Perturbative quantum gravity with the Immirzi parameter

TL;DR

This paper investigates perturbative quantum gravity using the Holst action with the Immirzi parameter, analyzing its quantum corrections, renormalization behavior, and implications for fermions and stability.

Contribution

It provides the first one-loop effective action computation for the Holst action, revealing the renormalization of the Immirzi parameter in the presence of fermions.

Findings

Pure gravity is on-shell finite at one-loop.

The Immirzi parameter is renormalized by fermion-induced four-fermion interactions.

The Holst action remains stable under radiative corrections with minimal coupling.

Abstract

We study perturbative quantum gravity in the first-order tetrad formalism. The lowest order action corresponds to Einstein-Cartan plus a parity-odd term, and is known in the literature as the Holst action. The coupling constant of the parity-odd term can be identified with the Immirzi parameter of loop quantum gravity. We compute the quantum effective action in the one-loop expansion. As in the metric second-order formulation, we find that in the case of pure gravity the theory is on-shell finite, and the running of Newton's constant and the Immirzi parameter is inessential. In the presence of fermions, the situation changes in two fundamental aspects. First, non-renormalizable logarithmic divergences appear, as usual. Second, the Immirzi parameter becomes a priori observable, and we find that it is renormalized by a four-fermion interaction generated by radiative corrections. We compute its beta function and discuss possible implications. The sign of the beta function depends on whether the Immirzi parameter is larger or smaller than one in absolute value, and the values plus or minus one are UV fixed-points (we work in Euclidean signature). Finally, we find that the Holst action is stable with respect to radiative corrections in the case of minimal coupling, up to higher order non-renormalizable interactions.