Fixed Points of Quantum Gravity
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PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 20 21 MAY 2004 Fixed Points of Quantum Gravity Daniel F. Litim* Theory Division, CERN, CH-1211 Geneva 23 and School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom (Received 15 December 2003; published 19 May 2004) Euclidean quantum gravity is studied with renormalization group methods. Analytical results for a nontrivial ultraviolet fixed point are found for arbitrary dimensions and gauge fixing parameters in the Einstein-Hilbert truncation. Implications for quantum gravity in four dimensions are discussed. DOI: 10.1103/PhysRevLett.92.201301 PACS numbers: 04.60.–m, 11.10.Hi, 11.15.Tk Classical general relativity is acknowledged as the R2 g interactions [8] or noninteracting matter fields theory of gravitational interactions for distances suffi- [13]. Further indications for the existence of a fixed point ciently large compared to the Planck length. At smaller are based on dimensionally reduced theories [14] and on length scales, quantum effects are expected to become numerical studies within simplicial gravity [15]. For phe- important. The quantization of general relativity, how- nomenological applications, see [16]. ever, still poses problems. It has long been known that In this Letter, we study fixed points of quantum gravity quantum gravity is perturbatively nonrenormalizable, in the approach put forward in [6], amended by an ade- meaning that an infinite number of parameters have to quate optimization [4,5]. The main new result is the be fixed to renormalize standard perturbation theory. It existence of a nontrivial ultraviolet fixed point in the has been suggested that Einstein gravity may be non- Einstein-Hilbert truncation in dimensions higher than perturbatively renormalizable, a scenario known as 4, a region which previously has not been accessible. asymptotic safety [1]. This requires the existence of a Analytical results for the flow and its fixed points are nontrivial ultraviolet fixed point with at most a finite given for arbitrary dimension. The optimization ensures number of unstable directions. Then it would suffice to the maximal reliability of the result in the present trun- adjust a finite number of parameters, ideally taken from cation, thereby strengthening earlier findings in four di- experiment, to make the theory asymptotically safe. mensions. Implications of these results are discussed. Nonperturbative renormalizability has already been es- The Exact Renormalization Group is based on a mo- tablished for a number of field theories [2]. mentum cutoff for the propagating degrees of freedom The search for fixed points in quantum field theory and describes the change of the scale-dependent effective calls for a renormalization group study. A particularly action ÿk under an infinitesimal change of the cutoff scale useful approach is given by the Exact Renormalization k. Thereby it interpolates between a microscopic action in Group, based on the integrating out of momentum modes the ultraviolet and the full quantum effective action in the from a path integral representation of the theory [3]. The infrared, where the cutoff is removed. In its modern strength of the method is its flexibility when it comes to formulation, the renormalization group flow of ÿk with approximations. General optimization procedures are the logarithmic scale parameter t lnk is given by available [4], increasing the domain of validity and the @ ÿ 1Tr ÿ 2 Rÿ1@ R: (1) convergence of the flow. Consequently, the reliability of t k 2 k t results based on optimized flows is enhanced [5]. The trace stands for a sum over indices and a loop Explicit flow equations for Euclidean quantum gravity integration, and R (not to be confused with the Ricci have been constructed by Reuter [6], using background scalar) is an appropriately defined momentum cutoff at field techniques [6–8]. Diffeomorphism invariance under the momentum scale q2 k2. For quantum gravity, we local coordinate transformations is controlled by modi- consider the flow (1) for ÿkg in the Einstein-Hilbert fied Ward identities, similar to those known for non- truncation, retaining the volume element and the Ricci Abelian gauge theories [9]. In general, methods originally scalar as independent operators. Apart from a classical developed for gauge theories [10], with minor modifica- gauge fixing, the effective action is given by Z tions, can now be applied to quantum gravity. p 1 d So far most studies have been concerned with thep ÿk d x gÿR g2k: (2) g 16Gk Einstein-Hilbertp truncation based on the operators and gR g in the effective action, where g is the deter- In (2) we introduced the gravitational coupling constant minant of the metric tensor g and R g the Ricci scalar. Gk and the cosmological constant k. The ansatz (2) In four dimensions, the high energy behavior of quantum differs from the Einstein-Hilbert action in d Euclidean gravity is dominated by a nontrivial ultraviolet fixed dimensions in that the couplings have turned into point [7,11,12], which is stable under the inclusion of scale-dependent functions. Denoting G and the 201301-1 0031-9007=04=92(20)=201301(4)$22.50 2004 The American Physical Society 201301-1 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 20 21 MAY 2004 unrenormalized Newtonian coupling and cosmological g d ÿ 2g modulo subleading corrections. This en- constant at some reference scale k ,andZN;k the tails 0. Then the flow is trivially solved by gk dÿ2 ÿ2 wave function renormalization factor for the Newtonian g k= and k k= , the canonical scaling coupling, we introduce dimensionless renormalized cou- of the couplings as implied by (3). Here, denotes the plings as momentum scale where the canonical scaling regime is reached. In the infrared limit g=g k=d2 g kdÿ2G kdÿ2Zÿ1 GG; kÿ2 : (3) bound k k N;k k k becomes increasingly small for any dimension. Their flows follow from (1) by an appropriate projection In the remaining part, we discuss the fixed points of (4) onto the operators in (2). A scaling solution of the flow (1) and their implications. A first understanding is achieved in the truncation (2) corresponds to fixed points for the in the limit of vanishing cosmological constant, where couplings (3). Explicit momentum cutoffs have been provided for the 1 ÿ 4dg d ÿ 2g g : (6) fluctuations in the metric field in the Feynman gauge [6] 1 2 2 ÿ dg and for its component fields in a traceless transverse The flow (6) displays two fixed points: the Gaussian one at decomposition in a harmonic background field gauge g 0 and a non-Gaussian one at g 1= 4d.Inthe with the gauge fixing parameter [7]. In either case the vicinity of the nontrivial fixed point for large k,the tensor structure of the regulator is fixed, while the scalar gravitational coupling behaves as G g =kdÿ2.This part is left free. Here, we employ the optimized cutoff k behavior is similar to asymptotic freedom in Yang- R k2 ÿ q2 k2 ÿ q2 for the scalar part [4,5]. In opt Mills theories. The anomalous dimension of the graviton the setup (1)–(3), we have computed the flow equation for (5) remains finite and 0 for all g between the infrared arbitrary and arbitrary dimension. To simplify the and the ultraviolet fixed point, because g g < notation, a factor 1= is absorbed into the definition k g 0 for all d>ÿ2 and g g . At the fixed point, (3). Below we present explicit formulas only for the limit bound universal observables are the eigenvalues of the stability !1where the results take their simplest form. The matrix at criticality, i.e., the critical exponents. The uni- general case is discussed elsewhere. The functions are versal eigenvalue is given by @ g=@gj ÿ. We find P @ 1 ; d ÿ 2 t P 4 d 2g 2 ÿ d; 2d (7) 2 (4) G NG d 2 d ÿ 2gP2 g @tg d ÿ 2 g ; P2 4 d 2g for the Gaussian (G) and the non-Gaussian (NG) fixed points, respectively. The eigenvalues at criticality have with polynomials P1;2 ; g; d, opposite signs, the Gaussian one being infrared attractive 3 2 2 3 and the non-Gaussian one being ultraviolet attractive. P1 ÿ16 4 4 ÿ 10dg ÿ 3d g d g They are degenerate in two dimensions. Away from the 2 ÿ 3 ÿ 4 10dg d g d g 1 fixed points, the flow (6) can be solved analytically for d 2 d d ÿ 16g 8dg ÿ 3g; arbitrary scales k. With the initial condition g at k , P 8 2 ÿ ÿ dg2: the solution gk for any k is 2 ÿ1= ÿ1= d d=2ÿ1 gk G g ÿ gk NG k A numerical factor cd ÿ 2 2 4 originating : (8) mainly from the momentum trace in (1) is scaled into g g ÿ g the gravitational coupling g ! g=c , unless indicated d Figure 1 shows the crossover from the infrared to the otherwise. The graviton anomalous dimension is given by ultraviolet fixed point in the analytic solution (8) in four d ÿ 2 d 2g : (5) 1 2 6 d ÿ 2g ÿ 2 ÿ 2 g It vanishes for vanishing gravitational coupling, in two 4 and minus two dimensions, or for diverging .Ona nontrivial fixed point the vanishing of g implies 2 2 ÿ d and reflects the fact that the gravitational coupling IR UV is dimensionless in 2d. This behavior leads to modifica- 0 η tions of the graviton propagator at large momenta, e.g., [7]. The flows (4) are finite except on the boundary -2 2 gbound 2 ÿ 1 = 2d ÿ 4 derived from 1= 0.