Consider the $\phi^4$ theory for example.
In QFT, we do renormalized perturbation theory by defining the theory at a particular scale, see for example, eq. 12.30 of Peskin and Schroeder:
Then we can use the renormalization conditions to calculate the counterterms and beta functions, the details of which can be found in Peskin chapter 10 and 12.
In contrast, in statistical field theory, we use the Wilsonian approach, and renormalization is done through three steps (see P57 of David Tong's notes here:
- Integrate out high momentum modes, $\Lambda / \zeta<k<\Lambda$.
- Rescale the momenta $\mathbf{k}^{\prime}=\zeta \mathbf{k}$.
- Rescale the fields so that the gradient term remains canonically normalised.
The old view of QFT (Peskin method) tries to make sense of the theory up to arbitrary energy scale, so they define bare and renormalized fields and coupling constants and claim that only the renormalized ones are physical. While the modern view of QFT views the theory as an effective theory up to some certain cutoff, which is essentially using the statistical field theory viewpoint, but the question is, how to relate these two views?
In particular, because in QFT we have more freedom to do different renormalization schemes, is it possible to phrase the Wilsonian approach in Tong's notes in terms of Peskin's language?
Also, in Peskin's language, the scale $M$ is only useful in the renormalization conditions, while the infinities in the integrals are expressed in dimensional regularization $d=4-\epsilon$ through $\frac{1}{\epsilon}$, so you can not calculate at exactly $d=4$. In contrast in Wilsonian approach, there is no infinities, and it is perfectly fine to calculate everything at $d=4$. $\epsilon$ in dimensional regularization is only useful if you want to find the Wilson-Fisher fixed point, or to extrapolate the theory to $d=3$ ($\epsilon=1$). Clearly these two methods are very different fundamentally and it is hard to see why they just differ by a renormalization scheme.
I would like to know how to reconcile these two viewpoints, and in particular whether one can phrase one viewpoint in terms of another (even intuitively is very helpful). Where is rescaling of momenta and fields in Peskin's approach? How to define the counterterms in Wilsonian renormalization?
