From the conceptual idea of the RG to its actual implementation

Question

Everytime I want to understand a little more about the ideas behind Renormalization Group techniques, I get troubled by a gap between the general picture one usually presents (e.g. in books or pedagogical reviews/lecture notes) and its actual implementation. I know that there is no way of learning these techniques without working on a practial problem (or problems); however, it seems to me that there are so many ideas and concepts behind the problems which these techniques are used to, that sometimes it is hard to understand where these calculations are trying to get or to prove. In this sense, it would be nice to have a sketch of what is/are the problems one is trying to solve and what are some techniques to accomplish this.

Let me elaborate a little more with what I understand is the most common picture of the RG as presented by books. We want to study objects of the form: $$Z[g] = \int D\phi e^{-S[\phi,g]} \tag{1}\label{1}$$ where the action $S$ is supposed to be some sort of perturbation of a Gaussian; here the $g$ denotes a set of coupling parameters of the theory. Since (\ref{1}) is supposed to be a perturbation of a Gaussian, I'll rewrite it as: $$Z[g] = \int d\mu_{C,\alpha}(\phi)e^{-G[\phi,g]} \tag{2}\label{2}$$ for some Gaussian measure $\mu_{C}$ with covariance $C$, which is also assumed to have some coupling parameter dependence on $\alpha$. To keep things more well-behaved, I'm assuming we're working with imaginary time (Euclidean QFT), so $Z$ has the natural interpretation of the partition function of the underlying model.

It turns out to be more convenient to study the following object: $$Z[\varphi;g] = \int d\mu_{C,\alpha}e^{-G[\phi+\varphi,h]} \tag{3}\label{3}$$ instead of (\ref{2}). If we assume $C$ can be decomposed as: $$C = C_{1}+C_{2} \tag{4}\label{4}$$ then, (\ref{3}) becomes: $$Z[\varphi;g] = \int d\mu_{C_{1},\alpha_{1}}(\phi_{1})\int d\mu_{C_{2},\alpha_{2}}(\phi_{2})e^{-G[\phi_{1}+\phi_{2};g]} \equiv \int d\mu_{C_{1},\alpha_{1}}(\phi_{1})e^{-G'[\phi_{1}+\varphi;g]} \tag{5}\label{5}.$$ Reescaling (\ref{5}) we get: $$Z[\varphi;g] = \int d\mu_{C,\tilde{\alpha}}(\phi_{1})e^{-G'[\phi_{1}+\varphi;g]} \tag{6}\label{6}$$ which has the same form of (\ref{3}). The Renormalization Group transformation is the map $\mathscr{R}: G \to G'$.

Now it comes the more practical problems I wanted to discuss. If the desired object of study is (\ref{3}) and if we're interested in critical behavior, it would be natural to me to study the map $\mathscr{R}$ and look for a fixed point. I know this might be difficult in practice, and it seems to me that this is usually not what people study in practice. And this is the central problem to me: if it is difficult and not what people actually do, so what is the target then?

Some references discuss the so-called perturbation RG, which consists in expanding the term $e^{-G}$ in formal power series and trying to prove that the series converges or that each term of the expansion is finite. Proving that the power series converges proves that, under certain conditions, (\ref{3}) is well-defined mathematically as far as I know. And of course, to prove this convergence it is important to prove that each term of the series is finite. However, I don't know if this is the correct justification for such calculations, since it seems to me a purely mathematical justification, rather than a physical one. Also, I don't see why proving these bounds leads us to a practical result: in the general picture presented above, it was very clear that the idea was to evaluate $Z$, but when one is evaluating bounds it is not so clear to me what is the objective. Maybe evaluating two-point correlation function decays?

In summary, I'd like to understand what are the problems, in practice, with the general picture which is usually presented in abstract way as discussed above and how does it help to understand what one really does in a research activity. Also (and most importantly) what is one looking for in a typical research problem? What are the objects of interest, once obtaining an explicit form for $Z$ seems out of reach?

Remark: Examples are welcomed to clarify the main ideas.

Answer 1

Q: "What is one looking for in a typical RG research problem?"

One typically hopes to find that the combination of coarse-graining (e.g. by removing high wave number components) and rescaling results in a simpler action; for example, in many applications powers of $\phi$ greater than four become smaller and smaller (they "flow" to zero), and only second and fourth powers remain. This then provides a tractable calculation of the partition function (for example, by treating $\phi^4$ terms as perturbations of the Gaussian integral), and from that one can obtain correlation functions. These are not exact, but the expectation is that these will become more and more accurate as one approaches the critical point (the fixed point of the coarse-graining + rescaling operation).

Introduction to Renormalisation by João F. Melo works this out in some detail.