One-loop self-dual and N=4 super Yang-Mills

We conjecture a simple relationship between the one-loop maximally helicity violating gluon amplitudes of ordinary QCD (all helicities identical) and those of N = 4 supersymmetric Yang-Mills (all but two helicities identical). Because the amplitudes in self-dual Yang Mills have been shown to be the same as the maximally helicity violating ones in QCD, this conjecture implies that they are also related to the maximally helicity violating ones of N = 4 supersymmetric Yang-Mills. We have an explicit proof of the relation up to the six-point amplitude; for amplitudes with more external legs, it remains a conjecture. A similar conjecture relates amplitudes in self-dual gravity to maximally helicity violating N = 8 supergravity amplitudes.