Ultraviolet Fixed Point and Generalized Flow Equation of Quantum Gravity

TL;DR

This work develops an exact RG equation for the gravitational effective average action using the transverse-traceless (TT) decomposition, enabling IR cutoffs and flexible truncations in a generally covariant setting. In the Einstein–Hilbert truncation, the authors derive nonperturbative β-functions for the running couplings and demonstrate the existence of a non-Gaussian fixed point (NGFP) in 4D that is UV attractive along both fundamental directions, supporting Weinberg's asymptotic safety scenario. A key finding is the near universality of the product g_* λ_* across cutoffs and gauge choices, despite substantial scheme dependence of g_* and λ_* individually; the NGFP persists across reasonable truncations, indicating potential nonperturbative renormalizability of quantum gravity. The analysis of the high-momentum graviton propagator reveals η_N(g_*,λ_*) = -2, yielding a 1/p^4 propagator and a logarithmic short-distance behavior in position space, which corresponds to an effective dimensional reduction from 4 to 2 dimensions at sub-Planckian scales. These results provide a concrete nonperturbative framework for gravity with potentially testable implications for Planck-scale physics and black hole/cosmology scenarios, and lay groundwork for more general truncations including R^2 terms and matter fields.

Abstract

A new exact renormalization group equation for the effective average action of Euclidean quantum gravity is constructed. It is formulated in terms of the component fields appearing in the transverse-traceless decomposition of the metric. It facilitates both the construction of an appropriate infrared cutoff and the projection of the renormalization group flow onto a large class of truncated parameter spaces. The Einstein-Hilbert truncation is investigated in detail and the fixed point structure of the resulting flow is analyzed. Both a Gaussian and a non-Gaussian fixed point are found. If the non-Gaussian fixed point is present in the exact theory, quantum Einstein gravity is likely to be renormalizable at the nonperturbative level. In order to assess the reliability of the truncation a comprehensive analysis of the scheme dependence of universal quantities is performed. We find strong evidence supporting the hypothesis that 4-dimensional Einstein gravity is asymptotically safe, i.e. nonperturbatively renormalizable. The renormalization group improvement of the graviton propagator suggests a kind of dimensional reduction from 4 to 2 dimensions when spacetime is probed at sub-Planckian length scales.

Figures (11)

• Figure 1: $g_*$ as a function of $s$ and $\alpha$ from the approximation $\lambda_k=\lambda_*=0$, using (a) the cutoff type A, and (b) the cutoff type B, with the family of exponential shape functions (\ref{['H6']}) inserted.
• Figure 2: The exact $g_*$ as a function of $s$ and $\alpha$ from the combined $\lambda$-$g$ system, using (a) the cutoff type A, and (b) the cutoff type B, with the family of exponential shape functions (\ref{['H6']}) inserted.
• Figure 3: The exact $\lambda_*$ as a function of $s$ and $\alpha$ from the combined $\lambda$-$g$ system, using (a) the cutoff type A, and (b) the cutoff type B, with the family of exponential shape functions (\ref{['H6']}) inserted.
• Figure 4: (a) $s$-parametric plot of $(\lambda_*(s),g_*(s))$ in the range $1\le s\le 50$ for various values of $\alpha$. Each curve starts on the left at $s=1$, and ends on the right at $s=50$. (b) $\alpha$-parametric plot of $(\lambda_*(\alpha),g_*(\alpha))$ for various values of $s$. In both (a) and (b) the cutoff type B is used, with the family of exponential shape functions (\ref{['H6']}) inserted.
• Figure 5: $g_*$, $\lambda_*$, and $g_*\lambda_*$ as functions of $s$ for (a) $\alpha=0$, $1\le s\le 30$, (b) $\alpha=0$, $1\le s\le 5$, (c) $\alpha=1$, $1\le s\le 30$, and (d) $\alpha=1$, $1\le s\le 5$, using the cutoff type B with the family of exponential shape functions (\ref{['H6']}) inserted.