Dalitz plot analysis of Ds+ [D subscript s superscript +] →K+K-π+ [K superscript + K superscript - pi superscript +]

Dalitz plot analysis of Ds+ [D subscript s superscript +] →K

+K-π+ [K superscript + K superscript - pi superscript +]

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Citation

Sanchez, P. del Amo et al. (BABAR Collaboration) "Dalitz plot

analysis of Ds+→K+K-π+" Phys. Rev. D 83, 052001. © 2011 American

Physical Society

As Published

http://dx.doi.org/10.1103/PhysRevD.83.052001

Publisher

American Physical Society

Version

Final published version

Citable link

http://hdl.handle.net/1721.1/66173

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Article is made available in accordance with the publisher's

policy and may be subject to US copyright law. Please refer to the

publisher's site for terms of use.

Dalitz plot analysis of

D

! K

K

P. del Amo Sanchez,1J. P. Lees,1V. Poireau,1E. Prencipe,1V. Tisserand,1J. Garra Tico,2E. Grauges,2M. Martinelli,3a,3b D. A. Milanes,3a,3bA. Palano,3a,3bM. Pappagallo,3a,3bG. Eigen,4B. Stugu,4L. Sun,4D. N. Brown,5L. T. Kerth,5 Yu. G. Kolomensky,5G. Lynch,5I. L. Osipenkov,5H. Koch,6T. Schroeder,6D. J. Asgeirsson,7C. Hearty,7T. S. Mattison,7

J. A. McKenna,7A. Khan,8V. E. Blinov,9A. R. Buzykaev,9V. P. Druzhinin,9V. B. Golubev,9E. A. Kravchenko,9 A. P. Onuchin,9S. I. Serednyakov,9Yu. I. Skovpen,9E. P. Solodov,9K. Yu. Todyshev,9A. N. Yushkov,9M. Bondioli,10 S. Curry,10D. Kirkby,10A. J. Lankford,10M. Mandelkern,10E. C. Martin,10D. P. Stoker,10H. Atmacan,11J. W. Gary,11

F. Liu,11O. Long,11G. M. Vitug,11C. Campagnari,12T. M. Hong,12D. Kovalskyi,12J. D. Richman,12C. West,12 A. M. Eisner,13C. A. Heusch,13J. Kroseberg,13W. S. Lockman,13A. J. Martinez,13T. Schalk,13B. A. Schumm,13 A. Seiden,13L. O. Winstrom,13C. H. Cheng,14D. A. Doll,14B. Echenard,14D. G. Hitlin,14P. Ongmongkolkul,14 F. C. Porter,14A. Y. Rakitin,14R. Andreassen,15M. S. Dubrovin,15G. Mancinelli,15B. T. Meadows,15M. D. Sokoloff,15

P. C. Bloom,16W. T. Ford,16A. Gaz,16M. Nagel,16U. Nauenberg,16J. G. Smith,16S. R. Wagner,16R. Ayad,17,* W. H. Toki,17H. Jasper,18T. M. Karbach,18A. Petzold,18B. Spaan,18M. J. Kobel,19K. R. Schubert,19R. Schwierz,19 D. Bernard,20M. Verderi,20P. J. Clark,21S. Playfer,21J. E. Watson,21M. Andreotti,22a,22bD. Bettoni,22aC. Bozzi,22a

R. Calabrese,22a,22bA. Cecchi,22a,22bG. Cibinetto,22a,22bE. Fioravanti,22a,22bP. Franchini,22a,22bI. Garzia,22a,22b E. Luppi,22a,22bM. Munerato,22a,22bM. Negrini,22a,22bA. Petrella,22a,22bL. Piemontese,22aR. Baldini-Ferroli,23 A. Calcaterra,23R. de Sangro,23G. Finocchiaro,23M. Nicolaci,23S. Pacetti,23P. Patteri,23I. M. Peruzzi,23,†M. Piccolo,23

M. Rama,23A. Zallo,23R. Contri,24a,24bE. Guido,24a,24bM. Lo Vetere,24a,24bM. R. Monge,24a,24bS. Passaggio,24a C. Patrignani,24a,24bE. Robutti,24aS. Tosi,24a,24bB. Bhuyan,25V. Prasad,25C. L. Lee,26M. Morii,26A. J. Edwards,27

A. Adametz,28J. Marks,28U. Uwer,28F. U. Bernlochner,29M. Ebert,29H. M. Lacker,29T. Lueck,29A. Volk,29 P. D. Dauncey,30M. Tibbetts,30P. K. Behera,31U. Mallik,31C. Chen,32J. Cochran,32H. B. Crawley,32L. Dong,32 W. T. Meyer,32S. Prell,32E. I. Rosenberg,32A. E. Rubin,32A. V. Gritsan,33Z. J. Guo,33N. Arnaud,34M. Davier,34 D. Derkach,34J. Firmino da Costa,34G. Grosdidier,34F. Le Diberder,34A. M. Lutz,34B. Malaescu,34A. Perez,34 P. Roudeau,34M. H. Schune,34J. Serrano,34V. Sordini,34,‡A. Stocchi,34L. Wang,34G. Wormser,34D. J. Lange,35

D. M. Wright,35I. Bingham,36C. A. Chavez,36J. P. Coleman,36J. R. Fry,36E. Gabathuler,36R. Gamet,36 D. E. Hutchcroft,36D. J. Payne,36C. Touramanis,36A. J. Bevan,37F. Di Lodovico,37R. Sacco,37M. Sigamani,37 G. Cowan,38S. Paramesvaran,38A. C. Wren,38D. N. Brown,39C. L. Davis,39A. G. Denig,40M. Fritsch,40W. Gradl,40

A. Hafner,40K. E. Alwyn,41D. Bailey,41R. J. Barlow,41G. Jackson,41G. D. Lafferty,41J. Anderson,42R. Cenci,42 A. Jawahery,42D. A. Roberts,42G. Simi,42J. M. Tuggle,42C. Dallapiccola,43E. Salvati,43R. Cowan,44D. Dujmic,44

G. Sciolla,44M. Zhao,44D. Lindemann,45P. M. Patel,45S. H. Robertson,45M. Schram,45P. Biassoni,46a,46b A. Lazzaro,46a,46bV. Lombardo,46aF. Palombo,46a,46bS. Stracka,46a,46bL. Cremaldi,47R. Godang,47,xR. Kroeger,47

P. Sonnek,47D. J. Summers,47X. Nguyen,48M. Simard,48P. Taras,48G. De Nardo,49a,49bD. Monorchio,49a,49b G. Onorato,49a,49bC. Sciacca,49a,49bG. Raven,50H. L. Snoek,50C. P. Jessop,51K. J. Knoepfel,51J. M. LoSecco,51 W. F. Wang,51L. A. Corwin,52K. Honscheid,52R. Kass,52J. P. Morris,52N. L. Blount,53J. Brau,53R. Frey,53O. Igonkina,53

J. A. Kolb,53R. Rahmat,53N. B. Sinev,53D. Strom,53J. Strube,53E. Torrence,53G. Castelli,54a,54bE. Feltresi,54a,54b N. Gagliardi,54a,54bM. Margoni,54a,54bM. Morandin,54aM. Posocco,54aM. Rotondo,54aF. Simonetto,54a,54b R. Stroili,54a,54bE. Ben-Haim,55G. R. Bonneaud,55H. Briand,55G. Calderini,55J. Chauveau,55O. Hamon,55Ph. Leruste,55 G. Marchiori,55J. Ocariz,55J. Prendki,55S. Sitt,55M. Biasini,56a,56bE. Manoni,56a,56bA. Rossi,56a,56bC. Angelini,57a,57b

G. Batignani,57a,57bS. Bettarini,57a,57bM. Carpinelli,57a,57b,kG. Casarosa,57a,57bA. Cervelli,57a,57bF. Forti,57a,57b M. A. Giorgi,57a,57bA. Lusiani,57a,57cN. Neri,57a,57bE. Paoloni,57a,57bG. Rizzo,57a,57bJ. J. Walsh,57aD. Lopes Pegna,58

C. Lu,58J. Olsen,58A. J. S. Smith,58A. V. Telnov,58F. Anulli,59aE. Baracchini,59a,59bG. Cavoto,59aR. Faccini,59a,59b F. Ferrarotto,59aF. Ferroni,59a,59bM. Gaspero,59a,59bL. Li Gioi,59aM. A. Mazzoni,59aG. Piredda,59aF. Renga,59a,59b T. Hartmann,60T. Leddig,60H. Schro¨der,60R. Waldi,60T. Adye,61B. Franek,61E. O. Olaiya,61F. F. Wilson,61S. Emery,62

G. Hamel de Monchenault,62G. Vasseur,62Ch. Ye`che,62M. Zito,62M. T. Allen,63D. Aston,63D. J. Bard,63 R. Bartoldus,63J. F. Benitez,63C. Cartaro,63M. R. Convery,63J. Dorfan,63G. P. Dubois-Felsmann,63W. Dunwoodie,63

R. C. Field,63M. Franco Sevilla,63B. G. Fulsom,63A. M. Gabareen,63M. T. Graham,63P. Grenier,63C. Hast,63 W. R. Innes,63M. H. Kelsey,63H. Kim,63P. Kim,63M. L. Kocian,63D. W. G. S. Leith,63S. Li,63B. Lindquist,63S. Luitz,63

V. Luth,63H. L. Lynch,63D. B. MacFarlane,63H. Marsiske,63D. R. Muller,63H. Neal,63S. Nelson,63C. P. O’Grady,63 I. Ofte,63M. Perl,63T. Pulliam,63B. N. Ratcliff,63A. Roodman,63A. A. Salnikov,63V. Santoro,63R. H. Schindler,63

J. Schwiening,63A. Snyder,63D. Su,63M. K. Sullivan,63S. Sun,63K. Suzuki,63J. M. Thompson,63J. Va’vra,63

A. P. Wagner,63M. Weaver,63W. J. Wisniewski,63M. Wittgen,63D. H. Wright,63H. W. Wulsin,63A. K. Yarritu,63 C. C. Young,63V. Ziegler,63X. R. Chen,64W. Park,64M. V. Purohit,64R. M. White,64J. R. Wilson,64A. Randle-Conde,65

S. J. Sekula,65M. Bellis,66P. R. Burchat,66T. S. Miyashita,66S. Ahmed,67M. S. Alam,67J. A. Ernst,67B. Pan,67 M. A. Saeed,67S. B. Zain,67N. Guttman,68A. Soffer,68P. Lund,69S. M. Spanier,69R. Eckmann,70 J. L. Ritchie,70A. M. Ruland,70C. J. Schilling,70R. F. Schwitters,70B. C. Wray,70J. M. Izen,71X. C. Lou,71

F. Bianchi,72a,72bD. Gamba,72a,72bM. Pelliccioni,72a,72bM. Bomben,73a,73bL. Lanceri,73a,73b L. Vitale,73a,73bN. Lopez-March,74F. Martinez-Vidal,74A. Oyanguren,74J. Albert,75Sw. Banerjee,75

H. H. F. Choi,75K. Hamano,75G. J. King,75R. Kowalewski,75M. J. Lewczuk,75C. Lindsay,75 I. M. Nugent,75J. M. Roney,75R. J. Sobie,75T. J. Gershon,76P. F. Harrison,76T. E. Latham,76

M. R. Pennington,76,{E. M. T. Puccio,76H. R. Band,77S. Dasu,77K. T. Flood,77 Y. Pan,77R. Prepost,77C. O. Vuosalo,77and S. L. Wu77

(BABARCollaboration)

1_{Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Universite´ de Savoie, CNRS/IN2P3,} F-74941 Annecy-Le-Vieux, France

2_{Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain} 3a_{INFN Sezione di Bari, I-70126 Bari, Italy;}

3b_{Dipartimento di Fisica, Universita` di Bari, I-70126 Bari, Italy} 4_{University of Bergen, Institute of Physics, N-5007 Bergen, Norway}

5_{Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA} 6_{Ruhr Universita¨t Bochum, Institut fu¨r Experimentalphysik 1, D-44780 Bochum, Germany}

7_{University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1} 8_{Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom}

9_{Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia} 10_{University of California at Irvine, Irvine, California 92697, USA} 11_{University of California at Riverside, Riverside, California 92521, USA} 12_{University of California at Santa Barbara, Santa Barbara, California 93106, USA}

13_{University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA} 14

California Institute of Technology, Pasadena, California 91125, USA 15_{University of Cincinnati, Cincinnati, Ohio 45221, USA} 16_{University of Colorado, Boulder, Colorado 80309, USA} 17_{Colorado State University, Fort Collins, Colorado 80523, USA}

18_{Technische Universita¨t Dortmund, Fakulta¨t Physik, D-44221 Dortmund, Germany}

19_{Technische Universita¨t Dresden, Institut fu¨r Kern- und Teilchenphysik, D-01062 Dresden, Germany} 20_{Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France}

21_{University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom} 22a_{INFN Sezione di Ferrara, I-44100 Ferrara, Italy;}

22b_{Dipartimento di Fisica, Universita` di Ferrara, I-44100 Ferrara, Italy} 23_{INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy}

24a_{INFN Sezione di Genova, I-16146 Genova, Italy;}

24b_{Dipartimento di Fisica, Universita` di Genova, I-16146 Genova, Italy} 25_{Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India}

26_{Harvard University, Cambridge, Massachusetts 02138, USA} 27_{Harvey Mudd College, Claremont, California 91711} 28

Universita¨t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany 29_{Humboldt-Universita¨t zu Berlin, Institut fu¨r Physik, Newtonstr. 15, D-12489 Berlin, Germany}

30_{Imperial College London, London, SW7 2AZ, United Kingdom} 31_{University of Iowa, Iowa City, Iowa 52242, USA} 32_{Iowa State University, Ames, Iowa 50011-3160, USA} 33_{Johns Hopkins University, Baltimore, Maryland 21218, USA}

34_{Laboratoire de l’Acce´le´rateur Line´aire, IN2P3/CNRS et Universite´ Paris-Sud 11,} Centre Scientifique d’Orsay, B. P. 34, F-91898 Orsay Cedex, France 35_{Lawrence Livermore National Laboratory, Livermore, California 94550, USA}

36

University of Liverpool, Liverpool L69 7ZE, United Kingdom 37_{Queen Mary, University of London, London, E1 4NS, United Kingdom}

38_{University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom} 39_{University of Louisville, Louisville, Kentucky 40292, USA}

40_{Johannes Gutenberg-Universita¨t Mainz, Institut fu¨r Kernphysik, D-55099 Mainz, Germany} 41_{University of Manchester, Manchester M13 9PL, United Kingdom}

42_{University of Maryland, College Park, Maryland 20742, USA} 43_{University of Massachusetts, Amherst, Massachusetts 01003, USA}

44_{Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA} 45_{McGill University, Montre´al, Que´bec, Canada H3A 2T8}

46a_{INFN Sezione di Milano, I-20133 Milano, Italy;}

46b_{Dipartimento di Fisica, Universita` di Milano, I-20133 Milano, Italy} 47

University of Mississippi, University, Mississippi 38677, USA

48_{Universite´ de Montre´al, Physique des Particules, Montre´al, Que´bec, Canada H3C 3J7} 49a_{INFN Sezione di Napoli, I-80126 Napoli, Italy;}

49b_{Dipartimento di Scienze Fisiche, Universita` di Napoli Federico II, I-80126 Napoli, Italy} 50_{NIKHEF, National Institute for Nuclear Physics and High Energy Physics,}

NL-1009 DB Amsterdam, The Netherlands

51_{University of Notre Dame, Notre Dame, Indiana 46556, USA} 52_{Ohio State University, Columbus, Ohio 43210, USA}

53_{University of Oregon, Eugene, Oregon 97403, USA} 54a_{INFN Sezione di Padova, I-35131 Padova, Italy;}

54b_{Dipartimento di Fisica, Universita` di Padova, I-35131 Padova, Italy}

55_{Laboratoire de Physique Nucle´aire et de Hautes Energies, IN2P3/CNRS, Universite´ Pierre et Marie Curie-Paris6,} Universite´ Denis Diderot-Paris7, F-75252 Paris, France

56a_{INFN Sezione di Perugia, I-06100 Perugia, Italy;}

56b_{Dipartimento di Fisica, Universita` di Perugia, I-06100 Perugia, Italy} 57a_{INFN Sezione di Pisa, I-56127 Pisa, Italy;}

57b

Dipartimento di Fisica, Universita` di Pisa, I-56127 Pisa, Italy; 57c_{Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy} 58_{Princeton University, Princeton, New Jersey 08544, USA}

59a_{INFN Sezione di Roma, I-00185 Roma, Italy;}

59b_{Dipartimento di Fisica, Universita` di Roma La Sapienza, I-00185 Roma, Italy} 60_{Universita¨t Rostock, D-18051 Rostock, Germany}

61_{Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom} 62_{CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France}

63_{SLAC National Accelerator Laboratory, Stanford, California 94309 USA} 64_{University of South Carolina, Columbia, South Carolina 29208, USA}

65_{Southern Methodist University, Dallas, Texas 75275, USA} 66_{Stanford University, Stanford, California 94305-4060, USA} 67_{State University of New York, Albany, New York 12222, USA} 68_{Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel}

69_{University of Tennessee, Knoxville, Tennessee 37996, USA} 70_{University of Texas at Austin, Austin, Texas 78712, USA} 71_{University of Texas at Dallas, Richardson, Texas 75083, USA}

72a_{INFN Sezione di Torino, I-10125 Torino, Italy;}

72b_{Dipartimento di Fisica Sperimentale, Universita` di Torino, I-10125 Torino, Italy} 73a_{INFN Sezione di Trieste, I-34127 Trieste, Italy;}

73b_{Dipartimento di Fisica, Universita` di Trieste, I-34127 Trieste, Italy} 74_{IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain} 75_{University of Victoria, Victoria, British Columbia, Canada V8W 3P6} 76

Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 77_{University of Wisconsin, Madison, Wisconsin 53706, USA}

(Received 18 November 2010; published 3 March 2011)

We perform a Dalitz plot analysis of about 100 000 Dþs decays to KþK

þand measure the complex amplitudes of the intermediate resonances which contribute to this decay mode. We also measure the relative branching fractions of Dþs ! KþKþ

and Dþs ! KþKþK
. For this analysis we use a

*Now at Temple University, Philadelphia, PA 19122, USA

†_{Also with Universita` di Perugia, Dipartimento di Fisica, Perugia, Italy.</sub> ‡<sub>Also with Universita` di Roma La Sapienza, I-00185 Roma, Italy.} x_{Now at University of South AL, Mobile, AL 36688, USA.} k_{Also with Universita` di Sassari, Sassari, Italy.}

{_{Also with Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK.}

384 fb
1 data sample, recorded by the BABAR detector at the PEP-II asymmetric-energy eþe
collider running at center-of-mass energies near 10.58 GeV.

DOI:10.1103/PhysRevD.83.052001 PACS numbers: 13.25.Ft, 11.80.Et, 14.40.Be, 14.40.Lb

I. INTRODUCTION

Scalar mesons are still a puzzle in light meson spectros-copy. New claims for the existence of broad states close to threshold such as ð800Þ [1] and f0ð600Þ [2], have

reop-ened discussion about the composition of the ground state JPC¼ 0þþnonet, and about the possibility that states such as the a0ð980Þ or f0ð980Þ may be 4-quark states, due to

their proximity to the K
K threshold [3]. This hypothesis can be tested only through accurate measurements of the branching fractions and the couplings to different final states. It is therefore important to have precise information on the structure of the

and K
KS waves. In this context, Dþ_s mesons can shed light on the structure of the scalar amplitude coupled to s
s. The

S wave has been already extracted from BABAR data in a Dalitz plot analysis of Dþs !
þ


þ [4]. The understanding of the K
K S

wave is also of great importance for the precise mea-surement of CP violation in Bs oscillations using Bs!

J=c [5,6].

This paper focuses on the study of Dþs meson decay to

KþK

þ[7]. Dalitz plot analyses of this decay mode have been performed by the E687 and CLEO Collaborations using 700 events [8], and 14 400 events [9] respectively. The present analysis is performed using about 100 000 events.

The decay Dþ_s ! 
þ is frequently used in particle physics as the reference mode for Dþ_s decay. Previous measurements of this decay mode did not, however, account for the presence of the K
K S wave underneath the  peak. Therefore, as part of the present analysis, we obtain a precise measurement of the branching fraction BðDþ

s ! 
þÞ relative to BðDþs ! KþK

þÞ.

Singly suppressed (SCS) and doubly Cabibbo-suppressed (DCS) decays play an important role in studies of charmed hadron dynamics. The naive expectations for the rates of SCS and DCS decays are of the order of tan2Cand tan4C, respectively, where Cis the Cabibbo

mixing angle. These rates correspond to about 5.3% and 0.28% relative to their Cabibbo-favored (CF) counterpart. Because of the limited statistics in past experiments, branching fraction measurements of DCS decays have been affected by large statistical uncertainties [10]. A precise measurement of BðDþs!KþKþ

Þ

BðDþ

s!KþK

þÞ has been recently

performed by the Belle experiment [11]. In this paper we study the Dþs decay

Dþs ! KþK

þ (1)

and perform a detailed Dalitz plot analysis. We then mea-sure the branching ratios of the SCS decay

Dþs ! KþK
Kþ (2)

and the DCS decay

Dþs ! KþKþ

(3)

relative to the CF channel (1). The paper is organized as follows. Section II briefly describes the BABAR detector, while Sec. III gives details of event reconstruction. Section IV is devoted to the evaluation of the selection efficiency. Section V describes a partial-wave analysis of the KþK
system, the evaluation of the Dþ_s ! 
þ branching fraction, and the K
K S-wave parametrization. Section VI deals with the description of the Dalitz plot analysis method and background description. Results from the Dalitz plot analysis of Dþs ! KþK

þ are given in

Sec. VII. The measurements of the Dþs SCS and DCS

branching fractions are described in Sec. VIII, while Sec.IXsummarizes the results.

II. THEBABAR DETECTOR AND DATASET The data sample used in this analysis corresponds to an integrated luminosity of 384 fb
1 recorded with the BABAR detector at the SLAC PEP-II collider, operated at center-of-mass energies near the ð4SÞ resonance. The BABAR detector is described in detail elsewhere [12]. The following is a brief summary of the components important to this analysis. Charged particle tracks are detected, and their momenta measured, by a combination of a cylindrical drift chamber and a silicon vertex tracker, both operating within a 1.5 T solenoidal magnetic field. Photon energies are measured with a CsI(Tl) electro-magnetic calorimeter. Information from a ring-imaging Cherenkov detector, and specific energy-loss measure-ments in the silicon vertex tracker and cylindrical drift chamber are used to identify charged kaon and pion candidates.

III. EVENT SELECTION ANDDþ_s ! KþK

þ RECONSTRUCTION

Events corresponding to the three-body Dþ_s ! KþK

þ decay are reconstructed from the data sample having at least three reconstructed charged tracks with net charge 1. We require that the invariant mass of the KþK

þ system lie within the mass interval ½1:9–2:05 GeV=c2_{. Particle identification is applied to}

the three tracks, and the presence of two kaons is required. The efficiency that a kaon is identified is 90% while the rate that a kaon is misidentified as a pion is 2%. The three tracks are required to originate from a common vertex, and the 2 fit probability (P1) must be greater than 0.1%.

We also perform a separate kinematic fit in which the Dþ_s mass is constrained to its known value [10]. This latter fit will be used only in the Dalitz plot analysis.

In order to help in the discrimination of signal from background, an additional fit is performed, constraining the three tracks to originate from the eþe
luminous region (beam spot). The 2_{probability of this fit, labeled as P}

2, is

expected to be large for most of the background events, when all tracks originate from the luminous region, and small for the Dþs signal, due to the measurable flight

distance of the latter. The decay

Dsð2112Þþ! Dþs (4)

is used to select a subset of event candidates in order to reduce combinatorial background. The photon is required to have released an energy of at least 100 MeV into the electromagnetic calorimeter. We define the variable

m¼ mðKþK

þÞ
mðKþK

þÞ (5) and require it to be within 2Dþs with respect to

m_Dþ

s where mDþs ¼ 144:94  0:03stat MeV=c

2 _and

Dþs ¼ 5:53  0:04stat MeV=c

2 _{are obtained from a}

Gaussian fit of the m distribution.

Each Dþ_s candidate is characterized by three variables: the center-of-mass momentum p in the eþe
rest frame, the difference in probability P₁
P₂, and the signed decay distance dxy¼dp_jpxyxy_j where d is the vector joining the beam

spot to the Dþ_s decay vertex and pxyis the projection of the

Dþs momentum on the xy plane. These three variables

are used to discriminate signal from background events: in fact signal events are expected to be characterized by larger values of p [13], due to the jetlike shape of the eþe
! c
c events, and larger values of d_xyand P₁
P₂, due to the measurable flight distance of the Dþs meson.

The distributions of these three variables for signal and background events are determined from data and are

shown in Fig. 1. The background distributions are estimated from events in the Dþs mass-sidebands, while

those for the signal region are estimated from the Dþs

signal region with sideband subtraction. The normalized probability distribution functions are then combined in a likelihood-ratio test. A selection is performed on this variable such that signal to background ratio is maximized. Lower sideband, signal, and upper sideband regions are defined between ½1:911–1:934 GeV=c2_,

½1:957–1:980 GeV=c2_{, and ½2:003–2:026 GeV=c}2_,

re-spectively, corresponding to ð
10;
6Þ, ð
2; 2Þ, and ð6; 10Þ regions, where  is estimated from the fit of a Gaussian function to the Dþs lines shape.

We have examined a number of possible background sources. A small peak due to the decay Dþ!
þD0

where D0 _{! K}þ_{K
is observed. A Gaussian fit to this}

KþK
spectrum gives _D0_!Kþ_K
¼ 5:4 MeV=c2. For

events within 3:5_D0_!Kþ_K
of the D0 mass, we plot the

mass difference mðKþK

þÞ ¼ mðKþK

þÞ
mðKþK
Þ and observe a clean Dþ signal. We remove events that satisfy mðKþK

þÞ < 0:15 GeV=c2_{. The}

surviving events still show a D0 ! KþK
signal which does not come from this Dþdecay. We remove events that satisfy mðKþK
Þ > 1:85 GeV=c2_.

Particle misidentification, in which a pion
þ_mis is wrongly identified as a kaon, is tested by assigning the pion mass to the Kþ. In this way we identify the back-ground due to the decay Dþ! K

þ
þ which, for the most part, populates the higher mass Dþs ! KþK

þ

sideband. However, this cannot be removed without bias-ing the Dþs Dalitz plot, and so this background is taken into

account in the Dalitz plot analysis.

We also observe a clean peak in the distribution of the mass difference mðK

þ_mis
þÞ
mðK

þ_misÞ. Combining mðK

þ_misÞ with each of the
0 meson candidates in the event, we identify this contamination as due to Dþ!
þD0_{ð! K}

þ
0_{Þ with a missing}
0_{. We remove events}

that satisfy mðK

þ

mis
þÞ
mðK

þmisÞ < 0:15 GeV=c2.

0 0.01 0.02 0.03 0.04 0.05 3 4 5 p* (GeV/c) Probability (a) 0 0.02 0.04 0.06 0.08 0.1 0.12 -0.1 0 0.1 0.2 0.3 d_xy (cm) (b) 0 0.02 0.04 0.06 0.08 0 0.5 1 P₁ - P₂ (c)

FIG. 1 (color online). Normalized probability distribution functions for signal (solid) and background events (hatched) used in a likelihood-ratio test for the event selection of Dþs ! KþK

þ: (a) the center-of-mass momentum p, (b) the signed decay distance dxy, and (c) the difference in probability P1
P2.

Finally, we remove the Dþs candidates that share one or

two daughters with another Dþs candidate; this reduces the

number of candidates by 1.8%, corresponding to 0.9% of events. We allow there to be two or more nonoverlapping multiple candidates in the same event. The resulting KþK

þ mass distribution is shown in Fig. 2(a). This distribution is fitted with a double-Gaussian function for the signal, and a linear background. The fit gives a Dþ_s mass of 1968:70 0:02_stat MeV=c2_{, }

1¼ 4:96 

0:06_stat MeV=c2_{, }

2=1 ¼ 1:91  0:06stat where 1 (2)

is the standard deviation of the first (second) Gaussian, and errors are statistical only. The fractions of the two Gaussians are f1 ¼ 0:80  0:02 and f2¼ 0:20  0:02.

The signal region is defined to be within 2_Dþ s of

the fitted mass value, where Dþs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f1 2 1þ f2 2 2 q ¼ 6:1 MeV=c2_{is the observed mass resolution (the simulated}

mass resolution is 6 MeV=c2_{). The number of signal}

events in this region (Signal), and the corresponding purity [defined as Signal/ðSignal þ BackgroundÞ], are given in TableI.

For events in the Dþs ! KþK

þ signal region, we

obtain the Dalitz plot shown in Fig.2(b). For this distribu-tion, and for the Dalitz plot analysis (Sec.VI), we use the track parameters obtained from the Dþ_s mass-constrained fit, since this yields a unique Dalitz plot boundary.

In the KþK
threshold region, a strong ð1020Þ signal is observed, together with a rather broad structure. The f0ð980Þ and a0ð980Þ S-wave resonances are, in fact,

close to the KþK
threshold, and might be expected to

contribute in the vicinity of the ð1020Þ. A strong

Kð892Þ0 _{signal can also be seen in the K}

þ _system,

but there is no evidence of structure in the Kþ
þ mass. IV. EFFICIENCY

The selection efficiency for each Dþ_s decay mode ana-lyzed is determined from a sample of Monte Carlo (MC) events in which the Dþ_s decay is generated according to phase space (i.e. such that the Dalitz plot is uniformly populated). The generated events are passed through a detector simulation based on the GEANT4toolkit [14], and subjected to the same reconstruction and event selection procedure as that applied to the data. The distribution of the selected events in each Dalitz plot is then used to determine the reconstruction efficiency. The MC samples used to compute these efficiencies consist of 4:2 106 generated events for Dþs ! KþK

þ and Dþs ! KþKþ

, and

0:7 106 for Dþs ! KþK
Kþ.

For Dþs ! KþK

þ, the efficiency distribution is fitted

to a third-order polynomial in two dimensions using the expression,

ðx; yÞ ¼ a0þ a1x0þ a3x02þ a4y02þ a5x0y0

þ a6x03þ a7y03; (6)

where x¼ m2ðKþK
Þ, y ¼ m2ðK

þÞ, x0¼ x
2, and y0¼ y
1:25. Coefficients consistent with zero have been omitted. We obtain a good description of the efficiency with 2_{=NDF¼ 1133=ð1147
7Þ ¼ 0:994}

(where NDF refers to the number of degrees of freedom). The efficiency is found to be almost uniform in K

þand KþK
mass, with an average value of 3:3% (Fig.3).

V. PARTIAL-WAVE ANALYSIS OF THEKþK
ANDK

þTHRESHOLD REGIONS

In the KþK
threshold region both a0ð980Þ and f0ð980Þ

can be present, and both resonances have very similar parameters which suffer from large uncertainties. In this

0 2000 4000 6000 8000 1.9 1.95 2 2.05 m(K+K-π+) (GeV/c2) events/1 MeV/c 2 Signal (a) 0.5 1 1.5 2 1 1.5 2 2.5 3 3.5 m2(K+K-) (GeV2/c4) m 2 (K - π + ) (GeV 2 /c 4 ) (b)

FIG. 2 (color online). (a) KþK

þ mass distribution for the Dþs analysis sample; the signal region is as indicated; (b) Dþs ! KþK

þDalitz plot.

TABLE I. Yields and purities for the different Dþs decay modes. Quoted uncertainties are statistical only.

Dþs decay mode Signal yield Purity (%)

KþK

þ 96307 369 95

KþK
Kþ 748 60 28

section we obtain model-independent information on the KþK
S wave by performing a partial-wave analysis in the KþK
threshold region.

Let N be the number of events for a given mass interval I¼ ½m_Kþ_K
; m_Kþ_K
þ dm_Kþ_K
. We write the

corre-sponding angular distribution in terms of the appropriate spherical harmonic functions as

dN d cos ¼ 2
XL k¼0 hY0 kiYk0ðcosÞ; (7)

where L¼ 2‘max, and ‘max is the maximum orbital

angular momentum quantum number required to describe the KþK
system at mKþK
(e.g. ‘max¼ 1 for an S-,

P -wave description);  is the angle between the Kþ

direction in the KþK
rest frame and the prior direction of the KþK
system in the Dþ_s rest frame. The normal-izations are such that

Z1
1Y

0

kðcosÞYj0ðcosÞd cos ¼

kj

2
; (8)

and it is assumed that the distribution dN

d cos has been

efficiency corrected and background subtracted.

Using this orthogonality condition, the coefficients in the expansion are obtained from

hY0 ki ¼ Z1
1Y 0 kðcosÞ dN d cosd cos; (9) where the integral is given, to a good approximation, by P_N

n¼1Yk0ðcosnÞ, where n is the value of  for the n-th

event.

Figure 4 shows the KþK
mass spectrum up to_{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}1:5 GeV=c2 weighted by Y0_kðcosÞ ¼

ð2k þ 1Þ=4
p

PkðcosÞ for k ¼ 0, 1, and 2, where Pk is

the Legendre polynomial function of order k. These distributions are corrected for efficiency and phase space, and background is subtracted using the Dþs sidebands.

The number of events N for the mass interval I can be expressed also in terms of the partial-wave amplitudes describing the KþK
system. Assuming that onlyS- and P -wave amplitudes are necessary in this limited region, we can write: 0 20000 40000 60000 80000 100000 120000 0.95 1 1.05 1.1 1.15 m(K+K-) (GeV/c2) Events/4 MeV/c 2 (a)〈Y 0 0〉 0 5000 10000 15000 20000 25000 0.95 1 1.05 1.1 1.15 m(K+K-) (GeV/c2) (b)〈Y01〉 0 20000 40000 60000 80000 0.95 1 1.05 1.1 1.15 (c)〈Y02〉 m(K+K-) (GeV/c2)

FIG. 4. KþK
mass spectrum in the threshold region weighted by (a) Y0

0, (b) Y01, and (c) Y02, corrected for efficiency and phase space, and background subtracted.

0.5 1 1.5 2 1 1.5 2 2.5 3 3.5 0.02 0.026 0.032 0.038 0.044 0.05 m2(K+K-) (GeV2/c4) m 2 (K - π + ) (GeV 2 /c 4 ) (a) 0 0.01 0.02 0.03 0.04 1 1.5 2 2.5 3 3.5 m2(K+K-) (GeV2/c4) Efficiency (b) 0 0.01 0.02 0.03 0.04 0.5 1 1.5 2 m2(K-π+) (GeV2/c4) Efficiency (c)

FIG. 3. (a) Dalitz plot efficiency map; the projection onto (b) the KþK
, and (c) the K

þaxis.

dN

d cos ¼ 2
jSY

0

0ðcosÞ þ P Y10ðcosÞj2: (10)

By comparing Eqs. (7) and (10) [15], we obtain ffiffiffiffiffiffiffi 4
p hY0 0i ¼ jSj2þ jP j2; ffiffiffiffiffiffiffi 4
p hY0 1i ¼ 2jSjjP j cosSP; ffiffiffiffiffiffiffi 4
p hY0 2i ¼ 2ffiffiffi 5 p jP j2_; (11)

where _SP ¼ _S
_P is the phase difference between theS- and P -wave amplitudes. These equations relate the interference between theS wave [f0ð980Þ, and/or a0ð980Þ,

and/or nonresonant] and the P wave [ð1020Þ] to the prominent structure inhY0

1i [Fig.4(b)]. ThehY10i

distribu-tion shows the same behavior as for Dþs ! KþK
eþe

decay [16]. ThehY0

2i distribution [Fig.4(c)], on the other

hand, is consistent with the ð1020Þ line shape.

The above system of equations can be solved in each interval of KþK
invariant mass forjSj, jP j, and _SP, and the resulting distributions are shown in Fig. 5. We observe a threshold enhancement in the S wave [Fig. 5(a)], and the expected ð1020Þ Breit-Wigner (BW) in the P wave [Fig. 5(b)]. We also observe the expectedS-P relative phase motion in the ð1020Þ region [Fig.5(c)].

A.P -wave/S-wave ratio in the ð1020Þ region The decay mode Dþs ! ð1020Þ
þis used often as the

normalizing mode for Dþs decay branching fractions,

typi-cally by selecting a KþK
invariant mass region around the ð1020Þ peak. The observation of a significant S-wave contribution in the threshold region means that this con-tribution must be taken into account in such a procedure.

In this section we estimate theP -wave/S-wave ratio in an almost model-independent way. In fact integrating the distributions ofpffiffiffiffiffiffiffi4
pq0hY0 0i and ffiffiffiffiffiffiffi 5
p pq0hY0 2i (Fig.4)

in a region around the ð1020Þ peak yields RðjSj2_þ

jP j2_Þpq0_dm

KþK
and

R

jP j2_pq0_dm

KþK
, respectively,

where p is the Kþ momentum in the KþK
rest frame, and q0is the momentum of the bachelor
þin the Dþ_s rest frame.

The S-P interference contribution integrates to zero, and we define theP -wave and S-wave fractions as

f_P-wave ¼ R jP j2_pq0_dm KþK
R ðjSj2_{þ jP j}2_Þpq0_dm KþK
; (12) f_S-wave¼ R jSj2_pq0_dm KþK
R ðjSj2_{þ jP j}2_Þpq0_dm KþK
¼ 1
fP-wave: (13) The experimental mass resolution is estimated by comparing generated and reconstructed MC events, and is ’ 0:5 MeV=c2 at the  mass peak. Table II gives the

0 10000 20000

Events/4 MeV/c

(a)

|S|

(b)

|P|

Events/0.5 MeV/c

Phase difference (degree)

(c)

φ

m(K

K

) (GeV/c

)

S-wave phase (degree)

(d)

φ

FIG. 5. Squared (a) S- and (b) P -wave amplitudes; (c) the phase difference _SP; (d) _S obtained as explained in the text. The curves result from the fit described in the text.

resultingS-wave and P -wave fractions computed for three KþK
mass regions. The last column of TableIIshows the measurements of the relative overall rate ( N

Ntot) defined as

the number of events in the KþK
mass interval over the number of events in the entire Dalitz plot after efficiency-correction and background-subtraction.

B.S-wave parametrization at the KþK
threshold In this section we extract a phenomenological descrip-tion of theS wave assuming that it is dominated by the f0ð980Þ resonance while the P wave is described entirely

by the ð1020Þ resonance. We also assume that no other contribution is present in this limited region of the Dalitz plot. We therefore perform a simultaneous fit of the three distributions shown in Figs.5(a)–5(c)using the following model: dN_S2 dm_Kþ_K
¼ jCf0ð980ÞAf0ð980Þj 2_; dN_P2 dm_Kþ_K
¼ jCAj 2_; dN_SP dmKþK
¼ argðAf0ð980Þe i _Þ
argðA Þ; (14)

where C_, C_f0ð980Þ, and are free parameters and A ¼

F_rF_D m2


m2
im

 4pq (15) is the spin 1 relativistic BW parametrizing the ð1020Þ with  expressed as ¼ r _p p_r _2Jþ1_M r m  Fr2: (16)

Here q is the momentum of the bachelor
þ in the KþK
rest frame. The parameters in Eqs. (15) and (16) are defined in Sec.VIbelow.

For Af0ð980Þwe first tried a coupled channel BW (Flatte´)

amplitude [17]. However, we find that this parametrization is insensitive to the coupling to the

channel. Therefore, we empirically parametrize the f0ð980Þ with the following

function:

Af0ð980Þ¼

1

m2₀
m2
im00KK

; (17)

where _KK ¼ 2p=m, and obtains the following parameter values:

m₀¼ ð0:922  0:003_statÞ GeV=c2_;

0 ¼ ð0:24  0:08statÞ GeV:

(18) The errors are statistical only. The fit results are super-imposed on the data in Fig.5.

In Fig. 5(c), theS-P phase difference is plotted twice because of the sign ambiguity associated with the value of _SP extracted from cos_SP. We can extract the mass-dependent f0ð980Þ phase by adding the mass-dependent

ð1020Þ BW phase to the _SP distributions of Fig.5(c). Since the KþK
mass region is significantly above the f₀ð980Þ central mass value of Eq. (18), we expect that the S-wave phase will be moving much more slowly in this region than in the ð1020Þ region. Consequently, we re-solve the phase ambiguity of Fig.5(c)by choosing as the physical solution the one which decreases rapidly in the ð1020Þ peak region, since this reflects the rapid forward BW phase motion associated with a narrow resonance. The result is shown in Fig.5(d), where we see that theS-wave phase is roughly constant, as would be expected for the tail of a resonance. The slight decrease observed with increas-ing mass might be due to higher mass contributions to the S-wave amplitude. The values of jSj2_{(arbitrary units) and}

phase values are reported in Table III, together with the corresponding values ofjP j2_.

In Fig. 6(a) we compare the S-wave profile from this analysis with the S-wave intensity values extracted from Dalitz plot analyses of D0 _!
K0_Kþ_{K
[18] and}

D0_{! K}þ_K

0 _{[19]. The four distributions are}

normal-ized in the region from threshold up to 1:05 GeV=c2_{. We}

observe substantial agreement. As the a0ð980Þ and f0ð980Þ

mesons couple mainly to the u
u=d
d and s
s systems, re-spectively, the former is favored in D0 _!
K0_Kþ_K
and

the latter in Dþs ! KþK

þ. Both resonances can

con-tribute in D0! KþK

0. We conclude that the S-wave projections in the K
K system for both resonances are consistent in shape. It has been suggested that this feature supports the hypothesis that the a0ð980Þ and f0ð980Þ are

4-quark states [20]. We also compare the S-wave profile from this analysis with the
þ

S-wave profile extracted from BABAR data in a Dalitz plot analysis of Dþ_s !
þ


þ[4] [Fig.6(b)]. The observed agreement supports the argument that only the f₀ð980Þ is present in this limited mass region.

C. Study of theK

þS wave at threshold We perform a model-independent analysis, similar to that described in the previous sections, to extract the K
S-wave behavior as a function of mass in the threshold region up to 1:1 GeV=c2_{. Figure7shows the K}

þ_mass

spectrum in this region, weighted by Y0

kðcosÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2k þ 1Þ=4
p

PkðcosÞ, with k ¼ 0, 1, and 2, corrected

TABLE II. S-wave and P -wave fractions computed in three KþK
mass ranges around the ð1020Þ peak. Errors are statis-tical only.

mKþK
(MeV=c2) fS-wave(%) fP-wave(%) NtotN (%) 1019:456 5 3:5 1:0 96:5 1:0 29:4 0:2 1019:456 10 5:6 0:9 94:4 0:9 35:1 0:2 1019:456 15 7:9 0:9 92:1 0:9 37:8 0:2

for efficiency, phase space, and with background from the Dþ_s sidebands subtracted;  is the angle between the K
direction in the K

þ rest frame and the prior direction of the K

þ system in the Dþs rest frame. We observe

thathY0

0i and hY02i show strong
Kð892Þ0resonance signals,

and that the hY0

1i moment shows evidence for S-P

interference.

We use Eqs. (11) to solve forjSj and jP j. The result for the S wave is shown in Fig. 7(d). We observe a small

S-wave contribution which does not allow us to measure the expected phase motion relative to that of the
Kð892Þ0

resonance. Indeed, the fact that jSj2 _{goes negative}

indi-cates that a model including onlyS- and P -wave compo-nents is not sufficient to describe the K

þ system.

VI. DALITZ PLOT FORMALISM

An unbinned maximum likelihood fit is performed in which the distribution of events in the Dalitz plot is used to

TABLE III. S- and P -wave squared amplitudes (in arbitrary units) and the S-wave phase. The S-wave phase values, corresponding to the mass 0.988 and 1:116 GeV=c2_{, are missing because} the hY0

2i distribution [Fig.4(c)] goes negative or j cosSPj > 1 and so Eqs. (11) cannot be solved. Quoted uncertainties are statistical only.

mKþK
(GeV=c2) jSj2 (arbitrary units) jP j2(arbitrary units) S (degrees)

0.988 22 178 3120
133  2283 0.992 18 760 1610 2761 1313 92 5 0.996 16 664 1264 1043 971 84 7 1 12 901 1058 3209 882 81 4 1.004 13 002 1029 5901 915 82 3 1.008 9300 964 13 484 1020 76 3 1.012 9287 1117 31 615 1327 80 2 1.016 6829 1930 157 412 2648 75 8 1.02 11 987 2734 346 890 3794 55 6 1.024 5510 1513 104 892 2055 86 5 1.028 7565 952 32 239 1173 75 2 1.032 7596 768 15 899 861 74 2 1.036 6497 658 10 399 707 77 2 1.04 5268 574 7638 609 72 3 1.044 5467 540 5474 540 72 3 1.048 5412 506 4026 483 72 3 1.052 5648 472 2347 423 71 3 1.056 4288 442 3056 421 70 3 1.06 4548 429 1992 384 73 3 1.064 4755 425 1673 374 70 4 1.068 4508 393 1074 334 75 4 1.072 3619 373 1805 345 75 4 1.076 4189 368 840 312 70 5 1.08 4215 367 770 297 71 5 1.084 3508 345 866 294 71 5 1.088 3026 322 929 285 75 4 1.092 3456 309 79 240 37 90 1.096 2903 300 488 256 75 6 1.1 2335 282 885 248 68 5 1.104 2761 284 341 231 57 10 1.108 2293 273 602 231 77 5 1.112 1913 238 269 186 74 8 1.116 2325 252 57 198 1.12 1596 228 308 194 78 7 1.124 1707 224 233 188 67 10 1.128 1292 207 270 176 66 9 1.132 969 197 586 172 60 6 1.136 1092 196 553 170 67 6 1.14 1180 193 316 167 48 11 1.144 1107 187 354 170 68 8 1.148 818 178 521 164 64 7

determine the relative amplitudes and phases of intermedi-ate resonant and nonresonant stintermedi-ates.

The likelihood function is written as L ¼ YN n¼1  f_sig ðx; yÞ P i;j cicjAiðx; yÞAjðx; yÞ P i;j cicjIAiAj þ ð1
fsigÞ P i kiBiðx; yÞ P i k_iI_Bi  ; (19) where

(i) N is the number of events in the signal region; (ii) x¼ m2_ðKþ_{K
Þ and y ¼ m}2_ðK

þ_Þ;

(iii) fsig is the fraction of signal as a function of the

KþK

þ invariant mass, obtained from the fit to the KþK

þ mass spectrum [Fig.2(a)];

(iv) ðx; yÞ is the efficiency, parametrized by a third order polynomial (Sec.IV);

(v) the Aiðx; yÞ describe the complex signal amplitude

contributions;

(vi) the Biðx; yÞ describe the background probability

density function contributions;

(vii) kiis the magnitude of the i-th component for the

background. The k_i parameters are obtained by fitting the sideband regions;

(viii) I_AiA j ¼

R

A_iðx; yÞA_jðx; yÞðx; yÞdxdy and I_Bi¼ R

Biðx; yÞdxdy are normalization integrals.

Numerical integration is performed by means of Gaussian quadrature [21];

(ix) ciis the complex amplitude of the i-th component

for the signal. The ciparameters are allowed to vary

during the fit process.

The phase of each amplitude (i.e. the phase of the corresponding ci) is measured with respect to the

KþK
ð892Þ0 amplitude. Following the method described in Ref. [22], each amplitude Aiðx; yÞ is represented by the

product of a complex BW and a real angular term T depending on the solid angle :

Aðx; yÞ ¼ BWðmÞ  TðÞ: (20)

For a Dsmeson decaying into three pseudoscalar mesons

via an intermediate resonance r (Ds! rC, r ! AB),

BWðMABÞ is written as a relativistic BW:

FIG. 6 (color online). (a) Comparison between K
K S-wave intensities from different charmed meson Dalitz plot analyses. (b) Comparison of the K
KS-wave intensity from Dþs ! KþK

þwith the
þ

S-wave intensity from Dþs !
þ


þ.

0 5000 10000 15000 20000 0.6 0.7 0.8 0.9 1 1.1 m(K-π+) (GeV/c2) events/10 MeV/c 2 (a)〈Y00〉 -1000 0 1000 2000 3000 0.6 0.7 0.8 0.9 1 1.1 m(K-π+) (GeV/c2) (b)〈Y01〉 0 5000 10000 15000 20000 0.6 0.7 0.8 0.9 1 1.1 m(K-π+) (GeV/c2) (c)〈Y02〉 -2000 0 2000 4000 0.6 0.7 0.8 0.9 1 1.1 m(K-π+) (GeV/c2) (d)|S|2

FIG. 7. K

þmass spectrum in the threshold region weighted by (a) Y0

0, (b) Y10, and (c) Y20, corrected for efficiency, phase space, and background subtracted. (d) The K

þ mass dependence ofjSj2_.

BWðMABÞ ¼

FrFD

M2

r
M2AB
iABMr

; (21)

where AB is a function of the invariant mass of system

AB (MAB), the momentum pAB of either daughter in the

AB rest frame, the spin J of the resonance and the mass M_r, and the width _r of the resonance. The explicit expression is AB¼ r _p AB p_r _2Jþ1_M r M_AB  Fr2; (22) pAB¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðM2 AB
M2A
M2BÞ2
4M2AMB2 q 2MAB : (23)

The form factors F_r and F_D attempt to model the underlying quark structure of the parent particle and the intermediate resonances. We use the Blatt-Weisskopf penetration factors [23] (Table IV), which depend on a single parameter R representing the meson ‘‘radius.’’ We assume R_Dþ

s ¼ 3 GeV

1 _{for the D}

s and Rr¼ 1:5 GeV
1

for the intermediate resonances; qAB is the momentum of

the bachelor C in the AB rest frame:

q_AB¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðM2 Dsþ M 2 C
M2ABÞ2
4M2DsM 2 C q 2MAB : (24)

prand qrare the values of pAB and qAB when mAB ¼ mr.

The angular terms TðÞ are described by the following expressions: Spin 0: TðÞ ¼ 1; Spin 1: TðÞ ¼ M2 BC
M2AC
ðM2 Ds
M 2 CÞðMB2
MA2Þ M2 AB ; Spin 2: TðÞ ¼ a2 1
1 3a2a3; (25) where a₁ ¼ M2 BC
MAC2 þ ðM2 Ds
M 2 CÞðM2A
MB2Þ M2 AB ; a2¼ M2AB
2M2Ds
2M 2 Cþ ðM2 Ds
M 2 CÞ2 M2 AB ; a3¼ M2AB
2M2A
2M2Bþ ðM2 A
M2BÞ2 M2 AB : (26)

Resonances are included in sequence, starting from those immediately visible in the Dalitz plot projections. All allowed resonances from Ref. [10] have been tried, and we reject those with amplitudes consistent with zero. The goodness of fit is tested by an adaptive binning 2_.

The efficiency-corrected fractional contribution due to the resonant or nonresonant contribution i is defined as follows: fi¼ jcij2 R jAiðx; yÞj2dxdy R jP j cjAjðx; yÞj2dxdy : (27)

The fido not necessarily add to 1 because of interference

effects. We also define the interference fit fraction between the resonant or nonresonant contributions k and l as:

fkl¼ 2R<½c_RkclAkðx; yÞAlðx; yÞdxdy jP j cjAjðx; yÞj2dxdy : (28)

Note that fkk¼ 2fk. The error on each fi and fkl is

evaluated by propagating the full covariance matrix ob-tained from the fit.

Background parametrization

To parametrize the Dþs background, we use the Dþs

sideband regions. An unbinned maximum likelihood fit is performed using the function:

L ¼Y NB n¼1 P i k_iB_i P i kiIBi  ; (29)

where NB is the number of sideband events, the ki

parameters are real coefficients floated in the fit, and the Bi parameters represent Breit-Wigner functions that are

summed incoherently.

The Dalitz plot for the two sidebands shows the presence of ð1020Þ and
Kð892Þ0_{(Fig.8). There are further}

struc-tures not clearly associated with known resonances and due to reflections of other final states. Since they do not have definite spin, we parametrize the background using an incoherent sum ofS-wave Breit-Wigner shapes.

VII. DALITZ PLOT ANALYSIS OFDþ_s ! KþK

þ Using the method described in Sec.VI, we perform an unbinned maximum likelihood fit to the Dþs ! KþK

þ

decay channel. The fit is performed in steps, by adding

TABLE IV. Summary of the Blatt-Weisskopf penetration form factors. qr and pr are the momenta of the decay particles in the parent rest frame.

Spin Fr FD 0 1 1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þðRrprÞ2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þðRrpABÞ2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þðRDþsqrÞ 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þðRDþ sqABÞ 2 p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9þ3ðRrprÞ2_þðRrprÞ4 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9þ3ðRrpABÞ2_þðRrpABÞ4 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9þ3ðRDþsqrÞ 2_þðRDþ sqrÞ 4 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9þ3ðRDþ sqABÞ 2_þðRDþ sqABÞ 4 p

resonances one after the other. Most of the masses and widths of the resonances are taken from Ref. [10]. For the f0ð980Þ we use the phenomenological model described in

Sec.V B. The
Kð892Þ0 _{amplitude is chosen as the}

refer-ence amplitude.

The decay fractions, amplitudes, and relative phase val-ues for the best fit obtained, are summarized in TableV

where the first error is statistical, and the second is system-atic. The interference fractions are quoted in Table VI

where the error is statistical only. We observe the following features.

(i) The decay is dominated by the
Kð892Þ0Kþ and ð1020Þ
þ_amplitudes.

(ii) The fit quality is substantially improved by leaving the
Kð892Þ0 _{parameters free in the fit. The fitted}

parameters are

mK
ð892Þ0¼ ð895:60:2_stat0:3_sysÞ MeV=c2;

K
ð892Þ0¼ ð45:10:4_stat0:4_sysÞ MeV:

(30)

We notice that the width is about 3 MeV lower than that in Ref. [10]. However this measurement is

0.5 1 1.5 2 1 1.5 2 2.5 3 3.5 m2(K+K-) (GeV2/c4) m 2 (K - π + ) (GeV 2 /c 4 ) (a) 0 200 400 600 1 1.5 2 2.5 3 3.5 m2(K+K-) (GeV2/c4) Events/0.027 GeV 2 /c 4 (b) 0 100 200 0.5 1 1.5 2 m2(K-π+) (GeV2/c4) Events/0.02 GeV 2 /c 4 _(c)

FIG. 8 (color online). (a) Dalitz plot of sideband regions projected onto (b) the KþK
and (c) the K

þaxis.

TABLE V. Results from the Dþs ! KþK

þDalitz plot analysis. The table gives fit fractions, amplitudes, and phases from the best fit. Quoted uncertainties are statistical and systematic, respectively.

Decay mode Decay function (%) Amplitude Phase (radians)

Kð892Þ0_Kþ _{47:9 0:5  0:5 1. (Fixed) 0. (Fixed)} ð1020Þ
þ _{41:4 0:8  0:5 1:15 0:01  0:26 2:89 0:02  0:04} f0ð980Þ
þ 16:4 0:7  2:0 2:67 0:05  0:20 1:56 0:02  0:09
K₀ð1430Þ0_Kþ _{2:4 0:3  1:0 1:14 0:06  0:36 2:55 0:05  0:22} f0ð1710Þ
þ 1:1 0:1  0:1 0:65 0:02  0:06 1:36 0:05  0:20 f0ð1370Þ
þ 1:1 0:1  0:2 0:46 0:03  0:09
0:45  0:11  0:52 Sum 110:2 0:6  2:0 2_{=NDF 2843=ð2305
14Þ ¼ 1:24}

TABLE VI. Fit fractions matrix of the best fit. The diagonal elements fi correspond to the decay fractions in Table V. The off-diagonal elements give the fit fractions of the interference fkl. The null values originate from the fact that any S-P interference contribution integrates to zero. Quoted uncertainties are statistical only.

fkl (%) K
ð892Þ0Kþ ð1020Þ
þ f0ð980Þ
þ K
0ð1430Þ0Kþ f0ð1710Þ
þ f0ð1370Þ
þ
Kð892Þ0_Kþ _{47:9 0:5
4:36  0:03
2:4  0:2 0.
0:06  0:03 0:08 0:08} ð1020Þ
þ _{41:4 0:8 0.
0:7  0:2 0. 0.} f0ð980Þ
þ 16:4 0:7 4:1 0:6
3:1  0:2
4:5  0:3
K₀ð1430Þ0_Kþ _{2:4 0:3 0:48 0:08
0:7  0:1} f0ð1710Þ
þ 1:1 0:1 0:86 0:06 f0ð1370Þ
þ 1:1 0:1

consistent with results from other Dalitz plot analy-ses [9].

(iii) The f₀ð1370Þ contribution is also left free in the fit, and we obtain the following parameter values:

mf0ð1370Þ¼ ð1:22  0:01stat 0:04sysÞ GeV=c

2_;

_f0ð1370Þ¼ ð0:21  0:01_stat 0:03_sysÞ GeV: (31) These values are within the broad range of values measured by other experiments [10].

(iv) A nonresonant contribution, represented by a con-stant complex amplitude, was included in the fit function. However, this contribution was found to be consistent with zero, and therefore is excluded from the final fit function.

(v) In a similar way contributions from the K₁ð1410Þ, f0ð1500Þ, f2ð1270Þ, and f02ð1525Þ are found to be

consistent with zero.

(vi) The replacement of the K₀ð1430Þ by the LASS parametrization [24] of the entire K
S wave does not improve the fit quality.

(vii) The fit does not require any contribution from the ð800Þ [1].

The results of the best fit [2_{=NDF¼ 2843=}

ð2305
14Þ ¼ 1:24] are superimposed on the Dalitz plot projections in Fig. 9. Other recent high statistics charm Dalitz plot analyses at BABAR [25] have shown that a significant contribution to the 2_{=NDF can arise from}

imperfections in modelling experimental effects. The nor-malized fit residuals shown under each distribution (Fig.9) are given by Pull¼ ðNdata
NfitÞ=

ffiffiffiffiffiffiffiffiffiffi Ndata

p

. The data are well reproduced in all the projections. We observe some disagreement in the K

þprojection below 0:5 GeV2_=c4_.

It may be due to a poor parametrization of the background in this limited mass region. A systematic uncertainty takes such effects in account (Sec.VII A). The missing of a K
S-wave amplitude in the K

þ _{low mass region may be}

also the source of such disagreement.

0 5000 10000 15000 20000 25000 1 1.5 2 2.5 3 3.5 m2(K+K-) (GeV2/c4) events/0.027 GeV 2/c 4 0 1000 2000 3000 4000 5000 6000 7000 1 1.05 1.1 m2(K+K-) (GeV2/c4) events/0.0024 GeV 2 /c 4 0 2000 4000 6000 8000 0.5 1 1.5 2 m2(K-π+) (GeV2/c4) events/0.02 GeV 2/c 4 0 1000 2000 3000 4000 0.5 1 1.5 2 m2(K+π+) (GeV2/c4) events/0.02 GeV 2 /c 4 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3

FIG. 9. Dþs ! KþK

þ: Dalitz plot projections from the best fit. The data are represented by points with error bars, the fit results by the histograms.

Another way to test the fit quality is to project the fit results onto the hY0

ki moments, shown in Fig.10 for the

KþK
system and Fig. 11 for the K

þ system. We observe that the fit results reproduce the data projections for moments up to k¼ 7, indicating that the fit describes the details of the Dalitz plot structure very well. The K

þ hY0

3i and hY50i moments show activity in the
Kð892Þ 0

region which the Dalitz plot analysis relates to interference between the
Kð892Þ0_Kþ _{and f}

0ð1710Þ
þ decay

ampli-tudes. This seems to be a reasonable explanation for the failure of the model-independent K

þ analysis (Sec.V C), although the fit still does not provide a good description of the hY0

3i and hY50i moments in this mass

region.

We check the consistency of the Dalitz plot results and those of the analysis described in Sec.V B. We compute the amplitude and phase of the f0ð980Þ=S wave relative to

the ð1020Þ=P wave and find good agreement. A. Systematic errors

Systematic errors given in TableVand in other quoted results take into account:

(i) Variation of the R_r and R_Dþ

s constants in the

Blatt-Weisskopf penetration factors within the range ½0–3 GeV
1_{and½1–5 GeV}
1_{, respectively.}

(ii) Variation of fixed resonance masses and widths within the1 error range quoted in Ref. [10]. (iii) Variation of the efficiency parameters within1

uncertainty.

(iv) Variation of the purity parameters within 1 uncertainty.

(v) Fits performed with the use of the lower/upper sideband only to parametrize the background. (vi) Results from fits with alternative sets of signal

amplitude contributions that give equivalent Dalitz plot descriptions and similar sums of fractions.

(vii) Fits performed on a sample of 100 000 events selected by applying a looser likelihood-ratio cri-terion but selecting a narrower ( 1_Dþ

s) signal

region. For this sample the purity is roughly the same as for the nominal sample (’ 94:9%).

B. Comparison between Dalitz plot analyses ofDþ_s ! KþK

þ

Table VII shows a comparison of the Dalitz plot fit fractions, shown in TableV, with the results of the analyses performed by the E687 [8] and CLEO [9] Collaborations. The E687 model is improved by adding a f₀ð1370Þ ampli-tude and leaving the
Kð892Þ0 _{parameters free in the fit.}

0 1000 2000 3000 4000 5000 6000 1 1.5 m(K+K-) (GeV/c2) 〈Y0 0〉 events/10 MeV/c 2 -500 0 500 1000 1500 1 1.5 m(K+K-) (GeV/c2) 〈Y0 1〉 0 1000 2000 3000 4000 5000 1 1.5 m(K+K-) (GeV/c2) 〈Y0 2〉 -200 -100 0 100 200 300 400 500 1 1.5 m(K+K-) (GeV/c2) 〈Y0 3〉 -400 -200 0 200 400 1 1.5 m(K+K-) (GeV/c2) 〈Y0 4〉 -200 -100 0 100 200 300 1 1.5 m(K+K-) (GeV/c2) 〈Y0 5〉 -300 -200 -100 0 100 200 1 1.5 m(K+K-) (GeV/c2) 〈Y0 6〉 -200 -100 0 100 200 1 1.5 m(K+K-) (GeV/c2) 〈Y0 7〉 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 0 500 1000 1500 1. 1.025 0 100 200 300 400 1. 1.025 0 500 1000 1. 1.025

FIG. 10. KþK
mass dependence of the spherical harmonic moments,hY0

ki, obtained from the fit to the Dþs ! KþK

þDalitz plot compared to the data moments. The data are represented by points with error bars, the fit results by the histograms. The insets show an expanded view of the ð1020Þ region.

We find that the
Kð892Þ0width [Eq. (30)] is about 3 MeV lower than that in Ref. [10]. This result is consistent with the width measured by CLEO-c Collaboration (K
ð892Þ0 ¼

45:7 1:1 MeV).

What is new in this analysis is the parametrization of the KþK
S wave at the KþK
threshold. While E687 and CLEO-c used a coupled channel BW (Flatte´) amplitude [17] to parametrize the f0ð980Þ resonance, we

use the model-independent parametrization described in Sec.V B. This approach overcomes the uncertainties that affect the coupling constants g

and gKK of the f0ð980Þ,

and any argument about the presence of an að980Þ meson

decaying to KþK
. The model, described in this paper, returns a more accurate description of the event distribu-tion on the Dalitz plot (2_{=¼ 1:2) and smaller f}

0ð980Þ

and total fit fractions with respect to the CLEO-c result. In addition the goodness of fit in this analysis is tested by an adaptive binning 2_{, a tool more suitable when}

most of the events are gathered in a limited region of the Dalitz plot.

Finally, we observe that the phase of the ð1020Þ am-plitude (166 1 2) is consistent with the E687 result (178 20 24) but is roughly shifted by 180respect to the CLEO-c result (
8 4 4).

0 500 1000 1500 2000 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 0〉 events/10 MeV/c 2 -1500 -1000 -500 0 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 1〉 0 500 1000 1500 2000 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 2〉 -2000 -1500 -1000 -500 0 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 3〉 0 500 1000 1500 2000 2500 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 4〉 -2500 -2000 -1500 -1000 -500 0 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 5〉 -500 0 500 1000 1500 2000 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 6〉 -2000 -1000 0 0.55 0.75 0.95 1.15 1.35 1.55 m(K-π+) (GeV/c2) 〈Y0 7〉 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3 Pull 0 3 -3

FIG. 11. K

þmass dependence of the spherical harmonic moments,hY0

ki, obtained from the fit to the Dþs ! KþK

þDalitz plot compared to the data moments. The data are represented by points with error bars, the fit results by the histograms.

TABLE VII. Comparison of the fitted decay fractions with the Dalitz plot analyses performed by E687 and CLEO-c Collaborations.

Decay fraction (%)

Decay mode BABAR E687 CLEO-c

Kð892Þ0_Kþ _{47:9 0:5  0:5 47:8 4:6  4:0 47:4 1:5  0:4} ð1020Þ
þ _{41:4 0:8  0:5 39:6 3:3  4:7 42:2 1:6  0:3} f0ð980Þ
þ 16:4 0:7  2:0 11:0 3:5  2:6 28:2 1:9  1:8
K0ð1430Þ0Kþ 2:4 0:3  1:0 9:3 3:2  3:2 3:9 0:5  0:5 f0ð1710Þ
þ 1:1 0:1  0:1 3:4 2:3  3:5 3:4 0:5  0:3 f0ð1370Þ
þ 1:1 0:1  0:2    4:3 0:6  0:5 Sum 110:2 0:6  2:0 111.1 129:5 4:4  2:0 2_=NDF 2843 ð2305
14Þ¼ 1:2 50:233 ¼ 1:5 178 117¼ 1:5 Events 96307 369 701 36 12226 22

VIII. SINGLY-CABIBBO-SUPPRESSED Dþ

s ! KþK
Kþ, AND

DOUBLY-CABIBBO-SUPPRESSEDDþ_s ! KþK


DECAY In this section we measure the branching ratio of the SCS decay channel (2) and of the DCS decay channel (3) with respect to the CF decay channel (1). The two channels are reconstructed using the method described in Sec. III

with some differences related to the particle-identification of the Dþs daughters. For channel (2) we require the

identification of three charged kaons while for channel (3) we require the identification of one pion and two kaons having the same charge. We use both the Dþs identification

and the likelihood ratio to enhance the signal with respect to background as described in Sec.III.

The ratios of branching fractions are computed as BðDþ s ! KþK
KþÞ BðDþ s ! KþK

þÞ ¼NDþs!KþK
Kþ NDþs!KþK

þ Dþs!KþK

þ Dþs!KþK
Kþ (32) and BðDþ s ! KþKþ

Þ BðDþ s ! KþK

þÞ ¼NDþs!KþKþ

N_Dþ s!KþK

þ Dþs!KþK

þ _Dþ s!KþKþ

: (33) Here the N values represent the number of signal events for each channel, and the  values indicate the corresponding detection efficiencies.

To compute these efficiencies, we generate signal MC samples having uniform distributions across the Dalitz plots. These MC events are reconstructed as for data events, and the same particle-identification criteria are applied. Each track is weighted by the data-MC discrep-ancy in particle-identification efficiency obtained indepen-dently from high statistics control samples. A systematic uncertainty is assigned to the use of this weight. The generated and reconstructed Dalitz plots are divided into 50 50 cells and the Dalitz plot efficiency is obtained as

the ratio of reconstructed to generated content of each cell. In this way the efficiency for each event depends on its location on the Dalitz plot. By varying the likelihood-ratio criterion, the sensitivity S of Dþs ! KþK
Kþ is

maxi-mized. The sensitivity is defined as S¼ Ns=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nsþ Nb

p

, where s and b indicate signal and background. To reduce systematic uncertainties, we then apply the same likelihood-ratio criterion to the Dþ_s ! KþK

þ decay. We then repeat this procedure to find an independently optimized selection criterion for the Dþs ! KþKþ

to

Dþs ! KþK

þratio.

The branching ratio measurements are validated using a fully inclusive eþe
! c
c MC simulation incorporating all known charmed meson decay modes. The MC events are subjected to the same reconstruction, event selection, and analysis procedures as for the data. The results are found to be consistent, within statistical uncertainty, with the branching fraction values used in the MC generation.

A. Study ofDþ_s ! KþK
Kþ

The resulting KþK
Kþ mass spectrum is shown in Fig. 12(a). The Dþ_s yield is obtained by fitting the mass spectrum using a Gaussian function for the signal, and a linear function for the background. The resulting yield is reported in TableI.

The systematic uncertainties are summarized in TableVIIIand are evaluated as follows:

(i) The effect of MC statistics is evaluated by random-izing each efficiency cell on the Dalitz plot accord-ing to its statistical uncertainty.

(ii) The selection made on the Dþs candidate m is

varied to2:5Dþs and1:5Dþs .

(iii) For particle identification we make use of high statistics control samples to assign 1% uncertainty to each kaon and 0.5% to each pion.

(iv) The effect of the likelihood-ratio criterion is studied by measuring the branching ratio for different choices. 0 100 200 300 400 1.9 1.95 2 2.05 m(K+K-K+) (GeV/c2) events/1.5 MeV/c 2 (a) 1 1.5 2 1 1.5 2 m2(K+K-) (GeV2/c4) m 2 (K + K - ) (GeV 2 /c 4 ) (b) -20 0 20 40 60 80 100 1 1.1 m(K+K-) (GeV/c2) events/1 MeV/c 2 (c) -20 -10 0 10 20 1 1.1 m2(K+K-) (GeV2/c4) events/4 MeV 2 /c 4 (d)〈Y 0 1〉 0 25 50 75 1. 1.02

FIG. 12 (color online). (a) KþK
Kþmass spectrum showing a Dþs signal. The curve is the result of the fit described in the text. (b) Symmetrized Dalitz plot, (c) KþK
mass spectrum (two combinations per event), and (d) thehY0

1i moment. The insert in (c) shows an expanded view of the ð1020Þ region. The Dalitz plot and its projection are background subtracted and efficiency corrected. The curve results from the fit described in the text.

We measure the following branching ratio: BðDþ s ! KþK
KþÞ BðDþ s ! KþK

þÞ ¼ ð4:0  0:3stat 0:2systÞ  10
3: (34) A Dalitz plot analysis in the presence of a high level of background is difficult, therefore we can only extract empirically some information on the decay. Since there are two identical kaons into the final state, the Dalitz plot is symmetrized by plotting two combinations per event (½m2_ðK
Kþ

1 Þ; m2ðK
K2þÞ and ½m2ðK
Kþ2Þ;

m2_ðK
Kþ

1 Þ). The symmetrized Dalitz plot in the Dþs !

KþK
Kþsignal region, corrected for efficiency and back-ground subtracted, is shown in Fig. 12(b). It shows two bands due to the ð1020Þ and no other structure, indicating a large contribution via Dþ_s ! ð1020ÞKþ. To test the possible presence of f₀ð980Þ, we plot, in Fig. 12(d), the distribution of thehY0

1i moment;  is the angle between the

Kþdirection in the KþK
rest frame and the prior direc-tion of the KþK
system in the Dþs rest frame. We observe

the mass dependence characteristic of interference be-tweenS- and P -wave amplitudes, and conclude that there is a contribution from Dþ_s ! f₀ð980ÞKþ decay, although its branching fraction cannot be determined in the present analysis.

An estimate of the ð1020ÞKþfraction can be obtained from a fit to the KþK
mass distribution [Fig.12(c)]. The mass spectrum is fitted using a relativistic BW for the ð1020Þ signal, and a second order polynomial for the background. We obtain:

BðDþ

s ! KþÞ  Bð ! KþK
Þ

BðDþ

s ! KþK
KþÞ

¼ 0:41  0:08stat 0:03syst: (35)

The systematic uncertainty includes the contribution due to m and the likelihood-ratio criteria, the fit model, and the background parametrization.

B. Study ofDþ_s ! KþKþ

Figure13(a)shows the KþKþ

mass spectrum. A fit with a Gaussian signal function and a linear background function gives the yield presented in TableI. To minimize systematic uncertainty, we apply the same likelihood-ratio criteria to the KþKþ

and KþK

þ final states, and correct for the efficiency evaluated on the Dalitz plot. The branching ratio which results is

BðDþ s ! KþKþ

Þ BðDþ s ! KþK

þÞ ¼ ð2:3  0:3stat 0:2systÞ  10
3: (36) This value is in good agreement with the Belle measure-ment:BðDþs!KþKþ

Þ

BðDþ

s!KþK

þÞ¼ ð2:29  0:28  0:12Þ  10

3_[11].

TableIX lists the results of the systematic studies per-formed for this measurement; these are similar to those used in Sec.VIII A. The particle-identification systematic is not taken in account because the final states differ only in the charge assignments of the daughter tracks.

The symmetrized Dalitz plot for the signal region, cor-rected for efficiency and background subtracted, is shown in Fig. 13(b). We observe the presence of a significant

TABLE VIII. Summary of systematic uncertainties on the measurement of the Dþs ! KþK
Kþ branching ratio.

Uncertainty BðDþs!KþK
KþÞ BðDþ s!KþK

þÞ MC statistics 2.6% m 0.3% Likelihood-ratio 3.5% PID 1.5% Total 4.6% 0 50 100 150 200 250 1.9 1.95 2 2.05 m(K+K+π-) (GeV/c2) events/3 MeV/c 2

(a)

(b)

(c)

FIG. 13 (color online). (a) KþKþ

mass spectrum showing a Dþs signal. (b) Symmetrized Dalitz plot for Dþs ! KþKþ

decay. (c) Kþ

mass distribution (two combinations per event). The Dalitz plot and its projection are background subtracted and efficiency corrected. The curves result from the fits described in the text.