Search for lepton-flavor-violating tau-lepton decays to ℓγ at Belle

Charged lepton flavor violation is forbidden in the Standard Model but possible in several new physics scenarios. In many of these models, the radiative decays τ± → ℓ±γ (ℓ = e, μ) are predicted to have a sizeable probability, making them particularly interesting channels to search at various experiments. An updated search via τ± → ℓ±γ using full data of the Belle experiment, corresponding to an integrated luminosity of 988 fb−1, is reported for charged lepton flavor violation. No significant excess over background predictions from the Standard Model is observed, and the upper limits on the branching fractions, B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B} $$\end{document}(τ± → μ±γ) ≤ 4.2 × 10−8 and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{B} $$\end{document}(τ± → e±γ) ≤ 5.6 × 10−8, are set at 90% confidence level.

Charged lepton flavor violation (CLFV) is forbidden in the Standard Model but occurs with a yet unobservably small probability, O(10 −40 ), via neutrino oscillations [1].However, it is enhanced in theories beyond the Standard Model (BSM) such as Minimal Supersymmetric Standard Model, grand unified theories and seesaw mechanisms [2][3][4].Several BSM models predict CLFV processes occurring at an observable level in experiments.An observation of CLFV would be a clear signature of BSM, making the search for this phenomenon one of the high-priority physics tasks.
The Belle detector was a large-solid-angle magnetic spectrometer consisting 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 (ECL) comprising CsI(Tl) crystals.All these components are located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.An iron flux-return located outside of the coil is instrumented with resistive plate chambers to detect K 0 L mesons and muons (KLM).The detector is described in detail elsewhere [10].

Event selection
Photon candidates are selected from ECL clusters that are consistent with an electromagnetic shower but not associated with any charged tracks.This analysis uses a photon with energy from 100 MeV to 6 GeV, and is thus sensitive to the photon energy resolution over a broad energy range.We have revised the photon-energy calibration method using the e + e − → µ + µ − γ events for the first time at Belle.The photon energy resolution is evaluated by subtracting the recoil energy of the µ + µ − system from the photon energy measured in the ECL for data and MC simulation.Figure 1 shows the energy resolution obtained as a function of the reconstructed photon energy in the e + e − → µ + µ − γ events.The calibrated resolution in simulation agrees with that in data as well as is compatible with the test-beam result [16].This is a major improvement with respect to the previous analysis [5].Muon candidates are identified using a likelihood ratio, L µ , which is based on the difference between the range of the track calculated from the particle momentum and that measured in the KLM.This ratio includes the value of χ 2 formed from the KLM hit locations with respect to the extrapolated track.The muon identification efficiency for the selection applied L µ > 0.95 is 90%, with a pion misidentification probability of 0.8% [17].Identification of electrons uses an analogous likelihood ratio, L e , based on specific ionization from the CDC, the ratio of the energy deposited in the ECL to the momentum measured by the CDC and SVD combined, the shower shape in the ECL, hit information from the ACC, and matching between the position of the charged track and the ECL cluster.The electron identification efficiency for the selection applied L e > 0.9 is 95%, with a pion misidentification probability of 0.07% [18].
We follow a blind analysis approach in this search, where the data in the interesting kinematic region remain hidden until the selection criteria and background estimation strategy are finalized.All selection criteria are optimized in order to maximize the search sensitivity, / N bkg , where is the overall signal efficiency and N bkg is the number of background events.Since we use all Υ(nS) resonance data with different center-of-mass energy √ s, some of the selection variables are scaled by √ s.The following preselection criteria are applied in this search.Exactly two oppositely charged track are required to make the event's net charge zero to suppress q q events.Candidate events are retained if both tracks have p CM ≤ 0.43 √ s GeV/c, p T ≥ 0.1 GeV/c and −0.866 < cos θ track < 0.956 in order to reduce e + e − γ, µ + µ − γ, and two-photon events Here, θ track is the polar angle of the track in the laboratory frame.For the search of τ ± → e ± γ decays, the tracks that go through gaps between ECL crystals must be rejected to avoid misidentification of electrons.Thus, the tracks are required to lie within the ECL acceptance, cos θ track ∈ [−0.907, −0.652] ∪ [−0.602, 0.829] ∪ [0.854, 0.956].Photons are required to have an energy E γ > 0.1 GeV within the region, −0.625 < cos θ γ < 0.846, where θ γ is the polar angle of the photon in the laboratory frame.
A τ + τ − pair event is divided into two hemispheres in the CM frame using a thrust vector [19]: signal-and tag-side tau.The signal-side tau decays to a muon (electron) and a photon for the τ ± → µ ± γ (τ ± → e ± γ) search.The number of photons in the signal side should be exactly one, which must have E γ > 0.5 GeV and −0.602 < cos θ γ < 0.829 to suppress misreconstructed photons.
The tag-side tau is assumed to undergo one-prong decays such as τ → eν ν, µν ν, πν, and ρν.If the track in the tag side is identified as an electron or a muon, the event is classified as a leptonic channel.Otherwise, the event is classified as a π or ρ channel.If there are no photons in the tag side, the event is classified as a π channel.Otherwise, it is a ρ channel.In order to reduce the µ + µ − γ (e + e − γ) contamination, an extra muon (electron) is vetoed using the criterion, L µ < 0.1 (L e < 0.1) for τ ± → µ ± γ (τ ± → e ± γ) search.
After preselecting events, the following selection criteria are applied to further suppress background events.The total visible energy in the CM frame, E CM total / √ s, is required to be smaller than 0.93 for the leptonic channel, 0.86 for the π channel, and 0.94 for the ρ channel.Since the energy of neutrinos is different for these channels, the quantitative criteria are accordingly changed for them.For the ρ channel, an energy sum of the two charged tracks and the photon in the signal side, E CM sum / √ s, is also required to be smaller than 0.86 due to extra π 0 in the tag side, while no such requirement is applied for other channels.These requirements further suppress the e + e − γ and µ + µ − γ events.The cosine of the angle between the two tracks, cos θ track(sig,tag) , and that between the track and the photon in the signal side, cos θ γ , are required to be cos θ track(sig,tag) < 0.0, and 0.4 < cos θ γ < 0.8, respectively, to reject τ + τ − background events that contain π 0 's from tau decays.
The missing momentum is calculated by subtracting the sum of the three-momenta of all charged tracks and photons from the sum of the beam momenta in laboratory frame.Its magnitude | p miss | is required to be greater than 0.4 GeV/c.The cosine of the polar angle of p miss is required to be −0.866< cos θ miss < 0.956.A criterion on the cosine of the angle between p miss and the tag-side track, 0.4 < cos θ miss,track(tag) < 0.98 (0.4 < cos θ miss,track(tag) < 0.99) for τ ± → µ ± γ (τ ± → e ± γ) search is also required.These requirements can suppress e + e − γ and µ + µ − γ events.We define the missing-mass-squared on the tag side as , where E CM γ (E CM tag ) is the sum of the energy of the signal (tag) side in the CM frame, to reduce background events.Here, the natural unit c = 1 is used in the formula throughout the paper.Since τ + τ − events are produced back-to-back in the CM frame and there are no neutrinos in the signal side for τ ± → ± γ events, the energy of tag-side tau in the CM frame is taken as that of the signal-side tau, E CM γ and the missing momentum of tag-side tau is taken as that of the whole event.Figure 2 shows the distribution of m 2 ν .The signal distribution is distinct from background due to the kinematic difference.Since the distribution depends on the number of neutrinos, a quantitative criterion is accordingly adjusted for each channel; the specific requirements are 0.0 GeV 2 /c 4 < m 2 ν < 2.8 GeV 2 /c 4 for the leptonic channel, −0.1 GeV 2 /c 4 < m 2 ν < 1.2 GeV 2 /c 4 for the π channel, and −0.3 GeV 2 /c 4 < m 2 ν < 1.5 GeV 2 /c 4 for the ρ channel in order to reduce τ + τ − background events.
In order to improve search sensitivity, two more variables are introduced.The first one is an energy asymmetry between the lepton and the photon in the signal side, The signal events are two-body decays, while the main background arises from three-body decays, τ ± → ± ν ν τ .Thus, the energy asymmetry should be larger in background events.We apply a requirement of 65.The second variable, ξ CM τ (tag),track(tag) is defined as follows.The missing mass squared against a charged track in the tag-side tau is written as where  where E CM beam = √ s/2 and p CM γ is the sum of the lepton and photon momenta in the CM frame.Figure 4 shows the two-dimensional distribution of ∆E/ √ s vs. M bc .The signal events have M bc ∼ m τ and ∆E/ √ s ∼ 0 and in order to select them, an elliptical region around their expected values is adopted as follows:

Belle
) Here, σ high/low M bc and σ high/low ∆E/ √ s are the widths on the higher/lower side of the peak obtained by fitting the signal distribution to an asymmetric Gaussian function [5].The estimated resolutions are σ for τ ± → e ± γ events.The overall signal efficiency estimated using the above signal region is 3.7% for τ ± → µ ± γ and 2.9% for τ ± → e ± γ.

Signal and background estimation
To estimate the number of events in the signal region, we perform an unbinned maximumlikelihood fit with probability density functions (PDFs) depending on M bc and ∆E/ √ s.The likelihood function is defined in terms of the signal PDF (S), background PDF (B), and the number of signal events (s) and background events (b) as where N is the total number of observed events, i denotes the event index, and s and b are the free parameters.The fit is performed to candidate events in the signal region defined by Eq. (2.6).The signal PDF is obtained by smoothening the corresponding MC distribution and the background PDF uses the function described below.Since the distributions of M bc and ∆E/ √ s are well modeled for the τ + τ − and µ + µ − background events, the corresponding PDFs are determined using MC simulation.The PDFs of e + e − γ events are extracted from the data by applying an electron identification requirement, L e > 0.1, to the track in the tag side.This is the same approach as in the previous publication [5].Since M bc and ∆E/ √ s are almost independent of each other, the background PDF is written as As the background events do not exhibit any peak and are rather flat in the M bc distribution, a constant function is applicable to f (M bc ).In order to determine the g(∆E/ √ s) distribution, the requirement on M bc is relaxed until enough statistics have been obtained.The background MC events with M bc ∈ [1.74, 1.83] GeV/c 2 for τ + τ − events and M bc ∈ [1.60, 1.90] GeV/c 2 for µ + µ − γ events are used in the case of τ ± → µ ± γ search.For the τ ± → e ± γ search, the background MC events with M bc ∈ [1.70, 1.88] GeV/c 2 for τ + τ − events and M bc ∈ [1.73, 1.85] GeV/c 2 for e + e − γ events are used.The ∆E/ √ s distribution for τ + τ − background is described by a sum of Landau and exponential functions for both τ ± → µ ± γ and τ ± → e ± γ searches.The distribution for µ + µ − γ and e + e − γ is described by a sum of Landau and Gaussian functions [5].
The total background PDFs (B tot 0 , B tot 1 ) are obtained by combining each background function: where B τ τ , B µµγ , and B eeγ are the PDFs for τ + τ − , µ + µ − , and e + e − γ background events, and C 0 to C 3 are the free parameters determined by a fit.The fit is performed to the sideband data defined as .00] GeV/c 2 for the τ ± → e ± γ search.Figure 5 shows ∆E/ √ s distributions in the sideband.After performing the fit, we obtain C 0 = 19.3 ± 1.8, C 1 = 1.0 ± 0.7 for the τ ± → µ ± γ search, and C 2 = 19.7 ± 1.9, C 3 = 0.2 ± 0.7 for the τ ± → e ± γ search.The τ + τ − background events are dominant for both search channels and consistent with the MC expectation.The expected number of background events is 5.8 ± 0.4 for the τ ± → µ ± γ search and 5.1 ± 0.4 for the τ ± → e ± γ search.
The total number of observed events is 5 in both the τ ± → µ ± γ and τ ± → e ± γ searches, as shown in Figure 4.By using the aforementioned signal and background PDFs, we perform the likelihood fit defined in Eq. (3.1).The results of the likelihood fit are s = −0.3+1.8 −1.3 , b = 5.3 +3.2 −2.3 for τ ± → µ ± γ, and s = −0.5 +4.4 −3.6 , b = 5.5 +5.2 −4.1 for τ ± → e ± γ.We estimate the systematic uncertainties associated with track and photon reconstruction efficiencies, photon energy calibration, luminosity, trigger efficiencies, and background PDF modeling.A summary of these systematic uncertainties is given in Table 1.
The uncertainty in track reconstruction efficiencies is estimated with partially reconstructed D * + → D 0 π + , D 0 → K 0 S π + π − events.The systematic uncertainty of 0.35% is assigned per track, and thus a total uncertainty of 0.7% is estimated for our analysis.The efficiencies of photon reconstruction are estimated with radiative Bhabha events.The efficiencies in MC simulation agree with that in data, and the associated uncertainty is 2.0%.As discussed earlier, the uncertainty due to photon energy calibration is estimated with e + e − → µ + µ − γ events, and amounts to 3.2%.The uncertainty in the integrated luminosity is 1.4%.The trigger efficiencies are evaluated by comparing the data sideband and MC simulation, and estimated to be 2.1% for τ ± → µ ± γ and 3.4% for τ ± → e ± γ analysis.These are the uncertainties related to overall signal efficiency.The uncertainty due to background PDF modeling is evaluated by varying the fixed PDF parameters.By changing each of the fixed parameters by ±1σ, the number of signal events obtained from the fit is checked, and the relative difference from the nominal value is assigned as the systematic uncertainty.The estimated uncertainty is 3.3% for τ ± → µ ± γ and 3.7% for τ ± → e ± γ.The uncertainties due to limited MC statistics and particle identification are negligible compared to the other uncertainties described above.

Result
Since no significant excess of the signal events is observed in data, the upper limits at the 90% confidence level (CL) are evaluated using toy MC simulations.We generate toy signal and background events based on their PDFs while fixing the number of background events ( b) and varying the number of signal events (s).For every assumed s, 10,000 pseudoexperiments are generated following Poisson statistics with the means s and b for signal and background, respectively.In order to obtain the expected (observed) upper limits on the branching fraction at 90% CL, the s value that gives a 90% probability for s larger than zero (fitted signal yield) is taken: s90 .The method to incorporate the systematic uncertainties into a branching fraction discussed in Ref. [20] is adopted in this analysis: the uncertainties related to overall signal efficiency and background PDF modeling are treated separately.The likelihood defined in Eq. (3.1) is convolved with a Gaussian function of width equal to the systematic uncertainty, so the s and b values are smeared accordingly.The uncertainties inflate the upper limits on the branching fraction by ∼2-3%; this effect is not large and consistent with the past results [5].The expected upper limits on the branching fraction B(τ ± → ± γ) at 90% CL is calculated as B(τ ± → µ ± γ) < 4.9 × 10 −8 and B(τ ± → e ± γ) < 6.4 × 10 −8 .Our expected limits are 1.6-1.8times more stringent compared to the previous Belle results [5].

Summary
In this paper, a search conducted for the charged-lepton-flavor-violating decays, τ ± → µ ± γ and τ ± → e ± γ, at the Belle experiment is reported.It uses 988 fb −1 of data, about twice the size used in the previous Belle analysis [5].In addition, requirements with new observables of energy asymmetry and beam-energy-constrained mass are introduced to further reduce background events.The selection is optimized by taking into account the different tagside modes to maximize search sensitivities.Lastly, the photon energy is calibrated using radiative muon events.Thanks to those improvements and 1.9 times data, our expected limits are 1.6-1.8times more stringent compared to the previous Belle results [5].With the absence of signal in any modes, the upper limits are set on branching fractions: B(τ ± → µ ± γ) < 4.2 × 10 −8 and B(τ ± → e ± γ) < 5.6 × 10 −8 at the 90% confidence level.The observed limit on the τ ± → µ ± γ decay is the most stringent to date.

Figure 1 .
Figure 1.Energy resolution as a function of the reconstructed photon energy in the e + e − → µ + µ − γ events.Black (Blue) points are the photon energy resolution with (without) the calibration applied.Error bars are the statistical uncertainties.
tag) ] and p CM track(tag) = [E CM track(tag) , p CM track(tag) ] are the four-momenta of tag-side tau and track in the CM frame.For the ρ channel, photons in the tag side are considered in the calculation of the four-momentum of tag-side track.Substituting E CM τ (tag) = √ s/2, m τ (tag) = m τ ∼ 1.78 GeV/c 2 and m 2 miss.track(tag)= m 2 miss.γ.track(tag) , where m 2 miss.γ.track(tag) is a missing mass squared of the event against the lepton and the photon
Figure 5. ∆E/ √ s distribution in the sideband.The black points with error bars are the data and red curves show the fit result of the background PDF.

Table 1 .
Systematic uncertainties (in %) considered in this analysis.