Second-Order Radiative Corrections to the Triangle Anomaly. I

Lowest-order radiative corrections to the pion-photon-photon vertex are calculated in the pseudoscalar $\pi N\overline{N}$ theory. Causal analysis of two- and three-particle exchange processes is used, in the context of Schwinger's source theory, to calculate the spectral weight function for the vertex function. Evaluated at zero momentum transfer squared, this vertex function gives the so-called triangle anomaly in the relation between the pseudoscalar and axial-vector couplings, in second order. Unlike previous authors, we find a nonzero result, giving a total anomaly through this order proportional to $1+\frac{\alpha }{2\pi }$. This discrepancy is due to the fact that we have normalized the pseudoscalar form factor at zero momentum transfer squared, rather than at $\frac{3m^2}{\ln \left(\frac{{\mu }^2}{m^2}\right)}$, where $\mu$ is the fictitious photon mass. Independent of the choice of normalization point, there exist radiative corrections to the low-energy theorem for ${\pi }^0$ decay.