Current Algebras and Pole Dominance: A Consistent Treatment of theAρπSystem

A new technique for obtaining consistent solutions to the current-algebra equations for vertex functions in the pole-dominance approximation is developed and applied to the $A\pi \rho$ system. We consider matrix elements of the retarded commutator of two currents (or of one current and $D_A\equiv {\partial }_{\mu }{A}^{\mu }$) taken between single-particle states (the $\pi$, $A$, and $\rho$ mesons) and the vacuum. The absorptive parts of such amplitudes contain terms proportional to $\delta (q^2-{m_{\pi }}^2)$ and $\delta ({\Delta }^2-{m_{\rho }}^2)$, for example; we demonstrate that erroneous results are obtained if unsubtracted dispersion relations (UDR) at either fixed $q^2$ or ${\Delta }^2$ are assumed for these matrix elements. We therefore write UDR at fixed $\mu \equiv \alpha q^2+(1-\alpha ){\Delta }^2$, where $\alpha$ is an arbitrary number (unequal to zero or one). We are then able to derive consistent solutions to the current-algebra equations. Several sum rules involving the $\rho \to \pi \pi$ and $A\to \rho \pi$ couplings are derived, as well as the first Weinberg sum rule. Our method, and the difficulties of earlier calculations based upon fixed-$q^2$ UDR, are discussed in detail. A careful discussion of the relevant experimental quantities is given. These include the $\rho$ and $A$ widths, the pion form factor, and the structure-dependent radiative pion decay. There is good order-of-magnitude agreement between theory and experiment.