Approaches to Quantum Gravity

122 R. Percacci~G10.80.60.40.2–0.2 –0.1 0.1 0.2~ΛFig. 8.2. The flow in the Einstein–Hilbert truncation, see eqs. (8.23) and (8.24).The nontrivial FP at ˜ = 0.171, ˜G = 0.701 is UV-attractive with eigenvalues−1.69 ± 2.49i. The Gaussian FP is attractive along the ˜-axis with eigenvalue−2 and repulsive in the direction (0.04, 1.00) with eigenvalue 2.Lauscher & Reuter [22] and Reuter & Saueressig [37] have studied the gaugeandcutoff-dependence of the FP in the Einstein–Hilbert truncation. The dimensionlessquantity ′ = G (the cosmological constant in Planck units) and thecritical exponents have a reassuringly weak dependence on these parameters. Thishas been taken as a sign that the FP is not an artifact of the truncation. Lauscher &Reuter [23] have also studied the ERGE including a term R 2 in the truncation.They find that in the subspace of ˜Q spanned by ˜, ˜G, 1/ξ, the non-Gaussian FP isvery close to the one of the Einstein–Hilbert truncation, and is UV-attractive in allthree directions. More recently, the FP has been shown to exist if the Lagrangiandensity is a polynomial in R of order up to six (Codello, Percacci and Rahmede, inpreparation). In this truncation the UV critical surface is three dimensional.There have been also other generalizations. Niedermaier [28] considered the RGflow for dimensionally reduced d = 4 gravity, under the hypothesis of the existenceof two Killing vectors. This subsector of the theory is parametrized by infinitelymany couplings, and has been proved to be asymptotically safe.Matter couplings have been considered by Percacci & Perini [31; 32]. Considerthe general action∫Ɣ k (g μν ,φ)= d 4 x √ (g − 1 )2 gμν ∂ μ φ∂ ν φ − V (φ 2 ) + F(φ 2 )R , (8.25)where V and F are arbitrary functions of φ 2 , analytic at φ 2 = 0. This actionhas a so-called Gaussian-Matter FP, meaning that only the coefficients of theφ-independent terms in (8.25) (namely g (0) and g (1) ) are nonzero. The critical surfacehas dimension four and there are no marginal operators. In the presence ofother, minimally coupled matter fields, the dimension of the critical surface can be