Accordiohedra as positive geometries for generic scalar field theories

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 λ₃φ³+λ₄φ⁴, 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.