(Anomalous?) Wick rotation in Fujikawa's method

[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?

 

[eljose79]

Hello, thank you for bringing up this interesting topic. The use of Wick rotation in the derivation of the chiral anomaly is indeed a crucial step in regulating the divergent integral. To understand why this is the case, let's first take a closer look at the integral in question:

\int \frac{d^4k}{(2\pi)^4} e^{-k^\mu k_\mu}

As you have correctly pointed out, without the Wick rotation, this integral is clearly divergent. This is because the integrand contains a Gaussian factor, which does not decrease fast enough as k approaches infinity. In other words, the integral is dominated by large values of k, which leads to the divergence.

So why does the Wick rotation help in regulating this integral? To answer this question, we need to understand the concept of analytic continuation. In physics, we often encounter integrals that are divergent in some region of the integration variables. However, these integrals can sometimes be analytically continued to a different region where they are well-behaved and convergent. This is exactly what happens in the case of the chiral anomaly integral.

By performing a Wick rotation, we are essentially rotating the integration contour in the complex plane. This allows us to analytically continue the integral to a different region, where it becomes convergent. In particular, the Wick rotation transforms the integral into:

\int \frac{d^4k}{(2\pi)^4} e^{-k^0 k_0}

where k^0 is now interpreted as the Euclidean time component. This integral is now well-behaved and can be evaluated using standard techniques. Once we have obtained the result, we can then analytically continue back to the Minkowski space by rotating the integration contour back.

In summary, the Wick rotation is an essential step in regulating the divergent integral in the derivation of the chiral anomaly. It allows us to analytically continue the integral to a different region where it becomes convergent, and then back to the Minkowski space to obtain the final result. I hope this helps to clarify your doubts.