Bootstrapping the three-loop hexagon

TL;DR

This work advances a bootstrap approach to the three-loop hexagon remainder function $R_6^{(3)}$ in planar N=4 SYM by constraining its symbol with near-collinear OPE data, multi-Regge kinematics, and final-entry conditions. Starting from a general degree-6 symbol built from a nine-letter alphabet, the authors fix most of the structure using OPE and MRK constraints, leaving only a small number of constants, with all-loop requirements ultimately forcing a vanishing remaining parameter in key regimes. In MRK, the remainder function simplifies to expressions built from classical polylogarithms, and the results precisely match Lipatov’s LL and all-loop $3\to3$ predictions, providing a stringent consistency check of the symbol-based bootstrap and the Wilson loop–amplitude duality. The analysis also shows that, away from MRK, the full three-loop remainder cannot be reduced to classical polylogarithms, underscoring the need for a richer functional basis. Together, these results offer new, nontrivial predictions for the three-loop six-point amplitude and outline a path toward reconstructing the complete analytic form from the symbol.

Abstract

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar N=4 super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol's entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3-->3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

Figures (1)

• Figure 1: The four different Wilson loops entering the definition of the ratio (\ref{['logratio']}). The reference square is shown by the dashed line. The top and bottom Wilson loops are obtained by replacing a sequence of edges by the corresponding part of the square.