On the complex branch points in $πN$ scattering amplitude and the multiple poles feature of resonances

A simple heuristic argument to understand the existence of complex branch points in the $\pi N$ scattering amplitude is presented. It is based on a hypothesis that the singularity structure of the $\pi N$ scattering amplitude is a smooth varying function of the pion mass. We then show that the two-pole structure found to correspond to the Roper resonance could just a simple direct mathematical consequence of including additional Riemann surface in the analysis. Our study indicates that it is always possible to have multiple poles, either two or four etc., in different Riemann sheet, to be associated with a resonance. The poles in all Riemann sheets should be looked for to determine whether the two-pole feature of the Roper resonance $N^*(1440)$ is a manifestation of the "exact degeneracy" discussed here but masked by numerical indeterminacy, or an "accidental" one. The determination of the multiplicity of a pole could provide some information of the analytical structure of the numerator of the pole term, as the numerator of the pole term is related to resonance form factor.