\section{Introduction}
\cleqn
\clfn

The study of the top quark at a future
$e^+e^-$ linear collider in the $t\bar{t}$ threshold region
will be one of the unique testing grounds for unraveling
the properties of the top quark.
So far there have been a number of studies of the cross sections
in the $t\bar{t}$ threshold region, both theoretical and 
experimental \cite{r7}-\cite{r26}.
It has been recognized that the threshold region is rich in
physics and also is apt for extracting various physical parameters 
efficiently ($m_t$, $\alpha_s$, $\Gamma_t$, $m_H$, $g_{tH}$, etc.).

While most of the previous analyses were concerned with the production
process of the top quark, one may also analyze the decay process
in detail and extract some important physics information.
Especially the fact that
$t$ and $\bar{t}$ are produced highly polarized in the
threshold region is potentially quite advantageous 
for studying the electroweak properties 
of the top quark through its decay processes.
Detailed studies of the decay of free polarized top quarks have
already been available including the full ${\cal O}(\alpha_s)$ 
corrections \cite{r27,r28}.
In the threshold region, however, these precise
analyses do not apply directly
because of the existence of corrections unique to this region,
which connect the production and decay
processes of the top quark.
Namely, these are 
the corrections due to gluon exchange between $t$
and $\bar{b}$ ($\bar{t}$ and $b$) or between $b$ and $\bar{b}$.

This type of corrections arises when the produced particles decay
quickly into many particles, and are referred to
as ``final-state interactions'', or
``rescattering corrections'', or ``non-factorizable corrections''
in the literature.
They generally vanish in inclusive cross 
sections \cite{r21,r40,r22},
but modify the shape of differential distributions 
in a non-trivial manner \cite{r40,cracow,r26}.
The size of the corrections is at the 10\% level 
in the $t\bar{t}$ threshold region, so that it is 
inevitable to incorporate their
effects in the precision studies of top-quark decays near threshold.
The same kind of effects have recently been studied in $W$ 
pair-production \cite{my,bcb}.
%and decay processes near threshold.

The first study of the top-quark decay in the threshold
region was given as a part of the results in Ref.\ \cite{r26},
where the mean value $\langle n \ell \rangle$
of the charged lepton $l^+$ four-momentum projection on an arbitrarily chosen
four-vector $n$ in semi-leptonic top decays is proposed as an experimentally
observable quantity sensitive
to top quark polarization and is calculated including the
final-state interactions.
Clearly, and also admittedly in that paper, the calculation of
the differential distribution of $l^+$ including the
final-state interactions has been demanded.

In this paper, we calculate the final-state interaction corrections
to the differential 
energy-angular distribution of $l^+$ in semi-leptonic
top quark decays in the $t\bar{t}$ threshold region.
We find that these corrections deform 
the $l^+$ distribution non-trivially at the 10\% level.
Combining with other previously known results, we write
down explicitly the formula for
the energy-angular distribution of $l^+$ including
all ${\cal O}(\alpha_s)={\cal O}(\beta )$ corrections near 
$t\bar{t}$ threshold.
%Key techniques in our calculation are
%(1)soft-gluon approximation and
%(2)decompostion of spinor structure into production and decay parts
%by introduction of the polarization vector of
%top quark.

Another main achievement of this paper is the presentation of
physical descriptions of the final-state interactions which enable us 
to understand the qualitative features of our numerical results.
It turns out that one may explain various effects of the
final-state interactions on the
differential distributions by following the classical picture
that there is
an attractive force between $t$ and $\bar{b}$ (or between $\bar{t}$ and $b$).
Such descriptions
would be useful since
the systematic calculation of the final-state interactions
based on quantum field theory is
rather complicated, involving box and pentagon-type diagrams, and 
it is not easy to make physical sense
directly out of the obtained final expression.
To our knowledge these qualitative explanations have never 
been put forth although corresponding theoretical calculations 
and numerical studies have been available partly.

In section 2 we introduce some notations necessary for later 
sections.
Section 3 gives physical descriptions of the effects of
final-state interactions.
The results of the systematic calculations of final-state interaction
corrections to the $l^+$ energy-angular distribution, as well as
the complete formula for the distribution including all 
${\cal O}(\alpha_s)={\cal O}(\beta )$ corrections, are given in section 4.
A detailed account of our calculational methods will be published
elsewhere.
Section 5 shows various numerical results and a comparison with the
qualitative picture.
Conclusion and discussion are presented in section 6.

%First serious study of top threhold physics for a heavy top quark
%($m_t > 100$~GeV)
%was initiated by Fadin and Khoze.
%They addressed how to include the large decay width of top quark properly 
%in the calculation of cross sections.
%The method was improved by incorporating the running of $\alpha_s$
%and was applied to the study of total cross section and 
%various differential cross sections of top quark.
%It has been found that in the threshold region
%the total cross section, top-momentum distribution,
%and forward-backward asymmetry of top quark provide independent
%and important information.
%All ${\cal O}(\alpha_s)={\cal O}(\beta )$ corrections have been
%calculated to these quantities theoretically.
%Also serious experimental feasibility studies have been performed
%and it was found that indeed that various physical parameters 
%($m_t$, $\alpha_s$, $\Gamma_t$, $m_H$, $g_{tH}$)
%can be extracted in the threshold region.
