EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCbPAPER2011010 CERNPHEP2011194 July 13, 2019
Measurement of the  oscillation frequency in decays
Accepted by Phys. Lett. B
The  oscillation frequency is measured with 36 pb of data collected in collisions at = 7 TeV by the LHCb experiment at the Large Hadron Collider. A total of 1381 and signal decays are reconstructed, with average decay time resolutions of 44 fs and 36 fs, respectively. An oscillation signal with a statistical significance of 4.6 is observed. The measured oscillation frequency is = 17.63 0.11 (stat) 0.02 (syst) ps.
R. Aaij,
B. Adeva,
M. Adinolfi,
C. Adrover,
A. Affolder,
Z. Ajaltouni,
J. Albrecht,
F. Alessio,
M. Alexander,
G. Alkhazov,
P. Alvarez Cartelle,
A.A. Alves Jr,
S. Amato,
Y. Amhis,
J. Anderson,
R.B. Appleby,
O. Aquines Gutierrez,
F. Archilli,
L. Arrabito,
A. Artamonov ,
M. Artuso,
E. Aslanides,
G. Auriemma,
S. Bachmann,
J.J. Back,
D.S. Bailey,
V. Balagura,
W. Baldini,
R.J. Barlow,
C. Barschel,
S. Barsuk,
W. Barter,
A. Bates,
C. Bauer,
Th. Bauer,
A. Bay,
I. Bediaga,
K. Belous,
I. Belyaev,
E. BenHaim,
M. Benayoun,
G. Bencivenni,
S. Benson,
J. Benton,
R. Bernet,
M.O. Bettler,
M. van Beuzekom,
A. Bien,
S. Bifani,
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,
S. Brisbane,
M. Britsch,
T. Britton,
N.H. Brook,
H. Brown,
A. BüchlerGermann,
I. Burducea,
A. Bursche,
J. Buytaert,
S. Cadeddu,
J.M. Caicedo Carvajal,
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,
M. Charles,
Ph. Charpentier,
N. Chiapolini,
K. Ciba,
X. Cid Vidal,
G. Ciezarek,
P.E.L. Clarke,
M. Clemencic,
H.V. Cliff,
J. Closier,
C. Coca,
V. Coco,
J. Cogan,
P. Collins,
F. Constantin,
G. Conti,
A. Contu,
A. Cook,
M. Coombes,
G. Corti,
G.A. Cowan,
R. Currie,
B. D’Almagne,
C. D’Ambrosio,
P. David,
I. De Bonis,
S. De Capua,
M. De Cian,
F. De Lorenzi,
J.M. De Miranda,
L. De Paula,
P. De Simone,
D. Decamp,
M. Deckenhoff,
H. Degaudenzi,
M. Deissenroth,
L. Del Buono,
C. Deplano,
O. Deschamps,
F. Dettori,
J. Dickens,
H. Dijkstra,
P. Diniz Batista,
S. Donleavy,
A. Dosil Suárez,
D. Dossett,
A. Dovbnya,
F. Dupertuis,
R. Dzhelyadin,
C. Eames,
S. Easo,
U. Egede,
V. Egorychev,
S. Eidelman,
D. van Eijk,
F. Eisele,
S. Eisenhardt,
R. Ekelhof,
L. Eklund,
Ch. Elsasser,
D.G. d’Enterria,
D. Esperante Pereira,
L. Estève,
A. Falabella,
E. Fanchini,
C. Färber,
G. Fardell,
C. Farinelli,
S. Farry,
V. Fave,
V. Fernandez Albor,
M. FerroLuzzi,
S. Filippov,
C. Fitzpatrick,
M. Fontana,
F. Fontanelli,
R. Forty,
M. Frank,
C. Frei,
M. Frosini,
S. Furcas,
A. Gallas Torreira,
D. Galli,
M. Gandelman,
P. Gandini,
Y. Gao,
JC. Garnier,
J. Garofoli,
J. Garra Tico,
L. Garrido,
C. Gaspar,
N. Gauvin,
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,
S. Gregson,
B. Gui,
E. Gushchin,
Yu. Guz,
T. Gys,
G. Haefeli,
C. Haen,
S.C. Haines,
T. Hampson,
S. HansmannMenzemer,
R. Harji,
N. Harnew,
J. Harrison,
P.F. Harrison,
J. He,
V. Heijne,
K. Hennessy,
P. Henrard,
J.A. Hernando Morata,
E. van Herwijnen,
E. Hicks,
W. Hofmann,
K. Holubyev,
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. JeanMarie,
F. Jing,
M. John,
D. Johnson,
C.R. Jones,
B. Jost,
S. Kandybei,
M. Karacson,
T.M. Karbach,
J. Keaveney,
U. Kerzel,
T. Ketel,
A. Keune,
B. Khanji,
Y.M. Kim,
M. Knecht,
S. Koblitz,
P. Koppenburg,
A. Kozlinskiy,
L. Kravchuk,
K. Kreplin,
M. Kreps,
G. Krocker,
P. Krokovny,
F. Kruse,
K. Kruzelecki,
M. Kucharczyk,
S. Kukulak,
R. Kumar,
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,
R. Le Gac,
J. van Leerdam,
J.P. Lees,
R. Lefèvre,
A. Leflat,
J. Lefrancois,
O. Leroy,
T. Lesiak,
L. Li,
L. Li Gioi,
M. Lieng,
M. Liles,
R. Lindner,
C. Linn,
B. Liu,
G. Liu,
J.H. Lopes,
E. Lopez Asamar,
N. LopezMarch,
J. Luisier,
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,
D. Martinez Santos,
A. Massafferri,
Z. Mathe,
C. Matteuzzi,
M. Matveev,
E. Maurice,
B. Maynard,
A. Mazurov,
G. McGregor,
R. McNulty,
C. Mclean,
M. Meissner,
M. Merk,
J. Merkel,
R. Messi,
S. Miglioranzi,
D.A. Milanes,
M.N. Minard,
S. Monteil,
D. Moran,
P. Morawski,
R. Mountain,
I. Mous,
F. Muheim,
K. Müller,
R. Muresan,
B. Muryn,
M. Musy,
J. MylroieSmith,
P. Naik,
T. Nakada,
R. Nandakumar,
J. Nardulli,
I. Nasteva,
M. Nedos,
M. Needham,
N. Neufeld,
C. NguyenMau,
M. Nicol,
S. Nies,
V. Niess,
N. Nikitin,
A. OblakowskaMucha,
V. Obraztsov,
S. Oggero,
S. Ogilvy,
O. Okhrimenko,
R. Oldeman,
M. Orlandea,
J.M. Otalora Goicochea,
P. Owen,
B. Pal,
J. Palacios,
M. Palutan,
J. Panman,
A. Papanestis,
M. Pappagallo,
C. Parkes,
C.J. Parkinson,
G. Passaleva,
G.D. Patel,
M. Patel,
S.K. Paterson,
G.N. Patrick,
C. Patrignani,
C. PavelNicorescu,
A. Pazos Alvarez,
A. Pellegrino,
G. Penso,
M. Pepe Altarelli,
S. Perazzini,
D.L. Perego,
E. Perez Trigo,
A. PérezCalero Yzquierdo,
P. Perret,
M. PerrinTerrin,
G. Pessina,
A. Petrella,
A. Petrolini,
B. Pie Valls,
B. Pietrzyk,
T. Pilar,
D. Pinci,
R. Plackett,
S. Playfer,
M. Plo Casasus,
G. Polok,
A. Poluektov,
E. Polycarpo,
D. Popov,
B. Popovici,
C. Potterat,
A. Powell,
T. du Pree,
J. Prisciandaro,
V. Pugatch,
A. Puig Navarro,
W. Qian,
J.H. Rademacker,
B. Rakotomiaramanana,
M.S. Rangel,
I. Raniuk,
G. Raven,
S. Redford,
M.M. Reid,
A.C. dos Reis,
S. Ricciardi,
K. Rinnert,
D.A. Roa Romero,
P. Robbe,
E. Rodrigues,
F. Rodrigues,
P. Rodriguez Perez,
G.J. Rogers,
S. Roiser,
V. Romanovsky,
J. Rouvinet,
T. Ruf,
H. Ruiz,
G. Sabatino,
J.J. Saborido Silva,
N. Sagidova,
P. Sail,
B. Saitta,
C. Salzmann,
M. Sannino,
R. Santacesaria,
R. Santinelli,
E. Santovetti,
M. Sapunov,
A. Sarti,
C. Satriano,
A. Satta,
M. Savrie,
D. Savrina,
P. Schaack,
M. Schiller,
S. Schleich,
M. Schmelling,
B. Schmidt,
O. Schneider,
A. Schopper,
M.H. Schune,
R. Schwemmer,
A. Sciubba,
M. Seco,
A. Semennikov,
K. Senderowska,
I. Sepp,
N. Serra,
J. Serrano,
P. Seyfert,
B. Shao,
M. Shapkin,
I. Shapoval,
P. Shatalov,
Y. Shcheglov,
T. Shears,
L. Shekhtman,
O. Shevchenko,
V. Shevchenko,
A. Shires,
R. Silva Coutinho,
H.P. Skottowe,
T. Skwarnicki,
A.C. Smith,
N.A. Smith,
K. Sobczak,
F.J.P. Soler,
A. Solomin,
F. Soomro,
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,
N. Styles,
V.K. Subbiah,
S. Swientek,
M. Szczekowski,
P. Szczypka,
T. Szumlak,
S. T’Jampens,
E. Teodorescu,
F. Teubert,
C. Thomas,
E. Thomas,
J. van Tilburg,
V. Tisserand,
M. Tobin,
S. ToppJoergensen,
M.T. Tran,
A. Tsaregorodtsev,
N. Tuning,
A. Ukleja,
P. Urquijo,
U. Uwer,
V. Vagnoni,
G. Valenti,
R. Vazquez Gomez,
P. Vazquez Regueiro,
S. Vecchi,
J.J. Velthuis,
M. Veltri,
K. Vervink,
B. Viaud,
I. Videau,
X. VilasisCardona,
J. Visniakov,
A. Vollhardt,
D. Voong,
A. Vorobyev,
H. Voss,
K. Wacker,
S. Wandernoth,
J. Wang,
D.R. Ward,
A.D. Webber,
D. Websdale,
M. Whitehead,
D. Wiedner,
L. Wiggers,
G. Wilkinson,
M.P. Williams,
M. Williams,
F.F. Wilson,
J. Wishahi,
M. Witek,
W. Witzeling,
S.A. Wotton,
K. Wyllie,
Y. Xie,
F. Xing,
Z. Yang,
R. Young,
O. Yushchenko,
M. Zavertyaev,
L. Zhang,
W.C. Zhang,
Y. Zhang,
A. Zhelezov,
L. Zhong,
E. Zverev,
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, AnnecyLeVieux, France
Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, ClermontFerrand, France
CPPM, AixMarseille Université, CNRS/IN2P3, Marseille, France
LAL, Université ParisSud, CNRS/IN2P3, Orsay, France
LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France
Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
MaxPlanckInstitut für Kernphysik (MPIK), Heidelberg, Germany
Physikalisches Institut, RuprechtKarlsUniversität 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
Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
Nikhef National Institute for Subatomic Physics and Vrije Universiteit, Amsterdam, Netherlands
Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland
Faculty of Physics & Applied Computer Science, Cracow, Poland
Soltan Institute for Nuclear Studies, Warsaw, Poland
Horia Hulubei National Institute of Physics and Nuclear Engineering, BucharestMagurele, 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
PhysikInstitut, Universität Zürich, Zürich, Switzerland
NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine
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
CCIN2P3, CNRS/IN2P3, LyonVilleurbanne, France, associated member
Pontifícia Universidade Católica do Rio de Janeiro (PUCRio), Rio de Janeiro, Brazil, 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
Institució Catalana de Recerca i Estudis Avancats (ICREA), Barcelona, Spain
Hanoi University of Science, Hanoi, Viet Nam
1 Introduction
After the observation of  mixing and the measurement of its strength in 1987 [1], it took a further 19 years for the  frequency to be measured for the first time [2],[3]. This is mainly due to the fact that the  oscillation frequency is 35 times larger than that for the  system, posing a considerable challenge for the decay time resolution of detectors. For the LHCb experiment, the ability to resolve these fast  oscillations is a prerequisite for many physics analyses. In particular it is essential for the study of the timedependent asymmetry of decays [4]. The oscillation frequency in the  system is given by the mass difference between the heavy and light mass eigenstates, (we use units with = 1). In this letter, we report a measurement of by the LHCb experiment with data collected in 2010.
The LHCb spectrometer covers the pseudorapidity range 2 to 5. In this region, hadrons are produced with a large Lorentz boost and have an average flight path of 7 mm. The LHCb detector consists of several components arranged along the LHC beam line. The vertex detector (VELO) surrounds the collision point, followed by a first Ring Imaging Cherenkov (RICH) counter, a tracking station, a dipole magnet, three more tracking stations, a second RICH detector, a calorimeter system and a muon detector. The calorimeter system consists of a scintillating pad detector (SPD), a preshower detector, an electromagnetic calorimeter and a hadronic calorimeter. A detailed description of the detector can be found in Ref. [5]. The precise spatial resolution of the VELO results in an impact parameter resolution of 20–50 m in the and directions^{1}^{1}1LHCb uses a righthanded Cartesian coordinate system with the direction pointing inside the LHC ring, the direction pointing upwards and the direction running along the beamline from the interaction point towards the spectrometer. for charged particles with transverse momenta in the range relevant for daughter tracks used in this analysis. The and resolution in the position of the primary vertex reconstruction is about 15 m while the resolution is about 80 m. This excellent performance results in the decay time resolution needed to observe the fast  oscillations. The invariant mass resolution provided by the tracking system and the / separation given by the two RICH detectors provide clean meson signals with small background. The particle identification capabilities of the RICH together with the calorimeter and muon systems allow the initial flavour of the to be tagged using charged kaons, electrons and muons, respectively.
In the next section, the data sample used and the analysis strategy are introduced. This is followed by descriptions of the analysis of the invariant mass and decay time distributions, and the flavour tagging. Finally, we discuss the fit result for the oscillation frequency and the associated systematic uncertainties.
2 Data sample and analysis strategy
The analysis uses candidates reconstructed in four flavourspecific decay modes,^{2}^{2}2Unless explicitly stated, inclusion of chargeconjugated modes is implied. namely , , and . To avoid double counting, candidates that pass the selection criteria of one mode are not considered for the following modes. All reconstructed decays are flavourspecific final states, thus the flavour of the at the time of its decay is given by the charges of the final state particles of the decay. A combination of tagging algorithms is used to identify the flavour at production. The algorithms provide for each event a tagging decision as well as an estimate of the probability that this decision is wrong (mistag probability). These algorithms have been optimized and calibrated using large event samples of flavourspecific and decays and a sample of decays.
The analysis is based on a data set of 36 pb of collisions at TeV collected in 2010. The first trigger level is implemented in hardware, while the second trigger level is based on software. Trigger conditions were progressively tightened over the duration of the data taking period to cope with the rapidly increasing instantaneous luminosities delivered by the LHC. In the hardware trigger, the events used in this analysis were selected by requiring a cluster with a minimum transverse energy in the hadronic calorimeter. The applied threshold was increased from 2.5 to 3.6 throughout the data taking period. A cut on the number of hits in the SPD detector was applied to reject very high occupancy events. The software trigger for the first 2.4 pb of data required a good quality displaced vertex reconstructed from two tracks with transverse momenta of at least 500 . For the remaining data, a twolevel software trigger was applied. A good quality track with large impact parameter with respect to the primary vertex was required with and momentum [6]. For events passing these criteria, a good quality displaced vertex was required, formed out of two tracks with and and with a mass variable in the range 2 to 7 [7].
Some of the offline event selection criteria are optimized individually for each of the four decay modes under study. In this way specific features such as the masses of the intermediate and resonances or the helicity angle distribution of the can be used. The selection criteria common to all decay modes exploit the long lifetime by applying cuts on the impact parameters of the daughter tracks, on the angle of the reconstructed momentum relative to the line between the reconstructed primary vertex and the vertex and on the decay time. Additional cuts are applied on the and of the candidate and its decay products as well as on particle identification variables and on track and vertex quality. Finally, cuts on the impact parameter significance of the reconstructed and its distance of closest approach to the primary vertex are applied. The reconstructed mass is required to be consistent with the PDG value [8]. After this selection, a total of about 14,400 candidates remain in the invariant mass window of [4.80, 5.85] and in the invariant mass window of [5.00, 5.60] .
An unbinned likelihood method is employed to fit simultaneously the invariant mass and decay time distributions of the four decay modes. The probability density functions (PDFs) for the signal and for the background in each of the four modes can be written as
(1) 
where is the reconstructed invariant mass of the candidate, is its reconstructed decay time and is the eventbyevent estimate of the decay time resolution given by the event reconstruction algorithm. The tagging decision can be 0 (no tag), (different flavour at production and decay) or (same flavour at production and decay). The predicted eventbyevent mistag probability can take values between 0 and 0.5. The terms and describe the invariant mass distribution and the decay time distribution, respectively. is a conditional probability depending on and . The terms and are required to ensure the proper relative normalization of for signal and background [9]. These terms are determined directly from the data, using the measured distribution in the upper invariant mass sideband for the background PDF and the sideband subtracted distribution in the invariant mass signal region for the signal PDF.
3 Fit to the invariant mass distributions
The invariant mass of each candidate is determined in a vertex fit using a constraint on the mass. The invariant mass spectra for the four decay modes after all selection criteria are shown in Fig. 1. The four distributions are fit simultaneously taking into account contributions from signal, combinatorial background and decay backgrounds. The signals are described by Gaussian distributions. The fit constrains the mean of the Gaussian distributions to be the same for all four decay modes, whereas it allows the width to be different for the and the modes, respectively. The combinatorial backgrounds are described by exponential functions. Their parameters are allowed to vary individually for the four decay modes. An alternative parameterization of the combinatorial backgrounds by a first order polynomial is used as part of the systematic studies.
The decay backgrounds include partially reconstructed decays, as well as fully and partially reconstructed and decays with one misidentified daughter particle. Their shapes are derived from a large simulated event sample, where all selection cuts were applied on generator level quantities. The invariant mass spectra were then smeared with a Gaussian distribution to take into account effects of detector resolution. This approach was validated by comparing the results with those from a full simulation including a detailed description of the detector response. The relative normalization factors for the different decay backgrounds are parameters in the fit. They are constrained to be the same for the three decay modes.
The fit returns a value of = 5364.7 0.7 ,
about 1.5 below the PDG value [8]. This mass shift is
attributed to imperfections in the detector
alignment and magnetic field calibration. A dedicated study on the momentum
scale resulted in
a correction for this effect [10]. This calibration procedure is however not used for the
analysis presented here as the momentum scale correction largely cancels in the
calculation of . The mass templates describing decay backgrounds
are shifted according to the observed bias. The fit gives signal mass
resolutions of = 18.1 for the modes and = 12.7 for the mode, respectively. The signal yields extracted from the fit are summarized in
Table 1. For the remainder of the analysis, the
invariant mass range is limited to [, 5.85 GeV/] and [, 5.60 GeV/] for the and modes,
respectively.
The lower cut of this asymmetric mass window is chosen to reject all background candidates from partial
reconstructed decays. The only remaining decay backgrounds are thus due to
misidentified and decays. The candidates in the high mass sidebands provide a
clean sample of combinatorial background. Including them in the fit permits
to determine the decay time distribution and tagging behaviour of
this background contribution.
The parameters derived in the fit to the mass distributions are
fixed for the remainder of the analysis.
Decay mode  Signal yield 

515 25  
338 27  
283 27  
245 46  
Total  1381 65 
4 Fit to the decay time distribution
Ignoring detector resolution effects, selection biases and flavour tagging, the distribution of the decay time of the signal is described by
(2) 
where is the decay width and the decay width difference between the heavy and the light mass eigenstates. In the fit is fixed to its PDG value of 0.09 [8]. As part of the evaluation of systematic uncertainties on , the assumed value of is varied within its current uncertainty between 0 and 0.2 . The step function restricts the PDF to positive decay times.
The true decay time is convolved with the decay time resolution function of the detector. An eventbyevent estimate of the decay time resolution is calculated by the fitting algorithm, which reconstructs the decay vertex of the and computes its decay length and decay time. No constraint on the mass is applied in the computation of the decay time in order to minimize sensitivity to the knowledge of the momentum scale of the experiment. The decay time uncertainty calculated by the fitting algorithm does not include possible effects from an imperfect understanding of the detector material or its spatial alignment. To correct for such effects, the calculated eventbyevent decay time uncertainties, , are multiplied by a constant scale factor . The value of is determined from data, using a sample of fake candidates formed by a prompt and a from the primary vertex. The contamination due to secondary from decays is estimated and statistically subtracted using the measured impact parameter distribution. The distribution of decay times for this fake sample, each divided by its calculated eventbyevent uncertainty, is fitted with a Gaussian function and is taken as the resulting standard deviation. Using the full sample of fake candidates, a value of =1.3 is obtained. This value is used as the nominal scale factor in the analysis. Studying different regions of phase space of the fake candidates separately, values for between 1.2 and 1.4 are obtained. This variation is taken into account for evaluating the systematic uncertainties on . Including the nominal scale factor =1.3, the average decay time resolution is 44 fs for the sample and 36 fs for the sample. The decay time resolution is taken into account in the PDF by convolving Eq. 2 with a Gaussian with mean zero and standard deviation 1.3 .
The shape of the decay time distribution is distorted by trigger and offline selection criteria which require several particles with large impact parameter with respect to the primary vertex. This is accounted for in the PDF by introducing an acceptance function , derived from a full detector simulation. Determining from simulation is deemed acceptable since it cancels to first order in the determination of . The untagged signal decay time PDF becomes
(3) 
The decay time distributions for the decay backgrounds from and decays are described in the same way as that for signal candidates, using the PDG values for their lifetimes and =0. The shape of the decay time distribution for the combinatorial background is described by the sum of two exponential functions multiplied by a second order polynomial. The parameters of these functions are derived from the high mass sidebands. Figure 2 illustrates the results of the lifetime fit. Within its statistical uncertainty the reconstructed lifetime agrees with the PDG value [8].
5 Flavour tagging
To determine the flavour of the candidate at production we exploit the fact that quarks are predominantly produced in quarkantiquark pairs. The quark which is not part of the meson gives rise to an oppositeside hadron. For oppositeside hadron decay candidates, the charge of displaced muons, electrons and kaons and a decay vertex charge estimate are combined using a neural network to form a single oppositeside tagging decision. The tagging decision has a probability to be wrong which is called the mistag probability, . For each event an estimate, , of the mistag probability, is determined based upon topological and kinematic properties of the event, including the number of primary vertices, the number of tagging particle candidates, the impact parameter of the tagging particle and of the candidate with respect to the primary vertex, and the and of the selected tagging particle and the candidate. The optimization of the tagging algorithms and an initial calibration of are performed in an independent analysis using large event samples of and decays. More details on the individual tagging algorithms and this calibration procedure can be found in Ref. [11].
The and events used in the optimization and calibration were collected using different trigger and selection criteria than for the and events used in the analysis described here. As trigger and selection cuts can bias the distributions of the event properties used by the tagging algorithms, this could result in a biased estimate for the and events. Therefore, a recalibration is performed using a sample of 6,000 events, which have a similar topology to the and events, and were collected using the same trigger and similar selection cuts. This event sample is used to perform a measurement of the  flavour oscillation using a very similar method to that described here. In that measurement the true event mistag probability, , is parameterized as a linear function of using the relationship , where is the mean of the distribution of the values obtained from the initial tagger optimization. The parameters and are determined as part of the maximum likelihood fit of the  oscillation signal and found to be consistent with the original calibration. As a byproduct of this recalibration procedure the  oscillation frequency is measured. The resulting value of = 0.499 0.032 (stat) 0.003 (syst) ps, though statistically less precise, is in good agreement with the PDG value of = 0.507 0.004 ps [8] and provides a valuable cross check of the procedure.
The statistical power of the tagging is determined by the “effective” tagging efficiency for signal events and is defined as
(4) 
where the signal tagging efficiency is a free parameter in the fit of the oscillation frequency described in the next section. is the probability for being a signal event as determined by the invariant mass and decay time PDFs. The index runs over all candidates.
6 Measurement of the oscillation frequency
To determine the oscillation frequency, , the decay time PDF for signal candidates with tagging information is modified in the following way:
(5)  
The decay time PDF for untagged signal events is given by Eq. (3) multiplied by an additional factor . The calibration parameters and of the mistag probability are identical for all signal and decay background components. Within Gaussian constraints they are set to the values found in the calibration described in the previous section. The signal tagging efficiency for the and modes are two separate parameters in the fit. The same values of are however used for signal and decay background components in each of these two categories. In the description of the combinatorial background a separate parameter for the tagging efficiency is introduced for each of the four modes. In addition, tagging asymmetry parameters are introduced in the PDFs for the combinatorial background, to allow for a different number of events tagged as or in each mode. As expected the fit results for these asymmetries are compatible with zero.
The fit for the oscillation frequency is performed simultaneously to all four decay modes and gives = 17.63 0.11 ps (statistical uncertainty only). Signal tagging efficiencies of = (23.6 1.3) % and = (17.6 3.2) % are found for the and modes, respectively. The combined effective tagging efficiency for all four modes is = (3.8 2.1) %. The likelihood profile as a function of the assumed oscillation frequency is shown in Fig. 3. The statistical significance of the signal is evaluated to be 4.6 by comparing the likelihood value at the minimum of the fit with that found in the limit = .
To illustrate the oscillation pattern, we define the time dependent mixing asymmetry as
(6) 
where and are the number of background subtracted signal candidates with a given decay time and tagging decision and , respectively. Note, that this definition of the asymmetry does not include any information on the mistag probabilities and therefore does not use the full information of the likelihood fit. Despite the limited size of the sample, the oscillation pattern is clearly visible when the asymmetry is plotted in bins of the decay time modulo (Fig. 4). In an ideal scenario of perfect tagging and perfect decay time resolution the amplitude of this oscillation would be 1.0. The observed amplitude is reduced due to the performance of the tagging algorithm by a factor 0.41. Another reduction of 0.65 occurs due to the limited decay time resolution.
7 Systematic uncertainties
The dominant source of systematic uncertainty is due to the knowledge of the absolute decay time scale of the experiment. This uncertainty is dominated by the knowledge of the scale. A relative uncertainty of 0.1% on the scale and thus on the decay length is assigned based on comparisons of detector surveys and a software alignment using reconstructed tracks. This leads to a systematic uncertainty of 0.018 ps on . A second contribution to the decay time scale is due to the momentum scale of the experiment. From an independent analysis of the mass scale using various known resonances an uncertainty of the uncalibrated momentum scale of less than 0.1% is estimated. This uncertainty partially cancels as it enters both the reconstructed mass and the momentum. The resulting relative uncertainty on the decay time is 0.02%, which translates to an absolute systematic uncertainty of 0.004 ps on .
The next largest systematic uncertainty is related to the description of the combinatorial background in the fit to the mass spectra. It is evaluated by replacing the exponential function by a first order polynomial. Based on the shift in the value obtained for , a systematic uncertainty of 0.010 ps is assigned. Finally, based on variations of the decay time resolution scale factor within its estimated uncertainty from 1.2 to 1.4, a systematic uncertainty of 0.006 ps is assigned on . These contributions to the systematic uncertainty on are summarized in Table 2.
Various other possible sources of systematic effects have been studied, such as the decay time resolution model, the decay time acceptance, releasing parameters of the invariant mass and decay time PDF in the mixing fit, different parameterizations of the invariant mass of the decay backgrounds and variations of the value of . They are found to be negligible.
Source  Uncertainty [ps] 

Momentum scale  0.004 
scale  0.018 
Comb. background mass shape  0.010 
Decay time resolution  0.006 
Total systematic uncertainty  0.022 
8 Conclusion
A measurement of the  oscillation frequency is performed using and decays collected in 36 pb of collisions at = 7 TeV in 2010. The result is found to be
(7) 
This is in good agreement with the previous best measurement of = 17.77 0.10 (stat) 0.07 (sys) ps, reported by the CDF collaboration [3]. As a by product of the analysis we also determine a value for the  oscillation frequency = 0.499 0.032 (stat) 0.003 (syst) ps. Our results are completely dominated by statistical uncertainties and thus significant improvements are expected with larger data sets.
Acknowledgements
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 CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, 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 and the Region Auvergne.
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