Ward Identities, Unsubtracted Dispersion Relations, and Feynman Graphs

We study the interplay of commutators involving a conserved current, Feynman graphs, and dispersion relations in an attempt to formulate general rules for when the invariants involved in the decomposition of matrix elements of retarded products of currents obey unsubtracted dispersion relations. Our main conclusions are two: (1) The assumption of unsubtracted dispersion relations is incorrect when one of the currents is conserved and the other is the source of a strongly interacting particle on its mass shell. (2) When one current is conserved and the other is the source of a system which interacts only weakly or electromagnetically, the assumption may be correct to lowest order in the nonstrong interactions. These relations are, however, on a much different and less firm footing than the sum rules using the hypothesis of partially conserved axial-vector current (like that of Adler and Weisberger). In particular, assuming the unsubtracted dispersion relations, we find that the result (e.g., the sum rules of Cabibbo and Radicati) has a structure such that, to all orders in any field theory, the $N^{*}\left(\omega \right)$ intermediate-state graphs project to zero, that is, they fail to contribute to the charge radius of the nucleon (pion), whereas, calculated dispersively, they do contribute. We also show that the real part of amplitudes of type (2) has a fixed power behavior in energy.