Exact Solution of the One-Photon-ExchangeNDEquations

Abstract

The exact solution of the single-Yukawa-meson-exchange $\frac{N}{D}$ equations is obtained in the zero-meson-mass (Coulomb) limit. The solution sums an infinite series of infrared-divergent ladder-graph fragments into finite, unitary partial-wave amplitudes. As with the Schrödinger equation for the hydrogen atom, one finds an infinite number of bound states (zeros of the $D$ function) with an accumulation point at threshold. These bound-state poles of the scattering amplitude arise from the long-range force in much the same way as dynamical bound states arise generally in dispersion theory, thus allowing a discussion of the long-range force rather naturally in the usual dispersion-theoretic terms. The bound-state poles are neither so deeply bound nor so dense as those of the hydrogen atom, thus providing some understanding of the role of the one-photon-exchange force relative to the (long-range) multiple-photon-exchange forces. The possibilities for extending the technique to the relativistic one-photon case and the question of electromagnetic corrections to the strong interactions are briefly discussed. Finally, some possible approaches to including higher order photon exchanges are considered.