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.

arXiv commentv2:29 pages, 3 figures, version to appear in JHEP

References in corpus (43)

• Causality, Analyticity and an IR Obstruction to UV Completion
• Elastic $\pi \pi$ scattering to two loops
• The EFT-Hedron
• Positive Moments for Scattering Amplitudes
• New positivity bounds from full crossing symmetry
• A Mellin space approach to the conformal bootstrap
• Positivity Bounds for Scalar Theories
• Sharp Boundaries for the Swampland
• Conformal Bootstrap in Mellin Space
• Extremal Effective Field Theories
• Gravitational Effective Field Theory Islands, Low-Spin Dominance, and the Four-Graviton Amplitude
• Crossing Symmetric Dispersion Relations in QFTs
• Dispersive CFT Sum Rules
• Bootstrapping QCD: the Lake, the Peninsula and the Kink
• AdS Bulk Locality from Sharp CFT Bounds
• The Inflationary Wavefunction from Analyticity and Factorization
• Into the EFThedron and UV constraints from IR consistency
• On the Polyakov-Mellin bootstrap
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• Mellin space bootstrap for global symmetry
• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
• On critical exponents without Feynman diagrams
• Dispersion relations and exact bounds on CFT correlators
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Generalized positivity bounds on chiral perturbation theory
• Relative entropy in scattering and the S-matrix bootstrap
• Applications of dispersive sum rules: $\epsilon$-expansion and holography
• Quantum field theory and the Bieberbach conjecture
• Dual S-matrix Bootstrap I: 2D Theory
• On the Positive Geometry of Conformal Field Theory
• S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem
• Dispersion Formulas in QFTs, CFTs, and Holography
• Selection rules for the S-Matrix bootstrap
• Swampland Conditions for Higher Derivative Couplings from CFT
• Froissart bound for/from CFT Mellin amplitudes
• Scalar Fields Near Compact Objects: Resummation versus UV Completion
• The Low Energy Expansion for Pion-Pion Scattering and Crossing Symmetry in Dispersion Relations
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• Positive geometry in the diagonal limit of the conformal bootstrap
• On the N-pion extension of the Lovelace-Shapiro model
• Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values
• Charging Up the Functional Bootstrap
• The geometry of optimal functionals

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• Triple crossing positivity bounds for multi-field theories
• Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics
• QFT, EFT and GFT
• Bell inequalities in 2-2 scattering
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• Celestial insights into the S-matrix bootstrap
• NNLO Positivity Bounds on Chiral Perturbation Theory for a General Number of Flavours
• 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