Bootstrapping an NMHV amplitude through three loops

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ 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 $R_6$ at one higher loop, which works through three loops for $V$ (four loops for $R_6$).