New analysis of ηπ tensor resonances measured at the COMPASS experiment

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

New

analysis

of

ηπ

tensor

resonances

measured

at

the

COMPASS

experiment

JPAC

Collaboration

A. Jackura

a

,

b

,

∗

,

1

,

C. Fernández-Ramírez

c

,

2

,

M. Mikhasenko

d

,

3

,

A. Pilloni

e

,

1

,

V. Mathieu

a

,

b

,

4

,

J. Nys

f

,

4

,

5

,

V. Pauk

e

,

1

,

A.P. Szczepaniak

a

,

b

,

e

,

1

,

4

,

G. Fox

g

,

4 a_Physics_Dept.,_Indiana_University,_Bloomington,_IN_47405,_USA

b_Center_for_Exploration_of_Energy_and_Matter,_Indiana_University,_Bloomington,_IN_47403,_USA

c_Instituto_de_Ciencias_Nucleares,_Universidad_Nacional_Autónoma_de_México,_Ciudad_de_México_04510,_Mexico d_Universität_Bonn,_{Helmholtz-Institut}_für_{Strahlen- und}_Kernphysik,₅₃₁₁₅_Bonn,_Germany6

e_Theory_Center,_Thomas_Jefferson_National_Accelerator_Facility,_Newport_News,_VA_23606,_USA f_Dept._of_Physics_and_Astronomy,_Ghent_University,₉₀₀₀_Ghent,_Belgium

g_School_of_Informatics_and_Computing,_Indiana_University,_Bloomington,_IN_47405,_USA

COMPASS

Collaboration

M. Aghasyan

ae

,

R. Akhunzyanov

n

,

M.G. Alexeev

af

,

G.D. Alexeev

n

,

A. Amoroso

af

,

ag

,

V. Andrieux

ai

,

aa

,

N.V. Anﬁmov

n

,

V. Anosov

n

,

A. Antoshkin

n

,

K. Augsten

n

,

y

,

W. Augustyniak

aj

,

A. Austregesilo

v

,

C.D.R. Azevedo

h

,

B. Badełek

ak

,

F. Balestra

af

,

ag

,

M. Ball

j

,

J. Barth

k

,

R. Beck

j

,

Y. Bedfer

aa

,

J. Bernhard

s

,

p

,

K. Bicker

v

,

p

,

E.R. Bielert

p

,

R. Birsa

ae

,

M. Bodlak

x

,

P. Bordalo

r

,

8

,

F. Bradamante

ad

,

ae

,

A. Bressan

ad

,

ae

,

M. Büchele

o

,

V.E. Burtsev

ah

,

W.-C. Chang

ab

,

C. Chatterjee

m

,

M. Chiosso

af

,

ag

,

I. Choi

ai

,

A.G. Chumakov

ah

,

S.-U. Chung

v

,

9

,

A. Cicuttin

ae

,

10

,

M.L. Crespo

ae

,

10

,

S. Dalla Torre

ae

,

S.S. Dasgupta

m

,

S. Dasgupta

ad

,

ae

,

O.Yu. Denisov

ag

,

∗∗

,

L. Dhara

m

,

S.V. Donskov

z

,

N. Doshita

am

,

Ch. Dreisbach

v

,

W. Dünnweber

11

,

R.R. Dusaev

ah

,

M. Dziewiecki

al

,

A. Efremov

n

,

18

,

P.D. Eversheim

j

,

M. Faessler

11

,

A. Ferrero

aa

,

M. Finger

x

,

M. Finger jr.

x

,

H. Fischer

o

,

C. Franco

r

,

N. du Fresne von Hohenesche

s

,

p

,

J.M. Friedrich

v

,

∗∗

,

V. Frolov

n

,

p

,

E. Fuchey

aa

,

12

,

F. Gautheron

i

,

O.P. Gavrichtchouk

n

,

S. Gerassimov

u

,

v

,

J. Giarra

s

,

F. Giordano

ai

,

I. Gnesi

af

,

ag

,

M. Gorzellik

o

,

24

,

A. Grasso

af

,

ag

,

M. Grosse Perdekamp

ai

,

B. Grube

v

,

T. Grussenmeyer

o

,

A. Guskov

n

,

D. Hahne

k

,

G. Hamar

ae

,

D. von Harrach

s

,

F.H. Heinsius

o

,

R. Heitz

ai

,

F. Herrmann

o

,

N. Horikawa

w

,

13

,

N. d’Hose

aa

,

C.-Y. Hsieh

ab

,

14

,

S. Huber

v

,

S. Ishimoto

am

,

15

,

A. Ivanov

af

,

ag

,

Yu. Ivanshin

n

,

18

,

T. Iwata

am

,

V. Jary

y

,

R. Joosten

j

,

P. Jörg

o

,

E. Kabuß

s

,

A. Kerbizi

ad

,

ae

,

B. Ketzer

j

,

G.V. Khaustov

z

,

Yu.A. Khokhlov

z

,

16

,

Yu. Kisselev

n

,

F. Klein

k

,

J.H. Koivuniemi

i

,

ai

,

V.N. Kolosov

z

,

K. Kondo

am

,

K. Königsmann

o

,

I. Konorov

u

,

v

,

*

Correspondingauthor.

**

Correspondingauthors.

E-mailaddress:gerhard.mallot@cern.ch(G.K. Mallot).

1 _Supported_by_{U.S. Dept.}_of_Energy,_Oﬃce_of_Science,_Oﬃce_of_Nuclear_Physics_under_contracts_{DE-AC05-06OR23177,}_{DE-FG0287ER40365.}

2 _Supported_by_PAPIIT-DGAPA_(UNAM,_Mexico)_Grant_{No. IA101717,}_by_CONACYT_(Mexico)_Grant_{No. 251817}_and_by_Red_Temática_CONACYT_de_Física_en_Altas_Energías (RedFAE,Mexico).

3 _Also_a_member_of_the_COMPASS_{Collaboration.}

4 _Supported_by_National_Science_Foundation_Grant_PHY-1415459.

5 _Supported_as_an_{‘FWO-aspirant’}_by_the_Research_Foundation_Flanders_{(FWO-Flanders).} 6 _Supported_by_BMBF_–_{Bundesministerium}_für_Bildung_und_Forschung_(Germany). https://doi.org/10.1016/j.physletb.2018.01.017

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V.F. Konstantinov

z

,

A.M. Kotzinian

ag

,

20

,

O.M. Kouznetsov

n

,

Z. Kral

y

,

M. Krämer

v

,

P. Kremser

o

,

F. Krinner

v

,

Z.V. Kroumchtein

n

,

7

,

Y. Kulinich

ai

,

F. Kunne

aa

,

K. Kurek

aj

,

R.P. Kurjata

al

,

I.I. Kuznetsov

ah

,

A. Kveton

y

,

A.A. Lednev

z

,

7

,

E.A. Levchenko

ah

,

M. Levillain

aa

,

S. Levorato

ae

,

Y.-S. Lian

ab

,

21

,

J. Lichtenstadt

ac

,

R. Longo

af

,

ag

,

V.E. Lyubovitskij

ah

,

A. Maggiora

ag

,

A. Magnon

ai

,

N. Makins

ai

,

N. Makke

ae

,

10

,

G.K. Mallot

p

,

S.A. Mamon

ah

,

B. Marianski

aj

,

A. Martin

ad

,

ae

,

J. Marzec

al

,

J. Matoušek

ad

,

ae

,

x

,

H. Matsuda

am

,

T. Matsuda

t

,

G.V. Meshcheryakov

n

,

M. Meyer

ai

,

aa

,

W. Meyer

i

,

Yu.V. Mikhailov

z

,

M. Mikhasenko

j

,

E. Mitrofanov

n

,

N. Mitrofanov

n

,

Y. Miyachi

am

,

A. Nagaytsev

n

,

F. Nerling

s

,

D. Neyret

aa

,

J. Nový

y

,

p

,

W.-D. Nowak

s

,

G. Nukazuka

am

,

A.S. Nunes

r

,

A.G. Olshevsky

n

,

I. Orlov

n

,

M. Ostrick

s

,

D. Panzieri

ag

,

22

,

B. Parsamyan

af

,

ag

,

S. Paul

v

,

J.-C. Peng

ai

,

F. Pereira

h

,

M. Pešek

x

,

M. Pešková

x

,

D.V. Peshekhonov

n

,

N. Pierre

s

,

aa

,

S. Platchkov

aa

,

J. Pochodzalla

s

,

V.A. Polyakov

z

,

J. Pretz

k

,

17

,

M. Quaresma

r

,

C. Quintans

r

,

S. Ramos

r

,

8

,

C. Regali

o

,

G. Reicherz

i

,

C. Riedl

ai

,

N.S. Rogacheva

n

,

D.I. Ryabchikov

z

,

v

,

A. Rybnikov

n

,

A. Rychter

al

,

R. Salac

y

,

V.D. Samoylenko

z

,

A. Sandacz

aj

,

C. Santos

ae

,

S. Sarkar

m

,

I.A. Savin

n

,

18

,

T. Sawada

ab

,

G. Sbrizzai

ad

,

ae

,

P. Schiavon

ad

,

ae

,

T. Schlüter

19

,

K. Schmidt

o

,

24

,

H. Schmieden

k

,

K. Schönning

p

,

23

,

E. Seder

aa

,

A. Selyunin

n

,

L. Silva

r

,

L. Sinha

m

,

S. Sirtl

o

,

M. Slunecka

n

,

J. Smolik

n

,

A. Srnka

l

,

D. Steffen

p

,

v

,

M. Stolarski

r

,

O. Subrt

p

,

y

,

M. Sulc

q

,

H. Suzuki

am

,

13

,

A. Szabelski

ad

,

ae

,

aj

,

T. Szameitat

o

,

24

,

P. Sznajder

aj

,

M. Tasevsky

n

,

S. Tessaro

ae

,

F. Tessarotto

ae

,

A. Thiel

j

,

J. Tomsa

x

,

F. Tosello

ag

,

V. Tskhay

u

,

S. Uhl

v

,

B.I. Vasilishin

ah

,

A. Vauth

p

,

J. Veloso

h

,

A. Vidon

aa

,

M. Virius

y

,

S. Wallner

v

,

T. Weisrock

s

,

M. Wilfert

s

,

J. ter Wolbeek

o

,

24

,

K. Zaremba

al

,

P. Zavada

n

,

M. Zavertyaev

u

,

E. Zemlyanichkina

n

,

18

,

N. Zhuravlev

n

,

M. Ziembicki

al

h_University_of_Aveiro,_Dept._of_Physics,_3810-193_Aveiro,_Portugal

i_Universität_Bochum,_Institut_für_{Experimentalphysik,}₄₄₇₈₀_Bochum,_Germany25,26

j_Universität_Bonn,_{Helmholtz-Institut}_für_{Strahlen- und}_Kernphysik,₅₃₁₁₅_Bonn,_Germany25

k_Universität_Bonn,_{Physikalisches}_Institut,₅₃₁₁₅_Bonn,_Germany25

l_Institute_of_Scientiﬁc_Instruments,_AS_CR,₆₁₂₆₄_Brno,_Czech_Republic27

m_Matrivani_Institute_of_Experimental_Research_&_Education,_Calcutta-700_030,_India28

n_Joint_Institute_for_Nuclear_Research,₁₄₁₉₈₀_Dubna,_Moscow_region,_Russia18

o_Universität_Freiburg,_{Physikalisches}_Institut,₇₉₁₀₄_Freiburg,_Germany25,26

p_CERN,₁₂₁₁_Geneva_23,_Switzerland

q_Technical_University_in_Liberec,₄₆₁₁₇_Liberec,_Czech_Republic27

r_LIP,_1000-149_Lisbon,_Portugal29

s_Universität_Mainz,_Institut_für_Kernphysik,₅₅₀₉₉_Mainz,_Germany25

t_University_of_Miyazaki,_Miyazaki_889-2192,_Japan30

u_Lebedev_Physical_Institute,₁₁₉₉₉₁_Moscow,_Russia

v_Technische_Universität_München,_Physik_Dept.,₈₅₇₄₈_Garching,_Germany25,11

w_Nagoya_University,₄₆₄_Nagoya,_Japan30

x_Charles_University_in_Prague,_Faculty_of_Mathematics_and_Physics,₁₈₀₀₀_Prague,_Czech_Republic27

y_Czech_Technical_University_in_Prague,₁₆₆₃₆_Prague,_Czech_Republic27

z_State_Scientiﬁc_Center_Institute_for_High_Energy_Physics_of_National_Research_Center_‘Kurchatov_{Institute’,}₁₄₂₂₈₁_Protvino,_Russia aa_IRFU,_CEA,_Université_{Paris-Saclay,}₉₁₁₉₁_{Gif-sur-Yvette,}_France26

ab_Academia_Sinica,_Institute_of_Physics,_Taipei_11529,_Taiwan31

ac_Tel_Aviv_University,_School_of_Physics_and_Astronomy,₆₉₉₇₈_Tel_Aviv,_Israel32

ad_University_of_Trieste,_Dept._of_Physics,₃₄₁₂₇_Trieste,_Italy ae_Trieste_Section_of_INFN,₃₄₁₂₇_Trieste,_Italy

af_University_of_Turin,_Dept._of_Physics,₁₀₁₂₅_Turin,_Italy ag_Torino_Section_of_INFN,₁₀₁₂₅_Turin,_Italy

ah_Tomsk_Polytechnic_University,₆₃₄₀₅₀_Tomsk,_Russia33

ai_University_of_Illinois_at_{Urbana-Champaign,}_Dept._of_Physics,_Urbana,_IL_61801-3080,_USA34

aj_National_Centre_for_Nuclear_Research,_00-681_Warsaw,_Poland35

ak_University_of_Warsaw,_Faculty_of_Physics,_02-093_Warsaw,_Poland35

al_Warsaw_University_of_Technology,_Institute_of_{Radioelectronics,}_00-665_Warsaw,_Poland35

am_Yamagata_University,_Yamagata_992-8510,_Japan30

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Articlehistory: Received10July2017

Receivedinrevisedform20November2017 Accepted8January2018

Availableonlinexxxx Editor:M.Doser

We presentanew amplitudeanalysis ofthe

ηπ

D-wave inthereaction

π

−p→

ηπ

−p measuredby COMPASS. Employingananalyticalmodel basedontheprinciplesofthe relativistic S-matrix,weﬁnd two resonances that can be identiﬁed with the a2(1320) and the exciteda₂(1700), and perform a

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1±6)MeVand= (112±1±8)MeV,andfortheexcitedstatea₂weobtainM= (1720±10±60)MeV and= (280±10±70)MeV,respectively.

1. Introduction

The spectrum of hadrons contains a number of poorly de-termined or missing resonances, the better knowledge of which is of key importance for improving our understanding of Quan-tumChromodynamics(QCD),thefundamentaltheoryofthestrong interaction. Active research programs in this direction are being pursuedatvariousexperimentalfacilities,includingtheCOMPASS andLHCbexperimentsatCERN [1–4],CLAS/CLAS12andGlueX at JLab [5–7], BESIII at BEPCII [8], BaBar, andBelle [9]. In order to connecttheexperimentalobservableslikeangularandmomentum distributionsoffinal-stateparticleswiththecorrespondingdegrees offreedom ofthestronginteraction anamplitude analysisofthe experimentaldatais required.Traditionally, themass-dependence ofpartial-waves is described by a coherentsum ofBreit–Wigner amplitudesand,ifneeded,aphenomenologicalbackground.While generallyprovidingagoodfittothedata,suchaprocedure, how-ever,violatesfundamental principlesof S-matrixtheory.In order tobetterconstraintheformoftheamplitude,morereliable reac-tion modelswhich fulfill theprinciples ofunitarity and analytic-ity (which originate from probability conservation and causality,

7 _Deceased.

8 _Also_at_Instituto_Superior_Técnico,_Universidade_de_Lisboa,_Lisbon,_Portugal. 9 _Also_at_Dept._of_Physics,_Pusan_National_University,_Busan_609-735,_Republic_of KoreaandatPhysicsDept.,BrookhavenNationalLaboratory,Upton,NY11973,USA.

10 _Also_at_Abdus_Salam_ICTP,₃₄₁₅₁_Trieste,_Italy.

11 _Supported_by_the_DFG_cluster_of_excellence_‘Origin_and _Structure_of_the Uni-verse’(www.universe-cluster.de)(Germany).

12 _Supported_by_the_Laboratoire_{d’excellence}_P2IO_(France). 13 _Also_at_Chubu_University,_Kasugai,_Aichi_487-8501,_Japan.

14 _Also_at_Dept._of_Physics,_National_Central_University,₃₀₀_Jhongda_Road,_Jhongli 32001,Taiwan.

15 _Also_at_KEK,_1-1_Oho,_Tsukuba,_Ibaraki_305-0801,_Japan.

16 _Also_at_Moscow_Institute_of_Physics_and_Technology,_Moscow_Region,_141700, Russia.

17 _Present_address: _RWTH _Aachen _University, _III._{Physikalisches} _Institut, ₅₂₀₅₆ Aachen,Germany.

18 _Supported_by_CERN-RFBR_Grant_12-02-91500. 19 _Present_address:_LP-Research_Inc.,_Tokyo,_Japan.

20 _Also_at_Yerevan_Physics_Institute,_Alikhanian_Br._Street,_Yerevan,_Armenia,_0036. 21 _Also _at _Dept. _of _Physics, _National _Kaohsiung _Normal _University, _Kaohsiung County824,Taiwan.

22 _Also_at_University_of_Eastern_Piedmont,₁₅₁₀₀_Alessandria,_Italy. 23 _Present_address:_Uppsala_University,_Box_516,₇₅₁₂₀_Uppsala,_Sweden. 24 _Supported_by_the _DFG_Research_Training _Group_Programmes ₁₁₀₂_and ₂₀₄₄ (Germany).

25 _Supported_by _BMBF _– _{Bundesministerium} _für _Bildung _und _Forschung (Ger-many).

26 _Supported_by_FP7,_{HadronPhysics3,}_Grant₂₈₃₂₈₆_(European_Union). 27 _Supported_by_MEYS,_Grant_LG13031_(Czech_Republic).

28 _Supported_by_B._Sen_fund_(India).

29 _Supported_by_FCT_–_Fundação_para_a_Ciência_e_Tecnologia,_COMPETE_and_QREN, GrantsCERN/FP116376/2010,123600/2011 andCERN/FIS-NUC/0017/2015 (Portu-gal).

30 _Supported _by _MEXT _and _JSPS, _Grants _18002006, _20540299, _18540281 _and 26247032,theDaikoandYamadaFoundations(Japan).

31 _Supported_by_the_Ministry_of_Science_and_Technology_(Taiwan). 32 _Supported_by_the_Israel_Academy_of_Sciences_and_Humanities_(Israel). 33 _Supported _by _the _Russian _Federation _program _“Nauka” _(Contract _No. 0.1764.GZB.2017)(Russia).

34 _Supported_by_the_National_Science_Foundation,_Grant_no._PHY-1506416_(USA). 35 _Supported_by_NCN,_Grant_{2015/18/M/ST2/00550}_(Poland).

respectively) should be applied. When resonances dominate the spectrum, which is the case studied here, unitarity is especially important since it constrains resonance widths and allows us to determine thelocation ofresonancepolesin thecomplexenergy planeofthemultivaluedpartialwaveamplitudes.

In2014,COMPASSpublishedhigh-statisticspartial-wave analy-ses ofthe

π

−p

→

η

()

_π

−_{p reaction,} _at _p

beam

=

191 GeV[2].The waveswithoddangularmomentumbetweenthetwopseudoscalar particles in the ﬁnal state have manifestly spin-exotic quantum numbers andwere foundto exhibit structuresthatmay be com-patiblewithahybridmeson[10,11].Theevenangular-momentum wavesshow strongsignalsofnon-exoticresonances.Inparticular, the D-waveof

ηπ

,withIG

(

JP C

)

=

1−

(

2++

)

,isdominatedbythe peak of the a2

(

1320

)

and its Breit–Wigner parameters were ex-tractedandpresentedinRef. [2].The D-wavealsoexhibitsahint oftheﬁrstradialexcitation,thea₂

(

1700

)

[12].

In this letter we present a new analysis of the

ηπ

D-wave

based on an analytical model constrained by unitarity, which extends beyond the simple Breit–Wigner parameterization. Our modelbuildsonamoregeneralframeworkforasystematic anal-ysis of peripheral meson production, which is currently under development [13–15].Using the 2014COMPASS measurement as input, the modelisﬁtted tothe resultsof themass-independent analysisthatwasperformedin40MeVwidebinsofthe

ηπ

mass. The a2 anda2 resonance parameters are extractedin the single-channel approximation and the coupled-channel effects are esti-matedbyincludingthe

ρπ

ﬁnalstate.Wedeterminethestatistical uncertainties by meansofthebootstrap method[16–20],and as-sess thesystematicuncertainties inthepolepositions by varying model-dependentparametersinthereactionamplitude.

2. Reactionmodel

We consider the peripheral diffractive production process

π

p

→

ηπ

p (Fig. 1(a)), which is dominated by Pomeron (

P

) ex-changeathighenergiesoftheincomingbeamparticle.Thisallows usto assumefactorizationofthe “top”vertex, so thatthe

π

P

→

ηπ

amplituderesemblesan ordinaryhelicityamplitude[21].Itis a function of s and t1, the

ηπ

invariant mass squared and the invariantmomentumtransfersquaredbetweentheincomingpion andthe

η

,respectively.Italsodependsont,themomentum trans-ferbetweenthenucleontargetandrecoil.IntheGottfried–Jackson (GJ)frame [22],thePomeronhelicityin

π

P

→

ηπ

equals the

ηπ

total angular momentum projection M, and the helicity ampli-tudes aM

(s,

t,t1

)

canbe expandedinpartialwavesaJ M

(s,

t)with

total angular momentum J

=

L. The allowed quantum numbers ofthe

ηπ

partial wavesare JP

=

1−, 2+,3−

,

. . .

.Theexchanged Pomeron has natural parity. Parity conservation relates the am-plitudeswithoppositespinprojectionsaJ M

= −

aJ−M [23].Thatis,

theM

=

0 amplitudeisforbiddenandthetwoM

= ±

1 amplitudes aregiven,uptoasign,byasinglescalarfunction.

The assumptionaboutthePomerondominance canbe quanti-ﬁed by themagnitudeofunnatural partialwaves.In theanalysis ofRef. [2],themagnitudeofthe L

=

M

=

0 wave, whichalso ab-sorbs other possible reduciblebackgrounds, was estimatedto be

<

1%.Weareunabletofurtheraddressthenatureoftheexchange fromthedataofRef.[2],sincetheanalysiswasperformedata

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sin-Fig. 1. (a)Reactiondiagramofπ−p→ηπ−p viaPomeronexchange.(b)Unitarity diagram:theπP→ηπ amplitudeisexpandedinpartialwavesinthes-channelof theηπsystem,aJ M(s),with J=L andt→teff.Unitarityrelatestheimaginarypart

oftheamplitudetoﬁnalstateinteractionsthatincludeallkinematicallyallowed intermediatestatesn.

glebeamenergyandintegratedoverthemomentumtransfert.36

Analyzes suchasRef.[24]suggestthat f exchangecouldalso con-tribute.Sinceinouranalysiswedonotdiscriminatebetween dif-ferentnatural-parity exchanges,weconsideraneffectivePomeron whichmaybe amixtureofpurePomeronand f .Thepatternsof azimuthaldependenceinthecentralproductionofmesons[25–29]

indicatethat at low momentum transfer,t

∼

0, the Pomeron be-havesasa vector [30,31],whichis inagreement withthestrong dominanceofthe

|

M

|

=

1 componentintheCOMPASSdata.37

The COMPASS mass-independent analysis [2] is restricted to partialwaveswithL

=

1 to6 and

|

M

|

=

1 (exceptforL

=

2 where alsothe

|

M

|

=

2 waveistakenintoaccount).Thelowest-mass ex-changesinthecrossedchannelsof

π

P

→

ηπ

correspondtothe a (in the t1 channel) and the f (in the u1 channel) trajectories, thushigherpartialwavesarenotexpectedtobesigniﬁcantinthe

ηπ

massregionofinterest,

√

s

<

2 GeV.Systematicuncertainties dueto truncationofhigherwaveswerefound tobe negligiblein Ref.[34].

Inordertocomparewiththepartial-waveintensitiesmeasured in Ref. [2], which are integrated over t from tmin

= −

1

.

0 GeV2 totmax

= −

0

.

1 GeV2, we use an effective value for the momen-tumtransferteff

= −

0

.

1 GeV2 andaJ M

(s)

≡

aJ M

(s,

teff

)

.Theeffect of a possible teff dependence is taken into account in the esti-mateofthesystematicuncertainties. Thenatural-parity exchange partial-waveamplitudesaJ M

(s)

can beidentiﬁed withthe

ampli-tudes Aε=1

LM

(s)

asdeﬁnedinEq. (1)ofRef.[2],where

ε

= +

1 isthe

reﬂectivityeigenvaluethatselectsthenatural-parityexchange. Inthefollowingweconsiderthesingle, J

=

2,

|

M

|

=

1 natural-parity partial wave, which we denote by a(s), and ﬁt its mod-ulus squared to the measured (acceptance-corrected) number of events[2]:

36 _For_example,_Ref._[24]_suggested_a_dominance_of_f2_exchanges_for_a2

(1320) pro-duction.Toprobethis,oneshouldanalyzethet andtotalenergydependences.We noteherethatCOMPASShaspublisheddatainthe3πchannel,whicharebinned bothin3πinvariantmassandmomentumtransfert[3],whichmaygivefurther insightintotheproductionprocess.

37 _At_{low t,}_the_Pomeron_trajectory_passes_through _J₌_1,_while_at_larger, posi-tive t,thetrajectoryisexpectedtopassthough J=2 whereitwouldrelatetothe tensorglueball[32,33]. d

σ

d

√

s

∝

I

(

s

)

=

tmax

tmin dt p

|

a

(

s

,

t

)

|

2

≡

N

p

|

a

(

s

)

|

2

.

(1)

Here, I(s) is the intensity distribution of the D wave, p

=

λ

1/2

_(s,

_m2

η

,

m2π

)/(

2

√

s) the

ηπ

breakup momentum, and q

=

λ

1/2

_(s,

_m2

π

,

teff

)/(

2

√

s), which will be used later, is the

π

beam momentuminthe

ηπ

restframewith

λ(x,

y,z)

=

x2

₊

_y2

₊

_z2

₋

2xy

−

2xz

−

2 yz beingtheKälléntrianglefunction.Sincethe physi-calnormalizationofthecrosssectionisnotdeterminedinRef.[2], theconstant

N

ontherighthandsideofEq.(1)isafree parame-ter.

Inprinciple,oneshouldconsiderthecoupled-channelproblem involving all the kinematically allowed intermediate states (see

Fig. 1(b)).Forthe2++ states,thePDGreportsthe3

π

(

ρπ

, f2

π

) and

ηπ

ﬁnal states as dominant decay channels [12]. Far from thresholds, a narrow peak inthe data is generated by a pole in the closest unphysical sheet, regardless of the number of open channels. The residues (related to the branching ratios) depend ontheindividualcouplingsofeach channeltotheresonance,and thereforetheirextractionrequirestheinclusionofalltherelevant channels. However,thepolepositionisexpectedtobeessentially insensitivetotheinclusionofmultiplechannels.Thisiseasily un-derstoodintheBreit–Wignerapproximation,wherethetotalwidth extractedforagivenstateisindependentofthebranchingsto in-dividualchannels. Thus, wheninvestigatingthe poleposition,we restrict the analysisto theelastic approximation,where only

ηπ

canappearintheintermediatestate.Wewillelaborateonthe ef-fects of introducing the

ρπ

channel, which is known to be the dominant one of thedecay of a2

(

1320

)

[12], aspart of the sys-tematicchecks.

Intheresonanceregion,unitaritygivesconstraintsforboththe

ηπ

interactionandproduction.Denotingthe

ηπ

→

ηπ

scattering

D-waveby f

(s)

,unitarityandanalyticitydeterminetheimaginary partofbothamplitudesabovethe

ηπ

thresholdsth

= (

mη

+

mπ

)

2:

Ima

ˆ

(

s

)

=

ρ

(

s

) ˆ

f∗

(

s

)

a

ˆ

(

s

),

(2)

Im

ˆ

f

(

s

)

=

ρ

(

s

)

| ˆ

f

(

s

)

|

2

,

(3)

with

ρ

(s)

=

2p5

_/

√

_{s being} _the _two-body _phase _space_factor _that absorbs the barrier factors of the D-wave. From the analysis of kinematicalsingularities[35–37]itfollowsthattheamplitudea(s)

appearing in Eq.(1) haskinematical singularitiesproportional to

K

(s)

=

p2q,and f

(s)

hassingularitiesproportional to p4.The re-ducedpartialwavesin Eqs.(2)and(3)are freefromkinematical singularities,anddeﬁnedby e.g. a(s)

ˆ

=

a(s)/K

(s)

,

ˆ

f

(s)

=

f

(s)/

p4_. NotethatEq.(2)istheelasticapproximationofFig. 1(b).

Wewrite

ˆ

f inthestandard N-over-D form,

ˆ

f

(s)

=

N(s)/D(s), with N

(s)

absorbing singularities fromexchange interactions, i.e.

“forces”actingbetween

ηπ

alsoknownasleft-handcuts,andD(s)

containing the right-hand cuts that are associated with direct-channelthresholds.UnitarityleadstoarelationbetweenD and N,

Im D

(s)

= −

ρ

(s)N(s)

,withthegeneralonce-subtractedintegral so-lution D

(

s

)

=

D0

(

s

)

−

s

π

∞

sth ds

ρ

(

s

₎

_N

₍

_s

₎

s

(

s

−

s

)

.

(4) Here,thefunction D0

(s)

isrealfors

>

sth andcanbe

parameter-izedas

D0

(

s

)

=

c0

−

c1s

−

c2

c3

−

s

.

(5)

Note that the subtraction constant hasbeen absorbed into c0 of D0

(s)

.TherationalfunctioninEq.(5)isasumovertwoso-called

(5)

Castillejo–Dalitz–Dyson (CDD) poles [38], with the ﬁrst pole lo-catedat s

= ∞

(CDD_∞) andthe second one at s

=

c3. The CDD polesproducerealzerosoftheamplitude

ˆ

f andtheyalsoleadto polesof

ˆ

f inthecomplexplane(secondsheet).Sincethesepoles are introduced via parameters likec1, c2,rather thanbeing gen-erated through N (cf. Eq. (4)), they are commonly attributed to genuineQCDstates,i.e. statesthatdonotoriginate fromeffective, long-rangeinteractionssuch aspionexchange[39].Inordertoﬁx thearbitrarynormalization ofN

(s)

and D(s), weset c0 to

O(

1

)

, sinceitisexpectedtobeoftheorderofthea2masssquared ex-pressedinunitsofGeV2.Onealsoexpectsc1 tobeapproximately equaltotheslopeoftheleading Reggetrajectory[40].Thequark model[41] andlatticeQCD [42] predicttwo statesinthe energy regionofinterest,so weuseonlytwo CDDpoles.It followsfrom Eq.(4)that thesingularitiesof N(s) (whichoriginatefromthe ﬁ-niterangeoftheinteraction)willalsoappearonthesecondsheet in D(s),together withtheresonancepolesgeneratedby theCDD terms.WeuseasimplemodelforN(s),wheretheleft-handcutis approximatedbyahigher-orderpole,

ρ

(

s

)

N

(

s

)

=

g

λ

5/2

₍

_s

_,

_m2 η

,

m2π

)

(

s

+

sR

)

n

.

(6)

Here, g and sR effectivelyparameterize thestrength and inverse

range of the exchange forces in the D-wave, respectively. The powern

=

7 is ourmodel forthe left-handsingularities in N(s). Thisincludesthe effectsofthe ﬁniterangeofinteraction,i.e. the

regularizationofthethresholdsingularitiesduetoK

(s)

=

p2q.The parameterizationof N(s)removesthe kinematical1

/s singularity

in

ρ

(s)

.Therefore,dynamicalsingularitiesonthesecondsheetare eitherassociatedwiththeparticlesrepresentedbytheCDDpoles, or the exchange forces parameterized by the higher order pole in N

(s)

.

Thegeneralparameterizationfora(s)

ˆ

,whichisconstrainedby unitarityinEq.(2),isobtainedfollowingsimilarargumentsandis givenbyaratiooftwofunctions

ˆ

a

(

s

)

=

n

(

s

)

D

(

s

)

,

(7)

where D(s) is given by Eq. (4) and brings in the effects of

ηπ

ﬁnal-stateinteractions,whilen(s)describestheexchange interac-tionsin theproductionprocess

π

P

→

ηπ

andcontains the asso-ciated left-handsingularities. Inboth theproduction process and theelasticscatteringnoimportantcontributionsfromlight-meson exchangesareexpectedsincethelightestresonancesinthet1 and u1 channelsarethea2 and f2 mesons,respectively.Therefore,the numeratorfunctioninEq.(7)isexpectedtobeasmoothfunction ofs inthe complexplane nearthe physicalregion,withone ex-ception: the CDD poleat s

=

c3 produces a zeroin a(s)

ˆ

.Since a zerointheelasticscatteringamplitude doesnotingeneralimply azerointheproductionamplitude,wewriten(s)as

n

(

s

)

=

1 c3

−

s np

j ajTj

(

ω

(

s

)),

(8)

where the function to the right of the pole is expected to be analytical in s near the physical region. We parameterize it us-ing the Chebyshev polynomials Tj, with

ω

(s)

=

s/(s

+ )

ap-proximatingtheleft-hand singularitiesinthe productionprocess,

π

P

→

ηπ

.The realcoeﬃcientsaj aredeterminedfromtheﬁtto

the data. In the analysis, we ﬁx

=

1 GeV2. We choose an ex-pansioninChebyshevpolynomialsasopposedto asimplepower

series in

ω

to reduce the correlations between the aj

param-eters. Since we examine the partial-wave intensities integrated over the momentum transfer t, we assume that the expansion coeﬃcients are independent of t. The only t-dependence comes fromtheresidualkinematicaldependenceonthebreakup momen-tum q.

AcommentontherelationbetweentheN-over-Dmethodand the K -matrix parameterization is worth making. If one assumes that there are no left-hand singularities, i.e. let N(s) be a con-stant, then Eq.(4) is identical to that of the standard K -matrix

formalism [43]. Hence we can relate both approaches through

K−1

(s)

=

D0

(s)

. Itis alsoworth notingthat theparameterization inEq.(5)automaticallysatisﬁescausality,i.e. therearenopoleson thephysicalenergy-sheet.

3. Methodology

Weﬁtourmodeltotheintensitydistributionfor

π

−p

→

ηπ

−p

inthe D-wave(56datapoints)[2],asdeﬁnedinEq.(1),by min-imizing

χ

2_. _We _ﬁx _the _overall _scale,

_{N =}

₁₀6 _(see _Eq.₍₁₎_), _and ﬁtthecoeﬃcientsaj (seeEq.(8)), whichare thenexpectedtobe O

(

1

)

, and also the parameters in the D(s) function. In the first stepweobtainthebestfitforagiventotalnumberofparameters, andinthesecondstepweestimatethestatisticaluncertainties us-ing the bootstrap technique [16–20].That is to say, we generate 105 _pseudodata _sets, _each _data_point _being_resampled _according toa Gaussiandistributionhavingasmeanandstandarddeviation the original value and error, andwe repeat the fit for each set. Inthisway,weobtain105 _different_values_for_the_fit_parameters, andwetakethemeansandstandarddeviationsasexpectedvalues andstatisticaluncertainties, respectively.Theuseofthebootstrap methodallowsustodeterminethecorrelationsbetweenthepole positions andtheproductionparameters,providedas supplemen-tal material. As expected, the production parameters are highly correlated amongeachother,buttheir correlationswiththepole positions are rather low. This justifies the choice of Chebyshev polynomials;similarstudieswithastandardpolynomialexpansion showedlargercorrelationsbetweenproductionandresonance pa-rameters.

In order to assess the systematic uncertainties we study the dependence of the pole parameters on variations of the model. Speciﬁcally, we change i) thenumber ofCDD poles from1to 3,

ii)thetotalnumberoftermsnp intheexpansionofthe

numera-torfunctionn(s)inEq.(8),iii)thevalueofsR intheleft-hand-cut

model,i v)thevalueofteffofthetotalmomentumtransfered,and v) the addition of the

ρπ

channel to study coupled-channel ef-fects.

InordertodeterminesR,we scanthemodelwithvarious

val-ues of sR, ranging from 0.1 to 10.0 GeV2, and ﬁnd that values

near sR

=

1

.

5 GeV2 give a minimum in

χ

2. This choice is also

justified by phenomenological studies where the finite range of strong interactions is ofthe order of 1 GeV. The fit withCDD_∞ only, shown in Fig. 2(a), for sR

=

1

.

5 GeV2 and np

=

6 (with a

total of 9 parameters), captures neither the dip at 1

.

5 GeV nor the bump at 1

.

7 GeV. In contrast, the ﬁt with two CDD poles (11parameters),showninFig. 2(b),capturesbothfeatures,giving a

χ

2

_/

_d.o.f.

₌

₈₆

_.

₁₇

_/(

₅₆

₋

₁₁

₎

₌

₁

_.

_91. _The

_χ

2

_/

_{d.o.f. is}_somewhat large, dueto thesmall statisticaluncertainties ofthe data. How-ever, the residuals do not show anysystematic deviation, which supports the quality of the ﬁt (see residuals normalized bin-by-bin to the corresponding uncertainty in Fig. 2). The parameters corresponding to the best ﬁt with two CDD poles are given in

Table 1.The addition ofanother CDD poledoes notimprove the fit, as a fit to the intensity only is incapable of indicating any further resonances.Specifically the residueofthe additionalpole

(6)

Fig. 2. Intensitydistributionandﬁtstothe JP C₌₂++_wave_for_different_number_of_CDD_poles,_(a)_using_only_CDD_∞_and_(b)_using_CDD_∞_and_the_CDD_pole_at_s₌_c3_._Red

linesarefitresultswithI(s)givenbyEq.(1).DataistakenfromRef.[2].Theinsetshowsthea2region.Theerrorbandscorrespondtothe3σ(99.7%)confidencelevel.The lowerplotshowstheresidualsnormalizedbin-by-bintothecorrespondinguncertainty.Thedashedlinesindicatethe3σdeviations.(Forinterpretationofthecolorsinthe figure(s),thereaderisreferredtothewebversionofthisarticle.)

Table 1

Bestﬁtdenominatorand productionparametersfortheﬁt withtwoCDDpoles, sR=1.5 GeV2,N=106,c0= (1.23)2,and thenumberofexpansionparameters np=6,leadingtoχ2/d.o.f.=1.91.Denominatoruncertaintiesaredeterminedfrom

abootstrapanalysisusing105_random_ﬁts._We_report_no_{uncertainties}_on_the pro-ductionparametersastheyarehighlycorrelated.

Denominator parameters Productionparameters [GeV−2_] c0 1.5129(ﬁxed) GeV2 _a0 ₀_.₄₇₁ c1 0.532±0.006 GeV−2 _a1 ₀_.₁₃₄ c2 0.253±0.007 GeV2 _a2 ₋₁ .484 c3 2.38±0.02 GeV2 _a3 ₀ .879 g 113±1 GeV4 _a4 ₂_.₆₁₆ a5 −3.652 a6 1.821

turns out to be compatible with zero, leaving the other ﬁt pa-rameters unchanged. We associate no systematic uncertainty to that.

As discussed earlier, an acceptable numerator function n(s)

shouldbe“smooth”intheresonanceregion,i.e. withoutsigniﬁcant peaksordipson thescaleofthe resonancewidths.The parame-tersciandg ofthedenominatorfunctionarerelatedtoresonance

parameters, while sR controls thedistant second-sheet

singulari-tiesduetoexchangeforces.Theexpansioninn(s),showninFig. 3

forsR

=

1

.

5 GeV2 andtwoCDDpoles,hasasingularity occurring

ats

= −

1

.

0 GeV2_because_of_the_deﬁnition_of

_ω

_(s)

_and_our_choice of

.38Forvariationsinn(s)betweennp

=

3 andnp

=

7,weﬁnd

thatthepolepositionsarerelativelystable,whichwediscusslater inoursystematicestimates.

Thedependence onteff isexpected to affectmostly the over-all normalization. Indeed,the variation from teff

= −

1

.

0 GeV2 to

−

0

.

1 GeV2 _gives_less_than_{2% difference}_for_the_a

2

(

1700

)

parame-ters,and

<

1

h

forthea2

(

1320

)

,andcanbeneglected compared totheotheruncertainties.

38 _Note_that_the_production_term_is_not_well_constrained_below_s_∼_{1 GeV}2_,_as_the phase-spaceandbarrierfactorshighlysuppressthenear-thresholdbehavior.The singularityats= −1 GeV2_,_however,_persists_for_each_n

psolution.

Fig. 3. Amplitudenumeratorfunction|np

j ajTj(ω(s))|fordifferentvaluesofnp.

Theabsolutevalueistakenasthereisaphaseambiguitybecausewefitonlythe intensity∼ |a(s)|2.Notethateachcurveisanindependentfitforaspecificnumber oftermsnp.Thecurvesfornp=4,5,and6allcoincideintheresonanceregion,as

shownintheinset. 4. Results

Thisanalysisallows ustoextract the

ηπ

→

ηπ

elastic ampli-tudeinthe D-wave.Byconstruction,theamplitudehasazeroat

s

=

c3. Fig. 4 shows the real and imaginary parts of

ˆ

f

(s)

, with the3

σ

errorbandsestimatedbythebootstrapanalysis.Resonance poles areextractedby analyticallycontinuingthedenominator of the

ηπ

elasticamplitude tothesecond Riemannsheet (II)across theunitaritycutusingDII

(s)

=

D

(s)

+

2i

ρ

(s)N

(s)

.Byconstruction, noﬁrst-sheetpolesare present.Weﬁndthreesecond-sheetpoles in the energy range of

(

mπ

+

mη

)

≤

√

s

≤

3 GeV, two of which canbeidentiﬁedasresonances,asshowninFig. 5fornp

=

6 and sR

= {

1

.

0

,

1

.

5

,

2

.

0

,

2

.

5

}

GeV2.

The mass and width are deﬁned as m

=

Re

√

sp and

=

−

2Im

√

sp, respectively, where sp is the pole position in the s

plane. Two ofthe poles found can be identiﬁed as thea2

(

1320

)

anda₂

(

1700

)

resonances,respectively[12].The lighterofthetwo

(7)

Fig. 4. Thereducedηπ→ηπpartialamplitudeintheD-wave, ˆf(s)=N(s)/D(s). Shownarethereal(red)andimaginary(blue)partsasafunctionoftheηπ invari-antmasswith3σ errorband(whichisvisibleinthe[1.5,2.0]GeV regiononly). Thenodeintheimaginarypartat1.7 GeVisduetothehighcorrelationbetween therealandimaginaryparts.

corresponds to the a2

(

1320

)

. For sR

=

1

.

5 GeV2, the pole has

mass and width m

= (

1307

±

1

)

MeV and

= (

112

±

1

)

MeV, respectively. The nominal value is the best-ﬁt pole position, and theuncertaintyisthestatisticaldeviationdeterminedinthe boot-strapanalysis.ValuesofsR between1.0and2.5 GeV2leadtopole

deviationsofatmost

m

=

2 MeVand

=

3 MeV.The heav-ierpole corresponds to the excited a₂

(

1700

)

. For sR

=

1

.

5 GeV2,

the resonance has mass and width m

= (

1720

±

10

)

MeV and

= (

280

±

10

)

MeV,respectively.Themaximal deviationsforthe different sR values are

m

=

40 MeV and

=

60 MeV. The a2

(

1320

)

and a2

(

1700

)

poles (see Fig. 5) are found to be sta-bleundervariationsof sR,whichmodulatesthe left-handcut.As

expected, there is a third pole that dependsstrongly on sR and

reﬂectsthesingularityin N(s)modeledasapole.Itsmassranges from1

.

4 to3

.

3 GeV,anditswidthvariesbetween1

.

3 and1

.

8 GeV assR changes from1 GeV2 to2

.

5 GeV2.In thelimit g

→

0,this

pole movesto

−

sR as expected, while the other two migrateto

therealaxisabovethreshold[44].

Changingthenumberofexpansion termsbetweennp

=

3 and np

=

7 does not in any signiﬁcant way affect the a2

(

1320

)

or a₂

(

1700

)

pole positions. The maximal deviations are

m(a2

)

=

5 MeV,

(a

2

)

=

7 MeV and

m(a2

)

=

40 MeV,

(a

2

)

=

30 MeVbetweenthreeandseventermsinthen(s)expansion.

Inordertodemonstratethatcoupled-channeleffectsdonot in-ﬂuencethepolepositions,weconsideranextensionofthemodel toincludea second channelalso measuredby COMPASS,

ρπ

[3], andsimultaneouslyﬁtthe

ηπ

[2]andthe

ρπ

[3]ﬁnalstates.The branchingratioofthe a2

(

1320

)

is saturatedatthelevelof

∼

85%

bythe

ηπ

and3

π

channels[12],withthe

ρπ

S-wavehavingthe dominant contribution.For simplicitywe consider the

ρ

to be a stableparticlewithmass 775MeV,theﬁnite widthofthe

ρ

be-ing relevant only for

√

s

<

1 GeV. Theamplitude isthen a

ˆ

j

(s)

=

k[D

(s)

]−jk1

(s)

nk

(s)

. The denominator is now a 2

×

2 matrix,

whose diagonal elements are of the form givenby Eq. (4),with the appropriate phase space for each channel. The off-diagonal term is parameterized as a single real constant. The production elements nk

(s)

are asinEq.(8),withindependentcoeﬃcientsfor

each channel. We alsoperformeda K -matrixcoupled-channel ﬁt and obtained results very similar to our main modelusing CDD poles, ascan be seenin Fig. 6.The coupled-channel effects pro-duce acompetitionbetweentheparameters inthenumeratorsto ﬁtthebumpat1.6GeVin

ηπ

andthedipat1

.

8 GeVin

ρπ

atthe same time. The

ρπ

ﬁtprefers notto have anyexcited a₂

(

1700

)

, whichconverselyisevidentinthe

ηπ

data.Therefore,the uncer-tainty in the a₂

(

1700

)

poleposition increases, asit is practically unconstrainedby the

ρπ

data.Note,however,thatinRef.[3]the dipat

√

s

∼

1

.

8 GeV inthe

ρπ

dataist-dependent,whileweuse thet-integratedintensity,soitmaybeexpectedthattheeffectsof thea₂ aresuppressedinourcombinedﬁt.

We ﬁnd the following deviations in the pole positions rela-tive tothe single-channelﬁt:

m(a

2

)

=

2 MeV,

(a

2

)

=

3 MeV,

m(a

₂

)

=

20 MeV and

(a

₂

)

=

10 MeV. These deviations are rathersmallandwequotethemwithin oursystematic uncertain-ties.

5. Summaryandoutlook

Wedescribethe2++ waveof

π

p

→

ηπ

p reactionina single-channel analysisemphasizingunitarityandanalyticityofthe am-plitude. These fundamental S-matrixprinciples signiﬁcantly con-strainthepossibleformoftheamplitudemakingtheanalysismore stable than standard ones that use sums of Breit–Wigner reso-nanceswithphenomenologicalbackgroundterms.

The robustness of the model allows us to reliably reproduce the data,and toextract pole positions byanalytical continuation tothecomplexs-plane. Weusethesingle-energypartialwavesin Ref. [2]toextract thepole positions.We ﬁndtwopoles that can be identiﬁed as thea2

(

1320

)

andthe a₂

(

1700

)

resonances,with poleparameters

m

(

a2

)

= (

1307

±

1

±

6)MeV, m

(

a2

)

= (

1720

±

10

±

60)MeV,

(

a2

)

= (

112

±

1

±

8)MeV,

(

a2

)

= (

280

±

10

±

70)MeV, wherethe ﬁrstuncertaintyisstatistical(fromthebootstrap anal-ysis)andthesecondonesystematic.Thesystematicuncertaintyis obtainedaddinginquadraturethe differentsystematiceffects re-latedtotheﬁtmodel,i.e. thedependenceonthenumberofterms

Fig. 5. Locationofsecond-sheetpolepositionswithtwoCDDpoles,np=6,andwithsRvariedfrom1.0 GeV2to2.5 GeV2.Polesareshownwith2σ(95.5%)conﬁdencelevel

(8)

Fig. 6. Coupled-channelD-waveﬁt,(a) usingamodelbasedonCDDpoles,(b)usingthestandard K -matrixparameterization.Bothparameterizationsgivepolepositions consistentwiththesingle-channelanalysis.Theηπ dataistakenfromRef.[2]andtheρπdatafromRef.[3].

in the expansion of the numerator function n(s), on sR, on teff (negligible),andonthecoupled-channeleffects.Thea2 resultsare consistentwiththepreviousa2

(

1320

)

resultsfoundinRef.[2].

The third pole found tends to

−

sR in the limit of vanishing

coupling,indicatingthatthispolearisesfromthetreatmentofthe exchangeforces,andnotfromtheCDDpolesthataccount forthe resonances.

Inthefuturethisanalysiswillbeextendedtoalsoincludethe

η

π

channel[45],wherealargeexoticP -waveisobserved[2].

Additionalmaterialisavailableonlineassupplementalmaterial andthroughaninteractivewebsite[46,47].

Acknowledgements

WeacknowledgetheLillyEndowment,Inc.,throughitssupport fortheIndiana University PervasiveTechnologyInstitute, andthe IndianaMETACytInitiative.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttps://doi.org/10.1016/j.physletb.2018.01.017.

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