EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-008 LHCb-PAPER-2012-040 February 5, 2013

Amplitude analysis and branching fraction measurement of $\kern
2.59189pt\overline{\kern-2.59189ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$

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

An amplitude analysis of the final state structure in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
decay mode is performed using $1.0~{}\rm fb^{-1}$ of data collected by the
LHCb experiment in 7 TeV center-of-mass energy $pp$ collisions produced by the
LHC. A modified Dalitz plot analysis of the final state is performed using
both the invariant mass spectra and the decay angular distributions. Resonant
structures are observed in the $K^{+}K^{-}$ mass spectrum as well as a
significant non-resonant S-wave contribution over the entire $K^{+}K^{-}$ mass
range. The largest resonant component is the $\phi(1020)$, accompanied by
$f_{0}(980)$, $f_{2}^{\prime}(1525)$, and four additional resonances. The
overall branching fraction is measured to be $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})=(7.70\pm 0.08\pm 0.39\pm 0.60)\times 10^{-4}$, where the first
uncertainty is statistical, the second systematic, and the third due to the
ratio of the number of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
to $B^{-}$ mesons produced. The mass and width of the $f_{2}^{\prime}(1525)$
are measured to be $1522.2\pm 2.8^{+5.3}_{-2.0}~{}\rm MeV$ and $84\pm
6^{+10}_{-~{}5}~{}\rm MeV$, respectively. The final state fractions of the
other resonant states are also reported.

Submitted to Physical Review D

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

LHCb collaboration

R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17,f, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, C. Baesso54, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32,
I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, A. Berezhnoy29, R. Bernet37, M.-O. Bettler44, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52,15, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14, O. Deschamps5, F. Dettori39, A. Di Canto11, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V.
Egorychev28, S. Eidelman31, D. van Eijk38, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, I. El Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella14,e, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick35, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, P. Garosi51, J. Garra Tico44, L. Garrido33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T.
Huse49, N. Hussain52, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12,
J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach35,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B.
Khanji20, Y.M. Kim47, O. Kochebina7, V. Komarov36,29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18,35, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, Y. Li3, L. Li Gioi5, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc26, O. Maev27,35, J. Magnin1, M. Maino20, S.
Malde52, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe35,
C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35,e, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis50, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, S. Playfer47, M. Plo Casasus34,
F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro36, W. Qian3, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45,
A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, V. Rives
Molina33, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, A. Romero Vidal34, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M.
Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E.
Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M.
Savrie16,e, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49,35, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35, T. Szumlak24, S.
T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, D.
Tonelli35, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36,
M.T. Tran36, A. Tsaregorodtsev6, P. Tsopelas38, N. Tuning38, M. Ubeda
Garcia35, A. Ukleja25, D. Urner51, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, G. Veneziano36, M. Vesterinen35, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, H. Voss10, C. Voß55,
R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K.
Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,p,
F.F. Wilson46, J. Wishahi9, M. Witek23,35, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.

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

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

##  1 Introduction

The study of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays to
$J/\psi h^{+}h^{-}$, where $h$ is either a pion or kaon, has been used to
measure mixing-induced $C\\!P$ violation in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays [1, 2, 3, 4, 5, 6,
7].‡‡‡Mention of a particular mode implies use of its charge conjugate
throughout this paper. In order to best exploit these decays a better
understanding of the final state composition is necessary. This study has been
reported for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ channel [8]. Here we perform a similar analysis for
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$. While a large $\phi(1020)$ contribution is well known [9]
and the $f_{2}^{\prime}(1525)$ component has been recently observed [10] and
confirmed [11], other components have not heretofore been identified including
the source of S-wave contributions [12]. The tree-level Feynman diagram for
the process is shown in Fig. 1.

Figure 1: Leading order diagram for $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$.

In this paper the $J/\psi K^{+}$ and $K^{+}K^{-}$ mass spectra and decay
angular distributions are used to study resonant and non-resonant structures.
This differs from a classical “Dalitz plot” analysis [13] since the $J/\psi$
meson has spin-1, and its three helicity amplitudes must be considered.

##  2 Data sample and detector

The event sample is obtained using $1.0~{}\rm fb^{-1}$ of integrated
luminosity collected with the LHCb detector [14] using $pp$ collisions at a
center-of-mass energy of 7 TeV. The detector is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. Components include 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 has momentum§§§We work
in units where $c=1$. resolution $\Delta p/p$ that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$. The impact parameter (IP) is defined as
the minimum distance of approach of the track with respect to the primary
vertex. For tracks with large transverse momentum with respect to the proton
beam direction, the IP resolution is approximately 20$\,\upmu\rm m$. 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 trigger [15] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction. Events selected for this analysis are triggered by a
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, where the $J/\psi$ is required at the
software level to be consistent with coming from the decay of a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson by use either of IP
requirements or detachment of the $J/\psi$ from the primary vertex. Monte
Carlo simulations are performed using Pythia [16] with the specific tuning
given in Ref. [17], and the LHCb detector description based on Geant4 [18,
*Allison:2006ve] described in Ref. [20]. Decays of $B$ mesons are based on
EvtGen [21].

##  3 Signal selection and backgrounds

We select $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi K^{+}K^{-}$ candidates trying to simultaneously maximize the signal
yield and reduce the background. Candidate $J/\psi\rightarrow\mu^{+}\mu^{-}$
decays are combined with a pair of kaon candidates of opposite charge, and
then requiring that all four tracks are consistent with coming from a common
decay point. To be considered a $J/\psi\rightarrow\mu^{+}\mu^{-}$ candidate,
particles identified as muons of opposite charge are required to have
transverse momentum, $p_{\rm T}$, greater than 500 MeV, and form a vertex with
fit $\chi^{2}$ per number of degrees of freedom (ndf) less than 11. These
requirements give rise to a large ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ signal over a small background [22]. Only candidates with a dimuon
invariant mass between $-$48 MeV to +43 MeV relative to the observed $J/\psi$
mass peak are selected. The asymmetric requirement is due to final-state
electromagnetic radiation. The two muons are subsequently kinematically
constrained to the known $J/\psi$ mass [9].

Our ring-imaging Cherenkov system allows for the possibility of positively
identifying kaon candidates. Charged tracks produce Cherenkov photons whose
emission angles are compared with those expected for electrons, pions, kaons
or protons, and a likelihood for each species is then computed. To identify a
particular species, the difference between the logarithm of the likelihoods
for two particle hypotheses (DLL) is computed. There are two criteria used:
loose corresponds to DLL$(K-\pi)>0$, while tight has DLL$(K-\pi)>10$ and
DLL$(K-p)>-3$. Unless stated otherwise, we require the tight criterion for
kaon selection.

We select candidate $K^{+}\kern-1.60004ptK^{-}$ combinations if each particle
is inconsistent with having been produced at the primary vertex. For this test
we require that the $\chi^{2}$ formed by using the hypothesis that the IP is
zero be greater than 9 for each track. Furthermore, each kaon must have
$p_{\rm T}>250$ MeV and the scalar sum of the $p_{\rm T}$ of the kaon
candidates must be greater than 900 MeV. To select $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates we further require
that the two kaon candidates form a vertex with $\chi^{2}<10$, and that they
form a candidate $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex
with the $J/\psi$ where the vertex fit $\chi^{2}$/ndf $<5$. We require that
this $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex be more
than $1.5$ mm from the primary vertex, and the angle between the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ momentum vector and the vector
from the primary vertex to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex must be less than 11.8
mrad.

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate invariant
mass distribution is shown in Fig. 2. The vertical lines indicate the signal
and sideband regions, where the signal region extends to $\pm 20$ MeV around
the nominal $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass [9]
and the sidebands extend from 35 MeV to 60 MeV on either side of the peak. The
small peak near 5280 MeV results from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays, and will be subject to
future investigation.

Figure 2: Invariant mass spectrum of $J/\psi K^{+}K^{-}$ combinations. The
vertical lines indicate the signal (black-dotted) and sideband (red-dashed)
regions.

The background consist of combinations of tracks, which have a smooth mass
shape through the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\kern-1.60004ptK^{-}$ region, and peaking contributions caused by
the reflection of specific decay modes where a pion is misidentified as a
kaon. The reflection background that arises from the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$,
where the $\pi^{+}$ is misidentified as a $K^{+}$, is determined from the
number of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ candidates in the
control region $25-200$ MeV above the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.

For each of the candidates in the $J/\psi K^{+}K^{-}$ control region, we
reassign each of the two kaons in turn to the pion mass hypothesis. The
resulting $J/\psi K\pi$ invariant mass distribution is shown in Fig. 3. The
peak at the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass has
$906\pm 51$ candidates, determined by fitting the data to a Gaussian function
for the signal, and a polynomial function for the background. From these
events we estimate the number in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal region, based on a
simulation of the shape of the reflected distribution as a function of $J/\psi
K^{-}K^{+}$ mass. Using simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(892)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*}_{2}(1430)$ samples, we calculate
$309\pm 17$ reflection candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ peak. This number is used as a
constraint in the mass fit described below.

Figure 3: Invariant mass distribution for $J/\psi K^{+}K^{-}$ candidates
$25-200$ MeV above the $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$
mass, reinterpreted as $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi K^{\mp}\pi^{\pm}$
events. The fit is to a signal Gaussian whose mass and width are allowed to
vary as well as the polynomial background.

To determine the number of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal candidates we perform a
fit to the candidate $J/\psi K^{+}K^{-}$ invariant mass spectrum shown in Fig.
4.

Figure 4: Fit to the invariant mass spectrum of $J/\psi K^{+}K^{-}$
combinations. The dotted (black) line is the combinatorial background, the
dashed (red) shape shows the misidentified $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi
K^{-}\pi^{+}$decays, and the solid (blue) curve shows the total. The vertical
dashed lines indicate the signal region.

The fit function is the sum of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal component,
combinatorial background, and the contribution from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
reflections. The signal is modeled by a double-Gaussian function with a common
mean. The combinatorial background is described by a linear function. The
reflection background is constrained as described above. The mass fit gives
19,195$\pm$150 signal together with $894\pm 24$ combinatorial background
candidates within $\pm 20$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.

We use the decay $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}$ as the normalization channel for branching fraction
determinations. The selection criteria are similar to those used for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\kern-1.60004ptK^{-}$,
except for particle identification as here a loose kaon identification
criterion is used. Figure 5 shows the $J/\psi K^{-}$ mass distribution. The
signal is fit with a double-Gaussian function and a linear function is used to
fit the combinatorial background. There are 342,786$\pm$661 signal and
10,195$\pm$134 background candidates within $\pm 20$ MeV of the $B^{-}$ peak.

Figure 5: Fit to the invariant mass spectrum of $J/\psi K^{-}$ candidates. The
dotted line shows the combinatorial background and the solid (blue) curve is
the total.

##  4 Analysis formalism

One of the goals of this analysis is to determine the intermediate states in
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ decay within the context of an isobar model [23, 24], where we sum
the resonant and non-resonant components testing if they explain the invariant
mass squared and angular distributions. We also determine the absolute
branching fractions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(1020)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{2}^{\prime}(1525)$ final states and the mass and width of the
$f_{2}^{\prime}(1525)$ resonance. Another important goal is to understand the
S-wave content in the $\phi(1020)$ mass region.

Four variables completely describe the decay of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
with $J/\psi\rightarrow\mu^{+}\mu^{-}$. Two are the invariant mass squared of
$J/\psi K^{+}$, $s_{12}\equiv m^{2}(J/\psi K^{+})$, and the invariant mass
squared of $K^{+}K^{-}$, $s_{23}\equiv m^{2}(K^{+}K^{-})$. The other two are
the $J/\psi$ helicity angle, $\theta_{J/\psi}$, which is the angle of the
$\mu^{+}$ in the $J/\psi$ rest frame with respect to the $J/\psi$ direction in
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and the
angle between the $J/\psi$ and $K^{+}K^{-}$ decay planes, $\chi$, in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame. To simplify
the probability density function (PDF), we analyze the decay process after
integrating over the angular variable $\chi$, which eliminates several
interference terms.

###  4.1 The model for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$

In order to perform an amplitude analysis a PDF must be constructed that
models correctly the dynamical and kinematic properties of the decay. The PDF
is separated into two components, one describing signal, $S$, and the other
background, $B$. The overall PDF given by the sum is

$\displaystyle F(s_{12},s_{23},\theta_{J/\psi})$ $\displaystyle=$
$\displaystyle\frac{1-f_{\rm com}-f_{\rm refl}}{{\cal{N}}_{\rm
sig}}\varepsilon(s_{12},s_{23},\theta_{J/\psi})S(s_{12},s_{23},\theta_{J/\psi})$
$\displaystyle+$ $\displaystyle B(s_{12},s_{23},\theta_{J/\psi}),$

where $\varepsilon$ is the detection efficiency. The background is described
by the sum of combinatorial background, $C$, and reflection, $R$, functions

$B(s_{12},s_{23},\theta_{J/\psi})=\frac{f_{\rm com}}{{\cal{N}}_{\rm
com}}C(s_{12},s_{23},\theta_{J/\psi})+\frac{f_{\rm refl}}{{\cal{N}}_{\rm
refl}}R(s_{12},s_{23},\theta_{J/\psi}),$ (2)

where $f_{\rm com}$ and $f_{\rm refl}$ are the fractions of the combinatorial
background and reflection, respectively, in the fitted region. The fractions
$f_{\rm com}$ and $f_{\rm refl}$ obtained from the mass fit are fixed for the
subsequent analysis.

The normalization factors are given by

$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(s_{12},s_{23},\theta_{J/\psi})S(s_{12},s_{23},\theta_{J/\psi})\,ds_{12}\,ds_{23\,}d\cos\theta_{J/\psi},$
$\displaystyle{\cal{N}}_{\rm com}$ $\displaystyle=$
$\displaystyle\int\\!C(s_{12},s_{23},\theta_{J/\psi})\,ds_{12\,}ds_{23\,}d\cos\theta_{J/\psi},$
(3) $\displaystyle{\cal{N}}_{\rm refl}$ $\displaystyle=$
$\displaystyle\int\\!R(s_{12},s_{23},\theta_{J/\psi})\,ds_{12}\,ds_{23\,}d\cos\theta_{J/\psi}.$

This formalism similar to that used by Belle in their analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow K^{-}\pi^{+}\chi_{c1}$
[25], and later used by LHCb for the analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ [8].

The invariant mass squared of $J/\psi K^{+}$ versus $K^{+}K^{-}$ is shown in
Fig. 6 for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi K^{+}K^{-}$ candidates. No structure is seen in
$m^{2}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})$. There are however
visible horizontal bands in the $K^{+}K^{-}$ mass squared spectrum, the most
prominent of which correspond to the $\phi(1020)$ and $f_{2}^{\prime}(1525)$
resonances. These and other structures in $m^{2}(K^{+}K^{-})$ are now
examined.

Figure 6: Distribution of $m^{2}(K^{+}K^{-})$ versus $m^{2}(J/\psi K^{+})$ for
$\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ candidate decays within
$\pm 20$ MeV of the $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$
mass. The horizontal bands result from the $\phi(1020)$ and
$f_{2}^{\prime}(1525)$ resonances.

The signal function is given by the coherent sum over resonant states that
decay into $K^{+}K^{-}$, plus a possible non-resonant S-wave
contribution¶¶¶The interference terms between different helicities are zero
because we integrate over the angular variable $\chi$.

$S(s_{12},s_{23},\theta_{J/\psi})=\sum_{\lambda=0,\pm
1}\left|\sum_{i}a^{R_{i}}_{\lambda}e^{i\phi^{R_{i}}_{\lambda}}\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{J/\psi})\right|^{2},$
(4)

where $\mathcal{A}_{\lambda}^{R_{i}}(s_{12},s_{23},\theta_{J/\psi})$ describes
the decay amplitude via an intermediate resonance state $R_{i}$ with helicity
$\lambda$. Note that the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ has
the same helicity as the intermediate $K^{+}K^{-}$ resonance. Each $R_{i}$ has
an associated amplitude strength $a_{\lambda}^{R_{i}}$ and a phase
$\phi_{\lambda}^{R_{i}}$ for each helicity state $\lambda$. The amplitude for
resonance $R$, for each $i$, is given by

$\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi})=F_{B}^{(L_{B})}\;A_{R}(s_{23})\;F_{R}^{(L_{R})}\;T_{\lambda}(\theta_{KK})\Big{(}\frac{P_{B}}{m_{B}}\Big{)}^{L_{B}}\;\Big{(}\frac{P_{R}}{\sqrt{s_{23}}}\Big{)}^{L_{R}}\;\Theta_{\lambda}(\theta_{J/\psi}),$
(5)

where $P_{R}$ is the momentum of either of the two kaons in the di-kaon rest
frame, $m_{B}$ is the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
mass, $P_{B}$ is the magnitude of the $J/\psi$ three-momentum in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and
$F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson and $R_{i}$ resonance
decay form factors. The orbital angular momenta between the $J/\psi$ and
$K^{+}K^{-}$ system is given by $L_{B}$, and the orbital angular momentum in
the $K^{+}K^{-}$ decay is given by $L_{R}$; the latter is the same as the spin
of the $K^{+}K^{-}$ system. Since the parent $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ has spin-0 and the $J/\psi$ is
a vector, when the $K^{+}K^{-}$ system forms a spin-0 resonance, $L_{B}=1$ and
$L_{R}=0$. For $K^{+}K^{-}$ resonances with non-zero spin, $L_{B}$ can be 0, 1
or 2 (1, 2 or 3) for $L_{R}=1(2)$ and so on. We take the lowest $L_{B}$ as the
default value and consider the other possibilities in the systematic
uncertainty.

The Blatt-Weisskopf barrier factors $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$
[26] are

$\displaystyle F^{(0)}$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle
F^{(1)}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{1+z_{0}}}{\sqrt{1+z}},$
(6) $\displaystyle F^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{z_{0}^{2}+3z_{0}+9}}{\sqrt{z^{2}+3z+9}}.$

For the $B$ meson $z=r^{2}P_{B}^{2}$, where $r$, the hadron scale, is taken as
5.0 GeV-1; for the $R$ resonance $z=r^{2}P_{R}^{2}$, and $r$ is taken as 1.5
GeV-1 [27]. In both cases $z_{0}=r^{2}P_{0}^{2}$ where $P_{0}$ is the decay
daughter momentum at the pole mass; for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum is used, while for the
$R$ resonances the kaon momentum is used.

In the helicity formalism, the angular term, $T_{\lambda}(\theta_{KK})$ is
defined as

$T_{\lambda}(\theta_{KK})=d^{J}_{\lambda 0}(\theta_{KK}),$ (7)

where $d$ is the Wigner $d$-function, $J$ is the resonance spin, $\theta_{KK}$
is the helicity angle of the $K^{+}$ in the $K^{+}K^{-}$ rest frame with
respect to the $K^{+}K^{-}$ direction in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ rest frame, and may be
calculated directly from the other variables as

$\cos\theta_{KK}=\frac{\left[m^{2}(J/\psi K^{+})-m^{2}(J/\psi
K^{-})\right]m(K^{+}K^{-})}{4P_{R}P_{B}m_{B}}.$ (8)

The $J/\psi$ helicity dependent term $\Theta_{\lambda}(\theta_{J/\psi})$ is
defined as

$\displaystyle\Theta_{\lambda}(\theta_{J/\psi})$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}\theta_{J/\psi}}\;\;\;\;\;\;\;\;\;\text{for}\;\;\lambda=0$
(9) $\displaystyle=$
$\displaystyle\sqrt{\frac{1+\cos^{2}\theta_{J/\psi}}{2}}\;\;\text{for}\;\;|\lambda|=1.$

The mass squared shape of each resonance, $R$ is described by the function
$A_{R}(s_{23})$. In most cases this is a Breit-Wigner (BW) amplitude. When a
decay channel opens close to the resonant mass, complications arise, since the
proximity of the second threshold distorts the line shape of the amplitude.
The $f_{0}(980)$ can decay to either $\pi\pi$ or $KK$. While the $\pi\pi$
channel opens at much lower masses, the $K^{+}K^{-}$ decay channel opens near
the resonance mass. Thus, for the $f_{0}(980)$ we use a Flatté model [28] that
takes into account these coupled channels.

We describe the BW amplitude for a resonance decaying into two spin-0
particles, labeled as 2 and 3, as

$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma(s_{23})}~{},$ (10)

where $m_{R}$ is the resonance mass, $\Gamma(s_{23})$ is its energy-dependent
width that is parametrized as

$\Gamma(s_{23})=\Gamma_{0}\left(\frac{P_{R}}{P_{R_{0}}}\right)^{2L_{R}+1}\left(\frac{m_{R}}{\sqrt{s_{23}}}\right)F^{2}_{R}~{}.$
(11)

Here $\Gamma_{0}$ is the decay width when the invariant mass of the daughter
combinations is equal to $m_{R}$.

The Flatté mass shape is parametrized as

$A_{R}(s_{23})=\frac{1}{m_{R}^{2}-s_{23}-im_{R}(g_{\pi\pi}\rho_{\pi\pi}+g_{KK}\rho_{KK})},$
(12)

where the constants $g_{\pi\pi}$ and $g_{KK}$ are the $f_{0}(980)$ couplings
to $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ final states, respectively. The $\rho$
factors are given by Lorentz-invariant phase space

$\displaystyle\rho_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sqrt{1-\frac{4m^{2}_{\pi^{\pm}}}{s_{23}}}+\frac{1}{3}\sqrt{1-\frac{4m^{2}_{\pi^{0}}}{s_{23}}},$
(13) $\displaystyle\rho_{KK}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{\pm}}}{s_{23}}}+\frac{1}{2}\sqrt{1-\frac{4m^{2}_{K^{0}}}{s_{23}}}.$
(14)

For non-resonant processes, the amplitude
$\mathcal{A}(s_{12},s_{23},\theta_{J/\psi})$ is constant over the variables
$s_{12}$ and $s_{23}$, but has an angular dependence due to the $J/\psi$
decay. The amplitude is derived from Eq. (5), assuming that the non-resonant
$K^{+}K^{-}$ contribution is an S-wave (i.e. $L_{R}=0$, $L_{B}=1$) and is
uniform in phase space (i.e. $A_{R}=1$),

$\mathcal{A}(s_{12},s_{23},\theta_{J/\psi})=\frac{P_{B}}{m_{B}}\sqrt{\sin^{2}\theta_{J/\psi}}.$
(15)

###  4.2 Detection efficiency

The detection efficiency is determined from a phase space simulation sample
containing $3.4\times 10^{6}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
events with $J/\psi\rightarrow\mu^{+}\mu^{-}$. We also use a separate sample
of $1.3\times 10^{6}$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ events.
The $p$ and $p_{\rm T}$ distributions of the generated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are weighted to match
the distributions found using $J/\psi\phi$ data. The simulation is also
corrected by weighting for difference between the simulated kaon detection
efficiencies and the measured ones determined by using a sample of
$D^{*+}\rightarrow\pi^{+}(D^{0}\rightarrow K^{-}\pi^{+})$ events.

Next we describe the efficiency in terms of the analysis variables. Both
$s_{12}$ and $s_{13}$ range from 12.5 $\rm GeV^{2}$ to 24.0 $\rm GeV^{2}$,
where $s_{13}$ is defined below, and thus are centered at $s_{0}=18.25~{}{\rm
GeV}^{2}$. We model the detection efficiency using the dimensionless symmetric
Dalitz plot observables

$x=\left(s_{12}-s_{0}\right)/\left(1~{}{\rm
GeV}^{2}\right),~{}~{}~{}~{}~{}~{}y=\left(s_{13}-s_{0}\right)/\left(1~{}{\rm
GeV}^{2}\right),$ (16)

and the angular variable $\theta_{J/\psi}$. The observables $s_{12}$ and
$s_{13}$ are related to $s_{23}$ as

$s_{12}+s_{13}+s_{23}=m^{2}_{B}+m^{2}_{J/\psi}+m^{2}_{K^{+}}+m^{2}_{K^{-}}.$
(17)

To parametrize this efficiency, we fit the $\cos\theta_{J/\psi}$ distributions
of the $J/\psi K^{+}K^{-}$ and $J/\psi\phi$ simulation samples in bins of
$m^{2}(K^{+}K^{-})$ with the function

$\varepsilon_{2}(s_{23},\theta_{J/\psi})=\frac{1+a\cos^{2}\theta_{J/\psi}}{2+2a/3},$
(18)

giving values of $a$ as a function of $m^{2}(K^{+}K^{-})$. The resulting
distribution, shown in Fig. 7, is described by an exponential function

$a(s_{23})=\exp(a_{1}+a_{2}s_{23}),$ (19)

with $a_{1}=-0.76\pm 0.18$ and $a_{2}=(-1.02\pm 0.15)~{}\rm GeV^{-2}$.
Equation (18) is normalized with respect to $\cos\theta_{J/\psi}$.

Figure 7: Exponential fit to the efficiency parameter $a(s_{23})$. The point
near the $\phi(1020)$ meson mass is determined more precisely due to the use
of a large simulation sample.

The efficiency in $\cos\theta_{J/\psi}$ depends on $s_{23}$, and is observed
to be independent of $s_{12}$. Thus the detection efficiency can be expressed
as

$\varepsilon(s_{12},s_{23},\theta_{J/\psi})=\varepsilon_{1}(x,y)\times\varepsilon_{2}(s_{23},\theta_{J/\psi}).$
(20)

After integrating over $\cos\theta_{J/\psi}$, Eq. (20) becomes

$\int_{-1}^{+1}\varepsilon(s_{12},s_{23},\theta_{J/\psi})d\cos\theta_{J/\psi}=\varepsilon_{1}(x,y),$
(21)

and is modeled by a symmetric fifth order polynomial function given by

$\displaystyle\varepsilon_{1}(x,y)$ $\displaystyle=$ $\displaystyle
1+\epsilon^{\prime}_{1}(x+y)+\epsilon^{\prime}_{2}(x+y)^{2}+\epsilon^{\prime}_{3}xy+\epsilon^{\prime}_{4}(x+y)^{3}+\epsilon^{\prime}_{5}xy(x+y)$
(22)
$\displaystyle+\epsilon^{\prime}_{6}(x+y)^{4}+\epsilon^{\prime}_{7}xy(x+y)^{2}+\epsilon^{\prime}_{8}x^{2}y^{2}+\epsilon^{\prime}_{9}(x+y)^{5}$
$\displaystyle+\epsilon^{\prime}_{10}xy(x+y)^{3}+\epsilon^{\prime}_{11}x^{2}y^{2}(x+y),$

where $\epsilon^{\prime}_{i}$ are the fit parameters.

Figure 8: Parametrized detection efficiency as a function of
$m^{2}(K^{+}K^{-})$ versus $m^{2}(J/\psi K^{+})$. The z-axis scale is
arbitrary.

The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-}$ phase space simulation sample is modeled with the polynomial
function. The fitted function is shown in Fig. 8, and the projections of the
fit are shown in Fig. 9. The efficiency is well described by the
parametrization.

Figure 9: Projections of the invariant mass squared (a) $K^{+}K^{-}$ and (b)
$J/\psi K^{+}$ from the simulation used to measure the efficiency parameters.
The points represent the generated event distributions and the curves the
polynomial fit.

For the region within $\pm 20$ MeV of the $\phi(1020)$ mass, the
$\cos\theta_{KK}$ acceptance is used separately, due to the large number of
signal events. Here the $\cos\theta_{KK}$ distribution shows a variation in
efficiency, which can be parametrized using the efficiency function

$A(\theta_{KK})=\frac{1+\epsilon^{\prime}_{12}\cos^{2}\theta_{KK}}{1+\epsilon^{\prime}_{12}/3},$
(23)

where the parameter $\epsilon^{\prime}_{12}$ is measured from a fit to the
simulated $J/\psi\phi$ sample with $\varepsilon_{1}(x,y)\times
A(\theta_{KK})$, giving $\epsilon^{\prime}_{12}=-0.099\pm 0.010$, as shown in
Fig. 10.

Figure 10: Distribution of $\cos\theta_{KK}$ for the $J/\psi\phi$ simulated
sample fitted with $\varepsilon_{1}(x,y)\times A(\theta_{KK})$, within $\pm
20$ MeV of the $\phi(1020)$ mass.

The mass resolution is $\sim 0.7$ MeV at the $\phi(1020)$ mass peak, which is
added to the fit model by increasing the Breit-Wigner width of the
$\phi(1020)$ to 4.59 MeV.

###  4.3 Background composition

The shape of the combinatorial background is modeled as

$C(s_{12},s_{23},\theta_{J/\psi})=\left[C_{1}(s_{12},s_{23})\frac{P_{B}}{m_{B}}+\frac{c_{0}}{(m^{2}_{0}-s_{23})^{2}+m^{2}_{0}\Gamma_{0}^{2}}\right]\times\left(1+\alpha\cos^{2}\theta_{J/\psi}\right),$
(24)

where $C_{1}(s_{12},s_{23})$ is parametrized as

$C_{1}(s_{12},s_{23})=1+c_{1}(x+y)+c_{2}(x+y)^{2}+c_{3}xy+c_{4}(x+y)^{3}+c_{5}xy(x+y),$
(25)

with $c_{i}$, $m_{0}$, $\Gamma_{0}$ and $\alpha$ as the fit parameters. The
variables $x$ and $y$ are defined in Eq. (16).

Figure 11 shows the mass squared projections from the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass sidebands with the fit
projections overlaid. The $\chi^{2}/\rm ndf$ of the fit is 291/305. The value
of $\alpha$ is determined by fitting the $\cos\theta_{J/\psi}$ distribution of
background, as shown in Fig. 12, with a function of the form
$1+\alpha\cos^{2}\theta_{J/\psi}$, yielding $\alpha=-0.14\pm 0.08$.

Figure 11: Invariant mass squared projections of (a) $K^{+}K^{-}$ and (b)
$J/\psi K^{+}$ from the background Dalitz plot of candidates in the $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}$ mass sidebands. Figure 12:
Distribution of $\cos\theta_{J/\psi}$ from the background sample fit with the
function $1+\alpha\cos^{2}\theta_{J/\psi}$.

The reflection background is parametrized as

$R(s_{12},s_{23},\theta_{J/\psi})=R_{1}(s_{12},s_{23})\times\left(1+\beta\cos^{2}\theta_{J/\psi}\right),$
(26)

where $R_{1}(s_{12},s_{23})$ is modeled using the simulation; the projections
of $s_{12}$ and $s_{23}$ are shown in Fig. 13.

Figure 13: Projections of the reflection background in the variables (a)
$m^{2}(K^{+}K^{-})$ and (b) $m^{2}(J/\psi K^{+})$, obtained from $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}(892)$ and $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*}_{2}(1430)$ simulations. Figure 14:
Distribution of $\cos\theta_{J/\psi}$ for the reflection fit with the function
$1+\beta\cos^{2}\theta_{J/\psi}$.

The $J/\psi$ helicity angle dependent part of the reflections is modeled as
$1+\beta\cos^{2}\theta_{J/\psi}$, where the parameter $\beta$ is obtained from
a fit to the simulated $\cos\theta_{J/\psi}$ distribution, shown in Fig. 14,
giving $\beta=-0.19\pm 0.01$.

##  5 Final state composition

###  5.1 Resonance models

The resonances that are likely to contribute are produced from the $s\bar{s}$
system in Fig. 1, and thus are isoscalar ($I=0$). The $K^{+}K^{-}$ system in
the decay $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi K^{+}K^{-}$ can, in principle, have zero or any positive integer
angular momentum. Both the $P$-parity and $C$-parity of $K^{+}K^{-}$ pair in a
state of relative angular momentum $L$ are given by $(-1)^{L}$. Therefore the
allowed resonances decaying to $K^{+}K^{-}$ are limited to
$J^{PC}=0^{++},~{}1^{--},~{}2^{++},~{}...,$ with isospin $I=0$. In the
kinematically accessible mass range up to 2 GeV, resonances with
$J^{PC}=3^{--}$ or higher are not expected and thus the subsequent analysis
only uses spins up to $J=2$. Possible resonance candidates are listed in Table
1. There could also be a contribution from non-resonant events which we assume
to be S-wave and evenly distributed over the available phase space.

Table 1: Possible resonance candidates in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$ decay mode. Spin | Helicity | Resonance | Amplitude
---|---|---|---
0 | 0 | $f_{0}(980)$ | Flatté
0 | 0 |  $f_{0}(1370)$, $f_{0}(1500)$, $f_{0}(1710)$ | BW
1 | $0,\pm 1$ |  $\phi(1020)$, $\phi(1680)$ | BW
2 | $0,\pm 1$ |  $f_{2}(1270)$, $f_{2}^{\prime}(1525)$, $f_{2}(1640)$, | BW
|  |  $f_{2}(1750)$, $f_{2}(1950)$ |

To study the resonant structures of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
we use 20, 425 candidates with an invariant mass within $\pm 20$ MeV of the
observed $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak.
This includes both signal and background, with $94\%$ signal purity. We begin
our analysis considering only the resonance components
$\phi(1020),~{}f_{2}^{\prime}(1525)$ and a non-resonant component, established
in our earlier measurement [10], and add resonances until no others are found
with more than two standard deviation statistical significance (2$\sigma$).
The significance is estimated from the fit fraction divided by its statistical
uncertainty. Our best fit model includes a non-resonant component and 8
resonance states: $\phi(1020)$, $f_{0}(980)$, $f_{0}(1370)$,
$f_{2}^{\prime}(1525)$, $f_{2}(1640)~{}(\rm|\lambda|=1)$,
$\phi(1680)~{}(\rm|\lambda|=1),$∥∥∥The $f_{2}(1640)~{}(\lambda=0)$ and
$\phi(1680)~{}(\lambda=0)$ components have less than two standard deviation
significance when added separately to the fit, and therefore are not included
in the best fit model. $f_{2}(1750)$, and $f_{2}(1950)$. Most of the
resonances considered here are well established except for the modes
$f_{2}(1640)$, $f_{2}(1750)$, and $f_{2}(1950)$. Although the existence of
$f_{2}(1640)$ is not confirmed yet [9], the right shoulder of
$f_{2}^{\prime}(1525)$ fits better when we add this state. The presence of
multiple broad overlapping resonances in this region may indicate a failure of
the isobar model used in this analysis, but with the present data sample
alternative descriptions are not feasible. Indeed, the situation is not clear
for the resonance states in the vicinity of 1750 MeV. The PDG lists a spin-0
resonance, $f_{0}(1710)$, around 1.72 GeV of $K^{+}K^{-}$ invariant mass [9].
The Belle collaboration observed a resonance in the vicinity of 1.75 GeV with
$J^{PC}=(\rm even)^{++}$ in their study of $\gamma\gamma\rightarrow
K^{+}K^{-}$ [29], but could not establish its spin. A state of mass
1767$\pm$14 MeV was seen by the L3 collaboration decaying into
$K^{0}_{S}K^{0}_{S}$ with $J=2$ [30]. We find that our data are better fit
including the $f_{2}(1750)$ mode. If we substitute either the $f_{0}(1710)$ or
$f_{0}(1750)$ resonance the fit is worsened, as the $\rm-ln\mathcal{L}$
increase by 59 and 7 units, respectively.

In the same analysis of $\gamma\gamma\rightarrow K^{+}K^{-}$, Belle also
observed the $f_{2}(1950)$ [29] resonance. We include this state in our best
fit model. Furthermore, we do not expect significant contributions from the
$f_{2}(1270)$ and $f_{0}(1500)$ resonances, since the PDG branching fractions
are much larger in the $\pi^{+}\pi^{-}$ final state than in $K^{+}K^{-}$ [9]
and we did not see significant contributions from these two resonances in the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ final state [8]. Therefore, these two resonances are not
considered in the best fit model. However, we add these states, in turn, to
the best fit model in order to test for their possible presence.

The masses and widths of the BW resonances are listed in Table 2. When used in
the fit they are fixed to the central values, except for the
$f_{2}^{\prime}(1525)$, whose mass and width are allowed to vary.

Table 2: Breit-Wigner resonance parameters. Resonance | Mass (MeV) | Width (MeV) | Source
---|---|---|---
$\phi(1020)$ | $1019.46$ | $\pm$ | $0.02$ | $4.26$ | $\pm$ | $0.04$ | PDG [9]
$f_{2}(1270)$ | $1275$ | $\pm$ | $1$ | $185$ | $\pm$ | $3$ | PDG [9]
$f_{0}(1370)$ | $1475$ | $\pm$ | $6$ | $113$ | $\pm$ | $11$ | LHCb [8]
$f_{0}(1500)$ | $1505$ | $\pm$ | $6$ | $109$ | $\pm$ | $7$ | PDG [9]
$f_{2}^{\prime}(1525)$ | $1525$ | $\pm$ | $5$ | $73$ | $\pm$ | $6$ | PDG [9]
$f_{2}(1640)$ | $1639$ | $\pm$ | $6$ | $99$ | $\pm$ | $60$ | PDG [9]
$\phi(1680)$ | $1680$ | $\pm$ | $20$ | $150$ | $\pm$ | $50$ | PDG [9]
$f_{0}(1710)$ | $1720$ | $\pm$ | $6$ | $135$ | $\pm$ | $8$ | PDG [9]
$f_{2}(1750)$ | $1737$ | $\pm$ | $9$ | $151$ | $\pm$ | $33$ | Belle [29]
$f_{2}(1950)$ | $1980$ | $\pm$ | $14$ | $297$ | $\pm$ | $13$ | Belle [29]

The $f_{0}(980)$ is described by a Flatté resonance shape, see Eq. (12). The
parameters describing the function are the mass, and the couplings
$g_{\pi\pi}$ and $g_{KK}$, which are fixed in the fit from the previous
analysis of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ [8]. The parameters are $m_{0}=939.9\pm 6.3$ MeV,
$g_{\pi\pi}=199\pm 30$ MeV and $g_{KK}/g_{\pi\pi}=3.0\pm 0.3$. All background
and efficiency parameters are fixed in the fit.

To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as

$\mathcal{L}=\prod_{i=1}^{N}F(s^{i}_{12},s^{i}_{23},\theta^{i}_{J/\psi}),$
(27)

where $N$ is the total number of candidates, and $F$ is the total PDF defined
in Eq. (4.1). The PDF normalization is accomplished by first normalizing the
$J/\psi$ helicity dependent part by analytical integration, and then for the
mass dependent part using numerical integration over 400$\times$800 bins.

The fit determines the relative values of the amplitude strengths,
$a_{\lambda}^{R_{i}}$, and phases, $\phi_{\lambda}^{R_{i}}$, defined in Eq.
(4). We choose to fix $a_{0}^{\phi(1020)}=1$. As only relative phases are
physically meaningful, one phase in each helicity grouping must be fixed. In
addition, because $J/\psi K^{+}\kern-1.60004ptK^{-}$ is a self-charge-
conjugate mode and does not determine the initial $B$ flavor, the signal
function is an average of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$. If we consider no
$K^{+}\kern-1.60004ptK^{-}$ partial-waves of a higher order than D-wave, then
we can express the differential decay rate
($d\Gamma/dm_{KK}\,d\cos\theta_{KK\,}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$) derived from Eq. (4) in terms of S-, P-, and D-waves including
helicity 0 and $\pm 1$ components. The differential decay rates for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $B^{0}_{s}$, respectively
are

$\displaystyle\frac{d\overline{\Gamma}}{dm_{KK}\,d\cos\theta_{KK\,}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=$ $\displaystyle\hskip 11.38109pt\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}+{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{KK}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{KK}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{J/\psi}$
$\displaystyle+\left|{\cal A}^{s}_{P_{\pm 1}}e^{i\phi^{s}_{P_{\pm
1}}}\sqrt{\frac{1}{2}}\sin\theta_{KK}+{\cal A}^{s}_{D_{\pm
1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{KK}\cos\theta_{KK}\right|^{2}\frac{1+\cos^{2}\theta_{J/\psi}}{2},$
(28)

and

$\displaystyle\frac{d\Gamma}{dm_{KK}\,d\cos\theta_{KK\,}d\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}=$ $\displaystyle\hskip 11.38109pt\left|{\cal
A}^{s}_{S_{0}}e^{i\phi^{s}_{S_{0}}}-{\cal
A}^{s}_{P_{0}}e^{i\phi^{s}_{P_{0}}}\cos\theta_{KK}+{\cal
A}^{s}_{D_{0}}e^{i\phi^{s}_{D_{0}}}\left(\frac{3}{2}\cos^{2}\theta_{KK}-\frac{1}{2}\right)\right|^{2}\sin^{2}\theta_{J/\psi}$
$\displaystyle+\left|{\cal A}^{s}_{P_{\pm 1}}e^{i\phi^{s}_{P_{\pm
1}}}\sqrt{\frac{1}{2}}\sin\theta_{KK}-{\cal A}^{s}_{D_{\pm
1}}e^{i\phi^{s}_{D_{\pm
1}}}\sqrt{\frac{3}{2}}\sin\theta_{KK}\cos\theta_{KK}\right|^{2}\frac{1+\cos^{2}\theta_{J/\psi}}{2},$
(29)

where ${\cal A}^{s}_{k_{\lambda}}$ and $\phi^{s}_{k_{\lambda}}$ are the sum of
amplitudes and reference phases, for the spin-$k$ resonance group,
respectively. The decay rate for $B^{0}_{s}$ is similar to that of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, except
$\theta_{K^{+}\kern-1.12003ptK^{-}}$ and $\theta_{J/\psi}$ are now changed to
$\pi-\theta_{K^{+}\kern-1.12003ptK^{-}}$ and $\pi-\theta_{J/\psi}$
respectively, as a result of using $K^{-}$ and $\mu^{-}$ to define the
helicity angles and hence the signs change in front of the ${\cal
A}^{s}_{P_{0}}$ and ${\cal A}^{s}_{D_{\pm 1}}$ terms.

Summing Eqs. (28) and (29) results in cancellation of the interference
involving $\lambda=0$ terms for spin-1, and the $\lambda=\pm 1$ terms for
spin-2, as they appear with opposite signs for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $B^{0}_{s}$ decays.
Therefore we have to fix one phase in the spin-1 ($\lambda=0$) group
($\phi^{s}_{P_{0}}$) and one in the spin-2 ($\lambda=\pm 1$) group
($\phi^{s}_{D_{\pm 1}}$). The other phases in each corresponding group are
determined relative to that of the fixed resonance.

###  5.2 Fit results

The goodness of fit is calculated from 3D partitions of $s_{12}$, $s_{23}$ and
$\cos\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. We use the
Poisson likelihood $\chi^{2}$ [31] defined as

$\chi^{2}=2\sum_{i=1}^{N}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(30)

where $n_{i}$ is the number of candidates in the three dimensional bin $i$ and
$x_{i}$ is the expected number of candidates in that bin according to the
fitted likelihood function. An adaptive binning algorithm is used, requiring a
minimum of 25 entries in each bin. The associated number of degrees of freedom
(ndf) is $N-k-1$, where $k$ is the number of free parameters in the likelihood
function. The $\chi^{2}/\text{ndf}$ and the negative of the logarithm of the
likelihood, $\rm-ln\mathcal{L}$, of the fits are given in Table 3. Starting
values of parameters are varied in order to ensure that global likelihood
minimums are found rather than local minimums.

Table 3: $\chi^{2}/\text{ndf}$ and $\rm-ln\mathcal{L}$ of different resonance models. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$
---|---|---
Best fit | 29,275 | 649/545 = 1.1908
Best fit + $f_{2}(1270)$ | 29,273 | 644/541 = 1.1911
Best fit + $f_{0}(1500)$ | 29,274 | 647/543 = 1.1915

Attempts to add one more resonance such as $f_{2}(1270)$ and $f_{0}(1500)$
improve the $-\rm ln\mathcal{L}$ marginally, but the $\chi^{2}/\rm ndf$ are
worse than the best fit model. We retain only those resonances that are more
than $2\sigma$ significant, except for the $f_{2}(1750)$ where we allow the
$|\lambda|=1$ component, since the $\lambda=0$ component is significant. For
models with one more resonance, the additional components never have more than
2$\sigma$ significance. Figure 15 shows the projection of $m^{2}(K^{+}K^{-})$
for the best fit model, the $m^{2}(J/\psi K^{+})$ and $\cos\theta_{J/\psi}$
projections are displayed in Fig. 16. The projection of the $K^{+}K^{-}$
invariant mass spectrum is shown in Fig. 17.

Figure 15: Dalitz plot fit projection of $m^{2}(K^{+}K^{-})$ using a
logarithmic scale. The points with error bars are data, the (black) dotted
curve shows the combinatorial background, the (red) dashed curve indicates the
reflection from the misidentified $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
decays, and the (blue) solid line represents the total. Figure 16: Dalitz plot
fit projections of (a) $m^{2}(J/\psi K^{+})$ and (b) $\cos\theta_{J/\psi}$.
The points with error bars are data, the (black) dotted curve shows the
combinatorial background, the (red) dashed curve indicates the reflection from
the misidentified $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow
J/\psi K^{-}\pi^{+}$ decays, and the (blue) solid line represents the total
fit results. Figure 17: Dalitz fit projection of $m(K^{+}K^{-})$. The points
represent the data, the dotted (black) curve shows the combinatorial
background, and the dashed (red) curve indicates the reflection from
misidentified $\kern 1.61993pt\overline{\kern-1.61993ptB}{}^{0}\rightarrow
J/\psi K^{-}\pi^{+}$ decays. The largest three resonances $\phi(1020)$,
$f_{2}^{\prime}(1525)$ and $f_{0}(980)$ are shown by magenta, brown and green
long-dashed curves, respectively; all other resonances are shown by thin black
curves. The dashed (cyan) curve is the non-resonant contribution. The dot-
dashed (black) curve is the contribution from the interferences, and the solid
(blue) curve represents the total fit result.

While a complete description of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decay is given in terms of the fitted amplitudes and phases,
knowledge of the contribution of each component can be summarized by defining
a fit fraction, ${\cal{F}}^{R}_{\lambda}$. To determine
${\cal{F}}^{R}_{\lambda}$ we integrate the squared amplitude of $R$ over the
Dalitz plot. The yield is then normalized by integrating the entire signal
function over the same area. Specifically,

${\cal{F}}^{R}_{\lambda}=\frac{\int\left|a^{R}_{\lambda}e^{i\phi^{R}_{\lambda}}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi})\right|^{2}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}{\int
S(s_{12},s_{23},\theta_{J/\psi})~{}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}.$
(31)

Note that the sum of the fit fractions is not necessarily unity due to the
potential presence of interference between two resonances. Interference term
fractions are given by

${\cal{F}}^{RR^{\prime}}_{\lambda}=\mathcal{R}e\left(\frac{\int
a^{R}_{\lambda}\;a^{R^{\prime}}_{\lambda}e^{i(\phi^{R}_{\lambda}-\phi_{\lambda}^{R^{\prime}})}\mathcal{A}_{\lambda}^{R}(s_{12},s_{23},\theta_{J/\psi}){\mathcal{A}_{\lambda}^{R^{\prime}}}^{*}(s_{12},s_{23},\theta_{J/\psi})ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}{\int
S(s_{12},s_{23},\theta_{J/\psi})~{}ds_{12}\;ds_{23}\;d\cos\theta_{J/\psi}}\right),$
(32)

and

$\sum_{\lambda}\left(\sum_{R}{\cal{F}}^{R}_{\lambda}+\sum^{R\neq
R^{\prime}}_{RR^{\prime}}{\cal{F}}^{RR^{\prime}}_{\lambda}\right)=1.$ (33)

If the Dalitz plot has more destructive interference than constructive
interference, the sum of the fit fractions will be greater than unity.
Conversely, the sum will be less than one if the Dalitz plot exhibits
constructive interference. Note that interference between different spin-$J$
states vanishes because the $d^{J}_{\lambda 0}$ angular functions in
$\mathcal{A}^{R}_{\lambda}$ are orthogonal.

The determination of the statistical uncertainties of the fit fractions is
difficult because they depend on the statistical uncertainty of every fitted
magnitude and phase. Therefore we determine the uncertainties from simulated
experiments. We perform 500 experiments: each sample is generated according to
the model PDF, input parameters are taken from the fit to the data. The
correlations of fitted parameters are also taken into account. For each
experiment the fit fractions are calculated. The distributions of the obtained
fit fractions are described by Gaussian functions. The r.m.s. widths of the
Gaussian functions are taken as the statistical uncertainties on the
corresponding parameters. The fit fractions and phases of the contributing
components are given in Table 4, while the fit fractions of the interference
terms are quoted in Table 5.

Table 4: Fit fractions (%) and phases of contributing components. For P- and D-waves $\lambda$ represents the helicity. Component | Fit fraction (%) | Phase (degree)
---|---|---
$\phi(1020)$, $\lambda=0$ | $32.1\pm 0.5\pm 0.8$ | 0(fixed)
$\phi(1020)$, $|\lambda|=1$ | $34.6\pm 0.5\pm 1.3$ | 0(fixed)
$f_{0}(980)$ | $12.0\pm 1.8^{+2.8}_{-2.5}$ | $-294\pm 8\pm 25$
$f_{0}(1370)$ | $1.2\pm 0.3^{+0.3}_{-1.2}$ | $-81\pm 8\pm 8$
$f_{2}^{\prime}(1525)$, $\lambda=0$ | $9.9\pm 0.7^{+2.4}_{-1.6}$ | 0(fixed)
$f_{2}^{\prime}(1525)$, $|\lambda|=1$ | $5.1\pm 0.9^{+1.8}_{-1.4}$ | 0(fixed)
$f_{2}(1640)$, $|\lambda|=1$ | $1.5\pm 0.7^{+0.7}_{-0.9}$ | $165\pm 27^{+13}_{-44}$
$\phi(1680)$, $|\lambda|=1$ | $3.4\pm 0.3^{+4.4}_{-0.3}$ | $106\pm 14^{+260}_{-210}$
$f_{2}(1750)$, $\lambda=0$ | $2.6\pm 0.5^{+1.0}_{-0.6}$ | $238\pm 8\pm 9$
$f_{2}(1750)$, $|\lambda|=1$ | $1.8\pm 1.0^{+2.2}_{-1.8}$ | $45\pm 30^{+16}_{-70}$
$f_{2}(1950)$, $\lambda=0$ | $0.4\pm 0.2^{+0.2}_{-0.4}$ | $46\pm 17^{+110}_{-~{}20}$
$f_{2}(1950)$, $|\lambda|=1$ | $1.7\pm 0.5^{+2.5}_{-1.7}$ | $-53\pm 26^{+150}_{-~{}80}$
Non-resonant | $6.0\pm 1.6^{+2.0}_{-2.2}$ | $-39\pm 6\pm 23$
Total | $112.3$ |

Table 5: Fit fractions matrix for the best fit in units of %. The diagonal elements correspond to the decay fractions in Table 4. The off-diagonal elements give the fit fractions of the interference. The null values originate from the fact that any interference contribution between different spin-$J$ state integrates to zero. Here the resonances are labeled by their masses in MeV and the subscripts denote the helicities. | $1020_{0}$ | $1020_{1}$ | 980 | 1370 | $1525_{0}$ | $1525_{1}$ | $1640_{1}$ | $1680_{1}$ | $1750_{0}$ | $1750_{1}$ | $1950_{0}$ | $1950_{1}$ | NR
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1020_{0}$ | 32.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$1020_{1}$ |  | 34.6 | 0 | 0 | 0 | 0 | 0 | 2.0 | 0 | 0 | 0 | 0 | 0
980 |  |  | 12.0 | -2.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4.7
1370 |  |  |  | 1.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.5
$1525_{0}$ |  |  |  |  | 9.9 | 0 | 0 | 0 | -4.5 | 0 | 0.9 | 0 | 0
$1525_{1}$ |  |  |  |  |  | 5.1 | -0.9 | 0 | 0 | 2.5 | 0 | -1.8 | 0
$1640_{1}$ |  |  |  |  |  |  | 1.5 | 0 | 0 | -2.4 | 0 | 0.7 | 0
$1680_{1}$ |  |  |  |  |  |  |  | 3.4 | 0 | 0 | 0 | 0 | 0
$1750_{0}$ |  |  |  |  |  |  |  |  | 2.6 | 0 | -0.4 | 0 | 0
$1750_{1}$ |  |  |  |  |  |  |  |  |  | 1.8 | 0 | -2.2 | 0
$1950_{0}$ |  |  |  |  |  |  |  |  |  |  | 0.4 | 0 | 0
$1950_{1}$ |  |  |  |  |  |  |  |  |  |  |  | 1.7 | 0
NR |  |  |  |  |  |  |  |  |  |  |  |  | 6.0

Table 6 shows a comparison of the fit fractions when a different
parametrization is used for the $f_{0}(980)$ resonance shape. The BES
$f_{0}(980)$ functional form is the same as ours with the parameters
$m_{0}=965\pm 10$ MeV, $g_{\pi\pi}=165\pm 18$ MeV and
$g_{KK}/g_{\pi\pi}=4.21\pm 0.33$ [32]. The BaBar collaboration assumes that
the non-resonant S-wave is small and consistent with zero [33]. The BaBar
functional form is different and parametrized as

$A_{R}(s_{23})=\frac{1}{m^{2}_{R}-s_{23}-im_{R}\Gamma_{R}\rho_{KK}},$ (34)

with $\rho_{KK}=2P_{R}/\sqrt{s_{23}}$. The parameters are $m_{R}=922\pm 3$ MeV
and $\Gamma_{R}=240\pm 80$ MeV, taken from BaBar’s Dalitz plot analysis of
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ [34]. The $f_{0}(980)$ fraction is
smaller in the BaBar parametrization, while the total S-wave fraction is
consistent in the three different parameterizations. In all cases, the
dominant component is the $\phi(1020)$, the second largest contribution is the
$f_{2}^{\prime}(1525)$, and the third the $f_{0}(980)$ resonance. There are
also significant contributions from the $f_{0}(1370)$, $f_{2}(1640)$,
$\phi(1680)$, $f_{2}(1750)$, $f_{2}(1950)$ resonances, and non-resonant final
states. The amount of $f_{0}(980)$ is strongly parametrization dependent, so
we treat these three models separately and do not assign any systematic
uncertainty based on the use of these different $f_{0}(980)$ shapes. Therefore
we refrain from quoting a branching fraction measurement for the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$.

The determination of the parameters of the $f_{2}^{\prime}(1525)$ resonance
are not dependent on the $f_{0}(980)$ parametrization. The parameters of the
$f_{2}^{\prime}(1525)$ are determined to be:

$\displaystyle m_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
1522.2\pm 2.8^{+5.3}_{-2.0}~{}\rm MeV,$
$\displaystyle\Gamma_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
84\pm 6^{+10}_{-~{}5}~{}\rm MeV.$

Whenever two or more uncertainties are quoted, the first is the statistical
and the second systematic. The latter will be discussed in Section 5.6. These
values are the most accurate determinations of the $f_{2}^{\prime}(1525)$
resonant parameters [9]. Note that our determination of the mass has the same
uncertainty as the current PDG average.

Table 6: Comparison of the fit fractions (%) with the LHCb, BES and BaBar $f_{0}(980)$ parameterizations described in the text. For P- and D-waves, $\lambda$ represents the helicity. Component | LHCb | BES | BaBar
---|---|---|---
$\phi(1020)$, $\lambda=0$ | $32.1\pm 0.5~{}$ |  $32.1\pm$0.5 | $32.0\pm 0.5$
$\phi(1020)$, $|\lambda|=1$ | $34.6\pm 0.5~{}$ | $34.6\pm 0.5~{}$ | $34.5\pm 0.5$
$f_{0}(980)$ | $12.0\pm 1.8~{}$ | $9.2\pm 1.4$ | $4.8\pm 1.0$
$f_{0}(1370)$ | $1.2\pm 0.3$ | $1.2\pm 0.3$ | $1.3\pm 0.3$
$f_{2}^{\prime}(1525)$, $\lambda=0$ | $9.9\pm 0.7$ | $9.8\pm 0.7$ | $9.5\pm 0.7$
$f_{2}^{\prime}(1525)$, $|\lambda|=1$ | $5.1\pm 0.9$ | $5.1\pm 0.9$ | $4.9\pm 0.9$
$f_{2}(1640)$, $|\lambda|=1$ | $1.5\pm 0.7$ | $1.5\pm 0.7$ | $1.5\pm 0.7$
$\phi(1680)$, $|\lambda|=1$ | $3.4\pm 0.3$ | $3.4\pm 0.3$ | $3.4\pm 0.3$
$f_{2}(1750)$, $\lambda=0$ | $2.6\pm 0.5$ | $2.5\pm 0.5$ | $2.2\pm 0.5$
$f_{2}(1750)$, $|\lambda|=1$ | $1.8\pm 1.0$ | $1.8\pm 1.0$ | $1.9\pm 1.0$
$f_{2}(1950)$, $\lambda=0$ | $0.4\pm 0.2$ | $0.4\pm 0.2$ | $0.4\pm 0.2$
$f_{2}(1950)$, $|\lambda|=1$ | $1.7\pm 0.5$ | $1.8\pm 0.5$ | $1.8\pm 0.5$
Non-resonant S-wave | $6.0\pm 1.4$ | $4.7\pm 1.2$ | $8.6\pm 1.7$
Interference between S-waves |  $-$5.5 |  $-$1.7 |  $-$1.1
Total S-wave | 13.7 | 13.4 | 13.6
-ln$\mathcal{L}$ | 29,275 | 29,275 | 29,281
$\chi^{2}/\rm ndf$ | 649/545 | 653/545 | 646/545

###  5.3 $K^{+}K^{-}$ S-wave in the $\phi(1020)$ mass region

It was claimed by Stone and Zhang [12] that in the decay of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$, the
$K^{+}K^{-}$ system can have S-wave contributions under the $\phi(1020)$ peak
of order 7% of the total yield. In order to investigate this possibility we
calculate the S-wave fractions as given by the fit in 4 MeV mass intervals
between $990<m(K^{+}\kern-1.60004ptK^{-})<1050$ MeV. The resulting behavior is
shown in Fig. 18. Here we show the result from our preferred model and also
from the alternative $f_{0}(980)$ parameterizations discussed above. The
observation of significant S-wave fractions in this region means that this
contribution must be taken into account when measuring $C\\!P$ violation in
the $\phi$ mass region. The total S-wave fraction as a function of the mass
interval around the $\phi$ mass is also shown in Fig. 19.

Figure 18: S-wave fraction as a function of $m(K^{+}K^{-})$ starting from 990
MeV up to 1050 MeV in 4 MeV mass intervals. The squares (blue), triangles
(red), and circles (green) represent the LHCb, BES and BaBar parameterizations
of $f_{0}(980)$, respectively. The experimental statistical uncertainties are
only shown for the LHCb model; they are almost identical for the other cases.
The experimental mass resolution is not unfolded. Figure 19: S-wave fractions
in different $m(K^{+}K^{-})$ intervals centered on the $\phi$ meson mass. The
squares (blue), triangles (red), and circles (green) represent the LHCb, BES
and BaBar parameterizations of $f_{0}(980)$, respectively. The experimental
statistical uncertainties are only shown for the LHCb model; they are almost
identical for the other cases. The experimental mass resolution is not
unfolded.

Using a time dependent analysis of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi(1020)$,
LHCb reported $(2.2\pm 1.2\pm 0.07)\%$ [3] of S-wave within $\pm 12$ MeV of
the $\phi(1020)$ mass peak. We measure the S-wave fraction within the same
mass window as a consistent, and more precise $(1.1\pm 0.1^{+0.2}_{-0.1})\%$.
CDF measured the S-wave fraction as $(0.8\pm 0.2)$% for $m(K^{+}K^{-})$ within
about $\pm$9.5 MeV of the $\phi$ mass [6], while ATLAS quotes (2$\pm$2)% for
an 11 MeV interval [7]. These results are consistent with ours. The D0
collaboration, however, claimed a (14.7$\pm$3.5)% S-wave fraction within
approximately $\pm$10 MeV of the $\phi$ meson mass [5], in disagreement with
all of the other results.

###  5.4 Helicity angle distributions

The decay angular distributions or the helicity angle distributions are
already included in the signal model via Eqs. (8) and (9). In order to test
the fit model we examine the $\cos\theta_{J/\psi}$ and $\cos\theta_{KK}$
distributions in two different $K^{+}K^{-}$ mass regions: one is the
$\phi(1020)$ region defined within $\pm 12$ MeV of the $\phi(1020)$ mass peak
and the other is defined within one full width of the $f_{2}^{\prime}(1525)$
mass. The background-subtracted efficiency-corrected distributions are shown
in Figs. 20 and 21. The distributions are in good agreement with the fit
model.

Figure 20: Background-subtracted efficiency-corrected $\cos\theta_{J/\psi}$
helicity distributions: (a) in $\phi(1020)$ mass region ($\chi^{2}/\rm
ndf=54.4/40$), (b) in $f_{2}^{\prime}(1525)$ mass region ($\chi^{2}/\rm
ndf=34.4/40$).

Figure 21: Background-subtracted efficiency-corrected $\cos\theta_{KK}$
helicity distributions: (a) in $\phi(1020)$ mass region ($\chi^{2}/\rm
ndf=57.4/40$), (b) in $f_{2}^{\prime}(1525)$ mass region ($\chi^{2}/\rm
ndf=43.4/40$). The distributions are compatible with expectations for spin-1
and spin-2, respectively.

###  5.5 Angular moments

The angular moment distributions provide an additional way of visualizing the
effects of different resonances and their interferences, similar to a partial
wave analysis. This technique has been used in previous studies [8, 35].

We define the angular moments $\langle Y_{l}^{0}\rangle$ as the efficiency-
corrected and background-subtracted $K^{+}K^{-}$ invariant mass distributions,
weighted by orthogonal and normalized spherical harmonic functions
$Y_{l}^{0}(\cos\theta_{KK})$,

$\langle
Y_{l}^{0}\rangle=\int_{-1}^{1}d\Gamma(m_{KK},\cos\theta_{KK})Y_{l}^{0}(\cos\theta_{KK})d\cos\theta_{KK}.$
(35)

If we assume that no $K^{+}K^{-}$ partial-waves of a higher order than D-wave
contribute, then we can express the differential decay rate, derived from Eq.
(4) in terms of S-, P-, and D-waves including helicity 0 and $\pm 1$
components as

$\displaystyle\frac{d\overline{\Gamma}}{dm_{KK}\,d\cos\theta_{KK}}=$
$\displaystyle\hskip 11.38109pt2\pi\left|{\cal
S}_{0}Y_{0}^{0}(\cos\theta_{KK})+{\cal
P}_{0}e^{i\Phi_{P_{0}}}Y_{1}^{0}(\cos\theta_{KK})+{\cal
D}_{0}e^{i\Phi_{D_{0}}}Y_{2}^{0}(\cos\theta_{KK})\right|^{2}$
$\displaystyle+2\pi\left|{\cal P}_{\pm 1}e^{i\Phi_{P_{\pm
1}}}\sqrt{\frac{3}{8\pi}}\sin\theta_{KK}+{\cal D}_{\pm 1}e^{i\Phi_{D_{\pm
1}}}\sqrt{\frac{15}{8\pi}}\sin\theta_{KK}\cos\theta_{KK}\right|^{2},$ (36)

where ${\cal S}_{\lambda}$, ${\cal P}_{\lambda}$, ${\cal D}_{\lambda}$ and
$\Phi_{k_{\lambda}}$ are real-valued functions of $m_{KK}$, and we have
factored out the S-wave phase. We can then calculate the angular moments

$\displaystyle\sqrt{4\pi}\langle Y_{0}^{0}\rangle$ $\displaystyle=$
$\displaystyle{\cal S}_{0}^{2}+{\cal P}_{0}^{2}+{\cal D}_{0}^{2}+{\cal P}_{\pm
1}^{2}+{\cal D}_{\pm 1}^{2}$ $\displaystyle\sqrt{4\pi}\langle
Y_{1}^{0}\rangle$ $\displaystyle=$ $\displaystyle 2{\cal S}_{0}{\cal
P}_{0}\cos\Phi_{P_{0}}+\frac{4}{\sqrt{5}}{\cal P}_{0}{\cal
D}_{0}\cos(\Phi_{P_{0}}-\Phi_{D_{0}})$ (37)
$\displaystyle+8\sqrt{\frac{3}{5}}{\cal P}_{\pm 1}{\cal D}_{\pm
1}\cos(\Phi_{P_{\pm 1}}-\Phi_{D_{\pm 1}})$ $\displaystyle\sqrt{4\pi}\langle
Y_{2}^{0}\rangle$ $\displaystyle=$ $\displaystyle\frac{2}{\sqrt{5}}{\cal
P}_{0}^{2}+2{\cal S}_{0}{\cal D}_{0}\cos\Phi_{D_{0}}+\frac{2\sqrt{5}}{7}{\cal
D}_{0}^{2}-\frac{1}{\sqrt{5}}{\cal P}_{\pm 1}^{2}+\frac{\sqrt{5}}{7}{\cal
D}_{\pm 1}^{2}$ $\displaystyle\sqrt{4\pi}\langle Y_{3}^{0}\rangle$
$\displaystyle=$ $\displaystyle 6\sqrt{\frac{3}{35}}{\cal P}_{0}{\cal
D}_{0}\cos(\Phi_{P_{0}}-\Phi_{D_{0}})+\frac{6}{\sqrt{35}}{\cal P}_{\pm 1}{\cal
D}_{\pm 1}\cos(\Phi_{P_{\pm 1}}-\Phi_{D_{\pm 1}})$
$\displaystyle\sqrt{4\pi}\langle Y_{4}^{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{6}{7}{\cal D}_{0}^{2}-\frac{4}{7}{\cal D}_{\pm 1}^{2}.$

The angular moments for $l>4$ vanish. Figures 22 and 23 show the distributions
of the angular moments for the fit model around $\pm 30$ MeV of the
$\phi(1020)$ mass peak and above the $\phi(1020)$, respectively. In general
the interpretation of these moments is that $\langle Y^{0}_{0}\rangle$ is the
efficiency-corrected and background-subtracted event distribution, $\langle
Y^{0}_{1}\rangle$ the sum of the interference between S-wave and P-wave, and
P-wave and D-wave amplitudes, $\langle Y^{0}_{2}\rangle$ the sum of P-wave,
D-wave and the interference of S-wave and D-wave amplitudes, $\langle
Y^{0}_{3}\rangle$ the interference between P-wave and D-wave amplitudes, and
$\langle Y^{0}_{4}\rangle$ the D-wave. As discussed in Section 5.1, the
average of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ cancels the interference terms
that involve ${\cal P}_{0}$ and ${\cal D}_{\pm}$. This causes the angular
moments $\langle Y^{0}_{1}\rangle$ and $\langle Y^{0}_{3}\rangle$ to be zero
when averaging over $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays. We observe that the
fit results well describe the moment distributions, except for the $\langle
Y^{0}_{1}\rangle$ and $\langle Y^{0}_{4}\rangle$ values below 1.2 GeV. This
may be the result of statistical fluctuations or imperfect modeling.

Figure 22: Dependence of the spherical harmonic moments of $\cos\theta_{KK}$
as a function of the $K^{+}K^{-}$ mass around the $\phi(1020)$ mass peak after
efficiency corrections and background subtraction. The points with error bars
are the data and the solid curves are derived from the fit model. Figure 23:
Dependence of the spherical harmonic moments of $\cos\theta_{KK}$ as a
function of the $K^{+}K^{-}$ mass above 1050 MeV, after efficiency corrections
and background subtraction. The points with error bars are the data and the
solid curves are derived from the fit model.

###  5.6 Systematic uncertainties

The sources of the systematic uncertainties on the results of the Dalitz plot
analysis are summarized in Table 7. The uncertainties due to the background
parametrization are estimated by comparing the results from the best fit model
with those when the background shape parameters are obtained from a fit to the
lower sideband region only. The uncertainties in the efficiency are estimated
by comparing the fit results when the efficiency parameters are changed by
their statistical uncertainties and are added in quadrature. The effect on the
fit fractions of changing the efficiency function is evaluated using a similar
method to that used previously [8]. Briefly, we change the efficiency model by
increasing the minimum IP $\chi^{2}$ requirement from 9 to 12.5 on both of the
kaon candidates. This has the effect of increasing the $\chi^{2}$ of the fit
to the angular distributions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ data by 1 unit. The new efficiency function is then applied to the
data with the original minimum IP $\chi^{2}$ selection of 9, the likelihood is
re-evaluated and the uncertainties are estimated by comparing the results with
the best fit model. The largest variations among these two efficiency
categories are included in the uncertainty.

Table 7: Absolute systematic uncertainties on the fit results.  Item | Efficiency | Background | Fit model | Total |
---|---|---|---|---|---
$m_{f_{2}^{\prime}(1525)}$ (MeV) | 1.2 | 0.4 | ${}^{+5.2}_{-1.5}$ | ${}^{+5.3}_{-2.0}$ |
$\Gamma_{f_{2}^{\prime}(1525)}$ (MeV) | 4.7 | 0.5 | ${}^{+8.6}_{-1.8}$ | ${}^{+9.8}_{-5.0}$ |
Fit fractions (%) |
$\phi(1020)~{}\lambda=0$ | 0.8 | 0 | ${}^{+0.06}_{-0.04}$ | $\pm 0.8$ |
$\phi(1020)~{}|\lambda|=1$ | 1.3 | 0 | ${}^{+0.4}_{-0.1}$ | $\pm 1.3$ |
$f_{0}(980)$ | 1.7 | 0.4 | ${}^{+2.1}_{-1.7}$ | ${}^{+2.8}_{-2.5}$ |
$f_{0}(1370)$ | 0.3 | 0.02 | ${}^{+0.2}_{-1.2}$ | ${}^{+0.3}_{-1.3}$ |
$f_{2}^{\prime}(1525)~{}\lambda=0$ | 1.5 | 0.2 | ${}^{+1.8}_{-0.5}$ | ${}^{+2.4}_{-1.6}$ |
$f_{2}^{\prime}(1525)~{}|\lambda|=1$ | 1.1 | 0.4 | ${}^{+1.3}_{-0.8}$ | ${}^{+1.8}_{-1.4}$ |
$f_{2}(1640)~{}|\lambda|=1$ | 0.1 | 0.1 | ${}^{+0.7}_{-0.9}$ | ${}^{+0.7}_{-0.9}$ |
$\phi(1680)~{}|\lambda|=1$ | 0.3 | 0.1 | ${}^{+4.4}_{-0.1}$ | ${}^{+4.4}_{-0.3}$ |
$f_{2}(1750)~{}\lambda=0$ | 0.3 | 0.1 | ${}^{+1.0}_{-0.5}$ | ${}^{+1.0}_{-0.6}$ |
$f_{2}(1750)~{}|\lambda|=1$ | 1.6 | 0.3 | ${}^{+1.5}_{-2.9}$ | ${}^{+2.2}_{-3.3}$ |
$f_{2}(1950)~{}\lambda=0$ | 0.1 | 0.04 | ${}^{+0.2}_{-0.4}$ | ${}^{+0.2}_{-0.4}$ |
$f_{2}(1950)~{}|\lambda|=1$ | 2.1 | 0.6 | ${}^{+1.1}_{-3.1}$ | ${}^{+2.5}_{-3.8}$ |
Non-resonant | 1.7 | 0.4 | ${}^{+0.9}_{-1.5}$ | ${}^{+2.0}_{-2.2}$ |
S-wave within $\pm$12 MeV | 0.02 | 0.02 | ${}^{+0.2}_{-0.1}$ | ${}^{+0.2}_{-0.1}$ |
of $\phi(1020)$ peak (%) |  |  |  |  |
Phases (degrees) |
$f_{0}(980)$ | 19 | 8 | ${}^{+14}_{-~{}8}$ | ${}^{+25}_{-22}$ |
$f_{0}(1370)$ | 6 | 1 | ${}^{+6}_{-5}$ | $\pm 8$ |
$f_{2}(1640)~{}|\lambda|=1$ | 7 | 11 | ${}^{+~{}0}_{-42}$ | ${}^{+13}_{-44}$ |
$\phi(1680)~{}|\lambda|=1$ | 1 | 1 | ${}^{+260}_{-210}$ | ${}^{+260}_{-210}$ |
$f_{2}(1750)~{}\lambda=0$ | 2 | 2 | ${}^{+9}_{-9}$ | $\pm 9$ |
$f_{2}(1750)~{}|\lambda|=1$ | 12 | 10 | ${}^{+~{}2}_{-70}$ | ${}^{+16}_{-70}$ |
$f_{2}(1950)~{}\lambda=0$ | 13 | 1 | ${}^{+108}_{-~{}14}$ | ${}^{+110}_{-~{}20}$ |
$f_{2}(1950)~{}|\lambda|=1$ | 77 | 26 | ${}^{+130}_{-~{}~{}1}$ | ${}^{+150}_{-~{}80}$ |
Non-resonant | 18 | 7 | ${}^{+12}_{-~{}5}$ | ${}^{+23}_{-20}$ |

We estimate additional uncertainties by comparing the results when one more
resonance is added to the best fit model. The uncertainties due to the line
shape of the contributing resonances with fixed mass and width parameters are
estimated by varying them individually in the fit according to their combined
statistical and systematic uncertainties added in quadrature. We compare the
results with the best fit and add them in quadrature to estimate the
uncertainties due to the line shape.

Another source of systematic uncertainty is the value we choose for $L_{B}$,
the orbital angular momentum in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay. If $L_{R}$ equals zero
then $L_{B}$ equals zero. If, however, $L_{R}$ is 1 then $L_{B}$ can either be
0 or 1, and if $L_{R}$ is 2, $L_{B}$ can be 1, 2 or 3. For our best fit we
don’t allow multiple values for $L_{B}$, but choose the lowest allowed value.
To estimate the systematic uncertainties due to the choice of $L_{B}$, we
repeat the fit changing the default value of $L_{B}$, in turn, to each higher
allowed value and compare the fit results with the best fit. The differences
are grouped into the fit model category, and we assign the largest variations
as the systematic uncertainties. These later two categories often give in
asymmetric uncertainties.

##  6 Absolute branching fractions

Branching fractions are measured from ratios of the decay rates of interest
normalized to the well established decay mode $B^{-}\rightarrow J/\psi K^{-}$.
This decay mode, in addition to having a well measured branching fraction, has
the advantage of having two muons in the final state and hence the same
triggers as the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay.
However, we require knowledge of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/B^{-}$ production ratio. For
this we assume isospin invariance and use the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}/\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production ratio
$f_{s}/f_{d}=0.256\pm 0.020$, given in Ref. [36, *Aaij:2011jp]. The branching
fractions are calculated using

$\mathcal{B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi X)=\frac{N_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}/\epsilon_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}}{N_{B^{-}}/\epsilon_{B^{-}}}\times\mathcal{B}(B^{-}\rightarrow
J/\psi K^{-})\times\frac{1}{f_{s}/f_{d}},$ (38)

where $X$ indicates a specific $K^{+}K^{-}$ state, $N$ represents the yield of
the decay of interest, and $\epsilon$ corresponds to the overall efficiency.
We form an average of
$\mathcal{B}(B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-})=(10.18\pm 0.42)\times 10^{-4}$ using the recent Belle [38] and
BaBar [39] measurements, corrected to take into account different rates of
$B^{+}B^{-}$ and $B^{0}\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pair
production from $\Upsilon(4S)$ using
$\frac{\Gamma(B^{+}B^{-})}{\Gamma(B^{0}\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0})}=1.055\pm 0.025$ [9].

The detection efficiency is obtained from simulation and is a product of the
geometrical acceptance of the detector, the combined reconstruction and
selection efficiency and the trigger efficiency. The efficiency also includes
the efficiency of the Dalitz plot model for the case of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$,
where the best fit model is used. The detection efficiencies and their various
correction factors are given in Table 8. To ensure that the $p$ and $p_{\rm
T}$ distributions of the generated $B$ meson are correct we weight the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ simulations using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi(1020)$
data and the $B^{-}$ simulations using $B^{-}\rightarrow J/\psi K^{-}$ data.
Since the control channel has a different number of charged tracks than the
decay channel, we weight the simulations with the tracking efficiency ratio by
comparing the data and simulations in bins of the track’s $p$ and $p_{\rm T}$.
we further weight the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\kern-1.60004ptK^{-}$ simulation, using the PDG value of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime, $(1.497\pm
0.015)\times 10^{-12}~{}\rm s$ [9], as input.

Table 8: Detector efficiencies determined from simulation and the correction factors. Item | $J/\psi K^{+}K^{-}$ | $J/\psi K^{-}$
---|---|---
Detection efficiency (%) | $1.061\pm 0.004$ | $2.978\pm 0.011$
Correction factors
Tracking efficiency | $0.999\pm 0.010$ | $1.003\pm 0.010$
PID | $0.819\pm 0.008$ | $0.974\pm 0.005$
$p$ and $p_{\rm T}$ | $1.077\pm 0.005$ | $1.053\pm 0.005$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime | $0.993\pm 0.015$ | -
Total efficiency (%) | $0.887\pm 0.004\pm 0.018$ | $3.065\pm 0.012\pm 0.038$

The resulting branching fractions are

$\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})~{}~{}$ $\displaystyle=$ $\displaystyle~{}~{}\\!(7.70\pm 0.08\pm
0.39\pm 0.60)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))~{}$ $\displaystyle=$ $\displaystyle(10.50\pm 0.13\pm 0.64\pm
0.82)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))$ $\displaystyle=$ $\displaystyle~{}~{}\\!(2.61\pm
0.20^{+0.52}_{-0.46}\pm 0.20)\times 10^{-4},$

where the branching fractions ${\cal{B}}(\phi(1020)\rightarrow
K^{+}K^{-})=(48.9\pm 0.5)\%$ and ${\cal{B}}(f_{2}^{\prime}(1525)\rightarrow
K^{+}K^{-})=(44.4\pm 1.1)\%$ are used [9]. Here the first uncertainty in each
case is statistical, the second is systematic and the third reflects the
uncertainty due to $f_{s}/f_{d}$. Note that these are the time-integrated
branching fractions. Results on the polarization fractions of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi(1020)$ from a time-dependent analysis will be forthcoming in a
separate publication [40]. The ratio of $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))/\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi(1020))$
is consistent with our previous result [10], D0 [11], and the Belle result
[41]. The current PDG value of $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))=(1.4\pm 0.5)\times 10^{-3}$ is dominated by the CDF
measurement [42]. Our measured value is in good agreement with this
measurement and also the most recent yet unpublished values measured by CDF
[43] and Belle [41]. The Belle collaboration has also recently reported the
branching fraction of $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-})$
[41], where $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020)$ is excluded.

Table 9: Relative systematic uncertainties on branching fractions (%). Item | $J/\psi K^{+}K^{-}$ | $J/\psi\phi(1020)$ | $J/\psi f_{2}^{\prime}(1525)$
---|---|---|---
Tracking efficiency | 1.0 | 1.0 | 1.0
Material and physical effects | 2.0 | 2.0 | 2.0
PID efficiency | 1.0 | 1.0 | 1.0
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ $p$ and $p_{\rm T}$ distributions | 0.5 | 0.5 | 0.5
$B^{-}$ $p$ and $p_{\rm T}$ distributions | 0.5 | 0.5 | 0.5
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime | 1.5 | 1.5 | 1.5
Efficiency function | 0.02 | 0.02 | 0.02
$\mathcal{B}(\phi(1020)\rightarrow K^{+}K^{-})$ | - | 1.0 | -
$\mathcal{B}(f_{2}^{\prime}(1525)\rightarrow K^{+}K^{-})$ | - | - | 2.5
$\mathcal{B}(B^{-}\rightarrow J/\psi K^{-})$ | 4.1 | 4.1 | 4.1
Contributions from Dalitz analysis
Efficiency | - | 3.1 | 12.3
Background | - | 0 | 1.3
Fit model | - | ${}^{+0.7}_{-0.2}$ | ${}^{+14.6}_{-11.2}$
Sum in quadrature of items above | 5.0 | ${}^{+6.1}_{-6.0}$ | ${}^{+20.0}_{-17.7}$
$f_{s}/f_{d}$ | $7.8$ | $7.8$ | $7.8$

The systematic uncertainty on the branching fraction has several contributions
listed in Table 9. Since the branching fractions are measured with respect to
$B^{-}\rightarrow J/\psi K^{-}$ which has a different number of charged tracks
than the decays of interest, a 1% systematic uncertainty is assigned due to
differences in the tracking performance between data and simulation. Another
2% uncertainty is assigned for the additional kaon which is due to decay in
flight, large multiple scatterings and hadronic interactions along the track.
Using the PDG value for the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime [9] as input gives
rise to an additional 1.5% systematic uncertainty. Small uncertainties are
introduced if the simulation does not have the correct $B$ meson kinematic
distributions. We are relatively insensitive to any of these differences in
the $B$ meson $p$ and $p_{\rm T}$ distributions since we are measuring the
relative rates. By varying the $p$ and $p_{\rm T}$ distributions we see at
most a change of 0.5%. There is a 1% systematic uncertainty assigned for the
relative particle identification efficiencies. An uncertainty of 0.02% is
included due to the change of the efficiency function Eq. (20). Three
additional uncertainties are considered in the branching fractions of
$\mathcal{B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))$ and $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))$ as these are measured from the fit fractions of the
Dalitz plot analysis. The total systematic uncertainty is obtained by adding
each source of systematic uncertainty in quadrature as they are uncorrelated.

##  7 Conclusions

We have determined the final state composition of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
decay channel using a modified Dalitz plot analysis where we include the decay
angle of the $J/\psi$. The largest contribution is the $\phi(1020)$ resonance,
along with other S-, P- and D-wave $K^{+}\kern-1.60004ptK^{-}$ states, and a
non-resonant $K^{+}\kern-1.60004ptK^{-}$ contribution. All of the components
are listed in Table 4. The mass and width of the $f_{2}^{\prime}(1525)$
resonance are measured as

$\displaystyle m_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
1522.2\pm 2.8^{+5.3}_{-2.0}~{}\rm MeV,$
$\displaystyle\Gamma_{f_{2}^{\prime}(1525)}$ $\displaystyle=$ $\displaystyle
84\pm 6^{+10}_{-~{}5}~{}\rm MeV.$

We also observe a significant S-wave component that is present over the entire
$K^{+}K^{-}$ mass region. Within $\pm$12 MeV of the $\phi(1020)$ mass it is
$(1.1\pm 0.1^{+0.2}_{-0.1})\%$ of the yield, and can affect precision $C\\!P$
violation measurements [12]. Finally we determine the absolute branching
fractions

$\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
K^{+}K^{-})~{}~{}$ $\displaystyle=$ $\displaystyle~{}~{}\\!(7.70\pm 0.08\pm
0.39\pm 0.60)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\phi(1020))~{}$ $\displaystyle=$ $\displaystyle(10.50\pm 0.13\pm 0.64\pm
0.82)\times 10^{-4},$ $\displaystyle\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{2}^{\prime}(1525))$ $\displaystyle=$ $\displaystyle~{}~{}\\!(2.61\pm
0.20^{+0.52}_{-0.46}\pm 0.20)\times 10^{-4},$

where the first uncertainty in each case is statistical, the second is
systematic and the third due to $f_{s}/f_{d}$. These results provide a good
understanding of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
final state in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays
over the entire kinematically allowed region. The $J/\psi
f_{2}^{\prime}(1525)$ results supersede those of Ref. [10]. This decay mode
offers the opportunity for additional measurements of $C\\!P$ violation [44,
*Zhang:2012zk].

## 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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