Measurement of the ω → π+π−π0 Dalitz plot distribution: The WASA-at-COSY Collaboration

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Measurement

of

the

ω

→

π

+

π

−

π

0 Dalitz

plot

distribution

The

WASA-at-COSY

Collaboration

P. Adlarson

a

,

1

,

W. Augustyniak

b

,

W. Bardan

c

,

M. Bashkanov

d

,

F.S. Bergmann

e

,

M. Berłowski

f

,

H. Bhatt

g

,

A. Bondar

h

,

i

,

M. Büscher

j

,

2

,

3

,

H. Calén

a

,

I. Ciepał

k

,

H. Clement

l

,

m

,

E. Czerwi ´nski

c

,

K. Demmich

e

,

R. Engels

j

,

A. Erven

n

,

W. Erven

n

,

W. Eyrich

o

,

P. Fedorets

j

,

p

,

K. Föhl

q

,

K. Fransson

a

,

F. Goldenbaum

j

,

A. Goswami

j

,

r

,

K. Grigoryev

j

,

s

,

4

,

C.-O. Gullström

a

,

L. Heijkenskjöld

a

,

∗

,

V. Hejny

j

,

N. Hüsken

e

,

L. Jarczyk

c

,

T. Johansson

a

,

B. Kamys

c

,

G. Kemmerling

n

,

5

,

F.A. Khan

j

,

G. Khatri

c

,

6

,

A. Khoukaz

e

,

O. Khreptak

c

,

D.A. Kirillov

t

,

S. Kistryn

c

,

H. Kleines

n

,

5

,

B. Kłos

u

,

W. Krzemie ´n

f

,

P. Kulessa

k

,

A. Kup´s ´c

a

,

f

,

A. Kuzmin

h

,

i

,

K. Lalwani

v

,

D. Lersch

j

,

B. Lorentz

j

,

A. Magiera

c

,

R. Maier

j

,

w

,

P. Marciniewski

a

,

B. Maria ´nski

b

,

H.-P. Morsch

b

,

P. Moskal

c

,

H. Ohm

j

,

E. Perez del Rio

l

,

m

,

7

,

N.M. Piskunov

t

,

D. Prasuhn

j

,

D. Pszczel

a

,

f

,

K. Pysz

k

,

A. Pyszniak

a

,

c

,

J. Ritman

j

,

w

,

x

,

A. Roy

r

,

Z. Rudy

c

,

O. Rundel

c

,

S. Sawant

g

,

j

,

∗∗

,

S. Schadmand

j

,

I. Schätti-Ozerianska

c

,

T. Sefzick

j

,

V. Serdyuk

j

,

B. Shwartz

h

,

i

,

K. Sitterberg

e

,

T. Skorodko

l

,

m

,

y

,

M. Skurzok

c

,

J. Smyrski

c

,

V. Sopov

p

,

R. Stassen

j

,

J. Stepaniak

f

,

E. Stephan

u

,

G. Sterzenbach

j

,

H. Stockhorst

j

,

H. Ströher

j

,

w

,

A. Szczurek

k

,

A. Trzci ´nski

b

,

R. Varma

g

,

M. Wolke

a

,

A. Wro ´nska

c

,

P. Wüstner

n

,

A. Yamamoto

z

,

J. Zabierowski

aa

,

M.J. Zieli ´nski

c

,

J. Złoma ´nczuk

a

,

P. ˙Zupra ´nski

b

,

M. ˙Zurek

j

a_Division_of_Nuclear_Physics,_Department_of_Physics_and_Astronomy,_Uppsala_University,_Box_516,₇₅₁₂₀_Uppsala,_Sweden b_Department_of_Nuclear_Physics,_National_Centre_for_Nuclear_Research,_ul._Hoza_69,_00-681,_Warsaw,_Poland

c_Institute_of_Physics,_Jagiellonian_University,_prof._Stanisława_{Łojasiewicza}_11,_30-348_Kraków,_Poland

d_School_of_Physics_and_Astronomy,_University_of_Edinburgh,_James_Clerk_Maxwell_Building,_Peter_Guthrie_Tait_Road,_Edinburgh_EH9_3FD,_United_Kingdom e_Institut_für_Kernphysik,_{Westfälische}_{Wilhelms-Universität}_Münster,_{Wilhelm-Klemm-Str.}_9,₄₈₁₄₉_Münster,_Germany

f_High_Energy_Physics_Department,_National_Centre_for_Nuclear_Research,_ul._Hoza_69,_00-681,_Warsaw,_Poland g_Department_of_Physics,_Indian_Institute_of_Technology_Bombay,_Powai,_Mumbai_400076,_Maharashtra,_India h_Budker_Institute_of_Nuclear_Physics_of_SB_RAS,₁₁_akademika_Lavrentieva_prospect,_Novosibirsk,_630090,_Russia i_Novosibirsk_State_University,₂_Pirogova_Str.,_Novosibirsk,_630090,_Russia

j_Institut_für_Kernphysik,_{Forschungszentrum}_Jülich,₅₂₄₂₅_Jülich,_Germany

k_The_Henryk_{Niewodnicza´}_nski_Institute_of_Nuclear_Physics,_Polish_Academy_of_Sciences,₁₅₂_{Radzikowskiego}_St,_31-342_Kraków,_Poland l_{Physikalisches}_Institut,_{Eberhard-Karls-Universität}_Tübingen,_Auf_der_Morgenstelle_14,₇₂₀₇₆_Tübingen,_Germany

m_Kepler_Center_for_Astro_and_Particle_Physics,_Eberhard_Karls_University_Tübingen,_Auf_der_Morgenstelle_14,₇₂₀₇₆_Tübingen,_Germany n_{Zentralinstitut}_für_Engineering,_Elektronik_und_Analytik,_{Forschungszentrum}_Jülich,₅₂₄₂₅_Jülich,_Germany

o_{Physikalisches}_Institut,_{Friedrich-Alexander-Universität}_{Erlangen–Nürnberg,}_{Erwin-Rommel-Str. 1,}₉₁₀₅₈_Erlangen,_Germany

p_Institute_for_Theoretical_and_Experimental_Physics,_State_Scientiﬁc_Center_of_the_Russian_Federation,_Bolshaya_{Cheremushkinskaya 25,}₁₁₇₂₁₈_Moscow,_Russia q_II._{Physikalisches}_Institut,_{Justus-Liebig-Universität}_Gießen,_{Heinrich-Buff-Ring}_16,₃₅₃₉₂_Giessen,_Germany

r_Department_of_Physics,_Indian_Institute_of_Technology_Indore,_Khandwa_Road,_Indore_452017,_Madhya_Pradesh,_India

*

Correspondingauthor.

**

Correspondingauthorat:DepartmentofPhysics,IndianInstituteofTechnologyBombay,Powai,Mumbai400076,Maharashtra,India. E-mailaddresses:lena.heijkenskjold@physics.uu.se(L. Heijkenskjöld),siddhesh.sawant@iitb.ac.in(S. Sawant).

1 _Present_address:_Institut_für_Kernphysik,_{Johannes-Gutenberg-Universität}_Mainz,_{Johann-Joachim-Becher}_Weg_45,₅₅₁₂₈_Mainz,_Germany. 2 _Present_address:_Peter_Grünberg_Institut,_PGI-6_{Elektronische}_{Eigenschaften,}_{Forschungszentrum}_Jülich,₅₂₄₂₅_Jülich,_Germany.

3 _Present_address:_Institut_für_{Laser- und}_{Plasmaphysik,}_{Heinrich-Heine}_Universität_Düsseldorf,_{Universitätsstr.}_1,₄₀₂₂₅_Düsseldorf,_Germany. 4 _Present_address:_III._{Physikalisches}_Institut_B,_{Physikzentrum,}_RWTH_Aachen,₅₂₀₅₆_Aachen,_Germany.

5 _Present_address:_Jülich_Centre_for_Neutron_Science_JCNS,_{Forschungszentrum}_Jülich,₅₂₄₂₅_Jülich,_Germany. 6 _Present_address:_Department_of_Physics,_Harvard_University,₁₇_Oxford_St.,_Cambridge,_MA_02138,_USA. 7 _Present_address:_INFN,_Laboratori_Nazionali_di_Frascati,_Via_E._Fermi,_40,₀₀₀₄₄_Frascati_(Roma),_Italy.

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

(2)

s_High_Energy_Physics_Division,_Petersburg_Nuclear_Physics_Institute,_Orlova_{Rosha 2,}_Gatchina,_Leningrad_district_188300,_Russia

t_Veksler_and_Baldin_Laboratory_of_High_Energiy_Physics,_Joint_Institute_for_Nuclear_Physics,_{Joliot-Curie 6,}₁₄₁₉₈₀_Dubna,_Moscow_region,_Russia u_August_Chełkowski_Institute_of_Physics,_University_of_Silesia,_{Uniwersytecka}_4,_40-007,_Katowice,_Poland

v_Department_of_Physics,_Malaviya_National_Institute_of_Technology_Jaipur,_302017,_Rajasthan,_India

w_JARA-FAME,_Jülich_Aachen_Research_Alliance,_{Forschungszentrum}_Jülich,₅₂₄₂₅_Jülich,_and_RWTH_Aachen,₅₂₀₅₆_Aachen,_Germany x_Institut_für_{Experimentalphysik I,}_{Ruhr-Universität}_Bochum,_{Universitätsstr.}_150,₄₄₇₈₀_Bochum,_Germany

y_Department_of_Physics,_Tomsk_State_University,₃₆_Lenina_Avenue,_Tomsk,_634050,_Russia z_High_Energy_Accelerator_Research_Organisation_KEK,_Tsukuba,_Ibaraki_305-0801,_Japan aa_Department_of_{Astrophysics,}_National_Centre_for_Nuclear_Research,_Box_447,_90-950_Łód´z,_Poland

B. Kubis

ab

,

S. Leupold

ac

ab_{Helmholtz-Institut}_für_{Strahlen- und}_Kernphysik,_Rheinische_{Friedrich-Wilhelms-Universität}_Bonn,_Nußallee_14–16,₅₃₁₁₅_Bonn,_Germany ac_Division_of_Nuclear_Physics,_Department_of_Physics_and_Astronomy,_Uppsala_University,_Box_516,₇₅₁₂₀_Uppsala,_Sweden

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received10October2016

Receivedinrevisedform29January2017 Accepted8March2017

Availableonline25April2017 Editor:V.Metag

Keywords:

Decaysofothermesons Meson–mesoninteractions Lightmesons

Using theproduction reactions pd_→3_He

_ω

_and _pp_→_pp

_ω

_,_the _Dalitz_plot_distribution_for _the

_ω

_→

π

+

π

−

π

0 _decay _is _studied _with _the _WASA _detector _at _COSY, _based _on _a _combined _data _sample _of

(4.408±0.042)×104_events._The_Dalitz_plot_density_is_parametrised_by_a_product_of_the_{P -wave}_phase

spaceandapolynomialexpansioninthenormalisedpolarDalitzplotvariables Z and φ.Fortheﬁrst time,adeviationfrompureP -wavephasespaceisobservedwithasigniﬁcanceof4.1

σ

.Thedeviation isparametrisedbyalinearterm1+2

α

Z ,with

α

determinedtobe+0.147±0.036,consistentwiththe expectationsof

ρ

-meson-typeﬁnal-stateinteractionsoftheP -wavepionpairs.

1. Introduction

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

ω

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

The

ω

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

ω

→

π

+

π

−

π

0_, _with _a _branching _ratio _of _BR

₌

(

89

.

2

±

0

.

7

)

%. By now it is well established that the

ω

meson hasspin-parity JP

=

1− [3]. As a consequence,the combination ofBose,isospin,andparitysymmetryofthestronginteraction de-mandsthat for the decay

ω

→

π

+

π

−

π

0 _every _pion_pair _is_in _a state of odd relative orbital angular momentum. Given the lim-itedphase spaceofthedecayone cansafelyassume the P -wave

to be the dominant partial wave.8 If a pion pairis in a P -wave

state,then the third pionwill be in P -wave state relativeto the pair.This “ P -wave phase space”distribution has beenconﬁrmed experimentally. Historically the P -wave dominance of the decay hasactually beenusedtopindownthequantumnumbersofthe

ω

meson[5–8].

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

ω

→

π

+

π

−

π

0_._However,_a_pion_pair_in_a _{P -wave} showsavery strongﬁnal-state interaction.The two-pion P -wave

phase shift is dominated by the

ρ

meson, and is now known

very accurately [9–11]; thisis essential in particular for theoret-ical studies of these decays using dispersion theory, which use

8 _Genuine_{F -wave}_corrections_have_been_modelled_{theoretically,}_and_found_to_be tiny[4].

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

φ

→

π

+

π

−

π

0 _one_can_see_the

_ρ

_meson_as_a_resonance_in_the cor-respondingDalitzplot[13,14].Inthedecay

ω

→

π

+

π

−

π

0_there_is notenoughenergyforapionpairtoreachthe

ρ

resonancemass; yetalreadyfortheavailableinvariantmassesthetwo-pionP -wave

phaseshiftissigniﬁcantly differentfromzero.Infact,every theo-reticalapproachthatdealswiththedecay

ω

→

π

+

π

−

π

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

ρ

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

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

ω

→

π

+

π

−

π

0 _Dalitz _plot_measurement from 1966had 4208

±

75 signal events [21].Due to the limited statistics,ﬁtswithapure P -wavephasespacecouldnotbe distin-guishedfromadistortionbytheﬁnal-stateinteractions,i.e. by in-termediate

ρπ

states.Surprisinglytherewerenofurtherdedicated Dalitzplotstudiesofthe

ω

→

π

+

π

−

π

0_decay._In_the_present_work wewillrevealthattheuniversalﬁnal-stateinteractionsofthepion pairsareindeedpresentinthe

ω

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

Thesecond motivationforaprecisemeasurement ofthe

ω

→

π

+

π

−

π

0 _Dalitz _plot _consists _in _improving _hadronic _input _for the theoretical assessment of the hadronic light-by-light scatter-ingcontributiontotheanomalousmagneticmomentofthemuon. Thelargestindividualcontributionisgivenbythelightesthadronic intermediate state,the so-called

π

0 _pole_term, _whose_strength _is determinedbythecorrespondingsinglyanddoublyvirtual transi-tionformfactors.Oneofthefewpossibilitiestogainexperimental access to the doubly virtual

π

0 _transition_form _factor _with _high precision consistsinstudyingvector mesonconversion decays,in

(3)

particular

ω

→

π

0

+

−_—which_is _intimately_linked, _through dis-persion relations, to the

ω

→

π

+

π

−

π

0 _decay _amplitude _[22,23,

12]. However, these theoretical descriptions of the

ω

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

ω

→

π

0

_μ

+

_μ

− _taken_by _the _NA60 _{collaboration} _[25,26]_,

which may violate very fundamental theoretical bounds [27,28]. Inall thesestudies, the

ω

→

π

+

π

−

π

0 _decay_amplitude _is _a po-tential looseend, asit is so far only theoretically modelled, not experimentallytested.Incombinationwiththepreciselymeasured

φ

→

π

+

π

−

π

0_Dalitz _plot_information,_it_could_be _used_to_further constraintheamplitude analysisofe+e−

→

π

+

π

−

π

0_,_and_hence the

π

0_transition_form_factor_in_a_wider_range_[29]_.

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

ω

→

π

+

π

−

π

0 _Dalitz _plot_distribution_has_been_suggested_[4]_,_as ageneralisationofthecommonlyusedoneforthedecay

η

→

3

π

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

One wayto describe athree-particle Dalitz decaydistribution istouseinvariantmassesofparticlepairs[3].Thiswouldbe par-ticularlyusefulforreadingoffresonancemassesifthedecaywas mediatedbyoneorseveralresonances.However, thereareno in-termediateresonancesinthekinematicallyaccessibleenergyrange of the decay

ω

→

π

+

π

−

π

0 _that _would _be _compatible _with_the symmetriesofthestrong interaction.Forourcaseofinterest,we ﬁrst split offthe P -wave phase space(see, e.g., Refs. [19,4]) and parametrisetherestbyapolynomialdistributionfollowing[4]. De-notingthe polarisationvector of the

ω

mesonby

(

Pω

,

λ

ω

)

and

themomentaof theoutgoingpionsby P₊, P₋, and P0, westart withthemostgeneral matrixelement compatiblewiththe sym-metries,

M

=

i

ε

μναβ

μP₊νP₋αPβ0

F

.

(1)

The dynamics of the ﬁnal-state interactions is encoded in the scalarfunction

F

[19,4].Aftersummationoverthehelicity

λ

ω of

the

ω

mesononeobtainsaDalitzplotdistributionproportionalto

λω

|M|

2

_∝

_P

_|F|

2 ₍₂₎

withthepure P -wavephase-spacedistribution

P

=

m2₊m2₋m2₀

+

2

(

P₊P₋

)(

P₋P0

)(

P0P+

)

−

m2₊

(

P₋P0

)

2

−

m2₋

(

P+P0

)

2

−

m20

(

P+P−

)

2

.

(3) Notethat forthe

P

termwe canaccount for“kinematic”isospin violations due to the difference between the masses of the un-chargedandcharged pions,m0 andm±, respectively.For the re-maining distribution

F

, whichcovers thedynamics of the ﬁnal-stateinteraction,weignoreisospinbreakingeffects.

The quantity

F

and therefore also

|

F|

2 wouldbe a constant iftherewerenoﬁnal-stateinteractionsbetweentheproduced pi-ons.Inreality

|

F|

2 isnotaconstant,butrelativelyﬂat.Insteadof parameterising

|

F|

2 by invariant massesof pion pairs we follow Ref.[4]andutilisenormalisedvariables X andY ,whichhavetheir originatthecentreoftheDalitzplot.Theyaredeﬁnedby

X

=

√

3 T+

−

T− Qω

,

Y

=

3T0 Qω

−

1

,

(4) with Qω

=

T+

+

T₋

+

T0

.

(5)

Here Ti arethekinetic energies ofthepionsin the

ω

restframe (centre-of-massframeofthethree-pionsystem).Finallyone intro-ducespolarcoordinatesby

Table 1

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

α×103 _β_×₁₀3 Uppsala[20] 202 – Bonn[4] 84. . .102 – JPAC[12] 94 – Uppsala 190 54 Bonn 74. . .90 24. . .30 JPAC 84 28 X

=

√

Z cos

φ,

Y

=

√

Z sin

φ .

(6)

Theexpansionfor

|

F|

2_,_valid_in_the_isospin_limit,_reads

|F

|

2

₍

_Z

_{, φ)}

₌

_N

_·

_G

₍

_Z

_{, φ) ,}

₍₇₎ where

N

isanormalisationconstantand

G

containstheexpansion in Z and

φ

[4]:

G

(

Z

, φ)

=

1

+

2

α

Z

+

2

β

Z3/2sin 3

φ

+

2

γ

Z2

+

O

Z5/2

.

(8)

The Dalitz plotdistribution can thenbe ﬁtted usingthisformula toextractthe“Dalitzplotparameters”

α

,

β

,

γ

,. . . .Thefitresults to thetheory predictionsofRefs. [4,20,12],ifEq.(8)istruncated atorderZ (oneparameterfit)oratorder Z3/2_(two_parameter_fit), are shownin Table 1. The reproduction of the theoretical Dalitz plot distributions is improved significantly in all cases when in-cludingtheterm

∝ β

.

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

α

.Thisreﬂectsthe factthat thepion–

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

ρ

-mesonresonance;anexperimentalresultpointingtoanegative value of

α

wouldbe spectacularinthesense thatit wouldbeat oddswiththeuniversalityoftheﬁnal-stateinteractions.

Refs. [4,12] are based on dispersion theory: both employ the pion–pion P -wave scattering phase shift as input and describe rescatteringofallthreeﬁnal-statepionsconsistentlytoallorders. Thedispersiveformalismcan bechosenwithonlyonesingle free parameter(a“subtractionconstant”),basedonreasonable assump-tions onthehigh-energy behaviouroftheamplitude;this param-etercanbetakentobetheoverallnormalisationandﬁxed exper-imentally from the

ω

→

π

+

π

−

π

0 _partial _width. _The _energy de-pendence ofthedecayamplitude isthenfullypredictedfromthe pion–pionphaseshiftalone.TheDalitzplotparameterscitedfrom Ref. [12]in

Table 1

areobtainedfromsucha scenario.InRef. [4], varioussourcesofuncertaintyrelatedtothepion–pioninteraction havebeenconsidered,leadingtoanestimateofthetheoretical er-rorinthepredictionoftheDalitzplotparameters.Furthermore,an analysisof datafortheanalogous

φ

→

π

+

π

−

π

0 _Dalitz _plot_[13] demonstratedthataonce-subtracteddispersiverepresentation de-scribes suchvery-high-accuracydatawell, butnot perfectly, lead-ing to theneed tointroduce asecond subtraction [4].With such a second free parameter also used in the

ω

→

π

+

π

−

π

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

φ

→

π

+

π

−

π

0 _data _analysis,_it _was _found that the uncertainty range for the theoretical prediction of the Dalitzplotparametersisonlymoderatelyincreased[4].Theranges quotedin

Table 1

reﬂectthisfull,combineduncertaintyestimate.

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

ω

-

ρ

-

π

interaction is ﬁtted to the decay width of

ω

→

π

+

π

−

π

0 _and cross-checkedwiththe decaywidthof

ω

→

π

0

_γ

_._The _Lagrangian

(4)

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

Whileall theoryapproachesagreeon thesignof

α

,the value predictedby the Lagrangian approach[20] isvery different from the valuesobtained by dispersion theory [4,12]. The same holds trueforthe

β

parameter: thelarger valuesfromRef. [20] reﬂect a rather strong energy dependence of the matrix element. The same qualitative difference can be observed for the electromag-netictransitionformfactorof

ω

→

π

0

+

−_[24,23]_:_also_here_the Lagrangianapproachprovidesamuchstrongerenergydependence. Atechnicalreasonforthismightbe foundinthefactthat essen-tiallyﬁeld strengths instead ofvector potentialsare used forthe constructionofinteraction termsin theLagrangian approach[19, 24,20]. If the results of this low-energy Lagrangian were boldly extrapolatedto highenergies—beyond its limit of applicability—, thenonewouldﬁndthat thereactionamplitudeswouldnot con-verge. In contrast, modest high-energy constraints are automati-cally encodedinthe dispersiveapproaches. Apparentlythis leads to smaller energy variations of the reaction amplitudes even in thelow-energyregimethatisofrelevanceforthe

ω

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

2. Theexperiment

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

ω

was produced in the pd

→

3He

ω

reaction and in the pp

→

pp

ω

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

∼

35μm).

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

When the

ω

mesons were produced using the pd

→

3_He

_ω

reaction, two different proton kinetic energies were used: TA

=

1

.

450GeV and TB

=

1

.

500GeV.The cross section ofthe reaction is84(10) nbatthelower energy[32]andwas studiedpreviously bytheCELSIUS/WASAcollaboration.Triggersselecteventswithat leastone trackin theFDwith ahigh energydepositin thethin plasticscintillator layers.Thisconditionallows foran eﬃcient se-lectionof 3_He _ions _and _provides _an _unbiased _data _sample _of

_ω

meson decays.The proton beam energy was chosen so that the

3_He _produced _in _the _pd

_→

3_He

_ω

_reaction _stops _in _the _second thick scintillator layer of the FD. The correlation plot

E

−

E

from a thin layer and the ﬁrst thick layer of the FD is shown in Fig. 1 (top). The band corresponding to the 3He ion is well separated from the bands for other particles and allows a clear identiﬁcationof3He.The3He fromthereactionofinteresthas ki-neticenergies up to 700 MeV andscatteringangles rangingfrom 0◦ to10◦.

Fig. 1. ParticleidentificationintheForwardDetectorisperformedusingthe corre-lationofenergydepositsintheplasticdetectorlayers.(top)For3_He_{identification:} thecorrelationbetweentheenergydepositsinathin(0.5 cm)layerandthefirst subsequentthicklayer(11 cm).Onlytheregionselectedintheanalysisisshownin thefigure,withthebandfrom3_He_particles_clearly_visible_inside_it._(bottom)_For protonidentification:thecorrelationbetweentheenergydepositinthefirstthick layerandthesummedenergydepositsinallthicklayers(11or15 cm).The correla-tionbandcorrespondingtoenergydepositsmadebyprotonsissurroundedbythe blackline.Alsovisibleisalower,near-horizontal,bandwhichispopulatedbyfast protonspunchingthroughthefirstthicklayer,depositingenergyof20 MeV,and un-dergoingnuclearinteractioninoneofthesubsequentthicklayers,theredepositing anindefiniteamountofenergy.

The pp

→

pp

ω

experimentwas performedat TC

=

2

.

063 GeV beamkineticenergy,corresponding to60 MeV centre-of-mass ex-cessenergy andcross section 5.7 μb[33]. In the pp collision ex-periment, the selected events were required, at trigger level, to containatleasttwo tracksreaching thesecond thicklayer ofthe plasticscintillatorsintheFD, atleasttwohitsinthe PSB,andat leastoneclusterintheSEC.Intheoﬄineanalysis, pairsoftracks correspondingtothe

E

−

E protonbands,shownin

Fig. 1

(bot-tom),indifferentthicklayersoftheFDareselectedasprotonpair candidates.

Fortheparticlesmeasured intheCD,acommonanalysis pro-cedure is usedfor all threedata sets.Events are selected ifthey contain atleast onepairofopposite charge particletracksin the MDC with scattering angles greater than 30◦ and at least two neutralclusterswithenergydepositabove20 MeVintheSEC. Rel-ative time between the tracks is checked to minimise pile ups. The charged particle tracks are assigned the charged pion mass. Combinations of the all the measured charged and neutral par-ticle tracksin the selected eventsare testedusing a constrained kinematicﬁtassumingtheconservationofenergyandmomentum

(5)

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

with the pd

→

3He

π

+

π

−

γ γ

or pp

→

pp

π

+

π

−

γ γ

hypothesis, respectively. The combinations with p-values less than 0.05 are rejected. Forthe casewhen morethan one trackcombinationin aneventfulﬁlsthiscriteria,thecombinationgivinglarger p-value

isselected.Finally,furtherbackgroundsuppressionisachievedby

applying a kinematic ﬁt with the contending hypothesis pd

→

3_He

_π

+

_π

−_or_pp

_→

_pp

_π

+

_π

−_,_{respectively.}_If_the_resulting_p-value

islargerthanfortheﬁrstﬁt,theeventisrejected.

Themissingmassdistributions,MM

(

3He

)

andMM

(

pp

)

,forthe threedatasetsareshownin

Fig. 2

.Themissingmasses,calculated from the variables corrected by the kinematic ﬁt,are equivalent to the invariant mass of the

π

+

π

−

γ γ

system. The observed

ω

peakpositionisshiftedfromthenominalvalue ofthe

ω

massby

+

0.7 MeVforMM

(

3_He

₎

_in_the_two _{pd data}_sets _and_by

₊

_{1.1 MeV} for MM

(

pp

)

. The observed shifts correspond to deviations from thenominalbeamenergyby0.55 MeVand0.75 MeV,respectively, whichiswell withintheuncertaintyofthe absoluteenergyscale ofCOSY.Toreproducetheexperimental

ω

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

ω

peak,theresolutionfromsimulated detec-torresponseswereadjusted.

Both thebackgroundshape andthe

ω

peak contentare ﬁtted simultaneouslyto theexperimental distributionusing the follow-ingﬁtfunction: H

(

μ

)

=

NSHω

(

μ

)

+

a0

+

a1

μ

+

a2

μ

2

+

a3

μ

3

×

H3π

(

μ

),

(9)

where

μ

=

MM

(

3He

)

or MM

(

pp

)

. Hω

(

μ

)

and H3π

(

μ

)

represent

reconstructed distributions of simulated signal and background andcorrespondtoeventsthathavepassedthroughthesame anal-ysisstepsastheexperimentaldata.Hω

(

μ

)

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

μ

used forthe ﬁt is

[

0

.

640

,

0

.

832

]

GeV

/

c2 forset A,

[

0

.

640

,

0

.

856

]

GeV

/

c2 forset B, and

[

0

.

608

,

0

.

824

]

GeV

/

c2 forset C.Thelimitsoftheserangesare shownbythedashedlinesin

Fig. 2

,wheretheresultoftheseﬁts tothefulldatasamplesaregiven.Theresultingnumberofevents is:14600(200)forset A,13500(200)forset B,and16000(300)for set C.

3. Dalitzplot

TheDalitzplotdensityisrepresentedusingatwo-dimensional histogramin the Z and

φ

variables, deﬁned in Eq.(6). The size oftheselectedbinsisdeterminedby theexperimentalresolution

of Z and

φ

and the statistics of the collected data sample. The

numberofeventsineachbinshouldbesuﬃcientfordetermining

Fig. 3. TheArabicnumeralsshowthebinnumberstobeusedwhenpresentingthe resultingDalitzplots.Thecolourplotshowsthekinematicallyallowedregionofthe

ω→π+π−π0_reaction_with_ω_nominal_mass_as_well_as_the_density_distribution from P -wavedynamics.TheRomannumeralsdisplaythesectorsusedin consis-tencychecks.

thesignalyieldandtocarryouta

χ

2 _ﬁt_of_the_Dalitz_plot_density parametrisation. The

φ

variablerange

[−

π

, π

]

isdivided intosix binstopreservethethreefoldisospinsymmetryandtobesensitive toapossiblesin 3

φ

dependence.The Z variablerange

[

0

,

1

]

isalso divided intosix bins.Onlythe 21bins fullycontainedinside the kinematic limitsof the decayare used. Fig. 3introduces the bin numberingusedforthepresentationoftheresults.

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

φ

=

π

/

2 to the onesat

φ

= −

π

/

6 and

−

2

π

/

3.Thepictureshowsalsoseven sec-torsI–VIIthatareusedtotesttheconsistencyoftheﬁtresults.

ForeachDalitz plotbin,theexperimentalmissingmass distri-bution isconstructedandthenumberofentriesinthe

ω

peakis extractedby ﬁttinga simulated

ω

→

π

+

π

−

π

0 _signal _along_with backgroundcontributionsusingEq.(9).

Since the P -wavedistribution reproduces thegeneral features ofthe

ω

→

π

+

π

−

π

0 _Dalitz_plot_very_well _and_the_deviations_are expected to be small, the eﬃciency correction is obtained using signal simulationwiththe P -wave. Theeﬃciency, i,isextracted usingtheratio i

=

Ni

/

NiG.NGi isthenumberofeventswith gen-eratedkinematicvariablescorrespondingtobini intheDalitzplot.

Ni is the content of the bin i when the reconstructed values of the kinematic variables are used for events passing all analysis steps. Theextractedeﬃciencies forthe threedatasetsareshown in Fig. 4.Forthe pd data setsthe overalleﬃciencyis11%, while

for pp it is 20 times lower.This low eﬃciency for the pp data

(6)

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

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

→

3_He

_ω

_reaction,_there isonlyonedoublycharged3He stoppinginthedetector.Another causefortheloweﬃciencyisthelargercentre-of-massvelocityin

the pp reaction,which decreases the average emission anglefor

decayparticles,inparticularforthechargedpions.Thepionswill bemoreoftenemitted atangles below30◦ andwillthereforebe rejectedintheanalysisprocedure.

TheDalitzplotparameters(

α

,

β

,

. . .

)andnormalisationfactors forthethreedatasets(

N

A,

N

B,

N

C)aredeterminedby minimis-ingthefollowing

χ

2

₌

_χ

2

A

+

χ

B2

+

χ

C2function,where

χ

2_A

=

i

˜

Ni A

−

N

A

·

Hi

(

α

, β, . . .)

˜

σ

i A

2

.

(10)

˜

Ni and

σ

˜

iaretheeﬃciencycorrectedexperimentalDalitzplotbin contentanderror,respectively. Hiisgivenbyanintegral overbin

i: Hi

(

α

,

β,

. . .)

=

i

P(

Z

,

φ)G(

Z

,

φ)

d Zd

φ

.

P(

Z

,

φ)

is the P -wave phase space term given by Eq. (3), calculated using the nomi-nalmassofthe

ω

mesonof782.65 MeV,and

G(

Z

,

φ)

isgivenby Eq.(8).

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

G =

1) andwithout detector smearing.The extracted parameters arefoundto beconsistentwithzeroandthereforetheprocedure doesnotintroduceanybiasatthepresentstatisticalaccuracy.

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

The method of background subtraction for the missing mass

μ

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

π

+

π

−

π

0 productionbackgroundeventsandthe

ω

signal generatedusinga

P -wavephasespacedistribution.Thebackgroundisobtainedfrom the

a0

+

a1

μ

+

a2

μ

2

+

a3

μ

3

×

H3π

(

μ

)

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

μ

distributionswiththenumberofevents similarasintheexperimentarethensubjectedtothesame back-groundsubtractionastheexperimental data.Thecombinedﬁtof

Fig. 5. TheexperimentalDalitzplotdistributionafterapplyinganeﬃciency correc-tion.Therelationbetweenthebinnumbersusedhereandthebinsofthe two-dimensionalDalitzplotisshowninFig. 3.Circlescorrespondtoset A,squaresto set B,andtrianglestoset C.Thesolidredlineisthestandardﬁtresult(withα

parameter),andthedashedlineisP -waveonly.

Table 2

Dalitzplotbincontentforthethreedatasets.Therelativenormalisationbetween thesetsisbasedonthenormalisationfactors(NA,NB,NC)obtainedfrom

individ-ualﬁtsoftheαparametertothethreedatasets.Theoverallnormalisationfactor isarbitrary.

bin# set A set B set C

1 5.51(34) 6.09(33) 6.76(45) 2 6.71(35) 6.58(34) 5.63(38) 3 5.86(35) 5.30(36) 6.23(42) 4 6.07(37) 6.68(38) 6.29(43) 5 5.31(36) 5.17(35) 5.59(39) 6 6.48(36) 5.73(34) 5.41(41) 7 4.24(29) 4.55(29) 3.91(35) 8 4.63(30) 4.41(29) 4.83(31) 9 4.47(31) 4.03(32) 4.54(33) 10 4.23(33) 4.64(36) 4.59(38) 11 4.03(33) 4.72(34) 4.25(36) 12 4.96(32) 4.85(31) 5.19(42) 13 2.25(23) 2.09(22) 2.22(28) 14 3.36(25) 3.36(25) 3.51(28) 15 2.53(24) 2.90(27) 2.55(24) 16 3.66(30) 3.90(34) 2.93(32) 17 2.52(28) 2.86(30) 2.93(37) 18 3.98(28) 3.66(28) 3.55(38) 19 2.14(21) 2.38(21) 2.60(21) 20 2.26(26) 1.89(27) 2.19(32) 21 2.63(24) 2.33(23) 2.27(32)

theDalitz plotparametrisationtothesamplesA andB withonly the

α

parameter gives

α

= (

10

±

35

)

·

10−3 _and

_χ

2

₌

₃₆

_/

_39._For set C

α

= (

25

±

59

)

·

10−3 _and

_χ

2

₌

₂₄

_/

_19. _Therefore _the back-groundsubtractionproceduredoesnotintroduceanyexperimental bias.

Onecan alsostudythebias andaccuracyoftheeﬃciency de-terminationbyconsidering X - orY -dependentcorrectionsforthe eﬃciency:

i

→

i

· (

1

+ ξ

AX

)

or

i

→

i

· (

1

+ ζ

AY

)

,where

ξ

A,...,

ζ

A,...aresingle parametersforeachdataset.Fitstoseparate data sets show that all

ζ

coeﬃcientsare consistent withzeroand do not change the value of the

χ

2_. _On _the _other _hand,

_ξ

B and

ξ

C werefoundtosignificantlydeviatefromzero,althoughwith oppo-sitesigns.Applyingthesetwocorrectionstotheefficiencybeforea fitoftheDalitzplotparametrisationyieldsasignificantlyreduced

χ

2_value._However,_the_determined_values_of_the_Dalitz_parameters arenotaffected,e.g.

α

= (

147

±

35

)

·

10−3and

α

= (

147

±

36

)

·

10−3 without and with correction, respectively. This comes from the fact that the ﬁtted parametrisationis preserving isospin symme-try. In conclusion, we apply the X -dependentcorrections to the eﬃciencycorrectionsofdatasets BandC,asitensuresthe antic-ipatedchargesymmetryoftheDalitzplotandleadstoadecrease

(7)

Table 3

TheresultingDalitzplotparametersafteraindividualﬁtstothethreedatasets, whereatmosttwoparameterswereusedintheﬁt.

Data set α×103 _β_×₁₀3 _χ2_/d.o.f.

A – – 28.7/20 142(59) – 22.2/19 102(66) 109(87) 20.7/18 B – – 35.4/20 146(59) – 28.5/19 154(69) −21(92) 28.5/18 C – – 26.5/20 154(69) – 20.8/19 149(78) 14(102) 20.8/18 Table 4

Dalitzplotparametersfromsimultaneousﬁtstothethreedatasets,whereatmost twoparameterswereusedintheﬁt.

α×103 _β_×₁₀3 _χ2_/d.o.f.

– – 90.6/60

147(36) – 71.5/59

133(41) 37(54) 71.0/58

ofthe

χ

2 _as_well _as_the_correct _statistical_{significance} _when fit-tingthe Dalitz parameters. Theresulting efficiencycorrected and normalisedDalitzplotbincontentsareprovidedin

Table 2

.

The extracted Dalitz plot parameters and goodness of ﬁt for each data set separately are reported in Table 3. There is a sig-niﬁcantdecreaseofthe

χ

2_value_when_including_the

_α

_parameter and the results from the three data sets are consistent. The re-sults of the ﬁts for all data sets combined are givenin Table 4. The p-valuesigniﬁcantlyimproves afterincludingthe

α

parame-ter,while inclusion ofan additional parameter doesnot improve the p-value any further. The eﬃciency corrected Dalitz plots for thethreedatasetsareshownin

Fig. 5

,wheretheyarecompared tothe P -wavedistributionaswell astheﬁtwiththe

α

parame-ter.Weconsidertheresultwiththe

α

parameterourmainfinding. The differencebetween theresults fromthefirst andsecond fits in

Table 4

indicates theonsetofdynamicsinthe reactionontop ofthe P -wavephasespacedistribution.Thisfollowstheexpected behaviourofanincreasetowardstheedgesofphasespacedueto theattractive

π π

ﬁnal-stateinteraction (approachingthe

ρ

reso-nance),yieldingapositivevalueforthe

α

parameter.

Fig. 6showsatestofdataconsistencyforthesameDalitzplot sectors,markedwithRomannumeralsin

Fig. 3

.Arithmetical aver-agesofthenormalisedresiduals withrespect tothe

α

parameter ﬁtfrombinscorresponding tothesamesectorsarecalculatedfor the separate data sets and forall three data sets combined. The errorbars correspond to thecalculated rootmean square values, which are expectedto be 1for a random datasample with cor-rectlyestimateduncertainties.

Herefollowsashortsummaryofthechecksforsystematic ef-fectsreportedinthissection.Twoinput–outputcheckswere per-formed,whichveriﬁedthattheparametrisationprocedureaswell asthemethodofbackgroundsubtractionsdoesnot introduceany bias at the present statistical accuracy. The eﬃciency correction wascheckedandadjustedtoensurechargesymmetryintheDalitz plot.Lastly,theconsistencybetweenthethreedatasetsaswellas thedifferentsectorsoftheDalitzplotwaschecked.Theresultfrom thesechecksisthat theaccuracy oftheDalitz plotparametersis dominatedbystatisticuncertainty.

Fig. 6. Arithmeticaveragesand rootmeansquareofthenormalisedresidualsin separateDalitzplotsectorsforalldatasets(crosses)andfortheseparatedatasets (A–circles,B–squares,C–triangles)forthestandardﬁt.

Fig. 7. Comparisonofourresultfortheαparameter(shadedarea)withthethree theoreticalpredictions[20,4,12].

4. Summaryanddiscussion

Fortheﬁrsttime adeviationfromapure P -wavedistribution in

ω

→

π

+

π

−

π

0_is_observed_and_quantiﬁed_by_the_{determination} of theparameter

α

= (

147

±

36

)

·

10−3,i.e. a positivevalue with 4

.

1

σ

signiﬁcance.

Fig. 7

comparestheexperimentalresultofthe

α

parametertothetheoreticalpredictions.Theexperimental

α

value isclearlyinthevicinityofthepredictionsmadebydispersion the-ory[4,12]andtheeffective-Lagrangianapproach[20].Howeverthe experimental uncertaintyis still toosizeableto allowfordeﬁnite conclusions concerning the validity of these contrasting predic-tions.Thesystematiceffectswerestudiedbycomparingthreedata sets usingtwo production reactions, which differ signiﬁcantly in resolution and acceptance. The chosen Z ,

φ

parametrisation to-gether withisospinsymmetry allowsformoretestsofsystematic effects,andtheprecision oftheresultisdominatedbythe statis-ticaluncertainty.

Acknowledgements

This work was supported inpart by theEU Integrated Infras-tructure Initiative HadronPhysics Project under contract number RII3-CT-2004-506078;bytheEuropeanCommissionunderthe7th Framework Programme through the Research Infrastructures ac-tion of the Capacities Programme, Call:

FP7-INFRASTRUCTURES-2008-1, Grant Agreement N. 227431; by the Polish National

Science Centre through the grants DEC-2013/11/N/ST2/04152,

2011/01/B/ST2/00431, 2011/03/B/ST2/01847, and the Foundation

for Polish Science (MPD), co-ﬁnanced by the European Union

within the European Regional Development Fund. We gratefully acknowledge thesupport givenby theSwedish ResearchCouncil,

(8)

theKnutandAliceWallenbergFoundation,andthe Forschungszen-trumJülichFFEFundingProgram.Thisworkis basedonthePhD thesesofLenaHeijkenskjöldandSiddheshSawant.

References

[1]G.Colangelo,M.Hoferichter,B.Kubis,M.Procura,P.Stoffer,Phys.Lett.B738 (2014)6–12,arXiv:1408.2517.

[2]B.C. Magli ´c,L.W.Alvarez,A.H. Rosenfeld,M.L. Stevenson,Phys. Rev.Lett. 7 (1961)178–182;

B.C.Magli ´c,L.W. Alvarez,A.H.Rosenfeld, M.L.Stevenson, Adv.Exp.Phys. 5 (1976)106.

[3]K.A.Olive,etal.,ParticleDataGroup,Chin.Phys.C38(2014)090001. [4]F.Niecknig,B. Kubis, S.P. Schneider,Eur. Phys. J.C 72(2012) 2014, arXiv:

1203.2501.

[5]M.L.Stevenson,L.W.Alvarez,B.C.Magli ´c,A.H.Rosenfeld,Phys.Rev.125(1962) 687–690.

[6]C.Alff,etal.,Phys.Rev.Lett.9(1962)325–327. [7]N.-h.Xuong,G.R.Lynch,Phys.Rev.128(1962)1849–1867.

[8]J.S.Danburg,M.A.Abolins,O.I.Dahl,D.W.Davies,P.L.Hoch,J.Kirz,D.H.Miller, R.K.Rader,Phys.Rev.D2(1970)2564–2588.

[9]B.Ananthanarayan,G.Colangelo,J.Gasser,H.Leutwyler,Phys.Rep.353(2001) 207–279,arXiv:hep-ph/0005297.

[10]G.Colangelo,J.Gasser,H.Leutwyler,Nucl.Phys.B603(2001)125–179,arXiv: hep-ph/0103088.

[11]R.García-Martín,R.Kami ´nski,J.R.Peláez,J.RuizdeElvira,F.J.Ynduráin,Phys. Rev.D83(2011)074004,arXiv:1102.2183.

[12]I.V.Danilkin,C.Fernández-Ramírez,P.Guo,V.Mathieu,D.Schott,M.Shi,A.P. Szczepaniak,Phys.Rev.D91(2015)094029,arXiv:1409.7708.

[13]A.Aloisio,etal.,KLOE,Phys.Lett.B561(2003)55–60,arXiv:hep-ex/0303016; A.Aloisio,etal.,KLOE,Phys.Lett.B609(2005)449(Erratum).

[14]R.R.Akhmetshin,etal.,Phys.Lett.B642(2006)203–209. [15]M.Gell-Mann,D.Sharp,W.G.Wagner,Phys.Rev.Lett.8(1962)261. [16]F.Klingl,N.Kaiser,W.Weise,Z.Phys. A356(1996)193–206,arXiv:hep-ph/

9607431.

[17]M.Harada,K.Yamawaki,Phys.Rep.381(2003)1–233,arXiv:hep-ph/0302103. [18]P.D.Ruiz-Femenía,A.Pich,J.Portolés,J.HighEnergyPhys.07(2003)003,arXiv:

hep-ph/0306157.

[19]S.Leupold,M.F.M.Lutz,Eur.Phys.J.A39(2009)205–212,arXiv:0807.4686. [20]C.Terschlüsen,B.Strandberg,S.Leupold,F.Eichstädt,Eur.Phys.J.A49(2013)

116,arXiv:1305.1181.

[21]S.M.Flatté,D.O.Huwe,J.J.Murray,J.Button-Shafer,F.T.Solmitz,M.L.Stevenson, C.Wohl,Phys.Rev.145(1966)1050–1061.

[22]G.Köpp,Phys.Rev.D10(1974)932–940.

[23]S.P. Schneider,B. Kubis, F.Niecknig,Phys.Rev.D 86(2012) 054013,arXiv: 1206.3098.

[24]C.Terschlüsen,S.Leupold,Phys.Lett.B691(2010)191–201,arXiv:1003.1030. [25]R.Arnaldi,etal.,NA60,Phys.Lett.B677(2009)260–266,arXiv:0902.2547. [26]R.Arnaldi,etal.,NA60,Phys.Lett.B757(2016)437–444.

[27]B.Ananthanarayan,I.Caprini,B.Kubis,Eur.Phys.J.C74(2014)3209,arXiv: 1410.6276.

[28]I.Caprini,Phys.Rev.D92(2015)014014,arXiv:1505.05282.

[29]M.Hoferichter,B.Kubis,S.Leupold,F.Niecknig,S.P.Schneider,Eur.Phys.J.C 74(2014)3180,arXiv:1410.4691.

[30]H.H.Adam,etal.,WASA-at-COSY,arXiv:nucl-ex/0411038,2004.

[31]C.Bargholtz,etal.,CELSIUS/WASA,Nucl.Instrum.Methods,Sect. A594(2008) 339–350,arXiv:0803.2657.

[32]K. Schönning, et al., CELSIUS/WASA,Phys. Rev.C79 (2009) 044002,arXiv: 0902.3905.