Recursion relations, generating functions, and unitarity sums in N=4 SYM theory

We prove that the MHV vertex expansion is valid for any NMHV tree amplitude of N = 4 SYM. The proof uses induction to show that there always exists a complex deformation of three external momenta such that the amplitude falls off at least as fast as 1/z for large z. This validates the generating function for n-point NMHV tree amplitudes. We also develop generating functions for anti-MHV and anti-NMHV amplitudes. As an application, we use these generating functions to evaluate several examples of intermediate state sums on unitarity cuts of 1-, 2-, 3- and 4-loop amplitudes. In a separate analysis, we extend the recent results of arXiv:0808.0504 to prove that there exists a valid 2-line shift for any n-point tree amplitude of N = 4 SYM. This implies that there is a BCFW recursion relation for any tree amplitude of the theory.