MHV diagrams in momentum twistor space

We show that there are remarkable simplifications when the MHV diagram formalism for N = 4 super Yang-Mills is reformulated in momentum twistor space. The vertices are replaced by unity while each propagator becomes a dual superconformal 'R-invariant' whose arguments may be read off from the diagram, and include an arbitrarily chosen reference twistor. The momentum twistor MHV rules generate a formula for the full, all-loop planar integrand for the super Yang-Mills S-matrix that is manifestly dual superconformally invariant up to the choice of a reference twistor. We give a general proof of this reformulation and illustrate its use by computing the momentum twistor NMHV and (NMHV)-M-2 tree amplitudes and the integrands of the MHV and NMHV 1-loop and the MHV 2-loop planar amplitudes.