A new functional flow equation for Einstein-Cartan quantum gravity

We construct a special-purpose functional flow equation which facilitates non-perturbative renormalization group (RG) studies on theory spaces involving a large number of independent field components that are prohibitively complicated using standard methods. Its main motivation are quantum gravity theories in which the gravitational degrees of freedom are carried by a complex system of tensor fields, a prime example being Einstein-Cartan theory, possibly coupled to matter. We describe a sequence of approximation steps leading from the functional RG equation of the Effective Average Action to the new flow equation which, as a consequence, is no longer fully exact on the untruncated theory space. However, it is by far more “user friendly” when it comes to projecting the abstract equation on a concrete (truncated) theory space and computing explicit beta-functions. The necessary amount of (tensor) algebra reduces drastically, and the usually very hard problem of diagonalizing the pertinent Hessian operator is sidestepped completely. In this paper we demonstrate the reliability of the simplified equation by applying it to a truncation of the Einstein-Cartan theory space. It is parametrized by a scale dependent Holst action, depending on a O(4) spin-connection and the tetrad as the independent field variables. We compute the resulting RG flow, focusing in particular on the running of the Immirzi parameter, and compare it to the results of an earlier computation where the exact equation had been applied to the same truncation. We find consistency between the two approaches and provide further evidence for the conjectured non-perturbative renormalizability (asymptotic safety) of quantum Einstein-Cartan gravity. We also investigate a duality symmetry relating small and large values of the Immirzi parameter (γ→1/γ) which is displayed by the beta-functions in the absence of a cosmological constant.