The geometry of multiloop Feynman Integrals from the Scattering Facet
The Scattering Facet (SF) of the Cosmological Polytope is a positive geometry defined from a Feynman diagram and encoding its combinatorics. Its canonical differential form reproduces the result of the integration over the energy components of loop momenta in scalar Feynman integrals, and is therefore closely related to the Loop-Tree Duality (LTD). In this article, we present a novel description of the SF as a base-fiber geometry. The fibers are a family of polytopes called deformed graphical zonotopes, which have a natural interpretation in terms of the causal evolution of the underlying Feynman diagram, and capture interesting information on scattering amplitudes. This decomposition of the SF induces a description of the canonical form leading to the so-called causal representation. Moreover, it provides new mathematical tools that allow us to characterize all its regular triangulations. Canonical forms obtained from these triangulations correspond to spanning tree representations, matching the original LTD amplitudes expressed as a sum over spanning trees. This geometrical viewpoint provides us with new methods to obtain the causal and spanning tree representations, without the need of performing any integration.