Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

Abstract

We further exploit the relation between tropical Grassmannians and Gr(4, n) cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in N = 4 planar super Yang-Mills theory at higher multiplicity n ≥ 8. As a mathematical foundation that provides access to square-root symbol letters in principle for any n, we analyse infinite mutation sequences in cluster algebras with general coefficients. First specialising our analysis to the eight-particle amplitude, and comparing it with a recent, closely related approach based on scattering diagrams, we find that the only additional letters the latter provides are the two square roots associated to the four-mass box. In combination with a tropical rule for selecting a finite subset of variables of the infinite Gr(4, 9) cluster algebra, we then apply our results to obtain a collection of 3, 078 rational and 2, 349 square-root letters expected to appear in the nine-particle amplitude. In particular these contain the alphabet found in an explicit 2-loop NMHV symbol calculation at this multiplicity.

Truncated cluster algebras and Feynman integrals with algebraic letters

“…for one-mass heptagon kinematics with n = 8, our method gives a co-dimension 2 boundary of G + (4, 8)/T which has 100 + 1 facets, where we have 100 g-vectors and 1 limit ray (the subset from differential operators of [49] is smaller). Since the computation for G + (4, n)/T cluster algebra becomes very difficult beyond n = 8 (there are recent results for n = 9 using a subset of all Plücker coordinates [30]), it is crucial to develop both methods for studying higher-point DCI integrals. It is also an interesting mathematical problem to systematically classify the boundaries of G + (4, n)/T (see [28]) and study their relevance for Feynman integrals.…”

“…This means that in addition to the truncation, new ingredients are needed in the context of Grassmanian cluster algebras to explain these and more algebraic letters. A solution to both problems has been proposed using tropical positive Grassmannian [4] and related tools for n = 8 [24][25][26][27][28][29] and very recently for n = 9 [30,31] 2 . Another method for explaining the alphabet has been proposed using Yangian invariants or the associated collections of plabic graphs [39][40][41][42].…”

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.…”

Comments on all-loop constraints for scattering amplitudes and Feynman integrals

“…There are 70 allowed unfrozen letters before 1389 and 71 before 1489 . As discussed in [41,42] using the tropical Grassmannian, naively there can be thousands of rational letters before each last entry. Our computation, however, drastically reduces this number, showing that the possible pairs are in fact very limited.…”

“…As already seen for one-loop N 2 MHV, amplitudes with n ≥ 8 generally involve letters that cannot be expressed as rational functions of Plücker coordinates of the kinematics G(4, n)/T ; more non-trivial algebraic letters appear in computations based on Q equations [31] for two-loop NMHV amplitudes for n = 8 and n = 9 [32,33], requiring extension of Grassmannian cluster algebras to include algebraic letters. Solutions to both problems has been proposed using tropical positive Grassmannian [34] and related tools for n = 8 [35][36][37][38][39][40] and n = 9 [41,42], as well as using Yangian invariants or the associated collections of plabic graphs [43][44][45][46]. On the other hand, N = 4 SYM has been an extremely fruitful laboratory for the study of Feynman integrals (c.f.…”