EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-284 LHCb-PAPER-2012-020 October 9, 2012

First observation of the decay $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$

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

A discovery of the rare decay $B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
presented. This decay is observed for the first time, with 5.2 $\sigma$
significance. The observation is made using $pp$ collision data, corresponding
to an integrated luminosity of 1.0 fb-1, collected with the LHCb detector. The
measured branching fraction is (2.3 $\pm$ 0.6 (stat.) $\pm$ 0.1
(syst.))$\times 10^{-8}$, and the ratio of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions is measured to be 0.053 $\pm$ 0.014
(stat.) $\pm$ 0.001 (syst.).

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, V.
Balagura28, W. Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W.
Barter44, A. Bates48, C. Bauer10, 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,35, 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, 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. Derkach14,35, O. Deschamps5,
F. Dettori39, 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, F. Eisele11,
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. Fitzpatrick47, 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, D. Gascon33, 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, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,35,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, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, 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. Maciuc10, O. Maev27,35, J. Magnin1, S.
Malde52, R.M.D. Mamunur35, 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.
Mathe12, 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, 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 Navarro33,
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, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues48,51, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35,
V. Romanovsky32, A. Romero Vidal34, M. Rosello33,n, 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, D.
Savrina28, 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, 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, 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, C.
Voß55, H. Voss10, 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, F.F. Wilson46, J. Wishahi9, M. Witek23, 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

##  1 Introduction

The ratio of Cabibbo-Kobayshi-Maskawa matrix [1] elements
$|V_{\text{td}}|/|V_{\text{ts}}|$ has been measured in $B$ mixing processes,
where it is probed in box diagrams through the ratio of $B^{0}$ and
$B^{0}_{s}$ mixing frequencies [2, 3, 4, 5]. The ratio of these matrix
elements has also been measured using the ratio of branching fractions of
$b\\!\rightarrow s\gamma$ and $b\\!\rightarrow d\gamma$ decays, where
radiative penguin diagrams mediate the transition [6, 7, 8]. These
measurements of $|V_{\text{td}}|/|V_{\text{ts}}|$ are consistent, within the
(dominant) $\sim$10% uncertainty on the determination from radiative decays.
The decays $b\\!\rightarrow s\mu^{+}\mu^{-}$ and $b\\!\rightarrow
d\mu^{+}\mu^{-}$ offer an alternative way of measuring
$|V_{\text{td}}|/|V_{\text{ts}}|$ which is sensitive to different classes of
operators than the radiative decay modes [9]. These
$b\\!\rightarrow(s,d)\mu^{+}\mu^{-}$ transitions are flavour-changing neutral
current processes which are forbidden at tree level in the Standard Model
(SM). In the SM, the branching fractions for $b\\!\rightarrow
d\ell^{+}\ell^{-}$ transitions are suppressed relative to $b\\!\rightarrow
s\ell^{+}\ell^{-}$ processes by the ratio
$|V_{\text{td}}|^{2}/|V_{\text{ts}}|^{2}$. This suppression does not
necessarily apply to models beyond the SM, and
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decays111Charge conjugation is
implicit throughout this paper. may be more sensitive to the effect of new
particles than $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays. In the SM,
the ratio of branching fractions for these exclusive modes

$R\equiv{{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}~{}/~{}{{\cal
B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})}$ (1)

is given by $R=V^{2}f^{2}$, where $V=|V_{\text{td}}|/|V_{\text{ts}}|$ and $f$
is the ratio of the relevant form factors and Wilson coefficients, integrated
over the relevant phase space. A difference between the measured value of $R$
and $V^{2}f^{2}$ would indicate a deviation from the minimal flavour violation
hypothesis [10, 11], and would rule out certain approximate flavour symmetry
models [12].

No $b\\!\rightarrow d\ell^{+}\ell^{-}$ transitions have previously been
detected, and the observation of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay would therefore be the first
time such a process has been seen. The predicted SM branching fraction for
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is (2.0 $\pm$ 0.2)$\times 10^{-8}$
[13]. The most stringent limit to date is $\cal
B$($B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$) $<6.9\times 10^{-8}$ at 90%
confidence level [14]. The analogous $b\\!\rightarrow s\ell^{+}\ell^{-}$
decay, $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, has been observed with a
branching fraction of (4.36$~{}\pm~{}$0.15$~{}\pm~{}$0.18) $\times~{}10^{-7}$
[15].

This paper describes the search for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay using $pp$ collision data,
corresponding to an integrated luminosity of 1.0 fb-1, collected with the LHCb
detector. The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction is
measured with respect to that of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$, and the ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions is also determined.

The LHCb detector [16] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$. The experiment is designed for the study of
particles containing $b$ or $c$ quarks. The apparatus includes a high
precision tracking system, consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, and a large-area silicon-strip
detector located upstream of a dipole magnet. The dipole magnet has a bending
power of about $4{\rm\,Tm}$. Three stations of silicon-strip detectors and
straw drift-tubes are placed downstream of the magnet. The combined tracking
system has a momentum resolution $\Delta p/p$ that varies from 0.4% at momenta
of 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The tracking system gives an impact
parameter resolution of 20$\,\upmu\rm m$ for tracks with a high transverse
momentum ($p_{\rm T}$). 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 either
multi-wire proportional chambers or triple gaseous electron multipliers.

In the present analysis, events are first required to have passed a hardware
trigger which selects high-$p_{\rm T}$ single muons or dimuons. In the first
stage of the subsequent software trigger, a single high impact parameter and
high-$p_{\rm T}$ track is required. In the second stage of the software
trigger, events are reconstructed and then selected for storage based on
either the (partially) reconstructed $B$ candidate or the dimuon candidate
[17, 18].

To produce simulated samples of signal and background decays, $pp$ collisions
are generated using Pythia 6.4 [19] with a specific LHCb configuration [20].
Decays of hadronic particles are described by the EvtGen package [21] in which
final state radiation is generated using Photos [22]. The interaction of the
generated particles with the detector and the detector response are
implemented using the Geant4 toolkit [23, *Agostinelli:2002hh], as described
in Ref. [25].

The small branching fractions of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ signal decays necessitate good control of the backgrounds
and the use of suitably constrained models to fit the invariant-mass
distributions. The decay
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$ (hereafter denoted
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$) is
used to extract both the shape of the signal mass peaks and, in the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ case, the invariant mass
distribution of the misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
events. These misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ events
form the main residual background after the application of the selection
requirements.

##  2 Event selection

The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates are selected by combining pairs of oppositely
charged muons with a charged pion or kaon. The selection includes requirements
on the impact parameters of the final-state particles and $B$ candidate, the
vertex quality and displacement of the $B$ candidate, particle identification
(PID) requirements on the muons and a requirement that the $B$ candidate
momentum vector points to one of the primary vertices in the event. The rate
of events containing more than one reconstructed candidate is 1 in
$\sim$20,000 for $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$. No restriction is therefore placed on the number of candidates
per event.

The pion identification requirements select a sample of pions with an
efficiency of $\sim$70% and a kaon rejection of 99%. The kaon identification
requirements allow the selection of a mutually exclusive sample with similar
efficiencies. The muon identification requirements have an efficiency of
$\sim$80%, with a pion rejection of $\sim$99.5%. The PID requirements have a
momentum dependent efficiency which is measured from data, in bins of
momentum, pseudorapidity and track multiplicity. The efficiency of the hadron
PID requirements is measured from a sample of
$D^{*+}\rightarrow(D^{0}\rightarrow K^{-}\pi^{+})\pi^{+}$ candidates that
allows the hadrons to be unambiguously identified based on their kinematics.
The muon PID efficiencies are measured using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates, using a tag and probe method.

The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$
resonances, where ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu},{\psi{(2S)}}\rightarrow\mu^{+}\mu^{-}$, are excluded using a veto on
the dimuon mass. This veto has a total width of 200 (150)
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the nominal
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ($\psi(2S)$) mass [26], and
takes into account the radiative tail of these decays. Candidates where the
dimuon mass is poorly measured have a correlated mismeasurement in the
$h\mu\mu$ mass. The veto therefore includes a component which shifts with
$h\mu\mu$ mass to exclude such candidates. Several other backgrounds are
considered: combinatorial backgrounds, where the particles selected do not
originate from a single decay; peaking backgrounds, where a single decay is
selected but with one or more particles misidentified; and partially
reconstructed backgrounds, where one or more final-state particles from a $B$
decay are not reconstructed. These backgrounds are each described below.

###  2.1 Combinatorial backgrounds

A boosted decision tree (BDT) [27] which employs the AdaBoost algorithm[28] is
used to separate signal candidates from the combinatorial background.
Kinematic and geometric properties of the $B^{+}$ candidate and final state
particles, $B^{+}$ candidate vertex quality and final state particle track
quality are input variables to the BDT.

The BDT is trained on a simulated $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$
signal sample, and a background sample taken from sidebands in the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ invariant mass distributions. These invariant masses are
denoted $M_{\pi^{+}\mu^{+}\mu^{-}}$ and $M_{K^{+}\mu^{+}\mu^{-}}$,
respectively. The background sample consists of 20% of the candidates with
$M_{\pi^{+}\mu^{+}\mu^{-}}$ or $M_{K^{+}\mu^{+}\mu^{-}}$ $>$ 5500
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This sample is not used for any of
the subsequent analysis. Signal candidates are required to have a BDT output
which exceeds a set value. This value is determined by simulating an ensemble
of datasets with the expected signal and background yields, and choosing the
cut value which gives the best statistical significance for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ signal yield. The same method is
used to select the optimal set of PID requirements. The BDT output
distribution for simulated $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ events
and for mass sideband candidates is shown in Fig. 1. A cut on the BDT output
$>$ 0.325 reduces the expected combinatorial background from 652 $\pm$ 11 to 9
$\pm$ 2 candidates in a $\pm$60 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
window around the nominal $B$ mass, while retaining 68% of signal events.
Assuming the SM branching fraction and the single event sensitivity defined in
Sect. 4, 21 $\pm$ 3 $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ signal events
are expected in the data sample.

Figure 1: BDT output distribution for simulated
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ events (black solid line) and
candidates taken from the mass sidebands in the data (red dotted line). Both
distributions are normalised to unit area. The vertical line indicates the
chosen cut value of 0.325.

###  2.2 Peaking and partially reconstructed backgrounds

Backgrounds from fully reconstructed $B^{+}$ decays with one or more
misidentified particles have a peaking mass structure. After applying the PID
requirements, the fraction of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
candidates misidentified as $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
0.9%, giving a residual background expectation of 6.2 $\pm$ 0.3 candidates.
This expectation is computed by weighting $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates, isolated using a kaon PID requirement,
according to the PID efficiency obtained from the $D^{*+}$ calibration sample.
The only other decay found to give a significant peaking background in the
search for $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ is
$B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$, where both a $\pi^{+}$ and a
$\pi^{-}$ are misidentified as muons. For $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ decays, the only significant peaking background is
$B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$, which includes the contribution
from $B^{+}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\rightarrow K^{+}\pi^{-})\pi^{+}$.
The expected background levels from $B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$
($B^{+}\\!\rightarrow K^{+}\pi^{+}\pi^{-}$) decays are computed to be 0.39
$\pm$ 0.04 (1.56 $\pm$ 0.16) residual background candidates, using simulated
events.

Backgrounds from decays that have one or more final state particles which are
not reconstructed have a mass below the nominal $B$ mass, and do not extend
into the signal window. However, in the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ case, these backgrounds overlap
with the misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ component
described above, and must therefore be included in the fit. In the
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ case such partially reconstructed
backgrounds are negligible.

###  2.3 Control channels

The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay candidates are isolated by
replacing the pion PID criteria with a requirement to select kaons. In
addition, instead of the dimuon mass vetoes described above, the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates are required to have dimuon mass within $\pm$50
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass (the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass resolution is 14.5
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). The remainder of the selection is
the same as that used for $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$. This
minimises the systematic uncertainty on the ratio of branching fractions,
although the selection is considerably tighter than that which would give the
lowest statistical uncertainty on the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ event yield. The
${B^{+}\rightarrow(J\\!/\\!\psi\rightarrow\mu^{+}\mu^{-})\pi^{+}}$ candidates
(denoted ${B^{+}\rightarrow J\\!/\\!\psi\pi^{+}}$), which are discussed below,
are selected using the same BDT, the pion PID criteria, and the above window
on the dimuon invariant mass. There is no significant peaking background for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
For $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays the only significant peaking background is misidentified
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events.

##  3 Signal yield determination

The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$, $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yields
are determined from a simultaneous unbinned maximum likelihood fit to four
invariant mass distributions which contain:

  1. 1.

Reconstructed $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates;

  2. 2.

Reconstructed $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates, with the kaon attributed to have the pion mass;

  3. 3.

Reconstructed $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates; and

  4. 4.

Reconstructed $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates.

The signal probability density functions (PDFs) for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$, $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay
modes are modelled with the sum of two Gaussian functions. The PDFs for all of
these decay modes share the same mean, widths and fraction of the total PDF
between the two Gaussians. The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ PDF
is adjusted for the difference between the widths of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
distributions, which is observed to be at the percent level in simulation. The
peaking backgrounds described in Sect. 2.2 are taken into account in the fit
by including PDFs with shapes determined from simulation. The combinatorial
backgrounds are modelled with a single exponential PDF, with the exponent
allowed to vary independently for each distribution. The partially
reconstructed candidates are modelled using a PDF consisting of an exponential
distribution cut-off at a threshold mass, with the transition smeared by the
experimental resolution. The shape parameters are again allowed to vary
independently for each distribution. The misidentified
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates are modelled with a Crystal Ball function [29], as it describes the
shape well. In order to describe the relevant background components for
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$, the fit is performed in the mass
range 4900 $<$ $M_{\pi^{+}\mu^{+}\mu^{-}}$ $<$ 7000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To avoid fitting the partially
reconstructed background for $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$, which
is irrelevant for the analysis, the fit is performed in the mass range 5170
$<$ $M_{K^{+}\mu^{+}\mu^{-}}$ $<$ 7000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

###  3.1 Reconstructed
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates

The reconstructed $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates are shown in the $M_{K^{+}\mu^{+}\mu^{-}}$
distribution in Fig. 2(a). The fitted
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield
is 106,230 $\pm$ 330. This large event yield determines the lineshape for the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ signal distributions, and provides the normalisation for
the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction.

###  3.2 Reconstructed
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates with the pion mass hypothesis

The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates reconstructed under the pion mass hypothesis provide the lineshape
for the misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates
that are a background to the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$
signal. The equivalent background from
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ in the $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ sample is negligible.

Figure 2: Invariant mass distribution for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates under the (a) $K^{+}\mu^{+}\mu^{-}$ and (b) $\pi^{+}\mu^{+}\mu^{-}$
mass hypotheses with the fit projections overlaid. In the legend, “part. reco”
refers to partially reconstructed background. The fit models are described in
the text.

The PID requirements used in the selection have a momentum dependent
efficiency and therefore change the mass distribution of any backgrounds with
candidates that have misidentified particles. In order to correct for this
effect, the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates are reweighted according to the PID efficiencies
derived from data, as described in Sect. 2.2. This adjusts the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
invariant mass distribution to remove the effect of the kaon PID requirement
used to isolate $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$, and to reproduce the effect of the pion PID requirement used to
isolate $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$. In addition, there is a
difference in the lineshapes of the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ invariant mass distributions under
the pion mass hypothesis. This effect arises from the differences between the
two decay modes’ dimuon energy and hadron momentum spectra, and is therefore
corrected by reweighting
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates in terms of these variables. The $M_{\pi^{+}\mu^{+}\mu^{-}}$
distribution after both weighting procedures have been applied is shown in
Fig. 2(b).

###  3.3 Reconstructed $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates

The yield of misidentified $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
candidates in the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ invariant mass
distribution is constrained to the expectation given in Sect. 2.2. Performing
the fit without this constraint gives a yield of 5.6 $\pm$ 6.4 misidentified
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ candidates. The yields for the
peaking background components are constrained to the expectations given in
Sect. 2.2. For both the $M_{\pi^{+}\mu^{+}\mu^{-}}$ and
$M_{K^{+}\mu^{+}\mu^{-}}$ distributions, the exponential PDF used to model the
combinatorial background has a step in the normalisation at 5500
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to account for the data used for
training the BDT.

The $M_{\pi^{+}\mu^{+}\mu^{-}}$ and $M_{K^{+}\mu^{+}\mu^{-}}$ distributions
are shown in Figs 3 and 4, respectively. The fit gives a
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ signal yield of 25.3
${}^{+6.7}_{-6.4}$, and a $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ signal
yield of 553 ${}^{+24}_{-25}$.

Figure 3: Invariant mass distribution of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates with the fit projection
overlaid (a) in the full mass range and (b) in the region around the $B$ mass.
In the legend, “part. reco.” and “combinatorial” refer to partially
reconstructed and combinatorial backgrounds respectively. The discontinuity at
5500 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is due to the removal of data
used for training the BDT.

Figure 4: Invariant mass distribution of $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates with the fit projection overlaid (a) in the
full mass range and (b) in the region around the $B$ mass. In the legend,
“combinatorial” refers to the combinatorial background.

###  3.4 Cross check of the fit procedure

The fit procedure was cross-checked on
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decays, accounting for the background from
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays.
The resulting fit is shown in Fig. 5. The shape of the combined
${B^{+}\rightarrow J\\!/\\!\psi\pi^{+}}$ and ${B^{+}\rightarrow J\\!/\\!\psi
K^{+}}$ mass distribution is well reproduced. The ${B^{+}\rightarrow
J\\!/\\!\psi K^{+}}$ yield is not constrained in this fit. The fitted yield of
1024 $\pm$ 61 candidates is consistent with the expectation of 958 $\pm$ 31
$\mathrm{(stat.)}$ candidates. This expectation is again computed by weighting
the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates, which are isolated using a kaon PID requirement, according to the
PID efficiency derived from $D^{*+}\rightarrow(D^{0}\rightarrow
K^{-}\pi^{+})\pi^{+}$ events.

Figure 5: Invariant mass distribution of ${B^{+}\rightarrow
J\\!/\\!\psi\pi^{+}}$ candidates with the fit projection overlaid. In the
legend, “part. reco.” and “combinatorial” refer to partially reconstructed and
combinatorial backgrounds respectively. The fit model is described in the
text.

##  4 Determination of branching fractions

The $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction is given by

$\displaystyle{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})$
$\displaystyle=$ $\displaystyle\frac{{\cal
B}({B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}})}{N_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}}{\frac{\epsilon_{{B^{+}\rightarrow J\\!/\\!\psi
K^{+}}}}{{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}}}N_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$
(2) $\displaystyle=$ $\displaystyle\alpha\cdot
N_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}\,,$ (3)

where $\cal B$$(X)$, $N_{X}$ and $\epsilon_{X}$ are the branching fraction,
the number of events and the total efficiency, respectively, for decay mode
$X$, and $\alpha$ is the single event sensitivity. The total efficiency
includes reconstruction, trigger and selection efficiencies. The ratio
$\epsilon_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ is
determined to be 1.60 $\pm$ 0.01 using simulated events, where the uncertainty
is due to the limited sizes of the simulated samples only. Other sources of
systematic uncertainty are discussed in Sect. 5. The difference in
efficiencies between $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
${B^{+}\rightarrow J\\!/\\!\psi K^{+}}$ events is largely due to the mass
vetoes used to remove the charmonium resonances, and the different PID
requirements. The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$ branching fraction is (6.02 $\pm$
0.20)$\times 10^{-5}$ [26]. Together with the other quantities in Eq. 2, this
gives a single event sensitivity of $\alpha$ = (9.1 $\pm$ 0.1)$\times
10^{-10}$, where the uncertainty is due to the limited sizes of the simulated
samples only.

The ratio of $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ branching fractions is given by

$R=\frac{N_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}{N_{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}\frac{\epsilon_{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}\,,$
(4)

where simulated events give ${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$
= 1.15 $\pm$ 0.01.

##  5 Systematic uncertainties

Two sources of systematic uncertainties are considered: those affecting the
determination of the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ signal yields, and those affecting
only the normalisation.

Uncertainties in the shape parameters for the misidentified
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ PDF in the fit are taken into
account by including Gaussian constraints on their values. The most
significant sources of uncertainty in the determination of these shape
parameters arise from the procedure for correcting the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ mass
shape to match that of the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay,
and the correction for the hadron PID requirements. The uncertainty on the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ yield determined with the fit takes
these shape parameter uncertainties into account, and they are therefore
included in the statistical rather than the systematic uncertainty. These
uncertainties affect the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ yield at
below the one percent level. None of these effects give rise to any
significant uncertainty for the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
decay.

Uncertainties on the two efficiency ratios
$\epsilon_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ and
${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$
affect the conversion of the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ yield
into a branching fraction, and the measurement of the ratio of branching
fractions $R$. The largest systematic uncertainty on these efficiency ratios
is the choice of form factors used to generate the simulated events. Using an
alternative set of form factors changes the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ efficiency by 3%, and this
difference is taken as a systematic uncertainty. For the ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, the alternative form factors are used for both
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$, giving a systematic uncertainty of 1.7%. To estimate the
uncertainty arising from the PID efficiency, the ratio of corrected yields
between the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ decay modes is measured, varying the PID requirements. The
largest resulting difference with respect to the nominal value is 1.1%, which
is taken as the systematic uncertainty.

The systematic uncertainty arising from the knowledge of the trigger
efficiency is determined using
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates in the data. Taking the events which pass the trigger independently
of the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidate, the fraction of these events which also pass the trigger based on
the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidate provides a determination of the trigger efficiency. The efficiency
determined in this way is compared to that calculated in simulated events
using the same method, and the difference is taken as the systematic
uncertainty. This gives a 1.4% uncertainty on
$\epsilon_{B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}/\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ and
${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$.

For all decays under consideration, there are small differences between the
distributions of some reconstructed quantities in the data and in the
simulated events. These differences are assessed by comparing the
distributions of data and simulated events for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates. The simulation is corrected to match the data where it disagrees,
and the resulting 0.4% difference between the raw and corrected ratio of
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ efficiencies is taken as a
systematic uncertainty. The statistical uncertainty from the limited
simulation sample size is 0.7%. When normalising to
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, the
measured $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $J/\psi\rightarrow\mu^{+}\mu^{-}$ branching fractions
contribute an uncertainty of 3.5% to the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction. The systematic
uncertainties are summarised in Table 1.

Table 1: Summary of systematic uncertainties. Source |  $\cal B$($B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$) (%) |  $\frac{{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}{{\cal B}(B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-})}$ (%)
---|---|---
Form factors | 3.0 | 1.7
Trigger efficiency | 1.4 | 1.4
PID performance | 1.1 | 1.1
Data simulation differences | 0.4 | 0.4
Simulation sample size | 0.7 | 0.7
$\cal B$($B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}$) | 3.5 | –
Total | 5.0 | 2.6

##  6 Results and conclusion

The statistical significance of the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$
signal observed in Fig. 3 is computed from the difference in the minimum log-
likelihood between the signal-plus-background and background-only hypotheses.
Both the statistical and systematic uncertainties on the shape parameters
(which affect the significance) are taken into account. The fitted yield
corresponds to an observation of the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay with 5.2 $\sigma$
significance. This is the first observation of a $b\\!\rightarrow
d\ell^{+}\ell^{-}$ transition. Normalising the observed signal to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay,
using the single event sensitivity given in Sect. 4, the branching fraction of
the $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay is measured to be

${\cal
B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})=(2.3~{}\pm~{}0.6~{}\mathrm{(stat.)}~{}\pm~{}0.1~{}\mathrm{(syst.)})\times
10^{-8}\,.$

This is compatible with the SM expectation of (2.0 $\pm$ 0.2)$\times 10^{-8}$
[13]. Given the agreement between the present measurement and the SM
prediction, contributions from physics beyond the SM can only modify the
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ branching fraction by a small
amount. A significant improvement in the precision of both the experimental
measurements and the theoretical prediction will therefore be required to
resolve any new physics contributions.

Taking the measured $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ yield and
${\epsilon_{{B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}}}}/{\epsilon_{B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}}}$,
the ratio of $B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ branching fractions is measured to
be

$\frac{{\cal B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}{{\cal
B}(B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-})}=0.053~{}\pm~{}0.014~{}\mathrm{(stat.)}~{}\pm~{}0.001~{}\mathrm{(syst.)}\,.$

In order to extract $|V_{\text{td}}|/|V_{\text{ts}}|$ from this ratio of
branching fractions, the SM expectation for the ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions is calculated using the EvtGen
package [21], which implements the calculation in Ref. [30]. This calculation
has been updated with the expressions for Wilson coefficients and power
corrections from Ref. [31], and formulae for the $q^{2}$ dependence of these
coefficients from Refs. [32, 33]. Using this calculation, and form factors
taken from Ref. [34] (“set II”), the integrated ratio of form factors and
Wilson coefficients is determined to be $f=0.87$. Neglecting theoretical
uncertainties, the measured ratio of
$B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-}$ and $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ branching fractions then gives

$|V_{\text{td}}|/|V_{\text{ts}}|=\frac{1}{f}\sqrt{\frac{{\cal
B}(B^{+}\\!\rightarrow\pi^{+}\mu^{+}\mu^{-})}{{\cal B}(B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-})}}=0.266~{}\pm~{}0.035~{}\mathrm{(stat.)}~{}\pm~{}0.003~{}\mathrm{(syst.)},$

which is compatible with previous determinations [5, 6, 7, 8]. An additional
uncertainty will arise from the knowledge of the form factors. As an estimate
of the scale of this uncertainty, the “set IV” parameters available in Ref.
[34] change the value of $|V_{\text{td}}|/|V_{\text{ts}}|$ by 5.1%. This
estimate is unlikely to cover a one sigma range on the form factor
uncertainty, and does not take into account additional sources of uncertainty
beyond the form factors. A full theoretical calculation taking into account
such additional uncertainties, which also accurately determines the
uncertainty on the ratio of form factors, would allow a determination of
$|V_{\text{td}}|/|V_{\text{ts}}|$ with comparable precision to that from
radiative penguin decays.

## 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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