\section{QCD corrections}
\label{qcd}
\subsection{Continuum cross section}
In this section we calculate the complete order $\alpha_{s}$ QCD corrections
to the cross section for $\gamma\gamma\rightarrow t\bar{t}$ production.
The Born cross section ${\sigma}_{\gamma\gamma\rightarrow t\bar{t}}^0$
for open production vanishes
at threshold (see eq. (\ref{bornwq}))
as a consequence of S-wave phase space suppression.
The order ${\alpha}_s$ corrections are
nececcary for a smoth matching between resonance and open flavor regions.
Both the gluon bremsstrahlung amplitude squared and the interference
between Born and one-loop graphs contribute in order $\alpha_s$. The gluon
bremsstrahlung cross section is given by the process
\begin{equation}
\gamma\left(k_1
\right)+\gamma\left(k_2\right)\to g\left(k_3\right)+Q\left(p_1\right)
+\overline{Q}\left(p_2\right).
\label{reac}
\end{equation}
Six Feynman diagrams contributing
to the amplitude.
Let us now briefly sketch some technical ingredients that go into our
calculation.
For the $O(\alpha_{s})$ tree contributions in (\ref{reac}) we introduce the
variables
\cite{bkns}
\begin{equation}
s=\left(k_1+k_2\right)^2 \hspace{1cm}
t_1=\left(k_2-p_2\right)^2-m_t^2 \hspace{1cm}
u_1=\left(k_1-p_2\right)^2-m_t^2
\end{equation}
and $s_{4}=(k_{3}+p_{1})^{2}-m_{t}^{2}=s+t_{1}+u_{1}$.
Averaging over the initial spins the cross section for the hard gluon
bremsstrahlung can be written in the following form
\begin{equation}
\frac{d\sigma^H}{dt_1\,du_1}=\frac{\alpha_s\alpha^2}{8\pi s^2}
\frac{s_4}{s_4+m_t^2}\int_{0}^{\pi}d\theta_{1}\sin\theta_{1}
\int_{0}^{\pi}d\theta_{2}\,\,
R_{QED}
\label{sigh}
\end{equation}
$R_{QED}$ denotes the square of the matrix element of reaction
({\ref{reac}}) (without charges, color- and spin average factors).
We checked that the square of the
matrix element for hard radiation agrees with the expression found by
Gunion and Kunszt
\cite{guku}.
The integration over the final state gluon is most easily performed in the
quark-gluon CM system over the solid angle $d\Omega=\sin\theta_{1}
d\theta_{1}\,d\theta_{2}$ where $\theta_{1}$ and $\theta_{2}$ denote the polar
and azimuthal angle of the gluon.
In order to perform the angular integration in (\ref{sigh}) we follow the
procedure as outlined in \cite{bkns}.
Note that
collinear divergences are absent in the tree contributions (\ref{reac}) as a
consequence of the nonvanishing
quark mass.
The integration $d\Omega$ over the final state gluon is performed analytically
and requires the evaluation of integrals of the following form
\begin{equation}
I^{\left(k,l\right)}=\int\limits_0^{\pi}d\cos\theta_1\int\limits_0^{\pi}
d\theta_2\,\left(a+b\cos\theta_1\right)^{-k}
\left(A+B\cos\theta_1+C\sin\theta_1\cos\theta_2\right)^{-l},\label{integral}
\end{equation}
where $a,\ b,\ A,\ B$ and $C$ are functions of the external
kinematical variables $s,\ t_1,\ u_1$ and $m_t^2$ and $k,l$ are positive ore
negative integers.
The neceassary sets of integrals can be found in \cite{bkns}.
Note that the reduction to this basic set of integrals is achieved only after
very involved partial fractioning of the tree graph matrix elements.
Let us now turn to the virtual and soft contributions.
The soft cross section is defined by the condition
$0\leq s_4\leq\Delta$ in the bremsstrahlung diagrams of (\ref{reac}).
The pole terms appearing in (\ref{reac}) originate from infrared singularities
and cancel when the virtual
and soft contributions are combined.
For the sum of the virtual- and soft-gluon cross section $\sigma^{V+S}$
we make use of the result of \cite{bkns}.
After average over the initial spins $\sigma^{V+S}$ can be written as
\begin{equation}
\frac{d\sigma^{V+S}}{dt_1\,du_1}=\frac{2\alpha_s\alpha^2}{s^2}
\delta\left(s+t_1+u_1\right)\left[F^{V+S}_{QED}\left(t_1,u_1\right)
+F^{V+S}_{QED}\left(u_1,t_1\right)\right]
\label{sigvs}
\end{equation}
where $F^{V+S}_{QED}$ can be found in Appendix D of \cite{bkns}.
The remaining two dimensional integration in (\ref{sigh}) and
(\ref{sigvs}) are evaluated numeri\-cally.
In our numerical program the
$\ln^i\left(\Delta\right)$ ($i$=0,1) terms in $\sigma^{V+S}$
were rewritten
as integrals over $s_4$ \cite{schuler}:
\begin{eqnarray}
1&=&\int\limits_{\Delta}^{s_4^{max}}\frac{ds_4}{s_4^{max}-\Delta}\\
\ln\frac{\Delta}{m_t^2}&=&\int\limits_{\Delta}^{s_4^{max}}
\frac{s_4^{max}}{s_4^{max}-\Delta}\left[\ln\frac{s_4^{max}}{m_t^2}-
\frac{s_4^{max}-\Delta}{s_4}\right]
\end{eqnarray}
In this way the virtual+soft piece and the hard piece can directly
be added. The result is a flat $s_4$ distribution for the
total order $\alpha_s$ correction $\sigma^{V+S+H}$.
We checked that the total cross section is independent of
$\Delta$ in the limit $\Delta\to 0$.
Near threshold the exact analytical behaviour can be extracted.
The Taylor expansion in $\beta$ of the correction factor is given by
\input{entwicklung}
It starts with an inverse power of $\beta$, familiar from Sommerfelds
rescattering corrections. Combined with the phase space factor it
leads to a step-function-like threshold behaviour. The next term in the
expansion is identical to the correction factor for bound state production.
%while the $\frac{1}{\beta}$ term is in this case absorbed in the Coulomb
%potential.
The cross section for open $t\bar{t}$ production
can be cast into the form
\input{skalen_funktionen}
where $\fggn$ and $\fgge$ depend on $\frac{s}{4m_t^2}$ only. A plot of
$\fgge$ is shown in Fig. \ref{s_funktionen}a.
The ratio $\fgge/\fggn$ is shown in Fig. \ref{s_funktionen}b.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild8.eps,width=12cm}}
\end{center}
\vskip1.0cm
\begin{center}
\mbox{\epsfig{file=bild19.eps,width=12cm}}
\end{center}
\caption{(a) Plot of $\fgge$ and (b) $\fggn/\fggn$ versus ${s}/{4 m^2_t}$}
\label{s_funktionen}
\end{figure}
In Tab. \ref{f_tabelle} a set of characteristic results for $\fgge$
is displayed. These are in perfect agreement with those from Ref. \cite{krzu}.
\input{tab_f}