Question

Integrals of the form $\int\frac{d^Dk}{(2\pi)^D} \frac{1}{k^{2n}}$ in $D=4-2\varepsilon$ dimensions?

Answer 1

In a massless theory we often get integrals of the form

∫dDk(2π)D1k2n(*)(*)∫dDk(2π)D1k2n

where D=4−2εD=4−2ε. I have tried to calculate this in two ways in Minkowski space and in Euclidean space. With the former I would be integrating

∫dDk(2π)D1(k2+iε′)n∫dDk(2π)D1(k2+iε′)n

but in this case I appear to get a factor of 1/(ε′)ϵ1/(ε′)ϵin my final answer (for the case of n=2n=2) in which I can't take ε′→0ε′→0. In the Euclidean case I get an intergral of the form:

∫∞0tε/2−1∫0∞tε/2−1

which as far as I can tell diverges for ε>0ε>0 (which it is). Thus my question is as follows: What is the standard and easiest way to calculate (*) ideally based only on Feynman parametrization, Schwinger parametrization and guassian integrals

Answer 2

$See Collins "Renormalization" for a discussion of dimensional regularization, including this particular integral.$