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Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Phil. Trans. R. Soc. A (2011) 369, 2759–2778 doi:10.1098/rsta.2011.0103 REVIEW Renormalization group and the Planck scale B Y D ANIEL F. L ITIM* Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK I discuss the renormalization group approach to gravity, and its link to Weinberg’s asymptotic safety scenario, and give an overview of results with applications to particle physics and cosmology. Keywords: quantum gravity; Planck scale; renormalization group; asymptotic safety; low-scale quantum gravity 1. Introduction Einstein’s theory of general relativity is the remarkably successful classical theory of the gravitational force, characterized by Newton’s coupling constant GN = 6.67 × 10−11 m3 kg−1 s−2 and a small cosmological constant L. Experimentally, its validity has been confirmed over many orders of magnitude in length scales ranging from the sub-millimetre regime up to Solar System size. At larger length scales, the standard model of cosmology including dark matter and dark energy components fits the data well. At shorter length scales, quantum effects are expected to become important. An order-of-magnitude estimate for the quantum scale of gravity—the Planck scale—is obtained by dimensional analysis, leading to the Planck length Pl ≈ h̄GN /c 3 of the order of 10−33 cm, with c the speed of light. In particle physics units this translates into the Planck mass MPl ≈ 1019 GeV. (1.1) While this energy scale is presently out of reach for Earth-based particle accelerator experiments, fingerprints of Planck-scale physics can nevertheless become accessible through cosmological data from the very early Universe. From a theory perspective, it is widely expected that a fundamental understanding of Planck-scale physics requires a quantum theory of gravity. It is well known that the standard perturbative quantization programme faces problems, and a fully satisfactory quantum theory, even outside the framework of local quantum physics, is presently not to hand. In the past 15 years, however, a significant body of work has been devoted to re-evaluate the physics of the Planck scale within conventional settings. Much of this renewed interest is fuelled by *[email protected] One contribution of 11 to a Theme Issue ‘New applications of the renormalization group in nuclear, particle and condensed matter physics’. 2759 This journal is © 2011 The Royal Society Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2760 D. F. Litim Steven Weinberg’s seminal proposal, more than 30 years ago, that a quantum theory of gravitation may dynamically evade the virulent divergences encountered in standard perturbation theory [1]. This scenario, known as asymptotic safety, implies that gravity achieves a non-trivial ultraviolet (UV) fixed point under its renormalization group (RG) flow. If so, this would incorporate gravity alongside the set of well-understood quantum field theories whose UV behaviour is governed by a fixed point, e.g. Yang–Mills theory. In this paper, I discuss the RG approach to gravity. I recall some of the issues of perturbative quantum gravity (§2), introduce the RG à la Wilson to access the physics at the Planck scale (§3), review key results in four dimensions (§4), evaluate applications within low-scale quantum gravity (§5) and conclude (§6). 2. Perturbative quantum gravity A vast body of work has been devoted to the perturbative quantization programme of gravity. I recall a very small selection of these in order to prepare for the subsequent discussion, and I use particle physics units h̄ = c = kB = 1 throughout. Classical general relativity is described by the classical action 1 d4 x det gmn (−R(gmn ) + 2L), S= (2.1) 16pGN where I have chosen a Euclidean signature, R denotes the Ricci scalar, and L is the cosmological constant term. The main point to be stressed is that the fundamental coupling of gravity GN in equation (2.1) carries a dimension that sets a mass scale, with mass dimension [GN ] = 2 − d in d-dimensional space–time. This structure distinguishes gravity in a profound manner from the other fundamentally known interactions in Nature, all of which have dimensionless coupling constants from the outset. Consequently, the effective dimensionless coupling of gravity that organizes its perturbative expansion is given by geff ≡ GN E 2 (2.2) (in four dimensions), where E denotes the relevant energy scale. While the effective coupling (2.2) remains small for energies below the Planck scale E MPl , it grows large in the Planckian regime where an expansion in geff may become questionable. Within the Feynman diagrammatic approach, this behaviour translates into the (with loop order) increasing degree of divergence of perturbative diagrams involving gravitons [1]. This structure is different from standard quantum field theories and relates to the classification of interactions as super-renormalizable, renormalizable or ‘dangerous’, depending on whether their canonical mass dimension is positive, vanishing or negative. The degree of divergences implied by the negative mass dimension of Newton’s coupling is mirrored in the perturbative non-renormalizability of Einstein gravity, which has been established at the one-loop level [2] in the presence of matter fields, and at the two-loop level [3] within pure gravity. While this state of play looks discouraging from a field theory perspective, it does not rule out a quantum field theoretical description of gravity altogether. There are a few indicators available to support this view. Firstly, at low energies, Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2761 a weak-coupling analysis of quantum gravity effects is possible within an effective theory approach [4], which operates a UV cutoff at the Planck scale; see [5] for a review. Secondly, higher-order derivative operators appear to stabilize perturbation theory [6]. Including all fourth-order derivative operators, it has been proven by Stelle [7] that gravity is renormalizable to all orders in perturbation theory. This striking difference with Einstein–Hilbert gravity highlights the stabilizing effect of higher derivative terms. Unfortunately, the resulting theory is not compatible with standard notions of perturbative unitarity and has therefore not been considered as a candidate for a fundamental theory of gravity. The role of higher-order derivative operators has further been clarified in Gomis & Weinberg [8] using cohomology techniques. Interestingly, the theory remains unitary once all higher derivative operators are retained, whereas renormalizability is at best achieved in a very weak sense owing to the required infinitely many counter-terms. 3. Renormalization group The RG comes into play when the running of couplings with energy is taken into account. As in any generic quantum field theory, quantum fluctuations modify the strength of couplings. If the metric field remains the fundamental carrier of the gravitational force, the fluctuations of space–time itself should modify the gravitational interactions with energy or distance. For Newton’s coupling, this implies that GN becomes a running coupling GN → G(k) = GN Z −1 (k) as a function of the RG momentum scale k, where Z (k) denotes the wave function renormalization factor of the graviton. Consequently, the dimensionless coupling geff in (2.2) should be replaced by the running coupling g = G(k)k 2 , (3.1) which evolves with the RG scale. In particular, the UV behaviour of standard perturbation theory is significantly improved, provided that equation (3.1) remains finite in the high-energy limit. This is the asymptotic safety scenario as advocated by Weinberg [1]; see Niedermaier [6] and Niedermaier & Reuter [9] for extensive accounts of the scenario, and Litim [10,11] and Percacci [12] for brief overviews. The intimate link between a fundamental definition of quantum field theory and RG fixed points was stressed by Wilson some 40 years ago [13,14]. For gravity, the fixed-point property becomes visible by considering the Callan– Symanzik-type RG equation for (3.1), which in d dimensions takes the form [10] (see also [6,9]) (3.2) vt g ≡ bg = (d − 2 + h)g, with h = −vt ln Z (k) the graviton anomalous dimension, which in general is a function of all couplings of the theory including matter, and t = ln k. This simple structure arises provided the underlying effective action is local in the metric field. From the RG equation (3.2), one concludes that the gravitational coupling may display two types of fixed points. The non-interacting (Gaussian) fixed point corresponds to g∗ = 0 and entails the vanishing of the anomalous dimension h = 0. In its vicinity, gravity stays classical and G(k) ≈ GN . On the other hand, Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2762 D. F. Litim a non-trivial RG fixed point with g∗ = 0 can be achieved implicitly, provided that the anomalous dimension reads h∗ = 2 − d. (3.3) The significance of equation (3.3) is that the graviton anomalous dimension precisely counterbalances the canonical dimension of Newton’s coupling GN . This pattern is known from other gauge systems at a critical point away from their canonical space–time dimensionality [10], e.g. U(1) Higgs theory in three dimensions [15]. In consequence, the dimensionful, renormalized coupling scales as G(k) ≈ g∗ /k d−2 and becomes small in the UV limit where 1/k → 0. This pattern is at the root of the non-perturbative renormalizability of quantum gravity within a fixed-point scenario. Much work has been devoted to checking by explicit computation whether or not the gravitational couplings achieve a non-trivial UV fixed point. A versatile framework to address this question is provided by modern (functional) RG methods, based on the infinitesimal integrating-out of momentum degrees of freedom from a path integral representation of the theory à la Wilson [16–20]. This is achieved by adding a momentum cutoff to the action, quadratic in the propagating fields. In consequence, the action (2.1) becomes a scale-dependent effective or flowing action Gk , 1 d √ (−R(gmn ) + 2Lk ) + · · · + Sk,gf + Sk,gh + Sk,matter , (3.4) Gk = d x g 16pGk which in the context of gravity contains a running gravitational coupling, a running cosmological constant Lk , a gauge-fixing term, ghost contributions, matter interactions and the dots indicate possible higher derivative operators in the metric field. Upon varying the RG scale k, the effective action interpolates between a microscopic theory GL at the UV scale k = L (not to be confused with the running cosmological constant Lk ) and the macroscopic quantum effective action G, where all fluctuations are taken into account (k → 0). The variation of equation (3.4) with RG scale k is given by an exact functional differential equation [18,21] 1 1 1 vt R k = , (3.5) vt Gk = Tr (2) 2 2 G + Rk k which relates the change of the scale-dependent gravitational action with the full field-dependent propagator of the theory (full line) and the scale dependence vt Rk of the momentum cutoff (the insertion). Here, the trace stands for a momentum integration, and a sum over all propagating degrees of freedom f = (gmn , ghosts, matter fields), which minimally contains the metric field and its ghosts. The function Rk (not to be confused with the Ricci scalar) denotes the infrared (IR) momentum cutoff. As a function of (covariant) momenta q 2 , the momentum cutoff obeys Rk (q 2 ) → 0 for k 2 /q 2 → 0, Rk (q 2 ) > 0 for q 2 /k 2 → 0 and Rk (q 2 ) → ∞ for k → L (for examples and plots of Rk , see Litim [22]). The functional flow (3.5) is closely linked to other exact functional differential equations, such as the Callan– Symanzik equation [23] in the limit Rk → k 2 , and the Wilson–Polchinski equation by means of a Legendre transformation [12,17,19]. In a weak-coupling expansion, equation (3.5) reproduces standard perturbation theory to all loop orders [24,25]. Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2763 The strength of the formalism is that it is not bounded to the weak-coupling regime, and systematic approximations—such as the derivative expansion, vertex expansions or mixtures thereof—are available to access domains with strong coupling and/or strong correlation. Systematic uncertainties can be assessed [26], and the stability and convergence of approximations is enhanced by optimization techniques [22,27–30]. By construction, the flow (3.5) is both UV and IR finite. Together with the boundary condition GL it may serve as a definition for the theory. In renormalizable theories, the continuum limit is performed by removing the scale L, 1/L → 0, and the functional GL → G∗ approaches a fixed-point action at short distances (see the proposal in Manrique & Reuter [31] for a construction in gravity). In perturbatively renormalizable theories, G∗ is ‘trivial’ and given mainly by the classical action. In perturbatively non-renormalizable theories, the existence (or non-existence) of G∗ has to be studied on a case-by-case basis. A viable fixed-point action G∗ in quantum gravity should at least contain those diffeomorphism-invariant operators that display relevant or marginal scaling in the vicinity of the UV fixed point. A fixed-point action qualifies as fundamental if RG trajectories k → Gk emanating from its vicinity connect with the correct long-distance behaviour for k → 0 and stay well defined (finite, no poles) at all scales [1]. The operator trace in equation (3.5) is evaluated by using flat or curved backgrounds together with heat kernel techniques or plain momentum integration. To ensure diffeomorphism symmetry within this set-up, the background field formalism is used by adding a non-propagating background field ḡmn [21,30,32–34]. This way, the extended effective action Gk [gmn , ḡmn ] becomes gauge-invariant under the combined symmetry transformations of the physical and the background field. A second benefit of this is that the background field can be used to construct a covariant Laplacian −D̄ 2 (or similar) to define a mode cutoff at the RG momentum scale k 2 = −D̄ 2 . This implies that the mode cutoff Rk will depend on the background fields, which is controlled by an equation similar to the flow equation itself [34,35]. The background field is then eliminated from the final equations by identifying it with the physical mean field. This procedure dynamically readjusts the background field and implements the requirements of background independence for quantum gravity. An alternative technique that employs a bi-metric approximation has been put forward in Manrique & Reuter [36] and Manrique et al. [37,38]. For a general evaluation of the different implementations of a momentum cutoff in gauge theories, see [30,32,34]. 4. Fixed points of quantum gravity In this section, I give a brief summary of fixed points found so far, and refer to Litim [12] for a more detailed overview of results prior to 2008. The search for fixed points in quantum gravity starts by restricting the running effective action Gk to a finite set of operators Oi (f) with running couplings gi , Gk = gi (k)Oi (f), (4.1) i Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2764 D. F. Litim 0.20 =4 0.15 0.10 0.05 =3 =2 0 0 0.2 0.4 0.6 0.8 1.0 Figure 1. Dependence of the UV fixed point on space–time dimensionality. √ including, for example, the Ricci scalar gR and the cosmological constant √ g. The flow equations for the couplings gi are obtained from equation (3.5) by projection onto some subspace of operators. Convergence and stability of approximations are checked by increasing the number of operators retained, and further improved through optimized choices of the momentum cutoff [22,29]. The first set of flow equations that carry a non-trivial UV fixed point has been derived in Reuter [21] for the Einstein–Hilbert theory in Feynman gauge. As a function of the number of space–time dimensions, the flow equation reproduces the well-known fixed point in d = 2 + e dimensions [1,39–41] (figure 1). More importantly, the equations display a non-trivial UV fixed point in the four-dimensional theory [21,42]. Both the Ricci scalar and the cosmological constant term are relevant operators at the fixed point. Their scaling is strongly correlated, leading to a complex conjugate pair of universal scaling exponents. This result has subsequently been confirmed within a more general background field gauge by means of a trace-less transverse decomposition of the metric field [43]. A complete analytical understanding of the Einstein–Hilbert approximation has been achieved in Litim [44], leading to analytical flow equations and closed expressions for the (unique) UV fixed point and its universal eigenvalues. Key for this was the use of (optimized) momentum cutoffs that allow analytical access and improve convergence and stability of results [22,27]. Figure 2 shows the universal scaling exponents q = q + iq at the UV fixed point in four dimensions (left-hand side) for various (optimized) choices of the momentum cutoff [27,45]. Also on display is the invariant t = l∗ (g∗ )2/(d−2) . The variations in either of these are small. The dependence on the gravitational gauge-fixing parameter a is moderate and controlled by an independent fixed point in the gauge-fixing sector, i.e. Landau– de Witt gauge (a = 0) [46]. Figure 3 shows the RG running of couplings along the separatrix connecting the non-trivial UV fixed point with the Gaussian fixed point in the IR. At the scale k ≈ LT , the RG flow displays a cross-over from Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2765 Review. RG and the Planck scale (a) q ¢/q ¢opt 1.10 1.05 1.00 (b) q ¢¢/qopt ¢¢ 1.05 1.00 |q | / |qopt| (c) 1.05 1.00 t /topt (d) 1.00 0.98 0.96 4 5 6 7 8 9 10 11 Figure 2. Ultraviolet fixed point in the d-dimensional Einstein–Hilbert theory. Comparison of universal eigenvalues q , q , |q| and the invariant t = l∗ (g∗ )2/(d−2) for different Wilsonian momentum cutoffs and various dimensions, normalized to the result for Ropt (Rmexp , circles; Rexp , squares; Rmod , triangles; Ropt , inverted triangles). Adapted from Fischer & Litim [45]. (Online version in colour.) perturbative IR scaling to fixed-point scaling, where the anomalous dimension grows large. The dynamical scale LT is of the order of the fundamental Planck scale, and a consequence of the UV fixed point. In gravity, the scale LT plays a role analogous to that of LQCD in quantum chromodynamics. Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2766 D. F. Litim 2 1 0 –1 –2 –3 –4 –4 –2 0 2 4 Figure 3. Running couplings along the separatrix in four dimensions; adapted from Litim [44]. (Online version in colour.) In a formidable tour-de-force computation, the space of operators has been √ 2 gR interactions [47,48]. The perturbatively marginal R2 extended to include coupling turned out to become relevant at a UV fixed point. Otherwise, it added only minor corrections to the fixed point in the Einstein–Hilbert theory. With Litim [27,44], these computations became feasible analytically and have been √ gfk (R), which contain a general function of the implemented for operators Ricci scalar, in Codello et al. [49,50] and Machado & Saueressig [51]. A nontrivial UV fixed point has been found by Taylor-expanding fk (R) to high orders in the Ricci scalar, with results known up to eighth order in the eigenvalues [50] and up to tenth order in the fixed point [52]. Additionally, the fixed point is remarkably stable. The universal eigenvalues converge rapidly, confirming all √ n gR from previous results. Most importantly, it turned out that operators n ≥ 3 onwards become irrelevant at the UV fixed point [49], a first indicator that an asymptotically safe UV fixed point of the fundamental theory may, in fact, have a finite number of relevant operators. This is most welcome as otherwise a fixed point with infinitely many relevant couplings is likely to spoil the predictive power. All fourth-order derivative operators have been included both at the oneloop order and beyond. The re-evaluation of perturbative one-loop results by means of a functional flow emphasized the significance of, for example, quadratic divergences that are normally suppressed within dimensional regularization [53–55]. Two types of UV fixed points are found. The first one is non-trivial in all couplings [56,57], with three relevant directions at the fixed point. The second one is perturbative in the Weyl coupling [53,54] but non-trivial in Newton’s coupling, with two relevant and two marginally relevant directions. In either case Newton’s coupling takes a non-trivial fixed point in the UV. These results hence indicate that higher derivative operators are compatible with gravity developing a non-trivial fixed point. Once this is achieved, the difference between the Einstein– Hilbert approximation and the fourth-order approximation is qualitatively small. In the above, the RG running of the ghost sector is expected to be sub-leading. This has been recently confirmed in Groh & Saueressig [58] and in Eichhorn & Gies [59], and for the scalar curvature–ghost coupling in Eichhorn et al. [60]. Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2767 An important extension deals with the inclusion of matter couplings. In Yang– Mills theory, for example, it is well known that the property of asymptotic freedom can be spoiled by too many species of fermions. In the same vein it has to be checked whether the UV fixed point of gravity remains stable under the inclusion of matter, and vice versa. By now, this question has been analysed for minimally coupled scalar matter [61], non-minimally coupled scalar fields [62,63] and generic free matter coupled to gravity [53,64]. These findings are consistent with earlier results using different techniques in the limit of many matter fields [65,66]. The conclusion is that the gravitational fixed point generically persists under the inclusion of matter (including the Standard Model (SM) of particle physics and its main extensions) unless a significant imbalance between bosonic and fermionic matter fields is chosen. Also, non-minimally coupled scalar matter leads to slight deformation of the gravitational fixed point with matter achieving a weakly coupled regime, the so-called Gaussian matter fixed point. Fermions have been included recently in Zanusso et al. [67], as well as gravitational Yukawa systems [68]. In the latter case, the coupled system displays a non-trivial UV fixed point both in the Yukawa as well as in the gravitational sector. This new fixed point may become of great interest as a stabilizer for the SM Higgs. On the one-loop level, the interplay between an SM Higgs and asymptotically safe gravity has been addressed in Shaposhnikov & Wetterich [69], detailing conditions under which no new physics is required to bridge the energy range from the electroweak scale all the way up to equation (1.1). In a similar spirit, it has been argued that the consistency of Higgs inflation models is enhanced provided that gravity becomes asymptotically safe [70]. The impact of gravitational fluctuations on gauge theories has seen renewed interest initiated by a one-loop study within effective theory [71]. Asymptotic freedom is sustained by gravity, leading to a vanishing [35,72–75] or non-vanishing one-loop correction [35,71,76–78], depending on the regularization. In the present framework (3.4) and (3.5), the gravity-induced corrections have been studied in Daum et al. [78] and in Folkerts et al. [35]. In either case, the sign of the graviton contributions is fixed and asymptotic freedom of Yang–Mills theory persists in the presence of gravitational fluctuations, also including a cosmological constant. Furthermore, Folkerts et al. [35] evaluate generic regularizations and clarify the non-universal nature of the one-loop coefficient, also covering generic field theory-based UV scenarios for gravity, including asymptotically safe gravity and gravitational shielding. An interesting consequence of a non-trivial gravitational contribution to the running of an Abelian charge is the appearance of a combined UV fixed point in the U(1)–gravity system providing a mechanism to calculate the fine-structure constant [79]. In three dimensions, a fixed-point study has been reported within topological massive gravity, which includes a Chern–Simons (CS) term in addition to Newton’s coupling and the cosmological constant [80]. On the one-loop level, the CS term has a vanishing beta function and the gravitational sector displays both a Gaussian and a non-trivial fixed point for positive CS coupling, which is qualitatively in accord with the fixed point in figure 1 (no CS term). An interesting structural link has been noted between the gravitational fixed point [44] (Ricci scalar, no cosmological constant) and a UV fixed point in nonlinear sigma models [81]. In either system, the beta function of the relevant coupling displays a UV fixed point whose coordinates are non-universal. However, Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2768 D. F. Litim the systems display the same universal scaling, with identical exponent q = 2d(d − 2)/(d + 2) for all d ≥ 2 dimensions. This coincidence highlights that nonlinear sigma models have a non-trivial UV dynamics on their own, possibly quite similar to gravity itself, which is worthy of being explored further [82]. Fixed points in more complex (gauged and non-gauged) nonlinear sigma models have recently been obtained in [83,84]. It would be very helpful to understand which mechanism implies the gravitational fixed point by, for example, restricting the dynamical content. In fact, a two Killing vector reduction of four-dimensional gravity (with scalar matter and photons) has been shown to be asymptotically safe [85]. This topic has also been studied by reducing gravity to the dynamics of its conformal sector [86,87], including higher derivative terms and effects from the conformal anomaly [88]. Although a UV fixed point is visible in most approximations, its presence depends more strongly on, for example, matter fields, meaning that more work is required to settle this question for pure gravity. Implications of a gravitational fixed point have been studied in the context of black holes, astrophysics and IR gravity, and cosmology. The RG-induced running GN → G(r) of Newton’s coupling has interesting implications for the existence of RG-improved black hole space–times [89] as well as for their dynamics [90]. If gravity weakens according to its asymptotically safe RG flow, results point towards the existence of a smallest black hole with critical mass Mc of the order of MPl . This pattern persists for Schwarzschild black holes in higher dimensions [91], rotating black holes [92] and black holes within higher derivative gravity [93], showing that the existence of smallest black holes is a generic and stable prediction of this framework. Low-energy implications of the RG running of couplings have equally been studied, including applications to galaxy rotation curves [94,95]. Recent studies have re-evaluated gravitational collapse [96], exploring the one-loop running couplings, and the RG-induced non-local low-energy corrections to the gravitational effective action [97]. It has also been conjectured that gravity may possess an IR fixed point. In this case, Newton’s coupling grows large at large distances and may contribute to an accelerated expansion of the Universe, a scenario that can be tested against cosmological data such as Type Ia supernovae [98–100]. The intriguing idea that an asymptotically safe UV fixed point may control the beginning of the Universe has been implemented in Einstein gravity with an ideal fluid [101,102], and in the context of f (R) theories of gravity [52,103– 105]. The RG scale parameter is linked with cosmological time and therefore leads to RG-improved cosmological equations [101,106,107], which may even generate entropy [108,109]. The framework put forward in Hindmarsh et al. [106] includes scalar fields and exploits the Bianchi identity, leading to general conditions of existence for cosmological fixed points (with cosmic time) whose solutions include, for example, inflating UV safe fixed points of gravity coupled to a scalar field. The proposal that Lorentz symmetry can be broken by gravity on a fundamental level has been put forward in Horava [110], where space and time scale differently in the UV with a relative dynamical exponent z = 1. Provided z = 3 (in four dimensions) gravity is power-counting renormalizable in perturbation theory and Lorentz symmetry should emerge as a low-energy phenomenon [111]. Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2769 Interestingly, Horava–Lifshitz gravity with z = 3 also requires an asymptotically safe fixed point for gravity with an anomalous dimension h = 0 compensating for power-counting non-renormalizability. The RG scaling in the vicinity of an asymptotically safe fixed point implies scale invariance. On the other hand, it is often assumed that a quantum theory of gravity should, in one way or another, induce a minimal length. Different aspects of this question and interrelations with other approaches to quantum gravity have been addressed in Reuter & Schwindt [112,113], and more recently in Basu & Mattingly [114], Percacci & Vacca [115] and Calmet et al. [116]. The findings indicate that the fixed point implicitly induces a notion of minimal length, related to the RG trajectory and the fundamental scale where gravity crosses over from classical to fixed-point scaling. It has also been argued that RG corrections to gravity can modify the dispersion relation of massive particles, leading to indirect bounds on the one-loop coefficients of the gravitational beta functions [117]. Finally, it is worth noting that standard quantum field theories (without gravity) may become asymptotically safe in their own right. This was first exemplified for perturbatively non-renormalizable Gross–Neveu models in three dimensions with functional RG methods [118] and in the limit of many fermion flavours [119]. In four dimensions, this possibility has been explored for variants of the SM Higgs [120], for Yukawa-type interactions [121,122], and for strongly coupled gauge theories [123]. It has also been conjectured that the electroweak theory without a dynamical Higgs field could become asymptotically safe [124], based on structural similarities with quantum general relativity. In summary, there is an increasing amount of evidence for the existence of non-trivial UV fixed points in four-dimensional quantum field theories, including gravity and matter. 5. Extra dimensions Next, I turn to fixed points of quantum gravity in d = 4 + n dimensions, where n denotes the number of extra dimensions [44,45,125], and potential signatures thereof in models with a low quantum gravity scale [126–130]. There are several motivations for this. Firstly, particle physics models where gravity lives in a higher-dimensional space–time have raised enormous interest by allowing the fundamental Planck scale to be as low as the electroweak scale [131–133], M∗ ≈ 1–10 TeV. (5.1) This way, the notorious hierarchy problem of the SM is circumnavigated. If realized in Nature, this opens the exciting possibility that Planck-scale physics becomes accessible to experiment via, for example, high-energetic particle collisions at the Large Hadron Collider (LHC). The main point of these models is that the four-dimensional Planck scale (1.1) is no longer fundamental but a 2 ∼ M∗2 (M∗ L)n to the Planck scale (5.1) of the derived quantity, related as MPl higher-dimensional theory and the size of the compact extra dimensions L. Secondly, the critical dimension of gravity—the dimension where the gravitational coupling has vanishing canonical mass dimension—is two. Hence, for any dimension above the critical one, the canonical dimension of Newton’s coupling is negative. From an RG point of view, this means that four dimensions Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2770 D. F. Litim (a) (b) Figure 4. Log–log plot (schematic) of the dimensionless running gravitational coupling in a scenario with large extra dimensions of size ∼ L and a dynamical Planck scale LT with LT L 1. The fixed-point behaviour in the deep UV enforces a softening of gravity. (a) At the scale k ≈ LT the coupling displays a cross-over from fixed-point scaling to classical scaling in the higher-dimensional theory. (b) At the compactification scale k ≈ 1/L, the n compactified spatial dimensions are no longer available for gravity to propagate in, and the running coupling displays a cross-over from (4 + n)-dimensional to four-dimensional scaling. appear not to be special. Continuity in the dimension suggests that a UV fixed point, if it exists in four dimensions, should persist towards higher dimensions. Furthermore, Einstein gravity has d(d − 3)/2 propagating degrees of freedom, which rapidly increases with increasing dimensionality. It is important to understand whether these additional degrees of freedom spoil or support the UV fixed point detected in the four-dimensional theory. The local structure of quantum fluctuations, and hence local RG properties of a quantum theory of gravity, are qualitatively similar for all dimensions above the critical one, modulo topological effects for specific dimensions. Therefore, one should expect to find similarities in the UV behaviour of gravity in four and higher dimensions. This expectation has been confirmed in Litim [44] for d-dimensional Einstein– Hilbert gravity. The stability of the result has subsequently been tested through an extended fixed points search in higher-dimensional gravity for general cutoffs Rk [45], also probing the stability of the fixed point against variations of the gauge fixing [125], and the results are displayed in figure 2. The cutoff variations are very moderate, and smaller than the variation with gauge-fixing parameter. The structural stability of the fixed point also strengthens the findings in the four-dimensional case. Thus, the following picture emerges. In the setting with compact extra dimensions, the running dimensionless coupling g displays a cross-over from fixedpoint scaling to d-dimensional classical scaling at the cross-over scale LT of the order of M∗ (figure 4a). By construction, this has to happen at energies way above 1/L. At lower energies, close to the compactification scale ∼ 1/L, the n compactified spatial dimensions are no longer available for gravity to propagate Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2771 Review. RG and the Planck scale (a) (b) M* (TeV) 6 5 4 3 2 1 0 1 2 3 4 5 6 7 (TeV) 8 9 10 1 2 3 4 5 6 7 (TeV) 8 9 10 Figure 5. The 5s : pp → + − (D8) discovery contours in M∗ at the LHC as a function of cutoff L on Eproton for an assumed integrated luminosity of 10 fb−1 : (a) thick-dashed line, n = 5; thick solid line, n = 3; and (b) thick-dashed line, n = 6; thick solid line, n = 4. Note that the limit L → ∞ can be performed, and the levelling-off at M∗ ≈ L reflects the underlying fixed point (thin-dashed lines show a ±10% variation about the transition scale LT = M∗ ). Adapted from Litim & Plehn [126]. in, and the running coupling displays a cross-over from (4 + n)-dimensional to four-dimensional scaling (figure 4b). High-energy particle colliders such as the LHC are sensitive to the electroweak energy scale, and hence to the regime of figure 4a, provided that the fundamental Planck scale is of the order of the electroweak scale in (5.1). A generic prediction of large extra dimensions is a tower of massive Kaluza–Klein gravitons [134–136]. In particular, the exchange of virtual gravitons in Drell–Yan processes will lead to deviations in SM reference processes such as pp → + − [137]. Gravitational Drell–Yan production is mediated through tree-level graviton exchange and via graviton loops. Within effective theory, the corresponding effective operators are UV divergent (in two or more extra dimensions) and require a UV cutoff. Within fixed-point gravity, the graviton is dressed by its anomalous dimension, leading to finite and cutoffindependent results for cross sections [126]; see Gerwick & Plehn [138] for a comparison of different UV completions. Figure 5 displays the 5s discovery reach to detect the fundamental scale of gravity assuming M∗ = LT and an integrated luminosity of 10 fb−1 at the LHC. Note that the curves are given as functions of an energy cutoff L. In effective theory, these curves continue to grow with L, meaning that the cutoff cannot be removed. Here, the levelling-off is a dynamical consequence of the underlying fixed point, and the amplitude is independent of any cutoff. Figure 6 shows the differential cross sections for di-muon production within asymptotically safe gravity, with cross-over scale LT = M∗ [129]. For all dimensions considered, the enhanced di-lepton production rate sticks out above SM backgrounds, leading to the conclusion that Drell–Yan production is very sensitive to the quantum gravity scale. Further collider signatures of fixed-point gravity in warped and large extra dimensions have been addressed in Hewett & Rizzo [128] within a form-factor approximation [129]. Modifications to the semiclassical production of Plancksize black holes at the LHC induced by the fixed point have been analysed in Falls et al. [91] and Burschil & Koch [139]. The main new effect is an Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2772 D. F. Litim 1 –1 10–1 fixed point 10–2 10–3 Standard Model 10–4 10–5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Figure 6. Differential cross sections for di-muon production at the LHC as a function of the di-lepton invariant mass mmm within asymptotically safe gravity for M∗ = LT = 5 TeV and n = 2 (magenta), n = 3 (blue) and n = 6 (red) extra dimensions, in comparison with SM background (dashed line). Adapted from Gerwick et al. [129]. (Online version in colour.) additional suppression of the production cross section for mini-black holes. Some properties of Planck-size black holes have also been evaluated within a one-loop approximation, provided the gravitational RG flow is driven by many species of particles [140]. Graviton loop corrections to electroweak precision observables within asymptotically safe gravity with extra dimensions have recently been obtained in Gerwick [130]. In summary, experimental signatures for the quantization of gravity within the asymptotic safety scenario are in reach for particle colliders, provided the fundamental scale of gravity is as low as the electroweak scale in (5.1). 6. Conclusion RG methods have become a key tool in the attempt to understand gravity at shortest, and possibly at largest, distances. It is conceivable that Planckian energies (1.1) can be assessed with the help of running gravitational couplings, such as equation (3.1), replacing equation (2.2) from perturbation theory. If gravity becomes asymptotically safe, its UV fixed point acts as an anchor for the underlying quantum fluctuations. The increasing amount of evidence for a gravitational fixed point with and without matter and its significance for particle physics and cosmology are very promising and certainly warrant more far-reaching investigations. This review is based on invited plenary talks at the conference The Exact RG 2010 (Corfu, September 2010), at the workshop Critical Behavior of Lattice Models in Condensed Matter and Particle Physics (Aspen Center for Physics, June 2010), at the IV Workshop Hot Topics in Cosmology (Cargese, May 2010), at the INT workshop New Applications of the RG Method (Seattle, February 2010), at the conference Asymptotic Safety (Perimeter Institute, November 2009) and at Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2773 the XXV Max Born Symposium The Planck Scale (Wrocl/ aw, June 2009) and on an invited talk at GR19: the 19th International Conference on General Relativity and Gravitation (Mexico City, July 2010). I thank Dario Benedetti, Mike Birse, Denis Comelli, Fay Dowker, Jerzy Kowalski-Glikman, Yannick Meurice, Roberto Percacci, Nikos Tetradis, Roland Triary and Shan-Wen Tsai for their invitations and hospitality, and my collaborators and colleagues for discussions. This work was supported by the Royal Society, and by the Science and Technology Research Council (grant no. ST/G000573/1). References 1 Weinberg, S. 1979 Ultraviolet divergences in quantum theories of gravitation. In General relativity: an Einstein centenary survey (eds S. W. Hawking & W. Israel), p. 790. Cambridge, UK: Cambridge University Press. 2 ’t Hooft, G. & Veltman, M. J. G. 1974 One loop divergencies in the theory of gravitation. Ann. Poincaré Phys. Theor. A20, 69–94. 3 Goroff, M. H. & Sagnotti, A. 1985 Quantum gravity at two loops. Phys. Lett. B 160, 81–86. (doi:10.1016/0370-2693(85)91470-4) 4 Donoghue, J. F. 1994 Leading quantum correction to the Newtonian potential. Phys. Rev. Lett. 72, 2996–2999. (doi:10.1103/PhysRevLett.72.2996) 5 Burgess, C. P. 2004 Quantum gravity in everyday life: general relativity as an effective field theory. Living Rev. Relativ. 7, 5. (http://www.livingreviews.org/lrr-2004-5) 6 Niedermaier, M. 2007 The asymptotic safety scenario in quantum gravity: an introduction. Class. Quantum Gravity 24, R171. (doi:10.1088/0264-9381/24/18/R01) 7 Stelle, K. S. 1977 Renormalization of higher derivative quantum gravity. Phys. Rev. D 16, 953–969. (doi:10.1103/PhysRevD.16.953) 8 Gomis, J. & Weinberg, S. 1996 Are non-renormalizable gauge theories renormalizable? Nucl. Phys. B 469, 473–487. (doi:10.1016/0550-3213(96)00132-0) 9 Niedermaier, M. & Reuter, M. 2006 The asymptotic safety scenario in quantum gravity. Living Rev. Relativ. 9, 5. (http://www.livingreviews.org/lrr-2006-5) 10 Litim, D. F. 2006 On fixed points of quantum gravity. In A Century of Relativity Physics: Ere 2005, XXVIII Spanish Relativity Meeting, Oviedo, Spain, 6–10 September 2006. AIP Conference Proceedings, vol. 841, pp. 322–329. New York, NY: American Institute of Physics. (doi:10.1063/1.2218188). 11 Litim, D. F. 2008 Fixed points of quantum gravity and the renormalisation group. In From Quantum to Emergent Gravity: Theory and Phenomenology, Trieste, Italy, 11–15 June 2007. Proceedings of Science PoS(QG-Ph)024. (http://pos.sissa.it/archive/conferences/043/024/ QG-Ph_024.pdf) 12 Percacci, R. 2009 Asymptotic safety. In Approaches to quantum gravity (ed. D. Oriti), pp. 111– 128. Cambridge, UK: Cambridge University Press. 13 Wilson, K. G. 1971 Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B 4, 3174–3183. (doi:10.1103/PhysRevB.4.3174) 14 Wilson, K. G. 1971 Renormalization group and critical phenomena. II. Phase space cell analysis of critical behavior. Phys. Rev. B 4, 3184–3205. (doi:10.1103/PhysRevB.4.3184) 15 Bergerhoff, B., Freire, F., Litim, D. F., Lola, S. & Wetterich, C. 1996 Phase diagram of superconductors from non-perturbative flow equations. Phys. Rev. B 53, 5734–5757. (doi:10.1103/PhysRevB.53.5734) 16 Wilson, K. G. & Kogut, J. B. 1974 The renormalization group and the epsilon expansion. Phys. Rep. 12, 75–200. (doi:10.1016/0370-1573(74)90023-4) 17 Polchinski, J. 1984 Renormalization and effective Lagrangians. Nucl. Phys. B 231, 269–295. (doi:10.1016/0550-3213(84)90287-6) 18 Wetterich, C. 1993 Exact evolution equation for the effective potential. Phys. Lett. B 301, 90–94. (doi:10.1016/0370-2693(93)90726-X) 19 Berges, J., Tetradis, N. & Wetterich, C. 2002 Non-perturbative renormalization flow in quantum field theory and statistical physics. Phys. Rep. 363, 223–386. (doi:10.1016/ S0370-1573(01)00098-9) Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2774 D. F. Litim 20 Morris, T. R. 1994 The exact renormalization group and approximate solutions. Int. J. Mod. Phys. A 9, 2411–2450. (doi:10.1142/S0217751X94000972) 21 Reuter, M. 1998 Non-perturbative evolution equation for quantum gravity. Phys. Rev. D 57, 971–985. (doi:10.1103/PhysRevD.57.971) 22 Litim, D. F. 2000 Optimisation of the exact renormalisation group. Phys. Lett. B 486, 92–99. (doi:10.1016/S0370-2693(00)00748-6) 23 Symanzik, K. 1970 Small distance behavior in field theory and power counting. Commun. Math. Phys. 18, 227–246. (doi:10.1007/BF01649434) 24 Litim, D. F. & Pawlowski, J. M. 2002 Perturbation theory and renormalization group equations. Phys. Rev. D 65, 081701. (doi:10.1103/PhysRevD.65.081701) 25 Litim, D. F. & Pawlowski, J. M. 2002 Completeness and consistency of renormalization group flows. Phys. Rev. D 66, 025030. (doi:10.1103/PhysRevD.66.025030) 26 Litim, D. F. & Zappalà, D. 2011 Ising exponents from the functional renormalization group. Phys. Rev. D 83, 085009. (doi:10.1103/PhysRevD.83.085009) 27 Litim, D. F. 2001 Optimised renormalization group flows. Phys. Rev. D 64, 105007. (doi:10.1103/PhysRevD.64.105007) 28 Litim, D. F. 2001 Mind the gap. Int. J. Mod. Phys. A 16, 2081–2088. (doi:10.1142/S0217751X 01004748) 29 Litim, D. F. 2002 Critical exponents from optimised renormalisation group flows. Nucl. Phys. B 631, 128–158. (doi:10.1016/S0550-3213(02)00186-4) 30 Pawlowski, J. M. 2007 Aspects of the functional renormalization group. Ann. Phys. 322, 2831– 2915. (doi:10.1016/j.aop.2007.01.007) 31 Manrique, E. & Reuter, M. 2009 Bare action and regularized functional integral of asymptotically safe quantum gravity. Phys. Rev. D 79, 025008. (doi:10.1103/PhysRevD. 79.025008) 32 Litim, D. F. & Pawlowski, J. M. 1999 On gauge invariant Wilsonian flows. In The exact renormalization group (eds A. Krasnitz, P. S. De Sa & R. Potting), p. 168. Singapore: World Scientific. 33 Freire, F., Litim, D. F. & Pawlowski, J. M. 2000 Gauge invariance and background field formalism in the exact renormalisation group. Phys. Lett. B 495, 256–262. (doi:10.1016/ S0370-2693(00)01231-4) 34 Litim, D. F. & Pawlowski, J. M. 2002 Wilsonian flows and background fields. Phys. Lett. B 546, 279–286. (doi:10.1016/S0370-2693(02)02693-x) 35 Folkerts, S., Litim, D. F. & Pawlowski, J. M. Submitted. Asymptotic freedom of Yang–Mills theory with gravity. (http://arxiv.org/abs/1101.5552) 36 Manrique, E. & Reuter, M. 2010 Bimetric truncations for quantum Einstein gravity and asymptotic safety. Ann. Phys. 325, 785–815. (doi:10.1016/j.aop.2009.11.009) 37 Manrique, E., Reuter, M. & Saueressig, F. 2010 Matter induced bimetric actions for gravity. Ann. Phys. 326, 440–462. (doi:10.1016/j.aop.2010.11.003) 38 Manrique, E., Reuter, M. & Saueressig, F. 2011 Bimetric renormalization group flows in quantum Einstein gravity. Ann. Phys. 326, 463–485. (doi:10.1016/j.aop.2010.11.006) 39 Gastmans, R., Kallosh, R. & Truffin, C. 1978 Quantum gravity near two dimensions. Nucl. Phys. B 133, 417–434. (doi:10.1016/0550-3213(78)90234-1) 40 Christensen, S. M. & Duff, M. J. 1978 Quantum gravity in 2 + e dimensions. Phys. Lett. B 79, 213–216. (doi:10.1016/0370-2693(78)90225-3) 41 Aida, T. & Kitazawa, Y. 1997 Two-loop prediction for scaling exponents in (2 + e)dimensional quantum gravity. Nucl. Phys. B 491, 427–460. (doi:10.1016/S0550-3213(97) 00091-6). 42 Souma, W. 1999 Non-trivial ultraviolet fixed point in quantum gravity. Prog. Theor. Phys. 102, 181–195. (doi:10.1143/PTP.102.181) 43 Lauscher, O. & Reuter, M. 2002 Ultraviolet fixed point and generalized flow equation of quantum gravity. Phys. Rev. D 65, 025013. (doi:10.1103/PhysRevD.65.025013) 44 Litim, D. F. 2004 Fixed points of quantum gravity. Phys. Rev. Lett. 92, 201301. (doi:10.1103/PhysRevLett.92.201301) Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2775 45 Fischer, P. & Litim, D. F. 2006 Fixed points of quantum gravity in extra dimensions. Phys. Lett. B 638, 497–502. (doi:10.1016/j.physletb.2006.05.073) 46 Litim, D. F. & Pawlowski, J. M. 1998 Flow equations for Yang–Mills theories in general axial gauges. Phys. Lett. B 435, 181–188. (doi:10.1016/S0370-2693(98)00761-8) 47 Lauscher, O. & Reuter, M. 2002 Flow equation of quantum Einstein gravity in a higherderivative truncation. Phys. Rev. D 66, 025026. (doi:10.1103/PhysRevD.66.025026) 48 Lauscher, O. & Reuter, M. 2002 Is quantum Einstein gravity nonperturbatively renormalizable? Class. Quantum Gravity 19, 483. (doi:10.1088/0264-9381/19/3/304) 49 Codello, A., Percacci, R. & Rahmede, C. 2008 Ultraviolet properties of f (R)-gravity. Int. J. Mod. Phys. A 23, 143–153. (doi:10.1142/S0217751X08038135) 50 Codello, A., Percacci, R. & Rahmede, C. 2009 Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation. Ann. Phys. 324, 414–469. (doi:10.1016/j.aop.2008.08.008) 51 Machado, P. F. & Saueressig, F. 2008 On the renormalization group flow of f (R)-gravity. Phys. Rev. D 77, 124045. (doi:10.1103/PhysRevD.77.124045) 52 Bonanno, A., Contillo, A. & Percacci, R. 2010 Inflationary solutions in asymptotically safe f (R) gravity. (http://arxiv.org/abs/1006.0192) 53 Codello, A. & Percacci, R. 2006 Fixed points of higher derivative gravity. Phys. Rev. Lett. 97, 221301. (doi:10.1103/PhysRevLett.97.221301) 54 Niedermaier, M. R. 2009 Gravitational fixed points from perturbation theory. Phys. Rev. Lett. 103, 101303. (doi:10.1103/PhysRevLett.103.101303) 55 Niedermaier, M. R. 2010 Gravitational fixed points and asymptotic safety from perturbation theory. Nucl. Phys. B 833, 226–270. (doi:10.1016/j.nuclphysb.2010.01.016) 56 Benedetti, D., Machado, P. F. & Saueressig, F. 2009 Asymptotic safety in higher-derivative gravity. Mod. Phys. Lett. A 24, 2233–2241. (doi:10.1142/S0217732309031521) 57 Benedetti, D., Machado, P. F. & Saueressig, F. 2010 Taming perturbative divergences in asymptotically safe gravity. Nucl. Phys. B 824, 168–191. (doi:10.1016/j.nuclphysb.2009.08.023) 58 Groh, K. & Saueressig, F. 2010 Ghost wave-function renormalization in asymptotically safe quantum gravity. J. Phys. A 43, 365403. (doi:10.1088/1751-8113/43/36/365403) 59 Eichhorn, A. & Gies, H. 2010 Ghost anomalous dimension in asymptotically safe quantum gravity. Phys. Rev. D 81, 104010. (doi:10.1103/PhysRevD.81.104010) 60 Eichhorn, A., Gies, H. & Scherer, M. M. 2009 Asymptotically free scalar curvature– ghost coupling in quantum Einstein gravity. Phys. Rev. D 80, 104003. (doi:10.1103/ PhysRevD.80.104003) 61 Percacci, R. & Perini, D. 2003 Constraints on matter from asymptotic safety. Phys. Rev. D 67, 081503. (doi:10.1103/PhysRevD.67.081503) 62 Narain, G. & Percacci, R. 2010 Renormalization group flow in scalar–tensor theories. I. Class. Quantum Gravity 27, 075001. (doi:10.1088/0264-9381/27/7/075001) 63 Narain, G. & Rahmede, C. 2010 Renormalization group flow in scalar–tensor theories. II. Class. Quantum Gravity 27, 075002. (doi:10.1088/0264-9381/27/7/075002) 64 Percacci, R. & Perini, D. 2003 Asymptotic safety of gravity coupled to matter. Phys. Rev. D 68, 044018. (doi:10.1103/PhysRevD.68.044018) 65 Tomboulis, E. 1977 1/N expansion and renormalization in quantum gravity. Phys. Lett. B 70, 361–364. (doi:10.1016/0370-2693(77)90678-5) 66 Smolin, L. 1982 A fixed point for quantum gravity. Nucl. Phys. B 208, 439–466. (doi:10.1016/ 0550-3213(82)90230-9) 67 Zanusso, O., Zambelli, L., Vacca, G. P. & Percacci, R. 2010 Gravitational corrections to Yukawa systems. Phys. Lett. B 689, 90–94. (doi:10.1016/j.physletb.2010.04.043) 68 Vacca, G. P. & Zanusso, O. 2010 Asymptotic safety in Einstein gravity and scalar-fermion matter. Phys. Rev. Lett. 105, 231601. (doi:10.1103/PhysRevLett.105.231601) 69 Shaposhnikov, M. & Wetterich, C. 2010 Asymptotic safety of gravity and the Higgs boson mass. Phys. Lett. B 683, 196–200. (doi:10.1016/j.physletb.2009.12.022) 70 Atkins, M. & Calmet, X. 2011 Remarks on Higgs inflation. Phys. Lett. B 697, 37–40. (doi:10.1016/j.physletb.2011.01.028) Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2776 D. F. Litim 71 Robinson, S. P. & Wilczek, F. 2006 Gravitational correction to running of gauge couplings. Phys. Rev. Lett. 96, 231601. (doi:10.1103/PhysRevLett.96.231601) 72 Pietrykowski, A. R. 2007 Gauge dependence of gravitational correction to running of gauge couplings. Phys. Rev. Lett. 98, 061801. (doi:10.1103/PhysRevLett.98.061801) 73 Toms, D. J. 2007 Quantum gravity and charge renormalization. Phys. Rev. D 76, 045015. (doi:10.1103/PhysRevD.76.045015) 74 Ebert, D., Plefka, J. & Rodigast, A. 2008 Absence of gravitational contributions to the running Yang–Mills coupling. Phys. Lett. B 660, 579–582. (doi:10.1016/j.physletb.2008.01.037) 75 Toms, D. J. 2008 Cosmological constant and quantum gravitational corrections to the running fine structure constant. Phys. Rev. Lett. 101, 131301. (doi:10.1103/PhysRevLett.101.131301) 76 Tang, Y. & Wu, Y. L. 2010 Gravitational contributions to the running of gauge couplings. Commun. Theor. Phys. 54, 1040–1044. (doi:10.1088/0253-6102/54/6/15) 77 Toms, D. J. 2010 Quantum gravitational contributions to quantum electrodynamics. Nature 468, 56–59. (doi:10.1038/nature09506) 78 Daum, J.-E., Harst, U. & Reuter, M. 2010 Non-perturbative QEG corrections to the Yang– Mills beta function. (http://arxiv.org/abs/1005.1488) 79 Harst, U. & Reuter, M. 2011 QED coupled to QEG. (http://arxiv.org/abs/1101.6007) 80 Percacci, R. & Sezgin, E. 2010 One loop beta functions in topologically massive gravity. Class. Quantum Gravity 27, 155009. (doi:10.1088/0264-9381/27/15/155009) 81 Codello, A. & Percacci, R. 2009 Fixed points of nonlinear sigma models in d > 2. Phys. Lett. B 672, 280–283. (doi:10.1016/j.physletb.2009.01.032) 82 Percacci, R. 2009 Asymptotic safety in gravity and sigma models. In Workshop on Continuum and Lattice Approaches to Quantum Gravity, Brighton, UK, 17–19 September 2008. Proceedings of Science PoS(CLAQG08)002. (http://pos.sissa.it/archive/conferences/079/ 002/CLAQG08_002.pdf) 83 Percacci, R. & Zanusso, O. 2010 One loop beta functions and fixed points in higher derivative sigma models. Phys. Rev. D 81, 065012. (doi:10.1103/PhysRevD.81.065012) 84 Fabbrichesi, M., Percacci, R., Tonero, A. & Zanusso, O. 2011 Asymptotic safety and the gauged SU(N ) nonlinear s-model. Phys. Rev. D 83, 025016. (doi:10.1103/PhysRevD.83.025016) 85 Niedermaier, M. R. 2003 Dimensionally reduced gravity theories are asymptotically safe. Nucl. Phys. B 673, 131–169. (doi:10.1016/j.nuclphysb.2003.09.015) 86 Reuter, M. & Weyer, H. 2009 Background independence and asymptotic safety in conformally reduced gravity. Phys. Rev. D 79, 105005. (doi:10.1103/PhysRevD.79.105005) 87 Reuter, M. & Weyer, H. 2009 Conformal sector of quantum Einstein gravity in the local potential approximation: non-Gaussian fixed point and a phase of unbroken diffeomorphism invariance. Phys. Rev. D 80, 025001. (doi:10.1103/PhysRevD.80.025001) 88 Machado, P. F. & Percacci, R. 2009 Conformally reduced quantum gravity revisited. Phys. Rev. D 80, 024020. (doi:10.1103/PhysRevD.80.024020) 89 Bonanno, A. & Reuter, M. 2000 Renormalization group improved black hole space-times. Phys. Rev. D 62, 043008. (doi:10.1103/PhysRevD.62.043008) 90 Bonanno, A. & Reuter, M. 2006 Spacetime structure of an evaporating black hole in quantum gravity. Phys. Rev. D 73, 083005. (doi:10.1103/PhysRevD.73.083005) 91 Falls, K., Litim, D. F. & Raghuraman, A. 2010 Black holes and asymptotically safe gravity. (http://arxiv.org/abs/1002.0260). 92 Reuter, M. & Tuiran, E. 2010 Quantum gravity effects in the Kerr spacetime. Phys. Rev. D 83, 044041. (doi:10.1103/PhysRevD.83.044041) 93 Cai, Y.-F. & Easson, D. A. 2010 Black holes in an asymptotically safe gravity theory with higher derivatives. J. Cosmol. Astropart. Phys. 2010, 002. (doi:10.1088/1475-7516/2010/ 09/002) 94 Reuter, M. & Weyer, H. 2006 On the possibility of quantum gravity effects at astrophysical scales. Int. J. Mod. Phys. D 15, 2011–2028. (doi:10.1142/S0218271806009443) 95 Bonanno, A. 2009 Astrophysical implications of the asymptotic safety scenario in quantum gravity. In Workshop on Continuum and Lattice Approaches to Quantum Gravity, Brighton, UK, 17–19 September 2008. Proceedings of Science PoS(CLAQG08)008. (http://pos.sissa.it/archive/conferences/079/008/CLAQG08_008.pdf) Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 Review. RG and the Planck scale 2777 96 Casadio, R., Hsu, S. D. H. & Mirza, B. 2011 Asymptotic safety, singularities, and gravitational collapse. Phys. Lett. B 695, 317–319. (doi:10.1016/j.physletb.2010.10.060) 97 Satz, A., Codello, A. & Mazzitelli, F. D. 2010 Low energy quantum gravity from the effective average action. Phys. Rev. D 82, 084011. (doi:10.1103/PhysRevD.82.084011) 98 Bonanno, A. & Reuter, M. 2002 Cosmology with selfadjusting vacuum energy density from a renormalization group fixed point. Phys. Lett. B 527, 9–17. (doi:10.1016/S03702693(01)01522-2) 99 Bentivegna, E., Bonanno, A. & Reuter, M. 2004 Confronting the IR fixed point cosmology with high redshift observations. J. Cosmol. Astropart. Phys. 2004, 001. (doi:10.1088/ 1475-7516/2004/01/001) 100 Reuter, M. & Saueressig, F. 2005 From big bang to asymptotic de Sitter: complete cosmologies in a quantum gravity framework. J. Cosmol. Astropart. Phys. 2005, 012. (doi:10.1088/ 1475-7516/2005/09/012) 101 Bonanno, A. & Reuter, M. 2002 Cosmology of the Planck era from a renormalization group for quantum gravity. Phys. Rev. D 65, 043508. (doi:10.1103/PhysRevD.65.043508) 102 Bonanno, A. & Reuter, M. 2004 Cosmological perturbations in renormalization group derived cosmologies. Int. J. Mod. Phys. D 13, 107–122. (doi:10.1142/S0218271804003809) 103 Weinberg, S. 2010 Asymptotically safe inflation. Phys. Rev. D 81, 083535. (doi:10.1103/ PhysRevD.81.083535) 104 Tye, S.-H. H. & Xu, J. 2010 Comment on asymptotically safe inflation. Phys. Rev. D 82, 127302. (doi:10.1103/PhysRevD.82.127302) 105 Contillo, A. 2010 Evolution of cosmological perturbations in an RG-driven inflationary scenario. (http://arxiv.org/abs/1011.4618). 106 Hindmarsh, M., Litim, D. F. & Rahmede, C. 2011 Asymptotically safe cosmology. (http://arxiv. org/abs/1101.5401). 107 Koch, B. & Ramirez, I. 2011 Exact renormalization group with optimal scale and its application to cosmology. Class. Quantum Gravity 28, 055008. (doi:10.1088/0264-9381/ 28/5/055008) 108 Bonanno, A. & Reuter, M. 2011 Entropy production during asymptotically safe inflation. Entropy 13, 274–292. (doi:10.3390/e13010274) 109 Bonanno, A. & Reuter, M. 2007 Entropy signature of the running cosmological constant. J. Cosmol. Astropart. Phys. 2007, 024. (doi:10.1088/1475-7516/2007/08/024) 110 Horava, P. 2009 Quantum gravity at a Lifshitz point. Phys. Rev. D 79, 084008. (doi:10.1103/PhysRevD.79.084008) 111 Horava, P. 2011 General covariance in gravity at a Lifshitz point. (http://arxiv.org/abs/ 1101.1081). 112 Reuter, M. & Schwindt, J.-M. 2006 A minimal length from the cutoff modes in asymptotically safe quantum gravity. J. High Energy Phys. 2006, 070. (doi:10.1088/1126-6708/ 2006/01/070) 113 Reuter, M. & Schwindt, J.-M. 2007 Scale-dependent metric and causal structures in quantum Einstein gravity. J. High Energy Phys. 2007, 049. (doi:10.1088/1126-6708/ 2007/01/049) 114 Basu, S. & Mattingly, D. 2010 Asymptotic safety, asymptotic darkness, and the hoop conjecture in the extreme UV. Phys. Rev. D 82, 124017. (doi:10.1103/PhysRevD.82.124017) 115 Percacci, R. & Vacca, G. P. 2010 Asymptotic safety, emergence and minimal length. Class. Quantum Gravity 27, 245026. (doi:10.1088/0264-9381/27/24/245026) 116 Calmet, X., Hossenfelder, S. & Percacci, R. 2010 Deformed special relativity from asymptotically safe gravity. Phys. Rev. D 82, 124024. (doi:10.1103/PhysRevD.82.124024) 117 Girelli, F., Liberati, S., Percacci, R. & Rahmede, C. 2007 Modified dispersion relations from the renormalization group of gravity. Class. Quantum Gravity 24, 3995–4008. (doi:10.1088/ 0264-9381/24/16/003) 118 Gawedzki, K. & Kupiainen, A. 1985 Renormalizing the nonrenormalizable. Phys. Rev. Lett. 55, 363–365. (doi:10.1103/PhysRevLett.55.363) 119 de Calan, C., Faria da Veiga, P. A., Magnen, J. & Seneor, R. 1991 Constructing the threedimensional Gross–Neveu model with a large number of flavor components. Phys. Rev. Lett. 66, 3233–3236. (doi:10.1103/PhysRevLett.66.3233) Phil. Trans. R. Soc. A (2011) Downloaded from http://rsta.royalsocietypublishing.org/ on May 10, 2017 2778 D. F. Litim 120 Gies, H., Jaeckel, J. & Wetterich, C. 2004 Towards a renormalizable standard model without fundamental Higgs scalar. Phys. Rev. D 69, 105008. (doi:10.1103/PhysRevD.69.105008) 121 Gies, H. & Scherer, M. M. 2010 Asymptotic safety of simple Yukawa systems. Eur. Phys. J. C 66, 387–402. (doi:10.1140/epjc/s10052-010-1256-z) 122 Gies, H., Rechenberger, S. & Scherer, M. M. 2010 Towards an asymptotic safety scenario for chiral Yukawa systems. Eur. Phys. J. C 66, 403–418. (doi:10.1140/epjc/s10052-010-1257-y) 123 Pica, C. & Sannino, F. 2010 UV and IR zeros of gauge theories at the four loop order and beyond. Phys. Rev. D 83, 035013. (doi:10.1103/PhysRevD.83.035013) 124 Calmet, X. 2010 Asymptotically safe weak interactions. (http://arxiv.org/abs/1012.5529). 125 Fischer, P. & Litim, D. F. 2006 Fixed points of quantum gravity in higher dimensions. In Albert Einstein Century Int. Conf., Paris, France, 18–22 July 2005. AIP Conference Proceedings, vol. 861, pp. 336–343. New York, NY: American Institute of Physics. (doi:10.1063/1.2399593) 126 Litim, D. F. & Plehn, T. 2008 Signatures of gravitational fixed points at the LHC. Phys. Rev. Lett. 100, 131301. (doi:10.1103/PhysRevLett.100.131301) 127 Litim, D. F. & Plehn, T. 2007 Virtual gravitons at the LHC. In The 15th Int. Conf. on Supersymmetry and the Unification of Fundamental Interactions (SUSY07), Karlsruhe, Germany, 26 July–1 August 2007. (http://arxiv.org/abs/0710.3096) 128 Hewett, J. & Rizzo, T. 2007 Collider signals of gravitational fixed points. J. High Energy Phys. 2007, 009. (doi:10.1088/1126-6708/2007/12/009) 129 Gerwick, E., Litim, D. F. & Plehn, T. 2011 Asymptotic safety and Kaluza–Klein gravitons at the LHC. Phys. Rev. D 83, 084048. (doi:10.1103/PhysRevD.83.084048) 130 Gerwick, E. 2010 Asymptotically safe gravitons in electroweak precision physics. (http://arxiv.org/abs/ 1012.1118). 131 Arkani-Hamed, N., Dimopoulos, S. & Dvali, G. R. 1998 The hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429, 263–272. (doi:10.1016/S0370-2693(98)00466-3) 132 Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S. & Dvali, G. R. 1998 New dimensions at a millimeter to a Fermi and superstrings at a TeV. Phys. Lett. B 436, 257–263. (doi:10.1016/S0370-2693(98)00860-0) 133 Randall, L. & Sundrum, R. 1999 A large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 3370–3373. (doi:10.1103/PhysRevLett.83.3370) 134 Giudice, G. F., Rattazzi, R. & Wells, J. D. 1999 Quantum gravity and extra dimensions at high-energy colliders. Nucl. Phys. B 544, 3–38. (doi:10.1016/S0550-3213(99)00044-9) 135 Han, T., Lykken, J. D. & Zhang, R. J. 1999 On Kaluza–Klein states from large extra dimensions. Phys. Rev. D 59, 105006. (doi:10.1103/PhysRevD.59.105006) 136 Hewett, J. L. 1999 Indirect collider signals for extra dimensions. Phys. Rev. Lett. 82, 4765–4768. (doi:10.1103/PhysRevLett.82.4765) 137 Ellis, R. K., Stirling, W. J. & Webber, B. R. 1996 QCD and collider physics. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, no. 8. Cambridge, UK: Cambridge University Press. 138 Gerwick, E. & Plehn, T. 2009 Extra dimensions and their ultraviolet completion. In Workshop on Continuum and Lattice Approaches to Quantum Gravity, Brighton, UK, 17–19 September 2008. Proceedings of Science PoS(CLAQG08)009. (http://pos.sissa.it/archive/conferences/ 079/009/CLAQG08_009.pdf) 139 Burschil, T. & Koch, B. 2010 Renormalization group improved black hole space-time in large extra dimensions. Zh. Eksp. Teor. Fiz. 92, 219–225. 140 Calmet, X., Hsu, S. D. H. & Reeb, D. 2008 Quantum gravity at a TeV and the renormalization of Newton’s constant. Phys. Rev. D 77, 125015. (doi:10.1103/PhysRevD.77.125015) Phil. Trans. R. Soc. A (2011)