1 Introduction

\belowpdfbookmarkTitle pagetitle

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

CERN-PH-EP-2013-008 LHCb-PAPER-2012-040 February 5, 2013

Amplitude analysis and branching fraction measurement of

The LHCb collaborationAuthors are listed on the following pages.

An amplitude analysis of the final state structure in the decay mode is performed using of data collected by the \lhcbexperiment in 7 TeV center-of-mass energy collisions produced by the LHC. A modified Dalitz plot analysis of the final state is performed using both the invariant mass spectra and the decay angular distributions. Resonant structures are observed in the mass spectrum as well as a significant non-resonant S-wave contribution over the entire mass range. The largest resonant component is the , accompanied by , , and four additional resonances. The overall branching fraction is measured to be , where the first uncertainty is statistical, the second systematic, and the third due to the ratio of the number of \Bsbto mesons produced. The mass and width of the are measured to be and , respectively. The final state fractions of the other resonant states are also reported.

Submitted to Physical Review D

© CERN on behalf of the \lhcbcollaboration, license CC-BY-3.0.

LHCb collaboration

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, L. Anderlini, J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov , M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, C. Baesso, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, 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, A. Berezhnoy, 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, S. Cunliffe, 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, A. Di Canto, 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, 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, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, 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, S. Hall, 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, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, 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, O. Kochebina, V. 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, Y. Li, L. Li Gioi, 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, M. Maino, S. Malde, 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, K. Petridis, 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, V. Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S. Roiser, V. Romanovsky, A. Romero Vidal, 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, 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, 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, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, 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, H. Voss, C. 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

National Center for Nuclear Research (NCBJ), 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

Massachusetts Institute of Technology, Cambridge, MA, United States

1 Introduction

The study of decays to , where is either a pion or kaon, has been used to measure mixing-induced \CPviolation in decays [1, 2, 3, 4, 5, 6, 7].Mention of a particular mode implies use of its charge conjugate throughout this paper. In order to best exploit these decays a better understanding of the final state composition is necessary. This study has been reported for the channel [8]. Here we perform a similar analysis for . While a large contribution is well known [9] and the component has been recently observed [10] and confirmed [11], other components have not heretofore been identified including the source of S-wave contributions [12]. The tree-level Feynman diagram for the process is shown in Fig. 1.

Figure 1: Leading order diagram for .

In this paper the and mass spectra and decay angular distributions are used to study resonant and non-resonant structures. This differs from a classical “Dalitz plot” analysis [13] since the meson has spin-1, and its three helicity amplitudes must be considered.

2 Data sample and detector

The event sample is obtained using of integrated luminosity collected with the \lhcbdetector [14] using collisions at a center-of-mass energy of 7 TeV. The detector is a single-arm forward spectrometer covering the pseudorapidity range , designed for the study of particles containing \bquarkor \cquarkquarks. Components include a high precision tracking system consisting of a silicon-strip vertex detector surrounding the interaction region, 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 momentum§§§We work in units where . resolution that varies from 0.4% at 5\gevto 0.6% at 100\gev. The impact parameter (IP) is defined as the minimum distance of approach of the track with respect to the primary vertex. For tracks with large transverse momentum with respect to the proton beam direction, the IP resolution is approximately 20\mum. Charged hadrons are identified using two ring-imaging Cherenkov detectors. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.

The trigger [15] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage that applies a full event reconstruction. Events selected for this analysis are triggered by a decay, where the is required at the software level to be consistent with coming from the decay of a meson by use either of IP requirements or detachment of the from the primary vertex. Monte Carlo simulations are performed using \pythia [16] with the specific tuning given in Ref. [17], and the \lhcbdetector description based on \geant [18, *Allison:2006ve] described in Ref. [20]. Decays of mesons are based on \evtgen[21].

3 Signal selection and backgrounds

We select candidates trying to simultaneously maximize the signal yield and reduce the background. Candidate decays are combined with a pair of kaon candidates of opposite charge, and then requiring that all four tracks are consistent with coming from a common decay point. To be considered a candidate, particles identified as muons of opposite charge are required to have transverse momentum, , greater than 500 MeV, and form a vertex with fit per number of degrees of freedom (ndf) less than 11. These requirements give rise to a large signal over a small background [22]. Only candidates with a dimuon invariant mass between 48 MeV to +43 MeV relative to the observed mass peak are selected. The asymmetric requirement is due to final-state electromagnetic radiation. The two muons are subsequently kinematically constrained to the known mass [9].

Our ring-imaging Cherenkov system allows for the possibility of positively identifying kaon candidates. Charged tracks produce Cherenkov photons whose emission angles are compared with those expected for electrons, pions, kaons or protons, and a likelihood for each species is then computed. To identify a particular species, the difference between the logarithm of the likelihoods for two particle hypotheses (DLL) is computed. There are two criteria used: loose corresponds to DLL, while tight has DLL and DLL. Unless stated otherwise, we require the tight criterion for kaon selection.

We select candidate combinations if each particle is inconsistent with having been produced at the primary vertex. For this test we require that the formed by using the hypothesis that the IP is zero be greater than 9 for each track. Furthermore, each kaon must have MeV and the scalar sum of the of the kaon candidates must be greater than 900 MeV. To select candidates we further require that the two kaon candidates form a vertex with , and that they form a candidate vertex with the where the vertex fit /ndf . We require that this vertex be more than  mm from the primary vertex, and the angle between the momentum vector and the vector from the primary vertex to the vertex must be less than 11.8 mrad.

The candidate invariant mass distribution is shown in Fig. 2. The vertical lines indicate the signal and sideband regions, where the signal region extends to MeV around the nominal mass [9] and the sidebands extend from 35 MeV to 60 MeV on either side of the peak. The small peak near 5280 MeV results from decays, and will be subject to future investigation.

Figure 2: Invariant mass spectrum of combinations. The vertical lines indicate the signal (black-dotted) and sideband (red-dashed) regions.

The background consist of combinations of tracks, which have a smooth mass shape through the region, and peaking contributions caused by the reflection of specific decay modes where a pion is misidentified as a kaon. The reflection background that arises from the decay , where the is misidentified as a , is determined from the number of \Bdbcandidates in the control region  MeV above the \Bsbmass peak.

For each of the candidates in the control region, we reassign each of the two kaons in turn to the pion mass hypothesis. The resulting invariant mass distribution is shown in Fig. 3. The peak at the mass has candidates, determined by fitting the data to a Gaussian function for the signal, and a polynomial function for the background. From these events we estimate the number in the signal region, based on a simulation of the shape of the reflected distribution as a function of mass. Using simulated and samples, we calculate reflection candidates within MeV of the peak. This number is used as a constraint in the mass fit described below.

Figure 3: Invariant mass distribution for candidates MeV above the mass, reinterpreted as events. The fit is to a signal Gaussian whose mass and width are allowed to vary as well as the polynomial background.

To determine the number of \Bsbsignal candidates we perform a fit to the candidate invariant mass spectrum shown in Fig. 4.

Figure 4: Fit to the invariant mass spectrum of combinations. The dotted (black) line is the combinatorial background, the dashed (red) shape shows the misidentified decays, and the solid (blue) curve shows the total. The vertical dashed lines indicate the signal region.

The fit function is the sum of the signal component, combinatorial background, and the contribution from the reflections. The signal is modeled by a double-Gaussian function with a common mean. The combinatorial background is described by a linear function. The reflection background is constrained as described above. The mass fit gives 19,195150 signal together with combinatorial background candidates within MeV of the mass peak.

We use the decay as the normalization channel for branching fraction determinations. The selection criteria are similar to those used for , except for particle identification as here a loose kaon identification criterion is used. Figure 5 shows the mass distribution. The signal is fit with a double-Gaussian function and a linear function is used to fit the combinatorial background. There are 342,786661 signal and 10,195134 background candidates within MeV of the peak.

Figure 5: Fit to the invariant mass spectrum of candidates. The dotted line shows the combinatorial background and the solid (blue) curve is the total.

4 Analysis formalism

One of the goals of this analysis is to determine the intermediate states in decay within the context of an isobar model [23, 24], where we sum the resonant and non-resonant components testing if they explain the invariant mass squared and angular distributions. We also determine the absolute branching fractions of and final states and the mass and width of the resonance. Another important goal is to understand the S-wave content in the mass region.

Four variables completely describe the decay of with . Two are the invariant mass squared of , , and the invariant mass squared of , . The other two are the helicity angle, , which is the angle of the in the rest frame with respect to the direction in the rest frame, and the angle between the and decay planes, , in the rest frame. To simplify the probability density function (PDF), we analyze the decay process after integrating over the angular variable , which eliminates several interference terms.

4.1 The model for

In order to perform an amplitude analysis a PDF must be constructed that models correctly the dynamical and kinematic properties of the decay. The PDF is separated into two components, one describing signal, , and the other background, . The overall PDF given by the sum is

where is the detection efficiency. The background is described by the sum of combinatorial background, , and reflection, , functions

(2)

where and are the fractions of the combinatorial background and reflection, respectively, in the fitted region. The fractions and obtained from the mass fit are fixed for the subsequent analysis.

The normalization factors are given by

(3)

This formalism similar to that used by Belle in their analysis of  [25], and later used by LHCb for the analysis of  [8].

The invariant mass squared of versus is shown in Fig. 6 for candidates. No structure is seen in . There are however visible horizontal bands in the mass squared spectrum, the most prominent of which correspond to the and resonances. These and other structures in are now examined.

Figure 6: Distribution of versus for candidate decays within MeV of the mass. The horizontal bands result from the and resonances.

The signal function is given by the coherent sum over resonant states that decay into , plus a possible non-resonant S-wave contributionThe interference terms between different helicities are zero because we integrate over the angular variable .

(4)

where describes the decay amplitude via an intermediate resonance state with helicity . Note that the has the same helicity as the intermediate resonance. Each has an associated amplitude strength and a phase for each helicity state . The amplitude for resonance , for each , is given by

(5)

where is the momentum of either of the two kaons in the di-kaon rest frame, is the mass, is the magnitude of the three-momentum in the rest frame, and and are the meson and resonance decay form factors. The orbital angular momenta between the and system is given by , and the orbital angular momentum in the decay is given by ; the latter is the same as the spin of the system. Since the parent has spin-0 and the is a vector, when the system forms a spin-0 resonance, and . For resonances with non-zero spin, can be 0, 1 or 2 (1, 2 or 3) for and so on. We take the lowest as the default value and consider the other possibilities in the systematic uncertainty.

The Blatt-Weisskopf barrier factors and [26] are

(6)

For the meson , where , the hadron scale, is taken as 5.0 GeV; for the resonance , and is taken as 1.5 GeV [27]. In both cases where is the decay daughter momentum at the pole mass; for the \Bsbdecay the \jpsimomentum is used, while for the resonances the kaon momentum is used.

In the helicity formalism, the angular term, is defined as

(7)

where is the Wigner -function, is the resonance spin, is the helicity angle of the in the rest frame with respect to the direction in the rest frame, and may be calculated directly from the other variables as

(8)

The helicity dependent term is defined as

(9)

The mass squared shape of each resonance, is described by the function . In most cases this is a Breit-Wigner (BW) amplitude. When a decay channel opens close to the resonant mass, complications arise, since the proximity of the second threshold distorts the line shape of the amplitude. The can decay to either or . While the channel opens at much lower masses, the decay channel opens near the resonance mass. Thus, for the we use a Flatté model [28] that takes into account these coupled channels.

We describe the BW amplitude for a resonance decaying into two spin-0 particles, labeled as 2 and 3, as

(10)

where is the resonance mass, is its energy-dependent width that is parametrized as

(11)

Here is the decay width when the invariant mass of the daughter combinations is equal to .

The Flatté mass shape is parametrized as

(12)

where the constants and are the couplings to and final states, respectively. The factors are given by Lorentz-invariant phase space

(13)
(14)

For non-resonant processes, the amplitude is constant over the variables and , but has an angular dependence due to the decay. The amplitude is derived from Eq. (5), assuming that the non-resonant contribution is an S-wave (i.e. , ) and is uniform in phase space (i.e. ),

(15)

4.2 Detection efficiency

The detection efficiency is determined from a phase space simulation sample containing events with . We also use a separate sample of events. The and \ptdistributions of the generated mesons are weighted to match the distributions found using data. The simulation is also corrected by weighting for difference between the simulated kaon detection efficiencies and the measured ones determined by using a sample of events.

Next we describe the efficiency in terms of the analysis variables. Both and range from 12.5 to 24.0 , where is defined below, and thus are centered at . We model the detection efficiency using the dimensionless symmetric Dalitz plot observables

(16)

and the angular variable . The observables and are related to as

(17)

To parametrize this efficiency, we fit the distributions of the and simulation samples in bins of with the function

(18)

giving values of as a function of . The resulting distribution, shown in Fig. 7, is described by an exponential function