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

In summary, the Callan-Symanzik equation generates a flow in the space of Lagrangian couplings, while Wilson's RG group defines a flow in infinite dimensional action space. The two methods agree to first order, but the reason for this is not clear.
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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.
 
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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.
 
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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?
 

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

1. What is the relationship between Wilson's RG and the Callan-Symanzik Equation?

The relationship between Wilson's renormalization group (RG) and the Callan-Symanzik equation is that Wilson's RG provides a framework for understanding the behavior of physical systems at different length scales, while the Callan-Symanzik equation is a mathematical expression that describes how the physical observables of a system change as a function of the energy scale at which they are measured.

2. How do Wilson's RG and the Callan-Symanzik Equation relate to quantum field theory?

Wilson's RG and the Callan-Symanzik equation are both important concepts in quantum field theory. Wilson's RG is used to study the behavior of quantum field theories at different length scales, while the Callan-Symanzik equation is used to calculate the running of coupling constants in quantum field theories.

3. What is the significance of Wilson's RG and the Callan-Symanzik Equation in particle physics?

In particle physics, Wilson's RG and the Callan-Symanzik equation are used to calculate the behavior of physical observables at different energy scales. This is important because it allows physicists to make predictions about the behavior of particles at high energies, which is crucial for understanding the fundamental laws of nature.

4. How are Wilson's RG and the Callan-Symanzik Equation used in practical applications?

Wilson's RG and the Callan-Symanzik equation have many practical applications in physics, including predicting the behavior of particles at high energies, studying phase transitions in condensed matter systems, and understanding the behavior of critical phenomena. They are also used in the development of new theoretical models and in the interpretation of experimental data.

5. What are the limitations of using Wilson's RG and the Callan-Symanzik Equation?

While Wilson's RG and the Callan-Symanzik equation are powerful tools for understanding physical systems, they also have limitations. These include the difficulty of applying them to systems with strong interactions, the need for simplifying assumptions in order to make calculations, and the fact that they are only applicable to systems at thermal equilibrium.

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