Measurement of the I=1/2 Kπ S -wave amplitude from Dalitz plot analyses of η c →K K ¯ π in two-photon interactions
aa r X i v : . [ h e p - e x ] J a n B A B AR -PUB-15/008SLAC-PUB-16422 Measurement of the I = 1 / Kπ S -wave amplitude from Dalitz plot analyses of η c → KKπ in two-photon interactions
J. P. Lees, V. Poireau, and V. Tisserand
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),Universit´e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
A. Palano ∗ INFN Sezione di Bari and Dipartimento di Fisica, Universit`a di Bari, I-70126 Bari, Italy
G. Eigen
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
D. N. Brown and Yu. G. Kolomensky
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
H. Koch and T. Schroeder
Ruhr Universit¨at Bochum, Institut f¨ur Experimentalphysik 1, D-44780 Bochum, Germany
C. Hearty, T. S. Mattison, J. A. McKenna, and R. Y. So
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
V. E. Blinov abc , A. R. Buzykaev a , V. P. Druzhinin ab , V. B. Golubev ab , E. A. Kravchenko ab ,A. P. Onuchin abc , S. I. Serednyakov ab , Yu. I. Skovpen ab , E. P. Solodov ab , and K. Yu. Todyshev ab Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 a ,Novosibirsk State University, Novosibirsk 630090 b ,Novosibirsk State Technical University, Novosibirsk 630092 c , Russia A. J. Lankford
University of California at Irvine, Irvine, California 92697, USA
J. W. Gary and O. Long
University of California at Riverside, Riverside, California 92521, USA
A. M. Eisner, W. S. Lockman, and W. Panduro Vazquez
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
D. S. Chao, C. H. Cheng, B. Echenard, K. T. Flood, D. G. Hitlin, J. Kim,T. S. Miyashita, P. Ongmongkolkul, F. C. Porter, and M. R¨ohrken
California Institute of Technology, Pasadena, California 91125, USA
Z. Huard, B. T. Meadows, B. G. Pushpawela, M. D. Sokoloff, and L. Sun
University of Cincinnati, Cincinnati, Ohio 45221, USA
J. G. Smith and S. R. Wagner
University of Colorado, Boulder, Colorado 80309, USA
D. Bernard and M. Verderi
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
D. Bettoni a , C. Bozzi a , R. Calabrese ab , G. Cibinetto ab , E. Fioravanti ab , I. Garzia ab , E. Luppi ab , and V. Santoro a INFN Sezione di Ferrara a ; Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara b , I-44122 Ferrara, Italy A. Calcaterra, R. de Sangro, G. Finocchiaro, S. Martellotti, P. Patteri, I. M. Peruzzi, M. Piccolo, and A. Zallo
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
S. Passaggio and C. Patrignani † INFN Sezione di Genova, I-16146 Genova, Italy
B. Bhuyan
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
U. Mallik
University of Iowa, Iowa City, Iowa 52242, USA
C. Chen, J. Cochran, and S. Prell
Iowa State University, Ames, Iowa 50011, USA
H. Ahmed
Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia
M. R. Pennington
Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
A. V. Gritsan
Johns Hopkins University, Baltimore, Maryland 21218, USA
N. Arnaud, M. Davier, F. Le Diberder, A. M. Lutz, and G. Wormser
Laboratoire de l’Acc´el´erateur Lin´eaire, IN2P3/CNRS et Universit´e Paris-Sud 11,Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France
D. J. Lange and D. M. Wright
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
J. P. Coleman, E. Gabathuler, D. E. Hutchcroft, D. J. Payne, and C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom
A. J. Bevan, F. Di Lodovico, and R. Sacco
Queen Mary, University of London, London, E1 4NS, United Kingdom
G. Cowan
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
Sw. Banerjee, D. N. Brown, and C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
A. G. Denig, M. Fritsch, W. Gradl, K. Griessinger, A. Hafner, and K. R. Schubert
Johannes Gutenberg-Universit¨at Mainz, Institut f¨ur Kernphysik, D-55099 Mainz, Germany
R. J. Barlow ‡ and G. D. Lafferty University of Manchester, Manchester M13 9PL, United Kingdom
R. Cenci, A. Jawahery, and D. A. Roberts
University of Maryland, College Park, Maryland 20742, USA
R. Cowan
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
R. Cheaib and S. H. Robertson
McGill University, Montr´eal, Qu´ebec, Canada H3A 2T8
B. Dey a , N. Neri a , and F. Palombo ab INFN Sezione di Milano a ; Dipartimento di Fisica, Universit`a di Milano b , I-20133 Milano, Italy L. Cremaldi, R. Godang, § and D. J. Summers University of Mississippi, University, Mississippi 38677, USA
P. Taras
Universit´e de Montr´eal, Physique des Particules, Montr´eal, Qu´ebec, Canada H3C 3J7
G. De Nardo and C. Sciacca
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche,Universit`a di Napoli Federico II, I-80126 Napoli, Italy
G. Raven
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop and J. M. LoSecco
University of Notre Dame, Notre Dame, Indiana 46556, USA
K. Honscheid and R. Kass
Ohio State University, Columbus, Ohio 43210, USA
A. Gaz a , M. Margoni ab , M. Posocco a , M. Rotondo a , G. Simi ab , F. Simonetto ab , and R. Stroili ab INFN Sezione di Padova a ; Dipartimento di Fisica, Universit`a di Padova b , I-35131 Padova, Italy S. Akar, E. Ben-Haim, M. Bomben, G. R. Bonneaud, G. Calderini, J. Chauveau, G. Marchiori, and J. Ocariz
Laboratoire de Physique Nucl´eaire et de Hautes Energies,IN2P3/CNRS, Universit´e Pierre et Marie Curie-Paris6,Universit´e Denis Diderot-Paris7, F-75252 Paris, France
M. Biasini ab , E. Manoni a , and A. Rossi a INFN Sezione di Perugia a ; Dipartimento di Fisica, Universit`a di Perugia b , I-06123 Perugia, Italy G. Batignani ab , S. Bettarini ab , M. Carpinelli ab , ¶ G. Casarosa ab , M. Chrzaszcz a , F. Forti ab ,M. A. Giorgi ab , A. Lusiani ac , B. Oberhof ab , E. Paoloni ab , M. Rama a , G. Rizzo ab , and J. J. Walsh a INFN Sezione di Pisa a ; Dipartimento di Fisica, Universit`a di Pisa b ; Scuola Normale Superiore di Pisa c , I-56127 Pisa, Italy A. J. S. Smith
Princeton University, Princeton, New Jersey 08544, USA
F. Anulli a , R. Faccini ab , F. Ferrarotto a , F. Ferroni ab , A. Pilloni ab , and G. Piredda a INFN Sezione di Roma a ; Dipartimento di Fisica,Universit`a di Roma La Sapienza b , I-00185 Roma, Italy C. B¨unger, S. Dittrich, O. Gr¨unberg, M. Heß, T. Leddig, C. Voß, and R. Waldi
Universit¨at Rostock, D-18051 Rostock, Germany
T. Adye and F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
S. Emery, G. Hamel de Monchenault, and G. Vasseur
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
D. Aston, C. Cartaro, M. R. Convery, J. Dorfan, W. Dunwoodie, M. Ebert, R. C. Field, B. G. Fulsom,M. T. Graham, C. Hast, W. R. Innes, P. Kim, D. W. G. S. Leith, S. Luitz, V. Luth, D. B. MacFarlane,D. R. Muller, H. Neal, B. N. Ratcliff, A. Roodman, M. K. Sullivan, J. Va’vra, and W. J. Wisniewski
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
M. V. Purohit and J. R. Wilson
University of South Carolina, Columbia, South Carolina 29208, USA
A. Randle-Conde and S. J. Sekula
Southern Methodist University, Dallas, Texas 75275, USA
M. Bellis, P. R. Burchat, and E. M. T. Puccio
Stanford University, Stanford, California 94305, USA
M. S. Alam and J. A. Ernst
State University of New York, Albany, New York 12222, USA
R. Gorodeisky, N. Guttman, D. R. Peimer, and A. Soffer
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
J. L. Ritchie and R. F. Schwitters
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen and X. C. Lou
University of Texas at Dallas, Richardson, Texas 75083, USA
F. Bianchi ab , F. De Mori ab , A. Filippi a , and D. Gamba ab INFN Sezione di Torino a ; Dipartimento di Fisica, Universit`a di Torino b , I-10125 Torino, Italy L. Lanceri and L. Vitale
INFN Sezione di Trieste and Dipartimento di Fisica, Universit`a di Trieste, I-34127 Trieste, Italy
F. Martinez-Vidal and A. Oyanguren
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
J. Albert, A. Beaulieu, F. U. Bernlochner, G. J. King, R. Kowalewski,T. Lueck, I. M. Nugent, J. M. Roney, and N. Tasneem
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
T. J. Gershon, P. F. Harrison, and T. E. Latham
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
R. Prepost and S. L. Wu
University of Wisconsin, Madison, Wisconsin 53706, USA
We study the processes γγ → K S K ± π ∓ and γγ → K + K − π using a data sample of 519 fb − recorded with the B A B AR detector operating at the SLAC PEP-II asymmetric-energy e + e − colliderat center-of-mass energies at and near the Υ ( nS ) ( n = 2 , ,
4) resonances. We observe η c decays toboth final states and perform Dalitz plot analyses using a model-independent partial wave analysistechnique. This allows a model-independent measurement of the mass-dependence of the I = 1 / Kπ S -wave amplitude and phase. A comparison between the present measurement and those fromprevious experiments indicates similar behaviour for the phase up to a mass of 1.5 GeV /c . Incontrast, the amplitudes show very marked differences. The data require the presence of a new a (1950) resonance with parameters m = 1931 ± ±
22 MeV /c and Γ = 271 ± ±
29 MeV.
PACS numbers: 13.25.Gv, 14.40.Pq, 14.40.Df, 14.40.Be ∗ Also at: Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
I. INTRODUCTION
Scalar mesons are still a puzzle in light meson spec-troscopy: they have complex structure, and there aretoo many states to be accommodated within the quarkmodel without difficulty [1]. In particular, the structureof the I = 1 / Kπ S -wave is a longstanding problem. Inrecent years many experiments have performed accuratestudies of the decays of heavy-flavored hadrons produc-ing a Kπ system in the final state. These studies in-clude searches for CP violation [2], and searches for, andobservation of, new exotic resonances [3] and charmedmesons [4]. However, the still poorly known structure ofthe I = 1 / Kπ S -wave is a source of large systematic un-certainties. The best source of information on the scalarstructure of the Kπ system comes from the LASS ex-periment, which studied the reaction K − p → K − π + n [5].Partial wave analysis of the Kπ system reveals a largecontribution from the I = 1 / Kπ S -wave amplitudeover the mass range studied. In the description of the I = 1 / Kπ mass of about 1.5GeV /c the K ∗ (1430) resonant amplitude is added co-herently to an effective-range description in such a waythat the net amplitude actually decreases rapidly at theresonance mass. The I = 1 / S -wave amplitude rep-resentation is given explicitly in Ref. [6]. In the LASSanalysis, in the region above 1.82 GeV /c , the S -wavesuffers from a two-fold ambiguity, but in both solutionsit is understood in terms of the presence of a K ∗ (1950)resonance. It should be noted that the extraction of the I = 1 / S -wave amplitude is complicated by the presenceof an I = 3 / Kπ system has beenextracted from Dalitz plot analysis of the decay D + → K − π + π + where, in order to fit the data, thepresence of an additional resonance, the κ (800), wasclaimed [7]. Using the same data, a Model IndependentPartial Wave Analysis (MIPWA) of the Kπ system wasdeveloped for the first time [8]. This method allows theamplitude and phase of the Kπ S -wave to be extractedas functions of mass (see also Refs. [9] and [10]). How-ever in these analyses the phase space is limited to massvalues less than 1.6 GeV /c due to the kinematical limitimposed by the D + mass. A similar method has beenused to extract the π + π − S -wave amplitude in a Dalitzplot analysis of D + s → π + π − π + [11].In the present analysis, we consider three-body η c de-cays to KKπ and obtain new information on the Kπ † Now at: Universit`a di Bologna and INFN Sezione di Bologna,I-47921 Rimini, Italy ‡ Now at: University of Huddersfield, Huddersfield HD1 3DH, UK § Now at: University of South Alabama, Mobile, Alabama 36688,USA ¶ Also at: Universit`a di Sassari, I-07100 Sassari, Italy I = 1 / S -wave amplitude extending up to a mass of 2.5GeV /c . We emphasize that, due to isospin conservationin the η c hadronic decay to ( Kπ ) K , the ( Kπ ) amplitudemust have I = 1 / I = 3 / B A B AR experiment first performed a Dalitzplot analysis of η c → K + K − π and η c → K + K − η using anisobar model [12]. The analysis reported the first obser-vation of K ∗ (1430) → Kη , and observed that η c decays tothree pseudoscalars are dominated by intermediate scalarmesons. A previous search for charmonium resonancesdecaying to K S K ± π ∓ in two-photon interactions is re-ported in Ref. [13]. We continue these studies of η c decaysand extract the Kπ S -wave amplitude by performing aMIPWA of both η c → K S K ± π ∓ and η c → K + K − π finalstates.We describe herein studies of the KKπ system pro-duced in two-photon interactions. Two-photon eventsin which at least one of the interacting photons is notquasi-real are strongly suppressed by the selection crite-ria described below. This implies that the allowed J P C values of any produced resonances are 0 ± + , 2 ± + , 3 ++ ,4 ± + ... [14]. Angular momentum conservation, parity con-servation, and charge conjugation invariance imply thatthese quantum numbers also apply to the final state ex-cept that the KKπ state cannot be in a J P = 0 + state.This article is organized as follows. In Sec. II, a briefdescription of the B A B AR detector is given. Section IIIis devoted to the event reconstruction and data selectionof the K S K ± π ∓ system. In Sec. IV, we describe studiesof efficiency and resolution, while in Sec. V we describethe MIPWA. In Secs. VI and VII we perform Dalitzplot analyses of η c → K S K ± π ∓ and η c → K + K − π decays.Section VIII is devoted to discussion of the measured Kπ S -wave amplitude, and finally results are summarized inSec. IX. II. THE B A B AR DETECTOR AND DATASET
The results presented here are based on data collectedwith the B A B AR detector at the PEP-II asymmetric-energy e + e − collider located at SLAC, and correspondto an integrated luminosity of 519 fb − [15] recordedat center-of-mass energies at and near the Υ ( nS ) ( n =2 , ,
4) resonances. The B A B AR detector is described indetail elsewhere [16]. Charged particles are detected,and their momenta are measured, by means of a five-layer, double-sided microstrip detector, and a 40-layerdrift chamber, both operating in the 1.5 T magnetic fieldof a superconducting solenoid. Photons are measuredand electrons are identified in a CsI(Tl) crystal elec-tromagnetic calorimeter. Charged-particle identificationis provided by the measurement of specific energy lossin the tracking devices, and by an internally reflecting,ring-imaging Cherenkov detector. Muons and K L mesonsare detected in the instrumented flux return of the mag-net. Monte Carlo (MC) simulated events [17], with re-constructed sample sizes more than 10 times larger thanthe corresponding data samples, are used to evaluate thesignal efficiency and to determine background features.Two-photon events are simulated using the GamGamMC generator [18]. III. RECONSTRUCTION AND SELECTION OF η c → K S K ± π ∓ EVENTS
To study the reaction γγ → K S K ± π ∓ (1)we select events in which the e + and e − beam particlesare scattered at small angles, and hence are undetectedin the final state. We consider only events for whichthe number of well-measured charged-particle tracks withtransverse momentum greater than 0.1 GeV /c is exactlyequal to four, and for which there are no more than fivephoton candidates with reconstructed energy in the elec-tromagnetic calorimeter greater than 100 MeV. We ob-tain K S → π + π − candidates by means of a vertex fit ofpairs of oppositely charged tracks which requires a χ fit probability greater than 0.001. Each K S candidate isthen combined with two oppositely charged tracks, andfitted to a common vertex, with the requirements thatthe fitted vertex be within the e + e − interaction regionand have a χ fit probability greater than 0.001. We se-lect kaons and pions by applying high-efficiency particleidentification criteria. We do not apply any particle iden-tification requirements to the pions from the K S decay.We accept only K S candidates with decay lengths fromthe main vertex of the event greater than 0.2 cm, andrequire cos θ K S > .
98, where θ K S is defined as the anglebetween the K S momentum direction and the line join-ing the primary and the K S vertex. A fit to the π + π − mass spectrum using a linear function for the backgroundand a Gaussian function with mean m and width σ gives m = 497 .
24 MeV /c and σ = 2 . /c . We select the K S signal region to be within ± σ of m and reconstructthe K S K S PDG massvalue [19].Background arises mainly from random combinationsof particles from e + e − annihilation, from other two-photon processes, and from events with initial-state pho-ton radiation (ISR). The ISR background is dominatedby J P C = 1 −− resonance production [20]. We discrim-inate against K S K ± π ∓ events produced via ISR by re-quiring M ≡ ( p e + e − − p rec ) >
10 GeV / c , where p e + e − is the four-momentum of the initial state and p rec is the four-momentum of the K S K ± π ∓ system.The K S K ± π ∓ mass spectrum shows a prominent η c signal. We define p T as the magnitude of the vector sumof the transverse momenta, in the e + e − rest frame, of thefinal-state particles with respect to the beam axis. Sincewell-reconstructed two-photon events are expected tohave low values of p T , we optimize the selection as a func-tion of this variable. We produce K S K ± π ∓ mass spectra (GeV/c) T p0 0.1 0.2 0.3 0.4 0.5 e v en t s / ( M e V / c ) FIG. 1: Distributions of p T for γγ → K S K ± π ∓ . The dataare shown as (black) points with error bars, and the signalMC simulation as a (red) histogram; the vertical dashed lineindicates the selection applied to select two-photon events. with different p T selections and fit the mass spectra toextract the number of η c signal events ( N s ) and the num-ber of background events below the η c signal ( N b ). Wethen compute the purity, defined as P = N s / ( N s + N b ),and the significance S = N s / √ N s + N b . To obtain thebest significance with the highest purity, we optimize theselection by requiring the maximum value of the prod-uct of purity and significance, P · S , and find that thiscorresponds to the requirement p T < .
08 GeV /c .Figure 1 shows the measured p T distribution in com-parison to the corresponding p T distribution obtainedfrom simulation of the signal process. A peak at low p T isobserved indicating the presence of the two-photon pro-cess. The shape of the peak agrees well with that seenin the MC simulation. Figure 2 shows the K S K ± π ∓ mass spectrum in the η c mass region. A clear η c sig-nal over a background of about 35% can be seen, to-gether with a residual J/ψ signal. Information on thefitting procedure is given at the end of Sec. IV. We de-fine the η c signal region as the range 2.922-3.039 GeV /c ( m ( η c ) ± . . ± . /c and 3.117-3.175 GeV /c (within 3.5-5 Γ), respectively as indicated (shaded) inFig. 2.Details on data selection, event reconstruction, resolu-tion, and efficiency measurement for the η c → K + K − π decay can be found in Ref. [12]. The η c signal region forthis decay mode contains 6710 events with a purity of(55 . ± . ) ) (GeV/c ± π ± K S0 m(K2.8 3 3.2 ) e v en t s / ( M e V / c FIG. 2: The K S K ± π ∓ mass spectrum in the η c mass regionafter requiring p T < .
08 GeV /c . The solid curve shows thetotal fitted function, and the dashed curve shows the fittedbackground contribution. The shaded areas show signal andsideband regions. IV. EFFICIENCY AND RESOLUTION
To compute the efficiency, MC signal events are gen-erated using a detailed detector simulation [17] in whichthe η c decays uniformly in phase space. These simulatedevents are reconstructed and analyzed in the same man-ner as data. The efficiency is computed as the ratio ofreconstructed to generated events. Due to the presence oflong tails in the Breit-Wigner (BW) representation of theresonance, we apply selection criteria to restrict the gen-erated events to the η c mass region. We express the effi-ciency as a function of the invariant mass, m ( K + π − ) [21],and cos θ , where θ is the angle, in the K + π − rest frame,between the directions of the K + and the boost from the K S K + π − rest frame.To smooth statistical fluctuations, this efficiency mapis parametrized as follows. First we fit the efficiency as afunction of cos θ in separate intervals of m ( K + π − ), usingLegendre polynomials up to L = 12: ǫ (cos θ ) = X L =0 a L ( m ) Y L (cos θ ) , (2)where m denotes the K + π − invariant mass. For eachvalue of L , we fit the mass dependent coefficients a L ( m )with a seventh-order polynomial in m . Figure 3 showsthe resulting fitted efficiency map ǫ ( m, cos θ ). We ob-tain χ /N cells = 217 /
300 for this fit, where N cells is the ) GeV/c - π + m(K1 1.5 2 θ c o s − − FIG. 3: Fitted detection efficiency in the cos θ vs. m ( K + π − )plane. Each interval shows the average value of the fit forthat region. number of cells in the efficiency map. We observe a signif-icant decrease in efficiency in regions of cos θ ∼ ± K ± mesons withlaboratory momentum less than about 200 MeV /c , and π ± and K S ( → π + π − ) mesons with laboratory momentumless than about 100 MeV /c (see Fig. 9 of Ref. [6]). Theseeffects result from energy loss in the beampipe and inner-detector material.The mass resolution, ∆ m , is measured as the differ-ence between the generated and reconstructed K S K ± π ∓ invariant-mass values. The distribution has a root-mean-squared value of 10 MeV /c , and is parameterized bythe sum of a Crystal Ball [22] and a Gaussian function.We perform a binned fit to the K S K ± π ∓ mass spectrumin data using the following model. The background isdescribed by a second-order polynomial, and the η c res-onance is represented by a nonrelativistic BW functionconvolved with the resolution function. In addition, weallow for the presence of a residual J/ψ contribution mod-eled with a Gaussian function. Its parameter values arefixed to those obtained from a fit to the K S K ± π ∓ massspectrum for the ISR data sample obtained by requiring | M | < /c . The fitted K S K ± π ∓ mass spectrumis shown in Fig. 2. We obtain the following η c parame-ters: m = 2980 . ± . /c , Γ = 33 ± ,N η c = 9808 ± , (3)where uncertainties are statistical only. Our measuredmass value is 2.8 MeV /c lower than the world aver-age [19]. This may be due to interference between the η c amplitude and that describing the background in thesignal region [23]. V. MODEL INDEPENDENT PARTIAL WAVEANALYSIS
We perform independent MIPWA of the K S K ± π ∓ and K + K − π Dalitz plots in the η c mass region using un-binned maximum likelihood fits. The likelihood functionis written as L = N Y n =1 (cid:20) f sig ( m n ) ǫ ( x ′ n , y ′ n ) P i,j c i c ∗ j A i ( x n , y n ) A ∗ j ( x n , y n ) P i,j c i c ∗ j I A i A ∗ j +(1 − f sig ( m n )) P i k i B i ( x n , y n , m n ) P i k i I B i (cid:21) (4)where • N is the number of events in the signal region; • for the n -th event, m n is the K S K ± π ∓ or the K + K − π invariant mass; • for the n -th event, x n = m ( K + π − ), y n = m ( K S π − ) for K S K ± π ∓ ; x n = m ( K + π ), y n = m ( K − π ) for K + K − π ; • f sig is the mass-dependent fraction of signal ob-tained from the fit to the K S K ± π ∓ or K + K − π mass spectrum; • for the n -th event, ǫ ( x ′ n , y ′ n ) is the efficiencyparametrized as a function of x ′ n = m ( K + π − ) for K S K ± π ∓ and x ′ n = m ( K + K − ) for K + K − π , and y ′ n = cos θ (see Sec. IV); • for the n -th event, the A i ( x n , y n ) describe the com-plex signal-amplitude contributions; • c i is the complex amplitude for the i -th signal com-ponent; the c i parameters are allowed to vary dur-ing the fit process; • for the n -th event, the B i ( x n , y n ) describe the back-ground probability-density functions assuming thatinterference between signal and background ampli-tudes can be ignored; • k i is the magnitude of the i -th background compo-nent; the k i parameters are obtained by fitting thesideband regions; • I A i A ∗ j = R A i ( x, y ) A ∗ j ( x, y ) ǫ ( x ′ , y ′ ) d x d y and I B i = R B i ( x, y )d x d y are normalization integrals.Numerical integration is performed on phase spacegenerated events with η c signal and backgroundgenerated according to the experimental distribu-tions. In case of MIPWA or when resonances havefree parameters, integrals are re-computed at each minimization step. Background integrals and fitsdealing with amplitudes having fixed resonance pa-rameters are computed only once.Amplitudes are described along the lines described inRef. [24]. For an η c meson decaying into three pseu-doscalar mesons via an intermediate resonance r of spin J (i.e. η c → Cr , r → AB ), each amplitude A i ( x, y ) is rep-resented by the product of a complex Breit-Wigner (BW)function and a real angular distribution function repre-sented by the spherical harmonic function √ πY J (cos θ ); θ is the angle between the direction of A , in the rest frameof r , and the direction of C in the same frame. This formof the angular dependence results from angular momen-tum conservation in the rest frame of the η c , which leadsto the production of r with helicity 0.It follows that A i ( x, y ) = BW ( M AB ) √ πY J (cos θ ) . (5)The function BW ( M AB ) is a relativistic BW functionof the form BW ( M AB ) = F η c FM r − M AB − iM r Γ tot ( M AB ) (6)where M r is the mass of the resonance r , and Γ tot ( M AB )is its mass-dependent total width. In general, this massdependence cannot be specified, and a constant valueshould be used. However, for a resonance such as the K ∗ (1430), which is approximately elastic, we can use thepartial width Γ AB , and specify the mass-dependence as:Γ AB = Γ r (cid:18) p AB p r (cid:19) J +1 (cid:18) M r M AB (cid:19) F (7)where p AB = q ( M AB − M A − M B ) − M A M B M AB . (8)and p r is the value of p AB when M AB = M r .The form factors F η c and F attempt to model theunderlying quark structure of the parent particle andthe intermediate resonances. We set F η c to a constantvalue, while for F we use Blatt-Weisskopf penetrationfactors [25] (Table I), that depend on a single parameter R representing the meson “radius”, for which we assume R = 1 . − . The a (980) resonance is parameter-ized as a coupled-channel Breit-Wigner function whoseparameters are taken from Ref. [26].To measure the I = 1 / Kπ S -wave we make use ofa MIPWA technique first described in Ref. [8]. The Kπ S -wave, being the largest contribution, is taken as thereference amplitude. We divide the Kπ mass spectruminto 30 equally-spaced mass intervals 60 MeV wide, andfor each interval we add to the fit two new free parame-ters, the amplitude and the phase of the Kπ S -wave inthat interval. We fix the amplitude to 1.0 and its phaseto π/ TABLE I: Summary of the Blatt-Weisskopf penetration formfactors. Spin F √ R r p r ) √ R r p AB ) √ R r p r ) +( R r p r ) √ R r p AB ) +( R r p AB ) which we choose interval 14, corresponding to a mass of1.45 GeV /c . The number of associated free parametersis therefore 58.Due to isospin conservation in the hadronic η c and K ∗ decays, the ( Kπ ) K amplitudes are combined with posi-tive signs, and so therefore are symmetrized with respectto the two K ∗ K modes. In particular we write the Kπ S -wave amplitudes as A S -wave = 1 √ a K + π − j e iφ K + π − j + a K π − j e iφ K π − j ) , (9)where a K + π − ( m ) = a K π − ( m ) and φ K + π − ( m ) = φ K π − ( m ), for η c → K K + π − [21] and A S -wave = 1 √ a K + π j e iφ K + π j + a K − π j e iφ K − π j ) , (10)where a K + π ( m ) = a K − π ( m ) and φ K + π ( m ) = φ K − π ( m ), for η c → K + K − π . For both decay modesthe bachelor kaon is in an orbital S -wave with respectto the relevant Kπ system, and so does not affect theseamplitudes. The second amplitude in Eq.(9) is reducedbecause the K is observed as a K S , but the same re-duction factor applies to the first amplitude through thebachelor K , so that the equality of the three-body am-plitudes is preserved.Other resonance contributions are described as above.The K ∗ (1430) K contribution is symmetrized in the sameway as the S -wave amplitude.We perform MC simulations to test the ability ofthe method to find the correct solution. We generate η c → K S K ± π ∓ event samples which yield reconstructedsamples having the same size as the data sample, ac-cording to arbitrary mixtures of resonances, and extractthe Kπ S -wave using the MIPWA method. We find thatthe fit is able to extract correctly the mass dependenceof the amplitude and phase.We also test the possibility of multiple solutions bystarting the fit from random values or constant parametervalues very far from the solution found by the fit. We find only one solution in both final states and conclude thatthe fit converges to give the correct S -wave behaviour fordifferent starting values of the parameters.The efficiency-corrected fractional contribution f i dueto resonant or non-resonant contribution i is defined asfollows: f i = | c i | R | A i ( x n , y n ) | d x d y R | P j c j A j ( x, y ) | d x d y . (11)The f i do not necessarily sum to 100% because of inter-ference effects. The uncertainty for each f i is evaluatedby propagating the full covariance matrix obtained fromthe fit.We test the quality of the fit by examining a large sam-ple of MC events at the generator level weighted by thelikelihood fitting function and by the efficiency. Theseevents are used to compare the fit result to the Dalitzplot and its projections with proper normalization. Inthese MC simulations we smooth the fitted Kπ S -waveamplitude and phase by means of a cubic spline. Wemake use of these weighted events to compute a 2-D χ over the Dalitz plot. For this purpose, we dividethe Dalitz plot into a grid of 25 ×
25 cells and consideronly those containing at least five events. We compute χ = P N cells i =1 ( N i obs − N i exp ) /N i exp , where N i obs and N i exp are event yields from data and simulation, respectively. VI. DALITZ PLOT ANALYSIS OF η c → K S K ± π ∓ Figure 4 shows the Dalitz plot for the candidates inthe η c signal region, and Fig. 5 shows the correspondingDalitz plot projections. Since the width of the η c mesonis 32 . ± . K ∗ (1430) resonance. We also observe further bandsalong the diagonal. Isospin conservation in η c decay re-quires that the ( KK ) system have I=1, so that thesestructures may indicate the presence of a or a reso-nances. Further narrow bands are observed at the posi-tion of the K ∗ (892) resonance, mostly in the K S π − pro-jection; these components are consistent with originatingfrom background, as will be shown.The presence of background in the η c signal region re-quires precise study of its structure. This can be achievedby means of the data in the η c sideband regions, for whichthe Dalitz plots are shown in Fig. 6.In both regions we observe almost uniformly popu-lated resonant structures mostly in the K S π − mass, espe-cially in the regions corresponding to the K ∗ (892)‘¡ and K ∗ (1430) resonances. The resonant structures in K + π − mass are weaker. The three-body decay of a pseudoscalarmeson into a spin-one or spin-two resonance yields a non-uniform distribution (see Eq. 5) in the relevant resonanceband on the Dalitz plot. The presence of uniformly pop-ulated bands in the K ∗ (892) and K ∗ (1430) mass regions,0 ) /c ) (GeV ± π ± (K m0 2 4 6 ) / c ) ( G e V ± π S ( K m FIG. 4: Dalitz plot for η c → K S K ± π ∓ events in the signal region. ) /c ) (GeV ± π ± (K m0 2 4 6 ) / c e v en t s / ( . G e V (a) ) /c ) (GeV ± π S0 (K m0 2 4 6 ) / c e v en t s / ( . G e V (b) ) /c ) (GeV ± K S0 (K m0 2 4 6 8 ) / c e v en t s / ( . G e V (c) FIG. 5: The η c → K S K ± π ∓ Dalitz plot projections on (a) m ( K ± π ∓ ), (b) m ( K S π ± ), and (c) m ( K S K ± ). The superim-posed curves result from the MIPWA described in the text. The shaded regions show the background estimates obtained byinterpolating the results of the Dalitz plot analyses of the sideband regions. indicates that these structures are associated with back-ground. Also, the asymmetry between the two K ∗ modesin background may be explained as being due to interfer-ence between the I = 0 and I = 1 isospin configurationsfor the K ∗ ( → Kπ ) K final state produced in two-photonfusion.We fit the η c sidebands using an incoherent sum of am- plitudes, which includes contributions from the a (980), a (1450), a (1320), K ∗ (892), K ∗ (1430), K ∗ (1430), K ∗ (1680), and K ∗ (1950) resonances. To better constrainthe sum of the fractions to one, we make use of the chan-nel likelihood method [27] and include resonances untilno structure is left in the background and an accuratedescription of the Dalitz plots is obtained.1 ) /c ) (GeV ± π ± (K m0 2 4 6 8 ) / c ) ( G e V ± π S ( K m (a) ) /c ) (GeV ± π ± (K m0 2 4 6 8 ) / c ) ( G e V ± π S ( K m (b) FIG. 6: Dalitz plots for the η c → K S K ± π ∓ sideband regions: (a) lower, (b) upper. To estimate the background composition in the η c sig-nal region we perform a linear mass dependent interpo-lation of the fractions of the different contributions, ob-tained from the fits to the sidebands, and normalizedusing the results from the fit to the K S K ± π ∓ mass spec-trum. The estimated background contributions are indi-cated by the shaded regions in Fig. 5. A. MIPWA of η c → K S K ± π ∓ We perform the MIPWA including the resonanceslisted in Table II. In this table, and in the remainderof the paper, we use the notation ( Kπ ) K or K ∗ K torepresent the corresponding symmetrized amplitude. Af-ter the solution is found we test for other contributions,including spin-one resonances, but these are found to beconsistent with zero, and so are not included. This sup-ports the observation that the observed K ∗ (892) struc-tures originate entirely from background. We find a dom-inance of the Kπ S -wave amplitude, with small contribu-tions from a π amplitudes and a significant K ∗ (1430) K contribution.The table lists also a significant contribution from the a (1950) π amplitude, where a (1950) + → K S K + is a newresonance. We also test the spin-2 hypothesis for thiscontribution by replacing the amplitude for a → K S K + with an a → K S K + amplitude with parameter values leftfree in the fit. In this case no physical solution is foundinside the allowed ranges of the parameters, and the ad-ditional contribution is found consistent with zero. Thisnew state has isospin one, and the spin-0 assignment ispreferred over that of spin-2.A fit without this state gives a poor description of thehigh mass K S K + projection, as can be seen in Fig. 7(a).We obtain − L = − χ /N cells = 1 .
33 for this fit. We then include in the MIPWA a new scalarresonance decaying to K S K + with free parameters. Weobtain ∆(log L ) = 61 and ∆ χ = 38 for an increase offour new parameters. We estimate the significance for the a (1950) resonance using the fitted fraction divided byits statistical and systematic errors added in quadrature,and obtain 2 . σ . Since interference effects may also con-tribute to the significance, this procedure gives a conser-vative estimate. The systematic uncertainties associatedwith the a (1950) state are described below. The fittedparameter values for this state are given in Table III. Wenote that we obtain χ /N cells = 1 .
17 for this final fit, in-dicating good description of the data. The fit projectionson the three squared masses from the MIPWA are shownin Fig. 5, and they indicate that the description of thedata is quite good.We compute the uncorrected Legendre polynomial mo-ments h Y L i in each K + π − , K S π − and K S K + mass in-terval by weighting each event by the relevant Y L (cos θ )function. These distributions are shown in Fig. 8 as func-tions of Kπ mass after combining K + π − and K S π − , andin Fig. 9 as functions of K S K + mass. We also com-pute the expected Legendre polynomial moments fromthe weighted MC events and compare with the exper-imental distributions. We observe good agreement forall the distributions, which indicates that the fit is ableto reproduce the local structures apparent in the Dalitzplot.We compute the following systematic uncertainties onthe I = 1 / Kπ S -wave amplitude and phase. The dif-ferent contributions are added in quadrature. • Starting from the solution found by the fit, we gen-erate MC simulated events which are fitted usinga MIPWA. In this way we estimate the bias intro-duced by the fitting method.2
TABLE II: Results from the η c → K S K ± π ∓ and η c → K + K − π MIPWA. Phases are determined relative to the ( Kπ S -wave) K amplitude which is fixed to π/ /c . η c → K S K ± π ∓ η c → K + K − π Amplitude Fraction (%) Phase (rad) Fraction (%) Phase (rad)( Kπ S -wave) K ± ± ± ± a (980) π ± ± ± ± ± ± a (1450) π ± ± ± ± ± ± ± ± a (1950) π ± ± − ± ± ± ± − ± ± a (1320) π ± ± ± ± ± ± ± ± K ∗ (1430) K ± ± ± ± ± ± ± ± ± ± ± ± − L − − χ /N cells ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 ) e v en t s / ( M e V / c (a) ) ) (GeV/c - K + m(K1 1.5 2 2.5 ) e v en t s / ( M e V / c (b) FIG. 7: The mass projections (a) K S K ± from η c → K S K ± π ∓ and (b) K + K − from η c → K + K − π . The histograms show theMIPWA fit projections with (solid, black) and without (dashed, red) the presence of the a (1950) + → K S K ± resonance. Theshaded regions show the background estimates obtained by interpolating the results of the Dalitz plot analyses of the sidebandregions.TABLE III: Fitted a (1950) parameter values for the two η c decay modes.Final state Mass (MeV /c ) Width (MeV) η c → K S K ± π ∓ ± ±
76 265 ± ± η c → K + K − π ± ±
23 274 ± ± ± ±
22 271 ± ± • The fit is performed by interpolating the Kπ S -wave amplitude and phase using a cubic spline. • We remove low-significance contributions, such asthose from the a (980) and a (1320) resonances. • We vary the signal purity up and down according to its statistical uncertainty. • The effect of the efficiency variation as a functionof
KKπ mass is evaluated by computing separateefficiencies in the regions below and above the η c mass.These additional fits also allow the computation ofsystematic uncertainties on the amplitude fraction andphase values, as well as on the parameter values for the a (1950) resonance; these are summarized in Table IV.In the evaluation of overall systematic uncertainties, alleffects are assumed to be uncorrelated, and are added inquadrature.The measured amplitude and phase values of the I =1 / Kπ S -wave as functions of mass obtained from theMIPWA of η c → K S K ± π ∓ are shown in Table V. Interval3 ) ) (GeV/c ± π ± m(K0.5 1 1.5 2 2.5 ) w e i gh t s u m / ( M e V / c Y ) ) (GeV/c ± π ± m(K0.5 1 1.5 2 2.550 − Y ) ) (GeV/c ± π ± m(K0.5 1 1.5 2 2.550 − Y ) ) (GeV/c ± π ± m(K0.5 1 1.5 2 2.5 ) w e i gh t s u m / ( M e V / c Y ) ) (GeV/c ± π ± m(K0.5 1 1.5 2 2.520 − Y ) ) (GeV/c ± π ± m(K0.5 1 1.5 2 2.540 − − Y FIG. 8: Legendre polynomial moments for η c → K S K ± π ∓ as functions of Kπ mass, and combined for K ± π ∓ and K S π ∓ ; thesuperimposed curves result from the Dalitz plot fit described in the text. ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 3 ) w e i gh t s u m / ( M e V / c Y ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 320 − − Y ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 340 − − Y ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 3 ) w e i gh t s u m / ( M e V / c − − Y ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 320 − Y ) ) (GeV/c ± K S0 m(K1 1.5 2 2.5 320 − − Y FIG. 9: Legendre polynomial moments for η c → K S K ± π ∓ as a function of K S K ± mass, the superimposed curves result fromthe Dalitz plot fit described in the text.
414 of the Kπ mass contains the fixed amplitude and phasevalues. B. Dalitz plot analysis of η c → K S K ± π ∓ using anisobar model We perform a Dalitz plot analysis of η c → K S K ± π ∓ us-ing a standard isobar model, where all resonances aremodeled as BW functions multiplied by the correspond-ing angular functions. In this case the Kπ S -wave isrepresented by a superposition of interfering K ∗ (1430), K ∗ (1950), non-resonant (NR), and possibly κ (800) con-tributions. The NR contribution is parametrized as anamplitude that is constant in magnitude and phase overthe Dalitz plot. In this fit the K ∗ (1430) parameters aretaken from Ref. [12], while all other parameters are fixedto PDG values. We also add the a (1950) resonance withparameters obtained from the MIPWA analysis.For the description of the η c signal, amplitudes areadded one by one to ascertain the associated increase ofthe likelihood value and decrease of the 2-D χ . Table VIsummarizes the fit results for the amplitude fractions andphases. The high value of χ /N cells = 1 .
82 (to be com-pared with χ /N cells = 1 .
17) indicates a poorer descrip-tion of the data than that obtained with the MIPWAmethod. Including the κ (800) resonance does not im-prove the fit quality. If included, it gives a fit fraction of(0 . ± . K ∗ (892) contribution is consistentwith originating entirely from background. Other spin-1 K ∗ resonances have been included in the fit, but theircontributions have been found to be consistent with zero.We note the presence of a sizeable non-resonant con-tribution. However, in this case the sum of the frac-tions is significantly lower than 100%, indicating impor-tant interference effects. Fitting the data without theNR contribution gives a much poorer description, with − L = − χ /N cells = 2 . η c → K S K ± π ∓ Dalitz plot is notwell-described by an isobar model in which the Kπ S -wave is modeled as a superposition of Breit-Wigner func-tions. A more complex approach is needed, and theMIPWA is able to describe this amplitude without theneed for a specific model. VII. DALITZ PLOT ANALYSIS OF η c → K + K − π The η c → K + K − π Dalitz plot [12] is very similar tothat for η c → K S K ± π ∓ decays. It is dominated by uni-formly populated bands at the K ∗ (1430) resonance posi-tion in K + π and K − π mass squared. It also shows abroad diagonal structure indicating the presence of a or a resonance contributions. The Dalitz plot projectionsare shown in Fig. 10. The η c → K + K − π Dalitz plot analysis using the isobarmodel has been performed already in Ref. [12] . It wasfound that the model does not give a perfect descriptionof the data. In this section we obtain a new measurementof the Kπ S -wave by making use of the MIPWA method.In this way we also perform a cross-check of the resultsobtained from the η c → K S K ± π ∓ analysis, since analysesof the two η c decay modes should give consistent results,given the absence of I=3/2 Kπ amplitude contributions. A. MIPWA of η c → K + K − π We perform a MIPWA of η c → K + K − π decays us-ing the same model and the same mass grid as for η c → K S K ± π ∓ . As for the previous case we obtain a bet-ter description of the data if we include an additional a (1950) resonance, whose parameter values are listed inTable III. We observe good agreement between the pa-rameter values obtained from the two η c decay modes.The table also lists parameter values obtained as theweighted mean of the two measurements. Table II givesthe fitted fractions from the MIPWA fit.We obtain a good description of the data, as evidencedby the value χ /N cells = 1 .
22, and observe the a (1950)state with a significance of 4 . σ . The fit projectionson the K + π , K − π , and K + K − squared mass distri-butions are shown in Fig. 10. As previously, there isa dominance of the ( Kπ S -wave) K amplitude, witha significant K ∗ (1430) K amplitude, and small contri-butions from a π amplitudes. We observe good agree-ment between fractions and relative phases of the am-plitudes between the η c → K S K ± π ∓ and η c → K + K − π decay modes. Systematic uncertainties are evaluated asdiscussed in Sec. VI.A.We compute the uncorrected Legendre polynomial mo-ments h Y L i in each K + π , K − π and K + K − mass in-terval by weighting each event by the relevant Y L (cos θ )function. These distributions are shown in Fig. 11 asfunctions of Kπ mass, combined for K + π and K − π ,and in Fig. 12 as functions of K + K − mass. We also com-pute the expected Legendre polynomial moments fromthe weighted MC events and compare with the experi-mental distributions. We observe good agreement for allthe distributions, which indicates that also in this casethe fit is able to reproduce the local structures apparentin the Dalitz plot. VIII. THE I = 1 / Kπ S -WAVE AMPLITUDEAND PHASE Figure 13 displays the measured I = 1 / Kπ S -wave amplitude and phase from both η c → K S K ± π ∓ and η c → K + K − π . We observe good agreement between theamplitude and phase values obtained from the two mea-surements.5 TABLE IV: Systematic uncertainties on the a (1950) parameter values from the two η c decay modes. η c → K S K ± π ∓ η c → K + K − π Effect Mass Width Fraction (%) Mass Width Fraction (%)(MeV /c ) (MeV) (MeV /c ) (MeV)Fit bias 11 22 0.5 1 10 0.5Cubic spline 24 79 0.6 14 9 0.2Marginal components 70 72 0.0 2 8 0.3 η c purity 3 16 1.0 18 26 0.4Efficiency 11 8 0.2 1 15 0.2Total 76 110 1.3 23 30 0.8TABLE V: Measured amplitude and phase values for the I = 1 / Kπ S -wave as functions of mass obtained from the MIPWAof η c → K S K ± π ∓ and η c → K + K − π . The first error is statistical, the second systematic. The amplitudes and phases in themass interval 14 are fixed to constant values. η c → K S K ± π ∓ η c → K + K − π N Kπ mass Amplitude Phase (rad) Amplitude Phase (rad)1 0.67 0 . ± . ± .
215 0 . ± . ± .
290 0 . ± . ± .
337 3 . ± . ± . . ± . ± . − . ± . ± .
600 0 . ± . ± .
216 3 . ± . ± . . ± . ± .
180 0 . ± . ± .
500 0 . ± . ± .
098 1 . ± . ± . . ± . ± .
214 0 . ± . ± .
500 0 . ± . ± .
031 3 . ± . ± . . ± . ± .
194 0 . ± . ± .
237 0 . ± . ± .
162 0 . ± . ± . . ± . ± .
129 0 . ± . ± .
190 0 . ± . ± . − . ± . ± . . ± . ± .
102 0 . ± . ± .
273 0 . ± . ± .
053 0 . ± . ± . . ± . ± .
062 0 . ± . ± .
213 0 . ± . ± .
046 0 . ± . ± . . ± . ± .
081 0 . ± . ± .
221 0 . ± . ± .
089 0 . ± . ± . . ± . ± .
042 0 . ± . ± .
226 0 . ± . ± .
102 0 . ± . ± . . ± . ± .
112 0 . ± . ± .
358 0 . ± . ± .
084 0 . ± . ± . . ± . ± .
053 0 . ± . ± .
166 0 . ± . ± .
039 1 . ± . ± . . ± . ± .
105 1 . ± . ± .
288 0 . ± . ± .
056 1 . ± . ± . .
000 1 .
570 1 .
000 1 . . ± . ± .
059 1 . ± . ± .
132 0 . ± . ± .
076 1 . ± . ± . . ± . ± .
053 2 . ± . ± .
277 0 . ± . ± .
047 2 . ± . ± . . ± . ± .
065 2 . ± . ± .
180 0 . ± . ± .
052 2 . ± . ± . . ± . ± .
089 1 . ± . ± .
619 0 . ± . ± .
043 1 . ± . ± . . ± . ± .
067 1 . ± . ± .
655 0 . ± . ± .
063 1 . ± . ± . . ± . ± .
059 2 . ± . ± .
251 0 . ± . ± .
063 2 . ± . ± . . ± . ± .
085 2 . ± . ± .
284 0 . ± . ± .
080 2 . ± . ± . . ± . ± .
067 2 . ± . ± .
207 0 . ± . ± .
095 2 . ± . ± . . ± . ± .
055 2 . ± . ± .
092 0 . ± . ± .
075 2 . ± . ± . . ± . ± .
065 2 . ± . ± .
268 0 . ± . ± .
075 2 . ± . ± . . ± . ± .
083 2 . ± . ± .
169 0 . ± . ± .
078 2 . ± . ± . . ± . ± .
117 2 . ± . ± .
137 0 . ± . ± .
070 2 . ± . ± . . ± . ± .
104 2 . ± . ± .
241 0 . ± . ± .
123 2 . ± . ± . . ± . ± .
125 2 . ± . ± .
168 0 . ± . ± .
133 2 . ± . ± . . ± . ± .
118 2 . ± . ± .
321 0 . ± . ± .
076 2 . ± . ± . . ± . ± .
187 2 . ± . ± .
199 0 . ± . ± .
098 2 . ± . ± . The main features of the amplitude (Fig. 13(a)) can beexplained by the presence of a clear peak related to the K ∗ (1430) resonance which shows a rapid drop around1.7 GeV /c , where a broad structure is present whichcan be related to the K ∗ (1950) resonance. There is someindication of feedthrough from the K ∗ (892) background.The phase motion (Fig. 13(b)) shows the expected be-havior for the resonance phase, which varies by about π in the K ∗ (1430) resonance region. The phase shows adrop around 1.7 GeV /c related to interference with the K ∗ (1950) resonance.We compare the present measurement of the Kπ S -wave amplitude from η c → K S K ± π ∓ with measure-ments from LASS [5] in Fig. 14(a)(c) and E791 [8] inFig. 14(b)(d). We plot only the first part of the LASSmeasurement since it suffers from a two-fold ambiguity6 ) /c ) (GeV π + (K m0 2 4 6 ) / c e v en t s / ( . G e V (a) ) /c ) (GeV π - (K m0 2 4 6 ) / c e v en t s / ( . G e V (b) ) /c ) (GeV - K + (K m0 2 4 6 8 ) / c e v en t s / ( . G e V (c) FIG. 10: The η c → K + K − π Dalitz plot projections, (a) m ( K + π ), (b) m ( K − π ), and (c) m ( K + K − ). The superim-posed curves result from the MIPWA described in the text. The shaded regions show the background estimates obtained byinterpolating the results of the Dalitz plot analyses of the sideband regions. ) ) (GeV/c π m(K0.5 1 1.5 2 2.5 ) w e i gh t s u m / ( M e V / c Y ) ) (GeV/c π m(K0.5 1 1.5 2 2.540 − − Y ) ) (GeV/c π m(K0.5 1 1.5 2 2.540 − − Y ) ) (GeV/c π m(K0.5 1 1.5 2 2.5 ) w e i gh t s u m / ( M e V / c − Y ) ) (GeV/c π m(K0.5 1 1.5 2 2.520 − − Y ) ) (GeV/c π m(K0.5 1 1.5 2 2.510 − Y FIG. 11: Legendre polynomial moments for η c → K + K − π as functions of Kπ mass, combined for K + π and K − π . Thesuperimposed curves result from the Dalitz plot fit described in the text. above the mass of 1.82 GeV /c . The Dalitz plot fits ex-tract invariant amplitudes. Consequently, in Fig. 14(a),the LASS I = 1 / Kπ scattering amplitude values havebeen multiplied by the factor m ( Kπ ) /q to convert toinvariant amplitude, and normalized so as to equal thescattering amplitude at 1.5 GeV /c in order to facilitatecomparison to the η c results. Here q is the momentum ofeither meson in the Kπ rest frame. For better compar-ison, the LASS absolute phase measurements have beendisplaced by − . c in Table III of Ref. [8] bythe Form Factor F D , for which the mass-dependence ismotivated by theoretical speculation. This yields am-plitude values corresponding to the E791 Form Factorhaving value 1, as for the η c analyses. In Fig. 14(d), theE791 phase measurements have been displaced by +0 . η c mea-surements.While we observe similar phase behavior among the7 ) ) (GeV/c - K + m(K1 1.5 2 2.5 3 ) w e i gh t s u m / ( M e V / c Y ) ) (GeV/c - K + m(K1 1.5 2 2.5 320 − − Y ) ) (GeV/c - K + m(K1 1.5 2 2.5 320 − − Y ) ) (GeV/c - K + m(K1 1.5 2 2.5 3 ) w e i gh t s u m / ( M e V / c − − Y ) ) (GeV/c - K + m(K1 1.5 2 2.5 310 − Y ) ) (GeV/c - K + m(K1 1.5 2 2.5 320 − − Y FIG. 12: Legendre polynomial moments for η c → K + K − π as a function of K + K − mass. The superimposed curves result fromthe Dalitz plot fit described in the text. ) GeV/c π m(K ) A m p lit ud e / ( M e V / c (a) ) GeV/c π m(K P h a s e (r a d ) (b) FIG. 13: The I = 1 / Kπ S -wave amplitude (a) and phase (b) from η c → K S K ± π ∓ (solid (black) points) and η c → K + K − π (open (red) points); only statistical uncertainties are shown. The dotted lines indicate the Kη and Kη ′ thresholds. three measurements up to about 1.5 GeV /c , we observestriking differences in the mass dependence of the ampli-tudes. IX. SUMMARY
We perform Dalitz plot analyses, using an isobarmodel and a MIPWA method, of data on the decays η c → K S K ± π ∓ and η c → K + K − π , where the η c mesonsare produced in two-photon interactions in the B A B AR ) GeV/c π m(K ) A m p lit ud e / ( M e V / c (a) ) GeV/c π m(K ) A m p lit ud e / ( M e V / c (b) ) GeV/c π m(K P h a s e (r a d ) − (c) ) GeV/c π m(K P h a s e (r a d ) − − (d) FIG. 14: The I = 1 / Kπ S -wave amplitude measurements from η c → K S K ± π ∓ compared to the (a) LASS and (b) E791results: the corresponding I = 1 / Kπ S -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 LASSdata are plotted in the region having only one solution. experiment at SLAC. We find that, in comparison withthe isobar models examined here, an improved descrip-tion of the data is obtained by using a MIPWA method.We extract the I = 1 / Kπ S -wave amplitude andphase and find good agreement between the measure-ments for the two η c decay modes. The Kπ S -wave isdominated by the presence of the K ∗ (1430) resonancewhich is observed as a clear peak with the correspondingincrease in phase of about π expected for a resonance. Abroad structure in the 1.95 GeV /c mass region indicatesthe presence of the K ∗ (1950) resonance.A comparison between the present measurement andprevious experiments indicates a similar trend for thephase up to a mass of 1.5 GeV /c . The amplitudes, on the other hand, show very marked differences.To fit the data we need to introduce a new a (1950)resonance in both η c → K S K ± π ∓ and η c → K + K − π de-cay modes, and their associated parameter values are ingood agreement. The weighted averages for the parame-ter values are: m ( a (1950)) = 1931 ± ±
22 MeV /c , Γ( a (1950)) = 271 ± ±
29 MeV (12)with significances of 2.5 σ and 4.2 σ respectively, includ-ing systematic uncertainties. These results are, however,systematically limited, and more detailed studies of the I = 1 KK S -wave will be required in order to improve9 TABLE VI: Results from the η c → K S K ± π ∓ Dalitz plot anal-ysis using an isobar model. The listed uncertainties are sta-tistical only.Amplitude Fraction % Phase (rad) K ∗ (1430) K ± K ∗ (1950) K ± − ± ± ± a (980) π ± ± a (1450) π ± ± a (1950) π ± − ± a (1320) π ± − ± K ∗ (1430) K ± − ± ± − L − χ /N cells the precision of these values. X. ACKNOWLEDGEMENTS
We are grateful for the extraordinary contributionsof our PEP-II colleagues in achieving the excellent lu- minosity and machine conditions that have made thiswork possible. The success of this project also re-lies critically on the expertise and dedication of thecomputing organizations that support B A B AR . The col-laborating institutions wish to thank SLAC for itssupport and the kind hospitality extended to them.This work is supported by the US Department of En-ergy and National Science Foundation, the Natural Sci-ences and Engineering Research Council (Canada), theCommissariat `a l’Energie Atomique and Institut Na-tional de Physique Nucl´eaire et de Physique des Partic-ules (France), the Bundesministerium f¨ur Bildung undForschung and Deutsche Forschungsgemeinschaft (Ger-many), 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 RussianFederation, Ministerio de Economia y Competitividad(Spain), and the Science and Technology Facilities Coun-cil (United Kingdom). Individuals have received supportfrom the Marie-Curie IEF program (European Union),the A. P. Sloan Foundation (USA) and the BinationalScience Foundation (USA-Israel). The work of A. Palanoand M. R. Pennington was supported (in part) by theU.S. Department of Energy, Office of Science, Office ofNuclear Physics under contract DE-AC05-06OR23177. [1] G.’t Hooft et al. , Phys. Lett. B , 424 (2008); W. Ochs,J.Phys. G , 043001 (2013).[2] B. Aubert et al. ( B A B AR Collaboration), Phys. Rev. D , 032005 (2005); B. Aubert et al. ( B A B AR Collabo-ration), Phys. Rev. D , 034023 (2008); B. Aubert etal. ( B A B AR Collaboration), Phys. Rev. D , 012004(2008); A. Poluektov et al. (Belle Collaboration), Phys.Rev. D , 112002 (2010).[3] K. Chilikin et al. (Belle Collaboration), Phys. Rev. D , 074026 (2013); R. Aaij et al. (LHCb Collaboration),Phys. Rev. Lett. , 222002 (2014).[4] R. Aaij et al. (LHCb Collaboration), Phys. Rev. D ,072003 (2014).[5] D. Aston et al. (LASS Collaboration), Nucl. Phys. B ,493 (1988).[6] B. Aubert et al. ( B A B AR Collaboration), Phys. Rev. D , 112001 (2009).[7] E. M. Aitala et al. (E791 Collaboration), Phys. Rev. Lett. , 121801 (2002).[8] E. M. Aitala et al. (E791 Collaboration), Phys. Rev. D , 032004 (2006); Erratum-ibid. D , 059901 (2006).[9] G. Bonvicini et al. (CLEO Collaboration), Phys. Rev. D , 052001 (2008).[10] J. M. Link et al. (FOCUS Collaboration), Phys. Lett. B , 1 (2007).[11] B. Aubert et al. ( B A B AR Collaboration), Phys. Rev. D , 032003 (2009).[12] B. Aubert et al. ( B A B AR Collaboration), Phys. Rev. D , 112004 (2014).[13] P. del Amo Sanchez et al. ( B A B AR Collaboration), Phys.Rev. D , 012004 (2011).[14] C. N. Yang, Phys. Rev. , 242 (1950). [15] J. P. Lees et al. ( B A B AR Collaboration), Nucl. Instrum.Methods Phys. Res., Sect. A , 203 (2013).[16] B. Aubert et al. ( B A B AR Collaboration), Nucl. Instrum.Methods Phys. Res., Sect. A , 1 (2002); ibid. ,615 (2013).[17] The B A B AR detector Monte Carlo simulation is basedon Geant4 [S. Agostinelli et al. , Nucl. Instrum. Meth-ods A , 250 (2003)] and EvtGen [D. J. Lange, Nucl.Instrum. Methods A , 152 (2001)].[18] B. Aubert et al. ( B A B AR Collaboration), Phys. Rev. D , 092003 (2010).[19] K. A. Olive et al. (Particle Data Group), Chin. Phys. C , 090001 (2014).[20] B. Aubert et al. ( B A B AR Collaboration), Phys. Rev. D , 092002 (2008).[21] The use of charge conjugate reactions is implied, wherenot explicitly expressed, throughout the paper.[22] M. J. Oreglia, Ph.D. Thesis, SLAC-R-236 (1980); J. E.Gaiser, Ph.D. Thesis, SLAC-R-255 (1982); T. Skwar-nicki, Ph.D. Thesis, DESY-F31-86-02 (1986).[23] M. Ablikim et al. (BESIII Collaboration), Phys. Rev.Lett. , 222002 (2012).[24] S. Kopp et al. (CLEO Collaboration), Phys. Rev. D ,092001 (2001).[25] J. Blatt and V. Weisskopf, Theoretical Nuclear Physics,New York: John Wiley & Sons (1952).[26] A. Abele et al. (Crystal Barrel Collaboration), Phys. Rev.D , 3860 (1998).[27] P. E. Condon and P. L. Cowell, Phys. Rev. D9