# How to reproduce the result of perturbative $\phi^4$ from exact RG

I am studying Exact RG and I have a question.

I think, it should be possible to reproduce the flow equation of $$\lambda_4$$ term in perturbative RG, from exact RG. However, from Peskin, equation of $$\lambda_4$$ is like $$$$\frac{d\lambda_4}{dt} \sim \lambda_4^2$$$$ when I use $$t=\log \Lambda$$, where $$\Lambda$$ is the cutoff, and the Polchinski equation is like $$$$\frac{d S_{I,\Lambda}}{d t} \sim -\frac{1}{2} \left[\left(\frac{\delta S_{I,\Lambda}}{\delta \phi}\right)^2 - \left(\frac{\delta^2 S_{I,\Lambda}}{\delta \phi^2}\right) \right],$$$$ according to "An Introduction to Exact RG" from Sathiapalan. However, if I equate the terms in Polchinski equation of order $$\phi^4$$, naively I won't get terms like $$\lambda_4^2$$. Does anyone know what I messed up?

• Which page/equation in P&S? Commented Jan 6 at 12:40
• Page 404 for Wilsonian RG and Page 422 for Stueckelberg-Petermann (ordinary) renormalization. It is natural that it becomes the order of \lambda^2 from diagrammatical consideration.
– HQMA
Commented Jan 6 at 15:20
• I am sorry to comment on my own question, but this paper seems to derive beta function of $\lambda_4$, so I will take a look into it. Tim R Morris, "The Exact Renormalization Group and Approximate Solutions"
– HQMA
Commented Jan 7 at 1:31