\section{$l^+$ Energy-Angular Distribution}
\cleqn
\clfn

We present the formulas for the charged lepton energy-angular distribution
in the decay of a top quark that is produced via $\epem \to \ttbar$
near threshold.
In the following, $x_l=2E_l/m_t$ and $\Omega_l$, respectively, denote
the normalized energy and solid angle of the charged lepton as defined in the
rest frame of the parent top quark.
For simplicity we neglect the decay of $W^-$ in our calculations.

\subsection{Factorizable Part}

\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=8.cm
  \epsffile{born.xfig.eps}\\
  \caption[]{
        \label{bornfig}Born-type diagram for the process
$e^+e^- \to t\bar{t} \to bl^+\nu\bar{b}W^-$.
}
\end{center}
\end{figure}
It is well-known that the
contribution of the Born-type (reducible) diagram (Fig.~\ref{bornfig}) 
to the
differential distribution of $t$ and $l^+$ has a form where the
production and decay processes of the top quark
are factorized:
\bea
\frac{d\sigma_{\mbox{\scriptsize{Born}}}(\epem \! \to \ttbar \to bl^+\nu\bar{b}W^-)}
{d^3\vc{p}_t dx_l d\Omega_l}
=
\frac{d\sigma_{\mbox{\scriptsize{Born}}}(e^+e^- \! \to t\bar{t})}
{d^3\vc{p}_t} \times
\frac{1}{\Gamma_t} \,
\frac{d\Gamma_{t \to bl^+\nu}
(\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}) }
{dx_l d\Omega_l} .
\label{born}
\eea
The above factorized form holds true even including 
${\cal O}(\alpha_s)={\cal O}(\beta )$ corrections to each vertex
and propagator in the Born-type diagram.
Here, $\width$\cite{r42,r53} and
$d\Gamma_{t \to bl^+\nu}(\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}})
/ dx_l d\Omega_l$\cite{r44,r27},
respectively, are the width of a free top quark and
the charged lepton ($l^+$) energy-angular distribution in
the decay of a free polarized top quark,
both including full \oalfs corrections.
Near threshold, the top-quark production cross section for longitudinally
polarized \epem beams is given by\cite{r26}
\bea
&&
\frac{d\sigma_{\mbox{\scriptsize{Born}}}(e^+e^- \! \to t\bar{t})}
{d^3{\bf p}_t} 
= \frac{d\sigma^0_{t\bar{t}}}
{d^3{\bf p}_t} \times
\left[
1 + 2 C_{FB}(\chi ) \varphi_{\rm _R}(p,E) \cos \theta_{te}
\right] ,
\\
&&
\frac{d\sigma^0_{t\bar{t}}}
{d^3\vc{p}_t} 
= \frac{N_c\alpha^2\Gamma_t}{4\pi m_t^4} \,
\left( 1 - P_{e^+}P_{e^-} \right) ( a_1 + \chi a_2 )
\left( 1 - \frac{4C_F\alpha_s}{\pi} \right)
\left| \nrg \right|^2 .
\label{dstt0}
\eea
Here, $\alpha$ is the fine structure constant, and $N_c=3$.
$\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}
= {\cal P}_\| \hat{\vc{n}}_\| + {\cal P}_\bot \hat{\vc{n}}_\bot + 
{\cal P}_{\rm N} \hat{\vc{n}}_{\rm N}$ 
represents the polarization vector of a top quark produced via the
Born-type diagram (Fig.~\ref{bornfig}) near threshold.
The components are given by
\begin{eqnarray}
{\cal P}_\|(\vc{p}_t ,E,\chi) &=& C_\|^0(\chi)
+ C_\|^1(\chi)\, \varphi_{\rm _R}(p ,E)\,\cos\theta_{te}\,
 \label{thr_long}\\
{\cal P}_\bot(\vc{p}_t ,E,\chi) &=& C_\bot(\chi)\,
\varphi_{\rm _R}(p ,E)\,
\sin\theta_{te}\,
\label{thr_perp}\\
{\cal P}_{\rm N}(\vc{p}_t ,E,\chi) &=& C_{\rm N}(\chi)
\varphi_{\rm _I}(p ,E)
\sin\theta_{te}\,
        \label{thr_norm} .
\end{eqnarray}

\subsection{Final-state Interaction Corrections}
Corrections due to the final-state interactions (rescattering corrections)
that originate from the irreducible diagrams (a)-(d) in Fig.~\ref{fsifig}
are important particularly in the threshold region.
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=16.cm
  \epsffile{fsi.xfig.eps}\\
  \caption[]{
        \label{fsifig}
Diagrams for final-state interactions for 
$e^+e^- \to t\bar{t} \to bl^+\nu\bar{b}W^-$:
(a)Coulomb-gluon exchange between $t$ and $\bar{b}$,
(b)Coulomb-gluon exchange between $\bar{t}$ and ${b}$,
(c)Coulomb-gluon exchange between $b$ and $\bar{b}$, and
(d)transverse-gluon exchange between $b$ and $\bar{b}$.
}
\end{center}
\end{figure}
In fact their contributions are counted as 
${\cal O}(\alpha_s)={\cal O}(\beta )$ corrections to the leading
threshold enhancement\cite{r40}.
We list the
effect of each diagram on the $l^+$ energy-angular 
distribution.
We chose Coulomb-gauge for the QCD part in our calculations.

The contribution of diagram (a) 
(exchange of one Coulomb-gluon between $t$ and $\bar{b}$)
can be regarded, after integrating over $\bar{b}W^-$ phase-space, as
a correction to the production process of $t$.
Thus, the production cross section 
$d\sigma_{\mbox{\scriptsize{Born}}}(e^+e^- \! \to t\bar{t})/d^3\vc{p}_t$
and the polarization vector 
$\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}$
of the top quark
receive corrections by this diagram, whereas the decay distribution
$d\Gamma_{t \to bl^+\nu}(\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}})
/ dx_l d\Omega_l$
remains unaffected (except for the modification of 
$\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}$).
In fact the contribution of this diagram can be incorporated by 
the following substitutions in Eq.\ (\ref{born}):
\bea
\frac{d\sigma_{\mbox{\scriptsize{Born}}}(e^+e^- \! \to t\bar{t})}
{d^3{\bf p}_t} 
\rightarrow
\frac{d\sigma_{\mbox{\scriptsize{Born}}}(e^+e^- \! \to t\bar{t})}
{d^3{\bf p}_t} 
\times
\left( 1 + \delta_a \right),
~~~~~
\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}
\rightarrow
\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}
+ \delta \mbox{\boldmath$\cal P$}_a
\label{diagrama1}
\eea
with
\bea
\delta_a &=& {\frac{1}{2}}
\left[ 
\psi_1(p,E) + \kappa C^0_\| \psi_{\rm _R}(p,E) \cos \theta_{te}
\right] ,
\label{diagrama2}
\\
\delta \mbox{\boldmath$\cal P$}_a &=&
{\frac{1}{2}} \left[ 1 - (C^0_\|)^2 \right]
\kappa \, \psi_{\rm _R}(p,E) \cos \theta_{te} \cdot \hat{\vc{n}}_\| .
\label{diagrama3}
\eea

Diagram (b) (exchange of one Coulomb-gluon between $\bar{t}$ and $b$)
in Fig.\ \ref{fsifig} gives a correction that connects the production
and decay processes of the top quark. 
In fact one may incorporate the contribution of this diagram by
multiplying Eq.\ (\ref{born}) by a factor $[1+\xi (p,E,x_l,\cos\theta_{lt})]$,
where
\bea
\xi (p,E,x_l,\cos\theta_{lt})
= C_F \cdot 4\pi\alpha_s 
\int {\textstyle \frac{d^3\vc{q}}{(2\pi)^3} } \,
\frac{1}{|\vc{q}\! -\! \vc{p}_t|^3} \, \mbox{Re}
\left[ 
\frac{\tilde{G}^*(q,E)}{\tilde{G}^*(p,E)} 
\, {\displaystyle \int^{2\pi}_0} \!
{\textstyle \frac{d\phi_{bl}}{2\pi} } \,
\frac{|\vc{q}\! -\! \vc{p}_t|}
{\hat{\vc{n}}_b \! \cdot \! (\vc{q}\! -\! \vc{p}_t) + i\epsilon}
\right] .
\label{xi1}
\eea
$\hat{\vc{n}}_b$ denotes the unit vector in the direction of $b$.
After integration over $d\Omega_{\vc{q}}d\phi_{bl}$,
one may reduce the expression to a one-parameter integral form as
\bea
\xi (p,E,x_l,\cos\theta_{lt})
= C_F \cdot 4\pi\alpha_s 
\int^\infty_0 \! dq 
\left\{
w_{\rm _R} \,
\mbox{Re} \biggl[ \frac{\tilde{G}^*(q,E)}{\tilde{G}^*(p,E)} \biggl]
\, + \, w_{\rm _I} \,
\mbox{Im} \biggl[ \frac{\tilde{G}^*(q,E)}{\tilde{G}^*(p,E)} \biggl]
\right\}
\label{onepmint}
\eea
with
\bea
&&
w_{\rm _R} = \frac{1}{4\pi^2} \, \frac{q}{p^2-q^2}
\left\{
\theta ( z_+^2 \! -1 ) \cosh^{-1} |z_+| 
- \theta ( z_-^2 \! -1 ) \cosh^{-1} |z_-| 
\right\} ,
\\
&&
w_{\rm _I} = \frac{1}{4\pi^2}\, \frac{q}{|p^2-q^2|}
\left\{
\pi \theta \left(
( 1 + z_+ )( 1 + z_- )
\right) 
- \theta ( 1 - z_+^2 ) \cos^{-1} (z_+)
- \theta ( 1 - z_-^2 ) \cos^{-1} (z_-)
\right\} ,
\nonumber \\
\\
&&
z_\pm = \frac{q/p \pm \cos\theta_{Wl} \cos\theta_{lt}}{\sin\theta_{Wl} \sin\theta_{lt}},
~~~
\cos\theta_{Wl} =  \frac{1+y}{1-y} - \frac{2y}{x_l(1-y)},
~~~
\cos\theta_{lt} = \frac{\vc{p}_t\cdot\vc{p}_l}{|\vc{p}_t||\vc{p}_l|} .
\label{zangles}
\eea
Here, 
$\theta_{Wl}$ represents
the angle between $W^+$ and $l^+$ in the $t$ rest frame (given as
a function of $x_l$);
$\theta_{lt}$ represents the angle between $t$ and $l^+$ in the \ttbar c.m.\ frame
($0 \le \theta_{Wl},\theta_{lt} \le \pi$).
The inverse functions
$\cosh^{-1}$ and $\cos^{-1}$ in the above formulas take their values within
$[0,\infty)$ and $[0,\pi]$, respectively.
$\theta (x)$ is the step function.
It is understood that the principal value should be taken
in the integration of the
$w_{\rm _R}$-term as $p \to q$.

Two noteworthy properties of $\xi (p,E,x_l,\cos\theta_{lt})$ are:
(1) its $l^+$-angular dependence enters only through $\cos\theta_{lt}$ and is
independent of the angle from the $e^-$ beam direction or from the top-quark
polarization vector, and
(2) it is purely determined by the QCD interaction and free of the
coupling parameters of electroweak interactions (except $y$).

As a non-trivial cross-check 
of the formula (\ref{onepmint}), we integrated 
$\xi \times \Gamma_t^{-1} d\Gamma_{t \to bl^+\nu}/ dx_l d\Omega_l$
over the
lepton energy-angular variables $\int dx_l d\Omega_l$ analytically
and reproduced
the one-parameter integral formula\cite{r40} for the final-state
interaction correction (from diagram (b)) to the top-quark three-momentum 
distribution.

The contribution of diagram (c) 
(exchange of one Coulomb-gluon between $b$ and $\bar{b}$)
vanishes within our approximation. 
In fact one can show quite generally that 
the contribution of diagram (c) vanishes at \oalfs
{\it provided} one calculates a cross section where the top-quark energy is 
integrated out; for example, the top-quark three-momentum distribution.  
This is not the case when one considers a cross section that depends 
explicitly on the top-quark energy; for example, the top-quark four-momentum 
distribution. Then the diagram in question {\it does} contribute.

It can be shown using similar techniques that the contribution of diagram (d) 
(exchange of one transverse-gluon between $b$ and $\bar{b}$)
gets canceled when it is added 
to that of the corresponding real-gluon emission diagram
(interference of diagrams (e) and (f) in Fig.~\ref{gemissionfig}).
\begin{figure}
\begin{center}
  \leavevmode
  \epsfxsize=16.cm
  \epsffile{gemission.xfig.eps}\\
  \caption[]{
        \label{gemissionfig}
Diagrams for real-gluon emission process
$e^+e^- \to t\bar{t} \to bl^+\nu\bar{b}W^- g$.
}
\end{center}
\end{figure}
This cancellation is consistent with the same cancellation that was
found in the calculation of top-quark momentum distribution\cite{r40}.
The contribution from each of these diagrams comes from the gluon momentum
region $|\vc{k}| \simlt \alpha_s^2 m_t$ and
is in fact logarithmically divergent due to soft-gluon singularity.
\medbreak

Now let us compare our formulas and the argument given in the previous
section.
In subsections 3.b and 3.c, the $V\! -\! A$ nature of the
electroweak interaction in $\bar{t}$ decay played an essential role, while
this was not the case in subsections 3.a and 3.d.
We find that correspondingly the
$\cos\theta_{te}$ term of $\delta_a$
and $\delta \mbox{\boldmath$\cal P$}_a$ 
contain electroweak coupling parameters
(through $C^0_\|$) while the symmetric term of $\delta_a$ and
$\xi$ are independent of these electroweak parameters.
More precisely, the $\cos\theta_{te}$ term of $\delta_a$
and $\delta \mbox{\boldmath$\cal P$}_a$ 
have the forms anticipated in subsections 3.b and 3.c, respectively, 
if the function $\psi_{\rm _R}(p,E)$ is positive.
Indeed, the numerical evaluation in Ref.\ \cite{r26} shows that 
$\psi_{\rm _R}(p,E) \simgt 0$ holds in the entire threshold region.
Besides, an additional coefficient $\left[ 1 - (C^0_\|)^2 \right]$
in $\delta \mbox{\boldmath$\cal P$}_a$ can be understood 
within our previous argument
in the extreme cases $C^0_\| = \pm 1$.
Namely, if the top quark is 100\% polarized, there will be no
contamination from the opposite spin so that the correction should
disappear.

\subsection{Formula Including Full \oalfs Corrections}
In summary, the energy-angular distribution of $l^+$ including full
${\cal O}(\alpha_s) = {\cal O}(\beta)$ corrections can be cast into a form
\bea
\frac{d\sigma(\epem \! \to \ttbar \to bl^+\nu\bar{b}W^-)}
{d^3\vc{p}_t dx_l d\Omega_l}
=
\frac{d\sigma(e^+e^- \! \to t\bar{t})}
{d^3\vc{p}_t} \times
\frac{1}{\Gamma_t} \,
\frac{d\Gamma_{t \to bl^+\nu}
(\mbox{\boldmath$\cal P$}) }
{dx_l d\Omega_l}  \times \left( 1 + \xi \right)
\label{fullformula}
\eea
with
\bea
&&
\frac{d\sigma(e^+e^- \! \to t\bar{t})}
{d^3\vc{p}_t} 
= \frac{d\sigma^0_{t\bar{t}}}
{d^3\vc{p}_t} 
\times
\left[
1 + \frac{1}{2}\psi_1 + 
( 2 C_{FB} \varphi_{\rm _R} + \frac{\kappa}{2}C_\|^0\psi_{\rm _R} )
\cos \theta_{te}
\right] ,
\\
&&
\mbox{\boldmath$\cal P$}
=
\mbox{\boldmath$\cal P$}_{\mbox{\scriptsize{Born}}}
+ \delta \mbox{\boldmath$\cal P$}_a .
\eea
The above distribution is obtained as the sum of the cross sections for
$e^+e^- \to t\bar{t} \to bl^+\nu \bar{b}W^-$ 
and
$e^+e^- \to t\bar{t} \to bl^+\nu \bar{b}W^-g$. 
An independent emission of a gluon from the $t$ or $\bar{t}$ side has been
included in Eq.\ (\ref{born}), while the interference of both
has been incorporated in conjunction with the final-state interaction 
diagram (d) in subsection 4.b.

Experimentally the top-quark four-momentum $p_t^\mu$
will necessarily be reconstructed from the
$\bar{b}W^-$ system in the study of the
$l^+$ distribution.
For the case with a gluon in the final state, we assign the 
``top-quark momentum'' as
\bea
\begin{array}{llll}
\mbox{Case (A)}: & p_t \equiv (p_{e^-}+p_{e^+})
- (p_{\bar{b}} + p_{W^-} + p_g)
& \mbox{if} &
(p_{\bar{b}} + p_{W^-} + p_g)^2 - m_t^2 \simlt m_t\width  ,
\\
\mbox{Case (B)}: & p_t \equiv (p_{e^-}+p_{e^+})
- (p_{\bar{b}} + p_{W^-}) &
\mbox{if} & (p_{\bar{b}} + p_{W^-})^2 - m_t^2 \simlt m_t\width .
\end{array}
\nonumber
\eea
(See, however, the discussion in section 6.)
In other kinematical configurations the cross section is suppressed.
Experimentally there will be a corresponding cut in the $\bar{b}W^-$
invariant mass.
If both conditions in 
(A) and (B) are satisfied simultaneously, the gluon should
necessarily be soft, and there will be no difference within our
approximation between the cross 
sections corresponding to the above two assignments of the top-quark
momentum.

