Flow equation of quantum Einstein gravity in a higher derivative truncation

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 $(R^2)$. 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 (FP) properties of the 3-dimensional flow are investigated, and they are confronted with those of the 2-dimensional Einstein-Hilbert flow. The non-Gaussian FP predicted by the latter is found to generalize to a FP on the enlarged theory space. In order to test the reliability of the $R^2$-truncation near this FP 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 suprisingly high precision. Due to the consistency of the results it appears increasingly unlikely that the non-Gaussian FP 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 FP is not afflicted with this problem. Also the Gaussian FP of the Einstein-Hilbert truncation is analyzed; it turns out that it does not generalize to a corresponding FP on the enlarged theory space.