Ward Identities forηDecay in Perturbation Theory

The process $\eta \to 3\pi$ is known to violate a simple prediction of partial conservation of axial-vector current. In many models, $\eta \to 3\pi$ proceeds through a fermion loop with electromagnetic corrections. The first radiative corrections to fermion loops give rise to divergent double-loop integrals with forms like $\int \frac{d^{4}r{d}^{4}k}{}{[r^2-m^2]}^2{[{\mathrm{}(r-k)\mathrm{}}^2-m^2]}^2$. As with single linearly divergent integrals, when a meaning is ascribed to them it may not always be possible to shift the origin of integration without changing the value of these integrals. Such integrals appear when one tries to check, in perturbation theory, Ward identities and low-energy theorems which follow from the formal manipulation of the equations of motions. They can cause anomalies similar to the one in the axial-vector-current two-photon vertex. We study some applications to the $\eta \pi \sigma$ vertex, to the process $\eta \to 3\pi$, and to the corrections of order ${\alpha }^2$ to ${\pi }^0\to 2\gamma$. No anomalies which can be related to $\eta$ decay are discovered.