- #1

- 2,018

- 810

I am working on using dimensinal regularization of a loop integral in QED. I don't think the specific application to QED is important, but I will say that the original integral is in 4D Minkowsi space-time.

The example in the text starts with a Wick rotation of an integral in Minkowski space, changing the time coordinate by \(\displaystyle p^0 \to i p^4\). The problem is now to integrate over 4D Euclidean space. The metric is therefore a Kronecker delta rather than the usual space-time metric but for some reason the text still uses the Minkowski metric, \(\displaystyle \eta ^{ \mu \nu }\). In that spirit I'm just going to copy the text's notation.

So. To regularize the integral the text says

1) Relable the number of dimensions in the integral to d, which we will eventually take in the limit to 4. (I don't believe we need this comment.)

2) Drop all the terms in the integrand that are odd powers in p.

3) Replace the terms that are even powers of p with

\(\displaystyle p^{ \mu } p^{ \nu } \to p^2 \eta ^{ \mu \nu } / d\)

and

\(\displaystyle p^{ \mu } p^{ \nu } p^{ \rho } p^{ \sigma } \to (p^2)^2 \left ( \eta ^{ \mu \nu } \eta ^{ \rho \sigma } + \eta ^{ \mu \rho } \eta ^{ \nu \sigma } + \eta ^{ \mu \sigma } \eta ^{ \nu \rho } \right )/(d(d + 2))\)

1) I'm okay with this, though d is allowed to be complex. I really can't grok a complex "number" of dimensions but hey, I'm trying to evaluate a divergent integral. Welcome to the crazy world of QFT. (Party)

2) Why drop the odd powers?

3) Okay, the first one I can only work with but it doesn't seem too extreme. The second, I guess I could start out with \(\displaystyle p^{ \mu } p^{ \nu } p^{ \rho } p^{ \sigma } = \left ( p^{ \mu } p^{ \nu } \right ) \left ( p^{ \rho } p^{ \sigma } \right ) + \left ( p^{ \mu } p^{ \rho } \right ) \left ( p^{ \nu } p^{ \sigma } \right ) + \left ( p^{ \mu } p^{ \sigma } \right ) \left ( p^{ \nu } p^{ \rho } \right ) \)

but the coefficient wouldn't work out. (These two substitutions bear a resemblance to the trace theorems of Dirac gamma matrices. (See pages 6 - 8.) Is there, perhaps, a reason for this?)

Given the starting points I can pretty much follow what is going on, though there is a Gamma function identity that I don't know how to derive. But other than that it's not really all that bad. I just can't figure out how to start it.

-Dan