Applications of a Coulomb-Modifieds-Wave Dispersion Relation

Abstract

Low-energy proton-proton and neutron-neutron scattering are compared through the use of a Coulomb-modified dispersion relation. Basing the analysis upon a Jost-Bargmann potential, for which the $s$-wave amplitude in the uncharged case contains a single interaction pole at $q=\frac{1}{2}i\mu$, it is shown that the corresponding amplitude in the charged case contains a sequence of branch points at $q=\frac{1}{2}\mathrm{ni}\mu$. However, a small region near the highly singular $n=1\mathrm{}$ point still dominates the dispersion relation. The low-energy phase shifts obtained from the solution of the dispersion relation are in excellent agreement with those obtained directly from the Schrödinger equation both for the charged and uncharged cases. In a similar analysis based upon a Yukawa potential, the convergence of the method is much slower as higher Born-approximation singularities are fed in. A technique is presented for extracting the singular parts of individual Born terms arising from superpositions of Yukawa potentials, and, serves as the basis for computing the input to the dispersion relation. This procedure does not require explicit integration of the terms of the Born series. It is also shown that the integral equation used in this paper, which was originally derived by Wong and Noyes by studying the analytic properties of the effective-range function, is identical to the $D$ equation in the $\frac{N}{D}$ method.