ϕp amplitudes from the positive tropical Grassmannian: Triangulations of extended diagrams

Abstract

The global Schwinger formula, introduced by Cachazo and Early as a single integral over the positive tropical Grassmannian, provides a way to uncover properties of scattering amplitudes which are hard to see in their standard Feynman diagram formulation. In a recent work, Cachazo and one of the authors extended the global Schwinger formula to general ϕp theories. When p=4, it was conjectured that the integral decomposes as a sum over cones which are in bijection with noncrossing chord diagrams, and further that these can be obtained by finding the zeroes of a piecewise linear function, H(x). In this note we give a proof of this conjecture. We also present a purely combinatorial way of computing ϕp amplitudes by triangulating a trivial extended version of noncrossing (p−2)-chord diagrams, called extended diagrams, and present a proof of the bijection between triangulated extended diagrams and Feynman diagrams when p=4. This is reminiscent of recent constructions using Stokes polytopes and accordiohedra. However, the ϕp amplitude is now partitioned by a new collection of objects, each of which characterizes a polyhedral cone in the positive tropical Grassmannian in the form of an associahedron or of an intersection of two associahedra. Moreover, we comment on the bijection between extended diagrams and double-ordered biadjoint scalar amplitudes. We also conjecture the form of the general piecewise linear function, Hϕp(x), whose zeroes generate the regions in which the ϕp global Schwinger formula decomposes into. Published by the American Physical Society 2024