Completeness from Gravitational Scattering

We prove that symmetry in the presence of gravity implies a version of the completeness hypothesis. For a broad class of theories, we demonstrate that the existence of finitely many charged particles logically necessitates the existence of infinitely many charged particles populating the entire charge lattice. Our conclusions follow from the consistency of perturbative gravitational scattering and require the following ingredients: 1) a weakly coupled ultraviolet completion of gravity, 2) a nonabelian symmetry $G$, gauged or global, whose Cartan subgroup generates the abelian charge lattice, and 3) a spectrum containing some finite set of charged representations, in the simplest cases taken to be a single particle in the fundamental. Under these conditions, the abelian charge lattice is completely filled by single-particle states for $G=SO(N)$ with $N\geq 5$ and $G=SU(N)$ with $N\geq 3$, which in turn implies completeness for other symmetry groups such as $Spin(N)$, $Sp(N)$, and $E_8$. Curiously, a corollary of our results is that the $SU(5)$ and $SO(10)$ grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.

💡 Research Summary

This paper presents a rigorous, bottom-up proof of a version of the “completeness hypothesis” using the consistency of gravitational scattering amplitudes. The core idea is that under specific conditions, the existence of a finite set of charged particles logically forces the existence of an infinite tower of charged particles, completely populating the allowed charge lattice.

The argument hinges on three key assumptions: 1) a weakly-coupled, tree-level ultraviolet completion of gravity (motivated by examples like string theory), 2) an exact non-abelian symmetry group G (which can be gauged or global), and 3) a finite initial set of charged particle representations, often taken to be a single particle in the fundamental representation.

The authors’ primary tool is the dispersion relation for the four-point scattering amplitude A(s,t). Crucially, graviton exchange contributes a term A(s,t) ~ -8πGs²/t near the forward limit. This corresponds to a non-zero Wilson coefficient c₂(t) = -8πG/t in the dispersion relation. Under the assumption of tree-level unitarization (where the boundary contribution at infinity b₂(t) vanishes for t<0), the dispersion relation mandates that there must be states exchanged in either the s-channel or u-channel for any scattering process with a graviton in the t-channel.

The authors leverage this mandate to construct an iterative “completeness algorithm.” Starting from an initial finite set of charged states Q (e.g., the graviton and a fundamental particle), they consider all possible pairwise gravitational scattering where the total charge in the t-channel is neutral (so the graviton can contribute). Applying the dispersion relation forces the existence of new charged states in the s- or u-channel. These new states are added to Q. The spectrum Q must be invariant under the symmetry group G, so the authors also “orbit” known charges using the Weyl group and outer automorphisms of G to generate all charges within the same representation. This process of scattering new combinations and orbiting is repeated ad infinitum.

The success of this algorithm depends critically on the non-abelian structure of G. For abelian groups or small groups like SO(3) and SO(4), the algorithm fails to generate the full lattice. However, for sufficiently large non-abelian groups—specifically SO(N) with N≥5 and SU(N) with N≥3—the authors prove that the algorithm necessarily populates the entire abelian charge lattice (generated by the Cartan subgroup H) with single-particle states. This result is stronger than conventional swampland conjectures about completeness, which typically allow multi-particle states to realize high charges; here, single-particle states are mandated due to the tree-level unitarization assumption.

The results are then extended to imply completeness for other groups like Spin(N), Sp(N), and E₈. A striking corollary is noted for phenomenologically relevant Grand Unified Theories (GUTs). The authors find that the minimal field content required to embed the Standard Model in SU(5) GUTs (representations 5, 10, and 24) and SO(10) GUTs (16, 10, and 45) is precisely the minimal set needed to bootstrap charge completeness using their methodology. This suggests an intriguing alignment between mathematical consistency and phenomenological necessity.

In summary, the paper provides a novel, amplitude-based derivation of spectral completeness from first principles, demonstrating that gravitational interactions and non-abelian symmetry powerfully constrain the allowed particle spectrum in a theory of quantum gravity.