My current way of viewing Wilsonian RGE applied to QFT:
(1) We start with a lagrangian that accurately models dynamics up to a scale $\Lambda_0$.
(2) We fix $\Lambda_0$ as a cutoff to regulate the infinities characteristics to the infinite degrees of freedom of a QFT. This defines the theory dynamically.
(3) We define the theory numerically by choosing parameters (e.g., couplings) in such a way that the predicted values for observables are consistently related to the observed results in a scale $\mu$, through some renormalization scheme.
(4) Then we chose to perform a sort of reparametrization, integrating an upper continuous subset of the energy degrees of freedom $(\Lambda,\Lambda_0]$, and redefining the original lagrangian so that the integral over this subset does not need to be done again and the cutoff from now on becomes $\Lambda$ -- this amounts, at the of the day, to just reverse the order of some operations. This may cause the introduction of terms whose form was not in the original Lagrangian, which are effective interactions.
(5) We make sure that observables do not depend on where we are on 'the group', imposing that they do not depend on $\Lambda$, which, in turn, sets a dependence $g=g(\Lambda)$.
Questions:
(i) I'm basically only interested in RGE for HEP applications so that wilsonian RGE is not that important, but I want to fully understand the equivalence between it and non-Wilsonian dimensionally regulated RGE. In RGE with dimensional regularization, there is obviously no $\Lambda$, and what plays a similar role is the auxiliary dimension-full constant $\mu$, which is introduced to keep couplings dimensionless. What exactly is the relation between the two? How do I see that Wilsonian and dim. reg. RGE are equivalent?
(ii) On dim. reg. RGE, we really don't talk at all about those changes in the lagrangian. Where did they go?
I understand the underlying conceptual difference, that Wilsonian RGE assumes the known fact that every theory is at most an effective one, while this fact is hidden in non-Wilsonian RGE, which assumes renormalizable ultraviolet-complete theories in the beginning. But this doesn't help me close the gaps I mention above.
Feel free to answer the questions in whatever way you find appropriate, and any insights are greatly appreciated.
