A proper fixed functional for four-dimensional Quantum Einstein Gravity

Published for SISSA by Springer Received: June 11, 2015 Accepted: August 3, 2015 Published: August 24, 2015 Maximilian Demmel,a,b Frank Saueressigb and Omar Zanussoc a PRISMA Cluster of Excellence & Institute of Physics (THEP), University of Mainz, Staudingerweg 7, D-55099 Mainz, Germany b Institute for Mathematics, Astrophysics and Particle Physics (IMAPP), Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands c Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany E-mail: , , Abstract: Realizing a quantum theory for gravity based on Asymptotic Safety hinges on the existence of a non-Gaussian fixed point of the theory’s renormalization group flow. In this work, we use the functional renormalization group equation for the effective average action to study the fixed point underlying Quantum Einstein Gravity at the functional level including an infinite number of scale-dependent coupling constants. We formulate a list of guiding principles underlying the construction of a partial differential equation encoding the scale-dependence of f (R)-gravity. We show that this equation admits a unique, globally well-defined fixed functional describing the non-Gaussian fixed point at the level of functions of the scalar curvature. This solution is constructed explicitly via a numerical double-shooting method. In the UV, this solution is in good agreement with results from polynomial expansions including a finite number of coupling constants, while it scales proportional to R2 , dressed up with non-analytic terms, in the IR. We demonstrate that its structure is mainly governed by the conformal sector of the flow equation. The relation of our work to previous, partial constructions of similar scaling solutions is discussed. Keywords: Models of Quantum Gravity, Nonperturbative Effects, Renormalization Group ArXiv ePrint: 1504.07656 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP08(2015)113 JHEP08(2015)113 A proper fixed functional for four-dimensional Quantum Einstein Gravity Contents 1 2 4 2 Isolated global solutions: a toy model 8 3 The flow equation for fk (R) 3.1 Projecting the RG flow 3.2 Smoothing the staircase 3.3 Conformally reduced flow equation 10 11 14 15 4 Analytical properties of the stationary flow equation 4.1 Singularity structure 4.2 Asymptotic IR behavior of the solutions 16 16 17 5 Numerical construction of the fixed function 5.1 Implementing regularity at the fixed singularities 5.2 Extending the local solutions away from r = 0 5.3 Properties of the complete fixed function 19 19 21 24 6 Conclusions and discussion 26 A Fixed functions of the conformally reduced flow equation A.1 Fixed functions from the shooting method A.2 Fixed functions from bootstrapping method 28 29 31 1 Introduction Renormalization group (RG) fixed points are crucial for determining the properties of the underlying quantum field theory. They are lighthouses on theory space because in their vicinity one can explore the corresponding quantum field theory via scaling techniques and, eventually, via conformal field theory methods. Fixed points also play a crucial role in providing a general notion of renormalizability and ultraviolet completion of a quantum theory. In fact, in the vicinity of an ultraviolet (UV) fixed point the RG flow can be linearized and characterized by directions along which it is attracted to or repelled from the fixed point for increasing energies, thus providing a generalized notion of relevant and irrelevant deformations of the theory. The UV completeness of the theory is then ensured by finding a given RG trajectory that is attracted towards the fixed point at high energies, implying that any UV cutoff can be removed in such a way that all essential dimensionless –1– JHEP08(2015)113 1 Introduction 1.1 Quantum gravity via Asymptotic Safety 1.2 From fixed points to fixed functions 1.1 Quantum gravity via Asymptotic Safety The RG flow of any theory admits a GFP at which the theory itself becomes noninteracting. In the neighborhood of the Gaussian point it is thus possible to use the methods of perturbation theory to investigate the UV completion of the theory under consideration. These methods have been applied very successfully to Yang-Mills theories thanks to the property of asymptotic freedom. However, it has been known for many years that the GFP cannot provide a UV completion mechanism of General Relativity [1–3]. The question whether gravity can be formulated as a consistent and predictive (asymptotically safe) quantum field theory then becomes closely intertwined with the question whether there are other fixed points of the RG on the gravitational theory space which could provide such a UV completion. Indeed it has been conjectured already in the late seventies that there exists a suitable NGFP on the gravitational theory space which could render gravity asymptotically safe [4, 5]. The primary tool for investigating the Asymptotic Safety mechanism has been a Wilsonian-type functional renormalization group equation (FRGE) [6–8]   −1 1 (2) k∂k Γk = Tr Γk + Rk k∂k Rk , 2 (2) (1.1) in which Γk denotes an effective “average” action, Γk is its second variation with respect to the fields, and Rk is a suitable infrared-regulator which ensures that the r.h.s. of the equation is finite and peaked around the renormalization group scale k. The effective average action Γk represents an effective action of the system in which the UV modes have been integrated out in the path-integral using the scale k as a reference, thus providing a natural effective action for processes occurring at energies E ≃ k. In the limit k → 0 the IR cutoff Rk vanishes and the effective average action coincides with the full effective action of the system Γ = Γk=0 . The flow (1.1) is exact in the sense that no approximation is used in its derivation. –2– JHEP08(2015)113 coupling constants remain finite. Such an UV complete theory is also predictive if the dimension of the critical hypersurface that is attracted towards the fixed point in the UV is finite. In this case, only a finite number of experiments is required to pinpoint the specific RG trajectory that completes gravity in the deep UV regime. Depending on whether the fixed point corresponds to a free (Gaussian fixed point) or interacting theory (non-Gaussian fixed point), the RG trajectories terminating at the fixed point in the UV are termed asymptotically free or asymptotically safe, respectively. While it is well known that the Gaussian fixed point (GFP) of gravity does not provide a suitable mechanism to renormalize the theory, the possibility that the theory possesses at least one suitable non-Gaussian fixed point (NGFP) is not speculative and has been explored under various convincing approximations. In this work we investigate the Asymptotic Safety mechanism for gravity relaxing some of the (...truncated)