# Accordiohedra as positive geometries for generic scalar field theories

## TL;DR

This paper demonstrates that the accordiohedron, a specific convex polytope, serves as the positive geometry encoding tree-level planar amplitudes for a scalar field theory with cubic and quartic interactions.

## Contribution

It introduces the accordiohedron as a new positive geometry for scalar field theories with polynomial interactions, extending the geometric approach to scattering amplitudes.

## Key findings

- The accordiohedron provides a geometric realization of planar scattering amplitudes.
- A unique planar scattering form is associated with the accordiohedron.
- The approach generalizes positive geometries to more complex scalar theories.

## Abstract

We build upon the prior works of [1-3] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction is given by $\lambda_3\phi^{3}\ +\lambda_4 \phi^{4}$, 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.