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 planar $$ \mathcal{N} $$ N = 4 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.

Symbol alphabets from tensor diagrams

“…In such a case our algorithm would suggest associating W to the facet F , when it might be more appropriate to associate D instead. The fact that we have not encountered any apparent inconsistency in our calculations for C † (4, 9), which furthermore are corroborated by the independent work of [43], suggests that such worries may be postponed to higher (k, n), if not indefinitely.…”

“…We thank N. Henke for independently corroborating our results for C † (4, 9); see[43] 9. This data, which is the "raw input" to our computation, was first computed in[11] for(3,6) and (3, 7) (see also[44]),[7,45] for(3,8),[45] for(3,9),[7-9, 15, 45] for (4, 8),[46] for(3,10) and[45] for(4,9) 10.…”

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 higherpoint 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].…”

“…30 , W 33 , W 35 , W 36 , W 38 } will produce new factors, thus for bootstrapping Ω L , we only need the other 33 rational letters. It is remarkable that this is exactly the smaller rational alphabet obtained from the parity-invariant G + (4, 8)/T as Number of free parameters left after using constrains on the left column for bootstrapping Ω 3 .…”