I recently followed courses on QFT and the subject of renormalization was discussed in some detail, largely following Peskin and Schroeder's book. However, the chapter on the renomalization group is giving me headaches and I think there are many conceptual issues which are explained in a way that is too abstract for me to understand.
A key point in the renormalization group flow discussion seems to be the fact that the relevant and marginal operators correspond precisely to superrenormalizable and renormalizable operators. I.e., the non-renormalizable couplings die out and the (super)renormalizble couplings remain. In my lectures the remark was that this explains why the theories studied ($\phi^4$, QED) seem (at least to low energies) to be renormalisable QFTs. This is rather vague to me, and I don't know how to interpret this correspondance between relevant/marginal and renormalizable/superrenomalizable theories.
If I understand well, flowing under the renormalization group corresponds to integrating out larger and larger portions of high momentum / small distance states. Is there a natural way to see why this procedure should in the end give renormalizable QFTs?
Also, it appears that the cut-off scale $\Lambda$ is a natural order of magnitude for the mass. Since the mass parameter grows under RG flow for $\phi^4$-theory, after a certain amount of iterations we will have $m^2 \sim \Lambda^2$. But what does it mean to say that $m^2 \sim \Lambda^2$ only after a large number of iterations? The remark is that effective field theory at momenta small compared to the cutoff should just be free field theory with negligible nonlinear interaction.
Also, there is a remark that a renormalized field theory with the cutoff taken arbitrarily large corresponds to a trajectory that takes an arbitrary long time to evolve to a large value of the mass parameter. It would be helpful to get some light shed on such remarks.