\section{Preparations}
\cleqn
\clfn

In this section we introduce some notations to be used in this paper.

We consider longitudinally polarized \epem beams throughout
our analyses.
$P_{e^{\pm}}$ denotes the longitudinal polarization of $e^{\pm}$,
and we set
\begin{equation}\label{chi}
\chi={P_{e^+}-P_{e^-}\over1-P_{e^+}P_{e^-}} .
\end{equation}
We choose a reference coordinate system in the \ttbar c.m.\ frame.
Three orthonormal basis vectors are defined as
\begin{eqnarray}
\hat{\bf n}_{\|} &=& \frac{{\bf p}_{e^-}}{|{\bf p}_{e^-}|} \nonumber\\
\hat{\bf n}_{\rm N} &=& {{\bf p}_{e^-} \! \times {\bf p}_t \over
        |{\bf p}_{e^-} \! \times {\bf p}_t|}               \label{basis}\\
\hat{\bf n}_\bot &=& \hat{\bf n}_{\rm N} \times \hat{\bf n}_{\|} ~,  
\nonumber
\end{eqnarray}
where $\vc{p}_{e^-}$ and $\vc{p}_t$ represent the $e^-$ and $t$ momentum,
respectively. 

Our conventions for the fermion
vector and axial-vector couplings to the $Z$ boson are
\begin{equation}
v_f = 2 I^3_f - 4 q_f \sin^2\theta_{\rm W} , \qquad  a_f = 2 I^3_f ,
\end{equation}
respectively.
Certain combinations of these couplings will be useful below:
\begin{eqnarray}
a_1 &=& q_e^2 q_t^2 + (v_e^2 + a_e^2) v_t^2 d^2 +
        2 q_e q_t v_e v_t d \nonumber \\
a_2 &=& 2 v_e a_e v_t^2 d^2 + 2 q_e q_t a_e v_t d \nonumber \\
a_3 &=& 4 v_e a_e v_t a_t d^2 + 2 q_e q_t a_e a_t d \label{coupl}\\
a_4 &=& 2 (v_e^2 + a_e^2) v_t a_t d^2 + 2 q_e q_t v_e a_t d, \nonumber\\
d &=& {1\over 16 \sin^2\theta_{\rm W}\cos^2\theta_{\rm W}}\,{s\over s - M_Z^2},
    \nonumber
\end{eqnarray}
and 
\\
\parbox{75.ex}{
\begin{eqnarray*}
& &\hspace{5.ex}C_\|^0 (\chi) =
-{a_2 + \chi a_1 \over a_1 + \chi a_2} ,\hspace{6.9ex}
  C_\|^1 (\chi) = \left( 1-\chi^2 \right) {a_2 a_3 - a_1 a_4   \over
        \left(a_1 + \chi a_2 \right)^2} ,\\
& &\hspace{5.ex}C_\bot(\chi)  = -{1\over 2} \,
{a_4 + \chi a_3 \over a_1 + \chi a_2} ,
    \qquad C_{\rm N}(\chi) =-{1 \over 2}\, {a_3
    + \chi a_4 \over a_1 + \chi a_2}\, =\, - C_{\rm FB}(\chi) .
\end{eqnarray*}}
\hfill
\parbox{5.ex}{
\begin{eqnarray} \label{coefs} \end{eqnarray} }\\
Finally, we define 
\bea
y = \frac{M_W^2}{m_t^2}, ~~~~~~~~
\kappa = \frac{1-2y}{1+2y}.
\label{ykappa}
\eea

The QCD enhancement of the top-quark production cross section 
near threshold is incorporated through the
$S$-wave and $P$-wave Green's functions, $\nrg$ and $\tilde{F}(p,E)$,
of the non-relativistic Schr\"{o}dinger equations in the presence
of the QCD potential.
These functions are defined by
\bea
&&
\left[
- \frac{\nabla^2}{m_t} + V_{QCD}(r) -
\left( E+i\Gamma_t \right)
\right] \, G(\vc{x},E) = \delta^3(\vc{x}),
\label{sse2}
\\
&&
\left[
- \frac{\nabla^2}{m_t} + V_{QCD}(r) -
\left( E+i\Gamma_t \right)
\right] \, F^k(\vc{x},E) = -i \partial^k \delta^3(\vc{x}),
\label{pwse2}
\eea
and
\bea
\tilde{G}(p,E) &=& \int d^3\vc{x} \, e^{-i\vc{p} \cdot \vc{x} } \, 
G(\vc{x},E) ,
\\
p^k \,
\tilde{F}(p,E) &=& \int d^3\vc{x} \, e^{-i\vc{p} \cdot \vc{x} } \, 
F^k(\vc{x},E) .
\label{ftpgf2}
\eea
One may obtain the Green's functions either by first solving the 
Schr\"{o}dinger 
equations in coordinate space and taking the Fourier transforms of the
solutions \cite{r9},
or by solving the Schr\"{o}dinger equations directly in
momentum space \cite{r35}.

Various known \oalfs corrections to threshold cross sections can also be 
expressed  in terms of the above Green's functions.
In Ref.\ \cite{r26}
the following functions have been defined, which are
solely determined from QCD, to represent various independent
corrections.
We will pursue the same conventions in this paper.
\bea
&&
\varphi(p,E) =
{(1-{C_F \alpha_{s}/ \pi})\over (1-{2 C_F \alpha_{s}/ \pi})}\,
        {p \over m_t}\,
        {\tilde{F}^* \!(p,E) \over \tilde{G}^* \!(p,E)}  ,
\\
&&
\varphi_{\rm _R} = \mbox{Re}\,\varphi , ~~~~~
\varphi_{\rm _I} = \mbox{Im}\,\varphi ,
\\
&&
\psi_1(p,E) = - \, C_F \! \cdot \! 4\pi \alpha_s
\int \! \mbox{$\frac{d^3\vc{q}}{(2\pi)^3}$} \,
\frac{1 \,}{|\vc{q}\! -\! \vc{p}_t|^3} \,
2 \, \mbox{Im}
\biggl[ \frac{\tilde{G}(q,E)}{\tilde{G}(p,E)} \biggl] 
\cdot \frac{\pi}{2}
\label{psi1}
\\
&&
\psi_{\rm _R}(p,E) = - \, C_F \! \cdot \! 4 \pi \alpha_s \,
\mbox{Pr.}\int \mbox{$\frac{d^3\vc{q}}{(2\pi)^3}$} \,
\, \frac{1 \,}{|\vc{q}\! -\! \vc{p}_t|^3} \,
\frac{\vc{p}_t\! \cdot \! (\vc{q}\! -\! \vc{p}_t)}{|\vc{p}_t|\, 
|\vc{q}\! -\! \vc{p}_t|}
\,
2 \,
\mbox{Re}
\biggl[ \frac{\tilde{G}(q,E)}{\tilde{G}(p,E)} \biggl] .
\label{psi2}
\eea
Here $C_F=4/3$ is a color factor; $p=|\vc{p}_t|$ and $q=|\vc{q}|$.
Eqs.\ (\ref{psi1}) and (\ref{psi2})\footnote{
One may render Eqs.\ (\ref{psi1}) and (\ref{psi2}) to one-parameter
integral forms by integrating over $d\Omega_{\vc{q}}$ explicitly~\cite{r40}.
}
differ slightly in their integrands from those defined
in Ref.\ \cite{r26}.
The differences, however, can be regarded as higher order corrections
which are beyond the scope of our analysis.

