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Positivity, low twist dominance and CSDR for CFTs
by Agnese Bissi, Aninda Sinha
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Aninda Sinha |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2209.03978v3 (pdf) |
| Date accepted: | 2023-02-09 |
| Date submitted: | 2022-12-02 05:06 |
| Submitted by: | Sinha, Aninda |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider a crossing symmetric dispersion relation (CSDR) for CFT four point correlation with identical scalar operators, which is manifestly symmetric under the cross-ratios $u,v$ interchange. This representation has several features in common with the CSDR for quantum field theories. It enables a study of the expansion of the correlation function around $u=v=1/4$, which is used in the numerical conformal bootstrap program. We elucidate several remarkable features of the dispersive representation using the four point correlation function of $\Phi_{1,2}$ operators in 2d minimal models as a test-bed. When the dimension of the external scalar operator ($\Delta_\sigma$) is less than $\frac{1}{2}$, the CSDR gets contribution from only a single tower of global primary operators with the second tower being projected out. We find that there is a notion of low twist dominance (LTD) which, as a function of $\Delta_\sigma$, is maximized near the 2d Ising model as well as the non-unitary Yang-Lee model. The CSDR and LTD further explain positivity of the Taylor expansion coefficients of the correlation function around the crossing symmetric point and lead to universal predictions for specific ratios of these coefficients. These results carry over to the epsilon expansion in $4-\epsilon$ dimensions. We also conduct a preliminary investigation of geometric function theory ideas, namely the Bieberbach-Rogosinski bounds.
Author comments upon resubmission
We have incorporated the following changes in response to the referee comments.
Referee 1:
Referee 1 had accepted our paper and had some minor changes suggested. We have incorporated both. We have split eq 2.4 and have added an appendix A to address these suggestions. We have also added appropriate references.
Referee 2:
i) To clarify what we mean by LTD we have added a paragraph on pg 6. This should prevent point that the referee had confusions with.
ii) As suggested by the referee, we have looked at the tail contribution to the Taylor coefficients using the OPE convergence results. This has led to appendix B and a new plot in appendix B.
iii) The referee could not see how the present approach could lead to any new insights for situations where the correlator was not already fully known. To prevent this misimpression, we have added appendices D.1 and D.2. Here we have illustrated how the locality constraints arising from the CSDR can be gainfully used to get insights about the OPE coefficients in situations where the full correlator is not known.
iv) Considering the new analyses in points ii and iii, we have made some minor rewording throughout the text.
We do wish to emphasize that our approach was to point out a different crossing symmetric representation which seems to be better suited to address the observations in section 2 and exhibited better convergence properties than the s-channel OPE expansion. We do believe that it is a worthwhile enterprise to try to understand properties of known CFT correlators and if there are unifying explanations for such properties. In our opinion, even once you have the answer from somewhere, it is still worthwhile to understand the mathematical structure of the answer to see if there is a unifying physical picture that explains these properties (which for instance may potentially lead to a better representation). In the S-matrix bootstrap literature, there is a plethora of papers (some of which have already appeared in SciPost) which follow this philosophy, and our approach was very much in that spirit. Nevertheless, the referee’s points are well taken, and we have done our utmost best to address them.
We hope that with the new additions, the paper can be accepted for publication.
List of changes
Referee 1:
Referee 1 had accepted our paper and had some minor changes suggested. We have incorporated both. We have split eq 2.4 and have added an appendix A to address these suggestions. We have also added appropriate references.
Referee 2:
i) To clarify what we mean by LTD we have added a paragraph on pg 6. This should prevent point that the referee had confusions with.
ii) As suggested by the referee, we have looked at the tail contribution to the Taylor coefficients using the OPE convergence results. This has led to appendix B and a new plot in appendix B.
iii) The referee could not see how the present approach could lead to any new insights for situations where the correlator was not already fully known. To prevent this misimpression, we have added appendices D.1 and D.2. Here we have illustrated how the locality constraints arising from the CSDR can be gainfully used to get insights about the OPE coefficients in situations where the full correlator is not known.
iv) Considering the new analyses in points ii and iii, we have made some minor rewording throughout the text.
Published as SciPost Phys. 14, 083 (2023)
Reports on this Submission
Report
The resubmitted paper has been considerably improved . The Authors
have taken into account all my remarks
and in the added appendices they have answered, at least in part, to the remarks of the other Referee. In my opinion this paper could be now published on SciPost.