The Belle Collaboration

# Observation of $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ and search
for $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$

Y. Jin  Department of Physics, University of Tokyo, Tokyo 113-0033  H. Aihara
Department of Physics, University of Tokyo, Tokyo 113-0033  D. Epifanov
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  I. Adachi  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The Graduate
University for Advanced Studies), Hayama 240-0193  S. Al Said  Department of
Physics, Faculty of Science, University of Tabuk, Tabuk 71451  Department of
Physics, Faculty of Science, King Abdulaziz University, Jeddah 21589  D. M.
Asner  Brookhaven National Laboratory, Upton, New York 11973  V. Aulchenko
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  T. Aushev  Moscow Institute of Physics
and Technology, Moscow Region 141700  R. Ayad  Department of Physics, Faculty
of Science, University of Tabuk, Tabuk 71451  V. Babu  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  I. Badhrees  Department of Physics,
Faculty of Science, University of Tabuk, Tabuk 71451  King Abdulaziz City for
Science and Technology, Riyadh 11442  S. Bahinipati  Indian Institute of
Technology Bhubaneswar, Satya Nagar 751007  V. Bansal  Pacific Northwest
National Laboratory, Richland, Washington 99352  P. Behera  Indian Institute
of Technology Madras, Chennai 600036  M. Berger  Stefan Meyer Institute for
Subatomic Physics, Vienna 1090  V. Bhardwaj  Indian Institute of Science
Education and Research Mohali, SAS Nagar, 140306  T. Bilka  Faculty of
Mathematics and Physics, Charles University, 121 16 Prague  J. Biswal  J.
Stefan Institute, 1000 Ljubljana  A. Bobrov  Budker Institute of Nuclear
Physics SB RAS, Novosibirsk 630090  Novosibirsk State University, Novosibirsk
630090  G. Bonvicini  Wayne State University, Detroit, Michigan 48202  A.
Bozek  H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342  M.
Bračko  University of Maribor, 2000 Maribor  J. Stefan Institute, 1000
Ljubljana  M. Campajola  INFN - Sezione di Napoli, 80126 Napoli  Università di
Napoli Federico II, 80055 Napoli  L. Cao  Institut für Experimentelle
Teilchenphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe  D.
Červenkov  Faculty of Mathematics and Physics, Charles University, 121 16
Prague  V. Chekelian  Max-Planck-Institut für Physik, 80805 München  A. Chen
National Central University, Chung-li 32054  B. G. Cheon  Hanyang University,
Seoul 133-791  K. Chilikin  P.N. Lebedev Physical Institute of the Russian
Academy of Sciences, Moscow 119991  H. E. Cho  Hanyang University, Seoul
133-791  K. Cho  Korea Institute of Science and Technology Information,
Daejeon 305-806  Y. Choi  Sungkyunkwan University, Suwon 440-746  S. Choudhury
Indian Institute of Technology Hyderabad, Telangana 502285  D. Cinabro  Wayne
State University, Detroit, Michigan 48202  S. Cunliffe  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  S. Di Carlo  LAL, Univ. Paris-Sud,
CNRS/IN2P3, Université Paris-Saclay, Orsay  Z. Doležal  Faculty of Mathematics
and Physics, Charles University, 121 16 Prague  T. V. Dong  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The
Graduate University for Advanced Studies), Hayama 240-0193  D. Dossett  School
of Physics, University of Melbourne, Victoria 3010  S. Eidelman  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  P.N. Lebedev Physical Institute of the Russian
Academy of Sciences, Moscow 119991  J. E. Fast  Pacific Northwest National
Laboratory, Richland, Washington 99352  T. Ferber  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  B. G. Fulsom  Pacific Northwest
National Laboratory, Richland, Washington 99352  R. Garg  Panjab University,
Chandigarh 160014  V. Gaur  Virginia Polytechnic Institute and State
University, Blacksburg, Virginia 24061  N. Gabyshev  Budker Institute of
Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State University,
Novosibirsk 630090  A. Garmash  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  A. Giri
Indian Institute of Technology Hyderabad, Telangana 502285  P. Goldenzweig
Institut für Experimentelle Teilchenphysik, Karlsruher Institut für
Technologie, 76131 Karlsruhe  B. Golob  Faculty of Mathematics and Physics,
University of Ljubljana, 1000 Ljubljana  J. Stefan Institute, 1000 Ljubljana
D. Greenwald  Department of Physics, Technische Universität München, 85748
Garching  O. Grzymkowska  H. Niewodniczanski Institute of Nuclear Physics,
Krakow 31-342  J. Haba  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  K. Hayasaka  Niigata University, Niigata 950-2181  H.
Hayashii  Nara Women’s University, Nara 630-8506  M. T. Hedges  University of
Hawaii, Honolulu, Hawaii 96822  W.-S. Hou  Department of Physics, National
Taiwan University, Taipei 10617  K. Huang  Department of Physics, National
Taiwan University, Taipei 10617  T. Iijima  Kobayashi-Maskawa Institute,
Nagoya University, Nagoya 464-8602  Graduate School of Science, Nagoya
University, Nagoya 464-8602  K. Inami  Graduate School of Science, Nagoya
University, Nagoya 464-8602  G. Inguglia  Institute of High Energy Physics,
Vienna 1050  A. Ishikawa  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  M. Iwasaki  Osaka City University, Osaka 558-8585  Y.
Iwasaki  High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
W. W. Jacobs  Indiana University, Bloomington, Indiana 47408  H. B. Jeon
Kyungpook National University, Daegu 702-701  S. Jia  Beihang University,
Beijing 100191  D. Joffe  Kennesaw State University, Kennesaw, Georgia 30144
K. K. Joo  Chonnam National University, Kwangju 660-701  J. Kahn  Ludwig
Maximilians University, 80539 Munich  A. B. Kaliyar  Indian Institute of
Technology Madras, Chennai 600036  G. Karyan  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  T. Kawasaki  Kitasato University,
Sagamihara 252-0373  H. Kichimi  High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801  C. Kiesling  Max-Planck-Institut für Physik, 80805
München  D. Y. Kim  Soongsil University, Seoul 156-743  H. J. Kim  Kyungpook
National University, Daegu 702-701  K. T. Kim  Korea University, Seoul 136-713
S. H. Kim  Hanyang University, Seoul 133-791  K. Kinoshita  University of
Cincinnati, Cincinnati, Ohio 45221  P. Kodyš  Faculty of Mathematics and
Physics, Charles University, 121 16 Prague  S. Korpar  University of Maribor,
2000 Maribor  J. Stefan Institute, 1000 Ljubljana  D. Kotchetkov  University
of Hawaii, Honolulu, Hawaii 96822  P. Križan  Faculty of Mathematics and
Physics, University of Ljubljana, 1000 Ljubljana  J. Stefan Institute, 1000
Ljubljana  R. Kroeger  University of Mississippi, University, Mississippi
38677  P. Krokovny  Budker Institute of Nuclear Physics SB RAS, Novosibirsk
630090  Novosibirsk State University, Novosibirsk 630090  R. Kulasiri
Kennesaw State University, Kennesaw, Georgia 30144  R. Kumar  Punjab
Agricultural University, Ludhiana 141004  A. Kuzmin  Budker Institute of
Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State University,
Novosibirsk 630090  Y.-J. Kwon  Yonsei University, Seoul 120-749  K. Lalwani
Malaviya National Institute of Technology Jaipur, Jaipur 302017  J. S. Lange
Justus-Liebig-Universität Gießen, 35392 Gießen  J. Y. Lee  Seoul National
University, Seoul 151-742  S. C. Lee  Kyungpook National University, Daegu
702-701  C. H. Li  Liaoning Normal University, Dalian 116029  L. K. Li
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049
Y. B. Li  Peking University, Beijing 100871  L. Li Gioi  Max-Planck-Institut
für Physik, 80805 München  J. Libby  Indian Institute of Technology Madras,
Chennai 600036  K. Lieret  Ludwig Maximilians University, 80539 Munich  Z.
Liptak  University of Hawaii, Honolulu, Hawaii 96822  D. Liventsev  Virginia
Polytechnic Institute and State University, Blacksburg, Virginia 24061  High
Energy Accelerator Research Organization (KEK), Tsukuba 305-0801  P.-C. Lu
Department of Physics, National Taiwan University, Taipei 10617  T. Luo  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of
Modern Physics, Fudan University, Shanghai 200443  J. MacNaughton  University
of Miyazaki, Miyazaki 889-2192  M. Masuda  Earthquake Research Institute,
University of Tokyo, Tokyo 113-0032  T. Matsuda  University of Miyazaki,
Miyazaki 889-2192  D. Matvienko  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  P.N.
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991
M. Merola  INFN - Sezione di Napoli, 80126 Napoli  Università di Napoli
Federico II, 80055 Napoli  K. Miyabayashi  Nara Women’s University, Nara
630-8506  H. Miyata  Niigata University, Niigata 950-2181  R. Mizuk  P.N.
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991
Moscow Institute of Physics and Technology, Moscow Region 141700  T. Mori
Graduate School of Science, Nagoya University, Nagoya 464-8602  R. Mussa  INFN
- Sezione di Torino, 10125 Torino  E. Nakano  Osaka City University, Osaka
558-8585  M. Nakao  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  K. J. Nath  Indian Institute of Technology Guwahati, Assam
781039  Z. Natkaniec  H. Niewodniczanski Institute of Nuclear Physics, Krakow
31-342  M. Nayak  Wayne State University, Detroit, Michigan 48202  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  M. Niiyama  Kyoto
University, Kyoto 606-8502  N. K. Nisar  University of Pittsburgh, Pittsburgh,
Pennsylvania 15260  S. Nishida  High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced
Studies), Hayama 240-0193  S. Ogawa  Toho University, Funabashi 274-8510  H.
Ono  Nippon Dental University, Niigata 951-8580  Niigata University, Niigata
950-2181  Y. Onuki  Department of Physics, University of Tokyo, Tokyo 113-0033
P. Pakhlov  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991  Moscow Physical Engineering Institute, Moscow 115409
G. Pakhlova  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991  Moscow Institute of Physics and Technology, Moscow
Region 141700  B. Pal  Brookhaven National Laboratory, Upton, New York 11973
S. Pardi  INFN - Sezione di Napoli, 80126 Napoli  H. Park  Kyungpook National
University, Daegu 702-701  S.-H. Park  Yonsei University, Seoul 120-749  S.
Patra  Indian Institute of Science Education and Research Mohali, SAS Nagar,
140306  S. Paul  Department of Physics, Technische Universität München, 85748
Garching  T. K. Pedlar  Luther College, Decorah, Iowa 52101  R. Pestotnik  J.
Stefan Institute, 1000 Ljubljana  L. E. Piilonen  Virginia Polytechnic
Institute and State University, Blacksburg, Virginia 24061  V. Popov  P.N.
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991
Moscow Institute of Physics and Technology, Moscow Region 141700  E. Prencipe
Forschungszentrum Jülich, 52425 Jülich  M. V. Purohit  University of South
Carolina, Columbia, South Carolina 29208  A. Rostomyan  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  G. Russo  INFN - Sezione di Napoli,
80126 Napoli  D. Sahoo  Tata Institute of Fundamental Research, Mumbai 400005
Y. Sakai  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801  SOKENDAI (The Graduate University for Advanced Studies), Hayama
240-0193  M. Salehi  University of Malaya, 50603 Kuala Lumpur  Ludwig
Maximilians University, 80539 Munich  S. Sandilya  University of Cincinnati,
Cincinnati, Ohio 45221  L. Santelj  High Energy Accelerator Research
Organization (KEK), Tsukuba 305-0801  T. Sanuki  Department of Physics, Tohoku
University, Sendai 980-8578  V. Savinov  University of Pittsburgh, Pittsburgh,
Pennsylvania 15260  O. Schneider  École Polytechnique Fédérale de Lausanne
(EPFL), Lausanne 1015  G. Schnell  University of the Basque Country UPV/EHU,
48080 Bilbao  IKERBASQUE, Basque Foundation for Science, 48013 Bilbao  J.
Schueler  University of Hawaii, Honolulu, Hawaii 96822  C. Schwanda  Institute
of High Energy Physics, Vienna 1050  Y. Seino  Niigata University, Niigata
950-2181  K. Senyo  Yamagata University, Yamagata 990-8560  O. Seon  Graduate
School of Science, Nagoya University, Nagoya 464-8602  M. E. Sevior  School of
Physics, University of Melbourne, Victoria 3010  V. Shebalin  University of
Hawaii, Honolulu, Hawaii 96822  C. P. Shen  Beihang University, Beijing 100191
J.-G. Shiu  Department of Physics, National Taiwan University, Taipei 10617
B. Shwartz  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  F. Simon  Max-Planck-
Institut für Physik, 80805 München  J. B. Singh  Panjab University, Chandigarh
160014  A. Sokolov  Institute for High Energy Physics, Protvino 142281  E.
Solovieva  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Moscow 119991  S. Stanič  University of Nova Gorica, 5000 Nova Gorica  M.
Starič  J. Stefan Institute, 1000 Ljubljana  Z. S. Stottler  Virginia
Polytechnic Institute and State University, Blacksburg, Virginia 24061  J. F.
Strube  Pacific Northwest National Laboratory, Richland, Washington 99352  T.
Sumiyoshi  Tokyo Metropolitan University, Tokyo 192-0397  M. Takizawa  Showa
Pharmaceutical University, Tokyo 194-8543  J-PARC Branch, KEK Theory Center,
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
Theoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198  U.
Tamponi  INFN - Sezione di Torino, 10125 Torino  K. Tanida  Advanced Science
Research Center, Japan Atomic Energy Agency, Naka 319-1195  F. Tenchini
Deutsches Elektronen–Synchrotron, 22607 Hamburg  K. Trabelsi  LAL, Univ.
Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay  M. Uchida  Tokyo
Institute of Technology, Tokyo 152-8550  T. Uglov  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991  Moscow Institute
of Physics and Technology, Moscow Region 141700  Y. Unno  Hanyang University,
Seoul 133-791  S. Uno  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  P. Urquijo  School of Physics, University of Melbourne,
Victoria 3010  Y. Usov  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  R. Van
Tonder  Institut für Experimentelle Teilchenphysik, Karlsruher Institut für
Technologie, 76131 Karlsruhe  G. Varner  University of Hawaii, Honolulu,
Hawaii 96822  A. Vinokurova  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  V.
Vorobyev  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991  A. Vossen  Duke
University, Durham, North Carolina 27708  B. Wang  Max-Planck-Institut für
Physik, 80805 München  C. H. Wang  National United University, Miao Li 36003
M.-Z. Wang  Department of Physics, National Taiwan University, Taipei 10617
P. Wang  Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049  S. Watanuki  Department of Physics, Tohoku University, Sendai
980-8578  E. Won  Korea University, Seoul 136-713  S. B. Yang  Korea
University, Seoul 136-713  H. Ye  Deutsches Elektronen–Synchrotron, 22607
Hamburg  J. H. Yin  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049  C. Z. Yuan  Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049  Y. Yusa  Niigata University,
Niigata 950-2181  Z. P. Zhang  University of Science and Technology of China,
Hefei 230026  V. Zhilich  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  V.
Zhukova  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Moscow 119991

###### Abstract

We present the first measurements of branching fractions of rare tau-lepton
decays, $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$ ($\ell=e$ or
$\mu$), using a data sample corresponding to 562 fb-1 collected at a center-
of-mass energy of 10.58 GeV with the Belle detector at the KEKB asymmetric-
energy $e^{+}e^{-}$ collider. The
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ decay is observed for the
first time with 7.0$\sigma$ significance. The partial branching fraction
determined by the structure-dependent mechanisms mediated by either a vector
or an axial-vector current for the mass region $M_{\pi ee}>1.05$ GeV/$c^{2}$
is measured to be
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})[M_{\pi^{-}e^{+}e^{-}}>1.05~{}{\rm
GeV}/c^{2}]=(5.90\pm 0.53\pm 0.85\pm 0.11)\times 10^{-6}$, where the first
uncertainty is statistical, the second is systematic, and the third is due to
the model dependence. In the full phase space, due to the different detection
efficiencies for the structure-dependent mechanisms mediated by axial-vector
and vector currents, the branching fraction varies from
$\mathcal{B}_{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})=(1.46\pm
0.13\pm 0.21)\times 10^{-5}$ to
$\mathcal{B}_{V}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})=(3.01\pm
0.27\pm 0.43)\times 10^{-5}$, respectively. An upper limit is set on the
branching fraction of the $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$
decay,
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-})<1.14\times
10^{-5}$, at the 90% confidence level.

###### pacs:

13.35.-r, 14.60.Fg, 13.66.Jn

##  I I. Introduction

The hadronic final states of the tau-lepton decays provide a clean laboratory
to study the dynamics of strong interactions in the energy region below the
tau mass. The world’s largest sample of tau leptons, collected with the Belle
detector, allows measurement of rare tau decay branching fractions to a new
level of sensitivity. We present the first measurements of branching fractions
of rare tau-lepton decays
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$ ($\ell=e$ or $\mu$) CC
, whose theoretical predictions are calculated as
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})\in[1.4,2.8]\times
10^{-5}$ and
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-})\in[0.03,1.0]\times
10^{-5}$ Roig .

Figure 1: Feynman diagrams for the dominant amplitudes for
$\tau^{-}\to\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$ Roig . The dot indicates known
(fully data-driven) hadronization effects. The solid square (triangle)
represents the structure-dependent contribution mediated by the vector (axial-
vector) current.

These decays are interesting because they involve the $\gamma^{*}W^{*}\pi$
vertex with two gauge bosons off their mass shells. The relevant amplitudes
for the decays $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$ can be
written as a sum of five contributions, as shown in Fig. 1 explanation . The
diagrams (a), (b), and (c) are structure-independent while the diagram (d) and
(e) are structure-dependent. In all amplitudes, the lepton pair is produced by
a virtual photon. In the case of a real photon realpho , the $\gamma W\pi$
vertex plays an important role in determining the radiative corrections to the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}$ decay RC  and in the evaluation of the
contributions of the hadronic light-by-light scattering to the muon anomalous
magnetic moment g2 . The measured branching fractions of the decay modes of
interest can be used to validate the Resonance Chiral Theory RCT , a semi-
phenomenological approach to describe strong interactions at energies below
the tau mass. In addition, rare decays of $\tau$ often serve as a probe of
physics beyond the Standard Model sterile . Another important impact of a
precise measurement of
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-})$ is useful
input for a reliable background estimation in searches for various lepton
flavor and lepton number violating decays.

##  II II. Data Set and simulations

In this paper, we report branching fraction measurements for the rare decays
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ and
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ with data recorded at a
center-of-mass (CM) energy of $\sqrt{s}=10.58$ GeV by the Belle experiment at
the KEKB asymmetric-energy $e^{+}e^{-}$ collider KEKB . The data used in this
analysis correspond to an integrated luminosity of $562~{}{\rm fb}^{-1}$. The
Belle detector is a general purpose large-solid-angle spectrometer consisting
of a silicon vertex detector, a central drift chamber (CDC), an array of
aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-
of-flight scintillation counters (TOF), and an electromagnetic calorimeter
(ECL) located inside a superconducting solenoid coil that provides a 1.5 T
magnetic field. Outside the coil, an iron flux-return yoke is instrumented to
detect $K^{0}_{L}$ mesons and to identify muons (KLM). A detailed description
of the Belle detector can be found elsewhere thebelledetector . In this
analysis, we use the data set collected with an inner configuration that
comprises a 1.5 cm radius beampipe, a 4-layer silicon detector and a small-
cell inner drift chamber.

To optimize the event selection criteria and to determine the detection
efficiency for the signal events, Monte Carlo (MC) samples are employed. The
MC samples are generated using EVTGEN EVTGEN  for hadronic processes, KKMC KK
for fermion pairs, and AAFH AAFH  for two-photon production of fermion pairs.
The $\tau$ lepton decays are carried out by TAUOLA TAUOLA . The final-state
radiation of charged particles is simulated by PHOTOS PHOTOS . For the signal
modes $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$, we employ
formulas given in Ref. Roig  and implement them into the TAUOLA generator. For
the $\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ mode, the largest background for the
$\tau^{-}\to\pi^{-}\nu_{\tau}e^{+}e^{-}$ signal, we use the pion form factor
measured by the Belle collaboration Hayashii . For the $\pi^{0}$ decay, in
addition to $\pi^{0}\to\gamma\gamma$, the $\pi^{0}$ Dalitz decay,
$\pi^{0}\to\gamma e^{+}e^{-}$ (1.17%), is implemented via the pythia package
pythia . The response of the detector is simulated by a GEANT3-based program
GEANT3 . The optimization of selection criteria using MC samples is
implemented by maximizing $S/\sqrt{S+B}$, where $S$ ($B$) is the number of
signal (background) events.

##  III III. Event selection and reconstruction

The selection proceeds in two stages, aimed at suppressing background
processes while retaining a high efficiency for the decay mode of interest. At
the first stage, we select $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events and
substantially reject background from the other processes that occur at
$\Upsilon(4S)$. The second stage proceeds to select one $\tau$ decaying into
the $\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$ final state, where a vertex fit is
performed to the $\ell^{+}\ell^{-}$ pair, and the other $\tau$ decaying into
one-prong modes.

###  III.1 A. Selection of $\tau^{+}\tau^{-}$ events

Events with four tracks and zero net charge are selected. The distance of
closest approach of each charged particle to the interaction point (IP) must
be less than 5 cm along the beam direction (the $z$ axis) and less than 1 cm
in the transverse plane (the $x-y$ plane). Each charged particle must have
transverse momentum larger than 0.1 GeV/$c$; and at least one particle must
have transverse momentum larger than 0.5 GeV/$c$. The total energy deposited
in the ECL must be less than 10 GeV. A vertex fit is performed using all
charged particles to determine the position of the primary vertex of the
event, which is required to be near the interaction point within 0.5 cm in the
$x-y$ plane and 2.5 cm along the $z$ axis. The mean position of the
interaction point itself is monitored using generic multi-track events in the
same data taking period with an accuracy of 10 $\mu$m. Photons, reconstructed
from the clusters in the ECL not associated with tracks, are selected with
energy thresholds of 50 MeV in the barrel region
($32^{\circ}<\theta_{\gamma}<130^{\circ}$) and 100 MeV in the endcap region to
eliminate beam-background photons.

In order to reduce the remaining background from radiative Bhabha events,
continuum $e^{+}e^{-}\to q\bar{q}$ (where $q=u,d,s,c$) production, and two-
photon processes, we require the following conditions. In the CM frame, the
sum of the magnitudes of charged particle momenta must be less than 10
GeV/$c$, and its sum with photon energies must exceed 3 GeV. Opening angles
for every pair of tracks must be less than $175^{\circ}$. A tau-pair event is
accompanied by missing four-momentum due to neutrinos, defined as $p_{\rm
miss}=p_{\rm init}-\sum_{i}p_{{\rm trk},i}-\sum_{i}p_{\gamma,i}$, where
$p_{\rm init}$ is the four-momentum of the colliding $e^{+}e^{-}$ beams, and
$\sum_{i}p_{{\rm trk},i}$ and $\sum_{i}p_{\gamma,i}$ are the sums of the four-
momenta of charged particles (assumed to be pions) and photons, respectively.
Only events satisfying $1~{}{\rm GeV}/c^{2}<M_{\rm miss}<7~{}{\rm GeV}/c^{2}$
and $30^{\circ}<\theta_{\rm miss}<150^{\circ}$ criteria are selected, where
$M_{\rm miss}$ and $\theta_{\rm miss}$ are the mass and polar angle of the
missing four-momentum in the CM frame.

We use the thrust variable to reduce $e^{+}e^{-}\to q\bar{q}$ backgrounds and
to provide two disjoint hemispheres for each event. The thrust $T$ is defined
as the maximum value of
${\sum_{i}|\vec{p}_{i}\cdot\vec{n}_{T}|}/{\sum_{i}|\vec{p}_{i}|}$ with respect
to the thrust axis $\vec{n}_{T}$, where $\vec{p}_{i}$ is the CM momentum of
the $i$th particle, either charged or neutral. We require the thrust $T$ to be
in the range [0.85, 0.99]. Events are then divided into two hemispheres, in
the CM frame, by the plane perpendicular to the thrust axis. We require three
charged particles in the signal hemisphere and one charged particle in the
other.

###  III.2 B. Selection of $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$
events

To select $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ events, we require
one charged pion, one electron, and one positron in the signal hemisphere. In
order to identify a pion, pion (${\cal L}_{\pi}$) and kaon (${\cal L}_{K}$)
likelihoods are constructed from the ACC response, the specific ionization
($dE/dx$) in the CDC, and the flight-time measurement in the TOF. A likelihood
ratio ${\cal P}_{K/\pi}={\cal L}_{K}/({\cal L}_{\pi}+{\cal L}_{K})$ is formed
and we require that the pion track satisfies $\mathcal{P}_{K/\pi}<0.6$. To
select electrons, a likelihood ratio is required to satisfy ${\cal
P}_{e}={\cal L}_{e}/({\cal L}_{e}+{\cal L}_{X})>0.5$, where the electron
(${\cal L}_{e}$) and non-electron (${\cal L}_{X}$) likelihood functions
include information on the $dE/dx$ in the CDC, the ratio of the energy of the
cluster in the ECL to the momentum of the track, the transverse shape of the
ECL shower, the matching of the track with the ECL cluster, and the signal
amplitude in the ACC. A pion candidate is required to have ${\cal P}_{e}<0.2$
and a momentum larger than 0.2 GeV/c in both the CM and lab frame. To recover
from bremsstrahlung, four-momenta of photon candidates with a direction within
0.05 radians of the electron track and with an energy lower than that of the
electron in the CM frame are added to the four-momentum of the electron track.
In the signal hemisphere, after recovery, at most one photon is allowed and
its energy must be below 300 MeV. To reduce $e^{+}e^{-}\to q\bar{q}$
backgrounds, we use the variable

$\displaystyle x=\frac{2E_{\rm beam}E_{\pi ee}-m_{\tau}^{2}\cdot c^{4}-M_{\pi
ee}^{2}\cdot c^{4}}{2|p_{\tau}|\cdot|p_{\pi ee}|\cdot c^{2}},$ (1)

where $E_{\rm beam}=\sqrt{s}/2$ is the beam energy, $E_{\pi ee}$, $p_{\pi
ee}$, and $M_{\pi ee}$ are the energy, momentum, and invariant mass of the
$\pi ee$ system in the CM frame. In the case of tau-lepton events, the
variable $x$ is equal to $\cos\theta_{\tau-{\pi ee}}$, the cosine of the angle
between the momentum of the $\tau$ and that of the $\pi ee$ system, assuming a
massless tau neutrino. Therefore, the absolute value of $x$ is required not to
exceed one.

The main background to $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ is
from $\tau^{-}\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$ decays, which have the
largest branching fraction among all tau-lepton decay modes. The mode
$\tau^{-}\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$ has a similar final state if one
of the photons from $\pi^{0}\to\gamma\gamma$ converts into $e^{+}e^{-}$ pair
in the detector or if the $\pi^{0}$ decays into its Dalitz mode
($\pi^{0}\to\gamma e^{+}e^{-}$). To suppress background due to external photon
conversion, the transverse position of the $e^{+}e^{-}$ vertex ($R_{xy}$) with
respect to IP must be less than 1.2 cm, and the longitudinal position of the
$e^{+}e^{-}$ vertex must be in the range [$-1$ cm, 1.5 cm]. In addition, we
veto $\pi^{0}$ Dalitz decays using the photon that survives the aforementioned
requirement and require the invariant mass of the $e^{+}e^{-}\gamma$ system,
$M_{ee\gamma}$, to fall outside of the range 110 MeV/$c^{2}$
$<M_{ee\gamma}<$165 MeV/$c^{2}$ [$\pm 3.8$ standard deviation ($\sigma$)].

Even after these requirements, the dominant background is still from
$\tau^{-}\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$, and its reliable estimation is
essential for the signal extraction. The $M_{\pi ee}$ range from 0 to 1
GeV/$c^{2}$ is chosen as a control region to validate the MC, where there is a
large background from the
$\tau^{-}\rightarrow\rho^{-}(\to\pi^{-}\pi^{0})\nu_{\tau}$ contribution, while
we use the higher mass region, 1.05 GeV/$c^{2}$ $<M_{\pi ee}<$ 1.8
GeV/$c^{2}$, as the signal region. According to MC simulation the signal
region we choose is efficient for the events with the structure-dependent
mechanism (see Fig. 1 (d),(e)), while the structure-independent signal events
(see Fig. 1 (a)-(c)) are located mostly in the region with $M_{\pi ee}<$ 1.05
GeV/$c^{2}$. As a result, this study is sensitive only to the structure-
dependent contributions to the branching fraction of the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ decay. The signal region is
blinded until we finalize the selection conditions and the corrections to the
simulated samples. Several corrections, such as the efficiencies of the
particle identification, tracking, and $\pi^{0}$ reconstruction are applied to
the MC samples.

Figure 2: Distribution of $\pi e^{+}e^{-}$ invariant mass ($M_{\pi ee}$) in
the control region for the $\tau^{-}\to\pi^{-}\nu_{\tau}e^{+}e^{-}$
candidates. Circles with statistical error bars represent the experimental
data. The histograms show the expectations from simulation for the
$\tau^{-}\to\pi^{-}\nu_{\tau}e^{+}e^{-}$ signal (white) and various background
components (other colors, as explained in the legend). The MC samples are
corrected for known efficiency differences between data and simulation, and
are normalized to the luminosity of the data sample.

Figure 2 shows the $M_{\pi ee}$ distribution in the control region, where
yields from data and the expected background estimated from MC simulation
agree; 10243 events are observed in the control region with a background
expectation of 10083$\pm$504 events, where the uncertainty includes all
systematic uncertainties discussed in section IV A. The MC result shows that
this control region is accurately described by only background events within
1$\sigma$, i.e., we don’t observe a notable contribution from the structure-
independent signal events in the control region.

###  III.3 C. Selection of
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ events

To select $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ events, we
require one charged pion and two oppositely-charged muons in the signal
hemisphere. The sum of energies of all photons in the signal hemisphere is
required to be less than 300 MeV and less than six photons should be
reconstructed in both hemispheres. This requirement is more stringent than in
section III. B, since photon emission is negligible in this mode. The pion
candidate is required to satisfy $\mathcal{P}_{K/\pi}<0.6$ (not kaon) and
${\cal P}_{e}<0.8$ (not electron). To select muon candidates, we employ a
likelihood ratio ${\cal P}_{\mu}={\cal L}_{\mu}/({\cal L}_{\mu}+{\cal
L}_{\pi}+{\cal L}_{K})$, where ${\cal L}_{\mu}$, ${\cal L}_{\pi}$, and ${\cal
L}_{K}$ are likelihood functions. The likelihood for a muon, ${\cal L}_{\mu}$
is calculated from two variables: the difference between the penetration
lengths determined from the momentum of the particle and measured by the KLM,
and the $\chi^{2}$ of the KLM hits with respect to the track extrapolated from
the CDC to the KLM. A stringent condition ${\cal P}_{\mu}>$ 0.97 is imposed
upon muon candidates to suppress background events from
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$, and the transverse
momenta of the muons are required to exceed 720 MeV/$c$ to ensure that they
reach the KLM detector. Furthermore, muon candidates are required to have
${\cal P}_{K/\pi}<0.8$ to suppress the $\tau^{-}\rightarrow
K^{-}\pi^{+}\pi^{-}\nu_{\tau}$ mode. To suppress background from hadronic
processes, the event is required to have a thrust value larger than 0.9. In
addition, the pseudomass pseudo  of the $\pi\mu\mu$ system, defined as

$\displaystyle m^{*}=\biggl{[}2\cdot E_{\pi\mu\mu}(E_{\rm
beam}-E_{\pi\mu\mu})/c^{4}+M^{2}_{\pi\mu\mu}$
$\displaystyle-2\cdot|p_{\pi\mu\mu}|\cdot(E_{\rm
beam}-E_{\pi\mu\mu})/c^{3}\biggr{]}^{\frac{1}{2}},$ (2)

is required to be less than 1.8 GeV/$c^{2}$, where $E_{\pi\mu\mu}$,
$p_{\pi\mu\mu}$, and $M_{\pi\mu\mu}$ are the energy, momentum, and invariant
mass of the $\pi\mu\mu$ system, respectively. A loose requirement on the
invariant mass of the $\mu^{+}\mu^{-}$ system, $M(\mu^{+}\mu^{-})<0.85$
GeV/$c^{2}$, is applied to further reduce this background.

The remaining background to this decay mode is dominated by the decay
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}(\pi^{0})\nu_{\tau}$, where two
charged pions are misidentified as muons. This misidentification happens
mainly due to pion decays in flight. Therefore, many of these misidentified
tracks do not originate from the IP. We define the signal region as
$R_{xy}<$0.15 cm and the control region as $R_{xy}>$ 0.20 cm, where $R_{xy}$
is the radial position of the reconstructed $\mu^{+}\mu^{-}$ vertex with
respect to the IP.

Figure 3: Distribution of transverse position of the $\mu^{+}\mu^{-}$ vertex
$(R_{xy})$ in the control region for the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ candidates. Circles with
statistical error bars represent the experimental data. The histograms show
the expectations from MC simulation for the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ signal (white) and
various background components (other colors, as explained in the legend). See
the text for more details.

Figure 3 shows the $R_{xy}$ distribution for the control region after applying
all the selection criteria, where the MC events are normalized to the
luminosity and a small correction for the muon misidentification is applied. A
total of 505 events are observed in the control region with a background
expectation of 477$\pm$23 events, where the uncertainty includes all
uncertainties discussed in section IV B. The background is dominated by
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}(\pi^{0})\nu_{\tau}$ decays. The data
are well reproduced by the expected background from MC within 1$\sigma$. The
MC shows that the signal ($\tau^{-}\to\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$)
contribution in this control region is small (about 2.3%) even assuming the
maximum value of the theoretical predictions for the branching fraction of the
signal mode Roig .

##  IV IV. Results

The branching fractions of
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-}$ decays are calculated
according to the formula

$\displaystyle\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\ell^{+}\ell^{-})=\frac{N_{\rm
obs}-N_{\rm bkg}}{2\cdot\sigma_{\tau\tau}\cdot\mathcal{L}\cdot\epsilon_{\rm
sig}},$ (3)

where $\sigma_{\tau\tau}=(0.919~{}\pm~{}0.003)$ nb tautau  is the cross
section of $\tau\tau$ production at $\sqrt{s}=10.58$ GeV, $\mathcal{L}$ the
luminosity of experimental data, $\epsilon_{\rm sig}$ the detection efficiency
of signal events, $N_{\rm obs}$ the number of observed events from
experimental data and $N_{\rm bkg}$ the number of background events.

###  IV.1 A. Measurement of the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ branching fraction

The $M_{\pi ee}$ distribution of the selected sample is shown in Fig. 4. After
applying all selection criteria, 1365 events are observed in the signal region
with a background expectation of 954 $\pm$ 45 events. The uncertainty of
background expectation takes into account all possible sources discussed
later. The main background comes from the
$\tau^{-}\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$ decay, in which the $\pi^{0}$
decays either to two photons (about 24% of the total background), or to
$e^{+}e^{-}\gamma$ (about 56% of the total background). Both cases are caused
by one unreconstructed photon, which is common in both the signal and the
control region. The MC prediction is checked by the data in the control
region.

Figure 4: Distribution of $\pi e^{+}e^{-}$ invariant mass ($M_{\pi ee}$) for
the $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ candidates in the signal
region. Circles with statistical error bars represent the experimental data.
The white histogram is the expected signal using the theoretical model in Ref.
Roig .

A clear excess of data events over the background expectation is observed. The
background-only hypothesis is rejected with a 7.0$\sigma$ significance. Once
we select signal events in the limited $M_{\pi ee}$ mass region, two types of
branching fractions are evaluated, the partial branching fraction in the
limited mass region $M_{\pi ee}>1.05$ GeV/$c^{2}$, and the branching fraction
in the full $M_{\pi ee}$ mass region. The detection efficiency in the $M_{\pi
ee}>1.05$ GeV/$c^{2}$ region is $(6.75\pm 0.13)$%, where the dominant error is
due to the model dependence of vector and axial-vector components. Taking this
number into account, the corresponding partial branching fraction for the
$M_{\pi ee}>1.05$ GeV/$c^{2}$ is obtained to be
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})[M_{\pi^{-}e^{+}e^{-}}>1.05~{}{\rm
GeV}/c^{2}]=(5.90\pm 0.53\pm 0.85\pm 0.11)\times 10^{-6}$, where the first
uncertainty is statistical, the second is systematic, and the third is due to
the model dependence. The extension of this branching fraction to the full
$M_{\pi ee}$ mass region depends on the assumed relative contribution of
vector and axial-vector current of the structure-dependent terms. The
detection efficiency of the $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$
events in this case varies between two extreme cases, $(1.32\pm 0.05)$%
assuming vector current only and $(2.73\pm 0.10)$% assuming axial-vector
current only. Taking these numbers into account, we obtain
$\mathcal{B}_{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})=(1.46\pm
0.13\pm 0.21)\times 10^{-5}$ with an assumption of pure axial-vector current
and $\mathcal{B}_{V}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})=(3.01\pm
0.27\pm 0.43)\times 10^{-5}$ with pure vector current, where the first
uncertainty is statistical and the second is systematic.

The systematic uncertainty of the measured branching fraction takes into
account all sources and is estimated to be 14.4% in total. The uncertainty due
to the track reconstruction efficiency is estimated to be 4.7% using partially
reconstructed $D^{*}\rightarrow D^{0}\pi$ with
$D^{0}\rightarrow\pi^{-}\pi^{+}K^{0}_{S}$ events. The uncertainty due to
particle identification is estimated to be 11.1%, including contributions from
the $K/\pi$ separation and lepton identification. The former is investigated
with a control sample of $D^{*+}\rightarrow D^{0}\pi^{+}_{\rm
slow},D^{0}\rightarrow K^{-}\pi^{+}$ decays; the latter is studied with the
$\gamma\gamma\rightarrow\ell^{+}\ell^{-}$ and
$J/\psi\rightarrow\ell^{+}\ell^{-}$ processes. The uncertainty on the
luminosity is obtained to be 1.4% using Bhabha events where final-state
electrons are required to be detected in the barrel region. Since the
luminosity is used for the estimation of the number of tau-lepton pairs and
the number of background events, its contribution to the uncertainty
$\bigtriangleup\mathcal{B}/\mathcal{B}$ is 4.7%. The uncertainty associated
with the trigger efficiency is investigated by a dedicated trigger-simulation
program and found to be 1.2%. The uncertainty due to the $\pi^{0}$
reconstruction is investigated with a control sample of
$\tau^{-}\rightarrow\pi^{-}\pi^{0}\nu_{\tau}$, $\pi^{0}\rightarrow
e^{+}e^{-}\gamma$ events and estimated to be 1.9%. The uncertainty arising
from the limited size of MC samples for the study of background contamination
and efficiency of signal events is estimated to be 3.7% via its binomial
variation. The uncertainties of the branching fractions of the background
modes are also taken into account and found to be 4.4%. Finally, the
uncertainty of the $\tau\tau$ production cross section at $\Upsilon(4S)$,
0.3%, is also included.

###  IV.2 B. Upper limit of the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ branching fraction

The $R_{xy}$ distribution of the selected samples is shown in Fig. 5. After
applying all selection criteria, 2578 events are observed in the signal region
with a background expectation of 2244 $\pm$ 109 events. The main backgrounds
are from the $\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ decay (81.9%
of the total background) and
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\pi^{0}\nu_{\tau}$ decay (8.3% of the
total background). The total systematic uncertainty of the background
expectation is estimated to be 4.9% including contributions from tracking
(1.4%), particle identification (3.7%), luminosity (1.4%), trigger (0.3%), MC
sizes (1.7%), accuracies of branching fractions of background modes (1.0%),
$\tau\tau$ cross section (0.3%), and $\pi\rightarrow\mu$ misidentification
calibration (1.5%). The $\pi\rightarrow\mu$ misidentification is determined by
a control sample of $\tau^{\pm}\rightarrow\pi^{\pm}\pi^{-}\pi^{+}\nu_{\tau}$
events. As a result, the excess (signal yield) is 334 $\pm$51$\pm$109 with a
significance of 2.8$\sigma$, where the first (second) uncertainty is
statistical (systematic). We determine the upper limit for the branching
fraction of this decay.

Figure 5: Distribution of transverse position of the $\mu^{+}\mu^{-}$ vertex
$({\rm R_{xy}})$ for the $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$
candidates. Circles with statistical error bars represent the experimental
data. The white histogram is the expected signal assuming
$\mathcal{B}=1.0\times 10^{-5}$.

The detection efficiency of the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$ events is ($4.14\pm
0.16$)%. Taking into account the systematic uncertainty of the background
expectation and the statistical uncertainty of the observed data events, an
upper limit at 90% confidence level (CL) is set on this decay mode to be
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-})<1.14\times
10^{-5}$.

##  V V. Conclusion

Using 562 $\rm fb^{-1}$ of data collected with the Belle detector at the KEKB
asymmetric-energy $e^{+}e^{-}$ collider, the
$\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-}$ decay is observed for the
first time with 7.0$\sigma$ significance to reject the background-only
hypothesis. The partial branching fraction for the mass region $M_{\pi
ee}>1.05$ GeV/$c^{2}$ is measured to be
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})[M_{\pi^{-}e^{+}e^{-}}>1.05~{}{\rm
GeV}/c^{2}]=(5.90\pm 0.53\pm 0.85\pm 0.11)\times 10^{-6}$, where the first
uncertainty is statistical, the second is systematic, and the third is due to
the model dependence. The extension of the branching fraction to the full
phase space depends on the assumed model and ranges from
$\mathcal{B}_{A}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})=(1.46\pm
0.13\pm 0.21)\times 10^{-5}$ to
$\mathcal{B}_{V}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}e^{+}e^{-})=(3.01\pm
0.27\pm 0.43)\times 10^{-5}$, where the former corresponds to a model with
pure axial-vector current and the latter with pure vector current. This is the
smallest decay rate of tau lepton determined to date. An upper limit is set on
the branching fraction of $\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-}$
to be
$\mathcal{B}(\tau^{-}\rightarrow\pi^{-}\nu_{\tau}\mu^{+}\mu^{-})<1.14\times
10^{-5}$, at 90% CL. The obtained results are consistent with the theoretical
predictions Roig  and can help to constrain relevant form factors.

##  VI Acknowledgements

The authors would like to thank Pablo G. Roig and Gabriel L. Castro from
CINVESTAV for the extensive help and fruitful discussions. We thank the KEKB
group for the excellent operation of the accelerator; the KEK cryogenics group
for the efficient operation of the solenoid; and the KEK computer group, and
the Pacific Northwest National Laboratory (PNNL) Environmental Molecular
Sciences Laboratory (EMSL) computing group for strong computing support; and
the National Institute of Informatics, and Science Information NETwork 5
(SINET5) for valuable network support. We acknowledge support from the
Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of
Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-
Lepton Physics Research Center of Nagoya University; the Australian Research
Council including grants DP180102629, DP170102389, DP170102204, DP150103061,
FT130100303; Austrian Science Fund (FWF); the National Natural Science
Foundation of China under Contracts No. 11435013, No. 11475187, No. 11521505,
No. 11575017, No. 11675166, No. 11705209; Key Research Program of Frontier
Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the
CAS Center for Excellence in Particle Physics (CCEPP); the Shanghai Pujiang
Program under Grant No. 18PJ1401000; the Ministry of Education, Youth and
Sports of the Czech Republic under Contract No. LTT17020; the Carl Zeiss
Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster
Universe, and the VolkswagenStiftung; the Department of Science and Technology
of India; the Istituto Nazionale di Fisica Nucleare of Italy; National
Research Foundation (NRF) of Korea Grants No. 2015H1A2A1033649, No.
2016R1D1A1B01010135, No. 2016K1A3A7A09005 603, No. 2016R1D1A1B02012900, No.
2018R1A2B3003 643, No. 2018R1A6A1A06024970, No. 2018R1D1 A1B07047294;
Radiation Science Research Institute, Foreign Large-size Research Facility
Application Supporting project, the Global Science Experimental Data Hub
Center of the Korea Institute of Science and Technology Information and
KREONET/GLORIAD; the Polish Ministry of Science and Higher Education and the
National Science Center; the Grant of the Russian Federation Government,
Agreement No. 14.W03.31.0026; the Slovenian Research Agency; Ikerbasque,
Basque Foundation for Science, Spain; the Swiss National Science Foundation;
the Ministry of Education and the Ministry of Science and Technology of
Taiwan; and the United States Department of Energy and the National Science
Foundation.

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