Abstract
We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar 𝜙3 theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude \( {m}_n^1 \) (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.
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Salvatori, G. 1-loop amplitudes from the Halohedron. J. High Energ. Phys. 2019, 74 (2019). https://doi.org/10.1007/JHEP12(2019)074
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DOI: https://doi.org/10.1007/JHEP12(2019)074