Threshold Factors in Partial-Wave Dispersion Relations

If the solution of an approximate partial-wave dispersion relation is required to satisfy unitarity and possess the correct threshold behavior, additional poles (representing effective forces) are introduced into the amplitude. These poles are explicitly exhibited. To enforce $l$-wave threshold conditions, approximately $l$ arbitrary parameters appear (the precise number depending upon the properties of the amplitude considered). This difficulty is in principle independent of the method of solution. Using the nucleon exchange force as input, the $J=\frac{3}{2}$ amplitudes for $\pi -N$ scattering are obtained by numerical solution of the $\frac{N}{D}$ equations. The solutions are extremely sensitive to the arbitrary parameters. It appears that, in the present formulation, partial-wave dispersion relations do not provide a reliable means of calculating detailed properties of the amplitudes.