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\begin{document}

\title{Study of Michel parameters in $\tau$ decays} 

\noaffiliation
\affiliation{Department of Physics, University of Tokyo, Tokyo 113-0033}
\affiliation{Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090}
\author{H.~Aihara}\affiliation{Department of Physics, University of Tokyo, Tokyo 113-0033} % Tokyo
\author{D.~Epifanov}\affiliation{Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090}
\author{J.~Sasaki}\affiliation{Department of Physics, University of Tokyo, Tokyo 113-0033} % Tokyo
\author{N.~Shimizu}\affiliation{Department of Physics, University of Tokyo, Tokyo 113-0033} % Tokyo 
 
\pacs{13.35.Dx, 13.66.De, 13.66.Jn, 14.60.Fg} 

\maketitle

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In the Standard Model (SM), $\tau$ decays due to the charged weak interaction 
described by the exchange of $W^{\pm}$ with a pure vector coupling to only 
left-handed chirality fermions.
Decays of $\tau$ such as $\tau^-\to\ell^-\bar{\nu_{\ell}}\nu_{\tau}$, 
$\tau^-\to\ell^-\bar{\nu_{\ell}}\nu_{\tau}\gamma$ and 
$\tau^-\to\ell^-\ell'^+\ell'^-\bar{\nu_{\ell}}\nu_{\tau}$~$(\ell,\ell' = e,\mu)$ 
(unless specified otherwise, charge-conjugated decays are implied throughout 
the document), are of the special interest because electroweak couplings in 
these decays can be probed without disturbance from the strong interaction. 
This makes them an ideal system to study the Lorentz structure of the 
charged weak current. 
The most general, Lorentz invariant, derivative-free and 
lepton-number-conserving four-lepton point interaction 
Lagrangian can be written as \cite{Fetscher:1986uj}: 
\begin{eqnarray}
&& {\cal L} = \frac{4G_F}{\sqrt{2}}\sum_{\begin{subarray}{c} N=S,V,T \\ i,j=L,R \end{subarray}} g_{ij}^{N} \biggl[\bar{\Psi}_i(l)\Gamma^N \Psi_n(\nu_l)\biggr]\biggl[\bar{\Psi}_m(\nu_{\tau})\Gamma_N \Psi_j(\tau)\biggr];~~~\Gamma^{S}=1,~\Gamma^{V}=\gamma^{\mu},~\Gamma^{T}=\frac{i}{2\sqrt{2}}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}). \nonumber 
\end{eqnarray} 
The $\Gamma_N$ matrices ($\gamma^\nu$ are Dirac matrices) define the properties 
of the two currents under a Lorentz transformation with $N=S,V,T$ for scalar, 
vector and tensor interactions, respectively. The indices $i$ and $j$ label 
the right- or left-handedness (R, L) of the charged leptons. 
Ten non-trivial terms are characterized by ten complex coupling 
constants $g_{ij}^{N}$; those with $g^{T}_{RR}$ and $g^{T}_{LL}$ are 
identically zero. In the SM, the only non-zero coupling constant 
is $g^{V}_{LL}=1$, this property is also known as (V-A)$\otimes$(V-A) 
Lorentz structure of the matrix element. As the couplings can be complex, 
with arbitrary overall phase, there are 19 independent parameters. 
The total strength is determined by the Fermi constant $G_F$, 
this constrains the coupling constants to be $|g^S_{ij}|\leq 2$, 
$|g^V_{ij}|\leq 1$ and $|g^T_{ij}|\leq 1/\sqrt{3}$. 
  
In leptonic $\tau$ decay, $\tau^-\to\ell^-\bar{\nu_{\ell}}\nu_{\tau}$, where 
neutrinos are not detected and the spin of the outgoing charged lepton 
is not determined, only four Michel parameters $\rho$, $\eta$, 
$\xi$ and $\delta$ are experimentally accessible. They are bilinear combinations 
of the $g_{ij}^{N}$ coupling constants \cite{Fetscher:1993ki} and appear in the 
predicted energy spectrum of the charged lepton.
In the $\tau$ rest frame, neglecting radiative corrections, this spectrum is given by \cite{pdg}: 
\begin{eqnarray}
&& \frac{d\Gamma(\tau^{\mp})}{dx\,d\Omega_\ell} = \frac{4G_F^2 m_\tau E^4_{\rm max}}{(2\pi)^4}\sqrt{x^2-x^2_0}\biggl(x(1-x)+\frac{2}{9}\rho(4x^2-3x-x^2_0)+\eta x_0(1-x) \nonumber \\
&& \mp\frac{1}{3}P_{\tau}\cos\theta_{\ell}\xi\sqrt{x^2-x^2_0}\biggl[1-x+\frac{2}{3}\delta\bigl(4x-4+\sqrt{1-x^2_0}\bigr)\biggr]\biggr), \nonumber \\
&& x=E_\ell/E_{\rm max},~~E_{\rm max}=m_\tau\bigl(1+m^2_\ell/m^2_\tau\bigr)/2,~~x_0=m_\ell/E_{\rm max}, \nonumber 
\end{eqnarray} 
where $P_\tau$ is the $\tau$ polarization, $\Omega_\ell$ is the solid angle of the outgoing lepton and 
$\theta_{\ell}$ is the angle between the $\tau$ spin and the lepton momentum. 
In the SM, the (V-A) charged weak current is characterized by $\rho=3/4$, $\eta=0$, $\xi=1$ and $\delta=3/4$. 
In the radiative leptonic decay, $\tau^-\to\ell^-\bar{\nu_{\ell}}\nu_{\tau}\gamma$, three additional 
parameters, $\bar{\eta}$, $\eta^{\prime \prime}$ and $\xi\kappa$ can be extracted \cite{Arbuzov:0}. 
Michel formalism for the five-lepton $\tau$ decay, $\tau^-\to\ell^-\ell'^+\ell'^-\bar{\nu_{\ell}}\nu_{\tau}$, 
can be found in the recent paper \cite{Pablo}. 

Measurement of $\xi$, $\delta$ and $\xi\kappa$ parameters requires 
knowledge of the $\tau$ spin direction. In experiments at $e^+e^-$ colliders 
with unpolarized $e^\pm$ beams, the average polarization of a single 
$\tau$ is zero. However, spin-spin correlations between 
the $\tau^+$ and $\tau^-$ produced in the reaction $e^+e^-\to\tau^+\tau^-$ 
can be exploited \cite{Tsai:1}. 
The main idea of the method is to consider events where both taus 
decay to the selected final states. One tau decays to the signal mode while 
the opposite tau, which decays via  $\tau^+\to\pi^+\pi^0\bar{\nu}_\tau$, 
serves as a spin analyser. We choose the $\tau^+\to \pi^+\pi^0\bar{\nu}_\tau$ 
(it is characterized by $\xi_\rho$ parameter, in the SM $\xi_\rho=1$) decay 
mode because it has the largest branching fraction as well as properly studied 
dynamics. To write the total differential cross section we follow the approach 
developed in Refs.~\cite{Fetscher:1,Tamai:1}. The differential cross section 
is used to construct the probability density function (PDF), and Michel parameters 
are extracted in the unbinned maximum likelihood fit of the selected events. 

Further, we describe the construction of PDF for ordinary leptonic decay. 
The total differential cross section for the $(\tau^-\to\ell^-\bar\nu_\ell\nu_\tau;~\tau^+\to\pi^+\pi^0\bar\nu_\tau)$
(or, briefly, $(\ell^-;\,\rho^+)$) events in the full nine-dimensional phase space, 
$\frac{d\sigma}{d\vec{z}}\bigl(\vec{\Theta}\bigr)\,(\vec{\Theta}=\{1,\,\rho,\,\eta,\,\xi_\rho\xi,\,\xi_\rho\xi\delta\},\,\vec{\Theta}_{\rm SM}=\{1,\,3/4,\,0,\,1,\,3/4\})$, is used to construct the PDF for the measurement vector 
$\vec{z}=\{p_{\ell},\,\cos{\theta_{\ell}},\,\phi_{\ell},\,p_{\rho},\,\cos{\theta_{\rho}},\,\phi_{\rho},\,m_{\pi\pi},\,\cos{\tilde{\theta}_{\pi}},\,\tilde{\phi}_{\pi}\}$. The main background processes, $(\ell^-;\,\pi^+\pi^0\pi^0)$, $(\pi^-;\,\rho^+)$ and $(\rho^-;\,\rho^+)$ 
with the fractions, $\lambda_{3\pi}$, $\lambda_\pi$ and $\lambda_\rho$, respectively, are included in 
the PDF analytically. The remaining background with the fraction $\lambda_{\rm other}$ is described 
by the ${\cal P}^{\rm MC}_{\rm bg}(\vec{z})$ PDF, which is evaluated from the large Monte Carlo 
sample. The total PDF for the $(\ell^-;\,\rho^+)$ events is written as: 
\[ 
{\cal P}(\vec{z}) =
\frac{\varepsilon(\vec{z})}{\bar{\varepsilon}}\biggl((1-\lambda_{3\pi}-\lambda_\pi-\lambda_\rho-\lambda_{\rm other})\frac{S(\vec{z}\,|\,\vec{\Theta})}{\int\frac{\varepsilon(\vec{z})}{\bar{\varepsilon}}S(\vec{z}\,|\,\vec{\Theta})d\vec{z}}+ 
\]
\begin{equation}
 +\lambda_{3\pi}\frac{B_{3\pi}(\vec{z}\,|\,\vec{\Theta})}{\int\frac{\varepsilon(\vec{z})}{\bar{\varepsilon}}B_{3\pi}(\vec{z}\,|\,\vec{\Theta})d\vec{z}}+\lambda_\pi\frac{B_{\pi}(\vec{z})}{\int\frac{\varepsilon(\vec{z})}{\bar{\varepsilon}}B_{\pi}(\vec{z})d\vec{z}}+\lambda_{\rho}\frac{B_{\rho}(\vec{z})}{\int\frac{\varepsilon(\vec{z})}{\bar{\varepsilon}}B_{\rho}(\vec{z})d\vec{z}}+\lambda_{\rm other}{\cal P}^{\rm MC}_{\rm bg}(\vec{z})\biggr), \nonumber
\end{equation}
where
$S(\vec{z}\,|\,\vec{\Theta})$,~$B_{3\pi}(\vec{z}\,|\,\vec{\Theta})$, 
$B_{\pi}(\vec{z})$ and $B_{\rho}(\vec{z})$ are the cross sections 
for the $(\ell^-;\,\rho^+)$, $(\ell^-;\,\pi^+\pi^0\pi^0)$, $(\pi^-;\,\rho^+)$ 
and $(\rho^-;\,\rho^+)$ events, respectively; $\varepsilon(\vec{z})$ is 
the detection efficiency for signal events in the full phase space; and 
$\bar{\varepsilon}=\int\varepsilon(\vec{z})S(\vec{z}\,|\,\vec{\Theta}_{\rm SM})d\vec{z}/\int S(\vec{z}\,|\,\vec{\Theta}_{\rm SM})d\vec{z}$ 
is an average signal detection efficiency. 
There are several corrections that must be incorporated in the procedure 
to take into account real experimental situation. Physics corrections 
include electroweak higher-order corrections to the $e^+ e^-\to\tau^+\tau^-$ 
cross section \cite{Jadach:1985ac,Arbuzov:1997pj}. 
Apparatus corrections include the effect of the finite detection efficiency and resolution, 
the effect of the external bremsstrahlung for $(e^-;\,\rho^+)$ events, 
and the $e^\pm$ beam energy spread. 
 
Study of Michel parameters $\rho$, $\eta$, $\xi_\rho\xi$ and $\xi_\rho\xi\delta$ 
in leptonic $\tau$ decays using a $485~{\rm fb}^{-1}$ data sample collected at Belle 
showed that the statistical uncertainties of Michel parameters are already of the 
order of $10^{-3}$, see Table~\ref{tab} \cite{Epifanov:2017kly}. 
Although systematic uncertainties coming from the physical and apparatus corrections 
as well as from the normalisation are below 1\%, there are still relatively large 
systematic uncertainties from the experimental corrections to the detection efficiency. 
Currently, the largest contribution, $(1\div 3)$\%, comes from the trigger efficiency 
correction. The expected statistical uncertainties of Michel parameters at Belle-II 
(with the total planned luminosity integral of $50~{\rm ab}^{-1}$) are already of the 
order of $10^{-4}$.
%, see Table~\ref{tab}. 
\begin{table}[htbp] 
\centering
\caption{Statistical uncertainties of Michel parameters, $\rho$, $\eta$, $\xi_\rho\xi$, 
$\xi_\rho\xi\delta$, $\bar{\eta}$ and $\xi_\rho\xi\kappa$ in ordinary and radiative 
leptonic $\tau$ decays \cite{Shimizu:2017dpq} at Belle (with a 485~fb$^{-1}$ 
(703~fb$^{-1}$) data sample for ordinary (radiative) leptonic decays).}\label{tab} 
\begin{tabular}{lcccccc}
\hline
Exp.  & $(\Delta\rho)_{\rm stat}$,$10^{-4}$ & $(\Delta\eta)_{\rm stat}$,$10^{-4}$ & $(\Delta\xi_\rho\xi)_{\rm stat}$,$10^{-4}$ & $(\Delta\xi_\rho\xi\delta)_{\rm stat}$,$10^{-4}$ & $(\Delta\bar{\eta})_{\rm stat}$ & $(\Delta\xi_\rho\xi\kappa)_{\rm stat}$ \\
\hline
Belle    & $13$ & $62$ & $39$ & $25$ & $1.5$ & $0.4$ \\
%Belle II & $1.3$ & $6.2$ & $3.9$ & $2.5$ & $0.18$ & $0.05$ \\ 
\hline 
\end{tabular} 
\end{table}
At Belle II the systematic uncertainties will be the dominant ones. 
To improve them high- and uniform-efficiency two-track trigger is needed. 
%Higher statistics at Belle II will allow us to tabulate the other efficiency 
%corrections with much better accuracy than at Belle. 
%
%Preliminary result on Michel parameters $\bar{\eta}$ and $\xi_\rho\xi\kappa$ 
%in radiative leptonic $\tau$ decays with a 703~fb$^{-1}$ Belle data sample: 
%$\bar{\eta} = -1.3\pm 1.5\pm 0.8$ and $\xi\kappa = 0.5\pm 0.4\pm 0.2$,
%where the first error is statistical and the second is systematic \cite{Shimizu:2017dpq}. 

\begin{thebibliography}{15}

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%``Muon Decay: Complete Determination of the Interaction and Comparison with the Standard Model,''
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%%CITATION = PHLTA,B173,102;%% 

\bibitem{Fetscher:1993ki} W.~Fetscher and H.~J.~Gerber, 
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\bibitem{Arbuzov:0} A.~B.~Arbuzov and T.~V.~Kopylova,
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%%CITATION = doi:10.1007/JHEP09(2016)109;%% 

\bibitem{Pablo} A.~Flores-Tlalpa, G.~Lopez Castro and P.~Roig, 
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%%CITATION = doi:10.1007/JHEP04(2016)185;%% 

\bibitem{Tsai:1} Y.~-S.~Tsai, 
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\bibitem{Fetscher:1} W.~Fetscher,~Phys.~Rev.~D~{\bf 42} (1990) 1544. 

\bibitem{Tamai:1} K.~Tamai,~Nucl.~Phys.~B~{\bf 668} (2003) 385.

\bibitem{Jadach:1985ac} S.~Jadach and Z.~Was, 
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%%CITATION = APPOA,B15,1151;%% 

\bibitem{Arbuzov:1997pj} A.~B.~Arbuzov, G.~V.~Fedotovich,
E.~A.~Kuraev, N.~P.~Merenkov, V.~D.~Rushai and L.~Trentadue, 
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\bibitem{Epifanov:2017kly} D.~A.~Epifanov [Belle Collaboration],
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%%CITATION = doi:10.1016/j.nuclphysbps.2017.03.033;%%
 
\bibitem{Shimizu:2017dpq} N.~Shimizu {\it et al.} [Belle Collaboration],
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arXiv:1709.08833 [hep-ex].
%%CITATION = ARXIV:1709.08833;%%

\end{thebibliography}
\end{document}
