Positivity and geometric function theory constraints on pion scattering

A bstract 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, π + π − → π 0 π 0 ) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.