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I am currently studying Quantum Field Theory, but I cannot find any analytic calculation for the beta fucntion in $O(N)$ models such as $\phi^4$ theory. Can anyone provide me some hints on the calculation?

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May be you should start with the paper arXiv:hep-th/0607177v2 : "Renormalizable Expansion for Nonrenormalizable Theories: I. Scalar Higher Dimensional Theories", by D.I. Kazakov, G.S. Vartanov, and look the references therein.

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