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PoS(QG-Ph)024 http://pos.sissa.it/ ce. imensions. We also compare e of the Einstein-Hilbert theory, y and the role and implications of sation group techniques employed in menological implications for collider Brighton, BN1 9QH, U.K. ive Commons Attribution-NonCommercial-ShareAlike Licen ∗ [email protected] We review the asymptotic safety scenarioan underlying for ultraviolet quantum fixed gravit point. We discuss renormali findings with recent lattice simulations andexperiments. evaluate pheno the fixed point search, analyseand the provide main an picture overview at of the the exampl key results in four and higher d Plenary talk. ∗ Copyright owned by the author(s) under the terms of the Creat c From Quantum to Emergent Gravity: TheoryJune and 11-15 Phenomenology 2007 Trieste, Italy Daniel F. Litim Fixed Points of andRenormalisation the Group Department of and Astronomy, UniversityE-mail: of Sussex,

PoS(QG-Ph)024 can in fact Λ t, is characterised of the effective theory, with problem by pointing out that otic safety as first laid out in Λ s of gravity and the structure of s [27]-[29]. totic safety scenario (Sec. 2) and umes that the cutoff is the carrier of the fundamental y well exist, and be renormalisable . and higher dimensions (Sec. 5), and egimes of the theory. This is similar ty scenario for gravity is to provide t field configurations dominating the ated in the light of the underlying ap- lusions (Sec. 9). are at the root of fixed point searches quantum field theories. f the ultraviolet completion as long as fixed point in a way which circumnav- vity [30]. There, a systematic study of et cutoff ntly “anti-screening” to allow for this es an interacting ultraviolet fixed point on renormalisation group studies in four hniques [19, 20], renormalisation group ulations (Sec. 7), and phenomenological trivial high-energy fixed point of gravity red within standard perturbation theory. f so, the high energy behaviour of gravity eory of gravity cannot be accommodated tum theory for gravity. The well-known up y in four dimensions faces problems has eory may require a radical deviation from n theory in higher derivative gravity [25], bative non-renormalisability [1]. This sce- string theory. It remains an interesting and open 2 . g . e renormalisation group or lattice implementations of the . g . e , and that the high-energy behaviour of gravity, in this limi ∞ → Λ expansions in the matter fields [26], and lattice simulation N Some time ago added a new perspective to this The asymptotic safety scenario goes one step further and ass In this contribution, we review the key elements of the asymp We summarise the basic set of ideas and assumptions of asympt It is commonly believed that an understanding of the dynamic of the order of the Planck scale (see [31] for a recent review) Λ challenge to prove, or falsify, that a consistent quantum th for within the otherwise very successful framework of local a quantum theory of gravity inon terms a of non-perturbative the level, metric despite field it’s maynario, notorious ver since pertur then known asfor “asymptotic gravity under safety”, the necessitat renormalisation group (RG)is [1]-[4]. governed I by near-conformal scaling inigates the vicinity the of virulent the ultraviolet (UV) divergences encounte and higher dimensions [5]-[18], dimensional reductionstudies tec in lower dimensions [1], [21]-[24], 4d perturbatio [1] (see [2]-[4] for reviews).for a The aim path-integral of baseddegrees the framework of asymptotic in freedom, safe both which in the thein classical metric spirit and to field in effective the field quantum theoryquantum r approaches effects to is quantum possible gra without anthe explicit relevant energy knowledge scales o are much lower than the ultraviol Indications in favour of an ultraviolet fixed point are based large- implications are indicated (Sec. 8). We close with some conc introduce renormalisation group techniques (Sec.in 3) quantum which gravity. The fixedthe point phase structure diagram in of four gravity (Sec. (Sec.proximations. 4) 6) Results are are discussed compared and with evalu recent lattice sim 2. Asymptotic Safety raised the suspicion that a consistent formulationthe of concepts the th of local quantum field theory, fact that the perturbative quantisation program for gravit space-time at shortest distances requires an explicit quan Fixed Points of Quantum Gravity and the Renormalisation Gro 1. Introduction by an interacting fixed point.gravitational path It integral is at expectedlimit that high to the become energies feasible. relevan are If predomina so,may it exist is and conceivable should that be a visible non- within be removed, PoS(QG-Ph)024 η 0. in d = (2.2) (2.1) given reads − η ) 2 ) 0 µ µ ( ( ]= g G G [ rmalisability of s with 2 if the anomalous 0 , and µ > in of classical general g ) = d 2 µ − 0 which also entails d 0 in = hysics along some renormali- 1 at s local in the metric field. In ∗ vity is characterised by univer- 6= lisation group equation for the = ( g . ∗ g ) g β )= ualitatively different types of fixed µ , and the dimensionless coupling as 0 up flows. The existence of a fixed ( er-balances the canonical dimension µ g G ( ) ractions. No a priori assumptions are e fixed point. In fact, although the low- ) rs have to be fixed, ideally taken from oint s “asymptotically safe” and connected , it is expected that (a finite number of) rgy fixed point of QCD. Then the high- ve non-renormalisability of the theory. G . The graviton anomalous dimension η onds to µ up ation group sense, at the ultraviolet fixed 2 Z ies provides for a definition of the theory ( s 3 1 ) + m uld emerge as a “low-energy phenomenon” kg 0 is a function of all couplings of the theory − G 2 µ Z ) en simplify the task of identifying the set of relevant, 11 ( − − G ··· d , 3 )= 10 g · or interacting ultraviolet fixed points. µ ( = ( )= ; the dots denoting further gravitational and matter ( η . Then the Callan-Symanzik equation for µ ) d denotes the renormalisation scale. The graviton wave G ( G µ µ − Z µ ( G d 2. It is commonly believed that a negative mass dimension 67428 g 2 . ln is normalised as d 6 > µ d ) µ d d µ = )= µ ( ≡ N G ··· − g , Z G β ∗ = g ( η 1, we have canonical scaling since η ≪ , following [15] (see also [2, 3]). Its canonical dimension i . This structure is at the root for the non-perturbative reno ) 0 G G µ ( g ; the momentum scale ) is given by µ ) ( . Consequently, the gaussian regime corresponds to the doma µ G 0 ( 2 µ G − Z d < µ Then, in order to connect the ultraviolet with the infrared p µ 1 We illustrate this scenario with a discussion of the renorma The fixed point implies that the high-energy behaviour of gra For infrared fixed points, universality considerations oft )= 1 µ dimensions and hence negative for ( We introduce the renormalised coupling as for the relevant coupling is responsible for the perturbati d gravitational coupling sation group trajectory, a finiteexperiment. number In of this light, initial classical paramete generalof relativity a wo fundamental quantum field theory in the metric field. couplings. Hence, the anomalousof dimension Newton’s precisely coupling count by Newton’s for all sal scaling laws, dictatedmade by about the which invariants residual are the high-energyenergy relevant inte operators physics at is th dominated by thefurther Einstein-Hilbert invariants will action become relevant, in the renormalis related to point. with the low-energy behaviourpoint by together finite with finite renormalisation renormalisation gro at group arbitrary trajector energy scales. Fixed Points of Quantum Gravity and the Renormalisation Gro theory, analogous to the well-known perturbativeenergy high-ene behaviour of the relevant gravitational couplings i marginal and irrelevant operators. This is not applicable f function renormalisation factor Here we have assumedgeneral, a the fundamental graviton action anomalous for dimension gravity which i g In its vicinity with including matter fields. Thepoints. RG equation The non-interacting (2.1) (gaussian) displays fixed two point q corresp relativity. In turn, (2.1) can display an interacting fixed p dimension takes the value PoS(QG-Ph)024 . e 1 2 0, ∼ . ] = p = ) 2 2 e ∗ (3.1) (2.3) [ → p ≈ η within ( 2 oper pair 2 contrast k ) G µ / q 0 for large ( 2 q ϕ 6= ) y suppressed ∗ 0 leads to q , characteristic ( g ) k 6= µ 0 for 2 ∗ R | nsion at e y ) 0. The weakness of > q . − 6= ) has mass dimension g − with momenta smaller x nt at .  q ∗ ( | 2 e ( ( g φ e ϕ ϕ k · ln R R ansform to position space, J ity is expected to become, 1 2 ∼ . In turn, for large anomalous Z 2 ) = y − m gravity can be assessed once grows large. A strong coupling , k p ]+ x 0, (ii) S ) ( an gauge coupling becomes weak ϕ ∼ ∆ µ [ istically, this can be seen from the G , see [43]). Note that the Wilsonian ews). Wilsonian flows are based on k ( → k S 2 G ontrasted with R onset of fixed point scaling. Neverthe- n, if (2.3) corresponds to a non-trivial e modifications of gravity. ∆ q tor al) renormalisation group, which is used an ultraviolet fixed point pled because of the tensorial structure, scales as up / − pendent gravitational couplings are avail- 2 ] m other gauge theories at criticality and away from k 2 ϕ ∗ − [ d g S ventional superconductors with the charged scalar field µ − 0 for t the [32, 33]. In Yang-Mills theories abov Consequently, at an interacting fixed point where 4  2 )= and implications thereof have been discussed in [34, 35, 36] → +Higgs theory, the abelian charge µ d exp ) ) ( . 1 q − ( ( G 4 1 implies that the magnetic field penetration depth and the Co k ren U ] , where the propagation of fields R = = k ϕ ∗ Γ η η D [ Z becomes arbitrarily small in its vicinity. This is in marked (for examples and plots of ln G 0, the dimensionful coupling obeys (i) Λ -dimensional d ]= k J . In three dimensions, a non-perturbative infrared fixed poi → → R [ 2 k e k µ ) Z in the vicinity of a fixed point the propagator is additionall η d ln for + 4 − ∞ , we have the standard perturbative behaviour − 2 η d → → ) = ( q possibly modulo logarithmic corrections. After Fourier tr in momentum space. Here we have evaluated the anomalous dime 2 η ( e k 2 ) β / 2 R d / − η is suppressed. A Wilsonian cutoff is realised by adding ) − 0, the anomalous dimension implies the scaling 2 k Integer values for anomalous dimensions are well-known fro Whether or not a non-trivial fixed point is realised in quantu As a final comment, we point out that asymptotically safe grav 1 , with p ( 2 d 6= 2 ( , the dimensionful coupling − − ∗ p µ Then, for small g [32]. The fixed pointdescribing the belongs Cooper pair. to The the integer value universality class of con 4 dimension of QCD in four dimensionsbecause where of the a dimensionless non-interacting non-abeli ultravioletinfrared fixed fixed point. point for In tur Fixed Points of Quantum Gravity and the Renormalisation Gro quantum gravity within a fixed point scenario. their canonical dimension. In the for bosonic fields in two-dimensional systems. 3. Renormalisation Group explicit renormalisation group equations for theable. scale-de To that end, we recall thebelow set-up for of Wilson’s the (function case ofthe quantum notion gravity of (see a cutoff [37]-[42] effectivethan for action revi the coupling in (2.3) is a dimensional effect, and should be c behaviour of this type would imply interesting long distanc in an essential way,dressed two-dimensional graviton at propagator whose high scalar part, energies. neglecting Heur ∼ the Schwinger functional and the requirement that correlation length scale with the samefour universal dimensions, exponent ultraviolet a fixed points with this corresponds to a logarithmic behaviour for the propaga for the dimensionful gravitational coupling. In the case of to (2.2). Hence, (2.3) indicatesless, that at gravity the weakens fixed point at the the theory remains non-trivially cou and (iii) PoS(QG-Ph)024 = k f the (3.2) utoff ative Z ln t ∂ and the full Λ [43]-[51],[42]. k can be expressed → dent effective ac- and taking advan- R k k momentum squared ] and for small mo- k R onally, controlled by R for introduced in the previ- Λ Γ µ theories or gravity. Then the able to maximise the physics are required to integrate (3.2). ariant variables rather than the ent functions. Convergence is . k tion he benefits and shortcomings of beyed in the the physical limit wers of the fields (vertex func- auge theories [51]. s essentially local in momentum Γ ted choices of presently employ option (b). rivative expansion), and (d) com- t , ∂ k n theory is fully reproduced to all ll momenta and summation over all R t . Ward-Takahashi identities are main- ∂ Γ and 1 . It obeys an exact functional differential k up − J Γ i  k , the Schwinger functional obeys ϕ depends on it. k R h k ∆ + = Γ ) − By construction, the flow equation (3.2) is well- 2 5 φ ( k k , → Γ φ . A number of comments are in order: k are preserved by either (a) imposing modified Ward  k → ) q Γ R k Tr ( φ 1 2 is compatible with the symmetry. In general, this is not the -point functions of R δφ n k 1 2 ) = S p k ∆ ( − Γ [47, 48]. The expansions (b)-(d) are genuinely non-perturb t ]) ∂ k δφ J with a one-loop type integral over the full field-dependent c . This is illustrated in Fig. 1. The endpoint is independent o [ / R k 0 k k Z = Γ Γ k 2 ln Γ δ − Systematic approximations for ≡ ≡ φ Γ ) · q if the insertion J takes the role of the renormalisation group scale Global or local (gauge) symmetries of the underlying theory , . We also introduce its Legendre transform, the scale-depen k p R k k ( The stability and convergence of approximations is, additi ]( 0, or by (b) introducing background fields into the regulator J The integrand of (3.2) is peaked for field configurations with ln φ [ ) → , and suppressed for large momenta [due to condition (i) on = 2 sup 2 ( k k t k Γ ; J [43, 49]. Here, powerful optimisation techniques are avail ]= ≈ i k φ k 2 [ S binations thereof. The iterative structure of perturbatio These include (a) perturbation theory,tions), (b) (c) expansion expansions in in powers po of derivative operators (de orders, independently of effective action Finiteness and interpolation property. defined and finite, and interpolates between an initial condi regularisation, whereas the trajectories and lead, via (3.2), tothen coupled checked flow by equations extending the for approximation the to coeffici higher order Stability. identities which ensure thatwhen standard Ward identities are o Locality. content and the reliability of the result through well-adap R q case for non-linear symmetries suchrequirements as of in gauge non-Abelian symmtry gauge for menta [due to condition (ii)].and Therefore, field the space flow [43]-[45]. equation i Approximations. These ideas have been explicitely tested in scalar [50] and g Symmetries. tained for all as Ward-Takahashi identities for tage of the background field method, orgauge by fields (c) or using the gauge-cov metric field.these For options, a see detailed [37]. discussion of For t gravity, most implementations k ∆ Γ t • • • • • ∂ propagator. Here, the tracefields, Tr and denotes an integration over a equation introduced by Wetterich [46] Fixed Points of Quantum Gravity and the Renormalisation Gro momentum scale ous section. Under infinitesimal changes tion which relates the change in −h PoS(QG-Ph)024 , ∞ (3.4) (3.3) ective → Λ r quantum

integrating−out w energies. classical 0. a) Flow con- action fixed point general relativity → k rturbatively renormal-  ϕ k · EH . Γ ] Γ k k φ ∂ [ R ∗ ≈ k t can be defined without explicit , given mainly by the classical Γ δφ is simple, mainly given by the Γ ∂ 0 δ in (3.3) can be removed, 1 Γ al underlying (3.1). It should be f the strongly coupled low energy in the space of all action functionals − ying a possible high energy fixed Λ ting the Einstein-Hilbert action at low Z  k sing only the (finite) initial condition k Γ h as quantum gravity, proving the ex- R ]+ is more difficult. For gravity, illustrated IR analogous to standard reasoning within cannot be removed. Still, the flow equa- UV ϕ + ∗ up , such as in QCD. In this case, illustrated ) Λ ales and lower energies S Γ Λ + 2 ( k via a functional integro-differential equation φ < Γ [ p Γ  k S − Tr b) 6 2 1  k dk at high energies (“bottom-up”). 0 exp ∗ . Λ Γ Z ren

integrating−out ] + ϕ Λ at high energies in the ultraviolet with the full quantum eff quantum action D Γ classical effective action [ S = the hierarchy of Dyson-Schwinger equations. Z . The physical theory described by Γ = g . Γ e − e with the above properties qualifies as fundamental action fo In renormalisable theories, the cutoff k ∗ is given by the classical action Γ Γ Γ Γ ∗ k ∂ Γ S , which upon integration matches with the known physics at lo ≈ k ∗ remains well-defined for arbitrarily short distances. In pe 0 Γ ∗ Γ Γ → IR UV Λ Wilsonian flows for scale-dependent effective actions Γ , and the (finite) flow equation (3.2) at low energies in the infrared (“top-down”). b) Flow connec Λ reference to an underlying path integral representation, u Integral representation. Γ gravity. In non-renormalisable theories the cutoff tion allows to access the physics at all scales In principle, any which is at the basis of in Fig. 1b), experiments indicateEinstein that Hilbert the theory. low energy The theory point challenge action consists in identif This provides an implicit regularisationcompared with of the the standard representation path for integr effective field theory [31]. isable theories, in Fig. 1a), the highaction, energy and behaviour the of challenge the consists theory in is deriving simple the physics o istence (or non-existence) of a short distance limit limit. In perturbatively non-renormalisable theories suc and Renormalisability. Γ p a) • • action energies with a fundamental fixed point action necting a fundamental classical action (schematically); arrows point towards smaller momentum sc Figure 1: Fixed Points of Quantum Gravity and the Renormalisation Gro PoS(QG-Ph)024 , 2 gh µν ¯ the S D g . (3.5) − g . e 2), possi- ]= -dimensional k , d Λ open a door for  [ , a ghost term gf matter S responding flow is no S ein-Hilbert action with optimised scalar cutoff hen eliminated from the + becomes gauge-invariant ry within this set-up, we gh ] and above, the RG running or vertex functions, are ob- ravity [5]. A Wilsonian ef- S with a running gravitational V regularisation. This high- ∗ k µν ximation which captures the ¯ + . This implies that the mode . ) a functional Callan-Symanzik g Λ M 2 , , gf ¯ k µν D uation and functional flows (3.2). ≈ S ruct a covariant Laplacean µν g G -propagating background field ¯ ( − g k + [ R k l and the background field. A second uge-fixing term . This procedure, which dynamically s of “background independence” for much smaller than the = rs, derivatives, or functions of kground field flows, see [56]. Finally, Γ l constant term and employ a momen- ng heat kernel techniques. Here, well- ··· sibly, non-local operators in the metric h as k 2 opic of the following sections. . In (3.2), this corresponds to the choice ty, and give an overview of extensions. k up 2 + k φ to (3.2). All couplings in (3.5) become run- 2 Λ k . For 2 k ∼ )+ (with canonical mass dimension The well-known Callan-Symanzik equation de- 7 g k ( Λ R −  µν g det p x d . Altogether, should contain the Ricci scalar d k Z Γ k matter driven by a mass insertion [43, 49] lead to substantial algebraic simplifications, and S G d 1 k dk π R k 16 , which does not fulfill condition (i). Consequently, the cor 2 = , the gravitational sector is well approximated by the Einst k ∗ k with the tensorial structure of [7] and variants thereof, an Γ M k , a running cosmological constant )= R k , and similarily for the gravity-matter couplings. At 2 0 will depend on the background fields. The background field is t q G ( = k k k R longer local in momentum space,lights and a requires crucial difference an between additional theIn U Callan-Symanzik eq this light, the flowequation with equation momentum-dependent (3.2) mass could term insertion be [52] interpreted as R scribes a flow Link with Callan-Symanzik equation. G Now we are in a position to implement these ideas for quantum g In this section, we discuss the main picture in a simple appro A few technical comments are in order: To ensure gauge symmet • ≈ k Planck scale fective action for gravity cutoff G final equations by identifying itreadjusts with the the physical background mean field, field quantum implements gravity. the For requirement a detailed evaluationwe of note Wilsonian that bac the operator tracesadapted Tr choices in for (3.2) are evaluated usi [5, 37], [53]-[56]. This way, the extended effective action Fixed Points of Quantum Gravity and the Renormalisation Gro salient features of an asymptoticWe consider safety the scenario Einstein-Hilbert for theory gravi withtum a cutoff cosmologica under the combined symmetry transformations ofbenefit the of physica this is that the background field can be used to const systematic fixed point searches, which we discuss next. 4. Fixed Points coupling take advantage of the background field formalism and add a non of gravitational couplings becomes important. This is the t or similar, to define a mode cutoff at momentum scale and matter interactions and explicit flow equations for the coefficient functions suc Ricci scalar, the Ricci tensor, thefield. Riemann The tensor, and, effective action pos should also contain a standard ga ning couplings as functions of the momentum scale tained by appropriate projections after inserting (3.5) in bly higher order interactions in the metric field such as powe PoS(QG-Ph)024 d α 1, r and = (4.4) (4.3) (4.1) (4.2) (4.5) k 0 and partial k , G ) ) C = a diverges. Z λ ∗ 1 η g d − 2. At a non- 2 d ) ± 4 classical force λ at the gaussian, 2 = − d d d − -functions derived 1 ( -functions is , amended by stablity 1 . β − 2 k ( β 2 1 2 − R 2 ) ) d 1 2 − λ = d -variation and optimisation 2 + − k θ k − g R d − ) ( Λ g 1 1 2 2 2 ( ] ] . full − ] − k 7 ) 11 13 [ d c ref reproduces the 4 [ [ ( tional Einstein-Hilbert action in )= = , as are the full ) d g n renormalisation is set to 2 k in the Einstein-Hilbert theory with the ; ) λ π vitational and cosmological constants 2 ′′ λ , and , the anomalous dimension . nal coupling, and for ( k θ mension, we find + i 2 up to remove phase space factors. This does R ) 384 , and reflects the fact that the gravitational d 1 0 3 2 ( ± 2 , ( . . . λ d bound 0. In the domain of classical scaling ′ 2 2 − / g bound 7 4 3 d 2 g 2 ¯ ) g θ g ′′ G − / = − + 2 d − − − − − θ ) = 2 ) = d d at the fixed point are ) → k 5 4 0 1 + . . . g π ( . Results indicate the range covered under 2 5 ( = , 8 , which are connected under the renormalisation d k g d θ 2 2 3 1 4 . Then the coupled system of 1 ) ∗ , 2 matter R − − − k θ G 0 )( − )( S η ) ( Z g 2 d 2 = ) λ 2 = limits the domain of validity of the approximation. g 2 ( )( − 3 0 7 + − θ ∗ . . . ) ) 2 d | 2 d -variation. d bound 2 2 1 − 2 k g ( λ ( ′ g α + ( d 2 ∂ − − − Γ bound θ + ( ≡ implies d / 0 and find two fixed points, the gaussian one at g < 5 1 4 2 ( d g . . . k g = depend on ( ) d 1 1 1 − β = bound G d + -variation with optimised d ∂β λ g c g 2 g 2 g 4 α λ ) ( − [43, 49], and a harmonic background field gauge with paramete + = d 2 / k and scaling exponents 1 ) ) ) ) g λ full with ) = − 2 c a b g 2 d ) = d q = d d ; b − c g and the cutoff function ∗ , g − / k , g 2 g 1). In all cases the fixed point is stable. The variation with α R λ k ( ( ) = ) = ( = → g g η θ , , α and ) g 2 λ λ α q ( ( g λ − β β 2 k ≡ ≡ ( The variation of 4 g λ t t ∼ ∂ ∂ ) 2 We first consider the case q are approximately constant, and (3.5) reduces to the conven ( k k The anomalous dimension vanishes for vanishing gravitatio considerations [43, 49], is weaker than the We have rescaled trivial fixed point the vanishing of not alter the fixed point structure. The scaling and law in the non-relativistic limit [57]. For the anomalous di where it is understood that in a specific limit introduced in [11]. The ghost wave functio Fixed Points of Quantum Gravity and the Renormalisation Gro variation of both The full flow (3.2) is finitefrom (no it. poles) Therefore and the well-defined curve for all R and the effective action is given by (3.5) with gauge fixing parameter Table 1: a non-gaussian one at in Feynman gauge ( coupling is dimensionless in two dimensions. At group. The universal eigenvalue Λ at the non-gaussian fixed point. euclidean dimensions. The dimensionlessare renormalised gra PoS(QG-Ph)024 . ) ) ). 2. c − λ , 2 and ( ) − ) (4.6) b R d λ ; k 2 . bound µν R with the k − g g 2 1 R 1 0, we find a < 2 and -dependence det )( ] ] ] ] ] ] ] > . k ) . ] 6= − 1 0 in (3.5). The R 8 λ 2 p λ 16 16 17 17 17 17 17 [ λ ( ) ref [ [ [ [ [ [ [ + 6= R 0 – the eigenvalues d 2 k d g ) 2 2 Λ 1 )( θ √ 4 2 9 4 1 + . . . . 4 − − 4 3 4 4 d − − − − θ 4 ( and d − − − − 2 e eigenvalues are real [15]. ( ) 1 − 1 4 4 θ d . Note that leads to two branches of real − − 2 nstant term with eigenvalues 8 ) d . 2 λ )) = ( ) tructure and the scaling exponents λ d rlying operators β , results are summarised in Tab. 1. 2 0. On a non-gaussian fixed point, λ 6 9 0 28 2 5 1 6 8 5 . . 3 dure or the momentum cutoff func- ( , [5, 7, 11]. It is therefore noteworthy d . . . . . √ 0 θ 0 ( . ponents 3 2 1 1 1 − > − 25 26 g up with the correct infrared behaviour. g of the expansion, including the invariants reof. Furthermore, the 4 , . 1 ven by 2 ws: For non-vanishing . i = genvalues reflect that the interactions at ( e λ up 8 d ∗ ( + 1 ) = ( 4 g λ ∗ d β g , = ∗ )= Feynman gauge under partial variations of λ 3 λ η , . ( ) ( at the fixed point – are a complex conjugate pair, 9 4 ) 0 a 8 1 3 3 5 7 4 ) ′′ 4 g ...... − θ λ 2 3 2 2 2 2 2 in four dimensions. The reason for this is that the − , we find 1 . ) 4, the vanishing of 0. Only the second branch corresponds to an ultraviolet ≡ d 3 λ 3 167 2 ( 4 dimensions. Consequently, we find a unique ultraviolet ) > > − , with 6, using g √ , we find 0 1 ) , ∗ 2 1 2 g d = g λ d λ ··· + = ( ( d 2 = = 0 d ≡ d > ′′ 4 . g g 2 would have to change sign at least twice, which is impossible = cannot change sign under the renormalisation group flow (4.3 1 θ 2 1 )( λ 3 . In i g ′ 7 4 7 9 5 4 4 4 = g p ...... ( η 1 − θ 64 or ; d)-h) Landau-deWitt background field gauge with optimised 1 1 2 2 2 2 1 j g − k 1 − and 1 2 . 0, g R d = 4 from scaling exponents with the order 2 -functions is fairly non-trivial, ( ∂ 5 3 i for > = ∗ 2 / β d − i R g ∗ g = λ d λ β µν ′ ∂β g − θ i 2 2 3 4 5 6 2 2 and 2 d det 4 1 cannot change sign either. Hence, to connect a fixed point at p = with = η R 0. Furthermore, ∗ ) ) ) ) ) ) ) ) ′′ ∗ f λ c e a b g h d θ < λ . This pattern changes for lower and higher dimensions, wher 2. Evaluating (4.2) for and The first term vanishes in ± η . The variation of 4 µν ′ > µν g 2 d θ , see Tab. 1 and 2. For the Einstein-Hilbert theory in 4 g d -dependence of the k + Next we allow for a non-vanishing dynamical cosmological co At this point it is important to check whether the fixed point s R = α det det λ 2 , p p 1 d Table 2: however, R coupled system exhibits the gaussian fixed point Feynman gauge and optimised Non-trivial fixed points of (4.2) and (4.3) are found as follo of the stability matrix gaussian fixed point at R θ fixed point which is connected under the renormalisation gro This can be seen as follows: At Fixed Points of Quantum Gravity and the Renormalisation Gro An interesting property of this system is that the scaling ex depend on technical parameters such astion the gauge fixing proce non-trivially vanishing for all fixed points with either Therefore, we have a unique physically relevant solution gi that scaling exponents only depend mildly on variations the Consequently, the fixed point have modified the scaling behaviour of the unde stability matrix, albeit real, is not symmetric. Complex ei The fixed point 2 PoS(QG-Ph)024 3 R . We µν g α 5.2 5.0 1.3 det p R ; data from [13, 14]. k R y largest uncertainty variation with conclude that the fixed point is . Once more, the fixed point and re, we should expect similarities has been presented in [13, 14], also pecifically in more than four dimen- tion of non-interacting matter fields the stability of the fixed point under y of space-time. In four dimensions ge fixing parameters (using either Feyn- k ersists in higher dimensions. Further tive. Hence, from a renormalisation ial. Continuity in the dimension sug- R of gravity, are qualitatively similar for arting from the operator mensions. Interestingly, this pattern is ensions, should persist towards higher ing exponents [17, 18]. This is an im- upports the view introduced above. A ass dimension – is two. For any dimen- 1), and the regulator ffects for specific dimensions. Secondly, is on the level of a few percent and smaller cture of quantum fluctuations, and hence ff the UV fixed point being finite. Finally, 4.6). An extended systematic search for up picture [16]. This structural stability also cluding higher powers of the Ricci scalar cal dimension of gravity – the dimension ≤ e gauge fixing parameter (see Tab. 3). The α ≤ 10 . α -independent for all technical purposes, with the presentl k , ammended by stability considerations, is smaller than the R 5 6 7 8 9 10 k R 5.31 – 6.06 7.79 – 8.76 10.4 – 11.6 13.2 – 14.7 16.1 – 17.9 19.1 – 2 2.69 – 3.114.54 4.26 – – 5.16 4.78 6.52 6.43 – – 7.46 6.89 8.43 8.19 – – 9.46 9.34 10.3 10.5 – – 11.4 12.1 12.1 13.1 – – 13.2 1 13.9 – 1 The variation of scaling exponent with dimensionality, gau | ′ ′′ -variation, covering various classes of cutoff functions, d θ θ θ | k It is interesting to discuss fixed points of quantum gravity s R man gauge, or harmonic background field gauge with 0 Table 3: Fixed Points of Quantum Gravity and the Renormalisation Gro strengthens the results in the four-dimensional case, and s conclude from the weak variation thatstudies the including fixed point higher derivative indeed operators p confirm this The fixed points in higher-dimensional gravity for general cuto sion above the critical one,group point the of canonical view, dimension the is four-dimensionalgests theory nega that is an not ultraviolet spec fixeddimensions. point, More generally, if one it expects that exists thelocal local in renormalisation stru four group dim properties of aall dimensions quantum above theory the critical one, modulothe topological e dynamics of the metricand field depends above, on the the metric dimensionalit fieldin is the ultraviolet fully behaviour dynamical. ofrealised gravity Hence, in in once the four mo results and [11], higher see di the analytical fixed point ( where the gravitational coupling has vanishing canonical m 5. Extra Dimensions and higher, couplings become irrelevantportant with first negative indication scal for the set of relevant operators at has been confirmed in [10]. sions. The motivation for this is that, first of all, the criti we mention that the stability of the fixed point under the addi is smaller than the dependence on gauge fixing parameters. We fully stable and than the variation with arising through the gauge fixingextensions sector. beyond In the Tab. Einstein-Hilbert 2, approximation, we in discuss the scaling exponents come out very stable. Furthermore, st both in Feynman gauge [8, 16] and in Landau-DeWitt gauge [17] testing the stability of the result against variations of th variation with PoS(QG-Ph)024 d e 4 0, λ = where d η ) 4 / is larger λ λ ( g 0 cannot be 0 or 1 bound = g = η η / 0.5 0 and is interpreted as a → 0. 2 k → and the cosmological constant k 0.4 k λ d 4 g n-vanishing cosmological constant g hase portrait of the Einstein-Hilbert the ultraviolet. In the infrared limit, es that the line 1 me sign along any trajectory. In the = urn, the cosmological constant may t fixed point. Some trajectories ter- k out the ultraviolet fixed point reflects e quantum gravity is discussed below ared fixed point, the gravitational cou- Λ ve or negative cosmological constant at 0.3 ) line indicates the bound up trix lead to a positive cosmological con- int are indicated by dots (red). The separatrix ories for realistic cosmologies [60]. This es are characterised by whether 0 can neither be crossed (see Sec. 4). Thus, = , the flow displays a crossover from ultravio- Pl η 0.2 M 11 changes sign only across the lines ≈ k η 0.1 . Since g 0 , linked to the present approximation. The two fixed points ar ) λ ( 0.1 - bound g 0 and by the sign of 0.01 0.01 0.03 0.02 0.02 - - bound g d The phase diagram for the running gravitational coupling 4 g 0. Arrows indicate the direction of the RG flow with decreasin In this section, we discuss the main characteristics of the p = η / we conclude that the graviton anomalousphysical dimension domain has which the includes sa thepling ultraviolet and is the positive infr and thechange anomalous sign dimension on negative. trajectories Inminate emmenating t at from the the boundary ultraviole 1 Figure 2: Fixed Points of Quantum Gravity and the Renormalisation Gro or smaller phase transition boundary between cosmologies withlarge positi distances. Trajectories in thestant vicinity at of large the scales separa and are, therefore, candidate traject disconnected regions of renormalisation group trajectori connected by a separatrix.the The complex rotation nature of the of separatrix the ab eigenvalues. At crossed. Slowness of the flow implies that the line in four dimensions. The Gaussianconnects and the the ultraviolet two fixed fixed po points (full green line). The full (red phenomenological application of these findings(see in Sec. 8). low-scal 6. Phase Diagram theory [9, 11] (see Fig. 2). Finiteness of the flow (3.2) impli modifies the flow mainly in thethe crossover region separatrix rather leads than to in a vanishing cosmological constant let dominated running to infrared dominated running. The no PoS(QG-Ph)024 3 8 our = θ / explicit 1 GeV is no = . Rather, the 19 ν L 10 ∼ ≈ can be significantly under the renormal- n higher dimensions Pl ∗ . Therefore we expect η M M d − 2 k ∗ hysics of black holes [59], g ≈ pling in this scenario has been ) fects due to a fixed point at high k l aspects of quantum space-times ( e been put forward based on Regge he exciting possibility that particle in low-scale quantum gravity mod- Planck scale and the physics at particle colliders G us dimension scaling exponent has been measured ’s coupling to the critical point, with tudies of the spectral dimension (see ut-off flow [9], except for the location ulations [29]. In the Regge calculus otically safe quantum gravity behaves, bility of random walks on the triangu- lity of space-time has been measured as ve dimensionality displays a cross-over geometrical considerations on the lattice antisation of gravity [69, 70, 71]. up uum limit has been given in [27] using the dimensional bulk whereas matter fields are n + 4 12 = is of the order of the electroweak scale, this scenario d , where the gravitational coupling displays a cross-over ∗ ∗ M sets the fundamental scale for gravity, leading to the rela- M ∗ , and should be contrasted with the RG result extra dimensions are compact with radius ≈ 3 1 M . If k n ∗ ≈ const. to fixed point scaling 2 at small scales of the order of the Planck scale. This behavi M ν ≈ ≈ ≪ ) [28], a behaviour which is in qualitative agreement with the in the corresponding limit, see (4.5). d k L 1 ( d 1 / 1 2 − G d = = for the four-dimensional Planck scale. Consequently, ν n ν ) L 0 which is non-universal. Similar phase diagrams are found i ∗ provided 1 = M ( Pl 2 . The result reads η ∗ 1 ν / M M 4 at large scales to -dimensional Planck mass − ∼ n ≈ = 2 Pl + d ∗ The renormalisation group running of the gravitational cou Within the causal dynamical triangulation approach, globa Lattice implementations for gravity in four dimensions hav | The phenomenology of a gravitational fixed point covers the p 4 M g β = g tion lower than lifts the hierarchy problem ofcolliders could the establish standard experimental model evidence for and the opens qu t studied in [13, 14, 64]energies and set is in summarised at momentum in scales Fig. 3. The main ef d have been assessed in [29]. There,a the effective function dimensiona of the length scalelated by manifolds. evaluating The the key return result proba isfrom that the measured effecti compares nicely with the cross-over of the graviton anomalo from perturbative scaling ∂ of the line 1 Fixed Points of Quantum Gravity and the Renormalisation Gro picture agrees very well with numerical results for a sharp c [13, 14]. 7. Lattice calculus techniques [27]-[28] andapproach, causal a critical dynamical point which triang allows forEinstein a Hilbert lattice contin action with fixedin cosmological the constant. four-dimensional simulation A based on varying Newton RG fixed point result isation group (see Sec. 2),[2, and 3, 58]). with These renormalisation findings group corroboratein s the an claim essential that way, asympt two-dimensional at short distances. 8. Phenomenology cosmology [60, 61, 62],[64, modified 65, dispersion 66]. relations [63], Inels this [67, section 68]. we concentrate There, on gravity the propagates later in with [11] as discussed in Sec. 4.lead In the to large-dimensional the limit, estimate confined to a four-dimensional brane. The four-dimensional longer fundamental as soon as the PoS(QG-Ph)024 1, 6 , a ∗ (8.1) M | ≪ η ≈ | 1 | es in the k 0 5 k η = G G | s that (8.1) n from numerical of the graviton, η 4 η k . In that case, however, 3 cutoff leads to a power- d ln − 2 ∗ . In a fixed point scenario, -dimensional classical scaling ) 7 ) M Λ n m . (schematically). b) Classical-to- , = ∗ = 2 s + ension, the dilepton production n ( d M 4 G nario with large extra dimensions of 7 (from right to left). ( G , ough virtual gravitons at the Large − ange is described by an amplitude rnative matching has been proposed 1 ion is small, the propagator is well 2 r of ··· − , n erator in the effective action, involving 1 ≈ 1 tering processes at particle colliders as , we are lead to the dressed propagator  up 2 . In this case, (8.1) is ultraviolet divergent η ∗ = ∗ on the cutoff m m n M a) In the infrared (IR) regime where M 2 +  and the anomalous dimensions 1 to − 0 s 1. The fixed point behaviour in the deep ultraviolet n 9 7 5 3 1 1 ≪ k ) ∗ ∗ 0.4 0.8 0.6 0.2 G in (8.1), setting ≈ ≫ M dm | ≪ 13 ) M 2 L η s ∞ k / ∗ | 0 Λ √ M Z ( ( k is a function of the energy-momentum tensor, and 4 4 ∗ G − ∗ ν 1 ln and ν dimensions and of the graviton propagator [69, 72]. The integration over M . This propagator is used within effective theory settings, 2 measures the effective number of dimensions. At T M ∗ → 1 n ) µ ) µ − ) − M 2 m 2 + ∼ ) k T / , d 2 / ( n 4 n s S n 1 ( π m + G ≈ ( = 2 2 G Γ ) k + d n − s = dln , which we take as continuous, reflects that gravity propagat + µν ∗ S / UV (fixed point) 4 m g at momentum scale T M ) = ( displays a crossover from 4-dimensional to = µν η ln m 2 in the vicinity of an UV fixed point. The central observation i d , T − s is improved due to the non-trivial anomalous dimension 2 d ( = + k 2 S G k / L 2 T n / + ) G n ∗ 2 IR ( 4) m M = ln1 a) schematically b) numerically + = g s ( d . The slope dln The scale-dependence of the gravitational coupling in a sce , where L ≈ with fundamental Planck scale 2 due to the Kaluza-Klein modes [69]. Regularisation by an UV / T ) 1 L · ≥ m We illustrate this at the example of dilepton production thr S (4.4). Evaluating g ≈ ∼ , n . s k = ( IR ( ln g . law dependence of the amplitude the behaviour of for the Kaluza-Klein masses higher-dimensional bulk. Ifapproximated the by graviton anomalous dimens and applicable if the relevant momentum transfer is is a function of the scalar part at G classical-to-quantum crossover takes place from Fixed Points of Quantum Gravity and the Renormalisation Gro Figure 3: size the coupling e integrations of the flow equation; Hadron Collider (LHC) [64].amplitude is To generated through lowest an order effective dimension–8four in op fermions canonical and dim aA graviton [69]. Tree–level graviton exch becomes finite even in the UVin limit [65], of based the on integration. the substitution An alte quantum crossover at the respective Planck scale for that signatures of this cross-overlong should as these be are visible sensitive in to momentum scat transfers of the orde enforces a softening of gravitational coupling (see text). PoS(QG-Ph)024 ] is for TeV (D8) Λ − l + Λ[ l → -1 parton : pp becomes σ E 10 fb n=6 n=4 5 D on M Λ can be performed, 10% variation in the ∞ ± → Λ t 1 2 3 4 5 6 7 8 9 10 ] on the partonic energy [72], 6 5 4 3 2 1 0 lanck scale at the LHC, and TeV [ Λ

D M s quantum gravity in which the he topics discussed here, and the . To estimate uncertainties in the he metric field. ransition towards fixed point scal- conclude that the large anomalous ] ∗ g statistical errors. This translates excess can be observed, taking into . It is nicely seen that nvergence, and uncertainties which TeV (D8) M − , as a function of a cutoff l Λ ven in the quantum regime. We have tal Planck scale is as low as the elec- + Λ[ ) le, strongly supporting this scenario. σ l mptotic safety could even be observed ite dilepton production rate. n to classical general relativity as a “low → ial cutoff f asymptotic safety are equally seen in > dvances based on renormalisation group -1 fixed point [64]. In either case the mini- p. nks more deeply in the future. Finally, up p fixed points and scaling exponents de- + : pp σ 4 10 fb n=5 n=3 5 = parton d E ). a) Effective theory: the sensitivity to the cutoff 1 14 − (100fb atthe LHC( 1 D 1 2 3 4 5 6 7 8 9 10 ] − 6 5 4 3 2 1 0 M TeV [

D M ] n=6 n=4 n=3 n=2 n=1 TeV (D8) when fixed point scaling is taken into account. − reflects the gravitational fixed point, thin lines show a l Λ[ + for the fundamental Planck scale l ∞ Λ → D ≈ : pp → M σ 5 D Λ M contour; plot from [72]. b) Renormalisation group: the limi for discovery contours in D Λ σ M -1 -1 The 5 100 fb 10 fb 2 4 6 8 10 12 a) effective theory b) renormalisation group ] I thank Peter Fischer and Tilman Plehn for collaboration on t The asymptotic safety scenario offers a genuine path toward In Fig. 4 we show the discovery potential in the fundamental P 8 6 4 2 10 TeV [ D M account the leading standard modelinto backgrounds a maximum and reach assumin independent of setting the partonic signal cross section to zero for tected in four- and higher-dimensional gravity is remarkab metric field remains the fundamentalreviewed carrier the of ideas behind the this physics set-up e inand the light lattice of studies. recent a The stability of renormalisation grou Furthermore, underlying expansions show goodarise numerical through co approximations are moderate.troweak If scale, the signs fundamen for thein quantisation collider of gravity experiments. and It asy lattice is studies. intriguing that It key aspects will o be interesting toenergy evaluate phenomenon” these of a li fundamental quantum field theory in t Acknowledgements organisers for the invitation to a very stimulating worksho 9. Conclusions asymptotically safe gravity is a natural set-up which leads RG set-up, we allow for aing 10% sets variation in. in Consistency the is scale checked by where introducing the an t artific Figure 4: Fixed Points of Quantum Gravity and the Renormalisation Gro transition scale; plot from [64]. (8.1) remains UV divergent due todimensions the Kaluza-Klein in modes. asymptotically safe We gravity provides for a fin compare effective theory studies [72] withmal a signal gravitational cross sections have been computed for which a 5 reflected in the and the leveling-off at an assumed integrated luminosity of 10fb PoS(QG-Ph)024 , 77 , ing 638 , Class. (2003) n-Hilbert (2003) tum gravity (1999) 181 68 (1998) 971 67 Class. Quant. able? 102 , Living 57 , hep-th/0207028. , Phys. Rev. D , AIP Conf. Proc. , Phys. Lett. B , Int. J. Mod. Phys. A (2005) 035 d point in quantum duction, 0502 , Phys. Rev. D , Phys. Rev. D , Phys. Rev. D , 322 (2006). , JHEP , Eds. S.W. Hawking and W. Israel, (2004) 201301 [hep-th/0312114]. 841 , Prog. Theor. Phys. 92 y Press, 0709.3851 [hep-th]. up , AIP Conf. Proc. Flow equation of quantum Einstein gravity in a 15 (2002) 025026 [hep-th/0205062]. , Phys. Rev. Lett. , 0805.2909 [hep-th]. 66 Ultraviolet properties of f(R)-gravity Investigating the Ultraviolet Properties of Gravity with a On the group flow of f(R)-gravity A fixed point for truncated quantum Einstein gravity The Asymptotic Safety Scenario in Quantum Gravity in ’Approaches to Quantum Gravity: Towards a New Understand flow of quantum gravity in the Einstei , Phys. Rev. D (2002) 065016 [hep-th/0110054]. Ultraviolet fixed point and generalized flow equation of quan Is quantum Einstein gravity nonperturbatively renormaliz Proper time flow equation for gravity Asymptotic safety of gravity coupled to matter Fixed points of quantum gravity in extra dimensions Fixed points of quantum gravity in higher dimensions Constraints on matter from asymptotic safety 65 Gauge and cutoff function dependence of the ultraviolet fixe The asymptotic safety scenario in quantum gravity: An intro , 5 (2006). (2002) 025013 [hep-th/0108040]. (2002) 483 [hep-th/0110021]; General Relativity: An Einstein centenary survey 9 65 19 Asymptotic Safety, Fixed points of quantum gravity On fixed points of quantum gravity Non-trivial ultraviolet fixed point in quantum gravity Nonperturbative Evolution Equation for Quantum Gravity , Phys. Rev. D (2007) R171 [gr-qc/0610018]. , gr-qc/0006008. 24 , 336 (2006) [hep-th/0606135]. (2008) 143 [0705.1769 [hep-th]]; (2008) 124045 [0712.0445 [hep-th]]. Wilsonian Renormalization Group Equation 23 081503 [hep-th/0207033]; 044018 [hep-th/0304222]. [hep-th/0410191]. [hep-th/9907027]; gravity truncation higher-derivative truncation Phys. Rev. D Quant. Grav. 497 (2006) [hep-th/0602203]. 861 [hep-th/9605030]. of Space, Time and Matter’ ed. D. Oriti, Cambridge Universit Rev. Relativity Grav. Cambridge University Press (1979), p. 790. [9] M. Reuter and F. Saueressig, [7] O. Lauscher and M. Reuter, [8] O. Lauscher and M. Reuter, [6] W. Souma, [5] M. Reuter, [4] R. Percacci, [1] S. Weinberg, in [3] M. Niedermaier and M. Reuter, [2] M. Niedermaier, [19] P. Forgacs and M. Niedermaier, [18] P. F. Machado and F. Saueressig, [11] D. F. Litim, [12] A. Bonanno and M. Reuter, [10] R. Percacci and D. Perini, [13] P. Fischer and D. F. Litim, [16] D. Litim, unpublished notes. [17] A. Codello, R. Percacci and C. Rahmede, [14] P. Fischer and D. F. Litim, [15] D. F. Litim, Fixed Points of Quantum Gravity and the Renormalisation Gro References PoS(QG-Ph)024 , , 79 133 , , Phys. , Living on (1994) JHEP (2002) (2006) (2006) eld 72 97 73 0212 , Nucl. Phys. B dimensional (2003) 085015 , Phys. Rev. Lett. , Phys. Lett. B , Nucl. Phys. B y safe 68 , JHEP (1992) 507; (2006) 041501 ty tra dimensions 45 73 ffective field theory (2000) 124008 The Exact Renormalization , Phys. Rev. D , Phys. Rev. Lett. 61 , Phys. Rev. Lett. , in (2005) 002 [hep-ph/0410142]. . , Phys. Rev. D 0501 Phase diagram of superconductors , Phys. Rev. D , Phys. Rev. D up erich, , JHEP , Phys. Rev. D Phase transition of N component Spectral dimension of the universe 16 Scaling exponents in quantum gravity near two-dimensions (1996) 4273 [cond-mat/9502039]. Quantum Gravity Near Two-Dimensions Emergence of a 4D world from causal quantum gravity 11 (1996) 4588 [cond-mat/9605185]. (1997) 427 [hep-th/9609077]. Quantum gravity in large dimensions Non-perturbative gravity and the spin of the lattice gravit 76 Quantum Gravity In Two + Epsilon Dimensions On gauge invariant Wilsonian flows 491 Critical fluctuations in superconductors and the magnetic fi Fixed Points of Higher Derivative Gravity Two-loop prediction for scaling exponents in (2+epsilon)- Dimensionally reduced gravity theories are asymptoticall (1993) 280 [hep-th/9206081]. Phases of 4-d simplicial quantum gravity , Phys. Rev. Lett. On the renormalization of truncated quantum Einstein gravi (2004) 124007 [hep-th/0407039]. , Int. J. Mod. Phys. A (1996) 5734 [hep-ph/9503334]; , Nucl. Phys. B Leading quantum correction to the Newtonian potential On the gravitational scaling dimensions Ultraviolet fixed points in gauge and SUSY field theories in ex Quantum gravity in everyday life: General relativity as an e 393 70 Renormalizable extra-dimensional models 53 , 131301 (2004) [hep-th/0404156]; Further evidence for a gravitational fixed point (2004) 5 [gr-qc/0311082]. 93 7 Renormalizability of gauge theories in extra dimensions , Eds. Krasnitz et al, World Sci (1999) 168 [hep-th/9901063] (2003) 020 [hep-th/0209100]. (2003) 131 [hep-th/0304117]. (2005) 171301 [hep-th/0505113]. Group 0303 [hep-th/0305208]. Rev. Lett. 95 penetration depth Phys. Rev. B B. Bergerhoff, D. F. Litim, S.superconductors Lola and C. Wetterich, 2996 [gr-qc/9310024]. Rev. Rel. Phys. Rev. D H. W. Hamber and R. M. Williams, 044031 [hep-th/0512003]. 221301 [hep-th/0607128]. [hep-th/0511177]. 673 [hep-th/9912246]; (1978) 417 066 [hep-th/0207143], (1978) 213. Nucl. Phys. B quantum gravity [37] D. F. Litim and J. M. Pawlowski, [35] H. Gies, [36] T. R. Morris, [30] J. F. Donoghue, [33] I. F. Herbut and Z. Tesanovic, [32] B. Bergerhoff, F. Freire, D. F. Litim, S. Lola and C. Wett [34] D. I. Kazakov, [31] C. P. Burgess, [28] H. W. Hamber and R. M. Williams, [29] J. Ambjorn, J. Jurkiewicz and R. Loll, [26] R. Percacci, [27] H. W. Hamber, [21] R. Gastmans, R. Kallosh and C. Truffin, Fixed Points of Quantum Gravity and the Renormalisation Gro [20] M. Niedermaier, [22] S. M. Christensen and M. J. Duff, [23] H. Kawai, Y. Kitazawa and M. Ninomiya, [24] T. Aida and Y. Kitazawa, [25] A. Codello and R. Percacci, PoS(QG-Ph)024 56 , , Phys. (2002) s ges , Phys. (2002) , Phys. (2001) 059 631 view (2002) 635 546 52 0111 (2007) 2831 , Phys. Rev. D , 90 (1993). 322 301 (2000) 92 , JHEP , hep-ph/0611146. , Commun. Math. Phys. anck scale in arbitrary , Nucl. Phys. B B486 , Phys. Lett. B ows theories (2001) 105007 (2005) 005 [hep-th/0503096]. , Centr. Eur. Sci. J. Phys. 1 (2002) , Acta Phys. Slov. 64 , Annals Phys. , Phys. Lett. B 0507 Infrared behaviour and fixed points in , , Phys. Lett. up (2001) 2081 [hep-th/0104221]. , JHEP 16 , Phys. Rev. D Renormalisation group flows for gauge theories in , 256 (2000) [hep-th/0009110]. 17 (2002) 223 [hep-ph/0005122]. Gauge invariance and background field formalism in the 495 363 (2004) 152002 [hep-th/0312324]; Non-perturbative renormalisation flow in quantum field 93 Perturbation theory and renormalisation group equations Flow equations for Yang-Mills theories in general axial gau Wilsonian flows and background fields Completeness and consistency of renormalisation group flow , Int. J. Mod. Phys. A Exact renormalization group equations: An introductory re , Phys. Rept. Gluon condensation in nonperturbative flow equations , Phys. Lett. B (2002) 049 [hep-th/0203005]. Derivative expansion and renormalisation group flows , gr-qc/0609060. 0209 , Phys. Rev. Lett. Mind the gap (1998) 181 [hep-th/9802064]; Aspects of the functional renormalisation group Normalization conventions for Newton’s constant and the Pl Small Distance Behavior In Field Theory And Power Counting Exact evolution equation for the effective potential 435 , JHEP Universality and the renormalisation group Convergence and stability of the renormalisation group Critical exponents from optimised renormalisation group fl Optimised renormalisation group flows Optimisation of the exact renormalisation group Lectures on the functional renormalization group method (2002) 081701 [hep-th/0111191]. (2002) 025030 [hep-th/0202188]. (2001) 91 [hep-th/0002034]. Introduction to the functional RG and applications to gauge 65 66 348 (1970) 227. [hep-th/0208117]. Rept. theory and statistical physics [hep-th/0103195]. 279 [hep-th/0208216]. spacetime dimension Rev. D [hep-th/0005245]; Rev. D 128 [hep-th/0203006]; (1997) 7893 [hep-th/9708051]. 1 [hep-th/0110026]. [hep-th/0512261]. axial gauges [hep-th/0111159]. Phys. Lett. B Landau gauge QCD 18 exact renormalisation group [45] D. F. Litim, Fixed Points of Quantum Gravity and the Renormalisation Gro [38] C. Bagnuls and C. Bervillier, [39] J. Berges, N. Tetradis and C. Wetterich, [44] D. F. Litim, [50] D. F. Litim, [46] C. Wetterich, [40] J. Polonyi, [57] S. P. Robinson, [48] D. F. Litim and J. M. Pawlowski, [47] D. F. Litim and J. M. Pawlowski, [49] D. F. Litim, [42] J. M. Pawlowski, [54] D. F. Litim and J. M. Pawlowski, [41] H. Gies, [43] D. F. Litim, [51] J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal [55] F. Freire, D. F. Litim and J. M. Pawlowski, [56] D. F. Litim and J. M. Pawlowski, [53] M. Reuter and C. Wetterich, [52] K. Symanzik, PoS(QG-Ph)024 100 0510 in a , Phys. 0708 , Phys. Rev. , Phys. Rev. D y , JHEP , Phys. Lett. B y , JCAP (2007) 009 , 4765 (1999), 0712 82 , Phys. Rev. Lett. tra dimensions , JHEP , 0710.3096 [hep-ph]. New dimensions at a millimeter to a Dvali, , Phys. Rev. Lett. (2007) 3995 [gr-qc/0607030]. up 24 Modified dispersion relations from the The hierarchy problem and new dimensions at a , 257 (1998) [hep-ph/9804398]. 436 18 Confronting the IR Fixed Point Cosmology with High Quantum gravity and extra dimensions at high-energy (2005) 012 [hep-th/0507167]. Graviton collider effects in one and more large extra (2004) 001 [arXiv:astro-ph/0303150]. On Kaluza-Klein states from large extra dimensions Virtual Gravitons at the LHC Renormalization group improved black hole spacetimes 0509 , Class. Quant. Grav. 0401 , Phys. Lett. B , 455 (2005) [hep-ph/0408320]. From big bang to asymptotic de Sitter: Complete cosmologies , 263 (1998) [hep-ph/9803315]. Fractal spacetime structure in asymptotically safe gravit , 3 (1999) [hep-ph/9811291]. Entropy signature of the running cosmological constant Quantum gravity effects near the null black hole singularit , JCAP 706 Signatures of gravitational fixed points at the LHC 429 , JCAP 544 Collider Signals of Gravitational Fixed Points Indirect collider signals for extra dimensions , Nucl. Phys. B (2000) 043008 [hep-th/0002196]. , Phys. Lett. B Renormalization group and black hole production in large ex , Nucl. Phys. B 62 (2008) 334 [0707.4644 [hep-ph]]. , 105006 (1999), [hep-ph/9811350]. (1999) 084011 [gr-qc/9811026]; 59 (2005) 050 [hep-th/0508202]. Rev. D quantum gravity framework 60 (2007) 024 [0706.0174 [hep-th]]. Redshift Supernova Data renormalization group of gravity, (2008) 131301 [0707.3983 [hep-ph]]; [0707.3182 [hep-ph]]. [hep-ph/9811356]. Fermi and superstrings at a TeV D dimensions colliders 663/4 millimeter Fixed Points of Quantum Gravity and the Renormalisation Gro [58] O. Lauscher and M. Reuter, [60] M. Reuter and F. Saueressig, [59] A. Bonanno and M. Reuter, [61] A. Bonanno and M. Reuter, [62] E. Bentivegna, A. Bonanno and M. Reuter, [63] F. Girelli, S. Liberati, R. Percacci and C. Rahmede, [64] D. F. Litim and T. Plehn, [65] J. Hewett and T. Rizzo, [66] B. Koch, [70] T. Han, J. D. Lykken and R. J. Zhang, [71] J. L. Hewett, [72] G. F. Giudice, T. Plehn and A. Strumia, [69] G. F. Giudice, R. Rattazzi and J. D. Wells, [68] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. [67] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali,