EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2012-006 LHCb-PAPER-2011-038 January 26, 2012

Searches for Majorana neutrinos in $B^{-}$ decays

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

Searches for heavy Majorana neutrinos in $B^{-}$ decays in final states
containing hadrons plus a $\mu^{-}\mu^{-}$ pair have been performed using 0.41
fb-1 of data collected with the LHCb detector in proton-proton collisions at a
center-of-mass energy of 7 TeV. The $D^{+}\mu^{-}\mu^{-}$ and
$D^{*+}\mu^{-}\mu^{-}$ final states can arise from the presence of virtual
Majorana neutrinos of any mass. Other final states containing $\pi^{+}$,
$D_{s}^{+}$, or $D^{0}\pi^{+}$ can be mediated by an on-shell Majorana
neutrino. No signals are found and upper limits are set on Majorana neutrino
production as a function of mass, and also on the $B^{-}$ decay branching
fractions.

Keywords: LHC, Majorana, Neutrino, $B$ Decays
PACS: 14.40.Nd, 13.35.Hb, 14.60.Pq
Submitted to Physics Review D

The LHCb Collaboration

R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, 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, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, F. Constantin26, A. Contu52, A. Cook43, M. Coombes43, G.
Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De
Cian37, F. De Lorenzi12, 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, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, E.
Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, 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, N. Gauvin36, 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, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
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. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefran$c$cois7, O. Leroy6, T.
Lesiak23, L. 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, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, 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, J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35, A.
Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17,
G.D. Patel49, M. Patel50, S.K. Paterson50, 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.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, 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, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, 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, 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, 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, K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, 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, P.
Urquijo53, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P.
Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46,
M. Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47,
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
25Soltan Institute for Nuclear Studies, 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
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated to 6
56Institut 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

Leptons constitute a crucially important sector of elementary particles. Half
of the leptons are neutrinos. Yet we do not know if they are Dirac or Majorana
particles, the latter case characterized by being their own antiparticles [1].
Since the observation of neutrino oscillations has indisputably established
that neutrinos have non-zero mass, it is possible to distinguish the two types
experimentally. Finding neutrinoless double $\beta$ decay has long been
advocated as a premier demonstration of the possible Majorana nature of
neutrinos [2]. The Feynman diagram is shown in Fig. 1. We also show the
fundamental quark and lepton level process. An impressive lower limit from
neutrinoless double $\beta$ decays in nuclei has already been obtained on the
half-life of $\cal{O}$$(10^{25})$ years [3] for coupling to $e^{-}$.

Figure 1: (a) Diagram of neutrinoless double $\beta$ decay when two neutrons
in a nucleus decay simultaneously. (b) The fundamental diagram for changing
lepton number by two units.

Similar processes can occur in $B^{-}$ decays. The diagram is shown in Fig.
2(a). In this reaction there is no restriction on the mass of the Majorana
neutrino as it acts as a virtual particle. In this paper, unlike in
neutrinoless double beta decays, a like-sign dimuon is considered rather than
two electrons. The only existing limit is from a recent Belle measurement [4]
using the $B^{-}\rightarrow D^{+}\mu^{-}\mu^{-}$ channel. We consider only
final states where the $c\overline{d}$ pair forms a final-state meson, either
a $D^{+}$ or a $D^{*+}$, so the processes we are looking for are
$B^{-}\rightarrow D^{(*)+}\mu^{-}\mu^{-}$. In this paper mention of a specific
reaction also implies inclusion of the charge conjugate reaction.

Figure 2: Feynman diagrams for $B$ decays involving an intermediate heavy
neutrino ($N$). (a) $B^{-}\rightarrow D^{(*)+}\mu^{-}\mu^{-}$, (b)
$B^{-}\rightarrow\pi^{+}(D_{s}^{+})\mu^{-}\mu^{-}$, and (c) $B^{-}\rightarrow
D^{0}\pi^{+}\mu^{-}\mu^{-}$.

There are other processes involving $b$-quark decays that produce a light
neutrino that can mix with a heavy neutrino, designated as $N$. The heavy
neutrino can decay as $N\rightarrow W^{+}\mu^{-}$. In Fig. 2(b) we show the
annihilation processes $B^{-}\rightarrow\pi^{+}(D_{s}^{+})\mu^{-}\mu^{-}$,
where the virtual $W^{+}$ materializes either as a $\pi^{+}$ or $D_{s}^{+}$.
These decays have been discussed in the literature [5, 6, *Zhang:2010um].

We note that it is also possible for the $B^{-}\rightarrow
D^{(*)+}\mu^{-}\mu^{-}$ decay modes shown in Fig. 2(a) to proceed by a Cabibbo
suppressed version of the process in Fig. 2(b) where the virtual $W^{+}$ forms
$D^{(*)+}$. Similarly, the decay modes shown in Fig. 2(b) could be produced
via Cabibbo suppressed versions of the process in Fig. 2(a). Here the
$\pi^{+}\mu^{-}\mu^{-}$ final state requires a $b\rightarrow u$ quark
transition while for the $D_{s}^{+}\mu^{-}\mu^{-}$ final state, one of the
virtual $W^{-}$ must couple to a $\overline{s}$ quark rather than a
$\overline{d}$.

The lifetimes of $N$ are not predicted. We assume here that they are long
enough that the natural decay width is narrower than our mass resolution which
varies between 2 and 15 MeV111In this paper we use units where the speed of
light, $c$, is set equal to one. depending on mass and decay mode. For
$B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$, we can access the Majorana mass
region between approximately 260 and 5000 MeV while for $B^{-}\rightarrow
D_{s}^{+}\mu^{-}\mu^{-}$, the Majorana mass region is between 2100 and 5150
MeV. In the higher mass region, the $W^{+}$ may be more likely to form a
$D_{s}^{+}$ meson than a $\pi^{+}$. The
$B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ search was first performed by Mark-II
[8] and then by CLEO [9]. LHCb also performed a similar search using a smaller
0.04 fb-1 data sample [10] giving an upper limit of $5.8\times 10^{-8}$ at 95%
confidence level (CL). The decay of $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$
has never been investigated.

Finally, in Fig. 2(c) we show how prolific semileptonic decays of the $B^{-}$
can result in the $D^{0}\pi^{+}\mu^{-}\mu^{-}$ final state. This process has
never been probed [11]. We benefit from the higher value of the CKM coupling
$|V_{cb}|$ relative to $|V_{ub}|$ in the annihilation processes shown in Fig.
2(b). The accessible region for Majorana neutrino mass is between 260 and 3300
MeV. For all the modes considered in this paper, we search only for decays
with muons in the final state, though electrons, and $\tau$ leptons in cases
where sufficient energy is available, could also be produced. Searches have
also been carried out looking for like-sign dileptons in hadron collider
experiments [12, *Chatrchyan:2011wba, *Abulencia:2007rd, *Acosta:2004kx].

##  2 Data sample and signal

We use a data sample of 0.37 fb-1 collected with the LHCb detector [16] in the
first half of 2011 and an additional 0.04 fb-1 collected in 2010 at a center-
of-mass energy of 7 TeV.

The detector elements are placed along the beam line of the LHC starting with
the vertex detector, a silicon strip device that surrounds the proton-proton
interaction region having its first active layer positioned 8 mm from the beam
during collisions. It provides precise locations for primary $pp$ interaction
vertices, the locations of decays of long-lived particles, and contributes to
the measurement of track momenta. Further downstream, other devices used to
measure track momenta include a large area silicon strip detector located in
front of a 4 Tm dipole magnet, and a combination of silicon strip detectors
and straw-tube drift chambers placed behind. Two Ring Imaging Cherenkov (RICH)
detectors are used to identify charged hadrons. An electromagnetic calorimeter
is used for photon detection and electron identification, followed by a hadron
calorimeter, and a system that distinguishes muons from hadrons. The
calorimeters and the muon system provide first-level hardware triggering,
which is then followed by a software high level trigger.

Muons are triggered on at the hardware level using their penetration through
iron and detection in a series of tracking chambers. Projecting these tracks
through the magnet to the primary event vertex allows a determination of their
transverse momentum, $p_{\rm T}$. Events from the 2011 data used in this
analysis were triggered on the basis of a single muon having a $p_{\rm T}$
greater than 1480 MeV, or two muons with their product $p_{\rm T}$ greater
than 1.69 GeV2. To satisfy the higher level trigger, the muon candidates must
also be detached from the primary vertex.

Candidate $B^{-}$ decays are found using tracking information, and particle
identification information from the RICH and muon systems. The identification
of pions, kaons and muons is based on combining the information from the two
RICH detectors, the calorimeters and the muon system. The RICH detectors
measure the angles of emitted Cherenkov radiation with respect to each charged
track. For a given momentum particle this angle is known, so a likelihood for
each hypothesis is computed. Muon likelihoods are computed based on track hits
in each of the sequential muon chambers. In this analysis we do not reject
candidates based on sharing hits with other tracks. This eliminates a possible
bias that was present in our previous analysis [10]. Selection criteria are
applied on the difference of the logarithm of the likelihood between two
hypotheses. The efficiencies and the mis-identification rates are obtained
from data using $K_{S}$, $D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow
K^{-}\pi^{+}$ and $J/\psi\rightarrow\mu^{+}\mu^{-}$ event samples that provide
almost pure pion, kaon, and muon sources.

Efficiencies and and rejection rates depend on the momentum of the final state
particles. For the RICH detector generally the pion or kaon efficiencies
exceed 90% and the rejection rates are of the order of 5% [17]. The muon
system provides efficiencies exceeding 98% with rejection rates on hadrons of
better than 99%, depending on selection criteria [18]. Tracks of good quality
are selected for further analysis. In order to ensure that tracks have good
vertex resolution we insist that they all have $p_{\rm T}>300$ MeV. For muons
this requirement varies from 650$-$800 MeV depending on the final state. All
tracks must be inconsistent with having been produced at the primary vertex
closest to the candidate $B^{-}$ meson’s decay point. The impact parameter
(IP) is the minimum distance of approach of the track with respect to the
primary vertex. Thus we form the IP $\chi^{2}$ by testing the hypothesis that
the IP is equal to zero, and require it to be large; the values depend on the
decay mode and range from 4 to 35.

Table 1: Charm and charmonium branching fractions Particle | Final state | Branching fraction (%)
---|---|---
$D^{0}$ | $K^{-}\pi^{+}$ |  3.89$\pm$ | 0.05 [3]
$D^{+}$ | $K^{-}\pi^{+}\pi^{+}$ | 9.14$\pm$ | 0.20 [3]
$D_{s}^{+}$ | $K^{-}K^{+}\pi^{+}$ | 5.50$\pm$ | 0.27 [19]
$D^{*+}$ | $\pi^{+}D^{0}$ | 67.7$\pm$ | 0.5 [3]
$\psi(2S)$ | $\pi^{+}\pi^{-}J/\psi$ | 32.6$\pm$ | 0.5 [3]
$J/\psi$ | $\mu^{+}\mu^{-}$ | 5.93$\pm$ | 0.06 [3]

##  3 Normalization channels

Values for branching fractions will be normalized to well measured channels
that have the same number of muons in the final state and equal track
multiplicities. The first such channel is $B^{-}\rightarrow J/\psi K^{-}$. Its
branching fraction is ${\cal{B}}(B^{-}\rightarrow J/\psi K^{-})=(1.014\pm
0.034)\times 10^{-3}$ [3]. We use the $J/\psi\rightarrow\mu^{+}\mu^{-}$ decay
mode. The product branching fraction of this normalization channel is
$(6.013\pm 0.021)\times 10^{-5}$, and is known to an accuracy of $\pm$2%. The
charm meson decay modes used in this paper are listed in Table 1, along with
their branching fractions and those of the charmonium decays in the
normalization channels.

To select the $J/\psi K^{-}$ normalization channel, the $p_{\rm T}$
requirement is increased to 1100 MeV for the $K^{-}$ and 750 MeV for the
muons. To select $B^{-}$ candidates we further require that the three tracks
form a vertex with a $\chi^{2}<7$, and that this $B^{-}$ candidate points to
the primary vertex at an angle not different from its momentum direction by
more than 4.47 mrad, and that the impact parameter $\chi^{2}$ of the $B^{-}$
is less than 12. The same requirements will be used for the
$\pi^{+}\mu^{-}\mu^{-}$ selection. The total efficiency for
$\mu^{+}\mu^{-}K^{-}$ is (0.99$\pm$0.01)%, where the $\mu^{+}\mu^{-}$ come
from $J/\psi$ decay.

The invariant mass of $K^{-}\mu^{+}\mu^{-}$ candidates is shown in Fig. 3(a).
In this analysis the $\mu^{+}\mu^{-}$ invariant mass is required to be within
50 MeV of the $J/\psi$ mass. We use a Crystal Ball function (CB) to describe
the signal [20], a Gaussian distribution for the partially reconstructed
background events, and a linear distribution for combinatorial background. The
CB function provides a convenient way to describe the shape of the
distribution, especially in the mass region below the peak where radiative
effects often produce an excess of events that falls away gradually, a so
called “radiative tail”. The CB function is

$f(m;\alpha,n,{m}_{0},\sigma)=\left\\{\begin{array}[]{l
l}\exp\left(-\frac{(m-m_{0})^{2}}{2\sigma^{2}}\right)&\quad\mbox{for
$\frac{m-m_{0}}{\sigma}>-\alpha$}\\\
A\cdot\left(b-\frac{m-m_{0}}{\sigma}\right)^{-n}&\quad\mbox{for
$\frac{m-m_{0}}{\sigma}\leq-\alpha$}\\\ \end{array}\right.$ (1)

where

$A=\left(\frac{n}{|\alpha|}\right)^{n}\cdot\exp\left(-\frac{|\alpha|^{2}}{2}\right)$
$b=\frac{n}{|\alpha|}-|\alpha|.$

The measured mass of each candidate is indicated as $m$, while $m_{0}$ and
$\sigma$ are the fitted peak value and resolution, and $n$ and $\alpha$ are
parameters used to model the radiative tail. We use the notation $\sigma$ in
the rest of this paper to denote resolution values found from CB fits.

Using an unbinned log-likelihood fit yields 47,224$\pm$222 $B^{-}\rightarrow
J/\psi K^{-}$ events. Within a $\pm 2\sigma$ signal window about the peak
mass, taken as the signal region, there are 44,283 of these events. The number
of signal events in this window is also determined using the total number of
events and subtracting the number given by the background fit. The difference
is 119 events, and this is taken as the systematic uncertainty of 0.3%. The
width of the signal peak is found to be 19.1$\pm$0.1 MeV. Monte Carlo
simulations are based on event generation using Pythia [21], followed by a
Geant-4 [22] based simulation of the LHCb detector [23]. The $J/\psi K^{-}$
mass resolution is 20% larger than that given by the LHCb simulation. All
simulated mass resolutions in this paper are increased by this factor.

Figure 3: Invariant mass of (a) candidate $J/\psi K^{-}$ decays, and (b)
candidate $J/\psi K^{-}\pi^{+}\pi^{-}$ decays. The data are shown as the
points with error bars. Both the partially reconstructed background and the
combinatorial background are shown, although the combinatorial background is
small and barely visible. The solid curve shows the total. In both cases the
candidate $\mu^{+}\mu^{-}$ is required to be within $\pm$50 MeV of the
$J/\psi$ mass, and in (b) the dimuon pair is constrained to have the $J/\psi$
mass.

For final states with five tracks, we change the normalization channel to
$B^{-}\rightarrow\psi(2S)K^{-}$, with
$\psi(2S)\rightarrow\pi^{+}\pi^{-}J/\psi$, and
$J/\psi\rightarrow\mu^{+}\mu^{-}$. The branching fraction for this channel is
${\cal{B}}(B^{-}\rightarrow\psi(2S)K^{-})=(6.48\pm 0.35)\times 10^{-4}$ [3].
Events are selected using a similar procedure as for $J/\psi K^{-}$ but adding
a $\pi^{+}\pi^{-}$ pair, that must have an invariant mass when combined with
the $J/\psi$ which is compatible with the $\psi(2S)$ mass, and that forms a
consistent vertex with the other $B^{-}$ decay candidate tracks. The total
efficiency for $\mu^{+}\mu^{-}\pi^{+}\pi^{-}K^{-}$ is (0.078$\pm$0.002)%,
without inclusion of the $\psi(2S)$ or $J/\psi$ branching fractions. The
$B^{-}$ candidate mass plot is shown in Fig. 3(b). Here the $\mu^{+}\mu^{-}$
pair is constrained to the $J/\psi$ mass. (In what follows, whenever the final
state contains a ground-state charm meson, its decay products are constrained
to their respective charm masses.)

The data are fitted with a CB function for signal, a Gaussian distribution for
partially reconstructed background and a linear function for combinatorial
background. There are 767$\pm$29 signal events in a $\pm 2\sigma$ window about
the peak mass. The difference between this value and a count of the number of
events in the signal region after subtracting the background implies a 0.7%
systematic uncertainty on the yield.

##  4 Analysis of $B^{-}\rightarrow D^{+}\mu^{-}\mu^{-}$ and
$D^{*+}\mu^{-}\mu^{-}$

Decay diagrams for $B^{-}\rightarrow D^{(*)+}\mu^{-}\mu^{-}$ are shown in Fig.
2(a). Since the neutrinos are virtual, the process can proceed for any value
of neutrino mass. It is also possible for these decays to occur via a Cabibbo
suppressed process similar to the ones shown in Fig. 2(b), where the virtual
$W^{+}$ materializes as a $c\overline{d}$ pair. If this occurred we would
expect the Cabibbo allowed $D_{s}^{+}\mu^{-}\mu^{-}$ final state to be about
an order of magnitude larger. The search for Majorana neutrinos in this
channel are discussed in Section 6. The $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$
and $D^{*+}\rightarrow\pi^{+}D^{0},~{}D^{0}\rightarrow K^{-}\pi^{+}$ channels
are used. The decay products of the $D^{+}$ and $D^{0}$ candidates are
required to have invariant masses within $\pm$25 MeV of the charm meson mass,
and for $D^{*+}$ candidate selection the mass difference
$m(\pi^{+}K^{-}\pi^{+})-m(K^{-}\pi^{+})$ is required to be within $\pm 3$ MeV
of the known $D^{*+}-D^{0}$ mass difference.

The $D^{(*)+}\mu^{-}\mu^{-}$ candidate mass spectra are shown in Fig. 4. No
signals are apparent. The $B^{-}$ mass resolution is 15.7$\pm$0.5 MeV for the
$D^{+}$ channel and 14.1$\pm$0.6 MeV for the $D^{*+}$ channel. The background
has two components, one from mis-reconstructed $B$ decays that tends to peak
close to the $B^{-}$ mass, called “peaking backgrounds”, and random track
combinations that are parameterized by a linear function. To predict the
combinatorial background in the signal region we fit the data in the sidebands
with a straight line. In the $D^{+}$ mode we observe six events in the signal
region, while there are five in the $D^{*+}$ mode. The combinatorial
background estimates are 6.9$\pm$1.1 and 5.9$\pm$1.0 events, respectively.
Peaking backgrounds are estimated from mis-identification probabilities,
determined from data, coupled with Monte Carlo simulation. For these two
channels peaking backgrounds are very small. The largest, due to
$B^{-}\rightarrow D^{+}\pi^{-}\pi^{-}$, is only 0.04 events.

The total efficiencies for $D^{+}\mu^{-}\mu^{-}$ and $D^{*+}\mu^{-}\mu^{-}$
are $(0.099\pm 0.007)$% and $(0.066\pm 0.005)$%, respectively; here the charm
branching fractions are not included. The systematic errors are listed in
Table 2 for this mode and other modes containing charm mesons that will be
discussed subsequently. Trigger efficiency uncertainties are evaluated from
differences in the 2010 and 2011 data samples. The largest systematic
uncertainties are due to the branching fractions of the normalization channels
and the trigger efficiencies. The uncertainty on the background is taken into
account directly when calculating the upper limits as explained below. Other
uncertainties arise from errors on the charmed meson branching fractions. For
these final states the uncertainty due to different final state track momenta
with respect to the normalization mode are very small, on the order of 0.2%.
Other channels have uncertainties due to varying efficiencies as a function of
Majorana mass, and these are entered in the row labeled “Efficiency modeling”.
The detector efficiency modeling takes into account the different acceptances
that could be caused by having different track momentum spectra. For example,
the track momenta depend on the Majorana neutrino mass for on-shell neutrinos.
These uncertainties are ascertained by simulating the detector response at
fixed Majorana masses and finding the average excursion from a simple fit to
the response and the individually simulated mass points. This same method is
used for other modes.

Figure 4: Invariant mass spectrum for (a) $B^{-}\rightarrow D^{+}\mu^{-}\mu^{-}$ candidates, and (b) $B^{-}\rightarrow D^{*+}\mu^{-}\mu^{-}$ candidates. The solid lines show the linear fits to the data in the mass sidebands.  Table 2: Systematic uncertainties for $B^{-}\rightarrow DX\mu^{-}\mu^{-}$ modes.  Source | Systematic uncertainty (%)
---|---
Common to all modes |
${\cal{B}}(B^{-}\rightarrow\psi(2S)K^{-})$ | 5.4
${\cal{B}}(\psi(2S)\rightarrow J/\psi\pi^{+}\pi^{-})$ | 1.5
${\cal{B}}(J/\psi\rightarrow\mu^{+}\mu^{-})$ | 1.0
Uncertainty in signal shape | 3.0
Yield of reference channel | 0.7
$\mu$ PID | 0.6
Mode specific | $D_{s}^{+}$ | $D^{0}\pi^{+}$ | $D^{+}$ | $D^{*+}$
Trigger | 4.9 | 9.3 | 5.5 | 4.8
Efficiency modeling | 10.0 | 6.7 |  |
PID ($K/\pi$) | 1.0 |  |  |
Charm decay ${\cal{B}}$’s | 4.9 | 1.3 | 2.2 | 1.5
Total | 13.8 | 13.2 | 8.8 | 8.2

To set upper limits on the branching fraction the number of events $N_{\rm
obs}$ within $\pm 2\sigma$ of the $B^{-}$ mass are counted. The distributions
of the number of events ($N$) are Poisson with the mean value of ($S+B$),
where $S$ indicates the expectation value of signal and $B$ background. For a
given number of observed events in the signal region, the upper limit is
calculated using the probability for $N\leq N_{\rm obs}$:

$P(N\leq N_{\rm obs})=\sum_{N\leq N_{\rm obs}}\frac{(S+B)^{N}e^{-(S+B)}}{N!}.$
(2)

A limit at 95% CL for branching fraction calculations is set by having
$P(N\leq N_{\rm obs})=0.05.$ The systematic errors are taken into account by
varying the calculated $S$ and $B$, assuming Gaussian distributions.

The upper limits on the branching fractions at 95% CL are measured to be

$\displaystyle{\cal{B}}(B^{-}\rightarrow D^{+}\mu^{-}\mu^{-})$
$\displaystyle<$ $\displaystyle 6.9\times 10^{-7}~{}{\rm and}$
$\displaystyle{\cal{B}}(B^{-}\rightarrow D^{*+}\mu^{-}\mu^{-})$
$\displaystyle<$ $\displaystyle 2.4\times 10^{-6}~{}.$

The limit on the $D^{+}$ channel is more stringent than a previous limit from
Belle of $1\times 10^{-6}$ at 90% CL [4], and the limit on the $D^{*+}$
channel is the first such result.

##  5 Analysis of $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$

The selection of $\pi^{+}\mu^{-}\mu^{-}$ events uses the same criteria as
described for $J/\psi K^{-}$ in Section 3, except for like-sign rather than
opposite-sign dimuon charges and pion rather than kaon identification. The
invariant mass distribution of $\pi^{+}\mu^{-}\mu^{-}$ candidates is shown in
Fig. 5. The mass resolution for this final state is 20.3$\pm$0.2 MeV. An
interval of $\pm$2$\sigma$ centered on the $B^{-}$ mass is taken as the signal
region.

Figure 5: Invariant mass distribution of $\pi^{+}\mu^{-}\mu^{-}$. The
estimated backgrounds are also shown. The curve is the sum of the peaking
background and the combinatoric background.

There are 7 events in the signal region, but no signal above background is
apparent. The peaking background, estimated as 2.5 events, is due to
misidentified $B^{-}\rightarrow J/\psi K^{-}$ or $J/\psi\pi^{-}$ decays; the
shape is taken from simulation. The combinatorial background is determined to
be 5.3 events from a fit to the $\pi^{+}\mu^{-}\mu^{-}$ mass distribution
excluding the signal region. The total background in the signal region then is
$7.8\pm 1.3$ events.

Since the putative neutrinos considered here decay into $\pi^{+}\mu^{-}$, and
are assumed to have very narrow widths, more sensitivity is obtained by
examining this mass distribution, shown in Fig. 6, for events in the $B^{-}$
signal region.

Figure 6: Invariant mass distribution of $\pi^{+}\mu^{-}$ in the $\pm 2\sigma$
region of the $B^{-}$ mass with both peaking and combinatorial background
superimposed. The peaking background at 3100 MeV is due to misidentified
$B^{-}\rightarrow J/\psi X$ decays.There are two combinations per event.

There is no statistically significant signal at any mass. There are three
combinations in one mass bin near 2530 MeV; however two of the combinations
come from one event, while it is possible to only have one Majorana neutrino
per $B^{-}$ decay. Upper limits at 95% confidence level on the existence of a
massive Majorana neutrino are set at each $\pi^{+}\mu^{-}$ mass by searching a
signal region whose width is $\pm 3\sigma_{N}$, where $\sigma_{N}$ is the mass
resolution, at each possible Majorana neutrino mass, $M_{N}$. This is done in
very small steps in $\pi^{+}\mu^{-}$ mass and so produces a continuous curve.
If a mass combination is found anywhere in the $\pm 3\sigma_{N}$ interval it
is considered as part of the observed yield. To set upper limits the mass
resolution and the detection efficiency as a function of $\pi^{+}\mu^{-}$ mass
need to be known. Monte Carlo simulation of the mass resolution as a function
of the Majorana neutrino mass is shown in Fig. 7, along with resolutions of
other channels. The overall efficiencies for different values of $M_{N}$ are
shown in Fig. 8. A linear interpolation is used to obtain values between the
simulated points.

Figure 7: Majorana mass resolutions for the three $B^{-}$ decays as a function
of Majorana mass.  Figure 8: Detection efficiencies for the three $B^{-}$
decays as a function of Majorana mass. Charm meson decay branching fractions
are not included.

Many systematic errors in the signal yield cancel in the ratio to the
normalization channel. The remaining systematic uncertainties are listed in
Table 3. The largest sources of error are the modeling of the detector
efficiency (5.3%) and the measured branching fractions
${\cal{B}}\left(B^{-}\rightarrow J/\psi K^{-}\right)$ (3.4%), and
${\cal{B}}\left(J/\psi\rightarrow\mu^{+}\mu^{-}\right)$ (1.0%).

Table 3: Systematic uncertainties for $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ measurement.  Selection criteria | Systematic uncertainties (%)
---|---
$K/\pi$ PID | 1.0
$\mu$ PID | 0.6
Muon selection | 0.6
Trigger | 1.0
Yields of reference channel | 0.4
Efficiency modeling | 5.3
${\cal{B}}(B^{-}\rightarrow J/\psi K^{-}$) | 3.4
${\cal{B}}(J/\psi\rightarrow\mu^{+}\mu^{-}$) | 1.0
Total | 6.7

To set upper limits on the branching fraction, the number of events $N_{\rm
obs}$ at each $M_{N}$ value (within $\pm 3\sigma_{N}$) are counted, and the
procedure described in the last section applied. Estimated background levels
are taken from Fig. 6. Figure 9(a) shows the upper limit on
${\cal{B}}(B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-})$ as a function of $M_{N}$ at
95% CL. For most of the neutrino mass region, the limits on the branching
ratio are $<8\times 10^{-9}$. Assuming a phase space decay of the $B^{-}$ we
also determine

${\cal{B}}(B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-})<1.3\times 10^{-8}~{}{\rm
at~{}95\%~{}CL.}$

These limits improve on previous results from CLEO $({<1.4\times
10^{-6}}~{}{\rm at~{}90\%~{}CL})$ [9], and LHCb $({<5.8\times 10^{-8}}~{}{\rm
at~{}95\%~{}CL})$ [10].

Figure 9: Upper limits at 95% CL as a function of the putative Majorana
neutrino mass, (a) for ${\cal{B}}(B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-})$ as a
function of the $\pi^{+}\mu^{-}$ mass, (b) for ${\cal{B}}(B^{-}\rightarrow
D_{s}^{+}\mu^{-}\mu^{-}$) as a function of the $D_{s}^{+}\mu^{-}$ mass, and
(c) for ${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-})$ as a function
of the $\pi^{+}\mu^{-}$ mass.

##  6 Analysis of $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$

The process $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ is similar to
$B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$, with the difference being that the
heavy neutrino can decay into $D_{s}^{+}\mu^{-}$. Here we consider only
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decays. Our analysis follows a
similar procedure used for the $\pi^{+}\mu^{-}\mu^{-}$ channel. Candidate
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decays are selected by having an
invariant mass within $\pm$25 MeV of the $D_{s}^{+}$ mass. A Majorana neutrino
candidate decay is then looked for by by having the $D_{s}^{+}$ candidate
decay tracks form a vertex with an opposite-sign muon candidate. Then this
neutrino candidate must form a vertex with another muon of like-sign to the
first one consistent with a $B^{-}$ decay detached from the primary vertex.
The invariant mass spectrum of $D_{s}^{+}\mu^{-}\mu^{-}$ candidates is shown
in Fig. 10. The mass resolution is 15.5$\pm$0.3 MeV.

Figure 10: Invariant mass spectrum for $B^{-}\rightarrow
D_{s}^{+}\mu^{-}\mu^{-}$ candidates. The line shows the fit to the data
excluding the $B^{-}$ mass signal region.

There are 12 events within the $B^{-}$ candidate mass region; it appears that
there is a dip in the number of events here. An unbinned fit to the data in
the sidebands gives an estimate of 22 events. The fluctuation at the $B^{-}$
mass, therefore, is about two standard deviations. Peaking background
contributions at the level of current sensitivity are negligible
($\sim$3$\times 10^{-4}$); thus only combinatorial background is considered.

After selecting the events in the $B^{-}$ signal region, we plot the
$D_{s}^{+}\mu^{-}$ invariant mass distribution, which is shown in Fig. 11. A
background estimate is made using the sideband data in $B^{-}$ candidate mass
(see Fig. 10), by fitting to a 4th order polynomial. The background estimated
from the sidebands is also shown in the figure. The normalization is absolute
and in agreement with the data. The data in the signal region is consistent
with the background estimate. The systematic error due to the fitting
procedure is estimated using the difference between this fit and the one
obtained using a 6th order polynomial.

Figure 11: Invariant mass spectrum of $D_{s}^{+}\mu^{-}$ from
$B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ events in the signal region with
the background estimate superimposed (solid curve). There are two combinations
per event.

The overall efficiencies for different values of $M_{N}$ are shown in Fig. 8.
As done previously, during the scan over the accessible Majorana neutrino mass
region we use a $\pm$3$\sigma_{N}$ mass window around a given Majorana mass.
The resolution is plotted in Fig. 7 as a function of $M_{N}$. Systematic
uncertainties are listed in Table 2.

Again we provide upper limits as a function of the Majorana neutrino mass,
shown in Fig. 9(b), only taking into account combinatorial background in this
case as the peaking background is absent. For neutrino masses below 5 GeV, the
limits on the mass dependent branching fractions are mostly $<6\times
10^{-7}$. We also determine an upper limit on the total branching fraction.
Since the background estimate of 22 events exceeds the observed level of 12
events we use the $CL_{s}$ method for calculating the upper limit [24].
Assuming a phase space decay of the $B^{-}$ we find

${\cal{B}}(B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-})<5.8\times 10^{-7}~{}{\rm
at~{}95\%~{}CL.}$

##  7 Analysis of $B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-}$

A prolific source of neutrinos is semileptonic $B^{-}$ decay. Majorana
neutrinos could be produced via semileptonic decays as shown in Fig. 2(c).
Here the mass range probed is smaller than in the case of
$\pi^{+}\mu^{-}\mu^{-}$ due to the presence of the $D^{0}$ meson in the final
state. The sensitivity of the search in this channel is also limited by the
need to reconstruct the $D^{0}\rightarrow K^{-}\pi^{+}$ decay. We do not
explicitly veto $D^{*+}\rightarrow\pi^{+}D^{0}$ decays as this would introduce
an additional systematic uncertainty. The invariant mass distribution of
$D^{0}\pi^{+}\mu^{-}\mu^{-}$ is shown in Fig. 12. The mass resolution is
14.4$\pm$0.2 MeV.

Figure 12: Invariant mass distribution of $D^{0}\pi^{+}\mu^{-}\mu^{-}$. The
solid line shows a linear fit to the data in the sidebands of the $B^{-}$
signal region.

Peaking backgrounds are essentially absent; the largest source is
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{-}\pi^{+}$ which contributes only 0.13
events in the signal region. The combinatorial background, determined by a
linear fit to the sidebands of the $B^{-}$ signal region, predicts 35.9
events, while the number observed is 33.

The $\pi^{+}\mu^{-}$ invariant mass for events within two standard deviations
of the $B^{-}$ mass is shown in Fig. 13. The background shape is estimated by
a 5th order polynomial fit to the sideband data (see Fig. 12) and also shown
on the figure. The systematic error on this background is estimated using a
7th order polynomial fit.

Figure 13: Invariant mass distribution of $\pi^{+}\mu^{-}$ for
$B^{-}\rightarrow D^{0}\mu^{-}\mu^{-}\pi^{+}$ in the signal region and with
estimated background distribution superimposed. There are two combinations per
event.

The $\pi^{+}\mu^{-}$ mass resolution is shown in Fig. 7. The $M_{N}$ dependent
efficiencies are shown in Fig. 8. They vary from 0.2% to 0.1% over most of the
mass range. Systematic errors are listed in Table 2. The largest sources of
error are the trigger, and the $M_{N}$ dependent efficiencies.

The upper limits for ${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-})$
as a function of the $\pi^{+}\mu^{-}$ mass are shown in Fig. 9(c). For
Majorana neutrino masses $<$ 3.0 GeV, the upper limits are less than
$1.6\times 10^{-6}$ at 95% CL. The limit on the branching fraction assuming a
phase space decay is

${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-})<1.5\times
10^{-6}~{}{\rm at~{}95\%~{}CL.}$

##  8 Conclusions

A search has been performed for Majorana neutrinos in the $B^{-}$ decay
channels, $D^{(*)+}\mu^{-}\mu^{-}$, $\pi^{+}\mu^{-}\mu^{-}$,
$D_{s}^{+}\mu^{-}\mu^{-}$, and $D^{0}\pi^{+}\mu^{-}\mu^{-}$ that has only
yielded upper limits. The $D^{(*)+}\mu^{-}\mu^{-}$ channels may proceed via
virtual Majorana neutrino exchange and thus are sensitive to all Majorana
neutrino masses. They also could occur via the same annihilation process as
the other modes, though this would be Cabibbo suppressed. The other channels
provide limits for neutrino masses between 260 and 5000 MeV. The bounds are
summarized in Table 4. These limits are the most restrictive to date.

Table 4: Summary of upper limits on branching fractions. Both the limits on the overall branching fraction assuming a phase space decay, and the range of limits on the branching fraction as a function of Majorana neutrino mass ($M_{N}$) are given. All limits are at 95% CL. Mode |  ${\cal{B}}$ upper limit | Approx. limits as function of $M_{N}$
---|---|---
$D^{+}\mu^{-}\mu^{-}$ | $6.9\times 10^{-7}$ |
$D^{*+}\mu^{-}\mu^{-}$ | $2.4\times 10^{-6}$ |
$\pi^{+}\mu^{-}\mu^{-}$ | $1.3\times 10^{-8}$ | $(0.4-1.0)\times 10^{-8}$
$D_{s}^{+}\mu^{-}\mu^{-}$ | $5.8\times 10^{-7}$ | $(1.5-8.0)\times 10^{-7}$
$D^{0}\pi^{+}\mu^{-}\mu^{-}$ | $1.5\times 10^{-6}$ | $(0.3-1.5)\times 10^{-6}$

Our search has thus far ignored the possibility of a finite neutrino lifetime.
Figure 14 shows the relative detection efficiency as a function of Majorana
neutrino lifetime, for (a) $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ for a mass
of 3 GeV, (b) $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ for a mass of 3 GeV,
and (c) $B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-}$ for a mass of 2 GeV. All
sensitivity is lost for lifetimes longer than $10^{-10}$ s to $10^{-11}$ s,
depending on the decay mode. Note that for the $D^{(*)+}\mu^{-}\mu^{-}$ final
states the detection efficiency is independent of the neutrino lifetime, since
the neutrino acts a virtual particle.

Figure 14: Relative efficiencies as a function of Majorana neutrino lifetime
for (a) $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ for a mass of 3 GeV, (b)
$B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ for a mass of 3 GeV, and (c)
$B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-}$ for a Majorana neutrino mass of
2 GeV. Where the error bars are not visible, they are smaller than radii of
the points.

Our upper limits in the $\pi^{+}\mu^{-}\mu^{-}$ final state can be used to
establish neutrino mass dependent upper limits on the coupling $|V_{\mu 4}|$
of a heavy Majorana neutrino to a muon and a virtual $W$. The matrix element
has been calculated in Ref. [5]. The results are shown in Fig. 15 as a
function of $M_{N}$.

Figure 15: Upper limits on $|V_{\mu 4}|^{2}$ at 95% CL as a function of the
Majorana neutrino mass from the $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$
channel.

A model dependent calculation of ${\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{+}\mu^{-}\mu^{-})$ can also be used to extract $|V_{\mu 4}|$ [11],
but the $\pi^{+}\mu^{-}\mu^{-}$ mode is more sensitive. For the
$D^{(*)+}\mu^{-}\mu^{-}$ channels upper limits cannot be extracted until there
is a theoretical calculation of the hadronic form-factor similar to those
available for neutrinoless double $\beta$ decay.

## 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.

## References

  * [1] E. Majorana, Teoria simmetrica dell$\,{}^{\prime}\\!$elettrone e del positrone, Nuovo Cim. 14 (1937) 171
  * [2] F. T. Avignone III, S. R. Elliott, and J. Engel, Double beta decay, Majorana neutrinos, and neutrino mass, Rev. Mod. Phys. 80 (2008) 481, arXiv:0708.1033
  * [3] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021
  * [4] BELLE collaboration, O. Seon et al., Search for lepton-number-violating $B^{+}\rightarrow D^{-}\ell\ell^{\prime}$ decays, Phys. Rev. D84 (2011) 071106, arXiv:1107.0642
  * [5] A. Atre, T. Han, S. Pascoli, and B. Zhang, The search for heavy Majorana neutrinos, JHEP 05 (2009) 030, arXiv:0901.3589
  * [6] G. Cvetic, C. Dib, S. K. Kang, and C. Kim, Probing Majorana neutrinos in rare $K$ and $D$, $D_{s}$, $B$, $B_{c}$ meson decays, Phys. Rev. D82 (2010) 053010, arXiv:1005.4282
  * [7] J.-M. Zhang and G.-L. Wang, Lepton-number violating decays of heavy mesons, Eur. Phys. J. C71 (2011) 1715, arXiv:1003.5570
  * [8] A. J. Weir et al., Upper limits on $D^{\pm}$ and $B^{\pm}$ decays to two leptons plus $\pi^{\pm}$ or $K^{\pm}$, Phys. Rev. D41 (1990) 1384
  * [9] CLEO collaboration, K. W. Edwards et al., Search for lepton-flavor-violating decays of $B$ mesons, Phys. Rev. D65 (2002) 111102, arXiv:hep-ex/0204017
  * [10] LHCb collaboration, R. Aaij et al., Search for the lepton number violating decays $B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ and $B^{+}\rightarrow K^{-}\mu^{+}\mu^{+}$, arXiv:1110.0730, Accepted for publication in Phys. Rev. Lett.
  * [11] D. Delepine, G. Lopez Castro, and N. Quintero, Lepton number violation in top quark and neutral $B$ meson decays, Phys. Rev. D84 (2011) 096011, arXiv:1108.6009
  * [12] ATLAS Collaboration, G. Aad et al., Inclusive search for same-sign dilepton signatures in $pp$ collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, JHEP 1110 (2011) 107, arXiv:1108.0366
  * [13] CMS Collaboration, S. Chatrchyan et al., Search for new physics with same-sign isolated dilepton events with jets and missing transverse energy at the LHC, JHEP 1106 (2011) 077, arXiv:1104.3168
  * [14] CDF Collaboration, A. Abulencia et al., Inclusive search for new physics with like-sign dilepton events in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96-TeV, Phys. Rev. Lett. 98 (2007) 221803, arXiv:hep-ex/0702051
  * [15] CDF Collaboration, D. Acosta et al., Inclusive search for anomalous production of high $p_{T}$ like-sign lepton pairs in $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 93 (2004) 061802, arXiv:hep-ex/0405063
  * [16] LHCb collaboration, A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
  * [17] LHCb RICH Collaboration, A. Powell, Reconstruction and PID performance of the LHCb RICH detectors, Nucl. Instrum. Meth. A639 (2011) 260
  * [18] LHCb collaboration, G. Graziani, S. R., and S. A., Study of the LHCb muon detector performance using 2010 beam data, LHCb-PUB-2011-027
  * [19] CLEO collaboration, J. P. Alexander et al., Absolute measurement of hadronic branching fractions of the $D_{s}^{+}$ meson, Phys. Rev. Lett. 100 (2008) 161804, arXiv:0801.0680
  * [20] T. Skwarnicki, A study of the radiative cascade transitions between the $\Upsilon^{\prime}$ and $\Upsilon$ Resonances,  DESY-F31-86-02, Ph.D. thesis, Institute of Nuclear Physics, Krakow, Poland.
  * [21] T. Sj$\ddot{\rm o}$strand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [22] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
  * [23] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, Journal of Physics: Conference Series 331 (2011), no. 3 032023
  * [24] A. L. Read, Presentation of search results: The CL(s) technique, J. Phys. G G28 (2002) 2693