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On Tree-Level Unitarity in Theories of Massive Spin-2 Bosons

We analyze the tree-level 2->2 scattering of massive spin-2 bosons in a theory with only relevant and marginal operators and extract the sum rules on the coupling constants and masses required to achieve tree-level unitarity to very high energy. We do this for four illustrative cases. In the first, we include massive spin-1 and spin-0 bosons in our theory, but do not require gauge invariance. For this case, we find that it is, in fact, possible to construct a theory where all 2->2 scattering processes are tree-level unitary to very high energy. In the second case, we consider a theory that only includes massive spin-2 and spin-0 bosons. In the absence of spin-1 bosons, we find that it is impossible to unitarize the spin-2 scattering. In the third and fourth cases, we reintroduce the spin-1 bosons as gauge bosons and demand that all interactions are gauge invariant. We take the spin-2 bosons to transform under the adjoint representation of one or two gauge groups. For both of these cases, we find that unitarization is unachievable.