Quantum field theory and the Bieberbach conjecture
Parthiv Haldar, Aninda Sinha, Ahmadullah Zahed
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> 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| <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false" form="prefix">|</mml:mo> </mml:mrow> </mml:math> , fixed t <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>t</mml:mi> </mml:math> , the upper bound reads |\mathcal{M}(s,t)|\lesssim |s^2| <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mstyle mathvariant="script"> <mml:mi>ℳ</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mo>≲</mml:mo> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:msup> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false" form="prefix">|</mml:mo> </mml:mrow> </mml:math> . 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.