Fixed Variable Dispersion Relations for the Pion-Nucleon System

It is shown that an earlier study by Hamilton and Woolcock of fixed momentum-transfer dispersion relations may be complemented by a study of fixed energy dispersion relations. Two main results are obtained. First, by demanding that the two types of relation give the same value for the amplitude, nontrivial restrictions are obtained on the amplitude (the ${f}_{0}\ensuremath{-}N\ensuremath{-}N$ coupling constants and the values of certain integrals over the high-energy $\ensuremath{\pi}N$ amplitude are obtained). Secondly, the fixed energy relation enables one to discuss quantitatively the validity of the "CGLN" approximate method which Hamilton and Woolcock had used to calculate the partial-wave amplitudes at low energies from the fixed momentum-transfer dispersion relation alone. The terms neglected by this approximation are evaluated, and found to be large except at low energies (the authors had themselves suggested that this might be the case). Even when these terms are included, undesirable cancellations occur, and the conclusion in fact is that fixed variable relations are not suitable for calculating the partial-wave amplitudes (except at low energies), but only for providing sum rules.