problem using the Vilkovisky-DeWitt method. In a manifestly gauge invariant as well as gauge fixing independent formulation of the effective action he finds that the quantum gravity contributions to the running charge vanish.

All of these computations employ the dimensional regularization scheme. In [@ymg-ebert], Ebert, Plefka and Rodigast pointed out that its use might be problematic since it is insensitive to quadratic divergences, and it is precisely such quadratic divergences that are responsible for the non-zero result obtained in [@ymg-robwil; @ymg-robthesis]. Using a cutoff regularization instead they found that all gravitational quadratic divergences cancel so that there is again no correction to the beta function. Thereafter Tang and Wu [@ymg-tangwu] argued that the use of a cutoff regularization is not permissible here since it does not respect gauge invariance. Performing a calculation in a scheme which both retains quadratic divergences and preserves gauge invariance (“loop regularization”) they obtained a non-zero gravitational correction to the one-loop beta function. Furthermore, Toms [@ymg-toms2] demonstrated that, with a cosmological constant included, also dimensional regularization yields a non-vanishing gravitational correction, albeit of a different type.

In the present paper we shall analyze the running of the gauge coupling constant in the framework of the Asymptotic 