Spectral-function sum rules in asymptotically free theories

We show that the necessary and sufficient condition to derive Weinberg's second spectral-function sum rule within the framework of a Lagrangian theory invariant under a local non-Abelian gauge group $G$ and the global chiral $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$ group is that $G$ should commute with $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$. The $\mathrm{SU}\mathrm{}\left(3\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(3\right)\mathrm{}$ spectral-function sum rules for currents and sum rules involving spectral functions of scalar and pseudoscalar densities are also discussed.