Positivity and Geometric Function Theory Constraints on Pion Scattering

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.

v2:29 pages, 3 figures, version to appear in JHEP

References in corpus (15)

• Causality, Analyticity and an IR Obstruction to UV Completion
• Sharp Boundaries for the Swampland
• Conformal Bootstrap in Mellin Space
• Gravitational Effective Field Theory Islands, Low-Spin Dominance, and the Four-Graviton Amplitude
• AdS Bulk Locality from Sharp CFT Bounds
• The Inflationary Wavefunction from Analyticity and Factorization
• Into the EFThedron and UV constraints from IR consistency
• Quantum field theory and the Bieberbach conjecture
• S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem
• Dispersion Formulas in QFTs, CFTs, and Holography
• Swampland Conditions for Higher Derivative Couplings from CFT
• Scalar Fields Near Compact Objects: Resummation versus UV Completion
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• Charging Up the Functional Bootstrap
• The geometry of optimal functionals

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• QFT, EFT and GFT
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• Crossing antisymmetric Polyakov blocks + Dispersion relation
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• Non-Forward UV/IR Relations
• Locality and Analyticity of the Crossing Symmetric Dispersion Relation
• Positivity, low twist dominance and CSDR for CFTs
• Dispersion relations, knot polynomials and the $q$-deformed harmonic oscillator
• Positivity bounds at one-loop level: the Higgs sector
• A Celestial Route to AdS Bulk Locality
• Bootstrapping leading hadronic muon anomaly