On page 43 of Manohar notes on effective field theories, he argues that since all the integrals in EFT are scale less when expanded in terms of the IR parameter, they all vanish. To me it seems absolutely correct for the examples he's given on the same page, but if one goes to higher order terms in the IR expansion of the EFT one faces integrals that are either only UV divergent or only IR divergent, and they don't seem to be zero but rather infinity! Can someone help me with this problem please?
How cam I deal with scale less integrals of higher orders!
Or how can I rewrite all the scale less integrals in the (5.21) format? Like how to write $$\int 1/k^8 \, d^4k$$ or $$\int k^2\, d^4k$$ in the (5.41) format?
The paper: https://arxiv.org/abs/1804.05863