* Lyu  _et al._ (2022) Z. Lyu, N. Jiang, and K. Yagi,  _Phys. Rev. D_ 105, 064001 (2022), arXiv:2201.02543 [gr-qc] .
  * Tsujikawa  _et al._ (2008) S. Tsujikawa, K. Uddin, S. Mizuno, R. Tavakol, and J. Yokoyama,  _Phys. Rev. D_ 77, 103009 (2008), arXiv:0803.1106 [astro-ph] .
  * Lovelock (1971) D. Lovelock,  _J. Math. Phys._ 12, 498 (1971).
  * Fernandes  _et al._ (2021) P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne,  _Phys. Rev. D_ 104, 044029 (2021), arXiv:2107.00046 [gr-qc] .
  * Tsujikawa (2022) S. Tsujikawa, arXiv:2205.09932 [gr-qc] .


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-018 LHCb-PAPER-2013-065 March 19, 2014

Measurements of the $B^{+}$, $B^{0}$, $B^{0}_{s}$ meson and $\mathchar
28931\relax^{0}_{b}$ baryon lifetimes

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

Measurements of $b$-hadron lifetimes are reported using $pp$ collision data,
corresponding to an integrated luminosity of 1.0 fb-1, collected by the LHCb
detector at a centre-of-mass energy of $7$$\mathrm{\,Te\kern-1.00006ptV}$.
Using the exclusive decays
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ the
average decay times in these modes are measured to be

$\tau_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ | = $1.637\ \pm$ 0.004 $\pm$ 0.003 ${\rm\,ps}$,
---|---
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$ | = $1.524\ \pm$ 0.006 $\pm$ 0.004 ${\rm\,ps}$,
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ | = $1.499\ \pm$ 0.013 $\pm$ 0.005 ${\rm\,ps}$,
$\tau_{\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax}$ | = $1.415\ \pm$ 0.027 $\pm$ 0.006 ${\rm\,ps}$,
$\tau_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$ | = $1.480\ \pm$ 0.011 $\pm$ 0.005 ${\rm\,ps}$,

where the first uncertainty is statistical and the second is systematic. These
represent the most precise lifetime measurements in these decay modes. In
addition, ratios of these lifetimes, and the ratio of the decay-width
difference, $\Delta\Gamma_{d}$, to the average width, $\Gamma_{d}$, in the
$B^{0}$ system, $\Delta\Gamma_{d}/\Gamma_{d}=-0.044\pm 0.025\pm 0.011$, are
reported. All quantities are found to be consistent with Standard Model
expectations.

Submitted to JHEP

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

LHCb collaboration

R. Aaij40, B. Adeva36, M. Adinolfi45, A. Affolder51, Z. Ajaltouni5, J.
Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez
Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L.
Anderlini17,g, J. Anderson39, R. Andreassen56, M. Andreotti16,f, J.E.
Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli37, A.
Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,n, M. Baalouch5, S.
Bachmann11, J.J. Back47, A. Badalov35, V. Balagura30, W. Baldini16, R.J.
Barlow53, C. Barschel38, S. Barsuk7, W. Barter46, V. Batozskaya27, 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,i, P.M. Bjørnstad53, T. Blake47,
F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W.
Bonivento15,37, S. Borghi53, A. Borgia58, M. Borsato7, T.J.V. Bowcock51, E.
Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J. Bressieux38, D.
Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51, A. Bursche39,
G. Busetto21,r, J. Buytaert37, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez35,p, A. Camboni35, P. Campana18,37, D. Campora
Perez37, A. Carbone14,d, G. Carboni23,l, R. Cardinale19,j, A. Cardini15, H.
Carranza-Mejia49, L. Carson49, K. Carvalho Akiba2, G. Casse51, L. Castillo
Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles8, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco37, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, I. Counts55, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz
Torres59, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, J. Dalseno45, P.
David8, P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De
Capua53, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach54, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, S.
Donleavy51, F. Dordei11, M. Dorigo38, P. Dorosz25,o, A. Dosil Suárez36, D.
Dossett47, A. Dovbnya42, F. Dupertuis38, P. Durante37, R. Dzhelyadin34, A.
Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52, V. Egorychev30, S. Eidelman33,
S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5,
Ch. Elsasser39, S. Esen11, A. Falabella16,f, C. Färber11, C. Farinelli40, S.
Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M.
Ferro-Luzzi37, S. Filippov32, M. Fiore16,f, M. Fiorini16,f, C. Fitzpatrick37,
M. Fontana10, F. Fontanelli19,j, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,37,g, E. Furfaro23,l, A. Gallas Torreira36, D. Galli14,d,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, J. Garra Tico46, L.
Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T.
Gershon47, Ph. Ghez4, A. Gianelle21, S. Giani’38, V. Gibson46, L. Giubega28,
V.V. Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes1,a,
H. Gordon37, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, P. Griffith44, L. Grillo11, O. Grünberg60, B. Gui58, E. Gushchin32,
Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, T.W.
Hafkenscheid62, S.C. Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S.
Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60,
J. He37, T. Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, M. Heß60, A. Hicheur1, D. Hill54, M. Hoballah5,
C. Hombach53, W. Hulsbergen40, P. Hunt54, N. Hussain54, D. Hutchcroft51, D.
Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten55, R. Jacobsson37, A. Jaeger11,
E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, N. Jurik58, M. Kaballo9, S. Kandybei42, W.
Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20,
C. Khurewathanakul38, S. Klaver53, 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.
Kucharczyk20,25,37,k, V. Kudryavtsev33, K. Kurek27, 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, M. Liles51, R. Lindner37, C. Linn11, F. Lionetto39, B. Liu15, G.
Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-March38, P. Lowdon39,
H. Lu3, D. Lucchesi21,r, J. Luisier38, H. Luo49, E. Luppi16,f, O. Lupton54, F.
Machefert7, I.V. Machikhiliyan30, F. Maciuc28, O. Maev29,37, S. Malde54, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, U. Marconi14, P.
Marino22,t, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, A. Martín
Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2, A.
Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, A. Mazurov16,37,f, M.
McCann52, J. McCarthy44, A. McNab53, R. McNulty12, B. McSkelly51, B.
Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N.
Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, M. Morandin21, P.
Morawski25, A. Mordà6, M.J. Morello22,t, R. Mountain58, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N. Neufeld37, A.D.
Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,q, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin31, T. Nikodem11, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,e, G.
Onderwater62, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A.
Oyanguren35, B.K. Pal58, A. Palano13,c, M. Palutan18, J. Panman37, A.
Papanestis48,37, M. Pappagallo50, L. Pappalardo16, C. Parkes53, C.J.
Parkinson9, G. Passaleva17, G.D. Patel51, M. Patel52, C. Patrignani19,j, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A. Pellegrino40, G.
Penso24,m, M. Pepe Altarelli37, S. Perazzini14,d, E. Perez Trigo36, P.
Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen63, G. Pessina20, K.
Petridis52, A. Petrolini19,j, E. Picatoste Olloqui35, B. Pietrzyk4, T.
Pilař47, D. Pinci24, A. Pistone19, 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, A. Pritchard51, C.
Prouve45, V. Pugatch43, A. Puig Navarro38, G. Punzi22,s, W. Qian4, B.
Rachwal25, J.H. Rademacker45, B. Rakotomiaramanana38, M. Rama18, M.S. Rangel2,
I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, S. Reichert53, M.M.
Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, A.B. Rodrigues1, E.
Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero
Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P.
Ruiz Valls35, G. Sabatino24,l, J.J. Saborido Silva36, N. Sagidova29, P.
Sail50, B. Saitta15,e, V. Salustino Guimaraes2, B. Sanmartin Sedes36, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,l, M. Sapunov6, A.
Sarti18, C. Satriano24,n, A. Satta23, M. Savrie16,f, D. Savrina30,31, 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,f, Y.
Shcheglov29, T. Shears51, L. Shekhtman33, O. Shevchenko42, V. Shevchenko61, A.
Shires9, R. Silva Coutinho47, G. Simi21, M. Sirendi46, N. Skidmore45, T.
Skwarnicki58, N.A. Smith51, E. Smith54,48, E. Smith52, J. Smith46, M. Smith53,
H. Snoek40, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D. Souza45, B. Souza
De Paula2, B. Spaan9, A. Sparkes49, F. Spinella22, P. Spradlin50, F. Stagni37,
S. Stahl11, O. Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B.
Storaci39, S. Stracka22,37, M. Straticiuc28, U. Straumann39, R. Stroili21,
V.K. Subbiah37, L. Sun56, W. Sutcliffe52, S. Swientek9, V. Syropoulos41, M.
Szczekowski27, P. Szczypka38,37, D. Szilard2, T. Szumlak26, S. T’Jampens4, M.
Teklishyn7, G. Tellarini16,f, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, L.
Tomassetti16,f, 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, A. Ustyuzhanin61,
U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez Gomez18, P.
Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M.
Veltri17,h, G. Veneziano38, M. Vesterinen11, B. Viaud7, D. Vieira2, X.
Vilasis-Cardona35,p, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A.
Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, J.A. de Vries40, R. Waldi60,
C. Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, 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. Wimberley57, J. Wishahi9, W.
Wislicki27, M. Witek25, G. Wormser7, S.A. Wotton46, S. Wright46, S. Wu3, K.
Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,b, F. Zhang3, L. Zhang58, 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
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61National Research Centre Kurchatov Institute, Moscow, Russia, associated to
30
62KVI - University of Groningen, Groningen, The Netherlands, associated to 40
63Celal Bayar University, Manisa, Turkey, associated to 37

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy

##  1 Introduction

Within the framework of heavy quark expansion (HQE) theory [1, 2, 3, 4, 5, 6,
7], $b$-hadron observables are calculated as a perturbative expansion in
inverse powers of the $b$-quark mass, $m_{b}$. At zeroth order the lifetimes
of all weakly decaying $b$ hadrons are equal, with corrections appearing at
order $1/m_{b}^{2}$. Ratios of $b$-hadron lifetimes can be theoretically
predicted with higher accuracy than absolute lifetimes since many terms in the
HQE cancel. The latest theoretical predictions and world-average values for
the $b$-hadron lifetimes and lifetime ratios are reported in Table 1. A
measurement of the ratio of the $\mathchar 28931\relax^{0}_{b}$ baryon
lifetime, using the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ decay mode111Charge conjugation is implied throughout this
paper, unless otherwise stated., to that of the $B^{0}$ meson lifetime has
recently been made by the LHCb collaboration [8] and is not yet included in
the world average.

In this paper, a measurement of the lifetimes of the $B^{+}$, $B^{0}$ and
$B^{0}_{s}$ mesons and $\mathchar 28931\relax^{0}_{b}$ baryon is reported
using $pp$ collision data, corresponding to an integrated luminosity of 1.0
fb-1, collected in 2011 with the LHCb detector at a centre-of-mass energy of
$7$$\mathrm{\,Te\kern-1.00006ptV}$. The lifetimes are measured from the
reconstructed $b$-hadron decay time distributions of the exclusive decay modes
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$. Collectively, these are referred to as
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays. In
addition, measurements of lifetime ratios are reported.

As a result of neutral meson mixing the decay time distribution of neutral
$B^{0}_{q}$ mesons ($q\in\\{s,d\\}$) is characterised by two parameters,
namely the average decay width $\Gamma_{q}$ and the decay width difference
$\Delta\Gamma_{q}$ between the light (L) and heavy (H) $B^{0}_{q}$ mass
eigenstates. The summed decay rate of $B^{0}_{q}$ and $\overline{B}^{0}_{q}$
mesons to a final state $f$ is given by [9, 10, 11]

$\langle\Gamma(B^{0}_{q}(t)\rightarrow
f)\rangle\equiv\Gamma(B^{0}_{q}(t)\rightarrow
f)+\Gamma(\overline{B}^{0}_{q}(t)\rightarrow f)=R^{f}_{q,\rm
L}e^{-\Gamma_{q,\rm L}t}+R^{f}_{q,\rm H}e^{-\Gamma_{q,\rm H}t},$ (1)

where terms proportional to the small flavour specific asymmetry, $a_{\rm
fs}^{q}$, are ignored [12]. Therefore, for non-zero $\Delta\Gamma_{q}$ the
decay time distribution of neutral $B^{0}_{q}$ decays is not purely
exponential. In the case of an equal admixture of $B^{0}_{q}$ and
$\overline{B}^{0}_{q}$ at $t=0$, the observed average decay time is given by
[11]

$\tau_{B_{q}^{0}\rightarrow
f}=\frac{1}{\Gamma_{q}}\frac{1}{1-y_{q}^{2}}\left(\frac{1+2\mathcal{A}^{f}_{\Delta\Gamma_{q}}y_{q}+y_{q}^{2}}{1+\mathcal{A}^{f}_{\Delta\Gamma_{q}}y_{q}}\right),$
(2)

where $y_{q}\equiv\Delta\Gamma_{q}/(2\Gamma_{q})$ and
$\mathcal{A}^{f}_{\Delta\Gamma_{q}}\equiv(R_{q,\rm H}^{f}-R_{q,\rm
L}^{f})/(R_{q,\rm H}^{f}+R_{q,\rm L}^{f})$ is an observable that depends on
the final state, $f$. As such, the lifetimes measured are usually referred to
as effective lifetimes. In the $B^{0}_{s}$ system, where
$\Delta\Gamma_{s}/\Gamma_{s}=0.159\pm 0.023$ [13], the deviation from an
exponential decay time distribution is non-negligible. In contrast, in the
$B^{0}$ system this effect is expected to be small as
$\Delta\Gamma_{d}/\Gamma_{d}$ is predicted to be $(42\pm 8)\times 10^{-4}$ in
the Standard Model (SM) [14, 15]. Both the BaBar [16, 17] and Belle [18]
collaborations have measured $|\Delta\Gamma_{d}/\Gamma_{d}|$ and the current
world average is $|\Delta\Gamma_{d}/\Gamma_{d}|=0.015\pm 0.018$ [13]. A
deviation in the value of $\Delta\Gamma_{d}$ from the SM prediction has
recently been proposed [19] as a potential explanation for the anomalous like-
sign dimuon charge asymmetry measured by the D0 collaboration [20]. In this
paper, $\Delta\Gamma_{d}/\Gamma_{d}$ is measured from the effective lifetimes
of $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*}(892)^{0}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays, as proposed in Ref. [21].

The main challenge in the measurements reported is understanding and
controlling the detector acceptance, reconstruction and selection efficiencies
that depend upon the $b$-hadron decay time. This paper is organised as
follows. Section 2 describes the LHCb detector and software. The selection
criteria for the $b$-hadron candidates are described in Sec. 3. Section 4
describes the reconstruction efficiencies and the techniques used to correct
the decay time distributions. Section 5 describes how the efficiency
corrections are incorporated into the maximum likelihood fit that is used to
measure the signal yields and lifetimes. The systematic uncertainties on the
measurements are described in Sec. 6. The final results and conclusions are
presented in Sec. 7.

Table 1: Theoretical predictions and current world-average values [13] for
$b$-hadron lifetimes and lifetime ratios.

Observable Prediction World average $\tau_{B^{+}}$[${\rm\,ps}$ ] – $1.641\pm
0.008$ $\tau_{B^{0}}$[${\rm\,ps}$ ] – $1.519\pm 0.007$
$\tau_{B^{0}_{s}}$[${\rm\,ps}$ ] – $1.516\pm 0.011$ $\tau_{\mathchar
28931\relax^{0}_{b}}$[${\rm\,ps}$ ] – $1.429\pm 0.024$
$\tau_{B^{+}}/\tau_{B^{0}}$ $1.063\pm 0.027$ [22, 23, 15] $1.079\pm 0.007$
$\tau_{B^{0}_{s}}/\tau_{B^{0}}$ $1.00\phantom{0}\pm 0.01\phantom{0}$ [24, 25,
23, 15] $0.998\pm 0.009$ $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{B^{0}}$
0.86–0.950 [3, 26, 27, 28, 29, 30, 23, 31, 32] $0.941\pm 0.016$

##  2 Detector and software

The LHCb detector [33] 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 (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector (TT) 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 provides a momentum, $p$, measurement
with relative uncertainty 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 charged particles with high transverse momentum, $p_{\rm
T}$. Charged hadrons are identified using two ring-imaging Cherenkov detectors
[34]. 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 [35]. The right-handed coordinate system adopted has the $z$-axis
along the beam line and the $y$-axis along the vertical. The trigger [36]
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction.

Two distinct classes of tracks are reconstructed using hits in the tracking
stations on both sides of the magnet, either with hits in the VELO (long
track) or without (downstream track). The vertex resolution of $b$-hadron
candidates reconstructed using long tracks is better than that for candidates
reconstructed using downstream tracks. However, the use of long tracks
introduces a dependence of the reconstruction efficiency on the $b$-hadron
decay time.

In the simulation, $pp$ collisions are generated using Pythia 6.4 [37] with a
specific LHCb configuration [38]. Decays of hadronic particles are described
by EvtGen [39], in which final state radiation is generated using Photos [40].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [41, *Agostinelli:2002hh] as
described in Ref. [43].

##  3 Candidate selection

The reconstruction of each of the
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays is
similar and commences by selecting ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays. Events passing the hardware trigger
contain dimuon candidates with high transverse momentum. The subsequent
software trigger is composed of two stages. The first stage performs a partial
event reconstruction and requires events to have two well-identified
oppositely charged muons with an invariant mass larger than
$2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The selection at this stage has
a uniform efficiency as a function of decay time. The second stage performs a
full event reconstruction, calculating the position of each $pp$ interaction
vertex (PV) using all available charged particles in the event. The average
number of PVs in each event is approximately $2.0$. Their longitudinal ($z$)
position is known to a precision of approximately $0.05\rm\,mm$. If multiple
PVs are reconstructed in the event, the one with the minimum value of
$\chi^{2}_{\rm IP}$ is associated with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate, where $\chi^{2}_{\rm
IP}$ is the increase in the $\chi^{2}$ of the PV fit if the candidate
trajectory is included. Events are retained for further processing if they
contain a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ pair that forms a vertex that is
significantly displaced from the PV. This introduces a non-uniform efficiency
as function of decay time.

The offline sample of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson
candidates is selected by requiring each muon to have $p_{\rm T}$ larger than
500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate to be displaced from
the PV by more than three times its decay length uncertainty. The invariant
mass of the two muons, $m(\mu^{+}\mu^{-})$, must be in the range
$[3030,3150]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The $b$-hadron candidate selection is performed by applying kinematic and
particle identification criteria to the final-state tracks, the details of
which are reported in Sec. 3.1 to 3.5. No requirements are placed on variables
that are highly correlated to the $b$-hadron decay time, thereby avoiding the
introduction of additional biases. All final-state particles are required to
have a pseudorapidity in the range $2.0<\eta<4.5$. In addition, the
$z$-position of the PV ($z_{\rm PV}$) is required to be within $100\rm\,mm$ of
the nominal interaction point, where the standard deviation of the $z_{\rm
PV}$ distribution is approximately $47\rm\,mm$. These criteria cause a
reduction of approximately $10\%$ in signal yield but define a fiducial region
where the reconstruction efficiency is largely uniform.

The maximum likelihood fit uses the invariant mass,
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$, and proper decay time,
$t$, of each $b$-hadron candidate. The decay time of the $b$-hadron candidate
in its rest frame is derived from the relation $t=m\,l/q$, where $m$ is its
invariant mass and the decay length, $l$, and the momentum, $q$, are measured
in the experimental frame. In this paper, $t$ is computed using a kinematic
decay-tree fit (DTF) [44] involving all final-state tracks from the $b$-hadron
candidate with a constraint on the position of the associated PV. Unlike in
the trigger, the position of each PV is calculated using all available charged
particles in the event after the removal of the $b$-hadron candidate final-
state tracks. This is necessary to prevent the final-state tracks from biasing
the PV position towards the $b$-hadron decay vertex and helps to reduce the
tails of the decay-time resolution function. This prescription does not bias
the measured lifetime using simulated events. The $\chi^{2}$ of the fit,
$\chi^{2}_{\rm DTF}$, is useful to discriminate between signal and background.
In cases where there are multiple $b$-hadron candidates per event, the
candidate with the smallest $\chi^{2}_{\rm DTF}$ is chosen. The $z$-position
of the displaced $b$-hadron vertices are known to a precision of approximately
$0.15\rm\,mm$.

Studies of simulated events show that in the case of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
($B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$)
decays, imposing requirements on $\chi^{2}_{\rm DTF}$ introduces a dependence
of the selection efficiency on the decay time if the $K^{+}$ and $\pi^{-}$
($K^{+}$ and $K^{-}$) tracks are included in the DTF. If no correction is
applied to the decay time distribution, the measured lifetime would be biased
by approximately $-2\rm\,fs$ relative to the generated value. Using simulated
events it is found that this effect is correlated to the opening angle between
the $K^{+}$ and $\pi^{-}$ ($K^{+}$ and $K^{-}$) from the $K^{*0}$ ($\phi$)
decay. No effect is observed for the muons coming from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay due to the larger opening
angle in this case. To remove the effect, the calculation of $\chi^{2}_{\rm
DTF}$ for the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channels is performed with an alternative DTF in which the assigned track
parameter uncertainties of the kaon and pion are increased in such a way that
their contribution to the $b$-hadron vertex position is negligible.

Candidates are required to have $t$ in the range $[0.3,14.0]{\rm\,ps}$. The
lower bound on the decay time suppresses a large fraction of the prompt
combinatorial background that is composed of tracks from the same PV, while
the upper bound is introduced to reduce the sensitivity to long-lived
background candidates. In the case of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, the lower bound is increased to
0.45${\rm\,ps}$ to compensate for the worse decay time resolution in these
modes.

In events with multiple PVs, $b$-hadron candidates are removed if they have a
$\chi^{2}_{\rm IP}$ with respect to the next best PV smaller than $50$. This
requirement is found to distort the decay time distribution, but reduces a
source of background due to the incorrect association of the $b$ hadron to its
production PV.

The invariant mass is computed using another kinematic fit without any
constraint on the PV position but with the invariant mass of the
$\mu^{+}\mu^{-}$ pair, $m(\mu^{+}\mu^{-})$, constrained to the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [45]. Figures 1 and 2 show
the $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ distributions for the
selected candidates in each final state and Table 2 gives the corresponding
signal yields.

\begin{overpic}[scale={0.75}]{figs/BuJpsiKFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ }
\put(81.0,31.0){\scriptsize$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ }
\end{overpic}\begin{overpic}[scale={0.75}]{figs/BdJpsiKstarFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ }
\put(81.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ }
\end{overpic}\begin{overpic}[scale={0.75}]{figs/Bd2JpsiKSFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ }
\put(81.0,31.0){\scriptsize$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ } \end{overpic}

Figure 1: Distributions of the (left) mass and (right) decay time of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ candidates and their associated
residual uncertainties (pulls). The data are shown by the black points; the
total fit function by the black solid line; the signal contribution by the red
dashed line and the background contribution by the blue dotted line.

\begin{overpic}[scale={0.75}]{figs/BsJpsiPhiFlat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ }
\put(81.0,31.0){\scriptsize$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ }
\end{overpic}\begin{overpic}[scale={0.75}]{figs/Lambdab0Flat_Data_1bin_IPzWeight_TS1_TS1_S1_20bins_cFit.pdf}
\put(11.0,31.0){\scriptsize$\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$} \put(81.0,31.0){\scriptsize$\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$} \end{overpic}

Figure 2: Distributions of the (left) mass and (right) decay time of
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$\mathchar 28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates and their associated residual
uncertainties (pulls). The data are shown by the black points; the total fit
function by the black solid line; the signal contribution by the red dashed
line and the background contribution by the blue dotted line.

Table 2: Estimated event yields for the five $b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ channels selected using the criteria described in Sec. 3.1 to 3.5. Channel | Yield
---|---
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $229\,434\pm 503$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | $70\,534\pm 312$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ | $17\,045\pm 175$
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ | $18\,662\pm 152$
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax$ | $3\,960\pm\ \,89$

###  3.1 Selection of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays

The $B^{+}$ candidates are reconstructed by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with a charged
particle that is identified as a kaon with $p_{\rm T}$ larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p$ larger than
$10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant mass,
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})$, must be in the range
$[5170,5400]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the lower bound
is chosen to remove feed-down from incompletely reconstructed
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays. The $\chi^{2}_{\rm DTF}$ of the fit, which has 5 degrees of freedom,
is required to be less than 25. Multiple $B^{+}$ candidates are found in less
than $0.02\%$ of selected events.

###  3.2 Selection of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays

The $K^{*0}$ candidates are reconstructed by combining two oppositely charged
particles that are identified as a kaon and a pion. The pion and $K^{*0}$ must
have $p_{\rm T}$ greater than $0.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, respectively. The invariant mass,
$m(K^{+}\pi^{-})$, must be in the range
$[826,966]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The $B^{0}$ candidates are reconstructed by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $K^{*0}$ candidates. The
invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-})$,
must be in the range $[5150,5340]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
where the upper bound is chosen to remove the contribution from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}$ decays. The $\chi^{2}_{\rm DTF}$ of the fit, which
has 3 degrees of freedom, is required to be less than 15. Multiple $B^{0}$
candidates are found in $2.2\%$ of selected events.

###  3.3 Selection of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays

The $K^{0}_{\rm\scriptscriptstyle S}$ candidates are formed from the
combination of two oppositely charged particles that are identified as pions
and reconstructed as downstream tracks. This is necessary since studies of
simulated signal decays demonstrate that an inefficiency depending on the
$b$-hadron decay time is introduced by the reconstruction of the long-lived
$K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar 28931\relax$ particles using
long tracks. Even so, it is found that the acceptance of the TT still depends
on the origin of the tracks. This effect is removed by further tightening of
the requirement on the position of the PV to $z_{\rm PV}>-50\rm\,mm$.

For particles produced close to the interaction region, this effect is
suppressed by the requirements on the fiducial region for the PV, which is
further tightened by requiring that , to account for the additional acceptance
introduced by the TT.

The downstream pions are required to have $p_{\rm T}$ greater than
$0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p$ greater than
$2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $K^{0}_{\rm\scriptscriptstyle S}$
candidate must have $p_{\rm T}$ greater than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be well separated from the $B^{0}$
decay vertex, to suppress potential background from
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
where the kaon has been misidentified as a pion. The $\chi^{2}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$ vertex fit must be less than 25 and the
invariant mass of the dipion system, $m(\pi^{+}\pi^{-})$, must be within
$15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass [45]. For subsequent stages of the
selection, $m(\pi^{+}\pi^{-})$ is constrained to the known
$K^{0}_{\rm\scriptscriptstyle S}$ mass.

The invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S})$, of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidate combination must be in the range
$[5150,5340]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the upper bound
is chosen to remove the contribution from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays. The $\chi^{2}_{\rm DTF}$ of the
fit, which has 6 degrees of freedom, is required to be less than 30. Multiple
$B^{0}$ candidates are found in less than $0.4\%$ of selected events.

###  3.4 Selection of
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decays

The $\phi$ candidates are formed from two oppositely charged particles that
have been identified as kaons and originate from a common vertex. The
$K^{+}K^{-}$ pair is required to have $p_{\rm T}$ larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant mass of the
$K^{+}K^{-}$ pair, $m(K^{+}K^{-})$, must be in the range
$[990,1050]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The $B^{0}_{s}$ candidates are reconstructed by combining the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate with the $K^{+}K^{-}$
pair, requiring the invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-})$, to be in the range
$[5200,5550]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $\chi^{2}_{\rm
DTF}$ of the fit, which has 3 degrees of freedom, is required to be less than
15. Multiple $B^{0}_{s}$ candidates are found in less than $2.0\%$ of selected
events.

###  3.5 Selection of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays

The selection of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates follows a similar approach to that
adopted for $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays. Only downstream protons and
pions are used to reconstruct the $\mathchar 28931\relax$ candidates. The
pions are required to have $p_{\rm T}$ larger than
$0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, while pions and protons must have
$p$ larger than $2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $\mathchar
28931\relax$ candidate must be well separated from the $\mathchar
28931\relax^{0}_{b}$ decay vertex and have $p_{\rm T}$ larger than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $\chi^{2}$ of the $\mathchar
28931\relax$ vertex fit must be less than 25 and $m(p\pi^{-})$ must be within
$6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\mathchar
28931\relax$ mass [45]. For subsequent stages of the selection, $m(p\pi^{-})$
is constrained to the known $\mathchar 28931\relax$ mass.

The invariant mass, $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar
28931\relax)$, of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\mathchar 28931\relax$ candidate combination must be in the range
$[5470,5770]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $\chi^{2}_{\rm
DTF}$ of the fit, which has 6 degrees of freedom, is required to be less than
30. Multiple $\mathchar 28931\relax^{0}_{b}$ candidates are found in less than
$0.5\%$ of selected events.

##  4 Dependence of efficiencies on decay time

Section 3 described the reconstruction and selection criteria of the
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decays and
various techniques that have been used to minimise the dependence of selection
efficiencies upon the decay time. After these steps, there remain two effects
that distort the $b$-hadron decay time distribution. These are caused by the
VELO-track reconstruction efficiency, $\varepsilon_{\rm VELO}$, and the
combination of the trigger efficiency, $\varepsilon_{\rm trigger}$, and
offline selection efficiency, $\varepsilon_{\rm selection|trigger}$. This
section will describe these effects and the techniques that are used to
evaluate the efficiencies from data control samples.

###  4.1 VELO-track reconstruction efficiency

The largest variation of the efficiency with the decay time is introduced by
the track reconstruction in the VELO. The track finding procedure in the VELO
assumes that tracks originate approximately from the interaction region [33,
46]. In the case of long-lived $b$-hadron candidates this assumption is not
well justified, leading to a loss of reconstruction efficiency for charged
particle tracks from the $b$-hadron decay.

\begin{overpic}[scale={0.39}]{figs/data_online.pdf} \put(79.0,59.7){(a)}
\end{overpic}\begin{overpic}[scale={0.39}]{figs/data_offline.pdf}
\put(79.0,59.7){(b)} \end{overpic}

Figure 3: VELO-track reconstruction efficiency for kaon tracks reconstructed
using the (a) online and (b) offline algorithms as a function of the kaon
$\rho$, as defined in Eq. (3). The red solid lines show the result of an
unbinned maximum likelihood fit using the parameterisation in Eq. (4) to the
background subtracted data (black points).

The distance of closest approach of the track to the $z$-axis is defined as

$\rho$ $\displaystyle\equiv$
$\displaystyle\frac{\left|(\boldsymbol{d}-\boldsymbol{v})\cdot(\boldsymbol{p}\times\boldsymbol{\hat{z}})\right|}{\left|\boldsymbol{p}\times{\boldsymbol{\hat{z}}}\right|},$
(3)

where $\boldsymbol{p}$ is the momentum of the final-state track from a
$b$-hadron candidate decaying at point $\boldsymbol{d}$,
$\boldsymbol{\hat{z}}$ is a unit vector along the $z$-axis and
$\boldsymbol{v}$ is the origin of the VELO coordinate system. During data
taking the position of the LHCb VELO is monitored as a function of time and is
centred around the LHC beam line. Using a control sample of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates where the $K^{+}$ is reconstructed as a downstream track, the VELO-
track reconstruction efficiency, $\varepsilon_{\rm VELO}(\mbox{$\rho$})$, is
computed as the fraction of these tracks that are also reconstructed as long
tracks. From samples of simulated $b$-hadron decays, it is observed that
$\varepsilon_{\rm VELO}(\mbox{$\rho$})$ can be empirically parameterised by

$\varepsilon_{\rm VELO}(\mbox{$\rho$})=a(1+c\mbox{$\rho$}^{2}),$ (4)

where the parameters $a$ and $c$ are determined from a fit to the unbinned
efficiency distribution.

Table 3: VELO reconstruction efficiency in data for kaon tracks reconstructed with the online and offline algorithms. In both cases, the correlation coefficient between $a$ and $c$ is 0.2. | $a$ |  $c$ [$\rm\,mm^{-2}$]
---|---|---
Online | $0.9759\pm 0.0005$ | $-0.0093\pm 0.0007$
Offline | $0.9831\pm 0.0004$ | $-0.0041\pm 0.0005$

Figure 3 shows the VELO-track reconstruction efficiency obtained using this
method and Table 3 shows the corresponding fit results. Since different
configurations of the VELO reconstruction algorithms are used within the LHCb
software trigger (online) and during the subsequent processing (offline), it
is necessary to evaluate two different efficiencies. The stronger dependence
of the online efficiency as a function of $\rho$ is due to the additional
requirements used in the first stage of the software trigger such that it
satisfies the required processing time.

Applying the same technique to a simulated sample of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
yields qualitatively similar behaviour for $\varepsilon_{\rm
VELO}(\mbox{$\rho$})$. Studies on simulated data show that the efficiency for
kaons and pions from the decay of $\phi$ and $K^{*0}$ mesons is smaller than
for the kaon in $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ decays, due to the small opening between the particles in the
$\phi$ and $K^{*0}$ decays, as discussed in Sec. 3. In addition, there are
kinematic differences between the calibration $B^{+}$ sample and the signal
samples. Scaling factors on the efficiency parameters are derived from
simulation to account for these effects, and have typical sizes in the range
$[1.04,1.65]$, depending on the decay mode and final-state particle being
considered.

The distortion to the $b$-hadron candidate decay time distribution caused by
the VELO-track reconstruction is corrected for by weighting each $b$-hadron
candidate by the inverse of the product of the per-track efficiencies. The
systematic effect introduced by this weighting is tested using simulated
samples of each channel. The chosen efficiency depends on whether the particle
is reconstructed with the online or offline variant of the algorithm. Studies
on simulated data show that tracks found by the online tracking algorithm are
also found by the offline tracking efficiency. For example, the efficiency
weight for each $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ candidate takes the form

$\displaystyle w_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ $\displaystyle=$ $\displaystyle
1/\left(\varepsilon^{\mu^{+}}_{\rm VELO,online}\ \varepsilon^{\mu^{-}}_{\rm
VELO,online}\ \varepsilon^{K^{+}}_{\rm VELO,offline}\
\varepsilon^{\pi^{-}}_{\rm VELO,offline}\right),$ (5)

since the two muons are required to be reconstructed online, while the kaons
and the pions are reconstructed offline.

In the case of the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channels, since no VELO information is used when
reconstructing the $K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax$ particles, the candidate weighting functions take the form
$w=1/\left(\varepsilon^{\mu^{+}}_{\rm VELO,online}\ \varepsilon^{\mu^{-}}_{\rm
VELO,online}\right)$.

###  4.2 Trigger and selection efficiency

The efficiency of the second stage of the software trigger depends on the
$b$-hadron decay time as it requires that the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson is significantly
displaced from the PV. A parameterisation of this efficiency,
$\varepsilon_{\rm trigger}(t)$, is obtained for each
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ decay mode by
exploiting a corresponding sample of
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ candidates that
are selected without any displacement requirement. For each channel, the
control sample corresponds to approximately $40\%$ of the total number of
signal candidates. A maximum likelihood fit to the unbinned invariant mass
distribution $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ is performed
to determine the fraction of signal decays that survive the decay-time biasing
trigger requirements as a function of decay time.

The same technique is used to determine the decay time efficiency of the
triggered candidates caused by the offline selection, $\varepsilon_{\rm
selection|trigger}(t)$, which is introduced by the requirement on the
detachment of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in the
sample used to reconstruct the $b$-hadron decays. The combined selection
efficiency, $\varepsilon_{\rm selection}(t)$, is given by the product of
$\varepsilon_{\rm trigger}(t)$ and $\varepsilon_{\rm selection|trigger}(t)$.

Figure 4 shows $\varepsilon_{\rm selection}(t)$ obtained for the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ channel
as a function of decay time. The efficiencies obtained for the other
$H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ channels
are qualitatively similar. Studies using simulated events show that the
efficiency drop below $0.5{\rm\,ps}$ is caused by the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ displacement requirement. The
dip near $1.5{\rm\,ps}$ appears because the PV reconstruction in the software
trigger is such that some final-state tracks of short-lived $b$-hadron decays
may be used to reconstruct an additional fake PV close to the true $b$-hadron
decay vertex. As a result the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson does not satisfy the
displacement requirement, leading to a decrease in efficiency.

The efficiency parameterisation for each channel is used in the fit to measure
the corresponding $b$-hadron lifetime. An exception is made for the $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channel where, owing to its smaller event yield,
$\varepsilon_{\rm selection}(t)$ measured with
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays is used instead. The validity of
this approach is checked using large samples of simulated events.

Figure 4: Combined trigger and selection efficiency, $\varepsilon_{\rm
selection}(t)$, for $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates.

##  5 Maximum likelihood fit

For each channel, the lifetime is determined from a two-dimensional maximum
likelihood fit to the unbinned $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)$ and $t$ distributions. The full probability density function (PDF)
is constructed as ${\cal P}=f_{s}({\cal S}_{m}\times{\cal
S}_{t})+(1-f_{s})({\cal B}_{m}\times{\cal B}_{t})$, where $f_{s}$ is the
signal fraction, determined in the fit, and ${\cal S}_{m}\times{\cal S}_{t}$
and ${\cal B}_{m}\times{\cal B}_{t}$ are the
($m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$, $t$) PDFs for the signal
and background components, respectively. A systematic uncertainty is assigned
to the assumption that the PDFs factorise.

The signal mass PDF, ${\cal S}_{m}$, is modelled by the sum of two Gaussian
functions. The free parameters in the fit are the common mean, the width of
the narrower Gaussian function, the ratio of the second to the first Gaussian
width and the fraction of the first Gaussian function. The background mass
distribution, ${\cal B}_{m}$, is modelled by an exponential function with a
single free parameter.

The signal $b$-hadron decay time distribution is described by an exponential
function with decay constant given by the $b$-hadron lifetime,
$\tau_{H_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X}$. The
signal decay time PDF, ${\cal S}_{t}$, is obtained by multiplying the
exponential function by the combined $t$-dependent trigger and selection
efficiency described in Sec. 4.2. From inspection of events in the sidebands
of the $b$-hadron signal peak, the background decay time PDF, ${\cal B}_{t}$,
is well modelled by a sum of three exponential functions with different decay
constants that are free in the fit. These components originate from a
combination of prompt candidates, where all tracks originate from the same PV,
and long-lived candidates where tracks from the associated PV are combined
with other tracks of long-lived particles. For each channel the exponential
functions are convolved with a Gaussian resolution function with width
$\sigma$ and mean $\Delta$, an offset of the order of a few femtoseconds that
is fixed in the fit. Using a sample of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ background events, the decay
time resolution for $H_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X$ channels reconstructed using long tracks has been measured to be
approximately $45\rm\,fs$ [47]. For
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, which use downstream tracks to
reconstruct the $K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar 28931\relax$
particles, a similar study of an event sample composed of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons combined with two
downstream tracks, reconstructed as either a $K^{0}_{\rm\scriptscriptstyle S}$
or $\mathchar 28931\relax$, has determined the resolution to be $65\rm\,fs$.
The systematic uncertainties related to the choice of resolution model are
discussed in Sec. 6.

The negative log-likelihood, constructed as

$-\ln{\cal L}=-\alpha\sum_{\mathrm{events}\;i}{w_{i}\ln{{\cal P}}},$ (6)

is minimised in the fit, where the weights $w_{i}$ correspond to the per-
candidate correction for the VELO reconstruction efficiency described in Sec.
4.1. The factor $\alpha=\sum_{i}w_{i}/\sum_{i}w_{i}^{2}$ is used to include
the effect of the weights in the determination of the uncertainties [48].
Figures 1 and 2 show the result of fitting this model to the selected
candidates for each channel, projected onto the corresponding
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ and $t$ distributions.

As a consistency check, an alternative fit procedure is developed where each
event is given a signal weight, $W_{i}$, determined using the sPlot [49]
method with $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ as the
discriminating variable and using the mass model described above. A weighted
fit to the decay time distribution using the signal PDF is then used to
measure the $b$-hadron lifetime. In this case, the negative log-likelihood is
given by Eq. (6) where $w_{i}$ is replaced with $W_{i}w_{i}$ and
$\alpha=\sum_{i}(W_{i}w_{i})/\sum_{i}(W_{i}w_{i})^{2}$. The difference between
the results of the two fitting procedures is used to estimate the systematic
uncertainty on the background description.

##  6 Systematic uncertainties

The systematic effects affecting the measurements reported here are discussed
in the following and summarised in Tables 4 and 5.

The systematic uncertainty related to the VELO-track reconstruction efficiency
can be split into two components. The first uncertainty is due to the finite
size of the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ sample, reconstructed using downstream kaon tracks, which is used
to determine the per-candidate efficiency weights and leads to a statistical
uncertainty on the $\varepsilon_{\rm VELO}(\mbox{$\rho$})$ parameterisation.
The lifetime fits are repeated after varying the parameters by $\pm 1\sigma$
and the largest difference between the lifetimes is assigned as the
uncertainty. The second uncertainty is due to the scaling factors, which are
used to correct the efficiency for phase-space effects, obtained from
simulated events. The fit is repeated using the unscaled efficiency and half
of the variation in fit results is assigned as a systematic uncertainty. These
contributions, of roughly the same size, are added in quadrature in Table 4.

A number of additional consistency checks are performed to investigate
possible mismodelling of the VELO-track reconstruction efficiency. First,
$\varepsilon_{\rm VELO}(\mbox{$\rho$})$ is evaluated in two track momentum and
two track multiplicity bins and the event weights recalculated. Using both
data and simulated events, no significant change in the lifetimes is observed
after repeating the fit with the updated weights and, therefore, no systematic
uncertainty is assigned. Secondly, to assess the sensitivity to the choice of
parameterisation for $\varepsilon_{\rm VELO}(\mbox{$\rho$})$ (Eq. 4), the
results are compared to those with linear model for the efficiency. The effect
is found to be negligible and no systematic uncertainty is applied. Thirdly,
the dependence of the VELO-track reconstruction efficiency on the azimuthal
angle, $\phi$, of each track is studied by independently evaluating the
efficiency in four $\phi$ quadrants for both data and simulation. No
dependence is observed. Finally, the efficiency is determined separately for
both positive and negative kaons and found to be compatible.

The techniques described in Sec. 4 to correct the efficiency as a function of
the decay time are validated on simulated data. The lifetime is fit in each
simulated signal mode and the departure from the generated lifetime,
$\Delta\tau$, is found to be compatible with zero within the statistical
precision of each simulated sample. The measured lifetimes in the data sample
are corrected by each $\Delta\tau$ and a corresponding systematic uncertainty
is assigned, given by the size of the statistical uncertainty on the fitted
lifetime for each simulated signal mode.

The assumption that $m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ is
independent of the decay time is central to the validity of the likelihood
fits used in this study. It is tested by re-evaluating the signal weights of
the alternative fit in bins of decay time and then refitting the entire sample
using the modified weights. The systematic uncertainty for each decay mode is
evaluated as the sum in quadrature of the lifetime variations, each weighted
by the fraction of signal events in the corresponding bin.

For each signal decay mode, the effect of the limited size of the control
sample used to estimate the combined trigger and selection efficiency is
evaluated by repeating the fits with $\varepsilon_{\rm selection}(t)$ randomly
fluctuated within its statistical uncertainty. The standard deviation of the
distribution of lifetimes obtained is assigned as the systematic uncertainty.

The alternative likelihood fit does not assume any model for the decay time
distribution associated with the combinatorial background. Therefore, the
systematic uncertainty associated to the modelling of this background is
evaluated by taking the difference in lifetimes measured by the nominal and
alternative fit methods.

The fit uses a double Gaussian function to describe the
$m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ distribution of signal
candidates. This assumption is tested by repeating the fit using a double-
sided Apollonios function [50] where the mean and width parameters are varied
in the fit and the remaining parameters are fixed to those from simulation.
The differences in lifetime with respect to the default results are taken as
systematic uncertainties.

As described in Sec. 5 the dominant background in each channel is
combinatorial in nature. It is also possible for backgrounds to arise due to
misreconstruction of $b$-hadron decays where the particle identification has
failed. The presence of such backgrounds is checked by inspecting events in
the sidebands of the signal and re-assigning the mass hypotheses of at least
one of the final-state hadrons. The only contributions that have an impact are
$\mathchar 28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ decays in the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ channel
where a proton is misidentified as a kaon and the cross-feed component between
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays where pion and protons are misidentified.
In the case of $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays, the fit is repeated including a contribution of $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}pK^{-}$ decays. The two-dimensional PDF is determined from simulation,
while the yield is found to be $6\%$ from the sidebands of the $B^{0}_{s}$
sample. This leads to the effective lifetime changing by $0.4\rm\,fs$, which
is assigned as a systematic uncertainty. A similar procedure is used to
determine the invariant mass shape of the cross-feed background between
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, while the decay time is modelled with the
exponential distribution of the corresponding signal mode. A simultaneous fit
to both data samples is performed in order to constrain the yield of the
cross-feed and the resulting change in lifetime of $-0.3\rm\,fs$ and
$+1.1\rm\,fs$ for $B^{0}$ and $\mathchar 28931\relax^{0}_{b}$, respectively,
is assigned as a systematic uncertainty.

Another potential source of background is the incorrect association of signal
$b$ hadrons to their PV, which results in an erroneous reconstruction of the
decay time. Since the fitting procedure ignores this contribution, a
systematic uncertainty is evaluated by repeating the fit after including in
the background model a component describing the incorrectly associated
candidates. The background distribution is determined by calculating the decay
time for each $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ decay with respect to a randomly chosen PV from the previous
selected event. In studies of simulated events the fraction of this background
is less than $0.1\%$. Repeating the fit with a $1\%$ contribution results in
the lifetime changing by $0.1\rm\,fs$ and, therefore, no systematic
uncertainty is assigned.

The measurement of the effective lifetime in the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channel is integrated over the angular distributions of the final-state
particles and is, in the case of uniform angular efficiency, insensitive to
the different polarisations of the final state [47]. To check if the angular
acceptance introduced by the detector geometry and event selection can affect
the measured lifetime, the events are weighted by the inverse of the angular
efficiency determined in Ref. [47]. Repeating the fit with the weighted
dataset leads to a shift of the lifetime of $-1.0\rm\,fs$, the same as is
observed in simulation. The final result is corrected by this shift, which is
also assigned as a systematic uncertainty. The $B^{0}_{s}$ effective lifetime
could also be biased due to a small $C\\!P$-odd S-wave component from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
decays that is ignored in the fit. For the $m(K^{+}K^{-})$ mass range used
here (Sec. 3), Ref. [51] indicates that the S-wave contribution is $1.1\%$.
The effect of ignoring such a component is evaluated by repeating the fit on
simulated experiments with an additional $1\%$ $C\\!P$-odd component. A change
in the lifetime of $-1.2\rm\,fs$ is observed, which is used to correct the
final lifetime and is also taken as a systematic uncertainty. Finally, as
described in Sec. 3, only events with a decay time larger than $0.3{\rm\,ps}$
are considered in the nominal fit. This offset leads to a different relative
contribution of the heavy and light mass eigenstates such that the lifetime
extracted from the exponential fit does not correspond to the effective
lifetime defined in Eq. (2). A correction of $-0.3\rm\,fs$ is applied to
account for this effect.

The presence of a production asymmetry between $B^{0}$ and $\overline{B}^{0}$
mesons could bias the measured
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ effective lifetime, and therefore
$\Delta\Gamma_{d}/\Gamma_{d}$, by adding additional terms in Eq. (2). The
production asymmetry is measured to be $A_{\rm P}(B^{0})=(0.6\pm 0.9)\%$ [52],
the uncertainty of which is used to estimate a corresponding systematic
uncertainty on the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ lifetime of $1.1\rm\,fs$. No
uncertainty is assigned to the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
lifetime since this decay mode is flavour-specific222Flavour-specific means
that the final state is only accessible via the decay of a $B^{0}_{(s)}$ meson
and accessible by a meson originally produced as a $\overline{B}^{0}_{(s)}$
only via oscillation. and the production asymmetry cancels in the untagged
decay rate. For the $B^{0}_{s}$ system, the rapid oscillations, due to the
large value of $\Delta m_{s}=17.768\pm 0.024{\rm\,ps^{-1}}$ [53], reduce the
effect of a production asymmetry, reported as $A_{\rm P}(B^{0}_{s})=(7\pm
5)\%$ in Ref. [52], to a negligible level. Hence, no corresponding systematic
uncertainty is assigned.

There is a $0.02\%$ relative uncertainty on the lifetime measurements due to
the uncertainty on the length scale of LHCb [53], which is mainly determined
by the VELO modules $z$ positions. These are evaluated by a survey, having an
accuracy of $0.1\rm\,mm$ over the full length of the VELO ($1000\rm\,mm$), and
refined through a track-based alignment. The alignment procedure is more
precise for the first track hits, that are less affected by multiple
scattering and whose distribution of $z$ positions have an RMS of
$100\rm\,mm$. In this region, the differences between the module positions
obtained from the survey and track-based alignment are within $0.02\rm\,mm$,
which is taken as systematic uncertainty. The systematic uncertainty related
to the momentum scale calibration affects both the $b$ hadron candidate mass
and momentum and, therefore, cancels when computing the decay time.

The systematic uncertainty related to the choice of $45\rm\,fs$ for the width
of the decay-time resolution function ($65\rm\,fs$ in the case of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$) is evaluated by changing the width by $\pm
15\rm\,fs$ and repeating the fit. This change in width is larger than the
estimated uncertainty on the resolution and leads to a negligible change in
the fit results. Consequently, no systematic uncertainty is assigned.
Furthermore, to test the sensitivity of the lifetimes to potential
mismodelling of the long tails in the resolution, the resolution model is
changed from a single Gaussian function to a sum of two or three Gaussian
functions with parameters fixed from simulation. Repeating the fit with the
new resolution model causes no significant change to the lifetimes and no
systematic uncertainty is assigned. The lifetimes are insensitive to the
offset, $\Delta$, in the resolution model.

Several consistency checks are performed to study the stability of the
lifetimes, by comparing the results obtained using different subsets of the
data in terms of magnet polarity, data taking period, $b$-hadron and track
kinematic variables, number of PVs in the event and track multiplicity. In all
cases, no trend is observed and all lifetimes are compatible with the nominal
results.

Table 4: Statistical and systematic uncertainties (in femtoseconds) for the values of the $b$-hadron lifetimes. The total systematic uncertainty is obtained by combining the individual contributions in quadrature. Source | $\tau_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ | $\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$ | $\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax}$ |  $\tau_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$
---|---|---|---|---|---
Statistical uncertainty | 3.5 | 6.1 | 12.8 | 26.5 | 11.4
VELO reconstruction | 2.0 | 2.3 | 0.9 | 0.5 | 2.3
Simulation sample size | 1.7 | 2.3 | 2.9 | 3.7 | 2.4
Mass-time correlation | 1.4 | 1.8 | 2.1 | 3.0 | 0.7
Trigger and selection eff. | 1.1 | 1.2 | 2.0 | 2.0 | 2.5
Background modelling | 0.1 | 0.2 | 2.2 | 2.1 | 0.4
Mass modelling | 0.1 | 0.2 | 0.4 | 0.2 | 0.5
Peaking background | – | – | 0.3 | 1.1 | 0.4
Effective lifetime bias | – | – | – | – | 1.6
$B^{0}$ production asym. | – | – | 1.1 | – | –
LHCb length scale | 0.4 | 0.3 | 0.3 | 0.3 | 0.3
Total systematic | 3.2 | 3.9 | 4.9 | 5.7 | 4.6
Table 5: Statistical and systematic uncertainties (in units of $10^{-3}$) for the lifetime ratios and $\Delta\Gamma_{d}/\Gamma_{d}$. For brevity, $\tau_{B^{0}}$ $(\tau_{\overline{B}^{0}})$ corresponds to the measurement of $\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$ $(\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\overline{K}^{*0}})$. The total systematic uncertainty is obtained by combining the individual contributions in quadrature. Source | $\tau_{B^{+}}/\tau_{B^{0}}$ | $\tau_{B^{0}_{s}}/\tau_{B^{0}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{B^{0}}$ | $\tau_{B^{+}}/\tau_{B^{-}}$ | $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern 0.63004pt\overline{\kern-0.63004pt\mathchar 28931\relax}^{0}_{b}}$ | $\tau_{B^{0}}/\tau_{\overline{B}^{0}}$ |  $\Delta\Gamma_{d}/\Gamma_{d}$
---|---|---|---|---|---|---|---
Statistical uncertainty | 5.0 | 8.5 | 18.0 | 4.0 | 35.0 | 8.0 | 25.0
VELO reconstruction | 1.6 | 1.7 | 1.1 | – | – | – | 4.1
Simulation sample size | 2.0 | 2.2 | 2.8 | 2.1 | 5.3 | 3.0 | 6.3
Mass-time correlation | 1.6 | 1.2 | 2.3 | – | – | – | 4.7
Trigger and selection eff. | 1.1 | 1.8 | 1.5 | – | – | – | 4.0
Background modelling | 0.3 | 0.1 | 1.5 | 0.2 | 3.0 | 1.4 | 3.8
Mass modelling | 0.2 | 0.4 | 0.2 | 0.1 | 0.2 | 0.2 | 0.8
Peaking background | – | 0.3 | 0.7 | – | – | – | 0.5
Effective lifetime bias | – | 1.0 | – | – | – | – | –
$B^{0}$ production asym. | – | – | – | – | – | 8.5 | 1.9
Total systematic | 3.2 | 3.7 | 4.4 | 2.1 | 6.1 | 9.1 | 10.7

The majority of the systematic uncertainties described above can be propagated
to the lifetime ratio measurements in Table 7. However, some of the
uncertainties are correlated between the individual lifetimes and cancel in
the ratio. For the first set of ratios and for $\Delta\Gamma_{d}/\Gamma_{d}$,
the systematic uncertainty from the VELO-reconstruction efficiency weights and
the LHCb length scale are considered as fully correlated. For the second set
of ratios, other systematic uncertainties, as indicated in Table 5, cancel,
since the ratio is formed from lifetimes measured using the same decay mode.
In contrast to the situation for the measurement of the $B^{0}$ lifetime in
the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
mode, the $B^{0}$ production asymmetry does lead to a systematic uncertainty
on the measurement of
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}/\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}}$ since terms like $A_{\rm P}\cos(\Delta m_{d}t)$ do
not cancel in the decay rates describing the decays of $B^{0}$ and
$\overline{B}^{0}$ mesons to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}$ final states. The effect of candidates where the
flavour, via the particle identification of the decay products, has not been
correctly assigned is investigated and found to be negligible.

##  7 Results and conclusions

The measured $b$-hadron lifetimes are reported in Table 6. All results are
compatible with existing world averages [13]. The reported $\tau_{\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax}$ is smaller by approximately $2\sigma$ than a
previous measurements from LHCb [8]. With the exception of the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channel, these are the single most precise
measurements of the $b$-hadron lifetimes. The $B^{0}_{s}$ meson effective
lifetime is measured using the same data set as used in Ref. [47] for the
measurement of the $B^{0}_{s}$ meson mixing parameters and polarisation
amplitudes in $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays. The $B^{0}_{s}$ meson effective lifetime computed from
these quantities is compatible with the lifetime reported in this paper and a
combination of the two results is, therefore, inappropriate.

Table 6: Fit results for the $B^{+}$, $B^{0}$, $B^{0}_{s}$ mesons and
$\mathchar 28931\relax^{0}_{b}$ baryon lifetimes. The first uncertainty is
statistical and the second is systematic.

Lifetime Value [${\rm\,ps}$ ]
$\tau_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$
1.637 $\pm$ 0.004 $\pm$ 0.003
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}}$
1.524 $\pm$ 0.006 $\pm$ 0.004
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}$ 1.499 $\pm$ 0.013 $\pm$ 0.005
$\tau_{\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax}$ 1.415 $\pm$ 0.027 $\pm$ 0.006
$\tau_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}$ 1.480 $\pm$ 0.011 $\pm$ 0.005

Table 7 reports the ratios of the $B^{+}$, $B^{0}_{s}$ and $\mathchar
28931\relax^{0}_{b}$ lifetimes to the $B^{0}$ lifetime measured in the
flavour-specific $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ channel. This decay mode provides a better normalisation than
the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ channel due to the lower statistical
uncertainty on the $B^{0}$ meson lifetime and due to the fact that the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
lifetime only depends quadratically on $\Delta\Gamma_{d}/\Gamma_{d}$, as shown
in Eq. (7). The statistical and systematic uncertainties from the absolute
lifetime measurements are propagated to the ratios, taking into account the
correlations between the systematic uncertainties. All ratios are consistent
with SM predictions [24, 25, 22, 30, 23, 31, 32, 15] and with previous
measurements [13]. Furthermore, the ratios $\tau_{B^{+}}/\tau_{B^{-}}$,
$\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern
0.70004pt\overline{\kern-0.70004pt\mathchar 28931\relax}^{0}_{b}}$ and
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}/\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}}$ are reported. Measuring any of these different from
unity would indicate a violation of $C\\!PT$ invariance or, for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays, could also indicate that $\Delta\Gamma_{d}$ is non-zero and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ is not
$100\%$ flavour-specific. No deviation from unity of these ratios is observed.

Table 7: Lifetime ratios for the $B^{+}$, $B^{0}$, $B^{0}_{s}$ mesons and
$\mathchar 28931\relax^{0}_{b}$ baryon. The first uncertainty is statistical
and the second is systematic.

Ratio Value
$\tau_{B^{+}}/\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ 1.074 $\pm$ 0.005 $\pm$ 0.003
$\tau_{B^{0}_{s}}/\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ 0.971 $\pm$ 0.009 $\pm$ 0.004 $\tau_{\mathchar
28931\relax^{0}_{b}}/\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}$ 0.929 $\pm$ 0.018 $\pm$ 0.004 $\tau_{B^{+}}/\tau_{B^{-}}$ 1.002
$\pm$ 0.004 $\pm$ 0.002 $\tau_{\mathchar 28931\relax^{0}_{b}}/\tau_{\kern
0.70004pt\overline{\kern-0.70004pt\mathchar 28931\relax}^{0}_{b}}$ 0.940 $\pm$
0.035 $\pm$ 0.006
$\tau_{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}/\tau_{\overline{B}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\overline{K}^{*0}}$ 1.000 $\pm$ 0.008 $\pm$ 0.009

The effective lifetimes of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays are used to measure
$\Delta\Gamma_{d}/\Gamma_{d}$. Flavour-specific final states such as
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ have
$\mathcal{A}_{\Delta\Gamma_{d}}^{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}}=0$, while
$\mathcal{A}_{\Delta\Gamma_{d}}^{B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}}=\cos(2\beta)$ to a good approximation
in the SM, where
$\beta\equiv\arg\left[-(V_{cd}V^{*}_{cb})/(V_{td}V^{*}_{tb})\right]$ is one of
the CKM unitarity triangle angles. Hence, the two effective lifetimes can be
expressed as

$\displaystyle\tau_{B^{0}\rightarrow J/\psi K^{*0}}$
$\displaystyle=\frac{1}{\Gamma_{d}}\frac{1}{1-y_{d}^{2}}\left(1+y_{d}^{2}\right),$
(7) $\displaystyle\tau_{B^{0}\rightarrow J/\psi K^{0}_{S}}$
$\displaystyle=\frac{1}{\Gamma_{d}}\frac{1}{1-y_{d}^{2}}\left(\frac{1+2\cos(2\beta)y_{d}+y_{d}^{2}}{1+\cos(2\beta)y_{d}}\right).$
(8)

Using the effective lifetimes reported in Table 6 and
$\beta=(21.5^{+0.8}_{-0.7})^{\circ}$ [13], a fit of $\Delta\Gamma_{d}$ and
$\Gamma_{d}$ to the expressions in Eq. (7) and Eq. (8) leads to

$\displaystyle\Gamma_{d}$ $\displaystyle=\phantom{+}0.656\pm 0.003\pm
0.002{\rm\,ps^{-1}},$ (9) $\displaystyle\Delta\Gamma_{d}$
$\displaystyle=-0.029\pm 0.016\pm 0.007{\rm\,ps^{-1}},$ (10)

where the first uncertainty is statistical and the second is systematic. The
correlation coefficient between $\Delta\Gamma_{d}$ and $\Gamma_{d}$ is $0.43$
when including statistical and systematic uncertainties. The combination gives

$\frac{\Delta\Gamma_{d}}{\Gamma_{d}}=-0.044\pm 0.025\pm 0.011,$ (11)

consistent with the SM expectation [14, 15] and the current world-average
value [13].

## 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); 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 indebted to the communities behind the multiple open source
software packages we depend on. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).

## References

  * [1] V. A. Khoze and M. A. Shifman, Heavy quarks, Sov. Phys. Usp. 26 (1983) 387
  * [2] M. A. Shifman and M. Voloshin, Preasymptotic effects in inclusive weak decays of charmed particles, Sov. J. Nucl. Phys. 41 (1985) 120
  * [3] M. A. Shifman and M. Voloshin, Hierarchy of lifetimes of charmed and beautiful hadrons, Sov. Phys. JETP 64 (1986) 698
  * [4] I. I. Bigi, N. Uraltsev, and A. Vainshtein, Nonperturbative corrections to inclusive beauty and charm decays: QCD versus phenomenological models, Phys. Lett. B293 (1992) 430, arXiv:hep-ph/9207214
  * [5] I. I. Bigi, The QCD perspective on lifetimes of heavy-flavor hadrons, arXiv:hep-ph/9508408
  * [6] N. Uraltsev, Heavy quark expansion in beauty and its decays, arXiv:hep-ph/9804275
  * [7] M. Neubert, B decays and the heavy-quark expansion, Adv. Ser. Direct. High Energy Phys. 15 (1998) 239, arXiv:hep-ph/9702375
  * [8] LHCb collaboration, R. Aaij et al., Precision measurement of the $\mathchar 28931\relax^{0}_{b}$ baryon lifetime, Phys. Rev. Lett. 111 (2013) 102003, arXiv:1307.2476
  * [9] K. Hartkorn and H. Moser, A new method of measuring $\Delta\Gamma/\Gamma$ in the $B_{s}^{0}$ anti-$B_{s}^{0}$ system, Eur. Phys. J. C8 (1999) 381
  * [10] I. Dunietz, R. Fleischer, and U. Nierste, In pursuit of new physics with $B_{s}$ decays, Phys. Rev. D63 (2001) 114015, arXiv:hep-ph/0012219
  * [11] R. Fleischer and R. Knegjens, Effective lifetimes of $B_{s}$ decays and their constraints on the $B_{s}^{0}$-$\bar{B}_{s}^{0}$ mixing parameters, Eur. Phys. J. C71 (2011) 1789, arXiv:1109.5115
  * [12] U. Nierste, CP asymmetry in flavor-specific B decays, arXiv:hep-ph/0406300
  * [13] Heavy Flavor Averaging Group, Y. Amhis et al., Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of early 2012, arXiv:1207.1158, updated results and plots available at: http://www.slac.stanford.edu/xorg/hfag/
  * [14] A. Lenz and U. Nierste, Theoretical update of $B_{s}-\bar{B}_{s}$ mixing, JHEP 06 (2007) 072, arXiv:hep-ph/0612167
  * [15] A. Lenz and U. Nierste, Numerical updates of lifetimes and mixing parameters of B mesons, arXiv:1102.4274
  * [16] BaBar collaboration, B. Aubert et al., Limits on the decay-rate difference of neutral $B$ mesons and on CP, T, and CPT violation in $B^{0}\overline{B}^{0}$ oscillations, Phys. Rev. Lett. 92 (2004) 181801, arXiv:hep-ex/0311037
  * [17] BaBar collaboration, B. Aubert et al., Limits on the decay-rate difference of neutral $B$ mesons and on CP, T, and CPT violation in $B^{0}\overline{B}^{0}$ oscillations, Phys. Rev. D70 (2004) 012007, arXiv:hep-ex/0403002
  * [18] Belle collaboration, T. Higuchi et al., Search for time-dependent CPT violation in hadronic and semileptonic B decays, Phys. Rev. D85 (2012) 071105, arXiv:1203.0930
  * [19] G. Borissov and B. Hoeneisen, Understanding the like-sign dimuon charge asymmetry in $p\bar{p}$ collisions, Phys. Rev. D87 (2013) 074020, arXiv:1303.0175
  * [20] D0 collaboration, V. M. Abazov et al., Study of CP-violating charge asymmetries of single muons and like-sign dimuons in $p\overline{p}$ collisions, Phys. Rev. D89 (2014) 012002, arXiv:1310.0447
  * [21] T. Gershon, $\Delta\Gamma_{d}$: a forgotten null test of the standard model, J. Phys. G38 (2011) 015007, arXiv:1007.5135
  * [22] M. Beneke et al., The $B^{+}-B^{0}_{d}$ lifetime difference beyond leading logarithms, Nucl. Phys. B639 (2002) 389, arXiv:hep-ph/0202106
  * [23] E. Franco, V. Lubicz, F. Mescia, and C. Tarantino, Lifetime ratios of beauty hadrons at the next-to-leading order in QCD, Nucl. Phys. B633 (2002) 212, arXiv:hep-ph/0203089
  * [24] M. Beneke, G. Buchalla, and I. Dunietz, Width Difference in the $B_{s}-\bar{B_{s}}$ System, Phys. Rev. D54 (1996) 4419, arXiv:hep-ph/9605259
  * [25] Y.-Y. Keum and U. Nierste, Probing penguin coefficients with the lifetime ratio $\tau(B_{s})/\tau(B_{d})$, Phys. Rev. D57 (1998) 4282, arXiv:hep-ph/9710512
  * [26] N. Uraltsev, On the problem of boosting nonleptonic b baryon decays, Phys. Lett. B376 (1996) 303, arXiv:hep-ph/9602324
  * [27] I. I. Bigi, M. A. Shifman, and N. Uraltsev, Aspects of heavy quark theory, Ann. Rev. Nucl. Part. Sci. 47 (1997) 591, arXiv:hep-ph/9703290
  * [28] D. Pirjol and N. Uraltsev, Four fermion heavy quark operators and light current amplitudes in heavy flavor hadrons, Phys. Rev. D59 (1999) 034012, arXiv:hep-ph/9805488
  * [29] M. Voloshin, Reducing model dependence of spectator effects in inclusive decays of heavy baryons, Phys. Rev. D61 (2000) 074026, arXiv:hep-ph/9908455
  * [30] C. Tarantino, Beauty hadron lifetimes and B meson CP violation parameters from lattice QCD, Eur. Phys. J. C33 (2004) S895, arXiv:hep-ph/0310241
  * [31] F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, $\mathchar 28931\relax^{0}_{b}$ lifetime puzzle in heavy quark expansion, Phys. Rev. D68 (2003) 114006, arXiv:hep-ph/0303235
  * [32] F. Gabbiani, A. I. Onishchenko, and A. A. Petrov, Spectator effects and lifetimes of heavy hadrons, Phys. Rev. D70 (2004) 094031, arXiv:hep-ph/0407004
  * [33] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
  * [34] M. Adinolfi et al., Performance of the LHCb RICH detector at the LHC, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [35] A. A. Alves Jr et al., Performance of the LHCb muon system, JINST 8 (2013) P02022, arXiv:1211.1346
  * [36] R. Aaij et al., The LHCb trigger and its performance in 2011, JINST 8 (2013) P04022, arXiv:1211.3055
  * [37] T. Sjöstrand, S. Mrenna, and P. Skands, Pythia 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [38] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
  * [39] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
  * [40] P. Golonka and Z. Was, Photos Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [41] Geant4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [42] Geant4 collaboration, S. Agostinelli et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
  * [43] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. Phys. : Conf. Ser. 331 (2011) 032023
  * [44] W. D. Hulsbergen, Decay chain fitting with a Kalman filter, Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [45] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition
  * [46] O. Callot, FastVelo, a fast and efficient pattern recognition package for the Velo, LHCb-PUB-2011-001
  * [47] LHCb collaboration, R. Aaij et al., Measurement of $C\\!P$-violation and the $B^{0}_{s}$-meson decay width difference with $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ and $B_{s}^{0}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays, Phys. Rev. D87 (2013) 112010, arXiv:1304.2600
  * [48] Y. Xie, sFit: a method for background subtraction in maximum likelihood fit, arXiv:0905.0724
  * [49] M. Pivk and F. R. Le Diberder, sPlot: a statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [50] D. M. Santos and F. Dupertuis, Mass distributions marginalized over per-event errors, submitted to Nucl. Instrum. Meth. A (2013) arXiv:1312.5000
  * [51] LHCb collaboration, R. Aaij et al., Amplitude analysis and branching fraction measurement of $\overline{B}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$, Phys. Rev. D87 (2013) 072004, arXiv:1302.1213
  * [52] LHCb collaboration, R. Aaij et al., First measurement of time-dependent $CP$ violation in $B_{s}^{0}\rightarrow K^{+}K^{-}$ decays, JHEP 10 (2013) 183, arXiv:1308.1428
  * [53] LHCb collaboration, R. Aaij et al., Precision measurement of the $B^{0}_{s}-\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation frequency $\Delta m_{s}$ in the decay $B^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$, New J. Phys. 15 (2013) 053021, arXiv:1304.4741