Phil. Trans. R. Soc. A (2011) 369, 2759–2778doi:10.1098/rsta.2011.0103
REVIEW
Renormalization group and the Planck scaleBY DANIEL F. LITIM*
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’sasymptotic safety scenario, and give an overview of results with applications to particlephysics and cosmology.
Einstein’s theory of general relativity is the remarkably successful classical theoryof 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 scalesranging from the sub-millimetre regime up to Solar System size. At larger lengthscales, the standard model of cosmology including dark matter and dark energycomponents fits the data well. At shorter length scales, quantum effects areexpected to become important. An order-of-magnitude estimate for the quantumscale of gravity—the Planck scale—is obtained by dimensional analysis, leadingto the Planck length �Pl ≈
√hGN/c3 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 particleaccelerator experiments, fingerprints of Planck-scale physics can neverthelessbecome accessible through cosmological data from the very early Universe. Froma theory perspective, it is widely expected that a fundamental understandingof Planck-scale physics requires a quantum theory of gravity. It is well knownthat the standard perturbative quantization programme faces problems, and afully satisfactory quantum theory, even outside the framework of local quantumphysics, is presently not to hand. In the past 15 years, however, a significantbody of work has been devoted to re-evaluate the physics of the Planckscale 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’.
Steven Weinberg’s seminal proposal, more than 30 years ago, that a quantumtheory of gravitation may dynamically evade the virulent divergences encounteredin standard perturbation theory [1]. This scenario, known as asymptotic safety,implies that gravity achieves a non-trivial ultraviolet (UV) fixed point under itsrenormalization group (RG) flow. If so, this would incorporate gravity alongsidethe set of well-understood quantum field theories whose UV behaviour is governedby a fixed point, e.g. Yang–Mills theory.
In this paper, I discuss the RG approach to gravity. I recall some of the issuesof perturbative quantum gravity (§2), introduce the RG à la Wilson to accessthe 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 quantizationprogramme of gravity. I recall a very small selection of these in order to preparefor the subsequent discussion, and I use particle physics units h = c = kB = 1throughout. Classical general relativity is described by the classical action
S = 116pGN
∫d4x
√det gmn(−R(gmn) + 2L), (2.1)
where I have chosen a Euclidean signature, R denotes the Ricci scalar, and L is thecosmological constant term. The main point to be stressed is that the fundamentalcoupling 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 structuredistinguishes gravity in a profound manner from the other fundamentally knowninteractions in Nature, all of which have dimensionless coupling constants fromthe outset. Consequently, the effective dimensionless coupling of gravity thatorganizes its perturbative expansion is given by
geff ≡ GNE2 (2.2)
(in four dimensions), where E denotes the relevant energy scale. While theeffective coupling (2.2) remains small for energies below the Planck scaleE � MPl, it grows large in the Planckian regime where an expansion in geffmay become questionable. Within the Feynman diagrammatic approach, thisbehaviour translates into the (with loop order) increasing degree of divergenceof perturbative diagrams involving gravitons [1]. This structure is different fromstandard quantum field theories and relates to the classification of interactionsas super-renormalizable, renormalizable or ‘dangerous’, depending on whethertheir canonical mass dimension is positive, vanishing or negative. The degreeof divergences implied by the negative mass dimension of Newton’s coupling ismirrored in the perturbative non-renormalizability of Einstein gravity, which hasbeen established at the one-loop level [2] in the presence of matter fields, and atthe two-loop level [3] within pure gravity.
While this state of play looks discouraging from a field theory perspective, itdoes 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,
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a weak-coupling analysis of quantum gravity effects is possible within aneffective theory approach [4], which operates a UV cutoff at the Planck scale;see [5] for a review. Secondly, higher-order derivative operators appear tostabilize perturbation theory [6]. Including all fourth-order derivative operators,it has been proven by Stelle [7] that gravity is renormalizable to all ordersin perturbation theory. This striking difference with Einstein–Hilbert gravityhighlights the stabilizing effect of higher derivative terms. Unfortunately, theresulting theory is not compatible with standard notions of perturbative unitarityand has therefore not been considered as a candidate for a fundamental theory ofgravity. The role of higher-order derivative operators has further been clarified inGomis & Weinberg [8] using cohomology techniques. Interestingly, the theoryremains unitary once all higher derivative operators are retained, whereasrenormalizability is at best achieved in a very weak sense owing to the requiredinfinitely many counter-terms.
3. Renormalization group
The RG comes into play when the running of couplings with energy is taken intoaccount. As in any generic quantum field theory, quantum fluctuations modifythe strength of couplings. If the metric field remains the fundamental carrierof the gravitational force, the fluctuations of space–time itself should modifythe gravitational interactions with energy or distance. For Newton’s coupling,this implies that GN becomes a running coupling GN → G(k) = GNZ−1(k) as afunction of the RG momentum scale k, where Z (k) denotes the wave functionrenormalization factor of the graviton. Consequently, the dimensionless couplinggeff in (2.2) should be replaced by the running coupling
g = G(k)k2, (3.1)
which evolves with the RG scale. In particular, the UV behaviour of standardperturbation theory is significantly improved, provided that equation (3.1)remains finite in the high-energy limit. This is the asymptotic safety scenarioas 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 briefoverviews. The intimate link between a fundamental definition of quantum fieldtheory 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])
vtg ≡ bg = (d − 2 + h)g, (3.2)
with h = −vt ln Z (k) the graviton anomalous dimension, which in general is afunction of all couplings of the theory including matter, and t = ln k. This simplestructure arises provided the underlying effective action is local in the metricfield. From the RG equation (3.2), one concludes that the gravitational couplingmay display two types of fixed points. The non-interacting (Gaussian) fixedpoint corresponds to g∗ = 0 and entails the vanishing of the anomalous dimensionh = 0. In its vicinity, gravity stays classical and G(k) ≈ GN. On the other hand,
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a non-trivial RG fixed point with g∗ �= 0 can be achieved implicitly, provided thatthe anomalous dimension reads
h∗ = 2 − d. (3.3)
The significance of equation (3.3) is that the graviton anomalous dimensionprecisely counterbalances the canonical dimension of Newton’s coupling GN.This pattern is known from other gauge systems at a critical point away fromtheir canonical space–time dimensionality [10], e.g. U(1) Higgs theory in threedimensions [15]. In consequence, the dimensionful, renormalized coupling scalesas G(k) ≈ g∗/kd−2 and becomes small in the UV limit where 1/k → 0. This patternis at the root of the non-perturbative renormalizability of quantum gravity withina fixed-point scenario.
Much work has been devoted to checking by explicit computation whether ornot the gravitational couplings achieve a non-trivial UV fixed point. A versatileframework to address this question is provided by modern (functional) RGmethods, based on the infinitesimal integrating-out of momentum degrees offreedom 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 thepropagating fields. In consequence, the action (2.1) becomes a scale-dependenteffective or flowing action Gk ,
Gk =∫
ddx√
g[
116pGk
(−R(gmn) + 2Lk) + · · ·]
+ Sk,gf + Sk,gh + Sk,matter, (3.4)
which in the context of gravity contains a running gravitational coupling, arunning cosmological constant Lk , a gauge-fixing term, ghost contributions,matter interactions and the dots indicate possible higher derivative operatorsin the metric field. Upon varying the RG scale k, the effective action interpolatesbetween a microscopic theory GL at the UV scale k = L (not to be confused withthe running cosmological constant Lk) and the macroscopic quantum effectiveaction G, where all fluctuations are taken into account (k → 0). The variationof equation (3.4) with RG scale k is given by an exact functional differentialequation [18,21]
vtGk = 12
Tr1
G(2)k + Rk
vtRk = 12
, (3.5)
which relates the change of the scale-dependent gravitational action with the fullfield-dependent propagator of the theory (full line) and the scale dependence vtRkof the momentum cutoff (the insertion). Here, the trace stands for a momentumintegration, and a sum over all propagating degrees of freedom f = (gmn, ghosts,matter fields), which minimally contains the metric field and its ghosts. Thefunction Rk (not to be confused with the Ricci scalar) denotes the infrared (IR)momentum cutoff. As a function of (covariant) momenta q2, the momentum cutoffobeys Rk(q2) → 0 for k2/q2 → 0, Rk(q2) > 0 for q2/k2 → 0 and Rk(q2) → ∞ fork → L (for examples and plots of Rk , see Litim [22]). The functional flow (3.5) isclosely linked to other exact functional differential equations, such as the Callan–Symanzik equation [23] in the limit Rk → k2, and the Wilson–Polchinski equationby 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].
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The strength of the formalism is that it is not bounded to the weak-couplingregime, and systematic approximations—such as the derivative expansion, vertexexpansions or mixtures thereof—are available to access domains with strongcoupling and/or strong correlation. Systematic uncertainties can be assessed [26],and the stability and convergence of approximations is enhanced by optimizationtechniques [22,27–30].
By construction, the flow (3.5) is both UV and IR finite. Together withthe boundary condition GL it may serve as a definition for the theory. Inrenormalizable theories, the continuum limit is performed by removing the scaleL, 1/L → 0, and the functional GL → G∗ approaches a fixed-point action atshort distances (see the proposal in Manrique & Reuter [31] for a constructionin gravity). In perturbatively renormalizable theories, G∗ is ‘trivial’ and givenmainly 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 thosediffeomorphism-invariant operators that display relevant or marginal scaling inthe vicinity of the UV fixed point. A fixed-point action qualifies as fundamentalif RG trajectories k → Gk emanating from its vicinity connect with the correctlong-distance behaviour for k → 0 and stay well defined (finite, no poles) at allscales [1].
The operator trace in equation (3.5) is evaluated by using flat orcurved backgrounds together with heat kernel techniques or plain momentumintegration. To ensure diffeomorphism symmetry within this set-up, thebackground field formalism is used by adding a non-propagating background fieldgmn [21,30,32–34]. This way, the extended effective action Gk [gmn, gmn] becomesgauge-invariant under the combined symmetry transformations of the physicaland the background field. A second benefit of this is that the background fieldcan be used to construct a covariant Laplacian −D2 (or similar) to define a modecutoff at the RG momentum scale k2 = −D2. This implies that the mode cutoff Rkwill depend on the background fields, which is controlled by an equation similar tothe flow equation itself [34,35]. The background field is then eliminated from thefinal equations by identifying it with the physical mean field. This proceduredynamically readjusts the background field and implements the requirementsof background independence for quantum gravity. An alternative techniquethat 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 differentimplementations 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 toLitim [12] for a more detailed overview of results prior to 2008. The search forfixed points in quantum gravity starts by restricting the running effective actionGk to a finite set of operators Oi(f) with running couplings gi ,
Gk =∑
i
gi(k)Oi(f), (4.1)
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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 ofapproximations are checked by increasing the number of operators retained, andfurther improved through optimized choices of the momentum cutoff [22,29].
The first set of flow equations that carry a non-trivial UV fixed point hasbeen derived in Reuter [21] for the Einstein–Hilbert theory in Feynman gauge.As a function of the number of space–time dimensions, the flow equationreproduces 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 thefour-dimensional theory [21,42]. Both the Ricci scalar and the cosmologicalconstant term are relevant operators at the fixed point. Their scaling is stronglycorrelated, leading to a complex conjugate pair of universal scaling exponents.This result has subsequently been confirmed within a more general backgroundfield gauge by means of a trace-less transverse decomposition of the metricfield [43].
A complete analytical understanding of the Einstein–Hilbert approximationhas been achieved in Litim [44], leading to analytical flow equations and closedexpressions for the (unique) UV fixed point and its universal eigenvalues. Key forthis was the use of (optimized) momentum cutoffs that allow analytical access andimprove convergence and stability of results [22,27]. Figure 2 shows the universalscaling exponents q = q′ + iq′′ at the UV fixed point in four dimensions (left-handside) for various (optimized) choices of the momentum cutoff [27,45]. Also ondisplay 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 andcontrolled 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 alongthe separatrix connecting the non-trivial UV fixed point with the Gaussian fixedpoint in the IR. At the scale k ≈ LT , the RG flow displays a cross-over from
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Figure 2. Ultraviolet fixed point in the d-dimensional Einstein–Hilbert theory. Comparison ofuniversal eigenvalues q′, q′′, |q| and the invariant t = l∗(g∗)2/(d−2) for different Wilsonianmomentum cutoffs and various dimensions, normalized to the result for Ropt (Rmexp, circles; Rexp,squares; Rmod, triangles; Ropt, inverted triangles). Adapted from Fischer & Litim [45]. (Onlineversion in colour.)
perturbative IR scaling to fixed-point scaling, where the anomalous dimensiongrows large. The dynamical scale LT is of the order of the fundamental Planckscale, and a consequence of the UV fixed point. In gravity, the scale LT plays arole analogous to that of LQCD in quantum chromodynamics.
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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 beenextended to include
∫√gR2 interactions [47,48]. The perturbatively marginal R2
coupling turned out to become relevant at a UV fixed point. Otherwise, it addedonly minor corrections to the fixed point in the Einstein–Hilbert theory. WithLitim [27,44], these computations became feasible analytically and have beenimplemented for operators
∫√gfk(R), which contain a general function of the
Ricci scalar, in Codello et al. [49,50] and Machado & Saueressig [51]. A non-trivial UV fixed point has been found by Taylor-expanding fk(R) to high ordersin 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 isremarkably stable. The universal eigenvalues converge rapidly, confirming allprevious results. Most importantly, it turned out that operators
∫√gRn from
n ≥ 3 onwards become irrelevant at the UV fixed point [49], a first indicatorthat an asymptotically safe UV fixed point of the fundamental theory may,in fact, have a finite number of relevant operators. This is most welcome asotherwise a fixed point with infinitely many relevant couplings is likely to spoilthe predictive power.
All fourth-order derivative operators have been included both at the one-loop order and beyond. The re-evaluation of perturbative one-loop results bymeans of a functional flow emphasized the significance of, for example, quadraticdivergences that are normally suppressed within dimensional regularization[53–55]. Two types of UV fixed points are found. The first one is non-trivialin all couplings [56,57], with three relevant directions at the fixed point. Thesecond one is perturbative in the Weyl coupling [53,54] but non-trivial in Newton’scoupling, with two relevant and two marginally relevant directions. In either caseNewton’s coupling takes a non-trivial fixed point in the UV. These results henceindicate that higher derivative operators are compatible with gravity developing anon-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].
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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 freedomcan be spoiled by too many species of fermions. In the same vein it has to bechecked whether the UV fixed point of gravity remains stable under the inclusionof matter, and vice versa. By now, this question has been analysed for minimallycoupled scalar matter [61], non-minimally coupled scalar fields [62,63] and genericfree matter coupled to gravity [53,64]. These findings are consistent with earlierresults using different techniques in the limit of many matter fields [65,66].The conclusion is that the gravitational fixed point generically persists underthe inclusion of matter (including the Standard Model (SM) of particle physicsand its main extensions) unless a significant imbalance between bosonic andfermionic matter fields is chosen. Also, non-minimally coupled scalar matter leadsto slight deformation of the gravitational fixed point with matter achieving aweakly coupled regime, the so-called Gaussian matter fixed point. Fermions havebeen included recently in Zanusso et al. [67], as well as gravitational Yukawasystems [68]. In the latter case, the coupled system displays a non-trivial UVfixed point both in the Yukawa as well as in the gravitational sector. This newfixed point may become of great interest as a stabilizer for the SM Higgs. On theone-loop level, the interplay between an SM Higgs and asymptotically safe gravityhas been addressed in Shaposhnikov & Wetterich [69], detailing conditions underwhich no new physics is required to bridge the energy range from the electroweakscale all the way up to equation (1.1). In a similar spirit, it has been arguedthat the consistency of Higgs inflation models is enhanced provided that gravitybecomes asymptotically safe [70].
The impact of gravitational fluctuations on gauge theories has seen renewedinterest initiated by a one-loop study within effective theory [71]. Asymptoticfreedom is sustained by gravity, leading to a vanishing [35,72–75] or non-vanishingone-loop correction [35,71,76–78], depending on the regularization. In the presentframework (3.4) and (3.5), the gravity-induced corrections have been studied inDaum et al. [78] and in Folkerts et al. [35]. In either case, the sign of the gravitoncontributions is fixed and asymptotic freedom of Yang–Mills theory persists inthe presence of gravitational fluctuations, also including a cosmological constant.Furthermore, Folkerts et al. [35] evaluate generic regularizations and clarifythe non-universal nature of the one-loop coefficient, also covering generic fieldtheory-based UV scenarios for gravity, including asymptotically safe gravity andgravitational shielding. An interesting consequence of a non-trivial gravitationalcontribution to the running of an Abelian charge is the appearance of a combinedUV fixed point in the U(1)–gravity system providing a mechanism to calculatethe fine-structure constant [79].
In three dimensions, a fixed-point study has been reported within topologicalmassive gravity, which includes a Chern–Simons (CS) term in addition toNewton’s coupling and the cosmological constant [80]. On the one-loop level,the CS term has a vanishing beta function and the gravitational sector displaysboth a Gaussian and a non-trivial fixed point for positive CS coupling, which isqualitatively in accord with the fixed point in figure 1 (no CS term).
An interesting structural link has been noted between the gravitational fixedpoint [44] (Ricci scalar, no cosmological constant) and a UV fixed point innonlinear sigma models [81]. In either system, the beta function of the relevantcoupling displays a UV fixed point whose coordinates are non-universal. However,
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the systems display the same universal scaling, with identical exponent q =2d(d − 2)/(d + 2) for all d ≥ 2 dimensions. This coincidence highlights thatnonlinear sigma models have a non-trivial UV dynamics on their own, possiblyquite similar to gravity itself, which is worthy of being explored further [82]. Fixedpoints in more complex (gauged and non-gauged) nonlinear sigma models haverecently been obtained in [83,84].
It would be very helpful to understand which mechanism implies thegravitational fixed point by, for example, restricting the dynamical content. Infact, a two Killing vector reduction of four-dimensional gravity (with scalar matterand photons) has been shown to be asymptotically safe [85]. This topic has alsobeen 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 dependsmore strongly on, for example, matter fields, meaning that more work is requiredto settle this question for pure gravity.
Implications of a gravitational fixed point have been studied in the context ofblack holes, astrophysics and IR gravity, and cosmology. The RG-induced runningGN → G(r) of Newton’s coupling has interesting implications for the existenceof 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 pointtowards the existence of a smallest black hole with critical mass Mc of the orderof 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 predictionof this framework.
Low-energy implications of the RG running of couplings have equallybeen studied, including applications to galaxy rotation curves [94,95]. Recentstudies have re-evaluated gravitational collapse [96], exploring the one-looprunning couplings, and the RG-induced non-local low-energy corrections to thegravitational effective action [97]. It has also been conjectured that gravitymay possess an IR fixed point. In this case, Newton’s coupling grows largeat large distances and may contribute to an accelerated expansion of theUniverse, a scenario that can be tested against cosmological data such as TypeIa supernovae [98–100].
The intriguing idea that an asymptotically safe UV fixed point may controlthe beginning of the Universe has been implemented in Einstein gravity withan 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 thereforeleads to RG-improved cosmological equations [101,106,107], which may evengenerate entropy [108,109]. The framework put forward in Hindmarsh et al.[106] includes scalar fields and exploits the Bianchi identity, leading to generalconditions of existence for cosmological fixed points (with cosmic time) whosesolutions include, for example, inflating UV safe fixed points of gravity coupledto a scalar field.
The proposal that Lorentz symmetry can be broken by gravity on afundamental level has been put forward in Horava [110], where space and timescale differently in the UV with a relative dynamical exponent z �= 1. Provided z =3 (in four dimensions) gravity is power-counting renormalizable in perturbationtheory and Lorentz symmetry should emerge as a low-energy phenomenon [111].
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Interestingly, Horava–Lifshitz gravity with z �= 3 also requires an asymptoticallysafe fixed point for gravity with an anomalous dimension h �= 0 compensating forpower-counting non-renormalizability.
The RG scaling in the vicinity of an asymptotically safe fixed point impliesscale invariance. On the other hand, it is often assumed that a quantum theory ofgravity should, in one way or another, induce a minimal length. Different aspectsof this question and interrelations with other approaches to quantum gravityhave been addressed in Reuter & Schwindt [112,113], and more recently in Basu& Mattingly [114], Percacci & Vacca [115] and Calmet et al. [116]. The findingsindicate that the fixed point implicitly induces a notion of minimal length, relatedto the RG trajectory and the fundamental scale where gravity crosses over fromclassical to fixed-point scaling. It has also been argued that RG corrections togravity can modify the dispersion relation of massive particles, leading to indirectbounds on the one-loop coefficients of the gravitational beta functions [117].
Finally, it is worth noting that standard quantum field theories (withoutgravity) may become asymptotically safe in their own right. This was firstexemplified for perturbatively non-renormalizable Gross–Neveu models in threedimensions with functional RG methods [118] and in the limit of many fermionflavours [119]. In four dimensions, this possibility has been explored for variantsof the SM Higgs [120], for Yukawa-type interactions [121,122], and for stronglycoupled gauge theories [123]. It has also been conjectured that the electroweaktheory 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 ofnon-trivial UV fixed points in four-dimensional quantum field theories, includinggravity and matter.
5. Extra dimensions
Next, I turn to fixed points of quantum gravity in d = 4 + n dimensions, wheren denotes the number of extra dimensions [44,45,125], and potential signaturesthereof in models with a low quantum gravity scale [126–130]. There are severalmotivations for this. Firstly, particle physics models where gravity lives in ahigher-dimensional space–time have raised enormous interest by allowing thefundamental 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-scalephysics becomes accessible to experiment via, for example, high-energetic particlecollisions at the Large Hadron Collider (LHC). The main point of these modelsis that the four-dimensional Planck scale (1.1) is no longer fundamental but aderived quantity, related as M 2
Pl ∼ M 2∗ (M∗L)n to the Planck scale (5.1) of thehigher-dimensional theory and the size of the compact extra dimensions L.
Secondly, the critical dimension of gravity—the dimension where thegravitational coupling has vanishing canonical mass dimension—is two. Hence,for any dimension above the critical one, the canonical dimension of Newton’scoupling is negative. From an RG point of view, this means that four dimensions
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Figure 4. Log–log plot (schematic) of the dimensionless running gravitational coupling in a scenariowith large extra dimensions of size ∼ L and a dynamical Planck scale LT with LT L 1. Thefixed-point behaviour in the deep UV enforces a softening of gravity. (a) At the scale k ≈ LT thecoupling displays a cross-over from fixed-point scaling to classical scaling in the higher-dimensionaltheory. (b) At the compactification scale k ≈ 1/L, the n compactified spatial dimensions are nolonger 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 UVfixed point, if it exists in four dimensions, should persist towards higherdimensions. Furthermore, Einstein gravity has d(d − 3)/2 propagating degrees offreedom, which rapidly increases with increasing dimensionality. It is importantto understand whether these additional degrees of freedom spoil or support theUV fixed point detected in the four-dimensional theory. The local structure ofquantum fluctuations, and hence local RG properties of a quantum theory ofgravity, are qualitatively similar for all dimensions above the critical one, modulotopological effects for specific dimensions. Therefore, one should expect to findsimilarities 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 throughan extended fixed points search in higher-dimensional gravity for general cutoffsRk [45], also probing the stability of the fixed point against variations of thegauge fixing [125], and the results are displayed in figure 2. The cutoff variationsare very moderate, and smaller than the variation with gauge-fixing parameter.The structural stability of the fixed point also strengthens the findings in thefour-dimensional case.
Thus, the following picture emerges. In the setting with compact extradimensions, the running dimensionless coupling g displays a cross-over from fixed-point scaling to d-dimensional classical scaling at the cross-over scale LT of theorder of M∗ (figure 4a). By construction, this has to happen at energies wayabove 1/L. At lower energies, close to the compactification scale ∼ 1/L, the ncompactified spatial dimensions are no longer available for gravity to propagate
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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 solidline, n = 3; and (b) thick-dashed line, n = 6; thick solid line, n = 4. Note that the limit L → ∞ canbe performed, and the levelling-off at M∗ ≈ L reflects the underlying fixed point (thin-dashed linesshow 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 tofour-dimensional scaling (figure 4b). High-energy particle colliders such as theLHC are sensitive to the electroweak energy scale, and hence to the regimeof figure 4a, provided that the fundamental Planck scale is of the order ofthe electroweak scale in (5.1). A generic prediction of large extra dimensionsis a tower of massive Kaluza–Klein gravitons [134–136]. In particular, theexchange of virtual gravitons in Drell–Yan processes will lead to deviations in SMreference processes such as pp → �+�− [137]. Gravitational Drell–Yan productionis mediated through tree-level graviton exchange and via graviton loops. Withineffective theory, the corresponding effective operators are UV divergent (in twoor more extra dimensions) and require a UV cutoff. Within fixed-point gravity,the graviton is dressed by its anomalous dimension, leading to finite and cutoff-independent results for cross sections [126]; see Gerwick & Plehn [138] for acomparison of different UV completions.
Figure 5 displays the 5s discovery reach to detect the fundamental scale ofgravity 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 effectivetheory, these curves continue to grow with L, meaning that the cutoff cannot beremoved. Here, the levelling-off is a dynamical consequence of the underlyingfixed point, and the amplitude is independent of any cutoff. Figure 6 showsthe differential cross sections for di-muon production within asymptotically safegravity, with cross-over scale LT = M∗ [129]. For all dimensions considered, theenhanced di-lepton production rate sticks out above SM backgrounds, leadingto the conclusion that Drell–Yan production is very sensitive to the quantumgravity scale.
Further collider signatures of fixed-point gravity in warped and large extradimensions have been addressed in Hewett & Rizzo [128] within a form-factorapproximation [129]. Modifications to the semiclassical production of Planck-size black holes at the LHC induced by the fixed point have been analysedin Falls et al. [91] and Burschil & Koch [139]. The main new effect is an
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Figure 6. Differential cross sections for di-muon production at the LHC as a function of the di-leptoninvariant 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. Someproperties of Planck-size black holes have also been evaluated within a one-loopapproximation, provided the gravitational RG flow is driven by many species ofparticles [140]. Graviton loop corrections to electroweak precision observableswithin asymptotically safe gravity with extra dimensions have recently beenobtained in Gerwick [130].
In summary, experimental signatures for the quantization of gravity withinthe asymptotic safety scenario are in reach for particle colliders, provided thefundamental 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 gravityat shortest, and possibly at largest, distances. It is conceivable that Planckianenergies (1.1) can be assessed with the help of running gravitational couplings,such as equation (3.1), replacing equation (2.2) from perturbation theory. Ifgravity becomes asymptotically safe, its UV fixed point acts as an anchorfor the underlying quantum fluctuations. The increasing amount of evidencefor a gravitational fixed point with and without matter and its significance forparticle physics and cosmology are very promising and certainly warrant morefar-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 andParticle Physics (Aspen Center for Physics, June 2010), at the IV Workshop Hot Topics inCosmology (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
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the XXV Max Born Symposium The Planck Scale (Wrocl/aw, June 2009) and on an invited talk atGR19: the 19th International Conference on General Relativity and Gravitation (Mexico City, July2010). 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 theirinvitations and hospitality, and my collaborators and colleagues for discussions. This work wassupported by the Royal Society, and by the Science and Technology Research Council (grant no.ST/G000573/1).
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