# Looking for references about “operator mixing” (and anomalous dimensions)

I am looking for some pedagogic references 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!..}

-

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

## 1 Answer

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

-

1
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 at 11:51
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9595924019813538, "perplexity_flag": "middle"}