On Positive Geometry and Scattering Forms for Matter Particles

TL;DR

This paper explores positive geometry and scattering forms for matter particle amplitudes, introducing open associahedra polytopes and discussing color-kinematics duality in bi-color scalar theories.

Contribution

It introduces a recursive construction of open associahedra for matter amplitudes and extends positive geometry concepts to theories with fundamental matter particles.

Findings

Constructed open associahedra polytopes for bi-color scalar amplitudes.

Demonstrated the computation of amplitudes via canonical forms of these polytopes.

Explored the duality between color factors and wedge products for matter particles.

Abstract

We initiate the study of positive geometry and scattering forms for tree-level amplitudes with matter particles in the (anti-)fundamental representation of the color/flavor group. As a toy example, we study the bi-color scalar theory, which supplements the bi-adjoint theory with scalars in the (anti-)fundamental representations of both groups. Using a recursive construction we obtain a class of unbounded polytopes called open associahedra (or associahedra with certain facets at infinity) whose canonical form computes amplitudes in bi-color theory, for arbitrary number of legs and flavor assignments. In addition, we discuss the duality between color factors and wedge products, or "color is kinematics", for amplitudes with matter particles as well.