EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-214 LHCb-PAPER-2012-014 September 20, 2012
Measurement of the branching fraction and angular amplitudes
The LHCb collaboration†††Authors are listed on the following pages.
A sample of signal events obtained with fb of collisions at = 7 TeV collected by the LHCb experiment is used to measure the branching fraction and polarization amplitudes of the decay, with . The mass spectrum of the candidates in the peak is dominated by the contribution. Subtracting the non-resonant component, the branching fraction of is , where the first uncertainty is statistical and the second is systematic. A fit to the angular distribution of the decay products yields the polarization fractions and .
Submitted to Physical Review D (R)
R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva, M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov , M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bjørnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A. Büchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz , K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, D. Craik, R. Currie, C. D’Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Suárez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. Färber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. Göbel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa Gándara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graugés, G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Grünberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, O. Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lefèvre, A. Leflat, J. Lefrançois, O. Leroy, T. Lesiak, L. Li, Y. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. Märki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Martín Sánchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G. McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. Müller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K. Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. Pérez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pilař, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, M. Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S. T’Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Voß, H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin.
Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
Center for High Energy Physics, Tsinghua University, Beijing, China
LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France
CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France
Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
School of Physics, University College Dublin, Dublin, Ireland
Sezione INFN di Bari, Bari, Italy
Sezione INFN di Bologna, Bologna, Italy
Sezione INFN di Cagliari, Cagliari, Italy
Sezione INFN di Ferrara, Ferrara, Italy
Sezione INFN di Firenze, Firenze, Italy
Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
Sezione INFN di Genova, Genova, Italy
Sezione INFN di Milano Bicocca, Milano, Italy
Sezione INFN di Roma Tor Vergata, Roma, Italy
Sezione INFN di Roma La Sapienza, Roma, Italy
Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland
AGH University of Science and Technology, Kraków, Poland
Soltan Institute for Nuclear Studies, Warsaw, Poland
Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania
Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia
Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia
Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia
Institute for High Energy Physics (IHEP), Protvino, Russia
Universitat de Barcelona, Barcelona, Spain
Universidad de Santiago de Compostela, Santiago de Compostela, Spain
European Organization for Nuclear Research (CERN), Geneva, Switzerland
Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
Physik-Institut, Universität Zürich, Zürich, Switzerland
Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands
NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine
University of Birmingham, Birmingham, United Kingdom
H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
Department of Physics, University of Warwick, Coventry, United Kingdom
STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom
School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
Imperial College London, London, United Kingdom
School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom
Department of Physics, University of Oxford, Oxford, United Kingdom
Syracuse University, Syracuse, NY, United States
Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to
Institut für Physik, Universität Rostock, Rostock, Germany, associated to
P.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia
Università di Bari, Bari, Italy
Università di Bologna, Bologna, Italy
Università di Cagliari, Cagliari, Italy
Università di Ferrara, Ferrara, Italy
Università di Firenze, Firenze, Italy
Università di Urbino, Urbino, Italy
Università di Modena e Reggio Emilia, Modena, Italy
Università di Genova, Genova, Italy
Università di Milano Bicocca, Milano, Italy
Università di Roma Tor Vergata, Roma, Italy
Università di Roma La Sapienza, Roma, Italy
Università della Basilicata, Potenza, Italy
LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
Hanoi University of Science, Hanoi, Viet Nam
Interpretations of measurements of time-dependent violation in and decays have thus far assumed the dominance of the colour-suppressed tree-level process. However, there are contributions from higher order (penguin) processes (see Fig. 1) that cannot be calculated reliably in QCD and could be large enough to affect the measured asymmetries. It has been suggested that the penguin effects can be determined by means of an analysis of the angular distribution of , where the penguin diagram is not suppressed relative to the tree-level one, and flavour symmetry arguments can be used to determine the hadronic parameters entering the observables .
In this paper the meson will be written as , while for other resonances the mass will be given in parentheses. Furthermore, mention of any specific mode implies the use of the charge conjugated mode as well, and pairs will be simply written as . The decay has already been observed by the CDF experiment , which reported . Under the assumption that the light quark (,) is a spectator of the quark decay, the branching fraction can be approximated as
In this paper, of data taken in 2011 are used to determine , to study the angular properties of the decay products of the meson, and to measure the resonant contributions to the spectrum in the region of the meson. The measurement of the branching fraction uses the decay as a normalization mode.
The LHCb detector  is a single-arm forward spectrometer covering the pseudo-rapidity range . The detector includes a high precision tracking system consisting of a silicon-strip vertex detector located around the interaction point, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about , and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system has a momentum resolution that varies from 0.4 % at 5 to 0.6 % at 100. Two ring-imaging Cherenkov detectors (RICH) are used to determine the identity of charged particles. The separation of pions and kaons is such that, for efficiencies of the rejection power is above . Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage called High Level Trigger (HLT) that applies a full event reconstruction. Events with muon final states are triggered using two hardware trigger decisions: the single-muon decision (one muon candidate with transverse momentum ), and the di-muon decision (two muon candidates with and such that ). All tracks in the HLT are required to have a . The single muon trigger decision in the HLT selects events with at least one muon track with an impact parameter mm with respect to the primary vertex and . The di-muon trigger decision, designed to select mesons, also requires a di-muon mass () .
Simulated events are used to compute detection efficiencies and angular acceptances. For this purpose, collisions are generated using Pythia 6.4  with a specific LHCb configuration . Decays of hadronic particles are described by EvtGen  in which final state radiation is generated using Photos . The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [10, *Agostinelli:2002hh] as described in Ref. .
The selection of decays first requires the reconstruction of a candidate. The vertex is required to be separated from any primary vertex (PV) by a distance-of-flight significance greater than 13. Subsequently, the muons from the decay are combined with the and candidates to form a good vertex, where the di-muon mass is constrained to the mass. A is required for each of the four daughter tracks. Positive muon identification is required for the two tracks of the decay, and the kaons and pions are selected using the different hadron probabilities based on combined information given by the RICH detectors. The candidate momentum is required to be compatible with the flight direction as given by the vector connecting the PV with the candidate vertex. An explicit veto to remove events is applied, as they otherwise would pollute the upper sideband of the mass spectrum.
Following this initial selection, several geometrical variables are combined into a single discriminant geometrical likelihood variable (GL). This multivariate method is described in Refs. [13, 14]. The geometrical variables chosen to build the GL are: the candidate minimum impact parameter with respect to any PV in the event, the decay time of the candidate, the minimum impact parameter of the four daughter tracks with respect to all PV in the event (defined as the difference between the of the PV built with and without the considered track), the distance of closest approach between the and trajectories reconstructed from their decay products, and the of the candidate. The GL was tuned using simulated signal passing the selection criteria, and background from data in the mass sidebands with a value for the kaon particle identification variable in a range which does not overlap with the one used to select the data sample for the final analysis.
The mass spectrum in the channel is dominated by the resonance but contains a non-negligible S-wave contribution, originating from and non-resonant pairs . To determine it is therefore important to measure the S-wave magnitude in both channels. The spectrum is analyzed in terms of a non-resonant S-wave and several resonances parameterized using relativistic Breit-Wigner distributions with mass-dependent widths, following closely . The considered waves are: a non-resonant S-wave amplitude interfering with the resonance, for the P-wave and for the D-wave. F-wave and G-wave components are found to be negligible in the fit. In bins of the mass, a fit is made to the candidate mass distribution to determine the yield. As shown in Fig. 2, a fit is then made to the and yields as a function of the mass without any efficiency correction. The S and P-wave components dominate in the window around the mass, where the contribution is above 90%. A more exact determination of this contribution using this method would require mass-dependent angular acceptance corrections. For the branching fraction calculation, the fraction of candidates is determined from a different full angular and mass fit, which is described next.
The angular and mass analysis is based on an unbinned maximum likelihood fit which handles simultaneously the mass () and the angular parameters of the decays and the background. Each of these three components is is modelled as a product of probability density functions (), with the angle between the kaon momentum in the rest frame of the and the direction of motion of the in the rest frame of the . The polar and azimuthal angles (, ) describe the direction of the in the coordinate system defined in the rest frame, where the axis is the direction of motion of the meson, the axis is normal to the plane formed by the axis and the kaon momentum, and the axis is chosen so that the component of the kaon momentum is positive.
The function describing the mass distribution of both signal peaks is the sum of two Crystal Ball (CB) functions , which are a combination of a Gaussian and a power law function to describe the radiative tail at low masses,
The starting point of the radiative tail is governed by a transition point parameter . The mean and width of the Gaussian component are and . The values of the , , , and parameters are constrained to be the same for the and peaks. The difference in the means between the and the distributions, (), is fixed to the value taken from Ref. . The mass of the background is described by an exponential function.
where , and are the decay amplitudes corresponding to longitunally and transversely polarized vector mesons. is the S-wave amplitude and () the relative phase between the longitudinal and parallel amplitudes. The convention is used hereafter. The differential is . The polarization fractions are normalized according to
which satisfy .
The parameters , and describing the P-wave are left floating in the fit. The amplitude and the phase depend on , but this dependence is ignored in the fit, which is performed in a mass window of , and they are just treated also as floating parameters. A systematic uncertainty is later associated with this assumption. The angular distribution of observed events is parameterized as a product of the expression in Eq. S0.Ex6 and a detector acceptance function, , which describes the efficiency to trigger, reconstruct and select the events. Simulation studies have shown almost no correlation between the three one-dimensional angular acceptances , and . Therefore the global acceptance factorizes as , where is parameterized as a fifth degree polynomial, as a second degree polynomial and as a sinusoidal function. A systematic uncertainty due to this factorization hypothesis is later evaluated. The angular distribution for the background component is determined using the upper sideband of the mass spectrum, defined as the interval .
Figure 3 shows the projection of the fit in the mass axis, together with the projections in the angular variables in a window of around the mass. The number of candidates corresponding to and decays is found to be and , respectively.
|Value and statistical error|
|Background angular model|
|Total systematic error|
|Value and statistical error|
|Total systematic error|
Tables 1 and 2 summarize the measurements of the angular parameters, together with their statistical and systematic uncertainties. The correlation coefficient given by the fit between and is for decays. The results for the decay are in good agreement with previous measurements [16, 4, 19, 20]. Based on this agreement, the systematic uncertainties caused by the modelling of the angular acceptance were evaluated by summing in quadrature the statistical error on the measured parameters with the uncertainties on the world averages ( and ) . The angular analysis was repeated with two additional acceptance descriptions, one which uses a three-dimensional histogram to describe the efficiency avoiding any factorization hypothesis, and another one based on a method of normalization weights described in Ref. . A very good agreement was found in the values of the polarization fractions computed with all the three methods. For the parameter , uncertainties caused by the finite size of the simulation sample used for the acceptance description, as well as from the studies with several acceptance models, are included. The systematic uncertainty caused by the choice of the angular for the background is shown for the decay but it was found to be negligible for .
Also included in Tables 1 and 2 is the uncertainty from the assumption of a constant as a function of . This assumption can be relaxed by adding an extra free parameter to the angular . This addition makes the fit unstable for the small size of the sample, but can be used in the control channel . The differences found in the parameters with the two alternate parameterizations are used as systematic uncertainties. The parameters fit to for the and to (where the error corresponds to the positive one, being symmetrized) for the . These parameters could in principle affect the efficiency corrections, but it was found that the effect of different values of on the overall efficiency is negligible. A simulation study of the fit pulls has shown that the errors on and of the decays are overestimated by a small amount () since they do not follow exactly a Gaussian distribution, therefore the decision was taken to quote an uncertainty which corresponds to an interval containing of the generated experiments, rather than giving an error corresponding to a log-likelihood interval of . A slight bias observed in the pulls of in decays was accounted for by adding a systematic uncertainty corresponding to 6% of the statistical error.
The ratio of the two branching fractions is obtained from
where is the probability of the quark to hadronize to mesons, is the efficiency ratio, is the ratio of angular corrections, is the ratio of fractions and is the ratio of signal yields. The value of has been taken from Ref. . The efficiencies in the ratio are computed using simulation and receive two contributions: the efficiency of the offline reconstruction (including geometrical acceptance) and selection cuts, and the trigger efficiency on events that satisfy the analysis offline selection criteria. The systematic uncertainty in the efficiency ratio is negligible due to the similarity of the final states. Effects due to possible differences in the decay time acceptance between data and simulation were found to affect the efficiency ratio by less than 1 per mille. On the other hand, since the efficiency depends on the angular distribution of the decay products, correction factors and are applied to account for the difference between the angular amplitudes used in simulation and those measured in the data. The observed numbers of and decays, denoted by and , correspond to the number of and decays with a mass in a window around the nominal mass. This includes mostly the