlytested by comparingthe correlation functionsobtainedusingtheexactsolutionprovidedbyCATSwith theapproximatesolutionevaluatedusingtheLednickýmodel.The deviationsareonthepercentlevelandareneglected.

Anotherassumption,whichtheLednickýmodelisbasedon,is a Gaussian profile of the source. The EPOS [34] transportmodel predicts a non-Gaussian emission profile [35], andthe effects of short livedresonances are included. This source was adopted in CATS,by tuningitswidthsuch astodescribethep–p correlation function,andthepredictedC(k^∗)foralloftheNDandNFmodels, showninFig.3,were comparedto the–correlation function inpp collisionsat√

s=^{13 TeV.Thedeviationsin}

χ

²comparedto thecaseofaGaussiansourcearewithintheuncertainty,justifying theuseofaGaussiansource.

Fig. 3.Exclusionplotforthe–scatteringparametersobtainedusingthe– correlationsfrom pp collisionsat √

s=^{7 and13 TeVaswellas}p–Pb collisions at√

sNN=⁵.02 TeV.Thedifferentcolorsrepresenttheconfidencelevelofexclud- ingasetofparameters,giveninnσ.TheblackhashedregioniswheretheLednický modelproducesanunphysicalcorrelation.Thetwomodelsdenotedbycoloredstars arecompatiblewithhypernucleidata,whiletheredcrosscorrespondstothepre- liminaryresultofthelatticecomputationperformedbytheHALQCDcollaboration.

Fordetailsregardingthe regionat slightlynegative f₀⁻¹ andd0<4,compatible withaboundstate,refertoFig.4.

Toquantifythe uncertainties of f₀⁻¹ andd0,andestimate the confidence levelof each parameterset, a MonteCarlomethod is used. In the currentwork the approach described in [49] is fol- lowed,whichiscloselyrelatedtotheBootstrapmethod.Thestrat- egy istouse theLednickýmodelto performa scan overthepa- rameterspacespannedby f₀⁻¹∈ [−²,5] ^{fm−1 andd}0∈ [⁰,18] ^fm andrefitthe–correlationusingEq. (5) whenfixing thescat- teringparameterstoaspecificvalue(f₀⁻¹,d₀)i.Thecorresponding

χ

_i² is evaluatedby takingall datasets(pp at √

s=^{7 and13 TeV} andp–Pb at√

sNN=⁵.02 TeV)intoaccount.Thedifferentscatter- ingparameterscanbecomparedbyfindingthelowest(best)

χ

_best²

andevaluating

χ

_i²=

χ

_i²−

χ

_best² foreachparameterset.Thisob- servable, andthe associated (f₀⁻¹,d0)i, can be directly linked to the confidencelevel [49]. This can be achievedeither by assum- ing normallydistributed uncertainties of (f₀⁻¹,d0), orinvokinga moresophisticatedMonteCarlostudy,liketheBootstrapmethod.

Thelatterisusedinthecurrentanalysis.

The resulting exclusion plot ispresented in Fig. 3, where the color code corresponds to the confidencelevel n

σ

for a specific choice ofscatteringparameters.In thecomputation onlythe sta- tisticaluncertaintiesaretakenintoaccount,asthesystematicun- certainties are negligible according to the Barlow criterion [38].

Thepredictedscatteringparameters ofalldiscussedpotentialsare

Fig. 4.Theregionofthe1σconfidencelevelfromFig.3,displayedinthe(B,d0) plane.The inner (dark) regioncorresponds tothe statisticaluncertainty ofthe method,whiletheouter(light)regionincludesthesystematicvariations.Thered starcorrespondstotheparameterswiththelowestχ².

highlightedwithdifferentmarkers andthe phasespaceregionin which the Lednický model produces an unphysical correlation is specified by the black hatched area. In this region the effective rangeexpansionbreaks downandtheLednickýequation leadsto a negative correlation function. While theSTAR result [24] is lo- catedinthisregion,all theoreticalmodels excludethepossibility of a repulsive – interaction withlarge effective range.More- overare-analysisoftheSTARdata [20] demonstratedthatamore realistic treatment of the residual correlations leads to an inver- sion of the signof the scattering length, that corresponds to an attractivepotential. Theimposed limiton thescatteringlengthis f₀⁻¹>0.8 fm⁻¹ [20]. This result can be tested within the cur- rent work, and Fig. 3 demonstrates that the ALICE data can ex- tendthoseconstraints.Inparticulartheregioncorrespondingtoa strongly attractive or a very weakly binding short-range interac- tion (small |^f₀⁻¹| ^andsmalld0) isexcluded by the data, whilea shallowattractivepotential (large f₀⁻¹)isinverygoodagreement withtheexperimentalresultsobtainedfromthisanalysis.A– boundstate wouldcorrespond tonegative f₀⁻¹ andsmalld0 val- ues.Thepresentdataarecompatiblewithsuchascenario,butthe availablephasespaceisstronglyconstrained.TheHKMYY[22],FG [21] andHALQCD[50] valuesareofparticularinterest,asthefirst two models are tuned to describe the modern hypernuclei data, while the latter is the latest state-of-the-art lattice computation fromtheHALQCDcollaboration.Thelatticeresultsarepreliminary andpredictthescatteringparameters f₀⁻¹=¹.45±⁰.25 fm⁻¹ and d₀=⁵.16±⁰.82 fm[50].Allthreemodelsarecompatiblewiththe ALICEdata,providingfurthersupportforashallowattractive– interactionpotential.

Apossibleboundstateisinvestigatedwithintheeffective-range expansion by computing the corresponding binding energy from therelation[51,52]

B

=

¹ md²₀

1

−

1

+

^2d0f₀⁻¹

₂

.

(6)

Thisrelationisonlyvalidforboundstates,whicharecharacterized bynegative f₀⁻¹values.Further,thebindingenergyhastobeareal number,thustheexpression1+^2d0f₀⁻¹ hastobepositive,which impliesthat at leastone of the parameters f₀⁻¹ ord0 hasto be smallinabsolutevalue.WiththeserestrictionsEq. (6) transforms the observables in the exclusion plot (Fig. 3) from (f₀⁻¹,d0) to (B,d₀),considering only theparameter spacecompatiblewith aboundstate.ThisisdoneinFig.4,whereonlythe1

σ

confidence regionisshown,asitcorresponds totheuncertaintyof B.The darkregionmarksthestatisticaluncertaintyofthefit.Theallowed

binding energy, independent of d0, is B=³.2⁺₋¹₂^._.⁶₄(stat) MeV, where the central value corresponds to the lowest

χ

² and the uncertainties aredetermined basedonthelowest andhighestal- lowed B valueswithinthe 1

σ

confidenceregion. Howeverthe systematicuncertainties relatedto thesource sizes arenot taken into account, neither any possible biases related to the fit pro- cedure. Thus thecomputation ofthe exclusionplots (Figs. 3and 4) was repeated121 times,wherein eachre-iteration thesource sizes relatedtothedatasets arevariedwithin theassociatedun- certainties, the fit ranges within k^∗∈ {⁴²⁰,460,500} ^MeV/c and the bin widths of the experimental correlations are chosen as 12, 16 and 20 MeV/c. The resulting fluctuationsof the 1

σ

con- fidence region are marked in Fig. 4by the light region and rep- resent thetotal uncertainty. Assuming the latteris the quadratic sumofthestatisticalandsystematicuncertainty,thefinalresultis B=³.2⁺₋¹₂^._.⁶₄(stat)⁺₋¹₁^._.⁸₀(syst) MeV.

5. Summary

In this Letter, new data on p–p and – correlations in pp collisions at√

s=^{13 TeV and}p–Pb collisionsat√

sNN=⁵.02 TeV are presented.Together withthe resultsfroma pioneeringstudy ontwo-baryoncorrelationsinpp at√

s=^{7 TeV,thesedataallow} for a detailed studyof the –interaction with unprecedented precision.

Eachdatasetisanalyzedseparatelyby extractingthep–pand – correlation functions. The formerare used to constrain the size of the source r₀, which is assumedto be the same for p–p and–pairs.The–interactionistheninvestigatedby test- ing thecombinedcompatibilityofalldatasetstodifferentmodel predictionsandscatteringparameters.TheHKMYYandFGmodels, which are tuned to hypernuclei data,and thelattice calculations performed by the HAL QCD collaboration predict a shallow at- tractive interaction potential. The ALICE data manifest very good agreement with these predictions. Nevertheless, the data is also compatible with the existence of a bound state, givena binding energyofB=³.2⁺₋¹₂^._.⁶₄(stat)⁺₋¹₁^._.⁸₀(syst) MeV.TheRun3oftheLHC is expected to further increase the statistical significance of the –correlation function andallow the scatteringparameters to beconstraintevenmorepreciselyinthefuture.

Acknowledgements

TheALICEcollaborationisgratefultotheHALQCDcollaboration forprovidinglatticeresultsregardingthe–interaction.Weare particularlythankfultoProf.TetsuoHatsudaandProf.KenjiSasaki fortheprecioussuggestionsandstimulatingdiscussions.

The ALICECollaboration would like to thank all its engineers andtechniciansfortheirinvaluablecontributionstotheconstruc- 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 following fundingagenciesfortheir supportin buildingandrun- ningtheALICEdetector:A.I.AlikhanyanNationalScienceLabora- tory(YerevanPhysicsInstitute)Foundation (ANSL),State Commit- teeofScienceandWorldFederationofScientists(WFS),Armenia;

Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung,Austria;MinistryofCommunicationsandHighTech- nologies, National Nuclear Research Center, Azerbaijan; Conselho NacionaldeDesenvolvimentoCientíficoeTecnológico(CNPq),Uni- versidade Federaldo RioGrandedo Sul(UFRGS),Financiadorade EstudoseProjetos(Finep)andFundaçãodeAmparoàPesquisado