Explicit Conditions for Diagnosing Tree-Level Unitarity

TL;DR

This paper derives explicit conditions for tree-level unitarity in theories with particles up to spin 1, enabling diagnosis from particle content without full Lagrangian reconstruction.

Contribution

It provides a comprehensive set of coupling conditions for tree unitarity applicable to theories with massive and massless particles up to spin 1, simplifying unitarity analysis.

Findings

All four-point amplitudes with canceled high-energy growth are on-shell constructible.

Tree unitarity conditions are fully captured up to five-point amplitudes.

The results imply theories without scalars need an infinite tower of vectors and fermions for unitarity.

Abstract

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 of a system using only its particle content in the mass basis, without reconstructing the full Lagrangian. We show that all four-point amplitudes whose high-energy growth is canceled by tree unitarity conditions are on-shell constructible, thereby motivating the recursive construction of four-point amplitudes. By examining their high-energy growth, we derive tree unitarity conditions for four-point amplitudes. Imposing these conditions to simplify the Lagrangian structure, we use the St\"uckelberg formulation to derive the tree unitarity conditions arising from all higher-point amplitudes. We show that all tree unitarity conditions are fully captured up to five-point amplitudes, ensuring no necessity of examining higher-point ones. We apply our results to systematically examine tree unitarity conditions in the dark sector with a massive dark photon and dark matter particles of spin up to 1, and extract the essential features for mass generation in the massive dark photon case. In addition, we show that our results allow us to conclude that theories without scalars require an infinite tower of vectors and fermions for tree unitarity. Finally unitarity and related issues in the Higgs portal VDM are discussed in brief.