Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes

Understand

In this note we make a field-theoretical derivation of a series of new recursion relations by a one-parameter deformation of kinematic variables for tree and one-loop amplitudes of bi-adjoint $\phi^3$ theory.

• Tree amplitudes are given by canonical forms/functions of associahedra realized in kinematic space by Arkani-Hamed, Bai, He and Yan (ABHY); the construction has been extended to generalized associahedra, where type $\mathcal{B}$/$\mathcal{C}$ polytopes compute tadpole diagrams and type $\mathcal{D}$ polytopes compute one-loop planar $\phi^3$ amplitudes.
• The new recursions are natural generalizations of the formula we found in 1810.08508, and are shown to work for all "C-independent" ABHY polytopes.
• Geometrically, the formula indicates triangulation of the generalized associahedron by projecting the whole polytope onto its boundary determined by the deformation.