How tropical are seven- and eight-particle amplitudes?

Abstract

We study tropical Grassmanians Tr(k, n) in relation to cluster algebras, and assess their applicability to n-particle amplitudes for n = 7, 8. In N = 4 super Yang-Mills theory, we first show that while the totally positive part of Tr(4, 7) may encompass the iterated discontinuity structure of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too small for the Next-to-MHV helicity configuration. Then, using Tr(4, 8) we propose a finite set of 356 cluster A-coordinates expected to contain the rational symbol letters of the eight-particle MHV amplitude, and discuss how the remaining square-root letters may be obtained from limits of infinite mutation sequences. Finally, we use a triangulation of the totally positive part of Tr(3, 8) to obtain the associated generalised biadjoint scalar amplitude in a form containing a near-minimal amount of spurious poles.

Lifting heptagon symbols to functions

“…As indicated, there are really 14 such conditions, obtained by applying the dihedral group D 7 . Later these conditions were associated with edges in a certain "{a 1 , a 2 , a 3 } tropical fan" for the Grassmannian Gr(3,7) [52][53][54][55].…”

“…Actually, the conditions (7.1) are also obeyed by the NMHV amplitude through three loops, but are violated by the NMHV four-loop amplitude [54,55]. The violation starts in the 5-6 slot, and in that slot it has a simple form, in that the violations in all components are related to each other, where x and y are arbitrary letters.…”

“…There are further constraints deeper into the symbol, which can be described physically as extended Steinmann relations [43,44] or mathematically as cluster adjacency conditions [45][46][47][48][49][50]. Recently, patterns in the symbols of seven-and higher-point amplitudes have been associated with tropicalizations of the associated Grassmannian [51][52][53][54][55].…”

Symbol alphabets from plabic graphs

“…These naturally select certain finite subsets of cluster variables, including those in the eight-particle alphabet, and the square roots of four-mass box type make a natural appearance as well. At the same time, an equivalent but dual description, involving certain fans associated to the tropical totally positive Grassmannian [14], appeared simultaneously in [15,16]. Moreover [15] proposed a construction that precisely computes the 18 algebraic letters of the eight-particle symbol alphabet by (roughly speaking) analyzing how the simplest candidate fan is embedded within the (infinite) Gr (4,8) cluster fan.…”

“…Of course it would be very interesting to thoroughly study the numerous plabic graphs relevant to m = 4, n = 8 that have intersection number 1. In particular it would be interesting to see if they encode all 180 of the rational (i.e., Gr(4, 8) cluster variable) JHEP10(2020)128 symbol letters of [12], and whether they generate additional cluster variables such as those obtained from the constructions of [13,15,16].…”