Subtractions in Dispersion Relations

Abstract

Assuming the unitarity bound in the physical region on a partial-wave amplitude for an elastic scattering process, and assuming further that the amplitude satisfies a dispersion relation, it is shown that the necessity for subtractions in order to make the integral over the left-hand cut converge implies an oscillatory behavior of the discontinuity across the left-hand cut, the violence of the oscillations increasing with increasing number of subtractions. A theorem given by Sugawara and Kanazawa is provided with a rigorous proof and is applied to pion-nucleon and kaon-nucleon dispersion relations.