I'm trying to understand how to perform the CW mechanism (http://www.scholarpedia.org/article/Coleman-Weinberg_mechanism) to scalar theories at two-loop order. More specifically how to find the effective potential, $V_{eff}$, for the classical background, $\phi_{cl}$, obtained by path integrating out quantum fluctuations, $\delta\phi$.
At one-loop, $V_{eff}$ can be obtained by calculating the Feynman diagrams in Fig. 1.1 in https://pure.mpg.de/rest/items/item_3187817_1/component/file_3187818/content. (An alternative approach is to calculate the functional determinant, but I have a feeling the diagrammatic way is more useful at two-loop.)
My question now is: which diagrams should I compute at two-loop? What are the vertices (and why those specific vertices)? How many external legs (of $\delta\phi$)?
In Peskin and Shroeder Ch. 11.2 they say that only "the sunrise" and "the 8" diagram contribute. Why is this the case? And how many external legs should there be? Is this only the case in $\phi^4$-theory (in $d = 4 - \epsilon$)? What will differ in e.g. the $\phi^6$-model in $d = 3 - \epsilon$?
