Unitarization from Geometry

TL;DR

This paper investigates the conditions for maintaining unitarity in scattering amplitudes of dimensionally reduced theories, deriving sum rules that constrain the spectra of internal manifolds and establishing bounds on Kaluza-Klein excitation gaps.

Contribution

It derives sum rules ensuring unitarity in Kaluza-Klein theories and proves bounds on eigenvalue gaps for scalar Laplacians on special holonomy manifolds.

Findings

Sum rules relate spectra and couplings of Kaluza-Klein states.

Upper bounds on eigenvalue ratios for scalar Laplacians.

Constraints apply to Calabi-Yau compactifications.

Abstract

We study the perturbative unitarity of scattering amplitudes in general dimensional reductions of Yang-Mills theories and general relativity on closed internal manifolds. For the tree amplitudes of the dimensionally reduced theory to have the expected high-energy behavior of the higher-dimensional theory, the masses and cubic couplings of the Kaluza-Klein states must satisfy certain sum rules that ensure there are nontrivial cancellations between Feynman diagrams. These sum rules give constraints on the spectra and triple overlap integrals of eigenfunctions of Laplacian operators on the internal manifold and can be proven directly using Hodge and eigenfunction decompositions. One consequence of these constraints is that there is an upper bound on the ratio of consecutive eigenvalues of the scalar Laplacian on closed Ricci-flat manifolds with special holonomy. This gives a sharp bound on the allowed gaps between Kaluza-Klein excitations of the graviton that also applies to Calabi-Yau compactifications of string theory.