Consistency Conditions on S-Matrix of Spin 1 Massless Particles

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.

Figures (4)

• Figure 1: Terms in $M_n^{(1,2)}$ with particle 1 in three amplitudes, where dots denote other external particles and dashed lines are off-shell propagators, $\alpha$ and $\beta$ are short for $\alpha_{n-1}$ and $\beta_{n-1}$. In the second line, we use $n_\beta=n\oplus1^\alpha$ for internal legs.
• Figure 2: Other terms in $M_n^{(1,2)}$ where dots denote other external particles and dashed lines are off-shell propagators. In the second line we further factorize the left amplitude by deforming the pair $(1,n)$.
• Figure 3: Terms in $M_n^{(1,n)}$ with particle 1 in three amplitudes, where dots denote other external particles and dashed lines are off-shell propagators, $\alpha$ and $\beta$ are short for $\alpha_{n-1}$ and $\beta_{n-1}$. In the second line we use $2_\alpha=1^\beta\oplus2$ for internal legs, then the left amplitude is the same as that in A.
• Figure 4: Other terms in $M_n^{(1,n)}$ where dots denote other external particles and dashed lines are off-shell propagators. In the second line we further factorize the right amplitude by deforming the pair $(1,n)$, then the second line is the same as that in B.