EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-078 LHCb-PAPER-2013-017 June 19, 2013

Differential branching fraction

and angular analysis of

the decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$

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

The determination of the differential branching fraction and the first angular
analysis of the decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ are
presented using data, corresponding to an integrated luminosity of
$1.0\mbox{\,fb}^{-1}$, collected by the LHCb experiment at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The differential branching fraction
is determined in bins of $q^{2}$, the invariant dimuon mass squared.
Integration over the full $q^{2}$ range yields a total branching fraction of
${\cal
B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})=\left(7.07\,^{+0.64}_{-0.59}\pm
0.17\pm 0.71\right)\times 10^{-7}$, where the first uncertainty is
statistical, the second systematic, and the third originates from the
branching fraction of the normalisation channel. An angular analysis is
performed to determine the angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$,
and $A_{9}$. The observables are consistent with Standard Model expectations.

Submitted to JHEP

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

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso59, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S.
Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van
Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M.
Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58, V. Bocci24, A.
Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V.
Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J.
Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51,
I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M. Charles54, Ph.
Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz25, K. Ciba37, X. Cid
Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J.
Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S. Cunliffe52, R.
Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De
Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11,
F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A.
Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34,
A. Dziurda25, A. Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S.
Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L.
Eklund50,37, I. El Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C.
Färber11, G. Fardell49, C. Farinelli40, S. Farry51, V. Fave38, D. Ferguson49,
V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32,
M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, 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. Göbel59, D. Golubkov30, A.
Golutvin52,30,37, A. Gomes2, H. Gordon54, M. Grabalosa Gándara5, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening54, S. Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E.
Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C.
Haen37, S.C. Haines46, S. Hall52, T. Hampson45, S. Hansmann-Menzemer11, N.
Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, V. Heijne40,
K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, M. Holtrop40, 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, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, B. Mc Skelly51, J. McCarthy44, A. McNab53, R. McNulty12, B.
Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N.
Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, M.J.
Morello22,s, R. Mountain58, 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. Oyanguren35,o, B.K. Pal58, 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. Phan58, E. Picatoste Olloqui35,
B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F.
Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10,
B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51,
C. Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35,o, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50,
B. Saitta15,d, V. Salustino Guimaraes2, C. Salzmann39, B. Sanmartin Sedes36,
M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k,
M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D.
Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M.
Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R.
Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K.
Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34,
I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L.
Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva
Coutinho47, T. Skwarnicki58, N.A. Smith51, E. Smith54,48, M. Smith53, M.D.
Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45, B. Souza De Paula2, B.
Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone58, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K.
Subbiah37, L. Sun56, S. Swientek9, V. Syropoulos41, M. Szczekowski27, P.
Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F.
Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6,
P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36,
S. Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37,
B. Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D.
Volyanskyy10, D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10,
R. Waldi60, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wishahi9, M. Witek25, S.A.
Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, F. Xing54, Z. Xing58,
Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
A. Zhokhov30, L. Zhong3, A. Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

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

##  1 Introduction

The $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ ($\phi\\!\rightarrow
K^{+}K^{-}$) decay111The inclusion of charge conjugated processes is implied
throughout this paper. involves a $b\rightarrow s$ quark transition and
therefore constitutes a flavour changing neutral current (FCNC) process. Since
FCNC processes are forbidden at tree level in the Standard Model (SM), the
decay is mediated by higher order (box and penguin) diagrams. In scenarios
beyond the SM new particles can affect both the branching fraction of the
decay and the angular distributions of the decay products.

The angular configuration of the $K^{+}K^{-}\mu^{+}\mu^{-}$ system is defined
by the decay angles $\theta_{K}$, $\theta_{\ell}$, and $\Phi$. Here,
$\theta_{K}$ ($\theta_{\ell}$) denotes the angle of the $K^{-}$ ($\mu^{-}$)
with respect to the direction of flight of the $B^{0}_{s}$ meson in the
$K^{+}K^{-}$ ($\mu^{+}\mu^{-}$) centre-of-mass frame, and $\Phi$ denotes the
relative angle of the $\mu^{+}\mu^{-}$ and the $K^{+}K^{-}$ decay planes in
the $B^{0}_{s}$ meson centre-of-mass frame [1]. In contrast to the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$, the final state of the decay
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ is not flavour specific. The
differential decay rate, depending on the decay angles and the invariant mass
squared of the dimuon system is given by

$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{4}\Gamma}{{\rm d}q^{2}{\rm d}\cos{\theta_{\ell}}{\rm
d}\cos{\theta_{K}}{\rm d}\Phi}=\frac{9}{32\pi}$
$\displaystyle\bigl{[}S_{1}^{s}\sin^{2}\theta_{K}+S_{1}^{c}\cos^{2}\theta_{K}$
$\displaystyle+S_{2}^{s}\sin^{2}\theta_{K}\cos
2\theta_{\ell}+S_{2}^{c}\cos^{2}\theta_{K}\cos 2\theta_{\ell}$
$\displaystyle+{S_{3}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\Phi+{S_{4}}\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\Phi$
$\displaystyle+{A_{5}}\sin
2\theta_{K}\sin\theta_{\ell}\cos\Phi+{A_{6}}\sin^{2}\theta_{K}\cos\theta_{\ell}$
$\displaystyle+{S_{7}}\sin 2\theta_{K}\sin\theta_{\ell}\sin\Phi+{A_{8}}\sin
2\theta_{K}\sin 2\theta_{\ell}\sin\Phi$
$\displaystyle+{A_{9}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\Phi\bigr{]},$ (1)

where equal numbers of produced $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are assumed [2]. The
$q^{2}$-dependent angular observables $S_{i}^{(s,c)}$ and $A_{i}$ correspond
to $C\\!P$ averages and $C\\!P$ asymmetries, respectively. Integrating Eq. 1
over two angles, under the assumption of massless leptons, results in three
distributions, each depending on one decay angle

$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\cos\theta_{K}}$
$\displaystyle=\frac{3}{4}(1-{F_{\rm
L}})(1-\cos^{2}\theta_{K})+\frac{3}{2}{F_{\rm L}}\cos^{2}\theta_{K},$ (2)
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\cos\theta_{\ell}}$
$\displaystyle=\frac{3}{8}(1-{F_{\rm
L}})(1+\cos^{2}\theta_{\ell})+\frac{3}{4}{F_{\rm
L}}(1-\cos^{2}\theta_{\ell})+\frac{3}{4}{A_{6}}\cos\theta_{\ell},$ (3)
$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\Phi}$
$\displaystyle=\frac{1}{2\pi}+\frac{1}{2\pi}{S_{3}}\cos
2\Phi+\frac{1}{2\pi}{A_{9}}\sin 2\Phi,$ (4)

which retain sensitivity to the angular observables $F_{\rm
L}(=S_{1}^{c}=-S_{2}^{c})$, $S_{3}$, $A_{6}$, and $A_{9}$. Of particular
interest is the $T$-odd asymmetry $A_{9}$ where possible large
$C\\!P$-violating phases from contributions beyond the SM would not be
suppressed by small strong phases [1].

This paper presents a measurement of the differential branching fraction and
the angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ in six bins
of $q^{2}$. In addition, the total branching fraction is determined. The data
used in the analysis were recorded by the LHCb experiment in 2011 in $pp$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and correspond to an
integrated luminosity of $1.0\mbox{\,fb}^{-1}$.

##  2 The LHCb detector

The LHCb detector [3] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
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.
The LHCb trigger system [4] 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.

Simulated signal event samples are generated to determine the trigger,
reconstruction and selection efficiencies. Exclusive samples are analysed to
estimate possible backgrounds. The simulation generates $pp$ collisions using
Pythia 6.4 [5] with a specific LHCb configuration [6]. Decays of hadronic
particles are described by EvtGen [7] in which final state radiation is
generated using Photos [8]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [9,
*Agostinelli:2002hh] as described in Ref. [11]. Data driven corrections are
applied to the simulated events to account for differences between data and
simulation. These include the IP resolution, tracking efficiency, and particle
identification performance. In addition, simulated events are reweighted
depending on the transverse momentum ($p_{\rm T}$) of the $B^{0}_{s}$ meson,
the vertex fit quality, and the track multiplicity to match distributions of
control samples from data.

##  3 Selection of signal candidates

Signal candidates are accepted if they are triggered by particles of the
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ ($\phi\\!\rightarrow K^{+}K^{-}$)
final state. The hardware trigger requires either a high transverse momentum
muon or muon pair, or a high transverse energy ($E_{\rm T}$) hadron. The first
stage of the software trigger selects events containing a muon (or hadron)
with $\mbox{$p_{\rm T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
($\mbox{$E_{\rm T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and a minimum
IP with respect to all primary interaction vertices in the event of
$80\,\upmu\rm m$ ($125\,\upmu\rm m$). In the second stage of the software
trigger the tracks of two or more final state particles are required to form a
vertex that is significantly displaced from all primary vertices (PVs) in the
event.

Candidates are selected if they pass a loose preselection that requires the
kaon and muon tracks to have a large $\chi^{2}_{\rm IP}$ ($>9$) with respect
to the PV. The $\chi^{2}_{\rm IP}$ is defined as the difference between the
$\chi^{2}$ of the PV reconstructed with and without the considered particle.
The four tracks forming a $B^{0}_{s}$ candidate are fit to a common vertex,
which is required to be of good quality ($\chi^{2}_{\rm vtx}<30$) and well
separated from the PV ($\chi^{2}_{\rm FD}>121$, where FD denotes the flight
distance). The angle between the $B^{0}_{s}$ momentum vector and the vector
connecting the PV with the $B^{0}_{s}$ decay vertex is required to be small.
Furthermore, $B^{0}_{s}$ candidates are required to have a small IP with
respect to the PV ($\chi^{2}_{\rm IP}<16$). The invariant mass of the
$K^{+}K^{-}$ system is required to be within
$12{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\phi$ mass [12].

To further reject combinatorial background events, a boosted decision tree
(BDT) [13] using the AdaBoost algorithm [14] is applied. The BDT training uses
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
$({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-})$
candidates as proxy for the signal, and candidates in the
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ mass sidebands
($5100<m(K^{+}K^{-}\mu^{+}\mu^{-})<5166{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and
$5566<m(K^{+}K^{-}\mu^{+}\mu^{-})<5800{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
as background. The input variables of the BDT are the $\chi^{2}_{\rm IP}$ of
all final state tracks and of the $B^{0}_{s}$ candidate, the angle between the
$B^{0}_{s}$ momentum vector and the vector between PV and $B^{0}_{s}$ decay
vertex, the vertex fit $\chi^{2}$, the flight distance significance and
transverse momentum of the $B^{0}_{s}$ candidate, and particle identification
information of the muons and kaons in the final state.

Several types of $b$-hadron decays can mimic the final state of the signal
decay and constitute potential sources of peaking background. The resonant
decays $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ and $B^{0}_{s}\\!\rightarrow\psi{(2S)}\phi$ with
$\psi{(2S)}\\!\rightarrow\mu^{+}\mu^{-}$ are rejected by applying vetoes on
the dimuon mass regions around the charmonium resonances,
$2946<m(\mu^{+}\mu^{-})<3176{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$3592<m(\mu^{+}\mu^{-})<3766{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To
account for the radiative tails of the charmonium resonances the vetoes are
enlarged by $200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to lower
$m(\mu^{+}\mu^{-})$ for reconstructed $B^{0}_{s}$ masses below
$5316{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In the region
$5416<m(B^{0}_{s})<5566{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ the vetoes
are extended by $50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to higher
$m(\mu^{+}\mu^{-})$ to reject a small fraction of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ decays that
are misreconstructed at higher masses. The decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ ($K^{*0}\\!\rightarrow K^{+}\pi^{-}$) can be
reconstructed as signal if the pion is misidentified as a kaon. This
background is strongly suppressed by particle identification criteria. In the
narrow $\phi$ mass window, $2.4\pm 0.5$ misidentified $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ candidates are expected within $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $B^{0}_{s}$ mass of
$5366{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [12]. The resonant decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ can
also constitute a source of peaking background if the $K^{+}$ ($K^{-}$) is
misidentified as $\mu^{+}$ ($\mu^{-}$) and vice versa. Similarly, the decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
($K^{*0}\rightarrow K^{+}\pi^{-}$) where the $\pi^{-}$ ($\mu^{-}$) is
misidentified as $\mu^{-}$ ($K^{-}$) can mimic the signal decay. These
backgrounds are rejected by requiring that the invariant mass of the
$K^{+}\mu^{-}$ ($K^{-}\mu^{+}$) system, with kaons reconstructed under the
muon mass hypothesis, is not within $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass of
$3096{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [12], unless both the kaon and
the muon pass stringent particle identification criteria. The expected number
of background events from double misidentification in the $B^{0}_{s}$ signal
mass region is $0.9\pm 0.5$. All other backgrounds studied, including
semileptonic $b\rightarrow c\,\mu^{-}\bar{\nu}_{\mu}(c\rightarrow
s\,\mu^{+}\nu_{\mu})$ cascades, hadronic double misidentification from
$B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}(D^{-}_{s}\rightarrow\phi\pi^{-})$, and
the decay $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax(1520)\,\mu^{+}\mu^{-}$, have been found to be negligible.

##  4 Differential branching fraction

Figure 1 shows the $\mu^{+}\mu^{-}$ versus the $K^{+}K^{-}\mu^{+}\mu^{-}$
invariant mass of the selected candidates. The signal decay
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ is clearly visible in the
$B^{0}_{s}$ signal region. The determination of the differential branching
fraction is performed in six bins of $q^{2}$, given in Table 1, and
corresponds to the binning chosen for the analysis of the decay
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ [15]. Figure 2 shows the
$K^{+}K^{-}\mu^{+}\mu^{-}$ mass distribution in the six $q^{2}$ bins. The
signal yields are determined by extended unbinned maximum likelihood fits to
the reconstructed $B^{0}_{s}$ mass distributions. The signal component is
modeled by a double Gaussian function. The resolution parameters are obtained
from the resonant
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decay. A $q^{2}$-dependent scaling factor, determined with simulated
$B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ events, is introduced to account
for the observed $q^{2}$ dependence of the mass resolution. The combinatorial
background is described by a single exponential function. The veto of the
radiative tails of the charmonium resonances is accounted for by using a scale
factor. The resulting signal yields are given in Table 1. Fitting for the
signal yield over the full $q^{2}$ region, $174\pm 15$ signal candidates are
found. A fit of the normalisation mode
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ yields
$(20.36\pm 0.14)\times 10^{3}$ candidates.

Figure 1: Invariant $\mu^{+}\mu^{-}$ versus $K^{+}K^{-}\mu^{+}\mu^{-}$ mass.
The charmonium vetoes are indicated by the solid lines. The vertical dashed
lines indicate the signal region of $\pm
50{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ around the known $B^{0}_{s}$ mass
in which the signal decay $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ is
visible.

Figure 2: Invariant mass of $B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ candidates in six bins of invariant dimuon mass squared. The fitted signal component is denoted by the light blue shaded area, the combinatorial background component by the dark red shaded area. The solid line indicates the sum of the signal and background components.  Table 1: Signal yield and differential branching fraction ${\rm d}{\cal B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})/{\rm d}q^{2}$ in six bins of $q^{2}$. Results are also quoted for the region $1<q^{2}<6{\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ where theoretical predictions are most reliable. The first uncertainty is statistical, the second systematic, and the third from the branching fraction of the normalisation channel. $q^{2}$ bin $({\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})$ | $N_{\rm sig}$ | ${\rm d}{\cal B}/{\rm d}q^{2}~{}(10^{-8}\mathrm{\,Ge\kern-1.00006ptV}^{-2}c^{4})$
---|---|---
$0.10<q^{2}<\parbox{0.0pt}{2.00}$ | $25.0\,^{+5.8}_{-5.2}$ | $4.72\,^{+1.09}_{-0.98}\pm 0.20\pm 0.47$
$2.00<q^{2}<\parbox{0.0pt}{4.30}$ | $14.3\,^{+4.9}_{-4.3}$ | $2.30\,^{+0.79}_{-0.69}\pm 0.11\pm 0.23$
$4.30<q^{2}<\parbox{0.0pt}{8.68}$ | $41.2\,^{+7.5}_{-7.0}$ | $3.15\,^{+0.58}_{-0.53}\pm 0.12\pm 0.31$
$10.09<q^{2}<\parbox{0.0pt}{12.90}$ | $40.7\,^{+7.7}_{-7.2}$ | $4.26\,^{+0.81}_{-0.75}\pm 0.26\pm 0.43$
$14.18<q^{2}<\parbox{0.0pt}{16.00}$ | $23.8\,^{+5.9}_{-5.3}$ | $4.17\,^{+1.04}_{-0.93}\pm 0.24\pm 0.42$
$16.00<q^{2}<\parbox{0.0pt}{19.00}$ | $26.6\,^{+5.7}_{-5.3}$ | $3.52\,^{+0.76}_{-0.70}\pm 0.20\pm 0.35$
$1.00<q^{2}<\parbox{0.0pt}{6.00}$ | $31.4\,^{+7.0}_{-6.3}$ | $2.27\,^{+0.50}_{-0.46}\pm 0.11\pm 0.23$

The differential branching fraction of the signal decay in the $q^{2}$
interval spanning from $q^{2}_{\rm min}$ to $q^{2}_{\rm max}$ is calculated
according to

$\displaystyle\frac{{\rm d}{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})}{{\rm d}q^{2}}$
$\displaystyle=\frac{1}{q^{2}_{\rm max}-q^{2}_{\rm min}}\frac{N_{\rm
sig}}{N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}}\frac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}}{\epsilon_{\phi\mu^{+}\mu^{-}}}{\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi){\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}),~{}$ (5)

where $N_{\rm sig}$ and $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$
denote the yields of the $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
candidates and $\epsilon_{\phi\mu^{+}\mu^{-}}$ and
$\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi}$ denote their
respective efficiencies. Since the reconstruction and selection efficiency of
the signal decay depends on $q^{2}$, a separate efficiency ratio
$\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}/\epsilon_{\phi\mu^{+}\mu^{-}}$ is determined for every $q^{2}$
bin. The branching fractions used in Eq. 5 are given by ${\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(10.50\pm 1.05\right)\times 10^{-4}$ [16] and ${\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})=\left(5.93\pm 0.06\right)\times 10^{-2}$
[12]. The resulting $q^{2}$-dependent differential branching fraction ${\rm
d}{\cal B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})/{\rm d}q^{2}$ is shown
in Fig. 3. Possible contributions from $B^{0}_{s}$ decays to
$K^{+}K^{-}\mu^{+}\mu^{-}$, with the $K^{+}K^{-}$ pair in an S-wave
configuration, are neglected in this analysis. The S-wave fraction is expected
to be small, for the decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
it is measured to be $(1.1\pm 0.1\,^{+0.2}_{-0.1})\%$ [16] for the
$K^{+}K^{-}$ mass window used in this analysis.

The total branching fraction is determined by summing the differential
branching fractions in the six $q^{2}$ bins. Using the form factor
calculations described in Ref. [17] the signal fraction rejected by the
charmonium vetoes is determined to be $17.7\%$. This number is confirmed by a
different form factor calculation detailed in Ref. [18]. No uncertainty is
assigned to the vetoed signal fraction. Correcting for the charmonium vetoes,
the branching fraction ratio ${\cal
B}\left(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}\right)/{\cal
B}\left(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi\right)$ is measured to be

$\displaystyle\frac{{\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})}{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}$
$\displaystyle=\left(6.74\,^{+0.61}_{-0.56}\pm 0.16\right)\times 10^{-4}.$

The systematic uncertainties will be discussed in detail in Sec. 4.1. Using
the known branching fraction of the normalisation channel the total branching
fraction is

$\displaystyle{\cal B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})$
$\displaystyle=\left(7.07\,^{+0.64}_{-0.59}\pm 0.17\pm 0.71\right)\times
10^{-7},$

where the first uncertainty is statistical, the second systematic and the
third from the uncertainty on the branching fraction of the normalisation
channel.

Figure 3: Differential branching fraction ${\rm d}{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-})/{\rm d}q^{2}$. Error bars
include both statistical and systematic uncertainties added in quadrature.
Shaded areas indicate the vetoed regions containing the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ resonances.
The solid curve shows the leading order SM prediction, scaled to the fitted
total branching fraction. The prediction uses the SM Wilson coefficients and
leading order amplitudes given in Ref. [2], as well as the form factor
calculations in Ref. [17]. $B^{0}_{s}$ mixing is included as described in Ref.
[1]. No error band is given for the theory prediction. The dashed curve
denotes the leading order prediction scaled to a total branching fraction of
$16\times 10^{-7}$ [19].

###  4.1 Systematic uncertainties on the differential branching fraction

The dominant source of systematic uncertainty on the differential branching
fraction arises from the uncertainty on the branching fraction of the
normalisation channel
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$),
which is known to an accuracy of $10\%$ [16]. This uncertainty is fully
correlated between all $q^{2}$ bins.

Many of the systematic uncertainties affect the relative efficiencies
$\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi}/\epsilon_{\phi\mu^{+}\mu^{-}}$ that are determined using
simulation. The limited size of the simulated samples causes an uncertainty of
$\sim 1\%$ on the ratio in each bin. Simulated events are corrected for known
discrepancies between simulation and data. The systematic uncertainties
associated with these corrections (e.g. tracking efficiency and performance of
the particle identification) are typically of the order of $1\text{--}2\%$.
The correction procedure for the impact parameter resolution has an effect of
up to $5\%$. Averaging the relative efficiency within the $q^{2}$ bins leads
to a systematic uncertainty of $1\text{--}2\%$. Other systematic uncertainties
of the same magnitude include the trigger efficiency and the uncertainties of
the angular distributions of the signal decay
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$. The influence of the signal mass
shape is found to be $0.5\%$. The background shape has an effect of up to
$5\%$, which is evaluated by using a linear function to describe the mass
distribution of the background instead of the nominal exponential shape.
Peaking backgrounds cause a systematic uncertainty of $1\text{--}2\%$ on the
differential branching fraction. The size of the systematics uncertainties on
the differential branching fraction, added in quadrature, ranges from
$4\text{--}6\%$. This is small compared to the dominant systematic uncertainty
of $10\%$ due to the branching fraction of the normalisation channel, which is
given separately in Table 1, and the statistical uncertainty.

##  5 Angular analysis

The angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ are
determined using unbinned maximum likelihood fits to the distributions of
$\cos{\theta_{K}}$, $\cos{\theta_{\ell}}$, $\Phi$, and the invariant mass of
the $K^{+}K^{-}\mu^{+}\mu^{-}$ system. The detector acceptance and the
reconstruction and selection of the signal decay distort the angular
distributions given in Eqs. 2–4. To account for this angular acceptance
effect, an angle-dependent efficiency is introduced that factorises in
$\cos{\theta_{K}}$ and $\cos{\theta_{\ell}}$, and is independent of the angle
$\Phi$, i.e.
$\epsilon(\cos{\theta_{K}},\cos{\theta_{\ell}},\Phi)=\epsilon_{K}(\cos{\theta_{K}})\cdot\epsilon_{\ell}(\cos{\theta_{\ell}})$.
The factors $\epsilon_{K}(\cos{\theta_{K}})$ and
$\epsilon_{\ell}(\cos{\theta_{\ell}})$ are determined from fits to simulated
events. Even Chebyshev polynomial functions of up to fourth order are used to
parametrise $\epsilon_{K}(\cos{\theta_{K}})$ and
$\epsilon_{\ell}(\cos{\theta_{\ell}})$ for each bin of $q^{2}$. The point-to-
point dissimilarity method described in Ref. [20] confirms that the angular
acceptance effect is well described by the acceptance model.

Taking the acceptances into account and integrating Eq. 1 over two angles,
results in

$\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\cos\theta_{K}}$
$\displaystyle=\epsilon_{K}(\cos{\theta_{K}})\biggl{[}\frac{3}{4}(1-{F_{\rm
L}})(1-\cos^{2}\theta_{K})\,\xi_{1}+\frac{3}{2}{F_{\rm
L}}\cos^{2}\theta_{K}\,\xi_{2}\biggr{]},$ (6) $\displaystyle\frac{1}{{\rm
d}\Gamma/{\rm d}q^{2}}\frac{{\rm d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm
d}\cos\theta_{\ell}}$
$\displaystyle=\parbox{0.0pt}{$\epsilon_{\ell}(\cos{\theta_{\ell}})$}\biggl{[}\frac{3}{8}(1-{F_{\rm
L}})(1+\cos^{2}\theta_{\ell})\,\xi_{3}+\frac{3}{4}{F_{\rm
L}}(1-\cos^{2}\theta_{\ell})\,\xi_{4}$
$\displaystyle\hphantom{=\epsilon_{K}(\cos{\theta_{K}})}+\frac{3}{4}{A_{6}}\cos\theta_{\ell}\,\xi_{3}\biggr{]},$
(7) $\displaystyle\frac{1}{{\rm d}\Gamma/{\rm d}q^{2}}\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}\,{\rm d}\Phi}$
$\displaystyle=\biggl{[}\frac{1}{2\pi}\xi_{1}\xi_{3}+\frac{1}{2\pi}F_{\rm
L}(\xi_{2}\xi_{4}-\xi_{1}\xi_{3})$
$\displaystyle\hphantom{={}}+\frac{1}{2\pi}{S_{3}}\cos
2\Phi\,\xi_{2}\xi_{3}+\frac{1}{2\pi}{A_{9}}\sin
2\Phi\,\xi_{2}\xi_{3}\biggr{]}.$ (8)

The terms $\xi_{i}$ are correction factors with respect to Eqs. 2–4 and are
given by the angular integrals

$\displaystyle\xi_{1}$
$\displaystyle=\frac{3}{8}\int_{-1}^{+1}(1+\cos^{2}\theta_{\ell})\epsilon_{\ell}(\cos{\theta_{\ell}}){\rm
d}\cos{\theta_{\ell}},$ $\displaystyle\xi_{2}$
$\displaystyle=\frac{3}{4}\int_{-1}^{+1}(1-\cos^{2}\theta_{\ell})\epsilon_{\ell}(\cos{\theta_{\ell}}){\rm
d}\cos{\theta_{\ell}},$ $\displaystyle\xi_{3}$
$\displaystyle=\frac{3}{4}\int_{-1}^{+1}(1-\cos^{2}\theta_{K})\epsilon_{K}(\cos{\theta_{K}}){\rm
d}\cos{\theta_{K}},$ $\displaystyle\xi_{4}$
$\displaystyle=\frac{3}{2}\int_{-1}^{+1}\cos^{2}\theta_{K}\epsilon_{K}(\cos{\theta_{K}}){\rm
d}\cos{\theta_{K}}.$ (9)

Three two-dimensional maximum likelihood fits in the decay angles and the
reconstructed $B^{0}_{s}$ mass are performed for each $q^{2}$ bin to determine
the angular observables. The observable $F_{\rm L}$ is determined in the fit
to the $\cos{\theta_{K}}$ distribution described by Eq. 6. The
$\cos{\theta_{\ell}}$ distribution given by Eq. 7 is used to determine
$A_{6}$. Both $S_{3}$ and $A_{9}$ are measured from the $\Phi$ distribution,
as described by Eq. 8. In the fit of the $\Phi$ distribution a Gaussian
constraint is applied to the parameter $F_{\rm L}$ using the value of $F_{\rm
L}$ determined from the $\cos{\theta_{K}}$ distribution. The constraint on
$F_{\rm L}$ has negligible influence on the values of $S_{3}$ and $A_{9}$. The
angular distribution of the background events is fit using Chebyshev
polynomial functions of second order. The mass shapes of the signal and
background are described by the sum of two Gaussian distributions with a
common mean, and an exponential function, respectively. The effect of the veto
of the radiative tails on the combinatorial background is accounted for by
using an appropriate scale factor.

The measured angular observables are presented in Fig. 4 and Table 2. The
$68\%$ confidence intervals are determined using the Feldman-Cousins method
[21] and the nuisance parameters are included using the plug-in method [22].

Figure 4: a) Longitudinal polarisation fraction $F_{\rm L}$, b) $S_{3}$, c) $A_{6}$, and d) $A_{9}$ in six bins of $q^{2}$. Error bars include statistical and systematic uncertainties added in quadrature. The solid curves are the leading order SM predictions, using the Wilson coefficients and leading order amplitudes given in Ref. [2], as well as the form factor calculations in Ref. [17]. $B^{0}_{s}$ mixing is included as described in Ref. [1]. No error band is given for the theory predictions.  Table 2: Results for the angular observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ in bins of $q^{2}$. The first uncertainty is statistical, the second systematic.  $q^{2}$ bin $({\mathrm{\,Ge\kern-0.80005ptV^{2}\\!/}c^{4}})$ | $F_{\rm L}$ | $S_{3}$ | $A_{6}$ | $A_{9}$
---|---|---|---|---
$0.10<q^{2}<\parbox{0.0pt}{2.00}$ | $0.37\,^{+0.19}_{-0.17}\pm 0.07$ | $-0.11\,^{+0.28}_{-0.25}\pm 0.05$ | $0.04\,^{+0.27}_{-0.32}\pm 0.12$ | $-0.16\,^{+0.30}_{-0.27}\pm 0.09$
$2.00<q^{2}<\parbox{0.0pt}{4.30}$ | $0.53\,^{+0.25}_{-0.23}\pm 0.10$ | $-0.97\,^{+0.53}_{-0.03}\pm 0.17$ | $0.47\,^{+0.39}_{-0.42}\pm 0.14$ | $-0.40\,^{+0.52}_{-0.35}\pm 0.11$
$4.30<q^{2}<\parbox{0.0pt}{8.68}$ | $0.81\,^{+0.11}_{-0.13}\pm 0.05$ | $0.25\,^{+0.21}_{-0.24}\pm 0.05$ | $-0.02\,^{+0.20}_{-0.21}\pm 0.10$ | $-0.13\,^{+0.27}_{-0.26}\pm 0.10$
$10.09<q^{2}<\parbox{0.0pt}{12.90}$ | $0.33\,^{+0.14}_{-0.12}\pm 0.06$ | $0.24\,^{+0.27}_{-0.25}\pm 0.06$ | $-0.06\,^{+0.20}_{-0.20}\pm 0.08$ | $0.29\,^{+0.25}_{-0.26}\pm 0.10$
$14.18<q^{2}<\parbox{0.0pt}{16.00}$ | $0.34\,^{+0.18}_{-0.17}\pm 0.07$ | $-0.03\,^{+0.29}_{-0.31}\pm 0.06$ | $-0.06\,^{+0.30}_{-0.30}\pm 0.08$ | $0.24\,^{+0.36}_{-0.35}\pm 0.12$
$16.00<q^{2}<\parbox{0.0pt}{19.00}$ | $0.16\,^{+0.17}_{-0.10}\pm 0.07$ | $0.19\,^{+0.30}_{-0.31}\pm 0.05$ | $0.26\,^{+0.22}_{-0.24}\pm 0.08$ | $0.27\,^{+0.31}_{-0.28}\pm 0.11$
$1.00<q^{2}<\parbox{0.0pt}{6.00}$ | $0.56\,^{+0.17}_{-0.16}\pm 0.09$ | $-0.21\,^{+0.24}_{-0.22}\pm 0.08$ | $0.20\,^{+0.29}_{-0.27}\pm 0.07$ | $-0.30\,^{+0.30}_{-0.29}\pm 0.11$

###  5.1 Systematic uncertainties on the angular observables

The dominant systematic uncertainty on the angular observables is due to the
angular acceptance model. Using the point-to-point dissimilarity method
detailed in Ref. [20], the acceptance model is shown to describe the angular
acceptance effect for simulated events at the level of $10\%$. A cross-check
of the angular acceptance using the normalisation channel
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ shows
good agreement of the angular observables with the values determined in Refs.
[23] and [24]. For the determination of the systematic uncertainty due to the
angular acceptance model, variations of the acceptance curves are used that
have the largest impact on the angular observables. The resulting systematic
uncertainty is of the order of $0.05\text{--}0.10$, depending on the $q^{2}$
bin.

The limited amount of simulated events accounts for a systematic uncertainty
of up to $0.02$. The simulation correction procedure (for tracking efficiency,
impact parameter resolution, and particle identification performance) has
negligible effect on the angular observables. The description of the signal
mass shape leads to a negligible systematic uncertainty. The background mass
model causes an uncertainty of less than $0.02$. The model of the angular
distribution of the background can have a large effect since the statistical
precision of the background sample is limited. To estimate the effect, the
parameters describing the background angular distribution are determined in
the high $B^{0}_{s}$ mass sideband
($5416<m(K^{+}K^{-}\mu^{+}\mu^{-})<5566{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
using a relaxed requirement on the $\phi$ mass. The effect is typically
$0.05\text{--}0.10$. Peaking backgrounds cause systematic deviations of the
order of $0.01\text{--}0.02$. Due to the sizeable lifetime difference in the
$B^{0}_{s}$ system [24] a decay time dependent acceptance can in principle
affect the angular observables. The deviation of the observables due to this
effect is studied and found to be negligible. The total systematic
uncertainties, evaluated by adding all components in quadrature, are small
compared to the statistical uncertainties.

##  6 Conclusions

The differential branching fraction of the FCNC decay
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ has been determined. The results are
summarised in Fig. 3 and in Table 1. Using the form factor calculations in
Ref. [17] to determine the fraction of events removed by the charmonium
vetoes, the relative branching fraction ${\cal
B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)$ is
determined to be

$\displaystyle\frac{{\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})}{{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}$
$\displaystyle=$ $\displaystyle\left(6.74\,^{+0.61}_{-0.56}\pm
0.16\right)\times 10^{-4}.$

This value is compatible with a previous measurement by the CDF collaboration
of ${\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(11.3\pm 1.9\pm 0.7\right)\times 10^{-4}$ [25] and a recent
preliminary result which yields ${\cal
B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})/{\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(9.0\pm 1.4\pm 0.7\right)\times 10^{-4}$ [26]. Using the
branching fraction of the normalisation channel, ${\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)=\left(10.50\pm 1.05\right)\times 10^{-4}$ [16], the total
branching fraction of the decay is determined to be

$\displaystyle{\cal B}(B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-})$
$\displaystyle=$ $\displaystyle\left(7.07\,^{+0.64}_{-0.59}\pm 0.17\pm
0.71\right)\times 10^{-7},$

where the first uncertainty is statistical, the second systematic, and the
third from the uncertainty of the branching fraction of the normalisation
channel. This measurement constitutes the most precise determination of the
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ branching fraction to date. The
measured value is lower than the SM theory predictions that range from
$14.5\times 10^{-7}$ to $19.2\times 10^{-7}$ [19, 27, 28, 29]. The
uncertainties on these predictions originating from the form factor
calculations are typically of the order of $20\text{--}30\%$.

In addition, the first angular analysis of the decay
$B^{0}_{s}\rightarrow\phi\mu^{+}\mu^{-}$ has been performed. The angular
observables $F_{\rm L}$, $S_{3}$, $A_{6}$, and $A_{9}$ are determined in bins
of $q^{2}$, using the distributions of $\cos{\theta_{K}}$,
$\cos{\theta_{\ell}}$, and $\Phi$. The results are summarised in Fig. 4, and
the numerical values are given in Table 2. All measured angular observables
are consistent with the leading order SM expectation.

## 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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