Measurement of the
I ¼ 1=2 Kπ S-wave amplitude from Dalitz plot analyses
of
η
c→ K ¯Kπ in two-photon interactions
J. P. Lees,1V. Poireau,1V. Tisserand,1E. Grauges,2A. Palano,3,*G. Eigen,4D. N. Brown,5Yu. G. Kolomensky,5H. Koch,6 T. Schroeder,6 C. Hearty,7 T. S. Mattison,7 J. A. McKenna,7R. Y. So,7 V. 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,13J. 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,17A. Zallo,17
S. Passaggio,18a C. Patrignani,18a,18bB. Bhuyan,19U. Mallik,20C. Chen,21J. Cochran,21S. Prell,21H. Ahmed,22 M. R. Pennington,23A. V. Gritsan,24N. Arnaud,25M. Davier,25F. Le Diberder,25A. M. Lutz,25G. Wormser,25D. J. Lange,26
D. M. Wright,26J. P. Coleman,27E. Gabathuler,27D. E. Hutchcroft,27D. J. Payne,27C. Touramanis,27A. J. Bevan,28 F. Di Lodovico,28R. Sacco,28G. Cowan,29Sw. Banerjee,30D. N. Brown,30C. L. Davis,30 A. G. Denig,31M. Fritsch,31 W. Gradl,31K. Griessinger,31A. Hafner,31K. R. Schubert,31R. J. Barlow,32,†G. D. Lafferty,32R. Cenci,33A. Jawahery,33 D. A. Roberts,33R. Cowan,34R. Cheaib,35 S. H. Robertson,35B. Dey,36a N. Neri,36a F. Palombo,36a,36bL. Cremaldi,37 R. Godang,37,‡ D. J. Summers,37P. Taras,38G. De Nardo,39 C. Sciacca,39G. Raven,40C. P. Jessop,41J. M. LoSecco,41 K. Honscheid,42R. Kass,42A. Gaz,43aM. Margoni,43a,43bM. Posocco,43aM. Rotondo,43aG. Simi,43a,43bF. Simonetto,43a,43b R. Stroili,43a,43bS. Akar,44E. Ben-Haim,44M. Bomben,44G. R. Bonneaud,44G. Calderini,44J. Chauveau,44G. Marchiori,44 J. Ocariz,44M. Biasini,45a,45b E. Manoni,45a A. Rossi,45a G. Batignani,46a,46b S. Bettarini,46a,46bM. Carpinelli,46a,46b,§ G. Casarosa,46a,46bM. Chrzaszcz,46aF. Forti,46a,46bM. A. Giorgi,46a,46bA. Lusiani,46a,46cB. Oberhof,46a,46bE. Paoloni,46a,46b M. Rama,46aG. Rizzo,46a,46bJ. J. Walsh,46aA. J. S. Smith,47F. Anulli,48aR. Faccini,48a,48bF. Ferrarotto,48aF. Ferroni,48a,48b A. Pilloni,48a,48b G. Piredda,48aC. Bünger,49S. Dittrich,49O. Grünberg,49M. Heß,49T. Leddig,49C. Voß,49R. Waldi,49 T. Adye,50F. F. Wilson,50S. Emery,51G. Hamel de Monchenault,51G. Vasseur,51D. Aston,52C. Cartaro,52M. R. Convery,52 J. Dorfan,52W. Dunwoodie,52M. Ebert,52R. C. Field,52B. G. Fulsom,52M. T. Graham,52C. Hast,52W. R. Innes,52P. Kim,52 D. W. G. S. Leith,52S. Luitz,52V. Luth,52D. B. MacFarlane,52D. R. Muller,52H. Neal,52B. N. Ratcliff,52A. Roodman,52 M. K. Sullivan,52 J. Va’vra,52W. J. Wisniewski,52 M. V. Purohit,53J. R. Wilson,53A. Randle-Conde,54S. J. Sekula,54 M. Bellis,55P. R. Burchat,55E. M. T. Puccio,55M. S. Alam,56J. A. Ernst,56R. Gorodeisky,57N. Guttman,57D. R. Peimer,57 A. Soffer,57S. M. Spanier,58J. L. Ritchie,59R. F. Schwitters,59J. M. Izen,60X. C. Lou,60F. Bianchi,61a,61bF. De Mori,61a,61b A. Filippi,61a D. Gamba,61a,61bL. Lanceri,62L. Vitale,62F. Martinez-Vidal,63A. Oyanguren,63J. Albert,64A. Beaulieu,64
F. U. Bernlochner,64G. J. King,64R. Kowalewski,64T. Lueck,64I. M. Nugent,64 J. M. Roney,64N. Tasneem,64 T. J. Gershon,65P. F. Harrison,65 T. E. Latham,65R. Prepost,66and S. L. Wu66
(The 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
3INFN 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
5Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 6
Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany
7University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 8a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
8bNovosibirsk State University, Novosibirsk 630090, Russia 8c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
9University of California at Irvine, Irvine, California 92697, USA 10
University of California at Riverside, Riverside, California 92521, USA
11University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA 12
California Institute of Technology, Pasadena, California 91125, USA
13University of Cincinnati, Cincinnati, Ohio 45221, USA 14
University of Colorado, Boulder, Colorado 80309, USA
15Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France 16a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
17INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy 18a
INFN Sezione di Genova, I-16146 Genova, Italy
18bDipartimento di Fisica, Università di Genova, I-16146 Genova, Italy 19
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
20University of Iowa, Iowa City, Iowa 52242, USA 21
Iowa State University, Ames, Iowa 50011, USA
22Physics Department, Jazan University, Jazan 22822, Saudi Arabia 23
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
24Johns Hopkins University, Baltimore, Maryland 21218, USA 25
Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France
26
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
27University of Liverpool, Liverpool L69 7ZE, United Kingdom 28
Queen Mary, University of London, London, E1 4NS, United Kingdom
29University of London, Royal Holloway and Bedford New College, Egham,
Surrey TW20 0EX, United Kingdom
30University of Louisville, Louisville, Kentucky 40292, USA 31
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
32University of Manchester, Manchester M13 9PL, United Kingdom 33
University of Maryland, College Park, Maryland 20742, USA
34Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge,
Massachusetts 02139, USA
35McGill University, Montréal, Québec, Canada H3A 2T8 36a
INFN Sezione di Milano, I-20133 Milano, Italy
36bDipartimento di Fisica, Università di Milano, I-20133 Milano, Italy 37
University of Mississippi, University, Mississippi 38677, USA
38Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 39
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy
40
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, Netherlands
41
University of Notre Dame, Notre Dame, Indiana 46556, USA
42Ohio State University, Columbus, Ohio 43210, USA 43a
INFN Sezione di Padova, I-35131 Padova, Italy
43bDipartimento di Fisica, Università di Padova, I-35131 Padova, Italy 44
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France
45a
INFN Sezione di Perugia, I-06123 Perugia, Italy
45bDipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy 46a
INFN Sezione di Pisa, I-56127 Pisa, Italy
46bDipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy 46c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
47Princeton University, Princeton, New Jersey 08544, USA 48a
INFN Sezione di Roma, I-00185 Roma, Italy
48bDipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy 49
Universität Rostock, D-18051 Rostock, Germany
50Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 51
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
52SLAC National Accelerator Laboratory, Stanford, California 94309 USA 53
University of South Carolina, Columbia, South Carolina 29208, USA
54Southern Methodist University, Dallas, Texas 75275, USA 55
Stanford University, Stanford, California 94305, USA
56State University of New York, Albany, New York 12222, USA 57
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
58University of Tennessee, Knoxville, Tennessee 37996, USA 59
University of Texas at Austin, Austin, Texas 78712, USA
60University of Texas at Dallas, Richardson, Texas 75083, USA 61a
INFN Sezione di Torino, I-10125 Torino, Italy
61bDipartimento di Fisica, Università di Torino, I-10125 Torino, Italy 62
63IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain 64
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
65Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 66
University of Wisconsin, Madison, Wisconsin 53706, USA (Received 8 November 2015; published 20 January 2016)
We study the processesγγ → K0SKπ∓ andγγ → KþK−π0using 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 observe ηc decays to both final states and perform Dalitz plot analyses using a model-independent partial wave analysis technique. This allows a model-independent measurement of the mass-dependence of theI ¼ 1=2 Kπ S-wave amplitude and phase. A comparison between the present measurement and those from previous experiments indicates similar behavior for the phase up to a mass of 1.5 GeV=c2. In contrast, the amplitudes show very marked differences. The data require the presence of a newa0ð1950Þ resonance with parameters m ¼ 1931 14 22 MeV=c2 andΓ ¼ 271 22 29 MeV.
DOI:10.1103/PhysRevD.93.012005
I. INTRODUCTION
Scalar mesons are still a puzzle in light meson spec-troscopy: they have complex structure, and there are too many states to be accommodated within the quark model without difficulty [1]. In particular, the structure of the I ¼ 1=2 Kπ S-wave is a long-standing problem. In recent years many experiments have performed accurate studies of the decays of heavy-flavored hadrons producing aKπ system in the final state. These studies include searches forCP violation[2], and searches for, and observation of, new exotic resonances [3] and charmed mesons [4]. However, the still poorly known structure of the I ¼ 1=2 Kπ S-wave is a source of large systematic uncertain-ties. The best source of information on the scalar structure of the Kπ system comes from the LASS experiment, which studied the reaction K−p → K−πþn [5]. Partial wave analysis of theKπ system reveals a large contribu-tion from theI ¼ 1=2 Kπ S-wave amplitude over the mass range studied. In the description of the I ¼ 1=2 scalar amplitude up to a Kπ mass of about 1.5 GeV=c2 the K
0ð1430Þ resonant amplitude is added coherently to an
effective-range description in such a way that the net amplitude actually decreases rapidly at the resonance mass. The I ¼ 1=2 S-wave amplitude representation is given explicitly in Ref.[6]. In the LASS analysis, in the region above 1.82 GeV=c2, the S-wave suffers from a twofold ambiguity, but in both solutions it is understood in terms of the presence of aK0ð1950Þ resonance. It should be noted that the extraction of the I ¼ 1=2 S-wave
amplitude is complicated by the presence of anI ¼ 3=2 contribution.
Further information on theKπ system has been extracted from Dalitz plot analysis of the decay Dþ → K−πþπþ where, in order to fit the data, the presence of an additional resonance, the κð800Þ, was claimed [7]. Using the same data, a model independent partial wave analysis (MIPWA) of theKπ system was developed for the first time[8]. This method allows the amplitude and phase of theKπ S-wave to be extracted as functions of mass (see also Refs.[9]and [10]). However in these analyses the phase space is limited to mass values less than1.6 GeV=c2due to the kinematical limit imposed by theDþmass. A similar method has been used to extract theπþπ−S-wave amplitude in a Dalitz plot analysis ofDþs → πþπ−πþ [11].
In the present analysis, we consider three-bodyηcdecays toK ¯Kπ and obtain new information on the KπI ¼ 1=2 S-wave amplitude extending up to a mass of2.5 GeV=c2. We emphasize that, due to isospin conservation in the ηc hadronic decay toðKπÞ ¯K, the ðKπÞ amplitude must have I ¼ 1=2, and there is no I ¼ 3=2 contribution. The BABAR experiment first performed a Dalitz plot analysis ofηc → KþK−π0 and η
c → KþK−η using an isobar model [12].
The analysis reported the first observation ofK0ð1430Þ → Kη, and observed that ηc decays to three pseudoscalars
are dominated by intermediate scalar mesons. A previous search for charmonium resonances decaying toK0SKπ∓in two-photon interactions is reported in Ref.[13]. We con-tinue these studies ofηcdecays and extract theKπ S-wave amplitude by performing a MIPWA of bothηc→ K0SKπ∓ andηc → KþK−π0final states.
We describe herein studies of theK ¯Kπ system produced in two-photon interactions. Two-photon events in which at least one of the interacting photons is not quasireal are strongly suppressed by the selection criteria described below. This implies that the allowed JPC values of any
*Also at Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA.
†Present address: University of Huddersfield, Huddersfield HD1 3DH, United Kingdom.
‡Present address: University of South Alabama, Mobile, Alabama 36688, USA.
produced resonances are 0þ, 2þ, 3þþ, 4þ... [14]. Angular momentum conservation, parity conservation, and charge conjugation invariance imply that these quan-tum numbers also apply to the final state except that the K ¯Kπ state cannot be in a JP¼ 0þ state.
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 of the K0SKπ∓ system. In Sec. IV, we describe studies of efficiency and resolution, while in Sec.V we describe the MIPWA. In Secs.VIandVIIwe perform Dalitz plot analyses ofηc→ K0SKπ∓andηc → KþK−π0decays. SectionVIIIis devoted to discussion of the measuredKπ S-wave amplitude, and finally results are summarized in Sec.IX.
II. THEBABAR DETECTOR AND DATA SET The results presented here are based on data collected with the BABAR detector at the PEP-II asymmetric-energy eþe− collider located at SLAC, and correspond to an
integrated luminosity of519 fb−1 [15]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 [16]. Charged particles are detected, and their momenta are measured, by means of a five-layer, double-sided microstrip detector, and a 40-layer drift chamber, both operating in the 1.5 T magnetic field of a superconducting solenoid. Photons are measured and electrons are identified in a CsI(Tl) crystal electromagnetic calorimeter. Charged-particle identification is provided by the measurement of specific energy loss in the tracking devices, and by an internally reflecting, ring-imaging Cherenkov detector. Muons and K0L mesons are detected in the instrumented flux return of the magnet. Monte Carlo (MC) simulated events[17], with reconstructed sample sizes more than 10 times larger than the corresponding data samples, are used to evaluate the signal efficiency and to determine background features. Two-photon events are simulated using the GamGam MC generator[18].
III. RECONSTRUCTION AND SELECTION OFηc→ K0
SKπ∓ EVENTS
To study the reaction γγ → K0
SKπ∓ ð1Þ
we select events in which theeþ ande−beam particles are scattered at small angles, and hence are undetected in the final state. 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 electromag-netic calorimeter greater than 100 MeV. We obtainK0S→ πþπ− candidates by means of a vertex fit of pairs of
oppositely charged tracks which requires aχ2fit probability greater than 0.001. Each K0S 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 a χ2 fit probability greater than 0.001. 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 K0S decay. We accept only K0S candidates with decay lengths from the main vertex of the event greater than 0.2 cm, and require cosθK0
S > 0.98, where θK0Sis defined as
the angle between theK0Smomentum direction and the line joining the primary and theK0S vertex. 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=c2andσ ¼ 2.9 MeV=c2. We select the
K0
Ssignal region to be within2σ of m and reconstruct the
K0
S4-vector by adding the three-momenta of the pions and
computing the energy using theK0SPDG mass value[19]. Background arises mainly from random combinations of particles from eþe− annihilation, from other two-photon processes, and from events with initial-state two-photon radiation (ISR). The ISR background is dominated by JPC¼ 1−− resonance production [20]. We discriminate
against K0SKπ∓ events produced via ISR by requiring M2
rec≡ðpeþe−−precÞ2> 10 GeV2=c4, where peþe− is the
four-momentum of the initial state and prec is the
four-momentum of theK0SKπ∓ system.
The K0SKπ∓ mass spectrum shows a prominent ηc signal. We definepT as the magnitude of the vector sum of the transverse momenta, in theeþe−rest frame, of the final-state particles with respect to the beam axis. Since well-reconstructed two-photon events are expected to have low values ofpT, we optimize the selection as a function of this variable. We produceK0SKπ∓mass spectra with different pT selections and fit the mass spectra to extract the number
of ηc signal events (Ns) and the number of background events below the ηc signal (Nb). We then compute the purity, defined asP ¼ Ns=ðNsþ NbÞ, and the significance S ¼ Ns=pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNsþ Nb. To obtain the best significance with
the highest purity, we optimize the selection by requiring the maximum value of the product of purity and signifi-cance,P · S, and find that this corresponds to the require-mentpT < 0.08 GeV=c.
Figure1shows the measuredpTdistribution in compari-son to the corresponding pT distribution obtained from simulation of the signal process. A peak at low pT is observed indicating the presence of the two-photon process. The shape of the peak agrees well with that seen in the MC simulation. Figure2shows theK0SKπ∓mass spectrum in theηcmass region. A clearηc signal over a background of about 35% can be seen, together with a residualJ=ψ signal. Information on the fitting procedure is given at the end of Sec.IV. We define theηcsignal region as the range
2.922–3.039 GeV=c2 [mðηcÞ 1.5Γ], which contains 12849 events with a purity of 64.3% 0.4%. Sideband regions are defined by the ranges 2.785 − 2.844 GeV=c2 and3.117 − 3.175 GeV=c2(within3.5 − 5 Γ), respectively as indicated (shaded) in Fig.2.
Details on data selection, event reconstruction, resolu-tion, and efficiency measurement for the ηc → KþK−π0 decay can be found in Ref.[12]. The ηc signal region for this decay mode contains 6710 events with a purity of (55.2% 0.6%).
IV. EFFICIENCY AND RESOLUTION To compute the efficiency, MC signal events are gen-erated using a detailed detector simulation[17]in which the ηcdecays 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. Due to the presence of long tails in the Breit-Wigner (BW) representation of the resonance, we apply selection criteria to restrict the generated events to the ηc mass region. We express the efficiency as a function of
the invariant mass,mðKþπ−Þ[21], and cosθ, where θ is the angle, in theKþπ−rest frame, between the directions of the Kþ and the boost from theK0
SKþπ− rest frame.
To smooth statistical fluctuations, this efficiency map is parametrized as follows. First we fit the efficiency as a function of cosθ in separate intervals of mðKþπ−Þ, using Legendre polynomials up toL ¼ 12:
ϵðcos θÞ ¼X12
L¼0
aLðmÞY0Lðcos θÞ; ð2Þ
wherem denotes the Kþπ−invariant mass. For each value ofL, we fit the mass dependent coefficients aLðmÞ with a seventh-order polynomial inm. Figure3shows the result-ing fitted efficiency mapϵðm; cos θÞ. We obtain χ2=Ncells¼
217=300 for this fit, where Ncells is the number of cells
in the efficiency map. We observe a significant decrease in efficiency in regions of cosθ ∼ 1 due to the impossibility of reconstructingK mesons with laboratory momentum less than about200 MeV=c, and π andK0Sð→πþπ−Þ mesons with laboratory momentum less than about100 MeV=c (see Fig. 9 of Ref.[6]). These effects result from energy loss in the beampipe and inner-detector material.
The mass resolution,Δm, is measured as the difference between the generated and reconstructedK0SKπ∓ invari-ant-mass values. The distribution has a root-mean-squared value of10 MeV=c2, and is parameterized by the sum of a crystal ball [22] and a Gaussian function. We perform a binned fit to the K0SKπ∓ mass spectrum in data using the following model. The background is described by a second-order polynomial, and the ηc resonance is
(GeV/c) T p 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 events/(5 MeV/c)
FIG. 1. Distributions of pT for γγ → K0SKπ∓. The data are shown as (black) points with error bars, and the signal MC simulation as a (red) histogram; the vertical dashed line indicates the selection applied to select two-photon events.
) 2 ) (GeV/c ± π ± K S 0 m(K 2.8 3 3.2 ) 2 events/(5 MeV/c 0 200 400 600 800 1000
FIG. 2. TheK0SKπ∓mass spectrum in theηcmass region after requiring pT< 0.08 GeV=c. The solid curve shows the total fitted function, and the dashed curve shows the fitted background contribution. The shaded areas show signal and sideband regions.
0.02 0.04 0.06 0.08 0.1 0.12 2 ) GeV/c -π + m(K 1 1.5 2 θ cos −1 −0.5 0 0.5 1
FIG. 3. Fitted detection efficiency in the cosθ vs mðKþπ−Þ plane. Each interval shows the average value of the fit for that region.
represented by a nonrelativistic BW function convolved with the resolution function. In addition, we allow for the presence of a residual J=ψ contribution modeled with a Gaussian function. Its parameter values are fixed to those obtained from a fit to the K0SKπ∓ mass spectrum for the ISR data sample obtained by requiring jM2recj <
1 GeV2=c4. The fittedK0
SKπ∓mass spectrum is shown in
Fig.2. We obtain the followingηc parameters: m ¼ 2980.8 0.4 MeV=c2; Γ ¼ 33 1 MeV;
Nηc ¼ 9808 164; ð3Þ
where uncertainties are statistical only. Our measured mass value is 2.8 MeV=c2 lower than the world average [19]. This may be due to interference between the ηc amplitude and that describing the background in the signal region[23].
V. MODEL INDEPENDENT PARTIAL WAVE ANALYSIS
We perform independent MIPWA of the K0SKπ∓ and KþK−π0 Dalitz plots in the ηc mass region using unbinned maximum likelihood fits. The likelihood function is written as L ¼YN n¼1 fsigðmnÞϵðx0n; y0nÞ P i;jcicjPAiðxn; ynÞAjðxn; ynÞ i;jcicjIAiAj þ ð1 − fsigðmnÞÞ P ikiBPiðxn; yn; mnÞ ikiIBi ð4Þ where:
(i) N is the number of events in the signal region; (ii) for thenth event, mnis theK0SKπ∓or theKþK−π0
invariant mass;
(iii) for thenth event, xn ¼ m2ðKþπ−Þ, yn¼ m2ðK0Sπ−Þ for K0SKπ∓; xn¼ m2ðKþπ0Þ, yn¼ m2ðK−π0Þ for KþK−π0;
(iv) fsig is the mass-dependent fraction of signal
ob-tained from the fit to theK0SKπ∓orKþK−π0mass spectrum;
(v) for the nth event, ϵðx0n; y0nÞ is the efficiency para-metrized as a function of x0n ¼ mðKþπ−Þ for K0
SKπ∓ and x0n¼ mðKþK−Þ for KþK−π0, and
y0
n¼ cos θ (see Sec. IV);
(vi) for the nth event, the Aiðxn; ynÞ describe the com-plex signal-amplitude contributions;
(vii) ci is the complex amplitude for the ith signal component; the ci parameters are allowed to vary during the fit process;
(viii) for thenth event, the Biðxn; ynÞ describe the back-ground probability-density functions assuming that interference between signal and background ampli-tudes can be ignored;
(ix) ki is the magnitude of theith background compo-nent; the ki parameters are obtained by fitting the sideband regions;
(x) IAiA j ¼
R
Aiðx; yÞAjðx; yÞϵðx0; y0Þdxdy and IBi¼
R
Biðx; yÞdxdy are normalization integrals.
Numeri-cal integration is performed on phase space gener-ated events withηcsignal and background generated according to the experimental distributions. In case of MIPWA or when resonances have free parame-ters, integrals are recomputed at each minimization step. Background integrals and fits dealing with amplitudes having fixed resonance parameters are computed only once.
Amplitudes are described along the lines described in Ref.[24]. For anηcmeson decaying into three pseudoscalar mesons via an intermediate resonance r of spin J (i.e. ηc→ Cr, r → AB), each amplitude Aiðx; yÞ is represented
by the product of a complex Breit-Wigner (BW) function and a real angular distribution function represented by the spherical harmonic functionpffiffiffiffiffiffi2πY0Jðcos θÞ; θ is the angle between the direction ofA, in the rest frame of r, and the direction ofC in the same frame. This form of the angular dependence results from angular momentum conservation in the rest frame of theηc, which leads to the production of r with helicity 0. It follows that Aiðx; yÞ ¼ BWðMABÞ ffiffiffiffiffiffi 2π p Y0 Jðcos θÞ: ð5Þ
The functionBWðMABÞ is a relativistic BW function of the form
BWðMABÞ ¼M2 FηcF
r− M2AB− iMrΓtotðMABÞ
ð6Þ whereMris the mass of the resonancer, and ΓtotðMABÞ is its mass-dependent total width. In general, this mass dependence cannot be specified, and a constant value should be used. However, for a resonance such as theK0ð1430Þ, which is approximately elastic, we can use the partial width ΓAB, and specify the mass-dependence as:
ΓAB ¼ Γr pAB pr 2Jþ1 Mr MAB F2 ð7Þ where pAB¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðM2 AB− M2A− M2BÞ2− 4M2AM2B p 2MAB ; ð8Þ
andpris the value ofpAB whenMAB¼ Mr.
The form factors Fηc and F attempt to model the underlying quark structure of the parent particle and the intermediate resonances. We set Fηc to a constant value, while forF we use Blatt-Weisskopf penetration factors[25] (TableI), that depend on a single parameterR representing
the meson“radius”, for which we assume R ¼ 1.5 GeV−1. The a0ð980Þ resonance is parameterized as a coupled-channel Breit-Wigner function whose parameters are taken from Ref. [26].
To measure the I ¼ 1=2 Kπ S-wave we make use of a MIPWA technique first described in Ref. [8]. The Kπ S-wave, being the largest contribution, is taken as the reference amplitude. We divide theKπ mass spectrum into 30 equally spaced mass intervals 60 MeV wide, and for each interval we add to the fit two new free parameters, the amplitude and the phase of theKπ S-wave in that interval. We fix the amplitude to 1.0 and its phase to π=2 at an arbitrary point in the mass spectrum, for which we choose interval 14, corresponding to a mass of1.45 GeV=c2. The number of associated free parameters is therefore 58.
Due to isospin conservation in the hadronicηc andK decays, theðKπÞ ¯K amplitudes are combined with positive signs, and so therefore are symmetrized with respect to the two K¯K modes. In particular we write the Kπ S-wave amplitudes as AS-wave¼ 1ffiffiffi 2 p ðaKþπ− j eiϕ Kþπ− j þ a¯K0π− j eiϕ ¯K0π− j Þ; ð9Þ where aKþπ−ðmÞ ¼ a¯K0π−ðmÞ and ϕKþπ−ðmÞ ¼ ϕ¯K0π−ðmÞ, for ηc→ ¯K0Kþπ− [21]and AS-wave¼ 1ffiffiffi 2 p ðaKþπ0 j eiϕ Kþπ0 j þ aK−π0 j eiϕ K−π0 j Þ; ð10Þ where aKþπ0ðmÞ ¼ aK−π0ðmÞ and ϕKþπ0ðmÞ ¼ ϕK−π0ðmÞ, for ηc → KþK−π0. For both decay modes the bachelor kaon is in an orbitalS-wave with respect to the relevant Kπ system, and so does not affect these amplitudes. The second amplitude in Eq.(9)is reduced because the ¯K0is observed as a K0S, but the same reduction factor applies to the first amplitude through the bachelor ¯K0, so that the equality of the three-body amplitudes is preserved.
Other resonance contributions are described as above. The K2ð1430Þ ¯K contribution is symmetrized in the same way as the S-wave amplitude.
We perform MC simulations to test the ability of the method to find the correct solution. We generateηc → K0SKπ∓event samples which yield reconstructed samples having the same size as the data sample, according to arbitrary mixtures of resonances, and extract the Kπ S-wave using the MIPWA
method. We find that the fit is able to extract correctly the mass dependence of the amplitude and phase.
We also test the possibility of multiple solutions by starting the fit from random values or constant parameter values very far from the solution found by the fit. We find only one solution in both final states and conclude that the fit converges to give the correct S-wave behavior for different starting values of the parameters.
The efficiency-corrected fractional contribution fi due to resonant or nonresonant contribution i is defined as follows: fi¼jcij 2R jA iðxn; ynÞj2dxdy R jPjcjAjðx; yÞj2dxdy : ð11Þ
Thefido not necessarily sum to 100% because of interfer-ence effects. The uncertainty for each fi is evaluated by propagating the full covariance matrix obtained from the fit. 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. In these MC simulations we smooth the fittedKπ S-wave amplitude and phase by means of a cubic spline. We make use of these weighted events to compute a 2Dχ2 over the Dalitz plot. For this purpose, we divide the Dalitz plot into a grid of 25 × 25 cells and consider only those containing at least five events. We computeχ2¼PNcells
i¼1 ðNiobs− NiexpÞ2=Niexp,
where Niobs and Niexp are event yields from data and
simulation, respectively.
VI. DALITZ PLOT ANALYSIS OF ηc→ K0 SKπ∓
Figure4shows the Dalitz plot for the candidates in theηc signal region, and Fig.5 shows the corresponding Dalitz
TABLE I. Summary of the Blatt-Weisskopf penetration form factors. Spin F 0 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þðRrprÞ2 1þðRrpABÞ2 p 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9þ3ðRrprÞ2þðRrprÞ4 9þ3ðRrpABÞ2þðRrpABÞ4 p ) 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. 4. Dalitz plot for ηc→ K0SKπ∓ events in the signal region.
plot projections. Since the width of the ηc meson is 32.3 1.0 MeV, no mass constraint can be applied.
The Dalitz plot is dominated by the presence of horizontal and vertical uniform bands at the position of theK0ð1430Þ resonance. We also observe further bands along the diago-nal. Isospin conservation inηcdecay requires that theðK ¯KÞ system have I=1, so that these structures may indicate the presence ofa0ora2resonances. Further narrow bands are observed at the position of theKð892Þ resonance, mostly in theK0Sπ− projection; these components are consistent with originating from background, as will be shown.
The presence of background in the ηc signal region requires precise study of its structure. This can be achieved by means of the data in theηcsideband regions, for which the Dalitz plots are shown in Fig. 6.
In both regions we observe almost uniformly populated resonant structures mostly in theK0Sπ− mass, especially in the regions corresponding to the Kð892Þ and K2ð1430Þ resonances. The resonant structures in Kþπ− mass are weaker. The three-body decay of a pseudoscalar meson into a spin-one or spin-two resonance yields a nonuniform distribution [see Eq.(5)] in the relevant resonance band on the Dalitz plot. The presence of uniformly populated bands in theKð892Þ and K2ð1430Þ mass regions, indicates that these structures are associated with background. Also, the asymmetry between the twoKmodes in background may be explained as being due to interference between theI ¼ 0 andI ¼ 1 isospin configurations for the Kð→ KπÞ ¯K final state produced in two-photon fusion.
We fit the ηc sidebands using an incoherent sum of amplitudes, which includes contributions from the a0ð980Þ, a0ð1450Þ, a2ð1320Þ, Kð892Þ, K0ð1430Þ,
K
2ð1430Þ, Kð1680Þ, and K0ð1950Þ resonances. To better
constrain the sum of the fractions to one, we make use of the channel likelihood method[27]and include resonances until no structure is left in the background and an accurate description of the Dalitz plots is obtained.
To estimate the background composition in theηcsignal region we perform a linear mass dependent interpolation of the fractions of the different contributions, obtained from
the fits to the sidebands, and normalized using the results from the fit to theK0SKπ∓mass spectrum. The estimated background contributions are indicated by the shaded regions in Fig.5.
A. MIPWA ofηc→ K0 SKπ∓
We perform the MIPWA including the resonances listed in TableII. In this table, and in the remainder of the paper, we use the notation ðKπÞ ¯K or K¯K to represent the corresponding symmetrized amplitude. After the solution ) 4 /c 2 ) (GeV ± π ± (K 2 m ) 4 /c 2 events/(0.08 GeV (a) ) 4 /c 2 ) (GeV ± π S 0 (K 2 m ) 4 /c 2 events/(0.08 GeV (b) ) 4 /c 2 ) (GeV ± K S 0 (K 2 m 0 2 4 6 0 2 4 6 0 2 4 6 8 ) 4 /c 2 events/(0.1 GeV (c) 0 200 400 0 200 400 600 0 100 200 300 400
FIG. 5. Theηc→ K0SKπ∓ Dalitz plot projections on (a)m2ðKπ∓Þ, (b) m2ðK0SπÞ, and (c) m2ðK0SKÞ. The superimposed curves result from the MIPWA 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.
) 4 /c 2 ) (GeV ± π ± (K 2 m ) 4 /c 2 ) (GeV ± π S 0 (K 2 m (a) ) 4 /c 2 ) (GeV ± π ± (K 2 m 0 2 4 6 8 0 2 4 6 8 ) 4 /c 2 ) (GeV ± π S 0 (K 2 m (b) 0 2 4 6 8 0 2 4 6 8
FIG. 6. Dalitz plots for the ηc→ K0SKπ∓ sideband regions: (a) lower, (b) upper.
is found we test for other contributions, including spin-one resonances, but these are found to be consistent with zero, and so are not included. This supports the observation that the observed Kð892Þ structures originate entirely from background. We find a dominance of the Kπ S-wave amplitude, with small contributions from a0π amplitudes and a significantK2ð1430Þ ¯K contribution.
The table lists also a significant contribution from the a0ð1950Þπ amplitude, where a0ð1950Þþ → K0SKþis a new
resonance. We also test the spin-2 hypothesis for this contribution by replacing the amplitude for a0→ K0SKþ with ana2→ K0SKþ amplitude with parameter values left free in the fit. In this case no physical solution is found inside the allowed ranges of the parameters, and the additional contribution is found consistent with zero. This new state has isospin one, and the spin-0 assignment is preferred over that of spin-2.
A fit without this state gives a poor description of the high massK0SKþprojection, as can be seen in Fig.7(a). We obtain −2 log L ¼ − 4252.9 and χ2=N
cells¼ 1.33 for this fit. We
then include in the MIPWA a new scalar resonance decaying toK0SKþ with free parameters. We obtain Δðlog LÞ ¼ 61 andΔχ2¼ 38 for an increase of four new parameters. We estimate the significance for thea0ð1950Þ resonance using the fitted fraction divided by its statistical and systematic errors added in quadrature, and obtain2.5σ. Since interfer-ence effects may also contribute to the significance, this procedure gives a conservative estimate. The systematic uncertainties associated with the a0ð1950Þ state are described below. The fitted parameter values for this state are given in TableIII. We note that we obtainχ2=Ncells¼ 1.17 for this final fit, indicating good description of the data. The fit projections on the three squared masses from the MIPWA are shown in Fig. 5, and they indicate that the description of the data is quite good.
We compute the uncorrected Legendre polynomial moments hY0Li in each Kþπ−, K0Sπ− and K0SKþ mass interval by weighting each event by the relevant Y0
Lðcos θÞ function. These distributions are shown in
Fig. 8 as functions of Kπ mass after combining Kþπ−
TABLE II. Results from the ηc→ K0SKπ∓ and ηc→ KþK−π0 MIPWA. Phases are determined relative to the ðKπS-wave) ¯K amplitude which is fixed toπ=2 at 1.45 GeV=c2.
ηc→ K0SKπ∓ ηc→ KþK−π0
Amplitude Fraction (%) Phase (rad) Fraction (%) Phase (rad)
ðKπS-wave) ¯K 107.3 2.6 17.9 fixed 125.5 2.4 4.2 fixed
a0ð980Þπ 0.8 0.5 0.8 1.08 0.18 0.18 0.0 0.1 1.7 … a0ð1450Þπ 0.7 0.2 1.4 2.63 0.13 0.17 1.2 0.4 0.7 2.90 0.12 0.25 a0ð1950Þπ 3.1 0.4 1.2 −1.04 0.08 0.77 4.4 0.8 0.8 −1.45 0.08 0.27 a2ð1320Þπ 0.2 0.1 0.1 1.85 0.200.20 0.6 0.2 0.3 1.75 0.23 0.42 K 2ð1430Þ ¯K 4.7 0.9 1.4 4.92 0.05 0.10 3.0 0.8 4.4 5.07 0.09 0.30 Total 116.8 2.8 18.1 134.8 2.7 6.4 −2 log L −4314.2 −2339 χ2=N cells 301=254 ¼ 1.17 283.2=233 ¼ 1.22 ) 2 ) (GeV/c ± K S 0 m(K ) 2 events/(20 MeV/c (a) ) 2 ) (GeV/c -K + m(K 1 1.5 2 2.5 1 1.5 2 2.5 ) 2 events/(20 MeV/c (b) 0 50 100 150 200 0 50 100
FIG. 7. The mass projections (a)K0SKfromηc→ K0SKπ∓and (b)KþK−fromηc→ KþK−π0. The histograms show the MIPWA fit projections with (solid, black) and without (dashed, red) the presence of thea0ð1950Þþ→ K0SKresonance. The shaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sideband regions.
andK0Sπ−, and in Fig.9 as functions ofK0SKþ mass. We also compute the expected Legendre polynomial moments from the weighted MC events and compare 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. We compute the following systematic uncertainties on theI ¼ 1=2 Kπ S-wave amplitude and phase. The different contributions are added in quadrature.
(i) Starting from the solution found by the fit, we generate MC simulated events which are fitted using a MIPWA. In this way we estimate the bias intro-duced by the fitting method.
(ii) The fit is performed by interpolating theKπ S-wave amplitude and phase using a cubic spline.
(iii) We remove low-significance contributions, such as those from the a0ð980Þ and a2ð1320Þ resonances. (iv) We vary the signal purity up and down according to
its statistical uncertainty.
(v) The effect of the efficiency variation as a function of K ¯Kπ mass is evaluated by computing separate effi-ciencies in the regions below and above theηcmass.
These additional fits also allow the computation of systematic uncertainties on the amplitude fraction and phase values, as well as on the parameter values for the a0ð1950Þ resonance; these are summarized in Table IV.
In the evaluation of overall systematic uncertainties, all effects are assumed to be uncorrelated and are added in quadrature.
The measured amplitude and phase values of theI ¼ 1=2 Kπ S-wave as functions of mass obtained from the MIPWA ofηc → K0SKπ∓are shown in TableV. Interval 14 of the Kπ mass contains the fixed amplitude and phase values.
B. Dalitz plot analysis ofηc→ K0 SKπ∓
using an isobar model
We perform a Dalitz plot analysis of ηc→ K0SKπ∓ using a standard isobar model, where all resonances are modeled as BW functions multiplied by the corresponding angular functions. In this case theKπ S-wave is represented by a superposition of interfering K0ð1430Þ, K0ð1950Þ, nonresonant (NR), and possiblyκð800Þ contributions. The NR contribution is parametrized as an amplitude that is constant in magnitude and phase over the Dalitz plot. In this fit theK0ð1430Þ parameters are taken from Ref.[12], while all other parameters are fixed to PDG values. We also add thea0ð1950Þ resonance with parameters obtained from the MIPWA analysis.
For the description of theηcsignal, amplitudes are added one by one to ascertain the associated increase of the likelihood value and decrease of the 2D χ2. Table VI summarizes the fit results for the amplitude fractions and
) 2 ) (GeV/c ± π ± m(K ) 2
weight sum/(30 MeV/c
0 0 Y ) 2 ) (GeV/c ± π ± m(K 0 1 Y ) 2 ) (GeV/c ± π ± m(K 0 2 Y ) 2 ) (GeV/c ± π ± m(K ) 2
weight sum/(30 MeV/c
0 3 Y ) 2 ) (GeV/c ± π ± m(K 0 4 Y ) 2 ) (GeV/c ± 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 0 5 Y 0 50 100 150 200 250 300 350 −50 0 50 −50 0 0 50 −20 0 20 40 −40 −20 0
FIG. 8. Legendre polynomial moments for ηc→ K0SKπ∓ as functions of Kπ mass, and combined for Kπ∓ and K0Sπ∓; the superimposed curves result from the Dalitz plot fit described in the text.
TABLE III. Fitted a0ð1950Þ parameter values for the two ηc decay modes.
Final state Mass (MeV=c2) Width (MeV)
ηc→ K0SKπ∓ 1949 32 76 265 36 110
ηc→ KþK−π0 1927 15 23 274 28 30
phases. The high value ofχ2=Ncells¼ 1.82 (to be compared withχ2=Ncells¼ 1.17) indicates a poorer description of the data than that obtained with the MIPWA method. Including the κð800Þ resonance does not improve the fit quality. If included, it gives a fit fraction of 0.8% 0.5%.
The Dalitz plot analysis shows a dominance of scalar meson amplitudes, with small contributions from spin-two resonances. The Kð892Þ contribution is consistent with originating entirely from background. Other spin-1 K resonances have been included in the fit, but their con-tributions have been found to be consistent with zero. We note the presence of a sizeable nonresonant contribution. However, in this case the sum of the fractions is signifi-cantly lower than 100%, indicating important interference effects. Fitting the data without the NR contribution gives a much poorer description, with −2 log L ¼ −4115 and χ2=N
cells¼ 2.32.
We conclude that the ηc→ K0SKπ∓ Dalitz plot is not well-described by an isobar model in which theKπ S-wave is modeled as a superposition of Breit-Wigner functions. A more complex approach is needed, and the MIPWA is able to describe this amplitude without the need for a spe-cific model.
VII. DALITZ PLOT ANALYSIS OFηc→ KþK−π0
Theηc → KþK−π0Dalitz plot[12]is very similar to that for ηc → K0SKπ∓ decays. It is dominated by uniformly populated bands at the K0ð1430Þ resonance position in Kþπ0 and K−π0 mass squared. It also shows a broad
diagonal structure indicating the presence of a0 or a2 resonance contributions. The Dalitz plot projections are shown in Fig. 10.
Theηc → KþK−π0Dalitz plot analysis using the isobar model has been performed already in Ref. [12]. It was
TABLE IV. Systematic uncertainties on the a0ð1950Þ parameter values from the two ηc decay modes.
ηc→ K0SKπ∓ ηc→ KþK−π0
Effect Mass (MeV=c2) Width (MeV) Fraction (%) Mass (MeV=c2) Width (MeV) Fraction (%)
Fit bias 11 22 0.5 1 10 0.5 Cubic spline 24 79 0.6 14 9 0.2 Marginal components 70 72 0.0 2 8 0.3 ηcpurity 3 16 1.0 18 26 0.4 Efficiency 11 8 0.2 1 15 0.2 Total 76 110 1.3 23 30 0.8 ) 2 ) (GeV/c ± K S 0 m(K ) 2
weight sum/(30 MeV/c
0 0 Y ) 2 ) (GeV/c ± K S 0 m(K 0 1 Y ) 2 ) (GeV/c ± K S 0 m(K 0 2 Y ) 2 ) (GeV/c ± K S 0 m(K ) 2
weight sum/(30 MeV/c
0 3 Y ) 2 ) (GeV/c ± K S 0 m(K 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 1 1.5 2 2.5 3 0 5 Y 0 20 40 60 80 −20 −10 0 10 20 −40 −20 0 20 −20 −10 0 10 20 −20 0 20 40 −20 −10 0 10 20
FIG. 9. Legendre polynomial moments forηc→ K0SKπ∓ as a function ofK0SK mass, the superimposed curves result from the Dalitz plot fit described in the text.
found that the model does not give a perfect description of the data. In this section we obtain a new measurement of the Kπ S-wave by making use of the MIPWA method. In this way we also perform a cross-check of the results obtained
from theηc→ K0SKπ∓analysis, since analyses of the two ηc decay modes should give consistent results, given the
absence ofI ¼ 3=2 Kπ amplitude contributions. A. MIPWA of ηc→ KþK−π0
We perform a MIPWA of ηc → KþK−π0 decays using the same model and the same mass grid as for ηc→ K0SKπ∓. As for the previous case we obtain a
better description of the data if we include an additional a0ð1950Þ resonance, whose parameter values are listed in
Table III. We observe good agreement between the parameter values obtained from the twoηc decay modes. The table also lists parameter values obtained as the weighted mean of the two measurements. Table II gives the fitted fractions from the MIPWA fit.
We obtain a good description of the data, as evidenced by the valueχ2=Ncells¼ 1.22, and observe the a0ð1950Þ state
with a significance of4.2σ. The fit projections on the Kþπ0, K−π0, and KþK− squared mass distributions are shown
in Fig. 10. As previously, there is a dominance of the
TABLE VI. Results from theηc→ K0SKπ∓Dalitz plot analysis using an isobar model. The listed uncertainties are statistical only.
Amplitude Fraction % Phase (rad)
K 0ð1430Þ ¯K 40.8 2.2 0. K 0ð1950Þ ¯K 14.8 1.7 −1.00 0.07 NR 18.0 2.5 1.94 0.09 a0ð980Þπ 10.5 1.2 0.94 0.12 a0ð1450Þπ 1.7 0.5 2.94 0.13 a0ð1950Þπ 0.7 0.2 −1.76 0.24 a2ð1320Þπ 0.2 0.2 −0.53 0.42 K 2ð1430Þ ¯K 2.3 0.7 −1.55 0.11 Total 88.8 4.3 −2 log L −4290.7 χ2=N cells 467=256 ¼ 1.82
TABLE V. Measured amplitude and phase values for theI ¼ 1=2 KπS-wave as functions of mass obtained from the MIPWA of ηc→ K0SKπ∓andηc→ KþK−π0. The first error is statistical, the second systematic. The amplitudes and phases in the mass interval 14
are fixed to constant values.
ηc→ K0SKπ∓ ηc→ KþK−π0
N Kπ mass Amplitude Phase (rad) Amplitude Phase (rad)
1 0.67 0.119 0.100 0.215 0.259 0.577 1.290 0.154 0.350 0.337 3.786 1.199 0.857 2 0.73 0.103 0.043 0.113 −0.969 0.757 1.600 0.198 0.124 0.216 3.944 0.321 0.448 3 0.79 0.158 0.086 0.180 0.363 0.381 1.500 0.161 0.116 0.098 1.634 0.584 0.448 4 0.85 0.232 0.128 0.214 0.448 0.266 1.500 0.125 0.118 0.031 3.094 0.725 0.448 5 0.91 0.468 0.075 0.194 0.091 0.191 0.237 0.307 0.213 0.162 0.735 0.326 0.255 6 0.97 0.371 0.083 0.129 0.276 0.156 0.190 0.528 0.121 0.055 −0.083 0.178 0.303 7 1.03 0.329 0.071 0.102 0.345 0.164 0.273 0.215 0.191 0.053 0.541 0.320 0.638 8 1.09 0.343 0.062 0.062 0.449 0.196 0.213 0.390 0.146 0.046 0.254 0.167 0.144 9 1.15 0.330 0.070 0.081 0.687 0.167 0.221 0.490 0.135 0.089 0.618 0.155 0.099 10 1.21 0.450 0.059 0.042 0.696 0.156 0.226 0.422 0.092 0.102 0.723 0.242 0.267 11 1.27 0.578 0.048 0.112 0.785 0.208 0.358 0.581 0.113 0.084 0.605 0.186 0.166 12 1.33 0.627 0.047 0.053 0.986 0.153 0.166 0.643 0.106 0.039 1.330 0.264 0.130 13 1.39 0.826 0.047 0.105 1.334 0.155 0.288 0.920 0.153 0.056 1.528 0.161 0.160 14 1.45 1.000 1.570 1.000 1.570 15 1.51 0.736 0.031 0.059 1.918 0.153 0.132 0.750 0.118 0.076 1.844 0.149 0.048 16 1.57 0.451 0.025 0.053 2.098 0.202 0.277 0.585 0.099 0.047 2.128 0.182 0.110 17 1.63 0.289 0.029 0.065 2.539 0.292 0.180 0.366 0.079 0.052 2.389 0.230 0.213 18 1.69 0.159 0.036 0.089 1.566 0.308 0.619 0.312 0.074 0.043 1.962 0.195 0.150 19 1.75 0.240 0.034 0.067 1.962 0.331 0.655 0.427 0.093 0.063 1.939 0.150 0.182 20 1.81 0.381 0.031 0.059 2.170 0.297 0.251 0.511 0.094 0.063 2.426 0.156 0.277 21 1.87 0.457 0.035 0.085 2.258 0.251 0.284 0.588 0.098 0.080 2.242 0.084 0.210 22 1.93 0.565 0.042 0.067 2.386 0.255 0.207 0.729 0.114 0.095 2.427 0.098 0.254 23 1.99 0.640 0.044 0.055 2.361 0.228 0.092 0.777 0.119 0.075 2.306 0.102 0.325 24 2.05 0.593 0.046 0.065 2.329 0.235 0.268 0.775 0.134 0.075 2.347 0.107 0.299 25 2.11 0.614 0.057 0.083 2.421 0.230 0.169 0.830 0.134 0.078 2.374 0.105 0.199 26 2.17 0.677 0.067 0.117 2.563 0.218 0.137 0.825 0.140 0.070 2.401 0.127 0.189 27 2.23 0.788 0.085 0.104 2.539 0.228 0.241 0.860 0.158 0.123 2.296 0.131 0.297 28 2.29 0.753 0.097 0.125 2.550 0.234 0.168 0.891 0.167 0.133 2.320 0.131 0.273 29 2.35 0.646 0.096 0.118 2.315 0.241 0.321 0.994 0.202 0.076 2.297 0.153 0.197 30 2.41 0.789 0.184 0.187 2.364 0.336 0.199 0.892 0.322 0.098 2.143 0.292 0.393
(Kπ S-wave) ¯K amplitude, with a significant K2ð1430Þ ¯K amplitude, and small contributions from a0π amplitudes. We observe good agreement between fractions and relative phases of the amplitudes between theηc → K0SKπ∓ and ηc → KþK−π0 decay modes. Systematic uncertainties are
evaluated as discussed in Sec. VI. A.
We compute the uncorrected Legendre polynomial moments hY0Li in each Kþπ0, K−π0 and KþK− mass interval by weighting each event by the relevant Y0
Lðcos θÞ function. These distributions are shown in
Fig. 11 as functions of Kπ mass, combined for Kþπ0 andK−π0, and in Fig.12as functions ofKþK−mass. We also compute the expected Legendre polynomial moments from the weighted MC events and compare with the experimental distributions. We observe good agreement for all the distributions, which indicates that also in this
case the fit is able to reproduce the local structures apparent in the Dalitz plot.
VIII. THEI ¼ 1=2 Kπ S-WAVE AMPLITUDE AND PHASE
Figure 13 displays the measured I ¼ 1=2 Kπ S-wave amplitude and phase from both ηc → K0SKπ∓ and ηc→ KþK−π0. We observe good agreement between the
amplitude and phase values obtained from the two measurements.
The main features of the amplitude [Fig. 13(a)] can be explained by the presence of a clear peak related to the K
0ð1430Þ resonance which shows a rapid drop around
1.7 GeV=c2, where a broad structure is present which can
be related to the K0ð1950Þ resonance. There is some ) 2 ) (GeV/c π m(K ) 2
weight sum/(30 MeV/c
0 0 Y ) 2 ) (GeV/c π m(K 0 1 Y ) 2 ) (GeV/c π m(K 0 2 Y ) 2 ) (GeV/c π m(K ) 2
weight sum/(30 MeV/c
0 3 Y ) 2 ) (GeV/c π m(K 0 4 Y ) 2 ) (GeV/c 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 0 5 Y 0 20 40 60 80 100 120 140 160 180 −40 −20 0 20 −40 −20 0 20 −20 0 20 −20 −10 0 10 20 −10 0 10 20 30
FIG. 11. Legendre polynomial moments for ηc→ KþK−π0 as functions of Kπ mass, combined for Kþπ0 and K−π0. The superimposed curves result from the Dalitz plot fit described in the text.
) 4 /c 2 ) (GeV 0 π + (K 2 m ) 4 /c 2 events/(0.08 GeV (a) ) 4 /c 2 ) (GeV 0 π -(K 2 m ) 4 /c 2 events/(0.08 GeV (b) ) 4 /c 2 ) (GeV -K + (K 2 m 0 2 4 6 0 2 4 6 0 2 4 6 8 ) 4 /c 2 events/(0.1 GeV (c) 0 100 200 300 0 100 200 300 0 50 100 150 200
FIG. 10. Theηc→ KþK−π0Dalitz plot projections, (a)m2ðKþπ0Þ, (b) m2ðK−π0Þ, and (c) m2ðKþK−Þ. The superimposed curves result from the MIPWA 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.
indication of feed through from the Kð892Þ background. The phase motion [Fig.13(b)] shows the expected behavior for the resonance phase, which varies by about π in the K
0ð1430Þ resonance region. The phase shows a drop
around 1.7 GeV=c2 related to interference with the K
0ð1950Þ resonance.
We compare the present measurement of theKπ S-wave amplitude from ηc → K0SKπ∓ with measurements from LASS [5] in Figs. 14(a) and 14(c) and E791 [8] in Figs. 14(b) and 14(d). We plot only the first part of the LASS measurement since it suffers from a twofold ambi-guity above the mass of1.82 GeV=c2. The Dalitz plot fits
extract invariant amplitudes. Consequently, in Fig.14(a), the LASS I ¼ 1=2 Kπ scattering amplitude values have been multiplied by the factor mðKπÞ=q to convert to invariant amplitude, and normalized so as to equal the scattering amplitude at 1.5 GeV=c2 in order to facilitate comparison to theηc results. Here q is the momentum of either meson in theKπ rest frame. For better comparison, the LASS absolute phase measurements have been dis-placed by−0.6 rad before plotting them in Fig. 14(c). In Fig. 14(b) the E791 amplitude has been obtained by multiplying the amplitude c in Table III of Ref. [8] by the form factor F0D, for which the mass-dependence is
2 ) GeV/c π m(K ) 2 Amplitude/(60 MeV/c (a) 2 ) GeV/c π m(K 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 Phase (rad) (b) 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5
FIG. 13. The I ¼ 1=2 KπS-wave amplitude (a) and phase (b) from ηc→ K0SKπ∓ [solid (black) points] and ηc→ KþK−π0 [open (red) points]; only statistical uncertainties are shown. The dotted lines indicate theKη and Kη0 thresholds.
) 2 ) (GeV/c -K + m(K ) 2
weight sum/(30 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 −20 −10 0 0 2 Y ) 2 ) (GeV/c -K + m(K ) 2
weight sum/(30 MeV/c −20 −10 0 10 20 0 3 Y ) 2 ) (GeV/c -K + m(K −10 0 10 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 forηc→ KþK−π0as a function ofKþK−mass. The superimposed curves result from the Dalitz plot fit described in the text.
motivated by theoretical speculation. This yields amplitude values corresponding to the E791 form factor having value 1, as for the ηc analyses. In Fig. 14(d), the E791 phase measurements have been displaced by þ0.9 rad, again in order to facilitate comparison to the ηc measurements.
While we observe similar phase behavior among the three measurements up to about 1.5 GeV=c2, we observe striking differences in the mass dependence of the amplitudes.
IX. SUMMARY
We perform Dalitz plot analyses, using an isobar model and a MIPWA method, of data on the decays ηc→ K0
SKπ∓ and ηc → KþK−π0, where the ηc mesons are
produced in two-photon interactions in the BABAR experi-ment at SLAC. We find that, in comparison with the isobar models examined here, an improved description of the data is obtained by using a MIPWA method.
We extract theI ¼ 1=2 Kπ S-wave amplitude and phase and find good agreement between the measurements for the twoηc decay modes. TheKπ S-wave is dominated by the
presence of theK0ð1430Þ resonance which is observed as a clear peak with the corresponding increase in phase of aboutπ expected for a resonance. A broad structure in the 1.95 GeV=c2 mass region indicates the presence of the
K
0ð1950Þ resonance.
A comparison between the present measurement and previous experiments indicates a similar trend for the phase up to a mass of1.5 GeV=c2. The amplitudes, on the other hand, show very marked differences.
To fit the data we need to introduce a new a0ð1950Þ resonance in bothηc→ K0SKπ∓andηc → KþK−π0decay modes, and their associated parameter values are in good agreement. The weighted averages for the parameter values are:
mða0ð1950ÞÞ ¼ 1931 14 22 MeV=c2;
Γða0ð1950ÞÞ ¼ 271 22 29 MeV ð12Þ
with significances of2.5σ and 4.2σ respectively, including systematic uncertainties. These results are, however, sys-tematically limited, and more detailed studies of theI ¼ 1 2 ) GeV/c π m(K ) 2 Amplitude/(60 MeV/c 0 0.2 0.4 0.6 0.8 1 (a) 2 ) GeV/c π m(K ) 2 Amplitude/(60 MeV/c 0 0.5 1 1.5 2 2.5 3 (b) 2 ) GeV/c π m(K −1 0 1 2 3 4 5 (c) 2 ) GeV/c 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 −10.5 1 1.5 2 2.5 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 (d)
Phase (rad) Phase (rad)
FIG. 14. TheI ¼ 1=2 KπS-wave amplitude measurements from ηc→ K0SKπ∓compared to the (a) LASS and (b) E791 results: the correspondingI ¼ 1=2 KπS-wave phase measurements compared to the (c) LASS and (d) E791 measurements. Black dots indicate the results from the present analysis; square (red) points indicate the LASS or E791 results. The LASS data are plotted in the region having only one solution.
K ¯K S-wave will be required in order to improve the precision of these values.
ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II 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 (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 (USA), and the Binational Science Foundation (USA-Israel). The work of A. Palano and M. R. Pennington was supported (in part) by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Contract No. DE-AC05-06OR23177.
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