Quantum field theory and the Bieberbach conjecture

Journal Article · Thu Jul 08 00:00:00 EDT 2021 · SciPost Physics

DOI:https://doi.org/10.21468/scipostphys.11.1.002· OSTI ID:2280773

Haldar, Parthiv ^[1]; Sinha, Aninda ^[2]; Zahed, Ahmadullah ^[2]

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1. Indian Institute of Science, Bangalore (India); Stony Brook University
2. Indian Institute of Science, Bangalore (India)

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 Φ⁴ 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.

• Research Organization: Stony Brook University, NY (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: FG02-88ER40388
• OSTI ID: 2280773
• Journal Information: SciPost Physics, Journal Name: SciPost Physics Journal Issue: 1 Vol. 11; ISSN 2542-4653
• Publisher: SciPost Foundation
• Country of Publication: United States
• Language: English

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