Two-loop octagons, algebraic letters and Q¯ equations

Abstract

We compute the symbol of the first two-loop amplitudes in planar N ¼ 4 SYM with algebraic letters, the eight-point Next to Maximally Helicity Violating (NMHV) amplitude (or the dual octagon Wilson loops). We show how to applyQ equations of [S. Caron-Huot and S. He, J. High Energy Phys. 07 (2012) 174] for computing the differential of two-loop n-point NMHV amplitudes and present the result for n ¼ 8 explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.

Symbol alphabets from plabic graphs

“…As often the case for amplitudes, guesses and expectations are valuable but explicit computations are king. Recently the two-loop eight-particle NMHV amplitude in SYM theory was computed [12], and it was found to have a 198-letter symbol alphabet that can be taken to consist of 180 cluster variables on Gr (4,8) and an additional 18 algebraic letters that involve square roots of four-mass box type. (Evidence for the former was presented in [9] based on an analysis of the Landau equations; the latter are consistent with the Landau analysis but less constrained by it.)…”

“…(Evidence for the former was presented in [9] based on an analysis of the Landau equations; the latter are consistent with the Landau analysis but less constrained by it.) The result of [12] provided the first concrete new data on symbol alphabets in SYM theory in over eight years. We will refer to this as "the eight-particle alphabet" in this paper since (turning again to hopeful speculation) it may turn out to be the complete symbol alphabet for all eight-particle amplitudes in SYM theory at all loop order.…”

“…In section 2 we introduce our construction with a simple example and then complete the analysis for all graphs relevant to Gr(4, 6) in section 3. In section 4 we consider an example where the construction yields non-cluster variables of Gr (3,6) and in section 5 we apply it to graphs that precisely reproduce the algebraic functions on Gr(4, 8) that appear in the symbol of [12].…”

“…1256 3478 a 9 1234 5678 , (5.9) which precisely supplies the ninth, missing letter! To summarize: the union of the nine face weights associated to the graph (5.1), and the nine associated to its square-move partner, multiplicatively span the nine-dimensional space of √ ∆ 1357 -containing symbol letters in the eight-particle alphabet of [12].…”

Rooting out letters: octagonal symbol alphabets and algebraic number theory

“…We are grateful to Herbert Gangl and Francesco Moriello for discussion of alternate methods, to Mark Spradlin for motivation, and to Erik Panzer for help developing our integration strategy. We are also grateful to Song He for discussing soon to be published results [77] with close relevance to this work. This work was supported in part by the…”