Triangulations for ABHY Polytopes and Recursions for Tree and Loop Amplitudes

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. When projecting onto one facet, our recursion gives, e.g. "soft-limit triangulation" and "forward-limit triangulation" for tree and one-loop level. But we also find a lot of new formulae from our recursion relation, by projecting onto lower dimensional facets.