This is in continuation of my previous question, IR divergence and renormalization scale in dimensional regularization.
Lubos gave a nice answer there but I want to get to a very specific example which is not found in usual QFT - (in QFT all examples I have seen are such that the number of momenta powers are the same in the denominator and the numerator).
Consider the 3 dimensional integrals in A.5 (page 19) of arXiv:1301.7182. I guess one can say that these are UV divergent when 3>2(ν1+ν2) but the gap can be arbitrarily large depending on the values of ν1 and ν2 - right? Then how do the authors justify using d=3+ϵ expansion get the UV divergence?
Also I guess these are IR divergent if 3<2ν1..right?
How does one define the $\epsilon$ to get this IR divergence in the $\epsilon$-expansion?
Also by looking at this A.5 is it clear that $d=3$ is somehow the "critical dimension"? (...the terminology is from Lubos's comments in the previous question...)