Odd and even partial waves of ηπ- and η'π- in π-p→η(')π-

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

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

a_{UniversitätBielefeld,Germany} b_{UniversitätBochum,Germany}

c_{UniversitätBonn,Helmholtz-InstitutfürStrahlen- undKernphysik,53115Bonn,Germany}9 d_{UniversitätBonn,PhysikalischesInstitut,53115Bonn,Germany}9

e_{InstituteofScientificInstruments,ASCR,61264Brno,CzechRepublic}10

f_{MatrivaniInstituteofExperimentalResearch&Education,Calcutta–700030,India}11 g_{JointInstituteforNuclearResearch,141980Dubna,MoscowRegion,Russia}12 h_{UniversitätErlangen–Nürnberg,PhysikalischesInstitut,91054Erlangen,Germany}9 i_{UniversitätFreiburg,PhysikalischesInstitut,79104Freiburg,Germany}9,16 j_{CERN,1211Geneva23,Switzerland}

k_{TechnicalUniversityinLiberec,46117Liberec,CzechRepublic}10 l_{LIP,1000-149Lisbon,Portugal}13

m_{UniversitätMainz,InstitutfürKernphysik,55099Mainz,Germany}9 n_{UniversityofMiyazaki,Miyazaki889-2192,Japan}14

o_{LebedevPhysicalInstitute,119991Moscow,Russia}

p_{Ludwig-Maximilians-UniversitätMünchen,DepartmentfürPhysik,80799Munich,Germany}9,15 q_{TechnischeUniversitätMünchen,PhysikDepartment,85748Garching,Germany}9,15

r_{NagoyaUniversity,464Nagoya,Japan}14

s_{CharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublic}10 t_{CzechTechnicalUniversityinPrague,16636Prague,CzechRepublic}10

u_{StateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘KurchatovInstitute’,142281Protvino,Russia} v_{CEAIRFU/SPhNSaclay,91191Gif-sur-Yvette,France}16

w_{TelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israel}17 x_{UniversityofTrieste,DepartmentofPhysics,34127Trieste,Italy}

y_{TriesteSectionofINFN,34127Trieste,Italy} z_{AbdusSalamICTP,34151Trieste,Italy}

aa_{UniversityofTurin,DepartmentofPhysics,10125Turin,Italy} ab_{UniversityofEasternPiedmont,15100Alessandria,Italy} ac_{TorinoSectionofINFN,10125Turin,Italy}

ad_{NationalCentreforNuclearResearch,00-681Warsaw,Poland}18 ae_{UniversityofWarsaw,FacultyofPhysics,00-681Warsaw,Poland}18

af_{WarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Poland}18 ag_{YamagataUniversity,Yamagata,992-8510,Japan}14

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

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 strongenhancementof

η



π

−over

ηπ

−isfoundfortheL=1,3,5 waves,whichcarrynon-qq quantum¯

numbers.TheL₌1 intensitypeaksat1.7 GeV/c2_in

_η



_π

−_{andat1.4 GeV/c}2_in

_ηπ

−_{,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.}

The

ηπ

and

η



π

mesonic systems are attractive for spectro-scopicstudies becauseany statewithodd angularmomentum L, whichcoincideswiththetotalspin J ,hasnon-qq (“exotic”)

¯

quan-tumnumbers JPC

=

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 JPG

₌

₁−−_{, where G-parity is used forthe charged system,} correspondingtoC

= +

1 sincetheisospinis1.Resultsfor charge-exchangeproductionof

η

()

_π

0_{aredifficulttorelatetothese}

obser-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 centrally pro-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)which con-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 presentdatarequiredcoincidencebetweenbeam defi-nitioncountersandtheRPD, andno vetofromtheSWnorfrom a small counter telescope for non-interacting beam particles far downstream(32 m)fromthetarget.Asampleof4

.

5

×

109_events

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”clustersintheelectromagnetic calorime-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,whichconfirmstheexclusivityofthe re-action.DetailsarefoundinRefs.[17,18].

Inordertoaccountfortheacceptanceofthespectrometerand theselectionprocedure,MonteCarlosimulations[15,19]were per-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. The ob-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 cos

ϑ

GJ,

where

ϑ

GJ 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 a}

4

(

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)ηπ−. Acceptances (continuous lines) refer to the kinematic ranges of the present analysis.

Fig. 2. Data(notacceptancecorrected)asafunctionoftheinvariantηπ−(a)andηπ−(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).A forward/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,whichistheeigenvalueof re-flectionabouttheproductionplane.Positive(negative)



ischosen tocorrespondto natural(unnatural)spin-parity oftheexchanged Reggeonwith JP

tr

=

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 amplitudes

ψ

_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

)

dτ

,

(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 ln

L

∼ − ¯

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:a fac-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-wavefactorforthetwo spin-less mesons is expressed by spherical harmonics. Thus, the full

η

(

π

−

π

+

π

0

₎

_π

−_{partial-waveamplitudesread}

ψ

_LM

(

τ

)

=

fη

(

pπ−

,

pπ+

,

pπ0

)

×

Y_LM

(ϑ

GJ

,

0

)

×



_{sin Mϕ} GJ for



= +

1 cos MϕGJ 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-body} spec-tra.Theresultswerefoundtobeconsistentwiththepresent anal-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-pseudoscalarPWAsuffersfromdiscrete ambi-guities[27,28,24].The observedinsignificanceofunnatural-parity transfercruciallyreducestheambiguities.Inthecaseof

ηπ

−,the remainingambiguitiesareresolved whenthe M

=

2 D-wave am-plitudeisintroduced.For

η



π

−,ambiguitiesoccurwhenthePWA is extended beyond the dominant L

=

1

,

2 and 4 waves. We re-solvethisbyrequiringcontinuousbehaviourofthedominant par-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

|

A_LM

|

2ofEq.(1).Feedthroughoftheorder of3%fromthedominanta2

(

1320

)

signalisobservedinthe L

=

4

ηπ

−distribution,asshowninlightcolourinFig. 3.Relative inten-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

− butwith rel-atively larger intensity of the shoulder. The G-wave distribution showsana4

(

2040

)

plusbumpshapeasobservedfor

ηπ

−.In

con-trasttotheG and I-waves,theodd F and H -waveshaveafactor of2–3moreintensitythaninthe

ηπ

− channel.Relativetothe to-talintensities observedinthetwochannels, theodd-L wavesare enhancedbyanorderofmagnitudein

η



π

−.TheF -wave distribu-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 -waveintensitiesdropdramaticallywithinthis re-gion,almost vanishingat1

.

8 GeV

/

c2 in

ηπ

− andat2 GeV

/

c2 in

η



π

−.Incontrast,theG- versusD-phasemotionsarealmost iden-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,a sim-plemodelincorporatingonly P and D-waveBreit–Wigner ampli-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.Thepresent ob-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+₋₂₀80MeV

/

c2

,

Γ (

a4

)

=

300+₋₁₀₀80 MeV

/

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 partialwavesfromthepartial-waveanalysisoftheηπ−datainmassbinsof40 MeV/c2_{width.Thelight-coloured}

partoftheL=4 intensitybelow1.5 GeV/c2_{isduetofeedthroughfromtheL=2 wave.Theerrorbarscorrespondtoachangeofthelog-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

Fig. 4. IntensitiesoftheL=1–6,M=1 partialwavesfromthepartial-waveanalysisoftheηπ−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 onthePWA re-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

/

IL

.

(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 oftensormeson de-cays,includingthepresentchannels[30,31],whereasforr

=

0 fm Eq.(6) is recovered.To demonstratethe sensitivityof RL 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 nn (n

¯

=

u

,

d)oftheincoming

π

− andtheoutgoingsystemare in-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)andηπ−(circles)systems.Forηπ−,thephase betweenthe P andD-wavesisill-definedintheregionofvanishing P -waveintensitybetween1.8and2.05 GeV/c2_{(shaded).Panel(b)showstherelativeM=2 versus} M=1 phaseoftheηπ− 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.Thelast col-umnlistsηπ−overηπ−yieldratiosderivedfromthescaledintensities(seetext, Eq.(8)).Thefirst(second)valueofRL correspondstorangeparameterr=0 fm (r=0.4 fm). L yield(ηπ−) yield(ηπ−) 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. Thisisinreasonableagreementwiththepresent measure-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 to as-cribetheseobservationstothedominant8

⊗

8 and1

⊗

8

charac-terofthe

ηπ

− and

η



π

− SU

(

3

)

flavour configurations,respectively.

Whentheformercouplestoanoctetintermediatestate,Bose sym-metrydemandsevenL,whereasthelattermaycoupletothe non-symmetric odd-L configurations. The importance of this relation was alreadypointedout inprevious discussions oftheexotic

π

1,

whereinparticularthehybrid(gqq)

¯

orthelowestmolecularstate (qqq

¯

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

forward/backward asymmetry, ascribed to L

=

1

,

3

,

5 amplitudes interferingwiththeeven-L ones,extendstohighermasses,where atransitiontorapidity-gapphenomena(centralproduction)is ex-pected.Inthe

ηπ

−data,theasymmetryismuchlesspronounced. Inconclusion,two strikingfeaturescharacterisethesystematic behaviourofpartialwavespresentedhere:

(i) TheevenpartialwaveswithL

=

2

,

4

,

6 showaclosesimilarity betweenthetwochannels, bothintheintensitiesasfunction ofmass–afterscalingbythephase-spaceandbarrierfactors –aswellasintheirphasebehaviour.

(ii) TheoddpartialwaveswithL

=

1

,

3

,

5,carryingnon-qq quan-

¯

tum numbers,are suppressed in

ηπ

− withrespect to

η



π

−, underliningtheimportanceofflavoursymmetry.

Acknowledgements

We gratefullyacknowledge the support of the CERN manage-mentandstaffaswellastheskills andeffortsofthe technicians ofthecollaborating institutions.Thisworkwas madepossible by thefinancialsupportofourfundingagencies.

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