Positive Geometries of S-matrix without Color

Abstract: In this note, we prove that the realization of associahedron discovered by Arkani-Hamed, Bai, He, and Yun (ABHY) is a positive geometry for tree-level S-matrix of scalars which have no color and which interact via cubic coupling. More in detail, we consider diffeomorphic images of the ABHY associahedron. The diffeomorphisms are linear maps parametrized by the right cosets of the Dihedral group on n elements. The set of all the boundaries associated with these copies of ABHY associahedron exhaust all the simple poles. We prove that the sum over the diffeomorphic copies of ABHY associahedron is a positive geometry and the total volume obtained by summing over all the dual associahedra is proportional to the tree-level S matrix of (massive or massless) scalar particles with cubic coupling. We then provide non-trivial evidence that the projection of the planar scattering forms parametrized by the Stokes polytope on these realizations of the associahedron leads to the tree-level amplitudes of scalar particles, which interact via quartic coupling. Our results build on ideas laid out in our previous works, leading to further evidence that a large class of positive geometries which are diffeomorphic to the ABHY associahedron defines an ``amplituhedron" for a tree-level S matrix of some local and unitary scalar theory. We also highlight a fundamental obstruction in applying these ideas to discover positive geometry for the one loop integrand when propagating states have no color.