EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-180 LHCb-PAPER-2012-017 4 July 2012

Measurement of the $\kern 3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}$
effective lifetime in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ final state

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

The effective lifetime of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson in the decay mode $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ is measured using 1.0 fb-1 of data collected in $pp$
collisions at $\sqrt{s}=7$ TeV with the LHCb detector. The result is $1.700\pm
0.040\pm 0.026\,\mathrm{ps}$ where the first uncertainty is statistical and
the second systematic. As the final state is $C\\!P$-odd, and $C\\!P$
violation in this mode is measured to be small, the lifetime measurement can
be translated into a measurement of the decay width of the heavy $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate, $\Gamma_{\rm
H}=0.588\pm 0.014\pm 0.009\,\rm{ps}^{-1}$.

Submitted to Physics Review Letters

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
C. Bauer10, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28,
K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S.
Benson47, J. Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14, G. Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P.
Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil
Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A.
Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31,
D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El
Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella16,e,14, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S.
Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U.
Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52,
A. Martín Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z.
Mathe12, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34, F. Polci8,
G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C.
Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A. Puig Navarro33,
W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S. Rangel2, I.
Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos
Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44,
S. Roiser35, V. Romanovsky32, A. Romero Vidal34, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, D. Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A.
Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M.
Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6,
P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27,
T. Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T.
Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, G. Veneziano36, M. Vesterinen35,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C.
Voß55, H. Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R.
Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J.
Wicht35, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M.
Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z.
Yang3, R. Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a,
F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.

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 Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

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

The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$, $f_{0}(980)\\!\rightarrow\pi^{+}\pi^{-}$, discovered by
LHCb [1] at close to the predicted rate [2], is important for $C\\!P$
violation [3] and lifetime studies. In this Letter, we make a precise
determination of the lifetime. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ final state is $C\\!P$-odd, and in the absence of $C\\!P$
violation, can be produced only by the decay of the heavy ($\rm{H}$), and not
by the light ($\rm{L}$), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate [4]. As the
measured $C\\!P$ violation in this final state is small [5], a measurement of
the effective lifetime, $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}$, can be translated into a measurement of the decay width,
$\Gamma_{\rm H}$. This helps to determine the decay width difference,
$\Delta\Gamma_{\rm s}=\Gamma_{\rm L}-\Gamma_{\rm H}$, a number of considerable
interest for studies of physics beyond the Standard Model (SM) [6,
*Lenz:2012az, *Bobeth:2011st]. Furthermore, this measurement can be used as a
constraint in the fit that determines the mixing-induced $C\\!P$-violating
phase in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays,
$\phi_{s}$, using the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final states, and
thus improve the accuracy of the $\phi_{s}$ determination [9, 5]. In the SM,
if sub-leading penguin contributions are neglected,
$\phi_{s}=-2\arg\left[\frac{V_{ts}^{\phantom{*}}V_{tb}^{*}}{V_{cs}^{\phantom{*}}V_{cb}^{*}}\right]$,
where the $V_{ij}$ are the Cabibbo-Kobayashi-Maskawa matrix elements, which
has a value of $-0.036\,^{+0.0016}_{-0.0015}$ rad [10]. Note that the LHCb
measurement of $\phi_{s}$ [5] corresponds to a limit on $\cos\phi_{s}$ greater
than 0.99 at 95% confidence level, consistent with the SM prediction.

The decay time evolution for the sum of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, via the
$b\\!\rightarrow c\overline{}cs$ tree amplitude, to a $C\\!P$-odd final state,
$f_{-}$, is given by [11, *Bigi:2000yz]

$\Gamma\left(B^{0}_{s}\rightarrow f_{-}\right)+\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
f_{-}\right)=\frac{\cal N}{2}e^{-\Gamma_{\rm
s}t}\,\Bigg{\\{}e^{\Delta\Gamma_{\rm
s}t/2}(1+\cos\phi_{s})+e^{-\Delta\Gamma_{\rm
s}t/2}(1-\cos\phi_{s})\Bigg{\\}}\,,$ (1)

where ${\cal N}$ is a time-independent normalisation factor and $\Gamma_{\rm
s}$ is the average decay width. We measure the effective lifetime by
describing the decay time distribution with a single exponential function

$\Gamma\left(B^{0}_{s}\rightarrow f_{-}\right)+\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow f_{-}\right)={\cal
N}e^{-t/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}}.$ (2)

Our procedure involves measuring the lifetime with respect to the well
measured $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime, in the
decay mode $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\\!\rightarrow K^{-}\pi^{+}$ (the
inclusion of charge conjugate modes is implied throughout this Letter). In
this ratio, the systematic uncertainties largely cancel.

The data sample consists of 1.0 fb-1 of integrated luminosity collected with
the LHCb detector [13] in $pp$ collisions at the LHC with 7
$\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass energy. The detector 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 and three stations of silicon-
strip detectors and straw drift-tubes placed downstream. Charged hadrons are
identified using two ring-imaging Cherenkov (RICH) detectors. Muons are
identified by a muon system composed of alternating layers of iron and
multiwire proportional chambers. The trigger consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by a
software stage that applies a full event reconstruction. The simulated events
used in this analysis are generated using Pythia 6.4 [14] with a specific LHCb
configuration [15], where decays of hadronic particles are described by EvtGen
[16], and the LHCb detector simulation [17] based on Geant4 [18,
*Agostinelli:2002hh].

The selection criteria we use for this analysis are the same as those used to
measure $\phi_{s}$ in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays [20]. Events are triggered by a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
decay, requiring two identified muons with opposite charge, transverse
momentum greater than 500 MeV (we work in units where $c=\hbar=1$), invariant
mass within 120 MeV of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
[21], and form a vertex with a fit $\chi^{2}$ less than 16.
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ candidates are
first selected by pairing an opposite sign pion combination with a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate that has a dimuon
invariant mass from -48 MeV to +43 MeV from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [21]. The pions are
required to be identified positively in the RICH detector, have a minimum
distance of approach with respect to the primary vertex (impact parameter) of
greater than 9 standard deviation significance, have a transverse momentum
greater than 250 MeV and fit to a common vertex with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with a $\chi^{2}$ less than 16.
Furthermore, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
candidate must have a vertex with a fit $\chi^{2}$ less than 10, flight
distance from production to decay vertex greater than 1.5 mm and the angle
between the combined momentum vector of the decay products and the vector
formed from the positions of the primary and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay vertices (pointing
angle) is required to be consistent with zero. Events satisfying this
preselection are then further filtered using requirements determined using a
Boosted Decision Tree (BDT) [22, *Hocker:2007ht]. The BDT uses nine variables
to differentiate signal from background: the identification quality of each
muon, the probability that each pion comes from the primary vertex, the
transverse momentum of each pion, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex fit quality, flight
distance from production to decay vertex and pointing angle. It is trained
with simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ signal events and two background samples from data, the
first with like-sign pions with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{\pm}\pi^{\pm}$ mass within $\pm 50$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass and the second from the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ upper mass sideband
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ mass between
200 and 250 MeV above the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass.

As the effective $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime is measured relative to that of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, we use the same
trigger, preselection and BDT to select ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ events, except for the hadron identification that is
applied independently of the BDT. The selected $\pi^{+}\pi^{-}$ and
$K^{-}\pi^{+}$ invariant mass distributions, for candidates with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass within $\pm
20$ MeV of the respective $B$ mass peaks are shown in Fig. 1. The background
distributions shown are determined by fitting the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass distribution
in bins of $\pi^{+}\pi^{-}$ ($K^{-}\pi^{+}$) mass. Further selections of $\pm
90$ $\mathrm{\,Me\kern-1.00006ptV}$ around the $f_{0}(980)$ mass and $\pm 100$
$\mathrm{\,Me\kern-1.00006ptV}$ around the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass are applied. The
$f_{0}(980)$ selection results in a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ sample that is greater than 99.4% $C\\!P$-odd at 95%
confidence level [24].

Figure 1: Invariant mass distributions of selected (a) $\pi^{+}\pi^{-}$ and
(b) $K^{-}\pi^{+}$ combinations (solid histograms) for events within $\pm 20$
MeV of the respective $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peaks. Backgrounds
(dashed histograms) are determined by fitting the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass in bins of
$\pi^{+}\pi^{-}$ ($K^{-}\pi^{+}$) mass. Regions between the arrows are used in
the subsequent analysis.

The analysis exploits the fact that the kinematic properties of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ decay are very similar to those of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. We can select
$B$ mesons in either channel using identical kinematic constraints and hence
the decay time acceptance introduced by the trigger, reconstruction and
selection requirements should almost cancel in the ratio of the decay time
distributions. Therefore, we can determine the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime, $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}$, relative to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ lifetime,
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$, from the variation of the ratio
of the $B$ meson yields with decay time

$R(t)=R(0)e^{-t(1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}-1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}})}=R(0)e^{-t\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}}\,\mathrm{,}$ (3)

where the width difference $\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}=1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}-1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$.

We test the cancellation of acceptance effects using simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ events. Both the
acceptances themselves and also the ratio exhibit the same behaviour. Due to
the selection requirements, they are equal to 0 at $t=0$, after which there is
a sharp increase, followed by a slow variation for $t$ greater then 1 ps.
Based on this, we only use events with $t$ greater than 1 ps in the analysis.
To good approximation, the acceptance ratio is linear between 1 and 7 ps, with
a slope of $a=0.0125\pm 0.0036$ ps-1 (see Fig. 2). We use this slope as a
correction to Eq. 3 when fitting the measured decay time ratio

$R(t)=R_{0}(1+at)e^{-t\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}}.$ (4)

Figure 2: Ratio of decay time acceptances between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays obtained from
simulation. The solid (blue) line shows the result of a linear fit.

Differences between the decay time resolutions of the decay modes could affect
the decay time ratio. To measure the decay time resolution, we use prompt
events containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. Such
events are found using a dimuon trigger, plus two opposite-charged tracks with
similar selection criteria as for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$) events, apart from any decay time biasing requirements
such as impact parameters and $B$ flight distance, additionally including that
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass be within
$\pm 20$ $\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) mass. To describe the decay time
distribution of these events, we use a triple Gaussian function with a common
mean, and two long lived components, modelled by exponential functions
convolved with the triple Gaussian function. The events are dominated by zero
lifetime background with the long lived components comprising less than 5% of
the events. We find the average effective decay time resolution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays to be $41.0\pm
0.9$ fs and $44.1\pm 0.2$ fs respectively, where the uncertainties are
statistical only. This difference was found not to bias the decay time ratio
using simulated experiments.

In order to determine the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime, we determine the yield of $B$ mesons for both
decay modes using unbinned maximum likelihood fits to the $B$ mass
distributions in 15 bins of decay time of equal width between 1 and 7 ps. We
perform a $\chi^{2}$ fit to the ratio of the yields as a function of decay
time and determine the relative lifetime according to Eq. 4. We obtain the
signal and peaking background shape parameters by fitting the time-integrated
dataset. In each decay time bin, we use these shapes and determine the
combinatorial background parameters from the upper mass sidebands,
$5450<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0})<5600$ MeV and
$5450<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<5550$ MeV. With this approach,
the combinatorial backgrounds are re-evaluated in each bin and we make no
assumptions on the shape of the background decay time distributions. This
method was tested with high statistics simulated experiments and found to be
unbiased.

Figure 3: Invariant mass distributions of selected (a)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ and (b)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$ candidates. The
solid (blue) curves show the total fits, the long dashed (purple) curves show
the respective $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signals, and the
dotted (gray) curve shows the combinatorial background. In (a) the short
dashed (blue-green) curve shows the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ background and the dash dotted (green) curve shows the
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{-}$ $\pi^{+}$ reflection. In (b) the short dashed (red) curve near
5370 MeV shows the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ background.

The time-integrated fits to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass spectra are shown in Fig. 3.
The signal distributions are described by the sum of two Crystal Ball
functions [25] with common means and resolutions for the Gaussian core, but
different parameters describing the tails

$f(m;\mu,\sigma,n_{l,r},\alpha_{l,r})=\begin{cases}\left(\frac{n_{l}}{|\alpha_{l}|}\right)^{n_{l}}\cdot\exp\left(\frac{-|\alpha_{l}|^{2}}{2}\right)\cdot\left(\frac{n_{l}}{|\alpha_{l}|}-|\alpha_{l}|-\frac{|m-\mu|}{\sigma}\right)^{-n_{l}},&\text{if
$\frac{m-\mu}{\sigma}\leq-\alpha_{l}$,}\\\
\left(\frac{n_{r}}{|\alpha_{r}|}\right)^{n_{r}}\cdot\exp\left(\frac{-|\alpha_{r}|^{2}}{2}\right)\cdot\left(\frac{n_{r}}{|\alpha_{r}|}-|\alpha_{r}|-\frac{|m-\mu|}{\sigma}\right)^{-n_{r}},&\text{if
$\frac{m-\mu}{\sigma}\geq\alpha_{r}$,}\\\
\exp(\frac{-(m-\mu)^{2}}{2\sigma^{2}}),&\text{otherwise,}\end{cases}$ (5)

where $\mu$ is the mean and $\sigma$ the width of the core, while $n_{l,r}$
are the exponent of the left and right tails, and $\alpha_{l,r}$ are the left
and right transition points between the core and tails. The left hand tail
accounts for final state radiation and interactions with matter, while the
right hand tail describes non-Gaussian detector effects only seen with
increased statistics. The combinatorial backgrounds are described by
exponential functions. All parameters are determined from data. There are
$4040\pm 75$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $131\,920\pm 400$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal decays. The
decay time distributions, determined using fits to the invariant mass
distributions in bins of decay time as described above, are shown in Fig. 4.
These are made by placing the fitted signal yields at the average $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay time within the
bin rather than at the centre of the decay time bin. This procedure corrects
for the exponential decrease of the decay time distributions across the bin.
The subsequent decay time ratio distribution is shown in Fig. 5, and the
fitted reciprocal lifetime difference is
$\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=-0.070\pm
0.014\,\mathrm{ps}^{-1}$, where the uncertainty is statistical only. Taking
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ to be the mean $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime $1.519\pm 0.007$ ps [21],
we determine $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}=1.700\pm 0.040\,\mathrm{ps}$.

Figure 4: Decay time distributions for (a) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and (b) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. In (b) the error
bars are smaller than the points. Figure 5: Decay time ratio between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, and the fit for
$\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}$.

Sources of systematic uncertainty on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime are investigated and listed in Table 1. We first
investigate our assumptions about the signal and combinatorial background mass
shapes. The relative change of the determined $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime between fits with double Crystal Ball functions and
double Gaussian functions for the signal models is 0.001 ps, and between fits
with exponential functions and straight lines for the combinatorial background
models is 0.010 ps. The different particle identification criteria used to
select $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\rightarrow\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow\mu^{+}\mu^{-}K^{-}\pi^{+}$
decays could affect the acceptance cancellation between the modes. In order to
investigate this effect, we loosen and tighten the particle identification
selection for the kaon, modifying the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal yield by $+2$%
and $-20$% respectively, and repeat the analysis. The larger difference with
respect to the default selection, 0.007 ps, is assigned as a systematic
uncertainty. We also assign half of the relative change between the fit
without the acceptance correction and the default fit, 0.018 ps, as a
systematic uncertainty. Potential statistical biases of our method were
evaluated with simulated experiments using similar sample sizes to those in
data. An average bias of 0.012 ps is seen and included as a systematic
uncertainty. The observed bias vanishes in simulated experiments with large
sample sizes. As a cross-check, the analysis is performed with various decay
time bin widths and fit ranges, and consistent results are obtained. The
possible $C\\!P$-even component, limited to be less than 0.6% at 95%
confidence level [24], introduces a 0.001 ps systematic uncertainty. Using the
PDG value for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime
[21] as input requires the propagation of its error as a systematic
uncertainty. All the contributions are added in quadrature and yield a total
systematic uncertainty on the lifetime of 0.026 ps (1.5%). Thus the effective
lifetime of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final
state in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, when
describing the decay time distribution as a single exponential is

$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=1.700\pm 0.040\pm
0.026\,{\rm ps}\,.$ (6)

Table 1: Summary of systematic uncertainties on the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ effective lifetime. Source | Uncertainty (ps)
---|---
Signal mass shape | $0.001$
Background mass shape | $0.010$
Kaon identification | $0.007$
Acceptance | $0.018$
Statistical bias | $0.012$
$C\\!P$-even component | $0.001$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime [21] | $0.009$
Sum in quadrature | $0.026$

Given that $\phi_{s}$ is measured to be small, and the decay is given by a
pure $b\\!\rightarrow c\overline{}cs$ tree amplitude, we may interpret the
inverse of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ effective lifetime as a measurement of $\Gamma_{\rm H}$ with
an additional source of systematic uncertainty due to a possible non-zero
value of $\phi_{s}$. For $\cos\phi_{s}=0.99$, $\Gamma_{\rm s}=0.6580$ ps-1 and
$\Delta\Gamma_{\rm s}=0.116$ ps-1 [5],
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}$ changes by 0.002
ps. This is added in quadrature to the systematic uncertainties on
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}$ to obtain the final
systematic uncertainty on $\Gamma_{\rm H}$.

In summary, the effective lifetime of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson in the $C\\!P$-odd
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final state has been
measured with respect to the well measured $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime in the final state
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. The analysis exploits the
kinematic similarities between the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays to determine
an effective lifetime of

$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=1.700\pm 0.040\pm
0.026\,\mathrm{ps},$

corresponding to a width difference of

$\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=-0.070\pm 0.014\pm
0.001\,\mathrm{ps}^{-1},$

where the uncertainties are statistical and systematic respectively. This
result is consistent with, and more precise than, the previous measurement of
$1.70\,^{+0.12}_{-0.11}\pm 0.03$ ps from CDF [26]. Interpreting this as the
lifetime of the heavy $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
eigenstate, we obtain

$\Gamma_{\rm H}=0.588\pm 0.014\pm 0.009\,\mathrm{ps}^{-1}.$

This value of $\Gamma_{\rm H}$ is consistent with the value $0.600\pm 0.013$
ps-1, calculated from the values of $\Gamma_{\rm s}$ and $\Delta\Gamma_{\rm
s}$ in Ref. [5].

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.

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# A Tunnel Diode Oscillator for High Sensitivity Broad Temperature Range
Susceptibility Measurements

A. Sirusi  J. Adams  M. Lewkowitz  R. Sun  N. S. Sullivan
<EMAIL_ADDRESS>Department of Physics, University of Florida,
Gainesville, FL32611, USA.

###### Abstract

We report the design and operation of a versatile tunnel diode oscillator for
high sensitivity ( $\approx 50$ ppb) measurements of the magnetic
susceptibility of samples over a wide temperature range (1.7-100 K) that can
be used with a simple liquid helium storage dewar. The design allows for the
application of an electric field across the sample to search for
magnetoelectric effects.

##  I Introduction

We needed a versatile and robust system for measuring magnetic
susceptibilities over a wide temperature range ( 1.7 - 100 K) with high
sensitivity. A number of tunnel diode oscillators have been built for limited
purposes, either with high sensitivity (better than 100 ppm) over a limited
range of temperatures, [1, 2] or with low sensitivity ($\geq 1000$ ppm) but
capable of functioning over a broad range of external parameters (temperature
and pressure). [3, 4, 5, 6, 7]

The design reported here maintains the tunnel diode at a fixed low temperature
(2 - 4K) while the sample is located in a separate resonant coil of Helmholtz
design that can be varied in temperature  up to 100 K. The Helmholtz design
allows the placement of electrodes across the sample (typically mm in
dimension) providing static electric fields (up to $10^{5}$ V/m). AC electric
fields applied at low frequency can be used to detect magnetoelectric effects
synchronously with the electric excitation. This feature is important as it
has been shown that the magnetoelectric effects can have a strong frequency
dependence. [8]

##  II Design Considerations

Figure 1 provides a schematic representation of the system. The tunnel diode
and its biasing circuitry is located on a small copper clad circuit board (TD
block in Fig. 1) at the lowest end of the interior cryostat and maintained at
a fixed low temperature using a hard pressed thermal contact via a spring
loaded clip to bath B containing superfluid helium. A compression force of the
order of 100 N can provide a cooling power of a few tens of mW at helium
temperatures.[9] The superfuid helium bath B is supplied by a flow limiting
impedance (Z $\approx 10^{10}$ cm-3) that is connected to a normal (4.2 K)
helium bath, and has a maximum cooling capacity at T $\approx 2$ K of $\sim$
100 mW. The volume flow ${\dot{V}}_{4}=\Delta P/(\eta Z)$ where $\eta$ is the
viscosity of the helium (in Poise) and $\Delta P\approx~{}10^{5}$ Pa is the
pressure drop across Z.[10] The filter F screens out small particles of frozen
air that might be suspended in the liquid helium and could block the impedance
Z. The entire assembly can be inserted in simple wide neck liquid helium
storage dewar and thereby keeping the helium consumption to a low value.

Figure 1:  Schematic representation of the low temperature apparatus (not to
scale). The tunnel diode is held at constant temperature by a spring contact
(red) at the bottom of the inner container which is the wall of the pumped
liquid helium bath B. The sample S is located between Helmholtz windings L
which allow for two electrodes E that apply an electric field across the
sample. The thermal board TB is maintained at a fixed temperature by means of
a feedback heater, the thermal connection to the TB block being very weak. The
outer dimension was 2.25 in and the entire assembly could be inserted into a
simple wide neck liquid helium storage dewar for ease of operation.

Maintaining the tunnel diode (General Electric type BD4)[11] at a fixed
temperature maximizes the stability of the frequency and amplitude of the
oscillations and allows one to carefully select the operating point for which
the frequency variations are a minimum with respect to bias voltage as noted
by Clover and Wolf[3]). The sample cell was located at the center of a quasi-
Helmholtz radiofrequency coil L that was thermally anchored to an independent
thermal board whose temperature could be carefully regulated by a local
heater. The sample S was contained in a small Kel-F container and thermally
bonded to the thermal board with N-grease.[12] The inductance L was resonated
with a silver-dipped mica capacitor C located on the TD block and, depending
on choice of the coil L could be operated between 2 and 120 MHz. The sample
cell block was shielded by a copper cylinder that minimizes external noise and
provides additional thermal isolation for the sample.

The fixed temperature tunnel diode block is separated from the sample cell
block by a 10 cm length of semi-rigid cryogenic coaxial cable with stainless-
steel outer conductor and silver plated beryllium-copper inner conductor[13]
that provides adequate thermal isolation to maintain the sample board at a
temperature between 1.7 K and 100 K (with a precision of better than 1 %) by
using a regulated feedback circuit, while the tunnel diode block is held at
liquid helium temperatures . The sample cell and tunnel diode block
temperatures were measured using calibrated Cernox thermometers.[14] A small
amount of 4He exchange gas was added to the sample space to provide additional
cooling.

As shown in figure 2, the open Helmholtz coil design allowed one to introduce
small electrodes to place a static electric field (up to 100 kV.m-1) across
the sample to detect magnetoelectric effects. In addition to a static field a
small AC field can be applied and used in conjuction with FM detection for
high resolution, ($\sim$ hundreds of ppb), detection of magnetoelectric
effects.

Figure 2:  Picture of Kel-F sample cell showing the Helmholtz coil pair (5 mm
diameter) and the electrodes for generating an electric field across the
sample (typically 1 mm in lateral dimension. The sample is thermalized by
applying Apiezon N-grease which makes contact with the thermal board.

##  III Performance

In order to test the sensitivity we used a commercial sample of MgB2 [15] that
had a known magnetic susceptibility change at the superconducting transition
temperature of 40.05$\pm$ 0.10 K (in good agreement with the values for the
critical temperature reported in the literature.[16] The fractional change in
oscillator frequency

$\frac{\Delta f}{f}=-\frac{1}{2}\eta\chi^{{}^{\prime}}+O(Q^{-2}\chi^{"})$ (1)

where $\eta$ is the sample filling factor and $\chi^{{}^{\prime}}$ and
$\chi^{"}$ are the real and imaginary components of the sample susceptibility,
respectively. is in good agreement with the estimated value of $\eta\sim 4$%.
The noise level in the low resolution channel was 12ppm and in the high
resolution AC channel was of the order of 0.01 ppm, comparable to the best
performance reported by Degrift[17] for a low current diode BD6 at a fixed
temperature. Similar sensitivity was reported by Van Riet and Van Gerver who
used a low temperature tunnel diode circuit to observe NMR in a micromolar
sample at 4K.[18]

Figure 3:  Frequency response measured for the magnetic susceptibility of
high-temperature superconductor MgB2. The fractional change in frequency
corresponds to a filling factor of 3.8%.

##  IV Conclusion

We have demonstrated the use of a novel cryogenic design for a tunnel diode
oscillator for measuring the magnetic susceptibility of small samples over a
wide temperature range (1.7-100 K). High sensitivity and high stability were
achieved by thermally separating the sample cell from the tunnel diode with
the latter held at fixed helium temperatures. The compact geometry allows on
to insert the complete system in a simple liquid helium dewar keeping the boil
off to less than 0.2 liquid liters per hour at the highest sample
temperatures.

###### Acknowledgements.

We would like to thank the University of Florida Cryogenics Facility and the
Machine and Electronic shops of the Physics Department for their assistance
with this project. This work was supported as part of the Center for Molecular
Magnetic Quantum Materials, an Energy Frontier Research Center funded by the
U.S. Department of Energy, Office of Science, Basic Energy Sciences under
Award # DE-SC0019330.

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