* (99) J. Steinhoff, _Spin gauge symmetry in the action principle for classical relativistic particles_ , 1501.04951.
  * (100) J. Vines, D. Kunst, J. Steinhoff and T. Hinderer, _Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin_ , _Phys. Rev. D_ 93 (2016) 103008 [1601.07529].
  * (101) Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, T. Scheopner, F. Teng et al., _Quantum Field Theory, Worldline Theory, and Spin Magnitude Change in Orbital Evolution_ , 2308.14176.
  * (102) M. Henneaux and C. Teitelboim, _Quantization of gauge systems_ (1992).
  * (103) G. Kälin and R.A. Porto, _Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics_ , _JHEP_ 11 (2020) 106 [2006.01184].
  * (104) S. De Angelis, R. Gonzo and P.P. Novichkov, _Spinning waveforms from KMOC at leading order_ , 2309.17429.
  * (105) A. Brandhuber, G.R. Brown, G. Chen, J. Gowdy and G. Travaglini, _Resummed spinning waveforms from five-point amplitudes_ , 2023.
  * (106) R. Aoude, K. Haddad, C. Heissenberg and A. Helset, _Leading-order gravitational radiation to all spin orders_ , 2023.
  * (107) C. Shi and J. Plefka, _Classical double copy of worldline quantum field theory_ , _Phys. Rev. D_ 105 (2022) 026007 [2109.10345].
  * (108) J.A.M. Vermaseren, _New features of FORM_ , math-ph/0010025.


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-062 LHCb-PAPER-2013-014

Searches for violation of lepton flavour and baryon number in tau lepton
decays at LHCb

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

Searches for the lepton flavour violating decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and the lepton flavour and baryon
number violating decays $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$ have been carried out using proton-
proton collision data, corresponding to an integrated luminosity of $1.0$
$\mbox{\,fb}^{-1}$, taken by the LHCb experiment at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. No evidence has been found for any
signal, and limits have been set at $90\%$ confidence level on the branching
fractions: ${\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})<8.0\times
10^{-8}$, ${\cal B}(\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-})<3.3\times
10^{-7}$ and ${\cal B}(\tau^{-}\rightarrow p\mu^{-}\mu^{-})<4.4\times
10^{-7}$. The results for the $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$ decay modes represent the first direct
experimental limits on these channels.

Submitted to Physics Letters B

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

LHCb collaboration

R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov 34, M.
Artuso57, E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C.
Baesso58, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S.
Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I.
Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz 25,
K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37,
H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach14, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry12, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V. Gligorov37, C.
Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon54, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F. Jing3, M. John54, D.
Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, U. Kerzel37, T. Ketel41, A.
Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P.
Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M.
Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, L. Martin54,
A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D. Martins Tostes2,
A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A.
Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54,
F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez58, S. Monteil5, D. Moran53, P. Morawski25, M.J. Morello22,s, R.
Mountain57, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,p, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren 35,o, B.K. Pal57, A. Palano13,b, M. Palutan18, J.
Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, K.
Petridis52, A. Petrolini19,i, A. Phan57, E. Picatoste Olloqui35, B. Pietrzyk4,
T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25,
A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36,
S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35,o, G. Sabatino24,k, J.J. Saborido
Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C.
Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C.
Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C.
Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M.
Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp52, N.
Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P.
Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A.
Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, F. Xing54, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11

aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oIFIC, Universitat de Valencia-CSIC, Valencia, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

##  1 Introduction

The observation of neutrino oscillations was the first evidence for lepton
flavour violation (LFV). As a consequence, the introduction of mass terms for
neutrinos in the Standard Model (SM) implies that LFV exists also in the
charged sector, but with branching fractions smaller than $\sim 10^{-40}$ [1,
2]. Physics beyond the Standard Model (BSM) could significantly enhance these
branching fractions. Many BSM theories predict enhanced LFV in $\tau^{-}$
decays with respect to $\mu^{-}$ decays111The inclusion of charge conjugate
processes is implied throughout this Letter., with branching fractions within
experimental reach [3]. To date, no charged LFV decays such as
$\mu^{-}\rightarrow e^{-}\gamma$, $\mu^{-}\rightarrow e^{-}e^{+}e^{-}$,
$\tau^{-}\rightarrow\ell^{-}\gamma$ and
$\tau^{-}\rightarrow\ell^{-}\ell^{+}\ell^{-}$ (with $\ell^{-}=e^{-},\mu^{-}$)
have been observed [4]. Baryon number violation (BNV) is believed to have
occurred in the early universe, although the mechanism is unknown. BNV in
charged lepton decays automatically implies lepton number and lepton flavour
violation, with angular momentum conservation requiring the change
$|\Delta(B-L)|=0$ or $2$, where $B$ and $L$ are the net baryon and lepton
numbers. The SM and most of its extensions [1] require $|\Delta(B-L)|=0$. Any
observation of BNV or charged LFV would be a clear sign for BSM physics, while
a lowering of the experimental upper limits on branching fractions would
further constrain the parameter spaces of BSM models.

In this Letter we report on searches for the LFV decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and the LFV and BNV decay modes
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$ at LHCb [5]. The inclusive $\tau^{-}$ production cross-
section at the LHC is relatively large, at about $80\,\upmu$b (approximately
$80\%$ of which comes from $D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau}$),
estimated using the $b\bar{b}$ and $c\bar{c}$ cross-sections measured by LHCb
[6, 7] and the inclusive $b\rightarrow\tau$ and $c\rightarrow\tau$ branching
fractions [8]. The $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and
$\tau\rightarrow p\mu\mu$ decay modes222In the following $\tau\rightarrow
p\mu\mu$ refers to both the $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$ channels. are of particular interest at
LHCb, since muons provide clean signatures in the detector and the ring-
imaging Cherenkov (RICH) detectors give excellent identification of protons.

This Letter presents the first results on the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ decay mode from a hadron collider
and demonstrates an experimental sensitivity at LHCb, with data corresponding
to an integrated luminosity of $1.0$$\mbox{\,fb}^{-1}$, that approaches the
current best experimental upper limit, from Belle, ${\cal
B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})<2.1\times 10^{-8}$ at 90%
confidence level (CL) [9]. BaBar and Belle have searched for BNV $\tau$ decays
with $|\Delta(B-L)|=0$ and $|\Delta(B-L)|=2$ using the modes
$\tau^{-}\rightarrow\mathchar 28931\relax h^{-}$ and $\bar{\mathchar
28931\relax}h^{-}$ (with $h^{-}=\pi^{-},K^{-}$), and upper limits on branching
fractions of order $10^{-7}$ were obtained [4]. BaBar has also searched for
the $B$ meson decays $B^{0}\rightarrow\mathchar 28931\relax_{c}^{+}l^{-}$,
$B^{-}\rightarrow\mathchar 28931\relax l^{-}$ (both having $|\Delta(B-L)|=0$)
and $B^{-}\rightarrow\bar{\mathchar 28931\relax}l^{-}$ ($|\Delta(B-L)|=2$),
obtaining upper limits at 90% CL on branching fractions in the range
$(3.2-520)\times 10^{-8}$ [10]. The two BNV $\tau$ decays presented here,
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$, have $|\Delta(B-L)|=0$ but they could have rather different
BSM interpretations; they have not been studied by any previous experiment.

In this analysis the LHCb data sample from 2011, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, is used. Selection criteria are
implemented for the three signal modes,
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$,
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$, and for the calibration and normalisation channel, which is
$D_{s}^{-}\rightarrow\phi\pi^{-}$ followed by $\phi\rightarrow\mu^{+}\mu^{-}$,
referred to in the following as
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$. These initial, cut-based
selections are designed to keep good efficiency for signal whilst reducing the
dataset to a manageable level. To avoid potential bias,
$\mu^{-}\mu^{+}\mu^{-}$ and $p\mu\mu$ candidates with mass within $\pm
30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}~{}(\approx 3\sigma_{m})$ of the
$\tau$ mass are initially blinded from the analysis, where $\sigma_{m}$
denotes the expected mass resolution. For the $3\mu$ channel, discrimination
between potential signal and background is performed using a three-dimensional
binned distribution in two likelihood variables and the mass of the $\tau$
candidate. One likelihood variable is based on the three-body decay topology
and the other on muon identification. For the $\tau\rightarrow p\mu\mu$
channels, the use of the second likelihood function is replaced by cuts on the
proton and muon particle identification (PID) variables. The analysis strategy
and limit-setting procedure are similar to those used for the LHCb analyses of
the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$
channels [11, 12, *Junk_99].

##  2 Detector and triggers

The LHCb detector [5] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two RICH detectors. Photon, electron and
hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.

The trigger [14] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction. The hardware trigger selects muons with $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The software trigger requires
a two-, three- or four-track secondary vertex with a high sum of the $p_{\rm
T}$ of the tracks and a significant displacement from the primary $pp$
interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter chi-squared
(IP $\chi^{2}$), with respect to the $pp$ collision vertex, greater than 16.
The IP $\chi^{2}$ is defined as the difference between the $\chi^{2}$ of the
PV reconstructed with and without the track under consideration. A
multivariate algorithm is used for the identification of secondary vertices.

For the simulation, $pp$ collisions are generated using Pythia 6.4 [15] with a
specific LHCb configuration [16]. Particle decays are described by EvtGen [17]
in which final-state radiation is generated using Photos [18]. For the three
signal $\tau$ decay channels, the final-state particles are distributed
according to three-body phase space. The interaction of the generated
particles with the detector, and its response, are implemented using the
Geant4 toolkit [19, *Agostinelli:2002hh] as described in Ref. [21].

##  3 Signal candidate selection

The signal and normalisation channels have the same topology, the signature of
which is a vertex displaced from the PV, having three tracks that are
reconstructed to give a mass close to that of the $\tau$ lepton (or $D_{s}$
meson for the normalisation channel). In order to discriminate against
background, well-reconstructed and well-identified muon, pion and proton
tracks are required, with selections on track quality criteria and a
requirement of $p_{\rm T}$ $>300$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
Furthermore, for the $\tau\rightarrow p\mu\mu$ signal and normalisation
channels the muon and proton candidates must pass loose PID requirements and
the combined $p_{\rm T}$ of the three-track system is required to be greater
than $4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. All selected tracks are required
to have IP $\chi^{2}>9$. The fitted three-track vertex has to be of good
quality, with a fit $\chi^{2}<15$, and the measured decay time, $t$, of the
candidate forming the vertex has to be compatible with that of a heavy meson
or tau lepton ($ct>100\,\upmu\rm m$). Since the $Q$-values in decays of charm
mesons to $\tau$ are relatively small, poorly reconstructed candidates are
removed by a cut on the pointing angle between the momentum vector of the
three-track system and the line joining the primary and secondary vertices. In
the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ channel, signal candidates with
a $\mu^{+}\mu^{-}$ mass within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\phi$ meson mass are
removed, and to eliminate irreducible background near the signal region
arising from the decay
$D_{s}^{-}\rightarrow\eta(\mu^{+}\mu^{-}\gamma)\mu^{-}{\bar{\nu}_{\mu}}$,
candidates with a $\mu^{+}\mu^{-}$ mass combination below
$450{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are also rejected (see Section
6). Finally, to remove potential contamination from pairs of reconstructed
tracks that arise from the same particle, same-sign muon pairs with mass lower
than 250${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are removed in both the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$ channels. The signal regions are defined by $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\ (\approx 2\sigma_{m})$ windows
around the nominal $\tau$ mass, but candidates within wide mass windows, of
$\pm$400${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ decays and
$\pm$250${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $\tau\rightarrow
p\mu\mu$ decays, are kept to allow evaluation of the background contributions
in the signal regions. A mass window of $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is also used to define the signal
region for the $D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ channel, with
the $\mu^{+}\mu^{-}$ mass required to be within $\pm
20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $\phi$ meson mass.

##  4 Signal and background discrimination

After the selection each $\tau$ candidate is given a probability to be signal
or background according to the values of several likelihoods. For
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ three likelihoods are used: a
three-body likelihood, ${\rm\mathcal{M}_{3body}}$, a PID likelihood,
${\rm\mathcal{M}_{PID}}$, and an invariant mass likelihood. The likelihood
${\rm\mathcal{M}_{3body}}$ uses the properties of the reconstructed $\tau$
decay to distinguish displaced three-body decays from $N$-body decays (with
$N>3$) and combinations of tracks from different vertices. Variables used
include the vertex quality and its displacement from the PV, and the IP and
fit $\chi^{2}$ values of the tracks. The likelihood ${\rm\mathcal{M}_{PID}}$
quantifies the compatibility of each of the three particles with the muon
hypothesis using information from the RICH detectors, the calorimeters and the
muon stations; the value of ${\rm\mathcal{M}_{PID}}$ is taken as the smallest
one of the three muon candidates. For $\tau\rightarrow p\mu\mu$, the use of
${\rm\mathcal{M}_{PID}}$ is replaced by cuts on PID quantities. The invariant
mass likelihood uses the reconstructed mass of the $\tau$ candidate to help
discriminate between signal and background.

For the ${\rm\mathcal{M}_{3body}}$ likelihood a boosted decision tree [22] is
used, with the AdaBoost algorithm [23], and is implemented via the TMVA [24]
toolkit. It is trained using signal and background samples, both from
simulation, where the composition of the background is a mixture of
$b\bar{b}\rightarrow\mu\mu X$ and $c\bar{c}\rightarrow\mu\mu X$ according to
their relative abundance as measured in data. The ${\rm\mathcal{M}_{PID}}$
likelihood uses a neural network, which is also trained on simulated events.
The probability density function shapes are calibrated using the
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ control channel and
$J/\psi\rightarrow\mu^{+}\mu^{-}$ data for the ${\rm\mathcal{M}_{3body}}$ and
${\rm\mathcal{M}_{PID}}$ likelihoods, respectively. The shape of the signal
mass spectrum is modelled using
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ data. The
${\rm\mathcal{M}_{3body}}$ response as determined using the training from the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ samples is used also for the
$\tau\rightarrow p\mu\mu$ analyses.

For the ${\rm\mathcal{M}_{3body}}$ and ${\rm\mathcal{M}_{PID}}$ likelihoods
the binning is chosen such that the separation power between the background-
only and signal-plus-background hypotheses is maximised, whilst minimising the
number of bins. For the ${\rm\mathcal{M}_{3body}}$ likelihood the optimum
number of bins is found to be six for the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ analysis and five for
$\tau\rightarrow p\mu\mu$, while for the ${\rm\mathcal{M}_{PID}}$ likelihood
the optimum number of bins is found to be five. The lowest bins in
${\rm\mathcal{M}_{3body}}$ and ${\rm\mathcal{M}_{PID}}$ do not contribute to
the sensitivity and are later excluded from the analyses. The distributions of
the two likelihoods, along with their binning schemes, are shown in Fig. 1 for
the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ analysis.

\begin{overpic}[width=432.48048pt]{figs/supp/4a_new.pdf}
\put(40.0,120.0){\small{(a)}} \end{overpic}

\begin{overpic}[width=432.48048pt]{figs/supp/4b_new.pdf}
\put(40.0,120.0){\small{(b)}} \end{overpic}

Figure 1: Distribution of (a) ${\rm\mathcal{M}_{3body}}$ and (b)
${\rm\mathcal{M}_{PID}}$ for $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ where
the binning corresponds to that used in the limit calculation. The short
dashed (red) lines show the response of the data sidebands, whilst the long
dashed (blue) and solid (black) lines show the response of simulated signal
events before and after calibration. Note that in both cases the lowest
likelihood bin is later excluded from the analysis.

For the $\tau\rightarrow p\mu\mu$ analysis, further cuts on the muon and
proton PID hypotheses are used instead of ${\rm\mathcal{M}_{PID}}$ and are
optimised, for a $2\sigma$ significance, on simulated signal events and data
sidebands using the figure of merit from Ref. [25], with the distributions of
the PID variables corrected according to those observed in data. The expected
shapes of the invariant mass spectra for the
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ and $\tau\rightarrow p\mu\mu$
signals, with the appropriate selections applied, are taken from fits to the
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ control channel in data as
shown in Fig. 2. The signal distributions are modelled with the sum of two
Gaussian functions with a common mean, where the narrower Gaussian contributes
70% of the total signal yield, while the combinatorial backgrounds are
modelled with linear functions. The expected widths of the $\tau$ signals in
data are taken from simulation, scaled by the ratio of the widths of the
$D_{s}^{-}$ peaks in data and simulation. The data are divided into eight
equally spaced bins in the $\pm 20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
mass window around the nominal $\tau$ mass.

\begin{overpic}[width=432.48048pt]{figs/1a.pdf} \put(40.0,120.0){\small{(a)}}
\end{overpic}

\begin{overpic}[width=432.48048pt]{figs/1b.pdf} \put(40.0,120.0){\small{(b)}}
\end{overpic}

Figure 2: Invariant mass distribution of $\phi(\mu^{+}\mu^{-})\pi^{-}$ after
(a) the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ selection and (b) the
$\tau\rightarrow p\mu\mu$ selection and PID cuts. The solid (blue) lines show
the overall fits, the long dashed (green) and short dashed (red) lines show
the two Gaussian components of the signal and the dot dashed (black) lines
show the backgrounds.

##  5 Normalisation

To measure the signal branching fraction for the decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ (and similarly for $\tau\rightarrow
p\mu\mu$) we normalise to the
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ calibration channel using

$\displaystyle{{\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})}$
$\displaystyle\quad={{\cal
B}(D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-})}\times\frac{f^{D_{s}}_{\tau}}{{\cal
B}(D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau})}\times\frac{\rm{\epsilon\mathstrut_{cal}^{REC\&SEL}}}{\rm\epsilon\mathstrut_{sig}^{REC\&SEL}}\times\frac{\rm{\epsilon\mathstrut_{cal}^{TRIG}}}{\rm\epsilon\mathstrut_{sig}^{TRIG}}\times\frac{N_{\rm
sig}}{N_{\rm cal}}$ $\displaystyle\quad=\alpha\times N_{\rm sig}\,,$ (1)

where $\alpha$ is the overall normalisation factor and $N_{\rm sig}$ is the
number of observed signal events. The branching fraction ${\cal
B}(D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau})$ is taken from Ref. [26]. The
quantity $f^{D_{s}}_{\tau}$ is the fraction of $\tau$ leptons that originate
from $D_{s}^{-}$ decays, calculated using the $b\bar{b}$ and $c\bar{c}$ cross-
sections as measured by LHCb [6, 7] and the inclusive $b\rightarrow\tau$,
$c\rightarrow\tau$, $b\rightarrow D_{s}$ and $c\rightarrow D_{s}$ branching
fractions [8]. The corresponding expression for the $\tau\rightarrow p\mu\mu$
decay is identical except for the inclusion of a further term,
${\rm\epsilon\mathstrut_{cal}^{PID}}/{\rm\epsilon\mathstrut_{sig}^{PID}}$, to
account for the effect of the PID cuts.

The reconstruction and selection efficiencies, $\rm\epsilon^{REC\&SEL}$, are
products of the detector acceptances for the particular decays, the muon
identification efficiencies and the selection efficiencies. The combined muon
identification and selection efficiency is determined from the yield of
simulated events after the full selections have been applied. In the sample of
simulated events, the track IPs are smeared to describe the secondary-vertex
resolution of the data. Furthermore, the events are given weights to adjust
the prompt and non-prompt $b$ and $c$ particle production fractions to the
latest measurements [8]. The difference in the result if the weights are
varied within their uncertainties is assigned as a systematic uncertainty. The
ratio of efficiencies is corrected to account for the differences between data
and simulation in efficiencies of track reconstruction, muon identification,
the $\phi(1020)$ mass window cut in the normalisation channel and the $\tau$
mass window cut, with all associated systematic uncertainties included. The
removal of candidates in the least sensitive bins in the
${\rm\mathcal{M}_{3body}}$ and ${\rm\mathcal{M}_{PID}}$ classifiers is also
taken into account.

The trigger efficiency for selected candidates, $\rm\epsilon^{TRIG}$, is
evaluated from simulation while its systematic uncertainty is determined from
the difference between trigger efficiencies of $B^{-}\rightarrow J/\psi K^{-}$
decays measured in data and in simulation.

For the $\tau\rightarrow p\mu\mu$ channels the PID efficiency for selected and
triggered candidates, $\rm\epsilon^{PID}$, is calculated using data
calibration samples of $J/\psi\rightarrow\mu^{+}\mu^{-}$ and $\mathchar
28931\relax\rightarrow p\pi^{-}$ decays, with the tracks weighted to match the
kinematics of the signal and calibration channels. A systematic uncertainty of
1% per corrected final-state track is assigned [7], as well as a further 1%
uncertainty to account for differences in the kinematic binning of the
calibration samples between the analyses.

The branching fraction of the calibration channel is determined from a
combination of known branching fractions using

${\cal B}(D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-})=\frac{{\cal
B}(D_{s}^{-}\rightarrow\phi(K^{+}K^{-})\pi^{-})}{{\cal B}(\phi\rightarrow
K^{+}K^{-})}{\cal B}(\phi\rightarrow\mu^{+}\mu^{-})=(1.33\pm 0.12)\times
10^{-5}\,,$ (2)

where ${\cal B}(\phi\rightarrow K^{+}K^{-})$ and ${\cal
B}(\phi\rightarrow\mu^{+}\mu^{-})$ are taken from [8] and ${\cal
B}(D_{s}^{-}\rightarrow\phi(K^{+}K^{-})\pi^{-})$ is taken from the BaBar
amplitude analysis [27], which considers only the $\phi\rightarrow K^{+}K^{-}$
resonant part of the $D_{s}^{-}$ decay. This is motivated by the negligible
contribution of non-resonant $D_{s}^{-}\rightarrow\mu^{+}\mu^{-}\pi^{-}$
events seen in our data. The yields of
$D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-}$ candidates in data, $N_{\rm
cal}$, are determined from the fits to reconstructed
$\phi(\mu^{+}\mu^{-})\pi^{-}$ mass distributions, shown in Fig. 2. The
variations in the yields if the relative contributions of the two Gaussian
components are varied in the fits are considered as systematic uncertainties.
Table 1 gives a summary of all contributions to $\alpha$; the uncertainties
are taken to be uncorrelated.

Table 1: Terms entering in the normalisation factor $\alpha$ for
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$,
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$, and their combined statistical and systematic uncertainties.

$\begin{array}[]{c|r@{\hspace{1mm}\pm\hspace{1mm}}l|r@{\hspace{1mm}\pm\hspace{1mm}}l|r@{\hspace{1mm}\pm\hspace{1mm}}l}&\lx@intercol\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\tau^{-}\rightarrow
p\mu^{-}\mu^{-}\hfil\lx@intercol\\\ \hline\cr{\cal
B}(D_{s}^{-}\rightarrow\phi(\mu^{+}\mu^{-})\pi^{-})&\lx@intercol\hfil(1.33\hskip
2.84526pt\pm\hskip 2.84526pt&\lx@intercol 0.12)\times
10^{-5}\hfil\lx@intercol\\\ \hline\cr f^{D_{s}}_{\tau}&\lx@intercol\hfil
0.78\hskip 2.84526pt\pm\hskip 2.84526pt&\lx@intercol 0.05\hfil\lx@intercol\\\
\hline\cr{\cal
B}(D_{s}^{-}\rightarrow\tau^{-}\bar{\nu}_{\tau})&\lx@intercol\hfil
0.0561\hskip 2.84526pt\pm\hskip 2.84526pt&\lx@intercol
0.0024\hfil\lx@intercol\\\
\hline\cr\rm{\epsilon\mathstrut_{cal}}^{REC\&SEL}/\rm{\epsilon\mathstrut_{sig}}^{REC\&SEL}&1.49\hskip
2.84526pt\pm\hskip 2.84526pt&0.12&1.35\hskip 2.84526pt\pm\hskip
2.84526pt&0.12&1.36\hskip 2.84526pt\pm\hskip 2.84526pt&0.12\\\
\hline\cr\rm{\epsilon\mathstrut_{cal}}^{TRIG}/\rm{\epsilon\mathstrut_{sig}}^{TRIG}&0.753\hskip
2.84526pt\pm\hskip 2.84526pt&0.037&1.68\hskip 2.84526pt\pm\hskip
2.84526pt&0.10&2.03\hskip 2.84526pt\pm\hskip 2.84526pt&0.13\\\
\hline\cr\rm{\epsilon\mathstrut_{cal}}^{PID}/\rm{\epsilon\mathstrut_{sig}}^{PID}&\lx@intercol\hskip
28.45274pt\rm{n/a}\hfil\lx<EMAIL_ADDRESS>2.84526pt\pm\hskip 2.84526pt&0.07&1.42\hskip 2.84526pt\pm\hskip
2.84526pt&0.08\\\ \hline\cr N_{\rm cal}&48\,076\hskip 2.84526pt\pm\hskip
2.84526pt&840&\lx@intercol\hfil 8\,145\pm 180\hfil\lx@intercol\\\
\hline\cr\alpha&(4.34\hskip 2.84526pt\pm\hskip 2.84526pt&0.65)\times
10^{-9}&(7.4\hskip 2.84526pt\pm\hskip 2.84526pt&1.2)\times 10^{-8}&(9.0\hskip
2.84526pt\pm\hskip 2.84526pt&1.5)\times 10^{-8}\\\ \end{array}$

##  6 Background studies

The background processes for the decay
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ consist mainly of decay chains of
heavy mesons with three real muons in the final state or with one or two real
muons in combination with two or one misidentified particles. These
backgrounds vary smoothly in the mass spectra in the region of the signal
channel. The most important peaking background channel is found to be
$D_{s}^{-}\rightarrow\eta(\mu^{+}\mu^{-}\gamma)\mu^{-}{\bar{\nu}_{\mu}}$,
about $80\%$ of which is removed (see Section 3) by a cut on the dimuon mass.
The small remaining background from this process is consistent with the smooth
variation in the mass spectra of the other backgrounds in the mass range
considered in the fit. Based on simulations, no peaking backgrounds are
expected in the $\tau\rightarrow p\mu\mu$ analyses.

The expected numbers of background events within the signal region, for each
bin in ${\rm\mathcal{M}_{3body}}$, ${\rm\mathcal{M}_{PID}}$ (for
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$) and mass, are evaluated by fitting
the candidate mass spectra outside of the signal windows to an exponential
function using an extended, unbinned maximum likelihood fit. The small
differences obtained if the exponential curves are replaced by straight lines
are included as systematic uncertainties. For
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ the data are fitted over the mass
range $1600-1950$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, while for
$\tau\rightarrow p\mu\mu$ the fitted mass range is
$1650-1900$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, excluding windows
around the expected signal mass of $\pm
30$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $\mu^{-}\mu^{+}\mu^{-}$ and
$\pm 20$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $p\mu\mu$. The
resulting fits to the data sidebands for a selection of bins for the three
channels are shown in Fig. 3.

\begin{overpic}[width=455.24408pt]{figs/2a.pdf} \put(40.0,120.0){\small{(a)}}
\put(80.0,125.0){\tiny{${\rm\mathcal{M}_{3body}}$ $\in[0.65,1.0]$}}
\put(80.0,115.0){\tiny{${\rm\mathcal{M}_{PID}}$ $\in[0.725,1.0]$}}
\end{overpic}

\begin{overpic}[width=455.24408pt]{figs/2b.pdf} \put(40.0,120.0){\small{(b)}}
\put(60.0,120.0){\tiny{${\rm\mathcal{M}_{3body}}$ $\in[0.40,1.0]$}}
\end{overpic}

\begin{overpic}[width=455.24408pt]{figs/2c.pdf} \put(40.0,120.0){\small{(c)}}
\put(35.0,105.0){\tiny{${\rm\mathcal{M}_{3body}}$ $\in[0.40,1.0]$}}
\end{overpic}

Figure 3: Invariant mass distributions and fits to the mass sidebands in data
for (a) $\mu^{+}\mu^{-}\mu^{-}$ candidates in the four merged bins that
contain the highest signal probabilities, (b) ${\bar{p}}\mu^{+}\mu^{-}$
candidates in the two merged bins with the highest signal probabilities, and
(c) $p\mu^{-}\mu^{-}$ candidates in the two merged bins with the highest
signal probabilities.

##  7 Results

Tables 2 and 3 give the expected and observed numbers of candidates for all
three channels investigated, in each bin of the likelihood variables, where
the uncertainties on the background likelihoods are used to compute the
uncertainties on the expected numbers of events. No significant evidence for
an excess of events is observed. Using the $\textrm{CL}_{\textrm{s}}$ method
as a statistical framework, the distributions of observed and expected
$\textrm{CL}_{\textrm{s}}$ values are calculated as functions of the assumed
branching fractions. The aforementioned uncertainties and the uncertainties on
the signal likelihoods and normalisation factors are included using the
techniques described in Ref. [12, *Junk_99]. The resulting distributions of
$\textrm{CL}_{\textrm{s}}$ values are shown in Fig. 4.

The expected limits at $90\%~{}(95\%)$ CL for the branching fractions are

$\displaystyle{\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 8.3~{}(10.2)\times 10^{-8},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 4.6~{}(5.9)\times 10^{-7},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow p\mu^{-}\mu^{-})$ $\displaystyle<$
$\displaystyle 5.4~{}(6.9)\times 10^{-7},$

while the observed limits at $90\%~{}(95\%)$ CL are

$\displaystyle{\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 8.0~{}(9.8)\times 10^{-8},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-})$
$\displaystyle<$ $\displaystyle 3.3~{}(4.3)\times 10^{-7},$
$\displaystyle{\cal B}(\tau^{-}\rightarrow p\mu^{-}\mu^{-})$ $\displaystyle<$
$\displaystyle 4.4~{}(5.7)\times 10^{-7}.$

All limits are given for the phase-space model of $\tau$ decays. For
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$, the efficiency is found to vary by
no more than $20\%$ over the $\mu^{-}\mu^{-}$ mass range and by $10\%$ over
the $\mu^{+}\mu^{-}$ mass range. For $\tau\rightarrow p\mu\mu$, the efficiency
varies by less than $20\%$ over the dimuon mass range and less than $10\%$
with $p\mu$ mass.

In summary, a first limit on the lepton flavour violating decay mode
$\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ has been obtained at a hadron
collider. The result is compatible with previous limits and indicates that
with the additional luminosity expected from the LHC over the coming years,
the sensitivity of LHCb will become comparable with, or exceed, those of BaBar
and Belle. First direct upper limits have been placed on the branching
fractions for two $\tau$ decay modes that violate both baryon number and
lepton flavour, $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and
$\tau^{-}\rightarrow p\mu^{-}\mu^{-}$.

Table 2: Expected background candidate yields, with their systematic uncertainties, and observed candidate yields within the $\tau$ signal window in the different likelihood bins for the $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$ analysis. The likelihood values for ${\rm\mathcal{M}_{PID}}$ range from $0$ (most background-like) to $+1$ (most signal-like), while those for ${\rm\mathcal{M}_{3body}}$ range from $-1$ (most background-like) to $+1$ (most signal-like). The lowest likelihood bins have been excluded from the analysis.  ${\rm\mathcal{M}_{PID}}$ | ${\rm\mathcal{M}_{3body}}$ | Expected | Observed
---|---|---|---
|  $-$0.48 – | 0.05 |  345.0 $\pm$ | 6.7 | 409
| 0.05 – | 0.35 |  83.8 $\pm$ | 3.3 | 68
0.43 – 0.6 | 0.35 – | 0.65 |  30.2 $\pm$ | 2.0 | 35
| 0.65 – | 0.74 |  4.3 $\pm$ | 0.8 | 2
| 0.74 – | 1.0 |  1.4 $\pm$ | 0.4 | 1
|  $-$0.48 – | 0.05 |  73.1 $\pm$ | 3.1 | 64
| 0.05 – | 0.35 |  18.3 $\pm$ | 1.5 | 15
0.6 – 0.65 | 0.35 – | 0.65 |  8.6 $\pm$ | 1.1 | 7
| 0.65 – | 0.74 |  0.4 $\pm$ | 0.1 | 0
| 0.74 – | 1.0 |  0.6 $\pm$ | 0.2 | 2
|  $-$0.48 – | 0.05 |  45.4 $\pm$ | 2.4 | 51
| 0.05 – | 0.35 |  11.7 $\pm$ | 1.2 | 6
0.65 – 0.725 | 0.35 – | 0.65 |  5.3 $\pm$ | 0.8 | 3
| 0.65 – | 0.74 |  0.8 $\pm$ | 0.2 | 1
| 0.74 – | 1.0 |  0.4 $\pm$ | 0.1 | 0
|  $-$0.48 – | 0.05 |  44.5 $\pm$ | 2.4 | 62
| 0.05 – | 0.35 |  10.6 $\pm$ | 1.2 | 13
0.725 – 0.86 | 0.35 – | 0.65 |  7.3 $\pm$ | 1.0 | 7
| 0.65 – | 0.74 |  1.0 $\pm$ | 0.2 | 2
| 0.74 – | 1.0 |  0.4 $\pm$ | 0.1 | 0
|  $-$0.48 – | 0.05 |  5.9 $\pm$ | 0.9 | 7
| 0.05 – | 0.35 |  0.7 $\pm$ | 0.2 | 1
0.86 – 1.0 | 0.35 – | 0.65 |  1.0 $\pm$ | 0.2 | 1
| 0.65 – | 0.74 |  0.5 $\pm$ | 0.0 | 0
| 0.74 – | 1.0 |  0.4 $\pm$ | 0.1 | 0
Table 3: Expected background candidate yields, with their systematic uncertainties, and observed candidate yields within the $\tau$ mass window in the different likelihood bins for the $\tau\rightarrow p\mu\mu$ analysis. The likelihood values for ${\rm\mathcal{M}_{3body}}$ range from $-1$ (most background-like) to $+1$ (most signal-like). The lowest likelihood bin has been excluded from the analysis. | $\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ | $\tau^{-}\rightarrow p\mu^{-}\mu^{-}$
---|---|---
${\rm\mathcal{M}_{3body}}$ | Expected | Observed | Expected | Observed
$-$0.05 – | 0.20 |  37.9 $\pm$ | 0.8 | 43 |  41.0 $\pm$ | 0.9 | 41
0.20 – | 0.40 |  12.6 $\pm$ | 0.5 | 8 |  11.0 $\pm$ | 0.5 | 13
0.40 – | 0.70 |  6.76 $\pm$ | 0.37 | 6 |  7.64 $\pm$ | 0.39 | 10
0.70 – | 1.00 |  0.96 $\pm$ | 0.14 | 0 |  0.49 $\pm$ | 0.12 | 0

\begin{overpic}[width=455.24408pt]{figs/3a.pdf} \put(40.0,120.0){\small{(a)}}
\end{overpic}

\begin{overpic}[width=455.24408pt]{figs/3b.pdf} \put(40.0,120.0){\small{(b)}}
\end{overpic}

\begin{overpic}[width=455.24408pt]{figs/3c.pdf} \put(40.0,120.0){\small{(c)}}
\end{overpic}

Figure 4: Distribution of $\textrm{CL}_{\textrm{s}}$ values as functions of
the assumed branching fractions, under the hypothesis to observe background
events only, for (a) $\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-}$, (b)
$\tau^{-}\rightarrow\bar{p}\mu^{+}\mu^{-}$ and (c) $\tau^{-}\rightarrow
p\mu^{-}\mu^{-}$. The dashed lines indicate the expected curves and the solid
lines the observed ones. The light (yellow) and dark (green) bands cover the
regions of 68% and 95% confidence for the expected limits.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

## References

  * [1] M. Raidal et al., Flavour physics of leptons and dipole moments, Eur. Phys. J. C57 (2008) 13, arXiv:0801.1826
  * [2] A. Ilakovac, A. Pilaftsis, and L. Popov, Charged lepton flavor violation in supersymmetric low-scale seesaw models, Phys. Rev. D 87 (2013) 053014
  * [3] W. J. Marciano, T. Mori, and J. M. Roney, Charged lepton flavour violation experiments, Ann. Rev. Nucl. Part. Sci 58 (2008) 315
  * [4] Heavy Flavor Averaging Group, Y. Amhis et al., Averages of b-hadron, c-hadron, and tau-lepton properties as of early 2012, arXiv:1207.1158
  * [5] LHCb, A. Alves et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
  * [6] LHCb collaboration, R. Aaij et al., Measurement of $J/\psi$ production in $pp$ collisions at $\sqrt{s}$ = 7 TeV, Eur. Phys. J. C71 (2011) 1645, arXiv:1103.0423
  * [7] LHCb collaboration, R. Aaij et al., Prompt charm production in $pp$ collisions at $\sqrt{s}$ = 7 TeV, Nucl. Phys. B871 (2013) 1, arXiv:1302.2864
  * [8] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001
  * [9] Belle collaboration, K. Hayasaka et al., Search for lepton flavor violating $\tau$ decays into three leptons with 719 million produced $\tau^{+}\tau^{-}$ pairs, Phys. Lett. B687 (2010) 139, arXiv:1001.3221
  * [10] BaBar collaboration, P. del Amo Sanchez et al., Searches for the baryon- and lepton-number violating decays $B^{0}\rightarrow\mathchar 28931\relax_{c}^{+}\ell^{-}$, $B^{-}\rightarrow\mathchar 28931\relax\ell^{-}$, and $B^{-}\rightarrow\bar{\mathchar 28931\relax}\ell^{-}$, Phys. Rev. D83 (2011) 091101, arXiv:1101.3830
  * [11] LHCb collaboration, R. Aaij et al., First evidence for the decay $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$, Phys. Rev. Lett. 110 (2013) 021801, arXiv:1211.2674
  * [12] A. L. Read, Presentation of search results: the CL(s) technique, J. Phys. G28 (2002) 2693
  * [13] T. Junk, Confidence level computation for combining searches with small statistics, Nucl. Instrum. Meth. A434 (1999) 435, arXiv:hep-ex/9902006
  * [14] R. Aaij et al., The LHCb trigger and its performance, arXiv:1211.3055, to appear in JINST
  * [15] T. Sjöstrand, S. Mrenna and P. Skands, PYTHIA 6.4 Physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [16] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
  * [17] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
  * [18] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [19] GEANT4 collaboration, J. Allison et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [20] GEANT4 collaboration, S. Agostinelli et al., GEANT4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
  * [21] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, J. of Phys: Conf. Ser. 331 (2011) 032023
  * [22] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [23] R. E. Schapire and Y. Freund, A decision-theoretic generalization of on-line learning and an application to boosting, Jour. Comp. and Syst. Sc. 55 (1997) 119
  * [24] A. Hoecker et al., TMVA: Toolkit for multivariate data analysis, PoS ACAT (2007) 040, arXiv:physics/0703039
  * [25] G. Punzi, Sensitivity of searches for new signals and its optimization, in Statistical Problems in Particle Physics, Astrophysics, and Cosmology (L. Lyons, R. Mount, and R. Reitmeyer, eds.), p. 79, 2003.  arXiv:physics/0308063
  * [26] J. L. Rosner and S. Stone, Leptonic decays of charged pseudoscalar mesons, arXiv:1201.2401
  * [27] BaBar collaboration, P. del Amo Sanchez et al., Dalitz plot analysis of $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$, Phys. Rev. D83 (2011) 052001, arXiv:1011.4190