Critical collapse of a massive vector field

Critical gravitational collapse was first found by Choptuik[1] in simulations of a spherically symmetric massless scalar field. A natural question to pose is then how critical collapse behaves when the scalar field has a mass, since this will introduce a characteristic length that destroys the scale invariance of the field equations. This question was studied by Brady et. al.[2] The results of reference[2] show that there are two critical solutions: one which is essentially the Choptuik critical solution for the massless scalar field, and another which is a periodic solution first found by Seidel and Suen [3]. At first it might seem puzzling that the Choptuik solution can be a critical solution for both the massless and massive scalar field. The resolution of this conundrum is that as the singularity is approached in the Choptuik critical solution, the amplitude of the scalar field remains bounded while its gradient diverges. In the stress energy tensor, the mass terms are associated with the amplitude of the field, while other terms are associated with its gradient. So as the singularity is approached the mass terms in the stress energy become negligible. Given the results of reference[2] one might conjecture that similar behavior occurs in the case of a spherically symmetric massive vector field: i.e. that there is a critical solution for which the mass of the vector field becomes negligible. However, this conjecture involves a paradox: a massless vector field is just a Maxwell field, and a spherically symmetric Maxwell field has no degrees of freedom. Therefore there is no gravitational collapse (and thus no critical solution) of a spherically symmetric massless vector field. What then is the critical behavior of a massive vector field?