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1-loop Amplitudes from the Halohedron

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 $m^1_n(1,\dots,n|1,\dots,n)$ 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.