EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-081 LHCb-PAPER-2014-010 May 7, 2014 / revised June 18,
2014

Measurement of the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ baryon lifetimes

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

Using a data sample of $pp$ collisions corresponding to an integrated
luminosity of 3 fb-1, the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ baryons are reconstructed in the $\mathchar
28932\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28938\relax^{-}$ decay modes and their lifetimes measured to
be

$\displaystyle\tau(\mathchar 28932\relax^{-}_{b})$ $\displaystyle=$
$\displaystyle 1.55\,^{+0.10}_{-0.09}~{}{\rm(stat)}\pm 0.03\,{\rm(syst)\,ps},$
$\displaystyle\tau(\mathchar 28938\relax^{-}_{b})$ $\displaystyle=$
$\displaystyle 1.54\,^{+0.26}_{-0.21}~{}{\rm(stat)}\pm 0.05\,{\rm(syst)\,ps}.$

These are the most precise determinations to date. Both measurements are in
good agreement with previous experimental results and with theoretical
predictions.

Published in Phys. Lett. B 736, 154-162 (2014)

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

LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson38, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38,
W. Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k,
M. Calvo Gomez36,o, A. Camboni36, P. Campana18,38, D. Campora Perez38, A.
Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, H. Carranza-
Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20, L.
Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A. Comerma-
Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38,
M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori42, A. Di Canto38, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, F.
Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A. Dzyuba30, S.
Easo49,38, U. Egede53, V. Egorychev31, S. Eidelman34, S. Eisenhardt50, U.
Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5, Ch. Elsasser40, S.
Esen11, T. Evans55, A. Falabella16,f, C. Färber11, C. Farinelli41, N.
Farley45, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira
Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M.
Firlej27, C. Fitzpatrick38, T. Fiutowski27, M. Fontana10, F. Fontanelli19,j,
R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J.
Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S. Gallorini22,
S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, E. Gersabeck11, M.
Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Gianì39, V. Gibson47,
L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, H. Gordon38, C. Gotti20, M. Grabalosa Gándara5, R. Graciani
Diaz36, L.A. Granado Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E.
Greening55, S. Gregson47, P. Griffith45, L. Grillo11, O. Grünberg62, B. Gui59,
E. Gushchin33, Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C.
Haen38, S.C. Haines47, S. Hall53, B. Hamilton58, T. Hampson46, X. Han11, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62,
J. He38, T. Head38, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A.
Hernando Morata37, E. van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M.
Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55, N. Hussain55, D.
Hutchcroft52, D. Hynds51, M. Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11,
J. Jalocha55, E. Jans41, P. Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P.
Koppenburg41,38, M. Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11,
M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k,
V. Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, B. Langhans38, T. Latham48, C.
Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A.
Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11,
Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G.
Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40,
H. Lu3, D. Lucchesi22,q, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, J.F. Marchand4, U.
Marconi14, C. Marin Benito36, P. Marino23,s, R. Märki39, J. Marks11, G.
Martellotti25, A. Martens8, A. Martín Sánchez7, M. Martinelli41, D. Martinez
Santos42, F. Martinez Vidal64, D. Martins Tostes2, A. Massafferri1, R.
Matev38, Z. Mathe38, C. Matteuzzi20, A. Mazurov16,f, M. McCann53, J.
McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B. Meadows57,55, F.
Meier9, M. Meissner11, M. Merk41, D.A. Milanes8, M.-N. Minard4, N. Moggi14, J.
Molina Rodriguez60, S. Monteil5, D. Moran54, M. Morandin22, P. Morawski26, A.
Mordà6, M.J. Morello23,s, J. Moron27, R. Mountain59, F. Muheim50, K. Müller40,
R. Muresan29, M. Mussini14, B. Muster39, P. Naik46, T. Nakada39, R.
Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21, S. Neubert38, N.
Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, M.
Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A. Novoselov35, A.
Oblakowska-Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44,
R. Oldeman15,e, G. Onderwater65, M. Orlandea29, J.M. Otalora Goicochea2, P.
Owen53, A. Oyanguren64, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M.
Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51, C. Parkes54, C.J.
Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A.
Pazos Alvarez37, A. Pearce54, A. Pellegrino41, M. Pepe Altarelli38, S.
Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-Terrin6, L.
Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E. Polycarpo2, A. Popov35, D.
Popov10, B. Popovici29, C. Potterat2, A. Powell55, J. Prisciandaro39, A.
Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39, G. Punzi23,r, W.
Qian4, B. Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama18,
M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G. Raven42, S. Reichert54, M.M.
Reid48, A.C. dos Reis1, S. Ricciardi49, A. Richards53, M. Rihl38, K.
Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P. Robbe7, A.B. Rodrigues1, E.
Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero
Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P.
Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P.
Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo64, B.
Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37, E.
Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K.
Senderowska27, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11,
M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34,
V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N.
Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P.
Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,o, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.

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 Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

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
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy

##  1 Introduction

Heavy baryons are systems of three quarks, among which at least one is $c$ or
$b$. The quarks are bound by the strong interaction, which is described by
quantum chromodynamics (QCD). Hadron lifetimes are among the most useful
inputs to tune the parameters of QCD models. A powerful approach for
theoretical predictions of $b$-hadron lifetime ratios is the heavy quark
expansion (HQE) framework [1] which allows calculations in powers of
$\Lambda_{\rm QCD}/m_{b}$, where $\Lambda_{\rm QCD}$ is the energy scale at
which QCD becomes non-perturbative and $m_{b}$ is the $b$-quark mass. With the
exception of the $b$ hadrons containing a $c$ quark, the predictions for the
various $b$-hadron lifetimes only start to differ at the order $\Lambda_{\rm
QCD}^{2}/m_{b}^{2}$ and are equal within several percent.

So far only the most abundantly produced $b$ baryon, the $\mathchar
28931\relax^{0}_{b}$ with quark content $udb$, has been studied in detail.
Early $\mathchar 28931\relax^{0}_{b}$ lifetime measurements [2, 3, 4, 5]
yielded values significantly smaller than the $B$-meson lifetime
determinations, casting doubt on the HQE and causing increased theoretical
activity [6, 7, 8, 9, 10, 11]. More recent determinations of the ratio between
the $\mathchar 28931\relax^{0}_{b}$ and ${B}^{0}$ lifetimes, for instance that
of Ref. [12], are in much better agreement with the original predictions.
However, less information exists on the strange $b$ baryons, which are less
abundantly produced than $\mathchar 28931\relax^{0}_{b}$ baryons. In
particular for the $\mathchar 28932\relax^{-}_{b}$ ($dsb$) and $\mathchar
28938\relax^{-}_{b}$ ($ssb$) baryons only a few theoretical lifetime
calculations are available [13, 7, 8]. Furthermore, most of the predictions
date back to the 1990s and have limited precision, with central values ranging
from 1.0 ps to 1.7 ps. New experimental data are needed to provide more
stringent constraints on the models.

The weakly decaying $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ baryons were observed for the first time at the Tevatron
experiments CDF [14, 15] and D0 [16, 17]. Prior to these first observations,
the average $\mathchar 28932\relax_{b}$ lifetime (including $\mathchar
28932\relax^{-}_{b}$ and $\mathchar 28932\relax^{0}_{b}$) was measured by the
LEP experiments DELPHI [18, 19] and ALEPH [20] using partially reconstructed
decays. So far the only exclusive lifetime measurement of the strange $b$
baryons $\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$
has been made by the CDF experiment [15, 21]. Recently, LHCb demonstrated its
ability to reconstruct a significant number of $\mathchar 28932\relax^{-}_{b}$
and $\mathchar 28938\relax^{-}_{b}$ baryons [22] and to measure precisely
$b$-hadron lifetimes [23].

In this Letter we present lifetime measurements of the $\mathchar
28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$ baryons reconstructed
in the $\mathchar
28932\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28932\relax^{-}$ and $\mathchar
28938\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28938\relax^{-}$ decay modes. The daughter particles are
reconstructed in the decay modes ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$, $\mathchar
28932\relax^{-}\rightarrow\mathchar 28931\relax{{\pi}^{-}}$, $\mathchar
28938\relax^{-}\rightarrow\mathchar 28931\relax{{K}^{-}}$ and $\mathchar
28931\relax\rightarrow p{{\pi}^{-}}$. Unless specified otherwise, charge-
conjugated states are implied throughout.

##  2 Detector and event samples

The LHCb detector [24] 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 [25] placed downstream of the
magnet. The combined tracking system provides a momentum measurement with a
relative uncertainty that varies from 0.4% at low momentum, $p$, to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter measurement
with a resolution of 20${\,\upmu\rm m}$ for charged particles with large
transverse momentum, $p_{\rm T}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[26]. 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 [27].

The trigger 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. For this measurement, events are first required to
pass the hardware trigger, which selects muons with high transverse momentum.
In the subsequent software stage, events are retained by two independent sets
of requirements. One demands a muon candidate with momentum larger than
6${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ that, combined with another oppositely
charged muon candidate, yields a dimuon mass larger than
2.7${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The other requires a muon
candidate with momentum larger than 8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and an impact parameter above $100{\,\upmu\rm m}$ with respect to all of the
primary $pp$ interaction vertices (PVs) in the event. Finally, for all
candidates, two muons are required to form a vertex that is significantly
displaced from the PVs.

The $\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$
lifetime measurements presented here are based on the combination of the two
data sets recorded in 2011 and 2012. During the year 2011 the LHCb detector
recorded $pp$ collisions at a centre-of-mass energy of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ corresponding to an integrated
luminosity of 1 fb-1. In 2012, it recorded approximately twice as much data at
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$. Between 2011 and 2012, an
improvement in the tracking algorithm of the vertex detector was introduced,
leading to different trigger and reconstruction efficiencies in the two data
sets. The polarity of the dipole magnet was periodically inverted so that
roughly half of the data was collected with each polarity.

Four million $\mathchar 28932\relax^{-}_{b}$ ($\mathchar 28938\relax^{-}_{b}$)
signal events, corresponding to approximately 135 $\mbox{\,fb}^{-1}$ (1700
$\mbox{\,fb}^{-1}$) of LHCb data, were simulated with each of the 2011 and
2012 data taking conditions. The $pp$ collisions are generated using Pythia
[28, *Sjostrand:2007gs] with a specific LHCb configuration [30]. Decays of
hadronic particles are described by EvtGen [31], in which final state
radiation is generated using Photos [32]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [33, *Agostinelli:2002hh] as described in Ref. [35].

##  3 Reconstruction and selection

The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$ decay is reconstructed from oppositely
charged particles that leave deposits in the vertex detector, the tracking
stations and the muon system. The hyperons in the $b$-baryon decay chains
($\mathchar 28932\relax^{-}$, $\mathchar 28938\relax^{-}$ and $\mathchar
28931\relax$) are long-lived; approximately 10$\%$ of all reconstructed
$b$-baryon candidates are reconstructed with all tracks leaving deposits in
the vertex detector. To retain as many candidates as possible, tracks that
have no vertex detector information are also considered for the reconstruction
of the hyperon decays.

The $\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$
candidates are selected through identical requirements except for the ranges
in which the baryon masses are reconstructed. In addition, for the $\mathchar
28938\relax^{-}_{b}$ case the charged track from the $\mathchar
28938\relax^{-}$ decay is required to be identified as a kaon by the particle
identification detectors, removing more than 95% of the background pions.

All final-state tracks are required to satisfy minimal quality criteria and
kinematic requirements. In order to reduce backgrounds from combinations of
random tracks, the decay vertices are required to be well reconstructed. The
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, $\mathchar 28932\relax^{-}$,
$\mathchar 28938\relax^{-}$ and $\mathchar 28931\relax$ candidates are
selected within mass windows of $\pm
60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $\pm
11{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $\pm
11{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\pm
6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, around the
corresponding known masses [36].

The hadronic final-state tracks are required to have large impact parameters
with respect to the PV associated with the $b$-baryon candidate. The
associated PV is chosen as the PV giving the smallest increase in the
$\chi^{2}$ of the PV fit when the $b$ baryon is included. The associated PV is
also required to be isolated with respect to other PVs and consistent with the
nominal interaction region.

The $b$-baryon mass is computed after a complete kinematic fit of the decay
chain [37] in which the masses of both daughter particles are constrained to
their known values [36]. No constraint is applied on the $\mathchar
28931\relax$ mass. The resulting $b$-baryon invariant mass is required to lie
in the range 5600–6000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$\mathchar 28932\relax^{-}_{b}$ candidates and
5800–6300${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $\mathchar
28938\relax^{-}_{b}$ candidates. The decay time of the $b$-baryon candidate,
$t$, is computed from the decay length, $d$, as

$t=\frac{d}{\beta\gamma c}=\frac{m}{p}\,d\,,$ (1)

where $m$ is the reconstructed mass and $p$ the reconstructed momentum of the
$b$-baryon candidate. The decay length itself is obtained from a refit of the
decay chain with no mass constraints in order to keep the correlation between
the reconstructed decay time and mass at a negligible level. Backgrounds are
further suppressed by requiring this decay chain fit to be of good quality.
The reconstructed decay time is required to lie in the range 0.3–14 ps. The
lower bound of this decay-time range helps to suppress background coming from
random combinations of tracks with real ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mesons produced at the PV. In less than 1% of the cases, more than one
candidate per event pass the selection criteria and only the candidate with
the best decay chain fit result is retained.

##  4 Resolution and efficiency

The decay time resolution is obtained by fitting the difference between the
reconstructed decay time, $t$, and the true decay time, $t_{\rm true}$, in
simulated events. The fit model is a single Gaussian function $G(t-t_{\rm
true},\bar{t},\sigma_{\rm res})$ where the mean, $\bar{t}$, and the width,
$\sigma_{\rm res}$, are left free. For both considered decay modes and both
data-taking periods, $\bar{t}$ is compatible with zero and $\sigma_{\rm res}$
is close to 50 fs.

A bias in the measured lifetime can arise from a non-uniform efficiency as a
function of the $b$-baryon decay time [23]. There are two types of
inefficiencies which alter the decay time distribution. The first affects
mostly candidates with small decay times and is induced by the requirements of
the trigger that reject predominantly short-lived $b$ baryons. The second
affects mostly candidates with large decay times and is due to the geometrical
detector acceptance, the reconstruction process and the selection criteria
that lead to a lower efficiency for long-lived $b$ baryons. Both effects are
estimated and corrected for using simulation. This approach is validated with
several techniques described in Sec. 6.

The two trigger selections used for these lifetime measurements include a
requirement on the decay length significance of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. In addition, one
selection also contains a requirement on the impact parameter of the muons
from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay. The two
requirements induce an inefficiency at low values of the reconstructed decay
time. To assess this effect, simulated events undergo an emulation of the
trigger. In addition, an unbiased trigger selection is used to remove the
contribution from the detector acceptance, the reconstruction and the
selection. The resulting efficiency as a function of the reconstructed decay
time is fitted with an empirical function of the form

$\displaystyle\varepsilon_{1}(t)={\rm erf}(\ a\cdot(t-t_{0})^{n}\ )\,,$
$\displaystyle{\rm where}$ $\displaystyle{\rm
erf}(u)=\frac{2}{\sqrt{\pi}}\int_{0}^{u}e^{-x^{2}}dx\,,$ (2)

and where $a$, $t_{0}$ and $n$ are free parameters. The distributions of the
decay products of the $b$ baryons depend on the decay mode and on the year of
data taking. This dependence slightly affects the shape of the efficiency as a
function of the reconstructed decay time. Thus separate efficiency functions
are obtained for the two decay modes, for the two data taking periods and for
the two trigger selections. The efficiency functions corresponding to the 2012
data taking conditions for the $\mathchar 28932\relax^{-}_{b}$ case are shown
in Fig. 1 as an example.

The dependence of the efficiency on the decay time due to the geometrical
detector acceptance, the reconstruction and the selection is found to be well
described with a linear function,

$\varepsilon_{2}(t)=1+\beta t\,.$ (3)

The free parameter $\beta$ is obtained by fitting a function proportional to
$\varepsilon_{2}(t)\cdot\int_{0}^{\infty}\exp(-t_{\rm true}/\tau_{\rm
gen})\,G(t-t_{\rm true},0,\sigma_{\rm res})\,dt_{\rm true}$ to the
reconstructed decay time distribution of simulated signal events that are
generated with a mean lifetime of $\tau_{\rm gen}$ and that are fully
reconstructed and selected. Separate values for $\beta$ are determined for the
two different decay modes and the two data-taking periods and are given in
Table 1. The efficiency functions corresponding to the 2012 data taking
conditions for the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ cases are shown in Fig. 2 as an example.

Figure 1: Efficiency for triggering simulated $\mathchar 28932\relax^{-}_{b}$
events as a function of the reconstructed decay time under the 2012 data-
taking conditions. Only a restricted decay-time range is shown in order to
emphasize the region where the effect is large. The left panel shows the
efficiency for events passing the trigger with only the requirement on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ vertex displacement. The right
panel shows the efficiency for events passing the trigger with requirements on
both the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ vertex displacement
and the muon impact parameter and required to not pass the other trigger. The
results of fits with functions proportional to that given in Eq. 2 are
overlaid.

Figure 2: Efficiency due to the detector acceptance, reconstruction and
selection of simulated $\mathchar 28932\relax^{-}_{b}$ (left) and $\mathchar
28938\relax^{-}_{b}$ (right) events as a function of the reconstructed decay
time under the 2012 data-taking conditions. The results of fits with functions
proportional to that given in Eq. 3 are overlaid.

Table 1: Fitted $\beta$ values (in ps-1) for the efficiency described in Eq. 3 extracted from simulated $\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$ decays. The quoted uncertainties are statistical only. Decay mode | 2011 conditions | 2012 conditions
---|---|---
$\mathchar 28932\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mathchar 28932\relax^{-}$ | $(-13.1\pm 4.8)\times 10^{-3}$ | $(-20.2\pm 5.0)\times 10^{-3}$
$\mathchar 28938\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mathchar 28938\relax^{-}$ | $(-23.3\pm 3.5)\times 10^{-3}$ | $(-19.3\pm 3.9)\times 10^{-3}$

##  5 Lifetime fit

The lifetime is extracted from a two-dimensional extended maximum likelihood
fit to the unbinned $b$-baryon mass and decay-time distributions. The mass and
decay time are computed with the techniques described in Sec. 3. Assuming a
negligible correlation between these two quantities, the two-dimensional
probability density functions for the signal and the background are each
written as the product of a mass term and a decay-time term.

For the mass distribution, the signal is described with a single Gaussian
function in which the mean and width are free parameters. Independent means
are used for the data recorded in 2011 and in 2012, since different
calibrations are applied. The background in the mass distribution is modelled
with an exponential function. The signal in the decay time distribution is
described with the product of the efficiency functions (described in Eqs. 2
and 3) and a convolution between an exponential function and a Gaussian
function describing the decay time resolution,

$S(t)=N\cdot\varepsilon_{1}(t)\cdot\varepsilon_{2}(t)\cdot\int_{0}^{\infty}e^{-t_{\rm
true}/\tau}G(t-t_{\rm true},0,\sigma_{\rm res})\,dt_{\rm true}\,,$ (4)

where $N$ is a normalisation parameter and $\tau$ the fitted lifetime. The
decay time resolution $\sigma_{\rm res}$ is fixed to the value obtained in
simulation, separately for each decay mode and each year of data taking. The
background in the decay time distribution is modelled with the sum of two
exponential functions that are also convolved with the fixed decay time
resolution function. With the exception of $\sigma_{\rm res}$, all background
parameters are left free in the fit. A study based on pseudo-experiments shows
that no observable bias to the measured lifetimes arises from the fit model
itself.

The fit is performed for all selected $b$-baryon candidates. Due to the low
signal yields, asymmetric uncertainties are calculated. Figure 3 shows the
invariant mass and decay time distributions and the projection of the fit
results for the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar
28938\relax^{-}_{b}$ baryons. Table 2 displays the fit result for the relevant
signal parameters.

Figure 3: Distributions of the reconstructed invariant mass (top) and decay
time (middle and bottom) of the $\mathchar
28932\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28932\relax^{-}$ (left) and $\mathchar
28938\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28938\relax^{-}$ (right) candidates. The middle (bottom)
panels show the decay time distributions of the candidates in the signal
(background) mass regions. The signal mass region is defined as
5773–5825${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ for $\mathchar
28932\relax^{-}_{b}$ and 6028–6073${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$
for $\mathchar 28938\relax^{-}_{b}$ candidates, as shown by the vertical
dotted lines in the mass distributions, whereas the background mass regions
include all other candidates. The results of the fits are overlaid.

As a consistency check for the fitting method, a measurement of the $\mathchar
28931\relax^{0}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\mathchar 28931\relax$ lifetime is performed using the same data set
and techniques as presented in this paper. The measured $\mathchar
28931\relax^{0}_{b}$ lifetime is consistent with the world average [36] and
with recent measurements from LHCb [23, 12].

Table 2: Fitted parameters with statistical uncertainties for the $\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$ signals. Parameter | $\mathchar 28932\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mathchar 28932\relax^{-}$ | $\mathchar 28938\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mathchar 28938\relax^{-}$
---|---|---
Signal yield | 313 | $\pm$ | 20 | 58 | $\pm$ | 8
Mass resolution | 8.5 | $\pm$ | 0.5 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | 7.5 | $\pm$ | 1.0 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
Lifetime ($\tau$) | 1.55 | ${}^{+}_{-}$ |  ${}^{0.10}_{0.09}$ ps | 1.54 | ${}^{+}_{-}$ |  ${}^{0.26}_{0.21}$ ps

##  6 Systematic uncertainties

Unless specified otherwise, the evaluation of the systematic uncertainties is
performed by varying in turn each fixed parameter of the fit within its
uncertainty and taking the change in the fit result. The total systematic
uncertainty is obtained as the quadratic sum of the individual contributions.
Distributions of results from fits to pseudo-experiments are used for the
leading contributions (efficiencies and modelling) in order to ensure that
they are not incorrectly estimated due to a statistical fluctuation of the
data. A summary of all contributions to the total systematic uncertainty is
given in Table 3.

Two contributions are considered as uncertainties due to the trigger
efficiency. One arises from the finite size of the simulation samples and is
taken into account by varying the parameters of the efficiency function
$\varepsilon_{1}$ within their uncertainties. The other is due to a potential
discrepancy between data and simulation. This second contribution is assessed
by repeating the fit using an efficiency obtained from a data sample of
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\rm\scriptscriptstyle S}}$ decays that are topologically
similar to the $b$-baryon decays of interest and reconstructed in data
collected by the same trigger. In this case, extracting the efficiency from
data is possible because a large sample, selected with triggers that do not
bias the decay time distribution, is available (see Ref. [23]).

Table 3: Systematic uncertainties on the lifetime measurements in fs. For the total uncertainty, all the contributions are summed in quadrature. Source | $\mathchar 28932\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mathchar 28932\relax^{-}$ | $\mathchar 28938\relax^{-}_{b}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\mathchar 28938\relax^{-}$
---|---|---
Trigger efficiency |  9.9 |  6.5
Reconstruction and selection efficiency | 29.0 | 45.0
Signal modelling |  5.9 | 11.4
Combinatorial background modelling |  3.0 |  3.0
Cross-feed background |  0.1 | 11.1
Detector length scale |  0.3 |  0.3
Total | 31.4 | 48.3

Two contributions are considered as uncertainties in determining the
reconstruction and selection efficiency. One arises from the finite size of
the simulation samples and is assessed by varying the parameter $\beta$ within
its statistical uncertainty. The other takes into account the quality of the
simulation of the geometrical detector acceptance, the reconstruction process
and the selection. For this second contribution, the $\beta$ parameter is
varied by $\pm 50\%$ to cover any possible discrepancy between data and
simulation [38]. The total systematic uncertainty related to the
reconstruction and selection efficiency, taken as the quadratic sum of the two
contributions, is larger for $\mathchar 28938\relax^{-}_{b}$ than for
$\mathchar 28932\relax^{-}_{b}$ decays due to the larger value of the $\beta$
parameter for $\mathchar 28938\relax^{-}_{b}$ in 2011 data.

Several alternative fits are performed to assess the systematic uncertainties
related to the signal modelling. In one fit, the Gaussian function describing
the signal model in the $b$-baryon mass distribution is replaced by the sum of
two Gaussian functions of common mean. The widths and the relative yields are
left free. To assess the effect of the decay time resolution function, the
widths of the corresponding Gaussian functions are varied by $\pm 10\%$. In
another alternative fit, this resolution function is taken as the sum of two
Gaussian functions instead of one, where the parameters are still taken from
simulation. This takes into account potential tails in the decay time
resolution distribution. All variations of the function describing the decay
time resolution change the fit result by a negligible amount. Therefore the
systematic uncertainty related to the signal modelling is dominated by the
signal description in the mass distribution.

The systematic uncertainties due to combinatorial background are taken into
account with three alternative fit models. In the first, the background in the
mass distribution is described with a linear function. In another fit the
background is modelled with two different exponential functions for the two
different years of data taking. As an alternative description of the
background in the decay time distribution, three exponential functions,
instead of two, are convolved with the Gaussian resolution.

The only other significant background expected is a cross-feed between the two
$b$-baryon decays. The rate and mass distribution of the cross-feed
backgrounds is determined by reconstructing simulated decays of one channel
under the hypothesis of the other. According to simulation, 0.24 (3.0)
$\mathchar 28938\relax^{-}_{b}$ ($\mathchar 28932\relax^{-}_{b}$) decays are
expected to be reconstructed as $\mathchar 28932\relax^{-}_{b}$ ($\mathchar
28938\relax^{-}_{b}$) over the full mass range. The effect of this background
on the lifetime measurement is determined by injecting simulated background
events into a fit of simulated signal events and taking the observed bias as
the systematic uncertainty.

The overall length scale of the vertex detector is known with a relative
precision of 0.02% [23]. As the measured decay length is directly proportional
to the overall length scale, this precision directly translates into a
relative uncertainty on the lifetime measurements.

##  7 Conclusion

Using data samples recorded during the years 2011 and 2012, corresponding to
an integrated luminosity of 3 fb-1, the lifetimes of the weakly decaying
$\mathchar 28932\relax^{-}_{b}$ and $\mathchar 28938\relax^{-}_{b}$ baryons
are measured to be

$\displaystyle\tau(\mathchar 28932\relax^{-}_{b})$ $\displaystyle=$
$\displaystyle 1.55\,^{+0.10}_{-0.09}~{}{\rm(stat)}\pm 0.03\,{\rm(syst)\,ps},$
$\displaystyle\tau(\mathchar 28938\relax^{-}_{b})$ $\displaystyle=$
$\displaystyle 1.54\,^{+0.26}_{-0.21}~{}{\rm(stat)}\pm 0.05\,{\rm(syst)\,ps}.$

These are the most precise lifetime measurements of these $b$ baryons to date.
Both measurements are in agreement with the previous experimental results, in
particular with the most recent ones from the CDF collaboration of
$\tau(\mathchar 28932\relax^{-}_{b})=1.32\pm 0.14$ ps and $\tau(\mathchar
28938\relax^{-}_{b})=1.66\pm 0.47$ ps [21]. The measurements also lie in the
range predicted by theoretical calculations [13, 7, 8].

## 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); NASU (Ukraine); STFC and the Royal Society (United Kingdom);
NSF (USA). We also acknowledge the support received from EPLANET, Marie Curie
Actions and 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 on which we
depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia).

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# A geometric characterization of range-kernel complementarity

Dimosthenis Drivaliaris  Department of Financial and Management Engineering
University of the Aegean
Chios, 82100
Greece<EMAIL_ADDRESS>and  Nikos Yannakakis  Department of
Mathematics
School of Mathematical and Physical Sciences
National Technical University of Athens
Iroon Polytexneiou 9
15780 Zografou
Greece<EMAIL_ADDRESS>
###### Abstract.

We show that a bounded linear operator on a Banach space with closed range has
range-kernel complementarity if and only if its generalized amplitude is less
than $\pi$. An application to the strong convergence of the iterations of a
bounded linear operator is also given.

###### 2000 Mathematics Subject Classification:

Primary 47A05 Secondary 47A10, 46C50

##  1\. Introduction

Let $X$ be a complex Banach space and $A:X\rightarrow X$ be a bounded linear
operator on $X$. If we denote by $R(A)$ and $N(A)$ the range and the kernel of
$A$ respectively and $R(A)$ is closed, then range-kernel complementarity is
the decomposition

(1.1) $X=R(A)\oplus N(A)\,.$

In a sense, range-kernel complementarity is the closest thing to $A$ being
invertible, since it is equivalent to $A$ being of the form
“invertible$\,\oplus\,0$”. Our aim in this short paper is its characterization
by a simple geometric condition.

Our approach combines two ingredients. The first is that (1.1) is possible if
and only if $A$ can be factorized as $TP$ with $T$ invertible and $P$ a
projection, see [11, Theorem 3]. The second is that (1.1) follows from a
suitable angle of $A$ being less than $\pi$, see [3, Theorem 3.4].

The idea behind the possible connection between range-kernel complementarity
and some geometric condition on $A$ originates from the following question. If
an angle condition is imposed on $A$ (for example, the well-known $Re\langle
Ax,x\rangle\geq c\|x\|^{2}$ of the Lax-Milgram Theorem which implies that $A$
is invertible) how far from invertible can $A$ be? The answer is interesting
and surprising; $A$ cannot be too bad as a quite natural angle condition
implies (1.1).

We conclude this paper with an application of our results to the problem of
strong convergence of the iterates $T^{n}$ of an asymptotically regular
operator $T$, for which $R(I-T)$ is closed.

For other characterizations of range-kernel complementarity we refer the
interested reader to [11] and the references therein.

##  2\. Main result

In order to formulate our geometric condition we first define the angle of $A$
with respect to a certain class of linear operators. This class
$\mathcal{E}_{A}$ for $A$ is defined as follows

$\mathcal{E}_{A}=\left\\{T\in B(X):T(N(A))=N(A)\text{ and }A+tT\text{ is
invertible for some }t>0\right\\}\,.$

######  Remark 2.1.

Note that in the terminology of [6, p. 49] if $T\in\mathcal{E}_{A}$, then
$A+tT,\,\,\,t\in\mathbb{C}$, is a regular linear pencil.

If $[\cdot,\cdot]$ is a semi-inner product in $X$ compatible with its norm,
then the cosine of $A$ with respect to $T\in\mathcal{E}_{A}$ is defined as

$\cos_{T}(A)=\inf\left\\{\frac{Re[Ax,Tx]}{\|Ax\|\,\|Tx\|}:x\notin N(A)\cup
N(T)\right\\}$

and the angle of $A$ with respect to $T$ is defined as

$\varphi_{T}(A)=\arccos\left(\cos_{T}(A)\right)\,.$

For $T=e^{i\theta}I\,,\,\,\,\theta\in[0,2\pi]$, the above is nothing more than
the angle of $A$ along the ray

$\rho_{\theta}=\left\\{0\right\\}\cup\left\\{z\in\mathbb{C}:\arg
z=\theta\right\\}$

which was defined by Gustafson and Krein in [7] and [10] respectively. For
each $\theta$ this angle $\varphi_{\theta}(A)$ measures the maximum turning
effect of $A$ along the ray $\rho_{\theta}$.

The amplitude of $A$, also introduced by Krein in [10], is then defined as

$\operatorname{am}(A)=\min\left\\{\phi_{T}(A):\,T=e^{i\theta}I,\,\,\,\theta\in[0,2\pi]\right\\}\,.$

The amplitude compares the maximum turning effect of $A$ along every possible
$\rho_{\theta}$ and provides the smallest one.

In [3] we proved that if $A$ has closed range, $R(A)+N(A)$ is closed and
$\operatorname{am}(A)<\pi$, then $A$ has range-kernel complementarity. The
converse of this is not true, since it may be proved that
$\operatorname{am}(A)<\pi$ implies that the spectrum of $A$ is confined to a
sector of the complex plane. In [4] the hypothesis that $R(A)+N(A)$ is closed
was dropped and a characterization of range-kernel complementarity was given,
using the angle of $A$ along a curve, for closed range operators whose
spectrum is not confined to a sector of $\mathbb{C}$, but still does not
separate $0$ from $\infty$. Hence, again, no complete characterization of
range-kernel complementarity was given.

Trying to amend these shortcomings and motivated by the amplitude of Krein we
define the generalized amplitude of $A$ with respect to $\mathcal{E}_{A}$ to
be

$\operatorname{Am}(A)=\inf\left\\{\varphi_{T}(A):T\in\mathcal{E}_{A}\right\\}\,.$

######  Remark 2.2.

(a) Obviously $\operatorname{Am}(A)\leq\operatorname{am}(A)$, for all
operators $A$.
(b) In the special case of an invertible operator $A$, we have that
$A\in\mathcal{E}_{A}$ and hence $\operatorname{Am}(A)=\varphi_{A}(A)=0$. This
simple example encapsulates a crucial difference between the generalized
amplitude and Krein’s one. The bilateral shift $A$ on $l^{2}$ is an invertible
operator and so $\operatorname{Am}(A)=0$, whereas its Krein amplitude is equal
to $\pi$, since its spectrum separates 0 from $\infty$. Note that by [4,
Theorem 4.1] the angle of $A$ along any curve is also equal to $\pi$.
(c) More generally, if $A:X\rightarrow X$ is invertible, then for the non-
invertible operator $A\oplus 0:X\oplus X\rightarrow X\oplus X$ we have that
$A\oplus I\in\mathcal{E}_{A}$ and hence $\operatorname{Am}(A\oplus
0)\leq\frac{\pi}{2}$. Taking $A$ to be the bilateral shift we get an example
of a non-invertible operator with $\operatorname{Am}(A)<\operatorname{am}(A)$.

Using this generalized amplitude we get the following characterization of
range-kernel complementarity.

######  Theorem 2.3.

Let $X$ be a Banach space and $A:X\rightarrow X$ be a bounded linear operator
with closed range. Then

$X=R(A)\oplus N(A)$

if and only if $\operatorname{Am}(A)<\pi$.

To get to the proof we start with five preparatory Lemmas.

######  Lemma 2.4.

If $\varphi_{T}(A)<\pi$, then there exists $c>0$ such that $\|Ax+tTx\|\geq
c\|Ax\|$, for all $x\notin N(A)$ and all $t\geq 0$.

###### Proof.

We have that $\varphi_{T}(A)<\pi$ implies that there exists $\delta>0$ such
that

$\frac{Re[Ax,Tx]}{\|Ax\|\,\|Tx\|}\geq-1+\delta\,,$

for all $x\notin N(A)\cup N(T)$ and hence

$Re[Ax,Tx]+\|Ax\|\,\|Tx\|\geq\delta\,\|Ax\|\,\|Tx\|\,,$

for all $x\notin N(A)\cup N(T)$ .

So for $t\geq 0$ and $x\notin N(A)\cup N(T)$ we get that

$\displaystyle Re[Ax+tTx,Tx]+\|Ax+tTx\|\,\|Tx\|\geq$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq
Re[Ax,Tx]+t\|Tx\|^{2}+(\|Ax\|-t\|Tx\|)\|Tx\|$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=Re[Ax,Tx]+\|Ax\|\,\|Tx\|$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq\delta\|Ax\|\,\|Tx\|\,.$

Thus, for $c=\frac{\delta}{2}$, we get $\|Ax+tTx\|\geq c\|Ax\|$, for all
$x\notin N(A)\cup N(T)$ and $t\geq 0$. Since this inequality obviously holds
if $x\in N(T)$ the proof is complete. ∎

######  Lemma 2.5.

Assume $A\in B(X)$ has closed range and $\varphi_{T}(A)<\pi$, for some $T\in
B(X)$. If $N(A)=\left\\{0\right\\}$ and $A+t_{0}T$ is invertible, for some
$t_{0}>0$, then $A+tT$ is invertible, for any $|t-t_{0}|<\mu$, where $\mu>0$
is independent of $t_{0}$.

###### Proof.

Since the range of $A$ is closed, there exists $k_{A}>0$ such that

$\|Ax\|\geq k_{A}\operatorname{dist}(x,N(A))$

and thus, since $N(A)=\left\\{0\right\\}$,

$\|Ax\|\geq k_{A}\|x\|\,,\text{ for all }x\in X\,.$

Let $\mu=c\,k_{A}>0$, where $c$ is the constant from Lemma 2.4. Since
$N(A)=\left\\{0\right\\}$, we get that

(2.1) $\|Ax+tTx\|\geq\mu\|x\|\,,\text{ for all }x\in X\text{ and }t\geq 0\,.$

So $A+tT$ is injective for all $t\geq 0$ .

Hence it is enough to show that if $|t-t_{0}|<\mu$ then $A+tT$ is also onto.
So fix such a $t$ and let $L=\frac{|t-t_{0}|}{\mu}<1$. For $y$ in $X$, define
$F:X\rightarrow X$ by

$Fh=(A+t_{0}T)^{-1}\left(y+(t_{0}-t)h\right),\text{ for every }h\in X\,.$

Then, using (2.1), we have

$\displaystyle\|Fh_{1}-Fh_{2}\|$
$\displaystyle=\|(A+t_{0}T)^{-1}\left(y+(t_{0}-t)h_{1}\right)-(A+t_{0}T)^{-1}\left(y+(t_{0}-t)h_{2}\right)\|$
$\displaystyle\leq\frac{1}{\mu}\,\|(t_{0}-t)(h_{1}-h_{2})\|$
$\displaystyle=L\,\|h_{1}-h_{2}\|\,.$

Hence $F$ is a contraction and thus there exists a unique $x\in X$ such that
$Fx=x$, which implies that $(A+t_{0}T)x=y+(t_{0}-t)x$. Hence $(A+tT)x=y$ and
so $A+tT$ is onto as was required. ∎

Next we show that if some angle of $A$ is less than $\pi$, then the sum of
$R(A)$ and $N(A)$ is both closed and direct.

######  Lemma 2.6.

Assume $\varphi_{T}(A)<\pi$, for some $T\in\mathcal{E}_{A}$. Then $R(A)+N(A)$
is closed and $R(A)\cap N(A)=\left\\{0\right\\}$.

###### Proof.

If $x\notin N(A)$, $y\in N(A)$ and $t>0$, from Lemma 2.4 we get that

$\left\|A\left(x+\frac{1}{t}\,y\right)+tT\left(x+\frac{1}{t}\,y\right)\right\|\geq
c\|Ax\|$

and thus

$\|Ax+tTx+Ty\|\geq c\|Ax\|\,.$

Letting $t\rightarrow 0$ we have that

$\|Ax+Ty\|\geq c\|Ax\|\,.$

Since $T(N(A))=N(A)$, we get that the sum $R(A)+N(A)$ is closed.

To conclude note that if $Ax\in N(A)$, then from the above inequality $Ax=0$
and so $R(A)\cap N(A)=\left\\{0\right\\}$. ∎

The final two results that we will need for the proof of Theorem 2.3 are the
following:

######  Lemma 2.7.

[11, Corollary p. 129] If

$R(A)\cap N(A)=\left\\{0\right\\}\,,$

then $R(A)+N(A)$ is closed if and only if $R(A^{2})$ is closed.

######  Lemma 2.8.

[4, Corollary 5.2] If $P$ is a projection, then $\varphi_{I}(P)<\pi$.

We may now proceed with the proof of our main result.

###### Proof of Theorem 2.3.

Assume that $X=R(A)\oplus N(A)$. Then, by [11, Theorem 3], there exist an
invertible operator $S$ and a projection $P$ such that $A$ can be factorized
as $A=SP$. In particular $S=A|_{R(A)}\oplus I_{N(A)}$ with respect to
$X=R(A)\oplus N(A)$ and $P$ is the projection onto $R(A)$ parallel to $N(A)$.
Hence $S(N(A))=N(A)$ and $SP=PS$ and so, by Lemma 2.8, we get that

$\varphi_{S}(A)=\varphi_{I}(P)<\pi\,.$

Thus $\operatorname{Am}(A)<\pi$.

For the converse if $\operatorname{Am}(A)<\pi$, then there exists
$T\in\mathcal{E}_{A}$ such that $\varphi_{T}(A)<\pi$.

We first assume that $N(A)=\left\\{0\right\\}$. Since $T\in\mathcal{E}_{A}$ we
have that $A+t_{0}T$ is invertible for some $t_{0}>0$ and hence, by Lemma 2.5,
we get that so is $A+tT$, for $|t-t_{0}|<\mu$. Since $\mu>0$ is independent of
$t_{0}$, continuing this way we get that $A$ is invertible and hence we have
range-kernel complementarity.

If $N(A)\neq\left\\{0\right\\}$ and since by Lemma 2.6 we have that $R(A)\cap
N(A)=\left\\{0\right\\}$, in order to conclude (see [9, Proposition 38.4]) it
is enough to show that $R(A^{2})=R(A)$. By Lemma 2.7 we have that $R(A^{2})$
is closed and hence if

$R\left(A|_{R(A)}\right)=R(A^{2})\neq R(A)\,,$

then $A|_{R(A)}$ has trivial kernel, closed range but is not invertible. Hence
by the first part of the proof

$\varphi_{T}(A|_{R(A)})=\pi\,.$

But $\varphi_{T}(A|_{R(A)})\leq\varphi_{T}(A)$ and thus we have a
contradiction. Hence $R(A^{2})=R(A)$ and the proof is complete. ∎

######  Remark 2.9.

If the operator $T$ for which $\varphi_{T}(A)<\pi$ is invertible, then things
are much simpler. In particular then $\varphi(T^{-1}A)<\pi$ and hence, by [3,
Theorem 3.4], we have that

$X=R(T^{-1}A)\oplus N(T^{-1}A)\,.$

Therefore by [11, Theorem 3] there exist an invertible operator $S$ and a
projection $P$ such that $T^{-1}A=SP$ and thus $A=TSP$ which, again by [11,
Theorem 3], gives

$X=R(A)\oplus N(A)$

as required.

If $0$ is an isolated point of the spectrum of $A$, then with respect to the
Riesz decomposition $A$ is of the form “invertible$\,\oplus$ quasinilpotent” .
Since range-kernel complementarity is equivalent to the “quasinilpotent part”
of $A$ being $0$ it is quite expected that quasinilpotent operators with
closed range should have amplitude equal to $\pi$.

######  Corollary 2.10.

Let $X$ be a Banach space and $Q$ be a quasinilpotent operator with closed
range. Then $\operatorname{Am}(Q)=\pi$.

###### Proof.

If $\operatorname{Am}(Q)<\pi$ then by Theorem 2.3

$X=R(Q)\oplus N(Q)$

which is impossible since by [5] the range chain of a quasinilpotent operator
does not terminate. ∎

######  Remark 2.11.

As $\operatorname{Am}(Q)\leq\operatorname{am}(Q)$, for quasinilpotent
operators with closed range the generalized amplitude is equal to Krein’s.
Moreover since $\sigma(Q)$ does not separate 0 from $\infty$ by [4, Theorem
4.1] the angle of $Q$ along any curve is also equal to $\pi$.

##  3\. An application

In this section we present a simple application of our main result to the
problem of strong convergence of the iterates $T^{n}$ of a bounded linear
operator $T$ defined on a Banach space $X$. Recall that $T$ is called
asymptotically regular if, for all $x\in X$,

$(T^{n}-T^{n+1})x\rightarrow 0,\text{ as }n\rightarrow+\infty\,.$

######  Remark 3.1.

If $A$ is nilpotent then it is asymptotically regular and hence the asymptotic
regularity of $A$ does not imply that $\operatorname{Am}(A)<\pi$.

The proof of our result follows the idea of [1, Theorem 4.1].

######  Theorem 3.2.

Let $X$ be a complex Banach space and $T\in B(X)$ be an asymptotically regular
operator, such that $R(I-T)$ is closed. The sequence of iterates $T^{n}x$
converges, for all $x\in X$, as $n\rightarrow+\infty$ if and only if
$\operatorname{Am}(I-T)<\pi$.

###### Proof.

Since $R(I-T)$ is closed if $\operatorname{Am}(I-T)<\pi$ we have from Theorem
2.3 that

$X=R(I-T)\oplus N(I-T)\,.$

Moreover by the fact that $T$ is asymptotically regular we have that
$T^{n}z\rightarrow 0$, as $n\rightarrow+\infty$, for all $z\in R(I-T)$. Hence
if $x\in X$, then $x=y+z$ with $y\in N(I-T)$ and $z\in R(I-T)$ and so

$T^{n}x=T^{n}y+T^{n}z\rightarrow y\text{ as }n\rightarrow+\infty\,.$

The converse follows again from Theorem 2.3 since the strong convergence of
$T^{n}$ implies by [8, Theorem 3] that $X=R(I-T)\oplus N(I-T)$. ∎

######  Remark 3.3.

(a) It may be easily seen from the above proof that $T^{n}$ converges strongly
to the projection onto $R(I-T)$ parallel to $N(I-T)$.
(b) Note that, by [1, Theorem 4.1], $R(I-T)$ is closed if and only if the
convergence of $T^{n}$ is uniform.
(c) The connection between range-kernel complementarity and the strong
convergence of the iterates $T^{n}$ was used in the well-known paper of
Halperin [8] in order to prove the convergence of the iterates of
$T=P_{1}P_{2}...P_{n}$, for $P_{1},...,P_{n}$ orthogonal projections on a
Hilbert space. This was later generalized for norm-one projections on a Banach
space by Bruck and Reich in [2, Theorem 2.1] and by Badea and Lyubich in [1,
Main Theorem, p. 25], with suitable additional conditions .

Recall that we say that a contraction $T$ is primitive if $T$ satisfies the
Katznelson-Tzafriri spectral condition, i.e. if

$\sigma(T)\subseteq\left\\{1\right\\}\cup\left\\{t\in\mathbb{C}:|t|<1\right\\}\,.$

We have the following Corollary of Theorem 3.2 for primitive contractions on a
reflexive Banach space.

######  Corollary 3.4.

If $X$ is a reflexive Banach space and $T$ is a primitive contraction with
$R(I-T)$ closed, then $\operatorname{Am}(I-T)<\pi$.

###### Proof.

To see this note that $T$ is asymptotically regular and by [1, Theorem 4.1]
its iterates $T^{n}$ converge strongly. Hence the result follows by Theorem
3.2. ∎

######  Remark 3.5.

(a) Since $T$ in Corollary 3.4 is a contraction $\sigma(I-T)$ does not
separate 0 from $\infty$ and hence, by [4, Theorem 4.1], we have that the
Krein amplitude of $I-T$ is also less than $\pi$.
(b) Corollary 3.4 combined with [1, Main Theorem] shows the following: If $T$
is a convex combination of products of orthoprojections on a complex Banach
space $X$, $R(I-T)$ is closed, $X$ is uniformly convex or uniformly smooth or
reflexive, and the orthoprojections are of class (D) (for the definition see
[1]), then $\operatorname{Am}(I-T)<\pi$. We would like to note here that since
such a $T$ may be quasinilpotent with closed range, the generalized amplitude
of such a $T$, by Corollary 2.10, may be equal to $\pi$.

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