MiS Preprint Repository

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of $SU(N)\times SU(\tilde{N})$. 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 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 $n\le 9$ 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 give a preview of a companion paper introducing new objects called i<>chirotopal tropical Grassmannians, their connection to generalized color orderings, and to CEGM integral formulas.