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Exact effective action for the O(N) vector model in the large N limit
by Han Ma, Sung-Sik Lee
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|Authors (as registered SciPost users):||Han Ma|
|Preprint Link:||https://arxiv.org/abs/2107.05654v3 (pdf)|
|Date submitted:||2023-03-01 16:22|
|Submitted by:||Ma, Han|
|Submitted to:||SciPost Physics|
We present the solution of the exact RG equation at the critical fixed point of the interacting O(N) vector model in the large $N$ limit. Below four dimensions, the exact effective action at the fixed point is a transcendental function of two leading scaling operators with infinitely many derivatives.
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In this paper the quantum RG formalism, developed earlier by one the authors, is applied to the O(N) model, for which the Wilsonian effective action is computed at large N.
On the positive side, I think that the quantum RG is an interesting reformulation of the Wilsonian RG, and as such it can provide a fresh look at old problems.
On the negative side, the formalism and notation is not the most immediate, and at the same time, there is the impression that not much new is achieved, therefore discouraging further readings on this formalism. Therefore, it would be important to compare their results to existing literature.
As stated in the summary, and stressed by the title, the main result of the paper is equation (12), which is a closed expression for the Wilsonian effective action for the vector O(N) model in the large N limit. From the way it is presented, it would seem that this is a completely new result achieved thanks to the quantum RG formalism, but in fact it is well known that for the O(N) model in the large N limit one can compute exactly the free energy with or without sources. This can be achieved in various ways, most famously by introducing the Hubbard-Stratonovich field (e.g. Coleman, Jackiw and Politzer, Phys.Rev.D 10 (1974) 2491, or many books and reviews), but also by changing variables to bilocal fields (as in Jevicki and Sakita, NuclearPhysicsB185 (1981) 89-100, similarly to the more recent treatment of the SYK model), or by the 2PI formalism of Cornwall, Jackiw and Tomboulis, Phys.Rev.D 10 (1974) 2428-2445 (see https://inspirehep.net/literature/659724 for a good review of its application to the O(N) model at large N). All these methods apply straightforwardly also to the O(N) model with a modified free propagator, such as that with UV and IR cutoffs used to derive the exact RG equations. And indeed equation (12), with its logarithm and squared tadpole terms, looks rather familiar, although with a slightly heavier notation.
Therefore, it would be very important to compare this results with previously known effective actions, and explain what, if anything, it is gained by the quantum RG formalism.
Before seeing such a comparative discussion I cannot recommend the manuscript for publication.
The authors computed and solved the exact RG equations of the large N O(N) vector model, using the quantum RG technique developed by one of the authors previously. The result in this paper is novel and interesting, and is a nice demonstration of the usage of the quantum RG method.
1. In the calculation, the authors used the large z limit which seems to me that they are looking at the RG flow that is close to a fixed point. Can the authors comment on the error in this approximation, and it what sense is the result exact?
2.Can the authors comment on the effects of 1/N corrections?