Covariant Pauli-Villars regularization of quantum gravity at the one-loop order

We study a regularization of the Pauli-Villars kind of the one-loop gravitational divergences in any dimension. The Pauli-Villars fields are massive particles coupled to gravity in a covariant and nonminimal way, namely, one real tensor and one complex vector. The gauge is fixed by means of the unusual gauge fixing that gives the same effective action as in the context of the background field method. Indeed, with the background field method it is simple to see that the regularization effectively works. On the other hand, we show that in the usual formalism (nonbackground) the regularization cannot work with each gauge fixing. In particular, it does not work with the usual one. Moreover, we show that, under a suitable choice of the Pauli-Villars coefficients, the terms divergent in the Pauli-Villars masses can be corrected by the Pauli-Villars fields themselves. In dimension four, there is no need to add counterterms quadratic in the curvature tensor to the Einstein action (which would be equivalent to the introduction of new coupling constants). The technique also works when matter is coupled to gravity. We discuss the possible consequences of this approach, in particular the renormalization of Newton’s coupling constant and the appearance of two parameters in the effective action, that seem to have physical implications. Whether our procedure can be extended to all loops or can make quantum gravity a finite theory are matters that deserve further investigation.