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Higher-Point Integrands in N=4 super Yang-Mills Theory
by Till Bargheer, Thiago Fleury, Vasco Gonçalves
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Submission summary
| Authors (as registered SciPost users): | Till Bargheer · Thiago Fleury |
| Submission information | |
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| Preprint Link: | scipost_202212_00058v1 (pdf) |
| Date submitted: | 2022-12-21 20:40 |
| Submitted by: | Fleury, Thiago |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We compute the integrands of five-, six-, and seven-point correlation functions of twenty-prime operators with general polarizations at the two-loop order in N=4 super Yang-Mills theory. In addition, we compute the integrand of the five-point function at three-loop order. Using the operator product expansion, we extract the two-loop four-point function of one Konishi operator and three twenty-prime operators. Two methods were used for computing the integrands. The first method is based on constructing an ansatz, and then numerically fitting for the coefficients using the twistor-space reformulation of N=4 super Yang-Mills theory. The second method is based on the OPE decomposition. Only very few correlator integrands for more than four points were known before. Our results can be used to test conjectures, and to make progresses on the integrability-based hexagonalization approach for correlation functions.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-3-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00058v1, delivered 2023-03-08, doi: 10.21468/SciPost.Report.6870
Strengths
3 - A link between twistor Feynman rules and graphical methods for the construction of higher-loop integrands for half-BPS n-point functions in N=4 SYM theory.
Weaknesses
There is no clear exposition of the twistor computation beyond the definitions, presumably because of want of space?
Report
The authors analyse n-point functions of gauge invariant composite operators in N=4 SYM theory in four dimensions.
In the literature, integrands of so-called half-BPS operators have received much attention. In particular, for four-point functions of this type the integrand has been constructed to high loop orders by the method of Lagrangian insertion. Building on experience from Feynman diagram computations the higher loop integrands were constructed on grounds of conformal symmetry and graph theory.
The authors construct ansaetze for the two-loop five-, six- and seven-point functions by the same method, and at three loops for the five-point case. Typically this leaves an array of unknown coefficients, and the idea is to fix these comparing to diagrammatic computations in terms of twistor Feynman rules. It would be nice to have at least some detail of that computation, though I do not require amendments in that respect in the understanding that the authors proceeded carefully and that the computations are very large.
However, in the aforementioned four-point integrands parity-odd parts could be ignored because they must integrate to zero in exactly four dimensions. For five points and more this is not obviously the case. The authors should highlight this fact in a prominent place, or provide/exclude the parity-odd part of the integrands, as has been done for example in preceding work by one of them and R. Pereira.
Second the referencing wrt. higher-point and non-planar "integrability" results is unfair.
Finally, wrt. the KOOO computation by OPE means I would suggest to insert a comment on 0104016.
In summary, a good and useful article in which these points at least should be put all right. I recommend the article for publication in SciPost subject to these corrections.
Requested changes
1 - Highlight the fact that parity-odd parts in the correlators are not given although they do not obviously integrate to zero, or provide these.
2 - Omitted references to be inserted.
Report #1 by Anonymous (Referee 1) on 2023-3-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202212_00058v1, delivered 2023-03-03, doi: 10.21468/SciPost.Report.6841
Strengths
1- clearly written
2- a clear statement of the results
3- ancillary .nb file
Weaknesses
1- technical paper
Report
The paper is well written and the results are clearly specified. They are not very general, but they are interesting since they are not easily or straightforwardly attainable with any other method. I would recommend the publication of the paper after some changes/comments have been addressed.
Requested changes
1- On page 4 the authors claim that up to 3 loops non planar contributions are absent. Do you have an understanding of this issue for correlators of higher weights? Do you have a physical interpretation for this?
2-Can you compare the results that you got for the correlator of the konishi operator and the 20' operator with the results presented in the paper arxiv:0104016?
3- is it possible to perform the same analysis of the paper on operators with different weights? what are the difficulties?
4- can the twistor approach be used also at strong coupling?
Author: Thiago Fleury on 2023-04-24 [id 3611]
(in reply to Report 1 on 2023-03-03)
Dear referee,
Thanks a lot for your comments. Regarding the required changes:
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We have added a new subsection 4.5 to the draft addressing your question. Shortly, for correlators of higher weights it is clear that non-planar contributions start at lower loops because of non-trivial color factors (there are calculations in the literature). About the vanishing of 3 loop non-planar contributions for 20’ operators, we do not have a rigorous argument, but it is reasonable that this statement is independent of n because a handle has to connect the inside and the outside of the cyclic graphs (the only possible tree-level connected graphs) and the absence of non-planar corrections for four-point functions is well stablished.
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We added the reference 0104016 and a comment at the end of Section 5. It is not possible to compare, because the result of 0104016 is one-loop and our result is a two-loop one. On the comment we have also mentioned the one-loop integrability calculation of the same quantity.
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It is possible to do the same analysis for different weights with a bootstrap approach, where the only difficulty is the increased size of the ansatz. In an upcoming work by Caron-Huot, Coronado and Mulhmann, it is shown, among other things, how to perform the computation for any weight using the twistor method. As cited in the paper, it is already known in the literature how to construct the external higher-weight operators using the twistor variables.
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We used the twistor approach only perturbatively in this work. The action in terms of the twistor variables was known in the literature, together with the set of Feynman rules. It works as a perturbative QFT where it is hard to extract strong-coupling information. Recently there have been new attempts to understand the string theory using twistor-like variables. The twistors in this case are expected to become world-sheet variables, and it is very different from our case. This is still work in progress.
In addition to the comments above, we have added several new footnotes. We hope that our manuscript can now be accepted for publication.
Sincerely,
Thiago (on behalf of the authors)
Author: Thiago Fleury on 2023-04-24 [id 3612]
(in reply to Report 2 on 2023-03-08)Dear referee,
Thanks a lot for your comments. Regarding the required changes:
We have added a paragraph on parity-odd terms at the end of the introduction. In addition, we have added the reference 1007.3246 (B. Eden, G. Korchemsky and E. Sokatchev; from correlation functions to scaterring amplitudes) where parity and the fate of the parity-odd part within the N=2 approach is discussed in the Appendix A. The paragraph highlights that the parity odd terms are total derivatives because they are generated by the topological term iFF˜ present in the chiral Lagrangian.
We added the reference 0104016 and a comment at the end of Section 5. It is not possible to compare, because the result of 0104016 is one-loop and our result is a two-loop one. On the comment we have also mentioned the one-loop integrability calculation of the same quantity. In addition to this reference we have added the following integrability reference: B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, “Colour-dressed hexagon tessellations for correlation functions and non-planar corrections,” [arXiv:1710.10212 [hep-th].
In addition to the comments above, we have added several new footnotes and the new subsection 4.5. We hope that our manuscript can now be accepted for publication. Sincerely, Thiago (on behalf of the authors)