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arXiv · 2108.10355

Positivity and Geometric Function Theory Constraints on Pion Scattering

Ahmadullah Zahed
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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.

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Ahmadullah Zahed. 2021-08-23. Positivity and Geometric Function Theory Constraints on Pion Scattering. https://doi.org/10.1007/jhep12(2021)036

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