On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms

TL;DR

This paper extends the geometric understanding of quartic scalar interactions in quantum field theory by linking Stokes polytopes, associahedra, and worldsheet forms, providing new geometric and algebraic tools for amplitude calculations.

Contribution

It demonstrates that the positive geometry of quartic amplitudes can be described by Stokes polytopes embedded in associahedra and develops worldsheet forms for $$ theory using these geometric structures.

Findings

Planar amplitudes are captured by Stokes polytopes within associahedra.

Kinematic space realizations of Stokes polytopes as boundaries of associahedra.

Construction of worldsheet forms for $$ theory related to scattering equations.

Abstract

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 worldsheet forms for $\phi^{4}$ theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint $\phi^3$ scalar amplitudes, scattering equations can be used as diffeomorphisms between certain $\frac{n-4}{2}$ forms on the worldsheet and $\frac{n-4}{2}$ forms on ABHY associahedron that generate quartic amplitudes.