Positivity and Geometric Function Theory Constraints on Pion Scattering

TL;DR

This paper explores the connection between geometric function theory and $O(N)$ symmetric pion scattering amplitudes, deriving new dispersion relations, sum rules, and bounds on amplitude coefficients using positivity and geometric inequalities.

Contribution

It introduces fully crossing symmetric dispersion relations in the $z$-variable for $O(N)$ models and derives novel sum rules and bounds on scattering amplitudes.

Findings

Derived fully crossing symmetric dispersion relations for $O(N)$ amplitudes.

Established sum rules and locality constraints from these dispersion relations.

Obtained bounds on Taylor coefficients of pion scattering amplitudes.

Abstract

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with $O(N)$ global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the $z$-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for $O(N)$ model in $z$-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the $O(N)$ model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, $\pi^+\pi^-\to \pi^0\pi^0$) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.