\section{Top quark production rate}
\subsection{Born approximation}
\label{born_niveau}
The Born cross section for the process $\gamma\gamma\rightarrow t\bar{t}$ 
is given by 
\begin{equation}
 \sigma_{\gamma\gamma\rightarrow t\bar{t}}^0 \left(s\right) 
 =\frac{\sigma_{++}+\sigma_{+-}+\sigma_{-+}+\sigma_{--}}{4}
\label{bornwq}
\end{equation}
where
$$
\beta =\wurzel
$$
\begin{displaymath}
\sigma_{++}=\sigma_{--}=\frac{2 \pi \beta {\alpha^2} Q^4 N_c}{s}
\left[ 2\left(1+{\beta}^2\right) 
+ \frac{\left(1-{\beta}^4\right)}{\beta} \ln \left(\frac{1+\beta}
           {1-\beta}\right) \right]
\end{displaymath}
\begin{displaymath}
\sigma_{+-}=\sigma_{-+}=\frac{2 \pi \beta {\alpha^2} Q^4 N_c}{s}
\left[ -10+2{\beta}^2+\frac{5-{\beta}^4}{\beta}
 \ln\left(\frac{1+\beta}{1-\beta}\right) \right]
\end{displaymath}

The number of colors is denoted by $N_c = 3$, 
the electric charge by $Q={2}/{3}$ and 
the plus and minus indices refer to the photon helicities $\pm 1$.
The cross sections for different photon helicities is shown in Fig. 
\ref{born}.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild5.eps,width=12cm}}
\end{center}
\caption{$\sigma_{\gamma\gamma\rightarrow t\bar{t}}^0$ for $m_t = 150\, GeV$ 
and 
equal photon helicities (solid), opposite photon helicities (dashed) and 
unpolarized photons (dotted).}
\label{born}
\end{figure}
\noindent
To obtain the effective cross section for top pair production the $\gamma\gamma$ cross section is folded with the $\gamma\gamma$ luminosity:
\begin{displaymath}
{\sigma}^{eff}_{\gamma\gamma\rightarrow t\bar{t}}\left(s_{ee}\right)
=\sum_{\pgse\pgsz}
\int\limits^{\frac{x}{x+1}}_{\frac{2 m_t}{\sqrt{s_{ee}}}}\! dz\, \lgg
\sigma_{\gamma\gamma\rightarrow t\bar{t}}(z^2 s_{ee})
\label{effekt}
\end{displaymath}
The cross sections for 
different 
helicity configurations are weighted with the corresponding 
luminosities, i.~e. $\sigma_{++}$ and $\sigma_{--}$ are multiplied with the 
two $h_{\gamma\gamma}=0$ contributions and $\sigma_{+-}$ and $\sigma_{-+}$ 
with 
the $h_{\gamma\gamma}=\pm 2$ contributions of the luminosity. 
Whenever polarization of the incoming electron and photon is used 
to improve the luminosity this weighted 
summation has to be performed.
The effective cross section is shown in Fig. \ref{faltung_born}.
An appropriate choice of helicities enhances the effective cross section 
by more than a factor two. 
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild6.eps,width=12cm}}
\end{center}
\caption{Effective cross section divided by ${\sigma}_{pt}=\frac{4\pi
{\alpha}^2}{3 s_{ee}}$, $E_{CM}$ denotes the $\eplus\eminus$ cms energy, 
($m_t$ = 150 GeV).}
\label{faltung_born}
\end{figure}
Close to the threshold the enhancement is even stronger since 
only the S-wave appears and the P-wave is strongly suppressed. 
This can be seen directly by expanding the matrix element in powers 
of $\beta$. The leading term of $\sigma_{++}$ resp. $\sigma_{--}$ is linear, 
the leading term of $\sigma_{+-}$ resp. $\sigma_{-+}$ however 
starts with 
${\beta}^3$. In Fig. \ref{faltung_schwelle} this threshold region 
is expanded indicating a enhancement by an order of magnitude. 
Top pair production starts above an $\eplus\eminus$ energy of about 380 GeV 
which corresponds to a maximal photon photon energy of about 300 GeV.
\begin{figure}
\begin{center}
\mbox{\epsfig{file=bild7.eps,width=12cm}}
\end{center}
\caption{Effective cross section close above the threshold divided by 
${\sigma}_{pt}=\frac{4\pi
{\alpha}^2}{3 s_{ee}}$, $E_{CM}$ denotes the $\eplus\eminus$ cms energy, 
($m_t$ = 150 GeV).}
\label{faltung_schwelle}
\end{figure}





