A Relationship between Wilson's RG and the Callan-Symanzik Equation

[gsschandu]

TL;DR Summary

Question about the equivalence or relationship between the callan-symanzik normalization scale and Wilson's RG scale parameter

I have taken a Quantum Field Theory course recently in which we first derived the Callan-Symanzik equation and then discussed Wilson's Renormalization. However, I don't think I have a clear understanding of the procedures and how they relate to each other. For the sake of this question, let's restrict ourselves to massless theories. Let's also say we have the following normalization scheme:$$ \Gamma^{(2)}(p^2)|_{p^2=0} = 0 $$
$$\Gamma^{(4)}(p_1,p_2,p_3,p_4)|_{p_i = \mu e_i} = \lambda_{(\mu)}$$where $e_i$ are a set of four vectors such that $e_1+e_2+e_3+e_4=0$. The normalization scheme depends on the scale normalization $μ$ and the associated equation for the correlation functions when the normalization scheme is changed is the Callan-Symanzik equation.

$$(\mu \frac{\partial}{\partial \mu} + \beta(\lambda) \frac{\partial}{\partial \lambda} + n \gamma (\lambda)) \langle \phi(x_1)...\phi(x_2)\rangle= 0$$

On the other hand, we have Wilson's approach to renormalization. We start of with an action and a cut-off, integrate out to get the transformed action. This changes the cut-off of the new action so we perform a variable change to return the cut-off to its original value. So if $L$ is the parameter of the RG flow, we have the following relationship between correlation functions:$$ \langle\phi(x_1)...\phi(x_2)\rangle_{A^{'}} = Z^{\frac{n}{2}}(L) \langle \phi(Lx_1)... \phi(Lx_n) \rangle_{A} $$where we have $RGL[A]=A^{'}$. We obtain the same relationship in the case of a normalization scheme. I am pretty sure the parameter $L$ given by the ratio of cut-offs in Wilsons RG somehow is the same as choosing a normalizing a scheme $μ$ and both these methods are equivalent. However, I am not sure how to rigorously justify it.

I've also heard that $\gamma(λ)$ is the anomalous dimension of Wilsons RG but don't know why. So any help is appreciated.

 

[Son Goku]

Wilson's RG group is about generating effective actions from a given bare action defined at some cutoff scale, defined as a flow in the space of Lagrangian couplings.
The Callan-Symanzik equation is about how physical parameters, typically in a continuum theory, change with the energy scale.

The Callan-Symanzik equation generates a flow within a finite set of parameters, the physical ones of the theory. However Wilson's group defines a flow in infinite dimensional action space. So they can be similar up to one-loop restricting oneself to the physical parameters but in general they are not.

See Montvay and Munster's Lattice book for a deeper discussion of their differences.

 

[gsschandu]

I see, so both methods are different in general. But why do these methods even agree to first order and why is $\frac{d-2}{2}+\gamma$ from the Callan-Symanzik equation agree with the anomalous dimension from Wilson's RG?