\section{Conclusion and Discussion}
\cleqn

In this paper we 
considered the differential distribution of $l^+$ from the semi-leptonic
decay of
top quark, where the parent top quark is
produced 
in $e^+e^- \to t\bar{t}$ near threshold.
Particularly, we have calculated the final-state interaction
corrections (rescattering corrections)
to the energy-angular distribution of lepton in the
top quark decay.
Also we have explicitly
written down the $l^+$ energy-angular distribution 
${d\sigma(\epem \! \to \ttbar \to bl^+\nu\bar{b}W^-)}/
{d^3\vc{p}_t dx_l d\Omega_l}$
including the full ${\cal O}(\alpha_s)={\cal O}(\beta )$ corrections
near threshold.

We presented numerical studies of the various effects of the final-state
interaction corrections.
All numerical results can be understood qualitatively from intuitive
pictures.
Attractive forces between $t$ and $\bar{b}$ and between $\bar{t}$ and $b$
modify not only the
momentum distribution of $bW^+$ or $\bar{b}W^-$ system but also
the top-quark polarization vector and 
the lepton energy-angular distribution.
\begin{itemize}
\item
The effect of Coulomb-gluon exchange between $\bar{b}$ and $t$,
when integrated over $\bar{b}W^-$ phase-space,
can be regarded as a correction to the top-quark production process.
The effect can be incorporated by modifying the top-quark production
cross section and the top-quark polarization vector.
The top-quark momentum distribution is shifted to take a smaller
average momentum due to the attraction by $\bar{b}$.
Also since $\bar{b}$ is emitted preferrably in the ${t}$ spin direction,
the attraction generates a $\cos \theta_{te}$ distribution of 
the top quark
as well as modifies the top-quark polarization vector.
\item
The Coulomb interaction between $b$ and $\bar{t}$ causes a 
non-factorizable correction with respect 
to the production and decay processes of the top quark.
The lepton distribution is deformed in favor of the kinematical 
configurations
``small $E_l$ and emitted in $\bar{t}$ direction'' or
``large $E_l$ and emitted in $t$ direction'',
which can be understood as originating from the attraction of $b$ in the
direction of $\bar{t}$.
\item
Corrections from the gluon
exchange between $b$ and $\bar{b}$ turn out to vanish
when the top-quark energy is integrated out.
\end{itemize}

Without the non-factorizable effect $\xi$, the $l^+$ angular distribution
is dependent only on the polar angle from 
the polarization vector of the parent top quark
in its rest frame.
The final-state interaction brings in another direction into the problem,
the direction of $\bar{t}$, which is a completely new feature
in comparison to the decays of free polarized top quarks.
\medbreak

In subsection 4.c, we assumed a perfect assignment of top-quark
momentum in cases (A) and (B) for defining the distribution formula,
Eq.(\ref{fullformula}).
In real experiments, however,
miss-assignment of top-quark momentum will be inevitable
whenever there is real gluon radiation in final state
because of the typical jet-clustering algorithm that will be used.
For instance, when a gluon is indistinguishable from a $b$-jet, the 
top-quark momentum reconstructed by clustering
may be off-shell; rather grouping 
the gluon on the other side (with $\bar{b}$) would 
result in an on-shell momentum.\footnote{
There is no ambiguity of assigning the gluon to the production of \ttbar
since real gluon radiation in the top-quark production process 
is suppressed near threshold.
}
Ref.\cite{r40} studied how this miss-assignment alters the top-quark
three-momentum distribution near threshold and found that the
correction is less than a few percent;
the clustering algorithm assumed in that paper, 
however, is somewhat unrealistic.
The effect of the miss-assignment was also studied in Ref.\cite{sch}
in the open-top region ($\sqrt{s} \gg 2m_t$) using a Monte Carlo
generator and with more realistic experimental assumptions.
It was shown that the effects on the top-quark invariant-mass
distribution and on the angular distributions are substantial at
$\sqrt{s} = 400$~GeV (for $m_t = 175$~GeV) and increase in 
magnitude and in complexity as the c.m.\ energy is raised.
Clearly, in our case, we need more 
detailed studies to see how the similar effect may influence our
results.
For this purpose, 
studies based on a Monte Carlo generator that produces 
fully-differential distribution including full \oalfs corrections
near threshold would be necessary.
\medbreak

The authors wish to thank
M.~Je\.{z}abek and J.H.~K\"{u}hn for enlightening discussion.

