Positivity and Geometric Function Theory Constraints on Pion Scattering

Understand

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.