# Infinite series of diagrams = one-loop RG equations？

In the RG of $$\rm Fe$$ based superconductor, the author wrote something like Infinite series of diagrams (RPA)= one-loop RG equations. I wonder if it is a universal theorem.

I don't have a background in high-energy QFT. So I opened up Srednicki for some Hints. It seems the counter-term RG does the same thing as RPA in condensed matter physics, in the sense they both derive self-energy and sum over the same type of diagrams. And the Wilsonian RG is very similar to the RG we learned in condensed matter physics.

Are they basically the same thing? They seems different because the former sum over all diagrams and integrates out all momentum. But the latter only sums over finite diagrams and integrates out the momentum shell.

The ref:https://arxiv.org/abs/0902.4188

Short: no, infinite series of diagrams are not directly related to 1-loop RG

Let me remind you some keypoints of Wilsonian RG. We start from the path integral, $$\mathcal{Z}(g)=\int\mathcal{D}\Phi\,\exp\left\{-S[\Phi;g]\right\},$$ where $$\Phi$$ denotes field and $$g$$ parameters (not only couplings). At first step we perform decimation of modes, $$\Phi=\Phi^{<}+\Phi^{>}.$$ We can do it in many different ways, but Wilsonian RG implies mommentum-based decimation. Next, we integrate out the "larger field", $$\mathcal{Z}=\int D[\Phi^{<},\Phi^{>}]\,e^{-S[\Phi^{<}+\Phi^{>};g]}=\int\mathcal{D}\Phi^{<}\,\exp\{-S^{<}_{\Lambda}[\Phi^{<};g^{<}]\},$$ where $$\Lambda$$ is the momentum scale and $$S^{<}_{\Lambda}[\Phi^{<};g]$$ is the effective low energy action. Next step is the rescaling. Our new action $$S^{<}_{\Lambda}$$ depends on new couplings $$g^{<}$$. We express this quantities via the initial couplings $$g$$. The second step is the rescaling. We rescale wave vectors, fields and couplings. Performing this two steps, we obtain RG transformation, $${\bf g}'=\mathcal{R}(b;{\bf g}),\tag{*}$$ where $$b$$ denotes the scale. $$\mathcal{R}$$ is very complicated non-linear function and more over this map is non-invertible. It is general description of RG approach and Wilsonian RG corresponds to momentum shell RG, where decimation of field performed on a energy scale $$\Lambda$$. Investigation of eq. $$(*)$$ tells about fixed points of RG flow and allows to find critical points, compute critical exponents and so on.

Now let us consider RPA. One of the canonical examples for RPA computations is the ground state energy of electron gas. You start from the action that contains electron fields $$\psi$$ and $$\overline{\psi}$$ and electrons interact via Coulomb force. The action looks like $$S[\psi,\overline{\psi}]=\sum_p\overline{\psi}_{p\sigma}\left(-i\omega_n+\frac{{\bf p}^2}{2m}-\mu\right)\psi_{p\sigma}+\sum_{pp'q}\overline{\psi}_{p+q,\sigma}\overline{\psi}_{p'-q,\sigma'}V(q)\psi_{p'\sigma'}\psi_{p\sigma},$$ where $$V=V(q)$$ is Coulomb interaction. You would like to consider the free energy, $$F=-T\ln\mathcal{Z},\quad \mathcal{Z}=\int\mathcal{D}[\psi,\overline{\psi}]e^{-S[\psi,\overline{\psi}]}.$$ And as usual, you can not do it exactly and have to develop perturbation theory. The perturbation theory gives series of diagrams. The properties of initial problem dictates that not all the diagrams are very important and at this step RPA appears.

In Wilsonian RG, you also obtain series of diagrams. Not all the diagrams are important, only several types of diagram causes renormalization (for instance, in QED we have only three important diagrams). For me, here the similarity between RG and RPA is over. In both approaches you deal with infinite diagram series and intrinsic structure of an initial problem says which diagrams important and which diagrams are not important.

In RG you look for diagrams that causes interaction and affect initial action parameters. For electron gas, RPA arises from the fact that ring-like diagrams are more important (see diagram $$F^{(2),1}$$ at the figure). Almost always an infinite series of diagrams can be written in a compact way. This is provided due to the Dyson equation & Bethe-Salpeter equation. In a given infinite series of diagram, you can almost always extract several subclasses of diagrams. In RPA for electron gas, you simply extract ring-like diagrams and focus on them because they are the most important.

Hope this helps.

References:

1. "Introduction to the Functional Renormalization Group", Peter Kopietz , Lorenz Bartosch , Florian Schütz
2. "Condensed Matter Field Theory", A. Altland & B. Simons

• Thanks for answering. I also used to think that their similarity ends in their identical Feynman diagrams by cumulate expansion. However, although this one loop RG = RPA may not be so general. Such behavior happens actually so I wonder whether some physics hide in there. For example, in Dzyaloshinskii's "Parquet solution for a flat Fermi surface" 1996. The calculation shows infinite bubbles (or Parquet) = one loop RG under some simplication. And Shankar's famous Review of Modern Physics paper mentioned that too. Sep 7, 2023 at 10:12