# The positive geometry for $\phi^{p}$ interactions

## TL;DR

This paper extends the positive geometry framework to include tree-level planar amplitudes for $\, ext{phi}^p$ interactions (p>4), using accordiohedra in kinematic space, and provides a method to compute the amplitudes through weighted residues.

## Contribution

It introduces accordiohedra as the geometric objects for $\, ext{phi}^p$ interactions and develops a residue-based method to compute scattering amplitudes.

## Key findings

- Tree-level amplitudes for $\, ext{phi}^p$ theories are derived from accordiohedron geometry.
- A weighted sum of residues on accordiohedra computes the amplitudes.
- The method is demonstrated with explicit examples.

## Abstract

Starting with the seminal work of Arkani-Hamed et al arXiv:1711.09102, in arXiv:1811.05904, the "Amplituhedron program" was extended to analyzing (planar) amplitudes in massless $\phi^{4}$ theory. In this paper we show that the program can be further extended to include $\phi^{p}$ ($p>4$) interactions. We show that tree-level planar amplitudes in these theories can be obtained from geometry of polytopes called accordiohedron which naturally sits inside kinematic space. As in the case of quartic interactions the accordiohedron of a given dimension is not unique, and we show that a weighted sum of residues of the canonical form on these polytopes can be used to compute scattering amplitudes. We finally provide a prescription to compute the weights and demonstrate how it works in various examples.