We extend the hexagon function bootstrap to the
next-to-maximally-helicity-violating (NMHV) configuration for six-point
scattering in planar N=4 super-Yang-Mills theory at three loops.
Constraints from the Qˉ differential equation, from the operator product
expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge
factorization, lead to a unique answer for the three-loop ratio function. The
three-loop result also predicts additional terms in the OPE expansion, as well
as the behavior of NMHV amplitudes in the multi-Regge limit at one higher
logarithmic accuracy (NNLL) than was used as input. Both predictions are in
agreement with recent results from the flux-tube approach. We also study the
multi-particle factorization of multi-loop amplitudes for the first time. We
find that the function controlling this factorization is purely logarithmic
through three loops. We show that a function U, which is closely related to
the parity-even part of the ratio function V, is remarkably simple; only five
of the nine possible final entries in its symbol are non-vanishing. We study
the analytic and numerical behavior of both the parity-even and parity-odd
parts of the ratio function on simple lines traversing the space of cross
ratios (u,v,w), as well as on a few two-dimensional planes. Finally, we
present an empirical formula for V in terms of elements of the coproduct of
the six-gluon MHV remainder function R6 at one higher loop, which works
through three loops for V (four loops for R6).Comment: 69 pages, 12 figures, 1 table, 3 ancillary files; v2, minor typo's
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