Bjorken Limit and Pole Dominance. I. TheωρπSystem and the Algebra of Fields

The technique developed in previous work for solving the current-algebra equations for vertex functions in the pole-dominance approximation is modified and simplified. The Bjorken limit is used to incorporate and test assumptions about equal-time commutators (ETC) involving spatial components of currents. Our previous results for the $A\rho \pi$ system are shown to be consistent with the ETC derived from the algebra of fields but inconsistent with those of the quark model. We find that if $g_{\omega \rho \pi }\ne 0$, the ETC of $V_i$ with $D_A\left(\equiv {\partial }_{\mu }{A}^{\mu }\right)$ must not vanish. It is shown, however, that the vanishing of this commutator need not be included in the algebra of fields, since it rests on special assumptions about the form of the symmetry-breaking Lagrangian. In contradistinction to the work of Gell-Mann, Sharp, and Wagner, we find that the $\omega \rho \pi$ vertex function has a strong off-mass-shell dependence. As a consequence, for example, if strict partial conservation of the axial-vector current is assumed, then ${\pi }^0$ cannot decay into two photons. An explicit vertex function is obtained assuming the algebra of fields and applied to the decays ${\pi }^0\to 2\gamma$, $\omega \to \pi \gamma$, $\rho \to \pi \gamma$, and $\omega \to 3\pi$.