Generalized Color Orderings: CEGM Integrands and Decoupling Identities

In a recent paper, we defined generalized color orderings (GCO) and Feynman diagrams (GFD) to compute color-dressed generalized biadjoint amplitudes. In this work, we study the Cachazo-Early-Guevara-Mizera (CEGM) representation of generalized partial amplitudes and ``decoupling" identities. This representation is a generalization of the Cachazo-He-Yuan (CHY) formulation as an integral over the configuration space $X(k,n)$ of $n$ points on $\mathbb{CP}^{k-1}$ in generic position. Unlike the $k=2$ case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for $k>2$. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all $(3,n<9)$ integrands, all $(3,9)$ integrands up to four undetermined constants, and $95 \%$ of $(4,8)$ integrands up to 24 undetermined constants. $k=2$ partial amplitudes are known to satisfy identities. Among them, the so-called $U(1)$ decoupling identities are the simplest ones. These are characterized by a label $i$ and a color ordering in $X(2,|[n]\setminus \{i\}|)$. Here we introduce decoupling identities for $k>2$ determined combinatorially using GCOs. Moreover, we identify the natural analog of $U(1)$ identities as those characterized by a pair of labels $i\neq j$, and a pair of GCOs, one in $X(k,|[n]\setminus \{i\}|)$ and the other in $X(k-1,|[n]\setminus \{j\}|)$. We call them {\it double extension} identities. We also provide explicit connections among different ways of representing GCOs, such as configurations of lines, configurations of points, and reorientation classes of uniform oriented matroids (chirotopes).