Consistency Conditions on the S-Matrix of Massless Particles

TL;DR

The paper develops a program to constrain S-matrices of massless four-dimensional theories through constructibility, using the BCFW construction to relate four-particle amplitudes to three-particle data. It proves that three-particle amplitudes are fixed up to helicity-dependent constants and imposes a stringent four-particle test that many candidate theories fail, often yielding a trivial S-matrix. Among the main results, only spin-2 theories survive as nontrivial self-interactions, spin-1 theories require Lie-algebra structure constants, and spin-3/2 interactions with gravity align with linearized N=1 supergravity; YM and GR emerge as fully constructible and unique in their respective sectors. The authors also propose methods to extend the framework beyond strict constructibility, such as auxiliary fields and multi-channel deformations, and outline future directions, including applications to massive states and other spacetime dimensions.

Abstract

We introduce a set of consistency conditions on the S-matrix of theories of massless particles of arbitrary spin in four-dimensional Minkowski space-time. We find that in most cases the constraints, derived from the conditions, can only be satisfied if the S-matrix is trivial. Our conditions apply to theories where four-particle scattering amplitudes can be obtained from three-particle ones via a recent technique called BCFW construction. We call theories in this class constructible. We propose a program for performing a systematic search of constructible theories that can have non-trivial S-matrices. As illustrations, we provide simple proofs of already known facts like the impossibility of spin $s > 2$ non-trivial S-matrices, the impossibility of several spin 2 interacting particles and the uniqueness of a theory with spin 2 and spin 3/2 particles.

Figures (2)

• Figure 1: Factorization of a four-particle amplitude into two on-shell three-particle amplitudes. In constructible theories, four-particle amplitudes are given by a sum over simple poles of the 1-parameter family of amplitudes $M_{4}(z)$ times the corresponding residues. At the location of the poles the internal propagators go on-shell and the residues are the product of two on-shell three-particle amplitudes.
• Figure 2: The three different kinds of Feynman diagrams which exhibit different behavior as $z\rightarrow\infty$. They correspond to the ${\bf s}$-channel, ${\bf t}\,({\bf u})$-channel and the four-particle coupling respectively.