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Quantum field theory and the Bieberbach conjecture

Haldar, P and Sinha, A and Zahed, A (2021) Quantum field theory and the Bieberbach conjecture. In: SciPost Physics, 11 (1).

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Official URL: https://doi.org/10.21468/SCIPOSTPHYS.11.1.002


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 �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 |M(s, t)| ® |s2|. 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. Copyright P. Haldar et al.

Item Type: Journal Article
Publication: SciPost Physics
Publisher: SciPost Foundation
Additional Information: The copyright for this article belongs to Authors
Department/Centre: Division of Physical & Mathematical Sciences > Centre for High Energy Physics
Date Deposited: 20 Nov 2021 11:35
Last Modified: 20 Nov 2021 11:35
URI: http://eprints.iisc.ac.in/id/eprint/69891

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