- Email: [email protected]
Monte Carlo study of the measurement of the Michel parameters in the radiative decay of the τ at Belle
Nuclear Instruments and Methods in Physics Research A 824 (2016) 237–239
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Monte Carlo study of the measurement of the Michel parameters in the radiative decay of the τ at Belle N. Shimizu n, H. Aihara, D. Epifanov Department of Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
art ic l e i nf o
a b s t r a c t
Available online 29 December 2015
Michel parameters are kinematic values defined as bilinear combinations of the coupling terms in the general matrix element of the Fermi interaction where all possible Lorentz-invariant scalar, vector and tensor terms are included. The leptonic τ decays provide an excellent laboratory in which to measure the Michel parameters. Any deviation of these parameters from the Standard Model expectation would indicate New Physics. Two Michel parameters, η and ξκ , can only be measured in the radiative leptonic decay. These parameters have previously been measured in the radiative decay of the muon but as yet have not been measured in τ decays. We report the results of a feasibility study of the measurement of η and ξκ in radiative leptonic τ decays at Belle and Belle II. & 2015 Elsevier B.V. All rights reserved.
Keywords: Michel parameter Belle Tau Applications
1. Introduction The Standard Model (SM) is successful as a theory describing the properties of the elementary particles. However, it is known that the SM is incomplete and further precision tests are required to unveil the effects of New Physics (NP) beyond the SM. The measurement of the Michel parameters of the τ decay is an interesting test of the weak sector of the SM due to the larger mass of the τ compared with the electron and muon. The most general Lorentz-invariant derivative-free matrix element of the leptonic τ decay can be written as: i 4GF X N h N g ij ui ðl ÞΓ vn ðνl Þ um ðντ ÞΓ N uj ðτ Þ ; ð1Þ M ¼ pffiffiffi 2 N ¼ S;V;T i;j ¼ L;R
where GF is the Fermi constant, i and j are the chirality indices for the charged leptons, n and m are the chirality indices of the neu pffiffiffi S V T trinos, l ¼e or μ, Γ ¼ 1, Γ ¼ γ μ and Γ ¼ i γ μ γ ν γ ν γ μ =2 2 are, respectively, the scalar, vector and tensor Lorentz structures where γ μ are the Dirac matrices, and gijN are the corresponding dimen sionless couplings. In the SM, the τ -l ντ νl process only occurs through the decay of the τ into a left-handed neutrino via the emission of a W vector boson. Thus, the only nonzero coupling is g VLL ¼ 1. Experimentally only the squared matrix element is observable. That is, only bilinear combinations of the gijN can be measured. In the ordinary leptonic τ decay, the parameters ρ, η, δ and ξ are n
Corresponding author. E-mail address: [email protected] (N. Shimizu).
http://dx.doi.org/10.1016/j.nima.2015.12.037 0168-9002/& 2015 Elsevier B.V. All rights reserved.
measured [1]. In the radiative leptonic decay, whose Feynman diagrams are shown in Fig. 1, the spin state of the outgoing lepton is partially probed by the momentum of the photon. Therefore, three additional parameters ξκ , η [2] and η″ [3] can also be measured in this decay: 2 2 2 2 2 2 η ¼ gVRL þ gVLR þ 18 gSRL þ 2g TRL þ g SLR þ 2gTLR þ 2 gTRL þ gTLR ;
ð2Þ
2 2 2 2 2 2 ξκ ¼ g VRL g VLR þ 18 gSRL þ 2g TRL g SLR þ 2gTLR þ2 g TRL g TLR ;
ð3Þ
η″ ¼ R 24gVRL ðgSLRn þ6gTLRn Þ þ 24g VLR ðgSRLn þ 6gTRLn Þ 8ðgVRR gSLLn þ gVLL gSRRn Þ : ð4Þ In the SM, these values are zero. In muon decays, these Michel parameters have already been measured and are consistent with the SM expectation: η ðμÞ ¼ 0:014 7 0:09 [4], ξκ ðμÞ ¼ 0:0 70:1 and η″ðμÞ ¼ ð8 7 17Þ 10 3 [5]. The parameters have not yet been measured in the radiative τ decay. The Belle experiment on the asymmetric energy e þ e KEKB collider was one of two successful B-factory experiments [6]. Belle collected experimental data from 1999 to 2010. The center of mass pffiffi energy, s ¼ 10:58, was chosen to be the mass of Υ ð4SÞ. GeV. As the Υ ð4SÞ preferentially decays to BB, a huge number of B meson pair decays were recorded. For the purpose of this study, however, the B-factory can be also regarded as a τ-factory since a large sample of τ lepton decays (around 9 108 ττ events) was also collected. This dataset makes it possible to analyze radiative τ decays to a new level of precision.
238
N. Shimizu et al. / Nuclear Instruments and Methods in Physics Research A 824 (2016) 237–239
Fig. 1. Feynman diagrams for radiative leptonic decay of τ. The sum of the corresponding matrix elements contribute to the total decay rate.
The differential decay width for the radiative decay of a definite spin direction Sτ can be written as1:
τ with
dΓ ðτ -l γννÞ p ½A0 þ η A1 þ ½B0 þ ξκ B1 Sτ ;
ð5Þ
where Ai and Bi ði ¼ 0; 1Þ are known functions of the kinematics of the decay products. Eq. (5) shows that ξκ appears in the spindependent part of the decay width. However, these parameters can be measured by utilizing the well-known spin-spin correlation of the tau pair in the e e þ -τ þ τ reaction: dσ ðe e þ -τ ðS Þτ þ ðS þ Þ α2 β τ ¼ ðD0 þ Dij Si Sjþ Þ ði; j ¼ 1; 2; 3Þ; dΩ τ 64 E2τ
ð6Þ
where α is the fine structure constant, βτ and Eτ are the velocity and energy of the τ respectively, D0 is a form factor for the spinindependent part of the reaction and Dij is a tensor describing the spin-spin correlation [7]. Using the τ þ -ρ þ ντ -π þ π0 ντ decay, whose differential decay width is also written as dΓ ðτ þ -π þ π0 νÞ p A þ þ B þ Sτ þ (where again, A þ and B þ are form factors for the spin-independent and spin-dependent parts respectively), the total differential cross section of e þ e -τ τ þ -ðl ντ νl γÞðπ þ π0 ντ Þ (or shortly ðl γ; π þ π0 Þ) is formulated as:
dσ ðl γ; π þ π0 Þ ¼ D0 ½A0 þ A1 η A þ þ Dij ½B0 þ B1 ξκ i Bjþ ; dPS12
ð7Þ
where PS12 represents the twelve-dimension phase space of the full set of the kinematic variables x fpl ; Ωl ; pγ ; Ωγ ; pρ ; ~ g, where Ω ¼ f cos θ ; ϕ g (k ¼ l; γ ; ρ) and variables Ω ; m2 ; Ω ρ
ππ
π
k
k
k
with a tilde are defined in the π þ π0 rest frame.
2. Method Selection criteria were chosen to suppress background while maintaining high efficiency for signal events. A characteristic feature of the radiative decay is that the photon tends to be produced nearly collinear with the final state lepton direction. Distributions of the cosine of the angle between the lepton and photon ( cos θlγ ) for the selected events are shown in Fig. 2. In the electron mode, the fraction of the signal (eγ; ππ0 ) decay in the selected sample is only about 30%. This is due to the large external bremsstrahlung rate in the non-radiative leptonic τ decay events. In the case of the muon mode, however, the fraction of the (μγ; ππ0 ) signal decay is about 60%. Here, the main background comes from (μ; ππ0 ) events where either an additional photon is reconstructed from a fake cluster in the calorimeter or a high polar angle photon is emitted by the initial state e þ e (initial state radiation). 1 The Michel parameters that were measured precisely in the ordinary leptonic decay are set to the SM values [1].
Fig. 2. Distribution of the cosine of the angle between lepton and photon ( cos θlγ ). The empty histogram corresponds to the signal distribution while the shaded histograms represent background modes. (2a) τ-eννγ events; the largest background contribution (in red) is external bremsstrahlung. (2b) τ-μννγ events. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
For the observable x in each event, a total probability density function (PDF) is defined as: ! X X SðxÞεðxÞ B ðxÞεðxÞ þ ; ð8Þ λi R λi R i PðxÞ ¼ 1 ε ðxÞ dxSðxÞ dxB i ðxÞεðxÞ i i where SðxÞ is the signal PDF constructed from the total differential cross section given by Eq. (7), Bi ðxÞ is the PDF of the i-th background, λi is the fraction of i-th background and εðxÞ is the detection efficiency. From PðxÞ, a negative logarithmic likelihood
N. Shimizu et al. / Nuclear Instruments and Methods in Physics Research A 824 (2016) 237–239
function (NLL) is constructed and the best estimator of the Michel Parameters, η and ξκ , is obtained by minimizing the NLL. The efficiency, εðxÞ is a common multiplier in Eq. (8) and does not depend on the Michel parameters. Hence, it can be simplified in the NLL to a point where it is tractable to derive it from the Monte Carlo simulation. This is one of the essential features of the unbinned maximum likelihood method we use.
3. Result and conclusion A method to measure the Michel parameters η , ξκ and η″ in radiative leptonic decays of the τ was developed. A Monte Carlo data sample with statistics three times larger than that of the real Belle dataset was generated. The expected sensitivities at Belle were found to be σ η ¼ 0:64 and σ ξκ ¼ 0:12 (sensitivity on η″ is
239
under investigation). With the expected Belle pffiffiffiffiffiffi II dataset, these sensitivities will be improved by a factor of 50 [8].
References [1] [2] [3] [4] [5]
Chinese Physics C, 38, Muon decay parameters (2014) 090001. C. Fronsdal, H. Uberall, Physical Review 113 (1939) 654. A.B. Arbuzov, document in preparation. W. Eichenberger, et al., Nuclear Physics A 412 (1984) 523. ξκ and η″ of μ were calculated from PDG value based on W. Fetscher ETHZ-IMP PR-93-1 (1993). [6] A.J. Bevan, et al., European Physical Journal C 74 (2014) 3026. [7] Y.S. Tsai, Physical Review D 4 (1971) 2821. [8] T. Abe, et al., KEKREPORT 2010-1.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Monte Carlo study of the measurement of the Michel parameters in the radiative decay of the τ at Belle N. Shimizu n, H. Aihara, D. Epifanov Department of Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
art ic l e i nf o
a b s t r a c t
Available online 29 December 2015
Michel parameters are kinematic values defined as bilinear combinations of the coupling terms in the general matrix element of the Fermi interaction where all possible Lorentz-invariant scalar, vector and tensor terms are included. The leptonic τ decays provide an excellent laboratory in which to measure the Michel parameters. Any deviation of these parameters from the Standard Model expectation would indicate New Physics. Two Michel parameters, η and ξκ , can only be measured in the radiative leptonic decay. These parameters have previously been measured in the radiative decay of the muon but as yet have not been measured in τ decays. We report the results of a feasibility study of the measurement of η and ξκ in radiative leptonic τ decays at Belle and Belle II. & 2015 Elsevier B.V. All rights reserved.
Keywords: Michel parameter Belle Tau Applications
1. Introduction The Standard Model (SM) is successful as a theory describing the properties of the elementary particles. However, it is known that the SM is incomplete and further precision tests are required to unveil the effects of New Physics (NP) beyond the SM. The measurement of the Michel parameters of the τ decay is an interesting test of the weak sector of the SM due to the larger mass of the τ compared with the electron and muon. The most general Lorentz-invariant derivative-free matrix element of the leptonic τ decay can be written as: i 4GF X N h N g ij ui ðl ÞΓ vn ðνl Þ um ðντ ÞΓ N uj ðτ Þ ; ð1Þ M ¼ pffiffiffi 2 N ¼ S;V;T i;j ¼ L;R
where GF is the Fermi constant, i and j are the chirality indices for the charged leptons, n and m are the chirality indices of the neu pffiffiffi S V T trinos, l ¼e or μ, Γ ¼ 1, Γ ¼ γ μ and Γ ¼ i γ μ γ ν γ ν γ μ =2 2 are, respectively, the scalar, vector and tensor Lorentz structures where γ μ are the Dirac matrices, and gijN are the corresponding dimen sionless couplings. In the SM, the τ -l ντ νl process only occurs through the decay of the τ into a left-handed neutrino via the emission of a W vector boson. Thus, the only nonzero coupling is g VLL ¼ 1. Experimentally only the squared matrix element is observable. That is, only bilinear combinations of the gijN can be measured. In the ordinary leptonic τ decay, the parameters ρ, η, δ and ξ are n
Corresponding author. E-mail address: [email protected] (N. Shimizu).
http://dx.doi.org/10.1016/j.nima.2015.12.037 0168-9002/& 2015 Elsevier B.V. All rights reserved.
measured [1]. In the radiative leptonic decay, whose Feynman diagrams are shown in Fig. 1, the spin state of the outgoing lepton is partially probed by the momentum of the photon. Therefore, three additional parameters ξκ , η [2] and η″ [3] can also be measured in this decay: 2 2 2 2 2 2 η ¼ gVRL þ gVLR þ 18 gSRL þ 2g TRL þ g SLR þ 2gTLR þ 2 gTRL þ gTLR ;
ð2Þ
2 2 2 2 2 2 ξκ ¼ g VRL g VLR þ 18 gSRL þ 2g TRL g SLR þ 2gTLR þ2 g TRL g TLR ;
ð3Þ
η″ ¼ R 24gVRL ðgSLRn þ6gTLRn Þ þ 24g VLR ðgSRLn þ 6gTRLn Þ 8ðgVRR gSLLn þ gVLL gSRRn Þ : ð4Þ In the SM, these values are zero. In muon decays, these Michel parameters have already been measured and are consistent with the SM expectation: η ðμÞ ¼ 0:014 7 0:09 [4], ξκ ðμÞ ¼ 0:0 70:1 and η″ðμÞ ¼ ð8 7 17Þ 10 3 [5]. The parameters have not yet been measured in the radiative τ decay. The Belle experiment on the asymmetric energy e þ e KEKB collider was one of two successful B-factory experiments [6]. Belle collected experimental data from 1999 to 2010. The center of mass pffiffi energy, s ¼ 10:58, was chosen to be the mass of Υ ð4SÞ. GeV. As the Υ ð4SÞ preferentially decays to BB, a huge number of B meson pair decays were recorded. For the purpose of this study, however, the B-factory can be also regarded as a τ-factory since a large sample of τ lepton decays (around 9 108 ττ events) was also collected. This dataset makes it possible to analyze radiative τ decays to a new level of precision.
238
N. Shimizu et al. / Nuclear Instruments and Methods in Physics Research A 824 (2016) 237–239
Fig. 1. Feynman diagrams for radiative leptonic decay of τ. The sum of the corresponding matrix elements contribute to the total decay rate.
The differential decay width for the radiative decay of a definite spin direction Sτ can be written as1:
τ with
dΓ ðτ -l γννÞ p ½A0 þ η A1 þ ½B0 þ ξκ B1 Sτ ;
ð5Þ
where Ai and Bi ði ¼ 0; 1Þ are known functions of the kinematics of the decay products. Eq. (5) shows that ξκ appears in the spindependent part of the decay width. However, these parameters can be measured by utilizing the well-known spin-spin correlation of the tau pair in the e e þ -τ þ τ reaction: dσ ðe e þ -τ ðS Þτ þ ðS þ Þ α2 β τ ¼ ðD0 þ Dij Si Sjþ Þ ði; j ¼ 1; 2; 3Þ; dΩ τ 64 E2τ
ð6Þ
where α is the fine structure constant, βτ and Eτ are the velocity and energy of the τ respectively, D0 is a form factor for the spinindependent part of the reaction and Dij is a tensor describing the spin-spin correlation [7]. Using the τ þ -ρ þ ντ -π þ π0 ντ decay, whose differential decay width is also written as dΓ ðτ þ -π þ π0 νÞ p A þ þ B þ Sτ þ (where again, A þ and B þ are form factors for the spin-independent and spin-dependent parts respectively), the total differential cross section of e þ e -τ τ þ -ðl ντ νl γÞðπ þ π0 ντ Þ (or shortly ðl γ; π þ π0 Þ) is formulated as:
dσ ðl γ; π þ π0 Þ ¼ D0 ½A0 þ A1 η A þ þ Dij ½B0 þ B1 ξκ i Bjþ ; dPS12
ð7Þ
where PS12 represents the twelve-dimension phase space of the full set of the kinematic variables x fpl ; Ωl ; pγ ; Ωγ ; pρ ; ~ g, where Ω ¼ f cos θ ; ϕ g (k ¼ l; γ ; ρ) and variables Ω ; m2 ; Ω ρ
ππ
π
k
k
k
with a tilde are defined in the π þ π0 rest frame.
2. Method Selection criteria were chosen to suppress background while maintaining high efficiency for signal events. A characteristic feature of the radiative decay is that the photon tends to be produced nearly collinear with the final state lepton direction. Distributions of the cosine of the angle between the lepton and photon ( cos θlγ ) for the selected events are shown in Fig. 2. In the electron mode, the fraction of the signal (eγ; ππ0 ) decay in the selected sample is only about 30%. This is due to the large external bremsstrahlung rate in the non-radiative leptonic τ decay events. In the case of the muon mode, however, the fraction of the (μγ; ππ0 ) signal decay is about 60%. Here, the main background comes from (μ; ππ0 ) events where either an additional photon is reconstructed from a fake cluster in the calorimeter or a high polar angle photon is emitted by the initial state e þ e (initial state radiation). 1 The Michel parameters that were measured precisely in the ordinary leptonic decay are set to the SM values [1].
Fig. 2. Distribution of the cosine of the angle between lepton and photon ( cos θlγ ). The empty histogram corresponds to the signal distribution while the shaded histograms represent background modes. (2a) τ-eννγ events; the largest background contribution (in red) is external bremsstrahlung. (2b) τ-μννγ events. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
For the observable x in each event, a total probability density function (PDF) is defined as: ! X X SðxÞεðxÞ B ðxÞεðxÞ þ ; ð8Þ λi R λi R i PðxÞ ¼ 1 ε ðxÞ dxSðxÞ dxB i ðxÞεðxÞ i i where SðxÞ is the signal PDF constructed from the total differential cross section given by Eq. (7), Bi ðxÞ is the PDF of the i-th background, λi is the fraction of i-th background and εðxÞ is the detection efficiency. From PðxÞ, a negative logarithmic likelihood
N. Shimizu et al. / Nuclear Instruments and Methods in Physics Research A 824 (2016) 237–239
function (NLL) is constructed and the best estimator of the Michel Parameters, η and ξκ , is obtained by minimizing the NLL. The efficiency, εðxÞ is a common multiplier in Eq. (8) and does not depend on the Michel parameters. Hence, it can be simplified in the NLL to a point where it is tractable to derive it from the Monte Carlo simulation. This is one of the essential features of the unbinned maximum likelihood method we use.
3. Result and conclusion A method to measure the Michel parameters η , ξκ and η″ in radiative leptonic decays of the τ was developed. A Monte Carlo data sample with statistics three times larger than that of the real Belle dataset was generated. The expected sensitivities at Belle were found to be σ η ¼ 0:64 and σ ξκ ¼ 0:12 (sensitivity on η″ is
239
under investigation). With the expected Belle pffiffiffiffiffiffi II dataset, these sensitivities will be improved by a factor of 50 [8].
References [1] [2] [3] [4] [5]
Chinese Physics C, 38, Muon decay parameters (2014) 090001. C. Fronsdal, H. Uberall, Physical Review 113 (1939) 654. A.B. Arbuzov, document in preparation. W. Eichenberger, et al., Nuclear Physics A 412 (1984) 523. ξκ and η″ of μ were calculated from PDG value based on W. Fetscher ETHZ-IMP PR-93-1 (1993). [6] A.J. Bevan, et al., European Physical Journal C 74 (2014) 3026. [7] Y.S. Tsai, Physical Review D 4 (1971) 2821. [8] T. Abe, et al., KEKREPORT 2010-1.