Effective field theory for massive gravitons and gravity in theory space

Abstract We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan–Coleman–Wess–Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale M Pl and a mass m g , we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Λ parametrically above m g . Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, among them the need for the Fierz–Pauli form of the Lagrangian, the presence or absence of the van Dam–Veltman–Zakharov discontinuity in general backgrounds, and the onset of non-linear effects and the breakdown of the effective theory at large distances from heavy sources. The natural sizes of all non-linear corrections beyond the Fierz–Pauli term are easily determined. The cutoff scales as Λ ∼( m g 4 M Pl ) 1/5 for the Fierz–Pauli theory, but can be raised to Λ ∼( m g 2 M Pl ) 1/3 in certain non-linear extensions. Having established that these models make sense as effective theories, there are a number of new avenues for exploration, including model building with gravity in theory space and constructing gravitational dimensions.