Bootstrapping the three-loop hexagon

TL;DR

This paper determines the three-loop six-point MHV amplitude in planar N=4 super Yang-Mills theory using the symbol bootstrap approach, confirming predictions in the multi-Regge limit and providing explicit expressions for the remainder function.

Contribution

It introduces a symbol-based bootstrap method constrained by the OPE and multi-Regge limits to determine the three-loop remainder function, with minimal undetermined constants.

Findings

Confirmed the BFKL prediction in the multi-Regge limit.

Provided explicit form of the three-loop remainder function.

Validated the duality between Wilson loops and MHV amplitudes.

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.