Relation between Wilsonian renormalization and Counterterm Renormalization

Question

Wilsonian renormalization The answer by Heider in this link points out that when we integrate out high momentum Fourier modes, we end up with Wilsonian effective action (not the 1PI action). This is the modern way of understanding renormalization.

I want to relate it to the old way of understanding renormalization i.e., not from a path-integral calculation but as perturbative renomalization (in the Hamiltonian formulation of field theory).

Counterterm renormalization In this technique, (for example, in scalar $\phi^4-$theory) separates $\mathcal{L}$ into a renormalized and a counterterm part as $$\mathcal{L}=\mathcal{L}_{renorm}+\mathcal{L}_{ct}.$$ Now $\mathcal{L}_{renorm}$ contains $V_r(\phi_r)=\frac{1}{2}m_r^2\phi_r^2+\frac{\lambda_r}{4!}\phi_r^4$ and the counterterm contains $$V_{ct}=\frac{1}{2}\delta_m\phi_r^2+\frac{\delta_\lambda}{4!}\phi_r^4$$ which cancels certain divergences of one-loop diagrams. The potential $V(\phi_r)$ is now in terms of measured parameters $m_r^2$ and $\lambda_r$.

Question 1: When you integrate high frequency modes, you generate terms proportional to $\phi^2$ and $\phi^4$ which looks like counterterms. My question is, can we regard the $\mathcal{L}_{ct}$ (in this approach) to be same as the contributions coming from integrating out high momentum modes (in the path integral)?

Question 2 Moreover, counterterm renormalization is a loopwise renormalization process. Is it also the same in the Wilsonian picture? Moreover, these two ways of understanding renormalization should be consistent. Can someone explain the connection between Wilsonian idea of renormalization and counterterm renormalization.

Answer 1

I like to think about it as described in this article by Shankar. In counterterm renormalization, you are essentially studying Wilsonian renormalization, but with the intention of sending the cutoff to infinity at the end (as is necessary to have a continuum definition of a field theory). The UV "divergences" that show up in perturbation theory are the result of taking this limit sloppily. Counterterms are a mathematical construct that allow you to take this limit more carefully. The fact that the counterterms can be different depending on renormalization scheme, but the physics must be independent of the choice of scheme, is what leads to the idea of scale dependence and running couplings in the counterterm picture (this is why we are free to use MS, MS bar, or on shell, and not worry about conflicts in predictions). From this point of view, in counterterm renormalization, we integrate out the modes above some cutoff, which (as we know from the Wilsonian picture) gives finite results for the couplings as functions of the cutoff. We then use this result to define the continuum limit of the field theory, by pushing the cutoff to infinity, equivalent to taking it as much larger than any scales of interest. This leaves us with counterterms capturing the limiting behavior of the cutoff dependence necessary for the limit to exist, and an arbitrary mass scale capturing what the Wilsonian picture told us about the scaling behavior of the theory. For the other part of your question, yes, Wilsonian renormalization can be done loopwise. By symmetry, the Feynman diagrams which show up in perturbative field theoretic calculations will be identical to the Feynman diagrams generating corrections to the couplings from the fast modes, with the only difference being that the integrals are now convergent, taken over the momentum shell from the scaled down cutoff to the original cutoff.

I would like to add one additional note: the leftover information from Wilsonian renormalization of a perturbatively renormalizable field theory after the continuum limit has been taken can also be viewed as the information telling us about the anomaly of scale invariance. From this point of view, Wilsonian renormalization is a more careful way of handling how scale invariance behaves at the quantum level, by studying the proper way to apply a scale transformation to the path integral. This is analogous to how Fujikawa computed the chiral anomaly; he looked at the fact that the path integral measure broke the symmetry. In this case, it's not as "obvious" as the measure breaking the symmetry, it's the loop corrections which appear in the Wilsonian effective action which break the symmetry. The beta functions tell us exactly how it is broken (or preserved, in very special cases!).

Hope this helped.

Edit: It has been some time since I wrote the original answer, and I am not completely satisfied with it (some issues regarding these concepts were brought up again to me recently). The description here is, of course, very rough. The Wilsonian point of view gives us a philosophy to understand effective field theories and why renormalization works, but in involved calculations the old way is preferred. Ideas from the "Exact Renormalization Group" are the closest thing in existence to making the connection between the points of view precise, but even in those cases the computations in the Wilson scheme are simply intractable. At the end of the day, it comes down to an issue of philosophy and pragmatism. The best choice is to use counterterm renormalization to compute, and the Wilson view to interpret.

Answer 2

It is possible to do Wilsonian renormalization using couterterms. This is the main idea followed by the French school of constructive QFT around Feldman, Magnen, Rivasseau and Sénéor. You can learn about this point of view in the book "From Perturbative to Constructive Renormalization" by Vincent Rivasseau (especially Chapter II.4 about the so-called effective expansion). You can also find the book here.

The basic idea is to introduce a multiscale (Littlewood-Paley) decomposition of your field $\phi=\sum_{i=0}^{\infty}\phi_i$ where the Fourier modes of $\phi_i$ live in the shell $2^i<|p|<2^{i+1}$. In the presence of a UV cutoff (to be removed in the end) the sum stops at say some large number $n$. Now a $\phi^4$ vertex with bare coupling $g$ in the action becomes $$ g\ \phi(x)^4=\sum_{i_1,i_2,i_3,i_4=0}^{n} g\ \phi_{i_1}(x)\phi_{i_2}(x)\phi_{i_3}(x)\phi_{i_4}(x)\ . $$ This can be rewritten as $$ g\phi(x)^4=\sum_{i_1,i_2,i_3,i_4=0}^{n} g_{\max\{i_1,i_2,i_3,i_4\}}\ \phi_{i_1}(x)\phi_{i_2}(x)\phi_{i_3}(x)\phi_{i_4}(x)\ $$ $$ + \sum_{G} c_{G} \sum_{i_1,\ldots,i_4<i(G)} \phi_{i_1}(x)\phi_{i_2}(x)\phi_{i_3}(x)\phi_{i_4}(x)\ . $$ The $G$'s are divergent subgraphs (with four external legs) and $c_{G}$ is the corresponding counterterm. The subtraction scale of such a graph denoted by $i(G)$ is the minimal $i$ index of lines internal to $G$. Finally for the identity to hold one needs to introduce running couplings $g_i$ such that $g_n=g$ (the bare coupling) and determined by a (discrete) Wilsonian flow $$ g_{i-1}=g_i-\sum_{G} c_{G} $$ where the sum is over all graphs with $i(G)=i$.

Instead of expanding perturbation theory in the sole coupling $g$ one now has an expansion in all the $g_i$'s and $c_G$'s. The latter are determined by imposing the cancellation of all local parts ($\tau$ operation in old-fashioned BPHZ renormalization) of the divergent subgraphs $G$. The difference with BPHZ is as follows. In BPHZ, the subgraphs $G$ are subtracted regardless of the relative frequency ordering of the internal versus external lines. In the approach I described (which is just a reformulation of Wilsonian renormalization), these subgraphs are subtracted only when they are truly dangerous (produce UV divergences), namely, when the internal lines are of higher frequency than the external ones. The "useless" BPHZ renormalizations done when this condition is not satisfied not only have nothing to do with curing divergences but they create a new problem, that of renormalons.