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

• 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.

• 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. – Adam Oct 19 '13 at 0:49
• @Adam Thanks for the reply. What do you men by "statistical field theory"? I am thinking of QFTs. It would be great if you can give a reference which hopefully derives these results about anomalous dimension that you quoted. [...the $\phi^4$ term is a special case of what would be called a double-trace term for a n-component or a matrix valued field..I guess my question about the $\phi ^4$ term is meaningful even when one doesn't need to generalize to the full-fledged scenario but this issue comes up quite often in the context of AdS/CFT as the full fledged double-trace term...] – user6818 Oct 19 '13 at 3:35
• By statistical field theory I mean a theory with euclidean signature, whose lagrangian is typically of the form $\sum_a[(\nabla\phi_a)^2+r\phi_a^2]+(\sum_a\phi_a^2)^2$, where $\phi_a$ is a $N$ component vector. As I said, maybe I'm not talking about the same thing than you (even though in QFT and SFT, O(N) theory usually describe the above lagrangian) as I (still) have no idea what you mean by "full-fledged scenario" and "AdS/CFT as the full fledged double-trace term". – Adam Oct 19 '13 at 5:43
• This is discussed in many places. Purely CFT works are by Lang and Ruehl. There is even a book by Kleinert - critical properties of $\phi^4$. You can also have a look at Zinn-Justin, Critical phenomena. – John Oct 19 '13 at 8:43
• @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 Oct 29 '13 at 3:39