Generalized color orderings: CEGM integrands and decoupling identities

Abstract

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 CPk−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]∖{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≠j, and a pair of GCOs, one in X(k,|[n]∖{i}|) and the other in X(k−1,|[n]∖{j}|). We call them 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).