Hexagon functions and the three-loop remainder function

Understand

We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar N=4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios.

• The result can be expressed in terms of multiple Goncharov polylogarithms.
• We also employ a more restricted class of "hexagon functions" which have the correct branch cuts and certain other restrictions on their symbols.
• We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations.