Gravitational fixed points and asymptotic safety from perturbation theory

Abstract

The fixed point structure of the renormalization flow in Einstein gravity and higher derivative gravity is investigated in terms of the background effective action. Using a covariant operator cutoff that keeps track of powerlike divergences and the transversal-traceless decomposition a construction is proposed that renders the {\it regularized} one-loop effective action gauge independent on-shell. In combination with a `Wilsonian' matching condition nontrivial strictly positive fixed points for the dimensionless Newton constant $g$ and the cosmological constant $\lambda$ can then be identified already in one loop perturbation theory. The renormalization flow is asymptotically safe with respect to the nontrivial fixed points in both cases. In Einstein gravity a residual gauge dependence of the fixed points is unavoidable while in higher derivative gravity both the fixed point and the flow equations are universal. Along this flow spectral positivity of the Hessians can be satisfied, evading the traditional positivity problems. Dependence on $O(10)$ initial data is erased to accuracy $10^{-5}$ after $O(10)$ units of the renormalization mass scale and the flow settles on a $\lambda(g)$ orbit.