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$2π$ production in the Giessen coupled-channel model

We present a coupled-channel Lagrangian approach (GiM) to describe the $πN \to πN$, $2πN$ scattering in the resonance energy region. The $2πN$ production has been significantly improved by using the isobar approximation with $σN$ and $πΔ(1232)$ 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 $πN \to πN$, $π^0π^0 N$ reactions in the Roper resonance region. We obtain $R_{σN}(1440)=27^{+4}_{-9}$\,\% and $R_{σN}(1440)=12^{+5}_{-3}$\,\% for the $σN$ and $πΔ$ decay branching ratios of $N^*(1440)$ respectively. The extracted $πN$ inelasticities and reaction amplitudes are consistent with the results from other groups.