Einstein–Cartan gravity, Asymptotic Safety, and the running Immirzi parameter

• A functional RG equation for a first order formulation of gravity is constructed.
• The theory space constituted by tetrad and spin connection variables is explored.
• The RG equation is solved in a 3 dimensional truncation of theory space.
• The flow of Newton’s constant, the cosmological constant and the Immirzi parameter is analyzed.
• Evidence for the nonperturbative renormalizability of the theory is found.

In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein–Cartan theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (co-frame) which are invariant under both spacetime diffeomorphisms and local frame rotations. In the first part of the paper we develop a general methodology and corresponding calculational tools which can be used to analyze the flow equation for the pertinent effective average action for any truncation of this theory space. In the second part we apply it to a specific three-dimensional truncated theory space which is parametrized by Newton’s constant, the cosmological constant, and the Immirzi parameter. A comprehensive analysis of their scale dependences is performed, and the possibility of defining an asymptotically safe theory on this hitherto unexplored theory space is investigated. In principle Asymptotic Safety of metric gravity (at least at the level of the effective average action) is neither necessary nor sufficient for Asymptotic Safety on the Einstein–Cartan theory space which might accommodate different “universality classes” of microscopic quantum gravity theories. Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.