Measurement of the Angular and Lifetime Parameters of the Decays B0 d ! J=cK�0 and B0 s ! J=c� V.M. Abazov,36 B. Abbott,75 M. Abolins,65 B. S. Acharya,29 M. Adams,51 T. Adams,49 E. Aguilo,6 M. Ahsan,59 G. D. Alexeev,36 G. Alkhazov,40 A. Alton,64,* G. Alverson,63 G.A. Alves,2 M. Anastasoaie,35 L. S. Ancu,35 T. Andeen,53 B. Andrieu,17 M. S. Anzelc,53 M. Aoki,50 Y. Arnoud,14 M. Arov,60 M. Arthaud,18 A. Askew,49 B. Åsman,41 A. C. S. Assis Jesus,3 O. Atramentov,49 C. Avila,8 F. Badaud,13 L. Bagby,50 B. Baldin,50 D. V. Bandurin,59 P. Banerjee,29 S. Banerjee,29 E. Barberis,63 A.-F. Barfuss,15 P. Bargassa,80 P. Baringer,58 J. Barreto,2 J. F. Bartlett,50 U. Bassler,18 D. Bauer,43 S. Beale,6 A. Bean,58 M. Begalli,3 M. Begel,73 C. Belanger-Champagne,41 L. Bellantoni,50 A. Bellavance,50 J. A. Benitez,65 S. B. Beri,27 G. Bernardi,17 R. Bernhard,23 I. Bertram,42 M. Besançon,18 R. Beuselinck,43 V. A. Bezzubov,39 P. C. Bhat,50 V. Bhatnagar,27 C. Biscarat,20 G. Blazey,52 F. Blekman,43 S. Blessing,49 K. Bloom,67 A. Boehnlein,50 D. Boline,62 T. A. Bolton,59 E. E. Boos,38 G. Borissov,42 T. Bose,77 A. Brandt,78 R. Brock,65 G. Brooijmans,70 A. Bross,50 D. Brown,81 X. B. Bu,7 N. J. Buchanan,49 D. Buchholz,53 M. Buehler,81 V. Buescher,22 V. Bunichev,38 S. Burdin,42,† T. H. Burnett,82 C. P. Buszello,43 J.M. Butler,62 P. Calfayan,25 S. Calvet,16 J. Cammin,71 M.A. Carrasco-Lizarraga,33 E. Carrera,49 W. Carvalho,3 B. C. K. Casey,50 H. Castilla-Valdez,33 S. Chakrabarti,18 D. Chakraborty,52 K.M. Chan,55 A. Chandra,48 E. Cheu,45 F. Chevallier,14 D.K. Cho,62 S. Choi,32 B. Choudhary,28 L. Christofek,77 T. Christoudias,43 S. Cihangir,50 D. Claes,67 J. Clutter,58 M. Cooke,50 W. E. Cooper,50 M. Corcoran,80 F. Couderc,18 M.-C. Cousinou,15 S. Crépé-Renaudin,14 V. Cuplov,59 D. Cutts,77 M. Ćwiok,30 H. da Motta,2 A. Das,45 G. Davies,43 K. De,78 S. J. de Jong,35 E. De La Cruz-Burelo,33 C. De Oliveira Martins,3 K. DeVaughan,67 F. Déliot,18 M. Demarteau,50 R. Demina,71 D. Denisov,50 S. P. Denisov,39 S. Desai,50 H. T. Diehl,50 M. Diesburg,50 A. Dominguez,67 T. Dorland,82 A. Dubey,28 L. V. Dudko,38 L. Duflot,16 S. R. Dugad,29 D. Duggan,49 A. Duperrin,15 J. Dyer,65 A. Dyshkant,52 M. Eads,67 D. Edmunds,65 J. Ellison,48 V.D. Elvira,50 Y. Enari,77 S. Eno,61 P. Ermolov,38,xx H. Evans,54 A. Evdokimov,73 V.N. Evdokimov,39 A.V. Ferapontov,59 T. Ferbel,71 F. Fiedler,24 F. Filthaut,35 W. Fisher,50 H. E. Fisk,50 M. Fortner,52 H. Fox,42 S. Fu,50 S. Fuess,50 T. Gadfort,70 C. F. Galea,35 C. Garcia,71 A. Garcia-Bellido,71 G.A. Garcia-Guerra,33 V. Gavrilov,37 P. Gay,13 W. Geist,19 W. Geng,15,65 C. E. Gerber,51 Y. Gershtein,49,‡ D. Gillberg,6 G. Ginther,71 B. Gómez,8 A. Goussiou,82 P. D. Grannis,72 H. Greenlee,50 Z. D. Greenwood,60 E.M. Gregores,4 G. Grenier,20 Ph. Gris,13 J.-F. Grivaz,16 A. Grohsjean,25 S. Grünendahl,50 M.W. Grünewald,30 F. Guo,72 J. Guo,72 G. Gutierrez,50 P. Gutierrez,75 A. Haas,70 N. J. Hadley,61 P. Haefner,25 S. Hagopian,49 J. Haley,68 I. Hall,65 R. E. Hall,47 L. Han,7 K. Harder,44 A. Harel,71 J.M. Hauptman,57 J. Hays,43 T. Hebbeker,21 D. Hedin,52 J. G. Hegeman,34 A. P. Heinson,48 U. Heintz,62 C. Hensel,22,x K. Herner,72 G. Hesketh,63 M.D. Hildreth,55 R. Hirosky,81 J. D. Hobbs,72 B. Hoeneisen,12 M. Hohlfeld,22 S. Hossain,75 P. Houben,34 Y. Hu,72 Z. Hubacek,10 V. Hynek,9 I. Iashvili,69 R. Illingworth,50 A. S. Ito,50 S. Jabeen,62 M. Jaffré,16 S. Jain,75 K. Jakobs,23 C. Jarvis,61 R. Jesik,43 K. Johns,45 C. Johnson,70 M. Johnson,50 D. Johnston,67 A. Jonckheere,50 P. Jonsson,43 A. Juste,50 E. Kajfasz,15 D. Karmanov,38 P. A. Kasper,50 I. Katsanos,70 D. Kau,49 V. Kaushik,78 R. Kehoe,79 S. Kermiche,15 N. Khalatyan,50 A. Khanov,76 A. Kharchilava,69 Y.M. Kharzheev,36 D. Khatidze,70 T. J. Kim,31 M.H. Kirby,53 M. Kirsch,21 B. Klima,50 J.M. Kohli,27 E. V. Komissarov,36,xx J.-P. Konrath,23 A. V. Kozelov,39 J. Kraus,65 T. Kuhl,24 A. Kumar,69 A. Kupco,11 T. Kurča,20 V. A. Kuzmin,38 J. Kvita,9 F. Lacroix,13 D. Lam,55 S. Lammers,70 G. Landsberg,77 P. Lebrun,20 W.M. Lee,50 A. Leflat,38 J. Lellouch,17 J. Li,78,xx L. Li,48 Q. Z. Li,50 S.M. Lietti,5 J. K. Lim,31 J. G. R. Lima,52 D. Lincoln,50 J. Linnemann,65 V.V. Lipaev,39 R. Lipton,50 Y. Liu,7 Z. Liu,6 A. Lobodenko,40 M. Lokajicek,11 P. Love,42 H. J. Lubatti,82 R. Luna-Garcia,33,k A.L. Lyon,50 A. K.A. Maciel,2 D. Mackin,80 R. J. Madaras,46 P. Mättig,26 C. Magass,21 A. Magerkurth,64 P. K. Mal,82 H. B. Malbouisson,3 S. Malik,67 V. L. Malyshev,36 Y. Maravin,59 B. Martin,14 R. McCarthy,72 M.M. Meijer,35 A. Melnitchouk,66 L. Mendoza,8 P. G. Mercadante,5 M. Merkin,38 K.W. Merritt,50 A. Meyer,21 J. Meyer,22,x J. Mitrevski,70 R.K. Mommsen,44 N.K. Mondal,29 R.W. Moore,6 T. Moulik,58 G. S. Muanza,15 M. Mulhearn,70 O. Mundal,22 L. Mundim,3 E. Nagy,15 M. Naimuddin,50 M. Narain,77 N.A. Naumann,35 H.A. Neal,64 J. P. Negret,8 P. Neustroev,40 H. Nilsen,23 H. Nogima,3 S. F. Novaes,5 T. Nunnemann,25 V. O’Dell,50 D. C. O’Neil,6 G. Obrant,40 C. Ochando,16 D. Onoprienko,59 N. Oshima,50 N. Osman,43 J. Osta,55 R. Otec,10 G. J. Otero y Garzón,50 M. Owen,44 P. Padley,80 M. Pangilinan,77 N. Parashar,56 S.-J. Park,22,x S. K. Park,31 J. Parsons,70 R. Partridge,77 N. Parua,54 A. Patwa,73 G. Pawloski,80 B. Penning,23 M. Perfilov,38 K. Peters,44 Y. Peters,26 P. Pétroff,16 M. Petteni,43 R. Piegaia,1 J. Piper,65 M.-A. Pleier,22 P. L.M. Podesta-Lerma,33,{ V.M. Podstavkov,50 Y. Pogorelov,55 M.-E. Pol,2 P. Polozov,37 B. G. Pope,65 A. V. Popov,39 C. Potter,6 W. L. Prado da Silva,3 H. B. Prosper,49 S. Protopopescu,73 J. Qian,64 A. Quadt,22,x PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 0031-9007=09=102(3)=032001(7) 032001-1 � 2009 The American Physical Society B. Quinn,66 A. Rakitine,42 M. S. Rangel,2 K. Ranjan,28 P. N. Ratoff,42 P. Renkel,79 P. Rich,44 M. Rijssenbeek,72 I. Ripp-Baudot,19 F. Rizatdinova,76 S. Robinson,43 R. F. Rodrigues,3 M. Rominsky,75 C. Royon,18 P. Rubinov,50 R. Ruchti,55 G. Safronov,37 G. Sajot,14 A. Sánchez-Hernández,33 M. P. Sanders,17 B. Sanghi,50 G. Savage,50 L. Sawyer,60 T. Scanlon,43 D. Schaile,25 R. D. Schamberger,72 Y. Scheglov,40 H. Schellman,53 T. Schliephake,26 S. Schlobohm,82 C. Schwanenberger,44 A. Schwartzman,68 R. Schwienhorst,65 J. Sekaric,49 H. Severini,75 E. Shabalina,51 M. Shamim,59 V. Shary,18 A. A. Shchukin,39 R.K. Shivpuri,28 V. Siccardi,19 V. Simak,10 V. Sirotenko,50 P. Skubic,75 P. Slattery,71 D. Smirnov,55 G. R. Snow,67 J. Snow,74 S. Snyder,73 S. Söldner-Rembold,44 L. Sonnenschein,17 A. Sopczak,42 M. Sosebee,78 K. Soustruznik,9 B. Spurlock,78 J. Stark,14 V. Stolin,37 D.A. Stoyanova,39 J. Strandberg,64 S. Strandberg,41 M.A. Strang,69 E. Strauss,72 M. Strauss,75 R. Ströhmer,25 D. Strom,53 L. Stutte,50 S. Sumowidagdo,49 P. Svoisky,35 A. Sznajder,3 A. Tanasijczuk,1 W. Taylor,6 B. Tiller,25 F. Tissandier,13 M. Titov,18 V.V. Tokmenin,36 I. Torchiani,23 D. Tsybychev,72 B. Tuchming,18 C. Tully,68 P.M. Tuts,70 R. Unalan,65 L. Uvarov,40 S. Uvarov,40 S. Uzunyan,52 B. Vachon,6 P. J. van den Berg,34 R. Van Kooten,54 W.M. van Leeuwen,34 N. Varelas,51 E.W. Varnes,45 I. A. Vasilyev,39 P. Verdier,20 L. S. Vertogradov,36 M. Verzocchi,50 D. Vilanova,18 F. Villeneuve-Seguier,43 P. Vint,43 P. Vokac,10 M. Voutilainen,67,** R. Wagner,68 H.D. Wahl,49 M.H. L. S. Wang,50 J. Warchol,55 G. Watts,82 M. Wayne,55 G. Weber,24 M. Weber,50,†† L. Welty-Rieger,54 A. Wenger,23,‡‡ N. Wermes,22 M. Wetstein,61 A. White,78 D. Wicke,26 M. Williams,42 G.W. Wilson,58 S. J. Wimpenny,48 M.Wobisch,60 D. R. Wood,63 T. R. Wyatt,44 Y. Xie,77 C. Xu,64 S. Yacoob,53 R. Yamada,50 W.-C. Yang,44 T. Yasuda,50 Y.A. Yatsunenko,36 H. Yin,7 K. Yip,73 H. D. Yoo,77 S.W. Youn,53 J. Yu,78 C. Zeitnitz,26 S. Zelitch,81 T. Zhao,82 B. Zhou,64 J. Zhu,72 M. Zielinski,71 D. Zieminska,54 A. Zieminski,54,xx L. Zivkovic,70 V. Zutshi,52 and E.G. Zverev38 (D0 Collaboration) 1Universidad de Buenos Aires, Buenos Aires, Argentina 2LAFEX, Centro Brasileiro de Pesquisas Fı́sicas, Rio de Janeiro, Brazil 3Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil 4Universidade Federal do ABC, Santo André, Brazil 5Instituto de Fı́sica Teórica, Universidade Estadual Paulista, São Paulo, Brazil 6University of Alberta, Edmonton, Alberta, Canada, Simon Fraser University, Burnaby, British Columbia, Canada, York University, Toronto, Ontario, Canada, and McGill University, Montreal, Quebec, Canada 7University of Science and Technology of China, Hefei, People’s Republic of China 8Universidad de los Andes, Bogotá, Colombia 9Center for Particle Physics, Charles University, Prague, Czech Republic 10Czech Technical University, Prague, Czech Republic 11Center for Particle Physics, Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic 12Universidad San Francisco de Quito, Quito, Ecuador 13LPC, Université Blaise Pascal, CNRS/IN2P3, Clermont, France 14LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National Polytechnique de Grenoble, Grenoble, France 15CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 16LAL, Université Paris-Sud, IN2P3/CNRS, Orsay, France 17LPNHE, IN2P3/CNRS, Universités Paris VI and VII, Paris, France 18CEA, Irfu, SPP, Saclay, France 19IPHC, Université Louis Pasteur, CNRS/IN2P3, Strasbourg, France 20IPNL, Université Lyon 1, CNRS/IN2P3, Villeurbanne, France and Université de Lyon, Lyon, France 21III. Physikalisches Institut A, RWTH Aachen University, Aachen, Germany 22Physikalisches Institut, Universität Bonn, Bonn, Germany 23Physikalisches Institut, Universität Freiburg, Freiburg, Germany 24Institut für Physik, Universität Mainz, Mainz, Germany 25Ludwig-Maximilians-Universität München, München, Germany 26Fachbereich Physik, University of Wuppertal, Wuppertal, Germany 27Panjab University, Chandigarh, India 28Delhi University, Delhi, India 29Tata Institute of Fundamental Research, Mumbai, India 30University College Dublin, Dublin, Ireland 31Korea Detector Laboratory, Korea University, Seoul, Korea 32SungKyunKwan University, Suwon, Korea PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 032001-2 33CINVESTAV, Mexico City, Mexico 34FOM-Institute NIKHEF and University of Amsterdam/NIKHEF, Amsterdam, The Netherlands 35Radboud University Nijmegen/NIKHEF, Nijmegen, The Netherlands 36Joint Institute for Nuclear Research, Dubna, Russia 37Institute for Theoretical and Experimental Physics, Moscow, Russia 38Moscow State University, Moscow, Russia 39Institute for High Energy Physics, Protvino, Russia 40Petersburg Nuclear Physics Institute, St. Petersburg, Russia 41Lund University, Lund, Sweden, Royal Institute of Technology and Stockholm University, Stockholm, Sweden, and Uppsala University, Uppsala, Sweden 42Lancaster University, Lancaster, United Kingdom 43Imperial College, London, United Kingdom 44University of Manchester, Manchester, United Kingdom 45University of Arizona, Tucson, Arizona 85721, USA 46Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 47California State University, Fresno, California 93740, USA 48University of California, Riverside, California 92521, USA 49Florida State University, Tallahassee, Florida 32306, USA 50Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 51University of Illinois at Chicago, Chicago, Illinois 60607, USA 52Northern Illinois University, DeKalb, Illinois 60115, USA 53Northwestern University, Evanston, Illinois 60208, USA 54Indiana University, Bloomington, Indiana 47405, USA 55University of Notre Dame, Notre Dame, Indiana 46556, USA 56Purdue University Calumet, Hammond, Indiana 46323, USA 57Iowa State University, Ames, Iowa 50011, USA 58University of Kansas, Lawrence, Kansas 66045, USA 59Kansas State University, Manhattan, Kansas 66506, USA 60Louisiana Tech University, Ruston, Louisiana 71272, USA 61University of Maryland, College Park, Maryland 20742, USA 62Boston University, Boston, Massachusetts 02215, USA 63Northeastern University, Boston, Massachusetts 02115, USA 64University of Michigan, Ann Arbor, Michigan 48109, USA 65Michigan State University, East Lansing, Michigan 48824, USA 66University of Mississippi, University, Mississippi 38677, USA 67University of Nebraska, Lincoln, Nebraska 68588, USA 68Princeton University, Princeton, New Jersey 08544, USA 69State University of New York, Buffalo, New York 14260, USA 70Columbia University, New York, New York 10027, USA 71University of Rochester, Rochester, New York 14627, USA 72State University of New York, Stony Brook, New York 11794, USA 73Brookhaven National Laboratory, Upton, New York 11973, USA 74Langston University, Langston, Oklahoma 73050, USA 75University of Oklahoma, Norman, Oklahoma 73019, USA 76Oklahoma State University, Stillwater, Oklahoma 74078, USA 77Brown University, Providence, Rhode Island 02912, USA 78University of Texas, Arlington, Texas 76019, USA 79Southern Methodist University, Dallas, Texas 75275, USA 80Rice University, Houston, Texas 77005, USA 81University of Virginia, Charlottesville, Virginia 22901, USA 82University of Washington, Seattle, Washington 98195, USA (Received 1 October 2008; published 20 January 2009) We present measurements of the linear polarization amplitudes and the strong relative phases that describe the flavor-untagged decays B0 d ! J=cK�0 and B0 s ! J=c� in the transversity basis. We also measure the mean lifetime ��s of the B0 s mass eigenstates and the lifetime ratio ��s=�d. The analyses are based on approximately 2:8 fb�1 of data recorded with the D0 detector. From our measurements of the angular parameters we conclude that there is no evidence for a deviation from flavor SU(3) symmetry for these decays and that the factorization assumption is not valid for the B0 d ! J=cK�0 decay. DOI: 10.1103/PhysRevLett.102.032001 PACS numbers: 14.20.Mr, 13.25.Hw, 13.30.Eg, 14.40.Nd PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 032001-3 http://dx.doi.org/10.1103/PhysRevLett.102.032001 B mesons are fertile ground to study CP violation and search for evidence of new physics. There are elements, in addition to CP violation, involved in the theoretical de- scription of B meson decays, such as flavor SU(3) symme- try, factorization, and final-state strong interactions. To understand the role CP violation plays in these decays, it is essential to understand and isolate the effect of each of these elements in the B meson decays. Factorization states that the decay amplitude of Bmeson decays can be expressed as the product of two single current matrix elements [1] and this implies that the rela- tive strong phases are 0 (mod �) [2]. A different measured value for the strong phases would indicate the presence of final-state strong interactions. The B0 d meson can be formed by replacing the s quark with the d quark in the B0 s meson. From flavor SU(3) symmetry applied to the B0 d � B0 s system one expects that the theoretical descrip- tion is similar; in particular, the B0 d ! J=cK�0 and B0 s ! J=c� [3] decays, can be described in the transversity basis [2] by the relative strong phases �1 and �2, and by the three independent components A0, Ak, and A?. The components A0 and Ak represent the CP-even and A? the CP-odd contributions to the decay amplitude. Other observables of these decays are the lifetimes of both mesons, which allow us to compare with theoretical predictions of the lifetime ratio. Phenomenological models predict differences of about 1% [4,5] between the B0 d and B0 s lifetimes. Previous B meson lifetime measurements [6] are consistent with these predictions. In this Letter we report the measurements of the parame- ters that describe the time-dependent angular distributions of the decays B0 d ! J=cK�0 and B0 s ! J=c� in the trans- versity basis, where the initial B meson flavor is not determined (‘‘untagged’’). We study the B0 d and B 0 s mesons to verify the validity of the factorization assumption [2] and to check if flavor SU(3) symmetry [2] holds for these decays. We also report the lifetime ratio ��s=�d for these mesons and the width difference��s between the light and heavy B0 s mass eigenstates. The analyses were performed using data collected with the D0 detector [7] in Run II of the Fermilab Tevatron Collider during 2003–2007 with an integrated luminosity of approximately 2:8 fb�1 of p �p collisions at a center-of-mass energy of 1.96 TeV. In con- trast with the flavor-tagged analysis reported in Ref. [8], in this Letter we report a simultaneous analysis of both the B0 d and B0 s meson decays, carried out in such a way that a straightforward comparison between their angular and life- time parameters can be performed. We use the B0 s ! J=c�, J=c ! �þ��, � ! KþK� selection described in Ref. [9]. The decay B0 d ! J=cK�0, J=c ! �þ��,K�0 ! K��� is reconstructed using simi- lar selection criteria and algorithms as the B0 s channel because they have the same four-track topology in the final state. The differences are the requirement that the trans- verse momentum of the pion be greater than 0:7 GeV=c, the invariant mass for the ðJ=c ; K�0ð892ÞÞ pair be in the range 4:93–5:61 GeV=c2, and the selection of theK�0ð892Þ candidates by demanding the two-particle invariant mass between 850 and 930 MeV=c2. Because of the lack of charged particle identification, we assign the mass of the pion and kaon to the latter two tracks and use the combi- nation with invariant mass closest to the K�0 mass. The proper decay length (PDL), defined as in Refs. [10,11], for a given B0 d or B 0 s candidate is determined by measuring the distance traveled by each b-hadron can- didate in a plane transverse to the beam direction, and then applying a Lorentz boost correction. In the B0 d and B 0 s final selection, we require a PDL uncertainty of less than 60 �m. We find 334199 and 41691 candidates that pass the B0 d and B0 s selection criteria, respectively (see Fig. 1). We denote the set of the angular variables defined in the transversity basis, where the decays B0 d ! J=cK�0 and B0 s ! J=c� are studied, as ! ¼ f’; cos�; cosc g. The description of these decays in this basis gives us access to the three linear polarization amplitudes at production time, t ¼ 0, jA0ð0Þj, jAkð0Þj, and jA?ð0Þj, satisfying )2Mass (GeV/c 5 5.1 5.2 5.3 5.4 5.5 5.6 2 E ve nt s pe r 9 M eV /c 0 1000 2000 3000 4000 5000 )2Mass (GeV/c 5 5.1 5.2 5.3 5.4 5.5 5.6 2 E ve nt s pe r 9 M eV /c 0 1000 2000 3000 4000 5000 Total fit Prompt background Non prompt background -1D0, 2.8 fb(a) )2Mass (GeV/c 5.1 5.2 5.3 5.4 5.5 5.6 5.7 2 E ve nt s pe r 9 M eV /c 0 100 200 300 400 500 600 700 800 )2Mass (GeV/c 5.1 5.2 5.3 5.4 5.5 5.6 5.7 2 E ve nt s pe r 9 M eV /c 0 100 200 300 400 500 600 700 800 -1D0, 2.8 fb Total Fit Prompt background Non prompt background (b) FIG. 1 (color online). Invariant mass distribution for selected (a) B0 d and (b) B0 s candidate events. The points with error bars represent the data, and the curves represent the fit projections for the total and the background components. PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 032001-4 jA0j2 þ jAkj2 þ jA?j2 ¼ 1 [12]; and the CP conserving strong phases �1 � arg½A� kA?�, and �2 � arg½A� 0A?�. Since only the relative phases of the amplitudes can enter physics observables, we are free to fix the phase of one of them, and we choose to fix �0 � argðA0Þ ¼ 0. According to the standard model,CP-violation effects in the B0 s system are very small [13]. In this analysis, we assume CP conservation and express the differential decay rate for the untagged decay B0 s ! J=c� as [2]: d4P=ðd!dtÞ / e��Lt½jA0j2f1ð!Þ þ ReðA� 0AkÞf5ð!Þ þ jAkj2f2ð!Þ� þ e��HtjA?j2f3ð!Þ; (1) where �LðHÞ � 1=�LðHÞ is the inverse of the lifetime corre- sponding to the light (heavy) mass eigenstate. The mea- sured parameters, the width difference ��s � �L � �H and the mean lifetime ��s � 1= �� ¼ 2=ð�L þ �HÞ, are given in terms of these inverse lifetimes. The angular functions fið!Þ are defined in Ref. [2]. In this decay, we have access to the phase �k ¼ argðA� 0AkÞ, which is related to �1 and �2 by �k ¼ �2 � �1. In the B0 d system, there is evidence of interference between the P- and S-wave K� amplitudes [14], which is taken into account in this analysis. The differential decay rate for the untagged decay B0 d ! J=cK�0 is given by [2,14]: d4P=ðd!dtÞ / e��dtfcos2�½jA0j2f1ð!Þ þ jAkj2f2ð!Þ þ jA?j2f3ð!Þ � �ImðA� kA?Þf4ð!Þ þ ReðA� 0AkÞf5ð!Þ þ �ImðA� 0A?Þf6ð!Þ� þ sin2� � f7ð!Þ þ 1 2 sin2�½f8ð!Þ cosð�k � �sÞjAkj þ f9ð!Þ sinð�? � �sÞjA?j þ f10ð!Þ cosð�sÞjA0j�g; (2) where �d � 1=�d is the inverse of the B0 d lifetime, � ¼ þ1ð� ¼ �1Þ for KþðK�Þ; �, �s, and fið!Þ are defined in Refs. [2,14]. For the B0 d, ��d is expected to be zero [13]. An unbinned likelihood fit is performed to extract all the B0 d and B0 s parameters. For the jth B meson candidate, the inputs for the fit are the massmj, PDL ctj, PDL uncertainty ctj , and the angular variables!j. The likelihood function L for the untagged decays B0 d ! J=cK�0 and B0 s ! J=c�, is defined by L ¼ YN j¼1 ½fsF j s þ ð1� fsÞF j b�; (3) where N is the total number of selected events and fs is the fraction of signal events in the sample, a free parameter in the fit. F s is the product of the signal probability distribution functions (PDF) of mass, PDL, and transversity angles, and the angular acceptances, which are determined via Monte Carlo simulations. The mass and PDL signal dis- tributions are modeled for both decays in the same way. The mass distribution is modeled by a Gaussian function with free mean and width. The PDL distribution is de- scribed [10] by the convolution of an exponential, whose decay constant is one of the fit parameters with a resolution function represented by two weighted Gaussian functions centered at zero. The widths si ctj of each Gaussian with scale factors si (i ¼ 1, 2) are free parameters in the fit to allow for a possible misestimate of the PDL uncertainty. The transversity angular distributions are modeled by the corresponding normalized equations (1) and (2). The con- tribution where the mass of the K and � are misassigned in our data is estimated by using Monte Carlo studies to be about 13% and is taken into account. F b is the product of the background PDF of the same variables and the angular acceptance as in the signal. We separate the background contributions into two types. The prompt background accounts for directly produced J=c mesons combined with random tracks. Nonprompt back- ground is due to J=c mesons produced by a b hadron decay combined with tracks that come from either a multi- body decay of the same b hadron or from hadronization. The mass distribution for the background is modeled by two independent normalized negative-slope exponentials, one for the prompt and one for the nonprompt contribu- tions. The PDL distribution for the prompt background is parametrized by the resolution function described above. The PDL distribution for the nonprompt background is modeled by a sum of two exponential components for positive ct and one for negative ct that account for a mix of heavy flavor meson decays and their possible misrecon- struction. The angular distributions for the background components are modeled by a shape similar to that of the signal, but with an independent set of amplitudes and phases. The results of our measurements are summarized in Table I. Figures 1 and 2 show the mass and the PDL distributions for the B0 d and B0 s candidates, respectively, with the projected results of the fits. The parameters with TABLE I. Summary of measurements for the decays B0 d ! J=cK�0 and B0 s ! J=c�. The uncertainties are only statistical. Parameter B0 d B0 s Units jA0j2 0:587� 0:011 0:555� 0:027 � � � jAkj2 0:230� 0:013 0:244� 0:032 � � � �1 �0:38� 0:06 � � � rad �2 3:21� 0:06 � � � rad �k � � � 2:72þ1:12�0:27 rad � 1:414� 0:018 1:487� 0:060 ps ��s � � � 0:085þ0:072 �0:078 ps�1 Nsig 11195� 167 1926� 62 � � � PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 032001-5 the strongest correlations are the linear amplitudes for the B0 d, and the width difference and the mean lifetime for the B0 s . Table II summarizes the systematic uncertainties in our measurements for B0 d and B0 s decays. To study the system- atic uncertainty due to the model for the mass distributions, we vary the shapes of the mass distributions for back- ground by using two normalized first-order polynomials instead of the nominal two negative exponentials. We estimate the systematic uncertainty due to the resolution on the PDL by using one Gaussian function for the reso- lution model. The fitting code is tested for the presence of biases by generating 1300 pseudoexperiments for B0 d and 1000 for B0 s , each with the same statistics as our data samples. We generated the events following the PDL, mass, and transversity angular distributions described above. The differences between the input and output values are quoted as the systematic uncertainty due to the fitting. The systematic uncertainty for �k reported for this source is due to an intrinsic ambiguity for this parameter in Eq. (1). The pseudoexperiments produced also cover the other solution for �k. The contribution from the detector alignment uncertainty is taken from Ref. [11]. Other po- tential sources of systematic uncertainties have been in- vestigated and found to give negligible variations in the measured parameters. The systematic uncertainties for the ratio ��s=�d are obtained by finding the ratio of the lifetimes for each systematic variation on Table II and taking the difference between this value and the nominal ratio. In conclusion, we have measured the angular and life- time parameters for the time-dependent angular untagged decays B0 d ! J=cK�0 and B0 s ! J=c�, the lifetime ratio of both B mesons, and the width difference ��s for the B 0 s meson. From the measured lifetime parameters ��s and �d we obtain the ratio ��s=�d ¼ 1:052� 0:061ðstatÞ � 0:015ðsystÞ which is consistent with the theoretical predic- tion [5] and previous measurements [6]. The measurement of the width difference ��s ¼ 0:085þ0:072 �0:078ðstatÞ � 0:006ðsystÞ ps�1 is consistent with the theoretical predic- tion [5,13] and with the value reported in Refs. [6,15]. D0 also has a measurement of ��s in a flavor-tagged analysis of B0 s ! J=c� in Ref. [8]. Our measurements for the linear polarization ampli- tudes for the B0 d, taking into account the interference between the K� S wave and P wave, are jA0j2 ¼ 0:587� 0:011ðstatÞ � 0:013ðsystÞ and jAkj2 ¼ 0:230� 0:013ðstatÞ � 0:025ðsystÞ; and for B0 s : jA0j2 ¼ 0:555� 0:027ðstatÞ � 0:006ðsystÞ, and jAkj2 ¼ 0:244� 0:032ðstatÞ � 0:014ðsystÞ are consistent and competi- tive with those reported in the literature [6,14,16]. Our measurement of the strong phases �1 and �2 indi- cates the presence of final-state interactions for the decay TABLE II. Summary of systematic uncertainties in the measurement of angular and lifetime parameters. The total uncertainties are given combining individual uncertainties in quadrature. B0 d B0 s Source jA0j2 jAkj2 �1 (rad) �2 (rad) �d (ps) jA0j2 jAkj2 �k (rad) ��s (ps �1) ��s (ps) ��s=�d Mass background � � � 0.024 0.09 0.05 0.030 0.004 0.002 0.02 � � � 0.021 0.009 PDL resolution 0.013 0.008 0.02 0.03 0.013 0.005 0.003 � � � � � � 0.016 0.012 Fitting code 0.001 � � � � � � � � � 0.004 0.004 0.014 0.26 0.001 0.008 0.003 Alignment � � � � � � � � � � � � 0.007 � � � � � � � � � � � � 0.007 � � � Total 0.013 0.025 0.09 0.06 0.034 0.006 0.014 0.26 0.001 0.028 0.015 ct (cm) -0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 E ve nt s pe r 0. 00 7 cm 1 10 210 310 410 510 ct (cm) -0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 E ve nt s pe r 0. 00 7 cm 1 10 210 310 410 510 -1D0, 2.8 fb(a) Total fit Signal Background ct (cm) -0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 E ve nt s pe r 0. 00 5 cm 1 10 210 310 410 ct (cm) -0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 E ve nt s pe r 0. 00 5 cm 1 10 210 310 410 Total fit Lτc Hτc Background -1D0, 2.8 fb(b) FIG. 2 (color online). PDL distribution for selected (a) B0 d and (b) B0 s candidate events. The points with error bars represent the data, and the curves represent the fit projections for the total, signal, and background components. PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 032001-6 B0 d ! J=cK�0 [2] since �1 ¼ �0:38� 0:06ðstatÞ � 0:09ðsystÞ rad is 3:5 away from zero, where is the total uncertainty. From the comparison of the measured ampli- tudes and strong phases [17] for both decays we conclude that they are consistent with being equal for B0 d and B 0 s and hence there is no evidence for a deviation from flavor SU(3) symmetry. In our sample we find that the K� S-wave intensity, as described in Ref. [14], is ð4:0� 1:0Þ%. We thank the staffs at Fermilab and collaborating insti- tutions, and acknowledge support from the DOE and NSF (U.S.); CEA and CNRS/IN2P3 (France); FASI, Rosatom and RFBR (Russia); CNPq, FAPERJ, FAPESP, and FUNDUNESP (Brazil); DAE and DST (India); Colciencias (Colombia); CONACyT (Mexico); KRF and KOSEF (Korea); CONICET and UBACyT (Argentina); FOM (The Netherlands); STFC (United Kingdom); MSMT and GACR (Czech Republic); CRC Program, CFI, NSERC, and WestGrid Project (Canada); BMBF and DFG (Germany); SFI (Ireland); The Swedish Research Council (Sweden); CAS and CNSF (China); and the Alexander von Humboldt Foundation (Germany). *Visitor from Augustana College, Sioux Falls, SD, USA. †Visitor from The University of Liverpool, Liverpool, U.K. ‡Visitor from Rutgers University, Piscataway, NJ, USA. xVisitor from II. Physikalisches Institut, Georg-August- University, Göttingen, Germany. kVisitor from Centro de Investigacion en Computacion- IPN, Mexico City, Mexico. {Visitor from ECFM, Universidad Autonoma de Sinaloa, Culiacán, Mexico. **Visitor from Helsinki Institute of Physics, Helsinki, Finland. ††Visitor from Universität Bern, Bern, Switzerland. ‡‡Visitor from Universität Zürich, Zürich, Switzerland. xxDeceased. [1] T. E. Browder, K. Honscheid, and D. Pedrini, Annu. Rev. Nucl. Part. Sci. 46, 395 (1996). [2] A. S. Dighe, I. Dunietz, and R. Fleischer, Eur. Phys. J. C 6, 647 (1999), and references therein. [3] Unless explicitly stated, the appearance of a specific charge state will also imply its charge conjugate through- out the Letter. [4] E. Franco et al., Nucl. Phys. B633, 212 (2002). [5] A. Lenz, arXiv:0802.0977. [6] W.-M. Yao et al. (Particle Data Group), J. Phys. G 33, 1 (2006) and 2007 partial update for the 2008 edition. [7] V. Abazov et al. (D0 Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 565, 463 (2006). [8] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 101, 241801 (2008). [9] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 98, 121801 (2007). [10] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 94, 102001 (2005). [11] V. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 94, 042001 (2005). [12] Throughout the paper, if not explicit dependence on time is stated, we denote Aið0Þ � Ai for i ¼ f0; k;?g. [13] A. Lenz and U. Nierste, J. High Energy Phys. 06 (2007) 072. [14] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 71, 032005 (2005). [15] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 100, 121803 (2008). [16] K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 95, 091601 (2005). [17] Using the relation between these phases we obtain �k;B0 d ¼ 3:59� 0:06� 0:09 rad. PRL 102, 032001 (2009) P HY S I CA L R EV I EW LE T T E R S week ending 23 JANUARY 2009 032001-7