Odd and even partial waves of ηπ − and η π − in π −p → η ( ) π −p at 191 GeV / c

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Physics Letters B

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Odd and even partial waves of ηπ ⁻ and η ^ π ⁻ in π ⁻ p → η ^{( )} π ⁻ p at 191 GeV / c

COMPASS Collaboration

C. Adolph

^h

, R. Akhunzyanov

^g

, M.G. Alexeev

^aa

, G.D. Alexeev

^g

, A. Amoroso

^aa,ac

, V. Andrieux

^v

, V. Anosov

^g

, A. Austregesilo

^j,q

, B. Badełek

^ae

, F. Balestra

^aa,ac

, J. Barth

^d

, G. Baum

^a

, R. Beck

^c

, Y. Bedfer

^v

, A. Berlin

^b

, J. Bernhard

^m

, K. Bicker

^j,q

, E.R. Bielert

^j

, J. Bieling

^d

, R. Birsa

^y

, J. Bisplinghoff

^c

, M. Bodlak

^s

, M. Boer

^v

, P. Bordalo

^l,1

,

F. Bradamante

^x,y

, C. Braun

^h

, A. Bressan

^x,y,∗

, M. Büchele

ⁱ

, E. Burtin

^v

, L. Capozza

^v

, M. Chiosso

^aa,ac

, S.U. Chung

^q,2

, A. Cicuttin

^z,y

, M.L. Crespo

^z,y

, Q. Curiel

^v

, S. Dalla Torre

^y

, S.S. Dasgupta

^f

, S. Dasgupta

^y

, O.Yu. Denisov

^ac

, S.V. Donskov

^u

, N. Doshita

^ag

, V. Duic

^x

, W. Dünnweber

^p

, M. Dziewiecki

^af

, A. Efremov

^g

, C. Elia

^x,y

, P.D. Eversheim

^c

, W. Eyrich

^h

, M. Faessler

^p

, A. Ferrero

^v

, M. Finger

^s

, M. Finger Jr.

^s

, H. Fischer

ⁱ

, C. Franco

^l

,

N. du Fresne von Hohenesche

^m,j

, J.M. Friedrich

^q

, V. Frolov

^j

, F. Gautheron

^b

,

O.P. Gavrichtchouk

^g

, S. Gerassimov

^o,q

, R. Geyer

^p

, I. Gnesi

^aa,ac

, B. Gobbo

^y

, S. Goertz

^d

, M. Gorzellik

ⁱ

, S. Grabmüller

^q

, A. Grasso

^aa,ac

, B. Grube

^q

, T. Grussenmeyer

ⁱ

, A. Guskov

^g

, F. Haas

^q

, D. von Harrach

^m

, D. Hahne

^d

, R. Hashimoto

^ag

, F.H. Heinsius

ⁱ

, F. Herrmann

ⁱ

, F. Hinterberger

^c

, Ch. Höppner

^q

, N. Horikawa

^r,4

, N. d’Hose

^v

, S. Huber

^q

, S. Ishimoto

^ag,5

, A. Ivanov

^g

, Yu. Ivanshin

^g

, T. Iwata

^ag

, R. Jahn

^c

, V. Jary

^t

, P. Jasinski

^m

, P. Jörg

ⁱ

, R. Joosten

^c

, E. Kabuß

^m

, B. Ketzer

^q,6

, G.V. Khaustov

^u

, Yu.A. Khokhlov

^u,7

, Yu. Kisselev

^g

, F. Klein

^d

, K. Klimaszewski

^ad

, J.H. Koivuniemi

^b

, V.N. Kolosov

^u

, K. Kondo

^ag

, K. Königsmann

ⁱ

, I. Konorov

^o,q

, V.F. Konstantinov

^u

, A.M. Kotzinian

^aa,ac

, O. Kouznetsov

^g

, M. Krämer

^q

, Z.V. Kroumchtein

^g

, N. Kuchinski

^g

, F. Kunne

^v,∗

, K. Kurek

^ad

, R.P. Kurjata

^af

, A.A. Lednev

^u

, A. Lehmann

^h

, M. Levillain

^v

, S. Levorato

^y

, J. Lichtenstadt

^w

, A. Maggiora

^ac

, A. Magnon

^v

, N. Makke

^x,y

, G.K. Mallot

^j

, C. Marchand

^v

, A. Martin

^x,y

, J. Marzec

^af

, J. Matousek

^s

, H. Matsuda

^ag

, T. Matsuda

ⁿ

, G. Meshcheryakov

^g

, W. Meyer

^b

, T. Michigami

^ag

, Yu.V. Mikhailov

^u

, Y. Miyachi

^ag

, A. Nagaytsev

^g

, T. Nagel

^q

, F. Nerling

^m

, S. Neubert

^q

, D. Neyret

^v

, J. Novy

^t

, W.-D. Nowak

ⁱ

, A.S. Nunes

^l

, A.G. Olshevsky

^g

, I. Orlov

^g

, M. Ostrick

^m

, R. Panknin

^d

, D. Panzieri

^ab,ac

, B. Parsamyan

^aa,ac

, S. Paul

^q

, D.V. Peshekhonov

^g

,

S. Platchkov

^v

, J. Pochodzalla

^m

, V.A. Polyakov

^u

, J. Pretz

^d,8

, M. Quaresma

^l

, C. Quintans

^l

, S. Ramos

^l,1

, C. Regali

ⁱ

, G. Reicherz

^b

, E. Rocco

^j

, N.S. Rossiyskaya

^g

, D.I. Ryabchikov

^u

, A. Rychter

^af

, V.D. Samoylenko

^u

, A. Sandacz

^ad

, S. Sarkar

^f

, I.A. Savin

^g

, G. Sbrizzai

^x,y

, P. Schiavon

^x,y

, C. Schill

ⁱ

, T. Schlüter

^p,∗

, K. Schmidt

^i,3

, H. Schmieden

^d

, K. Schönning

^j

, S. Schopferer

ⁱ

, M. Schott

^j

, O.Yu. Shevchenko

^g,19

, L. Silva

^l

, L. Sinha

^f

, S. Sirtl

ⁱ

,

M. Slunecka

^g

, S. Sosio

^aa,ac

, F. Sozzi

^y

, A. Srnka

^e

, L. Steiger

^y

, M. Stolarski

^l

, M. Sulc

^k

, R. Sulej

^ad

, H. Suzuki

^ag,4

, A. Szabelski

^ad

, T. Szameitat

^i,3

, P. Sznajder

^ad

, S. Takekawa

^aa,ac

, J. ter Wolbeek

^i,3

, S. Tessaro

^y

, F. Tessarotto

^y

, F. Thibaud

^v

, S. Uhl

^q

, I. Uman

^p

, M. Virius

^t

,

*

Correspondingauthors.

E-mailaddresses:Andrea.Bressan@cern.ch(A. Bressan),Fabienne.Kunne@cern.ch(F. Kunne),tobias.schlueter@physik.uni-muenchen.de(T. Schlüter).

http://dx.doi.org/10.1016/j.physletb.2014.11.058

0370-2693/©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP³.

L. Wang

^b

, T. Weisrock

^m

, M. Wilfert

^m

, R. Windmolders

^d

, H. Wollny

^v

, K. Zaremba

^af

, M. Zavertyaev

^o

, E. Zemlyanichkina

^g

, M. Ziembicki

^af

, A. Zink

^h

aUniversitätBielefeld,Germany bUniversitätBochum,Germany

cUniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany⁹ dUniversitätBonn,PhysikalischesInstitut,53115Bonn,Germany⁹

eInstituteofScientificInstruments,ASCR,61264Brno,CzechRepublic¹⁰

fMatrivaniInstituteofExperimentalResearch&Education,Calcutta–700030,India¹¹ gJointInstituteforNuclearResearch,141980Dubna,MoscowRegion,Russia¹² hUniversitätErlangen–Nürnberg,PhysikalischesInstitut,91054Erlangen,Germany⁹ iUniversitätFreiburg,PhysikalischesInstitut,79104Freiburg,Germany^9,16 jCERN,1211Geneva23,Switzerland

kTechnicalUniversityinLiberec,46117Liberec,CzechRepublic¹⁰ lLIP,1000-149Lisbon,Portugal¹³

mUniversitätMainz,InstitutfürKernphysik,55099Mainz,Germany⁹ nUniversityofMiyazaki,Miyazaki889-2192,Japan¹⁴

oLebedevPhysicalInstitute,119991Moscow,Russia

pLudwig-Maximilians-UniversitätMünchen,DepartmentfürPhysik,80799Munich,Germany^9,15 qTechnischeUniversitätMünchen,PhysikDepartment,85748Garching,Germany^9,15

rNagoyaUniversity,464Nagoya,Japan¹⁴

sCharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublic¹⁰ tCzechTechnicalUniversityinPrague,16636Prague,CzechRepublic¹⁰

uStateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘KurchatovInstitute’,142281Protvino,Russia vCEAIRFU/SPhNSaclay,91191Gif-sur-Yvette,France¹⁶

wTelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israel¹⁷ xUniversityofTrieste,DepartmentofPhysics,34127Trieste,Italy

yTriesteSectionofINFN,34127Trieste,Italy zAbdusSalamICTP,34151Trieste,Italy

aaUniversityofTurin,DepartmentofPhysics,10125Turin,Italy abUniversityofEasternPiedmont,15100Alessandria,Italy acTorinoSectionofINFN,10125Turin,Italy

adNationalCentreforNuclearResearch,00-681Warsaw,Poland¹⁸ aeUniversityofWarsaw,FacultyofPhysics,00-681Warsaw,Poland¹⁸

afWarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Poland¹⁸ agYamagataUniversity,Yamagata,992-8510,Japan¹⁴

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

Articlehistory:

Received21August2014

Receivedinrevisedform24November2014 Accepted30November2014

Availableonline3December2014 Editor:M.Doser

Exclusive production of ηπ⁻ and η^π⁻ has been studied with a 191 GeV/c π⁻ beam impinging on a hydrogen target at COMPASS (CERN). Partial-wave analyses reveal different odd/even angular momentum(L)characteristicsintheinspectedinvariantmassrangeupto3 GeV/c².Astrikingsimilarity betweenthetwosystemsisobservedfortheL=²,4,6 intensities(scaledbykinematicalfactors)andthe relativephases.Theknownresonancesa2(1320)anda4(2040)areinlinewiththissimilarity.Incontrast, a strongenhancementofη^π⁻overηπ⁻isfoundfortheL=¹,3,5 waves,whichcarrynon-q^{q quantum}¯ numbers.TheL=1 intensitypeaksat1.7 GeV/c²inη^π⁻andat1.4 GeV/c²inηπ⁻,thecorresponding phasemotionswithrespecttoL=^{2 aredifferent.}

©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP³.

1 AlsoatInstitutoSuperiorTécnico,UniversidadedeLisboa,Lisbon,Portugal.

2 AlsoatDepartmentofPhysics,PusanNationalUniversity,Busan609-735,RepublicofKoreaandatPhysicsDepartment,BrookhavenNationalLaboratory,Upton,NY11973, USA.

3 SupportedbytheDFGResearchTrainingGroupProgramme1102“PhysicsatHadronAccelerators”.

4 AlsoatChubuUniversity,Kasugai,Aichi,487-8501,Japan.

5 AlsoatKEK,1-1Oho,Tsukuba,Ibaraki,305-0801,Japan.

6 Presentaddress:UniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany.

7 AlsoatMoscowInstituteofPhysicsandTechnology,MoscowRegion,141700,Russia.

8 Presentaddress:RWTHAachenUniversity,III.PhysikalischesInstitut,52056Aachen,Germany.

9 SupportedbytheGermanBundesministeriumfürBildungundForschung.

10 SupportedbyCzechRepublicMEYSGrantsME492andLA242.

11 SupportedbySAIL(CSR),Govt.ofIndia.

12 SupportedbyCERN-RFBRGrants08-02-91009and12-02-91500.

13 SupportedbythePortugueseFCT–FundaçãoparaaCiênciaeTecnologia,COMPETEandQREN,GrantsCERN/FP/109323/2009,CERN/FP/116376/2010and CERN/FP/123600/2011.

14 SupportedbytheMEXTandtheJSPSundertheGrantsNos.18002006,20540299and18540281;DaikoFoundationandYamadaFoundation.

15 SupportedbytheDFGclusterofexcellence‘OriginandStructureoftheUniverse’(www.universe-cluster.de).

16 SupportedbyEUFP7(HadronPhysics3,GrantAgreementnumber283286).

17 SupportedbytheIsraelScienceFoundation,foundedbytheIsraelAcademyofSciencesandHumanities.

18 SupportedbythePolishNCNGrantDEC-2011/01/M/ST2/02350.

19 Deceased.

The

ηπ

and

η

^

π

mesonic systems are attractive for spectro- scopicstudies becauseany statewithodd angularmomentum L, whichcoincideswiththetotalspin J ,hasnon-qq (“exotic”)¯ quan- tumnumbers J^PC=¹⁻⁺,3⁻⁺,5⁻⁺,. . .. The 1⁻⁺ state has been theprincipalcasestudiedsofar[1,2].

A comparison of

ηπ

and

η

^

π

should illuminate the role of flavoursymmetry.Since

η

and

η

^aredominantlyflavouroctetand singletstates,respectively, differentSU(3)flavour configurationsare formed by

ηπ

and

η

^

π

. These configurations are linked to odd or even L by Bose symmetry [3–5]. Indeed, experimentally the diffractivelyproduced P -wave(L= ^J=^{1) in}

η

^

π

⁻ was found to bemorepronouncedthanin

ηπ

⁻[6].Amoresystematicstudyof thetwosystemsintheoddandevenpartialwavesisdesirable.

Diffractive productionof

ηπ

⁻ and

η

^

π

⁻ was studied by pre- vious experiments with

π

⁻ beams in the 18 GeV/c–37 GeV/c range[6–9].Apartfromthewell-knownresonancesa₂(1320) and a₄(2040),resonancefeatureswereobservedfortheexotic P -wave inthe 1.4 GeV/c²–1.7 GeV/c² mass range.It has quantum num- bers J^PG=^{1−−, where G-parity is used forthe charged system,} correspondingtoC= +^{1 sincetheisospinis1.Resultsforcharge-} exchangeproductionof

η

^()

π

⁰aredifficulttorelatetotheseobser- vations[1].Criticaldiscussionsoftheresonancecharacterconcern apossibledynamicaloriginofthebehaviourofthe L=^{1 wavein} thesesystems[10,11,1].

The present study is performed with a 191 GeV/c

π

⁻ beam andin the region 0.1 (GeV/c)²<−^t<1 (GeV/c)², where t de- notesthe squaredfour-momentum transfer to the proton target.

This is within the range of Reggeon-exchange processes [12,13], wherediffractiveexcitationandmid-rapidity(“central”)production coexist.Theformercaninduceexclusiveresonanceproduction.The latterwill leadto a systemofthe leading andthe centrallypro- ducedmesonswith(almost)nointeractioninthefinalstate.

InthisLetter,thebehaviourofallpartialwaveswithL=^{1–6 in} the

η

^()

π

⁻ invariantmassrangeupto 3 GeV/c² isstudied.A pe- culiardifference between

ηπ

⁻ and

η

^

π

⁻ in theeven andodd-L wavesisobserved.

ThedatawerecollectedwiththeCOMPASSapparatusatCERN.

COMPASSisatwo-stagemagneticspectrometer withtrackingand calorimetryin both stages[14,15]. A beamofnegatively charged hadronsat191 GeV/c wasimpinging ona liquidhydrogentarget of40 cm lengthand35 mm diameter.Usingtheinformationfrom beamparticleidentificationdetectors,itwascheckedthat K⁻ and

¯

p admixtures to the 97%

π

⁻ beam are insignificant in the final sampleanalysed here. Recoilingtarget protonswere identified by theirtimeofflightandenergylossinadetector(RPD)whichcon- sistedoftwocylindricalringsofscintillatingcountersatdistances of12 cm and 78 cm fromthebeam axis,covering thepolar an- gle rangeabove 50^◦ as seen fromthe target centre.The angular rangebetween the RPDand the openingangle of the spectrom- eter of about ±^{10◦ was covered mostly by a large-area photon} andcharged-particle veto detector (SW), thus enriching the data recordingwithkinematicallycompleteevents[16].Thetriggerfor takingthe presentdatarequiredcoincidencebetweenbeamdefi- nitioncountersandtheRPD, andno vetofromtheSWnorfrom a small counter telescope for non-interacting beam particles far downstream(32 m)fromthetarget.Asampleof4.5×^109events wasrecordedwiththistriggerin2008.

For the analysis of the exclusively produced

π

⁻

η

and

π

⁻

η

^

mesonicsystems,the

η

wasdetected by itsdecay

η

→

π

⁻

π

⁺

π

⁰

(

π

⁰_→

γ γ

),andthe

η

^ byitsdecay

η

^→

π

⁻

π

⁺

η

(

η

_→

γ γ

).The preselectionforthecommonfinalstate

π

⁻

π

⁻

π

⁺

γ γ

required:

(a) three tracks withtotal charge −1 reconstructedin the spec- trometer,

(b) avertex,locatedinsidethetargetvolume,withoneincoming beamparticletrackandthethreeoutgoingtracks,

(c) exactlytwo“eligible”clustersintheelectromagneticcalorime- tersofCOMPASS(ECAL1,ECAL2),and

(d) thetotalenergyEtotoftheoutgoingparticleswithina10 GeV wide windowcentredon the6 GeV FWHM peak at191 GeV inthe Etotdistribution.

Clusterswereconsidered“eligible”iftheywerenotassociatedwith a reconstructedtrack, ifthecluster energywas above 1 GeVand 4 GeVin ECAL1andECAL2, respectively, andiftheir timing with respecttothebeamwaswithin±^{4 ns.}

Sharp

η

(

η

^) peaks of widths 3 MeV/c²–4 MeV/c² were ob- tainedinthe

π

⁻

π

⁺

π

⁰ and

π

⁻

π

⁺

η

massspectraafterkinematic fittingofthe

γ γ

systemswithin ±^{20 MeV}/c² windowsaboutthe respective

π

⁰ and

η

masses. For the present four-body analyses of the systems

π

⁻

π

⁻

π

⁺

π

⁰ and

π

⁻

π

⁻

π

⁺

η

, broad windows of 50 MeV/c² widthaboutthe

η

and

η

^ masseswereapplied tothe three-body

π

⁻

π

⁺

π

⁰ and

π

⁻

π

⁺

η

systems, respectively. In this way,acommontreatmentof

η

^() andthesmallnumberofnon-

η

^()

eventsbecomes possible inthe subsequentlikelihood fit.No sig- nificantdeviationsfromcoplanarity(required toholdwithin13^◦) areobservedforthemomentumvectorsofbeamparticle,mesonic systemandrecoilproton,whichconfirmstheexclusivityofthere- action.DetailsarefoundinRefs.[17,18].

Inordertoaccountfortheacceptanceofthespectrometerand theselectionprocedure,MonteCarlosimulations[15,19]wereper- formed for four-body phase-space distributions. The latter were weighted withthe experimental t distributions, approximatedby d

σ

/dt∝ |^t|^exp(−^b|^t|) ^{with slope parameter b}=⁸.0 (GeV/c)⁻² andb=⁸.45(GeV/c)⁻² for

η

^

π

⁻ and

ηπ

⁻,respectively. Theob- served weakmass-dependenceofthe slopeparameterwas found notto affectthe presentresults.Theoverall acceptancesfor

ηπ

⁻

and

η

^

π

⁻ in the present kinematic range and decay channels amountedto10% and14%,respectively.Duetothelargecoverage of forwardsolid angle by the COMPASS spectrometer, the accep- tancesvarysmoothlyovertherelevantregionsofphasespace,see Ref. [20].A test ofthe Monte Carlodescription was provided by comparison to a five-charged-track sample where

η

^ decays via

π

⁺

π

⁻

η

(

η

_→

π

⁺

π

⁻

π

⁰). The known branching ratioof

η

decay into

γ γ

and

π

⁻

π

⁺

π

⁰ was reproduced [18] leading toa conser- vativeestimateof8%fortheuncertaintyoftherelativeacceptance ofthetwochannelsdiscussedhere.

To visualise the gross features of the two channels, subsam- ples ofeventswereselectedwithtight ±^{10 MeV}/c² windows on the

η

and

η

^ masses. These contain 116 000 and 39 000 events, respectively,including5% backgroundfromnon-

η

^() events.These subsamplesareshownasfunctionofthe

ηπ

⁻and

η

^

π

⁻masses in Figs. 1(a)and(b),andadditionallyinthescatterplotsFigs. 2 (a) and (b) as a function of these invariant masses and of cosϑGJ, whereϑGJ istheanglebetweenthedirectionsofthe

η

^() andthe beamas seenin the centreof massof the

η

^()

π

⁻ system (polar anglein the Gottfried–Jackson frame). Thesedistributions are in- tegratedover |^t|^from0.1(GeV/c)² to1.0(GeV/c)² andover the azimuth

ϕ

GJ (measured withrespect to the reaction plane). The

ϕ

GJ distributionsare observedto followcloselyasin²

ϕ

GJ pattern throughoutthemassrangescoveredinbothchannels[18,20].

Several salientfeatures of the intensity distributions in Fig. 2 are noted before proceeding to the partial-wave analysis. In the

ηπ

⁻ data, the a2(1320) with its two-hump D-wave angular distribution is prominent, see also Fig. 1 (a). The D-wave pat- tern extends to 2 GeV/c² where interference with the a₄(2040) can be discerned. For higher masses, increasingly narrow for- ward/backward peaks are observed. This feature corresponds to the emergence of a rapidity gap. In terms of partial waves it

Fig. 1. Invariant mass spectra (not acceptance corrected) for (a)ηπ⁻and (b)η^π⁻. Acceptances (continuous lines) refer to the kinematic ranges of the present analysis.

Fig. 2. Data(notacceptancecorrected)asafunctionoftheinvariantηπ⁻(a)andη^π⁻(b)masses andofthecosineofthedecayangleintherespectiveGottfried–Jackson frameswherecosϑGJ=1 correspondsη^()emissioninthebeamdirection.Two-dimensionalacceptancescanbefoundinRef.[20].

indicates coherent contributions from larger angular momenta.

Forward/backward asymmetries (only weakly affected by accep- tance) occur for all masses in both channels, which indicates interferenceofoddandevenpartialwaves.Inthe

η

^

π

⁻ data,the a₂(1320)isclosetothethresholdenergyofthischannel(1.1 GeV), andthesignalisnotdominant,seealsoFig. 1(b).Aforward/back- wardasymmetricinterferencepattern,indicatingcoherent D- and P -wave contributions with mass-dependent relative phase, gov- ernsthe

η

^

π

⁻massrangeupto2 GeV/c².Inthea4(2040)region, well-localisedinterferenceisrecognised. As for

ηπ

⁻,narrow for- ward/backwardpeakingoccursathighermass,butinthiscasethe forward/backwardasymmetryisvisiblylargeroverthewholemass rangeof

η

^

π

⁻.

Thedataweresubjectedtoapartial-waveanalysis(PWA)using aprogramdevelopedatIllinoisandVES[21–23].Independentfits were carried out in 40 MeV/c² wide bins of thefour-body mass fromthresholdupto3 GeV/c² (so-calledmass-independentPWA).

Momentumtransferswerelimitedtotherangegivenabove.

An

η

^()

π

⁻ partial-wave is characterised by the angular mo- mentum L, theabsolute value ofthe magnetic quantum number M= |^m|^andthereflectivity



= ±^{1,whichistheeigenvalueofre-} flectionabouttheproductionplane.Positive(negative)



ischosen tocorrespondto natural(unnatural)spin-parity oftheexchanged Reggeonwith J_tr^P=^{1−or 2+or 3−}. . .(0⁻ or1⁺or2⁻. . . )trans- fertothebeamparticle[18,24].Thesetwoclassesareincoherent.

Ineachmassbin,thedifferentialcrosssectionasafunctionof four-body kinematic variables

τ

is taken to be proportional to a modelintensity I(

τ

)which isexpressedinterms ofpartial-wave amplitudesψ_LM^ (

τ

),

I

( τ ) = 



  

L,M

A^_LM

ψ

_LM^

( τ ) 



²

+

^non-

η

^()background

.

(1)

The magnitudesandphasesofthecomplexnumbers A_LM consti- tutethefreeparametersofthefit.Theexpectednumberofevents inabinis

N

¯ ∝



I

( τ )

a

( τ )

d

τ ,

(2)

where d

τ

is the four-body phase spaceelement and a(

τ

) desig- nates the efficiency of detector and selection. Following the ex- tended likelihoodapproach [25,24], fits are carriedout maximis- ing

lnL

∼ − ¯

^N

+



n

k=¹

ln I

( τ

k

),

(3)

where the sum runs over all observed events in the mass bin.

Inthisway,theacceptance-correctedmodelintensityisfittothe data.

Thepartial-waveamplitudesarecomposedoftwoparts:afac- tor fη ( fη^)that describesboth theDalitzplotdistributionofthe successive

η

(

η

^) decay [26] and the experimental peak shape, anda two-bodypartial-wave factorthat dependson theprimary

η

^()

π

⁻ decay angles. In this way, the four-body analysis is re- ducedtoquasi-two-body.Thepartial-wavefactorforthetwospin- less mesons is expressed by spherical harmonics. Thus, the full

η

(

π

⁻

π

⁺

π

⁰)

π

⁻partial-waveamplitudesread

ψ

_LM^

( τ ) =

^fη

(

p_π−

,

p_π+

,

p_π0

) ×

^Y_L^M

(ϑ

GJ

,

0

)

×



_{sin M}

ϕ

GJ for

 _{= +}

1

cos M

ϕ

GJ for

 _{= −}

1 (4)

and analogously for

η

^(

π

⁻

π

⁺

η

)

π

⁻. There are no M=^{0, and} therefore no L=^{0 waves for}



= +^{1. The fits require a weak}

L=^M=^0,



= −1 amplitude which contributes 0.5% (1.1%) to thetotal

ηπ

⁻(

η

^

π

⁻)intensity.Thisisotropicwaveisattributedto incoherentbackgroundcontaining

η

^(),whereasthenon-

η

^() back- groundamplitudeinEq.(1)isisotropicinfour-bodyphasespace.

An independent two-body PWA was carried out not taking intoaccount the decays ofthe

η

^(), butusing tight windowcuts (±^{10 MeV}/c²) onthe

η

^() peakintherespectivethree-bodyspec- tra.Theresultswerefoundtobeconsistentwiththepresentanal- ysis[18].

The above-mentioned azimuthal sin²

ϕ

GJ dependence is in agreement with a strong M=1 dominance, as was experienced earlier[6–9].NoM>1 contributionsareneededtofitthedatain thepresentt range,withtheexceptionofthe

ηπ

⁻ D-wavewhere statisticsallowstheextractionofasmallM=2 contribution.The final fit modelis restrictedto the coherent L=^{1–6, M}=^{1 plus} L=^{2, M}=^{2 partialwavesfrom naturalparitytransfer (}



= +¹⁾ andtheincoherentbackgroundsintroducedabove.

Incoherenceofpartialwavesofthesamenaturality,leading to additionaltermsinEq.(1),couldarisefromcontributionswithand withoutprotonhelicityflip,orfromdifferentt-dependencesofthe amplitudesoverthebroadt range.However,fortwopseudoscalars, incoherenceorpartialincoherence ofanytwopartialwaveswith M=^{1 canbe} accommodated by full coherence with appropriate choice ofphase [7]. ComparingPWA results fort above and be- low 0.3 (GeV/c)², no significant variation of the relative M=¹ amplitudeswitht isobserved[18].TheL=^2,M=2 contribution showsadifferentt-dependencebutdoesnotintroducesignificant incoherence.

Ingeneral,atwo-pseudoscalarPWAsuffersfromdiscreteambi- guities[27,28,24].The observedinsignificanceofunnatural-parity transfercruciallyreducestheambiguities.Inthecaseof

ηπ

⁻,the remainingambiguitiesareresolved whenthe M=^{2 D-waveam-} plitudeisintroduced.For

η

^

π

⁻,ambiguitiesoccurwhenthePWA is extended beyond the dominant L=¹,2 and 4 waves. We re- solvethisbyrequiringcontinuousbehaviourofthedominantpar- tialwavesandoftheBarreletzeros[24].Theacceptablesolutions agreewithinthestatisticaluncertaintieswiththesolutionselected here,whichistheonewiththesmallestL=3 contribution.

The results ofthe PWA are presented asintensities of all in- cluded partial waves in Figs. 3, 4, and as relative phases with respect to the L=^{2, M}=^{1 wavein Fig. 5. The plotted intensi-} tiesaretheacceptance-correctednumbersofeventsineachmass bin,asderivedfromthe|^ALM|^2ofEq.(1).Feedthroughoftheorder of3%fromthedominanta2(1320)signalisobservedinthe L=⁴

ηπ

⁻distribution,asshowninlightcolourinFig. 3.Relativeinten- sitiesintegratedovermassupto3 GeV/c²,takingintoaccountthe respective

η

^() decaybranchings,aregiveninTable 1.Theratioof thesummedintensitiesis I(

ηπ

⁻)/I(

η

^

π

⁻)=⁴.0±⁰.3.Thisratio isnot affected by luminosity,its erroris estimatedfromthe un- certaintyoftheacceptance.The

ηπ

⁻yieldislargerforalleven-L waves.Conversely,theodd-L yieldsarelargerinthe

η

^

π

⁻data.

The

ηπ

⁻ P -wave intensity shows a compact peak of 400 MeV/c² width, centred at a mass of 1.4 GeV/c². Beyond 1.8 GeV/c² it disappears. The D-wave intensity is a factor of twenty larger than the P -waveintensity. These observations re- semblethose atlower beamenergy[7,9].Asimilar P -wavepeak was observed in pn annihilation¯ at rest, where it appears with an intensitycomparableto that of the D-wave [29]. The present D-waveischaracterisedbyadominanta2(1320)peakandabroad shoulderthat extendstohighermassesandpossiblycontains the a2(1700). An M=^{2 D-wave intensity is found at the 5% level.}

The G-wave shows a peak consistent with the a₄(2040) and a broadbump centredatabout2.7 GeV/c². The F , H and I-waves (L=³,5,6)adopteachlessthan1%oftheintensityinthepresent

massrangebutaresignificantinthelikelihoodfitascanbejudged fromtheuncertaintiesgiveninTable 1.

The

η

^

π

⁻ P and D-waves have comparable intensities. The former peaks at 1.65 GeV/c², drops to almost zero at 2 GeV/c² and displays a broad second maximum around 2.4 GeV/c². The D-waveshowsatwo-partstructure similar to

ηπ

⁻ butwithrel- atively larger intensity of the shoulder. The G-wave distribution showsana₄(2040)plusbumpshapeasobservedfor

ηπ

⁻.Incon- trasttotheG and I-waves,theodd F and H -waveshaveafactor of2–3moreintensitythaninthe

ηπ

⁻ channel.Relativetotheto- talintensities observedinthetwochannels, theodd-L wavesare enhancedbyanorderofmagnitudein

η

^

π

⁻.TheF -wavedistribu- tionfeaturesabroadpeakaround2.6 GeV/c².

Phasemotionsinbothsystemscanbestbestudiedwithrespect totheD-wave,whichispresentwithsufficientintensityinthefull massrange.Therapidphaserotationscausedbythea2(1320)and a₄(2040)resonancesarediscernible.The P versus D-wavephases inbothsystemsarealmostthesamefromthe

η

^

π

⁻ thresholdup to 1.4 GeV/c² where a branching takes place. Giventhe similar- ity of the D-wave intensities after applying a kinematical factor (seebelow),itissuggestiveto ascribethedifferentrelativephase motionsin the1.4 GeV/c²–2.0 GeV/c² rangetothe P -wave.It is notedthatthe P -waveintensitiesdropdramaticallywithinthisre- gion,almost vanishingat1.8 GeV/c² in

ηπ

⁻ andat2 GeV/c² in

η

^

π

⁻.Incontrast,theG- versusD-phasemotionsarealmostiden- tical.Allphasedifferencestendtoconstantvaluesathighmasses, which is a wave-mechanical condition for narrow angular focus- ing.

Fits of resonance and background amplitudes to these PWA results (so-called mass-dependent fits) lead to strongly model- dependent resonance parameters. If these fits are restricted to massesbelow1.9 GeV/c²,comparabletopreviousanalyses,asim- plemodelincorporatingonly P and D-waveBreit–Wignerampli- tudes and a coherent D-wave background yields

π

1(1400)

ηπ

⁻

resonance parameters and

π

1(1600)

η

^

π

⁻ resonance parameters consistent with those of Refs. [7–9]. However, the inclusion of highermassesdemandsadditionalmodelamplitudes,inparticular additional D-waveresonancesandcoherent P -wavebackgrounds.

ThepresenceofacoherentbackgroundintheP -waveissuggested by the PWA resultsin Figs. 3, 4, 5 (a): The vanishing of the in- tensitiesaround2.0 GeV/c² isascribedtodestructiveinterference within this partial wave, and the relatively slow phase motion across the

η

^

π

⁻ P -wavepeak demands the additionalamplitude in order to dampen the

π

1(1600) phase rotation. Fitted P -wave resonance masses in both channels are found to be shifted up- wards by typically 200 MeV/c² whenintroducing constant-phase model backgrounds as in Ref. [23]. In the present Letter, we re- frainfromproposingresonanceparameters fortheexotic P -wave oreventheexoticF and H -wavesobservedhere.Thepresentob- servations at massesbeyondthe a₂(1320) andthe

π

1 structures mightstimulateextensionsofresonance-productionmodels,ase.g.

multi-Reggemodels[13].

For the distinct a2(1320) and a4(2040) resonances, mass- dependent fits using a standard relativistic Breit–Wigner param- eterisation, which for the a₂ includes also the

ρπ

decay in the parameterisationofthetotalwidth[6],givethefollowingresults:

m

(

a2

) =

¹³¹⁵

±

^{12 MeV}

/

c²

, Γ (

a2

) =

¹¹⁹

±

^{14 MeV}

/

c²

,

m

(

a4

) =

1900⁺₋⁸⁰₂₀MeV

/

c²

, Γ (

a4

) =

300⁺₋⁸⁰₁₀₀MeV

/

c²

,

B₂

≡

^N

(

a2

→ η

^

π

⁻

)

N

(

a₂

→ ηπ ) = (

⁵

±

²

)

%

,

B4

≡

^N

(

a4

→ η

^

π

⁻

)

N

(

a4

→ ηπ ) = (

23

±

7

)

%

.

(5)

Fig. 3. IntensitiesoftheL=^1–6,M=^{1 andL}=^2,M=^{2 partialwavesfromthe}partial-waveanalysisoftheηπ⁻datainmassbinsof40 MeV/c²width.Thelight-coloured partoftheL=4 intensitybelow1.5 GeV/c²isduetofeedthroughfromtheL=^{2 wave.Theerrorbarscorrespondtoachangeofthe}log-likelihoodbyhalfaunitanddo notincludeMCfluctuationswhichareontheorderof5%.

Here, N stands forthe integratedBreit–Wigner intensities ofthe given decay branches. The errors given above are dominated by the systematicuncertainty, which is estimatedby comparing fits

with and without coherent backgrounds, a₂(1700) or

π

1(1400). The masses and B₂ agree with the PDG values [26]. The decay branchingratioB₄ isextractedhereforthefirsttime.

Fig. 4. IntensitiesoftheL=1–6,M=1 partialwavesfromthepartial-waveanalysisoftheη^π⁻datainmassbinsof40 MeV/c² width(circles).Shownforcomparison (triangles)aretheηπ⁻resultsscaledbytherelativekinematicalfactorgiveninEq.(7).

For a detailed comparison of the results from the mass- independent PWA of both channels, their different phase spaces andangular-momentum barriers are taken into account. For the decay of pointlike particles, transition rates are expected to be proportionalto

g

(

m

,

L

) =

^q

(

m

) ×

^q

(

m

)

^2L (6) withbreak-upmomentumq(m)[30–32].Overlaid onthePWAre- sultsfor

η

^

π

⁻inFig. 4arethosefor

ηπ

⁻,multipliedineachbin bytherelativekinematicalfactor

c

(

m

,

L

) =

b

×

^g

(

m

,

L

)

g

(

m

,

L

) ,

(7)

whereg^() refersto

η

^()

π

⁻withbreak-upmomentumq^(),andthe factorb=⁰.746 accountsforthedecaybranchingsof

η

and

η

^into

π

⁻

π

⁺

γ γ

[26].

Byintegratingtheinvariant massspectraofeach partialwave, scaledby[^g()(m,L)]^−1,fromthe

η

^

π

⁻thresholdupto3 GeV/c², weobtainscaledyields I⁽_L^) andderivetheratios

RL

=

b

×

IL

/

I^_L

.

(8)

Asanalternativetotheangular-momentumbarrierfactorsq(m)^2L ofEq.(6),we havealsoused Blatt–Weisskopfbarrierfactors[33].

For the range parameter involved there, an upper limit of r= 0.4 fm wasdeducedfromsystematicstudies oftensormesonde- cays,includingthepresentchannels[30,31],whereasforr=^{0 fm} Eq.(6) is recovered.To demonstratethe sensitivityof R_L on the barrier model, the rangeof values corresponding to these upper andlowerlimitsisgiveninTable 1.

ThecomparisoninFig. 4revealsaconspicuousresemblanceof theeven-L partialwavesofbothchannels. Thisfeatureremainsif r=⁰.4 fm, but the values of RL increase with increasing r (Ta- ble 1). This similarity is corroborated by the relative phases as observedinFigs. 5(d)and(f).Theobservedbehaviourisexpected fromaquark-linepicturewhereonlythenon-strangecomponents n^{n (n}¯ =^u,d)oftheincoming

π

⁻ andtheoutgoingsystemarein- volved. The similar values of RL for L=²,4,6 suggest that the respectiveintermediate statescoupletothe sameflavourcontent oftheoutgoingsystem.

Fig. 5. PhasesΦLoftheM=1 partialwaveswithangularmomentumL relativetotheL=2,M=1 waveofηπ⁻(triangles)andη^π⁻(circles)systems.Forηπ⁻,thephase betweenthe P andD-wavesisill-definedintheregionofvanishing P -waveintensitybetween1.8and2.05 GeV/c²(shaded).Panel(b)showstherelativeM=2 versus M=^{1 phaseofthe}ηπ⁻ D-wave.

Table 1

Intensities(yields),integratedover themassrangeupto3 GeV/c², forthe par- tialwaveswith M=1 (and M=2 for L=2)relativetoL=2,M=1 inηπ⁻ (setto100).Theseyieldstakeintoaccountthedecaybranchingratiosofη^()into π⁻π⁺γ γ.Errorsarederivedfrom thelog-likelihoodfit anddonotincludethe commonuncertainty(8%)oftheacceptanceratioofthetwochannels.Thelastcol- umnlistsηπ⁻overη^π⁻yieldratiosderivedfromthescaledintensities(seetext, Eq.(8)).Thefirst(second)valueofRL correspondstorangeparameterr=^{0 fm} (r=⁰.4 fm).

L yield(ηπ⁻) yield(η^π⁻) RL

1 5.4±0.3 12.8±0.4 0.08–0.12

2 100 (fixed) 13.0±0.3 0.84–1.18

2, M=2 5.4±0.2

3 0.39±^{0.07 1.14}±^{0.13 0}.14–0.19

4 10.0±^{0.3 2.57}±^{0.18 0}.80–0.97

5 0.12±^{0.04 0.28}±^{0.10 0}.13–0.15

6 0.87±^{0.08 0.36}±^{0.05 0}.66–0.74

The quark-line estimate (see Eq. (3) in[31]) for thea2(1320) decaybranchingusingr=⁰.4 fm andtheisoscalarmixinganglein thequarkflavour basis,φ=³⁹.3^◦ [32],is B₂=³.9% for ourmass

value. Thisisinreasonableagreementwiththepresentmeasure- ment.Ananalogouscalculationforthea4(2040)yieldsB4=¹¹.8%, whichisbelowtheexperimentalvalue.Alargerrangeparameterr wouldimprovetheagreement.

On the other hand, the odd-L

η

^

π

⁻ intensities are enhanced by a factor 5–10 as compared to

ηπ

⁻, see Fig. 4, Table 1. The P -wavefits wellinto thetrendobservedforthe F and H -waves, which alsocarry exotic quantumnumbers. Itis suggestive toas- cribetheseobservationstothedominant8⊗^{8 and1}⊗^{8 charac-} terofthe

ηπ

⁻ and

η

^

π

⁻ SU(3)flavour configurations,respectively.

Whentheformercouplestoanoctetintermediatestate,Bosesym- metrydemandsevenL,whereasthelattermaycoupletothenon- symmetric odd-L configurations. The importance of this relation was alreadypointedout inprevious discussions oftheexotic

π

1, whereinparticularthehybrid(gq^q)¯ ^{orthelowestmolecularstate} (q^qq¯ ^q)¯ ^{has 1}⊗8 character[3–5].

A P -wave peak, consistent with quoted resonance parame- ters [26], appears in each channel. In the

η

^

π

⁻ channel, its rel- atively large contribution is directly visible in Fig. 2 (b). The