We build upon the prior works of Arkani et al. [J. High Energy Phys. 05 (2018) 096], Banerjee et al. [J. High Energy Phys. 08 (2019) 067], and Raman [arXiv:1906.02985] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, in which the interaction is given by λ 3 ϕ 3 þ λ 4 ϕ 4 , we show that a specific convex realization of a simple polytope known as the accordiohedron in kinematic space is the positive geometry for this theory. As in the previous cases, there is a unique planar scattering form in kinematic space, associated to each positive geometry which yields planar scattering amplitudes.

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We show that accordiohedra furnish polytopes which encode amplitudes for all massive scalar field theories with generic interactions. This is done by deriving integral formulas for the Feynman diagrams at the tree level and integrands at the one-loop level in the planar limit using the twisted intersection theory of convex realizations of the accordiohedron polytopes.