Hexagon functions and the three-loop remainder function

Abstract

We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N = 4 superYang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multiRegge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The nearcollinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.

Bootstrapping an NMHV amplitude through three loops

“…In this paper we will follow an alternative approach, the hexagon function bootstrap [26][27][28][29][30]. The philosophy of this program is to bypass integrands altogether and focus on infrared-finite quantities from the very beginning.…”

“…Further perspectives on multi-Regge factorization have been provided by Caron-Huot [55]. The factorization limit has a logarithmic ordering, which allows for the efficient recycling of lower-loop information to higher loops [28,29,56,57]. The recycling is aided by the recognition [56] that in the six-point case the functions relevant for the multi-Regge limit are single-valued harmonic polylogarithms (SVHPLs) [58].…”

“…The same ansatz could also be applied to the symbols of a pair of functions V (u, v, w) and V (u, v, w) entering the NMHV ratio function [27]. Those symbols define a class of functions of three variables, iterated integrals called hexagon functions [28]. The number of iterated integrations defines the weight of the hexagon function, which should be 2L for the L-loop contributions to R 6 , V andṼ .…”

“…The number of iterated integrations defines the weight of the hexagon function, which should be 2L for the L-loop contributions to R 6 , V andṼ . Given the hexagon-function ansatz, the near-collinear limit, multi-Regge behavior, and a few other physical constraints uniquely determine the full six-point remainder function at both three [28] and four loops [29]. The uniquenes of the solution, despite the existence of around 6000 unknown parameters in the inital four-loop ansatz, is a testament to the power of the boundary data.…”

“…We characterize the functions in terms of their weight-five {5, 1} coproduct components [63,64], which are essentially their first derivatives. This characterization makes use of a previous classification of hexagon functions through weight five [28]. There are several hundred free parameters (unknown rational numbers) in our initial ansatz.…”