On Positive Geometry and Scattering Forms for Matter Particles

Understand

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.