\section{Numerical Results}
\cleqn
\clfn

In this section 
we examine effects of the final-state interaction
corrections to the $l^+$ energy-angular distribution numerically and
compare the results with the qualitative argument given in
section 3.
We set $m_t=175$~GeV, $\alpha_s(M_Z)=0.118$, $P_{e^+}=0$ and
$\alpha=1/128$ in all our analyses.

We first examine the contribution of diagram (a) in Figs.~\ref{fsifig}
(Coulomb interaction between $t$ and $\bar{b}$) as given in
Eqs.(\ref{diagrama1})-(\ref{diagrama3}).
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=8.cm
  \epsffile{mdistfig1.ps}\\
  \caption[]{
        \label{momdist}Top-quark momentum distributions $d\sigma/d|\vc{p}_t|$
obtained from 
Eq.(\ref{diagrama1}) for various c.m.\ energies measured from
the lowest lying resonance, $\Delta E = \sqrt{s} - M_{1S}$.
Solid lines and dashed lines represent the distributions with
and without the final-state interaction between $t$ and $\bar{b}$
($\delta_a$), respectively.  We set $P_{e^-}=0$.
}
\end{center}
\end{figure}
Shown in Fig.\ref{momdist} are the top-quark 
momentum distribution $d\sigma / d|\vc{p}_t|$
for various c.m.\ energies measured 
from the lowest lying resonance,\footnote{
$M_{1S}$ is defined as the
real part of the position of the lowest lying
resonance pole in the complex energy
plane.
The energy measured from $M_{1S}$ is more convenient than the energy measured
from the threshold ($E=\sqrt{s}-2m_t$) when we compare different potentials
in literatures.
} 
$\Delta E = \sqrt{s} - M_{1S}$.
This is calculated from the
top-quark production cross section Eq.(\ref{diagrama1}).
As expected, the top-quark momentum is reduced.
The effects are half in magnitude as compared to the
final-state interaction corrections given in Refs.\cite{r40,r26}
since only the interaction between $t$ and $\bar{b}$ is included here.

We show angular distributions of top quarks in Figs.~\ref{angdist}.
\begin{figure}
\begin{center}\begin{tabular}{ccc}
  \epsfxsize 65mm \mbox{\epsffile[64  243  501  592]{angulardist1.ps}} 
  & \hspace{5mm} &
  \epsfxsize 65mm \mbox{\epsffile[64  243  500  592]{angulardist2.ps}} 
\end{tabular}\end{center}
\caption{\label{angdist}
Top-quark angular distribution $d\sigma / d\cos\theta_{te}$
obtained from 
Eq.(\ref{diagrama1}) for various electron polarizations $P_{e^-}$.
Solid lines and dashed lines represent the distributions with
and without the final-state interaction between $t$ and $\bar{b}$
($\delta_a$), respectively.  
}
\end{figure}
It can be seen that the final-state interaction increases the top-quark 
distribution in the
forward direction for $P_{e^-}=1$.
This is consistent with our argument in subsection 3.b since
in leading-order approximation
$t$ and $\bar{t}$ have their
spins aligned perfectly in the $\hat{\vc{n}}_\|$
direction for this $e^-$ polarization. 
Oppositely we see that the final-state interaction decreases the top-quark
distribution in the forward direction for $P_{e^-}=-1$.
We note that the top quark has a natural
polarization 
$\mbox{\boldmath$\cal P$} \simeq -0.4 \hat{\vc{n}}_\|$
for unpolarized $e^+e^-$ beams. 
Hence, the sign of the correction
is the same as in the $P_{e^-}=-1$ case.
Also we show corrections to the top-quark polarization vector
in Figs.~\ref{delP}.
\begin{figure}
\begin{center}\begin{tabular}{ccc}
  \epsfxsize 65mm \mbox{\epsffile[63  243  501  588]{delP1.ps}} 
  & \hspace{5mm} &
  \epsfxsize 65mm \mbox{\epsffile[63  243  501  588]{delP2.ps}} 
\end{tabular}\end{center}
\caption{\label{delP}
Final-state interaction correction to the $\hat{\vc{n}}_\|$ component
of the top-quark polarization vector, $\delta \mbox{\boldmath$\cal P$}_a$, for
$P_{e^-}=+ 0.8$ (solid line), 0 (dash line), $-0.8$ (dotdash line).
}
\end{figure}
Although the qualitative behavior meets our expectation, the magnitude of
the correction is rather small.
Note that $\delta \mbox{\boldmath$\cal P$}_a$ vanishes for $P_{e^-}=\pm 1$
since $C^0_\|=\pm 1$.

Next we investigate the irreducible (non-factorizable) correction
that stems from the Coulomb interaction between $\bar{t}$ and $b$.
The correction factor $\xi(p,E,x_l,\cos \theta_{lt})$ given
in Eqs.(\ref{onepmint})-(\ref{zangles})
depends on four parameters.
Of these, $x_l$ and $\cos \theta_{lt}$ specify the lepton
configuration.
Therefore, we will examine the dependence of $\xi$ on these 
two parameters
for several $(p, \Delta E)$ combinations.
We fix the top-quark momentum $p$ to be the peak momentum
of the distribution for
each $\Delta E$;
see the top-quark momentum distributions in Fig.~\ref{momdist}.
Shown in Figs.~\ref{xi} are 3-dimensional
plots of $\xi$ as a function of $x_l$ and $\cos \theta_{lt}$.
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=12.cm
  \epsffile[68  231  543  564]{xifig1.ps}\\
  \leavevmode
  \epsfxsize=12.cm
  \epsffile[68  231  543  564]{xifig2.ps}
  \caption[]{
        \label{xi}
Three-dimensional plots of $\xi$ as a function of $x_l$ ($x$-axis) and
$\cos\theta_{te}$ ($y$-axis) at the peak momenta of the 
$|\vc{p}_t|$-distribution in Fig.~\ref{momdist}:
(a)$\Delta E=0$~GeV and $|\vc{p}_t|=14.1$~GeV,~~
(b)$\Delta E=5$~GeV and $|\vc{p}_t|=26.8$~GeV.
}
\end{center}
\end{figure}
One sees that in the figures $\xi$ takes comparatively large
positive values for
either `small $x_l$ and $\cos \theta_{lt} \simeq -1$' or
`large $x_l$ and $\cos \theta_{lt} \simeq +1$'.
On the opposite side,
in the other two corners of the $x_l$-$\cos \theta_{lt}$ plane
$\xi$ becomes small or becomes negative for smaller $\Delta E$.
The typical magnitude of $\xi$ is 10-20\%, which would be 
reasonable size for an
${\cal O}(\alpha_s) = {\cal O}(\beta )$ correction.
These qualitative features hold also for 
$\xi$ at $p$ off the peak of top-quark momentum distribution.
These features of the correction factor $\xi$ are consistent
with our qualitative argument in section 3.

We made a cross-check of our numerical results for $\xi$ by numerically
integrating 
$\xi \times \Gamma_t^{-1} d\Gamma_{t \to bl^+\nu}/ dx_l d\Omega_l$
over the
lepton energy-angular vairables $\int dx_l d\Omega_l$ 
and comparing to the final-state interaction corrections to top-quark
momentum distribution given in Refs.\cite{r40,r26}.

