Dispersion-Theoretic Perturbation Series in Some Models

Abstract

The perturbation series of Dashen and Frautschi is derived via a "$D$-matrix" approach to all orders in the perturbation. Explicit formulas are given for the first and second orders, and their equivalence to the Rayleigh-Schrödinger perturbation series in potential theory is shown. A method is then given to carry out the evaluation of the series to any order. The first-order formula is examined in three models; the four-point interaction, two-particle scattering with a left-hand cut given by a single pole, and an augmented Lee model. The exact results of these models are shown to agree in each case with the theory of Dashen and Frautschi. Finally, we discuss differences between the Lee-model calculation presented here and Dashen's calculation of the neutron-proton mass difference.