Sum Rules from Charge-Charge-Density and Charge-Charge Commutators

The spectral-function approach to the broken-$\mathrm{SU}\left(3\right)\mathrm{}$-symmetry sum rules has two difficulties. One is the ambiguity with respect to the Schwinger terms, and the other is the derivation of the so-called second sum rules, which are not acceptable. By using commutators involving a charge operator, our approach is free of the first difficulty. In computation, we make the following approximation for the operator $V_K$ [which is an $\mathrm{SU}\left(3\right)\mathrm{}$ raising or lowering operator in the symmetry limit]: In the broken symmetry the operator $V_K$ still acts as a generator, to a good approximation, in an appropriately chosen infinite-momenta limit. For the $\mathrm{vector}\mathrm{meson}\to l+\overline{l}$ couplings, we are able to derive sum rules which are essentially equivalent to the first spectral-function sum rules but not to the second ones. Instead, we obtain other sum rules which enable us to determine the first-order $\omega -\phi$ mixing angle from the rates of $V\to l+\overline{l}$ decays. A sum rule for the $\omega \to 3\pi$, $\phi \to 3\pi$, and $K^{*}\to K\pi \pi$ couplings is also obtained. We also derive in our approach the Gell-Mann-Zachariasen relation and its $K^{*}$ analog which favors the existence of the $\kappa$ meson.