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\begin{titlepage}
\begin{flushright}
\hfill{YUMS 97-19}\\
\hfill{KEK-TH-526}\\
\hfill{DESY 97-131}\\
\hfill{August 13, 1997}
\end{flushright}
\vspace{1.0cm}

\begin{center}
{\Large\bf CP Violation at Electron-Positron Colliders\footnote{To appear
           in the Proceedings of the KEK meetings on `$CP$ violation and
           its origin' (1993-1997).} }\\
\hfill{}
\vskip 1.1cm
{M.S.~Baek and S.Y.~Choi} \\
\vskip 0.2cm
{\sl Department of Physics, Yonsei University, Seoul 120-749, Korea}\\
\vskip 0.2cm
{and}\\
\vskip 0.2cm
{K.~Hagiwara}\\
\vskip 0.2cm
{\sl Theory Group, KEK, 1-1 Oho, Tskuba, Ibaraki 305, Japan}
\end{center}


\vskip 1.2cm
\setcounter{footnote}{0}
%
\begin{abstract}
A detailed, model-independent study of $CP$ violation at present
and future electron-positron colliders is reported. 
Firstly, we investigate $CP$ violation
effects in  $W$ boson and $t$ quark pair production in $e^+e^-$ annihilation
and in two-photon fusion at a next $e^+e^-$ linear collider,
where Compton-backscattered laser light off the electrons
or positrons are employed as a powerful
polarized photon source. Secondly, noting that there do not exist
any direct measurements for the tau-lepton electric dipole moment (EDM),
we address the importance of performing its direct measurements
at $e^+e^-$ collisions off the $Z$-boson pole at TRISTAN, LEPII
and CLEOII. We present a rough comparison of the 
potential of these experiments in the $\tau$ EDM measurements. 
Finally, we report on our recent works for probing $CP$ violation 
in the semileptonic decays of the tau lepton,
which involve two different intermediate resonances with large ratios of 
widths to masses, and which can be most efficiently identified at the 
planned $B$ and proposed $\tau$-charm factories. 
All the $CP$-violation phenomena in the processes under 
consideration, if discovered, imply new $CP$-violation mechanisms 
completely different from the Standard Model $CP$ violation
mechanism through the complex phase of the Cabibbo-Kobayashi-Maskawa  
matrix.
\end{abstract}
%

\end{titlepage}

\newpage
\renewcommand{\thefootnote}{\alph{footnote}}
\input intro.tex 
\input nlc.tex
\input tau.tex
\input summary.tex
\input ack.tex
\input refer.tex

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$CP$ Violation at Next $e^+e^-$ Linear Colliders}
\label{sec:NLC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The possibility of having an NLC  has increased 
significantly in the last few years in parallel with a series
of international workshops called Physics and Experiments with
Linear $e^+e^-$ Colliders\cite{NLC}. There is general consensus for
a 500 GeV NLC  with an integrated luminosity
of the order of 10 fb$^{-1}$ for the first phase. With such experimental 
parameters, NLC produces a copious number of $W$ and top-quark pairs. 
With the expectations that non-KM $CP$ 
violation effects from new interactions beyond the SM are predominant 
in the most massive charged gauge boson and fermion systems, 
the $W$ boson and the $t$ quark, a lot of
works\cite{WbAb,CR,Chang,Atwood,PR,HPZH,GgDs,PkPm,Gounaris,Atwood2} have 
made important contributions to probing $CP$ violation through the production 
and decay processes of these two heavy particles in 
hadron-hadron and $e^+e^-$ collisions.
On the other hand, as one important spin-off of NLC 
highly-energetic photon beams from Compton backscattering of laser
light off electrons or positrons can be utilized to realize electron-photon
and photon-photon collisions. Therefore, recently non-KM $CP$ violation
has been probed in the $W$-pair and $t$-pair production processes
via two-photon fusion\cite{Wudka,Choi2,GbGc}. 

In this section we review a series of our recent works\cite{Choi1,Choi2}
on a detailed, model-independent study of $CP$ violation in $W$-boson 
pair production and top-quark pair production in the $e^+e^-$ and 
two-photon annihilation at an NLC.

The works prior to our works are classified into two categories according to 
their emphasized aspects: (i) the classification of spin and angular 
correlations of the decay products without electron beam 
polarization\cite{WbAb,CR,Chang,HPZH,GgDs,PkPm,KLY} and 
(ii) the use of a few typical $CP$-odd observables with electron beam 
polarization\cite{Atwood,PR}. 
In the first class, they have constructed
a tower of $CP$-odd observables according to their ranks. However, since
the $W$ boson and top quark  are spin-1 and spin-1/2, 
the number of $CP$-odd spin correlations appearing in the processes 
$e^+e^-(\gamma\gamma)\rightarrow W^+W^-$ and 
$e^+e^-(\gamma\gamma)\rightarrow t\bar{t}$ is finite so that 
{\it all the linearly-independent $CP$-odd correlations can be completely
defined}. Then, all the previously-considered correlations are 
expressed in terms of the complete set of linearly-independent 
$CP$-odd correlations. 
In the second class, it has been shown that electron beam polarization
is very crucial for a few specialized $CP$-odd correlations. 
Those works can be easily extended with the complete set of $CP$-odd
correlations in order to investigate which 
$CP$-odd correlations depend crucially on electron polarization and which correlations do not.
     
In order to make our predictions model-independent, we consider all  
possible leading $CP$-violating effects which can be
induced by any extensions of the SM at the weak interaction scale.  
To describe $CP$ violation in the $W$-boson pair production, 
we follow the effective field theory approach with a linear realization 
of the symmetry-breaking sector due to some new interactions which involve
the Higgs sector and the electroweak gauge bosons.
The SU(2)$_L\times$U(1)$_Y$ electroweak gauge symmetry is imposed 
in constructing the higher dimensional effective Lagrangian since there is 
no experimental sign of violation of this symmetry. 
There exist six relevant dimension-six $CP$-odd operators.
On the other hand, the relevant $CP$-odd corrections to the top-quark pair 
production processes are obviously EDM-type
couplings of top and anti-top quarks to a photon and $Z$.

We use two methods in detecting $CP$ violation. One
method makes use of the produced $W$ bosons and top-quarks by measuring
various spin correlations in their final decay products, and the other
method is to employ polarized photon beams to measure various
$CP$-odd polarization asymmetries of the initial states. In the $e^+e^-$ mode,
where the initial $e^+e^-$ state is (almost) $CP$-even due to the very
small electron mass\cite{Hikasa}, only the first method can be used, 
but in the $\gamma\gamma$ mode both methods can be employed. 
We concentrate on the second method in the
two-photon mode by making use of the Compton backscattered laser 
light off the electron or positron beam as a powerful photon source.
The polarization of the scattered high energy photon beams can be controlled 
by adjusting polarizations of initial electron beams and the laser light.
Nevertheless, we will simply consider purely linearly-polarized photon
beams in the present report.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Photon Spectrum}
\label{subsec:photon}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
Generally, a purely polarized photon beam state is a linear combination
of two helicity states and the photon polarization vector  
can be expressed in terms of two angles $\alpha$ and $\phi$
in a given coordinate system as
%
\begin{eqnarray}
|\alpha,\phi\rangle =-\cos(\alpha) e^{-i\phi}|+\rangle
           +\sin(\alpha) e^{i\phi}|-\rangle,
\end{eqnarray}
%
where $0\leq \alpha\leq \pi/2$ and $0\leq \phi\leq 2\pi$.
The degrees of circular and linear polarization are  
$\xi=\cos(2\alpha)$ and $\eta=\sin(2\alpha)$, respectively,
and the direction of maximal linear polarization is denoted
by the azimuthal angle $\phi$. 
For a partially polarized photon
beam it is necessary to re-scale $\xi$ and $\eta$ by its degree
of polarization.


%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% fig1.eps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\hbox to\textwidth{\hss\epsfig{file=fig1.eps,width=10cm}\hss}
\caption{The coordinate system in the colliding $\gamma\gamma$
         c.m. frame. The scattering angle, $\Theta$, and the
         azimuthal angles, $\phi_1$ and $\phi_2$, for the linear
         polarization directions measured from the scattering plane
         are described.}
\label{fig:fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

The state vector of the two-photon system in the c.m. frame, 
where one photon momentum is along the positive $z$ direction, is 
%
\begin{eqnarray}
|\alpha_1,\phi_1;\alpha_2,\phi_2\rangle
       =|\alpha_1,\phi_1\rangle|\alpha_2,-\phi_2\rangle. 
\label{eq:two-photon_wf}
\end{eqnarray}
%
The angle $\phi_1$ ($\phi_2$) is the azimuthal angle of 
the maximal linear polarization of the photon beam, whose momentum 
is in the positive (negative) $z$ direction, with respect to 
the direction of the $W^+(t)$ momentum in $\gamma\gamma
\rightarrow W^+W^- (\gamma\gamma \rightarrow t\bar{t})$
as shown in Fig.~\ref{fig:fig1}.

Generally, the angular dependence for $\gamma\gamma\rightarrow X$
with two linearly-polarized photon beams, of which the degrees of linear
polarization are $\eta$ and $\bar{\eta}$, is expressed in the form
%
\begin{eqnarray}
&& {\cal D}(\eta,\bar{\eta};\chi,\phi)
=\Sigma_{\rm unpol}-\frac{1}{2}Re\Bigg[
  \left(\eta{\rm e}^{-i\phi}+\bar{\eta}{\rm e}^{i\phi)}\right)
  {\rm e}^{-i\chi}\Sigma_{02}\Bigg]\nonumber\\
&&\hspace{0.5cm} +\frac{1}{2}Re\Bigg[
  \left(\eta{\rm e}^{-i\phi}-\bar{\eta}{\rm e}^{i\phi}\right)
  {\rm e}^{-i\chi}\Delta_{02}\Bigg]
  +\eta\bar{\eta}Re\Bigg[
  {\rm e}^{-2i\phi}\Sigma_{22}+{\rm e}^{-2i\chi}\Sigma_{00}
  \Bigg],
\label{linear}
\end{eqnarray}
%
with the subscripts, $0$ and $2$, representing the magnitude of the 
sum of two photon helicities of the initial two-photon system.
Here, two angular variables $\chi$ and $\phi$ are given by
%
\begin{eqnarray}
\chi=\phi_1-\phi_2,\qquad 
\phi=\phi_1+\phi_2.
\end{eqnarray}
%
The azimuthal angle difference, $\chi$, is independent of the final state, 
while the azimuthal angle sum, $\phi$, depends on the scattering plane, 
and both angles are invariant with respect to the Lorentz boost along 
the two-photon beam direction. 

%
\begin{table}[ht]
\caption{$CP$ and $CP\tilde{T}$ properties of the invariant functions 
        and the angular distributions.}
\label{tbl:gamma_symmetry}
%
%
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline\hline
 \mbox{ }\hskip 0.2cm $CP$ \mbox{ }\hskip 0.2cm   
&\mbox{ }\hskip 0.2cm $CP\tilde{T}$\mbox{ }\hskip 0.2cm   
&\mbox{ }\hskip 0.2cm Invariant functions \mbox{ }\hskip 0.2cm  
&\mbox{ }\hskip 0.2cm Angular dependences \mbox{ }\hskip 0.2cm  \\
\hline
even & even & $\Sigma_{\rm unpol}$     
            & { } \\
            \cline{3-4}
{ }  & { }  & ${\cal R}(\Sigma_{02})$  
            & $\eta\cos(\phi+\chi)+\bar{\eta}\cos(\phi-\chi)$  \\
            \cline{3-4}
{ }  & { }  & ${\cal R}(\Sigma_{22})$
            & $\eta\bar{\eta}\cos(2\phi)$ \\
            \cline{3-4}
{ }  & { }  & ${\cal R}(\Sigma_{00})$  
            & $\eta\bar{\eta}\cos(2\chi)$ \\ \hline 
even & odd  & ${\cal I}(\Delta_{02})$
            & $\eta\sin(\phi+\chi)+\bar{\eta}\sin(\phi-\chi)$  \\
            \cline{3-4}
{ }  & { }  & ${\cal I}(\Sigma_{22})$  
            & $\eta\bar{\eta}\sin(2\phi)$ \\ \hline
odd  & even & ${\cal I}(\Sigma_{02})$
            & $\eta\sin(\phi+\chi)-\bar{\eta}\sin(\phi-\chi)$  \\
            \cline{3-4}
{ }  & { }  & ${\cal I}(\Sigma_{00})$  
            & $\eta\bar{\eta}\sin(2\chi)$  \\ \hline
odd  & odd  & ${\cal R}(\Delta_{02})$  
            & $\eta\cos(\phi+\chi)-\bar{\eta}\cos(\phi-\chi)$  \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
%

It is useful to classify the invariant functions $\Sigma$ and
$\Delta$'s, which depend only on the scattering angle $\Theta$,
and the angular distributions according to 
their transformation properties under the discrete symmetries,
$CP$ and $CP\tilde{T}$\footnote{$\tilde{T}$ is the "naive" time
reversal operation which flips particle momenta and spins, but does
not interchange initial and final states}
as shown in Table~\ref{tbl:gamma_symmetry}. 
There exist three $CP$-odd functions; ${\cal I}(\Sigma_{02})$, 
${\cal I}(\Sigma_{00})$ and ${\cal R}(\Delta_{02})$. Here, ${\cal R}$
and ${\cal I}$ stand for real and imaginary parts, respectively.
While the first two terms are $CP\tilde{T}$-even, the last 
term ${\cal R}(\Delta_{02})$ is
$CP\tilde{T}$-odd. Since the $CP\tilde{T}$-odd term 
${\cal R}(\Delta_{02})$ requires an absorptive part in the amplitude,
it is generally expected to be smaller in magnitude than 
the $CP\tilde{T}$-even terms. 
We then can define two $CP$-odd asymmetries from ${\cal I}(\Sigma_{02})$ 
and ${\cal I}(\Sigma_{00})$. 
First, note that $\Sigma_{00}$  is independent of the azimuthal angle 
$\phi$ whereas $\Sigma_{02}$ is not. 
In order to improve the observability we may integrate 
${\cal I}(\Sigma_{02})$ over the azimuthal angle $\phi$ 
with an appropriate weight function. 
Without any loss of generality we can take $\eta=\bar{\eta}$. 
Then, ${\cal I}(\Sigma_{00})$ 
can be separated by taking the difference of the distributions 
at $\chi=\pm\pi/4$  and the ${\cal I}(\Sigma_{02})$ by taking 
the difference of the distributions at $\chi=\pm\pi/2$.
As a result we obtain the following two integrated $CP$-odd
asymmetries\footnote{The authors in Ref.\ \cite{GbGc} have 
also  considered ${\cal I}(\Sigma_{02})$, but not 
   ${\cal I}(\Sigma_{00})$}:
%
\begin{eqnarray}
\hat{A}_{02}=\left(\frac{2}{\pi}\right)
     \frac{{\cal I}(\Sigma_{02})}{\Sigma_{\rm unpol}},\qquad
\hat{A}_{00}=\frac{{\cal I}(\Sigma_{00})}{\Sigma_{\rm unpol}},
\end{eqnarray}
%
where the factor $(2/\pi)$ in the $\hat{A}_{02}$ stems from taking 
the average over the azimuthal angle $\phi$ with the weight function
${\rm sign}(\cos\phi)$.

The Compton backscattering process is  
characterized by two parameters $x$ and $y$:
%
\begin{eqnarray}
x=\frac{4E\omega_0}{m^2_e}
 \approx 15.3\left(\frac{E}{\rm TeV}\right)
             \left(\frac{\omega_0}{\rm eV}\right),\qquad
y=\frac{\omega}{E},
\end{eqnarray}
%
where $E$ is the electron beam energy and $\omega_0$ the incident
laser beam frequency.
On the average, the backscattered photon energies increase with $x$;
the maximum photon energy fraction is given by
$y_m=x/(1+x)$. Operation below the threshold\cite{GKS} for $e^+e^-$ 
pair production in collisions between the laser beam and the 
Compton-backscattered photon beam requires $x\leq 2(1+\sqrt{2})\approx
4.83$. 

In the two-photon collision case with Compton-backscattered photon beams,
only part of each laser linear polarization is transferred to the 
high-energy photon beam and the degrees of linear polarization 
transfer\footnote{The function $A_{\eta\eta}$ has been 
considered in Ref.~\cite{Kramer}.} are determined by two   
functions, ${\cal A}_\eta$ and ${\cal A}_{\eta\eta}$:  
%
\begin{eqnarray}
{\cal A}_\eta(\tau)=\frac{\langle \phi_0\phi_3\rangle_\tau}{\langle 
               \phi_0\phi_0\rangle_\tau},\qquad 
{\cal A}_{\eta\eta}(\tau)
     =\frac{\langle \phi_3\phi_3\rangle_\tau}{\langle 
               \phi_0\phi_0\rangle_\tau},
\end{eqnarray}
%
where $\phi_0(y)=\frac{1}{1-y}+1-y-4r(1-r)$ and $\phi_3(y)=2r^2$
with $r=y/x(1-y)$,  and 
$\tau$ is the ratio of the $\gamma\gamma$ 
c.m. energy squared $\hat{s}$ to the $e^+e^-$ collider energy squared 
$s$. ${\cal A}_\eta$ is for the collision of an 
unpolarized photon beam and a linearly polarized photon beam, and 
${\cal A}_{\eta\eta}$ for that of two linearly 
polarized photon beams.  $\langle \phi_i\phi_j\rangle_\tau$ ($i,j=0,3$) 
is defined as a normalized convolution integral for a fixed value of 
$\tau$. Folding the photon spectrum with the $\gamma\gamma\rightarrow X$
cross section yields two $CP$-odd asymmetries $A_{02}$ and
$A_{00}$, which depend crucially on the two-photon spectrum
and the two linear polarization transfers. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% fig3.eps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\hbox to\textwidth{\hss\epsfig{file=fig3.eps,width=12cm}\hss}
\caption{(a) the $\gamma\gamma$ luminosity spectrum and (b) the
          two linear polarization transfers, ${\cal A}_\eta$ (solid lines)
          and ${\cal A}_{\eta\eta}$ (dashed lines), 
          for $x=4E\omega_0/m^2_e=0.5$, $1$ and $4.83$.}
\label{fig:fig3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We first investigate the $\sqrt{\tau}$ dependence of the two-photon 
spectrum and the two linear polarization transfers,
${\cal A}_\eta$ and ${\cal A}_{\eta\eta}$ by varying the value of 
the parameter $x$. 
Three values of $x$ are chosen; $x=0.5$, $1$, and $4.83$.
Fig.~\ref{fig:fig3} clearly shows that the energy of two 
photons reaches higher ends for larger $x$ values but the maximum linear 
polarization transfers are larger for smaller $x$ values. 
The parameter $x$ should be kept as large as possible to
reach higher energies. However, larger $CP$-odd asymmetries can be
obtained for smaller $x$ values. Therefore, there should
exist a compromise value of $x$ for the optimal observability of 
$CP$ violation. In this regard, it is very crucial to have a laser with 
adjustable beam frequency.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$W$ pair production}
\label{subsec:WW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To describe $CP$ violation from new interactions among electroweak 
vector bosons in a model-independent way\cite{Model}, we assume that the operators 
with lowest energy dimension dominate the $CP$-odd 
amplitudes and that they respect the electroweak gauge invariance
which is broken spontaneously by an effective SU(2)$_L$-doublet scalar.
Considering $CP$-odd interactions of dimension six composed of electroweak 
gauge bosons and Higgs fields, we can construct an effective Lagrangian
${\cal L}_{\rm eff}$, which is the sum of the SM Lagrangian 
${\cal L}_{\rm SM}$ and the new Lagrangian ${\cal L}_{\rm new}$ defined as
%
\begin{eqnarray}
{\cal L}_{\rm new}=\frac{1}{\Lambda^2}
           \bigg[f_{B\tilde{B}}{\cal O}_{B\tilde{B}}
               + f_{B\tilde{W}}{\cal O}_{B\tilde{W}}
               + f_{W\tilde{W}}{\cal O}_{W\tilde{W}}
               + f_{\tilde{B}}{\cal O}_{\tilde{B}}
               + f_{\tilde{W}}{\cal O}_{\tilde{W}}
               + f_{WW\tilde{W}}{\cal O}_{WW\tilde{W}}\bigg],
\label{eq:effective}
\end{eqnarray}
%
where the six $CP$-odd dimension-six operators are as follows 
%
\begin{eqnarray}
\begin{array}{ll}
  {\cal O}_{B\tilde{B}}=g^{\prime 2}(\Phi^\dagger\Phi)
                        B_{\mu\nu}\tilde{B}^{\mu\nu}, &
  {\cal O}_{B\tilde{W}}=gg^\prime(\Phi^\dagger\sigma^I\Phi)
                        B_{\mu\nu}\tilde{W}^{I\mu\nu}, \\
  {\cal O}_{W\tilde{W}}
   =g^2(\Phi^\dagger\Phi)W^I_{\mu\nu}\tilde{W}^{I\mu\nu}, &
  {\cal O}_{\tilde{B}}
   =ig^\prime\left[(D_\mu\Phi)^\dagger(D_\nu\Phi)\right]
       \tilde{B}^{\mu\nu},\\
  {\cal O}_{\tilde{W}}
   =ig\left[(D_\mu\Phi)^\dagger\sigma^I(D_\nu\Phi)\right]
      \tilde{W}^{I\mu\nu}, &
  {\cal O}_{WW\tilde{W}}=g^3\epsilon^{IJK}\tilde{W}^{I\mu\nu}
                       W_\nu^{J\rho}W^K_{\rho\mu},
\end{array}
\label{eq:dimension-six}
\end{eqnarray}
%
with the Higgs-doublet $\Phi$,
$\tilde{W}^{I\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}
W^I_{\alpha\beta}$, and $\tilde{B}^{\mu\nu}
=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}
\tilde{B}_{\alpha\beta}$.
Table~\ref{tbl:dimension-six vertices}
shows which vertices already exist in the SM at tree level 
and which  new vertices appear from the new dimension-six $CP$-odd 
operators.
%
\begin{table}[htb]
\caption{Vertices relevant for the processes $e^+e^- \rightarrow W^+W^-$ and
         $\gamma\gamma\rightarrow W^+W^-$ in the effective Lagrangian 
         with the six dimension-six $CP$-odd operators.}
\label{tbl:dimension-six vertices}
%
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline\hline
 \mbox{ }\hskip 0.2cm Vertex\mbox{ }\hskip 0.2cm 
&\mbox{ }\hskip 0.2cm $\gamma WW$ \mbox{ }\hskip 0.2cm 
&\mbox{ }\hskip 0.2cm $ZWW$ \mbox{ }\hskip 0.2cm 
&\mbox{ }\hskip 0.2cm $\gamma\gamma WW$ \mbox{ }\hskip 0.2cm 
&\mbox{ }\hskip 0.2cm $HWW$ \mbox{ }\hskip 0.2cm 
&\mbox{ }\hskip 0.2cm $\gamma\gamma H$ \mbox{ }\hskip 0.2cm\\ 
\hline
SM &  {\large o} & {\large o} &  {\large o}  &  {\large o}  & x \\ 
\hline
\mbox{ }\hskip 0.2cm ${\cal O}_{B\tilde{B}}$\mbox{ }\hskip 0.2cm 
   &  x  & x &  x  &  x  &  {\large o}       \\ 
\hline
\mbox{ }\hskip 0.2cm ${\cal O}_{B\tilde{W}}$\mbox{ }\hskip 0.2cm 
   &  {\large o} & {\large o}  &  x  &  x  &  {\large o}       \\ 
\hline
\mbox{ }\hskip 0.2cm ${\cal O}_{W\tilde{W}}$\mbox{ }\hskip 0.2cm
   &  x  & x &  x  &  {\large o}  &  {\large o}       \\ 
\hline
\mbox{ }\hskip 0.2cm ${\cal O}_{\tilde{B}}$\mbox{ }\hskip 0.2cm 
   &  {\large o} & {\large o} &  x  &  x  &  x   \\ 
\hline
\mbox{ }\hskip 0.2cm ${\cal O}_{\tilde{W}}$\mbox{ }\hskip 0.2cm 
   & {\large o} & {\large o}  &  x  &  {\large o}  & x      \\ 
\hline
\mbox{ }\hskip 0.2cm ${\cal O}_{WW\tilde{W}}$\mbox{ }\hskip 0.2cm 
   &  {\large o} & {\large o}  &  {\large o} & x &  x       \\ 
\hline\hline
\end{tabular}
\end{center}
\end{table}
%

For the sake of an efficient analysis in the following, 
we define four new dimensionless form factors, 
$Y_i$ ($i=1$ to $4$), which are related with the coefficients, 
$f_i$'s ($i=B\tilde{B},B\tilde{W},W\tilde{W},\tilde{B},\tilde{W},WW\tilde{W}$) 
as
% 
\begin{eqnarray}
&&Y_1=\left(\frac{m_W}{\Lambda}\right)^2
      \bigg[f_{B\tilde{W}}
         +\frac{1}{4}f_{\tilde{B}}+f_{\tilde{W}}\bigg],\qquad 
  Y_2=\left(\frac{m_W}{\Lambda}\right)^2
      \frac{g^2}{4}f_{WW\tilde{W}},\nonumber\\
&&Y_3=\left(\frac{m_W}{\Lambda}\right)^2
      \bigg[f_{W\tilde{W}}+\frac{1}{4}f_{\tilde{W}}\bigg],\qquad
  Y_4=\left(\frac{m_W}{\Lambda}\right)^2
      \bigg[f_{B\tilde{B}}-f_{B\tilde{W}}-f_{W\tilde{W}}\bigg].
\end{eqnarray}
%
We note in passing that if all the coefficients, $f_i$, are of the 
similar size, then $Y_2$ would be about ten times smaller than the 
other form factors in size because of the factor $g^2/4\sim 0.1$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Electron-positron mode}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The angular dependence
$\Sigma_{L,R}(\Theta;\theta,\bar{\theta};\phi,\bar{\phi})$
of a sequential process $e^+e^-\rightarrow W^+W^-\rightarrow
(f_1\bar{f}_2)(f_3\bar{f_4})$ can be 
decomposed in terms of eighty-one orthogonal functions 
${\cal D}_\alpha$'s of the $W^+$ and $W^-$ decay products as
%
\begin{eqnarray}
\Sigma_{L,R}(\Theta;\theta,\bar{\theta};\phi,\bar{\phi})
  =
\sum_{\alpha=1}^{81}{\cal P}_{\alpha L,R}(\Theta)
     {\cal D}_\alpha(\theta,\bar{\theta};\phi,\bar{\phi}),
\label{eq:angular_dependence:W}
\end{eqnarray}
%
where $\Theta$ is the scattering angle between $e^-$ and $W^-$, 
and $\theta(\bar{\theta})$ and $\phi(\bar{\phi})$ are
the angular variables of the $W^-$ and $W^+$ decay products.
All the terms, ${\cal P}_{\alpha X}$ and ${\cal D}_\alpha$, can 
be divided into four categories under $CP$ and $CP\tilde{T}$: 
even-even, even-odd, odd-even, and odd-odd terms\cite{Choi1}. 
There exist {\it thirty-six} independent $CP$-odd terms among which 
{\it eighteen} terms are $CP\tilde{T}$-even and the other {\it eighteen} 
$CP\tilde{T}$-odd.

Including electron polarization, most of the  
works\cite{HPZH,GgDs,Chang,PkPm} prior to our works
have considered four special $CP$-odd and $CP\tilde{T}$-even asymmetries, 
of which two are 
essentially equivalent to the so-called triple vector products, and four 
new $CP$-odd and $CP\tilde{T}$-odd asymmetries in addition to the two 
conventional lepton energy asymmetries. 
Clearly, the present analysis shows that much more $CP$-odd asymmetries 
are available.

Observables which are constructed from the momenta of the
charged leptons originating from $W^+$ and $W^-$
decay are directly and most easily measurable in future experiments. 
Therefore, the $W$ leptonic decay channels together with the 
corresponding charge-conjugated ones are exclusively
used in the following analysis. 
The first set of observables under consideration 
involves the momentum of a lepton from $W^-$ decay 
correlated with the momentum of a lepton from
$W^+$ decay in the sequential process
%
\begin{eqnarray}
e^+(\vec{p}_{\bar{e}})+e^-(\vec{p}_{e})
   \rightarrow W^+ + W^-
   \rightarrow l^+(\vec{q}_+)+l^-(\vec{q}_-)+X,
\label{eq:inclusive;eeww}
\end{eqnarray}
%
As shown in Ref.~\cite{BN} a tower of $CP$-odd observables can be
in principle constructed, 
among which a few typical $CP$-odd observables\cite{BN} are listed
in the following:
%
\begin{eqnarray}
&& A_1=\hat{p}_e\cdot(\vec{q}_+\times\vec{q}_-),\nonumber\\
&& T_{ij}=(\vec{q}_--\vec{q}_+)_i(\vec{q}_-\times\vec{q}_+)_j
        +(i\leftrightarrow j),\nonumber\\
&& A_E=E_+-E_-,\qquad
   A_2=\hat{p}_e\cdot(\vec{q}_++\vec{q}_-),\nonumber\\
&&Q_{ij}=(\vec{q}_-+\vec{q}_+)_i(\vec{q}_--\vec{q}_+)_j
        +(i\leftrightarrow j)
        -\frac{2}{3}\delta_{ij}(\vec{q}^2_--\vec{q}^2_+).
\label{observable}
\end{eqnarray}
%
The observables $T$ and $A_1$ are $CP\tilde{T}$-even, whereas 
$Q$, $A_2$ and $A_E$ are $CP\tilde{T}$-odd.
Certainly, all the $CP$-odd observables can be expressed as a linear
combination of a fixed number of linearly-independent observables whose
classification depends only on the spins of the final particles\cite{Choi1}.
Because of the lack of space, we refer to the work\cite{Choi1} for
a more detailed explanation for the point and, following the same procedure 
as Ref.~\cite{BN}, we use the observables (\ref{observable}) in probing 
$CP$ violation in $e^+e^-\rightarrow W^+W^-$ in the present report. 


The statistical significance of a given observable $O_X$ is determined by
comparing its expectation $\langle O_X\rangle$ with the expectation 
variance $\langle O^2_{X}\rangle_{\rm SM}$ in the SM where
$\langle O_X\rangle_{SM}$ vanishes.
Quantitatively, an observation of any deviation from the SM expectation
with better than one-standard deviations requires
%
\begin{eqnarray}
\langle O_{X}\rangle\geq
        \sqrt{\frac{\langle O_{X}^2\rangle_{\rm SM}}{N_{WW}}},\qquad 
N_{WW}=\varepsilon\left[B_{X^+}\bar{B}_{X^-}\right]
        {\cal L}_{ee}\sigma(e^+e^-\rightarrow W^+W^-),
\label{eq:deviation;w}
\end{eqnarray}
%
where $N_{WW}$ is the number of events,  ${\cal L}_{ee}$ 
is the $e^+e^-$ integrated luminosity, 
$B_{X^\pm}$ are the branching fractions of $W^\pm\rightarrow l^\pm \nu_l$,
and $\varepsilon$ is the detection efficiency, which is assumed to be 
unity.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ylimit.eps 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[thb]
\hbox to\textwidth{\hss\epsfig{file=ylimit.eps,height=5.0cm}\hss}
\caption{(a) The 1-$\sigma$ allowed regions of 
             $Re(Y_1)$ and $Re(Y_2)$ through $A^l_1$ (solid) and 
             $T^l_{33}$ (long-dashed) with polarized electron beams 
             and with the $e^+e^-$ integrated luminosity 10 fb$^{-1}$ 
             at $\sqrt{s}=500$ GeV.  
         (b) The 1-$\sigma$ allowed regions of 
             $Im(Y_1)$ and $Im(Y_2)$ through $A^l_E$ (solid), 
             $A^l_2$ (long-dashed) and $Q^l_{33}$ (dashed) with 
             polarized electron beams and with the $e^+e^-$
             integrated luminosity 10 fb$^{-1}$ at $\sqrt{s}=500$ GeV.}
\label{fig:ylimit}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

In determining the 1-$\sigma$ allowed region of the $(Y_1,Y_2)$ plane 
we take the following set of experimental parameters:
%
\begin{eqnarray}
\sqrt{s}=0.5\ \ {\rm TeV},\qquad L_{ee}=10\ \ {\rm fb}^{-1}.
\label{exp_parameters}
\end{eqnarray}
% 
We note in Fig.~\ref{fig:ylimit} that the use of longitudinal 
electron beam polarization obviates the need for the simultaneous 
measurement of more than one distribution 
and it can greatly enhance the sensitivities to the $CP$-odd parameters
by using two or more $CP$-odd observables.
The 1-$\sigma$ optimal sensitivities to ($Re(Y_1)$, $Re(Y_2)$) and 
($Im(Y_1)$, $Im(Y_2)$) read 
%
\begin{eqnarray}
&& |Re(Y_1)|\leq 2.9\times 10^{-3},\qquad 
   |Re(Y_2)|\leq 3.7\times 10^{-4},\nonumber\\
&& |Im(Y_1)|\leq 3.9\times 10^{-3},\qquad 
   |Im(Y_2)|\leq 1.0\times 10^{-3}.
\label{optimal bound on real Y_1 and Y_2}
\end{eqnarray}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Two-photon mode}
\label{subsubsec:WW_PP}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In counting experiments where final $W$ polarizations are not
analyzed, we measure only the distributions summed over the final
$W$ polarizations, from which the explicit form of $\Sigma_{\rm unpol}$, 
$ \Sigma_{02}$, $\Delta_{02}$, $\Sigma_{22}$, and  $\Sigma_{00}$
in Eq.~(\ref{linear}) can be obtained.
First of all, we emphasize that ${\cal I}(\Sigma_{00})$ does not require
any identification of the scattering plane as mentioned before. 
Even if one excludes the $\tau^+\tau^-+\not\!{p}$ modes of 1\%, 
the remaining 99\% of the events can be used
to measure ${\cal I}(\Sigma_{00})$. On the other hand, the scattering
plane should be identified to measure ${\cal I}(\Sigma_{02})$. 
Nevertheless, it is worth noting that the charge of the decaying 
$W$ is not needed to extract ${\cal I}(\Sigma_{02})$. 
Therefore, all the modes except for the $l^+l^-+\not\!{p}$ modes 
(9\%) can be used for ${\cal I}(\Sigma_{02})$.

The $\gamma\gamma\rightarrow W^+W^-$ reaction has a much 
larger cross section than heavy fermion-pair production such as 
$\gamma\gamma\rightarrow t\bar{t}$ and, furthermore, the total cross 
section approaches a constant value at high energies. 
At $\sqrt{\hat{s}}=500$ GeV the total cross section is about 80~pb, 
while the $t\bar{t}$ cross section is about 1~pb. So, there exist no
severe background problems. 
In the following analysis we simply assume that all the $W$ pair events 
can be used. 

We present our numerical results for the following set of experimental
parameters:             
%
\begin{eqnarray}
\sqrt{s}=0.5\ \ {\rm and}\ \  1.0\ \ {\rm TeV},\qquad 
\kappa^2 L_{ee}=20\ \ {\rm fb}^{-1}.
\label{rr_exp}
\end{eqnarray}
%
Here, $\kappa$ is the $e$-$\gamma$ conversion coefficient in the Compton
backscattering. The parameter $x$, which is dependent on the laser 
frequency $\omega_0$, is treated as an adjustable parameter. 
Folding the photon luminosity spectrum and integrating the 
distributions over the final kinematic variables, we obtain the 
$x$-dependence of available event rates.

Separating the $CP$-odd asymmetries $A_a$ into three parts as
%
\begin{eqnarray}
A_a=R(Y_1)A^{Y_1}_a+R(Y_2)A^{Y_2}_a+R(Y_4)A^{Y_4}_a,
\end{eqnarray}
%
and considering each form factor separately,
we then obtain the $1$-$\sigma$ allowed upper bounds of the form factors
($i=1,2,4$)
%
\begin{eqnarray}
{\rm Max}(|R(Y_i)|_a)
   =\frac{\sqrt{2}}{|A^{Y_i}_a\sqrt{\varepsilon N_{\rm unpol}}|}, 
\end{eqnarray}
%
if no asymmetry is found.
Here, $\varepsilon$ is for the sum of $W$ branching fractions available, 
which is taken to be $100\%$ for $A_{00}$ and $91\%$ for $A_{02}$.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%fig4.eps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\hbox to\textwidth{\hss\epsfig{file=fig4.eps,height=3.5cm,width=8cm}\hss}
\vskip 0.3cm
\hbox to\textwidth{\hss\epsfig{file=fig5.eps,height=3.5cm,width=8cm}\hss}
\caption{The $x$ dependence of ${\rm Max}(|R(Y_1)|)$ and 
         ${\rm Max}(|R(Y_2)|)$ at $\sqrt{s}=0.5$ (solid) and 
         $1.0$ TeV (long-dashed) from (a) $A_{02}$ and 
         (b) $A_{00}$.}
\label{fig:fig4}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
Postponing the analysis of the $CP$-violation effects due to 
$Y_4$\cite{Choi1,Gounaris}, we present the analysis of the constraints on $Y_1$
and $Y_2$. Figs.~\ref{fig:fig4}(a) and (b) show the $x$ dependence of 
the 1-$\sigma$ sensitivities 
to $R(Y_1)$ and $R(Y_2)$, which are obtained from $A_{02}$ and $A_{00}$, 
respectively, for $\sqrt{s}=0.5$ TeV (solid) and $\sqrt{s}=1$ TeV 
(long-dashed). 
These figures and Table~\ref{tbl:1-sigma bounds;Y_1 and Y_2} clearly 
exhibit that
(i) the sensitivities, especially through $A_{00}$,
depend strongly on $x$, and  (ii) the optimal sensitivities on $R(Y_2)$ 
are very much improved as $\sqrt{s}$ increases from $0.5$ TeV to 
1 TeV, while those of $R(Y_1)$ are a little improved.
%
\begin{table}[hb]
\caption{The best $1$-$\sigma$ sensitivities to $R(Y_1)$ 
and $R(Y_2)$, and their corresponding $x$ values for $\sqrt{s}=0.5$ 
and $1$ TeV.}
%
\label{tbl:1-sigma bounds;Y_1 and Y_2}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline\hline
 ${\rm Asymmetry} $   &  \multicolumn{2}{c|}{$A_{02}$}  
                      &  \multicolumn{2}{c|}{$A_{00}$}  \\
\hline
    $\sqrt{s}$ (TeV)  & $0.5$  &  $1.0$  
                      & $0.5$  &  $1.0$ \\ \hline
       $x$            & $1.83$ &  $0.96$
                      & $0.75$ &  $0.31$\\ \hline
  ${\rm Max}(|R(Y_1)|)$  
      &\mbox{ }\hskip 0.2cm  $1.1\times 10^{-2}$\mbox{ }\hskip 0.2cm  
      &\mbox{ }\hskip 0.2cm  $5.0\times 10^{-3}$\mbox{ }\hskip 0.2cm 
      &\mbox{ }\hskip 0.2cm  $3.2\times 10^{-3}$\mbox{ }\hskip 0.2cm  
      &\mbox{ }\hskip 0.2cm  $2.2\times 10^{-3}$\mbox{ }\hskip 0.2cm \\ \hline
       $x$            & $2.09$ &  $1.23$
                      & $1.11$ &  $0.59$\\ \hline
  ${\rm Max}(|R(Y_2)|)$  
      &\mbox{ }\hskip 0.2cm $2.4\times 10^{-4}$\mbox{ }\hskip 0.2cm 
      &\mbox{ }\hskip 0.2cm $9.0\times 10^{-5}$\mbox{ }\hskip 0.2cm
      &\mbox{ }\hskip 0.2cm $2.6\times 10^{-4}$\mbox{ }\hskip 0.2cm 
      &\mbox{ }\hskip 0.2cm $1.1\times 10^{-4}$\mbox{ }\hskip 0.2cm\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Top-quark pair production}
\label{subsec:Top}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


An important property of a heavy top\cite{Top,PDG96} is that
it decays before it can form hadronic
bound states\cite{BDKKZ}. This implies in particular that spin effects, 
for instance polarization of, and spin correlations between $t$ and 
$\bar{t}$ quarks  can be analyzed through the distributions and angular
correlations of the weak decay products of the $t$ and $\bar{t}$
quarks. Moreover, these effects can be calculated in perturbation theory.
They provide an additional means for testing SM predictions and 
of searching for possible new physics effects in top quark production
and decay.

The $\gamma t\bar{t}$ vertex  consists of not only the SM tree-level 
vector and axial-vector coupling terms but also a magnetic dipole moment (MDM)
and an EDM coupling. Likewise, in addition to the tree-level SM $Zt\bar{t}$ 
coupling, we have the analogous $Z$ MDM and $Z$ EDM couplings,
of which the latter is called the top-quark weak dipole moment (WDM).  
The MDM-like couplings are present in the SM at the one-loop level.
On the other hand, the EDM-like couplings violate $CP$ and,
due to the structure of the SM, are only present perturbatively
in the SM at the three loop level\cite{Shabalin}. 
In some extensions to the SM such
as left-right models, multi-Higgs-doublet models (MHD), supersymmetric
SM, however, the EDM couplings may be present even at one-loop 
level\cite{Barr,Soni-Xu}. Neglecting the MDM couplings, we assume the 
$\gamma tt$ and $Ztt$ vertices to be given by 
%
\begin{eqnarray}
\Gamma^V_\mu=v_V\gamma_\mu+a_V\gamma_\mu\gamma_5
            +\frac{c_V}{2m_t}\sigma_{\mu\nu}\gamma_5q^\nu,\qquad
            V=\gamma, Z,
\label{eq:vertex}
\end{eqnarray}
%
with the vector and axial-vector couplings of the top-quark  given 
in the SM.
Here, $q$ is the four-momentum of the vector boson, $V(=\gamma, Z)$. 
Then, for $m_t=175$ GeV, the top-quark EDM and WDM, $d_{\gamma,Z}$,
are related with $c_{\gamma,Z}$ as  
%
\begin{eqnarray}
d_{\gamma,Z}=\frac{e}{m_t}c_{\gamma,Z}
            \approx 1.13\times 10^{-16}c_{\gamma,Z}({\rm e}cm).
\end{eqnarray}
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Electron-positron mode}
\label{subsubsec:Top_EP}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The angular dependence for the process $e^+e^-\rightarrow t\bar{t}
\rightarrow (X^+b)(X^-\bar{b})$, can be written as
%
\begin{eqnarray}
\Sigma_{L,R}(\Theta;\theta,\bar{\theta};\phi,\bar{\phi})
  =\sum^{16}_{i=1}{\cal P}_{iL,R}(\Theta)
       {\cal D}_i(\theta,\bar{\theta};\phi,\bar{\phi}).
\end{eqnarray}
%
where $\Theta$ is the scattering angle for $e^+e^-\rightarrow t\bar{t}$,
and $\theta(\bar{\theta})$ and $\phi(\bar{\phi})$ are the angle
variables for the $b$($\bar{b}$) in the inclusive decays and
for the $l^+$($l^-$) in the semileptonic decays\cite{CJK} of the top and
anti-top quarks, respectively.
The terms, ${\cal P}_\alpha$ and ${\cal D}_\alpha$, can thus be divided 
into four categories under $CP$ and $CP\tilde{T}$\cite{Choi2}.
There exist {\it six} independent $CP$-odd terms among which 
{\it three} terms are $CP\tilde{T}$-even, and the other {\it three}  
$CP\tilde{T}$-odd.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%re.eps 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\hbox to\textwidth{\hss\epsfig{file=polre.eps,height=3.5cm,width=8cm}\hss}
\vskip 0.3cm
\hbox to\textwidth{\hss\epsfig{file=unpolre.eps,height=3.5cm,width=8cm}\hss}
\caption{The 1-$\sigma$ allowed region of $Re(c_\gamma$) and 
         $Re(c_Z)$ through  (a) $A^b_{1}$ (solid) and 
         $T^b_{33}$ (long-dashed) and (b) $A^l_{1}$ (solid) and 
         $T^l_{33}$ (long-dashed) with the $e^+e^-$ integrated 
         luminosity 10 fb$^{-1}$ for the polarized electron beam (upper part)
         and 20 fb$^{-1}$ for the unpolarized electron beam (lower part), 
         respectively, at $\sqrt{s}=500$ GeV.}
\label{fig:polre}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

Including electron beam polarization, Poulose and Rindani\cite{PR}  
recently have considered two new $CP$-odd and $CP\tilde{T}$-even 
asymmetries, of which one asymmetry is essentially equivalent to 
the so-called triple vector product, and two new $CP$-odd and 
$CP\tilde{T}$-odd asymmetries in addition to the two conventional 
lepton energy asymmetries. 
Clearly, that we can use {\it six} more asymmetries among which 
{\it four} asymmetries are $CP$-odd and $CP\tilde{T}$-even and the other 
{\it two} terms are $CP$-odd and $CP\tilde{T}$-odd.

Observables which are constructed from the momenta of the
charged leptons and/or $b$ jets originating from $t$ and $\bar{t}$
decay are directly measurable in future experiments. 
We consider both the inclusive and exclusive semileptonic decays 
%
\begin{eqnarray}
t\rightarrow bX_{\rm had},\qquad
t\rightarrow bl^+\nu;\ \ l=e,\mu,\tau,
\end{eqnarray}
%
together with the corresponding charge-conjugated ones, and
we use the $CP\tilde{T}$-even $T_{33}$ and $A_1$ and the $CP\tilde{T}$-odd
$Q_{33}$, $A_2$ and $A_E$ as in the $W$-pair production case. All the
observables are constructed from the momenta
of the final $b$ and $\bar{b}$ in the inclusive decay mode and of the
final $l^+$ and $l^-$ in the exclusive semileptonic decay mode.

Inserting the values of the SM electron vector and axial-vector couplings,
we obtain for left-handed and right-handed electrons
%
\begin{eqnarray}
c_L=c_\gamma+0.64\delta_Z c_Z,\qquad
c_R=c_\gamma-0.55\delta_Z c_Z,
\label{eq:coupling_num}
\end{eqnarray}
%
where $\delta_Z=(1-m^2_Z/s)^{-1}$.  
For $m_t=175$ GeV and $m_Z=91.2$ GeV, $1\leq \delta_Z\leq 1.073$.
The $c_Z$ contribution to $c_L$ and $c_R$ is similar in size 
but different in sign.  Naturally, electron polarization plays
a crucial role in discriminating $c_\gamma$ and $c_Z$. 


%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%im.eps 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\hbox to\textwidth{\hss\epsfig{file=polim.eps,height=3.5cm,width=8cm}\hss}
\vskip 0.3cm
\hbox to\textwidth{\hss\epsfig{file=unpolim.eps,height=3.5cm,width=8cm}\hss}
\caption{The 1-$\sigma$ allowed region of $Im(c_\gamma)$ and 
         $Im(c_Z)$ through (a) $A^b_{E}$ (solid), $A^b_2$ (long-dashed) 
         and $Q^b_{33}$ (dashed) and 
         (b) $A^l_{E}$ (solid), $A^l_2$ (long-dashed) and $Q^l_{33}$ (dashed) 
         with the $e^+e^-$ integrated luminosity 10 fb$^{-1}$ for
         the polarized electron beam (upper part) and 20 fb$^{-1}$ for the 
         unpolarized electron beam (lower part) at $\sqrt{s}=500$ GeV.}
\label{fig:polim}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

Our numerical results are presented for the following set of experimental
parameters:
%
\begin{eqnarray}
\sqrt{s}=0.5\ \ {\rm TeV},\qquad
L_{ee}= \left\{
 \begin{array}{ll}
20 \ {\rm fb}^{-1} & \ \ {\rm for}\ \ {\rm unpolarized}\ \ {\rm electrons} \\
10 \ {\rm fb}^{-1} & \ \ {\rm for}\ \ {\rm polarized}\ \ {\rm electrons} 
\end{array}, \right.
\end{eqnarray}
%
The shadowed parts in Fig~\ref{fig:polre} show the 1-$\sigma$ allowed 
regions of $Re(c_\gamma$) and $Re(c_Z)$ through (a) $A^b_{1}$ 
and $T^b_{33}$ and (b) $A^l_{1}$ and $T^l_{33}$ with polarized electron 
beams (upper part), respectively, and those in Fig.~\ref{fig:polim} show 
the 1-$\sigma$ allowed regions of $Im(c_\gamma)$ and $Im(c_Z)$ 
through (a) $A^b_{E}$, $A^b_2$ and $Q^b_{33}$ and 
(b) $A^l_{E}$, $A^l_2$ and $Q^l_{33}$ with unpolarized 
electron beams (lower part), respectively. 
We observe several interesting properties from the figures:
(i) The allowed regions strongly depend on electron polarization.    
(ii) Even with unpolarized electrons and positrons, it is possible to obtain
      a closed region for the $CP$-odd parameters by using two or more 
      $CP$-odd asymmetries.  
(iii) With polarized electrons, the most stringent bounds on the
      $CP\tilde{T}$-even and $CP\tilde{T}$-odd parameters are obtained 
      through  $A^b_1$ and $A^b_E$ in the inclusive top-quark decay 
      mode, respectively.
Numerically, the 1-$\sigma$ allowed regions of  
$Re(c_\gamma)$, $Re(c_Z)$, $Im(c_\gamma)$, and $Im(c_Z)$ are 
%
\begin{eqnarray}
&&|Re(c_\gamma)|\leq 0.12,\qquad |Re(c_Z)|\leq 0.20,\\
&&|Im(c_\gamma)|\leq 0.16,\qquad |Im(c_Z)|\leq 0.27.
\end{eqnarray}
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Two-photon mode}
\label{subsubsec:Top_PP}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Extracting ${\cal I}(\Sigma_{00})$ in $\gamma\gamma\rightarrow t\bar{t}$
again does not require identifying the scattering plane, and 
although the $\tau^+\tau^-+\not\!{p}$ modes of 1\% is excluded, 
the remaining 99\% of the events 
can be used to measure ${\cal I}(\Sigma_{00})$.
On the other hand, ${\cal I}(\Sigma_{02})$ turns out to be zero
so that it is useless in determining ${\rm Re}(c_\gamma)$. 

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%rrtt.eps 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[ht]
 \hbox to\textwidth{\hss\epsfig{file=rrttsig00.eps,height=4cm,width=6cm}\hss}
 \caption{The $x$ dependence of the $Re(c_\gamma)$ upper bound, 
          Max($|Re(c_\gamma)|$), at $\sqrt{s}=0.5$ (solid) and 
          1 TeV (long-dashed), from $A_{00}$.} 
 \label{fig:rrttsig00}
 \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

We present our numerical results for the same set of experimental 
parameters as the process $\gamma\gamma\rightarrow W^+W^-$.              
Experimentally, $\gamma\gamma\rightarrow W^+W^-$ is the most 
severe background process against $\gamma\gamma\rightarrow t\bar{t}$.
In our analysis, we simply take for a numerical analysis a rather 
conservative value of the detection efficiency $\varepsilon=10\%$, 
even though more experimental 
analysis is required to estimate the efficiency precisely.  
 
It is clear from Fig.~\ref{fig:rrttsig00} and 
Table~\ref{tbl:1-sigma bounds;c_r} that the constraints on 
$Re(c_\gamma)$ through $A_{00}$ are very sensitive to $x$ and 
$\sqrt{s}$.
It is impressive that the doubling of the c.m. energy
enables us to improve the sensitivities almost by a factor of ten.
%
\begin{table}[ht]
\caption{The optimal 1-$\sigma$ sensitivities to $Re(c_\gamma)$ 
         and their corresponding $x$
         values for $\sqrt{s}=0.5$ and 1 TeV.}
\label{tbl:1-sigma bounds;c_r}
\begin{center}
\begin{tabular}{|c|c|c|}\hline\hline
   $\sqrt{s}$ (TeV)  &   0.5   &  1.0  \\ \hline
       $x$           &   3.43  &  0.85 \\ \hline
   $Re(c_\gamma)$    &   0.16  &  0.02 \\ 
\hline\hline
\end{tabular}
\end{center}
\end{table}
%  





















\newcommand{\prd}[1]{Phys.~Rev.~D{{\bf #1}}}
\newcommand{\prl}[1]{Phys.~Rev.~Lett.~{{\bf #1}}}
\newcommand{\plb}[1]{Phys.~Lett.~{{\bf #1B}}}
\newcommand{\npb}[1]{Nucl.~Phys.~{{\bf B#1}}}
\newcommand{\zpc}[1]{Z.~Phys.~{{\bf C#1}}}
\newcommand{\progtp}[1]{Prog.~Theor.~Phys.~{{\bf #1}}}
\newcommand{\jetpl}[1]{JETP Lett.~{{\bf #1}}}
\newcommand{\sjnp}[1]{Sov.~J.~Nucl.~Phys.~{{\bf #1}}}

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% 
\end{thebibliography}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$CP$ Violation in the Tau Lepton System}
\label{sec:tau}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The $\tau$ has the same interaction structure as the $e$ and $\mu$ in
the SM, apart from their masses. However for practical purposes\cite{Gomez}
the $\tau$ lepton, the most massive of the known leptons, behaves quite
differently from the $e$ and $\mu$ leptons in that (i)
the $\tau$ has hadronic decay modes (e.g. $\tau\rightarrow \pi\nu,
\rho\nu, a_1\nu, K^*, ...$) which allow an efficient measurement of its
polarization\cite{Tsai} and (ii) the coupling to the neutral
and charged Higgs bosons\cite{Grossman,Falk} and other 
scalar particles is expected to dominate those of the $e$ and $\mu$.
These features allow the $\tau$ to be a rather special experimental 
probe of new physics\cite{Nachtmann1,Kilan,Rindani,Tsai1,Huang,Mirkes}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Tau Lepton EDM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Recently, the OPAL\cite{OPAL} and ALEPH\cite{ALEPH} detector groups have 
demonstrated by detailed investigations of $Z\rightarrow\tau^+\tau^-$ at 
LEPI that sensitive $CP$ symmetry tests for the $\tau$ WDM at the few per mill 
level can be performed in high energy $e^+e^-$ collisions.
The OPAL group has employed two optimal genuine $CP$-odd observables,
of which one is $CP\tilde{T}$-even and the other one $CP\tilde{T}$-odd,
and the ALEPH group has used the $CP$-odd tensor observable $T_{33}$,
which can give information only on the real part of the $\tau$ WDM.
Specifically the upper limit on the real and imaginary parts
of the $\tau$ WDM obtained at OPAL are 
$|Re(d_Z)|\leq 7.8\times 10^{-18} e{\rm cm}$ and
$|Im(d_Z)|\leq 4.5\times 10^{-17} e{\rm cm}$ with 95\% confidence level,
while the upper limit on the real part of the $\tau$ WDM obtained at ALEPH
is $|Re(d_Z)|\leq 1.5\times 10^{-17} e{\rm cm}$.

Compared with the present constraints on the $\tau$ WDM,
those on the $\tau$ EDM are rather weak and, in contrast to the OPAL
and ALEPH measurements, the reported 
measurements\cite{PDG96} have used indirect methods based on the 
$CP$-even observables such as the differential cross section of 
$e^+e^-\rightarrow \tau^+\tau^-$\cite{Delaguila}, the 
partial decay widths of $Z\rightarrow \tau^+\tau^-$\cite{Grifols} and
$Z\rightarrow \tau^+\tau^-\gamma$\cite{Escribano}. 
We claim that {\it the measurements may not be regarded as genuine
$\tau$ EDM measurements and therefore the quoted values have to be 
replaced by those by  direct measurements through $CP$-odd 
observables}\cite{Nachtmann}. 
Surprisingly, no direct $\tau$ EDM measurements have been reported
in the literature.
Therefore, we strongly suggest the $\tau$ EDM to be measured directly
at the existing collider facilities TRISTAN, LEPII and CLEOII.

In the present report, first of all, we present a rough comparison of 
the potential of TRISTAN and LEPII in the $\tau$ EDM
measurements based on the following experimental parameters:
%
\begin{eqnarray}
\begin{array} {clll}
{\rm TRISTAN} & :  &  \sqrt{s}=\ \ 60 {\rm GeV},   
              & \ \ {\cal L}_{ee}=271 {\rm pb}^{-1},\\
{\rm LEPII}   & :  &  \sqrt{s}=180 {\rm GeV},
              & \ \ {\cal L}_{ee}=500 {\rm pb}^{-1}
\end{array}
\label{tau:exp_par}
\end{eqnarray}
%
In the above quoted values, we note that TRISTAN
has already accumulated the integrated luminosity 
$271$ pb$^{-}$\cite{Hanai}, but LEPII is assumed to eventually 
accumulate 500 pb$^{-1}$, which is the designed integrated
luminosity per year. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tristan_lep2.eps - TRISTAN & LEP II 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\hbox to\textwidth{\hss\epsfig{file=tristan_lep2.eps,
                   height=8cm,width=10cm}\hss}
\vskip 0.5cm
\caption{The 1-$\sigma$ allowed region (left part) of 
         $Re(c_\gamma)$ and $Re(c_Z)$
         through $A_1$ (solid) and $T_{33}$ (long-dashed), 
         and that (right part) of $Im(c_\gamma)$ and $Im(c_Z)$ through
         $A_E$ (solid) and $A_2$ (long-dashed) at TRISTAN and 
         LEPII with the experimental parameter values 
         (\ref{tau:exp_par}). The lines for $Q_{33}$ are overlapped
         with those for $A_E$.}
\label{fig:Re_EDM}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the sake of simplicity, we consider the simplest semileptonic decay
of the $\tau$ lepton, $\tau\rightarrow\pi\nu_\tau$ whose branching 
ratio is 11\%. Certainly, to obtain better constraints, all the 
reconstructable decay channels of the $\tau$ should be included.
Given the branching fraction (11\%), TRISTAN and LEPII yield
ninety-seven and twenty-three events of the sequential
process $e^+e^-\rightarrow \tau^+\tau^-\rightarrow
(\pi^+\bar{\nu}_\tau) (\pi^-\nu_\tau)$.
To make a rough estimate of the sensitivities to be obtained, we
use the same set of $CP$-odd observables as used in the top-quark
pair production at NLC in Section~\ref{sec:NLC}, that is to say, 
the $CP\tilde{T}$-even $A_1$ 
and $T_{33}$ and the $CP\tilde{T}$-odd $A_E$, $A_2$, and $Q_{33}$.
All the $CP$-odd observables are constructed from the momenta of the
electron in the initial state and two pions in the final state.

We exhibit in Fig.~\ref{fig:Re_EDM} the 1-$\sigma$ allowed 
region of the $CP$-violation parameters $c_\gamma$ and $c_Z$, which are 
numerically related with the $\tau$ EDM and WDM as follows
%
\begin{eqnarray}
d^{\gamma,Z}_\tau=1.1\times 10^{-14} c_{\gamma, Z} (e{\rm cm}).
\end{eqnarray}
%
Incidentally, it is clear that TRIATAN and LEPII can not compete with LEPI
in measuring the $\tau$ WDM as can be seen clearly by comparing the
bound in the figure and the values quoted in the first paragraph of the
present section. So, we take into account the LEPI measurements
and then we derive the 1-$\sigma$ allowed range for the 
$\tau$ EDM by taking the $x$-axis cut values. Before listing the 
numerical values, we note that the $CP\tilde{T}$-odd 
observables $A_E$ and $Q_{33}$ provide very similar constraints
on the imaginary part of the $\tau$ EDM so that their 1-$\sigma$
boundary lines can be hardly distinguished in Fig.~\ref{fig:Re_EDM}. 
Combined with the LEPI results, 
the 1-$\sigma$ sensititivies to the $\tau$ EDM are
%
\begin{eqnarray}
\begin{array} {clll}
{\rm TRISTAN} & : &  |Re(d_\tau^\gamma)|\leq 5.3\times 10^{-16} (e{\rm cm}),
                  &  |Im(d_\tau^\gamma)|\leq 1.1\times 10^{-15} (e{\rm cm}),\\
{\rm LEPII}   & : &  |Re(d_\tau^\gamma)|\leq 4.4\times 10^{-16} (e{\rm cm}),
                  &  |Im(d_\tau^\gamma)|\leq 2.8\times 10^{-15} (e{\rm cm}),
\end{array}
\end{eqnarray}
%
TRISTAN and LEPII can measure the real part of the $\tau$ EDM with 
similar sensitivities, which may become comparable with those quoted by 
the PDG group, in the case that all the reconstructable decay modes of the 
tau lepton are included.  

On the other hand, the $e^+e^-$ storage ring CESR has accumulated the 
integrated luminosity of about $3.5$ fb$^{-1}$ at the c.m. energy 
$\sqrt{s}=10.6$ GeV in the years between 1990 and 1994\cite{CLEOII}. 
It corresponds to the production of about $3.02\times 10^{6}$ $\tau$ pairs, 
which is about four-hundred times larger than the number of $\tau$ pairs 
at TRISTAN with its integrated luminosity 271 pb$^{-1}$. 
Therefore, a great improvement of the sensitivities to the $\tau$ EDM 
is expected. As a matter of fact, as shown in Fig.~\ref{fig:cleo}, 
the 1-$\sigma$ allowed range for $Re(c_\gamma)$ is about four times smaller 
than that at TRISTAN and LEPII, while the 1-$\sigma$ allowed range of
$Im(c_\gamma)$ is about fifteen times smaller than that at TRISTAN.
Quantitatively, the 1-$\sigma$ sensitivities to the $\tau$ EDM, if
the LEPI results on the $\tau$ WDM are included, are
%
\begin{eqnarray}
|Re(d^\gamma_\tau)|\leq 1.5\times 10^{-16} (e{\rm cm}),\qquad
|Im(d^\gamma_\tau)|\leq 6.7\times 10^{-17} (e{\rm cm}).
\end{eqnarray}
% 

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% decleounpol.eps - CLEOII 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\hbox to\textwidth{\hss\epsfig{file=cleo.eps,height=4cm,width=8cm}\hss}
\vskip 0.5cm
\caption{(a) The 1-$\sigma$ allowed region of 
         $Re(c_\gamma)$ and $Re(c_Z)$ through $A_1$ (solid) and 
         $T_{33}$ (long-dashed) and (b) that  of 
         $Im(c_\gamma)$ and $Im(c_Z)$ through $A_E$ (solid)
         and $Q_{33}$ (dot-dashed) with the $3.02\times 10^{6}$ 
         $\tau$ pairs at the CLEOII energy $\sqrt{s}=10.6$ GeV.}
\label{fig:cleo}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
 
The quoted indirect limits of the absolute value of the $\tau$ EDM is
$5\times 10^{-17} (e {\rm cm})$, which is still three times smaller
than the expected limits from the CLEOII measurements.
However, we should note that only the simplest decay, 
$\tau\rightarrow\pi\nu_\tau$, whose branching fraction is 11\%, 
has been considered in our analysis. Therefore, we expect that including 
other dominant decay modes such as $\tau\rightarrow \rho\nu_\tau, 
a_1\nu_\tau,\ldots$ enable us to easily improve the sensitivities 
more than five times. The more detailed work on the CLEOII measurements of
the $\tau$ EDM with all the reconstructable decay modes included
is in progress\cite{Choi4}. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$CP$ Violation in Semileptonic $\tau$ Decays}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

An observation of $CP$ violation in $\tau$ decays requires the existence 
of not only a $CP$ violating phase but also the interference of processes with 
different $CP$ phases. One can have in general a $CP$ violating phase 
between the $W$-exchange diagram and the charged-scalar-exchange 
diagram in models beyond the SM such as MHD and scalar
leptoquark (SLQ) models. On the other hand, two resonance states 
with large width-to-mass ratios\cite{PDG96}, which decay to the same 
final states, leads to an interference of the two processes with different 
$CP$ phases. 

The decay amplitudes of $\tau\rightarrow 3\pi\nu_\tau$ and
$\tau\rightarrow K\pi\nu_\tau$ have contributions from the two 
overlapping resonances ($a_1,\pi^\prime$) and ($K^*,K^*_0$)
with different spins and relatively large width-to-mass 
ratios. Here we should note that the parameters of the resonances
are not so accurately determined. In the $\tau$ decays, various 
phenomenological parameterizations\cite{Jadach,Tornqvist,Isgur,Decker}
of the form factors have been 
employed to analyze experimental data. Keeping in mind the
uncertainty of the resonance parameters, we simply adopt the
parameterization of the $\tau$-decay library TAUOLA\cite{Jadach} 
for the masses and widths of the resonances tabulated as follows
%
\begin{center}
\begin{tabular}{|c||c|c||c|c|}\hline\hline
resonances  &   $a_1$     &    $\pi^\prime$   &  $K^* $   &   $K^*_0$ \\ \hline
$J^P$       &   $1^+$     &    $0^-$          &  $1^-$    &   $0^+$   \\ \hline
mass (GeV)  &   $1.25$    &    $1.30$         &  $0.89$   &   $1.43$  \\ \hline
width (GeV) &   $0.6$     &    $0.3$          &  $0.05$   &   $0.29$  \\ 
 \hline\hline
\end{tabular}
\end{center}
%

The semileptonic decay modes of the $\tau$ are promising for 
the detection of $CP$ violation for the following reasons. 
First, no tagging of the other $\tau$ is necessary. 
Second, the decay modes can be measured not only at the CESR 
and CERN $e^+e^-$ Collider LEP but
also at the planned B factories and $\tau$-charm factories\cite{Gomez} 
where many $\tau$ leptons ($10^7$ to $10^8$) are expected to be produced.

There is, however, an experimental difficulty; Since the neutrinos 
escape detection, it is in general very difficult to reconstruct 
the $\tau$ rest frame. Nevertheless, there are two situations
where the $\tau$ rest frame can be actually reconstructed. 
One is $\tau$-pair production close to threshold where $\tau$ leptons
are produced at rest. This possibility can be realized at future
$\tau$-charm factories. The other is when both $\tau$ leptons decay 
semileptonically. In the latter case impact parameter methods allow us
to reconstruct the $\tau$ rest frame even for $\tau$'s in flight.
Obviously, the impact parameter method\cite{Kuhn} 
requires the full identification
of the decay product momenta. Therefore, we use the decay modes
$\tau^-\rightarrow \pi^-\pi^+\pi^-\nu_\tau$ in the $3\pi$ mode and 
$\tau^-\rightarrow (K^0_s\rightarrow\pi^+\pi^-)\pi^-\nu_\tau$ in
the $K\pi$ mode, of which
the branching fractions are 6.8\% and 0.33\%, respectively.
 
Generally, the matrix element for the semileptonic $\tau$ decays 
can be cast into the form
%
\begin{eqnarray}
M=\sqrt{2}G_F \Bigl[(1+\chi)\bar{u}(k,-)\gamma^\mu P_- u(p,\sigma)J_\mu
                 +\eta\bar{u}(k,-)P_+ u(p,\sigma)J_S\Bigr],
\label{decaym}
\end{eqnarray}
%
where $G_F$ is the Fermi constant, $p$ and $k$ are the four momenta 
of the $\tau$ lepton and the tau neutrino, respectively, and $\eta$ is 
a parameter determining the size of the scalar contribution.
$J_\mu$ and $J_S$ are the vector hadronic current and 
the scalar hadronic current given by
%
\begin{eqnarray}
\begin{array}{ll}
J^{3\pi}_\mu
     =\cos\theta_C\langle 3\pi|\bar{d}\gamma_\mu(1-\gamma_5)u|0\rangle, &
\ \ J^{3\pi}_S  
     =\cos\theta_C\langle 3\pi|\bar{d}(1+\gamma_5)u|0\rangle, \\
J^{K\pi}_\mu
     =\sin\theta_C\langle (K\pi)^-|\bar{s}\gamma_\mu u|0\rangle, & 
\ \ J^{K\pi}_S
     =\sin\theta_C\langle (K\pi)^-|\bar{s}u|0\rangle,
\end{array}
\end{eqnarray}
%
where $\theta_C$  is the Cabibbo angle.
The explicit form of $J_\mu$  is found in the $\tau$ decay library TAUOLA
and the hadronic scalar current $J_S$ can be determined by the
Dirac equation from the $J_\mu$ current.

We can now construct a $CP$-even sum $\Sigma$
and a $CP$-odd difference $\Delta$ of the differential $\tau^\pm$ 
decay rates:
%
\begin{eqnarray}
&& \Sigma=\frac{{\rm d}\Gamma}{{\rm d}\Phi_{X^-}}
                [\tau^-\rightarrow X^-\nu_\tau]
         +\frac{{\rm d}\Gamma}{{\rm d}\Phi_{X^+}}
                [\tau^+\rightarrow X^+\bar{\nu}_\tau],
          \nonumber\\ 
&& \Delta=\frac{{\rm d}\Gamma}{{\rm d}\Phi_{X^-}}
                [\tau^-\rightarrow X^-\nu_\tau]
         -\frac{{\rm d}\Gamma}{{\rm d}\Phi_{X^+}}
                [\tau^+\rightarrow X^+\bar{\nu}_\tau]\Bigr\},
\end{eqnarray}
%
where $X=3\pi, K\pi$ and $\Phi_{X^\pm}$ are the full kinematic variables
of the detectable final states $X^+$ and $X^-$, which should be
related with each other under $CP$ transformations.
Then, the $CP$-odd $\Delta$ is proportional to the imaginary
part of the parameter $\xi$, which is given by
%
\begin{eqnarray}
\xi_{3\pi}=\frac{m^2_{\pi^\prime}}{(m_u+m_d)m_\tau}
    \left(\frac{\eta}{1+\chi}\right),\qquad
\xi_{K\pi}=\frac{m^2_{K^*_0}}{(m_s-m_d)m_\tau}
    \left(\frac{\eta}{1+\chi}\right).
\end{eqnarray}
%
We re-emphasize that every $CP$ asymmetry requires not only 
a non-vanishing $Im(\xi)$ but also the interference between 
the longitudinal mode of the vector
meson and the scalar meson and the interference is proportional to the 
decay constants $f_{\pi^\prime}$ and $f_{K^*_0}$. The value of the 
$\pi^\prime$ decay constant, $f_{\pi^\prime}=0.02$-$0.08$ GeV 
estimated in Ref.~\cite{Isgur} and $f_{\pi^\prime}=0.02$ GeV quoted 
in TAUOLA are quitely likely
invalid because the mixing between the chiral pion field and a
massive pseudoscalar $q\bar{q}$ bound state should be considered. 
So, we have reconsidered $f_{\pi^\prime}$ in the chiral Lagrangian
framework and have shown that it actually vanishes in the chiral
limit due to the mixing of $\pi^\prime$ with the chiral pion 
field\cite{Choi3}.
$f_{\pi^\prime}$ is indeed proportional to the square of the pion mass
$m_\pi$ and so it may be much smaller than the value quoted in 
TAUOLA. This small $f_{\pi^\prime}$ value is also supported by
arguments from QCD sum rules. 
On the other hand, $f_{K^*_0}$ can be well determined 
phenomenologically by using the well-measured $K^*_0$ width 
and the QCD sum rule technique.
Considering possible uncertainties in our estimates we use in our
numerical analysis $f_{\pi^\prime}=(1\sim 5)\times 10^{-3}$ GeV
for the $\pi^\prime$ decay constant, which is a slightly broader range
than estimated, and $f_{K^*_0}=45$ MeV
for the $K^*_0$ decay constant, which is rather well estimated. 

As possible new sources of $CP$ violation detectable in the $\tau$ 
decays we consider new scalar-fermion interactions which preserve 
the symmetries of the SM. Under those conditions there exist
only four types of scalar-exchange models\cite{Davies}, 
which can contribute to the decays, $\tau\rightarrow 3\pi\nu_\tau$ and
$\tau\rightarrow (K\pi)\nu_\tau$.
One is the MHD model\cite{Grossman} and the other three are 
scalar-leptoquark (SLQ) models\cite{Davies,Hall}. 
The quantum numbers of the three leptoquarks under the gauge group 
${\rm SU}(3)_C\times {\rm SU}(2)_L\times {\rm U}(1)_Y$ are 
%
\begin{eqnarray}
&& \Phi_1=\left(3,3,\ \ \frac{7}{6}\right)\ \ ({\rm model}\ \ 
         {\rm I}),\nonumber\\
&& \Phi_2=\left(3,1,-\frac{1}{3}\right)\ \ ({\rm model}\ \ 
         {\rm II}),\nonumber \\
&& \Phi_3=\left(3,3,-\frac{1}{3}\right)\ \ ({\rm model}\ \ 
         {\rm III}),
\end{eqnarray}
%
respectively. The hypercharge $Y$ is defined to be $Q=I_3+Y$.

The constraints on the $CP$-violation parameters $\xi_{3\pi}$ and $\xi_{K\pi}$
depend upon the values of the $u$, $d$ and $s$ current quark masses,
which are taken to be
%
\begin{eqnarray}
m_u=5\ \ {\rm MeV},\qquad m_d=9\ \ {\rm MeV},\qquad  
m_s=320\ \ {\rm MeV}.
\end{eqnarray}
%
The present experimental constraints on the $CP$-violation parameters in the
MHD model, which have been extensively reviewed in Ref.\ \cite{Grossman},
are 
%
\begin{eqnarray}
  \begin{array}{l}
\Gamma(B\rightarrow X\tau\nu_\tau)\\
\Gamma(K^+\rightarrow\pi^+\nu\bar{\nu})
  \end{array}
\Rightarrow 
|Im(\xi_{MHD})|_{3\pi}<0.28, \ \
|Im(\xi_{MHD})|_{K\pi}<0.48,
\label{HDc}
\end{eqnarray}
%
for the charged Higgs mass $M_H=45$ GeV.
There are at present no direct constraints on the $CP$-violating 
parameters in the SLQ models. Assuming that the three leptoquark couplings 
to fermions are universal,  
we can roughly estimate the constraints on the SLQ $CP$-violating 
parameters. The constraints\cite{PDG96,Dohmen,Marciano} 
on the $CP$-violation parameters 
from various low energy experiments then are
%
\begin{eqnarray}
\Gamma(K_L\rightarrow \mu e)
   &\Rightarrow& 
   |Im(\xi^I_{SLQ})|_{3\pi} < 1.5\times 10^{-3}, \nonumber\\
\frac{\Gamma(\mu{\rm Ti}\rightarrow e{\rm Ti})}{\Gamma
      (\mu{\rm Ti}\rightarrow {\rm capture})}
   &\Rightarrow& 
   |Im(\xi^{II}_{SLQ})|_{3\pi} < 0.9\times 10^{-3}, \\
D\bar{D}\ \ {\rm mixing} 
   &\Rightarrow& \left\{\begin{array}{l} 
    |Im(\xi^I_{SLQ})|_{K\pi} < 4\times 10^{-2}, \\ 
    |Im(\xi^{II}_{SLQ})|_{K\pi}< 6\times 10^{-2},
        \end{array}\right. \nonumber\\
\Gamma(K\rightarrow \pi\nu\bar{\nu})
   &\Rightarrow&
    |Im(\xi^{III}_{SLQ})|_{K\pi}< 10^{-5},
\end{eqnarray}
%
Compared to the constraint on the $Im(\xi_{MHD})$,
the constraints on the SLQ $CP$-violation parameters
are much more severe. Especially, the constraint on $Im(\xi^{III}_{SLQ})$
is so severe that no $CP$-violation effects are expected to be observed.
Therefore we will not consider this type-III SLQ model any longer.

The following numerical analysis is made for the maximally allowed values 
of the $CP$-violation parameters and the number of $\tau$ leptons required 
to detect $CP$ violation is estimated by using 
the optimal $CP$-odd asymmetry\cite{Atwood}  
%
\begin{eqnarray}
w_{\rm opt}=\frac{\Delta}{\Sigma}.
\end{eqnarray}
%

%
\vskip 0.5cm
\begin{table}[ht]
\caption{ The maximal expected size of $\varepsilon_{\rm opt}$ and 
the number of $\tau$ leptons, $N$, for detection with 
the $\varepsilon_{\rm opt}$ at the $2\sigma$ level in the $3\pi$
and $K\pi$ modes for $f_{\pi^\prime}=1\sim 5$ MeV and $f_{K^*_0}=45$ MeV 
with the maximally-allowed values for the $CP$-violation parameters.} 
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline\hline
Model  & $\varepsilon^{3\pi}_{\rm opt}$(\%)
       & $\varepsilon^{K\pi}_{\rm opt}$(\%) 
       & $N^{3\pi}$                     & $N^{K\pi}$  \\ \hline
MHD    & $0.13\sim 0.67$                & $2.3$  
       & $(0.13\sim 3.3)\times 10^7$    & $2.3\times 10^6$  \\ \hline
SLQI   & $(0.7\sim 3.6)\times 10^{-3}$  & $0.2$  
       & $(0.5\sim 11)\times 10^{11}$   & $3.4\times 10^8$ \\ \hline
SLQII  & $(0.4\sim 2.2)\times 10^{-3}$  & $0.3$  
       & $(1.2\sim 33)\times 10^{11}$   & $1.5\times 10^8$ \\ 
\hline\hline
\end{tabular}
\end{center}
\label{tab:size}
\end{table}
%

Table~\ref{tab:size} shows the expected size of
$\varepsilon_{\rm opt}$, along with the number of $\tau$  
leptons, $N$, required to obtain the 2-$\sigma$ signal with the
optimal asymmetry, $\varepsilon_{\rm opt}$, 
in the MHD model and the two SLQ models for the $CP$-violation 
parameter values. The values of $\varepsilon_{\rm opt}$ and 
the corresponding $N$ values in Table~\ref{tab:size} show that 
the $CP$-violating effects from the MHD model can be detected with 
less than $10^7$ $\tau$ leptons, while still more than $10^{8}$ $\tau$ 
leptons are required to see $CP$ violation in the SLQ models. 
In light of the fact that about $10^7$ and $10^8$ $\tau$ leptons are 
produced yearly at $B$ and $\tau$-Charm factories, respectively, 
$CP$ violation from the MHD model can hopefully be observed.








