  We give the light-cone gauge calculation of the one-loop on-shell scattering
amplitudes for gluon-gluon scattering which violate helicity conservation. We
regulate infrared divergences by discretizing the p^+ integrations, omitting
the terms with p^+=0. Collinear divergences are absent diagram by diagram for
the helicity non-conserving amplitudes. We also employ a novel ultraviolet
regulator that is natural for the light-cone worldsheet description of planar
Feynman diagrams. We show that these regulators give the known answers for the
helicity non-conserving one-loop amplitudes, which don't suffer from the usual
infrared vagaries of massless particle scattering. For the maximal helicity
violating process we elucidate the physics of the remarkable fact that the loop
momentum integrand for the on-shell Green function associated with this
process, with a suitable momentum routing of the different contributing
topologies, is identically zero. We enumerate the counterterms that must be
included to give Lorentz covariant results to this order, and we show that they
can be described locally in the light-cone worldsheet formulation of the sum of
planar diagrams.
