$2\pi$ production in the Giessen coupled-channel model

We present a coupled-channel Lagrangian approach (GiM) to describe the $\pi N \to \pi N$, $2\pi N$ scattering in the resonance energy region. The $2\pi N$ production has been significantly improved by using the isobar approximation with $\sigma N$ and $\pi \Delta(1232)$ in the intermediate state. The three-body unitarity is maintained up to interference pattern between the isobar subchannels. The scattering amplitudes are obtained as a solution of the Bethe-Salpeter equation in the $K$ matrix approximation. As a first application we perform a partial wave analysis of the $\pi N \to \pi N$, $\pi^0\pi^0 N$ reactions in the Roper resonance region. We obtain $R_{\sigma N}(1440)=27^{+4}_{-9}$\,\% and $R_{\sigma N}(1440)=12^{+5}_{-3}$\,\% for the $\sigma N$ and $\pi \Delta$ decay branching ratios of $N^*(1440)$ respectively. The extracted $\pi N$ inelasticities and reaction amplitudes are consistent with the results from other groups.