Dalitz plot analysis of the decay B[superscript
±]→K[superscript ±]K[superscript ±]K[superscript #]
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Citation
Aubert, B. et al. “Dalitz plot analysis of the decay B[superscript
±]→K[superscript ±]K[superscript ±]K[superscript #]” Physical
Review D 74.3 (2006). © 2006 The American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevD.74.032003
Publisher
American Physical Society
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Final published version
Citable link
http://hdl.handle.net/1721.1/76202
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policy and may be subject to US copyright law. Please refer to the
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Dalitz plot analysis of the decay B
! K
K
K
B. Aubert,1R. Barate,1M. Bona,1D. Boutigny,1F. Couderc,1Y. Karyotakis,1J. P. Lees,1V. Poireau,1V. Tisserand,1 A. Zghiche,1E. Grauges,2A. Palano,3M. Pappagallo,3J. C. Chen,4N. D. Qi,4G. Rong,4P. Wang,4Y. S. Zhu,4G. Eigen,5 I. Ofte,5B. Stugu,5G. S. Abrams,6M. Battaglia,6D. N. Brown,6J. Button-Shafer,6R. N. Cahn,6E. Charles,6C. T. Day,6 M. S. Gill,6Y. Groysman,6R. G. Jacobsen,6J. A. Kadyk,6L. T. Kerth,6Yu. G. Kolomensky,6G. Kukartsev,6G. Lynch,6 L. M. Mir,6P. J. Oddone,6T. J. Orimoto,6M. Pripstein,6N. A. Roe,6M. T. Ronan,6W. A. Wenzel,6M. Barrett,7K. E. Ford,7
T. J. Harrison,7A. J. Hart,7C. M. Hawkes,7S. E. Morgan,7A. T. Watson,7K. Goetzen,8T. Held,8H. Koch,8 B. Lewandowski,8M. Pelizaeus,8K. Peters,8T. Schroeder,8M. Steinke,8J. T. Boyd,9J. P. Burke,9W. N. Cottingham,9 D. Walker,9T. Cuhadar-Donszelmann,10B. G. Fulsom,10C. Hearty,10N. S. Knecht,10T. S. Mattison,10J. A. McKenna,10 A. Khan,11P. Kyberd,11M. Saleem,11L. Teodorescu,11V. E. Blinov,12A. D. Bukin,12V. P. Druzhinin,12V. B. Golubev,12 A. P. Onuchin,12S. I. Serednyakov,12Yu. I. Skovpen,12E. P. Solodov,12K. Yu Todyshev,12D. S. Best,13M. Bondioli,13
M. Bruinsma,13M. Chao,13S. Curry,13I. Eschrich,13D. Kirkby,13A. J. Lankford,13P. Lund,13M. Mandelkern,13 R. K. Mommsen,13W. Roethel,13D. P. Stoker,13S. Abachi,14C. Buchanan,14S. D. Foulkes,15J. W. Gary,15O. Long,15
B. C. Shen,15K. Wang,15L. Zhang,15H. K. Hadavand,16E. J. Hill,16H. P. Paar,16S. Rahatlou,16V. Sharma,16 J. W. Berryhill,17C. Campagnari,17A. Cunha,17B. Dahmes,17T. M. Hong,17D. Kovalskyi,17J. D. Richman,17 T. W. Beck,18A. M. Eisner,18C. J. Flacco,18C. A. Heusch,18J. Kroseberg,18W. S. Lockman,18G. Nesom,18T. Schalk,18
B. A. Schumm,18A. Seiden,18P. Spradlin,18D. C. Williams,18M. G. Wilson,18J. Albert,19E. Chen,19A. Dvoretskii,19 D. G. Hitlin,19I. Narsky,19T. Piatenko,19F. C. Porter,19A. Ryd,19A. Samuel,19R. Andreassen,20G. Mancinelli,20 B. T. Meadows,20M. D. Sokoloff,20F. Blanc,21P. C. Bloom,21S. Chen,21W. T. Ford,21J. F. Hirschauer,21A. Kreisel,21
U. Nauenberg,21A. Olivas,21W. O. Ruddick,21J. G. Smith,21K. A. Ulmer,21S. R. Wagner,21J. Zhang,21A. Chen,22 E. A. Eckhart,22A. Soffer,22W. H. Toki,22R. J. Wilson,22F. Winklmeier,22Q. Zeng,22D. D. Altenburg,23E. Feltresi,23 A. Hauke,23H. Jasper,23B. Spaan,23T. Brandt,24V. Klose,24H. M. Lacker,24W. F. Mader,24R. Nogowski,24A. Petzold,24
J. Schubert,24K. R. Schubert,24R. Schwierz,24J. E. Sundermann,24A. Volk,24D. Bernard,25G. R. Bonneaud,25 P. Grenier,25E. Latour,25Ch. Thiebaux,25M. Verderi,25D. J. Bard,26P. J. Clark,26W. Gradl,26F. Muheim,26S. Playfer,26
A. I. Robertson,26Y. Xie,26M. Andreotti,27D. Bettoni,27C. Bozzi,27R. Calabrese,27G. Cibinetto,27E. Luppi,27 M. Negrini,27A. Petrella,27L. Piemontese,27E. Prencipe,27F. Anulli,28R. Baldini-Ferroli,28A. Calcaterra,28 R. de Sangro,28G. Finocchiaro,28S. Pacetti,28P. Patteri,28I. M. Peruzzi,28M. Piccolo,28M. Rama,28A. Zallo,28 A. Buzzo,29R. Capra,29R. Contri,29M. Lo Vetere,29M. M. Macri,29M. R. Monge,29S. Passaggio,29C. Patrignani,29 E. Robutti,29A. Santroni,29S. Tosi,29G. Brandenburg,30K. S. Chaisanguanthum,30M. Morii,30J. Wu,30R. S. Dubitzky,31
J. Marks,31S. Schenk,31U. Uwer,31W. Bhimji,32D. A. Bowerman,32P. D. Dauncey,32U. Egede,32R. L. Flack,32 J. R. Gaillard,32J. A. Nash,32M. B. Nikolich,32W. Panduro Vazquez,32X. Chai,33M. J. Charles,33U. Mallik,33
N. T. Meyer,33V. Ziegler,33J. Cochran,34H. B. Crawley,34L. Dong,34V. Eyges,34W. T. Meyer,34S. Prell,34 E. I. Rosenberg,34A. E. Rubin,34A. V. Gritsan,35M. Fritsch,36G. Schott,36N. Arnaud,37M. Davier,37G. Grosdidier,37
A. Ho¨cker,37F. Le Diberder,37V. Lepeltier,37A. M. Lutz,37A. Oyanguren,37S. Pruvot,37S. Rodier,37P. Roudeau,37 M. H. Schune,37A. Stocchi,37W. F. Wang,37G. Wormser,37C. H. Cheng,38D. J. Lange,38D. M. Wright,38C. A. Chavez,39 I. J. Forster,39J. R. Fry,39E. Gabathuler,39R. Gamet,39K. A. George,39D. E. Hutchcroft,39D. J. Payne,39K. C. Schofield,39 C. Touramanis,39A. J. Bevan,40F. Di Lodovico,40W. Menges,40R. Sacco,40C. L. Brown,41G. Cowan,41H. U. Flaecher,41
D. A. Hopkins,41P. S. Jackson,41T. R. McMahon,41S. Ricciardi,41F. Salvatore,41D. N. Brown,42C. L. Davis,42 J. Allison,43N. R. Barlow,43R. J. Barlow,43Y. M. Chia,43C. L. Edgar,43M. P. Kelly,43G. D. Lafferty,43M. T. Naisbit,43
J. C. Williams,43J. I. Yi,43C. Chen,44W. D. Hulsbergen,44A. Jawahery,44C. K. Lae,44D. A. Roberts,44G. Simi,44 G. Blaylock,45C. Dallapiccola,45S. S. Hertzbach,45X. Li,45T. B. Moore,45S. Saremi,45H. Staengle,45S. Y. Willocq,45
R. Cowan,46K. Koeneke,46G. Sciolla,46S. J. Sekula,46M. Spitznagel,46F. Taylor,46R. K. Yamamoto,46H. Kim,47 P. M. Patel,47C. T. Potter,47S. H. Robertson,47A. Lazzaro,48V. Lombardo,48F. Palombo,48J. M. Bauer,49L. Cremaldi,49
V. Eschenburg,49R. Godang,49R. Kroeger,49J. Reidy,49D. A. Sanders,49D. J. Summers,49H. W. Zhao,49S. Brunet,50 D. Coˆte´,50M. Simard,50P. Taras,50F. B. Viaud,50H. Nicholson,51N. Cavallo,52G. De Nardo,52D. del Re,52F. Fabozzi,52 C. Gatto,52L. Lista,52D. Monorchio,52D. Piccolo,52C. Sciacca,52M. Baak,53H. Bulten,53G. Raven,53H. L. Snoek,53
C. P. Jessop,54J. M. LoSecco,54T. Allmendinger,55G. Benelli,55K. K. Gan,55K. Honscheid,55D. Hufnagel,55 P. D. Jackson,55H. Kagan,55R. Kass,55T. Pulliam,55A. M. Rahimi,55R. Ter-Antonyan,55Q. K. Wong,55N. L. Blount,56
J. Brau,56R. Frey,56O. Igonkina,56M. Lu,56R. Rahmat,56N. B. Sinev,56D. Strom,56J. Strube,56E. Torrence,56 F. Galeazzi,57A. Gaz,57M. Margoni,57M. Morandin,57A. Pompili,57M. Posocco,57M. Rotondo,57F. Simonetto,57
PHYSICAL REVIEW D 74, 032003 (2006)
R. Stroili,57C. Voci,57M. Benayoun,58J. Chauveau,58P. David,58L. Del Buono,58Ch. de la Vaissie`re,58O. Hamon,58 B. L. Hartfiel,58M. J. J. John,58Ph. Leruste,58J. Malcle`s,58J. Ocariz,58L. Roos,58G. Therin,58P. K. Behera,59 L. Gladney,59J. Panetta,59M. Biasini,60R. Covarelli,60M. Pioppi,60C. Angelini,61G. Batignani,61S. Bettarini,61
F. Bucci,61G. Calderini,61M. Carpinelli,61R. Cenci,61F. Forti,61M. A. Giorgi,61A. Lusiani,61G. Marchiori,61 M. A. Mazur,61M. Morganti,61N. Neri,61E. Paoloni,61G. Rizzo,61J. Walsh,61M. Haire,62D. Judd,62D. E. Wagoner,62
J. Biesiada,63N. Danielson,63P. Elmer,63Y. P. Lau,63C. Lu,63J. Olsen,63A. J. S. Smith,63A. V. Telnov,63F. Bellini,64 G. Cavoto,64A. D’Orazio,64E. Di Marco,64R. Faccini,64F. Ferrarotto,64F. Ferroni,64M. Gaspero,64L. Li Gioi,64 M. A. Mazzoni,64S. Morganti,64G. Piredda,64F. Polci,64F. Safai Tehrani,64C. Voena,64M. Ebert,65H. Schro¨der,65 R. Waldi,65T. Adye,66N. De Groot,66B. Franek,66E. O. Olaiya,66F. F. Wilson,66S. Emery,67A. Gaidot,67S. F. Ganzhur,67
G. Hamel de Monchenault,67W. Kozanecki,67M. Legendre,67B. Mayer,67G. Vasseur,67Ch. Ye`che,67M. Zito,67 W. Park,68M. V. Purohit,68A. W. Weidemann,68J. R. Wilson,68M. T. Allen,69D. Aston,69R. Bartoldus,69P. Bechtle,69
N. Berger,69A. M. Boyarski,69R. Claus,69J. P. Coleman,69M. R. Convery,69M. Cristinziani,69J. C. Dingfelder,69 D. Dong,69J. Dorfan,69G. P. Dubois-Felsmann,69D. Dujmic,69W. Dunwoodie,69R. C. Field,69T. Glanzman,69
S. J. Gowdy,69M. T. Graham,69V. Halyo,69C. Hast,69T. Hryn’ova,69W. R. Innes,69M. H. Kelsey,69P. Kim,69 M. L. Kocian,69D. W. G. S. Leith,69S. Li,69J. Libby,69S. Luitz,69V. Luth,69H. L. Lynch,69D. B. MacFarlane,69 H. Marsiske,69R. Messner,69D. R. Muller,69C. P. O’Grady,69V. E. Ozcan,69A. Perazzo,69M. Perl,69B. N. Ratcliff,69 A. Roodman,69A. A. Salnikov,69R. H. Schindler,69J. Schwiening,69A. Snyder,69J. Stelzer,69D. Su,69M. K. Sullivan,69
K. Suzuki,69S. K. Swain,69J. M. Thompson,69J. Va’vra,69N. van Bakel,69M. Weaver,69A. J. R. Weinstein,69 W. J. Wisniewski,69M. Wittgen,69D. H. Wright,69A. K. Yarritu,69K. Yi,69C. C. Young,69P. R. Burchat,70A. J. Edwards,70 S. A. Majewski,70B. A. Petersen,70C. Roat,70L. Wilden,70S. Ahmed,71M. S. Alam,71R. Bula,71J. A. Ernst,71V. Jain,71 B. Pan,71M. A. Saeed,71F. R. Wappler,71S. B. Zain,71W. Bugg,72M. Krishnamurthy,72S. M. Spanier,72R. Eckmann,73
J. L. Ritchie,73A. Satpathy,73C. J. Schilling,73R. F. Schwitters,73J. M. Izen,74I. Kitayama,74X. C. Lou,74S. Ye,74 F. Bianchi,75F. Gallo,75D. Gamba,75M. Bomben,76L. Bosisio,76C. Cartaro,76F. Cossutti,76G. Della Ricca,76
S. Dittongo,76S. Grancagnolo,76L. Lanceri,76L. Vitale,76V. Azzolini,77F. Martinez-Vidal,77Sw. Banerjee,78 B. Bhuyan,78C. M. Brown,78D. Fortin,78K. Hamano,78R. Kowalewski,78I. M. Nugent,78J. M. Roney,78R. J. Sobie,78 J. J. Back,79P. F. Harrison,79T. E. Latham,79G. B. Mohanty,79H. R. Band,80X. Chen,80B. Cheng,80S. Dasu,80M. Datta,80
A. M. Eichenbaum,80K. T. Flood,80J. J. Hollar,80J. R. Johnson,80P. E. Kutter,80H. Li,80R. Liu,80B. Mellado,80 A. Mihalyi,80A. K. Mohapatra,80Y. Pan,80M. Pierini,80R. Prepost,80P. Tan,80S. L. Wu,80Z. Yu,80and H. Neal81
(BABARCollaboration)
1Laboratoire de Physique des Particules, F-74941 Annecy-le-Vieux, France 2Universitat de Barcelona, Facultat de Fisica Dept. ECM, E-08028 Barcelona, Spain
3Universita` di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy 4Institute of High Energy Physics, Beijing 100039, China 5University of Bergen, Institute of Physics, N-5007 Bergen, Norway
6Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 7University of Birmingham, Birmingham, B15 2TT, United Kingdom
8Ruhr Universita¨t Bochum, Institut fu¨r Experimentalphysik 1, D-44780 Bochum, Germany 9University of Bristol, Bristol BS8 1TL, United Kingdom
10University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 11Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
12Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia 13University of California at Irvine, Irvine, California 92697, USA 14University of California at Los Angeles, Los Angeles, California 90024, USA
15University of California at Riverside, Riverside, California 92521, USA 16University of California at San Diego, La Jolla, California 92093, USA 17University of California at Santa Barbara, Santa Barbara, California 93106, USA
18University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA 19California Institute of Technology, Pasadena, California 91125, USA
20University of Cincinnati, Cincinnati, Ohio 45221, USA 21University of Colorado, Boulder, Colorado 80309, USA 22Colorado State University, Fort Collins, Colorado 80523, USA 23Universita¨t Dortmund, Institut fu¨r Physik, D-44221 Dortmund, Germany
25Ecole Polytechnique, LLR, F-91128 Palaiseau, France 26University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 27Universita` di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy
28Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy 29Universita` di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy
30Harvard University, Cambridge, Massachusetts 02138, USA
31Universita¨t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany 32Imperial College London, London, SW7 2AZ, United Kingdom
33University of Iowa, Iowa City, Iowa 52242, USA 34Iowa State University, Ames, Iowa 50011-3160, USA 35Johns Hopkins University, Baltimore, Maryland 21218, USA
36Universita¨t Karlsruhe, Institut fu¨r Experimentelle Kernphysik, D-76021 Karlsruhe, Germany 37Laboratoire 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
38Lawrence Livermore National Laboratory, Livermore, California 94550, USA 39University of Liverpool, Liverpool L69 7ZE, United Kingdom 40Queen Mary, University of London, E1 4NS, United Kingdom
41University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom 42University of Louisville, Louisville, Kentucky 40292, USA
43University of Manchester, Manchester M13 9PL, United Kingdom 44University of Maryland, College Park, Maryland 20742, USA 45University of Massachusetts, Amherst, Massachusetts 01003, USA
46Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA 47McGill University, Montre´al, Que´bec, Canada H3A 2T8
48Universita` di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy 49University of Mississippi, University, Mississippi 38677, USA
50Universite´ de Montre´al, Physique des Particules, Montre´al, Que´bec, Canada H3C 3J7 51Mount Holyoke College, South Hadley, Massachusetts 01075, USA
52Universita` di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy
53NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands 54University of Notre Dame, Notre Dame, Indiana 46556, USA
55Ohio State University, Columbus, Ohio 43210, USA 56University of Oregon, Eugene, Oregon 97403, USA
57Universita` di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy
58Universite´s Paris VI et VII, Laboratoire de Physique Nucle´aire et de Hautes Energies, F-75252 Paris, France 59University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
60Universita` di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy
61Universita` di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy 62Prairie View A&M University, Prairie View, Texas 77446, USA
63Princeton University, Princeton, New Jersey 08544, USA
64Universita` di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy 65Universita¨t Rostock, D-18051 Rostock, Germany
66Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 67DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France
68University of South Carolina, Columbia, South Carolina 29208, USA 69Stanford Linear Accelerator Center, Stanford, California 94309, USA
70Stanford University, Stanford, California 94305-4060, USA 71State University of New York, Albany, New York 12222, USA
72University of Tennessee, Knoxville, Tennessee 37996, USA 73University of Texas at Austin, Austin, Texas 78712, USA 74University of Texas at Dallas, Richardson, Texas 75083, USA
75Universita` di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy 76Universita` di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy
77IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain 78University of Victoria, Victoria, British Columbia, Canada V8W 3P6 79Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
*Also at Laboratoire de Physique Corpusculaire, Clermont-Ferrand, France ‡Also with Universita` della Basilicata, Potenza, Italy
†Also with Universita` di Perugia, Dipartimento di Fisica, Perugia, Italy
DALITZ PLOT ANALYSIS OF THE DECAY B! KKK PHYSICAL REVIEW D 74, 032003 (2006)
80University of Wisconsin, Madison, Wisconsin 53706, USA 81Yale University, New Haven, Connecticut 06511, USA
(Received 29 April 2006; published 7 August 2006)
We analyze the three-body charmless decay B! KKK using a sample of 226:0 2:5 million
B Bpairs collected by the BABAR detector. We measure the total branching fraction and CP asymmetry to beB 35:2 0:9 1:6 106and ACP 1:7 2:6 1:5%. We fit the Dalitz plot distribution
using an isobar model and measure the magnitudes and phases of the decay coefficients. We find no evidence of CP violation for the individual components of the isobar model. The decay dynamics is dominated by the KKS-wave, for which we perform a partial-wave analysis in the region mKK <
2 GeV=c2. Significant production of the f
0980 resonance, and of a spin zero state near 1:55 GeV=c2are
required in the isobar model description of the data. The partial-wave analysis supports this observation.
DOI:10.1103/PhysRevD.74.032003 PACS numbers: 13.25.Hw, 11.30.Er, 11.80.Et
I. INTRODUCTION
Charmless decays of B mesons provide a rich laboratory for studying different aspects of weak and strong interac-tions. With recent theoretical progress in understanding the strong interaction effects, specific predictions for two-body pseudoscalar, B ! PP, and pseudoscalar-vector, B ! PV, branching fractions and asymmetries are available [1–4] and global fits to experimental data have been performed [5,6]. Improved experimental mea-surements of a comprehensive set of charmless B decays coupled with further theoretical progress hold the potential to provide significant constraints on the CKM matrix pa-rameters and to discover hints of physics beyond the Standard Model in penguin-mediated b ! s transitions.
We analyze the decay B ! KKK, dominated by the b ! s penguin-loop transition, using 226:0 2:5 million B Bpairs collected by the BABAR detector [7] at the SLAC PEP-II asymmetric-energy B factory [8] operating at the 4S resonance. BABAR has previously measured the total branching fraction and asymmetry in this mode [9] and the two-body branching fractions B ! K1020, B! K
c0 [10,11]. A comprehensive Dalitz plot analysis of B! K
KK has been pub-lished by the Belle collaboration [12].
II. EVENT SELECTION
We consider events with at least four reliably recon-structed charged-particle tracks consistent with having originated from the interaction point. All three tracks forming a B! KKK decay candidate are required to be consistent with a kaon hypothesis using a particle identification algorithm that has an average efficiency of 94% within the acceptance of the detector and an average pion-to-kaon misidentification probability of 6%.
We use two kinematic variables to identify the signal. The first is E E s0
p
=2, the difference between the reconstructed B candidate energy and half the energy of the ee initial state, both in the ee center-of-mass (CM) frame. For signal events the E distribution peaks near zero with a resolution of 21 MeV. We require the candidates to have jEj < 40 MeV. The second
variable is the energy-substituted mass mES
s0=2 p0 pB2=E20 p2B q
, where pB is the momentum of the B candidate and E0; p0 is the four-momentum of the ee initial state, both in the laboratory frame. For signal events the mES distribution peaks near the B mass with a resolution of 2:6 MeV=c2. We define a signal region (SR) with mES2 5:27; 5:29 GeV=c2and a sideband (SB) with mES2 5:20; 5:25 GeV=c2.
The dominant background is due to events from light-quark or charm continuum production, ee! q q, whose jetlike event topology is different from the more spherical B decays. We suppress this continuum background by requiring the absolute value of the cosine of the angle between the thrust axes of the B candidate and the rest of the event in the CM frame to be smaller than 0.95. Further suppression is achieved using a neural network with four inputs computed in the CM frame: the cosine of the angle between the direction of the B candidate and the beam direction; the absolute value of the cosine of the angle between the candidate thrust axis and the beam direction; and momentum-weighted sums over tracks and neutral clusters not belonging to the candidate, Pipi and P
ipicos2i, where pi is the track momentum and i is the angle between the track momentum direction and the candidate thrust axis.
Figure1shows the mES distribution of the 9870 events thus selected. The histogram is fitted with a sum of a Gaussian distribution and a background function having a probability density, Px / xp1 x2exp1 x2, where x 2mES= ps0 and is a shape parameter [13]. The binned maximum likelihood fit gives 2 104 for 100 bins and 21:1 1:6. The ratio of the integrals of the background function over the signal region and the sideband yields an extrapolation coefficient Rqq 0:231 0:007. The expected number of q q background events in the signal region is nSR
q q RqqnSB nSBB B 972 34, where nSB 4659 is the number of events in the sideband from which we subtract the number of non-signal B Bbackground events nSB
B B 431 19, estimated using a large number of simulated exclusive B decays. The
expected number of B B background events in the signal region is nSR
B B 276 20, with B
! DKdecays giving the largest contribution.
We use a kinematic fit, constraining the mass of the selected candidates to the mass of the B meson. The three-body decay kinematics is described by two indepen-dent di-kaon invariant-mass variables m2
23; m213 s23; s13, where we order the same-sign kaons such that s23 s13. The signal region contains 1769 B and 1730 B candidates whose Dalitz plot distribution is shown in Fig.2.
III. ISOBAR MODEL FIT
We perform an extended binned maximum likelihood fit to the event distribution in Fig. 2 by binning the folded Dalitz plot into 292 nonuniform rectangular bins and min-imizing the log of the Poisson likelihood ratio, 2
LLR=2 P292
i1i nSRi nSRi lnnSRi =i, where nSRi is the number of observed signal region events in the i-th bin, assumed to be sampled from a Poisson distribution with mean i. In the limit of large statistics, the 2
LLR function has a 2 distribution and can be used to evaluate the goodness-of-fit. The binning choice is a compromise between the available statistics and the need to capture the observed features of the event distribution.
The expected number of events in the i-th bin is modeled as i 2 Z i jMj2dm2 23dm213 Rq qnSBi nSBiB B n SR iB B; (1) where the first term is the expected signal contribution, given by twice the bin integral of the square of the matrix element multiplied by the signal selection efficiency determined as a function of m23; m13 using simulated signal events. It is parameterized as a two-dimensional histogram with 0:25 0:25 GeV2=c4bins. The efficiency is uniform except for the regions of low efficiency in the corners of the plot, corresponding to decay topologies with a low-momentum kaon in the final state. The integral is multiplied by two because we use a folded Dalitz plot.
We use the isobar model formalism [14,15] and describe the matrix elementM as a sum of coherent contributions, M PN
k1Mk. The individual contributions are symme-trized with respect to the interchange of same-sign kaons, 1 $ 2, and are given by
Mk keik
2
p Aks23PJkcos13 f1 $ 2g; (2)
where keikis a complex-valued decay coefficient,A
kis the amplitude describing a KK system in a state with angular momentum J and invariant mass ps23, PJ is the Legendre polynomial of order J, and the helicity angle 13 between the direction of the bachelor recoil kaon 1 and kaon 3 is measured in the rest frame of kaons 2 and 3.
The model includes contributions from the 1020 and c0 intermediate resonances, which are clearly visible in Fig. 2. Following Ref. [12], we introduce a broad scalar resonance, whose interference with a slowly varying non-resonant component is used to describe the rapid decrease in event density around mKK 1:6 GeV=c2. Evidence of a possible resonant S-wave contribution in this region has been reported previously [16,17], however its attribution is uncertain: the f01370 and f01500 resonances are known to couple more strongly to than to K K[18]; possible interpretations in terms of those
2 4 6 8 10 12 3 2 /c4) 2 V e G ( ) − K + K ( s 5 10 15 20 3 1 ) 4 c/ 2 Ve G( ) − K + K( s
FIG. 2. The Dalitz plot of the 1769 B and 1730 B candi-dates selected in the signal mESregion. The axes are defined in
the text. 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 S E (GeV/c2) m 0 50 100 150 200 250 300 350 400 450 ) 2 c/ Ve M 9. 0( /s t ne v E
FIG. 1. The mES distribution of the 9870 selected events,
shown as the data points with statistical errors. The solid histogram shows a fit with a sum of a Gaussian distribution (m0 5:2797 0:0007 GeV=c2, 2:64 0:07 MeV=c2,
N 2394 63) and a background function [13], shown as a dashed histogram. The shaded regions correspond to the signal region and the mESsideband defined in the text.
DALITZ PLOT ANALYSIS OF THE DECAY B! KKK PHYSICAL REVIEW D 74, 032003 (2006)
states [19] must account for the fact that no strong B ! Kf01370 or B! Kf
01500 signal is observed in B! K[12,20]. The contribution of the f
01710 resonance is included in the fit as a separate component and is found to be small. In the following, we designate the broad scalar resonance X01550 and determine its mass m0X0 and width 0X0 directly from the fit.
The contribution from a spin J resonance with mass m0 and total width 0 is described by a relativistic Breit-Wigner amplitude:
AJs
FJqKR
m20 s im00 s
: (3)
FJis the Blatt-Weisskopf centrifugal barrier factor [21] for angular momentum J: F0x 1 and F1x x=
1 x2 p , qh s=4 m2 h q
, and R represents the effective radius of the interaction volume for the resonance; we use R 4:0 GeV1 (0.8 fm) [22]. In the formulation of Eq. (3), only the centrifugal barrier factor for the decay of a spin J resonance into two pseudoscalar kaons is included; we have ignored the corresponding centrifugal barrier factor for the two-body decay of a B meson into a pseudoscalar kaon and a spin J resonance. The effect of this approxi-mation on the parameterization of B! K
1020, the only component with J > 0, is negligible. Unless other-wise specified, all resonance parameters are taken from Ref. [18]. The term s, parameterizing the mass de-pendence of the total width, is in general given by s P
iis, where the sum is over all decay modes of the resonance, and im2
0 0. The c0 has many decay modes, the decay modes of the f01710 are not well established, and decay modes other than KK of the possible X01550 resonance are unknown; in all these cases we set s 0 and neglect the mass dependence of the total width. For the 1020 resonance we use s 1s 2s, where 1 0B ! KK, 2 0B ! K0K0, and the mass dependence of the partial width for the two-body vector to pseudoscalar-pseudoscalar decay 1020 ! h h is pa-rameterized as 1;2s 1;2 q h q0h m s p F 2 1qhR F21q0hR 1 ; (4) where q0h m2 =4 m2h q . A large B! Kf 0980 signal measured in B ! K [12,20], and a recent measurement of gK=g, the ratio of the f0980 coupling constants to K Kand [23], motivate us to include an f0980 contribution using a coupled-channel amplitude parameterization:
Af0980s 1 m2 0 s im0g% gK%K ; (5) where % 2=3 1 4m2=s q 1=3 1 4m20=s q , %K 1=2 1 4m2 K=s q 1=2 1 4m2 K0=s q , and we use gK=g 4:21 0:25 0:21, m0 0:965 0:008 0:006 GeV=c2 and g 0:165 0:010 0:015 GeV=c2 [23].
We have investigated two theoretical models of the non-resonant component [24,25] and found that neither de-scribes the data adequately. We therefore include an S-wave nonresonant component expanded beyond the usual constant term as
MNR
NReiNR 2
p ei s23 ei s13: (6)
A fit to the m23> 2 GeV=c2 region of the folded Dalitz plot of Fig. 2, which is dominated by the nonresonant component, gives 0:140 0:019 GeV2c4, 0:02 0:06 GeV2c4, consistent with no phase varia-tion. In the following we fix 0, and incorporate the MNR contribution over the entire Dalitz plot, thus effec-tively employing the same parameterization as in Ref. [12]. We fit for the magnitudes and phases of the decay coefficients, the mass and width of the X01550, and the nonresonant component shape parameter . As the overall complex phase of the isobar model amplitude is arbitrary, we fix the phase of the nonresonant contribution to zero, leaving 14 free parameters in the fit. The number of de-grees of freedom is 292 14 278. We perform multiple minimizations with different starting points and find mul-tiple solutions clustered in pairs, where the solutions within each pair are very similar, except for the magnitude and phase of the c0decay coefficient. The twofold ambiguity arises from the interference between the narrow c0 and the nonresonant component, which is approximately con-stant across the resonance. The highest-likelihood pair has 2
LLR 346:4; 352:0; the second best pair has 2LLR 362:4; 368:7. The least significant components are the f0980 and the f01710. Their omission from the fit model degrades the best fit from 2
LLR 346:4 to 363.9 and 360.7, respectively.
The invariant-mass projections of the best fit are shown in Fig.3. The goodness-of-fit is 2 56 for 56 bins in the m23projection and 2 66 for 63 bins in the m13 projec-tion. The sharp peak in the B Bbackground distribution in the m23 projection is due to the contribution from the B! DK backgrounds. The fit gives 0:152 0:011 GeV2c4, m0X0 1:539 0:020 GeV=c2, and 0X0 0:257 0:033 GeV=c2. The fitted values of the shape parameter and the resonance mass are consistent with the values in Ref. [12], but our preferred value for the width is significantly larger. The results of the best isobar model fit are summarized in Table I, where we have also included the results for the c0component from the second solution in the highest-likelihood pair, labeled II
c0, and component fit fractions,
Fk R jMkj2ds23ds13 R jMj2ds 23ds13 ; (7)
where the integrals are taken over the entire Dalitz plot.
The sum of the component fit fractions is significantly larger than 1 due to large negative interference in the scalar sector [26].
IV. BRANCHING FRACTIONS AND ASYMMETRIES
To search for possible direct CP violation we extend the isobar model formalism by defining charge-dependent de-cay coefficients: keik keik 1 Ak 2 s ei k=2; (8)
where Akis the CP asymmetry of the k-th component, and k k
k. We modify the likelihood to be the product of the likelihoods for the two charges and repeat the fit. The phase of the nonresonant component is fixed to zero for both charges. The results are given in the last three columns of Table I in terms of the fitted CP asymmetry values, the symmetric 90% confidence level intervals around them, and the fitted phase differences between the charge-dependent decay coefficients. The asymmetry in-tervals are estimated by fitting Monte Carlo simulated samples generated according to the parameterized model of the nominal asymmetry fit. There is no evidence of statistically significant CP violation for any of the components.
Taking into account the signal Dalitz plot distribution, as described by the isobar model fit, the average signal effi-ciency is 0:282 0:011, where the uncertainty is evaluated using control data samples, and is primarily due to the uncertainties in tracking and particle identifica-tion efficiencies. The total branching fracidentifica-tion is BB! KKK 35:2 0:9 1:6 106, where the first er-ror is statistical and the second is systematic. The fit frac-tion of the isobar model terms that do not involve the c0 resonance is 95:0 0:6 1:1% for the best fit, giving BB! KKK 33:5 0:9 1:6 106 if intrin-sic charm contributions are excluded. The total asymmetry is ACPBB !KKKBB!KKK BB!K KKBB!K KK 1:7 2:6 1:5%.
The systematic error for the overall branching fraction is obtained by combining in quadrature the 3.9% efficiency TABLE I. The magnitudes and phases of the decay coefficients, fit fractions, two-body branching fractions, CP asymmetries, symmetric 90% confidence level CP asymmetry intervals around the nominal value, and the phase differences between the charge-dependent decay coefficients for the individual components of the isobar model fit.
Comp. (rad) F(%) F BB! KKK A A
min; Amax90% (rad) 1020 1:66 0:06 2:99 0:20 0:06 11:8 0:9 0:8 4:14 0:32 0:33 106 0:00 0:08 0:02 0:14; 0:14 0:67 0:28 0:05 f0980 5:2 1:0 0:48 0:16 0:08 19 7 4 6:5 2:5 1:6 106 0:31 0:25 0:08 0:72; 0:12 0:20 0:16 0:04 X01550 8:2 1:1 1:29 0:10 0:04 121 19 6 4:3 0:6 0:3 105 0:04 0:07 0:02 0:17; 0:09 0:02 0:15 0:05 f01710 1:22 0:34 0:59 0:25 0:11 4:8 2:7 0:8 1:7 1:0 0:3 106 0:0 0:5 0:1 0:66; 0:74 0:07 0:38 0:08 I c0 0:437 0:039 1:02 0:23 0:10 3:1 0:6 0:2 1:10 0:20 0:09 106 0:19 0:18 0:05 0:09; 0:47 0:7 0:5 0:2 II c0 0:604 0:034 0:29 0:20 6:0 0:7 2:10 0:24 106 0:03 0:28 — 0:4 1:3 NR 13:2 1:4 0 141 16 9 5:0 0:6 0:4 105 0:02 0:08 0:04 0:14; 0:18 0 1.0 1.5 2.0 2.5 3.0 3.5 3 2 ) (GeV/c2) − K + K ( m 0 50 100 150 200 ) 2 c/ Ve M 0 5(/ st ne v E 0.99 1.00 1.01 1.02 1.03 1.04 0 20 40 60 80 100 ) 2 c/ Ve M 2(/ st ne v E (a) 2.0 2.5 3.0 3.5 4.0 4.5 3 1 ) (GeV/c2) − K + K ( m 0 20 40 60 80 100 120 140 ) 2 c/ Ve M 0 5(/ st ne v E (b)
FIG. 3. The projected mKK invariant-mass distributions for the best fit: (a) m23 projection (the inset shows the fit
projection near the 1020 resonance), (b) m13 projection.
The histograms show the result of the fit with B B and q q
background contributions shown in dark and light gray, respec-tively.
DALITZ PLOT ANALYSIS OF THE DECAY B! KKK PHYSICAL REVIEW D 74, 032003 (2006)
uncertainty, a 1.1% uncertainty on the total number of BB pairs, a 0.7% uncertainty due to the modeling of B Bbackgrounds, and a 1.4% uncertainty arising from the uncertainty on the Rq q sideband extrapolation coefficient. The 1.5% systematic uncertainty for the asymmetry is due to possible charge asymmetry in kaon tracking and particle identification efficiencies, evaluated using data control samples. Where appropriate, the systematic uncertainties discussed above have been propagated to estimate the uncertainties on the leading isobar model fit results. We have also evaluated the systematic uncertainties due to the parameterization of resonance lineshapes by varying the parameters of all resonances within their respective uncer-tainties. Uncertainties arising from the distortion of narrow resonance lineshapes due to finite detector resolution and, for candidates containing a 1020 resonance produced in the q qcontinuum, due to the kinematic fit, have also been studied.
The values of the partial two-body branching fractions are summarized in the fifth column of TableI. Using the B1020 ! KK and Bc0! KK branching fractions from Ref. [18], we compute BB ! K1020 8:4 0:7 0:7 0:1 106 and BB! K
c0 1:84 0:32 0:14 0:28 104, where the last error is due to the uncertainty on the 1020 and c0 branching fractions. Both results are in agreement with previous measurements [10–12,27,28].
The partial branching fractions for B! Kf 0980 measured in the KKK and K final states are related by the ratio
R Bf0980 ! K K Bf0980 ! 3 4 IK I gK g; (9)
where 3/4 is an isospin factor, and IK=Iis the ratio of the integrals of the square of the f0980 amplitude given by Eq. (5) over the B ! KKK and B ! K Dalitz plots, and gK=gis the ratio of the f0980 coupling constants to K Kand . Using our results and those in Ref. [20], we get R 0:69 0:32, where we have combined the statis-tical and systematic errors of the two measurements in quadrature. This is consistent with R 0:92 0:07, which we get by evaluating the right-hand side of Eq. (9) using the values of the f0980 parameters reported by the BES collaboration [23].
V. PARTIAL-WAVE ANALYSES
We further study the nature of the dominant S-wave component by considering the interference between the low-mass and the high-mass scattering amplitudes in the region m232 1:1; 1:8 GeV=c2, m
13> 2 GeV=c2. The matrix element is modeled as
M Ss23 2
p eiSs23NR 2
p es13; (10)
where Sand S refer to the S-wave and are taken to be constant within each bin of the s23 variable and the non-resonant amplitude parameterization is taken from the fit to the high-mass region. The partial-wave expansion trun-cated at the S-wave describes the data adequately; the magnitude of the S-wave in each bin is readily determined. Because of the mass dependence of the nonresonant com-ponent, the phase of the S-wave can also be determined, albeit with a sign ambiguity and rather large errors for bins with a small number of entries or small net variation of the nonresonant component.
The results are shown in Fig.4, with the S-wave com-ponent of the isobar model fit overlaid for comparison. Continuity requirements allow us to identify two possible solutions for the phase; the solution labeled by black squares is consistent with a rapid counterclockwise motion in the Argand plot around mKK 1:55 GeV=c2, which is accommodated in the isobar model as the contri-bution of the X01550.
Isospin symmetry relates the measurements in B! KKK and B0 ! KKK0
S [29]. Our results for the KK S-wave can therefore be used to estimate a poten-tially significant source of uncertainty in the measurements of sin2 in B0! 1020K0
S[30,31] due to the contribu-tion of a CP-even S-wave amplitude. We perform a partial-wave analysis in the region m23KK 2 1:013; 1:027 GeV=c2, which we assume to be dominated by the low-mass P-wave, due to the contribution of the 1020 resonance, and a low-mass S-wave. The matrix element is modeled as 0 50 100 150 200 250 S ) 8 c/ 4 Ve G( / 2 ρ 1.00 1.15 0 250 750 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 ) 2 c / V e G ( ) − K + K ( m -3 -2 -1 0 1 2 3 S φ (a) (b)
FIG. 4. The results of the partial-wave analysis of the KK
S-wave: (a) magnitude squared, (b) phase. The discrete ambi-guities in the determination of the phase give rise to two possible solutions labeled by black and white squares. The curves corre-spond to the S-wave component from the isobar model fit. The inset shows the evidence of a threshold enhancement from the fits of the S-wave in the vicinity of the KK threshold and in the region around the 1020 resonance.
M S 2 p Ps23 2 p eiPs23cos 13; (11)
where the low-mass S-wave is taken to be constant over the small s23interval considered. The fit results for the P-wave are shown in Fig.5, with a Breit-Wigner fit of the 1020 resonance overlaid for comparison. For the S-wave we get 2
S 3:4 2:5 102 GeV4c8and compute its fraction in this region using Eq. (7) to be 9 6%.
We also consider the region 2mK< mKK < 1:006 GeV=c2, in the immediate vicinity of the KK threshold. The contribution of the 1020 resonance tail in this region is suppressed by the centrifugal barrier and is estimated to be smaller than 10%. We fit 2
S 6:1 1:6 102 GeV4c8 for the magnitude of the S-wave in this region. The fits in the vicinity of the KK threshold and in the region around the 1020 resonance indicate a threshold enhancement of the S-wave, which is accommodated in the isobar model by the con-tribution of the f0980 resonance as shown in the inset of Fig.4.
VI. CONCLUSIONS
In conclusion, we have measured the total branching fraction and the CP asymmetry in B! KKK. An isobar model Dalitz plot fit and a partial-wave analysis of the KK S-wave show evidence of large contributions from a broad X01550 scalar resonance, a mass-dependent nonresonant component, and an f0980 resonance. The ratio of B! Kf
0980 two-body branching fractions measured by BABAR in B! K and B! KKK is consistent with the measurement of gK=g by the BES collaboration, albeit with large errors. Our isobar model fit results are substantially different from those obtained in Ref. [12] due to the larger fitted width of the X01550 and the inclusion of the f0980 compo-nent in the isobar model. Our results for the BB! K1020 and BB! K
c0 branching fractions are in agreement with the previous results from BABAR [10,11], which they supersede, and from other experiments [12,27,28]. We have measured the CP asymmetries and the phase differences between the charge-dependent decay coefficients for the individual components of the isobar model and found no evidence of direct CP violation.
The ‘‘phase space plot’’ forms an essential part of the analysis presented in this paper. We wish to acknowledge its originator, the late Professor Richard Dalitz, upon whose contributions so much of the work of our collabo-ration rests.
ACKNOWLEDGMENTS
We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organiza-tions that support BABAR. The collaborating instituorganiza-tions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada) , IHEP (China), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), and PPARC (United Kingdom). Individuals have received support from CONACyT (Mexico), Marie Curie EIF (European Union), the A. P. Sloan Foundation, the Research Corporation, and the Alexander von Humboldt Foundation.
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