Convolution Approach to The πNN System

Abstract

The unitary NN−πNN model contains a serious theoretical flaw: unitarity is obtained at the price of having to use an effective πNN coupling constant that is smaller than the experimental one. This is but one aspect of a more general renormalization problem whose origin lies in the truncation of Hilbert space used to derive the equations. Here we present a new theoretical approach to the πNN problem where unitary equations are obtained without having to truncate Hilbert space. Indeed, the only approximation made is the neglect of connected three-body forces. As all possible dressings of one-particle propagators and vertices are retained in our model, we overcome the renormalization problems inherent in previous πNN theories. The key element of our derivation is the use of convolution integrals that have enabled us to sum all the possible disconnected time-ordered graphs. We also discuss how the convolution method can be extended to sum all the time orderings of a connected graph. This has enabled us to calculate the fully dressed NN one pion exchange potential. We show how such a calculation can be used to estimate the size of the connected three-body forces neglected in the new πNN equations. Early indications are that such forces may be negligible.