\section{Qualitative Description of Final-State Interaction Effects}
\cleqn
\clfn

In this section a qualitative understanding of the 
effects of final-state interactions is explained.
It is based on the classical picture that 
$t$ and $\bar{b}$ ($\bar{t}$ and $b$) attract
each other due to the interaction.
We will see that all the following qualitative features 
match well with our numerical
studies presented in section 5.

\subsection{Top Momentum Distribution}
First let us consider the effect of final-state interactions on
the top-quark momentum ($|\vc{p}_t|$) distribution, where
$\vc{p}_t$ is reconstructed from the $bW^+$ momenta at time
$\tau \to \infty$.
The energy of a $t\bar{t}$ system before its decay is given by
\bea
{\cal E} = E_t + E_{\bar{t}} + V( |\vc{r}_t - \vc{r}_{\bar{t}}|),
\eea
where
\bea
E_t = m_t + \frac{\vc{p}_t^2}{2m_t}, ~~~~~
E_{\bar{t}} = m_t + \frac{\vc{p}^2_{\bar{t}}}{2m_t}, 
~~~~~V(r)=-C_F \frac{\alpha_s}{r}.
\eea
Suppose $t$ decays first and 
let $a \sim (\alpha_s m_t)^{-1} \sim (m_t \Gamma_t)^{-1/2}$ be 
the typical distance between $t$ and $\bar{t}$ at the time of $t$ decay.
Then, just before the decay, the momenta of $t$ and $\bar{t}$ are given by
\bea
|\vc{p}_t| = |\vc{p}_{\bar{t}}| \sim
\sqrt{
m_t \left[ {\cal E} - 2m_t + |V(a)| \, \right]
} .
\eea
Their order of magnitude is 
$\alpha_s m_t \sim (\mbox{Bohr radius})^{-1}$.
If it were not for the final-state interactions between $\bar{t}$ and $b$,
and if $\bar{t}$, $b$ and $W^+$ travelled as free particles, 
the above momentum would be transferred to
the $bW^+$ system at time
$\tau \to \infty$:
\bea
|\vc{p}_b + \vc{p}_{W^+}|_{\tau \to \infty} \sim
\sqrt{
m_t \left[ {\cal E} - 2m_t + |V(a)| \, \right]
} .
\eea
Taking into account the final-state interaction
(Coulomb interaction) between $\bar{t}$ and $b$, 
the energy of $\bar{t}bW^+$ system is given by\footnote{
Here we neglect the interaction of $\bar{t}$ ($b$) and 
magnetic field generated by $b$ ($\bar{t}$).
This approximation is justified in the case of our interest;
see section 4.
}
\bea
{\cal E} = E_b + E_{W^+} + E_{\bar{t}} +
V( |\vc{r}_b - \vc{r}_{\bar{t}}|) ,
\\
E_b = |\vc{p}_b|, ~~~~~ E_{W^+} = \sqrt{ \vc{p}_{W^+}^2 + M_W^2 } .
\eea
As depicted in Fig.~\ref{forcebtbar}, the tracks of $b$ and $\bar{t}$
are deflected due to the attraction between the two particles, which
lose kinetic energy as $b$ flies off to infinity at the
speed of light.
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=5.cm
  \epsffile{forcebtbar.xfig.eps}\\
  \caption[]{
        \label{forcebtbar}Attractive Coulomb force between $\bar{t}$
        and $b$ (from $t$ decay). The momentum transfer 
        $\delta \vc{p}_b = - \delta \vc{p}_{\bar{t}}$ due to
        the attraction is indicated by thick arrows.}
\end{center}
\end{figure}
The classical equation of motion is given by
\bea
\frac{d\vc{p}_b}{d\tau} = - \frac{d\vc{p}_{\bar{t}}}{d\tau}
= - \frac{\partial}{\partial \vc{r}_b} \, 
V( |\vc{r}_b - \vc{r}_{\bar{t}}|) .
\eea
Substituting the free particle solution 
$\vc{r}_b = \vc{v} \tau + \vc{r}_0$ on the right-hand side and
noting $|\vc{r}_b - \vc{r}_{\bar{t}}| \simeq \tau$,
we may estimate the size of the momentum transfer due to the attractive force 
as\footnote{
It corresponds to solving the equation of motion by a series expansion
of $\alpha_s$.
}
\bea
|\delta \vc{p}_b| = |\delta \vc{p}_{\bar{t}}| \sim
|V( r_{min})| ,
\label{momtr}
\eea
where the minimum distance between $b$ and $\bar{t}$ is denoted as
$r_{min}$.
Typically $r_{min} \sim a$, and
we find from Eq.\ (\ref{momtr}) that the effect of final-state interactions
on the $|\vc{p}_t|$ distribution is of the order of
$|\delta \vc{p}_t| \sim \alpha_s^2 m_t$.
Obviously the effect is to reduce $|\vc{p}_t|$.

\subsection{Forward-Backward Asymmetric Distribution}
Next we consider the $\cos \theta_{te}$ distribution of the top quark.
($\theta_{te}$ denotes the angle between $t$ and $e^-$ 
in the \ttbar c.m.\ frame.)
It has been known that a forward-backward asymmetric distribution
of the top quark is generated by the final-state 
interactions \cite{r40,r26}.
We describe its mechanism here.

We consider the case where $\bar{t}$ decays first and examine the
interaction between $t$ and $\bar{b}$.
Approximately, 
the $t$ and $\bar{t}$ pair-produced near threshold in $e^+e^-$ collisions
have their spins parallel or anti-parallel to the $e^-$ beam direction and
the spins of $t$ and $\bar{t}$ are always oriented parallel to
each other.
In fact in leading order the polarization vector of the top quark is
given by\cite{r38}
$\mbox{\boldmath$\cal P$} = C_\|^0(\chi) \, \hat{\vc{n}}_\|$.
On the other hand, the decay of $\bar{t}$ occurs via a $V \! -\! A$
coupling, and
$\bar{b}$ is emitted preferably in the spin direction of the parent $\bar{t}$,
see Fig.\ \ref{tbardecay}.
More precisely,
the excess of the $\bar{b}$'s 
emitted in the $\bar{t}$ spin direction over those
emitted in the opposite direction is given by 
$\kappa$ defined in Eq.\ (\ref{ykappa}).
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=7.cm
  \epsffile{tbardecay.xfig.eps}\\
  \caption[]{
        \label{tbardecay}Typical configurations in the decay 
        of $\bar{t}$ with definite
        spin orientation.  Transverse $W^-$ ($W^-_T$) tend 
        to be emitted in the direction of the $\bar{t}$ spin orientation,
        while longitudinal $W^-$ ($W^-_L$) are emitted 
        in the opposite direction due to helicity conservation.
        For $m_t \simeq 175$~GeV, $\bar{t}$ decays mainly to
        $W^-_L$, hence $\bar{b}$ is emitted more in the $\bar{t}$ spin
        direction.}
\end{center}
\end{figure}
Suppose $t$ and $\bar{t}$ have their spins in the $\hat{\vc{n}}_\|$ direction.
Then $\bar{b}$ will be emitted dominantly in the $\hat{\vc{n}}_\|$
direction.
One can see from Fig.~\ref{spin}(a)
that in this case $t$ is always attracted to the forward direction 
($\bar{b}$ is attracted to the backward direction)
due to the attractive force between $t$ and $\bar{b}$.
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=12.cm
  \epsffile{tbbarfsi.xfig.eps}\\
  \caption[]{
        \label{spin}Attractive force between $t$ and $\bar{b}$
        when the $t$ and $\bar{t}$ spins are oriented in the
        (a)$\hat{\vc{n}}_\|$ direction, and in the
        (b)$-\hat{\vc{n}}_\|$ direction. The 
        momentum transfer  
        $\delta \vc{p}_b = - \delta \vc{p}_{\bar{t}}$ due to
        the attraction is indicated by thick black arrows.}
\end{center}
\end{figure}
The direction of the attractive force will be opposite
if $t$ and $\bar{t}$ have
their spins in the $-\hat{\vc{n}}_\|$ direction.
Thus, polarized top quarks will be pulled in a
definite (forward or backward) direction,
and we may expect 
that a forward-backward asymmetric distribution of
the top quark
$\sim \kappa \, C_\|^0(\chi) \cos \theta_{te}$ is
generated by the final-state interaction.

Incidentally, a forward-backward asymmetric distribution of the top quarks
is also generated by the interference
between the $S$-wave and $P$-wave $t\bar{t}$ pair-production 
amplitudes \cite{r11}.
It is formally a quantity of the same order as the final-state
interaction near threshold, since it arises as an 
${\cal O}(\alpha_s)={\cal O}(\beta )$ correction to the leading
spherically symmetric distribution.
Interestingly, 
we find that each $\cos \theta_{te}$ distribution
has quite a different physical explanation for
its generation mechanism, although it is a common feature that both 
originate from the interplay between QCD and electroweak interactions.

\subsection{Top Quark Polarization Vector}
The Coulomb attraction between $t$ and $\bar{b}$ also modifies
the top-quark polarization vector \cite{r26}.
As we have seen previously, in the $\bar{t}$ decay,
$\bar{b}$ tends to be emitted in the direction of the parent
$\bar{t}$'s spin direction (Fig.\ \ref{tbardecay}).
We then find from Fig.\ \ref{spin} 
that if the $t$ and $\bar{t}$ spins are oriented in the $\hat{\vc{n}}_\|$ 
direction, $t$ will be attracted to the forward direction
due to the attraction by $\bar{b}$, and oppositely attracted
to the backward direction if the $t$ and $\bar{t}$ spins are 
in the $-\hat{\vc{n}}_\|$ direction.
This means that in the forward region ($\cos \theta_{te} \simeq 1$)
the number of $t$'s with
spin in $\hat{\vc{n}}_\|$ direction increases whereas in the
backward region the number of 
those with spin in the opposite direction increases.
Or equivalently, the $\hat{\vc{n}}_\|$-component of
the top-quark polarization vector
increases in the forward region and decreases in the backward region.
We may thus conjecture that the top-quark polarization vector is modified
as 
$\delta \mbox{\boldmath$\cal P$} \sim \kappa \cos \theta_{te} \, 
\hat{\vc{n}}_\|$
due to the interaction between $t$ and $\bar{b}$.

\subsection{$l^+$ Energy-Angular Distribution}
Finally let us examine the effect of the Coulomb attraction between
$b$ and $\bar{t}$ on the $l^+$ energy-angular distribution in the
semi-leptonic decay of $t$.
The $b$-quark from $t$ decay
will be attracted in the direction of $\bar{t}$ due to
the Coulomb interaction between these two particles.
We show schematically typical configurations of the particles
in the top-quark semi-leptonic decay in Fig.\ \ref{semildecay}.
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=12.cm
  \epsffile{leptonconfig.xfig.eps}\\
  \caption[]{
        \label{semildecay}Typical configurations of the particles
        in semi-leptonic decay of $t$ when the $b$-quark
        is emitted in the $\bar{t}$ direction.
        Due to the boost by $W^+$, the energy-angle correlation of $l^+$
        will be either
``$E_l$ is small and $l^+$ emitted in $- \vc{p}_t$ direction'' or
``$E_l$ is large and $l^+$ emitted in $\vc{p}_t$ direction''.}
\end{center}
\end{figure}
It can be seen that if the probability for $b$ being emitted in
the $\bar{t}$ direction increases, correspondingly
the probability for particular $l^+$ energy-angular configurations 
increases.
These configurations are either
``$E_l$ is small and $l^+$ emitted in $- \vc{p}_t$ direction'' or
``$E_l$ is large and $l^+$ emitted in $\vc{p}_t$ direction''.
