Is the Nucleon aπNBound State?

The $P_{11}$ partial wave of the $\pi N$ system is characterized by a large absorption and a resonance (possibly inelastic) at 575-MeV laboratory pion energy (Roper resonance). In view of these features of the $P_{11}$ channel, the $\pi N$ system may be rendered sufficiently weak not to bind the nucleon. Using the experimental mass of the nucleon and the recently obtained extensive energy-dependent, complex phase-shift results on $\pi N$ scattering data, we examine here, through a nonperturbative $S$-matrix approach, the existence of a nucleon bound state in the $\pi N$ channel in the absence of forces arising from the inelastic states. The formalism is independent of any model for the contribution of the left-hand cut. We discuss two different possibilities concerning the nature of the Roper resonance: (a) It is an elastic resonance primarily due to the forces in the $\pi N$ channel; (b) it is an inelastic resonance mainly due to the forces from the inelastic states. In either case we find it unlikely that the $\pi N$ system by itself can sustain the nucleon bound state.