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Quantum field theory and the Bieberbach conjecture

by Parthiv Haldar, Aninda Sinha, Ahmadullah Zahed

Submission summary

As Contributors: Parthiv Haldar
Arxiv Link: (pdf)
Date submitted: 2021-04-30 11:09
Submitted by: Haldar, Parthiv
Submitted to: SciPost Physics
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 Szeg\"{o}'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.

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Submission 2103.12108v3 on 30 April 2021

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