Dalitz plot analyses of J/ψ→π+π−π0 , J/ψ→K+K−π0 , and J/ψ→Ks0K±π∓ produced via e+e− annihilation with initial-state radiation

Dalitz plot analyses of J=ψ → π

π

π

, J=ψ → K

K

π

_{, and}

J=ψ → K

s

K

π

produced via e

e

annihilation with initial-state radiation

J. P. Lees,1V. Poireau,1V. Tisserand,1E. Grauges,2A. Palano,3,†G. Eigen,4D. N. Brown,5Yu. G. Kolomensky,5M. Fritsch,6 H. Koch,6T. Schroeder,6C. Hearty,7a,7bT. S. Mattison,7bJ. A. McKenna,7bR. Y. So,7bV. E. Blinov,8a,8b,8cA. R. Buzykaev,8a

V. P. Druzhinin,8a,8bV. B. Golubev,8a,8b E. A. Kravchenko,8a,8b A. P. Onuchin,8a,8b,8cS. I. Serednyakov,8a,8b Yu. I. Skovpen,8a,8b E. P. Solodov,8a,8b K. Yu. Todyshev,8a,8b A. J. Lankford,9 J. W. Gary,10O. Long,10A. M. Eisner,11 W. S. Lockman,11W. Panduro Vazquez,11D. S. Chao,12C. H. Cheng,12B. Echenard,12K. T. Flood,12D. G. Hitlin,12

J. Kim,12T. S. Miyashita,12P. Ongmongkolkul,12F. C. Porter,12M. Röhrken,12Z. Huard,13B. T. Meadows,13 B. G. Pushpawela,13M. D. Sokoloff,13L. Sun,13,‡J. G. Smith,14S. R. Wagner,14D. Bernard,15M. Verderi,15D. Bettoni,16a

C. Bozzi,16a R. Calabrese,16a,16bG. Cibinetto,16a,16bE. Fioravanti,16a,16b I. Garzia,16a,16b E. Luppi,16a,16b V. Santoro,16a A. Calcaterra,17R. de Sangro,17G. Finocchiaro,17S. Martellotti,17P. Patteri,17I. M. Peruzzi,17M. Piccolo,17M. Rotondo,17 A. Zallo,17S. Passaggio,18C. Patrignani,18,§H. M. Lacker,19B. Bhuyan,20A. P. Szczepaniak,21,†U. Mallik,22C. Chen,23 J. Cochran,23S. Prell,23H. Ahmed,24M. R. Pennington,25A. V. Gritsan,26N. Arnaud,27M. Davier,27F. Le Diberder,27

A. M. Lutz,27G. Wormser,27D. J. Lange,28 D. M. Wright,28J. P. Coleman,29E. Gabathuler,29,* D. E. Hutchcroft,29 D. J. Payne,29C. Touramanis,29A. J. Bevan,30F. Di Lodovico,30R. Sacco,30G. Cowan,31Sw. Banerjee,32D. N. Brown,32 C. L. Davis,32A. G. Denig,33W. Gradl,33K. Griessinger,33A. Hafner,33K. R. Schubert,33R. J. Barlow,34,∥G. D. Lafferty,34 R. Cenci,35A. Jawahery,35D. A. Roberts,35R. Cowan,36S. H. Robertson,37B. Dey,38a N. Neri,38a F. Palombo,38a,38b

R. Cheaib,39L. Cremaldi,39R. Godang,39,** D. J. Summers,39P. Taras,40G. De Nardo,41 C. Sciacca,41G. Raven,42 C. P. Jessop,43J. M. LoSecco,43K. Honscheid,44R. Kass,44A. Gaz,45a M. Margoni,45a,45bM. Posocco,45a G. Simi,45a,45b

F. Simonetto,45a,45bR. Stroili,45a,45b S. Akar,46E. Ben-Haim,46 M. Bomben,46G. R. Bonneaud,46G. Calderini,46 J. Chauveau,46G. Marchiori,46 J. Ocariz,46M. Biasini,47a,47bE. Manoni,47a A. Rossi,47a G. Batignani,48a,48b S. Bettarini,48a,48bM. Carpinelli,48a,48b,††G. Casarosa,48a,48b M. Chrzaszcz,48a F. Forti,48a,48bM. A. Giorgi,48a,48b A. Lusiani,48a,48cB. Oberhof,48a,48bE. Paoloni,48a,48bM. Rama,48a G. Rizzo,48a,48bJ. J. Walsh,48a A. J. S. Smith,49 F. Anulli,50aR. Faccini,50a,50bF. Ferrarotto,50aF. Ferroni,50a,50bA. Pilloni,50a,50bG. Piredda,50a,*C. Bünger,51S. Dittrich,51 O. Grünberg,51M. Heß,51T. Leddig,51C. Voß,51R. Waldi,51T. Adye,52F. F. Wilson,52S. Emery,53G. Vasseur,53D. Aston,54 C. Cartaro,54M. R. Convery,54J. Dorfan,54W. Dunwoodie,54M. Ebert,54R. C. Field,54B. G. Fulsom,54M. T. Graham,54

C. Hast,54W. R. Innes,54P. Kim,54D. W. G. S. Leith,54S. Luitz,54D. B. MacFarlane,54 D. R. Muller,54 H. Neal,54 B. N. Ratcliff,54 A. Roodman,54M. K. Sullivan,54J. Va’vra,54W. J. Wisniewski,54M. V. Purohit,55J. R. Wilson,55 A. Randle-Conde,56S. J. Sekula,56M. Bellis,57P. R. Burchat,57E. M. T. Puccio,57M. S. Alam,58J. A. Ernst,58 R. Gorodeisky,59N. Guttman,59D. R. Peimer,59A. Soffer,59S. M. Spanier,60J. L. Ritchie,61R. F. Schwitters,61 J. M. Izen,62X. C. Lou,62F. Bianchi,63a,63b F. De Mori,63a,63b A. Filippi,63a D. Gamba,63a,63b L. Lanceri,64L. Vitale,64

F. Martinez-Vidal,65 A. Oyanguren,65J. Albert,66bA. Beaulieu,66b F. U. Bernlochner,66b G. J. King,66b R. Kowalewski,66b T. Lueck,66bI. M. Nugent,66b J. M. Roney,66b R. J. Sobie,66a,66bN. Tasneem,66b T. J. Gershon,67

P. F. Harrison,67 T. E. Latham,67 R. Prepost,68and S. L. Wu68 (BABAR Collaboration)

1

Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie,

CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France

2

Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain

3_{INFN Sezione di Bari and Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy}

4

University of Bergen, Institute of Physics, N-5007 Bergen, Norway

5_{Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA}

6

Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany

7a_{Institute of Particle Physics, Vancouver, British Columbia, Canada V6T 1Z1}

7b

University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

8a_{Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia}

8b

Novosibirsk State University, Novosibirsk 630090, Russia

8c_{Novosibirsk State Technical University, Novosibirsk 630092, Russia}

9

University of California at Irvine, Irvine, California 92697, USA

10_{University of California at Riverside, Riverside, California 92521, USA}

11

University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA

12_{California Institute of Technology, Pasadena, California 91125, USA}

13

University of Cincinnati, Cincinnati, Ohio 45221, USA

14_{University of Colorado, Boulder, Colorado 80309, USA}

15

16a_{INFN Sezione di Ferrara, I-44122 Ferrara, Italy} 16b

Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy

17_{INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy}

18

INFN Sezione di Genova, I-16146 Genova, Italy

19_{Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany}

20

Indian Institute of Technology Guwahati, Guwahati, Assam 781 039, India

21_{Indiana University, Bloomington, Indiana 47405, USA}

22

University of Iowa, Iowa City, Iowa 52242, USA

23_{Iowa State University, Ames, Iowa 50011, USA}

24

Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia

25_{Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA}

26

Johns Hopkins University, Baltimore, Maryland 21218, USA

27_{Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,}

Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France

28_{Lawrence Livermore National Laboratory, Livermore, California 94550, USA}

29

University of Liverpool, Liverpool L69 7ZE, United Kingdom

30_{Queen Mary, University of London, London E1 4NS, United Kingdom}

31

University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom

32

University of Louisville, Louisville, Kentucky 40292, USA

33_{Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany}

34

University of Manchester, Manchester M13 9PL, United Kingdom

35_{University of Maryland, College Park, Maryland 20742, USA}

36

Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA

37

Institute of Particle Physics and McGill University, Montréal, Québec, Canada H3A 2T8

38a_{INFN Sezione di Milano, I-20133 Milano, Italy}

38b

Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy

39_{University of Mississippi, University, Mississippi 38677, USA}

40

Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7

41_{INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Università di Napoli Federico II,}

I-80126 Napoli, Italy

42_{NIKHEF, National Institute for Nuclear Physics and High Energy Physics,}

NL-1009 DB Amsterdam, The Netherlands

43_{University of Notre Dame, Notre Dame, Indiana 46556, USA}

44

Ohio State University, Columbus, Ohio 43210, USA

45a_{INFN Sezione di Padova, I-35131 Padova, Italy}

45b

Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy

46_{Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie}

Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France

47a_{INFN Sezione di Perugia, I-06123 Perugia, Italy}

47b

Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy

48a_{INFN Sezione di Pisa, I-56127 Pisa, Italy}

48b

Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy

48c_{Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy}

49

Princeton University, Princeton, New Jersey 08544, USA

50a_{INFN Sezione di Roma, I-00185 Roma, Italy}

50b

Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy

51_{Universität Rostock, D-18051 Rostock, Germany}

52

Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom

53_{CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France}

54

SLAC National Accelerator Laboratory, Stanford, California 94309 USA

55_{University of South Carolina, Columbia, South Carolina 29208, USA}

56

Southern Methodist University, Dallas, Texas 75275, USA

57_{Stanford University, Stanford, California 94305, USA}

58

State University of New York, Albany, New York 12222, USA

59_{Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel}

60

University of Tennessee, Knoxville, Tennessee 37996, USA

61_{University of Texas at Austin, Austin, Texas 78712, USA}

62

63a_{INFN Sezione di Torino, I-10125 Torino, Italy} 63b

Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy

64_{INFN Sezione di Trieste and Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy}

65

IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain

66a_{Institute of Particle Physics, Victoria, British Columbia, Canada V8W 3P6}

66b

University of Victoria, Victoria, British Columbia, Canada V8W 3P6

67_{Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom}

68

University of Wisconsin, Madison, Wisconsin 53706, USA (Received 6 February 2017; published 10 April 2017)

We study the processes eþe−→ γISRJ=ψ, where J=ψ → πþπ−π0, J=ψ → KþK−π0, and J=ψ →

K0

SK
π∓ using a data sample of519 fb−1 recorded with the BABAR detector operating at the SLAC

PEP-II asymmetric-energyeþe−collider at center-of-mass energies at and near theΥðnSÞ (n ¼ 2; 3; 4)

resonances. We measure the ratio of branching fractionsR₁¼BðJ=ψ→K_BðJ=ψ→πþþK_π−−_ππ00_ÞÞandR2¼BðJ=ψ→K

0 SK
π∓Þ

BðJ=ψ→πþ_π−_π0_Þ. We

perform Dalitz plot analyses of the three J=ψ decay modes and measure fractions for resonances

contributing to the decays. We also analyze theJ=ψ → πþπ−π0 decay using the Veneziano model. We

observe structures compatible with the presence ofρð1450Þ in all three J=ψ decay modes and measure the

relative branching fraction:Rðρð1450ÞÞ ¼Bðρð1450Þ→K_{Bðρð1450Þ→π}þþK_π−−_ÞÞ¼ 0.307
0.084ðstatÞ
0.082ðsysÞ.

DOI:10.1103/PhysRevD.95.072007

I. INTRODUCTION

Charmonium decays, in particular radiative and hadronic decays of the J=ψ meson, have been studied extensively [1,2]. One of the motivations for these studies is to search for non-q¯q mesons such as glueballs or molecular states that are predicted by QCD to populate the low mass region of the hadron mass spectrum [3].

Previous studies ofJ=ψ decays to πþπ−π0show a clear signal of ρð770Þ production [4,5]. In addition there is an indication of higher mass resonance production inψð2SÞ decays[5]. This is not necessarily the case inJ=ψ decays, but neither does the ρð770Þ contribution saturate the spectrum. Attempts have been made to describe the J=ψ decay distribution with additional partial waves[6]. It was found that interference effects are strong and even after addingππ interactions up to ≈1.6 GeV=c2the description remained quite poor. Continuing to expand the partial wave basis to cover an even higher mass region would lead to a rather unconstrained analysis. On the other hand with the amplitudes developed in the Veneziano model, all partial waves are related to the same Regge trajectory, which gives a very strong constraint on the amplitude analysis [7].

While large samples ofJ=ψ decays exist, some branch-ing fractions remain poorly measured. In particular the J=ψ → Kþ_K−_π0_{branching fraction has been measured by} Mark II[8] using only 25 events.

Only a preliminary result exists, to date, on a Dalitz plot analysis of J=ψ decays to πþπ−π0 [9]. The BESII experiment [10] has performed an angular analysis of J=ψ → Kþ_K−_π0_{. The analysis requires the presence of a} broadJPC¼ 1−−state in theKþK−threshold region, which is interpreted as a multiquark state. However Refs.[11,12] explain it by the interference between the ρð1450Þ and ρð1700Þ. On the other hand, the decay ρð1450Þ → Kþ_K− appears as“not seen” according to the PDG listing[13]. No Dalitz plot analysis has been performed to date on the J=ψ → K0

SK
π∓ decay.

We describe herein a study of the J=ψ → πþπ−π0, J=ψ → Kþ_K−_π0_{, and J=ψ → K}0

SK
π∓ decays produced ineþe− annihilation via initial-state radiation (ISR), where only resonances withJPC¼ 1−− can be produced.

This article is organized as follows. In Sec. II, a brief description of the BABAR detector is given. SectionIII is devoted to the event reconstruction and data selection. In Sec.IV, we describe the study of efficiency and resolution, while Sec. V is devoted to the measurement of the J=ψ branching fractions. Section VI describes the Dalitz plot analyses while in Sec.VII, we report the measurement of theρð1450Þ branching fraction. Finally we summarize the results in Sec.VIII.

II. THE BABAR DETECTOR AND DATA SET The results presented here are based on data collected with the BABAR detector at the PEP-II asymmetric-energy

*_Deceased.

†_{Also at Thomas Jefferson National Accelerator Facility,}

Newport News, Virginia 23606, USA.

‡_{Present address: Wuhan University, Wuhan 43072, China.}

§_{Present address: Università di Bologna and INFN Sezione di}

Bologna, I-47921 Rimini, Italy.

¶_{Present address: University of Huddersfield, Huddersfield}

HD1 3DH, United Kingdom.

**_{Present address: University of South Alabama, Mobile,}

Alabama 36688, USA.

eþ_e− _{collider located at SLAC. The data sample} corre-sponds to an integrated luminosity of519 fb−1[14]recorded at center-of-mass energies at and near the ϒðnSÞ (n ¼ 2; 3; 4) resonances. The BABAR detector is described in detail elsewhere[15]. Charged particles are detected, and their momenta are measured, by means of a five-layer, double-sided microstrip detector, and a 40-layer drift cham-ber, both operating in the 1.5 T magnetic field of a super-conducting solenoid. Photons are measured and electrons are identified in a CsI(Tl) crystal electromagnetic calorim-eter (EMC). Charged-particle identification is provided by the specific energy loss in the tracking devices, and by an internally reflecting ring-imaging Cherenkov detector. Muons are detected in the instrumented flux return of the magnet. Monte Carlo (MC) simulated events [16], with sample sizes more than 10 times larger than the correspond-ing data samples, are used to evaluate signal efficiency and to determine background features.

III. EVENT RECONSTRUCTION AND DATA SELECTION

We study the following reactions: eþ_e− _{→ γ} ISRπþπ−π0; ð1Þ eþ_e− _{→ γ} ISRKþK−π0; ð2Þ eþ_e− _{→ γ} ISRK0SK
π∓; ð3Þ

where γ_ISR indicates the ISR photon.

For reactions (1) and (2), we consider only events for which the number of well-measured charged-particle tracks with transverse momenta greater than0.1 GeV=c is exactly equal to 2. The charged-particle tracks are fitted to a common vertex with the requirements that they originate from the interaction region and that the χ2 probability of the vertex fit be greater than 0.1%. We observe prominent J=ψ signals in both reactions and optimize the signal-to-background ratio using the data by retaining only selection criteria that do not remove significantJ=ψ signal. We require the energy of the less-energetic photon fromπ0decays to be greater than 100 MeV. Each pair of photons is kinematically fitted to aπ0requiring it to emanate from the primary vertex of the event, and with the diphoton mass constrained to the nominalπ0mass[13]. Due to the soft-photon background, we do not impose a veto on the presence of additional photons in the final state but we require exactly one π0 candidate in each event. Particle identification is used in two different ways. For reaction(1), we require two oppositely charged particles to be loosely identified as pions. For reaction(2), we loosely identify one kaon and require that neither track be a well-identified pion, electron, or muon.

For reaction(3), we consider only events for which the number of well-measured charged-particle tracks with

transverse momentum greater than 0.1 GeV=c is exactly equal to 4, and for which there are no more than five photon candidates with reconstructed energy in the EMC greater than 100 MeV. We obtainK0_S→ πþπ−candidates by means of a vertex fit of pairs of oppositely charged tracks, for which we require a χ2 fit probability greater than 0.1%. EachK0_S candidate is then combined with two oppositely charged tracks, and fitted to a common vertex, with the requirements that the fitted vertex be within the eþe− interaction region and have aχ2fit probability greater than 0.1%. We select kaons and pions by applying high-efficiency particle identification criteria. We do not apply any particle identification requirements to the pions from the K0_S decay. We accept only K0_S candidates with decay lengths from theJ=ψ candidate decay vertex greater than 0.2 cm, and require cosθ_K0

S > 0.98, where θK0Sis defined as

the angle between theK0_Smomentum direction and the line joining theJ=ψ and K0_S vertices. A fit to the πþπ− mass spectrum using a linear function for the background and a Gaussian function with meanm and width σ gives m ¼ 497.24 MeV=c2 _{and σ ¼ 2.9 MeV=c}2_{. We select the K}0 S signal region to be within
2σ of m and reconstruct the K0_S four-vector by summing the three-momenta of the pions and computing the energy using the knownK0_Smass[13]. The ISR photon is preferentially emitted at small angles with respect to the beam axis (see Fig. 1), and escapes detection in the majority of ISR events. Consequently, the ISR photon is treated as a missing particle.

We define the squared mass M2rec recoiling against the πþ_π−_π0_{, K}þ_K−_π0_{, and K}0

SK
π∓ systems using the four-momenta of the beam particles (p_e
) and of the

recon-structed final state particles: M2

rec≡ ðpe−þ peþ− ph₁− ph₂− ph₃Þ2; ð4Þ where theh_i indicate the three hadrons in the final states. This quantity should peak near zero for both ISR events and

(deg) ISR θ 0 50 100 events/(2 deg) 0 2000 4000 6000 8000

FIG. 1. (a) Distribution ofθISRfor events in theJ=ψ → πþπ−π0

for exclusive production of eþe−→ h₁h₂h₃. However, in the exclusive production the h₁h₂h₃ mass distribution peaks at the kinematic limit. We select the ISR reactions (in the following also defined as ISR regions) requiring

jM2

recj < 2 GeV2=c4 ð5Þ for reaction (1) and(2)and

jM2

recj < 1.5 GeV2=c4 ð6Þ for reaction (3).

We reconstruct the three-momentum of the ISR photon from momentum conservation as

pISR¼ pe−þ peþ− ph1− ph2− ph3: ð7Þ

TableIgives the ranges used to define the ISR signal regions for the threeJ=ψ decay modes. We show in Fig.1, for events in theJ=ψ → πþπ−π0ISR signal region, the distribution of θISR, the angle of the reconstructed ISR photon with respect to the e− beam direction in the laboratory system. We observe a narrow peak close to zero with a tail extending up to1400while background events fromJ=ψ sidebands are distributed over the full angular range. Since angular cover-age of the EMC starts atθ > 230, we improve the signal to background ratio for J=ψ events where θ_ISR> 230, by removing events for which no photon shower is found in the EMC in the expected angular region. Therefore, we require the difference between the predicted polar and azimuthal angles frompISRand the closest photon shower to bejΔθj < 0.1 rad and jΔϕj < 0.05 rad. We do not use the information on the energy since some photons may not be fully contained in the EMC.

For reaction (1) we define the helicity angle θ_h as the angle in theπþπ− rest frame between the direction of the πþ_{and the boost from theπ}þ_π−_{. We observe that residual} background from eþe−→ γπþπ− is concentrated at jcos θπj ≈ 1 and therefore we remove events having jcos θπj > 0.95. A very small J=ψ signal is observed in the events removed by this selection. No evidence is found for background from the ISR reactioneþe− → γISRKþK−. Figure 2 shows the M2rec distributions for the three reactions in the J=ψ signal regions, in comparison to the correspondingM2_recdistributions obtained from simulation.

A peak at zero is observed in all distributions indicating the presence of the ISR process. We observe some discrepancy for reactions (1) and (2) due to some inaccuracy in reconstructing slow π0 in the EMC. Figure 3 shows the πþ_π−_π0_,Kþ_K−_π0_{, andK}0

SK
π∓ mass spectra in the ISR region, before applying the efficiency correction. We observe strong J=ψ signals over relatively small back-grounds and no more than one candidate per event. We perform a fit to theπþπ−π0,KþK−π0, andK0_SK
π∓ mass spectra. Backgrounds are described by first-order poly-nomials, and each resonance is represented by a simple

TABLE I. Ranges used to define theJ=ψ signal regions, event

yields, and purities for the threeJ=ψ decay modes.

J=ψ Signal region Event Purity

decay mode (GeV=c2) yields %

πþ_π−_π0 _{3.028–3.149 20417 91.3
0.2} Kþ_K−_π0 _{3.043–3.138 2102 88.8
0.7} K0 SK
π∓ 3.069–3.121 3907 93.1
0.4 ) 4 /c 2 (GeV 2 rec M −10 −5 0 5 10 )) 4 /c 2 events/(0.2 (GeV 0 200 400 600 800 1000 1200 1400 0 π -π + π → ψ (a) J/ ) 4 /c 2 (GeV 2 rec M −10 −5 0 5 10 )) 4 /c 2 events/(0.2 (GeV 0 50 100 150 200 250 0 π -K + K → ψ (b) J/ ) 4 /c 2 (GeV 2 rec M −10 −5 0 5 10 )) 4 /c 2 events/(0.2 (GeV 0 100 200 300 400 500 600 700 ± π ± K S 0 K → ψ (c) J/

FIG. 2. Distributions of M2rec foreþe−→ γISRJ=ψ, where (a)

J=ψ → πþ_π−_π0_{, (b)J=ψ → K}þ_K−_π0_{, and (c)J=ψ → K}0

SK
π∓.

In each figure the data are shown as points with error bars, and the MC simulation is shown as a histogram.

Breit-Wigner function convolved with the corresponding resolution function (see Sec. IV). Figure 3 shows the fit result, and TableIIsummarizes the mass values and yields. We observe (not taking into account systematic uncertain-ties) aJ=ψ mass shift of þ2.9, þ4.1, and −2.2 MeV=c2for the three decay modes.

IV. EFFICIENCY AND RESOLUTION To compute the efficiency,J=ψ MC signal events for the three channels are generated using a detailed detector

simulation [16] in which the J=ψ decays uniformly in phase space. These simulated events are reconstructed and analyzed in the same manner as data. The efficiency is computed as the ratio of reconstructed to generated events. We express the efficiency as a function of the m₁₂ mass (πþπ− for J=ψ → πþπ−π0, KþK− for J=ψ → KþK−π0, and K0_SK
for J=ψ → K_S0K
π∓) and cosθ_h defined in Sec.III. To smooth statistical fluctuations, this efficiency is then parametrized as follows[17].

First we fit the efficiency as a function of cosθ_h in separate intervals ofm₁₂, in terms of Legendre polynomials up toL ¼ 12:

εðcos θhÞ ¼ X12 L¼0

aLðm12ÞY0Lðcos θhÞ: ð8Þ For each value ofL, we fit the mass dependent coefficients aLðm12Þ with a seventh-order polynomial in m12. Figure4 shows the resulting fitted efficiencyεðm₁₂; cos θ_hÞ for each of the three reactions. We observe a significant decrease in efficiency at low m₁₂ for cosθ ∼
1 and 1.1 < mðKþ_K−_{Þ < 1.5 GeV=c}2 _{due to the difficulty of} recon-structing low-momentum tracks (p < 200 MeV=c in the laboratory frame), which arise because of significant energy losses in the beampipe and inner-detector material. The mass resolution,Δm, is measured as the difference between the generated and reconstructed πþπ−π0, Kþ_K−_π0_{, and K}0

SK
π∓ invariant-mass values. These dis-tributions, for theJ=ψ decays having a π0in the final state, deviate from Gaussian shapes due to a low-energy tail caused by the response of the CsI calorimeter to photons. We fit the distributions using the sum of a Crystal Ball function [18] and a Gaussian function. The root-mean-squared values are 24.4 and22.7 MeV=c2for the J=ψ → πþ_π−_π0_{andJ=ψ → K}þ_K−_π0_{final states, respectively. The} mass resolution forJ=ψ → K0_SK
π∓is well described by a single Gaussian having aσ ¼ 9.7 MeV=c2.

V. J=ψ BRANCHING RATIOS

We compute the ratio of the branching fractions for J=ψ → Kþ_K−_π0 _{andJ=ψ → π}þ_π−_π0_{according to} R1¼BðJ=ψ → K þ_K−_π0_Þ BðJ=ψ → πþ_π−_π0_Þ ¼ NKþ_K−_π0 Nπþ_π−_π0 επþ_π−_π0 εKþ_K−_π0; ð9Þ ) 2 ) (GeV/c 0 π -π + π m( 2.8 3 3.2 ) 2 events/(5 MeV/c 0 200 400 600 800 1000 1200 1400 1600 1800 ) 2 ) (GeV/c 0 π -K + m(K 2.9 3 3.1 3.2 3.3 ) 2 events/(5 MeV/c 0 50 100 150 200 250 ) 2 ) (GeV/c ± π ± K S 0 m(K 3 3.05 3.1 3.15 3.2 ) 2 events/(2.5 MeV/c 0 100 200 300 400

FIG. 3. (a) The πþπ−π0, (b) KþK−K0, and K0_SK
π∓ mass

spectra in the ISR region. In each figure, the solid curve shows the total fit function and the dashed curve shows the fitted background contribution.

TABLE II. Results from the fits to the mass spectra and

efficiency corrections. Errors are statistical only. J=ψ decay

mode χ2=NDF

J=ψ mass

(MeV=c2) Signal yield 1=ε

πþ_π−_π0 _{90=105 3099.8
0.2 19560
164 15.57
1.05}

Kþ_K−_π0 _{129=95 3101.0
0.2 2002
48 18.31
0.63}

K0

whereN_πþ_π−_π0 andN_Kþ_K−_π0 represent the fitted yields for

J=ψ in the πþ_π−_π0 _{and K}þ_K−_π0 _{mass spectra, while} επþ_π−_π0 andε_Kþ_K−_π0 are the corresponding efficiencies. We

estimateε_πþ_π−_π0andε_Kþ_K−_π0for theJ=ψ signals by making

use of the 2D efficiency distributions described in Sec.IV. To remove the dependence of the fit quality on the efficiency functions we make use of the unfitted efficiency distributions. Due to the presence of non-negligible back-grounds in theJ=ψ signals, which have different distribu-tions in the Dalitz plot, we perform a sideband subtraction

by assigning a weightw ¼ f=εðm₁₂; cos θÞ, where f ¼ 1 for events in theJ=ψ signal region and f ¼ −1 for events in the sideband regions. The size of the sum of the two sidebands is taken to be the same as that of the signal region. Therefore we obtain the weighted efficiencies as

εhþ_h−_π0 ¼

P_N i¼1fi P_N

i¼1fi=εðm12; cos θiÞ; ð10Þ where N indicates the number of events in the signal þ sidebands regions. The resulting yields and efficiencies are reported in TableII.

We note that in Eq. (9)the number of charged-particle tracks and γ’s is the same in the numerator and in the denominator of the ratio, so that several systematic uncer-tainties cancel. We estimate the systematic unceruncer-tainties as follows. We modify the signal fitting function, describing theJ=ψ signals using the sum of two Gaussian functions. The uncertainty due to efficiency weighting is evaluated by computing 1000 new weights obtained by randomly modifying the weight in each cell of the εðm₁₂; cos θÞ plane according to its statistical uncertainty. The widths of the resulting Gaussian distributions yield the estimate of the systematic uncertainty for the efficiency weighting pro-cedure. These values are reported as the uncertainties on 1=ε in TableII. We assign a 1% systematic uncertainty for the identification of each of the two kaons, from studies performed using high statistics control samples. The con-tributions to the systematic uncertainties from different sources are given in TableIIIand combined in quadrature. We obtain R1¼BðJ=ψ → K þ_K−_π0_Þ BðJ=ψ → πþ_π−_π0_Þ ¼ 0.120
0.003ðstatÞ
0.009ðsysÞ: ð11Þ The PDG reportsBðJ=ψ →πþπ−π0Þ¼ð2.11
0.07Þ×10−2, while the branching fraction BðJ=ψ → KþK−π0Þ has been measured by Mark II [8] using 25 events, to be ð2.8
0.8Þ × 10−3_{. These values give a ratio R}PDG

1 ¼

0.133
0.038, in agreement with our measurement. 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2 ) GeV/c -π + π m( 0.5 1 1.5 2 2.5 h θ cos 0 -1 -0.5 0.5 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2 ) GeV/c -K + m(K 1 1.5 2 2.5 h θ cos -1 -0.5 0 0.5 1 0.05 0.06 0.07 0.08 0.09 0.1 2 ) GeV/c ± K S 0 m(K 1 1.5 2 2.5 3 h θ cos -1 -0.5 0 0.5 1 (a) (b) (c)

FIG. 4. Fitted detection efficiency in the cosθ_hvsm₁₂plane for

(a)J=ψ →πþπ−π0, (b)J=ψ →KþK−π0, and (c)J=ψ →K0_SK
π∓.

Each bin shows the average value of the fit in that region.

TABLE III. Fractional systematic uncertainties in the

evalu-ation of the ratios of branching fractions.

Effect R₁ (%) R₂ (%) Efficiency 7.5 7.0 Background subtraction 1.3 1.0 Particle identification 2.0 1.8 K0 S reconstruction 1.1 π0 _{reconstruction 3.0} Mass fits 0.8 0.8 Total 7.9 8.0

We perform a test of the R₁ measurement using a minimum bias procedure. We remove all the selections used to separate reactions (1) and (2), except for the requirements on M2_rec and obtain the events yield for J=ψ → πþ_π−_π0_{. To obtain the J=ψ → K}þ_K−_π0 _yield, we apply very loose identifications of the two kaons to remove the large background and the strong cross feed from theJ=ψ → πþπ−π0 final state. We observe a loss of theJ=ψ signal which is estimated by MC to be 3.6%. The ratios between the two minimum bias yields, corrected for the above efficiency loss gives directly the ratio of the two branching fractions which is in good agreement with the previous estimate.

Using a similar procedure as for the measurement ofR₁, correcting for unseen K0_S decay modes, we compute the ratio of the branching fractions for J=ψ → K0_SK
π∓ and J=ψ → πþ_π−_π0 _{according to} R2¼BðJ=ψ → K 0 SK
π∓Þ BðJ=ψ → πþ_π−_π0_Þ ¼NK0SK
π∓ Nπþ_π−_π0 επþ_π−_π0 εK0 SK
π∓ ¼ 0.265
0.005ðstatÞ
0.021ðsysÞ: ð12Þ Systematic uncertainties on the evaluation ofR₂include 0.46% per track for charged tracks reconstruction, 3% and 1.1% forπ0 andK0_S reconstruction, and 0.5% and 1% for the identification of pions and kaons, respectively. The contributions to the total systematic uncertainty are sum-marized in Table III.

The branching fraction BðJ=ψ → K0_SK
π∓Þ has been measured by Mark I [19], using 126 events, to be ð26
7Þ × 10−4_{. Using the above measurements we obtain} an estimate of R₂:

RPDG

2 ¼ 0.123
0.033; ð13Þ

which deviates by3.6σ from our measurement.

As a cross-check, using the aboveR₁andR₂ measure-ments and adding in quadrature statistical and systematic uncertainties, we compute

R3¼BðJ=ψ → K 0 SK
π∓Þ

BðJ=ψ → Kþ_K−_π0_Þ¼ 2.21
0.24 ð14Þ in agreement with the expected value of 2.

VI. DALITZ PLOT ANALYSIS

We perform Dalitz plot analyses of theJ=ψ → πþπ−π0, J=ψ → Kþ_K−_π0_{, and J=ψ → K}0

SK
π∓ candidates in the J=ψ mass region using unbinned maximum likelihood fits. The likelihood function is written as

L ¼YN n¼1  fsigðmnÞ · εðx0n; y0nÞ P i;jcic_PjAiðxn; ynÞAjðxn; ynÞ i;jcicjIAiAj þ ð1 − fsigðmnÞÞ P ik_PiBiðxn; ynÞ ikiIBi  ; ð15Þ where

(i) N is the number of events in the signal region; (ii) for the nth event, m_n is the πþπ−π0, KþK−π0, or

K0

SK
π∓ invariant mass;

(iii) for the nth event, x_n¼ m2ðπþπ0Þ, y_n¼ m2ðπ−π0Þ for πþπ−π0; x_n¼ m2ðKþπ0Þ, y_n¼ m2ðK−π0Þ for Kþ_K−_π0_{; x}

n¼ m2ðK
π∓Þ, yn¼ m2ðK0Sπ∓Þ for K0

SK
π∓;

(iv) fsig is the mass-dependent fraction of signal ob-tained from the fits to the πþπ−π0, KþK−π0, and K0

SK
π∓ mass spectra;

(v) for the nth event, εðx0_n; y0_nÞ is the efficiency para-metrized as a function x0_n¼ m₁₂ and y0_n ¼ cos θ_h (see Sec.IV);

(vi) for thenth event, the A_iðx_n; y_nÞ represent the complex signal-amplitude contributions described below; (vii) c_i is the complex amplitude of the ith signal

component; the c_i parameters are allowed to vary during the fit process;

(viii) for thenth event, the B_iðx_n; y_nÞ describe the back-ground probability-density functions assuming that interference between signal and background ampli-tudes can be ignored;

(ix) k_i is the magnitude of theith background compo-nent; the k_i parameters are obtained by fitting the sideband regions;

(x) I_AiA j ¼

R

Aiðx; yÞAjðx; yÞεðm12; cos θÞdxdy and IBi¼

R

Biðx; yÞdxdy are normalization integrals; numerical integration is performed on phase-space-generated events with J=ψ signal and background generated according to the experimental distributions. Parity conservation inJ=ψ → πþπ−π0restricts the pos-sible spin-parity of any intermediate two-body resonance to be JPC¼ 1−−; 3−−; …. Amplitudes are parametrized using Zemach’s tensors[20,21]. Except as noted, all fixed resonance parameters are taken from the Particle Data Group averages [13].

For reaction (1), we label the decay particles as J=ψ → πþ

1π−2π03: ð16Þ

Indicating withp_ithe momenta of the particles in theJ=ψ center-of-mass rest frame, for a resonanceR_jkdecaying as Rjk→ j þ k we also define the three-vectors tias the vector part of

tμ_i ¼ pμ_j− pμ_k− ðp_jμþ pμ_kÞm2j− m2k m2

withi; j; k cyclic. We make use of the p_ivectors to describe the angular momentumL between R_jk and particle i, and the t_i vectors to describe the spin of the R_jk resonance. Since the J=ψ resonance has spin-1 and needs to be described by a vector, the only way to obtain this result is to perform a cross-product between thep_iandt_i three-vectors. Indicating with ρ a generic spin-1 resonance, Table IV reports the list of amplitudes used to describe theJ=ψ decays. Due to Bose symmetry, the amplitudes are symmetrized with respect to the ρ charge. The Table also reports the expression for the nonresonant contribution (NR) which should also have the J=ψ quantum numbers.

For reaction(2), we label the decay particles as J=ψ → Kþ

1K−2π03: ð18Þ

In this case two separate contributions are listed in TableIV, one in which the intermediate resonance is aK
→ K
π0 and the other where the intermediate resonance is a ρ0_{→ K}þ_K−_{. The Table also lists the amplitude for the} K

2ð1430Þ
K∓ contribution. This decay mode can only occur in D-wave. To obtain this amplitude, we construct rank-2 tensorsT_i¼ tj_itk_i − jt_ij2δjk=3 to describe the spin-2 of theK₂ð1430Þ
resonance andP_i¼ p_ijpk_i− jp_ij2δjk=3 to describe the angular momentum between the K₂ð1430Þ
and theK∓. The two rank-2 tensors are then contracted into vectors k_i to obtain the spin of the J=ψ resonance. We obtain the components of k_i as kl_i¼Pλ¼3_λ¼1Tm;λ_i Pλ;n_i − Tn;λ

i Pλ;mi withl; m; n cyclic[22].

The amplitudes for reaction(3)are similar to those from reaction (2). In this case we label the decay particles as

J=ψ → K

1K0S2π∓3: ð19Þ

The efficiency-corrected fractional contribution f_i due to resonant or nonresonant contribution i is defined as follows: fi¼jcij 2R_jA iðxn; ynÞj2dxdy R jPjcjAjðx; yÞj2dxdy : ð20Þ The f_i do not necessarily sum to 100% because of interference effects. The uncertainty for eachf_iis evaluated

by propagating the full covariance matrix obtained from the fit.

Similarly, the efficiency-corrected interference fractional contributionf_ij, for i < j are defined as

fij ¼ R

2Re½c_RicjAiðxn; ynÞAjðxn; ynÞdxdy jPjcjAjðx; yÞj2dxdy : ð21Þ In all the Dalitz analyses described below we validate the fitting algorithms using MC simulations with known input amplitudes and phases. We also start the fitting procedure both on MC and data from random values. In all cases the fits converge towards one single solution.

A. Dalitz plot analysis of J=ψ → π+_π−_π0 1. Isobar model

We perform a Dalitz plot analysis of J=ψ → πþπ−π0 in the J=ψ signal region given in Table I. This region contains 20417 events withð91.3
0.2Þ% purity, defined as S=ðS þ BÞ, where S and B indicate the number of signal and background events, respectively, as determined from the fit to the πþπ−π0 mass spectrum shown in Fig. 3(a). Sideband regions are defined as the ranges 2.919–2.980GeV=c2_{and3.198–3.258GeV=c}2_{, respectively.}

TABLE IV. Amplitudes considered inJ=ψ → πþπ−π0,J=ψ → KþK−π0, andJ=ψ → K0_SK
π∓Dalitz plot analysis. BW indicates the

Breit-Wigner function.

J=ψ decay mode Decay Amplitude

πþ_π−_π0 _{ρπ BWρðm13Þðt2×p2Þ þ BWρðm23Þðt1×p1Þ þ BWρðm12Þðt3×p3Þ} NR ðt₁×p₁Þ þ ðt₂×p₂Þ þ ðt₃×p₃Þ K ¯Kπ K_{¯K BWK}ðm₁₃Þðt₂×p₂Þ þ BW_Kðm₂₃Þðt₁×p₁Þ K 2ð1430Þ ¯K BWK 2ðm13Þðk2Þ þ BWK2ðm23Þðk1Þ ρπ BW_ρðm₁₂Þðt₃×p₃Þ ) 4 /c 2 ) (GeV 0 π + π ( 2 m 0 2 4 6 8 ) 4 /c 2 ) (GeV 0 π -π( 2 m 0 2 4 6 8

FIG. 5. Dalitz plot for theJ=ψ → πþπ−π0events in the signal

Figure 5shows the Dalitz plot for the J=ψ signal region and Fig.6 shows the Dalitz plot projections. We observe that the decay is dominated by ρð770Þπ amplitudes which appear as nonuniform bands along the Dalitz plot boundaries.

We first perform separate fits to theJ=ψ sidebands with an incoherent sum of amplitudes using the method of the channel likelihood [23]. We find significant contributions from ρð770Þ resonances with uniform distributions of events along their bands, as well as from an incoherent uniform background. The resulting amplitude fractions are interpolated into theJ=ψ signal region and normalized to

the fitted purity. Figure 6 shows the projections of the estimated background contributions as shaded.

For the description of theJ=ψ Dalitz plot, amplitudes are added one at a time to ascertain the associated increase of the likelihood value and decrease of the 2Dχ2computed on theðmðπþπ−Þ; cos θ_hÞ plane. We test the quality of the fit by examining a large sample of MC events at the generator level weighted by the likelihood fitting function and by the efficiency. These events are used to compare the fit result to the Dalitz plot and its projections with proper normalization. The latter comparison is shown in Fig.6, and good agreement is obtained for all projections. We make use of these weighted events to compute a 2D χ2 over the Dalitz plot. For this purpose, we divide the Dalitz plot into a number of cells such that the expected population in each cell is at least five events. We compute χ2_¼PNcells

i¼1 ðNiobs− NiexpÞ2=Niexp, whereNiobs andNiexpare event yields from data and simulation, respectively.

We leave free in the fit theρð770Þ parameters and obtain results which are consistent with PDG averages[13]. We also leave free theρð1450Þ and ρð1700Þ parameters in the fit and obtain a significant improvement of the likelihood with the following resonances parameters:

mðρð1450ÞÞ ¼ 1429
41 MeV=c2_; Γðρð1450ÞÞ ¼ 576
29 MeV; mðρð1700ÞÞ ¼ 1644
36 MeV=c2_;

Γðρð1700ÞÞ ¼ 109
19 MeV: ð22Þ

We also test the presence of the isospin violating decay ω → πþ_π−_{. We notice that theωð782Þπ}0_{contribution has a} rather small fraction (0.08
0.03) but its fitted amplitude is (0.013
0.002). To obtain the statistical significance for this contribution, we remove theωð782Þπ0 amplitude. We obtainΔð−2 log LÞ ¼ 27.7 and Δχ2¼ 17 for the difference of two parameters which corresponds to a significance of 4.9σ. We also include the spin-3 ρ3ð1690Þπ contribution but it is found consistent with zero.

Table V summarizes the fit results for the amplitude fractions and phases. We note that theρð770Þπ amplitude ) 4 /c 2 ) (GeV -π + π ( 2 m 0 2 4 6 8 ) 4 /c 2 events/(0.05 GeV 0 200 400 600 800 (a) ) 4 /c 2 ) (GeV 0 π ± π ( 2 m 0 2 4 6 8 ) 4 /c 2 events/(0.05 GeV 0 500 1000 1500 (b)

FIG. 6. TheJ=ψ → πþπ−π0Dalitz plot projections. The

super-imposed curves result from the Dalitz plot analysis described in the text. The shaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.

TABLE V. Results from the Dalitz plot analysis of theJ=ψ → πþπ−π0channel. When two uncertainties are given, the first is statistical

and the second systematic. The error on the amplitude is only statistical.

Final state Amplitude Isobar fraction (%) Phase (radians) Veneziano fraction (%)

ρð770Þπ 1.0 114.2
1.1
2.6 0.0 133.1
3.3 ρð1450Þπ 0.513
0.039 10.9
1.7
2.7 −2.63
0.04
0.06 0.80
0.27 ρð1700Þπ 0.067
0.007 0.8
0.2
0.5 −0.46
0.17
0.21 2.20
0.60 ρð2150Þπ 0.042
0.008 0.04
0.01
0.20 1.70
0.21
0.12 6.00
2.50 ωð783Þπ0 _{0.013
0.002 0.08
0.03
0.02 2.78
0.20
0.31} ρ3ð1690Þπ 0.40
0.08 Sum 127.8
2.0
4.3 142.5
2.8 χ2_{=ν 687=519 ¼ 1.32 596=508 ¼ 1.17}

provides the largest contribution. We also observe an important contribution from theρð1450Þπ amplitude, while the contributions from higherρ0resonances are small. We also notice that theρð1700Þπ amplitude is significant even if the resulting fraction is very small, which can be attributed to the presence of important interference effects. To illustrate the contributions from higherρ states, we plot in Fig.7(a), a binned scatter diagram of the helicity

angle θ_π3 vs π₁π₂ mass for the three possible combina-tions. The curved bands on the top and bottom are reflections from the other combinations. Selecting events j cos θπj < 0.2 almost completely removes these reflec-tions and gives a more clear representation of the ππ mass spectrum, shown in Fig. 7(b) with a logarithmic scale for the sum of the three ππ mass combinations. We also compare the fit projections with the results from a fit where only the ρð770Þπ contribution is included. The distribution shows clearly the presence of higher excited ρ resonances contributing to the J=ψ → πþ_π−_π0 _decay.

The NR contribution has been included but does not improve the fit quality. The sum of the fractions is significantly different from 100%. Denoting by nð¼ 8Þ the number of free parameters in the fit, we obtainχ2=ν ¼ 687=519 (ν ¼ Ncells− n).

We compute the uncorrected Legendre polynomial moments hY0_Li in each πþπ− and π
π0 mass interval by weighting each event by the relevantY0_Lðcos θ_hÞ function. These distributions are shown in Figs. 8 and 9. We also compute the expected Legendre polynomial moments from the weighted MC events and compare with the experimen-tal distributions. We observe a reasonable agreement for all the distributions, which indicates that the fit is able to reproduce most of the local structures apparent in the Dalitz plot. We also notice a few discrepancies in the highππ mass region indicating the possible presence of additional unknown excited ρπ contributions not included in the present analysis.

Systematic uncertainty estimates for the fractions and relative phases are computed in different ways.

(i) The purity function is scaled up and down by its statistical uncertainty.

(ii) The parameters of each resonance contributing to the decay are modified within one standard deviation of their uncertainties in the PDG averages.

(iii) The Blatt-Weisskopf [24] factors entering in the relativistic Breit-Wigner function have been fixed to 1.5 ðGeV=cÞ−1 and varied between 1 and 4 ðGeV=cÞ−1_.

(iv) We make use of the efficiency distribution without the smoothing described in Sec.IV.

(v) To estimate possible bias, we generate and fit MC simulated events according to the Dalitz plot fitted results.

The different contributions are added in quadrature in TableV.

2. Veneziano model

The particular approach used in this analysis follows recent work described in Ref.[7]. The dynamical assump-tions behind the Veneziano model are the resonance dominance of the low-energy spectrum and resonance-Regge duality. The latter means that all resonances are ) 2 ) (GeV/c π π m( 1 2 3 π θ cos -1 -0.5 0 0.5 1 (a) ) 2 ) (GeV/c π π m( 1 2 3 ) 2 events/(0.03 GeV/c 1 10 2 10 3 10 _(b) ) 2 ) (GeV/c π π m( 1 2 3 ) 2 events/(0.03 GeV/c 1 10 2 10 3 10 _(c)

FIG. 7. (a) Binned scatter diagram of cosθ_π3 vsmðπ₁π₂Þ. (b)

and (c)ππ mass projection in the j cos θ_πj < 0.2 region for all the

threeππ charge combinations. The horizontal lines in (a) indicate

the cosθ_πselection. The dashed line in (b) is the result from the fit

with only theρð770Þπ amplitude. The fit in (b) uses the isobar

model and the shaded histogram shows the background

distri-bution estimated from theJ=ψ sidebands. The fit in (c) uses the

located on Regge trajectories and that Regge poles are the only singularities of partial waves in the complex angular momentum plane. Therefore, there are no “unaccounted for” backgrounds and the Veneziano amplitude is used to fully describe the given reaction. A single Veneziano amplitude of the type

An;m ¼_{Γðn þ m − αðsÞ − αðtÞÞ}Γðn − αðsÞÞΓðn − αðtÞÞ ð23Þ has “predetermined” resonance strengths. Here α is the Regge trajectory,s and t are the Mandelstam variables and n; m are integers. The position of resonances is determined ) 2 ) (GeV/c -π + π m( ) 2

weight sum/(0.075 GeV/c 0

100 200 300 400 500 600 0 0 Y ) 2 ) (GeV/c -π + π m( −40 −20 0 20 40 0 1 Y ) 2 ) (GeV/c -π + π m( −200 0 200 0 2 Y ) 2 ) (GeV/c -π + π m( ) 2

weight sum/(0.075 GeV/c

−40 −20 0 20 40 0 3 Y ) 2 ) (GeV/c -π + π m( −200 −100 0 0 4 Y ) 2 ) (GeV/c -π + π m( 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 −40 −20 0 20 40 0 5 Y

FIG. 8. Legendre polynomial moments forJ=ψ → πþπ−π0as a function ofπþπ−mass. The superimposed curves result from the

Dalitz plot analysis described in the text.

) 2 ) (GeV/c 0 π ± π m( ) 2

weight sum/(75 MeV/c 0 200 400 600 800 1000 1200 0 0 Y ) 2 ) (GeV/c 0 π ± π m( −150 −100 −50 0 50 0 1 Y ) 2 ) (GeV/c 0 π ± π m( −500 0 0 2 Y ) 2 ) (GeV/c 0 π ± π m( ) 2

weight sum/(75 MeV/c

0 50 100 0 3 Y ) 2 ) (GeV/c 0 π ± π m( −400 −200 0 200 0 4 Y ) 2 ) (GeV/c 0 π ± π m( 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 −60 1 2 3 −40 −20 0 20 0 5 Y

FIG. 9. Legendre polynomial moments forJ=ψ → πþπ−π0 as a function ofπ
π0 mass. The superimposed curves result from the

by poles of the amplitude, i.e. resonances in thesðtÞ-channel are determined by poles of the first (second)Γ function in the numerator, respectively. Resonance couplings are deter-mined by residues of the amplitude at the poles. In the model these are therefore determined by the properties of the Γ function and the form of the Regge trajectory. Which resonances are excited depends, however, on the quantum numbers of external particles. Thus the amplitude in Eq.(23) should be considered as a building block rather than a physical amplitude. The latter is obtained by forming a linear combination of theA_n;m’s with parameters that are reaction dependent, i.e. fitted to the data. e.g.

AX→abc ¼ X

n;m

cX→abcðn; mÞAn;m: ð24Þ In this analysis a modified set of amplitudes A_n;m, which incorporate complex trajectories were used. Unlike the isobar model, the Veneziano model describes an infinite number of resonances. The resonances are not independent, the correlation between resonance masses, m_R and spins JR is described by the Regge trajectory functionαðsÞ such that αðm2_RÞ ¼ J_R. Once the parameters c in Eq. (24) are determined by fitting data, it is possible to compute the coupling constants of resonances to the external particles. Weak resonances may not be apparent in the data. They however are analytically connected to other, stronger res-onances and determining the latter helps to constrain the couplings to the former. For example, the ρ₃ meson is expected to lie on the same Regge trajectory as theρ. Thus coupling of theρ in J=ψ → ρπ → 3π determines coupling of theJ=ψ to the ρ₃.

In the Veneziano model the complexity of the model is related to n which is related to the number of Regge trajectories included in the fit. The number of free param-eters also increases with n. The integer m in Eq. (23) is related to the number of daughter trajectories and it is restricted by 1 ≤ m ≤ n. The lower limit on m guarantees that J=ψ decay amplitude has the expected high-energy behavior and the upper limit eliminates double poles in overlapping channels. We fit the data varyingn from 1 to 8 and test the improvement in the likelihood function and the 2Dχ2. We find that no improvement is obtained withn > 7. Takingn ¼ 7 the model requires 19 free parameters. Using a modified expression of Eq. (20) we obtain the fractions given in TableV. We observe a reduction of theρð1450Þπ contribution by more than a factor of 10 compared to the results from the isobar model. However the ρð2150Þπ amplitude has a much larger contribution. We also observe a better fit quality as compared with the isobar model. The projection of the fit on the ππ mass in the j cos θ_πj < 0.2 region is shown in Fig.7(c).

We note that the isobar model gives a better description of theρð1450Þ region, while the Veneziano model describes better the high mass region. This may indicate that other resonances, apart from the low mass ρ resonances, are contributing to the J=ψ decay.

B. Dalitz plot analysis of J=ψ → K+_K−_π0 We perform a Dalitz plot analysis ofJ=ψ → KþK−π0in theJ=ψ signal region, defined in TableI, which contains 2102 events withð88.8
0.7Þ% purity, as determined from the fit shown in Fig.3(b). Figure10shows the Dalitz plot for theJ=ψ signal region and Fig.11shows the Dalitz plot

) 4 /c 2 ) (GeV 0 π + (K 2 m 0 2 4 6 ) 4 /c 2 ) (GeV 0 π -(K 2 m 0 2 4 6

FIG. 10. Dalitz plot for the J=ψ → KþK−π0 events in the

signal region. ) 4 /c 2 ) (GeV -K + (K 2 m ) 4 /c 2 events/(0.1 GeV 0 20 40 60 (a) ) 4 /c 2 ) (GeV 0 π ± (K 2 m 0 2 4 6 8 10 0 2 4 6 ) 4 /c 2 events/(0.07 GeV 0 200 400 600 (b)

FIG. 11. The J=ψ → KþK−π0 Dalitz plot projections. The

superimposed curves result from the Dalitz plot analysis de-scribed in the text. The shaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.

projections. We observe that the decay is dominated by the K_ð892Þ
K∓_{amplitude. We also observe a diagonal band} which we tentatively attribute to theρð1450Þ0π0amplitude. As in the previous section, we fit the J=ψ sideband regions to determine the background distribution. Due to the limited statistics and the low background, we take enlarged sidebands, defined as the ranges2.910–3.005 GeV=c2and 3.176–3.271 GeV=c2_{, respectively. Also in this case we fit} these sidebands using noninterfering amplitudes described by relativistic Breit-Wigner functions using the method of the channel likelihood [23]. The K¯K contributions are symmetrized with respect to the kaon charge. Sideband regions are dominated by the presence of Kð892Þ ¯K and K

2ð1430Þ ¯K amplitudes.

We fit theJ=ψ → KþK−π0Dalitz plot using the isobar model. Also in this case amplitudes are added one at a time to ascertain the associated increase of the likelihood value and decrease of the 2D χ2 computed on the

ðmðKþ_K−_{Þ; cos θ}

hÞ plane. The results from the best fit are summarized in Table VI. We observe the following features:

(i) The decay is dominated by the Kð892Þ
K∓ and ρð1450Þ0π0 amplitudes with smaller contribu-tions from the K₂ð1430Þ
K∓ and K₁ð1410Þ
K∓ amplitudes.

(ii) We fix the ρð1450Þ and ρð1700Þ mass and width parameters to the values obtained from the J=ψ → πþ_π−_π0 _{Dalitz plot analysis. This improves the} description of the data, in comparison with a fit where the masses and widths are fixed to the PDG values[13].

(iii) Kð1680ÞK, ρð1700Þ, ρð2100Þ, and NR have been tried but do not give significant contributions. We therefore assign the broad enhancement in theKþK− mass spectrum to the presence of theρð1450Þ resonance: the present data do not require the presence of an exotic contribution. In evaluating the fractions we compute systematic uncertainties in a similar way as for the analysis of the J=ψ → πþπ−π0 final state.

We compute the uncorrected Legendre polynomial moments hY0_Li in each KþK− and K
π0 mass interval by weighting each event by the relevantY0_Lðcos θÞ function. These distributions are shown in Figs. 12 and 13. We also compute the expected Legendre polynomial moments from the weighted MC events and compare these with the experimental distributions. We observe good agreement for all the distributions, which indicates that the fit is able to reproduce the local structures apparent in the Dalitz plot.

TABLE VI. Results from the Dalitz plot analysis of theJ=ψ →

Kþ_K−_π0 _{signal region. When two uncertainties are given, the}

first is statistical and the second systematic.

Final state Fraction (%) Phase (radians)

K_ð892Þ
K∓ _{92.4
1.5
3.4 0.0} ρð1450Þ0_π0 _{9.3
2.0
0.6 3.78
0.28
0.08} K_ð1410Þ
K∓ _{2.3
1.1
0.7 3.29
0.26
0.39} K 2ð1430Þ
K∓ 3.5
1.3
0.9 −2.32
0.22
0.05 Total 107.4
2.8 χ2_{=ν 132=137 ¼ 0.96} ) 2 ) (GeV/c -K + m(K ) 2

weight sum/(60 MeV/c 0 10 20 30 40 50 0 0 Y ) 2 ) (GeV/c -K + m(K 20 − 10 − 0 10 20 0 1 Y ) 2 ) (GeV/c -K + m(K 0 20 40 0 2 Y ) 2 ) (GeV/c -K + m(K ) 2

weight sum/(60 MeV/c ₋₂₀

10 − 0 10 20 0 3 Y ) 2 ) (GeV/c -K + m(K 40 − 20 − 0 20 0 4 Y ) 2 ) (GeV/c -K + m(K 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 −20 1 1.5 2 2.5 3 10 − 0 10 20 0 5 Y

FIG. 12. Legendre polynomial moments forJ=ψ → KþK−π0as a function ofKþK−mass. The superimposed curves result from the

C. Dalitz plot analysis of J=ψ → K0_SK
π∓ We perform a Dalitz plot analysis ofJ=ψ → K0_SK
π∓in the J=ψ signal region defined in Table I. This region contains 3907 events with ð93.1
0.4Þ% purity, as deter-mined from the fit shown in Fig.3(c). Figure14shows the Dalitz plot for theJ=ψ signal region and Fig.15shows the Dalitz plot projections.

As in the previous sections, we fit the J=ψ sideband regions to determine the background distribution using the channel likelihood [23]method.

We fit theJ=ψ → K0_SK
π∓ Dalitz plot using the isobar model. Amplitudes have been included one by one testing the likelihood values and the 2D χ2 computed on the ðmðK0

SK
Þ; cos θhÞ plane. The results from the best fit are summarized in Table VII. We observe the following features:

(i) The decay is dominated by the Kð892Þ ¯K, K

2ð1430Þ ¯K, and ρð1450Þ
π∓ amplitudes with a smaller contribution from the K₁ð1410Þ ¯K amplitude.

(ii) We obtain a significant improvement of the descrip-tion of the data by leaving free theKð892Þ mass and width parameters and obtain

mðK_ð892Þþ_{Þ ¼ 895.6
0.8 MeV=c}2_; ΓðK_ð892Þþ_{Þ ¼ 43.6
1.3 MeV;} mðK_ð892Þ0_{Þ ¼ 898.1
1.0 MeV=c}2_;

ΓðK_ð892Þ0_{Þ ¼ 52.6
1.7 MeV: ð25Þ}

The measured parameters for the chargedKð892Þþ are in good agreement with those measured in τ lepton decays[13].

(iii) We fix theρð1450Þ and ρð1700Þ parameters to the values obtained from theJ=ψ → πþπ−π0Dalitz plot analysis. This improves the description of the data in comparison with a fit where the masses and widths are fixed to the PDG values[13].

(iv) Kð1680Þ ¯K, ρð1700Þπ, ρð2100Þπ, and NR ampli-tudes have been tried but do not give significant contributions. ) 2 ) (GeV/c 0 π ± m(K ) 2

weight sum/(60 MeV/c 0 50 100 150 200 250 0 0 Y ) 2 ) (GeV/c 0 π ± m(K 50 − 0 50 0 1 Y ) 2 ) (GeV/c 0 π ± m(K 100 − 50 − 0 50 0 2 Y ) 2 ) (GeV/c 0 π ± m(K ) 2

weight sum/(60 MeV/c

0 20 40 0 3 Y ) 2 ) (GeV/c 0 π ± m(K 20 − 10 − 0 10 20 30 0 4 Y ) 2 ) (GeV/c 0 π ± m(K 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 30 − 20 − 10 − 0 10 0 5 Y

FIG. 13. Legendre polynomial moments forJ=ψ → KþK−π0as a function ofK
π0mass. The superimposed curves result from the

Dalitz plot analysis described in the text. The correspondingKþπ0and K−π0distributions are combined.

) 4 /c 2 ) (GeV ± π ± (K 2 m 0 2 4 6 ) 4 /c 2 ) (GeV ± π _S 0 (K 2 m 0 2 4 6

FIG. 14. Dalitz plot for the J=ψ → K0_SK
π∓ events in the

We therefore assign the broad enhancement in theK0_SK
mass spectrum to the presence of theρð1450Þ
resonance. In evaluating the fractions we compute systematic uncertainties in a similar way as for the analysis of the

J=ψ → πþ_π−_π0 _{and J=ψ → K}þ_K−_π0 _{final states. We} compute the uncorrected Legendre polynomial moments hY0

Li in each K0SK
, K
π∓, and K0Sπ∓ mass interval by weighting each event by the relevant Y0_Lðcos θÞ function. These distributions are shown in Fig. 16 as functions of the K0_SK
mass and in Fig.17 as functions of theKπ mass, combining the K0_Sπ∓ and K
π∓ distributions. We also compute the expected Legendre polynomial moments from the weighted MC events and compare them with the experimental distributions. We observe good agree-ment for all the distributions, which indicates that the fit is able to reproduce the local structures apparent in the Dalitz plot.

VII. MEASUREMENT OF THE ρð1450Þ0 RELATIVE BRANCHING FRACTION In the Dalitz plot analysis of J=ψ → KþK−π0, the data are consistent with the observation of the decay ρð1450Þ0_{→ K}þ_K−_{. This allows a measurement of its} relative branching fraction toρð1450Þ0→ πþπ−.

We notice that the Veneziano model gives a ρð1450Þ contribution which is 10 times smaller than the isobar model. No equivalent Veneziano analysis of the J=ψ → Kþ_K−_π0 _{decay has been performed, therefore we} perform a measurement of the ρð1450Þ relative branch-ing fraction usbranch-ing the isobar model only.

We have measured in Sec. V [Eq. (11)] the ratio R ¼ BðJ=ψ → Kþ_K−_π0_{Þ=BðJ=ψ → π}þ_π−_π0_{Þ and obtain} R ¼ 0.120
0.003
0.009. From the Dalitz plot analysis of J=ψ → πþπ−π0 and J=ψ → KþK−π0 we obtain the ρð1450Þ0 _{fractions whose systematic uncertainties are} found to be independent. From the Dalitz plot analysis ofJ=ψ → πþπ−π0 we obtain B1¼BðJ=ψ → ρð1450Þ 0_π0_{ÞBðρð1450Þ}0_{→ π}þ_π−_Þ BðJ=ψ → πþ_π−_π0_Þ ¼ ½ð10.9
1.7ðstatÞ
2.7ðsysÞÞ=3.% ¼ ð3.6
0.6ðstatÞ
0.9ðsysÞÞ%: ð26Þ

From the Dalitz plot analysis of J=ψ → KþK−π0 we obtain B2¼BðJ=ψ → ρð1450Þ 0_π0_{ÞBðρð1450Þ}0_{→ K}þ_K−_Þ BðJ=ψ → Kþ_K−_π0_Þ ¼ ð9.3
2.0ðstatÞ
0.6ðsysÞÞ%: ð27Þ We therefore obtain ) 4 /c 2 ) (GeV ± K S 0 (K 2 m 0 2 4 6 8 10 ) 4 /c 2 events/(0.1 GeV 0 50 100 (a) ) 4 /c 2 ) (GeV ± π S 0 (K 2 m 0 2 4 6 ) 4 /c 2 events/(0.07 GeV ₀ 200 400 600 800 (b) ) 4 /c 2 ) (GeV ± π ± (K 2 m 0 2 4 6 ) 4 /c 2 events/(0.07 GeV 0 200 400 600 (c)

FIG. 15. The J=ψ → K0_SK
π∓ Dalitz plot projections. The

superimposed curves result from the Dalitz plot analysis de-scribed in the text. The shaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.

TABLE VII. Results from the Dalitz plot analysis of theJ=ψ →

K0

SK
π∓ signal region. When two uncertainties are given, the

first is statistical and the second systematic.

Final state Fraction (%) Phase (radians)

K_{ð892Þ ¯K 90.5
0.9
3.8 0.0} ρð1450Þ
_π∓ _{6.3
0.8
0.6 −3.25
0.13
0.21} K 1ð1410Þ ¯K 1.5
0.5
0.9 1.42
0.31
0.35 K 2ð1430Þ ¯K 7.1
1.3
1.2 −2.54
0.12
0.12 Total 105.3
3.1 χ2_{=ν 274=217 ¼ 1.26}

Bðρð1450Þ0_{→ K}þ_K−_Þ Bðρð1450Þ0_{→ π}þ_π−_Þ ¼B_B2 1 ·R ¼ 0.307
0.084ðstatÞ
0.082ðsysÞ: ð28Þ VIII. SUMMARY

We study the processes eþe−→γ_ISRJ=ψ where J=ψ →πþ_π−_π0_{,J=ψ →K}þ_K−_π0_{, andJ=ψ →K}0

SK
π∓using a data sample of519 fb−1recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy eþe− ) 2 ) (GeV/c ± K S 0 m(K ) 2

weight sum/(24 MeV/c ₀

5 10 15 20 25 30 35 ₀ 0 Y ) 2 ) (GeV/c ± K S 0 m(K 20 − 10 − 0 10 20 0 1 Y ) 2 ) (GeV/c ± K S 0 m(K 0 20 0 2 Y ) 2 ) (GeV/c ± K S 0 m(K ) 2

weight sum/(24 MeV/c −20 10 − 0 10 20 0 3 Y ) 2 ) (GeV/c ± K S 0 m(K 20 − 0 0 4 Y ) 2 ) (GeV/c ± K S 0 m(K 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 1 1.5 2 2.5 3 −20 1 1.5 2 2.5 3 10 − 0 10 20 0 5 Y

FIG. 16. Legendre polynomial moments forJ=ψ → K0_SK
π∓as a function ofK0_SK
mass. The superimposed curves result from the

Dalitz plot analysis described in the text.

) 2 ) (GeV/c π m(K ) 2

weight sum/(25 MeV/c 0 50 100 150 200 250 0 0 Y ) 2 ) (GeV/c π m(K 0 20 40 ₀ 1 Y ) 2 ) (GeV/c π m(K 100 − 50 − 0 0 2 Y ) 2 ) (GeV/c π m(K ) 2

weight sum/(25 MeV/c −10 0 10 20 0 3 Y ) 2 ) (GeV/c π m(K 20 − 10 − 0 10 20 30 0 4 Y ) 2 ) (GeV/c π m(K 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 20 − 10 − 0 10 20 0 5 Y

FIG. 17. Legendre polynomial moments forJ=ψ → K0_SK
π∓as a function ofKπ mass. The superimposed curves result from the

collider at center-of-mass energies at and near theϒðnSÞ (n¼2;3;4) resonances. We measure the branching fractions: R1¼BðJ=ψ→K þ_K−_π0_Þ BðJ=ψ→πþ_π−_π0_Þ¼0.120
0.003ðstatÞ
0.009ðsysÞ, and R2¼BðJ=ψ→K 0 SK
π∓Þ BðJ=ψ→πþ_π−_π0_Þ¼0.265
0.005ðstatÞ
0.021ðsysÞ. We

perform Dalitz plot analyses of the threeJ=ψ decay modes and measure fractions for resonances contributing to the decays. We also perform a Dalitz plot analysis ofJ=ψ → πþ_π−_π0_{using the Veneziano model. We observe structures} compatible with the presence ofρð1450Þ0 in bothJ=ψ → πþ_π−_π0 _{and J=ψ → K}þ_K−_π0 _{and measure the ratio} of branching fractions: Rðρð1450Þ0Þ ¼Bðρð1450Þ_{Bðρð1450Þ}00→K_→πþþ_πK−−_ÞÞ¼

0.307
0.084ðstatÞ
0.082ðsysÞ.

ACKNOWLEDGMENTS

We are grateful for the extraordinary contributions of our PEP-II2 colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the U.S. Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Economia y Competitividad (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union), the A. P. Sloan Foundation (U.S.A.) and the Binational Science Foundation (U.S.A.–Israel). The work of A. Palano, M. R. Pennington and A. P. Szczepaniak were supported (in part) by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Contract No. DE-AC05-06OR23177. We acknowledge P. Colangelo for useful suggestions.

APPENDIX: INTERFERENCE FIT FRACTIONS AND VENEZIANO MODEL COEFFICIENTS The central values and statistical errors for the interference fit fractions are shown in Tables VIII, IX, and X, for the J=ψ → πþπ−π0, J=ψ → KþK−π0, and J=ψ → K0_SK
π∓, respectively. Table XI reports the fitted c_X→abcðn; mÞ coefficients with statistical uncertainties from the Veneziano model description of J=ψ → πþπ−π0.

TABLE VIII. Interference fit fractions (%) and statistical

uncertainties from the Dalitz plot analysis of J=ψ →πþπ−π0.

The amplitudes are: (A₀)ρð770Þπ, (A₁)ρð1450Þπ, (A₂)ρð1700Þπ,

(A₃)ρð2150Þπ, (A₄)ωð783Þπ0. The diagonal elements are the

same as the conventional fit fractions.

A0 A1 A2 A3 A4 A0 114.2
1.1 −10.4
0.8 0.7
0.1 0.1
0.1 −1.1
0.3 A1 10.9
1.7 −1.7
0.6 −0.2
0.1 0.0
0.0 A2 0.8
0.2 −0.07
0.02 0.0
0.0 A3 0.04
0.01 0.0
0.0 A4 0.08
0.03

TABLE IX. Interference fit fractions (%) and statistical

un-certainties from the Dalitz plot analysis ofJ=ψ → KþK−π0. The

amplitudes are: (A₀) Kð892Þ
K∓, (A₁) ρð1450Þ0π0, (A₂)

K_ð1410Þ
K∓_{, (A3)K}

2ð1430Þ
K∓. The diagonal elements are

the same as the conventional fit fractions.

A0 A1 A2 A3

A0 92.4
1.5 −5.5
0.6 −0.7
0.1 −0.9
0.2

A1 9.3
2.0 2.2
0.7 2.1
0.4

A2 2.3
1.1 3.3
0.9

A3 3.5
1.3

TABLE X. Interference fit fractions (%) and statistical

uncer-tainties from the Dalitz plot analysis ofJ=ψ → K0_SK
π∓. The

amplitudes are: (A₀) Kð892Þ ¯K, (A₁) ρð1450Þ
π∓, (A₂)

K

1ð1410Þ ¯K, (A3) K2ð1430Þ ¯K. The diagonal elements are the

same as the conventional fit fractions.

A0 A1 A2 A3

A0 90.5
0.9 −5.4
0.4 0.1
0.1 −1.3
0.2

A1 6.3
0.8 −0.1
0.5 1.9
0.3

A2 1.5
0.5 3.3
1.6

A3 7.1
1.3

TABLE XI. Fittedc_X→abcðn;mÞ coefficients with statistical

un-certainties from the Veneziano model description ofJ=ψ → πþπ−π0.

n m cX→abcðn; mÞ 1 1 0.5720
0.0016 2 1 0.7380
0.0027 3 1 0.1165
0.0014 2 4901
426 4 1 354
53 2 1781
49 5 1 −137.4
3.4 2 2087
245 3 −248
25 6 1 1869
86 2 −354
10 3 9.8
0.3 7 1 1084
132 2 63.5
13.7 3 −1.0
0.4 4 6259
335