Dispersion Relations for Fixed-Source Meson Theories: Effective-Range Relations

Dispersion relations for boson-fermion scattering have been shown to yield, in the static limit, coupled integral equations for the scattering amplitudes of a definite orbital angular momentum. The present investigation is devoted to the search for the "solutions" of these equations, or more accurately, to the problem of obtaining essentially real integral equations for the reciprocals of the scattering amplitudes, since such equations would form the basis for effective range approximations. The mathematical problem is that of determining an analytic continuation of the scattering amplitudes which has no zeros in the complex plane. It is shown that for all cases considered in I, an analytic continuation can be defined, either for the scattering amplitude in a definite channel, or for a simple linear combination of these, which has the basic properties of Wigner's $R$ functions, and which upon inversion yields the required equations. The inverse will, however, contain as many arbitrary constants (or in some instances functions) as there are zeros in the original amplitudes, as has been pointed out previously for simpler examples. It is shown that the procedures apply, without essential change, to the case that inelastic processes are permitted. No attempt is made in this paper to apply the resultant formulas to experiment.