Minimal Kinematics: An All k and n Peek into Trop⁺G(k,n)

Abstract

In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all k and n on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of n points on CP k−1 has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all k generalization of the one recently introduced by Early for k = 2 and uses a choice of cyclic ordering. We conjecture an explicit formula for m (k) n (I, I) which we have checked analytically through n = 10 for all k. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph C (1,2,...,k−2) n. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian Tr + G(k, n) in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the 90 608 planar arrays for Tr + G(4, 8) recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for k = 2 and uses the planar basis recently introduced by Early for all k. Based on numerical computations through n = 8 for all k, we conjecture that on the planarbasis kinematics m (k) n (I, I) evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For k = 2 these are the standard Catalan numbers.

Symbol alphabets from tensor diagrams

“…Together, this was taken as evidence supporting the notion 1 The connection between cluster algebras and singularities of amplitudes could extend well beyond SYM theory [5,6]. 2 The very same polytopes also appear in the study of P k−1 -generalized scattering equations for biadjoint amplitudes [13][14][15][16].…”

Algebraic branch points at all loop orders from positive kinematics and wall crossing

“…The notion of degenerate scattering diagrams has applications beyond planar gauge theories, specifically higher loop integrands of φ 3 . However, it is instead the cluster polytope picture that is more interesting for studying higher loop integrands of φ 3 [114,115] and generalized scattering amplitudes [116][117][118][119][120][121][122]. Both the higher loop integrands of φ 3 and generalized scattering amplitudes can be identified with the canonical rational function of the (degenerate) cluster polytopes discussed in section 4 [105].…”

Notes on worldsheet-like variables for cluster configuration spaces

“…Higher-k Grassmannian stringy integrals have been studied in [3,30], whose leading orders are equivalent to the higher-k CHY formulas (or CEGM generalized bi-adjoint amplitudes) [12,31]. Tropical Grassmannian [32][33][34][35][36], matroid subdivisions [37][38][39][40] and the planar collections of Feynman diagrams [41][42][43] are also found to be very useful to study them. Compared to higher-k Grassmannians, all finite-type cluster algebras have better factorization behaviors.…”