# 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
@Adam I guess Peskin-Schroeder gives the results for $3+1$. Can you kindly give a reference to all the many results that you quoted in your first comment?

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user user6818
@user6818: sure. For $d=3$, you can have a look at arXiv:1110.2665, table 1, where they quote quite a lot of different results. For dimension $4-\epsilon$, Zinn-Justin's book on critical phenomena gives $\eta$ up to three loops, chapter 28, equation 28.7. In $d=2+\bar \epsilon$, same book, equation 30.49. In large N, same book, equation 29.51.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user Adam
@Adam I couldn't locate much of anything in the arxv link of yours. It seems to be some simulation data and not any analytic result. By $\eta$ you mean the anomalous dimensions of $\phi$ and not $\phi^2$..right? I guess $\phi^2$ has an anomalous dimension ?( at least in large N at $d=3+1$?...though I am more interested in $d=2+1$...)[...I have been trying to learn the derivations from the Zinn-Justin-Moshe review...]

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.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user Adam
@Adam Thanks for your reply. So what is the statement about the anomalous dimension of $\phi^2$?

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user user6818
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