Importance of d-wave contributions in the charge symmetry breaking reaction $dd\rightarrow ^{4}He\pi^{0}$

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Physics Letters B

www.elsevier.com/locate/physletb

Importance of d-wave contributions in the charge symmetry breaking reaction dd → ^{4 He} π ⁰

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

, A. Bondar

^g,h

, M. Büscher

^i,j

, H. Calén

^a

, I. Ciepał

^k

, H. Clement

^l,m

, E. Czerwi ´nski

^c

, K. Demmich

^e

, R. Engels

ⁿ

, A. Erven

^o

, W. Erven

^o

, W. Eyrich

^p

,

P. Fedorets

^n,q

, K. Föhl

^r

, K. Fransson

^a

, F. Goldenbaum

ⁿ

, A. Goswami

^n,s

, K. Grigoryev

^n,t

, C.-O. Gullström

^a

, C. Hanhart

^n,u

, L. Heijkenskjöld

^a,1

V. Hejny

ⁿ

, N. Hüsken

^e

, L. Jarczyk

^c

, T. Johansson

^a

, B. Kamys

^c

, G. Kemmerling

^o,2

, G. Khatri

^c,3

, A. Khoukaz

^e

, O. Khreptak

^c

, D.A. Kirillov

^v

, S. Kistryn

^c

, H. Kleines

^o,2

B. Kłos

^w

, W. Krzemie ´n

^f

, P. Kulessa

^k

, A. Kup´s ´c

^a,f

, A. Kuzmin

^g,h

, K. Lalwani

^x

, D. Lersch

ⁿ

, B. Lorentz

ⁿ

, A. Magiera

^c

, R. Maier

^n,y,z

,

P. Marciniewski

^a

, B. Maria ´nski

^b

, H.-P. Morsch

^b

, P. Moskal

^c

, H. Ohm

ⁿ

, W. Parol

^k

,

E. Perez del Rio

^l,m,4

, N.M. Piskunov

^v

, D. Prasuhn

ⁿ

, D. Pszczel

^a,f

, K. Pysz

^k

, A. Pyszniak

^a,c

, J. Ritman

^n,y,z,aa

, A. Roy

^s

, Z. Rudy

^c

, O. Rundel

^c

, S. Sawant

^ab

, S. Schadmand

ⁿ

,

I. Schätti-Ozerianska

^c

, T. Sefzick

ⁿ

, V. Serdyuk

ⁿ

, B. Shwartz

^g,h

, K. Sitterberg

^e

, T. Skorodko

^l,m,ac

, M. Skurzok

^c

, J. Smyrski

^c

, V. Sopov

^q

, R. Stassen

ⁿ

, J. Stepaniak

^f

,

E. Stephan

^w

, G. Sterzenbach

ⁿ

, H. Stockhorst

ⁿ

, H. Ströher

^n,y,z

, A. Szczurek

^k

, A. Trzci ´nski

^b

, M. Wolke

^a

, A. Wro ´nska

^c

, P. Wüstner

^o

, A. Yamamoto

^ad

, J. Zabierowski

^ae

, M.J. Zieli ´nski

^c

, J. Złoma ´nczuk

^a

, P. ˙Zupra ´nski

^b

, M. ˙Zurek

^n,∗

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,EdinburghEH9 3FD,UnitedKingdomofGreat BritainandNorthernIreland

eInstitutfürKernphysik,WestfälischeWilhelms-UniversitätMünster,Wilhelm-Klemm-Str. 9,48149 Münster,Germany fHighEnergyPhysicsDepartment,NationalCentreforNuclearResearch,ul.Hoza69,00-681,Warsaw,Poland gBudkerInstituteofNuclearPhysicsofSBRAS,11akademikaLavrentievaprospect,Novosibirsk,630090,Russia hNovosibirskStateUniversity,2PirogovaStr.,Novosibirsk,630090,Russia

iPeterGrünbergInstitut,PGI-6ElektronischeEigenschaften,ForschungszentrumJülich,52425Jülich,Germany

jInstitutfürLaser- undPlasmaphysik,Heinrich-HeineUniversitätDüsseldorf,Universitätsstr. 1,40225Düsseldorf,Germany kTheHenrykNiewodnicza´nskiInstituteofNuclearPhysics,PolishAcademyofSciences,Radzikowskiego152,31-342Kraków,Poland lPhysikalischesInstitut,Eberhard-Karls-UniversitätTübingen,AufderMorgenstelle 14,72076Tübingen,Germany

mKeplerCenterfürAstro- undTeilchenphysik,PhysikalischesInstitutderUniversitätTübingen,AufderMorgenstelle14,72076Tübingen,Germany nInstitutfürKernphysik,ForschungszentrumJülich,52425Jülich,Germany

oZentralinstitutfürEngineering,ElektronikundAnalytik,ForschungszentrumJülich,52425Jülich,Germany

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

qInstituteforTheoreticalandExperimentalPhysicsnamedbyA.I.AlikhanovofNationalResearchCentre“KurchatovInstitute”,25BolshayaCheremushkinskaya, Moscow,117218,Russia

rII.PhysikalischesInstitut,Justus-Liebig-UniversitätGießen,Heinrich-Buff-Ring16,35392Giessen,Germany

sDepartmentofPhysics,IndianInstituteofTechnologyIndore,KhandwaRoad,Simrol,Indore- 453552,MadhyaPradesh,India

tHighEnergyPhysicsDivision,PetersburgNuclearPhysicsInstitutenamedbyB.P.KonstantinovofNationalResearchCentre“KurchatovInstitute”,1mkr.Orlova roshcha,LeningradskayaOblast,Gatchina,188300,Russia

uInstituteforAdvancedSimulation,ForschungszentrumJülich,52425Jülich,Germany

vVekslerandBaldinLaboratoryofHighEnergy Physics,JointInstituteforNuclearPhysics,6Joliot-Curie,Dubna,141980,Russia wAugustChełkowskiInstituteofPhysics,UniversityofSilesia,Uniwersytecka4,40-007,Katowice,Poland

xDepartmentofPhysics,MalaviyaNationalInstituteofTechnologyJaipur,JLNMargJaipur- 302017,Rajasthan,India yJARA-FAME,JülichAachenResearchAlliance,ForschungszentrumJülich,52425Jülich,Germany

zRWTHAachen,52056Aachen,Germany

aaInstitutfürExperimentalphysikI,Ruhr-UniversitätBochum,Universitätsstr.150,44780Bochum,Germany

https://doi.org/10.1016/j.physletb.2018.04.037

0370-2693/©^{2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

abDepartmentofPhysics,IndianInstituteofTechnologyBombay,Powai,Mumbai- 400076,Maharashtra,India acDepartmentofPhysics,TomskStateUniversity,36LeninaAvenue,Tomsk,634050,Russia

adHighEnergyAcceleratorResearchOrganisationKEK,Tsukuba,Ibaraki305-0801,Japan aeAstrophysicsDivision,NationalCentreforNuclearResearch,Box447,90-950Łód´z,Poland

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

Articlehistory:

Received8January2018

Receivedinrevisedform18April2018 Accepted19April2018

Availableonline23April2018 Editor:V.Metag

Keywords:

Chargesymmetrybreaking Deuteron–deuteroninteractions Pionproduction

Thisletterreportsafirstquantitativeanalysisofthecontributionofhigherpartialwavesinthecharge symmetrybreakingreactiondd→^4Heπ⁰usingtheWASA-at-COSYdetectorsetupatanexcessenergyof Q =^{60 MeV.Thedetermined}differentialcrosssectioncanbeparametrizedas dσ/d=^a+^bcos2θ^∗, whereθ^∗istheproductionangleofthepioninthecenter-of-masscoordinatesystem,andtheresultsfor the parametersarea=

1.55±⁰.46(stat)⁺₋⁰₀^._.³²₈(syst)



pb/sr andb=

13.1±².1(stat)⁺₋¹₂^._.⁰₇(syst)

 pb/sr.

Thedataarecompatiblewithvanishingp-wavesandasizabled-wavecontribution.Thisfindingshould stronglyconstrainthecontributionoftheisobartothedd→^4Heπ⁰reactionandis,therefore,crucial foraquantitativeunderstandingofquarkmasseffectsinnuclearproductionreactions.

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

1. Introduction

WithintheStandardModelofelementaryparticlesisospinsym- metryisviolatedviaquarkmassdifferencesaswellaselectromag- neticeffects [1–3].Onthehadroniclevelthisisreflected,forexam- ple,bytheproton–neutronmassdifference.Itisduetoquark-mass effectsthat theproton islighter thanthe neutronand, therefore, stable. The observationofisospin violation(IV) inhadronicreac- tionsinprincipleallowsonetostudytheeffectsofquark masses.

However,mostexperimentalsignaturesofIVaredominatedbythe pionmassdifferencemπ⁰−mπ^±,whichistoaverygoodapprox- imationofpurelyelectromagneticorigin.Anexceptionareobserv- ablesthatarecharge symmetrybreaking(CSB).Chargesymmetry, a subgroup ofisospin symmetry, is the invariance ofthe Hamil- tonian underrotation by 180^◦ around thesecond axis inisospin spacethatinterchangesup anddown quarks.The chargesymme- tryoperatordoesnotinterchangechargedandneutralpionstates, andthepionmassdifferencedoesnotenter(see,e.g., [4]).Onthe basis of theoretical approaches witha direct connection to QCD, likelatticeQCD andchiralperturbationtheory(ChPT),itis,there- fore,possibletolinkquark-masseffectstohadronicobservables.

While CSB observables have the advantage of being directly related to quark-mass differences, their smallness poses an ex- perimental challenge. First precision measurements of CSB were reportedforthereactiondd→^4He

π

⁰atbeamenergiesveryclose to the reaction threshold [5] and, at the same time, via a non- vanishing forward–backward asymmetry in np→^d

π

⁰ [6]. Both results triggered a series of theoretical investigations. The signal of the latter measurement was shown to be proportional to the quark-mass-inducedpartoftheproton–neutronmassdifferenceup tonext-to-leadingorderinChPT [7,8].Thisbecamepossiblebythe adaptionofChPTtopionproductionreactionsinRef. [9].Thefor- malismhasrecentlybeenpushedtonext-to-next-to-leadingorder fors-waves [10,11].Thecontributionof p-waveshasbeeninvesti- gatedinRef. [12].ForarecentreviewseeRef. [13].

*

Correspondingauthor.

E-mailaddress:m.zurek@fz-juelich.de(M. ˙Zurek).

1 Presentaddress:InstitutfürKernphysik,JohannesGutenberg-UniversitätMainz, Johann-Joachim-BecherWeg 45,55128Mainz,Germany.

2 Presentaddress:Jülich Centrefor Neutron ScienceJCNS,Forschungszentrum Jülich,52425Jülich,Germany.

3 Presentaddress:DepartmentofPhysics,HarvardUniversity,17 OxfordSt.,Cam- bridge,MA 02138,USA.

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

For the reaction dd→^4He

π

⁰ the four-nucleon interaction in initial and final state adds an additional facet. First steps to- wards a theoretical understanding of thisreaction were taken in Refs. [14,15]. Additional CSBeffectsfromsoftphotons intheini- tial state have been studied in Refs. [16,17]. The focus in that work hasbeenon s-wavesinthefinalstate,since noexperimen- talinformationonhigherpartialwaveswasavailableatthattime.

However,such informationisimportant,sinceitwillallowoneto constrainthecontributionfromtheresonancethatisknownto provide thebulk ofthe p-wavecontributions inthe isospincon- serving pp→^d

π

⁺ reaction [18–20] —withoutthis,aquantitative controlofhigherorderoperators forthereactionathandappears impossible.AfirstmeasurementwithWASAwasinconclusivedue tolimitedstatistics[21].Thus,therearenotheoreticalpredictions forhigherenergies and/orhigherpartialwavesyet.Inthispaper, dataarepresentedforthefirsttimethatquantifythecontribution ofhigherpartialwavestothereactiondd→^4He

π

⁰.

2. Experiment

The ten-week-long experiment was performed at the Cooler Synchrotron COSY [22] ofthe Institute forNuclear Physics atthe Forschungszentrum Jülich in Germany. The particles produced in the collisions of a deuteron beam with a momentum of p_d = 1.2 GeV/c ( Q =^{60 MeV)with frozen deuteron pellets were de-} tected inthe modified WASAfacility[23]. Thesetup consistedof forward and central detectors, where the ⁴He ejectiles and the photons from the

π

⁰ decaywere detected, respectively. Forthis experimenttheforwarddetectorwasoptimizedforatime-of-flight (TOF)measurement.Severallayersoftheoriginaldetectorwerere- moved to introduce a free flight path of more than 1.5 m. This modificationprovidesaccesstoanadditional,independentobserv- ableforenergycalibrationandparticleselection—intheprevious measurement [21] thesewerebasedonlyonthecorrelationofen- ergy lossesinthe detectorlayers. Thenewsetup consistedofan arrayofstrawtubesforprecisetrackingandthreelayersofplastic scintillators for energy reconstruction and particle identification:

two 3 mm thick layers of the forward window counter,used as startdetectors,andthe20 mm thicklayeroftheforwardvetoho- doscope,usedasastopdetector.Photonsfromthe

π

⁰decaywere detected in the centralelectromagnetic calorimeteranddiscrimi- nated fromchargedparticles by means ofa vetosignal fromthe plasticscintillatorbarrellocatedinsidethecalorimeter.

Themaintriggerrequiredahighenergydepositinatleastone element ofthefirst andthesecond layerofthe forwardwindow

counterandatleastoneclusteroriginatingfromaneutralparticle inthecentraldetector.

3. Analysis

The signature ofthe dd→^4He

π

⁰ reactionis a forward-going 4He particleandtwophotonsfromthedecayofthe

π

⁰.Theonly otherchannel with⁴He and twophotonsinthefinal state isthe double radiative capture reaction dd→^4He

γ γ

asan irreducible physicsbackground.Afurthersourceofbackgroundistheisospin symmetry conserving dd→^3Hen

π

⁰ reaction with a more than four orders of magnitudelarger cross section [24]. The suppres- sionofthisreactionischallengingsince³He and⁴He havesimilar, giventhedetectorresolution,energylossesintheforwardwindow counters.Comparedto dd→^3Hen

π

⁰,thedirecttwo photon pro- ductionindd→³Hen

γ γ

issuppressedbyafactorof

α

² (with

α

beingthefine-structureconstant)andcanbeneglected.

TheenergylossintheforwardwindowcountersandTOFhave beenused to reconstructthe kinetic energy ofthe outgoing ³He and⁴He particlesbymatchingtheirpatternstoMonteCarlosim- ulations. The full four-vectors havebeen obtained usingin addi- tiontheazimuthalandpolaranglesreconstructedbytheforward trackingdetector.Forthefurtheranalysisatleastonetrackinthe forwarddetectorandatleasttworeconstructedclustersofcrystals withenergydepositedby neutralparticlesinthecentraldetector havebeenrequired.

The final candidate events have been selected by means of a kinematicfit.Thepurposeofthefitwastoimprovetheprecision ofthemeasuredkinematicvariablesandtoserveasaselectioncri- terionforbackgroundreduction.Fortheassumedreactionhypoth- esisthemeasured variables were varied within the experimental uncertaintiesuntilcertainkinematicconstraintswerefulfilled,here theoverall momentum andenergy conservation.For every event thedd→^3Hen

γ γ

anddd→^4He

γ γ

hypotheseshavebeentested separately.No additional constraint onthe invariant massof the two photons has been imposed, in order to be able to measure thesignalyieldusingthetwo-photon invariant-massdistribution.

Incase ofmore than one track in theforward detector ormore thantwoneutralclustersinthecentral detector(causedbyevent pileup or low energy satellites of the main photon clusters) the combinationwiththesmallest

χ

² fromthefithasbeenchosen.

Thereductionofthedd→^3Hen

π

⁰ backgroundby fourorders ofmagnitudehas beenmainly achievedusing a cut onthe two- dimensional cumulative probability distribution from the kine- maticfits,analogouslyasdescribedinRef. [21].Thecut hasbeen optimizedbymaximizingthestatisticalsignificanceofthe

π

⁰ sig- nalinthefinalmissingmassplot.

Thefour-momentaobtainedfromthekinematicfitofthedd→

4He

γ γ

hypothesis havebeenused tocalculate themissingmass mX forthereactiondd→^{4He X asafunctionofthe}center-of-mass productionangleθ^∗ ofthe

π

⁰.InFig.1themissingmassspectra forthefourangular binswithin thedetectoracceptance (−⁰.9≤ cosθ^∗≤⁰.4)arepresented.Onasmoothbackgroundfromdouble radiative capture dd→^4He

γ γ

two significant peaks are visible.

One of these, originating from the signal reaction dd→^4He

π

⁰, islocatedatthe

π

⁰ massof 0.135 GeV/c².The other one corre- spondstomisidentifiedeventsfromthebackgroundreactiondd→

3Hen

π

⁰andisshiftedbythe³He−^{n bindingenergy.Themissing} massspectrahavebeenfittedwithalinearcombinationofthefol- lowinghigh-statisticsMonteCarlotemplates:(i)dd→^4He

γ γ

as- suminga3-bodyphase-spacedistribution,(ii) dd→^3Hen

π

⁰using themodelfrom[24],and(iii)thetwo-bodyreactiondd→^4He

π

⁰. Foreachcosθ^∗ bin,afitoftheMonteCarlotemplatestothedata hasbeenperformedwiththeconstraintthatthesumofthefitted templateshastofit theoverallmissingmassspectrum.As result,

Fig. 1. Missingmassforthedd→⁴HeX reactionforthefourangularbinsofthe productionangleofthepioninthecenter-of-masssystem.Thespectrumisfitted withalinearcombinationofthesimulatedsignalandbackgroundreactions:double radiativecapturedd→⁴Heγ γ(greendashedline),plusdd→³Henπ⁰(bluedotted line),plusdd→⁴Heπ⁰(redsolidline).Thefitexcludesthemissingmassregion below0.11 GeV/c².(Forinterpretationofthecolorsinthefigure(s),thereaderis referredtothewebversionofthisarticle.)

the

π

⁰ peak from the dd→^4He

π

⁰ reaction contains 336±⁴³ eventsintotal.

In thecourse of the fitthe Monte Carlotemplates have been modifiedintwoways.Inthemissingmassspectra,thebackground originatingfrommisidentified dd→^3Hen

π

⁰ is slightlyshifted in comparisontodata.Thisshiftcanbeattributedtosystematicdif- ferencesinthesimulateddetectorresponse for⁴He andmisiden- tified ³He. Witha cut efficiency closeto 10⁻⁴ the latter mainly originate fromthe tailsofthecorresponding distributions.Never- theless, the shape of background contribution is well described.

Therefore,thismismatchhasbeencompensatedbyintroducingan angle-dependent scaling factor in the missing mass m_X as free parameter. The obtained factors (from backward to forward an- gles) are within the range of 1.005–0.972. The second modifica- tion concerns the missingmass spectrum below 0.11 GeV/c² in the mostbackward angular bin.Thisregion is dominated by the dd→^4He

γ γ

reaction, which has been simulated using 3-body phasespace.Thismodel,however,underestimatesthecontribution inthatregion.Thedominatingbackgroundfromthedd→^3Hen

π

⁰

reactionathighermissingmassespreventsdescribingallcontribu- tionspreciselyenoughtoverifymoreadvancedmodels.Foracon- sistent descriptioninallangularbins,forthefinal fitthemissing mass rangebelow 0.11 GeV/c² hasbeen excluded in all angular bins.

For the final acceptance correction, the dd→^4He

π

⁰ gener- ator with the angular distribution obtained in this analysis has been used. The integrated luminosity has been calculated us- ing the dd→^3Hen

π

⁰ reaction, based on a measurement with WASA at pd=¹.2 GeV/c [24]. It equals to (37.2±³.7(norm)± 0.1(syst)) pb⁻¹, which is about 7.5 times larger than the value fromthepreviousmeasurementwithWASAreportedinRef. [21].

Thestabilityoftheresultshasbeentestedagainstvariationsof allselection cuts,accordingtomethoddescribed inRef. [25].Out ofthese,theonlystatisticallysignificanteffecthasbeenobserved with the variation of the cumulative probability distribution cut andaddedassystematicuncertainty.Thesensitivityoftheoverall fithasbeencheckedby varyingthe fitparameters,especially the linearscalingfactorinmX,andusingsmoothanalyticfunctionsto reproducetheshapeofbackgroundatlowmissingmasses.Nosig- nificantchangeintheresulthasbeenobservedwhilemaintaining

thegoodness-of-fitinthepeakregion.Thus,nosystematicuncer- tainty hasbeenassigned here.The erroron thenormalizationto thedd→^3Hen

π

⁰reactionhasbeenpropagatedtothefinalresult.

4. Results

Fig. 2 presents the obtained differential cross section. Since identicalparticles inthe initialstate require aforward–backward symmetric cross section, it has been fitted using the function d

σ

/d=^a+^bcos2θ^∗ resultingin:

a

= 

1

.

55

±

⁰

.

46

(

stat

)

⁺₋⁰₀^._.³²₈

(

syst

)



pb

/

sr

,

(1a)

b

= 

13

.

1

±

2

.

1

(

stat

)

⁺₋¹₂^._.⁰₇

(

syst

)



pb

/

sr

.

(1b)

Both parameters have an additional, common systematic uncer- taintyofabout10% fromnormalization.

Thetotalcrosssection obtainedastheintegralofthefunction fittedtotheangulardistributionamountsto:

σ

tot

= 

74

.

3

±

⁶

.

8

(

stat

)

⁺₋¹₁₀^.2_.1

(

syst

) ±

⁷

.

7

(

norm

)



pb

.

(2)

Fig.3showstheresultingmomentumdependenceofthereaction amplitude(p/_pπ0)

σ

totincludingthedatafromRef. [5].Here, pπ⁰ isthemomentumofthepionandp istheincident-deuteronmo- mentum,bothinthecenter-of-masssystem.

The cross sections are systematically smaller than the results reported in Ref. [21], however, consistent within errors. In view of the limited statistics a decisive analysis of this difference is difficult. As most probable reason our studies identified the im- plementation of nuclear interactions of ³He in the Monte Carlo simulations. Itwas foundthat this effectwas not properlytaken intoaccountinthe analysisofthe previousdata.Thisresultedin anincreased(simulated)detectionefficiencyforthenormalization reactionand,consequently,inatoolowluminosity.Astheeffectis thelargestforthestoppinglayer,theanalysisofthecurrentdata setislesssensitiveasitisbasedonaTOFmeasurementanddoes notrelyonenergycorrelationsonly.

Forafurtheranalysisofthedifferential crosssectioninterms ofpartialwavesinthefinalstate,theformalismfromRef. [26] has beenused.Consideringonlys- and p-waves theparameterb can bewrittenas:

b

= −

^pπ0 p

2

3

|

C

|

²p²_π0

,

(3)

whereC isthe p-waveamplitude. Notethat thesymmetry ofthe initialstaterequiresthatonlypartialwavesofthesameparityin- terfere.Upto thisorder, p-wavescontribute withanegative sign correspondingtoamaximumatθ^∗=^{90◦ intheangulardistribu-} tion.Theobservedminimumcanonlybeexplainedextendingthe formalism tod-wavesin thefinal state. Therefore,thesedata es- tablishforthefirsttime thepresenceofasizablecontributionof d-wavestothedd→^4He

π

⁰ reaction,whichhavesofarnot been consideredinthetheoreticalcalculations.

Aconsistentdescriptionthat includesd-waveshastoconsider termsup to fourthorder inpionmomentum. FollowingRef. [26]

thedifferentialcrosssectioncanbewrittenas:

d

σ

d

 =

^pπ0 p

2 3

 |

A0

|

²

+

2 Re

(

A^∗₀A2

)

P2

(

cos

θ

^∗

)

p²_π0

+ |

A2

|

²P²₂

(

cos

θ

^∗

)

p⁴_π0

+ |

C

|

²sin²

θ

^∗p²_π0

+ |

^B

|

^2sin2

θ

^∗cos²

θ

^∗p⁴_π0



.

(4)

Fig. 2. Angulardistributionofthedd→^4Heπ⁰reactionatQ=^{60 MeV.Theresult} ofthefituptosecondorderincosθ^∗isshownwithadottedcurve.Thesystematic errorsofthefitarepresentedasagrayband.Thehorizontalerrorbarsindicatethe binwidth.

Fig. 3. Thedd→⁴Heπ⁰ reactionamplitudesquared(p/p_π0)σtot asafunctionof η=p_π0/m_π0.Thecirclesrepresenttheresultsfrom[5],thesquarecorrespondsto thefinalresultforthetotalcrosssectionfromthiswork,andthetrianglerepresents thecrosssectionfromthepreviousWASAmeasurement[21].Notethattheresult from[21] hasbeenobtainedassumingpures-wave.Theerrorbarsshowthecom- binedstatisticalandsystematicuncertainties.FortheresultsobtainedwithWASA theerrorbarswithsubtractedcommonuncertaintyoriginatingfromnormalization arealsopresented.Thedottedcurveindicatesthemomentumdependenceofthe totalcrosssectionfromEq. (5) withthefittedamplitudesfromEq. (6).

Here, A₀ isthe s-waveamplitude, A₂ and B arethed-wave am- plitudes, and P2 is the second order Legendre polynomial. The correspondingexpressionforthetotalcrosssectionreads:

σ

tot

=

^pπ0 p

8

π

3

 |

^A0

|

²

+

²

3

|

^C

|

^2p2_π0

+

¹

5

|

^A2

|

^2p4_π0

+

² 15

|

^B

|

^2p4_π0

 .

(5) Since a full fit with four independent amplitudes and one rela- tive phaseisoutsidethescopeofthepresenteddata,quantitative resultscanonlybeobtainedusingadditionalconstraints.Anunbi- ased determinationoftheamplitudesisnot possibleunderthese circumstances,thus,thefocusisonthecorrelationsbetweenthem.

If one assumes that the amplitude A0 does not carry any momentum dependence, it can be extracted from the results in Ref. [5] where s-wave is by far dominating. The obtained value is|^A0|thr= (⁵.74±⁰.38(stat)) (pb/sr)^1/2,whichcanthenbeused as fixed parameter in the fit of the angular distribution at Q = 60 MeV.Furthermore,systematicstudiesofthebehavior ofthefit with respect to B and the relative phase δ between A₀ and A₂ (i.e., {^A∗₀^A2}= |^A0||^A2|^cosδ) show that the data are not sen- sitive to |^B| ^and δ, which have comparatively large errors and are consistent with zero. For example, the fit with the param-

eters |^A2|^, |^B|^, |^C| ^{free and} δ fixed to zero results in |^B|=



150⁺₋¹³⁰₄₂₀(stat)



(pb/sr)^1/2(GeV/c)⁻²,andthefitwith|^A2|^,δ,|^C| freeand|^B|^{fixed to zeroresultsin}δ=⁰±⁰.66(stat).Moreover, theparameters |^C|^and |^A2| ^{fromboth fits are consistent within} theuncertainties.Consequently,both|^B|^andδhavebeenfixedto zero.

From the final fit ofthe angular distribution at Q =^{60 MeV} withalldescribedconstraintsthefollowingamplitudeshavebeen extracted:

|

^A2

| = 

258⁺₋⁵⁰₄₂

(

stat

)

⁺₋⁴⁵₃₈

(

syst

)

⁺₋³⁷₁₂

(

norm

)

 (

pb

/

sr

)

^1/2

(

GeV

/

c

)

²

,

(6a)

|

^C

| = 

6⁺₋⁹₂₁

(

stat

)

⁺₋³₁₀

(

syst

)

⁺₋¹⁰₅

(

norm

)

 (

pb

/

sr

)

^1/2

GeV

/

c

.

(6b)

The asymmetric statistical errors are a consequence of the non- linearityofthefitfunction.

Fig.4showsacorrelationplotbetweentheparameters|^C|^and

|^A2|^{.Thecenterpointmarkedwithacrossshowstheresultfrom} Eq. (6).Theshadedareasindicatethe68%and95% confidencere- gions.Thedottedlineshowsthedependenceofthecentralvalues for|^C|^and|^A2|^on|^A0|^{—somevaluesfor}|^A0|^{areshownexplic-} itlyinthefigure.Theminimal total

χ

² asafunction ofthefixed valueof |^A0|^{ispresented inFig.5.At} |^A0|=⁵.81(pb/sr)^1/2 the p-wavecontributiongivenbytheparameter|^C|^{vanishes.A further} increase of |A0| still keeps |C| at 0at the cost ofthe goodness- of-fit. One can see that the fit to the data has the tendency to maximize |^A0| ^{and, thus, minimize} |^C|^{. This maximum value of}

|^A0|^{isconsistent withthe oneobtainedfrom}Ref. [5] supporting the assumption of a momentum independent s-wave amplitude.

Furthermore,when|^C|^{vanishes and}|^A0|^{hasitsmaximumvalue,} thecorrespondingminimal|^A2|^valuestillsignificantlydiffersfrom zero.Evenifone allows|^A0|^{todropwithincreasingmomentum,} thisiscompensated by largervalues of |^C|^{to maintain the total} crosssection. At the sametime the value of |^A2| ^{also increases,} i.e.,thed-wavecontributionwouldbecomeevenlarger.

5. Summary

Insummary, this letterreports for the first time a successful measurementofhigherpartialwavesinthedifferentialcrosssec- tionof thecharge symmetry violating reactiondd→^4He

π

⁰.The datawithaminimumatθ^∗=^{90◦ can beunderstoodonlybythe} presenceofasignificantd-wave contributioninthefinal state.At thesame time they are consistent witha vanishing p-wave. Ex- istingtheoreticalcalculationstodescribethereactiondd→^4He

π

⁰

withinChiralPerturbationTheoryarelimitedtos-wavepionpro- duction. There are first considerations to extend these efforts to p-wavesinthefinalstate,however,thepresenteddatashow that thisisnotsufficient.

It is well known from phenomenology aswell asstudies us- ingeffectivefield theorythat theisobarplays a crucialrolein pionproductionreactions,especiallyforpartialwaveshigherthan s-wave [18–20]. Sinceisospinconservationdoesnotallowforthe excitationofasingle inthedd state,theappearanceofpromi- nenthigherpartialwavesindd→^4He

π

⁰ mightpointatanisospin violatingexcitation oftheisobar.Thisindicates that atheoret- ical analysisof the datapresented in the lettershould allow for deepinsightsnot onlyintothe dynamicsofthenucleon–nucleon interaction butalso intothe role ofquark masses in hadron dy- namics.

Fig. 4. Correlationplotfortheparameters|C|and|A2|determinedfromthefitof theangulardistributionofdd→⁴Heπ⁰at Q=60 MeV.Thecenterpointmarked withthecrossshowstheresultfromEq. (6).Theshadedareasindicatethe68%and 95%confidenceregions.Thedottedlineshowstheinfluenceofavariationof|^A0| on|^C|^and|^A2|^{,withthecirclepoints} representingtheresultsfor theindicated valuesof|^A0|^.

Fig. 5. Minimaltotalχ²fromthefitoftheangulardistributionofdd→^4Heπ⁰at Q=60 MeV asafunctionofthefixedvalueofthe s-waveamplitude|A0|.The dottedlineindicatesthevalueof|A0|forwhichthep-wavecontributiongivenby theparameter|C|vanishes.Afurtherincreaseof|A0|stillkeeps|C|at0atthecost ofthegoodness-of-fit.

Acknowledgements

We would like to thank the technical staff of the COoler SYnchrotron COSY. We thank C. Wilkin for valuable discussions.

This work was supported in part by the EU Integrated Infras- tructure Initiative Hadron Physics Project under contract number RII3-CT-2004-506078;bytheEuropeanCommissionunderthe7th FrameworkProgrammethroughtheResearchInfrastructuresaction oftheCapacitiesProgramme,Call:FP7-INFRASTRUCTURES-2008-1, GrantAgreementNo.227431;bythePolishNationalScienceCen- trethroughthegrants2016/23/B/ST2/00784,2014/15/N/ST2/03179, DEC-2013/11/N/ST2/04152,andthe Foundation forPolish Science (MPD), co-financed by the European Union within the European Regional Development Fund. We acknowledge the support given by the SwedishResearch Council,the Knutand AliceWallenberg Foundation, and the Forschungszentrum Jülich FFE Funding Pro- gram.ThisworkisbasedonthePhDthesisofMaria ˙Zurek.

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