Conformal sector of quantum Einstein gravity in the local potential approximation: Non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance

We explore the nonperturbative renormalization group flow of quantum Einstein gravity (QEG) on an infinite dimensional theory space. We consider “conformally reduced” gravity where only fluctuations of the conformal factor are quantized and employ the local potential approximation for its effective average action. The requirement of “background independence” in quantum gravity entails a partial differential equation governing the scale dependence of the potential for the conformal factor which differs significantly from that of a scalar matter field. In the infinite dimensional space of potential functions we find a Gaussian as well as a non-Gaussian fixed point which provides further evidence for the viability of the asymptotic safety scenario. The analog of the invariant cubic in the curvature which spoils perturbative renormalizability is seen to be unproblematic for the asymptotic safety of the conformally reduced theory. The scaling fields and dimensions of both fixed points are obtained explicitly and possible implications for the predictivity of the theory are discussed. Spacetime manifolds with $R^d$ as well as $S^d$ topology are considered. Solving the flow equation for the potential numerically we obtain examples of renormalization group trajectories inside the ultraviolet critical surface of the non-Gaussian fixed point. The quantum theories based upon some of them show a phase transition from the familiar (low energy) phase of gravity with spontaneously broken diffeomorphism invariance to a new phase of unbroken diffeomorphism invariance; the latter phase is characterized by a vanishing expectation value of the metric.

Figure 1

The negative fixed point potential $-Y_{*}$ for $S^4$. The dashed line is the CREH potential with the same value for ${\lambda }_{*}$. The figure on the LHS is a detail of the one on the RHS; the latter shows that $Y_{*}$ coincides with the CREH potential asymptotically.

Figure 2

Left figure: The points in the shaded region correspond to the scaling dimensions $({\theta }^{\prime },{\theta }^{\prime \prime })$ realized by the “$n_{+}$-family.” The bold-faced part of the vertical axis represents the scaling fields with $n_{+}^{\prime \prime }=0$. The points above the parabola satisfy $n_{+}^{\prime }<4$. Right figure: The condition $n_{+}^{\prime }\ge 2$ is imposed in addition. It is satisfied by all points below the upper parabola.

Figure 3

The function $\theta (n)$ of (5.40) for $n$ real. It is positive only between $n=0$ and $n=2\omega \approx 3.38$.

Figure 4

The trajectory discussed in Subsection 6a. The diagrams (a) and (b) [(c) and (d)] show the potential for different values of $k$ above (below) $k_{\mathrm{T}}$, as indicated in the schematic diagram (e). The dotted (dashed) line is the fixed point (turning point) potential. The plot (b) [(d)] zooms into the diagram (a) [(c)] at small $\phi$. The plot (f) shows the dimensionful minimum ${\varphi }_0$ as a function of $k<k_{\mathrm{T}}$. Dimensionful quantities are in units of $k_{\mathrm{T}}$. The gray level of the various curves changes from black to gray for decreasing values of $k$.

Figure 5

The trajectory discussed in Subsection 6b. The diagrams (a), (b), and (c) show the potential for different values of $k$ above $k_{\mathrm{T}}$, the diagram (d) for $k$-values below $k_{\mathrm{T}}$. The dotted (dashed) line is the fixed point (turning point) potential. The plots (b) and (c) are zooms into the diagram (a) at smaller $\phi$ values.

Figure 6

The $S^4$ trajectory discussed in Subsection 6c. The diagram (a) shows the potential for different values of $k$ above $k_{\mathrm{T}}$. The plot (b) zooms into diagram (a) for small $\phi$ and shows the first order phase transition. The dotted (dashed gray) line is the fixed point (initial) potential. The gray level of the various curves changes from black to gray for decreasing values of $k$.