I'm trying to understand this paper by A. J. Millis (can't link to scihub because I'm on university's wifi and we live in a dystopia)

What has me utterly perplexed is something he says at the start of section 2: "One may question the validity of integrating out the low energy excitations ..."

You're damned sure I question it. This seems like the exact opposite of what renormalization is supposed to be. You are zooming in, not out, right? I've been trying to see if there's a way in which keeping high energy excitations could result in a divergent correlation length, but I can't. The uncertainty principle guarantees: high energy -> high momentum -> small space. I lack the ability to conceive of a way this wouldn't be true

Now, Millis says: "Chill out babe, if we get analytic results, how bad can this be?" and like... I guess?

Does that mean that the correlation length converges to a finite value?…