Explicit conditions for diagnosing tree-level unitarity

A bstract We explicitly present all coupling conditions required for tree-level unitarity (tree unitarity) in theories with a finite number of massive and massless particles of spin up to 1. They allow us to diagnose tree unitarity using only the particle content in the mass basis, without reconstructing the full Lagrangian. We show that all tree-unitary amplitudes except the four-scalar amplitude are on-shell constructible, allowing us to recursively construct four-point amplitudes with the four-scalar amplitude as a building block. From their high-energy behavior and Ward identities, we derive the coupling conditions for four-point amplitudes. Imposing these conditions, we use the Stückelberg formulation to derive those from all higher-point amplitudes and show that all tree unitarity conditions are fully captured up to five points. We apply our results to dark photon scenarios with dark matter particles of spin up to 1 and identify the essential features of their mass generation. We also examine simple theories without scalars in which tree unitarity requires an infinite tower of vectors or fermions. Finally, we briefly discuss tree unitarity and related issues in Higgs-portal vector dark matter.