Consistency Conditions on S-Matrix of Spin 1 Massless Particles

TL;DR

This paper provides a purely S-matrix theoretic proof that the BCFW recursion relations correctly determine the tree-level S-matrix for massless spin 1 particles, ensuring the proper factorization and physical consistency.

Contribution

It extends the four-particle test to all particles, establishing a purely S-matrix based proof for the validity of BCFW recursion relations for spin 1.

Findings

BCFW recursion relations determine the full tree-level S-matrix for massless spin 1 particles.

The n-particle tests confirm correct factorization and collinear limits of the S-matrix.

The proof does not rely on Lagrangian or Feynman diagram techniques.

Abstract

Motivated by new techniques in the computation of scattering amplitudes of massless particles in four dimensions, like BCFW recursion relations, the question of how much structure of the S-matrix can be determined from purely S-matrix arguments has received new attention. The BCFW recursion relations for massless particles of spin 1 and 2 imply that the whole tree-level S-matrix can be determined in terms of three-particle amplitudes (evaluated at complex momenta). However, the known proofs of the validity of the relations rely on the Lagrangian of the theory, either by using Feynman diagrams explicitly or by studying the effective theory at large complex momenta. This means that a purely S-matrix theoretic proof of the relations is still missing. The aim of this paper is to provide such a proof for spin 1 particles by extending the four-particle test introduced by P. Benincasa and F. Cachazo in arXiv:0705.4305[hep-th] to all particles. We show how n-particle tests imply that the rational function built from the BCFW recursion relations possesses all the correct factorization channels including holomorphic and anti-holomorphic collinear limits. This in turn implies that they give the correct S-matrix of the theory.