Triangular tessellations of one-loop scattering amplitudes in $ϕ^3$ theory

Inspired by the recent work of Nima Arkani Hamed and collaborators who introduced the notion of positive geometry to account for the structure of tree-level scattering amplitudes in bi-adjoint $ϕ^3$ theory, which led to one-loop descriptions of the integrands. Here we consider the one-loop integrals themselves in $ϕ^3$ theory. In order to achieve this end, the geometrical construction offered by Schnetz for Feynman diagrams is hereby extended, and the results are presented. The extension relies on masking the loop momentum variable with a constant and proceeding with the calculations. The results appear as a construction given in a diagrammatic manner. The significance of the resulting triangular diagrams is that they have a common side amongst themselves for the corresponding Feynman diagrams they pertain to. Further extensions to this mathematical construction can lead to additional insights into higher loops. A mathematica code has been provided in order to generate the final results given the initial parameters of the theory.
35 pages, 15 figures, 1 table, 4 mathematica notebooks given as ancillary files