# What's the relation between Wilson Renormalization Group (RG) in Statistical Mechanics and QFT RG?

What's the relation between Wilson Renormalization Group(RG) in Statistical Mechanics and QFT RG? For easier to compare, I choose scalar $\phi^4$ in both cases.

Wilson RG: Given $\phi^4$ model, $$Z=\int\mathcal{D}\phi(x)\exp[-\beta H[\phi(x)]]$$ $$\beta H[\phi(x)]=\int d^dx\left\{ \frac{k}{2}(\nabla\phi(x))^2 +\frac{t}{2}\phi^2(x)+\frac{u}{4!}\phi^4(x)\right\}$$ There is a natural UV-cutoff $\Lambda_0$ for statistical mechanics.

Integrating the momentum shell from $\Lambda=\Lambda_0 e^{-l}$ to $\Lambda_0$ and rescale, we can get the RG flow equation/Beta function:

$$\frac{du}{dl}=(4-d)u-\frac{3}{2}u^2\frac{K_d \Lambda_0^d}{(t+k\Lambda_0^2)^2} \tag 1$$ $$\frac{dt}{dl}=2t +\frac{u}{2}\frac{K_d\Lambda_0^d}{t+k\Lambda_0^2}\tag 2$$ with $K_d=S_d/(2\pi)^d$.

High energy QFT RG: Given $\phi^4$ model, $$\mathcal{L}=\frac{1}{2} (\partial_\nu \phi_R)^2+ \frac{1}{2}m_R^2\phi_R^2+\frac{\mu^\epsilon g_R}{4!} \phi^4_R + \frac{1}{2}(Z_\phi-1) (\partial_\nu \phi_R)^2+ \frac{1}{2}(Z_m Z_\phi -1)m_R^2\phi_R^2+(Z_g Z^2_\phi-1)\frac{\mu^\epsilon g_R}{4!} \phi^4_R$$ with $\epsilon=4-D$, $D$ the spacetime dimension, $\mu$ an arbitrary mass scale, $g_R, m_R$ dimensionless renormalized parameters.

Using dimensional regurization, momentum substraction scheme, we can get the RG flow equation in $D \rightarrow 4$ like this: $$\frac{\partial g_R}{\partial \ln\mu}=-\epsilon g_R+ \frac{3}{2}g_R^2+O(g_R^3) \tag 3$$ $$\frac{\partial m_R^2}{m_R^2 \partial \ln\mu}=\frac{K_4}{2}g_R + O(g_R^2)\tag 4$$

My questions:

1.Everyone says that Wilson RG is essentially same as QFT RG. No matter in technique or concept I can't understand this relation.

Firstly in concept, Wilson RG describes how the effective parameter flow as you view the system in a larger and larger size.

In QFT, we can compute any observable $\sigma$ $$\sigma = \sigma(m_0,g_0,\Lambda)$$ with $m_0$, $g_0$ the bare parameters, $\Lambda$ the UV-cutoff. However if you fix the bare parameters and make $\Lambda\rightarrow \infty$ then every observable $\sigma$ must be divergent. The only way out is using experiment to fix the some observables $\sigma$, then RG tells us how bare parameters should grow with cutoff, that is $m_0(\Lambda), g_0(\Lambda)$, to keep observables $\sigma(m_0(\Lambda),g_0(\Lambda),\Lambda)$ finite and independent with cutoff $\Lambda$. So what's the relation?

2.In technique, especially $(2),(4)$ are totally different. How to see explicitly the Wilson RG $(1),(2)$ and QFT RG $(3),(4)$ are essentially same? (or in weaker sense they can have same Wilson-Fisher fixed point? have same critical exponent?)

3.Above calculation is only $1$-loop. Even though you can show in $1$-loop they are essentially same, how to prove that they are same in higher loop? Because I heard that in QFT RG, the coefficent of $g_R^3$ and higher will depend on the regularization. It seems impossible Wilson RG is same as QFT RG.

PS: There is a relating question Relation between Wilson approach to renormalization group and 'standard' RG But our questions are totally different.

Beyond one-loop, the two methods seem completely different : in Wilson's approach, one generates new interactions in the lagrangian ($\phi^6$ etc.), which are needed to get the fixed points at order $\epsilon^2$ for instance. In the QFT RG, one never generate new interactions in the lagrangian, everything in captured in $g_R$, which is used to generate all vertex functions (corresponding to $\langle\phi^6\rangle$ and so on). In fact, what the perturbative RG is doing in practice is to project the RG flow onto one specific RG trajectory (called L in the reference above), which can be parametrized by the flow of $m^2_R$ and $g_R$ only (thus describing the flow of all other interactions in terms of $m^2_R$ and $g_R$).
As I said in a previous answer (see below) : In the Wilsonian approach, one starts from the microscopic scale $\Lambda$ and looks at what's going at smaller energy, whereas in the "standard" approach, one fixes than macroscopic scale and sends $\Lambda\to\infty$ in order to effectively probe smaller and smaller energy scales.