Anomalous dimensions in the $O(N)$ model

Question

• 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

Comments

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

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@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...]

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

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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".

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

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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.

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

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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

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@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

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@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

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@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

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@John In Peskin-Schroeder the closest thing I can see is their equation 13.47 and 13.50. But I can't see how these results about $\eta$ that Adam is quoting can be derived from there.

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

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That's why I said "very basic things can be found in Peskin..." and gave other references. You can also look in arxiv.org/abs/hep-th/0306133 which is a summary

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

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@John Yeah..I have been planning to take a look at that review on my own anyway. Is there anyway you can see how PS's 13.47 and 13.50 can be used to get the results that Adam is quoting?..at least for the case of d=3+1...and any insights about the second bullet point?...

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

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@user6818: The results in $d=4$ are pretty trivial, since the fixed point is gaussian, and $\eta=0$ for all N. There might be some logarithmic correction to the correlation function though.

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

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@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

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@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

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@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

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@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

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@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