Regge trajectories for UV completions of graviton scattering from polynomial boundedness

Abstract

We study graviton scattering amplitudes. Assuming they are UV completed by a theory of weakly coupled massive higher spins, we demonstrate that the UV completion must possess infinitely many Regge trajectories, and thus they are forced to have a stringy spectrum. We extend and simplify a previous proof by some of us for open-string like states to the case of external gravitons. In the present new proof, we trace the need for infinitely many trajectories to the constraint of polynomial boundedness, ultimately tied to causality. We further present numerical results based on the stringy ansatz of H\"aring-Zhiboedov, which illustrates how single-trajectory-like solutions actually emerge as extremal solutions of numerical bootstrap. In our numerics, these trajectories curiously show up as numerically very large \textit{sister} trajectories. We provide solid evidence that the solutions are spurious as they appear to admit a divergent limit for infinite ansatz size.