Quantum field theory and the Bieberbach conjecture

TL;DR

This paper uncovers a novel connection between geometric function theory, specifically the Bieberbach conjecture, and non-perturbative scattering amplitudes, leading to new bounds on Wilson coefficients and scattering amplitudes validated across multiple theories.

Contribution

It establishes a surprising link between geometric function theory and quantum scattering amplitudes, deriving new inequalities and bounds applicable to various quantum field theories.

Findings

Derived inequalities analogous to Bieberbach conjecture in scattering amplitudes.

Established bounds on Wilson coefficients consistent with known theories.

Provided bounds on scattering amplitude magnitude valid in multiple models.

Abstract

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.