aainstitutetext: Higher School of Economics (National Research University),
Moscow, Russiabbinstitutetext: P.N. Lebedev Physical Institute of the RAS,
Moscow, Russia

# A new method for measurement of the Michel parameters that describe the
daughter muon polarization in the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay

D. Bodrov b,a P. Pakhlov<EMAIL_ADDRESS>
###### Abstract

This paper provides a detailed description of the method for the first direct
measurement of all Michel parameters in the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay related to the
polarization of the daughter muon. An application of the suggested method in
the existing and future experiments at $e^{+}e^{-}$ colliders is considered.
We have performed a feasibility study for the future Super Charm-Tau Factory
and Belle II experiments. For the first one, the sensitivity to the Michel
parameters $\xi^{\prime}$, $\xi^{\prime\prime}$, $\eta^{\prime\prime}$,
$\alpha^{\prime}/A$, and $\beta^{\prime}/A$ is estimated. For the latter, only
one Michel parameter, $\xi^{\prime}$, for which the sensitivity is maximum, is
considered.

##  1 Introduction

The Standard Model (SM) has proven its validity in describing particle physics
in multiple experimental tests, at least within the energy ranges and
measurement accuracy available to experiment. However, due to the
incompleteness and internal contradictions of the SM, there is no doubt that
sooner or later, an experiment will show disagreement with the SM predictions;
thus, the New Physics (NP) will reveal itself.

SM postulates the maximum parity violation in weak decays via charged current
since only left-handed fermions participate in vector interaction. This
postulate is formulated to ensure consistency of the theory to experiments,
mainly the precise measurements in the muon decay. Deviations from this
behavior in other processes can be caused by changes in the couplings of
$W$-boson with fermions or by interactions mediated by new gauge or charged
Higgs bosons, thus pointing to NP Bryman:2021teu . Therefore, the conjecture
of the $V-A$ Lorentz structure of the charged currents needs to be verified
preferably in all possible processes caused by them, since in many SM
extensions, an admixture of right-handed currents is predicted nonuniversal
for various processes Herczeg:1985cx .

In $\tau$-decays, due to the much larger mass of the $\tau$ compared to the
muon, the relative contribution of New Physics processes with an “incorrect”
Lorentz structure can be enhanced Krawczyk:2004na ; Chun:2016hzs . The pure
leptonic $\tau$-decays provide a clean laboratory to test the structure of the
weak currents since there are no QCD-associated uncertainties in their
calculations.

The interaction of charged currents in $\tau$-decays in a general form that
allows for a non-SM contribution but does not violate Lorentz invariance can
be expressed in terms of so-called Michel parameters Michel:1949qe . At
present, four Michel parameters have been measured with high accuracy in
$\tau$-decays Zyla:2020zbs , and the obtained values are in agreement with the
SM. They completely describe the differential decay width, integrated over the
neutrino momenta and summed over the daughter lepton spin. Measurement of the
remaining Michel parameters requires knowledge of the daughter lepton
polarization, and it has not been done yet. The exception is two parameters
obtained in radiative leptonic $\tau$-decays by the Belle collaboration
Shimizu:2017dpq . However, this measurement still suffers from too large
uncertainties to constrain somehow the NP parameters.

Thus, for the complimentary test of the theory, it is necessary to find a
method that gives access to the daughter lepton polarization. The development
of Super flavor factories such as Belle II Kou:2018nap  and Super Charm-Tau
Factory Bondar:2013cja ; Luo:2018njj , thanks to their extremely high
luminosity and correspondingly enormous data statistics, opens up a
possibility to apply new methods based on the study of rare processes that
were not previously considered due to their practical inapplicability with the
modest statistics of the past experiments. In this work, we discuss such a
method based on a rare process of decay-in-flight of the daughter muon from
the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$111Charge conjugation is
implied unless otherwise is indicated. decay in the tracking detector, which
opens opportunities to measure the missing Michel parameters.

The idea to use muon decay to access its polarization in the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay was suggested in
Fetscher:1990su , where it was proposed to use stopped muons. Due to the
$P$-violation in the $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay, the
electron emission angle correlates with the muon spin, providing information
on its direction. Such a method with stopped muons suffers from a practical
disadvantage: it requires a specific sophisticated detector that should stop
energetic ($\sim 1\,{\mathrm{GeV}}$) $\mu^{+}$ while not depolarizing them
($\mu^{-}$ will be depolarized anyway Mann:1961zz ) and detect $\sim
50\,{\mathrm{MeV}}$ daughter electrons. Installation of such a detector is
impracticable in modern experiments.

Recently, it was proposed to use muon decay-in-flight in the tracking system
of the detector to measure the muon polarization from $\tau$ to obtain the
Michel parameter $\xi^{\prime}$ in the future experiment at the Super Charm-
Tau Factory Bodrov:2021vkn ; Bodrov:2021hfe . For such a measurement in the
experiment, it is necessary to reconstruct tracks of both the muon from the
$\tau$-decay and the secondary electron from the muon decay. With a typical
size of tracking detectors in modern $e^{+}e^{-}$ experiments $\sim 1$ m, the
probability of muon to decay-in-flight in the tracker is extremely small
($\sim 10^{-4}-10^{-3}$). Nevertheless, the specific signature of such events
(track kink) makes it possible to select signal events without a large
background. The main background, hadrons ($\pi^{\pm},~{}K^{\pm}$) decays-in-
flight, are mainly two-body, leading to a narrow line in the spectrum of the
daughter particle in the parent’s rest frame. Provided that the momentum
resolution in the tracker is good enough, such physical background can be
effectively vetoed. In this paper, we demonstrate that the large statistics of
the experiments at the SCTF and Belle II will allow to achieve a good accuracy
of measurement for all Michel parameters in the near future.

The proposed method is charge-universal compared to those that utilize stopped
muons, since in the latter, $\mu^{-}$ are depolarized in the matter after
stopping. Usage of decay-in-flight muons allows measuring the Michel
parameters with the same accuracy in both
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ and
$\tau^{+}\to\mu^{+}\nu_{\mu}\bar{\nu}_{\tau}$ decays. Thus, one can separately
analyze both charges to test CPT invariance or merge two samples under the CPT
invariance assumption to increase precision.

##  2 Method

Let us recall the general formalism for describing the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay of our interest. The most
general form of the Lorentz invariant, local, derivative-free, lepton-number
conservative Hamiltonian of the four-fermion interaction Michel:1949qe  leads
to the following matrix element of the
$\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$ decay ($\ell=e$ or $\mu$),
written in the form of helicity projections Scheck:1984md ; Mursula:1984zb ;
Fetscher:1986uj :

$\displaystyle
M=\dfrac{4G_{F}}{\sqrt{2}}\sum_{\begin{subarray}{c}\rho=S,~{}V,~{}T\\\
\varepsilon,~{}\mu=R,~{}L\end{subarray}}g^{\rho}_{\varepsilon\mu}\left\langle\bar{\ell}_{\varepsilon}\left|\Gamma^{\rho}\right|(\nu_{\ell})_{\alpha}\right\rangle\left\langle(\bar{\nu}_{\tau})_{\beta}\left|\Gamma_{\rho}\right|\tau_{\mu}\right\rangle\,,$
$\displaystyle\Gamma^{S}=1,\quad\Gamma^{V}=\gamma^{\mu},\quad\Gamma^{T}=\dfrac{1}{\sqrt{2}}\sigma^{\mu\nu}=\dfrac{i}{2\sqrt{2}}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})\,.$
(1)

Here $\rho=S,\,V,\,T$ and means scalar, vector, and tensor interactions;
$\varepsilon,\,\mu=L,\,R$ means left- and right-handed leptons. Each set of
indices $\rho$, $\varepsilon$, and $\mu$ uniquely determines the values of
$\alpha$ and $\beta$. Ten complex constants $g^{\rho}_{\varepsilon\mu}$
($g^{T}_{RR}=g^{T}_{LL}\equiv 0$) completely describe the weak interaction of
charged currents. For the case of the Standard Model, the only nonzero
constant is $g^{V}_{LL}=1$.

It is convenient to express the experimental observables in terms of the
Michel parameters, which are bilinear combinations of the coupling constants
$g^{\rho}_{\varepsilon\mu}$. In general, the experimentally observed
(integrated over neutrino momenta) differential decay width for
$\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$ ($\ell=e$ or $\mu$) can be
obtained from the matrix element (2):

$\displaystyle\dfrac{d^{2}\Gamma}{dx\,d\\!\cos{\theta}}=\dfrac{m_{\tau}}{4\,\pi^{3}}W_{\ell\tau}^{4}\,G^{2}_{F}\sqrt{x^{2}-x^{2}_{0}}\,\big{(}F_{IS}(x)\pm
F_{AS}(x)P_{\tau}\cos{\theta}+F_{T_{1}}(x)P_{\tau}\sin{\theta}\zeta_{1}$
$\displaystyle+F_{T_{2}}(x)P_{\tau}\sin{\theta}\zeta_{2}+(\pm
F_{IP}(x)+F_{AP}(x)P_{\tau}\cos{\theta})\zeta_{3}\big{)}\,.$ (2)

Here $x=E_{\ell}/W_{\ell\tau}$ and $x_{0}=m_{l}/W_{\ell\tau}$ are the reduced
energy and mass of the daughter lepton (the ratio of its energy and mass to
its maximum energy $W_{\ell\tau}=(m^{2}_{\ell}+m^{2}_{\tau})/2m_{\tau}$,
respectively), $\theta$ is the angle between the direction of the daughter
lepton momentum and the $\tau$ lepton polarization vector in its rest frame,
$\vec{\zeta}=(\zeta_{1},~{}\zeta_{2},~{}\zeta_{3})$ is the direction along
which the polarization of the daughter lepton is measured. We use the
conventional coordinate system $(x_{1},~{}x_{2},~{}x_{3})$: the
$\vec{x}_{3}$-axis is directed along with the daughter lepton momentum in the
$\tau$ lepton rest system, the $\vec{x}_{2}$-axis is directed perpendicular to
the $\vec{x}_{3}$-axis and the $\tau$ lepton polarization $\vec{P}_{\tau}$,
and $\vec{x}_{1}=\vec{x}_{2}\times\vec{x}_{3}$. The plus and minus signs in
(2) correspond to the $\tau^{+}$\- and $\tau^{-}$-decays, respectively. The
functions $F_{IS}$, $F_{AS}$, $F_{IP}$, $F_{AP}$, $F_{T_{1}}$, and $F_{T_{2}}$
are expressed in terms of the Michel parameters and depend only on $x$:

$\displaystyle F_{IS}(x)$ $\displaystyle=$ $\displaystyle
x(1-x)+\dfrac{2}{9}\rho(4x^{2}-3x-x_{0}^{2})+\eta x_{0}(1-x)\,,$
$\displaystyle F_{AS}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{3}\xi\sqrt{x^{2}-x_{0}^{2}}\left[1-x+\dfrac{2}{3}\delta\left(4x-3-\dfrac{x_{0}^{2}}{2}\right)\right],$
$\displaystyle F_{IP}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{54}\sqrt{x^{2}-x_{0}^{2}}\left[-9\xi^{\prime}\left(2x-3+\dfrac{x_{0}^{2}}{2}\right)+4\xi\left(\delta-\dfrac{3}{4}\right)\left(4x-3-\dfrac{x_{0}^{2}}{2}\right)\right],$
(3) $\displaystyle F_{AP}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{6}\left[\xi^{\prime\prime}\left(2x^{2}-x-x_{0}^{2}\right)+4\left(\rho-\dfrac{3}{4}\right)\left(4x^{2}-3x-x_{0}^{2}\right)+2\eta^{\prime\prime}x_{0}(1-x)\right],$
$\displaystyle F_{T_{1}}(x)$ $\displaystyle=$
$\displaystyle-\dfrac{1}{12}\left[2\left(\xi^{\prime\prime}+12\left(\rho-\dfrac{3}{4}\right)\right)(1-x)x_{0}+3\eta(x^{2}-x_{0}^{2})+\eta^{\prime\prime}(3x^{2}-4x+x_{0}^{2})\right],$
$\displaystyle F_{T_{2}}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{3}\sqrt{x^{2}-x_{0}^{2}}\left(3\dfrac{\alpha^{\prime}}{A}(1-x)+\dfrac{\beta^{\prime}}{A}(2-x_{0}^{2})\right).$

Here we have expanded the functions up to the quadratic term $x_{0}^{2}$ while
holding $\sqrt{x^{2}-x_{0}^{2}}$. Formulas (2) and (2) are well known and can
be found in Zyla:2020zbs . To calculate the partial decay width of
$\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$, one should integrate (2) over
$x$ and $\cos{\theta}$ and sum over the spin of the daughter lepton.

To simplify subsequent calculations, we rewrite formula (2):

$\displaystyle\dfrac{d^{2}\Gamma}{dx\,d\\!\cos{\theta}}=\dfrac{m_{\tau}}{4\,\pi^{3}}W_{\ell\tau}^{4}\,G^{2}_{F}\sqrt{x^{2}-x^{2}_{0}}\,\big{[}G_{0}+(\vec{G}\cdot\vec{\zeta})\big{]}\,,$
(4)

where $G_{0}$ is a scalar and $\vec{G}\equiv(G_{1},G_{2},G_{3})$ is a formal
vector formed by functions (2):

$\displaystyle G_{0}$ $\displaystyle=$ $\displaystyle F_{IS}(x)\pm
F_{AS}(x)P_{\tau}\cos{\theta}\,,$ $\displaystyle\vec{G}_{\phantom{0}}$
$\displaystyle=$
$\displaystyle\left(F_{T_{1}}(x)P_{\tau}\sin{\theta},~{}F_{T_{2}}(x)P_{\tau}\sin{\theta},~{}\pm
F_{IP}(x)+F_{AP}(x)P_{\tau}\cos{\theta}\right).$ (5)

In the muon decay, $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$, the Michel
parameters were measured to be consistent with the exact $V-A$ structure with
high precision Zyla:2020zbs ; therefore, we use the SM muon differential decay
width:

$\displaystyle\dfrac{d^{2}\Gamma}{dy\,d\Omega_{e}}=\dfrac{G_{F}^{2}m_{\mu}^{5}}{384\,\pi^{4}}y^{2}\left[(3-2y)\pm(2y-1)(\vec{n}_{e}\cdot\vec{\zeta})\right],$
(6)

where $y=2E_{e}/m_{\mu}$ is the ratio of the electron energy in the muon rest
frame to its maximum value, $\vec{\zeta}$ is the muon polarization direction,
$\vec{n}_{e}$ is the electron momentum direction in the muon rest frame,
$d\Omega_{e}$ is the electron solid angle element. Here plus (minus) sign
corresponds to the $\mu^{+}$ ($\mu^{-}$) decay. Since we do not register
neutrinos in either the $\tau$ lepton decay or the muon decay, the width of
the cascade decay $\tau^{-}\to(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu})\bar{\nu}_{\mu}\nu_{\tau}$ can be obtained by
simple muon spin convolution of expressions (4) and (6). However, the proposed
method uses muon decay in a tracking detector, implying muon motion before its
decay in a magnetic field, which not only rotates the muon momentum but also
drives its spin to precess. This should be taken into account when folding the
expressions (4) and (6). The spin evolution equations were obtained in
Bargmann:1959gz . Usually, a constant in time uniform magnetic field is used,
which simplifies the spin evolution to the ordinary precession around the axis
of the magnetic field. Neglecting the anomalous magnetic moment, the muon spin
rotates by the same angle as the momentum: $\phi(t)=\mp 2\mu_{\mu}Ht$, where
$t$ is the muon decay time in its rest frame, $H$ is the absolute value of the
magnetic field, and $\mu_{\mu}=e/2m_{\mu}$ is the muon magnetic moment. The
upper sign corresponds to a positive charge, and the lower sign corresponds to
a negative charge. For the convolution of (4) and (6) over the muon spin, one
should replace $\vec{\zeta}$ by $\vec{\zeta}_{0}$ in (4) and by
$\vec{\zeta}(t)=\boldsymbol{R}(\phi(t))\vec{\zeta_{0}}$ in (6), where the
matrix $\boldsymbol{R}(\phi)$ has the form:

$\displaystyle\boldsymbol{R}(\phi)\\!=\\!\begin{pmatrix}c+(1-c)h_{1}^{2}&-h_{3}s+(1-c)h_{1}h_{2}&h_{2}s+(1-c)h_{1}h_{3}\\\
h_{3}s+(1-c)h_{1}h_{2}&c+(1-c)h_{2}^{2}&-h_{1}s+(1-c)h_{2}h_{3}\\\
-h_{2}s+(1-c)h_{1}h_{3}&h_{1}s+(1-c)h_{2}h_{3}&c+(1-c)h_{3}^{2}\end{pmatrix}.$
(7)

Here $\vec{h}=(h_{1},~{}h_{2},~{}h_{3})$ is the magnetic field direction;
$c\equiv\cos{\phi}$ and $s\equiv\sin{\phi}$. The differential width of the
cascade decay $\tau^{-}\to(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu})\bar{\nu}_{\mu}\nu_{\tau}$ is calculated as the
spin convolution of the decay width (4) and decay width (6):

$\displaystyle\dfrac{d^{5}\Gamma}{dx\,d\\!\cos{\theta}\,dy\,d\Omega_{e}\,dt}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\,\dfrac{3}{\pi}y^{2}\,\sqrt{x^{2}-x_{0}^{2}}$
$\displaystyle\left[(3-2y)G_{0}\pm(2y-1)\vec{n}_{e}\boldsymbol{R}(\phi)\vec{G}\right]\dfrac{1}{\tau_{\mu}}\exp{\left(-\dfrac{t}{\tau_{\mu}}\right)}\,,$
(8)

where $\vec{n}_{e}$ is the electron momentum direction in the muon rest frame,
written in the coordinate system $(x_{1},~{}x_{2},~{}x_{3})$,
$\Gamma_{\tau\to\mu\nu\nu}$ is the partial width of the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay, $\mathcal{B}(\mu\to
e\nu\nu)$ is the branching ratio of the $\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}$ decay, $\tau_{\mu}$ is the muon lifetime. The
plus sign and the minus sign correspond to the $\tau^{+}$\- and
$\tau^{-}$-decays, respectively.

Expression (2) is applicable for $e^{+}e^{-}$ annihilation at any $\sqrt{s}$
with the proviso that we know the momentum of the $\tau$ lepton and its
polarization. Hereinafter, we assume that the momentum of the $\tau$ lepton is
known. In real experiment, due to the presence of undetected neutrinos, it is
necessary to take into account the uncertainty in the $\tau$ momentum by
integration over the region of possible $\tau$ directions.

Thus, we have obtained the differential width of the cascade
$\tau^{-}\to(\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu})\bar{\nu}_{\mu}\nu_{\tau}$
decay, taking into account the muon rotation in the detector magnetic field.
Further analysis of the obtained expression is carried out in the context of
its application in the conditions of a particular experiment. It is worth
noting that due to the rarity of muon decays in the detector, the measurement
accuracy of the Michel parameters $\xi^{\prime}$, $\xi^{\prime\prime}$,
$\eta^{\prime\prime}$, $\alpha^{\prime}/A$, and $\beta^{\prime}/A$ will be
significantly lower than those of $\rho$, $\eta$, $\xi$, and $\xi\delta$
within the same experiment; therefore, to simplify all further calculations,
the latter can be equated to their SM values.

##  3 Application in the experiments at $e^{+}e^{-}$ colliders

For the experimental implementation of the proposed method, three conditions
are necessary: a large high-purity $\tau$ sample, determination of the $\tau$
momentum, and, finally, knowledge of the $\tau$ polarization before its decay.
The latter can be abandoned for the measurement of $\xi^{\prime}$, which is
not related to the mother lepton polarization. All the conditions are fully
met at the future Super Charm-Tau Factory (SCTF) with polarized beam and
partially met at Belle II experiment, where tagging with the second $\tau$ in
the event helps to constrain the directions of the signal $\tau$ momentum and
polarization.

###  3.1 SCTF with polarized beam

The SCTF experiment with a polarized beam provides an ideal environment for
the simplest and the most accurate measurement of the Michel parameters, which
describe daughter muon polarization. In this experiment, $\tau$ leptons will
be produced almost at rest and with a known polarization. Moreover, the
$e^{+}e^{-}\to\tau^{+}\tau^{-}$ events can be effectively selected with a high
purity of the $\tau^{+}\tau^{-}$ sample.

The expression for the differential width of the cascade decay (2) can be
simplified by a transition to a new coordinate system
$(x_{1}^{\prime},~{}x_{2}^{\prime},~{}x_{3}^{\prime})$ defined at the muon
decay vertex. This coordinate system is obtained by rotation of the initial
coordinate system $(x_{1},~{}x_{2},~{}x_{3})$ by the angle $\phi$ around the
magnetic field vector, as shown in figure 1.

Figure 1: $(x_{1},~{}x_{2},~{}x_{3})$ and
$(x_{1}^{\prime},~{}x_{2}^{\prime},~{}x_{3}^{\prime})$ coordinates
illustration for the SCTF case of the cascade decay.

This transformation leads to $\vec{n}_{e}\to\vec{n}_{e}^{\prime}$ and
$y^{2}\,dy\,d\Omega_{e}\,dt\to|J|y^{\prime}{}^{2}\,dy^{\prime}\,d\Omega_{e}^{\prime}\,dt^{\prime}$,
where $t^{\prime}=t$ and $y^{\prime}=y$. The latter is due to the fact that
our transformation is a rotation. For the same reason, the Jacobian
determinant, $|J|$, is equal to unity.222The simplest way to prove it is to
move from spherical coordinates $(y,\,\Omega)$ to Cartesian coordinates
$(n_{1},\,n_{2},\,n_{3})$:
$\displaystyle|J|\\!=\\!\left|\dfrac{\partial(n_{1}^{\prime},\,n_{2}^{\prime},\,n_{3}^{\prime},\,t^{\prime})}{\partial(n_{1},\,n_{2},\,n_{3},\,t)}\right|=\begin{vmatrix}\boldsymbol{R}^{T}_{ij}&\dfrac{d\boldsymbol{R}^{T}_{ij}}{dt}n_{j}\\\
0&1\end{vmatrix}=\left|\boldsymbol{R}^{T}_{ij}\right|=1\,.$ After the
transformation, the differential decay width takes the following form:

$\displaystyle\dfrac{d^{5}\Gamma}{dx\,d\\!\cos{\theta}\,dy\,d\Omega_{e}^{\prime}\,dt}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\dfrac{3}{\pi}y^{2}\sqrt{x^{2}-x_{0}^{2}}$
$\displaystyle\left[(3-2y)G_{0}\pm(2y-1)(\vec{n}_{e}^{\prime}\cdot\vec{G})\right]\dfrac{1}{\tau_{\mu}}\exp{\left(-\dfrac{t}{\tau_{\mu}}\right)}\,.$
(9)

After we present the electron momentum in the new coordinate system for each
decay event, the time dependence disappears from angular distributions and
remains only in the exponential. Finally, integrating over $t$ and
substituting
$\vec{n}_{e}^{\prime}=(\sin{\theta_{e}}\cos{\psi_{e}},\,\sin{\theta_{e}}\sin{\psi_{e}},\,\cos{\theta_{e}})$,
we can derive the explicit dependence of the differential decay width on
$\cos{\theta_{e}}$ and $\psi_{e}$:

$\displaystyle\dfrac{d^{5}\Gamma}{dx\,d\\!\cos{\theta}\,dy\,d\\!\cos{\theta_{e}}\,d\psi_{e}}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\dfrac{3}{\pi}y^{2}\sqrt{x^{2}-x_{0}^{2}}\left[(3-2y)G_{0}\right.$
$\displaystyle\left.\pm(2y-1)(G_{1}\sin{\theta_{e}}\cos{\psi_{e}}+G_{2}\sin{\theta_{e}}\sin{\psi_{e}}+G_{3}\cos{\theta_{e}})\right].$
(10)

The muon longitudinal polarization, determined by the parameters
$\xi^{\prime}$ and $\xi^{\prime\prime}$, is frozen (up to $g_{\mu}-2$
corrections) into the $x_{3}^{\prime}$-axis, thus, directed along with the
muon momentum in the $\tau$ rest frame (which is well approximated by the
laboratory frame at SCTF). Finally, after the integration of (3.1) over
$\psi_{e}$, only the dependence on the muon longitudinal polarization remains:

$\displaystyle\dfrac{d^{4}\Gamma}{dx\,d\\!\cos{\theta}\,dy\,d\\!\cos{\theta_{e}}}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{6\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\,y^{2}\sqrt{x^{2}-x_{0}^{2}}$
$\displaystyle\left[(3-2y)G_{0}\pm(2y-1)G_{3}\cos{\theta_{e}}\right].$ (11)

Using the obtained formulas, we estimate the SCTF sensitivity to the Michel
parameters $\xi^{\prime}$, $\xi^{\prime\prime}$, $\eta^{\prime\prime}$,
$\alpha^{\prime}/A$, and $\beta^{\prime}/A$ measurements using the SCTF
parameters from Bondar:2013cja . In this project, it is planned to use a
highly polarized electron beam (the degree of polarization of the beam at the
interaction point is $\xi_{\text{beam}}=0.8$) and an unpolarized positron
beam. For the SCTF energies, as shown in Tsai:1994rc , the degree of $\tau$
lepton polarization and its direction with good accuracy are independent of
the $\tau$ lepton production angle. We thus fix the degree of the $\tau$
polarization to be $P_{\tau}\approx\xi_{\text{beam}}=0.8$ and its direction to
be along the electron beam. To simplify the estimates, we assume that full
SCTF statistics will be collected around the $\tau^{+}\tau^{-}$-pair
production threshold; thus, $\tau$ leptons are produced almost at rest.

Since there is no full detector simulation and reconstruction program, we use
a toy Monte Carlo (MC) simulation with an overall efficiency to reconstruct
the decay of interest defined as
$\eta=\eta_{\text{tag}}\,\omega_{\text{dec}}\,\eta_{\text{kink}}\eta_{\text{sel}}$.
Here $\eta_{\text{tag}}\approx 30\%$ is the selection efficiency of the
$\tau^{+}\tau^{-}$ events based on the estimation for the BES III experiment
Asner:2008nq . We consider only the simplest cases for the reconstruction
algorithm to find the kink: the muon decays in the drift chamber (DC) on the
first turn of its track and in $\geq\\!10\,\text{cm}$ from the outer walls.
These requirements ensure the reconstruction of the kink by the track
reconstruction algorithm with a typical efficiency $\eta_{\text{kink}}\approx
90\%$ for both mother and daughter tracks. We calculate the probability for
muon to decay with these requirements to be $\omega_{\text{dec}}\approx
3.2\cdot 10^{-4}$.

Our requirement for the decay vertex in the DC provides at least 10 hits for
the secondary electron; thus, its momentum and angle resolution will be quite
good. It was estimated in Bodrov:2021hfe  given the spatial hit resolution in
the DC of about $125\,\mu\text{m}$ Bondar:2013cja . Here we have confirmed
that such a resolution does not affect the accuracy of the Michel parameters
measurement.

The last factor that reduces the analyzed sample, $\eta_{\text{sel}}$, is
related to the need to suppress backgrounds that initially exceed the signal.
The expected background contamination from other processes rather than
$\tau^{+}\tau^{-}$, mainly $q\bar{q}$, was estimated in Asner:2008nq  to be
$\sim 6\%$, and we ignore it. The main background sources come from
$\tau^{+}\tau^{-}$-pairs events with kink candidates that imitate the signal
process. They are charged pion and kaon decays, and the particles elastic
scattering. The specific kinematics allow effectively discriminate background
processes from signal one: pion and kaon decay mainly to two monochromatic
particles producing a narrow line in the mother particle rest frame, while
elastic scattering conserves the momentum magnitude of the particle. The
efficiency of the discrimination is determined by the momentum resolution,
which is good enough; thus, we estimate that the backgrounds can be suppressed
to a negligible level with $\eta_{\text{sel}}\approx 80\%$ efficiency for the
signal Bodrov:2021hfe .

Taking into account the branching fraction of the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay, we obtain the number of
signal kink events for the $\tau$ decay sample, expected with the full SCTF
statistics, to be $N\approx 5\cdot 10^{5}$. Using the generated toy MC sample
with the statistics corresponding to the full data set at SCTF, we estimate
the sensitivity of a global fit to the SCTF data that includes all Michel
parameters. We separately analyze $\tau^{+}$ and $\tau^{-}$ samples and, for
each of them, estimate the precision of the Michel parameters measurement. We
perform a 5D unbinned likelihood fit of the simulated data on the $x$,
$\cos{\theta}$, $y$, $\cos{\theta_{e}}$, and $\psi_{e}$ variables with the fit
function given by (3.1). The free parameters of the fit are only those Michel
parameters that determine the muon polarization ($\xi^{\prime}$,
$\xi^{\prime\prime}$, $\eta^{\prime\prime}$, $\alpha^{\prime}/A$, and
$\beta^{\prime}/A$), while parameters $\rho$, $\eta$, $\xi$, and $\xi\delta$
are fixed to their SM values since they can be measured separately with
unprecedented statistical accuracy at SCTF. We obtained the following
statistical errors from the fit: $\sigma_{\xi^{\prime}}\approx 6\cdot
10^{-3}$, $\sigma_{\xi^{\prime\prime}}\approx 3\cdot 10^{-2}$,
$\sigma_{\eta^{\prime\prime}}\approx 2\cdot 10^{-2}$,
$\sigma_{\alpha^{\prime}/A}\approx 14\cdot 10^{-3}$, and
$\sigma_{\beta^{\prime}/A}\approx 7\cdot 10^{-3}$. The expected SCTF accuracy
is comparable to those achieved in the muon decay.

In case of a discrepancy with the SM prediction is observed, it would be
possible to find the Lorentz structure of new interactions. Otherwise, global
fit allows setting upper limits on the magnitude of the coupling constants
$g^{\rho}_{\varepsilon\mu}$. They are presented in table 1 together with the
current loose limits Zyla:2020zbs  for comparison.

We estimate the potential limits using the accuracy for the Michel parameters
$\xi^{\prime}$, $\xi^{\prime\prime}$, $\eta^{\prime\prime}$,
$\alpha^{\prime}/A$, and $\beta^{\prime}/A$ obtained in this work. For the
Michel parameters $\rho$, $\eta$, $\xi$, and $\xi\delta$, it was shown in work
Epifanov:2020elk  that the statistical precision will be at the $10^{-4}$
level. Actually, the Belle experiment has already achieved $10^{-3}$
statistical uncertainty. However, the systematic uncertainties dominate this
measurement at the percent level Epifanov:2017kly . That is why we set
$10^{-3}$ uncertainty for $\rho$, $\eta$, $\xi$, and $\xi\delta$, assuming
that systematic will be controlled at that level.

To set limits on the coupling constants magnitude
$|g^{\rho}_{\varepsilon\mu}|$, we use the alternative parametrization from
Fetscher:1986uj : $Q_{RR}$, $Q_{LR}$, $Q_{RL}$, $Q_{LL}$, $B_{LR}$, and
$B_{RL}$. The Michel parameters $\rho$, $\xi$, $\xi\delta$, $\xi^{\prime}$,
and $\xi^{\prime\prime}$ expressions in terms of these variables can be found
in Gagliardi:2005fg . For the sake of simplicity, we do not follow the full
procedure of finding boundaries and do not use information about $\eta$,
$\eta^{\prime\prime}$, $\alpha^{\prime}/A$, and $\beta^{\prime}/A$. Instead,
we refit our simulated data sample with new variables with constraints on
$\rho$, $\xi$, and $\xi\delta$ to extract the uncertainties, which are then
utilized to set limits. The values of variables are set to their SM
expectations. The result for $|g^{\rho}_{\varepsilon\mu}|$ constraints,
obtained for the $\tau^{-}$-decay sample, is shown in table 1. We do not show
results for $|g^{S}_{LL}|$ and $|g^{V}_{LL}|$ because they cannot be obtained
separately in $\tau$-decays measurements. The same result is for the
$\tau^{+}$-decay sample. We analyze the samples for both charges separately as
it will allow testing $CPT$-invariance, while for the muon decay, there is no
such an opportunity because not all Michel parameters were measured for both
signs.

Table 1: Here coupling constants $g^{\rho}_{\varepsilon\mu}$ are presented with 95% confidence level experimental limits and 90% confidence level limits from our estimations. Our results are shown only for the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ sample, while for the current experimental limits, results for both signs were averaged. | ref. Zyla:2020zbs  | SCTF |  | ref. Zyla:2020zbs  | SCTF |  | ref. Zyla:2020zbs  | SCTF
---|---|---|---|---|---|---|---|---
$|g^{S}_{RR}|$ | $<0.72$ | $<0.18$ | $|g^{V}_{RR}|$ | $<0.18$ | $<0.09$ | $|g^{T}_{RR}|$ | $\equiv 0$ | $\equiv 0$
$|g^{S}_{LR}|$ | $<0.95$ | $<0.18$ | $|g^{V}_{LR}|$ | $<0.12$ | $<0.05$ | $|g^{T}_{LR}|$ | $<0.079$ | $<0.03$
$|g^{S}_{RL}|$ | $<2.01$ | $<0.19$ | $|g^{V}_{RL}|$ | $<0.52$ | $<0.05$ | $|g^{T}_{RL}|$ | $<0.51$ | $<0.03$
$|g^{S}_{LL}|$ | $<2.01$ |  | $|g^{V}_{LL}|$ | $<1.005$ |  | $|g^{T}_{LL}|$ | $\equiv 0$ | $\equiv 0$

We demonstrate the ability of SCTF using the proposed method to discover New
Physics if it biases the Michel parameters. To illustrate this, we generated
Monte Carlo samples with a small contribution of the NP with a different
Lorentz structure to the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay.
In the first sample, we slightly shifted from the SM value the Michel
parameter $\xi^{\prime}$ ($\xi^{\prime\prime}=\xi^{\prime}=0.96$) that
determines the muon longitudinal polarization ($P_{L}$) and the probability of
any (right- or left-handed) $\tau$ lepton to decay to the right-handed muon:
$Q^{\mu}_{R}=(1-\xi^{\prime})/2$. Such a case can be implemented in the NP
scenario with an additional admixture of the scalar interaction that does not
change other Michel parameters. The points with error bars in figure 2
represent a projection onto $\cos{\theta_{e}}$ for the $y>0.75$
interval333Integration of expression (3.1) over $y$ in the whole interval
partially cancels the angular dependence of the muon polarization term; thus,
we choose the interval where the effect is maximum. for the simulated events
with $\xi^{\prime}=0.96$ corresponding to the full SCTF data sample. The solid
line corresponds to the SM value of $\xi^{\prime}=1$, and the dashed line
shows the fit function. Although the shown projection has reduced sensitivity
compared to 5D-fit, the difference between the data and the SM expectation is
clearly visible. The statistical significance of the deviation from the SM in
the fit is greater than $5\sigma$, pointing out that the NP leading to
$\xi^{\prime}<0.96$ can be revealed at SCTF.

Figure 2: The projection on $\cos{\theta_{e}}$ for $y>0.75$. Points with error
bars are the simulated data with $\xi^{\prime}=0.96$ at SCTF, the solid line
is the SM expectation, the dashed line is the fit function. The bottom plot
shows the subtraction of the solid line from the data points.

The second sample is dedicated to another NP scenario that affects the muon
transverse polarization, $P_{T_{2}}$. It is determined by two Michel
parameters, $\alpha^{\prime}/A$ and $\beta^{\prime}/A$. Non-identical equality
of the transverse polarization, $P_{T_{2}}$, to 0 (and thus
$\alpha^{\prime}/A\neq 0$ and $\beta^{\prime}/A\neq 0$) means the
$T$-violating NP contribution. In the simulation, we set $\alpha^{\prime}/A=0$
and $\beta^{\prime}/A=0.03$. The projection onto
$\sin{\theta_{e}}\sin{\psi_{e}}$ for $y>0.75$ for this sample is shown in
figure 3.

Figure 3: The projection on $\sin{\theta_{e}}\sin{\psi_{e}}$ for $y>0.75$.
Points with error bars are the simulated data at SCTF with
$\alpha^{\prime}/A=0$ and $\beta^{\prime}/A=0.03$, the solid line shows SM
expectation, the dashed line is a fit function. The bottom plot shows the
subtraction of the solid line from the data points.

Points with error bars show the simulated data corresponding to the full SCTF
data set. The solid line corresponds to the SM case ($\alpha^{\prime}/A=0$ and
$\beta^{\prime}/A=0$), and the dashed line shows the fit function. The fit
yields $5\sigma$ significance of the NP contribution; thus, SCTF is able to
observe the $T$-violating NP effect if it results in $\beta^{\prime}/A\gtrsim
0.03$.

###  3.2 Belle II experiment

From expression (2), we can conclude that to measure the Michel parameter
$\xi^{\prime}$, it is not necessary to know the $\tau$ lepton polarization,
while the parameters $\xi^{\prime\prime}$, $\eta^{\prime\prime}$,
$\alpha^{\prime}/A$, and $\beta^{\prime}/A$ cannot be measured without this
information. At the Belle II experiment, $\tau$ leptons are produced with zero
average polarization. However, it is possible to extract information about the
polarization of the signal $\tau$ using the decay of the second (tagging)
$\tau$ lepton in the event. This procedure complicates the calculations of the
theoretical function, and experimental analysis becomes difficult due to the
higher dimensional fit function. Moreover, the accuracy of such measurement
will be lower compared to the SCTF experiment since not all decays of the
tagging $\tau$ are useful, and besides, the information on the $\tau$ spin is
indirect (statistical). We thus consider the application of the method to
measure only the Michel parameter $\xi^{\prime}$ and show that its measurement
has comparable to the SCTF accuracy.

For $\xi^{\prime}$ measurement at Belle II case, we first average differential
decay width over $\tau$ polarization. For this, one cannot use expression (2)
because it is written in the coordinate system given by the $\tau$
polarization vector; instead, one should use the conventional for the studies
of the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ process with unpolarized beams
coordinate system. This system, $(\bar{x}_{1},~{}\bar{x}_{2},~{}\bar{x}_{3})$,
is defined as follows: the $\bar{x}_{3}$-axis is directed along with the
$\tau^{-}$ momentum in the CMS, the $\bar{x}_{2}$-axis is perpendicular to the
$\bar{x}_{3}$-axis and the direction of the electron beam in the CMS, and the
$\bar{x}_{1}$-axis forms with the $\bar{x}_{2}$\- and $\bar{x}_{3}$-axis
right-handed orientation. Figure 4 illustrates the bar coordinate system with
a schematic view of the cascade decay we study.

Figure 4: $(\bar{x}_{1},~{}\bar{x}_{2},~{}\bar{x}_{3})$ coordinates
illustration for the Belle II case of the cascade decay.

The differential decay width of the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay written in the coordinate
system $(\bar{x}_{1},~{}\bar{x}_{2},~{}\bar{x}_{3})$ has the following form:

$\displaystyle\dfrac{d^{2}\Gamma}{dx\,d\Omega_{\mu}}=\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\dfrac{3}{2\pi}\sqrt{x^{2}-x_{0}^{2}}\left[F_{IS}(x)\pm
F_{IP}(x)\,(\vec{n}_{\mu}\cdot\vec{\zeta})\right.$ $\displaystyle\left.\pm
F_{AS}(x)\,(\vec{n}_{\mu}\cdot\vec{s})+(\vec{B}\cdot\vec{s})\right],$ (12)

where

$\displaystyle\vec{B}=F_{T_{1}}(x)\,\vec{\zeta}+[F_{AP}(x)-F_{T_{1}}(x)]\,(\vec{\zeta}\cdot\vec{n}_{\mu})\,\vec{n}_{\mu}+F_{T_{2}}(x)\,[\vec{\zeta}\times\vec{n}_{\mu}]\,.$
(13)

Here $\vec{s}$ is the $\tau$ spin direction, and $\vec{n}_{\mu}$ is the
direction of muon emission in the $\tau$ rest frame, $d\Omega_{\mu}$ is the
muon solid angle element, and $\vec{\zeta}$ is the muon spin direction in its
rest frame. After averaging over $\tau$ lepton spin, expression (3.2)
simplifies to the following:

$\displaystyle\dfrac{d^{2}\Gamma}{dx\,d\Omega_{\mu}}=\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\dfrac{3}{2\pi}\sqrt{x^{2}-x_{0}^{2}}\left[F_{IS}(x)\pm
F_{IP}(x)\,(\vec{n}_{\mu}\cdot\vec{\zeta})\right].$ (14)

Here $F_{IP}(x)$ contains the Michel parameter $\xi^{\prime}$ and describes
the muon longitudinal polarization.

To obtain the differential decay width of the cascade decay, we should repeat
the folding procedure of the $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$
differential decay width (14) with the $\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}$ differential decay width (6) described in section
2. This leads to the following result:

$\displaystyle\dfrac{d^{5}\Gamma}{dx\,d\Omega_{\mu}\,dy\,d\Omega_{e}\,dt}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\dfrac{3}{2\pi^{2}}y^{2}\sqrt{x^{2}-x_{0}^{2}}\left[(3-2y)F_{IS}(x)\right.$
$\displaystyle\left.+(2y-1)F_{IP}(x)\vec{n}_{e}\boldsymbol{R}(\phi)\vec{n}_{\mu}\right]\dfrac{1}{\tau_{\mu}}\exp{\left(-\dfrac{t}{\tau_{\mu}}\right)}\,.$
(15)

We then repeat the procedure of the coordinate system rotation to the vertex
of the muon decay after propagation in the magnetic field, explained in the
previous section, and integrate over time:

$\displaystyle\dfrac{d^{4}\Gamma}{dx\,d\Omega_{\mu}\,dy\,d\Omega_{e}^{\prime}}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}\dfrac{3}{2\pi^{2}}y^{2}\sqrt{x^{2}-x_{0}^{2}}$
$\displaystyle\left[(3-2y)F_{IS}(x)+(2y-1)F_{IP}(x)(\vec{n}_{e}^{\prime}\cdot\vec{n}_{\mu})\right],$
(16)

which can be simplified to

$\displaystyle\dfrac{d^{3}\Gamma}{dx\,dy\,d\cos{\theta_{e}^{\prime}}}=\mathcal{B}(\mu\to
e\nu\nu)\dfrac{12\Gamma_{\tau\to\mu\nu\nu}}{1-3x_{0}^{2}}y^{2}\sqrt{x^{2}-x_{0}^{2}}$
$\displaystyle\left[(3-2y)F_{IS}(x)+(2y-1)F_{IP}(x)\cos{\theta_{e}^{\prime}}\right].$
(17)

Here we have introduced $\theta^{\prime}_{e}$, the angle between
$\vec{n}_{e}^{\prime}$ and $\vec{n}_{\mu}$. As it was noted before, since the
direction of the $\tau$ lepton cannot be precisely determined due to the
undetectable neutrino, additional averaging over the kinematically allowed
$\tau$ direction is required. This procedure is commonly used in this kind of
analysis.

Using the obtained theoretical expression, we estimate a sensitivity for the
Michel parameter $\xi^{\prime}$ measurement at the Belle II experiment. Again,
we use a toy Monte Carlo simulation with an overall efficiency to reconstruct
the decay of interest defined as
$\eta=\eta_{\text{tag}}\,\omega_{\text{dec}}\,\eta_{\text{kink}}\,\eta_{\text{sel}}$.
Here $\eta_{\text{tag}}$ is the selection efficiency of the $\tau^{+}\tau^{-}$
events at B-factories; its typical value at Belle is equal to 15%
Epifanov:2017kly .

The probability of the muon to decay inside CDC ($\geq 10\,\text{cm}$ from the
walls) and on the first turn is $\omega_{\text{dec}}\approx 2.8\cdot 10^{-4}$.
While the outer radius of the drift chamber is larger at Belle II, the muon
boost is also larger; thus, in total, the probability to decay within kink
reconstructable volume is slightly smaller than at SCTF. Since the kink
reconstruction and selection algorithms are the same as for the SCTF, the
efficiencies $\eta_{\text{kink}}\approx 90\%$ and $\eta_{\text{sel}}\approx
80\%$ are also supposed to be the same.

Finally, the number of reconstructed cascades $\tau^{-}\to(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu})\bar{\nu}_{\mu}\nu_{\tau}$ from the full expected
Belle II data sample is $N\approx 4.8\cdot 10^{5}$. We estimate the accuracy
of the Michel parameter $\xi^{\prime}$ measurement in the Belle II experiment
to be $\sigma_{\xi^{\prime}}\approx 7\cdot 10^{-3}$ for the separate analysis
of the $\tau^{+}$ and $\tau^{-}$ decay samples. This result is comparable to
one obtained for the SCTF.

##  4 Conclusion

In this work, the method of the first direct measurement of all Michel
parameters, which determine the polarization of the daughter muon from the
$\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ decay, was described in detail.
In addition, the feasibility study of this method application at the working
experiment Belle II and the future Super Charm-Tau Factory was carried out.
The potential accuracy of the Michel parameters measurement was estimated to
be comparable to those achieved in the muon decay. It was shown that the SCTF
with polarized beam is an optimal experiment for the precise measurement of
all Michel parameters in $\tau$-decays.

##  5 Acknowledgments

The authors would like to thank D. Epifanov for making critical remarks and
useful discussions. The work of D. Bodrov was carried out within the framework
of the Basic Research Program at the National Research University Higher
School of Economics (HSE). P. Pakhlov acknowledges the support by the Russian
Science Foundation under contract 22-22-00564.

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