I found the following paper online: http://www2.mathematik.hu-berlin.de/publ/pre/2013/P-2014-01.pdf. I also attach a screenshot of the paragraph I am confused by (page 63-64).

What is meant by the statement that when renormalizing the effective action flows in the theory space? I am especially confused why the effective action is expanded in operators which where not part of my theory before. E.g. in the paper $\lambda \phi^4$ is renormalized but suddenly the effective action has terms $\propto \phi^n $.

In my understand of renormalization the effective action $\Gamma(\phi)$ is the Legendre transformed of \begin{equation} W(J)=\int [d \phi] e^{-S_{bare}(\phi)-S_{ct}(\phi)+\phi J} \end{equation} where the countqerterms are are just terms $\propto \frac{1}{\epsilon^n}$ in dimensional regularization ensuring that each Feynman diagram is finite. In some sense they are just constants. So how do they give rise to new operators in $\Gamma( \phi)$.

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  • $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic
    Jul 16, 2022 at 14:13
  • $\begingroup$ Possible duplicate: Difference between 1PI effective action and Wilsonian effective action? $\endgroup$
    – Qmechanic
    Jul 16, 2022 at 14:15
  • $\begingroup$ [My answer to "Generating nonlinearities in renormalization group"][1] has a $0d$ example that may be instructive. ["Proof that Wilsonian renormalization only generates terms consistent with the symmetry of the action"][2] is also relevant. [1]: physics.stackexchange.com/questions/633610/… [2]: physics.stackexchange.com/questions/665144/… $\endgroup$
    – bbrink
    Jul 16, 2022 at 21:20
  • $\begingroup$ The missing piece of the connection to this question would be the relationship between the Wilsonian effective action and the 1PI effective action, which I believe are related by a Legendre transform. $\endgroup$
    – bbrink
    Jul 16, 2022 at 21:21


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