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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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Submission summary
| Authors (as registered SciPost users): | Han Ma |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2107.05654v4 (pdf) |
| Date accepted: | 2023-07-27 |
| Date submitted: | 2023-07-07 16:18 |
| Submitted by: | Ma, Han |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We present the Wilsonian effective action as a solution of the exact RG equation for the critical $O(N)$ vector model in the large $N$ limit. Below four dimensions, the exact effective action can be expressed in a closed form as a transcendental function of two leading scaling operators with infinitely many derivatives. From the exact solution that describes the RG flow from a UV theory to the fixed point theory in the IR, we obtain the mapping between UV operators and IR scaling operators. It is shown that IR scaling operators are given by sums of infinitely many UV operators with infinitely many derivatives.
List of changes
1. We added 2,3,4 paragraphs in the Introduction to explain (1) what is already known in the literature, (2) what is new in our paper and (3) how the quantum RG formalism is improved from the earlier method based on collective fields.
2. We added the last paragraph in Sec.IV to point out our new finding on the exact relation between the two leading scaling operators at the Wilson-Fisher fixed point.
3. We added Sec.VI to compute the 1/N corrections in the effective action.
Published as SciPost Phys. 15, 111 (2023)
Reports on this Submission
Report
The authors have partially addressed my concern by adding some comments and references in the introduction. I was rather hoping in a more explicit comparison of their equation (12) (notice that it is in overflow, which should be corrected) to other known expressions of the effective action, or at least to a more standard notation. However, this is a minor point which does not affect the content of the paper.
Author: Han Ma on 2023-07-27 [id 3839]
(in reply to Report 1 on 2023-07-17)We thank the referee for the comment. Please find the note attached, where we have shown the explicit relation between our action in Eq.(9) (and consequently Eq.(12)) and previously used effective action in terms of collective field. And then we emphasize that it is our formulation which keeps track of the RG flow and gives the relation between UV and IR scaling operators.
Simply speaking, starting from Eq.(9), we can integrate over the UV field $\phi$ and insert an auxiliary collective field $\varphi$ to get the familiar action:
$S= \int d^D r \left[ (\partial_r\varphi_r )^2+ X_r \varphi_r^2 \right]$
But the resulting action loses information in the UV. On the contrary, our effective action is always expressed in terms of the UV field and it can be used to study the relation between UV and IR scaling operators.
Attachment:
reply.pdf