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I am looking for an explanation about the idea of "operator mixing" and its associated concept about when anomalous dimension has to be thought of as a matrix.

For example this idea is slightly touched upon in this article though the link to anomalous dimension doesn't lead anywhere. Here they just introduce this notation of $\gamma_{kl}$ and leave it unexplained and undefined.

For some of its aspects that I want to learn about let me refer to this article. I would like to understand the meaning and derivation of the equation $12$ (..that thing called $\gamma_{\phi ^2 I}$..) in the beginning of the section "Perturbative Examples" (bottom of page 5) and the argument at the top of page $7$ and equation $18$.

{...also I would like to know if this is known by some other name since I was a bit surprised to not find these two concepts in various standard QFT books like even in Weinberg's!..}

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Eq. (18) is a general statement about scalars in CFTs. It expresses the fact that the OPE of two scalars consists of symmetric tensors (and no other operators), but otherwise that equation itself doesn't mean much. The (operator) functions $C_{\Delta}^{(\ell)}(x,\partial)$ are universal and can be calculated by looking at three-point functions, see for example Osborn's own paper hep-th/0011040 (but it's a useless exercise). – Vibert Apr 21 at 16:31
@DavidZaslavsky I edited the question to focus on the physics concepts of interest rather than references. Can it be reopend? BTW since in this conctext, the concepts of "operator mixing" and anomalous dimension are very closely related, it does not hurt that the OP asks about them simultaneously in one question. So this should not be a close reason either in this case. – Dilaton yesterday
@DavidZaslavski In addition, it would be helpful to leave a comment for the OP about what is wrong with this generally nice higher level question and how he can fix it. The question is certainly not less salvagable than the many homework questions we get and worth a comment about why it is closed ... – Dilaton yesterday
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I disagree with the closing of this question. It asks for definitions of terms, it mentions an equation in an article as an example of something to understand better, and it asks if they are known under other names. The upside of leaving it open is that someone might write an informative answer. What is the downside? – Mitchell Porter 23 hours ago
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@Dilaton "an example of what one wants to better understand from knowing these concepts" What concepts? That is the key issue. Right now the question asks to know "about the idea of 'operator mixing' and...anomalous dimension" but that is not a concept. What about operator mixing and anomalous dimensions does user6818 want to know? The definition? What they are used for? Some particular aspect of how they are used? That is what we need to wait for clarification on. – David Zaslavsky♦ 21 hours ago
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closed as not constructive by David Zaslavsky♦ yesterday

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1 Answer

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Have you tried Peskin and Schroeder? It has two entries for operator mixing.

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Yeah..I have seen that but as usual I find Peskin and Schroeder's exposition always kind of disparate and can't use it for anything more than an occasional reference. I am looking for something more substantial and pedagogic. – user6818 Apr 26 '12 at 20:22
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How about Zinn-Justin? From memory there is a whole chapter devoted to it, or at least something more substantial than P&S. – Michael Brown Mar 22 at 11:51

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