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  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
asked Oct 18, 2013 in Theoretical Physics by user6818 (960 points) [ no revision ]
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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
@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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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
@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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