Threshold Theorem for the Radius of theF2VNucleon Form Factor

Abstract

We develop a method for expanding the absorptive part of the sidewise dispersion relation for the nucleon ${F}_{2V}$ radius. We assume that a threshold expansion is valid and we are able to obtain the exact first-order terms in this expansion in the limit of the pion mass going to zero. We are able to prove that the first-order terms in this expansion come solely from the $\ensuremath{\pi}\ensuremath{-}N$ intermediate state. Thus, the expansion provides justification for the retention of only the $\ensuremath{\pi}\ensuremath{-}N$ intermediate state in a first attempt at describing the radius, as in Drell and Silverman. Furthermore, the expansion provides a handhold for attempting to estimate the corrections to the first-order term coming from the $2\ensuremath{\pi}\ensuremath{-}N$ and higher intermediate states. It must be emphasized that we are not providing justification for the assumption of threshold dominance. We are trying to follow that assumption to its logical conclusion. That is, if the threshold dominance idea is good, then a threshold expansion provides a logical way to calculate low-energy parameters. We have provided such an expansion for the ${F}_{2V}$ radius.