Current Algebras and Pole Dominance: A Consistent Treatment of the Aρπ System

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 Aπρ system. We consider matrix elements of the retarded commutator of two currents (or of one current and DA≡∂_μA^μ) taken between single-particle states (the π, A, and ρ mesons) and the vacuum. The absorptive parts of such amplitudes contain terms proportional to δ(q²-m_π²) and δ(Δ²-m_ρ²), for example; we demonstrate that erroneous results are obtained if unsubtracted dispersion relations (UDR) at either fixed q² or Δ² are assumed for these matrix elements. We therefore write UDR at fixed μ≡αq²+(1-α)Δ², where α 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 ρ→ππ and A→ρπ couplings are derived, as well as the first Weinberg sum rule. Our method, and the difficulties of earlier calculations based upon fixed-q² UDR, are discussed in detail. A careful discussion of the relevant experimental quantities is given. These include the ρ 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.