1-loop amplitudes from the Halohedron

Abstract

A bstract We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar 𝜙 3 theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude $$ {m}_n^1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> (l, ..., n |1, ... , 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.