Bare action and regularized functional integral of asymptotically safe quantum gravity

Investigations of quantum Einstein gravity (QEG) based upon the effective average action employ a flow equation which does not contain any ultraviolet (UV) regulator. Its renormalization group trajectories emanating from a non-Gaussian fixed point define asymptotically safe quantum field theories. A priori these theories are, somewhat unusually, given in terms of their effective rather than bare action. In this paper we construct a functional integral representation of these theories. We fix a regularized measure and show that every trajectory of effective average actions, depending on an IR cutoff only, induces an associated trajectory of bare actions which depend on a UV cutoff. Together with the regularized measure these bare actions give rise to a functional integral which reproduces the prescribed effective action when the UV cutoff is removed. In this way we are able to reconstruct the underlying microscopic (classical) system and identify its fundamental degrees of freedom and interactions. The bare action of the Einstein-Hilbert truncation is computed and its flow is analyzed as an example. Various conceptual issues related to the completion of the asymptotic safety program are discussed.

Figure 1

A typical nonsingular cutoff function (a) and its scale derivative (b). Dimensionful quantities are expressed in units of $k$.

Figure 2

Employing the optimized cutoff every complete solution to the $\Lambda$-free FRGE gives rise to a solution of the FRGE with UV cutoff, valid up to any value of $\Lambda$.

Figure 3

The bare parameters are shown in dependence on the effective ones, $g$ and $\lambda$, for two values of $Q$. Figure (a) $\check{g}$ for $Q=0$; Fig. (b) $\check{g}$ for $Q=-5\pi$; Fig. (c) $\check{\lambda }$ for $Q=0$; Fig. (d) $\check{\lambda }$ for $Q=-5\pi$. The results are displayed both as a 3D and a contour plot. The blue dot in the plots of the right column marks the NGFP.

Figure 4

The diagram (a) shows the phase portrait of the effective RG flow on the $(g,\lambda )$-plane. The other diagrams are its image on the $(\check{g},\check{\lambda })$-plane of bare parameters for three different values of $Q$, namely, (b) $Q=+1$, (c) $Q=-0.1167$, where ${\check{\lambda }}_{*}=0$ and (d) $Q=-1$, respectively.