Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators

TL;DR

The paper advances CHY-based one-loop amplitudes by deriving Kleiss-Kuijf relations for Parke-Taylor factors, uncovering non-planar one-loop factors and introducing non-planar CHY-graphs that efficiently encode subleading contributions. It validates the construction in the massless bi-adjoint $\Phi^3$ theory, establishing a CHY framework that reproduces quadratic propagators and decomposes amplitudes into planar, non-planar, and mixed sectors. Explicit 4- and 5-point examples illustrate butterfly-like non-planar CHY-graphs and reveal an intersection structure with planar results, while general N-point results provide a scalable non-planar CHY formulation. The work suggests broad implications for Yang–Mills, gravity, and BCJ-type dualities at one loop and motivates extending the formalism to higher loops and alternative loop-orderings.

Abstract

In this work we have studied the Kleiss-Kuijf relations for the recently introduced Parke-Taylor factors at one-loop in the CHY approach, that reproduce quadratic Feynman propagators. By doing this, we were able to identify the non-planar one-loop Parke-Taylor factors. In order to check that, in fact, these new factors can describe non-planar amplitudes, we applied them to the bi-adjoint $Φ^3$ theory. As a byproduct, we found a new type of graphs that we called the non-planar CHY-graphs. These graphs encode all the information for the subleading order at one-loop, and there is not an equivalent of these in the Feynman formalism.

Figures (5)

• Figure 1: Unitary cut on a two-loop diagram
• Figure 2: Vertex Color code in CHY-graphs for the $\Lambda-$algorithm.
• Figure 3: Edges Color code in CHY-graphs for the $\Lambda-$algorithm.
• Figure 4: Standard Feynman three-vertex in the bi-adjoint theory.
• Figure 5: Irreducible contribution to the one-loop four-point amplitude.