The positive geometry for ${\varphi }^p$ interactions

Abstract

Starting with the seminal work of Arkani-Hamed et al arXiv:1711.09102, in arXiv:1811.05904, the "Amplituhedron program" was extended to analyzing (planar) amplitudes in massless ${\varphi }^4$ theory. In this paper we show that the program can be further extended to include ${\varphi }^p$ ($p>4$) interactions. We show that tree-level planar amplitudes in these theories can be obtained from geometry of polytopes called accordiohedron which naturally sits inside kinematic space. As in the case of quartic interactions the accordiohedron of a given dimension is not unique, and we show that a weighted sum of residues of the canonical form on these polytopes can be used to compute scattering amplitudes. We finally provide a prescription to compute the weights and demonstrate how it works in various examples.

arXiv comment37 pages

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Cited by in corpus (20)

• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• Singular Solutions in Soft Limits
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Amplituhedra, and Beyond
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Celebrating Loday's Associahedron
• Weights and recursion relations for ${\varphi }^p$ tree amplitudes from the positive geometry
• Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory
• Constraining the weights of Stokes Polytopes using BCFW recursions for Phi^4
• Weights, Recursion relations and Projective triangulations for Positive Geometry of scalar theories
• On Polytopes and Generalizations of the KLT Relations
• Gauge and Gravity Amplitudes on the Celestial Sphere
• Open associahedra and scattering forms
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• One-loop integrand from generalised scattering equations
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes
• BCFW like recursion for Deformed Associahedron