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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 \begin{equation} \frac{d\lambda_4}{dt} \sim \lambda_4^2 \end{equation} when I use $t=\log \Lambda$, where $\Lambda$ is the cutoff, and the Polchinski equation is like \begin{equation} \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], \end{equation} 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?

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  • $\begingroup$ Which page/equation in P&S? $\endgroup$
    – Qmechanic ♦
    Commented Jan 6 at 12:40
  • $\begingroup$ 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. $\endgroup$
    – HQMA
    Commented Jan 6 at 15:20
  • $\begingroup$ 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" $\endgroup$
    – HQMA
    Commented Jan 7 at 1:31

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