Determination of $N^{\ast }$ amplitudes from associated strangeness production in p+p collisions

(1)

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Determination of N* amplitudes from associated strangeness production in p+p collisions

R. Münzer

^a^,^b^,^∗

, L. Fabbietti

^a^,^b^,^∗

, E. Epple

^c

, S. Lu

^b

, P. Klose

^b

, F. Hauenstein

^d

,

N. Herrmann

^e

, D. Grzonka

^d^,^f^,^g

, Y. Leifels

^h

, M. Maggiora

ⁱ

, D. Pleiner

^b

, B. Ramstein

^j

, J. Ritman

^d^,^f^,^g

, E. Roderburg

^d

, P. Salabura

^k

, A. Sarantsev

^l

, Z. Basrak

^m

, P. Buehler

ⁿ

, M. Cargnelli

ⁿ

, R. ˇCaplar

^m

, H. Clement

^o^,^p

, O. Czerwiakowa

^q

, I. Deppner

^e

, M. Dželalija

^r

, W. Eyrich

^s

, Z. Fodor

^t

, P. Gasik

^a^,^b

, I. Gašpari ´c

^m

, A. Gillitzer

^d^,^f^,^g

, Y. Grishkin

^u

,

O.N. Hartmann

^h

, K.D. Hildenbrand

^h

, B. Hong

^v

, T.I. Kang

^h^,^v

, J. Kecskemeti

^t

, Y.J. Kim

^h

, M. Kirejczyk

^q

, M. Kiš

^h

, P. Koczon

^h

, R. Kotte

^w

, A. Lebedev

^u

, A. Le Fèvre

^h

, J.L. Liu

^x

, V. Manko

^y

, J. Marton

ⁿ

, T. Matulewicz

^q

, K. Piasecki

^q

, F. Rami

^z

, A. Reischl

^e

, M.S. Ryu

^v

, P. Schmidt

ⁿ

, Z. Seres

^t

, B. Sikora

^q

, K.S. Sim

^v

, K. Siwek-Wilczy ´nska

^q

, V. Smolyankin

^u

, K. Suzuki

ⁿ

, Z. Tymi ´nski

^e

, P. Wagner

^z

, I. Weber

^r

, E. Widmann

ⁿ

, K. Wi´sniewski

^q

,

Z.G. Xiao

^aa

, T. Yamasaki

^ab^,^ac

, I. Yushmanov

^y

, P. Wintz

^d

, Y. Zhang

^ad

, A. Zhilin

^u

, V. Zinyuk

^e

, J. Zmeskal

ⁿ

aExcellenceClusterUniverse,TechnischeUniversität,München,Boltzmannstr.2,D-85748,Germany bPhysikDepartmentE62,TechnischeUniversitätMünchen,85748Garching,Germany

cYaleUniversity,NewHaven,CT,UnitedStates

dInstitutfürKernphysik,ForschungszentrumJülich,52428Jülich,Germany ePhysikalischesInstitutderUniversitätHeidelberg,Heidelberg,Germany

fJülichAachenResearchAlliance,ForcesandMatterExperiments(JARA-FAME),Germany gExperimentalphysikI,Ruhr-UniversitätBochum,44780Bochum,Germany

hGSIHelmholtzzentrumfürSchwerionenforschungGmbH,64291Darmstadt,Germany iIstitutoNazionalediFisicaNucleare(INFN)- SezionediTorino,10125Torino,Italy jInstitutdePhysiqueNucleaire,CNRS/IN2P3- Univ.ParisSud,F-91406OrsayCedex,France kSmoluchowskiInstituteofPhysics,JagiellonianUniversityofCracow,30-059Kraków,Poland lPetersburgNuclearPhysicsInstitute,Gatchina,Russia

mRu ¯drlaperBoškovi´cInstitute,Zagreb,Croatia

nStefan-Meyer-InstitutfürsubatomarePhysik,ÖsterreichischeAkademiederWissenschaften,Wien,Austria oPhysikalischesInstitutderUniversitätTübingen,AufderMorgenstelle14,72076Tübingen,Germany

pKeplerCenterforAstroandParticlePhysics,UniversityofTübingen,AufderMorgenstelle14,72076Tübingen,Germany qInstituteofExperimentalPhysics,FacultyofPhysics,UniversityofWarsaw,Warsaw,Poland

rFacultyofScience,UniversityofSplit,Split,Croatia

sFriedrich-Alexander-UniversitätErlangen-Nürnberg,91058Erlangen,Germany tWignerRCP,RMKI,Budapest,Hungary

uInstituteforTheoreticalandExperimentalPhysics,Moscow,Russia vKoreaUniversity,Seoul,RepublicofKorea

wInstitutfürStrahlenphysik,Helmholtz-ZentrumDresden-Rossendorf,Dresden,Germany xHarbinInstituteofTechnology,Harbin,China

yNationalResearchCentre‘KurchatovInstitute’,Moscow,Russia

zInstitutPluridisciplinaireHubertCurienandUniversitédeStrasbourg,Strasbourg,France aaDepartmentofPhysics,TsinghuaUniversity,Beijing,China

abDepartmentofPhysics,TheUniversityofTokyo,Tokyo,113-0033,Japan acRIKENNishinaCenter,RIKEN,Wako,351-0198,Japan

adInstituteofModernPhysics,ChineseAcademyofSciences,Lanzhou,China

*

Correspondingauthors.

E-mailaddresses:robert.muenzer@cern.ch(R. Münzer),laura.fabbietti@ph.tum.de(L. Fabbietti).

https://doi.org/10.1016/j.physletb.2018.08.068

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

(2)

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

Articlehistory:

Received12December2017 Receivedinrevisedform11July2018 Accepted28August2018

Availableonline10September2018 Editor:V.Metag

Keywords:

Partialwaveanalysis Resonance Hadrons Strangeness Scatteringlength

Hyperon–nucleoninteraction

Wepresentthefirstdeterminationoftheenergy-dependentamplitudesofN^∗resonancesextractedfrom their decay inK pairs in p+p→^pK⁺reactions. A combined Partial Wave Analysis of seven data samples with exclusively reconstructedp+p→^pK⁺ events measured by the COSY-TOF, DISTO, FOPI and HADESCollaborations infixedtarget experiments atkineticenergiesbetween2.14 to 3.5GeV is used to determine the amplitudeofthe resonant and non-resonant contributionsinto theassociated strangenessfinalstate.ThecontributionofsevenN^∗resonanceswithmassesbetween1650MeV/c²and 1900MeV/c²foranexcessenergybetween0 and600 MeVhasbeenconsidered.The–pcuspandfinal stateinteractionsforthep–channelarealsoincludedascoherentcontributionsinthePWA.TheN^∗ contributionisfoundtobedominantwithrespecttothephasespaceemissionofthepK⁺finalstate atallenergiesdemonstratingtheimportantroleplayedbybothN^∗andinterferenceeffectsinhadron–

hadroncollisions.

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

1. Introduction

Theproduction of strange hadronswithin nuclear matter is a keyingredientintheunderstandingoftheinnermoststructureof neutronstars(NS).Indeed,severaltheoretical modelspredictthat the production of strange hadronsis energeticallyfavourable al- readyatmoderatedensitiesofneutron-richmatter[1,2] andhence neutron stars with strange hadrons could appear. On the other hand,the appearance ofstrange hadronssoftens the equation of stateofNSexcludingtheexistence ofmassiveNSunlessastrong repulsive interaction is assumed for the NN system [3]. Since NSwithtwo solarmasseshavealreadybeenmeasuredwithhigh precision [4,5],thissituation translatesinto apuzzle that can be solved only studying hyperons andkaons production in hadron–

hadroncollisions. The bestenvironment tocarry out thiskindof studiesisprovidedbyhadron–hadroncollisionsatfewGeVkinetic energiesbecauseattheseenergieslargebaryonicdensities,similar tothosewithinNS,canbecreated.Ontheotherhandthereaction dynamicsattheseenergiesisdominated byhadronicresonances, thatneedthentobequantitativelyunderstood[6–13].

Forfinalstatescontaining pionsandnucleonsproducedinele- mentaryreactions, partial wave analysis (PWA) was already employed to correctly take into account interferences among resonances and determine the amplitude of the contributing waves [14–17]. For the contribution of resonances to final states with openstrangenessthereactionN^∗→^K+wasfirststudiedbyana- lyzingtheDalitzplotforthereactionp+p→^p+K⁺⁺uptokinetic energies of T=².5 GeV, butwithout accountingforinterference effects [13]. The HADES collaboration was the first to employ a PWAforthesearchforthekaonicboundstateppK⁻[18,19] inthe reaction p+p→^p+K⁺⁺ at a beam kinetic energy of 3.5 GeV. In thisreactionitwas found thatN^∗ contributeto themeasured fi- nalstateandinfluencethebackgroundforthekaonicboundstate [20,21].NoevidencefortheexistenceofppK⁻ boundstatescould befoundandupperlimitsfortheproductionofsuchstatesofthe orderofafew

μ

bwereextracted.Togetaconsistentdescriptionof theopen strangenessproduction,we furtherimprovethismethod anddevelop a framework that allows forthe simultaneous anal- ysisof seven different data sets measured in the p+p→^p+K⁺⁺ reactionbytheCOSY-TOF,HADES,DISTOandFOPIexperimentsin fixedtargetexperimentsatkineticenergiesinthelaboratoryframe varying from 2.14 to 3.5 GeV [20,22–27]. This is the very first jointPWAanalysisofdifferentdatasetsforthisreaction.Thisway, theenergy-dependentamplitudeofsevendifferentcontributingN^∗ resonancesdecayingintothe-K⁺ channel andfornon-resonant pK⁺finalstatescouldbeextractedforthefirsttime.

Asecondinterestingaspectisthestudyofthep–interaction.

Thisinteractionwaspreviouslyinvestigatedprimarilybymeansof scatteringexperiments[28–30].Thereactionp+p→^p+K⁺⁺offers thepossibilitytostudytheﬁnalstate interactionofthep–pair asanalternativetoscatteringexperiments[27,30–32].Sincesofar theresonances werenot treatedinacoherentway,a precisede- terminationoftheircontributionsandofthescatteringlengthsand effectiverangeswaschallenging.

The combined PWA presented in thiswork offers the unique possibilityto studythe interplaybetweentheN^∗ couplingto the

-K⁺channelandthep–ﬁnalstateinteraction.

2. Datasamplesandcombinedanalysis

TheexperimentaldataweremeasuredbytheCOSY-TOF,DISTO, FOPI andHADES Collaborations. Table 1 provides an overview of thedatasetsusedforthecombinedPWA,their beamenergyand numberofevents.Togetherwitheachexperimentaldataset,sim- ulations of the pK⁺ productionaccording to phase spacekine- matics, filtered through the detector simulation and analysed as theexperimentaldataareusedforthePWA.Thedetailsaboutthe reconstruction oftheexclusivepK⁺finalstate, achievedresolu- tion,efficiency,andpurityare explainedinthealreadypublished worksby thedifferentcollaborations [20,22–27]. The twoHADES data samples at the same kinetic energy correspond to two dif- ferentreconstruction analyses includingor excludingthe forward spectrometer[20].Thesedatasetsarecomplementaryanddonot shareanyreconstructed eventsbecauseofthe exclusiveselection ofthefinalstate.

The goalof thisPWA isto employ the seven datasamples in a combined analysis andextract the amplitudes of the different waves,characterisedby their quantumnumbers,leading to given final states. We use the Bonn-Gatchina PWA (BG-PWA) framework [15,16] to fit event-by-event the measured 4-momenta for the exclusivefinal state p+p→^p+K⁺⁺weighted withthecoher- ent superpositionofspecific participating waves.The bestchoice for thewaves used in thePWA isdetermined by comparingthe experimental datato thePWA output event-by-eventin termsof a log-likelihood parameter. In the specific case of the COSY-TOF datasample,onlytheregionofphasespacewithin|^cosθ_p^CM|<0.7, whereθ_p^CM istheprotonangleinthep–pcenterofmasssystem, was considered becauseofthe poordescription ofthetrigger ef- ficiencyin thesimulation forthe excluded region.Forthe DISTO datasamplesthe regioncorresponding tocosθ_p^CM>0.95 wasex- cludedfromthefittominimize thebiasintroduced bythedigiti- zationofthescintillation-fibersubdetectorusedfortrackingclose

(3)

Table 1

Listofavailablenumberofeventsforthereactionp+p→^p+^K⁺+ ^measured^by^the^COSY-TOF,^DISTO,^FOPI^and^HADESCollaborations.Thekineticbeamenergy,thetotal crosssectionandthereducedχ²valuesresultingfromdifferentPWAanalysesareshown(seetextfordetails).

Experiment T (GeV) Events/ndf σtot[μb] χ²/ndf (single) χ²/ndf (combined)

DISTO [22,23] 2.14 121000/644 19.0±³.3 0.52 1.52

COSY-TOF [25,27] 2.16 43662/712 19.7±3.5 1.69 0.44

DISTO [22,23] 2.5 304000/766 30.5±5.7 2.85 2.56

DISTO [22–24] 2.85 424000/555 38.7±7.9 7.68 3.55

FOPI [26] 3.1 903/226 43.1±⁹.3 1.21 0.91

HADES [20] 3.50 13155/528 48.0±¹¹.5 1.12 2.14

HADES [20] 3.50 8155/534 48.0±¹¹.5 1.38 1.86

tothetargetregion.Thesecutswerealsoaddedinthesimulations usedinthePWAanalysisprocedure.

This PWA allows to decompose the baryon-baryon scattering amplitude into separate sub-processes characterized by dif- ferentintermediate states.Withinthe BG-PWAframework thisis achievedby fittingevent-by-event theexperimental 4-vectors for agivenreactionmeasuredwithintheacceptanceofthespectrom- eterwithacoherentsuperpositionoftheparticipatingwaves.This coherentcocktail ofcontributing wavesisweighted withthe full scalephasespacesimulationsoftheconsideredfinalstatethatac- countsforthegeometricalacceptanceandreconstructionefficiency ofthespectrometer.

Within the BG-PWA, the production cross section of a three particle ﬁnal state with single particle four-momenta q₁_,₂_,₃ is parametrizedas[15]:

d

σ = (

2

π )

⁴

|

^A

|

² 4

|

k

| √

s d

3

(

P

,

q₁

,

q₂

,

q₃

) ,

(1) wherein P isthetotal four-vector,k is thebeammomentum,√

s thecenterofmassenergyofthereaction,d3 istheinfinitesimal phase-spacevolumeofthefinalstateand A isthetotaltransition amplitude oftheconsidered reaction.Both initial andfinalstates canbeseenasasuperpositionofeigenstateswithvariousangular momentum and A is the sumover all the transitionamplitudes Aα_tr betweentheseeigenstates[35]:

A

=

α

A^α_tr

(

s

)

Q_μⁱⁿ

1..μj

(

S

,

L

,

J

)

A_2b

(

i

,

S₂

,

L₂

,

J₂

)

Q_μ^ﬁn

1..μj

(

i

,

S₂

,

L₂

,

J₂

,

S

,

L

,

J

).

(2)

Theindex

α

runsoveralltheamplitudescontributingtothetran- sitionfromtheinitialtothefinalstate.Thefactors Qμⁱⁿ1..μj(S,L, J) and Qμ^fin1..μj(i,S₂,L₂,J₂,S,L,J) are thespin-momentum opera- torsoftheinitialandfinalstatesrespectivelyandtheindexes

μ

j refer to therank ofthe total angularmomentum J in thespin–

momentum operators Q . The index i refers to the two-particle sub-systemconsideredintheﬁnalstate.

The dependency of the amplitudes Aα_tr(s) upon the centre of massenergyisgivenby:

A^α_tr

(

s

) =

a^α₁

+

^a^α3

√

s

exp

ia^α₂

.

(3)

Therealparametersaα₁,aα₂ andaα₃ aredeterminedbytheﬁttothe experimentaldata.

The parametrization of the factor A2b depends on the final state. For the production of a N^∗ resonance, the final state is treatedasatwo-bodysystemcomposedofaprotonandtheN^∗.In thiscasethequantumnumbers S2,L2, J2 refers totheN^∗,while theS,L,J representthequantumnumbersoftheN^∗-protonsys- tem. Non resonant pK⁺ final states are also treated as a two particlesystemcomposedofap“particle”andaK⁺.Inthiscase S₂,L₂, J₂ are the spin, angular and total angularmomentum of

Table 2

N^∗resonancesincludedinthePWAwritteninthespectroscopicnotationwiththe correspondingmasses,widthsandbranchingratiosintheK-ﬁnalstates[33,34].

N^∗ J^P Mass (^GeV_c2 ) Width (^GeV_c2 ) K/tot(%)

1650 ¹₂⁻[33] 1.655 0.14 7±⁴

1710 ¹₂⁺[33] 1.710 0.23 15±¹⁰

1720 ³₂⁺[33] 1.720 0.25 4±¹

1875 ³₂⁻ 1.875 0.20 [33] 4±2 [34]

1880 ¹₂⁺ 1.870 0.24 [34] 2±^{1 [34]}

1895 ¹₂⁻ 1.895 0.09 [34] 18±5 [34]

1900 ³₂⁺ 1.900 0.26 [34] 11±^{9 [33]}

the p ‘particle’while S,L, J arethe quantum numbersof the p-K⁺system.

For the resonant case, the factor A2b is parametrized with a relativisticBreit–Wignerformula[36].

A^β_2b

=

¹

(

M²

−

^s

−

ⁱ

M

) ,

(4)

with M and asthepole massandwidth ofthecorresponding resonance.Forthepresentedanalysis,theN^∗ resonanceslistedin Table2havebeenconsideredwithﬁxed massesandﬁxed widths takenfrom[33,34].

Toobtainanacceptabledescriptionoftheexperimentaldatait isnecessarytoincludenon-resonantpartialwave amplitudes.We have includedtheseamplitudesin asimple formwhichprovides a correctbehaviournearthreshold.FortheS-wavethisformcor- respondstothewellknownWatson–Migdalparameterization.The resulting A_2b amplitudeis

A^β_2b

=

√

_s

i

1

−

¹₂^r^β^q²^a^β_p

+

^iqa^β_p^q^2L

/

F

q

,

r^β

,

L

,

⁽⁵⁾

whereq isthep–relativemomentum,a^β_p₋isthep–-scattering length, r^β is the effective rangeof the p– systemand the indexβ denotesthequantumnumberscombination.

F(q,r,L)istheBlatt–Weisskopffactorusedforthenormaliza- tion,itis1forL=^{0 and}^theêxplicit^form^forôther^partial^waves canbefoundin[15].Thevaluesofthescatteringlengthandeffec- tiverangecanbesetasfreeparametersinthePWAfitandhence be extracted within this analysis. This coherent approach differs from theanalysistechniques usually employed forthe extraction ofscatteringparameters[37] andshouldbeconsideredascomple- mentary.

AnotherintermediatechannelcontributingtopK⁺ﬁnalstate is the -N cusp, which appears at or above the -N threshold (2130MeV/c²)[38].Thecouplingbetweenthe-Nand-Nchan- nelsleadstoanenhancementofthecross-sectioninthep–ﬁnal state ina massrangecloseto theabovementioned threshold.In ordertoincludethecusp contributionintheBG-PWAframework,

(4)

Fig. 1. (Coloronline).Missingmassdistributions(MM)forthethreedifferentparticlesoftheﬁnalstate(p,,K⁺)areshown.Theexperimentaldatawithinthegeometrical acceptancearefromCOSY-TOFat2.16 GeV(bluesymbols),DISTOat2.85 GeV(greensymbols)andHADESat3.5 GeV(redsymbols)samples.Thecoloredlinesinthesame color-coderepresentthePWAresults(seetextfordetails).

newtransition wavesmust be addedto Eq.2. Since the cusp is locatedatthe -Nthreshold,the andNmustbe ina relative S-wavestate,whichmeansthatthespin-parityofthe-Nsystem iseither J^P=⁰⁺ôr¹⁺^[38].^The^resulting^p–systemthenmay appearinanS-wavestateincaseof J^P=⁰⁺ôrⁱⁿân^{s- or}^d-wave stateincaseof J^P=¹⁺^.^This^hasâlso^been^confirmed^byânânaly- sisofthe-NcuspcarriedoutbytheCOSY-TOFcollaboration[38].

Additionally,sincethecuspisaresonancestructureinanalogyto theN^∗, the Breit–Wigner parametrizationis used forA_2b (Eq. 4) where the mass and width are varied within 2.1–2.16 GeV and 0.01–0.03 GeV/c², respectively in the PWA ﬁt. This ﬁrst attempt can be also replaced by a more sophisticated parametrizationof thecuspcontributionlikeaFlatte’function,butthisisbeyondthe scopeofthisinvestigation.Indeedthecuspcontributionhasaneg- ligibleeffectonthedeterminationoftheN^∗contributions.

3. Results

First,thePWAwasperformedindividuallyforthedifferentdata samplestodeterminethecorrectstartvaluesoftheparametersfor theglobalfit.Thetotalnumberofavailabledegreesoffreedomfor eachdatasetislistedinTable1.Thetotalnumberoffreeparame- tersinthePWAfitcontainingallaccessibleN^∗isequalto345±^17, theerrorrefers tothesystematicvariationofthecontributingN^∗ consideredintheglobalfit.Thebest solutionofthePWAfitcor- respondstotheminimumofthelog-likelihoodobtainedbyfitting theexperimentaldatawiththePWAevent-by-event.

A comparison of the three missing mass spectra and CM, Gottfried–Jackson andHelicityangle distributions (for thedeﬁni- tion ofthese variables see [11]) obtainedfrom the experimental dataandfromthesinglePWA ﬁtswas carriedoutandthecorre- spondingreduced

χ

²valuesarelistedinTable1.Onlythestatisti- calerroroftheexperimentaldatahasbeenconsideredtoevaluate the

χ

² ofthe single PWA fits. As a second step, a simultaneous PWAofthreedatasampleswascarriedout.Thisintermediatestep allowed to determine the starting values for the global fit. The HADES, FOPI andDISTO(T =².5 GeV) samples were selected to account forboth the contributions from the -N cusp and from highermassresonances.Afterfindingasolutionthatdescribedthe threedatasamples,furtherdatasampleswereaddedstepwise.The starting valuesofeach new PWA fitwere takenfrom theresults oftheprevious fitstep.The systematicerrorofthe experimental sampleshavenot beenconsideredinthefitsince thelatterwere notavailableforallthedatasets.Toaccountforpossiblesystem- aticvariations of the kinematicdistributions we have considered allpermutations fortheexclusionof one ormoreN^∗ resonances from the list in Table 2 in the PWA fit. The five best solutions intermsoflog-likelihoodobtainedfromthissystematicvariation ofthe PWA fits were considered to extract the final results and

thePWAsystematicerrors.Asfarastheresonancesareconcerned, consideringthelistofsevenresonancesinTable2,theﬁvebestso- lutionscorrespondtothefollowingcombinations: 1)allseven N^∗ included,2)N^∗(1720)excluded,3)N^∗(1875)excluded,4)N^∗(1900) excludedand,5)N^∗(1900)andN^∗(1875)excluded.

Thereduced

χ

² valuesforthecombinedPWAlistedinTable1 were obtained by comparing the experimental data in the mass andanglevariables withthe averagevaluesofthe ﬁvebestPWA solutions,takingaserrorsthestatisticalerrorsoftheexperimental dataandthestandarddeviationoftheﬁvesolutionsforeachbin.

Byaddingadditionalsolutionsthe

χ

² didnotimprove.Thisjusti- fiesthechoiceofthefivebestsolutions.Amorerefinedtreatment ofsystematicuncertaintiesiscurrentunderdevelopment.

Fig. 1 shows the missing mass distributions (MM) for the threeﬁnal state particles p,andK⁺ forCOSY-TOF at2.16 GeV (bluesymbols),DISTOat2.85 GeV(greensymbols)andHADESat 3.5 GeV(redsymbols)datasamplesmeasuredwithintheirrespec- tive acceptances andarbitrarily normalized. The signature of the

-NcuspisvisibleintheCOSY-TOFandDISTOMM_K+ distribution around 2.13 GeV/c².Theerrors oftheexperimental dataare sta- tisticalonly. The linesinthe samecolor-code representthePWA resultsforthecorresponding datasets.The linewidthsrepresent the error bandsof the globalPWA fit expressed asthe standard deviationofthefivebestPWAsolutions.Fig.2showstheangular distributions ofthe threeparticlesmeasured inthefinalstate for different reference systems forthe same data samples discussed in Fig. 1. A similar quality is obtained forthe description of the kinematicvariablesofotherdatasamples.

The outputofeach PWA solutionprovides thestrengthof the individual waves with respect to the total measured yield. The resulting relative contributions ofthe resonant andnon-resonant wavescanbetranslatedintocrosssectionsfortheKdecaychan- nel multiplying the relative yield by the total production cross sectionforthepK⁺ﬁnalstate.

The total pK⁺ cross section for the different data sets was evaluated employing a phase space ﬁt of the existing measure- ments ofthe pK⁺channel asa function ofthe excessenergies [13,25,38–40]. The errorassociated to thepK⁺cross section of each datasample isextractedfromtheﬁt.A detaileddescription oftheextractionofthepK⁺crosssectionscanbefoundin[41].

In Fig. 3 the cross section for the different N^∗ channels de- cayinginto theK ﬁnal state isplottedversus its excess energy calculated asthe centerofmass energyofthe p–pcollidingsys- tem minus the sum ofthe proton and N* masses(√

s−^Mp,N^∗).

The standarddeviationoftheﬁvebestsolutions isshownby the blackverticalerrorbars,thegreenbandsshowtheerrororiginat- ingfromthecrosssectionnormalization.Thenon-vanishing cross section belowtherespectivethresholds isduetothelarge width ofalltheconsideredresonances(seeTable2).Therelativecontri-

(5)

Fig. 2. AngularcorrelationsforthepK⁺ﬁnalstate.Theupperindexattheangleindicatestherestframe(RF)inwhichtheangleisdisplayed.Thelowerindexnamesthe twoparticlesbetweenwhichtheangleisevaluated.CMstandsforthecenter-of-masssystem.BandTdenotethebeamandtargetvectors,respectively.Theobservables are:CMdistributions(cos

θ_X^{C M}

)ofthe(d),Proton(e)andKaon(f);Gottfried–Jacksondistributionscos θ_{K B}^{R F p K}_/_T

(g),cos

θ_{K B}^{R F K}_/_T

(h),cos

θ_{p B}^{R F p}_/_T

(i)andHelicityangle distributionscos

θ_{K p}^{R F p} (j),cos

θ_K^{R F p K}

(k)andcos θ_p^{R F K}

(l).TheexperimentaldatawithinthegeometricalacceptancearefromCOSY-TOFat2.16 GeV(bluesymbols), DISTOat2.85 GeV(greensymbols)andHADESat3.5 GeV(redsymbols)samples.Thecoloredlinesinthesamecolor-coderepresentthePWAresults.

Fig. 3. (Coloronline).CrosssectionsofthedifferentN^∗resonancesdecayingintothepK⁺ﬁnalstateobtainedfromthecombinedPWAasafunctionoftheexcessenergy.

Theexcessenergyiscalculatedasthecenterofmassenergyofthep–pcollidingsystemminusthesumoftheprotonandKaonmasses(√

s−^Mp,K+,).Theblackbars showthesystematicerrorsoriginatingfromtheﬁvedifferentPWAsolutionsandthegreenbandsrepresenttheerrorsduetothenormalizationtothetotalpK⁺cross section.

(6)

Fig. 4. (Coloronline).CrosssectionsoftheinitialstatewavesasafunctionoftheexcessenergyforthepK⁺ﬁnalstate.Theexcessenergyiscalculatedasthecenterof massenergyofthep–pcollidingsystemminusthesumoftheprotonandN*masses(√

s−^Mp,N^∗).Theerrorbarscorrespondtothestandarddeviationamongtheﬁvebest PWAsolutionsandthegreenbandreferstothenormalizationtothetotalpK⁺productioncrosssection.

Table 3

ScatteringlengthsextractedfromthecombinedPWAﬁtandreferencevaluesfrom previousmeasurements[30,31,42] andtheoreticalcalculations[43,44] (seetextfor details).

Source ¹S0a₋p[fm] ³S1a₋p[fm]

This work −¹.43±⁰.36±⁰.09 −¹.88±⁰.38±⁰.10 [30] −¹.8⁺₋²₄^._.³₂ −¹.6⁺₋¹₀^._.¹₈

[42] −².43⁺₋⁰₀^._.¹⁶₂₅ −¹.56⁰₋^.¹⁹₀_.₂₂

[31] – −².55⁺₋⁰₁^._.⁷²₃₉±⁰.6±⁰.3

χEFT LO [43] −¹.91 −¹.23

χEFT NLO [43] −².91 −¹.54

ESC08 [44] −².7 −¹.65

butionofthenon-resonantamplitudedecreasesfrom37%for2.14 GeVto 10%for3.5 GeV,sothat mostoftheyieldstemsfromN^∗ resonancesforallthemeasuredenergies. Thedominantcontribu- tionfromtheN^∗ resonancesisconsistent withtheresultsshown inRef. [13], exceptfor therelative contribution ofthe N^∗(1650), whichis decreasingasa function ofthebeam energyin[13].In thisworkwefoundanincrementoftheN^∗(1650)similarlytothe

N^∗(1710)andN^∗(1720).Thisdifferenceprobablyresultsfromne- glectinginterferenceinRef. [13].

The-Ncusp contributionvariesfrom10⁻³ to10⁻² withde- creasingenergywithrespecttotheN^∗ andisnotshowninFig.3.

The global PWA fit favors the -N cusp contribution of the s- ord-wave state J^P=¹⁺ ^with^respect ^to ^the^S-wave ^J^P=⁰⁺ âs shownbytheamplitudesinTable4.Theobtained-Ncuspyield isslightlydifferentfromthefindingsinRef. [38] whereatabeam energyof2.28 GeVthecontributionofthecusp wasfound equal to5%ofthetotalcrosssection,butneglectinginterferences.

Fig.4showsthecrosssectionsofthedifferentp+pinitialstates asafunctionofthepK⁺excessenergycalculatedasthecenter ofmassenergyofthep–pcollidingsystemminusthesumofthe proton and Kaon masses (√

s−^Mp,K⁺,). The error bars are associatedtothestandarddeviationoftheﬁvebestPWAsolutions, andthegreenbandreferstotheuncertaintyoftheexclusivepK⁺ productioncrosssection.Allextractedcross-sectionsasafunction oftheexcessenergyaresummarisedinTable4andTable5.

The non-resonant amplitude included in thisPWA is parametrized as a function of the scattering length and effective range for the p– ﬁnal state interaction. The interference of the non-

Table 4

ProductioncrosssectionsofthetotalpK⁺non-resonantcontributionandofthedifferentN^∗resonancesdecayingintothepK⁺finalstate obtainedfromtheglobalPWAasafunctionofthebeamkineticenergy.Thecrosssectionsrefertotheamplitudespriortothecoherent sumofthelatterandhencedonotconsiderinterferenceeffects.TheN^∗ crosssectionsarenotcorrectedforthebranchingratiointothe K⁺-finalstates.Thefirsterrorcorrespondstothesystematicerrorduetothefivebestsolutions,thesecondstemsfromthecrosssection normalisations.ThesystematicerrorofthePWAfittingprocedureisfoundtobenegligibleandhenceisnotshown.

3.500 GeV 3.100 GeV 2.85 GeV

pK⁺[μb] 5.1±1.0±1.2 6.3±1.2±1.4 6.5±1.1±1.3

N^∗(1650)→^pK⁺[μb] 8.6±⁰.6±².1 5.7±⁰.4±¹.2 8.1±⁰.4±¹.6 N^∗(1710)→^pK⁺[μb] 11.7±¹.0±².8 9.7±⁰.8±².1 10.2±¹.0±².1 N^∗(1720)→^pK⁺[μb] 2.4±¹.3±⁰.6 2.2±¹.2±⁰.5 1.6±⁰.8±⁰.3 N^∗(1875)→^pK⁺[μb] 1.5±¹.3±⁰.4 2.2±¹.9±⁰.5 1.2±¹.0±⁰.2 N^∗(1880)→pK⁺[μb] 14.9±0.2±3.6 13.7±0.4±3.0 8.4±0.4±1.7 N^∗(1895)→pK⁺[μb] 3.3±0.2±0.8 2.5±0.2±0.6 1.7±0.2±0.3 N^∗(1900)→pK⁺[μb] 0.2±0.2±0.0 0.3±0.3±0.1 0.2±0.2±0.0

−^N(1⁺S) [μb] 0.01±⁰.02±⁰.002 0.03±⁰.02±⁰.007 0.12±⁰.05±⁰.02

−^N(1⁺D) [μb] 0.13±⁰.02±⁰.03 0.2±⁰.04±⁰.05 0.5±⁰.08±⁰.1

2.5 GeV 2.157 GeV 2.14 GeV

pK⁺[μb] 7.2±1.1±1.3 7.5±0.6±1.3 7.1±0.6±1.2

N^∗(1650)→pK⁺[μb] 7.5±0.4±1.4 5.5±0.3±1.0 5.4±0.3±1.0 N^∗(1710)→^pK⁺[μb] 7.5±⁰.9±¹.4 3.3±⁰.5±⁰.6 3.5±⁰.5±⁰.6 N^∗(1720)→^pK⁺[μb] 1.3±⁰.7±⁰.2 0.8±⁰.4±⁰.1 0.7±⁰.3±⁰.1 N^∗(1875)→^pK⁺[μb] 0.7±⁰.6±⁰.1 0.2±⁰.1±⁰.0 0.2±⁰.1±⁰.0 N^∗(1880)→^pK⁺[μb] 4.8±⁰.3±⁰.9 1.7±⁰.1±⁰.3 1.5±⁰.1±⁰.3 N^∗(1895)→pK⁺[μb] 0.7±0.1±0.1 0.2±0.0±0.0 0.2±0.0±0.0 N^∗(1900)→pK⁺[μb] 0.2±0.2±0.0 0.1±0.1±0.0 0.1±0.1±0.0

−N(1⁺S) [μb] 0.12±0.04±0.02 0.16±0.04±0.03 0.13±0.03±0.02

−^N(1⁺D) [μb] 0.34±⁰.07±⁰.06 0.21±⁰.04±⁰.04 0.17±⁰.03±⁰.03