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 and in the radiative leptonic decay
Belle preprint 2017-20, KEK preprint 2017-29

\nameN. Shimizu75    \nameH. Aihara75    \nameD. Epifanov3,59    \nameI. Adachi15,11    \nameS. Al Said69,34    \nameD. M. Asner61    \nameV. Aulchenko3,59    \nameT. Aushev49    \nameR. Ayad69    \nameV. Babu70    \nameI. Badhrees69,33    \nameA. M. Bakich68    \nameV. Bansal61    \nameE. Barberio47    \nameV. Bhardwaj17    \nameB. Bhuyan19    \nameJ. Biswal29    \nameA. Bobrov3,59    \nameA. Bozek56    \nameM. Bračko45,29    \nameT. E. Browder14    \nameD. Červenkov4    \nameM.-C. Chang9    \nameP. Chang55    \nameV. Chekelian46    \nameA. Chen53    \nameB. G. Cheon13    \nameK. Chilikin40,48    \nameK. Cho35    \nameS.-K. Choi12    \nameY. Choi67    \nameD. Cinabro80    \nameT. Czank73    \nameN. Dash18    \nameS. Di Carlo80    \nameZ. Doležal4    \nameD. Dutta70    \nameS. Eidelman3,59    \nameJ. E. Fast61    \nameT. Ferber7    \nameB. G. Fulsom61    \nameR. Garg62    \nameV. Gaur79    \nameN. Gabyshev3,59    \nameA. Garmash3,59    \nameM. Gelb31    \nameP. Goldenzweig31    \nameD. Greenwald71    \nameE. Guido27    \nameJ. Haba15,11    \nameK. Hayasaka58    \nameH. Hayashii52    \nameM. T. Hedges14    \nameS. Hirose50    \nameW.-S. Hou55    \nameT. Iijima51,50    \nameK. Inami50    \nameG. Inguglia7    \nameA. Ishikawa73    \nameR. Itoh15,11    \nameM. Iwasaki60    \nameI. Jaegle8    \nameH. B. Jeon38    \nameS. Jia2    \nameY. Jin75    \nameK. K. Joo5    \nameT. Julius47    \nameK. H. Kang38    \nameG. Karyan7    \nameT. Kawasaki58    \nameC. Kiesling46    \nameD. Y. Kim65    \nameJ. B. Kim36    \nameS. H. Kim13    \nameY. J. Kim35    \nameK. Kinoshita6    \nameP. Kodyš4    \nameS. Korpar45,29    \nameD. Kotchetkov14    \nameP. Križan41,29    \nameR. Kroeger25    \nameP. Krokovny3,59    \nameR. Kulasiri32    \nameA. Kuzmin3,59    \nameY.-J. Kwon82    \nameJ. S. Lange10    \nameI. S. Lee13    \nameL. K. Li22    \nameY. Li79    \nameL. Li Gioi46    \nameJ. Libby20    \nameD. Liventsev79,15    \nameM. Masuda74    \nameM. Merola26    \nameK. Miyabayashi52    \nameH. Miyata58    \nameG. B. Mohanty70    \nameH. K. Moon36    \nameT. Mori50    \nameR. Mussa27    \nameE. Nakano60    \nameM. Nakao15,11    \nameT. Nanut29    \nameK. J. Nath19    \nameZ. Natkaniec56    \nameM. Nayak80,15    \nameM. Niiyama37    \nameN. K. Nisar63    \nameS. Nishida15,11    \nameS. Ogawa72    \nameS. Okuno30    \nameH. Ono57,58    \nameG. Pakhlova40,49    \nameB. Pal6    \nameC. W. Park67    \nameH. Park38    \nameS. Paul71    \nameT. K. Pedlar43    \nameR. Pestotnik29    \nameL. E. Piilonen79    \nameV. Popov49    \nameM. Ritter42    \nameA. Rostomyan7    \nameY. Sakai15,11    \nameM. Salehi44,42    \nameS. Sandilya6    \nameY. Sato50    \nameV. Savinov63    \nameO. Schneider39    \nameG. Schnell1,16    \nameC. Schwanda23    \nameY. Seino58    \nameK. Senyo81    \nameM. E. Sevior47    \nameV. Shebalin3,59    \nameT.-A. Shibata76    \nameJ.-G. Shiu55    \nameB. Shwartz3,59    \nameA. Sokolov24    \nameE. Solovieva40,49    \nameM. Starič29    \nameJ. F. Strube61    \nameK. Sumisawa15,11    \nameT. Sumiyoshi77    \nameU. Tamponi27,78    \nameK. Tanida28    \nameF. Tenchini47    \nameK. Trabelsi15,11    \nameM. Uchida76    \nameT. Uglov40,49    \nameY. Unno13    \nameS. Uno15,11    \nameY. Usov3,59    \nameC. Van Hulse1    \nameG. Varner14    \nameV. Vorobyev3,59    \nameA. Vossen21    \nameC. H. Wang54    \nameM.-Z. Wang55    \nameP. Wang22    \nameM. Watanabe58    \nameE. Widmann66    \nameE. Won36    \nameY. Yamashita57    \nameH. Ye7    \nameC. Z. Yuan22    \nameZ. P. Zhang64    \nameV. Zhilich3,59    \nameV. Zhukova40,48    \nameV. Zhulanov3,59    \nameA. Zupanc41,29    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 University of the Basque Country UPV/EHU, 48080 Bilbao Beihang University, Beijing 100191 Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Faculty of Mathematics and Physics, Charles University, 121 16 Prague Chonnam National University, Kwangju 660-701 University of Cincinnati, Cincinnati, Ohio 45221 Deutsches Elektronen–Synchrotron, 22607 Hamburg University of Florida, Gainesville, Florida 32611 Department of Physics, Fu Jen Catholic University, Taipei 24205 Justus-Liebig-Universität Gießen, 35392 Gießen SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193 Gyeongsang National University, Chinju 660-701 Hanyang University, Seoul 133-791 University of Hawaii, Honolulu, Hawaii 96822 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao Indian Institute of Science Education and Research Mohali, SAS Nagar, 140306 Indian Institute of Technology Bhubaneswar, Satya Nagar 751007 Indian Institute of Technology Guwahati, Assam 781039 Indian Institute of Technology Madras, Chennai 600036 Indiana University, Bloomington, Indiana 47408 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 Institute of High Energy Physics, Vienna 1050 Institute for High Energy Physics, Protvino 142281 University of Mississippi, University, Mississippi 38677 INFN - Sezione di Napoli, 80126 Napoli INFN - Sezione di Torino, 10125 Torino Advanced Science Research Center, Japan Atomic Energy Agency, Naka 319-1195 J. Stefan Institute, 1000 Ljubljana Kanagawa University, Yokohama 221-8686 Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe Kennesaw State University, Kennesaw, Georgia 30144 King Abdulaziz City for Science and Technology, Riyadh 11442 Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah 21589 Korea Institute of Science and Technology Information, Daejeon 305-806 Korea University, Seoul 136-713 Kyoto University, Kyoto 606-8502 Kyungpook National University, Daegu 702-701 École Polytechnique Fédérale de Lausanne (EPFL), Lausanne 1015 P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991 Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana Ludwig Maximilians University, 80539 Munich Luther College, Decorah, Iowa 52101 University of Malaya, 50603 Kuala Lumpur University of Maribor, 2000 Maribor Max-Planck-Institut für Physik, 80805 München School of Physics, University of Melbourne, Victoria 3010 Moscow Physical Engineering Institute, Moscow 115409 Moscow Institute of Physics and Technology, Moscow Region 141700 Graduate School of Science, Nagoya University, Nagoya 464-8602 Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602 Nara Women’s University, Nara 630-8506 National Central University, Chung-li 32054 National United University, Miao Li 36003 Department of Physics, National Taiwan University, Taipei 10617 H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342 Nippon Dental University, Niigata 951-8580 Niigata University, Niigata 950-2181 Novosibirsk State University, Novosibirsk 630090 Osaka City University, Osaka 558-8585 Pacific Northwest National Laboratory, Richland, Washington 99352 Panjab University, Chandigarh 160014 University of Pittsburgh, Pittsburgh, Pennsylvania 15260 University of Science and Technology of China, Hefei 230026 Soongsil University, Seoul 156-743 Stefan Meyer Institute for Subatomic Physics, Vienna 1090 Sungkyunkwan University, Suwon 440-746 School of Physics, University of Sydney, New South Wales 2006 Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451 Tata Institute of Fundamental Research, Mumbai 400005 Department of Physics, Technische Universität München, 85748 Garching Toho University, Funabashi 274-8510 Department of Physics, Tohoku University, Sendai 980-8578 Earthquake Research Institute, University of Tokyo, Tokyo 113-0032 Department of Physics, University of Tokyo, Tokyo 113-0033 Tokyo Institute of Technology, Tokyo 152-8550 Tokyo Metropolitan University, Tokyo 192-0397 University of Torino, 10124 Torino Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 Wayne State University, Detroit, Michigan 48202 Yamagata University, Yamagata 990-8560 Yonsei University, Seoul 120-749
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]:

(2)
(3)
(4)
(5)
{fmffile}

externalph1 {fmfchar*}(120,60) \fmflefttm3,antinu3 \fmflabeltm3 \fmflabelantinu3 \fmfrightg3,f3,fb3 \fmflabelfb3 \fmflabelf3 \fmflabelg3 \fmffermiontm3,v23,Wi3,antinu3 \fmfdashes,tension=1,label=Wi3,Wf3 \fmfphoton,tension=0v23,g3 \fmffermion,label= fb3,Wf3 \fmffermionWf3,f3 \fmfdotWi3,Wf3,v23 {fmffile}externalph2 {fmfchar*}(120,60) \fmflefttm4,antinu4 \fmflabeltm4 \fmflabelantinu4 \fmfrightg4,f4,fb4 \fmflabelfb4 \fmflabelf4 \fmflabelg4 \fmffermiontm4,Wi4 \fmffermion,tension=3Wi4,antinu4 \fmfdashes,tension=2.3,label=Wi4,Wf4 \fmffermion,label= fb4,Wf4 \fmffermionWf4,v24,f4 \fmfphoton,tension=0v24,g4 \fmfdotWi4,Wf4,v24 {fmffile}externalph3 {fmfchar*}(120,60) \fmflefttm4,antinu4 \fmflabeltm4 \fmflabelantinu4 \fmfrightg4,f4,fb4 \fmflabelfb4 \fmflabelf4 \fmflabelg4 \fmffermiontm4,Wi4 \fmffermion,tension=3Wi4,antinu4 \fmfdashes,tension=2.3,label=Wi4,vtx \fmfdashes,tension=2.3,label=,label.dist=-4.5mmvtx,Wf4 \fmffermion,label= fb4,Wf4 \fmffermionWf4,f4 \fmfphoton,tension=0vtx,g4 \fmfdotWi4,Wf4,vtx

Figure 1: Three Feynman diagrams of the tau radiative leptonic decay

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:

(6)
(7)
(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.

Name SM Spin Experimental Comments and Ref.
value correlation result [11]
0 no (ALEPH) [12]
no (CLEO) [13]
yes (CLEO) [13]
1 yes measured in leptonic decays (CLEO) [13]
1 yes measured in hadronic decays (CLEO) [13]
0 no not measured   from radiative decay (RD)
0 yes not measured from RD
0 no not measured        from RD, suppressed by
1 yes -
1 no -

 Experimental results represent average values obtained by PDG [11].

Table 1: Michel 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

(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:

(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

(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:

(12)

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

(13)

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

Figure 2: Kinematics of decay. Cones A and B are the surfaces that satisfy the c.m.s. conditions and . The direction of is constrained to lie on an arc defined by the intersection of cone B and the interior or exterior sector constrained by the reversal (i.e., mirror) cone A. The arc (shown in red) is parametrized by the angle .

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

(14)
(15)
(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

(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

(18)
(19)
(20)
(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.

(a)
(b)
Figure 3: Distribution of . Dots with uncertainties are experimental data and histograms are MC distributions. The MC histograms are scaled to the experimental one based on the yields just after the preselection. The red arrows indicate the selection window  MeV  MeV.   (a) candidates: the open histogram corresponds to the signal, the yellow () and green () histograms represent ordinary leptonic decay plus extra bremsstrahlung due to the detector material and radiative leptonic decay plus bremsstrahlung, respectively, and the blue () histogram represents other processes such as radiative Bhabha, two-photon, and productions.    (b) candidates: the open histogram corresponds to signal, the magenta () histogram represents ordinary leptonic decay plus beam background, the aqua () histogram represents ordinary leptonic decay plus ISR/FSR processes, the purple () histogram represents three-pion events where is misreconstructed as a tagging candidate, the green () histogram represents - background where is selected due to misidentification of pion as muon, the red () histogram represents 3- events where is selected by misidentification similarly to the - case, and the orange () histogram represents other processes (as in the electron mode).    In Eq. (17) and the categories mentioned in this caption, and for the electron and muon modes, respectively.
(a)
(b)
Figure 4: Distribution of : (a) candidates and (b) candidates. Dots with uncertainties are experimental data and histograms are MC distributions. The color of each histogram is explained in Fig. 3. The red arrows indicate the selection window  GeV  MeV.
(a)
(b)
Figure 5: Distribution of : (a) candidates and (b) candidates. Dots with uncertainties are experimental data and histograms are MC distributions. The color of each histogram is explained in Fig. 3. The red arrows indicate the selection condition cos and cos 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].

(a)
(b)
Figure 6: Final distribution of (a) photon energy and (b) for the decay candidates. Dots with uncertainties are experimental data and histograms are MC distributions. The color of each histogram is explained in Fig. 3.
(a)
(b)
Figure 7: Final distribution of (a) photon energy and (b)  for the decay candidates. Dots with uncertainties are experimental data and histograms are MC distributions. The color of each histogram is explained in Fig. 3.

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.

Item
391954 384880 35198 35973
(%)
Purity (%)

 The efficiency is determined based on the photon energy threshold of  MeV in the rest frame.

Table 2: Summary of event selection

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

(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:

(23)
(24)
(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 (