From twistor string theory to recursion relations

Witten's twistor string theory gives rise to an enigmatic formula [1] known as the "connected prescription'' for tree-level Yang-Mills scattering amplitudes. We derive a link representation for the connected prescription by Fourier transforming it to mixed coordinates in terms of both twistor and dual twistor variables. We show that it can be related to other representations of amplitudes by applying the global residue theorem to deform the contour of integration. For six and seven particles we demonstrate explicitly that certain contour deformations rewrite the connected prescription as the Britto-CachazoFeng- Witten representation, thereby establishing a concrete link between Witten's twistor string theory and the dual formulation for the S matrix of the N = 4 SYM recently proposed by Arkani-Hamed et al. Other choices of integration contour also give rise to "intermediate prescriptions.'' We expect a similar though more intricate structure for more general amplitudes.