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 Belle preprint 2017-20, KEK preprint 2017-29

# Measurement of the tau Michel parameters ¯η and ξκ in the radiative leptonic decay τ−→ℓ−ντ¯νℓγBelle preprint 2017-20, KEK preprint 2017-29

###### Abstract

We present a measurement of the Michel parameters of the lepton, and , in the radiative leptonic decay using 711 f of collision data collected with the Belle detector at the KEKB collider. The Michel parameters are measured in an unbinned maximum likelihood fit to the kinematic distribution of or . The measured values of the Michel parameters are and , 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.

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XXXX-XXXX

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C01, C07, C21

## 1 Introduction

In the Standard Model (SM), there are three flavors of charged leptons: , and . 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 decays is a powerful probe for the BSM contributions [1, 2].

The most general Lorentz-invariant derivative-free matrix element of leptonic decay  ***Unless otherwise stated, use of charge-conjugate modes is implied throughout the paper. is represented as [3]

{fmffile}tautree {fmfchar*}(90,60) \fmflefttm,antinu \fmflabeltm \fmflabelantinu \fmffermiontm,Wi \fmffermionWi,antinu \fmfdashes,label=Wi,Wf \fmffermion,label= fb,Wf \fmffermionWf,f \fmfrightf,fb \fmflabelfb \fmflabelf \fmfdotWi,Wf (1)

where is the Fermi constant, and are the chirality indices for the charged leptons, and are the chirality indices of the neutrinos, is or , , , and are, respectively, the scalar, vector and tensor Lorentz structures in terms of the Dirac matrices , and are the four-component spinors of a particle and an antiparticle, respectively, and are the corresponding dimensionless couplings. In the SM, decays into and a -boson, the latter decays into and right-handed ; i.e., the only non-zero coupling is . Experimentally, only the squared matrix element is observable and bilinear combinations of the are accessible. Of all such combinations, four Michel parameters, , , , and , can be measured in the leptonic decay of the when the final-state neutrinos are not observed and the spin of the outgoing lepton is not measured [4]:

 ρ = 34−34(∣∣gVLR∣∣2+∣∣gVRL∣∣2+2∣∣gTLR∣∣2+2∣∣gTRL∣∣2+R(gSLRgT∗LR+gSRLgT∗RL)), (2) η = 12R(6gVRLgT∗LR+6gVLRgT∗RL+gSRRgV∗LL+gSRLgV∗LR+gSLRgV∗RL+gSLLgV∗RR), (3) ξ = 4R(gSLRgT∗LR−gSRLgT∗RL)+∣∣gVLL∣∣2+3∣∣gVLR∣∣2−3∣∣gVRL∣∣2−∣∣gVRR∣∣2 (4) +5∣∣gTLR∣∣2−5∣∣gTRL∣∣2+14(∣∣gSLL∣∣2−∣∣gSLR∣∣2+∣∣gSRL∣∣2−∣∣gSRR∣∣2), ξδ = 316(∣∣gSLL∣∣2−∣∣gSLR∣∣2+∣∣gSRL∣∣2−∣∣gSRR∣∣2) (5) −34(∣∣gTLR∣∣2−∣∣gTRL∣∣2−∣∣gVLL∣∣2+∣∣gVRR∣∣2−R(gSLRgT∗LR+gSRLgT∗RL)).

The Feynman diagrams describing the radiative leptonic decay of the are presented in Fig. 1. The last amplitude is ignored because this contribution turns out to be suppressed by the very small factor  [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, , , and , become experimentally accessible:

 ¯η = ∣∣gVRL∣∣2+∣∣gVLR∣∣2+18(∣∣gSRL+2gTRL∣∣2+∣∣gSLR+2gTLR∣∣2)+2(∣∣gTRL∣∣2+∣∣gTLR∣∣2), (6) η′′ = R{24gVRL(gS∗LR+6gT∗LR)+24gVLR(gS∗RL+6gT∗RL)−8(gVRRgS∗LL+gVLLgS∗RR)}, (7) ξκ = ∣∣gVRL∣∣2−∣∣gVLR∣∣2+18(∣∣gSRL+2gTRL∣∣2−∣∣gSLR+2gTLR∣∣2)+2(∣∣gTRL∣∣2−∣∣gTLR∣∣2). (8)

Both and appear in spin-independent terms in the differential decay width. Since all terms in Eq. (6) are strictly non-negative, the upper limit on provides a constraint on each coupling constant. The effect of the nonzero value of is suppressed by a factor for an electron mode and about for a muon mode and so proves to be difficult to measure with the available statistics collected at Belle. In this study, we fix at its SM value ().

To measure , 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 is zero in experiments at colliders with unpolarized beams, the spin-spin correlation between the and in the reaction can be exploited to measure  [8].

According to Ref. [9], is related to another Michel-like parameter . Because the normalized probability that the decays into the right-handed charged daughter lepton is given by  [10], the measurement of 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 , the relevant parameters and have been already measured. Those of the have not been measured yet.

Using the statistically abundant data set of ordinary leptonic decays, previous measurements [12, 13] have determined the Michel parameters , , , and 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 of and by fixing , , , , and to the SM values. This represents the first measurement of the and parameters of the lepton.

## 2 Method

### 2.1 Unbinned maximum likelihood method

The differential decay width for the radiative leptonic decay of with a definite spin direction is given by

 dΓ(τ−→ℓ−ντ¯νℓγ)dE∗ℓdΩ∗ℓdE∗γdΩ∗γ=(A−0+¯ηA−1)+(\boldmathB−0+ξκ\boldmathB−1)⋅\boldmathS∗−, (9)

where and  () are known functions of the kinematics of the decay productsThe detailed formulae of , in Eq. (9) and , in Eq. (11) are given in the appendix. with indices ( is the function identifier), stands for a set of for a particle of the type , and the asterisk means that the variable is defined in the rest frame. Equation (9) shows that 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 leptons in the production:

 dσ(e−e+→τ−(% \boldmathS∗−)τ+(\boldmathS∗+))dΩτ=α2βτ64E2τ(D0+∑i,jDij(\boldmathS∗−)i(\boldmathS∗+)j), (10)

where is the fine structure constant, and are the velocity and energy of the in the center-of-mass system (c.m.s.), respectively, is the spin-independent part of the cross section, and is a tensor describing the spin-spin correlation (see Eq. (4.11) in Ref. [8]). For the partner , its spin information is extracted using the two-body decay whose differential decay width is

 dΓ(τ+→π+π0¯ντ)dΩ∗ρdm2d˜Ωπ=A++ξρ\boldmathB+⋅\boldmathS∗+; (11)

and are known functions for the spin-independent and spin-dependent parts, respectively; the tilde indicates variables defined in the rest frame and is the invariant mass of the system, . As mentioned before, we use the SM value: . Thus, the total differential cross section of (or, briefly, ) can be written as:

 dσ(ℓ−γ,π+π0)dE∗ℓdΩ∗ℓdE∗γdΩ∗γdΩ∗ρdm2d˜ΩπdΩτ∝βτE2τ[D0(A−0+A−1⋅¯η)A++∑i,jDij(\boldmathB−0+% \boldmathB−1⋅ξκ)i(\boldmathB+)j]. (12)

To extract the visible differential cross section, we transform the differential variables into ones defined in the c.m.s. using the Jacobian :

 J=∣∣∣∂(E∗ℓ,Ω∗ℓ)∂(Pℓ,Ωℓ)∣∣∣∣∣∣∂(E∗γ,Ω∗γ)∂(Pγ,Ωγ)∣∣∣∣∣∣∂(Ω∗ρ,Ωτ)∂(Pρ,Ωρ,Φ)∣∣∣=(P2ℓEℓP∗ℓ)(EγE∗γ)(mτPρEρP∗ρPτ), (13)

where the parameter denotes the angle along the arc illustrated in Fig. 2.

The visible differential cross section is, therefore, obtained by integration over :

 dσ(l−γ,π+π0)dPℓdΩℓdPγdΩγdPρdΩρdm2d˜Ωπ =∫Φ2Φ1dΦdσ(ℓ−γ,π+π0)dΦdPℓdΩℓdPγdΩγdPρdΩρdm2d˜Ωπ (14) (15) ≡S(\boldmathx), (16)

where is proportional to the probability density function (PDF) of the signal and 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 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 events, and the beam energy spread.

Accounting for the event-selection criteria and the contamination from identified backgrounds, the total visible (properly normalized) PDF for the observable in each event is given by

 P(\boldmathx)=(1−∑iλi)S(% \boldmathx)ε(\boldmathx)∫d\boldmath% xS(\boldmathx)ε(\boldmathx)+∑iλiBi(\boldmathx)ε(\boldmathx)∫d\boldmathxBi(\boldmathx)ε(% \boldmathx), (17)

where is the distribution of the category of background, is the fraction of this background, and is the selection efficiency of the signal distribution. The categorization of is explained later (see the caption of Fig. 3). In general, is evaluated as an integral of the 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 is a linear combination of the Michel parameters , the normalization of signal PDF becomes

 ∫d\boldmathx (S0(\boldmathx)+S1(\boldmathx)¯η+S2(\boldmathx)ξκ)ε(% \boldmathx) (18) =N0∫d\boldmathx(S0(\boldmathx)ε(\boldmathx% )N0)⋅S0(\boldmathx)+S1(\boldmathx)¯η+S2(\boldmath% x)ξκS0(\boldmathx) (19) =N0¯εNsel∑i:selS0(\boldmathxi)+S1(% \boldmathxi)¯η+S2(\boldmathxi)ξκS0(\boldmathxi) (20) =N0¯ε[1+⟨S1S0⟩¯η+⟨S2S0⟩ξκ], (21)

where is a normalization coefficient of the SM part defined by , represents a set of variables for the selected event of events, is an average selection efficiency, and the brackets indicate an average with respect to the selected SM distribution. We refer to and () as absolute and relative normalizations, respectively.

From , the negative logarithmic likelihood function (NLL) is constructed and the best estimators of the Michel parameters, and , are obtained by minimizing the NLL. The efficiency 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.

### 2.2 KEKB collider

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

### 2.3 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 mesons and to identify muons (KLM). The detector is described in detail elsewhere [20].

## 3 Event selection

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

### 3.1 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  cm along the beam axis and  cm in the transverse plane. The two-track transverse momentum must exceed  GeV/ and that of one track must exceed  GeV/.

• Total energy deposition in the ECL in the laboratory frame must be lower than 9 GeV.

• The opening angle of the two tracks must satisfy in the laboratory frame.

• The number of photons whose energy exceeds MeV in the c.m.s. must be fewer than five.

• For the four-vector of missing momentum defined by , the missing mass defined by must lie in the range GeV GeV, where and are the four-momenta of the beam and all detected particles, respectively.

• The polar angle of missing-momentum must satisfy in the laboratory frame.

### 3.2 Final selection

The candidates of the outgoing particles in , i.e., the lepton, photon, and charged and neutral pions, are assigned in each of the preselected events.

• The electron selection is based on the likelihood ratio cut, , where and are the likelihood values of the track for the electron and non-electron hypotheses, respectively. These values are determined using specific ionization () in the CDC, the ratio of ECL energy and CDC momentum , 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 , where the likelihood values are determined by the measured versus expected range for the hypothesis, and transverse scattering of the track in the KLM [22]. The reductions of the signal efficiencies with lepton selections are approximately 10% and 30% for the electron and muon, respectively. The pion candidates are distinguished from kaons using , where the likelihood values are determined by the ACC response, the timing information from the TOF, and in the CDC. The reduction of the signal efficiency with pion selection is approximately 5%.

• The candidate is formed from two photon candidates, where each photon satisfies  MeV, with an invariant mass of  MeV  MeV. Figure 3 shows the distribution of the invariant mass of the candidates. The reduction of the signal efficiency by the mass selection is approximately 50%. In addition, when more than two candidates are found, the event is rejected.

• The candidate is formed from a and a candidate, with an invariant mass of  GeV. Figure 4 shows the distribution of the invariant mass of the candidates. The reduction of the signal efficiency is approximately 3%.

• The c.m.s. energy of signal photon candidate must exceed  MeV if within the ECL barrel () or  MeV if within the ECL endcaps ( or ). As shown in Fig. 5, this photon must lie in a cone determined by the lepton-candidate direction that is defined by cos and cos for the electron and muon mode, respectively, where ( or ) 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 , which is a daughter of the candidate, form an invariant mass of the ( MeV  MeV), the event is rejected. The additional selection reduces the signal efficiency by .

• 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 candidate: an event should satisfy , where is the spatial angle between the system and the candidate. This selection reduces the signal efficiency by .

• 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 or the signal photon (denoted as ) 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 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 pair production and the successive decay of the 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 and the angle between the lepton and photon, , for the selected events are shown in Figs. 6 and 7 for and candidates, respectively.

In the electron mode, the fraction of the signal decay in the selected sample is about due to the large external bremsstrahlung rate in the non-radiative leptonic decay events. In the muon mode, the fraction of the signal decay is about ; here, the main background arises from ordinary leptonic decay () events where either an additional photon is reconstructed from beam background in the ECL or a photon is emitted by the initial-state . The information is summarized in Table 2.

As mentioned before, in the integration over in Eq. (15), the generated differential variables are varied according to the resolution function . 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 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.

## 4 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 that is close to unity; its extraction is described below. With this correction, Eq. (17) is modified to

 PEX(\boldmathx)=(1−∑iλi)⋅S(\boldmathx)εMC(\boldmathx)R(% \boldmathx)∫d\boldmathxS(\boldmathx)εMC(\boldmathx)R(\boldmathx)+∑iλiBi(\boldmathx)εMC(% \boldmathx)R(\boldmathx)∫d\boldmathxBi(\boldmathx)εMC(\boldmathx)R(% \boldmathx). (22)

The presence of in the numerator does not affect the NLL minimization, but its presence in the denominator does.

We evaluate as the product of the measured corrections for the trigger, particle identification, track, , and reconstruction efficiencies:

 R(\boldmathx)=RtrgRℓ(Pℓ,cosθℓ)Rγ(Pγ,cosθγ)Rπ(Pπ,cosθπ)Rπ0(Pπ0,cosθπ0), (23) Rℓ(Pℓ,cosθℓ)=Rtrk(Pℓ,cosθℓ)RLID(Pℓ,cosθℓ), (24) Rπ(Pπ,cosθπ)=Rtrk(Pπ,cosθπ)RπID(Pπ,cosθπ). (25)

The lepton identification efficiency correction is estimated using two-photon processes ( or ). Since the momentum of the lepton from the two-photon process ranges from the detector threshold to approximately  GeV in the laboratory frame, the efficiency correction factor can be evaluated for our signal process as a function of and .

The pion PID correction factor is obtained by the measurement of decay (where the subscript indicates “slow”). The small momentum of the pion from allows us to select this process. As a result, assuming the mass of meson, we can reconstruct even if this is missed.

The track reconstruction efficiency correction is extracted from events. Here, we count the number of events (