\subsection{Resonance production}
\label{resonanz}
Top production close to threshold proceeds through $\esn$
toponium resonances. With increasing mass the probability for a top quark
to decay weakly into $b+W$ before annihilating with $\bar{t}$
becomes more and
more important. The final state consists of a b-quark jet plus two leptons
or two light-quark jets, and a open top meson or baryon at rest which
subsequently decays weakly. For a top mass above 100 GeV the single quark decay
almost exclusively exhausts all other decay modes and raises the toponium
width quickly to the GeV range \cite{scenario}.
Beyond 130 GeV the width becomes larger than the level spacing.
The narrow spacing between the high radial excitations together with the
large top decay width will lead to an apparent contribution to R that comes
quite close to the QCD corrected parton value already within the resonance
region.
The resonance cross section for the process $\gamma\gamma\rightarrow$ $\esn$
can be obtained numerically by solving the Schr\"odinger equation for the
Greenfunction similary to the method proposed in \cite{fadin,peskin,thomas} for
$\eplus\eminus$ annihilation or
by summing Breit-Wigners (this is done earlier for $\eplus\eminus \rightarrow
\dse$ \cite{guesken,kwong}).The cross
section $\sigma \left(\gamma\gamma \rightarrow \esn \right)$
is related to $\sigma \left(\eplus\eminus \rightarrow ^3 S_1 \right)$
by applying the Breit-Wigner formula for both cases:
\input{breit_wigner}
The $\left(2s+1\right)$ factors are the multiplicities of the initial
spin states, to be replaced by 2 for unpolarized photons.
k denotes the momentum of one of
the initial state particles and hence $4 k^2 =s$ for the cases of interest.
The total toponium decay width assumed to be $2 {\Gamma} \left(
t \rightarrow b + W \right)$. The
resonance mass $M$ is approximately the same for $\dse$ and $\esn$ states.
${\Gamma}_i$ denotes the decay width into the initial state.
With an extra factor $\frac{1}{2}$ to account for identical particles in the
final state one finds:
\begin{displaymath}
\frac{\sigma \left( \gamma\gamma \to \esn\right)}{\sigma \left(\eplus\eminus\to{^3S_1} \right)} = \frac{2}{3}\frac{\Gamma \left(
\esn \rightarrow \gamma\gamma \right)}{\Gamma \left( ^3 S_1 \rightarrow
\eplus\eminus \right)}
\end{displaymath}
Using
\begin{displaymath}
\Gamma \left( \esn \rightarrow \gamma\gamma \right) = \frac{12 Q^4
{\alpha}^2 |R(0)|^2}{M^2} \qquad,\qquad \Gamma \left( ^3 S_1 \rightarrow
\eplus\eminus \right) = \frac{4 Q^2 {\alpha}^2 |R(0)|^2}{M^2}
\end{displaymath}
one obtains for the ratio
\begin{displaymath}
\frac{\sigma \left( \gamma\gamma \rightarrow \esn \right) }{ \sigma
\left( \eplus\eminus \rightarrow ^3 S_1 \right) } = 2 Q^2 = \frac{8}{9}
\end{displaymath}
This coincides with the ratio for
open top production close to the threshold.
Additional factors for the ratio between the two cross sections are
induced by QCD
corrections, to be considered in the following.