1-loop Amplitudes from the Halohedron

Abstract

We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar ${\varphi }^3$ theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude $m_n^1(1,\dots ,n|1,\dots ,n)$ from the canonical form of an Halohedron which lives in an abstract space. This space is just a step away from ordinary kinematical space at 1-loop, because it is composed by abstract variables associated to propagators of 1-loop Feynman diagrams. Such variables, however, are unbound from momentum conservation relations that would give problems such as double poles. As an application of our construction, we exploit a well known recursion formula for the canonical form of a polytope in order to produce an expression for the 1-loop integrand which would not be evident starting from Feynman diagrams.

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• The Amplituhedron
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• Scattering Equations and KLT Orthogonality
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Positive Geometries and Canonical Forms
• Scattering Amplitudes from Intersection Theory
• One-loop amplitudes on the Riemann sphere
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Scattering in Three Dimensions from Rational Maps
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• Scattering Equations: Real Solutions and Particles on a Line
• Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces
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• Propagators, BCFW Recursion and New Scattering Equations at One Loop
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• On-Shell Electroweak Sector and the Higgs Mechanism
• The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Cosmological Polytopes and the Wavefuncton of the Universe for Light States
• From the flat-space S-matrix to the Wavefunction of the Universe
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• Amplituhedra, and Beyond
• Stokes Polytopes and Intersection Theory
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Celebrating Loday's Associahedron
• Emergence of Unitarity and Locality from Hidden Zeros at One-Loop Order
• Open associahedra and scattering forms
• Emergent unitarity from the amplituhedron
• One-loop integrand from generalised scattering equations
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
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• Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars
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• A Note on Convex Realization of Halohedron
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