EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-267 LHCb-PAPER-2012-023 18 September 2012

Search for the rare decay $K_{\rm\scriptscriptstyle
S}^{0}\rightarrow\mu^{+}\mu^{-}$

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

A search for the decay $K_{\rm\scriptscriptstyle
S}^{0}\rightarrow\mu^{+}\mu^{-}$ is performed, based on a data sample of 1.0
fb-1 of $pp$ collisions at $\sqrt{s}$ = 7 TeV collected by the LHCb experiment
at the Large Hadron Collider. The observed number of candidates is consistent
with the background-only hypothesis, yielding an upper limit of
$\mathcal{B}(K_{\rm\scriptscriptstyle
S}^{0}\rightarrow\mu^{+}\mu^{-})<11(9)\times 10^{-9}$ at 95 (90)$\%$
confidence level. This limit is a factor of thirty below the previous
measurement.

Published in the Journal of High Energy Physics

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, 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, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P.
Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. 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, 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, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53, 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, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, 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 decay $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ is a
Flavour Changing Neutral Current (FCNC) transition that has not yet been
observed. This decay is suppressed in the Standard Model (SM), with an
expected branching fraction [1, 2]

${\cal B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})=(5.0\pm
1.5)\times 10^{-12},$

while the current experimental upper limit is $3.2\times 10^{-7}$ at 90$\%$
confidence level (C.L.) [3].

Although the dimuon decay of the $K^{0}_{\rm\scriptscriptstyle L}$ meson is
known to be ${\cal B}(K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-})=(6.84\pm 0.11)\times 10^{-9}$ [4], in agreement
with the SM, effects of new particles can still be observed in
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ decays. In the most
general case, the decay width of $K^{0}_{\rm\scriptscriptstyle
L,S}\rightarrow\mu^{+}\mu^{-}$ can be written as [5]

$\Gamma(K^{0}_{\rm\scriptscriptstyle
L,S}\rightarrow\mu^{+}\mu^{-})=\frac{m_{K}}{8\pi}\sqrt{1-\left(\frac{2m_{\mu}}{m_{K}}\right)^{2}}\left[|A|^{2}+\left(1-\left(\frac{2m_{\mu}}{m_{K}}\right)^{2}\right)|B|^{2}\right],$
(1)

where $A$ is an S-wave amplitude and $B$ a P-wave amplitude. These two
amplitudes have opposite $C\\!P$ eigenvalues, and in absence of $C\\!P$
violation ($K^{0}_{\rm\scriptscriptstyle S}$ $=K_{1}^{0}$,
$K^{0}_{\rm\scriptscriptstyle L}$ $=K_{2}^{0}$), $K^{0}_{\rm\scriptscriptstyle
L}$ decays would be generated only by $A$ while $K^{0}_{\rm\scriptscriptstyle
S}$ decays would be generated only by $B$. The decay width
$\Gamma(K^{0}_{\rm\scriptscriptstyle L}\rightarrow\mu^{+}\mu^{-})$ receives
long-distance111The long-distance scales correspond to masses below that of
the $c$ quark, while short-distance scales correspond to masses of the $c$
quark and above. contributions to $A$ from intermediate two-photon states, as
well as short distance contributions to the real part of $A$. In any model
with the same basis of effective FCNC operators as the SM, the contributions
from $B$ can be neglected for ${\cal B}(K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-})$. The decay width of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ depends on the
imaginary part of the short-distance contributions to $A$ and on the long-
distance contributions to $B$ generated by intermediate two-photon states.
Therefore, the measurement of ${\cal B}(K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-})$ in agreement with the SM does not necessarily
imply that ${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})$ has to agree with the SM. Contributions up to
one order of magnitude above the SM expectation are allowed [2]; enhancements
of the branching fraction above $10^{-10}$ are less likely. The study of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ has been suggested
as a possible way to look for new light scalars [1].

In addition, bounds on the upper limit of ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ close to
$10^{-11}$ could be very useful to discriminate among scenarios beyond the SM
if other modes, such as $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ (charge
conjugation is implied throughout this paper), were to indicate a non-standard
enhancement of the $s\rightarrow d\ell\bar{\ell}$ transition [2]. The KLOE
collaboration has searched for the related decay $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow e^{+}e^{-}$, which is affected by a larger helicity suppression
than the muonic mode, and set an upper limit on the branching fraction ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow e^{+}e^{-})<9\times 10^{-9}$ at
90% confidence level [6].

The LHC produces $\sim 10^{13}$ $K^{0}_{\rm\scriptscriptstyle S}$ per
$\mbox{\,fb}^{-1}$ inside the LHCb acceptance. In this paper, a search for
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ is presented using
1.0 fb-1 of $pp$ collisions at $\sqrt{s}$ = 7 TeV collected by LHCb in 2011.
Dimuon candidates are classified in bins of a multivariate discriminant, and
compared to background and signal expectations. The background present in the
signal region is a combination of combinatorial background and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays in which
both pions are misidentified as muons. The number of expected signal
candidates for a given branching fraction hypothesis is obtained by
normalising to the measured $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ rate. The results obtained by the measurements in
different bins are combined, and a limit is set using the
$\textrm{CL}_{\textrm{s}}$ method [7, 8]. The data in the signal region were
only analysed once the full analysis strategy was defined, including the
selection, the binning and the evaluation of systematic uncertainties.

The LHCb apparatus, and the aspects of the trigger relevant for this analysis
are presented in Sect. 2. Section 3 is devoted to the full signal selection
and to the definition of the multivariate method used as the main
discriminant. In Sect. 4 the different backgrounds for
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ decay are
described, as well as the expected background in the signal region. The
normalisation, required to convert the number of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ candidates to the branching fraction, is detailed
in Sect. 5. The systematic uncertainties are described in Sect. 6. The limit
setting procedure, together with the corresponding expected and observed
limits, is presented in Sect. 7, and conclusions are drawn in Sect. 8.

##  2 Experimental setup

The LHCb detector [9] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector (VELO) 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 a momentum resolution $\Delta p/p$ that
varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (${\rm
IP}$) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$) with respect to the beam direction. 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 consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. For this analysis, the events are first required to
pass a hardware trigger which selects at least one muon with $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the subsequent software
trigger [10], at least one of the final state tracks is required to be of good
quality and to have $\mbox{$p_{\rm
T}$}>1.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, an ${\rm IP}$ $>0.5$$\rm\,mm$
and the $\chi^{2}$ of the impact parameter (IP $\chi^{2}$) above $200$. The IP
$\chi^{2}$ is defined as the difference between the $\chi^{2}$ of the proton-
proton, $pp$, interaction point (primary vertex, PV) built with and without
the considered track. A prescale factor of two is applied to the lines
triggered by the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates. The $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates responsible for the trigger of both the hardware and software
levels are called TOS (trigger on signal).

Events with a reconstructed $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ candidate can also be triggered independently of
the signal candidate if some other combination of particles in the underlying
event passes the trigger. Such candidates are called TIS (trigger
independently of signal). The TIS and TOS categories are not exclusive as
muons from both the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates and from the underlying event can pass the trigger. There is
overlap between the two, which allows the determination of trigger
efficiencies from the data [11]. Finally, minimum bias candidates triggered by
a dedicated random trigger (MB) provide a negligible amount of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates. Instead
they allow the selection of a sample of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ useful to understand the distributions that the
signal would have in the case of no trigger bias.

For the simulation, $pp$ collisions are generated using Pythia 6.4 [12] with a
specific LHCb configuration [13]. Decays of hadronic particles are described
by EvtGen [14] in which final state radiation is generated using Photos [15].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [16, *Agostinelli:2002hh] as
described in Ref. [18].

##  3 Selection and multivariate classifier

The $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates are
reconstructed requiring two tracks with opposite curvature with hits in the
VELO and in the tracking stations. About 40% of the
$K^{0}_{\rm\scriptscriptstyle S}$ mesons with the two daughter tracks inside
the LHCb acceptance decay in the VELO detector. Those tracks are required to
be of high quality ($\chi^{2}<5$ per degree of freedom), to have an ${\rm IP}$
$\chi^{2}$ greater than 100 and a distance of closest approach of less than
0.3 $\rm\,mm$. The two tracks are required to be identified as muons [19]. The
reconstructed $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
candidates are required to have a proper decay time greater than 8.9
${\rm\,ps}$ and to point to the PV (${\rm IP}(K^{0}_{\rm\scriptscriptstyle
S})<400$ $\,\upmu\rm m$). The secondary vertex, SV, of the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidate is
required to be downstream of the PV. If more than one PV is reconstructed, the
PV associated to the $K^{0}_{\rm\scriptscriptstyle S}$ is the one that
minimises its ${\rm IP}$ $\chi^{2}$. Furthermore, $\Lambda\rightarrow
p\pi^{-}$ decays are vetoed via a requirement in the Armenteros-Podolanski
plane [20], by including cuts on the transverse momentum of the daughter
tracks with respect to the $K^{0}_{\rm\scriptscriptstyle S}$ flight direction
and on their longitudinal momentum asymmetry. The reconstructed
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ mass is required to
be in the range [450,1500] ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay is used
as a control channel and is reconstructed and selected in the same way as the
signal candidates, with the exception of the particle identification
requirements on the daughter tracks and the mass range, which is requested to
be between 400 and 600 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Figure 1 shows the mass spectrum for selected $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ candidates in the MB sample after applying the
set of cuts described above and in the $\pi\pi$ and $\mu\mu$ mass hypotheses:
the two mass peaks are separated by 40
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This separation, combined with the
LHCb mass resolution of about 4 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
such combinations of tracks, is used to discriminate the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ signal from
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays where both
pions are misidentified as muons.

Figure 1: Mass spectrum for selected $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ candidates in the MB sample. The black points
correspond to the mass reconstructed under the $\pi\pi$ mass hypothesis for
the daughters, while the red triangles correspond to the mass reconstructed
under the $\mu\mu$ mass hypothesis.

In order to further increase the background rejection, a boosted decision tree
(BDT) [21] with the AdaBoost algorithm [22] is used. The variables entering in
the BDT discriminant are:

  * •

the decay time of the $K^{0}_{\rm\scriptscriptstyle S}$ candidate, computed
using the distance between the SV and the PV, and the reconstructed momentum
of the $K^{0}_{\rm\scriptscriptstyle S}$ candidate;

  * •

the smallest muon IP $\chi^{2}$ of the two daughter tracks with respect to any
of the PVs reconstructed in the event;

  * •

the $K^{0}_{\rm\scriptscriptstyle S}$ IP $\chi^{2}$ with respect to the PV;

  * •

the distance of closest approach between the two daughter tracks;

  * •

the secondary vertex $\chi^{2}$, which adds complementary information with
respect to the distance of closest approach of the tracks, as it uses
information on the uncertainty of the vertex fit;

  * •

the angle of the decay plane in the $K^{0}_{\rm\scriptscriptstyle S}$ rest
frame with respect to the $K^{0}_{\rm\scriptscriptstyle S}$ flight direction,
which is isotropic for signal decays, but not necessarily for background
candidates;

  * •

variables used to discriminate against material interactions, as further
detailed below.

An important source of background consists of muons resulting from
interactions between the particles produced in the PV and the detector
material in the region of the VELO. The position of the SV of the background
candidates from the $K^{0}_{\rm\scriptscriptstyle S}$ mass sidebands in the
$x-z$ plane is shown in Fig. 2. The structures observed correspond to the
position of the material inside the VELO detector. To discriminate against
this background, two different approaches are used for the TIS and TOS trigger
categories, consisting of two different choices of variables for the BDT.

For the TOS category, two additional variables are included in the BDT, the
$p_{\rm T}$ of the $K^{0}_{\rm\scriptscriptstyle S}$ and a boolean matter veto
that uses the VELO geometry to assess whether a given decay vertex coincides
with a point in the detector material or not. Muons from material interactions
have a harder $p_{\rm T}$ spectrum than muons from other background sources
and hence are more likely to be selected by the trigger. The use of this
variable in the BDT provides 50% less background yield for the same signal
efficiency than simply applying the veto as a selection cut.

For the TIS category, the coordinates of the position of the SV in the
laboratory frame are used to deal with this background. As the simultaneous
use of the lifetime, $p_{\rm T}$ of the $K^{0}_{\rm\scriptscriptstyle S}$
meson, and the SV position allows the BDT to effectively compute the mass of
the candidate, a fake signal peak could be artificially created out of the
combinatorial background. Hence the $p_{\rm T}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$ meson is not used in the TIS analysis. This
second approach provides a factor of two less background yield for the same
signal efficiency than the matter veto (and $K^{0}_{\rm\scriptscriptstyle S}$
$p_{\rm T}$) for the TIS analysis, while, on the contrary, the matter veto
boolean variable gives a factor of four less background yield for the same
signal efficiency than the SV coordinates for the TOS analysis.

Figure 2: Position in the $x-z$ plane of the secondary vertices of the
background candidates found in the high mass sideband for (left) TIS
candidates and (right) TOS candidates. The lighter coloured areas correspond
to higher density of points.

Because of these different approaches and to take into account the biases on
the variable distributions introduced by the trigger, the data sample is split
in two subsamples according to the TIS and TOS categories, for which BDT
discriminants are optimised separately. In the TOS analysis, the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays are required
to have at least one of the daughters with a $p_{\rm T}$ above 1.3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in order to minimise the difference in
the momentum distributions with respect to the triggered
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates. The
candidates that are simultaneously TIS and TOS are analysed only as TIS
candidates to avoid counting them twice. Only one per mille of the TOS
candidates overlap with TIS candidates.

In addition, the BDT discriminants for both trigger categories are defined and
trained on data using $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ candidates as signal sample and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ candidates in the
upper mass sideband as background sample. For the background sample, the
region above 1100 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (above the $\phi$
resonance) is used to define the BDT settings and the region between 504 and
1000 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to train the BDT algorithm
chosen. For the signal sample, the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ TIS events are used to train the BDT for the TIS
category, while $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$
decays with both pions misidentified as muons and passing the same trigger
requirements as the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$
signal are used for the TOS category. In order to minimise the differences
between misidentified $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ events and $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ decays, tight muon identification requirements
(including cuts in the quality of the tracks or in the number of muon hits
shared by different tracks) are applied to the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ sample. These tight requirements are chosen such
that the efficiency of the trigger in the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ simulated decays is the same as in the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ simulated decays.

In addition, the TOS and TIS categories are further split in two equal-sized
subsamples, corresponding to the first and second halves of the data taking
period. This procedure prevents possible biases related to the use of the same
events in the mass sidebands both to train the BDT discriminant and to
evaluate the background in the signal region, while making maximal use of the
available data both for BDT training and background evaluation. Thus, in
total, four different samples are defined (two subsamples for the TIS trigger
category and two subsamples for the TOS trigger category) and combined as
described in Sect. 7.

Candidates with low values of the BDT response are not considered because of
the large amount of background in that region. This requirement provides about
$50\%$ signal efficiency and $99\%$ background rejection, depending on the
sample. The rest of the candidates are classified in ten bins of equal signal
efficiency, i.e. a total of forty bins are combined to get the
$\textrm{CL}_{\textrm{s}}$ limit.

##  4 Background

The search region is defined as the mass range
$[492,504]$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The background level is
calibrated by interpolating the observed yield from mass sidebands
($[470,492]$ and $[504,600]$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) to
the signal region. This is done by means of an unbinned maximum likelihood fit
in the sidebands, using a model with two components. The first component is a
power law that describes the tail of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ decays where both pions are misidentified as
muons; this model has been checked to be appropriate using MC simulation. The
second component is an exponential function describing the combinatorial
background. As an illustration, Fig. 3 shows the distribution of candidates
for all BDT bins and for TIS and TOS samples, respectively. The expected total
background yield in the most sensitive BDT bins of both samples ranges from 0
to 1 candidates.

Figure 3: Background model fitted to the data separated along (left) TIS and
(right) TOS trigger categories. The vertical lines delimit the search window.

Other sources of background, such as $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\mu^{-}\bar{\nu}_{\mu}$, $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}\gamma$, $K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-}\gamma$, $K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\pi^{+}\mu^{-}\bar{\nu}_{\mu}$ and $K^{0}_{\rm\scriptscriptstyle
L}\rightarrow\mu^{+}\mu^{-}$ decays, are negligible for the current analysis.
In the case of $K^{0}_{\rm\scriptscriptstyle L}\rightarrow\mu^{+}\mu^{-}$ and
$K^{0}_{\rm\scriptscriptstyle L}\rightarrow\mu^{+}\mu^{-}\gamma$, the
contributions have been evaluated using the ratio of the
$K^{0}_{\rm\scriptscriptstyle S}$ and $K^{0}_{\rm\scriptscriptstyle L}$
lifetimes and the proper time acceptance measured in data with the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays. The
contributions of the other decay modes have been determined using MC simulated
events.

##  5 Normalisation

A normalisation is required to translate the number of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ signal decays into
a branching fraction measurement. Two normalisations are determined
independently for TIS and TOS candidates. The ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ is computed
using

$\frac{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})}{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}=\frac{\epsilon_{\pi\pi}}{\epsilon_{\mu\mu}}\frac{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}},$ (2)

where, in a given BDT bin, $N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}$ is the observed number of signal decays,
$N_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}$ the number of
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays, and
$\epsilon_{\pi\pi}/\epsilon_{\mu\mu}$ the ratio of the corresponding
efficiencies. The efficiencies are factorised as
$\epsilon=\epsilon^{\text{SEL}}\epsilon^{\text{PID}}\epsilon^{\text{TRIG/SEL}}$
where:

  * •

$\epsilon^{\text{SEL}}$ is the offline selection efficiency. It includes the
geometrical acceptance, reconstruction and selection, i.e, it is the
probability for a $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$
($K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$) decay generated
in a $pp$ collision, to have been reconstructed and selected;

  * •

$\epsilon^{\text{PID}}$ is the efficiency of the muon identification for
reconstructed and selected $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ signal decays;

  * •

$\epsilon^{\text{TRIG/SEL}}=N^{\text{SEL\&PID\&TRIG}}/N^{\text{SEL\&PID}}$,
where TRIG denotes either the TIS or the TOS categories, is the trigger
efficiency for decays that would be offline selected. Under this definition,
trigger efficiencies can be determined from data using the procedure described
in Ref. [11].

The ratio of reconstruction and selection efficiencies between
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays is evaluated
in bins of $p_{\rm T}$ and rapidity of the $K^{0}_{\rm\scriptscriptstyle S}$
meson using simulated events reweighted in order to reproduce the
$K^{0}_{\rm\scriptscriptstyle S}$ $p_{\rm T}$ and rapidity spectra measured in
data [23]. The reconstruction and selection efficiency for
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays is between
60% and 85% (depending on which point in the phase space a given event is
from) of that of the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ decays due to difference in the material
interactions of the pions compared to muons.

The factor $\epsilon^{\text{PID}}$ is evaluated in bins of the BDT (both for
the TOS and TIS categories) by measuring the muon identification efficiency as
a function of $p$ and $p_{\rm T}$ using calibration muons. The sample of
calibration muons is obtained from a $J/\psi\rightarrow\mu^{+}\mu^{-}$ sample
in which positive muon identification is required for only one of the tracks.
The $p$ and $p_{\rm T}$ spectra of the pions from
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays in a MB
sample is later used to get the efficiency for $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ decays. The $\epsilon^{\text{PID}}$ efficiency is
between $68\%$ and $82\%$ (depending on the BDT bin and the sample). It is
measured with a precision between $1\%$ and $10\%$. For the ratio of trigger
efficiencies, different strategies are considered for the TIS and TOS samples.

For the TIS samples, the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ yield is normalised to the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ TIS yield. In this
case, the trigger efficiencies cancel in the ratio, because the probability to
trigger on the underlying event is independent of the decay mode of the
$K^{0}_{\rm\scriptscriptstyle S}$ meson. This cancellation is verified in
simulation. The normalisation expression for TIS decays reads

$\frac{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})}{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}^{\text{TIS}}}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{TIS}}},$ (3)

where $N^{\text{TIS}}_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}$ and
$N^{\text{TIS}}_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}$
are the number of TIS decays in a given BDT bin for signal and
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ modes respectively.
$N^{\text{TIS}}_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}$ is
found to be around 9000 for every BDT bin.

For the TOS sample, the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ yield is normalised to the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ yield from MB
triggers. The normalisation requires in this case an absolute determination of
the TOS trigger efficiency for $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$, $\epsilon^{\text{TOS/SEL}}_{\mu\mu}$, as well as
the knowledge of the average prescale factor of the MB trigger,
$s^{\text{MB}}$. The absolute TOS trigger efficiency for the signal is
computed using muons from $B^{+}\\!\rightarrow
J/\psi(\rightarrow\mu^{+}\mu^{-})K^{+}$ decays.222To avoid bias, it is
required that another object be the origin of the trigger and not the muons
alone, i.e. the muons from this sample are TIS. The $p$ and $p_{\rm T}$
spectra of the $B^{+}\\!\rightarrow J/\psi(\rightarrow\mu^{+}\mu^{-})K^{+}$
muons are reweighted in order to match those of pions from the
$K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decays. Trigger
unbiased $p$ and $p_{\rm T}$ spectra of the $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}$ decays can be obtained from the MB sample. The
TOS efficiency is found to be at the level of 20% for all BDT bins. The
normalisation expression for TOS decays reads

$\frac{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})}{{\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{s^{\text{MB}}}{\epsilon^{\text{TOS/SEL}}_{\mu\mu}}\frac{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}^{\text{TOS}}}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}},$ (4)

$N_{K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}$
being the number of $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$
decays from the MB trigger and $N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}}^{\text{TOS}}$ denoting the number of signal
decays from the TOS category. $N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}$ is found to be around 1000 for every
BDT bin.

The quantities

$\alpha_{\text{TIS}}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{{\cal
B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{TIS}}}$ (5)

and

$\alpha_{\text{TOS}}=\frac{\epsilon^{\text{SEL}}_{\pi\pi}}{\epsilon^{\text{SEL}}_{\mu\mu}}\frac{1}{\epsilon^{\text{PID}}_{\mu\mu}}\frac{s^{\text{MB}}}{\epsilon^{\text{TOS/SEL}}_{\mu\mu}}\frac{{\cal
B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})}{N_{K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-}}^{\text{MB}}}$ (6)

are called normalisation factors and are defined for each of the BDT bins. For
a given number $N$ of $K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-}$ signal decays, the corresponding value of ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ is then
$\alpha\times N$. Using the value of ${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\pi^{+}\pi^{-})$ from Ref. [4], the normalisation factors are in
the range $[6.6,16.2]\times 10^{-8}$ for the TIS category, and
$[0.9,7.8]\times 10^{-8}$ for the TOS category, depending on the BDT bin. From
the normalisation factors, around $2\times 10^{-4}$ ($6\times 10^{-5}$) SM
candidates are expected per BDT bin for the TOS (TIS) analysis.

##  6 Systematic uncertainties

The quantities considered in the determination of the branching fraction that
are affected by systematic uncertainties are listed below.

  * •

The background expectations per bin, obtained by comparing the results with
the model described in Sect. 4 to those computed: a) if the combinatorial
background is modelled by a linear function; b) if the mass range over which
the fit is performed is modified; c) repeating the fit excluding (together
with the signal region) the 12${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ left
and right windows neighbouring the search window and comparing the fit
prediction to the yields in those regions; no correlation is considered among
the different bins for this systematic uncertainty.

  * •

The ratios of reconstruction and selection efficiencies and absolute muon
identification efficiencies, for which systematic uncertainties are obtained
from the difference between different methods in the data reweighting of the
MC computed ratios and from the comparison to simulation respectively (around
20% for the ratios and 5% for muon identification efficiencies); no
correlation is considered among the different bins.

  * •

The branching fraction of the normalisation channel ${\cal
B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-})$ $=(69.20\pm
0.05)\%$ [4]; its uncertainty affects coherently the signal expectations of
the forty bins of the analysis.

  * •

The absolute TOS efficiency, for which the systematic uncertainty is obtained
from the comparison to simulation (around 15%, depending on the BDT bin); no
correlation is considered among the different bins.

  * •

The effective prescale factor of the MB sample, $s^{\text{MB}}=(2.70\pm
0.76)\times 10^{-6}$. The uncertainty is evaluated from the difference between
the prescale factor as measured in data and the value of the prescale as set
in the trigger system. This systematic uncertainty affects coherently the
signal expectations of the twenty bins of the TOS analysis.

The leading systematic uncertainties are those coming from the absolute TOS
efficiency and $s^{\text{MB}}$ factor for the TOS analysis and from the ratio
of reconstruction and selection efficiencies for the TIS analysis.

##  7 Results

The modified frequentist approach (or $\textrm{CL}_{\textrm{s}}$ method) [7,
8] is used to assess the compatibility of the observation with expectations as
a function of ${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})$.

Test statistics are built from pseudo-experiments for the signal plus
background and background-only hypotheses. For each pseudo-experiment a
product of likelihood ratios is computed depending on the expected number of
signal events for a given branching fraction, $s_{i}$, the expected number of
background events, $b_{i}$ and the observed number of events, $d_{i}$ for bin
$i$. The $\textrm{CL}_{\textrm{s+b}}$ ($\textrm{CL}_{\textrm{b}}$) is defined
as the probability for signal plus background (background only) generated
pseudo-experiments to have a test-statistic value larger than or equal to that
observed in the data. The $\textrm{CL}_{\textrm{s}}$ is defined as the ratio
of confidence levels
$\frac{\textrm{CL}_{\textrm{s+b}}}{\textrm{CL}_{\textrm{b}}}$. This ratio is
used to set the exclusion (upper) limit on the branching fraction, whereas
$1-\textrm{CL}_{\textrm{b}}$ is used as a $p$-value to claim evidence or
observation. A $95(90)\%$ confidence level exclusion corresponds to
$\textrm{CL}_{\textrm{s}}=0.05(0.1)$.

The values of $b_{i}$ are obtained from the fit of the mass sidebands, as
detailed in Sect. 4. The values of $s_{i}$ depend on the assumed branching
fraction, as well as on the normalisation factors computed in Sect. 5. The
uncertainties on the input parameters are taken into account by fluctuating
the signal and background expectations when generating the $b$ and $s+b$
ensembles. These fluctuations are performed via asymmetric Gaussian priors,
following the formula

$x^{\prime}_{i}=x_{i}\left(1+\frac{1}{2}r(s_{+}-s_{-})+\frac{1}{2}r^{2}(s_{+}+s_{-})\right)$
(7)

where $x_{i}$ is the central value of the parameter, $r$ is a random number
generated from a normal distribution and $s_{+}$ and $s_{-}$ are the relative
(signed) errors of $x_{i}$ [24]. Correlations are implemented by using the
same value of $r$ for the parameters that should fluctuate coherently.

Figure 4: $\textrm{CL}_{\textrm{s}}$ curves for (a) TIS, (b) TOS categories and for (c) the combined sample. The solid line corresponds to the observed $\textrm{CL}_{\textrm{s}}$. The dashed line corresponds to the median of the $\textrm{CL}_{\textrm{s}}$ for an ensemble of background-alone experiments. In each plot, two bands are shown. The green (dark) band covers $68\%$ ($1\sigma$) of the $\textrm{CL}_{\textrm{s}}$ curves obtained in the background only pseudo-experiments, while the yellow (light) band covers $95\%$ ($2\sigma$).  Table 1: Upper limits on ${\cal B}(K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-})$ for the TIS and the TOS categories separately, and for the combined analysis. The last entry in the table is the $p$-value of the background-only hypothesis. Quantity | TIS | TOS | Combined
---|---|---|---
Expected upper limit at 95 (90)% C.L. $[10^{-9}]$ | $42\,(33)$ | $13\,(10)$ | $11\,(9)$
Observed upper limit at 95 (90)% C.L. $[10^{-9}]$ | $24\,(19)$ | $15\,(12)$ | $11\,(9)$
$p$-value | 0.95 | 0.20 | $0.27$

The observed distribution of events is compatible with background
expectations, giving a $p$-value of $27\%$. In particular, in the last 4 bins
of the BDT output, corresponding to the most significant region of the
analysis, just one candidate is observed in each of the trigger categories, in
agreement with the background expectations. Figure 4 shows the expected and
observed $\textrm{CL}_{\textrm{s}}$ curves for the TIS category and for the
TOS category as well as for the combined measurement. The upper limit found is
11 (9)$\times 10^{-9}$ at 95 (90)% confidence level and is a factor of thirty
below the previous world best limit. Table 1 summarises the limits in the TIS,
TOS categories, and the combined result.

##  8 Conclusions

A search for $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\mu^{+}\mu^{-}$ has
been performed using 1.0 fb-1 of data collected at the LHCb experiment in
2011. This search profits from the $10^{13}$ $K^{0}_{\rm\scriptscriptstyle S}$
produced inside the LHCb acceptance and the powerful discrimination against
the $K^{0}_{\rm\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ decay in which
both pions are misidentified as muons, achieved thanks to the LHCb mass
resolution for two body decays of the $K^{0}_{\rm\scriptscriptstyle S}$ meson.
The candidates observed are consistent with the expected background, with the
$p$-value for the background only hypothesis being $27\%$. The measured upper
limit

${\cal B}(K^{0}_{\rm\scriptscriptstyle
S}\rightarrow\mu^{+}\mu^{-})<11(9)\times 10^{-9}$

at 95(90)% confidence level is an improvement of a factor of thirty below the
previous world best limit[3].

## Acknowledgements

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

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