Exploring the S matrix of massless particles

We use the recently proposed generalized on-shell representation for scattering amplitudes and a consistency test to explore the space of tree-level consistent couplings in four-dimensional Minkowski space-time. The extension of the notion of constructibility implied by the generalized on-shell representation, i.e. the possibility to reconstruct at tree level all the scattering amplitudes from the three-particle ones, together with the imposition of the consistency conditions at four-particle level, allow us to rediscover all the known theories characterized by three-particle couplings and propagator 1/P-2, and their algebra structure if any. Interestingly, this analysis seems to leave room for high-spin couplings, provided that at least the requirement of locality is weakened. We do not claim to have found tree-level consistent high-spin theories, but rather that our methods show signatures of them and very likely, with a suitable modification, they can be a good framework to perform a systematic search.