What is the meaning of the concepts of "operator mixing" (and anomalous dimensions)

Question

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!..}

This post imported from StackExchange Physics at 2016-09-07 14:42 (UTC), posted by SE-user user6818

Comments

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).

This post imported from StackExchange Physics at 2016-09-07 14:42 (UTC), posted by SE-user Vibert

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I'm not competent to answer the question properly. It seems like operator mixing is about extra operators introduced by renormalization counterterms, and you have a square matrix of RGE parameters because it's really about a set of operators, where all the others arise as counterterms when you start with any one of them (so the entry Mij in the matrix would be, parameter pertaining to counterterm j when you start with operator i) ... but I got all that from Scholarpedia.

This post imported from StackExchange Physics at 2016-09-07 14:42 (UTC), posted by SE-user Mitchell Porter

Answer 1

Anomalous dimensions and operator mixing can be best explained using the Wilson RGE and operator language (the discussion and notation mostly follows Chapter 23.6.2 of this book). To obtain the Wilson RGE one looks at the infinitesimal change of the action (or Lagrangian) due to integrating over an infinitesimally thin shell $d\Lambda$ of energy/momentum under the constraint that the generating functional (that describes the physics of the system)

\[Z[J] = \int\limits^{\Lambda}\mathcal{D}\phi\exp\left(i\int d^4x\sum\limits_nC_nO_n(\phi)\right)\]

does not changes by this infinitesimal RG transformation

\[\Lambda\frac{d}{d\Lambda}Z[J] = 0\]

This leads to a system of non-linear differential equations for the Wilson coefficients

\[\Lambda\frac{d}{d\Lambda}C_n = \beta_n(\{C_m\},\Lambda)\]

Linearizing this system of equations around a fixed point gives the linear differential equations (also called continuous RGEs)

\[\Lambda\frac{d}{d\Lambda} C_n = \gamma_{nm}C_m\]

where the matrix $\gamma_{mn}$ contains the so-called anomalous dimensions. They describe how the dimension of an operator deviates from its classical dimension $d_n$ (sometimes also called engineering dimensions).

Diagonalizing the matrix of anomalous dimensions $\gamma_{mn}$, the operators can be classified into irrelevant ($\lambda_n > 0$), relevant ($\lambda_n < 0$), and marginal ($\lambda_n = 0$) operators. Following the RG flow towards longer distances, the irrelevant operators lose their importance (they span the basin of attraction of the fixed point considered), the relevant operators grow in importance (they lead away from the fixed point around which the linearization is done), whereas the marginal operator neither grow not increase in first order (they potentially allow for cyclic behavior of the RG flow).

In principle, the classification into relevant, irrelevant, and marginal operators is only valid if the analysis of the fixed point is done in the eigenbasis of the matrix of anomalous dimensions. Otherwise, the non-zero off-diagonal elements lead to operator-mixing. This can lead to situations, where an operator classified as irrelevant by its classical scaling dimensions becomes important for the IR behavior of the system due to getting mixed with other relevant and marginal operators.

Also, if an eigenvalue of the anomalous dimension matrix has multiplicity >1, diagonalization may be impossible and the operators of the corresponding block remain mixed even after block-diagonalization.

Answer 2

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

This post imported from StackExchange Physics at 2016-09-07 14:42 (UTC), posted by SE-user Jay Bigman

Comments

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.

This post imported from StackExchange Physics at 2016-09-07 14:42 (UTC), posted by SE-user user6818

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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.

This post imported from StackExchange Physics at 2016-09-07 14:42 (UTC), posted by SE-user Michael Brown