Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

TL;DR

This paper introduces a new framework for biadjoint scalar amplitudes using generalized color orderings derived from arrangements of projective lines, revealing novel identities and extending to higher-dimensional projective spaces.

Contribution

It proposes a natural generalization of color orderings leading to color-dressed amplitudes and explores their properties, including new phenomena and identities for up to nine points.

Findings

Discovery of generalized decoupling identities

Definition of colorless generalized scalar amplitudes

Extension of color orderings to arbitrary projective spaces

Abstract

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are "color" decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of "local planarity" into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the "colorless" generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.