Analysis of Partial-Wave Dispersion Relations

Questions of existence and uniqueness of solutions of partial-wave dispersion relations are studied, with particular attention to the $\frac{N}{D}$ method. The interaction, assumed to be given, is represented by (i) the strengths and locations of unphysical singularities and (ii) the inelastic partial-wave cross section. A generalization of the $\frac{N}{D}$ method to include part (ii) of the interaction leads to a nonsingular integral equation for $\mathrm{Im}D$. This equation is amenable to the Fredholm theory only if there is a correlation between items (i) and (ii) of the interaction, and only if the increase of inelastic processes at high energies is not too rapid. Certain Cauchy integrals associated with (i) and (ii) must be nonzero at threshold if there is to exist a solution with the normal threshold momentum dependence. Thus, there is no solution for any model in which (i) is constructed from a few partial waves in the two crossed channels. For certain interactions the real part of the phase shift approaches a multiple of $\pi$ at large energy, just as in potential scattering. The Castillejo-Dalitz-Dyson (CDD) ambiguity is analyzed in some detail. A uniqueness theorem is proved which asserts that if a solution of a particular type exists, it is the only solution of the problem within the class usually considered. Thus the CDD ambiguity is partially bypassed. In certain cases the unique solution is found by the ordinary $\frac{N}{D}$ method without subtractions. Some useful results on principal value integrals are obtained. The discussion is carried out for the example of pion-nucleon scattering in the complex plane of $w$, the center-of-mass energy. The behavior of the amplitude near $w=\pm \mathrm{}(M-m)\mathrm{}$ is derived from crossing symmetry.