On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms

In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3, 4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2, 5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra. We then use these kinematic space geometric constructions to write world-sheet forms for 𝜙4 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint 𝜙3 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain $$ \frac{n-4}{2} $$ forms on the world-sheet and $$ \frac{n-4}{2} $$ forms on ABHY associahedron that generate quartic amplitudes.