$\Phi^p$ Amplitudes from the Positive Tropical Grassmannian: Triangulations of Extended Diagrams

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 $\phi^p$ theories. When $p=4$, it was conjectured that the integral decomposes as a sum over cones which are in bijection with non-crossing chord diagrams, and further that these can be obtained by finding the zeroes of a piece-wise linear function, $H(x)$. In this note we give a proof of this conjecture. We also present a purely combinatorial way of computing $\phi^p$ amplitudes by triangulating a trivial extended version of non-crossing $(p-2)$-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 $\phi^p$ 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 piece-wise linear function, $H^{\phi^p}(x)$, whose zeroes generate the regions in which the $\phi^p$ global Schwinger formula decomposes into.