1-loop Amplitudes from the Halohedron

TL;DR

The paper proves that the Halohedron is the correct $1$-loop Amplituhedron for planar $ \, ext{φ}^3 \, $ theory by showing how the partial amplitude $m^1_n(1, obreak obreak obreak obreak ,n|1, obreak obreak ,n)$ can be extracted from the Halohedron's canonical form in an abstract kinematic space with propagator variables $X_I$. By relaxing momentum conservation (via an $ ext{e}_{I}$-limit) and then substituting $X_I o s_I$, the $1$-loop integrand for the planar bi-adjoint theory emerges; a recursion by triangulating the Halohedron yields explicit $n=4$ results, including spurious poles that cancel in the cyclic sum. The work provides a geometric framework for loop amplitudes and outlines paths to higher-loop generalizations and alternative triangulations, deepening the connection between positive geometries and quantum field theory amplitudes.

Abstract

We recently proposed the Halohedron to be the 1-loop Amplituhedron for planar $φ^3$ theory. Here we prove this claim by showing how it is possible to extract the integrand for the partial amplitude $m^1_n(1,\dots,n|1,\dots,n)$ from the canonical form of an Halohedron which lives in an abstract space. This space is just a step away from ordinary kinematical space at 1-loop, because it is composed by abstract variables associated to propagators of 1-loop Feynman diagrams. Such variables, however, are unbound from momentum conservation relations that would give problems such as double poles. As an application of our construction, we exploit a well known recursion formula for the canonical form of a polytope in order to produce an expression for the 1-loop integrand which would not be evident starting from Feynman diagrams.

Figures (9)

• Figure 1: The topology of bubble diagrams make so that double poles arise
• Figure 2: The sequence of truncations that produces the 3-dimensional Halohedron.
• Figure 3: The usual mutation swaps an s-channel for a t-channel in a subdiagram. The move is represented also in terms of arcs on the annulus
• Figure 4: The new mutation swaps IR and UV tadpoles, the latter are distinguished by coloring the tadpole in black. In terms of arcs on the annulus, the mutation swaps a cut arc with the UV arc.
• Figure 5: A succession of mutations on planar loop diagrams