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arXiv:0810.3675v1 [hep-th] 21 Oct 2008
Fixed Points of Quantum Gravity and the
Renormalisation Group
Daniel F. Litim
Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, U.K.
E-mail: d.litim@sussex.ac.uk
We review the asymptotic safety scenario for quantum gravity and the role and implications of
an underlying ultraviolet fixed point. We discuss renormalisation group techniques employed in
the fixed point search, analyse the main picture at the example of the Einstein-Hilbert theory,
and provide an overview of the key results in four and higher dimensions. We also compare
findings with recent lattice simulations and evaluate phenomenological implications for collider
experiments.
From Quantum to Emergent Gravity: Theory and Phenomenology
June 11-15 2007
Trieste, Italy
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/
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Fixed Points of Quantum Gravity
Daniel Litim
1. Introduction
It is commonly believed that an understanding of the dynamics of gravity and the structure of
space-time at shortest distances requires an explicit quantum theory for gravity. The well-known
fact that the perturbative quantisation program for gravity in four dimensions faces problems has
raised the suspicion that a consistent formulation of the theory may require a radical deviation from
the concepts of local quantum field theory, e.g. string theory. It remains an interesting and open
challenge to prove, or falsify, that a consistent quantum theory of gravity cannot be accommodated
for within the otherwise very successful framework of local quantum field theories.
Some time ago Steven Weinberg added a new perspective to this problem by pointing out that
a quantum theory of gravity in terms of the metric field may very well exist, and be renormalisable
on a non-perturbative level, despite it’s notorious perturbative non-renormalisability [1]. This sce-
nario, since then known as “asymptotic safety”, necessitates an interacting ultraviolet fixed point
for gravity under the renormalisation group (RG) [1, 2, 3, 4, 5]. If so, the high energy behaviour
of gravity is governed by near-conformal scaling in the vicinity of the fixed point in a way which
circumnavigates the virulent ultraviolet (UV) divergences encountered within standard perturba-
tion theory. Indications in favour of an ultraviolet fixed point are based on renormalisation group
studies in four and higher dimensions [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], dimensional re-
duction techniques [19, 20], renormalisation group studies in lower dimensions [1, 21, 22, 23, 24],
four-dimensional perturbation theory in higher derivative gravity [25], large-N expansions in the
matter fields [26], and lattice simulations [27, 28, 29].
In this contribution, we review the key elements of the asymptotic safety scenario (Sec. 2) and
introduce renormalisation group techniques (Sec. 3) which are at the root of fixed point searches
in quantum gravity. The fixed point structure in four (Sec. 4) and higher dimensions (Sec. 5), and
the phase diagram of gravity (Sec. 6) are discussed and evaluated in the light of the underlying ap-
proximations. Results are compared with recent lattice simulations (Sec. 7), and phenomenological
implications are indicated (Sec. 8). We close with some conclusions (Sec. 9).
2. Asymptotic Safety
We summarise the basic set of ideas and assumptions of asymptotic safety as first laid out in
[1] (see [2, 3, 4, 5] for reviews). The aim of the asymptotic safety scenario for gravity is to provide
for a path-integral based framework in which the metric field is the carrier of the fundamental
degrees of freedom, both in the classical and in the quantum regimes of the theory. This is similar
in spirit to effective field theory approaches to quantum gravity [30]. There, a systematic study of
quantum effects is possible without an explicit knowledge of the ultraviolet completion as long as
the relevant energy scales are much lower than the ultraviolet cutoff Λ of the effective theory, with
Λ of the order of the Planck scale (see [31] for a recent review).
The asymptotic safety scenario goes one step further and assumes that the cutoff Λ can in fact
be removed, Λ → ∞, and that the high-energy behaviour of gravity, in this limit, is characterised
by an interacting fixed point. It is expected that the relevant field configurations dominating the
gravitational path integral at high energies are predominantly “anti-screening” to allow for this
limit to become feasible. If so, it is conceivable that a non-trivial high-energy fixed point of gravity
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Fixed Points of Quantum Gravity
Daniel Litim
may exist and should be visible within e.g. renormalisation group or lattice implementations of the
theory, analogous to the well-known perturbative high-energy fixed point of QCD. Then the high-
energy behaviour of the relevant gravitational couplings is “asymptotically safe” and connected
with the low-energy behaviour by finite renormalisation group flows. The existence of a fixed
point together with finite renormalisation group trajectories provides for a definition of the theory
at arbitrary energy scales.
The fixed point implies that the high-energy behaviour of gravity is characterised by univer-
sal scaling laws, dictated by the residual high-energy interactions. No a priori assumptions are
made about which invariants are the relevant operators at the fixed point. In fact, although the low-
energy physics is dominated by the Einstein-Hilbert action, it is expected that (a finite number of)
further invariants will become relevant, in the renormalisation group sense, at the ultraviolet fixed
point.1Then, in order to connect the ultraviolet with the infrared physics along some renormali-
sation group trajectory, a finite number of initial parameters have to be fixed, ideally taken from
experiment. In this light, classical general relativity would emerge as a “low-energy phenomenon”
of a fundamental quantum field theory in the metric field.
We illustrate this scenario with a discussion of the renormalisation group equation for the
gravitational coupling G, following [2] (see also [3, 4]). Its canonical dimension is [G] = 2−d in d
dimensions and hence negative for d > 2. It is commonly believed that a negative mass dimension
for the relevant coupling is responsible for the perturbative non-renormalisability of the theory.
We introduce the renormalised coupling as G(µ) = Z−1
g(µ) = µd−2G(µ); the momentum scale µ denotes the renormalisation scale. The graviton wave
function renormalisation factor ZG(µ) is normalised as ZG(µ0) = 1 at µ = µ0with G(µ0) given
by Newton’s coupling constant GN= 6.67428·10−11 m3
related to ZG(µ) is given by η = −µd
G(µ)G, and the dimensionless coupling as
kgs2. The graviton anomalous dimension η
dµlnZG. Then the Callan-Symanzik equation for g(µ) reads
βg≡ µdg(µ)
dµ
= (d−2+η)g(µ).
(2.1)
Here we have assumed a fundamental action for gravity which is local in the metric field. In
general, the graviton anomalous dimension η(g,···) is a function of all couplings of the theory
including matter fields. The RG equation (2.1) displays two qualitatively different types of fixed
points. The non-interacting (gaussian) fixed point corresponds to g∗= 0 which also entails η = 0.
In its vicinity with g(µ0) ≪ 1, we have canonical scaling since βg= (d−2)g, and
G(µ) = G(µ0)
(2.2)
for all µ < µ0. Consequently, the gaussian regime corresponds to the domain of classical general
relativity. In turn, (2.1) can display an interacting fixed point g∗?= 0 in d > 2 if the anomalous
dimension takes the value η(g∗,···) = 2−d; the dots denoting further gravitational and matter
couplings. Hence, the anomalous dimension precisely counter-balances the canonical dimension
of Newton’s coupling G. This structure is at the root for the non-perturbative renormalisability of
1For infrared fixed points, universality considerations often simplify the task of identifying the set of relevant,
marginal and irrelevant operators. This is not applicable for interacting ultraviolet fixed points.
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Daniel Litim
quantum gravity within a fixed point scenario.2Consequently, at an interacting fixed point where
g∗?= 0, the anomalous dimension implies the scaling
G(µ) =
g∗
µd−2
(2.3)
for the dimensionful gravitational coupling. In the case of an ultraviolet fixed point g∗?= 0 for large
µ, the dimensionful coupling G becomes arbitrarily small in its vicinity. This is in marked contrast
to (2.2). Hence, (2.3) indicates that gravity weakens at the onset of fixed point scaling. Neverthe-
less, at the fixed point the theory remains non-trivially coupled because of g∗?= 0. The weakness of
the coupling in (2.3) is a dimensional effect, and should be contrasted with e.g. asymptotic freedom
of QCD in four dimensions where the dimensionless non-abelian gauge coupling becomes weak
because of a non-interacting ultraviolet fixed point. In turn, if (2.3) corresponds to a non-trivial
infrared fixed point for µ → 0, the dimensionful coupling G(µ) grows large. A strong coupling
behaviour of this type would imply interesting long distance modifications of gravity.
As a final comment, we point out that asymptotically safe gravity is expected to become,
in an essential way, two-dimensional at high energies. Heuristically, this can be seen from the
dressed graviton propagator whose scalar part, neglecting the tensorial structure, scales as G(p2)∼
p−2(1−η/2)in momentum space. Here we have evaluated the anomalous dimension at µ2≈ p2.
Then, for small η, we have the standard perturbative behaviour ∼ p−2. In turn, for large anomalous
dimension η → 2−d in the vicinity of a fixed point the propagator is additionally suppressed
∼ (p2)−d/2possibly modulo logarithmic corrections. After Fourier transform to position space,
this corresponds to a logarithmic behaviour for the propagator G(x,y) ∼ln(|x−y|µ), characteristic
for bosonic fields in two-dimensional systems.
3. Renormalisation Group
Whether or not a non-trivial fixed point is realised in quantum gravity can be assessed once ex-
plicit renormalisation group equations for the scale-dependent gravitational couplings are available.
To that end, we recall the set-up of Wilson’s (functional) renormalisation group (see [37, 38, 39, 40,
41, 42, 43] for reviews), which is used below for the case of quantum gravity. Wilsonian flows are
based on the notion of a cutoff effective action Γk, where the propagation of fields φ with momenta
smaller than k is suppressed. A Wilsonian cutoff is realised by adding ∆Sk=1
within the Schwinger functional
2
?ϕ(−q)Rk(q)ϕ(q)
ln Zk[J] = ln
?
[Dϕ]ren.exp
?
−S[ϕ]−∆Sk[ϕ]+
?
J·ϕ
?
(3.1)
and the requirement that Rkobeys (i) Rk(q) → 0 for k2/q2→ 0, (ii) Rk(q) > 0 for q2/k2→ 0,
and (iii) Rk(q) → ∞ for k → Λ (for examples and plots of Rk, see [44]). Note that the Wilsonian
2Integer values for anomalous dimensions are well-known from other gauge theories at criticality and away from
their canonical dimension. In the d-dimensional U(1)+Higgs theory, the abelian charge e2has mass dimension [e2] =
4−d, with βe2 = (d −4+η)e2. In three dimensions, a non-perturbative infrared fixed point at e2
[32]. The fixed point belongs to the universality class of conventional superconductors with the charged scalar field
describing theCooper pair. Theinteger valueη∗=1impliesthatthemagnetic fieldpenetration depthand theCooper pair
correlation length scale with the same universal exponent at the phase transition [32, 33]. In Yang-Mills theories above
four dimensions, ultraviolet fixed points with η = 4−d and implications thereof have been discussed in [34, 35, 36].
∗?= 0 leads to η∗= 1
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Daniel Litim
momentum scale k takes the role of the renormalisation group scale µ introduced in the previ-
ous section. Under infinitesimal changes k → k−∆k, the Schwinger functional obeys ∂tlnZk=
−?∂t∆Sk?J; t = lnk. We also introduce its Legendre transform, the scale-dependent effective ac-
tion Γk[φ] = supJ(?J·φ −lnZk[J])−1
equation introduced by Wetterich [45]
2
?φRkφ, φ = ?ϕ?J. It obeys an exact functional differential
∂tΓk=1
2Tr
?
Γ(2)
k+Rk
?−1∂tRk,
(3.2)
which relates the change in Γkwith a one-loop type integral over the full field-dependent cutoff
propagator. Here, the trace Tr denotes an integration over all momenta and summation over all
fields, and Γ(2)
k[φ](p,q) ≡ δ2Γk/δφ(p)δφ(q). A number of comments are in order:
• Finiteness and interpolation property. By construction, the flow equation (3.2) is well-
defined and finite, and interpolates between an initial condition ΓΛfor k → Λ and the full
effective action Γ ≡ Γk=0. This is illustrated in Fig. 1. The endpoint is independent of the
regularisation, whereas the trajectories k → Γkdepend on it.
• Locality. The integrand of (3.2) is peaked for field configurations with momentum squared
q2≈ k2, and suppressed for large momenta [due to condition (i) on Rk] and for small mo-
menta [due to condition (ii)]. Therefore, the flow equation is essentially local in momentum
and field space [44, 47].
• Approximations. Systematic approximations for Γkand ∂tΓkare required to integrate (3.2).
These include (a) perturbation theory, (b) expansions in powers of the fields (vertex func-
tions), (c) expansion in powers of derivative operators (derivative expansion), and (d) com-
binations thereof. The iterative structure of perturbation theory is fully reproduced to all
orders, independently of Rk[48, 49]. The expansions (b) - (d) are genuinely non-perturbative
and lead, via (3.2), to coupled flow equations for the coefficient functions. Convergence is
then checked by extending the approximation to higher order.
• Stability. The stability and convergence of approximations is, additionally, controlled by
Rk[44, 46]. Here, powerful optimisation techniques are available to maximise the physics
content and the reliability through well-adapted choices of Rk[44, 46, 47, 51, 42]. These
ideas have been explicitly tested in e.g. scalar [50] and gauge theories [51].
• Symmetries. Global or local (gauge/diffeomorphism) symmetries of the underlying theory
can be expressed as Ward-Takahashi identities for n-point functions of Γ. Ward-Takahashi
identities are maintained for all k if the insertion ∆Skis compatible with the symmetry. In
general, this is not the case for non-linear symmetries such as in non-Abelian gauge theo-
ries or gravity. Then the requirements of gauge symmtry for Γ are preserved by either (a)
imposing modified Ward identities which ensure that standard Ward identities are obeyed in
the the physical limit when k → 0, or by (b) introducing background fields into the regulator
Rkand taking advantage of the background field method, or by (c) using gauge-covariant
variables rather than the gauge fields or the metric field [52]. For a discussion of benefits
and shortcomings of these options see [38, 42]. For gravity, most implementations presently
employ option (b) together with optimisation techniques to control the symmetry [14, 15].
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Daniel Litim
a)b)
S
Γ∗
k∂kΓk
k∂kΓk
Γ0≈ ΓΓ0≈ ΓEH
IR
UV
p
action
quantum
effective action
integrating−out
classical
IR
UV
p
action
integrating−out
fixed point
general relativity
classical
Figure 1: Wilsonian flows for scale-dependent effective actions Γkin the space of all action functionals
(schematically); arrows point towards smaller momentum scales and lower energies k → 0. a) Flow con-
necting a fundamental classical action S at high energies in the ultraviolet with the full quantum effective
action Γ at low energies in the infrared (“top-down”). b) Flow connecting the Einstein-Hilbert action at low
energies with a fundamental fixed point action Γ∗at high energies (“bottom-up”).
• Integral representation. The physical theory described by Γ can be defined without explicit
reference to anunderlying path integral representation, using only the (finite) initial condition
ΓΛ, and the (finite) flow equation (3.2)
Γ = ΓΛ+
?0
Λ
dk
k
1
2Tr
?
Γ(2)
k+Rk
?−1∂tRk.
(3.3)
This provides an implicit regularisation of the path integral underlying (3.1). It should be
compared with the standard representation for Γ via a functional integro-differential equation
e−Γ=
?
[Dϕ]ren.exp
?
−S[φ +ϕ]+
?δΓ[φ]
δφ
·ϕ
?
(3.4)
which is at the basis of e.g. the hierarchy of Dyson-Schwinger equations.
• Renormalisability. In renormalisable theories, the cutoff Λ in (3.3) can be removed, Λ→∞,
and ΓΛ→Γ∗remains well-defined for arbitrarily short distances. In perturbatively renormal-
isable theories, Γ∗is given by the classical action S, such as in QCD. In this case, illustrated
in Fig. 1a), the high energy behaviour of the theory is simple, given mainly by the classical
action, and the challenge consists in deriving the physics of the strongly coupled low energy
limit. In perturbatively non-renormalisable theories such as quantum gravity, proving the ex-
istence (or non-existence) of a short distance limit Γ∗is more difficult. For gravity, illustrated
in Fig. 1b), experiments indicate that the low energy theory is simple, mainly given by the
Einstein Hilbert theory. The challenge consists in identifying a possible high energy fixed
point action Γ∗, which upon integration matches with the known physics at low energies.
In principle, any Γ∗with the above properties qualifies as fundamental action for quantum
gravity. In non-renormalisable theories the cutoff Λ cannot be removed. Still, the flow equa-
tion allows to access the physics at all scales k < Λ analogous to standard reasoning within
effective field theory [31].
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Daniel Litim
• Link with Callan-Symanzik equation. The well-known Callan-Symanzik equation de-
scribes a flow kd
Rk(q2) = k2, which does not fulfill condition (i). Consequently, the corresponding flow is no
longer local in momentum space, and requires an additional UV regularisation. This high-
lights a crucial difference between the Callan-Symanzik equation and functional flows (3.2).
In this light, the flow equation (3.2) could be interpreted as a functional Callan-Symanzik
equation with momentum-dependent mass term insertion [53].
dkdriven by a mass insertion ∼ k2φ2. In (3.2), this corresponds to the choice
Now we are in a position to implement these ideas for quantum gravity [6]. A Wilsonian ef-
fective action for gravity Γkshould contain the Ricci scalar R(gµν) with a running gravitational
coupling Gk, a running cosmological constant Λk(with canonical mass dimension [Λk]= 2), possi-
bly higher order interactions in the metric field such as powers, derivatives, or functions of e.g. the
Ricci scalar, the Ricci tensor, the Riemann tensor, and, possibly, non-local operators in the metric
field. The effective action should also contain a standard gauge-fixing term Sgf, a ghost term Sgh
and matter interactions Smatter. Altogether,
Γk=
?
ddx?detgµν
?
1
16πGk(−R+2Λk)+···+Sgf+Sgh+Smatter
?
,
(3.5)
and explicit flow equations for the coefficient functions such as Gk, Λkor vertex functions, are ob-
tained by appropriate projections after inserting (3.5) into (3.2). All couplings in (3.5) become run-
ning couplings as functions of the momentum scale k. For k much smaller than the d-dimensional
Planck scale M∗, the gravitational sector is well approximated by the Einstein-Hilbert action with
Gk≈ Gk=0, and similarily for the gravity-matter couplings. At k ≈ M∗and above, the RG running
of gravitational couplings becomes important. This is the topic of the following sections.
A few technical comments are in order: To ensure gauge symmetry within this set-up, we
take advantage of the background field formalism and add a non-propagating background field ¯ gµν
[6, 38, 54, 55, 56, 57, 58]. This way, the extended effective action Γk[gµν, ¯ gµν] becomes gauge-
invariant under the combined symmetry transformations of the physical and the background field.
A second benefit of this is that the background field can be used to construct a covariant Laplacean
−¯D2, or similar, to define a mode cutoff at momentum scale k2= −¯D2. This implies that the mode
cutoff Rkwill depend on the background fields. The background field is then eliminated from the
final equations by identifying it with the physical mean field. This procedure, which dynamically
readjusts the background field, implements the requirements of “background independence” for
quantum gravity. For a detailed evaluation of Wilsonian background field flows, see [57]. Finally,
we note that the operator traces Tr in (3.2) are evaluated using heat kernel techniques. Here, well-
adapted choices for Rk[44, 46] lead to substantial algebraic simplifications, and open a door for
systematic fixed point searches, which we discuss next.
4. Fixed Points
In this section, we discuss the main picture in a simple approximation which captures the
salient features of an asymptotic safety scenario for gravity, and give an overview of extensions.
We consider the Einstein-Hilbert theory with a cosmological constant term and employ a momen-
tum cutoff Rkwith the tensorial structure of [8] and variants thereof, an optimised scalar cutoff
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Daniel Litim
θ′
θ′′
ref.
[8]
[12]
[14]
a)
b)
c)
1.1−2.3
1.4−2.0
1.5−1.7
2.5−7.0
2.4−4.3
3.0−3.2
Table 1: The variation of 4d scaling exponents θ1,2= θ′±iθ′′in the Einstein-Hilbert theory with the
gauge fixing parameter α and the cutoff function Rk. Results indicate the range covered under a) partial
variation of both α and Rk, b) full α-variation with optimised Rk, and c) full Rk-variation and optimisation
in Feynman gauge (α = 1). In all cases the fixed point is stable. The variation with Rk, amended by stablity
considerations [44, 46], is weaker than the α-variation.
Rk(q2) ∼ (k2−q2)θ(k2−q2) [44, 46], and a harmonic background field gauge with parameter α
in a specific limit introduced in [12]. The ghost wave function renormalisation is set to ZC,k= 1,
and the effective action is given by (3.5) with Smatter= 0. In the domain of classical scaling Gkand
Λkare approximately constant, and (3.5) reduces to the conventional Einstein-Hilbert action in d
euclidean dimensions. The dimensionless renormalised gravitational and cosmological constants
are
g = kd−2Gk≡ kd−2Z−1
where it is understood that g and λ depend on k. Then the coupled system of β-functions is
G(k)¯G ,
λ = k−2Λk
(4.1)
∂tλ ≡ βλ(λ,g) = −2λ +g
2d(d+2)(d−5)−d(d+2)g(d−1)g+
2(d −2)(d+2)g2
2(d −2)g−(1−2λ)2.
2+2)(4π)d/2−1to remove phase space factors. This does
1
d−2(1−4d−1
1
d−2(1−2λ)2
dλ)
2g−
(4.2)
∂tg ≡ βg(λ,g) = (d−2)g+
(4.3)
We have rescaled g → g/cdwith cd= Γ(d
not alter the fixed point structure. The scaling g → g/(384π2) reproduces the 4d classical force
law in the non-relativistic limit [59]. For the anomalous dimension, we find
η(λ,g;d) =
(d+2)g
g−gbound(λ),
gbound(λ;d) =(1−2λ)2
2(d −2).
(4.4)
Theanomalous dimension vanishes for vanishing gravitational coupling, and for d =±2. Ata non-
trivial fixed point the vanishing of βgimplies η∗= 2−d, and reflects the fact that the gravitational
coupling is dimensionless in two dimensions. At g = gbound, the anomalous dimension η diverges.
The full flow (3.2) is finite (no poles) and well-defined for all k, as are the full β-functions derived
from it. Therefore the curve g = gbound(λ) limits the domain of validity of the approximation.
We first consider the case λ = 0 and find two fixed points, the gaussian one at g∗= 0 and
a non-gaussian one at g∗= 1/(4d) < gbound(0), which are connected under the renormalisation
group. The universal eigenvalue ∂βg/∂g|∗= −θ at the fixed point are θ = 2−d at the gaussian,
and
θ = 2dd−2
d+2
(4.5)
at the non-gaussian fixed point.
8
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Daniel Litim
i
2
2
2
2
3
4
5
6
θ′
θ′′
θ3
θ4
−
−
−
−
ref.
[9]
[16]
[16]
[17]
[17]
[17]
[17]
[17]
a)
b)
c)
d)
e)
f)
g)
h)
2.1−3.4
1.4
1.7
1.4
2.7
2.9
2.5
2.4
3.1−4.3
2.8
3.1
2.3
2.3
2.5
2.7
2.4
8.4−28.8
25.6
3.5
26.9
2.1
1.6
1.8
1.5
−4.2
−3.9
−4.4
−4.1
Table 2: The variation of 4d scaling exponents with the order i of the expansion, including the invariants
? ?detgµνand? ?detgµνRifrom i=2···6, using a) Feynmangauge underpartial variations of Rk; b) and
c) Feynman gauge and optimised Rk; d)−h) Landau-deWitt backgroundfield gauge with optimised Rk.
Next we allow for a non-vanishing dynamical cosmological constant term Λk?= 0 in (3.5). The
coupled system exhibits the gaussian fixed point (λ∗,g∗) = (0,0) with eigenvalues −2 and d −2.
Non-trivial fixed points of (4.2) and (4.3) are found as follows: For non-vanishing λ ?= 0, we find a
non-trivially vanishing βgfor g = g0(λ), with g0(λ) =
for all d > 2. Evaluating (4.2) for g = g0(λ), we find βλ(λ,g0(λ)) =1
2dλ +d
fixed point λ∗=1
fixed points with either λ∗>1
fixed point which is connected under the renormalisation group with the correct infrared behaviour.
This can be seen as follows: At λ =1
however, η < 0. Furthermore, g cannot change sign under the renormalisation group flow (4.3).
Consequently, η cannot change sign either. Hence, to connect a fixed point at λ >1
gaussian fixed point at λ = 0, η would have to change sign at least twice, which is impossible.
Therefore, we have a unique physically relevant solution given by
1
4d(1−2λ)2. Note that g0(λ) < gbound(λ)
4(d −4)(d +1)(1−2λ)2−
2. The first term vanishes in d = 4 dimensions. Consequently, we find a unique ultraviolet
4and g∗=
2or1
1
64. In d > 4, the vanishing of βλleads to two branches of real
2> λ∗> 0. Only the second branch corresponds to an ultraviolet
2, we find η = d +2 > 0. On a non-gaussian fixed point,
2with the
λ∗=d2−d−4−?2d(d2−d−4)
2(d−4)(d+1)
An interesting property of this system is that the scaling exponents θ1and θ2– the eigenvalues
of the stability matrix ∂βi/∂gj(g1≡ g,g2≡ λ) at the fixed point – are a complex conjugate pair,
θ1,2= θ′±iθ′′with θ′=5
stability matrix, albeit real, is not symmetric. Complex eigenvalues reflect that the interactions at
the fixed point have modified the scaling behaviour of the underlying operators
? ?detgµν. This pattern changes for lower and higher dimensions, where eigenvalues are real [2].
Atthis point it is important to check whether the fixed point structure and the scaling exponents
depend on technical parameters such as the gauge fixing procedure or the momentum cutoff func-
tion Rk, see Tab. 1 and 2. For the Einstein-Hilbert theory in 4d, results are summarised in Tab. 1.
The α-dependence of the β-functions is fairly non-trivial, e.g. [6, 8, 12]. It is therefore noteworthy
that scaling exponents only depend mildly on variations thereof. Furthermore, the Rk-dependence
,
g∗=(√d2−d−4−√2d)2
2(d−4)2(d +1)2
.
(4.6)
3and θ′′=
√167
3
in four dimensions. The reason for this is that the
? ?detgµνR and
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Fixed Points of Quantum Gravity
Daniel Litim
d
θ′
θ′′
|θ|
56789 10
2.69 – 3.11
4.54 – 5.16
5.31 – 6.06
4.26 – 4.78
6.52 – 7.46
7.79 – 8.76
6.43 – 6.89
8.43 – 9.46
10.4 – 11.6
8.19 – 9.34
10.3 – 11.4
13.2 – 14.7
10.5 – 12.1
12.1 – 13.2
16.1 – 17.9
13.1 – 15.2
13.9 – 15.0
19.1 – 21.3
Table 3: The variation of scaling exponentwith dimensionality, gauge fixing parameters (using either Feyn-
man gauge, or harmonic background field gauge with 0 ≤ α ≤ 1), and the regulator Rk; data from [14, 15].
The Rk-variation, covering various classes of cutoff functions, is on the level of a few percent and smaller
than the variation with α.
is smaller than the dependence on gauge fixing parameters. We conclude that the fixed point is
fully stable and Rk-independent for all technical purposes, with the presently largest uncertainty
arising through the gauge fixing sector. In Tab. 2, we discuss the stability of the fixed point under
extensions beyond the Einstein-Hilbert approximation, including higher powers of the Ricci scalar
both in Feynman gauge [9, 16] and in Landau-DeWitt gauge [17]. Once more, the fixed point and
the scaling exponents come out very stable. Furthermore, starting from the operator? ?detgµνR3
and higher, couplings become irrelevant with negative scaling exponents [17, 18]. This is an im-
portant first indication for the set of relevant operators at the UV fixed point being finite. Finally,
we mention that the stability of the fixed point under the addition of non-interacting matter fields
has been confirmed in [11].
5. Extra Dimensions
It is interesting to discuss fixed points of quantum gravity specifically in more than four dimen-
sions. The motivation for this is that, first of all, the critical dimension of gravity – the dimension
where the gravitational coupling has vanishing canonical mass dimension – is two. For any dimen-
sion above the critical one, the canonical dimension is negative. Hence, from a renormalisation
group point of view, the four-dimensional theory is not special. Continuity in the dimension sug-
gests that an ultraviolet fixed point, if it exists in four dimensions, should persist towards higher
dimensions. More generally, one expects that the local structure of quantum fluctuations, and hence
local renormalisation group properties of a quantum theory of gravity, are qualitatively similar for
all dimensions above the critical one, modulo topological effects for specific dimensions. Secondly,
the dynamics of the metric field depends on the dimensionality of space-time. In four dimensions
and above, the metric field is fully dynamical. Hence, once more, we should expect similarities
in the ultraviolet behaviour of gravity in four and higher dimensions. Interestingly, this pattern is
realised in the results [12], see the analytical fixed point (4.6). An extended systematic search for
fixed points in higher-dimensional gravity for general cutoff Rkhas been presented in [14, 15], also
testing the stability of the result against variations of the gauge fixing parameter (see Tab. 3). The
variation with Rk, ammended by stability considerations, is smaller than the variation with α. We
conclude from the weak variation that the fixed point indeed persists in higher dimensions. Further
studies including higher derivative operators confirm this picture [16]. This structural stability also
strengthens the results in the four-dimensional case, and supports the view introduced above. A
10
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Fixed Points of Quantum Gravity
Daniel Litim
λ4d
g4d
?0.10 0.10.20.30.40.5
?0.02
?0.01
0
0.01
0.02
0.03
Figure 2: The phase diagram for the running gravitational coupling g4dand the cosmological constant λ4d
in four dimensions. The Gaussian and the ultraviolet fixed point are indicated by dots (red). The separatrix
connects the two fixed points (full green line). The full (red) line indicates the bound gbound(λ) where
1/η = 0. Arrows indicate the direction of the RG flow with decreasing k → 0.
phenomenological application of these findings in low-scale quantum gravity is discussed below
(see Sec. 8).
6. Phase Diagram
In this section, we discuss the main characteristics of the phase portrait of the Einstein-Hilbert
theory [10, 12] (see Fig. 2). Finiteness of the flow (3.2) implies that the line 1/η = 0 cannot be
crossed. Slowness of the flow implies that the line η = 0 can neither be crossed (see Sec. 4). Thus,
disconnected regions of renormalisation group trajectories are characterised by whether g is larger
or smaller gboundand by the sign of g. Since η changes sign only across the lines η = 0 or 1/η =0,
we conclude that the graviton anomalous dimension has the same sign along any trajectory. In the
physical domain which includes the ultraviolet and the infrared fixed point, the gravitational cou-
pling is positive and the anomalous dimension negative. In turn, the cosmological constant may
change sign on trajectories emmenating from the ultraviolet fixed point. Some trajectories ter-
minate at the boundary gbound(λ), linked to the present approximation. The two fixed points are
connected by a separatrix. The rotation of the separatrix about the ultraviolet fixed point reflects
the complex nature of the eigenvalues. At k ≈ MPl, the flow displays a crossover from ultraviolet
dominated running to infrared dominated running. The non-vanishing cosmological constant mod-
ifies the flow mainly in the crossover region rather than in the ultraviolet. In the infrared limit, the
separatrix leads to a vanishing cosmological constant Λk= λkk2→ 0 and is interpreted as a phase
transition boundary between cosmologies with positive or negative cosmological constant at large
distances. Trajectories in the vicinity of the separatrix lead to a positive cosmological constant at
large scales and are, therefore, candidate trajectories for realistic cosmologies [62]. This picture
11
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Fixed Points of Quantum Gravity
Daniel Litim
agrees very well with numerical results for a sharp cut-off flow [10], except for the location of
the line 1/η = 0 which is non-universal. Similar phase diagrams are found in higher dimensions
[14, 15].
7. Lattice
Lattice implementations for gravity in four dimensions have been put forward based on Regge
calculus techniques [27, 28] and causal dynamical triangulations [29]. In the Regge calculus ap-
proach, a critical point which allows for a lattice continuum limit has been given in [27] using the
Einstein Hilbert action with fixed cosmological constant. A scaling exponent has been measured
in the four-dimensional simulation based on varying Newton’s coupling to the critical point, with
∂gβg|∗= −1
[12] as discussed in Sec. 4. In the large-dimensional limit, geometrical considerations on the lattice
lead to the estimate ν =
RG fixed point result ν =
Within the causal dynamical triangulation approach, global aspects of quantum space-times
have been assessed in [29]. There, the effective dimensionality of space-time has been measured as
a function of the length scale by evaluating the return probability of random walks on the triangu-
lated manifolds. The key result is that the measured effective dimensionality displays a cross-over
from d ≈ 4 at large scales to d ≈ 2 at small scales of the order of the Planck scale. This behaviour
compares nicely with the cross-over of the graviton anomalous dimension η under the renormal-
isation group (see Sec. 2), and with renormalisation group studies of the spectral dimension (see
[3, 4, 60]). These findings corroborate the claim that asymptotically safe quantum gravity behaves,
in an essential way, two-dimensional at short distances.
ν. The result reads ν ≈1
3, and should be contrasted with the RG result ν = 1/θ =3
8
1
d−1[28], a behaviour which is in qualitative agreement with the explicit
1
2din the corresponding limit, see (4.5).
8. Phenomenology
The phenomenology of a gravitational fixed point covers the physics of black holes [61],
cosmology [62, 63, 64], modified dispersion relations [65], and the physics at particle colliders
[66, 67, 68]. In this section we concentrate on the later within low-scale quantum gravity mod-
els [69, 70]. There, gravity propagates in d = 4+n dimensional bulk whereas matter fields are
confined to a four-dimensional brane. The four-dimensional Planck scale MPl≈ 1019GeV is no
longer fundamental as soon as the n extra dimensions are compact with radius ∼ L. Rather, the
d = 4+n-dimensional Planck mass M∗sets the fundamental scale for gravity, leading to the rela-
tion M2
lower than MPlprovided 1/L ≪ M∗. If M∗is of the order of the electroweak scale, this scenario
lifts the hierarchy problem of the standard model and opens the exciting possibility that particle
colliders could establish experimental evidence for the quantisation of gravity [71, 72, 73].
The renormalisation group running of the gravitational coupling in this scenario has been
studied in [14, 15, 66] and is summarised in Fig. 3. The main effects due to a fixed point at high
energies set in at momentum scales k ≈ M∗, where the gravitational coupling displays a cross-over
from perturbative scaling G(k) ≈ const. to fixed point scaling G(k) ≈ g∗k2−d. Therefore we expect
Pl∼M2
∗(M∗L)nfor the four-dimensional Planck scale. Consequently, M∗can be significantly
12
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Fixed Points of Quantum Gravity
Daniel Litim
1
1
3
5
7
9
0123456
0.2
0.4
0.6
0.8
|η|
n = 1
n = 7
Gk
G0
a) schematicallyb) numerically
lng
UV (fixed point)
IR(d = 4+n)
IR(d = 4)
lnk
lnk
lnM∗
ln1/L
Figure 3: The scale-dependence of the gravitational coupling in a scenario with large extra dimension of
size ∼ L with fundamental Planck scale M∗and M∗L ≫ 1. The fixed point behaviour in the deep ultraviolet
enforces a softening of gravitational coupling (see text). a) In the infrared (IR) regime where |η| ≪ 1,
the coupling g = Gkkd−2displays a crossover from 4-dimensional to (4+n)-dimensional classical scaling
at k ≈ 1/L. The slope dlng/dlnk ≈ d −2 measures the effective number of dimensions. At k ≈ M∗, a
classical-to-quantum crossover takes place from |η| ≪ 1 to η ≈ 2−d (schematically). b) Classical-to-
quantum crossover at the respective Planck scale for Gkand the anomalous dimensions η from numerical
integrations of the flow equation; d = 4+n dimensions with n = 1,···,7 from right to left.
that signatures of this cross-over should be visible in scattering processes at particle colliders as
long as these are sensitive to momentum transfers of the order of M∗.
We illustrate this at the example of dilepton production through virtual gravitons at the Large
Hadron Collider (LHC) [66]. To lowest order in canonical dimension, the dilepton production
amplitude is generated through an effective dimension–8 operator in the effective action, involving
four fermions and a graviton [71]. Tree–level graviton exchange is described by an amplitude
A = S·T, where T = TµνTµν−
2πn/2
Γ(n/2)
M4
∗
is a function of the scalar part G(s,m) of the graviton propagator [71, 74]. The integration over
the Kaluza-Klein masses m, which we take as continuous, reflects that gravity propagates in the
higher-dimensional bulk. If the graviton anomalous dimension is small, the propagator is well
approximated by G(s,m) = (s+m2)−1. This propagator is used within effective theory settings,
and applicable if the relevant momentum transfer is ≪M∗. In this case, (8.1) is ultraviolet divergent
for n ≥ 2 due to the Kaluza-Klein modes [71]. Regularisation by an UV cutoff leads to a power-
law dependence of the amplitude S ∼ M−4
the behaviour of S is improved due to the non-trivial anomalous dimension η of the graviton,
e.g. (4.4). Evaluating η at momentum scale k2≈ s+m2, we are lead to the dressed propagator
G(s,m) ≈
becomes finite even in the UV limit of the integration. An alternative matching has been adapted in
[67, 68], based on the substitution G(k)→G(√s) in (8.1), setting G=M2−d
1
2+nTµ
µTν
νis a function of the energy-momentum tensor, and
S =
1
?∞
0
dm
M∗
?m
M∗
?n−1
G(s,m)
(8.1)
∗(Λ/M∗)n−2on the cutoff Λ. In a fixed point scenario,
Mn+2
∗
(s+m2)n/2+2in the vicinity of an UV fixed point. The central observation is that (8.1)
∗
. In that case, however,
13
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Fixed Points of Quantum Gravity
Daniel Litim
a) effective theoryb) renormalisation group
2
4
6
8
10
24681012
Λ[TeV]
MD[TeV]
5σ: pp → l+l− (D8)
10 fb-1
100 fb-1
n=6
n=4
n=3
n=2
n=1
0
1
2
3
4
5
6
123456789
Λ[TeV]
10
MD [TeV]
5σ: pp → l+l− (D8)
10 fb-1
n=5
n=3
0
1
2
3
4
5
6
123456789
Λ[TeV]
10
MD [TeV]
5σ: pp → l+l− (D8)
10 fb-1
n=6
n=4
Figure 4: The 5σ discoverycontours in MDat the LHC (d = 4+n), as a function of a cutoff Λ on Epartonfor
an assumed integrated luminosity of 10fb−1(100fb−1). a) Effective theory: the sensitivity to the cutoff Λ is
reflected in the MDcontour; plot from [74]. b) Renormalisation group: the limit Λ → ∞ can be performed,
and the leveling-off at MD≈ Λ reflects the gravitational fixed point, thin lines show a ±10% variation in the
transition scale; plot from [66].
(8.1) remains UV divergent due to the Kaluza-Klein modes. We conclude that the large anomalous
dimension in asymptotically safe gravity provides for a finite dilepton production rate.
In Fig. 4 we show the discovery potential in the fundamental Planck scale at the LHC, and
compare effective theory studies [74] with a gravitational fixed point [66]. In either case the mini-
mal signal cross sections have been computed for which a 5σ excess can be observed, taking into
account the leading standard model backgrounds and assuming statistical errors. This translates
into a maximum reach MDfor the fundamental Planck scale M∗. To estimate uncertainties in the
RG set-up, we allow for a 10% variation in the scale where the transition towards fixed point scal-
ing sets in. Consistency is checked by introducing an artificial cutoff Λ on the partonic energy [74],
setting the partonic signal cross section to zero for Eparton> Λ. It is nicely seen that MDbecomes
independent of Λ for Λ → ∞ when fixed point scaling is taken into account.
9. Conclusions
The asymptotic safety scenario offers a genuine path towards quantum gravity in which the
metric field remains the fundamental carrier of the physics even in the quantum regime. We have
reviewed the ideas behind this set-up in the light of recent advances based on renormalisation group
and lattice studies. The stability of renormalisation group fixed points and scaling exponents de-
tected in four- and higher-dimensional gravity is remarkable, strongly supporting this scenario.
Furthermore, underlying expansions show good numerical convergence, and uncertainties which
arise through approximations are moderate. If the fundamental Planck scale is as low as the elec-
troweak scale, signs for the quantisation of gravity and asymptotic safety could even be observed
in collider experiments. It is intriguing that key aspects of asymptotic safety are equally seen in
lattice studies. It will be interesting to evaluate these links more deeply in the future. Finally,
asymptotically safe gravity is a natural set-up which leads to classical general relativity as a “low
energy phenomenon” of a fundamental quantum field theory in the metric field.
Acknowledgements
I thank Peter Fischer and Tilman Plehn for collaboration on the topics discussed here, and the
organisers for their invitation to a very stimulating workshop.
14
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Fixed Points of Quantum Gravity
Daniel Litim
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