An Etude on Recursion Relations and Triangulations

TL;DR

This paper develops a recursion relation for tree-level scattering amplitudes in bi-adjoint phi^3 theory, connecting geometric structures called associahedra with amplitude calculations, and provides explicit all-multiplicity formulas.

Contribution

It introduces a novel recursion based on kinematic deformations and offers explicit triangulations of the associahedron for all multiplicities.

Findings

Derived a recursion relation for bi-adjoint phi^3 amplitudes.

Provided explicit all-multiplicity formulas using associahedron triangulations.

Connected geometric structures to amplitude calculations in a new way.

Abstract

Following~\cite{Arkani-Hamed:2017thz}, we derive a recursion relation by applying a one-parameter deformation of kinematic variables for tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. The recursion relies on properties of the amplitude that can be made manifest in the underlying kinematic associahedron, and it provides triangulations for the latter. Furthermore, we solve the recursion relation and present all-multiplicity results for the amplitude: by reformulating the associahedron in terms of its vertices, it is given explicitly as a sum of "volume" of simplicies for any triangulation, which is an analogy of BCFW representation/triangulation of amplituhedron for ${\cal N}=4$ SYM.