Nonperturbative Continuity in Graviton Mass versus Perturbative Discontinuity

We address the question whether a graviton could have a small nonzero mass. The issue is subtle for two reasons: there is a discontinuity in the mass in the lowest tree-level approximation, and, moreover, the nonlinear four-dimensional theory of a massive graviton is not defined unambiguously. First, we reiterate the old argument that for the vanishing graviton mass the lowest tree-level approximation breaks down since the higher order corrections are singular in the graviton mass. However, there exist nonperturbative solutions which correspond to the summation of the singular terms and these solutions are continuous in the graviton mass. Furthermore, we propose a nonlinear completion of the four-dimensional theory of massive gravity by embedding it into a five-dimensional generally covariant nonlinear model. We show that the exact solutions of the model are continuous in the mass, yet the perturbative expansion has the discontinuity in the leading order and the singularities in higher orders as in the four-dimensional case. The helicity-zero graviton state which is responsible for the perturbative discontinuity and singularities decouples from the matter in the limit of vanishing graviton mass in the full classical theory. Thus, we conclude that a small graviton mass is not in contradiction with observations.