Measurement of the tau Michel parameters $\bar{\eta}$ and $\xi\kappa$ in the radiative leptonic decay $\tau^- \rightarrow \ell^- \nu_{\tau} \bar{\nu}_{\ell}\gamma$

We present a measurement of the Michel parameters of the $\tau$ lepton, $\bar{\eta}$ and $\xi\kappa$, in the radiative leptonic decay $\tau^- \rightarrow \ell^- \nu_{\tau} \bar{\nu}_{\ell} \gamma$ using 711~f$\mathrm{b}^{-1}$ of collision data collected with the Belle detector at the KEKB $e^+e^-$ collider. The Michel parameters are measured in an unbinned maximum likelihood fit to the kinematic distribution of $e^+e^-\rightarrow\tau^+\tau^-\rightarrow (\pi^+\pi^0 \bar{\nu}_\tau)(\ell^-\nu_{\tau}\bar{\nu}_{\ell}\gamma)$ $(\ell=e$ or $\mu)$. The measured values of the Michel parameters are $\bar{\eta} = -1.3 \pm 1.5 \pm 0.8$ and $\xi\kappa = 0.5 \pm 0.4 \pm 0.2$, where the first error is statistical and the second is systematic. This is the first measurement of these parameters. These results are consistent with the Standard Model predictions within their uncertainties and constrain the coupling constants of the generalized weak interaction.

We present a measurement of the Michel parameters of the τ lepton,η and ξκ, in the radiative leptonic decay τ − → ℓ − ν τνℓ γ using 711 fb −1 of collision data collected with the Belle detector at the KEKB e + e − collider. The Michel parameters are measured in an unbinned maximum likelihood fit to the kinematic distribution of e + e − → τ + τ − → (π + π 0ν τ )(ℓ − ν τνℓ γ) (ℓ = e or µ). The measured values of the Michel parameters arē η = −1.3 ± 1.5 ± 0.8 and ξκ = 0.5 ± 0.4 ± 0.2, where the first error is statistical and the second is systematic. This is the first measurement of these parameters. These results are consistent with the Standard Model predictions within their uncertainties and constrain the coupling constants of the generalized weak interaction.

Introduction
In the Standard Model (SM), there are three flavors of charged leptons: e, µ, and τ . The SM has proven to be the fundamental theory in describing the physics of particles; nevertheless, precision tests may reveal the presence of physics beyond the Standard Model (BSM). In particular, a measurement of Michel parameters in leptonic and radiative leptonic τ decays is a powerful probe for the BSM contributions [1,2]. The most general Lorentz-invariant derivative-free matrix element of leptonic τ decay τ − → ℓ − ν τνℓ γ * is represented as [3]  where G F is the Fermi constant, i and j are the chirality indices for the charged leptons, n and m are the chirality indices of the neutrinos, ℓ is e or µ, Γ S = 1, Γ V = γ µ , and Γ T = i (γ µ γ ν − γ ν γ µ ) /2 √ 2 are, respectively, the scalar, vector and tensor Lorentz structures in terms of the Dirac matrices γ µ , u i and v i are the four-component spinors of a particle and an antiparticle, respectively, and g N ij are the corresponding dimensionless couplings. In the SM, τ − decays into ν τ and a W − -boson, the latter decays into ℓ − and right-handedν ℓ ; i.e., the only non-zero coupling is g V LL = 1. Experimentally, only the squared matrix element is observable and bilinear combinations of the g N ij are accessible. Of all such combinations, four Michel parameters, η, ρ, δ, and ξ, can be measured in the leptonic decay of the τ when the final-state neutrinos are not observed and the spin of the outgoing lepton is not measured [4]: The Feynman diagrams describing the radiative leptonic decay of the τ are presented in Fig. 1. The last amplitude is ignored because this contribution turns out to be suppressed by the very small factor (m τ /m W ) 2 [5]. As shown in Refs. [6,7], through the presence of a radiative photon in the final state, the polarization of the outgoing lepton is indirectly exposed; accordingly, three more Michel parameters,η, η ′′ , and ξκ, become experimentally accessible: η ′′ = ℜ 24g V RL (g S * LR + 6g T * LR ) + 24g V LR (g S * RL + 6g T * RL ) − 8(g V RR g S * LL + g V LL g S * RR ) , Experimental results represent average values obtained by PDG [11].
Bothη and η ′′ appear in spin-independent terms in the differential decay width. Since all terms in Eq. (6) are strictly non-negative, the upper limit onη provides a constraint on each coupling constant. The effect of the nonzero value of η ′′ is suppressed by a factor m 2 ℓ /m 2 τ ∼ 10 −7 for an electron mode and about 4 × 10 −3 for a muon mode and so proves to be difficult to measure with the available statistics collected at Belle. In this study, we fix η ′′ at its SM value (η ′′ = 0).
To measure ξκ, which appears in the spin-dependent part of the differential decay width, the knowledge of tau spin direction is required. Although the average polarization of a single τ is zero in experiments at e + e − colliders with unpolarized beams, the spin-spin correlation between the τ + and τ − in the reaction e + e − → τ + τ − can be exploited to measure ξκ [8].
According to Ref. [9], ξκ is related to another Michel-like parameter ξ ′ = −ξ − 4ξκ + 8ξδ/3. Because the normalized probability that the τ − decays into the right-handed charged daughter lepton Q τ R is given by Q τ R = (1 − ξ ′ )/2 [10], the measurement of ξκ provides a further constraint on the Lorentz structure of the weak current. The information on these parameters is summarized in Table 1.
In muon decay, through the direct measurement of electron polarization in µ + → e + ν eνµ , the relevant parameters ξ ′ and ξ ′′ = 16ρ/3 − 4η − 3 have been already measured. Those of the τ have not been measured yet.
Using the statistically abundant data set of ordinary leptonic decays, previous measurements [12,13] have determined the Michel parameters η, ρ, δ, and ξ to an accuracy of a few percent and shows agreement with the SM prediction. Taking into account this measured agreement, the smaller data set of the radiative decay and its limited sensitivity, we focus in this analysis only on the extraction ofη and ξκ by fixing η, ρ, δ, ξ, and ξ ρ to the SM values. This represents the first measurement of theη and ξκ parameters of the τ lepton. 5/28

Unbinned maximum likelihood method
The differential decay width for the radiative leptonic decay of τ − with a definite spin direction S * − is given by where A − i and B − i (i = 0, 1) are known functions of the kinematics of the decay products † with indices i = 0, 1 (i is the function identifier), Ω a stands for a set of {cosθ a , φ a } for a particle of the type a, and the asterisk means that the variable is defined in the τ − rest frame. Equation (9) shows that ξκ appears in the spin-dependent part of the decay width. This parameter can be measured by utilizing the well-known spin-spin correlation of the τ leptons in the e − e + → τ + τ − production: where α is the fine structure constant, β τ and E τ are the velocity and energy of the τ − in the center-of-mass system (c.m.s.), respectively, D 0 is the spin-independent part of the cross section, and D ij (i, j = 0, 1, 2) is a tensor describing the spin-spin correlation (see Eq. (4.11) in Ref. [8]). For the partner τ + , its spin information is extracted using the two-body decay τ + → ρ +ν τ → π + π 0ν τ whose differential decay width is A + and B + are known functions for the spin-independent and spin-dependent parts, respectively; the tilde indicates variables defined in the ρ + rest frame and m is the invariant mass of the ππ 0 system, m 2 = (p π + p π 0 ) 2 . As mentioned before, we use the SM value: ξ ρ = 1. Thus, the total differential cross section of e + e − → τ − τ + → (ℓ − ν τνℓ γ)(π + π 0ν τ ) (or, briefly, (ℓ − γ, π + π 0 )) can be written as: To extract the visible differential cross section, we transform the differential variables into ones defined in the c.m.s. using the Jacobian J: where the parameter Φ denotes the angle along the arc illustrated in Fig The visible differential cross section is, therefore, obtained by integration over Φ: where S(x) is proportional to the probability density function (PDF) of the signal and x denotes the set of twelve measured variables: There are several corrections that must be incorporated in the procedure to take into account the real experimental situation. Physics corrections include electroweak higher-order corrections to the e + e − → τ + τ − cross section [14,15,16,17,18]. Apparatus corrections include the effect of the finite detection efficiency and resolution, the effect of the external bremsstrahlung for (e − γ, π + π 0 ) events, and the e ± beam energy spread. Accounting for the event-selection criteria and the contamination from identified backgrounds, the total visible (properly normalized) PDF for the observable x in each event is given by where B i (x) is the distribution of the i th category of background, λ i is the fraction of this background, and ε(x) is the selection efficiency of the signal distribution. The categorization 7/28 of i is explained later (see the caption of Fig. 3). In general, B i (x) is evaluated as an integral of the i th background PDF multiplied by the inefficiency that depends on the variables of missing particles. The PDFs of the dominant background processes are described analytically one by one, while the remaining background processes are described by one common PDF, tabulated from Monte Carlo (MC) simulation. The denominator of the signal term in Eq. (17) represents normalization. Since S(x) is a linear combination of the Michel parameters S(x) = S 0 (x) + S 1 (x)η + S 2 (x)ξκ, the normalization of signal PDF becomes where N 0 is a normalization coefficient of the SM part defined by N 0 = dx S 0 (x), x i represents a set of variables for the i th selected event of N sel events,ε is an average selection efficiency, and the brackets indicate an average with respect to the selected SM distribution. We refer to N 0 and S i /S 0 (i = 1, 2) as absolute and relative normalizations, respectively.
From P (x), the negative logarithmic likelihood function (NLL) is constructed and the best estimators of the Michel parameters,η and ξκ, are obtained by minimizing the NLL. The efficiency ε(x) is a common multiplier in Eq. (17) and does not depend on the Michel parameters. This is one of the essential features of the unbinned maximum likelihood method. We validated our fitter and procedures using a MC sample generated according to the SM distribution. The optimal values of the Michel parameters are consistent with their SM expectations within the statistical uncertainties.

KEKB collider
The KEKB collider (KEK laboratory, Tsukuba, Japan) is an energy-asymmetric e + e − collider with beam energies of 3.5 GeV and 8.0 GeV for e + and e − , respectively. Most of the data were taken at the c.m.s. energy of 10.58 GeV, corresponding to the mass of the Υ(4S), where a huge number of τ + τ − as well as BB pairs were produced. The KEKB collider was operated from 1999 to 2010 and accumulated 1 ab −1 of e + e − collision data with the Belle detector. The achieved instantaneous luminosity of 2.11 × 10 34 cm −2 /s is the world record. For this reason, the KEKB collider is often called a B-factory but it is worth considering it also as a τ -factory, where O(10 9 ) τ pair events have been produced. The world largest sample of τ leptons collected at Belle provides a unique opportunity to study radiative leptonic decay of τ . In this analysis, we use 711 fb −1 of collision data collected at the Υ(4S) resonance energy [19]. 8/28

Belle detector
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, 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) 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 to detect K 0 L mesons and to identify muons (KLM). The detector is described in detail elsewhere [20].

Event selection
The event selection proceeds in two stages. At the preselection, τ + τ − candidates are selected efficiently while suppressing the beam background and other physics processes like radiative Bhabha scattering, two-photon interaction, and radiative µ + µ − pair production. The preselected events are then required to satisfy final selection criteria to enhance the purity of the signal events.

Preselection
• There must be exactly two oppositely charged tracks in the event. The impact parameters of these tracks relative to the interaction point are required to be within ±2.5 cm along the beam axis and ±0.5 cm in the transverse plane. The two-track transverse momentum must exceed 0.1 GeV/c and that of one track must exceed 0.5 GeV/c. • Total energy deposition in the ECL in the laboratory frame must be lower than 9 GeV. • The opening angle ψ of the two tracks must satisfy 20 • < ψ < 175 • in the laboratory frame. • The number of photons whose energy exceeds 80 MeV in the c.m.s. must be fewer than five. • For the four-vector of missing momentum defined by p miss = p beam − p obs , the missing mass M miss defined by M 2 miss = p 2 miss c 2 must lie in the range 1 GeV/c 2 ≤ M miss ≤ 7 GeV/c 2 , where p beam and p obs are the four-momenta of the beam and all detected particles, respectively.
• The polar angle of missing-momentum must satisfy 30 • ≤ θ miss ≤ 150 • in the laboratory frame.
• The electron selection is based on the likelihood ratio cut, where L e and L x are the likelihood values of the track for the electron and non-electron hypotheses, respectively. These values are determined using specific ionization (dE/dx) in the CDC, the ratio of ECL energy and CDC momentum E/P , the transverse shape of the cluster in the ECL, the matching of the track with the ECL cluster, and the light yield in the ACC [21]. The muon selection uses the likelihood ratio P µ = L µ /(L µ + L π + L K ) > 0.9, where the likelihood values are determined by the measured versus expected range for the µ hypothesis, and transverse scattering of the track in the KLM [22]. The reductions of the signal efficiencies with lepton selections are approximately 10% and 9/28 30% for the electron and muon, respectively. The pion candidates are distinguished from kaons using P π = L π /(L π + L K ) > 0.4, where the likelihood values are determined by the ACC response, the timing information from the TOF, and dE/dx in the CDC. The reduction of the signal efficiency with pion selection is approximately 5%. • The π 0 candidate is formed from two photon candidates, where each photon satisfies E γ > 80 MeV, with an invariant mass of 115 MeV/c 2 < M γγ < 150 MeV/c 2 . Figure 3 shows the distribution of the invariant mass of the π 0 candidates. The reduction of the signal efficiency by the mass selection is approximately 50%. In addition, when more than two π 0 candidates are found, the event is rejected. • The ρ + candidate is formed from a π + and a π 0 candidate, with an invariant mass of 0.5 GeV/c 2 < M π + π 0 < 1.5 GeV/c 2 . Figure 4 shows the distribution of the invariant mass of the ρ candidates. The reduction of the signal efficiency is approximately 3%. • The c.m.s. energy of signal photon candidate must exceed 80 MeV if within the ECL barrel (31. As shown in Fig. 5, this photon must lie in a cone determined by the lepton-candidate direction that is defined by cosθ eγ > 0.9848 and cosθ µγ > 0.9700 for the electron and muon mode, respectively, where θ ℓγ (ℓ = e or µ) is the angle between the lepton and the photon. The reductions of the signal efficiencies for the requirement on this photon direction are approximately 11% and 27% for the electron and muon mode, respectively. Furthermore, if the photon candidate and either of the photons from the π 0 , which is a daughter of the ρ + candidate, form an invariant mass of the π 0 (115 MeV/c 2 < M γγ < 150 MeV/c 2 ), the event is rejected. The additional selection reduces the signal efficiency by 1%. • The direction of the combined momentum of the lepton and photon in the c.m.s. must not belong to the hemisphere determined by the ρ candidate: an event should satisfy θ (ℓ − γ)ρ + > 90 • , where θ (ℓ − γ)ρ + is the spatial angle between the ℓ − γ system and the ρ candidate. This selection reduces the signal efficiency by 0.4%. • There must be no additional photons in the aforementioned cone around the lepton candidate; the sum of the energy in the laboratory frame of all additional photons that are not associated with the π 0 or the signal photon (denoted as E LAB extraγ ) should not exceed 0.2 GeV and 0.3 GeV for the electron and muon mode, respectively. The reductions of the signal efficiencies for the requirement on the E LAB extraγ are approximately 14% and 6% for the electron and muon mode, respectively.
These selection criteria are optimized using MC simulation (five times as large as real data) where e + e − → τ + τ − pair production and the successive decay of the τ are simulated by the KKMC [23] and TAUOLA [24,25] generators, respectively. The detector effects are simulated based on the GEANT3 package [26].
Distributions of the photon energy E γ and the angle between the lepton and photon, θ ℓγ , for the selected events are shown in Figs. 6 and 7 for τ − → e − ν τνe γ and τ − → µ − ν τνµ γ candidates, respectively.
In the electron mode, the fraction of the signal decay in the selected sample is about 30% due to the large external bremsstrahlung rate in the non-radiative leptonic τ decay events. In the muon mode, the fraction of the signal decay is about 60%; here, the main background arises from ordinary leptonic decay (τ − → ℓ − ν τνℓ ) events where either an additional photon 10/28 (a) τ − → e − ν τνe γ candidates: the open histogram corresponds to the signal, the yellow (i = 1) and green (i = 2) histograms represent ordinary leptonic decay plus extra bremsstrahlung due to the detector material and radiative leptonic decay plus bremsstrahlung, respectively, and the blue (i = 3) histogram represents other processes such as radiative Bhabha, two-photon, and e + e − → qq (q = u, d, s, c) productions.
is reconstructed from beam background in the ECL or a photon is emitted by the initial-state e + e − . The information is summarized in Table 2.
As mentioned before, in the integration over Φ in Eq. (15), the generated differential variables are varied according to the resolution function R. Thus, the kinematic variables can extend outside the allowed phase space. For the unphysical values, we assign zero to the integrand because this implies negative neutrino masses. If such discarded trials in the 11/28   ] (57.4%)    28.9 ± 0.8 57.4 ± 1.3 † The efficiency is determined based on the photon energy threshold of E * γ > 10 MeV in the τ rest frame.
integration exceed 20% of the total number of iterations, we reject the event. This happens for events that lie near the kinematical boundary of the signal phase space. The corresponding reduction of the efficiency is approximately 2% and 3% for the electron and muon mode, respectively. This additional decrease of the efficiency is not reflected in the values of Table 2.

Analysis of experimental data
When we fit the Michel parameters for the real experimental data, the difference in selection efficiency between real data and MC simulation must be taken into account by the correction factor R(x) = ε EX (x)/ε MC (x) that is close to unity; its extraction is described below. With this correction, Eq. (17) is modified to The presence of R(x) in the numerator does not affect the NLL minimization, but its presence in the denominator does. We evaluate R(x) as the product of the measured corrections for the trigger, particle identification, track, π 0 , and γ reconstruction efficiencies: R(x) = R trg R ℓ (P ℓ , cos θ ℓ )R γ (P γ , cos θ γ )R π (P π , cos θ π )R π 0 (P π 0 , cos θ π 0 ), R ℓ (P ℓ , cos θ ℓ ) = R trk (P ℓ , cos θ ℓ )R LID (P ℓ , cos θ ℓ ), R π (P π , cos θ π ) = R trk (P π , cos θ π )R πID (P π , cos θ π ).
The lepton identification efficiency correction is estimated using two-photon processes e + e − → e + e − ℓ + ℓ − (ℓ = e or µ). Since the momentum of the lepton from the two-photon process ranges from the detector threshold to approximately 4 GeV/c in the laboratory frame, the efficiency correction factor can be evaluated for our signal process as a function of P ℓ and cos θ ℓ .
The pion PID correction factor is obtained by the measurement of D * + → D 0 π + s → (K − π + )π + s decay (where the subscript s indicates "slow"). The small momentum of the pion from D * + allows us to select this process. As a result, assuming the mass of D 0 meson, we can reconstruct D * + even if this π + is missed.
The track reconstruction efficiency correction is extracted from τ + τ − → (ℓ + ν ℓντ )(π − π + π − ν τ ) events. Here, we count the number of events N 4 (N 3 ) in which four (three) charged tracks are reconstructed. The three-track event is required to have a negative net charge (π + is missing). Since the charged track reconstruction efficiency ε is included as, respectively, ε 4 and ε 3 (1 − ε) in N 4 and N 3 , the value of ε can be obtained by ε = N 4 /(N 4 + N 3 ). The 14/28  The π 0 reconstruction efficiency correction is obtained by comparing the ratio of the number of selected events of τ + τ − → (π + π 0ν τ )(π − π 0 ν τ ) and τ + τ − → (π + π 0ν τ )(π − ν τ ) between experiment and MC simulation. The momentum and angular dependence of the π 0 reconstruction efficiency is extracted by counting the number of events observed in a certain kinematic-variable cell of the π 0 phase space. By randomly choosing either of the photon daughters from the π 0 , the γ reconstruction efficiency correction is extracted in the same manner.
The trigger efficiency correction has the largest impact among these factors. In particular, for the electron mode, because of the similar structure of our signal events and Bhabha events (back-to-back topology of two-track events), many signal events are rejected by the Bhabha veto in the trigger. The veto of the trigger results in a spectral distortion and a large systematic uncertainty. The correction factor is extracted using the charged and neutral subtriggers (denoted as Z and N ), which provide completely independent signals. Since the trigger signal appears when at least one of the subtriggers fires (i.e., Z OR N ), its efficiency is given by ε trg = 1 − (1 − ε Z )(1 − ε N ), where ε Z = N Z&N /N N and ε N = N Z&N /N Z are the efficiencies of the charged and neutral subtriggers, respectively; N Z&N is the number of events where both subtriggers fire (i.e. Z AND N ), N Z (N N ) is a number of events triggered by Z (N ). Thus R trg is obtained as the ratio of ε trg between the experiment and MC simulation. Figure 8 shows the distribution of the momentum and the cosine of the polar angle of electron and muon events. In the figure, the effects of all corrections are seen mainly at cosθ e < −0.6 and cosθ µ < −0.6.

Evaluation of systematic uncertainties
In Table 3, we summarize the contributions of the identified sources of systematic uncertainties. The dominant source for the electron mode is the calculation of the relative normalizations. Due to the peculiarity of the signal PDF when m ℓ → m e , the convergence of the relative normalization coefficients is quite slow and results in a notable effect. For a 15/28 given number of MC events N , the errors of the relative normalizations A i /A 0 (i = 1, 2) are evaluated by σ 2 = Var(A i /A 0 )/N , where Var(X) represents the variance of a random variable X. The resulting systematic effect on the Michel parameter is estimated by varying the normalizations. The effect of the absolute normalization is estimated in the same way. The largest systematic uncertainty for the muon mode is due to the limited precision of the description of the background PDF that appears in Eq. (22). As mentioned before, the remaining background sources are described by a common PDF, which is tabulated utilizing a large τ + τ − generic MC sample. This effective description can generally discard information about correlations in the phase space and thereby give significant bias. The residuals of the fitted Michel parameters from the SM prediction obtained by the fit to the MC distribution are taken as the corresponding systematic uncertainties.
Other notable uncertainties come from the accuracy of the measured branching ratios. In particular, the uncertainties of the branching ratio of the radiative decay τ − → ℓ − ν τνℓ γ dominate the contribution. The systematic effects of the cluster merging in the ECL are evaluated as a function of the angle between the photon and lepton clusters at the front face of ECL (θ ECL ℓγ ). The limit θ ECL ℓγ → 0 represents the merger of the two clusters and the comparison of the distribution between experiment and MC gives us the corresponding bias. A systematic effect due to the detector resolution is evaluated by comparing Michel parameters obtained in the fit with and without account of the resolution function R(x − x ′ ).
The error of the measured correction factor R is estimated by varying the central values based on the uncertainty in each bin. Moreover, as can be observed in Fig. 8d in the muon mode, there is a notable disagreement of efficiency in the forward domain (cosθ LAB µ > 0.9). This is due to the contamination of backgrounds in the extraction of the correction factor of R µID . We excluded this region (reducing the statistics by 1.5%) and checked the shift of the refitted Michel parameters.
In the electron mode, we observe the disagreement of the photon reconstruction efficiency in the low-energy region (Fig. 8c). It could arise from a discrepancy in the simulation of extra bremsstrahlung. We excluded the events having a low energy photon E LAB γ < 150 MeV and compared the refitted values. Because this cut reduces the number of events by approximately 20%, the statistical fluctuation is also reflected in the shifts.
The effect of the beam-energy spread is estimated by varying the input of this value for the calculation of the PDF with respect to run-dependent uncertainties, and turns out to be negligible.
The effects from the next-to-leading-order (NLO) contribution were checked by adding the NLO formulae [27] to the signal PDF and refitting, and were found to be negligible.

Results
Because of the suppression of sensitivity due to the small mass of the electron, theη parameter is extracted only from the τ − → µ − ν τνµ γ mode. Using the 71171 selected τ − → µ − ν τνµ γ candidates,η and ξκ are simultaneously fitted to the kinematic distribution to bē Figure 9 shows the contour of the likelihood function for τ − → µ − ν τνµ γ events. 17/28 In the electron mode, ξκ is fitted by fixing theη value to the SM prediction ofη = 0 and the optimal value is extracted using 776834 events to be (ξκ) e = −0.4 ± 0.8 ± 0.9.
In Equations 26-29, the first error is statistical and second systematic. The obtained values are consistent with the SM prediction. Furthermore, the ξκ product is also obtained by fitting simultaneously to both electron and muon events as Here, the systematic uncertainty is estimated from 1/σ 2 comb = 1/σ 2 e + 1/σ 2 µ by assuming they are uncorrelated.
We also obtain the dependence of the E LAB extraγ selection on the fitted Michel parameters as shown in Fig. 10. In the extraction ofη, we use τ − → µ − ν τνµ γ while, for ξκ, we use the combined result for τ − → e − ν τνe γ and τ − → µ − ν τνµ γ decays. We observe stability of the fitted Michel parameters within uncertainties. Figure 11 shows the residual of the likelihood function ∆L = NLL min − NLL projected onto one axis. We observe a smooth and quadratic shape of the NLL around its minimum.

Measurement of the branching ratio
In addition to the Michel parameters, we have determined the branching ratios of the τ − → ℓ − ν τνℓ γ (ℓ = e, µ) decays.
Following the definition of Ref. [27], we distinguish between two types of radiative decays in the NLO approximation: the exclusive radiative decay implies that only one hard photon 18/28   is emitted in the event; in the inclusive radiative decay, at least one hard photon is emitted. Here, the hard photon energy threshold is 10 MeV in the τ − rest frame.
In Ref. [27], the precision measurement of the branching ratios of the radiative leptonic τ decays at BaBar is also discussed. While the measured branching ratios of both electron and muon modes agree with their leading-order (LO) theoretical predictions, the NLO exclusive branching ratio prediction for the τ − → e − ν τνe γ decay differs from the BaBar result by 3.5 standard deviations. This is explained by the insufficient accuracy of the current MC simulation of the radiative and doubly-radiative leptonic τ decays. Neither an NLO correction to the radiative leptonic decay, nor the doubly-radiative leptonic mode itself, are incorporated in the current version of the TAUOLA MC generator. As a result, the detection efficiency is not precisely evaluated for the radiative decay. As well, the background from the doubly radiative decay is not subtracted at all. Finally, the second photon emission might affect the efficiency of the photon veto and the shape of the neutral clusters in the calorimeter. Indeed, the ratio of the yield with two-photon emission to that with single-photon emission 19/28 is approximately 5% and 1% for the electron and muon modes, respectively. Thus, there is an experimentally notable impact of the two-photon emission on the electron mode.
In our measurement of the branching ratios, we do not take into account the up-to-date formalism of Ref. [27] since the main purpose of this study is a consistency check of our selection criteria and experimental efficiency corrections.

Method
The branching ratio is determined using where B(τ + → π + π 0ν τ ) = (25.52 ± 0.09)% [11] is the branching ratio of τ + → π + π 0ν τ decay, N obs is the number of observed events, f bg is the fraction of background events, σ τ τ = (0.919 ± 0.003) nb −1 is the cross section of the e + e − → τ + τ − process at Υ(4S) [28], L = (711 ± 10) fb −1 is the integrated luminosity recorded at Υ(4S), andε EX is the average detection efficiency of signal events. The efficiency,ε EX , is evaluated with help of the MC simulation. The correction factor, R(x) = ε EX (x)/ε MC (x), which is used to extract Michel parameters, is applied to compensate for the difference between experimental and MC efficiencies as follows: where S(x) is the PDF of the signal events andR is an average efficiency correction factor for the selected signal MC events. Here, the average MC efficiency,ε MC , is determined for the photon energy threshold of 10 MeV in the τ rest frame.

Event selection
We apply additional selection criteria to enhance the purity of the sample as well as to reduce systematic uncertainties. The extra-gamma-energy selection is released for the latter purpose but other selection criteria are common to those of Michel parameter measurement (see in Sec. 3). For the electron mode, we apply the following selection criteria: • The uncertainty of the lepton identification efficiency in the forward and backward regions of the detector is large due to the notable background contamination of the control sample; thus, the electron polar angle in the laboratory frame must lie in the region defined by θ LAB e < 126 • as shown in Figs. 12a and 12b. • The electron identification is less precise at small momenta, so we apply the momentum threshold P LAB e > 1.5 GeV/c as shown in Fig. 12c. • After the final selections (explained in Sec. 3), the dominant background arises from the external bremsstrahlung on the material of the detector. It is effectively suppressed by applying the requirement on the invariant mass of the electron-photon system, M eγ > 0.1 GeV/c 2 , as shown in Fig. 13. • The extra gamma energy in the laboratory frame, E LAB extraγ , must be less than 0.2 GeV. For the muon mode, we apply the following selection criteria: • The muon polar angle in the laboratory frame must satisfy 51  • The spatial angle between µ and γ in c.m.s. must satisfy cos θ µγ > 0.99.
• The extra gamma energy in laboratory frame, E LAB extraγ , must be smaller than 0.3 GeV.

Evaluation of systematic uncertainties
In Table 4, we summarize the sources of the systematic uncertainties of the branching ratios of the electron and muon modes. To estimate a systematic uncertainty from the efficiency 21/28 correction,R, we use the following method. The uncertainties of the R ℓID are determined by the finite statistics of e + e − → ℓ + ℓ − e + e − sample, a comparison of R ℓID from e + e − → ℓ + ℓ − e + e − and R ℓID from J/ψ → ℓ + ℓ − , and its time variation during the experiment. R πID values are estimated from the finite statistics of a D * + → D 0 (K − π + )π + sample, the fit of the reconstructed mass distribution of D * , and observation of time variation.
The systematic uncertainties of the R π 0 ID , R γID , R trg , and R trk values are estimated from a comparison betweenR and unity.
The statistical uncertainty of MC events is ignored because its fluctuation is small. The evaluation of the systematic uncertainty of the purity f bg is estimated based on sideband information. The sideband events are selected by the following criteria: M eγ < 0.1 GeV/c 2 and 0.90 <cosθ eγ < 0.94 (0.99) for the electron (muon) mode, where other selection criteria are common with those of the signal extraction. The difference of the background yield in the sideband region between MC simulation and the real experiment is 4.4% (5.5%) for the electron (muon) mode. Taking each fraction into account, we estimate that the resulting uncertainty is 1.3% and 1.5%.
The effect of detector response is estimated by varying selection cut parameters. The effect due to variation of the photon energy threshold is based on the energy resolution at the threshold, and found to be 5% [20]. The variation of other selection criteria is determined based on the propagation of the error matrix of the helix parameters. Of all the selection criteria, the requirement of M eγ > 0.1 GeV/c 2 has the largest impact.
As mentioned, to estimate the efficiency, we define the radiative events by the imposition of a photon energy threshold of E * γ = 10 MeV in the τ − rest frame. However, in the real experiment, we cannot precisely determine this energy because the τ − momentum is not directly reconstructed. Accordingly, there is a chance that an event that has a photon with an energy less than the threshold is also reconstructed as signal. This is only possible in a limited phase space, and such events are included in the selection with fractions of 1.1% and 0.3% for electron and muon events, respectively. We take these fractions as sources of systematic effects due to the uncertainty of the experimental E * γ threshold.
As summarized in Table 4, the dominant systematic contribution comes from the uncertainty of the efficiency correction for π 0 . This uncertainty is canceled when we measure the ratio of branching fractions Q = B(τ − → e − ν τνe γ)/B(τ − → µ − ν τνµ γ). Moreover, other Red, blue, and magenta lines represent the measured branching ratio of τ ± → ℓ ± ννγ, τ − → ℓ − ν τνℓ γ and τ + → ℓ + ν ℓντ γ, respectively. The bars represent uncertainties and are drawn only for the combined modes, where both statistical and systematic uncertainties are included. The orange band shows the BaBar measurement [30]. Black, green, and red lines are LO, NLO inclusive, and NLO exclusive theoretical predictions, respectively [27]. common systematic sources, namely R trk , R πID , the integrated luminosity, the branching ratio of τ + → π + π 0ν τ decay, and the cross section σ(e + e − → τ + τ − ), also cancel. The obtained ratio is where the first error is statistical and the second is systematic. In Table 7, we summarize the theoretical prediction and past experimental results for the ratio Q.

Conclusion
We present the measurement of Michel parametersη and ξκ of the τ using 711 fb −1 of data collected with the Belle detector at the KEKB e + e − collider. These parameters are extracted from the radiative leptonic decay τ − → ℓ − ν τνℓ γ which is tagged by τ + → ρ + (→ π + π 0 )ν τ decay of the partner τ + to exploit the spin-spin correlation in e + e − → τ + τ − . Due to the small sensitivity toη in the electron mode, this parameter is extracted only from τ − → µ − ν τνµ γ to giveη = −1.3 ± 1.5 ± 0.8. The product ξκ is measured using both decays τ − → ℓ − ν τνℓ γ (ℓ = e and µ) to be ξκ = 0.5 ± 0.4 ± 0.2. The first error is statistical and the second is systematic. This is the first measurement of both parameters for the τ lepton. These values are consistent with the SM expectation within uncertainty. For a consistency check of the procedure to measure the Michel parameters, we measure the branching ratio of τ − → ℓ − ν τνℓ γ decay. The obtained values are consistent with the LO theoretical prediction and support the measurement by BaBar, which is known to deviate from the SM exclusive branching ratio by 3.5σ. Accounting for the agreement between our result and the BaBar measurement [30], the implementation of the NLO formalism in the TAUOLA generator is required to carry out more precise measurements.