#### Towards apparent convergence in asymptotically safe quantum gravity

Eur. Phys. J. C
Towards apparent convergence in asymptotically safe quantum gravity
T. Denz 1
J. M. Pawlowski 0 1
M. Reichert 1
0 ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung mbH , Planckstr. 1, 64291 Darmstadt , Germany
1 Institut für Theoretische Physik, Universität Heidelberg , Philosophenweg 16, 69120 Heidelberg , Germany
The asymptotic safety scenario in gravity is accessed within the systematic vertex expansion scheme for functional renormalisation group flows put forward in Christiansen et al. (Phys Lett B 728:114, 2014), Christiansen et al. (Phy Rev D 93:044036, 2016), and implemented in Christiansen et al. (Phys Rev D 92:121501, 2015) for propagators and three-point functions. In the present work this expansion scheme is extended to the dynamical graviton four-point function. For the first time, this provides us with a closed flow equation for the graviton propagator: all vertices and propagators involved are computed from their own flows. In terms of a covariant operator expansion the current approximation gives access to , R, R2 as well as Rμ2ν and higher derivative operators. We find a UV fixed point with three attractive and two repulsive directions, thus confirming previous studies on the relevance of the first three operators. In the infrared we find trajectories that correspond to classical general relativity and further show non-classical behaviour in some fluctuation couplings. We also find signatures for the apparent convergence of the systematic vertex expansion. This opens a promising path towards establishing asymptotically safe gravity in terms of apparent convergence.
1 Introduction
A consistent formulation of quantum gravity (QG) is an
open problem, with many contenders being investigated in
great detail. In the past two decades, Weinberg’s
asymptotic safety scenario (AS) proposed in 1976 [
4
] has been
investigated with the help of non-perturbative
renormalisation group techniques. AS posits that QG, while being
perturbatively non-renormalisable, is non-perturbatively
renormalisable and features a non-trivial fixed point in the ultraviolet
(UV).
By now an impressive body of evidence has been
collected that supports this intriguing scenario: the application
of functional renormalisation group (FRG) techniques [
5
]
to QG [
6
] has allowed for a confirmation of the existence
of the non-trivial UV fixed point in basic Einstein-Hilbert
approximations [
6–8
]. Later works have improved on these
approximations and gone beyond Einstein-Hilbert [
1–3,9–
52
]; for an overview see reviews [
53–56
]. Furthermore, the
stability of the asymptotic safety setting for gravity-matter
systems, [
57–59,61–76
], as well as asymptotically safe
theories within a perturbative setup, [
77–82
], have also attracted
a lot of attention.
In [
3
] the systematic vertex expansion in quantum
gravity initiated in [
1,2
], see also [
26,33
], has been pushed to
the graviton three-point function, for the first time
including a dynamical graviton-scattering in the asymptotic safety
analysis. Here we extend the vertex expansion to the graviton
four-point function. Apart from the significant technical
challenge such an upgrade of the approximation has posed, we
think that this constitutes a necessary and significant progress
towards asymptotically safe gravity:
• As such it is an important step towards apparent
convergence of the vertex expansion in quantum gravity:
apparent convergence aims at the convergence of vertices
as well as observables in the order of a given
systematic expansion scheme; here we use the vertex expansion
scheme. Together with the investigation of the regulator
(in-)dependence of observables this provides a
systematic error estimate in the present approach, and should be
compared with apparent continuum scaling and
extrapolation on the lattice.
• The present approximation allows for the identification of
diffeomorphism-invariant structures in the vertex
expansion, i.e. R2 and Rμ2ν tensor structures as well as those of
higher derivative invariants. This is not only important for
getting access to the number of relevant directions at the
asymptotically safe UV fixed point in quantum gravity,
but can also provide non-trivial support and additional
information for computations within the standard
background field approximation.
• It is the first approximation in the asymptotic safety
approach to gravity where the flow of the pivotal
building block, the two-point correlation function or (inverse)
propagator, is closed: The flows of all involved vertex
functions are computed within given approximations. As
the propagator is the core object in the present approach,
we consider this an important milestone on the way
towards asymptotically safe quantum gravity.
As a main result of this paper, we find further significant
evidence for a non-trivial UV fixed point in quantum gravity.
This fixed point has three relevant directions and two
repulsive ones. The three relevant directions can be associated
with the cosmological constant (graviton mass parameter),
Newton’s coupling and the R2-coupling, see Sect. 5. We also
investigate the stability of this UV fixed point and observe
that the system is significantly less sensitive to the closure
of the flow equations than previous truncations. In addition,
we observe that the critical exponents also become less
sensitive to the details of the approximations. These are two
necessary signatures of apparent convergence. Furthermore,
we investigate the infrared (IR) behaviour of the system and
find trajectories connecting the UV fixed point with classical
general relativity in the IR.
This work is structured in the following way: In Sect. 2
we elaborate on the important property of background
independence of quantum gravity and its consequences for the
dynamical correlation functions. In Sect. 3, we briefly
introduce the functional renormalisation group and the
covariant vertex expansion used in the present work. Next, Sect. 4
presents our setup and the derivations of the flow equations
for the couplings. We especially focus on how the tensor
structures of higher curvature invariants are embedded in our
vertex expansion. In Sect. 5, we present our results, in
particular the non-trivial UV fixed point with three attractive
and two repulsive directions. Furthermore, we test the
stability of the UV fixed point with respect to change of the
precise truncation, and find that in almost all approximations
we obtain a similar UV fixed point. In Sect. 6, we investigate
the IR-behaviour of the UV-finite trajectories. Our analysis
confirms the classical IR fixed points found before in the
vertex expansion. In Sect. 7 we discuss the convergence of the
present vertex expansion scheme with an increasing number
of flowing n-point correlation functions. Finally, in Sect. 8
we summarise our results.
2 Diffeomorphism invariance and background independence
The quantum field theoretical formulation of quantum
gravity in terms of metric correlation functions necessitates the
introduction of a background metric g¯, and quantum
fluctuations are taken to be fluctuations about this background
metric. This begs the question of whether diffeomorphism
invariance and background independence of observables are
guaranteed in such a framework. While this is an important
question, its answer is not directly relevant for the
computations presented here. Hence, the present chapter may be
skipped in a first reading.
In the present work we perform computations in the linear
split, where the full metric g is given by g = g¯ +h. More
general splits, g = g¯ + f (g¯, h) have been considered for
example within the geometrical or Vilkovisky-deWitt approach,
e.g. [
26,83–85
], or the exponential split, e.g. [
40,44,68,86–
88
]. Observables, on the other hand, are background
independent. This property is encoded in the Nielsen (NI) or
splitWard identities (SWI) that relate derivatives of the effective
action [g¯, φ] w.r.t. the background metric g¯ to those w.r.t.
the graviton fluctuations h. Here we have introduced the
fluctuation superfield
(1)
φ = (h, c, c¯).
The (anti-) ghost fields, c and c¯, stem from the
FaddeevPopov gauge fixing procedure, see next section. The
effective action generates all one-particle-irreducible correlation
functions and as such encodes the symmetries of the theory.
Schematically, these identities read [
6,20,26,84,89–95
]
δ [g¯, h]
δg¯(x ) =
y
C[g¯, h](x , y)
δ [g¯, h]
δh(y)
+ N [g¯, h](x ),
(2)
where we have suppressed the ghost fields. In the
linear split we have C[g¯, h](x , y) = δ(x − y) and the
second term N [g¯, h] carries the information about the
nontrivial behaviour under diffeomorphism transformations of
the gauge fixing sector and the regularisation. In turn, in
the geometrical approach diffeomorphism invariance of the
effective action is achieved by a non-linear split with f (g¯, h)
leading to the non-trivial prefactor C[g¯, h] in (2). The term
N [g¯, h] then carries the deformation of the Nielsen identity
in the presence of a regularisation but does not spoil
diffeomorphism invariance.
In both cases the Nielsen identity is a combination of a
quantum equation of motion, the Dyson-Schwinger
equation, and the Slavnov-Taylor identity (STI) or
diffeomorphism constraint. The setup also entails that correlations of
the fluctuation fields are necessarily background-dependent.
This is easily seen by iterating (2). Moreover, in the
linear split, diffeomorphism invariance of the observables is
encoded in non-trivial STIs for the fluctuation correlation
functions, while in the geometrical formulation, the
nontrivial STIs are encoded in expectation values of f (g¯, h) and
its derivatives.
Due to (2) we have to deal with the peculiarity that
background independence and physical diffeomorphism
invariance of observables necessitate background-dependence and
non-trivial STIs for the correlation functions of the
fluctuation fields. This leads to seemingly self-contradictory
statements: in particular, for the quantum effective action [g¯, h]
it entails that physical diffeomorphism invariance of
observables is not achieved by diffeomorphism invariance w.r.t.
diffeomorphism transformations of the fluctuation fields. The
latter does not do justice to either diffeomorphism invariance
or background independence.
This peculiarity can easily be checked in a non-Abelian
gauge theory within the background field formulation: in a
fluctuation gauge invariant approximation to the effective
action, even two-loop universal observables such as the
twoloop β-function cannot be computed correctly. Indeed, in
this case it is well-known that only the non-trivial STIs for
the fluctuation gauge field elevate the auxiliary background
gauge invariance to the physical one holding for observables,
see e.g. [
96,97
].
The above considerations underline the importance of a
direct computation of correlation functions of the fluctuation
field h. Indeed, the corresponding set of flow equations for
(n) = (0,n) is closed in the sense that the flow diagrams
only depend on (n) with n ≥ 2. Here, (n,m) stands for the
n-th background field derivative and m-th fluctuation field
derivative of the effective action,
(3)
(n,m)[g¯, φ] :=
δn+m
[g¯, h] .
δg¯nδφm
In turn, the flows for pure background, (n,0), or mixed
background-fluctuation functions, (n,m) with m = 0,
necessitate the fluctuation correlation functions as an input: the
background correlation functions can be iteratively
computed in powers of the background metric. In other words,
the dynamics of the system is solely determined by the pure
fluctuation correlation functions.
For this reason, several approaches for computing
correlation functions of the fluctuation field have been put forward
in the last years. Some of these approaches are set up for
computing both correlation functions of the fluctuation field
as well as those of the background field: vertex expansion
[
1–3,33,71,73,74
] and bimetric approach [
20,24,37,90
].
Another set of approaches relies on utilising the Nielsen
or split Ward identities explicitly or implicitly, [
26,84,92–
95,98–102
].
So far, the (n) for n ≥ 2 have been computed directly
in only one approach, the vertex expansion, see [
1–3,33
] for
pure gravity and [
71,73,74
] for gravity-matter systems. A
mixture of vertex expansion and background approximation
has been used in [
33,67,72
]. Present results include (n) for
n = 0, 1, 2, 3, where higher vertices have been estimated by
lower ones, relying on approximate covariance of the
correlation functions. Such a structure has already been confirmed
in the perturbative and semi-perturbative regime of QCD, see
[103].
3 Effective action and functional renormalisation group
The set of (covariant) correlation functions of the metric,
g(x1) · · · g(xn) , defines a given theory of quantum gravity.
All observables can be constructed from these basic
building blocks. The correlation functions are generated from the
single metric effective action, [g] = [g, h = 0], which is
the free energy in a given metric background g = g¯ + h at
h = 0. Here we have restricted ourselves to a linear split. The
underlying classical action is the Einstein-Hilbert action,
1
SEH = 16π G N
d4x
det g 2
− R(g) + Sgf + Sgh,
where R(g) is the Ricci curvature scalar, while Sgf[g¯, h]
and Sgh[g¯, φ] describe the gauge-fixing and Faddeev-Popov
ghost parts of the action, respectively. The gauge fixing action
reads
1
Sgf[g¯, h] = 2α
d4x
det g¯ g¯μν Fμ Fν .
We employ a linear, de-Donder type gauge-fixing,
Fμ = ∇¯ ν hμν −
1 + β
4
∇¯ μhν ν .
In particular we use the harmonic gauge given by β = 1. This
choice simplifies computations considerably due to the fact
that the poles of all modes of the classical graviton propagator
coincide. We furthermore work in the limit of a vanishing
gauge parameter, α → 0. This is favourable because then the
gauge does not change during the flow since α = 0 is a RG
fixed point [
104
]. Moreover, it allows for a clear separation
of propagating and non-propagating degrees of freedom. The
ghost part of the action reads
Sgh[g¯, φ] =
d4x
det g¯ c¯μMμν cν ,
where c¯ and c denote the (anti-) ghost field and M is the
Faddeev-Popov operator deduced from (6). For β = 1 it is
given by
Mμν = ∇¯ ρ gμν ∇ρ + gρν ∇μ − ∇¯ μ∇ν .
(4)
(5)
(6)
(7)
(8)
The gauge fixing and ghost term in (5) and (7) introduce the
separate dependence on g¯ and h leading to the non-trivial
Nielsen identities in (2).
3.1 Flow equation
An efficient way of computing non-perturbative correlation
functions is the functional renormalisation group. In its form
for the effective action, see [
5,105,106
], it has been applied
to quantum gravity [
6
]. For reviews on the FRG approach to
gauge theories and gravity see e.g. [
53–56,89,107
]. The RG
flow of the effective action for pure quantum gravity is given
by
1
∂t k = 2 Tr
Here, ∂t ≡ k∂k denotes the scale derivative, where k is the
infrared cutoff scale. k(2) = k(0,2) stands for the second
fluctuation field derivative of the effective action, while Rk
is the regulator, which suppresses momenta below k. The
trace in (9) sums over internal indices and integrates over
space-time.
The introduction of cutoff terms leads to
regulatordependent modifications of STIs and NIs that vanish for
Rk → 0. The respective symmetry identities have hence
been named modified Slavnov-Taylor identities (mSTIs) and
modified Nielsen- or split Ward identities (mNIs/mSWIs).
The modification entails the breaking of the physical or
quantum diffeomorphism invariance in the presence of a
background covariant momentum cutoff. Still, background
diffeomorphism invariance is maintained in the presence of the
cutoff term.
3.2 Covariant expansion
The effective action k [g¯, φ] depends on the background
metric g¯ and the fluctuation superfield φ = (h, c, c¯), see (1),
separately. The functional flow equation (9) is accompanied
by the functional mSTIs & mNIs for the effective action that
monitor the breaking of quantum diffeomorphism invariance,
see (2) in Sect. 2.
In order to solve (9), we employ a vertex expansion around
a given background g¯, to wit
k [g¯, φ] =
∞ 1 (φi1 ...φin ) [g¯, 0]
n=0 n! k
n
l=1
φil ,
where the superscript fields in parentheses are a short-hand
notation for field derivatives, and where contracting over
super-indices i j occurring twice is implied. In this work, we
with
f (0) = 0,
are also taken into account. Without the constraint f (0) = 0,
Eq. (14) also includes R2 and Rμ2ν , more details on this basis
can be found in Sect. 4. Note that also
non-diffeomorphisminvariant terms are generated by the flow. In Sect. 4.3 we
include the full flow of the vertex functions up to the graviton
four-point function.
As discussed in Sect. 2, the expansion coefficients
k(φi1 ...φin ) = k(n) = k(0,n) satisfy mSTIs as well as mNIs
with k(n,m) being defined in (3). For the sake of
simplicity we now restrict ourselves to the gauge fixing used in the
present work, (6) with α = 0. Then the fluctuation graviton
propagator is transverse: it is annihilated by the gauge fixing
condition.
An important feature of the functional RG equations is
that for α = 0 the flow equations for the transverse vertices
k(n,T) are closed: the external legs of the vertices in the flow
are transverse due to the transverse projection of the flow,
the internal legs are transverse as they are contracted with
the transverse propagator. Schematically this reads
∂t k,T = Flow(Tn)[{ k(m,T)}].
(n)
In other words, the system of transverse fluctuation
correlation functions is closed and determines the dynamics of the
system. On the other hand, the mSTIs are non-trivial relations
for the longitudinal parts of vertices in terms of transverse
vertices and longitudinal ones. This leads us to the schematic
relation
k(n,L) = mSTI(n)[{ k(m,T)}, { k(m,L)}],
see [
108
] for non-Abelian gauge theories. In consequence,
the mSTIs provide no direct information about the transverse
correlation functions without further constraint. In the
perturbative regime this additional constraint is given by the
uniformity of the vertices, for a detailed discussion in
nonAbelian gauge theories see [
109
].
Accordingly, our task reduces to the evaluation of the
coupled set of flow equations for the transverse vertices k(n,T) .
Each transverse vertex can be parameterised by a set of
diffeomorphism-invariant expressions. Restricting ourselves
to local invariants and second order in the curvature we are
left with
R ,
R2 ,
Rμ2ν .
The square of the Weyl tensor C 2 is eliminated via the
Gauß-Bonnet term, which is a topological invariant.
Higherderivative terms, such as
(11)
(12)
(13)
(14)
discuss all invariants which are included in the
parameterisation of our vertices.
For the background vertices k(n,0) we use the following:
the NIs become trivial in the IR as we approach classical
gravity, as shown in Sect. 6. Moreover, for one of the two IR
fixed points this implies that the derivative with respect to a
background field is the same as a derivative with respect to
a fluctuation field. This allows us to impose the trivial NIs
in the IR, and all couplings are related. Then, the couplings
at k > 0 follow from the flow equation. However, for the
fluctuation couplings this amounts to solving a fine-tuning
problem in the UV, for more details see Sect. 6. The latter is
deferred to future work.
4 Flows of correlation functions
In this chapter we discuss the technical details of the
covariant expansion scheme used in the present work, including the
approximations used and their legitimisation. In our opinion,
a careful reading of this chapter is essential for a full
understanding of the results obtained in the present work. This
applies in particular to Sect. 4.1.
4.1 Covariant tensors and uniformity
The flows of the n-point correlation functions are generated
from the FRG Eq. (9) by taking n-th order fluctuation field
derivatives in a background g¯, (see Fig. 1). In order to solve
the flow equation, we employ a vertex ansatz [
1, 110
]
including the flow of all relevant vertices up to the graviton
fourpoint function. This vertex ansatz disentangles the couplings
of background and fluctuation fields by introducing
individual couplings n and Gn for each n-point function. These
individual couplings are introduced at the level of the
npoint correlators and replace the cosmological constant
and Newton’s coupling G N of the classical Einstein-Hilbert
action after performing the respective field derivatives. In
summary, for the flat background g¯ = δ our vertex ansatz
reads
(φ1...φn )( p) =
k
where
n
i=1
1
Zφ2i ( pi2)
Gnn2 −1( p)T (φ1...φn )( p;
T (φ1...φn )( p;
n ) = G N SE(φH1...φn )( p;
→
n ),
denote the tensor structures extracted from the classical
gauge-fixed Einstein-Hilbert action (4). The only flowing
parameter in these tensors T (φ1...φn ) is n , while Gn ( p)
carries the global scale- and momentum dependence of the
vern ),
(15)
(16)
tex. In the above equations, p = ( pφ1 , ..., pφn ) denotes the
momenta of the external fields φi of the vertex.
Apart from their flow equations, the n-point functions in
(15) also satisfy standard RG-equations, see e.g. [
89
]. These
RG-equations entail the reparameterisation invariance of the
theory under a complete rescaling of all scales including k.
With the parameterisation given in (15), this RG-running
is completely carried by the wave function
renormalisations Zφi ( pi2) of the fields φi , see e.g. [
2, 71, 110
].
Consequently, the Gn and n are RG-invariant, and hence are more
directly related to observables such as S-matrix elements.
This parameterisation of the vertices also ensures that the
wave function renormalisations never appear directly in the
flow equations, but only via the anomalous dimensions
ηφi ( pi2) := −∂t ln Zφi ( pi2) .
Gn ( p) is the gravitational coupling of the n-point
function, while n denotes the momentum-independent part of
the correlation function. In particular, 2 is related to the
graviton mass parameter M 2 := −2 2. Finally, all the
(17)
parameters Zφi , Gn, and n are scale-dependent, but we
have dropped the subscript k in order to improve readability.
In principle, all tensor structures, including
nondiffeomorphism-invariant ones, are generated by the flow,
but for our vertex functions we choose to concentrate on the
classical Einstein-Hilbert tensor structures in the presence of
a non-vanishing cosmological constant. Despite the
restriction to these tensor structures, the n-point functions have an
overlap with higher curvature invariants via the momentum
dependence of the gravitational couplings. For example, the
complete set of invariants that span the graviton wave
function renormalisation is given by
R ,
Rμν fμ(2ν)ρσ (∇)Rρσ ,
(18)
where the superscript indicates that is a covariant tensor
contributing to the two-point correlation function. Note also that
we now drop the restriction on f present in (14). Then,
this invariant naturally includes R2 and Rμ2ν as the lowest
order local terms. If we also allow for general
momentumdependencies, the corresponding covariant functions f are
given by given by
f R(22),μνρσ = δμν δρσ PR(22)(−∇2),
1
f R(2μ2)ν ,μνρσ = 2
δμρ δνσ + δμσ δνρ PR(2μ2)ν (−∇2).
(19)
The lowest order local terms, R2 and Rμ2ν , are given by
PR(22) = 1 and PR(2μ2)ν = 1, respectively. Note that (19) also
allows for non-local terms in the IR, i.e. anomaly-driven
terms with PR(22) = 1/∇2, see e.g. [
111
]. In turn, higher
curvature invariants do not belong to the set of the graviton wave
function renormalisation since they are at least cubic in the
graviton fluctuation field.
In the present work we resort to a uniform graviton
propagator in order to limit the already large computer-algebraic
effort involved. The uniform wave function renormalisation
is then set to be that of the combinatorially dominant
tensor structure, the transverse-traceless graviton wave function
renormalisation, thereby estimating the wave function
renormalisations of the other modes by the transverse-traceless
one. Such uniform approximations have been very
successfully used in thermal field theory. There, usually the tensor
structures transverse to the heat-bath are used as the
uniform tensor structure, for a detailed discussion see e.g. [
112
]
and references therein. This approximation is typically
supported by combinatorial dominance of this tensor structure
in the flow diagrams. Indeed, as already indicated above, the
transverse-traceless mode gives the combinatorially largest
contribution to the flow of the vertices computed here. Note
that such an approximation would get further support if the R
tensor structures dominate the flows, which indeed happens
in the present computation.
Within this approximation the R2 tensor structures drop
out on the left-hand side of the graviton flow, since R is
already quadratic in the transverse-traceless graviton
fluctuation field: in other words, the tensors defined by f R(22) in (19)
have no overlap with the transverse-traceless graviton.
The set of invariants that span the gravitational coupling
G3( p) is given by
R ,
Rμν fμ(3ν)ρσ (∇)Rρσ ,
Rμν Rρσ fμ(3ν)ρσ ωζ (∇)Rωζ .
(20)
Again, the invariants R2 and R3 can be excluded from this set
due to their order in transverse-traceless graviton fluctuation
fields. In consequence, G4( p) is the only coupling in our
setup that has overlap with R2 contributions and higher terms
in f R(42).
Furthermore, in Sect. 4.3 we show that the by far dominant
contribution to G3( p) in the momentum range 0 ≤ p2 ≤ k2
stems from the invariant R. All higher momentum
dependencies of the graviton three-point function are covered by
the momentum dependence of the graviton wave function
renormalisation. This was already observed in [
3
]. As already
briefly mentioned above, it gives further support to the
current uniform approximation: the assumption of uniformity
allows us to restrict ourselves to computing the
EinsteinHilbert tensor structure for the transverse-traceless graviton
as the combinatorially dominating tensor structure. The
striking momentum-independence of the actual numerical flows
supports a momentum-independent approximation of G3. In
terms of (19) it implies that the dominant tensor structure for
the transverse-traceless mode is given by f R(3) with PR(3) = 1.
The Rμ2ν tensor structure vanishes approximately, see (27).
In contrast to the situation for the two- and three point
function, the R2 invariant overlaps with our
transversetraceless projection for the graviton four-point function.
Indeed, its flow receives significant contributions from the
invariant R2. It follows that for the graviton four-point
function R is not the only dominant invariant in the momentum
range from p = 0 to p = k, as we show in Sect. 4.3. In
consequence we either have to disentangle contributions from
R and R2 tensor structures in terms of an additional
tensor structure or we resolve the momentum dependence of
G4( p). In the present work we follow the latter procedure,
see Sect. 4.5 for details.
4.2 Projection onto n-point functions
The flow equations for the couplings n and Gn are obtained
by the following projection onto the flow of the graviton
npoint functions ∂t k(n). We use the classical Einstein-Hilbert
tensor structures T (n)( p; n) as a basis for our
projection operators. Furthermore, we project onto the spin-two
transverse-traceless part of the flow, which is numerically
dominant. Moreover, it does not depend on the gauge. This
transverse-traceless projection operator is then applied to all
external graviton legs. The flow of the couplings n is then
extracted with the help of the momentum-independent part
of said tensor structures, namely n := T (n)(0; n)/ n.
For the couplings Gn we use Gn := T (n)( p; 0)/ p2.
Dividing by n and p2 ensures that the projection operators are
dimensionless and scale-independent.
In principle, the flow of any n-point function depends on
all external momenta pi , i ∈ {1, ..., n}, where e.g. pn can be
eliminated due to momentum conservation. For the two-point
function, the momentum configuration is trivial, and only
one momentum squared, p2, needs to be taken into account.
In contrast, this dependence becomes increasingly complex
for the higher n-point functions: The three-point function
depends on three parameters (two momenta squared and one
angle), the four-point function already depends on six
parameters, and so on. To simplify the computations, we use a
maximally symmetric (n − 1)-simplex configuration for all
n-point-functions, thereby reducing the momentum
dependence to a single scalar parameter. This symmetric
momentum configuration was already used for the graviton
threepoint function in [
3
]. In the context of Yang-Mills theories,
this approximation has been shown to be in good agreement
with lattice computations on the level of the flow of the
propagator [
109
]. Notably, in the symmetric momentum
configuration all external momenta have the same absolute value p,
and the same angles between each other. The scalar product
of any two momenta in this momentum configuration then
reads
pi · p j =
nδi j − 1 2
p ,
n − 1
where δi j denotes the Kronecker delta. Note that such a
symmetric momentum configuration only exists up to the (d
+1)point function, where d is the dimension of spacetime.
In the following, the expressions Flow(n) stand for the
dimensionless right-hand sides of the flow equations divided
by appropriate powers of the wave function renormalisations.
More explicitly, we define
(21)
(22)
Flowi(n)( p2) :=
∂t i(n)( p2)
n
Zφ2 ( p2)k2−n
,
where the index i represents the projection on some
tensor structure. In this work, we use the transverse-traceless
projection operator TT, the projection operators Gn and
n mentioned earlier for the graviton n-point functions, as
well as the transverse projection operator T for the ghost
propagator. Note that the objects Flowi(n) do not contain any
(23)
(24)
(25)
(26)
explicit factors of the wave function renormalisations Zφ .
Instead, their running appears via the anomalous dimensions
ηφ .
Last but not least, we choose to model the regulator
functions Rφi on the corresponding two-point functions at
vanishing mass, i.e.
Rφi ( pi2) =
(φi φi )( pi2) mφi =0 rφi ( pi2/ k2).
Here, rφi ( pi2/ k2) denotes the regulator shape function. For
all fields in this work, we choose the Litim regulator [
113
],
to wit
r (x ) =
We now investigate the momentum dependence of the flow of
the graviton n-point functions as defined in (22). We restrict
ourselves to the momentum range 0 ≤ p2 ≤ k2 as well as to
the transverse-traceless part of the graviton n-point functions.
The first non-trivial result is that the flows of the
graviton three- and four-point functions projected on the
tensor structure of the gravitational coupling and divided by
(− n2 ηh ( p2) − n + 2) are well described by a polynomial in
p2, provided that the couplings λn are small, i.e.
Flow(G3)( p2)
2
− 23 ηh ( p2) − 1 ≈ a0 + a1 p ,
Flow(G4)( p2) 2 4
−2ηh ( p2) − 2 ≈ b0 + b1 p + b2 p ,
with some constants ai and bi that depend on the evaluation
point in theory space. This momentum dependence is
displayed in Fig. 2. We emphasise that these equations only hold
in the momentum range 0 ≤ p2 ≤ k2, if the flow is generated
by Einstein-Hilbert vertices, and if the constant parts of the
vertices are small, i.e. |λn| 1. If the condition of small λn
is violated, then the flow as in (26) is non-polynomial. We
did not compute the flow generated by an action including
higher curvature terms, however, we suspect that the flow
will still be polynomial but possibly of a higher degree.
0.6
0.4
Fig. 2 Momentum dependence of the flow of the graviton three-point
function (left) and the graviton four-point function (right) divided by
n
(− 2 ηh ( p2) − n + 2) as defined in (26). The flows are evaluated at
(μ, λ3, λ4, g3, g4) = (−0.4, 0.1, −0.1, 0.7, 0.5) and λ6 = λ5 = λ3
as well as g6 = g5 = g4. The flows have such a simple polynomial
structure as long as all couplings λn remain small, i.e. |λn | 1.
Impor
It is important to note that the graviton three- and
fourpoint functions have a different highest power in p2. This
is a second non-trivial result for the following reasons: as
already mentioned before, the coupling g3( p2) has an
overlap with R and Rμ2ν , and higher derivative terms in f R(3μ2)ν ,
but not with any R2 tensor structures in f R(32), c.f. (19). For
example, the generation of Rμ2ν with P R(3μ2)ν = 1 would
manifest itself in a p4-contribution to the flow of the graviton
three-point function. Equation (26) and Fig. 2 show that such
a p4-contribution as well as higher ones are approximately
vanishing. This demonstrates in particular that the generation
of Rμ2ν is non-trivially suppressed. In other words,
(27)
f R(3μ2)ν ≈ 0,
where the superscript indicates the three-graviton vertex.
On the other hand, the projection on g4( p2) overlaps with
R, Rμ2ν , R2 tensor structures, and the related higher
derivatives terms in f R(4μ2)ν and f R(42). It also overlaps with curvature
invariants to the third power with covariant tensors such as
f R(4μ3)ν and similar ones. Note that it has no overlap with f R(43).
Similarly to possible p4-contributions for the
threegraviton vertex, p6-contributions and even higher powers in
p2 could be generated but are non-trivially suppressed. The
p4-contribution to the flow, which is described in (26) and
displayed in Fig. 2, could stem from either R2 or Rμ2ν tensor
structures. Now we use (27). It entails that the graviton
threepoint vertex does not generate the diffeomorphism invariant
term Rμ2ν although it has an overlap with it. This excludes
Rμ2ν as a relevant UV direction, which would otherwise be
generated in all vertices. This statement only holds if we
exclude non-trivial cancellations of which we have not seen
any signature. Accordingly we set
Flow(G4)(p2) / −2ηh(p2) − 2
Trilocal: 0.60p2 − 0.24p4
0.5
p2/k2
1
tantly, the inclusion of a p4 term in the left panel offers no significant
improvement. Note that the constant parts of the functions are irrelevant
for the beta functions since they are extracted from a different tensor
projection. For p2 > k2 the momentum dependence of the flows is not
polynomial anymore
(28)
f R(4μ2)ν ≈ 0,
and conclude that this p4-contribution or at least its
UVrelevant part stems solely from R2. It may be used to
determine f R(42).
In summary, the above statements about the
momentumdependencies are highly non-trivial and show that
R2contributions are generated while Rμ2ν and other higher
derivative terms are strongly suppressed. These non-trivial
findings also allow us to determine the most efficient way to
project precisely onto the couplings of different invariants.
This is discussed in Sect. 4.5.
We close this section with a brief discussion of the effect of
higher derivative terms on perturbative renormalisability and
the potential generation of massive ghost states. As already
discussed in [
114
] in a perturbative setup, it is precisely the
Rμ2ν term which makes the theory perturbatively
renormalisable. However, in this setup it gives rise to negative norm
states. On the other hand, the R2 term neither ensures
perturbative renormalisability, nor does it generate negative norm
states. This is linked to the fact that the R2 term does not
contribute to the transverse traceless part of the graviton
propagator. Consequently, the non-trivial suppression of Rμ2ν tensor
structures might be interpreted as a hint that we do not
suffer from massive ghost states. However, a fully conclusive
investigation requires the access to the pole structure of the
graviton propagator, and hence a Wick rotation. Progress in
the direction of real-time flows in general theories and gravity
has been made e.g. in [
25, 50, 115–121
].
4.4 Higher order vertices and the background effective
action
The results in the last section immediately lead to the
question about the importance of the higher-order covariant
tensor structures like e.g. f Rn which have no overlap with the
graviton n-point functions computed in this work. These are
potentially relevant for the flows of G5 and G6. These
tensors have been dropped in the current work, thus closing our
vertex expansion. However, we may utilise previous results
obtained within the background field approximation for
estimating their importance: first we note that R2 gives rise to
a new relevant direction, as we will show in Sect. 5.1. This
has also been observed for the background field
approximation [
10,14–16,31
]. There it has also been shown that the
critical dimensions of the Rn-terms approximately follow
their canonical counting [
31
]. Furthermore, our results so far
have sustained the qualitative reliability of the background
field approximation for all but the most relevant couplings.
Indeed, it is the background field-dependence of the regulator
that dominates the deviation of the background
approximation from the full analysis for the low order vertices, and in
particular the mass parameter μ of the graviton. This
fielddependence is less relevant for the higher order terms. Thus,
we may qualitatively trust the background field
approximation for higher curvature terms. This means that they are of
sub-leading importance and can be dropped accordingly.
Finally, the above findings together with those from the
literature suggest that an Einstein-Hilbert action is
generating a diffeomorphism-invariant R2-term but not an Rμ2ν term
in the diffeomorphism-invariant background effective action
k [g] = k [g, φ = 0]. Moreover, no higher derivative terms
are generated if a non-trivial wave function renormalisation
Zh ( p2) and graviton mass parameter μ = −2λ2 are taken
into account. Note that this only applies for an expansion
with p2 < k2. This is a very interesting finding as it provides
strong non-trivial support for the semi-quantitative reliability
of the background approximation in terms of an expansion in
R for spectral values smaller than k2 subject to a resolution
of the fluctuating graviton propagator: μ and Zh have to be
determined from the flows of the fluctuation fields or in terms
of the mNIs.
4.5 Flow equations for the couplings
In this section we derive the flow equations for the couplings
from the projected n-point functions.
The flow equations for μ and ηh ( p2) are extracted from
the transverse-traceless part of the flow of the graviton
twopoint function. We evaluate this two-point function at p2 =
0 for ∂t μ, and bilocally at −μk2 and p2 for ηh ( p2). The
algebraic equation for ηc( p2) can be obtained directly from
the transverse part of the flow of the ghost two-point function.
These equations are derived in the same fashion as in [
2,71
].
For details see also App. 1.
In the case of the couplings λn and gn( p2), we project onto
the flow of the graviton n-point functions. The flow equations
for the couplings λn are always obtained at p2 = 0, since
λn describes the momentum-independent part of the graviton
n-point functions.
In the case of the couplings gn( p2) it is technically
challenging to resolve the full momentum dependence in the flow.
Thus, we resort to a further approximation of the
momentumdependence. We have checked that this approximation holds
quantitatively. First we note that typically FRG-flows are
strongly peaked at q ≈ k due to the factor q3 from the
loop integration and the decay for momenta q k due to
∂t Rk (q2). This certainly holds for all the flows considered
here. From this we can infer that we extract the leading
contribution to the flow diagrams if we feed gn(k2) back into the
diagrams. In consequence we compute only the flow
equations for gn(k2), as they form a closed system of equations
within the given approximation.
Conveniently, the momentum dependence of the flow for
g3( p2) is trivial, see Fig. 2 in Sect. 4.3. Hence the
approximation discussed above is of quantitative nature, and we obtain
precisely the same equation as in [
3
].
In contrast, the flow of the graviton four-point function
exhibits a p4 contribution, implying a non-trivial g4( p2).
Here our approximation is necessary to simplify the
computation significantly. The flow equation for g4(k2) is obtained
from a bilocal momentum projection at p2 = 0 and p2 = k2,
and furthermore uses an approximation that relies on the
fact that the coupling λ4 remains small. We refer to this
equation as a bilocal equation. It is explicitly displayed in
Appendix 1, see (E5). Within our setup this equation gives the
best approximation of the vertex flows since it feeds back the
most important momentum information into the flow. This
further entails that the coupling g4(k2) includes information
about the invariants R and R2.
4.6 Disentangling R and R2 tensor structures
In this section we present projection operators that
disentangle contributions from R and R2 tensor structures to the
flows of the couplings gn( p2). In the present setup this only
allows us to switch off the R2 coupling and thus to check the
importance of the R2 coupling.
For the disentanglement, we have to pay attention to two
things: First of all, a local momentum projection at p2 = 0
is very sensitive to small fluctuations and in consequence
not very precise with regard to the whole momentum range
0 ≤ p2 ≤ k2. This was already discussed in [
2,3
] and is
explicitly shown in Appendix 1. Hence, we have to rely on
non-local momentum projections. Here the highest
polynomial power of p2, as indicated in (26), dictates the simplest
way of projecting on the p2-coefficient. The graviton
threepoint function is at most quadratic in the external momentum,
and consequently it is enough to use a bilocal projection at
p2 = 0 and p2 = k2. The resulting equation is displayed in
Appendix 1, see (E6).
The graviton four-point function, on the other hand, has
p4 as its highest momentum power, i.e. it is of the form
f ( p2) = b0 + b1 p2 + b2 p4,
see also (26). Thus a bilocal momentum projection would not
extract the p2 coefficient b1 alone. Instead, we use a trilocal
momentum projection at p2 = 0, p2 = k2/2, and p2 = k2
in order to solve the above equation for b1. I.e., we solve a
system of linear equations and obtain
b1 = −3 f (0) + 4 f (k2/2) − f (k2).
The resulting flow equation is again presented in Appendix 1,
see (E7).
For even higher order momentum contributions we would
have to use even more points of evaluation. These momentum
projections together with the observation of (26) guarantee
that we project precisely on the p2 coefficient in the whole
momentum range 0 ≤ p2 ≤ k2.
A natural upgrade of the current approximations amounts
to the introduction of a second tensor structure that is
orthogonal to the Einstein-Hilbert one in terms of these projections.
Within our uniformity assumption this is considered to be
sub-leading, and the momentum-dependence of g4( p2) takes
care of the contribution of the R2 tensor structure f R(42). While
the orthogonal projection on the respective flow is simple, its
back-feeding demands a two tensor structure approximation
of the three- and four-graviton vertex in the flow, the
implementation of which is deferred to future work.
Here, we only perform a further check of the relevance of
the R2 tensor structure. This sustains the fact that the
inclusion of the four-graviton vertex with its contribution of the R2
tensor structure leads to an additional UV-relevant direction.
To that end we generalise our ansatz for the graviton
fourpoint function such that we can extract a flow equation for
both the Einstein-Hilbert tensor structure as well as for the
R2 tensor structure. As already mentioned above, we cannot
feed the generated coupling back into the flows, since they
are given by vertices with Einstein-Hilbert tensor structures.
Instead we compute the fixed point value that arises only
from the Einstein-Hilbert tensor structures.
As the corresponding ansatz for the transverse-traceless
graviton four-point function we choose
k(4)( p2) = Zh2( p2)G4 C G4
4 4 + C Gp24 p2 + CωG44 4 p4 ,
(31)
which is precisely the vertex that emerges from the sum of
Einstein-Hilbert tensor structure and R2 tensor structure. The
related generating diffeomorphism-invariant action for this
(32)
(33)
four-graviton vertex is
1
S = SEH + 16π G N
d4x
det g
R2,
where SEH is defined as in (4). The flow of 4 is then obtained
by the trilocal momentum projection described below (29).
For b2 we obtain
b2 = 2 f (0) − 4 f (k2/2) + 2 f (k2).
The explicit form of the resulting flow equation for the
dimensionless coupling ω4 := 4k2 is given in Appendix 1, see
(E8). Note that in the present approximation, the flows do not
depend on the coupling ω4 since it does not feed back into
the vertices.
4.7 Computational details
The computations of correlation functions described in this
section involve contractions of very large tensor structures.
To give a rough estimate: the classical Einstein-Hilbert
threepoint vertex alone consists of around 200 terms, and the
classical graviton propagator of 7 terms. For the box diagram
of the flow of the graviton four-point function, displayed
in Fig. 1, this results in a total number of approximately
2004 · 74 ≈ 4 · 1012 terms, if no intermediate simplifications
are applied.
These contractions are computed with the help of the
symbolic manipulation systems FORM [
122,123
] and
Mathematica [124]. For individual tasks, we employ specialised and in
part self-developed Mathematica packages. In particular, we
use VertEXpand [
125
] and xPert [
126
] for the generation of
vertex functions, DoFun [
127
] to obtain symbolic flow
equations, and the FormTracer [
103,109,128
] to create optimised
FORM scripts to trace diagrams. Furthermore, we make use
of the self-developed framework TARDIS [
125
] facilitating an
automated and seamless usage of the aforementioned tools.
5 Asymptotic safety
In this section, we discuss the UV fixed point structure of our
system. We first present our best result, which includes the
tensor structures as presented in Sect. 4.1 and in particular in
(18) and (20). The underlying UV-relevant diffeomorphism
invariants turn out to be , R, and R2. The R2 coupling is
included via the momentum dependence of the gravitational
coupling g4( p2), see Sect. 4.5. As a main result we find an
attractive UV fixed point with three attractive directions. The
third attractive direction is related to the inclusion of the R2
coupling.
We further analyse the stability of this UV fixed point with
respect to the identification of the higher couplings. We also
analyse the previous truncation [
3
] and compare the stability
of both truncations. Here we find that the improvement of the
truncation increases the stability of the system. In particular,
we find a rather large area in the theory space of higher
couplings where the UV fixed point exists with three attractive
directions throughout.
Lastly, we discuss the importance of the R2 coupling. In
Sect. 4.6 we have constructed projection operators that
disentangle the contributions from R and R2 tensor structures.
This allows us to switch off the R2 coupling and compare
the stability of the reduced system to that of the full system.
We find that the reduced system is significantly less stable,
and that the area in the theory space of higher couplings
where the fixed point exists is rather small. This highlights
the importance of the R2 coupling.
5.1 UV fixed point
In this section we display the UV fixed point structure of our
full system. This means that we feed back the generated R2
coupling via the momentum dependence of the gravitational
coupling g4( p2), as discussed in Sect. 4.5. Fixed points are
by definition points where the flows of the dimensionless
couplings vanish. In consequence, we look for the roots of
the Eqs. (E1), (E4), (E6), and (E5). We use the identification
scheme g6 = g5 = g4 and λ6 = λ5 = λ3. We find a UV
fixed point at the values
μ∗, λ3∗, λ4∗, g3∗, g4∗ = (−0.45, 0.12, 0.028, 0.83, 0.57) .
(34)
The fixed point values are similar to those of the previous
truncation [
3
]. The biggest change concerns the graviton
mass parameter, which is now less negative and thus
further away from its pole. Moreover, it is remarkable that the
new couplings λ4 and g4 are close to their lower counterparts
λ3 and g3, but not at precisely the same values. Since we use
the difference between these couplings to parameterise the
breaking of diffeomorphism invariance, this is more or less
what we expected. This issue is further discussed in the next
section.
We do not have access to the full stability matrix of the UV
fixed point due to the unknown flow equations of the higher
couplings. For this reason, we discuss two different
approximations of the stability matrix. The main difference between
these two approximations concerns the order of taking the
derivatives and identifying the higher couplings, which is
explained in more detail in Appendix 1. We argue that in
a well converged approximation scheme the most relevant
critical exponents should not depend on the approximation
of the stability matrix. Thus, we can use the two different
approximations to judge the quality of the current level of
truncation. In this work, we define the critical exponents as
the eigenvalues of the stability matrix without a minus sign.
We call the critical exponents of the first approximation θ¯i ,
and the ones of the second approximation θ˜i . The critical
exponents using the first approximation are given by
θ¯i = (−4.7, −2.0 ± 3.1i, 2.9, 8.0),
(35)
while the critical exponents using the second approximation
are
θ˜i = (−5.0, −0.37 ± 2.4i, 5.6, 7.9).
(36)
Hence this fixed point has three attractive directions in both
approximations of the stability matrix. The third attractive
direction compared to the system of the graviton three-point
function [
3
] is related to the fact that the graviton four-point
function has an overlap with R2, which we feed back via the
momentum dependence of the gravitational coupling g4( p2).
The R2 coupling has also been relevant in earlier
computations with the background field approximation [
10,14–
16,31
]. In addition, note that the most attractive eigenvalue
is almost identical in both approximations of the stability
matrix. This is a positive sign towards convergence since it is
expected that the lowest eigenvalue is the first that converges,
c.f. Appendix 1.
Furthermore, the anomalous dimensions at the UV fixed
point read
(37)
(ηh∗(0), ηh∗(k2)) = (0.56, 0.079) ,
(ηc∗(0), ηc∗(k2)) = (−1.28, −1.53) ,
where we have chosen to display the anomalous dimensions
at the momenta that feed back into the flow. All anomalous
dimensions stay well below the reliability bound ηφi ( p2) <
2, as introduced in [
71
].
5.2 Stability
In the following we investigate the UV fixed point from the
previous section by varying the identification of the higher
couplings. Again we look for the roots of the Eqs. (E1), (E4),
(E6), and (E5). These equations however still depend on the
higher couplings g5, g6, λ5, and λ6. We have to identify these
couplings with the lower ones or set them to constants in order
to close the flow equations.
It is a natural choice to simply set these higher couplings
equal to lower ones, e.g. g6 = g5 = g3 and λ6 = λ5 = λ3,
as done in the previous section. The couplings would
fulfil this relation exactly in a fully diffeomorphism invariant
setup. However, such a diffeomorphism invariant setup is
not at hand. In fact, we can parameterise the breaking of
diffeomorphism invariance via these couplings, e.g. by writing
gn = g3 + gn . Here we have designated g3 as a reference
coupling since it is the lowest genuine gravitational coupling.
For this reason, it is also the most converged gravitational
coupling within this vertex expansion, thus justifying this
choice. In general we expect gn to be small and in
consequence we vary the identification of the higher couplings
only in this part of the theory space of higher couplings. The
quantity g4 is indeed small at the UV fixed point presented
in the last section, see (34). More precisely, it takes the value
| g4 /g3| ≈ 0.3 at this UV fixed point.
In this analysis we choose to identify
g5 = α1 g3,
g6 = α2 g3,
(38)
and λ6 = λ5 = λ3 for simplicity, and investigate the
existence of the UV fixed point as a function of the parameters α1
and α2. In Fig. 3 the area where an attractive UV fixed point
exists is displayed in blue. In the left panel, this is done for
the previous truncation (μ, λ3, g3) [
3
], and in the right panel
for the current truncation (μ, λ3, λ4, g3, g4). At the border
of the blue area the UV fixed point either vanishes into the
complex plane or loses its attractiveness. Remarkably, both
areas are rather large, suggesting that the existence of the
UV fixed point is quite stable. Even more conveniently, the
area increases with the improved truncation, suggesting that
the system is heading towards a converging limit. Note that
the number of attractive directions of the UV fixed point is
or loses its attractiveness. In both systems the area where the fixed point
exists is rather large and contains the identification gn>nmax = g3.
Conveniently, the area increases for the better truncation, indicating that the
system becomes more stable with an improvement of the truncation.
The number of attractive directions is uniformly two in the left panel
and three in the right panel
constant throughout the blue areas, namely two in the left
panel and three in the right panel.
We further analyse the fixed point values that occur within
the blue area in the right panel of Fig. 3. Interestingly, the
fixed point values are rather stable throughout the whole area
where the UV fixed point exists. More precisely, they stay
within the following intervals:
Hence, in particular the fixed point value of λ3 is already
confined to a very small interval, and also a very small number.
The latter is important since some of our approximations rely
on the fact that the λn are small, see Sect. 4.5. The fact that
λ4∗ is varying more strongly than λ3∗ is not surprising since
we expect λ3 to be better converged, being a lower coupling.
The fixed point values of g3 and g4 seem to try to
compensate the change induced by the identification. Thus, g3∗ and
g4∗ become larger towards the identification g6 = g5 = 0 and
smaller towards g6 = g5 = 2g3. The shape of the area in the
left panel in particular suggests the relation g4∗ < g3∗, which
is fulfilled by the improved truncation almost throughout the
whole area where the fixed point exists. This is indeed a
nontrivial prediction that has been fulfilled by our approximation
scheme.
Fig. 4 Plot of the existence of an attractive non-trivial UV fixed point
(blue) dependent on the higher couplings g5 and g6. Here, the trilocal
equation for the gravitational coupling g4 was used, which allows us to
switch off the R2 coupling. We found two different fixed points with
rather similar fixed point values. Each fixed point has its own area of
existence in the theory space of the higher couplings. The blue area
marks the unified area of both fixed points. Nevertheless, the area is
significantly smaller than the areas displayed in Fig. 3. This reflects the
importance of the R2 coupling
A further study of the dependence of the UV fixed
point properties on the choice of identification is given in
Appendix 1.
5.3 Importance of the R2 tensor structure
In the previous subsection we have fed back the R2
contributions to the flow via the momentum-dependent gravitational
coupling g4( p2). In order to check the quality of our
approximation and to investigate the influence of the R2 tensor
structure on the fixed point structure of the system, we switch off
the R2 contribution in this section. We do the latter by
projecting onto the p2 part of the flow via a trilocal momentum
projection scheme, cf. Sects. 4.3 and 4.6. This is both an
examination of the influence of R2 on the results presented
in the previous subsections, as well as a proof of concept
for disentangling the tensor structures of different invariants.
Our analysis in this subsection suggests that leaving out the
contribution of R2 leads to significantly less stable results.
In Fig. 4 we display the result for the same analysis as
in the previous section, but with the trilocal equation (E7)
for g4 instead. We find two fixed points with rather similar
fixed point values. However, we are only interested in
identifying the area in the theory space of the higher couplings
where at least one UV fixed point exits. Thus, we unify both
areas and obtain the blue area displayed in Fig. 4. This area
forms a rather narrow band whose total area is significantly
smaller than for the momentum dependent gravitational
coupling g4( p2), c.f. Fig. 3. The identification g6 = g5 = g3
also does not lie within these regions, but just outside of them.
Since we switched off the R2 contribution, a less stable fixed
point structure was to be expected, and consequently these
results highlight the importance of the R2 coupling.
6 IR behaviour
In this section, we discuss the IR behaviour of the present
theory of quantum gravity. We only consider trajectories that
lie within the UV critical hypersurface, i.e. trajectories that
are UV finite, and which end at the UV fixed point presented
in (34) for k → ∞. In this section we use the analytic flow
equations given in Appendix 1 for simplicity, and set the
anomalous dimensions to zero, i.e. ηφ = 0. This
approximation gives qualitatively similar results, as discussed in
Appendix 1.
In the IR, it is particularly interesting to examine the
background couplings g¯ and λ¯. In the limit k → 0 the regulator
vanishes by construction and the diffeomorphism invariance
of the background couplings is restored. Hence they become
observables of the theory. The flow equations for the
background couplings are displayed in Appendix 1.
In general we look for trajectories that correspond to
classical general relativity in the IR. This implies that the
quantum contributions to the background couplings vanish and
in consequence that they scale classically according to their
mass dimension. The classical scaling is described by
g¯, g3, g4 ∼ k2,
λ¯, μ, λ3, λ4 ∼ k−2 .
(39)
We use the classical scaling in the flow from the UV fixed
point to the IR in order to set the scale k in units of the Planck
mass MPl. We need to find a large enough regime where
g¯ ∼ k−2. This entails that Newton’s coupling is a constant
in this regime and sets the scale k via G N = MP−l2 = g¯k−2.
In Fig. 5, two exemplary trajectories are displayed. In the
left panel all couplings scale classically below the Planck
scale and reach their UV fixed point values shortly above
the Planck scale. All quantum contributions are suppressed
simply by the fact that μ → ∞. In the right panel on the other
hand some couplings exhibit a non-classical behaviour even
below the Planck scale, which is triggered by the graviton
mass parameter μ flowing towards the pole of the graviton
propagator at μ = −1. This entails that the dimensionful
graviton mass parameter M 2 = μk2 is vanishing in the IR.
This IR behaviour is analogous to the one observed in [
2
],
and recently also [
50
]. Remarkably, not only μ is behaving
non-classically but also λ3, even though it is not restricted
by any pole. However, in this scenario the numerics break
102
100
10−2
Fig. 5 Examples of UV finite trajectories from the UV fixed point
(34) towards the IR. In the left panel all couplings scale classically
below the Planck scale and reach their UV fixed point values shortly
above the Planck scale. In the right panel some couplings show
nonclassical behaviour even below the Planck scale, which is triggered by
the graviton mass parameter μ flowing towards the pole of the graviton
propagator at μ = −1. However, in this case the numerics break down
at k ≈ 0.02MPl due to competing orders of the factor (1 + μ) close to
the singularity at μ = −1. The trajectories in both panels correspond
to theories that behave like classical general relativity in the IR. Note
that some couplings are plotted shifted or with a minus sign in order to
keep them positive over the whole range
Table 1 Properties of the non-trivial UV fixed point for different orders
of the vertex expansion scheme, computed for momentum dependent
anomalous dimensions ηφi ( p2) and bilocally projected Newton’s
couplings gn (k2). The critical exponents θ¯i and θ˜i stem from two different
approximation of the stability matrix as discussed in Appendix 1. The
fixed points are computed with the identifications g6 = g5 = gmax and
λ6 = λ5 = λ3. We observe that the fixed point values are only varying
mildly between the different orders of the vertex expansion. Notably,
if we compare the critical exponents of the two approximations of the
stability matrix, we observe that the difference becomes smaller with
an increasing order of the vertex expansion. This is precisely what one
would expect of a systematic approximation scheme that is approaching
a converging limit
System
μ, g3, λ3
μ, g3, λ3, g4
μ, g3, λ3, λ4
μ, g3, λ3, g4, λ4
μ∗
− 0.57
− 0.53
− 0.58
− 0.45
λ∗
3
down at k ≈ 0.02MPl due to competing orders of the factor
(1 + μ) close to the singularity at μ = −1.
In the left panel we have tuned the background couplings
g¯ and λ¯ so that they are equal to the lowest corresponding
fluctuation coupling in the IR, i.e. g¯ = g3 and λ¯ = λ2 =
−μ/2 for k MPl. This is equivalent to solving a trivial
version of the Nielsen identities (NIs). Since all quantum
contributions are suppressed by the graviton mass parameter
going to infinity in the IR, μ → ∞, the NI in (2) reduces to
δ [g¯, h]
δg¯
=
δ [g¯, h]
δh
for μ → ∞ & k → 0 .
In consequence, we should see that all couplings coincide
in this limit, g¯ = gn and λ¯ = λn . This is not the case in
the left panel of Fig. 5 since we have only fine-tuned the
background couplings, and thus we have two further degrees
of freedom that could be used for fine-tuning, stemming from
the three dimensional UV critical hypersurface. We postpone
this fine-tuning problem to future work.
In summary, we find different types of trajectories that
correspond to classical general relativity in the IR. The main
difference lies in the behaviour of the graviton mass
parameter μ, which flows to infinity in one case and to minus one
in the other case. Both scenarios are equivalent to general
relativity in the end, in particular since only the background
couplings become observables in the limit k → 0.
7 Towards apparent convergence
In this section we discuss and summarise the findings of this
work concerning apparent convergence. On the one hand,
the order of our vertex expansion is not yet high enough
to fully judge whether the system approaches a converging
limit. Nevertheless, we have collected several promising first
hints that we want to present in the following.
In this work we have introduced two different
approximations to the stability matrix, as presented in Appendix 1. We
have argued that in a well converged approximation scheme
the most relevant critical exponents should not depend on the
approximation of the stability matrix. In Table 1 we display
the UV fixed point properties for different orders of the
vertex expansion. The first system is without the graviton
fourpoint function and exactly the same as in [
3
]. Then we look
at systems where we add either only an equation for g4(k2)
(c.f. (E5)), or only an equation for λ4 (c.f. (E4)). Lastly, we
display our best truncation including all couplings up to the
graviton four-point function, see Sect. 5.1. We observe that
the fixed point values of the couplings vary only mildly with
an improving truncation, although there is no clear pattern
to those variations. The most important piece of
information is the difference between the critical exponents from the
two different approximations of the stability matrix. While
the difference is rather large in the truncation of the
graviton three-point function, it is becoming smaller with each
improvement of the truncation. At the level of the graviton
four-point function, the critical exponents show only a small
difference. This is precisely what we expect, and thus we
interpret this as a sign that the system is approaching a
converging limit.
Another important piece of information comes from the
stability of the UV fixed point under different closures of
the flow equation. In a well converged expansion scheme,
the properties of the UV fixed point should be completely
insensitive to the details of the closure of the flow equation.
We have performed this analysis in Sect. 5.2. We observed
that the area in which the UV fixed point exists in the
theory space of higher couplings is indeed increasing with the
improvement of the truncation. Furthermore, we saw that the
UV fixed point values are confined to small intervals. We
again interpret this as a sign that the system is approaching
a converging limit.
In summary, we have already seen several signatures of
apparent convergence although we are only at the level of the
graviton four-point function within the present systematic
expansion scheme. This suggests that we are on a
promising path and that the present setup will eventually lead to a
converging limit.
8 Summary
We have investigated quantum gravity with a vertex
expansion and included propagator and vertex flows up to the
graviton four-point function. The setup properly
disentangles background and fluctuation fields and, for the first time,
allows to compare two genuine Newton’s couplings
stemming from different vertex flows. Moreover, with the current
truncation we have closed the flow of the graviton
propagator: all vertices and propagators involved are computed from
their own flows.
As a first non-trivial result we have observed that the
vertex flows of the graviton three-point and four-point functions,
in the sense of (26), are well described by a polynomial in p2
within the whole momentum range 0 ≤ p2 ≤ k2. The
projection used for the flows takes into account the R, R2 and
Rμ2ν tensor structures as well as higher order invariants with
covariant momentum dependencies. Importantly, it is
orthogonal to the R2 tensor structure for the graviton three-point
function, but includes it for the graviton four-point function.
We have shown that the highest momentum power
contributing to the graviton three-point function is p2. Therefore, Rμ2ν
and higher derivative terms do not contribute to the graviton
three-point function. Thus, in particular Rμ2ν is excluded as
a UV-relevant direction. On the other hand, the flow of the
graviton four-point function shows p4 as its highest
momentum power. Together with the three-point function result we
infer that R2 is UV-relevant and contributes to the graviton
four-point function. This is a very interesting and highly
nontrivial result.
At the moment, we cannot make final statements about
higher Rn terms directly from our analysis. Nonetheless,
predictions can be made with a combination of the results
presented here and previous ones obtained within the
background field approximation as well as the vertex expansion:
Firstly, our work sustains the qualitative reliability of
background field or mixed approximations for all but the most
relevant couplings. We have seen that the range of allowed
Newton’s couplings stemming from n-graviton vertices is
growing with the level of the approximation. Moreover, in
[
71, 74
] it has been shown that already the substitution of
the most relevant operator, the mass parameter μ, in a mixed
computation with that in the full vertex expansion stabilises
the results in a particular matter gravity system. Hence, this
gives us some trust in the qualitative results for higher Rn
terms in the background field approximation. In [31] the
f ( R)- potential has been computed polynomially up to R34,
and the relevance of these operators follows the
perturbative counting closely. Accordingly it is quite probable that
the higher Rn will turn out to be irrelevant in the full vertex
expansion as well.
Based on the above observations we have also constructed
projection operators that properly disentangle the
contributions of different diffeomorphism-invariant tensor structures.
This allowed us to switch off the R2 coupling in order to
analyse its importance for the system. In this case, we are led to an
unstable system, which highlights the importance of the R2
coupling for the asymptotic safety scenario. In the present
work we include the R2 contributions via the momentum
dependence of the gravitational coupling g4( p2), leading to
a very stable system in the UV.
In the full system with R2 contributions we found an
attractive UV fixed point with three attractive directions and
two repulsive directions. The third attractive direction can be
explained due to the overlap with R2, and is in agreement
with previous R2 studies in the background field
approximation [
10, 14–16, 31
]. We investigated the stability of this
UV fixed point with respect to changes of the identification
of the higher couplings and compared it to the stability of
the previous truncation without the graviton four-point
function. We characterised the stability via the area of existence
in the theory space of higher couplings, and remarkably this
area increased with the improved truncation. We interpret
this as a sign that the systematic approximation scheme is
approaching a converging limit.
Furthermore, we investigated the IR behaviour and found
trajectories that connect the UV fixed point with classical
general relativity. In particular, we found two different types
of such trajectories. In the first category all couplings,
including background and fluctuation couplings, scale classically
according to their mass dimension below the Planck scale.
In consequence the Nielsen identities become trivial in this
regime and we can solve them in the IR. In the second
category, the graviton mass parameter and the coupling λ3 scale
non-classically below the Planck scale, which is triggered by
the graviton mass parameter flowing towards the pole of the
graviton propagator μ → −1. In summary, the IR behaviour
was found to be very similar to [
2
], and recently also [
50
].
Lastly, we discussed signs of apparent convergence in the
present system by comparing the results to previous
truncations. As mentioned before, we observed that the present
system is more stable and less sensitive to the closure of the
flow equation, which is expected from a converging system.
We furthermore used two different approximations of the
stability matrix and argued that the critical exponents belonging
to the most attractive directions should not differ in a well
converged expansion. Indeed we found that the difference of
the critical exponents is decreasing with an improvement of
the truncation. We interpret this as a sign towards
convergence.
In the present approximation we have taken the , R
and R2 tensor structures into account. Furthermore, we have
shown that the higher derivative tensor structures and the
Rμ2ν tensor structure are suppressed. There are also very
strong indications for the irrelevance of the higher orders
in Rn. Altogether this suggests that the natural extension
of the present work towards apparent convergence consists
primarily of the inclusion of all tensor structures of the
vertices and propagators on the given level n = 4 of the vertex
expansion. In particular, this concerns the inclusion of the R2
tensor structure with the orthogonal projection devised in the
present work. Moreover, the different graviton modes should
be furnished with their separate dispersion or wave function
renormalisation. This is well in reach with the current
technical status of the programming code and its implementation.
Then, selected tensor structures of higher vertices could be
used for further tests of apparent convergence. We hope to
report on this in the near future.
Acknowledgements We thank N. Christiansen, A. Eichhorn, K. Falls,
S. Lippoldt and A. Rodigast for discussions. MR acknowledges funding
from IMPRS-PTFS. This work is supported by EMMI and by
ERCAdG-290623.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Approximations of the stability matrix
The stability matrix B is defined as the Jacobi matrix of
the flow equations for all couplings αi . Mathematically, it is
given by
Bi j := ∂α j ∂t αi .
(A1)
In this work, the critical exponents of a fixed point are defined
as the eigenvalues of the stability matrix evaluated at this
fixed point. In our setup, the stability matrix is infinite
dimensional since it is spanned by all couplings λn and gn( p2). Note
that one momentum dependent coupling alone would already
be enough to render the stability matrix infinite dimensional.
In the present work, only couplings up to order six appear
in the flow. We furthermore do not resolve the full
momentum dependence of the couplings. Thus, we have already
rendered the stability matrix finite. Nevertheless, the full
stability matrix is not known since the flows of the fifth- and
sixth order couplings are unknown and would depend on
further higher couplings anyway. In consequence, we have
to make an approximation of the stability matrix in order
to obtain the critical exponents. Note that we also have to
make an approximation of the flow itself in order to close it.
Naturally, these approximations are related.
In the following we present two different approximations
of the stability matrix. We further argue that these
approximations should give approximately the same values for the
most relevant critical exponents if the expansion scheme is
already well converged.
The approximation of the flow is related to its closure
and describes how the higher couplings are identified with
the lower ones. In the following, we call this process
identification scheme and denote it by |id.. The two different
approximations of the stability matrix are distinguished by
the sequence of taking the derivatives and applying this
identification scheme. In the first approximation, the
identification is performed before taking the derivatives:
B¯i j := ∂α j (∂t αi |id.).
(A2)
The critical exponents that correspond to this approximation
represent the critical exponents that belong to the computed
phase diagram of the theory.
In the second approximation, the identification is
performed after taking the derivatives, to wit
(A3)
Bi j := (∂α j ∂t αi )|id. .
This approximation is more closely related to the full stability
matrix in the sense that it respects the fact that the higher
couplings in the full system do not coincide with the lower
ones. Note that these two different approximations only differ
if we choose a non-trivial identification scheme, i.e. if the
higher couplings are functions of the lower ones.
If the expansion scheme is well converged, then the
contributions of the higher couplings to the flow of the lower
couplings are small, e.g. (∂λnmax+2 λnmax )|FP ≈ 0. In this case,
the most relevant eigenvalues of the stability matrices B¯ and
B coincide approximately. The stabilisation of the most
relevant eigenvalues was also observed in an expansion in Rn
in [
31
]. In consequence, a huge deviation in the most relevant
eigenvalues of both approximations would clearly indicate a
lack of convergence. For this reason, we use the comparison
of the different approximations as a first check of
convergence.
Appendix B: Background couplings
In this section we present the flow equations for the
background couplings g¯ and λ¯. They are in particular interesting
in the limit k → 0 where they become observables. In this
limit, the regulator term vanishes by construction and
diffeomorphism invariance is restored, which implies that these
couplings can be interpreted as physical observables only for
vanishing k.
For notational convenience we reintroduce the coupling
λ2 = −μ/2. Following [
44
], we compute the flow of the
background couplings with a curvature expansion on an
Einstein space. We use a York-decomposition [
129,130
] and
field redefinitions [
9,57
] to cancel the non-trivial Jacobians.
The resulting flow equations are given by
∂t g¯ = 2g¯ − g¯2 f R1 (λ2; ηφ ) ,
∂t λ¯ = −4 λ¯ + λ¯ ∂gtg¯ + g¯ f R0 (λ2; ηφ ),
¯
where the functions f R0 and f R1 read
1
f R0 (λ2; ηφ ) = 24π
×
(10 − 8λ2)(6 − ηh (k2))
1 − 2λ2
− 8(6 − ηc(k2)) ,
(B1)
(B2)
(B3)
1
f R1 (λ2; ηφ ) = 24π
×
,
Note that the background couplings are non-dynamical, i.e.
they do not influence any other coupling. Furthermore, the
background couplings only depend on the couplings of the
two-point function. Hence only the graviton mass parameter
μ (or equivalently λ2) and the anomalous dimensions ηh and
ηc directly affect them.
Appendix C: Dependence on the identification scheme
The flow of each n-point function depends on the couplings
of the (n + 1)-point function and the (n + 2)-point function,
see Fig. 1. For the highest couplings, we consequently do
not have a flow equation at hand. In our setup, these are the
couplings of the five- and six-point function, i.e. λ5, λ6, g5,
and g6. In order to close the flow of our system, we need to
make an ansatz for these higher order couplings. A natural
choice is one that is close to diffeomorphism invariance, i.e.
to identify these couplings with a lower order coupling.
In our setup, there are two lower order couplings that
correspond to a (partly) diffeomorphism invariant identification
scheme, e.g. λ5 can be identified with λ3 or λ4. In a well
converged expansion scheme, the details of the identification
should not matter and lead to similar results. In this section,
we compare the results for different identification schemes
in order to evaluate the stability of our expansion scheme.
The properties of the non-trivial UV fixed point for
different identifications schemes are displayed in Table 2. In all
identification schemes except for the identification gn>4 →
g4 and λn>4 → λ4 we find an attractive UV fixed point. In
this case we can see that the fixed point has just vanished in
the complex plane. For all other identifications we observe
that the fixed point values and the critical exponents vary
only mildly. Especially the number of attractive directions is
consistently three with the first approximation to the stability
matrix, c.f. Appendix 1. With the second approximation the
Table 2 Properties of the non-trivial UV fixed point for different
identification schemes, i.e. different closures of the flow equations, as
discussed in Appendix 1. The flow equations are computed with
momentum dependent anomalous dimensions ηφi and bilocally projected
Newton’s couplings gn (k2). The critical exponents θ¯i and θ˜i stem from
two different approximation of the stability matrix as discussed in
Appendix 1. An attractive UV fixed point is found in most identification
schemes with mildly varying fixed point values. In the first
approximation of the stability matrix we always find three attractive directions,
while in the second approximation of the stability matrix we find one
or three attractive directions, since the real part of one complex pair of
eigenvalues is quite close to zero. These results suggest that the present
system is rather stable under change of the closure of the flow equations.
In the case of the single identification without a physical UV fixed point
we found that it had in fact just vanished in the complex plane
Identification scheme
gn>4 → g3, λn>4 → λ3
gn>4 → g4, λn>4 → λ3
gn>4 → g4, λn>4 → λ4
μ∗
λ∗
3
λ∗
4
g∗
3
g∗
4
number of attractive directions varies from one to three since
the real part of one complex conjugated pair is close to zero.
In conclusion, this analysis suggests that our system is
rather stable with respect to different identification schemes.
Only one particular identification scheme has led to the
disappearance of the attractive UV fixed point. This constitutes
further support for our results in Sect. 5.2, where we found
that our full system is very stable with respect to the
identification of g5 and g6.
Appendix D: Possible issues of a local momentum projection
In this section we want to point out some possible issues of
a local momentum projection. A local momentum
projection is for example a derivative expansion about a certain
momentum, usually p = 0.
The full solution of a flow equation includes a full
resolution of the momentum dependence of all vertex flows.
For higher n-point functions this task is computationally
extremely challenging due to the high number of momentum
variables. We have already argued in Sect. 4.2 that this task
can be tremendously simplified with a symmetric
momentum configuration. We have further shown in Sect. 4.3 that the
quantity Flow(Gn)/(− n2 ηh ( p2) − n + 2) is polynomial in p2, at
least for n = 3, 4. Thus it is possible to consistently project on
each coefficient of this polynomial in the whole momentum
range 0 ≤ p2 ≤ k2 by employing a non-local momentum
projection. In contrast, a local momentum projection scheme
does not capture the correct momentum dependence over the
whole momentum range 0 ≤ p2 ≤ k2 in general since it
is sensitive to local momentum fluctuations. Furthermore, it
is very challenging to project on the p4 coefficient or even
higher momentum order coefficients due to IR singularities.
All these statements are explicitly exemplified in Fig. 6.
0.6
0.4
0.2
0
0
Flow(G3)(p2) / − 23 ηh(p2) − 1
Bilocal: −0.57p2
Projection at p2 = 0: −0.80p2
Fig. 6 Fit of a local momentum projection at p2 = 0 to the momentum
dependence of the flow of the graviton three-point function (left) and
the graviton four-point function (right) divided by (− n2 ηh ( p2) − n + 2)
as defined in (26). The flows are evaluated at (μ, λ3, λ4, g3, g4) =
(−0.4, 0.1, −0.1, 0.7, 0.5) and λ6 = λ5 = λ3 as well as g6 = g5 = g4,
i.e. the same values as in Fig. 2 where a non-local momentum
projection was used. In comparison to the non-local momentum projection,
Flow(G4)(p2) / −2ηh(p2) − 2
Trilocal: 0.60p2 − 0.24p4
Projection at p2 = 0: 0.66p2
0.8
the local momentum projection does not capture the correct
momentum dependence in the whole momentum range 0 ≤ p2 ≤ k2 since it
is sensitive to local momentum fluctuations. Furthermore, it is
technically very challenging to project on the p4-term due to IR singularities.
Note again that the constant parts of the flows are irrelevant for the beta
functions since they are extracted from a different tensor projection
On the other hand, the local momentum projection at
p = 0 has the advantage that it allows for analytic flow
equations, as discussed in Appendix 1. Analytic flow equations
are more easily evaluated in the whole theory space, but, as
the discussion above suggests, one should be be mindful of
the fact that they easily introduce a large error.
We use the analytic flow equations in Sect. 6 precisely
for the reason that they can easily be evaluated in the whole
theory space. Thus we show now that the fixed point
properties in this analytic system are qualitatively similar to the full
system, despite the error that is introduced by the analytic
equations.
The properties of the UV fixed point for different
approximations are displayed in Table 3. Truncation 1 corresponds
to our full system, i.e. with momentum dependent anomalous
dimensions and bilocally evaluated gravitational couplings.
Truncation 4 corresponds to the system used in Sect. 6, i.e.
without anomalous dimensions and with gravitational
couplings from a derivative expansion. Truncation 2 and 3 are
in between those truncations, i.e. with anomalous dimension
but gravitational couplings from a derivative expansion and
without anomalous dimensions but with bilocally evaluated
gravitational couplings, respectively.
We observe that the UV fixed point exists in all
truncations, and that the properties of this fixed point vary only
mildly. The fixed point values are all located within a small
region, with the exception of our simplest truncation. There,
the couplings λ3 and λ4 have a different sign compared to
the other truncations. Considering the critical exponents we
always find three attractive directions with the first
approximation of the stability matrix and in three out of four cases
three attractive directions with the second approximation of
the stability matrix as well. These results suggest that it is an
acceptable approximation to use the analytic flow equations
if one is only interested in the qualitative behaviour of the
system.
Appendix E: Derivation of flow equations
We obtain the flow equations for the individual coupling
constants by projecting onto the flow of the graviton n-point
functions, as explained in Sect. 4.5.
The equations for the graviton mass parameter μ and for
the graviton anomalous dimension ηh are extracted from the
transverse-traceless part of the flow of the graviton two-point
function. For p2 = 0, we obtain
∂t μ = (ηh (0) − 2) μ +
for the flow of the graviton mass parameter.
(E1)
where the projection dependent constant C n is implicitly
defined via n ◦ TnT ◦ T (n)(0; 1) = C n . Here, ◦ denotes
the pairwise contraction of indices.
As discussed in Sect. 4.5, the gravitational couplings
gn( p2) are momentum dependent. In order to simplify the
computation we make an approximation of the full
momentum dependence. This approximation exploits the fact that
the flows are peaked at p2 = k2 and consequently we set
the feed back on the right-hand side of the flow equation to
gn( p2) ≈ gn(k2). This closes the flow equation for gn(k2)
and thus we only solve this equation. The easiest way to
obtain the flow equation for gn(k2) is a bilocal projection
at p = 0 and p = k. The resulting equation for g3(k2) is
precisely the same as in [
3
]. For g4(k2) we obtain
∂t g4(k2) = 2g4(k2) + 2ηh (k2)g4(k2) − C4g4(k2)λ4(ηh (k2)
− ηh (0)) + C Gp24 −1(Flow(G4)(k2) − Flow(G4)(0)) .
The derivation of this equation is based on the assumption
that λ4 is small.
In Sect. 4.6 we have laid out a strategy to disentangle
contributions from different tensor structures, in
particular those of R and R2. The flow equations for the gn are
obtained by a projection onto the p2 part of Flow(n) divided
by − n2 ηh p2 − n + 2 , see Sect. 4.3 and SectG. 4.6. The
graviton three-point function is at most quadratic in the
external momentum, and consequently it is again enough to use
We obtain an equation for the graviton anomalous
dimension ηh ( p2) by evaluating the flow of the graviton two-point
function bilocally at p2 and −μk2, to wit
.
The ghost anomalous dimension ηc( p2) can be directly
obtained from the transverse flow of the ghost two-point
function, to wit
ηc( p2) = −
Flow(Tc¯c)( p2)
.
In case of the higher order couplings, we employ the
projection operators described in Sect. 4.2. For the couplings λn,
this leads to
∂t λn =
n
2 ηh (0) + (n − 4) −
n − 2 ∂t gn
2 gn
λn
gn1− n2
+ C n
Flow(n)(0),
n
(E2)
(E3)
(E4)
(E5)
Table 3 Properties of the non-trivial UV fixed points for different
approximations within the full system. In the truncations 3 and 4, we set
ηφi = 0, while in the truncations 1 and 2 we use momentum dependent
anomalous dimensions. In truncations 2 and 4, the couplings g3,4 are
computed via a derivative expansion at p = 0, while in the truncations
1 and 3 the couplings g3,4(k2) are evaluated with a bilocal projection
between p = 0 and p = k. The quality of the truncation decreases
from 1 to 4. The fixed point values are obtained with the identification
1
2
3
4
0.12
0.076
0.049
− 0.060
λ∗
4
a bilocal projection at p2 = 0 and p2 = k2. Consequently,
the flow equation for g3 is quantitatively equivalent to the
previous one if λ3 is small. The graviton four-point function,
on the other hand, has p4 as its highest momentum power,
and thus we use a trilocal momentum projection at p2 = 0,
p2 = k2/2, and p2 = k2. The flow equations of g3 and g4
are then given by
(1 + η3) ∂t g3 = 2g3 − 2g3C3 (∂t λ3 + 2λ3)
(1 + η4) ∂t g4 = 2g4 − g4C4 (∂t λ4 + 2λ4)
1
1
23 ηh (k2) + 1 − 23 ηh (0) + 1
Flow(G3)(k2)
Flow(G3)(0)
23 ηh (k2) + 1 − 23 ηh (0) + 1
2
+ C Gp23 √g3
,
(E6)
scheme λn>4 = λ3 and gn>4 = g4. The critical exponents θ¯i and θ˜i stem
from two different approximation of the stability matrix as discussed in
Appendix 1. The fixed point properties from different approximations
are qualitatively very similar. In particular, all fixed points exhibit three
relevant directions when the first approximation of the stability matrix
is used. Using the second approximation of the stability matrix also
results in three relevant directions in three out of four cases
θ˜i
traction of indices. Note that the constants ηn are chosen in
such a way that ηn = 0 for vanishing anomalous dimensions.
Analogously, we can obtain a flow equation for the R2
coupling of the graviton four-point function ω4 by using a
trilocal momentum projection, as explained in Sect. 4.6. We
evaluate the flows at the same momenta as for the trilocal
flow equation of g4. The equation for ω4 then reads
(1 + ηω) ∂t ω4 = 2ω4 −
∂t g4 ⎛ C G44 λ4 + C Gp24 + CωG44 ω4
g4 ⎝
ηh (k2) + 1
−2
C G44 λ4 + 21 C Gp24 + 41 CωG44 ω4
ηh (k2/2) + 1
C G44 λ4 ⎞
+ ηh (0) + 1⎠
− CCωGG4444 (∂t λ4 + 2λ4)
1 2
ηh (k2) + 1 − ηh (k2/2) + 1
4 3
+ ηh (k2/2) + 1 − ηh (0) + 1
1
+ C Gp24
Flow(G4)(0)
−3
ηh (0) + 1
,
Flow(G4)(k2) Flow(G4)(k2/2)
− ηh (k2) + 1 + 4 ηh (k2/2) + 1
where
η3 =
C3λ3 − 23 ηh (k2)
23 ηh (k2) + 1
− 3C4λ4
η4 = ηh (0) + 1 + 4
C3λ3
− 23 ηh (0) + 1
C4λ4 − 21 ηh (k2/2)
ηh (k2/2) + 1
,
The constants C are implicitly defined via Gn ◦ TnT ◦
T (n)( p2; n ) = C Gnn n + C Gp2n p2, and we use the
abbreviation Cn := C Gnn /C Gp2n . Again, ◦ denotes the pairwise
con
C4λ4 − ηh (k2)
ηh (k2) + 1
.
ηh (k2/2) 2ηh (k2)
ηω = ηh (k2/2) + 1 − ηh (k2) + 1.
The constants C are again defined via the contraction Gn ◦
TnT ◦T (n)( p2; n ) = C Gnn n +C Gp2n p2 +CωGnn 4 p4. Again,
ηω is defined such that ηω = 0 for vanishing anomalous
dimensions.
In the previous paragraphs we introduced abbreviations
for constants that arise from the projection scheme. The
explicit values of these constants are:
5
C 3 = 192π 2 ,
9
C G33 = − 4096π 2 ,
171
C Gp23 = 32768π 2 ,
If we want to obtain analytic flow equations for the
gravitational couplings gn, which are significantly less accurate,
as discussed in Appendix 1, then we have to apply a
partial derivative with respect to p2 and evaluate the result at
p2 = 0. The resulting equations are given by
ngn
∂t gn = 2gn 2+ ng−n2−2n2
+ n − 2 C Gp2n
ηh(0) + Cnλnηh(0)
Flow(Gn) (0) ,
(E10)
where denotes the dimensionless derivative with respect to
p2. These equations remain completely analytic if we use
a Litim-shaped regulator [
113
] and approximate the
anomalous dimensions as constant, ηφ q2 ≈ const.. We present
the explicit analytic flow equations in the next Appendix.
Appendix F: Analytic flow equations
All analytic flow equations are derived at p2 = 0 (see e.g.
(E10)) and with a Litim-shaped regulator [
113
]. The
anomalous dimensions in the momentum integrals are
approximated as constant, i.e. ηφi (q2) ≈ ηφi . It is usually a good
approximation to use the anomalous dimensions evaluated
at k2 [
71
]. The analytic flow equations are then given by
∂t μ = (ηh(0) − 2) μ
1 g4
+ 12π (1 + μ)2 (3(ηh − 8) − 8λ4(ηh − 6))
∂t λ3 =
1 g3
− 180π (1 + μ)3 21(ηh − 10)
−120λ3(ηh − 8) + 320λ23(ηh − 6)
g3 (ηc − 10),
+ 5π
3 1 ∂t g3
2 ηh(0) − 1 − 2 g3
λ3
(F1)
(E9)
−
2g32
− 15(1 + μ)5
2002365 2125764
∂t g4 =2 1 + ηh (0) + 6815761 ηh (0)λ4 g4 + 6815761π
2g3g4
+ 9(1 + μ)4
− 2 (353519805 + 4λ3 (742510961λ3
6783386859(ηh − 10)
− (140 · 86837935λ3 + 457106270λ4) (ηh − 8)
− 8 (731880777λ4 − 220800215) λ32
g32
+ 45(1 + μ)5
.
(F5)
1. N. Christiansen , D.F. Litim , J.M. Pawlowski , A. Rodigast , Phys. Lett. B 728 , 114 ( 2014 ). arXiv: 1209 .4038 [hep-th]
2. N. Christiansen , B. Knorr , J.M. Pawlowski , A. Rodigast , Phys. Rev. D 93 , 044036 ( 2016 ). arXiv: 1403 .1232 [hep-th]
3. N. Christiansen , B. Knorr , J. Meibohm , J.M. Pawlowski , M. Reichert , Phys. Rev. D 92 , 121501 ( 2015 ). arXiv: 1506 .07016 [hep-th]
4. S. Weinberg, in General Relativity: An Einstein Centenary Survey , vol. 790 , ed. by S.W. Hawking , W. Israel (Cambridge University Press, Cambridge, 1979 )
5. C. Wetterich , Phys. Lett. B 301 , 90 ( 1993 ). arXiv: 1710 .05815 [hep-th]
6. M. Reuter , Phys. Rev. D 57 , 971 ( 1998 ). arXiv:hep-th/9605030
7. W. Souma , Prog. Theor. Phys . 102 , 181 ( 1999 ). arXiv:hep-th/9907027 [hep-th]
8. M. Reuter , F. Saueressig , Phys. Rev. D 65 , 065016 ( 2002 ). arXiv:hep-th/0110054 [hep-th]
9. O. Lauscher , M. Reuter , Phys. Rev. D 65 , 025013 ( 2002a ). arXiv:hep-th/0108040
10. O. Lauscher , M. Reuter , Phys. Rev. D 66 , 025026 ( 2002b ). arXiv:hep-th/0205062
11. D.F. Litim , Phys. Rev. Lett . 92 , 201301 ( 2004 ). arXiv:hep-th/0312114 [hep-th]
12. P. Fischer , D.F. Litim , Phys. Lett. B 638 , 497 ( 2006 ). arXiv:hep-th/0602203 [hep-th]
13. A. Codello , R. Percacci , Phys. Rev. Lett . 97 , 221301 ( 2006 ). arXiv:hep-th/0607128
14. A. Codello , R. Percacci , C. Rahmede , Int. J. Mod. Phys. A 23 , 143 ( 2008 ). arXiv: 0705 .1769 [hep-th]
15. P.F. Machado , F. Saueressig , Phys. Rev. D 77 , 124045 ( 2008 ). arXiv: 0712 .0445 [hep-th]
16. A. Codello , R. Percacci , C. Rahmede, Annals Phys . 324 , 414 ( 2009 ). arXiv: 0805 .2909 [hep-th]
17. M.R. Niedermaier , Phys. Rev. Lett . 103 , 101303 ( 2009 )
18. D. Benedetti , P.F. Machado , F. Saueressig , Mod. Phys. Lett. A 24 , 2233 ( 2009 ). arXiv: 0901 .2984 [hep-th]
19. A. Eichhorn , H. Gies , M.M. Scherer , Phys. Rev. D 80 , 104003 ( 2009 ). arXiv:0907 . 1828 [hep-th]
20. E. Manrique, M. Reuter, Ann. Phys. 325 , 785 ( 2010 ). arXiv: 0907 .2617 [gr-qc]
21. G. Narain, R. Percacci , Class. Quant. Grav. 27 , 075001 ( 2010 ). arXiv: 0911 .0386 [hep-th]
22. A. Eichhorn , H. Gies, Phys. Rev. D 81 , 104010 ( 2010 ). arXiv: 1001 .5033 [hep-th]
23. K. Groh , F. Saueressig , J. Phys. A 43 , 365403 ( 2010 ). arXiv: 1001 .5032 [hep-th]
24. E. Manrique, M. Reuter , F. Saueressig , Ann. Phys. 326 , 463 ( 2011a ). arXiv:1006 .0099 [hep-th]
25. E. Manrique, S. Rechenberger , F. Saueressig , Phys. Rev. Lett . 106 , 251302 ( 2011b ). arXiv:1102 .5012 [hep-th]
26. I. Donkin, J. M. Pawlowski , ( 2012 ), arXiv: 1203 .4207 [hep-th]
27. S. Nagy , J. Krizsan , K. Sailer , JHEP 07 , 102 ( 2012 ). arXiv: 1203 .6564 [hep-th]
28. D. Benedetti , F. Caravelli , JHEP 06 , 017 ( 2012 ), [Erratum: JHEP10 , 157 ( 2012 )], arXiv: 1204 .3541 [hep-th]
29. S. Rechenberger , F. Saueressig , Phys. Rev. D 86 , 024018 ( 2012 ). arXiv: 1206 .0657 [hep-th]
30. J.A. Dietz , T.R. Morris , JHEP 01 , 108 ( 2013a ). arXiv:1211 .0955 [hep-th]
31. K. Falls , D. Litim , K. Nikolakopoulos , C. Rahmede, ( 2013 ), arXiv: 1301 .4191 [hep-th]
32. D. Benedetti , Europhys. Lett. 102 , 20007 ( 2013 ). arXiv: 1301 .4422 [hep-th]
33. A. Codello , G. D'Odorico , C. Pagani , Phys. Rev. D 89 , 081701 ( 2014 ). arXiv: 1304 .4777 [gr-qc]
34. J.A. Dietz , T.R. Morris , JHEP 07 , 064 ( 2013b ). arXiv:1306 .1223 [hep-th]
35. S. Nagy , B. Fazekas , L. Juhasz , K. Sailer , Phys. Rev. D 88 , 116010 ( 2013 ). arXiv: 1307 .0765 [hep-th]
36. N. Ohta , R. Percacci , Class. Quant. Grav. 31 , 015024 ( 2014 ). arXiv: 1308 .3398 [hep-th]
37. D. Becker , M. Reuter, Ann. Phys. 350 , 225 ( 2014 ). arXiv: 1404 .4537 [hep-th]
38. K. Falls, JHEP 01 , 069 ( 2016 ). arXiv: 1408 .0276 [hep-th]
39. K. Falls , D.F. Litim , K. Nikolakopoulos , C. Rahmede , Phys. Rev. D 93 , 104022 ( 2016a ). arXiv:1410 .4815 [hep-th]
40. K. Falls, Phys. Rev. D 92 , 124057 ( 2015a ). arXiv:1501 .05331 [hep-th]
41. A. Eichhorn, JHEP 04 , 096 ( 2015 ). arXiv: 1501 .05848 [gr-qc]
42. K. Falls, ( 2015b ), arXiv:1503 .06233 [hep-th]
43. N. Ohta , R. Percacci , G.P. Vacca , Phys. Rev. D 92 , 061501 ( 2015 ). arXiv: 1507 .00968 [hep-th]
44. H. Gies , B. Knorr , S. Lippoldt, Phys. Rev. D 92 , 084020 ( 2015 ). arXiv: 1507 .08859 [hep-th]
45. D. Benedetti , Gen. Rel. Grav. 48 , 68 ( 2016 ). arXiv: 1511 .06560 [hep-th]
46. N. Ohta , R. Percacci , G.P. Vacca , Eur. Phys. J. C 76 , 46 ( 2016 ). arXiv: 1511 .09393 [hep-th]
47. H. Gies , B. Knorr , S. Lippoldt , F. Saueressig , Phys. Rev. Lett . 116 , 211302 ( 2016 ). arXiv:1601 . 01800 [hep-th]
48. K. Falls , D. F. Litim , K. Nikolakopoulos , C. Rahmede, ( 2016b ), arXiv:1607 .04962 [gr-qc]
49. K. Falls , N. Ohta , Phys. Rev. D 94 , 084005 ( 2016 ). arXiv: 1607 .08460 [hep-th]
50. J. Biemans , A. Platania , F. Saueressig , Phys. Rev. D 95 , 086013 ( 2017 ). arXiv: 1609 .04813 [hep-th]
51. C. Pagani , M. Reuter , Phys. Rev. D 95 , 066002 ( 2017 ). arXiv: 1611 .06522 [gr-qc]
52. N. Christiansen, ( 2016 ), arXiv: 1612 .06223 [hep-th]
53. M. Niedermaier , M. Reuter , Living Rev. Rel. 9 , 5 ( 2006 )
54. R. Percacci, in Approaches to quantum gravity, ed. by D. Oriti ( 2007 ), pp. 111 - 128 . arXiv: 0709 .3851 [hep-th]
55. D.F. Litim , Phil. Trans. R. Soc. Lond. A 369 , 2759 ( 2011 ). arXiv: 1102 .4624 [hep-th]
56. M. Reuter , F. Saueressig , New J. Phys. 14 , 055022 ( 2012 ). arXiv: 1202 .2274 [hep-th]
57. D. Dou , R. Percacci , Class. Quant. Grav. 15 , 3449 ( 1998 ). arXiv:hep-th/9707239 [hep-th]
58. R. Percacci , D. Perini , Phys. Rev. D 67 , 081503 ( 2003 ). arXiv:hep-th/0207033
59. J.E. Daum , U. Harst , M. Reuter , Gen. Relativ. Gravit. ( 2010 ). https://doi.org/10.1007/s10714-010-1032-2
60. J.E. Daum , U. Harst , M. Reuter , Gen. Rel. Grav. 43 , 2393 ( 2011 ). arXiv: 1005 .1488 [hep-th]
61. S. Folkerts , D.F. Litim , J.M. Pawlowski , Phys. Lett. B 709 , 234 ( 2012 ). arXiv: 1101 .5552 [hep-th]
62. U. Harst, M. Reuter , JHEP 05 , 119 ( 2011 ). arXiv: 1101 .6007 [hepth]
63. A. Eichhorn , H. Gies , New J. Phys. 13 , 125012 ( 2011 ). arXiv: 1104 .5366 [hep-th]
64. A. Eichhorn, Phys. Rev. D 86 , 105021 ( 2012 ). arXiv: 1204 .0965 [gr-qc]
65. P. Donà , R. Percacci , Phys. Rev. D 87 , 045002 ( 2013 ). arXiv: 1209 .3649 [hep-th]
66. T. Henz , J.M. Pawlowski , A. Rodigast , C. Wetterich , Phys. Lett. B 727 , 298 ( 2013 ). arXiv: 1304 .7743 [hep-th]
67. P. Donà , A. Eichhorn , R. Percacci , Phys. Rev. D 89 , 084035 ( 2014 ). arXiv: 1311 .2898 [hep-th]
68. R. Percacci , G.P. Vacca , Eur. Phys. J. C 75 , 188 ( 2015 ). arXiv: 1501 .00888 [hep-th]
69. P. Labus , R. Percacci , G.P. Vacca , Phys. Lett. B 753 , 274 ( 2016a ). arXiv:1505 .05393 [hep-th]
70. K.-Y. Oda , M. Yamada , Class. Quant. Grav. 33 , 125011 ( 2016 ). arXiv: 1510 .03734 [hep-th]
71. J. Meibohm , J.M. Pawlowski , M. Reichert , Phys. Rev. D 93 , 084035 ( 2016 ). arXiv: 1510 .07018 [hep-th]
72. P. Donà , A. Eichhorn , P. Labus , R. Percacci , Phys. Rev. D 93 , 044049 ( 2016 ), [Erratum: Phys. Rev. D 93 , no. 12 , 129904 ( 2016 )], arXiv: 1512 .01589 [gr-qc]
73. J. Meibohm , J.M. Pawlowski , Eur. Phys. J. C 76 , 285 ( 2016 ). arXiv: 1601 .04597 [hep-th]
74. A. Eichhorn , A. Held , J.M. Pawlowski , Phys. Rev. D 94 , 104027 ( 2016 ). arXiv:1604 . 02041 [hep-th]
75. T. Henz , J.M. Pawlowski , C. Wetterich , Phys. Lett. B 769 , 105 ( 2017 ). arXiv:1605 . 01858 [hep-th]
76. A. Eichhorn , S. Lippoldt, Phys. Lett. B 767 , 142 ( 2017 ). arXiv: 1611 .05878 [gr-qc]
77. D.F. Litim , F. Sannino , JHEP 12 , 178 ( 2014 ). arXiv: 1406 .2337 [hep-th]
78. J.K. Esbensen , T.A. Ryttov , F. Sannino , Phys. Rev. D 93 , 045009 ( 2016 ). arXiv: 1512 .04402 [hep-th]
79. A. Codello , K. Langaeble , D.F. Litim , F. Sannino , JHEP 07 , 118 ( 2016 ). arXiv: 1603 .03462 [hep-th]
80. A.D. Bond , D.F. Litim , Eur. Phys. J. C 77 , 429 ( 2017 ). arXiv: 1608 .00519 [hep-th]
81. E. Mølgaard , F. Sannino , Phys. Rev. D 96 , 056004 ( 2017 ). arXiv: 1610 .03130 [hep-ph]
82. B. Bajc , F. Sannino , JHEP 12 , 141 ( 2016 ). arXiv: 1610 .09681 [hep-th]
83. V. Branchina , K.A. Meissner , G. Veneziano, Phys. Lett. B 574 , 319 ( 2003 ). arXiv:hep-th/0309234 [hep-th]
84. J. M. Pawlowski , ( 2003 ), arXiv:hep-th/0310018 [hep-th]
85. M. Demmel , F. Saueressig , O. Zanusso , Annals Phys . 359 , 141 ( 2015 ). arXiv: 1412 .7207 [hep-th]
86. H. Kawai , Y. Kitazawa , M. Ninomiya , Nucl. Phys. B 393 , 280 ( 1993 ). arXiv:hep-th/9206081 [hep-th]
87. A. Nink, Phys. Rev. D 91 , 044030 ( 2015 ). arXiv: 1410 .7816 [hepth]
88. M. Demmel , A. Nink , Phys. Rev. D 92 , 104013 ( 2015 ). arXiv: 1506 .03809 [gr-qc]
89. J.M. Pawlowski , Annals Phys . 322 , 2831 ( 2007 ). arXiv:hep-th/0512261 [hep-th]
90. E. Manrique, M. Reuter , F. Saueressig , Annals Phys . 326 , 440 ( 2011c ). arXiv:1003 .5129 [hep-th]
91. I.H. Bridle , J.A. Dietz , T.R. Morris , JHEP 03 , 093 ( 2014 ). arXiv: 1312 .2846 [hep-th]
92. J.A. Dietz , T.R. Morris , JHEP 04 , 118 ( 2015 ). arXiv: 1502 .07396 [hep-th]
93. M. Safari , Eur. Phys. J. C 76 , 201 ( 2016 ). arXiv: 1508 .06244 [hepth]
94. T.R. Morris , JHEP 11 , 160 ( 2016 ). arXiv: 1610 .03081 [hep-th]
95. R. Percacci , G.P. Vacca , Eur. Phys. J. C 77 , 52 ( 2017 ). arXiv: 1611 .07005 [hep-th]
96. B.S. DeWitt , Phys. Rev . 162 , 1195 ( 1967 )
97. L.F. Abbott , Acta Phys. Polon. B 13 , 33 ( 1982 )
98. T.R. Morris , A.W.H. Preston , JHEP 06 , 012 ( 2016 ). arXiv: 1602 .08993 [hep-th]
99. P. Labus, T.R. Morris , Z.H. Slade , Phys. Rev. D 94 , 024007 ( 2016b ). arXiv:1603 .04772 [hep-th]
100. M. Safari , G.P. Vacca , JHEP 11 , 139 ( 2016 ). arXiv: 1607 .07074 [hep-th]
101. M. Safari , G.P. Vacca , Phys. Rev. D 96 , 085001 ( 2017 ). arXiv: 1607 .03053 [hep-th]
102. C. Wetterich, ( 2016 ), arXiv: 1607 .02989 [hep-th]
103. M. Mitter , J.M. Pawlowski , N. Strodthoff , Phys. Rev. D 91 , 054035 ( 2015 ). arXiv: 1411 .7978 [hep-ph]
104. D.F. Litim , J.M. Pawlowski , Phys. Lett. B 435 , 181 ( 1998 ). arXiv:hep-th/9802064 [hep-th]
105. U. Ellwanger, Proceedings, Workshop on Quantum field theoretical aspects of high energy physics : Bad Frankenhausen, Germany, September 20-24 , 1993 , Z. Phys . C62 , 503 ( 1994 ), arXiv:hep-ph/9308260 [hep-ph]
106. T.R. Morris , Int. J. Mod. Phys. A 9 , 2411 ( 1994 ). arXiv:hep-ph/9308265
107. H. Gies, Lect. Notes Phys . 852 , 287 ( 2012 ). arXiv:hep-ph/0611146 [hep-ph]
108. C.S. Fischer , A. Maas , J.M. Pawlowski , Annals Phys . 324 , 2408 ( 2009 ). arXiv:0810 . 1987 [hep-ph]
109. A.K. Cyrol , L. Fister , M. Mitter , J.M. Pawlowski , N. Strodthoff , Phys. Rev. D 94 , 054005 ( 2016 ). arXiv:1605 . 01856 [hep-ph]
110. C.S. Fischer , J.M. Pawlowski , Phys. Rev. D 80 , 025023 ( 2009 ). arXiv: 0903 .2193 [hep-th]
111. E. Mottola, Non-perturbative gravity and quantum chromodynamics . Proceedings, 49th Cracow School of Theoretical Physics , Zakopane, Poland, May 31-June 10, 2009 , Acta Phys . Polon. B 41 , 2031 ( 2010 ), arXiv: 1008 .5006 [gr-qc]
112. D. Schnoerr , I. Boettcher , J.M. Pawlowski , C. Wetterich, Ann. Phys. 334 , 83 ( 2013 ). arXiv: 1301 .4169 [ cond-mat. quant-gas]
113. D.F. Litim , Phys. Lett. B 486 , 92 ( 2000 ). arXiv:hep-th/0005245 [hep-th]
114. K.S. Stelle , Gen. Rel. Grav. 9 , 353 ( 1978 )
115. S. Floerchinger, JHEP 1205 , 021 ( 2012 ). arXiv: 1112 .4374 [hepth]
116. K. Kamikado , N. Strodthoff , L. von Smekal , J. Wambach, Eur. Phys. J. C 74 , 2806 ( 2014 ). arXiv: 1302 .6199 [hep-ph]
117. R. - A. Tripolt , N. Strodthoff , L. von Smekal , J. Wambach, Phys. Rev. D 89 , 034010 ( 2014 ). arXiv: 1311 .0630 [hep-ph]
118. J.M. Pawlowski , N. Strodthoff , Phys. Rev. D 92 , 094009 ( 2015 ). arXiv: 1508 .01160 [hep-ph]
119. N. Strodthoff, Phys. Rev. D 95 , 076002 ( 2017 ). arXiv: 1611 .05036 [hep-th]
120. A. Bonanno , M. Reuter , JHEP 02 , 035 ( 2005 ). arXiv:hep-th/0410191 [hep-th]
121. C. Wetterich, Phys. Lett. B 773 , 6 ( 2017 ). arXiv: 1704 .08040 [grqc]
122. J. A. M. Vermaseren , ( 2000 ), arXiv:math-ph/0010025 [math-ph]
123. J. Kuipers , T. Ueda , J.A.M. Vermaseren , J. Vollinga , Comput. Phys. Commun . 184 , 1453 ( 2013 ). arXiv: 1203 .6543 [cs.SC]
124. Wolfram Research Inc, “Mathematica” 10 , 2 ( 2015 )
125. T. Denz , A. Held , J. M. Pawlowski , A . Rodigast, (in preparation)
126. D. Brizuela , J. M. Martin-Garcia , G. A. M. Marugan , ( 2008 ), https://doi.org/10.1007/s10714-009-0773-2, arXiv: 0807 . 0824
127. M.Q. Huber , J. Braun , Comput. Phys. Commun . 183 , 1290 ( 2012 ). arXiv: 1102 .5307 [hep-th]
128. A.K. Cyrol , M. Mitter , N. Strodthoff , Comput. Phys. Commun . 219 , 346 ( 2017 ). arXiv: 1610 .09331 [hep-ph]
129. J.J.W. York , J. Math. Phys. 14 , 456 ( 1973 )
130. K.S. Stelle , Phys. Rev. D 16 , 953 ( 1977 )