EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-087 LHCb-PAPER-2013-012 10 June 2013

First observation of the decay $B^{0}_{s}\rightarrow\phi\kern
4.14793pt\overline{\kern-4.14793ptK}{}^{*0}$

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

The first observation of the decay $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is reported. The analysis is
based on a data sample corresponding to an integrated luminosity of 1.0 fb-1
of $pp$ collisions at $\sqrt{s}=7\,$$\mathrm{\,Te\kern-1.00006ptV}$, collected
with the LHCb detector. A yield of ${30\pm 6}$
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ decays is found in the mass
windows
${1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${746<M(K^{-}\pi^{+})<1046{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$. The
signal yield is found to be dominated by $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, and the corresponding
branching fraction is measured to be ${{\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(1.10\pm
0.24\,\mathrm{(stat)}\pm 0.14\,\mathrm{(syst)}\pm
0.08\left(f_{d}/f_{s}\right)\right)\times 10^{-6}}$, where the uncertainties
are statistical, systematic and from the ratio of fragmentation fractions
$f_{d}/f_{s}$ which accounts for the different production rate of $B^{0}$ and
$B^{0}_{s}$ mesons. The significance of $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal is 6.1 standard
deviations. The fraction of longitudinal polarization in
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decays is found to be ${f_{0}=0.51\pm 0.15\,\mathrm{(stat)}\pm
0.07\,\mathrm{(syst)}}$.

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. Artuso57,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso58, 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. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,p, 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, 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, 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,
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. Gandini57,
Y. Gao3, J. Garofoli57, 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. Göbel58, 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. Grünberg59, B. Gui57, E. Gushchin32, Yu.
Guz34,37, T. Gys37, C. Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C.
Haines46, S. Hall52, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T.
Harnew45, J. Harrison53, T. Hartmann59, 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, 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. Lanfranchi18,37, 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, 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,p, 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, 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, A. McNab53, R.
McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53, P.
Morawski25, M.J. Morello22,r, R. Mountain57, 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,o, 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, B.K. Pal57, 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. Phan57, 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, A.
Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,q, 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, 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, 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. Skwarnicki57, 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. Stone57, 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. 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. Voß59, H. Voss10, R. Waldi59, R. Wallace12, S. Wandernoth11, J.
Wang57, 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, Z.
Xing57, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang57, 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
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut 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
pUniversità di Padova, Padova, Italy
qUniversità di Pisa, Pisa, Italy
rScuola Normale Superiore, Pisa, Italy

##  1 Introduction

The measurement of $C\\!P$ asymmetries in flavour-changing neutral-current
processes provides a crucial test of the Standard Model (SM). In particular,
loop-mediated (penguin) decays of $B$ mesons are sensitive probes for physics
beyond the SM. Transitions between the quarks of the third and second
generation ($b\rightarrow s$) or between the quarks of the third and first
generation ($b\rightarrow d$) are complementary since SM $C\\!P$ violation is
tiny in $b\rightarrow s$ transitions and an observation of $C\\!P$ violation
would indicate physics beyond the SM. For $b\rightarrow d$ transitions the SM
branching fraction is an order of magnitude smaller than $b\rightarrow s$ due
to the relative suppression of $|V_{td}|^{2}/|V_{ts}|^{2}$. It is particularly
useful to have experimental information on pairs of channels related by
$d\leftrightarrow s$ exchange symmetry to test that the QCD contribution to
the decay is independent of the initial $B^{0}$ or $B^{0}_{s}$ meson.

The BaBar and Belle experiments have performed measurements of $b\rightarrow
sq\overline{q}$ processes, such as $B^{0}\rightarrow\phi
K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}\rightarrow\eta^{\prime}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{0}\rightarrow f_{0}K^{0}_{\rm\scriptscriptstyle S}$ [1, 2, 3], and of
$b\rightarrow dq\overline{q}$ penguin diagrams, such as $B^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{+}\rightarrow K^{+}K^{0}_{\rm\scriptscriptstyle S}$ [4, 5]. These modes
contain pseudo-scalar or scalar mesons in their final state whereas
$B^{0}_{(s)}\rightarrow VV^{\prime}$ decays, where $V$ and $V^{\prime}$ are
light vector mesons, provide a valuable additional source of information
because the angular distributions give insight into the physics of hadronic
$B$ meson decays and the interplay between the strong and weak interactions
they involve. From the V$-$A structure of the weak interaction and helicity
conservation in the strong interaction, the final state of these decays is
expected to be highly longitudinally polarized. This applies to both tree and
penguin decays. The BaBar and Belle experiments have confirmed that
longitudinal polarization dominates in $b\rightarrow u$ tree processes such as
$B^{0}\rightarrow\rho^{+}\rho^{-}$ [6, 7], $B^{+}\rightarrow\rho^{0}\rho^{+}$
[8, 9] and $B^{+}\rightarrow\omega\rho^{+}$ [10]. However, measurements of the
polarization in decays with both tree and penguin contributions, such as
$B^{0}\rightarrow\rho^{0}K^{*0}$ and $B^{0}\rightarrow\rho^{-}K^{*+}$ [11] and
in $b\rightarrow s$ penguin decays, $B^{0}\rightarrow\phi K^{*0}$ [12, 13],
$B^{0}_{s}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
[14] and $B^{0}_{s}\rightarrow\phi\phi$ [15, 16, *Aaij:2013qha], indicate a
low value of the longitudinal polarization fraction comparable with, or even
smaller than, the transverse fraction.

The $B^{0}_{(s)}\rightarrow VV^{\prime}$ decays can be described by models
based on perturbative QCD, or QCD factorization and SU(3) flavour symmetries.
Whilst some authors predict a longitudinal polarization fraction
$f_{0}\mathord{\sim}0.9$ for tree-dominated and $\mathord{\sim}0.75$ for
penguin decays [18, *Suzuki:2002yk, 20], other studies have proposed different
mechanisms such as penguin annihilation [21, 22] and QCD rescattering [23] to
accommodate smaller longitudinal polarization fractions $\mathord{\sim}0.5$,
although the predictions have large uncertainties. A review on the topic of
polarization in $B$ decays can be found in Ref. [24].

There are only two other $B^{0}_{(s)}\rightarrow VV^{\prime}$ penguin modes
that correspond to $b\rightarrow d$ loops. The first is the $B^{0}\rightarrow
K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. The BaBar
collaboration reported the discovery of this channel with $6\,\sigma$
significance and a measurement of its branching fraction ${\cal
B}(B^{0}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=(1.28\,{}^{+0.35}_{-0.30}\pm
0.11)\times 10^{-6}$ [25]. This is in tension with the results of the Belle
collaboration that published an upper limit of ${\cal B}(B^{0}\rightarrow
K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<0.8\times 10^{-6}$ at
the $90\%$ confidence level [26]. The BaBar publication also reported a
measurement of the longitudinal polarization ${f_{0}=0.80^{+0.12}_{-0.13}}$
[25], which is large compared to those from $B^{0}\rightarrow\phi K^{*0}$
($f_{0}=0.494\pm 0.036$ [13]), $B^{0}_{s}\rightarrow\phi\phi$ ($f_{0}=0.365\pm
0.025$ [16]) and $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ ($f_{0}=0.31\pm 0.13$ [14]).

The mode $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is the other $b\rightarrow d$
penguin decay into vector mesons that has not previously been observed. This
decay is closely linked to $B^{0}\rightarrow\phi K^{*0}$, differing in the
spectator quark and the final quark in the loop, as shown in Fig. 1.111Both
the decays $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow\phi K^{*0}$
could also have contributions from QCD singlet-penguin amplitudes [21]. From
the aforementioned relation between $b\rightarrow s$ and $b\rightarrow d$
transitions, their relative branching fractions should scale as
$|V_{td}|^{2}/|V_{ts}|^{2}$ and their polarization fractions are expected to
be very similar. Moreover, since both decays share the same final state,
except for charge conjugation, $B^{0}\rightarrow\phi K^{*0}$ is the ideal
normalization channel for the determination of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
branching fraction. The $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay is also related to
$B^{0}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$,
since their loop diagrams only differ in the spectator quark (${s\text{
instead of }d}$), although it has been suggested that S-wave interference
effects might break the SU(3) symmetry relating two channels [27]. Finally, it
is also interesting to explore the relation of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decay with the $B^{0}\rightarrow\rho^{0}K^{*0}$ mode since the penguin loop
diagrams of these modes are related by the $d\leftrightarrow s$ exchange. The
$B^{0}\rightarrow\rho^{0}K^{*0}$ decay also has a $b\rightarrow u$ tree
diagram, but it is expected that the penguin contribution is dominant, since
the branching fraction is comparable to that of the pure penguin
$B^{0}\rightarrow\phi K^{*0}$ decay.

The most stringent previous experimental limit on the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
branching fraction is ${{\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<1.0\times 10^{-3}}$ at the $90\%$
confidence level [24], whereas calculations based on the QCD factorization
framework predict a value of ${(0.4\,{}^{+0.5}_{-0.3})\times 10^{-6}}$ [21]
while in perturbative QCD a value of ${(0.65\,{}^{+0.33}_{-0.23})\times
10^{-6}}$ [28] is obtained. The precise determination of the branching
fraction tests these models and provides a probe for physics beyond the SM.

The study of the angular distributions in the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ channel provides a measurement of
its polarization. In Ref. [28], a prediction of
$f_{0}=0.712\,{}^{+0.042}_{-0.048}$ is made for the longitudinal polarization
fraction, using the perturbative QCD approach, that can be compared to the
experimental result.

In this paper the first observation of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay, with $\phi\rightarrow
K^{+}K^{-}$ and $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow
K^{-}\pi^{+}$, is reported and the determination of its branching fraction and
polarizations are presented. The study is based on data collected by the LHCb
experiment at CERN from the $\sqrt{s}=7\,$$\mathrm{\,Te\kern-1.00006ptV}$
proton-proton collisions of LHC beams. The dataset corresponds to an
integrated luminosity of 1.0 fb-1.

Figure 1: Feynman diagrams for the $B^{0}_{s}\rightarrow\phi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ and the $B^{0}\rightarrow\phi
K^{*0}$ decays.

##  2 Detector and software

The LHCb detector [29] 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
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors [30]. 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 [31].

The trigger [32] 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. The software trigger used in this analysis requires
a two-, three- or four-track secondary vertex with a high sum of the $p_{\rm
T}$ of the tracks and significant displacement from the primary $pp$
interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter $\chi^{2}$
($\chi^{2}_{{\rm IP}}$) with respect to all primary interactions greater than
16. The $\chi^{2}_{{\rm IP}}$ is defined as the difference between the
$\chi^{2}$ of a PV reconstructed with and without the considered track. A
multivariate algorithm [33] is used for the identification of secondary
vertices consistent with the decay of a $b$ hadron.

In the simulation, $pp$ collisions are generated using Pythia 6.4 [34] with a
specific LHCb configuration [35]. Decays of hadronic particles are described
by EvtGen [36], in which final state radiation is generated using Photos [37].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [38, *Agostinelli:2002hh] as
described in Ref. [40].

##  3 Signal selection

Signal $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates are formed from
$\phi\rightarrow K^{+}K^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow K^{-}\pi^{+}$
decays.222Inclusion of charge conjugated processes is implied in this work,
unless otherwise stated. The pairs of charged particles in the
$\phi\rightarrow K^{+}K^{-}$ and the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow K^{-}\pi^{+}$
candidates must combine to give invariant masses
${1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${746<M(K^{-}\pi^{+})<1046{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$,
consistent with the known $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ masses [24]. Each of the four
tracks is required to have $\mbox{$p_{\rm
T}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{{\rm IP}}$ $>9$.

Kaons and pions are distinguished by use of a log-likelihood algorithm that
combines information from the RICH detectors and other properties of the event
[30]. The final state particles are identified by requiring that the
difference in log-likelihoods of the kaon and pion mass hypotheses is
$\mathrm{DLL}_{K\pi}$ $>2$ for each kaon candidate and $<0$ for the pion
candidate. In addition, the difference in log-likelihoods of the proton and
kaon hypotheses, $\mathrm{DLL}_{pK}$, is required to be $<0$ for the kaon from
the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. This suppresses
background from $\mathchar 28931\relax^{0}_{b}$ decays. This requirement is
not necessary for the kaons from the $\phi$ candidate owing to the narrow
$K^{+}K^{-}$ invariant mass window.

The $K^{-}\pi^{+}$ pair that forms the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidate is required to
originate from a common vertex with a $\chi^{2}$ per number of degrees of
freedom ($\chi^{2}/{\rm ndf}$) $<9$, and to have a positive cosine of the
angle between its momentum and the reconstructed $B^{0}_{(s)}$ candidate
flight direction, calculated with the $B^{0}_{(s)}$ decay vertex and the best
matching primary vertex. The $K^{-}\pi^{+}$ combination is also required to
have $\mbox{$p_{\rm T}$}>900{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The same
conditions are imposed on the $\phi$ candidate.

The $B^{0}_{(s)}$ candidates are also required to fulfil some minimal
selection criteria: the $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates must form a vertex
with $\chi^{2}/{\rm ndf}<15$; the distance of closest approach between their
trajectories must be less than $0.3\,\rm\,mm$; and they must combine to give
an invariant mass within
${4866<M(K^{+}K^{-}K^{-}\pi^{+})<5866{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$.

In addition, a geometrical-likelihood based selection (GL) [41, 42] is
implemented using as input variables properties of the $B^{0}_{(s)}$ meson
candidate. These are

  * •

the $B^{0}_{(s)}$ candidate impact parameter (${\rm IP}$) with respect to the
closest primary vertex;

  * •

the decay time of the $B^{0}_{(s)}$ candidate;

  * •

the $p_{\rm T}$ of the $B^{0}_{(s)}$ candidate;

  * •

the minimum $\chi^{2}_{{\rm IP}}$ of the four tracks with respect to all
primary vertices in the event; and

  * •

the distance of closest approach between the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\phi$ candidates’
trajectories reconstructed from their respective daughter tracks.

The GL is trained to optimize its discrimination power using representative
signal and background samples. For the signal a set of
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
simulated events is used. For the background a sample of events where, in
addition to the signal selections, other than those on the masses,
requirements of
$999.5<M(K^{+}K^{-})<1012.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or
${1026.5<M(K^{+}K^{-})<1039.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ for
the $\phi$ candidate and
${M(K^{+}K^{-}K^{-}\pi^{+})>5413{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ for
the four-body mass are applied. The selection of only the high-mass
$B^{0}_{(s)}$ sideband is motivated by the nature of the background in that
region, which is purely combinatorial, whereas the low-mass sideband contains
partially reconstructed $B$ meson decays that have topological similarities to
the signal.

##  4 Suppression of background from other $b$-hadron decays

A small background from $B^{0}_{s}\rightarrow\phi\phi$ decays, where one of
the kaons from the $\phi$ is misidentified as a pion, is found to contaminate
the signal. Candidate $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays are therefore required to
be outside of the window defined by
${1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${5324<M(K^{+}K^{-}K^{+}K^{-})<5424{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$
in the $K^{+}K^{-}$ and $K^{+}K^{-}K^{+}K^{-}$ invariant masses when the mass
hypothesis for the sole pion of the decay is switched into a kaon. In
simulated events this selection removes $0.12\%$ of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
signal decays and does not affect the $B^{0}\rightarrow\phi K^{*0}$ decay
mode. Other possible reflections, such as $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, are found to be
negligible.

In order to remove background from $B^{0}_{s}\rightarrow
D_{s}^{\mp}(\phi\pi^{\mp})K^{\pm}$ decays when the $\pi^{\mp}$ and the
$K^{\pm}$ mesons form a
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{\bf{--}}\scalebox{0.4}{)}}{K}^{*0}$
candidate, events with the invariant mass of the $K^{+}K^{-}\pi^{\mp}$ system
within
${1953.5<M(K^{+}K^{-}\pi^{\mp})<1983.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$,
consistent with the known $D^{+}_{s}$ mass [24], are excluded.

Background from $b$-hadron decays containing a misidentified proton has also
been considered. For candidate $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, the kaon with the largest
$\mathrm{DLL}_{pK}$ is assigned the proton mass and the four-body invariant
mass recomputed. The largest potential background contribution arises from
$\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow K^{+}K^{-}\overline{}p\pi^{+}$ where the
antiproton is misidentified as the kaon originating from the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ meson, and $\mathchar
28931\relax^{0}_{b}\rightarrow K^{+}K^{-}K^{-}p$, where the proton is
misidentified as the pion originating from the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ meson. Simulation shows that
these decays produce wide four-body mass distributions which peak around
$5450{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$5500{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. This background
contribution is considered in the fit model discussed below. Other
$B^{0}_{(s)}$ decay modes containing a $\mathchar 28931\relax\rightarrow
p\pi^{-}$ decay or background from $\mathchar 28931\relax^{+}_{c}\rightarrow
pK^{-}\pi^{+}$ decays are found to be negligible.

##  5 Fit to the four-body mass spectrum

The sample of $1277$ candidates, selected as described in Sections 3 and 4,
contains many $B^{0}\rightarrow\phi K^{*0}$ decays whereas only a small
contribution from $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays is anticipated. Both
signals are parametrized with identical shapes, differing only in the mass
shift of $87.13{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ between the $B^{0}$
and $B^{0}_{s}$ mesons [24] which is fixed in the fit. The signal shapes are
described by the sum of Crystal Ball (CB) [43] and Gaussian functions that
share a common mean. The CB function, which contains most of the signal, is a
combination of a Gaussian function with a power law tail, accounting for the
intrinsic detector resolution and the radiative tail toward low masses,
respectively. The Gaussian shape describes events reconstructed with worse
mass resolution, which produce a contamination of $B^{0}\rightarrow\phi
K^{*0}$ decays in the region of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal peak. The dependence
between the Gaussian and CB resolutions, $\sigma_{{\rm G}}$ and $\sigma_{{\rm
CB}}$, respectively, is found to be

$\sigma_{{\rm G}}=\sqrt{\sigma_{{\rm
CB}}^{2}+(24.74{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})^{2}},$ (1)

from a data sample of $25\times 10^{3}$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays. This channel is topologically very similar to the signal and is almost
background free. The fit to this sample also provides the power law exponent
of the CB function tail, which is subsequently fixed in the
$B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$ and
$B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})$ mass models. The parameter that
governs the transition from the Gaussian shape to the power law function in
the CB function is unrestrained in the fit. The other unrestrained fit
parameters include: the central $B$ meson mass, the width of the CB function,
the fractional yield contained in the Gaussian function and the total signal
yield.

In addition to the $B^{0}$ and $B^{0}_{s}$ signal shapes, three more
components are included. The first accounts for partially reconstructed $B$
meson decays into $\phi$ and $K$ or $K^{*}$ excited states where a pion has
been lost. This is described by a convolution of the ARGUS shape [44] with a
Gaussian distribution. The second contribution is due to $\mathchar
28931\relax^{0}_{b}\rightarrow K^{+}K^{-}K^{-}p$ and $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}^{0}_{b}\rightarrow
K^{+}K^{-}\overline{}p\pi^{+}$ decays and is modelled with a histogram
obtained from simplified simulations. The third contribution is an exponential
function to account for combinatorial background.

The data passing the selection criteria are fitted using an extended unbinned
maximum likelihood fit. The invariant mass distribution of the candidates,
together with the fit contribution, is shown in Fig. 2. The yields of
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ and
${B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})}$ decays are $30\pm 6$ and
$1000\pm 32$, respectively. The fit model is validated with $10,000$ pseudo-
experiments, generated with simplified simulations, which show that the signal
yields are unbiased. Table 1 summarizes the signal and background
contributions resulting from the fit. A likelihood ratio test is employed to
assess the statistical significance of the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal yield. This is
performed using $\sqrt{2{\rm ln}(\mathcal{L}_{\rm s+b}/\mathcal{L}_{\rm b})}$,
where $\mathcal{L}_{\rm s+b}$ and $\mathcal{L}_{\rm b}$ are the maximum values
of the likelihoods for the signal-plus-background and background-only
hypotheses, respectively.333The applicability of this method has been verified
from the parabolic behaviour of the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal yield profile of
$-2\ln\mathcal{L}_{\rm s+b}$ about its minimum. This calculation results in
$6.3\,\sigma$ significance for the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal. The fit gives
$\sigma_{\rm CB}=15.0\pm 1.1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the
invariant mass resolution. Integration in a $\pm
30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window yields $26.4\pm 5.7$
signal candidates and $8.2\pm 1.3$ background events, composed of $5.4\pm 0.2$
from ${B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})}$, $2.1\pm 1.3$ from
$\mathchar 28931\relax^{0}_{b}$ and $0.7\pm 0.4$ from combinatorial
contributions.

In order to explore systematic effects in the signal yield originating in the
fit model two effects were considered. First, the amount of
${B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})}$ events under the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal is governed by the
$24.74{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ factor in Eq. 1. Similarly,
the contamination of misidentified $\mathchar 28931\relax^{0}_{b}$ decays
under the signal is controlled by a tail that is parametrized. An extended
likelihood is built by multiplying the original likelihood function by
Gaussian distributions of these two nuissance parameters with standard
deviations of $20\%$ of their nominal values at which they are centered. The
corresponding systematic uncertainty in the signal yield is obtained by
performing a fit that maximizes this modified likelihood. The systematic
contribution is calculated subtracting the statistical uncertainty in
quadrature and found to be $\pm 1.2$ events. Including this uncertainty
results in a significance of $6.2\sigma$. Effects of other systematic
uncertainties, discussed in Sect. 9, have negiglible impact in the signal
significance.

Figure 2: Four-body $K^{+}K^{-}K^{-}\pi^{+}$ invariant mass distribution. The points show the data, the blue solid line shows the overall fit, the solid dark red shaded region is the $B^{0}_{s}\rightarrow\phi\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ signal, the light blue shaded region corresponds to the $B^{0}\rightarrow\phi K^{*0}$ signal, the grey dotted line is the combinatorial background and the green dashed line and magenta dashed-dotted lines are the partially reconstructed and misidentified $\mathchar 28931\relax^{0}_{b}$ backgrounds. Table 1: Results of the fit to the sample of selected candidates. Contribution | Yield
---|---
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | $\>\>30\pm 6$
$B^{0}\rightarrow\phi K^{*0}$ | $1000\pm 32$
Partially reconstructed background | $\>\>218\pm 15$
$\mathchar 28931\relax^{0}_{b}$ background | $\>\>13\pm 8$
Combinatorial background | $\>\>10\pm 6$

##  6 Determination of the S-wave contribution

The $B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$ signal is expected to be
mainly due to $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, although there are
possible non-resonant contributions and $K^{+}K^{-}$ and $K^{-}\pi^{+}$ pairs
from other resonances. To estimate the S-wave contributions, it is assumed
that the effect is the same for $B^{0}\rightarrow\phi K^{*0}$ and
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decays, therefore allowing the larger sample of $B^{0}\rightarrow\phi K^{*0}$
decays to be used. The effect of this assumption is considered as a source of
systematic uncertainty in Sect. 8.

The $K^{+}K^{-}$ invariant mass distribution for $\phi$ candidates within a
$\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window of the known $B^{0}$
mass is described by a relativistic spin-1 Breit-Wigner distribution convolved
with a Gaussian shape to account for the effect of resolution. A linear term
is added to describe the S-wave contribution. The purity resulting from this
fit is $0.95\pm 0.01$ in a $\pm 7{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
window around the known $\phi$ mass.

The $K^{+}\pi^{-}$ pairs are parametrized by the incoherent sum of a
relativistic spin-1 Breit-Wigner amplitude and a shape that describes non-
resonant and $K^{*0}(1430)$ S-wave contributions introduced by the LASS
experiment [13, 45]. The fraction of events from $K^{*0}$ decays within a $\pm
150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window around the $K^{*0}$ mass
results in a purity of $0.89\pm 0.02$. When combining the $K^{+}K^{-}$ and
$K^{+}\pi^{-}$ contributions, the total $\phi K^{*0}$ purity is found to be
$0.84\pm 0.02$. This purity can be translated into a p-value, quantifying the
probability that the entire ${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$
signal is due to decays other than $\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. After combining with the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ significance the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is
observed with $6.1\,\sigma$ significance.

Figure 3: Invariant mass distributions for (left) $K^{+}K^{-}$ and (right)
$K^{\mp}\pi^{\pm}$ pairs in a $\pm 30{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$
window around the (top) $B^{0}_{s}$ and (bottom) $B^{0}$ mass. The solid blue
line is the overall fit, the green dashed line corresponds to $B^{0}$ cross-
feed into the $B^{0}_{s}$ mass window, the red dotted line is the S-wave
contribution and the light blue is the combinatorial background.

##  7 Determination of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ branching fraction

The branching fraction is calculated with the $B^{0}\rightarrow\phi K^{*0}$
channel as normalization. Both decays pass the same selection and share almost
identical topologies. However, since the two decay channels can have different
polarizations, their angular distributions may differ which would cause a
difference in their detection efficiencies. A factor

$\lambda_{f_{0}}=\frac{\epsilon^{B^{0}\rightarrow\phi
K^{*0}}}{\epsilon^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}=\frac{1-0.29f_{0}^{B^{0}\rightarrow\phi
K^{*0}}}{1-0.29f_{0}^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}$

is calculated, where $\epsilon^{B^{0}\rightarrow\phi K^{*0}}$ and
$\epsilon^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ are the efficiencies for the
$B^{0}\rightarrow\phi K^{*0}$ and $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays reconstruction,
$f_{0}^{B^{0}\rightarrow\phi K^{*0}}$ and
$f_{0}^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ their longitudinal polarization
fractions, determined in Sect. 9 for the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mode, and the factor 0.29 is
obtained from simulation.

The value of ${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ is computed from

${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\lambda_{f_{0}}\times\frac{f_{d}}{f_{s}}\times{\cal
B}(B^{0}\rightarrow\phi K^{*0})\times\frac{N_{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{N_{B^{0}\rightarrow\phi
K^{*0}}},$ (2)

where $N_{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ and $N_{B^{0}\rightarrow\phi
K^{*0}}$ are the numbers of $B^{0}_{s}$ and $B^{0}$ decays, respectively, and
$f_{d}/f_{s}=3.75\pm 0.29$ [46] is the ratio of hadronization factors needed
to account for the different production rates of $B^{0}$ and $B^{0}_{s}$
mesons. With the values given in Table 2, the result,

${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=(1.10\pm 0.24)\times 10^{-6},$

is obtained, where only the statistical uncertainty is shown.

Table 2: Input values for the branching fraction computation. Parameter | Value
---|---
$\lambda_{f_{0}}$ | $1.01\pm 0.06$
$N_{B^{0}\rightarrow\phi K^{*0}}$ | $1000\pm 32\>\>\>\>$
$N_{B^{0}_{s}\rightarrow\phi\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ | $30\pm 6\>\>$
${\cal B}(B^{0}\rightarrow\phi K^{*0})$ |  $(9.8\pm 0.6)\times 10^{-6}$ [24]

As a cross-check, a different decay mode,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, with
$J/\psi\rightarrow\mu^{+}\mu^{-}$, has been used as a normalization channel.
Special requirements were imposed to harmonize the selection of this reference
with that for the signal. The obtained result is fully compatible with the
$B^{0}\rightarrow\phi K^{*0}$ based value.

##  8 Systematic uncertainties on the branching fraction

Four main sources of systematic effects in the determination of the branching
fraction are identified: the fit model, the dependence of the acceptance on
the longitudinal polarization, the purity of the signal and the uncertainty in
the relative efficiency of $B^{0}_{s}$ and $B^{0}$ detection.

Alternatives to the fit model discussed in Sect. 5 give an uncertainty of $\pm
1.2$ in the signal yield. This results in a relative systematic uncertainty of
$\pm 0.04$ on the branching fraction.

The systematic uncertainty in the acceptance correction factor
$\lambda_{f_{0}}$ originates from the uncertainties of the longitudinal
polarization fractions, $f_{0}$, in the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow\phi K^{*0}$
channels and is found to be $\pm 0.06$.

As described in Sect. 6 an S-wave contribution of $0.16\pm 0.02$ was found in
the $K^{+}K^{-}$ and $K^{-}\pi^{+}$ mass windows of the $B^{0}\rightarrow\phi
K^{*0}$ candidates. The uncertainty caused by the assumption that this
fraction is the same in $B^{0}$ and $B^{0}_{s}$ decays is estimated to be
$50\%$ of the S-wave contribution. This results in a $\pm 0.08$ contribution
to the systematic uncertainty. This uncertainty also accounts for uncanceled
interference terms between the $K^{*0}$, the $\phi$ and their corresponding
S-waves. These contributions are linear in the sine or cosine of polarization
angles [13] and cancel after integration. The dependence of the acceptance on
the angles violates this cancellation contributing $\pm 0.04$ to the total
$\pm 0.08$ S-wave uncertainty.

The $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow\phi K^{*0}$
final states are very similar and a detector acceptance efficiency ratio $\sim
1$ is expected. However, small effects, such as the mass shift
$M(B^{0}_{s})-M(B^{0})$, translate into slightly different $p_{\rm T}$
distributions for the daughter particles. This results in an efficiency ratio
of $1.005$, as determined from simulation. The deviation of $\pm 0.005$ from
unity is taken as a systematic uncertainty that is propagated to the branching
fraction.

Finally, the uncertainty in the knowledge of the $B^{0}\rightarrow\phi K^{*0}$
decay branching fraction of $\pm 0.6\times 10^{-6}$ is also accounted for and
results in a relative uncertainty of 0.06 in the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decay branching fraction.

A summary of the systematic uncertainties is shown in Table 3. The final
result for the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay branching fraction is

${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(1.10\pm
0.24\,\mathrm{(stat)}\pm 0.14\,\mathrm{(syst)}\pm
0.08\left(\frac{f_{d}}{f_{s}}\right)\right)\times 10^{-6},$

which corresponds to a ratio with the $B^{0}\rightarrow\phi K^{*0}$ decay
branching fraction of:

$\frac{{\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal B}(B^{0}\rightarrow\phi
K^{*0})}=0.113\pm 0.024\,\mathrm{(stat)}\pm 0.013\,\mathrm{(syst)}\pm
0.009\left(\frac{f_{d}}{f_{s}}\right).$

Table 3: Sources of systematic uncertainty in the branching fraction measurement. The total uncertainty is the addition in quadrature of the individual sources. Source | Relative uncertainty in $\cal B$
---|---
Fit model | $0.04\phantom{0}$
$f_{0}$ | $0.06\phantom{0}$
Purity | $0.08\phantom{0}$
Acceptance | $0.005$
$\cal B$($B^{0}\rightarrow\phi K^{*0}$) | $0.06\phantom{0}$
Total | $0.12\phantom{0}$

##  9 Polarization analysis

The $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$
decay proceeds via two intermediate spin-1 particles. The angular distribution
of the decay is described by three transversity amplitudes $A_{0}$,
$A_{\parallel}$ and $A_{\perp}$ [47]. These can be obtained from the
distribution of the decay products in three angles $\theta_{1}$, $\theta_{2}$
and $\varphi$, defined in the helicity frame. The convention for the angles is
shown in Fig. 4. A flavour-averaged and time-integrated polarization analysis
is performed assuming that the $C\\!P$-violating phase is zero and that an
equal amount of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are produced. Under
these assumptions, the decay rate dependence on the polarization angles can be
written as

$\displaystyle\frac{{\rm d}^{3}\Gamma}{{\rm d}{\rm cos}\theta_{1}\,{\rm d}{\rm
cos}\theta_{2}\,{\rm d}\varphi}$ $\displaystyle\propto$
$\displaystyle|A_{0}|^{2}\cos^{2}\theta_{1}\cos^{2}\theta_{2}+|A_{\parallel}|^{2}\frac{1}{2}\sin^{2}\theta_{1}\sin^{2}\theta_{2}\cos^{2}\varphi$
$\displaystyle+$
$\displaystyle|A_{\perp}|^{2}\frac{1}{2}\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin^{2}\varphi+|A_{0}||A_{\parallel}|\cos\delta_{\parallel}\frac{1}{2\sqrt{2}}\sin
2\theta_{1}\sin 2\theta_{2}\cos\varphi.$

Additional terms accounting for the S-wave and interference contributions, as
in Ref. [13], are also considered. These terms are set to the values obtained
for the $B^{0}\rightarrow\phi K^{*0}$ sample.

The polarization fractions are defined from the amplitudes as:
$f_{j}=|A_{j}|^{2}/(|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2})$ (with
$j=0,\parallel,\perp$). In addition to the polarization fractions the cosine
of the phase difference between $A_{0}$ and $A_{\parallel}$,
$\cos\delta_{\parallel}$, is accessible in this study.

Figure 4: Definition of the angles in $B^{0}_{s}\rightarrow\phi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ decays where $\theta_{1}$
($\theta_{2}$) is the $K^{+}$ ($K^{-}$) emission angle with respect to the
direction opposite to the $B^{0}_{s}$ meson in the $\phi$ ($\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$) rest frame and $\varphi$ is the
angle between the $\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ and
$\phi$ decay planes in the $B^{0}_{s}$ rest frame.

The determination of the angular amplitudes depends on the spectrometer
acceptance as a function of the polarization angles $\theta_{1}$ and
$\theta_{2}$. The acceptance was found not to depend on $\varphi$. A
parametrization of the acceptance as a function of $\theta_{1}$ and
$\theta_{2}$ is calculated using simulated data and is used to correct the
differential decay rate by scaling Eq. 9. Additionally, a small correction for
discrepancies in the $p_{\rm T}$ spectrum and the trigger selection of the $B$
mesons between simulation and data is introduced.

The data in a $\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window around
the $B^{0}_{s}$ mass are fitted to the final angular distribution. The fit
accounts for two additional ingredients: the tail of the $B^{0}\rightarrow\phi
K^{*0}$ decays, that are polarized with a longitudinal polarization fraction
of $f_{0}=0.494$ [13], and the combinatorial background, parametrized from the
distributions of events in the high-mass $B$ sideband
$5450<M(K^{+}K^{-}K^{-}\pi^{+})<5840{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
after relaxing the selection requirements. The latter accounts for both the
combinatorial and misidentified $\mathchar 28931\relax^{0}_{b}$ backgrounds.

The systematic uncertainties in the determination of the angular parameters
are calculated modifying the analysis and computing the difference with the
nominal result. Three elements are considered.

  * •

The uncertainty in the S-wave fraction. This is computed modifying the S-wave
contribution by $50\%$ of its value. This covers within $2\,\sigma$ an S-wave
fraction from 0 to $30\%$, consistent with that typically found in decays of
$B$ mesons to final states containing a $K^{*0}$ meson.

  * •

The spectrometer acceptance. This contribution is calculated comparing the
results considering or neglecting the above-mentioned $p_{\rm T}$ and trigger
corrections to the acceptance.

  * •

The combinatorial background. The background model derived from the $B$ mass
sideband is replaced by a uniform angular distribution.

Figure 5: Result of the fit to the angular distribution of the
$B^{0}_{s}\rightarrow\phi\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$
candidates in (left) $\cos\theta_{1}$ and (right) $\cos\theta_{2}$. The red
dotted line corresponds to the combinatorial background under the $B^{0}_{s}$
signal, the green dashed line is the $B^{0}\rightarrow\phi K^{*0}$ signal in
the $B^{0}_{s}$ region and the grey dotted-dashed line corresponds to the sum
of the S-wave and the interference terms.

The different contributions to the systematic uncertainty are given in Table 4
and the one-dimensional projections of the angular distributions are shown
Fig. 5. Other possible systematic sources, such as the uncertainty in the
polarization parameters of the $B^{0}\rightarrow\phi K^{*0}$, are found to be
negligible.

Considering all the above, the values obtained are

$\displaystyle f_{0}$ $\displaystyle=\phantom{-}0.51\pm
0.15\,\mathrm{(stat)}\pm 0.07\,\mathrm{(syst)},$ $\displaystyle f_{\parallel}$
$\displaystyle=\phantom{-}0.21\pm 0.11\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)},$ $\displaystyle\cos\delta_{\parallel}$
$\displaystyle=-0.18\pm 0.52\,\mathrm{(stat)}\pm 0.29\,\mathrm{(syst)}.$

These results for the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay are consistent with the
values measured in $B^{0}\rightarrow\phi K^{*0}$ decays of $f_{0}=0.494\pm
0.036$, $f_{\parallel}=0.212\pm 0.035$ and $\cos\delta_{\parallel}=-0.74\pm
0.10$ [13].

Table 4: Systematic uncertainties of the angular parameters. Effect | $\Delta f_{0}$ | $\Delta f_{\parallel}$ | $\Delta\cos\delta_{\parallel}$
---|---|---|---
S-wave | $0.07\phantom{0}$ | $0.02\phantom{0}$ | $0.29\phantom{0}$
Acceptance | $0.007$ | $0.005$ | $0.002$
Combinatorial background | $0.02\phantom{0}$ | $0.01\phantom{0}$ | $0.01\phantom{0}$
Total | $0.07\phantom{0}$ | $0.02\phantom{0}$ | $0.29\phantom{0}$

##  10 Summary and conclusions

A total of $30\pm 6$ $B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$
candidates have been observed within the mass windows
$1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$746<M(K^{-}\pi^{+})<1046{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The result
translates into a significance of $6.2\,\sigma$. The analysis of the
$K^{+}K^{-}$ and the $K^{-}\pi^{+}$ mass distributions is consistent with
$(84\pm 2)\%$ of the signal originating from resonant $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mesons. The significance of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
resonant contribution is calculated to be $6.1\,\sigma$. The branching
fraction of the decay is measured to be

${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(1.10\pm
0.24\,\mathrm{(stat)}\pm 0.14\,\mathrm{(syst)}\pm
0.08\left(\frac{f_{d}}{f_{s}}\right)\right)\times 10^{-6},$

using the $B^{0}\rightarrow\phi K^{*0}$ decay as a normalization channel. This
result is roughly three times the theoretical expectation in QCD factorization
of $(0.4\,{}^{+0.5}_{-0.3})\times 10^{-6}$ [21] and larger than the
perturbative QCD value of ${(0.65\,{}^{+0.33}_{-0.23})\times 10^{-6}}$ [28],
although the values are compatible within $1\,\sigma$. The result is also
higher than the expectation of ${\cal B}(B^{0}\rightarrow\phi
K^{*0})\times|V_{td}|^{2}/|V_{ts}|^{2}$. Better precision on both the
theoretical and experimental values would allow this channel to serve as a
probe for physics beyond the SM.

An angular analysis of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay results in the polarization
fractions and phase difference

$\displaystyle f_{0}$ $\displaystyle=\phantom{-}0.51\pm
0.15\,\mathrm{(stat)}\pm 0.07\,\mathrm{(syst)},$ $\displaystyle f_{\parallel}$
$\displaystyle=\phantom{-}0.21\pm 0.11\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)},$ $\displaystyle\cos\delta_{\parallel}$
$\displaystyle=-0.18\pm 0.52\,\mathrm{(stat)}\pm 0.29\,\mathrm{(syst)}.$

The small value obtained for the longitudinal polarization fraction follows
the trend of the $b\rightarrow s$ penguin decays $B^{0}\rightarrow\phi
K^{*0}$, $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}_{s}\rightarrow\phi\phi$. The comparison with the decay
$B^{0}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$,
where $f_{0}=0.80^{+0.12}_{-0.13}$ [25], shows a $2\,\sigma$ discrepancy. This
is very interesting since the loop-mediated amplitudes of each decay differ
only in the flavour of the spectator quark. The result is also compatible with
the longitudinal polarization fraction $f_{0}=0.40\pm 0.14$ measured in
$B^{0}\rightarrow\rho^{0}K^{*0}$ decays [11], the penguin amplitude of which
is related to $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ by $d\leftrightarrow s$ exchange.
Finally, the result is smaller than the prediction of perturbative QCD,
$f_{0}=0.712\,{}^{+0.042}_{-0.048}$, given in Ref. [28].

## 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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