The "tetrad only" theory space: Nonperturbative renormalization flow and Asymptotic Safety

We set up a nonperturbative gravitational coarse graining flow and the corresponding functional renormalization group equation on the as to yet unexplored "tetrad only" theory space. It comprises action functionals which depend on the tetrad field (along with the related background and ghost fields) and are invariant under the semi-direct product of spacetime diffeomorphisms and local Lorentz transformations. This theory space differs from that of Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric constitutes the fundamental variable and because of the additional symmetry requirement of local Lorentz invariance. It also differs from "Quantum Einstein Cartan Gravity" (QECG) investigated recently since the spin connection is not an independent field variable now. We explicitly compute the renormalization group flow on this theory space within the tetrad version of the Einstein-Hilbert truncation. A detailed comparison with analog results in QEG and QECG is performed in order to assess the impact the choice of a fundamental field variable has on the renormalization behavior of the gravitational average action, and the possibility of an asymptotically safe infinite cutoff limit is investigated. Implications for nonperturbative studies of fermionic matter coupled to quantum gravity are also discussed. It turns out that, in the context of functional flow equations, the "hybrid calculations" proposed in the literature (using the tetrad for fermionic diagrams only, and the metric in all others) are unlikely to be quantitatively reliable. Moreover we find that, unlike in perturbation theory, the non-propagating Faddeev-Popov ghosts related to the local Lorentz transformations may not be discarded but rather contribute quite significantly to the beta functions of Newton's constant and the cosmological constant.