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Just like the vector bosons in Abelian and non-Abelian gauge theories, gravitons can attain mass by spontaneous local symmetry breaking. The question is whether this can happen in a Lorentz-invariant way. We consider the use of four scalar fields that break coordinate reparametrization invariance, by playing the role of preferred flat coordinates x, y, z, and t. In the unbroken representation, the theory has a (negative) cosmological constant, which is tuned to zero by the scalars in the broken phase. Massive spin 2 bosons and a single massive scalar survive. The theory is not renormalizable, so at best it can be viewed as an effective field theory for massive spin 2 particles. One may think of applications in cosmology, but a more tantalizing idea is to apply it to string theory approaches to QCD: if the gluon sector is to be described by a compactified 26 or 10 dimensional bosonic string theory, then the ideas considered here could be used to describe the mechanism that removes a massless or tachyonic scalar and provides mass to the spin 2 glueball states. The delicate problem of removing indefinite metric and/or negative energy states is addressed. The scalar particle has negative metric, so that unitarity demands that only states with an even number of them are allowed. Various ways are considered to adapt the matter section of the theory such that matter only couples to positive metric states, and we succeed in suppressing the main contributions to unitarity-violating amplitudes, but the exact restoration of unitarity in the spinless sector will continue to be a delicate issue in theories of this sort.

Unitarity in the Brout-Englert-Higgs Mechanism for Gravity