Unitarization from geometry

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 higherdimensional 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 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.

Soft theorems from compactification

“…Poincaré invariance of the vacuum, we impose the conditions [89] g µν = η µν ; A µ = 0 ; e φ = 1 (A. 84) and expand the metric around such a background as…”

“…The lagrangian for the massive spin 2 KK modes, by the procedure of compactification, has been considered in [77,83] ignoring their interaction with the gauge field, (see also Ref. [84] regarding the interaction with the massless fields). This gives the free Fierz-Pauli lagrangian (A.88).…”

Positivity constraints on interacting pseudo-linear spin-2 fields

“…These infinite dimensional symmetry groups generically enforce an infinite number of closely spaced states [17] and so are not useful descriptions in situations where there may exist a gap, such that there is a low energy effective description with only a finite number of spin-2 states. Such situations do occur in the condensed matter context [18] and it is interesting to explore this possibility in the Lorentz invariant context.…”

“…Crucially the massive gravity models of [12] describe the unique leading terms in a Lorentz invariant effective field theory (EFT) with the highest possible cutoff. Massive spin-2 states clearly play an important role in low energy descriptions of Kaluza-Klein and other braneworld constructions [15][16][17]. They also arise in the condensed matter context, for example as effective descriptions of the gapped collective excitation in fractional quantum Hall systems [18].…”

Bootstrap bounds on closed Einstein manifolds

“…When R = 0, these consistency conditions are the sum rules responsible for the good highenergy behavior of the four-point tree amplitudes of massive Kaluza-Klein excitations of the graviton in general relativity (GR) with Ricci-flat compact extra dimensions [12]. In these cases the sums correspond to sums over the exchanged particles in the four-point amplitude.…”