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\begin{titlepage}
%\begin{flushright}
%KEK-CP-049 \\
%KEK Preprint 97-42 \\
%June 1997 \\
%H/D \\
%September 13, 1996 
%\end{flushright}
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{\Large 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ with a $t\bar{t}$ production}
\end{center}
\vskip 0.5cm
\begin{center}
{\Large F. Yuasa\footnote{fukuko.yuasa@kek.jp}, Y.
Kurihara\footnote{kurihara@minami.kek.jp} and S. Kawabata\footnote{kawabata@minami.kek.jp\\\\PACS numbers:13.65.+i, 14.65.Ha\\
Keywords: top quark, semi-leptonic, 6body, linear colliders, GRACE \\
}} \\
High Energy Accelerator Research Organization\\
Oho 1-1, Tsukuba, Ibaraki 305, Japan
\end{center}

\begin{abstract}
The cross section of $e^+e^- \to b \bar{b} u \bar{d} \mu^-
\bar{\nu}_\mu$ process with a complete set of tree diagrams, 232
diagrams in the unitary gauge, was calculated at the energy range of
$\sqrt{s}$ = 340 - 500 GeV by using {\tt GRACE} system. 
A main contribution to the cross section comes from $t\bar{t}$
production, where $t$ and $\bar{t}$ decay
into $bu\bar{d}$ and $\bar{b} \mu^- \bar{\nu}_{\mu}$, respectively.
It was found that the interference between  the diagrams 
with $t\bar{t}$ production and those with single-$t$ through
$ WW $ pair production amounts to 10\% at the
$t \bar{t}$ threshold energy region.
In the energy region above twice of the top quark mass, more than 95\% of
the cross section comes from the $t\bar{t}$ diagrams. 
%The complete tree level cross sections of  
%$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
%process, which has 232 diagrams in unitary gauge,
%are calculated at $\sqrt{s}$ = 340 -500 GeV using {\tt GRACE} system. 
%A main contribution comes from $t\bar{t}$ production, $t$ ($\bar{t}$) decay
%into $bu\bar{d}$ ($\bar{b} \mu^- \bar{\nu}_{\mu}$), respectively.
%It is found that the destructive interference between  the diagrams 
%through $t\bar{t}$ production and one through single-$t$ via
%$ WW $ is around 10\% at the  $t \bar{t}$ threshold region.
%Above twice of the  top quark mass, the contribution 
%from non $t\bar{t}$ diagrams is less than 5\%.

\end{abstract}
\end{titlepage}

\section{Introduction}

The top quark physics\cite{top1, top2, top3, top}
is one of the most interesting targets in the near
future $e^+e^-$ linear collider experiments\cite{lc}. 
Since the top quark mass is around 174
GeV\cite{fermi1, fermi2}, 
by the linear colliders at the energy $ {\sqrt{s}} = 340 - 500 $
GeV the top physics  can  be studied precisely.
For example, due to the large top decay width, $t$ decays into $bW $ 
without non perturbative QCD interaction at the long
distance. This means that the study of perturbative QCD can be
done clearly by using the $t\bar{t}$ production process\cite{fadin,peskin}.
\par
So far, the investigation of the top quark physics at the future $e^+e^-$
linear colliders has made a great progress both from the theoretical
and experimental points of view.
At the $t \bar{t}$ threshold region,  the $t\bar{t}$ cross section is
expected to depend on several parameters as follows:
\begin{eqnarray*}
\sigma_{t\bar{t}} ( \sqrt{s}; m_t, \Gamma_t, \alpha_s(m_Z), m_H, \beta_H),
\end{eqnarray*}
where $\beta_H$ is the top YUKAWA coupling. Fujii {\it et al.} have
studied quantitatively on the determination of above physical
parameters under the realistic conditions.
A precise $t\bar{t}$ cross section scan was discussed
including the initial state radiation, beamstrahlung, and beam
energy spread\cite{top1}.
Above the $t\bar{t}$ threshold region, the future $e^+e^-$ linear colliders can provide
opportunities to investigate a new physics related the top quark. 
For example, the studies on the top quark couplings have been made by Comas
{\it et al.}\cite{top2}. 
\par
The $t\bar{t}$ pair production process decays mainly into:
6-jets ($e^+e^- \to b\bar{b}qq'qq'$), 
4-jets plus 1 charged lepton ($e^+e^- \to b\bar{b}qq'l{\nu}$), 
and 2-jets plus 2 charged leptons ($e^+e^- \to b\bar{b}l{\nu}l{\nu}$),
where $q=u,c$, $q'=d,s$, and $l=e, \mu, \tau$.  
%Among these three, in this paper we will concentrate on the second,
In this paper we concentrated on the second,
the semi-leptonic process $e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$.
This semi-leptonic (4-jets plus 1 charged lepton) process has several
advantages in
the event selection for the
top quark physics both at the $t \bar{t}$ threshold region and above.
Firstly, since the semi-leptonic process
has less than 6-jets, it is less suffered from the
backgrounds due to the wrong combination of jets.  
Secondly, these semi-leptonic events are produced with enough
statistics because 
$Br(t\bar{t} \to b\bar{b}qq'l\nu)$ is about 44\% and 
$Br(t\bar{t} \to b\bar{b}qq'qq')$ is 45\% after QCD correction. 
%Secondly semi-leptonic process is 
%easier to distinguish which three 
%$t\bar{t}$ production from backgrounds than 6-jets.
Moreover, when we use the electric charge of the lepton as tag,  
we can identify the
hemisphere where 3-jets, the decay products of the top quark,  
appear.
\par 
%At $\sqrt{s}$ = 340 - 500 GeV, the main contribution to the total
Although the main contribution to the total
cross section at $\sqrt{s}$ = 340  - 500 GeV comes from the
diagrams through the $t \bar{t}$ production,
those from the other diagrams are not negligible small at all.
To study the $t\bar{t}$ production process precisely, an accurate
estimation of the other diagrams
%is desirable with a good precision. 
is essentially required. 
We have calculated the cross
section of  $e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$
process with a full set of Feynman diagrams, 232 diagrams
in unitary gauge. 
Generally speaking, as the number of final state particles increases, 
the calculation of the cross section becomes more complicated and more
tedious because there exist a lot of Feynman diagrams
and there occur interferences among these diagrams.
In order to cope with this complex calculation, we have used an
automatic Feynman diagrams calculation package, {\tt
GRACE}\cite{grace}.
All the results obtained for the process $e^+e^- \to
b\bar{b}u\bar{d}\mu^- \bar{\nu}_{\mu}$ are immediately applicable to the
other semi-leptonic processes such as $e^+e^- \to b\bar{b}c\bar{s}
\mu^- \bar{\nu}_\mu$ when we neglect the difference of mass between
$u,d$ and $c,s$. 
%Furthermore, we have classified 232 diagrams into several groups and 
%have calculated the interference between groups of diagrams.
\par
In addition  to an accurate calculation of the total cross section,
the effects of interferences among diagrams have been studied.
These interferences change 
the $t {\bar{t}}$ threshold shape destructively.
%at the $t\bar{t}$ threshold energy region.
Also we present the invariant mass and angular distributions of
the final state particles at $\sqrt{s}$=500 GeV. 
In order to enhance the signals of $t\bar{t}$ production process
against the background, 
we studied on the effective  
cuts in the invariant mass distributions such as $M_{bu\bar{d}}$ and $M_{b\bar{b}}$
and the cut for the angle of the final lepton in these distributions.
Recently, the similar calculations of $e^+e^- \to$ six fermions are
performed by  two groups. Montagna {\it et al.}\cite{mont}
reported a full calculation of the six-fermion process
in the $e^+e^-$ linear colliders, 
where the higgs boson  with the intermediate mass  are produced, 
by the program package {\tt ALPHA}\cite{alpha} for the matrix elements
calculations and 
{\tt HIGGSPV/WWGENPV}\cite{higgspv} for Monte Carlo event generation.
Accomando {\it et al.}\cite{ball} have been calculating a semi-leptonic process such as
$e^+e^- \to b\bar{b}qq'l\nu$ by program package {\tt PHACT}\cite{phact}.
%and discuss the effect by cut of invariant
%mass of $ b, u$, and $\bar{d}$ to the distributions.
\par
%The paper is organized as follows. 
In Section 2, we present the details of the computational parameters
for the calculations in {\tt GRACE} at first.
Secondly, the numerical results are shown: the total cross sections
(Section 2.1), effects of the interferences among diagrams (Section
2.2), and the mass and angular distributions of interest (Section 2.3).
%Finally, Section 3 summarizes the results.
\par
\section{Cross Section Calculation}
The cross sections have been calculated based on a complete set of the
tree level diagrams using {\tt GRACE} system, 
a program package for an automatic Feynman diagram calculation and
the event generation.
When the initial and final states are specified, {\tt GRACE} generates the
matrix elements in terms of helicity amplitudes. 
%From the matrix elements generated by {\tt GRACE}, 
The total cross section is computed by the Monte
Carlo integration of the matrix elements over the phase space\cite{bases}.
%with a kinematics library to get the good convergence of integration. 
As for the kinematics, we only have to
prepare the suitable program code to the physics process of interest.  
Usually, as the number of the final state particles increases, 
it becomes more and more difficult to prepare such a program code of
kinematics that may catch every singularities appeared in all diagrams
and give a good convergence of the multidimensional Monte Carlo integration.
%the program code of the kinematics becomes more and more 
%difficult to obtain a good convergence of the multi dimensional 
%Monte Carlo integration.
We have developed a kinematics code for the process  
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$, which makes 
the calculation of the total cross sections converge rapidly enough to a
good accuracy of better than 1 \% at ${\sqrt{s}} = 340 - 500 $ GeV. 
\par
The cross sections has been calculated with the parameters shown in Table 1.
%The cross sections has been calculated with the parameters: 
%\begin{eqnarray*}
%m_Z&=&91.187~{\rm GeV}, \\
%\Gamma_Z&=&2.49~{\rm GeV}, \\
%m_W&=&80.22~{\rm GeV}, \\
%\Gamma_W&=&2.052~{\rm GeV}, \\
%{\rm sin}^2\theta_W&=&1-\frac{m^2_W}{m^2_Z}, \\
%\alpha&=&1/128.07 \\
%\end{eqnarray*}
%We used a fixed width for both $W$ and $Z$ bosons. 
The widths of $W$ and $Z$ bosons are taken to be the fixed values.
%as in the table.
%The $W$ boson width of 2.052 GeV corresponds to the tree-level width with
%above parameters.
The QCD correction is not included in the calculations.
The quark masses are set as in Table 2.
%Quark masses (and a width) are set as;
%\begin{eqnarray*}
%m_t&=&174~{\rm GeV}, \\
%\Gamma_t&=&1.558~{\rm GeV} \\
%m_b&=&4.1~{\rm GeV}, \\
%m_u&=&2~{\rm MeV}, \\
%m_d&=&5~{\rm MeV} \\
%\end{eqnarray*}
The $t$ quark width of 1.558 GeV corresponds to
the decay width of $t \to b W$, which means
that the decay branching ratio of $t \to b W$ channel is 100\%.
\par
The total cross section of  $e^+e^- \to t \bar{t}$ process is calculated
to be $5.77 \times 10^{-1}$ pb  at CM energy of 500 GeV by {\tt GRACE}. 
This is consistent with
the cross section of  $e^+e^- \to b \bar{b} W^+ W^-$ processes of
$5.78 \times 10^{-1}$ pb calculated by {\tt GRACE}. The cross section
of 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
process only with the $t \bar{t}$ production diagrams is calculated to be
$2.13 \times 10^{-2}$ pb ( in Table 3) and is also consistent with 
$\sigma_{t \bar{t}} \times Br(W \to u \bar{d}) \times Br(W \to \mu^-
\bar\nu^{\mu} ) =2.14 \times 10^{-2}$ pb. 
Here, we took $Br(t \to bW)$ = 100 \%.
\par
\subsection{Results of the numerical calculation}

Among all 232 diagrams (unitary gauge) of the process, 
the $t \bar{t}$ production diagrams of Fig.1 are dominant
over the others. 
We have divided the {\it major background diagrams} into three categories:
the diagrams with $W^+ W^- \gamma$ (hereafter $WW\gamma$) in Fig.2,
those with $W^+ W^- Z$ (hereafter $WWZ$)
in Fig.3, and those with single-$t$ through $W^+W^-$ pair production 
 (hereafter $tWW$) in Fig.4.  
\par
The total cross sections with a full set of  diagrams 
at the CM energy points of 340 - 500 GeV are summarized in Table 3 and
Fig.5 {\tt a)} and {\tt b)}.
Both in {\tt a)} and {\tt b)} of Fig.5, solid lines show the
numerical result of the total cross section with all the diagrams. 
The result with only the dominant 
$t \bar{t}$ diagrams are shown by dashed line in both {\tt a)
} and {\tt b)}.
Besides results with all the diagrams and that with
$t \bar{t}$ diagrams, we also showed those with $t \bar{t}$ and the major
background diagrams. 
Dotted lines in {\tt a)} and {\tt b)} show the result with $t
\bar{t}$  and $ WW \gamma$ and that with $t \bar{t}$ and $ WWZ$,  
respectively.
Dot-dashed lines in {\tt a)} and {\tt b)} show the result with 
$t \bar{t}$, $ WW \gamma$, and $tWW$
and that with $t \bar{t}$, $ WWZ $, and $tWW$, respectively.
These results shown by dotted lines and dot-dashed lines include the
interferences among selected diagrams.
\par
As shown in Fig.5, cross sections with both $t \bar{t}$ and the major
background diagrams show different behaviors from that with all 
the diagrams.
The difference between them is about 3\% at $\sqrt s$ = 500 GeV.
This means that the effect from the interference between $t \bar{t}$
and the rest diagrams except $WW\gamma$, $WWZ$, and $tWW$ are also
not negligible and important.
The contribution from the background diagrams to the total
cross section, ${(\sigma_{all} - \sigma_{t \bar{t}})}/{\sigma_{t\bar{t}}}$, 
is less than 5\% in total above 
the energy of twice of the top quark mass. 
%As shown in Fig.5, as the CM energy goes up, the contribution
%from the diagrams both with $W^+W^- Z$ and with tWW $ through WW
%becomes significant and the ratio,
%${\sigma_{t \bar{t} + WWZ + {\tt single}-t}}/{\sigma_{t
%\bar{t}}}$, 
%is about 8\%. 
\par
%A part of diagrams of the 
%$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
%process is shown in Figs.$1 \sim 4$. 
%To check the calculations, we look at
%only $t \bar{t}$ production diagrams.
\subsection{Interference between diagrams}
Due to the large decay width of the top quark, at the  $t \bar{t}$ threshold
region it decays so immediately that the non perturbative
QCD effect at the long distance does not take part.
Thus, the cross section of $ \sigma_{t \bar{t}}$ can be
calculated by means of the perturbative QCD theory\cite{fadin}.
By measuring the cross section of the $t \bar{t}$ production around the 
$t \bar{t}$ threshold region at the future linear colliders, 
a clear test of the perturbative QCD theory is possible.  
To determine the QCD parameters like $\alpha_{s}$, 
it is essential to measure of the cross section accurately.
%from the experiments at $t \bar{t}$ threshold region.
In our calculation, the effects from initial state
radiation, beamstrahlung, and beam energy spread were neglected.
These effects to the shape of $t\bar{t}$ cross section has been well
investigated by Fujii {\it el al.}\cite{top1, top2, top3}.
Besides these effects, it is important to take 
the effect such as the interferences between $t \bar{t}$ diagrams
and the other background diagrams into account.
%In the following, we present some details concerning the effects from the
%interference between diagrams.
%As it has been investigated that 
%at $t \bar{t}$ threshold region the shape of the cross section changes
%rapidly\cite{top1, top2, top3}, 
%it is important to take into account
%of effects such as the interference between $t \bar{t}$ diagrams
%and the other background diagrams.
\par
%In {\tt GRACE}, we can choose any combination of diagrams to see the
%interference between them. 
%In Table 2, we show the cross sections for $t {\bar{t}}$,
%single-$t$, and $t {\bar{t}}$ plus single-$t$ process.
%From these numerical results, 
%we can get the effects from the interference between diagrams (Fig.7).
%Background from single-$t$ via W W pair production 
%diagrams is biggest at ${\sqrt s} =340$ GeV.
It is found that there are
destructive interferences between diagrams with  $t {\bar{t}}$ production
and those with single-$t$ through WW pair production around the $t \bar{t}$
threshold region.
%In Fig.7, we plot the destructive interference calculated by the formulae:  
%${\sigma_{t\bar{t} + {\tt single}-t}}/{(\sigma_{t\bar{t}} + \sigma_{{\tt single}-t})}$.
In order to see the behavior of these destructive interferences the values of 
${\sigma_{t\bar{t} + tWW}}/{(\sigma_{t\bar{t}} +
\sigma_{tWW})}$ are plotted at several energy points in Fig.6.
%is large at CM energy around twice of the top quark mass.
The magnitude of the destructive interference is as large as 10 \% at 
$\sqrt {s}$ = 340 GeV.
Then it drops rapidly as CM energy goes up.
%until $\sqrt{s}$ = 350 GeV.
In the energy region above twice of the top quark mass, this destructive
interference becomes less than 2 \% and above $\sqrt{s}$ = 360 GeV it becomes 
negligible small.

\par
When we generate Monte Carlo events with the luminosity 1 ${\rm
fb}^{-1}$ in each energy points, the destructive interference of 10 \%
is not negligible compared to the statistical errors at the $t\bar{t}$
threshold region.
For example, at $\sqrt{s}$ = 350 GeV
the total cross section of $\sigma_{t \bar{t}}$ is $ 1.90
\times 10^{-1}$ pb and the statistical error is estimated to be a few \%.
%This behavior of the destructive interference makes a deep effect to the
%shape of the cross section at the $t \bar{t}$ threshold region.
%On the other hand, there seems no remarkable interference between
%$ W W \gamma$ type diagrams and $ W W Z$ type diagrams as in Table 4. 
%From this result, for the analysis of $t \bar{t}$ threshold scanning, 
%we have to consider that there exists a strong 
%destructive interference between the diagrams
%through $t \bar{t}$ and ones through single-$t$ via $W W$ pair.
\par

\subsection{Invariant mass and angular distributions}

Above the $t \bar{t}$ threshold energy region, there open the opportunities of
measuring the top production and decay couplings. 
For this analysis,  the process of semi-leptonic 4-jets plus 1 charged
lepton is useful to distinguish $t$ from $\bar{t}$ by the electric charge of the lepton. 
%since it is straightforward to think $W$ decays a lepton and a
%nutorino.   
Generally, in order to select the semi-leptonic $t\bar{t}$ production
events,  firstly
an isolated charged lepton and 4 jets are required in the event. 
Then two jets out of the four are to be identified as $b$ jets in the
vertex detector. 
From the electric charge of lepton, we can determine the hemisphere
where 3 jets from the top quark appear. 
If the invariant
mass of 2 jets in this hemisphere and one of $b$ jets becomes nearly equal to the mass of the top quark, we
accept it as the $t\bar{t}$ production event. 
\par
The invariant mass distribution of $b$, $u$ and
$\bar{d}$, and that of $b$ and $\bar{b}$ 
at $ \sqrt{s} = 500$ GeV are shown in Fig.7 {\tt a)} and {\tt b)}, respectively.
In Fig.7, histograms show the invariant mass distributions for the diagrams
of $t \bar{t}$ production and crosses show
those for all the diagrams.
%those with $W^+W^- \gamma$
%(circles) and those with  $W^+W^- \gamma$ plus single-$t$ (crosses),
%of while, Fig.7 shows the distributions for the diagrams 
%$t \bar{t}$ (solid line), $W^+W^- Z$ (circles) and $W^+W^- Z$ plus 
%single-$t$ (crosses).   
%In each figures,
%{\tt a)} shows the invariant mass distributions of $b$, $u$, and $\bar{d}$,
%{\tt b)} that of $u$ and $\bar{d}$,
%and {\tt c)} that of $b$ and $\bar{b}$.
%\end{itemize}
%\begin{itemize}
%\item a) shows the invariant mass distributions of $b$, $u$, and $\bar{d}$,
%\item b) shows the invariant mass distributions of $u$ and $\bar{d}$,
%\item c) shows the invariant mass distributions of $b$ and $\bar{b}$.
%\end{itemize}
As shown in Fig.7 {\tt a)}, if we take only those
events around the
top quark mass on the  $M_{bu\bar{d}}$
distribution, the contribution from the background diagrams becomes
negligibly small.
At the energy of $\sqrt{s}$ = 500GeV, when the top quark mass cut,
{\it e.g.}
148 GeV $\leq M_{bu\bar{d}} \leq$ 204 GeV is applied, the
contribution from the background diagrams is reduced from about 5\% to less than 3\%.
In Fig.7 {\tt b)}, crosses shows that the contribution from
$WW\gamma$ diagrams can be seen at the low-end 
and the contribution from $WWZ$ diagrams can be seen at the mass
of Z boson.
\par
For the analysis of the top couplings, it should be noticed that
the top quark decays immediately before it hadronizes and its spin
information 
is transferred to the final state particles\cite{top2}. 
The angles of the final state particles are important parameters in order to
reconstruct the production angle of the top quark.
Here, we show the angular distributions of final $\mu^{-}$ in Fig.8.
%, where show the angular distributions of $\mu^-$.
%\end{itemize}
%\begin{itemize}
%\item a) shows the angular distributions of $u$,
%\item b) shows the angular distributions of $b$,
%\item c) shows the angular distributions of $\mu^-$.
%\end{itemize}
The backgrounds in the forward angle region, 
%from $W^+W^-\gamma$, $W^+W^-Z$  and single-$t$
from all the diagrams are significant in the distribution of the 
$\mu^-$ angular distribution, at $ \sqrt{s} = 500$ GeV.
%
For example, when  the top quark mass cut such as
148 GeV $\leq M_{bu\bar{d}} \leq$ 204 GeV is applied,
%With this cut, the background in
%the invariant mass distribution have been removed. 
the effects from the background
in the angular distribution of $\mu^-$ can be reduced to less than 2\%
but still exists as shown in Fig.8 {\tt b)}.
%{\tt b)} of  Fig.8 and it is estimated less than 2\%.
%Therefore, in detection of $\mu^-$ lepton for the selection of
%4-jets plus 1 charged lepton events, we have to treat the forward angle
%events taking into account of these effects from background diagrams.
%In order to reduce the background to the selection of the events with 
%the $t\bar{t}$ production by the lepton tag, 
%after the cut of the invariant mass of $M_{bu\bar{d}}$,
%we should estimate the contribution of the background to the angular
%distribution of the final state lepton correctly. 

\section{Conclusion}

The total cross section of 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ process 
at the energy range of $\sqrt{s}$ = 340 - 500 GeV 
was calculated with a complete set of tree diagrams by using
{\tt GRACE}. 
%in complete tree level where there are 232 Feynman diagrams 
%in unitary gauge.
%The numerical calculation converges with a good accuracy of Monte
%Carlo integration les than 1\%.
Among all diagrams, the main contribution to the total cross section
comes from the  $t\bar{t}$ production 
diagrams. 
The background to the 
$t{\bar{t}}$ production from all the diagrams is below 5\% above 
the $t{\bar{t}}$ threshold. 
%
The contributions from the major background diagrams such as
$WW\gamma$, $WWZ$, and single-$t$ through $WW$ pair production are also calculated.
Compared these results to those with all diagrams, it is found that
the interference between $t \bar{t}$ and  the rest diagrams except the major backgrounds is not negligible.
It is also found that the diagrams of  $t {\bar{t}}$ production and
those with single-$t$ through WW pair
production make a strong destructive interference (as large as 10\%)
around the $t {\bar{t}}$ threshold region. 
This destructive interference has a deep effect on the shape
of the $t {\bar{t}}$ cross section at the $t\bar{t}$ threshold region. 
\par
We calculated the distributions of the invariant masses and the angular
distribution of the final state particles at $\sqrt{s} = 500$ GeV. 
%From the distribution of $M_{bu\bar{d}}$,
%when we apply the cut of the top quark mass, we can remove the effect from the
%backgrounds.
Angular distributions of $\mu^{-}$ shows a large effect of 
backgrounds at the forward angle. 
%therefore we should get rid of the forward angle event
%when we detect $\mu$. 
Even when we apply the cut in the invariant mass distribution of $b$,
$u$, and $\bar{d}$ to enhance the top quark signal, 
the background from the diagrams with $WW\gamma$, 
$WWZ$ and single-$t$ through WW pair did not disappear in 
the angular distribution of $\mu^-$ 
\par
%From these results, for the analysis of the top quark physics around 
%$t \bar{t}$ threshold region, the precise calculation including all
%232 Feynman diagrams is essential
%in terms of the $t{\bar{t}}$ threshold scanning and 
%of the event selection of the process.
As a conclusion, for the analysis of the top quark physics around the
$t \bar{t}$ threshold region and above, the accurate calculation including full
set of 232 Feynman diagrams is essential.
%in terms of the $t{\bar{t}}$ threshold scanning and 
%of the event selection of the process.

\par
\section*{Acknowledgements}
The authors wish to acknowledge to Minami-Tateya Collaboration.
Especially, we are indebted to Prof. Y.Shimizu, Prof. T.Ishikawa and 
Dr. J.Fujimoto for valuable suggestions and encouragements.
The authors also wish to thank Dr. E. Accomando and Prof. A.
Ballestrero for useful discussions and comparisons with their work are
gratefully acknowledged.
One of the authors (F.Y.) thanks Dr. T.Omori for fruitful discussions. 
This work is supported in part by Ministry of Education, Science, and
Culture, Japan under Grant-in-Aid for International Scientific Research
Program (No.09044359). 
%%%%%%%%%%%%%%%%%%%%%%%%%% ref %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{top1}
K.Fujii, T.Matsui, and Y.Sumino, Phys. Rev. {\bf D50} (1994) 4341.\\
K.Fujii, Proceedings of the Second Workshop on JLC, Tsukuba, Japan, 1990.
\bibitem{top2}
P.Igo-Kemenes, M. Martinez, R. Miquel, and S. Orteu,
Proceedings of the Workshop on Physics and Experiments with Linear 
Colliders (LCWS93), Waikoloa, Hawaii, USA, 1993.\\
P.Comas, R.Miquel, M.Martinez, and S.Orteu,
Proceedings of the Workshop on Physics and Experiments with Linear 
Colliders (LCWS95), Morioka-Appi, Iwate, JAPAN, 1995.
\bibitem{top3}
Proceedings of the Workshop $e^+e^-$ {\it Collisions at 500 GeV: The
Physics Potential}, Munich-Annecy-Hamburg, ed. P.M.Zerwas, 
Reports DESY 92-123A, 327 and  93-123C.
\bibitem{top}
Raymond Frey {\it et al.},
 4 Apr 1997, Top Quark Physics: Future Measurements.
\bibitem{lc}
JLC Group, KEK Reports 92-16, 1992, JLC-I.\\
TESLA collaboration (R. Brinkmann for the collaboration),
DESY-M-95-08N, Jun 1995. 3pp. Presented at 16th IEEE Particle Accelerator
Conference (PAC 95) and International Conference on High Energy
Accelerators (IUPAP), Dallas, Texas, 1-5 May 1995. 
Published in IEEE PAC 1995:674-676. \\
The NLC Design Group, Zeroth-Order Design Report for the Next Linear
Collider, SLAC-474, May 1996. 
\bibitem{fermi1}
F. Abe {\it et al.} (CDF Collaboration),
Phys. Rev. Lett. {\bf 74} (1995) 2626.
\bibitem{fermi2}
S. Abachi {\it et al.} (D0 Collaboration),
Phys. Rev. Lett. {\bf 74} (1995) 2632.
\bibitem{fadin}
V.S. Fadin and V.A. Khoze, JETP Lett. {\bf 46} (1987) 525.\\
V.S. Fadin and V.A. Khoze, Sov. J. Nucl. Phys. {\bf 48} (1988) 669.
\bibitem{peskin}
M. Strasseler and M.Peskin, Phys. Rev. D {\bf 43} (1991) 1500.
\bibitem{grace} 
T. Ishikawa, T. Kaneko, K. Kato, S. Kawabata, Y. Shimizu and H. Tanaka, 
KEK Report 92-19, 1993, The {\tt GRACE} manual Ver. 1.0.\\
H. Tanaka, {\sl Comput. Phys. Commun.} {\bf 58} (1990) 153, and
H. Tanaka, T. Kaneko and Y. Shimizu, {\sl Comput. Phys. Commun.} 
{\bf 64}(1991) 149. 
\bibitem{mont}
G. Montagna, M. Moretti, O. Nicrosini and F. Piccinini, 
 16 May 1997, Six-Fermion Calculation of
Intermediate-mass Higgs Boson Production at Future $e^+e^-$ Colliders.
\bibitem{alpha}F. Caravaglios and M. Moretti, Phys. Lett. {\bf B358}
(1995) 332.
\bibitem{higgspv}G.Montagna, O.Nicrosini and F.Piccinini, {\sl Comput.
Phys. Commun.} {\bf 90} (1995) 141.\\
D.G. Charlton, G.Montagna, O.Nicrisini and F.Piccinini, {\sl Comput.
Phys. Commun.} {\bf 99} (1997) 355. 
\bibitem{ball}E. Accomando, S. Ballestrero and M. Pizzio, 
Torino preprint DFTT 74/96,  2 Jun 1997.\\
E. Accomando, talk given at the ECFA/DESY LC Workshop, DESY, Hamburug,
November 1996.
\bibitem{phact}A. Ballestrero, E.Maina, Phys. Lett. {\bf B350} (1995) 225.
\bibitem{bases}
S. Kawabata, {\sl Comput. Phys. Commun.} {\bf 41}(1986) 127.\\
S. Kawabata, {\sl Comput. Phys. Commun.} {\bf 88}(1995) 309.
\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%% table 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{center}
%\small
\normalsize
\begin{tabular}{|c|c|} \hline
$m_Z$ & 91.187 GeV \\ \hline
$\Gamma_Z$ & 2.49 GeV  \\ \hline
$m_W$ & 80.22 GeV \\ \hline
$\Gamma_W$ & 2.052 GeV \\ \hline
${\rm sin}^2\theta_W$ & $1-\frac{m^2_W}{m^2_Z}$ \\ \hline
$\alpha$ & 1/128.07 \\ \hline
\end{tabular}
~\\
~\\  
\normalsize
Table 1  The parameters
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%% table 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
%\small
\normalsize
\begin{tabular}{|c|c|} \hline
$m_t$ & 174 GeV \\ \hline
$\Gamma_t$ & 1.558 GeV \\ \hline
$m_b$ & 4.1 GeV \\ \hline
$m_u$ & 2 MeV \\ \hline
$m_d$ & 5 MeV \\ \hline
\end{tabular}
~\\
~\\  
\normalsize
Table 2  The mass and width parameters
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%% table 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\begin{center}
%\small
\tiny
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
${\sqrt s}$ & all diagrams & $t\bar{t}$ & $t\bar{t}+WW\gamma$ 
&$t\bar{t}+WW\gamma$+$tWW$
& $t\bar{t}+WWZ$ &$t\bar{t}+WWZ$+$tWW$ \\ \hline
340&$6.87(2)\times10^{-4}$&$4.462(3)\times10^{-4}$&
    $4.936(9)\times10^{-4}$&$6.80(2)\times10^{-4}$& 
    $5.91(1)\times10^{-4}$&$7.80(2)\times10^{-4}$     \\   \hline
350&$6.45(1)\times10^{-3}$&$6.187(4)\times10^{-3}$&
    $6.235(6)\times10^{-3}$&$6.427(9)\times10^{-3}$&
    $6.354(9)\times10^{-3}$&$6.57(1)\times10^{-3}$     \\   \hline
360&$1.497(2)\times10^{-2}$&$1.463(1)\times10^{-2}$&
    $1.467(1)\times10^{-2}$&$1.494(2)\times10^{-2}$&
    $1.483(2)\times10^{-2}$&$1.509(2)\times10^{-2}$     \\   \hline
370&$1.906(4)\times10^{-2}$&$1.864(1)\times10^{-2}$&
    $1.869(2)\times10^{-2}$&$1.902(3)\times10^{-2}$&
    $1.887(3)\times10^{-2}$&$1.919(3)\times10^{-2}$     \\   \hline
380&$2.142(4)\times10^{-2}$&$2.100(1)\times10^{-2}$&
    $2.104(2)\times10^{-2}$&$2.145(3)\times10^{-2}$&
    $2.127(4)\times10^{-2}$&$2.164(3)\times10^{-2}$     \\   \hline
390&$2.298(5)\times10^{-2}$&$2.244(1)\times10^{-2}$&
    $2.247(2)\times10^{-2}$&$2.290(4)\times10^{-2}$&
    $2.268(4)\times10^{-2}$&$2.316(3)\times10^{-2}$     \\   \hline
400&$2.383(5)\times10^{-2}$&$2.329(1)\times10^{-2}$&
    $2.332(2)\times10^{-2}$&$2.385(4)\times10^{-2}$&
    $2.364(5)\times10^{-2}$&$2.409(4)\times10^{-2}$     \\   \hline
450&$2.424(4)\times10^{-2}$&$2.344(2)\times10^{-2}$&
    $2.354(3)\times10^{-2}$&$2.432(4)\times10^{-2}$&
    $2.397(4)\times10^{-2}$&$2.467(5)\times10^{-2}$     \\   \hline
500&$2.232(4)\times10^{-2}$&$2.130(1)\times10^{-2}$&
    $2.150(3)\times10^{-2}$&$2.242(5)\times10^{-2}$&
    $2.216(5)\times10^{-2}$&$2.308(6)\times10^{-2}$     \\   \hline
\end{tabular}
~\\
~\\  
\normalsize
Table 3  The total cross sections in pb.
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%% table 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{center}
%%\small
%\footnotesize
%\begin{tabular}{|c|c|c|c|c|} \hline
%CM energy & all diagrams & $t\bar{t}$ & $t\bar{t}+W^+W^-Z$ 
%&$t\bar{t}+W^+W^-Z$+single $t$ \\ \hline
%340&$6.87(2)\times10^{-4}$&$4.462(3)\times10^{-4}$&
%    $5.91(1)\times10^{-4}$&$7.80(2)\times10^{-4}$     \\   \hline
%350&$6.45(1)\times10^{-3}$&$6.187(4)\times10^{-3}$&
%    $6.354(9)\times10^{-3}$&$6.57(1)\times10^{-3}$     \\   \hline
%360&$1.497(2)\times10^{-2}$&$1.463(1)\times10^{-2}$&
%    $1.483(2)\times10^{-2}$&$1.509(2)\times10^{-2}$     \\   \hline
%370&$1.906(4)\times10^{-2}$&$1.864(1)\times10^{-2}$&
%    $1.887(3)\times10^{-2}$&$1.919(3)\times10^{-2}$     \\   \hline
%380&$2.142(4)\times10^{-2}$&$2.100(1)\times10^{-2}$&
%    $2.127(4)\times10^{-2}$&$2.164(3)\times10^{-2}$     \\   \hline
%390&$2.298(5)\times10^{-2}$&$2.244(1)\times10^{-2}$&
%    $2.268(4)\times10^{-2}$&$2.316(3)\times10^{-2}$     \\   \hline
%400&$2.383(5)\times10^{-2}$&$2.329(1)\times10^{-2}$&
%    $2.364(5)\times10^{-2}$&$2.409(4)\times10^{-2}$     \\   \hline
%450&$2.424(4)\times10^{-2}$&$2.344(2)\times10^{-2}$&
%    $2.397(4)\times10^{-2}$&$2.467(5)\times10^{-2}$     \\   \hline
%500&$2.232(4)\times10^{-2}$&$2.130(1)\times10^{-2}$&
%    $2.216(5)\times10^{-2}$&$2.308(6)\times10^{-2}$     \\   \hline
%\end{tabular}
%~\\
%~\\  
%Table 2  The total cross sections in pb.
%\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%% table 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{center}
%%\small
%\scriptsize
%\begin{tabular}{|c|c|c|c|c|} \hline
%${\sqrt s}$  & $t\bar{t}$ & single $t$ & $t\bar{t}$ + single $t$ &
%interference \\ \hline
%340&$4.462(3)\times10^{-4}$&
%    $2.521(6)\times10^{-4}$&$6.32(1)\times10^{-4}$ & 0.905  \\   \hline
%341&$5.165(3)\times10^{-4}$&
%    $2.582(6)\times10^{-4}$&$7.05(1)\times10^{-4}$ & 0.910 \\   \hline
%342&$6.059(4)\times10^{-4}$&
%    $2.639(6)\times10^{-4}$&$7.97(1)\times10^{-4}$ & 0.916   \\   \hline
%343&$7.232(5)\times10^{-4}$&
%    $2.696(6)\times10^{-4}$&$9.17(1)\times10^{-4}$ & 0.924   \\   \hline
%344&$8.840(6)\times10^{-4}$&
%    $2.756(6)\times10^{-4}$&$1.081(1)\times10^{-3}$ & 0.932   \\   \hline
%345&$1.117(1)\times10^{-3}$&
%    $2.811(6)\times10^{-4}$&$1.315(2)\times10^{-3}$ & 0.941  \\   \hline
%346&$1.483(1)\times10^{-3}$&
%    $2.874(6)\times10^{-4}$&$1.683(2)\times10^{-3}$ & 0.951    \\   \hline
%347&$2.112(1)\times10^{-3}$&
%    $2.930(6)\times10^{-4}$&$2.312(3)\times10^{-3}$ & 0.961  \\   \hline
%348&$3.210(2)\times10^{-3}$&
%    $2.989(6)\times10^{-4}$&$3.407(3)\times10^{-3}$ & 0.971   \\   \hline
%349&$4.705(3)\times10^{-3}$&
%    $3.047(7)\times10^{-4}$&$4.905(4)\times10^{-3}$ & 0.979   \\   \hline
%350&$6.187(4)\times10^{-3}$&
%    $3.095(7)\times10^{-4}$&$6.384(5)\times10^{-3}$ & 0.987   \\   \hline
%360&$1.463(1)\times10^{-2}$&
%    $3.702(8)\times10^{-4}$&$1.488(1)\times10^{-2}$ & 0.992   \\   \hline
%370&$1.864(1)\times10^{-2}$&
%    $4.315(9)\times10^{-4}$&$1.896(2)\times10^{-2}$ & 0.996   \\   \hline
%380&$2.100(1)\times10^{-2}$&
%    $4.91(1)\times10^{-4}$&$2.137(2)\times10^{-2}$ & 0.994   \\   \hline
%390&$2.244(1)\times10^{-2}$&
%    $5.52(1)\times10^{-4}$&$2.287(2)\times10^{-2}$ & 0.995   \\   \hline
%400&$2.329(1)\times10^{-2}$&
%    $6.11(1)\times10^{-4}$&$2.377(2)\times10^{-2}$ & 0.995   \\   \hline
%450&$2.344(2)\times10^{-2}$&
%    $8.85(2)\times10^{-4}$&$2.417(3)\times10^{-2}$ & 0.994   \\   \hline
%500&$2.130(1)\times10^{-2}$&
%    $1.106(3)\times10^{-3}$&$2.221(2)\times10^{-2}$ & 0.991   \\   \hline
%\end{tabular}
%~\\
%~\\  
%Table 2  The total cross sections in pb.
%\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%% table 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{center}
%\small
%\begin{tabular}{|c|c|c|c|} \hline
%CM energy & $W^+W^-\gamma$ & $W^+W^-Z$ &$W^+W^-\gamma$+$W^+W^-Z$ \\ \hline
%340&$4.748(8)\times10^{-5}$&
%    $1.419(3)\times10^{-4}$&$1.849(5)\times10^{-4}$     \\   \hline
%350&$5.44(1)\times10^{-5}$&
%    $1.691(4)\times10^{-4}$&$2.181(6)\times10^{-4}$     \\   \hline
%360&$6.19(1)\times10^{-5}$&
%    $1.981(4)\times10^{-4}$&$2.541(7)\times10^{-4}$     \\   \hline
%370&$7.01(1)\times10^{-5}$&
%    $2.291(5)\times10^{-4}$&$2.929(7)\times10^{-4}$     \\   \hline
%380&$7.91(1)\times10^{-5}$&
%    $2.619(6)\times10^{-4}$&$3.341(8)\times10^{-4}$     \\   \hline
%390&$8.88(1)\times10^{-5}$&
%    $2.965(7)\times10^{-4}$&$2.289(4)\times10^{-2}$     \\   \hline
%400&$9.92(2)\times10^{-5}$&
%    $3.328(8)\times10^{-4}$&$4.239(4)\times10^{-4}$     \\   \hline
%450&$1.642(3)\times10^{-4}$&
%    $5.46(1)\times10^{-4}$&$6.98(1)\times10^{-4}$     \\   \hline
%500&$2.542(4)\times10^{-4}$&
%    $8.13(2)\times10^{-4}$&$1.051(2)\times10^{-3}$     \\   \hline
%\end{tabular}
%~\\
%~\\  
%Table< 4  The total cross sections in pb.
%\end{center}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%% fig 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfig{file=tt.eps,height=10cm}}
\caption{A part of Feynman diagrams for the 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
process. Diagrams with the $t \bar{t}$ production are shown.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfig{file=wwa.eps,height=14cm}}
\caption{A part of Feynman diagrams for the 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
process. Diagrams with the $W^+ W^- \gamma$ production are shown.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfig{file=wwz.eps,height=14cm}}
\caption{A part of Feynman diagrams for the 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
process. Diagrams with the $W^+ W^- Z$ production are shown.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\mbox{\epsfig{file=st.eps,height=14cm}}
\caption{A part of Feynman diagrams for the 
$e^+e^- \to b \bar{b} u \bar{d} \mu^- \bar{\nu}_\mu$ 
process. Diagrams with the single $t$ production through $WW$ pair production
are shown.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\mbox{\epsfig{file=roots-roots2.ps,height=14cm}}
\caption{
{\tt a)} The total cross sections with all diagrams (solid line),
with $t \bar{t}$ diagrams (dashed line), with $t \bar{t}$ and 
$WW\gamma$ (dotted line), and with $t \bar{t}$, 
$WW\gamma$, and $tWW$ (dot-dashed line).
{\tt b)} The total cross sections with all diagrams (solid line),
with $t \bar{t}$ diagrams (dashed line), with $t \bar{t}$ and 
$WWZ$ (dotted line), and with $t \bar{t}$,
$WWZ$, and $tWW$ (dot-dashed line).
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\begin{center}
%\mbox{\epsfig{file=roots2.ps,height=10cm}}
%\caption{The total cross sections with all diagrams (solid line),
%$t \bar{t}$ diagrams (dashed line), $t \bar{t}$ plus 
%$W^+W^- Z$ diagrams (dotted line) and $t \bar{t}$ plus both
%$W^+W^- Z$ and single-$t$ production diagrams (dot-dashed line).
%}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\mbox{\epsfig{file=interference.ps,height=10cm}}
\caption{The effect of interference between diagrams with 
$t \bar{t}$ production and ones with single-$t$ through WW pair production.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\mbox{\epsfig{file=mass.all.ps,height=7cm}}
\caption{Invariant mass distributions of {\tt a)} $bu\bar{d}$ and {\tt
b)} $b\bar{b}$. Histograms show the contributions from $t\bar{t}$ diagrams only.
Crosses show the contributions from all the diagrams.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\begin{center}
%\mbox{\epsfig{file=mass.wwz.500.ps,height=14cm}}
%\caption{Invariant mass distributions of {\tt a)} $bu\bar{d}$, {\tt
%b)} $u\bar{d}$,
%and {\tt c)} $b\bar{b}$. Histograms show the $t\bar{t}$ diagrams only.
%Circles show the contributions from $t\bar{t}$ and $W^+W^-Z$ diagrams. 
%Crosses show the contributions from $t\bar{t}$,  $W^+W^-Z$, and single-$t$ diagrams.
%}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\mbox{\epsfig{file=cos.all.500.ps,height=7cm}}
\caption{Both {\tt a)} and {\tt b)} show the angular distributions of $\mu$. 
{\tt b)} shows the angular distribution of $\mu$ with the cut of the top quark mass.
Histograms show the contributions from $t\bar{t}$ diagrams only.
Crosses show the contributions from all the diagrams.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%% fig 11 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}
%\begin{center}
%\mbox{\epsfig{file=cos.wwz.500.ps,height=14cm}}
%\caption{Angular distributions of {\tt a)} $b$, {\tt b)} $u$,
%and {\tt c)} $\mu$. Histograms show the $t\bar{t}$ diagrams  only.
%Circles show the contributions from $t\bar{t}$ and $W^+W^-Z$ diagrams.
%Crosses show the contributions from 
%$t\bar{t}$,  $W^+W^-Z$, and single-$t$ diagrams.
%{\tt d)} shows the angular distribution of $\mu$ with the cut of the top quark mass.
%}
%\end{center}
%\end{figure}
\end{document}















