$Φ^p$ 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 $φ^p$ 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 $φ^p$ 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 $φ^p$ 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, $H^{φ^p}(x)$, whose zeroes generate the regions in which the $φ^p$ global Schwinger formula decomposes into.

Figures (30)

• Figure 1: The space of planar metric trees with $n=5$ leaves.
• Figure 2: The global Schwinger parameterization for $n=5$, viewed as a projection of $\textrm{Trop}^+G(2,5)$, which unifies five 2-dimensional Schwinger integrals into a single 2-dimensional integral, parametrized by two tropical variables $x_1$ and $x_2$. The red lines on the plane define the domains where the tropical potential $F_5(x)$ becomes linear.
• Figure 3: Triangulation of an extended diagram for $n=16$ in $\phi^4$ and its associated tree. The triangulating chords are shown in red, and the colored regions correspond to different cubic subamplitudes.
• Figure 4: Non-crossing chord diagrams for $n=6$. On the right, the chord $\theta_{36}$ surrounds the chord $\theta_{45}$ and therefore the condition $x_0<x_1$ is imposed.
• Figure 5: Non-crossing chord diagrams for $n=8$. In the second diagram $\theta_{58}$ surrounds $\theta_{67}$ and, therefore, $x_2<x_3$. In the third diagram $\theta_{38}$ surrounds both $\theta_{45}$ and $\theta_{67}$ so we have $x_0<x_1$ and $x_0<x_3$. In the fourth diagram $\theta_{36}$ surrounds $\theta_{45}$, so we have $x_0<x_1$. Finally, in the fifth diagram $\theta_{38}$ surrounds $\theta_{47}$, which at the same time surrounds $\theta_{56}$, so $x_0<x_1<x_2$.

Theorems & Definitions (19)

• Definition 2.1
• Conjecture 2.2: Conjecture 6.2 of Cachazo:2022voc
• Definition 2.3
• Theorem 2.4
• proof
• Definition 2.5
• Definition 2.6
• Lemma 2.1
• proof
• Definition 2.7