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A remarkable connection between BCFW recursion relations and constraints onthe S-matrix was made by Benincasa and Cachazo in 0705.4305, who noted thatmutual consistency of different BCFW constructions of four-particle amplitudesgenerates non-trivial (but familiar) constraints on three-particle couplingconstants --- these include gauge invariance, the equivalence principle, andthe lack of non-trivial couplings for spins >2. These constraints can also bederived with weaker assumptions, by demanding the existence of four-pointamplitudes that factorize properly in all unitarity limits with complexmomenta. From this starting point, we show that the BCFW prescription can beinterpreted as an algorithm for fully constructing a tree-level S-matrix, andthat complex factorization of general BCFW amplitudes follows from thefactorization of four-particle amplitudes. The allowed set of BCFW deformationsis identified, formulated entirely as a statement on the three-particle sector,and using only complex factorization as a guide. Consequently, our analysisbased on the physical consistency of the S-matrix is entirely independent offield theory. We analyze the case of pure Yang-Mills, and outline a proof forgravity. For Yang-Mills, we also show that the well-known scaling behavior ofBCFW-deformed amplitudes at large z is a simple consequence of factorization.For gravity, factorization in certain channels requires asymptotic behavior~1/z^2.

Constructing the Tree-Level Yang-Mills S-Matrix Using Complex Factorization