Some Connections between Sum Rules and Symmetries

The question of the existence of consistency conditions arising from the combination of sum rules with internal symmetries is discussed further. The sum rules considered are the forward-scattering Compton and photoproduction rules. A new photoproduction rule is stated. The notion of the support of a sum rule is introduced: viz., the set of states such that saturation with these states does not generate null solutions for low-energy parameters. Different sum rules generally have distinct supports because of differences in the consistency conditions from rule to rule. Applications of saturation methods to the axial $\frac{D}{F}$ ratio, full and anomalous magnetic moments, and the ${\pi }^0\to 2\gamma$ lifetime are discussed.