On the field-theoretical formulation of the low-energy pion-nucleon scattering problem

The Low-type equations for low-energy pion-nucleon scattering which take into account the effects of nucleon recoil as well as the heavy-meson and antinucleon degrees of freedom are reduced to the Lippmann-Schwinger-type equation with the energy-dependent potential. This potential contains a crossing term dependent on the πN-scattering t-matrix we are looking for. It is demonstrated that nonlinearity of the resulting equation which stems from the crossing term is inherent to the πN-scattering problem and arises in formulations of the latter within the framework of multichannel-scattering theory equations as well as four-dimensional Bethe-Salpeter equations. On the basis of the Lippmann-Schwinger-type equation suggested in the present work, the contribution of ρ-meson exchange to πN-scattering P-waves in the 0-300 MeV energy region is separately investigated. The ρ-exchange contribution turns out to be essential in describing the experimentally observed behaviour of the πN-scattering P11 phase shift, while for other P-waves the satisfactory description of experimental data is not destroyed by this contribution. (orig.)