# What is the meaning of the concepts of “operator mixing” (and anomalous dimensions) [closed]

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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## closed as not a real question by Manishearth♦Jun 9 '13 at 19:12

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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 '13 at 16:31
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 May 21 '13 at 22:38
@Mitchell this doesn't ask for a definition, it asks for an explanation, but it is not very specific about what it wants explained. Among the downsides of leaving it open are (1) that people use this as a justification to ask other unfocused questions, and (2) any answers posted may no longer apply later when the question is improved, and (3) it diminishes the site's usefulness as a place to find answers to practical questions. – David Z May 22 '13 at 0:43
@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 Z May 22 '13 at 0:50
If I were a mod I would take the well founded from a physics point of view opinions, words, and needs of knowledgable about physics people here much more serious, and adapt the SE rules as best as I can (and this is allowed as Shog9 said meta.stackexchange.com/a/124915/184300) to make the site a good place for acadmics, researchers, and (univrsity) students of physics and astronomy, as the targetted audience is still described in the About) again . We have the freedom to make Physics SE a good place for this audience again, but our moderators make no use of this freedom. – Dilaton Jun 11 '13 at 12:13

## 1 Answer

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
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 '13 at 11:51