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Phil. Trans. R. Soc. A (2011) 369, 2759–2778
doi:10.1098/rsta.2011.0103
REVIEW
Renormalization group and the Planck scale
B Y D ANIEL F. L ITIM*
Department of Physics and Astronomy, University of Sussex,
Brighton BN1 9QH, UK
I discuss the renormalization group approach to gravity, and its link to Weinberg’s
asymptotic safety scenario, and give an overview of results with applications to particle
physics and cosmology.
Keywords: quantum gravity; Planck scale; renormalization group; asymptotic safety;
low-scale quantum gravity
1. Introduction
Einstein’s theory of general relativity is the remarkably successful classical theory
of the gravitational force, characterized by Newton’s coupling constant GN =
6.67 × 10−11 m3 kg−1 s−2 and a small cosmological constant L. Experimentally,
its validity has been confirmed over many orders of magnitude in length scales
ranging from the sub-millimetre regime up to Solar System size. At larger length
scales, the standard model of cosmology including dark matter and dark energy
components fits the data well. At shorter length scales, quantum effects are
expected to become important. An order-of-magnitude estimate for the quantum
scale of gravity—the Planck
scale—is obtained by dimensional analysis, leading
to the Planck length Pl ≈ h̄GN /c 3 of the order of 10−33 cm, with c the speed of
light. In particle physics units this translates into the Planck mass
MPl ≈ 1019 GeV.
(1.1)
While this energy scale is presently out of reach for Earth-based particle
accelerator experiments, fingerprints of Planck-scale physics can nevertheless
become accessible through cosmological data from the very early Universe. From
a theory perspective, it is widely expected that a fundamental understanding
of Planck-scale physics requires a quantum theory of gravity. It is well known
that the standard perturbative quantization programme faces problems, and a
fully satisfactory quantum theory, even outside the framework of local quantum
physics, is presently not to hand. In the past 15 years, however, a significant
body of work has been devoted to re-evaluate the physics of the Planck
scale within conventional settings. Much of this renewed interest is fuelled by
*[email protected]
One contribution of 11 to a Theme Issue ‘New applications of the renormalization group in nuclear,
particle and condensed matter physics’.
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D. F. Litim
Steven Weinberg’s seminal proposal, more than 30 years ago, that a quantum
theory of gravitation may dynamically evade the virulent divergences encountered
in standard perturbation theory [1]. This scenario, known as asymptotic safety,
implies that gravity achieves a non-trivial ultraviolet (UV) fixed point under its
renormalization group (RG) flow. If so, this would incorporate gravity alongside
the set of well-understood quantum field theories whose UV behaviour is governed
by a fixed point, e.g. Yang–Mills theory.
In this paper, I discuss the RG approach to gravity. I recall some of the issues
of perturbative quantum gravity (§2), introduce the RG à la Wilson to access
the physics at the Planck scale (§3), review key results in four dimensions (§4),
evaluate applications within low-scale quantum gravity (§5) and conclude (§6).
2. Perturbative quantum gravity
A vast body of work has been devoted to the perturbative quantization
programme of gravity. I recall a very small selection of these in order to prepare
for the subsequent discussion, and I use particle physics units h̄ = c = kB = 1
throughout. Classical general relativity is described by the classical action
1
d4 x det gmn (−R(gmn ) + 2L),
S=
(2.1)
16pGN
where I have chosen a Euclidean signature, R denotes the Ricci scalar, and L is the
cosmological constant term. The main point to be stressed is that the fundamental
coupling of gravity GN in equation (2.1) carries a dimension that sets a mass scale,
with mass dimension [GN ] = 2 − d in d-dimensional space–time. This structure
distinguishes gravity in a profound manner from the other fundamentally known
interactions in Nature, all of which have dimensionless coupling constants from
the outset. Consequently, the effective dimensionless coupling of gravity that
organizes its perturbative expansion is given by
geff ≡ GN E 2
(2.2)
(in four dimensions), where E denotes the relevant energy scale. While the
effective coupling (2.2) remains small for energies below the Planck scale
E MPl , it grows large in the Planckian regime where an expansion in geff
may become questionable. Within the Feynman diagrammatic approach, this
behaviour translates into the (with loop order) increasing degree of divergence
of perturbative diagrams involving gravitons [1]. This structure is different from
standard quantum field theories and relates to the classification of interactions
as super-renormalizable, renormalizable or ‘dangerous’, depending on whether
their canonical mass dimension is positive, vanishing or negative. The degree
of divergences implied by the negative mass dimension of Newton’s coupling is
mirrored in the perturbative non-renormalizability of Einstein gravity, which has
been established at the one-loop level [2] in the presence of matter fields, and at
the two-loop level [3] within pure gravity.
While this state of play looks discouraging from a field theory perspective, it
does not rule out a quantum field theoretical description of gravity altogether.
There are a few indicators available to support this view. Firstly, at low energies,
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a weak-coupling analysis of quantum gravity effects is possible within an
effective theory approach [4], which operates a UV cutoff at the Planck scale;
see [5] for a review. Secondly, higher-order derivative operators appear to
stabilize perturbation theory [6]. Including all fourth-order derivative operators,
it has been proven by Stelle [7] that gravity is renormalizable to all orders
in perturbation theory. This striking difference with Einstein–Hilbert gravity
highlights the stabilizing effect of higher derivative terms. Unfortunately, the
resulting theory is not compatible with standard notions of perturbative unitarity
and has therefore not been considered as a candidate for a fundamental theory of
gravity. The role of higher-order derivative operators has further been clarified in
Gomis & Weinberg [8] using cohomology techniques. Interestingly, the theory
remains unitary once all higher derivative operators are retained, whereas
renormalizability is at best achieved in a very weak sense owing to the required
infinitely many counter-terms.
3. Renormalization group
The RG comes into play when the running of couplings with energy is taken into
account. As in any generic quantum field theory, quantum fluctuations modify
the strength of couplings. If the metric field remains the fundamental carrier
of the gravitational force, the fluctuations of space–time itself should modify
the gravitational interactions with energy or distance. For Newton’s coupling,
this implies that GN becomes a running coupling GN → G(k) = GN Z −1 (k) as a
function of the RG momentum scale k, where Z (k) denotes the wave function
renormalization factor of the graviton. Consequently, the dimensionless coupling
geff in (2.2) should be replaced by the running coupling
g = G(k)k 2 ,
(3.1)
which evolves with the RG scale. In particular, the UV behaviour of standard
perturbation theory is significantly improved, provided that equation (3.1)
remains finite in the high-energy limit. This is the asymptotic safety scenario
as advocated by Weinberg [1]; see Niedermaier [6] and Niedermaier & Reuter [9]
for extensive accounts of the scenario, and Litim [10,11] and Percacci [12] for brief
overviews. The intimate link between a fundamental definition of quantum field
theory and RG fixed points was stressed by Wilson some 40 years ago [13,14].
For gravity, the fixed-point property becomes visible by considering the Callan–
Symanzik-type RG equation for (3.1), which in d dimensions takes the form [10]
(see also [6,9])
(3.2)
vt g ≡ bg = (d − 2 + h)g,
with h = −vt ln Z (k) the graviton anomalous dimension, which in general is a
function of all couplings of the theory including matter, and t = ln k. This simple
structure arises provided the underlying effective action is local in the metric
field. From the RG equation (3.2), one concludes that the gravitational coupling
may display two types of fixed points. The non-interacting (Gaussian) fixed
point corresponds to g∗ = 0 and entails the vanishing of the anomalous dimension
h = 0. In its vicinity, gravity stays classical and G(k) ≈ GN . On the other hand,
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D. F. Litim
a non-trivial RG fixed point with g∗ = 0 can be achieved implicitly, provided that
the anomalous dimension reads
h∗ = 2 − d.
(3.3)
The significance of equation (3.3) is that the graviton anomalous dimension
precisely counterbalances the canonical dimension of Newton’s coupling GN .
This pattern is known from other gauge systems at a critical point away from
their canonical space–time dimensionality [10], e.g. U(1) Higgs theory in three
dimensions [15]. In consequence, the dimensionful, renormalized coupling scales
as G(k) ≈ g∗ /k d−2 and becomes small in the UV limit where 1/k → 0. This pattern
is at the root of the non-perturbative renormalizability of quantum gravity within
a fixed-point scenario.
Much work has been devoted to checking by explicit computation whether or
not the gravitational couplings achieve a non-trivial UV fixed point. A versatile
framework to address this question is provided by modern (functional) RG
methods, based on the infinitesimal integrating-out of momentum degrees of
freedom from a path integral representation of the theory à la Wilson [16–20].
This is achieved by adding a momentum cutoff to the action, quadratic in the
propagating fields. In consequence, the action (2.1) becomes a scale-dependent
effective or flowing action Gk ,
1
d √
(−R(gmn ) + 2Lk ) + · · · + Sk,gf + Sk,gh + Sk,matter , (3.4)
Gk = d x g
16pGk
which in the context of gravity contains a running gravitational coupling, a
running cosmological constant Lk , a gauge-fixing term, ghost contributions,
matter interactions and the dots indicate possible higher derivative operators
in the metric field. Upon varying the RG scale k, the effective action interpolates
between a microscopic theory GL at the UV scale k = L (not to be confused with
the running cosmological constant Lk ) and the macroscopic quantum effective
action G, where all fluctuations are taken into account (k → 0). The variation
of equation (3.4) with RG scale k is given by an exact functional differential
equation [18,21]
1
1
1
vt R k =
,
(3.5)
vt Gk = Tr (2)
2
2
G + Rk
k
which relates the change of the scale-dependent gravitational action with the full
field-dependent propagator of the theory (full line) and the scale dependence vt Rk
of the momentum cutoff (the insertion). Here, the trace stands for a momentum
integration, and a sum over all propagating degrees of freedom f = (gmn , ghosts,
matter fields), which minimally contains the metric field and its ghosts. The
function Rk (not to be confused with the Ricci scalar) denotes the infrared (IR)
momentum cutoff. As a function of (covariant) momenta q 2 , the momentum cutoff
obeys Rk (q 2 ) → 0 for k 2 /q 2 → 0, Rk (q 2 ) > 0 for q 2 /k 2 → 0 and Rk (q 2 ) → ∞ for
k → L (for examples and plots of Rk , see Litim [22]). The functional flow (3.5) is
closely linked to other exact functional differential equations, such as the Callan–
Symanzik equation [23] in the limit Rk → k 2 , and the Wilson–Polchinski equation
by means of a Legendre transformation [12,17,19]. In a weak-coupling expansion,
equation (3.5) reproduces standard perturbation theory to all loop orders [24,25].
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The strength of the formalism is that it is not bounded to the weak-coupling
regime, and systematic approximations—such as the derivative expansion, vertex
expansions or mixtures thereof—are available to access domains with strong
coupling and/or strong correlation. Systematic uncertainties can be assessed [26],
and the stability and convergence of approximations is enhanced by optimization
techniques [22,27–30].
By construction, the flow (3.5) is both UV and IR finite. Together with
the boundary condition GL it may serve as a definition for the theory. In
renormalizable theories, the continuum limit is performed by removing the scale
L, 1/L → 0, and the functional GL → G∗ approaches a fixed-point action at
short distances (see the proposal in Manrique & Reuter [31] for a construction
in gravity). In perturbatively renormalizable theories, G∗ is ‘trivial’ and given
mainly by the classical action. In perturbatively non-renormalizable theories,
the existence (or non-existence) of G∗ has to be studied on a case-by-case basis.
A viable fixed-point action G∗ in quantum gravity should at least contain those
diffeomorphism-invariant operators that display relevant or marginal scaling in
the vicinity of the UV fixed point. A fixed-point action qualifies as fundamental
if RG trajectories k → Gk emanating from its vicinity connect with the correct
long-distance behaviour for k → 0 and stay well defined (finite, no poles) at all
scales [1].
The operator trace in equation (3.5) is evaluated by using flat or
curved backgrounds together with heat kernel techniques or plain momentum
integration. To ensure diffeomorphism symmetry within this set-up, the
background field formalism is used by adding a non-propagating background field
ḡmn [21,30,32–34]. This way, the extended effective action Gk [gmn , ḡmn ] becomes
gauge-invariant under the combined symmetry transformations of the physical
and the background field. A second benefit of this is that the background field
can be used to construct a covariant Laplacian −D̄ 2 (or similar) to define a mode
cutoff at the RG momentum scale k 2 = −D̄ 2 . This implies that the mode cutoff Rk
will depend on the background fields, which is controlled by an equation similar to
the flow equation itself [34,35]. The background field is then eliminated from the
final equations by identifying it with the physical mean field. This procedure
dynamically readjusts the background field and implements the requirements
of background independence for quantum gravity. An alternative technique
that employs a bi-metric approximation has been put forward in Manrique &
Reuter [36] and Manrique et al. [37,38]. For a general evaluation of the different
implementations of a momentum cutoff in gauge theories, see [30,32,34].
4. Fixed points of quantum gravity
In this section, I give a brief summary of fixed points found so far, and refer to
Litim [12] for a more detailed overview of results prior to 2008. The search for
fixed points in quantum gravity starts by restricting the running effective action
Gk to a finite set of operators Oi (f) with running couplings gi ,
Gk =
gi (k)Oi (f),
(4.1)
i
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D. F. Litim
0.20
=4
0.15
0.10
0.05
=3
=2
0
0
0.2
0.4
0.6
0.8
1.0
Figure 1. Dependence of the UV fixed point on space–time dimensionality.
√
including,
for
example,
the
Ricci
scalar
gR and the cosmological constant
√
g. The flow equations for the couplings gi are obtained from equation (3.5)
by projection onto some subspace of operators. Convergence and stability of
approximations are checked by increasing the number of operators retained, and
further improved through optimized choices of the momentum cutoff [22,29].
The first set of flow equations that carry a non-trivial UV fixed point has
been derived in Reuter [21] for the Einstein–Hilbert theory in Feynman gauge.
As a function of the number of space–time dimensions, the flow equation
reproduces the well-known fixed point in d = 2 + e dimensions [1,39–41] (figure 1).
More importantly, the equations display a non-trivial UV fixed point in the
four-dimensional theory [21,42]. Both the Ricci scalar and the cosmological
constant term are relevant operators at the fixed point. Their scaling is strongly
correlated, leading to a complex conjugate pair of universal scaling exponents.
This result has subsequently been confirmed within a more general background
field gauge by means of a trace-less transverse decomposition of the metric
field [43].
A complete analytical understanding of the Einstein–Hilbert approximation
has been achieved in Litim [44], leading to analytical flow equations and closed
expressions for the (unique) UV fixed point and its universal eigenvalues. Key for
this was the use of (optimized) momentum cutoffs that allow analytical access and
improve convergence and stability of results [22,27]. Figure 2 shows the universal
scaling exponents q = q
+ iq
at the UV fixed point in four dimensions (left-hand
side) for various (optimized) choices of the momentum cutoff [27,45]. Also on
display is the invariant t = l∗ (g∗ )2/(d−2) . The variations in either of these are small.
The dependence on the gravitational gauge-fixing parameter a is moderate and
controlled by an independent fixed point in the gauge-fixing sector, i.e. Landau–
de Witt gauge (a = 0) [46]. Figure 3 shows the RG running of couplings along
the separatrix connecting the non-trivial UV fixed point with the Gaussian fixed
point in the IR. At the scale k ≈ LT , the RG flow displays a cross-over from
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Review. RG and the Planck scale
(a)
q ¢/q ¢opt
1.10
1.05
1.00
(b)
q ¢¢/qopt
¢¢
1.05
1.00
|q | / |qopt|
(c)
1.05
1.00
t /topt
(d)
1.00
0.98
0.96
4
5
6
7
8
9
10
11
Figure 2. Ultraviolet fixed point in the d-dimensional Einstein–Hilbert theory. Comparison of
universal eigenvalues q
, q
, |q| and the invariant t = l∗ (g∗ )2/(d−2) for different Wilsonian
momentum cutoffs and various dimensions, normalized to the result for Ropt (Rmexp , circles; Rexp ,
squares; Rmod , triangles; Ropt , inverted triangles). Adapted from Fischer & Litim [45]. (Online
version in colour.)
perturbative IR scaling to fixed-point scaling, where the anomalous dimension
grows large. The dynamical scale LT is of the order of the fundamental Planck
scale, and a consequence of the UV fixed point. In gravity, the scale LT plays a
role analogous to that of LQCD in quantum chromodynamics.
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2
1
0
–1
–2
–3
–4
–4
–2
0
2
4
Figure 3. Running couplings along the separatrix in four dimensions; adapted from Litim [44].
(Online version in colour.)
In a formidable tour-de-force
computation, the space of operators has been
√ 2
gR interactions [47,48]. The perturbatively marginal R2
extended to include
coupling turned out to become relevant at a UV fixed point. Otherwise, it added
only minor corrections to the fixed point in the Einstein–Hilbert theory. With
Litim [27,44], these computations
became feasible analytically and have been
√
gfk (R), which contain a general function of the
implemented for operators
Ricci scalar, in Codello et al. [49,50] and Machado & Saueressig [51]. A nontrivial UV fixed point has been found by Taylor-expanding fk (R) to high orders
in the Ricci scalar, with results known up to eighth order in the eigenvalues [50]
and up to tenth order in the fixed point [52]. Additionally, the fixed point is
remarkably stable. The universal eigenvalues converge rapidly, confirming
all
√ n
gR from
previous results. Most importantly, it turned out that operators
n ≥ 3 onwards become irrelevant at the UV fixed point [49], a first indicator
that an asymptotically safe UV fixed point of the fundamental theory may,
in fact, have a finite number of relevant operators. This is most welcome as
otherwise a fixed point with infinitely many relevant couplings is likely to spoil
the predictive power.
All fourth-order derivative operators have been included both at the oneloop order and beyond. The re-evaluation of perturbative one-loop results by
means of a functional flow emphasized the significance of, for example, quadratic
divergences that are normally suppressed within dimensional regularization
[53–55]. Two types of UV fixed points are found. The first one is non-trivial
in all couplings [56,57], with three relevant directions at the fixed point. The
second one is perturbative in the Weyl coupling [53,54] but non-trivial in Newton’s
coupling, with two relevant and two marginally relevant directions. In either case
Newton’s coupling takes a non-trivial fixed point in the UV. These results hence
indicate that higher derivative operators are compatible with gravity developing a
non-trivial fixed point. Once this is achieved, the difference between the Einstein–
Hilbert approximation and the fourth-order approximation is qualitatively small.
In the above, the RG running of the ghost sector is expected to be sub-leading.
This has been recently confirmed in Groh & Saueressig [58] and in Eichhorn &
Gies [59], and for the scalar curvature–ghost coupling in Eichhorn et al. [60].
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An important extension deals with the inclusion of matter couplings. In Yang–
Mills theory, for example, it is well known that the property of asymptotic freedom
can be spoiled by too many species of fermions. In the same vein it has to be
checked whether the UV fixed point of gravity remains stable under the inclusion
of matter, and vice versa. By now, this question has been analysed for minimally
coupled scalar matter [61], non-minimally coupled scalar fields [62,63] and generic
free matter coupled to gravity [53,64]. These findings are consistent with earlier
results using different techniques in the limit of many matter fields [65,66].
The conclusion is that the gravitational fixed point generically persists under
the inclusion of matter (including the Standard Model (SM) of particle physics
and its main extensions) unless a significant imbalance between bosonic and
fermionic matter fields is chosen. Also, non-minimally coupled scalar matter leads
to slight deformation of the gravitational fixed point with matter achieving a
weakly coupled regime, the so-called Gaussian matter fixed point. Fermions have
been included recently in Zanusso et al. [67], as well as gravitational Yukawa
systems [68]. In the latter case, the coupled system displays a non-trivial UV
fixed point both in the Yukawa as well as in the gravitational sector. This new
fixed point may become of great interest as a stabilizer for the SM Higgs. On the
one-loop level, the interplay between an SM Higgs and asymptotically safe gravity
has been addressed in Shaposhnikov & Wetterich [69], detailing conditions under
which no new physics is required to bridge the energy range from the electroweak
scale all the way up to equation (1.1). In a similar spirit, it has been argued
that the consistency of Higgs inflation models is enhanced provided that gravity
becomes asymptotically safe [70].
The impact of gravitational fluctuations on gauge theories has seen renewed
interest initiated by a one-loop study within effective theory [71]. Asymptotic
freedom is sustained by gravity, leading to a vanishing [35,72–75] or non-vanishing
one-loop correction [35,71,76–78], depending on the regularization. In the present
framework (3.4) and (3.5), the gravity-induced corrections have been studied in
Daum et al. [78] and in Folkerts et al. [35]. In either case, the sign of the graviton
contributions is fixed and asymptotic freedom of Yang–Mills theory persists in
the presence of gravitational fluctuations, also including a cosmological constant.
Furthermore, Folkerts et al. [35] evaluate generic regularizations and clarify
the non-universal nature of the one-loop coefficient, also covering generic field
theory-based UV scenarios for gravity, including asymptotically safe gravity and
gravitational shielding. An interesting consequence of a non-trivial gravitational
contribution to the running of an Abelian charge is the appearance of a combined
UV fixed point in the U(1)–gravity system providing a mechanism to calculate
the fine-structure constant [79].
In three dimensions, a fixed-point study has been reported within topological
massive gravity, which includes a Chern–Simons (CS) term in addition to
Newton’s coupling and the cosmological constant [80]. On the one-loop level,
the CS term has a vanishing beta function and the gravitational sector displays
both a Gaussian and a non-trivial fixed point for positive CS coupling, which is
qualitatively in accord with the fixed point in figure 1 (no CS term).
An interesting structural link has been noted between the gravitational fixed
point [44] (Ricci scalar, no cosmological constant) and a UV fixed point in
nonlinear sigma models [81]. In either system, the beta function of the relevant
coupling displays a UV fixed point whose coordinates are non-universal. However,
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the systems display the same universal scaling, with identical exponent q =
2d(d − 2)/(d + 2) for all d ≥ 2 dimensions. This coincidence highlights that
nonlinear sigma models have a non-trivial UV dynamics on their own, possibly
quite similar to gravity itself, which is worthy of being explored further [82]. Fixed
points in more complex (gauged and non-gauged) nonlinear sigma models have
recently been obtained in [83,84].
It would be very helpful to understand which mechanism implies the
gravitational fixed point by, for example, restricting the dynamical content. In
fact, a two Killing vector reduction of four-dimensional gravity (with scalar matter
and photons) has been shown to be asymptotically safe [85]. This topic has also
been studied by reducing gravity to the dynamics of its conformal sector [86,87],
including higher derivative terms and effects from the conformal anomaly [88].
Although a UV fixed point is visible in most approximations, its presence depends
more strongly on, for example, matter fields, meaning that more work is required
to settle this question for pure gravity.
Implications of a gravitational fixed point have been studied in the context of
black holes, astrophysics and IR gravity, and cosmology. The RG-induced running
GN → G(r) of Newton’s coupling has interesting implications for the existence
of RG-improved black hole space–times [89] as well as for their dynamics [90].
If gravity weakens according to its asymptotically safe RG flow, results point
towards the existence of a smallest black hole with critical mass Mc of the order
of MPl . This pattern persists for Schwarzschild black holes in higher dimensions
[91], rotating black holes [92] and black holes within higher derivative gravity [93],
showing that the existence of smallest black holes is a generic and stable prediction
of this framework.
Low-energy implications of the RG running of couplings have equally
been studied, including applications to galaxy rotation curves [94,95]. Recent
studies have re-evaluated gravitational collapse [96], exploring the one-loop
running couplings, and the RG-induced non-local low-energy corrections to the
gravitational effective action [97]. It has also been conjectured that gravity
may possess an IR fixed point. In this case, Newton’s coupling grows large
at large distances and may contribute to an accelerated expansion of the
Universe, a scenario that can be tested against cosmological data such as Type
Ia supernovae [98–100].
The intriguing idea that an asymptotically safe UV fixed point may control
the beginning of the Universe has been implemented in Einstein gravity with
an ideal fluid [101,102], and in the context of f (R) theories of gravity [52,103–
105]. The RG scale parameter is linked with cosmological time and therefore
leads to RG-improved cosmological equations [101,106,107], which may even
generate entropy [108,109]. The framework put forward in Hindmarsh et al.
[106] includes scalar fields and exploits the Bianchi identity, leading to general
conditions of existence for cosmological fixed points (with cosmic time) whose
solutions include, for example, inflating UV safe fixed points of gravity coupled
to a scalar field.
The proposal that Lorentz symmetry can be broken by gravity on a
fundamental level has been put forward in Horava [110], where space and time
scale differently in the UV with a relative dynamical exponent z = 1. Provided z =
3 (in four dimensions) gravity is power-counting renormalizable in perturbation
theory and Lorentz symmetry should emerge as a low-energy phenomenon [111].
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Interestingly, Horava–Lifshitz gravity with z = 3 also requires an asymptotically
safe fixed point for gravity with an anomalous dimension h = 0 compensating for
power-counting non-renormalizability.
The RG scaling in the vicinity of an asymptotically safe fixed point implies
scale invariance. On the other hand, it is often assumed that a quantum theory of
gravity should, in one way or another, induce a minimal length. Different aspects
of this question and interrelations with other approaches to quantum gravity
have been addressed in Reuter & Schwindt [112,113], and more recently in Basu
& Mattingly [114], Percacci & Vacca [115] and Calmet et al. [116]. The findings
indicate that the fixed point implicitly induces a notion of minimal length, related
to the RG trajectory and the fundamental scale where gravity crosses over from
classical to fixed-point scaling. It has also been argued that RG corrections to
gravity can modify the dispersion relation of massive particles, leading to indirect
bounds on the one-loop coefficients of the gravitational beta functions [117].
Finally, it is worth noting that standard quantum field theories (without
gravity) may become asymptotically safe in their own right. This was first
exemplified for perturbatively non-renormalizable Gross–Neveu models in three
dimensions with functional RG methods [118] and in the limit of many fermion
flavours [119]. In four dimensions, this possibility has been explored for variants
of the SM Higgs [120], for Yukawa-type interactions [121,122], and for strongly
coupled gauge theories [123]. It has also been conjectured that the electroweak
theory without a dynamical Higgs field could become asymptotically safe [124],
based on structural similarities with quantum general relativity.
In summary, there is an increasing amount of evidence for the existence of
non-trivial UV fixed points in four-dimensional quantum field theories, including
gravity and matter.
5. Extra dimensions
Next, I turn to fixed points of quantum gravity in d = 4 + n dimensions, where
n denotes the number of extra dimensions [44,45,125], and potential signatures
thereof in models with a low quantum gravity scale [126–130]. There are several
motivations for this. Firstly, particle physics models where gravity lives in a
higher-dimensional space–time have raised enormous interest by allowing the
fundamental Planck scale to be as low as the electroweak scale [131–133],
M∗ ≈ 1–10 TeV.
(5.1)
This way, the notorious hierarchy problem of the SM is circumnavigated.
If realized in Nature, this opens the exciting possibility that Planck-scale
physics becomes accessible to experiment via, for example, high-energetic particle
collisions at the Large Hadron Collider (LHC). The main point of these models
is that the four-dimensional Planck scale (1.1) is no longer fundamental but a
2
∼ M∗2 (M∗ L)n to the Planck scale (5.1) of the
derived quantity, related as MPl
higher-dimensional theory and the size of the compact extra dimensions L.
Secondly, the critical dimension of gravity—the dimension where the
gravitational coupling has vanishing canonical mass dimension—is two. Hence,
for any dimension above the critical one, the canonical dimension of Newton’s
coupling is negative. From an RG point of view, this means that four dimensions
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2770
D. F. Litim
(a)
(b)
Figure 4. Log–log plot (schematic) of the dimensionless running gravitational coupling in a scenario
with large extra dimensions of size ∼ L and a dynamical Planck scale LT with LT L 1. The
fixed-point behaviour in the deep UV enforces a softening of gravity. (a) At the scale k ≈ LT the
coupling displays a cross-over from fixed-point scaling to classical scaling in the higher-dimensional
theory. (b) At the compactification scale k ≈ 1/L, the n compactified spatial dimensions are no
longer available for gravity to propagate in, and the running coupling displays a cross-over from
(4 + n)-dimensional to four-dimensional scaling.
appear not to be special. Continuity in the dimension suggests that a UV
fixed point, if it exists in four dimensions, should persist towards higher
dimensions. Furthermore, Einstein gravity has d(d − 3)/2 propagating degrees of
freedom, which rapidly increases with increasing dimensionality. It is important
to understand whether these additional degrees of freedom spoil or support the
UV fixed point detected in the four-dimensional theory. The local structure of
quantum fluctuations, and hence local RG properties of a quantum theory of
gravity, are qualitatively similar for all dimensions above the critical one, modulo
topological effects for specific dimensions. Therefore, one should expect to find
similarities in the UV behaviour of gravity in four and higher dimensions.
This expectation has been confirmed in Litim [44] for d-dimensional Einstein–
Hilbert gravity. The stability of the result has subsequently been tested through
an extended fixed points search in higher-dimensional gravity for general cutoffs
Rk [45], also probing the stability of the fixed point against variations of the
gauge fixing [125], and the results are displayed in figure 2. The cutoff variations
are very moderate, and smaller than the variation with gauge-fixing parameter.
The structural stability of the fixed point also strengthens the findings in the
four-dimensional case.
Thus, the following picture emerges. In the setting with compact extra
dimensions, the running dimensionless coupling g displays a cross-over from fixedpoint scaling to d-dimensional classical scaling at the cross-over scale LT of the
order of M∗ (figure 4a). By construction, this has to happen at energies way
above 1/L. At lower energies, close to the compactification scale ∼ 1/L, the n
compactified spatial dimensions are no longer available for gravity to propagate
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2771
Review. RG and the Planck scale
(a)
(b)
M* (TeV)
6
5
4
3
2
1
0
1
2
3
4
5
6 7
(TeV)
8
9 10
1
2
3
4
5 6 7
(TeV)
8
9 10
Figure 5. The 5s : pp → + − (D8) discovery contours in M∗ at the LHC as a function of cutoff L
on Eproton for an assumed integrated luminosity of 10 fb−1 : (a) thick-dashed line, n = 5; thick solid
line, n = 3; and (b) thick-dashed line, n = 6; thick solid line, n = 4. Note that the limit L → ∞ can
be performed, and the levelling-off at M∗ ≈ L reflects the underlying fixed point (thin-dashed lines
show a ±10% variation about the transition scale LT = M∗ ). Adapted from Litim & Plehn [126].
in, and the running coupling displays a cross-over from (4 + n)-dimensional to
four-dimensional scaling (figure 4b). High-energy particle colliders such as the
LHC are sensitive to the electroweak energy scale, and hence to the regime
of figure 4a, provided that the fundamental Planck scale is of the order of
the electroweak scale in (5.1). A generic prediction of large extra dimensions
is a tower of massive Kaluza–Klein gravitons [134–136]. In particular, the
exchange of virtual gravitons in Drell–Yan processes will lead to deviations in SM
reference processes such as pp → + − [137]. Gravitational Drell–Yan production
is mediated through tree-level graviton exchange and via graviton loops. Within
effective theory, the corresponding effective operators are UV divergent (in two
or more extra dimensions) and require a UV cutoff. Within fixed-point gravity,
the graviton is dressed by its anomalous dimension, leading to finite and cutoffindependent results for cross sections [126]; see Gerwick & Plehn [138] for a
comparison of different UV completions.
Figure 5 displays the 5s discovery reach to detect the fundamental scale of
gravity assuming M∗ = LT and an integrated luminosity of 10 fb−1 at the LHC.
Note that the curves are given as functions of an energy cutoff L. In effective
theory, these curves continue to grow with L, meaning that the cutoff cannot be
removed. Here, the levelling-off is a dynamical consequence of the underlying
fixed point, and the amplitude is independent of any cutoff. Figure 6 shows
the differential cross sections for di-muon production within asymptotically safe
gravity, with cross-over scale LT = M∗ [129]. For all dimensions considered, the
enhanced di-lepton production rate sticks out above SM backgrounds, leading
to the conclusion that Drell–Yan production is very sensitive to the quantum
gravity scale.
Further collider signatures of fixed-point gravity in warped and large extra
dimensions have been addressed in Hewett & Rizzo [128] within a form-factor
approximation [129]. Modifications to the semiclassical production of Plancksize black holes at the LHC induced by the fixed point have been analysed
in Falls et al. [91] and Burschil & Koch [139]. The main new effect is an
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2772
D. F. Litim
1
–1
10–1
fixed point
10–2
10–3
Standard Model
10–4
10–5
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Figure 6. Differential cross sections for di-muon production at the LHC as a function of the di-lepton
invariant mass mmm within asymptotically safe gravity for M∗ = LT = 5 TeV and n = 2 (magenta),
n = 3 (blue) and n = 6 (red) extra dimensions, in comparison with SM background (dashed line).
Adapted from Gerwick et al. [129]. (Online version in colour.)
additional suppression of the production cross section for mini-black holes. Some
properties of Planck-size black holes have also been evaluated within a one-loop
approximation, provided the gravitational RG flow is driven by many species of
particles [140]. Graviton loop corrections to electroweak precision observables
within asymptotically safe gravity with extra dimensions have recently been
obtained in Gerwick [130].
In summary, experimental signatures for the quantization of gravity within
the asymptotic safety scenario are in reach for particle colliders, provided the
fundamental scale of gravity is as low as the electroweak scale in (5.1).
6. Conclusion
RG methods have become a key tool in the attempt to understand gravity
at shortest, and possibly at largest, distances. It is conceivable that Planckian
energies (1.1) can be assessed with the help of running gravitational couplings,
such as equation (3.1), replacing equation (2.2) from perturbation theory. If
gravity becomes asymptotically safe, its UV fixed point acts as an anchor
for the underlying quantum fluctuations. The increasing amount of evidence
for a gravitational fixed point with and without matter and its significance for
particle physics and cosmology are very promising and certainly warrant more
far-reaching investigations.
This review is based on invited plenary talks at the conference The Exact RG 2010 (Corfu,
September 2010), at the workshop Critical Behavior of Lattice Models in Condensed Matter and
Particle Physics (Aspen Center for Physics, June 2010), at the IV Workshop Hot Topics in
Cosmology (Cargese, May 2010), at the INT workshop New Applications of the RG Method (Seattle,
February 2010), at the conference Asymptotic Safety (Perimeter Institute, November 2009) and at
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Review. RG and the Planck scale
2773
the XXV Max Born Symposium The Planck Scale (Wrocl/ aw, June 2009) and on an invited talk at
GR19: the 19th International Conference on General Relativity and Gravitation (Mexico City, July
2010). I thank Dario Benedetti, Mike Birse, Denis Comelli, Fay Dowker, Jerzy Kowalski-Glikman,
Yannick Meurice, Roberto Percacci, Nikos Tetradis, Roland Triary and Shan-Wen Tsai for their
invitations and hospitality, and my collaborators and colleagues for discussions. This work was
supported by the Royal Society, and by the Science and Technology Research Council (grant no.
ST/G000573/1).
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