Extraction of Resonances from Meson-Nucleon Reactions

We present a pedagogical study of the commonly employed Speed-Plot (SP) and Time-delay (TD) methods for extracting the resonance parameters from the data of two particle coupled-channels reactions. Within several exactly solvable models, it is found that these two methods find poles on different Riemann sheets and are not always valid. We then develop an analytic continuation method for extracting the nucleon resonances within a dynamical coupled-channel formulation of the $\pi N$ and $\gamma N$ reactions. The main focus is on resolving the complications due to the coupling with the unstable $\pi \Delta, \rho N, \sigma N$ channels which decay into $\pi \pi N$ states. By using the results from the considered exactly solvable models, explicit numerical procedures are presented and verified. As a first application of the developed analytic continuation method, we present the nucleon resonances in the $S_{11}$ and $S_{31}$ partial waves extracted within a recently developed coupled-channels model of $\pi N$ reactions. The results from this realistic $\pi N$ model, which includes $\pi N$, $\eta N$, $\pi\Delta$, $\rho N$, and $\sigma N$ channels, also show that the simple pole parametrization of the resonant propagator using the poles extracted from SP and TD methods works poorly.