Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 016, 44 pages      arXiv:2212.11243      https://doi.org/10.3842/SIGMA.2024.016

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

Freddy Cachazo a, Nick Early b and Yong Zhang a
a) Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
b) Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Received August 25, 2023, in final form February 04, 2024; Published online February 21, 2024

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.

Key words: generalized color orderings; generalized Feynman diagrams; generalized decoupling identities; CEGM integral formulas; generalized biadjoint amplitudes.

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