On causality, unitarity and perturbative expansions

Recently a novel scheme for studying hadronic interactions beyond the threshold region was introduced by two of the authors [1]. The main objective of that work is a controlled realization of the causality and unitarity condition in a perturbative application of the chiral Lagrangian. The starting point are partial-wave dispersion relations. A generalized potential is constructed from the chiral Lagrangian in the subthreshold region and analytically extrapolated to higher energies. The partial-wave scattering amplitudes are obtained as solutions of non-linear integral equations. For quantum field theoretic systems, where it is already difficult to compute the interaction, our method proves very efficient. The purpose of the present study is an illustration of the method for a schematic system where the exact solution is known [2]. We consider the quantum mechanics of two particles defined by a superposition of a long-range and a shortrange Yukawa potentials. Typically the long-range forces are rather weak and perturbative but the short-range forces are strong and nonperturbative. For example in nucleonnucleon scattering the central part of the pion-exchange potential has a coupling constant that is three times smaller than its critical value. We renormalize the short-range force as to arrive at a closer correspondence to effective field theories. The analytic properties of the scattering amplitude generated by a Yukawa potential is well understood. A key observation is the known analytic structure of the generalized potential U(q).