Bootstrapping six-gluon scattering in planar ${\cal N}=4$ super-Yang-Mills theory

TL;DR

This work develops the hexagon function bootstrap for six-gluon scattering in planar ${\cal N}=4$ SYM, constraining finite amplitudes using boundary data from the near-collinear OPE, multi-Regge factorization, and dual super-Wilson-loop equations. It defines hexagon functions as the natural function space for these amplitudes and executes a constraint-driven procedure to fix the remainder function ${R_6}$ for MHV and the NMHV ratio via $V$ and $\tilde V$ up to four and three loops, respectively. The authors present explicit multi-loop results, including the appearance of the irreducible MZV $\zeta_{5,3}$ at weight eight, and observe consistent shapes across loop orders and agreement with strong-coupling limits along key kinematic lines. Overall, the paper demonstrates a powerful, boundary-data-driven framework for determining highly constrained scattering amplitudes and points toward all-orders structure and extensions to other amplitudes or theories.

Abstract

We describe the hexagon function bootstrap for solving for six-gluon scattering amplitudes in the large $N_c$ limit of ${\cal N}=4$ super-Yang-Mills theory. In this method, an ansatz for the finite part of these amplitudes is constrained at the level of amplitudes, not integrands, using boundary information. In the near-collinear limit, the dual picture of the amplitudes as Wilson loops leads to an operator product expansion which has been solved using integrability by Basso, Sever and Vieira. Factorization of the amplitudes in the multi-Regge limit provides additional boundary data. This bootstrap has been applied successfully through four loops for the maximally helicity violating (MHV) configuration of gluon helicities, and through three loops for the non-MHV case.

Figures (3)

• Figure 1: Space of cross ratios $(u,v,w)$ for six-gluon scattering. Various limits are described in the text.
• Figure 2: The remainder function on the line $(u,u,u)$, plotted at two, three, and four loops and at strong coupling, after normalizing the coefficient functions by their values at the point $(1,1,1)$R64.
• Figure 3: The three-loop functions $R_6^{(3)}$, $V^{(3)}$ and $\tilde{V}^{(3)}/i$, plotted on the plane $u+v+w=1$R63NMHV3L.