EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-046 LHCb-PAPER-2013-007 May 16, 2013

First measurement of the $C\\!P$-violating phase in
$B^{0}_{s}\rightarrow\phi\phi$ decays

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

A first flavour-tagged measurement of the time-dependent $C\\!P$-violating
asymmetry in $B^{0}_{s}\rightarrow\phi\phi$ decays is presented. In this decay
channel, the $C\\!P$-violating weak phase arises due to $C\\!P$ violation in
the interference between $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing and the $b\rightarrow
s\bar{s}s$ gluonic penguin decay amplitude. Using a sample of $pp$ collision
data corresponding to an integrated luminosity of $1.0\;\mbox{\,fb}^{-1}$ and
collected at a centre-of-mass energy of $7\;\rm TeV$ with the LHCb detector,
$880\ B^{0}_{s}\rightarrow\phi\phi$ signal decays are obtained. The
$C\\!P$-violating phase is measured to be in the interval
$\left[-2.46,-0.76\right]\;\rm rad$ at 68% confidence level. The p-value of
the Standard Model prediction is $16\%$.

Submitted to Phys. Rev. Lett.

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

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen59, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso56, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso57, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M.
Benayoun8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R.
Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani12, T.
Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11,
S. Blusk56, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53,
A. Borgia56, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van
den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton56, N.H.
Brook45, H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J.
Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G.
Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52,
K. Carvalho Akiba2, G. Casse51, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25, K. Ciba37, X.
Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D. Craik47, S. Cunliffe52, R. Currie49,
C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis59, I. De Bonis4, K. De
Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De Paula2, W. De
Silva59, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, D.
Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M.
Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A.
Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C.
Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F.
Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini56, Y. Gao3, J. Garofoli56, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel57, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O. Grünberg58, B.
Gui56, E. Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou56, G.
Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T. Hampson45, S. Hansmann-
Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann58, J. He37,
V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4,
W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D.
Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11,
E. Jans40, P. Jaton38, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, B.
Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37, I.R.
Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I.
Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L.
Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9,
M. Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38,
D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18,37, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, R. McNulty12, A. Mcnab53, B. Meadows59,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez57, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain56, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal56, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan56, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A.
Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W.
Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42,
N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35,o, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50,
B. Saitta15,d, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki56, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff59, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone56,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9,
V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß58, H. Voss10, R. Waldi58, R. Wallace12,
S. Wandernoth11, J. Wang56, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing56, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang56, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

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

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

The $B^{0}_{s}\rightarrow\phi\phi$ decay is forbidden at tree level in the
Standard Model (SM) and proceeds via a gluonic $b\rightarrow s\bar{s}s$
penguin process. Hence, this channel provides an excellent probe of new heavy
particles entering the penguin quantum loops [1, 2, 3]. Generally, $C\\!P$
violation in the SM is governed by a single phase in the Cabibbo-Kobayashi-
Maskawa quark mixing matrix [4, *Cabibbo:1963yz]. The interference between the
$B^{0}_{s}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation
and decay amplitudes leads to a $C\\!P$ asymmetry in the decay time
distributions of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons, which is characterised
by a $C\\!P$-violating weak phase. The SM predicts this phase to be small. Due
to different decay amplitudes the actual value is dependent on the $B^{0}_{s}$
decay channel. For $B^{0}_{s}\rightarrow J/\psi\phi$, which proceeds via a
$b\rightarrow c\bar{c}s$ transition, the SM prediction of the weak phase is
given by $-2\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)=-0.036\pm
0.002\;\rm rad$ [6]. The LHCb collaboration recently measured the weak phase
in this decay to be $0.068\pm 0.091(\rm stat)\pm 0.011(\rm syst)\;\rm rad$
[7], which is consistent with the SM and places stringent constraints on
$C\\!P$ violation in $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillations [8]. In the SM,
the phase in the $B^{0}_{s}\rightarrow\phi\phi$ decay, $\phi_{s}$, is expected
to be close to zero due to a cancellation of the phases arising from
$B^{0}_{s}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
oscillations and decay [9]. Calculations using QCD factorization provide an
upper limit of 0.02 $\rm\,rad$ for $|\phi_{s}|$ [1, 2, 3].

In this Letter, we present the first measurement of the $C\\!P$-violating
phase in ${B^{0}_{s}\rightarrow\phi\phi}$ decays. Charge conjugate states are
implied. The result is based on $pp$ collision data corresponding to an
integrated luminosity of $1.0\;\mbox{\,fb}^{-1}$ and collected by the LHCb
experiment in 2011 at a centre-of-mass energy of $7\;\rm TeV$. This data
sample was previously used for a time-integrated measurement of the
polarisation amplitudes and triple product asymmetries in the same decay mode
[10]. The analysis reported here improves the selection efficiency, measures
the $B^{0}_{s}$ decay time and identifies the flavour of the $B^{0}_{s}$ meson
at production. This allows a study of $C\\!P$ violation in the interference
between mixing and decay to be performed. It is necessary to disentangle the
$C\\!P$-even longitudinal ($A_{0}$), $C\\!P$-even transverse
($A_{\parallel}$), and $C\\!P$-odd transverse ($A_{\perp}$) polarisations of
the $\phi\phi$ final state by measuring the distributions of the helicity
angles [10].

The LHCb detector is a forward spectrometer at the Large Hadron Collider
covering the pseudorapidity range $2<\eta<5$ and is described in detail in
Ref. [11]. Events are selected by a hardware trigger, which selects hadron or
muon candidates with high transverse energy or momentum ($p_{\rm T}$),
followed by a two stage software trigger [12]. In the software trigger,
$B^{0}_{s}\rightarrow\phi\phi$ candidates are selected either by identifying
events containing a pair of oppositely charged kaons with an invariant mass
close to that of the $\phi$ meson or by using a topological $b$-hadron
trigger. In the simulation, $pp$ collisions are generated using Pythia 6.4
[13], with a specific LHCb configuration [14]. Decays of hadronic particles
are described by EvtGen [15] and the detector response is implemented using
the Geant4 toolkit [16, *Agostinelli:2002hh] as described in Ref. [18].

The $B^{0}_{s}\rightarrow\phi\phi$ decays are reconstructed by combining two
$\phi$ meson candidates that decay into the $K^{+}K^{-}$ final state. Kaon
candidates are required to have $p_{\rm T}>0.5$ $\rm\,GeV\\!/\\!{\it c}$, and
an impact parameter (IP) $\chi^{2}$ larger than 16 with respect to the primary
vertex (PV), where the IP $\chi^{2}$ is defined as the difference between the
$\chi^{2}$ of the PV reconstructed with and without the considered track.
Candidates must also be identified as kaons using the ring-imaging Cherenkov
detectors [19], by requiring that the difference in the global likelihood
between the kaon and pion mass hypotheses ($\Delta\ln{\cal
L}_{{\rm\mathit{K}}{\rm\mathit{\pi}}}\equiv\ln{\cal L}_{\mathit{K}}{}-\ln{\cal
L}_{\mathit{\pi}}{}$) be larger than $-5$. Both $\phi$ meson candidates must
have a reconstructed mass, $m_{K\kern-1.12003ptK}$, of the kaon pair within
$20~{}\rm\,MeV\\!/\\!{\it c}^{2}$ of the known mass of the $\phi$ meson, a
transverse momentum $(p_{\rm T}^{\phi})$ larger than 0.9 $\rm\,GeV\\!/\\!{\it
c}$ and a product, $p_{\rm T}^{\phi 1}p_{\rm T}^{\phi 2}>2\;{\rm
GeV}^{2}/c^{2}$. The $\chi^{2}$ per degree of freedom (ndf) of the vertex fit
for both $\phi$ meson candidates and the $B^{0}_{s}$ candidate is required to
be smaller than 25. Using the above criteria, $17\,575$ candidates are
selected in the invariant four-kaon mass range
$5100<m_{K\kern-1.12003ptK\kern-1.12003ptK\kern-1.12003ptK}<5600~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

A boosted decision tree (BDT) [20] is used to separate signal from background.
The six observables used as input to the BDT are: $p_{\rm T}$, $\eta$ and
$\chi^{2}/{\rm ndf}$ of the vertex fit for the $B^{0}_{s}$ candidate and the
cosine of the angle between the $B^{0}_{s}$ momentum and the direction of
flight from the closest primary vertex to the decay vertex, in addition to the
smallest $p_{\rm T}$ and the largest track $\chi^{2}/{\rm ndf}$ of the kaon
tracks. The BDT is trained using simulated $B^{0}_{s}\rightarrow\phi\phi$
signal events and background from the data where at least one of the $\phi$
candidates has invariant mass in the range $20<|{\it m}_{\it
K\kern-1.12003ptK}-{\it m}_{\phi}|<25\;\rm\,MeV\\!/\\!{\it c}^{2}$.

The sPlot technique [21, 22] is used to assign a signal weight to each
$B^{0}_{s}\rightarrow\phi\phi$ candidate. Using the four-kaon mass as the
discriminating variable, the distributions of the signal components for the
$B^{0}_{s}$ decay time and helicity angles can be determined in the data
sample. The sensitivity to $\phi_{s}$ is optimised taking into account the
signal purity and the flavour tagging performance. The final selection of
$B^{0}_{s}\rightarrow\phi\phi$ candidates based on this optimisation is
required to have a BDT output larger than 0.1, $\Delta\ln{\cal
L}_{{\rm\mathit{K}}{\rm\mathit{\pi}}}>-3$ for each kaon and $|{\it
m}_{\mathit{K}\kern-1.12003ptK}-{\mathit{m}}_{\phi}|<15\;\rm\,MeV\\!/\\!{\it
c}^{2}$ for each $\phi$ candidate.

In total, 1182 $B^{0}_{s}\rightarrow\phi\phi$ candidates are selected. Figure
1 shows the four-kaon invariant mass distribution for the selected events.
Using an unbinned extended maximum likelihood fit, a signal yield of $880\pm
31$ events is obtained. In this fit, the $B^{0}_{s}\rightarrow\phi\phi$ signal
component is modelled by two Gaussian functions with a common mean. The width
of the first Gaussian component is measured to be $12.9\pm
0.5\;\rm\,MeV\\!/\\!{\it c}^{2}$, in agreement with the expectation from
simulation. The relative fraction and width of the second Gaussian component
are fixed from simulation to values of 0.785 and
29.5${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, in order to
ensure a good quality fit. Combinatorial background is modelled using an
exponential function which is allowed to vary in the fit. Contributions from
specific backgrounds such as $B^{0}\rightarrow\phi K^{*0}$, where
$K^{*0}\rightarrow K^{+}\pi^{-}$, are found to be negligible.

Figure 1: Invariant $K^{+}K^{-}K^{+}K^{-}$ mass distribution for selected
$B^{0}_{s}\rightarrow\phi\phi$ candidates. The total fit (solid line) consists
of a double Gaussian signal component together with an exponential background
(dotted line).

An unbinned maximum likelihood fit is performed to the decay time, $t$, and
the three helicity angles, $\Omega=\\{\theta_{1},\theta_{2},\Phi\\}$, of the
selected $B^{0}_{s}\rightarrow\phi\phi$ candidates, each of which is re-
assigned a signal sPlot weight based on the four-kaon invariant mass,
$m_{K\kern-1.12003ptK\kern-1.12003ptK\kern-1.12003ptK}$[21, 22]. The
probability density function (PDF) consists of signal components, which
include detector resolution and acceptance effects, and are factorised into
separate terms for the decay time and the angular observables.

The $B^{0}_{s}$ decay into the $K^{+}K^{-}K^{+}K^{-}$ final state can proceed
via combinations of intermediate vector ($\phi$) and scalar ($f_{0}(980)$)
resonances and scalar non-resonant $K^{+}K^{-}$ pairs. Thus the total decay
amplitude is a coherent sum of $P$-wave (vector-vector), $S$-wave (vector-
scalar) and $SS$-wave (scalar-scalar) contributions. The differential decay
rate of the decay time and helicity angles is described by a sum of 15 terms,
corresponding to five polarisation amplitudes and their interference terms,

$\frac{\mathrm{d}^{4}\Gamma}{\mathrm{d}\negthinspace\cos\theta_{1}\,\mathrm{d}\negthinspace\cos\theta_{2}\,\mathrm{d}\Phi\,\mathrm{d}t}\propto\sum^{15}_{i=1}K_{i}(t)f_{i}(\Omega)\,.$
(1)

The angular functions $f_{i}(\Omega)$ for the $P$-wave terms are derived in
Ref. [23] and the helicity angles of the two $\phi$ mesons are randomly
assigned to $\theta_{1}$ and $\theta_{2}$. The time-dependent functions
$K_{i}(t)$ can be written as [23]

$K_{i}(t)=N_{i}e^{-\Gamma_{s}t}[c_{i}\cos(\Delta m_{s}t)+d_{i}\sin(\Delta
m_{s}t)+a_{i}\cosh(\tfrac{1}{2}\Delta\Gamma_{s}t)+b_{i}\sinh(\tfrac{1}{2}\Delta\Gamma_{s}t)],$
(2)

where $\Delta\Gamma_{s}=\Gamma_{\rm L}-\Gamma_{\rm H}$ is the decay width
difference between the light (L) and heavy (H) $B^{0}_{s}$ mass eigenstates,
$\Gamma_{s}$ is the average decay width, $\Gamma_{s}=(\Gamma_{\rm
L}+\Gamma_{\rm H})/2$, and $\Delta m_{s}$ is the $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ oscillation frequency. The
coefficients $N_{i}$, $a_{i}$, $b_{i}$, $c_{i}$ and $d_{i}$ can be expressed
in terms of $\phi_{s}$ and the magnitudes, $|A_{i}|$, and phases,
$\delta_{i}$, of the five polarisation amplitudes at $t=0$. The three $P$-wave
amplitudes, denoted by $A_{0},A_{\parallel},A_{\perp}$, are normalised such
that $|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2}=1$, with the strong
phases $\delta_{1}$ and $\delta_{2}$ defined as
$\delta_{1}=\delta_{\perp}-\delta_{\parallel}$ and
$\delta_{2}=\delta_{\perp}-\delta_{0}$. The $S$ and $SS$-wave amplitudes and
their corresponding phases are denoted by $A_{S}$, $A_{SS}$ and $\delta_{S}$,
$\delta_{SS}$, respectively. For a $B^{0}_{s}$ meson produced at $t=0$, the
coefficients in Eq. 2 and the angular functions
$f_{i}(\theta_{1},\theta_{2},\Phi)$ are given in Table 1, where
$\delta_{2,1}=\delta_{2}-\delta_{1}$.

Table 1: Coefficients of the time-dependent terms and angular functions
defined in Eqs. 1 and 2. Amplitudes are defined at $t=0$.
$\begin{array}[]{c|c|c|c|c|c|c}i&N_{i}&a_{i}&b_{i}&c_{i}&d_{i}&f_{i}\\\
\hline\cr
1&|A_{0}|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}&4\cos^{2}\theta_{1}\cos^{2}\theta_{2}\\\
2&|A_{\parallel}|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}&\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{+}\cos
2\Phi)\\\
3&|A_{\perp}|^{2}&1&\cos\phi_{s}&0&-\sin\phi_{s}&\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{-}\cos
2\Phi)\\\
4&|A_{\parallel}||A_{\perp}|&0&-\cos\delta_{1}\sin\phi_{s}&\sin\delta_{1}&-\cos\delta_{1}\cos\phi_{s}&-2\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin
2\Phi\\\
5&|A_{\parallel}||A_{0}|&\cos(\delta_{2,1})&-\cos(\delta_{2,1})\cos\phi_{s}&0&\cos(\delta_{2,1})\sin\phi_{s}&\sqrt{2}\sin
2\theta_{1}\sin 2\theta_{2}\cos\Phi\\\
6&|A_{0}||A_{\perp}|&0&-\cos\delta_{2}\sin\phi_{s}&\sin\delta_{2}&-\cos\delta_{2}\cos\phi_{s}&-\sqrt{2}\sin
2\theta_{1}\sin 2\theta_{2}\sin\Phi\\\
7&|A_{SS}|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}&\frac{4}{9}\\\
8&|A_{S}|^{2}&1&\cos\phi_{s}&0&-\sin\phi_{s}&\frac{4}{3}(\cos\theta_{1}+\cos\theta_{2})^{2}\\\
9&|A_{S}||A_{SS}|&0&\sin(\delta_{S}{-}\delta_{SS})\sin\phi_{s}&\cos(\delta_{SS}{-}\delta_{S})&\sin(\delta_{SS}{-}\delta_{S})\cos\phi_{s}&\frac{8}{3\sqrt{3}}(\cos\theta_{1}+\cos\theta_{2})\\\
10&|A_{0}||A_{SS}|&\cos\delta_{SS}&-\cos\delta_{SS}\cos\phi_{s}&0&\cos\delta_{SS}\sin\phi_{s}&\frac{8}{3}\cos\theta_{1}\cos\theta_{2}\\\
11&|A_{\parallel}||A_{SS}|&\cos(\delta_{2,1}{-}\delta_{SS})&-\cos(\delta_{2,1}{-}\delta_{SS})\cos\phi_{s}&0&\cos(\delta_{2,1}{-}\delta_{SS})\sin\phi_{s}&\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\cos\Phi\\\
12&|A_{\perp}||A_{SS}|&0&-\cos(\delta_{2}-\delta_{SS})\sin\phi_{s}&\sin(\delta_{2}{-}\delta_{SS})&-\cos(\delta_{2}{-}\delta_{SS})\cos\phi_{s}&-\frac{4\sqrt{2}}{3}\sin\theta_{1}\sin\theta_{2}\sin\Phi\\\
\hbox{\multirowsetup$13$}&\hbox{\multirowsetup$|A_{0}||A_{S}|$}&\hbox{\multirowsetup$0$}&\hbox{\multirowsetup$-\sin\delta_{S}\sin\phi_{s}$}&\hbox{\multirowsetup$\cos\delta_{S}$}&\hbox{\multirowsetup
$-\sin\delta_{S}\cos\phi_{s}$}&\frac{8}{\sqrt{3}}\cos\theta_{1}\cos\theta_{2}\\\
&&&&&&\times(\cos\theta_{1}+\cos\theta_{2})\\\
\hbox{\multirowsetup$14$}&\hbox{\multirowsetup$|A_{\parallel}||A_{S}|$}&\hbox{\multirowsetup$0$}&\hbox{\multirowsetup$\sin(\delta_{2,1}-\delta_{S})\sin\phi_{s}$}&\hbox{\multirowsetup$\cos(\delta_{2,1}{-}\delta_{S})$}&\hbox{\multirowsetup$\sin(\delta_{2,1}-\delta_{S})\cos\phi_{s}$}&\frac{4\sqrt{2}}{\sqrt{3}}\sin\theta_{1}\sin\theta_{2}\\\
&&&&&&\times(\cos\theta_{1}+\cos\theta_{2})\cos\Phi\\\
\hbox{\multirowsetup$15$}&\hbox{\multirowsetup$|A_{\perp}||A_{S}|$}&\hbox{\multirowsetup$\sin(\delta_{2}-\delta_{S})$}&\hbox{\multirowsetup$\sin(\delta_{2}-\delta_{S})\cos\phi_{s}$}&\hbox{\multirowsetup
0}&\hbox{\multirowsetup$-\sin(\delta_{2}-\delta_{S})\sin\phi_{s}$}&-\frac{4\sqrt{2}}{\sqrt{3}}\sin\theta_{1}\sin\theta_{2}\\\
&&&&&&\times(\cos\theta_{1}+\cos\theta_{2})\sin\Phi\\\ \end{array}$

Assuming that $C\\!P$ violation in mixing and direct $C\\!P$ violation are
negligible, the differential distribution for a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson is obtained by changing
the sign of the coefficients $c_{i}$ and $d_{i}$. The PDF is invariant under
the transformation
$(\phi_{s},\Delta\Gamma_{s},\delta_{\parallel},\delta_{\perp},\delta_{S},\delta_{SS})\rightarrow(\pi-\phi_{s},-\Delta\Gamma_{s},-\delta_{\parallel},\pi-\delta_{\perp},-\delta_{S},-\delta_{SS})$.
This two-fold ambiguity is resolved in the fit as Gaussian constraints are
applied for the $B^{0}_{s}$ average decay width and decay width difference to
the values measured in $B^{0}_{s}\rightarrow J/\psi\phi$ decays,
$\Gamma_{s}=0.663\pm 0.008\;\rm ps^{-1}$ and $\Delta\Gamma_{s}=0.100\pm
0.017\;\rm ps^{-1}$, with a correlation coefficient
$\rho(\Delta\Gamma_{s},\Gamma_{s})=-0.39$ [7]. Similarly, the $B^{0}_{s}$
oscillation frequency $\Delta m_{s}$ is constrained to the value $\Delta
m_{s}=17.73\pm 0.05\;\rm ps^{-1}$ [24].

A correction factor is multiplied to the interference terms in Table 1 between
the $P$ and $S$-wave (and the $P$ and $SS$-wave) contributions to account for
the finite $m_{KK}$ mass window considered in the amplitude integration. This
factor is calculated from the interference between the different $m_{KK}$
lineshapes of the vector and scalar contributions. The validity of the fit
model has been extensively tested using simulated data samples.

The acceptance as a function of the helicity angles is not completely uniform
due to the forward geometry of the detector and the momentum cuts placed to
the final state particles. A three-dimensional acceptance function is
determined using simulation. The acceptance factors are included in the fit as
a normalisation of the PDF for each of the angular terms. The acceptance
function varies by less than 20% across the phase-space.

The event reconstruction, trigger and offline selections introduce a decay
time dependent acceptance. In particular for short decay times, the acceptance
vanishes due to the trigger, which requires tracks with significant
displacement from any PV. Therefore, the decay time acceptance is determined
using simulation and incorporated by multiplying the signal PDF with a binned
acceptance histogram. The fractions of different triggers are found to be in
agreement between data and simulation.

The parameters of a double Gaussian function used to model the decay time
resolution are determined from simulation studies. A single Gaussian function
with a resolution of $40\,\rm\,fs$ is found to have a similar effect on
physics parameters and is applied to the data fit.

The $\phi_{s}$ measurement requires that the meson flavour be tagged as either
a $B^{0}_{s}$ or $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson
at production. To achieve this, both the opposite side (OS) and same side kaon
(SSK) flavour tagging methods are used [25, 26]. In OS tagging the
$\bar{b}$-quark hadron produced in association with the signal $b$-quark is
exploited through the charge of a muon or electron produced in semileptonic
decays, the charge of a kaon from a subsequent charmed hadron decay, and the
momentum-weighted charge of all tracks in an inclusively reconstructed decay
vertex. The SSK tagging makes use of kaons formed from the $s$-quark produced
in association with the $B^{0}_{s}$ meson. The kaon charge identifies the
flavour of the signal $B^{0}_{s}$ meson.

The event-by-event mistag is the probability that the decision of the tagging
algorithm is incorrect and is determined by a neural network trained on
simulated events and calibrated with control samples [25]. The value of the
event-by-event mistag is used in the fit as an observable and the
uncertainties on the calibration parameters are propagated to the statistical
uncertainties of the physics parameters, following the procedure described in
Ref. [7]. For events tagged by both the OS and SSK methods, a combined tagging
decision is made. The total tagging power is
$\varepsilon_{tag}\mathcal{D}^{2}=(3.29\pm 0.48)\%$, with a tagging efficiency
of $\varepsilon_{tag}=(49.7\pm 5.0)\%$ and a dilution
$\mathcal{D}=(1-2\omega)$ where $\omega$ is the average mistag probability.
Untagged events are included in the analysis as they increase the sensitivity
to $\phi_{s}$ through the $b_{i}$ terms in Eq. 2.

The total $S$-wave fraction is determined to be $(1.6\,^{+2.4}_{-1.2})\%$
where the double $S$-wave contribution $A_{SS}$ is set to zero, since the fit
shows little sensitivity to $A_{SS}$. A fit to the two-dimensional mass,
$m_{K\kern-1.12003ptK}$, for both kaon pairs, where background is subtracted
using sidebands is performed and yields a consistent $S$-wave fraction of
$(2.1\pm 1.2)\%$.

The results of the fit for the main observables are shown in Table 2. Figure 2
shows the distributions for the decay time and helicity angles with the
projections for the best fit PDF overlaid. The likelihood profile for the
$C\\!P$-violating weak phase $\phi_{s}$, shown in Fig. 3, is not parabolic. To
obtain a confidence level a correction is applied due to a small under-
coverage of the likelihood profile using the method described in Ref. [27].
Including systematic uncertainties (discussed below) and assuming the values
of the polarisation amplitudes and strong phases observed in data, an interval
of $\left[-2.46,-0.76\right]\;\rm rad$ at $68\%$ confidence level is obtained
for $\phi_{s}$. The polarisation amplitudes and phases, shown in Table 2,
differ from those reported in Ref. [10] as $\phi_{s}$ is not constrained to
zero.

Table 2: Fit results with statistical and systematic uncertainties. A 68%
statistical confidence interval is quoted for $\phi_{s}$. Amplitudes are
defined at $t=0$.

Parameter Value $\sigma_{\text{stat.}}$ $\sigma_{\text{syst.}}$
$\phi_{s}$[rad] (68 % CL) $[-2.37,-0.92]$ 0.22 $|A_{0}|^{2}$ 0.329 0.033 0.017
$|A_{\perp}|^{2}$ 0.358 0.046 0.018 $|A_{\mathrm{S}}|^{2}$ 0.016
${}^{+0.024}_{-0.012}$ 0.009 $\delta_{1}$ [rad] 2.19 0.44 0.12 $\delta_{2}$
[rad] $-1.47$ 0.48 0.10 $\delta_{\mathrm{S}}$ [rad] 0.65 $~{}^{+0.89}_{-1.65}$
0.33

Figure 2: One-dimensional projections of the $B^{0}_{s}\rightarrow\phi\phi$
fit for (a) decay time, (b) helicity angle $\Phi$ and the cosine of the
helicity angles (c) $\theta_{1}$ and (d) $\theta_{2}$. The data are marked as
points, while the solid lines represent the projections of the best fit. The
$C\\!P$-even $P$-wave, the $C\\!P$-odd $P$-wave and $S$-wave components are
shown by the long dashed, short dashed and dotted lines, respectively.

The uncertainties related to the calibration of the tagging and the assumed
values of $\Gamma_{s}$, $\Delta\Gamma_{s}$ and $\Delta m_{s}$ are absorbed
into the statistical uncertainty, described above. Systematic uncertainties
are determined and the sum in quadrature of all sources is reported in Table 2
for each observable. To check that the background is properly accounted for,
an additional fit is performed where the angular and time distributions are
parameterised using the $B^{0}_{s}$ mass sidebands. This gives results in
agreement with those presented here and no further systematic uncertainty is
assigned. The uncertainty due to the modelling of the $S$-wave component is
evaluated by allowing the $SS$-wave component to vary in the fit. The
difference between the two fits leads to the dominant uncertainty on
$\phi_{s}$ of 0.20 $\rm\,rad$. The systematic uncertainty due to the decay
time acceptance is found by taking the difference in the values of fitted
parameters between the nominal fit, using a binned time acceptance, and a fit
in which the time acceptance is explicitely paramaterised. This is found to be
0.09 $\rm\,rad$ for $\phi_{s}$. Possible differences in the simulated decay
time resolution compared to the data are studied by varying the resolution
according to the discrepancies observed in the $B^{0}_{s}\rightarrow
J/\psi\phi$ analysis [7]. This leads to a systematic uncertainty of 0.01
$\rm\,rad$ for $\phi_{s}$. The distributions of maximum $p_{\rm T}$ and
$\chi^{2}/{\rm ndf}$ of the final state tracks and the $p_{\rm T}$ and $\eta$
of the $B^{0}_{s}$ candidate are reweighted to better match the data. From
this, the angular acceptance is recalculated, leading to small changes in the
results (0.02 $\rm\,rad$ for $\phi_{s}$), which are assigned as systematic
uncertainty. Biases in the fit method are studied using simulated pseudo-
experiments that lead to an uncertainty of 0.02 $\rm\,rad$ for $\phi_{s}$.
Further small systematic uncertainties (0.02 $\rm\,rad$ for $\phi_{s}$) are
due to the limited number of events in the simulation sample used for the
determination of the angular acceptance and to the choice of a single versus a
double Gaussian function for the mass PDF, which is used to assign the signal
weights. The total systematic uncertainty on $\phi_{s}$ is 0.22 $\rm\,rad$,
significantly smaller than the statistical uncertainty.

Figure 3: Negative $\Delta$ln likelihood scan of $\phi_{s}$. Only the
statistical uncertainty is included.

In summary, we present the first study of $C\\!P$ violation in the decay time
distribution of hadronic $B^{0}_{s}\rightarrow\phi\phi$ decays. The
$C\\!P$-violating phase, $\phi_{s}$, is restricted to the interval of
$\left[-2.46,-0.76\right]\;\rm rad$ at 68% C.L. The p-value of the Standard
Model prediction [9] is 16%, taking the values of the strong phases and
polarisation amplitudes observed in data and assuming that systematic
uncertainties are negligible. The precision of the $\phi_{s}$ measurement is
dominated by the statistical uncertainty and is expected to improve with
larger LHCb data sets.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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