Motivated by recent evidence indicating that quantum Einstein gravity (QEG) might be nonperturbatively renormalizable, the exact renormalization group equation of QEG is evaluated in a truncation of theory space which generalizes the Einstein-Hilbert truncation by the inclusion of a higher-derivative term ${(R}^2).\mathrm{}$ The beta functions describing the renormalization group flow of the cosmological constant, Newton’s constant, and the $R^2$ coupling are computed explicitly. The fixed point properties of the 3-dimensional flow are investigated, and they are confronted with those of the 2-dimensional Einstein-Hilbert flow. The non-Gaussian fixed point predicted by the latter is found to generalize to a fixed point on the enlarged theory space. In order to test the reliability of the $R^2$ truncation near this fixed point we analyze the residual scheme dependence of various universal quantities; it turns out to be very weak. The two truncations are compared in detail, and their numerical predictions are found to agree with a surprisingly high precision. Because of the consistency of the results it appears increasingly unlikely that the non-Gaussian fixed point is an artifact of the truncation. If it is present in the exact theory QEG is probably nonperturbatively renormalizable and “asymptotically safe.” We discuss how the conformal factor problem of Euclidean gravity manifests itself in the exact renormalization group approach and show that, in the $R^2$ truncation, the investigation of the fixed point is not afflicted with this problem. Also the Gaussian fixed point of the Einstein-Hilbert truncation is analyzed; it turns out that it does not generalize to a corresponding fixed point on the enlarged theory space.

• Received 8 May 2002

DOI:https://doi.org/10.1103/PhysRevD.66.025026