Closed set of operators under renormalization

While reading the article http://inspirehep.net/record/61135, I came across the concept of "closed set under renormalization". The definition they give is the following.

In any renormalizable field theory, let $$A_1$$, ..., $$A_n$$ be a set of monomials in the fundamental fields and their derivatives. Change the original Lagrangian of the theory like $$\begin{equation} \cal{L}\rightarrow\cal{L}+\underset{a}{\sum}A_{a}{\cal J}_{a}\,, \end{equation}$$ where $$\cal{J}$$'s are arbitrary functions of space and time. Let us define $$\Gamma_a^n$$ as the Fourier transform of the variational derivative of the $$n$$-point Green's function in the fundamental field with respect to $$\cal{J}_a$$, evaluated at $$\cal{J}_a$$ equals zero. If we add appropriate counterterms in the Lagrangian, we can make the $$\Gamma$$'s finite. If these counterterms are also in the set $$A_1$$, ..., $$A_n$$, we say that the set is closed under renormalization.

I have tried 'Renormalization' by Collins to look for this concept, but I couldn't find much information there.

EDIT: Just for your information, the comment on page 145 around Eq.6.2.13 was all I could find in 'Renormalization'.

Q1: In the reference, they frequently use the expression "simple power counting shows that the following set of operators are closed under renormalization". It is hard for me to guess what kind of power counting they have in mind. For example in the massive $$\lambda\phi^4$$ theory, they say that the following operators are closed under renormalization: $$\begin{equation} \left\{ g_{\mu\nu}\phi^{2},g_{\mu\nu}\partial_{\lambda}\phi\partial^{\lambda}\phi,g_{\mu\nu}\phi\square\phi,\partial_{\mu}\phi\partial_{\nu}\phi,\phi\partial_{\mu}\partial_{\nu}\phi,g_{\mu\nu}\phi^{4}\right\}\,. \end{equation}$$ What could be the logic to arrive at this conclusion?

Q2: They also say as a consequence of the BPH(Bogoliubov, Parasuik, and Hepp) theorem, given a set of operators closed under renormalization, we can find a set of cut-off independent functions $$R_a^n$$, such that $$\begin{equation} \Gamma_{a}^{n}=\underset{b}{\sum}c_{ab}R_{b}^{n}\,, \end{equation}$$ where $$c$$'s are constant, possibly cut-off-dependent, coefficients. This statement seems very important when proving finiteness of $$\Gamma$$'s using Ward identities. It is the first time for me to see this. For example, I couldn't find information on this on textbooks such as Peskin, Schwartz. Could anyone guide me where I could find out more about this? (I have tried to read the original paper(https://link.springer.com/content/pdf/10.1007%2FBF01773358.pdf), but it is quite an old paper and I find it very hard to follow.) I would also appreciate any hints or intuitions to understand this relation.

• It seem to me that the statement about closed set of operators is equivalent to saying that the number of counter-terms, necessary for making a theory finite, is limited. – Vladimir Kalitvianski Dec 11 '18 at 11:05
• @VladimirKalitvianski Thanks for the comment. Yes, that's their statement. What kind of criterion would there be to actually identify the closed set, e.g., the example in Q1? – Gould67 Dec 12 '18 at 7:50
• It is the famous "power counting". Read more about it, look at examples. – Vladimir Kalitvianski Dec 12 '18 at 9:21
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