Positivity and geometric function theory constraints on pion scattering

Ahmadullah Zahed

doi:10.1007/jhep12(2021)036

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, π+π−→ π0π0) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

Highlights

This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with O(N ) global symmetry
We ask here if such dispersion relation can be written for theories with O(N ) global symmetry? Do these correspondences exist for theories with global O(N ) symmetry? The answer turns out to be yes! We write down three sets of fully crossing symmetric dispersion relations for three specific combinations of isospin amplitudes
We have applied the techniques in geometric function theory, namely typically real functions to 2-2 scattering amplitudes with global O(N ) symmetry

Summary

Typically real functions

A schlicht function f (z) (normalized f (0) = 0 and f (0) = 1, see [4] for a quick overview), which is univalent inside the unit disk |z| < 1 with bp ∈ R, is typically real function. In |z| < 1 a regular function F (z) is typically real if and only if it has the Robertson representation:. The kernel H(s1; s2, s3) is a univalent typically real function. We will show below from these two facts that the full amplitude can be recast as Robertson representation; it is a typically real function. In [3] it was shown that F0(z, a) is a typically real function from the positivity of the Discs G0 s1; s(2+) (s1, a). We know from [4] that Kernel is a univalent function inside the unit disk.

Geometric function theory constraints on pion amplitudes

Bounds on Taylor coefficients of physical amplitudes

Summary and future directions

Inversion formulas and sum rules

Findings

The 2-loop chiral perturbation theory