  Loop-level scattering amplitudes for massless particles have singularities in
regions where tree amplitudes are perfectly smooth. For example, a $2\to4$
gluon scattering process has a singularity in which each incoming gluon splits
into a pair of gluons, followed by a pair of $2\to2$ collisions between the
gluon pairs. This singularity mimics double parton scattering because it occurs
when the transverse momentum of a pair of outgoing gluons vanishes. The
singularity is logarithmic at fixed order in perturbation theory. We exploit
the duality between scattering amplitudes and polygonal Wilson loops to study
six-point amplitudes in this limit to high loop order in planar ${\cal N}=4$
super-Yang-Mills theory. The singular configuration corresponds to the limit in
which a hexagonal Wilson loop develops a self-crossing. The singular terms are
governed by an evolution equation, in which the hexagon mixes into a pair of
boxes; the mixing back is suppressed in the planar (large $N_c$) limit. Because
the kinematic dependence of the box Wilson loops is dictated by (dual)
conformal invariance, the complete kinematic dependence of the singular terms
for the self-crossing hexagon on the one nonsingular variable is determined to
all loop orders. The complete logarithmic dependence on the singular variable
can be obtained through nine loops, up to a couple of constants, using a
correspondence with the multi-Regge limit. As a byproduct, we obtain a simple
formula for the leading logs to all loop orders. We also show that, although
the MHV six-gluon amplitude is singular, remarkably, the transcendental
functions entering the non-MHV amplitude are finite in the same limit, at least
through four loops.
