Scattering Amplitudes and Simple Canonical Forms for Simple Polytopes

We provide an efficient recursive formula to compute the canonical forms of arbitrary $d$-dimensional simple polytopes, which are convex polytopes such that every vertex lies precisely on $d$ facets. For illustration purposes, we explicitly derive recursive formulae for the canonical forms of Stokes polytopes, which play a similar role for a theory with quartic interaction as the Associahedron does in planar bi-adjoint $\phi^3$ theory. As a by-product, our formula also suggests a new way to obtain the full planar amplitude in $\phi^4$ theory by taking suitable limits of the canonical forms of constituent Stokes polytopes.