Application of General Unitarity to Pion-Nucleon Scattering

The $\pi -\pi -N$ intermediate state is included in the unitarity condition of pion-nucleon scattering. The general unitarity relations then couple the amplitudes for $\pi -N$ and $\pi -\pi -N$ scattering, and one-pion production or absorption. The kinematical structure of these amplitudes is discussed and a formal decomposition into partial waves is outlined. The dynamical approach is based on a system of coupled partial wave dispersion relations. The dynamical singularities of $\pi -N$ scattering can be taken from the Mandelstam representation, whereas perturbation-theoretic approximations must be made for the other amplitudes. The dispersion relations can be solved by the generalized $\frac{N}{D}$ method. As an example, a one-pion exchange approximation for the production and absorption amplitudes is considered. It is shown that amplitudes can be chosen which do not have any kinematical singularities. The $\frac{N}{D}$ method here takes a more simple form by setting up integral equations for the $N$ functions rather than for the $D$ functions. Integrations over contours in the complex plane are then avoided.