Analytic result for the two-loop six-point NMHV amplitude in $ \mathcal{N} = {4} $ super Yang-Mills theory

Abstract

We provide a simple analytic formula for the two-loop six-point ratio function of planar N = 4 super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral Ω (2) , also plays a key role in a new representation of the remainder function R (2) 6 in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) × (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) × (parity even) part. The second non-polylogarithmic function, the loop integralΩ (2) , characterizes this sector. Both Ω (2) andΩ (2) can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.

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.…”

“…The analytic solution for the two-loop remainder function R (2) 6 (u, v, w) [61,62], after it was simplified dramatically using the symbol [4], provided the inspiration for an ansatz for the symbol of the remainder function at higher loops [26]. 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].…”

“…In particular, we will compute V andṼ through three loops, entirely from physical constraints. A similar exercise was performed previously at two loops [27]. However, at that time fewer constraints were available, and so an explicit evaluation of two-loop integrals for special kinematics had to be performed as well, in order to fix all the unknown parameters.…”

“…The six-point NMHV ratio function can be written in terms of two functions, a parityeven function V (u, v, w) and a parity-odd functionṼ (y u , y v , y w ) as follows [12,27]:…”

“…The fundamental assumption in this paper, which was also used at two loops [27], is that V (L) andṼ (L) are weight 2L hexagon functions, with even and odd parity respectively. The same basic assumption for the (parity-even) remainder function R (L) 6 [26,28] results in a consistent solution through four loops [28,29].…”