EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-143 LHCb-PAPER-2013-036 11 October 2013
Observation of - mixing and
measurement of mixing frequencies
using semileptonic decays
The LHCb collaboration†††Authors are listed on the following pages.
The and mixing frequencies, and , are measured using a data sample corresponding to an integrated luminosity of 1.0 fb collected by the LHCb experiment in collisions at a centre of mass energy of TeV during 2011. Around 1.8 candidate events are selected of the type ( anything), where about half are from peaking and combinatorial backgrounds. To determine the decay times, a correction is required for the momentum carried by missing particles, which is performed using a simulation-based statistical method. Associated production of muons or mesons allows us to tag the initial-state flavour and so to resolve oscillations due to mixing. We obtain \linenomath
The hypothesis of no oscillations is rejected by the equivalent of 5.8 standard deviations for and 13.0 standard deviations for . This is the first observation of mixing to be made using only semileptonic decays.
To be published in Eur. Phys. J. C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
R. Aaij, 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, S. Amerio, Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, F. Andrianala, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, 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, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, 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, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C. D’Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. Déléage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Suárez, D. Dossett, A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. Färber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. Göbel, D. Golubkov, A. Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa Gándara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graugés, G. Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Grünberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A. Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, 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, K. Kurek, 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, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi, P. Marino, R. Märki, J. Marks, G. Martellotti, A. Martens, A. Martín Sánchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mordà, M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. Müller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, 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, A. Oyanguren, B.K. Pal, A. Palano, T. Palczewski, 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, E. Perez Trigo, A. Pérez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pilař, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, 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, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H. Schindler, 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, M. Sirendi, T. Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, 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. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek, V. Syropoulos, 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, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez, P. Vazquez Regueiro, C. Vázquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, K. Vervink, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Voß, H. Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, 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, A. Zhokhov, 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 Padova, Padova, Italy
Sezione INFN di Pisa, Pisa, 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, Faculty of Physics and Applied Computer Science, 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
Massachusetts Institute of Technology, Cambridge, MA, United States
University of Cincinnati, Cincinnati, OH, United States
University of Maryland, College Park, MD, United States
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
Celal Bayar University, Manisa, Turkey, 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
Institute of Physics and Technology, Moscow, Russia
Università di Padova, Padova, Italy
Università di Pisa, Pisa, Italy
Scuola Normale Superiore, Pisa, Italy
and mesons propagate as superpositions of particle and antiparticle flavour states. For a flavour-specific decay process111In this paper, charge conjugate modes are always implied. such as , particle-antiparticle mixing lends a sinusoidal component to the decay rates [1, 2]. To measure mixing, the flavour state of the meson must be observed to change, which requires knowledge of the state from at least two points in time. The experimentally accessible times to determine the flavour are at production and decay. Neglecting violation in mixing, the decay rate at a proper decay time simplifies to
where and are the width and mass differences222The mass difference is measured here as an angular frequency, in units of inverse time. of the two mass eigenstates, and is the average decay width . The positive sign applies when the meson decays with the same flavour as its production and the negative sign when the particle decays with opposite flavour to its production, later referred to as “even” and “odd”. In this study, a sample of semileptonic decays obtained with the LHCb detector is used to measure the mixing frequencies and for the and systems. These quantities have previously been measured to high precision, usually in the combination of several channels, relying heavily on hadronic decay modes (see for example Refs. [3, 4] and our recent results, Refs. [5, 6, 7]). To date no observation of mixing has been made using only semileptonic decay channels.
2 Experimental setup
The LHCb detector  is a single-arm forward spectrometer covering the pseudorapidity range , designed for the study of particles containing or quarks. The detector consists of several dedicated subsystems, organized successively further from the interaction region. A silicon-strip vertex detector surrounds the interaction region and approaches to within 8 mm of the proton beams. The first of two ring-imaging Cherenkov (RICH) detectors comes next, followed by the remainder of the tracking system, which comprises, in order: a large-area silicon-strip detector; a dipole magnet with a bending power of about ; and three multilayer tracking stations, each with central silicon-strip detectors and peripheral straw drift tubes. After this comes the second RICH detector, the calorimeter and the muon stations.
The combined high-precision tracking system provides a momentum measurement with relative uncertainty that varies from 0.4 % at 5 GeV to 0.6 % at 100 GeV, and impact parameter333The impact parameter is the distance of closest approach of a track to a primary interaction vertex. resolution of 20 for tracks with high transverse momentum. By combining information from the two RICH detectors  charged hadrons can be identified across a wide range in momentum, around 2 to 150 GeV. The calorimeter system consists of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter, allowing identification of photon, electron and hadron candidates. Muons that pass through the calorimeters are detected using a system of alternating layers of iron and multiwire proportional chambers . Triggering of events is performed in two stages : a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which performs full event reconstruction.
3 Data selection and reconstruction
The LHCb dataset used in this analysis corresponds to an integrated luminosity of 1.0 fb collected in collisions at a centre of mass energy of TeV during the 2011 physics run at the LHC. Where simulation is required, Pythia 6.4  is used, 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 the detector response are implemented using the Geant4 toolkit [16, *Agostinelli:2002hh] as described in Ref. . Input to EvtGen is taken from the best knowledge of branching fractions () and form factors at the time of the simulation . The same reconstruction and selection is applied on simulated and detector data.
A sample of events is selected in which a candidate forms a vertex with a muon candidate. A cut-based selection is applied to enhance the fraction of real mesons in this sample that arise from semileptonic decays. Vertex and track reconstruction qualities, momenta, invariant masses, flight distances and particle identification (PID) variables are used. The selection was initially optimized on simulated data to maximize the signal significance, , where () denotes the number of selected signal (background) candidates. The most important cuts for this analysis are those on the PID and invariant masses. Combined information from the RICH detectors, muon stations, calorimeters and tracking allows us to place stringent requirements on a log-likelihood based PID parameter for each final-state particle separately, ensuring at least 99 % purity in the muon sample, and suppressing peaking backgrounds such as decays, where a pion has been misidentified as a kaon. To allow a simultaneous measurement of and , a broad mass window for the system is used to cover both the and masses, GeV, where is the known mass of the meson . Decays of the type are additionally suppressed by requiring that the invariant mass of the two kaons GeV, and combinatorial background with slow collinear pions is similarly removed with the mass requirement MeV.
Simulation studies indicate that the selected sample is dominated by , and decays, where no specific intermediate states are required other than those mentioned, and where at least one neutrino will occur together with any number of the other particles in the parentheses. These additional particles are ignored and so a clear mass peak cannot be reconstructed. For simplicity, to quantify the measured mass, , within its possible range, we define a “normalized mass”, , relative to the known masses of the , , and :
We require , where the lower cut mainly removes low-mass combinatorial background candidates. The invariant mass distribution and the normalized mass distribution () of the selected candidates are shown in Fig. 1, in which the and peaks can clearly be seen over the combinatorial background.
Determination of the initial-state flavour is performed using the standard LHCb flavour-tagging algorithms, which are described in detail elsewhere [19, 6, 5]. These algorithms rely on the reconstruction of particles that were produced in association with, and are flavour-correlated with, the signal -meson. The correlations arise either from fragmentation, which often produces a kaon or pion of specific charge correlated with the signal, or from “opposite-side” decays, where the decay products of the partner quark are reconstructed (e.g. a muon). A neural network combines tagging decisions for the best tagging power.
A hypothesis is required for the nature of the reconstructed candidate, either or , in order to choose the tagging algorithms to be applied and to select the appropriate mass with which to calculate . A split around the midpoint between the and peaks is used. For the hypothesis all available tags are used. For the hypothesis only opposite-side tags are used, to reduce the difference between and tagging performance and thus better constrain the background (see Secs. 5 and 6). The flavour-tagged dataset comprises 594,845 selected candidates.
Two techniques are employed to measure the mixing frequencies: (a) multidimensional log-likelihood maximization, simultaneously fitting and ; (b) model-independent Fourier analysis, used as a cross-check, which determines with good precision, but with a very poor precision. Both methods use a common determination of the proper decay time and so share a portion of the corresponding systematic effects.
4 Proper decay-time distributions
To obtain the -meson decay times, a correction is applied for the momentum lost due to missing particles, using a -factor method as employed in many previous measurements (see, for example, Refs.  and ). The -factor  is a simulation-based statistical correction, where the average missing momentum in a simulated sample is used to correct the reconstructed momentum as a function of the reconstructed mass (as shown in Fig. 2). In this study we use a fourth-order polynomial to parameterize as a function of the normalized mass ( from Eq. 2), which allows us to use the same correction for and . With this approach, both and exhibit residual biases of around %; these biases are known to good precision from the full simulation and are corrected in the final results.
The experimental resolution of the proper decay time () reduces the visibility of the oscillations, smearing Eq. 1 with a resolution function , where is the true decay time and is the measured value. The limited performance of the tagging also reduces the visibility of the oscillations. Our selection requirements include variables that are correlated with the decay time, leading to a time-dependent efficiency function, . Thus Eq. 1 becomes \linenomath
where is the tagging efficiency and is the mistag probability (the fraction of tags that assign the wrong flavour). We parameterize the time-dependent efficiency with an empirical “acceptance” function. Specifically Gaussian functions are used as motivated by data and full simulation studies , , where is the Gaussian function and the parameters are determined from fits to the data (typical values are ps and ps).
The -factor is a relative correction for the average missing momentum at a given value of ; as shown in Fig. 2, the range of missing momenta is broad and varies from about 70 % at to zero at . This large relative uncertainty on the corrected momentum () dominates the decay time resolution, i.e. . Hence is approximately proportional to (as seen in Fig. 3) and the decay time resolution worsens as decay time increases. This dependence is determined and parameterized from the full simulation. We may choose between a parameterization in terms of either the generated (“true”) decay time, using a numerical convolution, or in terms of the measured decay time, using analytical methods; the latter is the default approach. The resolution dependence is well-fitted with second or third order polynomials.
5 Multivariate fits to the data
A binned, multidimensional, log-likelihood fit to the data is made, using the Root and embedded RooFit fitting frameworks [23, 24]. In order to improve the resolution on the fitted value of , the sample is divided into two subsamples about normalized mass (with this value determined using fast-simulation “pseudo-experiment” studies), and the two subsamples are fitted simultaneously as described below. There are 101,000 bins over the mass, the measured decay time (), the normalized mass ( and ), and the tagging result (even and odd). Seven categories of signal and background are assigned component probability density functions (PDFs) whose fractions and shape parameters are left free in the fits to the data. The backgrounds are categorized in terms of their shapes in the mass and decay-time observables. Using the distribution we separate out peaking components from combinatorial background components. Each of these categories can be further divided into two based on their decay-time shape. We use the term “prompt” to describe fake candidates containing particles exclusively produced in the primary interaction, and the term “detached” for candidates that contain at least one daughter of a secondary decay and which therefore tend to exhibit a significantly larger lifetime. Candidates for the signal -decays of interest must be both detached and peaking. The signal-like decays are usually grouped together in the fit; however, we separate the specific background contribution of within the peak and fit that directly. These components are shown in together in Fig. 4 and separately in different regions in Figs. 5 and 6.
Each mass PDF is a Gaussian function or a Chebychev polynomial (Fig. 4), and each background decay-time PDF is a simple exponential with an appropriate acceptance function as previously described (Fig. 6). For the signal decay-time shape we use the model described in Eq. 3, with one instance for each peak. The majority of our sensitivity arises from the mixing asymmetry, whose time-dependent fit in the signal regions is shown in Fig. 7. Any odd/even asymmetry is assumed to be constant as a function of time for prompt backgrounds and for backgrounds that are known not to mix (, etc.). Generic detached backgrounds are allowed to have a time-dependent asymmetry varying as an arbitrary quadratic polynomial.
The proportion of with respect to is fixed to with a uncertainty, using the ratio of known fragmentation functions and branching fractions . Based on the full LHCb simulation, this ratio is corrected by to account for differences in the reconstruction and tagging efficiencies, with the full value of this correction taken as a systematic uncertainty. We fix using the result of a recent LHCb analysis , and is fixed to zero.
Only the signal mass shapes and the parameters of interest, and , are shared between the two subsamples in , which are fitted simultaneously. The goodness of the fit is verified with a local density method , which finds a -value of .
6 Fit results and systematic uncertainties
Table 1 gives the fitted values for some important quantities. In principle the signal lifetimes are also measured, but these have very large systematic uncertainties and so no results are quoted. The systematic uncertainties on and are first discussed before the final results are given.
Several sources of systematic uncertainty on the main measured quantities, and , are considered, as summarized in Table 2. The majority of the systematic uncertainties are obtained from the data.
The -factor: the -factor correction is a simulation-based method, and so differences between the simulation and reality that modify the visible and invisible momenta potentially invalidate the correction. Such differences could for example be in branching fractions or form factors. Large-scale pseudo-experiment studies are combined with full simulations to vary these underlying distributions within their uncertainties and examine biases produced on the fitted values. Small relative uncertainties are found, for and % for , representing the ultimate limit of this technique without further knowledge of the various sub-decays.
Detector alignment: momentum scale, decay-length scale, and track position uncertainties arise from known alignment uncertainties and result in variations in reconstructed masses and lifetimes as functions of decay opening angle. These uncertainties have been studied using detector survey data and various control modes; they are well determined and small in comparison to the statistical uncertainties .
Model bias: a correction has been made for the 1 % residual frequency bias seen in full simulation studies, as discussed in Sec. 4. This is taken directly from simulation and half of the correction is assigned a systematic uncertainty.
Signal proper-time model: the fit is repeated with two different time-resolution models. (a) When the resolution is parameterized as a function of true rather than measured decay time, using full numerical convolution, a (0.09, 0.002) ps variation is seen in (, ). (b) When a time-independent (average) resolution is used, a 0.001 ps variation is seen in (this method is not applicable to the measurement of due to many factors; crucially, within the time frame of any single oscillation the decay time resolution worsens by an appreciable fraction of the oscillation period, seen in Figs. 3 and 7). With other modifications to the signal model (resolutions and acceptances) a larger variation in of ps is found.
Quantity Normalized mass region Low- High- Fit fraction of: - signal 0.32470.0029 0.36040.0023 - signal 0.07810.0017 0.09680.0022 - prompt 0.04100.0026 0.04440.0018 - prompt 0.01960.0018 0.03110.0024 Mistag probability : - signal 0.3470.054 0.3330.021 - signal 0.35670.0063 0.33190.0065 Total candidates 368,965 225,880 Table 1: A selection of fitted parameter values, for which statistical uncertainties only are given. The signal fraction includes contributions from any detached production. When the omitted fractions (of combinatorial background components) are included, the total fraction sums to unity within each region separately.
Other models and binning: the order of the Chebychev polynomial is varied, Crystal Ball functions are used for the mass peak shapes, and the background parameterizations and the binning schemes are varied. Out of these modifications, the binning scheme has the largest effect. Resulting uncertanties of ps and ps are assigned to and , respectively.
Assumptions on decays: The measurement is sensitive to , the integrated mixing probability, which in turn is sensitive to the non-mixing -background. We hold constant several -background parameters in the baseline fit, determined from the full simulation. Many features of the background fit are varied to evaluate systematic variations, including the fraction, the lifetime, and the corrections for relative tagging performance. The largest uncertainty arises from tagging performance corrections and for this a ps uncertainty is assigned to . It is possible to leave one or more of these parameters free during the fit, but the loss in statistical precision is prohibitive.
|Source of uncertainty||Method||Systematic uncertainty|
|Values of||Data refit||n/a||0.0004|
|Signal proper-time model||Data refit||0.09||0.007|
|Other models and binning||Data refit||0.05||0.001|
|(, efficiency, tagging)||Data refit||n/a||0.008|
|Total||Sum in quadrature||0.15||0.013|
For cross-checks the data are split by LHCb magnet polarity and LHCb trigger strategies; no variations beyond the expected statistical fluctuations are observed. We obtain \linenomath
To obtain a measure for the significance of the observed oscillations, the global likelihood minimum for the full fit is compared with the likelihood of the hypotheses corresponding to the edges of our search window ( or ps). Both would result in almost flat asymmetry curves (cf. Fig. 7) corresponding to no observed oscillations. We reject the null hypothesis of no oscillations by the equivalent of standard deviations for oscillations and standard deviations for oscillations.
7 Fourier analysis
The full fit as described above was performed in the time domain, but measurement of the mixing frequency can also be made directly in the frequency domain as a cross-check, using well-established Fourier transform techniques [28, 29, 30]. The cosine term in Eq. 3 has a different sign for the odd and even samples, where the lifetime, acceptance, and other features are shared; this simplifies the analysis in the frequency domain. Any Fourier components not arising from mixing are suppressed by subtracting the odd Fourier spectrum from the even spectrum and no parameterizations of the background shapes, signal shapes, or decay-time resolution are required, allowing a model-independent measurement of the mixing frequencies. We search for the peak in the subtracted Fourier spectrum, shown in Fig. 8. Extensive fast simulation pseudo-experiments have shown that the value of is obtained reliably and with a reasonable precision using this method; however is heavily biased and has a large uncertainty, and so a result is not quoted. Since residual components of the Fourier spectrum are of much lower frequency than the component, and several complete oscillation periods of are observable, the search for a spectral peak is relatively free from complications. For , however, the relatively low frequency is similar to that of many other features of the data, and only a single oscillation period is observed; therefore the determination of is difficult with this simple model-independent approach.
Taking the spectrum for events in a 25 MeV bin around the mass, we find a clear and separated peak (Fig. 8, right). The rms width of the peak is 0.4 ps, around a peak value of ps; the rms can be used as a model-independent proxy for the statistical uncertainty. To further evaluate the expected statistical fluctuation in the peak value, we perform a large set of fast simulation pseudo-experiments taking the result of the multivariate fit as a model for signal and background. The uncertainty found from the simulation studies is 0.32 ps, slightly smaller than given by the rms. We report ps in order to be model-independent. Systematic uncertainties arise from the detector alignment and the -factor correction method, common to both measurement techniques, as quantified previously in Sec. 6.
The mixing frequencies for neutral mesons have been measured using flavour-specific semileptonic decays. To correct for the momentum lost to missing particles, a simulation-based kinematic correction, known as the -factor, was adopted. Two techniques were used to measure the mixing frequencies: a multidimensional simultaneous fit to the mass distribution, the decay-time distribution, and tagging information; and a simple Fourier analysis. The results of the two methods were consistent, with the first method being more precise. We obtain \linenomath
We reject the hypothesis of no oscillations by 5.8 standard deviations for and 13.0 standard deviations for . This is the first observation of - mixing to be made using only semileptonic decays.
We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on.
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