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

TL;DR

This paper demonstrates how the BCFW recursion relations and complex factorization principles can be used to construct the tree-level Yang-Mills S-matrix, revealing fundamental constraints and behaviors without relying on specific field theories.

Contribution

It provides a field-theory-independent framework for constructing the tree-level S-matrix using complex factorization and BCFW deformations, emphasizing physical consistency and factorization properties.

Findings

BCFW recursion can fully construct the tree-level Yang-Mills S-matrix.

Complex factorization of amplitudes follows from four-particle amplitude properties.

Large z scaling behavior in Yang-Mills amplitudes is explained by factorization.

Abstract

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