# Anomalous dimensions in the $O(N)$ model

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• Is there any statement known about the anomalous dimensions of the $O(N)$ model in various dimensions and/or in the large-N limit?

• If a $\phi^4$ ("double-trace") term is coupled to an $O(N)$ model then is there an argument as to why this quartic term is ignorable?

[..I believe that there are analogous statements known for higher bosonic spin fields too - at least for the second question of mine..]

I would be happy to see some pedagogic references which hopefully derive these.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user user6818
The anomalous dimension $\eta$ is known in various limits, so it depends what you are interested in. In statistical field theories, $\eta=0$ for all $N$ if $d\geq 4$. It is also zero in large $N$ in all dimension, and its $1/N$ correction is known. There are results in perturbation theories in $\epsilon=4-d$ and $\bar \epsilon=d-2$ up to large order (7-loop near dimension $4$), etc. I can find references somewhere. But I don't understand what you mean by "double-trace". But maybe I didn't understand your question, because you tagged adS/CFT and all that, and I don't know what the connection.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user Adam
The very basic things can be found in Peskin-Schroder

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user John
@Adam By the "full fledged" scenario I mean the two famous generalizations of this model, one where the field is a matrix and the other where it is a symmetric traceless rank-s tensor. These two forms are very common in the current discussions of AdS/CFT

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user user6818
@user6818: I don't know anything about that. But if you still want results for the plain vanilla O(N) model, it should not be difficult to find some references, when you tell me in which cases you are interested in (dimension and/or value of N).

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user Adam
@Adam It would be great if you can give any reference which derives the $\eta$ results that you quoted. (...also any help about my second bullet point?..)

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user user6818
@user6818: Well, in d=3 (=2+1 in euclidean time), there is not much analytical results (only $\epsilon=1$, which need to be resummed numerically anyway). In the O(N) model, $\eta$ usually refers to the behavior of $\langle \phi(x)\phi(0)\rangle$, which in fourier behaves like $1/p^{2-\eta}$. Of course, every operator has a scaling dimension, but the anomalous dimension is usually this one. For analytical results, have a look at Zinn-Justin's book, at the equation I referred to.
@Adam Thanks for your reply. So what is the statement about the anomalous dimension of $\phi^2$?
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