Quantum field theory and the Bieberbach conjecture

Understand

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.