Measurement of the $\omega \rightarrow \pi ^{+}\pi ^{-}\pi ^{0}$ Dalitz plot distribution

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Measurement of the ω _→ π ⁺ π ⁻ π ⁰ Dalitz plot distribution

The WASA-at-COSY Collaboration

P. Adlarson

^a,1

, W. Augustyniak

^b

, W. Bardan

^c

, M. Bashkanov

^d

, F.S. Bergmann

^e

, M. Berłowski

^f

, H. Bhatt

^g

, A. Bondar

^h,i

, M. Büscher

^j,2,3

, H. Calén

^a

, I. Ciepał

^k

, H. Clement

^l,m

, E. Czerwi ´nski

^c

, K. Demmich

^e

, R. Engels

^j

, A. Erven

ⁿ

, W. Erven

ⁿ

, W. Eyrich

^o

, P. Fedorets

^j,p

, K. Föhl

^q

, K. Fransson

^a

, F. Goldenbaum

^j

, A. Goswami

^j,r

,

K. Grigoryev

^j,s,4

, C.-O. Gullström

^a

, L. Heijkenskjöld

^a,∗

, V. Hejny

^j

, N. Hüsken

^e

, L. Jarczyk

^c

, T. Johansson

^a

, B. Kamys

^c

, G. Kemmerling

^n,5

, F.A. Khan

^j

, G. Khatri

^c,6

, A. Khoukaz

^e

, O. Khreptak

^c

, D.A. Kirillov

^t

, S. Kistryn

^c

, H. Kleines

^n,5

, B. Kłos

^u

, W. Krzemie ´n

^f

,

P. Kulessa

^k

, A. Kup´s ´c

^a,f

, A. Kuzmin

^h,i

, K. Lalwani

^v

, D. Lersch

^j

, B. Lorentz

^j

, A. Magiera

^c

, R. Maier

^j,w

, P. Marciniewski

^a

, B. Maria ´nski

^b

, H.-P. Morsch

^b

, P. Moskal

^c

, H. Ohm

^j

,

E. Perez del Rio

^l,m,7

, N.M. Piskunov

^t

, D. Prasuhn

^j

, D. Pszczel

^a,f

, K. Pysz

^k

, A. Pyszniak

^a,c

, J. Ritman

^j,w,x

, A. Roy

^r

, Z. Rudy

^c

, O. Rundel

^c

, S. Sawant

^g,j,∗∗

, S. Schadmand

^j

,

I. Schätti-Ozerianska

^c

, T. Sefzick

^j

, V. Serdyuk

^j

, B. Shwartz

^h,i

, K. Sitterberg

^e

, T. Skorodko

^l,m,y

, M. Skurzok

^c

, J. Smyrski

^c

, V. Sopov

^p

, R. Stassen

^j

, J. Stepaniak

^f

,

E. Stephan

^u

, G. Sterzenbach

^j

, H. Stockhorst

^j

, H. Ströher

^j,w

, A. Szczurek

^k

, A. Trzci ´nski

^b

, R. Varma

^g

, M. Wolke

^a

, A. Wro ´nska

^c

, P. Wüstner

ⁿ

, A. Yamamoto

^z

, J. Zabierowski

^aa

, M.J. Zieli ´nski

^c

, J. Złoma ´nczuk

^a

, P. ˙Zupra ´nski

^b

, M. ˙Zurek

^j

aDivisionofNuclearPhysics,DepartmentofPhysicsandAstronomy,UppsalaUniversity,Box516,75120Uppsala,Sweden bDepartmentofNuclearPhysics,NationalCentreforNuclearResearch,ul.Hoza69,00-681,Warsaw,Poland

cInstituteofPhysics,JagiellonianUniversity,prof.StanisławaŁojasiewicza11,30-348Kraków,Poland

dSchoolofPhysicsandAstronomy,UniversityofEdinburgh,JamesClerkMaxwellBuilding,PeterGuthrieTaitRoad,EdinburghEH93FD,UnitedKingdom eInstitutfürKernphysik,WestfälischeWilhelms-UniversitätMünster,Wilhelm-Klemm-Str.9,48149Münster,Germany

fHighEnergyPhysicsDepartment,NationalCentreforNuclearResearch,ul.Hoza69,00-681,Warsaw,Poland gDepartmentofPhysics,IndianInstituteofTechnologyBombay,Powai,Mumbai400076,Maharashtra,India hBudkerInstituteofNuclearPhysicsofSBRAS,11akademikaLavrentievaprospect,Novosibirsk,630090,Russia iNovosibirskStateUniversity,2PirogovaStr.,Novosibirsk,630090,Russia

jInstitutfürKernphysik,ForschungszentrumJülich,52425Jülich,Germany

kTheHenrykNiewodnicza´nskiInstituteofNuclearPhysics,PolishAcademyofSciences,152RadzikowskiegoSt,31-342Kraków,Poland lPhysikalischesInstitut,Eberhard-Karls-UniversitätTübingen,AufderMorgenstelle14,72076Tübingen,Germany

mKeplerCenterforAstroandParticlePhysics,EberhardKarlsUniversityTübingen,AufderMorgenstelle14,72076Tübingen,Germany nZentralinstitutfürEngineering,ElektronikundAnalytik,ForschungszentrumJülich,52425Jülich,Germany

oPhysikalischesInstitut,Friedrich-Alexander-UniversitätErlangen–Nürnberg,Erwin-Rommel-Str. 1,91058Erlangen,Germany

pInstituteforTheoreticalandExperimentalPhysics,StateScientificCenteroftheRussianFederation,BolshayaCheremushkinskaya 25,117218Moscow,Russia qII.PhysikalischesInstitut,Justus-Liebig-UniversitätGießen,Heinrich-Buff-Ring16,35392Giessen,Germany

rDepartmentofPhysics,IndianInstituteofTechnologyIndore,KhandwaRoad,Indore452017,MadhyaPradesh,India

*

Correspondingauthor.

**

Correspondingauthorat:DepartmentofPhysics,IndianInstituteofTechnologyBombay,Powai,Mumbai400076,Maharashtra,India.

E-mailaddresses:lena.heijkenskjold@physics.uu.se(L. Heijkenskjöld),siddhesh.sawant@iitb.ac.in(S. Sawant).

1 Presentaddress:InstitutfürKernphysik,Johannes-Gutenberg-UniversitätMainz,Johann-Joachim-BecherWeg45,55128Mainz,Germany.

2 Presentaddress:PeterGrünbergInstitut,PGI-6ElektronischeEigenschaften,ForschungszentrumJülich,52425Jülich,Germany.

3 Presentaddress:InstitutfürLaser- undPlasmaphysik,Heinrich-HeineUniversitätDüsseldorf,Universitätsstr.1,40225Düsseldorf,Germany.

4 Presentaddress:III.PhysikalischesInstitutB,Physikzentrum,RWTHAachen,52056Aachen,Germany.

5 Presentaddress:JülichCentreforNeutronScienceJCNS,ForschungszentrumJülich,52425Jülich,Germany.

6 Presentaddress:DepartmentofPhysics,HarvardUniversity,17OxfordSt.,Cambridge,MA02138,USA.

7 Presentaddress:INFN,LaboratoriNazionalidiFrascati,ViaE.Fermi,40,00044Frascati(Roma),Italy.

http://dx.doi.org/10.1016/j.physletb.2017.03.050

0370-2693/©^{2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

sHighEnergyPhysicsDivision,PetersburgNuclearPhysicsInstitute,OrlovaRosha 2,Gatchina,Leningraddistrict188300,Russia

tVekslerandBaldinLaboratoryofHighEnergiyPhysics,JointInstituteforNuclearPhysics,Joliot-Curie 6,141980Dubna,Moscowregion,Russia uAugustChełkowskiInstituteofPhysics,UniversityofSilesia,Uniwersytecka4,40-007,Katowice,Poland

vDepartmentofPhysics,MalaviyaNationalInstituteofTechnologyJaipur,302017,Rajasthan,India

wJARA-FAME,JülichAachenResearchAlliance,ForschungszentrumJülich,52425Jülich,andRWTHAachen,52056Aachen,Germany xInstitutfürExperimentalphysik I,Ruhr-UniversitätBochum,Universitätsstr.150,44780Bochum,Germany

yDepartmentofPhysics,TomskStateUniversity,36LeninaAvenue,Tomsk,634050,Russia zHighEnergyAcceleratorResearchOrganisationKEK,Tsukuba,Ibaraki305-0801,Japan aaDepartmentofAstrophysics,NationalCentreforNuclearResearch,Box447,90-950Łód´z,Poland

B. Kubis

^ab

, S. Leupold

^ac

abHelmholtz-InstitutfürStrahlen- undKernphysik,RheinischeFriedrich-Wilhelms-UniversitätBonn,Nußallee14–16,53115Bonn,Germany acDivisionofNuclearPhysics,DepartmentofPhysicsandAstronomy,UppsalaUniversity,Box516,75120Uppsala,Sweden

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

Articlehistory:

Received10October2016

Receivedinrevisedform29January2017 Accepted8March2017

Availableonline25April2017 Editor:V.Metag

Keywords:

Decaysofothermesons Meson–mesoninteractions Lightmesons

Using theproduction reactions pd→^3Heωand pp→^ppω,the Dalitzplotdistributionfor theω_→ π⁺π⁻π⁰ decay is studied with the WASA detector at COSY, based on a combined data sample of (4.408±⁰.042)×^{104events.TheDalitzplotdensityis}parametrisedbyaproductoftheP -wavephase spaceandapolynomialexpansioninthenormalisedpolarDalitzplotvariables Z and φ.Forthefirst time,adeviationfrompureP -wavephasespaceisobservedwithasignificanceof4.1σ.Thedeviation isparametrisedbyalinearterm1+²αZ ,withαdeterminedtobe+⁰.147±⁰.036,consistentwiththe expectationsofρ-meson-typefinal-stateinteractionsoftheP -wavepionpairs.

©^{2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Thepresentworkandforeseeablefollow-upsarebasedontwo motivations:1.Tocheckandimproveonourunderstandingofthe importance of hadronic final-state interactions for the structure anddecaysofhadrons. 2.ToimprovetheStandardModelpredic- tionforthegyromagneticratioofthemuon[1].Thepresentwork accomplishesthe first taskconcerning the Dalitz decay ofthe

ω

mesoninto three pions; italso constitutesa significant step for- ward towards providing improved hadronic input for the second task.In thefollowingweshall discussthe twotasksinmore de- tail.

The

ω

-mesonresonance was discovered in1961 [2]. Its main decay branch is

ω

→

π

⁺

π

⁻

π

⁰, with a branching ratio of BR= (89.2±⁰.7)%. By now it is well established that the

ω

meson hasspin-parity J^P =^{1− [3]. As a} consequence,the combination ofBose,isospin,andparitysymmetryofthestronginteractionde- mandsthat for the decay

ω

_→

π

⁺

π

⁻

π

⁰ every pionpair isin a state of odd relative orbital angular momentum. Given the lim- itedphase spaceofthedecayone cansafelyassume the P -wave to be the dominant partial wave.⁸ If a pion pairis in a P -wave state,then the third pionwill be in P -wave state relativeto the pair.This “ P -wave phase space”distribution has beenconfirmed experimentally. Historically the P -wave dominance of the decay hasactually beenusedtopindownthequantumnumbersofthe

ω

meson[5–8].

Ifthepions,once produced inthedecay, didnot interactfur- ther,then solelythe P -wavephase spacewouldshapethe Dalitz plotofthedecay

ω

→

π

⁺

π

⁻

π

⁰.However,apionpairina P -wave showsavery strongfinal-state interaction.The two-pion P -wave phase shift is dominated by the

ρ

meson, and is now known very accurately [9–11]; thisis essential in particular fortheoret- ical studies of these decays using dispersion theory, which use

8 GenuineF -wavecorrectionshavebeenmodelledtheoretically,andfoundtobe tiny[4].

the phase shifts as input directly [4,12]. In the similar decay φ→

π

⁺

π

⁻

π

⁰ onecanseethe

ρ

mesonasaresonanceinthecor- respondingDalitzplot[13,14].Inthedecay

ω

→

π

⁺

π

⁻

π

⁰thereis notenoughenergyforapionpairtoreachthe

ρ

resonancemass;

yetalreadyfortheavailableinvariantmassesthetwo-pionP -wave phaseshiftissignificantly differentfromzero.Infact,everytheo- reticalapproachthatdealswiththedecay

ω

→

π

⁺

π

⁻

π

⁰ includes this non-trivial phase shift and/or the

ρ

meson in one way or theother;see,e.g.,Refs.[15–19,4,20,12]andreferencestherein.In practicethisleadstoanincreaseofpopulationtowardsthebound- ariesoftheDalitzplot,superimposedwiththepure P -wavephase space, whichdropstowards theboundaries.Thisincrease ofpop- ulationisonthelevelofabout20%[19,4],andoughttobetested experimentally.

Interestingly this has not been achieved so far. The highest statistics of a dedicated

ω

→

π

⁺

π

⁻

π

⁰ Dalitz plotmeasurement from 1966had 4208±^{75 signal events [21].Due to the limited} statistics,fitswithapure P -wavephasespacecouldnotbedistin- guishedfromadistortionbythefinal-stateinteractions,i.e. byin- termediate

ρπ

states.Surprisinglytherewerenofurtherdedicated Dalitzplotstudiesofthe

ω

→

π

⁺

π

⁻

π

⁰decay.Inthepresentwork wewillrevealthattheuniversalfinal-stateinteractionsofthepion pairsareindeedpresentinthe

ω

Dalitzdecay.Intheanalysispre- sentedherewehaveproducedanacceptance-correctedDalitzplot and extracted experimental values for parameters describing the density distribution.This constitutes the firsttask spelled out in thebeginningofthisintroduction.

Thesecond motivationforaprecisemeasurement ofthe

ω

→

π

⁺

π

⁻

π

⁰ Dalitz plot consists in improving hadronic input for the theoretical assessment of the hadronic light-by-light scatter- ingcontributiontotheanomalousmagneticmomentofthemuon.

Thelargestindividualcontributionisgivenbythelightesthadronic intermediate state,the so-called

π

⁰ poleterm, whosestrength is determinedbythecorrespondingsinglyanddoublyvirtualtransi- tionformfactors.Oneofthefewpossibilitiestogainexperimental access to the doubly virtual

π

⁰ transitionform factor with high precision consistsinstudyingvector mesonconversion decays,in

particular

ω

→

π

⁰⁺⁻—whichis intimatelylinked, throughdis- persion relations, to the

ω

→

π

⁺

π

⁻

π

⁰ decay amplitude [22,23, 12]. However, these theoretical descriptions of the

ω

transition form factor (see also Ref. [24]) fail to describe the very precise data on

ω

→

π

⁰

μ

⁺

μ

⁻ takenby the NA60 collaboration [25,26], which may violate very fundamental theoretical bounds [27,28].

Inall thesestudies, the

ω

→

π

⁺

π

⁻

π

⁰ decayamplitude is a po- tential looseend, asit is so far only theoretically modelled, not experimentallytested.Incombinationwiththepreciselymeasured φ→

π

⁺

π

⁻

π

⁰Dalitz plotinformation,itcouldbe usedtofurther constraintheamplitude analysisofe⁺e⁻→

π

⁺

π

⁻

π

⁰,andhence the

π

⁰transitionformfactorinawiderrange[29].

Recently, an easy-to-use polynomial parametrisation of the

ω

→

π

⁺

π

⁻

π

⁰ Dalitz plotdistributionhasbeensuggested[4],as ageneralisationofthecommonlyusedoneforthedecay

η

→³

π

⁰

(whichhasa similarcrossingsymmetry). Inthepresentworkwe utilise the same parametrisation andcompare to recent theoret- ical approaches [4,20,12] that have provided predictions for the correspondingDalitzplotparameters.

One wayto describe athree-particle Dalitz decaydistribution istouseinvariantmassesofparticlepairs[3].Thiswouldbepar- ticularlyusefulforreadingoffresonancemassesifthedecaywas mediatedbyoneorseveralresonances.However, therearenoin- termediateresonancesinthekinematicallyaccessibleenergyrange of the decay

ω

→

π

⁺

π

⁻

π

⁰ that would be compatible withthe symmetriesofthestrong interaction.Forourcaseofinterest,we first split offthe P -wave phase space(see, e.g., Refs. [19,4]) and parametrisetherestbyapolynomialdistributionfollowing[4].De- notingthe polarisationvector of the

ω

mesonby

(_Pω,λ_ω) and themomentaof theoutgoingpionsby P₊, P₋, and P₀, westart withthemostgeneral matrixelement compatiblewiththesym- metries,

M

=

ⁱ

ε

_μναβ

^μP₊^νP₋^αP^β₀F

.

(1) The dynamics of the final-state interactions is encoded in the scalarfunctionF ^{[19,4].Aftersummationoverthehelicity} λω of the

ω

mesononeobtainsaDalitzplotdistributionproportionalto



λω

|

M

|

²

∝

P

|

F

|

² (2)

withthepure P -wavephase-spacedistribution P

=

^m2₊^m2₋^m2₀

+

²

(

P₊P₋

)(

P₋P₀

)(

P₀P₊

)

−

^m2₊

(

P₋P₀

)

²

−

^m2₋

(

P₊P₀

)

²

−

^m2₀

(

P₊P₋

)

²

.

(3) Notethat fortheP ^{termwe canaccount for}“kinematic”isospin violations due to the difference between the masses of the un- chargedandcharged pions,m0 andm_±, respectively.For the re- maining distribution F^{, whichcovers thedynamics of the final-} stateinteraction,weignoreisospinbreakingeffects.

The quantity F and therefore also |F|² wouldbe a constant iftherewerenofinal-stateinteractionsbetweentheproducedpi- ons.Inreality|F|^{2 isnotaconstant,butrelativelyflat.Insteadof} parameterising |F|² by invariant massesof pion pairs we follow Ref.[4]andutilisenormalisedvariables X andY ,whichhavetheir originatthecentreoftheDalitzplot.Theyaredefinedby X

= √

3 T₊

−

^T−

Q_ω

,

Y

=

^3T0

Q_ω

−

¹

,

(4)

with

Q_ω

=

^T+

+

^T−

+

^T0

.

(5)

Here T_i arethekinetic energies ofthepionsin the

ω

restframe (centre-of-massframeofthethree-pionsystem).Finallyoneintro- ducespolarcoordinatesby

Table 1

TheDalitzplotparametersfromfitstothetheoreticalpredictionsofRefs.[4,20,12], whereatmosttwoparameterswereusedinthefit.

α×¹⁰³ β×¹⁰³

Uppsala[20] 202 –

Bonn[4] 84. . .102 –

JPAC[12] 94 –

Uppsala 190 54

Bonn 74. . .90 24. . .30

JPAC 84 28

X

= √

Z cos

φ,

Y

= √

Z sin

φ .

(6)

Theexpansionfor|F|^{2,validintheisospinlimit,reads}

|

F

|

²

(

Z

, φ) =

N

·

G

(

Z

, φ) ,

(7) whereN isanormalisationconstantandG containstheexpansion in Z andφ[4]:

G

(

Z

, φ) =

¹

+

²

α

Z

+

²

β

Z^3/2sin 3

φ +

²

γ

Z²

+

O



Z^5/2



.

(8) The Dalitz plotdistribution can thenbe fitted usingthisformula toextractthe“Dalitzplotparameters”

α

,β,

γ

,. . . .Thefitresults to thetheory predictionsofRefs. [4,20,12],ifEq.(8)istruncated atorderZ (oneparameterfit)oratorder Z^3/2(twoparameterfit), are shownin Table 1. The reproduction of the theoretical Dalitz plot distributions is improved significantly in all cases when in- cludingtheterm∝ β^.

It is worth to point out thequalitative similarities and differ- ences of the theoretical approaches that provide predictions for the Dalitz plot parameters. All three approachesagree on a pos- itive andsizable valuefor

α

.Thisreflectsthe factthat thepion–

pion P -wave phase shift is dominated by the appearance of the

ρ

-mesonresonance;anexperimentalresultpointingtoanegative value of

α

wouldbe spectacularinthesense thatit wouldbeat oddswiththeuniversalityofthefinal-stateinteractions.

Refs. [4,12] are based on dispersion theory: both employ the pion–pion P -wave scattering phase shift as input and describe rescatteringofallthreefinal-statepionsconsistentlytoallorders.

Thedispersiveformalismcan bechosenwithonlyonesingle free parameter(a“subtractionconstant”),basedonreasonableassump- tions onthehigh-energy behaviouroftheamplitude;thisparam- etercanbetakentobetheoverallnormalisationandfixedexper- imentally from the

ω

→

π

⁺

π

⁻

π

⁰ partial width. The energy de- pendence ofthedecayamplitude isthenfullypredictedfromthe pion–pionphaseshiftalone.TheDalitzplotparameterscitedfrom Ref. [12]inTable 1areobtainedfromsucha scenario.InRef. [4], varioussourcesofuncertaintyrelatedtothepion–pioninteraction havebeenconsidered,leadingtoanestimateofthetheoreticaler- rorinthepredictionoftheDalitzplotparameters.Furthermore,an analysisof datafortheanalogousφ→

π

⁺

π

⁻

π

⁰ Dalitz plot[13]

demonstratedthataonce-subtracteddispersiverepresentationde- scribes suchvery-high-accuracydatawell, butnot perfectly,lead- ing to theneed tointroduce asecond subtraction [4].With such a second free parameter also used in the

ω

→

π

⁺

π

⁻

π

⁰ decay amplitude, theenergydependenceinprinciplecannot beentirely predicted anymore. However, estimating the size ofsuch a sec- ond constantfromthe φ→

π

⁺

π

⁻

π

⁰ data analysis,it was found that the uncertainty range for the theoretical prediction of the Dalitzplotparametersisonlymoderatelyincreased[4].Theranges quotedinTable 1reflectthisfull,combineduncertaintyestimate.

Ref. [20] is based on an effective Lagrangian for the lightest pseudoscalarandvectormesons.Thestrengthoftheinitial

ω

-

ρ

-

π

interaction is fitted to the decay width of

ω

→

π

⁺

π

⁻

π

⁰ and cross-checkedwiththe decaywidthof

ω

→

π

⁰

γ

.The Lagrangian

provides the kernel fora Bethe–Salpeter equation that generates thetwo-pionrescattering.Incontrasttothedispersiveapproaches, crossed-channel rescattering of the three-pion system is not in- cluded.

Whileall theoryapproachesagreeon thesignof

α

,the value predictedby the Lagrangian approach[20] isvery different from the valuesobtained by dispersion theory [4,12]. The same holds trueforthe β parameter: thelarger valuesfromRef. [20] reflect a rather strong energy dependence of the matrix element. The same qualitative difference can be observed for the electromag- netictransitionformfactorof

ω

→

π

⁰⁺⁻[24,23]:alsoherethe Lagrangianapproachprovidesamuchstrongerenergydependence.

Atechnicalreasonforthismightbe foundinthefactthat essen- tiallyfield strengths instead ofvector potentialsare used forthe constructionofinteraction termsin theLagrangian approach[19, 24,20]. If the results of this low-energy Lagrangian were boldly extrapolatedto highenergies—beyond its limit of applicability—, thenonewouldfindthat thereactionamplitudeswouldnotcon- verge. In contrast, modest high-energy constraints are automati- cally encodedinthe dispersiveapproaches. Apparentlythis leads to smaller energy variations of the reaction amplitudes even in thelow-energyregimethatisofrelevanceforthe

ω

decays.—At presentitisnotpossibletoobtainaserioustheoreticaluncertainty estimate forthe Lagrangian approach [20]. Additional interaction termshavebeen neglected therein,whichwere considered to be small,butitisnotclearyethowsmalltheyare.

2. Theexperiment

The experimental data was collected using the WASA setup, where the

ω

was produced in the pd→^3He

ω

reaction and in the pp→^pp

ω

reaction.The WASAdetector[30,31]isan internal target experimentat theCooler Synchrotron (COSY) storagering, ForschungszentrumJülich,Germany. TheCOSYprotonbeaminter- actswith an internal target consisting of smallpellets of frozen hydrogenordeuterium(diameter∼^35μm).

The WASAdetector consists of a Central Detector (CD)and a ForwardDetector(FD),coveringscatteringanglesof20^◦–169^◦ and 3^◦–18^◦, respectively. The CD is used to measure decay products ofthemesons. Acylindricalstraw chamber (MDC)is placed ina magneticfieldof1 T,providedbyasuperconductingsolenoid.The electromagnetic calorimeter (SEC) consists of 1012 CsI(Na) crys- tals whichare read out by photomultipliers. A plastic scintillator barrel(PSB)isplacedbetweentheMDCandtheSEC,allowingpar- ticleidentification andaccurate timing for chargedparticles. The FDconsistsofthirteenlayersofplasticscintillatorsforenergyand time determination anda straw tube trackerproviding a precise trackdirection.

When the

ω

mesons were produced using the pd→^3He

ω

reaction, two different proton kinetic energies were used: TA= 1.450GeV and TB =¹.500GeV.The cross section ofthe reaction is84(10) nbatthelower energy[32]andwas studiedpreviously bytheCELSIUS/WASAcollaboration.Triggersselecteventswithat leastone trackin theFDwith ahigh energydepositin thethin plasticscintillator layers.Thisconditionallows foran efficientse- lectionof ³He ions and provides an unbiased data sample of

ω

meson decays.The proton beam energy was chosen so that the 3He produced in the pd→^3He

ω

reaction stops in the second thick scintillator layer of the FD. The correlation plot E− ^E from a thin layer and the first thick layer of the FD is shown in Fig. 1 (top). The band corresponding to the ³He ion is well separated from the bands for other particles and allows a clear identificationof³He.The³He fromthereactionofinteresthaski- neticenergies up to 700 MeV andscatteringangles rangingfrom 0^◦ to10^◦.

Fig. 1. ParticleidentificationintheForwardDetectorisperformedusingthecorre- lationofenergydepositsintheplasticdetectorlayers.(top)For³Heidentification:

thecorrelationbetweentheenergydepositsinathin(0.5 cm)layerandthefirst subsequentthicklayer(11 cm).Onlytheregionselectedintheanalysisisshownin thefigure,withthebandfrom³Heparticlesclearlyvisibleinsideit.(bottom)For protonidentification:thecorrelationbetweentheenergydepositinthefirstthick layerandthesummedenergydepositsinallthicklayers(11or15 cm).Thecorrela- tionbandcorrespondingtoenergydepositsmadebyprotonsissurroundedbythe blackline.Alsovisibleisalower,near-horizontal,bandwhichispopulatedbyfast protonspunchingthroughthefirstthicklayer,depositingenergyof20 MeV,andun- dergoingnuclearinteractioninoneofthesubsequentthicklayers,theredepositing anindefiniteamountofenergy.

The pp→^pp

ω

experimentwas performedat T_C =².063 GeV beamkineticenergy,corresponding to60 MeV centre-of-massex- cessenergy andcross section 5.7 μb[33]. In the pp collision ex- periment, the selected events were required, at trigger level, to containatleasttwo tracksreaching thesecond thicklayer ofthe plasticscintillatorsintheFD, atleasttwohitsinthe PSB,andat leastoneclusterintheSEC.Intheofflineanalysis, pairsoftracks correspondingtotheE− ^{E protonbands,showninFig. 1(bot-} tom),indifferentthicklayersoftheFDareselectedasprotonpair candidates.

Fortheparticlesmeasured intheCD,acommonanalysispro- cedure is usedfor all threedata sets.Events are selected ifthey contain atleast onepairofopposite charge particletracksin the MDC with scattering angles greater than 30^◦ and at least two neutralclusterswithenergydepositabove20 MeVintheSEC.Rel- ative time between the tracks is checked to minimise pile ups.

The charged particle tracks are assigned the charged pion mass.

Combinations of the all the measured charged and neutral par- ticle tracksin the selected eventsare testedusing a constrained kinematicfitassumingtheconservationofenergyandmomentum

Fig. 2. Missing mass distributions after the full analysis procedure as well as the result of the fit Eq.(9). (a): TA=¹.450 GeV. (b): TB=¹.500 GeV. (c): TC=².063 GeV.

with the pd→^3He

π

⁺

π

⁻

γ γ

or pp→^pp

π

⁺

π

⁻

γ γ

hypothesis, respectively. The combinations with p-values less than 0.05 are rejected. Forthe casewhen morethan one trackcombinationin aneventfulfilsthiscriteria,thecombinationgivinglarger p-value isselected.Finally,furtherbackgroundsuppressionisachievedby applying a kinematic fit with the contending hypothesis pd→

3He

π

⁺

π

⁻orpp→^pp

π

⁺

π

⁻,respectively.Iftheresultingp-value islargerthanforthefirstfit,theeventisrejected.

Themissingmassdistributions,MM(³He)andMM(pp),forthe threedatasetsareshowninFig. 2.Themissingmasses,calculated from the variables corrected by the kinematic fit,are equivalent to the invariant mass of the

π

⁺

π

⁻

γ γ

system. The observed

ω

peakpositionisshiftedfromthenominalvalue ofthe

ω

massby +^{0.7 MeVforMM}(³He)inthetwo pd datasets andby +^{1.1 MeV} for MM(pp). The observed shifts correspond to deviations from thenominalbeamenergyby0.55 MeVand0.75 MeV,respectively, whichiswell withintheuncertaintyofthe absoluteenergyscale ofCOSY.Toreproducetheexperimental

ω

peakposition,themiss- ing mass distributions fromsimulated data were shifted accord- ingly.Toalsoreachagreementbetweenexperimentandsimulation forthewidthofthe

ω

peak,theresolutionfromsimulateddetec- torresponseswereadjusted.

Both thebackgroundshape andthe

ω

peak contentare fitted simultaneouslyto theexperimental distributionusing thefollow- ingfitfunction:

H

( μ ) =

^NSH_ω

( μ ) + 

a0

+

^a1

μ ₊

a2

μ

²

₊

a3

μ

³

^ _×

H3π

( μ ),

(9) where

μ

=^MM(³He) or MM(pp). Hω(

μ

) and H3π(

μ

) represent reconstructed distributions of simulated signal and background andcorrespondtoeventsthathavepassedthroughthesameanal- ysisstepsastheexperimentaldata.Hω(

μ

)isnormalisedsuchthat thefitgivesdirectlythenumberofsignalevents,NS,andthere- latederror.The other parameters fittedarea0,a1,a2,anda3 (in caseof pd data a₃ is set to 0). The rangein

μ

used forthe fit is[⁰.640,0.832]^GeV/c² forset A, [⁰.640,0.856]^GeV/c² forset B, and[⁰.608,0.824]^GeV/c² forset C.Thelimitsoftheserangesare shownbythedashedlinesinFig. 2,wheretheresultofthesefits tothefulldatasamplesaregiven.Theresultingnumberofevents is:14600(200)forset A,13500(200)forset B,and16000(300)for set C.

3. Dalitzplot

TheDalitzplotdensityisrepresentedusingatwo-dimensional histogramin the Z and φ variables, defined in Eq.(6). The size oftheselectedbinsisdeterminedby theexperimentalresolution of Z and φ and the statistics of the collected data sample. The numberofeventsineachbinshouldbesufficientfordetermining

Fig. 3. TheArabicnumeralsshowthebinnumberstobeusedwhenpresentingthe resultingDalitzplots.Thecolourplotshowsthekinematicallyallowedregionofthe ω→π⁺π⁻π⁰reactionwithωnominalmassaswellasthedensitydistribution from P -wavedynamics.TheRomannumeralsdisplaythesectorsusedinconsis- tencychecks.

thesignalyieldandtocarryouta

χ

² fitoftheDalitzplotdensity parametrisation. Theφ variablerange [−

π

,

π

] ^{isdivided intosix} binstopreservethethreefoldisospinsymmetryandtobesensitive toapossiblesin 3φdependence.The Z variablerange[⁰,1]^isalso divided intosix bins.Onlythe 21bins fullycontainedinside the kinematic limitsof the decayare used. Fig. 3introduces the bin numberingusedforthepresentationoftheresults.

A smallshiftoftheDalitz plotalong the Y axisisdueto the massdifferencebetweentheneutralandchargedpions.Itismost visible in Fig. 3 when comparing the regions atφ=

π

/2 to the onesatφ= −

π

/6 and−²

π

/3.Thepictureshowsalsoseven sec- torsI–VIIthatareusedtotesttheconsistencyofthefitresults.

ForeachDalitz plotbin,theexperimentalmissingmassdistri- bution isconstructedandthenumberofentriesinthe

ω

peakis extractedby fittinga simulated

ω

→

π

⁺

π

⁻

π

⁰ signal alongwith backgroundcontributionsusingEq.(9).

Since the P -wavedistribution reproduces thegeneral features ofthe

ω

→

π

⁺

π

⁻

π

⁰ Dalitzplotverywell andthedeviationsare expected to be small, the efficiency correction is obtained using signal simulationwiththe P -wave. Theefficiency, _i,isextracted usingtheratio _i=^Ni/N_i^G.N^G_i isthenumberofeventswithgen- eratedkinematicvariablescorrespondingtobini intheDalitzplot.

Ni is the content of the bin i when the reconstructed values of the kinematic variables are used for events passing all analysis steps. Theextractedefficiencies forthe threedatasetsareshown in Fig. 4.Forthe pd data setsthe overallefficiencyis11%, while for pp it is 20 times lower.This low efficiency for the pp data sample has the followingwell-understood causes.In most cases,

Fig. 4. TheresultingefficienciesforeachDalitzplotbinforthethreedatasets.The relationbetweenthebinnumbersusedhereandthebinsofthetwo-dimensional DalitzplotisshowninFig. 3.Thesolidlinecorrespondstoset A,thedashedline toset B,andthedottedlineistheacceptanceforthepp dataset C,whichismul- tipliedbyafactorof10.

thetwo fastprotontracksdepositonly afractionof their kinetic energyinthe detector,leading toa lowerprecision ofthekinetic energydeterminationandan asymmetricresolutionfunction.The eventsfrom the tails will likely be rejected by the kinematic fit procedure.Ontheother handforthe pd→^3He

ω

reaction,there isonlyonedoublycharged³He stoppinginthedetector.Another causeforthelowefficiencyisthelargercentre-of-massvelocityin the pp reaction,which decreases the average emission anglefor decayparticles,inparticularforthechargedpions.Thepionswill bemoreoftenemitted atangles below30^◦ andwillthereforebe rejectedintheanalysisprocedure.

TheDalitzplotparameters(

α

,β,. . .)andnormalisationfactors forthethreedatasets(NA,NB,NC)aredeterminedbyminimis- ingthefollowing

χ

²=

χ

²_A+

χ

_B²₊

χ

_C²function,where

χ

²_A

= 

i



N

˜

_{i A}

−

NA

·

^Hi

( α , β, . . .)

˜ σ

i A



2

.

(10)

N˜i and

σ

˜iaretheefficiencycorrectedexperimentalDalitzplotbin contentanderror,respectively. Hiisgivenbyanintegral overbin i: H_i(

α

,β,. . .)=

iP(^Z,φ)G(^Z,φ)d Zdφ. P(^Z,φ) is the P -wave phase space term given by Eq. (3), calculated using the nomi- nalmassofthe

ω

mesonof782.65 MeV,andG(^Z,φ) isgivenby Eq.(8).

Theparametrisation procedure ofthe Dalitz plot istestedus- ing10⁶signaleventssimulatedwithP -wavephasespaceonly(i.e.

G =^{1) andwithout detector smearing.The extracted parameters} arefoundto beconsistentwithzeroandthereforetheprocedure doesnotintroduceanybiasatthepresentstatisticalaccuracy.

The three independent data sets and the Dalitz plot symme- triesallowfordetailedchecks oftheexperimental efficiency and thebackgroundsubtractionproceduresincethebackgrounddistri- butionsandefficienciesaredifferentinthecorrespondingbins.

The method of background subtraction for the missing mass

μ

distributions istested by preparingsimulated distributions af- terfull detector reconstruction, consistingof a sumof

π

⁺

π

⁻

π

⁰

productionbackgroundeventsandthe

ω

signal generatedusinga P -wavephasespacedistribution.Thebackgroundisobtainedfrom the

a0+^a1

μ

+^a2

μ

²₊a3

μ

^3_×H3π(

μ

)distributionswithai de- termined from the fits using Eq. (9) and by setting the average signal-to-background ratio to be the same asin the experimen- taldata.Thegenerated

μ

distributionswiththenumberofevents similarasintheexperimentarethensubjectedtothesameback- groundsubtractionastheexperimental data.Thecombinedfitof

Fig. 5. TheexperimentalDalitzplotdistributionafterapplyinganefficiencycorrec- tion.Therelationbetweenthebinnumbersusedhereandthebinsofthetwo- dimensionalDalitzplotisshowninFig. 3.Circlescorrespondtoset A,squaresto set B,andtrianglestoset C.Thesolidredlineisthestandardfitresult(withα parameter),andthedashedlineisP -waveonly.

Table 2

Dalitzplotbincontentforthethreedatasets.Therelativenormalisationbetween thesetsisbasedonthenormalisationfactors(NA,NB,NC)obtainedfromindivid- ualfitsoftheαparametertothethreedatasets.Theoverallnormalisationfactor isarbitrary.

bin# set A set B set C

1 5.51(34) 6.09(33) 6.76(45)

2 6.71(35) 6.58(34) 5.63(38)

3 5.86(35) 5.30(36) 6.23(42)

4 6.07(37) 6.68(38) 6.29(43)

5 5.31(36) 5.17(35) 5.59(39)

6 6.48(36) 5.73(34) 5.41(41)

7 4.24(29) 4.55(29) 3.91(35)

8 4.63(30) 4.41(29) 4.83(31)

9 4.47(31) 4.03(32) 4.54(33)

10 4.23(33) 4.64(36) 4.59(38)

11 4.03(33) 4.72(34) 4.25(36)

12 4.96(32) 4.85(31) 5.19(42)

13 2.25(23) 2.09(22) 2.22(28)

14 3.36(25) 3.36(25) 3.51(28)

15 2.53(24) 2.90(27) 2.55(24)

16 3.66(30) 3.90(34) 2.93(32)

17 2.52(28) 2.86(30) 2.93(37)

18 3.98(28) 3.66(28) 3.55(38)

19 2.14(21) 2.38(21) 2.60(21)

20 2.26(26) 1.89(27) 2.19(32)

21 2.63(24) 2.33(23) 2.27(32)

theDalitz plotparametrisationtothesamplesA andB withonly the

α

parameter gives

α

= (¹⁰±³⁵)·^{10−3 and}

χ

²=³⁶/39.For set C

α

= (²⁵±⁵⁹)·^{10−3 and}

χ

²=²⁴/19. Therefore the back- groundsubtractionproceduredoesnotintroduceanyexperimental bias.

Onecan alsostudythebias andaccuracyoftheefficiencyde- terminationbyconsidering X - orY -dependentcorrectionsforthe efficiency:

i→

i· (¹+ ξAX) or

i→

i· (¹+ ζAY),where ξA,..., ζA,...aresingle parametersforeachdataset.Fitstoseparate data sets show that all ζ coefficientsare consistent withzeroand do not change the value of the

χ

². On the other hand, ξB and ξC werefoundtosignificantlydeviatefromzero,althoughwithoppo- sitesigns.Applyingthesetwocorrectionstotheefficiencybeforea fitoftheDalitzplotparametrisationyieldsasignificantlyreduced

χ

²value.However,thedeterminedvaluesoftheDalitzparameters arenotaffected,e.g.

α

_{= (}147±³⁵)·^10−3and

α

= (¹⁴⁷±³⁶)·¹⁰⁻³ without and with correction, respectively. This comes from the fact that the fitted parametrisationis preserving isospin symme- try. In conclusion, we apply the X -dependentcorrections to the efficiencycorrectionsofdatasets BandC,asitensurestheantic- ipatedchargesymmetryoftheDalitzplotandleadstoadecrease