Search for Lepton Flavor Violating tau- Decays into \ell-K0s and \ell-K0sK0s

We have searched for the lepton-flavor-violating decays tau- ->ell-K0s and ell-K0sK0s (ell = e or mu), using a data sample of 671 fb^-1 collected with the Belle detector at the KEKB asymmetric-energy e^+e^- collider. No evidence for a signal was found in any of the decay modes, and we set the following upper limits for the branching fractions: B(tau^- ->e^-K0s)<2.6 x 10^-8, B(tau^- ->\mu^-K0s)<2.3 x 10^-8, B(tau^- ->e^-K0sK0s)<7.1 x 10^-8 and B(tau^- ->mu^-K0sK0s)<8.0 x 10^-8 at the 90% confidence level.


Abstract
We have searched for the lepton-flavor-violating decays τ − → ℓ − K 0 S and ℓ − K 0 S K 0 S (ℓ = e or µ), using a data sample of 671 fb −1 collected with the Belle detector at the KEKB asymmetric-energy e + e − collider. No evidence for a signal was found in any of the decay modes, and we set the following upper limits for the branching fractions:

INTRODUCTION
Lepton flavor violation (LFV) in charged lepton decays is forbidden in the Standard Model (SM) or highly suppressed if neutrino mixing is included. However, LFV appears in various extensions of the SM. In particular, the lepton-flavor-violating decays τ − → ℓ − K 0 S and τ − → ℓ − K 0 S K 0 S (where ℓ = e or µ ) are enhanced in supersymmetric and many other models [1][2][3][4][5][6]. Some of these models predict branching fractions which, for certain combinations of model parameters, can be as high as 10 −7 ; this level is already accessible in high-statistics B-factory experiments. Previously, we obtained 90% confidence level (C.L.) upper limits for the τ − → ℓ − K 0 S branching fractions (B) using 281 fb −1 of data; the results were B(τ − → e − K 0 S ) < 5.6 × 10 −8 and B(τ − → µ − K 0 S ) < 4.9 × 10 −8 [7]. The BaBar collaboration has recently obtained 90% C.L. upper limits of B(τ − → e − K 0 S ) < 3.3 × 10 −8 and B(τ − → µ − K 0 S ) < 4.0 × 10 −8 using a data sample of 469 fb −1 [8]. The most restrictive existing upper limits at the 90% C.L. were set by the CLEO experiment using 13.9 fb −1 of data [9]. In this paper, we present a search for the lepton-flavor-violating decays τ − → ℓ − K 0 S and ℓ − K 0 S K 0 S (ℓ = e or µ) [ †] using 671 fb −1 of data collected at the Υ(4S) resonance and 60 MeV below with the Belle detector at the KEKB asymmetric-energy e + e − collider [10]. The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer 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 comprised of CsI(Tl) crystals (ECL), all located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron fluxreturn located outside of the coil is instrumented to detect K 0 L mesons and to identify muons (KLM). The detector is described in detail elsewhere [11].
Leptons are identified using likelihood ratios calculated from the responses of various detector subsystems. For electron identification, the likelihood ratio is defined as P(e) = L e /(L e + L x ), where L e and L x are the likelihoods for electron and non-electron hypotheses, respectively, determined using the ratio of the energy deposit in the ECL to the momentum measured in the SVD and CDC, the shower shape in the ECL, the matching between the position of the charged track trajectory and the cluster position in the ECL, the hit information from the ACC, and the dE/dx information in the CDC [12]. For muon identification, the likelihood ratio is defined as P(µ) = L µ /(L µ + L π + L K ), where L µ , L π and L K are the likelihoods for the muon, pion and kaon hypotheses, respectively, based on the matching quality and penetration depth of associated hits in the KLM [13]. For this measurement, we use hadron identification likelihood variables based on the hit information from the ACC, the dE/dx information in the CDC, and the particle time-of-flight from the TOF. To distinguish hadron species, we use likelihood ratios, P(i/j) = L i /(L i + L j ), where L i (L j ) is the likelihood for the detector response to a track with flavor hypothesis i (j).
In order to optimize the event selection and estimate the signal efficiency, we use Monte Carlo (MC) samples. The signal and background events from generic τ + τ − decays are generated by KKMC/TAUOLA [14]. The signal MC samples are generated by KKMC assuming a phase space model for the τ decay. Other backgrounds, including BB and continuum e + e − → qq (q = u, d, s, c) events, Bhabha events, and two-photon processes, are generated by EvtGen [15], BHLUMI [16], and AAFH [17], respectively. The Belle detector response [ †] Unless otherwise stated, charge-conjugate decays are included throughout this paper.
is simulated by a GEANT 3 [18] based program. The event selection is optimized modeby-mode since the backgrounds are mode dependent. All kinematic variables are calculated in the laboratory frame unless otherwise specified. In particular, variables calculated in the e + e − center-of-mass (CM) system are indicated by the superscript "CM".

DATA ANALYSIS
We search for τ + τ − events, in which one τ (signal side) decays into ℓK 0 S or ℓK 0 S K 0 S , while the other τ (tag side) decays into a final state with one charged track, any number of additional photons and neutrinos. We reconstruct each K 0 S meson candidate from a π + π − pair. By selecting decays into one charged track on the tag side, we reduce background from BB and qq events. All charged tracks and photons are required to be reconstructed within a fiducial volume, defined by −0.866 < cos θ < 0.956, where θ is the polar angle with respect to the direction opposite to the e + beam. We select charged tracks with momenta transverse to the e + beam p t > 0.1 GeV/c and photons with energies E γ > 0.1 GeV.
Candidate τ -pair events are required to have four or six charged tracks with zero net charge for the ℓK 0 S and ℓK 0 S K 0 S modes, respectively. Events are separated into two hemispheres corresponding to the signal (three-prong and five-prong for the ℓK 0 S and ℓK 0 S K 0 S modes, respectively) and tag (one-prong) sides by the plane perpendicular to the thrust axis [19]. We require one or two K 0 S candidates for the ℓK 0 S and ℓK 0 S K 0 S modes, respectively. The K 0 S is reconstructed from two oppositely charged tracks on the signal side that have an invariant mass 0.482 GeV/c 2 < M π + π − < 0.514 GeV/c 2 , assuming the pion mass for both tracks. The π + π − vertex is required to be displaced from the interaction point (IP) in the direction of the pion pair momentum [20]. In order to avoid fake K 0 S candidates from photon conversions (i.e., γ → e + e − ), the invariant mass reconstructed by assigning the electron mass to the tracks, is required to be greater than 0.2 GeV/c 2 . The electron and muon identification criteria are P(e) > 0.9 with momentum p > 0.3 GeV/c and P(µ) > 0.9 with p > 0.6 GeV/c, respectively. In order to take into account the emission of bremsstrahlung photons from the electron, the momentum of each electron track is reconstructed by adding the momentum of every photon within 0.05 rad of the track direction. The electron (muon) identification efficiency for the ℓK 0 S modes is 92% (87%) and that for the ℓK 0 S K 0 S modes is 79% (81%). The difference of efficiencies between ℓK 0 S and ℓK 0 S K 0 S is due to the different signal momentum distributions. The probability to misidentify a pion as an electron and a muon is below 0.5% and 3%, respectively.
In order to suppress background from qq events, the following requirements on the number of the photon candidates on the signal and tag side (n SIG γ and n TAG γ ) are imposed: n SIG γ ≤ 1 and n TAG γ ≤ 3. For the ℓK 0 S modes only, we also require n TAG γ ≤ 1 if the track of the tag side is a lepton to reduce the background, in particular from D + → ℓ + νK 0 S (→ π 0 π 0 ). To ensure that the missing particles are neutrinos rather than photons or charged particles that fall outside the detector acceptance, we impose additional requirements on the missing momentum vector, p miss , calculated by subtracting the vector sum of the momenta of all tracks and photons from the sum of the e + and e − beam momenta. We require that the magnitude of p miss be greater than 0.4 GeV/c and that its direction point into the fiducial volume of the detector. Since neutrinos are emitted only on the tag side, the direction of p miss should lie within the tag side of the event. The cosine of the opening angle between p miss and the tag-side track in the CM system, cos θ CM tag−miss , should be 0.0 < cos θ CM tag−miss for both modes (see Figs. 1 (a) and 2 (a)). For the ℓK 0 S modes, we also require that cos θ CM tag−miss < 0.99 to reduce background from Bhabha, µ + µ − and two-photon events, as radiated photons from the tag-side track result in missing momentum if they overlap with the ECL clusters associated with the tag-side track.
To reject the qq background, the magnitude of the thrust is required to be larger than  Fig. 1 (b) and 2 (b)). The invariant mass reconstructed from the charged track and any photon on the tag side is required to be less than 1.0 and 1.777 GeV/c 2 for the ℓK 0 S and ℓK 0 S K 0 S modes, respectively. For the ℓK 0 S modes, we impose a kaon veto L(K/π) < 0.6 if the track on the tag side is a hadron, to suppress e + e − → qq background; due to the conservation of strangeness by the strong interaction, the K 0 S in such events is often accompanied by another kaon.
All kinematic distributions for the ℓK 0 S modes shown in Fig. 1 are in reasonable agreement between data and background MC while those for the ℓK 0 S K 0 S modes shown in Fig. 2 clearly differ. This difference between the data and background MC in Fig. 2

originates from our poor knowledge of the branching fractions
3) × 10 −4 × 1/4 and the dynamics of these decays [21]. Since the final estimate of the background uses information from the data, this discrepancy does not directly affect our results.
Finally, to suppress backgrounds from generic τ + τ − and qq events, we apply a selection based on the magnitude of the missing momentum p miss and the missing mass squared m 2 miss . The latter is defined as E 2 miss −p 2 miss , where E miss = E total −E vis , E total is the sum of the beam energies and E vis is the total visible energy. We apply different selection criteria depending on the type of one-prong tag: the number of emitted neutrinos is two if the tagging track is an electron or muon (leptonic tag) while it is one if the tagging track is a hadron (hadronic tag). The requirements are listed in Table I (see also Fig. 3).

RESULTS
Signal candidates are examined in two-dimensional plots of the ℓK 0 S and ℓK 0 S ), and the difference of their energy from the beam energy in the CM system, ∆E. A signal event should have M sig close to the τ -lepton mass and ∆E close to zero. For both modes, the M sig and ∆E resolutions are parameterized from the MC distributions around the peak region using asymmetric Gaussian shapes to take into account initial state radiation. The widths of these Gaussians are shown in Table II.
To evaluate the branching fractions, we use an elliptical signal region that contains 90% of the signal MC events satisfying all selection criteria. The shape of the signal region is chosen to minimize its area and therefore obtain the highest sensitivity. We blind the data in the signal region until all selection criteria are finalized so as not to bias our choice of selection criteria. Figure 4 shows scatter-plots for data events and signal MC samples distributed over ±20σ in the M sig − ∆E plane. As MC simulation shows, the dominant background in the signal region comes from events with a fake lepton from a pion. Therefore, we estimate the number of expected background by multiplying the number of data events in the signal region with selected hadrons (P(ℓ) ≤ 0.9) by the fake lepton ratio. The latter is calculated as the number of events in the data with P (ℓ) > 0.9 divided by the number of events in the data with P (ℓ) ≤ 0.9 in the sideband region. For the ℓK 0 S modes we define the sideband region as the box inside the two horizontal lines (see Fig. 4 (a) and (b)) with the signal region excluded since real leptons from D + → ℓ + νK 0 S populate the region below the ∆E signal one. For the ℓK 0 S K 0 S modes, events that lie within a ±20σ region but outside the signal region are treated as sideband events (see Fig. 4 (c) and (d)). The final signal efficiency and the number of expected background events in the signal region for each mode are summarized in Table III.
The dominant systematic uncertainties on the detection sensitivity come from K 0 S reconstruction and tracking efficiencies. These are 4.5% per K 0 S candidate and 1.0% per track. Other sources of systematic uncertainties are: lepton identification (2.2−2.7)%, MC statistics (0.8−1.0)%, trigger efficiency (0.01−0.4)%, and integrated luminosity (1.4%). The uncertainty from B(K 0 S → π + π − ) is negligible. All these uncertainties are added in quadrature to provide total systematic uncertainties that range from 6.6% to 11.3%.
Finally, we examine the blinded region and find no data events in the signal region for any of the decay modes (see Fig. 4). Therefore, we set the following upper limits on the (MeV/c 2 ) and σ high/low ∆E (MeV)). Here σ high (σ low ) means the standard deviation on the higher (lower) side of the peak.
respectively. Data is indicated by the solid circles. The filled boxes show the MC signal distribution with arbitrary normalization. The elliptical signal regions shown by the solid curves are used for evaluating the signal yield. In (a) and (b), the region between the horizontal solid lines excluding the signal region is used as a sideband. The signal efficiency (ε), the number of the expected background events (N BG ) estimated from the sideband data, the total systematic uncertainty (σ syst ), the number of observed events in the signal region (N obs ), 90% C.L. upper limit on the number of signal events including systematic uncertainties (s 90 ) and 90% C.L. upper limit on the branching fraction for each individual mode. branching fractions based on the Feldman-Cousins method [22]. The 90% C.L. upper limit on the number of signal events (s 90 ) is obtained using the POLE program [23], based on the number of expected background events, observed data and the systematic uncertainty. The upper limit on the branching fraction is then given by where ε is the signal efficiency, B(K 0 S → π + π − ) = (69.20 ± 0.5)% [21], and n is 1 and 2 for the ℓK 0 S and ℓK 0 S K 0 S modes, respectively. The value N τ τ = 6.17 × 10 8 is obtained from the product of the integrated luminosity and the cross section of τ -pair production 0.919 ± 0.003 nb [24]. The resulting upper limits on the branching fractions at the 90% C.L. are For the ℓK 0 S modes, these results improve the existing upper limits by about a factor of 2, compared to our previously published limits [7]. For the ℓK 0 S K 0 S modes, these results improve the upper limits by factors of 31 and 43 for the eK 0 S K 0 S and µK 0 S K 0 S , respectively, compared to the previously published limits obtained by the CLEO experiment [9]. SUMMARY We have searched for the lepton-flavor-violating decays τ − → ℓ − K 0 S and ℓ − K 0 S K 0 S (ℓ = e or µ) using data collected with the Belle detector at the KEKB e + e − asymmetric-energy collider. We find no signal for any decay modes. The following upper limits on branching fractions at the 90% confidence level are obtained: B(τ − → e − K 0 S ) < 2.6 × 10 −8 , B(τ − → µ − K 0 S ) < 2.3 × 10 −8 , B(τ − → e − K 0 S K 0 S ) < 7.1 × 10 −8 and B(τ − → µ − K 0 S K 0 S ) < 8.0 × 10 −8 . These results are currently the most stringent upper limits for the ℓK 0 S and the ℓK 0 S K 0 S modes. These limits can be used to constrain new physics scenarios beyond the Standard Model.