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gsschandu
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- 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.
$$ \Gamma^{(2)}(p^2)|_{p^2=0} = 0 $$
$$\Gamma^{(4)}(p_1,p_2,p_3,p_4)|_{p_i = \mu e_i} = \lambda_{(\mu)}$$
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.