Dispersion Relations for Finite Momentum-Transfer Pion-Nucleon Scattering

The field-theoretical derivation of dispersion relations for forward pion-nucleon scattering has been generalized to apply to the case of a fixed finite momentum transfer. The generalization is facilitated by use of the special Lorentz frame in which the sum of the momenta of the initial and final nucleons is zero. In this reference system the relations between dispersive and absorptive parts of the scattering amplitude are independent of momentum transfer and are similar in form to the forward-angle relations. At energies below the minimum energy necessary to allow a particular momentum transfer, the scattering amplitude has no direct physical meaning; it is interpreted as an analytic continuation of the physical amplitude to scattering angles corresponding to $cos\theta <-1\mathrm{}$. The resulting equations are expressed in terms of the amplitudes for individual angular momenta and are given in two forms, corresponding to the inclusion or neglect of nucleon recoil.