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^4ϕ4 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||s|, fixed tt, the upper bound reads |\mathcal{M}(s,t)|\lesssim |s^2||ℳ(s,t)|≲|s2|. 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.
Highlights
On the QFT side, we have focused on results motivated by the Bieberbach conjecture but which one can derive by using the crossing symmetric dispersion relation
We have examined a potentially remarkable correspondence between aspects of geometric function theory and quantum field theory
By no stretch of the imagination is our examination of the vast mathematics literature on univalent functions exhaustive
Summary
The Bieberbach conjecture is about how fast the Taylor expansion coefficients of a holomorphic univalent function, of a single complex variable z, on the unit disc (|z| < 1). This famous conjecture was put forth by Bieberbach in 1916 [1] and resisted a complete proof until 1985 when it was proved by de Branges [2]. At least around a ∼ 0, it is true that the amplitude, and not just the kernel in the crossing symmetric dispersion relation, is univalent. Various explorations associated with the main text providing the analysis with wholesomeness have been placed in the appendices
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