Loops in twistor space

We elucidate the one-loop twistor-space structure corresponding to momentum-space maximally helicity-violating diagrams. We also discuss the infrared divergences, and argue that only a limited set of maximally helicity-violating diagrams contain them. We show how to introduce a twistor-space regulator corresponding to dimensional regularization for the infrared-divergent diagrams. We also evaluate explicitly the ``holomorphic anomaly'' pointed out by Cachazo, Svr {c} {c}{}ek, and Witten, and use the result to define modified differential operators which can be used to probe the twistor-space structure of one-loop amplitudes.