Abstract
We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N=4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N^3LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. We also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.
Highlights
With light-like edges [9, 11,12,13,14,15,16]
The first such function, the remainder function, R6(u, v, w), is defined to be the maximally-helicity-violating (MHV) all-gluon amplitude divided by the BDS ansatz [18, 19]
Hexagon functions are defined by two conditions [36]: 1. Their derivatives with respect to the cross ratios can be expanded in terms of just nine hexagon functions of one lower weight, n − 1
Summary
As in past work at one, two, and three loops [10, 35, 39, 90], we describe the six-point amplitude using an on-shell superspace [20,21,22,23]. (The one-loop quantity V (1) vanishes because there is no parity-odd weight-two hexagon function.) It is convenient to introduce some other functions E and E, which are closely related to V and V , but defined more directly in terms of the NMHV amplitude, rather than its ratio to the MHV amplitude. First recall that the MHV amplitude can be expressed in terms of two quantities, the BDS ansatz [17] and the remainder function R6 [18, 19]: AMHV = ABDS × exp(R6). We normalize the NMHV superamplitude by the BDS-like ansatz, and define new functions E(u, v, w) and E(u, v, w) as the coefficients of the R-invariants: ANMHV AB6 DS−like [(1) + (4)]E(u, v, w) + [(2) + (5)]E(v, w, u) + [(3) + (6)]E(w, u, v).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
