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Quantum field theory and the Bieberbach conjecture

by Parthiv Haldar, Aninda Sinha, Ahmadullah Zahed

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Submission summary

Authors (as registered SciPost users): Parthiv Haldar · Aninda Sinha · Ahmadullah Zahed
Submission information
Preprint Link: scipost_202106_00007v1  (pdf)
Date accepted: 2021-07-06
Date submitted: 2021-06-04 07:04
Submitted by: Haldar, Parthiv
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Complex Variables
  • High-Energy Physics - Theory
Approach: Theoretical


An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges' theorem) and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in one-loop $\phi^4$ theory, tree level string theory as well as in the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large $|s|$, fixed $t$, the upper bound reads $|\mathcal{M}(s,t)|\lesssim |s^2|$. We discuss how Szegö's theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.

Published as SciPost Phys. 11, 002 (2021)

List of changes

All the changes made to the draft in response to the various referee comments are written in blue colour to highlight them. The changes are as follows.

Changes in reference to referee report-1

1. It is true that the case pointed out by the referee is indeed possible. In fact, for our analysis, we really do not need strict positiveness for the absorptive part. We changed the statement of the lemma 3.1 to $\mathcal{A}$ being “non-negative” from “positive” as was before.

Changes in reference to referee report-3

1. In (1.5) $W_{1,0}$ has been changed to calligraphic $\mathcal{W}_{1,0}$.

2. On top of page 5, the typo of "equivalent" has been corrected to "univalent".

3. In reference to point 4, the coefficients $\omega_{j,0}$ are also bounded. In literature, their properties are treated under the name of logarithmic coefficients. We have added a brief account of these logarithmic coefficients and the bounds satisfied by them in section 2.2.1.

4. In reference to point 5, we have rewritten the discussion on the Szego theorem in section 2.5. In particular, we removed the confusing words and added a clearer explanation for delineating the content of the theorem.

5. Regarding point 6, the partial wave expansion converges for the given range of $a$ and $s_1$. The analyticity domain for the s-channel absorptive part was determined in the ref. [19] of the paper. Using that result, we could show that our domains of interest actually fall within the said analyticity domain. We have provided the details in the proof of lemma 3.1.

6. Regarding point 8, a clearer explanation added in the draft. We have expanded the comment before the previous eq 5.1. Hopefully, the derivation is now clear. The results in the draft are correct.

7. Regarding point 9, the caption has been corrected to contain “Black line is the exact answer.”

8. Regarding point 10, the captions for figure 5 and figure 6 have been updated as follows

Figure 5: “Bounds on amplitude, as in theorem (6.1), are satisfied by Tree level type II string amplitude and 1-loop $\phi^4$--amplitude.”
Figure 6: “Bounds on amplitude, as in theorem (6.1), are satisfied by Tree level type II string amplitude and 1-loop $\phi^4$-amplitude. These bounds on amplitude valid for complex $\tilde{z}$.”

Changes in reference to referee report-4

1. Regarding point 1, we have added the information to the draft.

2. Regarding point 2, for the given range of $a$ and $s_1$, the partial wave expansion converges. The analyticity domain for the s-channel absorptive part was determined in the ref. [19] of the paper. Using that result, we could show that our domains of interest actually fall within the said analyticity domain. We have provided the details in the proof of lemma 3.1.

3. Regarding point 5, we have changed “physical” to “analytic” and we have explained in footnote 9 what we mean by the term “analytic”.

Regarding typos
1. The "brunch cuts" -> "branch cuts" above equation (3.1) [ now page 10] : corrected.
2. "dispersion relations"-> "dispersion relation" above equation (3.3) [now page 10]: corrected
3. Regarding point c, we have added a line “(we will set $\mu=4m^2=4$ here)” in the paragraph following eq.(3.4)
4. Regarding point d, we have added the definition of $\alpha$ below equation (3.15), $\alpha=(d-3)/2$.

Reports on this Submission

Anonymous Report 2 on 2021-6-21 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202106_00007v1, delivered 2021-06-21, doi: 10.21468/SciPost.Report.3093


As discussed in the original report, the authors present their results in the context of other recent developments in the literature in a clear and well structured way. They review the literature, compare their bounds with previously known results and do a good job in summarizing and presenting future directions. Their methods are original and creative, and open up several research questions for the future.

In this resubmission, they do well in addressing questions from the reports.


I have no extra weakness to point out.


As reported originally, the authors, inspired by results in geometric function theory, derive a number of analytic bounds on the scattering amplitudes of (mostly) massive particles in higher dimensional QFTs. This is a new complementary approach to the higher dimensional S-matrix Bootstrap program and EFT program, and the results obtained by the authors provide a significant contribution to these fields. It also brings attention to important results in mathematics that could play a key role in future developments in the study of QFTs.

Their results raise several questions and open a number of future directions to be explored, such as "what is the role of univalent scattering amplitudes in QFT?". They derive and present their results in a clear, intelligible way. Their arguments are provided with sufficient detail, making the text instructive to the reader. The work is well motivated in the introduction, which provides a context for the results to be developed. Finally, the results and future directions are well summarized in the conclusions. Therefore, this work easily meets expectations and acceptance criteria for this journal.

Requested changes

There are no extra requested changes

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2021-6-20 (Invited Report)


I am satisfied with the response from the authors and the changes incorporated by them in the new version of the draft. I therefore recommend that the paper be published.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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