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$\mathbf{\pi N \longrightarrow \pi N}$

The $\pi N \longrightarrow \pi N$ interaction is very good modeled within our resonance picture. Therefore we get contributions like $\pi N \rightarrow R \rightarrow \pi N$ which take more than one time step and which we will consider later as resonance production processes. Here we want to consider the background terms which implement immediate $\pi N \longrightarrow \pi N$ reactions. On the one hand side we get background terms due to non-propagated resonances. They do not appear as intermediate states in pion nucleon processes. Therefore we model them by including their contribution to the cross section as a direct interaction:

$\begin{displaymath} \sigma^{\mathsf{BG}}_{res}=\sigma_{\mbox{\begin{scriptsize}A... ...\begin{scriptsize}Propagated resonances\end{scriptsize}}} \; . \end{displaymath}$

On the other hand side we also get a background term due to a lack of strength of the resonance cross sections at low energies. We define

$\begin{displaymath} \sigma^{\mathsf{BG}}_{low}=\sigma_{exp}-\sigma_{\mbox{\begin{scriptsize}All resonances\end{scriptsize}}} \, . \end{displaymath}$

Therefore the full direct cross section is given by

$\begin{displaymath} \sigma_{\pi N \longrightarrow \pi N}=\sigma^{\mathsf{BG}}_{low}+\sigma^{\mathsf{BG}}_{res} \, . \end{displaymath}$

Oliver Buss 2005-03-16