Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation

We consider two simple field-theoretic models, (a) spinor electrodynamics and (b) the $\ensuremath{\sigma}$ model with the Polkinghorne axial-vector current, and show in each case that the axial-vector current satisfies a simple anomalous divergence equation exactly to all orders of perturbation theory. We check our general argument by an explicit calculation to second order in radiative corrections. The general argument is made tractable by introducing a cutoff, but to check the validity of this artifice, the second-order calculation is carried out entirely in terms of renormalized vertex and propagator functions, in which no cutoff appears.