Fixed Points of Quantum Gravity and the Renormalisation Group
ABSTRACT 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. Comment: 18 pages, 4 figures. Plenary talk. To appear in the proceedings of "From Quantum to Emergent Gravity: Theory and Phenomenology", June 11-15 2007, Trieste, Italy
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ABSTRACT: We discuss the existence and properties of a nontrivial fixed point in f(R)-gravity, where f is a polynomial of order up to six. Within this seven-parameter class of theories, the fixed point has three ultraviolet-attractive and four ultraviolet-repulsive directions; this brings further support to the hypothesis that gravity is nonperturbatively renormalizabile.International Journal of Modern Physics A 06/2007; · 1.13 Impact Factor
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ABSTRACT: Within the exact renormalisation group, the scaling solutions for O(N) symmetric scalar field theories are studied to leading order in the derivative expansion. The Gaussian fixed point is examined for d>2 dimensions and arbitrary infrared regularisation. The Wilson–Fisher fixed point in d=3 is studied using an optimised flow. We compute critical exponents and subleading corrections-to-scaling to high accuracy from the eigenvalues of the stability matrix at criticality for all N. We establish that the optimisation is responsible for the rapid convergence of the flow and polynomial truncations thereof. The scheme dependence of the leading critical exponent is analysed. For all N⩾0, it is found that the leading exponent is bounded. The upper boundary is achieved for a Callan–Symanzik flow and corresponds, for all N, to the large-N limit. The lower boundary is achieved by the optimised flow and is closest to the physical value. We show the reliability of polynomial approximations, even to low orders, if they are accompanied by an appropriate choice for the regulator. Possible applications to other theories are outlined.Nuclear Physics B 01/2002; · 4.33 Impact Factor
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ABSTRACT: A new exact renormalization group equation for the effective average action of Euclidean quantum gravity is constructed. It is formulated in terms of the component fields appearing in the transverse-traceless decomposition of the metric. It facilitates both the construction of an appropriate infrared cutoff and the projection of the renormalization group flow onto a large class of truncated parameter spaces. The Einstein-Hilbert truncation is investigated in detail and the fixed point structure of the resulting flow is analyzed. Both a Gaussian and a non-Gaussian fixed point are found. If the non-Gaussian fixed point is present in the exact theory, quantum Einstein gravity is likely to be renormalizable at the nonperturbative level. In order to assess the reliability of the truncation a comprehensive analysis of the scheme dependence of universal quantities is performed. We find strong evidence supporting the hypothesis that 4-dimensional Einstein gravity is asymptotically safe, i.e. nonperturbatively renormalizable. The renormalization group improvement of the graviton propagator suggests a kind of dimensional reduction from 4 to 2 dimensions when spacetime is probed at sub-Planckian length scales. Comment: 99 pages, latex, 11 figures, corrected some typosPhysical Review D 08/2001; · 4.69 Impact Factor
arXiv:0810.3675v1 [hep-th] 21 Oct 2008
Fixed Points of Quantum Gravity and the
Daniel F. Litim
Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, U.K.
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
From Quantum to Emergent Gravity: Theory and Phenomenology
June 11-15 2007
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
Fixed Points of Quantum Gravity
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 . 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 , large-N expansions in the
matter fields , 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
 (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 . 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  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
Fixed Points of Quantum Gravity
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  (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
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)
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.
Fixed Points of Quantum Gravity
quantum gravity within a fixed point scenario.2Consequently, at an interacting fixed point where
g∗?= 0, the anomalous dimension implies the scaling
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
ln Zk[J] = ln
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 ). 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
. 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
Fixed Points of Quantum Gravity
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 
?φRkφ, φ = ?ϕ?J. It obeys an exact functional differential
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  and gauge theories .
• 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 . 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].