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

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 $M_{\pi e e}>1.05$ 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.

We present the first measurements of branching fractions of rare tau-lepton decays, τ − → π − ντ + − ( = e or µ), using a data sample corresponding to 562 fb −1 collected at a center-ofmass energy of 10.58 GeV with the Belle detector at the KEKB asymmetric-energy e + e − collider. The τ − → π − ντ e + e − decay is observed for the first time with 7.0σ 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 Mπee > 1.05 GeV/c 2 is measured to be B(τ − → π − ντ e + e − )[M π − e + e − > 1.05 GeV/c 2 ] = (5.90 ± 0.53 ± 0.85 ± 0.11) × 10 −6 , where the first uncertainty is statistical, the second is systematic, and the third is due to the model dependence.
. The dot indicates known (fully datadriven) 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 γ * W * π vertex with two gauge bosons off their mass shells. The relevant amplitudes for the decays τ − → π − ν τ + − can be written as a sum of five contributions, as shown in Fig. 1 [3]. 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 [4], the γW π vertex plays an important role in determining the radiative corrections to the τ − → π − ν τ decay [5] and in the evaluation of the contributions of the hadronic light-by-light scattering to the muon anomalous magnetic moment [6]. The measured branching fractions of the decay modes of interest can be used to validate the Resonance Chiral Theory [7], a semi-phenomenological approach to describe strong interactions at energies below the tau mass. In addition, rare decays of τ often serve as a probe of physics beyond the Standard Model [8]. Another important impact of a precise measurement of B(τ − → π − ν τ + − ) is useful input for a reliable background estimation in searches for various lepton flavor and lepton number violating decays.

II. DATA SET AND SIMULATIONS
In this paper, we report branching fraction measurements for the rare decays τ − → π − ν τ e + e − and τ − → π − ν τ µ + µ − with data recorded at a center-of-mass (CM) energy of √ s = 10.58 GeV by the Belle experiment at the KEKB asymmetric-energy e + e − collider [9]. The data used in this analysis correspond to an integrated luminosity of 562 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 barrellike 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 [10]. 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 [11] for hadronic processes, KKMC [12] for fermion pairs, and AAFH [13] for twophoton production of fermion pairs. The τ lepton decays are carried out by TAUOLA [14]. The final-state radiation of charged particles is simulated by PHOTOS [15]. For the signal modes τ − → π − ν τ + − , we employ formulas given in Ref. [2] and implement them into the TAUOLA generator. For the τ − → π − π 0 ν τ mode, the largest background for the τ − → π − ν τ e + e − signal, we use the pion form factor measured by the Belle collaboration [16]. The response of the detector is simulated by a GEANT3-based program [17]. The optimization of selection criteria using MC samples is implemented by maximizing S/ √ S + B, where S (B) is the number of signal (background) events.

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 − → τ + τ − events and substantially reject background from the other processes that occur at Υ(4S). The second stage proceeds to select one τ decaying into the π − ν τ + − final state, where a vertex fit is performed to the + − pair, and the other τ decaying into one-prong modes.
A. Selection of τ + τ − 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 µm. 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 • < θ γ < 130 • ) 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 − → qq (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 • . A tau-pair event is accompanied by missing four-momentum due to neutrinos, defined as where p init is the four-momentum of the colliding e + e − beams, and i p trk,i and i p γ,i are the sums of the four-momenta of charged particles (assumed to be pions) and photons, respectively. Only events satisfying 1 GeV/c 2 < M miss < 7 GeV/c 2 and 30 • < θ miss < 150 • criteria are selected, where M miss and θ 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 − → qq backgrounds and to provide two disjoint hemispheres for each event. The thrust T is defined as the maximum value of where p i is the CM momentum of the ith 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.
To select τ − → π − ν τ e + e − events, we require one charged pion, one electron, and one positron in the signal hemisphere. In order to identify a pion, pion (L π ) and kaon (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 P K/π = L K /(L π + L K ) is formed and we require that the pion track satisfies P K/π < 0.6. To select electrons, a likelihood ratio is required to satisfy P e = L e /(L e + L X ) > 0.5, where the electron (L e ) and non-electron (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 P e < 0.2 and a momentum larger than 0.2 GeV/c in both the CM and lab frame. To recover from bremsstrahlung, fourmomenta 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 − → qq backgrounds, we use the variable where E beam = √ s/2 is the beam energy, E πee , p πee , and M πee are the energy, momentum, and invariant mass of the πee system in the CM frame. In the case of tau-lepton events, the variable x is equal to cos θ τ −πee , the cosine of the angle between the momentum of the τ and that of the πee system, assuming a massless tau neutrino. Therefore, the absolute value of x is required not to exceed one.
The main background to τ − → π − ν τ e + e − is from τ − → π − π 0 ν τ decays, which have the largest branching fraction among all tau-lepton decay modes. The mode τ − → π − π 0 ν τ has a similar final state if one of the photons from π 0 → γγ converts into e + e − pair in the detector or if the π 0 decays into its Dalitz mode (π 0 → γe + 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 π 0 Dalitz decays using the photon that survives the aforementioned requirement and require the invariant mass of the e + e − γ system, M eeγ , to fall outside of the range 110 MeV/c 2 < M eeγ <165 MeV/c 2 [±3.8 standard deviation (σ)].
Even after these requirements, the dominant background is still from τ − → π − π 0 ν τ , and its reliable estimation is essential for the signal extraction. The M πee 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 τ − → ρ − (→ π − π 0 )ν τ contribution, while we use the higher mass region, 1.05 GeV/c 2 < M πee < 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 πee < 1.05 GeV/c 2 . As a result, this study is sensitive only to the structure-dependent contributions to the branching fraction of the τ − → π − ν τ 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 π 0 reconstruction are applied to the MC samples. Figure 2 shows the M πee 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±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 1σ, i.e., we don't observe a notable contribution from the structure-independent signal events in the control region.
C. Selection of τ − → π − ντ µ + µ − events To select τ − → π − ν τ µ + µ − 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 P K/π < 0.6 (not kaon) and P e < 0.8 (not electron). To select muon candidates, we employ a likelihood ratio P µ = L µ /(L µ + L π + L K ), where L µ , L π , and L K are likelihood functions. The likelihood for a muon, L µ 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 χ 2 of the KLM hits with respect to the track extrapolated from the CDC to the KLM. A stringent condition P µ > 0.97 is imposed upon muon candidates to suppress background events from τ − → π − π + π − ν τ , 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 P K/π < 0.8 to suppress the τ − → K − π + π − ν τ mode. To suppress background from hadronic processes, the event is required to have a thrust value larger than 0.9. In addition, the pseu-domass [18] of the πµµ system, defined as is required to be less than 1.8 GeV/c 2 , where E πµµ , p πµµ , and M πµµ are the energy, momentum, and invariant mass of the πµµ system, respectively. A loose requirement on the invariant mass of the µ + µ − system, M (µ + µ − ) < 0.85 GeV/c 2 , is applied to further reduce this background.
The remaining background to this decay mode is dominated by the decay τ − → π − π + π − (π 0 )ν τ , 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 µ + µ − vertex with respect to the IP. µ + µ − vertex (Rxy) in the control region for the τ − → π − ντ µ + µ − candidates. Circles with statistical error bars represent the experimental data. The histograms show the expectations from MC simulation for the τ − → π − ντ µ + µ − 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±23 events, where the uncertainty includes all uncertainties discussed in section IV B. The background is dominated by τ − → π − π + π − (π 0 )ν τ decays. The data are well reproduced by the expected background from MC within 1σ. The MC shows that the signal (τ − → π − ν τ µ + µ − ) 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 [2].

IV. RESULTS
The branching fractions of τ − → π − ν τ + − decays are calculated according to the formula where σ τ τ = (0.919 ± 0.003) nb [19] is the cross section of τ τ production at √ s = 10.58 GeV, L the luminosity of experimental data, sig the detection efficiency of signal events, N obs the number of observed events from experimental data and N bkg the number of background events.
A. Measurement of the τ − → π − ντ e + e − branching fraction The M πee 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 ± 45 events. The uncertainty of background expectation takes into account all possible sources discussed later. The main background comes from the τ − → π − π 0 ν τ decay, in which the π 0 decays either to two photons (about 24% of the total background), or to e + e − γ (about 56% of the total background).
A clear excess of data events over the background expectation is observed. The background-only hypothesis is rejected with a 7.0σ significance. Once we select signal events in the limited M πee mass region, two types of branching fractions are evaluated, the partial branching fraction in the limited mass region M πee > 1.05 GeV/c 2 , and the branching fraction in the full M πee mass region. The detection efficiency in the M πee > 1.05 GeV/c 2 region is (6.75 ± 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 πee > 1.05 GeV/c 2 is obtained to be B(τ − → π − ν τ e + e − )[M π − e + e − > 1.05 GeV/c 2 ] = (5.90 ± 0.53 ± 0.85 ± 0.11) × 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 πee mass region depends on the assumed relative contribution of vector and axial-vector current of the structure-dependent terms. The detection efficiency of the τ − → π − ν τ e + e − events in this case varies between two extreme cases, (1.32 ± 0.05)% assuming vector current only and (2.73 ± 0.10)% assuming axial-vector current only. Taking these numbers into account, we obtain B A (τ − → π − ν τ e + e − ) = (1.46 ± 0.13 ± 0.21) × 10 −5 with an assumption of pure axial-vector current and B V (τ − → π − ν τ e + e − ) = (3.01 ± 0.27 ± 0.43) × 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 * → D 0 π with D 0 → π − π + K 0 S events. The uncertainty due to particle identification is estimated to be 11.1%, including contributions from the K/π separation and lepton identification. The former is investigated with a control sample of D * + → D 0 π + slow , D 0 → K − π + decays; the latter is studied with the γγ → + − and J/ψ → + − 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 B/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 π 0 reconstruction is investigated with a control sample of τ − → π − π 0 ν τ , π 0 → e + e − γ 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 τ τ production cross section at Υ(4S), 0.3%, is also included.