EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-063 LHCb-PAPER-2013-015 April 16, 2013

Measurement of the effective $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ lifetime

The LHCb collaboration†††Authors are listed on the following pages.

This paper reports the first measurement of the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime and an updated measurement
of its time-integrated branching fraction. Both measurements are performed
with a data sample, corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$ of $pp$ collisions, recorded by the LHCb experiment in 2011
at a centre-of-mass energy of $7\>\mathrm{\,Te\kern-1.00006ptV}$. The results
are: $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}=1.75\pm
0.12\>(\text{stat})\pm 0.07\>(\text{syst})\>\text{ps}$ and ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})=(1.97\pm 0.23)\times 10^{-5}$. For
the latter measurement, the uncertainty includes both statistical and
systematic sources.

Published in Nucl. Phys. B

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A.
Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez58, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain57, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal57, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan57, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

In the Standard Model (SM), $C\\!P$ violation arises through a single phase in
the CKM quark mixing matrix [1, *Cabibbo:1963yz]. In decays of neutral $B$
mesons ($B$ stands for a $B^{0}$ or $B^{0}_{s}$ meson) to a final state
accessible to both $B$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$, the
interference between the amplitude for the direct decay and the amplitude for
decay via oscillation leads to time-dependent $C\\!P$ violation. A measurement
of the time-dependent $C\\!P$ asymmetry in the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode allows for a determination of the $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing phase $\phi_{d}$. In the SM
it is equal to $2\beta$ [3], where $\beta$ is one of the angles of the
unitarity triangle in the quark mixing matrix. This phase has already been
well measured by the $B$ factories [4, 5], but further improvements are still
necessary to conclusively resolve possible small tensions with the other
measurements constraining the unitarity triangle [6, *Charles:2004jd]. The
latest average composed by the Heavy Flavour Averaging Group (HFAG) is
$\sin\phi_{d}=0.682\pm 0.019$ [8]. To achieve precision below the percent
level, knowledge of the doubly Cabibbo-suppressed higher order perturbative
corrections, originating from penguin topologies, becomes mandatory. These
contributions are difficult to calculate reliably and therefore need to be
determined directly from experimentally accessible observables.

From a theoretical perspective, the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode is the most promising candidate for this task. It is related to the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ mode through the interchange of all
$d$ and $s$ quarks ($U$-spin symmetry, a subgroup of $SU(3)$) [9], leading to
a one-to-one correspondence between all decay topologies of these two modes,
as illustrated in Fig. 1. Moreover, the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ penguin topologies are not CKM suppressed relative to the tree diagram, as
is the case for their $B^{0}$ counterparts. A further discussion regarding the
theory of this decay and its potential use in LHCb is given in Ref. [10,
*DeBruyn:2010ge].

To determine the parameters related to the penguin contributions in these
decays, a time-dependent $C\\!P$ violation study of the $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ mode is required. The determination
of its branching fraction, previously measured by CDF [12] and LHCb [13], was
an important first step, allowing a test of the $U$-spin symmetry assumption
that lies at the basis of the proposed approach. The second step towards the
time-dependent $C\\!P$ violation study is the measurement of the effective
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime, formally defined as [14]

$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}^{\text{eff}}\equiv\frac{\int_{0}^{\infty}t\>\langle\Gamma(B_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S})\rangle\>\mathrm{d}t}{\int_{0}^{\infty}\langle\Gamma(B_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})\rangle\>\mathrm{d}t}\>,$ (1)

where

$\displaystyle\langle\Gamma(B_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})\rangle$ $\displaystyle=$
$\displaystyle\Gamma(B^{0}_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})+\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}(t)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})$ (2) $\displaystyle=$ $\displaystyle
R_{\mathrm{H}}e^{-\Gamma_{\mathrm{H}}t}+R_{\mathrm{L}}e^{-\Gamma_{\mathrm{L}}t}$
(3)

is the untagged decay time distribution, under the assumption that $C\\!P$
violation in $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing can be neglected [8].
Due to the non-zero decay width difference
$\Delta\Gamma_{s}\equiv\Gamma_{\mathrm{H}}-\Gamma_{\mathrm{L}}=0.106\pm
0.013\>\text{ps}^{-1}$ [15] between the heavy and light $B^{0}_{s}$ mass
eigenstates, the effective lifetime does not coincide with the $B^{0}_{s}$
lifetime $\tau_{B^{0}_{s}}\equiv 1/\Gamma_{s}=1.513\pm 0.011\>\text{ps}$ [15],
where $\Gamma_{s}=(\Gamma_{\mathrm{H}}+\Gamma_{\mathrm{L}})/2$ is the average
$B^{0}_{s}$ decay width. Instead, it depends on the decay mode specific
relative contributions $R_{\mathrm{H}}$ and $R_{\mathrm{L}}$. These two
parameters also define the $C\\!P$ observable

$\mathcal{A}_{\Delta\Gamma_{s}}\equiv\frac{R_{\mathrm{H}}-R_{\mathrm{L}}}{R_{\mathrm{H}}+R_{\mathrm{L}}}\>,$
(4)

which allows the effective lifetime to be expressed as [14]

$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}^{\text{eff}}=\frac{\tau_{B^{0}_{s}}}{1-y_{s}^{2}}\frac{1+2\>\mathcal{A}_{\Delta\Gamma_{s}}y_{s}+y_{s}^{2}}{1+\mathcal{A}_{\Delta\Gamma_{s}}y_{s}}\>,$
(5)

where $y_{s}\equiv\Delta\Gamma_{s}/2\Gamma_{s}$ is the normalised decay width
difference. For the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode, the value of $\mathcal{A}_{\Delta\Gamma_{s}}$ depends on the penguin
contributions, and in particular on their relative weak phase $\phi_{s}$ [9].
Using the latest estimates on the size of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ penguin contributions [16] gives $\mathcal{A}_{\Delta\Gamma_{s}}=0.944\pm
0.066$ and the SM prediction

$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}^{\text{eff}}\Big{|}_{\text{SM}}=1.639\pm 0.022\>\text{ps}\>.$ (6)

Effective lifetime measurements have been performed for the $B^{0}_{s}$
$\rightarrow$ $K^{+}$$K^{-}$ [17] and $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$f_{0}(980)$ [18] decay modes.

Figure 1: Decay topologies contributing to the $B_{d(s)}$$\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ channel: (left) tree diagram and (right) penguin diagram.

This paper presents the first measurement of the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime, as well as an update of the
time-integrated branching fraction measurement in Ref. [13], performed with a
data sample, corresponding to an integrated luminosity of
$1.0\>\mbox{\,fb}^{-1}$ of $pp$ collisions, recorded at a centre-of-mass
energy of $7\>\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb experiment in 2011.

The LHCb detector [19] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$) with
respect to the beam direction. Charged hadrons are identified using two ring-
imaging Cherenkov detectors [20]. 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.

Events are selected by a trigger system [21] consisting of a hardware trigger,
which requires muon or hadron candidates with high $p_{\rm T}$, followed by a
two-stage software trigger. In the first stage a partial event reconstruction
is performed. For this analysis, events are required to have either two
oppositely charged muons with combined mass above
$2.7\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or at least one muon or one
high-$p_{\rm T}$ track ($\mbox{$p_{\rm
T}$}>1.8\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) with a large impact parameter
with respect to all $pp$ interaction vertices (PVs). In the second stage a
full event reconstruction is performed and only events containing
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$
$\mu^{+}$$\mu^{-}$ candidates are retained.

The signal simulation samples used for this analysis are generated using
Pythia 6.4 [22] with a specific LHCb configuration [23]. Decays of hadronic
particles are described by EvtGen [24] in which final state radiation is
generated using Photos [25]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [26,
*Agostinelli:2002hh] as described in Ref. [28].

##  2 Data samples and initial selection

Candidate $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ decays are reconstructed in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$ $\mu^{+}\mu^{-}$
and $K^{0}_{\rm\scriptscriptstyle S}$ $\rightarrow$ $\pi^{+}\pi^{-}$ final
state. Candidate ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $\rightarrow$
$\mu^{+}\mu^{-}$ decays are required to form a good quality vertex and have a
mass in the range $[3030,3150]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
This interval corresponds to about eight times the $\mu^{+}\mu^{-}$ mass
resolution at the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass and
covers part of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ radiative
tail. The selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate is
required to satisfy the trigger decision at both software trigger stages. The
$K^{0}_{\rm\scriptscriptstyle S}$ selection requires two oppositely charged
particles reconstructed in the tracking stations placed on either side of the
magnet, both with hits in the vertex detector (‘long
$K^{0}_{\rm\scriptscriptstyle S}$’ candidate) or without (‘downstream
$K^{0}_{\rm\scriptscriptstyle S}$’ candidate). The long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ $\rightarrow$ $\pi^{+}\pi^{-}$ candidates
are required to form a good quality vertex and have a mass within
$35\>(64)\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass [29]. Moreover, to remove contamination
from $\mathchar 28931\relax$ $\rightarrow$ $p$$\pi^{-}$ decays, the
reconstructed $p$$\pi^{-}$ mass of the long (downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ candidates is required to be more than
$6\>(10)\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ away from the known
$\mathchar 28931\relax$ mass [29]. Furthermore, the
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have a flight
distance that is at least five times larger than its uncertainty.

Candidate $B$ mesons are selected from combinations of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidates with mass
$m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle
S}}$ in the range $[5180,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
reconstructed mass and decay time are obtained from a kinematic fit [30] that
constrains the masses of the $\mu^{+}\mu^{-}$ and $\pi^{+}\pi^{-}$ pairs to
the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ masses [29], respectively, and constrains
the $B$ candidate to originate from the PV. In case the event has multiple
PVs, all combinations are considered. The $\chi^{2}$ of the fit, which has
eight degrees of freedom, is required to be less than $72$ and the estimated
uncertainty on the $B$ mass must not exceed
$30\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Candidates are required to
have a decay time larger than $0.2\>{\rm\,ps}$. To remove misreconstructed
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{*0}$
background that survives the requirement on the $K^{0}_{\rm\scriptscriptstyle
S}$ flight distance, the mass of the long $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ candidates computed under the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{\pm}$$\pi^{\mp}$ mass hypotheses must not be within
$20\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}$ mass [29].

##  3 Multivariate selection

The loose selection described above does not suppress the combinatorial
background sufficiently to isolate the small $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ signal. The initial selection is therefore followed by a multivariate
analysis, based on a neural network (NN) [31]. The NN classifier’s output is
used as the final selection variable.

The NN is trained entirely on data, using the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ signal as a proxy for the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ decay. The training sample is taken from the mass windows
$[5180,5340]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5390,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, thus avoiding the
$B^{0}_{s}$ signal region. A normalisation sample consisting of one quarter of
the candidates, selected at random, is left out of the NN training to allow an
unbiased measurement of the $B^{0}$ yield. The signal and background weights
are determined using the _sPlot_ technique [32] and obtained by performing an
unbinned maximum likelihood fit to the mass distribution of the candidates
surviving the loose selection criteria. The fitted probability density
function (PDF) is defined as the sum of a $B^{0}$ signal component and a
combinatorial background. The parametrisation of the individual components is
described in more detail in the next section.

Due to the differences in the distributions of the input variables of the NN,
as well as the different initial signal to background ratio, the multivariate
selection is performed separately for the $B$ candidate samples containing
long and downstream $K^{0}_{\rm\scriptscriptstyle S}$ candidates. In the
remainder of this paper, these two datasets will be referred to as the long
and downstream $K^{0}_{\rm\scriptscriptstyle S}$ sample, respectively. The NN
classifiers use information about the candidate kinematics, vertex and track
quality, impact parameter, particle identification information from the RICH
and muon detectors, as well as global event properties like track and
interaction vertex multiplicities. The variables that are used in the NN are
chosen to avoid correlations with the reconstructed $B$ mass.

Final selection requirements on the NN classifier outputs are chosen to
optimise the expected sensitivity to the $B^{0}_{s}$ signal observation. The
expected signal and background yields entering the calculation of the figure
of merit [33] are obtained from the normalisation sample by scaling the number
of fitted $B^{0}$ candidates, and by counting the number of events in the mass
ranges $[5180,5240]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5400,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. After
applying the final requirement on the NN classifier output associated with the
long (downstream) $K^{0}_{\rm\scriptscriptstyle S}$ sample, the multivariate
selection rejects, relative to the initial selection, 98.7% (99.6%) of the
background while keeping 71.5% (50.2%) of the $B^{0}$ signal. Due to the worse
initial signal to background ratio, the final requirement on the NN classifier
output is much tighter in the downstream $K^{0}_{\rm\scriptscriptstyle S}$
sample than in the long $K^{0}_{\rm\scriptscriptstyle S}$ sample.

After applying the full selection, the $B$ candidate can still be associated
with more than one PV in about 1% of the events. Likewise, about $0.1\%$ of
the selected events have several candidates sharing one or more tracks. In
these cases, respectively one of the surviving PVs and one of the candidates
is used at random.

##  4 Event yields

Long $K^{0}_{\rm\scriptscriptstyle S}$

Downstream $K^{0}_{\rm\scriptscriptstyle S}$

Figure 2: Fitted $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate mass distributions and
their associated residual uncertainties (pulls) for the (left) long and
(right) downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples, after applying
the final requirement on the NN classifier outputs.

For the candidates passing the NN requirements, the ratio of $B^{0}_{s}$ and
$B^{0}$ yields is determined from an unbinned maximum likelihood fit to the
mass distribution of the reconstructed $B$ candidates. The fitted PDF is
defined as the sum of a $B^{0}$ signal component, a $B^{0}_{s}$ signal
component and a combinatorial background. The $B^{0}_{s}$ component is
constrained to have the same shape as the $B^{0}$ PDF, shifted by the known
$B^{0}_{s}$–$B^{0}$ mass difference [34]. The mass lineshapes of the $B$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ modes in both data and simulation
exhibit non-Gaussian tails on both sides of their signal peaks due to final
state radiation, the detector resolution and its dependence on the decay
angles. Each individual signal shape is parametrised by a double-sided Crystal
Ball (CB) function [35]. The parameters describing the CB tails are taken from
simulation; all other parameters are allowed to vary in the fit. The
background contribution is described by an exponential function.

The results of the fits are shown in Fig. 2, and the fitted yields are listed
in Table 1. The $B^{0}$ yield is determined in the normalisation sample and
scaled to the full sample, whereas the $B^{0}_{s}$ yield is obtained directly
from the full sample. The scaled $B^{0}$ yield, obtained from the unbiased
sample, differs from the corresponding fit result in the full sample by
$-211\pm 211$ events for the long $K^{0}_{\rm\scriptscriptstyle S}$ sample and
by $213\pm 273$ events for the downstream $K^{0}_{\rm\scriptscriptstyle S}$
sample. Both results are in good agreement, showing that the NN is not
overtrained. The yield ratios obtained from the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ samples are compatible with each other and
are combined using a weighted average.

Table 1: Signal yields from the unbinned maximum likelihood fits to the $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate mass distributions. The uncertainties are statistical only. The yield ratio is calculated from the quantities highlighted in boldface, where the fitted $B^{0}$ yield is first multiplied by a factor of four.  Sample | Yield | Long $K^{0}_{\rm\scriptscriptstyle S}$ | Downstream $K^{0}_{\rm\scriptscriptstyle S}$
---|---|---|---
Normalisation |  $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $\textbf{2205}\ \pm\ \textbf{47}$ | $\textbf{3651}\ \pm\ \textbf{61}$
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $21\ \pm\ 5$ | $49\ \pm\ 8$
Background | $56\ \pm\ 11$ | $110\ \pm\ 16$
Full |  $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $9031\ \pm\ 96$ | $\text{14,391}\ \pm\ 122$
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ | $\textbf{115}\ \pm\ \textbf{12}$ | $\textbf{158}\ \pm\ \textbf{15}$
Background | $287\ \pm\ 23$ | $490\ \pm\ 32$
Yield ratio $R\equiv N_{B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{Full}}/4N_{B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{Norm}}$ | $0.0131\ \pm\ 0.0014$ | $0.0108\ \pm\ 0.0010$
Average yield ratio $R$ | $0.0116\pm 0.0008$

##  5 Decay time distribution

Following the procedure explained in Ref. [36], the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime is determined by fitting a
single exponential function $g(t)\propto\exp(-t/\tau_{\text{single}})$ to the
decay time distribution of the $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ signal candidates. In this analysis, the exponential shape parameter
$\tau_{\text{single}}$ is determined from a two-dimensional unbinned maximum
likelihood fit to the mass and decay time distribution of the reconstructed
$B$ candidates. The fitted PDF is again defined as the sum of a $B^{0}$ signal
component, a $B^{0}_{s}$ signal component and a combinatorial background. The
freely varying parameters in the fit are the signal and background yields, and
the parameters describing the acceptance, mass and background decay time
distributions.

The decay time distribution of each of the two signal components needs to be
corrected with a decay time resolution and acceptance model to account for
detector effects. The shape of the acceptance function affecting the
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ mode is, like the lineshape of its
mass distribution, assumed to be identical to that of the $B^{0}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ component. The acceptance function is
obtained directly from the data using the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ mode. Contrary to the $B^{0}_{s}$ system, the $B^{0}$ system has a
negligible decay width difference $\Delta\Gamma_{d}$ [29]. The decay time
distribution of the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ channel is therefore fully described by a single exponential function with
known lifetime $\tau_{B^{0}}=1.519\>\text{ps}$ [8]. Hence, fixing the $B^{0}$
lifetime to its known value allows the acceptance parameters to be determined
from the fit.

From simulation studies it is found that the decay time acceptance of both
signal components is well modelled by the function

$f_{\text{Acc}}\>(t)=\frac{1+\beta\>t}{1+(\lambda\>t)^{-\kappa}}\>.$ (7)

The parameter $\beta$ describes the fall in the acceptance at large decay
times [15]. The parameters $\kappa$ and $\lambda$ model the turn-on curve,
caused by the use of decay time biasing triggers, the initial selection
requirements and, most importantly, the NN classifier outputs.

The decay time resolution for the signal and background components is
determined from candidates that have an unphysical, negative decay time. Due
to the requirement of $0.2\>\text{${\rm\,ps}$}$ on the decay time of the $B$
candidates applied in the initial selection, such events are not present in
the analysed data sample. Instead, a second sample, that is prescaled and does
not have the decay time requirement, is used. This sample consists primarily
of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons produced at the PV
which are combined with a random $K^{0}_{\rm\scriptscriptstyle S}$ candidate.
The decay time distribution for these events is a good measure of the decay
time resolution and is modelled by the sum of three Gaussian functions sharing
a common mean. Two of the Gaussian functions parametrise the inner core of the
resolution function, while the third describes the small fraction of outliers.

Long $K^{0}_{\rm\scriptscriptstyle S}$

Downstream $K^{0}_{\rm\scriptscriptstyle S}$

Figure 3: Fitted $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate decay time distributions
and their associated residual uncertainties (pulls) for the (left) long and
(right) downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples, after applying
the final requirement on the NN classifier outputs.

Long $K^{0}_{\rm\scriptscriptstyle S}$

Downstream $K^{0}_{\rm\scriptscriptstyle S}$

Figure 4: Fitted $B$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ candidate decay time distributions
and their associated residual uncertainties (pulls) for the (left) long and
(right) downstream $K^{0}_{\rm\scriptscriptstyle S}$ candidates in the
$B^{0}_{s}$ signal region $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}\in[5340,5390]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, after applying
the final requirement on the NN classifier outputs.

The background decay time distributions are studied directly using the data.
Their shape is obtained from background candidates that are isolated using the
background weights determined by the _sPlot_ technique, and cross-checked
using the high mass sideband. The exact values of the shape parameters are
determined in the nominal fit. Because of the differences induced by the
multivariate selection, the background decay time distribution of the long and
downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples cannot be parametrised
using the same background model. For the long $K^{0}_{\rm\scriptscriptstyle
S}$ sample, the background is modelled by two exponential functions,
describing a short-lived and a long-lived component, respectively. In the
downstream $K^{0}_{\rm\scriptscriptstyle S}$ sample such a short-lived
component is not present due to the tighter requirement on the NN classifier
output. Its decay time distribution is better described by a single
exponential shape corrected by the acceptance function in Eq. (7) with
independent parameters $(\kappa^{\prime},\alpha^{\prime},\beta^{\prime})$. The
parameter $\beta^{\prime}$ is set to zero because we also fit the lifetime of
the single exponential function itself, and the combination of both parameters
would result in ambiguous solutions.

The decay time distributions resulting from the two-dimensional fits are shown
in Figs. 3 and 4 for candidates in the full mass range
$m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}\in[5180,5520]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and in the
$B^{0}_{s}$ signal region $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle
S}}\in[5340,5390]\>{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively.
The fitted values are $\tau_{\text{single}}=1.54\pm 0.17\>\text{ps}$ and
$\tau_{\text{single}}=1.96\pm 0.17\>\text{ps}$ for the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ sample, respectively. The $1.7\sigma$
difference between both results is understood as a statistical fluctuation.
The two main fit results are therefore combined using a weighted average,
leading to

$\tau_{\text{single}}=1.75\pm 0.12\>\text{ps}\>,$

where the uncertainty is statistical only. The event yields obtained from the
two-dimensional fits are compatible with the results quoted in Table 1.

##  6 Corrections and systematic uncertainties

A number of systematic uncertainties affecting the relative branching fraction
$\cal B$($B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$)/$\cal B$($B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$) and the effective lifetime are considered. The sources affecting the
ratio of branching fractions are discussed first, followed by those
contributing to the effective lifetime measurement.

The largest systematic uncertainty on the yield ratio comes from the mass
shape model, and in particular from the uncertainty on the fraction of the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ component’s high mass tail extending
below the $B^{0}_{s}$ signal. The magnitude of this effect is studied by
allowing both tails of the CB shapes to vary in the fit. The largest observed
deviation in the yield ratios is 3.4%, which is taken as a systematic
uncertainty. The mass resolution, and hence the widths of the CB shapes, is
assumed to be identical for the $B^{0}$ and $B^{0}_{s}$ signal modes, but
could in principle depend on the mass of the reconstructed $B$ candidate. This
effect is studied by multiplying the CB widths of the $B^{0}_{s}$ signal PDF
by different scale factors, obtained by comparing $B^{0}$ and $B^{0}_{s}$
signal shapes in simulation. The largest observed difference in the yield
ratios is 1.4%, which is taken as a systematic uncertainty. Varying the
$B^{0}_{s}$–$B^{0}$ mass difference within its uncertainty has negligible
effect on the yield ratios.

The selection procedure is designed to be independent of the reconstructed $B$
mass. Simulated data is used to check this assumption, and to evaluate the
difference in selection efficiency arising from the different shapes of the
$B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ and $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ decay time distributions. The ratio of total selection efficiencies is
equal to $0.968\pm 0.007$, and is used to correct the yield ratio.

The stability of the multivariate selection is verified by comparing different
training schemes and optimisation procedures, as well as by calculating the
yield ratios for different subsets of the long and downstream
$K^{0}_{\rm\scriptscriptstyle S}$ sample. All of these tests give results that
are compatible with the measured ratio.

The corrections and systematic uncertainties affecting the branching fraction
ratio are listed in Table 2. The total systematic uncertainty is obtained by
adding all the uncertainties in quadrature.

Table 2: Corrections and systematic uncertainties on the yield ratio. Source | Value
---|---
Fit model | $1.000\ \pm\ 0.034$
$B^{0}_{s}$ mass resolution | $1.000\ \pm\ 0.014$
Selection efficiency | $0.968\ \pm\ 0.007$
Total correction $f_{\text{corr}}^{{\cal B}}$ | $0.968\ \pm\ 0.034$

The main systematic uncertainties affecting the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime arise from modelling the
different components of the decay time distribution. Their amplitudes are
evaluated by comparing the results from the nominal fit to similar fits using
alternative parametrisations. All tested fit models give compatible results.
The largest observed deviations in $\tau_{\text{single}}$ are 3.9% due to
modelling of the background decay time distribution, 0.47% due to the
acceptance function and 0.39% due to the reconstructed $B$ mass description,
all of which are assigned as systematic uncertainties. Variations in the decay
time resolution model are found to have negligible impact on
$\tau_{\text{single}}$.

The assumed value of the $B^{0}$ lifetime has a significant impact on the
shape of the acceptance function, and the $\beta$ parameter in particular,
which in turn affects the fitted value of $\tau_{\text{single}}$. This effect
is studied by varying the $B^{0}$ lifetime within its uncertainty [29]. The
largest observed deviation in $\tau_{\text{single}}$ is 0.52%, which is taken
as a systematic uncertainty.

The fit method is tested on simulated data using large sets of pseudo-
experiments, which have the same mass and decay time distributions as the
data. Different datasets are generated using the fitted two-dimensional signal
and background distributions, and $\tau_{\text{single}}$ is then again fitted
to these pseudo-experiments. The fit result is compared with the input value
to search for possible biases. From the spread in the fitted values and the
accompanying residual distributions, a small bias is found. This bias is
attributed to the limited size of the background sample, and the resulting
difficulty to constrain the background decay time parameters. A correction
factor of $1.002\pm 0.002$ is assigned to account for this potential bias.

Due to the presence of a non-trivial acceptance function, the result of
fitting a single exponential to the untagged $B^{0}_{s}$ decay time
distribution does mathematically not agree with the formal definition of the
effective lifetime in Eq. (1), as explained in Ref. [36]. The size and sign of
the difference between $\tau_{\text{single}}$ and
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}$ depend on the values of
$\tau_{B^{0}_{s}}$, $y_{s}$, $\mathcal{A}_{\Delta\Gamma_{s}}$, and the shape
of the acceptance function. The difference is calculated with pseudo-
experiments that sample the acceptance parameters, $\tau_{B^{0}_{s}}$ and
$y_{s}$ from Gaussian distributions related to their respective fitted and
known values. Since $\mathcal{A}_{\Delta\Gamma_{s}}$ is currently not
constrained by experiment, it is sampled uniformly from the interval
$[{-1},1]$. The average difference between
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}$ and
$\tau_{\text{single}}$, obtained using the acceptance function affecting the
long (downstream) $K^{0}_{\rm\scriptscriptstyle S}$ sample, is found to be
$-0.001\>\text{ps}$ $(-0.003\>\text{ps})$. A correction factor of $0.999\pm
0.001$ is assigned to account for this bias.

The presence of a production asymmetry between the $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons could potentially alter
the measured value of the effective lifetime, but even for large estimates of
the size of this asymmetry, the effect is found to be negligible.

Finally, the systematic uncertainties in the momentum and the decay length
scale propagate to the effective lifetime. The size of the former contribution
is evaluated by recomputing the decay time while varying the momenta of the
final state particles within their uncertainty. The systematic uncertainty due
to the decay length scale mainly comes from the track-based alignment. Both
effects are found to be negligible.

The stability of the fit is verified by comparing the nominal results with
those obtained using different fit ranges, or using only subsets of the long
and downstream $K^{0}_{\rm\scriptscriptstyle S}$ samples. All these tests give
compatible results.

The corrections and systematic uncertainties affecting the effective
$B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime are listed in Table 3. The
total systematic uncertainty is obtained by adding all the uncertainties in
quadrature.

Table 3: Corrections and systematic uncertainties on the effective $B^{0}_{s}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime. Source | Value
---|---
Background model | $1.000\ \pm\ 0.039$
Acceptance model | $1.000\ \pm\ 0.005$
Mass model | $1.000\ \pm\ 0.004$
$B^{0}$ lifetime | $1.000\ \pm\ 0.005$
Fit method | $1.002\ \pm\ 0.002$
Effective lifetime definition | $0.999\ \pm\ 0.001$
Total correction $f_{\text{corr}}^{\text{eff}}$ | $1.001\ \pm\ 0.040$

##  7 Results and conclusion

Using the measured ratio $R=0.0116\pm 0.0008$ of $B^{0}_{s}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}_{\rm\scriptscriptstyle
S}$ and $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ yields, the correction factor
$f_{\text{corr}}^{{\cal B}}=0.968\pm 0.034$, and the ratio of hadronisation
fractions $f_{s}/f_{d}=0.256\pm 0.020$ [37], the ratio of branching fractions
is computed to be

$\displaystyle\frac{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})}{{\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})}$ $\displaystyle=$ $\displaystyle
R\times f_{\text{corr}}^{{\cal B}}\times\frac{f_{d}}{f_{s}}$ $\displaystyle=$
$\displaystyle 0.0439\pm 0.0032\>\text{(stat)}\pm 0.0015\>\text{(syst)}\pm
0.0034\>(f_{s}/f_{d})\>,$

where the quoted uncertainties are statistical, systematic, and due to the
uncertainty in $f_{s}/f_{d}$, respectively.

Using the known $B^{0}$ $\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}$ branching fraction [29], the ratio of branching fractions can
be converted into a measurement of the time-integrated $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ branching fraction. Taking into
account the different rates of $B^{+}$$B^{-}$ and $B^{0}$$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pair production at the $\mathchar
28935\relax{(4S)}$ resonance $\Gamma(B^{+}{}B^{-})/\Gamma(B^{0}{}\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0})=1.055\pm 0.025$ [29], the above
result is multiplied by the corrected value ${\cal
B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0})=(8.98\pm 0.35)\times 10^{-4}$ and gives

$\displaystyle{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S})=$ $\displaystyle\left[1.97\pm
0.14\>\text{(stat)}\pm 0.07\>\text{(syst)}\pm 0.15\>(f_{s}/f_{d})\pm
0.08\>({\cal B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}))\right]\times 10^{-5}\>,$

where the last uncertainty comes from the $B^{0}$ $\rightarrow$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$$K^{0}$ branching fraction. This
result is compatible with, and more precise than, previous measurements [12,
13], and supersedes the previous LHCb measurement. The branching fraction is
consistent with expectations from $U$-spin symmetry [13].

Using $\tau_{\text{single}}=1.75\pm 0.12\>\text{ps}$ and the correction factor
$f_{\text{corr}}^{\text{eff}}=1.001\pm 0.040$, the effective $B^{0}_{s}$
$\rightarrow$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{0}_{\rm\scriptscriptstyle S}$ lifetime is given by

$\displaystyle\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}{}K^{0}_{\rm\scriptscriptstyle S}}^{\text{eff}}$ $\displaystyle=$
$\displaystyle f_{\text{corr}}^{\text{eff}}\times\tau_{\text{single}}$
$\displaystyle=$ $\displaystyle 1.75\pm 0.12\>(\text{stat})\pm
0.07\>(\text{syst})\>\text{ps}\>.$

This is the first measurement of this quantity. The result is compatible with
the SM prediction given in Eq. (6).

## 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 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); ANCS/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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