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PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 20 21 MAY 2004

Fixed Points of

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 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 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 ˆ lnk is given by available [4], increasing the domain of validity and the @ ÿ ˆ 1Tr ÿ 2† ‡ R†ÿ1@ 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 ÿk‰gŠ 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 ˆ d x g‰ÿR g†‡2kŠ: (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  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 ˆ d ÿ 2†g modulo subleading corrections. This en- constant at some reference scale k ˆ ,andZN;k the tails  ˆ 0. Then the flow is trivially solved by gk ˆ dÿ2 ÿ2 wave function renormalization factor for the Newtonian g k=† and k ˆ  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=†d‡2 g ˆ kdÿ2G  kdÿ2Zÿ1 GG;  ˆ 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 2†g g ˆ : (6) fluctuations in the metric field in the Feynman gauge [6] 1 ‡ 2 2 ÿ d†g 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 ˆ 0 and a non-Gaussian one at g ˆ 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 in Yang- R ˆ 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 ˆÿ. We find P  @  ˆ 1 ; d ÿ 2  t P ‡ 4 d ‡ 2†g  ˆ 2 ÿ d;  ˆ 2d (7) 2 (4) G NG d ‡ 2 d ÿ 2†gP2 g  @tg ˆ d ÿ 2 ‡ †g ˆ ; P2 ‡ 4 d ‡ 2†g 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 ˆÿ16 ‡ 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 3†g; arbitrary scales k. With the initial condition g at k ˆ , P ˆ 8 2 ÿ  ÿ dg†‡2: the solution gk for any k is 2     ÿ1= ÿ1= d d=2ÿ1 gk G g ÿ gk NG k A numerical factor cd ˆ ÿ 2 ‡ 2† 4† originating ˆ : (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 ‡ 2†g  ˆ : (5) 1 2 6 d ÿ 2†g 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  ˆ 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= ˆ 0. -4 -2 0 2 4 It signals a breakdown of the truncation (2). Some trajec- ln(k/MPI) tories terminate at g  gbound. The boundary is irrelevant as soon as the couplings enter the domain of canonical FIG. 1 (color online). Running of couplings according to (6) scaling: the limit of large jj implies that  ˆÿ2 and and (8). 201301-2 201301-2 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 20 21 MAY 2004 dimensions (and with c4 in g reinserted). The corre- Thus, disconnected regions of renormalization group sponding crossover momentum scale is associated with trajectories are characterized by whether g is larger or ÿ1=2 the Planck mass, MPl ˆ G† . More generally, (8) is a smaller gbound and by the sign of g. Since  changes sign solution at  ˆ 0 for an arbitrary gauge fixing parameter only across the lines  ˆ 0 or 1= ˆ 0, we conclude that and a regulator. In these cases, the exponents  and the the graviton anomalous dimension has the same sign fixed point g turn into functions of the latter. It is along any trajectory. In the physical domain which in- reassuring that the eigenvalues display only a mild de- cludes the ultraviolet and the infrared fixed points, the pendence on these parameters. gravitational coupling is positive and the anomalous di- Now we proceed to the nontrivial fixed points implied mension negative. In turn, the cosmological constant may by the simultaneous vanishing of g and , given in (3). change sign on trajectories emanating from the ultraviolet 1 2 A first consequence of P2 ˆ 0 is g ˆ  ÿ 2† =d, fixed point. Some trajectories terminate at the boundary which, when inserted into P1=g ˆ 0, reduces the fixed gbound †, linked to the present truncation (cf. Fig. 2). The point condition to a quadratic equation with two real two fixed points are connected by a separatrix. In Fig. 2, solutionsp ;g† Þ 0 as long as d  d‡, where d ˆ it has been given explicitly in four dimensions (with the 1 2 1  17†. The two branches of fixed points are charac- factor c4 in g reinstalled). Integrating the flow starting in 1 terized by  being larger or smaller than 2 . The branch the vicinity of the ultraviolet fixed point and fine-tuning 1 with   2 displays an unphysical singularity at four the initial condition leads to the trajectory which runs dimensions and is therefore discarded. Hence, into the Gaussian fixed point. The rotation of the separa- p trix about the ultraviolet fixed point reflects the complex d2 ÿ d ÿ 4 ÿ 2d d2 ÿ d ÿ 4†  ˆ ; nature of the eigenvalues (10). At k  MPl,theflow dis-  2 d ÿ 4† d ‡ 1† (9) plays a crossover from ultraviolet dominated running to d ‡ † †d=2ÿ1 infrared dominated running. For the running couplings 2ÿ 2 2 4 2 g ˆ : and  this behavior is displayed in Fig. 3. The nonvanish- d2 ÿ d ÿ 4 ing cosmological constant modifies the flow mainly in the In (9), we have reinserted the numerical factor cd.The crossover region rather than in the ultraviolet. A similar solutions (9) are continuous and well defined for all d  behavior is expected for operators beyond the truncation d‡  2:56 and become complex for lower dimensions. (2). This is supported by the stability of the fixed point The critical exponents associated with (9) are derived under R2 g† corrections [8], and by the weak dependence from the stability matrix at criticality. In the most inter- on the gauge fixing parameter. In the infrared limit, the esting case d ˆ 4, the two eigenvaluesp are a complex separatrix leads to a vanishing cosmological constant 0 00  2 conjugate pair     i ˆ 5  i 167†=3,or k ˆ kk ! 0. Therefore, it is interpreted as a phase 0 ˆ 1:667;00 ˆ 4:308: (10) transition boundary between cosmologies with positive or negative cosmological constant at large distances. This The eigenvalues remain complex for all dimensions picture agrees very well with numerical results for a 2:56 4 has revealed that the fixed 223 (2002); J. Polonyi, Centr. Eur. Sci. J. Phys. 1,1 (2002); D. F. Litim and J. M. Pawlowski, hep-th/9901063. point exists up to some finite dimension, where  reached the boundary of the domain of validity [12]. No [4] D. F. Litim, Phys. Lett. B 486, 92 (2000); Phys. Rev. D definite conclusion could be drawn for larger dimensions. 64, 105007 (2001); Int. J. Mod. Phys. A 16, 2081 (2001). [5] D. F. Litim, Nucl. Phys. B631, 128 (2002); J. High Energy Based on an optimized flow, the main new result here is Phys. 11 (2001) 059; Acta Phys. Slov. 52, 635 (2002). that a nontrivial ultraviolet fixed point exists within the [6] M. Reuter, Phys. Rev. D 57, 971 (1998). domain of the validity of (4) for arbitrary dimension. [7] O. Lauscher and M. Reuter, Phys. Rev. D 65, 025013 Furthermore, the fixed point is smoothly connected to (2002). its 4d counterpart and shows only a weak dependence on [8] O. Lauscher and M. Reuter, Classical Quantum Gravity the gauge fixing parameter. In the limit of arbitrarily 19, 483 (2002); Phys. Rev. D 66, 025026 (2002). large dimensions, this leads to  ! 1=2 and g ! [9] M. Reuter and C. Wetterich, Phys. Rev. D 56, 7893 2 cd= 2d † in the explicit solution (9). Similar results are (1997); F. Freire, D. F. Litim, and J. M. Pawlowski, obtained for an arbitrary gauge fixing parameter. Note Phys. Lett. B 495, 256 (2000). [10] D. F. Litim and J. M. Pawlowski, Phys. Lett. B 435,181 that  approaches its boundary value very slowly, in- creasing from 1=4 to 0.4 for d ranging from 4 to 40. Using (1998); J. High Energy Phys. 09 (2002) 049; Phys. Lett. B 546, 279 (2002); J. M. Pawlowski, Acta Phys. Slov. 52, (9) and the definition for g , we derive g  †= bound bound  475 (2002); Int. J. Mod. Phys. A 16, 2105 (2001); H. Gies, g ˆ 2d†= d ÿ 2† at the fixed point. Hence, g stays clear Phys. Rev. D 66, 025006 (2002); 68, 085015 (2003); by at least a factor of 2 from the boundary where 1= V. Branchina, K. A. Meissner, and G. Veneziano, Phys. vanishes, and the ultraviolet fixed point resides within the Lett. B 574, 319 (2003). domain of validity of (4) even in higher dimensions. This [11] W. Souma, Prog. Theor. Phys. 102, 181 (1999). stability of the fixed point also strengthens the result in [12] M. Reuter and F. Saueressig, Phys. Rev. D 65, 065016 lower dimensions, including d ˆ 4. (2002). In summary, we have found analytic results for the flow [13] R. Percacci and D. Perini, Phys. Rev. D 67, 081503 and a nontrivial ultraviolet attractive fixed point of quan- (2003); 68, 044018 (2003). tum gravity. Maximal reliability of the present truncation [14] P. Forgacs and M. Niedermaier, hep-th/0207028; is guaranteed by the underlying optimization. The fixed M. Niedermaier, J. High Energy Phys. 12 (2002) 066; Nucl. Phys. B673, 131 (2003). point is remarkably stable with only a mild dependence on [15] H.W. Hamber, Phys. Rev. D 61, 124008 (2000); 45,507 the gauge fixing parameter. Furthermore, it extends to (1992). dimensions higher than d ˆ 4, a region which previously [16] A. Bonanno and M. Reuter, Phys. Rev. D 62, 043008 has not been accessible. The qualitative structure of the (2000); 65, 043508 (2002); Phys. Lett. B 527, 9 (2002); phase diagram in the Einstein-Hilbert truncation is E. Bentivegna, A. Bonanno, and M. Reuter, J. Cosmol. equally robust. In four dimensions the results match Astropart. Phys. 01, 001 (2004).

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