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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.

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The main point that you have to always keep in mind is that relevant/irrelevant coupling constants are defined with respect to a fixed point. The standard/naive power counting is done assuming that the fixed point controlling the RG flow is the gaussian. This is true for massless QED and $\phi^4$ theories in four dimensions in the infrared, and for QCD in the UV. However, this is not true in the opposite limits ! QED and $\phi^4$ do not have a UV fixed point, meaning the the theories are not asymptotically safe, and all 'irrelevant' coupling constants grow in the UV. On the other hand, QCD flows to strong coupling in the IR, though it seems that it still flows toward another fixed point.

All in all, what it means is that if the theory flows to a fixed point in the IR/UV, only the few relevant operators have to be fixed in order to describe all the physics at lower/higher energy.

In the case of $\phi^4$ in $d=4$, the mass is a relevant operator at the gaussian fixed point. It means that as one integrates more and more degrees of freedom, the mass drives the system away from the gaussian fixed point (it introduces a length scale in the system). But in order for the system to approach the gaussian fixed point, the mass needs to be small (otherwise, the system will go away from the gaussian fixed point before it even approaches it). This means that under the RG transformation, it will take a long time to have the mass grow up to the cut-off scale (at which point the RG flow stops).

The best reference I know that discusses the relationship between the perturbative RG (as done in QED) and the Wilsonian point of view is given in this review on the (non perturbative) RG, and in particular the section 2.6 "Perturbative renormalizability, RG flows, continuum limit, asymptotic freedom and all that".

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Thanks for the answer. Some aspects are still rather mysterious to me, but I'd need to read more on the subject to understand it. One question: why precisely does the RG flow stop at the cut-off scale? The scale as I know it is a scale for the momentum of the field, so how does it relate to the coupling constant? –  Peter Mar 25 at 22:35
    
@Peter: what I mean is that when one lowers the cut-off (in a Wilsonian point of view) below the scale given by the mass, the flow stop. This is because the mass is then a high energy scale in the propagators compare to the typical momentum scale (of the order of the cut-off), and therefore the loops are suppressed. –  Adam Mar 25 at 23:04

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