(Anomalous?) Wick rotation in Fujikawa's method

1. Aug 27, 2010

Riposte

In the derivation of the chiral anomaly using the path integral (Fujikawa's method) one encounters a divergent sum which needs to be regulated. The usual prescription is to insert a factor of $$e^\frac{(i\gamma^\mu D_\mu)^2}{M^2}$$ while taking $$M \rightarrow \infty$$.

This eventually leads to (anomaly part)* $$\int \frac{d^4k}{(2\pi)^4} e^{-k^\mu k_\mu}$$. At this point textbooks (and Fujikawa's own paper) glibly Wick rotate the timelike component of k to get a simple convergent integral. However without the Wick rotation, the integral is clearly divergent, and therefore the Wick rotation does not give the same integral.

I suspect the Wick rotation is somehow part of the regularization, but can anyone justify this?