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HigherPoint Integrands in N=4 super YangMills 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  

Preprint Link:  scipost_202212_00058v1 (pdf) 
Date submitted:  20221221 20:40 
Submitted by:  Fleury, Thiago 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We compute the integrands of five, six, and sevenpoint correlation functions of twentyprime operators with general polarizations at the twoloop order in N=4 super YangMills theory. In addition, we compute the integrand of the fivepoint function at threeloop order. Using the operator product expansion, we extract the twoloop fourpoint function of one Konishi operator and three twentyprime 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 twistorspace reformulation of N=4 super YangMills 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 integrabilitybased hexagonalization approach for correlation functions.
Current status:
Reports on this Submission
Anonymous Report 2 on 202338 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202212_00058v1, delivered 20230308, doi: 10.21468/SciPost.Report.6870
Strengths
3  A link between twistor Feynman rules and graphical methods for the construction of higherloop integrands for halfBPS npoint 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 npoint functions of gauge invariant composite operators in N=4 SYM theory in four dimensions.
In the literature, integrands of socalled halfBPS operators have received much attention. In particular, for fourpoint 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 twoloop five, six and sevenpoint functions by the same method, and at three loops for the fivepoint 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 fourpoint integrands parityodd 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 parityodd part of the integrands, as has been done for example in preceding work by one of them and R. Pereira.
Second the referencing wrt. higherpoint and nonplanar "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 parityodd parts in the correlators are not given although they do not obviously integrate to zero, or provide these.
2  Omitted references to be inserted.
Anonymous Report 1 on 202333 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202212_00058v1, delivered 20230303, 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?
2Can 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 20230424 [id 3611]
(in reply to Report 1 on 20230303)
Dear referee,
Thanks a lot for your comments. Regarding the required changes:

We have added a new subsection 4.5 to the draft addressing your question. Shortly, for correlators of higher weights it is clear that nonplanar contributions start at lower loops because of nontrivial color factors (there are calculations in the literature). About the vanishing of 3 loop nonplanar 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 treelevel connected graphs) and the absence of nonplanar corrections for fourpoint functions is well stablished.

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 oneloop and our result is a twoloop one. On the comment we have also mentioned the oneloop integrability calculation of the same quantity.

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 CaronHuot, 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 higherweight operators using the twistor variables.

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 strongcoupling information. Recently there have been new attempts to understand the string theory using twistorlike variables. The twistors in this case are expected to become worldsheet 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 20230424 [id 3612]
(in reply to Report 2 on 20230308)Dear referee,
Thanks a lot for your comments. Regarding the required changes:
We have added a paragraph on parityodd 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 parityodd 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 oneloop and our result is a twoloop one. On the comment we have also mentioned the oneloop 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, “Colourdressed hexagon tessellations for correlation functions and nonplanar corrections,” [arXiv:1710.10212 [hepth].
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)