59. τ -Lepton Decay Parameters

159. τ -Lepton Decay Parameters
Revised February 2024 by A. Stahl (RWTH Aachen U.).
The purpose of the measurements of the decay parameters (also known as Michel parameters)of the τ is to determine the structure (spin and chirality) of the current mediating its decays.

59.1 Leptonic Decays:
The Michel parameters are extracted from the energy spectrum of the charged daughter lepton` = e, µ in the decays τ → `ν` ντ . Ignoring radiative corrections, neglecting terms of order (m` /mτ )2√ 2and (mτ / s) , and setting the neutrino masses to zero, the spectrum in the laboratory frame readsdΓG2 m5= τ ` 3τdx
192 πm`
f2 (x) − Pτ [ξg1 (x) + ξδg2 (x)]f0 (x) + ρf1 (x) + ηmτ(59.1)

withf0 (x) = 2 − 6 x2 + 4 x3(59.2)

3
f1 (x) = − 94 + 4x2 − 32
9 x

g1 (x) = − 32 + 4x − 6x2 + 83 x3

f2 (x) = 12 (1 − x)2264 3
g2 (x) = 94 − 163 x + 12x − 9 x .

√The quantity x is the fractional energy of the daughter lepton `, i.e., x = E` /E`,max ≈ E` /( s/2)
and Pτ is the polarization of the tau leptons. The integrated decay width is given byΓ =G2τ ` m5τ
192 π 3

1 + 4η

m`mτ



.

(59.3)The situation is similar to muon decays µ → eνe νµ . The generalized matrix element with theγ and their relations to the Michel parameters ρ, η, ξ, and δ have been described incouplings gεµ
the “Note on Muon Decay Parameters”. The Standard Model expectations are 3/4, 0, 1, and 3/4,respectively. For more details, see [1].
More information can be extracted from the decays, if the polarization of the daughter lepton
is measured. Belle presented a first result (see [2]). A new parameter ξ 0 is introduced. Its relation
to the couplings is given by (see [3]):ξ0 = 1 − 2X 1
ω=L,R

4

SgRω

2

V+ gRω

2

T+ 3 gRL

2.

(59.4)

T by about a factor of two.
The new measurement improves the limit on gRL
Additional Michel parameters can be defined in radiative decays τ → `ν` ντ γ, see [4]. A firstmeasurement is presented in [5]. The experimental precision does not provide a significant impact
on the knowledge about the couplings, yet.

59.2 Hadronic Decays:
In the case of hadronic decays τ → hντ , with h = π, ρ, or a1 , the ansatz is restricted to purely
vectorial currents. The matrix element isGτ h X√gλ h Ψ ω (ντ ) | γ µ | Ψλ (τ ) i Jµh
2 λ=R,LS. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:20am(59.5)

259. τ -Lepton Decay Parameterswith the hadronic current Jµh . The neutrino chirality ω is uniquely determined from λ. The
spectrum depends only on a single parameter ξhdn Γ= f (~x) + ξh Pτ g (~x) ,dx1 dx2 . . . dxn(59.6)

with f and g being channel-dependent functions of the n observables ~x = (x1 , x2 , . . . , xn ) (see [6]).
The parameter ξh is related to the couplings throughξh = |gL |2 − |gR |2 .

(59.7)

ξh is the negative of the chirality of the τ neutrino in these decays. In the Standard Model, ξh = 1.
Also included in the Data Listings for ξh are measurements of the neutrino helicity which coincidewith ξh , if the neutrino is massless (ASNER 00 [7], ACKERSTAFF 97R [8], AKERS 95P [9],ALBRECHT 93C [10], and ALBRECHT 90I [11]).

59.3 Combination of Measurements:
The individual measurements are combined, taking into account the correlations between theparameters. In a first fit, universality between the two leptonic decays, and between all hadronicdecays, is assumed. A second fit is made without these assumptions. The results of the two fits
are provided as OUR FIT in the Data Listings below in the tables whose title includes “(e ormu)” or “(all hadronic modes)”, and “(e)”, “(mu)” etc., respectively. The measurements show
good agreement with the Standard Model. The χ2 values with respect to the Standard model
predictions are 24.1 for 41 degrees of freedom and 26.8 for 56 degrees of freedom, respectively. The
correlations are reduced through this combination to less than 20%, with the exception of ρ and η
which are correlated by +23%, for the fit with universality and by +70% for τ → µνµ ντ .

59.4 Model-independent Analysis:
κ without further model
From the Michel parameters, limits can be derived on the couplings gελVassumptions. In the Standard model gLL = 1 (leptonic decays), and gL = 1 (hadronic decays) andall other couplings vanish. First, the partial decay widths have to be compared to the Standard
Model predictions to derive limits on the normalization of the couplings Ax = G2τ x /G2F with Fermi’s
constant GF :Ae = 1.0022 ± 0.0041 ,Aµ = 0.983 ± 0.017 ,Aπ = 0.9957 ± 0.0049 .

(59.8)

Then limits on the couplings (95% CL) can be extracted (see [12] and [13]). Without the assumptionof universality, the limits given in Table 59.1 are derived.

59.5 Model-dependent Interpretation:
More stringent limits can be derived assuming specific models. For example, in the frameworkof a two Higgs doublet model, the measurements correspond to a limit of mH ± > 2.0 GeV × tan βon the mass of the charged Higgs boson, or a limit of 500 GeV on the mass of the second W bosonin left-right symmetric models for arbitrary mixing (both 95% CL). See [13] and [14].

31st May, 2024

359. τ -Lepton Decay Parameters
γ . 95% confidence level experimental limits. The
Table 59.1: Coupling constants gεµlimits include the quoted values of Ae , Aµ , and Aπ and assume Aρ = Aa1 = 1.

τ → eνe ντ
S | < 0.70|gRR

V | < 0.17|gRR

T |≡0|gRR

S | < 0.99|gLR

V | < 0.13|gLR

T | < 0.082|gLR

S | < 2.01|gRL

V | < 0.52|gRL

T | < 0.51|gRL

S | < 2.01|gLL

V | < 1.004|gLLT |≡0|gLL

S | < 0.72|gRR

V | < 0.18|gRR

T |≡0|gRR

S | < 0.95|gLR

V | < 0.12|gLR

T | < 0.079|gLR

S | < 2.01|gRL

V | < 0.52|gRL

T | < 0.27|gRL

S | < 2.01|gLL

V | < 1.006|gLLT |≡0|gLL

τ → µνµ νττ → πντ|gRV | < 0.15|gLV | > 0.984

τ → ρντ|gRV | < 0.10|gLV | > 0.995

τ → a1 ντ|gRV | < 0.16|gLV | > 0.987

References[1] F. Scheck, Phys. Rept. 44, 187 (1978); W. Fetscher and H.J. Gerber in Precision Tests ofthe Standard Model, edited by P. Langacker, World Scientific, 1993; A. Stahl, Physics with τ
Leptons, Springer Tracts in Modern Physics, 160, 1, (2000).
[2] D. Bodrov et al. (Belle), Phys. Rev. D 108, 1, 012003 (2023), [arXiv:2303.10574].
[3] D. Bodrov and P. Pakhlov, JHEP 10, 035 (2022), [arXiv:2203.12743].
[4] A. Arbuzov and T. Kopylova, JHEP 109 (2016).
[5] N. Shimizu et al. (Belle), Prog. Theo. Exp. Phys. 2, 023C01 (2018).
[6] M. Davier et al., Phys. Lett. B 306, 411 (1993).
[7] D. Asner et al. (CLEO), Phys. Rev. D 61, 012002 (2000), [hep-ex/9902022].
[8] K. Ackerstaff et al. (OPAL), Z. Phys. C 75, 593 (1997).
[9] R. Akers et al. (OPAL), Z. Phys. C 67, 45 (1995).
[10] H. Albrecht et al. (ARGUS), Z. Phys. C 58, 61 (1993).
[11] H. Albrecht et al. (ARGUS), Phys. Lett. B 250, 164 (1990).
[12] K. Ackerstaff et al. (OPAL), Eur. Phys. J. C 8, 3 (1999), [hep-ex/9808016].
[13] A. Stahl, Nucl. Phys. B Proc. Suppl. 76, 173 (1999).
31st May, 2024

459. τ -Lepton Decay Parameters[14] M.-T. Dova, J. Swain and L. Taylor, Phys. Rev. D 58, 015005 (1998), [hep-ph/9712384];
T. Hebbeker and W. Lohmann, Z. Phys. C 74, 399 (1997); A. Pich and J. P. Silva, Phys. Rev.
D 52, 4006 (1995), [hep-ph/9505327].

31st May, 2024