1-loop Amplitudes from the Halohedron

TL;DR

This paper proves that the Halohedron geometric object encodes the 1-loop integrand for planar $\

Contribution

It demonstrates how the Halohedron's canonical form can be used to derive 1-loop integrands, providing a new geometric perspective beyond traditional Feynman diagrams.

Findings

Halohedron encodes 1-loop integrand for planar $\

Abstract

We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar $\phi^3$ theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude $m^1_n(1,\dots,n|1,\dots,n)$ 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.