Scattering Amplitudes and BCFW Recursion in Twistor Space

A number of recent advances in our understanding of scattering amplitudes have been inspired by ideas from twistor theory. While there has been much work studying the twistor space support of scattering amplitudes, this has largely been done by examining the amplitudes in momentum space. In this paper, we construct the actual twistor scattering amplitudes themselves. This gives a formulation of scattering amplitudes inN = 4 super Yang-Mills in which superconformal symmetry and its breaking is manifest, and uses the progress in on-shell methods in momentum space to build our understanding of how to construct quantum field theory in twistor space. We show that the recursion relations of Britto, Cachazo, Feng and Witten have a natural twistor formulation. Together with the three-point seed amplitudes, this allows us in principle to recursively construct general tree amplitudes in twistor space. The twistor space BCFW recursion is tractable, and we obtain explicit formulae for n-particle MHV and NMHV super-amplitudes, their CPT conjugates (whose representations are distinct in our chiral framework), and the eight particle N2MHV super-amplitude. We also give simple closed form formulae for the gravity MHV and MHV amplitudes. For NkMHV, the amplitudes are given by 2n− 4 integrals in the form of Hilbert transforms of a product of n − k − 2 purely geometric, superconformally invariant twistor delta functions, dressed by certain sign operators. These sign operators subtly violate conformal invariance, even for tree-level amplitudes in N = 4 super Yang-Mills, and we trace their origin to a topological property of split signature space-time. We develop the twistor transform to relate our work to the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed, Cachazo, Cheung and Kaplan. 1 ar X iv :0 90 3. 20 83 v2 [ he pth ] 4 A ug 2 00 9